JOHANNES KEPLER
UNIVERSITY LINZ
Altenberger Str. 69
4040 Linz, Austria
www.jku.at
DVR 0093696
Submitted by
Malak Ahmad, BSc.
Submitted at
Institute of Polymer Product
Engineering
Supervisor
Univ.-Prof. Dr. Zoltán Major
Co-Supervisor
DI Veronika M. Miron
March 2019
3D PRINTED OPTICAL
COMPONENTS FOR
THz-SPECTROSCOPY
Master Thesis
To obtain the academic degree of
Diplom-Ingenieurin
In the master’s Program
Management in Polymer Technology
Malak Ahmad 2/70
STATUTORY DECLARATION
I hereby declare that the thesis submitted is my own unaided work, that I have not used other than
the sources indicated, and that all direct and indirect sources are acknowledged as references.
This printed thesis is identical with the electronic version submitted.
Place, Date
Signature
Malak Ahmad 3/70
Table of Contents
Acknowledgement ....................................................................................................................... 4
Abstract ....................................................................................................................................... 5
1. Introduction ........................................................................................................................... 6
2. Related Work and Background ............................................................................................. 9
2.1. Terahertz Spectroscopy ................................................................................................ 9
2.2. Optical Components for THz applications .................................................................... 12
Optical Concepts .............................................................................................. 12
Simple Lenses .................................................................................................. 14
Axicons ............................................................................................................. 16
Spiral Phase Plates .......................................................................................... 19
Helical Axicons ................................................................................................. 21
2.3. 3D Printing Technology ............................................................................................... 22
2.4. 3D Printing for THz Applications .................................................................................. 25
3D Printable Materials for THz Applications ...................................................... 25
3D Printed Optical Components for THz Applications ....................................... 27
3. Methods .............................................................................................................................. 29
3.1. Material Selection and Characterization ...................................................................... 29
3.2. Design of the Optical Components .............................................................................. 31
Simple Lenses .................................................................................................. 31
Axicon .............................................................................................................. 34
Spiral Phase Plate ............................................................................................ 35
Helical Axicons ................................................................................................. 36
3.3. 3D Printed Optical Component Characterization ......................................................... 38
4. Results and Discussion....................................................................................................... 42
4.1. Material Selection and Characterization using THz-TDS ............................................. 42
4.2. Optical Component Characterization with Optical Coherent Tomography .................... 45
4.3. Optical Component Characterization at 0.14 THz ........................................................ 46
Simple Lenses .................................................................................................. 46
Axicons ............................................................................................................. 50
Spiral Phase Plates .......................................................................................... 52
Spiral Phase Plates and Axicons ...................................................................... 54
Helical Axicons ................................................................................................. 57
5. Conclusion .......................................................................................................................... 62
6. References ......................................................................................................................... 65
7. List of Tables ...................................................................................................................... 68
8. List of Figures ..................................................................................................................... 69
Malak Ahmad 4/70
Acknowledgement
At first I want to thank my supervisor Univ.-Prof. Dipl.-Ing. Dr. Zoltàn Major, the head of the Institute
of Polymer Product Engineering for giving me the opportunity to finish my master study at the
institute and his constant guidance and valuable advices.
I would also like to extend my thanks to Dr. Sandrine van Frank and Dr. Bettina Heise from
RECENDT GmbH for their valuable inputs and guidance specifically in the field of optics. Without
them this work would not have been realized. Also I would like to extend my thanks to RECENDT
GmbH for their cooperation and support in this project.
Also I want to thank my co-supervisor Dipl.-Ing. Veronika Miron for her valuable instructions to
finish the writing of my master thesis and for her availability for any coming questions. Finally I
would like to extend my thanks to my family for their love and constant support during the years
of my study.
Malak Ahmad 5/70
Abstract
In this thesis, the possibility of manufacturing Terahertz (THz) optical components using the 3D
printing technology is discussed through different aspects. The material selection is depending on
two factors: the material’s printability and the material’s fingerprint at the THz region 0.1-10 THz.
These factors were found in acrylic photo polymers printable with the PolyJet technology, which
also gives sufficient layer resolution of 16 µm. From the “Vero material family”, the VeroClear
photo polymer with 1.1 cm-1 absorption coefficient was chosen to fabricate THz optical
components at 0.14 THz. The following THz components were fabricated: Simple lenses including
bi-convex and plano-convex lenses, axicons with 5° and 11° base angles, spiral phase plates with
one and two steps, and helical axicons with 5° base angle including one and two steps, and 11°
base angle including one and two steps. To reach the optimal design for the optical components,
a ray tracing simulation was performed in the software ZEMAX. Then the simulated optical
components were converted to physical models by 3D printing using PolyJet technology. The 3D
printed THz optical components were characterized using a 0.14 THz sub Gaussian source and
a 32x32 pixels detector. These optical components were used to generate Gaussian beams,
Laguerre Gaussian beams and Bessel beams with zero, first and second orders. Focus was given
on the Bessel beams as they are commonly used in optics because of their special features to be
non-diffractive and self-healing, and because they have a lot of potential for THz applications as
well.
The testing of the optical components was successful. The fitting curves for the experimentally
generated Gaussian beams were aligned perfectly with the theoretical curve. As for the
experimentally generated Bessel beams, the fitting curves were similar to the theoretical curve
with some differences in the amplitude and the number of outer rings, due to the low sampling rate
of the used detector in the optical component characterization.
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1. Introduction
Terahertz (THz) spectrum has been around for a few decades now, it has a lot of potential to be
exploited in different aspects of life such as medicine and industry. However, because of the
difficulty of generating and detecting THz waves, it was not possible to make any use of it outside
of research labs until now. In the recent years, a lot of developments in technologies of generating
and detecting the THz waves were made, which allows to leave the confines of the laboratories
as a very promising technology to be used in medical and industrial applications. [1] [2]
The THz spectrum is part of the electromagnetic spectrum, laying between microwaves and
infrared. The THz spectrum has a frequency range between 0.1-10 THz which corresponds to
3 mm-30 µm wave length. This spectrum has special penetration capacity for many non-
conductive materials such as wood, plastic, clothing, paper, and biological tissue. Additionally, it
is safe, non-destructive, and non-invasive because it has low photon energy (4 meV). Due to the
unique features of the THz region, it has a variety of applications: medical THz imaging allows to
detect cancer and tooth cavities in early stages; in material characterization and non-destructive
testing defects can be detected from small scale to industrial scale, and THz spectroscopy is also
used in security scanning [1] [3].
The traditional way to generate THz optical components is the substrate manufacturing which is
a method that depends on removing the material from a bulk to fabricate different components.
Due to the substrate manufacturing disadvantages, which include the high manufacturing cost,
the fabrication process of THz optical components using this method is unfavorable for the industry
and the academy. Therefore, THz applications benefit from the 3D printing revolution to produce
customized THz optics with low prices and acceptable surface finishing.
In the recent decades 3D printing technology (additive manufacturing) gained wide use in different
aspects of life such as medical, automotive, optical industry, etc. due to its fast and low-cost
manufacturing process, and low waste generation, compared to the substrate manufacturing. The
main steps in 3D printing are to design a 3D model of the part (CAD), convert the CAD file to an
STL file, slice the STL file with a software to thin cross-sectional layers and create the code for the
printer, and in the end the physical part is printed layer by layer. The most common methods for
3D printing are fused deposit modeling (FDM), selective laser sintering (SLS), 3D ink jet printing,
and stereo lithography (SLA) [2] [4].
The following chart (Figure 1.1) contains the main steps to fabricate 3D optical components at
0.14 THz frequency:
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Figure 1.1: Work flow of 3D printed THz optical components fabrication.
Material selection for 3D printed THz optical components is very important because it should
combine the transparency in the THz region and the printability with good resolution. Recent
studies show that many polymers are transparent in THz region, which make them ideal materials
for THz applications. In addition, many polymers are 3D printed with good resolution depending
on the 3D printing technology. The THz optical component resolution depend on the designed
wavelength, THz is submillimeter wave length therefore the micro-resolution of 3D printing is
acceptable to fabricate good quality THz optical components. In this work the acrylate based Vero
photo polymers were chosen to fabricate THz optical components at 0.14 THz frequency, because
they are transparent in the THz region and they can be printed using the PolyJet technology with
16 µm resolution. [5].
THz-time domain spectroscopy (THz-TDS) was used to characterize samples at (0.1-1) THz
frequency at room temperature. The THz-TDS generates THz pulses in a time domain spectrum
(picosecond) for the samples and for the air, then a Fourier transformation is applied to generate
the absorption coefficient vs frequency curve and the refractive index vs frequency curve for the
8 samples. The VeroClear polymer was chosen due to its low absorption (1.1 cm-1) at 0.14 THz
frequency and its transparency in the visible light region.
The 3D printed VeroClear optical component at 0.14 THz are, simple lenses (bi-convex and plano-
convex lenses), axicons with 5° and 11° base angels (γ), spiral phase plates (SPPs) with one and
two steps (L), and helical axicons. Simple lenses are used to generate Gaussian beams with point
focus. Axicons are refractive conical lenses and have a thickness gradient in radial direction which
is used to generate zero order Bessel beams. SPP is a diffractive optical component that have a
thickness gradient along the azimuthal direction used to generate Laguerre-Gaussian beams.
Helical axicons are optical components that have a thickness gradient in both radial and azimuthal
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directions used to generate higher order Bessel beams. In this work, the focus will be on Bessel
beams generation due to their non-diffractive property, self-healing ability, and line focus
generation (axial resolution) compared to Gaussian beams.
Before the 3D printing process, a ray tracing simulation was done to optimize the optical
component’s design using the software ZEMAX. The simulation was done at 0.14 THz frequency
(2.14 mm wavelength), which corresponds to the frequency of the THz source that was used to
test the lenses. The optimum designs that were generated from these simulations are two simple
lenses with 50 mm diameter, one is bi-convex with a 84.8 mm focus point (ƒ) and the other is
plano-convex with 104 mm ƒ, two axicons with 25 mm diameter, one with γ=5° and ƒ=56 mm, and
the other axicon with γ = 11° and ƒ = 23.5 mm, two SPPs with 25 mm diameter, one with L=1 and
the other with L=2. In addition, four helical axicons with 25 mm diameter, the first two with L=1 and
different γ (5°, 11°), the second two helical axicons with L=2 and different γ (5°, 11°). After ZEMAX
simulations, ten optical components were 3D printed at 0.14 THz frequency using PolyJet
technology with 16 µm resolution.
The 3D printed optical components were characterized using a THz imaging system consisting of
a sub-THz Gaussian source at 0.14 THz frequency and a 2D detector (Camera) with 32x32 pixels
of 1,5x1,5 mm2. In the optical component’s characterization, the focus point of the simple lenses
(bi-convex and plano-convex) and axicons with γ=5°and γ=11° are tested, and four types of beams
are generated using the 3D printed optical components as follow:
I. Gaussian beams generated using 3D printed VeroClear simple lenses.
II. Vortex beams generated using 3D printed VeroClear SPP with L=1 and L=2.
III. Zero order Bessel beams (J0(x)) generated using 3D printed VeroClear axicons with γ=5°
and γ=11°.
IV. Higher order Bessel beams (J1(x) and J2(x)) generated using 3D printed VeroClear helical
axicons, and a combination between the SPPs (L=1, L=2) and the axicons (γ= 5°, γ=11°).
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2. Related Work and Background
In this chapter the Terahertz (THz) radiation and it’s applications are explained, an overview of the
basics of certain optical components and their generated beams (simple lenses, axicons, spiral
phase plates, and helical axicons) are provided, 3D printing technology and the fingerprint of 3D
printable plastics in the THz region is discussed, and an overview of the state of the art research
in the Bessel beams generation by using 3D optical components in THz region is given.
2.1. Terahertz Spectroscopy
The Terahertz (THz) spectrum, is part of the electromagnetic spectrum, laying between
microwaves and the infrared region. It has a frequency range between 0.1-10 THz which
corresponds to 3 mm-30 µm wave length as shown in Figure 2.1. The THz radiation is also called
submillimeter waves and is invisible to the naked eye. The THz region was not known as
microwaves or infrared, because its generation and detection methods were limited. This changed
in the 1970s when the first developed THz sources were operated at a frequency range between
0.1-3 THz. These sources were called backward wave oscillators. In the 1990s, more photonic
and electronic devices were developed and used to generate and detect THz radiation, such as
the time domain THz technology which uses an ultrafast laser to generate THz waves [1] [6].
