University of the Witwatersrand

Doctoral Thesis

Beam Shaping and Amplifiers

Author:
Mr. Arthur Justin Harrison

Supervisor:
Dr. Darryl Naidoo

Co-Supervisor:
Prof. Andrew Forbes

A thesis submitted in fulfillment of the requirements
for the degree of Doctor of Philosophy

in the

Structured Light Laboratory
School of Physics

June 9, 2023

http://www.wits.ac.za
https://structured-light.org/
https://www.wits.ac.za/physics/




iii

Declaration of Authorship
I, Mr. Arthur Justin Harrison, declare that this thesis titled, “Beam Shaping and Amplifiers”
and the work presented in it are my own. I confirm that:

• This work was done wholly or mainly while in candidature for a research degree at this
University.

• Where any part of this thesis has previously been submitted for a degree or any other
qualification at this University or any other institution, this has been clearly stated.

• Where I have consulted the published work of others, this is always clearly attributed.

• Where I have quoted from the work of others, the source is always given. With the
exception of such quotations, this thesis is entirely my own work.

• I have acknowledged all main sources of help.

• Where the thesis is based on work done by myself jointly with others, I have made clear
exactly what was done by others and what I have contributed myself.

Signed:

Date: June 9, 2023





v

“I... a universe of atoms, an atom in the universe”

Richard P. Feynman





vii

UNIVERSITY OF THE WITWATERSRAND

Abstract

Doctor of Philosophy

Beam Shaping and Amplifiers

by Mr. Arthur Justin Harrison

Tailored or structured laser light has received much attention from the photonics community
globally. Structured forms of laser light have the potential to enhance many aspects of laser-
enabled processes across a myriad of applications, such as industrial manufacturing processes,
optical communication, quantum computing, medical sciences, and space sciences. While the
laser itself is well over 60 years old, the range of obtainable "structured" laser outputs has been
restricted for many decades since its inception. With recent advances in micro-mechanical
electro-optics, liquid crystal display, and advanced lithographic processing technologies, it is
possible for us to create highly complex forms of structured light with exotic spatial, phase,
and polarization characteristics. As we make the inevitable transitions towards updating the
existing laser technology with structured light systems, there are certain parameters that
need to be met such as average laser power, which is crucial for performing micro and macro
material processing, accounting for 15% of the global demand for lasers. Therefore, in addition
to the selection of structured light, there is a clear need for obtaining high-power structured
light which generally involves the need for optical power amplifiers.

This dissertation explores the generation, power scaling, and characterization of structured
light for application areas that demand high-power laser outputs. To fully characterize the
process of power scaling of structured laser light, an accurate 3D numerical model is required.
Here we present a new 3D model for an end-pumped cylindrical rod Master Oscillator Power
Amplifier (MOPA) system, using contemporary analytical expressions and novel approxima-
tions which demonstrate significant improvements over current comparative 3D modelling
approaches. Then, we explore the amplification of structured light fields, in the form of
higher-order Laguerre-Gaussian modes, using a novel polarization-based dual-pass MOPA.
This system was specifically developed to maximize amplification efficiency while maintain-
ing the purity and complex structures of high-order Laguerre-Gaussian modes and showed
excellent agreement with the 3D simulated results. Finally, we investigate the thermally-
induced aberrations resulting from end-pumping bulk solid-state gain media. We show that
amplification of higher-order Laguerre-Gaussian modes in this aberrated system, specifically
those possessing orbital angular momentum, results in the separation of the phase singulari-
ties, otherwise known as "vortex splitting". We fully study this effect and describe the cause
and rectification of this phenomenon.

HTTP://WWW.WITS.AC.ZA




ix

Acknowledgements
I extend my gratitude to the Council for Scientific and Industrial Research (CSIR) and the
National Laser Center (NLC) for providing me with financial support and the opportunity to
pursue my passion for science. I would like to thank my supervisor, Dr. Darryl Naidoo, for
his patience, willingness to share knowledge, and friendship during this time. I am grateful
for the freedom you gave me to explore science in my own way, thank you very much. I would
like to acknowledge and thank my co-supervisor, Prof. Andrew Forbes, your dedication to
science and your students is inspirational, it has been an honour to be part of your exceptional
group. I want to express my appreciation to my colleagues at the NLC, Dr. Hencharl Strauss,
Dr. Chemist Mabena, Dr. Attie Hendriks, Mr. Ameeth Sharma, and Dr. Nokwazi Mphuthi,
for your valuable discussions, support, and motivation during this time. Thank you to my
student peers at the University of the Witwatersrand for their support.

I would like to give a special acknowledgement to my wife, Melissa Harrison, for her love and
support every day, I couldn’t have done this without you. Throughout this journey, we have
shared all the hardships, including a global pandemic, as well as beautiful and unforgettable
memories of getting married. It has been a roller coaster of emotions, and I love you so much.
Levi, our cat, I would like to thank you for waking us up every morning at 4 am for no reason.
Your contributions have not gone unnoticed.

I’m thankful for the love and support of my parents, Henda and Arthur, and my in-laws,
Michelle and Michele. I love you all. To my sister Nanine and her husband Renier, it is
always a great time when we are together. My brother, Harley, you are such an inspiring
young man, and I am incredibly proud of the gentleman you have become. Never change.
Emma and Kyle, thank you for all of your support. I wish you both a lifetime of love and
happiness.

And finally, to all my dearest friends - Craig, Simone, Preeban, Murali, Greg, Matt, Max,
and all the boys at MSMK - thank you for all the memories, love and support.





xi

Contents

Declaration of Authorship iii

Abstract vii

Acknowledgements ix

1 Introduction 1
1.1 Transverse Resonator Modes . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

1.1.1 The Paraxial Helmholtz Equation . . . . . . . . . . . . . . . . . . . . 1
1.1.2 Optical Resonators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2
1.1.3 The Fundamental Gaussian Mode . . . . . . . . . . . . . . . . . . . . 4

1.2 Higher-Order Transverse Modes . . . . . . . . . . . . . . . . . . . . . . . . . . 5
1.2.1 Laguerre-Gaussian Modes . . . . . . . . . . . . . . . . . . . . . . . . . 6

OAM beams . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
1.2.2 Hermite-Gaussian Modes . . . . . . . . . . . . . . . . . . . . . . . . . 7

Mode converters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
1.2.3 Vector Transverse Modes . . . . . . . . . . . . . . . . . . . . . . . . . 9

Vector Vortex Beams . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
1.3 Beam Shaping . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

1.3.1 Static Beam shaping . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
1.3.2 Dynamic Beam shaping . . . . . . . . . . . . . . . . . . . . . . . . . . 11

Spatial Light Modulators . . . . . . . . . . . . . . . . . . . . . . . . . 11
Digital Micromirror Devices . . . . . . . . . . . . . . . . . . . . . . . . 11

1.4 High-Power Higher-Order Transverse Modes . . . . . . . . . . . . . . . . . . . 12
1.4.1 Intra-Cavity Generation . . . . . . . . . . . . . . . . . . . . . . . . . . 13

Pump Beam Shaping . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
Off-Axis pumping . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
Mirror-Defects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
Interferometrically . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
Spherical Aberration . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
Birefingence and Thermal-effects . . . . . . . . . . . . . . . . . . . . . 14
Spatial-Light Modulator . . . . . . . . . . . . . . . . . . . . . . . . . . 14
Geometric Phase Optics . . . . . . . . . . . . . . . . . . . . . . . . . . 15
Integrated Mode Emitters . . . . . . . . . . . . . . . . . . . . . . . . . 15

1.4.2 Extra-Cavity Generation . . . . . . . . . . . . . . . . . . . . . . . . . . 15
Spiral Phase Plates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
Spatial-Light Modulators . . . . . . . . . . . . . . . . . . . . . . . . . 16
Digital-Micromirrored-Devices . . . . . . . . . . . . . . . . . . . . . . 16



xii

Modal Phase Plate . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
1.5 Solid-State Amplifiers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

1.5.1 Basic Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
1.5.2 MOPAs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
1.5.3 Amplification of Higher-Order Transverse modes . . . . . . . . . . . . 19

1.6 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

2 3D modelling of End-Pumped MOPAs 21
2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
2.2 Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

2.2.1 Pump Beam Modelling . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
Contra-Propagation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
Co-Propagation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26

2.2.2 Absorption Cross-Section . . . . . . . . . . . . . . . . . . . . . . . . . 26
2.2.3 Gain and Absorption coefficients . . . . . . . . . . . . . . . . . . . . . 27
2.2.4 Temperature-Gradient . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
2.2.5 Change in refractive-index . . . . . . . . . . . . . . . . . . . . . . . . . 30
2.2.6 Beam Propagation Method . . . . . . . . . . . . . . . . . . . . . . . . 31

Initialisation of Pump and Seed Fields . . . . . . . . . . . . . . . . . . 31
Iterative procedure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32

2.3 Experimental Setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
2.4 Results and Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35

2.4.1 MMFC Diode Vs FT-Gaussian Spatial Transformation Comparison . . 35
2.4.2 Absorption cross-section calibration . . . . . . . . . . . . . . . . . . . 36
2.4.3 SG of order n fitted to FT-Gaussian for ∆T (r, z) calculation . . . . . 37
2.4.4 FT-Gaussian vs SG thermal gradient . . . . . . . . . . . . . . . . . . 39
2.4.5 Change in Refractive Index . . . . . . . . . . . . . . . . . . . . . . . . 39
2.4.6 Amplification Performance . . . . . . . . . . . . . . . . . . . . . . . . 40

2.5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43

3 Amplification of higher-order Laguerre-Gaussian modes using a dual-pass
MOPA system 45
3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45
3.2 Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47

3.2.1 End-pumped MOPA designs . . . . . . . . . . . . . . . . . . . . . . . 47
3.2.2 Overlap Integral . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49

3.3 Experimental Setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49
3.3.1 LGl

p mode selection using the SLM . . . . . . . . . . . . . . . . . . . . 50
3.3.2 Dual-pass MOPA system . . . . . . . . . . . . . . . . . . . . . . . . . 51

3.4 Results and Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52
3.4.1 Amplified LGl

p Mode Outputs . . . . . . . . . . . . . . . . . . . . . . . 52
3.4.2 Output Power Results . . . . . . . . . . . . . . . . . . . . . . . . . . . 52
3.4.3 Pump-Seed Intensity Overlap . . . . . . . . . . . . . . . . . . . . . . . 55

3.5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56

4 Aberration-induced vortex splitting in amplified orbital angular momentum
beams 57



xiii

4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57
4.2 Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58

4.2.1 Wavefront Aberrations . . . . . . . . . . . . . . . . . . . . . . . . . . . 58
4.2.2 Modal Decomposition . . . . . . . . . . . . . . . . . . . . . . . . . . . 59
4.2.3 Gouy Phase in astigmatic systems . . . . . . . . . . . . . . . . . . . . 60

4.3 Experimental Setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60
4.4 Results and Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61

