Investigation on the spectral coherence of bulk generated supercontinuum implementing a novel cross-correlation technique

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Investigation on the spectral coherence of bulk generated

supercontinuum implementing a novel cross-correlation technique

Debanjan Show

Master Thesis April 2021

Department of Physics and Mathematics

University of Eastern Finland

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Debanjan Show Investigation on the spectral coherence of bulk generated supercontinuum implementing a novel cross-correlation technique, 46 pages University of Eastern Finland

Master’s Degree Programme in Photonics Supervisors Dr. Atri Halder

Dr. Matias Koivurova

Abstract

Investigation on the coherence properties of supercontinuum (SC) light is being a significant research field due to the presence of an enormous amount of applications depending on the coherence characteristics. Examining the coherence statistics of such ultrashort light pulses requires a fruitful technique, and this thesis work pro- vides an experimental scheme to study the spectral coherence properties of the supercontinuum light generated in bulk medium using a novel cross-correlation technique.

This thesis work enlightens the theoretical description of SC generation, correlation statistics and pulse characterization techniques. In the experimental part, we in- ject ultrashort femtosecond laser pulses on two 5 mm thick sapphire plates for the generation of SC light, and using a Mach–Zehnder (MZ) type interferometer the intensity normalized cross-correlation functions are obtained. From the obtained cross-correlation data, we demonstrate the overall degree of spectral coherence behavior of the generated SC light as a function of spatial positions. Moreover, we examine the spectral coherence properties at four pumping conditions, and the results suggested that spectral coherence properties become higher with the high pumping.

To reveal, the reason behind the higher spectral coherence, we examine spectral coherence properties of the filtered SC light by filtering out the pump wavelength and results are included in this thesis work. Investigations show newly generated filtered SC light is partially coherent.

Keywords: Ultrafast optics; Nonlinear optics; Supercontinuum; Correlation functions; Spectral coherence; Cross-correlation technique; Interferometry.

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Preface

First and foremost, I say Thank You to the Dept. of Physics and Mathematics and Institute of Photonics for giving me the opportunity to pursue my study in excellent and diverse research environments with great minds here at the University of Eastern Finland. I want to convey the deepest and endless gratitude to my supervisor Dr.

Atri Halder for his immense support, supervision, and guidance. His unique and easy way of sharing thoughts, dedication, and passion for hands-on work, motivated and engaged me throughout the whole process. I am extremely pleased to work with him. I want to deliver my sincere thanks to my co-supervisor Dr. Matias Koivurova for his valuable comments on my work.

I like to thank Prof. Jari Turunen for allowing me to work in his excellent research group. I acknowledge Dr. Pertti Pääkkönen and Mr. Tommi Itkonen for their support in all technical and laboratory-related issues. I am also thankful to Dr. Noora Heikkilä for her concern and constant help throughout the program. I acknowledge all professors of the Institute of Photonics for their constant support in study. Also, kind regards to Prof. Kallol Bhattacharya from whom I was inspired to do experimental work.

The journey of coming to UEF would have been impossible without the constant support from my friends Aniruddha, Akib, Rajarshi who kept in touch with me in every messed-up situation and enlightened my mood with fun. My journey here at Joensuu would be incomplete if I do not mention a few names. Thanks to Nandita Di, Srijoyee Di, Ria Di, Dipanjan Da, Niladri for their companionship during my stay. I would also like to thank Madona who engaged in deep thoughts with me while walking throughout forest roads and the neighborhood of Joensuu. Many thanks to Aravinth, Masoud, Mwita for their constant help, along with lovely food and fun time during my stay in Joensuu. Thanks to all my classmates at UEF for helping me out in my studies.

My indebt and heartfelt thanks to my parents for their constant love, blessing, and care. Also, I would like to thank them for giving me the strength to fight against all the odds throughout my life. Heartiest thanks to my elder brother for his concern and support. His precious line ’never stop learning’ always motivates me to try and learn new things in life. I am thankful to my sister for visualizing me the essence of dedication and hard work, which immensely helps in life. This journey would be

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rather severe without their continuous encouragement and stable support. Finally, I would like to thank all my family members in Sahoo Parivar and others relatives for their constant engagement during my stay in Joensuu.

In an endnote, I wish to dedicate my thesis to Joensuu, where I spent beautiful and unforgettable memories for the past two years with lovely peoples surrounded by scenic beauty. The visualization of dancing color effects of Aurora Borealis in the midnight sky above the lake named Aavaranta is one precious incident which will be engraved in my memory forever. I am ever grateful to Joensuu for its midnight rays in the summertime and countless snowflakes during winter days. Kiittos Joensuu, glitz forever.

Joensuu, 30th April, 2021 Debanjan Show

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Chapter I

Introduction

Optical fields, whether natural or artificial, posses some degree of random fluctuations. These fluctuation characteristics are governed by statistical approaches, which forms the base of optical coherence. The coherence phenomenon defines the randomness of amplitude and phase between fields at two instants (either in space, time, or frequency domain) [1]. In nature, all light fields show partially coherent behavior. The partial coherent nature of sources contributes to a wide variety of practical applications such as beam shaping, ghost imaging, particle trapping, microscopy, astronomy, and many more [2].

The coherence theory of light is an active research field in modern physics for a few decades. The concept of coherence was first tailored by Verdet with his observations on coherence properties of sunlight in the mid-1860s. An overview of coherence statistics with incoherent sources was interpreted by van Cittert and Zernike [3]. Later Zernike formed a bridge between two concepts, correlation statistics and fringe visibility based on Young’s double slit experiment [4]. Optical coherence theory reveals fundamental properties of classical and quantum fields [1].

Classical optics was formulated based on wave constituents, while quantum optics relied on particle behavior of light. Correlation statistics is a prominent tool for defining the coherence theory. Space-time correlation formalism is defined as mu- tual coherence function (MCF), coined by Wolf [5]. The space-frequency domain correlation function, named cross-spectral density (CSD) function, was founded by Mandel and Wolf [6]. These functions were originally formulated considering statistically stationary fields. Correlation functions are fruitful to develop second and higher-order coherence theory [1]. Higher-order correlation statistics are necessary

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to explain quantum coherence theory [7].

