LASER BEAM SHAPING OPTICAL SYSTEM DESIGN METHODS AND THEIR
APPLICATION IN EDGE-EMITTING SEMICONDUCTOR
LASER-BASED LIDAR SYSTEMS
Except where reference is made to the work of others, the work described in this
dissertation is my own or was done in collaboration with my advisory committee. This
dissertation does not include proprietary or classified information.
Mert Serkan
Certification of Approval:
Thaddeus A. Roppel Hulya Kirkici, Chair
Associate Professor Associate Professor
Electrical and Computer Engineering Electrical and Computer Engineering
John Y. Hung Lloyd S. Riggs
Professor Professor
Electrical and Computer Engineering Electrical and Computer Engineering
George T. Flowers
Interim Dean
Graduate School
LASER BEAM SHAPING OPTICAL SYSTEM DESIGN METHODS AND THEIR
APPLICATION IN EDGE-EMITTING SEMICONDUCTOR
LASER-BASED LIDAR SYSTEMS
Mert Serkan
A Dissertation
Submitted to
the Graduate Faculty of
Auburn University
in Partial Fulfillment of the
Requirements for the
Degree of
Doctor of Philosophy
Auburn, Alabama
December 17, 2007
iii
LASER BEAM SHAPING OPTICAL SYSTEM DESIGN METHODS AND THEIR
APPLICATION IN EDGE-EMITTING SEMICONDUCTOR
LASER-BASED LIDAR SYSTEMS
Mert Serkan
Permission is granted to Auburn University to make copies of this dissertation at its
discretion, upon request of individuals or institutions and at their expense. The author
reserves all publication rights.
Signature of Author
Date of Graduation
iv
VITA
Mert Serkan, son of Varol Serkan and Eyyube Sevgi Serkan, was born on February
26, 1979 in Istanbul, Turkey. He graduated from Huseyin Avni Sozen Anatolian High
School in 1996. He received the Bachelor of Science degree in Electronics and
Telecommunications Engineering from Istanbul Technical University, Turkey in 2000.
After graduation, he worked as a Research and Development Engineer in Nortel
Networks/Netas for about a year. He started his M. Sc. degree in Satellite
Communications and Remote Sensing Graduate Program at Istanbul Technical
University in 2001. He worked as a Research Assistant in Satellite Ground Receiving
Station at Istanbul Technical University between 2001 and 2003. He earned his M. Sc.
degree in 2003 and started his Ph. D. degree at Auburn University, AL in 2003. He
worked as a Teaching/Research Assistant at Auburn University between 2003 and 2007.
v
DISSERTATION ABSTRACT
LASER BEAM SHAPING OPTICAL SYSTEM DESIGN METHODS AND THEIR
APPLICATION IN EDGE-EMITTING SEMICONDUCTOR
LASER-BASED LIDAR SYSTEMS
Mert Serkan
Doctor of Philosophy, December 17, 2007
(M. Sc. Istanbul Technical University, Turkey, 2003)
(B. Sc. Istanbul Technical University, Turkey, 2000)
148 Typed Pages
Directed by Hulya Kirkici
LIDAR (Light Detection And Ranging) systems are employed for numerous
applications such as remote sensing, military applications, optical data storage, display
technology, and material processing. Furthermore, they are superior to other active
remote sensing tools such as RADAR systems, considering their higher accuracy and
more precise resolution due to their much shorter wavelengths and narrower beamwidth.
Several types of lasers can be utilized as the radiation source of several LIDAR systems.
Semiconductor laser-based LIDAR systems have several advantages such as low cost,
compactness, broad range of wavelengths, and high PRFs (Pulse Repetition Frequency).
vi
However, semiconductor lasers have different origins and angles of divergence in the two
transverse directions, resulting in the inherent astigmatism and elliptical beam shape.
Specifically, elliptical beam shape is not desirable for several laser-based applications
including LIDAR systems specifically designed to operate in the far-field region. In this
dissertation, two mirror-based and two lens-based beam shapers are designed to
circularize, collimate, and expand an edge-emitting semiconductor laser beam to a
desired beam diameter for possible application in LIDAR systems. Additionally, most
laser beams including semiconductor laser beams have Gaussian irradiance distribution.
For applications that require uniform illumination of an extended target area, Gaussian
irradiance distribution is undesirable. Therefore, a specific beam shaper is designed to
transform the irradiance distribution from Gaussian to uniform in addition to
circularizing, collimating, and expanding the semiconductor laser beam.
For the design of beam shapers, aperture sizes of the surfaces are preset for desired
power transmission and allowed diffraction level, surface parameters of the optical
components and the distances between these surfaces are determined. Design equations
specific to these beam shaping optical systems are determined using geometrical optics
principles. It is also verified that the geometrical optics is a valid approximation to the
physical optics for each designed system. The design equations are derived analytically
and MATLAB codes are developed to solve for the unknown quantities of the optical
systems. ZEMAX and MATLAB software programs are used to analyze the designed
systems with particular design examples and the results are presented.
vii
ACKNOWLEDGMENTS
First and foremost the author would like to thank Dr. Hulya Kirkici for her guidance
and support. He would like to extend his thanks to Dr. Thaddeus A. Roppel, Dr. John Y.
Hung, Dr. Lloyd S. Riggs, and Dr. Anwar Ahmed for their valuable suggestions and
constructive comments on his dissertation. He thanks his parents Varol and Eyyube
Sevgi and his brother Tevfik Murat for always pushing him in the right direction and for
all the years of encouragement. He thanks Yusuf Bahadir Anafarta for his guidance and
support. He thanks Selda Taskin for giving him the support, help, and encouragement he
needed. His thanks also go to all the professors, classmates, and friends in Auburn.
viii
Style manual or journal used IEEE Transactions on Geoscience and Remote Sensing
(together with the style known as ?auphd?).
Computer software used MS Word.
ix
TABLE OF CONTENTS
LIST OF TABLES............................................................................................................xii
LIST OF FIGURES..........................................................................................................xiv
CHAPTER 1 INTRODUCTION ........................................................................................1
CHAPTER 2 RESEARCH OBJECTIVES AND DISSERTATION OUTLINE ...............6
CHAPTER 3 BACKGROUND ........................................................................................11
3.1 Basic Principles of Laser Beams and Their Propagation Characteristics............11
3.2 Beam Shaping Optical System Design in the General Sense ..............................20
3.3 Literature Review.................................................................................................23
3.3.1 LIDAR Applications..................................................................................23
3.3.2 LIDAR System Design..............................................................................30
CHAPTER 4 METHODOLOGY......................................................................................34
4.1 Geometrical Ray Optics Approximation to Physical Optics Propagation...........34
4.2 Aspheric Surfaces and Their Usage in Beam Shaping Optical Systems .............36
4.3 Implementation of ZEMAX in Design and Analysis...........................................38
4.4.1 Overview and Design Approach................................................................41
4.4.2 First Design Method ..................................................................................43
4.4.3 Second Design Method..............................................................................53
4.5 Aspherical Lens-Based Optical System Design for Circularization, Collimation,
and Expansion of Elliptical Laser Beams..................................................................57
x
4.5.1 Overview and Design Approach................................................................57
4.5.2 Two-Lens Design Method.........................................................................58
4.5.3 Single-Lens Design Method ......................................................................66
4.6 Reshaping an Elliptically Diverging Gaussian Laser Beam into a Circular,
Collimated, and Expanded Uniform Beam with Aspherical Lens Design ................71
4.6.1 Overview and Design Approach................................................................71
4.6.2 Design Method...........................................................................................73
CHAPTER 5 DESIGN EXAMPLES AND RESULTS....................................................82
5.1 Off-Axis Mirror-Based Optical System Design for Circularization, Collimation,
and Expansion of Elliptical Laser Beams..................................................................82
5.1.1 Calculated Parameters of the Examples for the First Design Method.......82
5.1.2 Calculated Parameters of the Example for the Second Design Method....84
5.1.3 Results........................................................................................................84
5.2 Aspherical Lens-Based Optical System Design for Circularization, Collimation,
and Expansion of Elliptical Laser Beams..................................................................89
5.2.1 Calculated Parameters of the Example for the Two-Lens Design Method89
5.2.2 Calculated Parameters of the Example for the Single-Lens Design Method
............................................................................................................................90
5.2.3 Results........................................................................................................91
5.3 Reshaping an Elliptically Diverging Gaussian Laser Beam into a Circular,
Collimated, and Expanded Uniform Beam with Aspherical Lens Design ................99
5.3.1 Calculated Parameters of the Design Example..........................................99
5.3.2 Results......................................................................................................100
xi
CHAPTER 6 CONCLUSIONS.......................................................................................106
BIBLIOGRAPHY...........................................................................................................112
APPENDIX.....................................................................................................................119
xii
LIST OF TABLES
Table 4.1. General mirror equations for the first mirror-based design method ................48
Table 4.2. Global coordinate equations of the mirrors for the first mirror-based design
method...............................................................................................................................51
Table 4.3. Slopes of the rays for the first mirror-based design method............................52
Table 4.4. General mirror and ray equations for the second mirror-based design method...
...........................................................................................................................................55
Table 4.5. Global coordinate equations of the mirrors for the second mirror-based design
method...............................................................................................................................56
Table 5.1. Parameters of the first example designed with the first mirror-based design
method...............................................................................................................................83
Table 5.2. Parameters of the second example designed with the first mirror-based design
method...............................................................................................................................84
Table 5.3. Parameters of the example designed with the second mirror-based design
method...............................................................................................................................84
Table 5.4. ? parameter calculations of each studied mirror-based design example,
separately for perpendicular and parallel transverse directions........................................85
Table 5.5. Analytical calculations of the input and output beam for both o10 - o30 and
o12 - o46 beam examples..................................................................................................86
Table 5.6. Output beam results of all studied mirror-based design examples ..................87
Table 5.7. Two-lens system design example parameters..................................................90
xiii
Table 5.8. Single-lens system design example parameters...............................................90
Table 5.9. ? parameter calculations for the first lens-based design method example
separately for perpendicular and parallel transverse directions........................................92
Table 5.10. ? parameter calculations for the second lens-based design method example
separately for perpendicular and parallel transverse directions........................................92
Table 5.11. ? parameter calculations of the studied design example separately for
perpendicular and parallel transverse directions...............................................................99
Table 5.12. Input variables and values used in the design example..................................99
Table 5.13. Calculated first lens front surface and second lens back surface function
parameters .......................................................................................................................100
xiv
LIST OF FIGURES
Figure 1.1(a) Free-space radiation pattern of a semiconductor laser with an elliptical
beam profile.........................................................................................................................1
Figure 1.1(b) Illustration of propagation of a semiconductor laser beam...........................2
Figure 1.1(c) Semiconductor laser beam in two transverse directions (x ? perpendicular, y
? parallel) as a function of propagation distance (z - optical axis) .....................................2
Figure 3.1 The characteristic parameters of a Gaussian beam..........................................18
Figure 3.2. Transverse modes for lasers with circular symmetry.....................................19
Figure 3.3. Transverse modes for lasers with rectangular symmetry................................19
Figure 3.4. Beam profile plot of a laser beam with o46 divergence angle in the
perpendicular transverse direction, o12 divergence angle in the parallel transverse
direction, and neglected astigmatism................................................................................20
Figure 3.5. Block diagram of a coherent edge-emitting semiconductor laser based
Doppler LIDAR system design example ..........................................................................30
Figure 4.1. Illustration of the definition of surface sag.....................................................37
Figure 4.2. Illustration of the general design approach.....................................................42
Figure 4.3. x-z plane illustration of the first design method.............................................44
Figure 4.4. y-z plane illustration of the first design method.............................................44
Figure 4.5. Illustration of the hyperbola function .............................................................45
Figure 4.6. Optical path of the ?chief ray? of the beam....................................................46
Figure 4.7. Illustration of the parabola function................................................................47
Figure 4.8. Illustration of the first hyperbolic mirror in its local coordinates...................48
xv
Figure 4.9. Illustration of the second hyperbolic mirror in its local coordinates..............49
Figure 4.10. Illustration of the parabolic mirror in its local coordinates ..........................50
Figure 4.11. x-z plane illustration of the second design method.......................................54
Figure 4.12. y-z plane illustration of the second design method.......................................54
Figure 4.13. A schematic diagram of two-lens system.....................................................59
Figure 4.14. A schematic representation of the surfaces in the x-z plane.........................60
Figure 4.15. A schematic representation of the surfaces in the y-z plane.........................64
Figure 4.16. A schematic diagram of single lens system..................................................67
Figure 4.17. A schematic representation of the surfaces in the x-z plane.........................68
Figure 4.18. A schematic representation of the surfaces in the y-z plane.........................68
Figure 4.19. Illustration of the general design approach in the x-z plane.........................74
Figure 4.20. Illustration of the back surface of the first lens, together with its tangent and
normal lines and the incident and refracted angle definitions...........................................80
Figure 5.1. MATLAB plot demonstrating the representative beam propagation in the y-z
plane for the first design method first example.................................................................83
Figure 5.2. First mirror-based design method o10 - o30 beam example for the stated
image sizes (a) input (4 ?m x 10 ?m) (b) output (40 cm x 40 cm) (c) output at 0.5 km
distance (40 cm x 40 cm) (d) output at 1 km distance (40 cm x 40 cm) ......................88
Figure 5.3. First mirror-based design method o12 - o46 beam example for the stated
image sizes (a) input (4 ?m x 10 ?m) (b) output (40 cm x 40 cm) (c) output at 0.5 km
distance (40 cm x 40 cm) (d) output at 1 km distance (40 cm x 40 cm) ......................88
Figure 5.4. Second mirror-based design method o12 - o46 beam example for the stated
image sizes (a) input (4 ?m x 10 ?m) (b) output (40 cm x 40 cm) (c) output at 0.5 km
distance (40 cm x 40 cm) (d) output at 1 km distance (40 cm x 40 cm) ......................88
Figure 5.5. (a) Ray tracing results for the first lens-based design method in the x-z plane
...........................................................................................................................................93
xvi
Figure 5.5. (b) Ray tracing results for the first lens-based design method in the y-z plane
...........................................................................................................................................93
Figure 5.6. (a) Ray tracing results for the second lens-based design method in the x-z
plane..................................................................................................................................94
Figure 5.6. (b) Ray tracing results for the second lens-based design method in the y-z
plane..................................................................................................................................94
Figure 5.7. (a) Ray tracing results obtained with ZEMAX for the two-lens system in the
x-z plane............................................................................................................................95
Figure 5.7. (b) Ray tracing results obtained with ZEMAX for the two-lens system in the
y-z plane............................................................................................................................95
Figure 5.8. (a) Ray tracing results obtained with ZEMAX for the single-lens system in the
x-z plane............................................................................................................................95
Figure 5.8. (b) Ray tracing results obtained with ZEMAX for the single-lens system in
the y-z plane......................................................................................................................96
Figure 5.9. Two-lens system ZEMAX plots (a) input beam (window size: 4 ?m x 10
?m) (b) output beam at the system exit plane (window size: 8 cm x 8 cm) (c) output
beam at 1 km distance (window size: 8 cm x 8 cm).......................................................98
Figure 5.10. Single-lens system ZEMAX plots (a) input beam (window size: 4 ?m x 10
?m) (b) output beam at the system exit plane (window size: 8 cm x 8 cm) (c) output
beam at 1 km distance (window size: 8 cm x 8 cm).......................................................98
Figure 5.11. (a) Ray tracing results in x-z plane.............................................................101
Figure 5.11. (b) Ray tracing results in the y-z plane.......................................................102
Figure 5.12. (a) Irradiance distribution at the laser output plane....................................102
Figure 5.12. (b) Irradiance distribution at the input reference plane...............................103
Figure 5.12. (c) Irradiance distribution at the output reference plane.............................103
Figure 5.13. Irradiance distributions at five different locations in between the two lenses
in the parallel and the perpendicular directions ..............................................................104
Figure A.1. Lens Data Editor display..............................................................................120
Figure A.2. Surface Properties dialogue box display......................................................122
xvii
Figure A.3. 3D Layout Diagram Settings dialogue box display.....................................125
Figure A.4. Spot Diagram Settings dialogue box display...............................................126
Figure A.5. FFT PSF Settings dialogue box display.......................................................127
Figure A.6. Physical Optics Propagation Settings dialogue box display........................130
1
CHAPTER 1
INTRODUCTION
Semiconductor lasers have advantages in some applications over other lasers such as
CO2 and Nd-YAG due to low cost, compactness, electronic compatibility, broad range of
wavelengths, and high Pulse Repetition Frequency (PRF) values. However, they have
different origins and angles of divergence in the two transverse directions, resulting in the
inherent astigmatism and elliptical beam shape, as illustrated in Figure 1.1 (x ?
perpendicular transverse direction, y ? parallel transverse direction). For some
applications, this elliptical beam is acceptable. However, when a minimum beam spread
is needed in both directions, it becomes necessary to reshape the beam into a circular
form. In addition, some applications require a circular laser beam due to the usage of
various optical components such as fiber optics.
