Characterisation of polarisedsupercontinuum generation and its

focal field

A thesis submitted for the degree of

Doctor of Philosophy

by

Brendan James Chick

Centre for Micro-Photonics

Faculty of Engineering and Industrial Sciences

Swinburne University of Technology

Melbourne, Australia

Declaration

I, Brendan James Chick, declare that this thesis entitled :

Characterisation of polarised supercontinuum generation and its focal

field

is my own work and has not been submitted previously, in whole or in part, inrespect of any other academic award.

Brendan James Chick

Centre for Micro-PhotonicsFaculty of Engineering and Industrial ScienceSwinburne University of TechnologyAustralia

Dated this day, June 3, 2010

i

Abstract

Since the first investigation of supercontinuum generation in microstructured optical

fibre almost a decade ago, an enormous interest has developed in its application.

Supercontinuum generation, the construction of broadband light from nonlinear

and dispersive optical processes is a unique type of radiation that has the design

functionality to enhance a broad range of applications. The temporal and spectral

characteristics of a supercontinuum make it an ideal source in microscopy, as

these features can provide a means to simultaneously optically image with different

carrier frequencies or simultaneously optical record in a spectrally selective storage

medium. These applications all involve the diffraction and interference of the

supercontinuum field and what needs to be understood is how such a field behaves

under these conditions. The investigation in this thesis identifies the supercontinuum

characteristics which are important to the diffraction by a lens and how these

characteristics will affect the measurement of the optical properties in microscopic

applications.

To achieve this goal there are two major areas of investigation; supercontinuum

generation and optical diffraction theory. A theoretical and experimental inves-

tigation into supercontinuum generation is first presented, which investigates the

polarisation properties of supercontinuum generation in highly birefringent photonic

crystal fibre with two zero dispersion wavelengths. It is shown that the polarisation

state of the incident ultrashort optical pulse maintains its polarisation state as

it propagates through the optical fibre. The temporal and spectral properties of

the principal axes are determined not only by the phase mismatch and the group

velocity mismatch between the two fundamental linear polarised modes, but are

affected by the different higher order dispersion coefficients. The balance between

i

the nonlinearity induced by the Kerr effect and the second order dispersion initiates

the formation of a high order soliton which then shifts spectrally toward the infra-

red frequencies. This formation sets the condition for the emission of dispersive

waves which shifts toward the lower visible frequencies. However, the dispersion

parameters associated with the two fundamental modes produces different high

order solitons and phase matching conditions, which determine the wave-numbers

for the dispersive waves. The larger of the two dispersion terms enlarges the initial

compression of the ultrashort pulse creating a high order soliton with a significantly

smaller temporal width, which under conditions of Raman scattering the shift of the

soliton is further. Experimentally, it is confirmed that the two fundamental modes of

the photonic crystal fibre have different spectral and temporal features. The degree

of polarisation also confirms that the supercontinuum spectrum is highly polarised

with the degradation attributed to the depolarisation caused by the objective lens.

The processes of nonlinearity and dispersion act as phase shifts onto an ultra-

short pulse. When superimposed through the diffraction by a lens of low numerical

aperture, the temporal phase associated with the field couples with the spatial phase

incurred by the lens. This coupling changes the way the field correlates, which is

analysed through the degree of coherence of the field. Fluctuations occur in the

temporal coherence of the field because of enlarged variations in spatial phase, which

are associated with the conditions of destructive interference, which imposes zero

intensity locations in the focal region of the lens. These variations are quantified

through the coherence time of the field and is most dramatic for a nonstationary

observation frame which is affected by the path difference between the rays at the

extremities of the lens and the rays along the optical axis.

The significant phase contribution that affects the temporal coherence of the

SC field is the initial formation of the high order soliton. The compression of the

ultrashort pulse and the formation of the high order soliton increases the bandwidth

of the field altering the coherence time. After this point in the evolution the

coherence is constructed by the interference from dispersive waves and the fission

of the high order solitary waves. The two dominant processes which influence the

temporal coherence in the focal region are the third order dispersion effect and

ii

the Raman scattering. However, the interference of the temporal phase from these

effects and the other higher order dispersion and third order nonlinear effects couple

with the spatial phase from the diffraction by the lens increasing the complexity of

the degree of coherence.

Specifically, the coherence time in the case of a nonstationary observation frame

can be enhanced by a factor of 3 and occurs at the zero intensity locations within

the focal region. Furthermore, it is shown that such an enhancement in the degree

of coherence can be controlled by the pulse evolution through the photonic crystal

fibre, in which nonlinear and dispersive effects such as the soliton fission process

provides the key phase evolution necessary for dramatically changing the coherence

time of the focused electromagnetic wave.

An extension to this theory can be developed by an investigation into vectorial

effects in polarisation, which are achieved through vectorial diffraction theory. This

theoretical treatment gives insight into the coherence fluctuations introduced by a

supercontinuum in a high numerical diffraction system. Under such conditions and

due to the increased refraction at the extremities of the lens the incident polarisation

state rotates to transfer energy from this state to the orthogonal transverse field and

the longitudinal field, which is known as depolarisation. For a supercontinuum with

a horizontal polarisation state the coherence times along the x−, y− and z−axes

are different and change with increased numerical aperture. The coherence time

for the x−axis increases with numerical aperture and the y−axis decreases with

numerical aperture, which is due to the transfer of energy because of depolarisation.

The influence of numerical aperture is evident along the optical axis (z), which

shows the most significant change in coherence time. The mean coherence time as

a function of numerical aperture decreases by an order of magnitude and is due to

the superposition conditions no longer forming points of destructive interference.

Since the field is a vector field containing three polarisation components, the

theory for the degree of coherence is extended to incorporate cross correlation effects

within these vectorial components which is calculated through a coherency matrix.

The use of this matrix provides insight into interesting correlation effects between

co-propagating vector fields such as the coupled modes (linear polarised modes)

iii

of a photonic crystal fibre. An investigation is presented on the coherence times

for the supercontinuum field generated by cross coupling into the photonic crystal

fibre. The coherence times under cross coupling conditions show that the degree of

coherence of the two coupled modes from the fibre are different, which is due to the

difference in phase associate with each mode.

The effect of temporal phase from a supercontinuum and the spatial phase

inherent from diffraction by a lens, are important to many experimental applications

of supercontinuum generation. The manifestation of these temporal and spatial

phase effects result in a modification of the focal region and the bandwidth of

the field. Applications involving supercontinuum generation must first understand

the generation of the supercontinuum and the modification imposed by the optical

system.

iv

Acknowledgements

At the start of 2006 I was given the opportunity to enrol in a PhD at Swinburne

University or the University of Newcastle. At the time, I think the major reason

for coming to Swinburne was my supervisor Dr. John Holdsworth. However, John

made it clear to me that coming to Swinburne and working for Prof. Min Gu would

not be easy, but then I suppose a PhD is never simple.

First I would like to thank my three supervisors Prof. Min Gu, Dr. James Chon

and Prof. Richard Evans for all the effort that they have put into my research

development. Prof. Min Gu’s tireless contribution has developed my ability to

conduct and convey scientific research in a professional manner. I thank Dr. James

Chon for imparting his valuable guidance and knowledge throughout my research.

To Prof. Richard Evans for scientific contribution in the initial stages and the

difficulties of my PhD.

A PhD would never run smoothly without the help of administrative staff and

technicians. I would like to thank Ms. Johanna Lamborn and Ms. Katherine Cage

with all the administrative issues associated with my research. I thank Mr. Mark

Kivinen for all the custom made opto-mechanics and the enlightening conversations

every now and then of a morning. I would like to thank Dr. Daniel Day and Dr.

Dru Morrish for their endless support and for imparting their scientific knowledge.

During my PhD I have gained a valuable group of colleagues but within that

a valuable group of friends. I would like to thank both Dr. Peter Zijlstra and

Dr. Michael Ventura for their scientific contributions in knowing what to do and

more importantly what not do during my PhD. I would also like to thank Ms.

Elisa Nicoletti and Dr. Joel Van Embden for their extensive insight, which may

v

not always be scientific but has contributed. To my colleagues in the Centre for

Micro-Photonics I would like to thank you all for providing a constructive and open

environment to conduct research.

Finally, I would like to thank my family and friends. To my parents for their

encouragement and their guidance throughout my PhD. Thanks go to my twin

brother Joel for helping me through my PhD and for finishing your PhD before me.

Most importantly I would like to thank my best friend and partner, Skye. I could

not have done my PhD without you. Thank you for understanding my complicated

mind and for all your support.

Brendan James Chick

June 3, 2010

vi

Contents

Declaration i

Abstract i

Acknowledgements v

Contents vii

List of Figures x

List of Tables xviii

1 Introductory Literature Review 1

1.1 Introduction to Supercontinuum Generation . . . . . . . . . . . . . . 2

1.1.1 Nonlinear photonic crystal fibre . . . . . . . . . . . . . . . . . 3

1.1.2 Dispersion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

1.1.3 Birefringence . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

1.1.4 Nonlinearity . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

1.1.4.1 Self phase modulation (SPM) . . . . . . . . . . . . . 10

1.1.4.2 Cross phase modulation (XPM) . . . . . . . . . . . . 10

1.1.4.3 Self steepening . . . . . . . . . . . . . . . . . . . . . 11

1.1.4.4 Raman scattering . . . . . . . . . . . . . . . . . . . . 11

1.1.5 The nonlinear Schrodinger equation . . . . . . . . . . . . . . . 12

1.1.6 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

1.2 Introduction to Diffraction Theory . . . . . . . . . . . . . . . . . . . 16

1.2.1 Fresnel Diffraction . . . . . . . . . . . . . . . . . . . . . . . . 17

1.2.2 Vectorial Diffraction . . . . . . . . . . . . . . . . . . . . . . . 18

1.3 Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

vii

1.4 This thesis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

2 Theory 27

2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

2.2 Nonlinear pulse propagation . . . . . . . . . . . . . . . . . . . . . . . 27

2.2.1 Maxwell’s equations . . . . . . . . . . . . . . . . . . . . . . . 27

2.2.2 Slow varying envelope equation . . . . . . . . . . . . . . . . . 30

2.2.3 Optical properties of photonic crystal fibre . . . . . . . . . . . 31

2.2.4 Nonlinear Schrodinger equation . . . . . . . . . . . . . . . . . 32

2.2.5 Coupled mode nonlinear Schrodinger equation . . . . . . . . . 35

2.3 Diffraction theory: low numerical aperture . . . . . . . . . . . . . . . 38

2.3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38

2.3.2 Huygen-Fresnel principle . . . . . . . . . . . . . . . . . . . . . 38

2.3.3 Fresnel approximation . . . . . . . . . . . . . . . . . . . . . . 38

2.3.4 Fresnel diffraction by a circular lens . . . . . . . . . . . . . . . 40

2.4 Diffraction theory: high numerical aperture . . . . . . . . . . . . . . . 41

2.4.1 The Debye integral . . . . . . . . . . . . . . . . . . . . . . . . 41

2.4.2 Evaluation of the vectorial diffraction formula . . . . . . . . . 42

2.5 Coherence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45

3 Pulse Propagation in Nonlinear Photonic Crystal Fibre 47

3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47

3.2 Photonic crystal fibre characteristics . . . . . . . . . . . . . . . . . . 48

3.3 Nonlinear and dispersion effects . . . . . . . . . . . . . . . . . . . . . 50

3.4 Supercontinuum generation . . . . . . . . . . . . . . . . . . . . . . . 57

3.4.1 Experimental study . . . . . . . . . . . . . . . . . . . . . . . . 59

3.5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61

4 Fresnel Diffraction 65

4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65

4.2 Numerical methodology . . . . . . . . . . . . . . . . . . . . . . . . . 66

4.3 Ultrashort hyperbolic secant pulse . . . . . . . . . . . . . . . . . . . . 68

4.4 Nonlinear and dispersive phase . . . . . . . . . . . . . . . . . . . . . 70

4.5 Supercontinuum generation . . . . . . . . . . . . . . . . . . . . . . . 75

viii

4.6 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83

5 Vectorial Diffraction 85

5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85

5.2 Three-dimesional coherence matrix . . . . . . . . . . . . . . . . . . . 86

5.3 Vectorial diffraction of a supercontinuum . . . . . . . . . . . . . . . . 93

5.3.1 Linear Polarisation . . . . . . . . . . . . . . . . . . . . . . . . 93

5.3.2 Coupled mode propagation . . . . . . . . . . . . . . . . . . . . 97

5.4 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100

6 Conclusion 103

6.1 Thesis conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103

6.2 Future work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105

Bibliography 107

Appendix A: Numerical Code for the CMNLS A–1

A.1 Split step Fourier method . . . . . . . . . . . . . . . . . . . . . . . . A–1

A.2 Matlab Script . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A–2

Appendix B: Numerical Code for Diffraction Theory B–1

B.3 Diffraction theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . B–1

B.4 Matlab Script - Scalar diffraction . . . . . . . . . . . . . . . . . . . . B–1

B.5 Matlab Script - Scalar diffraction . . . . . . . . . . . . . . . . . . . . B–2

Appendix C: Supplementary movies C–1

Author’s Publications

ix

List of Figures

1.1.1 Illustration depicting the structure of a PCF. Λ is the pitch or

periodicity, d is the hole diameter and φ is the core diameter. . . . . . 3

1.1.2 A comparison between the effective refractive index of two PCF’s

with different d to Λ ratios and the material dispersion of fused silica.

The parameters in the calculation were d = 0.6 µm and Λ = 1.2 µm

(dashed line), d = 1 µm and Λ = 1.2 µm (dot dashed line). . . . . . . 5

1.1.3 A comparison between the dispersion of two PCF’s with different d to

Λ ratios and the material dispersion of fused silica. The parameters

in the calculation were d = 0.6 µm and Λ = 1.2 µm (dashed line),

d = 1 µm and Λ = 1.2 µm (dot dashed line). . . . . . . . . . . . . . . 6

1.1.4 The effect of β2 dispersion on an optical pulse after propagation for

10 m. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

1.1.5 The effect of β3 on an ultrashort pulse propagating along a 3 m PCF.

Except for where specified the coefficient β3 = 5 × 10−3 ps3/m. . . . . 8

1.1.6 Illustration depicting the introduction of birefringence into a PCF

structure. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

1.1.7 Raman response function modelled using Eqs. (1.1.8), (1.1.9) and

(1.1.10). (a) Raman transfer function, (b) The real (R(ω)) and

imaginary (S(ω)) components of the Raman response function in the

frequency domain. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

1.2.1 Diffraction by a lens of NA = 0.1 for an incident wavefront with

λ = 0.78 µm. S/S0 is the normalised intensity, z and r are the axial

and radial dimensions, respectively. . . . . . . . . . . . . . . . . . . . 17

1.2.2 The geometric illistration of vectorial diffraction53 of a incident

electric field (Eix) in the x direction. Ei

r and Eiφ are the polar

components of Eix.

53 . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

x

1.2.3 Diffraction by a lens of NA = 1 for an incident wavefront with λ =

0.78 µm. (a) the xy plane; (b) the xz plane and (c) the yz plane. . . 21

2.2.1 The formation of the third order soliton. The parameters in the

simulation were (a) β2 = −0.005 ps2/m and γ = 0.095 W/m; (b)

β2 = −0.005 ps2/m, β3 = 5 × 10−4 ps3/m and γ = 0.095 W/m; and

(c) β2 = −0.005 ps2/m, γ = 0.095 W/m and R (t) determined by Eq.

(1.1.7). All other terms were neglected. . . . . . . . . . . . . . . . . . 34

2.2.2 Ultrashort (∆t = 0.05 ps) pulse propagation using the CMNLSE.

(a) y−polarised mode and (b) the x−polarised mode. The pa-

rameters used in the simulation were βj2 = −0.005 ps2/m, βk2 =

−0.005 ps2/m, γ = 0.095 W/m and a ∆β1 = 0 ps/m. . . . . . . . . . 37

2.2.3 Ultrashort (∆t = 0.05 ps) pulse propagation using the CMNLSE.

(a) y−polarised mode and (b) the x−polarised mode. The pa-

rameters used in the simulation were βj2 = −0.005 ps2/m, βk2 =

−0.005 ps2/m, γ = 0.095 W/m and a ∆β1 = −2 ps/m. . . . . . . . . 37

2.3.1 Illustration of mutual interference caused by the superposition of the

primary wavefront and secondary spherical waves.53 . . . . . . . . . . 39

2.4.1 Illustration of the geometry of vectorial diffraction.53 . . . . . . . . . 42

3.2.1 The geometry as defined in the simulation using a refractive index

profile resolution for the PCF of 256× 256 pixels and a supercell size

10 × 10 unit cells. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48

3.2.2 The dispersion coefficients related to the mode propagation constant

β. (a) shows the first- and second-order dispersion coefficients for the

two fundamental modes. (b) shows the phase mismatch (∆β0) and

the group velocity mismatch (∆β1) between these modes. . . . . . . . 49

3.3.1 The effects of TOD originating from a PCF pumped with an ultra-

short pulse with a pulse duration of 0.1 ps and a peak power of

1000 W . (a) the time domain and (b) the frequency domain. . . . . . 50

3.3.2 The effects of RS originating from a PCF pumped with an ultra-short

pulse with a pulse duration of 0.1 ps and a power of 1000 W . (a) the

time domain and (b) the frequency domain. . . . . . . . . . . . . . . 51

xi

3.3.3 The effects of TOD originating from a PCF pumped with an ultra-

short pulse with a pulse duration of 0.1 ps and in each mode of

power of 750 W . The coupled polarisation state was 45. (a) the

time domain of the y−polarised mode (b) the frequency domain of

y−polarised mode (c) the time domain of x−polarised mode and (d)

the frequency domain of x−polarised mode. . . . . . . . . . . . . . . 52

3.3.4 The effects of RS originating from a PCF pumped with an ultra-short

pulse with a pulse duration of 0.1 ps and in each mode of power of

750 W . The coupled polarisation state was 45. (a) the time domain

of the y−polarised mode (b) the frequency domain of y−polarised

mode (c) the time domain of x−polarised mode and (d) the frequency

domain of x−polarised mode. . . . . . . . . . . . . . . . . . . . . . . 53

3.3.5 The effects of TOD originating from a PCF pumped with an ultra-

short pulse with a pulse duration of 0.1 ps and a fibre length of 0.3 m.

(a) the time domain and (b) the frequency domain. . . . . . . . . . . 54

3.3.6 The effects of RS originating from a PCF pumped with an ultra-short

pulse with a pulse duration of 0.1 ps and a fibre length of 0.3 m. (a)

the time domain and (b) the frequency domain. . . . . . . . . . . . . 54

3.3.7 The effects of TOD originating from a PCF pumped with an ultra-

short pulse with a pulse duration of 0.1 ps and a fibre length of 0.3 m.

The coupled polarisation state was 45. (a) the time domain of the

y−polarised mode (b) the frequency domain of y−polarised mode (c)

the time domain of x−polarised mode and (d) the frequency domain

of x−polarised mode. . . . . . . . . . . . . . . . . . . . . . . . . . . . 55

3.3.8 The effects of RS originating from a PCF pumped with an ultra-short

pulse with a pulse duration of 0.1 ps and a fibre length of 0.3 m.

The coupled polarisation state was 45. (a) the time domain of the

y−polarised mode (b) the frequency domain of y−polarised mode (c)

the time domain of x−polarised mode and (d) the frequency domain

of x−polarised mode. . . . . . . . . . . . . . . . . . . . . . . . . . . . 56

xii

3.4.1 Theoretically obtained spectra of propagation within a 130 mm NL-

PCF with a 87 fs pulse. Figures (a), (b) and (c) are the spectra for

the y−polarised output field with (d), (e) and (f) for the x−polarised

output field. θ is the input polarisation angle with respect to the

y−axis. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57

3.4.2 Modulation instability gain for the y− and the x−modes. . . . . . . . 58

3.4.3 Theoretically obtained spectral and temporal profile of 87 fs pulsed

propagation within a 130 mm nonlinear PCF. Figures (a), (b) and

(c) are the spectra for the y−polarised output field with (d), (e) and

(f) for the x−polarised output field. . . . . . . . . . . . . . . . . . . . 60

3.4.4 Optical arrangement used in this study. GT - Glan Tomson, WP -

Wave Plate, Spec - Spectrograph and SA - Spectrum Anaylser . . . . 61

3.4.5 Spectral properties of the polarised modes of the nonlinear PCF.

The perpendicular (blue) and parallel polarised (red) states are with

reference to the output orientation of the laser. . . . . . . . . . . . . 62

3.4.6 Degree of polarisation for the fast and the slow axes of the fibre. . . . 63

4.2.1 An illustration of pulse diffraction by a low numerical aperture (NA)

lens. (a) shows how the path length and the NA affect the pulse

distribution as the temporal envelope passes through the focus. (b)

shows the observation frames of the intensity profile in the focus. . . . 67

4.3.1 The temporal effects of a focused hyperbolic secant ultrashort pulse

propagating through the focus of a low NA (0.1) objective. (a) On

axis diffraction centred at the focal point (the full temporal evolution

of the hyperbolic secant on the axis is described in Appendix C). (b)

On axis diffraction centred at u0 = 5π. (c) Radial and axial diffraction

pattern centred at the focal point (the full temporal evolution of

the hyperbolic in the radial and axial direction is described in

Appendix C). (d) The intensity matrix used to obtain the temporal

and axial intensity information for the stationary and nonstationary

observation frames. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68

xiii

4.3.2 The coherence time of a focused hyperbolic secant ultrashort pulse

for the stationary and the non-stationary cases. (a) Axial and

radial distribution of the coherence time for the 0.1 NA lens for the

stationary case; (b) Axial and radial distribution of the coherence

time for the 0.1 NA lens for the non-stationary case; (c) Effect of NA

on the coherence time on the axis for the stationary case; and (d)

Effect of NA on the coherence time on the axis for the non-stationary

case. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69

4.3.3 The coherence time illustrating the effect of the variation in temporal

phase through the addition of chirp through the chirp parameter

C(ps2) for the stationary (a) and nonstationary (b) observation

frames. τ0 is the initial coherence time before the objective. . . . . . . 71

4.3.4 Mean frequency distribution of a focused hyperbolic secant ultrashort

pulse in the axial and radial plane of a 0.1 NA lens for stationary (a)

and non-stationary (b) cases. . . . . . . . . . . . . . . . . . . . . . . 72

4.4.1 The effects of RS on the coherence time for a focused electromagnetic

wave by a lens of NA = 0.1, originating from a PCF pumped with

an ultra-short pulse with a pulse duration of 0.1 ps and a power of

1000 W (relating to the field in Fig. 3.3.2). (a) stationary observation

frame and (b) a nonstationary observation frame. . . . . . . . . . . . 72

4.4.2 The effects of TOD on the coherence time for a focused electromag-

netic wave by a lens of NA = 0.1, originating from a PCF pumped

with an ultra-short pulse with a pulse duration of 0.1 ps and a power of

1000 W (relating to the field in Fig. 3.3.1). (a) stationary observation

frame and (b) a nonstationary observation frame. . . . . . . . . . . . 73

4.4.3 The effects of RS on the coherence time for a focused electromagnetic

wave by a lens of NA = 0.1, originating from a PCF pumped with an

ultra-short pulse with a pulse duration of 0.1 ps and a fibre length of

0.3 m (relating to the field in Fig. 3.3.6). (a) stationary observation

frame and (b) a nonstationary observation frame. . . . . . . . . . . . 74

xiv

4.4.4 The effects of TOD on the coherence time for a focused electromag-

netic wave by a lens of NA = 0.1, originating from a PCF pumped

with an ultra-short pulse with a pulse duration of 0.1 ps and a fibre

length of 0.3 m (relating to the field in Fig. 3.3.5). (a) stationary

observation frame and (b) a nonstationary observation frame. . . . . 74

4.5.1 The temporal effects of a SC propagating through the focus of a

low NA (0.1) objective. (a) On axis diffraction centred at the focal

point (the full temporal evolution of the SC on the axis is described

in Appendix C). (b) On axis diffraction centred at u0 = 5π. (c)

Radial and axial diffraction pattern centred at the focal point (the

full temporal evolution of the SC in the radial and axial direction

is described in Appendix C). (d) Complete axial and temporal

diffraction field. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75

4.5.2 The coherence time within a focused SC for the stationary and the

non-stationary cases. (a) the axial and radial distribution of the

coherence time for the 0.1 NA lens for the stationary case; (b) the

axial and radial distribution of the coherence time for the 0.1 NA

lens for the non-stationary case; (c) the effect of NA on the coherence

time on the axis for the stationary case; (d) the effect of NA on the

coherence time on the axis for the non-stationary case. . . . . . . . . 77

4.5.3 Propagation of an ultrashort hyperbolic secant pulse through a

nonlinear PCF. (a) field propagation as a function of fibre length;

(b) coherence time for the stationary observation frame in the focal

region of a 0.1 NA lens for different length fibre and (c) coherence time

for the nonstationary observation frame in the focal region of a 0.1 NA

lens for different length fibre. The peak input power to the photonic

crystal fibre is 2500 W with a pulse duration of 100 fs.(1) represents

the cross section used for Fig. 4.5.2c (blue) and (2) represents the

cross section used for Fig. 4.5.2d (blue) . . . . . . . . . . . . . . . . . 79

xv

4.5.4 Propagation of an ultrashort hyperbolic secant pulse through a

nonlinear PCF. (a) variation of output temporal envelope by varying

the input power. (b) the coherence time of the stationary observation

frame of the focal region of a 0.1 NA lens for different for the field

obtained from different input powers. (c) the coherence time of the

nonstationary observation frame of the focal region of a 0.1 NA lens

for different for the field obtained from different input powers. . . . . 80

4.5.5 Mean frequency distribution of the focused SC in the axial and radial

plane of a 0.1 NA lens for stationary (a) and non-stationary (b) cases. 82

5.2.1 A comparison between the coherence times for a lens of NA = 1 and

0.1 with hyperbolic secant ultrashort pulse with a width of 0.1 ps. . . 92

5.3.1 The coherence time of the diffraction by a lens of varying numerical

aperture along the x (a), y (b) and z (c) axes. These coherence

times are calculated for the autocorrelation of the electric field in the

direction of the Ei (Ex). (d) the coherence times for the diffraction by

a lens of NA = 1 along the x axis, which contains the autocorrelation

and cross-correlation coherence times with respect to the Ex and Ez

fields. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94

5.3.2 The mean coherence time of a SC as a function of NA for the x, y

and z−axes. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95

5.3.3 The coherence time of the autocorrelation of the diffraction by a lens

of NA = 1 the electric field Ex with variation in the fibre length. . . . 96

5.3.4 The power dependence of coherence time in the focus of a NA = 1

lens for input fields generated by the nonlinear PCF of varying input

power. The coherence time is for a linear polarised field orientated

along the x direction. . . . . . . . . . . . . . . . . . . . . . . . . . . . 96

5.3.5 The PCF output field for an incident polarisation state at 45.(a)

the horizontal (x) polarisation state, (b) the vertical (y) polarisation

state, (c) the horizontal (x) polarisation state as a function of fibre

length, and (c) the vertical (y) polarisation state as a function of fibre

length. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97

xvi

5.3.6 The coherence time for the autocorrelations and cross correlations

calculated for the diffraction by a lens of NA = 1 along the optical

axis for the SC field generated in Fig 5.3.5. . . . . . . . . . . . . . . . 98

5.3.7 The coherence time for the autocorrelations and cross correlations

calculated for the diffraction by a lens of NA = 1 as a function of

fibre length along the optical axis. (a) coherence time produced

by the autocorrelation of Ex; (b) coherence time produced by the

cross correlation Ex and Ey; and (c) coherence time produced by the

autocorrelation of Ey. . . . . . . . . . . . . . . . . . . . . . . . . . . . 99

5.3.8 The power dependence of coherence time in the focus of a NA =

1 lens for input fields generated by the nonlinear PCF of varying

input power. The coherence matrix is for a linear polarised field

orientated at 45 to the x direction. (a) coherence time produced

by the autocorrelation of Ex; (b) coherence time produced by the

cross correlation Ex and Ey; and (c) coherence time produced by the

autocorrelation of Ey. . . . . . . . . . . . . . . . . . . . . . . . . . . . 101

xvii

List of Tables

1.1.1 Parameters for the intermediate broadening model35 . . . . . . . . . . 13

3.2.1 Dispersion data for the polarised mode of the nonlinear fibre. . . . . . 49

5.2.1 Contributions to the field E for the x− y− and z−axes. . . . . . . . . 88

xviii

Chapter 1

Introductory Literature Review

At the forefront in the development of optical devices is the ability to generate,

propagate and process radiation, which is highlighted in the result of last years Nobel

prize with Kao awarded half the prize for his work in fibre optic communications.