Generation of short pulses developed into a crucial research areas after the invention of a mode-locked laser in the mid-1960s [8]. The establishment of high-power lasers drew attention towards the new phenomena, defined as nonlinear phenomena [9]. Ultrashort pulses are an example of non-stationary light fields, enhancing a popular and widely applicable modernistic research topic. Ultrashort pulses are involved in many applications such as micromachining [10], supercontinuum generation [11] based on leading features, i.e., pulse energy. In this thesis work, we mostly concentrate on SC pulses and their coherence properties. High power nar- row time domain laser pulses can generate a large-scale spectral broadening while passes through nonlinear materials using intensity-induced nonlinearity. This spectral broadening is named white-light continuum or SC generation [12,13], first iden- tified in 1970 utilizing bulk glass material [14]. Also, spectral broadening exhibited in different glasses and crystals [15] validated unique and versatile natural phenomena of SC generation in the early days of research. Over the years, SC generations have progressed extensively and achieved in photonic crystal fibers [12], microstruc- tured fibers [16] with high-efficiency. Fiber-based SC sources are formed in nonlinear materials such as telluride and chalcogenide glasses that covers spectral ranges from near to mid-infrared [13]. SC pulses draw much attention due to their broad- spectrum. SC pulses show remarkable applications in optics and photonics, such as high-resolution optical coherence tomography [17], stable frequency comb generations [18]. All applications are mostly based on the coherence properties of SC pulses.

The research on the second-order coherence theory of stationary light has been evolving steadily for decades, whereas the coherence characteristic of non-stationary light fields is a recently advanced research area [19]. Detection and experimental investigation of coherence properties of such short pulses is quite a challenging research topic, as high-speed detectors are not available. From the 1980s experimental techniques are employed to characterize the statistical behavior of non-stationary light [20]. The intensity auto-correlation is a technique that is engaged to investigate the pulse length, but it is not successful as a similar pulse shape is founded from auto-correlation traces [21]. Complete coherence characterizations of SC pulses require the knowledge of two-time MCF and two-frequency CSD, respectively. There are no such measurement techniques available by which we can measure MCF and

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CSD functions directly. In practice, one can measure individual pulses, utilizing nonlinear pulse measurement techniques, such as FROG (frequency resolved optical gating) or SPIDER (spectral phase interferometry for direct electric-field re- construction), and then construct two-point correlation functions, which suggests an easy way to get quantitative information about the coherence properties of such pulsed fields [22]. Dudley and Coen have also defined a spectrally resolved coherence function that also gives a similar estimation [23]. Spatial and temporal coherence measurement of SC light generated from various solid materials were experimen- tally inspected by Zeylikovich and Alfano in 2003. In this scheme they implemented diffraction-grating based interferometer and measured interference fringes between zero and first diffraction order of the SC pulses emanating from the one nonlinear medium [11]. In recent times, a field auto-correlation technique is proposed to measure the temporal behavior of pulses. But in the case of pulses with varying spectral phase, the field auto-correlation cannot estimate the pulse length. A novel technique, named cross-correlation, is introduced, which is employed to determine spectral coherence properties [22].

The main aspiration of this thesis is to develop an interferometric setup that can measure spectral coherence properties of supercontinuum fields generated through bulk media. This is accomplished by measuring cross-correlation signals of two supercontinuum pulses generated in two identical crystals. Chapter 2 describes nonlinear optical phenomena which are related to SC generations from both fibers and bulk materials. This chapter also includes practical realizations of SC generations in bulk materials. Chapter 3 introduces a concept of coherence properties depending on the second-order coherence theory for stationary and non-stationary fields. In this chapter, complex field representations are described. Then we present space-time and space-frequency domains correlation functions. Correlation functions dependent on spatial angles are also demonstrated. This chapter includes correlation behav- iors of simulated supercontinuum fields in angular space-frequency domains. Basic measurement concepts of ultrashort pulses and quantify coherence based on these, are described in Chapter 4. Chapter 5 depicts the experimental setup regarding cross-correlation measurements and portrays the main results. Finally, the overall conclusions of the entire works and future prospects are summarized in Chapter 6.

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Chapter II

Theory beneath the SC generation process

Supercontinuum (SC) light, known as the white-light continuum, is a beautiful pattern to visualize. SC light generates an ultrabroad spectrum between the interaction of an intense laser beam and a transparent medium [11,13]. The generation process occurs due to the nonlinear light and matter interactions. In 1983, Fork et al. first reported SC generation with femtosecond pulses in thin jet of ethylene glycol [24].

Later, with the invention of Kerr lens mode-locking, a new dimension was achieved in SC generation [25]. The availability of a wide variety of lasers, optical fibers, and nonlinear materials enhances the diversity of SC applications like ultrafast spec- troscopy [13], optical microscopy [26], telecommunication [27], optical clock [28]. SC light is achievable in various materials like liquids, optical fibers, bulk materials [14], but the most common and effective medium is photonic crystal fiber [12]. In general, the spectral broadening may occur in solid, liquid, and gases [29–31] from a high energy pulsed laser. SC generation may be attainable with both continuous and pulsed lasers. But the pulsed lasers, which have a high peak power, shows more prominent nonlinear phenomena, needed for SC generations [32].

The generation of SC in a fiber occurs from self-phase modulation, material dispersion, and soliton generation [13]. Numerical-modeling can be done utilizing the split-step Fourier method by solving the nonlinear Schrodinger equations [33].

It helps to calculate and demonstrate the temporal and spectral evolution of pulses.

Simulation performed in optical fibers for SC field was reported in Ref. [23]. In the case of optical fibers, the confinement of an energy are basically based on the effective area of fibers and one do not need to consider diffraction, which reduces the dimensionality of the problem. As a consequence simulations and calculations

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performed in fibers for SC are rather simple [13]. Femtosecond filamentation process are mostly convenient for the generation of SC light in a transparent bulk material.

The generation of SC in bulk media requires a more complex method. This chapter involves background about nonlinear phenomena and dispersion mechanism related to SC generations both in fibers and bulk materials. Also, at the end practically achievable SC generations process is included.

2.1 Physical background of nonlinear effects

Electrons can form a cloud structure when moving randomly between various orbitals around the nucleus. Electron occupation can be estimated via wavefunction. The transformation of an electron between energy levels will produce a photon. The photon encompasses discrete quantized energy levels U. It is related to wavelength λ and Planck constant h via U = hc/λ, where c refers to the speed of light. Now, if we assume that there is an external force impinging on an atom, then a strange phenomena can occur. It can be described by the oscillator model. We can consider interactions between electric field vector and electron as a forced vibration, as one can assume bound electrons are mass attached to the spring. This external force induces a dipole moment which shows polarization properties inside the material.

Polarization of a material is defined as a dipole moment per unit volume. If the field is low, then the polarization P(t) behaves linearly with the applied field E(t) and it is represented as [9],

P(t) = ϵ₀χ⁽¹⁾E(t), (2.1)

where ϵ₀ specifies the vacuum permittivity and χ⁽¹⁾ stands for linear susceptibility.