4.4.1 Pump beam distortion . . . . . . . . . . . . . . . . . . . . . . . . . . . 61
4.4.2 Analysing the thermal aberrations . . . . . . . . . . . . . . . . . . . . 63
4.4.3 How does this affect OAM beams ? . . . . . . . . . . . . . . . . . . . . 63
4.4.4 Analyzing the Splitting . . . . . . . . . . . . . . . . . . . . . . . . . . 64
4.4.5 Modal content . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65
4.4.6 Amplification Results . . . . . . . . . . . . . . . . . . . . . . . . . . . 69

4.5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70

5 Conclusion 73
5.1 Future Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74

5.1.1 Amplification of a propagation invariant vector flat-top beam . . . . . 74
5.1.2 Further Power-Scaling with Slab MOPAs . . . . . . . . . . . . . . . . 77
5.1.3 Thermal Beam Shaping . . . . . . . . . . . . . . . . . . . . . . . . . . 78
5.1.4 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78

Bibliography 81





xv

List of Figures

1.1 Schematic diagram showing the basic components and working principle of a
laser. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

1.2 A schematic diagram of a Gaussian beam (left) confocal resonator design, con-
sisting of two mirrors (magenta) with equal curvatures ρ separated by length L,
with Figure insets (bottom) showing the transverse intensity profile at various
positions inside the resonator. . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

1.3 Grid plot showing the first nine Laguerre-Gaussian LGl
p modes of equation

1.9, for the l, p=0, 1, 2 indices, with a) depicting the normalized transverse
intensity and corresponding b) phase profiles. . . . . . . . . . . . . . . . . . . 6

1.4 An illustration of the azimuthally varying phase profile of an LG mode con-
taining topological charges of a) l = 1 and b) l = 2. The red arrow indicates
the direction of the 0→ 2π magnitude phase ramp. . . . . . . . . . . . . . . . 7

1.5 Grid plots showing the first nine Hermite-Gaussian HGn,m modes of equation
1.10, for the n,m=0, 1, 2 indices, with a) depicting the normalized transverse
intensity and corresponding b) phase profiles. . . . . . . . . . . . . . . . . . . 8

1.6 Schematic diagram showing the transformation from an HG1,0 to LG1
0 mode

using a π/2 astigmatic mode converter configuration. . . . . . . . . . . . . . . 8
1.7 The poles of the HOPS represent the RCP and LCP basis states and the op-

posing equatorial points are the radially and azimuthally polarization states,
respectively. All intermediate points between the poles and equator are ellip-
tically polarized states [50] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

1.8 3D SEM cross-section pictures of a) Higher-order LG phase plate, b) Fresnel
lens and c), Bragg grating coupler. Images courtesy of Raith Nanofabrication. 10

1.9 a)A liquid-crystal spatial light modulator screen is shown with a zoomed-in
view of individual pixels [76]. b)Individual micromirrors in "on" and "off"
states. Reprinted with permission from SPIE [77] . . . . . . . . . . . . . . . . 12

1.10 a) Schematic diagram showing an input beam with intensity I0 passing through
an externally pumped solid-state crystal (magenta) to emerge with an increased
output intensity I(z). While b) illustrates the two gain regions of an amplifier,
above and below the saturation intensity Isat, under steady-state pumping
conditions. Where Lsat is the length of the crystal required to reach Isat. . . . 17

1.11 Schematic diagram depicting the a) side- and b) end-pumping MOPA config-
urations. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

2.1 Illustration of the spatial evolution of a typical MMFC pump beam (red) prop-
agating through a crystal material (magenta cylinder), with co- and contra-
propagating seed beams (green arrows). . . . . . . . . . . . . . . . . . . . . . 24



xvi

2.2 Initial Gaussian beam modulated by the phase of DOE1 that transforms into a
FT beam ΨFT(r) at the Fourier plane of lens f , used for a contra-propagation
simulation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

2.3 FT beam ΨFT(r) modulated by the phase transformation of DOE2 that re-
verses the transformation process back to the original Gaussian beam ΨG(r)

at the Fourier plane of lens f , used for a co-propagation simulation. . . . . . 26
2.4 Absorption cross-section of Nd:YAG crystal (black) and emission spectrum of

the pump diode (red) with resulting weighted overlap integral (blue shaded area). 27
2.5 Schematic diagram of an end-pumped (blue arrows) laser rod (magenta) of

radius R and length L mounted inside a copper heatsink (brown) with red
arrows indicating the radial direction of heat flow. . . . . . . . . . . . . . . . 29

2.6 The schematic diagram illustrates the split-step iterative procedure for the
pump beam (red) and seed beam (green) in a contra-propagating configuration,
where the pump beam travels from right to left and the seed beam travels from
left to right. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32

2.7 Schematic diagram of the single-pass end-pumped experimental configuration. 34
2.8 The figure illustrates a comparison between the FT-Gaussian beam shaping

profiles (represented by dash-dot lines) and the measured MMFC diode out-
put (indicated by markers) at various points along the free-space propagation
corresponding to the crystal length, specifically at z = 0, L/4, L/2, 3L/4, and
L. The inset of the figure displays the experimental 2D intensity profiles of the
MMFC pump beam measured at z = 0, L/4, 3L/4, and L, labeled as a, b, c,
and d, respectively. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35

2.9 Plot a) shows the experimentally measured (± 0.1 nm error bars) central wave-
length λ0 (left axis, black) and spectra bandwidth ∆λ (right axis, magenta),
with the optimal values used in the simulation (asterisk). Plots b) and c)
are experimentally measured (black asterisk) and simulated (magenta circles)
values of the transmitted and input pump beam powers, respectively. . . . . . 36

2.10 2D simulation, using the calibrated absorption cross-section calculation method,
of the absorption into the crystal for the Gaussian-FT evolving pump beam at
maximum pump power Pp= 38 W. . . . . . . . . . . . . . . . . . . . . . . . . 38

2.11 Gaussian-FT pump beam (solid) compared to the "best fit" super-Gaussian
SGn (dotted) used in the calculation of ∆T (r, z). . . . . . . . . . . . . . . . . 38

2.12 3D (top row), 2D (middle row) and 1D (bottom row) visualisation of the ther-
mal gradient ∆T (r, z) inside the Nd:YAG crystal using the SGn approximations
of the FT-Gaussian transformation pump beam. . . . . . . . . . . . . . . . . 40

2.13 3D (top row), 2D (middle row) and 1D (bottom row) visualisation of the ther-
mal gradient ∆T (r, z) inside the Nd:YAG crystal as a result of end-pumping
by a non-diverging super-Gaussian beam (left column) versus the FT-Gaussian
transformation pump beam (right column). . . . . . . . . . . . . . . . . . . . 41

2.14 Comparative plot showing the change in refractive index ∆n(r, z) caused by
the non-diverging super-Gaussian beam of order n=20 (dotted lines) and the
FT-Gaussian transformation pump beam (solid lines), respectively, at z = 0,
L/4, L/2, 3L/4 & L positions of the crystals. . . . . . . . . . . . . . . . . . . 42

2.15 The red, magenta and black plot colors represent the 3D SG, 2D SG and FT-
Gaussian pump beam models, respectively. . . . . . . . . . . . . . . . . . . . 42



xvii

3.1 Schematic diagrams showing the various solid-state crystal (magenta) end-
pumped MOPA configurations to achieve single-pass and/or dual-pass ampli-
fication. a) Single and b) dual-pass MOPA configurations for an unpolar-
ized input beam using a dichroic mirror (yellow rectangles). Diagrams c) and
d) are dual-pass configurations of linearly polarized input beams, requiring
polarization-based optics such as half-wave plates, quarter-wave plates (thin
red rectangle), and polarizing beam splitter cubes (light blue square). The
Fibre-coupled diode laser pump beam (shaded red) is imaged using a single
lens (light blue) into the MOPA system. The output coupler (grey rectangle)
of the laser resonator is shown and the broken line indicates subsequent beam
shaping and/or optical components. . . . . . . . . . . . . . . . . . . . . . . . 48

3.2 Schematic diagram showing a) the master oscillator and beam shaping setup
using the spatial light modulator with 4f spatial filtering, and b) a detailed
optical layout of the novel dual-pass end-pumped MOPA setup used in the
experiment. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49

3.3 Experimental data showing the spatial profiles of the various LGl
p modes se-

lected using complex-amplitude shaping on a spatial-light modulator before
undergoing dual-pass amplification, with the radial cross-section shown as the
green profile. The Laguerre polynomial orders are given in the insets. . . . . 50

3.4 Schematic diagram depicting a) the spatial evolution and beam size of the fibre-
coupled diode laser pump beam at the entrance (ωi) and exit (ωf ) faces of the
Nd:YAG crystal rod, and b) the simulated "unfolded" layout of the dual-pass
MOPA experiment. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51

3.5 Spatial profiles of the various LGl
p modes after dual-pass amplification at 38

W of pump power, with the radial cross-section shown in green. . . . . . . . . 53
3.6 a) Numerically simulated amplification results and b) the experimentally mea-

sured output powers for single (solid lines) and dual-pass (dotted lines) ampli-
fication of the LGl

p modes with incident powers Ps = 100 mW at input pump
powers ranging from Pp = 6 W - 38 W. . . . . . . . . . . . . . . . . . . . . . 54

3.7 Simulated results showing the trend of the normalized output powers versus
the normalized overlap integrals of each LGl

p mode. . . . . . . . . . . . . . . . 56

4.1 Schematic diagram showing the experimental layout with figure insets (top)
showing the pump intensity profile (not to scale) measured at various z-positions
of the Nd:YAG crystal (pink) with (green) ellipticity values (ϵ). The circled
figure inset (bottom right), schematically depicts the cross-section of the fixed
pump ωp (red circle) and varying seed ωs (blue circles) beam sizes at the
pumped face of the crystal (z = 0 mm). . . . . . . . . . . . . . . . . . . . . . 59

4.2 Ray Optics Simulation depicting the difference in the incident angles (αA &

αB) between the outer rays (points A & B) of the pump beam (yellow) normal
to the dichroic mirror (grey), resulting in a distorted pump beam at the focal
plane. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62

4.3 A bar plot showing the Zernike coefficients Cl
n of the thermally-induced wave-

front aberrations for incident pump powers Pp of 25 W (black), 40 W (purple)
and 60 W (blue), respectively. The figure inset shows the corresponding recon-
structed phase profiles of the wavefront aberration for the pump powers ϕPp

.
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62



xviii

4.4 Experimental results (top row) of the LG2
0 OAM modes with β = 0.5 (left grid),

0.8 (middle grid) and 1 (right grid) after interaction with the thermally-induced
pump Pp = 60 W aberration, measured at positions O1 (left column), O2

(middle column) and O3 (right column) with corresponding simulated results
of the beam intensities (middle row) and phases (bottom row). . . . . . . . . 64

4.5 Graph showing the simulated (solid lines) behaviour of Γ through the focus
of lens L2 for β = 0.5 (purple), 0.8 (yellow) and 1 (red), with experimentally
measured Γ values plotted as error bars. An example (right) of the algorithm
used to locate (red crosses) the intensity minima (top) with correlation to the
phase singularities (bottom) used to calculate dsep. . . . . . . . . . . . . . . . 65

4.6 Computational modal decomposition results (top) in the HGn,m basis of LG2
0

modes for β = 0.5, 0.6, 0.7, 0.8, 0.9, 1 aberrated by phase ϕ60 W, with insets
showing the corresponding split OAM spatial profiles. (bottom) Traces of the
main constituent HGn,m modes in the bar plot increase (solid lines left axis)
and decrease (dashed lines right axis) as a function of β. . . . . . . . . . . . . 66