Figure 1.1(a) Free-space radiation pattern of a semiconductor laser with an elliptical
beam profile.
2
Figure 1.1(b) Illustration of propagation of a semiconductor laser beam.
Figure 1.1(c) Semiconductor laser beam in two transverse directions (x ? perpendicular, y
? parallel) as a function of propagation distance (z - optical axis).
3
Lasers are used in several scientific and commercial applications. One application
that the author is concentrated on in this study is Light Detection And Ranging (LIDAR)
systems. LIDARs can be employed for several applications such as remote sensing,
military applications, optical data storage, display technology, and material processing.
Nevertheless, they are superior to several other remote sensing tools such as radar
systems specifically in atmospheric studies since LIDARs operate at much shorter
wavelengths. Hence, they are capable of higher accuracy and more precise resolution
than radars perform within the microwave range of the electromagnetic spectrum.
On the other hand, most LIDAR systems are not easily transportable for commercial
purposes due to large-scale lasers such as CO2 or Nd-YAG lasers [1]. These LIDARs
that use lasers with kW and MW output power levels are preferred over LIDARS that
use semiconductor lasers. High output power levels are essential especially for
atmospheric studies due to the intense background noise from sunlight and multiple
atmospheric scattering. Recent scientific and technological advances in pulsed
semiconductor lasers increase the usage of them more efficiently in LIDARs for other
appications. Semiconductor lasers have higher output power levels and higher PRF
values nowadays. Semiconductor laser-based LIDARs have also improved light
emission efficiency, electronic control compatibility, and reduced system size and cost
[2]. However, highly divergent elliptical beam profile of semiconductor lasers is not
suitable for LIDAR systems specifically utilized in long-range applications such as
atmospheric studies.
For LIDAR systems, the laser beam must be expanded to reduce the beam
divergence caused by diffraction. By expansion, the author refers to increasing the beam
4
waist radius. Expansion also increases the size of the beam spot on the target plane.
Thus, a larger area can be scanned within a shorter time.
For most LIDAR systems a circular beam profile is required. This is due to the
usage of optical components such as fiber optics. In addition to this, LIDARs might be
used in the far-field region [3]. If the laser source has a non-circular beam shape, at the
far field the difference between the radii of the beam in the two directions causes highly
elliptical beam shape after long-range propagation. Therefore, circularization is essential
for such applications.
A minimum beam spread is desired for LIDAR systems that require long-range
propagation. For such applications, collimating the beam limits the beam spread. By
collimation, the author refers to making the beam parallel in the diffraction limit. A
collimated beam also allows the receiver to have low Field-Of-View (FOV) value, which
reduces background noise and susceptibility to multiple scattering. To summarize, a
beam shaper must be designed to address three aspects before transmitting the beam long
distances:
? expansion,
? circularization,
? collimation.
Astigmatism is important for the optical systems designed to focus the laser beam to
a small spot [4- 6]. On the contrary, for applications such as most LIDAR systems where
the beam is expanded to cm to m levels and collimated, astigmatism in ?m levels can be
neglected.
5
Several other applications where lasers are used can be listed as display technology,
lithography, laser printing, and material processing. Most laser output beams possess
Gaussian irradiance distribution and this is undesirable for listed applications since such
applications require uniform illumination of an extended target area [7]. Therefore, a
beam shaper is required to transform Gaussian irradiance distribution of such beams to
uniform distribution. Many studies in the literature cannot be directly applied to highly
divergent edge-emitting semiconductor lasers. This is due to the assumption of these
studies that the laser beam is already collimated. Moreover, most of them assume that
the beam has a circular profile. Therefore, an optical system designed to transform
irradiance distribution of such lasers in addition to circularizing, collimating and
expanding it to a desired output beam diameter is also of interest in this research.
6
CHAPTER 2
RESEARCH OBJECTIVES AND DISSERTATION OUTLINE
As stated in the previous chapter, edge-emitting semiconductor lasers have
advantages over other lasers such as CO2 and Nd-YAG such as low cost, compactness,
electronic compatibility, broad range of wavelengths, and high Pulse Repetition
Frequency (PRF) values in LIDAR systems. However, CO2 and Nd-YAG are still
preferred in LIDAR systems especially employed in atmospheric studies due to the high
power levels. With recent advances in semiconductor laser technology, moderately high
output power levels and higher PRFs are available making such lasers feasible for
LIDAR systems. In addition, advances in detector technology improve the receiver?s
ability to process the weak return signal. This allows a LIDAR system to operate with
the lower power level of pulsed semiconductor lasers.
LIDAR systems utilizing semiconductor lasers are much more compact and easier to
transport. However, elliptical beam profile of such lasers must be corrected in long-range
applications such as atmospheric studies. An appropriate optical beam shaper must be
designed for this purpose. On the other hand, most of the research on beam shaping
optics topic is based on other applications such as laser printing and optical data storage.
Indeed, these studies are not directly applicable to semiconductor laser-based LIDAR
systems. Thus, there is ample room for an academic study for the understanding of this
problem and designing beam shapers for semiconductor laser-based LIDAR systems. As
7
a result, the main contribution of this dissertation is to design the optics required to
circularize, collimate, and expand the elliptical semiconductor laser beam, used as the
radiation source of a LIDAR sensor, and transmit it towards the target. Atmospheric
studies are chosen as the main application type for the design examples throughout the
dissertation because beam shaping is a fundamental and inevitable step for edge-emitting
semiconductor laser-based LIDAR systems utilized for such applications. On the other
hand, several other applications such as material processing, cutting, medical procedures,
and optical data processing utilize laser beams, and it is also usually required to reshape
the beam for these applications. For these applications, a beam shaper is required to
transform Gaussian irradiance distribution of such beams to uniform distribution, as
stated in the previous chapter. This type of beam shaper is different than the one
designed for LIDAR system, which does not alter the Gaussian beam profile. Both types
of beam shapers are discussed and such beam shapers are developed by the author. For
some other industrial applications such as CD players and robotics, beam shaping may
not be that much of importance due to short distance propagation and low-power level
requirements. Therefore, such applications are out of scope of this research.
As mentioned earlier, since semiconductor laser-based LIDAR systems are still not
very common, there are not many optical design studies about beam shapers specific to
these systems either. Therefore, the author intends to create a scientific resource for
feasibility of such LIDAR systems and designing necessary optics for these systems.
Two different off-axis mirror systems and two different lens systems are developed to
circularize, collimate, and expand the elliptical beam of edge-emitting semiconductor
laser-based LIDAR systems are developed. In addition to these systems, a separate lens-
8
based beam shaper is developed to transform the irradiance distribution of such lasers
from Gaussian to uniform in addition to circularizing, collimating, and expanding the
beam. Even though this beam shaper is not specifically designed for LIDAR systems, it
can still be employed in a LIDAR system where illuminating the receiver detector (or
detector array) with a uniform beam is desired, similar to the system discussed in [8].
The author presents the principles and tools used to design these systems explicitly in
order to create a guideline for the ones working with similar problems or further
improving the proposed designs. The equations specific to these systems are derived
analytically and computer package programs such as ZEMAX and MATLAB are utilized
to simulate the optical designs. These equations are derived based on optics principles
and geometrical relationships. These analytical equations are placed in equation systems
and solved with MATLAB codes to determine the parameters of the optical components.
The author expects that these beam-shaping systems can be used while manufacturing an
edge-emitting semiconductor laser-based LIDAR system and be easily modified to
address other issues related to laser beam shaping. These systems can be commercialized
as the next step since they will be much more compact and lighter than the conventional
LIDARs due to the advantages of semiconductor lasers. The author also expects that this
dissertation and the analysis methods developed will be a valuable resource for the
scientific community interested in LIDAR systems and/or beam shaper systems.
The background information for this dissertation together with literature survey of
beam shaping methods and LIDAR applications is presented in Chapter 3. Basic
principles of lasers with an emphasis on semiconductor lasers are introduced in Section
3.1. The details of laser beam physics are not covered here because this research focuses
9
on the beam shaping methods of edge-emitting semiconductor lasers. The irradiance
distribution of such lasers is assumed to have a Gaussian profile. Therefore, Gaussian
beam analysis is also presented in this section. The design methods discussed in this
dissertation are based on geometrical ray optics approximating the physical optics
propagation. As a result of this, physical optics propagation principles are not covered in
this section.
Beam shaping might be either circularization, collimation, and expansion of a
divergent elliptical beam or transformation of the irradiance distribution to another
distribution type such as Gaussian distribution to uniform distribution. Section 3.2
briefly discusses the beam shaping in both senses and provides examples from the
literature.
The main application type that this research focuses on is remote sensing using
LIDAR systems. Therefore, Section 3.3 is devoted to discussion of such systems,
providing an overview and literature review for the LIDAR systems and their
applications, including a brief discussion of edge-emitting semiconductor laser based
LIDAR system design. An example of a coherent LIDAR system design is also
presented in this section.
Chapter 4 presents the general methodology for the design and analysis of the beam
shaper systems. Section 4.1 discusses the geometrical ray optics approximation to
physical optics propagation. The optical surfaces of the designed systems are mostly
designed aspheric surfaces. Therefore, aspheric surfaces and conic sections are briefly
discussed in Section 4.2.
10
ZEMAX and MATLAB software programs are used both in design and analysis of
the optical systems discussed in this dissertation. ZEMAX is a software tool for optical
design [9, 10]. Section 4.3 discusses how ZEMAX is involved in the design and analysis
steps of the optical systems.
As mentioned earlier, two mirror-based systems are developed: an off-axis
hyperbolic/parabolic mirror system and an off-axis parabolic mirror system. Both design
methods consist of three aspheric mirrors. The design approach and the equations
specific to these mirror-based systems are presented in Section 4.4. Two design methods
using aspheric lenses are also designed. The first system consists of two aspheric lenses;
whereas, the second system has a single aspheric lens. The design approach and the
equations specific to the lens-based beam-shaper systems are presented in Section 4.5.
Section 4.6 presents the design approach and the equations of the beam shaping system of
two aspheric lenses developed to circularize, collimate, and expand a semiconductor laser
beam and to transform its irradiance distribution from Gaussian to a uniform distribution.
Chapter 5 provides design examples to the systems presented in Chapter 4. Results
of these examples are also presented and discussed. The sixth chapter provides a
summary to the design methods presented, concludes this dissertation, and discusses the
issues left for future study.
11
CHAPTER 3
BACKGROUND
This chapter presents the background information for this dissertation together with
literature survey of beam shaping methods and LIDAR applications. Basic principles of
laser beams and their propagation characteristics are discussed briefly in Section 3.1.
Section 3.2 briefly discusses the beam shaping types and methods and provides examples
from the literature. Section 3.3 is devoted to discussion of the LIDAR systems. Section
3.3.1 provides an overview and literature review for the LIDAR systems and their
applications. In Section 3.3.2, edge-emitting semiconductor laser based LIDAR system
design is briefly discussed. An example of a coherent LIDAR system design is also
presented in this section.
3.1 Basic Principles of Laser Beams and Their Propagation Characteristics
LASER is the acronym used for ?Light Amplification by the Stimulated Emission of
Radiation?. Lasers are optical sources that emit coherent, nearly monochromatic, and
unidirectional beams [11]. Here, coherent beam term is used for a beam having constant
frequency and phase for a significantly long duration. It is nearly monochromatic since
no laser beam can have an infinitely narrow linewidth. However, laser beams can be
assumed as monochromatic when compared with several other light sources.
12
Lasers mainly consist of three parts, namely, the active medium, the optical cavity,
and the pumping mechanism. The active medium is the part where the lasing starts and
sustained in the form of gases, liquids, solids, or semiconductor junctions. The optical
cavity is a cavity resonator mostly designed with reflective surfaces to provide the optical
feedback mechanism to amplify radiation generated in the active medium. Proper design
of the cavity along with the active medium and the pumping mechanism, directly affects
the spectral purity and directionality of the laser beam [12]. The pumping medium is
responsible for supplying the necessary energy to the active medium to excite the
electrons from the ground energy state to a higher energy state.
When the atomic spacing is large in a material, such as in gases, the electron energies
are discrete and the energy states are formed. With an increase in the number of neighbor
atoms, the number of split energy states is increased while the energy differences in the
adjacent states are reduced. In the semiconductor crystals, there are approximately
322 cm10 atoms, and the spacing of energy states is on the order of 1810? eV. This
energy spacing is much smaller than the bandgap of the semiconductor material;
therefore, the energy states are considered to be continuous forming energy bands [13].
In order for the lasers to operate, it is necessary to create a population inversion
between the valence and conduction bands. Population inversion basically means having
more electrons in an upper energy level compared with the number of electrons in a
lower energy level [14]. Such a population inversion can be created by external pumping
(lasers, electron beams, or flashlamps) or by electron injection (with a PN-junction) [14].
Electrons and photons exchange energy in the absorption and emission steps of lasing
such that the total energy is conserved.
13
This research is interested in the type of semiconductor lasers termed as edge-
emitting semiconductor lasers. The output beam of such lasers emerges in the plane of
PN-junction throughout a horizontal cleaved construction, as illustrated in Figure 1.1(a).
The rectangular slab of semiconductor material has two polished surfaces called cleaved
facets, functioning as resonator mirrors. The best method to define such lasers? output
beam and propagation is approximating the radiating field as the superposition of
fundamental and higher modes, where each mode is most conveniently expressed as the
product of a Hermite or a Laguerre polynomial and a Gaussian function [15]. The
simplest, and the lowest order mode, is called the fundamental Gaussian beam, and is
also the most stable and preferred beam type in many applications. A complete
understanding of laser beam propagation requires a treatment based on Maxwell
Equations. Therefore, in this section, only the basics of the fundamental Gaussian beam
analysis are presented, as presented in [16].
The scalar wave equation for the electric field in free space, which is sufficient to
model diffraction and interference, is given by:
( ) ( ) 0,1, 2222 =???+? tEtctE rr (3.1)
In order to model monochromacity of the laser beams the solutions to Equation 3.1
considered are in the following form:
( ) ( ) jwtetE ?= rr ?, (3.2)
When Equation 3.2 is substituted in 3.1, the Helmholtz Equation for ( )r? is
obtained:
14
( ) ( ) 022 =+? rr ?? k (3.3)
where k is the wave number (or phase constant) of the wave.
Ideally, a laser beam propagates as a unidirectional wave with a finite cross-sectional
area. However, a spherical wave is not unidirectional and a plane wave has an infinite
cross-sectional area. Therefore, neither plane nor spherical waves can be considered as
laser beams. Hence, a solution in the form as given in Equation 3.4 is sought:
( ) ( ) kz0 jerr ?? = (3.4)
which differs from a plane wave by the fact that its amplitude is not a constant. This
equation represents a unidirectional wave with a finite cross-sectional area. The solution
in Equation 3.4 must satisfy the Helmholtz Equation given in Equation 3.3. Thus,
Equation 3.4 is substituted in Equation 3.3, yielding in the following equation:
( ) ( ) 0020222222 =+??
?
?
???
?
?
?+
?
?+
?
? kzkz rr jj eke
zyx ?? (3.5)
( )r0? term is called ?slowly varying part? and kzje term is called ?rapidly varying part?.
Slowly varying part means that:
zkz ?
?<<
?
? 0
2
0
2 ??
(3.6)
Using the expression given in Equation 3.6, the following approximation can be
made:
( ) kzkzkzr jjj ekzjkekzjkzez ?
?
??
?
? ?
?
??
???
?
???
? ?
?
?+
?
?=
?
?
0
20
0
20
2
0
2
02
2
22 ?????? (3.7)
, and therefore we have:
15
( ) 02 02
2
2
2
???
?
?
???
?
?
?+
?
?+
?
? r?
zjkyx (3.8)
The solutions to Equation 3.8 must be sought. This equation yields to:
02 002 =??+? zjkT ?? (3.9)
which is the paraxial wave equation and 2
2
2
2
2
yxT ?
?+
?
?=? .
A solution to Equation 3.9 must be sought, having the following form:
( ) ( ) ( ) ( )zjpzqyxjk eAe 20 22 +=r? (3.10)
where A is a constant and q(z) and p(z) are to be determined.
q(z) is given by:
( ) ( ) )(
11
2 zw
j
zRzq ?
?+= (3.11)
The designation z = 0 is arbitrary, therefore the plane z = 0 is chosen so that 0R is
infinitely large. As a result of this, Equation 3.11 yields to:
2
00
1
w
j
q ?
