EFFECT OF A THIN OPTICAL KERR MEDIUM ON A LAGUERRE-GAUSSIAN

BEAM AND THE APPLICATIONS

By

WEIYA ZHANG

A dissertation submitted in partial fulfillment ofthe requirements for the degree of

DOCTOR OF PHILOSOPHY

WASHINGTON STATE UNIVERSITYDepartment of Physics

DECEMBER 2006

c©Copyright by WEIYA ZHANG, 2006All Rights Reserved

c©Copyright by WEIYA ZHANG, 2006All Rights Reserved

To the Faculty of Washington State University:

The members of the Committee appointed to examine the dissertation of

WEIYA ZHANG find it satisfactory and recommend that it be accepted.

Chair

ii

ACKNOWLEDGMENTS

We acknowledge the financial support of NSF (ECS-0354736), the Summer Doctoral Fel-

lows Program provided by Washington State University, and Wright Patterson Air Force

Base.

I would like to thank my adviser, Mark Kuzyk, who has always been encouraging and

patient, for teaching me how to be a scientist by exploring “crazy” ideas. I also thank the

faculty and the staff of the Physics Department for their help during my education. I am

grateful to my classmates and friends at WSU, for the many joys that we have shared.

Finally, I would like to thank my parents, sisters, and my wife, Wen. My endeavour in

the world of physics would have been meaningless without their love.

iii

EFFECT OF A THIN OPTICAL KERR MEDIUM ON A

LAGUERRE-GAUSSIAN BEAM AND THE APPLICATIONS

Abstract

by Weiya Zhang, Ph.D.Washington State University

December 2006

Chair: Mark G. Kuzyk

Using a generalized Gaussian beam decomposition method we determine the propaga-

tion of a Laguerre-Gaussian (LG) beam after it has passed through a thin nonlinear optical

Kerr medium. The orbital angular momentum per photon of the beam is found to be con-

served while the component beams change. We apply our theory to using LG10 beams to

measure the nonlinear refractive index coefficient of the medium with high sensitivity, such

as the Z-scan and I-scan techniques, and to a new optical limiting geometry.

We test the validity of the theory and demonstrate the applications experimentally us-

ing a dye-doped polymer, disperse red 1 (DR1) doped poly(methyl methacrylate) (PMMA)

(DR1/PMMA). In order to do that, we investigate the mechanisms of the nonlinear re-

fractive index change in DR1/PMMA (trans-cis-trans photoisomerization and photore-

orientation) by a three-state model and a holographic volume index gratings recording

experiment, and determine the conditions under which DR1/PMMA acts as an optical

Kerr medium.

iv

Contents

Acknowledgments iii

Abstract iv

List of Figures ix

List of Tables xvi

1 Introduction 1

2 Theory 15

2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

2.2 A review of the Laguerre-Gaussian beams . . . . . . . . . . . . . . . . . . 16

2.3 Optical Kerr effect and trans-cis-trans photoisomerization and photoreori-

entation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

2.3.1 Optical Kerr effect . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

2.3.2 Mechanisms of trans-cis-trans photoisomerization and photoreorien-

tation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24

2.4 Effect of a thin optical Kerr medium on an LG beam . . . . . . . . . . . . 37

2.4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37

2.4.2 Field of the beam immediately after the sample . . . . . . . . . . . 38

2.4.3 Propagation of the beam after the sample . . . . . . . . . . . . . . 41

v

2.4.4 Examples assuming small nonlinear phase distortion . . . . . . . . . 43

2.5 Application: Z scan . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46

2.5.1 Review of the traditional Z scan using a LG00 beam . . . . . . . . . 46

2.5.2 Z scan using a LG10 beam . . . . . . . . . . . . . . . . . . . . . . . 49

2.5.3 Effect of the aperture size: the off-axis normalized transmittance . . 53

2.6 Application: optical limiting . . . . . . . . . . . . . . . . . . . . . . . . . . 54

2.6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54

2.6.2 Effect of the position of the nonlinear thin film . . . . . . . . . . . . 57

2.6.3 Large nonlinear phase distortion . . . . . . . . . . . . . . . . . . . . 60

2.7 Application: Measuring the nonlinear refractive index . . . . . . . . . . . . 71

2.7.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71

2.7.2 I scan . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72

2.7.3 ∆Φmax scan to measure samples with large n2 . . . . . . . . . . . . 77

3 Experiment 89

3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89

3.2 Generating the higher order Laguerre Gaussian beams . . . . . . . . . . . . 90

3.2.1 The principles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90

3.2.2 Making the hologram . . . . . . . . . . . . . . . . . . . . . . . . . . 94

3.2.3 Examining the phase singularity . . . . . . . . . . . . . . . . . . . . 96

3.3 Fabricating the DR1/PMMA Samples . . . . . . . . . . . . . . . . . . . . 97

3.3.1 Solvent-polymer-dye method . . . . . . . . . . . . . . . . . . . . . . 97

3.3.2 polymerization-with-dye method . . . . . . . . . . . . . . . . . . . . 98

3.4 Recording of high efficiency holographic volume index gratings in DR1/PMMA102

3.4.1 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102

3.4.2 Experimental setup . . . . . . . . . . . . . . . . . . . . . . . . . . . 106

3.5 Experiments with the LG10 beam . . . . . . . . . . . . . . . . . . . . . . . 108

vi

4 Results and discussion 115

4.1 Properties of DR1/PMMA . . . . . . . . . . . . . . . . . . . . . . . . . . . 115

4.1.1 Absorption spectrum of DR1/PMMA . . . . . . . . . . . . . . . . . 116

4.1.2 Recording of high efficiency holographic volume index gratings in

DR1/PMMA . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117

4.1.3 Conditions for DR1/PMMA as optical Kerr media . . . . . . . . . . 124

4.2 Z-scan measurement using a LG10 beam . . . . . . . . . . . . . . . . . . . . 125

4.3 I-scan measurement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 130

4.4 Optical limiting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133

5 Conclusion 140

Appendices 144

A Generalized Gaussian Beam Decomposition 144

A.1 General Derivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 144

A.2 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 150

A.2.1 LG00 beam . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 150

A.2.2 LG10 beam . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 152

B Simplifying the normalized Z-scan transmittance T 158

B.1 LG00 beam . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 158

B.2 LG10 beam . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 160

C Evaluating the normalized optical limiting transmittance T 164

C.1 LG00 beam . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 164

C.2 LG10 beam . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 167

D Intensity and and power of an LG beam 173

D.1 Intensity of an LG beam . . . . . . . . . . . . . . . . . . . . . . . . . . . . 174

vii

D.2 Power of an LG beam . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 176

viii

List of Figures

1.1 The intensity profiles (upper) and the wavefront (lower) of a fundamental

gaussian beam ((a) and (c)) and a LG10 beam ((b) and (d)). . . . . . . . . 2

1.2 “Self-lensing” of a fundamental gaussian beam when it traverses a thin op-

tical Kerr medium. O.T., optical thickness. Upper: a fundamental gaussian

beam, whose radial intensity distribution is shown to the left, traverses a

thin optical Kerr medium. Lower: (a) when n2 < 0, the medium resembles

a concave lens; (b) when n2 > 0, the medium resembles a concave lens. . . 5

1.3 Ray diagram of the effect of a positive lens on the propagation of a funda-

mental gaussian beam. (a) If the lens is placed before the minimum beam

waist, the far-field pattern of the beam is more spread out; (b) If the lens

is placed after the beam waist, the far-field pattern of the beam is more

confined. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

1.4 The optical thickness of a thin optical Kerr medium illuminated by a LG10

beam. O.T., optical thickness. Upper: a LG10 beam, whose radial intensity

distribution is shown to the left, traverses a thin optical Kerr medium.

Lower: the optical thickness of the medium when (a) n2 < 0 and (b) n2 > 0. 7

2.1 Intensity profiles of some Laguerre Gaussian beams of different orders. l is

the angular mode number and p is the radial moder number. . . . . . . . 18

ix

2.2 Comparison of the phase profiles of a LG00 beam and a LG1

0 beam near their

beam waists. (A) The phase profile on the transverse plane of the LG00 beam

(the transverse plane is also the plane of equal phase in the LG00 beam.);

(B) The phase profile on the transverse plane of the LG10 beam; (C) The

plane of equal phase of the LG10 beam. . . . . . . . . . . . . . . . . . . . . 19

2.3 Isomers of the DR1 molecule. . . . . . . . . . . . . . . . . . . . . . . . . . 25

2.4 Schematic energy diagram of the photoisomerization process. Path 1: trans

isomers with absorption cross section σt jump to the excited state by ab-

sorbing photons; Path 2: molecules in the trans excited state relax to the cis

ground state with a quantum yield (or probability) of Φtc; Path 3: at room

temperature, cis isomers relax to the trans isomer thermally with a rate of

γ; Path 4: cis isomers with absorption cross section σc jump to the excited

state by absorbing photons; Path 5: molecules in the cis excited state relax

towards the trans ground state with a quantum yield (or probability) of Φct. 26

2.5 The dynamics of ∆n in DR1/PMMA at short time scales as calculated

from Eq.(2.39) with the following parameters: γ = 1 s−1 , ξtcI = 0.01 s−1,

ηtp = 2, and ηc = 1. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34

2.6 Configuration of the LG beam propagation problem. . . . . . . . . . . . . 38

2.7 Schematic diagram of the Z scan experiment. L: lens, S: sample, A: aperture,

and D: detector. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46

2.8 A typical Z-scan trace for positive (solid line) and negative (dotted line)

∆Φ0. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48

2.9 A typical LG10 Z-scan trace for positive (solid line) and negative (circles)

∆Φ0. The T = 1 level is indicated by the dashed line. . . . . . . . . . . . . 50

2.10 Comparison of a typical LG10 Z-scan trace with a typical LG0

0 Z-scan trace.

The values of ∆Φ0 are chosen such that the major peaks (valleys) of the

two traces almost overlap. Also shown is the T=1 line. . . . . . . . . . . . 51

x

2.11 The Z-scan normalized transmittance for a LG10 beam as a function of trans-

verse coordinate R. (∆Φ0 = 0.1) . . . . . . . . . . . . . . . . . . . . . . . 54

2.12 The Z-scan normalized transmittance for a LG10 beam for R = 0, R = 0.05

and R = 0.1. (∆Φ0 = 0.1) . . . . . . . . . . . . . . . . . . . . . . . . . . . 55

2.13 Illustration of the transmittance of the optical limiter. . . . . . . . . . . . 56

2.14 Schematic diagram of optical limiting using the LG beam. L1: focusing

lens, S: nonlinear thin film, L2: Fourier transform lens, A: small aperture,

D: optical component to be protected, f2: focal length of L2. . . . . . . . 57

2.15 Typical curves of normalized optical limiting transmittance T vs. position

Z. ∆Φmax = −0.1. The circled line is for the LG00 beam, the solid line is

for the LG10 beam, and the dashed line shows T = 1. . . . . . . . . . . . . 59

2.16 The normalized optical limiting transmittance T versus the maximum non-

linear phase distortion ∆Φmax in the sample when the incident beam is a

LG00 beam. The position Z of the sample for each of the curve is indicated

by the number along the curve. A sample of negative n2 is assumed. . . . 63

2.17 The normalized optical limiting transmittance T versus the maximum non-

linear phase distortion ∆Φmax in the sample when the incident beam is a

LG10 beam. The position Z of the sample for each of the curves is indicated

by the number along that curve. . . . . . . . . . . . . . . . . . . . . . . . 69

2.18 The maximum nonlinear phase distortion ∆Φmax as a function of the nor-

malized transmittance T . The incident beam is a LG00 beam and the position

of the sample is Z=-3 and Z=3 for the upper and lower curve, respectively.

The dots are the calculated results and the lines are the linear fits. . . . . 73

xi

2.19 The maximum nonlinear phase distortion ∆Φmax as a function of the nor-

malized transmittance T . The incident beam is a LG10 beam and the po-

sition of the sample is Z=-1.73 and Z=1.73 for the upper and lower curve,

respectively. The dots are the calculated results and the lines are the linear

fits. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74

2.20 The maximum nonlinear phase distortion ∆Φmax as a function of the nor-

malized transmittance T . The incident beam is a LG10 beam and the po-

sition of the sample is Z=-8.55 and Z=8.55 for the upper and lower curve,

respectively. The dots are the calculated results and the lines are the linear

fits. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75

2.21 Solid lines: selected curves of the normalized transmittance T versus the

maximum nonlinear phase distortion ∆Φmax in the sample when the incident

beam is a LG10 beam. The position Z of the sample for each of the curves

is indicated by the number along the curve. Dotted line: the coordinates of

the valleys of the T − ∆Φmax curves. . . . . . . . . . . . . . . . . . . . . . 78

2.22 The sample position Z versus the T coordinate of the valley of the corre-

sponding T − ∆Φmax curve. The arrows represent the useful range of the

Z − T curve for determining the position from the transmittance. . . . . . 79

2.23 The sample position Z versus the T coordinate of the valley of the corre-

sponding T −∆Φmax curve. Circles: calculated results. Line: best fit using

an inverse Gauss function (see text for details). . . . . . . . . . . . . . . . 80

2.24 Normalized transmittance, T , versus the maximum nonlinear phase distor-

tion, ∆Φmax, in the sample when the incident beam is a LG10 beam for

selected curves whose Z are between 0.61 and 3.49. The position, Z, of the

sample for each of the curve is indicated by the number along the curve. . 84

xii

2.25 Normalized transmittance, T , versus the maximum nonlinear phase distor-

tion, ∆Φmax, in the sample when the incident beam is a LG00 beam for

selected curves whose Z values are larger than 0. The position, Z, of the

sample for each of the curve is indicated by the number along the curve. . 85

3.1 Schematic diagram of a hologram that converts a LG00 beam into a LG1

0

beam. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91

3.2 Typical holographic pattern that converts a LG00 beam to a LG1

0 beam. . 94

3.3 The multiple orders of beams generated by the binary amplitude hologram. 95

3.4 Schematic diagram of the interference experiment to exam the phase dislo-

cation of a LG10 beam. M: mirror; BS: beam splitter; DP: dove prism. . . . 96

3.5 Typical self-interference pattern of a LG10 beam with a dove prism placed

in one arm. The three-prong fork in the center is evidence that the angular

mode number l of the incident beam is 1 (or -1). . . . . . . . . . . . . . . . 97

3.6 The alumina-filled column used to remove the inhibitor from the MMA. . . 99

3.7 Diagram of the squeezer that is used to press thick polymer films. . . . . . 101

3.8 Diagram of the diffraction of a light beam in an index grating. . . . . . . 103

3.9 Illustration of forming the index grating by two-beam coupling. . . . . . . 104

3.10 Setup of the holographic volume index grating recording in DR1/PMMA

and the in-situ diffraction efficiency measurement system. . . . . . . . . . 106

3.11 Schematic diagram of the setup for the experiments using a LG10 beam

( Z-scan measurement, I-scan measurement and optical limiting). WP:

half wave plate, P1,P2: polarizers, CGH: computer generated hologram,

AP1,AP2: apertures, M1,M2: mirrors, L1-L4: lenses, PH: pin hole, BS:

beam splitter, D1,D2: detectors. . . . . . . . . . . . . . . . . . . . . . . . . 109

4.1 Absorption spectrum of DR1/PMMA. The arrow shows the wavelength

which is used in our experiments. OD, optical density. . . . . . . . . . . . 116

xiii

4.2 Diffraction efficiency as a function of time. . . . . . . . . . . . . . . . . . . 117

4.3 n1 as a function of time in the grating recording experiment. Upper: the

data and the best-fit with a single exponential onset function, Lower: the

data and the best-fit with a biexponential onset function. . . . . . . . . . 119

4.4 Saturation values n1 as a function of the amplitude of intensity modulation

at the front surface of the sample. . . . . . . . . . . . . . . . . . . . . . . 122

4.5 Experimental (squares) and theoretical (solid curve) results of the Z-scan

of a DR1/PMMA sample using a LG10 beam. Also shown is the theory for

a LG00 Z-scan trace (dashed curve). . . . . . . . . . . . . . . . . . . . . . . 126

4.6 ∆Φ0 vs. power of the incident beam. The circles are the experimental data.

The line is a linear fit of the data. Also shown is a data point (the square)

obtained for a beam power higher than the range within which the sample

responds like an optical Kerr medium. . . . . . . . . . . . . . . . . . . . . 129

4.7 Normalized transmittance T as a function of the maximum beam intensity

at the front surface of a DR1/PMMA sample placed at Z = −1.6. The

circles are the experimental data. The line is a linear fit of the data. . . . 131

4.8 The maximum nonlinear phase distortion ∆Φmax as a function of the nor-

malized transmittance T . The incident beam is a LG10 beam and the position

of the sample is Z=-1.6. The dots are the calculated results and the line is

the linear fit. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132

4.9 Optical limiting using a LG10 beam and a DR1/PMMA sample placed at

Z = 0.6. The circles are the experimental data showing the normalized

transmittance (T) as a function of the maximum beam intensity (Imax, bot-

tom axis) at the front surface of the sample. The curve shows the theory for

T vs. the magnitude of the maximum nonlinear phase distortion (|∆Φmax|,

top axis) on the sample assuming the sample is an optical Kerr medium. . 134

xiv

4.10 Power transfer curve of the optical limiting system with sample position

Z = 0.6. The dots are the experimental data, and the line shows the

response of the system with no optical limiting. . . . . . . . . . . . . . . . 135

4.11 Comparison of the optical limiting effect with the sample in front of the

beam focus (negtive Z) and behind the beam focus (positive Z). Left: ex-

perimental results of the normalized transmittance (T) as a function of

the maximum beam intensity (Imax) in DR1/PMMA, where the squares are

data at Z= 0.6, and the circles are data at Z= −7. Right: calculated results

of the normalized transmittance (T) as a function of the magnitude of the

maximum nonlinear phase distortion in the sample (|∆Φmax|), assuming an

optical Kerr medium. The solid line is for Z= 0.6, and the broken line is

for Z= −7. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 136

xv

List of Tables

4.1 Time constants determined from a biexponential onset function fit to grating data.121

4.2 Radius of beam waist, ω0, obtained by Z-scan curve fit. . . . . . . . . . . . . . 130

xvi

Chapter 1

Introduction

In this dissertation we study the interaction between a nonlinear optical Kerr medium in

the form of a thin film and a special, somewhat mysterious, laser beam called a “twisted

light beam”, which has a spiral wave front, a dark center in the transverse plane, and

carries orbital angular momentum (OAM).

It is well known that laser cavities can produce laser beams of different modes, each

of which is a solution of the wave equation under the constrain of the cavity resonator.1–3

In cylindrical coordinates (r, φ, z), a complete set of solutions, known as the Laguerre-

Gaussian (LG) beams,1 can be obtained. Each member of the set is characterized by two

mode numbers, the angular mode number, l (l = 0,±1,±2, ...), and the transverse radial

mode number, p (p = 0, 1, 2, ...), written as LGlp.

Among the LG beams, the LG00 beam is probably familiar to most readers. It is more

frequently referred to as the fundamental gaussian beam, or just a gaussian beam when

there is no ambiguity, due to the gaussian distribution (∼ exp(−r2)) of its intensity in

the beam’s transverse plane (r, φ plane). Its phase profile has no φ dependence, and

resembles that of a plane wave near the beam waist. The high order LG beams are like

the fundamental gaussian beam in many aspects: their beam radii all reach a minimum at

the beam focus and diverge when getting away from the focal point. Meanwhile, there are

1

Figure 1.1: The intensity profiles (upper) and the wavefront (lower) of a fundamentalgaussian beam ((a) and (c)) and a LG1

0 beam ((b) and (d)).

many differences between the different modes, among which the most predominant ones are

the intensity and phase profiles. For example (see Fig. 1.1), the LG10 beam’s phase profile

has a factor of exp(iφ), making the phase at the center (r = 0) undetermined and forming

a screw dislocation,4 and the wavefront is screw-shaped, unlike the flat wavefront of a

plane wave. Corresponding to the phase singularity, the intensity profile is characterized

by a null in the center, in contrast to the bright center of the fundamental LG00 beam.

Sometimes, the non-fundamental LG beams are referred to as “twisted light beams” due

to their twisted wavefront. In section 2.2, we have a more detailed review of the properties

of the LG beams.

Historically the fundamental Gaussian beam LG00 has been the most commonly stud-

ied LG mode in both theory and experiment, probably because among all the modes it

is the only one that resembles a plane wave near the beam waist, and it is widely avail-

able though commercial lasers. However, recently higher order LG beams, especially those

2

with higher angular mode number l, are attracting more attention, because of their in-

triguing properties associated with the non-plane-wave-like phase and intensity profiles.

For example:

• The unique intensity profiles of the high order LG beams allow them to be used as

optical levitators5 and optical tweezers that can trap small particles having not only

an elevated refractive index but also a lower refractive index than the surrounding

medium.6

• The spiral interference pattern formed by LG beams can be used to control the

rotation of the trapped small particles.7–9

• The well-defined null throughout the beam axis helps to align the beam.10

• The screw-like phase singularity makes LG beams suitable for the study of optical

vortices,11, 12 which are named from their similarity to the vortices in fluids.

• Some standard experiments have been revisited by researchers with high LG beams,

such as scattering13 and double slit interference.14

Among the many studies using LG beams, one must mention the work by Allen et

al, who revealed that the LG beam possesses well defined orbital angular momentum

(OAM) of lh̄ per photon.15 Their work triggered a series of investigations involving the

OAM carried by the LG beams. In the mechanical aspect, experiments have been done

to transfer OAM from LG beams to microscopic particles,16 to rotate the microscopic

particles using LG beams like optical spanners,17 and to observe the rotational Doppler

effect.18 In the area of nonlinear optics, several wave mixing processes have been inves-

tigated using high-order LG beams, including second-harmonic generation,19–21 four wave

mixing,22 and parametric down conversion.23–26 In particular, the parametric down con-

version experiments used the OAM state of the photons in high order LG beams to realize

3

multi-dimensional entanglement of quantum state,24, 27 which provides a practical route

to multi-dimensional quantum computation and communication. Within this perspective,

some techniques to measure28, 29 and to store30 the OAM information carried by the pho-

tons have been proposed and tested. There are also studies of OAM spectra of beams31

and imaging with OAM.32

One of our research efforts in the Nonlinear Optics Laboratory (NOL) at Washington

State University (WSU) is to search for new phenomena that result from the interaction

between intense light and a material, and to apply such phenomena to build novel all-

optical devices. As such, when the rising importance of high-order LG beams caught

our attention, we were immediately motivated to study their interaction with a nonlinear

material.

In conventional (linear) optics, the properties of a medium, such as the refractive index

and the absorption coefficient, are assumed to be constants for beams of given frequency.

The presence of one beam can not alter the properties of the media nor can it mediate

the interaction between beams. In a linear medium, light beams obey the principle of

superposition, i.e., the response to multiple input beams is a sum of the responses to each

one of the beams. In general linear optics works well if the electric field strength of the

incident beam is weak compared to the internal fields. However, when the electric field

strength of the incident beam gets sufficiently strong, superposition fails. For example, the

refractive index or the absorption coefficient of a medium may become intensity dependent,

so a strong beam may be able to manipulate the behavior of a weak beam, and under proper

configurations, two input beams may be combined together and become one beam whose

frequency is the sum or difference of the two input beam frequencies. Such phenomena

are beyond the scope of linear optics, but are the subject of nonlinear optics.

In this work we focus on the nonlinear optical phenomena that result from the in-

teraction between beams and media whose refractive indices are intensity-dependent. In

general the way that the refractive index of a nonlinear optical medium depends on the

4

intensity can be of any form. But the simplest form of the dependence is called the optical

Kerr type, which requires that the refractive index depend linearly on the intensity, or

n = n0 + n2I, where n0 is the conventional refractive index and n2 is called the nonlin-

ear refractive index coefficient. Materials that have an optical Kerr type refractive index

are called optical Kerr media. Because of its simple mathematical form, the optical Kerr

medium is often used as a simple case in theoretical derivations. And many materials act

as optical Kerr media under proper conditions.

Figure 1.2: “Self-lensing” of a fundamental gaussian beam when it traverses a thin opticalKerr medium. O.T., optical thickness. Upper: a fundamental gaussian beam, whose radialintensity distribution is shown to the left, traverses a thin optical Kerr medium. Lower:(a) when n2 < 0, the medium resembles a concave lens; (b) when n2 > 0, the mediumresembles a concave lens.

A well studied phenomenon is the “self-lensing” effect of a fundamental gaussian beam

when it traverses a thin optical Kerr medium. As illustrated in Fig. 1.2, when an intense

LG00 beam propagate through an optical Kerr medium, the refractive index, and therefore

5

the optical thickness (or the optical path length) of the medium is changed according to

the beam’s transverse intensity distribution, which is a gaussian function (shown in the

top left in the figure). If n2 > 0, the beam induces a higher optical thickness in the center

than in the periphery like a convex lens, causing the beam to converge, or “self-focuse”; If

n2 < 0, the sample will have a lower optical thickness in the center than in the periphery

like a concave lens, causing the beam to diverge, or “self-defocuse”.

Figure 1.3: Ray diagram of the effect of a positive lens on the propagation of a fundamentalgaussian beam. (a) If the lens is placed before the minimum beam waist, the far-fieldpattern of the beam is more spread out; (b) If the lens is placed after the beam waist, thefar-field pattern of the beam is more confined.

The “self-lensing” effect has several important applications. For example, using ray

optics, it’s easy to show that depending on the position of the nonlinear sample, i.e., the

introduced lens, with respect to the location of the beam waist, the far-field pattern of

the beam can appear to be either dilated or constricted. The case of a positive lens is

illustrated as an example in Fig. 1.3. Based on this phenomenon, a high-sensitivity n2

measurement technique called Z-scan was developed.33, 34 If the induced lens causes the

beam to focus more tightly, then an increase in beam intensity will result in less light in

the beam center in the far field. Thus optical limiting can be achieved by, for example,

placing an aperture around the beam axis in the far field and only observing the light

6

through the aperture.35, 36

Figure 1.4: The optical thickness of a thin optical Kerr medium illuminated by a LG10

beam. O.T., optical thickness. Upper: a LG10 beam, whose radial intensity distribution is

shown to the left, traverses a thin optical Kerr medium. Lower: the optical thickness ofthe medium when (a) n2 < 0 and (b) n2 > 0.

Following the example of the LG00 beam, one would naturally speculate about the

consequence of a high-order LG beam transversing an optical Kerr medium. However, the

intensity profile of a high-order LG beam is more complex than the fundamental gaussian

beam. For example, Fig. 1.4 shows the transverse intensity profile of a LG10 beam, as well

as the optical thickness of an optical Kerr medium under the illumination of the beam.

Clearly, the sample neither acts as a concave nor a convex lens. As such, a simple ray

diagram can not be used to determine the far field profile after a nonlinear sample as was

the case of the fundamental gaussian beam. And whether or not high-order LG beams

can be used in the Z-scan measurement or optical limiting is unclear. Furthermore, since

high-order LG beams may carry OAM, it is natural to wonder if the OAM of the beam will

7

be changed by the nonlinear interaction. These questions, to the best of our knowledge,

have never been previously discussed and are worth exploring.

A large portion of this dissertation is thus devoted to developing a theory that can

answer the above questions. In doing so, we build on the existing study of the fundamental

gaussian beam. First, some of the theoretical approaches that have been implemented to

study the case of the fundamental gaussian beam can be generalized to study the case

of LG beams of arbitrary orders. To be specific, our theory is developed with a method

which is a generalization of the gaussian beam decomposition method used by Weaire, et.

al..37 Secondly, as one of the modes of all LG beams, the fundamental gaussian beam

should conform to the generalized theory. Therefore, the case of the fundamental gaussian

beam can always be used as an initial test of the validity of the generalized theory. Third,

a comparative study of the case of the fundamental gaussian beam and high-order LG

beams would be helpful in determining the merits and shortcomings of each, especially

when considering their applications.

We also carry out experiments to test the validity of our theory as well as to demonstrate

the proposed applications. We use disperse red 1 (DR1) doped poly(methyl methacrylate)

(PMMA) (DR1/PMMA) samples as the optical Kerr medium in our experiments. As a dye

doped polymer, DR1/PMMA has the advantages of low cost, ease of fabricating thin films,

and ease of mechanical processing, compared to inorganic materials. Previously in the NLO

lab, we found that DR1/PMMA can show big nonlinear intensity-dependent refractive

index change with off-resonant beams (i.e., where the material is transparent), and we have

successfully demonstrated several nonlinear optical processes that require an intensity-

dependent refractive index.38–40 In this work, however, we apply theoretical modeling

and experimental measurement to determine the conditions under which the DR1/PMMA

sample can be treated as an optical Kerr medium. We then design experiments that at

least approximately obey these conditions for LG beams.