4.7 Differences in Near-Field and Far-Field. In the top figure insets, we see the
transverse intensity and phase profiles of the simulated and reconstructed aber-
rated beam, alongside their modal weightings. The middle and bottom plots
show the evolution of the Gouy phase of the three different mode orders that
possess relevant weightings. Blue(green) lines connecting blue(green) and or-
ange lines show the difference between the phases, and the light blue(green)
lines on the bottom are the length of each corresponding line showing the
module of the difference between the phases. . . . . . . . . . . . . . . . . . . 67

4.8 The output powers of the LG2
0 higher-order OAM modes (marked with black

asterisks) and a comparison with the fundamental Gaussian LG0
0 mode (marked

with black crosses) were measured experimentally for β values of 0.5 (red), 0.8
(yellow), and 1 (purple). An exponential fit was applied to the experimental
data points for both the LG2

0 (represented by solid lines) and the LG0
0 modes

(represented by dashed lines). The right figure insets display the transverse
beam intensity profiles at the maximum pump power Pp = 60 W for the LG2

0

(left column) and LG0
0 (right column) modes for different β values, including

the ellipticity values (marked in green) for the LG0
0 outputs. . . . . . . . . . . 69

5.1 Intensity plot of the VFT beam (magenta) produced by the superposition of a
vortex LG1

0 mode (red) and a Gaussian LG0
0 mode (blue), with equal amplitude

weighting α = 0.5. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75
5.2 Mach-Zehnder interferometer comprising of two polarizing beam splitters (PBS)

to used to create a VFT beam at the output. . . . . . . . . . . . . . . . . . . 76
5.3 Amplification outputs of the end-pumped solid-state amplifier system. The

solid lines and shaded region represents the simulated power levels where the
upper and lower bands account for ±5% fluctuations in power measurements.
The experimental data, shown as asterisks, are for the input VFT power of a)
100 mW and b) 1 W, while the figure insets are the intensity profiles, with 1D
trace in green, of beams at the maximum amplified power. . . . . . . . . . . . 77

5.4 Figure illustrating multi-pass MOPA systems utilizing a) thin disk [266] and
b) slab gain medium [267] geometries to achieve high gain amplification. . . . 78



xix

List of Tables

2.1 Summary of Nd:YAG properties: The parameters marked with superscripts
’a’ and ’b’ were obtained from references [176] and [157], respectively, while
parameter ’c’ was calculated using the method outlined in reference [156]. . . 34

3.1 Summary of Nd:YAG crystal properties used in the simulation. . . . . . . . . 52
3.2 Experimentally measured beam quality M2 factors of the LGl

p modes Before
and After dual-pass amplification at Pp = 38 W. Theoretically, the beam
quality values scale according to M2 = 2p+ |l|+ 1. . . . . . . . . . . . . . . . 53

3.3 A summary of the simulated and experimentally measured single (SP) and
dual-pass (DP) amplified output powers Ps of the various LGl

p modes at maxi-
mum pump power Pp = 38 W with the relative % error between the simulation
and experimental data in the last two columns. The data has been arranged in
descending order of the simulated dual-pass output power Ps. The theoretical
M2 values are provided in the column adjacent to the corresponding LGl

p modes. 55





xxi

List of Publications
1. Harrison, J., Forbes, A. & Naidoo, D. "Amplification of higher-order Laguerre-Gaussian

modes using a dual-pass MOPA system." Optics Express. ISSN: 1094-4087. https:

//opg.optica.org/oe/abstract.cfm?doi=10.1364/OE.483373 (2023).

2. Harrison, J., Buono, W. T., Forbes, A. & Naidoo, D. "Aberration-induced vortex split-
ting in amplified orbital angular momentum beams." Optics Express. (In print).

https://opg.optica.org/oe/abstract.cfm?doi=10.1364/OE.483373
https://opg.optica.org/oe/abstract.cfm?doi=10.1364/OE.483373




xxiii

For my loving wife, Melissa.





1

Chapter 1

Introduction

1.1 Transverse Resonator Modes

This section presents a concise mathematical derivation of the paraxial Helmholtz approx-
imation from Maxwell’s electromagnetic wave equation, with the Kirchoff integral equation
that defines the eigenmode solutions of an optical resonator. These concepts lead to the fun-
damental transverse eigenmode solution in free space, also known as the Gaussian mode. The
importance and properties of the Gaussian mode are emphasized, along with key concepts
related to optical modes in general that will be significant throughout the dissertation.

1.1.1 The Paraxial Helmholtz Equation

In 1864, on the back of his famous collated equations for electromagnetism, Sir James Clerk
Maxwell derived one of the most important and elegant equations in modern physics, the
electromagnetic wave equation, the mathematical framework describing light propagation in
space and time [1]:

(
∇2 − 1

c2
∂2

∂t2

)
Ẽ(r, t) = 0, (1.1)

where, ∇2 = ∂2

∂x2 + ∂2

∂y2 + ∂2

∂z2 is the Laplacian operator in cartesian coordinates, c is the
speed of light in a vaccum and Ẽ(r, t) is the electromagnetic field vector. A popular textbook
solution to the wave equation is a plane wave travelling in the z-direction with the electric-field
vector oscillating in the x̂ and ŷ directions:

Ẽ(z, t) = (Ẽxx̂+ Ẽyŷ)e
−i(kz+wt), (1.2)

where k, w, z and t are the wavenumber, angular frequency, position and time, respectively.
Ẽx = E0

xe
iϕ and Ẽy = E0

ye
iϕ are the complex amplitudes with phase ϕ, which define the

polarization state of the electromagnetic field. Although plane waves do not exist physically,
they do share some resemblance with real laser beams, however, unlike plane waves, laser light
cannot exist as an infinite uniform "sheet" of propagating light, but rather must be spatially
confined.

To obtain solutions of the wave equation, which describes laser light that is spatially confined,
we start by separating the spatial and time-dependent variables of the electromagnetic field
so that Ẽ(r, t) ≡ Ẽ(r)Ẽ(t) = Ẽ(r)e−iwt. Substituting into equation 1.1, one can easily obtain



2 Chapter 1. Introduction

the time-independent Helmholtz equation:

(
∇2 + k2

)
Ẽ(r) = 0, (1.3)

where k = w2/c2 with the complex spatial variable Ẽ(r) = Ẽ0(r)e−ikz. Next, we assume
that the amplitude and phase of Ẽ0(r) vary slowly over some distance z which is much larger
than the wavelength, such that ∂2Ẽ0(r)

∂z2 ≪ k ∂Ẽ0(r)
∂z ≪ k2Ẽ0(r). Now, from the Helmholtz

equation 1.3, applying the slowly varying z approximation and after some algebraic steps [2]
one arrives at the paraxial Helmholtz equation [3]:(

∇2
⊥ − 2ik

∂

∂z

)
Ẽ(r) = 0, (1.4)

where ∇2
⊥ = ∂2

∂x2 +
∂2

∂y2 are the transverse partial derivatives perpendicular to the propagation
direction. The paraxial wave equation is a partial differential equation that describes the
propagation of laser beams in the direction of the optical axis. The solution of equation
1.4 provides a mathematical representation of a laser beam’s electric field Ẽ(r) and intensity
distribution I(r) = |Ẽ(r)|2 as it propagates in free space. In the following section, we discuss
the basic concept of the optical resonator and how it gives rise to natural solutions to the
paraxial wave equation.

1.1.2 Optical Resonators

Radiative energy conservation in atoms is described by the three Einstein coefficients: stimu-
lated emission, spontaneous emission, and absorption [4]. His seminal work revolutionised our
understanding of light interactions with atoms and set the stage for over a century of scientific
advancement. Perhaps the most important, was the LASER, the abbreviation for Light Am-
plification by Stimulated Emission Radiation, first demonstrated by Theodore Maiman and
featured as a single-page in the Nature Journal on August 8, 1960, titled; "Stimulated Opti-
cal Radiation in Ruby" [5]. More than 60 years on, the laser has evolved into a tremendous
variety of complex instruments that have advanced all aspects of the natural and medical
sciences, industrial engineering, and global communication, each requiring lasers with unique
spatial, temporal, spectral, and power characteristics.

Regardless of their complexity or output characteristics, all lasers share three fundamental
components: a gain medium, an optical cavity, and an external energy source, schematically
depicted in Figure 1.1.

A common example of a gain medium is a transparent solid-state material (magenta) doped
with rare-earth ions that can be energized by an external energy source, typically the emission
from a semiconductor diode laser (blue wavey arrows). To explain the fundamental concept
of a laser we simplify the number of energy levels in the rare-earth ions to only two (i.e. El &

Eu), as indicated on the crystal of Figure 1.1. Einstein stated that if an electron (blue dots)
is promoted from a lower energy level El to a higher Eu energy level via absorption (blue
arrow) of a photon, then it will return "spontaneously" in time to the lower state El (green
arrow), and in doing so, emit a photon of energy ∆E = Eu - El (red wavey arrow) to obey
the conservation of energy law. Furthermore, Einstein stated that if a spontaneously emitted
photon encounters an excited electron residing in Eu, then it can instantaneously "stimulate"



1.1. Transverse Resonator Modes 3

e

e

e

e

e

e

Figure 1.1: Schematic diagram showing the basic components and working
principle of a laser.

that electron to transition to the lower energy state (yellow arrow) and produce an identical
photon replica of itself, thus growing exponentially as it transits the crystal volume.

An example of an optical cavity is the Fabry-Perot resonator, which typically consists of two
mirrors - one highly reflective and one partially reflective - separated by a distance L. For a
ray of light that enters a Fabry-Perot resonator, it will remain confined between the mirrors
after an infinite number of round-trip reflections. Such optical systems are generally referred
to as stable resonators. In laser resonator design, the g-parameter is used to determine the
conditions for a stable two-mirror resonator to exist [6]:

gi = 1− L

ρi
& 0 < g1g2 < 1, (1.5)

where i = 1, 2 represents each mirror, ρi is the radius of mirror curvature (ρ = ∞ for flat
mirrors), and L is the distance between the mirrors. All combinations of mirror geometries
and cavity lengths, of which there are many, that satisfy the above constraint are considered
to be stable resonators, detailed examples can be found in reference. Outside of the constraint
values 0 < g1g2 < 1 are the unstable resonators, meaning that an on-axis launched ray of
light will eventually be ejected from the system. Stability diagrams, where the x and y axes
are hyperbolic functions of the g1 and g2 parameters, provide an excellent way to visualize
resonator stability regions and assist in the cavity design process and selection of mirror
curvature combinations and cavity length [7].

In a stable resonator with an excited gain medium, only one spontaneously emitted on-axis
photon is required to seed the lasing action. Each round trip adds more photons to the
electromagnetic field, thus increasing the total optical power contained inside the resonator
cavity. Assuming that the external energy source supplies a constant source of photons, the
optical power in the resonator will reach an asymptote when the rate at which electrons
populate Eu is equal to the rate at which they are stimulated to El, at which point, the
system is in "steady-state" and the optical power transmitted through the partially reflective
mirror is constant.