?= (3.12)
Equation 3.10 is a solution for Equation 3.9 if p(z) and q(z) satisfy:
1=dzdq (3.13)
q
j
dz
dp = (3.14)
These equations have the solutions:
( ) zqzq += 0 (3.15)
16
( ) ??
?
?
???
? +=
0
0ln
q
zqjzp
q
j
dz
dp = (3.16)
where ( )00 qq = and it is assumed that p(0)=0.
If Equation 3.11 is substituted in Equation 3.10, the following equation is
determined:
( ) ( ) ( ) ( ) ( ) )(22 2222222 zwyxzRyxjkzqyxjk eee +?++ = (3.17)
Solving Equation, 3.15 it is found that:
( )
2
001
1
wzjRze
zjp
??++= (3.18)
where 0R and 0w are the values of R and w at z = 0.
Using Equation 3.15, the following equation can be derived:
( ) ( )
( )
( )220
22
0
2
0
2
0
2
0
0
0
0 1
1
111
111
wz
wzzwj
wjz
wj
qz
q
zqzq ??
????
??
??
+
+=
+=+=+= (3.19)
When Equations 3.11 and 3.19 are equated to each other: the ?radius of curvature? at
an arbitrary point is determined as:
( ) zzzzR 20+= (3.20)
the spot size (radius) at an arbitrary plane is found as:
( ) 2020 1 zzwzw += (3.21)
where 0z is called the Rayleigh range and given by:
?
? 20
0
wz = (3.22)
( )zq defined in Equation 3.11 is termed as the complex radius of curvature.
17
Substituting Equations 3.20- 3.22 in Equation 3.18, the following equation is
determined:
( ) ( )zjzjp e
zzzjz
e ??
+
=+=
2
0
20 1
1
1
1
(3.23)
where ( ) ( )01tan zzz ?=?
Substituting Equations 3.17 and 3.23 in Equation 3.10:
( ) ( ) ( ) ( ) ( ) )(2
2
0
20
22222
1
zwyxzRyxjk
zj
ee
zz
Ae +?+?
+
=
?
? r (3.24)
is determined. As a result, the solution to the Helmholtz Equation can be determined as:
( ) ( )[ ] ( ) ( ) ( ) )(2
2
0
2
22222
1
zwyxzRyxjk
zkzj
ee
zz
Ae +?+?
+
=
?
? r (3.25)
The plane z = 0 is called the beam waist plane and the radius 0w at this plane is the
minimum beam radius. The relationship between the beam spot radius at Rayleigh range
and the beam waist radius is related such that:
( ) 200 wzw = (3.26)
The smaller the spot radius, the smaller the Rayleigh range, thus the greater the rate
of growth of the spot size from the waist with propagation in z direction. The distance
equal to 010z is considered to be the separating plane between the near-field and the far-
field regions. The beam approximates to two asymptotic lines in the far-field region.
These asymptotic lines intersect the propagation axis forming a virtual point source in the
beam waist plane.
Finally, the divergence of a Gaussian beam in the far-field region is given by the
beam divergence half angle:
18
( )
0
00
0 , zz
wz
w
z
zw >>=??
?
?? (3.27)
The characteristic parameters of a Gaussian beam are demonstrated in Figure 3.1.
Figure 3.1: The characteristic parameters of a Gaussian beam.
As mentioned earlier, the lowest order transverse mode is the fundamental mode and
denoted as TEM00 [17]. Higher order modes are expressed as the superposition of
fundamental Gaussian function and Laguerre polynomial for lasers with circular
symmetry, whereas, the superposition of Gaussian function and Hermite polynomial is
used to express lasers with rectangular symmetry. Higher modes are denoted as TEMmn
where m = 1, 2, 3, ? and n = 1, 2, 3, ?The beam spots for modes with circular and
rectangular symmetry are displayed in Figure 3.2 and Figure 3.3, respectively [17].
19
Figure 3.2: Transverse modes for lasers with circular symmetry
Figure 3.3: Transverse modes for lasers with rectangular symmetry
20
3.2 Beam Shaping Optical System Design in the General Sense
Beam shaping is basically modifying a laser beam to another form as required by a
specific application. This might be circularizing and collimating a divergent elliptical
beam or transforming a laser beam?s irradiance distribution such as from Gaussian to
uniform. These two types of beam shaping may also be handled simultaneously if
required.
Figure 3.4: Beam profile plot of a laser beam with o46 divergence angle in the
perpendicular transverse direction, o12 divergence angle in the parallel transverse
direction, and neglected astigmatism.
Edge-emitting semiconductor lasers have inherent astigmatism and elliptical beam
profile, as stated earlier. Elliptical beam emitted from the laser source forms a circular
beam at a specific target plane and as it further propagates it becomes a beam having an
elliptical beam profile again due to the different divergence angles in the two transverse
directions (see Figure 3.4). When designing a beam shaper, the divergence angles at the
output of the beam shaper must be made equal, satisfying that the beam radii in the two
21
transverse directions are the same at any arbitrary target plane after propagating through
the optical system.
One method to transform an elliptical beam to a circular beam is to use reflective
optics such as mirrors. Reflective optics such as mirrors is preferred over the refractive
optics such as lenses in some cases to avoid light feedback to the laser due to imperfect
antireflection coatings used in refractive systems [18]. In these instances, the mirrors are
designed as off-axis mirrors to avoid obscuration through the system. Two different
mirror systems, off-axis hyperbolic/parabolic and off-axis parabolic mirror systems for
possible applications in LIDAR systems have been investigated [19].
Refractive optics might be preferred to reflective optics in some cases because it is
possible to design an on-axis lens transmitter / receiver system without obscuration
caused the optical components. A ?cruciform? cylindrical lens, which is a combination
of two separate cylindrical lenses with their symmetry axes perpendicular to one another,
developed for this purpose has been presented in [20]. Also in [21], a pair of cylindrical
lenses is crossed to control the focus and the length of the image of the source. After
traveling through this optical system, the diverging laser beam becomes an effectively
thin plane of light. The cylindrical lenses are mounted so as to be rotatable
simultaneously or with respect to each other. A single aspherical lens approach has also
been used for circularization and collimation purposes in [22]. In [23], two optical
system designs using aspherical lenses for possible application in LIDAR systems are
presented. The first system consists of two lenses; whereas, the second one has a single
aspherical lens.
22
Another widely used method is to employ an anamorphic prism pair [24]. In this
case, the beam entering the prism is collimated and the prism pair serves to change the
beamwidth in one transverse direction to make the beamwidth equal in both transverse
directions, thus, circularizing the beam. Depending on the orientation of the prisms, the
beam size may be either reduced or expanded and the amount of change is variable
depending on the angular position of the prisms.
Both reflective [25? 27] and refractive [7, 27- 29] beam shapers to transform the
beam?s irradiance profile to a different profile have been proposed in the literature. In
most of these studies, the input beam is assumed to be a collimated Gaussian beam and
the irradiance distribution is converted to uniform distribution. Moreover, the refractive
systems discussed in these studies assume that the input beam has a circular profile
having the same beam radius and divergence angle values in the two transverse directions
[7, 27- 29]. In [30], a two aspheric lens-based beam shaper is presented to convert the
Gaussian irradiance distribution of an elliptically divergent edge-emitting semiconductor
laser beam into a uniform-square distribution while circularizing and collimating it. The
output beam radius is also expanded to a desired value.
Several analytical functions with a uniform central region to an almost null region
have been studied in the literature [31]. The most common ones are Super Gaussian,
Flattened Gaussian, and Fermi-Dirac functions. These functions allow for a gradual
transition from the central uniform intensity region of the output beam to a null region at
the edge of the beam, reducing the diffraction effects in regions of sharply changing
intensity. The diffraction effects become more and more significant as the beam
propagates after exiting the beam shaper.
23
3.3 Literature Review
3.3.1 LIDAR Applications
LIDAR systems are used in numerous remote-sensing applications. Only a few of
them are discussed here. One of the main application areas of LIDAR systems is
atmospheric research such as measuring the chemical constituents in the atmosphere [32].
DIfferential-Absorption LIDAR (DIAL) systems are based on the fact that the absorption
of light by the gases and particles in the atmosphere is different at different wavelengths
[33]. This type of LIDAR uses two different laser wavelengths one of which is absorbed
by the target particle, termed as the ?on wavelength;? whereas, the other one is not
absorbed and is termed as the ?off wavelength?. The wavelength separation between the
on and off wavelengths is minimized to avoid differences in the atmospheric backscatter
at the two wavelengths [34]. The returned signals at both wavelengths are compared in
intensity to determine the concentration of the target. It is mainly used to measure the
concentration of ozone, water vapor, and aerosols.
Another common atmospheric application of LIDAR systems is measurement of the
wind velocity. In order to measure the velocity of an arbitrary target, Doppler LIDAR is
utilized, operating based on the Doppler principle [35]. There is an increase in the
wavelength if the target is moving away from the LIDAR whereas the wavelength is
decreased if the target is moving towards the LIDAR. The wind velocity is measured by
measuring the Doppler shifts caused by dust and aerosol particles carried by the wind.
The coherent (heterodyning) LIDAR is a receiver technique that can be applied to
determine the Doppler shift as explained in [36]. This technique does not require
24
determining the full spectrum of the returned signal; instead Doppler shift of only the
peak frequency is measured. This is advantageous to non-coherent receiver systems
since only a few photons are enough for measurement. The heterodyning technique is
utilized at the detector, where the returned light is mixed with the beam from ?the local
oscillator? (LO). The frequency difference, caused by the Doppler shift, provides the
wind velocity. Numerous studies of coherent LIDAR can be found in the literature [1,
37- 42].
The detection mechanism utilized at the receiver is as of importance as the type of
the detector. By combining a weak optical signal with a strong, frequency-stable Local
Oscillator (LO) laser beam, radio frequency beats are produced at the frequency
difference between the two optical beams [40]. A LIDAR employing such coherent
(heterodyne) detection is called a coherent LIDAR, as stated earlier. Compared with
incoherent (direct) detection LIDARs, coherent detection LIDAR has the advantages of
greater sensitivity and the straightforward measurement of the velocity of a target [41].
The platform choice of a LIDAR system depends on the application and the laser
type. Most LIDAR systems are placed on the ground or airborne platforms. On the other
hand, the first spaceborne LIDAR system designed for atmospheric studies was named
Lidar In-space Technology Experiment (LITE) [43] and was designed with the capability
to make measurements of aerosols and specifically clouds in the stratosphere and
troposphere [44]. Due to their better space and time resolution characteristics than the
other satellite-based passive remote sensing tools possess, LIDARs are excellent systems
to investigate the clouds. By enabling accurate measurement of cloud height and
25
structure, LITE provides the necessary means for the numerical models used in climate
study and weather forecasting [45].
In another study, a CO2 LIDAR, operating at 10.59 ?m, is used since multiple
scattering occurring in clouds is eliminated [46]. The sensitivity of the LIDAR is not
altered by natural background light because of its extremely narrow optical passband.
This LIDAR is mainly used to measure the drop size distribution in water clouds,
averaged over the penetration depth of the LIDAR. Further, it is shown in [47] that a
longer wavelength covers a wider range of size distributions. Another application is
discriminating between water and ice clouds. The CO2 LIDAR uses the ratio of
backscattered signal at two wavelengths for this discrimination process.
LIDAR systems are also suitable for the spatially continuous measurement of air
pollution, and it is desirable to develop a LIDAR which can cover a wide area [48].
However, airborne LIDAR is not useful for the instantaneous measurement over an area
and its running cost is great compared to a ground-based LIDAR. For these reasons, a
large-scale station-type LIDAR is developed [48]. This LIDAR has a wide-area
observation capability and it can respond to rapidly varying phenomena such as stack
plume dispersion by the use of a pulsed laser with a high PRF.
In general, most LIDARs and conventional radars are based on similar fundamentals.
The biggest advantage of the LIDAR systems is that the LIDARs operate at much shorter
wavelengths so that they are capable of higher accuracy and more precise resolution than
the radars. This is the main reason that the LIDARs are preferred to radars in
atmospheric studies that especially include interaction with significantly small particles.
LIDARs can also be operated as multi-wavelength tools as radars do. In [49], a LIDAR
26
working at 7 wavelengths within the range 0.38 ? 1.0 ?m is presented. This system is
used to determine the atmospheric aerosols at different regions such as continental
background and industrial areas, deserts, and ocean regions. The obtained data are used
in radiation transfer calculations for the atmosphere to improve atmosphere correction
methods in observing the Earth from satellites and aircrafts.
There are several applications of LIDAR systems other than atmospheric research.
One application type is the Laser RangeFinder (LRF), which displays the distance of a
target [50]. A low-cost, handheld LRF system based on a pulsed GaAs laser diode for
automotive speed detection and law enforcement is presented in [51]. LRF systems have
several advantages over microwave Doppler police radar systems such as narrow angle
beam adjustment with proper optical system design and fewer false speed-readings. In
addition to this, they are harder to detect than the radar detectors are. It is shown in [52]
that the laser diode LIDAR systems are also promising as sensors for autonomous
targeting guided weapons due to their ranging and 3D profiling capabilities improving
target recognition and reducing false alarms. Profiling also permits determination of
optimum aim point on the target improving the destruction capability.
LIDAR systems can also be used in on-board automotive applications for monitoring
the target distance to the host vehicle and their relative velocities [53]. These systems
can be utilized in Adaptive Cruise Control (ACC), Collision Warning / Collision
Avoidance (CW/CA), Lane Keeping Assistance (LKA), Pre-Crash Sensing (PCS),
pedestrian detection, reversing aid, and blind zone checking applications. Each system
needs to recognize different targets such as the relevant vehicle for ACC, the lane mark
for LKA, and in-front objects for PCS. Moreover, LIDAR and Global Positioning
27
System (GPS) units can be utilized in coordination for highly accurate automatic
navigation [54].
LIDAR systems are also used in several civil planning applications. In [55], an
automatic LIDAR based approach is proposed to update building information in a GIS
(Geographic Information System) database in a dense urban area. An up-to-date building
inventory database is a crucial requirement for a reliable damage assessment specifically
for earthquake-prone and dynamically changing urban areas such as Tokyo and Istanbul.
Aerial photography for these urban areas has unavoidable drawbacks such as shadows of
tall buildings over the scene and leaning of the buildings due to the perspective
projection. An important application is to carry out two LIDAR surveying flights before
and after an earthquake, assessing the damage by a change detection process. In addition,
reconstruction of building geometry in an urban environment has several other important
applications such as urban planning, surveillance, wireless network planning, and virtual
tourism [56].
In [57], two airborne LIDAR systems are utilized for ocean and terrain remote
sensing purposes. In one configuration, the LIDAR uses two laser transmitters with
different wavelengths to perform experiments over the ocean, and in the other only one
transmitter is used to perform experiments over the terrain. During airborne experiments,
the laser pulse time-of-flight information, as a function of the LIDAR platform position,
are used to derive 2-D ocean and land profiles.
Airborne LIDAR bathymetry has also proven its worth as a hydrographic survey tool
[58]. Airborne LIDAR bathymetry is especially superior to conventional methods in
shallow water regions that are costly, hazardous, and time consuming for ship operations.
28
Water depth is calculated from the time difference of laser returns reflected from the sea
and seabed. In most systems, an infrared transmitter is used for surface detection
whereas a blue-green transmitter is utilized for seabed detection.
Scanning Hydrographic Operational Airborne LIDAR Survey (SHOALS) system,
including a bathymetric LIDAR component, a topographic LIDAR component, and a
digital imagery capability, produces high-density measurements of the coastal zone. This
enables engineers and scientists to quantify volumes of navigation channel shoaling, track
movement of sand placed for beach nourishment, aid in coral reef mapping, and provide
depths and navigation hazard locations for nautical charting [59]. Another application,
observing and characterizing the spectral behavior of sea ice roughness using Airborne
Topographic Mapper (ATM) LIDAR system, is defined in [60]. Surface roughness is a
basic input for the classification of the sea ice development stages. The ATM LIDAR
measures the round-trip travel time of the laser pulse between the aircraft and the surface,
and combines this information with GPS and on-board laser gyro. Height distributions
and roughness spectral densities are derived from surface profiles measured over a
variety of sea ice types.
A pulsed laser-based LIDAR together with a GPS and an Inertial Navigation System
(INS) unit provides a very efficient remote sensing imaging tool. In [61], a pulsed laser
beam is directed out of the aircraft and when this laser pulse is intercepted by a target on
earth, it is reflected back to the aircraft. The time interval between the laser pulse leaving
the aircraft and reflecting from the target back to the sensor is measured very accurately.
In post-flight data processing, the time interval measurements are converted to distance
and the calculations are finalized along with the data from the GPS and the INS units.
