ultrashort pulse characterization and coherent time-frequency light processing

Ultrashort Pulse Characterization and CoherentTime-Frequency Light Processing

by

Aleksandr S. Radunsky

Submitted in Partial Fulfillmentof the

Requirements for the DegreeDoctor of Philosophy

Supervised byProfessor Ian A. Walmsley

Co-supervised byProfessor Robert W. Boyd

The Institute of OpticsArts, Sciences and Engineering

Edmund A. Hajim School of Engineering and Applied Sciences

University of RochesterRochester, New York

2013

In loving memory of my grandparentsFedor and Mira Pekurovsky

iii

Biographical Sketch

The author was born in Minsk, Belarus, but in 1993 his family moved toRochester, NY where he attended Brighton High School. He studied at the Uni-versity of Rochester from 1996 to 2000, graduating magna cum laude with aBachelor of Science in Optics degree with highest distinction.

As an undergraduate sophomore, he worked in Professor Turan Erdogan’sgroup. The next year Prof. Ergodan left the Institute of Optics to establish hisown company, Semrock. For his senior honors project, the author joined Profes-sor Dennis G. Hall’s group, only to see in the subsequent year Prof. Hall leavethe Institute of Optics, becoming Associate Provost of Research at VanderbiltUniversity.

He started doctoral studies at the Institute of Optics in the fall of 2000, joiningthe research group of Ian A. Walmsley the following year (just in time to witnessProf. Walsmley’s departure to Oxford). In the summer of 2002 he was granteddissertation in absentia status and relocated to the Clarendon Laboratory in thePhysics department of the Oxford University, where he pursued research in thearea of ultrashort pulse metrology and low coherence interferometry.

iv

Publications

1. J. Greffet, M. De la Cruz-Gutierrez, P. Ignatovich, and A. S. Radunsky,“Influence of spatial coherence on scattering by a particle,” Journal of TheOptical Society of America A-Optics Image Science and Vision, vol. 20,pp. 2315–2320, Dec 2003.

2. E. Kosik, A. S. Radunsky, I. Walmsley, and C. Dorrer, “Interferometrictechnique for measuring broadband ultrashort pulses at the sampling limit,”Optics Letters, vol. 30, p. 326–328, Feb 1 2005.

3. A. S. Radunsky, E. Williams, I.A. Walmsley, P. Wasylczyk, W. Wasilewski,A. U’Ren, and M. Anderson, “Simplified spectral phase interferometry fordirect electric-field reconstruction by using a thick nonlinear crystal,” OpticsLetters, vol. 31, p. 1008–1010, Apr 1 2006.

4. P. Wasylczyk, A. U’Ren, P. J. Mosley, J. Lundeen, M. P. A. Branderhorst,S.-P. Gorza, A. Monmayrant, A. Radunsky, and I. A. Walmsley, “A shortperspective on long crystals: broadband wave mixing and its applicationto ultrafast quantum optics,” Journal of Modern Optics, vol. 54, no. 13-15,p. 1939–1958, 2007.

5. K. Banaszek, A. S. Radunsky, and I. A. Walmsley, “Blind dispersioncompensation for optical coherence tomography,” Optics Communications,vol. 269, p. 152–155, Jan 1 2007.

6. A. S. Radunsky, I. A. Walmsley, S.-P. Gorza, and P. Wasylczyk, “Com-pact spectral shearing interferometer for ultrashort pulse characterization,”Optics Letters, vol. 32, p. 181–183, Jan 15 2007.

7. S.-P. Gorza, A. S. Radunsky, P. Wasylczyk, and I. A. Walmsley, “Tailoringthe phase-matching function for ultrashort pulse characterization by spec-tral shearing interferometry,” Journal of The Optical Society of AmericaB-Optical Physics, vol. 24, p. 2064–2074, Sep 2007.

Patent

• I. A. Walmsley, A. S. Radunsky, and S.-P. Gorza, “Ultra-short optical pulsemeasurement using a thick nonlinear crystal.” Patent published Oct 2009.US 7599067.

• I. A. Walmsley, A. S. Radunsky, and S.-P. Gorza, “Ultra-short optical pulsemeasurement using a thick nonlinear crystal.” Patent published Dec 2011.EP 1886107 B1.

v

Acknowledgments

Long gone are the times (if they ever truly existed) when the scientific pursuitof knowledge was conducted in solitude, the intellectual and social isolation of theresearcher being accepted as the norm of scientific exploration.

It is certain that the work described in this dissertation was not done alone andthe author has benefited tremendously from the ties and all the human connectionsand associations made during his doctorate years.

I am greatly indebted to Ian Walmsley for his guidance and leadership through-out this part of my life-span. His incredible mastery of concentrating on the es-sential elements of any problem as well as his ability to see questions in a greatercontext and the talent to convey the crucial concepts and ideas in a clear, preciseand elegant manner have been a continuous source of wonder and learning throughmy graduate years. Doing good physics with beautiful style, as well as naturallypossessing that implicit ability to lift those around him above themselves is a rarecombination indeed.

It was wonderful to have worked with and it is my pleasure to acknowledge thehelp and detailed insight provided by Simon-Pierre Gorza during my concludingyear at Oxford. I am very much indebted to Konrad Banaszek and I have learneda great deal from collaborating with him during my early years.

Ian draws very talented people into his circle, and I was fortunate to haveworked with Alfred U’Ren, Piotr Wasylczuk, Wojtek Wasilweski, Matt Anderson,Christophe Dorrer (who is to be credited with initiating me into the field). Eachcontributed in some unique way to the build up of my professional character, notto mention that many have become personal friends.

I consider myself very lucky to have been the youngest member of the coreof Rochester students who have followed Ian to England. We have spent nu-merous nights in the lab with Pablo Londero. His constant willingness to listento every excruciating aspect of my projects and generously offer his advice andexpertise was always useful and enjoyable. It was impossible to know and notto love Ben Brown, whose incredible combination of scientific, interpersonal andorganizational skills, along with a true generosity of character, have been greatlymissed by everyone. Ellen Kosik-Williams has transformed from a fellow “spider-person” into an inspirational reminder of all the good things awaiting you in the“after-PhD” land. Manuel de la Cruz-Gutierrez was a constant reminder of whattrue character, focused effort and determination is all about.

I am sincerely grateful to my undergraduate teachers and mentors: Profes-sor Tom Brown, Professor Turan Erdogan, Dr. Mark Froggatt, Professor DennisHall, Professor Susan Houde-Walter and Professor Robert Knox for my academicfoundation and undergraduate research experience.

I am (deeply) grateful to Prof. Robert Boyd for extending his support andbecoming my co-adviser upon my return to the United States. It was a plea-sure to feel ‘adopted’ into the deeply talented and exciting NLO group. Prof.

vi

Miguel Alonso has been ever so helpful and considerately supportive during thedissertation finishing process.

I am very fond of all the Rochester fellow graduate classmates—AaronSchweinsburg, Alexis Spilman, Fatih Yaman, Fei Lu, Fillipp Ignatovich, HemaRoychowdhury, Nick Usechack, Mayer Landau, Yujun Deng—as well as seniorstudents—Sharon Weiss, Michael Beversluis, Kedar Khare, Jason Porter, JasonNeiser, Alberto Martino, Giovanni Piredda—for keeping in touch, not abandon-ing and “leaving me overboard” during my Oxford years. Len Zheleznyak andMichael Fisher provided essential and unwavering faith and support in the latestages of the authoring process.

I sincerely thank Betsy Benedict, Joan Christian, Gayle Thompson for notforgetting about my presence oversees. I thank people above and Lissa Cotterfor ensuring the wheel of the academic and administrative matters is revolvingsmoothly. Noelene Votens always has been absolutely outstanding in processingthe “remote” purchase requests and Maria Schnitzler was very helpful with thenecessary financial paperwork throughout my time overseas.

POA Librarian, Pat Suluoff and LLE Librarian, Linda Clement, have beenconsistently outstanding in assisting me with locating the most obscure and un-obtainable references in the most timely fashion, keeping me in continuous aweand admiration of their limitless prowess and efficiency.

Oxford became my “home away from home” and it could even seem quite afun place at times, not in the least due to a fine range of “non-Rochesterian”characters that came to represent the group as the initial “home” crew had beengradually drifting back to the “USofA”. I enjoyed greatly the international diver-sity of the group and thought it was a great personal asset to have the culturalexchange opportunity of comparing my world views and perceptions with alter-nate perspectives and attitudes. I am also proud to have finally understood that“a few pints in the pub on Friday night” ritual is indeed an essential part ofsuccessful research. The “mature”—Christine Silberhorn, Laura Corner, AntoineMonmayrant, Robert Davies, Zhongyang Wang—as well as the “less mature”members of the group—Adam Wyatt, Daryl Achilles, Matthijs Branderhorst, Pe-ter Mosley, Alex Dicks, Lijian Zhang and the rest of the “young blood”—havebeen all wonderful to share with the daily effort of research.

Susan Witney provided the crucial administrative assistance in the ClarendonLaboratory and seemed to have been running the group at times, in as far asbeing the only person really to know Ian’s schedule. Rob Harris made productiveevery visit to the student machine workshop and Alan Francis has always been areliable and helpful source at the Stores.

Through these years I have been sustained by my family and I express my es-pecial gratitude for the unwavering faith, continuous understanding and steadfastsupport by my parents and my sister as well as my (greatly missed) grandparentsall of whom made many personal sacrifices to the benefit of their only son, brotherand grandson.

vii

“We are warmed by the awareness of the ties that bind us to our people.

We know that we are one with the rest. But that the rest may know it, we

must learn to express it.”

— Antoine de Saint-Exupery, “Flight to Arras” (1942)

viii

Abstract

Over the past several decades ultrafast laser science and technology has evolved

into an extensive and diverse yet still one of the most rapidly growing and develop-

ing areas of optics. This evolution has been one of mutual interdependence. Each

current generation of technological innovations not only solves the specific prob-

lems it was designed for, but uncovers new application opportunities and enables

the exploration of new basic research areas. In turn, these new challenges will give

rise to the next generation of technological improvements born of the currently

existing technologies and the advances in fundamental scientific knowledge and

understanding.

Ultrashort pulse characterization has always been an essential part of this

ultrafast optics evolution. The thesis makes yet another contribution to it by

describing the principle, design, construction, development and operation of a

novel interferometric ultrashort pulse characterization device. It consists of a new

implementation of spectral-shearing interferometry for reconstructing the electric

field of ultrashort pulses, requiring only a single optical element to encode the

temporal field of the pulse under test. The technique relies on an asymmetric

group velocity matching type II sum frequency generation process in a single long

ABSTRACT ix

nonlinear crystal. We analyze the performance of the device for a wide range of

experimentally available input pulse parameters. The device — potential building

block for the future generations of ultrashort diagnostics — proves a practical,

elegant, compact, robust, and sensitive option for complete amplitude and phase

ultrashort pulse characterization.

As the femtosecond systems of increasingly larger bandwidth become a

widespread reality, the detrimental effects of dispersion require careful consider-

ation. Dispersive pulse distortion degrades longitudinal resolution of broadband

interferometric imaging methods such as optical coherence tomography and low-

coherence interferometry. We address the issue with a novel signal processing

dispersion compensation method. This numerical technique improves the axial

resolution without a priori knowledge of the material dispersive properties of the

sample under consideration. The dispersion compensation is based on the gen-

eralized temporal fourth order field autoconvolution function computed from the

readily available experimental interferometric scans and has an intuitive depiction

in the time-frequency phase-space via the Wigner distribution function formalism.

x

Contributors and Funding Sources

This work was supervised by a dissertation committee consisting of ProfessorsIan Walmsley (advisor), Robert Boyd (co-advisor), Miguel Alonso of the Instituteof Optics, and Professor Roman Sobolewski of the Department of Electrical andComputer Engineering.

The theoretical and numerical analysis of Chapter 3 was conducted in col-laboration with Simon-Pierre Gorza and Piotr Wasylczyk and published in thearticles [6] and [7] (as indicated in the ‘Publications’ section on page (iv)). Theexperimental results of Chapter 4 were conducted with multiple collaborators —Ellen Williams, Piotr Wasylczyk, Wojtek Wasilewski, Alfred U’Ren, Matt An-derson and Simon-Pierre Gorza — and published in the articles [3] and [6]. Thework included in Chapter 5 was conducted in collaboration with Konrad Banaszekand published in an article [5]. All other work conducted for the dissertation wascompleted by the student independently. The author’s research was supported byNational Science Foundation grants.

xi

Concise Table of Contents:

