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o 228

RAHUL DUTTA

TEMPORAL, SPECTRAL, AND SPATIAL COHERENCE OF OPTICAL PULSE TRAINS


This thesis considers the coherence properties of optical pulse trains in temporal, spectral, and spatial domains. Realistic optical fields are studied through analytical models and numerical analyses. The evaluation of the

two-time coherence function for pulse trains is elucidated with an appropriate experimental

arrangement and the corresponding analytical formulation. Moreover, preliminary

experimental results, in agreement with the theoretical predictions, are reported.

RAHUL DUTTA

RAHUL DUTTA

Temporal, spectral, and

spatial coherence of optical

pulse trains

Publications of the University of Eastern Finland


No 228

Academic Dissertation

To be presented by permission of the Faculty of Science and Forestry for public

examination in the Auditorium F100 in Futura Building at the University of

Eastern Finland, Joensuu, on June 17, 2016,

at 12 o’clock noon.

Institute of Photonics

Grano Oy

Jyvaskyla, 2016

Editors: Prof. Pertti Pasanen, Prof. Jukka Tuomela,

Prof. Matti Vornanen, Prof. Pekka Toivanen

Distribution:

University of Eastern Finland Library / Sales of publications

[email protected]

http://www.uef.fi/kirjasto

ISBN: 978-952-61-2152-9 (printed)

ISSNL: 1798-5668

ISSN: 1798-5668

ISBN: 978-952-61-2153-6 (pdf)

ISSNL: 1798-5668

ISSN: 1798-5676

Author’s address: University of Eastern Finland

Department of Physics and Mathematics

P. O. Box 111

80101 JOENSUU

FINLAND

email: [email protected]

Supervisors: Professor Jari Turunen, DSc (Tech)

University of Eastern Finland


P. O. Box 111

80101 JOENSUU

FINLAND


Professor Ari T. Friberg, PhD, DSc (Tech)

University of Eastern Finland


P. O. Box 111

80101 JOENSUU

FINLAND


Professor Goery Genty, PhD

Tampere University of Technology

Department of Physics

P. O. Box 692

33101 TAMPERE

FINLAND


Reviewers: Professor Katarzyna Chałasinska-Macukow, PhD

University of Warsaw

Faculty of Physics

Pasteura 5

02-093 WARSAW

POLAND


Associate Professor Miro Erkintalo, PhD

University of Auckland

Department of Physics

Private Bag 92019

1142 AUCKLAND

NEW ZEALAND


Opponent: Professor Jurgen Jahns, PhD

FernUniversitat in Hagen

Chair of Micro- and Nanophotonics

Universitatsstr. 27/PRG

58097 HAGEN

GERMANY


ABSTRACT

The work presented in this thesis deals with the coherence of op-

tical pulse trains in time, frequency, and space domains. This the-

sis, based on the second-order coherence theory of nonstationary

light, primarily consists of theoretical investigations which result in

useful insights in the coherence characterization of pulsed optical

sources. Analytical models that physically represent supercontin-

uum pulse trains, broadband axicon fields, and slightly disturbed

mode-locked pulse trains are proposed. Numerically simulated re-

alizations of supercontinuum pulse trains are also considered in

this study. In addition, an experimental setup with corresponding

analytical analysis is presented to illustrate the evaluation of the

two-time coherence function of pulse trains and it is shown that

for that one needs to perform time-resolved measurements of the

interference fringes. Moreover, information conveyed by the time-

integrated measurements of the interference fringes produced by

supercontinuum light and slightly perturbed mode-locked laser is

explained through analytical and numerical investigations. An ex-

perimental demonstration of the time-integrated measurements is

reported for an almost fully coherent femtosecond pulse train.

Universal Decimal Classification: 535.374, 535.41, 537.87

INSPEC Thesaurus: optics; light; light coherence; optical pulse genera-

tion; lasers; lenses; pulse measurement; light interferometry; Michelson

interferometers; simulation; numerical analysis

Yleinen suomalainen asiasanasto: optiikka; valo; koherenssi; interferenssi

- - fysiikka; simulointi; numeeriset menetelmt

Preface

First of all I wish to express my gratitude to my supervisors Prof.

Jari Turunen and Prof. Ari Friberg for their thorough guidance and

instructions during the years of research. I am also grateful to my

third supervisor Prof. Goery Genty for his advice and for the sim-

ulation data which made our fifth article possible.

I am thankful to the present and former heads of the department

of Physics and Mathematics, Prof. Timo Jaaskelainen, Prof. Seppo

Honkanen, and especially to Prof. Pasi Vahimaa, for providing me

the opportunity to work and to be a part of this department. I thank

the reviewers Dr. Miro Erkintalo and Prof. Katarzyna Chałasinska-

Macukow for their insightful comments and suggestions in a very

quick time. I am grateful to Prof. Jurgen Jahns for accepting to be

my opponent on short notice. I thank my co-authors Minna and

Kimmo for their contributions. I am thankful to Matias and Henri

for sharing their valuable practical knowledge while performing

experiments together.

I am grateful to my nearest and dearest friend Debashis for his

constant support for the last 10 years as a true friend. I thank Gau-

rav and Samriddhi to be in my side in recent years during which we

have come close to each other through various real-life discussions.

I also owe my sincere thanks to Nilabha, Subhajit, and Gaurav for

making my first two years of stay in Finland memorable and for all

the crazy days we spent together during our masters study. I would

further like to thank Hasanur, Somnath, Amar, Manisha, Murad,

Bisrat, Arif, Vishal, Rizwan, and all others for the good time we

have had in or outside the university.

Last, but certainly not the least, my deepest gratitude goes to

my grandparents, parents, elder brother, and sister-in-law for their

love and all the support they have provided in every sphere of my

life. I convey my heartiest love to my little niece Sanvi.

Joensuu, May 17, 2016 Rahul Dutta

LIST OF PUBLICATIONS

This thesis consists of the present review of the author’s work on

the coherence of optical pulse trains and the following publications

by the author:

I R. Dutta, M. Korhonen, A. T. Friberg, G. Genty, and J. Tu-

runen, “Broadband spatiotemporal Gaussian Schell-model

pulse trains,” J. Opt. Soc. Am. A 31, 637–643 (2014).

II R. Dutta, K. Saastamoinen, J. Turunen, and A. T. Friberg,

“Broadband spatiotemporal axicon fields,” Opt. Express 22,

25015–25026 (2014).

III R. Dutta, J. Turunen, and A. T. Friberg, “Michelson’s interfer-

ometer and the temporal coherence of pulse trains,” Opt. Lett.

40, 166–169 (2015).

IV R. Dutta, A. T. Friberg, G. Genty, and J. Turunen, “Two-time

coherence of pulse trains and the integrated degree of tempo-

ral coherence,” J. Opt. Soc. Am. A 32, 1631–1637 (2015).

V R. Dutta, J. Turunen, G. Genty, and A. T. Friberg, “Tempo-

ral coherence characterization of supercontinuum pulse trains

using Michelson’s interferometer,” Appl. Opt. 55, B72–B77

(2016).

Throughout the overview, these papers will be referred to by Ro-

man numerals.

Some parts of the work have been presented by the author in the

following national and international conferences:

Physics Days (Joensuu, Finland, 2012) (poster)

Progress in Electromagnetics Research Symposium (PIERS) (Stockholm,

Sweden, 2013) (oral)

Electromagnetic Optics with Random Light (Joensuu, Finland, 2014)

(poster)

Japan-Finland Joint Symposium on Optics in Engineering (Joensuu, Fin-

land, 2015) (poster)

Correlation Optics (Chernivtsi, Ukraine, 2015) (oral)

AUTHOR’S CONTRIBUTION

The original research papers included in this thesis are results of

group efforts. The author has performed the majority of the analyt-

ical calculations and numerical simulations reported in them and in

chapters 3–5. Numerical contributions of K. Saastamoinen to paper

II are gratefully acknowledged. The author performed the exper-

iments described in chapter 6 in collaboration with M. Koivurova.

The author wrote paper V and participated actively in writing all

other manuscripts. All the manuscripts were completed in cooper-

ation with the co-authors.

Contents

1 INTRODUCTION 1

2 PARTIALLY COHERENT OPTICAL

PULSE TRAINS 5

2.1 Coherence theory of nonstationary light . . . . . . . . 6

2.1.1 Complex representation of the field . . . . . . 6

2.1.2 Second-order coherence in space-time domain 7

2.1.3 Second-order coherence in space-frequency do-

main . . . . . . . . . . . . . . . . . . . . . . . . 9

2.2 Coherence of Supercontinuum light . . . . . . . . . . 10

2.2.1 Temporal and spectral coherence of SC light . 10

2.2.2 Spatial coherence of supercontinuum light . . 13

2.3 Measurement techniques . . . . . . . . . . . . . . . . . 13

2.3.1 Michelson interferometry . . . . . . . . . . . . 13

2.3.2 Frequency-resolved optical gating . . . . . . . 15

3 GAUSSIAN SCHELL-MODEL

PULSE TRAINS 19

3.1 Partially coherent model sources . . . . . . . . . . . . 19

3.1.1 Field model in frequency domain . . . . . . . 20

3.1.2 Field model in time domain . . . . . . . . . . . 23

3.2 Pulse propagation in ABCD systems . . . . . . . . . . 25

4 PULSE TRAINS GENERATED

BY AXICONS 29

4.1 Axicon fields in space-frequency domain . . . . . . . 32

4.2 Spatiotemporal axicon fields . . . . . . . . . . . . . . . 33

4.2.1 Spectrally coherent illumination . . . . . . . . 34

4.2.2 Spectrally incoherent illumination . . . . . . . 36

5 PULSE TRAIN CHARACTERIZATION

BY MICHELSON INTERFEROMETER 39

5.1 Time-resolved measurements with Michelson

interferometer . . . . . . . . . . . . . . . . . . . . . . . 40

5.2 Application to different models . . . . . . . . . . . . . 44

5.2.1 Gaussian Schell-model for mode-locked pulse

trains . . . . . . . . . . . . . . . . . . . . . . . . 44

5.2.2 Analytical model for SC pulse trains . . . . . 48

5.2.3 Simulated realizations of SC light . . . . . . . 50

6 EXPERIMENTAL PERSPECTIVES

AND RESULTS 53

6.1 Temporal coherence measurement of pulse

trains using Michelson’s interferometer . . . . . . . . 53

6.2 Measurement of pulse trains using FROG . . . . . . . 57

7 CONCLUSIONS 61

7.1 Summary with conclusions . . . . . . . . . . . . . . . 61

7.2 Potential future work . . . . . . . . . . . . . . . . . . . 63

REFERENCES 76

1 Introduction

It is now well established that light possesses the characteristics of

both particles and waves, i.e., it is dualistic in nature. Whereas

classical optics has its foundation on the wave properties of light,

quantum optics is based on its particle nature.

The wave nature of light was first proposed by Hooke [1] and

subsequently it was put on a more systematic basis by Huygens [2].

Later the experimental demonstration of the interference of light by

Young [3] confirmed the view that light consists of waves. The wave

properties of optical fields can be completely explained with the

help of the classic equations by Maxwell describing light as electro-

magnetic waves [4]. Thereafter, interference phenomena produced

by different kinds of light sources have been extensively studied

and this has led to the conclusion that the ability to interfere corre-

sponds to a basic property of light sources, called coherence. This

property varies from source to source. The coherence characteristics

of light sources are studied in the framework of optical coherence

theory, which deals with the statistical nature of light, whether nat-

ural or manmade.

The concept of coherence was put forward by Verdet when he

observed that “the vibrations of sunlight are in unison” [5], and

by Michelson through his interpretation of the correlation between

the visibility of the interference fringes and the energy distribution

in spectral lines [6, 7]. It was van Cittert and Zernike who deter-

mined the correlation function of the field created by an incoherent

light source through measuring intensity distributions at the source

plane [8, 9]. Later on, Zernike proposed the term “degree of coher-

ence” and its equivalence with the visibility of interference fringes

in Young’s experiment [9].

The general theory of optical coherence in the classical contexts

was formulated by Wolf in the space-time domain [10–14] and by

Mandel and Wolf [15,16] in the space-frequency domain. The quan-

Rahul Dutta: Temporal, spectral, and spatial coherence of pulse trains

tum theory of optical coherence was put forward by Glauber in

1963 [17, 18]. In much of these formalisms, the optical fields have

been considered as stationary in nature, i.e., their statistical proper-

ties do not depend on the origin of time. Although some random-

ness is inherent in all real optical sources, this approximation works

quite well in many cases.

While the coherence theory of stationary light is well-established

[10], for nonstationary optical fields it has not been developed sig-

nificantly. The study on the stochastic processes involved in nonsta-

tionary sources was initiated by Bertolotti et al. [19] using a spec-

tral domain approach. This technique was further extended to in-

vestigate the spatial coherence properties of nonstationary optical

fields [20]. These works provide the basic theoretical foundation,

known as the second-order coherence theory of nonstationary light.

