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APPLICATION OF BESSEL BEAMS IN THE

HUMAN EYE

DIPESH BHATTARAI

BOptom, MPH

Submitted in fulfilment of the requirements for the degree of

Doctor of Philosophy

School of Optometry and Vision Science

Institute of Health and Biomedical Innovation

Faculty of Health

Queensland University of Technology

2019

Application of Bessel beams in the human eye i

Keywords

Bessel beams, Gaussian beams, non-diffracting, self-reconstructing, phakometry,

fixation stability, scatter, retinal imaging, retinal image intensity.

ii Application of Bessel beams in the human eye

Abstract

Gaussian beams used in the study of the human eye suffer from diffraction and

scattering while passing through ocular media, resulting in weak signal-to-noise ratios.

The poor signals create difficulties in accurate understanding of shape and position of

ocular surfaces, and during ocular imaging. Bessel beams, being resistant to diffraction

and capable of self-reconstructing, have advantages over Gaussian beams in that their

use might result in decreased scattering artefacts, improved penetration in ocular

media, and increased ocular image quality. Bessel beams provide lower temporal

variation of aberration in an adaptive optics system than Gaussian beams, which might

be due to improved fixation stability of the eye. The effectiveness of Bessel beams

over Gaussian beams in determining information about ocular optics, in fixation

stability and in imaging the retina of the human eye is unknown. This study explored

the application of Bessel beams in determining ocular optics, in fixation stability, and

in imaging retinal structures of the eye.

This study consisted of three experiments. In Experiment 1 (Chapter 4), Bessel

beams were applied to phakometry, which is the determination of ocular lens surface

curvatures and lens refractive index through measuring sizes of the Purkinje images

formed by surface reflections. I intended to use a highly obstructed Bessel beam during

phakometry to produce an arc rather than a ring to improve the identification of

Purkinje images. Therefore, in a preliminary experiment (Chapter 3) I investigated

whether the highly obstructed Bessel beam possessed self-reconstructing and non-

diffracting properties. Even after blocking a major portion of the Bessel beam,

including its central lobe, the remaining beam retained self-reconstructing and non-

diffracting properties during propagation. In Experiment 1, the accuracy of the Bessel

phakometer was assessed using a model eye, and six healthy participants were

recruited to assess repeatability (inter- and intra-observer) and Purkinje images

brightnesses for Bessel and conventional Gaussian phakometers. The lens parameters

of the model eye determined by the Bessel phakometer were similar to those provided

by the manufacturer. The Bessel phakometer produced brighter Purkinje images and

better inter-observer repeatability than those of the Gaussian phakometer.

Application of Bessel beams in the human eye iii

In Experiment 2 (Chapter 5), I investigated the fixation stability of 16 healthy

participants using a monitor-based bulls eye/cross hair combination, Gaussian beams

(monitor-based images and laser beams), and Bessel beams (monitor-based images

and laser beams), as fixation targets. An EyeTribe tracker sampled eye positions at 30

Hz. Standard deviations of fixation positions along horizontal (𝜎𝑥) and vertical

meridians (𝜎𝑦) and areas of bivariate contour ellipses (BCEAs) encompassing 68.2

percent of the highest density eye position samples were calculated, and statistical

significances of fixation differences between targets were determined. Monitor-based

images of Bessel beams provided better fixation targets than the bull’s eye/cross hair

combination, monitor-based Gaussian images and laser Gaussian beams. There were

no significant differences in the fixation stability between monitor-based images of

Bessel beams and laser Bessel beams targets, and between laser Bessel beams and the

bull’s eye/cross hair combination, monitor-based Gaussian images and laser Gaussian

beams targets. This indicates that the shape of a Bessel beam, rather than its

propagation properties, is responsible for the improvement in fixation stability over

that achieved with other targets. Ophthalmic imaging instruments that require stable

fixation can benefit by using Bessel beams in the form of monitor-based images as

fixation targets. It remains unclear whether ophthalmic imaging instruments that

require stable fixation would benefit by using Bessel laser beams to provide both

illuminating beams and fixation targets.

In Experiment 3 (Chapter 6), I built a retinal imaging set up to investigate the

amount of light reaching the retinal area being imaged for Bessel and Gaussian beams.

The Bessel and Gaussian beam images formed at the retina were imaged using a

science camera conjugate to the retinal plane. The intensity of each image thus

acquired, referred as “retinal image intensity”, was used as a measure of the amount

of light reaching the retinal area being imaged. After dilating the pupil with 1 percent

tropicamide, retinal images of right eyes were acquired for 10 participants each from

below 35 years (young group) and above 59 years (older group). The retinal image

intensities for Bessel and Gaussian beams were compared between young and older

groups, and between without-cataract and early-cataract groups within the older group.

Bessel beams provided higher retinal image intensities than Gaussian beams in both

age groups, and in both without- and early-cataract groups. The Bessel to Gaussian

retinal image intensity ratio was similar for both age groups. The early-cataract group

iv Application of Bessel beams in the human eye

had significantly higher Bessel to Gaussian retinal image intensity ratio than the

without-cataract group. The signal to noise ratio of retinal images can be improved

using Bessel beams as they provide higher amounts of light reaching the retinal area

being imaged than Gaussian beams. This improvement is more apparent among

participants with early-cataract than those without-cataract.

In conclusion, Bessel beams were applied in phakometry, fixation stability and

retinal imaging of the human eye. Use of Bessel beams in phakometry produces

brighter Purkinje lens images and better inter-observer repeatability for lens radii of

curvature than Gaussian beams. Monitor-based images of Bessel beams provide better

fixation stability than a bull’s eye/cross hair combination, monitor-based Gaussian

images and laser Gaussian beams targets. The amount of light reaching the retinal area

being imaged for Bessel beams is higher than for Gaussian beams, and this effect is

strong among participants with early cataract.

Application of Bessel beams in the human eye v

Table of Contents

Keywords..............................................................................................................................i

Abstract ...............................................................................................................................ii

Table of Contents ................................................................................................................. v

List of Figures ...................................................................................................................viii

List of Tables ...................................................................................................................... xi

List of Abbreviations.......................................................................................................... xii

Statement of Original Authorship ......................................................................................xiii

Acknowledgements ............................................................................................................ xv

Chapter 1: INTRODUCTION .......................................................................... 1

Chapter 2: LITERATURE REVIEW .............................................................. 5

2.1 Gaussian beams .......................................................................................................... 5

2.2 Bessel beams .............................................................................................................. 6 2.2.1 Non-diffracting property ................................................................................... 7 2.2.2 Self-reconstructing property ............................................................................. 9 2.2.3 Production of Bessel beams ............................................................................ 11

2.3 Application of Bessel beams to the eye ..................................................................... 13

2.4 Phakometry .............................................................................................................. 14

2.5 Fixation stability ...................................................................................................... 16

2.6 Retinal imaging ........................................................................................................ 18 2.6.1 Modalities for retinal imaging ......................................................................... 18 2.6.2 State-of-the-art of retinal imaging technology ................................................. 23 2.6.3 Challenges in retinal imaging ......................................................................... 43

2.7 Aims and hypotheses ................................................................................................ 45

2.8 Ethics approval......................................................................................................... 47

Chapter 3: PROPAGATION PROPERTIES OF OBSTRUCTED BESSEL

BEAMS......................................................................................... 49

3.1 Introduction ............................................................................................................. 49

3.2 Methods ................................................................................................................... 49 3.2.1 Experimental setup ......................................................................................... 50 3.2.2 Computer simulation ...................................................................................... 52

3.3 Results ..................................................................................................................... 53

3.4 Conclusion ............................................................................................................... 53

Chapter 4: PHAKOMETRY WITH BESSEL BEAMS ................................ 55

4.1 Introduction ............................................................................................................. 55

4.2 Methods ................................................................................................................... 56 4.2.1 Participants..................................................................................................... 57 4.2.2 Production of a Bessel beam ........................................................................... 58 4.2.3 Bessel phakometer .......................................................................................... 60

vi Application of Bessel beams in the human eye

4.2.4 Gaussian phakometer ..................................................................................... 62 4.2.5 Analysis of Purkinje images ........................................................................... 63 4.2.6 Merit function ................................................................................................ 64 4.2.7 Determination of repeatability of phakometers ............................................... 64 4.2.8 Determination of Purkinje image brightness ................................................... 65

4.3 Results ..................................................................................................................... 66

4.4 Discussion ............................................................................................................... 68

Chapter 5: FIXATION STABILITY WITH BESSEL BEAMS ................... 71

5.1 Introduction ............................................................................................................. 71

5.2 Methods ................................................................................................................... 72 5.2.1 Participants .................................................................................................... 72 5.2.2 Fixation targets .............................................................................................. 73 5.2.3 Instrumentation and eye-movement recording ................................................ 74 5.2.4 Luminances of laser beam targets ................................................................... 76 5.2.5 Tasks and procedure ....................................................................................... 77 5.2.6 Data collection and analysis ........................................................................... 77

5.3 Results ..................................................................................................................... 80

5.4 Discussion ............................................................................................................... 85

Chapter 6: RETINAL IMAGING WITH BESSEL BEAMS........................ 89

6.1 Introduction ............................................................................................................. 89

6.2 Methods ................................................................................................................... 90 6.2.1 Participants .................................................................................................... 91 6.2.2 Grading of cataract ......................................................................................... 93 6.2.3 Production and propagation properties of the Bessel beam .............................. 95 6.2.4 Instrumentation .............................................................................................. 98 6.2.5 Linearity of the camera output ...................................................................... 100 6.2.6 Task and image acquisition .......................................................................... 101 6.2.7 Measurement of straylight ............................................................................ 102 6.2.8 Data collection and analysis ......................................................................... 102

6.3 Results ................................................................................................................... 104

6.4 Discussion ............................................................................................................. 111

Chapter 7: CONCLUSIONS AND FUTURE DIRECTIONS..................... 114

7.1 Introduction ........................................................................................................... 114 7.1.1 Phakometry with Bessel beams .................................................................... 114 7.1.2 Fixation stability with Bessel beams ............................................................. 115 7.1.3 Retinal imaging with Bessel beams .............................................................. 116

7.2 Future directions .................................................................................................... 117

References ........................................................................................................... 121

Appendices .......................................................................................................... 134

: Presentations and publications arising from this thesis ................................ 134

: Published paper 1 ....................................................................................... 136

: Published paper 2 ....................................................................................... 143

: Changes to MATLAB code for Bessel and Gaussian phakometers .............. 151

: Steps in MATLAB to determine Purkinje image brightness ........................ 152

Application of Bessel beams in the human eye vii

: MATLAB code for extracting fixation positions and pupil diameter ............ 154

: MATLAB code for plotting the retinal image intensity distribution ............. 160

: MATLAB code for determining pixel greyscale values of retinal images .... 161

viii Application of Bessel beams in the human eye

List of Figures

Figure 1.1: Cross sectional images of (a) a Gaussian beam and (b) a Bessel

beam ........................................................................................................... 2

Figure 2.1: A Gaussian beam profile along the propagation distance (z).................... 6

Figure 2.2: Spatial spectrum of a Bessel beam generated using an axicon ................. 7

Figure 2.3: Intensity at the centre of a Bessel beam (continuous line) and at the

centre of a Gaussian beam (dashed line), as a function of propagation

distance ....................................................................................................... 8

Figure 2.4: Self-reconstruction of a Bessel beam at various distances after

being obstructed by a rectangular obstacle ................................................ 10

Figure 2.5: Generation of Bessel beams using an annular slit .................................. 12

Figure 2.6: Simulated Purkinje images in a perfectly aligned model eye ................. 15

Figure 2.7: Different target shapes used by Thaler et al. to determine the effect

of target shape on stability of fixational eye movements ............................ 16

Figure 2.8: Time frequency plots of the Zernike tip aberration coefficient .............. 17

Figure 2.9: Flood illumination ophthalmoscope ...................................................... 19

Figure 2.10: Confocal scanning laser ophthalmoscope ............................................ 20

Figure 2.11: Spectral/Fourier domain based OCT. .................................................. 21

Figure 2.12: Image of the retina at 4-deg eccentricity .............................................. 23

Figure 2.13: Basic layout of adaptive optics system ................................................ 24

Figure 2.14: Wavefront correction principle by manipulating OPL ......................... 26

Figure 2.15: Segmented deformable mirrors ........................................................... 26

Figure 2.16: Continuous deformable mirrors ........................................................... 27

Figure 2.17: Neumatic LC-SLMs operating in reflection mode ............................... 28

Figure 2.18: The Hartmann-Shack spots distribution on the CCD ........................... 29

Figure 2.19: Aberrated eye ..................................................................................... 32

Figure 2.20: Zernike polynomial function pyramid ................................................. 34

Figure 2.21: Principle of the Hartmann-Shack aberrometer ..................................... 36

Figure 2.22: Different sources of scattering in the human eye ................................. 39

Figure 2.23: Stimulus layout for retinal straylight measurement in C-Quant ........... 40

Figure 2.24: Schematic diagram of a double pass system ........................................ 41

Figure 2.25: Forward scatter principle using Hartmann-Shack sensor ..................... 43

Figure 3.1: Experimental setup for testing properties of obstructed Bessel

beams ....................................................................................................... 50

Application of Bessel beams in the human eye ix

Figure 3.2: (a) Phase plate of an axicon, projected on the SLM, (b)

Experimental Bessel beam, (c) Simulated Bessel beam .............................. 51

Figure 3.3: Images of a Bessel beam with the second obstruction condition at

z0, zmin/2, zmin and z∞ .................................................................................. 51

Figure 3.4: Images of a Bessel beam with the third obstruction condition at z0,

zmin/2, zmin and z∞ ....................................................................................... 52

Figure 4.1: Phakometers, (top) Bessel and (bottom) Gaussian, developed at

QUT research lab. ..................................................................................... 56

Figure 4.2: Diagram of geometrical quantities associated in production of

Bessel beam in the Bessel phakometer....................................................... 58

Figure 4.3: Cross-sectional image of Bessel beam generated through lens with

spherical aberration ................................................................................... 59

Figure 4.4: Bessel phakometer ................................................................................ 60

Figure 4.5: Gaussian phakometer ............................................................................ 62

Figure 4.6: Purkinje images of a participant’s eye obtained with the Bessel

phakometer and edges through which ellipses were fitted. ......................... 63

Figure 5.1: Fixation targets ..................................................................................... 74

Figure 5.2: Experimental setup for fixation stability with Bessel beam .................... 75

Figure 5.3: Photograph of experimental setup for fixation stability with Bessel

beam ......................................................................................................... 75

Figure 5.4: Box-and-whisker plots of σx for the fixation targets ............................... 83

Figure 5.5: Box-and-whisker plots of σy for the fixation targets ............................... 83

Figure 5.6: Box-and-whisker plots of BCEAs for the fixation targets ...................... 85

Figure 6.1: Standard photographs of (left) grade 1, (middle) 2 and (right) 3

nuclear cataract provided by the WHO cataract grading group ................... 94

Figure 6.2: Spatial spectrum of a Bessel beam generated using an axicon................ 95

Figure 6.3: Cross sectional intensities of the (a & c) Bessel and (b & d)

Gaussian beams images ............................................................................. 97

Figure 6.4: Experimental setup for retinal imaging with Bessel beams .................... 98

Figure 6.5: Photograph of experimental setup for retinal imaging with Bessel

beams ........................................................................................................ 99

Figure 6.6: Scatterplot of normalised input power and pixel output intensity

response. ................................................................................................. 101

Figure 6.7: Intensity distribution of thresholded pixels of (a) Bessel and (b)

Gaussian retinal images at fovea for participant P1 .................................. 105

Figure 6.8: (Left) Bessel and (right) Gaussian retinal image intensities among

all participants ......................................................................................... 108

Figure 6.9: Bessel and Gaussian retinal image intensities in (left) young and

(right) older age groups ........................................................................... 109

x Application of Bessel beams in the human eye

Figure 6.10: Ratio of Bessel and Gaussian retinal image intensities in (left)

early- and (right) without-cataract groups ................................................ 110

Figure 6.11: Scatter plot of the straylight parameter and the ratio of Bessel and

Gaussian retinal image intensities ........................................................... 111

Figure 7.1: Experimental setup for adaptive optics retinal imaging using Bessel

beam ....................................................................................................... 118

Figure 7.2: Screenshot of ‘colormap gray’ image in MATLAB ............................. 152

Application of Bessel beams in the human eye xi

List of Tables

Table 4.1: Clinical tests and inclusion criteria of healthy participants ...................... 57

Table 4.2: Model eye lens parameters according to the manufacturer's

specifications and the Bessel phakometer .................................................. 67

Table 4.3: Participants lens parameters obtained from Gaussian and Bessel

phakometers. RI & F are reported for λ = 555 nm ...................................... 67

Table 4.4: Intra-observer repeatability for Gaussian and Bessel phakometers .......... 68

Table 4.5: Inter-observer repeatability for Gaussian and Bessel phakometers .......... 68

Table 4.6: Brightnesses of Purkinje images obtained from Gaussian and Bessel

phakometers .............................................................................................. 68

Table 5.1: Clinical tests and inclusion criteria of healthy participants ...................... 73

Table 5.2: Luminances of laser beams with neutral density optical filters ................ 77

Table 5.3: Shapiro-Wilk test p-values to assess the normality distribution of

SDs and BCEAs ........................................................................................ 79

Table 5.4: Standard deviations (degrees) of fixation positions along horizontal

meridian (σx) for the targets ....................................................................... 81

Table 5.5: Standard deviations (degrees) of fixation positions along vertical

meridian (σy) for the targets ....................................................................... 82

Table 5.6: Bivariate contour ellipse areas (BCEAs) (degrees2) for the fixation

targets ....................................................................................................... 84

Table 5.7: Average standard deviations of pupil diameter among participants

for all the targets combined ....................................................................... 87

Table 6.1: Clinical tests and inclusion criteria of participants .................................. 91

Table 6.2: Age, gender, BCVA and refractive error of the participants .................... 92

Table 6.3: Simplified cataract grading by the WHO cataract grading group ............. 93

Table 6.4: Shapiro-Wilk test p-values to assess the normality distribution of

Bessel and Gaussian retinal image intensities. ......................................... 103

Table 6.5: Intensities of Bessel and Gaussian retinal images, straylight

parameter and cataract among participants .............................................. 107

xii Application of Bessel beams in the human eye

List of Abbreviations

λ Wavelength

𝜎𝑥 Standard deviation along horizontal meridian

𝜎𝑦 Standard deviation along vertical meridian

AO Adaptive optics

AU Arbitrary unit

BCEA Bivariate contour ellipse area

BCVA Best corrected visual acuity

BS Beam splitter

CCD Charge-coupled device

F Power (optical)

f Focal length

HSWS Hartmann-Shack wavefront sensor

k Wave vector

LD Laser diode

LogMAR The logarithm of minimum angle of resolution in arc minutes

LSFM Light sheet fluorescence microscopy

MATLAB MathWorks Inc., Natick, MA, version R2011

OCT Optical coherence tomography

OLED Organic light-emitting diode

PI 1st Purkinje image

PII 2nd Purkinje image

PIII 3rd Purkinje image

PIV 4th Purkinje image

PMMA Poly (methyl methacrylate)

Ra Lens anterior radius of curvature

Rp Lens posterior radius of curvature

RI Lens equivalent refractive index

SD Standard deviation

SLM Spatial light modulator

sw Standard deviation of repeated measurements

VA Visual acuity

Application of Bessel beams in the human eye xiii

Statement of Original Authorship

The work contained in this thesis has not been previously submitted to meet

requirements for an award at this or any other higher education institution. To the best

of my knowledge and belief, the thesis contains no material previously published or

written by another person except where due reference is made.

Signature:

Date: April 2019

QUT Verified Signature

xiv Application of Bessel beams in the human eye

Application of Bessel beams in the human eye xv

Acknowledgements

This work was supported by Australian Research Council Discovery Project

(DP140101480; DAA), a Queensland University of Technology (QUT) HDR Tuition

Fee Sponsorship, and a School of Optometry and Vision Science Postgraduate

Scholarship.

First and foremost, acknowledgement must be made to my supervisor Professor

David A. Atchison. I would like to thank you for providing an opportunity to conduct

my PhD research in the Visual and Ophthalmic Optics Laboratory. Thank you for your

expertise, continual guidance, tolerance, patience, kindness, constructive advice, and

for being wonderful mentor during all stages of my PhD. My sincere thanks go to my

associate supervisor Dr Marwan Suheimat for helping me build the phakometer, the

fixation stability set up and the retinal camera. Thank you for teaching me basics of

Zemax and MATLAB. Thank you for your kind support, guidance and invaluable

advice during all stages of my PhD.

I am grateful to Associate Professor Andrew Lambert from University of New

South Wales School of Engineering and Information Technology for providing insight

on Bessel beams and helping me build the retinal camera.

I thank Hannah K. Maher, Meera Chandra, William Chelepy and Sarah K.

Halloran for helping me in the phakometry experiment.

I thank Dr Fan Yi and Anita Sathyanarayanan for helping me write MATLAB

codes for fixation stability and retinal imaging experiments, respectively.

I thank my friends from School of Optometry and Vision Science for their

support and encouragement during my candidature and to all my participants for the

time they devoted to my research. I would also like to thank B P Eye Foundation for

supporting me in pursuing my studies.

This thesis is dedicated to all the lives lost from the Nepal earthquake that

occurred on 25 April 2015. Finally, I am indebted to my family for their selfless

support throughout my PhD journey.

xvi Application of Bessel beams in the human eye

Chapter 1: INTRODUCTION 1

Chapter 1: INTRODUCTION

Better characterisations of the optics of human eyes and detections of the ocular

diseases can be achieved through better understanding of corneal, lenticular and retinal

structures and through improved fixation stability during ophthalmic procedures.

Typical light sources used in determining the optics, in fixation stability, and in

imaging the retinas of human eyes form Gaussian beams. Gaussian beams (Figure 1.1,

a) are non-localised beams which diffract and diverge after propagating through some

distance (Siegman, 1986). Along with diffraction, Gaussian beams scatter while

passing through media. This causes image signals to be weak and dispersed, and

generates ghost images creating difficulties in achieving accurate information about

the shape and position of ocular surfaces, and obtaining high quality retinal images

(Fahrbach, Simon, & Rohrbach, 2010; Rohrbach, 2009).

Bessel beams (Figure 1.1, b) are localised beams with transverse patterns that

remain stationary along the propagation distance, i.e., these beams are resistant to

diffractive spreading, and also have potential of self-reconstruction despite partial

perturbation while passing through inhomogeneous media (Durnin, Miceli, & Eberly,

1987; Fahrbach, et al., 2010; Nowack, 2012; Salo & Friberg, 2008; Turunen & Friberg,

2010). Bessel beams can be used as light sources in ocular studies, but till now they

have been used only in determining their feasibility as radiation sources in adaptive

optics systems (Lambert, Daly, deLestrange, & Dainty, 2011), including their

influence in temporal variation of aberration (Lambert, Daly, & Dainty, 2013) and

their influence on the axial resolution of aqueous-outflow imaging systems (Hong,

Shinoj, Murukeshan, Baskaran, & Aung, 2017).

The diffraction resistant and self-reconstructing properties of Bessel beams

result in decreased scattering artefacts, improved penetration in dense media, enhanced

depth of field and increased image quality than with Gaussian beams (Fahrbach &

Rohrbach, 2012; Fahrbach, et al., 2010; Rohrbach, 2009). This study explores the

application of Bessel beams in determining the optics, in fixation stability, and in

imaging of retinal structures of the human eyes and compares their effectiveness

relative to Gaussian beams.

2 Chapter 1: INTRODUCTION

Figure 1.1: Cross sectional images of (a) a Gaussian beam and (b) a Bessel beam. 𝑟0 is the central lobe

size of the Bessel beam.

Phakometry, a technique to determine shape and refractive index of the in-vivo

ocular lens by imaging reflections of a light source from the corneal and lens surfaces,

uses light sources that form Gaussian beams. The rough anterior lens surface causes

scattering and diffuse reflection of these beams, deteriorating the quality and lowering

the brightness of PIII and hence making accurate, repeatable estimate of lens

parameters difficult (Atchison & Smith, 2000; Navarro, Mendez-Morales, &

Santamaría, 1986; Tabernero, Benito, Nourrit, & Artal, 2006). The non-diffracting and

self-reconstructing properties of the Bessel beam might increase the specular reflected

signals from the lens surfaces and produce brighter, sharper PIIIs than Gaussian beams,

and thus give more accurate and repeatable estimations of lens surface curvatures.

Experiment 1 (Chapter 4) will assess Purkinje image brightness, accuracy, and

repeatability for a Bessel phakometer compared with those of a conventional Gaussian

phakometer. In this experiment, I will obstruct the majority of the Bessel beam to

produce an arc rather than a ring to improve the identification of Purkinje images. The

propagation properties of a Bessel beam after an obstruction have been studied before

(Anguiano-Morales, 2009; Anguiano-Morales, Méndez-Otero, Iturbe-Castillo, &

Chávez-Cerda, 2007; Bouchal, Wagner, & Chlup, 1998; MacDonald, Boothroyd,

Okamoto, Chrostowski, & Syrett, 1996; Zheng et al., 2013), but in linear media were

limited to the obstruction of the central lobe or another small proportion of the beam.

Therefore, a preliminary experiment (Chapter 3) will determine whether the desired

Chapter 1: INTRODUCTION 3

propagation properties of a Bessel beam, after the majority of it has been obstructed,

are retained in linear media.

Micro-saccades, micro-tremors, drift, flutter and nystagmus are involuntary eye

movements creating difficulties in stabilising fixation, necessary for obtaining optimal

results during ophthalmic and behavioural procedures (Elsner et al., 2013; Menz,

Sutter, & Menz, 2004; Sutter & Tran, 1992). A good fixation target is important to

minimising fixation instability. Thaler, Schutz, Goodale and Gegenfurtner (2013) have

found that a bull’s eye/cross hair combination (Figure 2.7), provided lower micro-

saccade rates than other target shapes such as circular shapes and crosses and

combinations of these two shapes. Lambert et al. (2013) reported that a Bessel beam,

when used as a fixation target in an adaptive optics system, reduced temporal variation

of aberration from that found with a Gaussian beam. This reduction might have been

due to improved stability of the eye through suppression of rapid eye movements, but

this has not been confirmed. Experiment 2 (Chapter 5) will use Bessel beams as

fixation targets and compare the fixation stability with that of a bull’s eye/cross hair

combination and Gaussian beams.

