2

3

ACKNOWLEDGEMENTS

Firstly, I would like to express sincere gratitude towards my advisor, Dr. Michael

Hart, for his support, guidance, advice, patience and wisdom during the past two

years. I would also like to acknowledge to my committee members, Dr. Tom

Milster and Dr. Olivier Guyon, for help and valuable comments. As well, I would

like to thank to all my lab colleagues Mr. Phil, Mr. Ryan, Ms. Charlotte, Ms.

Madison and Mr. Joseph for assistant of all laboratory experiments and wisdoms.

I would like to acknowledge to all classmate Mr. Jay, Ms. Chloë and many others,

for their friendship.

Finally, I would like to thank to my father, mother, and my sister for their

continuous encouragement, support, and love throughout my life.

4

TABLE OF CONTENTS

ACKNOWLEDGEMENTS .........................................................................3

TABLE OF CONTENTS .............................................................................4

LIST OF FIGURES .....................................................................................6

LIST OF TABLES .......................................................................................9

Abstract ......................................................................................................10

CHAPTER 1. INTRODUCTION ..............................................................11

CHAPTER 2. PRINCIPLES OF DYNAMIC ADJUSTABLE SHWFS ...14

2.1 system setup .............................................................................14

2.2 Fresnel lenslet array & centroid ...............................................17

2.3 SHWFS Self Generating Influence Function and

Reconstruction Matrix ...................................................................21

CHAPTER 3. EXPERIMENTS .................................................................28

3.1 Scale Factors and Condition Numbers .....................................28

3.2 Phase plate measurement .........................................................29

CHAPTER 4. CONCLUSION & FUTURE WORKS ..............................41

Appendix 1. Phase plate measurements with 11 × 11 lenslet array ...........43

Appendix 2. Phase plate measurements with 9 × 9 lenslet array ...............45

Appendix 3. Phase plate measurements with 7 × 7 lenslet array ...............48

5

Appendix 4. Phase plate measurements with 5 × 5 lenslet array ...............50

REFERENCE .............................................................................................52

6

LIST OF FIGURES

Figure 1. Dynamic adjustable SHWFS system scheme ........................................ 14

Figure 2. Dynamic adjustable SHWFS system setup ........................................... 16

Figure 3. Fresnel lenslet array phase maps containing (a) 11×11, (b) 9×9, (c) 7×7,

and (d) 5×5 lenses within 1080 x 1080 pixels ...................................................... 17

Figure 4. A center sub-aperture maps : (a) a sub-aperture of 11×11 lenslet array, (b)

profile of phase map (a), (c) a sub-aperture of 9×9 lenslet array, (d) profile of phase

map (c), (e) a sub-aperture of 7×7 lenslet array, (f) profile of phase map (e), (g) a

sub-aperture of 5×5 lenslet array, (h) profile of phase map (g) ............................ 18

Figure 5. Center spot images of (a) 11×11, (b) 9×9, (c) 7×7, and (d) 5×5 beam

pattern ................................................................................................................... 19

Figure 6. (a) Center spot profiles and (b) normalized the profiles ........................ 20

Figure 7. Applicable frequencies in Fourier domain to 7x7 lenslet array ............ 23

Figure 8. FBS phase maps for 7×7 lenslet array: (a) ξ=1, η=0 applied, (b) ξ=3, η=3

applied in Fourier domain ..................................................................................... 24

Figure 9. 7×7 lenslet phase map with FBS phase maps: (a) ξ=1, η=0 applied, (b)

ξ=3, η=3 applied in Fourier domain ..................................................................... 25

Figure 10. FBS phase maps for 7×7 lenslet array: (a) ξ=1, η=0 applied, (b) ξ=3, η=3

applied in Fourier domain and 7×7 lenslet phase map with FBS phase maps: (c)

frequency ξ=1, η=0 applied, (d) frequency ξ=3, η=3 applied in Fourier domain . 25

7

Figure 11. Maginified center 3×3 spots from figure 10 (a): red spots are reference

position, blue spots are deviated spots .................................................................. 26

Figure 12. Maginified center 3×3 spots from figure 10 (b): red spots are reference

position, blue spots are deviated spots .................................................................. 26

Figure 13. Scale factors of each reconstruction matrix ......................................... 28

Figure 14. (a) Phase plate, (b) reference data measured by 4D interferomenter .. 29

Figure 15. Wavefront measured by proposed system with (a) 11×11 lenslet array,

(b) 9×9 lenslet array, (c) 7×7 lenslet array, and (d) 5×5 lenslet array .................. 31

Figure 16. Wavefront measured by proposed system with 30ms, 18ms, 7ms, 3ms

and 2ms exposure time: (a) 11×11 lenslet array(30ms), (b) 9×9 lenslet array(30ms),