Since then a number of researches have been carried out employing

this framework [21–30].

Since the advent of ultrafast lasers, they have made substan-

tial inroads into a wide range of applications. While pulse energy

plays a significant role in some applications, such as micromachin-

ing [31], the broad bandwidth is the key feature in many others,

including optical communication [32], generation of supercontin-

uum light [33, 34], frequency combs [35, 36], and free-electron laser

sources [37, 38]. It is therefore necessary to analyse ultrafast light

sources both from the fundamental research and the applications

points of view. The coherence properties of pulsed lasers are one

of their primary characteristics that have considerable influence in

practical applications. Thus it is imperative to assess the coherence

of ultrafast pulse trains appropriately in theoretical and experimen-

tal contexts. This is the motivation behind the work presented in

this thesis. Although a number of investigations have lately been

performed to study the coherence properties of different pulsed

light sources, there is no comprehensive foundation such as exists

for stationary light fields.

The research presented in this thesis provides a contribution

towards the theoretical understanding of nonstationary light fields.

2 Dissertations in Forestry and Natural Sciences No 228

Introduction

This is primarily achieved by studying certain mathematical model

sources. Additionally, methods of measuring fields of this kind are

presented. Moreover, some preliminary experimental results are

reported in support of the theoretical claims.

The model sources and fields considered here for time-domain

broadband radiation are important in the sense that they physically

represent realistic optical sources. The models can be viewed as

extending the Gaussian Schell-model plane-wave representation for

nonstationary light [21]. This, in turn, is the temporal analogue

of the spatial Gaussian Schell-model for stationary light [10, 39],

obeyed by many real light sources. Other similar kinds of models

in both the space and time domains also exist.

Nondiffracting (or propagation-invariant) optical fields [40, 41]

have attracted much attention among researchers in recent decades

owing to their unique characteristics of maintaining approximately

unchanged transverse intensity profiles in propagation over long

distances. In general, Bessel beams represent nondiffracting fields

introduced by Durnin [42] as mathematical solutions to the free-

space wave equation. In practice they are realized with the help of

axicons [43, 44], optical resonators [45], computer holography [46],

or spatial light modulators [47]. Extensive theoretical and experi-

mental investigations of Bessel X-waves and pulsed Bessel beams

have been performed lately [48, 49].

The basic theoretical formalism regarding optical coherence of

nonstationary light is considered in chapter 2. Additionally, some

useful background on supercontinuum coherence as well as two

standard measurement techniques for the characterization of opti-

cal pulses are reviewed in brief. This chapter forms the coherence-

theoretical foundation of the work presented in this thesis.

In chapter 3, a new class of analytical model sources illustrat-

ing partially coherent broadband pulse trains is introduced. This

model is particularly inspired by supercontinuum light generated

in a nonlinear single-mode optical fiber. The corresponding results

from paper I are also discussed here.

Broadband nondiffracting wave fields are the topic of chapter 4.

Dissertations in Forestry and Natural Sciences No 228 3


We consider the spatiotemporal properties as well as the coherence

properties of the fields generated by two different kinds of axicons,

reflective and diffractive, under spectrally coherent and spectrally

incoherent (stationary) illumination. The key results from paper II

are presented here.

In chapter 5, we consider theoretical and experimental charac-

terization of the temporal coherence properties of pulsed optical

sources. In particular, we introduce an equal-path Michelson inter-

ferometer setup with tilted mirrors, proposed in paper III, to evalu-

ate the two-time temporal coherence function of pulse trains in view

of the second-order coherence theory of nonstationary light. Fur-

thermore, information conveyed by the standard time-integrated

Michelson interferrogram is assessed in the case of temporal coher-

ence measurements of pulsed light. Analytical models representing

practical sources play a pivotal role in these analyses, as discussed

in paper IV. Numerical simulations, described in paper V, are also

employed to affirm the conclusions.

Chapter 6 deals with measurements of the time-integrated tem-

poral coherence function associated with femtosecond pulse trains,

for comparison with the theoretical predictions. The pulse trains

are also characterized using the frequency-resolved optical gating

technique [50] to assist in the analysis.

Finally, the main results and conclusions of this thesis are sum-

marized in chapter 7. Further development of the theoretical mod-

els and experimental characterization techniques are also outlined

here as possible future extensions of the work.


2 Partially coherent optical

pulse trains

An optical pulse can be described as electromagnetic radiation of

finite duration as small as femtoseconds or even shorter (attosec-

onds). Optical pulses are generally realized in the form of a beam

generated from a pulsed laser propagating in a certain direction.

A short laser pulse is, in principle, fully coherent in both the tem-

poral and spectral domains. Realistic pulsed laser beams, which

consist of a train of pulses, differ from the ideal fully coherent case.

They can be almost coherent (when all the pulses are identical to

each other), incoherent (the pulses are not at all correlated to each

other), or something in between these two cases, called partially

coherent. Hence true characterization of the coherence properties

of pulsed sources is essential in fundamental and applied research.

Owing to the nonstationary characteristics of these sources, for in-

stance, time-dependent distribution of the (average) intensity, the

standard optical coherence theory of stationary light cannot be ap-

plied. One well-accepted approach to study optical pulses theo-

retically is the coherence theory of nonstationary light [20]. All

the analysis performed in this thesis is based on this framework.

Furthermore, nonstationary fields are here studied through scalar

optics which means that the vectorial characteristics of light, such

as polarization, are ignored in the analysis. Despite this limitation,

scalar approach performs satisfactorily in many cases in the field of

optics.

We begin this chapter with a brief discussion on the basic for-

malism of optical coherence theory that has been employed to an-

alyze nonstationary light, such as pulsed sources. We then recall

earlier investigations about the coherence properties of a certain

complex train of pulses, called supercontinuum light. This provides



background for the results presented in later parts of the thesis. We

conclude the chapter by describing concisely two standard methods

to characterize optical pulses.

2.1 COHERENCE THEORY OF NONSTATIONARY LIGHT

In this section we consider the analytical description of the second-

order coherence theory of nonstationary light, which is the foun-

dation of this thesis. We start with the complex representation of

the optical field and then move to the definition of the coherence

functions in space-time and space-frequency domains.

2.1.1 Complex representation of the field

In classical coherence theory, it is customary to express any real-

valued optical field in terms of a complex representation for the

sake of analytical simplicity. This representation encompasses the

amplitude and phase information of the real field.

We assume Er(ρ; t) is a real-valued function that depends on

the position ρ and the time t. This could be considered as a scalar

representation of any component of the electric field. If the function

is square-integrable, we can express it as a Fourier integral

Er(ρ; t) =∫ ∞

−∞Er(ρ; ω) exp(−iωt)dω, (2.1)

where ω is the angular frequency and the Fourier coefficients are

evaluated by

Er(ρ; ω) =1

2π

∫ ∞

−∞Er(ρ; t) exp(iωt)dt. (2.2)

Since the function Er(ρ; t) is real, the Fourier coefficients defined by

Eq. (2.2) obey the condition Er(ρ;−ω) = E∗r (ρ; ω), which reveals

that the negative frequency components do not carry any infor-

mation that is not already in the positive frequency components.

Hence, the negative frequency components can be removed. Now,


Partially coherent opticalpulse trains

we can define a function

E(ρ; t) = 2∫ ∞

0Er(ρ; ω) exp(−iωt)dω, (2.3)

which is called the complex analytic signal [51]. The real part of

this quantity represents the real-valued field defined in Eq. (2.1).

Further, we can express the complex analytic signal as a Fourier

transform pair as follows:

E(ρ; t) =∫ ∞

−∞E(ρ; ω) exp(−iωt)dω (2.4)

and

E(ρ; ω) =1

2π

∫ ∞

−∞E(ρ; t) exp(iωt)dt. (2.5)

It can readily be seen that, in the frequency domain, the com-

plex analytic signal is associated with the real optical field through

E(ρ; ω) = 2Er(ρ; ω) for ω ≥ 0, and E(ρ; ω) = 0 for ω < 0. As the

name suggests, the complex analytic signal has numerous analyti-

cal properties in the complex plane [10].

2.1.2 Second-order coherence in space-time domain

Realistic optical fields, stationary or nonstationary, are random in

nature. They exhibit fluctuations in the spatial and temporal prop-

erties and these characteristics can be studied through a statistical

approach, which is the basis of optical coherence theory. The co-

herence properties are defined as correlations of the field at two

different instants of time (or frequency) and at two different points

in space; thereby they are called second-order coherence functions.

In the space-time domain this function is known as the mutual co-

herence function (MCF).

Now, the MCF describing the correlation property of a nonsta-

tionary field at two spatial points ρ1 and ρ2 at two different instants

of time t1 and t2 is defined as

Γ(ρ1, ρ2; t1, t2) = 〈E∗(ρ1; t1)E(ρ2; t2)〉 , (2.6)



where the angular brackets denote the ensemble average

〈 f (ρ; t)〉 = limN→∞

1

N

N

∑n=1

fn(ρ; t). (2.7)

The functions fn(ρ; t) represent field realizations in our study. The

temporal intensity of the field can be expressed, through putting

ρ1 = ρ2 = ρ and t1 = t2 = t, as

I(ρ; t) = Γ(ρ, ρ; t, t) =⟨

|E(ρ; t)|2⟩

. (2.8)

Then the normalized version of the MCF, defined by

γ(ρ1, ρ2; t1, t2) =Γ(ρ1, ρ2; t1, t2)

√

I(ρ1; t1)I(ρ2; t2), (2.9)

is called the complex degree of coherence. The absolute value of

this function denotes the state of coherence and it satisfies the fol-

lowing inequality in all cases

0 ≤ |γ(ρ1, ρ2; t1, t2)| ≤ 1. (2.10)

At the lower limit, the field at points ρ1 and ρ2 and times t1 and t2 is

said to be completely incoherent due to total lack of correlations. At

the other extreme, the field at the two space-time points is termed as

fully coherent. Any value between these two limiting cases signifies

that the field is partially coherent.

If one is interested only in the temporal coherence of the field at

two separate instants of time and at a fixed spatial point, then the

MCF takes the following form (for brevity, the spatial dependence

is omitted)

Γ(t1, t2) = 〈E∗(t1)E(t2)〉 . (2.11)

This can be expressed in the average t = (t1 + t2)/2 and difference

∆t = t2 − t1 coordinates as

Γ(t, ∆t) = 〈E∗(t − ∆t/2)E(t + ∆t/2)〉 . (2.12)



The corresponding complex degree of temporal coherence is

γ(t, ∆t) =Γ(t, ∆t)

√

I(t − ∆t/2)I(t + ∆t/2). (2.13)

Similar formalisms can be obtained for the description of the spatial

coherence of the field at two different positions at a fixed time.

2.1.3 Second-order coherence in space-frequency domain

In analogy with space-time representation, cross-spectral density

(CSD) is the second-order correlation function in space-frequency

domain that describes the correlation of the field at two different

spatial points ρ1 and ρ2 and at two separate frequencies ω1 and ω2.

The CSD may be defined as follows

W(ρ1, ρ2; ω1, ω2) = 〈E∗(ρ1; ω1)E(ρ2; ω2)〉 , (2.14)

and similarly, the spectral density of the field is written as

S(ρ; ω) = W(ρ, ρ; ω, ω) =⟨

|E(ρ; ω)|2⟩

. (2.15)

Then the complex degree of coherence takes the form of

µ(ρ1, ρ2; ω1, ω2) =W(ρ1, ρ2; ω1, ω2)

√

S(ρ1; ω1)S(ρ2; ω2). (2.16)

It also satisfies the condition

0 ≤ |µ(ρ1, ρ2; ω1, ω2)| ≤ 1, (2.17)

indicating the field as being completely spectrally and spatially co-

herent, or fully incoherent, or partially coherent.

The coherence functions in two different domains are connected

by the Fourier transform relations

Γ(ρ1, ρ2; t1, t2) =∫∫ ∞

0W(ρ1, ρ2; ω1, ω2)

× exp [i(ω1t1 − ω2t2)] dω1dω2, (2.18)



and

W(ρ1, ρ2; ω1, ω2) =1

(2π)2

∫∫ ∞

−∞Γ(ρ1, ρ2; t1, t2)

× exp [−i(ω1t1 − ω2t2)] dt1dt2, (2.19)

which are known as the generalized Wiener-Khintchine relations

for nonstationary fields.