When THz waves pass through a material such as a polymer, ceramic, or tissue, they interact with
the material by creating weak material bond vibrations and deformations, so any variation in the
material’s thickness or in the chemical compositions will transfer information to the THz signal as
intensity difference which is representing the absorption coefficient, and as a phase difference
which is representing the refractive index. 2D and 3D internal images of parts can be created from
this information as it is shown in Figure 2.2 [7].
Figure 2.1: Electromagnetic spectrum with the THz region marked in red [8].
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The THz region was of growing interest in the last few years because of its special features:
- The penetration capacity for many non-conductive materials such as wood, plastic,
ceramics, cardboard, clothing, paper, and (dry) biological tissue. Depending on the
material, the THz radiation shows a penetration depth of several mm up to cm. In contrast,
it cannot penetrate far into metals or liquid water [3].
- Safe, non-destructive and non-invasive, because it has low photon energy (4 meV), in
contrast to the x-rays [3].
- The high sensitivity to water content [3].
Due to the unique features of the THz region, it has a variety of applications:
- In the field of medicine, medical THz imaging allows for example the detection of cancerous
tissue in early stage, specially the skin cancer. From a THz absorption image, the variation
of the THz absorption can be noticed because the defect tissue has a higher water content
compared to the normal tissue. Due to the THz radiation’s sensitivity to water content, the
defect tissue, which has the higher absorption, can be located. In addition, THz imaging
can potentially be used for early detection of tooth decay (see Figure 2.2) [9].
- For quality control, THz-spectroscopy is useful to detect defects on the scale of a few tens
of microns up to cm in many materials. For example, cracks and defects in silicon material
of solar panels or the coating, or defects in foams that are used in space shuttles [6].
- In security, it can be used to scan for hidden objects like weapons and explosives [3].
- In material characterization and non-destructive testing THz-spectroscopy is applied. In
the pharmaceutical industry, it can be used to check the tablet coating layer thickness, the
uniformity of the coating layers, and to identify material composition of tablets, without
damaging the active ingredient in these tablets [10].
Figure 2.2: Terahertz imaging applications [3] [45] [44].
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There are several types of sources that can be used to generate THz waves. One of the known
categories of generation methods are the laser pumped sources that include two types: The
continuous laser and the pulsed laser pump sources. Continuous laser pump sources contain a
photo-mixing device, which consists of low temperature gallium arsenide (GaAs) and has the
ability to produce continuous radiation up to 5 THz. Pulsed laser pump sources use
photoconductive switches to generate temporary THz pulses which are detected by another laser
pulse [2].
For THz detection a variety of detectors can be used, however they are generally divided in two
categories. Coherent detectors measure the THz amplitude and phase and are based on the
optoelectrical properties of the photoconductive switches. When an optical pulse hits the
photoconductive detector at the same time as the THz pulse, a photocurrent is created, that can
be measured. Alternatively, there are incoherent detectors that measure only the THz amplitude
[2].
For this work a pulsed laser pump source and coherent detector (2 photoconductive switches)
were used in the THz time domain spectroscopy (THz-TDS) for the generation and detection of
THz radiation as will be mentioned below.
THZ-time domain spectroscopy (THZ-TDS) is a method used for the generation and detection
of THz radiation (0.1-3 THz) using ultrafast laser pulses. The main components of the THz
spectroscopy are shown
Figure 2.3 [1] [11]:
Pulsed Femtosecond (fs) laser source: The most widely used sources in THz
spectroscopy is Ti sapphire lasers, at wavelength of 800 nm, which produce pulses < 100
fs. The laser pulse duration, wavelength and output power must suit for THz emitters and
receivers [1].
Beam splitter: The beam splitter is an optical device used to split the incident beam into
two or more portions of light beam. In this work, the THz-TDS used a beam splitter to split
the ultrafast pulse into two equal pulses travels into two different optical pathos [1] [11].
Optical delay line: The THz spectroscopy requires a delay for the first optical path relative
to second path in order to change the arrival time of the single used for detection. This
optical delay can be generated using a retro-reflecting mirror placed on a mechanical
scanner [1] [11].
THz transmitter: The most common way to generate THz radiation pulse is by photo
conductive antenna which use a gallium arsenide (GaAs) semiconductor as substrate
because of its special features (strong absorption at 800 nm and short free carrier lifetime).
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When the ultrafast pulse hits the antenna electrode, electron-hole pairs are created, then
it starts to accelerate which produces an electromagnetic field in the THz range [1] [11].
THz receiver: The common way to detect THz pulse radiation is also using a photo
conductive antenna which uses GaAs as substrate for the electrodes, but here the detector
receives two pulses: the generated THz pulse, and the ultrafast pulse from the laser
source. Here the amplitude of THz pulse is measured at each position of the delay line
(time) so at the end the THz time domain spectroscopy of the sample (THz pulse) is
obtained [1] [11].
Optics for collimating and focusing the THz beam [11].
The sample to be characterized [11].
Figure 2.3: Schematics of Terahertz time domain system [12].
2.2. Optical Components for THz applications
This section will discuss some optical concepts, provide basics knowledge of certain types of
optical components, simple lenses, axicons, spiral phase plates, and helical axicons, provide the
design equations to calculate the theoretical parameters for each component, describe the
generation of beams with these optics and their propagation behavior along the z-axis. Included
are Gaussian beams, Laguerre-Gaussian beams, and Bessel beams. Further on also the features
of these beams are discussed.
Optical Concepts
Transmittance represents the amount of light that passes through a sample without being
absorbed, reflected, or scattered. The transmittance value is described by the beer lambert law
[13], as follows:
(1)
T: Transmittance
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ɪ₀: Incident light intensity
ɪ: Transmitted light intensity.
α: Absorption coefficient.
L: Sample thickness.
The material transmittance is affected by two parameters: the material absorption and the sample
thickness. By increasing the material absorption coefficient the material transmittance will
decrease, and by increasing the sample thickness, the material transmittance will decrease.
Refraction represents the incident light that hits a surface with a certain angel (θi ≠ 0) and bends
when it passes through the surface with a certain angel (θt) because of the speed change [13].
The refraction law is represented by the following equation:
(2)
θi : The incident beam angel
θt : The transmitted beam angel(refracted angel)
ni : The incident medium refractive index
nt : The transmitting medium refractive index
This equation represents how much a beam bends when its crosses an interface between two
media. The refractive index characterizes the speed of light in a specific medium and is defined
by the following equation
(3)
n: Refractive index
c: Light speed in vacuum
v: Light speed in medium
Diffraction is the deviation of the light rays from moving in straight lines. This phenomenon occurs
when an opaque body is placed between a point source and a screen. This body creates a detailed
shadow consisting from dark and bright regions. If a transparent body is placed in front of a point
source and the light wave’s passes through this body are changed in phase or amplitude, then
diffraction occurs also [13].
Aberration is the deviation of light rays that happens when the light passes through a lens forming
a blurred (unclear) image of the object. There are two types of aberration [14] [13].
Chromatic aberration: An aberration that depends on the frequency of the light (colors).
The refractive index is often frequency dependent, leading to a color-dependent refraction
at the lens (see Figure 2.4) [14].
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Achromatic aberration: An aberration occurs if the incident light consists of only one
frequency (one color). There are several types of achromatic aberrations, including
spherical aberration. The spherical aberration occurs due to the spherical surface of the
lens (see Figure 2.5) [14].
Simple Lenses
A lens is a refractive component that reconstructs a transmitted energy distribution. When the light
waves hit a converging (thicker in the middle and thinner on the edges) lens’ surface, it bends
toward the optical axis forming the focal point as is shown in Figure 2.6. However, because of the
difference in refractive index for lens and the air, the light waves also can bend far away from the
optical axis when they hit a concave lens (thinner in the middle thicker on the edges). In this work,
two types of converging lenses will be designed: bi-convex and plano-convex lenses. The lens
maker equations that are used to design simple lenses [13] are as follows:
(4)
Figure 2.4: Chromatic aberration [50].
Figure 2.5: Spherical aberration [51].
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(5)
R1: Radius of the curvature, where the rays enter
R2: Radius of the curvature where the rays come out.
n: Index of refraction, the air is the medium, so we have 1
d: Lens thickness
ƒ: Lens focal length
ds: Source distance
di: Image distance
Figure 2.6: Converging lens (bi-convex) [15].
Using the lens maker equations, the focal point of the simple lens can be calculated as in (4). The
image positions can be calculated at different source positions only if the source rays are not
parallel. If the source rays are parallel, then the focal point is equal to the image position at any
source as in equation (5).
Gaussian light beams are beams that are generated by simple lenses. They have a Gaussian
intensity distribution which is radially symmetric. Figure 2.7 shows the Gaussian beam propagation
along the z-axis and intensity profiles at zero, z, and 2z propagation distances [16].
The Gaussian intensity profile is function of the beam waist (ω0) as shown in equation (6). The
highest beam intensity is found at the beam waist (z=0). The intensity starts to decrease and the
beam starts to diverge by increasing the propagation distance (z) (see Figure 2.7). Rayleigh range
(zR) represents the Gaussian beam’s propagation (spread out) in space which is the distance at
which the beam’s cross-section is doubled. This spread out happens because of the light
diffraction [16].
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(6)
(7)
I: Beam intensity
r: Radial distance
Z: Axial distance
ω0: Minimum beam waist at z=0
θ0: Divergence angel
Axicons
Axicons are refractive conical lenses that have a thickness gradient (increases toward the center)
in radial direction as is shown in equation (9) [17]. When the incident Gaussian beam hits the
axicon lens, the beam rays refract in the shape of conical waves, thus generating zero order
Bessel beams J0(x). Zero order Bessel beams have a circular core with concentric rings at the
cross-section intensity distribution profile. Figure 2.8 shows that an axicon generates a line of
focus (Zmax) which is called depth of focus (DOF). This is one of the features that Bessel beams
have. The Bessel beam depth of focus is a function of the axicon base angle γ and the incident
Gaussian beam waist (radius) ω0 as is shown in equation (8) [18] [19] [17].
Figure 2.7: Gaussian beam intensity propagation [49].
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Zmax = ω0
tan[(n−1) γ] (8)
ℎ(ρ ) = ρ tan(γ) (9)
ω0: beam waist
n: Axicon refractive index
γ: Axicon base angel
ρ: the radial direction
Bessel light beams are beams that have a conical shape of wave propagation. This way of
propagation gives Bessel beams special features like first limit from light diffraction phenomena,
long depth of focus (axial resolution) with invariant lateral resolution, and self-healing ability [19].
Self-healing ability means that when an obstacle is put in front of a Bessel beam, the outer rings
reconstruct the Bessel beam after the obstacle because the wave front has a cone shape. In
addition, this conical wave intersects in more than one point, which produces a line of focus (DOF).
In contrast to that, the Gaussian beams produce a point of focus [19].
Non diffractive means that the Bessel beam core will have the same size over a certain
propagation distance compared to a Gaussian beam core that starts to diverge (diffract) after the
focus point. These special features of Bessel beams make them very desirable for THz
applications compared to Gaussian beams [19].
Bessel beams have concentric rings as a cross section distribution profile, as it shown in Figure
2.9 Theoretically, it can contain an infinite number of rings but this will need an infinite amount of
power, so in reality (experimentally) it is impossible to generate perfect Bessel beams, but the
quasi Bessel beams can be generated over a finite axial distance with the same Bessel beam
properties [19]. Bessel beams can be described using the Helmholtz (Bessel) wave equation [17]:
(10)
ρ: The radial coordinate
Φ: The azimuthal coordinate (the rotation angel)
Figure 2.8: Zero order Bessel beam generated by an axicon lens [52].
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Z: The longitudinal coordinate
ℓ: The Bessel beam order
kz and kρ: The wave numbers
Jℓ: The wave numbersel function (solution)
Figure 2.9 shows different order Bessel beam functions ranging from Zero order Bessel beams
(J0(x)) to five order Bessel beam (J5(x)), n is the Bessel order which is equivalent to ℓ.
Zero order Bessel beams (J0(x)) have the highest intensity at the beam center (x=0) (see Figure
2.10). This produces a bright core at the center with concentric rings. The higher order beams
(J1(x), J2(x)...) have zero intensity at the beam center (x=0) which produces a dark core at the
center with outer concentric rings. In this work, the focus will be on the zero, the first and the
second order Bessel beam functions [19].
There is more than one way to generate zero order Bessel beams. For example, an annular slit
(ring) can be placed at the focus of a simple lens, so the source beam (Gaussian) will pass through
the slit, hit the simple lens and generate a zero order Bessel beam. But this method is inefficient
because a high percentage of the beam absorbed by the slit [19].
Figure 2.9: The half intensity profile of Bessel functions [53].
Figure 2.10: Higher order Bessel beam Intensity Profile [29].