4 Chapter 1. Introduction

1.1.3 The Fundamental Gaussian Mode

For any stable two-mirror resonator configuration (0 < g1g2 < 1), one can solve for the
steady-state transverse electric field distribution on each of the resonator mirrors i = 1, 2

using the Kirchhoff integral equation [6], which is a scalar solution to the paraxial Helmholtz
equation 1.4. Rewriting the electromagnetic field in cartesian coordinates (x, y) and in scalar
form, we can express the Kirchhoff integral equation as:

γẼi(x2, y2) = i
e−ikL

2Lgjλ

∫ ∫
Ẽi(x1, y1)×

exp

[
−iπ
2Lgjλ

(
G(x21 + y21 + x22 + y22)− 2(x1x2 + y1y2)

)]
dx1dy1,

(1.6)

where G = 2gigj − 1 is the "G-parameter" of the two mirror resonator (i, j = 1, 2 & i ̸= j)
and γ is a proportionality factor. If the transverse electric field distribution on each mirror
remains unchanged during a round trip inside the cavity and is self-repeating, then Ẽi(x, y) is
an eigensolution of the resonator. In steady-state, the transverse modes oscillating between
mirrors i↔ j experience diffraction losses equal to 1− |γ|2 per round trip [7]. Therefore, the
electric fields on either end of the mirror are identical, except for the proportionality factor γ
which affects only the amplitude. There are an infinite number of analytical solutions Ẽn(x, y)

& γn to equation 1.6, each corresponding to a unique eigenmode of the resonator, however,
the physical constraints of the resonator such as the gain medium shape, symmetry of the
excited gain region, thermal-effects, cavity length, mirror geometry, and curvature, etc., all
dictate the steady-state eigenmode solution.

The first steady-state eigensolution of equation 1.6 in a stable resonator cavity is the funda-
mental transverse mode, commonly referred to as the Gaussian beam [8]. The Gaussian laser
beam is radially symmetric, with its intensity distributed in a bell-shaped curve that peaks
on the optical axis, expressed mathematically as:

E(r, z) = E0
ω0

ω(z)
exp

(
− r2

ω2(z)

)
exp

(
−i kr2

2R(z)

)
exp(iξ(z)), (1.7)

where E0 is a constant defining the maximum electric field amplitude and ω0 is the beam waist
radius. The parameters that vary as a function of the longitudinal position z are the beam
radius ω(z), the wavefront radius of curvature R(z) and Gouy phase ξ(z), mathematically
written as:

ω(z) = ω0

√
1 +

(
z

zR

)2

, R(z) = z

[
1 +

(zR
z

)2]
, ξ(z) = arctan

(
z

zR

)
.

The parameter zR = πω2
0/λ is the Rayleigh length or Rayleigh range, which is the distance

from the beam waist where the beam radius increases by a factor of
√
2.

The schematic depicted in Figure 1.2 corresponds to a confocal stable Gaussian beam res-
onator, where both mirrors located at each end of the resonator have the same radius of
curvature, denoted by ρi, and are separated by a distance of L. As illustrated, a Gaussian
beam maintains its Gaussian shape while oscillating within the resonator mirrors, but its



1.2. Higher-Order Transverse Modes 5

Figure 1.2: A schematic diagram of a Gaussian beam (left) confocal res-
onator design, consisting of two mirrors (magenta) with equal curvatures ρ
separated by length L, with Figure insets (bottom) showing the transverse

intensity profile at various positions inside the resonator.

characteristics, such as beam radius and wavefront curvature radius, differ at different lon-
gitudinal positions. Upon reflection from a curved mirror surface (M1), the Gaussian beam
possesses the same wavefront curvature as the mirror itself, denoted by R(L/2) = ρi. This
property holds for mirrors of all curvatures, including flat (plane) mirrors that have a radius
of curvature extending to infinity, i.e., ρ = ∞. As z approaches zero in the confocal resonator
illustrated in Figure 1.2, the wavefront curvature (depicted by the red lines) increases so that
R(0) tends to infinity. This indicates that the wavefront becomes entirely flat at the beam
waist ω0 position.

The beam propagation factor commonly referred to as the "beam quality" or "M-squared
(M2)" is intrinsically linked to the half angle divergence (θ), indicated by the blue line in
Figure 1.2, of a laser beam so that

θ = M2 λ

πω0
(1.8)

The fundamental Gaussian mode has an M2=1 and is often referred to as being "diffraction-
limited". Since the M2 and divergence angle θ of equation 1.8 are directly proportional, a
unique characteristic of the Gaussian beam is that its radius ω(z) varies slowly resulting in
an extended depth of focus.

1.2 Higher-Order Transverse Modes

There exists an infinite number of solutions to the paraxial Helmholtz equation in a stable
resonator, which is determined by the geometry of the gain medium and cavity mirrors. These
include circularly, rectangular and elliptical symmetries solutions, corresponding to Laguerre-
Gaussian (LG), Hermite-Gaussian (HG) and Ince-Gaussian (IG) mode families, respectively,
and expressed in cylindrical (ρ, ϕ, z), cartesian (x, y, z) and elliptical (ξ, η, z) coordinates
[9], respectively. Here, the mathematical formulation of the higher-order circular LG and
rectangular HG mode families is provided.



6 Chapter 1. Introduction

1.2.1 Laguerre-Gaussian Modes

The generalized Laguerre-Gaussian LGl
p eigenmodes are a group of cylindrically symmetrical

solutions to the paraxial Helmholtz equation 1.4. These modes have found wide-ranging
applications in various fields, including optical tweezers [10–12], trapping [13–15], quantum
communication [16–19], free-space [20–23], underwater [24] optical communication, super-
resolution microscopy [25–27], and advanced material processing [28–30]. Mathematically,
the generalized LG function in cylindrical coordinates (ρ, ϕ, z) is expressed as:

LGl
p(ρ, ϕ, z) =

1

ω0

√
2p!

π (|l|+ p)!

(√
2ρ

ω(z)

)|l|

× Ll
p

[
2

(
ρ

ω(z)

)2
]

exp

[
−
(

ρ

ω(z)

)2
]
× exp [i(2p+ |l|+ 1)ξ(z)]

× exp

[
− ikρ2

2R(z)

]
exp [±ilϕ] ,

(1.9)

a) b)

Figure 1.3: Grid plot showing the first nine Laguerre-Gaussian LGl
p modes

of equation 1.9, for the l, p=0, 1, 2 indices, with a) depicting the normalized
transverse intensity and corresponding b) phase profiles.

where l and p are the azimuthal and radial integers of the Ll
p generalized Laguerre polynomial,

which quantifies the topological charge and number of bright intensity rings, respectively.
Figure 1.3, shows the a) transverse intensity and corresponding b) complex phase distributions
of the first 9 LGl

p permutations with azimuthal and radial integers l, p = 0, 1and2 at position
z = 0. In the top left boxes insets of Figure 1.3 a) and b), when the Laguerre polynomial
integers are set to zero (i.e. l = 0, p = 0 → LG0

0) we recover the fundamental Gaussian mode
with a uniform flat phase profile. The beam radius ω(z) and the M2 of the higher-order LGl

p

modes scale according to their azimuthal l and radial p indices as follows:

ω(l,p)(z) = ω0

√
2p+ |l|+ 1, M2

(l,p) = 2p+ |l|+ 1.

where ω0 is the beam waist radius of the fundamental Gaussian mode.



1.2. Higher-Order Transverse Modes 7

a) b)

Figure 1.4: An illustration of the azimuthally varying phase profile of an
LG mode containing topological charges of a) l = 1 and b) l = 2. The red

arrow indicates the direction of the 0→ 2π magnitude phase ramp.

OAM beams

The LGl
p function of equation 1.9 includes a complex phase term that varies with the azimuthal

angle exp[±ilϕ]. This complex phase imparts a quantized momentum of ±lℏ per photon when
the azimuthal index is not zero (|l| > 0), causing the wavefront to corkscrew as it propagates.
The topological charge, given by the magnitude of |l|, determines the number of 0 → 2π phase
ramps contained in a full azimuthal rotation of the transverse beam profile, which is illustrated
in Figure 1.4 for an l = 1 and l = 2 LG beam. These beams are known as orbital angular
momentum (OAM) or vortex beams, and they have a distinct annular intensity profile with
a zero-intensity centre where phase singularities exist. The sign of the topological charge,
±l, determines the rotational direction of the corkscrew, which can be either clockwise or
anti-clockwise, referred to as left (l > 0) and right-handedness (l < 0), respectively. The
detailed properties of OAM beams are extensively covered in good reviews [31, 32].

1.2.2 Hermite-Gaussian Modes

The paraxial Helmholtz equation 1.4 has rectangular symmetric solutions known as the
Hermite-Gaussian HGn,m modes. These modes have found widespread use in various ap-
plications, including free-space optical communication [33–38], optical trapping [39–41], non-
linear optics [42], and electron acceleration [43–45]. Mathematically, HG modes in cartesian
coordinates (x, y, z) are expressed as follows:

HGn,m(x, y, z) =
1

ω0

√
2

2n+mπ n!m!
Hn

( √
2

ω(z)
x

)
Hm

( √
2

ω(z)
y

)

exp

[
−x

2 + y2

ω(z)2

]
exp [iξn,m(z)] exp

[
− ik(x

2 + y2)

2R(z)

]
.

(1.10)

Here, Hj is the Hermite polynomial of order j for the n,m indicies. As before, ω(z) and
R(z) are the beam radius and wavefront radius of curvature at position z, while the Gouy
phase term ξn,m(z) = (n +m + 1) arctan(z/zR). Figure 1.5, shows the a) transverse spatial
intensity and b) complex phase distributions of the first 9 HGn,m permutations of the n,m
indices at position z = 0. As was the case of the LG modes, we recover the fundamental
Gaussian mode when the indices n,m are set to zero (i.e. HG0,0). The rectangular symmetry
of HGn,m modes results in an independent scaling of M2 and beam size along the x- and y-
directions based on the n and m indices, respectively. We can generalize the variations of the



8 Chapter 1. Introduction

a) b)

Figure 1.5: Grid plots showing the first nine Hermite-Gaussian HGn,m

modes of equation 1.10, for the n,m=0, 1, 2 indices, with a) depicting the
normalized transverse intensity and corresponding b) phase profiles.

HGn,m beam size and M2, with respect to the Gaussian as follows:

ωj(z) = ω0

√
2j + 1, M2

j = 2j + 1, for j = n,m,

where ω0 is the beam waist radius of the fundamental Gaussian mode. Here, ωj(z) is the axis
specific (j = x, y) beam radius as a function of distance z with corresponding M2

j value.