2.2 COHERENCE OF SUPERCONTINUUM LIGHT

Supercontinuum (SC) light shows the unique characteristics of a

wide spectrum ranging from the ultra-violet to the infrared region

combined with coherence properties that are strongly dependent

on the excitation conditions. In many applications, the wide band-

width and the coherence properties of SC radiation play significant

roles. For example, in frequency metrology and frequency combs,

optical pulse compression, and wavelength-division multiplexing

in telecommunication, high spectral and temporal coherence is es-

sential, whereas SC light with low-coherence is adequate in optical

coherence tomography and spectroscopy. Owing to the strong in-

fluence of temporal and spectral coherence in the applications of

SC light, these properties have attracted much attention of the re-

searchers in recent times. In this section, we consider the previous

results on the analysis of the coherence properties of SC light gen-

erated in a nonlinear fiber.

2.2.1 Temporal and spectral coherence of SC light

We begin with the first-order coherence of supercontinuum light in-

troduced by Dudley and Coen [52] for SC generated in a nonlinear

fiber. The modulus of the complex degree of first-order coherence,

also known as the Dudley–Coen coherence function, at frequency

ω over the whole SC spectrum is defined as

|g(1)12 (ω)| =

∣

∣

∣〈E∗

1(ω)E2(ω)〉pairs

∣

∣

∣

S(ω), (2.20)



where the angular brackets in the numerator denote statistical av-

eraging over different pairs of SC realizations and S(ω) is the en-

semble averaged spectrum. Due to its direct correspondence to the

readily measurable fringe visibility in a simple interference exper-

iment [53], this definition became popular among researchers and

has been used extensively in theoretical and experimental investiga-

tions to assess the coherence of supercontinuum light [54]. Another

quantity, called the spectrally averaged degree of coherence, was

also proposed as [52]

|g(1)12 | =∫ ∞

0|g(1)12 (ω)|S(ω)dω∫ ∞

0S(ω)dω

. (2.21)

This provides an overall measure that may take on any value be-

tween 0 and 1 implying spectrally (and temporally) incoherent, par-

tially coherent, or fully coherent SC light.

Recently the coherence of SC radiation has been analyzed by

means of the second-order coherence theory of nonstationary light,

which allows complete characterization of the spectral and tem-

poral coherence properties of pulse trains [55–57]. These investi-

gations unveiled a decomposition of the two-time mutual coher-

ence function and the two-frequency cross-spectral density func-

tion into quasi-coherent (qc) and quasi-stationary (qs) contributions

and showed that the relative weights of these contributions depend

on the pump pulse parameters. The studies also revealed that the

Dudley–Coen coherence function is directly related to the qc part

of the two-frequency correlation function and hence provides no

information about the qs contribution. These results pave the way

towards a complete coherence characterization of SC light but a di-

rect method of measuring the two-time correlation functions would

be desirable.

By representing the MCF of SC light in average and difference

time coordinates, we can thus write

Γ(t, ∆t) = Γqc(t, ∆t) + Γqs(t, ∆t). (2.22)



Similarly, the CSD can be expressed, in average ω = (ω1 + ω2)/2

and difference ∆ω = ω2 − ω1 coordinates, as

W(ω, ∆ω) = Wqc(ω, ∆ω) + Wqs(ω, ∆ω). (2.23)

The temporal intensity and the spectral density of SC light can like-

wise be decomposed into two parts,

I(t) = Iqc(t) + Iqs(t), (2.24)

S(ω) = Sqc(ω) + Iqs(ω), (2.25)

as can be seen in view of Eqs. (2.8) and (2.15).

Supercontinuum light, including noise effects, can be simulated

through numerically solving the nonlinear Schrodinger equations

that takes into account all the various nonlinear processes in the

fiber [54]. Figure 2.1 illustrates the decomposition of the mutual

coherence function of SC realizations produced by a pump pulse of

2 ps duration. The square region represents the qc part, while the

thin horizontal line denotes the qs contribution. A similar decom-

position follows for the cross-spectral density function.

−10 −5 0 5 10−0.2

0.1

0

0.1

0.2

0

0.2

0.4

0.6

0.8

1

t [ps]

∆t[p

s]

Figure 2.1: Absolute value of the normalized MCF, in average and difference coordinates,

for a supercontinuum ensemble numerically generated by 2 ps pump pulses.



2.2.2 Spatial coherence of supercontinuum light

Since SC light is usually produced in a single-spatial mode fiber, it

is typically regarded as spatially fully coherent. But the temporal

(spectral) coherence properties vary considerably depending on the

excitation conditions, and consequently the spatial coherence may

also be modified owing to the broad spectral bandwidth. Contrary

to the conventional belief, we have demonstrated that SC light is

highly (but not fully) spatially coherent. The analytical model is

described in the next chapter.

2.3 MEASUREMENT TECHNIQUES

In this section, we recall widely employed measurement techniques

to characterize optical pulses. As the essence of this thesis is the

coherence of pulse trains, we begin with the method which is used

to analyze the temporal coherence of pulsed light. We then consider

a more sophisticated way of measuring ultrashort optical pulses.

2.3.1 Michelson interferometry

Michelson’s interferometer is customarily used to characterize the

coherence of light (optical sources) in time domain [58]. It is the

standard tool to measure the coherence time of stationary light. Al-

though the classical Michelson interferometer is also often applied

to analyze temporal coherence of pulsed light, it does not always

lead to correct results due to nonstationarity. In this section we re-

strict ourselves to the stationary case only. Later, in chapter 5, we

consider temporal coherence assessment of nonstationary sources

using Michelson’s interferometer.

Let us consider a plane-wave field E0(t) incident onto the Michel-

son interferometer shown in Fig. 2.2 (here we have suppressed the

spatial dependence of the wave). The beamsplitter divides the field

into two equal parts, one moving towards mirror M1 and the other

towards mirror M2. On reflection and second passage through the

beamsplitter they subsequently meet at the detection plane. The



M1

M2

Observation plane

Scanning mirror

Lightsource

Figure 2.2: Standard setup of Michelson’s interferometer.

optical path difference between the two parts is 2∆z, where ∆z is

the axial shift of M2 with respect to M1. If we ignore the reflection

and absorption losses at the beamsplitter and at the two mirrors,

the output field in the detection plane can be written as

E(t, τ) =1

2[E0(t) + E0(t + τ)], (2.26)

where τ = 2∆z/c is the time delay between the two arms of the in-

terferometer. The intensity distribution at the detection plane then

takes on the form

I(t, τ) =1

4[⟨

|E0(t)|2⟩

+⟨

|E0(t + τ)|2⟩

+ 〈E∗0(t)E0(t + τ)〉

+ 〈E0(t)E∗0(t + τ)〉]

=1

4[I0(t) + I0(t + τ) + Γ0(t, τ) + Γ∗

0(t, τ)], (2.27)

where I0(t) and Γ0(t, τ) are the temporal intensity and the mutual

coherence function of the incident field.



Now we take the incident fields to be stationary in time, imply-

ing that I0(t) = I0(t + τ) = I0 and Γ0(t, τ) = Γ0(τ). The intensity

distribution then becomes

I(τ) =1

2I0 +ℜ[Γ0(τ)] =

1

2[I0 + |Γ0(τ)|cosarg[Γ0(τ)]] , (2.28)

where arg[Γ0(τ)] is the phase of Γ0(τ). Since the complex degree

of temporal coherence is given by γ0(τ) = Γ0(τ)/I0, we can rewrite

the intensity distribution as

I(τ) =I0

2[1 + |γ0(τ)|cosarg[γ0(τ)]]. (2.29)

Now, one can determine the absolute value of γ0(τ) from the visi-

bility of interference fringes, defined as

V(τ) = Imax(τ)− Imin(τ)

Imax(τ) + Imin(τ)= |γ0(τ)|, (2.30)

where Imax and Imin are the maximum and minimum values of the

interference pattern in the immediate neighborhood of τ. The phase

of the complex degree of temporal coherence can be evaluated from

the positions of the interference-fringe maxima and minima.

2.3.2 Frequency-resolved optical gating

The progress of generating ultrashort optical pulses has acceler-

ated in recent past. These pulses are important for fundamental

research and find increasing practical applications. Therefore, it is

crucial to characterize ultrafast sources properly. A significant num-

ber of different measurement methods have evolved over time that

are primarily based on autocorrelation [59]. Complete characteriza-

tion of ultrashort pulses has been possible only with the arrival of

the frequency-resolved optical gating (FROG) technique [50]. Other

means to study these pulses do exist, such as spectral phase in-

terferometry for direct electric-field reconstruction (SPIDER) [60].

However, FROG is the standard method and is routinely used in

the laboratory. In principle, FROG measures the spectrally resolved



signal beam in an autocorrelation arrangement. Hence, the optical

setup of FROG consists of an intensity autocorrelator and a spec-

trometer, as is illustrated in the second-harmonic generation (SHG)

FROG setup in Fig. 2.3. The SHG FROG is the most popular among

different FROG techniques that also include cross-correlation FROG

(XFROG), third-harmonic generation (THG) FROG, and polariza-

tion gating (PG) FROG [50].

Spectro-meter

Variabledelay

SHGcrystal

Beamsplitter

Incidentpulse

Figure 2.3: Schematic diagram of second-harmonic generation frequency-resolved optical

gating setup.

The basic idea of FROG is to combine a pulse with its delayed

replica in a nonlinear medium (SHG crystal in this case), generating

the desired signal in the presence of both pulses simultaneously.

Thus the SHG crystal acts as gating. The delay unit introduces a

variable delay in the incident pulse, as shown in Fig. 2.3. Finally the

spectrometer measures the spectrum of the field produced in the

SHG medium at each delay, i.e., a frequency-resolved measurement

is performed. The obtained spectrogram of the pulse, called FROG

spectrogram or trace, is sufficient to evaluate the complex electric

field in time and frequency domains.

Let us consider that E(t) is the electric field of the incident pulse

and E(t− τ) is its delayed replica. The FROG spectrogram can then



be analytically expressed as

I(ω, τ) =

∣

∣

∣

∣

∫ ∞

−∞E(t)E(t − τ) exp(iωt)dt

∣

∣

∣

∣

2

, (2.31)

where ω is the frequency and τ is the time delay. This expression

can be illustrated with a simple example.

Suppose we have a transform-limited Gaussian pulse expressed

as

E(t) = E0 exp(−t2/T2) exp(−iω0t), (2.32)

with the intensity distribution

I(t) = I0 exp(−2t2/T2) (2.33)

and the corresponding spectrum

S(ω) = S0 exp

[

− 2

Ω2(ω − ω0)

2

]

, (2.34)

where Ω = 2/T. Now the FROG trace can be written as

I(ω, τ) =π

2|V0|4T2 exp

[

− (ω − 2ω0)2

Ω2

]

exp

(

− τ2

T2

)

. (2.35)

Looking at the marginal τ = 0, we have a Gaussian spectrum but

due to SHG FROG, it is centered around 2ω0 instead of ω0 and its

width is Ω/√

2. On the other hand, the marginal ω = 2ω0 provides

a Gaussian intensity distribution with a width T/√

2, which is the

spread of the autocorrelation of I(t).

Reconstruction of the full information about the pulsed electric

field from the measured FROG trace constitutes a complex two-

dimensional phase-retrieval exercise which, however, is well known

for having a unique solution [50, 61].


3 Gaussian Schell-model

pulse trains

In this chapter we discuss an analytical model which is widely

employed to describe partially coherent wave fields in the frame-

work of optical coherence theory. This representation is known

as the Gaussian Schell-model (GSM). It is a mathematically con-

venient model initially used to study spatially partially coherent

stationary sources [39, 62–64]. Later it has been extended for the

analysis of the temporal and spectral coherence of nonstationary

fields [21]. Simple Gaussian functions aid to characterize the co-

herence features of optical fields, and many real light sources have

experimentally been demonstrated to obey the GSM [65–68]. Other

similar kind of models, including twisted GSM [69, 70], sinc Schell-

model pulses [71], cosine-Gaussian correlated Schell-model [72],

and many others [73–77], have further been put forward in the pro-

cess of more accurate description of realistic optical fields. How-

ever, the focus of this chapter is the GSM for nonstationary fields

that have broad or ultrabroad spectra, as discussed in paper I.

3.1 PARTIALLY COHERENT MODEL SOURCES

In this section we discuss a simple analytical model for partially

coherent pulsed optical sources. It is analogous to the GSM above,

consisting of a sequence of pulses with wide spectra. Thereby the

model represents broadband Gaussian Schell-model pulse trains.

This analysis can be regarded as an extension to the Gaussian Schell-

model plane wave pulses [21]. These broadband sources may ex-

hibit variable spectral coherence, although they possess full spa-

tial coherence at individual frequencies. This particular model is

inspired by the spectral coherence properties of supercontinuum



light produced in a nonlinear single-mode optical fiber. We con-

sider a single-spatial mode fiber whose mode profile is of Gaussian

form and it supports isodiffracting free-space propagation (i.e., the

Rayleigh range is independent of frequency) [78–81]. Moreover, the

spectrum and the spectral coherence function of the field are taken

to be Gaussians.