(c) 7×7 lenslet array(30ms), (d) 5×5 lenslet array(30ms), (e) 11×11 lenslet

array(18ms), (f) 9×9 lenslet array(18ms), (g) 7×7 lenslet array(18ms), (h) 5×5

lenslet array(18ms), (i) 11×11 lenslet array(7ms), (j) 9×9 lenslet array(7ms), (k)

7×7 lenslet array(7ms), (l) 5×5 lenslet array(7ms), (m) 11×11 lenslet array(3ms),

(n) 9×9 lenslet array(3ms), (o) 7×7 lenslet array(3ms), (p) 5×5 lenslet array(3ms),

(q) 11×11 lenslet array(2ms), (r) 9×9 lenslet array(2ms), (s) 7×7 lenslet array(2ms),

(t) 5×5 lenslet array(2ms)...................................................................................... 32

Figure 17. Pattern matching : cross-correlation correspondence vs iterations ..... 34

Figure 18. Pattern matching result: (a) masked reference data, (b) masked measured

data by using 11×11 lenslet array ......................................................................... 34

Figure 19. RMS error vs shutter speed ................................................................. 35

Figure 20. (a) FBS fitting applied to 4D interferometer phase map (b) phase map

after 120 FBS modes are removed from the 4D interferometer phase map ......... 37

8

Figure 21. FBS coefficients calculated by FBS fitting (black solid line) and

measured by the proposed WFS (blue solid line) and coefficients difference (red

dotted line) for (a) 11×11, (b) 9×9, (c) 7×7, and (d) 5×5 mode ............................ 38

Figure 22. Phase map generated by using coefficients difference (Fig 14 red dotted

line) for (a) 11×11, (b) 9×9, (c) 7×7, and (d) 5×5 mode ....................................... 39

9

LIST OF TABLES

Table 1. Device specifications .............................................................................. 16

Table 2. Specification of lenslet modes ................................................................ 18

Table 3. beam profile data .................................................................................... 20

Table 4. Number of FBS modes ........................................................................... 24

Table 5. Condition numbers .................................................................................. 29

Table 6. RMS values ............................................................................................. 36

10

Abstract

The Shack Hartmann wavefront sensor (SHWFS) is used to detect the incoming

aberrated wavefront for use with an adaptive optics system by using a 2-D lenslet

array to produce an array of spots and recording the spot deviations near the focal

plane. The size and number of lenslets within the array is closely related to the

exposure time and resolution of the WFS and depends on the irradiance of the

target, the atmospheric situation inside the FOV, and the optical performance of the

adaptive optics system. Since a WFS normally has a fixed number of lenslets and

aperture size, to make the WFS work properly, it requires a well-suited calibration

process using a planar wavefront. It produces difficulties in the optimization of the

system when varying circumstances are introduced. In this paper, we describe how

by delaying the phase at each pixel, an SLM (Spatial Light Modulator) is used to

build lenslets with different sizes and aperture counts. The multimode WFS has

calibration data for each mode before operating the system and depending on the

variable circumstances it is able to easily convert to another mode even as the

system operates. This paper describes the setup of a multimode wavefront sensor

using an SLM, the construction of a lenslet array using an SLM-generated Fresnel

lens, and the theory of calibration and reconstruction. We also prove the capability

of the WFS performance by measuring a wavefront aberrated by a phase plate.

11

CHAPTER 1. INTRODUCTION

A well-known system, Adaptive Optics(AO), helps to get high-quality images of

objects aberrated by random and turbulent media. It is widely used in large

astronomical imaging, retinal imaging, holographic coherent imaging, and free

space communication systems [8][9]. AO basically consists of four parts: a

computer, a science camera, a compensator, and a wavefront sensor (WFS.) Those

devices work by measuring the wavefront of an object, analyzing the object

information hidden in turbulence, and removing the distortion to obtain a high-

quality image.

Technologies for fast and precise optical measurement of a wavefront have

gained significant interest in the astronomical AO field since investigations of faint

star image capture have been increasing. Numerous efforts towards developing

different types of WFS such as the curvature sensor, shearing interferometer,

pyramid WFS, holographic WFS(HWFS), the Shack Hartmann WFS(SHWFS),

and hybrid wavefront sensor have been devoted to achieve extremely high speed,

large dynamic range, and sensitive measurement capability.[8-10] In spite of its

relatively low resolution, the SHWFS is the most widely used sensor that is

comprised of a two-dimensional lenslet array because of its compactness, wide

dynamic range, well-proved and well-developed algorithms to reconstruct

incidence wavefront, and interchangeability of the lenslet array in front of the

camera for different purposes and environments. The SHWFS produces an array of

spots indicating averaged tilt over each sub-aperture. It is straightforward to convert

12

these local tilts to the wavefront propagating to the array by recording each spot

position and comparing it to reference spot positions recorded with a plane

wavefront near the focal plane. There are different wavefront reconstruction

algorithms for the SHWFS such as linear integration, zonal reconstruction, and

modal reconstruction. [12].