As the electric field becomes more intense, this approximation breaks down. Then oscillations become a power series of applied field. So Eq. (2.1) modifies to,

P(t) =ϵ₀[︁

χ⁽¹⁾E(t) +χ⁽²⁾E²(t) +χ⁽³⁾E³(t) +...]︁

, (2.2)

whereχ⁽²⁾ andχ⁽³⁾ denote second-order and third-order susceptibilities, respectively.

Non-centrosymmetric materials show second-order nonlinear effects such as second harmonic generation, optical parametric oscillation and amplification, sum and difference frequency generation. Higher order effects are less prominent due to the stronger effect of the second-order susceptibility. For centrosymmetric materialsχ⁽²⁾

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is zero and thus third-order susceptibility becomes significant. As a consequence, third-order nonlinear effects like self-focusing, self-defocusing, and self-phase modulation takes place in the centrosymmetric materials [9]. All nonlinear effects can generate a new frequency of wave components, meaning different wavelengths. Now, in the following subsections we will discuss about few nonlinear phenomena which are responsible for supercontinuum generations from fibers and bulk materials.

2.1.1 Self-focusing

One of the most important third-order nonlinear effects is the optical Kerr effect.

This effect modifies the refractive index inside the material, and the variation of the refractive index is given as follows [9]

n =n₀+n₂I. (2.3)

The symbolsn₀ and n₂ are the linear and nonlinear refractive index of the medium, respectively. Here, I denotes the intensity of the incident light. The symbol n₂ depends onχ⁽³⁾, which shows positive values in the transparent region for dielectric media [13]. If an intense laser beam passes through a nonlinear material, then the intensity-dependent refractive index can produce an index gradient inside the material. It creates an instantaneous difference in the refractive index profile inside the material. It leads to the material act as a lens which focuses the beam, named as self-focusing [9,13]. For detailed descriptions, like critical power and self-focusing distances, reader may go through [13].

The self-focusing effect is efficient to produce filamentation effect [13], i.e., the bridge of parity between self-focusing and plasma defocusing effect, which is a part of SC generation in bulk media. Also, self-focusing shows important features in material processing, where optical damage can occur due to the high intensity of the self-focused beam. Kerr shutter effect [9] is one of the aspectful features that helps to calculate the pulse duration of ultrashort pulses by measuring the cross-correlation signal.

2.1.2 Self-phase modulation

Self-phase modulation (SPM) is also a third-order nonlinear phenomenon that induces a variation of phase shift to the propagating pulse due to modulation of the refractive index inside the material caused by the strong intensity of light. This

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phenomenon occurs in a temporal domain. The induced nonlinear phase shift ϕ_nl is expressed as [9,13],

ϕ_nl(t) = ω₀

c n₂I(t)z, (2.4)

where ω0 denotes the carrier frequency of the pulse, I(t) is the temporal intensity, and z is the propagating distance. This phase shift induces a change in frequencies of pulses that result in pulse broadening, as depicted in Fig. 2.1. Pulse is acquainted lower frequency shift (red shift) and higher frequency shift (blue shift) in the leading and trailing edge in the SPM process [9,13].

Nonlinear crystal n₂> 0

Blue shift

Red shift

z Input

pulse

Output pulse

Figure 2.1: Demonstration of schematic self-phase modulation and frequency shift through a nonlinear crystal injected with ultrashort pulses.

The amount of broadening depends on incident light peak energy and the initial chirping condition of the pulse. Chirping effect [32], i.e., frequency of a pulse is time-dependent. SPM plays an important role in the SC generation process, also widely applicable for optical parametric amplification (OPA), pulse compression in optical fibers [32]. Cross phase modulation (XPM) is an analogous phenomenon to SPM where various pulses show modulations of refractive indices, which ensures asymmetric spectral broadening [32].

2.1.3 Soliton generation

Optical soliton formation is a wave phenomenon that is generated due to balancing between SPM and dispersion. This balancing effect takes place in an anomalous dispersion region. In an optical soliton process, the pulse can propagate without any change in its temporal shape. The temporal shape of a femtosecond pulse shows the behavior of hyperbolic secant, so a higher-order soliton can be useful for modelling. Higher order soliton breaks down to several fundamental solitons as the

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instability arises in higher order solitons due to broadening. This splitting process is named soliton fission. During this splitting process extra energy is dissipated as a dispersive waves. Raman scattering plays a pivotal role in soliton fission which produces a shift of the center frequency towards lower frequencies [12]. Soliton takes part in the fiber generated SC [32].

2.1.4 Stimulated Raman scattering

Stimulated Raman scattering (SRS) is an inelastic process, where energy is coupled between an electromagnetic field and the vibrational modes of molecules. SRS ampli- fies the weak light at longer wavelengths. It leads towards red-shift, termed as Stokes scattering [32]. Also, it can make the spectrum broaden at shorter wavelengths due to four-wave mixing (FWM) [34]. SRS with soliton phenomena broadens the spectral regime, which generates the SC at longer wavelengths in anomalous dispersion region [32].

2.1.5 Four-wave mixing and modulation instability

Four-wave mixing is a third-order nonlinear phenomenon, which may also lead to spectral broadening with four distinct wavelengths are mixed in the process. This mechanism occurs when the phase-matching condition holds, which comes from the conservation of energy and momentum. With the phase-matching agreement, sidebands around a central region get amplified [9,32]. Modulation instability (MI) is a phenomenon which occurs due to the presence of SPM and XPM. MI is useful for noise amplification, wavelength enhancements. Also, it is responsible for spectral noise and coherence loss [35]. MI mostly occurs in the anomalous regime, but it can also occur in normal dispersion regime [32].

2.2 Dispersion mechanism

The refractive index of a material depends on the wavelength of light, which is termed dispersion. There are various kind of dispersion, such as material dispersion and waveguide dispersion. Interaction of electrons inside a material and an electromagnetic field causes material dispersion [3]. For example, the prism shows the dispersion, i.e., separation of colors observed from incident white light. Light trav- elling inside waveguides experiences a different type of dispersion named waveguide

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dispersion. It occurs due to the spatial distribution of modes inside the waveguide depending on wavelength [32]. Dispersion causes a variation both in phase and group velocity of light fields. Group velocity dispersion (GVD) is a process where pulses of different frequencies broadens the temporal pulse profile due to the movement of various wavelengths at different speeds.

GVD parameter [32] plays a crucial role to generate SC, mostly on a fiber material. Depending on the GVD parameter, light fields experience different dispersion regimes. A positive value of the GVD parameter shows normal dispersion, whereas a negative value refers to anomalous dispersion. In the positive dispersion region, high-frequency components of light fields move slower than the low-frequency components, while in anomalous dispersion the occurrence is the opposite. Zero dispersion wavelength [32] refers to the wavelength value, where no dispersion phenomenon takes place. It is important because it modifies the temporal pulse and the temporal pulse modifies the spectrum. These two happen simultaneously, and they strongly affect the generated SC [32].