1 Introduction 1

1.1 History and Overview of Ultrafast Science and Technology . . . . 1

1.2 History and Overview of Ultrashort Pulse Characterization . . . . 2

1.3 Thesis Outline . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

2 Theoretical Background and Mathematical Context 7

2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

2.2 First-Order (Linear) Signals and Transformations . . . . . . . . . 8

2.3 Second-Order (Bilinear ⊂ Nonlinear) Signals and Transformations 11

2.4 Fourth (Higher-Orders ⊂ Nonlinear ≡ Multilinear) Representations 16

2.5 Chapter Summary . . . . . . . . . . . . . . . . . . . . . . . . . . 17

3 Novel Pulse Characterization Method: Theory 18

3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

3.2 Simplified analysis in frequency domain . . . . . . . . . . . . . . . 22

3.3 Complete temporal analysis . . . . . . . . . . . . . . . . . . . . . 28

CONCISE TABLE OF CONTENTS: xii

3.4 Numerical Simulations . . . . . . . . . . . . . . . . . . . . . . . . 37

3.5 ARAIGNEE reconstruction algorithm . . . . . . . . . . . . . . . . 42

3.6 Chapter Summary . . . . . . . . . . . . . . . . . . . . . . . . . . 47

4 Novel Pulse Characterization Method: Experiments and Results 49

4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49

4.2 Experimental Implementation . . . . . . . . . . . . . . . . . . . . 50

4.3 Implementation Results and Discussion . . . . . . . . . . . . . . . 53

4.4 Error Analysis and Discussion . . . . . . . . . . . . . . . . . . . . 58

4.5 Chapter Summary . . . . . . . . . . . . . . . . . . . . . . . . . . 62

5 A Method for Dispersion Compensation 64

5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64

5.2 The dispersion compensation method . . . . . . . . . . . . . . . . 66

5.3 Chapter Summary . . . . . . . . . . . . . . . . . . . . . . . . . . 77

6 Conclusions 78

6.1 Thesis Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78

6.2 Future Directions . . . . . . . . . . . . . . . . . . . . . . . . . . . 79

6.3 Final word . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83

Bibliography 84

xiii

Table of Contents

1 Introduction 1

1.1 History and Overview of Ultrafast Science and Technology . . . . 1

1.2 History and Overview of Ultrashort Pulse Characterization . . . . 2

1.2.1 Pulse Characterization: Early pulse measurements . . . . . 2

1.2.2 Pulse Characterization: Unified formulation and classification 4

1.3 Thesis Outline . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

2 Theoretical Background and Mathematical Context 7

2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

Primitive concept: the field . . . . . . . . . . . . . . 7

2.2 First-Order (Linear) Signals and Transformations . . . . . . . . . 8

Signals . . . . . . . . . . . . . . . . . . . . . . . . . 8

Linear Transformations . . . . . . . . . . . . . . . . 9

Linear time-invariant and time-variant analysis . . . 9

2.3 Second-Order (Bilinear ⊂ Nonlinear) Signals and Transformations 11

Temporal domain . . . . . . . . . . . . . . . . . . . 11

CONTENTS xiv

Spectral Domain . . . . . . . . . . . . . . . . . . . 11

Time-Frequency Analysis . . . . . . . . . . . . . . . 11

Invertibility and Marginals . . . . . . . . . . . . . . 13

Modulation and Convolution . . . . . . . . . . . . . 14

WDF: localization . . . . . . . . . . . . . . . . . . . 14

WDF: time-frequency shift ‘covariance’ . . . . . . . 15

WDF: chirp convolution and multiplicaton . . . . . 15

2.3.1 Pulse Characterization scheme . . . . . . . . . . . . . . . . 16

2.4 Fourth (Higher-Orders ⊂ Nonlinear ≡ Multilinear) Representations 16

2.5 Chapter Summary . . . . . . . . . . . . . . . . . . . . . . . . . . 17

3 Novel Pulse Characterization Method: Theory 18

3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

3.1.1 (Conventional) SPIDER technique . . . . . . . . . . . . . 19

3.1.2 Motivation (for the novel method) . . . . . . . . . . . . . . 21

3.2 Simplified analysis in frequency domain . . . . . . . . . . . . . . . 22

3.3 Complete temporal analysis . . . . . . . . . . . . . . . . . . . . . 28

3.3.1 Dispersionless medium . . . . . . . . . . . . . . . . . . . . 29

3.4 Numerical Simulations . . . . . . . . . . . . . . . . . . . . . . . . 37

Wavelength tunability . . . . . . . . . . . . . . . . . 39

3.4.1 Effect of group-velocity dispersion . . . . . . . . . . . . . . 40

3.5 ARAIGNEE reconstruction algorithm . . . . . . . . . . . . . . . . 42

CONTENTS xv

3.6 Chapter Summary . . . . . . . . . . . . . . . . . . . . . . . . . . 47

4 Novel Pulse Characterization Method: Experiments and Results 49

4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49

4.2 Experimental Implementation . . . . . . . . . . . . . . . . . . . . 50

4.2.1 Measurements: Calibration . . . . . . . . . . . . . . . . . . 52

4.3 Implementation Results and Discussion . . . . . . . . . . . . . . . 53

4.3.1 Measurements: Proof of Principle . . . . . . . . . . . . . . 53

4.3.2 Measurements: Spectral Tunability and Bandwidth . . . . 56

4.4 Error Analysis and Discussion . . . . . . . . . . . . . . . . . . . . 58

Limitations of ARAIGNEE . . . . . . . . . . . . . . 61

4.5 Chapter Summary . . . . . . . . . . . . . . . . . . . . . . . . . . 62

5 A Method for Dispersion Compensation 64

5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64

5.2 The dispersion compensation method . . . . . . . . . . . . . . . . 66

5.2.1 Generalized autoconvolution: the principle . . . . . . . . . 67

5.2.2 Generalized autoconvolution: the details . . . . . . . . . . 71

5.2.3 Analysis in the time-frequency domain . . . . . . . . . . . 73

5.3 Chapter Summary . . . . . . . . . . . . . . . . . . . . . . . . . . 77

6 Conclusions 78

6.1 Thesis Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78

CONTENTS xvi

6.2 Future Directions . . . . . . . . . . . . . . . . . . . . . . . . . . . 79

ARAIGNEE Alternative geometries . . . . . . . . . 79

ARAIGNEE: Multiplexing advantages . . . . . . . . 81

ARAIGNEE: Quasi-phasematching . . . . . . . . . 81

ARAIGNEE: Practical and Numerical nuances . . . 81

ARAIGNEE: Extending the spectral range . . . . . 81

6.3 Final word . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83

Bibliography 84

xvii

List of Tables

Table Title Page

2.1 The LTI and LTV transfer filter pairs necessary for pulse charac-terization.) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

4.1 Experimental and theoretical spectral weighted phase error εφ . . 62

xviii

List of Figures

Figure Title Page

1.1 Classes of pulse characterization methods. . . . . . . . . . . . . . 4

2.1 Fourier transform duality of spectral and energy densities. . . . . 122.2 WDF time-frequency shift covariance . . . . . . . . . . . . . . . . 15

3.1 Conventional Spider encoding and reconstruction: (a: top row) Theconventional SPIDER schematic. The two delayed pulse replica up-convert with different frequency components of the chirped pulseand the resulting upconverted signal is measured in a spectrom-eter. (b: second row) The SPIDER inversion algorithm. After aFourier transform of the spectral interferogram into the pseudotimedomain, filtering of the ac sidelobe and inverse Fourier transformback to the frequency domain, the argument of the result is a ref-erence phase and the spectral phase difference between two pointsin the input spectrum separated in frequency by the spectral shearΩ. Subtraction of the reference phase and concatenation of theremainder results in a reconstruction of the pulse spectral phase. . 20

3.2 The absolute magnitudes of the collinear, type II PMF,sinc2(∆kL/2), ∆k(ωe, ωo) = ke(ωe + ωo) − ke(ωe) − ko(ωo) of aL=2 cm thick KDP crystal for two (different by 0.5) values of thepropagation angle, plotted as a function of frequency for ordinaryωo and extraordinary ωe input polarization components (black in-dicating perfect phasematching). The sum-frequency signals aredrawn on the diagonal axis, ωs = ωe + ωo, illustrating the spectralshear between the outputs. . . . . . . . . . . . . . . . . . . . . . . 24

3.3 The diagram of the modeled collinear upconversion for a type IIsum frequency generation nonlinear process. Ro is the o- (solid(red) line) and Re is the e-polarized (dashed (red) line) test pulses(or polarization components of the same pulse) respectively andB (solid (blue) line) the sum-frequency pulse. t0 is an arbitrarypre-delay between the two fundamental test pulses and L is thethickness of an appropriately cut nonlinear crystal. . . . . . . . . 30

LIST OF FIGURES xix

3.4 The absolute magnitude of the collinear, type II PMF tuned awayfrom perfect phasematching. . . . . . . . . . . . . . . . . . . . . . 35

3.5 Generation and evolution of the amplitude (top row) and the phase(bottom row) of the sum frequency signal B(t) as the two funda-mentals propagate through the crystal. Amplitude and phase of Ro

(solid red line), amplitude of Re (dotted red line), amplitude andphase of B (solid blue line). The three columns (from left to right)correspond to the entrance, center and exit planes of the crystal. . 37

3.6 Effect of a group velocity mismatch ∆k′bro

= 0.15∆k′bre

on theSF generation. The initial conditions are identical than for Fig-ure (3.5). Intensity profile and phase of the SF pulse (dashed-line)and the o-fundamental pulse (solid-line) at the output of the crys-tal. The shifted and scaled output SF pulse is also shown (circles). 39

3.7 Comparison of the amplitude and the phase of the SF-pulse at theoutput of a thick KDP crystal obtained by numerically solving thesystem (3.6) (solid-lines) and given by the model Eq. (3.22) (circles). 40

3.8 Block diagram of the phase retrieval procedure in ARAIGNEE. . 473.9 Scaling factor s (a) and quadratic phase factor acorr (b) as a func-

tion of the central wavelength of the unknown pulse for KDP. Theparameters used in Figure b) are: crystal thickness L=10mm, delayt0 = 600 fs. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48

4.1 Schematic of the ARAIGNEE device (top-view). WP, half-waveplate which splits linearly polarized input into ‘o-’ and ‘e-’ compo-nents; Q, quartz plate which introduces the pre-delay t0 (≃300 fs);MP, mutually tilted (by β) and longitudinally shifted (by d) mirrorpair which introduces the delay τ ≡ d/c (≃ 1.5 ps); PM, pick-offmirror; KDP, nonlinear crystal (with its optical axis set horizon-tally, in the plane of the diagram). (Dotted red line pulse-shape rep-resents ordinary polarization (perpendicular to the diagram plane)and solid red, and blue line pulse-shape represents extraordinarypolarization (in the plane of the diagram). . . . . . . . . . . . . . 51

4.2 Photograph of the compact ARAIGNEE design. WP, half-waveplate; MP, mirror pair; PM, pick-off mirror; FM, folding mirror (tospectrometer); Xtal, nonlinear crystal. . . . . . . . . . . . . . . . 52

4.3 Spectral amplitude (line with squares) of the test pulse and its sinu-soidally modulated spectral phase reconstructed by a conventionalSPIDER (solid line) and a novel long crystal (dashed line) method. 54

4.4 Spectral amplitude (line with squares) and phase (dots) measuredby the device and the theoretical spectral phase (solid line) afterpropagation through 10 cm of BK7 glass. . . . . . . . . . . . . . . 55

4.5 Measured (solid lines) and calculated (dashed lines) spectral phaseand measured spectral intensity for different central wavelengthsand bandwidths of the input pulse. . . . . . . . . . . . . . . . . . 57

LIST OF FIGURES xx

4.6 Left: Spectrum of the test pulse (dotted line) and its spectralphase returned by ARAIGNEE (dashed line) and conventional SPI-DER (solid line) for different central wavelengths. Right: Time-dependent intensity and phase measured by ARAIGNEE (circles)and SPIDER (solid line) from the data plotted on the left. . . . . 59

4.7 Spectral phase error εφ of the retrieved pulse calculated from a nu-merical simulation of the pulse propagation in the nonlinear crys-tal plotted as a function of the input pulse bandwidth (intensityFWHM) and central wavelength. Gaussian transform-limited in-put pulse is used. . . . . . . . . . . . . . . . . . . . . . . . . . . . 61

5.1 The basic schematic of an OCT interferometer. . . . . . . . . . . 655.2 Depth-profile reconstruction. . . . . . . . . . . . . . . . . . . . . . 70

6.1 Possible next generation ARAIGNEE device. First subfigurepresents the schematic of the apparatus and the second demon-strates the actual prototype. . . . . . . . . . . . . . . . . . . . . . 80

6.2 Phase Matching Function magnitude for a type II SFG in a thickKDP crystal demonstrating both extrema of the asymmetric groupvelocity matching. . . . . . . . . . . . . . . . . . . . . . . . . . . . 83

1

Chapter 1

Introduction

1.1 History and Overview of Ultrafast Scienceand Technology . . . . . . . . . . . . . . . 1

1.2 History and Overview of Ultrashort PulseCharacterization . . . . . . . . . . . . . . 2

1.3 Thesis Outline . . . . . . . . . . . . . . . 5

1.1 History and Overview of Ultrafast Science

and Technology

From the first demonstrations of laser mode-locking behavior in mid-1960’s, it was

not until 1974 that subpicosecond pulsed regime was entered for the first time [1]

and up until then a comprehensive overview of the entire subject area of pulsed

optical radiation was still possible [2]. The femtosecond regime proper (ie. optical

pulses of sub-100 fs duration, one femtosecond is one quadrillionth of a second)

was entered in 1982 [3] and since then there truly was no going back.

With the help of a commercial break-through, started in the early 1990’s by the

discovery and demonstration of the self-modelocking in the Ti:sapphire lasers [4,5],

the field has grown from some tens or so research laboratories using subpicosecond

1.2. HISTORY AND OVERVIEW OF ULTRASHORT PULSECHARACTERIZATION 2

pulses around the world just quarter of a century ago [6] to ∼ hundred of research

(academic and (corporate/industrial)) groups, ∼ thousands of publications and

dozens of textbooks, not to mention many, many commercial companies with

estimated market capitalization of several hundred millions [7].

The dramatic development of the ultrafast optical science and technology is

truly impossible to overstate.

Today, the use of ultrashort optical pulses in physics, chemistry, biology, en-

gineering and medicine has become widespread and has created its own research

areas and emerging applications such as femtochemistry and femtospectroscopy,

soliton fiber communications, two-photon confocal microscopy, ultrafast high-field

physics, and micromachining [8,9]. Fueled by these constant scientific and tech-

nological advances, the development and application of femtosecond laser systems

is showing no signs of slowing down [10]. In turn, there is as great of a need

as ever for accurate and reliable metrology and diagnostics of ultrashort optical

sources and systems.

1.2 History and Overview of Ultrashort Pulse

Characterization

1.2.1 Pulse Characterization: Early pulse measurements

Out of all the pulse metrological parameters, it is the temporal distribution

(temporal amplitude and phase) of the electromagnetic field that is of greatest


experimental interest. However, the temporal dimension is also the one presenting

the most diagnostic challenges. The photoelectric response times of the detectors

do not allow for them to resolve the temporal structure of the pulses on the sub-

picosecond scale.

In order to raise to this challenge, alternative all-optical pulse measurement

methods were beginning to be developed all together with the earliest develop-

ment of mode-locked sources [2,11]. For an extended period the (ultrafast) inten-

sity correlation techniques came to dominate how the pulse duration was mea-

sured. [12,13] However, the information extracted from such (ultrafast) intensity

autocorrelation functions is almost always insufficient for complete (both temporal

amplitude and phase) characterization of the underlying optical waveform. Start-

ing from the late 70’s, there started to appear more and more methods and tech-

niques for complete measurement of pulse characteristics (pulse temporal envelope

and phase). As a result, an enormous range and variety of (seemingly disconnected

and unrelated) methods and approaches has been invented and demonstrated over

the years.

(There seems to be an unfortunate similarity in how these historic develop-

ments seem to repeat themselves within different sub-disciplines. For example,

the field of nonlinear spectroscopy [14] and optical image/system aberration in-

strumentation [15] also seem to have experienced very similar periods of early ex-

pansion and unstructured growth, characterized by being overwhelmed with the

flood of different experimental techniques, instruments, concepts, terminologies.)


1.2.2 Pulse Characterization: Unified formulation andclassification

By analyzing the fundamental concepts underlying the widely diverse (pulse

characterization) methods and techniques, in the mid 1990’s it finally became pos-

sible to order and organize pulse characterization discipline into a classification

scheme, as all the methods were conveniently formulated in a linear filtering for-

malism [16–18]. A pulse measurement is viewed as a signal transmission through

a collection of well characterized linear filters (signal encoding stage) with subse-

quent recording of the output as a function of the filters’ parameters to be followed

by an appropriate inversion algorithm (signal reconstruction stage).

Such ‘wider-view’ analytical framework allows for a general categorization of

all methods for complete pulse characterization into three classes according to the

encoding method and the inversion algorithm they employ for reconstructing the

unknown pulse from the measurement. The method can be either interferometric

on non-interferometric, the latter being further split into tomographic and spec-

trographic classes, as shown in Figure 1.1.

PULSE CHARACTERIZATION

NON-INTERFEROMETRIC

TomographicSpectrographic

INTERFEROMETRIC

Non-Self-ReferencingSelf-Referencing

Figure 1.1: Classes of pulse characterization methods.