29
The GPS unit determines the aircraft longitude, latitude and altitude; whereas, the INS
unit determines the aircraft pitch, yaw, and roll providing the laser beam exit geometry
relative to the earth coordinates. Laser pulses may reflect from trees or penetrate holes in
the canopy and reflected from the ground level. Remote sensing techniques that utilize
such LIDAR systems are proved to provide unique data on forest canopy geometry and
subcanopy topography [62]. Most remote sensing systems, although providing images of
the horizontal organization of canopies, do not provide direct information on the vertical
distribution of canopy elements. In this study, a 3-D model is developed to understand
the relation between canopy structure and the LIDAR return signal.
Another important application of LIDAR systems is the vegetation status control.
Fluorescence LIDAR is an efficient tool used for this purpose giving information about
stresses [63]. Vegetation fluorescence is due to the pigments enclosed in the leaf and
mainly clorophyll is engaged in the complex electron-exchange, which is the basis of
photosynthesis. The resulting spectrum depends on the plant status allowing information
extraction on the content of the leaf pigments and the plant health.
In addition to vegetation studies, fluorescence LIDAR systems are also used to
investigate oil pollution at sea along with passive hyperspectral imagery [64]. Synthetic
Aperture Radar (SAR) tool is mainly used in oil pollution detection based on radiometric
properties induced by oil slicks in the resultant images [65]. However, this method
suffers from high false detection rates due to environmental parameters such as sea state
and wind that can induce similar radiometric properties as oil spills do in SAR images.
Location, extent, and volume of the spilled oil and its spatial distribution are the mainly
sought parameters. Data fusion from hyperspectral imagery and fluorescence LIDAR
30
sensors allows the high-resolution spatial distribution of oil thickness to be
geographically mapped.
3.3.2 LIDAR System Design
Even though LIDAR systems are efficient tools for applications such as
environmental and atmospheric remote sensing, they are not easily transportable for
commercial purposes [1]. On the other hand, recent technological advances in edge-
emitting semiconductor lasers and detectors permit realization of low cost, easily
transportable, and eye-safe LIDARs [66]. Such lasers also offer the advantage of several
available wavelengths for the LIDAR operation. Tuning in to several wavelengths can be
accomplished by varying the bulk temperature or the injected current into the active
junction of the laser [67].
A coherent edge-emitting semiconductor lased-based Doppler LIDAR system is
chosen as a design example in this section, as shown with a block diagram in Figure 3.5.
Figure 3.5: Block diagram of a coherent edge-emitting semiconductor laser based
Doppler LIDAR system design example.
Most LIDAR systems make use of ?J level pulse energies and PRFs of several
thousand Hertz [68]. While the eye-safe maximum permissible exposure levels for
31
visible and near infrared wavelengths are quite low, eye-safety can be achieved by
restricting pulse energies to about a ?J level or even less and expanding the laser beam
to a diameter of several centimeters [69]. The low pulse energy permits transmitted beam
energy densities that are within the eye safety standards. The reduced signal level
drawback due to low transmitted pulse energies is overcome by operating at higher PRFs.
Many pulses are averaged in reasonable time periods provided that the lasers can be
operated with kHz or higher PRFs. The energy of the beam is further reduced while it
propagates through the atmosphere. As a result, the return signal is generally weak and
can be best detected by photon counting and summing Avalanche Photodiode Detectors
(APD) operating in the Geiger mode; whereas, most conventional methods employed
bulky Photo-Multiplier Tube (PMT) detectors with low quantum efficiencies.
A photodiode is basically a semiconductor diode with a p-i-n structure that functions
as a light sensor. It can be used either with zero bias (photovoltaic mode) or reverse bias
(photoconductive mode). An APD is a special type of photodiode that amplifies the
photocurrent via an electron avalanche process [70]. A larger reverse bias voltage
generates a larger gain with the cost of increased noise level. Furthermore, the detectors,
based on a single photon avalanche diode operating in Geiger mode, utilize avalanche
multiplication to enhance detection of the returned LIDAR light [71]. In Geiger mode,
the diode is biased a few volts above the breakdown voltage. When a free carrier is
generated, it is accelerated and a large current surge is generated by impact ionization.
The current reaches saturation for a single-generated carrier [72]. Since the absorption of
a photon results in the generation of an electron hole pair, the start of a breakdown
indicates the absorption of a photon [73].
32
Even though APDs are quite effective with weak return signals, they have a small
active area that might not be sufficient for many applications. In order to enlarge the
active area of the detectors and enhance their sensitivity for LIDAR applications, arrays
of Geiger-mode APDs might also be integrated [74]. Two other drawbacks of the
Geiger-mode APDs are that they need a finite reset time after triggering and that the
carriers are generated not only by the signal but also by dark currents and background
radiation, sometimes killing the detector before the returned pulse reaches the detector.
Transmitter / receiver choice for a LIDAR system is also of great importance. Most
LIDAR systems employ a single optical system used for both transmitting and receiving
the laser beam. This optical system must be designed to satisfy the required output beam
characteristics for LIDARs using edge-emitting semiconductor lasers. For LIDARs using
edge-emitting semiconductor lasers, circularization is required since an elliptical beam
profile is not appropriate for an optical system that employs rotationally symmetric
optical components and / or fiber optics. Besides this, most LIDAR systems transmit the
laser pulses in the far field. Thus, ellipticity of the laser beam would cause highly
elliptical footprint on the target plane after long distance propagation. Collimation is
necessary since minimum beam spread is required for most LIDAR systems operating at
high altitudes. In addition to this, collimation allows the receiver to have low Field of
View values. This is required especially for LIDAR systems utilized in atmospheric
studies since a receiver with a low FOV value collects less background noise and is less
susceptible to multiple scattering caused by the atmosphere. Finally, expansion is needed
to reduce the beam divergence caused by diffraction (the minimum beam divergence
angle in the diffraction-limit is reduced with increasing beam waist radius). Expansion is
33
also required to achieve eye-safety without significantly reducing the total transmitted
power and larger target area scanning as well.
On the other hand, beam shaping optics and transmitter optics might also be chosen
as separate units. This is especially required for coherent LIDAR systems since a portion
of the laser source beam is split up with a beam splitter and used as the LO beam.
Moreover, the return signal might be further propagated to the detector unit with fiber
optics or other optical components. There are also systems with separate transmitter and
receiver units, as discussed in the literature [66].
The wavelength choice of a LIDAR system depends on the application type. For
instance, infrared region is mostly preferred in atmospheric studies since most gases and
atmospheric particles have strong absorption bands in this region. (The atmosphere
contains certain gases whose absorption resonances are so strong that some spectral
regions are nearly opaque.) One particular example is that the 700 ? 900 nm region is
useful for atmospheric water vapor measurements [67].
34
CHAPTER 4
METHODOLOGY
This chapter presents the general methodology for the design and analysis of the
beam shaper systems. Section 4.1 discusses the geometrical ray optics approximation to
physical optics propagation. The optical surfaces of the designed systems are mostly
designed aspheric surfaces. Therefore, aspheric surfaces and conic sections are briefly
discussed in Section 4.2. Section 4.3 discusses how ZEMAX is involved in the design
and analysis steps of the optical systems. Section 4.4, Section 4.5, and Section 4.6
present the design approaches and the equations specific to the developed beam shapers.
4.1 Geometrical Ray Optics Approximation to Physical Optics Propagation
Optical design of a beam shaper can be achieved using either physical or geometrical
ray optics [75]. It is customary to design an optical system with geometrical ray optics
instead of Gaussian beam analysis when ray optics is a good approximation to the
physical optics beam propagation. Rays generally characterize the direction of the flow
or propagation of radiant energy, except near the focus or an edge point where
interference and diffraction takes place. Thus, a ray is a mathematical concept rather than
a physical entity. They may also be defined as lines normal to the geometrical wavefront
of a beam. As an analogy, the direction of radiant energy propagation is given by the
Poynting vector in the electromagnetic theory [76].
35
The amount of divergence or convergence of a laser beam is measured by the
divergence angle, which is the angle subtended by the 21 e asymptotes of the beam
irradiance profile [12]. These asymptotes approach the real beam in the far field, which
is defined with the Rayleigh range. In the far field, for 0zz >> , the waves approximate
to two 21 e rays emanating from a point source centered at the waist [77] as shown in
Figure 1.1(b). In addition, the author also calculates a dimensionless parameter, ? ,
which provides a quantitative measure of the diffraction effects and the validity of the
geometrical ray optics design [76]. This parameter is defined as:
?
??
d
ww oi2= , (4.1)
where iw , ow , d , and ? are the incoming beam radius, the beam radius at the target
plane, the distance between the optical component and the target plane, and the
wavelength of the laser beam, respectively. This equation is derived for simple
geometries such as a circle. In general, for 1>>? , the diffraction effects are negligible.
Designing optical systems based on geometrical ray optics approach involves the
laws of reflection and refraction, Snell?s Law, ray tracing techniques, conservation of
energy within a bundle of rays, and the constant optical path length conditions. The
constant optical path length condition is based on Fermat?s Principle, which states that
each ray follows the path of shortest time passing through an optical system [78].
36
4.2 Aspheric Surfaces and Their Usage in Beam Shaping Optical Systems
The design methods that will be discussed in this dissertation basically consist of
lens and mirror design. These lens and mirror surface functions are mostly chosen to be
conic sections such as a parabola, a hyperbola, or an ellipse, rather than spherical
surfaces. This reduces the aberrations caused by spherical surfaces [79]. Conic sections
are basically aspheric surfaces defined with radius of curvature and conic constant
parameters. In general, an aspheric surface can also be defined with even polynomial
deformation terms together with radius of curvature and conic constant values. These
polynomial terms increase the accuracy of the representative surface function. However,
these even aspheric surfaces are harder to optimize and manufacture than conic sections.
In this section the mathematical equations for these conic sections are explained here
briefly [79, 80].
The focal length or the power of a spherical optical surface is defined by the radius
of curvature of the surface, which is correct only for paraxial rays. As the angle of the
rays with the propagation axis increases, this paraxial refraction equation does not give
accurate results. Due to this inaccuracy, a spherical surface aberrates a beam with high
field angle. The use of an aspheric surface permits selective variation in the refractive
power of the surface as a function of the coordinate location, reducing the aberrations
induced.
The most common general aspheric can be described as a power-series departure
from a simple base surface, such as a sphere. The base surface may be a conic section
too, but the aspheric departure is generally defined relative to a spherical reference
surface with the same radius of curvature.
37
An important parameter of an optical surface is the ?surface sag?, which is illustrated
in Figure 4.1, as reproduced from [81]. The exact definition of surface sag is given by:
22sag yRR ??= (4.2)
Figure 4.1: Illustration of the definition of surface sag.
The general surface equation for an even aspheric surfaces is given by:
( ) ?=++?+=
N
i
i
is rArCK
Cr
1
2
222
2
111
z , (4.3)
where sz is the sag in the direction of the propagation , r is the coordinate in one of the
transverse directions, C is the curvature (the reciprocal of the radius of curvature), K is
the conic constant, and iA2 are the coefficients of the polynomial deformation terms. K
and iA2 can have several solutions which are dependent on each other. Generally, i = 1
is not used and i = 2, 3, 4, ?, N are used where 2N is the polynomial term with the
highest degree for a particular even aspheric surface. However, from a numerical point
of view, it is desirable to use the least number of deformation terms meaning that the
surface shape is closer to a conic section, which will also simplify the surface
manufacturing.
38
A special class of aspheric surfaces is the conic sections, where all the polynomial
deformation terms are equal to 0; whereas, the conic constant is not equal to zero. For a
hyperbola 1?K (sometimes referred
to as oblate ellipse [81]) or 01 < ||? . First, the beam is collimated in both perpendicular
and parallel transverse directions at the front surface of the first lens ( 11S ) only. The
beam at this stage is still has the elliptical shape while propagating within the lens. Then
only the beam in the parallel transverse direction is allowed to diverge at the back surface
of the first surface ( 12S ) while the beam in the perpendicular transverse direction
propagates with no refraction. The second lens is positioned at a distance from the first
where the beam shape becomes circular. Then this surface ( 21S ) is designed to collimate
the beam only in the parallel perpendicular beam allowing the beam to continue to
propagate as a collimated circular beam in both transverse directions. The back surface
59
of the second lens ( 22S ) is designed to be flat, so the output beam has the desired circular,
collimated, and expanded beam shape.
Figure 4.13: A schematic diagram of two-lens system. Here Sij represents each refractive
surface of the lenses used in this design. S11 is rotationally symmetric, S12 is cylindrical
symmetric (flat in the perpendicular direction), S21 is a cylindrical symmetric (flat in the
perpendicular direction), and S22 is flat (in both directions) surfaces.
As a first step, the front surface of the first lens ( 11S ) is designed. Once the surface
parameters of this surface are determined, then the parameters of the back surface of this
lens ( 12S ) and the front surface of the second lens ( 21S ) are determined. As a design
constraint, the output beam radius fw is specified (to be approximately 2.0 cm).
Figure 4.14 shows a representation of the surfaces in the perpendicular transverse
direction (x-z plane) and variables used to calculate lens parameters for the front surface
of the first lens. In Figure 4.14 and the following equations, L is the directrix of the
conic section, e is eccentricity of the conic section, a is the semi-major axis parameter,
b is the semi-minor axis parameter, and 22 bac += . Also, the subscript 2,1=ij refers
to as follows: i is the first or second lens, and j is the first (front) or second (back)
60
surface, and 11sx is the sag of the ( 11S ) surface in perpendicular direction. Also, the
refractive index for air is designated as 0n and is equal to 1 and for glass (BK7) medium
it is designated as n and it is equal to 1.5095 (at 870 nm) in the equations derived
throughout this paper.
Figure 4.14: A schematic representation of the surfaces in the x-z plane.
The 11S surface is a rotationally symmetric surface, and only the x-z plane
calculations are shown in this section. The y-z plane calculations are the same as x-z
plane.
From a trigonometric relation:
( ) ( )
?
? = ?tan11
fwz (4.39)
From Fermat?s Principle, we have:
61
( )
2
11
2
01110 ?+=+ zwnnxdn fs (4.40)
Using ( ) 11111 sxzd ?= ? and Equation 4.40:
( )( ) ( )
2
11
2
01111110 ?? +=+? zwnnxxzn fss (4.41)
Solving simultaneously Equation 4.39 and 4.41 provides the numeric value of the
distance, 1d , between the source and the vertex of 11S . Using Equation 4.38, the radius
of curvature of the front surface is determined as 11S as ( ) ( ) 1||111111 5095.0 dRRR === ?
for the object and image distance values of 11 ds = , inf2 =s .
As stated earlier, Equation 4.38 is valid only for paraxial rays and the laser beam in
this study is a highly divergent elliptical beam. Therefore, determining only the radius of
curvature is not sufficient for the beam to propagate as desired within the lens. The 11S
surface is intended to be a rotationally symmetric hyperbolic surface as a design
constraint, and the hyperbolic surface definition is used to determine the conic constant of
the surface. It is known from the hyperbola definition that the locus of points whose
distance from the focus is proportional to the horizontal distance from the conic section
directrix. The ratio is given with the eccentricity of the hyperbola. Therefore:
( ) ( )( )
( ) 11111
22
11111 e
xLa
wcadz
s
f =
+?
+???? (4.42a)
Furthermore, the hyperbola parameters are:
1
2
1
1 a
bR = (4.42b)
11
2
1
2
1
2
11 Raabac +=+= (4.42c)
62
1
11
2
1
1
1
1 a
Raa
a
ce +== (4.42d)
11
2
1
2
1
1
2
1
1 Raa
a
c
aL
+
== (4.42e)
2
1
2
1
1 1 a
bK ??= (4.42f)
Substituting Equations 4.42b-e in Equation 4.42a, an equation with only one variable
1a is obtained, as given below:
( )
1
11
2
1
11
11
2
1
2
1
1
2
0
2
11
2
1111
a
Raa
x
Raa
aa
xRaaax
s
s +=
?
?
?
?
?
?
?
?
+
+
?
++?+
(4.43)
1a is found solving the above equation. Then, 1b is found solving Equation 4.42b.
Then the conic constant of the hyperbola is found using Equation 4.42f.
Next, the aperture diameter (physical diameter of the surface of the lenses in both
transverse directions) and the lens thickness are determined using the energy transmission
criteria of over 99% as stated earlier. For simplicity, the aperture sizes of both lenses are
decided to be the same. In general the energy transfer through an aperture is given by
( )zwa
T
a e
P
P 2221 ??= , where
TP is the total input power, aP is the transmitted power
through the aperture, a is the aperture radius, and ( )zw is the beam spot radius at a
position z. In our case, we define the parameters ha = defined and assumed both lenses
have the same aperture size. Rewriting the power transmission equation yields to:
63
???