Retinal imaging systems are now being used in resolving even the smallest

photoreceptor cells (Dubra et al., 2011). Retinal image quality, even after correction

of aberrations using adaptive optics, is still limited by inherent diffraction and

scattering properties of Gaussian beams while passing through ocular media,

especially with media opacities and with increasing age (Kuroda et al., 2002; Mwanza

et al., 2011; Zhou, Bedggood, & Metha, 2014). Gaussian beams while entering the eye

suffers from forward scatter, which reduces the amount of light reaching the retinal

area of concern. This reduces the amount of backscattered light from that area and

hence decreases the signal to noise ratio (Carpentras, Laforest, Künzi, & Moser, 2018;

Chen et al., 2016; Christaras, Ginis, Pennos, & Artal, 2016; Wanek, Mori, & Shahidi,

2007). While imaging biological tissues such as skin, Bessel beams reduce scattering

artefacts, and provide better image quality and penetration depth in dense media than

Gaussian beams (Fahrbach, et al., 2010; Rohrbach, 2009). Farhbach et al. (2010)

showed that Gaussian beams lose almost 40 percent of their energy in a scattering

condition, where around 50 percent of the beam field was disturbed by the scatter,

whereas under the same condition Bessel beams lose only 5 percent of their energy.

This property of Bessel beams might help in increasing the amount of light reaching

4 Chapter 1: INTRODUCTION

the retinal area being imaged. Experiment 3 (Chapter 6) will investigate the amount of

light reaching the retinal area being imaged when using Bessel and Gaussian

illuminations among young (< 35 years) and older (> 59 years) age groups, and among

without-cataract and early-cataract groups.

Chapter 2: LITERATURE REVIEW 5

Chapter 2: LITERATURE REVIEW

This chapter reviews the literature on Gaussian beams, Bessel beams with

respect to their non-diffracting and self-reconstructing properties, production of Bessel

beams, and the application of Bessel beams to the eye (phakometry, fixation stability

and retinal imaging). Challenges in phakometry, fixation stability and retinal imaging

are identified, and properties of Bessel beams that might help in overcoming these

challenges are discussed. The chapter concludes with the aims and hypotheses of this

thesis.

2.1 GAUSSIAN BEAMS

Gaussian beams are non-localised beams that diffract and diverge due to their

plane-wave components running out of phase as they propagate (Figure 2.1) (Siegman,

1986). The electric field component of a Gaussian beam can be represented as

(Paschotta, 2008)

𝐸(𝑟, 𝑧) = exp (−𝑟2

𝑤(𝑧)2 ) exp[𝑖𝜑(𝑧, 𝑟)] (2.1)

where 𝑟 and 𝑧 are the radial and longitudinal components of the electric field,

respectively, 𝑤(𝑧) is the beam radius and 𝜑(𝑧, 𝑟) is the phase evolution along the

beam, which is represented as

𝜑(𝑧, 𝑟) = 𝑘𝑧 − tan−1 𝑧

𝑧R+

𝑘𝑟2

2𝑅(𝑧) (2.2)

where 𝑘 = 2𝜋

λ is the wave number, λ is the wavelength, 𝑅(𝑧) is the curvature of the

wavefront and 𝑧𝑅 is the Rayleigh length. The 𝑧𝑅 is represented as

𝑧𝑅 = 𝜋𝑤0

2

λ (2.3)

where 𝑤0 is the incident beam waist. The 𝑧𝑅 shows the distance over which the

Gaussian beam cross-sectional area enlarges by the factor of two and is typically used

to characterise the spreading of Gaussian beams.

6 Chapter 2: LITERATURE REVIEW

Figure 2.1: A Gaussian beam profile along the propagation distance (z). 𝑤0 , incident beam radius; 𝑧𝑅,

Rayleigh length; θ, half-angle divergence. Modified from

https://www.cvilaseroptics.com/file/general/All_About_Gaussian_Beam_OpticsWEB.pdf

2.2 BESSEL BEAMS

A Bessel beam is formed by superposition of a set of plane waves with wave

vectors propagating on a cone and each propagating wave undergoing the same phase

shift (kzΔz) over a spatial propagation of Δz (Figure 2.2). The electric field component

of an ideal Bessel beam can be represented as (Bouchal, et al., 1998; Durnin, 1987;

McGloin & Dholakia, 2005)

𝐸(𝑟, 𝜙, 𝑧) = 𝐴0 exp(𝑖𝑘z𝑧) 𝐽n(𝑘r𝑟) exp(±𝑖𝑛𝜙) (2.4)

where Jn is an nth-order Bessel function, A0 is an amplitude of electric component of

the propagating light, kz is the longitudinal component of 𝑘, kr is the radial component

of 𝑘 (both kz and kr in cylinder coordinates system), and 𝑘 is the wave vector defined

as 𝑘 = √𝑘z2 + 𝑘r

2 =2𝜋

λ , λ is the wavelength of the light responsible for Bessel beam

creation, and r, ϕ and z are the radial, azimuthal and longitudinal components of the

electric field, respectively.


Figure 2.2: Spatial spectrum of a Bessel beam generated using an axicon with wave vectors of plane

waves on the surface of a cone. θ, opening angle of cone at the tip of the axicon; 𝛼, physical/wedge

angle of the axicon; k, wave vector; 𝑧𝑚𝑎𝑥, the maximum distance until which the central lobe of the

beam maintains its propagation-invariant property; kr, radial component of k; kz, longitudinal component

of k; 𝑤0, incident beam waist; 𝐷, diameter of the aperture. Modified from Litvin, McLaren, & Forbes

(2008).

The propagation of a Bessel beam, which is manifested as a ring (Figure 1.1, b)

in k-space due to its angular spectrum, is non-diffracting and self-repairing while that

of a Gaussian beam is universally affected by diffractive phenomena and once

obstructed does not reform (McGloin & Dholakia, 2005). The non-diffracting and self-

reconstructing properties of a Bessel beam are described below.

2.2.1 Non-diffracting property

Unlike a Gaussian beam, a Bessel beam remains localised and does not spread

with propagation distance (Figure 2.3) i.e., while propagating in the z direction, the

intensity (I) of the Bessel beam satisfies the equality equation as

𝐼(𝑥, 𝑦, 𝑧 ≥ 0) = 𝐼(𝑥, 𝑦) (2.5)

This indicates that the cross section of the beam is unchanged as it propagates,

and the beam can be considered as propagation invariant, or diffraction free (McGloin

& Dholakia, 2005).


Figure 2.3: Intensity at the centre of a Bessel beam (continuous line) and at the centre of a Gaussian beam (dashed line), as a function of propagation distance. The Gaussian beam suffers diffractive

spreading along with propagation distance while the Bessel beam, being resistant to diffractive

spreading, has constant intensity of beam centre over a significant distance. Image taken from Durnin

(1987).

An ideal Bessel beam requires infinite energy to propagate diffraction-free. An

experimentally approximated finite energy Bessel beam maintains its diffraction-

resistant property until a certain distance (𝑧𝑚𝑎𝑥) (Figure 2.2). The 𝑧𝑚𝑎𝑥 of a Bessel

beam generated using an axicon can be approximated as (McGloin & Dholakia, 2005)

𝑧𝑚𝑎𝑥 = 𝐷

2𝜃 (2.6)

where 𝐷 is the diameter of the aperture and 𝜃 (Figure 2.2) is the inclination

angle of the waves to the optical axis after passing through the axicon, which is

represented as (McGloin & Dholakia, 2005)

𝜃 = (𝑛 − 1)𝛼 (2.7)

where 𝛼 is the physical/wedge angle of the axicon (Figure 2.2) and 𝑛 is the

refractive index of the axicon material.


The separation between the wave-vectors of the Bessel beam increases beyond

𝑧𝑚𝑎𝑥, causing the beam to spread-out, for which its Rayleigh length (𝑧𝑅𝐵𝑒𝑠𝑠𝑒𝑙) can be

approximated as (Duocastella & Arnold, 2012)

𝑧𝑅𝐵𝑒𝑠𝑠𝑒𝑙 =2𝐷𝑟0

4λ (2.8)

where 𝑟0 is the central lobe size (Figure 1.1, b) of the beam, which is the radial distance

from the core to the first intensity minimum, and can be approximated as

𝑟0 = 2.405

𝑘 sin 𝜃 (2.9)

where the value 2.405 is derived from the first root of the zeroth-order Bessel function.

The 𝑟0 is equal to the Gaussian beam waist (𝑤0) and is usually much smaller than 𝐷,

which shows that the Bessel beam has much larger Rayleigh length than the Gaussian

beam of the same wavelength.

2.2.2 Self-reconstructing property

The central lobe of a Bessel beam transports only a small portion of its total

energy while the remaining energy is transported in rings (Fahrbach, et al., 2010;

McGloin & Dholakia, 2005; Turunen & Friberg, 2010). If an obstructing object is

placed in the centre of the beam (Figure 2.4), the energy from the rings is transported

to the centre of the beam and the initial profile is reformed beyond obstruction after a

distance (zmin) given by

𝑧min ≈ 𝑎𝑘

2𝑘z (2.10)

where a is the width of the obstruction measured from the beam centre and k is the

wave number.


Figure 2.4: Self-reconstruction of a Bessel beam at various distances after being obstructed by a

rectangular obstacle. (a) z = 0, (b) z = zmin/2, (c) z = zmin, (d) z = 4*zmin. Images taken from Bouchal,

Wagner, & Chlup (1998).

The non-diffracting and self-reconstructing properties of Bessel beams have

been applied in metrology (Häusler & Heckel, 1988), optical imaging with long focal

length (Arimoto, Saloma, Tanaka, & Kawata, 1992), thick media light sheet-based

microscopy (Fahrbach & Rohrbach, 2012), atom guiding (Arlt, Hitomi, & Dholakia,

2000), measuring non-linear refractive index using z-scan method (Hughes & Burzler,

1997), optical micromanipulation (Garces-Chavez, McGloin, Melville, Sibbett, &

Dholakia, 2002), optical interconnection (MacDonald, et al., 1996), medical imaging

(Lu & Greenleaf, 1992), and three-dimensional optical trapping (Tao & Yuan, 2004).

For instance, Garces-Chavez et al. used a single Bessel beam to trap particles in

spatially separated multiple sample cells within optical tweezers by using the beam’s

self-healing property (Garces-Chavez, et al., 2002).

The propagation properties of Bessel beams after passing through an obstruction

have been studied before (Anguiano-Morales, 2009; Anguiano-Morales, et al., 2007;


Bouchal, et al., 1998; MacDonald, et al., 1996; Zheng, et al., 2013), but in linear media

were limited to the obstruction of the central lobe or another small proportion of the

beam. Larger obstructions (e.g. >50 percent of the beam) have been considered for

nonlinear media (Butkus et al., 2002; Sogomonian, Klewitz, & Herminghaus, 1997).

Larger obstructions may occur in media with larger scatterers or with refractive index

inhomogeneity such as biological tissues and ocular media (Fahrbach, et al., 2010;

Garces-Chavez, et al., 2002).

2.2.3 Production of Bessel beams

Ideal Bessel beams have infinite number of rings and require infinite power to

be diffraction free over infinite propagation distance, and hence generating them is not

physically possible. Experimentally, limited by finite aperture and power, Bessel

beams that exhibit diffraction free properties over a limited distance can be generated

using various methods such as use of annular slits (Durnin, et al., 1987), axicon lenses

(Herman & Wiggins, 1991; Indebetouw, 1989; Pu, Zhang, & Nemoto, 2000), spherical

lenses with spherical aberration (Herman & Wiggins, 1991), distributed Bragg

reflectors (Williams & Pendry, 2005), axicon mirrors (Tiwari, Mishra, Ram, & Rawat,

2012), holographic techniques (Vasara, Turunen, & Friberg, 1989), and spatial light

modulators (SLM) (Davis, Carcole, & Cottrell, 1996). These methods commonly

utilise Gaussian beams as incident beams, resulting in the generation of a combination

of Bessel and Gaussian beam profiles and are hence known as Bessel-Gauss beams

(Gori, Guattari, & Padovani, 1987). The amplitude profile of a Bessel-Gauss beam at

a given spatial distance (𝑧) is represented as (Gori, et al., 1987)

𝑉(𝑟, 𝑧) = 𝐴𝑤0

𝑤(𝑧)𝑒

𝑖[(𝑘−𝛽2

2𝑘)𝑧−𝜑(𝑧)]

𝐽0 (𝛽𝑟

1+𝑖𝑧

𝐿

) 𝑒(

−1

𝑤2(𝑧)+

𝑖𝑘

2𝑅(𝑧))(𝑟2+𝛽2𝑧2

𝑘2) (2.11)

where 𝑟 and 𝑧 are the radial and longitudinal coordinates, 𝑤0 is the beam waist of the

incident Gaussian beam, A is a amplitude factor, 𝑘 is the wave number (represented as

Equation 2.4), 𝛽 = 𝑘 sin 𝜃 with 𝜃 being the opening angle of the cone, 𝐿 is the

Rayleigh range, and 𝑤(𝑧), 𝜑(𝑧) and 𝑅(𝑧) are the beam-width, the phase shift and the

radius of curvature for the Gaussian wavefront, respectively.


The Bessel-Gauss beam reduces to a Gaussian beam when 𝛽=0 and reduces to

an ideal Bessel beam when 𝑤0 approaches infinity. For the rest of this thesis, both ideal

Bessel beams and Bessel-Gauss beams will be referred as Bessel beams.

Initially Durnin et al. (1987) generated Bessel beams by illuminating an annular

slit (a ring) located in the back focal length of a positive lens using collimated Gaussian

beams (Figure 2.5). The opening angle of the cone, the vertex angle made by a cross

section through the apex and centre of the base, is given by

tan 𝜃 =𝑑

2𝑓 (2.12)

where 𝑑 is the diameter of the annular slit and 𝑓 is the focal length of the positive lens.

It is an inefficient method for generating the beams as most of the incident power is

obstructed by the slit.

Axicons, lenses with conical surfaces, were first described by McLeod (1954) as

the optical elements that form, along their axis, a continuous line image of a point

Figure 2.5: Generation of Bessel beams using an annular slit of diameter 𝑑 and slit width ∆𝑑, placed at

a focal length 𝑓 of a positive lens that has aperture radius 𝑅. The Bessel beam has a maximum diffraction

free distance 𝑧max (represented by the beginning of the geometrical shadow zone along the 𝑧 axis).

Image taken from Durnin, Miceli & Eberly (1987).


object and suggested using them for alignment. Since then, the optical properties of

axicons have been extensively studied (Fujiwara, 1962; McLeod, 1960; Sheppard,

1977; Sheppard & Choudhury, 1977; Sheppard & Wilson, 1978; Steel, 1960) including

their applications in generation of Bessel beams (Herman & Wiggins, 1991;

Indebetouw, 1989). An axicon refracts rays at approximately the same angle to

produce a conical wave, and hence creating the Bessel beams (Figure 2.2) (Herman &

Wiggins, 1991; McGloin & Dholakia, 2005). Using an axicon allows most of the

incident power to be utilised in the generation of Bessel beams, and hence is a more

efficient method than the use of an annular slit (McGloin & Dholakia, 2005). The

combination of two axicon lenses (dual-axicon lenses) can also be used to generate

Bessel beams with an advantage of achieving a range of focal depths by varying the

distance between the lenses instead of the lens-object distance (Belyi, Forbes, Kazak,

Khilo, & Ropot, 2010).

Details on generation of Bessel beams in this study using SLMs, axicons and

spherical lenses with spherical aberration are given in sections 3.2.1, 6.2 and 4.2,

respectively.

2.3 APPLICATION OF BESSEL BEAMS TO THE EYE

I am aware of only four studies that applied Bessel beams to the eye. Lambert et

al. (2011) investigated the feasibility of using active optics with Bessel beams for

wavefront sensing in an adaptive optics system. Bessel beams provided the sufficient

information to correct aberrations, although at a slightly slower dynamic correction

during loop closure than for Gaussian beams.

Lambert et al. (2013) investigated the effect of a Bessel beam, as a fixation target

in an adaptive optics system, in reducing the temporal variations of aberrations. The

Bessel beam reduced the temporal variations of tilt and coma considerably compared

with those for a Gaussian beam. They suggested that this was due to improved stability

of the eye through suppression of rapid eye movements. However, the study did not

monitor eye positions to verify whether there was improved fixation stability.

Kim et al. (2013) imaged human retinal microstructures with optical frequency

domain imaging systems. They compared the lateral resolution of images between

systems using Bessel beams, generated using dual-axicon lenses, and Gaussian beams.


Dual-axicon lenses are a combination of two axicon lenses that can achieve a range of

focal depths by varying the distance between the lenses instead of varying the lens-

object distance. The lateral resolutions were 6 µm and 15 µm with Bessel and Gaussian

beams, respectively. Images were less affected by spherical aberrations with the Bessel

beams system than with the Gaussian beams system. The study is available only as an

abstract, and details on methods and results are limited.

Hong et al. (2017) developed two aqueous-outflow-system imaging devices

based on light sheet fluorescence microscopy (LSFM); one was static LSFM with a

light source that formed Gaussian beams, and the other was digitally scanned LSFM

with a light source that formed Bessel beams. LSFM uses a thin plane of light to

optically section the tissues with detection axis perpendicular to the illumination axis

(Reynaud, Kržič, Greger, & Stelzer, 2008). A 176° apex angle plano-convex axicon

lens was used to produce Bessel beams. They injected a fluorescence dye into the

anterior chamber of enucleated porcine eyes and imaged the trabecular meshwork. The

axial resolution of images from the digitally scanned Bessel LSFM was 0.5 µm while

that from the static Gaussian LSFM was 6 µm. The digitally scanned Bessel LSFM

produced images with higher axial resolution, better signal-to-noise ratio, reduced

scattering and shadowing artefacts, and greater depth of focus than the static Gaussian

LSFM. They suggested that the better image quality with the Bessel LSFM was due to

the self-reconstructing property of Bessel beams, although it might have been at least

partly because the digitally scanned LSFM technique provided better control over

intensity profile and size of the illumination sheet and had fewer artefacts than the

static LSFM (Keller, Schmidt, Wittbrodt, & Stelzer, 2008).

2.4 PHAKOMETRY

Information on shapes of cornea and lens surfaces can be obtained by analysing

the reflections occurring at these surfaces (Atchison & Smith, 2000; Tscherning,

1924). When light enters the optical system of the eye, four main images of the light

source are formed through reflections at air-cornea, cornea-aqueous, aqueous-lens and

lens-vitreous interfaces; the respective images are known as 1st (PI), 2nd (PII), 3rd (PIII)

and 4th (PIV) Purkinje images. Phakometry is the technique of determining the shape

and refractive index of the in-vivo ocular lens by imaging these Purkinje images.


PI and PII have similar sizes and PII is difficult to see as it is at a similar location

but is much dimmer than PI, PIII has the largest size in the unaccommodated eye

(approximately twice PI) and PIV is slightly smaller than PI and is inverted with

respect to the other three images (Figure 2.6). PI, PII and PIV are formed near the pupil

plane, while PIII is formed in the vitreous in the unaccommodated eye. Upon

accommodation, PIII becomes smaller and moves forward, while PIV moves

backward slightly with little change in size. Using the Fresnel reflection equation for

normal incidence, the brightnesses of PII, PIII and PIV relative to PI are approximately

0.008, 0.013, and 0.013 (Atchison & Smith, 2000; Tabernero, et al., 2006). The quality

and brightness of PIII are deteriorated by a rough anterior lens surface causing diffuse

reflection and scattering of the Gaussian beam, and hence making accurate and

repeatable estimate of lens parameters difficult (Adnan, 2015; Atchison & Smith,

2000; Navarro, et al., 1986; Tabernero, et al., 2006). A Bessel beam remains stationary

and concise along the propagation distance, which might produce a brighter and

sharper PIII than a Gaussian beam and hence aid in accurate estimation of lens surface

curvature.

Figure 2.6: Simulated Purkinje images in a perfectly aligned model eye using a semicircular array of

LEDs as the source. Image taken from Tabernero, Benito, Nourrit, & Artal (2006) .


2.5 FIXATION STABILITY

Stable fixation is needed to obtain accurate findings for ophthalmic procedures

such as retinal imaging, multifocal electroretinograms, functional magnetic resonance

imaging and visual-evoked potential responses (Elsner, et al., 2013; Menz, et al., 2004;

Sutter & Tran, 1992). Involuntary eye movements in the form of micro-saccades,

micro-tremors, drift, and nystagmus create difficulties in achieving stable fixation, and

this becomes more challenging in conditions such as age-related macular degeneration,

diabetic maculopathy, Down syndrome, amblyopia, and Parkinson’s and Hodgkin’s

diseases (Kube, Schmidt, Toonen, Kirchhof, & Wolf, 2005; Peter, Baumgartner, &

Greenlee, 2010; Zhang et al., 2008).

A good choice of fixation targets is important to maximise fixation stability

(Lambert, et al., 2013; Steinman, 1965; Thaler, et al., 2013). Studies with monitor-

based targets suggest that shapes such as a circular point (Hirasawa, Okano, Koshiji,

Funaki, & Shoji, 2016; Rattle, 1969), a cross (Bellmann, Feely, Crossland, Kabanarou,

& Rubin, 2004), a bull’s eye/cross hair combination (Thaler, et al., 2013), and a “%”

optotype (Pirdankar & Das, 2016) provide good fixation stability. For instance, Thaler

et al. (2013) compared the fixation stability of participants using different fixation

target shapes that included circular shapes, crosses, and combinations of these two

basic shapes (Figure 2.7). The lowest micro-saccade rate occurred for the bull’s

eye/cross hair combination (target shape ABC in Figure 2.7).

Figure 2.7: Different target shapes used by Thaler et al. to determine the effect of target shape on

stability of fixational eye movements. Image taken from Thaler, Schutz, Goodale, & Gegenfurtner

(2013).


Bessel beams, with diffraction resistant and self-reconstructing properties,

improve image quality compared with conventional Gaussian beams by decreasing

scattering artefacts, improving depth of field and improving penetration in dense media

(Durnin, et al., 1987; Rohrbach, 2009). As mentioned in section 2.2, Lambert et al.

(2013) reported that a Bessel beam used as a fixation target in an adaptive optics

system reduced temporal variations of aberrations from those found with a

conventional Gaussian beam (Figure 2.8). They suggested that this was due to

improved stability of the eye through suppression of rapid eye movements. However,

the study did not compare the effectiveness of a Bessel beam, as a fixation target, in

improvement of fixation stability over that for the standard fixation target shapes such

as a bull’s eye/cross hair combination.

Figure 2.8: Time frequency plots of the Zernike tip aberration coefficient. Energy burst, perhaps due to

micro-saccades, are present with Gaussian beam (left) but not with the Bessel beam (right). The most

prominent energy burst is indicated by the arrow. Image taken from Lambert, Daly, & Dainty (2013).


2.6 RETINAL IMAGING

Retinal imaging plays a crucial role in the study of retinal structures, through

aiding in diagnosis and management of retinal and chronic systemic diseases

(Abramoff, Garvin, & Sonka, 2010; MacGillivray et al., 2014). Different modalities

used for retinal imaging include flood illumination ophthalmoscopy that illuminates

over larger regions than other modalities, scanning laser ophthalmoscopy (SLO)

involving a scanning beam over a small retinal region, and optical coherence

tomography (OCT) to obtain depth information (MacGillivray, et al., 2014).

2.6.1 Modalities for retinal imaging

Flood illumination ophthalmoscopy

Flood illumination ophthalmoscopy uses low coherence narrow spectral

bandwidth light source to illuminate ~ 1° isoplanatic patch of retina and images it with

a CCD camera, placed at a retinal conjugate plane (Figure 2.9) (Hampson, 2008;

Liang, et al., 1997). The low coherence light reduces speckle (a random granular

shaped phase pattern in the image plane resulting from interference among wavefronts

that are coherent but differ in phase, orientation, spacing and intensity) that originates

from scattering optical components (Goodman, 1976). The narrow spectral bandwidth

of illuminating light reduces the impact of chromatic aberration. However, the light

sources used for wavefront sensing and imaging are usually of different wavelengths,

hence requiring error adjustment for this chromatic discrepancy (Fernández et al.,

2005). There is a trade-off between image resolution and the maximum amount of light

that can be exposed to the eye while selecting the wavelength. The use of a short

wavelength decreases the width of the point spread function and improves the

resolution. However, for an exposure of long duration (> 600 seconds), the maximum

permissible exposure for a 514 nm laser is around 18 times lower than that for 633 nm

laser, hence limiting the amount of light illuminating the retina for lower wavelengths

(Seeber, 2007).


Figure 2.9: Flood illumination ophthalmoscope. The flash-lamp illuminates the retina. The aberrations

are corrected by a deformable mirror before capturing the image. Image redrawn from Hampson (2008).


Scanning laser ophthalmoscopy (SLO)

SLO uses a laser to scan across the retina and builds up point-by-point images.

The light passes through an aperture, a confocal pinhole, placed at a retinal conjugate

plane before reaching the detector (Figure 2.10) so that the scattered light from other

parts of retina can be avoided, and hence image contrast is increased (Hampson, 2008).

Unlike flood illumination ophthalmoscopy, it eliminates the error due to

chromatic aberration because it uses the same light source for wavefront sensing and

imaging. The aberrations are measured and corrected for each point of the total

imaging area, which increases the resolution.

Figure 2.10: Confocal scanning laser ophthalmoscope. The point-by-point image is built up after

scanning across the retinal area of interest. Image redrawn from Hampson (2008).


Optical coherence tomography (OCT)

OCT is an interferometry based imaging system that uses a low-coherence light

(Figure 2.11) (Hampson, 2008). The light beam is split into two parts, with one part

directed on to the retina and its reflection brought back to interfere with a reference

beam to produce interference patterns. The interference pattern depends on the

difference between the optical path length of the reference beam and the optical path

length of the beam reflected from the retina. The intensity profiles of these interference

patterns are used to construct axial A-scans with high axial resolution. Combining

these A-scans in a line provides a two-dimensional cross sectional image of the retina,

known as a B-scan (de Amorim Garcia Filho, Yehoshua, Gregori, Puliafito, &

Rosenfeld, 2013). The bandwidth of the light source used in an OCT determines the

axial resolution, while the diffraction from the pupil and the AO system performance

determines the transverse resolution (Drexler et al., 2001).

Figure 2.11: Spectral/Fourier domain based OCT. cl, coherence length; OPD, optical path difference;

λ, wavelength of the light source. Image redrawn from Podoleanu (2012).


Miller et al. (2003) reported the first use of an AO system in an OCT, operated

in a time-domain model, which achieves the axial images by changing the mirror

position of the reference beam and generates the depth information over time. The

time-domain AO-OCT provides axial resolution of around 3 – 14 µm and transverse

resolution of 3 – 5 µm (Hermann et al., 2004; Miller, et al., 2003) but has an issue of

image blur as it is affected by eye movement. Zhang et al. (Zhang, Rha, Jonnal, &

Miller, 2005) developed an AO-OCT based on spectral/Fourier domain model, which

uses a spectrometer to detect the OCT signal while keeping the reference arm static.

