NONLOCAL MODULATION AND DISPERSION A DISSERTATION

NONLOCAL MODULATION AND DISPERSION

A DISSERTATION

SUBMITTED TO THE DEPARTMENT OF ELECTRICAL

ENGINEERING

AND THE COMMITTEE ON GRADUATE STUDIES

OF STANFORD UNIVERSITY

IN PARTIAL FULFILLMENT OF THE REQUIREMENTS

FOR THE DEGREE OF

DOCTOR OF PHILOSOPHY

Steven Sensarn

May 2010

http://creativecommons.org/licenses/by-nc/3.0/us/

This dissertation is online at: http://purl.stanford.edu/ts817jz4801

© 2010 by Steven Sensarn. All Rights Reserved.

Re-distributed by Stanford University under license with the author.

This work is licensed under a Creative Commons Attribution-Noncommercial 3.0 United States License.

ii



http://purl.stanford.edu/ts817jz4801

I certify that I have read this dissertation and that, in my opinion, it is fully adequatein scope and quality as a dissertation for the degree of Doctor of Philosophy.

Stephen Harris, Primary Adviser


Martin Fejer, Co-Adviser


Jelena Vuckovic

Approved for the Stanford University Committee on Graduate Studies.

Patricia J. Gumport, Vice Provost Graduate Education

This signature page was generated electronically upon submission of this dissertation in electronic format. An original signed hard copy of the signature page is on file inUniversity Archives.

iii

Abstract

Quantum entanglement has been the subject of a growing field in quantum optics that

aims to study the physical properties of entangled systems and find new applications

for their use. Entangled photons have been proposed as a means to achieve long

distance quantum communications and cryptography, perform computational tasks

beyond the capabilities of classical computers, image beyond the classical limit, and

provide measurement sensitivities unobtainable with analogous classical light sources.

At the heart of these applications is the fact that entangled photons behave strik-

ingly different from classical light or even pairs of independent single photons. This

thesis reviews an interference effect known as nonlocal cancellation of dispersion and

proposes and demonstrates a new, analogous effect which we term as nonlocal mod-

ulation. These techniques allow one to undo the effect of one dispersive medium

or phase modulator with a similar system at a distant point in space, as observed

by correlations between entangled photons. We explore the use of nonlocal disper-

sion compensation, combined with chirped quasi-phase-matched nonlinear crystals, to

generate ultrashort entangled photons with high generation rates. To aid in the mea-

surement of ultrashort entangled photons, we propose and demonstrate a resonance

technique to increase the sensitivity of ultrafast sum-frequency-generation correlators.

Principal accomplishments of this work include: the first observation of nonlocal

modulation, the first observation of the radar-like “chirp and compress” technique

with entangled photons, and the first experimental demonstration of resonant sum

frequency generation using broadband biphotons.

iv

Acknowledgement

During my time as a PhD student, many people have provided invaluable support. I

am very grateful to all of them; without them, I could not have completed the work

summarized by this thesis.

I feel lucky to have been a part of Steve Harris’ group. It is extremely rare for a

professor to convey knowledge in a way that is so easy to understand; Steve definitely

has that gift. As a mentor, I could not have asked for better; Steve’s enthusiasm,

willingness to help, lively discussions, and frequent interaction with the group made

it easy to work through problems and find new ideas when I was stuck. Steve’s

excellence as a supervisor is also worth praising; many students do not have the

opportunity to travel and present at conferences, but Steve always made sure to send

his students to meetings. I am proud and grateful to be a student of Steve’s.

I thank Guang-Yu Yin for sharing his broad expertise in optics, electronics, and

other fields. He is a source of brilliant ideas for tackling problems in the lab, and he

is always available and willing to help the students. Dr. Yin has been an integral

contributor to all of my experiments, both in planning and in execution.

The remainder of the Harris group: Sunil Goda, Miro Shverdin, Irfan Ali-

Khan, Chinmay Belthangady, Pavel Kolchin, Shane Du, Chih-Sung Chu, and Sharon

Shwartz, all deserve recognition for helping me with my research. Sunil trained me

during my first year in the group, and we completed two successful experiments to-

gether (one of which is described in Appendix C). Irfan Ali-Khan worked with me

on resonantly-enhanced SFG (Chapter 6) and was integral in obtaining the results

of that experiment. He also participated in helpful discussions related to nonlocal

modulation (Chapter 4) and chirp and compress (Chapter 5). It has been a pleasure

v

working in the same room as Chinmay for most of my years in the group; he has

always helped me think through various problems that come up in my work. I am

thankful for the collaborative atmosphere of the Harris group and the teamwork it

embodies.

Others around Ginzton who have been a great help to me are Vivian Drew, Darla

Le-Grand-Sawyer and the Ginzton administrative office, Larry Randall, Tim Brand,

and other Ginzton staff who have helped with technical aspects of my work.

I thank my dissertation reading committee, Marty Fejer and Jelena Vuckovic, for

taking the time to review this thesis and for aiding me in my research. They are

both excellent professors, and I learned a great deal about lasers and quantum optics

from their courses at Stanford. Marty deserves a special thanks for sharing some of

his immense knowledge on periodic poling and nonlinear optics in general. He also

provided crystals from his group and served as a co-investigator for the chirp and

compress project.

I also owe gratitude to my family for supporting me during my studies. My mom

and dad have always had faith in me and have encouraged me to work my hardest in

school. The occasional visits home to Texas and now Connecticut have been a very

special and relaxing part of my years in graduate school. I thank my mom and dad

for believing in me and supporting me with anything I need. I also thank my sister

Melanie and brother-in-law Paul for their company and support. After they moved

to California, we spent enormous amounts of time together, and it has certainly made

me very happy.

My girlfriend Maria deserves thanks for her unconditional support and love, es-

pecially during stressful times. Without her to instill confidence in me, I would have

spent a lot of graduate school worrying.

I thank my friends and colleagues around Ginzton and outside who have provided

many impromptu, fascinating discussions about various scientific topics as well as

those conversations that had nothing to do with science.

Financial support was provided by the U.S. Air Force Office of Scientific Re-

search (AFOSR), the U.S. Army Research Office (ARO), and the Defense Advanced

vi

Research Projects Agency (DARPA). The AFOSR and ARO supported all of the ex-

periments presented in this thesis, including nonlocal modulation (Chapter 4), chirp

and compress (Chapter 5), resonantly-enhanced sum frequency generation (Chapter

6), and molecular modulation (Appendix C). DARPA provided financial support for

the nonlocal modulation project.

vii

Contents

Abstract iv

Acknowledgement v

1 Introduction 1

1.1 Biphotons and Time-Energy Entanglement . . . . . . . . . . . . . . . 1

1.2 Ultrashort Biphotons . . . . . . . . . . . . . . . . . . . . . . . . . . . 2

1.3 Applications of Biphotons . . . . . . . . . . . . . . . . . . . . . . . . 3

1.4 Thesis Outline . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

2 Theory of Down Conversion: ABCD Formalism 6

2.1 Classical Optical Parametric Amplification . . . . . . . . . . . . . . . 6

2.2 Spontaneous Parametric Down Conversion: Coupled Equations . . . . 8

2.3 Solution to the Coupled Equations: ABCD Coefficients . . . . . . . . 8

2.4 Defining the Signal and the Idler . . . . . . . . . . . . . . . . . . . . 10

3 Nonlocal Dispersion Cancellation 13

3.1 Dispersion with Classical Pulses . . . . . . . . . . . . . . . . . . . . . 13

3.2 Classical Intensity Correlation . . . . . . . . . . . . . . . . . . . . . . 15

3.3 Dispersion with Entangled Photons . . . . . . . . . . . . . . . . . . . 16

4 Nonlocal Modulation 18

4.1 Connection to Nonlocal Dispersion Cancellation . . . . . . . . . . . . 19

4.2 Other Types of Quantum Interference . . . . . . . . . . . . . . . . . . 20

viii

4.3 Experimental Description . . . . . . . . . . . . . . . . . . . . . . . . 20

4.4 Experimental Results . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

4.5 Theory of Nonlocal Modulation . . . . . . . . . . . . . . . . . . . . . 24

4.6 Fitting Theory to the Data . . . . . . . . . . . . . . . . . . . . . . . . 28

4.7 Summary of Nonlocal Modulation . . . . . . . . . . . . . . . . . . . . 29

5 Chirp and Compress 30

5.1 Powerful Biphotons . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30

5.2 Overview of Chirp and Compress . . . . . . . . . . . . . . . . . . . . 31



5.5 Compression and Crystal Orientation . . . . . . . . . . . . . . . . . . 36

5.6 Theory of Chirp and Compress . . . . . . . . . . . . . . . . . . . . . 36

5.7 Fitting Theory to the Data . . . . . . . . . . . . . . . . . . . . . . . . 39

5.8 Summary of Chirp and Compress . . . . . . . . . . . . . . . . . . . . 39

6 Resonantly-Enhanced Quantum SFG 41

6.1 Overview of Resonant Enhancement . . . . . . . . . . . . . . . . . . . 42

6.2 Theory of Resonant Enhancement . . . . . . . . . . . . . . . . . . . . 43

6.3 Cavity Enhancement Ratio . . . . . . . . . . . . . . . . . . . . . . . . 46



6.6 Summary of Resonant Enhancement . . . . . . . . . . . . . . . . . . 50

7 Conclusion 51

A Extended Theory of Down Conversion 54

A.1 High-Gain Parametric Down Conversion . . . . . . . . . . . . . . . . 54

A.2 Iterative Technique for Computing ABCDs . . . . . . . . . . . . . . . 56

B Theory of SFG with Biphotons 60

ix

C Molecular Modulation in a Hollow Fiber 63

C.1 Molecular Modulation . . . . . . . . . . . . . . . . . . . . . . . . . . 63

C.2 Overview of Molecular Modulation in a Hollow Fiber . . . . . . . . . 64

C.3 Experimental Description . . . . . . . . . . . . . . . . . . . . . . . . 64

C.4 Conversion Efficiency . . . . . . . . . . . . . . . . . . . . . . . . . . . 66