1.3. THESIS OUTLINE 5

Interferometric techniques, such as spectral phase interferometry for direct

electric field reconstruction (SPIDER), measure the time or frequency field corre-

lation function (which is factorizable, thus one-dimensional, for a non-stochastic

field) and have a non-iterative inversion/reconstruction algorithm. Spectrographic

methods (e.g. frequency resolved optical gating (FROG) [19,20]) necessarily sam-

ple the two-dimensional time-frequency (chronocyclic) phase-space distribution

function (typically the Wigner distribution function (WDF)) of a pulse realiza-

tion and require an iterative algorithm to recover the one-dimensional pulse am-

plitude and phase. Tomographic techniques (e.g. [21]) measure the sections of

the chronocyclic WDF parametrized by some rotation angle in the joint time-

frequency phase-space and have a unique, non-iterative, inversion algorithm to

reconstruct the pulse shape.

1.3 Thesis Outline

The organization of the ensuing material in this thesis is straightforward. Chap-

ter 2 introduces the basic concepts and presents the theoretical framework of the

subject. Chapter 3 provides the analysis and numerical modeling of our ultrashort

pulse characterization method while Chapter 4 describes its experimental imple-

mentation and presents the empirical results, tests and analysis. The details of

the novel dispersion compensation approach are given in Chapter 5 with Chap-

1.3. THESIS OUTLINE 6

ter 6 containing the concluding remarks along with the outline of the possible

research paths and directions for furthering the work.

7

Chapter 2

Theoretical Background andMathematical Context

It appears to me, therefore, that the study of electromagnetism inall its extent has now become of the first importance as a meansof promoting the progress of science.

J. C. Maxwell, A Treatise on Electricity and Magnetism (1873).

2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . 72.2 First-Order (Linear) Signals and Transformations . 82.3 Second-Order (Bilinear ⊂ Nonlinear) Signals and

Transformations . . . . . . . . . . . . . . . . . . . 112.4 Fourth (Higher-Orders ⊂ Nonlinear ≡ Multilinear)

Representations . . . . . . . . . . . . . . . . . . . . 162.5 Chapter Summary . . . . . . . . . . . . . . . . . . 17

2.1 Introduction

Primitive concept: the field The most essential physical quantity underlying

our fundamental understanding of all optical phenomena is that of a field. On

the most fundamental level, within the framework of Quantum Electrodynamics

(QED) (arguably, one of the most successful modern physics theories of past half

century) [22], the (quantum) field is mathematically described as an operator

valued vector function E(r, t) on the space-time. The action of the operator is

defined on the (bosonic) Fock space [23, §4.1] and this construction formalizes the

2.2. FIRST-ORDER (LINEAR) SIGNALS ANDTRANSFORMATIONS 8

concept of a light particle (a “photon” is an elementary unit of excitation of the

electromagnetic field) absent in the classical electromagnetic theory.

While the representational capacity of the fundamental mathematical toolbox

above is sufficient to describe (in a unified manner) almost all known interactions

of light and matter, it is not necessary, not effective, and beyond the scope of

this thesis. Nonetheless, we can make use of one of the mathematical structures

embedded in the mathematical toolkit above.

The mathematical language of the Hilbert space theory provides a convenient

and effective description of the dissertation’s scope.

Consequently, by focusing on the classical (deterministic) optical fields (as

well as suppressing the spatial and polarization degrees of freedom for simplicity of

notation) we restrict and simplify our theoretical treatment to wavefields described

by the Hilbert space of complex valued square integrable (in the Lebesgue sense)

functions f ∈ H := L2(R) of the real-valued time argument t ∈ R, and linear

transformations on L2(R).

2.2 First-Order (Linear) Signals and Transfor-

mations

Signals In accordance with common practice, we further restrict our attention

to the subspace of L2(R) consisting of so-called analytic (bandpass) signals [24]

— complex-valued signals f , such that their Fourier transform F (ω) is nonzero


only over the positive frequency half axis (≡ whose imaginary part is the Hilbert

transform of the real part).

The analytic signal f(t) := |f(t)| eiϕ(t), F (ω) := |F (ω)| eiφ(ω) is useful because

the functions of temporal ϕ and spectral phase φ, allow for a mathematically pre-

cise definition of the notions of time-varying frequency (instantaneous frequency)

∂tϕ(t) assigning a frequency value to a given time, and its dual quantity, the group

delay ∂ωφ(ω) giving a time (of arrival/occurence) value to a particular frequency.

Linear Transformations A general linear transformation (≡ (linear) filtering,

(linear) operator) fh−→ g is given by

g(t) =

∫h(t, t′)f(t′) dt′,

where h is the kernel (impulse response) of the transformation.

Linear time-invariant and time-variant analysis A linear time-invariant

(LTI) (≡ shift-invariant, temporally-homogeneous, stationary) transformation is

characterized by the impulse response of the form h = h(t − t′) and expressed

either as the convolution (in time domain) or a product of corresponding Fourier

transforms (in frequency domain):

g(t) =

∫h(t− t′)f(t′)dt′ G(ω) = H(ω)F (ω)


There is a Fourier dual class of (time-variant) filters to the LTI filters: lin-

ear frequency-invariant (LFI) filters, or modulators. The transformation is char-

acterized by operation of (pointwise) multiplication in the time domain and a

convolution in the frequency domain:

g(t) = m(t)f(t) G(ω) =

∫Γ(ω − ω′)F (ω)dω′ (2.1)

and formally has the impulse response h = m(t)δ(t− t′).

The essential examples of LTI and LFI filters are given by time and frequency

shifts:

f(t± τ)Ft→ω−−−→ e±iτωF (ω) (2.2)

e±iΩtf(t)Ft→ω−−−→ F (ω ∓ Ω)

In addition, in accordance with the experimental practice, it is also necessary

to introduce two more pairs of filters (cf. Table 2.1). (Parenthetically, we note

that while, strictly speaking, the impulse response function of the LTI Gaussian

Spectral Filter is not casual, this fact causes no operational problems, given the

understanding that all LTI filter response functions are defined up to an arbitrary

time delay constant which can be selected such as to eliminate the ‘acasuality’ of

the impulse response [25, Appendix A]).

2.3. SECOND-ORDER (BILINEAR ⊂ NONLINEAR) SIGNALSAND TRANSFORMATIONS 11

LTI LTV

Spectral Filter : H(ω) = e−(ω/∆ω)2 Time Gate: h(t) = e−(t/∆t)2

Dispersion: H(ω) = e−iφ′′ω2

Phase Modulation: h(t) = e−iϕ′′t2

Table 2.1: The LTI and LTV transfer filter pairs necessary for pulse characteriza-tion.)

2.3 Second-Order (Bilinear ⊂ Nonlinear) Sig-

nals and Transformations

The detected physical quantities in optics (energy-like quantities) are quadratic

(and more generally, bi-linear) functions of the electromagnetic field.

Temporal domain The ‘instantaneous intensity’ of the signal f is represented

by the temporal energy density |f(t)|2 (energy density with respect to time).

Furthermore, for ultrashort pulses, even this quantity is not temporally resolved

when measured with slow (integrating) detectors.

Spectral Domain Correspondingly, the |F (ω)|2 can be viewed as spectral en-

ergy density. It represents the output of a spectrometer.

The spectral and temporal densities are related by the following (non-invertible

between horizontal levels) diagram:

Time-Frequency Analysis The objective of time-frequency analysis is a for-

malized ‘intermediate’ representation of a function, combining its temporal as

well as spectral behavior and the Wigner distribution (≡ function, transforma-


|f(t)|2 Ft→ω←−−→F−1t←ω

F (ω) ∗ F (ω)

|· |2x ∗

x

f(t) ←−−→ F (ω)

∗

y |· |2y

f(t) ∗ f(t) ←−−→ |F (ω)|2

Figure 2.1: Fourier transform duality of spectral and energy densities.

tion, representation) is one of the most well known and studied members of many

possible time-frequency representations.

First introduced in the context of quantum mechanics [26], the Wigner distri-

bution has not only become an essential tool in Quantum Optics [27], but has also

found very successful applications in many other sub-branches of optics, including

both conventional (spatial) Fourier Optics [28–30], and signal analysis [31] (≡

(temporal) Fourier Optics) [32].

For a deterministic signal f , the real-valued Wigner function Wf can be defined

as:

Wf(t, ω) = Fτ→ω f(t + τ/2)f ∗(t− τ/2) (2.3)

=

∫f(t + τ/2)f ∗(t− τ/2)e−iωτdτ. (2.4)

It demonstrates (as all bilinear quantities do), the ‘quadratic’ superposition

principle) mathematically describing the observable interference phenomena. For


a superposition of signals f = af1 + bf2, (a, b ∈ C), the Wigner distribution

function is given by:

Waf1+bf2= |a|2Wf1

+ |b|2Wf2+ 2ℜab∗Wf1f2

And in addition to giving a visual (intuitive) representation of the time-varying

content of the analyzed signal, it also satisfies a number of important and useful

properties.

Invertibility and Marginals The original signal f , can be (conditionally, up

to a constant phase factor) reconstructed from the knowledge of the Wigner dis-

tribution function by the inverse Fourier transformation of Wf :

f(t)f ∗(0) = F−1ω→tWf(t/2, ω),

and it satisfies the so called ‘marginal’ conditions, interlinking it with temporal

and spectral energy densities:

∫Wf(t, ω) dω = |f(t)|2

∫Wf(t, ω) dt = |F (ω)|2 . (2.5)


Modulation and Convolution A new signal f formed by (pointwise) multi-

plication of two other signals f1, f2

f(t) = f1(t)f2(t) F (ω) =

∫F1(ω)F2(ω − ω′)dω′ (2.6)

has a Wigner distribution given by the convolution in the frequency domain:

W (t, ω) =

∫W1(t, ω)W2(t, ω − ω′)dω′. (2.7)

Symmetrically we have that a new signal which is a convolution of the two

signals

f(t) =

∫f1(t

′)f2(t− t′)dt′ F (ω) = F1(ω)F2(ω) (2.8)

has a Wigner distribution given by the convolution in the temporal domain:

W (t, ω) =

∫W1(t, ω)W2(t− t′, ω)dt′. (2.9)

WDF: localization The Wigner distribution function of a pure phase signal

f(t) = eiϕ(t) takes the following form:

f(t) = eiϕ(t) W (t, ω) ≃ δ(ω − ϕ′(t)) (2.10)


and becomes an exact equality for purely linear or quadratic phase.

Note that if ϕ(t) is a quadratic polynomial, then the algebraic relation

ϕ(t +

τ

2

)− ϕ

(t− τ

2

)= τ ϕ(t) (2.11)

holds exactly for all t and τ . From this, it is clear that the WDF provides the

ideal distribution for linear chirps (f(t) = eiϕ(t) = eiat2):

Wf(t, ω) =

∫f

(t +

τ

2

)f ∗

(t− τ

2

)e−iωτ dτ (2.12)

=

∫e−iτ(ω−ϕ′(t)) = δ(ω − ϕ′(t)).

WDF: time-frequency shift ‘covariance’ The time-frequency shifts of the

signal f , correspond to time-frequency shifts of the Wigner function Wf .

f(t) −−−→ f(t− t0) e−iω0t

yy

Wf(t, ω) −−−→ Wf (t− t0, ω − ω0)

Figure 2.2: WDF time-frequency shift covariance

WDF: chirp convolution and multiplicaton

eiat2f(t) −→ Wf(t, ω − at) (2.13)

eiat2 ∗ f(t) −→ Wf(t− ω/a, ω)

2.4. FOURTH (HIGHER-ORDERS ⊂ NONLINEAR ≡MULTILINEAR) REPRESENTATIONS 16

2.3.1 Pulse Characterization scheme

Walmsley and Wong [16] showed that all pulse characterization measurement can

be represented by a ’detection’ functional of the Wigner distribution:

D( pi ) =

∫ ∫dtdωW (t, ω)Wp(t, ω),

where Wp(t, ω) is the generalized window function (parametrized by the set pi =

Ω, τ ; ∆w, ∆t; φ′′, ϕ′′ (cf. Section 2.1) of the the instrument’s parameters .

2.4 Fourth (Higher-Orders ⊂ Nonlinear ≡ Mul-

tilinear) Representations

The higher-order time-frequency distributions, while becoming exceedingly com-

plicated objects to deal with, contain some useful properties and reveal some

deeper symmetries which prove beneficial certain applications (and which may

not be present in the linear or bilinear classes).

While the complete theory of these higher-order ’multi-linear time-frequency

representations’, ‘multi-linear correlation functions’ is a current research topic,

still being actively explored [33], in Chapter 5 we will examine a particular result

of a quartic (4th-order) time-frequency transformation to eliminate an unwanted

phase-characteristic of the interferometric signal.

2.5. CHAPTER SUMMARY 17

2.5 Chapter Summary

A very concise account is given of the general mathematical framework for repre-

senting and manipulating electromagnetic signals as well as describing the optical

interference phenomena in the time-frequency domain.

18

Chapter 3

Novel Pulse CharacterizationMethod: Theory

3.1 Introduction . . . . . . . . . . . . . . . . . . . 183.2 Simplified analysis in frequency domain . . . 223.3 Complete temporal analysis . . . . . . . . . . 283.4 Numerical Simulations . . . . . . . . . . . . . 373.5 ARAIGNEE reconstruction algorithm . . . . 423.6 Chapter Summary . . . . . . . . . . . . . . . 47

3.1 Introduction

The widespread use of ultrafast technology (cf. §1.1) and its enabling role in

numerous practical applications calls for the development of pulse characterization

instrumentation that is at least as robust as the current generation of laser sources.

As discussed previously in Chapter 1, all general schemes for measuring the

electric field envelope and phase of ultrashort optical pulses that rely on square-law

(≡ ‘envelope’, integrating, slow response) detectors must incorporate at least two

filters, one with a time-shift invariant response function, and one with a time-shift

variant response [16]. The specific arrangement of these filters analyzing the pulse,

and the method of reconstruction of the unknown pulse from the measured data

3.1. INTRODUCTION 19

allows the techniques to be separated into fundamentally distinguishable classes:

interferometric and non-interferometric and one rather well established and dis-

tinguished class of interferometric techniques, introduced in the next subsection,

will be considered in this Chapter.

3.1.1 (Conventional) SPIDER technique

Spectral phase interferometry for direct electric-field reconstruction (SPIDER)

[34,35] is a well-known accurate, precise, reliable and rapid self-referencing in-

terferometric technique for characterizing the ultrashort pulses (belonging to the

interferometric class (and more specifically, spectral shearing interferometric (SSI)

sub-class) of pulse characterization methods). Recording of the spectral interfero-

gram generated by a pair of spectrally shifted (≡ sheared) replicas of the measured

pulse is followed by a non-iterative algorithm to retrieve the pulse spectral ampli-

tude and phase [36] . Over the years, numerous variations of the original SPIDER

concept have been realized and multiple modifications to the original apparatus

and algorithm implemented, enabling complex pulse measurement, control and

optimization for a wider range of pulse power, repetition rates, bandwidths and

wavelengths [37–50].