The dissertation is organized as follows: Chapter 2 presents the theory and the princi-

8

ples of the applications. We start with a brief review of the Laguerre-Gaussian beam (Sec.

2.2) and the optical Kerr medium (Sec. 2.3), including a study of the mechanisms of the

optical nonlinearity in DR1/PMMA samples, namely, the trans-cis-trans photoisomeriza-

tion and photoreorientation mechanisms. Then in section 2.4, we present our theory on

the effect of a thin optical Kerr medium on a Laguerre-Gaussian beam. Subsequently, we

propose several applications based on our theory, including new methods to measure the

nonlinear refractive index coefficient (the Z-scan technique in section 2.5 and the I-scan

and ∆Φmax-scan techniques in section 2.7) and optical limiting (section 2.6). Chapter 3

describes the details of the experiments, including how to generate high order LG beams

(Sec. 3.2), sample fabrication (Sec. 3.3), a holographic volume index gratings recording

experiment for the purpose of studying the properties of the DR1/PMMA samples (Sec.

3.4), and most importantly, the experiments that implement LG beams (Sec. 3.5). In

chapter 4 we show the experimental results and discuss their implications. We first sum-

marize the properties of our DR1/PMMA samples, particularly the conditions under which

they can be treated as an optical Kerr medium (Sec. 4.1). We then test the validity of

our theory and show that the Z-scan (Sec. 4.2) and the I-scan (Sec. 4.3) techniques using

the LG10 beams can measure correctly the nonlinear refractive index coefficients. Finally

in Sec. 4.4 we demonstrate optical limiting in DR1/PMMA using a LG10 beam and discuss

limitations of our technique as well as show the advantages of using LG10 beams over LG0

0

beams. We concludes the dissertation with Chapter 5.

9

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14

Chapter 2

Theory

2.1 Introduction

The purpose of this dissertation is to study the propagation of light beams with orbital

angular momentum (OAM) in a nonlinear optical material. As such, the two key elements

in our theory are the Laguerre-Gaussian (LG) beam1 and the optical Kerr medium.2 Before

we elaborate on our theory, we spend the first two sections (2.2 and 2.3) of this chapter

reviewing these two concepts.

Because we use disperse red 1 (DR1) doped poly(methyl methacrylate) (PMMA)

(DR1/PMMA) samples as the optical Kerr media in our experiments, we also study in

section 2.3 the mechanisms of the optical nonlinearity in DR1/PMMA samples, namely,

trans-cis-trans photoisomerization and photoreorientation.3–5 In specific, we develop a

three-state model to formulate the mechanisms and point out the conditions under which

the DR1/PMMA samples can be treated as the optical Kerr media.

In section 2.4, we present our theory on the effect of a thin optical Kerr medium on

a Laguerre-Gaussian beam. After that, we propose several applications according to our

theory, including the methods to measure the nonlinear refractive index coefficient (the Z

scan technique in section 2.5 and the I scan and ∆Φmax scan technique in section 2.7) and

15

optical limiting (section 2.6).

2.2 A review of the Laguerre-Gaussian beams

Laguerre-Gaussian (LG) modes were discovered soon after the invention of the laser.1

Under the slowly varying envelope approximation, the paraxial wave solution of the wave

equation, U(~r, t) = A(~r) exp (−ikz) exp (i2πνt), can be obtained by solving the scalar

paraxial Helmholtz equation6

∇2TA− i2k

∂A

∂z= 0, (2.1)

where A(~r) is a slowly varying complex function of position which characterizes the am-

plitude of a wave component, ∇T = ∂2/∂x2 + ∂2/∂y2, k is the wave vector, and ν is the

frequency. For simplicity, we will not write exp (i2πνt) explicitly in the following.

In cylindrical coordinates (r, φ, z), a complete set of solutions, known as the LG beams,

can be obtained. Each member of the set is called a “mode” specified by two mode

numbers, the angular mode number and the transverse radial mode number. An LG beam

with angular mode number l (l = 0,±1,±2, ...) and transverse radial mode number p

(p = 0, 1, 2, ...) can be written as:

LGlp (r, φ, z) =

(

ω0

ω (z)

)

(√2r

ω (z)

)|l|

L|l|p

(

2r2

ω2 (z)

)

× exp

( −r2

ω2 (z)

)

exp

(

−i kr2z

2 (z2 + z2r )

)

× exp

(

i (2p+ |l| + 1) tan−1

(

z

zr

))

× exp (−ilφ) exp (−ikz) , (2.2)

where ω0 is the beam waist radius, zr = kω20/2 is the Rayleigh length, ω(z) = ω0(1+z2/z2

0)1

2

16

is the beam radius at z, and Llp is the associated Laguerre polynomial defined as7

Llp(x) =

p∑

m=0

(−1)m(p+ l)!

(p−m)! (l +m)!m!xm, l > −1. (2.3)

When l = 0 and p = 0, the LG beam becomes the familiar fundamental Gaussian beam

LG00 (r, φ, z) =

(

ω0

ω (z)

)

exp

( −r2

ω2 (z)

)

(2.4)

× exp

(

−i kr2z

2 (z2 + z2r )

)

exp

(

i tan−1

(

z

zr

))

exp (−ikz) .

High order (|l| > 0 or p > 0) LG beams share a few important properties with the

fundamental Gaussian beam, such as:

1. The intensity distributions in the transverse planes in one beam are similar regardless

of the beam propagation, with the beam radius, ω(z), as the scale factor at the

position z.

2. The beam divergence, or how the beam radius, ω(z), changes as a function of z,

is totally determined by its minimum value, ω0, which is reached at a place called

beam waist.

3. A thin convex or a concave lens can focus or defocus an LG beam without affecting

its mode numbers.

In contrast, there are important differences between the high order LG beams and the

fundamental Gaussian beam, among which are:

1. The intensity distributions in the transverse planes are different from one mode to

another. The typical patterns consist of concentric bright rings and/or dark rings

with a bright center (if l = 0) or a dark center (if l 6= 0). And p gives the number of

dark rings, which is the reason why it is called the radial mode number. Figure 2.1

shows the intensity distributions of some LG beams.

17

Figure 2.1: Intensity profiles of some Laguerre Gaussian beams of different orders. l is theangular mode number and p is the radial moder number.

2. While the fundamental gaussian beam has plane-wave-like wavefronts near the beam

waist, the high l order LG beams have screw shaped wavefronts, like a “twisted”

(fundamental gaussian) beam. Figure 2.2 shows the phase profiles of a LG00 beam

and a LG10 beam near their beam waists.

3. The Guoy phase, which is the extra on-axis phase retardation of the beam in com-

parison with a plane wave as the wave propagates, is described in a more general

form (2p+ |l| + 1) tan−1 (z/zr).

18

Figure 2.2: Comparison of the phase profiles of a LG00 beam and a LG1

0 beam near theirbeam waists. (A) The phase profile on the transverse plane of the LG0

0 beam (the transverseplane is also the plane of equal phase in the LG0

0 beam.); (B) The phase profile on thetransverse plane of the LG1

0 beam; (C) The plane of equal phase of the LG10 beam.

19

4. When l ≥ 1, the LG beam possesses well defined orbital angular momentum of lh̄

per photon.8

The phase term exp (−ilφ) of the LG beams for l > 0 accounts for many of of the above

differences. For example, it is the cause of the twisted wavefronts; it makes the on-axis

(r = 0) phase undefined, which makes it a singularity or a screw dislocation,9 forcing the

intensity to be zero at the center; and it is closely related to the orbital angular momentum

possessed by the LG beams.

Finally we introduce a notation that we will use in the rest of this dissertation. For a

given LG beam mode, zr (or ω0) alone is sufficient to characterize the relative amplitude

and phase of the electric field of the beam. When multiple beams are involved, it is

often necessary to specify the waist locations of each beam. For convenience we use

C · LGlp(r, φ, z − zw; zr) to describe an LG beam unambiguously, where zw is the waist

location on the z axis and C is a complex constant that gives the amplitude and the initial

phase.

2.3 Optical Kerr effect and trans-cis-trans photoiso-

merization and photoreorientation

In this section, we first review the optical Kerr effect and the intensity dependent refractive

index. In the second part we use a three-state ( (1) molecules in trans form parallel and (2)

perpendicular to the polarization of the incident beam; and (3) in the cis form) model to

describe the trans-cis-trans photoisomerization and molecule reorientation effect, in which

emphasis is placed on the conditions under which the response of the material can be

treated as the optical Kerr effect.

20

2.3.1 Optical Kerr effect

Historically different systems of units have been used in nonlinear optics, such as the

gaussian system and several versions of MKS systems.2, 6, 10 Although the expressions and

even some definitions are different from one to another, the physics content is the same.

In this brief review we mainly follow the convention in reference 6, which is one of the

versions using the MKS system.

It is well known that in linear optics, the polarization P (t) of a material system depends

linearly on the applied optical field E(t) i.e.,

P (t) = ǫ0χ(1)E (t) , (2.5)

where

P (t) =1

2

(

P (ω) eiωt + c.c.)

, (2.6)

E (t) =1

2

(

E (ω) eiωt + c.c.)

, (2.7)

and χ(1) is the linear susceptibility. For simplicity, we have treated P (t) and E(t) as

scalars and assumed that the material is lossless and dispersionless. In general, however,

the relation between the two is not necessarily linear. If expressed as a power series in the

electric field strength, the polarization can be written as

P (t) = ǫ0χ(1)E (t) + 2χ(2)E2 (t) + 4χ(3)E3 (t) + · · · , (2.8)

where, χ(n) describes the nth-order nonlinear effect and is called the nth-order nonlinear

optical susceptibility. The high-order terms in Eq. (2.8) are usually much smaller than

the first order susceptibility, so the linear relationship is a good approximation when the

electric field strength is sufficiently weak. When the electric field strength gets stronger,

however, it is necessary to include some of the higher order susceptibilities. It can be

21

shown that if the material possesses inversion symmetry, the even order nonlinear optical

susceptibilities vanish. Then to the lowest order nonlinear term

P (t) = ǫ0χ(1)E (t) + 4χ(3)E3 (t) . (2.9)

Substituting Eq. (2.7) into the above equation, we have

P (t) =1

2ǫ0χ

(1)E (ω) eiωt +1

2χ(3)E3 (ω) ei3ωt +

3

2χ(3)|E (ω) |2E (ω) eiωt + c.c.. (2.10)

One immediately sees that the induced polarization has a new frequency component 3ω,

which is the third-harmonic frequency of the applied field. Third harmonic generation

requires a material that responses in an optical cycle, which is not the case for the pho-

toisomerization/reorientation mechanisms that are the focus of this work. Here we focus

on the ω frequency component of the polarization, which can be expressed as

P (ω) =(

ǫ0χ(1) + 3χ(3)|E (ω) |2

)

E (ω) . (2.11)

Defining the effective susceptibility as

χeff = χ(1) +3χ(3)|E (ω) |2

ǫ0, (2.12)

then

P (ω) = ǫ0χeffE (ω) . (2.13)

The above equation together with the Maxwell equations can be solved using the usual

procedures,11 which then give us the refractive index of the material

n =√

1 + χeff . (2.14)

22

In the linear case, we have

n0 =√

1 + χ(1), (2.15)

with the help of Eq. (2.14), this can be rewritten as

n = n0

√

1 +3χ(3)|E (ω) |2

ǫ0n20

≈ n0 +3χ(3)|E (ω) |2

2ǫ0n0

. (2.16)

Recall the intensity of light is given by6, 11

I =1

2η|E (ω) |2, (2.17)

where

η =

√

µ

ǫ=η0

n0

(2.18)

is the impedance of the material and η0 = (µ0/ǫ0)1/2. After replacing |E (ω) |2 with the

intensity of the light, Eq. (2.16) becomes

n = n0 +3η0χ

(3)

ǫ0n20

I. (2.19)

Defining the nonlinear refractive index coefficient as

n2 ≡3η0

ǫ0n20

χ(3), (2.20)

then

n = n0 + n2I. (2.21)

23

The effective refractive index of the material is a linear function of the intensity of the

incident light. This effect is known as the optical Kerr effect.

The optical Kerr effect can be observed in many materials. For materials that posses

inversion symmetry including centro-symmetry, it is often the lowest-order nonlinear effect

that appears when the intensity of the incident beam is increased beyond the linear regime.

Due to the intensity-dependent feature, the refractive index of the material can be modified

by the incident optical beam. As a consequence, the propagation of the beam itself is

affected by the modified refractive index. A lot of interesting phenomena result from this

interaction. As shown later, the major part of this dissertation is devoted to the interaction

between a thin optical Kerr medium and an LG beam.

2.3.2 Mechanisms of trans-cis-trans photoisomerization and pho-

toreorientation

One of the nonlinear materials frequently used in the nonlinear optics lab at Washington

State University is disperse red 1 (DR1) dye doped poly(methyl methacrylate) (PMMA)

polymer (DR1/PMMA) due to its big nonlinear effect and easiness of synthesis and process-

ing. The big nonlinear effect of DR1/PMMA is due to the photo-induced trans-cis-trans

isomerization of DR1 molecules followed by reorientation in the direction perpendicular to

the polarization of the laser beam. In this section, we develop a simple theory to explain

the mechanisms of photoisomerization and photoreorientation. The theory is not intended

to be a precise description of all aspects of the real physical system. For example, the

geometry of a real sample is three-dimensional, but our theory is a highly idealized model

that approximates the dynamics of the real system. The purpose is to help understand the

experimental observations qualitatively without complex mathematics or numerical calcu-

lations. Emphasis is placed on the conditions under which the response of the material can

be treated as an optical Kerr effect. The model is an improvement of the one in reference 5

24

where the cis population is totally ignored and only the effects of photoreorientation (but

not photoisomerization ) are considered.

DR1 molecules and photoisomerization

Figure 2.3: Isomers of the DR1 molecule.

A DR1 molecule can exist in two geometric forms, or isomers, as shown in Fig. 2.3.

In the trans form, the two substituent groups are oriented on the opposite sides of the

nitrogen double bonds, while in the cis form, the two substituent groups are oriented on

the same side of the nitrogen double bonds.

Due to the difference in their shapes, the response of the two isomers to electric fields

and light are different. The trans isomer, having a shape like a cigar, is anisotropic in

25

response to the external field because it forms a larger dipole if the polarization of the

applied field is parallel to the axis of the “cigar” than if it is perpendicular. A cis isomer,

more like a ball, responds to the external field more isotropically.

The energy levels of the two isomers are slightly different. The trans isomer has lower

energy levels than the cis isomer, as indicated in Fig. 2.4. Therefore, most DR1 molecules

are in the trans form at the room temperature.

However, if an optical field at the proper wavelength is applied, a trans isomer can

jump to the excited state by absorbing a photon, where it either relaxes back to the trans

ground state or to the cis ground state. A cis isomer, once formed, decays to the trans

isomer through thermal relaxation. Or a cis isomer can be excited to a higher energy level

by absorbing a photon, then can relax to either the trans ground state or the cis ground

state. The process of trans to cis, then back to trans, is called photoisomerization. Fig.

2.4 shows a schematic energy diagram of the photoisomerization process.

Figure 2.4: Schematic energy diagram of the photoisomerization process. Path 1: transisomers with absorption cross section σt jump to the excited state by absorbing photons;Path 2: molecules in the trans excited state relax to the cis ground state with a quantumyield (or probability) of Φtc; Path 3: at room temperature, cis isomers relax to the transisomer thermally with a rate of γ; Path 4: cis isomers with absorption cross section σcjump to the excited state by absorbing photons; Path 5: molecules in the cis excited staterelax towards the trans ground state with a quantum yield (or probability) of Φct.

26

DR1/PMMA and photoreorientation

A polymer such as PMMA consists of many entangled long molecular chains. The en-

tanglement is statistically random, leaving many small empty space, or “voids” between

chains. The distribution of the sizes of these voids depend on the polymerization condi-

tions such as the amount and type of initiator (which chemically cause the polymer to

form), the temperature, and the pressure. An example in daily life is the sponge with a

lot of small pores. Just as the sponge can hold water, we can dope PMMA with molecules

such as DR1 through a special process (see the experimental part of this dissertation).

The DR1 molecules are then trapped inside those voids and may have limited freedom of

mobility depending on the size and shape of each individual void.

A fresh (having not been exposed to light) DR1/PMMA sample is usually homogeneous

with the orientation of the trans isomers evenly distributed. The refractive index of the

sample is therefore isotropic. If we pump a DR1/PMMA sample with a linearly polarized

light beam, two things happen. First, some of the trans isomers are converted to cis

isomers due to the photoisomerization. Second, the trans isomers whose long axis are

parallel to the polarization of the beam are more likely to be excited and converted to cis

isomers than those are not. The cis isomers, being smaller than trans isomers, move and

rotate much more easily in the PMMA voids than the trans isomers. As a consequence,

when a cis isomer relaxes back to the trans states, its orientation is not necessarily the

same as before. In the long run, more and more trans isomers that are oriented along the

polarization direction of the incident beam are depleted and converted to trans isomers

oriented in other directions. This is called photoreorientation.

Both the photoisomerization and the reorientation result in changes of the properties

of the material, including the mechanical5 and the optical properties.4, 12, 13 In this dis-

sertation, we focus on the change of the refractive index of the material as “seen” by the

incident beam.

27

The idealized three-state model

To catch the key dynamics of the processes without overly complicated mathematics, we

use three-state model to approximate the photoisomerizing system. The model has the

following approximations:

1. A DR1 molecule in DR1/PMMA can only be in one of the following three states:

(a) a trans isomer parallel to the polarization of the incident light (assuming linearly

polarized beam);

(b) a trans isomer perpendicular to both the polarization and the wave vector of

the incident beam;

(c) a cis isomer which is isotropic.

2. A trans isomer interacts with light only if it is oriented parallel to the polarization

of the incident light.

3. When relaxing back to the trans form from the cis form, a molecule has equal pos-

sibilities to be oriented in either of the two orientations.

4. An entropic process independent of the light intensity always tries to equalize the

populations of the trans isomers in both orientations.

By making the above assumptions, we mainly ignore the following facts about the real

material system:

1. The trans isomer can orient in all directions in the three-dimensional space, inter-

acting with the light differently depending on the orientation.

2. The cis isomer is not perfectly isotropic.

28

3. The dynamical behavior of the molecules, such as the entropic decay of the orienta-

tion of the trans isomers, are affected by their environment, i.e. the PMMA voids

surrounding them, and may vary form site to site.

Now let’s define the following quantities:

1. Ntp: the fraction of molecules in the trans form that is oriented parallel to the

polarization of the incident light beam.

2. Nc: the fraction of molecules in the cis form. The fraction of molecules in the trans

form oriented perpendicular to both the polarization and the wave vector of the

incident beam is thus Nts = 1 −Ntp −Nc.

3. I: the intensity of the light beam.

4. ξtc: the probability rate per unit intensity of light in the material that a trans isomer

will be converted into a cis isomer.

5. ξct: the probability rate per unit intensity that a cis isomer will be converted into a

trans isomer.

6. γ: the thermal relaxation rate of the cis isomer. 1/γ thus gives the lifetime of the

cis isomer in darkness.

7. β: the entropic decay rate of the anisotropy due to the trans isomer orientation.

With these definitions, we are ready to formulate the processes. But before that, we

point out that from Ntp(t) and Nc(t), we can determine the change of the refractive index

∆n(t) along the light’s polarization. Taking the differential of Eq. (2.14), it can be shown

that for small ∆χeff , ∆n is given by

∆n ≈ ∆χeff2n0

. (2.22)

29

But ∆χeff is connected to the change of the isomer populations Ntp and Nc by

∆χeff = χtp∆Ntp + χc∆Nc, (2.23)

where χtp and χc are the contributions to the total effective susceptibility from the trans

isomers parallel to the polarization of the incident beam and the cis isomers, respectively.

Therefore, we have

∆n ≈ χtp∆Ntp + χc∆Nc

2n0

(2.24)

= ηtp∆Ntp + ηc∆Nc,

where we have introduced the new constant coefficients ηtp and ηc for simplicity.

Assuming first-order kinetics, the dynamics of the photoisomerization and photoreori-

entation processes are governed by

dNtp

dt= −ξtcINtp +

1

2ξctINc +

1

2γNc + β(1 − 2Ntp −Nc), (2.25)

dNc

dt= ξtcINtp − ξctINc − γNc, (2.26)

where (1 − 2Ntp −Nc) is the population fraction difference between the parallel and per-

pendicular trans isomers. We assume that we start with a fresh sample, so the the initial

conditions are:

Ntp(t = 0) = 12, Nc(t = 0) = 0. (2.27)

Equations (2.25) and (2.26) can be solved rigorously, yielding general solutions charac-

terized by two exponentially decay functions with different time constants. Here we are

interested in the special case in which further approximations can be made according to

the properties of the DR1/PMMA samples under our experimental conditions.

30

Dynamics over short time scales

First we focus on a time scale that is short enough that only a small fraction of isomers

is converted (a few seconds). The entropic decay of the anisotropy of the trans isomer

orientation in DR1/PMMA is a slow process (hours) compared to the photoisomerization

process (seconds), which means β << γ. And at short time scales, the population fraction

difference between the parallel and perpendicular trans isomers is small, so (1−2Ntp−Nc)

is a small quantity. Under these conditions, it is reasonable to drop the last term in Eq.

(2.25), which yields the solution:

Ntp(t) =1

4

1 +ξtcI − (ξctI + γ)

√

(ξtcI)2 + (ξctI + γ)2

e−λ1t +1

4

1 − ξtcI − (ξctI + γ)√

(ξtcI)2 + (ξctI + γ)2

e−λ2t,

(2.28)

and

Nc(t) =ξtcI

2√

(ξtcI)2 + (ξctI + γ)2

(

e−λ2t − e−λ1t)

, (2.29)

where

λ1 =1

2

(

ξtcI + (ξctI + γ) +

√

(ξtcI)2 + (ξctI + γ)2

)

, (2.30)

and

λ2 =1

2

(

ξtcI + (ξctI + γ) −√

(ξtcI)2 + (ξctI + γ)2

)

. (2.31)

31

If the intensity I is low such that ξtcI << γ and ξctI << γ, then the above expressions

can be simplified, yielding:

Ntp(t) ≈ξtcI

4γe−λ1t +

(

1

2− ξtcI

4γ

)

e−λ2t, (2.32)

and

Nc(t) ≈ξtcI

2γ

(

e−λ2t − e−λ1t)

, (2.33)

where

λ1 ≈ (γ + ξctI) +1

2ξtcI, (2.34)

and

λ2 ≈1

2ξtcI. (2.35)

We see that the population dynamics of the isomers are characterized by the two exponen-

tial decay functions with time constants 1/λ1 and 1/λ2. 1/λ1, which is dominated by the

contribution from γ, is the time needed to build up the population equilibrium between

the trans isomers and the cis isomers through photoisomerization . 1/λ2 gives the time

scale for the trans isomers parallel to the polarization of the light to be totally depleted to

the perpendicular direction. Because we have ignored the entropic process that reverses

such a depletion, 1/λ2 will need to be modified if we include β into the equations, as will

be discussed shortly.

It’s obvious that 1/λ1 << 1/λ2 since we have assumed ξtcI << γ and ξctI << γ. If

we are interested only in the short time period within which λ1t is no greater than the

order of 1, then λ2t << 1. Also we assume λ1 = γ for simplicity. With these further

32

approximations, the above results become

Ntp(t) ≈1

2− ξtcI

4t− ξtcI

4γ

(

1 − e−γt)

, (2.36)

and

Nc(t) ≈ξtcI

2γ

(

1 − e−γt)

. (2.37)

Using Eq. (2.24) with Eqs. (2.36) and (2.37), the change of the refractive index of the

sample as “seen” by the incident beam is

∆n(t) ≈ (2ηc − ηtp)ξtcI

4γ

(

1 − e−γt)

− ηtpξtcI

4t. (2.38)

This shows that at any time instant t, ∆n depends linearly on the intensity I. Hence we

draw one important conclusion: on short time scales (t ∼ 1/γ) the material can be treated

as an optical Kerr medium if the intensity of the incident light beam is not too strong.

We rewrite Eq.(2.38) to better illustrate the two contributors to the change of the

refractive index as

∆n(t) ≈ − (ηtp − ηc)ξtcI

2γ

(

1 − e−γt)

− ηtp

(

ξtcI

4t− ξtcI

4γ

(

1 − e−γt)

)

, (2.39)

where the first term is the change of the refractive index due to photoisomerization and

the second is due to the photoreorientation. A plot of ∆n as well as the two components

as a function of time is shown in Fig. 2.5, where the values of the parameters are assumed

to be: γ = 1 s−1 , ξtcI = 0.01 s−1, ηtp = 2, and ηc = 1. The figure shows that at the

beginning, both mechanisms contribute to ∆n significantly, but in the long run (after

t > 1/γ), photoreorientation wins out.

33

-0.5 0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5

-0.020

-0.015

-0.010

-0.005

0.000

n (a

rbitr

ary

unit)

t(s)

total n n due to photoisomerization n due to photoreorientation

Figure 2.5: The dynamics of ∆n in DR1/PMMA at short time scales as calculated fromEq.(2.39) with the following parameters: γ = 1 s−1 , ξtcI = 0.01 s−1, ηtp = 2, and ηc = 1.

Dynamics at long time scales

For long time scales (t >> 1/γ), the entropic decay of the anisotropy of the trans isomer

orientation in DR1/PMMA plays an important role, so the β term in Eq.(2.25) must be

retained. Using Eq: (2.26), Eq.(2.25) can be written as

dNtp

dt= −1

2ξtcINtp −

1

2

dNc

dt+ β(1 − 2Ntp −Nc). (2.40)

We limit our discussion to low intensity such that ξtcI << γ and ξctI << γ. According

to Eq. (2.33), the population of the cis isomers is no more than ξtcI/(2γ) at any time.

34

Specifically, if t >> 1/γ, then

Nc(t) ≈ξtcI

2γe−λ2t << 1 (2.41)

and

dNc(t)

dt≈ −ξtcI

2Nc(t). (2.42)

So Nc is a small quantity compared with (1 − 2Ntp) when t >> 1/γ, as the latter is the

population difference between the parallel and the perpendicular trans isomers, which in-

creases with time and approaches 1. Also Nc is much smaller than Ntp until the majority of

the population of the parallel trans isomers are depleted. Thus in the following discussion,

we ignore the cis population, yielding from Eq. (2.40),

dNtp

dt= −1

2ξtcINtp + β(1 − 2Ntp). (2.43)

The solution of the above equation is easily obtained, giving

Ntp(t) =1

2− ξtcI

2 (4β + ξtcI)

(

1 − exp

(

−(

2β +ξtcI

2

)

t

))

. (2.44)

Using Eq. (2.24) and ignoring the population of the cis isomers, the change of the refractive

index of the sample as “seen” by the incident beam is

∆n(t) ≈ −ηtpξtcI

2 (4β + ξtcI)

(

1 − exp

(

−(

2β +ξtcI

2

)

t

))

. (2.45)

We see that in general, the way ∆n changes with I is not of the optical Kerr type, but

saturates in high intensity. Moreover, the saturation time constant also depends on the

intensity. However, there are two situations under which the material can be approximated

35

as an optical Kerr medium. First, if the intensity is very weak such that ξtcI << 4β, then

∆n(t) ≈ −ηtpξtcI

8β(1 − exp (−2βt)) . (2.46)

Secondly, if (2β + ξtcI/2) t << 1 or t << 1/ (2β + ξtcI/2), then

∆n(t) ≈ −ηtpξtcI

4t. (2.47)

We note that usually ξtcI << 4β is a more strict approximation than ξtcI << γ for

DR1/PMMA since γ >> β. Therefore the amplitude of ∆n using Eq. (2.46) which is

limited by ηtpξtcI/8β is fairly small. Using Eq. (2.47) the behavior overlaps with the short

time scale result given by Eq. (2.38), where the requirement of λ2t << 1 or t << 1/ξtcI/2

is now replaced by t << 1/ (2β + ξtcI/2).