C.5 Spatial Modes of the Sidebands . . . . . . . . . . . . . . . . . . . . . 68

C.6 Summary of Molecular Modulation in a Hollow Fiber . . . . . . . . . 69

Bibliography 70

x

List of Figures

1.1 Spontaneous parametric down conversion . . . . . . . . . . . . . . . . 1

2.1 Phase matching schemes . . . . . . . . . . . . . . . . . . . . . . . . . 11

3.1 Classical dispersion and correlation . . . . . . . . . . . . . . . . . . . 15

4.1 Schematic of nonlocal modulation. . . . . . . . . . . . . . . . . . . . . 19

4.2 Experimental setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

4.3 Driver circuit for 30 GHz phase modulators . . . . . . . . . . . . . . 21

4.4 Photo of frequency correlation setup . . . . . . . . . . . . . . . . . . 22

4.5 Frequency correlations with zero or one modulator running . . . . . . 23

4.6 Correlations with both modulators running . . . . . . . . . . . . . . . 24

5.1 Overview of chirp and compress . . . . . . . . . . . . . . . . . . . . . 31

5.2 Experimental schematic . . . . . . . . . . . . . . . . . . . . . . . . . 32

5.3 Signal photon spectrum . . . . . . . . . . . . . . . . . . . . . . . . . 33

5.4 Biphoton correlation measurements . . . . . . . . . . . . . . . . . . . 34

5.5 Correlation measurements with reversed crystal . . . . . . . . . . . . 35

6.1 Schematic of resonant enhancement . . . . . . . . . . . . . . . . . . . 42

6.2 Schematic of experiment . . . . . . . . . . . . . . . . . . . . . . . . . 47

6.3 SFG count rate versus detuning . . . . . . . . . . . . . . . . . . . . . 48

6.4 SFG count rate versus infrared input power . . . . . . . . . . . . . . 49

A.1 Photo of SLT crystal with 8 W pumping . . . . . . . . . . . . . . . . 55

A.2 Quasi-phase-matched crystal domains . . . . . . . . . . . . . . . . . . 56

xi

C.1 Experimental setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65

C.2 Sideband energy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67

C.3 Conversion efficiency . . . . . . . . . . . . . . . . . . . . . . . . . . . 68

C.4 Spatial modes of sidebands . . . . . . . . . . . . . . . . . . . . . . . . 69

xii

Chapter 1

Introduction

Pump

LaserPump

Photon

Signal

Photon

Idler

Photon

Nonlinear Crystal

Figure 1.1: Spontaneous parametric down conversion. Pump photons from amonochromatic laser divide into lower-energy signal and idler photons.

1.1 Biphotons and Time-Energy Entanglement

When a nonlinear crystal is pumped by a monochromatic laser, the pump photons

have a small but finite probability of splitting into a pair of lower-energy photons

called the signal and the idler in a process called spontaneous parametric down con-

version (Fig. 1.1). These photons, taken together, are often referred to as a bipho-

ton [1]. Because the photons are born at the same time and place in the crystal,

and because they must conserve energy with the pump photon, they share a quan-

tum mechanical relationship known as time-energy entanglement. Entanglement is

a nonlocal connection between quantum objects; the two photons must be described

1

CHAPTER 1. INTRODUCTION 2

together as a two-photon state rather than a separable product of independent single

photon states, even if they are separated by a large distance in space. In other words,

the two-photon—or biphoton—state takes the form

|ψ 〉 =

∫ ∫

F (ωs, ωi) |ωs, ωi 〉dωsdωi, (1.1)

where F (ωs, ωi) is the non-separable biphoton wavefunction, and the shorthand

|ωs, ωi〉 = a†s(ωs)a†i (ωi)|0〉 represents the creation of a signal photon at frequency

ωs and an idler photon at frequency ωi. For the case of monochromatic pumping,

F (ωs, ωi) = F ′(ωs)δ(ωp − ωs − ωi), and

|ψ 〉 =

∫

F ′(ωs) |ωs, ωp − ωs 〉dωs. (1.2)

These entangled photon pairs may be spatially separated into two channels and

their frequencies and arrival times at distant detectors measured. If Observer A

measures the frequency of her photon, she knows, by energy conservation, the exact

frequency of the photon measured by Observer B, since both photons are generated

from a monochromatic pump. If, instead, she measures the arrival time of her photon,

she can predict the detection time measured by Observer B to within a small window

that varies inversely with the photon bandwidth, as determined by the function F ′(ωs)

[2]. Much like a classical optical pulse, this temporal correlation window may be

lengthened by dispersion [3]. The temporal and spectral uncertainties of one photon,

given the corresponding measurement on the other, are not bounded by an uncertainty

principle; i.e., ∆t∆ω → 0. The ability to measure relative time and relative frequency,

with accuracies not limited by time-energy uncertainty, is the hallmark of time-energy

entanglement [4, 5, 6].

1.2 Ultrashort Biphotons

With a monochromatic pump, the generated signal and idler photon energies are

perfectly anti-correlated (ωi = ωp − ωs), and nonlinear processes such as two-photon

absorption and sum frequency generation (SFG) may be highly resonant, e.g. with


an atomic transition [7, 8] or an optical cavity mode [9]. For a fixed signal/idler

photon bandwidth, monochromatic pumping results in the highest degree of frequency

entanglement [10].

The degree of temporal entanglement depends on the bandwidths of the photons;

the larger the bandwidth, the shorter the temporal correlation width (in the absence of

dispersion) [2]. The largest theoretical bandwidth possible in spontaneous parametric

down conversion is equal to the pump frequency, and the shortest possible correlation

time is a single optical cycle [11]. Single-cycle biphotons maximize the two-photon

transition probability and, as it bears on either sum frequency generation or two

photon absorption, have an effective peak power of approximately 12π

~ω2p.

In addition to maximizing the efficiency of two-photon nonlinear processes, ul-

trashort biphotons have other nonclassical properties. Silberberg and colleages have

shown that, if these photons enter a nonlinear crystal phase matched so as to generate

the sum frequency field, the resulting field can be monochromatic and scale linearly

with input power [12, 13, 14]. With the rate of generated biphotons denoted as Rpair

and the characteristic temporal length of the biphoton as T , this nonclassical behav-

ior occurs when RpairT < 1. When RpairT > 1, sum frequency generation (SFG) is

in the classical regime and is characterized by generated power proportional to R2pair

and a linewidth of SFG that is equal to the convolution of the signal and idler line-

shapes. In the quantum regime, time-energy entangled photons with broad individual

bandwidths can be summed to produce a monochromatic field. Earlier theory and

experiments describe nonlinear processes with nonclassical light, and in particular,

the linear dependence of two photon processes on the incident power [15, 16, 17].

1.3 Applications of Biphotons

Entangled photons have been proposed for use in a broad range of applications in

quantum optics. In the realms of quantum communication and cryptography, en-

tangled photons are a natural choice for long-distance applications [18]. Quantum

teleportation uses entanglement to transport a quantum state over long distances

[19], and entangled photons form the basis of many quantum cryptographic protocols


and “quantum repeaters” that enable long-distance, secure communication [20].

Entanglement may be used in quantum computers to perform computational tasks

much more efficiently than currently possible using classical computers [21]. For

example, quantum computers can use entanglement to efficiently search an unsorted

database (Grover’s algorithm) or factor large integers into primes (Shor’s algorithm).

The latter algorithm may be used to crack current-day RSA encryption, which is

commonly used to secure credit card numbers, bank transactions, and other secret

data.

The nonclassical correlations inherent in position-momentum entangled photons

led to the birth of the quantum imaging field [22]. Spatial correlations between en-

tangled photons have been used to perform “quantum ghost imaging,” in which an

object in one channel is imaged using spatially-resolved detection in another channel

[23, 24]. Biphotons have been shown to enable optical lithography with higher reso-

lution than classical light of the same wavelength, and higher-order entangled photon

states improve the resolution further [25]. These high-order, so-called “N00N states”

also enable metrology and sensing beyond the classical limit [26].

Although this thesis focuses on time-energy entanglement, the ideas within apply

to biphotons entangled in other degrees of freedom, such as polarization and position-

momentum. Since spontaneous parametric down conversion is used to generate many

of these other entangled biphotons, they are time-energy entangled as well.

1.4 Thesis Outline

This thesis will begin with a theoretical description of biphotons as generated by

spontaneous parametric down conversion. The resulting ABCD formalism will be

used throughout the thesis with slight alterations as needed for the various physical

effects discussed.

A brief explanation of a quantum mechanical effect called nonlocal dispersion

cancellation will follow in Chapter 3. This effect is a consequence of time-energy

entanglement and allows cancellation of the dispersion experienced by one photon

passing through a medium by dispersing its partner photon with a compensating


medium at a separate location. This unique effect serves as the motivation for the

topics covered by this thesis.

Chapter 4 will introduce a new quantum mechanical effect called nonlocal modula-

tion with biphotons. The time-frequency analog of nonlocal dispersion compensation,

this effect allows distant phase modulators to interfere as observed by frequency cor-

relation between entangled photons. The basic theory of nonlocal modulation will be

covered and an experimental demonstration described.

The remainder of the thesis will focus on ultrashort biphotons. Analogous in many

ways to classical ultrashort pulses, these biphotons have high effective peak power in

two-photon nonlinear optical processes such as sum frequency generation and two-

photon absorption. Chapter 5 describes an experiment where ultrashort biphotons

are generated using chirped quasi-phase-matched nonlinear crystals. “Chirp and com-

press” allows broadband, ultrashort biphotons to be generated using long crystals.

At the heart of this generation technique is nonlocal dispersion cancellation, which

allows chirped biphotons to be subsequently compressed by dispersing only one of the

photons.

If single photon detectors with femtosecond temporal resolution were available,

ultrashort biphotons such as those produced by the chirp and compress technique

could be directly characterized using coincidence measurements. In the absence of

such detectors, other techniques are needed to measure the correlation functions of

ultrashort biphotons. In the experiment described in Chapter 5, we recombine the

photons into a nonlinear crystal to perform sum frequency generation (SFG) corre-

lation. Chapter 6 examines a technique to increase the efficiency of SFG correlation

with broadband biphotons. By resonating the sum frequency field in an optical cav-

ity, the SFG count rate can be improved. We describe an experiment improving the

efficiency of SFG by an order of magnitude.

Chapter 7 will summarize the above topics and discuss some of the challenges

present in the advancement of each.

Chapter 2

Theory of Down Conversion:

ABCD Formalism

In this chapter, we develop the ABCD formalism describing biphotons generated by

spontaneous parametric down conversion. The usual strategy for describing the field

operators mathematically involves quantizing the electric field using a harmonic os-

cillator analogy [27]. The result of such an exercise is that one may use equations

from classical electromagnetism where the electric fields have been replaced by appro-

priate field operators. (Equivalently, one may replace classical Poisson brackets with

the quantum-mechanical commutators.) We assume that fields may be quantized

in this manner, and we skip directly to the substitution step. All of our theoret-

ical predictions using this approach have been verified experimentally, as reported

by this thesis, and we believe the assumption to be valid, at least for bulk-crystal

and free-space experiments. We begin by recalling the coupled equations for optical

parametric amplification.