In the original (conventional) SPIDER, two time-delayed replicas of the un-

known pulse are mixed with a highly linearly chirped ancillary pulse in a thin non-

linear crystal. The unknown pulses undergo a spectral shear when they upconvert

with different quasi-monochromatic slices of the ancilla. The delay between the


upconverted pulses induces fringes on the spectrum measured after upconversion,

which allow the direct (non-iterative), unambiguous extraction of the interference

term using the Fourier transform-based algorithm, as (compactly) illustrated in

Figure 3.1.

Figure 3.1: Conventional Spider encoding and reconstruction: (a: top row) The conventionalSPIDER schematic. The two delayed pulse replica upconvert with different frequency compo-nents of the chirped pulse and the resulting upconverted signal is measured in a spectrometer.(b: second row) The SPIDER inversion algorithm. After a Fourier transform of the spectralinterferogram into the pseudotime domain, filtering of the ac sidelobe and inverse Fourier trans-form back to the frequency domain, the argument of the result is a reference phase and thespectral phase difference between two points in the input spectrum separated in frequency bythe spectral shear Ω. Subtraction of the reference phase and concatenation of the remainderresults in a reconstruction of the pulse spectral phase.


The recorded SPIDER signal—spectral interferogram S(ω)—is given by:

S(ω − ω0) =∣∣∣E(ω − ω0) + E(ω − ω0 − Ω)e−iωτ

∣∣∣2

(3.1)

=∣∣∣E(ω − ω0)

∣∣∣2

+∣∣∣E(ω − ω0 − Ω)

∣∣∣2

+

× 2E(ω − ω0)E(ω − ω0 − Ω) cos[φ(ω − ω0)− φ(ω − ω0 − Ω)− ωτ ]

with the pulse spectral density∣∣∣E(ω − ω0)

∣∣∣2

centered at ω0, τ representing the

delay between the two pulse replicas, and in the limit of small spectral shear Ω,

the spectral phase difference term approximated ≃ Ω φ′(ω) and then integrated

to yield φ(ω).

3.1.2 Motivation (for the novel method)

For complete characterization of pulses with durations of less than 100 fs it is

necessary to use nonlinear optical processes in order to effect time-variant filtering

with the appropriate bandwidth. Common wisdom dictates that thin nonlinear

crystals must be used (to satisfy the broad-bandwidth requirement).

However, recently it has been shown how for several non-interferometric

schemes, a single thick nonlinear crystal enables the combined action of both

types of filters to be achieved simultaneously, leading to important practical de-

vice simplifications.

In spectrography, for example, the narrow phase-matching bandwidth of a

thick second-harmonic generating crystal can be used as a spectral filter, thereby

3.2. SIMPLIFIED ANALYSIS IN FREQUENCY DOMAIN 22

enabling the sum-frequency generation process to act simultaneously as a time-

gate and a spectrometer [51]. This idea has been incorporated into a compact

format for frequency-resolved optical gating (FROG) [52] as well as for sonogram

measurement [53].

The aim of this Chapter is to show how a thick nonlinear crystal with an

appropriately tailored phase-matching function can be utilized in spectral shearing

interferometry pulse characterization [54,55]. In the proposed application, too,

the nonlinear interaction performs two functions (acts as a time-variant filter),

though the precise filter response is different than those in spectrography and

sonography approaches). The idea leads to a considerable simplification of the

measurement instrument.

3.2 Simplified analysis in frequency domain

In this section we demonstrate that the necessary filtering operator is achieved

by allowing the test pulse to propagate in a single long nonlinear crystal, whose

phase matching function is arranged so that the ordinary wave has a very large

acceptance bandwidth, whereas the extra-ordinary wave has a narrow bandwidth.

This means that there is no need to arrange for a chirped ancillary pulse, since the

narrowband ancilla is selected directly from the test pulse by the phase-matching

function (PMF) of the nonlinear crystal.

(We will informally refer to this scheme as ‘ARAIGNEE’ for now (the word


means ‘SPIDER’ in French), and properly introduce a spirited acronym in the next

chapter, when discussing the actual experimental device based on the principle).

Figure 3.2 shows the theoretical PMF magnitude for optical fields traveling at

two angles tilted by ±0.25 away from the cut angle through a 2 cm long type II

KDP crystal cut (θc = 67.76) for maximum collinear upconversion at 830 nm.

The particular combination of the crystal’s material, length and the wavelength

range (Figure 3.2 shows 800–860 nm) produces a nearly vertical PMF which is

simultaneously very broad along the ordinary axis and very narrow along the ex-

traordinary axis. Such highly asymmetric PMF shape is due to a group velocity

match between the o-fundamental input and the e-upconverted output field and a

group velocity mismatch between the e-fundamental and the e-upconverted fields.

A KDP crystal wavelength coverage is comparable to the tuning range of ultra-

short Ti:sapphire lasers, while other types of crystals have been found to satisfy

the above-mentioned requirement in different wavelength regions [56]. For an ul-

trashort pulse with the spectrum located in the 830 nm region, the unique shape

of the PMF shown in Figure 3.2 allows for the entire bandwidth of the o-wave to

convolve with a quasi-monochromatic portion of the e-wave spectrum as selected

by the phase matching function. The precise angle of propagation relative to

the crystal’s optic axis (OA) determines the exact monochromatic slice of the e-

wave bandwidth which upconverts with the entire o-wave bandwidth. Thus, if two

copies of a pulse are directed into the crystal, altering their respective propagation

angles produces a spectral shift between the upconverted outputs.


Figure 3.2: The absolute magnitudes of the collinear, type II PMF, sinc2(∆kL/2), ∆k(ωe, ωo) =ke(ωe + ωo)− ke(ωe)− ko(ωo) of a L=2 cm thick KDP crystal for two (different by 0.5) valuesof the propagation angle, plotted as a function of frequency for ordinary ωo and extraordinaryωe input polarization components (black indicating perfect phasematching). The sum-frequencysignals are drawn on the diagonal axis, ωs = ωe +ωo, illustrating the spectral shear between theoutputs.

(Continuing with the theme of analogies and interconnections, we note that

such asymmetric phase matching function has been used in the process of spon-

taneous parametric down-conversion for generation of non-classical light and its

unique properties found several applications in the two-photon quantum state

engineering [56].)

In a sufficiently long nonlinear crystal oriented for type II sum frequency gen-

eration (SFG) the incident pulse propagating as an ordinary wave (o-wave) has


a large acceptance bandwidth, whereas the extra-ordinary wave (e-wave) has a

much narrower acceptance bandwidth. This highly asymmetric PMF shape is

due to a group velocity (GV) match between the o-fundamental input and the

e-upconverted output and a group velocity mismatch between the e-fundamental

and the e-upconverted fields [56]. As a result, the ordinary test pulse is upcon-

verted with a single e-ray frequency resulting in its replication at the upconverted

frequency. The angle of propagation relative to the crystal optic axis determines

the frequency of the narrowband component of the e-wave which upconverts with

the entire spectrum of the o-wave, enabling the spectral shear necessary for SSI.

However, this simple argument does not provide enough details to enable the full

range of operation to be derived and we have therefore developed a more compre-

hensive wave mixing model.

In the frequency domain, the reasoning above can have a physically intuitive

description as follows. We represent the complex amplitude of the input test pulse,

E(t), in the spectral domain by a Fourier transformation, E(ω) =∫

E(t) eiωtdt.

We are interested in obtaining a spectrally shifted replica of the input: E(ω) −→

E(ω − Ω), where Ω designates the spectral shear. Considering a χ(2) nonlinear

crystal, we can approximate the sum frequency signal, Es(ω) for two arbitrary


input fields, E1(ω1) and E2(ω2) as

Es(ω)∝∫∫

δ(ω1+ω2−ω)E1(ω1)E2(ω2)Φ(ω1, ω2)dω1dω2

=

∫E1(ω − ω2)E2(ω2)Φ(ω − ω2, ω2)dω2, (3.2)

where the phase-matching function of the interaction in the crystal is represented

by

Φ(ω1, ω2) = eiT sin(T )/T, (3.3)

where T (ω1, ω2) = [k1(ω1) + k2(ω2)− ks(ω1 + ω2)]L/2, L is the interaction length

and kj is the propagation constant of the j = 1, 2, s field.

For a conventional SPIDER device, which uses a very thin crystal, the PMF

can be approximated as Φ(ω1, ω2) ≃ 1 over the pulse’s bandwidth, and the SFG

process described by Eq. (3.2) is then equivalent to a convolution of the two fun-

damental fields

Es(ω)= E1(ω) ∗ E2(ω) =

∫E1(ω − ω2)E2(ω2)dω2. (3.4)

Therefore, if one of the fundamental beams is a quasi-monochromatic ancilla and

it can be modeled as a delta function, E1 ≃ δ(ω−Ω), then Eq. (3.4) represents an

exact spectrally shifted replica of the broadband input pulse: Es(ω) = E2(ω−Ω).

In order to visualize the main idea of ARAIGNEE, we must return to the

starting Eq. (3.2) and consider the case when the phase-matching function


Φ(ω1, ω2) can be written as a direct product of two one-dimensional functions,

Φ(ω1, ω2) = Φ1(ω1)× Φ2(ω2). This factorization of the PMF means that the sum

frequency field of Eq. (3.2) is now also a simple convolution of the two inputs,

each modified by its respective PMF component:

Es(ω)=(Φ1(ω)E1(ω)

)∗

(Φ2(ω)E2(ω)

). (3.5)

As a result, if we can arrange for the factorization to be of the form Φ1(ω) ×

Φ2(ω) = δ(ω−Ω)× 1, the output pulse is still a replica of one of the input pulses

(Es(ω) = E2(ω−Ω)) although the only requirement on the other one (E1) is that

it contains the frequency Ω.

Figure 3.2 shows exactly such a situation, where the PMF magnitude (|Φ|2)

has been plotted for optical fields traveling at two angles tilted by ±0.25 away

from the cut angle through a 2 cm long type II KDP crystal cut for maximum

collinear upconversion at 830 nm. The particular combination of the crystal’s

material, the cutting angle and the wavelength range produces a nearly vertical

PMF which is simultaneously very broad along the ordinary axis and very narrow

along the extraordinary axis. Such highly asymmetric PMF shape is the result of

a group velocity match between the o-fundamental input and the e-upconverted

output field and a group velocity mismatch between the e-fundamental and the

e-upconverted fields. Because of this specific PMF, the entire bandwidth of an

o-pulse, with the spectrum located in the 830 nm region, convolves with a quasi-

3.3. COMPLETE TEMPORAL ANALYSIS 28

monochromatic portion of the e-pulse spectrum as selected by the phase matching

function. The precise angle of propagation relative to the crystal’s optic axis

(OA) determines the exact monochromatic slice of the e-wave bandwidth which

upconverts with the entire o-wave bandwidth. Thus, if two copies of a pulse are

directed into the crystal, altering their respective propagation angles produces the

spectral shift between the upconverted outputs required by the spectral shearing

interferometry.

3.3 Complete temporal analysis

We will analyze the upconversion process of a single input beam R. We consider

type II interaction of two fundamental waves Ro and Re – collinear polarization

components of R(t) – respectively o- and e- polarized, and the up-converted wave

B generated by the sum frequency (SF) process in a dispersive dielectric medium

with χ(2) nonlinear susceptibility. Assuming the temporal complex amplitude en-

velopes Ro, Re and B to be slowly varying, we can derive from Maxwell’s equations

the system of three nonlinear partial differential equations coupled parametrically

through the components χ(2)ijk of the nonlinear susceptibility tensor [57],

i∂zRo(t, z) + i∆k′bro

∂tRo(t, z)− k′′ro

2∂ttRo(t, z) = −κro

R∗e(t, z)B(t, z) e−i∆kz,

i∂zRe(t, z) + i∆k′bre

∂tRe(t, z)− k′′re

2∂ttRe(t, z) = −κre

R∗o(t, z)B(t, z) e−i∆kz,

i∂zB(t, z)− k′′b

2∂ttB(t, z) = −κbRo(t, z)Re(t, z) ei∆kz, (3.6)


where the effects of diffraction and spatial walk-off have been neglected. The

variable z is the propagation distance and the variable t is time in a reference frame

traveling at the group velocity of SF pulse (k′−1b ) at the frequency ωb = ωro

+ ωre

where ωroand ωre

are the carrier frequencies of the two fundamental waves; κl =

(ω2l /2klc

2)χ(2)l (l = ro, re, b) is the nonlinear coupling coefficient; ∆k = kro

+kre−kb

is the wave vector mismatch, a shorthand introduced for the full dependence on

its arguments ∆k(ωre, ωro

; ωb = ωro+ ωre

) = kre(ωre

) + kro(ωro

) − kb(ωre+ ωro

);

∆k′bj = k′

j − k′b the group velocity mismatch while k′

l = ∂kl/∂ωland k′′

l = ∂k2l /∂ω2

l

are the respective inverses of the group velocity and the group-velocity dispersion.

3.3.1 Dispersionless medium

Let us first consider solving the system Eq. (3.6) assuming a dispersionless medium

and thus neglecting all thek′′j2

∂tt terms. If the nonlinear interaction is weak, the

fundamental waves propagate undistorted in the medium and the coupled system

of Equations (3.6) becomes separable and can be reduced to a single equation for

the SF wave:

d

dzB(t, z) = iκbRo(t−

z

∆k′−1bro

)Re(t−z

∆k′−1bre

− t0) ei∆kz, (3.7)

where t0 is an additional external parameter that we have explicitly introduced

into the model to include the most general upconversion situation where there ex-

ists an arbitrary delay adjusting the synchronization between the two fundamental


pulses upon entering the nonlinear medium at z = 0. A sketch of the modeled

process is presented in Figure 3.3. The group velocities of the two fundamental po-

larization components are different in general, and in a negative (positive) crystal,

the e-polarized fundamental pulse propagates faster (slower) than the o-polarized

one. Therefore, the predelay t0 has to be positive (negative) if the two pulses are

to interact in the nonlinear medium. In anticipation of things to come, Figure 3.3

depicts a situation in which the extraordinary wave travels faster through the

crystal than the ordinary wave. We have also implied that the group velocity of

the generated SF beam matches that of slower ordinary field. The complex enve-

Figure 3.3: The diagram of the modeled collinear upconversion for a type II sum frequencygeneration nonlinear process. Ro is the o- (solid (red) line) and Re is the e-polarized (dashed(red) line) test pulses (or polarization components of the same pulse) respectively and B (solid(blue) line) the sum-frequency pulse. t0 is an arbitrary pre-delay between the two fundamentaltest pulses and L is the thickness of an appropriately cut nonlinear crystal.

lope of the SF wave at the output of the crystal of length L is formally obtained

by integration of the right-hand side of Eq. (3.7) over the length of the crystal

from z = 0 to z = L:

B(t, L) = iκb

∫ L

0

Ro(t−∆k′bro

z)Re(t−∆k′bre

z − t0) ei∆kzdz. (3.8)


For the sake of simplicity, we restrict our discussion to negative crystals but

our discussion can straightforwardly be applied to positive crystal. Assuming

that the fastest pulse Re(t) does not overlap either before or after the crystal

and entirely walks through the slowest one Ro(t), we can replace the integration

boundaries of Eq. (3.8) by ±∞. In the experiment configuration, this can be

achieved by choosing a pre-delay t0 such that it exceeds the time support widow

(∆T ) of the input light (i. e. the pulses do not temporally overlap when entering

the crystal), and also selecting a crystal length such that condition∣∣∆k′

rero

∣∣ L >

(∆T +|t0|) is also satisfied (i. e. the crystal is sufficiently long for the input pulses’

interaction length∣∣∆k′

rero

∣∣ L to ’fit’ completely inside the nonlinear medium, away

and removed from its edges).