Summary

We conclude this section by summarizing the conditions under which a DR1/PMMA sam-

ple can be treated as a optical Kerr medium.

1. If the intensity of the incident beam is very weak such that ξtcI << β, the sample

can be approximated by an optical Kerr medium at any time. However, the change

of refractive index is small because the population of both the cis isomers and the

reoriented trans isomers are rather small.

2. If the intensity of the incident beam is not very strong such that ξtcI << γ (but

possibly ξtcI > β.), the sample can be approximated by an optical Kerr medium for

the time range of t << 1/ (2β + ξtcI/2). The introduced change of the refractive

index in this case can grow substantially larger than the previous case.

36

2.4 Effect of a thin optical Kerr medium on an LG

beam

2.4.1 Introduction

When an intense light beam propagates through a nonlinear material with an intensity-

dependent refractive index, the beam will modify the refractive index of the material. As

a consequence, the propagation of the optical beam itself will be affected by the modified

refractive index. Such phenomenon have been studied extensively for the case that the

incident beam is the fundamental Gaussian beam.2, 14, 15 When the sign of the change of

the refractive index is positive(i.e. n2 > 0), the nonlinear sample acts as a convex lens,

causing the beam to converge, or “self-focus”; When the sign of the change of the refractive

index is negative(i.e. n2 < 0), the nonlinear sample acts as a concave lens, causing the

beam to diverge, or “self-defocus”. Furthermore, how the Gauss beam will change its

shape also depends on the position of the nonlinear sample with respect to the location of

the beam waist. For example, for a sample having positive intensity-dependent refractive

index, the far-field pattern of the beam will appear to be dilated if the sample is placed

before the beam waist, while it will appear to be contracted if the sample is placed after

the beam waist. Based on this phenomenon, a high-sensitivity n2 measurement technique

called Z-scan was proposed by Sheik-bahae, etc.15, 16

An interesting question pertains to what would happen if the incident beam is a high

order LG beam. In general, the transverse intensity profile of a high order LG beam is

more complex than the fundamental Gaussian beam. For example, the transverse intensity

profile of a LG10 beam is like a donut, so the resulting nonlinear refractive index change

would neither act as a concave nor a convex lens. As a consequence, how would the beam

reshape itself after the nonlinear sample?

A schematic diagram of this problem is shown in Fig. 2.6. An LG beam E(r, φ, z) =

37

Figure 2.6: Configuration of the LG beam propagation problem.

E0 ·LGl0p0

(r, φ, z; zr) is focused by a convex lens. The waist of the focused beam is located

at z = 0. A nonlinear sample of thickness d is placed at position z = zs along the optical

axis of the beam. Our purpose is to analyze the propagation of the beam after it passes

through the nonlinear sample. We assume the nonlinearity of the sample is of the optical

Kerr type, i.e. n = n0 + ∆n(I) with ∆n(I) = n2I.

Historically several approaches have been developed to investigate the propagation of

laser beams (mostly fundamental Gaussian beams) inside and through a nonlinear mate-

rial. Some of them can be modified to the case of the higher order LG beams. We choose

to follow the derivation procedures used in references 15 and 16 because it is simple yet

practical. We make essential modifications such that the method can apply to the LG

beams.

2.4.2 Field of the beam immediately after the sample

We assume that the sample is very “thin” such that the intensity pattern of the beam does

not change within the sample. Two conditions are required to guarantee this assumption.

(i): The sample thickness is much shorter than the beam’s diffraction length, or d <<

zr, so linear diffraction within the sample can be neglected. A phase shift of ∆Φl = n0kd

will be introduced due to the linear refractive index of the sample. But since ∆Φl is

constant in the beam’s transverse plane, it won’t affect the beam’s propagation other than

38

a trivial phase shift, so this effect will be ignored in the following discussion.

(ii): d << zr

∆Φmax, where ∆Φmax is the maximum value of the nonlinear phase distortion

∆Φ across the beam’s transverse plane due to the nonlinear refractive index change of the

sample. This is sometimes referred to as the “ external self-action” condition,16–18 which

assures that nonlinear refraction can be neglected within the sample. The nonlinear phase

distortion ∆Φ depends on the intensity distribution of the beam in the transverse plane

and will be taken into account when considering the beam propagation after the sample.

Under the above “thin” sample assumption, ∆Φ is governed by

d∆Φ

dz= ∆n(I)k. (2.48)

The intensity, I, varies in the sample due to absorption according to

dI

dz= −α(I)I, (2.49)

where α(I) is the absorption coefficient of the sample material. In general, α(I) includes

linear and nonlinear parts and can be written as:

α(I) = α+ ∆α(I), (2.50)

where α is the linear absorption coefficient and is a constant. ∆α(I) is the nonlinear

coefficient and depends on the intensity. We assume

∆α(I) = βI, (2.51)

where β is the first-order nonlinear absorption coefficient.

Using Eq. (2.49) - Eq. (2.51), we can solve for the intensity of the beam as a function

39

of distance of propagation through the sample,

I =Iincidente

−αz′

1 + q, (2.52)

where Iincident is the intensity at the input surface of the sample, z′ is the propagation

depth in the sample, and

q = βIincidentz′eff , (2.53)

where

z′eff =1 − e−αz

′

α. (2.54)

Substituting Eq. (2.52) into Eq. (2.48), we can solve the nonlinear phase shift due to the

sample:

∆Φ =kn2

βln (1 + q) . (2.55)

The natural logarithm in the above equation can be expanded about q. When q << 1,

which is satisfied when either β is very small or the sample is very thin, the total nonlinear

phase shift due to the sample is approximately:

∆Φ = kn2Iincident1 − e−αd

α. (2.56)

The intensity of the beam at the exit surface of the sample is obtained by replacing z′ with

the sample thickness d in Eq.(2.52):

I = Iincidente−αd, (2.57)

where q is dropped under the condition q << 1. Eq.(2.56) and Eq.(2.57) together deter-

40

mine the complex electric field immediately after the sample,

E ′ = Eincidente−αd

2 e−i∆Φ. (2.58)

When the incident beam is an LG beam, E(r, φ, z) = E0 · LGl0p0

(r, φ, z; zr), and the

sample is at z = zs on the z axis, Eq.(2.58) becomes:

E ′ (r, φ, zs) = E (r, φ, zs) e−αd

2 e−i∆Φ(r,φ,zs). (2.59)

Expressing Iincident in Eq.(2.56) with the electric field (refer to Appendix D.1), we find the

nonlinear phase distortion ∆Φ (r, φ, zs) obeys:

∆Φ (r, φ, z) =∆Φ0

1 + z2/z2r

(

2r2

ω2 (z)

)|l0|(

L|l0|p0

(

2r2

ω2 (z)

))2

exp

( −2r2

ω2 (z)

)

, (2.60)

where

∆Φ0 =π

λcǫ0n0n2|E0|2

1 − e−αd

α(2.61)

is a constant proportional to the maximum nonlinear phase change ∆Φmax(zs) in the

sample. This coefficient depends on the radial and angular mode numbers of the LG beam

as well as the position of the sample zs. When the incident beam is a LG00 beam, we

simply have ∆Φ0 = ∆Φmax(zs = 0).

2.4.3 Propagation of the beam after the sample

Eq.(2.59) gives the complex electric field of the beam immediately after it traverses the

sample. In principle the propagation of the beam thereafter can be analyzed using the

standard methods that evaluate field propagation in free space, for example, the Fresnel

diffraction integral and the angular spectrum method. However, an analytic result is

41

hard to obtain using these methods. The “Gaussian decomposition” method used by

Weaire19 could give analytic solution under certain approximations and provides a more

clear physical interpretation. Weaire used this method to analyze the propagation of a

fundamental gaussian beam that traverses a nonlinear sample. Here we generalize this

method to deal with the LG beams.

The exponential in Eq. (2.59) can be expanded in a Taylor series as

e−i∆Φ(r,φ,zs) =∞∑

m=0

(−i∆Φ (r, φ, zs))m

m!. (2.62)

The complex electric field of the incident beam after it passes through the sample can be

written as a summation of the electric fields of a series of LG beams of different modes as

E ′ (r, φ, z) =∞∑

m=0

pm∑

p=0

∞∑

l=−∞Cp,l,mLG

lp (r, φ, z − zwm; zrm) , (2.63)

where zwm and zrm are the waist location and the Rayleigh length, respectively, of the

corresponding beam mode and Cp,l,m is the amplitude and phase of the component beam.

These parameters are determined by letting z = zs in Eq. (2.63) and comparing it with

Eq. (2.59) with the exponential replaced by Eq. (2.62). The details of the decomposition

calculation are included in Appendix A. Here we write the result:

E ′ (r, φ, z) =∞∑

m=0

pm∑

p=0

Cp,mLGl0p (r, φ, z − zwm; zrm) , (2.64)

where

zwm = zr4m (m+ 1)Z

Z2 + (2m+ 1)2 , (2.65)

42

zrm = zr(2m+ 1) (Z2 + 1)

Z2 + (2m+ 1)2 , (2.66)

and Cp,m = Dp,m · Fp,m where Z is defined as Z = zs/zr, and

Fp,m = E0e−αd/2 (−i∆Φ0)

m

m! (2m+ 1)

√

(2m+ 1)2 + Z2

(1 + Z2)2m+1 exp

(

−ikzr4m (m+ 1)Z

Z2 + (2m+ 1)2

)

× exp(

i (2p0 + |l0| + 1) tan−1 (Z))

exp

(

−i (2p+ |l0| + 1) tan−1

(

Z

2m+ 1

))

,

(2.67)

and pm and Dp,m are determined through

pm∑

p=0

Dp,m · L|l0|p (x) =

xm|l0|(

L|l0|p0

(

x2m+1

)

)2m+1

(2m+ 1)2m+1

2|l0|

, (2.68)

where x is an arbitrary real variable. Our theoretical results are embodied in Eq. (2.64).

It is worth noting that all the component LG beams have the same angular mode

number l0 as that of the incident beam, which reflects the conservation of the photon’s

orbital angular momentum. Therefore the effect of the Kerr material on the incident LG

beam is to generate new LG beams of different radial modes. These results are important

in applications that leverage mode sensitivity.

2.4.4 Examples assuming small nonlinear phase distortion

The Taylor expansion in Eq. (2.62) always converges and the speed of convergence depends

on the value of ∆Φmax (zs). So does the expansion of Eq. (2.64). In practice, m only needs

to be summed up to a certain finite value in order to achieve a given precision. If the

nonlinear phase distortion is very small (e.g., ∆Φmax (zs) << 1) such that only a few

terms in the summation are needed to make a good approximation, we can write out the

result analytically.

43

To illustrate, assume that the incident beam is a LG00 beam and the maximum nonlinear

phase distortion in the sample at position Z is

|∆Φmax (Z) | =|∆Φ0|1 + Z2

<< 1. (2.69)

It is sufficient to keep the first two terms (m = 0 and m = 1) in Eq. (2.62) and neglect the

higher order terms, yielding ( see Appendix A: Example 1 for details of the derivation.)

E ′ (r, φ, z) ≈ C0,0LG00 (r, φ, z − zw0; zr0) + C0,1LG

00 (r, φ, z − zw1; zr1) , (2.70)

where

C0,0 = E0e−αd/2;

C0,1 = E0e−αd/2 (−i∆Φ0)

3

√

9 + Z2

(1 + Z2)3 exp

(

−ikzr8Z

Z2 + 9

)

× exp(

i tan−1 (Z))

exp

(

−i tan−1

(

Z

3

))

;

and

zw0 = 0;

zw1 = zr8Z

Z2 + 9;

zr0 = zr;

zr1 = zr3 (Z2 + 1)

Z2 + 9.

Next we show an example of the higher order LG beam. Assume the incident beam is

44

a LG10 beam and the maximum nonlinear phase distortion in the sample at position Z is

|∆Φmax (Z) | =|∆Φ0|

e · (1 + Z2)<< 1. (2.71)

Again we keep the first two terms in the Eq. (2.62) and neglect the higher order terms,

yielding ( see Appendix A: Example 2 for details of the derivation.)

E ′ (r, φ, z) ≈ F0,0LG10 (r, φ, z − zw0; zr0)

+2

3√

3F0,1LG

10 (r, φ, z − zw1; zr1)

− 1

3√

3F1,1LG

11 (r, φ, z − zw1; zr1) , (2.72)

where

F0,0 = E0e−αd/2;

F0,1 = E0e−αd/2 (−i∆Φ0)

3

√

9 + Z2

(1 + Z2)3 exp

(

−ikzr8Z

Z2 + 9

)

× exp(

i2 tan−1 (Z))

exp

(

−i2 tan−1

(

Z

3

))

;

F1,1 = E0e−αd/2 (−i∆Φ0)

3

√

9 + Z2

(1 + Z2)3 exp

(

−ikzr8Z

Z2 + 9

)

× exp(

i2 tan−1 (Z))

exp

(

−i4 tan−1

(

Z

3

))

,

45

and

zw0 = 0;

zw1 = zr8Z

Z2 + 9;

zr0 = zr;

zr1 = zr3 (Z2 + 1)

Z2 + 9.

This outgoing electric field includes the generated LG10 and LG1

1 beam.

2.5 Application: Z scan

Our theory applies to several important applications. In this section, we discuss the Z-scan

measurement.

2.5.1 Review of the traditional Z scan using a LG00 beam

Figure 2.7: Schematic diagram of the Z scan experiment. L: lens, S: sample, A: aperture,and D: detector.

The Z-scan measurement was first reported by Sheik etc. as a highly sensitive technique

to measure the optical nonlinearities using a single LG00 beam.16 Figure 2.7 shows the

46

schematic diagram of the Z-scan experiment. The placement of the incident beam and

the nonlinear sample is the same as in Figure 2.6. The incident beam is focused by a

convex lens. The waist of the beam is located at z = 0. An optically nonlinear sample of

thickness d is placed at position z = zs along the optical axis of the beam. In addition, a

small aperture is placed on axis of the beam in the far field. The power passed through

the aperture is recorded by a detector as a function of sample position z.

To better explain the Z-scan procedures, it’s useful to define the on-axis normalized

Z-scan transmittance15, 16

T (Z,∆Φ0) =|E ′ (r → 0, φ, z → ∞)|2

|E ′ (r → 0, φ, z → ∞) |∆Φ0=0|2, (2.73)

which characterizes the on axis light power transmitted though the small aperture in the

far field. Applying Eq. (2.70) we find (see Appendix B for details on the derivation.)

T (Z,∆Φ0) = 1 +4Z

(1 + Z2) (9 + Z2)∆Φ0 (2.74)

when |∆Φ0| << 1.

In a Z-scan, one measures the trace of the normalized transmittance T as a function

of the sample position z, which we call a Z-scan trace. Figure 2.8 shows a typical Z-scan

trace for positive (solid line) and negative (dotted line) ∆Φ0. A typical Z-scan trace has

a peak (maximum) and a valley (minimum). The positions of the peak and valley can be

obtained by solving the equation

dT (Z,∆Φ0)

dZ= 0, (2.75)

47

-6 -5 -4 -3 -2 -1 0 1 2 3 4 5 60.97

0.98

0.99

1.00

1.01

1.02

1.03

valley

peak

Tp-v

0= 0.1 0=-0.1

Nor

mal

ized

Tra

nsm

ittan

ce T

Z

Figure 2.8: A typical Z-scan trace for positive (solid line) and negative (dotted line) ∆Φ0.

yielding

Zpeak(valley) = ±

√

2√

13 − 5

3≈ ±0.859, (2.76)

where the sign is + (−) for the peak (valley) and − (+) for the valley (peak) when

∆Φ0 > 0 (∆Φ0 < 0). Substituting result (2.76) into Eq. (2.74), we can calculate the

difference between the peak and the valley

∆Tp−v = 0.406|∆Φ0|. (2.77)

The above relationship provides a handy way to determine ∆Φ0 from the Z-scan trace,

48

from which one can calculate the nonlinear refractive index n2, e.g., through Eq. (2.61).

If the experimental apparatus is able to resolve the normalized transmittance change of

∆Tp−v = 1%, then it can measure ∆Φ0 as small as 0.025, corresponding to wavefront

distortion of λ/250. Thus the Z-scan technique has very high sensitivity.

2.5.2 Z scan using a LG10 beam

Following the example of the LG00 Z-scan, we calculate the normalized transmittance T

for a LG10 incident beam. Using Eq. (2.72) and Eq. (2.73) we find (see Appendix B for

details on the derivation.)

T (Z,∆Φ0) = 1 +8Z (27 + 10Z2 − Z4)

(1 + Z2) (9 + Z2)3 ∆Φ0

+16

(9 + Z2)3 ∆Φ20. (2.78)

The last term can be dropped if |∆Φ0| << 1, yielding

T (Z,∆Φ0) = 1 +8Z (27 + 10Z2 − Z4)

(1 + Z2) (9 + Z2)3 ∆Φ0. (2.79)

This equation shows a similar relationship between T and ∆Φ0 as Eq. (2.74), sug-

gesting that it is possible to do a Z-scan experiment using the LG10 beam to measure the

nonlinearity of a thin sample. Figure 2.9 shows the theoretical typical LG10 Z-scan traces

for positive (solid line) and negative (dotted line) ∆Φ0. The shape of the Z-scan curve

using the LG10 beam differs from the traditional one using the LG0

0 beam in that the for-

mer has an extra peak and valley (indicated in the figure by the arrows). The extra valley

brings down the tail of the major peak to below T = 1 and the extra peak brings up the

tail of the major valley up above T = 1, while the tails in the LG00 Z-scan trace never cross

the T = 1 line. The differences are clearly seen in Figure 2.10 which shows a LG10 Z-scan

trace and a LG00 Z-scan trace simultaneously.

49

-10 -8 -6 -4 -2 0 2 4 6 8 100.98

0.99

1.00

1.01

1.02

0= 0.1 0=-0.1 T=1

Nor

mal

ized

Tra

nsm

ittan

ce T

Z

Figure 2.9: A typical LG10 Z-scan trace for positive (solid line) and negative (circles) ∆Φ0.

The T = 1 level is indicated by the dashed line.

The coordinates of the peaks and valleys can be calculated using Eq. (2.75) and Eq.

(2.79).

Zmajor peak(valley) = ±0.902 · sign(∆Φ0), (2.80)

Tmajor peak(valley) = 1 ± 0.145∆Φ0, (2.81)

and

Zminor peak(valley) = ∓5.24 · sign(∆Φ0), (2.82)

Tminor peak(valley) = 1 ± 0.0138∆Φ0, (2.83)

50

-10 -5 0 5 100.98

0.99

1.00

1.01

1.02

Z

Nor

mal

ized

Tra

nsm

ittan

ce T

LG01 Z-scan

LG00 Z-scan

T=1

Figure 2.10: Comparison of a typical LG10 Z-scan trace with a typical LG0

0 Z-scan trace.The values of ∆Φ0 are chosen such that the major peaks (valleys) of the two traces almostoverlap. Also shown is the T=1 line.

where sign(∆Φ0) = 1 if ∆Φ0 > 0 and sign(∆Φ0) = −1 if ∆Φ0 < 0.

The amplitudes of the minor peak and valley are much smaller than the major ones

in a Z-scan trace and are less important in determining ∆Φ0 and n2. However, they have

significance in other applications such as optical limiting, which will be discussed later.

The major peak and valley are important in the Z-scan measurement. We can use the

difference between them (∆Tp−v) to determine ∆Φ0, which then lets us calculate n2. From

Eq. (2.81) it’s easy to get

∆Tp−v = 0.290 |∆Φ0| . (2.84)

51

We point out that there is a subtle difference between the ∆Φ0 in Eq. (2.77) and

in Eq. (2.84). The ∆Φ0 in Eq. (2.77) equals ∆Φmax (Z = 0) (refer to Eq. (2.69)), the

maximum nonlinear phase distortion introduced by the LG00 beam in the transverse plane

of the sample. The ∆Φ0 in Eq. (2.84) is equal to e · ∆Φmax (Z = 0) (refer to Eq. (2.71)),

more than the maximum nonlinear phase distortion introduced by the LG10 beam in the

sample. So the ∆Φ0 in the LG00 case is a “real” phase distortion that indeed happens in

the sample, while the ∆Φ0 in the LG10 case is an “imaginary” phase distortion, bigger than

any “real” phase distortion in the sample.

In order to make a fair comparison of the sensitivity of the two Z-scan methods, we

express ∆Tp−v in terms of the “real” maximum nonlinear phase distortion. For the LG00

Z-scan,

∆Tp−v = 0.406 |∆Φmax(Z = 0)| , (2.85)

and for the LG10 Z-scan,

∆Tp−v = 0.789 |∆Φmax(Z = 0)| . (2.86)

Therefore, the sensitivity of the LG10 Z-scan is slightly higher than the LG0

0 Z-scan pro-

viding the same maximum nonlinear phase distortion is produced in the sample.

A more important difference between the two Z-scan measurements is that in the LG10

Z-scan experiment the detector is placed at the beam center where the intensity is the

weakest due to the screw phase dislocation while in the LG00 Z-scan experiment the center

intensity is the strongest. As a result the former shows a much bigger deviation from

the normal value if any phase or intensity distortion that destroys the symmetry of the

beam profile is present. This suggests that the LG10 Z-scan experiment is more sensitive

to changes of the n2 of the sample.

52

2.5.3 Effect of the aperture size: the off-axis normalized trans-

mittance

One may question how it is possible to measure the on-axis transmittance of a LG10 beam

since the on-axis intensity is always zero. In fact, what’s really being measured in the Z

scan is the power transmitted through a small aperture centered on the optical axis. Thus

not only the on-axis light but also some off-axis (but near the axis) light are collected and

measured. A follow-up question is then: does the off-axis light behave the same as the

on-axis light in terms of the normalized transmittance? This question can be answered by

examining the off-axis normalized transmittance. We use the LG10 beam as an example.

Using a procedure similar to what is described in Appendix B, we can calculate the off-axis

normalized transmittance for a LG10 beam, yielding

T (R,Z,∆Φ0) =

∣

∣

∣

∣

1 − i2

3∆Φ0 exp

[

−(

−1 +3 (1 + Z2)

9 + Z2+ i

8Z

9 + Z2

)

R2

]

(2.87)

× (1 + iZ)2

(3 + iZ)2 (1 + Z2)

(

1 +9 + Z2

(3 + iZ)2 − 3 (1 + Z2)

(3 + iZ)2 R2

)

∣

∣

∣

∣

∣

2

,

where, R ≡ r/ω(z → ∞) is the radial distance from the beam center normalized by the

beam radius at the far field.

Fig. 2.11 shows a 3D plot of the Z-scan normalized transmittance for a LG10 beam as

a function of Z as well as R. The important information from the figure is that T changes

very slowly as R increases. Therefore we expect that because the off-axis intensity profile

is relatively smooth, the size of the aperture has little effect on the accuracy of the Z-scan

result.

Fig. 2.12 further illustrates our conclusion, where the Z-scan normalized transmittance

for a LG10 beam at R = 0.05 and R = 0.1 are plotted in comparison with at R = 0. The

differences between the three curves are hardly distinguishable in the graph. The numerical

values show that the difference of T between R = 0 and R = 0.05 is less than 2%, and

53

-4-2

0

2

4Z

0

0.2

0.4

R

0.99

1

1.01T

-4-2

0

2

4Z

Figure 2.11: The Z-scan normalized transmittance for a LG10 beam as a function of trans-

verse coordinate R. (∆Φ0 = 0.1)

about 6% for R = 0.1. In the far field, the beam radius is very large, so a small aperture

usually corresponds to a very small R. In practice, we can treat all the light that passes

through the aperture as on-axis light.

2.6 Application: optical limiting

2.6.1 Introduction

With the development of laser technology, the output power and intensity of lasers has

been substantially increased. For example, lasers operating on the principle of chirped

pulse amplification (CPA) can produce ultrashort laser pulse up to the petawatt(1015

54

-10 -5 0 5 10

0.985

0.990

0.995

1.000

1.005

1.010

1.015

T

Z

R=0 R=0.05 R=0.1

Figure 2.12: The Z-scan normalized transmittance for a LG10 beam for R = 0, R = 0.05

and R = 0.1. (∆Φ0 = 0.1)

watt) level.20 Higher laser power can induce large effects and certainly enables more

novel applications. However, high powers and intensities inevitably lead to increasing the

possibilities of damaging optical components. It is often desirable to apply a “protector”

to those expensive components such that under normal power or intensity, the “protector”

passes all the light and the protected components work normally; if the power or intensity

of incident light is higher than the safe level, the “protector” blocks the dangerous light.

The optical limiter is one such “protector”.

Figure 2.13 illustrates the transmittance of an optical limiter. Without a limiter, the

output power equals the input power as shown by the dashed line. An ideal limiter (the

solid line) has no influence on the system under low incident power, but prohibits a further

increase of the output power when the incident power is more than a threshold value. In

55

practice, an ideal limiter can not be achieved. A practical limiter’s response approaches

that of the ideal limiter, e.g., the triangles in the figure.

0 1000

60

Pthreshold

O

utpu

t Pow

er (a

.u.)

Incident Power (a.u.)

no limiter ideal limiter practical limiter

Figure 2.13: Illustration of the transmittance of the optical limiter.

In this section, we discuss a new optical limiting geometry using the LG beam and

a thin nonlinear optical film. In fact, the valleys of the curves, and more generally, the

portions of the curves that fall below T = 1 in figure 2.9 imply the optical limiting

phenomenon. Recall that ∆Tp−v is proportional to the nonlinear phase distortion ∆Φ.

For those portions below T = 1, it means that the bigger ∆Φ becomes, the lower the value

of T will be. But ∆Φ is proportional to the intensity I. So the higher the intensity, I,

the lower the transmittance, T . The key requirement of an optical limiter is thus met,

suggesting that it’s possible to make an optical limiter using a setup similar to the Z-

scan setup in figure 2.7. The detector in the Z-scan setup is replaced with the optical

56

component to be protected. Alternatively (and sometimes more practically) a second lens

can be placed after the thin film as shown in figure 2.14, acting as a Fourier transform lens

to bring the far field closer.

Figure 2.14: Schematic diagram of optical limiting using the LG beam. L1: focusing lens,S: nonlinear thin film, L2: Fourier transform lens, A: small aperture, D: optical componentto be protected, f2: focal length of L2.

There are, however, questions to be answered before we put this new optical limiting

geometry into practice.

2.6.2 Effect of the position of the nonlinear thin film

One question pertains to the best place to put the thin film. In the Z-scan measurement,

the sample is scanned over a wide range of Z along the optical axis, so there is no such issue.

In the optical limiting application, however, the position of the sample must be fixed. So

sample placement is important. We therefore need to compare the limiting efficiency of

the thin film at different positions, Z, to obtain the optimal placement.

In order to make such comparisons meaningful, we should use the same illuminating

conditions at different values of Z. Specifically, the maximum intensity and therefore the

maximum nonlinear phase distortion in the sample should be kept the same. This ensures

that the sample is under the same “challenge” since the damage threshold of the material

57

is often determined by the maximum intensity it can tolerate. This should not be confused

with the Z-scan measurement, in which the maximum intensity in the sample has to change

when the sample is placed at different Z’s because the power of the beam is fixed but the

beam’s radius varies with position. In the optical limiting case, for the purpose of making

a fair comparison, we intentionally force the maximum intensity in the sample to be the

same even though the sample is at different Z’s. So the power of the incident beam must

be changed for each Z. One may argue that when comparing the limiting efficiencies, the

power of the incident beam should be kept the same. This is not appropriate because by

changing the intensity at the sample, e.g., using different focal lenses, we can get quite

different limiting efficiencies even if the power of the incident beam is fixed.

So we rewrite the normalized Z-scan transmittance T (Z,∆Φ0), using the maximum

nonlinear phase distortion at position Z, ∆Φmax (Z), as a parameter instead of using ∆Φ0

or the maximum nonlinear phase distortion at position Z = 0, ∆Φmax (Z = 0). And we

will compare the values of T at different positions using the same value of ∆Φmax (Z).

Since ∆Φmax (Z) will be treated as a constant, for convenience we write ∆Φmax (Z) as

∆Φmax. We call this new form of transmittance T (Z,∆Φmax) the normalized optical

limiting transmittance to distinguish it from the normalized Z-scan transmittance. Below

we show some examples.