Kao showed that by producing a high glass purity, fibre losses could be minimised

and allow the propagation of light to approach 100 km’s.1,2 These general principles

of generation, propagation and processing in applying radiation can be seen in all

photonic applications.

The advancement made by Kao could be regarded as the foundation for

the motivation of current fibre optic technology. The more recent technological

advancement in optical fibre technology has led to the use of a photonic crystal

structure to enhance the fibre guidance properties which leads to a modification of

the optical properties such as dispersion. The idea in the use of the photonic crystal

structure in the cladding of the fibre, known as a photonic crystal fibre (PCF)3

has led to many changes in optical properties and in essence is similar to Kao in

providing the ability to enhance propagation. The PCF has led to the improvement

and generation of many optical applications and none more fascinating than the

supercontinuum (SC).

SC generation is a remarkable source of radiation formed by the construction

of broadband light from the severity of nonlinear optical processes, which has

extraordinary temporal and spectral features. SC generation is an ideal source

1

Chapter 1

for many applications and it provides two key photonic capabilities; optical

imaging using many microscopy techniques simultaneously for biophotonic and

biomedical applications; optical processing information on a tremendous scale since

its bandwidth could provide multiple channels to carry information, leading to the

realisation of all-optical computing.

The realisation of the applications described involve the propagation and control

of light on the microscale and a method for the control is the use of diffraction. For

SC applications predominantly involving diffraction and interference, it is imperative

that a fundamental treatment should be constructed in these optical phenomena.

This chapter is divided into two major sections: an introductory literature review

of SC generation and an introductory literature review of diffraction by a lens.

The review on SC generation highlights the optical properties, which generates its

spectral and temporal characteristics and makes SC generation a desirable form of

radiation. The review is then turned toward diffraction to present the reasons why

diffraction has been a heavily investigated optical phenomenon. Finally, we highlight

the necessity for a fundamental study into the diffraction of complex light such as a

SC.

1.1 Introduction to Supercontinuum Generation

SC generation in microstructured optical fibre, also known as PCF4–6 has become

an imperative instrument for many applications.7–12 SC is generated in PCF by the

interplay between nonlinearity induced by the Kerr effect originating from the tight

modal confinement and tailored modal dispersion. SC generation caused by this

engineered dispersion and nonlinearity can be easily demonstrated with high power

continuous wave lasers or pulsed laser systems.

This section provides an overview of the design of PCFs and its particular optical

properties which generate the SC. The discussion involves the dispersion properties

which are important for polarisation maintaining PCFs, and the nonlinear properties

such as self phase modulation, cross phase modulation, self steepening and Raman

2

Chapter 1

Λ

d

φ

Figure 1.1.1 Illustration depicting the structure of a PCF. Λ is the pitch or periodicity, d is the holediameter and φ is the core diameter.

scattering.

1.1.1 Nonlinear photonic crystal fibre

PCFs are a type of optical fibre which have a periodic micro-structure of air holes

within the cladding to reduce the refractive index. The design freedom of PCFs allow

the tuning of the refractive index in the cladding, which alters the effective refractive

index of the guided mode changing its dispersion profile. The basic geometry of a

PCF is shown in Fig. 1.1.1.

A defect is placed at the centre of the crystal to form a waveguide which

guides under the conditions of total internal reflection. To determine the optical

characteristics of the PCF structure, the mode profile can be solved using algorithms

such as the finite difference method13, plane wave theory14–16 or the multipole

method.17–20 By solving for the mode, the solution gives information about the

effective refractive index and the effective modal area, which is used to determine

the dispersion and nonlinear characteristics, respectively.

Three structural properties of PCFs allow the modification of dispersion and

3

Chapter 1

nonlinearity; the air-fill fraction, a one-dimensional asymmetry and the core size.

Dispersion engineering by the manipulation of the air hole structure of a PCF21–23

allows the guidance of the optical wave to be influenced by different dispersion

effects. The development of PCFs has become widely used due to the pioneering

work by Russell and co-workers in 1996.3 The results showed a single mode core

optical fibre could be achieved for a wavelength region of 458 - 1550 nm. In

1997 Birks et al. extended their work to create the endlessly single mode PCF.21

Furthermore Mogilevtsev et al.24 showed that by tuning the PC structure the group

velocity could be altered to shift the zero dispersion wavelength (ZDW) to below

1.27 µm.24 These developments have led to an extensive range of PCFs with different

guidance properties.

The SC in PCFs was first demonstrated in 2000 by Ranka et al.5 Experimentally

it was presented that the modification of the photonic crystal lattice can shift the

dispersion profile into the visible wavelength region, creating the ability to couple

an ultrashort laser pulse within the anomalous dispersion region (the increase in

refractive index with wavelength). By doing so, this coupling enhances the nonlinear

effects which were relatively weak in the standard silica fibres. This research has

led to extensive studies on how the PCF structure influences the generation of a SC

both experimentally and theoretically.

1.1.2 Dispersion

The dispersion properties of a PCF are due to two components; the material and

the modal properties. The effect of material dispersion comes from the material’s

frequency dependent refractive index and is well approximated using the Sellmier

equation,25 which is given by

n2 (ω) = 1 +3

∑

j=1

Bjω2j

ω2j − ω2

, (1.1.1)

where ω is the frequency, ωj and Bj are the frequency and the oscillator strength of

the jth material resonance.

4

Chapter 1

0.4 0.8 1.2 1.61.25

1.3

1.35

1.4

1.45

1.5

Wavelength (µm)

n eff

d/Λ = 0.5

d/Λ = 0.83

Fused Silica

Figure 1.1.2 A comparison between the effective refractive index of two PCF’s with different d to Λratios and the material dispersion of fused silica. The parameters in the calculation were d = 0.6 µmand Λ = 1.2 µm (dashed line), d = 1 µm and Λ = 1.2 µm (dot dashed line).

The effective refractive index produced by the waveguide must be added to the

material contribution to obtain the dispersion properties. For standard optical fibres

the contribution from the waveguide effective refractive index is small. However for

a PCF the contribution from the waveguide is increased by the addition of air holes

in the cladding which reduces the effective refractive index. It is this property of

PCFs that allows the tailoring of the dispersion profile by modification of the air

hole diameter (d) and the pitch (Λ) of the photonic crystal. Figure 1.1.2 shows the

effect of changing the d to Λ ratio on the effective refractive index of a PCF and a

comparison to the material refractive index of silica.

The dispersion by an optical waveguide is determined by the mode propagation

constant β(ω) which is calculated by

β (ω) = n (ω)ω

c, (1.1.2)

where c is the speed of light. The two most dominant dispersion properties in

the pulse propagation are the 2nd and the 3rd order dispersion effects which can

be determined by expanding Eq. (1.1.2) into a high order Taylor series about the

5

Chapter 1

0.4 0.8 1.2 1.6−200

−100

0

100

Wavelength (µm)

β 2 (ps

2 km−

1 )

d/Λ = 0.5

d/Λ = 0.83

Fused Silica

Figure 1.1.3 A comparison between the dispersion of two PCF’s with different d to Λ ratios and thematerial dispersion of fused silica. The parameters in the calculation were d = 0.6 µm and Λ = 1.2 µm(dashed line), d = 1 µm and Λ = 1.2 µm (dot dashed line).

carrier frequency ω0, which is given by25

βj (ω) = β0j + β1j (ω − ω0) +β2j

2!(ω − ω0)

2 +β3j

3!(ω − ω0)

3 · · · , (1.1.3)

where j refers to the particular mode.

An important parameter for the generation of a SC is the second order dispersion

term (β2) as it determines the regime the pulse propagates within. For the case of

β2 > 0 the dispersion is in the normal dispersion region where the ‘red’ frequencies

travel faster then the ‘blue’ frequencies, and for the regime where β2 < 0 the

dispersion is in the anomalous dispersion region. The dispersion profile, in particular

the slope of β2, can be tailored by changing the PCF structure. Figure 1.1.3 shows

a comparison between the β2 profiles for two PCFs and the material dispersion of

fused silica. The balance of nonlinearity and dispersion, in particular the second

order term determines the pulse evolution and ultimately the effects which govern

SC generation.

The β2 term acts to broaden or compress an optical pulse depending which

regime the pulse is propagating within. Figure 1.1.4 shows the broadening of a

0.1 ps ultrashort pulse when affected by varying amounts of β2.

6

Chapter 1

−3 −1.5 0 1.5 30

0.25

0.5

0.75

1

Time (ps)

Inte

nsity

(W

)

β2 = 0 (ps2/m)

β2 = 0.005 (ps2/m)

β2 = 0.01 (ps2/m)

Figure 1.1.4 The effect of β2 dispersion on an optical pulse after propagation for 10 m.

The third order dispersion term β3 is an important parameter as it provides

the dominant dispersive component when considering the optical pulses in the

femtosecond regime and pumping near the ZDW. Figure 1.1.5 shows the effects

of β3 in the absence and presence of β2. This term places an asymmetric phase shift

on the pulse and becomes increasingly dominant with fibre length.

1.1.3 Birefringence

All optical fibres, whether they are single-mode or multi-mode support two

orthogonal linearly polarised (LP) modes for the same spatial modal distribution,

which is called birefringence. For standard single mode fibre the difference between

these LP modes is small and they are said to exhibit weak birefringence. A pivotal

development in SC generation was the use of highly birefringent PCF. When an

asymmetry is incorporated into an optical fibre, there is a difference between the

modal effective refractive indices for the two LP modes. The difference in the

modes can be exploited to maintain the incident polarisation direction. In 2005, Zhu

and Brown presented experimental and theoretical studies26,27 on the polarisation

properties of SC generation in a birefringent PCF, or also known as polarisation

maintaining PCF. It was shown that the polarisation state could be maintained

for polarised pulsed coupling aligned with a birefrigent axis of the fibre and was

7

Chapter 1

−1.5 −0.75 0 0.75 1.50

0.25

0.5

0.75

1

Time (ps)

Inte

nsity

(W

)

β2 = 0,β

3 = 0

β2 = 0

β2 = 0.02 (ps2/m)

Figure 1.1.5 The effect of β3 on an ultrashort pulse propagating along a 3 m PCF. Except for wherespecified the coefficient β3 = 5 × 10−3 ps3/m.

otherwise elliptically polarised.

The birefringence of an optical fibre can be calculated by

B = |nxeff − nyeff | =|βj − βk|

k0

, (1.1.4)

where x and y represent the two LP modes.

The difference between these two polarised modes can be enhanced by the

inclusion of an asymmetry in the photonic crystal structure. This is achieved by

modifying two holes either side of the core region or by making all the holes of the

photonic crystal slightly elliptical which is illustrated in Fig.1.1.6. Mathematically,

the birefringence of the optical wave guide creates a mismatch between the two

polarised fundamental modes propagation constants, which is given by

∆β (ω) = ∆β0 + ∆β1 (ω − ω0) +∆β2

2!(ω − ω0)

2 +∆β3

3!(ω − ω0)

3 · · · , (1.1.5)

where the 0 and 1st order coefficients represent the phase mismatch and group

velocity mismatch, respectively.

8

Chapter 1

Λ

d

φ

Λ

d

φ

Figure 1.1.6 Illustration depicting the introduction of birefringence into a PCF structure.

1.1.4 Nonlinearity

Along with the guidance properties of modal dispersion, nonlinearity plays an

important role in the generation of SC spectra. In 1999, Broderick et al. reported

that the effective modal area enhances the effective nonlinearity of an optical fibre.28

This enhancement leads to a range of nonlinear processes which combine with

dispersion to form the complex temporal spectral features that are seen in a SC

today. The nonlinearity in a PCF is due to the Kerr effect, which is the refractive

indices dependence on intensity and is determined by the following equation

n (ω) = n0 (ω) + n2 (ω) I, (1.1.6)

where n0 (ω) and n2 (ω) are the linear and nonlinear refractive indices, respectively

and I is the intensity.

To induce a nonlinear response from bulk silica, high peak intensity laser sources

are required. There exists a temporal and spatial requirement for the confinement

of the propagating light to be able to reach such high intensities. In PCF, this can

be achieved by reducing the core size of the structure. The nonlinear response is

quantified through the nonlinear coefficient γ which is given by28,29

γ =n2 (ω0)ω0

cAeff (ω0), (1.1.7)

9

Chapter 1

where Aeff (ω0) is the effective modal area of the PCF. From a numerical simulation

perspective, the nonlinear coefficient controls the strength of nonlinearity achievable

within the PCF. Optical nonlinearity is composed of an expanse of effects, with the

most relevant of these processes being self phase modulation (SPM), cross phase

modulation (XPM), self steeping and stimulated Raman scattering.

1.1.4.1 Self phase modulation (SPM)

An increase in the intensity modulates the refractive index, which changes the phase

on the field and is known as SPM. The first report of SPM in optical fibres was by

Stolen in 197829; he showed that with a single-mode silica fibre the pulse from a

mode-locked argon laser could produce defined frequency shifts, which were in good

agreement with theoretical predictions of SPM.

SPM is an important process in SC generation as it balances the chirp from β2

which assists in the formation of non dispersive waves known as optical solitons. The

formation and optical properties of solitons have been studied extensively in optical

communications as they provide a means of sending information unperturbed by

dispersion.

1.1.4.2 Cross phase modulation (XPM)

As described earlier (Section 1.1.3) there exists two fundamental LP modes in a

PCF. When light is coupled into a PCF in such a way that a portion of the power

is propagating in both LP modes, power from one mode can transfer to the other

LP mode which modulates the effective refractive index of that mode. This effect

is called XPM which is an important process as it can be used as a means of

introducing gain to a co-propagating wave and is not necessarily restricted to the

two fundamental LP modes in the case of multimode propagation. However, XPM

in the situation of low birefringent PCF can degrade the degree of polarisation of a

mode. The first report of XPM in silica fibre was shown by Chraplyvy and Stone in

198430 and a theoretical description was described by Agrawal. G.P. in 1987.31

10

Chapter 1

1.1.4.3 Self steepening

In the femtosecond regime third order nonlinear effects must be taken into

account. The two major third order nonlinear effects that induce phase shifts

onto an ultrashort pulse are self steepening and stimulated Raman scattering. Self

steepening is an asymmetric nonlinear phase shift which acts upon the effects of

SPM and XPM and was first described by De Martini et al. in 1967.32 Since the

speed of the optical pulse varies across its envelope due to the intensity dependent

refractive index, the trailing edge of the pulse steepens as it catches up with the

peak component of the pulse. The effects seen from self steepening is a gradual

asymmetric temporal shift across the pulse caused by the temporal gradient of the

intensity of the pulse.

1.1.4.4 Raman scattering

Raman scattering is the process where photons of a given energy are scattered by

a molecule and form a photon of lower energy. The process of Raman scattering

is an important optical property as it can transfer energy from a pump beam to a

Stokes beam. For silica the molecular vibration levels span a wide frequency range

and therefore the gain in the stokes beam occurs over a wide frequency range. The

Raman gain for fused silica has been extensively studied with the most significant

contributions to the field produced by Stolen et al. in 1989 on the Raman response

function in silica fibres33 and a theoretical description in optical fibres by Blow and

Wood in 1989.34

When simulating pulse propagation in silica optical fibres either the exper-

imentally measured response function or the theoretical approximate response

functions are used. The theoretical approximation used in this thesis was developed

by Hollenbeck and Cantrell 35 which uses a multiple-vibration-mode model and

is repeated here for completeness. The Raman response function using the

intermediate broadening model for silica optical fibres is given by

11

Chapter 1

0 0.5 1 1.5 2−2

−1

0

1

2

3

4

5

Time (ps)

Am

plitu

de (

A.U

.)

0 10 20 30 40 50−6

−4

−2

0

2

4

6

Frequency shift (THz)

Am

plitu

de (

A.U

.)

S(ω)R(ω)

a b

Figure 1.1.7 Raman response function modelled using Eqs. (1.1.8), (1.1.9) and (1.1.10). (a) Ramantransfer function, (b) The real (R(ω)) and imaginary (S(ω)) components of the Raman responsefunction in the frequency domain.

hR (t) =13

∑

i=1

A′i

ωv,ie−iγite−Γ2

it2/4 sin (ωv,it) Θ (t) , (1.1.8)

where the parameters related to the intermediate broadening model are shown in

Table 1.1.1 and Θ (t) is a unit step function. The Raman response function is shown

in Fig. 1.1.7. The Fourier transform of the Raman response function determines

the real (R(ω)) and imaginary (S(ω)) frequency components, where the imaginary

component is related to the Raman gain bandwidth. These two components are

shown in Fig. 1.1.7b and are described by the following equations35

S (ω) =13

∑

l=1

Al2

∫ ∞

0

(cos [(ωv,l − ω) t] − cos [(ωv,l + ω) t]) e−iγlte−Γ2lt2/4dt, (1.1.9)

R (ω) =13

∑

l=1

Al2

∫ ∞

0

(sin [(ωv,l − ω) t] + sin [(ωv,l + ω) t]) e−iγlte−Γ2lt2/4dt. (1.1.10)

1.1.5 The nonlinear Schrodinger equation

The theoretical description of pulse propagation has been understood for decades

and begins with the formulation of Maxwell’s equations to form what is called the

nonlinear Schrodinger equation (NLSE). Pulse propagation in fibre waveguides owes

12

Chapter 1

Table 1.1.1 Parameters for the intermediate broadening model35

Mode Ai ωv,l γi Γi# [TRads−1] [TRads−1] [TRads−1]

1 1 10.6029 1.6371 4.91032 11.4 18.8496 3.6578 10.40683 36.67 43.5896 5.4975 16.49344 67.67 68.3296 5.1054 15.31535 74 87.2734 4.2515 12.75466 4.5 93.6823 0.7700 2.30917 6.8 115.2650 1.3034 3.91138 4.6 130.3767 4.8698 14.60849 4.2 149.6033 1.8689 5.607710 4.5 157.4880 2.0197 6.060111 2.7 175.3009 4.7124 14.137212 3.1 203.5752 2.8585 8.576513 3 157.4880 5.0262 15.0796

Ai is the peak intensity of the mode, ωv,l is the frequencyof the component position, Γi is the Gaussian full widthat half maximum of the mode and γi is the Lorentzian fullwidth at half maximum of the mode

13

Chapter 1

its development to Agrawal as his texts25 have laid the foundation to the modern

interpretation of such formulation. The initial use of the NLSE for SC generation

was investigated by Husakou and Herrmann in the develop of an understanding of

soliton fission dynamics leading to SC generation.36–38 The formulation used in these

studies neglects Raman scattering, which as shown later, is a dominant nonlinear

process which influences the pulse propagation leading to SC generation.

The most significant contribution to the theoretical description of the NLSE

was developed by Kodama and Hasegawa.39 This derivation incorporated higher

order dispersion effects and nonlinar effects such as Raman scattering. A significant

contribution by Blow and Wood in 1989 was the description of a wave equation

for the modelling of transient stimulated Raman scattering.34 The model also

incorporated a method of numerical integration for the nonlinear response which has

become commonplace. Although the mathematical treatment as described in the

nonlinear integration formulation has been misprinted, this description has formed a

basis for the theoretical treatment of pulse propagation in optical fibres (a correction

to this formulation is provided by Cristiani et al. in 200440). A more accurate form

of the wave equation incorporating stimulated Raman scattering was provided later

by Mamyshev and Chernikov in 1990.41

The validity of the NLSE is limited since the theory involves a slow varying

envelope approximation (Section 2.2.2). The limit of this theory occurs as the

pulse width approaches the oscillation period of the carrier wave (sub-cycle regime).

Extensions to the slow varying envelope equation have been derived to extend the

NLSE into sub-cycle regime.42,43

The theoretical description in this thesis is based on the derivation provided

by Agrawal 25 in Chapter 2 with the description of stimulated Raman scattering as

provided by Hollenbeck and Cantrell in Section 1.1.4.4.

1.1.6 Discussion

As described in the preceding sections, SC generation is the formation of broadband

light from the involvement of all the discussed optical processes. In the anomalous

14

Chapter 1

dispersion regime, initially in the evolution of the ultrashort pulse along the PCF,

the phase shift incurred is a balance between the chirp from the influence of β2

and the chirp associated with SPM. The pulse compresses and forms a high order

soliton. Throughout the initial evolution, the soliton accumulates additional phase

and expands spectrally.

The ability to control the propagation of an ultrashort pulse was first investigated

in PCFs by Reeves et al.23 It was shown that by coupling a pulse into a PCF

within the anomalous dispersion regime that the effects from β2 could be balanced

by the chirp associated with self phase modulation (SPM). The key parameter

which determines the wavelength of the coupled laser pulse is the ZDW. The

extent of SC generation is enhanced in the anomalous dispersion regime, where

the carrier frequency of the coupled light is close to the ZDW. In this region four

wave mixing is the strongest causing strong Stokes and Anti-Stokes frequencies.44

In the femtosecond regime the influence of third order dispersion is the dominant

broadening mechanism and a high degree of curvature in dispersion is required

to enhance the third order term.45 Since the geometry of fibres with two ZDWs

inherent these characteristics, they are of particular interest in generating extensive

SC spectra.

At a particular point along the fibre, the phase accumulation makes the

high order soliton unstable and begins fissions into stable fundamental solitons.

Throughout this process the soliton sheds energy into dispersive waves which form

because the soliton is phase matched to the wave vector of the dispersive waves.

Since the intensity is now spread over several fundamental solitons which are

temporally independent, the effect of nonlinearity becomes less profound and higher

order dispersion effects begin to dominate. What forms is the red shifted radiation

from Raman solitons and a blue shifted dispersive wave46.