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Another more efficient way to generate zero order Bessel beams is, to use axicons (conical lenses)
that are illuminated by Gaussian beams. When the beam hits the axicon’s surface, the entire
incident beam is approximately converted to a Bessel beam [19] [20]. The axicons will be used in
this work to generate zero order Bessel beams as it will described more detailed in chapter 4.
Axicons can be fabricated by many methods. Micro optical axicons with 220 µm diameter are
fabricated using focused Xe ion beam (FIB) milling. These micro axicons are fabricated from
lithium niobate crystal (LN). The focused Xe ion beam milling is a rapid technique used to fabricate
complex 3 dimensional surfaces. Here the micro axicons are fabricated using focused 30 keV Xe
beam at 200 nA current with less than 20 nm as the surface resolution [21].
The higher order Bessel beams can be generated by many methods. In this work two methods
will be discussed in order to generate higher order Bessel beams. The first one is using axicons
that are illuminated by Laguerre Gaussian beams, the second one is using helical axicons that are
illuminated by Gaussian beams. To generate Laguerre Gaussian beams a special type of optics,
which is called spiral phase plates, will be used in this work.
Spiral Phase Plates
As mentioned before, to generate higher order Bessel beams, spiral phase plates (SPPs) can be
used with axicons. An SPP is a diffractive optical component consisting of two surfaces, one with
spiral surface shape, which has a thickness gradient along the azimuthal direction (θ) (see Figure
2.11). The other surface is flat with a height determined by the SPP thickness (h0). The varying
thickness of the spiral surface is given by equation (11), the azimuthal angel (θ) ranging between
0-2π. SPP contain a number of steps (L) which have the same step height calculated with equation
(12) [17] [22].
When the Gaussian incident beam hits the SPP, it generates an optical vortex with an orbital
angular momentum (OAM). This optical vortex is described by the Laguerre Gaussian mode which
has a donut shape intensity profile (see Figure 2.12). OAM beams have the unique feature that is
data transfer at one frequency, which make it desirable in communication systems [22] [23].
As is shown in equation (10), exp(iᶩΦ) represents the azimuthal phase variation which is used to
describe the vortex beams generated by SPP, where l represents the SPP topographic charge
Figure 2.11: Spiral Phase Plate [55].
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(Bessel order). In the zero order Bessel beams (ᶩ=0) the azimuthal phase variation will equal one
and the variation will be only in radial and longitudinal directions [17].
h(θ) =Lλθ
2π(n−1) (11)
ℎ = Lλ
n−1 (12)
h(θ): SPP varying thickness
θ: Azimuthally direction (rotation angel)
ρ: Radial direction
h: SPP step height
L: SPP topographic charge (number of steps)
λ: Wavelength
n: Refraction index
Gaussian Laguerre beams (vortex beam) generated by SPP are proportional to the number of
steps (topographic charge) L. By increasing the number of steps the radius of the doughnut shape
intensity profile increases (see Figure 2.12).
As it is shown in Figure 2.13, when Laguerre Gaussian beams hit the axicon lens, a higher order
Bessel beam will be generated. The higher order Bessel beam represents a combination between
the spiral wave shape (vortex) and the non-diffractive Bessel feature. The Bessel beam order
depends on the SPP’s step number. So to generate a first order Bessel beam a SPP with L=1 is
required, for second order Bessel beam generation a SPP with L=2 is required and so on [20].
Figure 2.12: SPP with L= 0, 1, and 2 steps clockwise direction [54].
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SPPs can be fabricated by many methods. SPPs with different step numbers are fabricated in
fused Silica using photolithography at a single wavelength of 532 nm. These spirals have a
number of steps ranging from 1 to 10 steps. These spirals are used to convert focused Gaussian
beams to focused Gaussian Laguerre beams. In the SPP’s fabrication process the fused silica
plate is coated with 3.2 µm of photo resist layer (AZ 4562). The plate is exposed to an Ar-ion laser
with 364 nm wavelength, then the resist patterns were converted to a silicon substrate by selective
reactive ion etching [22].
SPP with L>1 can be fabricated by compressing the behavior of SPP with one step into specific
angular regions according to the following relation θb = 2π/b, were b represents the absolute value
of L which called the split approach, for example SPP with L=2, will has θb = π and so on. This
approach was used in this work in the SPP with L=2 CAD design. PP SPP with L=10 where
designed and fabricated by using the split approach with stepped surface at 0.1 THz frequency.
Each step (module) was machined individually from the PP material and has a tongue and grove
in order to attach each step with the other, so finally one SPP with 10 steps will produced [23].
Helical Axicons
Helical axicons are used to generate higher order Bessel beams, with a Gaussian beam as input.
Helical axicon is an optical component having a spiral shaped surface on one side and a conical
shaped surface on the other side (see Figure 2.14). The helical axicons have a thickness gradient
in both radial and azimuthal direction. The thickness gradient in the azimuthal direction is
represented by the spiral shape of the one surface and the thickness gradient in the radial direction
is represented by the conical shape of the other surface [24].
When the Gaussian light beam hits the helical lens surface as it shown in Figure 2.14, it is
converted to higher order Bessel beams. The Bessel beam order depends on the number of steps
on the spiral surface (L). So if the helical axicons have one step L=1 then it will generate first order
Bessel beam and so on [24].
Figure 2.13: Generation of higher order Bessel beams using axicons [30].
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2.3. 3D Printing Technology
In the recent decades additive manufacturing (3D printing) gained wide use in in different aspects
of life such as medical industry, automotive industry, optical industry, etc. due to its fast and low
cost manufacturing, reduced number of the manufacturing steps, less power consumption in
manufacturing, less waste generation, less labor cost and because no tool (no mold) is needed,
which reduces the fabrication cost for small series compared to the injection molding process. In
contrast more conventional manufacturing techniques depend on removing the material from a
bulk using different techniques, such as milling, machining, or turning. Such processes can require
long time for part production, large waste materials during part production, high labor cost, and
expensive tools [4] [25].
3D printing can be categorized into dielectric and metallic according to the build material, bonding
or depositing according to the process, and active or passive according to the part functionality
[4]. The main steps in 3D printing are to design a 3D model of the part (CAD), convert the CAD
file to an STL file, slice the STL file to thin cross-sectional layers with a suitable software, each
layer gets converted to a G-code and sent to the 3D printer. Finally the part is printed layer by
layer (see Figure 2.15). The mechanism of 3D printing (layer by layer) allows the manufacturing
of complex shapes and internal structures with minimum waste of material and this is main
advantage of 3D printing [25] [2].
Many materials can be fabricated by additive manufacturing such as metals, polymers, ceramics,
and composites, with different material phases such as liquid, solid or melted as raw material,
depending on the additive manufacturing method [2].
Figure 2.14: Generation of higher order Bessel beam using helical axicon [34].
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The most common methods for additive manufacturing are Fused Deposit Modeling (FDM),
Selective Laser Sintering (SLS), 3D Ink Jet Printing, and Stereo Lithography (SLA) [2].
Fused deposit modeling (FDM) is a method that uses molten polymer as build material. The
printers consist mainly of an extrusion head and a build platform (see Figure 2.16). The material
is pulled from the spool and heated at the hot-end in the print-head. The melt then is pushed
through the nozzle. The head and platform combination is usually controlled in a way that allows
for movements in horizontal and vertical direction. The layers are solidified as they cool down
outside of the nozzle. The resolution for FDM corresponds to the formed layer thickness and the
nozzle diameter. Usually resolution values around 0.1 mm are given by 3D printer manufacturers.
The most common polymers used in FDM are Acrylonitrile-Butadiene-Styrene (ABS), Polylactic
Acid (PLA), High Impact Polystyrene (HIPS) and Polycarbonate (PC) [26].
Figure 2.16: Fused deposition modeling process [27].
The PolyJet Technology is a photo polymerization method where a photopolymer is cured by
ultraviolet (UV) light. A PolyJet printer consist of a jetting head and a build tray (see Figure 2.17).
The jetting head inject droplets of two materials, liquid photopolymer and the support material
parallel to the build tray, and then the UV light emitted by a lamp integrated to the jetting head
hardens it. The part is built layer by layer on a platform moving vertically. PolyJet has a resolution
(layer thickness) of 16 µm, which is an important advantage for optical components. The common
polymers used in PolyJet technology are acrylates [26].
Figure 2.15: Schematics of a 3D printing process [45].
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Selective laser sintering (SLS) uses a laser beam to melt powder (build material) locally and let
it solidify afterwards in a bulk (see Figure 2.18). The first step of SLS is to spread powder as layers
on the fabrication piston surface, then the laser causes sintering of the layers and filling of the
solid cross sections (hardening), then the fabrication piston goes down by the same value of the
layer thickness, then the powder material is spread again, during the next layer hardening the
previews layers are partially molted which produce a uniform part at the end. SLS has resolution
(layer thickness) between 50 µm and 100 µm. The common polymers used in SLS are polyamide
(PA12) and polyether-ether-ketone (PEEK) [26].
Figure 2.17: PolyJet process [47].
Figure 2.18: Selective laser sintering process [48].
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The additive manufacturing has two disadvantages that are the uncertain dimensional tolerance
and surface roughness (resolution). The dimensional tolerance is caused by the partial size and
the thermal shrinkage. These disadvantages vary from one 3D printing method to another, for
example, in SLS, the dimensional tolerance is medium compared to the PolyJet technology that
has tight tolerance, because SLS technology uses powder particles as raw material and laser
sintering causes thermal shrinkage. FDM has high dimensional tolerance because it uses molten
filaments as raw material which can have a high thermal shrinkage which can result in rough
surface (low resolution) [25] [4].
2.4. 3D Printing for THz Applications
THz imaging systems use tight dimensional tolerance devices and optical components because
there are some types of THz passive devices (lenses, antenna, waveguides, and filters) that are
dimensionally proportional to the designed wavelength, so any change in the devices’ dimensions
will cause shift to the required wavelength. In addition, the surface finishing is very important for
THz devices because the high surface roughness causes higher power losses [4].
The traditional method to fabricate such passive THz devices is with substrate manufacturing, but
due to its disadvantages, mentioned earlier, THz devices are manufactured with high costs, and
that was unfavorable for the industry and the academy. THz applications benefit from the 3D
printing revolution to produce customized THz optical components and passive devices with lower
price and precise dimensions. Lenses can be fabricated by 3D printing using dielectrics (resin, UV
curable polymer, ABS…), filters and waveguides can be printed by both dielectric and metallic
methods [4] [25].
Waveguides (light pipes) at THz frequencies can be manufactured by 3D printing instead of
substrate manufacturing using stereo lithography (SLA) of UV-polymer, and PolyJet of acrylic
resin. Reflector antenna can be also fabricated by 3D printing using SLS [4].
Luneburg lens and band pass filter were printed by ceramic stereolithographic (CSLA) at THz
frequency. PolyJet technology was also used to fabricate hollow-core electromagnetic crystal
waveguide at 0.14 - 0.22 THz and hollow-core antenna [4] [25].
Polyactide (PLA) diffraction gratings are fabricated at THz frequencies of 0.2 THz by 3D printing
using fused deposit molding (FDM) with 0.1 mm resolution (layer thickness). Aspherical lenses
(simple lenses) from VisiJet polymer are fabricated by 3D printing using Multi Jet modelling (pro
Jet HD3500 plus) [28].
3D Printable Materials for THz Applications
A recent study shows that many polymers are transparent in the terahertz region, which make
them ideal materials for THz applications [5] [29] [30]. In this section, the optical properties
(refractive index and absorption coefficient) of the common printable polymers at THz region 0.1-
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10 THz are discussed. It should be noted that by increasing the THz frequency, the absorption
coefficient will often increase, and the refractive index will almost be the same for the following
printable polymers.
Polyethylene (HDPE) and Polypropylene (PP), are thermoplastic polymers and have a low
refractive index (n), around 1.5, and a low absorption coefficient (α), maximum 1.5 cm-1 at 0.5-
10 THz, even for frequencies less than 0.5 THz the absorption for both polymers reach zero (highly
transparent) which make them perfect materials for THz components specially at lower
frequencies. HDPE is considered an ideal material for THz windows manufacturing because of its
optical properties (n, α), its mechanical strength, and its low cost. HDPE windows can be
manufactured by injection molding and extrusion, but for lenses fabrication machining (substrate
manufacturing) is used [5] [29].
HDPE simple lenses (aspheric) with a 50 mm diameter and a focal length of 25 mm were
fabricated at THz region 0.3-1 THz by machining [31].