2.1 Classical Optical Parametric Amplification

Spontaneous parametric down conversion is the same physical process as optical para-

metric amplification, except that the signal and idler fields are seeded by vacuum

fluctuations rather than a classical laser beam. Guided by Harris [28], we write

6

CHAPTER 2. THEORY OF DOWN CONVERSION: ABCD FORMALISM 7

classical coupled equations for parametric amplification of broadband fields in the

monochromatic- and non-depleted-pump regime:

∂Es(ω, z)

∂z= iǫ0dηs(ω)ωgsEpE

∗i (ωi, z) exp[i∆k(ω)z],

∂Ei(ωi, z)

∂z= iǫ0dηi(ωi)ωigiEpE

∗s (ω, z) exp[i∆k(ω)z]. (2.1)

To keep the notation concise, we use ωi = ωp − ω. The quantities Es(ω, z) and

Ei(ω, z) are positive-frequency, analytic signals, and we take the electric fields to

vary slowly in z. The real electric fields at the signal and idler are related to these

quantities as

Es(t, z) =

∫ ∞

−∞

1

2[Es(ω, z) + E∗

s (−ω, z)] expi[ks(ω)z − ωt]dω,

Ei(t, z) =

∫ ∞

−∞

1

2[Ei(ωi, z) + E∗

i (−ωi, z)] expi[ki(ωi)z − ωit]dωi. (2.2)

In Eqs. (2.1), ηj(ω) = 1nj(ω)

√

µ0

ǫ0is the characteristic impedance of the crystal

medium having refractive index nj(ω) at the signal or idler, d is the nonlinear co-

efficient, Ep is the monochromatic pump field amplitude, kj(ω) = nj(ω)ω/c is the

wave-vector, ∆k(ω) = kp(ωp) − [ks(ω) + ki(ωi)] is the wave-vector mismatch (with

an additional term K0 for quasi-phase-matched crystals), and gs and gi are spatial

coupling factors for the signal and idler beams (assumed Gaussian):

gs =2ApAi

ApAs + ApAi + AsAi,

gi =2ApAs

ApAs + ApAi + AsAi. (2.3)

Ap, As, and Ai are the spot sizes of the pump, signal, and idler beams, respec-

tively. When the confocal parameter of the pump beam is equal to the crystal length,


the single-mode parametric gain is near-maximum [28], and the resulting entangled

photon fields are well approximated by single-mode Gaussian beams.

2.2 Spontaneous Parametric Down Conversion:

Coupled Equations

To convert Eqs. (2.1) to the quantum optical domain for modeling down conversion,

we replace the electric field analytic signals with annihilation and creation operators,

properly normalized so that operators have units of√

photons/s; e.g., Es(ω, z) =

bs(ω, z)√

2~ωηs(ω)/As and E∗s (ω, z) = b†s(ω, z)

√

2~ωηs(ω)/As. After substitution,

the operator equations are

∂bs(ω, z)

∂z= iκ(ω)bpb

†i (ωi, z) exp[i∆k(ω)z],

∂bi(ωi, z)

∂z= iκ(ω)bpb

†s(ω, z) exp[i∆k(ω)z] (2.4)

κ(ω) = ǫ0d

[

8~ωpωωiηp(ωp)ηs(ω)ηi(ωi)ApAsAi

(ApAs + ApAi + AsAi)2

]1

2

(2.5)

Like the analytic signals Es(ω, z) and Ei(ωi, z), the operators bs(ω, z) and bi(ωi, z)

are envelopes which vary slowly in z. The conventional annihilation and creation

operators are related to these envelopes as aj(ω, z) = bj(ω, z) exp[ikj(ω)z]. (Since the

pump is a classical field, bp is a constant rather than an operator.)

2.3 Solution to the Coupled Equations:

ABCD Coefficients

If we assume that the parametric gain is low, we can take the driving fields on the

right-hand sides of Eqs. (2.4) to be independent of z and equal to their values at z = 0.

The equations are therefore decoupled and may be integrated separately, leading to

the following solution:


as(ω, L) = A1(ω)as(ω, 0) +B1(ω)a†i(ωi, 0),

a†i (ωi, L) = C1(ω)as(ω, 0) +D1(ω)a†i(ωi, 0), (2.6)

where L is the crystal length, and the ABCD coefficients are

A1(ω) = exp[iks(ω)L],

B1(ω) = ibpκ(ω)L exp

i

[

∆k(ω)

2+ ks(ω)

]

L

sinc

[

∆k(ω)L

2

]

,

C1(ω) = −ibpκ(ω)L exp

−i[

∆k(ω)

2+ ki(ωi)

]

L

sinc

[

∆k(ω)L

2

]

,

D1(ω) = exp[−iki(ωi)L], (2.7)

where sinc(x) = sin(x)/x. Equations (2.4) may also be solved without the low-gain

assumption. The resulting solution has the same form as Eqs. (2.6), but the ABCD

coefficients are different. See Appendix A for details.

We work in the Heisenberg picture, where the initial state is vacuum |0〉; the field

operators at z = 0 correspond to the vacuum field and satisfy aj(ω, 0)|0〉 = 0. They

also obey the commutation relations [aj(ω1, 0), a†k(ω2, 0)] = 12πδjkδ(ω1 − ω2) or, in the

time domain, [aj(t1, 0), a†k(t2, 0)] = δjkδ(t1 − t2).

The frequency-domain operators in Eq. (2.6) may be propagated through linear

optical systems, e.g. air or a dispersive medium, by multiplying the appropriate an-

nihilation operator by exp[ikj(ω)z]. Likewise, a lossy medium may be modeled by

multiplying the operator by a scalar field attenuation coefficient (whenever Langevin

terms can be neglected, e.g. at 0 K or even at room temperature). The time-

domain creation and annihilation operators are given by inverse Fourier transform,

aj(t, z) =∫ ∞−∞ aj(ω, z) exp(−iωt)dω, and are used to calculate observable quanti-

ties in the various experiments described in this thesis, e.g. the rate of generated

signal photons (equivalent to the pair generation rate) Rpair = 〈a†s(t, L)as(t, L)〉


and the second-order Glauber intensity correlation function (coincidence probabil-

ity) G(2)(t1, z1, t2, z2) = 〈a†i (t2, z2)a†s(t1, z1)as(t1, z1)ai(t2, z2)〉.By using Eqs. (2.6) and following the procedure of normal ordering, the pair

generation rate and coincidence probability are

Rpair =1

2π

∫ ∞

−∞|B1(ω)|2dω (2.8)

G(2)(t1, z1, t2, z2) =1

4π2

∫ ∞

−∞|B2(ω)|2dω

∫ ∞

−∞|C2(ω)|2dω

+1

4π2

∣

∣

∣

∣

∫ ∞

−∞A2(ω)C∗

2(ω) exp(iωτ)dω

∣

∣

∣

∣

2

(2.9)

In Eq. (2.9), τ = t2 − t1, and the signal and idler field operators have been

propagated to positions z1 and z2, yielding new ABCD coefficients denoted by the

subscript ‘2’.

In the above theory, the operators aj(ω, z) and a†j(ω, z) are positive-frequency

operators, analogous to the analytic signals used to describe the classical electric

fields. [Physically, the creation operator a†j(ω, z) represents the negative frequency

component −ω of the electric field.] To allow consistency with unbounded integrals

such as Fourier transforms, it is helpful to modify the ABCD coefficients in Eq. (2.7)

to reflect the constraints on the operators; we may take each of the ABCD coefficients

to be equal to zero for ω < 0.

The ABCD coefficients may be further constrained based on the experimental con-

figuration they model; different strategies for collecting the signal and idler photons

lead to different theoretical descriptions of the photons.

2.4 Defining the Signal and the Idler

For a particular experiment, the choices of phase matching configuration and the

method for collecting signal and idler photons—defining which photon is the signal


V

H(a) (b)PBS

50/50(c)

V

H

w > wp/2

w < wp/2

(d)

SIGNAL

IDLER

SIGNAL

IDLERSIGNAL

IDLER

SIGNAL

IDLER

Figure 2.1: Phase matching schemes: (a) type II and polarizing beam splitter, (b)type 0 or I and dichroic mirror, (c) type 0 or I and 50/50 beam splitter, (d) non-collinear.

and which is the idler—set additional constraints on the ABCD coefficients. Sev-

eral choices are available, outlined in Fig. 2.1. In a collinear, type II configuration

[Fig. 2.1(a)], the signal and idler photons are orthogonally polarized; a polarizing

beam splitter may be used to separate the photons and perform experiments. Be-

cause the signal and idler spectra may each span the frequency range 0 ≤ ω < ωp, the

ABCD coefficients should be constrained to be zero outside this range. For collinear,

type 0 or type I phase matching [Fig. 2.1(b)], the signal and idler have the same

polarization, and a natural choice to differentiate the photons is to define the signal

as the photon whose frequency lies in the range ωp/2 ≤ ω < ωp. In an experiment,

this might be realized using a dichroic mirror to separate the upper and lower halves

of the total biphoton spectrum (above and below degeneracy). With this definition,

the ABCD coefficients should be zero outside the range ωp/2 ≤ ω < ωp (noting that

the ABCD coefficients are expressed as a function of the signal frequency). If, in-

stead, one chooses to use a 50/50 beam splitter to separate the photons [Fig. 2.1(c)],

the coefficients should be constrained as in type II phase matching, and a single-field

(combined signal and idler) approach is appropriate, having only two coefficients (A

and B) instead of four (ABCD): see Chapter 4, “Nonlocal Modulation.” The ABCD

coefficients for non-collinear phase matching schemes, in which the signal and idler


are defined and collected based on their emission angles [Fig. 2.1(d)], should also be

constrained to the range 0 ≤ ω < ωp.

An alternate strategy to model a specific experimental configuration is to bound

the limits of all relevant integrals, e.g. replacing the integration limits in Eq. (2.8)

with ωp/2 ≤ ω < ωp for a type I phase-matched experiment. The advantage of

using the former technique—constraining the ABCD coefficients to be zero outside

the relevant frequency range—is that the same (infinite) integration limits may be

used for any experimental configuration.

Chapter 3

Nonlocal Dispersion Cancellation

An important consequence of time-energy entanglement, first noted by Franson [29,

30] and observed by Brendel and Baek [31, 32], is nonlocal cancellation of dispersion.

When the photons of an entangled pair are sent through different channels having

arbitrary dispersions, the dispersion in one channel may be negated by dispersion of

the opposite sign in the other channel. This effect results from quantum interference

and has no classical analog.

Nonlocal dispersion cancellation motivated our group to propose an analogous ef-

fect called nonlocal modulation, which is discussed in Chapter 4. Nonlocal dispersion

is also the enabling process that allows chirped biphotons, discussed in Chapter 5,

to be compressed into ultrashort biphotons using a single medium. In this chapter,

we introduce nonlocal dispersion by first exploring how dispersion affects classical

short pulses. Then, by comparison, we show how dispersion behaves differently with

entangled photons.