2.3 Femtosecond filamentation in SC generation

SC generation in bulk media occurs from the femtosecond filamentation process that consists of self-focusing, self-phase modulation, multiphoton absorption or ionization induced free-electron plasma, diffraction, and dispersion [13,36]. This process helps to propagate light fields at higher diffraction length through a channel [37], without any external guiding process [13]. The first stage of the filamentation process produces a self-focusing effect inside the nonlinear material, depending upon the intensity-dependent refractive index, described in Subsection 2.1.1. Spectral broadening mainly occurs based on the self-phase modulation mechanism described in Subsection 2.1.2.

Except these two main processes other mechanisms are invoked in SC generations as well. Self-focusing phenomenon focuses the light beam into a spot inside the material. The plane or a point at which the self focusing takes place out of a collimated Gaussian beam, is called nonlinear focus. The beam is destructed at nonlinear focus, and is seized due to multiphoton absorption and ionization effects.

These processes induce energy loss and produce free-electron plasma that causes absorptions and defocusing of the beam [13]. Limiting or clamping to a certain

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value of intensity is based on these combined effects. The clamping intensity [38]

depends on the multiphoton absorption parameter, see Ref. [39]. The higher value of the multiphoton absorption coefficient produces larger clamping intensity. It leads to a smaller limiting diameter of the beam at the nonlinear focus, which gives a broader spectral range [13].

Numerical studies for the filamentation process and nonlinear propagation through the bulk media are performed based on solving the unidirectional non-paraxial propagation equation for the single pulse under rotational symmetry in the spatial domain, see Refs. [13,37]. Chromatic dispersion plays an important role in the shape of SC generation [40,41], which reveals from numerical studies. The contribution of chromatic dispersion can be assessed based on effective three-wave mixing that generates new frequency components. Lower chromatic dispersion follows the phase matching condition for larger spectral components, hence it produces wider spectral broadening and vice versa [13]. Self-steeping [36], pulse splitting or compression [42], space-time focusing, and optical shocks [29,37] are the intermediate effects formed during the interplay processes between nonlinear effects and chromatic dispersion.

2.4 Practical considerations of SC generations

The physical phenomena behind the emergence of SC light are a complex interplay between linear and nonlinear effects. But the practical realization of SC generation is a simple process, where the femtosecond laser pulse of high energy focuses on a nonlinear crystal S using a suitable focusing lens L, as shown in Fig. 2.2. The

Pulsed laser HWP

P L S

Screen SC

Figure 2.2: Practical setup for SC generations in bulk material. HWP: half- wave plate, P: polarizer, L: focusing lens, S: nonlinear crystal, and SC: supercontinuum lights.

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SC itself is a brilliant white color, and the rings around it are caused by Raman scattering. Sapphire plate generates the SC spectra in visible and near-infrared regions [13,43]. The critical power is required to achieve a self-focusing effect for SC generations. It is related to the laser wavelength and nonlinear refraction coefficient of the material. Focusing the light beam with higher pulse energies can produce optical damage. Thus with proper configurations of input pulse energy and focusing condition (numerical aperture) helps to avoid the catastrophic avalanche ionization [44], which is mainly responsible for optical damage [13]. To have a control over the pump energy without affecting the polarization state, one can use HWP and a polarizer P, as depicted in Fig. 2.2. The detailed experimental setup used for SC generation will be discussed in Chapter 5.

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Chapter III

Correlation statistics of light

Coherence is a fundamental theory which plays an essential role in statistical optics. Optical fields oscillate randomly in nature. Optical fields can be constructed as a superposition of different plane waves with various amplitudes, phases, and frequencies. Coherence theory describes correlations between points in space and time domains. If we know the correlation function at a particular instant of position and time, then coherence theory gives a statistical estimate of field amplitudes after different distances at a definite time or current location with various times. In classical optics, coherence theory estimates the randomness of the field fluctuation, while, in quantum counterpart, it describes the statistical analysis of photon arrival [7]. In this chapter, the analytical signal representations are described. Also, the chapter includes correlation function formulations in time and frequency domains with lateral and angular spatial coordinates. Based on angular space-frequency correlation formulation, the simulated SC coherence properties are demonstrated.

3.1 Complex analytical signal representation

Complex field representation is a tool to present real optical fields in an analytically simple terms. Mostly it is convenient to use in the classical coherence theory espe- cially when fields treating with average values, for example in correlation functions.

Let us consider, the real light field u(r;t) propagates towards thez direction, where r = (x, y) is the lateral position vector and t is the time frame. Electric field in general is a vector quantity, but here we are taking into account as a scalar field due to simplicity also as we are not considered any polarization properties. Assuming the fieldu(r;t) is square-integrable, let us expressu(r;t) with its Fourier transform

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counterpart u(r;ω) as [45]

u(r;t) =

∫︂ ∞

−∞

u(r;ω) exp (−iωt) dω, (3.1) where i =√

−1 is the complex number, andω denotes the angular frequency of light field. Fourier coefficient can be determined by

u(r;w) = 1 2π

∫︂ ∞

−∞

u(r;t) exp (iωt) dt. (3.2) As,u(r;t) is a real-valued function and its Fourier counterpart u(r;ω) obeys condition, i.e.,

u(r;−ω) = u^∗(r;ω), (3.3)

which reveals that negative frequency component of the signal does not quantify as an extra information. So, we neglect the negative frequency component and as a consequence Eq. (3.1) becomes

A(r;t) =

∫︂ ∞ 0

A(r;ω) exp (−iωt) dω, (3.4) which is the definition of complex analytic signal. This representation is an useful tool to form correlation functions in any domains [45,46] for stationary and non- stationary fields.

3.2 Stationary light fields

Light fields whose random behavior is independent of the origin of time are termed as statistically stationary fields [45,47]. It is rather nonphysical, but it may be possible for fields become stationary when no fluctuations occur on the particular time scale of interest. Stationary light fields can be categorized as strict and wide-sense stationary fields. Strict-stationary fields are fields where all correlation functions are not dependent on origin of time, whereas, the wide-sense stationary counterpart, only has the second-order correlation function independent of time origin. Stationary light field coherence theory show an interesting features in single-mode laser [47], spectrum variation during light propagation [48,49]. Additionally, stationary coherence theory can be described the twisted beam effects [50], the improvement of the resolution of optical microscopes [51,52]. Statistical similarity functions of stationary fields in space-time and space-frequency domains are formulated in the following subsections, which are responsible for coherence property investigations.