LG00 beam

Substituting

∆Φ0 =(

1 + Z2)

∆Φmax (2.88)

into Eq. (2.74), we get

T (Z,∆Φmax) = 1 +4Z

9 + Z2∆Φmax. (2.89)

58

LG10 beam

Substituting

∆Φ0 = e(

1 + Z2)

∆Φmax (2.90)

into Eq. (2.79), we get

T (Z,∆Φmax) = 1 +8e · Z (27 + 10Z2 − Z4)

(9 + Z2)3 ∆Φmax. (2.91)

-20 -15 -10 -5 0 5 10 15 200.8

0.9

1.0

1.1

1.2

max= -0.1

Z

Nor

mal

ized

Tra

nsm

ittan

ce T

LG00

LG01

T=1

Figure 2.15: Typical curves of normalized optical limiting transmittance T vs. position Z.∆Φmax = −0.1. The circled line is for the LG0

0 beam, the solid line is for the LG10 beam,

and the dashed line shows T = 1.

Typical curves of normalized optical limiting transmittance T vs. position Z are shown

59

in figure 2.15, where the maximum nonlinear phase distortion ∆Φmax = −0.1 is selected as

an example, corresponding to material with negative nonlinear refractive index n2. When

∆Φmax > 0, it’s easy to verify that the curves are the mirror images of their negative

counterparts with maximum nonlinear phase distortion being −∆Φmax.

For the curve associated with the LG00 beam, Z > 0 and T < 1. Thus, the Z > 0

regime can be used for optical limiting if the nonlinear film has a negative n2. The most

sensitive place is Z = 3, where T reaches the valley with a value of (1 + 2∆Φmax/3), (or

0.933 in the figure) as ∆Φmax = −0.1. The extremum is solved by analyzing Eq. (2.89).

The curve of the LG10 beam has two regions whose normalized transmittance T falls

below 1. one is when Z is less than approximately −3.49 with a valley at (−8.55, 1 +

1.54∆Φmax). The other is between Z = 0 and approximately Z = 3.49 with a valley at

(1.73, 1 + 1.05∆Φmax). These values are obtained by analyzing Eq. (2.91). In the figure,

the extrema are (−8.55, 0.846) and (1.73, 0.895), respectively, as ∆Φmax = −0.1. Both

valleys are lower than the one for the LG00 beam.

It’s obvious that not only the extrema but also the shapes of the curves in figure 2.15

are different from the Z-scan traces in which ∆Φ0 is held constant. We emphasize that

only by keeping ∆Φmax constant does the curve reflect the true potential of the thin film

for optical limiting at different positions, as we discussed earlier. The normalized optical

limiting transmittance, rather than the normalized Z-scan transmittance, should be used

in evaluating where is the best place to put the thin film for optical limiting.

2.6.3 Large nonlinear phase distortion

In the Z-scan measurement, the nonlinear phase distortion is assumed to be very small,

e.g., ∆Φ << 1. The incident intensity of the beam is thus limited by this assumption. Eqs.

(2.74), (2.89), (2.79) and (2.91) are all derived under this assumption. However, in the op-

tical limiting application, the incident intensity usually varies over a much larger range and

60

the resulting nonlinear phase distortion is not necessarily small. Whether optical limiting

takes place when the incident intensity is high remains unknown. To answer this question,

a calculation of the normalized optical limiting transmittance, T , for arbitrary nonlinear

phase distortion is necessary. In Appendix C, we derive the normalized optical limiting

transmittance for arbitrary nonlinear phase distortion for the LG00 and LG1

0 beams. We

find that in general, a numerical calculation is required. We have developed Mathematica

codes to complete the numerical calculations which are included in Appendix C. Here we

present the calculated results.

LG00 beam

Figure 2.16 shows the series of curves of the normalized optical limiting transmittance, T ,

versus the maximum nonlinear phase distortion, ∆Φmax, in the sample, where each curve

corresponds to a different sample position, Z, indicated by the number along the curve.

The nonlinear refractive index of the sample, n2, is assumed to be negative. As shown in

the figure, the curves can be divided into several groups according to their behavior near

∆Φmax = 0:

• Group I (−∞ < Z ≤ −3): Initially T increases from 1 as |∆Φmax| increases from 0.

And for fixed value of ∆Φmax near ∆Φmax = 0, the value of T increases as the value

of Z increases.

• Group II (−3 ≤ Z ≤ 0): Initially T increases from 1 as |∆Φmax| increases from 0.

And for fixed value of ∆Φmax near ∆Φmax = 0 , the value of T decreases as the value

of Z increases.

• Group III (0 ≤ Z ≤ 1.11) and Group IV (1.11 ≤ Z ≤ 3): Initially T decreases from

1 as |∆Φmax| increases from 0. And for fixed value of ∆Φmax near ∆Φmax = 0, the

value of T decreases as the value of Z increases. Among them the curve of Z = 1.11

is special because it’s value of T reaches 0 when ∆Φmax ≈ −4.01.

61

-6 -4 -2 00.0

0.2

0.4

0.6

0.8

1.0

1.11

0.60.4

0.20

III

Maximum nonlinear phase distortion

-6 -4 -2 00

1

2

3

4

5

-2

-1

-0.75 0

-3

II

-6 -4 -2 01

2

4

6

-20

-12-9

-6-3

-3

Nor

mal

ized

Tra

nsm

ittan

ce T

I

Figure 2.16 (I-III)

62

-6 -4 -2 00.0

0.2

0.4

0.6

0.8

1.0

2012 9 6

3

Nor

mal

ized

Tra

nsm

ittan

ce T

V

Maximum nonlinear phase distortion

-6 -4 -2 00.0

0.2

0.4

0.6

0.8

1.0

3

2

2

3

3 2

1.11

IV

Figure 2.16: The normalized optical limiting transmittance T versus the maximum non-linear phase distortion ∆Φmax in the sample when the incident beam is a LG0

0 beam. Theposition Z of the sample for each of the curve is indicated by the number along the curve.A sample of negative n2 is assumed.

63

• Group V (3 ≤ Z < ∞): Initially T decreases from 1 as |∆Φmax| increases from 0.

And for fixed value of ∆Φmax near ∆Φmax = 0, the value of T increases as the value

of Z increases.

It’s easy to seen that the curves in Group III-IV are candidates for optical limiting

since their transmittance T ’s decrease as the nonlinear phase distortion |∆Φmax| increases.

The slope of T vs. |∆Φmax| curve depends on the position Z. For example, the curve of

Z = 3 has the most steep slope. In practice, we should choose the curve based upon the

requirement of the system. For example, curves with greater slope provide smaller limiting

threshold, while curves with smaller slope have wider transmittance range for the weak

intensity.

It’s important to point out that after T reaches the minimum value, it will turn back to

increase as |∆Φmax| further increases. Thus, optical limiting operates up to the minimum

value of T . The minimum value of T and the corresponding value of ∆Φmax also depend

on the position Z. For example, the curve of Z = 1.11 is the only one whose value of T

reaches 0 when ∆Φmax ≈ −4.01. In practice, there is usually an upper limit of the incident

intensity in the system, like the maximum intensity that a laser can emit. Furthermore,

the nonlinear sample has its own damage threshold which limits the maximum intensity it

can tolerate. Thus the maximum nonlinear phase distortion in a practical system is always

limited. Therefore, the turning back of the transmittance T can be avoided by selecting

the proper curves whose turning points are out of the range of operation.

Finally, we note that some of the curves in Group II can be used to perform a special

type of optical limiting. Curves like the one with Z = −0.75, which initially increase as

|∆Φmax| increases, soon turn back to decrease. These curves can be used to make a system

that enhances low intensity transmittance but limits high intensity transmittance.

64

LG10 beam

When the incident beam is a LG10 beam, the situation is more complex. Figure 2.17

shows the series of curves of the normalized optical limiting transmittance T versus the

maximum nonlinear phase distortion ∆Φmax in the sample, where each curve corresponds

to a different sample position Z indicated by the number along the curve. The nonlinear

refractive index of the sample n2 is assumed to be negative. As shown in the figure, the

curves can be divided into several groups according to their behavior near ∆Φmax = 0:

• Group I (−∞ < Z ≤ −8.55): Initially T decreases from 1 as |∆Φmax| increases from

0. And for fixed value of ∆Φmax near ∆Φmax = 0 , the value of T decreases as the

value of Z increases.

• Group II (−8.55 ≤ Z ≤ −7.42) and Group III (−7.42 ≤ Z < −3.49): Initially T

decreases from 1 as |∆Φmax| increases from 0. And for fixed value of ∆Φmax near

∆Φmax = 0, the value of T increases as the value of Z increases. The curve of

Z = −7.42 is special because it’s value of T reaches 0 when ∆Φmax ≈ −1.42.

• Group IV (−3.49 ≤ Z ≤ −1.73): Initially T increases from 1 as |∆Φmax| increases

from 0. And for fixed value of ∆Φmax near ∆Φmax = 0, the value of T increases as

the value of Z increases.

• Group V (−1.73 ≤ Z < 0): Initially T increases from 1 as |∆Φmax| increases from

0. And for fixed value of ∆Φmax near ∆Φmax = 0, the value of T decreases as the

value of Z increases.

• Group VI (0 ≤ Z ≤ 0.61) and Group VII (0.61 ≤ Z ≤ 1.73) : Initially T decreases

from 1 as |∆Φmax| increases from 0. And for fixed value of ∆Φmax near ∆Φmax = 0,

the value of T decreases as the value of Z increases. The curve of Z = 0.61 is special

because it’s value of T reaches 0 when ∆Φmax ≈ −3.20.

65

-3 -2 -1 00.0

0.2

0.4

0.6

0.8

1.0

1.2

III

-3.49-4

-4.5

-5

-6-7.42

Maximum nonlinear phase distortion

-3 -2 -1 00.0

0.2

0.4

0.6

0.8

1.0

II

-7.42

-7.42

-7.42

-8.55

-8.55

-8.55

Nor

mal

ized

Tra

nsm

ittan

ce T -6 -4 -2 0

0.0

0.2

0.4

0.6

0.8

1.0

I-40

-20 -12

-8.55

Figure 2.17 (I-III)

66

-6 -4 -2 00

1

2

3

4

5

6

V

0-0.5

-1

-1.5

-1.73

-2.5 -2.0 -1.5 -1.0 -0.5 0.01

2

3

4

5

6

IV-3.49

-1.73-3

-3

-3.49 -1.73

Maximum nonlinear phase distortion

Nor

mal

ized

Tra

nsm

ittan

ce T

Figure 2.17 (IV-V)

67

-6 -4 -2 00.0

0.4

0.8

1.2

1.6

2.0

2.4

VIII3.49

3

2.52

1.73

Nor

mal

ized

Tra

nsm

ittan

ce T

Maximum nonlinear phase distortion

-6 -4 -2 00.0

0.2

0.4

0.6

0.8

1.0

VII

1.73

0.61 0.610.90.9

1.73

0.90.6

1

-6 -4 -2 00.0

0.2

0.4

0.6

0.8

1.0

VI

0.610.6

10.4

0.20

Figure 2.17 (VI-VIII)

68

-3 -2 -1 01

3

5

7

9

X

40 20

8.55

-3 -2 -1 01

3

5

7

9

IX8.55

6.5

5

3.49

Maximum nonlinear phase distortion

Nor

mal

ized

Tra

nsm

ittan

ce T

Figure 2.17: The normalized optical limiting transmittance T versus the maximum non-linear phase distortion ∆Φmax in the sample when the incident beam is a LG1

0 beam. Theposition Z of the sample for each of the curves is indicated by the number along thatcurve.

69

• Group VIII (1.73 ≤ Z < 3.49): Initially T decreases from 1 as |∆Φmax| increases

from 0. And for fixed value of ∆Φmax near ∆Φmax = 0, the value of T increases as

the value of Z increases.

• Group IX (3.49 ≤ Z ≤ 8.55): Initially T increases from 1 as |∆Φmax| increases from

0. And for fixed value of ∆Φmax near ∆Φmax = 0, the value of T increases as the

value of Z increases.

• Group X (8.55 ≤ Z <∞): Initially T increases from 1 as |∆Φmax| increases from 0.

And for fixed value of ∆Φmax near ∆Φmax = 0, the value of T decreases as the value

of Z increases.

The curves in Group I-III and Group VI-VII can be implemented for optical limiting

since their transmittances decrease as the nonlinear phase distortion |∆Φmax| increases.

Compared to using a LG00 beam, using a LG1

0 beam to achieve optical limiting has more

choices. We can put the sample in front of the focus of the beam, utilizing the curves in

Group I-III, or put the sample behind the focus of the beam, making use of the curves in

Group VI-VII. This is more flexible than the LG00 beam case in which the sample has to be

placed behind the focus of the beam. This added flexibility is important when designing

certain optical systems. More importantly, some of the curves in the LG10 beam case have

steeper slopes than the LG00 beam case. For example, the curve of Z = −7.42 in the LG1

0

beam case decreases to 0 when ∆Φmax ≈ −1.42, while it’s counterpart in the LG00 beam

case, the curve of Z = 1.11 does not drop to 0 until ∆Φmax reaches approximately -4.01.

Thus using a LG10 beam can achieve a smaller limiting threshold than using a LG0

0 beam

provided the rest of the conditions are held the same.

As in the LG00 beam case, the limiting curves of the LG1

0 beam also turn back to

increase as |∆Φmax| further increases after T reaches the minimum value. Thus the same

kind of considerations should be taken when we design the optical limiting systems.

70

Some of the curves in Group V can be used to perform a special kind of optical limiting

we proposed in the case of the LG00 beam. Curves like the one with Z = −0.5, although

they initially increase as |∆Φmax| increases, soon turn back to decrease. These curves can

be used to make a system that enhances low intensity transmittance but limit the high

intensity transmittance.

2.7 Application: Measuring the nonlinear refractive

index

2.7.1 Motivation

The Z-scan technique is very useful for it’s high sensitivity and relatively simple setup

using a single beam. However, the requirement of moving the sample along the z axis has

drawbacks. One problem is the error due to re-alignment. When the sample is moved

to a new Z position, the sample must be re-aligned to be perpendicular to the incident

beam and the aperture may need to be re-aligned to the center of the far field. These

procedures not only cause errors but are also generally time-consuming. The problem

becomes more serious when the sample is not uniform. Then not only the sample needs

to be perpendicular to the optical axis, but also the exact same spot in the sample must

overlap with the beam. The latter is very difficult without a precise mechanical adjusting

system. Also, in order to move the sample along the Z axis, there must be enough clear

space to operate, which is not possible when making a small device. Another requirement

of Z scan is that |∆Φ0| must be less than 1, which could be a problem under certain

circumstances. For example, if we want to measure how the n2 of a sample with a slow

nonlinear mechanism increases with time, a larger incident intensity is used to let T (and

|∆Φ0|) increase at a larger rate, such that the resolution of n2 as a function of time is

good. The traditional Z scan technique can not handle this kind of measurement once

71

|∆Φ0| increases to larger than 1.

In this section, we explore alternative techniques to measure the nonlinear refractive

index that do not require moving the sample along the Z axis, yet retain as much as possible

the merits of the Z-scan technique, such as high sensitivity and single beam simplicity. One

of the new techniques, namely the ∆Φmax scan, can handle the situations when |∆Φ| > 1

.

2.7.2 I scan

The inspiration comes from Figure 2.16 and Figure 2.17, which are originally plotted to

examine the optical limiting abilities of the corresponding setups by showing the series of

curves of the normalized optical limiting transmittance T versus the maximum nonlinear

phase distortion ∆Φmax in the sample at different position Z. One of the features of the

curves is that at a given position Z, T is totally determined by the value of ∆Φmax. The

reverse is not always true because knowing the value of T generally does not suffice to give

the value of ∆Φmax. However, between ∆Φmax = 0 and the first turning point of each curve,

∆Φmax and T are in one-to-one correspondence. In fact, if ∆Φmax is small, T depends

on ∆Φmax almost linearly, and vice versa. Thus, if we know the sample position Z and

measure normalized T , we should be able to determine the nonlinear phase distortion in

the sample, which combined with other information such as intensity I gives the nonlinear

refractive index.

Figures (2.18) to (2.20) show the maximum nonlinear phase distortion ∆Φmax as a

function of the normalized transmittance T . The dots are calculated using our methods

developed in the previous section. The lines are the best linear fits. The incident beam is

a LG00 beam in Figure (2.18) and the sample positions are Z = ±3. The incident beam

in Figure (2.19) and Figure (2.20) is a LG10 beam and the positions of the sample are

Z = ±1.73 and Z = ±8.55, respectively. As can be seen in the figures, the lines fit the

72

0.93 0.94 0.95 0.96 0.97 0.98 0.99 1.00

-0.10

-0.08

-0.06

-0.04

-0.02

0.00

calculated result linear fit: max= 1.53*(T-1)

z=3

Max

imum

non

linea

r pha

se d

isto

rtion

Normalized Transmittance T

1.00 1.01 1.02 1.03 1.04 1.05 1.06 1.07

-0.10

-0.08

-0.06

-0.04

-0.02

0.00

calculated result linear fit: max= -1.48*(T-1)

z=-3

Figure 2.18: The maximum nonlinear phase distortion ∆Φmax as a function of the normal-ized transmittance T . The incident beam is a LG0

0 beam and the position of the sampleis Z=-3 and Z=3 for the upper and lower curve, respectively. The dots are the calculatedresults and the lines are the linear fits.

73

0.90 0.92 0.94 0.96 0.98 1.00

-0.10

-0.08

-0.06

-0.04

-0.02

0.00

Z= 1.73

calculated result linear fit: max= 0.99*(T-1)

1.00 1.02 1.04 1.06 1.08 1.10 1.12

-0.10

-0.08

-0.06

-0.04

-0.02

0.00

Normalized Transmittance T

Max

imum

non

linea

r pha

se d

isto

rtion

Z= -1.73

calculated result linear fit: max= -0.93*(T-1)

Figure 2.19: The maximum nonlinear phase distortion ∆Φmax as a function of the normal-ized transmittance T . The incident beam is a LG1

0 beam and the position of the sampleis Z=-1.73 and Z=1.73 for the upper and lower curve, respectively. The dots are thecalculated results and the lines are the linear fits.

74

1.00 1.02 1.04 1.06 1.08 1.10 1.12 1.14 1.16

-0.10

-0.08

-0.06

-0.04

-0.02

0.000.84 0.86 0.88 0.90 0.92 0.94 0.96 0.98 1.00

-0.10

-0.08

-0.06

-0.04

-0.02

0.00

Normalized Transmittance T

Z= 8.55

calculated result linear fit: max= -0.63*(T-1)

Max

imum

non

linea

r pha

se d

isto

rtion

Z= -8.55

calculated result linear fit: max= 0.67*(T-1)

Figure 2.20: The maximum nonlinear phase distortion ∆Φmax as a function of the normal-ized transmittance T . The incident beam is a LG1

0 beam and the position of the sampleis Z=-8.55 and Z=8.55 for the upper and lower curve, respectively. The dots are thecalculated results and the lines are the linear fits.

75

data very well. Examination of the numerical values shows that the maximum deviation

of the data from the linear fit lines is less than 4%.

Although the same kind of behavior occurs in many other positions of the sample, the

positions chosen in the figures give the most sensitive change of T for a given amount

of change of ∆Φmax among their classes. In other words, they provide the most high

sensitivity when measuring the nonlinear refractive index. Among the plotted curves,

using a LG10 beam and letting Z = 8.55, the highest sensitivity results. The linear fit gives

∆Φmax = −0.63(T − 1). If the detector is able to measure a change of T = 1%, then we

are able to determine a change of |∆Φmax| = 0.0063, equivalant to a precision of about

λ/1000, better than that of the traditional Z-scan.

In theory, one measurement of T other than T = 1 would allow us to determine the

corresponding ∆Φmax and then the nonlinear refractive index. In practice, we can change

∆Φmax over a range, e.g., by changing the intensity of the incident beam, (hence the name

“I scan” is given.) to get multiple measurements of T in order to minimize the errors. As

long as T is within the plotted region, the linear relationship holds well.

Compared with the Z-scan method, the I-scan method discussed here uses the same

kind of setup and provides competitive sensitivity, but does not require moving the sample

along the z axis. The price paid for this advantage is that the position of the sample must

be set exactly in order to make use of the corresponding ∆Φmax − T curve. Thus before

actually measuring the nonlinear refractive index of the sample, the position of the waist

and the Rayleigh length of the incident beam must be measured which then allows one to

determine the position Z of the sample. Alternatively a calibration curve relating ∆Φmax

to T at a certain position can be measured using a sample with known nonlinear refractive

index. The curve can then be used to measure the nonlinear refractive index of other

samples that are placed at the exact same position, obviating the need of knowing sample

position Z explicitly.

In the next section, we propose a method that determines the position Z of the sample at

76

the same time when T is measured. The procedure of pre-calibrating the system mentioned

above is then unnecessary.

2.7.3 ∆Φmax scan to measure samples with large n2

We start with Figure 2.21, a re-plot of Figure 2.17 which shows the normalized trans-

mittance T versus the maximum nonlinear phase distortion ∆Φmax in the sample when

the incident beam is a LG10 beam, with selected curves of different sample positions. The

position Z of the sample for each of the curves is indicated by the number along the curve.

The curves plotted in Figure 2.21 all show one and only one extreme point (valley) within

the plotted range. These valleys are crossed by the dotted curve which is the set of minima

of the T − ∆Φmax curves with Z ranging from -3.49 to -7.42.

One important feature immediately seen from Figure 2.21 is that the T values of the

valleys are in one-to-one correspondence with the position Z of the sample. In other words,

if we can measure the T value of the valley of the T −∆Φmax curve, we should be able to

deduce the position Z of the sample.

Experimentally, measuring the T value of the valley is possible. Although ∆Φmax is

not directly measurable, the increase or decrease of its value is controllable. For example,

if the nonlinear mechanism of the sample is a fast process, we can increase ∆Φmax by

increasing the intensity of the incident beam; If the nonlinear mechanism of the sample

is a slow process, we can fix the intensity of the incident beam and prolong the exposure

time, simply waiting for ∆Φmax to increase with time, assuming that the nonlinearity of

the sample builds up monotonously with the exposure time (which is often true). Initially

we shall see the decrease of T with ∆Φmax increasing. At some point, ∆Φmax reaches

and passes the valley. Subsequently T turns back and increases with ∆Φmax with further

increase of ∆Φmax. The value of T at the turning point is determined from the height of

the minimum. In practice, we can measure the T − I curve (normalized transmittance vs.

77

-3.0 -2.5 -2.0 -1.5 -1.0 -0.5 0.00.0

0.2

0.4

0.6

0.8

1.0

Nor

mal

ized

Tra

nsm

ittan

ce T

Maximum nonlinear phase distortion

-7.42

-6.5

-6

-5.75

-5.5

-5.25

-5

-4.75

-4.5-4.25

-3.75-4

Figure 2.21: Solid lines: selected curves of the normalized transmittance T versus themaximum nonlinear phase distortion ∆Φmax in the sample when the incident beam is aLG1

0 beam. The position Z of the sample for each of the curves is indicated by the numberalong the curve. Dotted line: the coordinates of the valleys of the T − ∆Φmax curves.

78

intensity) or the T−t curve (normalized transmittance vs. time) instead of the T−∆Φmax

curve. Although the shape of these curves are not the exactly the same, they all have one

valley, and the T coordinates of the valley are the same.

0.0 0.2 0.4 0.6 0.8 1.0-8

-7

-6

-5

-4

-3

Normalized Transmittance T

Po

sitio

n of

sam

ple

Z

Figure 2.22: The sample position Z versus the T coordinate of the valley of the corre-sponding T −∆Φmax curve. The arrows represent the useful range of the Z − T curve fordetermining the position from the transmittance.

Once the T value of the valley is obtained, the sample position Z can be found, e.g.,

using Figure 2.22, where Z is plotted as a function of T value of the valley of the corre-

sponding T − ∆Φmax curve. Note that for given resolution of T (which is determined by

the detecting system), the resolution of Z is not uniform over the plotted range. At both

ends, the Z−T curve is very steep, making the resolution of Z poor. However, the middle

part of the curve, roughly from Z = −6 to Z = −4 as indicated by the arrows in Figure

2.22, has smaller and nearly uniform slope, providing better resolution of Z. Therefore the

middle part of the curve is more useful in practice. We re-plot the middle part as shown in

79

Figure 2.23, in which we also show a best fit line with an analytic function. The function

0.2 0.4 0.6 0.8 1.0

-6.0

-5.5

-5.0

-4.5

-4.0

calculated

fitted

Posi

tion

of s

ampl

e Z

Normalized Transmittance T

Figure 2.23: The sample position Z versus the T coordinate of the valley of the corre-sponding T − ∆Φmax curve. Circles: calculated results. Line: best fit using an inverseGauss function (see text for details).

we choose is an inverse Gauss function defined as

y = y0 + ω

√

0.5 ln

(

A

x

)

. (2.92)

It is named such because it is the inverse function of the Gauss function

x = A exp

(

−2

(

y − y0

ω

)2)

. (2.93)

The fitting result from Figure 2.23 is

Z = −3.53 − 2.52√

0.5 ln (0.98/T ). (2.94)

80

Examination of the numerical values shows that the maximum deviation of the fit from

the calculated data is less than 0.3%. Therfore Eq. (2.94) gives a good approximation of

the value of Z given the value of T of the valley.

Now that the sample position Z is found, we can apply the method introduced in the

previous section, i.e., using the ∆Φmax - T curve at this Z to determine the values of ∆Φmax

from the measured T . From ∆Φmax we determine the nonlinear refractive index coefficient

n2. Following the name of Z scan, we name this method ∆Φmax scan because during the

process of searching the valley, |∆Φmax| is changed gradually from 0 to some value beyond

the valley point. As mentioned earlier, ∆Φmax can be changed by changing the intensity

I or the exposure time t, depending on the nonlinear mechanism of the sample. If it is

the intensity I, then we obtain a n2 − I curve from the measured T − I curve; if it is the

time t, then we obtain a n2 − t curve from the measured T − t curve. Thus, one inherent

advantage of the ∆Φmax scan is revealed: it can measure the speed of the development of

n2 (through the n2 − t curve) automatically for a slow nonlinear mechanism.

When we determine ∆Φmax from T , there is a major difference between the I-scan

method introduced in the previous section and the ∆Φmax scan. In the previous section,

the change of T is well inside the linear region of the ∆Φmax - T curve and ∆Φmax and T

are in one-to-one correspondence, so we can use the best fit formula to calculate ∆Φmax

from T easily. In the ∆Φmax scan, however, T has to extend out of the linear region

of the ∆Φmax - T curve in order to reach the valley point. Moreover, once passing the

valley, T is not in one-to-one correspondence with ∆Φmax because for each value of T there

could be two ∆Φmax values (see Figure 2.21). The issue of the one-to-one correspondence

is not a problem because during the measurement ∆Φmax is always allowed to increase

monotonously. So if the same value of T occurs twice, the first must correspond to the

smaller |∆Φmax| before the valley and the second corresponds to the bigger |∆Φmax| after

the valley. The correspondence can be identified unambiguously from the graph or by a

computer program. The nonlinear attribute of the ∆Φmax - T curves makes it difficult to

81

find an analytic function to fit the curve, especially when Z is also a parameter. We handle

this problem by calculating T as a function of ∆Φmax numerically and interpolating the

value of ∆Φmax for the experimental values of T . All these can be achieved automatically

by a computer program.

The procedures of the ∆Φmax scan are summarized as follows.

1. Place the sample to be measured somewhere between Z = −6 and Z = −4 along

the optical axis, using a setup similar to the Z-scan measurement with a LG10 beam.

2. Record the T − I (or the T − t) curve untile the valley is passed.

3. Calculate the sample position Z by substituting the T coordinate of the valley point

into Eq. (2.94).

4. Calculate T as a function of ∆Φmax with the Z from the above step. Make a two

column data table, and fill one column with ∆Φmax increasing with certain step and

the other column with the corresponding values of T .

5. Transfer the T − I (or the T − t) curve into the ∆Φmax− I (or the ∆Φmax− t) curve

by interpolating the above data table.

6. Calculate the n2 from the ∆Φmax and obtain the n2 value (or the n2 − t curve).

The advantages of the ∆Φmax-scan method are obvious in comparison with the Z-scan

or the I-scan method. There is no need to move the sample along the Z axis. It’s not nec-

essary to pre-determine the sample position accurately or use another sample with known

nonlinear refractive index to calibrate the system. In the case of a slow nonlinear mecha-

nism, the ∆Φmax-scan method can measure how the n2 increases with time automatically.