3.1 Dispersion with Classical Pulses

We begin by defining a transform-limited, classical short pulse having a Gaussian

intensity profile of temporal width T (the half-width at which the intensity drops to

a fraction 1/e2 of its peak value). For simplicity, we omit the z dependence of the

field and write the pulse envelope as

13

CHAPTER 3. NONLOCAL DISPERSION CANCELLATION 14

E0(t) = exp

(

− t2

T 2

)

. (3.1)

The real electric field oscillates at frequency ω0 and has the form E0(t) =

Re[E0(t) exp(−iω0t)]. When the pulse propagates through a dispersive medium hav-

ing group delay dispersion of the amount β (which scales as the length of the medium

and has units of s2), the output pulse may be calculated by first taking the Fourier

transform of E0(t), then multiplying by exp(iβω2), and finally taking the inverse

transform:

Eout(t) = F−1F [E0(t)] exp(iβω2)

= F−1

[

1

2π

∫ ∞

−∞E0(t) exp(iωt)dt

]

exp(iβω2)

= F−1

T

2√π

exp

[(

iβ − T 2

4

)

ω2

]

=

∫ ∞

−∞

T

2√π

exp

[(

iβ − T 2

4

)

ω2

]

exp(−iωt)dω

=T

√

T 2 − 4iβexp

[

t2

T 2 − 4iβ

]

. (3.2)

If the pulse is detected by a photodiode, the resulting photocurrent is proportional

to the intensity of the pulse. We define the photocurrent as Iout(t) = |Eout(t)|2, or

Iout(t) =T 2

√

T 4 + 16β2exp

[

− 2T 2

T 4 + 16β2t2

]

. (3.3)

Suppose two identical, transform-limited pulses pass through distant dispersive

media and are each detected by photodiodes (see Fig. 3.1). We call these pulses

“signal” and “idler.” If the dispersion coefficients of the two media are β1 and β2, the

output currents are


Is(t) =T 2

√

T 4 + 16β21

exp

[

− 2T 2

T 4 + 16β21

t2]

,

Ii(t) =T 2

√

T 4 + 16β22

exp

[

− 2T 2

T 4 + 16β22

t2]

. (3.4)

From Eqs. (3.4), the 1/e2 intensity half-widths of the pulses, given by the inverse

square roots of the coefficients multiplying t2, are broadened by dispersion. The peak

intensities of the pulses are decreased in a similar manner.

Correlation

t

!1

!2

Figure 3.1: Classical dispersion and correlation. Dispersion coefficients β1 andβ2 = −β1 chirp the pulses in opposite directions. Photodiode detectors measureintensity and ignore the direction of the chirp, and the resulting cross correlation ofthe photocurrents is broadened by dispersion.

3.2 Classical Intensity Correlation

The cross-correlation between the photocurrents for the broadened pulses is

(Is ⋆ Ii)(t) =

∫ ∞

−∞I∗s (τ)Ii(t+ τ)dτ

=T 3

√π

2√

T 4 + 8(β21 + β2

2)exp

[

− T 2

T 4 + 8(β21 + β2

2)t2

]

. (3.5)

Examination of Eq. (3.5) reveals that the correlation width and peak value depend

on the quantity β21 + β2

2 . If the dispersion in either channel is nonzero, regardless

of sign, the intensity correlation function broadens. This “square before you add”

behavior with classical pulses precludes one dispersive medium from reversing the


broadening of another; the classical correlation function is broader than either of the

pulses, as illustrated by Fig. 3.1.

3.3 Dispersion with Entangled Photons

If entangled photons—instead of classical pulses—are sent through distant disper-

sive media, Franson showed that one medium can cancel the other, as observed by

correlation.

To model the effect of group delay dispersion on the signal and idler photons,

the annihilation operators at the output of the crystal, given by Eqs. (2.6), may be

multiplied by the appropriate phase factors, e.g. exp[iβ1(ω − ωs0)2] for the signal

as(ω, L) and exp[iβ2(ωi − ωi0)2] for the idler ai(ωi, L), where ωs0 and ωi0 are the

center frequencies of the signal and idler, respectively. The phase factors may equiv-

alently be lumped into the ABCD coefficients: A2(ω) = A1(ω) exp[iβ1(ω − ωs0)2],

B2(ω) = B1(ω) exp[iβ1(ω − ωs0)2], C∗

2(ω) = C∗1(ω) exp[iβ2(ωi − ωi0)

2], and D∗2(ω) =

D∗1(ω) exp[iβ2(ωi − ωi0)

2]. For simplicity, we again ignore the z dependence of the

field.

In the quantum domain, the intensity correlation is described by the second-order

Glauber correlation function of Eq. (2.9):

G(2)(τ) =1

4π2

∫ ∞

−∞|B2(ω)|2dω

∫ ∞

−∞|C2(ω)|2dω

+1

4π2

∣

∣

∣

∣

∫ ∞

−∞A2(ω)C∗

2(ω) exp(iωτ)dω

∣

∣

∣

∣

2

,

where τ = t2 − t1. The first term in G(2)(τ) is a constant background level and

represents the coincidence probability for uncorrelated photons from different pairs.

The second term has a shape, as a function of the relative arrival time τ , that is

proportional to the amplitude-square of the biphoton wavefunction [introduced in

the frequency domain by Eq. (1.2)]. The product A2(ω)C∗2(ω) contains the factor

expi[β1(ω−ωs0)2+β2(ωi−ωi0)

2]. Noting that ωs0+ωi0 = ωp and using ωi = ωp−ω,


this factor is equal to exp[i(β1 + β2)(ω − ωs0)2]. If β2 = −β1, as in Fig. 3.1, the

coefficients cancel each other, and G(2)(τ) is completely unaffected by dispersion.

Nonlocal dispersion cancellation allows one to have complete control over the

biphoton state, as it bears on dispersion, by acting on only photon of the pair. In

contrast, a two-photon system without entanglement must modify both photons to

have full control over the measured correlation. In the next chapter, we discuss the

time-frequency analog to nonlocal dispersion cancellation, which we term as nonlocal

modulation.

Chapter 4

Nonlocal Modulation

In this chapter, we report the first observation of a time-frequency analog to nonlocal

dispersion cancellation and term this effect as nonlocal modulation [33]. Spontaneous

parametric down conversion is used to generate entangled photons that each have a

spectral width of about 280 cm−1. Sinusoidal phase modulators, each operating at a

frequency of 30 GHz (1 cm−1), are placed in the two beams, labeled Channel 1 and

Channel 2 (see Fig. 4.1). When either beam is dispersed by a prism or grating and

viewed in the frequency domain (i.e. as a function of position), this modulation is

completely hidden by the much broader spectral width of the photon. To observe

the modulation, we correlate in the frequency (spatial) domain. In the absence of

modulation, a photon detected at a particular position in Channel 1 coincides with a

photon of frequency ω2 = ωp − ω1 in Channel 2, and the correlation in the frequency

(spatial) domain is therefore a delta function δ(ω1 + ω2 − ωp). When synchronously

driven modulators are placed in the signal and idler channels, this correlation becomes

a distribution of discrete sidebands spaced by the modulation frequency. What is

strange and interesting is that these distant modulators now act cumulatively. For

example, if the two identical modulators have opposite phase, they negate each other

and act as if neither modulator were present. Conversely, if operated with the same

phase, they produce the same correlation as does a single modulator with twice the

modulation depth acting on only one of the photons.

18

CHAPTER 4. NONLOCAL MODULATION 19

!"#$

%&'(()*+,

%&'(()*+-

!&'.)+/01"*'203.

!,45,6

!-45-6

Figure 4.1: Schematic of nonlocal modulation.

4.1 Connection to Nonlocal Dispersion

Cancellation

To make the connection between nonlocal modulation and Franson’s nonlocal dis-

persion cancellation, dispersion is replaced by temporal modulation, and temporal

correlation is replaced by frequency correlation. Although the two nonlocal effects

are completely analogous, the experimental parameters differ strongly in realistic

cases. In the previous chapter, nonlocal dispersion was examined for the case of

monochromatically-pumped down conversion where the photons are perfectly anti-

correlated in frequency (ωi = ωp − ωs, with zero uncertainty). The analogous case

for nonlocal modulation has the photons perfectly correlated in time, again with

zero uncertainty. In real experiments, photons have finite bandwidths and therefore

finite temporal correlation widths. For crystal sources such as ours, the biphoton

width typically varies between 0.1 and 10 ps, but for cold-atom sources, the corre-

lation width can be as long as several hundred nanoseconds [34]. Finite temporal

correlation widths place an additional constraint on nonlocal modulation that is not

applicable to nonlocal dispersion with cw-pumped down conversion: modulation must

be performed slowly as compared to the biphoton correlation width [33].


4.2 Other Types of Quantum Interference

To avoid confusion, we mention a different type of quantum interference discovered

by Steinberg and colleagues [35, 36] that bears on Hong-Ou-Mandel interferometry.

Here, because it is not possible to determine which photon passed through a dispersive

medium, there is an interference of Feynman paths, and even-order dispersive terms

are not seen by the interferometer. Recognizing the importance of time-frequency

duality, Tsang and Psaltis have suggested the equivalent of the Steinberg interference

in the time domain [37, 38]. Since the photons meet on a single beam splitter in

these examples, the interference is a local effect, and classical analogies have been

demonstrated [39, 40].

!"#$$%&'()(

)*

!"#$$%&'*

+#,%- ../+0!(123

!*123

/.!4

/.!4

Figure 4.2: Experimental setup. Φ1(t) and Φ2(t) are 30 GHz sinusoidal modulatorssynchronously driven with variable relative phase (see text for details).

4.3 Experimental Description

A schematic of the experiment is shown in Fig. 4.2. We pump a 20-mm-long,

periodically-poled, magnesium-oxide-doped stoichiometric lithium tantalate crystal

(PPSLT, HC Photonics Corp.) with 0.8 W from a 532-nm cw laser (Coherent Verdi


V10). The nonlinear crystal is phase matched to produce 32-nm bandwidth, degener-

ate photon pairs at 1064 nm. The designed poling period of the crystal is 8.0008 µm

at room temperature, and the experimental phase matching temperature is 306 K.

All fields are polarized along the extraordinary axis of the crystal.

!

K V

K V

PM

PM(a) (b)

(d)

(e)

(f) (g) (h)(c)

Figure 4.3: Driver circuit for 30 GHz phase modulators: (a) Microwave DynamicsPLO-3072 30 GHz source, (b) Weinschel Associates 953K-3dB variable attenuator, (c)Picosecond Pulse Labs 5350-218 power divider, (d) Weinschel Associates 953K-10dBvariable attenuator, (e) Atm Inc. P1409-360 phase trimmer, (f) Nextec-RF NA00435amplifiers, (g) MegaPhase CA-V1K2 K to V coaxial adapters, (h) EOSPACE PM-DV5-40-PFU-PFU-106-LV-UL custom phase modulators.