Replacing Ro by its frequency representation in the spectral domain:

Ro(t−∆k′bro

z) =1

2π

∫ ∞

−∞

Ro(ω) ei∆k′bro

zω e−iωtdω, (3.9)

the equation (3.8) becomes:

B(t, L) =iκb

2π

∫ ∞

−∞

Ro(ω) e−iωt

∫ ∞

−∞

Re(t−∆k′bre

z − t0)

× ei(∆k+∆k′bro

ω)zdzdω. (3.10)

In order to perform the integration over z, the following variable change,


τ = t−∆k′bre

z − t0, is introduced and Eq. (3.10) becomes:

B(t, L) =iκb

2π∆k′bre

∫ ∞

−∞

Ro(ω) e−iωt

∫ ∞

−∞

Re(τ)

× ei(∆k+∆k′bro

ω)[t−τ−t0]/∆k′bre dτ dω. (3.11)

Performing the inner Fourier integral with respect to τ , we get:

∫ ∞

−∞

Re(τ) e−i[(∆k+∆k′bro

ω)/∆k′bre

]τdτ = Re

(−∆k + ∆k′

broω

∆k′bre

),

and to solve Eq. (3.11), Eq. (3.12) is expanded in a series around ω = 0:

Re(−∆k + ∆k′

broω

∆k′bre

)

= Re(−∆k

∆k′bre

)− ∆k′bro

∆k′bre

ω Re′(− ∆k

∆k′bre

) + O(ω2), (3.12)

where ′ denotes a derivative with respect to the argument of Re(ω), O(ω2) desig-

nating the neglected higher order expansion terms of the series.

In a crystal with a tailored PM function having a nearly vertical shape, as

depicted in Figure 3.2, there is a group velocity mismatch between the two fun-

damental pulses (i.e. ∆k′rore6= 0) but a group velocity match between the o-

fundamental and the SF pulses (i.e. ∆k′bro≈ 0). As a result,

∆k′bro

∆k′bre

≪ 1 and only

the first term of the right-hand of Eq. (3.12) can be kept. Therefore Eq. (3.11) is


approximated by:

B(t, L) ≈ iκb

2π∆k′bre

Re(−∆k

∆k′bre

) ei∆k[t−t0]/∆k′bre

×∫ ∞

−∞

Ro(ω) e−iωt ei∆k′bro

[t−t0]ω/∆k′bredω. (3.13)

We can easily calculate the remaining exact Fourier transform of Ro(ω), and the

sum frequency pulse at the output of the crystal is given in the temporal domain

by:

B(t, L) =iκb

∆k′bre

Re(−∆k

∆k′bre

) ei∆k[t−t0]/∆k′bre

× Ro(t[1−∆k′

bro

∆k′bre

] + t0∆k′

bro

∆k′bre

). (3.14)

Eq. (3.14) is the main result of the Chapter. It holds for an arbitrary mod-

ulation of the envelope and phase of the input pulse. While the full expression

appears awkward due to its detailed nature (inclusive of all the parameters), we

can rewrite it in a more obvious form:

B(t) ∝ ei∆ωtRo(s t−∆t). (3.15)

It shows that after emerging from the crystal, the output SF pulse B(t) is an exact

replica (amplitude and phase) of the input Ro(t) pulse (i.e. the fundamental pulse

that travels in the nonlinear crystal with almost the same group velocity as the


SF pulse). This replica is a scaled version of Ro(t) with the scaling factor s:

s = 1− ∆k′bro

∆k′bre

. (3.16)

There is also a small frequency shift ∆w and an absolute time shift ∆t and we

will detail these quantities in the discussion below.

The scaling factor is equal to s = 1 when the group velocities of B and

Ro are perfectly matched, corresponding to the situation depicted in Figure 3.2.

Eq. (3.16) agrees with the results reported in Ref. [58], however our theoretical

analysis is more general, showing that the result above holds even if the funda-

mental wave Re(t) is not approximated by a δ(t) function, that is as long as the

second order and terms in Eq. (3.12) can be neglected.

Incidentally, we note that there is an intuitive understanding of the scaling

factor s via the frequency-frequency PMF plot interpretation of the upconversion

process (cf. Figure 3.2). Figure 3.4 shows the same interaction but centered,

correspondingly at lower and higher center frequencies, away from 830 nm. The

imperfect phasematching can be seen as a ‘tilt’ of the PMF curve away from the

vertical direction. The projection of that tilt on the sum frequency dimension

results in the scaling of the upconverted bandwidth.

The spectrum of the replica is centered around the frequency ωb = ωro+

ωre−∆k/∆k′

breand its amplitude is proportional to the amplitude of the spectral

density constant component R(ω)∣∣ω=−∆k/∆k′

bre

.


(a) λc = 750nm, (spectral range shown: 60 nm)

(b) λc = 950nm, (spectral range shown: 60 nm)

Figure 3.4: The absolute magnitude of the collinear, type II PMF tuned away from perfectphasematching.


Thus, the SF process has a physical interpretation as a waveform transfer from

the o-wave to the SF-wave by mixing a quasi-monochromatic slice of the spectrum

of the e-wave with the whole spectrum of the o-wave. The actual frequency of the

quasi-monochromatic slice is defined by the phase matching function.

The exact derivation from the PM function is as follows. Taking into account

only the lowest order terms in the power expansion of the wavenumbers about the

center frequencies [56]:

0 = ∆k + k′ro

(ω1 − ωro) + k′

re(ω2 − ωre

)− k′b(ω3 − ωb), (3.17)

if ω1 = ωro, the central frequency of the spectrum that is entirely up-converted,

ω2 = ωre+ ∆ω and ω3 = ωro

+ ωre+ ∆ω, Eq. (3.17) leads to:

∆ω = −∆k/∆k′bre

. (3.18)

This result is in agreement with the simplified picture of the SF generation dis-

cussed in the previous section on the basis of the phase matching function of

continuous waves [54].

From Eq. (3.14) it can be seen that, in the reference frame traveling at the

group velocity k′−1b , the SF pulse is temporally shifted by

∆t = −t0∆k′bro

/∆k′rore

. (3.19)

3.4. NUMERICAL SIMULATIONS 37

This result has been previously derived from the assumption that the SF pulse is

generated in the crystal at the location where the two fundamental pulses meet in

the crystal, and agrees well with the experiments [58] which will be the subject

of the next Chapter.

3.4 Numerical Simulations

In order to confirm our theoretical results, we have numerically simulated the

propagation of the pulses in the nonlinear crystal by solving the system of Equa-

tions (3.6) with a standard split-step beam propagating method [59,60]. Figure 3.5

Figure 3.5: Generation and evolution of the amplitude (top row) and the phase (bottom row)of the sum frequency signal B(t) as the two fundamentals propagate through the crystal. Am-plitude and phase of Ro (solid red line), amplitude of Re (dotted red line), amplitude and phaseof B (solid blue line). The three columns (from left to right) correspond to the entrance, centerand exit planes of the crystal.

shows the temporal amplitude and phase of the SF pulse at three different loca-

tions in the nonlinear crystal. We choose the initial conditions to simulate a gaus-

sian linearly chirped pulse: Ro(t) = e−(t/T )2 ei0.5(t/T )2 and Re(t) = 0.7Ro(t − t0),


where the pre-delay t0 = 4.3T is such that the two pulses do not overlap at z = 0.

The amplitude of Re(t) is chosen below that of Ro(t) to have the convenience

of distinguishing the two beams during the simulations. We have also assumed

a perfect group velocity matching between the o-fundamental and the SF waves:

∆k′bro

= 0 and neglected the higher order group velocity dispersion effects. In Fig-

ure 3.5, the leftmost column indicates the arrangement of the two fundamental

beams just before they enter the crystal (z = 0). The group velocity of the Re(t)

is smaller than the group velocity of the Ro(t) and the e-wave “walks-through”

the o-wave while the upconversion process takes place. The middle column is a

“snapshot” at the location where the two fundamental pulses meet in the crys-

tal (z = t0/∆k′rero

) and it can be seen that the phase profile of B(t) is different

from the phase of Ro(t). Actually, the phase profile of B(t) is a reproduction of

the phase profile of the o-wave in the right part of Ro(t), i.e. in the part that

has already experienced an almost complete mixing with the e-fundamental wave.

Moreover, since the mixing is not complete, the SF pulse is delayed relative to the

o-pulse even as the two frames (Ro and B) travel at the same group velocity. The

last column of the Figure 3.5 shows the same amplitudes and phases but at the

output of the crystal (z = 2t0/∆k′rero

). The complex amplitude of Ro(t) has been

completely transferred to the SF wave B(t). This example shows that a complete

waveform transfer from the fundamental wave to the SF wave is possible only if

the pulse Re walks completely trough Ro in the nonlinear crystal. This require-


PSfrag repla ements

Phase(rad)Intensity(a.u.) (tt)=T

a)b)012345

0000 01

2222-2-2-2-2-4-4-4-4-6-6-6-6 66664444 00.81.62.43.240.20.40.60.8Figure 3.6: Effect of a group velocity mismatch ∆k′

bro

= 0.15∆k′

bre

on the SF generation. Theinitial conditions are identical than for Figure (3.5). Intensity profile and phase of the SF pulse(dashed-line) and the o-fundamental pulse (solid-line) at the output of the crystal. The shiftedand scaled output SF pulse is also shown (circles).

ment imposes a limit on the width of the time support window within which the

pulse can be accurately reconstructed by such process.

Wavelength tunability In the spectral region where the group velocity of the

o-fundamental and the SF pulse are not matched (i.e. ∆k′bro6= 0), we have

seen that the SF pulse replicates the fundamental o-pulse up to a known time

axis scaling factor s and a known time shift, that only depend on the crystal

properties. This result is illustrated in Figure 3.6 for a group velocity mismatch

∆k′bro

= 0.15∆k′bre

. The dashed-lines show the intensity and the phase profile of

the SF pulse at the output of the crystal in the same conditions than for Figure 3.5.

It can be seen that if the SF pulse is stretched by the theoretical factor s = 0.87

and delayed by L∆k′bro

+ ∆t = 0.645T , the complex amplitude of the SF pulse

(circles) is a perfect replica of the o-fundamental pulse (solid-line).


3.4.1 Effect of group-velocity dispersion

PSfrag repla ements

Phase(rad)Intensity(a.u.) t (ps)

a)b)0

-5-10-155

01

2

-0.2-0.2-0.4-0.4 00 0.20.2 0.40.400.81.62.43.24

00.20.4-0.2-0.4 0.60.8Figure 3.7: Comparison of the amplitude and the phase of the SF-pulse at the output of a thickKDP crystal obtained by numerically solving the system (3.6) (solid-lines) and given by themodel Eq. (3.22) (circles).

In dispersive materials, the simple transfer function relation (3.14) may no

longer be accurate because it neglects GVD (and high order dispersion) at both

fundamental and SF frequencies. However, to the first order, the GVD can

straightforwardly be taken into account, since it leads to an extra chirp of k′′effL.

The mathematical derivation of this extra chirp is as follow. Assuming that the

SF pulse is generated in the crystal at the location l where the two fundamental

pulses meet, which can be calculated from their group velocities:

l = t0/(k′re− k′

ro), (3.20)

the effective GVD is broken down into two contributions:

k′′effL = k′′

rol × 1

s+ k′′

b (L− l). (3.21)


The first term of the right-hand side (RSH) of Eq. (3.21) accounts for the chirp

acquired by the o-pulse while traveling in the crystal to the location l. The

factor 1/s has been added because of the temporal scaling factor that appears in

Eq. (3.14). The second term describes the chirp acquired by the SF pulse from

the location l to the end of the crystal. Therefore the frequency representation of

the SF pulse at the output of the crystal can be approximated by combining the

results of Equations (3.14) and (3.21) and taking into account the frequency shift

(∆ω) from the carrier frequency of the SF pulse:

B(ω, L) ≈ B0(ω, L) exp(i

2k′′

effL(ω −∆ω)2), (3.22)

where B0(ω, L) is the frequency representation of the temporal signal Eq. (3.14).

The accuracy of this latest expression has been verified by comparing Eq. (3.22)

with the numerical simulation of the system Eqs. (3.6). A typical example is shown

in Figure 3.7 for a pulse with a complex temporal shape resulting from a Gaussian

spectrum centered around 760 nm (14.5 nm bandwidth FWHM) with quadratic

and cubic spectral phase components (respectively 200 fs2 and 3 × 104 fs3). The

physical properties of KDP crystal cut for type II second harmonic generation

at 760 nm have been used, resulting in a non zero group velocity mismatch

∆k′bro

= 0.117∆k′bre

. The length L of the crystal has been set to 10mm and

the delay t0 to 600 fs. Eventually, the angle of propagation has been tilted by

4mrad (0.23) from the phase matching angle for second harmonic generation at

3.5. ARAIGNEE RECONSTRUCTION ALGORITHM 42

760 nm to produce a frequency shift ∆ω = 4.1 radTHz (∆λ = 1.26 nm). The solid

lines in Figure 3.7 show the amplitude and the phase of the SF pulse at the out-

put of the crystal resulting from the sum frequency generation in the crystal and

obtained by numerically solving the system (3.6). The curves represented by the

circles correspond to the amplitude and the phase given by our model (Eq. 3.22).

As can be seen, the agrement between the numerical solution and our model is

very good even if the GVD is not exactly taken into account.

Disregarding the details of the upconversion process, the most important fea-

ture of the Eq. (3.22) is that the SF pulse is linked to the input pulse by a simple

linear transformation that does not depend on the shape of the input pulse but

only on the physical properties of the nonlinear crystal. This feature is crucial

if we want to implement a spectral shearing interferometer device for the tempo-

ral characterization of the ultrashort optical pulses based on the SF generation

process described above.

3.5 ARAIGNEE reconstruction algorithm

Conventional SPIDER devices measure spectral interferogram generated by a pair

of temporally delayed and spectrally shifted replicas of the measured pulses. In

the novel ARAIGNEE method, the spectral shear Ω is obtained by adjusting the

angle of propagation of the pulses in the nonlinear crystal (as analyzed in detail

in the previous section).


From Eq.(3.22), the SSI interferogram S(ω) is obtained after the SF pulses are

passed through the spectrometer and has a form given by

S(ω) = |B1(ω) + B2(ω) exp(−iωτ)|2

= |B1(ω)|2 + |B2(ω|2 + 2|B1(ω)||B2(ω)|

× cos[φ0(ω

s)− φ0(

ω − Ω

s)− δφ(ω) + ωτ ], (3.23)

where the subscripts (1, 2) distinguish the two SF pulses, φ0(ω) is the spectral

phase of the measured pulse (the information we wish to recover), and Ω = ∆ω2−

∆ω1 is the spectral shear.