However, even though it is widely used in various fields with its numerous

advantages, it has several drawbacks when applied to different fields and

environments. The crucial part of the SHWFS is the lenslet array. To observe a faint

object with low photon flux and to take high-resolution images of it, telescopes

with AO systems systemically require larger entrance pupil diameters or a larger

aperture size for the SHWFS lenslet array to receive enough photons for proper

exposure. Since the lenslets have a fixed number and sub-aperture area, changing

the specification of the SHWFS lenslets is required to enable the system to measure

a target properly. It is a work-intensive process due to the installation process which

includes accurately positioning and rigidly mounting the lenslet array in front of

the camera, and precise calibration of the SHWFS, which means it is hard to

dynamically implement the module in rapidly changing challenging environments

that can cause inaccurate measurements. In addition, a technique to manufacture

precise lenslet that has same focal length and same size aperture at each sub-

aperture is needed. As a solution, in this paper we propose to use an adjustable

lenslet array using a spatial light modulator (SLM), a high-resolution phase

modulator that modulates phase at each pixel and has been used as a diffractive

optical element.

13

The usage of SLMs has been increasing in holography research and in AO

compensation. They have also been used as WFS using a modal reconstructor

method. [7] The Fresnel-encoded lenses in the SLM were introduced, and the

behavior of those lenses were revealed. [1][2] Reconfigurable Shack-Hartmann

sensors have been produced and tested. [3-6] However, past research had been done

with low-resolution due to low-quality equipment. We have shown improved

performance as the device used has been developed for a decade. The proposed

WFS has a reflective lenslet array consisting of square Fresnel lenses built on the

SLM surface. Since the size, number and focal length of the lenslet arrays can be

easily adjusted by programming the SLM driver, and with no need for difficult tasks

such as translating and replacing hardware, it can be applied to sensitive

environments depending on the wavelength and photon flux of an object. Running

the WFS requires only a calibration process which includes programming each

mode with different sizes and number of apertures, and every calibration

measurement is stored on the science computer to enable dynamic switching

between modes. This paper aims to prove that the proposed model successfully

reconstructs wavefront with different modes of lenslets built by the SLM, and can

measure faint stars and switch modes dynamically and has feasibility for use in

astronomical AO system.

14

CHAPTER 2. PRINCIPLES OF DYNAMIC

ADJUSTABLE SHWFS

2.1 system setup

Figure 1. Dynamically adjustable SHWFS system scheme

SLM has been used to implement various optical concepts as a crucial component

with its capability and to build lenslet array with low resolutions in previous studies.

[1-5] We experimented with a higher level such as more numbers of aperture,

smaller sub-aperture size, and shorter focal length. Figure 1 shows us the adjustable

SHWFS scheme. The SHWFS consists of an SLM, a polarizer, relay lenses with ½

magnification, and a camera. The SLM plays an important role in building the WFS

as a dynamic adjustable lenslet array. We were able to build a reflection type lenslet

array by inserting a set of Fresnel lenses into the SLM. The number and size of sub-

15

apertures on the SLM surface as well as the focal length of the array can be easily

and dynamically converted with simple programming. Since the SLM is reflection

type, the micro-lenses built in the SLM is located in front of a camera with relay

lenses.

Figure 2 is the real system that we used for the experiment. SHWFS are

comprised of an SLM, relay lenses, and a camera. It is beneficial for the detection

of the position of beam spots if the spot size is smaller, which is related to the f/#.

However, decreasing focal length increases the possibility of interference with the

incident beam by the camera. By using relay lenses, we were able to transfer the

focal plane to the camera surface and ½ magnification of the relay system helps to

image the SLM surface onto the camera surface fully. Lenses with 100mm and

50mm focal length were used for the relay system producing ½ magnification. A

2-axis translator helps to place the camera. In addition, due to Gaussian beam

profile after pinhole, we used a 200mm focal length lens to make beam profile

uniform. The specifications of the SLM and camera are listed on the following

Table 1.

To verify the performance of the WFS, we used a 638 nm laser and phase

plate to introduce a turbulence pattern to the wavefront. Matlab was used to control

the SLM, capture images and perform signal processing to reconstruct the

wavefront. The spots images on the camera are analyzed by using matched filter

algorithms to interpret spot deviations and then measure the wavefront.

16

Figure 2. Dynamic adjustable SHWFS system setup

Table 1. Device specifications

SLM Camera

PLUTO-NIR-010A

1920 1080 pixels

8 microns per pixel

GS3-U3-51S5M-C

2448 2048 pixels

3.45 microns per pixel

17

2.2 Fresnel lenslet array & centroid

Several modes of lenslet arrays were built to install on the SLM. The phase maps

consist of different numbers of lenses, and each mode has a different sub-aperture

size as seen in Figure 3. Each lens has a 75mm focal length at 638nm wavelength.