HDPE and PP 3D printed parts have a rough surface (low resolution) and large dimensional
tolerance due to:
the used 3D printing technology in HDPE and PP lenses fabrication. FDM is the suitable
technique for HDPE and PP 3D printed components but results, because of the materials’
high crystallinity in high shrinkage and surface roughness of the parts, which is not
acceptable for THz millimeter wave optics [29].
the high thermal expansion coefficient of HDPE and PP, which causes a high thermal
shrinkage when the printed filaments start to cool, which produces a large dimensional
tolerance [29].
Acrylonitrile butadiene styrene (ABS), Polyactic acid (PLA), and Polyamide 6.6 (Nylon) are
common printable thermoplastic polymers. They are printed using FDM technique. These
polymers have different refractive indices at 0.5 THz frequency, the lowest is 1.56 for ABS, Nylon
has 1.7, and PLA has 1.9. These polymers have 5 cm-1 absorption coefficient as the minimum
value at the same frequency, which means that these polymers are highly absorbing (opaque) at
THz frequencies, so they are not the right choice for THz optical components fabrication [29].
Polystyrene (PS) is a printable thermoplastic polymer which is printed by FDM technique (100 µm
resolution). PS has low refractive index n=1.56, and low absorption coefficient 0.5 cm-1 at 0.5 THz
frequency and its absorption increases slightly with THz frequency, which makes it a good option
for THz optics specially under 1 THz frequency [29]. PS has a low thermal expansion coefficient,
which produces smoother 3D printing optical components at THz frequencies than 3D printed
HDPE optics.
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Polytetrafluorethylene (PTFE) is a printable thermoplastic with high melting temperature, which
can be printed using SLA [32]. PTFE has a low refractive index of 1.43 and the low absorption
coefficient 0.1 cm-1 at 0.5 THz frequency [30]. 3D printed PTFE optical components (lenses) have
smoother surfaces compared to HDPE lenses due to the used 3D printing technology in PTFE
lenses fabrication which is SLS technique which have 50 µm -100 µm as resolution [33]. Therefore
PTFE (Teflon) is a good choice for THz optical components fabrication.
Polymethylmethacrylate (acrylic) is a transparent thermoplastic that has approximately 1.62
as index of refraction at 0.5 THz and 1 cm-1 as the absorption coefficient at the same frequency
[30]. This acrylic polymer can be fabricated by 3D printing using photo polymerization techniques.
Vero photo polymers (acrylic based polymers) are photo polymers consisting mainly of acrylic
compounds. The optical properties of VeroWhitePlus photo polymer were characterized using
Zomega-Z3 THz time-domain spectrometer (THz-TDS), the refractive index is 1.655 at 0.3 THz,
and the absorption coefficient is 1.5 cm-1 at the same frequency. PolyJet technology is used to
fabricate 3D printed THz optical components from this material [34].
Polyethylene Terephthalate (PET) is a thermoplastic polymer that has a relatively high refractive
index of 1.72 and high absorption coefficient in the THz region (opaque). Therefore PET is not
that suitable for THz optics fabrication [5].
3D Printed Optical Components for THz Applications
Bessel beams were generated at 0.3 THz (λ=1 mm) using 3D printed polymeric optical
components. These optical components are six axicons with different base angels
(5°,10°,15°,20°,25°, and 30° degrees), six SPPs with one to six steps (L), and six helical axicons
with a fixed base angel of 10° and the spiral steps L ranging from 1 to 6. These optical components
where fabricated using Objet30 3D printer with 42 µm resolution in x and y directions and 28 µm
resolution in the z direction. The axicons were used to generate zero order Bessel beams and the
helical axicons used to generate higher order Bessel beams ranging from 1 to 6, and the SPPs
were used for making a comparison between the vortex beam and Bessel vortex beam. The
material used for the optical components was “FullCure835VeroWhite” polymer, which has 1.665
as the refractive index and 1.5 cm-1 as the absorption coefficient at 0.3 THz frequency. All the
optical components have 52.8 mm as diameter. The maximum thicknesses for the six axicons are
2.31, 4.66, 7.07, 9.61, 12.31, and 15.24 mm. All six SPP have the same step (L) height, which is
1.53 mm according to equation (12). The generated Bessel beams are used in THz imaging
application because of their special features, such as long depth of focus with invariant lateral
resolution, which is important for defect detection for thick samples [17].
Higher order (vortex) Bessel beams were generated using the combination between Teflon
axicons with different γ [15°, 20°, 25°, and 30°] and SPP with L=1 at 0.5 THz (600 µm). The
generated vortex Bessel beams are compared in detail according to the light spot size and the
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diffraction-free range. Teflon material has 1.424 as the refractive index at a range of 0.4 to 0.8 THz.
The step height is 1.41 mm which is calculated according to equation (12). The axicons and the
SPP have the same diameter which is 19 mm. The experimental results show that the generated
vortex Bessel beams with larger γ have smaller main spot sizes, and shorter diffraction-free
ranges [33].
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3. Methods
This chapter discusses the reasons of the Vero photo polymers selection to fabricate THz optics.
The Vero photo polymers were characterized in the range of 0.1-1 THz frequency using THz-time-
domain-spectroscopy at room temperature. Ray-tracing simulations were performed using
ZEMAX software to predict the optimum design parameters at 0.14 THz frequency for simple
lenses, axicons, spiral phase plates, and helical axicons. Finally, the 3D printed optics were
characterized using a Tera Sense source with 0.14 THz frequency and a Tera Sense detector with
32x32 pixel resolution.
3.1. Material Selection and Characterization
For the fabrication of THz optical components there are two factors that should be satisfied by the
used material: a low absorption coefficient of the material between 0.1 and 10 THz and a high
resolution of the used 3D printing technique. This is why a selection of Vero photopolymers was
investigated for the THz application in this work because they can be 3D printed by PolyJet
technology with a 16 µm resolution, they have an acceptable absorption coefficient (1.5 cm-1) for
THz optical components, and they were already used to fabricate 3D printed THz optical
components (spiral phase plates, axicons, helical axicons) [17]. Eight materials were selected from
the Vero family: VeroClear, VeroWhitePlus, VeroBlack, DurusWhite, High Temperature,
VeroGray, VeroBlue, and ABS-like polymer (see Figure 3.1).
Vero photo polymers consist mainly from acrylic compounds (see Figure 3.2 and Figure 3.3) which
give them the high transparency and resistance to break [35] [36] [37] [38] [39] [40] [41].
Figure 3.1: Investigated samples of Vero photopolymers.
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Figure 3.2: Tricyclodecane Dimethanol Diacrylate.
Figure 3.3: Exo-1, 7,7trimethylbicyclo [2.2.1] hept2-yl acrylate.
The eight selected samples of Vero photo polymers had a thickness of 5 mm and were analyzed
using THZ-time domain spectroscopy in the range of 0.1 to 1 THz at room temperature. Every two
to three samples, air was measured as the reference sample.
In THz-TDS when the laser pulse is split into two pulses by the beam splitter, as it is explained in
chapter 3 section 3.1, one pulse (pump) is used to generate THz pulse that passes through our
sample and the other pulse used for the detection of the THz pulse (Figure 3.4). So at the end,
the THz-TDS generated THz pulses represented by the time domain signal (pico-second) for the
8 samples and for the air, a Fourier transformation is applied to convert the time domain signal to
frequency domain spectrum, then the absorption coefficients and the refractive indices for the 8
samples are extracted using a dedicated software (Teralyzer).
Figure 3.4: THz-time-domain-spectroscopy (TDS) setup.
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3.2. Design of the Optical Components
This section will provide the optical components’ ray tracing simulations using ZEMAX software at
a single wave length of 2.14 mm (0.14 THZ) for simple lenses (bi-convex, plano-convex), axicons,
SPPs, and helical axicons. ZEMAX is “a program which can model, analyze, and assist in the
design of optical systems” [42]. For the simple lenses, the achromatic (dispersion) behavior was
tested using multiply wavelength in the simulation. The simulation results were represented by the
beam shape (intensity distribution) at the focal point (length) and the beam amplitude profile at the
focal point. These results will help to choose the optimum design parameters. In ZEMAX
simulation, the physical steps (L) of the spiral surfaces were not visualized in the 3D image, but
only considered for the simulation, so the CAD software Siemens NX was used to create the STL
file of the spiral surfaces, in order to get the correct spiral shape for the 3D printing process.
Simple Lenses
After the material selection, simple bi-convex and plano-convex lenses were designed using the
software ZEMAX. These lenses were designed first with a single wavelength of 2.14 mm
(0.14 THz) and second with multiple wavelengths ranging between 0.3 and 3 mm corresponding
to 1 to 0.1 THz frequency to test the dispersion behavior. Table 3.1 shows the ray tracing
simulations for the two plano-convex lenses with a 100 mm diameter at a single wave length
(2.14 mm) and Table 3.2 shows the ray tracing simulation for the plano-convex lens and the bi-
convex lens with a 50 mm diameter at single and multiple wave length. The simulation dimensions
for these lenses are represented by the radius of the curvature (R), the lens thickness (d), and the
lens diameter (D). Table 3.2 shows the generated Gaussian beam intensity map (distribution) at
the ƒ of 100 mm diameter for plano-convex lenses, at this ƒ, the Gaussian beam has a wide point
shape with high beam absorption (low transmittance) compared to the 50 mm diameter plano-
convex lens which has a small point of focus and lower absorption at the same wave length
2.14 mm as it is shown in Table 3.2. Therefore, the 100 mm diameter plano-convex lenses were
excluded from further studies.
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Table 3.1: ZEMAX simulation of a plano-convex lens with 100 mm diameter.
Table 3.2 shows the generated Gaussian beam intensity map (distribution) at the ƒ for 50 mm
diameter plano-convex and bi-convex lenses. The Bi-convex lens has an ƒ of 84.8 mm which has
a bigger size and lower intensity compared to the plano-convex lens’ ƒ.
The simulations for 50 mm plano-convex and bi-convex lenses were repeated at multiple
wavelengths (0.3-3 mm) to test the chromatic aberrations (dispersion) behavior for theses lenses.
Table 3.2 shows small dispersion at multiple wavelengths. So, in conclusion the 50 mm plano-
convex and bi-convex lenses were chosen for 3D printing and further investigation.
Then the lens maker equations (4) was used to calculate the theoretical ƒ for the chosen lenses:
Plano-convex, ƒ=209 mm Plano-convex, ƒ=150 mm
Dimensions (mm)
R1=140, R2=∞, d=10, D=100
R1=100, R2=∞, d=14, D=100
Intensity map at lens ƒ for 2.14 mm wave length
Amplitude profile at lens ƒ
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On the one hand the 50 mm plano-convex lens has a theoretical ƒ of 104 mm, which is the same
ƒ of the simulated plano-convex lens (see Table 3.2). But on the other hand, the theoretical ƒ of
the bi-convex lens is 83 mm which is smaller than the simulated bi-convex lens’ ƒ (84.8 mm). The
difference in the theoretical and the experimental ƒ of the bi-convex lens is due to the aspheric
aberration that the bi-convex lens has.
Table 3.2: ZEMAX simulation of a bi-convex and a plano-convex lens with 50 mm diameter.
Bi-convex, ƒ=84.8 mm Plano-convex, ƒ=104 mm
Dimensions (mm)
R1 = 110, R2 = -110, d = 10, D= 50 R1 = 70, R2 = ∞, d = 5, D= 50
Intensity map at lens ƒ for 2.14 mm wave length
Intensity map at lens ƒ for multiple wave length
0.3-3 mm
Amplitude profile at ƒ for 2.14 mm wave length
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Axicon
Several axicons were designed with different diameters and base angels in order to produce zero
order Bessel beams using the software ZEMAX. The first chosen design parameter for the axicons
was a 50 mm diameter. The generated zero order Bessel beams (J0(x)) from these axicons have
a high number of outer rings in the ray tracing simulation results, which is difficult to detect using
a 32x32-pixel detector (low resolution).
The second choice for the axicon parameters was a diameter of 25 mm with the base angels
γ=20°, γ=15°, γ=11°, and γ=5°. The ray tracing simulation results for 25 mm diameter axicons with
high base angels (γ =20°, and γ =15°) were as follow: high beam absorption due to the high
axicons thicknesses, short depth of focus (DOF) for the generated Bessel beams, and the small
size of the generated Bessel beam rings. The detection process for these axicons are difficult
using a low resolution detector. In contrast to the ray tracing simulation, results for these axicons
with lower base angels (γ =11°, γ =5°) were represented by lower beam absorption due to the
lower thickness, low number of the generated Bessel beam outer rings, and acceptable size for
the generated beam outer rings. Therefore the best choice for axicon design parameters were a
diameter of 25 mm and base angles of 11° and 5° (see Table 3.3), to enable good axicon
characterization.