The spectral/Fourier-domain model allows higher speed imaging and higher sensitivity

than the time-domain model (Leitgeb, Hitzenberger, & Fercher, 2003; Nassif et al.,

2004; Podoleanu, 2012).


2.6.2 State-of-the-art of retinal imaging technology

Imaging modalities combined with an AO system improve resolution and

contrast of the image (Figure 2.12) (Liang, et al., 1997; Roorda et al., 2002). AO based

imaging systems are now being used to resolve the smallest photoreceptor cells of the

retina (Dubra, et al., 2011) which helps in studying the changes occurring at the cellular

levels in clinical trials (Talcott et al., 2011). In recent years AO based retinal imaging

has been mostly developed from a combination of AO-SLO and AO-OCT that

provides more information of structures than from the use of any single imaging

modality (Meadway, Girkin, & Zhang, 2013; Zawadzki et al., 2011). AO-SLO

provides a diffraction limited transverse image (en-face) of retinal structures while

AO-OCT provides high resolution axial retinal images.

Figure 2.12: Image of the retina at 4-deg eccentricityfor subject DM (a) without AO compensation, and

(b) with AO compensation. Image taken from Liang, Williams, & Miller (1997).


Adaptive optics principles

An AO system used in retinal imaging has three principal components: a

wavefront sensor, a wavefront corrector and a control system (Figure 2.13) (Hampson,

2008). Both the wavefront sensor and corrector are located in planes conjugate with

the eye’s pupil. A beam from a laser diode passes into the eye to form a small spot on

the retina which becomes the effective light source for the wavefront sensor. The

reflected wavefront is aberrated by the optical components of the eye. The wavefront

sensor captures the image of the pupil that has phase information of the wavefront.

The control system calculates the wave aberration, and changes the shape of the

wavefront corrector to compensate. A separate light source, that faces the same

aberration while passing through the optical components of the eye as that of the beam

used for wavefront sensing, and a separate camera are used for acquiring retinal

images.

Figure 2.13: Basic layout of adaptive optics system. (1) Wavefront sensor, (2) Corrector and (3) Control

computer. Image redrawn from Hampson (2008).


Wavefront sensor

There are three main techniques used for wavefront sensing in the eye: the ray-

tracing, the pyramid sensing, and the Hartmann-Shack wavefront sensing. Most

commercial AO systems used in retinal imaging use the latter technique. The

wavefront sensor I will be discussing in the subsequent sections of the thesis is the

Hartmann-Shack Wavefront Sensor (HSWS) unless specified otherwise. Details on

ocular aberrations and their measurement using the HSWS are described in the next

section.

Wavefront corrector

There are two main types of correctors: deformable mirrors and liquid crystal

spatial light modulators (LC-SLMs). Both correct the phase error of an aberrated

wavefront through phase conjugation (Hampson, 2008).

The electric field magnitude of the wavefront can be represented as:

𝐸 = 𝐴 exp (−𝑖∅) (2.13)

where 𝐴 is the amplitude and ∅ is the phase. ∅ is given by:

∅ =2𝜋 ∆𝑂𝑃𝐿

𝜆 (2.14)

where 𝜆 is the wavelength of light and ∆𝑂𝑃𝐿 is the optical path length error of the part

of wavefront:

∆𝑂𝑃𝐿 = 𝑛1∆𝑧1 + 𝑛2∆𝑧2 + 𝑛3∆𝑧3 + ⋯ (2.15)

where 𝑛𝑥 is the refractive index in a medium and ∆𝑧𝑥 is the distance error travelled by

the wavefront in that medium. The alteration of 𝑧 in deformable mirrors and 𝑛 in LC-

SLMs corrects the aberrated wavefront (Figure 2.14).


Figure 2.14: Wavefront correction principle by manipulating OPL. (a) Deformable mirrors change the

physical path length. (b) LC-SLMs change the refractive index. Image redrawn from Hampson (2008).

Deformable mirror

A deformable mirror consists of a mirrored surface and actuators (Figure 2.15

and Figure 2.16). The actuators deform the mirrored surface to correct the aberrated

wavefront. The mirrored surface is deformed into the opposite shape, but at half the

amplitude, of the aberrated wavefront (Hampson, 2008).

The mirrored surface is either segmented or continuous. A segmented surface

consists of individual mirrors attached to separate actuators that can move with one or

three degrees of freedom (Figure 2.15). The control algorithm for segmented surfaces

is simpler than for continuous surfaces and can be used to correct higher spatial

frequency aberrations. Segmented surfaces are easier to manufacture than continuous

surfaces. However, segmented surfaces have gaps between the segments which cause

difficulty in smooth approximation to the wavefront due to light loss and diffraction

effects, and hence are less suited to correct lower spatial frequency aberrations

(Hampson, 2008; Lombardo, Serrao, Devaney, Parravano, & Lombardo, 2012).

Figure 2.15: Segmented deformable mirrors: (a) piston only and (b) piston, tip and tilt. Image redrawn

from Hampson (2008).

Aberrated wave

Plane wave Plane wave

(a) (b)


In continuous surfaces, the surface deflection caused by the movement of

actuators is not restricted to the area directly above the given actuators (Figure 2.16).

The continuous surface provides necessary smooth approximation to wavefront to

correct higher spatial frequency aberrations. There are four types of continuous

deformable mirrors. The first type consists of actuators, attached to the surface

membrane, that expand to change shape of the membrane when a voltage is applied

(Figure 2.16, a). The second type consists of multiple layers of materials bonded

together as a thermocouple (bimorph) (Figure 2.16, b). The voltage applied to the

electrode causes the lower surface to expand, and hence changes the shape of the

mirror. The third type consists of a thin metallic membrane suspended over an array

of actuators working on the principle of electrostatic attraction and has high control

over the surface (Figure 2.16, c). The fourth type uses a magnetic field that attracts or

repels the mirror surface depending upon the need (Figure 2.16, d).

Figure 2.16: Continuous deformable mirrors: (a) mirror actuated by actuators that expand or contract,

(b) bimorph mirror, (c) membrane mirror, and (d) magnetic mirror. Image redrawn from Hampson

(2008).


Liquid crystal spatial light modulators

LC-SLMs are based on neumatic liquid crystal technology to correct phase

distortion which work in either transmission or reflection mode (Figure 2.17). The

voltage applied to the electrode changes the shape of the liquid crystal molecules to

alter the refractive index, and hence changes the phase of wavefront passing through

them. LC-SLMs have higher resolution than deformable mirrors (~ 1 million pixels

compared with ~ 100 actuators), allowing them to correct high frequency aberrations.

The LC-SLMs can only be used with linearly polarised light and have limited dynamic

range (Lombardo, et al., 2012).

Figure 2.17: Neumatic LC-SLMs operating in reflection mode. Voltage applied to an electrode changes

the alignment of liquid crystal molecules, altering the refractive index and hence modulating the phase

of the light. Image redrawn from Hampson (2008).


Control System

The control system is a computer based program that receives data from the

wavefront sensor and sends the necessary signals to the actuators for aberration

correction. The control algorithm considers geometry by which the sensor and

corrector are matched, correction operation loop, and phase reconstruction method.

Here I have presented a brief details about the control matrix of a HSWS with 32 𝑋 40

lenslets and an electromagnetic deformable mirror with 52 actuators, as sensing and

correcting devices, respectively.

The outcome of the HSWS measurement with lenslets give two data matrices

containing 𝑠𝑥𝑖𝑗 and 𝑠𝑦𝑖𝑗

, the projections of ∆𝑠𝑖𝑗 on 𝑥 and 𝑦 axes, where 𝑖 = (1,2, . . ,32)

and 𝑗 = (1,2, . .40) are the lenslet array indices (Figure 2.18) and can be given by:

𝑠𝑥𝑖𝑗= [ ]32×40 , 𝑠𝑦𝑖𝑗

= [ ]32×40 (2.16)

Figure 2.18: The Hartmann-Shack spots distribution on the CCD, generated by the two dimensional

lenslet array where 𝑑 is the diameter of the lenslet, and 𝑖 and 𝑗 represent the lenslet indices.


The slope vector 𝑆 of a wavefront measurement is built from 𝑠𝑥 and 𝑠𝑦,

𝑆 = [𝑠𝑥1, 𝑠𝑥2, . . , 𝑠𝑥𝑘 , 𝑠𝑦1, 𝑠𝑦2, . . 𝑠𝑦𝑘]1×2𝐾

𝑇 (2.17)

where 𝐾 is the number of valid lenslets, 𝐾 ≤ 32 × 40 and 𝑇 is the transposed matrix.

All the slope vectors are measured and built after poking each actuator one by one.

The vectors are assembled to create an interaction matrix 𝐼𝑀 given by:

𝐼𝑀 = [𝑠1, … , 𝑠𝑚 , … , 𝑠𝑀]2𝐾×𝑀

= (2.18)

where 𝑀 = 52 (number of actuators).

The voltage command vector 𝑉 has the voltages to be applied to the actuators of

the deformable mirror which is given by:

𝑉 = [𝑣1, 𝑣2, … , 𝑣𝑀]1×𝑀 (2.19)

Once the interaction matrix (𝐼𝑀) of the sensor and deformable mirror

configuration is known, the slope vector 𝑆 of the wavefront produced by the

deformable mirror in response to the applied voltages 𝑉 is given by:

𝑆 = 𝐼𝑀 × 𝑉 (2.20)


The control algorithm calculates the voltage command vector 𝑉 to be applied to

the deformable mirror so that the required phase distortion to correct aberrations is

achieved. Since 𝐼𝑀 is not an invertible square matrix, its pseudo inverse is used in

calculating the voltage command vector.

The AO system works in either of two modes: open-loop or closed-loop. In

open-loop mode, measurement and compensation of aberrations is done only once at

the beginning of the experiment so that corrector is placed at a fixed position

throughout the experiment. In closed mode, the sensor and corrector are kept in

feedback loop so that the phase distortion is measured and corrected throughout the

experiment.

Ocular aberrations

The retinal image formed by the ocular optical components is affected by optical

imperfections called aberrations.

Classification of aberrations

The aberrations of eye are widely classified as chromatic and monochromatic

aberrations. The variations in the refractive indices of the ocular optical components

with change in the wavelength of the polychromatic light cause their focal lengths to

be a function of wavelength resulting in chromatic aberrations. The equivalent

refractive index of the eye decreases as the wavelength increases and is approximately

1.3445 for 410 nm and 1.3303 for 694 nm (Thibos, Ye, Zhang, & Bradley, 1992).

Chromatic aberrations are classified into longitudinal and transverse chromatic

aberrations. Longitudinal chromatic aberration can be quantified either as the variation

in power with wavelength (chromatic difference of power) or as the variation in object

vergences of the source with wavelength for which the source is focused at the retina

(chromatic difference of refraction). The chromatic difference of refraction varies by

about 2.0 D across the visible spectrum (Atchison & Smith, 2000). Transverse

chromatic aberration is the variation in transverse displacement of the image principal

rays with wavelength (chromatic difference of position).


Monochromatic aberrations are those occurring at any wavelength. Refractive

errors are referred to as lower-order aberrations while other aberrations such as

spherical aberration and coma are known as higher-order aberrations (Atchison &

Smith, 2000). The aberrations I will be discussing in the subsequent sections of the

thesis are monochromatic aberrations unless specified otherwise.

Representation of aberrations

A wavefront is a uniform phase surface, orthogonal to light rays and usually

perpendicular to the direction of propagation. Aberrations of the eye can be

represented in three ways: wave aberrations, transverse aberrations and longitudinal

aberrations (Figure 2.19) (Atchison & Smith, 2000). Wave aberration, the most

commonly represented form, is the departure of the aberrated wavefront from its ideal

form when measured at the exit pupil of an optical system. Transverse aberration is

the departure of a ray from its ideal position at the image surface. Longitudinal

aberration is the deviation of intersection of a ray with the reference axis from its ideal

intersection position.

Figure 2.19: Aberrated eye: (a) image plane, (b) exit pupil, (c) ideal wavefront, (d) aberrated wavefront,

(e) wave aberration, (f) transverse aberration, (g) longitudinal aberration.

Zernike polynomials, a complete set of polynomials defined over a unit circular

pupil, are the most accepted way of representing wave aberrations of an optical system

(Atchison, 2004). Using Zernike polynomials, wavefront aberrations can be defined as


𝑊(𝜌, 𝜃) = ∑ ∑ 𝑐𝑛𝑚𝑍𝑛

𝑚(𝜌, 𝜃)𝑛𝑚=−𝑛

𝑛−|𝑚|=𝑒𝑣𝑒𝑛

𝑘𝑛=0 (2.21)

where 𝑊(𝜌, 𝜃) is a polar representation of the wave aberration, 𝑐𝑛𝑚is the coefficient of

Zernike polynomial 𝑍𝑛𝑚(𝜌, 𝜃), 𝜃 is the meridian in radians measured from positive

horizontal axis (to an observer’s right when looking at the patient) in the anti-

clockwise direction, 𝜌 is the relative distance from the pupil centre to the aberration

measuring point which ranges from 0 to 1, 𝑘 is the highest order of radial polynomial.

The Zernike polynomial function 𝑍𝑛𝑚 is defined as

𝑍𝑛𝑚(𝜌, 𝜃) = {

𝑁𝑛𝑚𝑅𝑛

∣𝑚∣(𝜌) cos(𝑚𝜃) , 𝑓𝑜𝑟 𝑚 ≥ 0

𝑁𝑛𝑚𝑅𝑛

∣𝑚∣(𝜌) sin(|𝑚|𝜃) , 𝑓𝑜𝑟 𝑚 < 0} (2.22)

where 𝑅𝑛∣𝑚∣ is a radial polynomial and 𝑁𝑛

𝑚 is a normalisation term. 𝑅𝑛∣𝑚∣ is given by

𝑅𝑛∣𝑚∣(𝜌) = ∑

(−1)𝑠(𝑛−𝑠)!

𝑠!{0.5(𝑛+|𝑚|)−𝑠}!{0.5(𝑛−|𝑚|)−𝑠}!𝜌𝑛−2𝑠(𝑛−|𝑚|)/2

𝑠=0 (2.23)

where 𝑛 is the radial index (highest radial polynomial power) and 𝑚 is the meridional

frequency index of the sinusoidal component.

The normalisation term 𝑁𝑛𝑚 is given as

𝑁𝑛𝑚 = √𝑛 + 1 𝑓𝑜𝑟 𝑚 = 0, 𝑎𝑛𝑑 𝑁𝑛

𝑚 = √2(𝑛 + 1) 𝑓𝑜𝑟 𝑚 ≠ 0 (2.24)

The total optical system aberrations are defined using the root mean square

(𝑅𝑀𝑆) aberration value of coefficient of Zernike polynomials as a measure of image

quality. 𝑅𝑀𝑆 is given by

𝑅𝑀𝑆 = √∑ (𝑐𝑛𝑚)2

𝑛>1,𝑚 (2.25)

The Zernike polynomials have orthonormality properties where changes of

higher-order terms do not affect lower-order coefficients and all polynomials except

the zero-order term have a mean value of 0 across the pupil. Zernike polynomial

functions are expressed as 𝑍𝑛𝑚 where order 𝑛 changes vertically and azimuthal

frequency 𝑚 changes horizontally. Figure 2.20 shows the Zernike polynomial terms

plotted as a function of position in the pupil for terms up to fifth-order.


Figure 2.20: Zernike polynomial function pyramidwhere order (𝑛) changes vertically and azimuthal

frequency (𝑚) changes horizontally. The Zernike polynomials with order terms 0, 1 and 2 are known

as low order aberrations and order terms ≥ 3 are known as high order aberrations. Image taken from

http://www.telescope-optics.net/monochromatic_eye_aberrations.htm

The first-, second- and third-row polynomial functions (𝑛 ≤ 2) are lower-order

aberrations. The first-row polynomial (𝑛 = 0), known as piston, and second row

polynomials (𝑛 = 1), known as prisms or tilts, are often ignored as they do not affect

image quality (Atchison, 2004).

The third-row polynomials (𝑛 = 2) are oblique astigmatism (𝑍2−2), rotationally

symmetrical defocus (𝑍20) and with/against the rule astigmatism (𝑍2

2). Second-order

aberrations represent traditional refractive errors and can be corrected using

conventional lenses.

The polynomial functions higher than the second-order (𝑛 ≥ 3) are higher-order

aberrations. The fourth-row polynomials (𝑛 = 3) are vertical trefoil (𝑍3−3), vertical

coma (𝑍3−1), horizontal coma (𝑍3

1) and oblique trefoil (𝑍33). The middle polynomial

function of fifth-row polynomials (𝑛 = 4) is rotationally symmetrical and is known as

spherical aberration. Vertical coma, horizontal coma and spherical aberration are

usually present in higher amounts than other higher-order aberrations in most eyes


(Thibos, Applegate, Schwiegerling, & Webb, 2002). Higher-order aberrations affect

retinal image quality significantly and limit the range of spatial frequencies and reduce

image contrast transmitted by the optics (Liang & Williams, 1997). Higher-order

aberrations increase as pupil size increases. For a 3.0 mm diameter pupil, aberrations

up to third-order should be corrected to achieve a well-corrected system with a 0.8

Strehl intensity ratio. For a 7.3 mm diameter pupil, an 0.8 Strehl ratio can only be

achieved after correcting aberrations up to eighth-order (Liang & Williams, 1997). It

is important to use the same pupil diameter while comparing aberrations of different

eyes, or the aberrations of an eye at different times (Atchison, 2004).

Measurement of aberrations

Aberrations of the eye can be measured objectively by ‘into-the-eye’

aberrometry and ‘out-of-the-eye’ aberrometry (Atchison, 2005). In ‘into-the-eye’

aberrometers, a narrow light beam is directed into the eye through a specific pupil

position to form an image on the retina. The retinal image is then re-imaged out of the

eye through the whole pupil and is compared with that of the reference image to

determine the aberrations. These aberrometers take sequential measurements at

different pupil positions. In ‘out-of-the-eye’ aberrometers, a narrow beam is projected

into the eye and the rays from retina to out of the eye are traced to determine the

aberrations (Atchison, 2005). These aberrometers take simultaneous measurements at

a number of pupil positions. All the existing techniques for aberrometry measure

transverse aberrations except for the retinoscopic technique that measures longitudinal

aberrations (Atchison, 2005).

The Hartmann-Shack aberrometer

The Hartmann-Shack aberrometer, the most commonly used type of

aberrometer, is an ‘out-of-the-eye’ aberrometer. It consists of three main components:

a monochromatic light source, a lenslet array and a light detector.

A narrow beam from a point light source is projected on to the retina. A part of

this passes back from the retina. The wavefront exiting the eye passes through a micro-

lenslet array that is at a pupil conjugate plane (Figure 2.21). When the wavefront passes

through the micro-lenslet array, it is broken into small light beams, each of which is


focused at the detector of a CCD camera to form a spot. When a plane wavefront from

a perfect eye passes through the micro-lenslet array, it forms a uniform grid of spots

that can also be used as a reference grid. For an aberrated eye, the aberrated wavefront

forms a distorted grid on the detector due to the deviations of spots from the reference

grid. The displacement of each spot is proportional to the local slope of the wavefront

(transverse aberration) at the corresponding pupil location. Mathematical integration

of the displacement information across all the micro-lenslets produces the wavefront

aberration (Atchison, 2005).

Figure 2.21: Principle of the Hartmann-Shack aberrometer. When a wavefront passes through a lenslet

array placed at a pupil conjugate plane, it forms a grid of spots in the detector plane at the focal plane

of the lenslet array. The wavefront passing through each micro-lens corresponds with a small region of

the pupil. (a) For a perfect eye, the plane wavefront forms a uniform reference grid on the detector. (b)

For an aberrated eye, the aberrated wavefront forms a distorted grid on the detector. The displacement

of each spot from a reference position is proportional to the local slope of the wavefront (transverse

aberration) at corresponding pupil location. Mathematical integration of this displacement information

across all the micro-lenslets produces the wavefront aberration.

The Hartmann-Shack aberrometer is faster, more reliable and less affected by

the scattering of light than most other aberrometers (Atchison, 2005). However it has

limited dynamic range, creating problem while measuring high aberrations due to spot


overlapping, and hence requires auxiliary optics to correct most of the defocus

(Atchison, 2005). The ingoing and ongoing beams from the point source of the

aberrometer have Gaussian beams properties that diffract and scatter while passing

through ocular media (Rohrbach, 2009; Siegman, 1986). The scattering of the

Gaussian beam in combination with poor retinal reflectivity cause poor signal-to-noise

ratio and ghost images creating difficulties in locating the accurate positions of the

spots in the detector of the CCD camera. The following section provides brief

information about the ocular scatter.

Ocular scatter

Scatter is a physical phenomenon, intrinsic to light propagation in media with

inhomogeneities, which causes the light to deviate from its theoretical straight

trajectory (van de Hulst, 1981). It is due to a combination of refraction, reflection and

diffraction. There are two types of scatter: forward scatter and backward scatter. When

a light beam is incident on a transparent object placed in a medium of a different

refractive index, some of the light is reflected in a backward direction, some is

refracted in a forward direction, and some is reflected inside the object a number of

times before being refracted backwards or forwards. Light outside and near the edge

of the object is diffracted in a forward direction. The angular distribution of scatter

depends upon the size and shape of a scattering particle, the difference in the refractive

indices between the media and the particle, the wavelength, the scale of inhomogeneity

induced by the particle relative to the wavelength, and the spatial regularity in the

inhomogeneity (Atchison & Smith, 2000).

Effects of scatter on eye

The forward-scattered light impinges on the retina producing a visual effect, like

a veil of light, known as straylight. This causes glare, and adversely affects visual

functions such as contrast sensitivity and colour discrimination, but hardly affects the

visual acuity and measurement of aberrations (Artal, 2017; Paulsson & Sjöstrand,

1980; Wanek, et al., 2007). Forward scatter reduces the contrast and signal to noise

ratio of the images of intraocular structures (Wanek, et al., 2007). The backward scatter

from the ocular structures reduces the light reaching the retina, but it is unlikely to


affect the visual function much at all unless operating at low light levels. The backward

scatter is used to assess the quality of ocular tissues using slit-lamp biomicroscopy and

Scheimpflug photography (Artal, 2017).

Scattering theory

For scattering particles that are mutually incoherent and independent, and whose

dimensions are much smaller than the wavelength of the incident light, Rayleigh

theory can be used to describe the scattering process. This theory assumes the

scattering particles to be polarisable. According to this theory, the electric field of the

incident radiation polarises the electronic structure of each particle into small dipoles.

These dipoles oscillate in time with the incident radiation. The dipoles absorb energy

from the incident field and re-radiate it. Rayleigh scattering predicts that scatter is

proportional to the inverse of the fourth power of the wavelength e.g. 410 nm light is

scattered seven times more strongly than 670 nm light (Atchison & Smith, 2000).

Mie theory can explain the scattering for spherical scattering particles of any

size. For large spheres, scattering is independent of the wavelength of the incident

light. Assuming a flat monochromatic incident wavefront and a spherical scattering

particle, the amount of light scattered is directly proportional to the amount of light

incident on cross section of the particle. For a given volume of homogenous media of

spherical scattering particles such as aerosols, the scattering is directly proportional to

the number of scattering particles and their total cross sectional area.

For scatterers that are not small compared to the wavelength but have a small

refractive index difference with the surrounding medium, Rayleigh-Gans or Rayleigh-

Deybe theory can be applied. This theory predicts that, for a spherical scattering

particle, forward-scatter increases and back-scatter decreases with increase in the

particle size.

Sources of scattering in the eye

Scatter is present at every discontinuity in the human eyes (Figure 2.22). The

scatter depends on age, pigmentation of the structures such as iris and retina, and

pathologies or surgical interventions affecting transparency (Pinero, Ortiz, & Alio,

2010). In healthy eyes, the cornea and lens have cells and connective tissue, which


contain inhomogeneities on the scale of the order of the wavelength of light, but

surprisingly have a high transparency (Atchison & Smith, 2000). However, any

opacity in them increases both forward and backward scatters (Braunstein et al., 1996;

Michael et al., 2009). The iris and sclera are not completely opaque so they are

potential source of forward scatter, depending upon the grade of pigmentation and

structural density (van den Berg, IJspeert, & De Waard, 1991). Some light reaching

the retina reflects and contributes to intraocular scatter. The aqueous and vitreous

contribute less than other parts, but pathological conditions such as cells in aqueous

and opacities or floaters in vitreous increase scatter considerably (Mura et al., 2011).

The eye has more forward scatter than backward scatter.

Psychophysical method for measuring intraocular forward scatter ‒ the

compensation comparison method

The C-Quant (Oculus Optikgeräte, Wetzler, Germany) uses the compensation

comparison method to measure straylight, a functional measure of the forward scatter

(Franssen, Coppens, & van den Berg, 2006; Pinero, et al., 2010). A flickering light

(straylight source) is presented in the peripheral ring field (Figure 2.23). A part of this

light is scattered by the eye into the right and left halves of a central test field. One half

of the central test field (field “a”) is also presented with a compensation light,

Figure 2.22: Different sources of scattering in the human eye: cornea, sclera, iris, lens, vitreous humour,

and retina. Image redrawn from Pinero, Ortiz, & Alio (2010).


modulated at the same frequency but in counter-phase with the straylight, while the

other half (field “b”) receives only the straylight. This results in two different flickers

in the central half fields. There are two consecutive stages. In the first stage, the

intensity of the straylight source is varied, while that of the compensation light is kept

constant. In the second stage, the intensity of the straylight source is kept constant,

while that of the compensation light is varied. The first stage determines a coarse

estimate of the straylight value while the final stage refines it. Depending upon the

balance between the flickers of the straylight and the compensation light, the field “a”

either flickers stronger or weaker than the field “b”. A participant choses the half that

is perceived to have stronger flicker. A maximum likelihood technique is used to fit a

psychometric curve to the participant’s responses. The instrument uses a two-

alternative-force-choice psychophysical measurement algorithm to obtain the

straylight parameter (𝑠), expressed in log units, which is half the value of the 50 percent

point of the psychometric curve.

Figure 2.23: Stimulus layout for retinal straylight measurement in C-Quant. A flickering straylight

source is presented in the peripheral ring field. The central test field is divided into two halves, each

of which is perceived as flickering, but with different brightness, due to the combination of flickering

straylight and compensation light. Participant choses the half that has stronger flicker. Image taken

from Franssen, Coppens, & van den Berg (2006).