There are, however, trade-offs. To be able to reach the valley point requires a relatively

large |∆Φmax|: when Z = −4, the ∆Φmax coordinate of the valley is about -0.5, and

when Z = −6, the ∆Φmax coordinate of the valley is about -1.2. Although |∆Φmax| can

82

be increased in several ways, such as using a thicker sample, larger intensity, or longer

exposure time for a slow nonlinear process, they all eventually have an upper limit, due

to the thin sample assumption, the damage threshold of the sample, or the time period in

which the nonlinearity can be treated as the optical Kerr effect. So materials that have a

larger n2 are more appropriately to be measured by the |∆Φmax|-scan method. Another

issue is that the resolution of n2 is not uniform for given resolution of T , which can be

seen from Figure 2.21, where around the valley in the T − ∆Φmax curve, ∆Φmax shows

the biggest uncertainty for given uncertainty of T , giving the lowest resolution of ∆Φmax

(and therefore n2) compared to other part of the curve. This should be kept in mind when

interpreting the final result.

Finally we note that the region between Z = −4 and Z = −6 is not the only one that

a ∆Φmax scan can be carried out. In principle any group of T − ∆Φmax curves that show

a clear turning point of T where |∆Φmax| increases from 0 can be used to do the ∆Φmax-

scan measurement, provided that the T coordinate of the turning point is in one-to-one

correspondence to Z. For example, the curves whose Z values are between 0.61 and 3.49

using a LG10 beam have the valleys as shown Figure 2.24, and the curves whose Z are

larger than 0 using a LG00 beam have the peaks as shown Figure 2.25. In practice the

selection of these groups should be governed by their overall performances, including the

resolution and the minimum |∆Φmax| required, which depend on the shape of the curves

and the location of the turning points, respectively. Among them, the one used as the

example in this section (i.e., the curves whose Z are between -4 and -6 using a LG10 beam)

has the least minimum |∆Φmax| requirement, yet gives good overall resolution.

83

-3.0 -2.5 -2.0 -1.5 -1.0 -0.5 0.00.0

0.2

0.4

0.6

0.8

1.0

1.2

1.5

1.73

2

2.25

2.5

2.75

33.25

Nor

mal

ized

Tra

nsm

ittan

ce T

Maximum nonlinear phase distortion

Figure 2.24: Normalized transmittance, T , versus the maximum nonlinear phase distortion,∆Φmax, in the sample when the incident beam is a LG1

0 beam for selected curves whose Zare between 0.61 and 3.49. The position, Z, of the sample for each of the curve is indicatedby the number along the curve.

84

-6 -5 -4 -3 -2 -1 01

2

3

4

5

-0.25

-3

-2.75

-2.5

-2.25

-2

-1.75

-1.5

-1.25-1

-0.75-0.5

Nor

mal

ized

Tra

nsm

ittan

ce T

Maximum nonlinear phase distortion

Figure 2.25: Normalized transmittance, T , versus the maximum nonlinear phase distortion,∆Φmax, in the sample when the incident beam is a LG0

0 beam for selected curves whose Zvalues are larger than 0. The position, Z, of the sample for each of the curve is indicatedby the number along the curve.

85

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[8] L. Allen, M. Beijersbergen, R. Spreeuw, and J. Woerdman, “Orbital angular momen-

tum of light and the trasformation of laguerre-gaussian laser modes,” Phys. Rev. A

45, 8185 (1992).

86

[9] J. Nye and M. V. Berry, “Dislocations in wave trains,” Proc. R. Soc. Lond. A. 336,

165 (1974).

[10] A. Yariv, Quantum electronics, 3rd ed. (Wiley, New York, 1989).

[11] J. D. Jackson, Classical electrodynamics, 3rd ed. (Wiley, [New York], 1999).

[12] Z. Sekkat, D. Morichere, M. Dumont, R. Loucifsaibi, and J. A. Delaire, “Photoiso-

merization of azobenzene derivatives in polymeric thin-films,” J. Appl. Phys. 71, 1543

(1992).

[13] R. Loucifsaibi, K. Nakatani, J. A. Delaire, M. Dumont, and Z. Sekkat, “Photoisomer-

ization and 2nd harmonic-generation in disperse red one-doped and one-functionalized

poly(methyl methacrylate) films,” Chem. Mater. 5, 229 (1993).

[14] J. A. Hermann, “Simple model for a passive optical power limiter,” J. Mod. Opt. 32,

7 (1985).

[15] M. Sheik-bahae, S. A. A., and V. S. E. W., “High-sensitivity, single-beam n2 mea-

surement,” Opt. Lett. 14, 955 (1989).

[16] M. Sheik-Bahae, A. A. Said, T. H. Wei, D. J. Hagen, and E. W. Van Stryland, “Sen-

sitive measurement of optical nonlinearities using a single beam,” IEEE J. Quantum

Electron. 26, 760 (1990).

[17] E. W. Van Stryland and M. Sheik-Bahae, “Z-scan,” in Characterization techniques

and tabulations for organic nonlinear optical materials, C. W. Dirk and M. G. Kuzyk,

eds., (Marcel Dekker, New York, 1998), p. 655.

[18] A. E. Kaplan, ““external” self-focusing of light by a nonlinear layer,” Radiophys.

Quant. Electron. 12, 692 (1969).

87

[19] D. Weaire, B. Wherrett, D. Miller, and S. Smith, “Effect of low-power nonlinear

refraction on laser-beam propagation in insb,” Opt. Lett. 4, 331 (1979).

[20] M. D. Perry, D. Pennington, B. C. Stuart, G. Tietbohl, J. A. Britten, C. Brown,

S. Herman, B. Golick, M. Kartz, J. Miller, H. T. Powell, M. Vergino, and V. Yanovsky,

“Petawatt laser pulses,” Opt. Lett. 24, 160 (1999).

88

Chapter 3

Experiment

3.1 Introduction

This chapter is devoted to introduce the experiments that we carry out in order to test

the validity of our theory, as well as to demonstrate the proposed applications.

First in Section 3.2, we describe how we generate high-order LG beams. This is impor-

tant because the-high order LG beams are usually not available from commercial lasers.

We also include in this section an experimental method that can identify the orbital an-

gular momentum carried by an LG beam.

Sample fabrication is another important factor to the success of our experiments. So

in Section 3.3 we explain how we synthesize the DR1/PMMA material, and how we make

the samples that are suitable for our experiments.

As pointed out in the theory part, DR1/PMMA acts like an optical Kerr medium only

under certain conditions. The time that the sample is illuminated can not be long, and

the beam intensity can not be too high. In order to get an estimation of this time scale

and the beam intensity, we measure the dynamics of the nonlinear refractive index change

in DR1/PMMA through a holographic volume index gratings recording experiment. The

details of the experiment are explained in Section 3.4.

89

Finally in Section 3.5, we describe the experiments that use the LG10 beams, including

the Z-scan measurement, the I-scan measurement, and optical limiting. As the setups

for the three experiments are similar, we explain them together and make notes wherever

there are differences.

3.2 Generating the higher order Laguerre Gaussian

beams

Several methods have been reported to produce higher-order Laguerre Gaussian beams,

including directly from specially designed laser cavities,1–4 by converting Hermite-Gaussian

modes using cylindrical-lens mode converters,5–7 by using spiral phase plates,8, 9 and by

computer-generated holograms.10–13 In our experiment, we use the computer-generated

hologram to generate the LG10 beam, because it is simple, demands only limited equip-

ments, yet gives reasonable efficiency and good beam quality. In this section, we first

review the principles of designing the computer-generated hologram, and then describe

how we make the hologram and generate the LG10 beam.

3.2.1 The principles

A hologram is a recording of the interference pattern generated by two light beams, namely,

the signal beam and the reference beam. The signal beam can be reconstructed by illumi-

nating the hologram with the reference beam. By using a computer, we can simulate the

interference pattern and produce the hologram without using real beams.

Consider the following hologram which converts a LG00 beam into a LG1

0 beam as shown

in Fig. 3.1. The hologram is placed in the xy plane at z = 0. The LG00 beam propagates

along the z axis. The wave vector of the LG10 beam lies in the xz plane and makes an

angle, α, with the z axis such that it can be better separated from the incident beam in

90

Figure 3.1: Schematic diagram of a hologram that converts a LG00 beam into a LG1

0 beam.

practice. Although it is not necessary, for simplicity we assume the waists of both beams

are at z = 0. The complex amplitude of the two beams at z = 0 can be written as

ELG00

= exp

(

− r2

ω20

)

, (3.1)

and

ELG10

=

√2r

ω0

exp

(

− r2

ω20

)

exp (−iφ) exp (−ikxx) exp (−iψ0) , (3.2)

respectively, where kx = k sin (α) is the x component of the wave vector k of the LG10

beam, ψ0 is a possible relative phase shift between the LG00 and the LG1

0 beam, r, φ and z

are the cylindrical coordinates, and we have ignored the time-dependent factor exp (i2πνt).

The conversion can be achieved if the hologram gives a complex transmittance function

T (r, φ) such that ELG00T (r, φ) = ELG1

0. The expression of T (r, φ) is easily obtained by using

Eqs. (3.1) and (3.2), yielding

T (r, φ) =

√2r

ω0

exp (−i (φ+ kxr cosφ+ ψ0)) . (3.3)

As can be seen, a pure conversion requires the hologram to be able to modify both the

91

amplitude and the phase of the incident beam, which is demanding but doable.

As a compromise, an amplitude hologram, which has a relaxed requirement, can be

used, by sacrificing the conversion efficiency and the mode purity of the converted beam.

For example, an amplitude transmittance function

T (r, φ) =(1 + cos (φ+ kxr cosφ+ ψ0))

2(3.4)

can also be written as

T (r, φ) =1

2+

1

4exp (−i (φ+ kxr cosφ+ ψ0)) +

1

4exp (i (φ+ kxr cosφ+ ψ0)) , (3.5)

where the second term has the phase modulation required by Eq. (3.3) and generates

the target beam. The first term generates a beam propagating along the direction of the

incident beam. The third term generates a beam whose wave vector is the mirror image

of that of the target beam with respect to the yz plane, and whose angular mode number

is −1, opposite to that of the target beam in sign.

Note that for an amplitude hologram, the transmittance function is real and generally

within the range of 0 to 1. Therefore, some “useless” terms in Eq. (3.5) do not generate

the desired mode. As a consequence, not all power of the transmitted light is converted to

the target beam and a significant amount is wasted on other beams. For the same reason,

the amplitude variation√

2r/ω0 in Eq. (3.3) should be modified with care such that it is

between 0 to 1. However, it has been shown that for all practical purposes, this variation

can be neglected for convenience at the price of slight mode impurity.10, 13

The fabrication is further simplified if the amplitude hologram is binary, i.e. the values

92

of T (r, φ) are either 1 or 0. For this purpose, Eq. (3.4) can be modified as follows

T (r, φ) =

0, 0 ≤ (1+cos(φ+kxr cosφ+ψ0))2

< 12;

1, 12≤ (1+cos(φ+kxr cosφ+ψ0))

2≤ 1.

(3.6)

This actually describes a square wave with ν = φ+ kxr cosφ+ ψ0 as the variable,

T (ν) =

0,(

2n+ 12

)

π < ν <(

2n+ 32

)

π;

1,(

2n− 12

)

π ≤ ν ≤(

2n+ 12

)

π,(3.7)

where n is an integer. Thus, using the Fourier series representation of a square wave, we

have

T (r, φ) =1

2+

∞∑

n=1

sinc(

nπ

2

)

cos (n (φ+ kxr cosφ+ ψ0)) , (3.8)

among which the term associated with n = 1 is the desired one.

The conversion efficiency of the binary amplitude hologram can be estimated using the

coefficient of the term. The ratio of the power of the generated LG10 beam to that of the

incident beam is approximately

(

1

2sinc

(π

2

)

)2

≈ 10%, (3.9)

where the factor 1/2 arises from the power of the LG10 mode being shared with the LG−1

0

beam that is generated by the same term.

The angle α in Fig. 3.1 determines how well the different generated beams separate

from each other in the far field. The larger it is, the less the target beam overlaps the

other beams. The minimum value of α must be larger than half the divergence angle of

the gaussian beam, thus we require α >> λ/πω0.

93

3.2.2 Making the hologram

The following Mathematica codes are used to calculate and plot the pattern of the binary

amplitude hologram.

l = 1; (*angular mode number *)

Lamda = 647*10^-9; (*wavelength*)

alpha = 0.1/180*Pi; (*inclined angle *)

kx = 2*Pi/Lamda*Sin[alpha]; (* x component of the wave vector *)

T[x_, y_] = Sign[Cos[kx*x + l*ArcTan[x, y] + Pi/2]] + 1;

(* transmittance function of the binary amplitude hologram *)

DensityPlot[T[x, y], {x, -0.0035, 0.0035}, {y, -0.0035, 0.0035},

PlotPoints -> 3000, Mesh -> False, Frame -> False] (* plot the pattern*)

Figure 3.2: Typical holographic pattern that converts a LG00 beam to a LG1

0 beam.

Figure 3.2 shows the typical pattern of a binary hologram. The pattern appears like

a grating but with a fork in the center, which is a result of the phase singularity. The

pattern is printed on an A4 paper with a laser printer and then reduced to about a 7 mm

94

Figure 3.3: The multiple orders of beams generated by the binary amplitude hologram.

by 7mm area onto a transparency film using a high resolution photocopy machine. We

then cut the transparency film to the appropriate size and attach it to an optical mount,

which is used to make fine adjustments of the hologram’s position.

We use the fundamental gaussian beam at 647 nm from a Coherent Innova 70C argon

CW laser by using the smaller output aperture of the laser. We then let the beam strike

the center of the hologram at normal incidence. In the far field we observe a pattern of

the different orders of the diffracted beams, which is shown in Figure 3.3. In the center

of the multiple beams is the fundamental gaussian beam (LG00), as can be judged by its

bright center. Nearby are the LG10 and LG−1

0 beams whose centers are dark. Also shown

are the LG±20 and the LG±3

0 beams. The LG±20 beams are weak, consistent with Eq. 3.8

in which the coefficient is zero when n is a even number. We use a screen with a window

of appropriate size as a spatial filter to let through the LG10 beam and block the other

beams.

95

3.2.3 Examining the phase singularity

The intensity profile of the generated LG10 beam is easily verified by looking at the pat-

tern on a screen or more precisely, using a beam profiler. The screw-like phase profile,

however, is not directly observable. Specifically we need to verify that the angular mode

number l of the beam is indeed equal to 1, or, equivalently, that the beam carries or-

bital angular momentum of h̄ per photon. This can be done by the following interference

experiment.10, 14

Figure 3.4: Schematic diagram of the interference experiment to exam the phase dislocationof a LG1

0 beam. M: mirror; BS: beam splitter; DP: dove prism.

The setup is basically a modified Mach-Zehnder interferometer as shown in Figure

3.4. The beam is separated into two paths by a beam splitter. One of the beams travels

through a dove prism, making the profile of the transmitted beam by reflection the mirror

image of the incident beam. As a result, the handedness of the phase screw of the beam

is changed, i.e., the LG10 beam becomes a LG−1

0 beam, and vice versa. The transmitted

96

beam is recombined with the beam from the other arm by the second beam splitter. The

two beams, now having opposite signs in their angular mode number l, interfere and give

us an pattern that can be used to characterize the value of l.

Figure 3.5: Typical self-interference pattern of a LG10 beam with a dove prism placed in

one arm. The three-prong fork in the center is evidence that the angular mode number lof the incident beam is 1 (or -1).

Figure 3.5 shows the typical interference pattern from the beam that we generated

using the binary amplitude hologram. The three-prong fork in the center indicates that

the angular mode number l of the incident beam is 1 (or -1).2

3.3 Fabricating the DR1/PMMA Samples

Two methods are used in our lab to make the DR1/PMMA samples. The solvent-polymer-

dye method is used to make thin (several micrometers) films that are suitable to mea-

sure the absorption spectrum, while the polymerization-with-dye method is used to make

thicker bulk samples (a few millimeters) and are used in our nonlinear optics experiments.

3.3.1 Solvent-polymer-dye method

The idea of this method is to make a homogeneous mixture of the polymer and the dye by

dissolving them in certain solvents which are subsequently removed through evaporation.

The ingredients are weighted by an electronic balance according to the following formula

(all the percentage are in weight):

97

1. The solution is made of 85% solvents and 15% solids.

2. The solvents include 67% propylene Glycol Methyl Ether Acetate (PGMEA) and

33% γ-Buterolactone.

3. The solids consist of the PMMA and the DR1 with the desired proportions, e.g. 1%

DR1 and 99% PMMA.

The ingredients are then put into a bottle with a magnetic stirrer in it in the following

order, stirring between each ingredient:

1. PGMEA,

2. DR1,

3. γ-Buterolactone,

4. PMMA.

We use a Corning micro slide as the substrate of the sample, clean it with methanol and

place it on a spin coater. After all the ingredients are thoroughly mixed, the solution is

filtered through a 0.2 µm filter using a syringe, deposited directly on the center of the

substrate, forming a small puddle. We then use the spin coater to cast the solution over

the surface of the substrate to form a thin film. The thickness of the film decreases with

faster spin speed or longer spin time. The substrate is then baked in a convection oven

at a temperature of 95 ◦C for one hour to evaporate the solvents. Finally, the sample is

cooled slowly to room temperature.

3.3.2 polymerization-with-dye method

The solvent-polymer-dye method works well to make thin films because the surface area is

large relative to the volume, allowing the solvents to evaporate. However, this method can

98

not be used to make bulk samples because the solvents would be trapped in the sample

and would form bubbles. Instead, we use the polymerization-with-dye method. The basic

idea is to mix the dye with the liquid monomer and then polymerize it to form the solid

polymer with embedded dye, eliminating the need for solvent.

Figure 3.6: The alumina-filled column used to remove the inhibitor from the MMA.

The monomer used to make PMMA is methyl methacrylate (MMA), and is commer-

cially available. It arrives in bottles with the inhibitor, which is a chemical added to prevent

the monomer from polymerization during transportation and storage. The inhibitor needs

to be removed to allow for polymerization. The inhibitor is removed by passing the MMA

through a column which is filled with alumina powders (see Fig. 3.6) and collecting the

resulting liquid one drop at a time. As a result, the inhibitor is trapped in the column and

99

pure MMA goes through. Because MMA can be polymerized by room light, we wrap the

container of the MMA with aluminum foil after the inhibitor is removed.

Next we add the desired amount of DR1 powder into the MMA liquid and use a

magnetic stirrer to mix them until the powder is thoroughly dissolved in the liquid. When

the concentration of the DR1 is high, greater than 1%, we find that using an ultrasonic

bath is needed to help the dye to dissolve into the solution.

Subsequently the plasticizer (dibutyl phthalate, 0.5%-1% by weight), the chain transfer

agent (butanethiol, 2.2 µl/ml solution), and the initiater (ter-butyl peroxide, 2.2 µl/ml

solution), are added into the solution sequentially. The plasticizer adds flexibility to the

polymer so that the polymer is easier to be shaped mechanically. It works by separating

the polymer chains, which increases the size of the “voids” in the polymer as a consequence.

The initiater starts the polymerization by making the monomer molecules chemically active

and links them together one by one like a chain, until a CTA molecule is met, which

terminates the growth of the chain. Thus, the amount of CTA limits the average length of

the polymer molecules. We note that all the operations must be conducted in the chemical

fume hood to avoid the strong unpleasant smells, but more importantly, to protect against

the associated respiration hazard and toxicity.

After the solution is thoroughly mixed, we filter it through a 0.2 µm filter by using

a syringe and inject it into test tubes. This removes possible undissolved solid particles,

ensuring that a homogeneous sample results. The test tubes are then sealed tightly using

their caps and placed vertically in a metal test tube rack. The rack together with the test

tubes are put into a convection oven which is set to 90 ◦C. After half an hour of heating,

we open the caps of the test tubes to allow the generated gases to escape and to reduce

the increased pressure. This step is helpful in decreasing bubbling in the polymer. It also

prevents the test tubes from exploding due to the high gas pressure inside. The caps are

then re-tightened and the samples are left in the oven for further polymerization.

After about 48 to 72 hours, when the sample should be totally polymerized, we remove

100

the test tubes from the oven and immediately put them in a box into a freezer. The sudden

temperature drop separates the polymer from the wall of the test tube so that it is easy

to remove the polymer undamaged by braking the test tube. The resulting sample, called

a preform, is a rod-shaped solid about 1.2 cm in diameter and 6-8 cm in length.

Figure 3.7: Diagram of the squeezer that is used to press thick polymer films.

To make film samples a squeezing process is used. The preform is cut into small chunks

about 2cm to 4 cm in length in the machine shop. A chunk is then sandwiched between

two pieces of glass plates with the sides of the cylinder touching the glass. The glass plates

together with the sample are fixed in a squeezer (as shown in Fig. 3.7) and are put into the

convection oven which is set at about 120 ◦C. After about half an hour, when the sample

becomes soft, we remove the squeezer from the oven and start to tighten the set screws

(There are a total of four of them, one at each corner of the square-shaped metal plate.)

to press down the sample. This must be done quickly as the sample is cooling down

and becoming stiff. Care must be taken not to apply too high a torque to the screws,

preventing the glass plates from breaking. Once the screws can not be tightened further,

we put the squeezer back into the oven for another half hour. This process is repeated

until the sample is close to the desired thickness. At this point, we insert the spacers of

101

the desired thickness in between the glass plates to control further squeezing. Once both

glass plates come in contact with the spacers, no more squeezing is necessary. The oven

temperature is then reduced to about 90 ◦C and the samples are allowed to relax at this

temperature for several hours. Finally we turn off the oven and let the sample cool down

slowly (in the oven) to room temperature, which often takes one night. The sample is then

removed from the squeezer and the glass plates. Samples with good surface quality are

obtained from this process. The thickness of the sample is decided by the spacer, usually

a few millimeters.

3.4 Recording of high efficiency holographic volume

index gratings in DR1/PMMA

Using the DR1/PMMA film made by the squeezing process, we have successfully recorded

high efficiency (up to 80% ) holographic volume index gratings using off-resonant writing

beams (such as 633 nm and 647 nm).15 In this section we introduce this experiment.

Although achieving such a high efficiency is significant, the main reason that we include

this experiment is that it allows us to determine the conditions under which the nonlinear

process in DR1/PMMA can be treated as the optical Kerr effect, as will be discussed

below.

3.4.1 Background

A grating is an optical component whose dielectric constant is periodic in space. When

only on the real part of the refractive index is periodic, the grating is an index grating. A

grating can diffract a light beam to some other direction at certain incident angles. The

theory on gratings can be found in many references.16, 17 Here we briefly summarize some

important points.

102

Assume that an index grating has a refractive index of the form

n(z) = n0 + n1 cos(Kx), (3.10)

where n0 is the normal refractive index of the medium, n1 is the amplitude of the periodic

modulation of the refractive index, andK is called the grating wave number. For simplicity,

the index variation is assumed to be sinusoidal. The grating spacing or period Λ is thus

equal to 2π/K. Figure 3.8 shows one such grating, where the thickness of the grating is

assumed to be d. When 2πλd/n0Λ2 >> 1, the grating is said to be a thick or volume

grating.17 Our discussion focuses on the volume grating.

Figure 3.8: Diagram of the diffraction of a light beam in an index grating.

The Bragg angle is defined by

θB = sin−1

(

λ

2n0Λ

)

. (3.11)

For a volume grating (refer to Fig. 3.8, where the wave vector is in the xz plane.), the

103

grating can diffract part or all of the energy of the incident beam to another direction

and form a new beam if and only if a beam is incident to the grating with the incident

angle θ1 = θB.16, 17 The angle between the diffracted beam and the normal of the incident

plane, θ2, also equals θB, but the direction of the x component of the wave vector of the

diffracted beam is opposite to that of the incident beam.

The diffraction efficiency, which is the ratio of the intensity of the diffracted beam to

that of the incident beam, can be proved to be16, 17

η = sin2

(

πn1d

λ cos θB

)

. (3.12)

Figure 3.9: Illustration of forming the index grating by two-beam coupling.

If the refractive index of the material is intensity dependent, such as it is in an optical

Kerr medium, an index grating can be formed by letting two coherent beams interference

inside the material.18 Figure 3.9 illustrates such a configuration, where the two coherent

beams intersect inside the material and form an interference pattern whose intensity is

104

sinusoidal in the x direction. Consequentially the refractive index of the material in the

intersection region is modulated by the intensity pattern and becomes periodic, forming

an index grating. It’s easy to show that the directions of the two beams automatically

satisfies the Bragg angle θB of the grating. In other words, if one of the beams is blocked,

the grating diffracts the other beam to the direction of the blocked beam. If we let one of

the beams (signal beam) to carry information, and the other (reference beam) is a plane

wave, then the information will be embedded in the formed grating and can be recovered

by letting the reference beam strike the grating at the Bragg angle. Therefore the grating

is like a hologram, and can be called a holographic grating.

A good holographic volume grating demands high diffraction efficiency, which in turn

requires relatively big refractive index modulation and grating thickness according to Eq.

(3.12). For materials with an intensity dependent refractive index, given the same inten-

sity of the light beam, bigger refractive index modulation means bigger nonlinear response.

The azo-dye doped polymer materials, including DR1/PMMA, are well known for hav-

ing big nonlinear refractive index change due to the trans-cis-trans photoisomerization

with subsequent molecular reorientation.19–22 Using these materials to record holographic

gratings have been studied extensively.23–25 The previous research had used the wave-

lengths at the strong resonant absorption band of the azo-dye in order to maximize the

photo-isomerization efficiency. However, because of azo-dye’s very high optical absorption

at these wavelengths, the thickness of the gratings are limited to several microns, which

makes a volume grating difficult.

Our group proposed the use of off-resonant writing beams (such as 633 nm and 647

nm) to generate volume gratings in DR1/PMMA.15 When absorption is weak, the light

can travel much deeper into the sample before being attenuated, so thick films (on the

order of mm) can be used to record the volume gratings. On the other hand, we find

that light at these wavelengths can still introduce a large change of refractive index. We

have observed diffraction efficiencies in excess of 80% in a 2 mm thick DR1/PMMA bulk

105

material by using 647 nm writing beams.15

3.4.2 Experimental setup

We build an in-situ diffraction efficiency measurement system to monitor the diffraction

efficiency of a grating while it is forming. As the grating results directly from the nonlinear

refractive index, the system also allows us to observe the dynamics of the underlying

nonlinear mechanisms, which in DR1/PMMA is mainly photoreorientation of the DR1

molecules. As such, the observed dynamics let us determine the time scales within which

we can treat the DR1/PMMA as a Kerr medium.

Figure 3.10: Setup of the holographic volume index grating recording in DR1/PMMA andthe in-situ diffraction efficiency measurement system.

The setup is illustrated in Figure 3.10. A laser beam of wavelength of 647 nm from

an Innova 90K Krypton/Argon mixed gas laser is expanded and collimated by a beam

expander so that the beam closely resembles a plane wave. A polarizer is used to ensure

106

that the beam is linearly polarized in the vertical direction. The beam is then split by a

beam splitter into two beams, which are subsequently directed by two mirrors, and intersect

inside the sample. Careful adjustments are made to ensure maximum overlap between the

two beams inside the sample. The cross sections of the beams are approximately 5 mm in

diameter. The crossing angle between the two beams is 35◦. The lengths of the two beam

paths are arranged to be the same such that they maintain coherence when they arrive

in the sample. A half wave plate and a polarizer combination are inserted in the path of

the beam that has higher power to make fine adjustments to its power. As a result, the

power and the intensity of the two beams are made the same when they interfere, which

maximizes the contrast of the interference pattern.

Two silicon photo-diode detectors are put behind the sample to monitor the power of

each beam while the grating is forming. The two lenses in front of the detectors ensure all

power of the beams is being collected. A shutter, controlled by a computer, is placed in

the path of one of the beams. During grating writing, the shutter is closed for 0.5 s every

40 s so that the detectors can measure the grating diffraction efficiency in this 0.5 s time

interval when the beam is off. The diffraction efficiency is calculated as the ratio of the

power of the diffracted beam to the sum of the power of the transmitted and the diffracted

beam. All the optical components are placed on an air isolated vibration damping table.