Table 3.3 represent the ray tracing simulation results for the chosen axicons. Axicon11° has a
shorter ƒ at 23.5 mm compared to the Axicon5° which has its ƒ at 56 mm. Also the generated zero
order Bessel beam (J0(x)) using Axicon11° has a shorter DOF (35-40 mm) and a smaller beam
ring size compared to the generated J0(x) using Axicon5° which has a higher DOF (85-90 mm)
and larger ring size. Table 3.3 shows also the shape of the generated J0(x) which is represented
by a needle shape in the center with outer rings. In order to compare the simulated DOF for the
axicons, the theoretical axicons DOF is calculated using equation (8). The theoretical Axicon5°
DOF is 85 mm, and the theoretical Axicon11° DOF is 38.4 mm. It can be noticed that both axicons
have a theoretical DOF in the range of the simulated DOF. Therefore, both axicons were 3D
printed.
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Table 3.3: Axicon5° and Axicon11° simulation by ZEMAX at 2.14 mm wavelength.
Spiral Phase Plate
As mentioned in chapter 3 section 3.4.4, SPPs convert the incident Gaussian beams to Laguerre-
Gaussian beams, but if an axicon lens is placed after the SPPs, a higher order Bessel beam is
generated. Two types of SPPs were designed using a ZEMAX simulation to determine the
geometry and then the CAD software Siemens NX to construct the 3D geometry with those
dimensions. The first SPP contains one step (L=1) with 3.175 mm height and the second SPP
contains two steps (L=2), each step having a 3.175 mm height. Table 3.4 shows the ZEMAX ray
tracing simulation results for SPPs with L=1 and L=2 at 2.14 mm wave length, represented by the
SPP intensity map and the beam amplitude profile, and also shows the SPPs 3D image
represented by hs (step height), h0 (SPP height), and D (diameter) using the software Siemens NX.
Axicon γ=5°, ƒ=56 mm Axicon γ=11°, ƒ=23.5 mm
Dimensions Thickness = 6 mm
D = 25 mm
Thickness = 7.4 mm
D= 25 mm
DOF
85-90 mm
35-40 mm
Intensity map at axicon ƒ
Amplitude at the ƒ
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Table 3.4: Spiral Phase Plate (L=1, L=2) simulation by ZEMAX at 2.14 mm wave length.
SPP L=1 SPP L=2
Dimensions
(mm)
D=25, h0=2, hs=3.175
D=25, D=25, h0=2, hs=3.175
Intensity map at 56 mm
Amplitude at 56 mm
The generated Laguerre-Gaussian beam (vortex beam) using SPP with L=2 has a bigger
doughnut (ring) size compared to the generated Laguerre beam using SPP with L=1 (see Table
3.4). Increasing the SPP step number (L) will increase the size of the generated vortex beam and
also increases the beam divergence. Therefore SPPs with L=1 and L=2 were chosen to be
designed and fabricated. These SPPs (L=1 and L=2) are used with the previous designed axicons
(γ=5° and γ=11°) to generate first order Bessel beams (J1(x)) and second order Bessel beams
(J2(x)).
Helical Axicons
As mentioned in chapter 3 section 3.4.5, the helical axicons are used also to generate higher order
Bessel beams. The designed helical axicons will have the same base angels of the previous
designed axicons (γ =5°, γ =11°) and the same step numbers of the previous designed SPPs (L=1,
L=2). So a comparison can be made between the generated Bessel beams using the helical
axicons and the generated Bessel beams using the combination between the axicons with (γ =5°,
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γ =11°) and SPPs with (L=1, L=2). At the end four helical axicons were designed at 2.14 mm
wavelength using ZEMAX simulation and Siemens NX software (see Table 3.5 and Table 3.6).
Table 3.5: Helical axicons (L=1) simulation by ZEMAX at 2.14 mm wave length.
Helical axicon (L=1, γ =5°)
at 56 mm Helical axicon (L=1, γ =11°)
at 23.5 mm
Dimensions
(mm)
D=25, thickness = 6 + 3.175 D=25, thickness = 7.4 + 3.175
Intensity map
Amplitude
The helical axicon thickness consists of the axicon thickness and the SPP step height, which was
drawn using Siemens NX software for the four helical axicons. Table 3.5 shows the ray tracing
simulation for two helical axicons with the same step number (L=1) and different base angels
(γ=5°, and γ=11°). Both helical axicons generate a vortex shape (ring) profile with outer rings,
which is called the first order Bessel beam (J1(x)). The J1(x) generated using helical axicon11°
with L=1 has a smaller ring size compared to the J1(x) generated using the helical axicon5° with
L=1 (see Table 3.5).
Table 3.6 shows the ray tracing simulation for two helical axicons with the same number of steps
(L=2) and different base angels (γ =5° and γ =11°). Both helical axicons with L=2 generate a vortex
beam shape (ring) with outer rings, which is called the second order Bessel beam (J2(x)) (see
Table 3.6). The J2(x) generated using helical axicon11° with L=2 has smaller ring size beam
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compared to the generate J2(x) using helical axicon5° with L=2 (see the intensity map in Table
3.6).
Table 3.6: Helical axicons (L=2) simulation by ZEMAX at 2.14 mm wave length.
Helical Axicon (L=2 and γ=5°) at 56 mm
Helical Axicon (L=2, and γ =11°) at 23.5 mm
Dimensions
(mm)
D=25, thickness = 6 + 3.175 D=25, thickness = 7.4 + 3.175
Intensity map
Amplitude
The J2(x) generated using helical axicons with L=2 have a bigger vortex (ring) size beam compared
to the J1(x) generated using helical axicons with L=1. Therefore the Bessel beam order is a
function of the SPP step number.
3.3. 3D Printed Optical Component Characterization
The simulated optical components were converted to physical components by 3D printing. The bi-
convex lens was printed two times, in order to test the polishing effect on the optical properties of
the bi-convex lens. One of them was polished mechanically using sandpapers ranging from 400-
2000 grid.
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To validate the suitability of the chosen 3D printing technology (PolyJet) for THz applications in
terms of the surface resolution, optical coherent tomography (ÓCT) was used to determine the
surface roughness of the plano-convex lens.
For the lenses’ characterization a THz imaging system was used, that consists of TeraSense
detector and TeraSense source operating at room temperature. From this characterization the
focus of the simple lenses (bi-convex and plano-convex) and axicons (γ= 5°, γ=11°) were tested
and the following beams were generated:
I. Vortex beams using SPP with L=1 and L=2.
II. Zero order Bessel beams (J0(x)) using axicons with γ= 5° and γ=11°.
III. Higher order Bessel beams (J1(x) and J2(x)) using two methods, firstly using a combination
between the SPPs (L=1, L=2) and the axicons (γ= 5°, γ=11°) and secondly using the helical
axicons which contains the same parameters (L, γ) of the SPPs and axicons.
The THz imaging system consist of:
A Sub-THz Gaussian source at 0.14 THz frequency. This source is represented by silicon
diodes fixed on a cupper heat sink (see Figure 3.5). The standard output power can reach up
to 100 mW [43].
100x100x55 mm³ Detector (Camera) with 48x48 mm² sensor size (32x32 pixel) (see Figure
3.5). This detector detects frequencies from 0.05-0.7 THz, and is fabricated from a GaAs
semiconductor using conventional optical lithography [43].
Figure 3.5: TeraSense source (left) and TeraSense detector (right).
In the optical components characterization other types of optics were used to optimize (enhance)
the characterization results (see Figure 3.6):
A collimating lens was placed in front of the source in order to get parallel beam rays to the
optical path. The parallel (collimated) beam is very important in optics characterization to get
the focus point of the optical components at the correct distance, and to reduce the beam
divergence when it comes out of the source.
Filtration lenses (optical filters) with different transmittance values were used to reduce the
beam intensity (saturation) of the images by absorbing a part of the beam. By using these
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filters smooth intensity beam curves with clear peaks were created, in order to determine the
correct optical components’ focal point (the highest intensity).
The applied procedure in the optical components characterization was as follow:
I. The source was fixed at a certain position. The detector was mounted on an immovable
part, which allowed the detector to move along the z-axis (axial distance).
II. A collimated lens with 40 mm focal length was placed on the front of the source (see Figure
3.6).
III. The 3D printed optical component (simple lens) was fixed at a certain distance from the
source (see Figure 3.6).
IV. The detector (camera) was moved along the axial distance, and several images were taken
at different positions from the camera.
V. Image processing software was used to analyze the images of the optical components and
create an intensity curve profiles for each image. The software tools that were used to do
the image analysis are ImageJ and MATLAB.
For the analysis of the simple lenses’ images (bi-convex and plano-convex) ImageJ was used.
ImageJ is a software used for analyzing 2D intensity profiles of light beams (Gaussian) that allows
the beam characterization by extract specific parameters from the intensity profile.
The beam profile plugin in ImageJ was used, this plugin provided the following information about
simple lenses images:
I. The intensity values along two axes, at two different angels (0°, 45°) (see Figure 3.7).
II. The standard deviation (σ) along each axis.
A Python script was used to calculate the average intensity profile for the 0° and 45° axes, so in
the end the beam intensity profile will be created for each image at different axial distance. Then
Figure 3.6: THz imaging system setup.
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the standard deviation was used to calculate the Gaussian beam waist according to the equation:
4*σ = 2*ω, which was applied by ImageJ.
σ: Standard deviation; ω: Beam waist
For axicons, SPPs, and helical axicons images MATLAB was used because it provides more
accurate and precise tools to analyze Bessel beam data. A script using MATLAB was designed to
calculate the average intensity for each image as follows:
I. The beam center of the image was determined (see Figure 3.8).
II. Integration from the center over 360° was applied.
III. Finally, the half beam profile intensity was created.
Figure 3.7: Gaussian beam Image analyzed by ImageJ a) 0° and b) 45°.
Figure 3.8: Bessel beam image analyzed by MATLAB
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4. Results and Discussion
This chapter includes two sections: the first is about the Vero photo polymer samples
characterization using THz-time-domain-spectroscopy (TDS) at the frequency range 0.1-1.1 THz
at room temperature, the second is about the fabricated optical components’ characterization
using a sub-THz source at 0.14 THz and a camera with 32x32 pixel resolution at room
temperature.
4.1. Material Selection and Characterization using THz-TDS
The Vero photo polymer samples were characterized using THz-TDS at the range of 0.1-1.1 THz
at room temperature. The Vero photo polymer samples are VeroClear, VeroBlue, VeroGray,
VeroBlack, High Temperature (High Temp), VeroWhitePlus, DurusWhite, and ABS-like photo
polymers. THz-TDS generates THz pulses represented by the time domain signal (pico-second)
for each sample including the reference sample (air) (see Figure 4.1). In the graph it can be
observed that the generated THz pulses for the eight samples have almost the same shape except
for the reference sample (air), which has a lot of peaks going up and down due to the high
sensitivity of THz-spectroscopy to the water content.
Figure 4.1: THz-Time domain spectrum of Vero photo polymers.
A numerical Fourier transformation was applied to convert the time domain signal to frequency
domain spectrum, then the absorption coefficients and the refractive indices for the eight samples
were extracted using the dedicated software Teralyzer. Finally two graphs were generated,
refractive index (N) vs frequency and absorption coefficient (α) vs frequency. Figure 4.2 shows
the refractive index of the eight samples as function of THz frequency ranging between 0.1-1.1,
indicating that the refractive index increases inversely proportional with the frequency. It needs to
be noted that the refractive index values range between 1.62-1.74 at different frequencies for the
different eight samples, so the refractive index is almost the same for the eight samples with
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DurusWhite showing the lowest and VeroBlue showing the highest values. Apparent is, that the
sample “High Temperature” has a much higher refractive index and absorption value than the
other samples, so it was excluded after these measurements.
Figure 4.2: Refractive index of Vero photo polymers at frequency range of 0.1-1.1 THz.
However, the absorption coefficient values range between 0-25 at different frequencies for the
eight samples (see Figure 4.3). Therefore, the material selection depends on the sample’s
absorption coefficients. The graph shows, that the absorption coefficient increases by increasing
the frequency (less transmittance) so the acceptable frequency for our samples should be in the
range between 0.1-0.2 THz to insure the fabrication of transparent optical components (less
absorption) in THz region. The high temperature photo polymer was excluded from our selection
because it has high absorption coefficient (low transparency) at the lowest THz frequencies 0.1-
0.2, which is around 5 cm-1.
Figure 4.3: Absorption coefficient of Vero photo polymers at frequency range of 0.1-1.1 THz.