?
???
? ??=
T
a
f P
P
w
h 1ln5.0 (4.44)
From this, the aperture radius is calculated as h = fw75.1 . for 99.78% power
transmission.
The thickness of the first lens is determined in order to design the back surface of the
first lens ( 12S ). The two surfaces, 11S and 12S , are highly curved surfaces. Therefore,
the thickness of the lens is calculated while making sure that 11S does not intersect with
12S at any x and y coordinate points. When h is substituted for x and hz for z in the
general hyperbola equation 12
2
2
2
=? bxaz in the x-z plane, the equation for minimum
thickness for the first lens is obtained as:
??
?
?
?
??
?
?
?
?+??
?
?
???
?=?= 112
1
111 b
haazt
h (4.45)
One should remember that 11S is a rotationally symmetric surface, and the beam is
collimated by this surface in both transverse directions, but not yet circularized.
Therefore, 12S is designed to allow the beam to propagate without any refraction in the
perpendicular direction while allowing it to diverge further in the parallel direction. That
mens, 12S is flat in perpendicular direction (see Figure 4.14), but is not flat in parallel
direction (see Figure 4.15). The second lens is positioned at a distance when the beam
profile becomes circular in both transverse directions (see Figure 4.13). The variables
used to calculate lens parameters in the parallel transverse direction (y-z plane) for both
surfaces are shown in Figure 4-15.
64
After exiting the 12S surface, the beam diverges with an angle ? in the parallel
direction (see Figure 4.15). The corresponding ray is extended so as to intersect the
optical axis forming the virtual object location for this ray on the left hand side of the first
lens. This distance between the intersection point (virtual object) and the vertex of the
12S surface is defined as 1q . As a result, there are five unknown quantities: 11y , 11sy , 1q ,
12sy , and 21sy .
Figure 4.15: A schematic representation of the surfaces in the y-z plane.
Five equations are needed to solve these five unknown quantities.
From Fermat?s principle:
( )211121101110 ss ydynnydn ++=+ (4.46)
From a trigonometric relation:
( )
111
11
||tan
syd
y
+=? (4.47)
Equation 4.46 and Equation 4.47 are solved together for 11y and 11sy . Three more
equations are needed to determine the remaining unknown quantities 1q , 12sy , and 21sy .
65
From a trigonometric relation:
( )
121
11
21122
11tan
sss
f
yq
y
yyd
yw
+=+?
?=? (4.48)
From Fermat?s principle:
( )
( ) ( ) ( )211221122012111
2120111
ywyydnyytn
nydnytn
fssss
ss
?++?++?
=++?
(4.49)
We need one more equation to solve for all the unknowns. This can be derived by
assuming the left side of 12S surface to be air and the right side of to be glass medium.
Then a ?real? ray originating from the location of the virtual object point on the left of the
lens would be collimated after exiting the 12S surface of the lens. Using this fact and the
Fermat?s principle, the following equation is derived:
( )212121101210 ss yqynnyqn ++=+ (4.50)
Then, Equations 4.48, 4.49, and 4.50 are solved simultaneously for 1q , 12sy , and
21sy . Again, from Equation 4.38, for the object distance of inf1 =s and the image
distance of 12 qs ?= the radius of curvature of the 12S surface is determined as
( ) 1||12 5095.0 qR = . For 21S surface, the radius of curvature of is calculated as
( ) ( )21||21 5095.0 dqR += for the object distance of 211 dqs += and the image distance of
inf2 =s .
One should remember that 12S , and 21S surfaces are both cylindrical (have refractive
powers in the y-z plane only and flat in the x-z plane). Using the hyperbola definition,
66
similar to the ones in Equation 4.42, the parameter 2a of 12S and the parameter 3a of 21S
are determined as follows:
( )( )
( )
( )
2
||122
2
2
12
||122
2
2
2
2
2
2
11
2
||122
2
2212
a
Raa
y
Raa
aa
yRaaay
s
s +=
??
?
?
?
??
?
?
?
+
+
?
++?+
(4.51)
( )( )
( )
( )
3
||213
2
3
21
||213
2
3
2
3
3
22
||213
2
3321
a
Raa
y
Raa
aa
wRaaay
s
fs +=
??
?
?
?
??
?
?
?
+
+
?
++?+
(4.52)
The conic constants of these two surfaces are determined as for the 11S surface.
Similarly from the hyperbola equation for the y-z plane, the minimum thickness of the
second lens is obtained as:
??
?
?
?
??
?
?
?
?+??
?
?
???
?=?= 112
3
332 b
haazt
h (4.53)
The lens thickness values must be chosen to satisfy the minimum condition
determined for both lenses with Equations 4.45 and 4.53. As mentioned earlier, no
calculations are needed for the back surface of the second lens since the beam is already
circular and collimated before interacting with this surface. Hence, this last surface is
chosen to be flat in both transverse directions.
4.5.3 Single-Lens Design Method
This design method uses a single asymmetrical lens. The front surface ( 1S ) is
anamorphic with two different refractive powers and surface functions in the two
67
transverse directions. The back surface ( 2S ) is a cylindrical surface (has refractive
power in the y-z plane and is flat in the x-z plane). Figure 4.16 shows the schematic
diagram of these surfaces. Similar to the two-lens system, the beam is collimated in the
perpendicular direction only at the 1S surface. However, the beam in the parallel
direction is allowed to propagate with a different divergence angle within the lens after
refraction at 1S surface. The 2S surface is flat in the perpendicular direction allowing
the beam propagate through with no refraction in this direction, while the surface in
parallel direction collimates the beam, so the output beam has the desired circular,
collimated, and expanded beam shape.
Figure 4.16: A schematic diagram of single lens system. Here, S1 is a hyperbolic surface
in the perpendicular direction and is elliptical in the parallel direction. The surface S2 is
an elliptical surface in the parallel direction and flat in the perpendicular direction.
Figure 4.17 displays the schematic representation of the surfaces in the perpendicular
transverse direction (x-z plane) including the lens variables used in the calculations. The
x-z plane calculations are the same as in the two-lens system. Therefore, the x-z plane
68
calculations are not repeated here. There are only a few notation changes as the surfaces
are defined by single subscript. The distance from the laser source to the vertex of the
front surface and the lens thickness are d and t respectively (no subscript).
Figure 4.17: A schematic representation of the surfaces in the x-z plane.
Figure 4.18: A schematic representation of the surfaces in the y-z plane.
69
Figure 4.18 shows the schematic representation of the surfaces in the parallel
transverse direction (y-z plane) and the variables used to calculate lens parameters.
Fermat?s Principle and trigonometric relationships are used to determine the surface
parameters. The parallel transverse direction divergence angle ( ||? ), the final output
beam radius ( fw ), and the thickness of the lens (t) are chosen quantities as the design
constraints. The distance d has already been found in the x-z plane calculations.
1y , 1sy , and 2sy must be calculated to determine the surface parameters. At the 1S
surface, the beam further diverges in the parallel direction, and the virtual object point is
formed on the optical axis between the laser source and the vertex of the first surface.
The distance between the virtual object point and the vertex of the front surface is
11 syq + . Under the circumstances, there are four unknown quantities 1y , 1q , 1sy , and
2sy meaning that four equations are needed to solve for them.
First equation is derived applying the Fermat?s principle for the 2S surface. For this,
it is assumed that the left side of this surface is glass whereas the right side of it is air.
Tracing the ray propagating on the optical axis and the representative 21 e ray from
virtual object point up to the exit plane we obtain have the following equation for
constant optical path length (OPL):
( ) ( ) 202221111 sfsss ynwytyqntyqn ++?++=++ (4.54)
From a trigonometric relation:
( )
1
1
||tan
syd
y
?=? (4.55)
70
The third equation also derived from the trigonometric relationship for the ray
diverged at the 2S surface and given by:
( )
1
1
12
1tan
q
y
yyt
yw
ss
f =
+?
?=? (4.56)
Finally, the last equation is determined again from Fermat?s Principle tracing the
rays from the laser source to the exit plane of the lens.
( ) ( ) ( ) 2021221212100 sssfs ynyytywnyydnntdn ++?+?++?=+ (4.57)
Once the unknown quantities are determined using these four equations, the radius of
curvature values are calculated using Equation 4.38. For 1S surface, the object distance
value of ds =1 and the image distance value of ( )112 syqs +?= , are used. For 2S
surface, the object distance value of ( )tyqs s ++= 111 and the image distance value of
inf2 =s , are used. The conic constant values are calculated from aspherical surface
equation (without any aspherical polynomial terms) for the both surfaces are determined
as follows.
The radius of curvature for the front and the back surfaces respectively:
( )
( ) dyq
dyqR
s
s
y 5095.1
5095.0
11
11
1 ?+
+= , (
( )||11 RR y = ) (4.58)
( )tyqR sy ++= 112 5095.0 , ( ( )||22 RR y = ) (4.59)
The conic constants for the front and back surfaces respectively:
( )yy
y
s KyC
yCy
1
2
1
2
1
2
11
1 111 +?+= , ( yy RC 11 1= ) (4.60)
71
( )yfy
fy
s KwC
wCy
2
22
2
2
2
2 111 +?+= , ( yy RC 22 1= ) (4.61)
The conic constant values for the two surfaces are determined solving Equation 4.60
for yK1 and Equation 4.61 for yK2 .
4.6 Reshaping an Elliptically Diverging Gaussian Laser Beam into a Circular,
Collimated, and Expanded Uniform Beam with Aspherical Lens Design
4.6.1 Overview and Design Approach
As mentioned before, high divergence and elliptical profile of edge-emitting
semiconductor lasers is undesirable for many LIDAR applications. In addition to high
divergence and elliptical beam shape drawback of such lasers, Gaussian irradiance
distribution is also undesirable for several applications such as display technology,
lithography, laser printing, and material processing because these applications require
uniform illumination of an extended area. Therefore, in this section, an optical system
design method, which utilizes two aspherical lenses to circularize and collimate an edge-
emitting semiconductor laser beam and to transform its irradiance distribution from
Gaussian to a uniform-square distribution, is presented. Circularization term is
commonly used in the literature for correcting ellipticity in the semiconductor laser
beams. In this section, circularization term is still used even though the beam has a
uniform-square profile at the output of the system.
As discussed in Section 3.2, there are several studies in the literature developed to
transform the beam?s irradiance profile to a different profile. However, in most of these
72
studies, the input beam is assumed to be a collimated Gaussian beam and the irradiance
distribution is converted to uniform distribution. Moreover, the refractive systems
discussed in these studies assume that the input beam has a circular profile. The method
discussed in this section is based on the same beam shaping principle [7, 27- 29], but it
assumes that the input beam is radiated by an edge-emitting semiconductor laser.
Astigmatism in ?m levels is neglected in this study since the output beam is collimated at
cm level beam radius, but if needed, the model can easily be modified to correct it. As a
result, in this section, a specific optical beam shaping system is designed to circularize,
collimate, and expand the beam of an edge-emitting semiconductor laser to a desired
diameter and to transform its irradiance distribution from Gaussian to uniform.
A square pulse function is chosen as the output beam distribution function for
simplicity. As explained in section 3.2, the abrupt change in the output irradiance
distribution function causes diffraction effects, which are neglected in this chapter.
Geometrical optics methods such as ray tracing, conservation of energy within a bundle
of rays, and constant optical path length of all rays are used to design this beam shaper. It
is quantitatively shown that geometrical method ray optics approach is a good
approximation to physical optics propagation of the beam for this design method by the
dimensionless ? parameter calculations, as explained in section 4.1. To accomplish the
design, the source beam profile is decoupled into two independent beam profiles
represented in two transverse directions, i.e. the perpendicular and parallel transverse
directions.
73
4.6.2 Design Method
As mentioned in the previous section, the beam profile is modeled as two
independent 2-D Gaussian beams propagating along the two orthogonal planes (the x-z
and the y-z planes). Therefore, the beam shaper is designed separately in these two
orthogonal planes to achieve a circular, collimated, and uniform beam at the output plane.
All surfaces, but the back surface of the second lens, are designed as aspheric
surfaces. The front surface of the first lens is chosen as a rotationally symmetric aspheric
surface and surface function is derived such that it collimates the beam in both transverse
directions at different beam radius values. The back surface of the first lens and the front
surface of the second lens are designed to circularize the beam and transform its
irradiance distribution simultaneously. The front surface of the second lens further
collimates the beam. The back surface of the second lens is flat in both transverse
directions and does not participate in beam shaping since at this surface the beam already
has the desired profile. A sketch of the beam shaper is illustrated only in the x-z plane in
Figure 4.19. The y-z plane sketch is not presented here because the surfaces and their
parameter designations used in y-z plane are the same with only difference of the y-
coordinate parameters of the surfaces.
The designed optical system is assumed to reshape the beam without power loss and
the conservation of energy principle along a bundle of rays is used to establish mapping
of the ray coordinates between the ?input reference? and the ?output reference? planes, as
shown in Figure 4.19 [27]. In order to employ this energy conservation, the input and
output beams must have the same planar phase-front. Therefore, the input reference
74
plane is defined after the front surface of the first lens where the beam is collimated by
this surface in both transverse directions (see Figure 4.19).
Figure 4.19: Illustration of the general design approach in the x-z plane.
The input beam has a fundamental Gaussian irradiance distribution and the output
beam is desired to have a uniform distribution represented by a ?square-pulse? beam
profile which has a non-zero constant value within a region along x and y coordinates,
and zero outside this region. The input irradiance distribution function is given by:
2
2
2
2
0
00 ?
?
?
?
???
??
???
?
???
??
= y
y
x
x
in eeII
(4.62)
The desired output irradiance distribution function is chosen as:
YXout AAI = (4.63)
75
where 0I is the arbitrary amplitude of input beam and XA and YA are the constant values
of the square-pulse function in the perpendicular and parallel transverse directions. 0I is
chosen to be unity (1) in this design method.
For these input and output beam profiles, conservation of energy principle provides a
relationship for any arbitrary ray between the ray heights 1x and 2X , which are
designated as the height coordinates in the perpendicular direction for the back surface of
the first lens and the front surface of the second lens respectively (see Figure 4.19), as
applied in [27]. Lower case x and z are used for the first lens and upper case X and Z are
used for the second lens coordinates. The energy conservation between the input
reference plane and output reference plane used in the design is given in Equations 4.64
and 4.65:
outinoutin EEdXdYIdxdyI =?= (4.64)
dyedxeE y
y
x
x
in ??
?
???
?
???
??
?
???
?
???
??
=
ymax
ymax
2
2xmax
xmax
2
2
00 = dYAdXAE
YXout ??
??
=
Ymax
Ymax
Xmax
Xmax
(4.65)
where inE and outE are the input and output energies, xmax and ymax are the ?working
aperture? values of the first lens, Xmax and Ymax are the ?working aperture? values of
the second lens, 0x and 0y are the radii of the beam after being collimated at the front
surface of the first lens in the perpendicular and parallel transverse directions,
respectively. Because the x and y coordinates are independent of each other and the beam
propagation is decoupled to two orthogonal planes, the x and y coordinate variables are
calculated independently in Equation 4.65.
76
Power transmission was given by ( )zwa
T
a e
P
P 2221 ??= in Equation 4.37 earlier. In this
design method, the aperture radius of the first lens is chosen such that the lens collects
99.78% of the input energy. xmax=a and ( ) 0xzw = values are substituted in Equation
4.37. From this, 01.75xmax x= and 01.75ymax y= values are found for 99.78% energy
transmission. Xmax must be equal to Ymax for a square output beam profile. Thus,
02YmaxXmax x== is arbitrarily chosen in this study.
Once 0y , xmax , ymax, Xmax, and Ymax values are determined, XA and YA
values can be found as follows:
???
?
???
?=
0
0 xmax22erf
Xmax4
2
x
xA
X
? (4.66)
???
?
???
?=
0
0 ymax22erf
Ymax4
2
y
yA
Y
? (4.67)
As a result, the output beam irradiance profile constant value ( outI ) that satisfies the
conservation of energy principle can be found as:
???
?
???
?
???
?
???
?==
00
00 ymax22erfxmax22erf
YmaxXmax8 yx
yxAAI
YXout
? (4.68)
For these input and output beam profiles, conservation of energy principle also
provides a relationship for any arbitrary ray between the ray heights 1x and 2X , which
are designated as the height coordinates in the perpendicular direction for the back
surface of the first lens and the front surface of the second lens respectively (also see
Figure 4.19), as in [27]. This relationship can be determined with the following
equations:
77
dXAdxe
X
X
x
x
x
?? =?
?
?
?
???
?? 21
0
00
2
2
(4.69)
dYAdye
Y
Y
y
y
y
?? =?