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3.2.1 Correlation formulation in lateral space-time domain

Let us consider two statistically stationary light fields V(r₁;t), and V(r₂;t +τ) interfering on a screen. Here, r₁, r₂ are defined as position coordinates, and τ represents time differences between two fields. It is easy to determine the correlation between these fields by measuring the intensity pattern. MCF Γ(r₁,r₂;τ) defines the correlation behavior between fields in the space-time domain and it can be written as [53]

Γ(r₁,r₂;τ) =⟨V^∗(r₁;t)V(r₂;t+τ)⟩. (3.5) Here, asterisk represent the complex conjugate of fields, and angle brackets specify the ensemble average with the form

f(r;t) = lim

N→∞

1 N

N

∑︂

n=1

f_n(r;t). (3.6)

The average intensity can be calculated with

I(r) = Γ(r,r; 0) =⟨|V(r;t)|²⟩, (3.7) considering r₁ =r₂ =r. The normalized MCF is represented as

γ(r₁,r₂;τ) = Γ(r₁,r₂;τ)

√︁I(r₁)I(r₂). (3.8)

It is termed as complex degree of coherence (CDC), and its absolute value charac- terizes coherence properties via the inequality,

0≤|γ(r₁,r₂;τ)| ≤1. (3.9) It describes the quantitative analysis between interfering fields. For example, fringe visibility can be measured with Young’s experiment, as fringe visibility is propor- tional to the absolute value of the complex degree of coherence [45]. The condition given in Eq. (3.9) refers to the partially coherent nature of fields. The lower and higher limit values of the inequality describe that the fields are perfectly uncorrelated and completely correlated, respectively.

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3.2.2 Correlation formulation in lateral space-frequency domain

The analysis of the light field propagations and scattering effects became quite complex in the space-time domain due to the presence of time retardations, and thus, it is useful to explore correlation functions in different domains. The correlation function in the space-frequency domain takes the form

W(r₁,r₂;ω) =⟨V^∗(r₁;ω)V(r₂;ω)⟩. (3.10) It is named as CSD function. Assumingr₁ = r₂ =r, the analogous form of space- time domain intensity function can be written as

S(r;ω) = W(r,r;ω) =⟨|V(r;ω)|²⟩, (3.11) termed as spectral density in space-frequency domain. In the space-frequency domain complex degree of coherence can be represented as

µ(r₁,r₂;ω) = W(r₁,r₂;ω)

√︁S(r1)S(r2). (3.12)

The absolute value of the complex degree of coherence µ(r1,r2;ω) defines the correlation behavior between fields in the space-frequency domain via the inequality

0≤|µ(r₁,r₂;ω)| ≤1. (3.13) The inequality in Eq. (3.13) defines the degree of coherence of a field at a frequency ω. The value 1 suggests that fields are completely spatially coherent, whereas the 0 value refers to completely uncorrelated fields. Fourier transform helps to connect space-time and space-frequency domains via

W(r₁,r₂;ω) = 1 2π

∫︂ ∞

−∞

Γ(r₁,r₂;τ) exp(iωτ)dτ, (3.14) which is the famous Wiener–Khintchine theorem [1].

3.3 Non-stationary light fields

In nature, all fields are non-stationary, i.e., the fluctuation has its starting and ending points. An example of such non-stationary fields are ultrashort pulses, which may have pulse lengths in the range of optical cycles [54,55]. In general, pulses fluctuate randomly, and their random behavior is not constant with time. Thus, pulses cannot be characterized correctly with the stationary coherence theory and a new formulation is necessary [56,57].

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3.3.1 Correlation formulation in lateral space-time domain

Let us consider a non-stationary light of wavefront W is incident upon pinholes P and Q, as shown in Fig. 3.1. Let us assume two space-time secondary sources E_P(r_P;t_P) andE_Q(r_Q;t_Q) emerging from two pinholes and interfering on a screen.

Here, r_P, r_Q, t_P, and t_Q are defined as lateral positions and times. Correlations between these fields can be determined by measuring the intensity pattern on the screen. The measured interference pattern on the screen is formulated as I_T =

Screen W

P Q

Figure 3.1: Diagram of a simplified Young’s pinhole setup.

⟨E_T^∗ET⟩, which is the expected value of square modulus of the superimposed fields.

Superimpose fields ET at the planeT take the form

ET =EP(rP;tP) exp(iϕP T) +EQ(rQ;tQ) exp(iϕQT). (3.15) The symbols ϕ_{P T} and ϕ_QT define the phase of light fields. The correlation function between fields E_P(r_P;t_P) and E_Q(r_Q;t_Q) at two instant positions and times in the space-time domain is called MCF Γ(r_P,r_Q;t_P, t_Q), and is presented as

Γ(r_P,r_Q;t_P, t_Q) =⟨E_P^∗(r_P;t_P)E_Q(r_Q;t_Q)⟩. (3.16) The interference pattern at the observation plane is related to the MCF as

I_T =I_P(r_P;t_P) +I_Q(r_Q;t_Q) + 2ℜ[Γ(r_P,r_Q;t_P, t_Q)], (3.17)

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whereℜspecifies the real part of the fields. The temporal intensity of the individual field acquires a form

I(r;t) = Γ(r,r;t, t) =⟨|E(r;t)|²⟩, (3.18) with the consideration ofrP =rQ=r, andtP =tQ=tfrom Eq. (3.16). Considering both Eqs. (3.16) and (3.18), the normalized MCF can be written as

γ(r_P,r_Q;t_P, t_Q) = Γ(r_P,r_Q;t_P, t_Q)

√︁IP(rP;tP)IQ(rQ;tQ). (3.19) The normalized MCF is called complex degree of coherence, and its absolute values characterize coherence properties of light fields. The absolute value of the degree of coherence is bounded as

0≤|γ(r_P,r_Q;t_P, t_Q)| ≤1. (3.20) The lower limit of the function signifies that it is completely incoherent, such as two independent source as there is lack of correlation between two points in space and time. Higher limit says the source is completely coherent. But in general there are no such sources present which shows complete coherence. In nature all sources are partially coherent, i.e., absolute value of complex degree coherence varies between 0 and 1 [1]. To get single numerical value for the degree of coherence one can compute the integral defined as [58]

γ

¯² =

∫︁∫︁

L

∫︁∫︁∞

−∞|Γ(r_P,r_Q;t_P, t_Q)|²dt_Pdt_Qd²r_Pd²r_Q

∫︁∫︁

L

∫︁∫︁∞

−∞I_P(r_P;t_P);I_Q(r_Q, t_Q)dt_Pdt_Qd²r_Pd²r_Q, (3.21) whereLrepresents the spatial extent of the field. Based on Eq. (3.21), one can find the overall degree of coherence, that is the square root of the computed integral.