Data acquisition and shutter control are achieved by Labview programing using a

PC with a National Instrument Lab-PC-1200/AI multi-functional I/O board (DAQ). We

configure the analog input of the DAQ in differential connection mode for non-referenced

signals. Two such channels are used to acquire the signals from the two detectors, which

have been amplified by a two-channel low noise amplifier. We set the digital I/O port A of

the DAQ in output mode, and use PA0 to send out the TTL signal to trigger the shutter.

The Labview program controls the cooperation of the hardware and also calculates the

diffraction efficiency of the grating, which is shown instantly on the computer screen and

also stored on a hard driver for later analysis.

107

3.5 Experiments with the LG10 beam

To demonstrate the applications that use high order LG beams, as well as to test the

validity of our theory, we carry out the Z-scan, I-scan, and optical limiting experiments.

The theory of these experiments have been introduced in Chapter 2. Their experimental

setups are similar to each other, as illustrated in Figure 3.11.

We use a Coherent Innova 70C argon CW laser as the light source and tune it to emit

light at a wavelength of 647 nm. By adjusting the size of the output aperture of the laser,

we are able to get the fundamental gaussian mode out of the laser cavity. The size “3”

aperture is used since it gives a good gaussian beam profile and reasonable beam power.

We use the “light” mode of the laser such that the laser keeps a constant output power by

automatically adjusting the cavity current. A half wave plate (WP) and a polarizer (P1)

combination is used to fine tune the beam power. The polarization of the beam is set to

be perpendicular to the optical table by the polarizer.

A computer-generated hologram (CGH) as described in Sec. 3.2 is used to convert the

LG00 beam to the LG1

0 beam. The CGH is mounted on a 2D translational stage which

makes the position of the CGH adjustable in the plane (x,y directions) perpendicular to

the optical axis. We carefully adjust the position of the CGH such that the incident beam

strikes the center of the fork pattern. This is done by monitoring the beam profile of the

newly generated LG10 beam at the far field. The position of the phase singularity, which

appears to be a dark hole in the intensity beam profile, should be in the center of the

beam if the alignment is good. The CGH also generates LG beams of other orders. We

use a aperture (AP1) to block the undesired beams, only letting through the LG10 beam.

As the different orders of beams are only separable in the far field, the distance between

the CGH and the aperture is long. We use several mirrors (not shown) to guide the beams

back and force on the optical table a few times to increase the propagation distance.

The LG10 beam is redirected by two mirrors (M1 and M2). A shutter is used to turn

108

Figure 3.11: Schematic diagram of the setup for the experiments using a LG10 beam ( Z-

scan measurement, I-scan measurement and optical limiting). WP: half wave plate, P1,P2:polarizers, CGH: computer generated hologram, AP1,AP2: apertures, M1,M2: mirrors,L1-L4: lenses, PH: pin hole, BS: beam splitter, D1,D2: detectors.

the beam on and off. Another polarizer (P2) is used to further purify the polarization

of the beam. The beam then goes through a spatial filter formed by a Fourier transform

lens (L1) and a pin hole (PH) to clean up the high frequency noise. Another lens (L2)

re-collimates the beam. A LG10 beam with the desired Rayleigh length can be made by

properly positioning L2. A long Rayleigh length (compared to the thickness of the sample

) helps to obey the thin sample assumption.

The sample is mounted on a 3D translational stage. The movement in the xy plane

is desired because we want the beam to strike a fresh sample spot each time to avoid the

hysteresis effect.15 Movement along the optical axis (z axis) is necessary for the Z-scan

measurement, but not for the I-scan measurement and the optical limiting experiment.

A beam splitter (BS) is placed after the sample so that the “open aperture” and

the “closed aperture” transmittance can be measured simultaneously.26 In the “closed

109

aperture” path, a small aperture (AP2) is placed on the axis of the beam in the far field,

only allowing a small portion of the near-axis light to get through. A lens (L4) collects

the transmitted light for a photo detector (D1), which measures the power of the collected

beam. In the “open aperture” path, however, all the power of the beam is recorded by a

photo detector (D2) with the help of a lens (L3). The “open aperture” measurement let

us determine if the nonlinear absorption plays a role in the Z-scan measurement.

The output of the two detectors are sent to a digital oscilloscope, which communicates

with a computer through IEEE-488. The measurements are synchronized by a shutter

controller, which opens and closes the shutter, and triggers the measurement by the oscil-

loscope. As such, the oscilloscope records the power of the beams as a function of time. If

we call the instant when the shutter is opened t = 0, then the normalized transmittance

at time t is obtained by dividing the power recorded at time t by that at time t = 0.

Finally we note how we measure the radius and the position of the beam waist. An

Ophir Optronics BeamStar CCD beam profiler allows us to measure the beam size quickly.

However, the beam waist is too small to be measured directly by the beam profiler because

of its limited resolution. Instead, we determine the beam size at several positions far away

from the beam waist where the beam size is big enough to be measured. A data fitting

routine uses the relationship ω2(z) = ω20{1 + [(z − zw)/zr]

2}, where zr = πω20/λ is the

Rayleigh length, then extrapolates to the waist radius ω0 and the waist position zw. With

these two parameters we are able to determine the normalized position Z along the beam.

110

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112

[17] P. Yeh, Introduction to photorefractive nonlinear optics (Wiley, New York, 1993).

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114

Chapter 4

Results and discussion

In this chapter, we first study the properties of the DR1/PMMA samples made in our

laboratory as we describe in Sec. 4.1, mainly by using the results obtained in the holo-

graphic volume index gratings recording experiment as discussed in Sec. 4.1.2. We focus

on the conditions under which the DR1/PMMA samples can be treated as an optical

Kerr medium. Next we present the results of measuring the nonlinear refractive index of

DR1/PMMA by using a LG10 beam, including the Z-scan measurement (Sec. 4.2) and the

I-scan measurement (Sec. 4.3). Finally in Sec. 4.4 we demonstrate optical limiting in

DR1/PMMA using a LG10 beam and make discussions.

4.1 Properties of DR1/PMMA

In this section, we study the properties of DR1/PMMA, mainly by studying the recording

process for making high efficiency holographic volume index gratings. Focus is placed on

the estimation of the order of magnitude of several important parameters, which let us

determine the experimental conditions under which our samples can be used as optical

Kerr media.

115

4.1.1 Absorption spectrum of DR1/PMMA

Fig. 4.1. shows the absorption spectrum of DR1/PMMA. The peak of the absorption

400 500 600 700 800 900

0.0

0.5

1.0

1.5

2.0

647 nm

Abs

orba

nce

(OD

)

Wavelength(nm)

Figure 4.1: Absorption spectrum of DR1/PMMA. The arrow shows the wavelength whichis used in our experiments. OD, optical density.

band is at a wavelength of about 490 nm, where photoisomerization efficiency is the

largest. Therefore many previous researchers have used light with wavelengths around

the absorption peak in their experiments.1, 2 However, at these wavelengths, the propa-

gation of light in DR1/PMMA is limited to several micrometers due to the strong optical

absorption, making some experiments which require thicker samples, such as recording of

volume index gratings, impossible.

In contrast, light of wavelengths off the main absorption peak can travel much deeper

into the sample before being absorbed. Although the photoisomerization efficiency intro-

duced by such off-resonance beams is weaker than by resonance beams, the effect in thick

samples may be appreciable. For example, it was found that a refractive index change of

116

8 × 10−5 can be introduced in DR1/PMMA with a weak pumping beam at a wavelength

of 633 nm.3 This suggests that with off-resonant beams one can use thick films to record

volume gratings. In our experiments, we use 647-nm wavelength beams, as indicated by

the arrow in Fig. 4.1.

4.1.2 Recording of high efficiency holographic volume index grat-

ings in DR1/PMMA

Using the in-situ diffraction efficiency measurement system, which has been introduced

in section 3.4.2, we are able to monitor the diffraction efficiency of the grating while it is

being recorded in a DR1/PMMA sample by two-beam interference. Fig. 4.2 shows a typical

0 1000 2000 3000 4000

0.0

0.2

0.4

0.6

0.8

1.0

Diffraction Efficiency vs. Time

Diff

ract

ion

Effic

ienc

y

Time(s)

Figure 4.2: Diffraction efficiency as a function of time.

curve of the diffraction efficiency as a function of time, where the sample is a 2.2mm-thick

2% w/w DR1/PMMA thick film and the power of each ∼ 5-mm diameter incident beam

117

before the sample is 69 mW. The absorption coefficient, α, is ∼ 5.6 cm−1 at a wavelength

of 647 nm. The diffraction efficiency is observed to increase approximately exponentially

as a function of time, and approaches an upper limit of about 80%. Although it is not

the focus of this dissertation, it is worth mentioning that such high efficiency holographic

volume index gratings are signigicant technologically and have important applications in

information recording.4

The diffraction efficiency is related to the degree of refractive index modulation in

DR1/PMMA. For simplicity, we treat the modulation as uniform such that we can use

Eq. 3.12. Such a simplification is, of course, not rigorous because the intensity of the

beams changes along the transverse beam profile. However, our purpose here is to have

an estimation of the dynamics of the underlying mechanism, and we are only interested in

the order of magnitude of the corresponding physical quantities. Solving Eq. 3.12 we find

the amplitude of the refractive index modulation to be,

n1 =arcsin

(√η)

λ cos θB

πd. (4.1)

In calculating the Bragg angle, we note that θB is defined in the medium. Therefore

the angle that we obtained in our experiment, which is measured in air, must be converted

using Snell’s law. Because of the low concentration of DR1, the linear refractive index of

DR1/PMMA can be approximated by that of the PMMA, which is about 1.5. The Bragg

angle of the grating is calculated to be about 11.6 ◦.

With Eq. 4.1 and the experimental parameters, we are able to convert efficiency - time

(η vs. t) curves to refractive index modulation - time (n1 vs. t) curves, which reveal the

dynamics of n1.

118

0 1000 2000 3000 40000

2

4

6

8

10

12

0 1000 2000 3000 40000

2

4

6

8

10

12

Data Best-fit with a biexponential function

Time (s)

Data Best-fit with a single exponential function

n 1 (x10

-5)

Figure 4.3: n1 as a function of time in the grating recording experiment. Upper: the dataand the best-fit with a single exponential onset function, Lower: the data and the best-fitwith a biexponential onset function.

Transient behavior of n1

Fig. 4.3 shows n1 as a function of time as determined from Eq. (4.1) and the data in Fig.

4.2. Also shown are the best fit curves with a single exponential onset function

n1 = A0 + A1

(

1 − exp

(

− t

τ

))

(4.2)

and a biexponential onset function

n1 = A0 + A1

(

1 − exp

(

− t

τ1

))

+ A2

(

1 − exp

(

− t

τ2

))

. (4.3)

The time constant of the single exponential decay function is found to be 773 seconds,

while the two time constants of the biexponential decay function are 91 and 1035 seconds.

119

It is obvious that the biexponential decay function fits the data better than the single

exponential decay function.

Our theory on the underlying mechanisms of the nonlinear refractive index in DR1 /

PMMA (see Section 2.3.2) indicates that the dynamics of the nonlinear refractive index

change is characterized by two time constants, one associated with photoisomerization

and the other associated with the photoreorientation. This seems to explain why the

biexponential decay function fits the data better than the single exponential decay function.

However, the life time of the cis isomers is about 4-5 seconds,5, 6 which limits the time

constant for photoisomerization to be within a few seconds. Such fast dynamics can not be

resolved by our measurement system because the data sampling rate is one per 40 seconds.

Therefore neither of the two time constants obtained by the biexponential decay function

fit is associated with photoisomerization. The scales of the two time constants suggest

that they correspond to photoreorientation, which is a much slower process.

However, why are there two time constants for one process? We believe this is because

the mobility of DR1 molecules varies over a wide range of values due to the complex

polymer backbone environment in which they are embedded. The geometric shape of the

polymer molecules is like long tangled chains. Each of them consists of tens to hundreds of

monomers. In a bulk polymer these chains of differing lengths entangle together randomly

and form the polymer backbone, leaving many empty spaces or “voids” between them.

The shapes of these “voids” are highly irregular and their sizes vary from site to site. The

mobility of the DR1 molecules, which live in these “voids”, is highly affected by the shape

and size of the voids. In general, there could be three broad classes of DR1 molecules:

(1) those that can rotate freely, (2) those that are totally restricted from rotating, and

(3) those that can rotate but must first overcome an energy barrier due to the void. As

such, the physical parameters of the DR1 molecules, such as the entropic decay rate due

to the anisotropy of the trans isomer orientation, β, can not be described by a single value,

but rather by a distribution. This explains why the biexponential decay function fits the

120

data in Fig. 4.3 better than the single exponential decay function. In fact, three or more

exponential decay function might fit the data even better, but a fit with too many free

parameters is meaningless.

Our theory in Section 2.3.2 is an idealized model which assumes single β value. To

develop a more realistic theory, which takes into account the inhomogeneity of β, is very

complex and beyond the scope of this dissertation. The main obstacle for developing a

theory is that there is no simple universal formula on the distribution of β. The distribution

is strongly dependent on the structure of the “voids”, which is sensitive to the conditions of

polymerization and sample processing, such as baking time, temperature, squeezing, and

even sample storage time. Moreover, the structure of the “voids” can be changed during

light pumping, because the DR1 molecules can apply forces to the “voids”. Sometimes

such a change is abrupt.7 Such inhomogeneity and time variation of the polymer properties

have also been observed by other authors.8

Nevertheless, fitting the data with the exponential onset function can give us an es-

timate of the order of magnitude of the time constant of photoreorientation. We do the

experiment as a function of writing beam intensity. Table 4.1 summarizes the time con-

stants as determined by fitting to Eq. (4.3), where the intensity is the amplitude of the

intensity modulation at the front surface of the sample, calculated from the intensity of

each writing beam. A quantitative analysis of how the time constants change with inten-

Table 4.1: Time constants determined from a biexponential onset function fit to grating data.

Intensity(w/cm2) τ1 (s) τ2 (s)

0.07 674 ± 94 5298 ± 7630.14 231 ± 15 3139 ± 610.21 279 ± 10 3209 ± 770.34 306 ± 8 4176 ± 530.69 91 ± 3 1035 ± 11

sity is not possible due to the large fluctuations of the data, though it is clear that higher

121

intensity results in shorter time constants. This is not surprising considering the inhomo-

geneity and time variation of the polymer properties we discussed above. However, the

data lets us draw one important conclusion here: the time constants of photoreorientation

in our DR1/PMMA samples are on the order of magnitude of 102 to 103 seconds under

the listed experimental intensity.

Saturation values of n1

Another feature that can be seen from Fig. 4.2 is that the diffraction efficiency saturates

at a sufficiently long time. The saturation value of η changes with the intensity of the

writing beams. We calculate the saturation value of n1 from the corresponding saturation

value of η and plot it against the amplitude of intensity modulation at the front surface

of the sample, as shown in Fig. 4.4.

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7

5

6

7

8

9

10

11

12

Data Best fit

Satu

ratio

n Va

lue

of n

1 x (1

0-5)

Amplitude of the intensity modulation (W/cm2)

Figure 4.4: Saturation values n1 as a function of the amplitude of intensity modulation atthe front surface of the sample.

122

In contrast to the time constants, the saturation values of n1 fit to our theory in

Section 2.3.2 within experimental uncertainty. Recall Eq. 2.45, which gives the change of

the refractive index of the sample as “seen” by the incident beam,

∆n(t) ≈ −ηtpξtcI

2 (4β + ξtcI)

(

1 − exp

(

−(

2β +ξtcI

2

)

t

))

, (4.4)

where ηtp is a constant coefficient, I is the intensity of the light beam, ξtc is the probability

rate per unit intensity of light in the material that a trans isomer will be converted into a cis

isomer, β is the entropic decay rate of the anisotropy due to the trans isomer orientation.

Since n1 ∝ ∆n, it’s easy to show that when t→ ∞,

n1 ∝ξtcI

4β + ξtcI. (4.5)

We fit the data in Fig. 4.4 with a function n1 = AI/(1+rI), where A and r are the fitting

parameters. A comparison with Eq. 4.5 reveals that r = ξtc/4β. The fit shown in Fig. 4.4

by the solid line, yields r = 11 ± 1 cm2/W.

It is interesting that the same theory based on a single value of β can fit the saturation

value of n1 well, but not the time constants. We propose an explanation as follows.

Earlier we mentioned that in general there are three broad classes of DR1 molecules: (1)

those that can rotate freely, (2) those that are totally restricted from rotation, and (3)

those that can rotate but must first overcome an energy barrier due to the “void”. The

inhomogeneity of the time constants is due to different β values of the class (1) and the

class (3) molecules. During molecular reorientation, while the class (1) molecules have little

effect on the “voids” that they live in, the class (3) molecules must apply a force to the

“voids” in order to complete reorientation. Under the influence of this force, the “voids”

tend to change their shape to facilitate the rotation of the DR1 molecules. Each molecule

may experience many reorientation cycles. So the class (3) molecules keep reshaping the

123

“voids” until the “voids” allow free rotation of the enclosed molecules, making them class

(1) molecules. Eventually, when all the class (3) molecules become class (1) molecules,

the polymer structure no longer changes, and an equilibrium is reached, appearing to be

the saturation of n1. As such, when n1 is saturated, there are only two classes of DR1

molecules in the sample, class (1) and class (2). And because only the class (1) molecules,

whose β are the same, take part in reorientation, the reorientation can be modeled by a

single value of β . This explains why the theory based on a single value of β does not work

on the transient behavior of n1, but works well on the saturation values of n1.

From the value of r we can estimate the order of magnitude of ξtcI/2 and 2β, two

quantities that contribute to the time constant of photoreorientation (refer to the exponent

in Eq. (4.4)). We take an intensity of I ≈ 100 mW/cm2 as an example. Since r = ξtc/4β ≈

11 ± 1 cm2/W, it’s easy to see ξtcI/4β ∼ 1, or equivalently, ξtcI/2 ∼ 2β. But the time

constant of photoreorientation, which is equal to 1/(2β + ξtcI/2), is on the order of 102 to

103 s, therefore ξtcI/2 and 2β are on the order of 10−2 to 10−3 s−1 at this intensity.

4.1.3 Conditions for DR1/PMMA as optical Kerr media

We finish this section by examining the experimental conditions under which our DR1/PMMA

samples can be treated as an optical Kerr medium.

At the end of Sec. 2.3.2, we proposed two cases under which a DR1/PMMA sample

can be treated as an optical Kerr medium:

1. If the intensity of the incident beam is weak such that ξtcI << β, the sample can be

approximated by an optical Kerr medium at any time.

2. If the intensity of the incident beam is not strong such that ξtcI << γ (but possibly

ξtcI > β.), the sample can be approximated as an optical Kerr medium for the time

range of t << 1/ (2β + ξtcI/2).

124

The second case can offer bigger nonlinear refractive index coefficient, n2, than the first

case, therefore the second case is experimentally more useful.

According to the results in this section, ξtcI/2 and 2β are both on the order of 10−2 to

10−3 s−1 if intensity I ∼ 100 mW/cm2. Therefore the first case can be satisfied if I << 100

mW/cm2. The second case involves γ, which is ∼10−1 s−1 because the life time of a cis

isomer is about 4-5 seconds.5, 6 So if the intensity is kept under about 102 mW/cm2, which

makes ξtcI about 10−2 to 10−3 s−1 or less, ξtcI << γ is satisfied. The other requirement,

that t << 1/ (2β + ξtcI/2), is met if we let t be within a few seconds, because the time

constant of photoreorientation, 1/ (2β + ξtcI/2), is on the order of 102 to 103 s. Thus the

two cases that our DR1/PMMA samples can be treated as optical Kerr media are:

1. If I << 100 mW/cm2, the material can be approximated as an optical Kerr medium

at any time.

2. If the intensity I ∼ 102 mW/cm2 or less, the sample can be approximated as an

optical Kerr medium for the time range of t << 102 s.

The above experimental conditions are guidelines on our experiments with LG beams,

which require an optical Kerr medium for the data to be properly interpreted.

4.2 Z-scan measurement using a LG10 beam

In this section, we present the results of the Z-scan measurement using a LG10 beam in

DR1/PMMA. The measured Z-scan curves also test the validity of our theory.

We use small-power incident beams such that the maximum intensity along the beam,

which is at the beam waist, is low. The time duration that the beam illuminates the

sample is set to be 3 seconds for each run. By doing so, we expect the sample to respond

as an optical Kerr medium.

125

-5 0 5 10 15 20 25 30 35 40 45 50 550.75

0.80

0.85

0.90

0.95

1.00

1.05

1.10

1.15

1.20

1.25

LG01 Z-scan

Experiment LG0

0 Z-scan

Nor

mal

ized

Tra

nsm

ittan

ce T

z (cm)

Figure 4.5: Experimental (squares) and theoretical (solid curve) results of the Z-scan of aDR1/PMMA sample using a LG1

0 beam. Also shown is the theory for a LG00 Z-scan trace

(dashed curve).

Figure 4.5 shows a typical Z-scan result using a LG10 beam. The power of the incident

beam is 65 µW , and the radius of the beam waist is 72 µm. The sample is a 1.4mm-

thick 2% DR1/PMMA sample. The squares are the experimental data. Each data point

represents the normalized transmittance at t = 3 s at a fixed z position, which is obtained

by dividing the transmittance (of the beam through the aperture) at t = 3 s by that at t =

0 s. The experiment is repeated multiple times with the beam striking a different position

of the sample in each run to avoid hysteresis effects. The mean value are plotted in the

figure and the standard deviation are used as the error bars. The solid curve is the best

fit using Eq. (2.78) with ∆Φ0 and zr (the Rayleigh length) as the free parameters. The

dashed curve is the best fit to a LG00 Z-scan trace (given by Eq. (2.74)) for comparison.

126

The shape of the Z-scan curve using the LG10 beam has two peaks and two valleys, in

contrast to the one using the LG00 beam, which has only one peak and one valley. The

extra peak and valley (indicated in the figure by the arrows), although small in magnitude,

bring the Z-scan curve cross T = 1, suggesting qualitative changes in the behavior of the

far field, i.e. from self-focusing to self-defocusing, and vice versa. Our theory clearly shows

these features and is confirmed by the experimental data. On the other side, the LG00 Z-

scan trace, whose tails never cross T = 1, can not fit the data. Thus the validity of our

theory is supported. The extra peak and valley have significance in other applications such

as optical limiting, which will be demonstrated in section 4.4.

Just as the traditional LG00 Z-scan, the LG1

0 Z-scan provides sensitive measurements

of optical nonlinearities. Recall that

∆Φ0 =π

λcǫ0n0n2|E0|2

1 − e−αd

α, (4.6)

where E0 is the amplitude of the electric field of a LG10 beam in the form E(r, φ, z) =

E0 · LG10(r, φ, z; zr). The power of such a beam, P, can be obtained through a surface

integral of the beam intensity over any cross section of the beam. In Appendix D.2, we

show

P =1

4πω2

0cǫ0n0|E0|2, (4.7)

where ω0 is the radius of beam waist. This, together with Eq.(4.6) gives

n2 =αλω2

0

4P (1 − e−αd)∆Φ0. (4.8)

Using the data in Figure 4.5 as an example, the value of ∆Φ0 is found to be -1.13 ± 0.05.

The absorption coefficient, α, is ∼5.6 cm−1 for the 2% w/w DR1/PMMA sample. With

these and other values that we mentioned earlier, we calculate n2 = (−1.5 ± 0.2) × 10−4

127

cm2/W. Note that besides using the fitting method, we can also estimate ∆Φ0 using the

difference between the major peak and valley, ∆Tp−v = 0.290|∆Φ0|, which might be less

accurate, but quicker.

We carry out the measurement several times with different incident beam powers. Each

measurement generates a Z-scan curve, from which a ∆Φ0 value corresponding to that

beam power is obtained by nonlinear curve fit. Instead of calculating n2 value individually

for each power, we plot ∆Φ0 as a function of beam power. According to Eq. (4.8), ∆Φ0

should be proportional to beam power,

∆Φ0 = sP, (4.9)

where, s is the coefficient,

s =4(

1 − e−αd)

αλω20

n2. (4.10)

It can be seen immediately that

n2 =αλω2

0

4 (1 − e−αd)s. (4.11)

Figure 4.6 shows the plot of ∆Φ0 vs. P, where the circles are the experimental data, and

the line is a linear best fit. The slope of the best-fit line, s, is −0.016 µW−1. Substituting

s into Eq. 4.11, we calculate n2 to be (−1.4± 0.1)× 10−4 cm2/W. This is consistent with

the result measured using the traditional LG00 beam Z-scan by Park9 as follows. Using the

data in Fig. 4.20 in Park’s dissertation,9 n2 is found to be −1.0 × 10−4 cm2/W (top right

figure) and −1.4 × 10−4 cm2/W (top left figure).

The fact that the linear fit fits the data (circles) well confirms that the sample acts

as an optical Kerr medium. To illustrate how the theory fails at higher intensities, we

make a measurement with beam power higher than the range where we expect the sample

128

-50 0 50 100 150 200 250 300 350

-6.0

-5.5

-5.0

-4.5

-4.0

-3.5

-3.0

-2.5

-2.0

-1.5

-1.0

-0.5

0.0

0.5

0

Power ( w)

, Data Linear fit

Figure 4.6: ∆Φ0 vs. power of the incident beam. The circles are the experimental data.The line is a linear fit of the data. Also shown is a data point (the square) obtained for abeam power higher than the range within which the sample responds like an optical Kerrmedium.

response to behave as an optical Kerr medium. The data is shown in Fig. 4.6 by the

square. The data point (square) is off the best-fit line and has a smaller absolute value,

which is as expected because it reflects the saturation of the change of the refractive index.

The maximum intensity of the beam at this power (300 µw) is ∼1.4 W/cm2, above the

intensity (∼ 0.1 W/cm2) under which the samples act as optical Kerr media.

The other free parameter in the Z-scan data fit is the Rayleigh length zr, from which

we can calculate the radius of the beam waist, ω0 = (zrλ/π)1/2. Table 4.2 lists the results,

where the left column is the power of the incident beam and the right column is the

calculated value of ω0. The results are consistent with the value of ω0 measured by a beam

129

Table 4.2: Radius of beam waist, ω0, obtained by Z-scan curve fit.

Power (µw) ω0(µm)

15 7633 7265 78130 73

Average 75 ± 3

profiler, which gives ω0 = 72µm. The systematically slightly higher values may be due to

photo thermal heating effects,9 and mode impurity of the beam.

To conclude this section, we have shown that the results of the Z-scan measurement in

DR1/PMMA support the validity of our theory of wave propagation of a LG10 beam after

it passes through a thin Kerr sample.

4.3 I-scan measurement

We demonstrate how to measure n2 by I-scan experiment.

Fig. 4.7 shows a typical I-scan curve, where the normalized transmittance (T ) is plotted

against the maximum beam intensity (Imax) at the front surface of a DR1/PMMA sample,

which is placed at Z = −1.6. For the purpose of comparison, we use the same sample

that we use in the Z-scan measurement and measure T at t = 3 s. Imax is obtained by

measuring the power of the incident beam and then using Eq. (D.27). The circles in the

figure are the experimental data, and the line is a best fit. The slope of the best fit line is

s1 = 1.38 ± 0.08 cm2/W.

The relationship between the maximum nonlinear phase distortion in the sample,

∆Φmax, and the normalized transmittance, T , at this Z value is calculated by our nu-

merical calculation routine, as shown in Fig. 4.8 by the dots. Also shown in Fig. 4.8 is a

130

0.00 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08

0.98

1.00

1.02

1.04

1.06

1.08

1.10

1.12

1.14

Nor

mal

ized

Tra

nsm

ittan

ce T

Maximum Intensity Imax (w/cm2)

Data Linear fit

Figure 4.7: Normalized transmittance T as a function of the maximum beam intensityat the front surface of a DR1/PMMA sample placed at Z = −1.6. The circles are theexperimental data. The line is a linear fit of the data.

best fit line with slope s2 = −0.93. Therefore we have

∆Φmax = s2(T − 1) = s2 · s1 · Imax. (4.12)

On the other hand,

∆Φmax =2π

λn2

1 − e−αd

αImax. (4.13)

131

1.00 1.02 1.04 1.06 1.08 1.10 1.12

-0.10

-0.08

-0.06

-0.04

-0.02

0.00

Normalized Transmittance T

Max

imum

non

linea

r pha

se d

isto

rtion

Numerical calculation Linear fit: max= -0.93*(T-1)

Z= -1.6

Figure 4.8: The maximum nonlinear phase distortion ∆Φmax as a function of the normal-ized transmittance T . The incident beam is a LG1

0 beam and the position of the sampleis Z=-1.6. The dots are the calculated results and the line is the linear fit.