Mode converters

The HG field of equation 1.10 does not have an azimuthal term exp[±ilϕ] and therefore does

Figure 1.6: Schematic diagram showing the transformation from an HG1,0

to LG1
0 mode using a π/2 astigmatic mode converter configuration.

not possess OAM. However, it is possible to convert, for example, an HG1,0 mode with a
spherical wavefront into an LG1

0 mode with a helical wavefront possessing OAM. This can
be accomplished using an astigmatic mode converter (AMC), which consists of a pair of
cylindrical lenses with focal length f , as shown in Figure 1.6. The input HG1,0 mode is
angled at 45◦ with respect to the cylindrical lens to induce a relative Gouy phase shift of π/2
between the two intensity lobes, resulting in an LG1

0 beam after the second cylindrical lens
placed exactly f

√
2 from the first lens [46]. Passing an HGn,m mode with indices m and n

through a π/2 mode converter results in an LGl
p mode with azimuthal index l = n−m and

radial index p = min(n,m). Additionally, the induced Gouy phase shift can vary from π/2



1.2. Higher-Order Transverse Modes 9

to π by varying the separation between the cylindrical lenses from f
√
2 to 2f , which reverses

the sign of the topological charge l in the output LG mode. The AMC is a vital tool for
generating and characterizing LG modes [47] and will be discussed more in the generation
section of the dissertation.

1.2.3 Vector Transverse Modes

The state of polarization of light is determined by the oscillation direction (linear, circular,
or elliptical) of the electric field vector as it propagates [48]. Laser beams can be classified
as scalar polarized beams, or "scalar beams", when all electric field vectors have the same
polarization state. On the other hand, if the polarization state of the laser varies across the
beam, it is known as a vector polarized beam, or "vector beam". Vector beams, or VBs,
can occur naturally within a laser resonator, as they are solutions to the vectorial paraxial
Helmholtz equation, but they can also be generated outside of a resonator by combining two
orthogonally polarized scalar transverse modes. To mathematically represent the polarization
distribution of a vector field, it is necessary to first establish an orthogonal basis that forms a
complete set of linearly independent vectors. When mathematically describing such a vector
field |Ψ⟩, it is beneficial to use Bra-Ket notation:

|Ψ⟩ =
√
α ψR |R⟩+

√
1− α ψL |L⟩ , (1.11)

where ψR and ψL are two complex scalar transverse modes, with normalized amplitude co-
efficient α. Here, we used the right and left circular polarization states (RCP and LCP),
represented as |R⟩ and |L⟩, respectively, as the basis vectors, analogous to how x̂ and ŷ are
used in cartesian coordinates. However, one could alternatively opt for horizontal and vertical
{|V⟩ , |H⟩} or diagonal and anti-diagonal {|D⟩ , |A⟩} basis states [48, 49].

Vector Vortex Beams

Figure 1.7: The poles of the HOPS represent
the RCP and LCP basis states and the opposing
equatorial points are the radially and azimuthally
polarization states, respectively. All intermediate
points between the poles and equator are ellipti-

cally polarized states [50]

As a special case of vector beams, we substi-
tute the transverse modes ψR, ψL of equa-
tion 1.11 with LGl

p modes, specifically with
ψR = LG∓l

0 & ψL = LG±l
0 . This unique su-

perposition state produces cylindrical vector
vortex beams (CVVB) [51]. The resulting
polarization distribution can be visually rep-
resented on a modified higher-order Poincaré
sphere (HOPS) [50], as shown in Figure 1.7.
Varying the weighting coefficient α from 0
to 1 moves the location on the HOPS lon-
gitudinally to the |L⟩ and |R⟩ scalar polar-
ization states, respectively, and the equator
when α = 0.5. The latitudinal variation is
achieved by adjusting the relative phase shift
ϕ between the two beams, allowing for full
motion around the HOPS.



10 Chapter 1. Introduction

1.3 Beam Shaping

Selecting the modes described in section 1.2 requires manipulation of the amplitude, phase,
and/or polarization properties of optical light, known as beam shaping. This can be performed
either inside or outside of a laser resonator, known as intra- and extra-cavity generation, re-
spectively [52, 53]. In general, beam shaping requires additional optical elements to modulate
the amplitude, phase and/or polarization of the electric field, which can be closely grouped as
being either static or dynamic in their beam shaping flexibility. There are other methods for
obtaining a desired beam shape, specifically for intra-cavity generation, that can not be cat-
egorized as singular optical elements but rather a result of manipulation of the fundamental
mode of the resonator. Examples of these include, resonator design, tailoring the pump-beam
shape, exploiting thermally-induced aberrations and birefringent properties of materials - ex-
amples will be provided in the generation section. Nevertheless, here, we only report on the
stand-alone singular static and dynamic beam-shaping devices.

1.3.1 Static Beam shaping

The term "static optical elements" refers to optical components that are designed to carry
out a distinct task. These components fall into three primary categories: beam integrators
[54], diffractive optical elements (DOEs) [55], and refractors [56], each of which has several
subcategories. DOEs are the contemporary approach to beam shaping in modern times and

a) b)

c)

Figure 1.8: 3D SEM cross-section pictures of a) Higher-order LG phase
plate, b) Fresnel lens and c), Bragg grating coupler. Images courtesy of

Raith Nanofabrication.

can be fabricated into complex structures with high precision, as highlighted in Figure 1.8,
some of which are used to tailor all properties of the electromagnetic field to achieve any
desired transverse mode. Well-known examples of DOEs include: diffractive gratings [57],
binary phase gratings [58], fresnel zone plates [59], as well as geometric phase elements such
as; spiral phase plates [60, 61], modal phase plates [62], Q-plates [63], S-plates [64], and meta-
surfaces [65]. The fabrication of DOEs often involves lithographic techniques, which typically
entails designing the DOE using computer software, transferring the desired pattern onto a
mask, exposing the substrate to light through the mask, developing the exposed photoresist
and etching the substrate to transfer the pattern into the material [66]. Over the past two



1.3. Beam Shaping 11

decades, lithography has undergone significant technological progress, largely propelled by
the need for silicon-integrated chips that are more compact, and efficient. The fabrication of
DOEs has benefited greatly from these advances, with feature resolutions well below 300 nm
readily achievable, resulting in the high-purity selection of higher-order transverse modes.

1.3.2 Dynamic Beam shaping

In contrast to static optical elements, dynamic beam-shaping devices are not limited to a fixed
beam transformation and can be digitally reprogrammed in real-time, thus allowing for on-
the-fly selection of arbitrary complex phase elements. Liquid-crystal spatial-light modulators
(SLMs) [67], digital micromirror devices (DMDs) [68] and deformable mirrors (DMs) [69], are
exemplary examples of dynamic optical field manipulating devices.

Spatial Light Modulators

A SLM is a liquid crystal display (LCD) with millions of liquid crystal (LC) cells that can be
independently controlled using an externally applied voltage, shown in Figure 1.9 a). Each
cell consists of small rice-shaped molecules that are aligned parallel to one another under zero
bias and collectively rotate when an external voltage is applied across the cell. The rotation
of the molecules corresponds to a specific phase retardance that modifies the phase of the
incident light. This allows for precise and controllable phase changes of up to 8π across the
beam’s spatial profile. Computer-generated holographic images (8-bit) are addressed onto
the SLM display as 256-level grey-scale images, ranging from black to white (i.e. 0 → 256)
corresponding to a 0 → 2π phase shift. Although SLM displays only modulate the phase of
the incident light, they can be used to perform amplitude and phase modulation [70] to select
pure transverse modes, such as the LG and HG modes in section 1.2. The diffraction efficiency
(DE) of an SLM display defines the percentage of input light transformed into the desired
output mode. The level of DE depends on the physical hardware (i.e. fill factor, flickering
and fringing effects), type of beam shaping (i.e. phase-only, or complex-amplitude and phase
modulation), phase modulation depth, and grating period and structure (i.e sinusoidal or
blazed) [71]. Newer SLM devices have ultra-high resolutions, smaller pixel pitch, and higher
refresh rates compared to traditional devices. They can achieve resolutions of 4160 x 2464,
pixel pitch of 3.1 µm, and refresh rates of 60 Hz [72], with some lower resolution devices
capable of refresh rates of 500 Hz [73]. The LC pixels are susceptible to thermal damage,
therefore, are limited to input optical powers of around 2 W/cm2 with newer water-cooled
versions capable of 200 W/cm2 [74]. Therefore, especially for high-power beam shaping,
careful selection of the SLM system to manage DE losses and mitigate thermal limitations is
required for correct and successful implementation of SLMs.

Digital Micromirror Devices

A DMD is a micromechanical system that comprises highly reflective mirrors, which are
comparable in resolution to the SLM displays. Each mirror or "pixel" can independently tilt
between +12◦ and −12◦, corresponding to "on" and "off" states, respectively, as depicted in
Figure 1.9 b). When the mirror is in an "on" state the light falling on the micromirror is
reflected along the optical axis and contributes to the "shaped beam", whereas an "off" state
is diverted away and discarded from the "shaped beam". Therefore, DMDs are referred to
as binary holographic amplitude shaping devices [75]. Although the DMD is an amplitude



12 Chapter 1. Introduction

a) b)

Figure 1.9: a)A liquid-crystal spatial light modulator screen is shown with
a zoomed-in view of individual pixels [76]. b)Individual micromirrors in "on"

and "off" states. Reprinted with permission from SPIE [77]

guiding device, it is possible to achieve complex field modulation of the phase and amplitude
through a binary Lee hologram, whereby the continuous greyscale holograms utilized in the
SLM are modified to binary values [68]. Modern DMD devices have comparable pixel pitch
and resolution to SLM devices, however, the mechanical nature of DMDs grant them a far
superior refresh rate, which can exceed 20 kHz, making them well suited for applications
requiring rapid wavefront corrections. On the contrary, the simple binary nature of the
DMDs systems produces very poor diffraction efficiencies compared to SLMs.

1.4 High-Power Higher-Order Transverse Modes

High-powered laser systems are highly utilized and adept at carrying out complicated op-
erations across multiple industries, including medical procedures, material processing (e.g.
cutting, welding, and drilling), additive manufacturing and military defence systems. Typi-
cally, commercially sold laser systems have a transverse profile that is restricted to either a
Gaussian (single-mode) output or a superposition of multiple modes (multi-mode) to generate
a uniform or flat-top profile at the working surface. The shapes produced by commercially
available laser systems represent only a tiny fraction of the possible spatial profiles achievable
through beam shaping, not to mention the additional polarization and phase degrees of free-
dom that are generally overlooked. Various studies have demonstrated unique light-matter
interactions resulting from vortex modes annular spatial profile, polarization structures, and
orbital angular momentum components. For instance, the polarization structure of a vector
vortex beam can be transferred directly to a material to generate a textured micro-structured
surface that is suitable for advanced material processing and creation of self-disinfecting sur-
faces [78]. Radially polarized vector vortex beams are highly effective in drilling micro-sized
holes and producing near-perfect tapered edges at the output [79]. Additionally, the orbital
angular momentum component of vortex beams can be transferred to a material, resulting
in the creation of chiral micro-needle structures [30]. These are just a few examples from
a small subset of higher-order transverse modes which demonstrate the benefits and vast



1.4. High-Power Higher-Order Transverse Modes 13

potential applications of high-power structured light, over traditional laser beam outputs.
Therefore, it is crucial to conduct research on the production of high-power structured light
using contemporary beam shaping methods. In the literature, one finds that the high-power
generation of higher-order transverse modes is focused primarily on the circular-symmetric
Laguerre-Gaussian modes, particularly scalar and vector OAM beams. Additionally, it is
customary to generate Hermite-Gaussian modes with the purpose of converting them into
Laguerre-Gaussian modes, therefore there is still a lot to explore regarding the benefits of
the other higher-order modes. This section summarizes some of the current developments
towards achieving high-power structured light, either directly generated from the cavity or
created outside the laser cavity using direct beam shaping as well as indirect approaches (i.e.
thermal lensing, cavity design, etc.)