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As mentioned in chapter 2, our focus will be at 0.14 THz frequency, so a shorter THz frequency
range was taken, which is between 0.1-0.26 THz, to characterize the eight samples. Figure 4.4
shows the refractive index for the other seven samples which ranges between 1.685-1.655, which
represent very small variation in the refractive index for the seven samples at 0.1-0.26 THz
frequency. However, in Figure 4.5 the absorption coefficient shows a higher variation for the seven
samples compared to the samples refractive index at the same frequency range. The graphs show
that VeroClear, VeroBlue, VeroGray, and VeroBlack have a lower absorption compared to the
other samples (VeroWhitePlus, Duruswhite, and ABS-like samples).
The noise visible in these graphs was generated through the material characterization using THz-
TDS. The refractive index and the absorption values at 0.14 THz were taken for the eight samples
(see
Figure 4.4: Refractive index of Vero photo polymers at frequency range of 0.1-0.26 THz.
Figure 4.5: Absorption coefficient of Vero photo polymers at frequency range of 0.1-0.26 THz.
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Table 4.1), in order to select the material with the lower absorption at 0.14 THz frequency. This
table shows that VeroClear, VeroBlue, VeroGray, and VeroBlack polymers have the lowest
absorption coefficients compared to the other samples which is 1.1 cm-1.
Table 4.1: Refractive indices and absorption coefficients for Vero photo polymers at 0.14 THz.
Material Sample thickness
(mm) Refractive index
(-) Absorption coefficient
(cm-1)
VeroClear 5.015 1.674 1.1
VeroWhite Plus 5.001 1.681 1.5
VeroBlack 5.038 1.673 1.1
DurusWhite 5.045 1.666 1.6
High Temperature 5.015 ≥1.74 4
VeroGray 5.045 1.678 1.1
VeroBlue 5.042 1.684 1.1
ABS-like 5.000 1.676 1.4
In summary, the THz-TSD was used to characterize the Vero Photo polymers samples in THz
range 0.1-1 by the refractive index and the absorption coefficient, to know which polymer is the
best choice for THz optical components fabrication (the lowest absorption). First, the high
temperature photo polymer was excluded because it has high absorption at range of 0.1-1 THz
frequency, then Vero White Plus, Durus White and ABS-like photo polymers were excluded
because they have higher absorption coefficients compared to the rest of the samples at 0.14 THz.
The rest of the samples (VeroClear, VeroBlue, VeroGray, and VeroBlack polymers) have the same
absorption coefficients, which is 1.1 cm-1 at 0.14 THz. Finally, the VeroClear photo polymer was
chosen to 3D-print the optical components because it is also transparent in the visible range which
makes the optical alignment easier in THz system characterization.
4.2. Optical Component Characterization with Optical Coherent Tomography
Optical Coherent Tomography (OCT) was used to determine the surface roughness of the plano-
convex lens to see, if the PolyJet technology with 16 µm layer resolution gives the required surface
smoothness for THz applications.
Figure 4.6 presents the surface roughness of the plano-convex lens. The black line represents the
surface of the lens from the side and has a stepwise inclination due to the layer-by-layer 3D
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printing. The height of each step was measured using ImagJ, then the average height was
calculated as follows:
𝑠𝑡𝑒𝑝1 + 𝑠𝑡𝑒𝑝2 + 𝑠𝑡𝑒𝑝3 + 𝑠𝑡𝑒𝑝4 + 𝑠𝑡𝑒𝑝5 + 𝑠𝑡𝑒𝑝6 + 𝑠𝑡𝑒𝑝7 + 𝑠𝑡𝑒𝑝8
8 = 16.95 µ𝑚
The calculated layer height of the plano-convex lens is 16.95 µm which is very close to the
theoretical resolution of the used PolyJet technology. The difference between the theoretical
resolution of the used PolyJet and the actual resolution of the 3D printed plano-convex lens using
the same PolyJet technology could be on the one hand due to the printing process itself an on the
other hand also due to the unclear steps in the OCT image. Therefore the measured step height
is not accurate 100%, but gives the confirmation, that the resolution is suitable for THz
applications.
Figure 4.6: OCT image of a plano-convex lens.
4.3. Optical Component Characterization at 0.14 THz
As mentioned earlier, the characterization of the optical components (simple lenses, axicons,
spiral phase plates, and helical axicons) were done using Sub-THz Gaussian source at 0.14 THz
frequency and a camera with 48x48 mm² resolution, and several images were taken for each
optical component at different axial distances. ImageJ and MATLAB were used to analyze the
optical components’ images in order to get the shape of the intensity beam profile, the lenses’
focal length, and the beam propagation along the z-axis.
Simple Lenses
4.3.1.1. Simple Lenses - Bi-Convex
Figure 4.7 shows multiple curves, each curve representing the images’ average intensity profile
for the bi-convex lens at different axial distance from the camera. The intensity beam shape for
every image has a Gaussian distribution and the image at 84 mm represents the highest intensity,
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which corresponds to the lens’ ƒ. The theoretical ƒ of the bi-convex lens is 83 mm according to
the lens maker equation, which is shorter than the experimental ƒ by 1 mm, this is due to:
- The divergence of the source beam (the beam is not perfectly collimated).
- The misalignment of the optical components in the characterization process.
- The aspheric abrasions (lens spherical shape).
As mentioned before, the Gaussian beam waist can be calculated using ImageJ equation, so for
each image at a certain axial distance, the beam waist is calculated in order to create the Gaussian
beam propagation curve (see Figure 4.8).
Figure 4.8 shows two types of scattered data, the red dots represent the data (images) of the
normal bi-convex lens. The blue dots represent the data of the polished bi-convex lens. For both
sets of data, a polynomial fitting was applied, because the polynomial fitting was the most similar
fitting to the bi-convex data distribution. After the polynomial fitting, Gaussian beam propagation
curves were created for both the normal and the polished bi-convex lenses. As mentioned before,
the lowest beam waist correspond to the lens’ ƒ. Therefore, the ƒ of the normal bi-convex lens is
around 85 mm, and the polished bi-convex had its lowest beam waist around 100 mm. The ƒ
difference between the normal and the polished bi-convex lenses is due to the changes that
happen to the lens’ radius of curvature (R1, R2) during the polishing process, which cause a shift
of ƒ. As calculated previously, the theoretical ƒ of the bi-convex lens is 83 mm, which is close
enough to the experimental value of ƒ=84 mm.
Figure 4.7: Bi-convex beam profile at different axial distances
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Figure 4.8: Gaussian beam propagation of an unpolished and polished bi-convex lens.
Figure 4.9 shows a Gaussian fitting represented by the red dotes which was applied to the intensity
beam profile image of the bi-convex lens at 84 mm ƒ axial distance. It can be seen that the 84 mm
intensity data of the bi-convex lens represented by the blue dots was fitting quite smoothly to the
Gaussian distribution.
4.3.1.2. Simple Lenses - Plano-Convex
Figure 4.10 shows multiple curves, each curve represents the image intensity profile for the plano-
convex lens at different axial distance from the camera. As it can be seen, the beam shape for
each image has a Gaussian distribution, and the image intensity profile at 104 mm represents the
highest intensity which corresponds to the lens’ ƒ. The theoretical ƒ of the plano-convex according
to the lens maker equation is 104 mm which is the same as the experimental ƒ.
Figure 4.9: Gaussian beam propagation of bi-convex lens and polished bi-convex lens.
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As mentioned before the Gaussian beam waist can be calculated using ImageJ equation, so for
each image at a certain axial distance the beam waist is calculated in order to create the Gaussian
beam propagation curve (see Figure 4.11).
Figure 4.11 shows scattered data (images) represented by the purple dots, a polynomial fitting
was applied to this data, because the polynomial distribution was the most similar distribution to
the plano-convex data. After the polynomial fitting, a Gaussian beam propagation curve for the
plano-convex lens was created. It shows that at smaller axial distance (z = 80 mm) the beam waist
size is large and it starts to decrease by increasing the axial distance (z) until it reaches the
minimum waist (ƒ) then it increases again. The lowest beam waist is at around 105 mm. The
plano-convex ƒ (104 mm) is within the minimum Gaussian beam waist.
Figure 4.12 shows a Gaussian fitting represented by the red dotes which was applied to the
intensity beam profile image of the Plano-convex at 104 mm (ƒ) axial distance. It can be seen that
Figure 4.10: Plano-convex beam profile at different propagation distances.
Figure 4.11: Gaussian beam propagation of plano-convex lens.
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the 104 mm intensity data of the plano-convex lens represented by the blue dots is aligned
perfectly and smoothly to Gaussian distribution.
The Gaussian fitting curves and the experimental ƒ for both bi-convex and plano-convex lenses
indicate that the 3D printed simple lenses at THz frequencies using PolyJet technology produce
an accurate simple lenses which generate perfect Gaussian beams with a precise focal point.
Axicons
4.3.2.1. Axicons – Axicon5°
Figure 4.13: Half beam intensity profile of Axicon5° at different axial distances (left) and fitting of
the zero order Bessel beam for Axicon5° (right).Figure 4.13 (left) shows multiple curves; each
curve representing a half beam intensity profile of Axicon5° at different axial distance from the
camera. As it mentioned, the MATLAB was used to generate these half beam intensity profiles.
Figure 4.13 (right) shows that the half beam intensity profile for every image has a zero order
Bessel beam distribution with one ring. As it can be seen, the image at 50 mm represents the
highest intensity which should be the Axicon5°’s ƒ, but in the simulation results the Axicon5°’s ƒ
was at 56 mm, so there is a 6 mm difference. In order to fit the Axicon5°’s half intensity beam
profile at 50 mm to the zero order Bessel beam (J0(x)), a MATLAB script was written.
Figure 4.13 (right) shows the Axicon5° half profile intensity at 50 mm represented by the red curve.
A zero order Bessel beam (J0(x)) fitting was applied to this 50 mm image which is represented by
the blue curve. This fitting was applied because its distribution curve is similar to the axicon5° half
beam intensity profile distribution. It can also be seen that, axicon5° half beam intensity profile has
a thicker Bessel beam core and one outer ring compared to the zero order Bessel fitting (J0) which
has a thinner core with multiple outer rings.
Figure 4.12: Fitting of Gaussian beam for a plano-convex lens.
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Figure 4.13: Half beam intensity profile of Axicon5° at different axial distances (left) and fitting of the zero order Bessel beam for Axicon5° (right).
4.3.2.2. Axicons – Axicon11°
Figure 4.14 (left) shows multiple curves, each curve representing a half beam intensity profile of
Axicon11° at different axial distances from the camera. As for the Axicon5° it can be seen that the
half beam intensity profile for each image has a zero order Bessel beam distribution with one outer
ring. It can also be seen that the image at 30 mm represent the highest intensity which should be
the focal point of Axicon11°. This differs from the simulation results of the Axicon11° where the ƒ
was at 23.5 mm. This results in a 6.5 mm difference. In order to fit the Axicon11° half intensity
beam profile at 23.5 mm to the zero order Bessel beam (J0(x)), a MATLAB script was written.
Figure 4.14 (right) shows the Axicon11°’s half profile intensity at 23.5 mm represented by the red
curve. A zero order Bessel beam (J0(x)) fitting was applied to this 23.5 mm image which is
represented by the blue curve. This fitting was applied because its distribution curve is similar to
the Axicon11° half beam intensity profile distribution. It can also be seen that the Axicon11°’s half
beam intensity profile has a thicker Bessel beam core and no clear outer rings compared to the
zero order Bessel beam fitting (J0) which has a thinner core with multiple outer rings.
Figure 4.14: Half beam intensity profile of Axicon11° at different axial distances (left) and fitting of the zero order Bessel beam for Axicon11° (right).
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The ƒ difference between the experimental results and the simulated results for both Axicon5° and
Axicon11° and the imperfectly matched zero order Bessel beam fitting, which is represented by
the Bessel core thickness and the Bessel beam outer rings are due to:
I. The divergence of the source beam (the beam is not perfectly collimated).
II. The misalignment of the optical components in the characterization process.
III. The poor sampling rate (low resolution).
In order to create the zero order Bessel beam propagation curves for both Axicon5° and
Axicon11°, the Bessel beam core radius was taken from the half beam intensity profiles at different
axial distances (see Figure 4.15).
Figure 4.15 represents the propagation of the generated zero order Bessel beams using Axicon5°
and Axicon11°. It can also be seen that both beams have the same beam radius at different axial
distances 20-60 mm, which mean that these beams are non-diffractive.