?
?
?
???
?? 21
0
00
2
2
(4.70)
Solving Equations 4.69 and 4.70 for 2X and 2Y provide the functions of ( )12 xX and
( )12 yY , as seen in Equations 4.71 and 4.72. As a result of these functions, for each 1x
and 1y values, the corresponding 2X and 2Y values are found.
( )
???
?
???
?
???
?
???
?
=
0
0
12 xmax
22erf
22erf
Xmax
x
x
x
xX (4.71)
( )
???
?
???
?
???
?
???
?
=
0
0
12 ymax
22erf
22erf
Ymax
y
y
y
yY (4.72)
It is assumed that the front surface of the first lens collimates the beam at a beam
radius of 20 fwx = cm (x-z plane derivation). The distance 1d is found solving
following equations (Equations 4.73 and 4.74 are obtained from a trigonometric
relationship and the constant optical path length condition, respectively):
( ) ( )
?
? = ?tan
0
11
xz (4.73)
( )( ) ( )
2
11
2
01111110 ?? +=+? zwnnxxzn fss (4.74)
( ) 11111 sxzd ?= ? (4.75)
78
Based on the chosen 0x value, the 0y value is found solving following equations:
( )21112001110 ss ydynnydn ++=+ (4.76)
( )
111
0
||tan
syd
y
+=? (4.77)
The radius of curvature value of the first lens front surface is found using the general
refraction equation that is only valid for paraxial rays for a given refractive index for
spherical surfaces given by Equation 4.38. Once the radius of curvature is determined,
the conic constant of this surface is determined using the hyperbola definition. As stated
earlier, this surface collimates the beam under the condition that the astigmatism is
neglected. As shown in Figure 4.19 (also in the equations below), L is designated as the
directrix of the conic section, e is eccentricity of the conic section, a is the semi-major
axis parameter, b is the semi-minor axis parameter, 22 bac += , and the subscript 1 in
the variables refers to ?first lens? parameters.
( ) ( )( )
( ) 11111
2
0
2
11111 e
xLa
xcadz
s
=+? +???? (4.78a)
1
2
1
1 a
bR = (4.78b)
11
2
1
2
1
2
11 Raabac +=+= (4.78c)
1
11
2
1
1
1
1 a
Raa
a
ce +== (4.78d)
11
2
1
2
1
1
2
1
1 Raa
a
c
aL
+
== (4.78e)
79
Substituting Equations 4.78b ? 4.78e in Equation 4.78a, the following equation is
obtained:
( )
1
11
2
1
11
11
2
1
2
1
1
2
0
2
11
2
1111
a
Raa
x
Raa
aa
xRaaax
s
s +=
?
?
?
?
?
?
?
?
+
+
?
++?+
(4.79)
1a is determined solving the above equation. Then, 1b is found solving Equation
4.78b. Then the conic constant of the hyperbola is found using Equation 4.80:
2
1
2
1
1 1 a
bK ??= (4.80)
The next step is to determine the parameters of the back surface of the first lens and
the front surface of the second lens.
In order for the output reference plane to have the same phasefront as the input
reference plane, it is necessary for all the rays traced between these planes to have the
same optical path length according to Fermat?s Principle [78]. Using this fact, the
following equations are only derived in terms of perpendicular transverse direction
because they are the same for the parallel direction. The origin of the z-axis is fixed at
the reference input plane, as seen in Figure 4.19.
The optical path length of a ray passing along the optical axis is:
( ) 22011oOPL ntdntn ++= (4.81)
The optical path length of an arbitrary ray crossing the x-coordinate at a point, 1x , is
given by:
( ) ( )[ ] ( )222112121221201OPL ZtdtnzZxXnnzx ?+++?+?+= (4.82)
80
where, ( )11, zx and ( )22,ZX are the coordinates of an arbitrary ray location at the back
surface of the first lens and the front surface of the second lens, respectively. The
constant optical path length condition requires that xOPLOPL0 = . Using this, an
equation for ( )12 zZ ? in terms of ( )12 xX ? is obtained, as in [27] and it is given by:
( ) ( ) ( ) ( )( )[ ]2
0
2
212
12
2
0
22
0
2
2002
12 nn
xXnnnndnnnndzZ
?
??+??=? m (4.83)
According to Snell?s Law the rays are refracted at the back surface of the first lens.
Also using geometrical relation shown in Figure 4.20, the following equations are
obtained:
???
?
???
?=
ir n
n ?? sinarcsin
0
(4.84)
( )
12
12tantan
zZ
xX
ir ?
?=?= ??? (4.85)
By substitution, Equation 4.86 is obtained:
( )
( ) ( )[ ] ???
?
?
?
?
?
??+?
?=
02
212
12
2
12
120arctan
ndzZxXn
xXn
i? (4.86)
Figure 4.20: Illustration of the back surface of the first lens, together with its tangent and
normal lines and the incident and refracted angle definitions.
81
This method allows determining the angle i? and the slope of any tangent line ( Tm )
on this surface for the coordinates 1x of each ray on the x-axis by sampling between 0
and xmax coordinate points. The slope equation is given by:
( )iTm ??= 90tan (4.87)
Once the Tm values between each ray point are calculated with the equations
presented above, these points are connected to form the surface function of the back
surface of the first lens. If the number of traced rays is increased sufficiently, the tangent
lines of the surface between these points become an accurate representation of the
surface. Using regular slope equation of a line, 1z coordinates of these traced rays are
calculated. After this, the 2Z coordinates of these rays are calculated using Equation
4.83. The 2X coordinates are calculated using Equation 4.71. As a result, a coordinate
table for both the back surface of the first lens and the front surface of the second lens is
determined, as applied in [27].
82
CHAPTER 5
DESIGN EXAMPLES AND RESULTS
This chapter provides design examples to the systems discussed in Section 4.4,
Section 4.5, and Section 4.6. The parameters of each design example are calculated with
MATLAB codes developed separately for each design method. The MATLAB and
ZEMAX results are presented and discussed separately for each design method in Section
5.1, Section 5.2, and Section 5.3.
5.1 Off-Axis Mirror-Based Optical System Design for Circularization, Collimation,
and Expansion of Elliptical Laser Beams
5.1.1 Calculated Parameters of the Examples for the First Design Method
Two examples are presented for the first design method. First one utilizes a laser
source with o10 and o30 divergence angles in the parallel and perpendicular transverse
directions. Such laser beam is named as o10 - o30 beam. Second example utilizes a laser
source with o12 and o46 divergence angles and the beam is named as o12 - o46 beam.
Table 5.1 shows the calculated values for the first example of this design method.
These calculated parameters are input in a ray tracing MATLAB code to check if the
system is feasible for the 1.4 rays. A MATLAB generated plot tracing 1.4 rays for the
system with the parameters in Table 5.1 is displayed in Figure 5.1.
83
Table 5.1: Parameters of the first example designed with the first mirror-based design
method
||? ?? ? a (cm) b (cm) d (cm) e (cm) g (cm)
fw
(cm)
o5 o15 o5 0.96 1.04 0.96 3.96 34.49 10
Figure 5.1: MATLAB plot demonstrating the representative beam propagation in the y-z
plane for the first design method first example.
In Figure 5.1, the second mirror seems to obscure a portion of the beam reflected
from the third mirror. One can increase the field angle to reduce the mirror obscuration.
However, this might also increase the aberration level. Displacing the parabolic mirror
around the virtual point source at the focal point of this mirror is another method to
prevent obscuration on the second mirror. The latter method is preferred in this method
to prevent obscuration providing an option to keep the overall system size small rather
than changing the system parameters that may increase the system size.
84
The calculated parameters for the o12 - o46 beam are presented in Table 5.2.
Table 5.2: Parameters of the second example designed with the first mirror-based design
method
||? ?? ? a (cm) b (cm) d (cm) e (cm) g (cm)
fw
(cm)
o6 o23 o3 0.75 0.75 0.75 4.89 20.51 10.00
5.1.2 Calculated Parameters of the Example for the Second Design Method
Only a laser beam with o46 divergence angle in the perpendicular transverse
direction and o12 divergence angle in the parallel transverse direction is used as a design
example for this design method. The parameters found for this system are given in Table
2-8. Obscuration problem does not exist for the design example using these tabulated
parameters. In case of obscuration, rotation and tilt of any mirror is also an option in this
design method as in the first design method.
Table 5.3: Parameters of the example designed with the second mirror-based design
method
||? ?? ? 1g (cm) 2g (cm) 3g (cm) fw (cm)
o6 o23 o5 2.42 9.36 95.41 10.00
5.1.3 Results
As explained in Section 4.1, geometrical ray optics design approach must be
validated that it is a good and valid approximation to physical propagation of a laser
85
beam through the considered optical system. For this purpose, the dimensionless
parameter, ? , which provides a quantitative measure of the diffraction effects and the
validity of the geometrical ray optics design, is used. As long as the 1>>? condition is
satisfied separately in both transverse directions for the first and second mirrors, the
geometrical rays are valid to use for design and analysis purposes. Geometrical rays
reflected from the third mirror are not used for design or analysis purposes so that the ?
parameter is not calculated for this collimating mirror. Calculated values of ? for the
first and second mirrors in each design example are presented in Table 5.4. These values
validate the geometrical ray optics approach for these examples.
Table 5.4: ? parameter calculations of each studied mirror-based design example,
separately for perpendicular and parallel transverse directions
Design Method I Design Method II
o10 - o30 beam o12 - o46 beam o12 - o46 beam
x-axis y-axis x-axis y-axis x-axis y-axis
0w (cm) 0.67 0.22 0.80 0.21 1.03 0.26
fw (cm) 1.67 1.65 3.15 3.07 1.03 1.00
d (cm) 3.75 3.75 5.54 5.54 7.08 7.37
First
mirror
? 2.15e4 7.05e3 3.29e4 8.20e3 1.08e4 2.53e3
0w (cm) 1.67 1.65 3.15 3.07 1.03 1.00
fw (cm) 10.00 9.94 10.36 10.10 10.34 10.12
d (cm) 31.08 31.08 17.75 17.75 86.76 86.76
Second
mirror
? 3.88e4 3.81e4 1.33e5 1.25e5 8.86e3 8.42e3
86
The wavelength and the total power of the laser beam are chosen to be 870 nm. The
beam diameter of the laser beam after collimation at the third mirror is chosen to be 20
cm. The beam characteristic values for o10 - o30 and o12 - o46 laser beams are also
calculated analytically and given in Table 5.5 (where fw , fz0 , f? are the expected beam
waist, the expected Rayleigh range, and the expected divergence half angle of the final
beam circularized and collimated beam respectively).
Table 5.5: Analytical Calculations of the input and output beam for both o10 - o30 and
o12 - o46 beam examples
o10 - o30 beam o12 - o46 beam
||0w 3.18 ?m 2.64 ?m
?0w 1.06 ?m 0.69 ?m
||0z 36.32 ?m 25.30 ?m
?0z 4.04 ?m 1.72 ?m
( )?||,fw 10.00 cm 10.00 cm
( )?||,0 fz 36.11 km 36.11 km
( )?||,f? 2.77 ?rad 2.77 ?rad
?Physical Optics Propagation? tool results for the first design method o10 - o30 and
the o12 - o46 beam examples, and the second design method o12 - o46 beam example are
displayed in Figure 5.2, Figure 5.3, and Figure 5.4 respectively. In these figures, the
input beam spots are determined at the laser source exit. The output beam spots are
determined after reflection from the collimating parabolic mirror (the ?exit plane?),
87
0.5km away from the exit plane, and 1km away from the exit plane respectively. The
figure image sizes are 4 ?m x 10 ?m for the input beams and 40 cm x 40 cm for the
output beams for each example.
The quantitative results are presented in Table 5.6. These values are given separately
for the two transverse directions because there is no such thing as perfect circularization.
Thus, this table also summarizes the circularization and collimation effectiveness for each
design example when compared to Table 5.5. The results presented in these tables reveal
that the beams are circularized and collimated satisfactorily.
Table 5.6: Output beam results of all studied mirror-based design examples
o10 - o30 beam first
design method
o12 - o46 beam first
design method
o12 - o46 beam second
design method
||fw 9.94 cm 10.10 cm 10.12 cm
?fw 10 cm 10.36 cm 10.34 cm
||0 fz 35.70 km 36.49 km 38.56 km
?fz0 36.12 km 37.46 km 36.79 km
||f? 2.79 ?rad 2.74 ?rad 2.74 ?rad
?f? 2.77 ?rad 2.67 ?rad 2.68 ?rad
88
Figure 5.2: First mirror-based design method o10 - o30 beam example for the stated
image sizes a) input (4 ?m x 10 ?m) b) output (40 cm x 40 cm) c) output at 0.5 km
distance (40 cm x 40 cm) d) output at 1 km distance (40 cm x 40 cm).
Figure 5.3: First mirror-based design method o12 - o46 beam example for the stated
image sizes a) input (4 ?m x 10 ?m) b) output (40 cm x 40 cm) c) output at 0.5 km
distance (40 cm x 40 cm) d) output at 1 km distance (40 cm x 40 cm).
Figure 5.4: Second mirror-based design method o12 - o46 beam example for the stated
image sizes a) input (4 ?m x 10 ?m) b) output (40 cm x 40 cm) c) output at 0.5 km
distance (40 cm x 40 cm) d) output at 1 km distance (40 cm x 40 cm).
89
As stated earlier, in the first design, the beam reflected at the second mirror
propagates to the collimating parabolic mirror with a divergence angle that is equal to the
input beam perpendicular divergence angle. Thus, one can expect that the level of
aberrations introduced by the optical system to increase with the increasing values of ?? ,
as seen comparing Figure 5.2 and Figure 5.3. These results reveal that the first design
method can not be efficiently used in the far field region for large ?? values. This
problem is less significant in the second design method because the beam reflected at the
second mirror propagates to the collimating parabolic mirror with a divergence angle that
is equal to the input beam parallel divergence angle. Given that generally ||? is quite low
compared to ?? , the aberrations introduced by the second method are less significant in
the second method. This is proved with the results displayed in Figure 5.4.
5.2 Aspherical Lens-Based Optical System Design for Circularization, Collimation,
and Expansion of Elliptical Laser Beams
5.2.1 Calculated Parameters of the Example for the Two-Lens Design Method
One example is presented for the first design method utilizing a o12 - o46 beam.
The beam (waist) radius fw at the exit of the optical system is a design constraint and
chosen as 2.0 cm. The other variables, namely the initial divergence angles, ||? , ?? , are
the known quantities, and are based on the characteristics of a laser commercially
available. The lens thicknesses 1t , 2t , and fw are analytically calculated values, and are
the input parameters of the MATLAB code calculating 1d , 2d , ?11R , ?12R , ||21R . The
90
remaining radii of curvatures are not calculated since the corresponding surfaces are
optically flat. All parameters are presented in Table 5.7.
Table 5.7: Two-lens system design example parameters
Given
Quantities Calculated quantities Constraints
?? ||? 1
t
(cm)
2t
(cm)
1d
(cm)
?11R , ||11R
(cm)
||12R
(cm)
||21R
(cm)
2d
(cm)
fw
(cm)
o23 o6 2.00 0.70 3.91 1.99 2.01 9.65 15.00 2.00
?11K , ||11K ||12K ||21K
-2.28 -2.28 -2.28
5.2.2 Calculated Parameters of the Example for the Single-Lens Design Method
One example is presented for the first design method utilizing a o12 - o46 beam.
The beam (waist) radius fw at the exit of the optical system is again a design constraint
and chosen as 2.0 cm. All parameters are presented in Table 5.8.
Table 5.8: Single-lens system design example parameters
Given
Quantities Calculated quantities Constraints
?? ||? d(cm) ?1R (cm) ||1R (cm) ||2R (cm) t (cm) fw (cm)
o23 o6 3.91 1.99 -2.48 -5.49 13.00 2.00
?1K ||1K ||2K
-2.28 0.04 -0.44
91
5.2.3 Results
To verify that the 1>>? condition is satisfied, the ? parameters for each surface,
separately in the x-z and y-z planes, are calculated based on the calculated surface
parameters and the surface distances for both design methods. These values are given in
Table 5.9 and 5.10 for the two-lens and single-lens systems, respectively. From the
values presented in these tables, it is conformed that the use of the geometrical ray optics
approach for these examples is a valid approximation.
Both design methods circularize and collimate an elliptical laser beam satisfactorily.
This can also be seen in ray tracing results in x-z and y-z planes for the first and second
design methods, presented in Figure 5.5 and Figure 5.6, respectively. These figures are
determined with a MATLAB code developed to simulate the beam propagation with the
ray tracing method. The ray tracing results obtained using ZEMAX are presented in
Figure 5.7 for two-lens system and in Figure 5.8 for single-lens system.