1.4.1 Intra-Cavity Generation

Pump Beam Shaping

Creating an annular pump shape is a straightforward and convenient method for selecting
circularly symmetric LG modes inside a cavity. This has been demonstrated successfully using
capillary fibers [80, 81], Dammann grating binary phase elements [82], and centre-punched
pump mirrors [83], to achieve scaler vortex beams with topological charges of l = 1 with a few
watts of average power output. Annular pumped optical parametric oscillators (OPOs) can
provide more dynamic and tunable scalar vortex emission, with topological charges ranging
from l = 1 to 6, tunable wavelengths over the mid-IR range of 2260-3576 nm, and average
power up to 8 W [84]. Digital micromirror devices (DMDs) have also been used to produce
pump beam shapes resembling the HGn,m modes of Figure 1.5, which result in direct lasing
of scalar HG90,0 modes that could then be transformed into LG90

0 modes using an astigmatic
mode converter (AMC) [85, 86].

Off-Axis pumping

Pumping the gain-medium off-axis is another simple method for inducing scalar HG emission
from a laser cavity [87]. This simple approach combined with the AMC, is an effective way of
generating scalar vortex beams with high topological charges and moderate powers. Recently,
a vertical external cavity surface emitting laser (VECSEL) demonstrated direct emission of
higher-order scalar HG modes with average powers exceeding 2 W [88]. This setup was later
modified to include an AMC inside the laser cavity, enabling the simultaneous emission of
both HG and LG scalar modes at low average power (20 mW) [89]. Another interesting effect
of off-axis pumping, under certain cavity length constraints, is degenerate laser cavities that
can emit multi-axis higher-order scalar HG modes that can be transformed into multi-axis
OAM beams using an external AMC [90].

Mirror-Defects

Circular defects on cavity mirrors can also be used to obtain a desired mode in a cavity. In
fact, the highest topological charge l scalar vortex emission from an intra-cavity resonator
was achieved using an output coupler mirror inscribed with circular markings generated by
ablating the coating layer, which produced l = 288 vortex emission [91]. The author of
a related study claims that this technique can support intra-cavity vortex emission with
topological charges of l > 1000.



14 Chapter 1. Introduction

Interferometrically

Recently, a Sagnac interferometric mode transforming vortex output coupler (VOC) was
demonstrated. The resonator emitted a 95% pure LG1

0 mode which is the highest continuous-
wave average power (31.1 W) scalar vortex emission achieved from an intra-cavity laser at
the time of writing [92].

Spherical Aberration

A novel bounce laser was developed using a simple cavity design with no additional optics, and
only exploited the usually unfavourable effects of spherical aberration to achieve an impressive
16 W average power of pure scalar LG1

0 emission [93]. Other approaches, which also exploited
spherical aberration, but used lenses, have demonstrated scalar LGl

0 mode emission up to
l = ±33 with 1.87 W of output power [94]. An improved version of this design allowed for
scalar LGl

0 emission up to l = ±95, but the power was limited to < 140 mW [95].

Birefingence and Thermal-effects

The LGl
0 mode exhibits both radial and azimuthal polarization states inside the cavity simul-

taneously. However, in the presence of a birefringent material, the S and P-polarization states
become decoupled. As a result, one state may suffer a significant loss, while the other can
continue to oscillate. This straightforward method of producing either radial or azimuthally
polarized vector vortex beams has been well-documented and demonstrated using a c-cut
Nd:YVO4 [96, 97]. In non-birefringent materials, thermal-lens effect-induced bifocusing may
occur at output powers in the few watts range. By adjusting the incident pump beam power
and the relative position of the crystal to the pump beam focus position, the fundamental
cavity oscillation can switch between radial and azimuthal polarization vector vortex modes
[98, 99]. When high-power pumping conditions are applied, a strong bifocusing effect can pro-
duce vector polarization states with high topological charges, ranging from l = 1 to 14 [100].
Recently, a stable, highly pure (95%), and high-power (20 W) radially polarized LGl

0 mode
emission at a wavelength of 2090 nm was achieved by utilizing annular pumping, thermally-
induced birefringence, and bifocusing effects. This was accomplished by using an end-pumped
Ho:YAG crystal with an intra-cavity lens on a translation stage, which was controlled by a
machine learning feedback loop [101].

Spatial-Light Modulator

Adding complex diffractive optical devices inside the laser cavity is a more direct approach to
selecting higher-order modes. One example of this is the digital laser, which utilized an SLM
as an end-mirror and applied a phase mask to control the resonator eigenmode, resulting in
arbitrary scalar LG mode outputs [102]. However, due to the low damage threshold of liquid
crystal cells and high diffraction losses, the digital laser is limited to producing only a few
mW’s of power. A few years later, this concept was improved to include the generation of
vector vortex beams by inserting a birefringent material into the cavity to separate the s- and
p-polarization states. A single phase-only SLM was then used to address each polarization
state independently and to select orthogonal HG modes that superimpose on the return pass
and produce vector vortex modes with l values of 1 and 2 [103].



1.4. High-Power Higher-Order Transverse Modes 15

Geometric Phase Optics

Recently, an intra-cavity metasurface was used to achieve the highest intra-cavity topological
charge output for vector vortex beam. By rotating the metasurface inside the cavity, the
laser cavity produced variable topological charge emissions ranging from l = 10 to l = 100

and demonstrated the first "metasurface assisted laser" [104]. The spatially varying half-wave
phase retarders or S-waveplates have been demonstrated to function inside the laser cavity
and delivered high-purity vector vortex beams (over 90%) with a moderate slope efficiency
(11%) with average output powers < 150 mWs [105]. For the high-power (> 30 W) generation
of scalar vortex beams, the concept of fabricating spiral phase structures directly onto the
mirrors is promising [106].

Integrated Mode Emitters

To enable the integration of next-generation telecommunications and quantum computing, it
is essential to reduce the size of OAM mode-emitting lasers with high topological charges and
tunable wavelength outputs. Various techniques can be employed for achieving this objective,
including microchip lasers [107, 108], VECSELS [89, 109], and complete micron-sized laser
systems [110–112].

1.4.2 Extra-Cavity Generation

Extra-cavity beam shaping devices can easily be added to existing laser systems as a separate
modular system to produce the desired optical modes, however, intra-cavity generated modes
are intrinsically of high-purity and with better output power. In this section, we highlight
the recent advances in the extra-cavity generation of scalar and vector LG and HG modes.

Spiral Phase Plates

A common method for generating scalar vortex beams is to impart the azimuthally varying
phase of the LGl

p function in equation 1.9, onto a Gaussian beam using a spiral phase plate
(SPP). SPPs are fabricated to achieve a fixed topological charge, and the higher the desired
topological charge the more challenging and expensive the fabrication becomes. However,
it is possible to stack or cascade multiple SPPs in series to generate a vortex beam with a
high topological charge. This was demonstrated by stacking 6 SPPs (3 times l = 2 and 3
times l = 4) to create a vortex beam with a total topological charge of l = 18. However,
the purity of the final OAM beam is compromised due to unoptimized spatial overlap as the
beam diverges. A liquid crystal cell-based reprogrammable SPP has been demonstrated as a
cost-effective alternative to expensive devices like SLMs for generating high OAM beams. The
SPP utilizes 24 pie-sliced segments of liquid crystal cells that allow for selective topological
charges of l = 1, 2, 3, 4, 6, 8, and 12, while still maintaining a high fill factor since there are
no individual pixels. This technology has the potential to be useful in preparing high OAM
beams [113]. The spiral phase mirror (SPM) is similar to the SPP but is reflective rather
than transmissive, and imparts the same azimuthally varying phase. With the use of an
aluminium mirror constructed with 125 segments, each imprinting a phase ramp of 160π, the
SPM has achieved the highest topological charge for an OAM beam, with l = 10000. This
is calculated by multiplying the number of segments by the phase ramp value, resulting in
125× 160π

2π = 10010 [114].



16 Chapter 1. Introduction

Spatial-Light Modulators

Traditional SLMs with a display resolution of 1920x1080 pixels have been used to generate
high-purity scalar vortex beams with a topological charge of up to l = 600 [115]. Newer
SLMs with a display resolution of 3840x2160 pixels, four times greater than previous models,
hav achieved topological charges of l = 1400 by following the same optimization process as
previous models [116]. SLMs have also been incorporated into Sagnac interferometer design,
to produce a highly controllable superposition vector vortex beam with dynamic control
over the polaization state. The system demonstrated "perfect" vector vortex beams with
topological charges ranging from l = 1 to 3 [117].

Digital-Micromirrored-Devices

OAM beams with topological charges as high as l = 90 can be generated through binary
amplitude modulation and super-pixel resolution optimization [118]. The deformable mirror
has the potential to handle much higher power beams (up to a few kW) than DMDs or SLMs.
Recently, a continuous surface deformable mirror demonstrated proof-of-concept dynamic
generation of OAM with topological charges ranging from l = 1 to 5 at low powers. [119].

Modal Phase Plate

To improve the sensitivity of detecting gravitational waves at LIGO, an LGl
p mode with l = 3

and radial modes (p = 3) generated at an average power of 130 W, with 97% purity, using a
single phase element made of fused silica and etched with its phase pattern of an LG3

3 mode
with a blazed grating. This is the highest average power extra cavity generation of a pure
higher-order LG mode.

1.5 Solid-State Amplifiers

As we have seen in the previous section, the generation of structured light both inside and
outside the laser cavity is a complex process that involves sensitive optical alignment and
highly-specialized optical components. Optical cavities with structured light outputs are gen-
erally limited to a narrow window of output power before thermally-induced aberrations cause
the cavity to become unstable and cease to lase. In the case of the extra-cavity beam shaping
systems, the static elements, specifically the DOEs, are made from transparent fused silica
which can sustain high average powers. However, they have very stringent input beam require-
ments to operate efficiently and are fabricated for only one beam shape. The dynamic sys-
tems, specifically the SLM, have arbitrary beam-shaping capabilities but are severely limited
to low-power beam selection, especially for producing pure modes where complex-amplitude
modulation is performed. To circumvent the low-power limitations of SLMs, while benefiting
from their unrivalled beam shaping abilities, is to power scale the low-power output using an
optical amplifier.