Highly birefringent PCF has become an important area of investigation in SC

generation as it provides a key insight into a number of dispersive and nonlinear

processes. In considering birefringence the scalar description of the NLSE is no

longer appropriate as vectorial effects caused by modal dispersion differences affect

pulse propagation47. The soliton formation and fission processes have been shown to

15

Chapter 1

be highly polarisation sensitive25. By coupling an input pulse at polarisation angles

other than the primary axes of the fibre, polarisation sensitive nonlinear effects can

be enhanced which could give more insight into the dynamics of SC generation.

The incorporation of PCFs with two ZDWs can enhance the spectral extent of

the SC. The change in slope of the second order dispersion term effectively changes

the influence of the third order dispersion, which has been shown by Gaeta to

produce ‘red’ shifted dispersive waves,48 enhancing the SC in the near infra-red

region. Experimentally, the effects of the two ZDWs have been investigated44,49–51

and confirm the results presented by Gaeta.

1.2 Introduction to Diffraction Theory

The origin of diffraction theory dates back centuries and its theoretical description

is still debated. Diffraction theory can owe its development to the pioneering

works of Huygens and Fresnel, who have constructed the basis for the optical

wave theory. The diffraction of a wave occurs when light propagating through

an aperture or around an obstruction changes its propagation direction. The effect

seen at a particular distance from the aperture is the superposition of the incident

electromagnetic waves causing interference.

The theoretical description of diffraction is summarised into two forms, Fresnel

diffraction and Fraunhofer diffraction also known as near-field and far-field diffrac-

tion, respectively. The diffraction of the electromagnetic field due to a lens is caused

by refraction where the curvature and the refractive index of the lens converge or

diverge the field. For a lens the diffraction is determined by Fresnel diffraction

of a circular aperture. This theory works well for lens focusing under low NA

conditions (lens systems with a long focal length), however for high NA (short focal

length) there exists a transfer of energy from the incident polarisation state to the

orthogonal transverse field and the longitudinal field component, which is known as

depolarisation.

This section serves as an introduction to Fresnel diffraction of a lens and the

16

Chapter 1

−300 −200 −100 0 100 200 300−15

−10

−5

0

5

10

15

z (µm)r

(µm

)

S/S

0 (no

rm. 1

0log

10)

−35

−30

−25

−20

−15

−10

−5

0

Figure 1.2.1 Diffraction by a lens of NA = 0.1 for an incident wavefront with λ = 0.78 µm. S/S0

is the normalised intensity, z and r are the axial and radial dimensions, respectively.

extension for the high NA condition known as vectorial diffraction theory. Presented

is a literature review of the research which has led to the currently accepted

diffraction theory.

1.2.1 Fresnel Diffraction

The diffraction of a polychromatic wave such as a SC wave is described by what

is known as the Huygens-Fresnel principle52 where the incoming wave produces a

set of secondary wavelets which superimpose and mutually interfere to form an Airy

pattern. Figure 1.2.1 shows the diffraction of an electromagnetic wave and what can

be seen is the diffraction shows zero intensity locations, known as singular points.

The mathematical expression for the Fresnel diffraction by a lens, under the

paraxial approximation is given by52,53

E (r, z, ω) = −iωNA2

ceik0z

∫ a

b

Ei (ω) J0 (k0rNAρ) e− 1

2ik0zNA2ρ2ρdρ, (1.2.1)

where Ei is the incident electromagnetic field upon the lens, k0 is the free space wave

number, z and r are the axial and radial dimensions, respectively. Using complex

analysis it is evident that the equation has singular points at discrete positions in

both the radial and axial directions. These points have been rigorously studied in

17

Chapter 1

the case of a generalised polychromatic wave and create spectral anomalies in the

focal region54,55. Gbur et al. showed that these singular points cause the spectrum

of a polychromatic wave to red or blue shift. This occurs because the singular point

exists in a discrete position but also changes with wavelength (frequency) implying

that the focal region has localised regions of wavelength dependent singularities.

However, these points are not limited to the focal region of a lens. They also occur

in many other optical systems involving diffraction and interference56,57. The effects

of phase singularities have been rigorously studied by Berry 58 and provide the key

initial knowledge of the effects of singularities (or Caustics).

Experimentally the effect of the singularities on the focal region have been verified

by Popescu and Dogariu in 200259, using a Michelson interferometer deviced from a

2×2 fibre coupler. A comparison is made between the light in the reference arm and

the test arm. The test arm comprises the focusing lens and a reflecting object, which

in this case is a spherical mirror. This experiment is pivotal to the understanding

of Fresnel diffraction under polychromatic wave illumination as it provides a direct

means of verifying the anomalous behaviour previously theoretically described.

1.2.2 Vectorial Diffraction

Vectorial diffraction was developed by Richards and Wolf in 1959 to describe

the changes in the distribution of the focal region of a high NA lens.60,61 The

polarisation of a field in general can be described as containing three polarised

vector field components relating to the spatial dimensions of the system, for example

the diffraction system shown in Fig. 1.2.2. The change in the focal distribution

occurs due to the phenomena known as depolarisation which is when an incident

polarisation state containing a single vector component (e.g. the electric field

vector Ex) undergoes a vector rotation which transfers energy from this state to an

orthogonal transverse field (Ey) and a longitudinal field (Ez). The validity of this

approach was further investigated by Wolf and Li in 1981 to verify such conditions

of high NA.62

Figure 1.2.2 shows the geometry of vectorial diffraction under conditions of a

18

Chapter 1

a

x(i)

y(j)

E0(i)

E0(ρ)E0(φ)

b

z(k)

bO

bC×

E0(ρ)

E0(φ)

α

ρ

θ

Figure 1.2.2 The geometric illistration of vectorial diffraction53 of a incident electric field (Eix) in

the x direction. Eir and Ei

φ are the polar components of Eix.53

19

Chapter 1

high NA. Depolarisation occurs because at the extremities of the lens there exists

an increase in refraction which causes the radial component of the incident vector

field to rotate. The transfer of energy due to depolarisation changes the effects of

singularities on the focal distribution which can be seen in Fig. 1.2.3.

The singularities that occurr in Fresnel diffraction by a low NA lens only exist

in certain directions in the focal region of the lens. Ganic et al. showed that the

spectral splitting emphasised by Gbur et al. for the low NA system54 do extend

to vectorial diffraction, but only for the particular directions and no longer existing

along the optical axis63.

In considering the diffraction by a lens, what has previously been investigated is

the effect of temporal phase associated with an ultrashort pulse.64,65 However, such

effects become more complicated when the temporal, spectral and phase complexity

of a SC is considered.

1.3 Applications

The temporal structure and the extensive bandwidth of the SC field have changed

the experimental design of applications such as optical microscopy. SC provides

multiple microscopy techniques in the one compact source and is a simple extension

to a conventional ultrafast laser and microscope. In 2004 Shi et al.12 presented

experimental evidence that SC generation could be applied to the conventional

confocal microscope and then in 2005 Isobe et al.11 was able to use the structured

temporal envelope of the SC to perform two photon microscopy.

The intuitive extension of SC generation is the application of coherent anti-Stokes

Raman scattering microscopy, where the SC field is used to pump the molecular

energy states of a chemical or biological sample. In 2003 Paulsen et al.8 presented a

study on the use of a SC and ultrashort pulse as a co-propagating pump and Stokes

beams to excite molecular vibrations of a chemical sample.

In optical coherence tomography, low coherence interferometry is used to section

biological tissue by both in vivo and in situ. The resolution of this imaging system

20

Chapter 1

−1.5 −0.75 0 0.75 1.5−1.5

−0.75

0

0.75

1.5

x (µm)

y (µ

m)

S/S

0 (10

log 10

)

−35

−30

−25

−20

−15

−10

−5

0

−3 −1.5 0 1.5 3−1.5

−0.75

0

0.75

1.5

z (µm)

x (µ

m)

S/S

0 (10

log 10

)

−30

−25

−20

−15

−10

−5

0

−3 −1.5 0 1.5 3−1.5

−0.75

0

0.75

1.5

z (µm)

y (µ

m)

S/S

0 (10

log 10

)

−35

−30

−25

−20

−15

−10

−5

0

a

b

c

Figure 1.2.3 Diffraction by a lens of NA = 1 for an incident wavefront with λ = 0.78 µm. (a) thexy plane; (b) the xz plane and (c) the yz plane.

21

Chapter 1

is limited by the bandwidth, since the resolution is inversely proportional to the

bandwidth. In 2001 Hartl et al.7 showed that by using SC generation predominantly

spanning the near infra-red region they could achieve an axial resolution of 2.5 µm.

The singular points within the diffraction plane of a lens are an important optical

property as they can be used as a signature to phase unwrap information. Typically,

the field of optical vortex metrology regards these points as an obstacle66–69 and

the main focus is their removal by adding a phase vortex map to the beam which

contains phase singularities of opposite phase. However, in 2005 by Wang et al. it

was shown that a random distribution of phase singularities could be used as method

for displacement measurement.70

Another application which would benefit from SC generation is optical data

storage where the trend is to encode data into spectral and polarisation properties

of an optical material. An example of this is the research completed by Zijstra et

al., where five-dimensional recording was achieved by surface plasmon mediation

in gold nanorods.71 The gold nanorods in this study photo-thermally melt under

single pulse laser illumination causing them to reshape, which changes their

absorption, fluorescence and polarisation characteristics. SC generation is ideal for

this application as it provides pulsed and spectral features capable of encoding these

modalities simultaneously.

It is quite clear that these applications involve diffraction and interference.

However, what is not understood is how the complexity of a broadband source

such as a SC is affected by diffraction and interference, and what is absent from the

literature is the behaviour of a SC field within the focal region.

1.4 This thesis

The major theme of the research completed in this thesis is the characterisation

of SC generation and its focal distribution. The use of SC generation has become

an important laser source because its optical features, whether it be temporal or

spectral, can be tailored to the experiment through the manipulation of dispersion

22

Chapter 1

and nonlinearity. However, the manipulation is not limited to just the fibre

characteristics and would be strongly dependent on the application where diffraction

and interference would implicate changes in spatial and temporal properties.

Although the characteristics of diffraction in an optical lens system have been studied

for a polychromatic wave such as an ultrashort pulse, no such investigation has been

completed for the complex field of a SC. It is evident that this study would have a

far outreaching influence as it has consequences that relate to photonics applications

such as microscopy.

The objectives of this thesis are to provide three main scientific contributions: the

development of a theoretical and experimental comparison into highly birefringent

PCF, an understanding of how a SC field is affected by Fresnel diffraction, and an

understanding of how a SC field is affected by vectorial diffraction.

Chapter 2 provides a detailed theoretical background to both SC generation

and diffraction by a lens. The development of the theory behind SC generation

begins with Maxwell’s equations to derive the wave equation. Two key equations

are derived to solve the modal properties and the propagation equation, where the

modal properties characterise the dispersion and the nonlinearity of the optical

fibre. These parameters are then used within the propagation equation to derive

the nonlinear Schrodinger equation (NLSE). The treatment continues to make a

modification to incorporate the two orthogonal LP modes of the PCF which leads

to the coupled mode nonlinear Schrodinger equation (CMNLSE).

The scientific contribution to SC generation is to provide a detailed description

of pulse propagation in highly birefringent PCF constructed with two ZDWs. This

is different to other previous studies in the way that the birefringence in the selected

PCF is much higher than previously experimentally and theoretically investigated.

Also, the contribution is also novel by the use of polarisation maintaining PCF with

two ZDWs.

Chapter 3 involves a theoretical and experimental study of pulse propagation in

highly birefringent PCF with two ZDWs. The CMNLSE derived in Section 2.2.5 is

applied to a commercial developed highly birefringent PCF (NLPCF-750, Crystal-

23

Chapter 1

Fibre72) which has two ZDWs. Experimentally an investigation was completed

which was verified by the theoretical observations. Soliton dynamics provides the

key characteristics which determine the temporal and spectral difference between

the linear polarised modes.

The diffraction by a lens of a field such as a SC field, is a pivotal step in

the scientific contribution, as it lays the theoretical optical foundation for the

investigations into the interaction between a SC and imaging samples. What is

required is an in depth study into the characteristics of the focal region of a

focused SC. These effects need to be quantified through scientifically appreciated

characteristics such as optical coherence. The effects of diffraction need to be

considered in both the classical optics viewpoint of Fresnel diffraction and the more

modern understanding of vectorial diffraction theory.

The next topic carried out in Chapter 4 investigates the modification of the SC

field through the Fresnel diffraction by a lens. Using the mathematical framework

described in Section 2.3.3 the focal distribution of the lens is characterised. It

is demonstrated that the spatial modification through phase modification couples

with the temporal phase of the SC field which modifies the radiation. These effects

are characterised through temporal and spatial correlations known as the degree

of coherence, which are quantified through the parameters of coherence time and

mean frequency. The coherence of the field can be observed from two viewpoints:

a stationary and a nonstationary observation frame. The coherence times are

influenced by the phase associated with the destructive interference around points

of singularity and become complicated for a nonstationary reference frame, which is

influenced by the path differences of the rays extending over the aperture.

Chapter 5 investigates the influence of diffraction by a high NA lens through

the mathematical description of vectorial diffraction developed in Section 2.4.2.

The theoretical model not only provides key insight into the effects produced by

spatial and temporal coupling between the lens and the incident field, but also

investigates the effects on the polarisation state. Since the high NA focusing

produces depolarisation, this should correlate to changes in the temporal profile of

the SC field. These effects are characterised though correlations which investigate

24

Chapter 1

polarisation coherence and are quantified through the coherency matrix. The

influence of depolarisation can be observed through the transverse and axial

directions of the focal region, where the optical axis shows the most significant

change, which is due to the superposition no longer forming points of destructive

interference.

Chapter 6 provides a summary of the conclusions drawn from the investigation.

The chapter highlights the key aspects of pulse propagation through highly

birefringent PCFs and the role of the two ZDWs and the modifications of the SC

field caused by diffraction of a lens. The implications of this study and the future

work that needs to be investigated are provided in Section 6.2.

25

Chapter 1

26

Chapter 2

Theory

2.1 Introduction

This chapter provides a basis for the theory behind nonlinear pulse propagation in

optical fibre. The theory used in this chapter was extensively presented by Agrawal

in 2002.25 The aim is to provide a concise description of how this theory has been

developed to lead to polarised pulse propagation. The outline here starts from

Maxwell’s equations to derive the wave equation for an electromagnetic field in a

dielectric waveguide. The transverse and longitudinal properties of the field are

separated to derive the formulae for the dispersion and the nonlinear parameters

which leads to what is known as the nonlinear Schrodinger equation. An extension

is then made to incorporate modal birefringence and how this relates to variations

in dispersion and nonlinearity, to derive the coupled-mode nonlinear Schrodinger

equation.

2.2 Nonlinear pulse propagation

2.2.1 Maxwell’s equations

The derivation starts from Maxwell’s equation which are given by

27

Chapter 2

∇× E = −∂B∂t, (2.2.1)

∇× H = J +∂D

∂t, (2.2.2)

∇ • D = ρ, (2.2.3)

∇ • B = 0, (2.2.4)

where E and D are the electric field intensity and density, respectively. H and B are

the magnetic intensity and density, respectively. For a dielectric material such as a

silica waveguide there are no free charges so the current density J = 0 and charge

density ρ = 0. D and E are related to E and H through the following equations

D = ǫ0E + P, (2.2.5)

B = µ0H + M, (2.2.6)

where ǫ0 and µ0 are the free space permittivity and permeability respectively. P

and M are the electric and magnetic induced polarisations. For a dielectric material

M = 0. From this point the wave equation for the electric field can be derived and

is much simpler using the Fourier transform relationship of Maxwell’s equations,

which leads to

∇× E = −iωB, (2.2.7)

∇× H = J + iωD, (2.2.8)

∇ • D = ρ, (2.2.9)

∇ • B = 0. (2.2.10)

By taking curl of Eq. (2.2.7) and using Eq. (2.2.8), the wave equation in the

frequency domain can be shown to be given by

28

Chapter 2

∇×∇× E = −ω2

c2E − µ0ω

2P. (2.2.11)

Eq. (2.2.11) can be simplified by using the vector identity

∇×∇× E = ∇ (∇ • E) −∇2E. (2.2.12)

Since ∇ • E = 0, Eq. (2.2.11) now becomes

∇2E =ω2

c2E + µ0ω

2P. (2.2.13)

P can be divided into two components, the linear contribution PL and a nonlinear

contribution PNL such that

P = PL + PNL. (2.2.14)

The linear and nonlinear components are related to the material’s susceptibilities

χ which are given by

PL (r, t) = ǫ0

∫ ∞

−∞

χ1 (t− t′)E (r, t) eiω0(t′)dt′, (2.2.15)

PNL (r, t) = ǫ0χ3E (r, t)

∫ t

−∞

R (t′) |E (r, t− t′) |2dt′, (2.2.16)

where R (t− t′) is the Raman response of fused silica which includes instantaneous

and delayed components defined by

R (t) = (1 − fR) δ (t) + fRhR (t) . (2.2.17)

Here fR is the Raman contribution and for fused silica is equal to 0.18. The

nonlinear response to the induced polarisation is treated as a small perturbation

29

Chapter 2

with the relative permittivity determined by

ǫ (ω) = 1 + χ1 + ǫNL, (2.2.18)

where ǫNL is the nonlinear permittivity. Eq. (2.2.13) can be simplified and is now

given by

∇2E + ǫ (ω)ω2

c2E = 0. (2.2.19)

2.2.2 Slow varying envelope equation

The electric field and the induced electric polarisation terms contain slow amplitude

and rapidly varying components and for this derivation are separated from the slow

varying field, which are given by

E (r, t) =1

2x

[

E (r, t) e−iω0t + · · ·]

, (2.2.20)

PL (r, t) =1

2x

[

PL (r, t) e−iω0t + · · ·]

, (2.2.21)

PNL (r, t) =1

2x

[

PNL (r, t) e−iω0t + · · ·]

. (2.2.22)

Here, x is the transverse spatial dimension of the propagating mode. The solution to

Eq. (2.2.11) depends on the transverse and the longitudinal components of the field,

and how they vary spatially and temporally. The transverse modal properties of the

field are treated to be invariant in the propagation direction (z) and are separated

from the longitudinal field components. By assuming a solution to be of the form

E (r, ω − ω0) = E (x, y)E (z, ω − ω0) eiβ0z, (2.2.23)

the solution to Eq. (2.2.19) becomes

30

Chapter 2

∂2E (x, y)

x2+∂2E (x, y)

y2+

[

ǫ (ω) k20 − β2

]

E (x, y) = 0, (2.2.24)

2iβ0∂E (z, ω − ω0)

∂z+

(

β2 − β20

)

E (z, ω − ω0) = 0, (2.2.25)

where β is the wave number of the fibre mode.

Equation (2.2.24) is used to determine the modal properties of the optical

fibre such as dispersion coefficients and the nonlinear cross section. By using the

coefficients obtained from Eq. (2.2.24) and using Eq. (2.2.25) the mathematical

formulae for nonlinear pulse propagation can be determined.

2.2.3 Optical properties of photonic crystal fibre

The dispersion and nonlinear properties of a photonic crystal fibre (PCF) are

determined by solving Eq. (2.2.24) to determine the wave number β. Solving Eq.

(2.2.24) involves an iterative procedure to determine β (ω) and the field distribution

E (x, y), which can be used to determine the effective refractive index neff (ω). A

numerical solution to Eq. (2.2.24) has been achieved by a wide range of numerical

models including the finite element method73,74, the multipole method17–20,the plane

wave expansion method14–16 and the finite difference method.13 A point to add

here is that the perturbations from the nonlinear component discussed earlier have

no effect on the modal distribution and are only considered for the longitudinal

propagating field.

The method that is used in the investigation in this thesis uses the plane wave

expansion method as this numerical method has been shown to be a fast and an

accurate method. The software package used to achieve this is RSoft Photonics

CAD Suite 8.1.0 75.

31

Chapter 2

2.2.4 Nonlinear Schrodinger equation

The wave number β (ω) can now be used to include the perturbations from the

nonlinear polarisation for which β (ω) is now given by

β (ω) = β (ω) + δβ (ω) . (2.2.26)

The change in the wave number δβ (ω) is related to the modal field and is given

by25

δβ (ω) = k0

∫∫ ∞

−∞δn|E (x, y) |2dxdy

∫∫ ∞

−∞|E (x, y) |2dxdy . (2.2.27)

The change in the effective refractive index is related to the nonlinear index by

δn (ω) = n2R (ω) |E (z, ω) |2 +iα

2k0

, (2.2.28)

where we have replaced χ3 with n2 and therefore δβ (ω) now becomes

δβ (ω) = k0γ (ω)R (ω) |E (z, ω) |2 +iα

2. (2.2.29)

The term γ (ω) is the nonlinear coefficient described by γ (ω) =n2ω0

cAeffand α (ω)

is the loss coefficient. Similarly to the expansion of the wave number in Section 1.1.2,

γ (ω) and α (ω) can also be expanded. The effective modal area Aeff is determined

by

Aeff =

(

∫∫ ∞

−∞|E (x, y) |2dxdy

)2

∫∫ ∞

−∞|E (x, y) |4dxdy . (2.2.30)

Equation (2.2.25) now can be solved where the approximation for β2 − β20 =

2β0

(

β − β0

)

is used and therefore Eq. (2.2.25) becomes

32

Chapter 2

∂E (z, ω)

∂z= i [β (ω) + δβ0 − β0]E (z, ω) . (2.2.31)

Substituting Eqs. (1.1.3) and (2.2.29) into Eq. (2.2.31) and taking the Fourier

transform leads to the generalised NLSE, which is given by

∂E (z, t)

∂z−

∑

m≥2

im+1

m!βm

∂mE (z, t)

∂tm

= iγ

(

1 + iτ0∂

∂t

)

E (z, t)

(∫ ∞

−∞

R (t′) |E (z, t− t′) |2dt′)

. (2.2.32)

The equation has been shifted in time to create a moving observation frame, also

known as a retard time given by the relation t = τ − β1z.

The important process in supercontinuum (SC) generation is the formation

of high order optical solitons which can be demonstrated with the nonlinear

Schrodinger equation (NLSE). Figure 2.2.1a shows the formation of the third order

soliton. The input optical pulse used in this model was an ultrashort hyperbolic

secant pulse (∆t = 0.05 ps) and a peak power of 200 W . The input ultrashort

pulse undergoes a transformation through phase caused by the balanced chirp

contributions from β2 and self phase modulation (SPM), which rapidly expands

the field spectrally.

The incorporation of β3 introduces a phase shift on the soliton which becomes

increasingly dominant with fibre length. The influence of β3 is to perturb the high

order soliton which introduces the nonlinear and dispersive waves resulting in effects

such as four wave mixing. Figure 2.2.1b shows the effects of β3 in the propagation

of a third order soliton.

Dispersive effects are not the only form of phase perturbations on the ultrashort

pulse. Intra-pulse Raman scattering is also a dominant effect which red shifts the

solitary waves. Figure 2.2.1c shows the effects of stimulated Raman scattering on

the formation of a soliton. As the soliton propagates it sheds energy into phase

matched dispersive waves and in shedding the energy forms a fundamental soliton.

33

Chapter 2

0 0.25 0.5 0.75 1−0.5

−0.25

0

0.25

0.5

Fibre length (m)

Tim

e (p

s)

S/S

0 (10

log 10

)

−40

−30

−20

−10

0

0 0.25 0.5 0.75 1300

325

350

375

400

425

450

Fibre length (m)F

requ

ency

(T

Hz)

S/S

0 (10

log 10

)

−40

−30

−20

−10

0

10

20

0 0.25 0.5 0.75 1−1

−0.5

0

0.5

1

Fibre length (m)

Tim

e (p

s)

S/S

0 (10

log 10

)

−40

−30

−20

−10

0

0 0.25 0.5 0.75 1300

325

350

375

400

425

450

Fibre length (m)

Fre

quen

cy (

TH

z)

S/S

0 (10

log 10

)

−40

−30

−20

−10

0

10

20

0 0.25 0.5 0.75 1−1

−0.5

0

0.5

1

Fibre length (m)

Tim

e (p

s)

S/S

0 (10

log 10

)

−40

−30

−20

−10

0

0 0.25 0.5 0.75 1300

350

400

450

500

Fibre length (m)

Fre

quen

cy (

TH

z)

S/S

0 (10

log 10

)

−40

−30

−20

−10

0

10

20

a

b

c

Figure 2.2.1 The formation of the third order soliton. The parameters in the simulation were (a)β2 = −0.005 ps2/m and γ = 0.095 W/m; (b) β2 = −0.005 ps2/m, β3 = 5 × 10−4 ps3/m andγ = 0.095 W/m; and (c) β2 = −0.005 ps2/m, γ = 0.095 W/m and R (t) determined by Eq. (1.1.7).All other terms were neglected.

34

Chapter 2

2.2.5 Coupled mode nonlinear Schrodinger equation

Similar to the procedure presented in Section 2.2.4, the coupled mode nonlinear

Schrodinger equation can be derived and as described earlier when considering

birefringent PCF there needs to be consideration for the two linearly polarised (LP)

modes. The major difference that exists between the modes is the effect of PNL.