The generated photons are filtered from the strong 532 nm pumping beam using

a four-prism setup and are then coupled into a polarization-maintaining fused-fiber

beam splitter (AFW PFC-64-1-50-L-P-7-1-F) which diverts the photons into Channels

1 and 2 with equal probability. The photons pass through identical sinusoidal phase

modulators (EOSPACE) driven at 30 GHz with modulation depths of about 1.5

radians. To set the modulation depth, we adjust the variable attenuators in the

microwave driver circuit and verify the depth by measuring the sideband amplitudes

of a 1064-nm Nd:YAG reference laser. The relative phase between the modulators

is controlled using a calibrated phase trimmer. The microwave circuit schematic is

detailed in Fig. 4.3.

Following the modulators are identical monochromators, each having a linear dis-

persion of 210 GHz/mm and a Gaussian instrument response function with a FWHM

bandwidth of 8.5 GHz. The monochromators each consist of a 1200-grooves/mm,

aluminum-coated grating (Thorlabs GR50-1210) operating at a deviation angle of


about 8 degrees. Lenses with focal lengths of -12.75 and 250 mm are used, in a

telescope arrangement, to magnify the beam before diffraction by the grating. A

750-mm lens focuses the diffracted beam through a 50-µm slit. To obtain frequency-

domain correlations, we fix the output slit in Channel 1 at x1 and scan the position

x2 in Channel 2. The photons transmitted through the monochromator slits are

coupled into multimode fibers and detected with single-photon counting modules

(SPCMs, id Quantique id400 and PerkinElmer SPCM-AQR-16-FC). A photograph

of the monochromators is shown in Fig. 4.4.

Figure 4.4: Photo of frequency correlation setup. Identical monochromators measurethe frequencies of the entangled photons. Red and blue lines show the beam path forChannel 1 and Channel 2, respectively.

4.4 Experimental Results

The primary experimental results of this work are shown in Figs. 4.5 and 4.6. For each

case, we set the monochromator slit in Channel 1 at an arbitrary position x1 near the


center of the generated 32-nm-wide spectrum and leave the position of this slit fixed

thereafter. The slit in Channel 2 is scanned over positions x2, and the coincidence

rate of the two detectors (with gate width T = 1.25 ns) is recorded as a function of

this position. For each position, the rate is averaged for 20 s.

With the pump frequency defined as ωp, and the position x2 proportional to the

frequency ω2, we express the coincidence rate as a function of relative frequency

∆ ≡ ω2 − (ωp − ω1). The scale of the frequency axis is calibrated by measuring the

sideband spacing of a single-mode 1064-nm laser modulated at 30 GHz, with the zero

position chosen (at the start of the experiment) as the location of the correlation peak

for unmodulated photon pairs.

!!"# !$# # $# !"##

%#

!##

!%#

"##&'(

)*+,-+./,-/01'2/0&3!!(

!0&456(!!"# !$# # $# !"##

%#

!##

!%#

"##&7(

!0&456(

(

Figure 4.5: Frequency correlation measurements (a) with both modulators turned offand (b) with the modulator in Channel 1 running at a modulation depth of 1.5. Dotsare data; curves are theoretical fits (see text). All data is approximately shot-noise-limited.

Figure 4.5 shows the experimental results without modulation and with modula-

tion in a single channel. In Fig. 4.5(a), both modulators are turned off by discon-

necting their 30 GHz drive signals. As expected by energy conservation, a single

correlation peak is observed. In Figure 4.5(b), Channel 1 is phase modulated as

exp[iδ cos(ωmt)] with a modulation depth of δ = 1.5, and Channel 2 is not modu-

lated. The frequency correlation is now distributed over a set of sidebands, having

Bessel function amplitudes J2n(δ), whose total area is equal to that of Fig. 4.5(a).

In Fig. 4.6(a), both modulators are turned on at a modulation depth of δ =


!!"# !$# # $# !"##

%#

!##

!%#

"##&'(

)*+,-+./,-/01'2/0&3!!(

!0&456(!!"# !$# # $# !"##

%#

!##

!%#

"##&7(

!0&456(

Figure 4.6: As in Fig. 4.5 but both modulators running (a) with the same phase and(b) with opposite phase.

1.5, and the cable length is adjusted so that they have the same phase. They now

act cumulatively (constructively interfere) to produce a set of sidebands having a

Bessel function distribution J2n(2δ). The frequency-domain correlation function of

two distant modulators is therefore the same as that which would be obtained by

correlating an unmodulated photon with a photon modulated at twice the modulation

depth.

In Fig. 4.6(b), the modulators are run at the same depth as in the previous

paragraph, but now the relative cable length is adjusted so that the modulators

are run in phase opposition. The modulators now destructively interfere, and no

sidebands are visible. The solid curves in Figs. 4.5 and 4.6 are theoretical fits to the

data.

4.5 Theory of Nonlocal Modulation

The theory of nonlocal modulation as measured in the frequency domain has been

developed by Harris [33] for the case of frequency correlation using ideal detectors

with perfect frequency (spatial) resolution. In the following paragraphs, we develop

the theory to allow for finite-resolution monochromators and detectors having both

arbitrary transmission functions and specified temporal gate widths.


Working in the Heisenberg picture, a nonlinear crystal of length L is pumped by

a monochromatic laser at frequency ωp. A positive-frequency field operator a(ω, z),

representing entangled photons, evolves inside the crystal and may be written in

terms of an envelope b(ω, z) which varies slowly along the propagation direction:

a(ω, z) = b(ω, z) exp[ik(ω)z]. The propagation equations describing entangled photon

generation are

∂b(ω, z)

∂z= iκ(ω)b†(ωp − ω, z) exp [i∆k(ω)z],

∂b†(ω, z)

∂z= −iκ(ω)b(ωp − ω, z) exp [−i∆k(ω)z]. (4.1)

where κ(ω) and ∆k(ω) are the coupling factor and wave-vector mismatch, respectively.

[The coupled equations in this section differ slightly from those in Eqs. (2.4) in that

they represent a single field rather than two distinct signal and idler fields. Only after

this field impinges on the beam splitter are the photons treated as separate operators

which propagate through their respective channels.] The solution for the output field

at z = L, expressed in terms of the vacuum field avac(ω) at the input of the crystal,

is

aout(ω) = A(ω)avac(ω) +B(ω)a†vac(ωp − ω), (4.2)

where, to preserve the commutation relations, the functions A(ω) and B(ω) satisfy

|A(ω)|2−|B(ω)|2 = 1 and A(ω)B(ωp−ω) = B(ω)A(ωp−ω). [We take the coefficients

A(ω) and B(ω) to be zero outside the range 0 ≤ ω < ωp; see Section 2.4.] For

simplicity, we ignore the z dependence of the field for the remainder of the theoretical

analysis.

The time-domain output field operator is related to its frequency-domain counter-

part [Eq. (4.2)] by the inverse Fourier transform, aout(t) =∫ ∞−∞ aout(ω) exp(−iωt)dω,

and is normalized so that the total rate of generated photons exiting the crystal is

Rout = 〈a†out(t)aout(t)〉. The generated photons are separated into two channels, de-

noted as Channel 1 and Channel 2, using a 50/50 beam splitter. The field operators at


the outputs of the beam splitter are a1(t) = a2(t) = 1√2aout(t). The photons are mod-

ulated by periodic phase modulators whose time-domain, Fourier-series transfer func-

tions are m1(t) =∑

k qk exp(−ikωmt) in Channel 1 and m2(t) =∑

l rl exp(−ilωmt)

in Channel 2, with Fourier transforms m1(ω) =∑

k qkδ(ω − kωm) and m2(ω) =∑

l rlδ(ω−lωm), respectively. With the ∗ symbol denoting convolution, the frequency-

domain modulated fields are a1(ω) = a1(ω) ∗m1(ω) and a2(ω) = a2(ω) ∗m2(ω). Sub-

stituting a1(ω), a2(ω), m1(ω), and m2(ω) into the expressions for a1(ω) and a2(ω)

yields

a1(ω) =1√2

∞∑

k=−∞

qk[

A(ω − kωm)avac(ω − kωm)

+B(ω − kωm)a†vac(ωp − ω + kωm)]

,

a2(ω) =1√2

∞∑

l=−∞

rl

[

A(ω − lωm)avac(ω − lωm)

+B(ω − lωm)a†vac(ωp − ω + lωm)]

. (4.3)

The modulated photons are frequency correlated by passing each through identical

monochromators whose output slits may be translated to select frequencies ω1 =

βx1 in Channel 1 and ω2 = βx2 in Channel 2, where the constant β is the linear

dispersion of the grating systems. The monochromators (spectral filters) have field

transmission functions H1(ω − βx1) and H2(ω − βx2). The filtered field operators in

Channels 1 and 2 are a1f(ω, x1) = a1(ω)H1(ω − βx1) and a2f(ω, x2) = a2(ω)H2(ω −βx2), respectively. The count rates at the outputs of the monochromators are given

by R1(x1) = 〈a†1f(t, x1)a1f(t, x1)〉 and R2(x2) = 〈a†2f(t, x2)a2f(t, x2)〉. These rates are

R1(x1) =1

4π

∞∑

k=−∞

|qk|2∫ ∞

−∞|B(ω − kωm)|2 |H1(ω − βx1)|2dω,

R2(x2) =1

4π

∞∑

l=−∞

|rl|2∫ ∞

−∞|B(ω − lωm)|2 |H2(ω − βx2)|2dω. (4.4)


Assuming a gate width T , the coincidence rate for the two detectors is related to

the second-order Glauber correlation function

G(2)(t1, x1, t2, x2) = 〈a†2f(t2, x2)a†1f(t1, x1)a1f(t1, x1)a2f(t2, x2)〉. (4.5)

With the assumption that the resolution of the monochromators is high, or equiv-

alently that the filter widths are small (as compared to the modulation frequency

ωm), it can be shown that the correlation function depends only on the difference of

the arrival times τ = t2 − t1, and the coincidence rate is

Rc(x1, x2) =

∫ T/2

−T/2

G(2)(τ, x1, x2)dτ. (4.6)

Equation (4.6) may be expanded using Wick’s theorem and shown to be given by

Rc(x1, x2) = R1(x1)R2(x2)T +

∫ ∞

−∞

∣

∣

∣

∣

∣

∞∑

k=−∞

qkrn−kFk(τ, x1, x2)

∣

∣

∣

∣

∣

2

dτ, (4.7)

where ∆ = β(x1 + x2) − ωp, n = ⌊∆/ωm + 12⌋, and

Fk(τ, x1, x2) =1

4π

∫ ∞

−∞A(ω − kωm)B(ωp − ω + kωm)

×H1(ω − βx1)H2(ωp − ω − βx2 + nωm)

× exp(iωτ)dω. (4.8)

The first term in Eq. (4.7) is the result of accidental coincidences between unpaired

photons in a gate width T . The second term is the coincidence rate between paired

photons and captures the modulation effects described in this chapter. To obtain

Eqs. (4.4)–(4.8), we have assumed that the transmission widths of the monochroma-

tors are small as compared to the modulation frequency and large as compared to

the inverse of the temporal gate width T . In our experiment these assumptions are

satisfied by factors of 3.5 and 11, respectively.