The wavelength and the total power of the laser beam are chosen to be 870 nm and
30 W respectively. This is a commercially available semiconductor laser with
approximately 2.64 x 0.69 ?m beam spot size at the laser output. The beam diameter of
the laser beam after collimation at the optical system output for both designs is chosen to
be 2.0 cm as stated earlier. Based on this laser beam, the system parameters are
calculated analytically and presented in Table 5.11 where fw , fz0 , f? are the expected
beam waist, the expected Rayleigh range, and the expected divergence half angle of the
final circularized and collimated beam respectively. ( )?||,f? = 13.89 ?rad is the diffraction
limit for a beam with a radius of 2.0 cm.
92
Table 5.9: ? parameter calculations for the first lens-based design method example
separately for perpendicular and parallel transverse directions
x-axis y-axis
11iw (cm) 2.00 0.42
12ow (cm) 2.00 0.42
d (cm) 2.00 2.00
First lens front
surface
? 1.44e5 6.25e3
12iw (cm) 2.00 0.42
21ow (cm) 2.00 2.00
d (cm) 15.00 15.00
First lens back
surface
? 1.92e4 4.00e3
21iw (cm) 2.00 2.00
22ow (cm) 2.00 2.00
d (cm) 0.70 0.70
Second lens
front surface
? 4.12e5 4.12e5
Table 5.10: ? parameter calculations for the second lens-based design method example
separately for perpendicular and parallel transverse directions
x-axis y-axis
1iw (cm) 2.00 0.41
2ow (cm) 2.00 2.00
d (cm) 13.00 13.00
Front surface
? 2.22e4 4.53e4
93
Figure 5.5(a): Ray tracing results for the first lens-based design method in the x-z plane.
Figure 5.5(b): Ray tracing results for the first lens-based design method in the y-z plane.
94
Figure 5.6(a): Ray tracing results for the second lens-based design method in the x-z
plane.
Figure 5.6(b): Ray tracing results for the second lens-based design method in the y-z
plane.
95
Figure 5.7(a): Ray tracing results obtained with ZEMAX for the two-lens system in the x-
z plane.
Figure 5.7(b): Ray tracing results obtained with ZEMAX for the two-lens system in the
y-z plane.
Figure 5.8(a): Ray tracing results obtained with ZEMAX for the single-lens system in the
x-z plane.
96
Figure 5.8(b): Ray tracing results obtained with ZEMAX for the single-lens system in the
y-z plane.
ZEMAX physical beam propagation results are presented in Figure 5.9 and Figure
5.10. In these figures, the input beam spots are determined at the laser source exit and the
output beam spots are determined at the optical system exit plane and 1 km away from
the exit plane. The figure image sizes are 4 ?m x 10 ?m for the input beam and 8 cm x
8 cm for the output beams. One important note is that the ZEMAX Physical Optics
Propagation tool may not provide precise results for large beam divergence angles and
highly curved refractive surfaces. For such design examples, the beam spots
recommended to be analyzed together with ray tracing results. In this design example,
the beam divergence angle is o46 in the perpendicular transverse direction and the
surfaces are highly curved. Therefore, the beam spots seem to be slightly elliptical in
Figure 5.9 and Figure 5.10, which are slightly different than the ray tracing results
presented in Figure 5.5 and Figure 5.6. The author believes that this minor difference is
attributed to the ZEMAX code?s limitations for highly divergent beams and highly
97
curved surfaces. Regardless of the minor deviations, MATLAB results and ZEMAX
results are in agreement.
In comparison, second design method seems to be more practical than the first
one since there is only one lens and it might be easier to align a single-lens system than to
align a two-lens system in a laboratory environment. However, two-lens system has
more degrees of freedom and, therefore, the optical parameters can be more widely
adjusted. Also considering the size of it, it may get bulky and hard to use in the
laboratory environment. The overall system length is smaller (21.61 cm for the two-lens
example, where as 16.91 cm for the single lens example) for the second method for these
particular examples. However, this does not have a significant meaning since the
distance between the lenses in the two-lens system is an input parameter and can be
selected differently, if desired.
On the other hand, the two-lens system becomes more practical for large difference
between the transverse divergence angles and large output beam spot, fw . If such
scenario is present, then, the lens thickness in the single-lens system method needs to be
increased to accommodate the requirements. The beam can be further expanded if
required in both design methods. Increasing the output beam?s diameter reduces the
divergence angle and increases the Rayleigh range of the output beam.
98
Figure 5.9: Two-lens system ZEMAX plots a) input beam (window size: 4 ?m x 10 ?m)
b) output beam at the system exit plane (window size: 8 cm x 8 cm) c) output beam at 1
km distance (window size: 8 cm x 8 cm).
Figure 5.10: Single-lens system ZEMAX plots a) input beam (window size: 4 ?m x 10
?m) b) output beam at the system exit plane (window size: 8 cm x 8 cm) c) output
beam at 1 km distance (window size: 8 cm x 8 cm).
99
5.3 Reshaping an Elliptically Diverging Gaussian Laser Beam into a Circular,
Collimated, and Expanded Uniform Beam with Aspherical Lens Design
5.3.1 Calculated Parameters of the Design Example
Table 5.11 displays the ? parameter calculations of the studied design example
separately for the perpendicular and parallel transverse directions design proving that the
geometrical ray optics is a good approximation to physical optics propagation. The input
values of the lenses and the beam characteristics are also pre-determined as shown in
Table 5.12.
Table 5.11: ? parameter calculations of the studied design example separately for
perpendicular and parallel transverse directions
0w (cm) fw (cm) d (cm) ?
Perpendicular transverse direction 1.00 2.00 10.00 14444
Parallel transverse direction 0.21 2.00 10.00 3033
Table 5.12: Input variables and values used in the design example.
Laser Parameters Lens Parameters Beam Size at Reference Planes
Wavelength Divergence Angles Lens Material Lens Thickness Values Input Plane Output Plane
o23=
??
50.11 =t cm 00.10 =x cm Xmax= 2.00 cm 870 nm
o
|| 6=?
BK7 glass
n=1.5095
at 870 nm 50.12 =t cm 21.00 =y cm Ymax= 2.00 cm
The distance 96.11 =d cm is determined by choosing 10 =x cm after the beam is
being collimated at the front surface of the first lens. Also the distance between the two
lenses is chosen as 102 =d cm. Based on the pre-determined 0x and 0y values, the
100
75.1xmax = cm and 37.0ymax = cm values are calculated as the working aperture
values of the first lens in the two transverse directions. The calculated values of the front
surface of the first lens parameter are 0.1=R cm (the radius of curvature value) and
28.2?=K (the conic constant value) in both transverse directions. The back surface of
the second lens is flat, as mentioned earlier. In Table 5.13, the calculated parameters of
the first lens back surface and the second lens front surface are tabulated for this design
example.
Table 5.13: Calculated first lens front surface and second lens back surface function
parameters
First Lens ? Back Surface Second Lens ? Front Surface
x-direction y-direction x-direction y-direction
Radius (cm) 2.34 0.36 7.43 5.47
Conic Constant -1.50 -7.70 -1.50 -2.50
4th order term (cm-3) -9.06e-2 n/a -2.89e-4 n/a
6th order term (cm-3) 1.16e-4 n/a -1.12e-5 n/a
8th order term (cm-3) 1.42e-6 n/a 2.98e-8 n/a
10th order term (cm-3) -7.29e-9 n/a 9.35e-12 n/a
12th order term (cm-3) 1.02e-11 n/a -1.06e-13 n/a
5.3.2 Results
Once the surface characteristics of both lenses are determined, the ray tracing is
computed by the MATLAB code. Figure 5.11(a) and Figure 5.11(b) display the two
lenses and the rays traced in x-z and y-z planes based on the model. Total of 21 rays are
traced, as in the previous section. As it can be seen in these figures, the input beam has
101
Gaussian irradiance distribution; whereas, the output beam has uniform beam with a
constant coordinate difference between each traced ray.
In Figure 5.12(a), Figure 5.12(b), and Figure 5.12(c), the beam profiles at the laser
output, the beam shaper input reference plane, and the output reference planes are plotted
using MATLAB. To test the design model, the normalized irradiance distribution of the
beam at five different locations in between the two lenses using the ?coordinate? maps of
the traced rays is plotted, as shown in Figure 5.13. These distributions are obtained back-
tracing the rays from the output reference plane to the input reference plane with the help
of coordinate transformation equations derived in Section 4.6. Because the model is
developed in two independent transverse directions, the irradiance distributions are also
plotted in these two transverse coordinates for positive values only.
Figure 5.11(a): Ray tracing results in x-z plane.
102
Figure 5.11(b): Ray tracing results in the y-z plane.
Figure 5.12(a): Irradiance distribution at the laser output plane.
103
Figure 5.12(b): Irradiance distribution at the input reference plane.
Figure 5.12(c): Irradiance distribution at the output reference plane.
Parallel Direction (y) Perpendicular Direction (x)
0
0.2
0.4
0.6
0.8
1
0 0.001 0.002 0.003 0.004 0.005
Aperture Coordinate y (m) (z=0 m)
No
rm
ali
ze
d I
rra
dia
nc
e
0
0.2
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1
0 0.005 0.01 0.015 0.02 0.025
Aperture Coordinate x (m) (z=0 m)
No
rm
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d I
rra
dia
nc
e
104
0
0.2
0.4
0.6
0.8
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0 0.002 0.004 0.006 0.008 0.01
Aperture Coordinate y (m) (z=0.03 m)
No
rm
ali
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d I
rra
dia
nc
e
0
0.2
0.4
0.6
0.8
1
0 0.005 0.01 0.015 0.02 0.025
Aperture Coordinate x (m) (z=0.03 m)
No
rm
ali
ze
d I
rra
dia
nc
e
0
0.2
0.4
0.6
0.8
1
0 0.005 0.01 0.015
Aperture Coordinate y (m) (z=0.055 m)
No
rm
liz
ed
Ir
ra
dia
nc
e
0
0.2
0.4
0.6
0.8
1
0 0.005 0.01 0.015 0.02 0.025
Aperture Coordinate x (m) (z=0.055 m)
No
rm
ali
ze
d I
rra
dia
nc
e
0
0.2
0.4
0.6
0.8
1
0 0.005 0.01 0.015 0.02
Aperture Coordinate y (m) (z=0.08 m)
No
rm
ali
ze
d I
rra
dia
nc
e
0
0.2
0.4
0.6
0.8
1
0 0.005 0.01 0.015 0.02 0.025
Aperture Coordinate x (m) (z=0.08 m)
No
rm
ali
ze
d I
rra
dia
nc
e
0
0.2
0.4
0.6
0.8
1
0 0.005 0.01 0.015 0.02 0.025
Aperture Coordinate y (m) (z=0.115 m)
No
rm
ali
ze
d I
rra
dia
nc
e
0
0.2
0.4
0.6
0.8
1
0 0.005 0.01 0.015 0.02 0.025
Aperture Coordinate x (m) (z=0.115 m)
No
rm
ali
ze
d I
rra
dia
nc
e
Figure 5.13: Irradiance distributions at five different locations in between the two lenses
in the parallel and the perpendicular directions.
105
As seen in Figure 5.13, the input-plane and the output-plane beam profiles have the
same shape as the ones in Figure 5.12b and Figure 5.12c. Figure 5.13 also reveals that
the beam changes its profile from Gaussian to uniform faster in the parallel direction (y-z
plane) than in the perpendicular direction (x-z plane). Finally, it is seen that the beam
profile is transformed to a uniformly distributed beam in both directions at the exit plane
of the system.
106
CHAPTER 6
CONCLUSIONS
Edge-emitting semiconductor lasers have advantages over other lasers (i.e. CO2 and
Nd-YAG) such as low cost, compactness, electronic compatibility, broad range of
wavelengths, and high Pulse Repetition Frequency (PRF) values. However, they have
different origins and angles of divergence in the two transverse directions, resulting in the
inherent astigmatism and elliptical beam shape.
For most LIDAR systems, a circular beam profile is required. Besides, some optical
components such as fiber optics require a circular beam profile. In addition to this,
LIDARs might be used at high altitudes so that the laser beam is propagated over long
distances. This raises the need for collimating the beam to reduce the beam spread.
Moreover, the collimated beam allows the receiver to have a low FOV value, which
reduces background noise and susceptibility to multiple scattering. In addition to
circularization and collimation, the laser beam might be expanded to reduce the beam
divergence due to diffraction. This also allows the designed system to achieve eye-safety
without significantly reducing the transmitted pulse power.
This dissertation discusses two mirror-based and two lens-based beam shaping
optical system designs that circularize, collimate, and expand edge-emitting
semiconductor laser beams with elliptical profile for possible application in LIDAR
systems.
107
All the design methods are based on geometrical ray optics approach rather than
physical optics propagation calculations. It is shown quantitatively that this approach is a
good and valid approximation to physical beam propagation for all the designs. In
addition, astigmatism in ?m levels is neglected since the beam is expanded to cm to m
levels and collimated in these designs.
Two examples of mirror-based design methods are developed. The first method uses
two cylindrical hyperbolic mirrors to circularize and expand the beam simultaneously.
The second one uses two cylindrical parabolic mirrors for the same purpose. The
circularized and expanded beam is collimated with a rotationally symmetric parabolic
mirror in both methods. The main difference between the two methods is that the first
one further increases the beam divergence angle in the parallel transverse direction to be
equal to the angle in perpendicular direction while the second one reduces the divergence
angle in the perpendicular direction to be equal to the angle in the other direction. The
design equations are derived analytically and presented in Section 4.4. A MATLAB code
is developed to solve the equations system for the parameters of the mirrors. ZEMAX
simulation results are presented in Section 5.1 to analyze the output beam for both design
methods. The output beam images, which are derived by physically propagating the
beam through the optical systems, reveal that the elliptical input beam is circularized,
expanded, and collimated effectively and the results are consistent with the analytical
calculations. It is further shown that the second design method provides more
satisfactory results than the first one especially for large beam divergence angle values in
the perpendicular transverse direction.
108
Two examples of lens-based design methods are developed. First method consists of
two lenses; whereas, the second one is a single-lens system. The design equations are
derived analytically and a MATLAB code is used to solve for the unknown quantities of
the optical systems. MATLAB and ZEMAX software programs are used to analyze these
systems for particular design examples. The output beam images, which are derived by
physically propagating the beam through the optical systems, reveal that the elliptical
input beam is circularized, expanded, and collimated effectively and the results are only
slightly different with the analytical calculations. The author believes that this minor
difference is attributed to the ZEMAX software?s limitations for highly divergent beams
and highly curved surfaces. Regardless of the minor deviations, MATLAB results and
ZEMAX results are in agreement. In author?s opinion, ZEMAX physical optics
propagation results for reflective surfaces are more accurate than for refractive surfaces.
Hence, this limitation is negligible for the mirror-based design method results. It is
shown that the lens thickness in the second method needs to be increased with the
increasing difference of the beam divergence angles and / or increasing final beam
diameter, which also increases the overall system size. In this case, first design method
becomes a more practical approach.
In addition to high divergence and elliptical beam shape drawback of the edge-
emitting semiconductor lasers, Gaussian irradiance distribution is also undesirable for
several applications such as display technology, lithography, laser printing, and material
processing because these applications require uniform illumination of an extended area.
An optical system design method, which utilizes two aspherical lenses to circularize and
collimate an edge-emitting semiconductor laser beam and to transform its irradiance
109
distribution from Gaussian to a uniform-square distribution, is presented in Section 4.6.
A square pulse function is chosen as the output beam distribution function for simplicity.
The abrupt change in the output irradiance distribution functions (such as in the square
pulse function) causes diffraction effects, which are neglected in this chapter.
Geometrical optics methods such as ray tracing, conservation of energy within a bundle
of rays, and constant optical path length of all rays, are used to design this beam shaper.
Results of a particular design example are presented in Section 5.3. A specific optical
beam shaping system is designed to circularize, collimate, and expand the beam of an
edge-emitting semiconductor laser to a desired diameter and to transform its irradiance
distribution from Gaussian to uniform. This is the main difference of this study with the
studies that are based on the same beam shaping principle in the literature since most of
those studies assume that the input beam is a circular and collimated beam. The distance
between the laser source and the vertex of the first surface of the beam shaper is also an
unknown quantity and solved for in the design.
To summarize, in this dissertation several optical systems are presented to reshape an
edge-emitting semiconductor laser beam as required by a particular application. The
main application type that the author concentrates on is LIDAR systems. A coherent
Doppler LIDAR system is discussed briefly in section 3.3.2. The detailed design
considerations of this and other LIDAR systems are left for future study.