S(ω) is a standard shearing interferogram consisting of fringes nominally

spaced in frequency at 2π/τ , while the phase term φ0(ω/s)−φ0([ω−Ω]/s)+δφ(ω)

manifests itself as a perturbation from the nominal fringe spacing. Since the two

SF pulses travel at different group velocity in the crystal, the time delay τ is not

the delay between these two pulses at the output of the crystal, but must be in-

terpreted as the delay between the pair of e- (or o-) polarized pulses in front of

the crystal. The extra time delay resulting from the birefringence as well as the

chromatic dispersion is included in the term δφ(ω) defined as:

δφ(ω) = (∆t1 −∆t2 + Ωk′′effL)ω, (3.24)

where the two first terms are defined by Eq. (3.19) and depend on the actual


angle of propagation of the two beams. The distortion of the interferogram S(ω)

resulting from the effective GVD experienced by the SF pulses while propagating

in the crystal is the latter term of Eq. (3.24), where the effective GVD has been

assumed to be identical for both beams.

The phase difference between the two SF pulses,

φ0(ω

s)− φ0(

ω − Ω

s)− δφ(ω) + ωτ, (3.25)

is extracted by the conventional Spider reconstruction algorithm (cf. Figure 3.1):

Fourier transforming the interferogram, filtering the peak around the pseudo-time

+τ and inverse Fourier transforming back to the frequency domain.

Exactly as for the conventional SPIDER interferogram, it can be seen

from (3.25) that knowledge of the shear and the reference phase ωτ−δφ(ω) is key

to recovering the spectral phase φ0(ω). It is simple to have access to the experi-

mental shear since each beam can be individually selected. The shear is therefore

given by the frequency shift between the two individual SF spectra. The reference

phase is usually obtained by recording an additional spectrogram without spectral

shear. In the ARAIGNEE implementation, zeroing the shear means cancelling the

angular tilt between the two beams and this cannot be done without changing the

delay between the pulses. As a result, the reference phase can not be recorded

at the SF frequencies. However, the reference phase can be extracted from the

interferogram between the pair of e- or o- pulses at the output of the crystal.


Extracting the reference phase from the interferogram with the same polarization

state as the SF pulses allows the spatial walk-off between the test pulse and the SF

e-wave signal to be minimized. The experimental procedure (as demonstrated in

next Chapter) requires, therefore, no intermediate alignment and the fundamental

and SF interferogram can be recorded either simultaneously [61] or separately [34].

For instance, in a KDP crystal, the phase matching as well as the group velocity

matching requirements are fulfilled for “oee” interaction around a wavelength of

830 nm. The reference phase is thus extracted from the phase difference between

the two fundamental e-pulses:

ωτ − δφ(ω) + δφcorr(ω), (3.26)

where δφcorr(ω) is defined as:

δφcorr(ω) = δφ(ω) + (∆k′1bre−∆k′2

bre)Lω. (3.27)

The linear phase obtained from calibration is simply subtracted from Eq. (3.25)

to give the phase difference φ0(ω/s)−φ0([ω−Ω]/s) plus a linear phase correction

−δφcorr(ω) that depends only on the geometry of the apparatus and the physical

properties of the crystal. The correction can be removed either before or after

the standard concatenation algorithm used to reconstruct φ(ω/s) from the phase

difference [62,36]. After concatenation, the correction function φcorr appears as a


quadratic phase. Indeed, if the integration approximation is used to evaluate this

function [62], then

φcorr ≈1

Ω

∫δφcorr(ω)dω = acorrω

2, (3.28)

with

acorr =1

2Ω[(∆k′1

bre−∆k′2

bre)L + ∆t1 −∆t2] +

1

2k′′

effL. (3.29)

It is better to remove the correction after the concatenation because although

δφcorr varies with the shear, acorr does not depend on the shear nor the exact

angle of propagation of the two beams, but only on the crystal properties, its

length L and the delay t0. This can be simply understood by pointing out that

for small angular tilt of the beams from the phase matching angle (θ), ∆ω, ∆t

and ∆k′bre

can be linearly expanded around θ so that both ratio [∆t1 − ∆t2]/Ω

and [∆k′1bre−∆k′2

bre]/Ω do not depend on Ω = ∆ω2 −∆ω1. Numerical evaluation

of (3.29) has shown that acorr does not vary more than 0.1% up to a shear of

40mrad fs−1 (3.6 nm at λ = 830 nm). Eventually after frequency scaling by the

factor s, the phase profile is the spectral phase profile of the unknown input pulse.

A measurement of the spectral density completes the pulse characterization. The

block diagram in Figure 3.8 summarizes the phase retrieval procedure.

The wavelength dependence of the scaling factor s for KDP crystal as well

as the quadratic phase factor acorr for a crystal length of 10mm and a delay

t0 = 600 fs have been plotted in Figure 3.9.

3.6. CHAPTER SUMMARY 47PSfrag repla ements Extra ted e-SFphase dieren e Extra ted e-fundamentalphase dieren eSubstra tion on atenation0(!s ) a orr!2Add the phase orre tion0(!s )Frequen y s aling0(!)Figure 3.8: Block diagram of the phase retrieval procedure in ARAIGNEE.

3.6 Chapter Summary

We presented a detailed analytical and numerical analysis of the pulsed fields in-

teraction in type II SFG and used it to explore the performance of a proposed

instrument, including reconstruction of the ultrashort pulses with spectral band-

widths and central frequencies at the limits of the theoretically accessible param-

eter ranges.


PSfrag repla ements

(µm)S alingfa tor(s) a)

0.75 0.8 0.85 0.90.850.90.9511.051.11.15PSfrag repla ements

(µm)a orr(ps2 ) b)1050.75 0.8 0.85 0.9-11-10-9-8-7-6-5

-4-3Figure 3.9: Scaling factor s (a) and quadratic phase factor acorr (b) as a function of the centralwavelength of the unknown pulse for KDP. The parameters used in Figure b) are: crystalthickness L=10mm, delay t0 = 600 fs.

49

Chapter 4

Novel Pulse CharacterizationMethod: Experiments and

Results

4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . 494.2 Experimental Implementation . . . . . . . . . . . . 504.3 Implementation Results and Discussion . . . . . . 534.4 Error Analysis and Discussion . . . . . . . . . . . . 584.5 Chapter Summary . . . . . . . . . . . . . . . . . . 62

4.1 Introduction

In the preceding Chapter we presented an extensive theoretical analysis of the op-

erational principle for the novel pulse characterization invention. In simple terms,

it was proposed that sending an unknown test femtosecond pulse through a long

nonlinear crystal at two respectively slightly different angles and then recording

the resultant spectral interferogram is sufficient for performing SSI reconstruction

on the test pulse. In this Chapter we describe the experimental implementation of

the foretold principle. The objective is to introduce a simple, compact and robust

implementation of spectral shearing interferometry (SSI) using a single nonlinear

crystal for both ancilla generation and upconversion and demonstrate its capa-

4.2. EXPERIMENTAL IMPLEMENTATION 50

bility and performance for a range of pulses of different bandwidths and center

wavelengths.

In keeping with established practice in the field, we name such implementation

ARAIGNEE (Another Ridiculous Acronym for Interferometric Geometrically-

simplified Non-iterative E-field Extraction) [63,55].

4.2 Experimental Implementation

Figure 4.1 displays the schematic of the latest compact ARAIGNEE design (pre-

ceded by several earlier designs which were used as proof of principle prototypes

and were not as reliable nor compact). The linearly (horizontally) polarized in-

put pulse to be measured passes through a zero-order λ/2-waveplate (optic axis

at 22.5 with respect to horizontal) and a quartz plate (10 mm thick, slow axis

horizontal), these two elements splitting it into the ordinary and extraordinary

polarizations, with the e-wave being pre-delayed by 317 fs (in the quartz plate)

with respect to the o-wave enabling a distortionless upconversion in the crystal

[64].

The beam is subsequently sent onto a pair of mirrors adjacent to each other

with a small offset d along the beam propagation direction, and a small mutual

horizontal tilt β that splits the incident beam into two beams (see detail in Figure

4.1). The two beams are directed into a KDP crystal (5 mm thick, cut for second

harmonic generation at 830 nm, optic axis horizontal) where each beam undergoes


Figure 4.1: Schematic of the ARAIGNEE device (top-view). WP, half-wave plate which splitslinearly polarized input into ‘o-’ and ‘e-’ components; Q, quartz plate which introduces thepre-delay t0 (≃300 fs); MP, mutually tilted (by β) and longitudinally shifted (by d) mirror pairwhich introduces the delay τ ≡ d/c (≃ 1.5 ps); PM, pick-off mirror; KDP, nonlinear crystal(with its optical axis set horizontally, in the plane of the diagram). (Dotted red line pulse-shaperepresents ordinary polarization (perpendicular to the diagram plane) and solid red, and blueline pulse-shape represents extraordinary polarization (in the plane of the diagram).

type II sum frequency generation. The resulting SFG pulses are spectrally shifted

(sheared) due to the angular offset (2β) of the fundamental beams in the crystal.

In our experiment, the angle β has been set to 0.25 resulting in a spectral shear

of about 0.8 nm. No great accuracy is necessary in setting the angle β since the

actual value of the spectral shear is easily measured from the two individually

recorded SFG spectra. The delay between the SFG pulses (≃ 1.5 ps), required

for recording the SPIDER interferogram, is achieved by setting the offset d to

≃ 225µm. At the output of the crystal, the two beams are overlapped with a 10

cm lens onto the entrance slit of a compact grating spectrometer (USB2000, Ocean

Optics). (If required, the existing quartz window and nonlinear KDP crystal set,

can be easily substituted with a different pair.) The entire arrangement easily fits


onto a 20×20 cm breadboard, is easy to align and produces a standard SPIDER

interferogram (encoding of the unknown spectral phase) (cf. Figure 4.2).

Figure 4.2: Photograph of the compact ARAIGNEE design. WP, half-wave plate; MP, mirrorpair; PM, pick-off mirror; FM, folding mirror (to spectrometer); Xtal, nonlinear crystal.

4.2.1 Measurements: Calibration

Two parameters must be known for the simple, non-iterative SSI reconstruction

algorithm: the spectral shear Ω and the time delay between the two input replicas

τ . The spectral shear value is easily obtained by independently measuring the

blue SFG spectrum from each beam and comparing the two spectrally shifted

profiles. The angle between the beams can be used to adjust the shear value

and is typically set in the 0.2 − 0.5 range in our experiments, leading to the

spectral shear values of 4−10 radTHz (0.35−0.9 nm at 415 nm). The delay value

4.3. IMPLEMENTATION RESULTS AND DISCUSSION 53

is established by first measuring the interferogram of the horizontally polarized

portions of the original input test pulse which pass through the crystal without

upconversion and then correcting the extracted τ by a factor which takes into

account the material dispersion in the long crystal experienced by the upconverted

pulses. Alternatively, the delay calibration is performed with the simultaneously

recorded interferogram from the two pairs of extraordinary red and blue pulses

[61]. In either case, the experimental procedure requires no intermediate alignment

as there is no spatial walk-off between the horizontal (extraordinary) components

of the test pulse and the upconverted e-wave signal.

4.3 Implementation Results and Discussion

4.3.1 Measurements: Proof of Principle

The performance (accuracy) [62] of the novel technique was established by its

comparison with a conventional SPIDER apparatus by using both devices to char-

acterize complex pulses sculpted in a spatial light modulator (SLM) [65]. (This

pulse shaper consists of a two lens 4-f Fourier processor setup: a pair of f=50cm

convex lenses, programable acousto-optic modulator (AOM) in the Fourier plane,

and a pair of 1200 lines/mm gratings in the conjugate ‘object’/‘image’ planes) [66].

Figure 4.3 displays the reconstruction results for a test pulse with a sinusoidally

modulated spectral phase profile which was obtained by sending the pulse through

the SLM. The recovered phases agree very well over the central bandwidth with


some discrepancy appearing in the regions of mid and lower spectral density. Low

signal level is responsible for the conventional SPIDER phase somewhat deviating

from an ideally sinusoidal profile recovered with a more sensitive (due to longer

crystal) ARAIGNEE.

Figure 4.3: Spectral amplitude (line with squares) of the test pulse and its sinusoidally modulatedspectral phase reconstructed by a conventional SPIDER (solid line) and a novel long crystal(dashed line) method.

The accuracy of the technique has also been verified against a ‘control’ spec-

tral phase by measuring the dispersion accumulated by a 70 fs pulse (FWHM)

centered at 830 nm after propagation through a test block of 10 cm BK7 glass. A

comparison between the theoretical curve and measured data is shown in Figure

4.4. The spectral phases agree very well over the entire non-zero spectral density

region of the test pulse and the measured value of group delay dispersion 4160 fs2

is consistent with the theoretical value of 4175.5 fs2 (0.3% discrepancy). The pre-

cision [67] of the measurements of the novel technique should be equivalent to

that of the conventional SPIDER device.


Figure 4.4: Spectral amplitude (line with squares) and phase (dots) measured by the device andthe theoretical spectral phase (solid line) after propagation through 10 cm of BK7 glass.


4.3.2 Measurements: Spectral Tunability and Bandwidth

Two femtosecond laser sources were used to evaluate the performance of the de-

vice:

1. Tunable “Mai-Tai” (Spectra-Physics) delivering ≃ 70 fs (spectral intensity

FWHM: ∆ω ≃ 35mrad fs−1, ∆λ ≃ 10 nm) pulses centered in the 750–

850 nm range, and

2. “Mira Seed” (Coherent) providing shorter and correspondingly broader

bandwidth (∆ω ≃ 80mrad fs−1, ∆λ ≃ 25 nm) pulses.

The input beam waists at the mirror-pair were ≃ 3−5mm (no beam expansion

or focusing was necessary). With ARAIGNEE’s refresh rate of a few Hz (which

is mostly limited by the integration time of the spectrometer), the average input

power could be attenuated to ≃ 1mW before signal dropped to the noise floor

level. The results of measurement and reconstruction of the spectral phase for

pulses of different bandwidth and center frequency are demonstrated in Figure 4.5.

The solid line plots show the retrieved spectral phase added to three distinct test

pulses after propagation through different blocks of the BK7 glass indicating a

very good agreement between the retrieved and calculated (from the Sellmeier’s

coefficients) data.

The plots in Figure 4.5 represent the following cases: a) pulses with the time

support close to the theoretical limit. Indeed, after propagation through a test

block of 10 cm BK7 glass, the output pulse of the Mai-Tai laser is stretched to


PSfrag repla ements

!(mradfs1)Intensity(a.u.) Phase(rad

)-100-100-100 -50-50-50 00 000 0 505050 1001001000.20.40.60.8

111

22334455

(a) λc = 830nm, ∆ω = 35mrad fs−1, 100mm BK7

PSfrag repla ements


)-100-100-100 -50-50-50 00 000 0 505050 1001001000.20.40.60.8 111 22

(b) λc = 760nm, ∆ω = 35mrad fs−1, 28.5mm BK7

PSfrag repla ements


)-100-100-100 -50-50-50 00 000 0 505050 1001001000.20.40.60.8

111

22334455

(c) λc = 830nm, ∆ω = 80mrad fs−1, 9.5mm BK7

Figure 4.5: Measured (solid lines) and calculated (dashed lines) spectral phase and measuredspectral intensity for different central wavelengths and bandwidths of the input pulse.