For fused silica there are three major contributions to χ3 (since χ3 is a 4th Rank

tensor) and they are of similar strength. It can be shown that the contribution PNL

forms the following equation

PNLj (r, t) = ǫ0χ

3Ej (r, t)

∫ t

−∞

R (t′)

[

|Ej (r, t) |2 +2

3|Ek (r, t− t′) |2

]

dt′

+ ǫ0χ3Ek (r, t− t′)

∫ t

−∞

1

3R (t′)E∗

j (r, t− t′)Ek (r, t− t′) dt′, (2.2.33)

where j, k = x or y. It can be shown that for a linear birefringent PCF

the propagation equation now becomes the coupled mode nonlinear Schrodinger

equation (CMNLSE) and is given by

∂Ej (t)

∂z+

1

2

(

∆β0 + ∆β1∂Ej (t)

∂t

)

−∑

m≥2

im+1

m!βmj

∂mEj (t)

∂tm

= iγ

(

1 + iτ0∂

∂t

)

Ej (t)

(

(1 − fR)

[

|Ej (t)|2 +2

3|Ek (t)|2

]

+ fRRj (z, t)

)

,

(2.2.34)

Rj (z, t) =

∫ t

−∞

hR (t− t′)(

|Ej (t)|2 + |Ek (t)|2)

dt′, (2.2.35)

where Ej and Ek are the field components with j and k = x or y (x 6= y), z

is a propagation coordinate, the time coordinate moving in a reference frame is

given by t = τ − (β1j + β1k) z/2, βm is the mth order propagation coefficient, ∆β0 =

(β0j − β0k) is the phase mismatch, ∆β1 = (β1j − β1k) is the group velocity mismatch,

γ and τ0 are the nonlinearity and optical shock coefficients respectively.

35

Chapter 2

The method for solving Eqs. (2.2.32) and (2.2.34) can be achieved by the split

step Fourier method. A description of the numerical implementation of this method

is presented in Appendix A.

For polarised pulse propagation the term governing the degree of polarisation

of the output is the group velocity mismatch ∆β1 which relates to the walk off

length between the two LP modes. If the group velocity mismatch was neglected,

co-propagating pulses in adjacent modes would interact for the duration of the

fibre length causing coupled polarisation effects. At this point, a method for

describing the spectral and temporal characteristics is introduced, which is through a

spectrogram. Experimentally the spectrogram can be obtained by using a frequency

resolved optical gating (FROG) system. The mathematical description of the

spectrogram is determined by

S (ω, τ) = |∫ ∞

−∞

E (t) g (t− τ) e−iωtdt|2, (2.2.36)

where g (t− τ) is a delayed gate pulse at a delay of τ . The spectrogram is used to

compare the temporal and spectral characteristics between the coupled fibre modes.

Figure 2.2.2 shows the spectrograms for the two LP modes of a PCF of 1 metre

length. The dispersion and nonlinear properties for the two modes are the same

and as ∆β1 = 0 ps/m the two modes have the same spectrogram. However, when

the group velocity mismatch between the modes is increased to ∆β1 = −2 ps/m

(Fig. 2.2.3) the spectrograms for the two modes are different since the group

velocity mismatch introduces a walk off length for which the two modes can interact

within and hence have a small time frame for which XPM effects can influence the

pulses. This situation arises in highly birefringent PCF’s as the modal mismatch is

enhanced.

36

Chapter 2

−2 −1 0 1 2300

320

340

360

380

400

Time (ps)

Fre

quen

cy (

TH

z)

S0 (

norm

. 10l

og10

)

−20

−18

−16

−14

−12

−10

−2 −1 0 1 2300

320

340

360

380

400

Time (ps)

Fre

quen

cy (

TH

z)

S0 (

norm

. 10l

og10

)

−20

−18

−16

−14

−12

−10a b

Figure 2.2.2 Ultrashort (∆t = 0.05 ps) pulse propagation using the CMNLSE. (a) y−polarised modeand (b) the x−polarised mode. The parameters used in the simulation were βj2 = −0.005 ps2/m,βk2 = −0.005 ps2/m, γ = 0.095 W/m and a ∆β1 = 0 ps/m.

−2 −1 0 1 2300

320

340

360

380

400

Time (ps)

Fre

quen

cy (

TH

z)

S0 (

norm

. 10l

og10

)

−20

−18

−16

−14

−12

−10

−2 −1 0 1 2300

320

340

360

380

400

Time (ps)

Fre

quen

cy (

TH

z)

S0 (

norm

. 10l

og10

)

−20

−18

−16

−14

−12

−10

a b

Figure 2.2.3 Ultrashort (∆t = 0.05 ps) pulse propagation using the CMNLSE. (a) y−polarised modeand (b) the x−polarised mode. The parameters used in the simulation were βj2 = −0.005 ps2/m,βk2 = −0.005 ps2/m, γ = 0.095 W/m and a ∆β1 = −2 ps/m.

37

Chapter 2

2.3 Diffraction theory: low numerical aperture

2.3.1 Introduction

The basic construction of any optical imaging system is the lens or the microscope

objective because it delivers the capability to optically image with magnification.

The knowledge that has developed the current understanding of how a lens performs,

known as diffraction theory was constructed by Huygens and Fresnel which has led

to the Huygens-Fresnel principle.53 This chapter serves as a theoretical background

to the understanding of the Huygens-Fresnel principle which intuitively leads to the

diffraction integral for the low numerical aperture (NA). The theoretical background

then extends to Debye theory leading to the vectorial diffraction by a high NA lens.

2.3.2 Huygen-Fresnel principle

As described in Section 1.2.1 the Huygens-Fresnel principle considers the primary

wave front being diffracted by an aperture as a source of secondary spherical wave

fronts. The diffraction at a point after the aperture the field is the superposition of

the primary wave front and the secondary spherical wave fronts which interfere.

Mathematically the Huygens-Fresnel principle can be described as

E (r2, z2) = C

∫∫

A

e−ikr

rE (r1) dA, (2.3.1)

where E (r1) is the primary wave, e−ikr/r describes a secondary spherical wavelet,

C is a constant and A is aperture as shown in Fig. 2.3.1.

2.3.3 Fresnel approximation

The development of the theoretical treatment of diffraction has had many important

contributions which have led to the current formulation. Contributions from

researchers such as Rayleigh, Fraunhofer, Somerfield and Kirchhoff have been

38

Chapter 2

Ei

Figure 2.3.1 Illustration of mutual interference caused by the superposition of the primary wavefrontand secondary spherical waves.53

instrumental in the development of this theory.52 The mathematical treatment in

this section will not show the historical development of diffraction but provides the

most significant formulation which leads to the final form of the diffraction integral.

The diffraction of an electromagnetic wave can be described by

E2 (x2, y2) =iω

2πc

∫∫ ∞

−∞

Ei (x1, y1)e−ik|r|

|r| cos (n, r) dx1dy1. (2.3.2)

The integral of Eq. (2.3.2) describes the superposition of the incident wave front

(Ei (x1, y1)) and a set of secondary spherical wave fronts. The cos (n, r) factor is the

vector component of r in the direction of n.

Equation (2.3.2) can be simplified by using a few assumptions. Firstly, the

directional cosine can be assumed to be unity since for the case of a low NA lens the

majority of r is in the direction n. Secondly, the r in the denominator is replaced

by z. Finally, the formulation makes an assumption for the vector r, which is as

follows

39

Chapter 2

r2 = z2 + (x2 − x1)2 + (y2 − y1)

2

= z2

[

1 +(x2 − x1)

2 + (y2 − y1)

z

]

,

r ≈ z

[

1 +(x2 − x1)

2 + (y2 − y1)2

2z2

]

, (2.3.3)

where the approximation of the form√

1 + x ≈ 1 + x2

is used.

Therefore the formula for the Fresnel diffraction of an electromagnetic wave is

given by

E (x2, y2) =ie−ikz

λz

∫∫ ∞

−∞

E (x1, y1) e− ik

2z[(x2−x1)2+(y2−y1)2]dx1dy1. (2.3.4)

The Fresnel diffraction of a circular aperture can be obtained from Eq. (2.3.4)

and is simplified by using a cylindrical coordinate system which is given by

E (r2) =iω

ce−ikze−

ikr22

2z

∫ a

0

E (r1) e−

ikr21

2z J0

(

kr1r2z

)

r1dr1, (2.3.5)

where a is the aperture radius.

2.3.4 Fresnel diffraction by a circular lens

The diffraction formula (Eq. (2.3.5)) can be modified to determine the diffraction

by a circular lens. The focal length f of the lens is assumed to be approximately

equal to the distance z. The scalar diffraction theory for a circular lens is given

by52,53

E (u, v, ω) = −iωNA2

ceiu/NA

2

∫ a

b

Ei (ω) J0 (vρ) e−1

2iuρ2ρdρ. (2.3.6)

40

Chapter 2

The dimensionless parameters u and v are given by u =ω

c(NA)2 z and v =

ω

c(NA) r respectively, where r and z are the radial and axial coordinated of the

lens image space. The parameters a and b are the aperture radius and the integral

lower bound for the lens, NA is the numerical aperture, J0 is a zero order Bessel

function of the first kind, ω is the angular frequency and c is the speed of light.

2.4 Diffraction theory: high numerical aperture

2.4.1 The Debye integral

The Debye approximations are used when a high NA objective is considered. The

assumptions in Debye theory is the field in the focal region is a superposition of

plane waves with their associated propagation vectors originating from within the

aperture. The Debye integral can be expressed as

E (x2, y2, z2) =i

λ

∫∫

Ω

Eie−is•RdΩ, (2.4.1)

where Ei is the incident electric field with s and R related to the geometric

representation shown in Fig. 2.4.1.

For a circular coordinate system the dot product of the unit vector s and the

vector R is given by

s • R = r2 sin θ cos (φ− ψ) + z2 cos θ. (2.4.2)

The integral over the solid angle can be replaced by

dΩ = sin θdθdφ. (2.4.3)

The incident electromagnetic field at the lens aperture is assumed to fully

41

Chapter 2

y

z

xp1

p2

θ

f

r

fs R

Ei

Figure 2.4.1 Illustration of the geometry of vectorial diffraction.53

spatially coherent and enters the aperture as a plane wave. The diffraction by a

circular lens under the Debye approximations53 is calculated by

E (r, ψ, z, ω) =i

λ

∫∫

Ω

Ei (θ, φ, ω)

× e−ikr sin θ cos(φ−ψ)−ikz cos θ sin θdθdφ. (2.4.4)

2.4.2 Evaluation of the vectorial diffraction formula

The treatment of the diffraction formula for a high NA lens begins with the formu-

lation described by Wolf and Richards.60,61 Consider an incident electromagnetic

wave at the back of a high NA lens described by

Ei (ω) =

Eix

Eiy

Eiz

e−iω0t, (2.4.5)

where the incident field is represented by its polarisation components.

42

Chapter 2

However, the incident field undergoes refraction as it propagates through the lens

which causes a vectorial rotation of the polarisation (Fig. 1.2.2). The unit vector

related to the radial component of the field (E(aρ)) is transformed into the angular

component θ (E(aθ)). The incident electric field as it propagates past the lens can

now be described by

Ei(θ, φ) = P (θ)

cosφaθ

sinφaφ

, (2.4.6)

where

aθ =

cos θ cosφi

cos θ sinφj

sin θk

, aφ =

− sinφi

cosφj

. (2.4.7)

Using Eqs. (2.4.6) and (2.4.7), the vectorial form of the refraction of the incident

field through the lens can be obtained. The incident field can be separated into its

spatial dimensions (θ, φ) and frequency components(ω) where the spatial component

is given by

Ei(θ, φ) = P (θ)

(cos θ + sin2 φ(1 − cos θ))i

cosφ sinφ(cos θ − 1)j

cosφ sin θk

. (2.4.8)

Here P (θ) =√

cos θ, which is the apodisation function of the lens determined by

the sine condition. The vectorial form of the diffraction by a lens for a spherical

coordinate system can now be obtained by substituting Eq. (2.4.8) into Eq. (2.4.4)

which can be shown to be given by

43

Chapter 2

E (r, ψ, z, ω) =i

λ

∫∫

Ω

Ei (ω)P (θ)

(cos θ + sin2 φ(1 − cos θ))i

cosφ sinφ(cos θ − 1)j

cosφ sin θk

× e−ikr sin θ cos(φ−ψ)−ikz cos θ sin θdθdφ. (2.4.9)

For a circular symmetric incident field the diffraction of the field can be simplified

to being only dependent on θ. The diffraction of the incident field with a horizontal

polarisation direction can be shown to be given by

Eh(v, u, ψ, ω) =

Ex(v, u, ψ, ω)

Ey(v, u, ψ, ω)

Ez(v, u, ψ, ω)

=iω

2c

[I0 + cos (2ψ)I2] i

sin (2ψ)I2j

2i cos (ψ)I1k

, (2.4.10)

where

I0

I1

I2

=

∫ α

0

Ei(ω) cos1/2 θ sin θ

(1 + cos θ)J0(v sin θ/ sinα)

(sin θ)J1(v sin θ/ sinα)

(1 − cos θ)J2(v sin θ/ sinα)

× eiu cos θ/ sin2 αdθ, (2.4.11)

where u and v represent the normalised axial and radial dimensionless parameters

of the imaging system given by u = kz2 sin2 α and v = kr2 sinα. Essentially what

occurs is the incident polarisation rotates slightly to increase the strength of the

orthogonal transverse and longitudinal field polarisation states. For an incident

polarisation state in the vertical direction the field components are given by

44

Chapter 2

Ev(v, u, ψ, ω) =

Ex(v, u, ψ, ω)

Ey(v, u, ψ, ω)

Ez(v, u, ψ, ω)

=iω

2c

sin (2ψ)I2i

[I0(v, u) − cos (2ψ)I2] j

2i sin (ψ)I1k

. (2.4.12)

The diffraction formula is now composed of the horizontal and vertical polarisa-

tion components determined by

E = aEh + bEv, (2.4.13)

where a and b are the polarisation coefficients.

When the NA of the lens is reduced, the vectorial diffraction theory under the

paraxial approximation for the horizontally polarisation state is given by

E(v, u, ω) = Ex(v, u, ω)

=iω

2cI0(v, u)i, (2.4.14)

which converges to scalar diffraction theory described in Section 2.3.3.

2.5 Coherence

The modification of the temporal and spatial behavior of a focused wave can be

characterised through the degree of coherence (g1(u0, v0, τ)) which is generalised

through the correlation between two points and is calculated by76

g1 (z1, t1 : z2, t2) =〈E∗ (z1, t1)E (z2, t2)〉

[〈|E (z1, t1) |2〉 〈|E (z2, t2) |2〉]1

2

, (2.5.1)

where z and t are the axial and temporal coordinates.

For a stationary beam (e.g. in a Mach Zehnder interferometer) within the

45

Chapter 2

diffraction field, the spatial parameters u0 and v0 remain constant; hence Eq. (2.5.1)

becomes an autocorrelation technique determined by

g1 (u0, v0, τ) =〈E∗ (u0, v0, t) ,E (u0, v0, t+ τ)〉

[〈|E (u0, v0, t) |2〉 〈|E (u0, v0, t) |2〉]1

2

, (2.5.2)

where g1 (u0, v0, τ) depends on the position (u0, v0) of the detector.

However, when considering the wave packet (in a nonstationary observational

frame) the calculation becomes spatiotemporal and is given by76

g1 (u0, v0, τ) =〈E∗

1 (u, v0, t)E1 (u+ u0, v0, t+ τ)〉[〈|E1 (u, v0, t) |2〉 〈|E1 (u, v0, t) |2〉]

1

2

, (2.5.3)

where the variables u0 and τ are related by c = u0/τ . This nonstationary frame of

reference has been investigated for nonstationary polychromatic waves.77–83 Using

Eqs. (2.5.2) and (2.5.3) the coherence time for the field can be calculated through76

τc (u0, v0) =

∫ ∞

−∞

|g1 (u0, v0, τ) |2dτ. (2.5.4)

46

Chapter 3

Pulse Propagation in Nonlinear

Photonic Crystal Fibre

3.1 Introduction

The ability to control supercontinuum (SC) radiation is reliant on the optical prop-

erties of the photonic crystal fibre (PCF) waveguide and the precise knowledge of its

modal characteristics of nonlinearity and dispersion. The dispersion characteristics

determine the regime which the ultrashort pulse propagates within and the ability

to maintain the polarisation state of the incident beam.

The motivation behind this chapter is to provide an understanding of pulse

propagation in a highly birefringent PCF. The aim is to present that under the

condition of high birefringence (≈ 10−3) the dispersion profiles of the two linear

polarised modes must be treated separately. The interesting consequence in using

highly birefringent PCF is that the structure enforces the incorporation of two zero

dispersion wavelengths (ZDWs), which could be important in generating extensive

spectra.

47

Chapter 3

x/Λ

y/Λ

−5 0 5

−5

0

5

ε

1

1.2

1.4

1.6

1.8

2

Figure 3.2.1 The geometry as defined in the simulation using a refractive index profile resolution forthe PCF of 256 × 256 pixels and a supercell size 10 × 10 unit cells.

3.2 Photonic crystal fibre characteristics

The numerical study presented in this chapter uses the coupled mode nonlinear

Schrodinger equation (CMNLSE) as described in Section 2.2.5 and Eq. (2.2.34).

The effective refractive index for the two fundamental propagating modes were

calculated using the plane wave expansion method as described in Section 2.2.3.

The dispersion coefficients and phase mismatch coefficients were determined by

the Taylor series expansion as described by Eqs. (1.1.3) and (1.1.5). The field

distributions can be used to determine γ, using the methods described by Hainberger

and Watanabe 84, and similarly with the optical shock coefficient τ0 described by

Blow and Wood and Karasawa et al..34,85 The changes in optical nonlinearity and

optical shock are insignificant between modes and for this study are 0.095 W/m and

0.57 fs, respectively. The phase mismatch and the group velocity mismatch are

± 6120.1 1/m and ± 3.984 ps/m, respectively and are determined by Eq. (1.1.5).

The PCF geometry is shown in Fig. 3.2.1. The birefringent axis is in the

y−direction with the two birefringent holes being 0.2 µm larger than the rest of

PCF air holes. Using R-Soft Photonics CAD Suite 8.10 a theoretical model was

developed to model the PCF structure. The pitch of the fibre was 1.2 µm with

an air hole size of 0.7 µm. The supercell size, which relates to a collection of unit

48

Chapter 3

0.5 0.65 0.8 0.95 1.1 1.25 1.4 1.55−0.05

0

0.05

0.1

0.15

0.2

0.25

0.3

Wavelength (µm)

β 2 (ps

2 /m)

0.5 0.65 0.8 0.95 1.1 1.25 1.4 1.554.95

4.96

4.97

4.98

4.99

5

5.01

5.02

β 1 (ps

/m)

[1x1

0−3 ]

β1y

β1x

β2y

β2x

0.5 0.65 0.8 0.95 1.1 1.25 1.4 1.551000

3000

5000

7000

9000

11000

13000

15000

Wavelength (µm)

∆β(1

/m)

0.5 0.65 0.8 0.95 1.1 1.25 1.4 1.55−9

−7

−5

−3

−1

1

3

5

∆β1(p

s/m

)

a b

Figure 3.2.2 The dispersion coefficients related to the mode propagation constant β. (a) shows thefirst- and second-order dispersion coefficients for the two fundamental modes. (b) shows the phasemismatch (∆β0) and the group velocity mismatch (∆β1) between these modes.

cells should be within the range of 6 × 6 to 10 × 10 unit cells. The most important

parameter used is the eigenvalue tolerance and for these calculations was set at 10−16

which means a precision of the order of 10−8.

Figure 3.2.2a shows β1 and β2 dispersion terms. The difference between the

two polarised modes is significant and conveys the importance of the group velocity

mismatch, which is shown in Fig. 3.2.2b. The β1 dispersion term is related to the

group velocity term by the relation β1 = 1/vg, which shows that the y-polarised

mode travels faster than the x-polarised mode and are called the fast axis and slow

axes, respectively. The dispersion properties used in this study are presented in

Table 3.2.1.

Table 3.2.1 Dispersion data for the polarised mode of the nonlinear fibre.

βm mode y (psm/m) mode x (psm/m)β2 −4.654 × 10−3 −8.215 × 10−3

β3 4.879 × 10−5 5.063 × 10−5

β4 −1.735 × 10−8 −2.118 × 10−8

β5 1.42 × 10−10 2.018 × 10−10

β6 1.802 × 10−13 1.356 × 10−13

β7 −6.964 × 10−15 −8.764 × 10−15

β8 3.095 × 10−17 3.979 × 10−17

β9 −4.649 × 10−20 −6.02 × 10−20

The high order dispersion properties are important as they lead to the generation

49

Chapter 3

−1 −0.5 0 0.5 10

0.06

0.12

0.18

0.24

0.3

Time (ps)

Fib

re le

ngth

(m

)

S/S

0 (10

log 10

)

−30

−20

−10

0

200 250 300 350 400 450 500 5500

0.06

0.12

0.18

0.24

0.3

Frequency (THz)

Fib

re le

ngth

(m

)

S/S

0 (10

log 10

)

−40

−30

−20

−10

0

10

20

30

a b

Figure 3.3.1 The effects of TOD originating from a PCF pumped with an ultra-short pulse with apulse duration of 0.1 ps and a peak power of 1000 W . (a) the time domain and (b) the frequencydomain.

of dispersive waves and ultimately determine the dispersive waves phase matched

wave vectors.

3.3 Nonlinear and dispersion effects

The theoretical analysis throughout this thesis involves the investigation into SC

generation. The dominant processes that form the SC need to be isolated in order

to understand the effects such a field has in an optical system, for example a

lens. During the propagation of an ultrashort pulse through a PCF the nonlinear

and dispersion effects become dominant at different stages. The two terms which

have the most influence in shifting radiation are intra-pulse Raman scattering

(RS) (see Section 1.1.4.4) and the third order dispersion (TOD) effect (see Section

1.1.2). SC generation in the anomalous dispersion regime is dominated by soliton

dynamics where the solitary wave forms due to second order dispersion and self

phase modulation. To develop an understanding of the processes which occur in a

highly birefringent PCF with two ZDWs, the dominate phase terms influencing the

propagation of the ultrashort pulse are investigated.

Figure 3.3.1 shows the propagation of an ultra-short pulse affected by the TOD

effect through the PCF for an input polarisation state orientated along the fast axis

(y−direction) of the PCF. The high order dispersion and third order nonlinear terms

have been neglected. The input peak power of 1000 W was chosen to illustrate the

50

Chapter 3

effects of soliton fission from a third order soliton.

The soliton fission dynamics are modified by dispersive waves generated at phase

matched frequencies associated with the propagation constant of the soliton and

nonlinear phase.86,87 It can be seen in Fig. 3.3.1 that energy from the initial soliton

is transferred to a resonant wave at a lower frequency. This is an important process

as it is the key component in formation of visible wavelengths within the spectrum

of a SC field. These dispersive waves do not shift with the fibre length after their

initial ejection and therefore the low wavelength component of the SC is fixed by

initial soliton dynamics.

RS is also a dominant effect in SC generation. After the formation of a high

order soliton, the soliton begins fissions into fundamental solitons which shift with

the Raman spectrum of the fibre, Fig. (3.3.2). The dynamics of these process are

determined by the nonlinear and second order dispersion characteristics which form

the soliton and subsequent pulse widths of the ejected fundamental solitons.88,89 The

temporal envelope shows the formation of these solitons and the eventual separation

from the remaining dispersive temporal features. The rate at which the soliton pulses

shifts has been formulated by Gordon in 198690 and is given by

∂νR∂z

∝ |β2|T 4

0

. (3.3.1)

Under conditions of strong birefringence the walk off length for energy transfer

−1 −0.5 0 0.5 10

0.06

0.12

0.18

0.24

0.3

Time (ps)

Fib

re le

ngth

(m

)

S/S

0 (10

log 10

)

−30

−20

−10

0

10

200 250 300 350 400 450 500 5500

0.06

0.12

0.18

0.24

0.3

Frequency (THz)

Fib

re le

ngth

(m

)

S/S

0 (10

log 10

)

−40

−30

−20

−10

0

10

20

30

a b

Figure 3.3.2 The effects of RS originating from a PCF pumped with an ultra-short pulse with a pulseduration of 0.1 ps and a power of 1000 W . (a) the time domain and (b) the frequency domain.

51

Chapter 3

−1 −0.5 0 0.5 10

0.06

0.12

0.18

0.24

0.3

Time (ps)

Fib

re le

ngth

(m

)

S/S

0 (10

log 10

)

−30

−20

−10

0

200 250 300 350 400 450 500 5500

0.06

0.12

0.18

0.24

0.3

Frequency (THz)

Fib

re le

ngth

(m

)

S/S

0 (10

log 10

)

−40

−30

−20

−10

0

10

20

30

−1 −0.5 0 0.5 10

0.06

0.12

0.18

0.24

0.3

Time (ps)

Fib

re le

ngth

(m

)

S/S

0 (10

log 10

)

−30

−20

−10

0

200 250 300 350 400 450 500 5500

0.06

0.12

0.18

0.24

0.3

Frequency (THz)

Fib

re le

ngth

(m

)

S/S

0 (10

log 10

)

−40

−30

−20

−10

0

10

20

30

a b

c d

Figure 3.3.3 The effects of TOD originating from a PCF pumped with an ultra-short pulse with apulse duration of 0.1 ps and in each mode of power of 750 W . The coupled polarisation state was 45.(a) the time domain of the y−polarised mode (b) the frequency domain of y−polarised mode (c) thetime domain of x−polarised mode and (d) the frequency domain of x−polarised mode.

between the linear polarised modes, occurs within a very short spatial window (≈mm) and does not have a strong influence on propagation. As can be seen with the

previous investigation dealing with TOD and RS effects, the temporal envelope is

slanted due to the difference between the effective group velocities of the two modes.