Optical methods

The Optical Quality Analysis System (OQAS) (Visiometrics SL, Terrasa, Spain)

is the only commercially available device that uses a double pass technique to measure

the scatter of the eye (Güell, Pujol, Arjona, Diaz-Douton, & Artal, 2004). The light

point source is imaged on the retina (Figure 2.24). A diffuse reflection of light from

the retina passes back through the optics of eye as a second pass and is focused on a

charge-coupled device camera. The external image is divided into two regions: a

central region within a circle of 1 min of arc radius with the highest intensity at the

circle centre, and a peripheral ring set between 12 and 20 minutes of arc from the circle

centre. The objective scattering index (OSI) is calculated as the ratio of integrated

intensity of light at the peripheral area to that of the central area. The higher the OSI

value, the higher the scatter.

Figure 2.24: Schematic diagram of a double pass system (the OQAS device).D, achromatic double lens;

AP, artificial pupil; BS, beam splitter; CCD, charge-coupled device. Image taken from Pinero, Ortiz, &

Ali (2010).


The OQAS uses infrared light (Güell, et al., 2004), which penetrates into the

deeper layer of retina even reaching the choroid, to measure the scatter objectively.

The intensity of light measured for the peripheral ring is likely to be dwarfed by the

back-scattered infrared light from the choroid, and hence largely invalidating the OSI

parameter. In addition, the visual performances evaluated using infrared light might

not be accurate or relevant as visual sensitivity peaks at 550 nm (van den Berg, 2010).

The Hartmann-Shack sensor is another optical method used to measure the

scatter of the eye. The displacements of the spots with respect to their ideal positions

carry information about the aberrations, and in addition the spread of the pixel values

at a small distance from the peaks of the point spread functions (PSFs) (scatter

distribution) carry information about the scatter (Figure 2.25). The bitmap image of

the Hartmann-Shack spot pattern is used to assess the scatter distribution. The scatter

distribution across an area of interest can be represented using the metric of mean

standard deviation of the pixel values of all lenslet PSFs neighbourhoods (𝑀𝑒𝑎𝑛_𝑆𝐷)

(Donnelly, Pesudovs, Marsack, Sarver, & Applegate, 2004) as

𝑀𝑒𝑎𝑛_𝑆𝐷 = ∑ √

∑ 𝑃(𝑖,𝑗)2𝑖,𝑗

𝑀−(

∑ 𝑃(𝑖,𝑗)𝑖,𝑗

𝑀)

2

𝑁

𝑁 (2.26)

where 𝑃(𝑖, 𝑗) is the intensity value of a pixel at a location (𝑖, 𝑗), 𝑀 is the total number

of pixels minus the central pixels (representing the spread due to aberrations)

surrounding the centroid in each pixel neighbourhood (a square perimeter surrounding

the centroid of each PSF of total pixels determined by average centroid spacing), and

N is the total number of PSFs in the Hartmann-Shack spot pattern.


The low sampling of microlenses (usually more than 100 µm apart) limits the

precision of the measurement by the Hartmann-Shack wavefront sensor. High

aberrations produce spread-out spots such that it is difficult to accurately locate their

centroids.

2.6.3 Challenges in retinal imaging

Regardless of the type of imaging modality used, light sources in retinal imaging

form Gaussian beams that suffer from aberrations, diffraction and scatter while passing

through the ocular media, creating challenges in obtaining high quality retinal images

(Rohrbach, 2009; Wanek, et al., 2007). Liang and Williams (1997) have reported that,

even after correction of defocus and astigmatism, high order aberrations reduced

retinal image contrast by the factor of 7. These aberrations, however, can be detected

and corrected using adaptive optics during retinal imaging to improve the resolution

and contrast of the image (Liang, et al., 1997; Roorda, et al., 2002). The image quality

Figure 2.25: Forward scatter principle using Hartmann-Shack sensor. Panel A shows cross-section

of key elements. Panel B shows locally affected lenslets as a result of local scattering from cataract. Panel C shows the magnified view of the affected lenslets. Panel D shows the frontal view of affected

lenslets as the darkened area. Panel E shows the view of two neighbouring lenslets images: the top

image is less affected by the scatter than the bottom image. Image taken from Donnelly, Pesudovs,

Marsack, Sarver & Applegate (2004).


obtained by using adaptive optics retinal imaging is still limited by diffraction and

scattering properties of Gaussian beams while passing through ocular media,

especially in the presence of scatterers such as cataracts and other media opacities.

Obtaining high quality retinal images is a challenge in patients who have abnormal

fixation due to conditions such as amblyopia and age-related macular degeneration

(Ditchburn & Ginsborg, 1953; Gonzalez, Wong, Niechwiej-Szwedo, Tarita-Nistor, &

Steinbach, 2012; Sivaprasad, Pearce, & Chong, 2011).

Scatter is present in every discontinuity in the human eyes. The light entering

the eye suffers from forward and backward scatters, and the quantity of scatter depends

on age, pigmentation of the structures such as iris and retina, and pathologies or

surgical interventions affecting transparency (Pinero, et al., 2010). The increase in age

and cataract increase the ocular scatter, and hence decreasing the amount of light

reaching the retinal area being imaged (De Waard, IJspeert, Van den Berg, & De Jong,

1992) (Kuroda, et al., 2002; Wanek, et al., 2007). Retinal imaging modalities utilise

the backscattered light from the retinal area being imaged. Forward scatter before

reaching the retina causes a significant reduction in the amount of light reaching the

retinal area being imaged. This reduces the amount of backscattered light from this

area, thereby decreasing the signal to noise ratio (Carpentras, et al., 2018; Wanek, et

al., 2007). Studies have been done for compensation of forward scatter in retinal

imaging using approaches such as point spread function reconstruction (Christaras, et

al., 2016) and structured illumination microscopy (Zhou, et al., 2014), but with limited

success.

The loss of light on the way to retinal area of concern due to forward scatter and

diffraction cannot be compensated by increasing the incident light because of the

limited amount of light that can be sent into the eye due to the laser safety standard

(Delori, Webb, & Sliney, 2007). Therefore, identifying approaches that negate the

detrimental effects of forward scatter and diffraction and hence increase the amount of

light reaching the retinal area being imaged will lead to a significant advancement in

the field of retinal imaging.

Bessel beams are localised beams with transverse patterns that remain stationary

along the propagation distance, i.e., these beams are resistant to diffractive spreading,

and also have potential of self-reconstruction despite partial perturbation while passing

through inhomogeneous media (Durnin, et al., 1987; Nowack, 2012). Farhbach et al.


(2010) showed that Gaussian beams lose almost 40 percent of their energy in a

scattering condition, where around 50 percent of the beam field was disturbed by the

scatter, whereas in the same case Bessel beams lose only 5 percent of their energy.

While imaging biological tissues such as skin, Bessel beams reduced scattering

artefacts, and provided better image quality and penetration depth in dense media than

the Gaussian beams (Fahrbach, et al., 2010; Rohrbach, 2009). Retinal imaging might

benefit from using Bessel beams as illuminating beams by increasing the amount of

light reaching the retinal area being imaged.

2.7 AIMS AND HYPOTHESES

In sections 2.4, 2.5 and 2.6, I identified some limitations of the use of Gaussian

beams in phakometry, fixation stability and retinal imaging.

During phakometry, the rough anterior lens surface causes scattering and diffuse

reflection of Gaussian beams, deteriorating the quality and lowering the brightness of

PIII and hence making accurate, repeatable estimate of lens parameters difficult. The

non-diffracting and self-reconstructing properties of the Bessel beam might increase

the specular reflected signals from the lens surfaces and produce brighter, sharper PIIIs

than Gaussian beams, and thus give more accurate and repeatable estimations of lens

surface curvatures.

Aim 1 (Experiment 1 and Chapters 3 and 4)

To determine the Purkinje image brightness, accuracy and repeatability of

Bessel phakometer compared with those of Gaussian phakometer.

Hypothesis

Using a Bessel beam rather than a Gaussian beam in phakometry will produce

brighter 3rd and 4th Purkinje images resulting in more accurate and repeatable

estimates of lens surface curvature.

Lambert et al. (2013) reported that a Bessel beam, when used as a fixation target

in an AO system, reduced temporal variation of aberration from that found with a

Gaussian beam; this could be due to improved fixation stability. Retinal imaging

instruments requiring fixation might benefit from using Bessel beams in the dual roles


of illuminating beams and fixation targets, thus eliminating the need of a screen for

targets and making instruments slimmer and lighter. There remains an outstanding

question of whether Bessel beams improve fixation stability compared with the

conventional targets i.e. Gaussian beams and monitor-based targets (a bull’s eye/cross

hair combination, and images of Gaussian beams).

Aim 2 (Experiment 2 and Chapter 5)

To determine the effectiveness of a Bessel beam, as a fixation target, in

improving fixation stability compared with that for conventional targets.

Hypothesis

A Bessel beam used as a fixation target will improve fixation stability over that

achieved with conventional targets.

The amount of light reaching the retinal area being imaged is limited by

diffraction and scattering properties of Gaussian beams while passing through ocular

media, especially with increasing age and in the presence of scatterers such as cataract

(Kuroda, et al., 2002; Wanek, et al., 2007). Bessel beams, being diffraction-resistant

and capable of self-reconstructing, are less affected by the scatters, present in the

biological tissues such as skin, than Gaussian beams (Fahrbach, et al., 2010). This

property of Bessel beams might provide higher amount of light reaching the retinal

area being imaged than Gaussian beams. Therefore, I built a retinal imaging set up to

investigate the following aim and hypothesis.

Aim 3 (Experiment 3 and Chapter 6)

To compare the amount of light reaching the retinal area being imaged between

Bessel and Gaussian beams.

Hypothesis

Using a Bessel beam rather than a Gaussian beam will provide higher amount

of light at the retinal area being imaged.


2.8 ETHICS APPROVAL

This study has ethical approval from “University Human Research Ethics

Committee” of QUT. The QUT ethics approval number is 1300000816 and is titled as

“Advanced methods for intraocular imaging”. The study adhered to the tenets of the

Declaration of Helsinki and informed consent was obtained from each participant.

Chapter 3: PROPAGATION PROPERTIES OF OBSTRUCTED BESSEL BEAMS 49

Chapter 3: PROPAGATION

PROPERTIES OF

OBSTRUCTED BESSEL

BEAMS

3.1 INTRODUCTION

In the study on phakometry using Bessel beams (Aim 1, Chapter 4), I intended

to obstruct the majority of the Bessel beam to produce an arc rather than a ring to

improve the identification of Purkinje images. Bessel beams are resistant to diffractive

spreading, and have potential of self-reconstruction despite partial perturbation during

propagation (Durnin, et al., 1987; Fahrbach, et al., 2010; Nowack, 2012; Salo &

Friberg, 2008; Turunen & Friberg, 2010). These propagation properties of the beam

after an obstruction have been studied before (Anguiano-Morales, 2009; Anguiano-

Morales, et al., 2007; Bouchal, et al., 1998; MacDonald, et al., 1996; Zheng, et al.,

2013), but in linear media were limited to the obstruction of the central lobe or another

small proportion of the beam. While larger obstructions (e.g. >50 percent of the beam)

may also occur in cases such as linear media with larger scatterers or media with

refractive index inhomogeneity including biological tissues (Fahrbach, et al., 2010;

Garces-Chavez, et al., 2002), the studies on propagation properties of the Bessel beam

after such obstructions were considered only for nonlinear media (Butkus, et al., 2002;

Sogomonian, et al., 1997). Therefore, a preliminary study was performed to determine

whether the desired propagation properties of a Bessel beam, after the majority of it

has been obstructed, are retained in linear media.

3.2 METHODS

The propagation properties of the obstructed Bessel beam were determined using

an experimental setup and a computer simulation. In the results that follow the

emphasis is on the structure of the beam, so the results are contrast enhanced for

visibility. It is a simple matter to confirm the amplitude in simulation.

50 Chapter 3: PROPAGATION PROPERTIES OF OBSTRUCTED BESSEL BEAMS

3.2.1 Experimental setup

An axicon with an opening angle of 0.120 was simulated using a liquid crystal

SLM (Holoeye, PLUTO-NIR-010-A) in phase-only mode (Figure 3.1). A laser diode

(514.5 nm) was shone onto the SLM, and a zeroth order Bessel beam was generated

(Figure 3.2). The SLM had a resolution of 1920 x 1080 pixels of 8 μm pitch with an

active area of 15.36 mm x 8.64 mm on a silicon micro-display.

There were three conditions. The first condition imaged the Bessel beam without

any obstruction (Figure 3.2). The second condition included a rectangular obstruction

(Figure 3.3, a) blocking the beam downwards from the upper margin of the second

ring. The obstruction was 60 cm from the SLM so that most of the diffraction orders

originating from the pixel structure of the SLM other than from the Bessel beam could

be avoided. This plane was referred to as z0. The third condition was the second

condition with an additional circular obstruction of 1.5 mm diameter placed at the

upper portion of the beam in the z0 plane (Figure 3.4, a).

Figure 3.1: Experimental setup for testing properties of obstructed Bessel beams. LD: laser diode; L1

& L2: lenses with focal lengths 10.9 mm and 200 mm, respectively, SLM: spatial light modulator; O:

obstruction; P1 & P2: Polarisers. The P1 was used to operate SLM in phase-only mode and P2 was used

to filter out the unprocessed beam after the SLM. The diffuser was placed at the required locations along

the z plane, and the camera was focused at the diffuser.


Figure 3.2: (a) Phase plate of an axicon, projected on the SLM, (b)

Experimental Bessel beam, (c) Simulated Bessel beam. The DC term is due

to the unfiltered reconstruction beam.

Figure 3.3: Images of a Bessel beam with the second obstruction condition at z0, zmin/2, zmin and z∞.

(a-d) experimental, (e-h) simulation where (e) is the phase plate projected on the SLM rather than

the intensity profile at z0. At z0, rectangular obstruction blocks the lower part of the beam including

the central lobe. The obstructed Bessel beam self-heals after the obstruction. At zmin/2 some of the

lower blocked area is reconstructed, while at the far-field, z∞, most of the beam is reconstructed

except for a region around the horizontal meridian.

52 Chapter 3: PROPAGATION PROPERTIES OF OBSTRUCTED BESSEL BEAMS

Figure 3.4: Images of a Bessel beam with the third obstruction condition at z0, zmin/2, zmin and z∞. (a-d)

experimental, (e-h) simulation where (e) is the phase plate projected on the SLM rather than the intensity

profile at z0. In addition to the obstruction of condition 2, a circular obstruction with diameter 1.5 mm

blocks some portion of the upper rings.

3.2.2 Computer simulation

Propagation properties were calculated with Fraunhofer diffraction using

MATLAB software. The non-diffracting property of an obstructed Bessel beam was

investigated by comparing the radii of the inner three rings of the beam for the first

and second obstruction conditions at optical infinity. To maintain compliance with the

experiment where only positive values may be used on an SLM, the same phase plates

that were projected on the SLM were used. A plane wave was shone through each

phase plate, and the beam profile, which corresponds to the Bessel beam, was

reconstructed at the same distances away from the obstruction. The DC term (i.e. the

bright central spot or + sign in Figure 3.2, c) is the unfiltered beam.


3.3 RESULTS

Figure 3.3 shows images of an obstructed Bessel beam for the second obstruction

condition at z0, zmin/2 (17.5 cm from z0), zmin and z∞ (optical infinity). By z∞ the beam

has reconstructed except for a small region around the horizontal meridian. Since

reconstruction occurs due to plane waves on opposite sides, an obstruction that covers

the full beam width in one dimension and more than half of it in the other (Figure 3.3)

does not allow for a part of the beam to reconstruct. Therefore, the reconstructed beam

at z∞ is missing cones on either side of the centre instead of being rotationally

symmetric (Bouchal, et al., 1998; Butkus, et al., 2002). To the nearest pixel, the radii

of the first three inner rings at z∞ were the same for the second obstruction condition

as for the non-obstructed condition. This shows that the obstructed beam is non-

diffracting.

Figure 3.4 shows the images of the beam for the third obstruction condition at

various propagation distances. The shadow of the additional obstruction can be seen

at zmin/2 (Figure 3.4, b), while it disappears at zmin (c). At z∞, the second and third

obstruction conditions attain the same size and shape (Figure 3.3 & Figure 3.4, d),

showing that the unobstructed part of the beam is able to self-reconstruct and is non-

diffracting.

The computer simulation shown in Figure 3.3 (e-h) and Figure 3.4 (e-h) supports

the experimentally observed reconstruction of the Bessel beam.

3.4 CONCLUSION

A Bessel beam has self-reconstructing and non-diffracting properties, even after

a major portion including its central lobe, is blocked. This can be a useful trait while

using the beam in media with multiple scatterers or with refractive index

inhomogeneity. This finding gave us the confidence to block a part of the beam to

shape it like an arc that provides easily distinguished Purkinje images for the

Experiment 1.

Chapter 4: PHAKOMETRY WITH BESSEL BEAMS 55

Chapter 4: PHAKOMETRY WITH

BESSEL BEAMS

4.1 INTRODUCTION

Phakometry is the technique of determining shape and refractive index of the in-

vivo ocular lens by imaging reflections of a light source from the cornea and lens

surfaces. The major reflections occur at air-cornea, cornea-aqueous, aqueous-lens and

lens-vitreous interfaces, forming images known respectively as the 1st (PI), 2nd (PII),

3rd (PIII) and 4th (PIV) Purkinje images. PI, PII and PIV are formed near the pupil

plane, while PIII is formed in the vitreous. Brightnesses of PII, PIII and PIV relative

to that of PI are approximately 0.008, 0.013, and 0.013, respectively (calculated using

Fresnel reflection for normal incidence) (Atchison & Smith, 2000). Light sources in

phakometry are typically Gaussian beams, which are non-localised beams that diffract

and scatter while passing through media (Rohrbach, 2009; Siegman, 1986). The rough

anterior lens surface causes scattering and diffuse reflection of beams, deteriorating

the quality and lowering the brightness of PIII and hence making accurate, repeatable

estimate of lens parameters difficult (Adnan, 2015; Atchison & Smith, 2000; Navarro,

et al., 1986; Tabernero, et al., 2006).

Bessel beams are localised beams with transverse patterns that remain stationary

along the propagation distance, i.e., these beams are resistant to diffractive spreading,

and also have potential of self-reconstruction despite partial perturbation while passing

through inhomogeneous media (Durnin, et al., 1987; Fahrbach, et al., 2010; Nowack,

2012). These properties of the Bessel beam might increase the specular reflected

signals from the lens surfaces and produce brighter, sharper PIIIs than Gaussian beams,

and thus give more accurate and repeatable estimations of lens surface curvatures.

Therefore, the hypothesis being tested in this chapter is that use of a Bessel beam, as

an illumination source, in phakometry will produce brighter Purkinje images resulting

in more accurate and repeatable estimates of lens surface curvature than that of a

Gaussian beam. Therefore, this study assessed Purkinje image brightness, accuracy,

and repeatability of a Bessel phakometer compared with those of a Gaussian

phakometer.

56 Chapter 4: PHAKOMETRY WITH BESSEL BEAMS

4.2 METHODS

I developed a phakometer using a Bessel beam for illumination (Figure 4.1, top)

by modifying an existing Gaussian phakometer (Figure 4.1, bottom) on a 450 mm ×

300 mm movable optical breadboard over a base containing forehead and chin rest in

the Visual and Ophthalmic Optics lab.

The accuracy of the Bessel phakometer was assessed using a model eye (OEMI-

7, Ocular imaging eye model). Phakometry was performed in the model eye using the

Bessel phakometer and differences between the values of lens parameters from the

Figure 4.1: Phakometers, (top) Bessel and (bottom) Gaussian, developed at QUT research lab. BS, beam

splitter; OLED, organic light-emitting diode; OC, occlude; LD, laser diode; L1 to L4, lenses. Further details on the two phakometers are provided in the following sections.


phakometer and manufacturer’s specification were calculated. Six healthy participants

(Table 4.1) were recruited to compare the repeatability in estimating lens parameters,

and Purkinje image brightness between Bessel and Gaussian phakometers. Data

collection for the Gaussian phakometer was completed before modifying it. Details on

participants, production of Bessel beam, phakometers setup (Bessel and Gaussian),

data collection and analysis of Purkinje images, determination of Purkinje image

brightness and calculation of lens parameters using merit function in MATLAB

software are described in the following sections.

4.2.1 Participants

Participants were recruited by sending an email invitation. A flyer was

developed providing necessary information about the study and shared via QUT email

database as an invitation to participate in the study. Six healthy participants (Table 4.1)

aged between 18-45 years (mean age 23.8 ± 4.1 years; 4 female, 2 male) met our

inclusion criteria, and hence were enrolled in the study. Routine clinical tests were

performed to determine the eligibility of participants. The study adhered to the tenets

of the Declaration of Helsinki and informed consent was obtained from each

participant.

Table 4.1: Clinical tests and inclusion criteria of healthy participants

Examination Instrument/Procedure Criteria

Visual acuity (VA) Bailey-Lovie log MAR

chart

6/6

Tear break up time Slit lamp biomicroscopy > 5 seconds

Intra ocular

pressure

iCare Tonometer ≤ 21 mm Hg

Refraction Shin-Nippon

autorefractometer

Astigmatism ≤ 2 D

Ocular health Slit lamp biomicroscopy

& history taking

No corneal and lenticular

opacities. No history of ocular

injury or surgery

General health History taking No diabetes and no

contraindicated condition for


pupil dilation (including

pregnancy, breastfeeding or past

history of anterior chamber

angle closure events)

4.2.2 Production of a Bessel beam

A Bessel beam was generated using a variation of the method described by

Herman & Wiggins (1991), by passing a collimated diode laser beam (Thorlabs

SFC635, wavelength 637 nm) through a high powered lens with a central obstruction

(Figure 4.2). The illuminating Gaussian beam, obstructed optimally from lens centre

and margin, when passed through the spherically aberrating lens produced a Bessel

beam (Figure 4.3) that satisfies Equation 2.5 and can be represented as

𝐸(𝑦, 𝑧)~𝐸0[𝜌𝛽(𝑧)] exp [𝑖𝑘 ∫ cos 𝛽(𝑧) 𝑑𝑧 +𝜋

4

𝑧

0] × 𝐽0[𝑘𝑦 sin 𝛽(𝑧)] (4.1)

where E0 is an amplitude of an electric field component of the propagating light, ρβ is

the ring radius for illuminating the point P of observing plane (Figure 4.2), β is the

angle at which rays cross the optical axis at point P of observing plane, k is the wave

number represented as in Equation 2.2, J0 is the zero-order Bessel function, and z is

the distance from the lens apex to the observing plane.

Figure 4.2: Diagram of geometrical quantities associated in production of Bessel beam in the Bessel

phakometer. R: radius of curvature of lens second surface, θ: angle of incidence, fC: focal length of

central rays, fM: focal length of marginal rays, z: distance from lens apex to observing plane (P), β: angle

at which rays cross the axis at P, ρβ: the ring radii for illuminating the point P, ρM: marginal radius, ρ:

radius of the furthest ring from optical axis which contributes in illuminating point P, C: central stop.

Image redrawn from Herman & Wiggins (1991).


I used a doublet rather than a singlet spherical lens to generate a Bessel beam.

However, for simplicity Figure 4.2 and Equation 4.1, that describe the model of

Herman and Wiggins (1991) for the singlet spherical lens, have been used to represent

my setup. I ran several experimental tests as described in Chapter 3 to confirm that the

generated beam was a Bessel beam. The generated Bessel beam was able to reconstruct

after partial obstruction and maintained its diffraction-resistant property over the

significant propagation distance.

The crossing of the optical axis by marginal rays at different angles than paraxial

rays caused the central ring pattern of image produced by the lens to change in size

along with axial position. However, the variation in size was not significant over small

distances so the local behaviour of the beam fulfilled the criteria to be referred to as a

Bessel beam with constant central ring pattern. A central obstruction was added to

block the Gaussian beam component and to regulate the intensity of the central ring

pattern of the Bessel beam that would otherwise oscillate with uneven intensity

variation (Figure 2.3). A contribution from light diffracted at the edge of the central

Figure 4.3: Cross-sectional image of Bessel beam generated through lens with spherical aberration.


obstruction and at the edge of the lens might interfere with the optical field along the

optic axis. So, this contribution was minimised by setting the size of the central

obstruction to 1/√3 of the diameter of the lens (full modelling details shown in

Herman & Wiggins (1991)).

4.2.3 Bessel phakometer

In the Bessel phakometer, a Bessel beam was generated after passing a 5.8 mm

diameter collimated laser beam (wavelength = 637 nm; Thorlabs SFC635, Newton,

NJ) with 0.05 mW output power through the 50 mm focal length lens (Figure 4.4, L2)

and the laser beam was re-collimated with the 50 mm focal length lens (L3). The

collimated Bessel beam was focused by a 100 mm focal length lens (L4) and shone

into the eye, and multiple reflections were recorded using a telecentric imaging system

(Figure 4.4). The phakometer was optimised so that most of the light reflected from

the ocular surfaces passed through the imaging path.

Figure 4.4: Bessel phakometer. The beam is generated by passing a laser diode beam through a powerful

doublet with central obstruction and shining it into the participant’s eye. Reflections from anterior

cornea (PI), anterior (PIII) and posterior (PIV) lens are recorded using a telecentric imaging system. BS: beam splitter, OLED: organic light-emitting diode for fixation target, OC: occluder, LD: laser

diode, L1 to L4: lenses.


A horizontal obstruction (OC) was introduced in the beam path (between L3 and

L4) to block 72 % of the beam to shape it like an arc that produces easy-to-identify

Purkinje images. The incident angle of the Bessel beam at the participant’s eye was

optimised in such a way that the Purkinje images were not symmetric with respect to

the vertical, and hence they could be easily identified. The partially blocked Bessel

beam still maintained the diffraction resistant and self-healing properties which was

confirmed from the preliminary experiment (Chapter 3) and was reconfirmed through

Zemax simulation. Chapter 3 provides the detailed information on the propagation

properties of Bessel beams after large obstructions, similar to the one (OC) used in this

experiment. A fixation target was presented by an Organic Light-Emitting Diode

(OLED) display (with viewing area 12.78 mm × 9 mm and pixel pitch 15 micrometre).

A beam splitter (BS2) reflected the target to the participant’s eye through L4. The

telecentric imaging system consisted of an IR-enhanced CCD camera (PixeLink) with

a 55 mm focal length telecentric lens, and was focused at the pupil plane of the eye

(260 mm from the camera) where PI and PIV were in focus with PIII slightly out of

focus. Images were captured after obtaining good alignment of PI, PIII and PIV. The

distance from L4 to the apex of the cornea (88 mm) and the radius of the collimated

Bessel beam (5.8 mm) were the object distance and size, respectively.