1.5.1 Basic Theory

Section 1.1.2 explains the concept of a laser resonator, which forms the basis of an optical
amplifier. However, in an optical amplifier, the resonator cavity mirrors are omitted, leaving
only the gain medium and an external energy source. For instance, consider a solid-state
gain medium with a length of L and two energy levels ∆E = Eu − El, each having an



1.5. Solid-State Amplifiers 17

e

e

e

e

e

e

e

e

e

e

e

e

a)

b)

Figure 1.10: a) Schematic diagram showing an input beam with intensity
I0 passing through an externally pumped solid-state crystal (magenta) to
emerge with an increased output intensity I(z). While b) illustrates the
two gain regions of an amplifier, above and below the saturation intensity
Isat, under steady-state pumping conditions. Where Lsat is the length of

the crystal required to reach Isat.

electron population density of Nu and Nl, respectively. By subjecting the gain medium to a
steady supply of external optical energy, population inversion is achieved, with most electrons
residing in the upper energy level (Nu ≫ Nl). When an input optical field of intensity I0

and wavelength λ = hc/∆E enters the excited gain medium at z = 0 (as illustrated in Figure
1.10 a)), it induces stimulated emission, leading to the exponential growth of the optical field
intensity I(z) as a function of z within the gain medium:

I(z) = I0 e
σem[Nu−Nl]z. (1.12)

The exponential growth of the optical field intensity I(z), as given by equation 1.12, depends
on the emission cross-section σem and the difference in electron densities ∆N = Nu −Nl. As
I(z) increases, it eventually reaches the saturation intensity limit Isat = hν/σabsτu, where h,
ν, and τu represent Planck’s constant, the optical frequency, and the decay lifetime of the
upper energy level Eu. At this point, the population inversion state is depleted, and the
electron densities balance out (Nu = Nl). The rate of electrons being excited to Nu through
absorption of the external energy source equals the rate of stimulated emission. Once I(z)
exceeds the saturation limit, the growth of the optical field is linear as a function of z:

I(z) = I0 + g0Isatz, (1.13)



18 Chapter 1. Introduction

where g0 is the small gain signal (g0 = σ∆N). Figure 1.10 b) demonstrates the two operational
regions of an amplifier, the small signal gain region when I(z) < Isat and the saturation gain
region when I(z) > Isat. Amplifiers are typically designed to operate above the Isat region
where efficient energy extraction occurs to minimise beam distortions caused by thermal effect
due to excessive build-up of heat.

1.5.2 MOPAs

Master oscillator power amplification (MOPA) is a system to power scale beams from an
oscillator through a power amplifier (PA). The technique of power scaling is modular in
nature, meaning that multiple MOPAs can be stacked in series to obtain a high laser power
[120]. When considering bulk-solid state crystals, an efficient way to excite or "pump" the gain
medium is using narrow-wavelength diode-lasers in either side- or end-pumping geometries,
as shown in Figure 1.11.

Figure 1.11: Schematic diagram depicting the a) side- and b) end-pumping
MOPA configurations.

Side-pumped PAs can be generalized as systems where the pump photons enter the gain
medium perpendicular with respect to the direction of the optical axis so that the crystal
is homogenously excited. Side-pumped configurations offer high amplification potential but
require a minimum threshold input seed power to amplify efficiently and are subjected to high-
intensity pump radiation, which increases the risk of thermally-induced phase aberrations
[121, 122]. In end-pumped systems, the pump photons are injected parallel to the optical
axis, resulting in spatial confinement to a small region to achieve optimal spatial overlap with
the optical field. End-pumped PAs are ideal for scaling power in the low to intermediate range
(less than 100 W) and function effectively as pre-amplifier PA stages before being injected
into side-pumped MOPAs to achieve high-power output [123]. The solid-state gain media
used in MOPAs are manufactured into various geometries [124], such as; cylindrical rods
[125] and single-crystal fibres (SCFs) [126], which support single or double-pass amplification,
and the rectangular slabs [127], thin-disks [128] which can support multi-pass amplification
for maximal energy extraction.



1.6. Motivation 19

1.5.3 Amplification of Higher-Order Transverse modes

MOPA systems have been implemented to amplify the output of both intra-and extra-cavity
generated higher-order transverse modes, specifically scalar and vector-polarized Laguerre-
Gaussian modes. However, most of the attention has been directed towards the amplification
of vortex beams. Here, we report a few of these examples from the literature that have
achieved high-average power outputs using a MOPA.

End-pumped cylindrical rod MOPAs have been used to demonstrate power scaling of LG1
0

vortex modes with scalar polarization [120], as well as radially [101] and azimuthally [69]
polarized vector vortex modes, achieving average output powers of 40 W, 10.7 W, and 100
W, respectively. Amplification of vortex modes with 3 and 5-lobe petal intensity structures
to 2.4 W and 1.8 W, respectively, demonstrated that complex spatial and phase structures
can be preserved in end-pumped cylindrical rod MOPAs [129]. For side-pumped cylindrical
rod MOPAs, examples range from low-power vector vortex outputs (< 1 W) with controllable
polarization and high-purity [130, 131] to very high-power radially and azimuthally polarized
vector vortex beams with 2.1 kW [132] and 4 kW [133] of average power outputs, using
multiple MOPA systems stacked in series. Recently, an end-pumped rectangular slab MOPA
was used to perform multi-pass amplification of a radially-polarized vector vortex beam to
achieve an average output power of 33.7 W [134]. Power scaling of vortex beams using
thin-disk multipass MOPAs has seen major improvements since it was first demonstrated,
where it achieved an impressive average power of 635 W [135]. Since then, improvements
were made to the MOPA design to increase the number of amplification passes from 20 to
30 [136] and then 44 passes, to achieve an impressive 1.7 kW average power output for a
radially polarized vortex beam [137]. To our knowledge, there is only one instance in the
literature where an arbitrary high-order scalar Laguerre-Gaussian mode generated with an
SLM has been amplified using a single end-pumped Rubidium MOPA stage [138], but at low
output powers < 30 mW. Therefore, based on the literature, the selection and amplification
of higher-order modes using the MOPA strategy have yielded high-average powers sufficient
for industrial applications. However, systems with dynamic selectivity, such as the SLM, are
still far behind in terms of high-power generation.

1.6 Motivation

As we have seen in the previous sections, the complexity of laser beams, accelerated by
advancements in beam-shaping technologies, has transcended from the simple fundamental
Gaussian mode to exotic forms of structured light with complex spatial, phase and polarization
properties. Potential growth for new and emerging application areas that may benefit from
structured light is coupled firstly with our ability to select them at will, and most importantly,
to obtain them at the average power levels that the application necessitates. In fact, there
are currently no commercially available laser systems that deliver on-demand pure higher-
order transverse modes. Nevertheless, there is extensive research to show that many if not
all laser-enables processes can benefit greatly from the ability to tailor light using an ad hoc
approach for each task at hand, where the spatial, phase and polarization degrees of freedom
should be added to the toolbox [30, 139, 140]. The digital spatial-light modulator (SLM)
is a highly versatile and flexible beam-shaping technology that has been shown to produce
structured light with high purity and efficiency, surpassing that of its digital micromirror
device counterpart. SLMs are presently constrained by their thermal limitations, which render



20 Chapter 1. Introduction

them inadequate for direct beam shaping in high-power experiments and applications. As a
result, they are predominantly employed for low-power fundamental research.

In this dissertation, SLM beam-shaping technology is utilized for the on-demand selection
of high-order Laguerre-Gaussian beams used in conjunction with power-scaling systems to
provide a route towards the realization of a system capable of delivering high-power struc-
tured light with no moving parts, reconfiguration or adjustments of any kind. As evident in
the literature, there is limited research regarding amplifying higher-order Laguerre-Gaussian
modes or higher-order modes in general. Presented in the format as separate research articles,
the upcoming chapters 2 to 4 cover various important topics which will be briefly summa-
rized here. In chapter 2, a three-dimensional model for an end-pumped rod MOPA system
is provided, utilizing a step-wise beam propagation method and contemporary analytical ex-
pressions for the gain saturation, thermal-gradient and thermal effects. We expand on current
modelling approaches, which treat the pump beam as a static shape, by using a Gaussian
to flat-top transformation allowing the step-wise model to include pump beam propagation
dynamics. Furthermore, using experimental spectral data, a calibration procedure is demon-
strated to establish the correct physical inputs of the model. In chapter 3, a novel dual-pass
end-pumped rod MOPA design is demonstrated and compared to conventional approaches.
The novel MOPA design is used to perform power scaling of the first nine LGl

p modes, hav-
ing identical beam sizes and input powers, generated via complex amplitude shaping using
an SLM. The developed three-dimensional model was used to analyse the single and dou-
ble pass amplification of each LGl

p mode in terms of beam quality, and correlation between
the output power and spatial overlap with the pump beam. In chapter 4, amplification of
higher-order LG2

0 modes is performed and the thermally-induced aberrations manifesting in
the end-pumped crystal rod are investigated using a Shack-Hartmann wavefront sensor and
analysed using the Zernike polynomials. The phase aberration induced the phenomena of
"vortex splitting" beam distortion. A computational modal decomposition was performed
on the aberrated LG2

0 beams for various input beam sizes. Interestingly, a case is presented
demonstrating the amalgamation of the distortion in the far field through a carefully induced
Gouy phase shift. The dissertation is concluded in chapter 5 with brief summary of each
chapter followed by a discussion on future work.



21

Chapter 2

3D modelling of End-Pumped
MOPAs

2.1 Introduction

Bulk solid-state Master Oscillator Power Amplifier (MOPA) systems have, for several decades,
been of considerable interest in industry to power scale low-power laser beams from a master
oscillator to higher powers [141]. In such a MOPA configuration, the amplifier laser gain
medium is excited using a pump source that is typically a narrow-wavelength diode-laser
(single or multi-mode) in either a side- or end-pumping configuration. In a side-pumped
configuration, pump light from stacked diode-laser bars is directed perpendicular to the optical
axis while the seed beam (output from the oscillator), which is to be amplified, is propagated
along the optical axis [142, 143]. Conversely, in end-pumped systems, the input pump source,
which is typically a multi-mode fibre-coupled (MMFC) diode-laser, impinges a laser gain
medium along the optical axis where the seed beam is co-axially propagated with the pump
beam for optimal spatial overlap. While the design complexity of master oscillators prove to
be considerably lower than MOPA systems in achieving high output powers, they possess high
optical intra-cavity intensities which may lead to a degradation in the output beam parameters
such as purity, power and polarization, especially if specialized intra-cavity optical components
such as wavelength tuning devices and electro-optical q-switches are required. MOPA systems
that contain a low-power oscillator and several sequential amplifiers enable the decoupling of
sensitive intra-cavity optical components in the oscillator to achieve the desired output beam
performance characteristics [120]. Such systems offer flexibility in pulse duration control,
wavelength tuning, output power control, and maintaining the beam quality of the output
from the oscillator. In addition, gradual power scaling reduces the thermal load on the laser
gain medium thereby reducing thermal distortions and preserving the spatial, temporal and
spectral properties of the amplified beam [144]. Such control is pertinent in applications such
as marking and laser shock peening [145].

The two MOPA design configurations above both have advantages and disadvantages. Side-
pumped systems offer higher amplification potential, however, since the entire gain medium
is pumped, the thermal load in the gain medium increases dramatically thereby potentially
distorting the seed beam due to thermally-induced phase aberrations [146–148]. End-pumped
systems offer full control of the thermally induced aberrations to maintain the seed beam
quality, however, at the cost of a lower amplification potential leading to a considerable



22 Chapter 2. 3D modelling of End-Pumped MOPAs

decrease in the output power of the amplified seed beam. In applications that employ high-
brightness laser beams, which require both the output laser beam power and its beam quality
to be maximised, both design configurations may be implemented, however, a prominent
advantage of end-pumped systems is that it may be applied to a variety of laser gain media
geometries such as thin-disks [128], cylindrical rods [149], fibre [150], and slab architectures
[151]. To ensure that both the output power and beam quality are maximised, a stringent
modelling approach is required to optimise the design strategy. For this purpose, we consider
cylindrical rod geometries as they prove to be cost-effective, controllable, robust and still
widely used as pre-amplifier stages for the low to intermediate power scaling of low-power
seed beams.