Table2 shows the number of pixels used to build a sub-aperture, the physical size

of the sub-aperture, and the f/# of the sub-aperture.

Figure 3. Fresnel lenslet array phase maps containing (a) 11×11, (b) 9×9, (c) 7×7,

and (d) 5×5 lenses within 1080 x 1080 pixels

18

Figure 4. A center sub-aperture maps : (a) a sub-aperture of 11×11 lenslet array,

(b) profile of phase map (a), (c) a sub-aperture of 9×9 lenslet array, (d) profile of

phase map (c), (e) a sub-aperture of 7×7 lenslet array, (f) profile of phase map (e),

(g) a sub-aperture of 5×5 lenslet array, (h) profile of phase map (g)

Lenslet

array

Pixels for a sub-

aperture

Sub-aperture

size

f/# of sub-

aperture

11×11 98×98 pixels 784×784 μm2 95.66

9×9 120×120 pixels 960×960 μm2 78.13

7×7 154×154 pixels 1232×1232 μm2 60.88

5×5 216×216 pixels 1728×1728 μm2 43.40

Table 2. Specification of lenslet modes

Once each mode was inserted into the SLM, spot arrays were introduced.

The images on Figure 5 are the center of the PSF of a spot array captured by the

camera using a ten times averaging process. The averaging helps to decrease

random intensity variations at the detection side which are due to a phase flicking

19

effect of the SLM. As listed on table 2 the f/# decreases when the sub-aperture size

increases. It can be visually detected when the diameter of the beam is decreasing,

and the irradiance of the spot on the camera increases due to a larger aperture.

Figure 5. Center spot images of (a) 11×11, (b) 9×9, (c) 7×7, and (d) 5×5 beam

pattern

Figure 6 and Table 3 quantitatively show the performance of the lenslet

array built by the SLM. Figure 6 (a) shows the beam profiles of the center spots. In

figure 6 (b), we normalized the beam profiles to make it easier to compare and to

measure the full width half maximum (FWHM) of the spots. The intensity and

FWHM can be measured in the focal regions. Table 3 shows the applied exposure

time for the camera, the peak intensity of the center beam, and the FWHM of the

20

beam spots. Shutter speed varied since the beam was too dim and was difficult to

be detected by the lenslet with the smallest sub-aperture. It is encouraging that even

if the exposure time of the camera decreases, the peak intensity increases. Due to

the f/# variation among the lenslet modes as discussed earlier, it was found that

measured FWHM of beam spot decreases when f/# decreases. By this analysis, it

has been proven that the SLM lenslet arrays are built properly and follow the optical

characteristics we expect.

Figure 6. (a) Center spot profiles and (b) normalized the profiles

Table 3. beam profile data

Lenslet

array

Exposure time on

camera Peak intensity FWHM

Data1 11×11 25 ms 56 0.0289 μm

Data2 9×9 25 ms 107 0.0247 μm

Data3 7×7 24.5 ms 233 0.0191 μm

Data4 5×5 8 ms 228 0.0142 μm

21

2.3 Built in influence matrix measurement

There are several methods for reconstructing wavefronts from spot deviations. We

implemented the modal reconstruction method using a Fourier basis set (FBS)

algorithm defined on a square array for generating the influence matrix and

reconstruction matrix. FBS shares a similar form with Zernike polynomials

characterizing the wavefront distortion. Even though Zernike polynomials are an

effective way of defining aberrations, since they restricted to a circular aperture, we

used FBS to reconstruct wavefront on our square aperture. FBS describes the

wavefront with a series of two-dimensional sine and cosine functions and their

frequencies following equation 1. 𝐶𝑠,𝑛,𝑚 and 𝐶𝑐,𝑛,𝑚 are the coefficients of the FBS

to be determined by the proposed WFS, and sine and cosine functions are the two-

dimensional functions. 𝑛 and 𝑚 are the number of modes in the x and y-directions.

W(x, y) = ∑∑𝐶𝑠,𝑛,𝑚 sin(𝑛𝑥) sin(𝑚𝑦)

𝑛𝑚

+ ∑∑𝐶𝑐,𝑛,𝑚 cos(𝑛𝑥) cos(𝑚𝑦)

𝑛𝑚

(1)

[ 𝐶𝑠,1,1

𝐶𝑠,1,2

⋮𝐶𝑠,𝑛,𝑚

𝐶𝑐,1,1

𝐶𝑐,1,2

⋮𝐶𝑐,𝑛,𝑚]

= S = RW,where W =

[ 𝑥1

𝑦1

⋮𝑥𝑙

𝑦𝑙 ]

(2)

22

R = VTƩ−1𝑈 (3)

F = UƩVT =

[ 𝑥1 𝑥1

𝑦1 𝑦1⋯

𝑥1

𝑦1

⋮ ⋱ ⋮𝑥𝑙

𝑦𝑙

𝑥𝑙

𝑦𝑙⋯

𝑥𝑙

𝑦𝑙 ]