4.2.4 Gaussian phakometer

The Gaussian phakometer (Figure 4.5) used in this study was the system

developed by Tabernero et al. (2006). The phakometer had 13 LEDs as an illumination

source with wavelength of 890 nm in a semicircular ring arrangement with 18.5 mm

radius (object size) at 80 mm (object distance) from the cornea. The Gaussian

phakometer used the same telecentric imaging system including the steps for capturing

Purkinje images as that of Bessel phakometer. Further details on the Gaussian

phakometer have been given elsewhere (Adnan et al., 2015).

Figure 4.5: Gaussian phakometer. Image taken from Adnan, Suheimat, Efron, Edwards, Pritchard,

Mathur, Mallen, & Atchison (2015).


4.2.5 Analysis of Purkinje images

The images were subjected to custom-built software in MATLAB (MathWorks

Inc., Natick, MA, version R2011). Ellipses were fitted from fitting ellipse mode by

tracing the edges of PI, PIII, PIV, the pupil and the limbus (Figure 4.6). The option of

logs of the images were used to enhance PIII. The fittings gave the centres and sizes

of Purkinje images. Lens anterior and posterior radii of curvatures (Ra and Rp) and lens

equivalent refractive index (RI) were calculated using a merit function algorithm in

MATLAB (Adnan, et al., 2015; Garner, 1997), and lens equivalent power (F) was

determined using the thick lens formula (Atchison & Smith, 2000).

Figure 4.6: Purkinje images of a participant’s eye obtained with the Bessel phakometer and edges

through which ellipses were fitted.


4.2.6 Merit function

The merit function (MF), an iterative method used for calculation of lens

parameters, is the sum of three components. One component MF1 is the square of the

difference between experimental (Vexp) and theoretical (Vthe) vitreous lengths obtained

from ray tracing into the eye to the retina. The 2nd component MF2 is the square of the

difference between measured (h3exp) and theoretical (h3the) heights of PIII, and the 3rd

component MF3 is the square of the difference between measured (h4exp) and

theoretical (h4the) heights of PIV. h3the and h4the are determined by ray tracing into and

then out of the eye following reflection from the front and back surfaces of lens,

respectively, and Vthe is determined by ray tracing after refraction at cornea and lens

into the eye to the retina. The MF is expressed as:

𝑀𝐹 = 𝑀𝐹1 + 𝑀𝐹2 + 𝑀𝐹3 = (𝑉the − 𝑉exp)2

+ (ℎ3the − ℎ3exp)2 + (ℎ4the −

ℎ4exp)2 (4.2)

The merit functions algorithm in MATLAB determined Ra, Rp and RI by

simultaneously varying them until the merit function reached a minimum value. The

merit function was considered to reach the minimum value when differences between

successive estimates were less than 0.01 percent or when the algorithm completed

2000 cycles, whichever occurred first.

Parameters used by the merit function were the Purkinje image heights, object

distance and size, anterior and posterior corneal mean radii of curvature (obtained from

an Oculus Pentacam), refractive error (determined from the Shin Nippon auto

refractometer), and corneal thickness, anterior chamber depth, lens thickness, vitreous

and axial length (all obtained from a Haag-Streit Lenstar). The changes required in the

MATLAB program during calculation of lens parameters when swapping between

phakometers (Bessel/Gaussian) are given in Appendix D.

4.2.7 Determination of repeatability of phakometers

Phakometry was conducted in six participants fulfilling the inclusion criteria of

healthy participants (Table 4.1). Separate sessions were conducted for the Bessel and

Gaussian phakometers. One drop of 1 percent cyclopentolate hydrochloride was

instilled in the right eyes to dilate pupils and paralyse accommodation forty minutes


before the first image was taken. For each participant, an image was taken by each of

two observers at 40 minutes (T40) and then at 50 minutes (T50) after instillation. The

observers analysed their corresponding images to determine Purkinje sizes, and

subsequently the lens radii of curvature, refractive index and equivalent power.

The distributions of differences between the two phakometers for lens

parameters were calculated using the average of four measurements for each

participant as a single observation. The standard deviation (sw) of repeated

measurements was used as the measure of repeatability (Bland & Altman, 1996):

s𝑤 = √ ∑𝝈𝟐

𝒏 (4.3)

where σ is the standard deviation of two repeated measurements for each participant

and n is the number of participants. Two types of repeatability were calculated for each

phakometer. Intra-observer repeatability was calculated from two repeated

measurements at T40 and T50, determined after averaging the values of both observers

at each time. Inter-observer repeatability was calculated from two repeated

measurements of two observers, determined after averaging the values of both times

for each observer.

4.2.8 Determination of Purkinje image brightness

The relative brightnesses of Bessel and Gaussian images were compared by

selecting the best 10 images for each phakometer. The CCD camera captured images

with greyscale values (reaching saturation at 255 arbitrary unit). In both phakometers,

the incident power of the beam and exposure time for the camera were optimised such

that PI was just saturated at 255 (arbitrary unit). This ensured the normalisation of PI

so that the brightnesses of PIII and PIV can be compared between two phakometers.

The greyscale values per pixel for 5×5 square of pixels corresponding to the brightest

area of each Purkinje image were averaged in MATLAB. The average brightness for

selected pixels corresponding to the respective Purkinje images subtracted from that

of background brightness was determined as the final Purkinje image brightness, and

was recorded in the datasheet. The steps performed in MATLAB to determine the

Purkinje image brightness are given in Appendix E.


4.3 RESULTS

The determined parameters of model eye lens were similar to those provided by

the manufacturer with the differences in Ra, Rp, RI and F of +1.18 mm, −0.18 mm,

+0.0053 and −0.55 D, respectively (Table 4.2). There were no significant differences

in mean Ra and Rp of participants between the two phakometers while the Bessel

phakometer gave smaller estimates of RI and F than the Gaussian phakometer (Table

4.3). The intra-observer repeatabilities for the Gaussian and Bessel phakometer were

similar (Table 4.4) and were less than 1 mm and 0.25 D for radii of curvature and lens

power, respectively. The inter-observer repeatabilities of Ra, Rp and RI for the Bessel

phakometer were almost half those (i.e., two times better) for the Gaussian phakometer

(Table 4.5). The brightnesses of PIII and PIV were about 3.5 and 2.5 times,

respectively, higher with the Bessel phakometer than with the Gaussian phakometer

(Table 4.6).


Table 4.2: Model eye lens parameters according to the manufacturer's specifications

and the Bessel phakometer

Manufacturer’s

specification

Bessel

phakometer

Difference

Lens parameters

Ra 11.99±0.13 mm 10.81 mm +1.18 mm

Rp ̶ 5.99±0.13 mm ̶ 5.81 mm ̶ 0.18 mm

RI (λ=637nm) 1.4907 1.4854 +0.0053

F (λ=637nm) 38.93 D 39.48 D ̶ 0.55 D

Material Polymethyl Methacrylate

(PMMA)

Central thickness 3.90 mm

Cornea

Ra 7.82 mm

Rp 4.14 mm

RI (λ=637nm) 1.4907

Material PMMA

Central thickness 0.55 mm

Aqueous

Anterior chamber

depth

2.95 mm

Material Distilled water

RI (λ=637nm) 1.3315

Vitreous

Vitreous length 18.40 mm

Material PMMA

RI (λ=637nm) 1.3315

Table 4.3: Participants lens parameters obtained from Gaussian and Bessel

phakometers. RI & F are reported for λ = 555 nm

Lens

parameters

Gaussian

(Mean ± SD)

Bessel

(Mean ± SD)

Mean Difference ± SD

(95% CI)

Ra (mm) +10.53±0.64 +10.91±0.33 ̶ 0.38±0.66 ( ̶1.08 to +0.31)

Rp (mm) ̶ 6.14±0.68 ̶ 6.32±0.24 +0.18±0.80 ( ̶ 0.66 to +1.02)

RI 1.4318±0.0351 1.4266±0.0362 +0.0052±0.0075 (+0.0027

to +0.0131)

F (D) 24.61±9.11 22.47±9.01 +2.14±0.17 (+1.96 to +2.31)


Table 4.4: Intra-observer repeatability for Gaussian and Bessel phakometers

Intra-observer repeatability

Lens parameters Gaussian Bessel

Ra (mm) 0.44 0.56

Rp (mm) 0.31 0.59

RI 0.0039 0.0048

F (D) 0.12 0.12

Table 4.5: Inter-observer repeatability for Gaussian and Bessel phakometers

Inter-observer repeatability

Lens parameters Gaussian Bessel

Ra (mm) 0.68 0.39

Rp (mm) 1.06 0.51

RI 0.0102 0.0041

F (D) 0.31 0.41

Table 4.6: Brightnesses of Purkinje images obtained from Gaussian and Bessel

phakometers

Brightness Gaussian (Mean ± SD) Bessel (Mean ± SD)

PIII 8.3±3.6 28.0±8.2

PIV 12.8±6.0 32.6±10.7

PI All the points on PI saturated at 255 All the point on PI

saturated at 255

PIII:PI 0.03 0.11

PIV:PI 0.05 0.13

4.4 DISCUSSION

I have built a phakometer using a Bessel beam for illumination and compared its

Purkinje image brightness and repeatability with an existing Gaussian phakometer.

The accuracy of the Bessel phakometer was similar to that reported by Barry, Dunne,

& Kirschkamp (2001), who found differences between values given by the

manufacturer and those determined by their Gaussian phakometer for Ra, Rp and F of

the lenses of model eyes to be within ranges of −0.10 to +0.02 mm, +0.10 to +0.55

mm, and +0.50 to +1.02 D, respectively. Rosales and Marcos (2006) reported higher

accuracy in Ra (+0.09 mm average difference between their value and manufacturer

specifications) and lower accuracy in Rp (+0.33 mm) with their Gaussian phakometer


compared to my Bessel phakometer. They used simulations of model eyes in Zemax

without accounting for experimental biases, which might have overestimated the real

accuracy of their instrument.

While there were no significant differences in mean Ra and Rp of participants

between the two phakometers, the Bessel phakometer gave smaller estimates of RI and

F than the Gaussian phakometer (Table 4.3). As the merit function algorithm was

based on paraxial ray tracing, different beam paths for the phakometers would have

contributed to the differences in parameter estimates. In the calculation using the merit

function algorithm, the raytracing into the eye to the retina to estimate theoretical

vitreous length was done assuming both beams would take the same path propagating

into the eye, whereas in reality they might take the different paths.

There was considerable improvement in inter-observer repeatability for the

Bessel phakometer over that of the Gaussian phakometer but no differences were seen

in intra-observer repeatability between the two phakometers. Rosales, Dubbelman,

Marcos, and Van der Heijde’s (2006) study with a Gaussian phakometer had similar

intra-observer repeatability to those of my phakometers. Mutti, Zadnik, & Adams

(1992) found intra-observer repeatability of lens power for their Gaussian phakometer

of 0.45 D (after conversion to my method of calculating repeatability), which is

considerably poorer than 0.12 D for both phakometers.

While different studies have reported that the diffuse reflection and scattering of

Gaussian beam from the rough anterior lens surface poses limitation in accurate

calculation of Ra in a Gaussian phakometer (Adnan, 2015; Navarro, et al., 1986;

Tabernero, et al., 2006), the Bessel phakometer produced PIII and PIV that were about

3.5 and 2.5 times brighter, respectively, than by the Gaussian phakometer (Table 4.6).

The increased brightnesses of PIII and PIV with the Bessel phakometer can be

attributed to the diffraction resistant nature of the localised beam along the propagation

distance, resulting in increased specular reflections from the lens surfaces. The average

brightness ratio between the Gaussian PIII and PI, and that between the Gaussian PIV

and PI were 0.03 and 0.05, respectively, and the average brightness ratio between the

Bessel PIII and PI, and that between the Bessel PIV and PI were 0.08 and 0.13,

respectively. The selection of a few brightest pixels, ignoring other dimmer pixels, of

PIII and PIV might have yielded these ratios to be higher than one would normally

expect (Atchison & Smith, 2000). However, the increase in the brightnesses of PIII


and PIV from Bessel phakometer compared to that from the Gaussian phakometer can

still be considered valid since the same calculation steps were followed for both

phakometers.

In summary, the Bessel phakometer provided accurate estimates of lens

parameters of a model eye, and produced brighter Purkinje lens images and better

inter-observer repeatability for lens radii of curvature than a Gaussian phakometer. It

had reasonable agreement with the Gaussian phakometer in estimating the radii of

curvature, and had similar intra-observer repeatability despite the brighter Purkinje

images. The lack of improvement in the latter was probably mostly due to the thick

fitting line in MATLAB obscuring the Bessel beam Purkinje images which were

smaller than those for the Gaussian beam setup. The repeatability and accuracy of the

Bessel beam can be improved by manipulating the illumination so that larger Purkinje

images are obtained.

Chapter 5: FIXATION STABILITY WITH BESSEL BEAMS 71

Chapter 5: FIXATION STABILITY WITH

BESSEL BEAMS

5.1 INTRODUCTION

Good selection of fixation targets is important in optimising fixation stability.

Studies with monitor-based targets suggest that shapes such as a circular point

(Hirasawa, et al., 2016; Rattle, 1969), a cross (Bellmann, et al., 2004), a bull’s

eye/cross hair combination (Thaler, et al., 2013), and a “%” optotype (Pirdankar &

Das, 2016) provide good fixation stability. Target colour, luminance and contrast have

little effect on fixation stability (Boyce, 1967; McCamy, Jazi, Otero-Millan, Macknik,

& Martinez-Conde, 2013; Steinman, 1965; Ukwade & Bedell, 1993).

Bessel beams, with diffraction resistant and self-reconstructing properties, have

potential for improving image quality during ophthalmic imaging (Suheimat et al.,

2017) compared with conventional Gaussian beams by decreasing scattering artefacts,

improving depth of field and penetration in dense media (Durnin, et al., 1987;

Rohrbach, 2009). Lambert et al. (2013) reported that a Bessel beam used as a fixation

target in an adaptive optics system reduced temporal variations of aberrations from

those found with a conventional Gaussian beam. They suggested that this was due to

improved stability of the eye through suppression of rapid eye movements.

Retinal imaging instruments requiring fixation might benefit from using Bessel

beams in the dual roles of both illuminating beams and fixation targets, thus

eliminating the need of a screen for targets and thus making instrument slimmer and

lighter. However, there remains an outstanding question of whether Bessel beams

improve fixation stability compared with conventional targets i.e. Gaussian beams and

monitor-based targets (a bull’s eye/cross hair combination, and images of Gaussian

beams).

Therefore, the hypothesis being tested in this chapter is that Bessel beams, when

used as fixation targets, will improve fixation stability over that for other fixation

targets. Accordingly, the primary aim of this study was to investigate the fixation

stability using Bessel beams as fixation targets compared with conventional targets. In

addition, this study investigated the variation in fixation stability with different number

72 Chapter 5: FIXATION STABILITY WITH BESSEL BEAMS

of rings on the Bessel beams, and assessed whether it is the propagation property or

the shape of the Bessel beams that plays role in improvement, if any, of fixation

stability.

5.2 METHODS

I developed a set up that recorded fixation eye positions, using the EyeTribe

tracker, of right eyes of participants when presented with seven fixation targets. There

were four monitor-based images: a bull’s eye/ cross hair, a Gaussian beam, a Bessel

beam with 4 rings, and a Bessel beam with 3 rings. There were three laser based

targets: a Gaussian beam, a Bessel beam with 4 rings, and a Bessel beam with 3 rings.

Details on participants, fixation targets, instrumentation and eye-movement recording,

target luminances, task and procedures, and data collection and analysis are described

below.

5.2.1 Participants



database as an invitation to participate in the study. Sixteen adult participants met our

inclusion criteria, and hence were enrolled in the study, with right eyes tested. Routine

clinical tests were performed to determine their eligibility (Table 5.1). Mean age was

27 years with an 18 to 42 years range. Unaided visual acuity was ‒0.03±0.02 log MAR

(range ‒0.08 to 0.00) in right eyes and tear break-up time > 5 s in both eyes. As Bessel

beams are affected differently by lenses than are Gaussian beams, which might affect

measurements, focussing lenses were not included in my set up and only emmetropes

were included (refractive errors within +0.50 D sphere). None of the participants had

any history of binocular problems, and all had normal near points of convergence. The

study adhered to the tenets of the Declaration of Helsinki and informed consent was

obtained from all participants.


Table 5.1: Clinical tests and inclusion criteria of healthy participants

Examination Instrument/Procedure Criterion

Visual acuity (VA) Bailey-Lovie log MAR

chart

> 0.0 log MAR

Tear break up time Slit lamp biomicroscopy > 5 seconds

Refraction Shin-Nippon

autorefractometer

Spherical equivalent

refraction within +0.50 D


& history taking

No corneal or lenticular

opacities, and no retinal

diseases

General health History taking No systemic conditions

leading to abnormal fixation,

such as Down syndrome,

Williams’s syndrome,

Parkinson’s disease and

Hodgkin’s disease.

5.2.2 Fixation targets

Participants were presented randomly with seven fixation targets (Figure 5.1):

monitor-based images of a bull’s eye/cross hair combination (Target A, Figure 5.1, a),

a Gaussian beam (Target B, b), a Bessel beam with 4 rings (Target C, c), and a Bessel

beam with 3 rings (Target D, d); Gaussian beam produced by a 637 nm laser (Target

E, b), Bessel laser beam with 4 rings (Target F, c), and Bessel laser beam with 3 rings

(Target G, d). The diameters of the outer circles of targets A, C and D were 13.1 mm,

and when presented at 50 cm subtended 1.5° visual angle at the eye. The total

diameters of targets B and the central lobe of the target A were 1.75 mm, and when

presented at 50 cm subtended 0.2° visual angle at the eye. Targets F and G were

truncated using a 13.1 mm aperture for 1.5° angular subtense at the eye. Target E was

truncated using a 1.75 mm aperture for 0.2° angular subtense at the eye. The diameters

of targets B and E were chosen to be 0.2°, which also matched the central fixating

lobes of other targets used in this experiment, because studies have shown that a 0.2°

diameter target with shape similar to that of Gaussian beam provided better fixation


stability than larger targets (Steinman, 1965; Thaler, et al., 2013). Colour of targets A,

B, C and D was matched with that of the monochromatic 637 nm laser beam targets

E, F and G. The red, green and blue components of the matched colour, in a computer

with 8-bit colour resolution, were represented numerically as 255, 48, and 0,

respectively.

Figure 5.1: Fixation targets. (a) A combination of bulls eye/cross hair combination - target A, (b)

Gaussian beam (image presented on a computer monitor - target B, and real beam as it appears to the

participants - target E), (c) Bessel beam with 4 rings (image presented on a computer monitor - target

C, and real beam as it appears to the participants - target F), (d) Bessel beam with 3 rings (image

presented on a computer monitor - target D, and real beam as it appears to the participants - target G).

5.2.3 Instrumentation and eye-movement recording

Targets A, B, C and D were presented to the eye on a 22-inch LCD monitor (Dell

P2214H) (Figure 5.2 & Figure 5.3) with 1920 horizontal × 1080 vertical spatial

resolution, 60 Hz temporal resolution, 0.248 mm pixel pitch, 476.08 mm horizontal ×

267.78 mm vertical active display, and 100 percent brightness at 500 mm from the eye.

To generate Bessel beams for targets F and G, a collimated laser beam was shone onto

a spatial light modulator (SLM) (Figure 5.2, iii) while simulating axicons with opening

angles of 0.12° and 0.24°, respectively. Target E, a Gaussian beam, followed the same

optical path as that of Bessel beams but without simulation on the SLM. Targets E, F

and G were presented using mirrors (vii & viii) next to the monitor targets A, B, C and

D with the participants moving between left and right side bite bars (arrow) - left bite

bar for laser beams and right bite bar for monitor targets. While viewing monitor

targets, mount viii obstructed the left eye from seeing them.


Figure 5.2: Experimental setup for fixation stability with Bessel beam (not to scale). (i) Laser diode, (ii)

Lens, (iii) Spatial light modulator, (iv) 0.6 neutral density filter, (v) Aperture, (vi) Polariser, (vii) & (viii) Mirrors, (ix) Eye tracker, (x) Monitor, (xi) Controller. The participants fixate at the laser targets

being reflected from the mirror (viii). The dotted arrow shows that the participants can be moved

towards the right to fixate the monitor targets.

Figure 5.3: Photograph of experimental setup for fixation stability with Bessel beam. (i) Laser diode,

(ii) SLM, (iii) ND filter, (iv) Aperture, (v & vi) Mirrors, (vii) Eye tracker, (viii) Bite-bars, (ix) Monitor

for displaying targets A, B, C and D, (x) Computer monitor as a controller.


The image based (pupil with corneal reflection) EyeTribe tracker was used to

record eye positions at 30 Hz sampling rate. It has a 0.10° spatial resolution and latency

< 20 ms. The tracker was calibrated with a 12-point (3 rows and 4 columns) grid at the

beginning of each session for each participant. The tracker rates the quality of

calibration on a scale of no stars to 5 stars, where no stars indicates “uncalibrated”, 4

stars indicates “good” and 5 stars indicates “perfect”; the quality was always 4 or 5

stars. The distances between points in the monitor were 140 mm (565 pixels), and 106

mm (427 pixels) in 𝑥 and 𝑦 dimensions, respectively. After calibration, the tracker

compensates for the head movements during fixation tasks and provides real eye

fixation positions, but with low accuracy for large head movements (Jung et al., 2016).

To avoid large head movements, each participant’s head was stabilised using a bite bar

(Porter, Queener, Lin, Thorn, & Awwal, 2006). Jung et al (2016) showed that the

translation head movement along the horizontal meridian (movement between left and

right bite bars) does not affect fixational position data, so the same calibration was

used for both monitor- and laser-based targets. The tracker uses infrared illumination

to determine the point of gaze and has two software packages: EyeTribe UI and

EyeTribe Server. EyeTribe Server makes the eye tracker ready to use, and EyeTribe

UI does the calibration and controls the features for tracking the eye. The tracker works

only when both eyes are open and provides the data in pixel values for gaze positions

of both eyes separately as .txt files using C++ programming language. The data from

right eyes only were analysed.

5.2.4 Luminances of laser beam targets

Luminances, measured with 0.2° aperture of a Topcon luminance colourimeter

BM-3A, for the central circles (0.20) of targets A, B, C and D were 60 cd/m2. The

brightnesses for the central lobes of targets E, F and G were matched subjectively by

an observer with that of target A by combining a laser power of 0.01 mW with neutral

density filters before the SLM: a 0.6 ND filter gave a satisfactory match for all targets

with luminances measured as 56-66 cd/m2 (Table 5.2). The filter was used with the

laser targets in the experiment.


Table 5.2: Luminances of laser beams with neutral density optical filters

ND filter optical density

0.4 0.6 0.8 1 1.2

Gaussian beam target E 87 66 40 25 16

Bessel beam with 4 rings target F 82 57 37 26 16

Bessel beam with 3 rings target G 84 56 38 24 15

5.2.5 Tasks and procedure

Participants were seated with their heads stabilised using a bite-bar with their

eyes at the same height as that of the fixation targets. They were instructed to open

both eyes and fixate the target centres as well as possible throughout the session

(visible only to right eyes as explained above). They were instructed to blink if needed

and refixate at the target centres afterwards. There were five runs for each participant,

in which each run presented seven targets in random order for 20 seconds each. A

practice run using one target was conducted before data collection.

5.2.6 Data collection and analysis

Data were exported to MATLAB and then to Microsoft Excel, R, and SPSS. The

fixation data samples between 333 ms (10 data samples) before and after blinks were

excluded to avoid data contamination from loss of tracking during blinks. The

MATLAB code for extracting data from .txt file so that they can be exported to

Microsoft Excel, R and SPSS is added in Appendix F. The MATLAB algorithm did

not exclude the blinks occurring during the first 333 ms of the start of the run, and

hence such data were deleted manually. The initial 450 data samples, discounting

excluded blink data, were included. Data from the five runs were combined for each

target so that there were 2,250 data samples. There were missing data for five

participants: target C and G of the third run for participant P2, target F and G of the

fourth run for participant P6, and targets B, D and G of the first run and target D of the

fifth run for participant P8. Participants P4 and P7 reported they could not perform the

first run properly. To maintain equal number of samples for each target, only runs that


had complete data for each target were considered for analysis, so there were only

three runs for P8 and four runs for P2, P4, P6 and P7.

The tracker provided data in pixel values. The values that were beyond ± 3

standard deviation were identified as outliers and removed for each run. The mean

values of x and y coordinates for each run were used as fixation centre (�̅�, �̅�), which

for a set of n points is given by (∑ 𝑥𝑘

𝑘=𝑛𝑘=1

𝑛,

∑ 𝑦𝑘𝑘=𝑛𝑘=1

𝑛), where (𝑥𝑘, 𝑦𝑘) is the kth point. For

each individual run, the values of x and y coordinates for each data point were re-

referenced as ∆𝑥𝑘 = (𝑥𝑘 − �̅�) and ∆𝑦𝑘 = (𝑦𝑘 − �̅�), respectively. The re-referenced

x and y coordinates for all runs for each target were combined. The pixel values for

referenced x and y coordinates were converted into visual angle (degree), where 1 pixel

corresponds to tan−1(0.248

500) ∗ (

180

𝜋) = 0.0284 degrees (or 1.71 min arc) at 50 cm. The

standard deviations of fixation positions for each participant and target along

horizontal (𝜎𝑥) and along vertical (𝜎𝑦) meridians were calculated as

𝜎𝑥 𝑜𝑟 𝑦 = √∑ (𝐴𝑖− 𝜇)2𝑁1

𝑁−1 (5.1)

where 𝑁 is the total number of combined re-referenced data, 𝐴𝑖 is the visual angle of

the 𝑖𝑡ℎpoint of 𝑁 points and 𝜇 is the mean visual angle of 𝑁 points for each target of

each participant for that meridian.

The area of bivariate contour ellipse (BCEA), encompassing 68.2 percent of the

highest density eye positions, is an extensively used and generally accepted technique

with good test-retest reliability and accuracy (Cesareo et al., 2015; Chen et al., 2011;

Steinman, 1965; Timberlake et al., 2005). It provides better estimates of the variability

of fixating eye positions than measures such as fixation quality score, proportion

(Crossland, Dunbar, & Rubin, 2009), mean (Steinman, 1965) and standard deviation,

which do not account for the elliptical nature of fixation distributions (Sansbury,

Skavenski, Haddad, & Steinman, 1973). BCEA for each target was calculated using

𝐵𝐶𝐸𝐴 = 𝜒2𝜋𝜎𝐻𝜎𝑉(1 − 𝜌2)0.5 (5.2)

where 𝜎𝐻 and 𝜎𝑉 are the standard deviations of eye position in horizontal and vertical

meridians, respectively, 𝜌 is the product-moment correlation of the two position

components, and 𝜒2 is 2.291 for 68.2 percent of the highest density points.