Comparing Eq. (4.12) and Eq. (4.13), we conclude

n2 =αλs1s2

2π (1 − e−αd). (4.14)

Substituting the experimental parameters into Eq. (4.14), we get n2 = (−1.4±0.1)×10−4

cm2/W, consistent with the Z-scan measurement.

Compared to Z-scan measurements, I-scan has an extra source of uncertainty associated

with determining the sample position Z. This uncertainty can be eliminated by calibrating

the system using a standard sample with known n2 at the same position Z. The advantage

of the I-scan measurement over the Z-scan measurement is that I-scan itself does not

require moving the sample, which eliminates the error due to sample re-alignment and

132

sample inhomogeneity. This advantage is, however, not reflected in our measurement with

DR1/PMMA sample, as can be seen from the error bars in Fig. 4.7, because we have to

move the sample to fresh spots in order to avoid the hysteresis effect of DR1/PMMA. With

samples that have no hysteresis effects, we expect to obtain data with less uncertainty,

and to complete the measurement faster. Nevertheless, we have demonstrated that I-scan

measurement can give the correct n2 values.

4.4 Optical limiting

In this section we demonstrate optical limiting in DR1/PMMA using a LG10 beam. When

interpreting the results, however, the readers should be aware of the following important

facts.

1. In order to show optical limiting, we must change the intensity of the incident beam

over a large range. However, the maximum available power of the LG10 beam gen-

erated by the fork hologram is limited to about 500 µW, not enough to show the

limiting threshold for exposure times less than 3 seconds. To overcome this power

limitation, we increase exposure times to 300 seconds so that the accumulated change

of refractive index of the sample is big enough to detect.

2. As a consequence of the longer exposure time, the DR1/PMMA samples do not

respond as optical Kerr media, but tend to saturate. As such, we do not expect that

our theory, which is based on optical Kerr media, can fit the experimental results

quantitatively. Instead, the theory should be used as a qualitative approximation to

predict the trends of the optical limiting curve and the regions of the sample position

where optical limiting happens.

Figure 4.9 shows the typical optical limiting result using a LG10 beam and a DR1/PMMA

sample, where the sample is placed at Z = 0.6. The circles are the experimental data,

133

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

0.0

0.2

0.4

0.6

0.8

1.0

1.2

| max|

Nor

mal

ized

tran

smitt

ance

T

Maximum Intensity Imax (W/cm2)

Data

0 1 2 3 4 5 6

0.0

0.2

0.4

0.6

0.8

1.0

1.2

Response if using optical Kerr media

Figure 4.9: Optical limiting using a LG10 beam and a DR1/PMMA sample placed at

Z = 0.6. The circles are the experimental data showing the normalized transmittance(T) as a function of the maximum beam intensity (Imax, bottom axis) at the front surfaceof the sample. The curve shows the theory for T vs. the magnitude of the maximumnonlinear phase distortion (|∆Φmax|, top axis) on the sample assuming the sample is anoptical Kerr medium.

showing the normalized transmittance (T) as a function of the maximum beam intensity

(Imax, bottom axis) at the front surface of the sample. The curve is calculated according

to Eq. (C.22) and shows T vs. the magnitude of the maximum nonlinear phase distortion

(|∆Φmax|, top axis) of the sample assuming the sample is an optical Kerr medium. The

scale of |∆Φmax| is adjusted such that at small intensity the curve overlaps the experimental

data (the dots).

The theory qualitatively predicts the shape of the optical limiting curve at this Z

position. At low beam intensity, the experimental data follows the theoretical curve.

134

At higher intensity, however, the saturation effect takes over, so the value of T is less

affected by increasing the beam intensity. If viewing the intensity axis as a time axis, the

experimental data lags more behind the calculated curve with longer time.

The consequence of this saturation is two fold. On the down side, the non-Kerr satu-

ration makes the limiting speed slower compared to non-saturation, i.e., to achieved the

same small transmittance T, the medium that saturates requires a higher intensity than

an optical Kerr medium. On the bright side, saturation can postpone, or even eliminate,

the reversal of T after T reaches the minimum value, which is evident in Fig. 4.9 near

T = 0.45 W/cm2. The latter is important because the reversal of T could result in total

failure of an optical limiter.

0 50 100 150 200 250

0

40

80

120

160

200

Input Power ( w)

Out

put P

ower

(arb

. uni

t)

Data Response without optical limiting

Figure 4.10: Power transfer curve of the optical limiting system with sample positionZ = 0.6. The dots are the experimental data, and the line shows the response of thesystem with no optical limiting.

135

The performance of the optical limiting system can be better illustrated by the power

transfer curve, which shows the transmitted power as a function of the input power. Fig.

4.10 shows such a curve based on the same experimental data set in Fig. 4.9. Also shown

for comparison is a line that represents the response of the system if there were no optical

limiting. The line is obtained by fitting the experimental data at low intensity. The

effect of the optical limiter is obvious: the output power is limited to a level below ∼ 40

independent of input power, compared to ∼ 160 or higher without the limiter.

0.0 0.2 0.4 0.6 0.8

0.0

0.2

0.4

0.6

0.8

1.0

0 2 4 6

0.0

0.2

0.4

0.6

0.8

1.0

Nor

mal

ized

tran

smitt

ance

T

| max|Imax (w/cm2)

Z=0.6 Z= -7

Z=0.6 Z= -7

Figure 4.11: Comparison of the optical limiting effect with the sample in front of the beamfocus (negtive Z) and behind the beam focus (positive Z). Left: experimental results ofthe normalized transmittance (T) as a function of the maximum beam intensity (Imax)in DR1/PMMA, where the squares are data at Z= 0.6, and the circles are data at Z=−7. Right: calculated results of the normalized transmittance (T) as a function of themagnitude of the maximum nonlinear phase distortion in the sample (|∆Φmax|), assumingan optical Kerr medium. The solid line is for Z= 0.6, and the broken line is for Z= −7.

136

The position corresponding to Z = 0.6 is not the only position for which a sample can

cause optical limiting. Our theory, based on an optical Kerr medium, also predicts optical

limiting if the sample is placed in front of the beam focus. Fig. 4.11 shows the optical

limiting effect with the sample at Z = -7 as an example. Also shown is the result with

the sample at Z = 0.6 for comparison. The left figure shows the experimental results of

the normalized transmittance (T) as a function of the maximum beam intensity (Imax) in

DR1/PMMA, where the squares are the data at Z= 0.6, and the circles are the data at

Z= −7. The right figure shows the calculated results of the normalized transmittance (T)

as a function of the magnitude of the maximum nonlinear phase distortion in the sample

(|∆Φmax|), assuming an optical Kerr medium. The solid line is for Z= 0.6, and the broken

line is for Z= −7. We note that due to the limited LG10 beam power in our experiment,

and the relatively big beam size at Z= −7, the available peak intensity Imax at Z = −7

is small, making a full set of data at this Z impossible. However, this data still gives us

some insight as discussed below.

It can be seen from Fig. 4.11 that again, the theory successfully predicts the trend of

the change of T . Not only does T decrease initially when Imax increases in both cases, but

also both the theoretical curve and the experimental data show a steeper decrease of T

when Z = −7 than when Z = −0.6. Thus our theory can be used as a guide even when

the nonlinear refractive index of the sample saturates. A hand-waiving but useful rule

is: under the influence of saturation, the change of T as a function of increased Imax lags

behind the theory when the material does not act like an optical Kerr medium.

If the incident beam is a fundamental gaussian (LG00) beam, optical limiting is achieved

only when the sample is placed behind the beam focus (assuming self-defocusing media).

So using a LG10 beam adds flexibility in system design by allowing the sample to be placed

either in front of or behind the beam focus.

137

Bibliography

[1] M. Dumont, Z. Sekkat, R. Loucifsaibi, K. Nakatani, and J. A. Delaire, “Photoisomer-

ization, photoinduced orientation and orientational relaxation of azo dyes in polymeric

films,” Nonlinear Opt. 5, 395 (1993).

[2] P. Rochon, D. Bissonnette, A. Natansohn, and S. Xie, “Azo polymers for reversible

optical storage .3. effect of film thickness on net phase retardation and writing speed,”

Appl. Opt. 32, 7277 (1993).

[3] Z. Sekkat, D. Morichere, M. Dumont, R. Loucifsaibi, and J. A. Delaire, “Photoiso-

merization of azobenzene derivatives in polymeric thin-films,” J. Appl. Phys. 71, 1543

(1992).

[4] W. Zhang, S. Bian, S. Kim, and M. Kuzyk, “High efficiency holographic volume index

gratings in dr1-dopped pmma,” Opt. Lett. 27, 1105 (2002).

[5] R. Loucifsaibi, K. Nakatani, J. A. Delaire, M. Dumont, and Z. Sekkat, “Photoisomer-

ization and 2nd harmonic-generation in disperse red one-doped and one-functionalized

poly(methyl methacrylate) films,” Chem. Mater. 5, 229 (1993).

[6] Z. Sekkat and W. Knoll, “Creation of second-order nonlinear-optical effects by pho-

toisomerization of polar azo dyes in polymeric films - theoretical-study of steady-state

and transient properties,” J. Opt. Soc. Am. B 12, 1855 (1995).

138

[7] S. Bian, D. Robinson, and M. Kuzyk, “Optical activated cantilever using photome-

chanical effects in dye-doped polymer fibers,” J. Opt. Soc. Am. B 23, 697 (2006).

[8] M. Dumont and A. E. Osman, “On spontaneous and photoinduced orientational mo-

bility of dye molecules in polymers,” Chem. Phys. 245, 437 (1999).

[9] J. J. Park, Photo-induced molecular reorientation and photothermal heating as mecha-

nisms of the intensity-dependent refractive index in dye-doped polymers, Doctoral dis-

sertation, Washington State University, 2006.

139

Chapter 5

Conclusion

In this work, we have developed a theory on the effect of a thin optical Kerr medium on an

LG beam, and applied the theory to several important applications, including techniques

that can measure nonlinear refractive index coefficients with high sensitivity and a new

optical limiting geometry. We have also carried out experiments to test the validity of our

theory and to demonstrate the applications with DR1/PMMA samples.

We have generalized the gaussian beam decomposition method1 and implemented it

to develop our theory. Unlike some other pure mathematical approaches, the generalized

gaussian beam decomposition method provides a clear physical picture about the effect

of the thin optical Kerr medium on the beam, i.e., generating LG beams with different

radial modes, but with the same angular momentum. This shows that the OAM carried

by the photons is conserved. The results are important in applications that leverage mode

sensitivity.

With our theory we have calculated the Z-scan curve using a LG10 beam in comparison

with using a LG00 beam. The shape of the Z-scan curve using the LG1

0 beam differs from

using the LG00 beam in that the former has an extra peak and valley besides a major peak

and valley that are common to both of them. This feature is the key criteria in testing the

validity of the theory. The difference between the major peak and valley, ∆Tp−v, is the

140

key parameter in the Z-scan measurement. And We have shown that the Z-scan technique

using a LG10 beam has advantages over the the traditional measurement using a LG0

0 beam,

such as having higher sensitivity to changes of the n2 of the sample.

We have also proposed new techniques that can measure n2, including the I-scan and

∆Φ-scan technique. The I-scan technique has similar sensitivity to that of the Z-scan

technique, but does not require moving the sample along the beam axis, thus reducing

experimental uncertainties and saves the space of the apparatus and experiment run time.

As a price, the I-scan technique requires knowing the sample position along the optical

axis or a calibration of the system with a standard sample with known n2. However, if

the medium has a big nonlinear phase change (∆Φmax ∼ 1) and still acts as an optical

Kerr medium, then the aforementioned price can be avoided by implementing the ∆Φ-scan

technique.

Another application that our theory has been applied to is optical limiting. We have

shown that optical limiting can be achieved by using a LG10 beam when the sample is

placed either in front of or behind the beam focus. If the incident beam is a fundamental

gaussian (LG00) beam, optical limiting is achieved only when the sample is placed behind

the beam focus (assuming a self-defocusing media). So using a LG10 beam adds flexibility

in system design. Furthermore, we have shown that using a LG10 beam can achieve smaller

limiting threshold than using a LG00 beam.

Our theory and the proposed applications have been tested by experiments with DR1

/ PMMA samples. Considerable effort in our research has been expended to understand

the mechanisms of the nonlinear refractive index change in DR1/PMMA, especially to find

the conditions under which DR1/PMMA can be treated as an optical Kerr medium. We

have developed a three-state model, which is an improvement over a previous two-state

model,2 to describe the dominant mechanisms of the nonlinear refractive index change

in DR1/PMMA: trans-cis-trans photoisomerization and photoreorientation. With a holo-

graphic volume index grating recording experiment, we have been able to determine the

141

experimental conditions, essentially the order of magnitude range of the beam intensity

and exposure time, under which the DR1/PMMA samples act as optical Kerr media.

Our theory agrees well the experimental results in the Z-scan and I-scan measurement,

and the measured n2 values are consistent with each other and with those obtained from

other independent experiments,3 which supports our theory as well as our understanding

of the DR1/PMMA material.

In spite of the unavoidable saturation of the nonlinear refractive index change, we have

demonstrated optical limiting in DR1/PMMA using a LG10 beam by using our theory as

a guide, and showed that it has advantages over using a LG00 beam.

Overall, this work has satisfied our intellectual curiosity about the consequence of a

high order LG beam transversing an optical Kerr medium. Besides those discussed here,

our theory can be applied to many other thin film applications. With some modifications,

the theory should also apply to nonlinear optical media whose refractive index change

is intensity dependent but not of an optical Kerr type, such as those having saturable

refractive index change.

142

Bibliography

[1] D. Weaire, B. Wherrett, D. Miller, and S. Smith, “Effect of low-power nonlinear re-

fraction on laser-beam propagation in insb,” Opt. Lett. 4, 331 (1979).

[2] S. Bian, D. Robinson, and M. Kuzyk, “Optical activated cantilever using photome-

chanical effects in dye-doped polymer fibers,” J. Opt. Soc. Am. B 23, 697 (2006).

[3] J. J. Park, Photo-induced molecular reorientation and photothermal heating as mecha-

nisms of the intensity-dependent refractive index in dye-doped polymers, Doctoral dis-

sertation, Washington State University, 2006.

143

Appendix A

Generalized Gaussian Beam

Decomposition

In this appendix, we show how we decompose the electric field, after it passes through the

nonlinear sample, into a superposition of a series of LG beams. We also includes examples

on how to apply the general results to specific incident beams.

A.1 General Derivation

In Sec. 2.4.2, we have shown that when the incident beam is an LG beam, E(r, φ, z) =

E0 · LGl0p0

(r, φ, z; zr), and the optical Kerr sample is at z = zs on the z axis, the complex

electric field of the incident beam immediately after it passes through the sample is:

E ′ (r, φ, zs) = E (r, φ, zs) e−αd

2 e−i∆Φ(r,φ,zs), (A.1)

where the nonlinear phase distortion ∆Φ (r, φ, zs) obeys:

∆Φ (r, φ, z) =∆Φ0

1 + z2/z2r

(

2r2

ω2 (z)

)|l0|(

L|l0|p0

(

2r2

ω2 (z)

))2

exp

( −2r2

ω2 (z)

)

, (A.2)

144

and

∆Φ0 =π

λcǫ0n0n2|E0|2

1 − e−αd

α. (A.3)

The exponential in Eq. (A.1) can be expanded in a Taylor series as

e−i∆Φ(r,φ,zs) =∞∑

m=0

(−i∆Φ (r, φ, zs))m

m!. (A.4)

On the other hand, the complex electric field of the incident beam after it passes through

the sample can always be written as a summation of the electric fields of a series of LG

beams of different modes as

E ′ (r, φ, z) =∞∑

m=0

pm∑

p=0

∞∑

l=−∞Cp,l,mLG

lp (r, φ, z − zwm; zrm) , (A.5)

where zwm and zrm are the waist location and the Rayleigh length, respectively, of the

corresponding beam mode and Cp,l,m is the amplitude and phase of the component beam.

Since when z = zs, Eq. (A.5) and Eq. (A.1) describe the same electric field, we can

determine the parameters in Eq. (A.5) by comparing these two equations.

The incident LG beam can be explicitly expressed by applying Eq.(A.4) and Eq.(A.2)

to Eq. (A.1):

E ′ (r, φ, zs) =∞∑

m=0

E0e−αd/2 (−i∆Φ0)

m

m!

1

(1 + z2s/z

2r )

2m+1

2

× exp

(

i (2p0 + |l0| + 1) tan−1

(

zszr

))

exp (−ikzs)

×(

2r2

ω2 (zs)

)2m+1

2|l0|(

L|l0|p0

(

2r2

ω2 (zs)

))2m+1

× exp

(− (2m+ 1) r2

ω2 (zs)

)

exp

(

−i kr2zs2 (z2

s + z2r )

)

× exp (−il0φ) . (A.6)

145

On the other side, with z = zs and the explicit form of LG beams, Eq. (A.5) becomes:

E ′ (r, φ, zs) =∞∑

m=0

pm∑

p=0

∞∑

l=−∞Cp,l,m

1√

(1 + z2sm/z

2rm)

× exp

(

i (2p+ |l| + 1) tan−1

(

zsmzrm

))

exp (−ikzsm)

×( √

2r

ωm (zsm)

)|l|

L|l|p

(

2r2

ω2m (zsm)

)

exp

( −r2

ω2m (zsm)

)

exp

(

−i kr2zsm2 (z2

sm + z2rm)

)

× exp (−ilφ) , (A.7)

where

zsm = zs − zwm (A.8)

is the displacement of the sample to the waist of the corresponding beam.

The factors involving φ in Eq. (A.6) and Eq. (A.7) must equal each other, which

immediately lets us determine that the only possible l value in Eq.(A.7) is l = l0. Thus

Eq.(A.7) is simplified as follows,

E ′ (r, φ, zs) =∞∑

m=0

pm∑

p=0

Cp,m1

√

(1 + z2sm/z

2rm)

× exp

(

i (2p+ |l0| + 1) tan−1

(

zsmzrm

))

exp (−ikzsm)

×( √

2r

ωm (zsm)

)|l0|

L|l0|p

(

2r2

ω2m (zsm)

)

exp

( −r2

ω2m (zsm)

)

exp

(

−i kr2zsm2 (z2

sm + z2rm)

)

× exp (−il0φ) . (A.9)

Next we require that the two exponential factors with an r dependence in Eq. (A.6)

146

and Eq. (A.9) are equal, i.e.:

exp

(− (2m+ 1) r2

ω2 (zs)

)

exp

(

−i kr2zs2 (z2

s + z2r )

)

= exp

( −r2

ω2m (zsm)

)

exp

(

−i kr2zsm2 (z2

sm + z2rm)

)

,

(A.10)

or equivalently,

2m+ 1

ω2 (zs)=

1

ω2m (zsm)

(A.11)

and

zsz2s + z2

r

=zsm

z2sm + z2

rm

. (A.12)

Recall that the beam radius ωm(z) and the Rayleigh length zrm are related by

ω2m (z) = ω2

0m

(

1 +z2

z2rm

)

, (A.13)

where zrm and the radius of the beam waist, ω0m, depend on each other through

zrm =πω2

0m

λ. (A.14)

So there are just two independent unknowns (for example, zrm and zsm) in the right sides

of Eqs. (A.11) and (A.12). Solving the equations simultaneously, the results are:

zrm = zr(2m+ 1) (Z2 + 1)

Z2 + (2m+ 1)2 (A.15)

and

zsm = zrZ (Z2 + 1)

Z2 + (2m+ 1)2 , (A.16)

147

where Z is defined as Z = zs/zr. Using Eq. (A.8), we have

zwm = zr4m (m+ 1)Z

Z2 + (2m+ 1)2 . (A.17)

Then we deal with the rest of the factors with an r dependence in Eq. (A.6) and Eq.

(A.9). We require

(

2r2

ω2 (zs)

)2m+1

2|l0|(

L|l0|p0

(

2r2

ω2 (zs)

))2m+1

=

pm∑

p=0

Dp,m

( √2r

ωm (zsm)

)|l0|

L|l0|p

(

2r2

ω2m (zsm)

)

,

(A.18)

where Dp,m is the coefficient. Applying Eq. (A.11) to the right hand side of the above

equation and simplifying the notation by introducing a new real variable

x = (2m+ 1)2r2

ω2 (zs), (A.19)

we have

(

x

2m+ 1

)2m+1

2|l0|(

L|l0|p0

(

x

2m+ 1

))2m+1

=

pm∑

p=0

Dp,mx|l0|2 L|l0|

p (x) . (A.20)

Eq. (A.20) can be further rearranged as:

pm∑

p=0

Dp,mL|l0|p (x) =

xm|l0|

(2m+ 1)2m+1

2|l0|

(

L|l0|p0

(

x

2m+ 1

))2m+1

. (A.21)

The right hand side of the above equation is a polynomial in x and the left hand side

is a superposition of the associated Laguerre polynomials L|l0|p (x). Since the associated

Laguerre polynomials L|l0|p (x) are linearly independent and complete, there is always one

and only one set of pm and Dp,m values that satisfies the above equation. We will show

some examples later.

148

Finally we compare the rest of the factors in Eq. (A.9) and Eq. (A.6) after Eq. (A.10)

and Eq. (A.18) are satisfied, yielding

Dp,mE0e−αd/2 (−i∆Φ0)

m

m!

1

(1 + z2s/z

2r )

2m+1

2

exp

(

i (2p0 + |l0| + 1) tan−1

(

zszr

))

exp (−ikzs)

= Cp,m1

√

(1 + z2sm/z

2rm)

exp

(

i (2p+ |l0| + 1) tan−1

(

zsmzrm

))

exp (−ikzsm) . (A.22)

Substituting Eq. (A.15) and Eq. (A.16) into the above equation and simplifying it, we get

Cp,m = Dp,m · Fp,m, (A.23)

where,

Fp,m = E0e−αd/2 (−i∆Φ0)

m

m! (2m+ 1)

√

(2m+ 1)2 + Z2

(1 + Z2)2m+1 exp

(

−ikzr4m (m+ 1)Z

Z2 + (2m+ 1)2

)

× exp(

i (2p0 + |l0| + 1) tan−1 (Z))

exp

(

−i (2p+ |l0| + 1) tan−1

(

Z

2m+ 1

))

.

(A.24)

Putting our results together, the complex electric field of the incident beam after it

passes through the sample can be written as a summation of the electric fields of a series

of LG beams of different modes as

E ′ (r, φ, z) =∞∑

m=0

pm∑

p=0

Cp,mLGl0p (r, φ, z − zwm; zrm) , (A.25)

where zwm and zrm are the waist location and the Rayleigh length, respectively, of the

corresponding beam mode and Cp,m is the amplitude and phase of the component beam.

zwm = zr4m (m+ 1)Z

Z2 + (2m+ 1)2 , (A.26)

149

zrm = zr(2m+ 1) (Z2 + 1)

Z2 + (2m+ 1)2 , (A.27)

and Cp,m = Dp,m · Fp,m where Z is defined as Z = zs/zr.

Fp,m = E0e−αd/2 (−i∆Φ0)

m

m! (2m+ 1)

√

(2m+ 1)2 + Z2

(1 + Z2)2m+1 exp

(

−ikzr4m (m+ 1)Z

Z2 + (2m+ 1)2

)

× exp(

i (2p0 + |l0| + 1) tan−1 (Z))

exp

(

−i (2p+ |l0| + 1) tan−1

(

Z

2m+ 1

))

,

(A.28)

and pm and Dp,m are determined through

pm∑

p=0

Dp,m · L|l0|p (x) =

xm|l0|(

L|l0|p0

(

x2m+1

)

)2m+1

(2m+ 1)2m+1

2|l0|

, (A.29)

where x is an arbitrary real variable. Our theoretical results are embodied in Eq. (A.25).

A.2 Examples

Now let’s see a few examples after we recall the expression of the associated Laguerre

polynomials. In general,

Lkn(x) =n∑

m=0

(−1)m(n+ k)!

(n−m)! (k +m)!m!xm, k > −1. (A.30)

A.2.1 LG00 beam

Assume the incident beam is a fundamental Gaussian beam, i.e., E(r, φ, z) = E0·LG00(r, φ, z; zr).

150

General result

Since p0 = 0 and l0 = 0, Eq. ( A.29) is given by

pm∑

p=0

Dp,m · L0p (x) = 1, (A.31)

yielding pm = 0 for all m’s and D0,m = 1. Therefore the decomposition result is

E ′ (r, φ, z) =∞∑

m=0

C0,mLG00 (r, φ, z − zwm; zrm) (A.32)

with

zwm = zr4m (m+ 1)Z

Z2 + (2m+ 1)2 , (A.33)

zrm = zr(2m+ 1) (Z2 + 1)

Z2 + (2m+ 1)2 , (A.34)

and

C0,m = E0e−αd/2 (−i∆Φ0)

m

m! (2m+ 1)

√

(2m+ 1)2 + Z2

(1 + Z2)2m+1 exp

(

−ikzr4m (m+ 1)Z

Z2 + (2m+ 1)2

)

× exp(

i tan−1 (Z))

exp

(

−i tan−1

(

Z

2m+ 1

))

. (A.35)

Eq. (A.32) shows that after a fundamental Gaussian beam passes through a thin optical

Kerr medium, a series of fundamental Gaussian beams with different beam waist radii and

locations are generated.

151

Small nonlinear phase distortion

When the nonlinear phase distortion ∆Φ is small, we can neglect the higher order terms

in Eq. (A.32) and still make a good approximation. E.g., if we just keep the terms m = 0

and m = 1, then

E ′ (r, φ, z) ≈ C0,0LG00 (r, φ, z − zw0; zr0) + C0,1LG

00 (r, φ, z − zw1; zr1) , (A.36)

where

C0,0 = E0e−αd/2;

C0,1 = E0e−αd/2 (−i∆Φ0)

3

√

9 + Z2

(1 + Z2)3 exp

(

−ikzr8Z

Z2 + 9

)

× exp(

i tan−1 (Z))

exp

(

−i tan−1

(

Z

3

))

;

zw0 = 0;

zw1 = zr8Z

Z2 + 9;

zr0 = zr;

zr1 = zr3 (Z2 + 1)

Z2 + 9. (A.37)

A.2.2 LG10 beam

Assume the incident beam E(r, φ, z) = E0 · LG10(r, φ, z; zr).

General result

Since p0 = 0 and l0 = 1, Eq. ( A.29) becomes

pm∑

p=0

Dp,m · L1p (x) =

xm

(2m+ 1)2m+1

2

, (A.38)

152

where

L1p(x) =

p∑

i=0

(−1)i(p+ 1)!

(p− i)! (1 + i)!i!xi (A.39)

according to Eq. (A.30). Comparing the highest order of x on both sides of Eq. (A.38),

we find pm = m. Next we solve for Dp,m. Define two n-dimensional vectors

L ={

L10 (x) , L1

1 (x) , L12 (x) , ..., L1

n−1 (x)}T

(A.40)

and

X ={

1, x, x2, ..., xn−1}T

, (A.41)

where T stands for transpose. Also define a n× n matrix H whose elements are

(H)pi =

(−1)i (p+1)!(p−i)!(1+i)!i! , p ≥ i;

0, p < i,(A.42)

where p, i = 0, 1, 2, ..., n − 1. Note H is a lower triangular matrix. Using Eq. (A.39) we

can write:

L = HX, (A.43)

from which we have

X = H−1L, (A.44)

153

where H−1 is the inverse of H. It’s easy to verify that H−1 is also a lower triangular

matrix, i.e., its elements(

H−1)

pi= 0 if p < i. Thus,

xm =m∑

i=0

(

H−1)

mi· L1

i (x) . (A.45)

Comparing Eq. (A.38) and Eq. (A.45) we find:

Dp,m =

(

H−1)

mp

(2m+ 1)2m+1

2

, (A.46)

or

Dp,m =

(

(

H−1)T)

pm

(2m+ 1)2m+1

2

. (A.47)

Once Dp,m is determined, the result can be expressed straightforwardly:

E ′ (r, φ, z) =∞∑

m=0

m∑

p=0

Cp,mLG1p (r, φ, z − zwm; zrm) , (A.48)

where

zwm = zr4m (m+ 1)Z

Z2 + (2m+ 1)2 , (A.49)

zrm = zr(2m+ 1) (Z2 + 1)

Z2 + (2m+ 1)2 , (A.50)

154

and Cp,m = Dp,m · Fp,m, where

Fp,m = E0e−αd/2 (−i∆Φ0)

m

m! (2m+ 1)

√

(2m+ 1)2 + Z2

(1 + Z2)2m+1 exp

(

−ikzr4m (m+ 1)Z

Z2 + (2m+ 1)2

)

× exp(

i2 tan−1 (Z))

exp

(

−i (2p+ 2) tan−1

(

Z

2m+ 1

))

(A.51)

and

Dp,m =

(

(

H−1)T)

pm

(2m+ 1)2m+1

2

(A.52)

with matrix H defined by Eq. (A.42).