Again, dispersive and nonlinear effects can be isolated for each mode. The power

injected into each mode is the same (1000 W ); however the second order dispersion

coefficients are different (Table 3.2.1). Figure 3.3.3 shows the TOD effect and what

can be seen is the difference in the effect of second order dispersion in both the

formation of solitary waves and the generation of dispersive waves. Since second

order dispersion is stronger for the x−mode, the pulse compresses more than the

y−mode and red shifts the ejected solitons to greater extent. This leads to the

dispersive waves at higher resonant frequencies (shorter wavelengths), which can be

seen in Fig. 3.3.3.

The Raman shifting of the co-propagating waves generated by the equal coupling

52

Chapter 3

−1 −0.5 0 0.5 10

0.06

0.12

0.18

0.24

0.3

Time (ps)

Fib

re le

ngth

(m

)

S/S

0 (10

log 10

)

−30

−20

−10

0

200 250 300 350 400 450 500 5500

0.06

0.12

0.18

0.24

0.3

Frequency (THz)

Fib

re le

ngth

(m

)

S/S

0 (10

log 10

)

−40

−30

−20

−10

0

10

20

30

−1 −0.5 0 0.5 10

0.06

0.12

0.18

0.24

0.3

Time (ps)

Fib

re le

ngth

(m

)

S/S

0 (10

log 10

)

−30

−20

−10

0

200 250 300 350 400 450 500 5500

0.06

0.12

0.18

0.24

0.3

Frequency (THz)

Fib

re le

ngth

(m

)

S/S

0 (10

log 10

)−40

−30

−20

−10

0

10

20

30

a b

c d

Figure 3.3.4 The effects of RS originating from a PCF pumped with an ultra-short pulse with a pulseduration of 0.1 ps and in each mode of power of 750 W . The coupled polarisation state was 45. (a)the time domain of the y−polarised mode (b) the frequency domain of y−polarised mode (c) the timedomain of x−polarised mode and (d) the frequency domain of x−polarised mode.

53

Chapter 3

−1 −0.5 0 0.5 10

250

500

750

1000

Time (ps)

Inpu

t pow

er (

W)

S/S

0 (10

log 10

)

−30

−20

−10

0

200 250 300 350 400 450 500 5500

250

500

750

1000

Frequency (THz)

Inpu

t pow

er (

W)

S/S

0 (10

log 10

)

−30

−20

−10

0

10

20

a b

Figure 3.3.5 The effects of TOD originating from a PCF pumped with an ultra-short pulse with apulse duration of 0.1 ps and a fibre length of 0.3 m. (a) the time domain and (b) the frequency domain.

−1 −0.5 0 0.5 10

250

500

750

1000

Time (ps)

Inpu

t pow

er (

W)

S/S

0 (10

log 10

)

−30

−20

−10

0

200 250 300 350 400 450 500 5500

250

500

750

1000

Frequency (THz)

Inpu

t pow

er (

W)

S/S

0 (10

log 10

)

−30

−20

−10

0

10

20

a b

Figure 3.3.6 The effects of RS originating from a PCF pumped with an ultra-short pulse with a pulseduration of 0.1 ps and a fibre length of 0.3 m. (a) the time domain and (b) the frequency domain.

of the fundamental modes can be seen in Fig. 3.3.4. The frequency shift associated

with the soliton formed in the x−mode shifts further than the soliton in the y−mode,

which reiterates the influence of the stronger compression from the larger second

order dispersion term. The difference between the dispersion coefficients for the

two fundamental modes is therefore important as it sets the conditions for the red

shifting of the fundamental solitons and the transfer of energy to dispersive waves.

The effects seen for ultra-short pulse propagation as a function of fibre length can

also be verified through an investigation into a variation in input power. Figure 3.3.5

shows the dependence of the temporal and spectral properties on the input pulse

pulse peak power and how it is affected by TOD. The coupled peak power determines

the initial soliton order N , which will then fission into the N fundamental solitons.

Fig. 3.3.5 confirms that the TOD effect initiates the growth of blue shifted radiation

54

Chapter 3

−1 −0.5 0 0.5 10

250

500

750

Time (ps)

Inpu

t pow

er (

W)

S/S

0 (10

log 10

)

−30

−20

−10

0

200 250 300 350 400 450 500 5500

250

500

750

Frequency (THz)

Inpu

t pow

er (

W)

S/S

0 (10

log 10

)

−30

−20

−10

0

10

20

−1 −0.5 0 0.5 10

250

500

750

Time (ps)

Inpu

t pow

er (

W)

S/S

0 (10

log 10

)−30

−20

−10

0

200 250 300 350 400 450 500 5500

250

500

750

Frequency (THz)

Inpu

t pow

er (

W)

S/S

0 (10

log 10

)

−30

−20

−10

0

10

20

a b

c d

Figure 3.3.7 The effects of TOD originating from a PCF pumped with an ultra-short pulse with apulse duration of 0.1 ps and a fibre length of 0.3 m. The coupled polarisation state was 45. (a) thetime domain of the y−polarised mode (b) the frequency domain of y−polarised mode (c) the timedomain of x−polarised mode and (d) the frequency domain of x−polarised mode.

for which after a particular power the spectral expansion begins to slow down.

Similarly to the investigation of the TOD effect, the effects of RS can also be

understood and is presented in Fig. 3.3.6. The figure shows the clear formation

and fission of solitons (Fig. 3.3.6a) and the extensive red shifted radiation. In

the absence of TOD there are no blue shifted dispersive waves and the spectra is

dominated by near infra-red radiation from soliton self frequency shift.

The power dependence of co-progating modes confirms the observations discussed

for the dependence of fibre length and are shown in Figs. 3.3.7 and 3.3.8. The

difference between the dispersion terms for the two creates a difference in the

soliton and dispersion dynamics which leads to a difference in temporal and spectral

behaviour. The figures depicting the power dependence show that the formation

and self frequency shift of the solitons occurs from a balance between the power

within the pulse and its pulse width. As the pulse propagates along the fibre the

pulse duration and peak power are adjusted to maintain solitary shape. The initial

55

Chapter 3

−1 −0.5 0 0.5 10

250

500

750

Time (ps)

Inpu

t pow

er (

W)

S

/S0 (

10lo

g 10)

−30

−20

−10

0

200 250 300 350 400 450 500 5500

250

500

750

Frequency (THz)

Inpu

t pow

er (

W)

S/S

0 (10

log 10

)

−30

−20

−10

0

10

20

−1 −0.5 0 0.5 10

250

500

750

Time (ps)

Inpu

t pow

er (

W)

S/S

0 (10

log 10

)

−30

−20

−10

0

200 250 300 350 400 450 500 5500

250

500

750

Frequency (THz)

Inpu

t pow

er (

W)

S/S

0 (10

log 10

)

−30

−20

−10

0

10

20

a b

c d

Figure 3.3.8 The effects of RS originating from a PCF pumped with an ultra-short pulse with a pulseduration of 0.1 ps and a fibre length of 0.3 m. The coupled polarisation state was 45. (a) the timedomain of the y−polarised mode (b) the frequency domain of y−polarised mode (c) the time domainof x−polarised mode and (d) the frequency domain of x−polarised mode.

56

Chapter 3

(a)

θ = 0

Inpu

t Pow

er (

mW

)

250 350 450 550

5

10

15

20

25

30

(d)

θ = 0Inpu

t Pow

er (

mW

)

250 350 450 550

5

10

15

20

25

30

(b)

θ = 45

250 350 450 550

(e)

θ = 45

Frequency (THz)250 350 450 550

(c)

θ = 90

250 350 450 550

(f)

θ = 90

250 350 450 550

Oup

ut P

ower

(dB

m)

−40

−35

−30

−25

−20

−15

−10

−5

0

Figure 3.4.1 Theoretically obtained spectra of propagation within a 130 mm NL-PCF with a 87 fspulse. Figures (a), (b) and (c) are the spectra for the y−polarised output field with (d), (e) and (f) forthe x−polarised output field. θ is the input polarisation angle with respect to the y−axis.

peak power is important because it sets the strength of the nonlinearity with in the

fibre which along with second order dispersion determines the temporal duration of

the soliton and hence the ability of the pulse to shift in frequency.

3.4 Supercontinuum generation

Equation (2.2.34) was used to calculate 87 fs pulses propagating with different

input polarisation orientations. A pulse width of 87 fs and a fibre length of

130 mm were chosen to coincide with experimental conditions. Figure 3.4.1 shows

the output spectra obtained from a nonlinear PCF pumped with 87 fs pulses at

input polarisation orientations of 0 (y−axis), 45 and 90 (x−axis) degrees.91 The

simulation shows that the polarisation state is maintained for light coupled into

either the x-polarised or y-polarised axis of the fibre. The x-polarised mode has

the most extensive spectra and this is because the pump wavelength undergoes a

stronger initial compression caused by the stronger second-order dispersion term.

57

Chapter 3

150 250 350 450 550 6500

50

100

150

200

250

Frequency (THz)

γ MI

y−modex−mode

Figure 3.4.2 Modulation instability gain for the y− and the x−modes.

For an input polarisation orientation of 45 degrees, the degree of polarisation is not

maintained due to an equal coupling between modes and the spectra are different

due to the relative strength of the second order dispersion.

The difference in the dispersion curves creates a change in the wave numbers

for the generation of dispersive waves which are determined by the phase matching

condition for each individual mode. This occurs because of the modulation insta-

bility which gives rise to gain and amplifying the nonlinear processes. Modulation

instability is determined by25

γMI = Im(

√

Q (Q+ 2γP0))

, (3.4.1)

where Q is the propagation constant containing only the even order coefficients

and P0 is the peak power of the input pulse. Figure 3.4.2 shows the modulation

instability gain for the two polarised modes and quantifies that the differences in

the dispersion of the two mode can change the output spectra.

When the pulse enters the nonlinear fibre, it undergoes a transformation to form

a high order soliton. The high order soliton then breaks up in a fission process which

converts the soliton into fundamental solitons.46 The order of the initial soliton and

the length at which soliton fission occurs is given by92

58

Chapter 3

N =

(

γP0T20

|β2|

)1/2

, (3.4.2)

Lf =T 2

0

|β2|1

N, (3.4.3)

where P0 is the peak power and T0 is the full width at half maximum.

The soliton order is higher along the y−axis in comparison to the x−axis, which

is due to the smaller second order dispersion term. However, the soliton fission

length is 0.0465 m and 0.035 m for the y−axis and x−axis, respectively, which

could be contributing to the more extensive spectra. The self frequency shift of

the soliton is inversely proportional to the fourth power of the temporal duration

of the soliton. The initially stronger compression by the second order dispersion in

the x−mode leads to further spectral expansion in comparison with the y−mode.

Although there are some differences between the spectral components between the

input polarisation orientations, the spectra are similar and hence both axes could

be used for polarised broadband applications.

Figure 3.4.3 confirms that the output spectra are different between the two

fundamental modes. The output spectra for the y−polarised mode travels at a

higher speed compared with the x−polarised mode which is due to the group velocity

mismatch. Figures 3.4.3 (a) and (f) show the difference in the pulse structure which

is due to the different dispersion properties and fission processes of the coupled

axes. The effects observed would be important to consider in time-resolved polarised

illumination applications since there is a delay between spectral features.

3.4.1 Experimental study

The experimental setup is shown in Fig. 3.4.4. A pulsed light beam from a

Ti:Sapphire laser was coupled into a nonlinear PCF (Crystal-fibre). Two Glan

Thomson polarisers were used to vary the input power and a half wave plate was

used to alter the input polarisation orientation. The output pulse was analysed

with a Glan Thomson polariser. Spectra were observed and recorded using an

59

Chapter 3

(a) θ = 0

Fre

quen

cy (

TH

z)

−1 −0.5 0 0.5 1

250

300

350

400

450

500

(d) θ = 0

Fre

quen

cy (

TH

z)

−1 −0.5 0 0.5 1

250

300

350

400

450

500

(b) θ = 45

−1 −0.5 0 0.5 1

(e) θ = 45

Time (ps)−1 −0.5 0 0.5 1

(c) θ = 90

−1 −0.5 0 0.5 1

(f) θ = 90

−1 −0.5 0 0.5 1

Oup

ut P

ower

(dB

m)

−40

−35

−30

−25

−20

−15

−10

−5

Figure 3.4.3 Theoretically obtained spectral and temporal profile of 87 fs pulsed propagation withina 130 mm nonlinear PCF. Figures (a), (b) and (c) are the spectra for the y−polarised output field with(d), (e) and (f) for the x−polarised output field.

Ando spectrometer and Princeton Instruments CCD (pixis 100). The pulsed

propagation spectra were obtained for different input polarisation orientations. The

characteristics of the fibre used in this experimental study are shown in Fig. 3.2.2

and Table 3.2.1.

Figure 3.4.5 shows the spectra for the two output modes of the PCF coupled

with 780 nm, 87 fs pulses and 15 mW average power. The spectra show the

high degree of polarisation for the input pulse orientations of 0 and 90 degrees. In

addition, a large degree of the red-shifted radiation attributed to stimulated Raman

scattering is present. The 0 and 90 degree spectra are different and confirm the

theoretically obtained results. For 0 degrees it is apparent that there is a blue-shift of

radiation and is attributed to the dispersive wave generation and four-wave mixing.

In addition, a small amount of radiation is coupled into the orthogonal mode which

is attributed to the depolarisation by the high numerical aperture input coupling.

The degree of polarisation (DOP), defined as DOP =(

I|| − I⊥)

/(

I|| + I⊥)

, is

shown in Fig. 3.4.6. The experimental curves show a strong degree of polarisation

60

Chapter 3

Ti:Sa

GT GT 12WP

SA

Fibre

GT

Spec

Figure 3.4.4 Optical arrangement used in this study. GT - Glan Tomson, WP - Wave Plate, Spec -Spectrograph and SA - Spectrum Anaylser

for all wavelengths except for the pump bandwidth. This confirms the high degree of

polarisation measured in Fig. 3.4.5. The theoretical degree of polarisation shows the

effects of depolarisation are not attributed to cross coupling and must be introduced

by the input coupling.

3.5 Conclusion

A highly birefringent PCF with two ZDWs is beneficial to produce extensive highly-

polarised optical spectra due to the nonlinear and dispersive properties inherent

from its geometry. A methodology for generating broadband pulsed light from a

two ZDWs PCF is presented. The theoretical and experimental observation shows

that the spectra maintain their linear polarisation state and that the extent of the

spectra is stronger at either of the fundamental mode axes.

Qualitatively, the dispersion properties of the the two fundamental polarised

modes of the PCF produce different SC spectral features and is due to the strength

of the second order dispersion term. This difference creates different order solitons

in each mode when coupled with the same power, which spectrally shift at different

rates. The dispersion polynomials for the two modes set the conditions for the

radiation of dispersion waves, which determines the extent of blue shifted radiation.

These characteristics are achieved because of the highly birefringent PCF.

61

Chapter 3

300 350 400 450 500 550−50

−45

−40

−35

−30

−25

−20In

tens

ity (

dB)

Frequency (THz)

θin

= 0o

300 350 400 450 500 550−50

−45

−40

−35

−30

−25

−20

Inte

nsity

(dB

)

Frequency (THz)

θin

= 45o

300 350 400 450 500 550−50

−45

−40

−35

−30

−25

−20

Inte

nsity

(dB

)

Frequency (THz)

θin

= 90o

a

b

c

Figure 3.4.5 Spectral properties of the polarised modes of the nonlinear PCF. The perpendicular(blue) and parallel polarised (red) states are with reference to the output orientation of the laser.

62

Chapter 3

300 350 400 450 5000.2

0.4

0.6

0.8

1

Frequency (THz)

Deg

ree

of p

olar

isat

ion

Parallel

Perpendicular

Fast axis (theory)

Figure 3.4.6 Degree of polarisation for the fast and the slow axes of the fibre.

63

Chapter 3

64

Chapter 4

Fresnel Diffraction

4.1 Introduction

The phase associated with an electromagnetic wave can affect the way by which

the field correlates in different optical phenomena. Within the last decade the

fundamental description of diffraction in optical systems such as a lens was heavily

investigated because of the influence of spatial phases which forms points of

destructive interference. This effect has been extensively studied and in particular

from two points of view: firstly from a interference through an interferometer such

as a Michelson Mach Zehnder interferometer, and secondly from diffraction by a

lens. The study in thesis will investigate the later case.

As stated earlier in Chapter 2 there has been recent interest in the quantitative

description and formulation of the diffraction of a polychromatic wave. Around

the point of destructive interference, the focused wave shows the behaviour of

red shifted and blue shifted radiation. Physically, the addition of temporal phase

onto an electromagnetic wave can change the superposition condition of the focal

distribution. Since there exists a temporal and spatial phase coupling through the

wavefront propagation in a lens, it is intuitive that a temporal phase variation of

supercontinuum (SC) field would affect the diffraction by the lens.

Presented in this chapter is an understanding of SC generation under conditions

65

Chapter 4

of Fresnel diffraction, which is considered because it is a simplified and accurate

description of the diffraction by a lens.52 Also presented in this chapter is the

coupling relationship between the temporal and spatial phase within a focused

SC field, and how the diffraction modification through the steep phase gradient

associated with the points of destructive interference enhances the degree of

coherence of a SC field in the focal region.

4.2 Numerical methodology

The diffraction of a polychromatic wave such as a SC wave is calculated using the

scalar diffraction theory, which can be given, under the paraxial approximation,

by Eq. (2.3.6).53 If b = 0, the diffraction is for the complete aperture and for a

non-zero b is a diaphragm. E (ω) is the Fourier transform of the SC wave using

the dispersion parameters, nonlinear parameters and method described in Chapter

3, and the Fourier transform of E1 (u, v, ω) is used to obtain the temporal profile

E1 (u, v, t).

An analytic solution (v = 0) for the diffraction field can be obtained by the

following equation

E1 (u, v, ω) = −ωNA2

uceiu/NA

2

E (ω)(

e−1

2ib2u − e−

1

2ia2u

)

. (4.2.1)

From Eq. (4.2.1) it can be seen that when a2u/2 = ±2nπ and if b = 0, the

equation is equal to 0u

which is a singularity. Since u is dependent on both ω and z

there exists a region of singularity.

As the electromagnetic field propagates through a lens it is diffracted and

modified through the spatial phase of the wavefront. The superposition of the

wavefront from the outer aperture to the inner aperture has an inherent path

difference (Fig. 4.2.1). The destructive interference at certain frequency components

of the SC wave produces a singularity or a null in intensity. These points occur

at discrete positions in both the axial and radial directions and can be determined

66

Chapter 4

through the parameter u = ωc(NA)2 z and occur at u = ±4nπ (where n is an integer,

for the radial direction this point is the zero of a zero order Bessel function of the first

kind). The parameters u0 and v0 (Fig. 4.2.1a) are defined as the normalised axial

and radial coordinates of the optical system and are given by u0 = 2πλ0

(NA)2 z and

v0 = 2πλ0

(NA) r, where NA is the numerical aperture, z and r are the axial and radial

dimensions (in µm), and λ0 is the centre wavelength of the original pulse coupled to

the nonlinear photonic crystal fibre (PCF) (the input pulse is a hyperbolic secant

pulse which is used to represent a mode-locked laser pulse).

The analysis used to understand how a field propagates through the focal

region depends on the method of its detection or observation (Fig. 4.2.1b). In

a conventional optical system with a single point detector, the intensity that is

collected depends on the diffraction for an axial (and radial) position (stationary

observer, S (t, z0) ) and evolves with time. The intuitive observation, however, would

be to view the focal plane from the side, where the intensity is both temporally

u (t, z)

v (t, r)

f

r

U (t)a

O(t=0,z=0)

b v (ω0)

u (ω0)

S (t′ = t − z0/c, z)

S (t, z0)

Nonstationary observer

Stationary observer

Figure 4.2.1 An illustration of pulse diffraction by a low numerical aperture (NA) lens. (a) showshow the path length and the NA affect the pulse distribution as the temporal envelope passes throughthe focus. (b) shows the observation frames of the intensity profile in the focus.

67

Chapter 4

0

0.2

0.4

0.6

0.8

1

u0 (axial)

S/S

0

−6π −3π 0 3π 6π0

0.01

0.02

0.03

0.04

u0 (axial)

S/S

0

0π 2π 4π 6π

u0 (axial)

v 0 (ra

dial

)

−6π −3π 0 3π 6π

−3π

− 3π

2

0

3π

2

3π

S/S

0 (10

log 10

)

−40

−30

−20

−10

0

−6π −3π 0 3π 6π−1

−0.5

0

0.5

1

Tim

e (p

s)

u0 (axial)

S/S

0 (10

log 10

)

−35

−30

−25

−20

−15

−10

−5

0

a b

c d

Figure 4.3.1 The temporal effects of a focused hyperbolic secant ultrashort pulse propagating throughthe focus of a low NA (0.1) objective. (a) On axis diffraction centred at the focal point (the fulltemporal evolution of the hyperbolic secant on the axis is described in Appendix C). (b) On axisdiffraction centred at u0 = 5π. (c) Radial and axial diffraction pattern centred at the focal point (thefull temporal evolution of the hyperbolic in the radial and axial direction is described in Appendix C).(d) The intensity matrix used to obtain the temporal and axial intensity information for the stationaryand nonstationary observation frames.

and axially dependent since the leading intensities of the pulse are modified by the

diffraction for an axial position and differs from the trailing intensities (S (t, z)),

which is referred to as a nonstationary observer. E1 (ω) is the intensity distribution

for the stationary observation frame where the intensity for the nonstationary

observation frame is obtained by taking the diagonal of the the matrix E1 (u, t)

for different v.

4.3 Ultrashort hyperbolic secant pulse

For an ultrashort pulse its phase is linear, a phase modification is expected from

the diffraction by the lens (Fig. 4.3.1a) and when the pulse encounters a point

68

Chapter 4

u0 (axial)

v 0 (ra

dial

)

−6π −3π 0 3π 6π

−3π

− 3π

2

0

3π

2

3π

τ c (ps

)

0.1

0.12

0.14

0.16

u0 (axial)

v 0 (ra

dial

)

−6π −3π 0 3π 6π

−3π

− 3π

2

0

3π

2

3π

τ c (ps

)

0.15

0.2

0.25

u0 (axial)

τ c (ps

)

−6π −3π 0 3π 6π0.09

0.11

0.13

0.15

0.17

NA = 0.1NA = 0.14NA = 0.2

u0 (axial)

τ c (ps

)

−6π −3π 0 3π 6π0.1

0.1175

0.135

0.1525

0.17

NA = 0.1NA = 0.14NA = 0.2

a b

c d

Figure 4.3.2 The coherence time of a focused hyperbolic secant ultrashort pulse for the stationaryand the non-stationary cases. (a) Axial and radial distribution of the coherence time for the 0.1 NAlens for the stationary case; (b) Axial and radial distribution of the coherence time for the 0.1 NA lensfor the non-stationary case; (c) Effect of NA on the coherence time on the axis for the stationary case;and (d) Effect of NA on the coherence time on the axis for the non-stationary case.

of destructive interference its temporal profile changes significantly (Fig. 4.3.1b)

in which the pulse envelope and spectrum are split. The phase modification also

extends to the radial direction (Fig. 4.3.1c) caused by the zero intensity location due

to the zero order Bessel function. For a stationary observer the temporal intensity

information is obtained for a constant axial position (data contained in the columns

of Fig. 4.3.1d). The nonstationary observation frame is complicated since the

intensity information is linked in both the temporal and axial coordinates and is

obtained through the diagonal of the matrix S(u, t) (diagonal of Fig. 4.3.1d). Both

observation frames may vary in the radial direction to obtain the three dimension

intensity information.

The temporal characteristics of the field are quantified through the coherence

time, which is mathematically described by Eqs. (2.5.1)-(2.5.3). For an ultrashort

69

Chapter 4

pulse the coherence time is 60 % larger than its initial coherence time (τ0 = 0.164 ps)

before the lens (Figs. 4.3.2a and 4.3.2b). The variation in the coherence time for

the stationary case is expected to be narrow for an ultrashort pulse due the narrow

bandwidth of the field. The observation frames are different because of the geometric

path difference (Figs. 4.3.2a and b), which is illustrated by changing the NA in the

non-stationary observation frame (Fig. 4.3.2d). The coherence time is not enhanced

for a hyperbolic secant because there is no temporal phase contribution. An increase

in chirp is effectively increasing the amount of phase which creates an enhancement

in the coherence time (Fig. 4.3.3). The spatial phase incurred because of diffraction

has a change of negative phase to positive phase across the focal plane and has a

greater path difference on the inside of the focal plane (the rays from the extremity

of the lens approach the focal length the further away from the lens), which explains

the asymmetric distribution in the coherence time with an increase in the chirp

parameter.