3.1.1 Field model in frequency domain

We begin the introduction of the field model in the frequency do-

main by assuming that the SC light is generated in a graded-index

fiber with a parabolic refractive-index distribution. The modal pro-

file in the space-frequency domain, as discussed in paper I, then

can be expressed as

E(ρ; ω) = E0 exp

(

− ω

ω0

ρ2

w2

)

, (3.1)

where E0 is a constant, ρ denotes the transverse position, ω is the

angular frequency, and w represents the 1/e modal half-width at

the peak frequency ω0 of the Gaussian spectrum. In this model, we

assume a spectral correlation function of the following form

W(ω1, ω2) = [S(ω1)S(ω2)]1/2

µ(ω1, ω2), (3.2)

with

S (ω) = S0 exp

[

− 2

Ω2(ω − ω0)

2

]

(3.3)

and

µ(ω1, ω2) = exp

[

− (ω1 − ω2)2

2Σ2

]

. (3.4)

Here Ω and Σ represent the spectral width and the spectral coher-

ence width of the field, respectively, and the ratio Σ/Ω is a measure

of the degree of spectral coherence.

To proceed with the analysis, it is preferable to use the average

and difference frequency coordinates defined as ω = (ω1 + ω2) /2


Gaussian Schell-modelpulse trains

and ∆ω = ω2 −ω1, respectively. Then the spectral correlation func-

tion can be written as

W(ω, ∆ω) = S0 exp

[

− 2

Ω2(ω − ω0)

2

]

exp

(

−T2

8∆ω2

)

, (3.5)

with a new constant T defined as

T2

4=

1

Ω2+

1

Σ2=

1

Ω2µ4, (3.6)

where

µ =

√

2

ΩT=

[

1 +

(

Ω

Σ

)2]−1/4

(3.7)

is called the integrated (or overall) degree of spectral coherence (see

paper I). In the case of full spectral coherence Σ → ∞, we clearly

have µ → 1, whereas in the quasi-stationary case Σ ≪ Ω, we may

approximate µ ≈√

Σ/Ω.

Now, the cross-spectral density (CSD) associated with the field

may be expressed as

W(ρ1, ρ2; ω1, ω2) = W(ω1, ω2)E∗(ρ1; ω1)E(ρ2; ω2). (3.8)

Using Eqs. (3.1) and (3.5), the CSD, in average and difference fre-

quency coordinates, takes the form

W(ρ1, ρ2; ω, ∆ω) = W0 exp

(

− ω

ω0

ρ21 + ρ2

2

w2

)

× exp

[

− 2

Ω2(ω − ω0)

2

]

exp

(

−T2

8∆ω2

)

× exp

(

−1

2

∆ω

ω0

ρ22 − ρ2

1

w2

)

, (3.9)

where W0 = S0 |E0|2. Correspondingly, the frequency-integrated

degree of spatial coherence, defined through

µ2(ρ1, ρ2) =

∫∫ ∞

0 |W(ρ1, ρ2; ω1, ω2)|2 dω1dω2∫ ∞

0S(ρ1; ω1)dω1

∫ ∞

0S(ρ2; ω2)dω2

, (3.10)



takes the form of

µ(ρ1, ρ2) = µ exp

[

−1

8

Ω2(

1 − µ4)

ω20

(

ρ21 − ρ2

2

)2

w4

]

. (3.11)

In this formula the exponential function is a result of the spectral

variation of the beam size. For full spectral coherence µ = 1, we

get µ(ρ1, ρ2) = µ. This result follows if ρ1 = ρ2, for instance when

the two points are symmetrically located with respect to the optical

axis. Considering an off-axis and an axial point, we obtain

µ(ρ, 0)/µ = exp

[

−1

8

Ω2(

1 − µ4)

ω20

( ρ

w

)4]

. (3.12)

0 0.5 1 1.5 20

0.2

0.4

0.6

0.8

1

ρ/w

µ(ρ)/

µ

Figure 3.1: Normalized integrated degree of spatial coherence between an axial and any

off-axis point for Ω/ω0 = 0.5 and µ4 = 0.1 (black), 0.5 (blue), 0.7 (green), and 0.85

(red). The normalized Gaussian spectral densities are denoted by dashed lines at ω = ω0

(black), ω = ω0 − Ω (blue), and ω = ω0 + Ω (red).

The variation of the integrated degree of spatial coherence for a

broad spectrum (Ω/ω0 = 0.5) is shown in Fig. 3.1. Although the

reduction of coherence is observable around ρ = w, it is notable

only in the region ρ > w. For smaller values of Ω/ω0, this effect is

less significant. The normalized spectral densities are also plotted

in the figure to assist comparison. Generally, it is obvious from the

curves that the spatial coherence remains very high in the region

where the spectral density has a significant value.



3.1.2 Field model in time domain

We proceed to analyse the model in the temporal domain. Using

the generalized Wiener-Khintchine theorem, the mutual coherence

function can be expressed, in average t = (t1 + t2) /2 and difference

∆t = t2 − t1 time coordinates, as

Γ(ρ1, ρ2; t, ∆t) = I0 exp

(

−ρ21 + ρ2

2

w2

)

exp (−iω0∆t)

× exp

[

Ω2

8

(

ρ21 + ρ2

2

ω0w2+ i∆t

)2]

× exp

[

2

T2

(

ρ21 − ρ2

2

2ω0w2− it

)2]

, (3.13)

where I0 = πS0 |E0|2 Ω2µ2. Thus the instantaneous spatiotemporal

intensity distribution becomes

I(ρ; t) = I0 exp

[

−2( ρ

w

)2]

× exp

[

1

2

(

Ω

ω0

)2( ρ

w

)4]

exp

(

−2t2

T2

)

. (3.14)

Thereby, on average the pulse has a Gaussian temporal shape at

each transverse position. The characteristic pulse length T depends

on spectral coherence. For complete coherence T = 2/Ω and in the

quasi-stationary case T ≈ 2/Σ.

The complex degree of temporal coherence takes on the form

γ(ρ1, ρ2; t, ∆t) = exp

[

−1

8

Ω2(

1 − µ4)

ω20

(

ρ21 − ρ2

2

)2

w4

]

× exp

(

− i

2

Ωµ2

ω0

ρ21 − ρ2

2

w2

t

T

)

exp

(

i

2

Ω

ω0µ2

ρ21 + ρ2

2

w2

∆t

T

)

× exp

(

− ∆t2

2Θ2

)

exp (−iω0∆t) , (3.15)

where

Θ = Tµ2

√

1 − µ4= T

Σ

Ω. (3.16)



Here Θ is a quantitative measure of the coherence time for plane-

wave case, i.e., w → ∞. The integrated (or overall) degree of coher-

ence, which is the same in the spectral and temporal domain, can

then be written, as shown in paper I, as

µ =

[

1 +

(

T

Θ

)2]−1/4

. (3.17)

The absolute value of the complex degree of coherence becomes

|γ(ρ1, ρ2; t, ∆t)| = exp

[

−1

8

Ω2(

1 − µ4)

ω20

(

ρ21 − ρ2

2

)2

w4

]

exp

(

− ∆t2

2Θ2

)

,

(3.18)

which is independent of the average time t and equals one for the

spectrally fully coherent case µ = 1 and when ρ1 = ρ2. Further, the

absolute value of the complex degree of temporal coherence is

|γ(ρ, ρ; ∆t)| = exp

(

− ∆t2

2Θ2

)

, (3.19)

which does not depend on space and is equal to the plane-wave

result. The magnitude of the complex degree of spatial coherence

between an axial and an off-axis point takes the form

|γ(ρ, 0; 0)| = exp

[

−1

8

Ω2(

1 − µ4)

ω20

( ρ

w

)4]

. (3.20)

The right-hand side of this equation is exactly the same as Eq. (3.12)

and thus we may write

|γ(ρ, 0; 0)| = µ(ρ, 0)/µ. (3.21)

Hence, the complex degree of spatial coherence in the temporal do-

main changes in the same manner as in the spectral domain and

therefore it leads to similar conclusions as in Fig. 3.1. In principle,

the spatial coherence in the time domain depends on the spectral

coherence of the field, but in the quasi-stationary case when µ ≪ 1



the width of the spectrum is crucial. Therefore, it is shown that

the SC light generated in a nonlinear single-mode optical fiber is

highly, but not fully, spatially coherent in the temporal domain.

Experimental verification of these results is, in principle, possible

but it is not that straightforward. One potential way is to mea-

sure the spectrally resolved interference patterns in the standard

Young’s double-slit experiment and subsequently the Friberg-Wolf

theorem [82] can be applied to characterize the spatial coherence in

time domain. Alternatively, a fully achromatic interferometer, such

as the wavefront-folding interferometer [83], could be employed to

measure the space-time domain degree of coherence directly.

3.2 PULSE PROPAGATION IN ABCD SYSTEMS

Next, the fields generated by the model sources introduced in the

previous section are propagated through optical systems that can

be represented by typical 2 × 2 ABCD matrices. If the CSD at the

plane z = 0 is considered to be of the form of Eq. (3.8), then at any

plane z > 0 the CSD may be expressed as

W(ρ1, ρ2, z; ω1, ω2) = W(ω1, ω2)E∗(ρ1, z; ω1)E(ρ2, z; ω2), (3.22)

where

E(ρ, z; ω) =ω

i2πcBexp (iωL/c) exp

(

iωD

2cBρ2

)

×∫ ∞

−∞E(ρ′; ω) exp

(

iωA

2cBρ′2)

exp

(

− iω

cBρ · ρ

′

)

d2ρ′ (3.23)

is the diffraction formula for spectral field propagation in a paraxial

optical system and L is the axial optical path length through the

system. It is advantageous for the analysis to express the fields

in terms of the well-known q parameter. The isodiffracting field

defined in Eq. (3.1) can be written in the form

E(ρ; ω) = E0 exp

(

iω

2cq0ρ2

)

, (3.24)



where q0 = −izR denotes the q parameter at the plane z = 0 and

zR = ω0w2/2c is the Rayleigh range. Then the field distribution at

any plane z > 0 becomes (see paper I)

E(ρ, z; ω) =E0q0

Aq0 + Bexp

(

iωL

c

)

exp

[

iω

2cq(z)ρ2

]

, (3.25)

where

q(z) =Aq0 + B

Cq0 + D(3.26)

is the q parameter at the output plane z of the system. Finally, the

CSD at any plane z > 0 takes on the form

W(ρ1,ρ2, z; ω, ∆ω) =S0|E0|2z2

R

A2z2R + B2

exp (i∆ωL)

× exp

− 2

Ω2(ω − ω0)

2 +i

2c

[

ρ22

q(z)− ρ2

1

q∗(z)

]

ω

× exp

−T2

8∆ω2 +

i

4c

[

ρ21

q∗(z)+

ρ22

q(z)

]

∆ω

. (3.27)

In the space-time domain at any plane z the MCF can be expressed,

using the Wiener–Khintchine theorem, as

Γ(ρ1,ρ2, z; t, ∆t) = I0Ω

Texp(−iω0∆t)

× exp

iω0

2c

[

ρ22

q(z)− ρ2

1

q∗(z)

]

× exp

−Ω2

8

[

1

2c

(

ρ22

q(z)− ρ2

1

q∗(z)

)

− ∆t

]2

× exp

− 2

T2

[

1

4c

(

ρ21

q∗(z)+

ρ22

q(z)

)

− tr

]2

, (3.28)

where

I0 =2πS0|E0|2z2

R

A2z2R + B2

, (3.29)



and

tr = t − L/c, (3.30)

is the retarded average time. The information about the variation of

the spatial coherence of the field propagated through any paraxial

optical system can be extracted from Eqs. (3.27) and (3.28) in both

frequency and time domains, respectively.


4 Pulse trains generated

by axicons

The analysis of the coherence properties of optical pulses is contin-

ued in this chapter by studying the optical fields created by an opti-

cal element, called the axicon [43,44]. An axicon produces long and

narrow focal lines along its optical axis. It converts incident plane

waves into conical waves [43,44], as shown in Fig 4.1, thereby gener-

ating approximately nondiffracting (propagation-invariant) fields,

for example Bessel beams [41, 42, 84]. This kind of an optical el-

ement is realized in practice in the form of refractive or reflective

conical surfaces [43]. A rotationally symmetric refractive axicon,

considered in Fig 4.1, bends the incident plane waves towards the

optical axis by an angle

θ = arcsin(n sin φ)− φ, (4.1)

where n is the refractive index and φ is the angle of the axicon,

respectively. The propagation-invariant range is L = R sin θ.