There have been several seminal papers that advanced the modelling of MOPA systems,
particularly end-pumped systems. These include the simulation of Laguerre-Gaussian beam
propagation in Raman, dye, and free-electron lasers [152], consideration of the transverse
intensity distribution of the pump and seed beam during amplification [153], and derivation
of an analytical expression for the temperature distribution of the pump beam [154]. The
temperature distribution inside the gain medium induces physical thermal expansion and
thermo-optical changes, which influence the propagation of the seed beam through the gain
medium. Over the last two decades, end-pumped isotropic materials, specifically YAG, have
been modelled using only analytical expressions for the steady state rate-equations [155],
temperature distributions of various pump beam profiles [156] and stress-strain effects leading
to seed beam distortions [157] using the beam propagation method [158, 159]. More complex
numerical models using finite element methods have also been investigated [160–162].

End-pumped systems typically employ a fiber-coupled diode laser as the pump source that is
focused to the crystal face or at some position along the length of the crystal. For multi-mode
sources (with core diameters >200 µm), the output rapidly evolves from a top-hat-shaped
beam at the exit face of the fiber to a Gaussian or bell-like transverse profile along the
propagation axis. As the pump beam is a fundamental component in end-pumped systems
as it dictates the overall performance of the amplifier and the gain achievable in the system,
it is necessary that not only the pump spatial profile at its highest fluence be characterized
but so too its dynamic propagation behaviour over the gain medium volume. Current 2D or
3D end-pumped models consider a pump beam output that is static in its spatial intensity
profile and is approximated as being either a uniform beam or top-hat [148, 159, 163, 164],
Gaussian beam [154, 160, 165, 166], super-Gaussian beam [158, 167–170] or an arbitrary
polynomial distribution [171]. With this, the static pump beam at the crystal face either
holds its transverse distribution in propagation over the crystal length (3D models) or is
reduced to two dimensions (2D models).

A second key characteristic of the pump source is its spectral behaviour. This describes
the amplification potential of the amplifier as the gain saturation is based on the output
wavelength of the pump source. The output wavelength of the pump laser is known to vary
based on the driving current applied to the diodes in the pump laser due to changes in
the temperature of the diodes. The vast majority of existing amplifier models consider a
single value for the pump absorption cross-section which is typically related to the maximum
absorption wavelength of the gain material. This critical factor is not static across the current
levels and if incorrectly selected, will lead to significant errors in the determination of the
temperature distribution, amplification potential and overall accuracy of the performance of



2.2. Theory 23

the amplifier.

In this chapter, we provide a new approach to modelling end-pumped Nd:YAG rod amplifiers
in three dimensions with a multi-mode fibre-coupled diode laser as the pump. Here we de-
scribe the pump beam as a complex field that mimics the spatial evolution of a multi-mode
fibre-coupled diode laser using a lossless phase-only Gaussian to Flat-top (FT) beam shaping
technique. Additionally, we describe an approach, with calibration steps, to accurately de-
termine the pump absorption cross-section for arbitrary driving currents. In our model, the
gain medium is treated as a 3D mesh (Nx,y,z) so that the pump transformation occurs as it
propagates longitudinally inside the crystal using a beam propagation method (BPM). This
approach enables the gain saturation, thermal gradient and change in refractive index to be
calculated at each Nz-interval using only analytical expressions in an iterative procedure. To
demonstrate the accuracy of the model, we simulate the amplification of a Gaussian beam
and verify the results experimentally.

2.2 Theory

An attractive approach to modelling end-pumped amplifiers in three dimensions is to make
use of a beam propagation method (BPM) [158]. Such an approach harnesses Fast Fourier
Transforms (FFTs) that are used to simulate accurate propagation of the pump and seed
beams along the gain medium which includes diffraction effects provided that the pump and
seed are complex fields that are solutions to Maxwell’s wave equation. For seed beams that
are eigenmodes, the electric fields are well-defined solutions to Maxwell’s equations, however,
this is not the case for MMFC diode-laser pump beams. Borrowing from the concepts of
lossless beam-shaping, the MMFC pump beam may be described as a metamorphosis of a
Gaussian mode to a FT beam using a phase-only diffractive element. The creation of the
Gaussian to FT phase-element is highly flexible and allows for full control over the desired
beam sizes, divergences and associated propagation distance between the two [172].

The energy-transfer between pump photons and the active ions or dopants in the laser crystal
determines the gain distribution, which results in amplification, and the temperature gradi-
ent which leads to thermal-expansion (stress-strain) and optical (change in refractive index)
variations in the crystal medium. Spatially, the gain and thermal distributions are propor-
tional to the spatial profile and intensity of the pump beam which necessitates the accurate
description of the pump beam evolution over the crystal length. An equally important factor
is the rate of energy-transfer from the pump photons to the crystal material, which is defined
as the absorption-cross section that dictates the magnitude of the gain and temperature dis-
tributions. Therefore, a comprehensive model must consider the spatial evolution and rate of
energy-transfer of the pump beam collectively.

In this section, we discuss the theory for a cubic Nd:YAG ([1 1 1] cut) laser medium. Here,
the amplifier characteristics, especially the thermal effects, may be understood through the
treatment of analytical equations which provides a holistic and quantitative insight into the
working parameters of the amplifier. In the subsections to follow, we will discuss the Gaussian
to FT pump beam shaping process, the treatment of the pump beam absorption cross-section,
and provide analytical expressions for the temperature distribution and the resulting change in
refractive index. This section will be concluded by outlining the BPM to which all analytical
expressions will be combined.



24 Chapter 2. 3D modelling of End-Pumped MOPAs

2.2.1 Pump Beam Modelling

The output of multi-mode fibre-coupled (MMFC) diode pump lasers do not possess a well-
defined phase structure which results in an output that is incoherent and highly divergent.
As such, simulation of its free-space propagation or propagation in complex media is a non-
trivial task. The spatial profile of the output of a MMFC pump beam is flat-top (FT) shaped
at the face of the fiber and rapidly diverges and transforms to a Gaussian-like intensity
profile along the propagation axis, as illustrated in Figure 2.1. It is well known that such a
pump beam may be focused to a Gaussian-like profile, however, beyond the focus, the spatial
intensity profile rapidly evolves to a beam with a central dip with a triangular-shaped profile
in between. To achieve a more stable behaviour, the exit face of the fiber may be imaged
to the desired size using an afocal telescope. Here the beam will evolve from a FT-shaped
beam to a Gaussian-like profile in a smooth and predictable manner. Through the stationary
phase approximation, this behaviour can be modelled by utilising a lossless phase-only beam
shaping technique which transforms a Gaussian beam to a FT beam, at the Fourier plane
of a lens [52, 172, 173]. This approach enables the MMFC pump beam to be modelled as a
complex field that mimics the divergence and spatial intensity transformation in free-space
propagation.

Pump Beam Propagation

Contra-propagationCo-propagation

Flat-top Gaussian

Figure 2.1: Illustration of the spatial evolution of a typical MMFC pump
beam (red) propagating through a crystal material (magenta cylinder), with

co- and contra-propagating seed beams (green arrows).

The phase-only transmission function ΦFT (r) and Fourier lens function Tlens(r) to perform
a Gaussian to FT beam transformation is expressed as:

ΦFT (r) =
πωiωf√
2fλ

∫ ρ

0

√
1− exp (−ρ2)dρ ,

with ρ =

√
2r

ωi
and,

(2.1)

Tlens(r) = exp

[
−ikr2

2f

]
. (2.2)

where λ is the wavelength, k is the wavenumber, r is the radial coordinate, ωi is the initial
Gaussian beam radius, ωf is the radius of the desired FT beam and f is the focal length of
the Fourier lens Tlens, defined by equation 2.2.

For inline end-pumped amplifiers, the seed beam can be made to either co- or contra-propagate



2.2. Theory 25

with respect to the propagation direction of the pump beam. In the co-propagating configu-
ration the seed and pump beams enter the crystal at the same end and propagate in the same
direction, whereas for the contra-propagating configuration, the seed beam and pump beam
propagate in opposite directions and enter the crystal at adjacent ends, as shown in Figure
2.1.

In the following section, we outline the mathematical steps for simulating the Gaussian to
FT pump shaping transformations in both co- and contra-propagation configurations.

Contra-Propagation

The contra-propagation configuration consists of a convergent seed beam that is directed at
a crystal on the opposite end of the highest pump fluence. At this end, the pump beam
has a large Gaussian-like profile and as the seed decreases in size as it traverses the length
of the crystal, it will arrive to its desired width on the opposite end of the crystal where
the pump beam is a tightly focused FT profile. To simulate this case, we create the phase
transformation ΦFT (r), by specifying the desired FT beam radius ωf , Gaussian beam radius
ωi, and the Fourier lens focal length f to produce the transmittance of a single phase-element
DOE1

DOE1 = Tlens(r)× exp [−iΦFT(r)] . (2.3)

For a Gaussian beam of radius wi and power Pp, the corresponding beam will be modulated
with the phase-transformation of DOE1

ΨG(r) =

√
2Pp

πω2
i

exp

[
−r2

ω2
i

]
×DOE1 , (2.4)

The modulated Gaussian beam ΨG(r) will transform gradually to a FT beam ΨFT of radius
wf during free-space propagation over the distance of the Fourier lens f , as depicted by
Figure 2.2. It is important to note that when simulating the contra-propagating case, the
pump beam propagation direction must be flipped so that it appears to travel backwards over
the length of the crystal. Therefore, Pp of the Gaussian beam ΨG(r) is equal to the absorbed
pump power measured after propagating through the crystal and not the total power that was
injected into the crystal. Therefore, the sign of the absorption coefficient in the simulation
must also change from an exponentially decreasing to an exponentially increasing function as
will be discussed later.

Figure 2.2: Initial Gaussian beam modulated by the phase of DOE1 that
transforms into a FT beam ΨFT(r) at the Fourier plane of lens f , used for

a contra-propagation simulation.



26 Chapter 2. 3D modelling of End-Pumped MOPAs

Co-Propagation

In the co-propagating configuration, the seed and pump beams propagate in the same direc-
tion. The divergent seed beam, at its desired beam width, is injected at the crystal end where
the pump beam is a tightly focused FT profile and exits the crystal where the pump beam
has a large Gaussian-like shape. To simulate this configuration, the pump shaping process
must be reversed from a Gaussian-FT to a FT-Gaussian transformation. To achieve this, the
complex-conjugate phase of the FT beam ΨFT(r) at the plane of the Fourier lens f , shown
on the right of Figure 2.2, is extracted and labelled as φFT(r). This is then used to create
the reverse phase-modulation transformation, which is defined as DOE2.

DOE2 = exp [iφFT(r)] . (2.5)

The phase profile DOE2 is then