𝑀1 𝑀2 ⋯ 𝑀𝑖

(4)

S is coefficients matrix that consists of sine and cosine coefficients and

comes from the WFS measurements, W multiplied by the modal reconstruct matrix

R. W is WFS response 1D matrix reshaped from 2D x and y spot deviations matrix.

The influence function F is built up by looking at the WFS response to each mode

and arranging the measured spot deviations into column vectors. Reconstruction

matrix R is calculated by using Singular-value decomposition (SVD) or

Pseudoinverse. Matrix V , U , and Ʃ are factorized matrix by SVD. V is an

orthonormal basis set in the domain of F, U is an orthonormal basis set in the range

of F, Ʃ is the set of scale factors that relate each basis vector of V to the

corresponding basis vector of U.

In order to build influence matrix for the WFS, it needs a process to form

incidence wavefront to be FBS modes and record the beam spot displacement on

camera, which means deforming a device such as other SLM or deformable mirror

is required as well as relating optics. In this paper, we will introduce built in

influence matrix measurement without the deforming device. The algorithm uses a

technique to shift PSF by tilting a wavefront. The PSF generated by a Fresnel lens

on SLM, is deviated as the tilt phase is overlapped on the numerical lens. We

23

achieve to build the influence function for the wavefront reconstruction by using

only SLM and combining phase maps into lenslet array.

Figure 7. Applicable frequencies in Fourier domain to 7x7 lenslet array

The required number of FBS modes is proportional to the number of apertures

of the lenslet array, which is directly related to Nyquist sampling theory. For

instance, theoretically, if a lenslet has a 7x7 array as seen in figure 7, it can be

considered that there are 49 required frequencies in the Fourier domain to generate

2D FBS phase maps for WFS calibration. 24 frequencies can be used to generate

it, since the 0,0 point is the DC term, and frequencies symmetric to the origin point

are conjugate to each other. Because sine and cosine functions are used, therefore,

48 FBS phase maps in total are built for the influence with a 7x7 lenslet array. In

this experiment, 5x5, 7x7, 9x9, and 11x11 lenslet arrays are generated, and the

number of FBS phase maps for each mode are listed in the following Table 4 below.

24

The amplitude of the FBS modes is pi/4 to prevent the introduction of high defocus

at spots, which increases the possibility of a centroid measurement error. We used

a matched filter algorithm to find spot position due to its robustness in dealing with

noise.

Figure 8 shows the two examples of 2D FBS phase maps and figure 9 illustrates

the phase map that the Fresnel lenslet phase map includes the same sinusoidal

pattern when the FBS phase maps are overlapped.

Lenslet array Number of FBS modes

11×11 120

9×9 80

7×7 48

5×5 24

Table 4. Number of FBS modes

Figure 8. FBS phase maps for 7×7 lenslet array: (a) ξ=1, η=0 applied, (b) ξ=3,

η=3 applied in Fourier domain

25

Figure 9. 7×7 lenslet phase map with FBS phase maps: (a) ξ=1, η=0 applied, (b)

ξ=3, η=3 applied in Fourier domain

Figure 10. FBS phase maps for 7×7 lenslet array: (a) ξ=1, η=0 applied, (b) ξ=3,

η=3 applied in Fourier domain and 7×7 lenslet phase map with FBS phase maps:

(c) frequency ξ=1, η=0 applied, (d) frequency ξ=3, η=3 applied in Fourier domain

26

Figure 11. Maginified center 3×3 spots from figure 10 (a): red spots are reference

position, blue spots are deviated spots

Figure 12. Maginified center 3×3 spots from figure 10 (b): red spots are reference

position, blue spots are deviated spots

27

Figure 10 shows the displacement of the spots in the camera when two phase

maps are applied to SLM. The red spots are the original reference point without

FBS phase maps, and the blue spots indicate the changed spots positions when the

calibration phase maps are applied. The figure 11 and figure 12 show magnified

center 9 spots of figure 10 (a) and (b). As seen in the center column of figure 11,

the red and blue spots indicate the same position since the center column of the

lenslet array and the peaks of FBS phase map, the cosine function, overlap. Since

the first and third columns of lenslet array is located in the slope of the FBS phase

map and low frequency is applied, first and third columns of figure 11 show that

the blue spots slightly are shifted left and right. FBS phase map contains 2D high

frequency as seen in figure 8 (b) and figure 9 (b), and the spots displaced diagonally

with large amount than the first examples as seen in figure 12. Through the

examples, it is verified that the self-generating influence function for modal

wavefront reconstruction method properly works without extra wavefront

deforming device. It is converted to reconstruction matrix, and FBS coefficients are

calculated by the matrix with WFS response matrix as described in equation 2. The

number of coefficients measured has the same number of FBS phase map used for

calibration. In the experiment, built and calculated influence matrix, F, and

reconstruction matrix, R.