R

φ

θ

n

L

z

Figure 4.1: Schematic diagram depicting how plane waves incident onto a rotationally

symmetric refractive axicon are converted into conical waves.



Alternatively, diffractive counterparts [85, 86] of the refractive

and reflective axicons are also applied to produce similar kinds of

monochromatic beams. A diffractive axicon may be considered as

a radially periodic structure derived from the refractive axicon by

applying the principle of zone construction [87]. When the various

types of axicons are illuminated with broadband light, they act dif-

ferently since their dispersive properties are not the same [41, 88].

In particular, diffractive axicons are highly chromatic, which can be

advantageous in certain applications [44].

In this chapter we discuss the nature of axicon fields generated

by reflective and diffractive axicons illuminated with polychromatic

light. While the optical fields created by reflective axicons possess

frequency-independent cone angles, leading to an approximation of

X waves [89], diffractive axicons produce wave fields whose trans-

verse patterns instead do not depend on frequency, thus resulting

in Bessel pulses [90, 91]. The axicons are irradiated with spectrally

fully coherent light to analyze the effect of dispersion on the spa-

tiotemporal shape of the generated fields. Further, spectrally inco-

herent illumination is applied to examine the spatiotemporal coher-

ence characteristics of the axicon fields.

We consider a broadband isodiffracting Gaussian field [78–81]

incident onto the axicon and assume that the axicon is illuminated

with the beam waist. The incident field can then be represented in

the frequency domain as

E(ρ′; ω) =

(

2

πw2

)1/2 ( ω

ωr

)1/2√

S(ω) exp

(

− ω

ωr

ρ′2

w2

)

, (4.2)

where ρ′ denotes the transverse coordinate in the plane z = 0 of the

axicon, ωr is a reference frequency, w stands for the beam width at

ω = ωr, and S(ω) is the spectral distribution given by

S(ω) =S0

Γ(2n)ωr

(

2nω

ωr

)2n

exp

(

−2nω

ωr

)

. (4.3)

Here the parameter n controls the width of the spectrum, ωr is the

peak frequency taken as the reference frequency in the expression


Pulse trains generatedby axicons

of the incident field, and Γ is the Gamma function. In this anal-

ysis we consider n ≥ 1. A spectrum of this kind is often used to

investigate broadband localized waves as it contains no negative

frequencies [92, 93].

Now, the spatial distribution of the spectral density of the inci-

dent field, defined in Eq. (2.15), can be expressed as

S(ρ′; ω) =2(2n)2nS0

πw2Γ(2n)ωr

(

ω

ωr

)2n+1

exp

[

−2ω

ωr

(

n +ρ′2

w2

)]

, (4.4)

having the maximum at

ωρ = ωrn + 1/2

n + ρ′2/w2. (4.5)

If the incident wave field is fully coherent in the space-frequency

domain, its space-time counterpart can be found using the Fourier-

transform relationship

E(ρ′; t) =∫ ∞

0E(ρ′; ω) exp (−iωt) dω, (4.6)

and the spatiotemporal intensity profile, defined in Eq. (2.8), is

I(ρ′; t) =I0

[

(

n + ρ′2/w2)2

+ (ωrt)2

]n+3/2, (4.7)

0 1 2 3 40

0.2

0.4

0.6

0.8

1(a)

ω/ωr

S(ρ′;ω

)/S(ρ′;ω

ρ)

−4 −2 0 2 40

0.2

0.4

0.6

0.8

1(b)

ωrt

I(ρ′;t

)/I(

ρ′;0

)

Figure 4.2: Normalized (a) spectral and (b) temporal intensity profiles of the incident field

at distances ρ′ = 0 (blue), ρ′ = w/√

2 (black), ρ′ = w (green), and ρ′ =√

2w (red).



where I0 = |E0|2. The spectral density and the temporal intensity

profiles are shown in Figs. 4.2(a) and 4.2(b), respectively. We see

that as the transverse distance increases, the incident field’s spec-

trum gets narrower and the intensity profile widens.

4.1 AXICON FIELDS IN SPACE-FREQUENCY DOMAIN

We begin the analysis of the axicon fields with the space-frequency

domain formulation, since it is more convenient from the analytical

aspect. We assume that the effect of the axicon can be described in

the thin-element approximation by defining a complex-amplitude

transmission function t(ρ′; ω) of the form

t(ρ′; ω) =√

η(ω) exp[

−i (ω/c) sin θ(ω)ρ′]

, (4.8)

where θ(ω) = θ for reflective axicons and for diffractive axicons

sin θ(ω) = 2πc/dω, with d being the radial period. Here η(ω) de-

notes the (frequency-dependent) reflection coefficient and diffrac-

tion efficiency for the two types of axicons.

Quite generally, the field exiting the axicon is t(ρ′ ; ω)E(ρ′; ω).

In the paraxial domain, the propagation of the field is described by

the Fresnel integral for the rotationally symmetric case,

E(ρ, z; ω) =ω

iczexp

[

iω

c

(

z +ρ2

2z

)]

×∫ ∞

0ρ′t(ρ′; ω)E(ρ′; ω)J0

(ω

czρρ′)

exp

(

iω

2czρ′2)

dρ′, (4.9)

where J0 is the Bessel function of the first kind. This integral is

evaluated with the help of the method of stationary phase [58, 94],

yielding the result

E(ρ, z; ω) = −i

√

η(ω)

π

√

z

zR

ω

csin θ(ω)

√

S(ω)

× exp

[

− ω

ωr

z2

w2sin2 θ(ω)

]

J0

[ω

cρ sin θ(ω)

]

× exp

iω

[

z

v(ω)+

ρ2

2cz

]

, (4.10)



where ρ is the radial coordinate at an arbitrary plane z, zR = ωrw2/2c

is the Rayleigh range of the incident field, and the phase velocity is

denoted by v(ω) = c/[1 − sin2 θ(ω)/2].

In the case of reflective axicons, the axcion angle is independent

of frequency, i.e., θ(ω) = θ = constant. From Eq. (4.10) the spectral

density then becomes

S(ρ, z; ω) =− η(ω)

π

z

zR

(ω

c

)2sin2 θS(ω) exp

(

−2ω

ωr

z2

w2sin2 θ

)

× J20

(ω

cρ sin θ

)

. (4.11)

For diffractive axicons sin θ(ω) = 2πc/dω, and from Eq. (4.10) we

now find that

E(ρ, z; ω) = −2πi

d

√

η(ω)

π

√

z

zR

√

S(ω) exp

(

−2π2cz2

d2zRω

)

× J0

(

2πρ

d

)

exp

[

iω

c

(

z +ρ2

2z

)]

exp

(

− i2π2cz

d2ω

)

, (4.12)

giving

S(ρ, z; ω) = −4π2

d2

η(ω)

π

z

zRS(ω) exp

(

−4π2cz2

d2zRω

)

J20

(

2πρ

d

)

(4.13)

for the spectral density.

The results presented in this analysis are not strongly dependent

on the precise form of η(ω). Hence we take η(ω) as constant, albeit

this is not always true especially for diffractive axicons in broad-

band illumination. In particular, from now on we assume η = 1.

4.2 SPATIOTEMPORAL AXICON FIELDS

This section is dedicated to the discussion of the axicon fields under

two different kinds of illumination. Firstly, coherent pulses are in-

cident onto the axicons and the corresponding spatiotemporal pro-

files of the ensuing fields are considered. Secondly, the axicons are

illuminated by stationary fields and the temporal coherence prop-

erties of the resulting fields are analyzed.



4.2.1 Spectrally coherent illumination

We proceed to examine the spatiotemporal properties of broadband

fields generated by reflective and diffractive axicons under the illu-

mination of spectrally coherent light (paper II).

The space-time fields generated by reflective axicons are found

from Eq. (4.10) through Fourier transformation. We find that

E(ρ, z; tr) =− i(2n)n

[

S0

πΓ(2n)

z

zR

ωr

c

]1/2

sin θ

×∫ ∞

0xn+1 J0

(

ωr

cρ sin θx

)

exp(−ax)dx, (4.14)

where x = ω/ωr is a scaled variable and a is defined as

a = n +( z

L

)2+ iωr

(

tr −ρ2

2cz

)

, (4.15)

where tr = t − z/v is the retarded time, v = c/(1 − sin2 θ/2) is the

phase velocity, and L = w/ sin θ denotes the range for propagation

invariance. There appears to be no analytical solution to Eq. (4.14)

in the off-axial case, while we find a simple expression for the on-

axis intensity distribution, as

I(0, z; tr) = I(0, z; 0)

(

n + z2/L2)2(n+2)

[

(n + z2/L2)2 + (ωrtr)2]n+2

, (4.16)

where

I(0, z; 0) =(2n)2nΓ2(n + 2) sin2 θ

(n + z2/L2)2(n+2)

S0

πΓ(2n)

ωr

c

z

zR. (4.17)

The half width at half maximum (HWHM) of I(0, z; tr) at tr = Tr is

given by

ωrTr =√

21/(n+2) − 1[

n + (z/L)2]

. (4.18)

Hence we can observe that the effective temporal width of the pulse

increases with the propagation distance. This effect is obvious from

the intensity distribution of the incident field in Fig. 4.2(b) which



−20 0 20−10

−6

−202

6

10(a)

ωrtr

ρ[µ

m]

−20 0 20−10

−6

−202

6

10(b)

ωrtr

ρ[µ

m]

Figure 4.3: Illustrating the amplitude of the field after the reflective axicon, at distances

(a) z = L/2, and (b) z = L.

illustrates the pulse width increase as ρ′ increases. The geometrical

explanation of this phenomenon is discussed in paper II.

Figure 4.3 depicts axial (2D) cross-sections of the field produced

by reflective axicons. In the initial propagation stage the field seems

like a distorted X wave, but towards the end of the propagation-

invariant range its profile takes the expected X-wave shape. From

Eq. (4.10) we see that the phase of the axicon field varies inversely

with propagation distance z. As a result, the phase term gives rise

to a spherical wave in the beginning and the field becomes more

planar on propagation. This explains the distortion present in the

axcion field.

Next we investigate the spatiotemporal properties of the fields

generated by diffractive axicons. In this case, Eq. (4.10) yields for

the axicon field

E(ρ, z; tr) = −4πi

d(2n)n

[

S0z

πωrΓ(2n)zR

]1/2

J0

(

2πρ

d

)

×[

b(z)

an(ρ, z; tr)

](n+1)/2

Kn+1

[

2√

an(ρ, z; tr)b(z)

]

, (4.19)

where Km denotes the modified Bessel function of the second kind

and order m, the parameter an is defined as

an(ρ, z; tr) = n + i

[

ωrtr −π

w/d

(ρ/d)2

z/L

]

, (4.20)



with tr = t − z/c, and

b(z) =( z

L

)2+ iπ

z

L

w

d. (4.21)

In this case no analytical expression for the pulse HWHM is found

even on-axis. The resulting field at two different propagation dis-

tances is illustrated in Fig. 4.4. Well-defined zeros are clearly visible

in the transverse direction, as expected for Bessel-type pulses. Cer-

tain off-axis trailing of the pulse’s temporal front end is observed,

an effect that is explained in paper II.

0 20 40 60 80 100−5

−3

−101

3

5(a)

ωrtr

ρ[µ

m]

0 20 40 60 80 100−5

−3

−101

3

5(b)

ωrtr

ρ[µ

m]

Figure 4.4: Illustrating the amplitude of the field after the diffractive axicon, at distances

(a) z = L/2, and (b) z = L.

4.2.2 Spectrally incoherent illumination

Here we consider that the axicons are illuminated with stationary

but spatially fully coherent fields. Focusing only on the axial case,

we may evaluate the mutual coherence function, for fields produced

by reflective axicons in spectrally uncorrelated illumination, from

Eqs. (2.6) and (4.14) with ρ = 0. The corresponding complex degree

of coherence then becomes

γ(0, z; τ) =

[

n + (z/L)2

n + (z/L)2 + iωrτ/2

]2n+3

. (4.22)

The HWHM value Θr of this profile is given by

ωrΘr = 2√

21/(2n+3) − 1[

n + (z/L)2]

. (4.23)



Following a similar procedure as with reflective axicons but using

Eq. (4.19) instead of Eq. (4.14), we obtain, for fields generated by

diffractive axicons, the axial complex degree of coherence

γ(0, z; τ) =

(

2n

2n + iωrτ

)n+1/2 K2n+1(2√

αnβ)

K2n+1(2√

2nβ), (4.24)

where αn(τ) = 2n + iωrτ and

β(z) =8π2c2z2

d2w2ω2r

. (4.25)

In this case it has not been possible to obtain an analytical solution

for the HWHM.

The on-axis complex degree of temporal coherence for the inci-

dent field and for the fields produced by both types of axicons at

two propagation distances are illustrated in Figs. 4.5(a) and 4.5(b).