The coherence time is related to the bandwidth ∆ν by τc = 1/∆ν. The mean

frequency µ can validate the effects observed in the temporal correlation, which is

illustrated in Fig. 4.3.4. The variation in the mean frequency caused by the change

in phase associated with the frequency dependent point of destructive interference

is similar to the previous literature presented by Gbur.54

4.4 Nonlinear and dispersive phase

To determine how the phase associated with electromagnetic field from a PCF

behaves in the focal region of a lens, the dominant phase terms can be isolated.

Third order dispersion (TOD) and stimulated Raman scattering (RS) are dominant

processes which act to perturb an optical soliton. To isolate these effects the coupled

mode nonlinear Schrodinger equation is reduced to the dependence of β2, SPM and

the coupled mode phase mismatch terms ∆β0 and ∆β1.

The incorporation of RS is shown in Fig. 4.4.1. In the initial stages of the

propagation, the temporal phase affecting the coherence time is the influence of

soliton formation. At a length corresponding to approximately the soliton fission

70

Chapter 4

u0 (axial)

(τc−

τ 0)/τ 0

−6π −3π 0 3π 6π−0.6

−0.4

−0.2

0

0.2

0.4

C = 0C = 0.1C = 1

u0 (axial)

(τc−

τ 0)/τ 0

−6π −3π 0 3π 6π−0.4

−0.3

−0.2

−0.1

0

0.1

C = 0C = 0.1C = 1

a

b

Figure 4.3.3 The coherence time illustrating the effect of the variation in temporal phase throughthe addition of chirp through the chirp parameter C

(

ps2)

for the stationary (a) and nonstationary (b)observation frames. τ0 is the initial coherence time before the objective.

71

Chapter 4

u0 (axial)

v 0 (ra

dial

)

−6π −3π 0 3π 6π

−3π

− 3

2π

0

3

2π

3π

−3π

− 3

2π

0

3

2π

3π

(µ−

µ 0)/µ 0

−2

−1

0

1

2

x 10−3

u0 (axial)

v 0 (ra

dial

)

−6π −3π 0 3π 6π

−3π

− 3

2π

0

3

2π

3π

−6π −3π 0 3π 6π

−3π

− 3

2π

0

3

2π

3π

(µ−

µ 0)/µ 0

−6

−4

−2

0

2

4

6x 10

−3

a

b

Figure 4.3.4 Mean frequency distribution of a focused hyperbolic secant ultrashort pulse in the axialand radial plane of a 0.1 NA lens for stationary (a) and non-stationary (b) cases.

− 8π − 4π 0 4π 8π0

0 .06

0 .12

0 .18

0 .24

0 .3

u0 (axial)

Fib

re le

ngth

(m

)

(τc −

τ 0)/τ 0

−0.5

0

0.5

1

− 8π − 4π 0 4π 8π0

0 .06

0 .12

0 .18

0 .24

0 .3

u0 (axial)

Fib

re le

ngth

(m

)

(τc −

τ 0)/τ 0

0

1

2

3

4

5a b

Figure 4.4.1 The effects of RS on the coherence time for a focused electromagnetic wave by a lensof NA = 0.1, originating from a PCF pumped with an ultra-short pulse with a pulse duration of 0.1 psand a power of 1000 W (relating to the field in Fig. 3.3.2). (a) stationary observation frame and (b)a nonstationary observation frame.

72

Chapter 4

− 8π − 4π 0 4π 8π0

0 .06

0 .12

0 .18

0 .24

0 .3

u0 (axial)

Fib

re le

ngth

(m

)

(τc −

τ 0)/τ 0

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

0.8

− 8π − 4π 0 4π 8π0

0 .06

0 .12

0 .18

0 .24

0 .3

u0 (axial)

Fib

re le

ngth

(m

)

(τc −

τ 0)/τ 0

0

1

2

3

4

a b

Figure 4.4.2 The effects of TOD on the coherence time for a focused electromagnetic wave by a lensof NA = 0.1, originating from a PCF pumped with an ultra-short pulse with a pulse duration of 0.1 psand a power of 1000 W (relating to the field in Fig. 3.3.1). (a) stationary observation frame and (b)a nonstationary observation frame.

length the coherence (Fig. 4.4.1) properties change significantly. The coherence time

becomes complicated by the radiative and shifting processes of soliton dynamics.

The steep phase gradient of the point of destructive interference caused by the

diffraction by the lens, contributes to a modification of the field through the removal

of frequency components which modifies the temporal coherence. The point of

destructive interference associated with the carrier frequency of the input pulse to

the PCF occurs at u0 = 4π and, as expected, the coherence time and bandwidth

shift around this point. Since RS shifts the soliton toward infra-red wavelengths and

the point of destructive interference is frequency dependent, the temporal change in

the field by these points should move axially outward, which would correspond to

an increase in coherence.

As described earlier in Section 3.3 the TOD effect is a significant contribution

in SC generation. The perturbation that the TOD effect places on the field is a

dominant phase contribution and should strongly affect the coherence of its focal

field. Figure 4.4.2 shows how the PCF field perturbed by third order dispersion,

relates to modifications in the focal plane of a lens of 0.1 NA. As expected the

stationary observation frame for both the field effected by RS and the TOD effect

shows strong spectral broadening which is caused by the rapid spectral expansion

of the field associated with soliton formation. The nonstationary reference frame is

different as it is affected by the path difference associated with the lens.

73

Chapter 4

−8π −4π 0 4π 8π0

250

500

750

1000

u0 (axial)

Inpu

t pow

er (

W)

(τc −

τ 0)/τ 0

−0.5

0

0.5

1

−8π −4π 0 4π 8π0

250

500

750

1000

u0 (axial)

Inpu

t pow

er (

W)

(τc −

τ 0)/τ 0

0

1

2

3

4a b

Figure 4.4.3 The effects of RS on the coherence time for a focused electromagnetic wave by a lensof NA = 0.1, originating from a PCF pumped with an ultra-short pulse with a pulse duration of 0.1 psand a fibre length of 0.3 m (relating to the field in Fig. 3.3.6). (a) stationary observation frame and(b) a nonstationary observation frame.

−8π −4π 0 4π 8π0

250

500

750

1000

u0 (axial)

Inpu

t pow

er (

W)

(τc −

τ 0)/τ 0

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

−8π −4π 0 4π 8π0

250

500

750

1000

u0 (axial)

Inpu

t pow

er (

W)

(τc −

τ 0)/τ 0

0

1

2

3

4a b

Figure 4.4.4 The effects of TOD on the coherence time for a focused electromagnetic wave by a lensof NA = 0.1, originating from a PCF pumped with an ultra-short pulse with a pulse duration of 0.1 psand a fibre length of 0.3 m (relating to the field in Fig. 3.3.5). (a) stationary observation frame and(b) a nonstationary observation frame.

The coherence times for the cases of RS and the TOD effect should also

convey the previous effects when considering a variation in input peak power

and is important as it determines the soliton order, which then relates to the

structure temporal and spectral properties. Figures 4.4.3 and 4.4.4 shows the power

dependence of both the stationary and nonstationary observation frames for the RS

and the TOD effect. For the stationary case, RS and the TOD effect are dominant

on the outside and inside (with respect to the origin) of the points of destructive

interference, respectively. The spectral shifting associated with the spatial phase

around the point of destructive interference causes an increase in coherence which is

amplified by the spectral shifting properties associated with RS and the TOD effect.

74

Chapter 4

0

0.5

1

1.5

2

2.5

u0 (axial)

S/S

0

−6π −3π 0 3π 6π

Diffraction fieldInput field

0.01

0.02

0.03

0.04

0.05

u0 (axial)

S/S

0

0 2π 4π 6π

u0 (axial)

v 0 (ra

dial

)

−6π −3π 0 3π 6π

−3π

− 3π

2

0

3π

2

3π

S/S

0 (10

log 10

)

−40

−30

−20

−10

0

u0 (axial)

Tim

e (p

s)

−6π −3π 0 3π 6π−1

−0.5

0

0.5

1

S/S

0 (10

log 10

)

−40

−30

−20

−10

0

a b

c d

Figure 4.5.1 The temporal effects of a SC propagating through the focus of a low NA (0.1) objective.(a) On axis diffraction centred at the focal point (the full temporal evolution of the SC on the axis isdescribed in Appendix C). (b) On axis diffraction centred at u0 = 5π. (c) Radial and axial diffractionpattern centred at the focal point (the full temporal evolution of the SC in the radial and axial directionis described in Appendix C). (d) Complete axial and temporal diffraction field.

4.5 Supercontinuum generation

The influence of the lens results in a superposition of amplitude and phase which

determines the diffracted focal distribution (Fig. 4.5.1a). Since the SC field contains

structured temporal components, as it encounters singularities, it is expected that

the temporal modification would be more significant.93 If the frequency distribution

of a temporal field coincides with the frequency dependence of the points of

destructive interference, a pulsed feature would be removed (Fig. 4.5.1b). Since

there is a path difference incurred across the aperture and an increased temporal

and spectral extent of the SC, the diffraction in the focal plane is more dramatic (Fig.

4.5.1c). The focal distribution on the axis (Fig. 4.5.1d) shows the complexity of the

points of destructive interference of the lens diffraction and how they manipulate

the SC field temporal structure.

75

Chapter 4

The correlation function of a field point essentially provides a measure of the

frequency component variation. The degree of coherence within a focused SC field is

expected to vary dramatically due to the removal of frequencies within the temporal

profile caused by the points of destructive interference and is quantified through

the coherence time τc (Figs. 4.5.2a-d). For the stationary observation frame, the

coherence time changes around the region of the points of destructive interference

(Fig. 4.5.2a) and in fact an enhancement of the coherence time occurs because of the

spectral redistribution that modifies the bandwidth. Compared with the coherence

time, τ0 = 0.005 ps, of the SC field before it is focused, τc at the points of destructive

interference is enhanced by a factor of 2. The coherence time in this situation is

symmetric with respect to the focal plane, which is physically expected since it is

contributed by a single axial position. In this case, the spatial phase contribution

from the lens diffraction is unchanged during the correlation measurement, since the

diffraction equation is symmetric with respect to the focal plane. This symmetry

holds for larger NA lenses and thus the coherence time shows little variation with

NA (Fig. 4.5.2c).

However, depending on the observational view the calculated coherence time

is different. For the nonstationary observation frame (e.g. in a time resolved

experiment), the coherence time shows a remarkable difference (Fig. 4.5.2b) and

is caused by the path difference incurred by the rays which pass the extremities of

the lens compared to rays on the optical axis. Further, the path difference is not

symmetric with respect to the focal plane. This effect can be confirmed by changing

the NA (Fig. 4.5.2d) where the coherence time changes dramatically. Such an effect

leads to the enhancement of the coherence time by a factor of 3 near the point of

destructive interference before the focal plane. This effect occurs because of the

variation of the path difference through the focal plane and the change in sign of

the spatial phase on either side of the focus. Though both observational frames are

valid in a laboratory measurement, the nonstationary observation frame has greater

consequences. The coherence time is strongly dependent on the temporal variance

of the input field as well as the spatial phase contribution from the lens diffraction.

This effect would have a strong impact on time resolved (or frequency resolved)

measurements and would rely on the characteristics of the SC field and the NA of

76

Chapter 4

u0 (axial)

v 0 (ra

dial

)

−6π −3π 0 3π 6π

−3π

− 3π

2

0

3π

2

3π

τ c (ps

)

0.005

0.007

0.009

0.011

u0 (axial)

v 0 (ra

dial

)

−6π −3π 0 3π 6π

−3π

− 3π

2

0

3π

2

3π

τ c (ps

)

0.006

0.01

0.014

0.018

u0 (axial)

τ c (ps

)

−6π −3π 0 3π 6π0.005

0.007

0.009

0.011

NA = 0.1NA = 0.14NA = 0.2

u0 (axial)

τ c (ps

)

−6π −3π 0 3π 6π

0.005

0.01

0.015

0.02

NA = 0.1NA = 0.14NA = 0.2

a b

c d

Figure 4.5.2 The coherence time within a focused SC for the stationary and the non-stationary cases.(a) the axial and radial distribution of the coherence time for the 0.1 NA lens for the stationary case;(b) the axial and radial distribution of the coherence time for the 0.1 NA lens for the non-stationarycase; (c) the effect of NA on the coherence time on the axis for the stationary case; (d) the effect ofNA on the coherence time on the axis for the non-stationary case.

77

Chapter 4

the lens.

The phase on the SC temporal profile is dependent on the physical origin of

nonlinear and dispersive effects that occur because of the nonlinear PCF. The

dominant effect in the initial pulse propagation through a PCF is the balance

between self phase modulation and second order dispersion, as the pulse proceeds

further into the fibre higher order dispersive effects become more dominant (Fig.

4.5.3a). The ultrashort pulse initially forms a higher order soliton and at a particular

point in the propagation, fissions into many fundamental solitons. The phase

contribution caused by these effects can be isolated by observing the coherence

time in the diffraction of a lens using an input field generated by a PCF with

varying length (Fig. 4.5.3b). Initially, the temporal coherence behaves similar to a

chirped hyperbolic secant pulse shape (partially coherent source) with a predictable

structure, but at a particular length corresponding to the fission length of the higher

order soliton the coherence time dramatically changes. This observation confirms

that an increase in phase complexity added to the original ultrashort pulse (with

linear phase25) is coupled with the spatial phase from the lens diffraction to modify

the correlation of the electromagnetic field.

The formation of high order solitons is power dependent due to SPM and it

would be expected that the temporal coherence in the focal region would change

dramatically. Fig. 4.5.4 shows the temporal coherence of focused electromagnetic

field produced by a PCF coupled with different input power pulses. It can be

seen that the stationary observation frame has a more predictable structure. As

the power of the input pulse increases the spectral and temporal features of the

output change (Fig. 4.5.4a). In the focal region these spectral features coincide with

frequency shifting property of the phase singularities which changes the temporal

coherence. This is evident in Fig. 4.5.4b as it can be seen that the strength of

the coherence time variation changes from side to side around the phase singularity.

The coherence time variation is much stronger on the inside of the phase singularity

which coincides with red shifted radiation (Gbur et al.54) which is understandable

since this would correspond to the removal blue shifted dispersive waves from the

SC spectra and hence the diffraction superposition of red shifted soliton. Similar to

78

Chapter 4

−1 −0.5 0 0.5 1

0.03

0.06

0.09

0.12

0.15

Time (ps)

Fib

re le

ngth

(m

)

S/S

0

1

2

3

4

5

6

7

8

−6π −3π 0 3π 6π

0.02

0.04

0.06

0.08

0.1

0.12

0.14

u0 (axial)

Fib

re le

ngth

(m

)

(τc−

τ 0)/τ 0

−0.5

0

0.5

1

−6π −3π 0 3π 6π

0.03

0.06

0.09

0.12

0.15

u0 (axial)

Fib

re le

ngth

(m

)

(τc −

τ0)/

τ 0

0

0.5

1

1.5

2

2.5

3

a

b

c

(1)

(2)

Figure 4.5.3 Propagation of an ultrashort hyperbolic secant pulse through a nonlinear PCF. (a) fieldpropagation as a function of fibre length; (b) coherence time for the stationary observation frame inthe focal region of a 0.1 NA lens for different length fibre and (c) coherence time for the nonstationaryobservation frame in the focal region of a 0.1 NA lens for different length fibre. The peak input powerto the photonic crystal fibre is 2500 W with a pulse duration of 100 fs.(1) represents the cross sectionused for Fig. 4.5.2c (blue) and (2) represents the cross section used for Fig. 4.5.2d (blue)

79

Chapter 4

−1 −0.5 0 0.5 10

500

1000

1500

2000

2500

Time (ps)

Inpu

t pow

er (

W)

S (

norm

.)

0.2

0.4

0.6

0.8

1

−6π −3π 0 3π 6π0

500

1000

1500

2000

2500

u0 (axial)

Inpu

t pow

er (

W)

(τc−

τ 0)/τ 0

−0.5

0

0.5

1

−6π −3π 0 3π 6π0

500

1000

1500

2000

2500

u0 (axial)

Inpu

t pow

er (

W)

(τc−

τ 0)/τ 0

0

1

2

3

4

a

b

c

Figure 4.5.4 Propagation of an ultrashort hyperbolic secant pulse through a nonlinear PCF. (a)variation of output temporal envelope by varying the input power. (b) the coherence time of thestationary observation frame of the focal region of a 0.1 NA lens for different for the field obtainedfrom different input powers. (c) the coherence time of the nonstationary observation frame of the focalregion of a 0.1 NA lens for different for the field obtained from different input powers.

80

Chapter 4

numerical simulation for the variation in fibre length, the nonstationary observation

frame becomes complicated due to the spatial and temporal coupling effect (Fig.

4.5.4c).

Statistically, SC generation varies from pulse to pulse due to fluctuations

created by noise94–96 such as spontaneous Raman scattering. The correlation and

therefore the coherence time in the focus would also vary at the single pulse level.

However, since the majority of applications involving a SC field involve the ensemble

measurement, these fluctuations would average out and should result in minimal

fluctuations in the coherence time.

Physically, the temporal coherence of a field is related to the bandwidth ∆ν by

τc = 1/∆ν. Though the definition of the bandwidth is not straightforward in the case

of the point of destructive interference, the mean frequency µ, introduced previously

for the description of focusing a polychromatic wave54, can be used to confirm the

temporal correlation and relative frequency shifting of the focused SC wave. From

both observational frames, the mean frequency would be related to the inverse of its

temporal coherence (Fig. 4.5.5). For a SC it is expected that the frequency shifting

would be much broader due to the increase in bandwidth. Since the increase in

bandwidth results in a wider region of singular points, it is also expected that the

spatial location of spectral shifting would be broader. However, Fig. 4.5.5a shows a

behaviour which is different from what is seen in previous literature.54 The frequency

shifting in the radial plane from the focal point moving radially outward becomes

less profound. The superposition of the diffraction field and the input SC in this

region makes the bandwidth narrower, causing a reduction in the magnitude of the

mean frequency. Specifically, for the stationary observation frame (Fig. 4.5.5a) the

mean frequency is symmetric about the focal point which is due to the symmetric

nature of the diffraction process and the spatial phase contribution remains constant.

However, for the nonstationary observation frame (Fig. 4.5.5b) the result becomes

asymmetric because of the observed phenomenon in Fig. 4.5.2b.

81

Chapter 4

u0 (axial)

v 0 (ra

dial

)

−6π −3π 0 3π 6π

−3π

− 3π

2

0

3π

2

3π

(µ−

µ 0)/µ 0

−0.1

−0.05

0

0.05

0.1

u0 (axial)

v 0 (ra

dial

)

−6π −3π 0 3π 6π

−3π

− 3π

2

0

3π

2

3π

(µ−

µ 0)/µ 0

−0.05

0

0.05

a

b

Figure 4.5.5 Mean frequency distribution of the focused SC in the axial and radial plane of a 0.1 NAlens for stationary (a) and non-stationary (b) cases.

82

Chapter 4

4.6 Conclusion

To summarise, it has been demonstrated that there exists a coupling between the

temporal and spatial phases that arise from the diffraction of a SC field by a lens.

The contribution of temporal phase from the input source superimposes with the

diffraction field of a low NA lens to modify the bandwidth of the input which alters

its correlation. At a particular point where the pulse incurs a phase in the evolution

through the PCF, which is attributed to soliton formation and fission, changes

the correlation from a simple predictable structure to a complex structure. These

effects can be observed from two different observation frames which gives rise to

significantly different coherence times. Consequently, for a nonstationary observer

with the addition of complex temporal input phase can enhance the coherence time

by a factor of 3.

The alteration of bandwidth is extremely important and would change the

excitation frequency range that can be applied in microscopy applications involving

multiple wavelength excitation. The interesting effects on the bandwidth and

temporal correlation at points of destructive interference could provide interesting

dynamics for applications such as optical vortex metrology where these singular

regions provide signatures for phase unwrapping. In all these applications, the SC

source provides the capability to tailor the temporal coherence and bandwidth within

the focal region for a particular application.

83

Chapter 4

84

Chapter 5

Vectorial Diffraction

5.1 Introduction

When an electromagnetic field is focused by a high numerical aperture (NA) lens,

energy is transferred from the incident polarisation state to the transverse orthogonal

and the longitudinal field components, which is called depolarisation. The transfer

of energy due to depolarisation is related to a change in coherence, which physically

can be quantified through the coherence time of each vectorial component and

the coherence time of the cross correlation between vectorial components. The

theoretical treatment of the coherence effects of a vector field have been previously

investigated by Wolf in 200397 and by Dennis in 2004.98 However, these studies

only investigate the frequency dependence of an incident polychromatic wave. The

extension that is made in this thesis is to investigate how these correlations influence

the temporal aspect of a propagating wave such as supercontinuum (SC) field in the

focal region of a high NA lens.

Physically, the spatial and temporal phase coupling which was presented in

Chapter 4 is not restricted to scalar fields and would manifest as a cross phase

coupling between vectorial field components. Birefringence is an important property

of a photonic crystal fibre (PCF) because it allows the capability of maintaining

the polarisation state by creating both strong modal guidance and spectrally

dependent vectorial field components. The complicated temporal phase associated

85

Chapter 5

with the birefringent modes of the SC field couples with the spatial phase from

the diffraction by the lens93, which would produce interesting correlations under

vectorial diffraction conditions.

The aim of this chapter is to provide a detailed theoretical description of the

degree of coherence of a SC field under vectorial diffraction conditions. We present

the coherence relationship between the field components produced by depolarisation

under high NA diffraction and the relationship between the SC fields produced by

a highly birefringent PCF when diffracted by a lens of high NA.

5.2 Three-dimesional coherence matrix

The characterisation of the degree of coherence for a vectorial field E (V, ω) begins

with the correlation equation given by Eq. (2.5.2), which extends to a coherence

matrix and is calculated by

g1mn (V, τ) =

〈E∗m (V, t) ,En (V, t+ τ)〉

[〈|Em (V, t) |2〉 〈|En (V, t) |2〉]1/2

=

g1xx(V, τ) g1

xy(V, τ) g1xz(V, τ)

g1yx(V, τ) g1

yy(V, τ) g1yz(V, τ)

g1zx(V, τ) g1

zy(V, τ) g1zz(V, τ)

, (5.2.1)

where m and n are the polarisation states in the spatial directions x, y, z and V

represents the collective dimensions of the diffraction volume. For m and n = x,

g1 represents the autocorrelation of the electric field component with a polarisation

orientation in the x−direction. For m 6= n, g1 represents the cross correlation of

the vector components of the field. Physically, this matrix quantifies the transfer of

energy between field components and provides the ability to analyse the polarisation

properties of the degree of coherence for the focal region.