Small nonlinear phase distortion

When the nonlinear phase distortion ∆Φ is small, we can neglect the higher order terms

in Eq. (A.48), which still leads to a good approximation. E.g., if we just keep the terms

of m = 0 and m = 1, then

H =

1 0

2 −1

, (A.53)

H−1 =

1 0

2 −1

, (A.54)

and

(

H−1)T

=

1 2

0 −1

. (A.55)

155

Using Eq. (A.52), we have

D0,0 = 1;

D0,1 =2

3√

3;

D1,0 = 0;

D1,1 =−1

3√

3. (A.56)

Substituting Eq. (A.56) into Eq. (A.48), we get,

E ′ (r, φ, z) ≈ F0,0LG10 (r, φ, z − zw0; zr0)

+2

3√

3F0,1LG

10 (r, φ, z − zw1; zr1)

− 1

3√

3F1,1LG

11 (r, φ, z − zw1; zr1) , (A.57)

where

F0,0 = E0e−αd/2;

F0,1 = E0e−αd/2 (−i∆Φ0)

3

√

9 + Z2

(1 + Z2)3 exp

(

−ikzr8Z

Z2 + 9

)

× exp(

i2 tan−1 (Z))

exp

(

−i2 tan−1

(

Z

3

))

;

F1,1 = E0e−αd/2 (−i∆Φ0)

3

√

9 + Z2

(1 + Z2)3 exp

(

−ikzr8Z

Z2 + 9

)

× exp(

i2 tan−1 (Z))

exp

(

−i4 tan−1

(

Z

3

))

,

156

and

zw0 = 0;

zw1 = zr8Z

Z2 + 9;

zr0 = zr;

zr1 = zr3 (Z2 + 1)

Z2 + 9. (A.58)

157

Appendix B

Simplifying the normalized Z-scan

transmittance T

This appendix shows how to simplify the expression of the normalized Z-scan transmittance

T (Z,∆Φ0) =|E ′ (r → 0, φ, z → ∞)|2

|E ′ (r → 0, φ, z → ∞) |∆Φ0=0|2, (B.1)

for select incident beams.

B.1 LG00 beam

Assume the incident beam is a LG00 beam

Ei (r, φ, z) = E0 · LG00 (r, φ, z; zr) , (B.2)

158

where we recall

LG00 (r, φ, z; zr) =

ω0

ω (z)exp

(

− r2

ω2 (z)− i

kr2z

2 (z2 + z2r )

)

exp

(

−i(

kz − tan−1

(

z

zr

)))

.

(B.3)

When the nonlinear phase distortion ∆Φ is small, using Eq. (A.36) in Appendix A the

electric field of the incident beam after passing through the nonlinear sample is

E ′ (r, φ, z) ≈ C0,0LG00 (r, φ, z − zw0; zr0) + C0,1LG

00 (r, φ, z − zw1; zr1) , (B.4)

where

C0,0 = E0e−αd/2;

C0,1 = E0e−αd/2 (−i∆Φ0)

3

√

9 + Z2

(1 + Z2)3 exp

(

−ikzr8Z

Z2 + 9

)

× exp(

i tan−1 (Z))

exp

(

−i tan−1

(

Z

3

))

;

zw0 = 0;

zw1 = zr8Z

Z2 + 9;

zr0 = zr;

zr1 = zr3 (Z2 + 1)

Z2 + 9. (B.5)

Therefore,

T (Z,∆Φ0) =

∣

∣

∣

∣

E ′ (r → 0, φ, z → ∞)

E ′ (r → 0, φ, z → ∞) |∆Φ0=0

∣

∣

∣

∣

2

=

∣

∣

∣

∣

1 +C0,1

C0,0

· LG00 (0, φ,∞− zw1; zr1)

LG00 (0, φ,∞− zw0; zr0)

∣

∣

∣

∣

2

. (B.6)

159

Substitute Eq.(B.3) and Eq.(B.5) into Eq. (B.6) and simplifying it, we get,

T (Z,∆Φ0) =

∣

∣

∣

∣

1 − i∆Φ01√

1 + Z2√

9 + Z2exp

(

i

(

tan−1 (Z) − tan−1

(

Z

3

)))∣

∣

∣

∣

2

, (B.7)

which can be also written as

T (Z,∆Φ0) =

∣

∣

∣

∣

1 − i∆Φ01

1 − iZ· 1

3 + iZ

∣

∣

∣

∣

2

. (B.8)

When ∆Φ0 << 1, we omit the higher order terms of ∆Φ0, therefore

T (Z,∆Φ0) ∼= 1 +4∆Φ0Z

(1 + Z2) (Z2 + 9). (B.9)

B.2 LG10 beam

Assume the incident beam is a LG10 beam, then

Ei (r, φ, z) = E0 · LG10 (r, φ, z; zr) , (B.10)

where

LG10 (r, φ, z; zr) =

ω0

ω (z)

(√2r

ω (z)

)

exp

(

− r2

ω2 (z)− i

kr2z

2 (z2 + z2R)

)

× exp (−iφ) exp

(

−i(

kz − 2tan−1

(

z

zR

)))

.

(B.11)

160

When the nonlinear phase distortion ∆Φ is small, using Eq.( A.57) in Appendix A the

electric field of the incident beam after passing through the nonlinear sample is

E ′ (r, φ, z) ≈ F0,0LG10 (r, φ, z − zw0; zr0)

+2

3√

3F0,1LG

10 (r, φ, z − zw1; zr1)

− 1

3√

3F1,1LG

11 (r, φ, z − zw1; zr1) , (B.12)

where

F0,0 = E0e−αd/2;

F0,1 = E0e−αd/2 (−i∆Φ0)

3

√

9 + Z2

(1 + Z2)3 exp

(

−ikzr8Z

Z2 + 9

)

× exp(

i2 tan−1 (Z))

exp

(

−i2 tan−1

(

Z

3

))

;

F1,1 = E0e−αd/2 (−i∆Φ0)

3

√

9 + Z2

(1 + Z2)3 exp

(

−ikzr8Z

Z2 + 9

)

× exp(

i2 tan−1 (Z))

exp

(

−i4 tan−1

(

Z

3

))

,

and

zw0 = 0;

zw1 = zr8Z

Z2 + 9;

zr0 = zr;

zr1 = zr3 (Z2 + 1)

Z2 + 9. (B.13)

161

Also recall

LG11 (r, φ, z; zr) =

(

ω0

ω (z)

)

(√2r

ω (z)

)

exp

( −r2

ω2 (z)− i

kr2z

2 (z2 + z2r )

)

× exp (−iφ) exp

(

−i(

kz − 4 tan−1

(

z

zr

)))(

2 − 2r2

ω2 (z)

)

. (B.14)

Therefore,

T (Z,∆Φ0) =

∣

∣

∣

∣

E ′ (r → 0, φ, z → ∞)

E ′ (r → 0, φ, z → ∞) |∆Φ0=0

∣

∣

∣

∣

2

=

∣

∣

∣

∣

∣

1 +

23√

3F0,1

F0,0

· LG10 (0, φ,∞− zw1; zr1)

LG10 (0, φ,∞− zw0; zr0)

−1

3√

3F1,1

F0,0

· LG11 (0, φ,∞− zw1; zr1)

LG10 (0, φ,∞− zw0; zr0)

∣

∣

∣

∣

∣

2

. (B.15)

Expressing the parameters explicitly and simplifying Eq. (B.15), we get,

T (Z,∆Φ0) =

∣

∣

∣

∣

1 − i∆Φ02

3 (9 + Z2)exp

(

−i2 tan−1

(

Z

3

))

× exp(

i2 tan−1 (Z))

(

1 + exp

(

−i2 tan−1

(

Z

3

)))∣

∣

∣

∣

2

, (B.16)

which can also be written as

T (Z,∆Φ0) =

∣

∣

∣

∣

∣

1 − i∆Φ02

3

1

(3 + iZ)2

(1 + iZ)2

(1 + Z2)

(

1 +Z2 + 9

(3 + iZ)2

)

∣

∣

∣

∣

∣

2

. (B.17)

Expanding the above equation, we get

T (Z,∆Φ0) = 1 +8Z (27 + 10Z2 − Z4)

(1 + Z2) (9 + Z2)3 ∆Φ0 +16

(9 + Z2)3 ∆Φ20. (B.18)

162

When ∆Φ0 << 1, we omit the higher order terms of ∆Φ0, then

T (Z,∆Φ0) ∼= 1 +8Z (27 + 10Z2 − Z4)

(1 + Z2) (9 + Z2)3 ∆Φ0. (B.19)

163

Appendix C

Evaluating the normalized optical

limiting transmittance T

This appendix shows how to evaluate the normalized optical limiting transmittance

T (Z,∆Φmax) =|E ′ (r → 0, φ, z → ∞)|2

|E ′ (r → 0, φ, z → ∞) |∆Φmax=0|2(C.1)

for select incident beams. In contrast to the calculation for the normalized Z-scan transmit-

tance in Appendix B, Φmax is used instead of Φ0 as a parameter. Another major difference

is that the exact expression for E ′ (r, φ, z), such as Eq. (A.25), Eq. (A.32) and Eq. (A.48),

must be used in the calculation of the normalized optical limiting transmittance since in

general the small nonlinear phase distortion approximation is not valid in optical limiting

applications.

C.1 LG00 beam

Applying Eq. (A.32) to Eq. (C.1) and noting that the only non-zero C is C0,0 when

∆Φmax = 0, we get,

164

T (Z,∆Φmax) =

∣

∣

∣

∣

∣

∞∑

m=0

C0,m

C0,0

· LG00 (r → 0, φ, (z → ∞) − zwm; zrm)

LG00 (r → 0, φ, (z → ∞) − zw0; zr0)

∣

∣

∣

∣

∣

2

. (C.2)

Recalling

LG00 (r, φ, z; zr) =

ω0

ω (z)exp

(

− r2

ω2 (z)− i

kr2z

2 (z2 + z2r )

)

exp

(

−i(

kz − tan−1

(

z

zr

)))

,

(C.3)

we have

LG00 (r, φ, z − zwm; zrm)

LG00 (r, φ, z − zw0; zr0)

=

ω0m

ωm(z−zwm)

ω0

ω(z)

·exp

(

− r2

ω2m(z−zwm)

− i kr2(z−zwm)

2((z−zwm)2+z2rm)

)

exp(

− r2

ω2(z)− i kr2z

2(z2+z2r )

)

×exp

(

−i(

k (z − zwm) − tan−1(

(z−zwm)zrm

)))

exp(

−i(

kz − tan−1(

zzr

))) , (C.4)

where zw0 = 0 and zr0 = zr are assumed. Letting r = 0 and z → ∞, the above equation

becomes

LG00 (r → 0, φ, (z → ∞) − zwm; zrm)

LG00 (r → 0, φ, (z → ∞) − zw0; zr0)

= exp (ikzwm) ·ω0m

ωm(z−zwm)

ω0

ω(z)

∣

∣

∣

∣

∣

z→∞

, (C.5)

which can be further simplified to be

LG00 (r → 0, φ, (z → ∞) − zwm; zrm)

LG00 (r → 0, φ, (z → ∞) − zw0; zr0)

= exp (ikzwm) · zrmzr

= exp (ikzwm) · (2m+ 1) (Z2 + 1)

Z2 + (2m+ 1)2 . (C.6)

165

The other factor in Eq. (C.2) is

C0,m

C0,0

=(−i∆Φ0)

m

m! (2m+ 1)

√

(2m+ 1)2 + Z2

(1 + Z2)2m+1 exp

(

−ikzr4m (m+ 1)Z

Z2 + (2m+ 1)2

)

× exp(

i tan−1 (Z))

exp

(

−i tan−1

(

Z

2m+ 1

))

=(−i∆Φ0)

m

m! (2m+ 1)

√

(2m+ 1)2 + Z2

(1 + Z2)2m+1 exp (−ikzwm)

× 1 + iZ√1 + Z2

(2m+ 1) − iZ√

(2m+ 1)2 + Z2

=(−i∆Φ0)

m (1 + iZ) ((2m+ 1) − iZ) exp (−ikzwm)

m! (2m+ 1) (1 + Z2)m+1 . (C.7)

For the LG00 beam, since ∆Φ0 = (1 + Z2) ∆Φmax, the above equation becomes

C0,m

C0,0

=(−i∆Φmax)

m (1 + iZ) ((2m+ 1) − iZ) exp (−ikzwm)

m! (2m+ 1) (1 + Z2). (C.8)

Substituting Eq. (C.6) and Eq. (C.8) into Eq. (C.2) and simplifying it, we have

T (Z,∆Φmax) =

∣

∣

∣

∣

∣

∞∑

m=0

(−i∆Φmax)m

m!

1 + iZ

(2m+ 1) + iZ

∣

∣

∣

∣

∣

2

. (C.9)

Eq. (C.9) concludes our derivation. As we can see, it involves a summation of infinite

terms, making further analytic results difficult. Fortunately, the summation is always

convergent for finite values of Z and ∆Φmax. In practice, we use numerical methods to

evaluate the value of T (Z,∆Φmax). For example, the Mathematica codes that calculate

the value of T (Z,∆Φmax) for given Z and ∆Φmax are attached as follows.

Mathematica codes

(* numerical calculation of the normalized optical limiting

transmittance T for LG(l=0, p=0) beam *)

166

(* The codes here minimize the redundant calculations *)

(* The precision of the result, or how many terms to be included in the sum

is controlled by the variable tt, which is simply set as 100 here. *)

Clear["Global‘*"];

tt = 100; (* tt: how many terms to be included to reach pre-given precision.*)

Tm := Function[{phim, Z},

tmp = 1; (* tmp: variable to record result. *)

t1 = 1 + I Z // N; (*t1: temporary variable.*)

For[m = 1, m <= tt, m++,

t1 = t1*(((-I)*phim)/m)//N;

t2 = t1/((2*m + 1) + I*Z)//N; (*t2: temporary variable.*)

tmp = tmp + t2//N;

];

Abs[tmp]^2 (* return the final result *)

]

(* an example of usage *)

Plot[Tm[-0.5Pi, z], {z, -10, 10},PlotRange ->{0, 1}];

C.2 LG10 beam

Applying Eq. (A.48) to Eq. (C.1) and noting that the only non-zero C is C0,0 when

∆Φmax = 0, we get,

T (Z,∆Φmax) =

∣

∣

∣

∣

∣

∞∑

m=0

m∑

p=0

Cp,mC0,0

·LG1

p (r → 0, φ, (z → ∞) − zwm; zrm)

LG10 (r → 0, φ, (z → ∞) − zw0; zr0)

∣

∣

∣

∣

∣

2

. (C.10)

167

Recalling

LG1p (r, φ, z; zr) =

(

ω0

ω (z)

)

(√2r

ω (z)

)

L1p

(

2r2

ω2 (z)

)

exp

( −r2

ω2 (z)− i

kr2z

2 (z2 + z2r )

)

× exp (−iφ) exp

(

−i(

kz − (2p+ 2) tan−1

(

z

zr

)))

, (C.11)

we have

LG1p (r, φ, z − zwm; zrm)

LG10 (r, φ, z − zw0; zr0)

=

ω0m

ωm(z−zwm)

ω0

ω(z)

· ω (z)

ωm (z − zm)· L1

p

(

2r2

ω2m (z − zm)

)

×exp

(

− r2

ω2m(z−zwm)

− i kr2(z−zwm)

2((z−zwm)2+z2rm)

)

exp(

− r2

ω2(z)− i kr2z

2(z2+z2r )

)

×exp

(

−i(

k (z − zwm) − (2p+ 2) tan−1(

(z−zwm)zrm

)))

exp(

−i(

kz − 2tan−1(

zzr

))) ,

(C.12)

where zw0 = 0 and zr0 = zr are assumed. Letting r = 0 and z → ∞, the above equation

becomes

LG00 (r → 0, φ, (z → ∞) − zwm; zrm)

LG00 (r → 0, φ, (z → ∞) − zw0; zr0)

= (−1)p L1p (0) exp (ikzwm)

ω0m ω (z)2

ω0 ωm (z − zwm)2

∣

∣

∣

∣

∣

z→∞

,

(C.13)

which can be further simplified if we write ω0, ω0m, etc. explicitly, yielding

LG00 (r → 0, φ, (z → ∞) − zwm; zrm)

LG00 (r → 0, φ, (z → ∞) − zw0; zr0)

= (−1)p L1p (0) exp (ikzwm)

(

zrmzr

) 3

2

. (C.14)

168

Furthermore,

zrmzr

=(2m+ 1) (Z2 + 1)

Z2 + (2m+ 1)2 , (C.15)

and

L1p (0) = p+ 1. (C.16)

Therefore,

LG00 (r → 0, φ, (z → ∞) − zwm; zrm)

LG00 (r → 0, φ, (z → ∞) − zw0; zr0)

= (−1)p (p+ 1) exp (ikzwm)

(

(2m+ 1) (Z2 + 1)

Z2 + (2m+ 1)2

)3

2

.

(C.17)

The other factor in Eq. (C.10)

Cp,mC0,0

=Dp,m

D0,0

· Fp,mF0,0

. (C.18)

It’s easy to verify that D0,0 = 1 and F0,0 = E0e−αd/2, thus

Cp,mC0,0

= Dp,m(−i∆Φ0)

m

m! (2m+ 1)

√

(2m+ 1)2 + Z2

(1 + Z2)2m+1 exp

(

−ikzr4m (m+ 1)Z

Z2 + (2m+ 1)2

)

× exp(

i2 tan−1 (Z))

exp

(

−i (2p+ 2) tan−1

(

Z

2m+ 1

))

= Dp,m(−i∆Φ0)

m

m! (2m+ 1)

√

(2m+ 1)2 + Z2

(1 + Z2)2m+1 exp (−ikzwm)

× (1 + iZ)2

1 + Z2

(

((2m+ 1) − iZ)2

(2m+ 1)2 + Z2

)p+1

= Dp,m(−i∆Φ0)

m (1 + iZ)2 ((2m+ 1) − iZ)2p+2 exp (−ikzwm)

m! (2m+ 1) (1 + Z2)m+ 3

2

(

(2m+ 1)2 + Z2)p+ 1

2

. (C.19)

169

For the LG10 beam, ∆Φ0 = e (1 + Z2) ∆Φmax, so the above equation becomes

Cp,mC0,0

= Dp,m(−i · e · ∆Φmax)

m (1 + iZ)2 ((2m+ 1) − iZ)2p+2 exp (−ikzwm)

m! (2m+ 1) (1 + Z2)3

2

(

(2m+ 1)2 + Z2)p+ 1

2

. (C.20)

Putting Eq. (C.17) and Eq. (C.20) into Eq. (C.10) and simplifying it, we have

T (Z,∆Φmax) =

∣

∣

∣

∣

∣

∞∑

m=0

m∑

p=0

Dp,m(−1)p (p+ 1)

√2m+ 1 (−i · e · ∆Φmax)

m

m!

×(

(2m+ 1)2 + Z2)p

(1 + iZ)2

((2m+ 1) + iZ)2(p+1)

∣

∣

∣

∣

∣

2

. (C.21)

Eq. (C.21) also has an infinite number of terms and we must use numerical methods

to evaluate it. For the purpose of efficient numerical calculation, we rewrite the above

equation as

T (Z,∆Φmax) =

∣

∣

∣

∣

∣

(1 + iZ)2∞∑

m=0

(i · e · ∆Φmax)m

m∑

p=0

Rpm

(

(2m+ 1)2 + Z2)p

((2m+ 1) + iZ)2(p+1)

∣

∣

∣

∣

∣

2

, (C.22)

where

Rpm =(

(

H−1)T)

pm

(−1)m+p (p+ 1)

m! (2m+ 1)m(C.23)

with the matrix H is defined in Eq. (A.42). We note that Rp,m is independent of Z and

∆Φmax. Finally, we show the example of the Mathematica code that calculates the value

of T (Z,∆Φmax) for given Z and ∆Φmax.

Mathematica codes

(* numerical calculation of the normalized optical limiting

transmittance T for LG(l=1, p=0) beam *)

(* The codes here minimize the redundant calculations *)

170

(* The precision of the result, or how many terms to be included in the sum

is controlled by the variable tt, which is simply set as 100 here. *)

Clear["Global‘*"];

tt = 100; (* tt: how many terms to be included to reach pre-given precision.*)

H = Table[((-1)^m*(p + 1)!)/((p - m)!*(1 + m)!*m!), {p, 0, tt}, {m, 0, tt}];

(* define the matrix H *)

G = Inverse[H]; (* G = the inverse of H *)

Q = Table[((-1)^(m + p)*(p + 1))/(m!*(2*m + 1)^m), {m, 0, tt}, {p, 0, tt}];

(* the other factors in R *)

R = G*Q//N; (* the matrix R *)

Clear[H, G, Q]; (* release the memory *)

Tm := Function[{phim, Z},

tmp = 0; (* tmp: variable to record result. *)

phim1 = I*phim*Exp[1]//N; (* phim1: temporary variable. *)

t1 = (1 + I*Z)^2/phim1//N; (* t1: temporary variable.*)

For[m = 0, m <= tt, m++,

t1 = t1*phim1//N;

t2 = (Z^2 + (2*m + 1)^2)/((2*m + 1) + I*Z)^2//N;

(* t2: temporary variable.*)

t3 = t1/(Z^2 + (2*m + 1)^2)//N; (* t3: temporary variable.*)

For[p = 0, p <= m, p++,

t3 = t3*t2//N;

tmp = tmp + t3*R[[m + 1]][[p + 1]] //N;

];

];

171

Abs[tmp]^2 (* return the final result *)

]

(* an example of usage *)

Plot[Tm[phi, 3], {phi, 0, -0.8}, PlotRange -> {0.9, 1.1}];

172

Appendix D

Intensity and and power of an LG

beam

We derive several useful expressions in regard to the intensity and power of an LG beam,

assuming the electric field of the beam is of the form

E (~r, t) =1

2

(

E (~r) eiω′t + c.c.

)

, (D.1)

where a linearly polarized monochromatic wave is assumed, ω′ is the angular frequency,

and

E (~r) = E0 · LGlp (r, φ, z)

= E0

(

ω0

ω (z)

)

(√2r

ω (z)

)|l|

L|l|p

(

2r2

ω2 (z)

)

× exp

( −r2

ω2 (z)

)

exp

(

−i kr2z

2 (z2 + z2r )

)

× exp

(

i (2p+ |l| + 1) tan−1

(

z

zr

))

× exp (−ilφ) exp (−ikz) , (D.2)

173

where l (l = 0,±1,±2, ...) is the angular mode number, p (p = 0, 1, 2, ...) is the transverse

radial mode number, ω0 is the beam waist radius, zr = kω20/2 is the Rayleigh length, ω(z) =

ω0(1 + z2/z20)

1/2 is the beam radius at z, and Llp is the associated Laguerre polynomial

defined as

Llp(x) =

p∑

m=0

(−1)m(p+ l)!

(p−m)! (l +m)!m!xm, l > −1. (D.3)

D.1 Intensity of an LG beam

The time-averaged intensity of the beam is1

I (~r) =|E (~r)|2

2η, (D.4)

where

η =η0

n(D.5)

is the impedance of the medium, n is the refractive index of the medium, and

η0 =

(

µ0

ǫ0

)1/2

(D.6)

is the impedance of free space. Since the speed of light in free space is

c =1√µ0ǫ0

, (D.7)

Eq. (D.4) can be written as

I (~r) =1

2cǫ0n |E (~r)|2 . (D.8)

174

Applying Eq. (D.2), the above equation becomes

I (r, φ, z) =1

2cǫ0n |E0|2

(

ω0

ω (z)

)2(2r2

ω2 (z)

)|l|(

L|l|p

(

2r2

ω2 (z)

))2

exp

( −2r2

ω2 (z)

)

(D.9)

Adopting normalized coordinates, R = r/ω(z) and Z = z/zr, also using ω2(z) = ω20(1 +

z2/z2r ), Eq. (D.9) can be written as

I (R,Z) =1

2cǫ0n |E0|2

(

1 + Z2)−1 (

2R2)|l| (

L|l|p

(

2R2))2

exp(

−2R2)

. (D.10)

Case I p = 0

We are interested in the case of p = 0. Since L|l|0 (2R2) = 1, Eq. (D.10) becomes

I (R,Z) =1

2cǫ0n |E0|2

(

1 + Z2)−1 (

2R2)|l|

exp(

−2R2)

. (D.11)

At fixed Z, the coordinate R of the extremes of the intensity are obtained by solving the

equation

∂I (R,Z)

∂R= 0, (D.12)

yielding, in general, two solutions, R = 0 and R = (|l|/2)1/2.

Case II LG00 beam

When the beam is a fundamental gaussian beam (l = 0), two of the extrema coincide at

R = 0, where the intensity is a maximum, yielding

Imax (Z) =1

2cǫ0n |E0|2

(

1 + Z2)−1

. (D.13)

175

When Z = 0, the intensity is the highest, yielding

Imax (0) =1

2cǫ0n |E0|2 . (D.14)

The intensity at r = ω(z) is

I (r = ω(z), z) =1

e2Imax (z) . (D.15)

Case III LG10 beam

When the beam is a LG10 beam, at R = 0 the intensity is a minimum and equal to zero.

At R = 1/√

2, i.e., r = ω(z)/√

2, the intensity is a maximum, given by

Imax (Z) =1

2ecǫ0n |E0|2

(

1 + Z2)−1

. (D.16)

When Z = 0, the intensity is the highest, yielding

Imax (0) =1

2ecǫ0n |E0|2 . (D.17)

The intensity at r = ω(z) is

I (r = ω(z), z) =2

eImax (z) . (D.18)

D.2 Power of an LG beam

The power, P, of an LG beam is obtained by a surface integral over any cross section of

the beam,

P ={

I (~r) dxdy. (D.19)

176

Applying Eq. (D.9), it yields

P =

∫ 2π

φ=0

∫ ∞

r=0

I (r, φ, z) rdrdφ (D.20)

=π

4cǫ0n |E0|2 ω0

2

∫ ∞

r=0

(

2r2

ω2 (z)

)|l|(

L|l|p

(

2r2

ω2 (z)

))2

exp

( −2r2

ω2 (z)

)

d

(

2r2

ω2 (z)

)

.

The integral can be integrated by using the following identity2

∫ ∞

0

e−xxkLkn (x)Lkm (x) dx =(n+ k)!

n!δm,n, (D.21)

yielding

P =1

4πω0

2cǫ0n |E0|2(p+ |l|)!

p!. (D.22)

Case I LG00 beam

The power of a LG00 beam is

P =1

4πω0

2cǫ0n |E0|2 , (D.23)

which can also be written as

P =1

2πω0

2Imax(Z)(

1 + Z2)

, (D.24)

or,

P =1

2πω0

2Imax (0) , (D.25)

by using Eq. (D.13) and Eq. (D.14), respectively.

177

Case II LG10 beam

The power of a LG10 beam is

P =1

4πω0

2cǫ0n |E0|2 , (D.26)

which can also be written as

P =e

2πω0

2Imax(Z)(

1 + Z2)

, (D.27)

or,

P =e

2πω0

2Imax (0) , (D.28)

by using Eq. (D.16) and Eq. (D.17), respectively.

178

Bibliography

[1] B. E. A. Saleh and M. C. Teich, Fundamentals of photonics, Wiley series in pure and

applied optics (Wiley, New York, 1991).

[2] G. B. Arfken and H. J. Weber, Mathematical methods for physicists, 5th ed. (Harcourt

Academic Press, San Diego, Calif., 2001), george B. Arfken, Hans J. Weber.

179