If we further assume that A(ω) and B(ω) are constant over the spectral region of

interest and are equal to A0 and B0, then Eq. (4.7) becomes

Rc(∆) = R1R2T + cnH(2)(nωm − ∆), (4.9)

where H(2)(ω) = |H1(ω)|2 ∗ |H2(ω)|2, and

R1 =1

4π|B0|2

∫ ∞

−∞|H1(ω)|2dω, (4.10)

R2 =1

4π|B0|2

∫ ∞

−∞|H2(ω)|2dω, (4.11)

cn =1

8π

∣

∣

∣

∣

∣

A0B0

∞∑

k=−∞

qkrn−k

∣

∣

∣

∣

∣

2

. (4.12)

By taking A(ω) and B(ω) to be constants over a frequency range much larger than

ωm, we assume that the modulation frequency is small as compared to the spectral

widths of the photons or, equivalently, that the photons are tightly correlated on the

time scale of the modulation period. The photons therefore arrive simultaneously at

the slowly-varying modulators and see correlated phase shifts, resulting in interfer-

ence. If the modulation frequency is larger than the bandwidths of the photons, the

product A(ω)B(ωp − ω) has a spectral width less than ωm, and the summation in

Eq. (4.7)—responsible for interference—reduces to a single term dependent on x1 and

x2. Because the modulation waveforms vary significantly within the correlation width

of the photons, the phase shifts are not correlated, and no interference is observed.

Slow modulation is therefore a necessary condition to observe nonlocal modulation.

4.6 Fitting Theory to the Data

The solid curves in Figs. 4.5 and 4.6 are theoretical fits to the data using Eq. (4.9)

shifted horizontally so as to match at their centers. The Fourier series coefficients

for sinusoidal phase modulators are Bessel functions with qk = Jk(−δ1) and rl =


Jl(−δ2), where δ1 and δ2 are the modulation depths in Channels 1 and 2, respec-

tively (|δ1| = |δ2| = 1.5 in our experiment). We model the monochromator response

functions in Channels 1 and 2 as Gaussians with FWHM bandwidths Γ: H1(ω) =

α1 exp [−2 ln(2)ω2/Γ2] and H2(ω) = α2 exp [−2 ln(2)ω2/Γ2]. (The monochromator in

Channel 1 is the mirror image of the one in Channel 2 which has a measured FWHM

bandwidth of 8.5 GHz.) The transfer functions include fitting parameters α1 and

α2 used in Figs. 4.5 and 4.6 to account for transmission losses and the difference in

detection efficiencies of the photon counters.

To obtain the constants A0 and B0, for each case in Figs. 4.5 and 4.6, we measure

the average value of R2 and use Eq. (4.11) to calculate |B0|. We obtain |A0| from

the commutator-preserving condition |A0|2 − |B0|2 = 1. For all curves, the fitting

parameters are taken as α21 = 1.20 × 10−2 and α2

2 = 5.59 × 10−4. These values are in

good agreement with loss measurements and estimates of the photon counter detection

efficiency, where we note that the id400 detector in Channel 1 has a detection efficiency

an order of magnitude larger than the SPCM-AQR-16-FC detector in Channel 2.

4.7 Summary of Nonlocal Modulation

In summary, this work reports the first observation of a quantum effect termed as

nonlocal modulation. We have experimentally shown how distant modulators, when

correlated in the frequency domain, may interfere constructively or destructively.

Though this work has dealt with the effects of synchronously-driven sinusoidal mod-

ulators, a more general statement for nonlocal modulation is that phase modulation

in Channel 1 of the form exp[iΦ1(t)] acts cumulatively with modulation exp[iΦ2(t)]

in Channel 2 so as to produce a frequency domain correlation proportional to the

square of the Fourier transform of expi[Φ1(t) + Φ2(t)]. For this relation to hold, it

is required that Φ1(t) and Φ2(t) both vary slowly as compared to the temporal width

of the two-photon wavefunction.

Chapter 5

Chirp and Compress

5.1 Powerful Biphotons

Temporal correlation determines the efficiency of biphoton nonlinear processes such

as two-photon absorption and sum frequency generation (SFG); if two photons are

highly correlated in time, they are more likely to reach the same atom at the same

time and interact in a two-photon process. The shortest possible correlation width

is a single optical cycle and is achieved when the bandwidths of the signal and idler

are large and the relative phase of the biphoton is properly corrected. As it bears on

either SFG or two photon absorption, a single-cycle biphoton will have an effective

peak power of approximately 12π

~ω2p (the total energy of the biphoton divided by the

temporal window in which it is delivered, e.g., to an absorbing atom).

In addition to being the most powerful two-photon state for nonlinear optics, the

biphoton also possess strong spectral correlation, due to time-energy entanglement.

As mentioned in Chapter 1, signal and idler photons generated by a monochromatic

pump have energies that are perfectly anti-correlated, and nonlinear processes such

as two-photon absorption and sum frequency generation (SFG), which reproduce the

pump frequency, may be resonant with atomic transitions and cavity modes.

30

CHAPTER 5. CHIRP AND COMPRESS 31

!"#$%&''"%

()%*)+,&-.&,)/

012

!3*%$&4

5*$3"6"7!"#$%&''&4

5*$3"6"7

89#$

Figure 5.1: Overview of “chirp and compress.” Biphotons are generated by a chirpedQPM crystal. The signal photon passes through a compressor and the idler througha variable delay. The photons are recombined in a summing crystal which serves asan ultrafast correlator.

5.2 Overview of Chirp and Compress

Ultrashort biphotons may be generated using chirped quasi-phase-matched (QPM)

nonlinear crystals, which allow engineering of the phase matching conditions, and

therefore the frequencies and bandwidths of the photons, simply by varying the local

poling period along the length of the crystal [41]. Because the biphotons are born

with a chirp, subsequent compression is necessary to achieve the shortest possible

correlation width. As a result of nonlocal dispersion cancellation, this compression is

additive and may be performed on either or both of the photons of the biphoton pair.

The amplitude of the biphoton wavepacket may then be measured using SFG as

an ultrafast correlator [2, 11, 12, 42, 43, 44, 45]. The chirp and compress technique,

outlined in Fig. 5.1, allows short biphotons to be generated using long crystals. In this

chapter, we report the generation of chirped biphotons compressed to a correlation

width of 130 fs, as compared to 0.7 ps for a non-chirped QPM crystal of the same

length.

We note the recent work by Nasr and colleagues who have demonstrated Hong-

Ou-Mandel interference using broadband biphotons generated with chirped QPM

crystals. However, the short correlation time is the result of immunity to even-order

terms in the relative phase and is not a measure of the actual width of the biphoton

wavepacket, which remains broad [45]. Recently, Brida et al. [46] have suggested


the use of optical fiber as a practical implementation of the compression proposed by

Harris [11]. We follow their work, using bulk SF6 glass to compress the biphoton.

!"#!

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!"#$

!%&

""%'!()

*+,-./+,

$+../.0(+123.04.+0

Figure 5.2: Schematic of chirp and compress experiment. Biphotons are generatedcollinearly, filtered from the cw pump laser, and separated with dichroic mirrors. Thesignal photon is dispersed by an SF6 glass block, while the idler channel has a variabledelay. The photons are recombined and focused into a PPLN crystal which generatesa monochromatic beam of sum frequency photons.


Our experimental setup is shown in Fig. 5.2. As in Chapter 4, a 20-mm-long

magnesium-oxide-doped stoichiometric lithium tantalate crystal is pumped by a

confocally-focused 532-nm cw laser. The biphotons are generated collinearly with the

pump beam, and all fields are polarized along the extraordinary axis of the nonlinear

crystal. In order to allow comparison between chirped/compressed and non-chirped

biphotons, the generating crystal contains two QPM gratings. In the first, the spatial

frequency of the domain reversals varies linearly from the input to the output end.

In the second, the grating periodicity is independent of position. We switch between

the two gratings by translating the crystal. The pump powers used are 7 and 6 W

for the chirped and periodic gratings, respectively, and are limited by thermal lens-

ing. The room-temperature (298 K) poling period varies from 8.0223 to 8.0481 µm

for the chirped grating, while the period for the non-chirped grating is 8.0008 µm.

The crystal is temperature tuned to generate signal photons with a center wavelength

of 1000 nm as measured by a CCD spectrograph (Jarrell Ash MonoSpec 27 with a


single-mode fiber coupled to the input slit). The corresponding idler photons are

centered at 1137 nm. The spectra for the chirped and non-chirped signal photons are

shown in Fig. 5.3, with phase matching temperatures of 301 and 320 K, respectively.

The generated photon bandwidths are about 250 cm−1 for the chirped grating and

50 cm−1 for the non-chirped grating.

980 1000 1020

0

0.2

0.4

0.6

0.8

! (nm)

Energ

y (

arb

. units) (a)

0

0.2

0.4

0.6

0.8

980 1000 1020

0

1

2

3

4

! (nm)

(b)

0

1

2

3

4

Figure 5.3: Measured spectrum of the signal photon as generated by (a) chirpedand (b) non-chirped QPM gratings in a 20 mm-long SLT crystal. Dotted curves aretheoretical fits (see text). We believe the broadening and asymmetry in (b), comparedto the plane-wave theory, is caused by focusing of the pump beam and thermal lensing.

After a collimating lens, the biphoton and pump beams are apertured using an

adjustable iris set to 2.5-mm diameter. The strong pump beam is removed from the

weak biphotons using a pair of filters (Semrock LP02-568RS-25 and Schott RG695).

After filtering, the biphoton flux measured by a silicon power meter is about 30 nW.

Since silicon has a much lower quantum efficiency at the idler wavelength (1137 nm)

than at the signal wavelength (1000 nm), almost all of the measured power corre-

sponds to signal photons. This implies a pair generation rate of about 1.5×1011

pairs/s.

The biphotons next enter a correlator which separates the signal from the idler.