The components of the field for a linear polarisation state with an arbitrary

polarisation angle under vectorial diffraction can be determined by Eqs. (2.4.10),

86

Chapter 5

(2.4.12) and (2.4.13). When combined these equations form the following set of

equations

Ex (V, t) =iω

2c(aI0 (V, t) + (a cos (2ψ) + b sin (2ψ)) I2 (V, t)) , (5.2.2)

Ey (V, t) =iω

2c(bI0 (V, t) + (a sin (2ψ) + b cos (2ψ)) I2 (V, t)) , (5.2.3)

Ez (V, t) =iω

2c(2i (a cos (ψ) + b sin (ψ)) I1 (V, t)) . (5.2.4)

87

Chap

ter5

Table 5.2.1 Contributions to the field E for the x− y− and z−axes.

axis ψ Eh(×iω2c

) Ev(×iω2c

) E(×iω2c

)

Ehx = I0 (z, t) + I2 (z, t) Ev

x = 0 Ex = aI0 (z, t) + aI2 (z, t)z 0 Eh

y = 0 Evy = I0 (z, t) − I2 (z, t) Ey = bI0 (z, t) − bI2 (z, t)

Ehz = 0 Ev

z = 0 Ez = 0

Ehx = I0 (y, t) − I2 (y, t) Ev

x = 0 Ex = aI0 (y, t) − aI2 (y, t)y 90 or 270 Eh

y = 0 Evy = I0 (y, t) + I2 (y, t) Ey = bI0 (y, t) + bI2 (y, t)

Ehz = 0 Ev

z = 2iI1 (y, t) Ez = b2iI1 (y, t)

Ehx = I0 (x, t) + I2 (x, t) Ev

x = 0 Ex = aI0 (x, t) + aI2 (x, t)x 0 or 180 Eh

y = 0 Evy = I0 (x, t) − I2 (x, t) Ey = bI0 (x, t) − bI2 (x, t)

Ehz = 2iI1 (x, t) Ev

z = 0 Ez = a2iI1 (x, t)

88

Chapter 5

The vectorial field components which contribute to the degree of coherence for

the x−, y− and z−axes are shown in Table 5.2.1. The degree of coherence for

a linear polarisation state with an arbitrary polarisation angle propagating in the

directions x, y and z can be determined in terms of the field components I0, I1 and

I2, which are given by

g1xx (x, τ) =

〈aI∗0 (x, t) + aI∗2 (x, t) , aI0 (x, t+ τ) + aI2 (x, t+ τ)〉[〈|aI0 (x, t) + aI2 (x, t) |2〉 〈|aI0 (x, t) + aI2 (x, t) |2〉]1/2

, (5.2.5)

g1xy (x, τ) =

〈aI∗0 (x, t) + aI∗2 (x, t) , bI0 (x, t+ τ) − bI2 (x, t+ τ)〉[〈|aI0 (x, t) + aI2 (x, t) |2〉 〈|bI0 (x, t) − bI2 (x, t) |2〉]1/2

, (5.2.6)

g1xz (x, τ) =

〈aI∗0 (x, t) + aI∗2 (x, t) , a2iI1 (x, t+ τ)〉[〈|aI0 (x, t) + aI2 (x, t) |2〉 〈|a2iI1 (x, t) |2〉]1/2

, (5.2.7)

g1yx (x, τ) =

〈bI∗0 (x, t) − bI∗2 (x, t) , aI0 (x, t+ τ) + aI2 (x, t+ τ)〉[〈|bI0 (x, t) − bI2 (x, t) |2〉 〈|aI0 (x, t) + aI2 (x, t) |2〉]1/2

, (5.2.8)

g1yy (x, τ) =

〈bI∗0 (x, t) − bI∗2 (x, t) , bI0 (x, t+ τ) − bI2 (x, t+ τ)〉[〈|bI0 (x, t) − bI2 (x, t) |2〉 〈|bI0 (x, t) − bI2 (x, t) |2〉]1/2

, (5.2.9)

g1yz (x, τ) =

〈bI∗0 (x, t) − bI∗2 (x, t) , a2iI1 (x, t+ τ)〉[〈|bI0 (x, t) − bI2 (x, t) |2〉 〈|a2iI1 (x, t) |2〉]1/2

, (5.2.10)

g1zx (x, τ) =

〈a2iI∗1 (x, t) , aI0 (x, t+ τ) + aI2 (x, t+ τ)〉[〈|a2iI1 (x, t) |2〉 〈|aI0 (x, t) + aI2 (x, t) |2〉]1/2

, (5.2.11)

g1zy (x, τ) =

〈a2iI∗1 (x, t) , bI0 (x, t+ τ) − bI2 (x, t+ τ)〉[〈|a2iI1 (x, t) |2〉 〈|bI0 (x, t) − bI2 (x, t) |2〉]1/2

, (5.2.12)

g1zz (x, τ) =

〈a2iI∗1 (x, t) , a2iI1 (x, t+ τ)〉[〈|a2iI1 (x, t) |2〉 〈|a2iI1 (x, t) |2〉]1/2

, (5.2.13)

89

Chapter 5

g1xx (y, τ) =

〈aI∗0 (y, t) − aI∗2 (y, t) , aI0 (y, t+ τ) − aI2 (y, t+ τ)〉[〈|aI0 (y, t) − aI2 (y, t) |2〉 〈|aI0 (y, t) − aI2 (y, t) |2〉]1/2

, (5.2.14)

g1xy (y, τ) =

〈aI∗0 (y, t) − aI∗2 (y, t) , bI0 (y, t+ τ) + bI2 (y, t+ τ)〉[〈|aI0 (y, t) − aI2 (y, t) |2〉 〈|bI0 (y, t) + bI2 (y, t) |2〉]1/2

, (5.2.15)

g1xz (y, τ) =

〈aI∗0 (y, t) − aI∗2 (y, t) , b2iI1 (y, t+ τ)〉[〈|aI0 (y, t) − aI2 (y, t) |2〉 〈|b2iI1 (y, t) |2〉]1/2

, (5.2.16)

g1yx (y, τ) =

〈bI∗0 (y, t) + bI∗2 (y, t) , aI0 (y, t+ τ) − aI2 (y, t+ τ)〉[〈|bI0 (y, t) + bI2 (y, t) |2〉 〈|aI0 (y, t) − aI2 (y, t) |2〉]1/2

, (5.2.17)

g1yy (y, τ) =

〈bI∗0 (y, t) + bI∗2 (y, t) , bI0 (y, t+ τ) + bI2 (y, t+ τ)〉[〈|bI0 (y, t) + bI2 (y, t) |2〉 〈|bI0 (y, t) + bI2 (y, t) |2〉]1/2

, (5.2.18)

g1yz (y, τ) =

〈bI∗0 (y, t) + bI∗2 (y, t) , b2iI1 (y, t+ τ)〉[〈|bI0 (y, t) + bI2 (y, t) |2〉 〈|b2iI1 (y, t) |2〉]1/2

, (5.2.19)

g1zx (y, τ) =

〈b2iI∗1 (y, t) , aI0 (y, t+ τ) − aI2 (y, t+ τ)〉[〈|b2iI1 (y, t) |2〉 〈|aI0 (y, t) − aI2 (y, t) |2〉]1/2

, (5.2.20)

g1zy (y, τ) =

〈b2iI∗1 (y, t) , bI0 (y, t+ τ) + bI2 (y, t+ τ)〉[〈|b2iI1 (y, t) |2〉 〈|bI0 (y, t) + bI2 (y, t) |2〉]1/2

, (5.2.21)

g1zz (y, τ) =

〈b2iI∗1 (x, t) , b2iI1 (x, t+ τ)〉[〈|b2iI1 (x, t) |2〉 〈|b2iI1 (x, t) |2〉]1/2

, (5.2.22)

g1xx (z, τ) =

〈aI∗0 (z, t) + aI∗2 (z, t) , aI0 (z, t+ τ) + aI2 (z, t+ τ)〉[〈|aI0 (z, t) + aI2 (z, t) |2〉 〈|aI0 (z, t) + aI2 (z, t) |2〉]1/2

, (5.2.23)

g1xy (z, τ) =

〈aI∗0 (z, t) + aI∗2 (z, t) , bI0 (z, t+ τ) − bI2 (z, t+ τ)〉[〈|aI0 (z, t) + aI2 (z, t) |2〉 〈|bI0 (z, t) − bI2 (z, t) |2〉]1/2

, (5.2.24)

g1yx (z, τ) =

〈bI∗0 (z, t) − bI∗2 (z, t) , aI0 (z, t+ τ) + aI2 (z, t+ τ)〉[〈|bI0 (z, t) − bI2 (z, t) |2〉 〈|aI0 (z, t) + aI2 (z, t) |2〉]1/2

, (5.2.25)

g1yy (z, τ) =

〈bI∗0 (z, t) − bI∗2 (z, t) , bI0 (z, t+ τ) − bI2 (z, t+ τ)〉[〈|bI0 (z, t) − bI2 (z, t) |2〉 〈|bI0 (z, t) − bI2 (z, t) |2〉]1/2

, (5.2.26)

g1xz (z, τ) = g1

zx (z, τ) = g1yz (z, τ) = g1

zy (z, τ) = g1zz (z, τ) = 0, (5.2.27)

respectively. The set of coherence functions can be use to determine the coherence

90

Chapter 5

times of the focus under conditions of vectorial diffraction, which are given by

τ cmn(V ) =

∫ ∞

−∞

|g1mn (V, τ) |2dτ

=

τ cxx(V ) τ cxy(V ) τ cxz(V )

τ cyx(V ) τ cyy(V ) τ cyz(V )

τ czx(V ) τ czy(V ) τ czz(V )

. (5.2.28)

For a horizontal polarisation state (a = 1, b = 0), the coherence times for the

x−, y− and z−axes are given by

τ cxx (x) =

∫ ∞

−∞

|g1xx (x, τ) |2dτ, (5.2.29)

τ cxz (x) =

∫ ∞

−∞

|g1xz (x, τ) |2dτ, (5.2.30)

τ czx (x) =

∫ ∞

−∞

|g1zx (x, τ) |2dτ, (5.2.31)

τ czz (x) =

∫ ∞

−∞

|g1zz (x, τ) |2dτ, (5.2.32)

τ cxx (y) =

∫ ∞

−∞

|g1xx (y, τ) |2dτ, (5.2.33)

τ cxx (z) =

∫ ∞

−∞

|g1xx (z, τ) |2dτ. (5.2.34)

The coherence times that are not defined in Eqs. (5.2.29) - (5.2.34) are equal

to zero, which are caused by the polarisation coefficient b = 0. When the NA is

below 0.7 the effects of depolarisation can be neglected and the terms I1 and I2 = 0.

Under these conditions the field E reduces to a scalar field determined by I0, where

the degree of coherence and the coherence time are given by

91

Chapter 5

(τc −

τ0)/

τ 0

−6π −3π 0 3π 6π−0.05

−0.025

0

0.025

0.05

u0 (z)

NA = 1NA = 0.1

Figure 5.2.1 A comparison between the coherence times for a lens of NA = 1 and 0.1 with hyperbolicsecant ultrashort pulse with a width of 0.1 ps.

g1mn (V, τ) = g1

xx (V, τ) =〈I0 (V, t) , I0 (V, t+ τ)〉

[〈|I0 (V, t) |2〉 〈|I0 (V, t) |2〉]1/2, (5.2.35)

τ cxx (V ) =

∫ ∞

−∞

|g1xx (V, τ) |2dτ, (5.2.36)

respectively. Equations (5.2.29) - (5.2.36) are used in Section 5.3.1 to characterise

the focus of a SC field under vectorial diffraction conditions.

Consider the general case of a hyperbolic secant with a pulse duration of 0.1 ps.

The coherence time for a NA = 0.1 under vectorial diffraction conditions is shown

in Fig. 5.2.1, which gives an identical result to the coherence time produced by

Fresnel diffraction. Under high NA vectorial diffraction conditions the coherence of

the field is no longer influenced by the point of destructive interference, which is due

to depolarisation.

The final mathematical analysis involves an incident field with a linear polarisa-

tion orientation at 45. The incident field is given by

Ei45 = Ehi + Evi =

1√2

Ehix

Ehiy

Ehiz

+1√2

Evix

Eviy

Eviz

. (5.2.37)

92

Chapter 5

When diffracted by a high NA the degree of coherence for the field E45 becomes

complicated. For this investigation the degree of coherence is calculated for only the

optical axis where the coherence matrix is determined by Eqs. (5.2.23) - (5.2.26).

The coherence times generated by the degree of coherence for E45 is given by

τ cmn (z) =

τ cxx (z) τ cxy (z)

τ cyx (z) τ cyy (z)

. (5.2.38)

The observation is along the optical axis, where the vectorial diffraction

contribution from the Ez component is zero and is why the coherence time has

only contributions from Ex and Ey. Equations (5.2.23) - (5.2.26) and (5.2.38) are

used in Section 5.3.2 to understand the influence of cross coupling in the degree of

coherence for a SC field in the focal region.

5.3 Vectorial diffraction of a supercontinuum

5.3.1 Linear Polarisation

For a linear polarisation state the degree of coherence and the coherence time are

determined by the theoretical derivations in Section 5.2. It is expected that the

coherence times for the electric field in the direction of the incident polarisation state

Ex would be influenced by the points of destructive interference. The coherence time

for the SC diffraction by a lens is shown in Fig. 5.3.1 for the x−, y− and z−axes.

The input polarisation state to the PCF is in the x−direction and the analysis

is for the autocorrelation of the field component Ex determined by Eqs. (5.2.29),

(5.2.33) and (5.2.34). Figure 5.3.1 shows three key effects: the influence of spatial

phase through the points of destructive interefence on the field; the reduction of the

coherence time with increased NA; and a lateral (x− and y−axes) and a longitudinal

(z−axis) shift in the coherence time. The gradual shift inward (y) and outward (x)

is due to the change in superposition of the wave as it passes through the lens. The

modification of the field by the spatial phase associated with the lens changes the

field Ex to becoming slightly asymmetric (over the xy−plane) which is only seen

93

Chapter 5

−4π −2π 0 2π 4π−0.4

0

0.4

0.8

1.2

1.6

v0 (x)

(τc xx

−τc 0)/

τc 0

NA = 0.1NA = 0.3NA = 0.7NA = 1

−4π −2π 0 2π 4π−0.4

0

0.4

0.8

1.2

1.6

v0 (y)

(τc xx

−τc 0)/

τc 0

NA = 0.1NA = 0.3NA = 0.7NA = 1

−8π −4π 0 4π 8π−0.4

0

0.4

0.8

1.2

1.6

u0 (z)

(τc xx

−τc 0)/

τc 0

NA = 0.1NA = 0.3NA = 0.7NA = 1

−4π −2π 0 2π 4π−0.4

0

0.4

0.8

1.2

1.6

2

2.4

v0 (x)

(τc m

n−τc 0)/

τ 0

mn = xxmn = xz = zxmn = zz

a b

c d

Figure 5.3.1 The coherence time of the diffraction by a lens of varying numerical aperture along thex (a), y (b) and z (c) axes. These coherence times are calculated for the autocorrelation of the electricfield in the direction of the Ei (Ex). (d) the coherence times for the diffraction by a lens of NA = 1along the x axis, which contains the autocorrelation and cross-correlation coherence times with respectto the Ex and Ez fields.

under higher NA conditions. Along the optical axis a more significant change occurs

and is due to the superposition condition no longer forming points of destructive

interference under higher NA conditions. The coherence time for a NA = 0.1 is

identical to the coherence time obtained for the stationary observation frame shown

in Fig. 4.5.2c.

For the transverse axis x the field component Ez 6= 0 leading to a coherence

matrix containing cross coupling correlation terms between Ex and Ez. Figure

5.3.1d shows the coherence times simulated using Eqs. (5.2.29) - (5.2.32). The

interesting observation is that the coherence time generated by the cross coupling

of the field components (mn = xz) is not a simple superposition of the coherence

times generated by the autocorrelations (mn = xx and mn = zz). This effect

is understandable because the correlation is dependent on the phase structure of

94

Chapter 5

0 0.2 0.4 0.6 0.8 10

0.1

0.2

0.3

0.4

NAM

ean

of τ

c xx

xyz

Figure 5.3.2 The mean coherence time of a SC as a function of NA for the x, y and z−axes.

each component of the incident field and the transfer energy due to depolarisation.

Effectively, the coherence time formed by the cross coupling between the polarisation

states is a measure of the longevity of the elliptical polarisation produced by

depolarisation.

The influence of the NA can be quantified by calculating the mean of the

coherence time as a function of NA and is shown in Fig. 5.3.2. By increasing

the NA, there exists a redistribution of energy within the Ex component from the

y−axis to the x−axis, which alters the degree of coherence of the field. Along the

optical axis the change in the degree of coherence is different as there no longer

exists points of destructive interference, leading to a change in the mean coherence

time by an order of magnitude from low NA (0 − 0.4) to the high NA (1).

The diffraction by a lens has been shown in Chapter 4 to be influenced by

phase. Since the SC field is generated by an accumulation of phase associated

with nonlinearity and dispersion, it becomes important to assess how the degree

of coherence changes with phase under vectorial diffraction conditions. A method

for analysing this change is by calculating the diffraction of the SC produced by

a variation in fibre length (Fig. 5.3.3) along the optical axis of the focal region

of a lens of NA = 1 and is shown in Fig. 5.3.3. As the initial ultrashort pulse

travels through the optical fibre it accumulates phase which changes the spectral

and temporal components. There exists a point in the evolution where the temporal

coherence dramatically changes which is due to the formation and annihilation of a

95

Chapter 5

0 2π 4π 6π0

0.03

0.06

0.09

0.12

0.15

u0 (z)

Fib

re le

ngth

(m

)

(τc xx

−τc 0)/

τc 0

−0.1

−0.05

0

0.05

0.1

0.15

Figure 5.3.3 The coherence time of the autocorrelation of the diffraction by a lens of NA = 1 theelectric field Ex with variation in the fibre length.

0 2π 4π 6π0

500

1000

1500

2000

2500

u0 (z)

Inpu

t pow

er (

W)

(τc xx

− τ

c 0)/τc 0

−0.04

−0.02

0

0.02

0.04

0.06

Figure 5.3.4 The power dependence of coherence time in the focus of a NA = 1 lens for input fieldsgenerated by the nonlinear PCF of varying input power. The coherence time is for a linear polarisedfield orientated along the x direction.

high order soliton and is similar to behavior discussed in Chapter 4. However, under

vectorial diffraction conditions the fluctuation in coherence is less profound, which

is attributed to depolarisation and the reduced influence of the points of destructive

interference.

Similar to Chapter 4, the degree of coherence is dependent on the input power

to the PCF and is shown in Fig. 5.3.4. For low input powers (0 − 500 W ) the

variation in coherence time is small, which occurs because the phase on the pulse

is dominated by dispersion effects and has little influence from soliton dynamics

since the soliton order is small (1 − 5). With increased power the coherence time

is expected to change due to the increased dominance of nonlinearity. The higher

input power increases the initial order of the soliton which then after fission changes

96

Chapter 5

the degree of coherence, which enhances the coherence time.

5.3.2 Coupled mode propagation

So far the analysis has been restricted to a linear incident polarisation state, which

emphasises the depolarisation inherent from a high NA lens. The coupled mode

nonlinear Schrodinger equation allows the ability to simulate a SC field with a

polarisation orientation at 45 which can occur in highly birefringent PCF. The

output spectrograms of the SC field emerging from a highly birefringent PCF is

shown in Fig. 5.3.5. The field was generated using the dispersion and nonlinear

parameters discussed in Chapter 3 with a pulse duration of 100 fs and a peak

power of 2500 W . Also shown is the propagation of the ultrashort pulse along the

fibre with a length of 0.15 m.

−1 −0.5 0 0.5 1250

300

350

400

450

500

550

Time (ps)

Fre

quen

cy (

TH

z)

S0 (

norm

. 10l

og10

)

−20

−15

−10

−5

−1 −0.5 0 0.5 1250

300

350

400

450

500

550

Time (ps)

Fre

quen

cy (

TH

z)

S0 (

norm

. 10l

og10

)

−20

−15

−10

−5

0

−1 −0.5 0 0.5 10

0.03

0.06

0.09

0.12

0.15

Time (ps)

Fib

re le

ngth

(m

)

S/S

0

0.5

1

1.5

2

2.5

3

3.5

−1 −0.5 0 0.5 10

0.03

0.06

0.09

0.12

0.15

Time (ps)

Fib

re le

ngth

(m

)

S/S

0

0.5

1

1.5

2

2.5

3

3.5

4

a b

c d

Figure 5.3.5 The PCF output field for an incident polarisation state at 45.(a) the horizontal (x)polarisation state, (b) the vertical (y) polarisation state, (c) the horizontal (x) polarisation state as afunction of fibre length, and (c) the vertical (y) polarisation state as a function of fibre length.

In the focal plane of the lens, these coupled modes should affect the coherence

matrix and the coherence times. Fig. 5.3.6 shows the coherence times produced

97

Chapter 5

−6π −3π 0 3π 6π−0.15

−0.075

0

0.075

0.15

u0 (z)

(τc m

n − τ

c 0)/τc 0

mn = xx

mn = xy = yx

mn = yy

Figure 5.3.6 The coherence time for the autocorrelations and cross correlations calculated for thediffraction by a lens of NA = 1 along the optical axis for the SC field generated in Fig 5.3.5.

from Eq. (5.2.38). It is evident that the coherence time for the cross correlation

between the modes is no longer the superposition between the autocorrelated fields

and occurs because of their non-constant relative phase.

The phase due to nonlinearity and dispersion can be isolated along the fibre

length to understand the influence of polarisation on the degree of coherence. Figure

5.3.7 shows the polarisation coherence times occurring due to the coupled modes of

the PCF along the optical axis as a function of the fibre length. The autocorrelations

behave in the same manner as depicted in Fig. 5.3.3 and Fig. 5.3.4, the degree of

coherence changes with input phase. The cross correlated degree of coherence in

Fig. 5.3.7b shows modulations which occur due to the differences in phase between

the fibres modes. As discussed in Chapter 3 and 4, the soliton fisson dynamics in SC

generation contains spectral expansion and contractions processes. The differences

between the soliton fluctuations of the fibre modes could be attributed to the

modulations shown in the cross correlated degree of coherence and the coherence

time in Fig. 5.3.7b. In both cases of a linear polarised SC field and a 45 polarised

SC field, the coherence time has a greater variance after the formation of the soliton

and the rapid spectral expansion of the SC field. After this point, the coherence

within the focal region becomes dominated by interference between dispersive waves

and the fundamental solitons.

The expansive spectral features of a SC can only be obtained by coupling an

98

Chapter 5

0 2π 4π 6π0

0.03

0.06

0.09

0.12

0.15

u0 (z)

Fib

re le

ngth

(m

)

(τc xx

−τc 0)/

τc 0

−0.05

0

0.05

0.1

0 2π 4π 6π0

0.03

0.06

0.09

0.12

0.15

u0 (z)

Fib

re le

ngth

(m

)

(τc xy

−τc 0)/

τc 0−0.1

−0.05

0

0.05

0.1

0.15

0.2

0 2π 4π 6π0

0.03

0.06

0.09

0.12

0.15

u0 (z)

Fib

re le

ngth

(m

)

(τc yy

−τc 0)/

τc 0

−0.05

0

0.05

a

b

c

Figure 5.3.7 The coherence time for the autocorrelations and cross correlations calculated for thediffraction by a lens of NA = 1 as a function of fibre length along the optical axis. (a) coherence timeproduced by the autocorrelation of Ex; (b) coherence time produced by the cross correlation Ex andEy; and (c) coherence time produced by the autocorrelation of Ey.

99

Chapter 5

ultrashort pulse of sufficient power to instigate the formation of solitary waves. The

effects of temporal phase on the focal region can also be analysed by observing the

change in coherence time as function of input power to the PCF (Fig. 5.3.8). At

low input powers, the output spectra of the PCF is dominated by the dispersion

of the fundamental solitons and under these conditions the phase accumulated

through propagation is relatively simple. The cross coupling between the focused

coupled modes in this case is shown as a small change in coherence time. With

the increase in input power the SC is formed by the amalgamation of nonlinear

and dispersive processes, and as expected rapidly expands the bandwidth. The

cross coupling between the focused coupled modes becomes complicated because

of the superposition of their differing phase, which results in subtle changes in the

coherence time (Fig. 5.3.8b).

5.4 Conclusions

In summary, the optical field components occurring because of depolarisation by

the diffraction of a high NA lens reduces the coherence time along the optical

axis which is attributed to the superposition of the wavefront no longer forming

points of destructive interference. Under conditions of vectorial diffraction, the mean

coherence time will change by an order of magnitude when the NA changes from a

low NA (0 - 0.4) to a high NA of 1. For the transverse axes the mean coherence time

increases and decreases in the x−direction and y−directions, respectively, which is

also due to depolarisation.

When considering the case of a vector field, the components of the field create

interesting cross coupling characteristics, which are determined by a coherence

matrix. When the SC modes of a highly birefringent PCF are focused by a high

NA objective, the coherence times produced by their autocorrelations are different

due to the phase differences between the modes. In addition, the coherence time

for the degree of coherence between these two modes (cross coupled coherence time)

is significantly different. In these cases of auto and cross correlation the temporal

phase is significantly contributing to the degree of coherence in the focal region, to

100

Chapter 5

0 2π 4π 6π0

500

1000

1500

2000

2500

u0 (z)

Inpu

t pow

er (

W)

(τc xx

− τ

c 0)/τc 0

−0.05

0

0.05

0 2π 4π 6π0

500

1000

1500

2000

2500

u0 (z)

Inpu

t pow

er (

W)

(τc xy

− τ

c 0)/τc 0

−0.05

0

0.05

0.1

0 2π 4π 6π0

500

1000

1500

2000

2500

u0 (z)

Inpu

t pow

er (

W)

(τc yy

− τ

c 0)/τc 0

−0.08

−0.06

−0.04

−0.02

0

0.02

0.04

0.06

a

b

c

Figure 5.3.8 The power dependence of coherence time in the focus of a NA = 1 lens for input fieldsgenerated by the nonlinear PCF of varying input power. The coherence matrix is for a linear polarisedfield orientated at 45 to the x direction. (a) coherence time produced by the autocorrelation of Ex;(b) coherence time produced by the cross correlation Ex and Ey; and (c) coherence time produced bythe autocorrelation of Ey.

101

Chapter 5

such an extent that phase difference between the two modes creates strong changes

in the coherence times.

102

Chapter 6

Conclusion

6.1 Thesis conclusion

The investigation in this thesis details the characterisation of polarised supercontin-

uum generation and its diffraction by a lens system. There are three major research

areas which have been covered, which are listed as follows

• The theoretical and experimental investigation of polarised supercontinuum

generation in a highly birefringent photonic crystal fibre.

• An investigation into the influence of Fresnel diffraction by a lens on a

supercontinuum field.

• An understanding of the diffraction of a supercontinuum field by a high

numerical aperture objective lens and the influence of polarisation on its degree

of coherence.

The theoretical understanding of supercontinuum generation was a crucial

component of the methodology of this thesis. Although the theoretical description

has been implemented previously through the coupled mode nonlinear Schrodinger

equation, the knowledge provided in this investigation is the study of highly

birefringent photonic crystal fibre with a birefringence approximately two orders

of magnitude larger then previously reported. The treatment in such a case is

103

Chapter 6

different to previously studied as the dispersion between the two linear polarised

modes is different producing different spectral features. An interesting consequence

of such high birefringence is the structure forces the inclusion of two zero dispersion

wavelengths. This inclusion is beneficial as it enhances the spectral extent of the

supercontinuum through high order dispersion and the enhancement of modulation

instability, which gives rise to nonlinear gain.