4.4. ERROR ANALYSIS AND DISCUSSION 58

160 fs FWHM, while the maximum time window for the 5 mm thick crystal and

10 mm thick quartz block used in the experiment is 317 fs, that is, only twice the

pulse duration FWHM; b) pulses with central wavelength of 760 nm, close to the

lower limit of the phase matching range (730 nm) and well outside the perfect

GV matching region. As can be seen, accounting for the theoretical scaling factor

s = 0.88, the agreement between the theory and the experiment is very good;

c) pulses of 80 mrad fs−1 bandwidth, corresponding to 30 fs transform-limited

pulse duration.

As an independent cross-check, the input pulse spectral phases were also mea-

sured (confirmed) with a conventional SPIDER apparatus and Fig. 4.6 shows (a

somewhat redundant set of results): the spectral phase (and corresponding tem-

poral phase) reconstruction for the “Mai-Tai” laser pulses of different central fre-

quency. Once again, the comparison with the spectral phase measured with a

conventional SPIDER apparatus [36] (solid lines) shows an excellent agreement

between the ARAIGNEE and the SPIDER measurements.

4.4 Error Analysis and Discussion

The effect of the pulse propagation, taking material dispersion into account, in a

20 mm nonlinear KDP crystal (with a predelay t0 = 1440 fs such that the o- and

e- pulses meet half way in the crystal) on the spectral phase reconstruction was

studied by solving numerically (using split-step algorithm) the system of three


Figure 4.6: Left: Spectrum of the test pulse (dotted line) and its spectral phase returned byARAIGNEE (dashed line) and conventional SPIDER (solid line) for different central wave-lengths. Right: Time-dependent intensity and phase measured by ARAIGNEE (circles) andSPIDER (solid line) from the data plotted on the left.


nonlinear coupled Equations (3.6). To quantify the errors in the reconstructed

phase, following [68], the spectral weighted phase error was used, defined as:

εφ =[∑

ω I(ω)2(φInput(ω)− φSpider(ω))2]1/2

[∑

ω I(ω)2]1/2. (4.1)

We demonstrate the parameter range over which the input pulse can be accurately

reconstructed by plotting this error for different central wavelengths and pulse

bandwidths in Figure 4.7. The phase error appears symmetric around the perfect

phase matching wavelength of 830 nm and for a transform-limited pulse remains

below 0.01 (a very conservative accuracy limit) up to a bandwidth of 120 mrad fs−1

(20 fs). The error does not exceed ∼ 0.01 up to 40 mrad fs−1 bandwidth for

the central wavelength between 740–900 nm, demonstrating the accurate pulse

reconstruction over the broad spectral range. The calculated spectral intensity

error [∑

ω |IInput(ω)−ISpider(ω)|2]1/2, where the intensities are normalized to unity,

is effectively negligible, being below 0.2 % over the entire parameter SPACE shown

in Figure 4.7.

In Table 4.1 we show the experimental spectral phase errors for the Figure

plots 4.5(a) - 4.5(c) along with the theoretical values predicted by the numerical

simulation of the SF generation in a 5 mm KDP crystal. The experimental spec-

tral phase errors for the plots (a)–(c) of Figure 4.5 are respectively 0.024, 0.005

and 0.017, while the theoretical values predicted by the numerical simulation of

the SF generation in a 5 mm KDP crystal are 0.022, 0.003 and 0.007. The com-

parison of the εφ values between the three cases is somewhat misleading as the εφ


PSfrag repla ements

! (mradfs1) (µm)

z = 0z = L

0.001 0.005 0.01 0.050 40 80 120160

0.740.780.820.860.9

Figure 4.7: Spectral phase error εφ of the retrieved pulse calculated from a numerical simula-tion of the pulse propagation in the nonlinear crystal plotted as a function of the input pulsebandwidth (intensity FWHM) and central wavelength. Gaussian transform-limited input pulseis used.

magnitude depends on the added dispersion (the amount of glass the beam has

traveled through), which is different for each of three cases. However, the com-

parison between the experimentally measured and numerically predicted εφ values

shows their nominal agreement, with the experimental values slightly exceeding

the ideal case results. We note that the errors are minimal on the absolute scale.

When matched with the results of Figure 4.5, the table provides a qualitative

understanding of the reconstruction performance metric εφ and can be used as

as an intuitive guide in physically interpreting the simulation results presented in

Figure 4.7.

Limitations of ARAIGNEE As mentioned before, the two fundamental

pulses must be predelayed so that they entirely walk through one another while

propagating in the crystal. This imposes the limits on the measured pulse dura-

4.5. CHAPTER SUMMARY 62! n 760 nm 830 nmth. exp. th. exp.35 (mrad=fs) 0.0034 0.0049 0.022 0.02480 (mrad=fs) - - 0.0069 0.017Table 4.1: Experimental (exp.) and theoretical (th.) spectral weighted phaseerror εφ for the three pulses presented in Fig. 4.5.

tions and bandwidths. For a given crystal length L, the maximum time support

window δt of the pulse to be measured accurately is δt ≃ |∆k′rore|L/2. This is

the equivalent of the requirement for a conventional SPIDER that the unknown

pulse is mixed in the nonlinear crystal with a quasi-monochromatic slice of the

stretched pulse. On the other hand, the maximum pulse bandwidth is set by the

requirement that all spectral components of the ordinary fundamental pulse travel

slower than the spectral components of the extraordinary fundamental pulse, i.e.

δω << |(|k′ro|− |k′

re|)|/|k′′

re|, where δω is the total spectral bandwidth of the pulse.

For a 5 mm KDP crystal, the maximum time window is 360 fs and the bandwidth

is limited to 480 mrad fs−1 for the central wavelength of 830 nm which agrees very

well with the error analysis results.

4.5 Chapter Summary

In summary, we have demonstrated a novel implementation of SSI for direct elec-

tric field reconstruction of ultrashort pulses. The device uses the asymmetric

group velocity matching condition present in type II upconversion to produce


spectrally sheared pulse replicas. The technique eliminates the need for an exter-

nally generated chirped pulse leading to a simplified construction and alignment

of the device with fewer components while retaining all the traditional advantages

of the SPIDER reconstruction algorithm.

64

Chapter 5

A Method for DispersionCompensation

5.1 Introduction . . . . . . . . . . . . . . . . . . . 645.2 The dispersion compensation method . . . . . 66

5.2.1 Generalized autoconvolution: theprinciple . . . . . . . . . . . . . . . . . 67

5.2.2 Generalized autoconvolution: the de-tails . . . . . . . . . . . . . . . . . . . 71

5.2.3 Analysis in the time-frequency domain 735.3 Chapter Summary . . . . . . . . . . . . . . . 77

5.1 Introduction

Since its inception twenty years ago [69], a variant of an optical ranging technique,

low-coherence optical interferometry, most commonly known as Optical Coher-

ence Tomography (OCT) has enjoyed a tremendous interest, gradually evolving

into an increasingly effective and promising non-invasive imaging tool, particu-

larly in biological and biomedical applications [70–72]. There is an enormous

amount of literature on the subject. Restricting ourselves to a purely academic

(non-commercial) domain, the success and the popularity of the technique can be


glimpsed from a topical search 1, producing over 10,200 peer reviewed articles on

“optical coherence tomography”. There is no doubt new progress continues and

will continue to be made.

While the comprehensive review of all OCT methods and extensions is im-

possible, in the simplest of terms, its principle of operation relies on using the

temporal (longitudinal) coherence length of a light beam as a gauge of the struc-

tural depth/distance information in the sample. The basic interferometer used in

the experiments is shown in Figure 5.1. The output of a broadband source is cou-

pled into a Michelson interferometer (typically with a single 50:50 beamsplitter).

The reference arm contains a retroreflecting mirror is longitudinally scanned in a

precisely controlled fashion. The other arm contains the sample under test. The

intensity of the interference signal as a function of the delay is recorded with a

photodetector and is used for the depth profile reconstruction of the sample.

t

( )ref 2E t t-

( )sampleE t

( )I t

Test sample

Broadband

Source

Figure 5.1: The basic schematic of an OCT interferometer.

The value of the depth resolution of OCT is inversely proportional to the spec-

1ISI Web of Science: Science Citation Index Expanded (http://portal.isiknowledge.com/)

5.2. THE DISPERSION COMPENSATION METHOD 66

tral bandwidth of the light source being employed for the sample illumination. For

a Gaussian spectrum it is given ∼λ2/∆λ, (√

2 ln(2)/πλ2/∆λ to be precise), where

∆λ is spectral full width half maximum (FWHM) of the source. An obvious way

of enhancing the depth resolution of the method is to use light sources of broader

spectral bandwidth — that is as long as the optical dispersion of the medium

can be neglected. The recent rapid development and commercial availability of

ultrabroad bandwidth light sources (e.g. Kerr-lens mode-locked lasers and super-

continuum light sources) has recently enabled a significant improvement in OCT

imaging resolution [73–76]. However, for the increasingly broadband light sources

employed to achieve ultrahigh axial resolution dispersion becomes highly relevant,

blurring out the interference modulation in the measured signal and thus leading

to the loss of resolution.

5.2 The dispersion compensation method

The aim of this chapter is to present a novel numerical method for compensating

dispersion effects in optical coherence tomography that does not require a priori

knowledge of dispersive properties of the sample [77]. Dispersion compensation

is accomplished by processing phase information present in standard interfero-

grams to calculate the generalized autoconvolution function. The operation of

the method can be conveniently visualized using the Wigner distribution function

formalism. The method is analogous to and yields results equivalent to that of


recently demonstrated quantum-optical coherence tomography, but without ex-

ploiting nonclassical states of optical radiation.

5.2.1 Generalized autoconvolution: the principle

Generally, two strategies have been developed to combat dispersion effects in OCT.

The first one consists of matching experimentally the amount of dispersion in both

the arms of the interferometer either with a simple introduction of an appropriate

compensation plate in the reference arm [78], or by a skilful design of the scanning

arrangement for the reference beam [79,80]. The second strategy is to postpro-

cess numerically complete fringe-resolved interferograms to compensate for the

dispersion-induced loss of resolution [81–84]. The numerical approach requires

usually a priori knowledge of the dispersive properties of the medium which enter

as the parameters of the compensation algorithm. The dispersion parameters of

the measured system can also be guessed by optimizing a criterion quantifying

the sharpness of the reconstructed profile [85]. An intriguing alternative to these

methods has been recently proposed [86] and demonstrated [87] in the form of

quantum-optical coherence tomography (QOCT), which uses pairs of photons en-

tangled in their spectral degree of freedom. When the photons are anticorrelated

in their frequencies, two-photon interference is insensitive to dispersion experi-

enced by one of the photons. This effect, known as dispersion cancelation [88],

is at the core of QOCT, and allows one to obtain experimentally dispersion-free

profiles without compensation. We discuss here mathematical properties of the re-


constructed depth profiles, which apply also to QOCT owing to the mathematical

equivalence of the two methods.

Before passing on to the detailed discussion of the method, we shall present

its principle of operation using an elementary example. The input for our sub-

sequently demonstrated method is the complex-valued envelope Γ(τ) of the full

analytical mutual coherence function parameterized by the delay (τ) of the ref-

erence beam. While this full ‘un-filtered’ interferogram is not typically used in

standard OCT, it is is still easily experimentally accessible [81,83,84].

The dispersion-compensated depth profile is obtained by calculating what we

call generalized autoconvolution of Γ(τ). This quantity is introduced and defined

as:

Ξw(τ) =

∫dτ ′e−2w2τ ′2Γ∗(τ + τ ′)Γ(τ − τ ′). (5.1)

The parameter w, not related directly to the dispersion characteristics of the

medium, is used to tune the performance of the method, and it will be typically

much smaller than the light bandwidth.

The autoconvolution function defined above can reveal the axial structure of a

sample with a better resolution than the interferogram itself, as it approximately

scales as the square of |Γ(τ)|). We illustrate this with Figure 5.2(a,b), where

we present a reconstruction of an exemplary depth profile for a pair of reflective

surfaces embedded in a dispersive material and illuminated with a broadband

light.


First, let us consider the scenario when no attempt to compensate for disper-

sion effects has been made. Figure 5.2(a) shows the absolute value of the envelope

|Γ(τ)| which is the most basic way to retrieve the depth profile. This profile is

severely broadened by dispersion. For comparison we also plot in this graph a

profile that would be obtained for a narrower bandwidth of the probe light, giving

the optimal resolution of |Γ(τ)| for this specific medium when no dispersion com-

pensation is included. These two plots should be contrasted with Figure 5.2(b)

which shows the autoconvolution function calculated according to Eq. (5.1) for

several values of the parameter w. It is seen that the autoconvolution function

reveals the location of the reflective surfaces as two sharp peaks with a resolution

comparable to the coherence length of the probe light itself. Additionally, the

convolution function contains a spurious artefact located half-way between the

peaks. Its presence is a result of numerical interference of signals reflected by the

two surfaces, but its magnitude can be quickly suppressed by increasing the value

of the parameter w. The suppression is accompanied by a slight broadening of

the genuine peaks in the depth profile, which is however significantly less severe

than that affecting |Γ(τ)|.


0 20 40 60 80

0

0.5

1.0

1.5

-0.5

-1.0

(b)

(a)

(c)

Dep

th p

rofi

le [

arb

. u

nit

s]Ω

[µm

]-1

Relative delay [ m]µ

0

0.5

1

-1.5

0

0.5

1

Figure 5.2: Reconstruction of a depth-profile of a pair of equally reflective surfaces precededby 2 cm of dispersive aqueous region with group velocity dispersion 15 fs2/mm: (a) standardinterferograms |Γ(τ)| for a coherence time T = 1.5 µm (solid line) and T = 5.2 µm (dashedline), the latter giving optimal resolution with no dispersion compensation; (b) autoconvolutionfunction Ξw(τ) calculated from Γ(τ) with T = 1.5 µm for w = 0.015 µm−1 (dotted line), w =0.06 µm−1 (dashed line), and w = 0.12 µm−1 (solid line); (c) the chronocyclic Wigner functionW (τ, Ω) of the complex interferogram envelope Γ(τ). For convenience, time and frequency havebeen expressed in length units using the vacuum speed of light. The profiles in graphs (a) and(b) have been renormalized to the same height in order to make the comparison of their widthseasier.