A dichroic mirror (Semrock LP02-1064RS-25) reflects the signal and transmits the

idler photons. A 10-degree angle of incidence is chosen to shift the transmission edge

of the mirror to the degenerate frequency of 1064 nm. The signal photon propagates

through a fixed path containing a removable 80-mm-long block of dispersive SF6 glass


that is used to compress the biphoton. The idler photon passes through a delay arm

which can be electronically scanned using an automated stage (Newport VP-25XL).

The photons are recombined using a second dichroic mirror and focused into a 1-

mm-long, periodically-poled, magnesium-oxide-doped lithium niobate crystal for SFG

(PPLN, Thorlabs SHG3-1). The PPLN crystal is heated to 433 K for phase matching

and has a calculated acceptance bandwidth of 1100 cm−1. Generated 532-nm SFG

photons are separated from the biphotons using filters (Schott BG39 and Semrock

LL01-532-12.5) and coupled through multi-mode fiber to a single-photon-counting

module (SPCM, PerkinElmer SPCM-AQR-16-FC). The SFG count rate, expressed

as a function of delay in the idler channel, is proportional to the second-order Glauber

intensity correlation function or, equivalently, to the square of the amplitude of the

biphoton wavepacket [11]. The measured signal transmission from the input of the

correlator to the output of the multi-mode fiber is 44%.

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'"

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&"

'"

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)5,

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Figure 5.4: Biphoton correlation measurements for chirped SLT crystal oriented with8.0223 µm poling period on the input edge. SFG count rate as a function of idlerdelay is shown (a) without and (b) with an 80-mm-long SF6 glass block in the signalchannel. Curves are theoretical fits (see text).


To demonstrate chirp and compress, we first orient the SLT crystal with the 8.0223 µm

poling period on the input edge. Figure 5.4 shows the measured SFG count rate as a


function of the idler delay, with and without the SF6 glass block in the signal channel.

The retroreflector in the idler channel is translated in 3-µm steps, delaying the idler

by 20 fs/step. At each reflector position, 5-second measurements of count rate are

repeated 12 times and averaged for a single data point. Each error bar shows the

standard deviation of the 12 measurements divided by√

12. The correlation width

in Fig. 5.4(b) is 130 fs.

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'"

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Figure 5.5: Biphoton correlation measurements for (a) chirped SLT crystal orientedwith 8.0481 µm poling period on the input edge and (b) non-chirped (periodic) grat-ing. Curves are theoretical fits (see text).

In contrast to the above orientation, we also measure the biphoton correlation with

the QPM grating reversed (8.0481 µm period at the input edge) and by using the

non-chirped grating (8.0008 µm period). Figure 5.5 shows the resulting correlation

data. Due to the reduced count rate in these correlations, 24 samples are used for

each data point with error bars depicting standard deviations divided by√

24. The

non-chirped biphoton in Fig. 5.5(b) has a width of about 0.7 ps.

Comparing Figs. 5.4(b) and 5.5(b), using a chirped nonlinear crystal and subse-

quent compression reduces the biphoton correlation width, and increases the peak

power of the biphoton, by a factor of 5.


5.5 Compression and Crystal Orientation

In the down conversion process, photons born at the output edge of the crystal ex-

perience no dispersion and arrive at distant detectors at the same time. In contrast,

photons born at the input edge have a relative group velocity of 1/Vr = 1/Vs − 1/Vi

and experience a relative group delay given by τD = L/Vr [47]. For a material with

positive group velocity dispersion (GVD) over the combined signal and idler spectra,

the idler photon leads the signal photon by τD. Because the photons can be born

anywhere inside the crystal, the relative delay for any given photon pair lies between

0 and τD, thereby giving rise to the wavepacket width. To compress the wavepacket,

the bandwidth ∆ω of the chirped biphotons must be large as compared to 2π/τD,

and the maximum ratio of the compressed to the uncompressed correlation width is

approximately 12πτD∆ω.

When either or both of the photons enters a dispersive medium with positive GVD,

compression may be achieved only with a particular orientation of the generating

crystal. If the photons that are phase matched at the output edge of the chirped

crystal have a larger wavelength difference than those at the input edge, these photons

will be more strongly dispersed by the medium. After an optimal propagation length,

the output-edge photons exit the medium with the same relative delay as the input-

edge photons, and the wavepacket width is therefore minimized. In our experiment,

orienting our crystal with the 8.0223-µm poling period at the input edge satisfies

the requirement for compression with positive GVD. In the opposite orientation, a

dispersive medium must have negative GVD to compress the biphoton.

5.6 Theory of Chirp and Compress

The solid curves in Figs. 5.4 and 5.5 are theoretical fits to the data. To produce

these curves, we follow the theory of Harris describing biphoton generation with a

chirped QPM crystal [11]. We take the spatial frequency of the domain reversals

to vary linearly with crystal length as K0 − ζz, where K0 is chosen to phase match

a particular pair of signal/idler frequencies at the input end of the crystal. The


parameter ζ defines the amount of chirp in the crystal and, in this experiment, is

|ζ | = 1.26 × 105 m−2. The signal frequency ωs = ω is defined to be in the range

ωp/2 ≤ ω < ωp. For a monochromatic pump at frequency ωp, and with ωi = ωp − ω,

the signal and idler annihilation operators at the output of a crystal of length L are

(see Chapter 2)

as(ω, L) = A1(ω)as(ω, 0) +B1(ω)a†i(ωi, 0),

a†i (ωi, L) = C1(ω)as(ω, 0) +D1(ω)a†i(ωi, 0), (5.1)

where, from Ref. [11], A1(ω) = exp[iks(ω)L], D1(ω) = exp[−iki(ωi)], C1(ω) =

B∗1(ω) exp[i(ks(ω) − ki(ωi))L], and

B1(ω) = −(−1)1

4

(

π

2ζ

)1

2

κ exp[iks(ω)L] exp

[−i∆k2(ω)

2ζ

]

×

erfi

[

(1 + i)∆k(ω)

2√ζ

]

− erfi

[

(1 + i)(∆k(ω) + ζL)

2√ζ

]

. (5.2)

In Eq. (5.2), κ is a constant proportional to the pump electric field, ∆k(ω) =

kp(ωp) − [ks(ω) + ki(ωi) + K0], and erfi is the imaginary error function. In general,

kj(ω) = nj(ω)ω/c; in our experiment, all fields are polarized along the same axis of the

nonlinear crystal and see the same refractive index n(ω). Time- and frequency-domain

field operators are related by Fourier transform: aj(t, z) =∫ ∞−∞ aj(ω, z) exp(−iωt)dω.

[In this chapter, the ABCD coefficients are taken to be zero outside the range ωp/2 ≤ω < ωp; see Section 2.4.]

The power spectral density of the signal photon, which also defines that of the

idler photon by energy conservation, is

Ss(ω) =1

2π|B1(ω)|2, (5.3)

with the total pair-generation rate given by R =∫ ∞−∞ Ss(ω)dω. The dotted curves

in Fig. 5.3 are calculated using Eq. (5.3) with the Sellmeier equation for SLT [48].


They are both scaled by the same factor to best fit Fig. 5.3(a). [The phase matching

temperatures used to calculate the curves in Figs. 5.3(a) and (b) are 311 and 328 K,

respectively.]

Equations (5.1) describe the photons at the exit face of the crystal; to propagate

the fields through arbitrary optical systems and model the correlation measurement,

as(ω, L) and ai(ωi, L) may be multiplied by transfer functions H(ω) and G(ωi, τ),

respectively, where τ represents the variable delay in the idler channel. The coeffi-

cients in Eqs. (5.1) become new coefficients A(ω) = H(ω)A1(ω), B(ω) = H(ω)B1(ω),

C(ω, τ) = G∗(ωi, τ)C1(ω), and D(ω, τ) = G∗(ωi, τ)D1(ω).

When the signal and idler photons are recombined and focused into a thin non-

linear crystal for correlation, the SFG photon rate is [11] (see also Appendix B):

Rsum(τ) = η1

[

R2 +

∣

∣

∣

∣

(

1

2π

)∫ ∞

−∞A(ω)C∗(ω, τ)dω

∣

∣

∣

∣

2]

, (5.4)

where the constant η1 is equal to its classical value. The first term in Eq. (5.4)

represents broadband SFG photons produced by the classical summing of uncorre-

lated photons from different pairs. For a monochromatic pump, the second term is

monochromatic and arises from SFG between entangled photons [11]. Dropping the

first term and denoting ψ(ω) = 12πA1(ω)C∗

1(ω), Eq. (5.4) becomes

Rsum(τ) = η1

∣

∣

∣

∣

∫ ∞

−∞H(ω)G(ωi, τ)ψ(ω)dω

∣

∣

∣

∣

2

. (5.5)

We assume that all of the dispersion caused by filters, mirrors, and lenses in the

system is limited to second order (GVD). Because GVD acting on the signal or idler

is additive in its effect on the biphoton (see Chapter 3), we lump all such dispersive

elements into H(ω) as a single, equivalent GVD term. H(ω), which also includes the

dispersion of the removable SF6 glass block in the signal channel, is therefore defined

as

H(ω) = exp

[

iβ

2(ω − ω0)

2 + ikg(ω)Lg

]

, (5.6)

where β is a constant representing the GVD of the optical system, ω0 is the center


frequency of the signal, and the kg(ω) = nSF6(ω)ω/c is the wave-vector for the signal

photon in the SF6 glass block of length Lg. For the case where the glass block is

present in the signal channel, Fig. 5.4(b), we take Lg = 80 mm; otherwise, Lg = 0.

Although all orders of dispersion are included using the Sellmeier equation for SF6

glass, the GVD term is dominant with a value of 143 fs2/mm at the 1000-nm signal

wavelength. The 80-mm glass length approximately satisfies the estimate required

for compression given in Ref. [11]: H(ω) ∼ exp[iτD(ω − ω0)2/(2∆ω)].

5.7 Fitting Theory to the Data

The solid curves in Figs. 5.4 and 5.5 are plots of Eq. (5.5) with G(ωi, τ) = exp(iωiτ).

Figures 5.4(a) and (b) use ζ = 1.26 × 105 m−2, while Fig. 5.5(a) uses ζ = −1.26 ×105 m−2. Figure 5.5(b) uses ζ = 10−6 m−2 to approximate the non-chirped grating.

The scale factor η1κ2 and the dispersion constant β are used as fitting parameters and

are chosen to minimize mean-square error for the data in Figs. 5.4 and 5.5(a), which

correspond to large-bandwidth biphotons that depend strongly on dispersion. Each

theory curve is independently time-shifted for best fit with the data and vertically

shifted by the dark-count level. The optimal values for the fitting parameters are

η1κ2 = 7.83 × 10−21 s/m2 and β = 3600 fs2.