An experimental understanding and conformation of the spectral difference was

also presented. The two linear polarised modes show the most extensive spectral

features as the pulse is restricted to a fundamental mode. Light coupled within

one of these modes shows minimal polarisation degradation with the change around

the pump wavelength attributed to the depolarisation by the lens coupling to the

photonic crystal fibre.

Theoretically, the diffraction of a supercontinuum by a lens is an important

scientific achievement and will have a strong impact in microscopic applications. The

theoretical investigation is different to other previously investigated research since a

treatment of a focused supercontinuum has not been studied before. Previous studies

have only dealt with ultrashort pulses, which do not contain the extensive bandwidth

and the phase complexity inherent in a supercontinuum field. The innovation in this

investigation has provided an insight into the spatiotemporal effects of focusing a

supercontinuum and how the diffraction points of destructive interference affect both

the spectral and temporal structure of the field. The spatial and temporal phase

coupling within a focused supercontinuum must also depend on the method by which

the field is measured or observed. Under conditions of a reference frame in motion,

the temporal coherence of the field around the points of destructive interference is

enhanced by a factor of 3, which is greater than the stationary reference frame.

These effects provide the knowledge which could have consequences in applications

involving time resolved interferometric measurements.

The diffraction of a supercontinuum by a high numerical aperture objective lens

is an intuitive extension of the previous discussed phenomena. The importance of

this study is evident in the effect of depolarisation caused by the severe refraction

by the lens as it reduces the degree of coherence of the field along the optical axis.

104

Chapter 6

The degree of coherence for polarised propagation becomes a complex system of

correlations between polarisation states and is determined by a vectorial coherency

matrix. When focusing by a high numerical aperture, the presence of points of

destructive interference are absent along the optical axis, which reduces the mean

coherence time by an order of magnitude when the numerical aperture is increased

from 0.1 to 1. When the incident field is 45 polarised, the coherency matrix involves

the superposition of orthogonal fields and for a supercontinuum (coupled modes) is

influenced by the phase differences between the fibre modes.

The theoretical and experimental investigation provided in this thesis gives

comprehensive study into the ability of applying a supercontinuum to wider photonic

applications. When considering the way supercontinuum is to be applied within an

experiment, a consideration must be made on the spectral, temporal and phase

complexity of the field and how this may effect a measurement.

6.2 Future work

This thesis has provided a comprehensive description of supercontinuum generation

and the diffraction by a lens. The effects that have been described could be verified

by using a near-field scanning optical microscope. The technique used to verify

the effects described in this thesis would have to carefully measure the interference

of the focal volume. The near-field scanning optical microscope would be ideal

for this measurement as it would provide a means to map the focal region with

high accuracy. The limit of this approach is the coupling efficiency, the chromatic

dispersion of the fibre probe and the inability to determine the polarisation state.

Another method which could possibly be used to verify these coherence properties

is molecule scattering or dipole scattering from nanoparticles. This method would

allow the ability to determine the polarisation state; however the scattering would

be spectrally dependent.

In using both the Fresnel and the vectorial diffraction theory the treatment in

this thesis assumes that there is no chromatic aberrations caused by the frequency

dependent refractive index of the lens material. Although this extension to the

105

Chapter 6

theory is not trivial, it could provide interesting information about the degree of

coherence as discussed previously, the frequency depend refractive index of the lens

and the influence of points of destructive interference. The theoretical description

in this thesis could also be extended to a refractive index mismatch between the

immersion oil and cover slip of a sample.99–102 Under such condition the temporal

coherence of the supercontinuum field would change significantly and would be

heavily dependent on frequency dependent refractive indices of both the oil and

cover glass.

The applications which supercontinuum generation and the investigation in this

thesis would benefit is microscopy such as coherent anti-stokes Raman scattering

microscopy, nonlinear microscopy and endoscopy, and optical data storage. With the

current optical recording media being revolutionised by five-dimensional recording71,

supercontinuum generation would have an immediate impact in which it could

provide a means of simultaneous optical recording thereby significantly increasing

the read/write access times of current disc technology.

106

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116

Numerical Code for the CMNLS

A.1 Split step Fourier method

The calculation described in Chapter 3 use the symmetrised split step Fourier

method. The solution to the coupled mode nonlinear Schrodinger equation is not

analytic for the combination of nonlinearity and dispersion, so the calculation is split

and each part is applied separately over small steps. Mathematically is described

as:

∂U

∂z=

(

D + N)

U (A-1)

D and N are the dispersion and nonlinear operators. The solution to Eq. A-1

can be approximated by:

U (z + h, t) ≈ ehDehNU (A-2)

As explained in Chapter 3 the dispersion operator which is the propagation

coefficient β is expanded into a high order polynomial and is best applied in the

frequency domain where the nonlinear component is applied in the time domain.

The above equation (Eq. A-1) is second order accurate in step size h. The accuracy

can be improved by using symmetrised scheme which is given by:

U (z + h, t) ≈ eh

2De

∫

z+h

z

ˆN(z′)dz′eh

2DU (z, t) (A-3)

A–1

Appendix - A–2

The nonlinear integration step is calculated using a 4th order Runge Kutta

method.

A.2 Matlab Script

The Matlab scripts used to calculate the SC field in chapter 3 include to major

functions; the main input script (cmsim.m) and the split step method (ssvbc.m).

The input script contains two sub function to calculate the Raman response function

(Raman quadv.m) and to generate the time and frequency domains (input par.m).

The main input script then calls on the split step script to propagate the pulse

with in the photonic crystal fibre. The split step method script has a five sub

functions (nonlinear.m, opt shock.m, int raman.m, rk.m, nl750 parameters 780.m)

to increase the speed of calculation and in the case of nl750 parameters 780.m to

input the dispersion data for the particular photonic crystal fibre.

function [] = cmsim()

% Matlab function which calculates the pulse propagation in o ptical fibre

% using the coupled mode nonlinear Schrodinger equation. Thi s script is the

% front end file which uses the script ssvbc function. To use th is file

% first a Raman spectrum be generated for the particular time a nd frequency

% parameters.

%%%%%%%%%%%%%%%% Input parameters %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

prompt = ’Time window (ps):’ ,...

’Time window accuracy (2ˆ):’,...

’Centre wavelength (micrometer):’,...

’Tolerance:’ ,...

’Peak power (Watts):’,...

’Pulse width (ps):’,...

’Input Polarisation Angle (fraction of pi):’ ,...

’Fibre Length (m):’,...

’Raman File:’;

dlg_title = ’Input’ ;

num_lines = 1;

ord = 11;

def = ’2’ ,num2str(ord), ’0.78’ ,’-6’, ’2500’ ,’0.1’ ,’0’ , ’0.15’ ,’yes’ ;

answer = inputdlg(prompt,dlg_title,num_lines,def);

T = str2double(char(answer(1)));

nt = 2ˆstr2double(char(answer(2)));

A–2

Appendix - A–3

lam = str2double(char(answer(3)));

tol = 10ˆ(str2double(char(answer(4))));

P0 = str2double(char(answer(5)));

T1 = str2double(char(answer(6)));

phi = str2double(char(answer(7))). * pi;

fibre_length = str2double(char(answer(8)));

file = char(answer(9));

[dt,t,v,w] = input_par(T,lam,nt);

gamma = 0.095; T0 = 0.57./1000;

if min(v) < 0;

fprintf( ’Error frequency rande below zero’)

return ;

else

end

% Raman file

if strcmp(file, ’yes’) == 1;

[R] = raman_quadv(T,nt);

dlmwrite([’raman_temp_’ ,num2str(T), ’_’ ,num2str(nt),’.dat’ ],R);

else

R = dlmread([’raman_temp_’ ,num2str(T), ’_’ ,num2str(nt), ’.dat’ ]);

end

ua = cos(phi). * sqrt(P0). * sech((t)/T1); % Input polarised pulse

ub = sin(phi). * sqrt(P0). * sech((t)/T1); % Input polarised pulse

sprintf([ ’Angle = ’,num2str(phi * 180/pi),...

’ Power = ’ ,num2str(P0), ’ FWHM = ’,num2str(T1)])

% Determines the intial step size from the nonlinear coeffici ent

dz = 0.00125./(P0 * gamma); nl = round(fibre_length./dz);

for n = 1:1:length(fibre_length);

% Calculates the propagation from given length

[ua,ub,dz] = ssvbc(ua,ub,dz,nl,nt,dt,gamma,R,T0,tol);

nl = round(fibre_step./dz);

% Output file

filename = [ ’BJC_NLPCF_’ ,num2str(P0), ’_’,...

num2str(phi(num). * 180./pi), ’_’ ,num2str(n), ’.dat’];

dlmwrite(filename,[abs(ua),angle(ua),abs(ub),angle(u b)],...

’delimiter’ ,’\t’ );

end

% Time and frequency dimensions

function [dt,t,v,w] = input_par(T,lam,nt)

dt = T/(nt-1); t = (-T/2:dt:T/2).’;

w = wspace(T,nt);

vs = fftshift(w/(2 * pi));

v0 = 3e2/(lam); v = vs+v0;

A–3

Appendix - A–4

function [u1a,u1b] = ssvbc(u0a,u0b,dz,nz,nt,dt,gamma,R,T0,tol)

fig = waitbar(0, ’Please Wait’, ’name’ , ’Processing’);

% Frequency domain

w = 2* pi * [(1:nt/2),(-nt/2:-1)]’/(dt * nt);

[ha,hb] = nl750_parameters_780(45,45,w);

% Intial dispersion operation

uafft = exp(ha. * dz/2). * fft(u0a);

ubfft = exp(hb. * dz/2). * fft(u0b);

% Creating total length to stop

% while loop

TL = dz * nz;

L = 0;

h = dz;

ph = figure(’position’ ,[500 500 100 100]);

rbh1 = uicontrol(ph,’Style’ ,’toggle button’,...

’String’, ’Cancel’,...

’Units’,’normalized’ ,...

’Position’,[.2 .2 .7 .6], ’value’ ,0);

% Complete till length complete

while L < TL;

if (TL - L < h)

h = TL - L;

else

end

L = L + h; % calculates current length

% Determines nonlinear integral for a the

% current step size and and half the integral

% half that size.

[k1a,k1b] = nonlinear(u0a,u0b,R,gamma,T0,h,w,2);

[k2a,k2b] = nonlinear(u0a,u0b,R,gamma,T0,h,w,4);

% Makes a comparison to determine local error and

% whether its with in tolerence

error = ((norm(abs(k1a-k2a).ˆ2)) +...

(norm(abs(k1b-k2b).ˆ2))) ...

./((norm(abs(k2a).ˆ2)) + ...

(norm(abs(k2b).ˆ2)));

% adjusts step size according to error

h1 = h. * ( tol./error)ˆ0.2;

k2a = k2a./(u0a+1e-25);

k2b = k2b./(u0b+1e-25);

% completes nonlinear step

u0a = ifft(uafft). * ( k2a);

A–4

Appendix - A–5

u0b = ifft(ubfft). * ( k2b);

% adds another nonlinear step

uafft = exp(ha. * h). * fft(u0a);

ubfft = exp(hb. * h). * fft(u0b);

waitbar(L./TL,fig,’Processing ’ );

h = h1;

val = get(rbh1,’value’);

if val == 1;

break ;

end

end

% removes the additonal dispersion step

u1a = ifft(exp(-ha. * dz/2). * uafft);

u1b = ifft(exp(-hb. * dz/2). * ubfft);

delete(fig)

delete(ph)

function [u1a,u1b] = nonlinear(u0a,u0b,R,gamma,T0,dz,w,iter)

% This function computes the nonlinear integration by the a

% fourth order Runge Kutta method

A = 1;

B = 2/3;

C = [1,0.5,0.5,1];

uva = u0a;

uvb = u0b;

u0a_abs = abs(u0a).ˆ2;

u0b_abs = abs(u0b).ˆ2;

[R0] = int_raman(R,u0a_abs+u0b_abs);

for is = 1:iter;

ka = zeros(size(u0a));

kb = zeros(size(u0b));

for in = 1:4;

uva = uva + C(in). * ka(:,in);

uvb = uvb + C(in). * kb(:,in);

uva_abs = abs(uva).ˆ2;

uvb_abs = abs(uvb).ˆ2;

[R1] = int_raman(R,uva_abs+uvb_abs);

[ts1] = opt_shock(uva_abs,uvb_abs,uva,w,R1,A,B);

[ts2] = opt_shock(uvb_abs,uva_abs,uvb,w,R1,A,B);

[ka(:,in+1)] = rk(u0a_abs,u0b_abs,uva_abs,uvb_abs,...

uva,A,B,gamma,T0,ts1,R0+R1,dz./iter);

[kb(:,in+1)] = rk(u0b_abs,u0a_abs,uvb_abs,uva_abs,...

uvb,A,B,gamma,T0,ts2,R0+R1,dz./iter);

end

A–5

Appendix - A–6

u1a = (u0a + (ka(:,2)+2. * ka(:,3)+2. * ka(:,4)+ka(:,5))./6);

u1b = (u0b + (kb(:,2)+2. * kb(:,3)+2. * kb(:,4)+kb(:,5))./6);

uva = u1a;

uvb = u1b;

clear ka;

clear kb;

end

function [ts] = opt_shock(u1a,u1b,ua,w,R1,A,B)

ts = ifft(j. * w. * fft(((A. * u1a+B. * u1b). * 0.82+0.18. * R1). * ua));

function [R1] = int_raman(R,u)

R1 = ifft(R. * f ft(u));

function [rk] = rk(u0a,u0b,u1a,u1b,ua,A,B,gamma,T0,ts1,R,dz)

rk = j * gamma* dz. * ((0.82. * A. * (u0a+u1a) + 0.82. * B. * (u0b+u1b) +...

0.18. * ( R)). * (ua) + j. * T0. * ts1);

function [R] = raman_quadv(T,nt)

% Raman scattering response using the mathematical

% desription in chapter 1

Ai = [1,11.40,36.67,67.67,74,4.5,...

6.8,4.6,4.2,4.5,2.7,3.1,3];

G = pi. * 3e8. * 1e-10. * [52.1,110.42,175.00,162.50,135.33,...

24.5,41.50,155,59.50,64.30,150,91.00,160];

g= pi. * 3e8. * 1e-10. * [17.37,38.81,58.33,54.17,45.11,...

8.17,13.83,51.67,19.83,21.43,50.00,30.33,53.33];

wi = 2. * 1e-10. * pi. * 3e8. * [56.25,100,231.25,362.5,...

463.00,497.00,611.5,691.67,793.67,835.50,930,1080,12 15];

w = wspace(T,nt);

s = zeros(nt,1); r = zeros(nt,1);

for n = 1:1:13;

s = s + Ai(n). * quadv(@(t) imag_R(t,w,wi(n),g(n),G(n)),0,T/2)./2;

r = r + Ai(n). * quadv(@(t) real_R(t,w,wi(n),g(n),G(n)),0,T/2)./2;

end

R = r + 1i. * s;

function [s] = imag_R(t,w,wi,g,G)

s = (cos((wi-w). * t ) - cos((wi+w). * t)). * exp(-g. * t). * exp(-G.ˆ2. * t.ˆ2./4);

function [r] = real_R(t,w,wi,g,G)

r = (sin((wi-w). * t ) + sin((wi+w). * t)). * exp(-g. * t). * exp(-G.ˆ2. * t.ˆ2./4);

A–6

Numerical Code for Diffraction

Theory

B.3 Diffraction theory

This appendix presents the numerical script used to calculate both Fresnel diffraction

by a lens and vectorial diffraction by a lens. The mathematical formulea used are

related to Eq. (2.3.6), (2.4.9) - (2.4.11).

B.4 Matlab Script - Scalar diffraction

1 function [E] = Calc_Field_low_NA()

2 % Calculates the diffraction field distribution in E(w,z)

3 % using scalar diffraction theory (paraxial regime)

4 c = 300; % Speed of light (micro/ps)

5 NA = 0.1; % Numerical aperture

6 T = 2; % time window width (ps)

7 lam = 0.780; % centre wavelength (micrometers)

8 nt = 2ˆ11; % temporal resolution

9 dt = T/(nt-1); % temporal step size (ps)

10 t = (-T/2:dt:T/2).’; % time domain (ps)

11 w = (wspace(T,nt)); % centred frequency domain

12 f = fftshift(w./(2 * pi)); % frequency (THz)

13 w0 = 2* pi * 3e2./lam; % center frequency (TRads)

14 f = f + c./lam; % center frequency (THz)

15 w = 2* pi. * f; % frequency domain (TRads)

16 k = w./c; % Wave number

17 z = (-600:1200/(2ˆ12-1):600).’; % Axial coordinate

18 E = zeros(nt,nt);

19

20 % Input field - SC

B–1

Appendix - B–2

21 s = dlmread(’BJC_NLPCF_2500_0.dat’);

22 s = s(:,1). * exp(1i. * s(:,2));

23 S = fftshift(fft(s));

24

25 % Resolution in r

26 N = 256;

27 a = 0;

28 b = 4* pi;

29 h = (b-a)/N;

30 v0 = (a:h:b). * l am./(2 * pi);

31

32 for p = 1:1:N;

33 v = k. * v0(p); % Norm. Radial Coord.

34 for q = 1:1:length(z);

35 kz = k * z(q);

36 u = NAˆ2. * kz; % Norm. axial Coord.

37 I0 = quadv(@(r) i0(r,v,u),0,1,1e-7);

38 E(:,q) = -1i. * S. * k. * NA.ˆ2. * I0. * exp(1i. * u./NA.ˆ2);

39 end

40 end

41

42 function [i0] = i0(r,v,u)

43 i0 = r. * besselj(0,v. * r). * exp(-1i. * r.ˆ2. * u./2);

B.5 Matlab Script - Scalar diffraction

1

2 function [Ex,Ey,Ez]=Calc_Field_sym(NA,method)

3 % This function calculates the symmetric form of vectorial di ffraction

4 N = 256; % Spatial resolution

5 alpha_1 = asin(NA); % Focusing angle

6 lam = 0.780; % Wavelength

7 T = 4; nt = 2ˆ12; dt = T/(nt-1); % Time width,Resolution and time step(ps)

8 t = (-T/2:dt:T/2).’; % Time domain(ps)

9 f = wspace(T,nt); % centered frequency domain

10 vs = fftshift(f/(2 * pi)); % frequency(THz)

11 v0 = 3e2/(lam); % center frequency(THz)

12 v = vs+v0; % centered frequency(THz)

13 k0 = 2 * pi * v./3e2; % Wave number

14 a = 0; b = 1.5; dx = (b-a)/(N-1);

15 x = (a:dx:b) + dx/2; % Displacement (x, micron)

16 a = 0; b = 1.5; dy = (b-a)/(N-1);

17 y = (a:dy:b) + dy/2; % Displacement (y, micron)

18 a = 0; b = 3; dz = (b-a)/(N-1);

19 z = (a:dz:b) + dz/2; % Displacement (z, micron)

20

21 %%%%%%%%%%%%%%%%%%%%%%%%%%% Field Distributions %%%%%%%%%%%%%%%%%%%%%%%%%%%

22

B–2

Appendix - B–3

23 Ex = zeros(nt,N);

24 Ey = zeros(nt,N);

25 Ez = zeros(nt,N);

26 Ex2 = zeros(nt,N);

27 Ey2 = zeros(nt,N);

28 Ez2 = zeros(nt,N);

29

30 % Input field vectors

31

32 if strcmp(source,’SC’ ) == 1;

33 u = dlmread([’BJC_NLPCF_2500_’,num2str(phi), ’_1.dat’ ]);

34 sx = u(:,1). * exp(1i. * u(:,2));

35 Sx = fftshift(fft(sx));

36 sy = u(:,3). * exp(1i. * u(:,4));

37 Sy = fftshift(fft(sy));

38 elseif strcmp(source, ’sech’ ) == 1;

39 sx = sech(t./0.05);

40 Sx = fftshift(fft(sx));

41 sy = sech(t./0.05);

42 Sy = fftshift(fft(sy));

43 end

44

45 for m = 0:1:N-1;

46 for n = 0:1:N-1;

47 if strcmp(method,’xy’) == 1;

48 z = 0;

49 u = k0 * z* sin(alpha_1)ˆ2;

50 psii = atan2(y(m+1),x(n+1));

51 r = sqrt(y(m+1).ˆ2+x(n+1).ˆ2);

52 v = k0 * r * sin(alpha_1);

53 elseif strcmp(method,’xz’) == 1;

54 u = k0 * z(n+1) * sin(alpha_1)ˆ2;

55 y = 0;

56 psii = atan2(y,x(m+1));

57 r = sqrt(y.ˆ2+x(m+1).ˆ2);

58 v = k0 * r * sin(alpha_1);

59 elseif strcmp(method,’yz’) == 1;

60 u = k0 * z(n+1) * sin(alpha_1)ˆ2;

61 x = 0;

62 psii = atan2(y(m+1),x);

63 r = sqrt(y(m+1).ˆ2+x.ˆ2);

64 v = k0 * r * sin(alpha_1);

65 elseif strcmp(method,’rz’) == 1;

66 u = k0 * z(n+1) * sin(alpha_1)ˆ2;

67 psii = atan2(y(m+1),x(m+1));

68 r = sqrt(y(m+1).ˆ2+x(m+1).ˆ2);

69 v = k0 * r * sin(alpha_1);

70 end

71 E0 = quadv(@(theta) e0(theta,v,u,alpha_1),0,alpha_1);

B–3

Appendix - B–4

72 E1 = quadv(@(theta) e1(theta,v,u,alpha_1),0,alpha_1);

73 E2 = quadv(@(theta) e2(theta,v,u,alpha_1),0,alpha_1);

74 Ex(:,n+1) = 1i. * Sx. * k0. * (E0 + cos(2. * psii). * E2)./2;

75 Ey(:,n+1) = 1i. * Sx. * k0. * (sin(2 * psii). * E2)./2;

76 Ez(:,n+1) = 1i. * Sx. * k0. * (2. * 1i. * cos(psii). * E1)./2;

77 Ex2(:,n+1) = 1i. * Sy. * k0. * (sin(2 * psii). * E2)./2;

78 Ey2(:,n+1) = 1i. * Sy. * k0. * (E0 - cos(2. * psii). * E2)./2;

79 Ez2(:,n+1) = 1i. * Sy. * k0. * (2. * 1i. * sin(psii). * E1)./2;

80 end

81 end

82

83 % vector integrals

84

85 function [e0]=e0(theta,v,u,alpha)

86 [e0] = (1+cos(theta)). * besselj(0,v. * sin(theta)./sin(alpha))...

87 . * sqrt(cos(theta)). * sin(theta). * exp(1i. * u. * cos(theta)./(sin(alpha).ˆ2));

88 function [e1]=e1(theta,v,u,alpha)

89 [e1] = sin(theta). * besselj(1,v. * sin(theta)./sin(alpha))...

90 . * sqrt(cos(theta)). * sin(theta). * exp(1i. * u. * cos(theta)./(sin(alpha).ˆ2));

91 function [e2]=e2(theta,v,u,alpha)

92 [e2] = (1-cos(theta)). * besselj(2,v. * sin(theta)./sin(alpha))...

93 . * sqrt(cos(theta)). * sin(theta). * exp(1i. * u. * cos(theta)./(sin(alpha).ˆ2));

B–4

Supplementary movies

In addition to this thesis is 4 supplementary movies which depict the evolution of a

hyperbolic secant ultrashort pulse (movies 1 and 2) and a supercontinuum (movies 3

and 4). Movies 1 and 2 show the evolution of a focused hyperbolic secant ultrashort

pulse and along the optical axis (movie 1) and the evolution over the radial and

axial dimensions (movie 2). The evolution of the hyperbolic secant ultrashort pulse

shows the influence of the points destructive interference. Movies 3 and 4 show the

evolution of a focused supercontinuum and along the optical axis (movie 3) and the

evolution over the radial and axial dimensions (movie 4). These movies also convey

the influence of the points of destructive interference, however they show how these

point can remove features within the temporal envelope of the supercontinuum.

C–1

Author’s Publications

Journal Articles

B. J. Chick, J. W. M. Chon and M. Gu, Polarization effects in a highly birefringent

nonlinear photonic crystal fiber with two-zero dispersion wavelengths. Opt. Express,

16:20099-20080, 2008.

B. J. Chick, J. W. M. Chon and M. Gu, Enhanced degree of temporal coherence

through temporal and spatial phase coupling within a focused supercontinuum. Opt.

Express, 17:20140-20148, 2009.

B. J. Chick, J. W. M. Chon and M. Gu, The effect of depolarization on the temporal

coherence within a focused supercontinuum. In preperation.

Conferences

B. J. Chick, J. W. M. Chon, R. Evans & M. Gu, Optical read out of nanoparticle

fluorescence using supercontinuum generation for optical data storage. Conference

on Lasers and Electro-Optics Europe - Technical Digest, Munich, Germany, June

17-22nd 2007.

B. J. Chick, J. W. M. Chon & M. Gu, Polarised pulse propagation in highly

birefringent photonic crystal fibre. International conference on Optics, Sydney,

Australia, July 7-10th, 2008.

B. J. Chick, J. W. M. Chon & M. Gu, High Numerical Aperture Diffraction of

a Supercontinuum. Sir Mark Oliphant conferences - Nanophotonics Down Under,

Melbourne, Australia, June 21-24th, 2009.