28

CHAPTER 3. EXPERIMENTS

3.1 Scale Factors and Condition Numbers

Figure 13. Scale factors of each reconstruction matrix

We studied scale factors computed by SVD to verify the right numbers of FSB

modes are used. The scale factors indicate strength that the WFS is able to detect

each FBS modes and the condition number is defined as ratio of maximum and

minimum value of scale factors. The scale factors and condition numbers are shown

in figure 13 and table5. In figure 13, the x-axis is a number of FBS modes comprised

of sine (odd number) and cosine (even number) functions. The black solid line is

29

scale factors of 11×11 lenslet array mode reconstruction matrix, red solid line is

scale factors of 9×9, green solid line is scale factors of 7×7, and the blue solid line

is scale factors of 5×5.

Lenslet array Condition number

11×11 5.1646

9×9 4.2947

7×7 3.1302

5×5 2.5284

Table 5. Condition numbers

3.2 Phase plate measurement

Figure 14. (a) Phase plate, (b) reference data measured by the 4D interferometer

The relationship between sub-aperture size and astronomical issues has been

studied and reveal that WFS’s with adjustable sub-apertures can apply to various

conditions. The previous studies found that smaller sub-apertures provide improved

measurement with a bright enough source and larger sub-apertures increases

30

performance with dim sources. [5] In this section, relevant results are provided to

show the performance of the developed SHWFS with a self-generating influence

function algorithm. The experiment was performed without any translating parts

and switching lenslet array modes by coding was used to achieve our goal.

A phase plate was measured by both the proposed system and a 4D

interferometer for these experiments. Since the phase plate has a 4 inches diameter,

we placed a mask on the area except for the 1cm x 1cm we were interested. The

figure 14 (b) shows us the phase map measured by the 4D interferometer. Tip and

tilt wavefront error were removed to analyze. To simulate various brightnesses of

stars, and to manipulate the number of photons incident on the camera, instead of

altering laser source intensity we varied the integration time of the camera from

0.2ms to 30ms with a 0.2ms interval. Lower integration time is analogous to a faint

star and higher shutter time analogous to a bright star. We didn’t use a threshold to

make the WFS sensitive to small amount photon.

Measuring wavefront with various integration time, we found that spots on

the camera are randomly decaying due to absorption of the phase plate, even if the

input beam is considerably uniform. It may fail to reconstruct wavefront or cause

huge measurement error. To overcome this issue, we generated various

reconstruction matrices that can be applied to images with some disappeared spots

and we considered the images that contain 80 percent of total spots, as adequate

data.

31

Figure 15. Wavefront measured by proposed system with (a) 11×11 lenslet array,

(b) 9×9 lenslet array, (c) 7×7 lenslet array, and (d) 5×5 lenslet array

Figure 15 and figure 16 shows the results measured by the proposed system

when the 11 × 11, 9 × 9, 7 × 7, and 5 × 5 lenslet arrays and with the shutter speed

varied between them. Figure 15 is wavefront measured at 30ms shutter speed time.

Row images of figure 16 indicate when 30ms, 18ms, 7ms, 3ms shutter, and 2ms

shutter speed were used and column images of figure 16 show wavefront when each

mode of the lenslets was used. The images contain relations between sub-aperture

size, resolution, and the brightness of the source and illustrate as the number of

apertures increase, the resolution of the reconstructed phase increases and the

shutter speed required for proper measurement increases.

32

Figure 16. Wavefront measured by proposed system with 30ms, 18ms, 7ms, 3ms

and 2ms exposure time: (a) 11×11 lenslet array(30ms), (b) 9×9 lenslet

array(30ms), (c) 7×7 lenslet array(30ms), (d) 5×5 lenslet array(30ms), (e) 11×11

lenslet array(18ms), (f) 9×9 lenslet array(18ms), (g) 7×7 lenslet array(18ms), (h)

5×5 lenslet array(18ms), (i) 11×11 lenslet array(7ms), (j) 9×9 lenslet array(7ms),

(k) 7×7 lenslet array(7ms), (l) 5×5 lenslet array(7ms), (m) 11×11 lenslet

array(3ms), (n) 9×9 lenslet array(3ms), (o) 7×7 lenslet array(3ms), (p) 5×5 lenslet

array(3ms), (q) 11×11 lenslet array(2ms), (r) 9×9 lenslet array(2ms), (s) 7×7

lenslet array(2ms), (t) 5×5 lenslet array(2ms)

33

The cross patterns at the center bottom of figure 15 (a) to (d), similar to

reference data in figure 14 (b), implies that the proposed system with the self-

calibration method has the capability of adequate measurement. When we applied

an 18ms shutter speed to the camera, as shown in figure 16 (e), degraded wavefront,

the WFS with 11 × 11 lenslet arrays couldn’t adequately reconstruct the wavefront

since the smaller sub-apertures weren’t able to collect enough photons and it

resulted in missing information on camera. However, figure 16 (f), (g), and (h)

show that the other WFS modes measure wavefront properly due to enough photons.