The axial degree of temporal coherence for the fields generated by

reflective axicons are seen to increase with propagation, whereas

for diffractive axicon the temporal coherence width in the on-axis

case decreases rapidly. These opposite behaviors can be well under-

stood on the basis of the axicon geometries and the axial spectral

densities characterizing the respective axicon fields, as discussed in

paper II. Here we have studied the axicon fields only under the illu-

−10 −5 0 5 100

0.2

0.4

0.6

0.8

1

(a)

ωrτ

|γ(0

,z;τ)|

−10 −5 0 5 100

0.2

0.4

0.6

0.8

1

(b)

ωrτ

|γ(0

,z;τ)|

Figure 4.5: Temporal coherence profiles of the incident field and the fields generated by

axicons at distances (a) z = L/2 and (b) z = L. In both figures, the incident field

is denoted by red curves, while green and blue curves are for reflective and diffractive

axicons, respectively.



mination of fully coherent and stationary (spectrally uncorrelated)

light. The analysis could be extended to partially temporally and

spectrally coherent axicon fields [95, 96].


5 Pulse train characterization

by Michelson interferometer

Temporal coherence evaluation of optical sources is one of the pri-

mary experimental tasks in photonics. For that Michelson’s inter-

ferometer is the classical tool. It is the conventional way to measure

the temporal coherence of stationary light sources [58]. But when

it comes to nonstationary light, such as pulsed optical sources, the

interpretation of the standard time-integrated Michelson interfero-

gram, though customarily employed [55,56,97–105], is not straight-

forward. The reason for that is that the relative temporal intensi-

ties of the two time-delayed interfering copies of the pulse change

with time, and hence the visibility of the interference fringes be-

comes time-dependent. Therefore time-integrated visibility is not

a true measure of the temporal coherence of pulsed light sources;

instead, time-resolved measurements of the interference fringe pat-

tern are necessary to determine the temporal coherence completely.

However, if the pulsed source is quasi-stationary in the sense that

the coherence time is only a small fraction of the pulse width, as

for example in excimer lasers, then a complete temporal coherence

characterization can be performed using the time-integrated mea-

surements. This entire description of coherence measurements is

schematically illustrated in Fig. 5.1.

In this chapter we consider an equal-path Michelson interferom-

eter and present a detailed analysis of how to measure the two-time

mutual coherence function of pulsed sources, as introduced in pa-

per III. Using specific analytical models, discussed in paper IV,

it is shown that the integrated degree of temporal coherence pro-

vides a true characterization only for quasi-stationary sources, but

for more coherent sources, for instance supercontinuum light, dis-

turbed mode-locked lasers, and free-electron lasers, time-resolved



Temporal coherencemeasurement of

light sources

Michelson’sinterferometer Non-stationary sources

(e.g., pulsed lasers)

Visibility becomestime-dependent

Time-integratedmeasurement is not correct

Quasi-stationary pulsedsources (e.g., excimer lasers)

Time-delay introduced by theinterferometer results in

change of the relative temporalintensity of two interfering

copies with time

Why?

Stationary sources

Standard tool

Often used, butnot always true

Exception

Figure 5.1: Schematic diagram representing the necessity of the time-resolved measure-

ments of interference fringes for pulsed light sources.

measurements are necessary. Further, in paper V, we employ nu-

merically simulated realizations of SC light to confirm the conclu-

sions drawn from the analytical models.

5.1 TIME-RESOLVED MEASUREMENTS WITH MICHELSON

INTERFEROMETER

Here we discuss a setup of equal-path Michelson’s interferometer

(see paper III) illustrated in Fig. 5.2 [105, 106]. In this arrangement

the mirrors are at the same central distances, but a small tilt angle

φ is introduced between the mirrors in opposite directions. This

modification gives rise to a transverse fringe pattern with an x-

dependent time delay τ = 2 sin(2φ)x/c. This kind of setup is

called balanced interferometer as the central path lengths of the

mirrors in both the arms are the same. On the other hand, an im-

balanced interferometer, where the path length difference between

the two arms corresponds to the laser repetition rate, is frequently

used to characterize the coherence of, for example, supercontinuum

sources [54]. This arrangement measures the coherence properties

of the field as a function of frequency providing information on


Pulse train characterizationby Michelson interferometer

the Dudley-Coen coherence function defined in Eq. (2.20). How-

ever, we are interested in the two-time coherence function and it

does not appear possible to measure this quantity with the imbal-

anced interferometer, for this reason the concept of time-resolved

balanced interferometry is proposed here.

Pulsedlaser

Intersectingwavefronts Observation plane

M2

M1

x

φ

φ

Figure 5.2: Setup of Michelson’s interferometer with tilted mirrors.

We consider a train of optical plane-wave pulses incident onto a

Michelson interferometer. A single pulse with temporal field E0(t)

is assumed to be an individual realization of an ensemble of succes-

sive pulses in the pulse train. This temporal single-pulse realization

may change from pulse to pulse, making the pulse train partially

coherent. The two-time mutual coherence function (MCF), defined

in Eq. (2.11), for the incident pulse train is denoted by Γ0(t1, t2).

Expressing it in terms of the average time t = (t1 + t2) /2 and the

time difference ∆t = t2 − t1, we have

Γ0(t, ∆t) = 〈E∗0(t − ∆t/2)E0(t + ∆t/2)〉

=√

I0(t − ∆t/2)I0(t + ∆t/2)γ0(t, ∆t), (5.1)

where I0(t) is the mean intensity of the pulse train and γ0(t, ∆t) is



its complex degree of temporal coherence.

If the field E0(t) is incident on the Michelson interferometer, the

output field can be represented as a superposition of its two time-

delayed replicas as

E(t, τ) =1

2[E0(t − τ/2) + E0(t + τ/2)] , (5.2)

where τ is the time delay introduced by the interferometer. The

mean temporal intensity I(t, τ) = 〈|E(t, τ)|2〉 of the entire ensemble-

averaged output pulse train then is

I(t, τ) =1

4[I0(t − τ/2) + I0(t + τ/2)] +ℜ [Γ0(t, τ)] , (5.3)

where ℜ denotes the real part. Defining φ0(t, τ) = arg [γ0(t, τ)], we

can write Eq. (5.3) in the form

I(t, τ) =1

4[I0(t − τ/2) + I0(t + τ/2)]

+√

I0(t − τ/2)I0(t + τ/2) |γ0(t, τ)| cos [φ0(t, τ)] . (5.4)

and the fringe visibility is given by

V(t, τ) =2√

I0(t − τ/2)I0(t + τ/2)

I0(t − τ/2) + I0(t + τ/2)|γ0(t, τ)| . (5.5)

Hence the visibility measurements provide the absolute value of

the two-time complex degree of coherence γ0(t, ∆t) and its phase

φ0(t, ∆t) can be determined from the positions of the fringes. For

that one needs to perform the measurement of the mean temporal

intensity I(t, τ) of the pulse train for different values of τ. This

can be implemented, for example, by recording temporal marginals

of spectrograms produced by FROG [50], which may also be used

to measure I0(t). However, the experiment is considerably more

complicated than the standard time-integrated Michelson interfer-

ometric measurement.

To investigate the time-integrated measurements, we introduce

the integrated temporal coherence function as the time integration



over the two-time MCF,

Γ(int)(∆t) =∫ ∞

−∞Γ(t, ∆t)dt, (5.6)

and, in a similar manner, the integrated complex degree of temporal

coherence is defined as

γ(int)(∆t) =

∫ ∞

−∞Γ(t, ∆t)dt

∫ ∞

−∞I(t)dt

. (5.7)

Now, the time-dependent interference pattern I(t, τ), in Eq. (5.3), is

integrated over time and we find

I(τ) =∫ ∞

−∞I0(t)dt +ℜ

∫ ∞

−∞Γ0(t, τ)dt

, (5.8)

and then using Eqs. (5.6) and (5.7) we obtain

I(τ)/E0 = 1 +∣

∣

∣γ(int)0 (τ)

∣

∣

∣cos

[

φ(int)0 (τ)

]

, (5.9)

where E0 =∫ ∞

−∞I0(t)dt, and

φ(int)0 (τ) = arg

[

γ(int)0 (τ)

]

. (5.10)

Hence the absolute value of γ(int)0 (τ) is found from the visibility

of the time-integrated interference fringes and the fringe positions

provide the information about the phase.

Next, we discuss a class of pulsed light sources for which time-

integrated measurements are sufficient. When the coherence time

is a small fraction of the pulse duration, the average intensity of the

pulse train may be approximated as√

I0(t − ∆t/2)I0(t + ∆t/2) ≈ I0(t). (5.11)

At the same time, we may also assume that γ0(t, ∆t) remains es-

sentially invariant during the coherence time. This is regarded as

a quasi-stationary situation. Then, the MCF of the incident pulse

train in Eq. (5.1) may be written as

Γ0(t, ∆t) = I0(t)γ0(∆t), (5.12)



and thereby, γ(int)(∆t) = γ0(∆t). Thus, for quasi-stationary pulsed

sources, the two-time MCF can be completely determined by mea-

suring I0(t) using FROG and obtaining γ0(∆t) from the conven-

tional Michelson interferogram.

5.2 APPLICATION TO DIFFERENT MODELS

In this section we discuss how the true two-time coherence function

differs from the time-integrated function for pulse trains. The anal-

ysis is carried out using analytical models physically representing

perturbed trains of mode-locked pulses and supercontinuum pulse

trains generated in a nonlinear fiber. Further, simulated SC realiza-

tions are also employed to confirm the conclusions.

5.2.1 Gaussian Schell-model for mode-locked pulse trains

Here we demonstrate practical feasibility of the theoretical descrip-

tion presented in the preceding section. We start with the simplest

case of plane-wave pulses and introduce a mathematical model

called the Gaussian Schell-model. It has been shown that partially

coherent fields generated by free-electron lasers [104,107] obey this

model.

The CSD of the incident pulse train, in the average frequency

ω = (ω1 + ω2) /2 and difference frequency ∆ω = ω2 − ω1 coordi-

nates, may generally be written as

W0(ω, ∆ω) =√

S0(ω − ∆ω/2)S0(ω + ∆ω/2)

× |µ0(ω, ∆ω)| exp [iϕ0(ω, ∆ω)] , (5.13)

where S0(ω) is the average spectral density and µ0(ω, ∆ω) is the

complex degree of spectral coherence of the pulse train. Further,

ϕ0(ω, ∆ω) = arg [µ0(ω, ∆ω)].

Now, we consider a chirped Gaussian Schell-model pulse train



having a spectrum of the Gaussian form, as in paper IV,

S0(ω) = S0 exp

[

−2 (ω − ω0)2

Ω2

]

, (5.14)

where ω0 is the center frequency and Ω is the width of the spec-

trum. For the Schell-model the absolute value of the complex de-

gree of spectral coherence depends only on the frequency differ-

ence [39]. The absolute value is likewise taken to be Gaussian

|µ0(ω, ∆ω)| = exp

(

−∆ω2

2Σ2

)

, (5.15)

where Σ denotes the width of the spectral coherence. The chirp

is described by introducing a phase term in the complex degree of

spectral coherence as follows

ϕ0(ω, ∆ω) = − 2C

Ω2(ω − ω0)∆ω, (5.16)

where C is the chirp strength. For the fully coherent case Σ → ∞,

it follows from Eqs. (5.13), (5.14), and (5.16), that the field in the

frequency domain may be expressed as

E0(ω) =√

S0 exp

[

−1 − iC

Ω2(ω − ω0)

2]

, (5.17)

which implies that all the pulses have an identical form.

Next, we transform to the time-domain representation using the

generalized Wiener–Khintchine theorem. The MCF in the average

and difference time coordinates then takes on the form

Γ0(t, ∆t) =√

I0(t − ∆t/2)I0(t + ∆t/2)

× |γ0(t, ∆t)| exp [iφ0(t, ∆t)] , (5.18)

where

I0(t) = I0 exp

(

−2t2

T2

)

(5.19)



with I0 = 2πΩS0/T,

|γ0(t, ∆t)| = exp

(

− ∆t2

2Θ2

)

, (5.20)

and

φ0(t, ∆t) = −2C

T2t∆t − ω0∆t. (5.21)

In this case, T and Θ denote the pulse duration and the coherence

time, respectively. Therefore, in the proposed Michelson setup, the

time-dependent interference pattern of Eq. (5.4) can be expressed

for this model pulse train as (see paper IV)

I(t, τ)/I0 =1

2

exp

[

− 2

T2

(

t − τ

2

)2]

+ exp

[

− 2

T2

(

t +τ

2

)2]

+ exp

(

−2t2

T2

)

exp

[

− τ2

2T2

(

1 +T2

Θ2

)]

× cos

[(

2C

T2t + ω0

)

τ

]

, (5.22)

and the time-integrated expression, found by integrating Eq. (5.22)

with respect to t, as

I(τ)/E0 = 1 + exp

(

− τ2

2T2β2

)

cos (ω0τ) . (5.23)

This result is identical with the interference pattern obtained for the

pulse train using the standard Michelson interferometer.