Spiral Phase Plates
Figure 4.16 shows multiple curves, each curve represents a half beam intensity profile of the SPPs
with L=1 (left) and L=2 (right) at different axial distance. As mentioned before, MATLAB was used
to create the half beam intensity profiles for the SPPs. It can be seen that at zero axial
(propagation) distance the intensity value is equal to zero for each image that represents the
center of the generated vortex (ring) beams.
The half beam profiles for the SPP with L=1 images, have a similar shape of the vortex beam
which was generated in ZEMAX simulations. Also, the figure shows that the half beam profiles
generated by the SPP with L=2 have a wider curve compared to the half beam profiles generated
by SPP with L=1 at the same propagation distance. It can be noticed that by increasing the axial
distances, the half beam intensity curve becomes wider and the intensity becomes lower. This is
an indication of the generated vortex beams divergence by increasing the axial distance. To
Figure 4.15: Zero order Bessel beam propagation of Axicon5° and Axicon11°.
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generate the propagation curves for the generated vortex beams, the beam outer radius for each
image was taken at different axial distances at the same intensity, which makes it easier to see
the increase in outer radius with increasing axial distance (see Figure 4.17).
Figure 4.16: Half beam intensity profile of SPP (L=1) (left) and SPP (L=2) (right) at different propagation distances.
It can also be seen that, for both beams the beam outer radius increases from 10-15 mm along
the axial distance, which indicates a diffraction (divergence) of the vortex beam along the axial
distance. It can also be noticed that the generated beam using the SPP with L=2 has a bigger
outer radius compared to the generated beam using the SPP with L=1 which indicates that by
increasing the SPP’s step number (L) the generated beam size becomes bigger.
On the one hand the SPPs with L=1 and L=2 generate diffractive vortex beams, on the other hand
the axicons with 5° and 11° base angles generate non-diffractive zero order Bessel beams.
Therefore, the combination between these two beams will generate a non-diffractive beam with
vortex shape intensity profile (doughnut shape) which is called the higher order Bessel beams
(J1(x), J2(x)). In the next section the higher order Bessel beams are generated using the
Figure 4.17: Vortex Beam propagation of spiral phase plates with L=1 and L=2.
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combination between the SPPs and the axicons. The characterization process will contain two of
the 3D printed optical components, one of the SPPs and one of the axicons.
Spiral Phase Plates and Axicons
As it mentioned, the first order and second order Bessel beams will be generated using a
combination between the SPPs with L=1 and L=2 and the axicons with 5° and 11° base angles.
So four combinations are characterized in this section.
4.3.4.1. Spiral Phase Plate with One Step L=1 and Axicons (5°, 11°)
The first combination includes the SPP with L=1 and Axicon5°. The results for this characterization
are shown in Figure 4.18 (left). This figure shows multiple curves, each curve represents a half
beam intensity profile which is generated by SPP with L=1 and Axicon5° at different propagation
distance. Figure 4.18 (right) represents the second combination which includes the SPP with L=1
and Axicon11°. This figure shows multiple curves, each curve represents a half beam intensity
profile generated by SPP with L=1 and Axicon11° at different propagation distance. Figure
4.18shows that at zero axial propagation distance the intensity value is zero for each image, which
represents the center of the generated beams. This indicates that the beam has a ring shaped
(vortex) intensity profile. Figure 4.18 (right) shows a smaller beam inner radius compared to Figure
4.18 (left), which indicates that the generated beam from the second combination (γ=11°) has a
smaller ring intensity profile compared to the first combination (γ=5°).
Figure 4.18: Half beam intensity profiles of Axicon5° + SPP L=1 (left) and Axicon11° + SPP L=1 (right) at different propagation distances.
Both graphs in Figure 4.18 have beam intensity profiles that are similar to the first order Bessel
beams which were generated in ZEMAX simulations, but without clear rings, due to the poor
sampling rate (low resolution camera). In order to generate the propagation curves for both beams,
the beam’s outer radius was taken for each image at different axial distances at the same intensity
(see Figure 4.19).
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Figure 4.19 shows a comparison between the generated vortex beam from the SPP with L=1 and
the generated first order Bessel beams from the first (SPP with L=1 and Axicon5°) and the second
(SPP with L=1 and Axicon11°) combinations. It can be seen that the vortex beam has a bigger
beam outer radius compared to the first order Bessel beams. Also the generated first order Bessel
beams are almost non-diffractive compared to the generated vortex beam.
When a comparison between the generated first order Bessel beams was done, it was noticed
that the first combination with the Axicon5° has a bigger beam outer radius compared to the
second combination with the Axicon11°. This difference is due to the axicon’s base angel (γ). By
increasing the axicon’s γ the generated Bessel beams have smaller ring sized intensity profiles.
In summary, the first order Bessel beams are generated by two methods, SPP with L=1 and
axicon5°, and SPP with L=1 and axicon11°. The generation of second order Bessel beams is
described in the next section.
4.3.4.2. Spiral Phase Plate with two steps L=2 and Axicons (5°, 11°)
The third combination includes the SPP with L=2 and Axicon5°. The results for this
characterization are shown in Figure 4.20 (left). This figure shows multiple curves, each curve
represents a half beam intensity profile which is generated by the SPP with L=2 and Axicon5° at
different propagation distances. Figure 4.20 (right) represents the forth combination which
includes the SPP with L=2 and Axicon11°. This figure shows multiple curves, each curve
represents a half beam intensity profile generated by SPP with L=2 and axicon11° at different
propagation distances.
Figure 4.20 shows that, at zero axial (z=0) distance the intensity value is equal to zero for each
image, which represents the center of the generated beams. This indicates that the beam has a
ring shaped (vortex) intensity profile. In addition, also small oscillations for each image that
represent the beam outer ring can be seen.
Figure 4.19: First order Bessel beam propagation of Axicon5° + SSP L=1 and Axiocon11° + SPP L=1 in comparison with the first order Bessel beam propagation of an SPP L=1.
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Both graphs in Figure 4.20 have a beam intensity profile that is similar to the second order Bessel
beams which were generated in ZEMAX simulations, but without clear rings, due to the poor
sampling rate (low resolution camera). In order to generate the propagation curves for both beams,
the beam’s outer radius for each image is taken at different axial distances at the same intensity
(see Figure 4.21).
Figure 4.20: Half beam intensity profiles of Axicon5° + SPP L=1 (left) and Axicon11° + SPP L=1 (right) at different propagation distances.
Figure 4.21 shows a comparison between the generated vortex beam from the SPP with L=2, and
the generated second order Bessel beams from the third (SPP with L=2 and Axicon5°) and the
forth (SPP with L=2 and Axicon11°) combinations. As it can be seen, the vortex beam has a bigger
outer beam radius compared to the second order Bessel beams. It can be observed that the
generated second order Bessel beams are almost non-diffractive compared to the generated
vortex beam.
When a comparison between the generated second order Bessel beams is done, it can be noticed
that the third combination which includes the Axicon5° has a bigger outer beam radius compared
to the forth combination which includes the Axicon11°. This difference is due to the axicon’s base
angel (γ). On the one hand when the axicon’s base angel (γ) increases for the same SPP number
of steps (L) a smaller Bessel beam is generated. On the other hand, if the axicon’s base angle is
fixed and the SPP number of steps (L) is increased, a bigger Bessel beam will generated (see
Figure 4.19 and Figure 4.21).
In summary, the second order Bessel beams are generated by two methods, SPP with L=2 and
Axicon5°, and SPP with L=2 and Axicon11°. Therefore, the SPP number of steps (L) determines
the Bessel beam order.
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Helical Axicons
In this section another method is used to generate higher order Bessel beams (J1(x), J2(x)).This
method is represented by using helical Axicons instead of the combination between the SPPs and
the Axicons. The helical axicons contain the same axicon base angels (5°,11°) with step numbers
L=1 and L=2. In the end, four helical axicons are characterized as it mentioned below.
4.3.5.1. Helical Axicons with One Step
In this section, helical axicons with one step are characterized. Figure 4.22 shows multiple curves,
each curve represents a half beam intensity profile generated by helical axicon with one step and
a 5° base angle (left) and an 11° base angle (right) at different propagation distances. As for the
experiments discussed before, MATLAB was used to create the half beam intensity profiles for
the helical axicons. Figure 4.22 shows that at zero axial propagation distance the intensity value
equals zero for each image, which represents the center of the generated beams. This indicates
that the beam has a ring shaped (vortex) intensity profile. It can also be seen that, these generated
beams have a similar intensity profile distribution of the first order Bessel beams (J1(x)), which
were generated in ZEMAX simulations, but without clear rings. In order to fit the generated beams
from the helical axicons with L=1 to the J1(x), a MATLAB script was written.
Figure 4.23 shows the fitting curves for the helical with L=1 to the J1(x). On the left side the half
profile intensity of the helical axicon with 5° base angel at 20 mm is represented by the red curve.
Figure 4.23 (right) shows the half profile intensity of the helical axicon with 11° base angel at
30 mm represented by the red curve. A J1(x) curve fit was applied to both curves which is
represented by the blue curve. This curve fit was applied because its distribution curve is similar
to the helical axicon with L=1 beam intensity profile distribution. The generated Bessel beam (red
curve) is similar to J1(x) fitting (blue curve) but it has a thicker beam core and one outer ring, due
to:
Figure 4.21: Bessel beam propagation of Axicon5° + SSP L=2 and Axiocon11° + SPP L=2 in comparison with the second order Bessel beam propagation of an SPP L=2.
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I. The low sampling rate (low resolution of the camera).
II. The source beam divergence.
III. The manual beam center determination.
Propagating curves for both beams are created along the z-axis, to allow a comparison between
the generated J1(x) using the helical axicons with L=1 and the combination between SPP with L=1
and axicons (5° and 11°). In order to generate the propagation curves for both beams, the outer
beam radius for each image was taken at different axial distances at the same intensity (see Figure
4.24 and Figure 4.25).
Figure 4.22: Half beam intensity profile of helical axicon with L=1 at different propagation distances Υ=5° (left) and Υ=11° (right).
Figure 4.23: Fitting of first order Bessel beams of helical axicons with L=1 Υ=5° (left) and Υ=11° (right).
From both figures the J1(x) which was generated using the helical axicons with L=1 are similar to
the J1(x) generated using the combination between SPP with L=1 and axicons (Υ=5° and Υ=11°).
It can be noticed that J1(x) generated by helical axicons have a smaller outer beam radius,
because in case of the helical axicons one optical component is used for J1(x) generation, but in
case of the combination between the SPP with L=1 and axicons (5° and 11°), two optical
components were used to generate J1(x). Therefore, the misalignment mistakes of optical
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components and beam divergence is bigger when the number of the characterized optical
components increase.
4.3.5.2. Helical Axicons with Two Steps
In this section, helical axicons with two steps (L=2) are characterized. Figure 4.26 shows multiple
half beam intensity profiles generated by helical axicons with two steps and a base angle of 5°
(left) and a base angle of 11° (right) at different propagation distances. It can be seen that at zero
axial (propagation) distance the intensity value is equal to zero for each image, which represents
the center of the generated beams, this indicates that the beam has a ring shaped (vortex) intensity
profile. It can also be seen that, these generated beams have a similar intensity profile distribution
of the second order Bessel beams (J2(x)), which were generated in ZEMAX simulations, but
without clear rings. The fitting of the half beam intensity profiles to the second order Bessel beams
J2(x) was performed in MATLAB.
Figure 4.24: Bessel beam propagation of helical axicon with 5° base angle and one step.
Figure 4.25: Bessel beam propagation of helical axicon with 11° base angle and one step.
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Figure 4.26: Half beam intensity profile of helical axicon with L=2 at different propagation distances Υ=5° (left) and Υ=11° (right).
Figure 4.23 shows the fitting curves for the helical axicons with L=2 to the J2(x)). The result for the
helical axicon with 5° base angle at 20 mm is depicted on the left, the result for the helical axicon
with 11° base angle at 30 mm can be seen on the right. A second order Bessel beam (J2(x)) fitting
was applied to both curves which is represented by the blue curves. This fitting was applied
because its distribution curve is similar to the helical axicon with L=2 beam intensity profile
distribution. The generated Bessel beam (red curve) is similar to J2(x) fitting (blue curve) but it has
a thicker beam core and one outer ring. Propagating curves for both beams were created along
the z-axis to make a comparison between the generated J2(x) from the helical axicons with L=2
and the combination between SPP with L=2 and axicons (5° and 11°). In order to generate the
propagation curves for both beams, the outer beam radius for each image was taken at different
axial distances at the same intensity (see Figure 4.28 and Figure 4.29).
Figure 4.27: Fitting of the second order Bessel beams of helical axicons with L=2 Υ=5° (left) and Υ=11° (right).