At figure 16 (i), the image includes less spot information than 80 percent of total

spots for 11 × 11 lenslet arrays, and there is no wavefront measurement. The result

with 9 × 9 lenslet arrays shown in figure 16 (j) includes error. The same inclination

occurred as shutter speed decreases.

We used a cross-correlation technique built in Matlab to match 4D

measurement data and wavefront measured by the proposed scheme and compare

those quantitatively. We made the decision to use measured wavefront by 11×11

lenslet array due to its higher resolution and masked and resized the wavefront as

seen in figure 16 (a) red mark, for the signal processing. Figure 17 illustrates the

cross-correlation correspondence versus iterations, which means that it shifts

smaller image, data measured by the proposed scheme, onto the larger image, which

is used as a reference image. The maximum value, indicated by the red spot on

figure 17, implies the best matching locations, and both images and figure 18 shows

the matched results.

34

Figure 17. Pattern matching : cross-correlation correspondence vs iterations

Figure 18. Pattern matching result: (a) masked reference data, (b) masked

measured data by using 11×11 lenslet array

We applied the same mask that we used during cross-correlation algorithm

to all the other phase maps measured by the proposed WFS with lenslet arrays and

subtracted the reference data to calculate the RMS error as shown in figure 19. The

x-axis is shutter speed of the camera, and the y-axis indicates RMS error. The red

35

solid line is data measured by using a 11×11 lenslet array imposed on the SLM at

different exposure times, the green solid line is measured by using a 9×9 array, the

blue solid line is measured by using a 7×7 array, and the black solid line is measured

by using a 5×5 lenslet array.

Figure 19, RMS curve, contains the meaningful information as we expected

that RMS error with larger sub-apertures dramatically decreases at the shorter

shutter speed and RMS error with smaller sub-apertures decreases more slowly as

a function of integration time.

Figure 19. RMS error vs shutter speed

The initial RMS error at 0ms shutter speed indicates RMS value what the

phase plate has. Due to smallest sub-aperture, at around 15ms the WFS with 11×11

lenslet array detects 80 percent of spots and measurements begin. The RMS error

slowly decreases as shutter speed increases (red solid line). The steps where the

36

RMS error rapidly drops after 15ms, mean as irradiance on camera increases

numbers of spots for wavefront reconstruction increases. Finally, the RMS error

lasts minimum over the other mode and matched filter points center of spots

precisely. WFS with 5×5 lenstet array, measures spots at shorter shutter speed time,

around 7ms, and there are less numbers of step due to its larger sub-apertures and

less number of apertures than when the 11×11 lenstet array was used (green solid

line). The 7×7 lenslet mode functions at around 4ms and the WFS with 5×5 lenslet

works at around 2ms (blue and black solid line). The RMS error dramatically

decreases for both mode due to its largest sub-apertures. It has a higher RMS error

than another mode at 30ms shutter speed. The results reveal that relation between

the sub-aperture size of lenslet array and capabilities of each lenslet modes to detect

a faint source.

Lenslet

array

(a)

Number of

modes for

calibration

(b)

Reconstructed

wavefront

RMS error

(Fig 12)

(c)

RMS error of

reconstructed

wavefront after

FBS fitting

(Fig 13)

(d)

RMS difference

between (b) and (c)

(e)

RMS error of

phase map

generated by

coefficient

differences (Fig 15)

N𝐹𝐵𝑆 σ1 σ2 √𝜎12 − 𝜎2

2 σ3

11×11 120 0.07412µm 0.04568µm 0.05837µm 0.06101µm

9×9 80 0.08023µm 0.04910µm 0.06345µm 0.06884µm

7×7 48 0.08461µm 0.05556µm 0.06381µm 0.06070µm

5×5 24 0.09235µm 0.06933µm 0.06101µm 0.06532µm

Table 6. RMS values

The results show that listed in table 6 (b) at 30ms, reconstructed wavefronts

introduce increasingly higher RMS error as numbers of the aperture of used lenslet

array decrease. It is directly related to the resolution of each mode as previous

research. The 11×11 lenslet array has the highest resolution, and it has the lowest

37

RMS error. On the other hand, 5×5 has the lowest performance, and it has the

highest RMS, but it functions at dim sources.