3.3.2 Correlation formulation in lateral space-frequency domain

With the analogy of Eq. (3.16), the correlation function in space-frequency domain between fieldsE(r_P;ω_P) and E(r_Q;ω_Q) is denoted as

W(r_P,r_Q;ω_P, ω_Q) =⟨E_P^∗(r_P;ω_P)E_Q(r_Q;ω_Q)⟩, (3.22)

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where ω_P and ω_Q are two different frequency components. This function termed as CSD, which is a counter part of MCF. Individual fields spectra can be calculated with the analogy of Eq. (3.18) and follows as

S(r;ω) = W(r,r;ω, ω) =⟨|E(r;ω)|²⟩, (3.23) which is named as spectral density of fields. Then the complex degree of coherence becomes

µ(rP,rQ;ωP, ωQ) = W(r_P,r_Q;ω_P, ω_Q)

√︁S_P(r_P;ω_P)S_Q(r_Q;ω_Q), (3.24) in the space-frequency domain, also termed as normalized CSD. Its absolute value tells about correlations in space and frequency domain via the inequality

0≤|µ(r_P,r_Q;ω_P, ω_Q)| ≤1. (3.25) If the absolute value of complex degree of coherence is zero then the fields are incoherent meaning that there are no spectral correlations between two fields. If it becomes 1, then the fields are fully mutually coherent. But in practice it always lies between 0 and 1. The overall degree of coherence can be computed as [58]

µ¯² =

∫︁∫︁

L

∫︁∫︁∞

0 |W(rP,rQ;ωP, ωQ)|²dωPdωQd²rPd²rQ

∫︁∫︁

L

∫︁∫︁∞

0 S_P(r_P;ω_P);S_Q(r_Q, ω_Q)dω_Pdω_Qd²r_Pd²r_Q. (3.26) Fourier relations entangles correlation functions between MCF and CSD as

Γ(r_P,r_Q;t_P, t_Q) =

∫︂ ∫︂ ∞ 0

W(r_P,r_Q;ω_P, ω_Q)

×exp [i(ω_Pt_P −ω_Qt_Q)]dω_Pdω_Q, (3.27) and

W(r_P,r_Q;ω_P, ω_Q) = 1 (2π)²

∫︂ ∫︂ ∞

−∞

Γ(r_P,r_Q;t_P, t_Q)

×exp[−i(ωPtP −ωQtQ)]dtPdtQ. (3.28) This is well-known Wiener–Khintchine theorem [47] for non-stationary light fields.

However, the normalized MCF and the normalized CSD are bridged via Friberg–

Wolf theorem [59].

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3.3.3 Correlation formulations in angular space-time and angular space-frequency domains

The correlation functions can also be presented in time and frequency domains with angular spaces. Let us present the angular field in time and frequency domains as

U(k;t) = 1 (2π)²

∫︂ ∞

−∞

E(r;t) exp(−ik·r)d²r, (3.29) and

U(k;ω) = 1 (2π)²

∫︂ ∞

−∞

E(r;ω) exp(−ik·r)d²r, (3.30) respectively. Here,k= (k_x,k_y) represent the transverse wavevector. The correlation functions in temporal and spectral domains for angular space can written as

T(kP,kQ;tP, tQ) =⟨U_P^∗(kP;tP)UQ(kQ;tQ)⟩, (3.31) and

G(k_P,k_Q;ω_P, ω_Q) = ⟨U_P^∗(k_P;ω_P)U_Q(k_Q;ω_Q)⟩, (3.32) respectively. Here, k_P and k_Q denote the angular spaces at two different instants.

Temporal intensity and power spectrum can be calculated from

I(k;t) =T(k,k;t, t) = ⟨|U(k;t)|²⟩, (3.33) and

S(k;ω) = W(k,k;ω, ω) =⟨|E(k;ω)|²⟩, (3.34) respectively in angular space, which can be used to normalize correlation functions.

The normalized correlation function can be written as, Ω(k_P,k_Q;t_P, t_Q) = T(k_P,k_Q;t_P, t_Q)

√︁I_P(k_P;t_P)I_Q(k_Q;t_Q), (3.35) in time domain, and as

g(kP,kQ;ωP, ωQ) = G(k_P,k_Q;ω_P, ω_Q)

√︁S_P(k_P;ω_P)S_Q(k_Q;ω_Q), (3.36)

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in frequency domain respectively. The absolute values of Eqs. (3.35) and (3.36) tell about coherence statistics in angular-time domain and angular-frequency domain, respectively. The overall degree of coherence can be computed from

Ω¯² =

∫︁∫︁∞

−∞|T(k,k;tP, tQ)|²dtPdtQ

∫︁∫︁∞

−∞I_P(k;t_P)I_Q(k;t_Q)dt_Pdt_Q, (3.37) in time domain, and

g

¯² =

∫︁∫︁∞

−∞|G(k,k;ω_P, ω_Q)|²dω_Pdω_Q

∫︁∫︁∞

−∞S_P(k;ω_P)S_Q(k, ω_Q)dω_Pdω_Q, (3.38) in frequency domain, respectively. Here, Eqs. (3.37) and (3.38) are presented considering only variations in temporal and spectral domains. The values always lies between 0 and 1. The angular correlation functions between temporal and spectral domains are also connected with Fourier relations [37].

3.4 Correlation behavior of simulated SC pulses

To examine the statistics of bulk generated SC fields, we need to form pulses at the output plane of the nonlinear crystal. To produce such pulses in bulk media, one can perform the numeric simulation of unidirectional non-paraxial propagation of pulse envelope [37]. To illustrate the coherence phenomena, the data of SC pulses are taken from the authors of the paper [37]. They chose 100 realization of SC pulses with input pulse energy 0.31 µJ. This energy is well above the threshold value for the SC generation at a central wavelength 800 nm on a 5 mm thick sapphire plate, considering the input beam diameter 15 µm and the FWHM of the input pulse of 100 fs. The relevant parameters such as linear and nonlinear refractive indices, GVD coefficient, which are needed for numerical simulation to be performed are given in Ref. [37].