The time-resolved and time-integrated interference patterns for

almost fully coherent and partially temporally coherent trains of

chirped pulses are illustrated in Fig. 5.3 as functions of the normal-

ized coordinates t/T and τ/T. In this plot, we consider C = 2 and

ω0T = 50. In the time-resolved pattern, fringes are inclined since

C 6= 0, with maxima and minima on lines 2Ct/T +ω0T = constant.

In the fully coherent case, fringes of high visibility are observed

when the two delayed pulses have a significant temporal overlap

(|τ/T| < 1). However, interference effects are not visible for large



−2 0 2

−2

−1

0

1

2

0.2

0.4

0.6

0.8

1

t/T

τ/

T

−2 −1 0 1 20

0.5

1

1.5

2

τ/T

I(τ)/

E0

−2 0 2

−2

−1

0

1

2

0.2

0.4

0.6

0.8

1

t/T

τ/

T

−2 −1 0 1 20

0.5

1

1.5

2

τ/T

I(τ)/

E0

Figure 5.3: Time-resolved (left) and time-integrated (right) interference patterns for

chirped trains of Gaussian Schell-model pulses. Top: almost coherent case (Θ/T = 0.9),

and bottom: partially coherent case (Θ/T = 0.3).

values of |τ/T|, where the X branches no longer overlap. In the par-

tially coherent case, things are a bit different. The fringe visibility

decreases quickly with increasing |τ/T|, even in the region where

the X branches still overlap. The time-integrated results are also

displayed in Fig. 5.3 (right column). The top panel in the right col-

umn demonstrates that a fully temporally coherent pulse train ap-

pears to be partially coherent in the time-integrated measurement:

the fringe visibility decreases for increasing values of |τ| because of

partial temporal overlap of the interfering pulses. A similar effect

is seen in the partially coherent case, but here the fringe visibility

decreases rapidly. Therefore, the time-integrated quantities do not

reveal the true coherence characteristics of pulses — for that time-

resolved measurements are required.



5.2.2 Analytical model for SC pulse trains

Application of the second-order coherence theory of nonstationary

light in the analysis of supercontinuum coherence [55, 56] unveiled

a decomposition of the two-time MCF and two-frequency CSD into

a quasi-coherent (qc) and a quasi-stationary (qs) contribution and

showed that the relative weights of these contributions depend on

the pump pulse parameters.

We introduce a simple analytical model representing SC light

generated in a nonlinear fiber. The qc and qs contributions are taken

into consideration. The model emulates the temporal coherence

properties of realistic SC sources. We may then write the total MCF

as a sum of J co-existing contributions

Γ(t, ∆t) =J

∑j=1

Γj(t, ∆t). (5.24)

Here

Γj(t, ∆t) =√

Ij(t − ∆t/2)Ij(t + ∆t/2)γj(t, ∆t), (5.25)

where Ij(t) is the temporal intensity and γj(t, ∆t) is the complex de-

gree of temporal coherence. Now, we assume the intensity profiles

of the contributions to be of the super-Gaussian form

Ij(t) = I0j exp

−2(t − tj)Mj

TMj

j

, (5.26)

where I0j is a constant amplitude, tj denotes the temporal origin

of the jth contribution, and Tj is temporal width of the jth pulse.

The shapes of the intensity profiles are governed by the parameters

Mj. Further, the complex degree of temporal coherence of each

contribution is assumed to be a Gaussian function

γj(t, ∆t) = exp

(

− ∆t2

2Θ2j

)

exp (−iω0∆t) , (5.27)

where Θj denotes the coherence time and can be chosen at will. To

represent realistic SC light, the simulation is to be performed so that



at least one qc contribution with Θj > Tj and one qs contribution

with Θj ≪ Tj are considered. On using Eqs. (5.25)–(5.27), the total

MCF in Eq. (5.24) assumes the form, as in paper V,

Γ(t, ∆t) =J

∑j=1

Gj(t, ∆t) exp (−iω0∆t) , (5.28)

where

Gj(t, ∆t) = I0j exp

−(

∆tj − ∆t/2)Mj +

(

∆tj + ∆t/2)Mj

TMj

j

× exp

(

− ∆t2

2Θ2j

)

, (5.29)

and ∆tj = t − tj.

−5 0 5

−2

0

2

0

0.5

1(a)

t

∆t

−5 0 5

−2

0

2

0

0.5

1(b)

t

τ

−2 0 20

0.5

1

1.5

2(c)

τ

I(τ)

Figure 5.4: Illustration of the analytical model for a supercontinuum pulse train: (a) mu-

tual coherence function Γ(t, ∆t), (b) time-resolved Michelson’s interference pattern I(t, τ),

and (c) time-integrated interference pattern I(τ).



This analytical model is illustrated in Fig. 5.4(a). We have in-

cluded two qc parts and one qs part in between them. The different

parameters for these contributions are listed in Table 5.1. More qc

or qs contributions with different values of the associated parame-

ters could be added.

Contribution M I0 T t Θ

1st quasi-coherent 4 2 1 −2.5 4

quasi-stationary 4 1 2 −0.15 0.1

2nd quasi-coherent 4 1 1 1.5 0.35

Table 5.1: Parameters (in dimensionless units) for the quasi-coherent and quasi-stationary

contributions in the analytical model for the SC pulse train in Fig. 5.4.

The time-resolved interference pattern is shown in Fig. 5.4(b).

High fringe visibility is observed in the interference pattern at large

values of |τ|, especially in the qc regions as long as the interfer-

ing pulses overlap in time. However, in the qs region the visibility

decreases quickly as |τ| increases. The time-integrated interference

pattern is plotted in Fig. 5.4(c). The curve primarily conveys the ex-

istence of a qs contribution, close to τ = 0, whereas at larger values

of |τ| the figure reveals also the presence of qc part. However, there

is no indication that exactly two qc contributions exist and the in-

formation about the temporal order of the qc and qs contributions

is lost.

5.2.3 Simulated realizations of SC light

Statistical ensembles of SC realizations are simulated in a 20 cm

long photonic crystal fiber pumped by 2 kW pulses at λ = 800

nm. Pulse propagation in the fiber is governed by the generalized

nonlinear Schrodinger equation [108, 109]. In each ensemble, 1000

pulses are considered. The input noise is included in the frequency

domain through the addition of a noise seed of one photon per

mode with random phase in each spectral discretization bin [56].

Two ensembles with pump-pulse durations (FWHM) of 200 fs



and 1 ps are studied in this section. The former represents a par-

tially coherent train of pulses with an overall degree of temporal

coherence, given in Eq. (3.17), of µ = 0.732, while the latter corre-

sponds to an almost incoherent pulse train (µ = 0.173).

−1 0 1 2 3−2

−1

0

1

2

0

0.5

1

replacements

(a)

t [ps]

∆t[p

s]

−2 −1 0 1 2 3−2

−1

0

1

2

0

0.5

1(d)

t [ps]∆

t[p

s]

−1−0.5 0 0.5 1 1.5 2 2.5 3−50

−30

−100

10

30

50

−6

−4

−2

0(b)

t [ps]

τ[f

s]

−2 −1 0 1 2 3−30

−20

−10

0

10

20

30

0.2

0.4

0.6

0.8

1(e)

t [ps]

τ[f

s]

−50 0 500

0.5

1

1.5

2(c)

τ [fs]

I(τ)

−50 0 500

0.5

1

1.5

2(f)

τ [fs]

I(τ)

Figure 5.5: Simulated results of SC realizations for 200 fs (left column) and 1 ps (right

column) pump pulses: (a,d) normalized MCF |γ0(t, ∆t)|, (b,e) time-resolved interference

pattern I(t, τ), and (c,f) time-integrated interference pattern I(τ). Note that in part (b),

I(t, τ) is plotted on a log scale to display finer details.

We begin with the 200 fs pulse realizations with the numerically



computed MCF shown in Fig. 5.5(a). The simulated time-resolved

results of Michelson’s experiment depicted in Fig. 5.5(b) feature sev-

eral distinguishable X patterns. Structures with high coherence are

manifested by high visibility of the fringes in the X patterns that ex-

tend over large delay values |τ|, revealing the quasi-coherent con-

tribution to the MCF. The visibility of the fringes calculated from

the interference pattern thus gives access to the full MCF. The time-

integrated result presented in Fig. 5.5(c), on the other hand, only

exhibits high visibility for small delay values |τ| corresponding to

the quasi-stationary part of the MCF. This profile does not provide

any information about the MCF’s coherent contribution.

For the ensemble corresponding to the longer 1 ps pump pulse

duration, a clear distinction among the qc and qs contributions

to the MCF can be made, as illustrated in Fig. 5.5(d). The small

square-shaped region denotes the qc part and the thin horizontal

line represents the qs part. Due to the dominant qs contribution,

the time-resolved interference fringes shown in Fig. 5.5(b) exhibit a

high contrast only in the vicinity of zero delay, except for the resid-

ual temporally coherent components at the pump’s leading edge.

Unlike in the 200 fs case, the qc and qs contributions can readily be

separated by inspection of the time-integrated interference pattern

shown in Fig. 5.5(c), qualitatively in the same way as in the analyt-

ical model of Fig. 5.4. Specifically, the qs contribution gives rise to

the strong modulation in the center of the integrated interference

pattern whilst the fringes in the broad pedestal correspond to the

qc contribution. The physical phenomena responsible for the gen-

eration the qc and the qs contributions in these two ensembles and

the corresponding effect in the interference patterns are explained

in paper V. These results show that a time-integrated measurement

may be sufficient to characterize the temporal coherence of nearly

incoherent SC pulses, whereas for partially coherent cases it clearly

hides part of the information.


6 Experimental perspectives

and results

In this chapter we discuss the temporal coherence measurements

of optical pulse trains using the standard Michelson interferome-

ter. Almost fully coherent chirped pulse trains emanating from a

femtosecond laser are investigated. As mentioned in the previous

chapter, complete temporal coherence characterization of sources of

this kind requires time-resolved measurements. If the means to per-

form time-resolved experiments are not available, time-integrated

measurements are the only option. Thus, the information conveyed

by the time-integrated techniques is considered here. Additionally,

the pulse trains are also measured using the frequency-resolved op-

tical gating (FROG) technique for completeness of the experimental

analysis. Here we present some preliminary experimental results,

while the ultimate objective is to implement comprehensive time-

resolved measurements.

6.1 TEMPORAL COHERENCE MEASUREMENT OF PULSE

TRAINS USING MICHELSON’S INTERFEROMETER

The classical Michelson interferometer is employed to measure the

integrated temporal coherence function of almost coherent trains of

pulses from a femtosecond laser. We consider a Michelson setup

with a small tilt φ of the fixed mirror M1, as shown (later in the

experimental part) in Fig. 6.1. In this situation, an additional x-

dependent delay is introduced as a result of the tilted wavefront.

Much as in section 5.1, the intensity distribution at the detection

plane may then be written as [cf., Eq. (5.3)]

I(t, τ) =1

2[I0(t + 2φx/c) + I0(t + τ) + Γ0(t + 2φx/c, t + τ)

+Γ∗0(t + 2φx/c, t + τ)]. (6.1)



Now, one may set τ to be equal to zero and 2φx/c then assumes the

role of time difference. Hence it is not necessary to scan the mirror

M2 to evaluate the temporal coherence function. Rather, one can

measure the visibility and the position of the interference fringes as

function of x to determine the absolute value and the phase of the

complex degree of coherence.

It is also of interest to scan mirror M2, and thereby change τ, in

the presence of tilt, especially if one finds difficulties in aligning the

interferometer perfectly. If the coherence time of the incident field

satisfies Tc ≫ λ0 = 2πc/ω0, the visibility in x direction is virtually

constant over several central fringes, but it depends on τ. When

τ is scanned, one can determine the absolute value of the complex

degree of coherence by measuring the spatial visibility.

Now we consider the mathematical description of the integrated

degree of coherence for nonstationary light as discussed in the pre-

vious chapter. Making use of Eqs. (5.7) and (5.9), we can predict the

results of our experimental measurements. As the pulse trains are

almost temporally coherent, the mutual coherence function can be

written as

Γ0(t, ∆t) = E∗0(t − ∆t/2)E0(t + ∆t/2), (6.2)

the integrated degree of temporal coherence becomes

γ(int)0 (τ) =

∫ ∞

−∞E∗

0(t)E0(t + τ)dt∫ ∞

−∞|E0(t)|2dt

, (6.3)

which is the normalized autocorrelation of the electric field. Thus,

the integrated degree of temporal coherence strongly depends on

the pulse duration. For any value τ 6= 0, |γ(int)0 (τ)| < 1, and so a

fully temporally coherent pulse train appears partially coherent in

time-integrated measurement.