We performed FBS fitting on the phase map measured by the 4D

interferometer to see how proposed WFS measure wavefront correctly. Figure 20

(a) shows the RMS values decreases and when we increase numbers of FBS mode

for fitting. Figure 20 (b) illustrate the phase map when 120 FBS modes are removed,

and the patterns what we detect on figure 14 (a) are trimmed. The 120, 80, 48 and

24 FBS modes were used for WFS calibration. Therefore, we focused on the RMS

values at the 120, 80, 48, 24 FBS modes used for fitting and the RMS values are

listed in table 6 (a) and (c). The RMS values between the WFS and the FBS fitting

has deviations listed at table 6 (d), but the differences over each lenslet modes seem

considerably linear.

Figure 20. (a) FBS fitting applied to 4D interferometer phase map (b) phase map

after 120 FBS modes are removed from the 4D interferometer phase map

38

Figure 21. FBS coefficients calculated by FBS fitting (black solid line) and

measured by the proposed WFS (blue solid line) and coefficients difference (red

dotted line) for (a) 11×11, (b) 9×9, (c) 7×7, and (d) 5×5 mode

39

We have studied the FBS coefficients calculated by FBS fitting and

calculated by the proposed WFS since it indicates the portion of each FBS that

affects to reconstruct phase map. The coefficients are illustrated in figure 21. The

black solid lines are coefficients measured by FBS fitting, the blue solid lines are

coefficients measured by WFS, and the red dotted line indicates the difference

between them. There are little differences on coefficients, but the graphs show us

that measured and fitted coefficients have similar values to each other at a

noticeable peak and over the FBS modes.

Figure 22. Phase map generated by using coefficients difference (Fig 14 red

dotted line) for (a) 11×11, (b) 9×9, (c) 7×7, and (d) 5×5 mode

By using the coefficient differences (red dotted line), we generated phase

maps that show the reconstructed wavefront differences between using FBS fitting

coefficients and WFS measuring coefficients and the RMS values are listed in table

6. It is encouraging results that the RMS values, table 6 (e), have similar values to

RMS differences between FBS fitting and WFS measurements, table 6 (d).

We have measured the phase plate with different modes of the lenslet array

and various shutter speed time. The results reveal that the lenslet with larger sup-

aperture has better performance at the dim source with its lower resolution and

lenslet with smaller sup-aperture has less functions at dim source, but it has higher

resolution with enough photons. The RMS values and FBS coefficients of both

40

fitting and proposes WFS prove that the self-generating influence function

algorithm and its reconstruction matrix correctly works. The differences on RMS

and coefficients might come from aberrations contained in optics what we used for

the 4D interferometer and the proposed system.

41

CHAPTER 4. CONCLUSION & FUTURE

WORKS

In this paper, we have demonstrated the generation of a WFS using an SLM

with related theory. We have proven that the proposed WFS can dynamically

change modes with every set of calibration data of the modes, and the WFS scheme

and new calibration algorithm properly work with our results. We have applied

calibration method, consisting of overlapping FBS modes on the SLM lenslet array

at each mode, to the system, so that we can get rid of calibration device deforming

wavefront, and also moving part for optical alignment. It is beneficial that the

lenslet specification can be easily changed, and that there are no moving parts.

Calibration, test, and measurement can be done at the same position without any

detaching.

There are future issues that still need to be solved. We conclude that the

RMS differences as measurement errors. We believe that the errors come from the

relay system, which causes coma aberrations at the edge of array spots. In future,

the coma corrected relay system will help the measurement precise and errors

removed. In addition, the relay lenses weren’t able to be placed for 4D

interferometer measurements. If applicable to setup an interferometer with

proposed SHWFS’s relay system, it will help to get correct measurement, and to

figure out the possible undetected errors.

The SLM device has phase flicking issues. In this paper, to overcome

flicking of the SLM random phase modulation, we used an averaging process for

42

the images. While electronically calibrating the SLM can reduce the flicking effect,

calibration has a dependency on a wavelength, which means that the calibration is

needed at each wavelength. The best solution is to find an SLM containing all

calibration data at a wavelength with lower random phase modulation. With well-

calibrated SLMs with low flicking, different wavelengths can be applied to the

system to test the feasibility of dynamically shifting the wavelength.

SLM based WFS has more advantage in astronomical research.

Astronomers are interested in a star with a different wavelength. However, all

transparent optics introduce chromatic aberration resulting in defocus when they

change bandpass filter. The SLM also has wavelength dependence, but it can

change focal length easily by coding.

As using SLM as compensator has been studied, the proposed system can

be compact and potable AO system itself by using sequence process. The first

sequence is that SLM is used by lenslet array to measure distortion and next process

the SLM is used by aberration corrector. In addition, the system can be hybrid WFS

by combining dOTF theory to use it in a different atmosphere.

.

43

Appendix 1.

Phase plate measurements with 11 × 11 lenslet array

44

45

Appendix 2.

Phase plate measurements with 9 × 9 lenslet array

46

47

48

Appendix 3.

Phase plate measurements with 7 × 7 lenslet array

49

50

Appendix 4.

Phase plate measurements with 5 × 5 lenslet array

51

52

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