The numerically simulated SC pulse spectra along different transverse wavevec- tors are visible in Fig. 3.2. Pulse spectra vary their shape with angular spaces, as it is illustrated in Fig. 3.2(a)–(d). Amplitudes of spectra decrease with increasing value of the transverse wavevector. Spectra of pulses are asymmetric and broader than that of the pump pulse due to the generation of new spectral components related to the nonlinear mechanism, which eventually forms SC. The correlations between spectral components of generated SC pulses were calculated using Eq. (3.32), and

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600 700 800 900 1000 0

5 10

600 700 800 900 1000 0

0.2 0.4 0.6 0.8

600 700 800 900 1000 0

0.1 0.2 0.3 0.4

600 700 800 900 1000 0

0.05 0.1

Figure 3.2: Demonstration of simulated SC pulse shape with variation of wavelengths at 0.31 µJ for four values of the transverse wavevectorkat the pump wavelength 800 nm. (a) k = 1 mm⁻¹, (b) k = 5 mm⁻¹, (c) k = 10 mm⁻¹, and (d)k = 20 mm⁻¹.

present in Fig. 3.3 (a)–(d). For convenience, representations are illustrated as functions of wavelength scalesλ_n= 2πc/ω_n, wheren= 1,2. As the transverse wavevector are increasing, i.e., it is more toward the off-axis position, the correlations between spectral points are reduced. The absolute values of normalized correlation functions are formed based on Eq. (3.36) and is depicted in Fig. 3.4 (a)–(d). Spectral coherence is maximum at the central wavelength 800 nm and reduces both in shorter and longer wavelengths. The overall degree of spectral coherence is computed using Eq. (3.38) and it values are 0.9994, 0.9922, 0.9789, and 0.9144 respectively for four stated transverse wavevector values 1 mm⁻¹, 5 mm⁻¹, 10 mm⁻¹, and 20 mm⁻¹, respectively. Based on these values, we estimated that the overall degree of coherence, i.e., the correlation statistics for entirety of spectral dimensions, is higher with lower angular space values, i.e., close to the optical axis. But it reduces as the position are moving towards off-axis. The increasing transverse wavevector mean that angles between spectral components are increasing in principle. So physically, as the diffraction angle is increasing, the coherence becomes lower, and vice versa.

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Figure 3.3: Demonstration of correlation function in spectral domain for four values of the transverse wavevector k at the pump wavelength 800 nm. (a)k

= 1 mm⁻¹, (b) k= 5 mm⁻¹, (c) k= 10 mm⁻¹, and (d)k= 20 mm⁻¹.

Figure 3.4: Demonstration of normalized correlation function in spectral domain for four values of the transverse wavevector k at the pump wavelength 800 nm. (a) k = 1 mm⁻¹, (b) k = 5 mm⁻¹, (c) k = 10 mm⁻¹, and (d) k = 20 mm⁻¹.

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Chapter IV

Review of pulse characterization techniques

Ultrashort pulses are electromagnetic radiations of small finite duration in femtosecond ranges or shorter like attoseconds. These pulses are generated by a pulsed laser.

Since pulsed light fields belong to the non-stationary light categories [60], the stationary light field of coherence theory is not applicable for the analysis. Characterization of these pulses, such as pulse duration, correlation properties has an enormous aspect of the scientific domain with increasing light fields for various applications [22]. But, identifications of these kinds of pulses are beyond the limit of standard electronic devices. The problem with examining short pulses is the measurement of a smaller temporal event. With the employment of different measurement techniques such as auto-correlation, cross-correlation, the pulse characterization is possible [21,61].

This chapter describes theories of these measurement processes.

4.1 Auto-correlation

Auto-correlation is a technique which attempts to measure the pulse duration. The auto-correlation measurement techniques can be categorized into two types such as field auto-correlation and intensity auto-correlation, depending on the measurement scheme. Now the following subsections briefly discuss about those two auto- correlation techniques, which may be employed in pulse characterization.

4.1.1 Field auto-correlation

Field auto-correlation is a technique may be employed for the measurement of the spectrum of a source. Also, it can estimate correlations between two identical temporal fields traces with a considerable delay. A simple Michelson interferometer

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scheme with various arm lengths can perform the field auto-correlation, as shown in Fig. 4.1. The light pulse splits into two paths via passing through a beam splitter (BS). Then, light pulses on both arms are reflected back from mirrors (M1 and M2) towards the beam splitter. Then pulses interfere and are imaged onto a CCD camera. The delay line introduces a path difference in one arm of the interferometer, as one can calculate auto-correlation functions against a time delay. Path differences between two arms are separations between two pulses. In this scheme, pulses are in the same state of polarization.

Laser source

BS

Delay line M2

CCD

Figure 4.1: Demonstration of Michelson interferometer setup implemented in field auto-correlation measurements. BS: beam splitter, M: mirrors, and CCD:

camera

The field auto-correlation functions can be described as A_F(r; ∆t) =

∫︂ ∞

−∞

E^∗(r;t)E(r;t−∆t)dt. (4.1) Here, ∆t represents the delay time between two pulses. Temporal pulse can be represented with a spectral counterpart via Fourier relation. Simple mathematical calculations can form

A_F(r; ∆t) =

∫︂ ∞ 0

|E(r;ω)|²exp(−iωt)dω, (4.2) which is the profound auto-correlation theorem. It is possible to find the auto- correlation of the pulsed signal from the Fourier transform of its spectral power. The temporal shape and duration cannot be estimated from the field auto-correlation,

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which is validated from Eq. (4.2). That is, every single pulse that has the same spectrum produces the same auto-correlation, even if their temporal pulse shapes are completely different.

Each pulses can be detected individually depending on the repetition rate. If the detector operating at a faster speed than the laser repetition rate, it records the single pulse trace. The ensemble average of the measured pulse traces modifies Eq.

(4.2) to

⟨A_F(r; ∆t)⟩=

∫︂ ∞ 0

⟨|E(r;ω)|²⟩exp(−iωt)dω =

∫︂ ∞ 0

S(r;ω) exp(−iωt)dω, (4.3) which contains power spectrum S(r;ω) information of the source. It can be employed to find the coherence time for stationary and quasi-stationary fields [62].

Field auto-correlation is related with MCF via

⟨A_F(r; ∆t)⟩=

∫︂ ∞

−∞

Γ(r;t,∆t)dt, (4.4)

which is applicable for measuring the coherence properties only for Schell-model sources [62].

4.1.2 Intensity auto-correlation

Early research showed that interferometric techniques are not suitable for temporal coherence measurements for non-stationary fields. Intensity auto-correlation [63]

measurement technique is employed to calculate the pulse length in the time domain scale. The basic outline of the intensity auto-correlation setup is given in Fig.

4.2. The light pulse is divided into two identical pulses with the help of a beam splitter. Then, these two replicas fall on the second harmonic generating crystal by passing through a focusing lens. Also, two copies of replicas are in the orthogonal state of polarization, and they are formed using a polarizer P, as depicted in Fig.