Next we discuss the experimental setup shown in Fig. 6.1 and

the corresponding measurements. The setup has two parts, one

is for the Michelson interferometry and the other is for the FROG

measurements. A Ti-sapphire laser operating at central wavelength

λ0 = 800 nm with a repetition rate 1 KHz is generating pulse trains


Experimental perspectivesand results

with the average pulse duration of 130 fs. An average output power

of a few mW is used in the experiments. The incident pulse train

Ti-sapphirelaser

CMOScamera

FROGsetup

Cylindricallens

Fresnelbiprism

ThickSHGcrystal

Imaginglens

Camera

B1 B2

M1

M2

x

xφ

τ = τ(x)

Figure 6.1: Schematic diagram of the experimental setup consisting of a Michelson in-

terferometer and a FROG measurement arrangement. Beam geometry inside the FROG

(GRENOUILLE) is also shown.

passes through the beamsplitter B1 which divides it to proceed to-

wards the two different parts of the setup. Here we concentrate

only on the Michelson interferometer part; the other part will be

discussed in the following section. As shown in Fig. 6.1, the pulse

train is now incident onto mirrors M1 and M2 through the beam-

splitter B2. We have used unpolarized beamsplitters in this exper-

iment. Mirror M1 is tilted by a very small angle φ and mirror M2

is set on a piezo stage which is employed to scan the mirror by a



step size of 1 µm. Then the time-integrated interference pattern is

measured by the CMOS camera. The measured interference pattern

for a fixed τ is shown in Fig. 6.2(a) and the corresponding average

intensity distribution is depicted in Fig. 6.2(b).

(a) (b)

Figure 6.2: Interferometric measurements (a) interference fringe pattern, and (b) corre-

sponding average intensity distribution.

Next we proceed to evaluate the absolute value of the com-

plex degree of temporal coherence from the spatial visibility of the

measured interference fringes. For accurate characterization one

needs to get rid of the envelop of the intensity distribution. This is

achieved with the help of a normalized quantity defined as

C(x) =I(x)− I1(x)− I2(x)

2[I1(x)I2(x)]1/2= |γ0(x)| cos(φ0 + κx), (6.4)

where I(x) is the measured intensity profile when both mirrors are

open, and I1(x) and I2(x) are the intensity profiles when only mir-

ror M1 or mirror M2 is open. The absolute value of the degree of

coherence is denoted by |γ0(x)|, whereas φ0 and κ = 2φ/c are the

phase of the normalized coherence function and the tilt parameter,

respectively.

The profiles obtained from the measured intensity distributions

are fitted with the sinusoidal curve resulting from the second equal-

ity in Eq. (6.4) to extract all the information about the coherence

function. The amplitude of the fitted curve provides the absolute

value of the degree of coherence and its phase can be obtained from

the lateral shift of the interference fringes. An example of the curve



fitting is provided in Fig. 6.3(a). We have only considered a few cen-

tral interference fringes to obtain more accurate results. We should

mention that the fitting procedure works well when the visibility is

significant. The evaluated absolute value of the complex degree of

temporal coherence is plotted in Fig. 6.3(b). From it one might eas-

−200 −100 0 100 200−1

−0.5

0

0.5

1(a)

x [µm]

C(x)

0 200 400 6000

0.5

1(b)

τ [ f s]

|γ0(τ)|

Figure 6.3: (a) Illustration of curve fitting. The blue curve indicates the normalized inter-

ference pattern and the red curve is the fitted result. (b) Absolute value of the measured

complex degree of temporal coherence.

ily conclude that the measured pulse trains are temporally partially

coherent, although they are almost fully coherent in reality. Hence

the experimental results are quite well in agreement with the the-

oretical prediction which revealed that fully coherent pulse trains

would appear partially coherent in time-integrated measurement.

In the next section we proceed to analyze what the time-integrated

measurements actually mean in practice.

6.2 MEASUREMENT OF PULSE TRAINS USING FROG

In this section we focus on the second part of our experimental

setup shown in Fig. 6.1, i.e., measurement of femtosecond pulse

trains using the frequency-resolved optical gating method. As dis-

cussed in section 2.3.2, the FROG technique can be used to char-

acterize ultrashort pulses in many aspects; here we are interested

to get the information about the temporal intensity and the phase

of pulse trains. In this experiment we have used a compact com-

mercial second-harmonic generation FROG setup (GRENOUILLE)



Figure 6.4: Experimental FROG traces associated with mode-locked femtosecond pulses:

(left) single pulse and (right) pulse train.

available in the market.

The pulse trains passing through beamsplitter B1 are directed

to the FROG setup with the help of a mirror, as shown in Fig. 6.1.

The incident pulse trains have been checked to be horizontally po-

larized since it is a prerequisite for our FROG setup to operate. The

measurements are performed with the help of FROG phase retrieval

software. Initially individual pulses are measured by isolating them

from the train with an electro-optic modulator. A measured FROG

trace of such a pulse is shown in Fig. 6.4(a). The pulse measure-

−400 −200 0 200 4000

0.5

1

τ [ f s]

A(τ)

Figure 6.5: Field autocorrelation evaluated from FROG measurements.

ments allow the direct construction of the two-time MCF by ensem-

ble averaging using the standard definitions. But the pulse train



turns out to be almost fully coherent. We then proceed to compare

the result with Michelson’s experiment presented in the previous

section. The FROG trace measured by averaging over a few pulse

trains is illustrated in Fig. 6.4(b). The corresponding temporal in-

tensity and phase are subsequently recorded. For reliable measure-

ment one needs to make sure that the FROG error is sufficiently

small. Then the autocorrelation of the field is calculated using

A(τ) =∫ ∞

−∞E∗

0(t)E0(t + τ)dt, (6.5)

where τ is time delay. The normalized field autocorrelation is de-

picted in Fig. 6.5. We see that full width at half maximum (FWHM)

of the field autocorrelation profile is approximately 160 fs, while

that of the time-integrated coherence function is around 170 fs.

Thus, we can conclude that the experimental results have vindi-

cated the theoretical explanation presented in the previous section.

Therefore, an almost coherent source appears to be partially coher-

ent in time-integrated measurement which provides only informa-

tion about the field autocorrelation.


7 Conclusions

The work presented in this thesis deals substantially with partially

coherent optical pulse trains. The coherence properties of such

sources in the temporal, spectral, and spatial domains have been

analyzed by means of the second-order coherence theory of non-

stationary light. Here theoretical investigations play a pivotal role

in the form of standard and newly proposed analytical models as

well as numerical analyses. Further, some preliminary experimen-

tal measurements of mode-locked femtosecond pulse trains were

performed to show consistence with theoretical claims. The out-

comes of these analyses, except the experimental results, have been

published in five peer-reviewed articles (papers I–V) included at

the end of this thesis.

7.1 SUMMARY WITH CONCLUSIONS

Paper I involves spatiotemporal coherence analysis of broadband

pulse trains. A simple analytical model of Gaussian Schell-type

is proposed, suggested by the coherence properties of supercon-

tinuum light generated in a nonlinear single-spatial mode optical

fiber. The optical field is considered spatially fully coherent at each

individual frequency, but the broadband spectrum and partial cor-

relations among different frequency components cause the field not

to be spatially fully coherent in the time domain. Hence, contrary

to popular belief, broadband sources (such as SC light) that are

completely spatially coherent in the frequency domain may have

reduced spatial coherence in the time domain. Further, the propa-

gation of model fields of this kind is considered in arbitrary (parax-

ial and rotationally symmetric) optical systems.

• Conclusion: Broadband sources such as supercontinuum light

can be represented by a Gaussian Schell-model for pulse trains.

Such optical sources are highly, but not fully, spatially coher-



ent in the time domain, in spite of being spatially fully coher-

ent at each frequency.

The investigation with broadband optical fields is continued in

paper II. Here we study the properties of broadband optical fields

generated by two different kinds of axicons (reflective and diffrac-

tive) under two different types of illuminations. The axicon fields

are a class of the broader, extensively researched localized fields.

For spectrally coherent illumination, the spatiotemporal shapes of

the ensuing axicon fields are evaluated. In this case, reflective axi-

cons produce X-wave-type pulses and diffractive axicons generate

slightly delayed Bessel-type pulses. For spectrally incoherent (sta-

tionary) illumination, the temporal coherence properties of the axi-

cons fields are studied. Here the degree of axial temporal coherence

of the ensuing fields exhibits faster decrease with propagation dis-

tance for the diffractive axicon as compared to the reflective one.

• Conclusion: Under coherent illumination axicons create well-

known localized field pulses. In stationary illumination the

temporal coherence properties of broadband fields generated

by reflective and diffractive axicons show opposite behavior

on propagation.

The core idea of papers III–V is to answer the following queries:

how to evaluate the actual two-time mutual coherence function of

temporally partially coherent pulse trains, what information is pro-

vided by the standard time-integrated measurements with Michel-

son’s interferometer for pulsed sources, and when can one say that

the time-integrated measurements are sufficient for true temporal

coherence characterization of pulsed light?

In paper III, we propose that an equal-path Michelson inter-

ferometer with tilted mirrors can be employed to evaluate the full

two-time MCF associated with temporally partially coherent pulse

trains by time-resolved measurement of interference patterns. We

also show that the time-integrated Michelson interferogram can

provide the full two-time MCF only when the pulse trains are quasi-

stationary, i.e., the coherence time is a tiny fraction of the pulse du-


Conclusions

ration. These conclusions are further justified with the help of spe-

cific realistic analytical models as discussed in paper IV. In this ar-

ticle, the two-time coherence function and the integrated coherence

function are evaluated for Gaussian Schell-model pulse trains phys-

ically representing slightly disturbed pulses from a mode-locked

laser and for a supercontinuum pulse train model which mimics

the properties of real SC pulses. Moreover, in paper V, a gener-

alized analytical model for SC pulse trains as well as numerically

simulated realizations of such pulse trains are investigated to as-

sess the information carried by time-resolved and time-integrated

measurements. It is concluded that time-resolved measurements

are necessary for complete temporal coherence characterization of

coherent pulsed sources, while time-integrated measurements are

sufficient for quasi-stationary sources.

In addition, using Michelson interferometry and the FROG tech-

nique we have experimentally demonstrated that an almost coher-

ent train of optical pulses appears to be partially coherent in time-

integrated measurements which only provide information on the

field autocorrelation. Thereby, we can conclude that temporal co-

herence characterization of substantially coherent pulsed sources

require time-resolved measurements, in agreement with our theo-

retical explanation.

• Conclusion: Temporal coherence characterization of quasi-

stationary trains of optical pulses is possible by means of

time-integrated Michelson interferograms, but partially co-

herent sources in general need to be characterized using time-

resolved measurements of the interference fringes.

7.2 POTENTIAL FUTURE WORK

As a continuation of this thesis, the two-time temporal coherence

function for supercontinuum light sources should be evaluated us-

ing time-resolved measurements. This could be implemented with

the help of the cross-correlation frequency-resolved optical gating

(XFROG) technique. Some preliminary experimental results have



been reported in [110]. Such measurements are highly valuable

since the exact knowledge of the temporal coherence of SC light is

critical in many applications. Likewise, experimental investigation

of the spatial and temporal coherence properties of supercontin-

uum light produced in bulk media, and the subsequent analysis in

the framework of the second-order coherence theory of nonstation-

ary light, are expected to be on the future agenda.

Furthermore, the setup proposed in chapter 5 can be used to

examine the temporal coherence of a temporally partially coher-

ent femtosecond laser so as to check consistency with the theoret-

ical explanation. The theoretical model put forward in chapter 3

could be extended to simulate more realistic cases, such as SC light

generation in nonlinear optical fibers with step-index profiles. The

propagation of the model beam may subsequently be evaluated in

realistic broadband optical systems to investigate the influence of

spatial coherence on various interactions.


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ISBN 978-952-61-2152-9ISSN 1798-5668


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RAHUL DUTTA

TEMPORAL, SPECTRAL, AND SPATIAL COHERENCE OF OPTICAL PULSE TRAINS


This thesis considers the coherence properties of optical pulse trains in temporal, spectral, and spatial domains. Realistic optical fields are studied through analytical models and numerical analyses. The evaluation of the

two-time coherence function for pulse trains is elucidated with an appropriate experimental

arrangement and the corresponding analytical formulation. Moreover, preliminary

experimental results, in agreement with the theoretical predictions, are reported.

RAHUL DUTTA

Dissertations in Forestry and Natural Sciences · Rahul Dutta: Temporal, spectral, and spatial...

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