Previous studies (Bellmann, et al., 2004; Pirdankar & Das, 2016; Steinman,

1965) have assumed BCEAs do not usually follow normal distributions so that they

have log-transformed the data before using parametric tests, although not always

determining if the transformation produced normal distributions. Shapiro-Wilk tests

indicated that our 𝜎 and BCEA distributions were not normally distributed for most

targets, even after logarithmic transformation (Table 5.3). Thus, non-parametric

Friedman tests were used to determine the statistical significance of targets differences

in unlogged 𝜎 and BCEAs. Post-hoc analysis using Wilcoxon signed-rank tests

determined the statistical significance of differences in 𝜎 and BCEAs between any two

targets; significance was set at p < 0.05 after Holm-Bonferroni adjustment.

Table 5.3: Shapiro-Wilk test p-values to assess the normality distribution of standard

deviation (SDs) and bivariate contour ellipse areas (BCEAs) Shapiro-Wilk (p-value*)

Target A Target B Target C Target D Target E Target F Target G

SD < 0.01 < 0.01 0.01 < 0.01 < 0.01 < 0.01 < 0.01

Log SD 0.05 0.31 0.28 < 0.01 0.05 0.07 0.23

BCEA < 0.01 < 0.01 < 0.01 < 0.01 < 0.01 < 0.01 < 0.01

Log BCEA 0.05 0.13 0.02 < 0.01 0.02 < 0.01 0.39

*p < 0.05 indicates non-normal distribution.


5.3 RESULTS

Table 5.4 and Table 5.5 show 𝜎𝑥 and 𝜎𝑦, respectively, of fixation positions of

each participant for the targets, and Figure 5.4 and Figure 5.5 box-and-whisker plots

of 𝜎𝑥 and 𝜎𝑦, respectively, of fixation positions for the targets. There was considerable

variability between participants, with most having 𝜎𝑥 and 𝜎𝑦 of fixation positions <

0.60 degree for the targets, but P2 had 𝜎𝑦 higher than 1.0 degree for all the targets and

P6 had 𝜎𝑦 higher than 1.0 degree for target B. 𝜎𝑦 was larger than 𝜎𝑥 by approximately

60 percent across the targets.

The average 𝜎𝑥 ranged from 0.26 degree (target C) to 0.35 degree (target B)

across targets. The target influence on 𝜎𝑥 was statistically significant (χ2(6) = 13.0, p

= 0.04), but no post-hoc pairwise target comparisons were significant following Holm-

Bonferroni adjustment.

The average 𝜎𝑦 ranged from 0.38 degree (target C) to 0.55 degree (target E)

across targets. The target influence on 𝜎𝑦 was statistically significant (χ2(6) = 36.8, p

< 0.001). The following post-hoc pairwise target comparisons were significant

following Holm-Bonferroni adjustment: target C with targets A (p = 0.019), B (p =

0.007) and E (p < 0.001), with lower 𝜎𝑦 for C than others; target D with targets B (p =

0.019) and E (p < 0.001), with lower 𝜎𝑦 for D than the others. In addition, the

difference between target A and D was close to significant at p = 0.052, with D having

the smaller 𝜎𝑦.


Table 5.4: Standard deviations (degrees) of fixation positions along horizontal

meridian (σx) for the targets Participant Target A Target B Target C Target D Target E Target F Target G

P1 0.29 0.21 0.21 0.35 0.20 0.26 0.26

P2 0.37 0.48 0.46 0.47 0.59 0.35 0.44

P3 0.33 0.51 0.25 0.30 0.33 0.27 0.23

P4 0.55 0.80 0.20 0.24 0.19 0.24 0.19

P5 0.23 0.20 0.19 0.21 0.26 0.26 0.29

P6 0.39 0.52 0.32 0.25 0.38 0.27 0.36

P7 0.44 0.32 0.32 0.32 0.48 0.48 0.33

P8 0.25 0.24 0.24 0.19 0.19 0.24 0.38

P9 0.26 0.23 0.24 0.19 0.26 0.21 0.24

P10 0.27 0.28 0.23 0.26 0.25 0.20 0.21

P11 0.32 0.22 0.29 0.23 0.25 0.19 0.25

P12 0.44 0.43 0.32 0.42 0.52 0.53 0.52

P13 0.29 0.43 0.24 0.21 0.25 0.18 0.18

P14 0.31 0.22 0.23 0.18 0.22 0.25 0.29

P15 0.28 0.26 0.27 0.28 0.28 0.30 0.24

P16 0.14 0.25 0.13 0.16 0.23 0.24 0.19

Mean+SD 0.32+0.10 0.35+0.17 0.26+0.07 0.27+0.09 0.31+0.12 0.28+0.10 0.29+0.10


Table 5.5: Standard deviations (degrees) of fixation positions along vertical meridian

(σy) for the targets Participant Target A Target B Target C Target D Target E Target F Target G

P1 0.30 0.22 0.21 0.21 0.24 0.34 0.30

P2 1.66 1.60 1.38 1.55 1.97 1.61 1.89

P3 0.16 0.23 0.16 0.17 0.25 0.24 0.16

P4 0.62 0.82 0.19 0.20 0.36 0.33 0.29

P5 0.37 0.65 0.41 0.35 0.63 0.35 0.42

P6 0.62 1.05 0.50 0.31 0.55 0.61 0.64

P7 0.43 0.40 0.32 0.31 0.35 0.34 0.37

P8 0.26 0.30 0.26 0.33 0.43 0.34 0.40

P9 0.26 0.25 0.25 0.31 0.37 0.43 0.35

P10 0.47 0.47 0.37 0.34 0.41 0.39 0.35

P11 0.94 0.73 0.72 0.64 0.60 0.58 0.64

P12 0.31 0.31 0.26 0.31 0.38 0.34 0.25

P13 0.31 0.48 0.22 0.27 0.66 0.47 0.24

P14 0.28 0.25 0.21 0.25 0.35 0.26 0.28

P15 0.26 0.33 0.24 0.22 0.71 0.25 0.28

P16 0.49 0.48 0.45 0.37 0.56 0.42 0.53

Mean+SD 0.48±0.37 0.54±0.37 0.38±0.30 0.38±0.33 0.55±0.41 0.46±0.33 0.46±0.40


Figure 5.4: Box-and-whisker plots of σx for the fixation targets. The lower and upper ends of the

whiskers represent minimum and maximum 𝜎𝑥, respectively. The lower and upper bounds of each box

represent 25th and 75th percentiles of the distribution.

Figure 5.5: Box-and-whisker plots of σy for the fixation targets.

Table 5.6 shows the BCEAs of each participant for the targets, and Figure 5.6

shows box-and-whisker plots of BCEAs of fixation positions for the targets. As for 𝜎𝑥

and 𝜎𝑦, there was considerable variability in BCEAs between participants, with most


having BCEAs < 2.0 degree2 for the targets, but P2 had BCEAs higher than 3.0 degree2

for all the targets and P6 had BCEAs higher than 3.0 degree2 for target B. BCEAs for

participant P6 were considerably different between targets, ranging from 0.56 degree2

(target D) to 3.36 degree2 (target B). The average BCEAs ranged from 0.78 degree2

(target D) to 1.31 degree2 (target E), and the target influence on BCEA was statistically

significant (χ2(6) = 34.4, p < 0.001). The following post-hoc pairwise target

comparisons were significant following Holm-Bonferroni adjustment: target C with

targets A (p = 0.022), B (p = 0.030) and E (p = 0.001), with lower BCEA for C than

the others; target D with targets A (p = 0.030), B (p = 0.004) and E (p = 0.001), with

lower BCEA for D than the others.

Table 5.6: Bivariate contour ellipse areas (BCEAs) (degrees2) for the fixation targets Participant Target A Target B Target C Target D Target E Target F Target G

P1 0.60 0.33 0.32 0.53 0.34 0.57 0.52

P2 4.38 5.22 4.43 4.58 7.45 3.55 5.83

P3 0.38 0.82 0.27 0.36 0.59 0.47 0.26

P4 1.00 1.81 0.27 0.34 0.48 0.55 0.35

P5 0.56 0.93 0.56 0.46 1.13 0.65 0.83

P6 1.71 3.36 1.14 0.56 1.50 1.17 1.60

P7 1.25 0.89 0.74 0.71 1.15 1.17 0.87

P8 0.46 0.51 0.45 0.43 0.57 0.58 0.98

P9 0.48 0.41 0.43 0.41 0.68 0.64 0.58

P10 0.91 0.95 0.61 0.61 0.70 0.54 0.53

P11 2.12 1.15 1.50 1.04 1.07 0.76 1.09

P12 0.72 0.88 0.60 0.93 1.40 1.03 0.87

P13 0.62 1.22 0.37 0.41 1.12 0.51 0.30

P14 0.62 0.39 0.35 0.32 0.55 0.47 0.57

P15 0.51 0.62 0.45 0.44 1.43 0.51 0.48

P16 0.49 0.86 0.41 0.43 0.79 0.67 0.64

Mean+SD 1.05±1.01 1.27±1.28 0.81±1.02 0.78±1.03 1.31±1.68 0.87±0.75 1.02±1.33


Figure 5.6: Box-and-whisker plots of BCEAs for the fixation targets.

5.4 DISCUSSION

The study compared the fixation stability as measured by 𝜎𝑥, 𝜎𝑦 and BCEAs for

participants viewing monitor and laser beam fixation targets. The fixation targets

included four computer monitor targets (targets A, B, C and D which were a bull’s

eye/cross hair combination, an image of a Gaussian beam, the images of Bessel beams

with four rings and with 3 rings, respectively) and three laser beam targets (targets E,

F and G which were a Gaussian beam and Bessel beams with 4 rings and with 3 rings,

respectively).

Target affected the three parameters significantly. There were significant post-

hoc differences between some of the target pairs for 𝜎𝑦 and BCEA, but not for 𝜎𝑥.

Monitor-based Bessel beam targets C and D provided significantly smaller 𝜎𝑦 and

BCEAs than the bull’s eye/cross hair combination (A) and the monitor- and laser-

based Gaussian beam targets (B, E).

The fixation stability along the horizontal meridian (𝜎𝑥) was better than along

the vertical meridian (𝜎𝑦) for all targets. Rattle (1969) also reported relatively better

fixation stability along the horizontal meridian than along the vertical meridian for

circular-shaped fixation targets, but Kosnik et al. (1987) reported better fixation


stability along vertical meridian than along the horizontal meridian with a square

shaped target; neither study indicated a reason for the meridional difference.

Thaler et al. (2013) and Steinman (1965) reported that a 0.2° diameter target with

shape similar to that of Gaussian beams provided better fixation stability than larger

targets. Therefore, the diameters of targets B and E were chosen to be 0.2° to match

the central fixating lobes of other targets used in this experiment.

There is large variation, ranging from 0.05 to 3.06 degree2, of average BCEAs

between studies for normal individuals (Chung, Kumar, Li, & Levi, 2015; Crossland

& Rubin, 2002; Dunbar, Crossland, & Rubin, 2010; Hirasawa, et al., 2016; Kosnik,

Fikre, & Sekuler, 1986; Kumar & Chung, 2014; Skavenski & Steinman, 1970). This

has been attributed to differences in experience of participants (Cherici, Kuang, Poletti,

& Rucci, 2012; Ditchburn, 1973), and variations between eye tracking devices

(Crossland & Rubin, 2002) including their spatial resolutions (Timberlake, Sharma,

Gobert, & Maino, 2003). Average BCEAs in this study were on the high side (0.78 –

1.31 degree2) and a contributor might be the relative naivety of the participants to the

task. Much eye movement research uses trained participants (Crossland & Rubin,

2002; Hirasawa, et al., 2016; Skavenski & Steinman, 1970; Steinman, 1965; Winterson

& Steinman, 1978) whose results are not representative of a clinical environment.

Cherici et al. (2012) reported that fixation stability, as measured by probability

distributions of fixation positions, was three times better for trained participants than

for untrained ones. Kosnik et al. (1986) reported that training naïve participants for 10

days improved fixation stability by 50 percent.

In this study, the BCEAs between participants varied by factors of 14 to 22,

larger than found by some studies (Crossland & Rubin, 2002; Dunbar, et al., 2010;

Kosnik, et al., 1986) that included trained participants. Morales et al. (2016) and

Macedo et al. (2007) reported that BCEAs of untrained participants varied by factors

of around 150 and 27, respectively.

Much of the inter-participant variation in the study results can be attributed to

participant P2. Accordingly, the plots of parameters across time and pupil size

variation (Table 5.7) were investigated, but nothing of particular interest was noted

e.g. variation in pupil size was not particularly high. I investigated results after removal

of his data. The Friedman test analysis showed only a few changes in significance.

However, I applied parametric statistics because log-transformation now showed most


distributions could not be considered to be significantly different from normal

distributions. Repeated measures with ANOVA for BCEA with post-hoc analyses

showed some significant differences not found in the original analysis, including

between laser Bessel beam F and Gaussian beam E. A two-way analysis of variance

omitting the bull’s eye/cross hair combination and using projection method

(monitor/laser) and target type (Bessel/Gaussian) as factors showed significant effect

for both with the monitor giving smaller results by a mean 0.07 log units (17%) and

the Bessel beams giving smaller results than the Gaussian beams by a mean 0.15 log

units (41%). These findings should be treated as tendencies only, given I cannot find

a valid reason to remove participant P2 from the main analysis.

Table 5.7: Average standard deviations (SDs) of pupil diameter among participants for

all the targets combined

Participant SD pupil diameter (mm)

P1 0.32

P2 0.33

P3 0.23

P4 0.21

P5 0.26

P6 0.21

P7 0.35

P8 0.25

P9 0.24

P10 0.20

P11 0.18

P12 0.24

P13 0.20

P14 0.26

P15 0.28

P16 0.35

A shortcoming of the study is the relatively low sampling rate of 30 Hz.

Crossland & Rubin (2002) reported BCEAs of three data files from an eye tracker with

250 Hz sampling rate to be 699, 833 and 1191 min arc2, and BCEAs when data were

under-sampled at 12.5 Hz were 707, 907 and 1187 min arc2 (different by 0 to 9

percent). These results showing BCEAs not being unduly affected by the low sampling

rate provides confidence to the results of this study. However, this limited us to


obtaining an overall picture of fixation stability rather than assessing fixational eye

movement components such as tremor, drift and micro-saccades.

There are two approaches of analysing fixation stability using BCEA (Castet &

Crossland, 2012): “fixation stability based on samples” describes the distribution

based on all eye position samples collected during the fixation trial, while “fixation

stability based on fixations” determines the mean position of each individual eye

fixation, where an individual eye fixation is defined as events underlying between

saccades, and describes the distribution of the fixations. This study can be directly

compared with all the studies mentioned above in the sense that all used the first

approach.

In summary, monitor-based images of Bessel beams provide better fixation

targets than a bull’s eye/cross hair combination, monitor-based Gaussian images and

laser Gaussian beams, but no claim can be made that laser Bessel beams provide better

fixation targets than laser Gaussian beams. There were no significant differences in

the fixation stability between monitor-based images of Bessel beams and laser Bessel

beams targets. This indicates that the shape of a Bessel beam, rather than its

propagation properties, is responsible for the improvement in fixation stability over

that achieved with other targets. The findings confirmed in part the hypothesis stated

in section 2.6 that a Bessel beam used as a fixation target will improve fixation stability

over that achieved with conventional targets. Ophthalmic imaging instruments that

require stable fixation can benefit by using Bessel beams in the form of monitor-based

images as fixation targets, but it remains unclear whether they would benefit by using

Bessel laser beams to provide both illuminating beams and fixation targets.

Chapter 6: RETINAL IMAGING WITH BESSEL BEAMS 89

Chapter 6: RETINAL IMAGING WITH

BESSEL BEAMS

6.1 INTRODUCTION

Retinal imaging modalities utilise the backscattered light from the retinal area

being imaged. Light sources in retinal imaging are typically Gaussian beams that suffer

from aberrations, diffraction and forward scatter while passing through the ocular

media, creating challenges in obtaining high quality retinal images (Liang & Williams,

1997; Wanek, et al., 2007). Aberrations can be detected and corrected using adaptive

optics during retinal imaging to improve the resolution and contrast of the image

(Liang, et al., 1997; Roorda, et al., 2002). In an aberration free eye, diffraction

determines the theoretical limit of image quality, which is inversely proportional to the

pupil size, and hence its effect can be compensated to some extent by increasing the

pupil size (Atchison & Smith, 2000). Forward scatter causes a significant reduction in

the amount of light reaching the retinal area being imaged. This reduces the amount of

backscattered light from the retinal area being imaged thereby decreasing the signal-

to-noise ratio (Carpentras, et al., 2018; Chen, et al., 2016; Christaras, et al., 2016;

Wanek, et al., 2007).

The loss of light on the way to the retinal area being imaged due to forward

scatter and diffraction cannot be compensated by increasing the incident light because

of the limited amount of light that can be sent into the eye due to the laser safety

standard (Delori, et al., 2007). Therefore, identifying approaches that negate the

detrimental effects of forward scatter and diffraction and hence increase the amount of

light reaching the retinal area being imaged will lead to a significant advancement in

the field of retinal imaging.

While imaging biological tissues such as skin, Bessel beams reduce scattering

artefacts, and provide better image quality and penetration depth in dense media than

Gaussian beams (Fahrbach, et al., 2010; Rohrbach, 2009). Farhbach et al. (2010) found

that Gaussian beams lost almost 40 percent of their energy in a scattering condition

where around 50 percent of the beam field was disturbed by the scatter, whereas under

the same condition Bessel beams lost only 5 percent of their energy. This property of

90 Chapter 6: RETINAL IMAGING WITH BESSEL BEAMS

Bessel beams might help in increasing the amount of light reaching the retinal area

being imaged. The hypothesis being tested in this chapter was that using a Bessel beam

rather than a Gaussian beam will provide higher amount of light at the retinal area

being imaged.

The quantity of ocular scatter depends on age, pigmentation of the structures

such as iris and retina, and pathologies or surgical interventions affecting transparency

(Guber et al., 2011; Pinero, et al., 2010). The increase in age and cataract increase the

scatter, hence decreasing the amount of light reaching the retinal area being imaged

(De Waard, et al., 1992; Kuroda, et al., 2002; Rozema, Van den Berg, & Tassignon,

2010; Wanek, et al., 2007). Provided that the primary hypothesis holds true, the

advantage of Bessel beams, being more resistant to the forward scatter than Gaussian

beams, might be emphasised in older eyes than young eyes, and in eyes with cataract

than eyes without cataract. Therefore, this study also compared the ratio of the amount

of light reaching the retinal area being imaged for Bessel and Gaussian illuminations

between young and older age groups, and between without-cataract and early-cataract

groups. Additionally, this study assessed the relationship between the psychophysical

measure of forward scatter and the ratio between the amount to light reaching the

retinal area of interest for Bessel and Gaussian beams.

6.2 METHODS

I built a retinal imaging set up and assessed the amount of light reaching the

retinal area being imaged for Bessel and Gaussian beams between young and older age

groups, and between without-cataract and early-cataract groups. The Bessel and

Gaussian beam images formed at the retina were imaged using a science camera

conjugate to the retinal plane. The intensity of each image thus acquired was used as

the measure of the amount of light reaching the retinal area being imaged and is

referred as retinal image intensity in this chapter, the abstract and section 7.1.3. Details

on participants, grading of cataract, production and propagation properties of the

Bessel beam, instrumentation, task and image acquisition, measurement of straylight,

and data collection and analysis are described below.


6.2.1 Participants



database as an invitation to participate in the study. Routine clinical tests were

performed to determine that they were eligible (Table 6.1). The study adhered to the

tenets of the Declaration of Helsinki and informed consent was obtained from all

participants.

Table 6.1: Clinical tests and inclusion criteria of participants

Examination Instrument/Procedure Criterion

Visual acuity Bailey-Lovie log MAR

chart

≤ 0.20 log MAR

Intra ocular

pressure

iCare tonometer ≤ 21 mm Hg

Anterior chamber

angle

Van Herrick test > 0.3


and history taking

No corneal opacities, glaucoma

and retinal diseases.

Cataract < Grade II (details in

page 91)

No history of ocular surgery

General health History taking No diabetes. No history of

allergic or abnormal reaction to

tropicamide.

Twenty adult participants (7 Male, 13 Female) were included of which 10 were

< 35 years of age (young group) and 10 were > 59 years of age (older group). Table

6.2 shows the ages, genders, BCVA and refraction. The mean age of the young group

was 26.1±6.6 years (range 18 to 34 years) and that of the older group was 63.3±2.9

years (range 60 to 70 years). Mean best corrected visual acuity (BCVA) of the young

group was ‒0.06±0.02 log MAR (range ‒0.10 to ‒0.02) and that of the older group was

‒0.01±0.09 log MAR (range ‒0.12 to 0.18) in right eyes.


Table 6.2: Age, gender, BCVA and refractive error of the participants

Participants Age

(years)

Gender BCVA (log

MAR)

Right eye refraction

(Dioptre)

Young group

P1 29 F ‒0.06 +0.48/‒0.56 x 1

P2 34 M ‒0.10 ‒0.50 x 163

P3 25 F ‒0.04 +0.62/‒0.72 x 1

P4 34 F ‒0.04 +1.81/‒0.62 x 10

P5 33 F ‒0.08 +0.67/‒0.29 x 170

P6 27 F ‒0.02 +0.48/‒0.60 x 23

P7 26 M ‒0.06 +0.69/‒0.34 x 136

P8 18 M ‒0.06 +0.70/‒0.59 x 179

P9 20 F ‒0.08 +0.10/‒0.02 x 37

P10 25 F ‒0.04 +0.04/‒0.49 x 99

Mean±SD 26.1±6.6 ‒0.06±0.02

Older group

P11 61 F ‒0.12 ‒0.09/‒0.31 x 139

P12 64 M ‒0.10 ‒1.23/‒0.74 x 95

P13 60 F ‒0.06 +1.13/‒0.21 x 178

P14 62 F ‒0.10 +2.06/‒0.72 x 88

P15 62 M ‒0.06 +0.96/‒0.69 x 67

P16 64 M 0.00 +0.41/‒0.40 x 92

P17 61 F 0.00 +1.57/‒1.61 x 15

P18 63 M ‒0.02 +0.55/‒0.11 x 153

P19 70 F 0.18 ‒1.70/‒0.31 x 16

P20 66 F 0.08 +1.93/‒2.62 x 93

Mean±SD 63.3±2.9 ‒0.01±0.09


6.2.2 Grading of cataract

A simplified cataract grading system developed by the World Health

Organisation (WHO) cataract grading group was used to grade the presence and

severity of cataract, based on its location in the lens (Table 6.3 and Figure 6.1)

(Thylefors, et al., 2002). Examinations of the lenses of right eyes were performed with

a slit lamp microscope (SM-70N, Takagi, Japan) at 10X magnification after dilating

the pupil (> 6 mm) with 1 percent tropicamide. Participants with grade 1 cataract,

irrespective of its location, were referred as “early-cataract”, and those without any

form of cataract were referred as “without-cataract”.

Table 6.3: Simplified cataract grading by the WHO cataract grading group

Grade Description

Nuclear cataract (NUC) – Standard photographs as shown in Figure 6.1 are used for

reference

Grade 0 < less than Grade NUC – 1

Grade NUC – 1 In nuclear zone, the anterior and posterior embryonal nuclei are

distinctly more opalescent than normally seen, but the central

clear zone is still easily distinguishable in its entirety

Grade NUC – 2 The nuclear zone is more uniformly opaque and the central clear

zone between the anterior and posterior nuclei is not clearly

visible

Grade NUC – 3 The nuclear zone is densely opaque with more or less uniform

nuclear opacity extending to the edge of the nuclear zone; nuclear

features are only partially visible, if at all

Grade 9 Cannot grade (due to corneal opacity or Morgagnian cataract)

Cortical cataract (COR) – Octants of lens circumference are used for grading

Grade 0 Cataract involves less than one eighth of the circumference

Grade COR – 1 Cataract involves one eighth, but less than a quarter, of the

circumference


Grade COR – 2 Cataract involves a quarter, but less than half, of the

circumference.

Grade COR – 3 Cataract involves half or more of the circumference

Grade 9 Cannot grade (due to corneal opacity or Morgagnian cataract)

Posterior sub-capsular cataract (PSC) – Vertical diameter of the cataract is used for

grading

Grade 0 Cataract with vertical diameter of less than 1 mm

Grade PSC - 1 Cataract with vertical diameter equal to or greater than 1 mm, but

less than 2 mm

Grade PSC - 2 Cataract with vertical diameter equal to or greater than 2 mm, but

less than 3 mm

Grade PSC - 3 Cataract with vertical diameter equal to or greater than 3 mm

Grade PSC - 4 Cannot grade (due to corneal opacity or Morgagnian cataract)

Figure 6.1: Standard photographs of (left) grade 1, (middle) 2 and (right) 3 nuclear cataract provided by

the WHO cataract grading group. Image taken from Thylefors (2002).


6.2.3 Production and propagation properties of the Bessel beam

A 3 mm beam waist (𝜔0) collimated narrow spectral bandwidth Gaussian beam

(wavelength 515 nm) was passed through a 1° physical angle (α) axicon (Thorlabs,

AX251A) to produce a Bessel beam (Figure 6.2). The axicon was used to generate

Bessel beams because it is a more power efficient method, that allows most of the

incident power to be utilised, than other methods such as an annular slit and SLM

(Bowman et al., 2011; Čižmár & Dholakia, 2009; McGloin & Dholakia, 2005). A

Bessel beam generated from an axicon has smoother on-axis intensity variation than

that from an annular slit (McGloin & Dholakia, 2005). The incident Gaussian beam

after passing through the axicon produced the Bessel beam whose intensity profile can

be represented as (Alexeev, Leitz, Otto, & Schmidt, 2010):

𝐼 (𝑟, 𝑧) = 2𝑘𝜋(tan2 𝛼)(𝑛 − 1)2𝑧𝐼0𝑒−2(𝑛−1) tan 𝛼/𝜔0 × 𝐽02(𝑘𝑟(𝑛 − 1) tan 𝛼) (6.1)

where 𝑟 and 𝑧 are the radial and longitudinal coordinates of the Bessel beam,

respectively, 𝜔0 and 𝐼0 are the beam waist and intensity of the Gaussian beam incident

onto the axicon, respectively, 𝐽0 is the zeroth-order Bessel function of the first kind, 𝑘

is the wave vector that can be calculated as 2𝜋 λ⁄ (wavelength of light), and 𝑛 is the

index of refraction of the axicon material.