4.2. Second-harmonic generation is a process where light fields produce frequency doubling by passing through a second-order nonlinear material. Some amount of delay is inserted in the superposition of two pulses by forming small path differences in one of the arms, as shown in Fig. 4.2. The second harmonic generation signal form on the nonlinear crystal and finally records onto a CCD camera as a function of delay, i.e., the path length difference between two arms. The intensity profile will

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Laser source

Delayline

BS M1

M3

M4 M5

L Nonlinear crystal CCD M2

P

Figure 4.2: Schematic diagram of an intensity auto-correlator. BS: beam splitter, M: mirror, P: polarizer, L: focusing lens, and CCD: camera

depend on delays between two pulses. For example, if the superposition between two pulses is at large separations, then the delay is more.

Mathematically, the intensity auto-correlation function takes the form A_I(r; ∆t) =

∫︂ ∞

−∞

I(r;t)I(r;t−∆t)dt. (4.5) It gives an approximate idea of pulse duration. The maximum value is at zero delay.

The signal is symmetric about zero, though the pulse shape may not be. The full pulse shape is not achievable via auto-correlation function as multiple pulses can form the same auto-correlation function. Also, it cannot predict the full width half maximum (FWHM) as the pulse shape is not known. But with simple measurements it can roughly estimate the time window.

4.2 Cross-correlation

For understanding the properties of pulses, a novel technique was proposed [22], known as cross-correlation. Cross-correlation is a technique, which is feasible and reliable to extract coherence properties of pulses. These technique is of two types, one is field cross-correlation, and another one is intensity cross-correlation.

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4.2.1 Field cross-correlation

Let us assume two pulses E_s(r;t) and E_l(r;t−∆t) from the same pulsed source form an interference effect, which is defined in field cross-correlation as,

XF(r; ∆t) =

∫︂ ∞

−∞

E_s^∗(r;t)El(r;t−∆t)dt, (4.6) where s̸=l. Field cross-correlation also be expressed with the spectral component

X_F(r; ∆t) =

∫︂ ∞ 0

E_s^∗(r;ω)E_l(r;ω) exp(−iωt)dω. (4.7) Equation (4.7) represents the well known cross-correlation theorem. This theorem can estimate unknown pulse characteristics if one pulse is known. This novel method is a process that is similar to spectral interferometry without a spectrometer, which helps to retrieve the amplitude and phase information of each pulse individually. But this technique has some limitations, i.e., we require a highly coherent known probe pulse to characterize unknown pulses. When both pulses are unknown, it is not possible to extract information from such measurements, but the ensemble average of this scheme can retrieve correlation behavior, what we are looking to find in this thesis. This kind of scheme can be used to examine the SC field properties [22] and we employ this technique in this thesis work to find spectral coherence properties of SC pulses.

Let us assume M different pulses are generated, thenM²−M possible pairs can be formed from that group. With all possible realization of pulses, the ensemble average of a cross-correlation function can be written as

⟨X_F(r; ∆t)⟩= 1 M²−M

∫︂ ∞ 0

M

∑︂

s̸=l

E_s^∗(r;ω)E_l(r;ω) exp(−iωt)dω. (4.8) With simple calculations and considering all possible fields, Eq. (4.8) modifies to

⟨X_F(r; ∆t)⟩= 1 M²−M

∫︂ ∞ 0

⎡

⎣

⃓

M

∑︂

m=1

E_m(r;ω)

⃓

2

−

M

∑︂

m=1

|E_m(r;ω)|²

⎤

⎦exp(−iωt)dω.

(4.9) With further simplifications it takes the form

⟨X_F(r; ∆t)⟩= 1 M −1

∫︂ ∞ 0

[︁M|⟨E(r;ω)⟩|²− ⟨|E(r;ω)|²⟩]︁

exp(−iωt)dω. (4.10)

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Performing measurements over a large ensemble makes the first term in Eq. (4.10) is significant, and it can approximate as,

⟨XF(r; ∆t)⟩ ≈

∫︂ ∞ 0

Sqc(r;ω) exp(−iωt)dω, (4.11) where Sqc = |⟨E(r;ω)⟩|² represents the spectrum of the quasi-coherent part of the field [22]. This is similar to the Dudley-Coen coherence measurements. Quasi- coherent part of the spectrum can be extracted by taking inverse Fourier transform of the cross-correlation function. It is possible to formulate CSD functions and normalized spectral degree of coherence as

|W_qc(r,r;ω_s, ω_l)| ≈√︂

S_qc(r;ω_s)

√︂

S_qc(r;ω_l), (4.12) and

|µ_qc(r,r;ω_s, ω_l)| ≈

√︁S_qc(r;ω_s)√︁

S_qc(r;ω_l)

√︁S(r;ω_s)√︁

S(r;ω_l) , (4.13) respectively, based on the quasi-coherent part of the spectrum.

The overall degree of coherence given in Eq. (3.26) is modified based on Eq.

(4.12) and further simplification yields as µ¯(r)≈

∫︁∞

0 S_qc(r;ω)dω

∫︁∞

0 S(r;ω)dω . (4.14)

Practical point of view, Eq. (4.14) suggests only ratio between coherent and overall power. Moreover, if we replace quasi-coherent part of the field with its inverse Fourier transform and using power spectrum ∫︁

S(r;ω) = √︁

Is(r)Il(r), we end up with the expression

µ¯(r)≈

∫︁∞ 0

∫︁∞

−∞⟨X_F(r; ∆t)⟩exp (iω∆t) d∆tdω

√︁I_s(r)I_l(r) . (4.15)

Performing further integral produces the simplified result as µ¯(r)≈ ⟨XF(r; 0)⟩

√︁I_s(r)I_l(r), (4.16)

from which we can compute the position dependent overall degree of spectral coherence. Spectral information can be retrieved with a one temporal measurement which helps to measure pulses with a simplified setup.

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Let us introduce an another important parameter, X_N, and it can be written as X_N = ⟨X_F(r = 0; ∆t)⟩

√︁I_s(r = 0)I_l(r = 0), (4.17) which is the intensity normalized cross-correlation functions. The value of the intensity normalized cross-correlation function is bounded between 0 and 1.

4.2.2 Intensity cross-correlation

Alike, intensity auto-correlation, the intensity cross-correlation can also be presented, and it is expressed as

X_I(r; ∆t) =

∫︂ ∞

−∞

I_s(r;t)I_l(r;t−∆t)dt. (4.18) If the probe pulse is much shorter compared to the signal pulse, then the Eq. (4.18) approximates to

X_I(r; ∆t)≈

∫︂ ∞

−∞

δ(r;t)I_l(r;t−∆t)dt=I_l(r; ∆t), (4.19) where δ(r;t) represents the Dirac delta function. This approximation can estimate the unknown pulse shape without knowing the shape of the probe pulse.

Investigation on the spectral coherence of bulk generated supercontinuum implementing a novel cross-correlation technique