As mentioned in Chapter 2, there is ample room for scientific studies about edge-
emitting semiconductor laser-based LIDAR systems. Such LIDAR systems require beam
shaping optics design for several applications such as atmospheric studies. Therefore, the
specific beam shapers designed for such LIDARs will be a strong contribution to the
110
scientific community. The discussed systems are developed by solving analytical
equations. Another strong contribution of this dissertation is the means that the equations
are derived and solved. The design methods will create a guideline for the ones that will
work on similar topics. Besides, the author uses ZEMAX software both in design and
analysis steps. Therefore, the author presents information about the usage of ZEMAX for
similar purposes.
Generally in optics, the spherical surfaces are preferred to manufacture lenses and
mirrors due to simplicity. However, the aberrations that these surfaces introduce increase
with increasing divergence angles of the laser sources. In order to define the optical
surfaces accurately with minimal aberrations, aspheric surfaces are chosen. These
surfaces are defined with radius of curvature and conic constant quantities. In order to
define these surfaces even more accurately, polynomial deformation terms might be
defined. The weakness of using such surfaces is that it is hard to physically manufacture
them. However, with advances in optical manufacturing technology, this will not be a
problem any more in the near future.
The off-axis mirror systems are comparatively harder to realize in a laboratory
environment than the on-axis lens systems due to the probable alignment problems in 3-D
space. Besides, those systems can be applied in LRF systems but not in coherent LIDAR
systems without modification. Therefore, the first system that the author plans to realize
as a future study is the two-lens system. This system, then, will be employed in a
coherent Doppler LIDAR system, as discussed in section 3.3.2.
Once, this system and required optics design is accomplished, the next step will be to
develop the signal processing unit to produce the LIDAR images of test target objects.
111
After successful data processing techniques are developed, necessary scanning
mechanism is designed depending on the application type, laser type, and system
operation altitude.
In conclusion, this dissertation introduces beam shaping optical system design
methods to circularize/collimate/expand an elliptically divergent beam of an edge-
emitting semiconductor laser and/or to transform such beam?s irradiance distribution
from Gaussian to uniform.
112
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Prentice Hall, Inc., New Jersey, 1988.
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[81] M. Geary, Introduction to Lens Design: With Practical ZEMAX Examples,
Willmann-Bell, 2002.
[82] http://www.zemax.com/kb, ZEMAX Users? Knowledge Base website
[83] K. Moore, ?What is a Point Spread Function?, ZEMAX Users? Knowledge Base,
2005.
[84] A. Locke, ?Exploring Physical Opics Propagation in ZEMAX?, ZEMAX Users?
Knowledge Base, 2005.
[85] A. Locke, ?Exploring Non-Sequential Mode in ZEMAX?, ZEMAX Users?
Knowledge Base, 2005.
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APPENDIX
ZEMAX is a software program, which can be used to design and analyze optical
systems. This appendix section is prepared as a ZEMAX tutorial for the first time users
based on the author?s experiences and on [82- 84]. It does not cover the details for
designing and analyzing an optical system or all of the tools and capabilities of ZEMAX.
For such purposes, the reader is encouraged to refer to the ZEMAX Manual [9]. Another
source to get used to design and analysis with ZEMAX is the ZEMAX Users? Knowledge
Base [82]. This resource contains articles written by ZEMAX specialists, instructors, and
users.
This tutorial is based on Sequential mode ZEMAX. In sequential mode, rays are
traced through a sequence of surfaces from the object surface to the image surface in the
defined order and each ray hits a surface only once. Non-sequential ZEMAX is out of
the scope of this tutorial and introductory information about this mode can be found in [9,
85].
Entering the System and Surface Parameters:
ZEMAX uses a spreadsheet format called the Lens Data Editor (LDE). Most of the
surface parameters are entered in this editor. The main parameters, as seen in Figure A.1,
are:
Surf: Type - the type of surface
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Radius - surface radius of curvature in lens units
Thickness - thickness between the vertex of the currently entered surface to the
vertex of the following surface in lens units
Glass - the material type (glass type, MIRROR, etc.) between the vertex of the
currently entered surface and the vertex of the following surface
Semi-Diameter - size of the surface aperture in lens units
Conic - the conic constant of the surface (zero for spherical surfaces)
Figure A.1: Lens Data Editor display
By default, three surfaces exist in the LDE window: the object surface (OBJ), stop
surface (STO), and image surface (IMA).
It is a better approach to enter the system parameters before entering any surface
parameter. The first system parameter is the system aperture that defines the size of the
beam that will be propagated through the system. System aperture can be defined by
different aperture types: Entrance Pupil Diameter (EPD), Image Space F/#, Object Space
NA, Float By Stop Size, Paraxial Working F/#, Object Cone Angle. The most common
one is the EPD, which is the diameter of the pupil in lens units as seen from object space.
The second issue that needs to be addressed is the illumination of the entrance pupil.
By default, the entrance pupil is always illuminated uniformly. This can be changed
using the Apodization Type. For instance, Gaussian apodization imparts an amplitude
variation over the pupil that is Gaussian in form. The Apodization Factor refers to the
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rate of decrease of the beam amplitude as a function of radial pupil coordinate and the
beam amplitude is normalized to unity at the center of the pupil.
Analysis of systems that collimates the input beam needs special attention. If Afocal
Image Space box is checked, ZEMAX will perform most analysis features in a manner
appropriate for optical systems with output beams in image space that are nominally
collimated.
Another system parameter that needs to be set is the field of view value. The field of
view is set on the Field Data dialog box. The default field entry is on-axis only.
ZEMAX supports 4 different models for defining fields: Angle, Object Height, Paraxial
Image Height, Real Image Height. The most common one is Angle that is entered in
degrees unit that the chief ray makes with respect to the object space z-axis. The
wavelengths are set on the Wavelength Data dialog box.
Lens unit must also be chosen before entering the parameters. There are four choices
for lens units in ZEMAX: millimeters, centimeters, inches, or meters. Lens unit defines
the unit of measure for dimensions in most of the spreadsheet editors. These dimensions
apply to data such as radii, thickness, entrance pupil diameters, and most other
parameters in ZEMAX. Afocal mode units may be microradians, milliradians, radians,
arc-seconds (1/3600 of a degree), arc-minutes (1/60 of a degree), or degrees.
Once the system settings have been defined, surface parameters can be entered into
the LDE. Each row in the LDE contains data that belong to a single surface. Stop
Surface can be chosen later by checking the Make Surface Stop box in the Surface
Properties dialogue box, as seen in Figure A.2. It is important to define the stop surface
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such that the entrance pupil is on the same axis as the object surface. This is specifically
of importance when designing off-axis systems.
Figure A.2: Surface Properties dialogue box display
Surface properties can be set by double clicking on the left most column of a surface
row, which invokes the Surface Properties dialogue box, as seen in Figure A.2. The
previous/next surface buttons in this box allow rapid navigation through all surfaces.
An aperture on a surface can be defined using the Aperture tab in this dialogue box.
Surface apertures are used to account for the effects of vignetting. The supported types
of surface apertures in ZEMAX are: no aperture, circular aperture, circular obscuration,
rectangular aperture, rectangular obscuration, elliptical aperture, elliptical obscuration,
spider obscuration, user defined aperture, user defined obscuration, and floating aperture.
The aperture and obscuration types define regions, which pass or stop rays, respectively.
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More than one aperture may be described at a given optical element by inserting a
dummy surface with zero thickness at the desired location, and then setting additional
apertures on that surface. This is useful for constructing complex apertures. For
instance, a circular aperture defines an annular region, which vignettes all rays striking
the surface inside of the minimum radius, and outside of the maximum radius. If a ray is
between the minimum and maximum radii, then the ray will be allowed to proceed. A
circular obscuration is the complement of the circular aperture such that if a ray is
between the minimum and maximum radii, then the ray will not be allowed to proceed.
After setting the surface specific settings, glass type for optical components such as
lenses and mirrors must be entered. ZEMAX automatically recognizes the entered glass
type as one of the many glasses that are compiled into the built-in Glass Catalog. For
mirrors, glass type is entered as MIRROR.
A Standard surface can be a plane, spherical, or conic aspheric surface. The only
parameters required besides the glass type are a radius (infinite value corresponds to a
planar surface), a thickness, and a conic constant (zero value corresponds to a sphere).
Other surface types use these same basic data, as well as other values. For instance, the
Even Asphere surface uses all the Standard surface parameters with additional values
describing the coefficients of a polynomial used to represent the deformation terms of the
surface. These values are called Parameters. Parameter meaning changes depending
upon the surface type selected. Another example is the Biconic surface. This surface has
different refractive power values in the two transverse directions. For this surface,
entered radius and conic constant parameters are designated values for the y direction.
Radius and conic constant values for the x direction are entered in the Parameter 1 and
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Parameter 2 boxes. Some surfaces such as the Binary Optic 1 surface type cannot be
described by only a few parameter values. This surface type requires up to several
hundred additional parameters. For such surfaces, a separate editor is used for the extra
data values. The extra data meaning is also different for each surface type.
Basic Analysis Tools:
ZEMAX has various analysis tools. The ones that will briefly be covered in this
tutorial are: 3D Layout diagram, Spot diagram, FFT PSF, and Physical Optics
Propagation tools.
3D Layout: To create the 3D Layout, ZEMAX projects the 3D coordinates of the
rotated optical model onto the 2D plane of the plot. Conceptually, the optical model is
first rendered in an unrotated 3D space, with +z oriented to the right, +y up, and +x away
from the viewer into the page. The coordinates and orientations of all surfaces and
objects are defined by the Global Coordinate Reference Surface. Global Reference
Surface is chosen by checking the Make Surface Global Coordinate Reference box in the
Surface Properties dialogue box. This 3D model is then rotated by the Rotation About X,
Y, and Z values specified in the settings. The resulting 3D model is then projected onto a
2D plane. 3D Layout settings are specified with the dialogue box, seen in Figure A.3.
Ray pattern can be chosen as XY fan, X fan, Y fan, Ring, List, Random or Grid.
Here with the List choice, a pre-defined set of rays are traced through the system. These
rays can be defined with a notepad file in the main ZEMAX directory called
?RAYLIST.txt?. Some of the rays might vignet while propagating through the system.
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These might be omitted in the layout by checking the Delete Vignetted box. Number of
rays that will be traced is also chosen in the settings menu.
Spot Diagram: The spot diagram reveals information about the aberrations of the
system imaging a point source. ZEMAX launches several rays from a point source,
traces them through the optical system, and plots the (x, y) coordinates of all the rays
relative to some common reference. If no aberrations are present, the image of the point
source also appears as a perfect image point. The spot diagram is a collection of points,
with each point representing a single ray. There is no interaction or interference between
the rays. Therefore, the spot diagram is very effective at showing the effects of the
geometric or ray aberrations.
Figure A.3: 3D Layout Diagram Settings dialogue box display
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Figure A.4: Spot Diagram Settings dialogue box display
The pattern of the traced rays and the ray density can be chosen in the Settings
Dialogue box, as seen in Figure A.4. The pattern may be either hexapolar, square, or
dithered. These terms refer to the pattern of rays as they appear in a pupil plane. There is
no ?best? pattern to use since each shows a different character of the spot diagram. The
ray density specifies the number of hexapolar rings to be traced if a hexapolar or dithered
pattern is selected, or the number of rays across the width and height if a rectangular
pattern is selected.
The GEO spot radius listed on the plot for each field point is the distance from the
reference point (which is either the chief ray at the primary wavelength, the centroid of
all the rays traced, or the middle of the spot cluster) to the ray, which is farthest away
from the reference point. On the other hand, the RMS spot radius is the root-mean-
square radial size. The distance between each ray and the reference point is squared, and
averaged over all the rays, and then the square root is taken. The RMS spot radius gives
a rough idea of the spread of the rays, since it depends upon every ray. The GEO spot
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radius only gives information about the one ray, which is farthest from the reference
point.
FFT PSF: The PSF of an optical system is the irradiance distribution that results
from a point source. Aberrations and diffraction effects caused by truncation spread the
image of this point source. The FFT decomposes a spatial distribution into a frequency
domain distribution. A grid of rays is traced from the source point to the exit pupil [83].
For each ray, the amplitude and optical path difference is used to compute the complex
amplitude of a point on the wavefront grid at the exit pupil. The FFT of this grid,
appropriately scaled, is then squared to yield the real valued PSF. Settings of this tool are
set in the dialogue box, as seen in Figure A.5.
Figure A.5: FFT PSF Settings dialogue box display
The size of the ray grid used to sample the pupil is set with Sampling value. The
Display size indicates what portion of the computed data will be drawn when a graphic
display is generated. The Display grid can be any size from 32 x 32 up to twice the
Sampling grid size. Smaller Display sizes will show less data, but at higher
magnification for better visibility. The Image Delta can be selected manually if a
different sampling distance is required. If the Image Delta is zero, ZEMAX uses the
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default spacing and sampling grids. If the Image Delta is greater than zero, then ZEMAX
scales the pupil sampling to yield the desired Image Delta size.
Physical Optics Propagation: Geometrical ray tracing is an incomplete description
of light propagation since propagation of the light is coherent [84]. Geometrical ray
tracing cannot model the interference of the rays and the diffraction effects. Rays
propagate along straight lines without interfering with one another; whereas; wavefronts
propagate while coherently interfering with themselves. ZEMAX has some ray based
diffraction computations, such as the diffraction MTF or PSF. However, these diffraction
computations make a simplifying approximation: that all the important diffraction effects
occur going from the exit pupil to the image. This is sometimes called the "single step"
approximation. Geometrical optics and the single step approximation work quite well for
the majority of traditional optical designs, where the beam is not near focus anywhere
except the final image. However, this model does not work accurately for several
important cases:
? When the beam comes to an intermediate focus, especially near optics that
truncate the beam (rays by themselves do not predict the correct distribution near
focus).
? When the diffraction effects far from focus are of interest (rays remain uniform in
amplitude and phase, wavefronts develop amplitude and phase structure).
? When the propagation length is long and the beam is nearly collimated
(collimated rays will remain collimated over any distance, real beams diffract and
spread).
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With Physical Optics Propagation tool, the wavefront of a beam is modeled using an
array of points. Each point in the array stores complex amplitude information about the
beam. The analysis computes the beam irradiance or phase on a plane tangent to the
chief ray at the point where the chief ray intercepts the surface. The data is shown after
the beam refracts into, or reflects from, the end surface. To propagate the beam from one
surface to another, either a Fresnel diffraction propagation or an angular spectrum
propagation algorithm is used. ZEMAX automatically chooses the algorithm that yields
the highest numerical accuracy. Beams in ZEMAX are always centered on the chief ray
for the selected field and wavelength.
Parameters necessary to use this tool are set in the POP Settings dialogue box, as
seen in Figure A.6. X-Width and Y-Width values specify the initial width in lens units,
whereas X-Sampling and Y-Sampling values specify the number of points used to sample
the beam. Sampling system of this tool uses a constant spacing near the waist, and a
linearly scaled spacing far from the waist. Although the total number of array points
remains constant, the array size and point spacings change as the beam propagates. If the
array width is very large at the beam waist relative to the waist size, then there are
relatively few points across the beam waist. This results in a smaller array size far from
the waist, with a relatively large number of points across the beam size. On the contrary,
if the array size is small at the waist, the array size grows large compared to the beam far
from the waist, leaving few sample points to represent the beam. This inverse
relationship is a necessary but frequently inconvenient product of the Fourier transform
theory used to model the diffraction. Therefore width and sample values must be
carefully adjusted as the beam propagates through the system. It must be checked that
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the beam is not under-sampled at any surface. If this is the case, width and sampling
values can be changed at appropriate intermediate surfaces in order to provide sufficient
data points.
Figure A.6: Physical Optics Propagation Settings dialogue box display
For rotationally symmetric systems, a single rotationally symmetric pilot beam and
propagation method is appropriate. When the beam is astigmatic or anamorphic, and/or
the optical components are cylindrical or toroidal, then the beam must be separated in the
X and Y propagation. Separation of X and Y direction propagation is accomplished by
checking Separate X, Y box in the Settings dialogue box.
The initial beam may be defined using any of these options:
Gaussian Waist - Beam waist radius and decenter values are entered.
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Gaussian Angle ? Beam far-field half divergence angle and decenter values are
entered.
Gaussian Size+Angle ? Beam size at any arbitrary point, far-field half divergence
angle and decenter values are entered.
Top Hat ? Beam waist radius and decenter values are entered. The irradiance
distribution is uniform rather than Gaussian.
User defined initial beam profiles are also possible.
In this tutorial, a quick start tutorial for ZEMAX software program is presented.
Basic analysis tools are also explained briefly. This tutorial is prepared based on the
author?s experiences using ZEMAX and [82- 84].