Figure 6.2: Spatial spectrum of a Bessel beam generated using an axicon with wave vectors of plane

waves on the surface of a cone. θ, opening angle of cone at the tip of the axicon; α, physical/wedge

angle of axicon; k, wave vector; zmax, the maximum distance until which the central lobe of the beam

maintains its propagation-invariant property, kr, radial component of k; kz, longitudinal component of

k; 𝑤0, incident beam waist; 𝐷, diameter of the aperture. This figure is a repeat of Figure 2.2. Modified

from Litvin, et al., (Litvin, et al., 2008).


All parts of the wave, after passing through the axicon, are inclined at the same

angle (𝜃) to the optical axis. 𝜃 is also known as the opening angle of cone and for my

set up it was 0.46° (calculated using Equation 2.7). The central lobe size of the beam

(𝑟0), which is the radial distance from the core to the first intensity minimum, was 0.03

mm (calculated using Equation 2.9). After propagating through the distance 𝑧max, the

maximum distance for which the central lobe of the beam maintains its propagation-

invariant property, the separation between the wave-vectors increases while

decreasing the beam intensity in the central axis. The 𝑧max for my set up was 375.47

mm (calculated using Equation 2.6). The beam intensity in the central axis eventually

attains a null value at the far field to shape into the annular beam. While the Bessel

beam traverses the eye optics, it attains its far field shape and forms an annular shape

image on the retina (Figure 6.3, a). The outer diameter of the annular beam at the far

field when combined with another focussing element such as human eye can be

approximated as (Duocastella & Arnold, 2012)

𝑑 = 2𝑓 tan 𝜃 (6.2)

where 𝑓 is the effective focal length of the human eye. Considering 𝑓 to be 16.67 mm

for the human eye, d at the retinal plane was about 260 micrometer (≈ 1 degree).

The Rayleigh length (𝑧𝑅𝐵𝑒𝑠𝑠𝑒𝑙) of the Bessel beam was 114.48 mm (calculated

using Equation 2.8). The Gaussian beam had a Rayleigh length of 3.81 mm (calculated

using Equation 2.3). Figure 6.3 shows the transverse intensity profile of my

experimental Bessel and Gaussian beams images obtained at the participant’s retinal

plane. The asymmetric intensity distribution observed in the far field of Bessel beam

might be due to the spatiotemporal astigmatism (Dallaire, McCarthy, & Piché, 2009)

induced by the imperfections inherent in the optical fibre, that is used in the

transmitting the beam before letting it propagate in the free space.


Figure 6.3: Cross sectional intensities of the (a & c) Bessel and (b & d) Gaussian beams images, orthogonal to the axis of the beams taken at the participant’s retinal plane. (a & b) Intensity distributions

along x-axis (horizontal) and y-axis (vertical), (c & d) intensity distributions along x-axis, (e & f) 3-D

intensity distributions of the Bessel and Gaussian beams, respectively.


6.2.4 Instrumentation

The retinal imaging setup (Figure 6.4 and Figure 6.5) was developed to compare

the amount of light reaching the retinal area of interest being imaged between Bessel

and Gaussian beams; this gives an indirect estimate of scatter.

Figure 6.4: Experimental setup for retinal imaging with Bessel beams. (i) sCMOS camera, (ii – iv, vii,

viii, xii – xiv) lenses, (v) polariser, (vi) beam splitter, (ix) 0.8 ND filter, (x) axicon, (xi) aperture, (xv)

laser diode. The arrow indicates that the axicon was flipped out of the illumination path for Gaussian

retinal imaging and flipped back into the illumination path for Bessel retinal imaging.


Illumination path

A high-coherence diode laser (Toptica photonics, Topmode) (515 nm) was

passed through a 7.5 mm focal length lens (Figure 6.5, xiv) to produce a collimated

beam with 1.4 mm waist diameter. It was then passed through a 10X achromatic

objective (xiii) and 75 mm focal length (xii) lenses to produce an expanded (12 mm

diameter) and collimated beam. The beam was then truncated to 6 mm diameter using

an aperture (xi) before passing through an axicon (Thorlabs, AX251A), placed at a

pupil conjugate plane (x), to generate a Bessel beam. The powers of the Bessel and

Gaussian beams, at the corneal plane, were measured using a digital handheld optical

power and energy meter (Thorlabs, PM100D) with photodiode power sensor

(Thorlabs, S120C), at the beginning of the session for each participant. The average

powers of the Bessel and Gaussian beams were 17.2 µW (range 16.1 to 19.1 µW) and

18.3 µW (range 17.4 to 20.1 µW), respectively. Two lenses (300 mm focal length

each) (viii & vii) imaged the axicon onto the participant’s pupil. The beam splitter (8

percent reflectance and 92 percent transmittance) (vi) shared the optical path between

Figure 6.5: Photograph of experimental setup for retinal imaging with Bessel beams. (i) sCMOS camera,

(ii – iv, vii, viii, xii – xiv) lenses, (v) polariser, (vi) beam splitter, (ix) 0.8 ND filter, (x) axicon, (xi)

aperture, (xv) laser diode. The arrow indicates that the axicon was flipped out of the illumination path

for Gaussian retinal imaging and flipped back into the illumination path for Bessel retinal imaging.

Participants were seated with their heads stabilised using bite-bars and right eyes aligned for the

imaging.


illumination and imaging arm, where it reflected the beam onto the pupil as well as

transmitted the reflected beam from the retina onto the camera. The retina, being

illuminated, was at the far field of the Bessel beam. As the Bessel beam approached

its far field, there was an increase in separation between the wave-vectors that

decreased the central beam intensity, eventually attaining a null value which

corresponded to an annular beam (Figure 6.3, a). The Gaussian beam followed the

same optical path as that of Bessel beam, but without the axicon, which was flipped

out of the illumination path.

Imaging path

The light reflected from the retina first passed through the beam splitter. A

polariser was then placed on the path and was rotated to remove the highly polarised

light reflected from the cornea and transmit only the depolarised light from the retina.

Three lenses, with focal lengths 300 mm, 75 mm and 150 mm (Figure 6.5, iv, iii and

ii, respectively) and placed in between the polariser and the camera, together with

16.67 mm focal length eye optical system formed a 4f system that imaged the retina

onto camera at approximately 36 times magnification. A 5.5 megapixel scientific-

grade sCMOS camera with 16.6 mm X 14.0 mm sensor size and 6.5 µm pixel size

(Andor, Zyla 5.5 sCMOS) placed at retinal conjugate plane, acquired the images. The

intensity measured as grayscale value (arbitrary unit (AU)) of each saturated pixel of

the camera was 4095.

6.2.5 Linearity of the camera output

The linearity of the image intensity acquired by the camera with respect to the

normalised input power was assessed. An 8 mm diameter collimated laser beam (515

nm) was shone onto the camera sensor and 78 images for different input powers were

acquired. For the first image, the power of the beam was adjusted such that none of

the pixels were saturated and the pixel that had the highest intensity was 4059 AU, just

below its saturation point. A digital handheld optical power and energy meter

(Thorlabs, PM100D) with photodiode power sensor (Thorlabs, S120C) measured the

power of the input beam. The power displayed on the power meter to acquire the first

image was 111.2 microwatt, and subsequently remaining images were acquired by


reducing the power at steps of about 1.4 microwatt until the power meter showed the

value immediately above 0.1 microwatt. The coordinates of the pixel that had the

highest intensity for the first image were located and the output intensity responses of

this pixel for all input powers were recorded. The linear regression fit of the pixel

intensity as a function of normalised input power (Figure 6.6) showed that output

intensity response of the pixel for the input power was linear (r = 0.99, p < 0.001).

Figure 6.6: Scatterplot of normalised input power and pixel output intensity response.

6.2.6 Task and image acquisition

One drop of 1 percent tropicamide was instilled in the participants’ right eyes to

dilate pupils. Pupil size of at least 6 mm, measured using pupil size measurement

function of COAS-HD aberrometer, was attained before imaging the retina.

Participants were seated with their heads stabilised using a bite-bar. Participants’ right

eyes were aligned with the illumination path, with left eyes being occluded, and were

instructed to fixate at the centre of the illuminating beam. Participants with spherical

equivalent refractive error > 0.50 dioptre or with cylinder > 1.0 D, determined using


COAS-HD aberrometer, in right eyes were corrected using corrective lenses in frame.

An exposure duration of 34 ms for image acquisition and a 0.8 neutral density filter

(Figure 6.5, ix) were used to avoid pixel saturation in the images. For each participant,

five images were taken each for Bessel and Gaussian beams illumination in the same

session with identical camera settings.

6.2.7 Measurement of straylight

The straylight was measured psychophysically using the C-Quant Straylight

Meter (Oculus Optikgeräte, Wetzler, Germany). The instrument uses a two-

alternative-force-choice psychophysical measurement algorithm to obtain the

straylight parameter as a measure of forward scatter, expressed in log units, which is

half the value of the 50 percent point of the psychometric curve. Details on the C-

Quant Straylight Meter are provided in section 2.5.2. To improve reliability, the test

was repeated until the expected standard deviation and the quality parameter values

given by the instruments were ≤ 0.08 and > 1.0, respectively.

6.2.8 Data collection and analysis

The acquired images for Bessel and Gaussian images formed at retina were

saved as *.tif files. The images were exported to MATLAB for analysis. For each

image, the noise threshold greyscale value (arbitrary unit (AU)) was determined,

which was obtained from the pixel with the maximum greyscale value from the bottom

left corner 40 𝑋 40 pixels, which corresponded to the camera sensor area outside the

retinal image. Only the thresholded pixels, those with greyscale values above the noise

threshold grayscale values, were used for analysis. The intensity of each image was

determined by summing the greyscale values of all the thresholded pixels. The

intensity distributions of thresholded pixels of Bessel and Gaussian retinal images

were acquired using the MATLAB code (Appendix G and H). An average intensity of

five images, obtained each from Bessel and Gaussian illuminations for each

participant, was determined. These values for Bessel and Gaussian retinal images of

each participant were normalised by multiplying the average Bessel retinal image

intensity by the ratio of Gaussian and Bessel beams powers, keeping the average

Gaussian retinal image intensity unchanged. These normalised Bessel and Gaussian


retinal images intensities were considered to represent the amount of light reaching the

retinal area being imaged. Shapiro-Wilk tests indicated that the Bessel and Gaussian

retinal image intensities were normally distributed in both age groups (Table 6.4). A

two-way mixed ANOVA was conducted to investigate the effects of Bessel and

Gaussian illuminations and two age groups on retinal image intensities. Statistical

significances of the differences in the ratios of Bessel and Gaussian retinal image

intensities between young and older group, and between without-cataract and early-

cataract group were determined using independent t-test.

Table 6.4: Shapiro-Wilk test p-values to assess the normality distribution of Bessel

and Gaussian retinal image intensities.

Shapiro-Wilk (p-value*)

Below 35 years Above 59 years

Bessel retinal

image intensity

Gaussian retinal

image intensity

Bessel retinal

image intensity

Gaussian retinal

image intensity

0.92 0.94 0.33 0.52

* p > 0.05 indicates normal distribution.

To identify the relationship between the ratio of Bessel and Gaussian retinal

image intensities and the straylight parameter, Pearson correlation analysis was

performed. The ratio of Bessel and Gaussian retinal image intensities was modelled as

dependent or response variable, while straylight value was set as the predictor variable.

The noise intensity of each image was calculated as the sum of the greyscale

values of the pixels less than or equal to the thresholded noise values. An average noise

intensity of five retinal images, obtained each from Bessel and Gaussian illuminations

for each participant, was determined. The average noise intensities for Bessel and

Gaussian retinal images were 5.6 X 108 AU±0.7 X 108 AU and 5.7 X 108 AU±0.3 X 108

AU, respectively. A paired t-test showed that the difference in dark noise intensities

between Bessel and Gaussian retinal images was not significant (p = 0.33).


6.3 RESULTS

Figure 6.7 shows the intensity distributions of (a) Bessel and (b) Gaussian retinal

images of participant P1. A Bessel beam illuminated the participant’s retina as an

annular beam, as the retina was at the far field of the beam where the central axis of

the beam attained a null value to shape it into this annular form. The thickness of the

annulus in the retina of an eye without any optical imperfections is half of the spot

diameter formed by the focused Gaussian beam at the retina.


Figure 6.7: Intensity distribution of thresholded pixels of (a) Bessel and (b) Gaussian retinal images at

fovea for participant P1. The outer diameter of the annular ring in (a) Bessel retinal image subtends ~

1° at the retina. d0 represents the diameter of the retinal spot formed by the focused Gaussian beam, which in an eye without aberrations and scattering is twice the thickness of the annular ring formed by

the Bessel beam. The colour bars represent greyscale (intensity) values of thresholded pixels.

d0/2

d0


Table 6.5 shows the Bessel and Gaussian retinal image intensities and their

ratios, straylight parameter and presence of cataract for each participant. For the young

group, the Bessel retinal image intensities ranged from 1.73 𝑋 108 arbitrary unit (AU)

to 5.13 𝑋 108 AU and the Gaussian retinal image intensities ranged from 1.29 𝑋 108

AU to 3.12 𝑋 108 AU. For older group, the Bessel retinal image intensities ranged from

0.71 𝑋 108 AU to 3.32 𝑋 108 AU and the Gaussian retinal image intensities ranged

from 0.30 𝑋 106 AU to 2.35 𝑋 108 AU. The retinal image intensities varied around 3

times and 2.5 times in young group and around 4.5 times and 8 times in older group

for Bessel and Gaussian beams, respectively.


Table 6.5: Intensities of Bessel and Gaussian retinal images, straylight parameter and

cataract among participants Participant Bessel image

intensity

(arbitrary units)

(𝑋 108)

Gaussian image

intensity

(arbitrary units)

(𝑋 108)

Ratio of Bessel

and Gaussian

retinal image

intensities

Straylight

parameter

(Log unit)

Cataract

Young group

P1 3.06 1.91 1.60 0.69 No

P2 1.73 1.29 1.34 0.99 No

P3 5.13 3.12 1.55 0.67 No

P4 2.48 1.69 1.47 0.78 No

P5 3.55 2.55 1.39 0.81 No

P6 4.01 2.57 1.56 0.99 No

P7 2.39 1.87 1.28 0.76 No

P8 3.74 2.97 1.26 0.75 No

P9 3.45 2.03 1.70 0.79 No

P10 3.91 2.44 1.60 0.80 No

Mean±SD 3.34±0.97 2.26±0.62 1.48±0.15 0.80±0.11

Older group

P11 1.65 1.47 1.13 0.89 No

P12 2.71 1.56 1.74 1.20 Early

P13 0.82 1.38 0.59 1.25 No

P14 3.32 2.35 1.42 1.13 No

P15 1.39 0.62 2.24 0.98 No

P16 2.06 1.22 1.69 1.04 Early

P17 0.71 0.32 2.21 1.47 Early

P18 1.68 1.29 1.31 1.52 No

P19 0.92 0.30 3.10 1.58 Early

P20 1.23 0.72 1.70 1.35 Early

Mean±SD 1.65±0.85 1.12±0.65 1.71±0.69 1.24±0.24

There was a significant main effect of illumination type on retinal image

intensities (F1,18 = 59.13, p < 0.001), with Bessel beams providing higher retinal image

intensities than Gaussian beams (Figure 6.8). There was a significant main effect of

age group on retinal image intensities (F1,18 = 18.06, p < 0.001), with retinal image

intensities being higher for the young than the older group. Figure 6.9 shows Bessel

and Gaussian retinal image intensities in each age group. Bessel retinal image

intensities were significantly higher than Gaussian retinal image intensities in both age


groups (p < 0.001 and p = 0.007 for young and older groups, respectively). In contrast

to the rest of the participants, P13 had lower Bessel retinal image intensity than

Gaussian retinal image intensity. The findings for P13 were checked carefully to rule

out any processing error. The ratio of Bessel and Gaussian retinal image intensities

among young group was 1.48±0.15 and among the older group was 1.71±0.69, and the

difference was not statistically significant (p = 0.30). The ratios of Bessel and Gaussian

retinal image intensities were 1.55±0.36 and 1.62±0.57 for males and females,

respectively, and the difference was not statistically significant (p = 0.76).

Figure 6.8: (Left) Bessel and (right) Gaussian retinal image intensities among all participants. The black

circles and error bars represent the means and ±1SDs, respectively, of retinal image intensities. The

small symbols are data for individual participants.


Figure 6.9: Bessel and Gaussian retinal image intensities in (left) young and (right) older age groups.

Other details are as for Figure 6.8.

Five older participants (P12, P16, P17, P19 and P20) had early cataract while the

rest did not have any form of cataract. The ratio of Bessel and Gaussian retinal image

intensities among older participants without- and early-cataract were 1.33±0.60 and

2.09±0.61, respectively (Figure 6.10), and with the difference being marginally

significant (p = 0.08). The ratios of Bessel and Gaussian retinal image intensities

among all participants (young and older) without-cataract was 1.42±0.35, which was

significantly different from those with early-cataract (p = 0.007).


Figure 6.10: Ratio of Bessel and Gaussian retinal image intensities in (left) early- and (right) without-

cataract groups. Other details are as for Figure 6.8.

The average straylight parameters for young and older groups were 0.80±0.11

and 1.24±0.24, respectively, which was statistically significant (p = 0.02).

Figure 6.11 shows a scatterplot comparing the ratio of Bessel and Gaussian

retinal image intensities to the straylight parameter. The linear regression (Equation

6.3) showed a weak positive correlation between them:

Ratio of intensities = 0.90 + 0.68 * straylight parameter (r = 0.39, p = 0.09)

(6.3)


6.4 DISCUSSION

The study assessed the amount of light reaching the retinal area being imaged

for Bessel and Gaussian beams between young (< 35 years) and older (> 59 years) age

groups, and between without-cataract and early-cataract groups. The study also

assessed the relationship between the forward scatter, as determined psychophysically,

and the ratio of the amount of light reaching the retinal area being imaged for Bessel

and Gaussian beams. Retinal image intensity was used as a measure of the amount of

light reaching the retinal area being imaged. The straylight parameter was used as a

psychophysical measure of the forward scatter.

Retinal image intensities for Bessel beams were higher than for Gaussian beams

in both age groups, and Bessel and Gaussian retinal image intensities for the young

group were around two times higher than for the older group. The ratios of Bessel

retinal image intensities to Gaussian retinal image intensities were similar for the two

age groups, but the ratios were higher in an early-cataract group than in a without-

Figure 6.11: Scatter plot of the straylight parameter and the ratio of Bessel and Gaussian retinal image

intensities. The dark grey area represents the 95% confidence interval.


cataract group. The straylight parameter was positively correlated with the ratio of

Bessel and Gaussian retinal image intensities.

Except for the differences in propagation properties between Bessel and

Gaussian beams, all other conditions such as pupil size, accommodation, camera

settings, and power, wavelength and angle of incidence of the incident beams were

similar. The retinal image intensity depended on the amount of light loss due to

absorption by ocular structures and on the amount of light loss due to scatter before

reaching the area being imaged. The proportions of light loss due to absorption in any

absorbing media such as lens and retina are same for both Bessel and Gaussian beams

of the same wavelengths (Zamboni-Rached, 2006). Therefore, the difference in Bessel

and Gaussian retinal image intensities is primarily due to Bessel beams being less

affected by scatter than Gaussian beams on the way to the retinal area being imaged

rather than to an absorption effect.

Signal-to-noise ratio is often used as a measure of retinal image quality (van

Velthoven, van der Linden, de Smet, Faber, & Verbraak, 2006). Direct calculation of

the signal-to-noise ratios of the images could not be performed as they did not contain

enough information about the retinal structures, which is necessary for the accurate

quantification of the signal. Chen et al. (2016) calculated the retinal image intensity

(total intensity minus the noise) from the raw data provided by optical coherence

tomography. They showed the retinal image intensity had strong positive correlation

with the image quality value, which was provided by the optical coherence

tomography based on the signal-to-noise ratio. The retinal image intensity in this study

could not be directly correlated with the signal-to-noise ratio. However, the similarity

in noise intensity between Bessel and Gaussian images, and higher Bessel retinal

image intensity than Gaussian retinal image intensity indicate that Bessel beams can

provide higher signal-to-noise ratios than Gaussian beams.

Increase in age and cataract increase ocular scatter (De Waard, et al., 1992;

Kuroda, et al., 2002; Rozema, et al., 2010). As Bessel beams are more scatter resistant

than Gaussian beams (Fahrbach, et al., 2010), it was expected that the advantage of

Bessel beams in increasing the retinal image intensity than Gaussian beams would be

greater in the older group than in the young group, and in the early-cataract group than

in the without-cataract group. The ratio of Bessel and Gaussian retinal image

intensities in early-cataract group being higher than that in without-cataract group


shows that the advantage of Bessel beams is emphasised in ocular media with large

scatterers, such as cataract, than in ocular media without cataract. The ratios of Bessel

and Gaussian retinal image intensities being similar between young and older groups

indicate that the advantage of Bessel beams is not emphasised with the increase in age.

However, caution should be applied while referring to these results because of the

small sample size of 10. A power analysis calculated using G*Power software

(Version 3.1.9.2) showed that the effect size of this study was 0.46 and the power was

16 %. To detect the same effect size with 80% power using t-test with α at 0.05, 75

people would be needed for each age group. This indicates that the effect of age is

small for older eyes without at least some degree of media opacification.

The positive correlation between the ratio of Bessel and Gaussian retinal image

intensities and the straylight parameter indicates that, unlike observed in age, the

advantage of Bessel beam is marginally emphasised with the increase in straylight

parameter.

In summary, Bessel beams provide higher amounts of light at the retinal area

being imaged, and hence are less affected by the scatter and diffraction in the ocular

media, than Gaussian beams. The ratios of the amount of light reaching the retinal area

being imaged for Bessel beams to that for Gaussian beams are similar in young and

older age groups. The ratio of the amount of light reaching the retinal area being

imaged for Bessel beams to that for Gaussian beams is higher for the eye with early

cataract than for the eye without cataract. The findings support the hypothesis that

using a Bessel beam rather than a Gaussian beam will provide higher amount of light

at the retinal area being imaged. The challenges faced by retinal imaging instruments

due to high light loss while passing through ocular media, especially in the presence

of partial media opacities such as early cataract, can be mitigated by using Bessel

beams for illumination.

114 Chapter 7: CONCLUSIONS AND FUTURE DIRECTIONS

Chapter 7: CONCLUSIONS AND

FUTURE DIRECTIONS

7.1 INTRODUCTION

I explored the application of Bessel beams in phakometry, fixation stability and

retinal imaging of the human eye. The three aims of the study were:

1. To determine the Purkinje image brightness, accuracy and repeatability of

Bessel phakometer compared with those of Gaussian phakometer (Chapters

3 and 4, Paper 1).

2. To determine the effectiveness of a Bessel beam, as a fixation target, in

improving fixation stability compared with that for conventional targets

(Chapter 5).

3. To compare the amount of backscattered light from retinal area being imaged

between Bessel and Gaussian beams (Chapter 6).

7.1.1 Phakometry with Bessel beams

I developed a phakometer using a Bessel beam for illumination as described in

Chapter 4. To produce easy-to-identify Purkinje images in the Bessel phakometer, I

intended to place a horizontal obstruction in the Bessel beam path to obstruct majority

of it and shape it like an arc. I conducted a preliminary study to determine whether the

desired propagation properties of a Bessel beam, after the majority of it has been

obstructed, are retained in linear media (Chapter 3). Even after blocking a major

portion of the Bessel beam, including its central lobe, the remaining beam during

propagation retained self-reconstructing and non-diffracting properties. This gave me

confidence to block a part of the beam as intended.

The accuracy of Bessel phakometer was assessed using a model eye of known

lens parameters. The determined lens parameters of the model eye by the phakometer

were similar to those provided by the manufacturer. The accuracy of the Bessel

phakometer was similar to those reported in the studies for the Gaussian phakometers

(Barry, et al., 2001; Rosales & Marcos, 2006). The repeatability in estimating lens

Chapter 7: CONCLUSIONS AND FUTURE DIRECTIONS 115

parameters and, brightnesses of 3rd and 4th Purkinje images of the Bessel phakometer

were compared with those of an existing Gaussian phakometer. The Bessel

phakometer provided improvement in inter-observer repeatability over that of the

Gaussian phakometer but there was no difference in intra-observer repeatability

between the two phakometers. Brightnesses of 3rd and 4th Purkinje images were

approximately three times higher with the Bessel phakometer than with the Gaussian

phakometer. I concluded that the Bessel phakometer provides similar accuracy in

estimating lens parameters, and produces brighter Purkinje images and better inter-

observer repeatability than that of the Gaussian phakometer.

The study confirmed in part the hypothesis stated in section 2.6 that using a

Bessel beam rather than a Gaussian beam will provide brighter 3rd and 4th Purkinje

images resulting in more repeatable (inter-observer) estimates of lens surface

curvature. The hypothesis was not confirmed for the accuracy and intra-observer

repeatability which were similar for both Bessel and Gaussian phakometers.

7.1.2 Fixation stability with Bessel beams

I investigated the fixation stability among 16 participants with seven fixation

targets: monitor-based images of a bull’s eye/cross hair combination, a Gaussian beam,

a Bessel beam with 4 rings, and a Bessel beam with 3 rings; laser Gaussian beam, a

Bessel beam with 4 rings and Bessel beams with 3 rings (Chapter 5). The results

presented in Chapter 5 showed that monitor-based images of Bessel beams provided

significantly smaller 𝜎𝑦 and BCEA than the bull’s eye/cross hair combination and the

monitor- and laser-based Gaussian beam targets. The fixation stability for the laser

Bessel beams and that for a bull’s eye/cross hair combination and the laser Gaussian

beams were similar. I found large variations (by factors of 14 to 22) in BCEAs between

participants, which might be due to naivety of the participants to the task. I concluded

that monitor-based images of Bessel beams provide better fixation targets than a bull’s

eye/cross hair combination and the monitor- and laser-based Gaussian beams, but no

claim can be made that laser Bessel beams provide better fixation targets than laser

Gaussian beams. It remains unclear whether ophthalmic imaging instruments that

require stable fixation would benefit by using Bessel laser beams to provide both

illuminating beams and fixation targets.


The hypothesis stated in section 2.6 for this study was that a Bessel beam used

as a fixation target will improve fixation stability over that achieved with conventional

targets. Bessel beam targets were of two types; monitor-based images of Bessel beams,

and laser Bessel beams. The findings confirmed in part the hypothesis, in that monitor-

based images of Bessel beams improved fixation stability over that achieved with

conventional targets but laser Bessel beams did not. The lack of significance between

monitor-based images of Bessel beams and laser Bessel beams indicates that the shape

of a Bessel beam, rather than its propagation properties, is responsible for this

improvement in fixation stability.