5.2.2 Generalized autoconvolution: the details

Let us now discuss in detail properties of the autoconvolution function Ξw(τ)

as a tool for reconstructing depth profiles. We consider a standard OCT setup

in which broadband light, characterized by the power spectrum S(ω) centered

around a frequency ω0, is split into two beams. One signal beam is reflected off

the sample, thus acquiring in the spectral domain the response function of the

sample (ω), whereas the second reference beam undergoes a controlled temporal

delay 2τ . Interference between these two beams yields the mutual coherence

function, given by:

Γ(τ)e−2iω0τ =

∫dω S(ω)(ω)e−2iωτ . (5.2)

In this formula, we separated out the phase factor rapidly oscillating with the

central optical frequency ω0, and denoted the resulting slowly-varying complex-

valued envelope as Γ(τ).

We shall model the spectral response function (ω) as composed of discrete

contributions coming from reflective surfaces within the sample characterized by

reflection coefficients rn: (ω) =∑

n rne2iφn(ω). The phase φn acquired by the

signal field reflected from the nth surface can be expanded around the central

frequency of the probe light up to the quadratic term:

φn(ω) ≈ φn(ω0) + (ω − ω0)τn +1

2(ω − ω0)

2Dn + . . . (5.3)


We shall incorporate the constant phase φn(ω0) into the reflection coefficient rn.

The parameter τn multiplying the linear term characterizes the position of the

nth reflective surface, whereas Dn describes dispersion affecting the component

reflected from that surface. We shall also assume a Gaussian spectrum of the

probe light S(ω) ∝ exp[−T 2(ω − ω0)2] with T characterizing its coherence time

(∼ inverse bandwidth). Throughout this chapter, we shall use half-width at 1/e-

maximum as a measure of the resolution. Using this model, the complex envelope

of the mutual coherence function is a sum of contributions from the surfaces

Γ(τ) =∑

n Γn(τ) given by:

Γn(τ) ∝ rn exp

(− (τ − τn)2

T 2 − iDn

). (5.4)

In standard OCT, the depth profile is retrieved directly from the interferogram

as the absolute value |Γ(τ)|, and consequently the nth surface is visualized as a

peak with a dispersion-broadened width√

T 2 + D2n/T 2.

Let us now turn to the analysis of the information about the depth profile

contained in the generalized autoconvolution function Ξw(τ). With the complex

envelope Γ(τ) given as a sum of terms calculated in Eq. (5.4), the generalized auto-

convolution function can be decomposed into a double sum Ξw(τ) =∑

mn Ξ(mn)w (τ)

of contributions obtained by inserting a product Γ∗m(τ+τ ′)Γn(τ−τ ′) into Eq. (5.1).

We shall analyze separately the diagonal terms with m = n, which as we shall

see reveal positions of the reflective surfaces, and then the cross-terms Ξ(mn)w (τ)


with m 6= n that are responsible for the artefacts in the reconstructed profile, like

the one we have seen in Figure 5.2(b). The complete analytical expressions are

rather complicated and we shall approximate them by performing an expansion

up to the leading order of w. This will give us an insight into relative scales of

parameters involved in the procedure.

For m = n, an explicit calculation yields the following expression for Ξ(nn)w (τ)

in the limit when w2 ≪ 1/(T 2 + D2n/T 2):

Ξ(nn)w (τ) ∝ |rn|2 exp

(− 2(τ − τn)2

T 2 + (Dnw)2

)(5.5)

This expression describes a Gaussian peak located at the position τn of the nth

reflective surface. The width of this peak is given by√

[T 2 + (Dnw)2]/2, and

in the limit when w → 0 it approaches the dispersion-free limit defined solely

by the coherence time of the light source, equal to T/√

2. Compared to the

standard interferogram envelope |Γ(τ)| in the absence of dispersion, the peak in

the autoconvolution function is narrower by a factor√

2; this narrowing is easily

understandable as Ξw(τ) is quadratic in Γ(τ).

5.2.3 Analysis in the time-frequency domain

The operation of the blind dispersion compensation method can be understood

intuitively with the help of the (chronocyclic) Wigner distribution function 2 [89],

2the term chronocyclic indicates the arguments of the Wigner function are from the comple-mentary time and frequency domains [32]


defined for the complex interferogram envelope Γ(τ) in the standard way as:

W (τ, Ω) =1

π

∫dτ ′ e−2iΩτ ′Γ∗(τ + τ ′)Γ(τ − τ ′) (5.6)

In Figure 5.2(c) we depict the Wigner function for the example in the previous

section. The Wigner function contains two peaks corresponding to the two re-

flective surfaces, and an oscillating interference pattern located half-way between

the peaks. This pattern is a signature of coherence between the two reflections.

Dispersion introduces time-frequency correlations which result in a tilt clearly

seen in Figure 5.2(c). In standard OCT the depth profile is retrieved as |Γ(τ)|

which is given by W (τ, Ω) integrated along the Ω axis and square-rooted. Then

the dispersion-induced tilt severely deteriorates the resolution. However, taking a

cross-section through the Wigner function along a horizontal line for Ω = 0, yields

a dispersion-free profile. The problem with this profile is that it contains strong

contributions from the interference terms [90,27] which were washed out in |Γ(τ)|

due to the integration over the frequency variable. The answer to this problem

is to carefully average the profile over a range of frequencies Ω. This is exactly

the purpose of the generalized autoconvolution function in Eq. (5.1), which can

be rewritten in the Wigner formalism as:

Ξw(τ) =

√π

2w2

∫dΩ W (τ, Ω)e−Ω2/2w2

. (5.7)


The averaging along the frequency axis over an interval defined by the parameter

w rapidly washes out the contribution from the interference pattern, while nearly

retaining the width of the genuine peaks. The width of the spectral interval must

be larger than the spacing of the interference pattern, which in turn is inversely

proportional to the separation between the peaks given by |τm − τn|.

The tuning of the parameter w can be discussed quantitatively by considering

the explicit expression for the cross-terms Ξ(mn)w (τ) with m 6= n. In order to keep

the interpretation of the mathematical expressions simple, we shall restrict our

attention to the regime when dispersion affecting contributions from two reflecting

surfaces is comparable, i.e. Dm ≈ Dn ≈ D. This is the case when the bulk of

dispersion comes from the medium preceding both the surfaces. In this regime, it

is possible to give a simple formula for the magnitude of the cross-terms:

|Ξ(mn)w (τ)| ∝ |r∗mrn| exp

(− 2(τ − τ )2

T 2 + (Dw)2

)

× exp

(−w2

2(τm − τn)2

)(5.8)

where the proportionality factor is the same as in Eq. (5.5) and τ = (τm + τn)/2.

This formula describes a structure located half-way between the positions of the

contributing surfaces. The magnitude of the structure is a function of w through

the multiplicative factor exp[−w2(τm − τn)2/2]. Its exponential dependence on w

allows us to suppress efficiently the spurious cross-terms in the autoconvolution

function by setting a non-zero value of w, with only a slight worsening of the


resolution exhibited in Eq. (5.5). The suppression of the cross-terms requires |w|

exceeding 1/|τm−τn| and therefore is more efficient for a larger separation between

the peaks.

The original inspiration for the presented dispersion compensation method

was nonclassical optics and we shall close by interconnecting the presented blind

dispersion-compensation method to QOCT. In the latter quantum case, the depth

profile is obtained from a scan of two-photon interference between a photon re-

flected off the sample and a twin photon which underwent a controlled temporal

delay. A straightforward calculation shows that the quantum optical fourth-order

coherence function measured in such an experiment [86] is identical with the au-

toconvolution function Ξw(τ). In the quantum-optical version, the parameter w

describes the bandwidth of the pump pulse producing entangled pairs of photons

which defines, through energy conservation, the strength of their spectral anti-

correlations. It is therefore a parameter of the experimental setup that needs to

be adjusted prior to a measurement, rather than tuned numerically in the data

postprocessing. This analogy indicates that quantum correlations and quantum

effects may not offer the originally claimed distinct advantages over classical sys-

tems. There is lots more to be discovered and understood about the higher-order

optical correlations and interference phenomena (both classical and nonclassical

alike).


5.3 Chapter Summary

In conclusion, we have shown that the generalized autoconvolution function cal-

culated from the complex interferogram envelope can reveal location of reflective

surfaces in a dispersive medium with a resolution reaching the coherence length of

the employed light itself. It can be anticipated that in the case of more complex

depth profiles, the oscillatory character of the artifacts will lead to similar suppres-

sion efficiency as that in the simple numerical example discussed here. For very

broadband light sources currently used in OCT setups, one needs to go beyond the

quadratic term in Eq. (5.3) and consider effects of higher-order dispersion. This

will introduce curvature in the Wigner function peaks generated by reflections

from the sample, leading to richer interference patterns. Nevertheless, it should

be possible to distinguish the sharp features in the Wigner function corresponding

to the positions of the reflective surfaces.

78

Chapter 6

Conclusions

6.1 Thesis Overview . . . . . . . . . 786.2 Future Directions . . . . . . . . . 796.3 Final word . . . . . . . . . . . . . 83

6.1 Thesis Overview

We have started with the consideration of some fundamental and general principles

of interference phenomena in Chapter 2 and demonstrated the concrete interfero-

metric experimental devices and methods in the subsequent chapters. In Chapters

3 and 4 we proposed and demonstrated a novel implementation of spectral shear-

ing interferometry (SSI) for reconstructing the electric field of ultrashort pulses

utilizing asymmetric group velocity matching in a long nonlinear crystal. The new

configuration eliminates the requirement for a linearly chirped auxiliary pulse com-

mon in all existing spectral shearing interferometry methods relying on nonlinear

conversion to produce a spectral shear. In Chapter 5 we presented a classical-

optics numerical method for reconstructing dispersion-compensated depth pro-

files that does not involve prior knowledge of the dispersive characteristics of the

6.2. FUTURE DIRECTIONS 79

medium. This method utilized the phase information contained in complete com-

plex interferograms to remove deleterious effects of dispersion and does not need

to rely on the low power non-classical light sources. The dispersion compensation

is performed by postprocessing data collected in a standard OCT setup with a

low-coherence light using a simple numerical algorithm straightforward in imple-

mentation.

6.2 Future Directions

Despite its commercialization, the presented ultrashort pulse characterization

work still leaves a lot of new questions and a substantial amount of future work

and unexplored possibilities ahead, some of which are presented below.

ARAIGNEE Alternative geometries The most straightforward direction to

consider is that of alternative ARAIGNEE geometries and configurations, which

might result in an even more “sexy” and user friendly devices. For example,

Ref. [91] demonstrated another device prototype shown in Figure 6.1. The pro-

totype has been engineered on a much more compact platform than any of the

prior laboratory setups used for the concept verification and feasibility tests of

Chapter 4. The spectral shear encoding optics has been minimized to fit in a

3× 6 in. casing—an exact replica of the compact Ocean Optics spectrometer it is

designed to directly connect to.


Quartz

Alignmentwindow

Blue filter

SpectrometerIs here!!!

Split MirrorPair

Crystal

Slit

Waveplate

Figure 6.1: Possible next generation ARAIGNEE device. First subfigure presents the schematicof the apparatus and the second demonstrates the actual prototype.


ARAIGNEE: Multiplexing advantages Some other previously demon-

strated SPIDER techniques, such as spatially encoded arrangement (SEA-

SPIDER) [92] can be combined with the single crystal design to reap the obvious

multiplexing advantages.

ARAIGNEE: Quasi-phasematching Some more fundamental issues remain

to be investigated, such as whether more complicated structures (i.e. nonlinear

crystals with quasi phase matched gratings) can be employed instead of the uni-

axial KDP crystal.

ARAIGNEE: Practical and Numerical nuances When modeling the pro-

cess of sum frequency generation in a type II crystal the effects of beam spatial

mode profile and beam spatial walk-off have not been investigated. While our

model seems to account and predict very well the results of the laboratory mea-

surements, the future might present situations where a more elaborate numerical

modeling will need to be developed to include the neglected geometrical effects in

the nonlinear interaction.

ARAIGNEE: Extending the spectral range The femtosecond pulse charac-

terization described in this dissertation has been performed in the ∼ 0.7−0.9 µm

spectral region with a simple angular tuning of the nonlinear crystal. The im-

portance of this spectral range is obvious due to the existence of the Ti:sapphire

Kerr-lens mode-locked lasers and the successful ARAIGNEE operation in that


spectral region is based on the fact that for a type II SFG in a KDP crystal, the

group velocity mismatch between the fundamental o-ray and upconverted e-ray

pair is negligible compared to the GVM between the fundamental e-ray and up-

converted e-ray pair: (k′ro− k′

b) ≃ 0 ≪ (k′re− k′

b). However, the ARAIGNEE

principle will apply equally well if the roles of the e- and o- beams are switched

and it is possible to find a frequency range where it is the extraordinary funda-

mental that is nearly group velocity matched to the extraordinary SFG (b-) beam:

(k′re− k′

b) ≃ 0≪ (k′ro− k′

b). These two limits are illustrated with the help of Fig-

ure 6.2 where we have plotted the absolute magnitude of the PMF for the crystal

with the upconversion parameters used for Figure 3.2 but on a wider 0.4−1.8 µm

spectral domain. The latter limit corresponds to the “horizontal” PMF curve in

the ∼ 1.6 − 2.0 µm spectral region, and the femtosecond pulses centered there

could be characterized with the same KDP crystal. The same general PMF shape

holds for type II SFG process in other nonlinear crystals. For example, the BBO

(β-Barium Borate) crystal can be used for pulse characterization centered around

∼ 1.2 µm and the KTP (Potasium Titanyl Phosphate) nonlinear crystal in the

∼ 1.46 µm region. These spectral regions are of interest to the work relying on

optical parametric oscillators and the spectral region ∼ 1.55 µm is obviously of

extreme importance to the telecommunications community [93] as well as of sig-

nificant interest to the biomedical optics as it provides high penetration depth in

the biological tissues [72,75].

6.3. FINAL WORD 83

0.6 0.8 1 1.2 1.4 1.6ΛeHµmL

0.6

0.8

1

1.2

1.4

1.6

ΛoHµmL

Figure 6.2: Phase Matching Function magnitude for a type II SFG in a thick KDP crystaldemonstrating both extrema of the asymmetric group velocity matching.

6.3 Final word

While the research process is frustrating at times, it can be said its beauty (and

ultimately, joy) is in its intrinsic ‘open-endedness’. Richard Feynman once said,

“Nature uses only the longest threads to weave her patterns, so each small piece

of her fabric reveals the organization of the entire tapestry” [94]. Extending the

metaphor, the weaving of fabric of science and technology is never ending and the

woven pattern is ever continuously evolving. This work is but a small contribution

to this grand pattern.

84

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“I am a part of all that I have met;

Yet all experience is an arch wherethro’

Gleams that untravell’d world, whose margin fades

For ever and for ever when I move.

How dull it is to pause, to make an end,

To rust unburnish’d, not to shine in use!

As tho’ to breathe were life. Life piled on life

Were all too little, and of one to me

Little remains: but every hour is saved

From that eternal silence, something more,

A bringer of new things; and vile it were

For some three suns to store and hoard myself,

And this gray spirit yearning in desire

To follow knowledge, like a sinking star,

Beyond the utmost bound of human thought.”

— Alfred Lord Tennyson, “Ulysses” (1842)

ultrashort pulse characterization and coherent time-frequency light processing

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