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From both figures J2(x) which is generated using the helical axicons with L=2 are similar to the
J2(x) generated using the combination between SPP with L=2 and Axicons (γ=5° and γ=11°). It
can be noticed that J2(x) generated by helical axicons have a smaller outer beam radius, because
for the helical axicons one optical component was used for the J2(x) generation. When two optical
components are used to generate J2(x), misalignment of the components can cause a higher beam
divergence.
Figure 4.28: Bessel beam propagation of a helical axicon with 5° base angle and two steps.
Figure 4.29: Bessel beam propagation of helical axicon with 11° base angle and two steps.
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5. Conclusion
The THz spectrum has unique features compared to microwaves, infrared, and X-rays, such as
the penetration capacity for many non-conductive materials as well as safe and non-distractive
radiation. A long time the THz region was not known as well as other types of the electromagnetic
spectrum due to its limitation in the generation and detection methods. However, in the 1990s
more new methods for the generation and detection of the THz spectrum were developed. Due to
its unique features, THz radiation has a variety of applications: medical THz imaging, defect
detection, material characterization, non-destructive testing, and security scanning.
3D printing is tool free process to build up a model layer by layer. Depending on the circumstances
it can reduces manufacturing time and costs compared to traditional manufacturing. In the recent
years, 3D printing was used widely in different areas, such as medicine, industry, and optics. For
optical components in THz applications the lower cost and the acceptable surfaces finishing
(depend on the 3D printing method) make 3D printing attractive for THz applications.
Many of the polymeric materials are transparent in THz region, which make them perfect for THz
optical component fabrication. Besides the transparency in the THz region there are other factors
that have to be taken in consideration when 3D printing these components: the polymer’s
printability and the resolution of the printing process. THz optics require a certain resolution
depending on the designed wavelength. As THz is with submillimeter wavelength a resolution in
the micro-millimeter range is necessary.
The PolyJet 3D-printing technology is one of the most suitable 3D printing methods for THz optics
because of its 16 µm layer resolution. Vero photo polymers are acrylate based polymers that are
3D printable with PolyJet technology and transparent in the THz region. THz-time domain
spectroscopy (TDS) was used to characterize eight Vero polymers at 0.1-1 THz frequency at room
temperature. Based on the results the material VeroClear was chosen to fabricate the THz optical
components at 0.14 THz, due to its low absorption coefficient (1.1 cm-1) at 0.14 THz frequency,
and its transparency in the visible light region.
Ten optical components were 3D printed with VeroClear: two simple lenses (bi-convex and plano-
convex lenses), two axicons with 5° and 11° base angels (γ), two spiral phase plates (SPPs) with
one and two steps (L), and four helical axicons with 5° and 11° base angles (γ) and with one and
two steps (L).
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Before 3D printing the components, a ray tracing simulation was done to optimize the optical
components’ designs using the software ZEMAX. The simulation was done at 0.14 THz frequency
(2.14 mm wavelength). The optimum design parameters that were generated from these
simulations are: two simple lenses with 50 mm diameter, one is bi-convex with 84.8 mm ƒ and the
other is plano-convex with 104 mm ƒ, two axicons with 25 mm diameter, one with γ=5° and
ƒ=56 mm, and the other axicon with γ=11° and ƒ=23.5 mm, two SPPs with 25 mm diameter, one
with L=1 and the other with L=2, and four helical axicons with 25 mm diameter, the first two with
L=1 and different γ (5°, 11°), the second two with L=2 and different γ (5°, 11°).
The 3D printed optical components were characterized using a THz imaging system consisting of
a Sub-THz Gaussian source at 0.14 THz frequency and a 100x100x55 mm³ Detector (Camera)
with 48x48 mm² sensor size. From the optical component’s characterization, four types of beams
were generated using the 3D printed optical components:
I. Gaussian beams were generated using 3D printed VeroClear simple lenses.
II. Vortex beams were generated using 3D printed VeroClear SPPs with L=1 and L=2.
III. Zero order Bessel beams (J0(x)) were generated using 3D printed VeroClear axicons with
γ= 5° and γ=11°.
IV. Higher order Bessel beams (J1(x) and J2(x)) were generated using 3D printed VeroClear
helical axicons, and a combination between the SPPs (L=1, L=2) and the axicons (γ= 5°,
γ=11°).
In this work the focus was on Bessel beams generation due to its non-diffractive property, self-
healing ability, and line focus generation (axial resolution) compared to Gaussian beams.
The experimental, theoretical, and simulated focus point of the simple lenses was almost the same
in each case. However, for the axicons the experimental focus was different from the simulated
focus with approximately 6 mm, due to the beam source divergence and the optical alignment
errors during the characterization process.
The fitting curves for the experimentally generated Gaussian beams aligned well to the theoretical
curve. For the experimentally generated Bessel beams (J0(x), J1(x), and J2(x)), the fitting curves
were similar to the theoretical curve with some differences in the amplitude, beam core size, and
the number of Bessel beams outer rings. The differences are likely to be due to beam source
divergence, optical alignment errors during the characterization process, and a poor sampling rate
(detector with low resolution). The experimentally generated Bessel beams are non-diffractive
compared to the experimentally generated vortex beams, but at the same time, these Bessel
beams suffer from small divergence compared to the theoretical Bessel beams, due to the beam
source divergence.
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Table 5.1 shows a brief summary of the main steps that were done in order to fabricate the THz
optical components to generate Bessel beams with different orders.
Table 5.1: The simulation, characterization, and the fitting curve results of selected THz optical components with 11° base angle to generate different order Bessel beams.
Axicon11° Helical Axicon11°, L=1 Helical Axicon11°, L=2
Lenses
simulations
Lenses
characterization
Fitting curves
In summary the fabrication of THz optical components using 3D printing technology is desirable
for THz applications, due to the lower production costs, the acceptable surface finishing, the ability
to fabricate customized THz optical component for certain wavelengths, and the unavailability of
THz commercial optical components.
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7. List of Tables
Table 3.1: ZEMAX simulation of a plano-convex lens with 100 mm diameter. ............................32
Table 3.2: ZEMAX simulation of a bi-convex and a plano-convex lens with 50 mm diameter. ....33
Table 3.3: Axicon5° and Axicon11° simulation by ZEMAX at 2.14 mm wavelength. ...................35
Table 3.4: Spiral Phase Plate (L=1, L=2) simulation by ZEMAX at 2.14 mm wave length. .........36
Table 3.5: Helical axicons (L=1) simulation by ZEMAX at 2.14 mm wave length. .......................37
Table 3.6: Helical axicons (L=2) simulation by ZEMAX at 2.14 mm wave length. .......................38
Table 4.1: Refractive indices and absorption coefficients for Vero photo polymers at 0.14 THz. 45
Table 5.1: The simulation, characterization, and the fitting curve results of selected THz optical components with 11° base angle to generate different order Bessel beams. .............................64
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8. List of Figures
Figure 1.1: Work flow of 3D printed THz optical components fabrication. .................................... 7
Figure 2.1: Electromagnetic spectrum with the THz region marked in red [8]. ............................. 9
Figure 2.2: Terahertz imaging applications [3] [45] [44]. .............................................................10
Figure 2.3: Schematics of Terahertz time domain system [12]. ..................................................12
Figure 2.4: Chromatic aberration [50]. ........................................................................................14
Figure 2.5: Spherical aberration [51]. .........................................................................................14
Figure 2.6: Converging lens (bi-convex) [15]. .............................................................................15
Figure 2.7: Gaussian beam intensity propagation [49]. ..............................................................16
Figure 2.8: Zero order Bessel beam generated by an axicon lens [52]. ......................................17
Figure 2.9: The half intensity profile of Bessel functions [53]. .....................................................18
Figure 2.10: Higher order Bessel beam Intensity Profile [29]. .....................................................18
Figure 2.11: Spiral Phase Plate [55]. ..........................................................................................19
Figure 2.12: SPP with L= 0, 1, and 2 steps clockwise direction [54]. ..........................................20
Figure 2.13: Generation of higher order Bessel beams using axicons [30]. ................................21
Figure 2.14: Generation of higher order Bessel beam using helical axicon [34]. ........................22
Figure 2.15: Schematics of a 3D printing process [45]. ..............................................................23
Figure 2.16: Fused deposition modeling process [27]. ...............................................................23
Figure 2.17: PolyJet process [47]. ..............................................................................................24
Figure 2.18: Selective laser sintering process [48]. ....................................................................24
Figure 3.1: Investigated samples of Vero photopolymers. ..........................................................29
Figure 3.2: Tricyclodecane Dimethanol Diacrylate. ....................................................................30
Figure 3.3: Exo-1, 7,7trimethylbicyclo [2.2.1] hept2-yl acrylate. ..................................................30
Figure 3.4: THz-time-domain-spectroscopy (TDS) setup. ..........................................................30
Figure 3.5: TeraSense source (left) and TeraSense detector (right). ..........................................39
Figure 3.6: THz imaging system setup. ......................................................................................40
Figure 3.7: Gaussian beam Image analyzed by ImageJ a) 0° and b) 45°. ..................................41
Figure 3.8: Bessel beam image analyzed by MATLAB ...............................................................41
Figure 4.1: THz-Time domain spectrum of Vero photo polymers................................................42
Figure 4.2: Refractive index of Vero photo polymers at frequency range of 0.1-1.1 THz. ...........43
Figure 4.3: Absorption coefficient of Vero photo polymers at frequency range of 0.1-1.1 THz. ...43
Figure 4.4: Refractive index of Vero photo polymers at frequency range of 0.1-0.26 THz. .........44
Figure 4.5: Absorption coefficient of Vero photo polymers at frequency range of 0.1-0.26 THz. .44
Figure 4.6: OCT image of a plano-convex lens. .........................................................................46
Figure 4.7: Bi-convex beam profile at different axial distances ...................................................47
Figure 4.8: Gaussian beam propagation of an unpolished and polished bi-convex lens. ............48
Figure 4.9: Gaussian beam propagation of bi-convex lens and polished bi-convex lens. ...........48
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Figure 4.10: Plano-convex beam profile at different propagation distances. ...............................49
Figure 4.11: Gaussian beam propagation of plano-convex lens. ................................................49
Figure 4.12: Fitting of Gaussian beam for a plano-convex lens. .................................................50
Figure 4.13: Half beam intensity profile of Axicon5° at different axial distances (left) and fitting of the zero order Bessel beam for Axicon5° (right). ........................................................................51
Figure 4.14: Half beam intensity profile of Axicon11° at different axial distances (left) and fitting of the zero order Bessel beam for Axicon11° (right). ......................................................................51
Figure 4.15: Zero order Bessel beam propagation of Axicon5° and Axicon11°. .........................52
Figure 4.16: Half beam intensity profile of SPP (L=1) (left) and SPP (L=2) (right) at different propagation distances. ...............................................................................................................53
Figure 4.17: Vortex Beam propagation of spiral phase plates with L=1 and L=2. .......................53
Figure 4.18: Half beam intensity profiles of Axicon5° + SPP L=1 (left) and Axicon11° + SPP L=1 (right) at different propagation distances. ...................................................................................54
Figure 4.19: First order Bessel beam propagation of Axicon5° + SSP L=1 and Axiocon11° + SPP L=1 in comparison with the first order Bessel beam propagation of an SPP L=1. .......................55
Figure 4.20: Half beam intensity profiles of Axicon5° + SPP L=1 (left) and Axicon11° + SPP L=1 (right) at different propagation distances. ...................................................................................56
Figure 4.21: Bessel beam propagation of Axicon5° + SSP L=2 and Axiocon11° + SPP L=2 in comparison with the second order Bessel beam propagation of an SPP L=2. ............................57
Figure 4.22: Half beam intensity profile of helical axicon with L=1 at different propagation distances Υ=5° (left) and Υ=11° (right). ......................................................................................................58
Figure 4.23: Fitting of first order Bessel beams of helical axicons with L=1 Υ=5° (left) and Υ=11° (right). ........................................................................................................................................58
Figure 4.24: Bessel beam propagation of helical axicon with 5° base angle and one step. ........59
Figure 4.25: Bessel beam propagation of helical axicon with 11° base angle and one step. ......59
Figure 4.26: Half beam intensity profile of helical axicon with L=2 at different propagation distances Υ=5° (left) and Υ=11° (right). ......................................................................................................60
Figure 4.27: Fitting of the second order Bessel beams of helical axicons with L=2 Υ=5° (left) and Υ=11° (right). .............................................................................................................................60
Figure 4.28: Bessel beam propagation of a helical axicon with 5° base angle and two steps. ....61
Figure 4.29: Bessel beam propagation of helical axicon with 11° base angle and two steps. .....61
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