7.1.3 Retinal imaging with Bessel beams

I developed a retinal imaging set up described in Chapter 6 to investigate the

amount of light reaching the retinal area being imaged for Bessel and Gaussian beams.

Retinal images of right eyes, after dilating the pupil with 1 percent tropicamide, were

acquired for 10 participants each from below 35 years (young group) and above 59

years (older group). Five participants in the older group had early cataract while the

remaining others were without cataract. The retinal image intensity was used as a

measure of the amount of light reaching the retinal area being imaged. The retinal

image intensities for Bessel and Gaussian beams were compared between young and

older groups, and between without-cataract and early-cataract groups. Bessel beams

provided higher retinal image intensities than Gaussian beams for participants in both

age groups, and in both without- and early-cataract groups. The Bessel to Gaussian

retinal image intensities ratio was similar for both age groups. Early-cataract group

had significantly higher Bessel to Gaussian retinal image intensities ratio than without-

cataract group.

The study confirmed the hypothesis stated in section 2.6 that using a Bessel beam

rather than a Gaussian beam will provide higher amount of light at the retinal area

being imaged. The ratio of the amount of light reaching the retinal area being imaged

for a Bessel beam to that for a Gaussian beam was higher for eyes with early cataract

than for eyes without cataract, but these ratios were similar for the young and the older

groups. This shows that the scatter-resistant advantage of Bessel beams is apparent

only for eyes with at least some degree of media opacification.


7.2 FUTURE DIRECTIONS

Experiment 1 (Chapter 4) showed that the Bessel phakometer had similar intra-

observer repeatability despite the brighter Purkinje images than the Gaussian

phakometer. The lack of improvement in the intra-observer repeatability was probably

mostly due to the thick ellipse fitting line in MATLAB obscuring the Bessel beam

Purkinje images which were smaller than those for the Gaussian beam setup. It is worth

investigating if this could be improved by manipulating the Bessel illumination so that

larger Purkinje images are obtained.

Historically phakometers have also been used in measurement of lens

decentration and tilt (Barry, et al., 2001; Nishi et al., 2010; Phillips, Rosskothen, Pérez-

Emmanuelli, & Koester, 1988; Tabernero, et al., 2006), where accuracy and

repeatability of lens decentration and tilt estimates are affected due to the deteriorated

quality and brightness of PIII and PIV by diffuse reflection and scattering of the

Gaussian beam. It can also be further investigated whether Bessel phakometers can

provide accurate and repeatable estimates of lens decentration and tilt.

Experiment 2 (Chapter 5) showed that the monitor-based images of Bessel

beams, as fixation targets, provided better fixation stability than a bull’s eye/cross hair

combination, monitor-based Gaussian images and laser Gaussian beams targets, while

laser Bessel beams targets were as good as the bull’s eye/cross hair combination target.

However, due to low sampling rate of data acquisition I could not assess fixational eye

movement components such as tremor, drift and micro-saccades. This work could be

extended by assessing the effectiveness of a Bessel beam, as a fixation target, in

reducing the fixational eye movement components compared to that with for

conventional targets. This can be done by using a set up similar to mine, but with an

eye tracker with high sampling size such as EyeLink 1000 plus (SR Research EyeLink,

Ontario, Canada) that has sampling rate up to 2,000 Hz.

Scatter affects the accuracy of the wavefront sensor (Neal, Topa, & Copland,

2001). The bitmap image of spread-out spots provided by Hartmann-Shack wavefront

sensor carry information about the effect of ocular scatter on the wavefront sensing

beam (Donnelly, et al., 2004), and the displacements of the spots with respect to their

ideal positions carry information about the aberrations (Figure 2.25). Bessel beams

provide sufficient information on aberration required for successful operation of

adaptive optics loop during imaging (Lambert, et al., 2011). The findings from this


study indicate that Bessel beams are less affected by ocular scatterers than Gaussian

beams during retinal imaging. The scatter-resistant property of Bessel beams might

produce less spread-out spots in the detector of a wavefront sensor, and might increase

signal-to-noise ratio to provide better accuracy in locating the positions of the spots

captured by the wavefront sensor. Initially I developed a flood illumination retinal

imaging set up with an adaptive optics system (Figure 7.1) to investigate the effect of

scatter on the spots in the detector of the wavefront sensor, and the accuracy of

aberration measurement using Bessel and Gaussian beams. Using this set up, I also

intended to acquire images with identifiable retinal structures to assess the retinal

image quality. The adaptive optics system had three principal components: a wavefront

sensor (Figure 7.1, iii), a wavefront corrector (Figure 7.1, ii) and a control system

(Figure 7.1, viii). The Hartmann-Shack wavefront sensor had 32 X 40 lenslets. The

deformable mirror with 52 actuators was used as a wavefront corrector. The underlying

principles behind the each component of the adaptive optics system were described in

section 2.6.2.

Figure 7.1: Experimental setup for adaptive optics retinal imaging using Bessel beam. (i) retinal camera,

(ii) deformable mirror, (iii) Hartmann-Shack wavefront sensor, (iv) model eye, (v) laser diode for

wavefront sensing, (vi) SLM, (vii) Laser diode for retinal imaging, (viii) Graphic user interface

photographs of control system of the adaptive optics system.


The deformable mirror of the system stopped working shortly before

experimentation. Timely replacement of the deformable mirror was not possible and

hence I could not carry out my investigations. In future, the deformable mirror can be

fixed or replaced and the intended investigations carried out.

Scanning Laser Ophthalmoscopes use a laser to scan across the retina and build

up point-by-point images, where the light passes through an aperture, a confocal

pinhole, placed at a retinal conjugate plane before reaching the detector (Roorda,

2010). Combined with adaptive optics system, these instruments provide higher

transverse resolution than other imaging modalities (Zhang, Li, Kang, He, & Chen,

2017). It is worth investigating if Bessel beams, being resistant to scattering and

diffraction, can provide better image resolution than currently achieved, particularly

in eyes with cataracts.

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134 Appendices

Appendices

: Presentations and publications arising from this thesis

Refereed journal papers

Suheimat M, Bhattarai D, Maher HK, Chandra M, Chelepy W, Halloran S, Lambert

AJ, Atchison DA (2017). Improvements to phakometry using Bessel beams.

Optometry and Vision Science 94(11), 1015-1021. [Based on Chapter 3 and Chapter

4, see published paper 1 in Appendix B]

Bhattarai D, Suheimat M, Lambert AJ, Atchison DA. Fixation stability with Bessel

beams. Optometry and Vision Science 94(2), 95-102. [Based on Chapter 5, see

published paper 2 in Appendix C]

Published presentations

Lambert A, Maher HK, Chandra M, Chelepy WA, Halloran SK, Bhattarai D, Atchison

DA, Suheimat M (2016). Improvements to Phakometry through use of Bessel beams.

In Imaging and Applied Optics, OSA Technical Digest (online) (Optical Society of

America, 2016), paper AOM3C.5.

Suheimat M, Maher HK, Chandra M, Chelepy WA, Halloran S K, Bhattarai D,

Lambert AJ, Atchison DA (2016). Phakometry using Bessel beams. In Rozema, JJ.

Proceedings of the 8th European Meeting on Visual and Physiological Optics. ISBN

978-90-5728-521-9. Antwerp, Belgium, 21-23 August, pp. 74-76.

Bhattarai D, Suheimat M, Lambert AJ, Atchison DA (2018). Fixation stability with

Bessel beams. The Association of Research in Vision and Ophthalmology, April 20 –

May 3, Hawaii, USA. Investigative Ophthalmology and Vision Science, 59 (9), 5791.

https://iovs.arvojournals.org/article.aspx?articleid=2693065

https://iovs.arvojournals.org/article.aspx?articleid=2693065

Appendices 135

Unpublished presentations

Bhattarai D, Suheimat M, Lambert AJ, Atchison DA (2017). Improvements to

phakometry using Bessel beams. 07 February, ARVO Asia, Brisbane, Australia.

Bhattarai D, Suheimat M, Atchison DA (2017). Fixation stability with Bessel beams.

24 August, IHBI Inspires, Brisbane, Australia.

Bhattarai D, Suheimat M, Atchison DA (2017). Application of Bessel beams in human

eye. 5 October, Adaptive Optics Workshop, University of New South Wales,

Canberra, Australia.

Suheimat M, Bhattarai D, Lambert AJ, Atchison DA (2018). Improving ophthalmic

devices using Bessel beams. 5 April, Scientific and Education Meeting in Optometry,

Melbourne, Australia.

Bhattarai D, Suheimat M, Lambert AJ, Atchison DA (2018). Application of Bessel

beams in the human eye. 25 April, The University of Auckland, Auckland, New

Zealand.

136 Appendices

: Published paper 1

Appendices 137

138 Appendices

Appendices 139

140 Appendices

Appendices 141

142 Appendices

Appendices 143

: Published paper 2

144 Appendices

Appendices 145

146 Appendices

Appendices 147

148 Appendices

Appendices 149

150 Appendices

Appendices 151

: Changes to MATLAB code for Bessel and Gaussian phakometers

The following changes were required in the MATLAB program during calculation of

lens parameters depending upon type of phakometer (Bessel/Gaussian) used:

In Subject_read.m file, the wavelength for Bessel beam was set as 637 nm while that

of the Gaussian beam was set as 890 nm i.e.,

325 lam = xxx; [xxx = 637 for Bessel, 890 for Gaussian phakometer]

In Double_Radii.m file, the object distance, object size and wavelength for Bessel

phakometer were set as 88 mm, 5.8 mm and 637 nm, respectively while for Gaussian

phakometer they were 80 mm, 18.5 mm, and 890 nm, respectively i.e.,

25 Pz = xxx; [xxx = 88 mm for Bessel, and 80 for Gaussian phakometer]

26 h0 = xxx; [xxx = 5.8 mm for Bessel, and 18.5 for Gaussian phakometer]

58 lam = xxx; [xxx = 637 nm for Bessel, and 890 nm for Gaussian phakometer]

152 Appendices

: Steps in MATLAB to determine Purkinje image brightness

The following steps were performed in MATLAB to determine the Purkinje

image brightness:

1) Import (drag and drop) the selected image file into command window of MATLAB

so that ‘Import Wizard’ window pops up. Untick ‘colormap’ and rename ‘cdata’

to desired file name and press ‘Finish’.

2) Double click the file name in ‘workspace’ so that ‘Variable-File name’ window

pops up. Type ‘imagesc(file name)’ in command window and press enter so that

colour image pops up. Type ‘colormap gray’ in command window and press enter

so that greyscale image pops up (Figure 7.2).

Figure 7.2: Screenshot of ‘colormap gray’ image in MATLAB.

.

Appendices 153

3) Zoom in as required and locate the relevant data point coordinates area (x1 to x2,

y1 to y2) of Purkinje images (5x5 in this case) as well as for nearby background to

measure ‘index’ (brightness) as required using ‘data cursor’. In command window

type ‘different file name’ = ‘file name(y2:y1,x2:x1)’ to locate required area for

Purkinje images and background brightness.

4) Calculate average brightness of the selected pixels by typing ‘file name for average

brightness’ = (sum(sum(different file name))/number of pixels selected) in

command window and press enter. Follow this process for both Purkinje image

and background brightness.

154 Appendices

: MATLAB code for extracting fixation positions and pupil

diameter

Below is the function to extract the various parameters associated with eye

tracking and timestamps. The function accepts a tracker file (.txt file) as input, and

outputs x and y coordinates, pupil size/diameter, pupil centre, and normalised

timestamps for both left and right eyes. However, for data analysis only x and y

coordinates and pupil size/diameter of the right eye are used.

function f = eyetrack(dinfo) allvalues = zeros(450,3*length(dinfo)); % matrix for x, y, and pupil diameter for the

right eye j = 0; for K = 1:length(dinfo)

fid = fopen(dinfo(K).name,'rb'); % Opens eye track file

dinfo(K).name Str=fgetl(fid); count=0; %Str=fgetl(fid); while Str~=-1, ind=strfind(Str,'tracker'); if isempty(ind)==0 % tracker data start ind=strfind(Str,'true'); if isempty(ind)==0 |isempty(ind)~=0 % not blinking data count=count+1; % divide into L/E strings and timestamps

ind_LE=strfind(Str,'"lefteye":{"avg":{"x":');ind_RE=strfind(Str,'"righteye');ind_TS=

strfind(Str,'"timestamp'); Str_LE=Str(ind_LE:ind_RE-1);Str_RE=Str(ind_RE:ind_TS-

1);Str_TS=Str(ind_TS:end); % reading left eye data ind_x=strfind(Str_LE,'"x":'); ind_x_end=strfind(Str_LE,',"y":'); ind_y_end=strfind(Str_LE,'},'); ind_ps=strfind(Str_LE,'"psize"'); ind_ps_end=strfind(Str_LE,',"raw"'); LE_avg_x(count)=str2num(Str_LE(ind_x(1)+4:ind_x_end(1)-1)); LE_pcenter_x(count)= str2num(Str_LE(ind_x(2)+4:ind_x_end(2)-1)); LE_avg_y(count)=str2num(Str_LE(ind_x_end(1)+5:ind_y_end(1)-1)); LE_pcenter_y(count)= str2num(Str_LE(ind_x_end(2)+5:ind_y_end(2)-1)); LE_psize(count)= str2num(Str_LE(ind_ps+8:ind_ps_end(1)-1));

Appendices 155

ind_x=strfind(Str_RE,'"x":'); ind_x_end=strfind(Str_RE,',"y":'); ind_y_end=strfind(Str_RE,'},'); ind_ps=strfind(Str_RE,'"psize"'); ind_ps_end=strfind(Str_RE,',"raw"'); RE_avg_x(count)=str2num(Str_RE(ind_x(1)+4:ind_x_end(1)-1)); RE_pcenter_x(count)=str2num(Str_RE(ind_x(2)+4:ind_x_end(2)-1)); RE_avg_y(count)=str2num(Str_RE(ind_x_end(1)+5:ind_y_end(1)-1)); RE_pcenter_y(count)= str2num(Str_RE(ind_x_end(2)+5:ind_y_end(2)-1)); RE_psize(count)= str2num(Str_RE(ind_ps+8:ind_ps_end(1)-1)); ind_ts_sp=strfind(Str_TS,' '); Str_TS=Str_TS(ind_ts_sp+1:end-4); ind_d=strfind(Str_TS,':'); h=str2num(Str_TS(1:ind_d(1)-1)); m=str2num(Str_TS(ind_d(1)+1:ind_d(2)-1)); s=str2num(Str_TS(ind_d(2)+1:end)); TimeStamp(count)=h*60*60+m*60+s; TimeStamp_normalized=TimeStamp-TimeStamp(1); end end Str=fgetl(fid); end %%%%%%%%%%%%%% Example of data plotting %%%%%%% this is to plot the right eye size change with a normalised time %%%%%%% stamp, you can change the plot content when you need to see changes %%%%%%% of different papremeters. %figure;plot(TimeStamp_normalized, RE_psize); %% Left eye x coordinate zero_positions = find(~LE_avg_x); LE_avg_x1 = LE_avg_x; for k = 1:size(zero_positions,2) LE_avg_x1(zero_positions(k)-10:zero_positions(k)+10)= zeros(1,21); end [row, col, new_LE_avg_x] = find(LE_avg_x1); new_LE_avg_x1 = new_LE_avg_x(1,1:450); new_TimeStamp = TimeStamp_normalized(1,1:size(new_LE_avg_x1,2)); %figure, plot(new_TimeStamp, new_LE_avg_x1) %% finding microsaccades delta_t = 1/30; for l=3:(size(new_LE_avg_x1,2)-2)

156 Appendices

velocity(l) = (new_LE_avg_x1(l+2) + new_LE_avg_x1(l+1) - new_LE_avg_x1(l-

1) - new_LE_avg_x1(l-2))/(6*delta_t); end threshold = std(velocity); microsaccadesLE = find(abs(velocity)>threshold); %% Left eye y coordinate zero_positions = find(~LE_avg_y); LE_avg_y1 = LE_avg_y; for k = 1:size(zero_positions,2) LE_avg_y1(zero_positions(k)-10:zero_positions(k)+10)= zeros(1,21); end [row, col, new_LE_avg_y] = find(LE_avg_y1); new_LE_avg_y1 = new_LE_avg_y(1,1:450); new_TimeStamp = TimeStamp_normalized(1,1:size(new_LE_avg_y1,2)); %figure, plot(new_TimeStamp, new_LE_avg_y1) %% Left eye x coordinate of centre zero_positions = find(~LE_pcenter_x); LE_pcenter_x1 = LE_pcenter_x; for k = 1:size(zero_positions,2) LE_pcenter_x1(zero_positions(k)-10:zero_positions(k)+10)= zeros(1,21); end [row, col, new_LE_pcenter_x] = find(LE_pcenter_x1); new_LE_pcenter_x1 = new_LE_pcenter_x(1,1:450); new_TimeStamp = TimeStamp_normalized(1,1:size(new_LE_pcenter_x1,2)); %figure, plot(new_TimeStamp, new_LE_pcenter_x1)

%% Left eye y coordinate of centre zero_positions = find(~LE_pcenter_y); LE_pcenter_y1 = LE_pcenter_y; for k = 1:size(zero_positions,2) LE_pcenter_y1(zero_positions(k)-10:zero_positions(k)+10)= zeros(1,21); end [row, col, new_LE_pcenter_y] = find(LE_pcenter_y1); new_LE_pcenter_y1 = new_LE_pcenter_y(1,1:450); new_TimeStamp = TimeStamp_normalized(1,1:size(new_LE_pcenter_y1,2)); %figure, plot(new_TimeStamp, new_LE_pcenter_y1)

%% Left eye pupil size zero_positions = find(~LE_psize); LE_psize1 = LE_psize; for k = 1:size(zero_positions,2) LE_psize1(zero_positions(k)-10:zero_positions(k)+10)= zeros(1,21); end [row, col, new_LE_psize] = find(LE_psize1); new_LE_psize1 = new_LE_psize(1,1:450); new_TimeStamp = TimeStamp_normalized(1,1:size(new_LE_psize1,2)); %figure, plot(new_TimeStamp, new_LE_psize1)

Appendices 157

%%

% Right eye x coordinate

zero_positions = find(~RE_avg_x); RE_avg_x1 = RE_avg_x; for k = 1:size(zero_positions,2) RE_avg_x1(zero_positions(k)-10:zero_positions(k)+10)= zeros(1,21); end [row, col, new_RE_avg_x] = find(RE_avg_x1); new_RE_avg_x1 = new_RE_avg_x(1,1:450); allvalues(:,j+1) = (new_RE_avg_x1)'; new_TimeStamp = TimeStamp_normalized(1,1:size(new_RE_avg_x1,2)); %figure, plot(new_TimeStamp, new_RE_avg_x1)

%% finding microsaccades delta_t = 1/30; for l=3:(size(new_RE_avg_x1,2)-2) velocity(l) = (new_RE_avg_x1(l+2) + new_RE_avg_x1(l+1) - new_RE_avg_x1(l-

1) - new_RE_avg_x1(l-2))/(6*delta_t); end threshold = std(velocity); microsaccadesRE = find(abs(velocity)>threshold); %%

% Right eye y coordinate zero_positions = find(~RE_avg_y); RE_avg_y1 = RE_avg_y; for k = 1:size(zero_positions,2) RE_avg_y1(zero_positions(k)-10:zero_positions(k)+10)= zeros(1,21); end [row, col, new_RE_avg_y] = find(RE_avg_y1); new_RE_avg_y1 = new_RE_avg_y(1,1:450); allvalues(:,j+2) = (new_RE_avg_y1)'; new_TimeStamp = TimeStamp_normalized(1,1:size(new_RE_avg_y1,2)); %figure, plot(new_TimeStamp, new_RE_avg_y1)

%% Right eye x coordinate of centre zero_positions = find(~RE_pcenter_x); RE_pcenter_x1 = RE_pcenter_x; for k = 1:size(zero_positions,2) RE_pcenter_x1(zero_positions(k)-10:zero_positions(k)+10)= zeros(1,21); end [row, col, new_RE_pcenter_x] = find(RE_pcenter_x1); new_RE_pcenter_x1 = new_RE_pcenter_x(1,1:450); new_TimeStamp = TimeStamp_normalized(1,1:size(new_RE_pcenter_x1,2)); %figure, plot(new_TimeStamp, new_RE_pcenter_x1)

%% Right eye y coordinate of centre zero_positions = find(~RE_pcenter_y);

158 Appendices

RE_pcenter_y1 = RE_pcenter_y; for k = 1:size(zero_positions,2) RE_pcenter_y1(zero_positions(k)-10:zero_positions(k)+10)= zeros(1,21); end [row, col, new_RE_pcenter_y] = find(RE_pcenter_y1); new_RE_pcenter_y1 = new_RE_pcenter_y(1,1:450); new_TimeStamp = TimeStamp_normalized(1,1:size(new_RE_pcenter_y1,2)); %figure, plot(new_TimeStamp, new_RE_pcenter_y1)

%%

% Right eye pupil size zero_positions = find(~RE_psize); RE_psize1 = RE_psize; for k = 1:size(zero_positions,2) RE_psize1(zero_positions(k)-10:zero_positions(k)+10)= zeros(1,21); end [row, col, new_RE_psize] = find(RE_psize1); new_RE_psize1 = new_RE_psize(1,1:450); allvalues(:,j+3) = (new_RE_psize1)'; new_TimeStamp = TimeStamp_normalized(1,1:size(new_RE_psize1,2)); %figure, plot(new_TimeStamp, new_RE_psize1) %% j = j+3; end f = allvalues; end

Below is the code to extract the pupil size and the x and y coordinates of right eye

using the above matlab function “eyetrack”. For each participant, the aforementioned

parameters are estimated for five runs for each of the seven targets – A, B, C, D, E, F

and G.

topFile = '<Name of the Directory>'; % Each folder in this directory is for one

participant containing the eye track .txt files for five runs of seven targets

a = dir(topFile);

a=a(~ismember({a.name},{'.','..'}));

for J = 15:16

cd(topFile);

cd(a(J).name);

dinfoA = dir('*A*.txt'); % 5 Target A files

eyetrackA = eyetrack(dinfoA);

dinfoB = dir('*B*.txt'); % 5 Target B files

eyetrackB = eyetrack(dinfoB);

dinfoC = dir('*C*.txt'); % 5 Target C files

Appendices 159

eyetrackC = eyetrack(dinfoC);

dinfoD = dir('*D*.txt'); % 5 Target D files

eyetrackD = eyetrack(dinfoD);

dinfoE = dir('*E*.txt'); % 5 Target E files

eyetrackE = eyetrack(dinfoE);

dinfoF = dir('*F*.txt'); % 5 Target F files

eyetrackF = eyetrack(dinfoF);

dinfoG = dir('*G*.txt'); % 5 Target G files

eyetrackG = eyetrack(dinfoG);

filename = strcat(a(J).name,'.xlsx')

% The eye track results are written as an excel file with the same name as the

participant. Each sheet of the file corresponds to one target

xlswrite(filename,eyetrackA,'TargetA')

xlswrite(filename,eyetrackB,'TargetB')

xlswrite(filename,eyetrackC,'TargetC')

xlswrite(filename,eyetrackD,'TargetD')

xlswrite(filename,eyetrackE,'TargetE')

xlswrite(filename,eyetrackF,'TargetF')

xlswrite(filename,eyetrackG,'TargetG')

end

160 Appendices

: MATLAB code for plotting the retinal image intensity distribution

B = imread('<Bessel Image file>'); % Bessel retinal image G = imread('<Gaussian Image file>'); % Gaussian retinal image G1=G; B1=B;

% Identify maximum noise value A=max(max(B1(2121:2160,1:40))); C=max(max(G1(2121:2160,1:40)));

% Set all pixel intensities less than max noise value as zero B1(B1<=A)=0; G1(G1<=C)=0; figure; % new figure h1=[]; h1(1) = subplot(1,1,1); colormap(hot) imagesc(B1,'Parent',h1(1)); % Display Bessel image with scaled colours caxis(h1,[0 3000]) colorbar(h1) set(gca,'FontSize',26, 'FontWeight', 'Bold') title('\fontsize{26}(a)') figure; h2=[]; h2(1) = subplot(1,1,1); colormap(hot) imagesc(G1,'Parent',h2(1)); % Display Gaussian image with scaled colours caxis(h2, [0 3000]) colorbar(h2) set(gca,'FontSize',26, 'FontWeight', 'Bold')

title('\fontsize{26}(b)')

Appendices 161

: MATLAB code for determining pixel greyscale values of retinal

images

The ‘brig’ function reads an image and calculates the sum of pixel intensities minus

noise.

function f = brig(Files)

C = zeros(1,6);

for k=1:5

B = imread(Files(k).name); % Image file names

A = max(max(B(2121:2160,1:40))); % Find maximum noise value

C(1,k) = sum(sum(B(B>A))); % Estimate the sum of the pixel intensities in an

image

end

C(1,6) = mean(C(1,1:5)); % Average of the pixel intensities sum (average over 5

files)

f = C;

end

Below is the code which uses the ‘brig’ function to calculate total image intensity for

both Bessel and Gaussian images for all participants in the young and older groups.

topFile = '<Name of directory>';

a = dir(topFile);


% Calculate the total image intensity for Bessel images in the young group

sumBesselB35 = zeros(10,6);

tmp = {};

for k = 1:10

f = fullfile(topFile, a(k).name, 'Bessel');

tmp{k,1} = a(k).name;

cd(f);

tiffFiles = dir('*.tif');

sumBesselB35(k,:) = brig(tiffFiles);

end

sumBesselB35 = [tmp, num2cell(sumBesselB35)];

% Calculate the total image intensity for Gaussian images in the young group

sumGaussianB35 = zeros(10,6);

tmp = {};

for k = 1:10

f = fullfile(topFile, a(k).name, 'Gaussian');


cd(f);

162 Appendices


sumGaussianB35(k,:) = brig(tiffFiles);

end

sumGaussianB35 = [tmp, num2cell(sumGaussianB60)];

topFile = '<Name of directory>';

a = dir(topFile);


% Calculate the total image intensity for Bessel images in the older group

sumBesselA60 = zeros(10,6);

tmp={};

for k = 1:10

f = fullfile(topFile, a(k).name, 'Bessel');


cd(f);


sumBesselA60(k,:) = brig(tiffFiles);

end

sumBesselA60 = [tmp, num2cell(sumBesselA60)];

% Calculate the total image intensity for Gaussian images in the older group

sumGaussianA60 = zeros(10,6);

tmp ={};

for k = 1:10

f = fullfile(topFile, a(k).name, 'Gaussian');


cd(f);


sumGaussianA60(k,:) = brig(tiffFiles);

end

sumGaussianA60 = [tmp, num2cell(sumGaussianA60)];

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