While the relatively-simple theory in this section is sufficient to predict the ex-

perimental results of Figs. 5.4 and 5.5, a more exact numerical method to calculate

ABCD coefficients for biphotons generated in chirped quasi-phase-matched crystals

may be found in Appendix A. The numerical method can be used, for example,

to model high-gain parametric down conversion and imperfect poling periods in the

crystal.

5.8 Summary of Chirp and Compress

In summary, this work reports a first experimental demonstration of the chirp and

compress technique for generating short biphotons. We achieve a compressed bipho-

ton correlation width of 130 fs, a factor of 5 shorter than a periodically-poled crystal


source. The efficiency of sum frequency generation increases by this same factor,

indicating an increase in the effective peak power of the biphoton. Calculation, as

outlined in the previous text, shows that this is the maximum compression that is

obtainable using our lithium tantalate crystal, pump wavelength, and a linear chirp.

Further improvements to the correlation width and peak power of the biphoton will

require third and higher-order dispersion control [49, 50]. [The small side-lobe in

Fig. 5.4(b) indicates the onset of third-order dispersion that must be compensated if

larger chirps are used.] Instead, it is probable that the chirping profile itself may be

designed to pre-compensate for higher-order dispersion further along in the optical

system.

Chapter 6

Resonantly-Enhanced Quantum

SFG

In the previous chapter, we used sum frequency generation as an ultrafast correlator

to measure the widths of our biphotons; however, the measured count rate was very

low: just 25 s−1 above the dark count level [Fig. 5.4(b)]. In this chapter, we explore a

technique to enhance the efficiency of SFG with biphotons—quantum SFG—by using

an optical cavity.

As discussed in Chapter 1, when two photons from the same entangled pair re-

combine in an SFG process, the generated field is monochromatic and scales linearly

with the input power. This occurs when the pair generation rate Rpair and biphoton

width T satisfy the condition RpairT < 1, and we call the process “quantum SFG.”

When RpairT > 1, the biphoton generation rate is sufficiently high—or the corre-

lation width sufficiently long—that two photons from different pairs may recombine

and generate broadband SFG photons (“classical SFG”). These photons have spectra

given by the convolution of the signal and idler spectra, and the generation rate scales

quadratically with input power.

As it bears on biphoton correlation, quantum SFG is sensitive to the biphoton

width and the relative delay between the signal and idler, whereas classical SFG is

a constant noise term. The technique demonstrated in this chapter enhances only

quantum SFG, which is useful for biphoton correlation.

41

CHAPTER 6. RESONANTLY-ENHANCED QUANTUM SFG 42

R =11 R =R2

SFG

crystal

Biphoton

pump

Figure 6.1: Schematic of resonant enhancement. The cavity is transparent to theincoming signal and idler photons. The operator asai denotes the incoming biphotonfield; aq is the resonant sum frequency field, and asum is the generated SFG outputfield.

6.1 Overview of Resonant Enhancement

This chapter describes, both theoretically and with a proof-of-principle experiment,

how one may resonate the sum frequency field (Fig. 6.1) so as to enhance the rate of

quantum SFG and the ratio of the quantum to the classical component. Time-energy

entangled signal and idler photons are generated by a parametric down-converter in

the low gain regime (not shown) with a monochromatic pump so that ωp = ωs + ωi.

These photons are summed in a nonlinear crystal placed inside a cavity whose qth

longitudinal mode (at frequency ωq) is detuned from the pump frequency by δω =

ωp − ωq. The input and output mirrors of the cavity are assumed to be transparent

over the broad biphoton bandwidth and highly reflecting at the sum frequency. The

essence of the quantum enhancement is that the work done by a time-varying dipole

moment against an electric field at its own frequency is increased by the presence

of the resonator. This enhancement occurs pair-by-pair, thus resulting in a linear

dependence of sum-frequency-generated power on the incident biphoton rate.

In the following, we show theoretically that, for a lossless crystal and an output

mirror reflectivity R, the sum frequency power is enhanced relative to the same crystal

without a resonator by a factor of 4/(1 − R). The quantum case has the unusual

property that the conversion efficiency is proportional to the effective peak power of

the biphoton ∼= ~ωp/T , while retaining the same resonance enhancement as a classical


monochromatic field.

6.2 Theory of Resonant Enhancement

Here, we deviate slightly from the theory of Chapter 2 and use the slowly-varying

envelope approximation to describe the signal and idler field operators; however,

since the generated dipole moment for quantum SFG is monochromatic regardless

of the bandwidths of the photons, the conclusions drawn from this theory apply

to single-cycle biphotons. The signal and idler photons are therefore expressed

as as(t, z) = as(t, z) exp [−i(ωs0t− ks0z)] and ai(t, z) = ai(t, z) exp [−i(ωi0t− ki0z)],

where operators with a tilde vary slowly with time and distance. We use a slowly

varying envelope formalism with the standing-wave cavity mode operator written

as aq(t, z) = aq(t) exp(−iωqt) sin(kqz), where kq = qπ/L. We project the broad-

band generated dipole moment operator, proportional to as(t, z)ai(t, z), against the

cavity mode. With the traveling wave SFG field emitted from mode q denoted by

asum(t, z) = asum(t) exp[−i(ωpt− kpz)], the equation for the evolution of aq(t) and its

relation to the SFG field is:

∂aq(t)

∂t+

Γ

2aq(t) = Pq(t) exp(−iδωt) + F(t)

Pq(t) = iκc2

L

∫ L

0

as(t, z)ai(t, z) exp(ikpz) sin(qπz

L

)

dz

asum(t) =√γ aq(t). (6.1)

Denoting the spacing of the cavity modes at the sum frequency by ∆ = c/(2Ln),

the decay rate Γ of photons in the cavity is determined by the mirror reflectivity R

and the single pass power loss ξ. With the output coupling rate γ = ∆(1 − R), the

total (power) decay rate is Γ = 2ξ∆+γ. We take all fields to be plane waves with cross

sectional area A and take the refractive index n and nonlinearity d to be independent

of frequency. With the cavity and summing crystal of the same length, the coupling

constant κc = (d/n2)(µ0~ωpωs0ωi0L/A)1/2. The quantity F(t) is a Langevin noise


operator that has contributions from an incoming wave at the right hand mirror (not

shown) and a macroscopic loss term. Both are negligible at room temperature. The

normalization is such that 〈a†q(t)aq(t)〉 is the total number of photons in the cavity

mode, and 〈a†sum(t)asum(t)〉 is the rate of sum photons exiting the cavity.

The solution of Eqs. (1) for the SFG field operator is

asum(t) =√γ exp

(

−Γ

2t

)∫ t

−∞exp

(

Γ

2t′)

Pq(t′) exp(−iδωt′)dt′. (6.2)

We assume that the signal and idler fields are not depleted in the summing crystal

and transform to the frequency domain using as(t, z) =∫ ∞−∞ as(ω, z) exp(−iωt)dω and

ai(t, z) =∫ ∞−∞ ai(ω, z) exp(−iωt)dω. Pq(t) becomes

Pq(t) = −κc

∫ ∞

−∞

∫ ∞

−∞Φ(ω1, ω2)as(ω1)ai(ω2)

× exp [−i(ω1 + ω2 − ωp)t] dω1dω2

Φ(ω1, ω2) = exp

[

−i∆k(ω1, ω2)L

2

]

sinc

[

∆k(ω1, ω2)L

2

]

,

(6.3)

where Φ(ω1, ω2) is a phase matching factor and ∆k(ω1, ω2) = kq − [ks(ω1) + ki(ω2)].

Equation (6.3) is substituted into Eq. (6.2) to obtain the SFG output field:

asum(t) = −κc

∫ ∞

−∞

∫ ∞

−∞

√γΦ(ω1, ω2)as(ω1)ai(ω2)

Γ/2 − i(ω1 + ω2 − ωp + δω)

× exp [−i(ω1 + ω2 − ωp + δω)t] dω1dω2. (6.4)

For simplicity, we have dropped the z dependence of the field operators, noting

that as(ω1) and ai(ω2) are the signal and idler operators at the input of the cavity,

and asum(t) is the sum-frequency operator at the output of the cavity. We further

ignore all z dependence of the sum-frequency field after leaving the cavity.

For monochromatically-pumped down conversion, the down-converted signal and

idler fields at the input of the summing crystal, as(ω) and ai(ωi), where ωi = ωp − ω,


can be described in terms of initial vacuum fields as0(ω) and ai0(ωi) as

as(ω) = A(ω)as0(ω) +B(ω)a†i0(ωi)

a†i (ωi) = C(ω)as0(ω) +D(ω)a†i0(ωi). (6.5)

Substituting Eqs. (6.5) into Eq. (6.4) and noting the commutator

[aj0(ω1), a†k0(ω2)] = 1

2πδjkδ(ω1 − ω2), we evaluate 〈a†sum(t)asum(t)〉 and find the

rate of SFG photons in mode q exiting the cavity to be

Rcav =(κc

π

)2∫ ∞

−∞

[

γ

Γ2 + 4δω2f(ω) +

γ

Γ2 + 4(ω − ωq)2g(ω)

]

dω, (6.6)

where f(ω) and g(ω) are

f(ω) =

∣

∣

∣

∣

∫ ∞

−∞A(Ω)C∗(Ω)Φ(Ω, ωp − Ω)dΩ

∣

∣

∣

∣

2

δ(ω − ωp),

g(ω) =

∫ ∞

−∞|B(Ω)|2 |C(Ω − ω + ωp)|2 |Φ(Ω, ω − Ω)|2 dΩ. (6.7)

The SFG rate in Eq. (6.6) consists of a quantum term containing the function

f(ω) and a classical term containing the function g(ω) . The quantum term is the

result of generation with correlated photons; it may be shown to scale linearly with

input power and has a monochomatic spectrum [11, 12]. The classical term comes

from SFG with uncorrelated photons and varies quadratically with input power. Its

spectrum is proportional to the convolution of the signal and idler spectra, multiplied

by the cavity lineshape.

In order to normalize the improvement that results from the use of a cavity, we

write the SFG rate for the traveling wave case with no cavity present. This rate is

Rtw =(κtw

2π

)2∫ ∞

−∞[f(ω) + g(ω)] dω, (6.8)

where κtw = κc/∆1/2, and kq in the function Φ(ω1, ω2) becomes kp(ω1 +ω2). As in the


cavity case, the SFG rate consists of a quantum term that varies linearly with input

power and a classical term that varies quadratically [11] (see also Appendix B).

6.3 Cavity Enhancement Ratio

The quantum SFG rate is enhanced by the presence of the cavity by an amount equal

to the ratio of the terms containing f(ω) in Eqs. (6.6) and (6.8). When the cavity

is tuned to resonate the pump frequency (δω = 0), the quantum SFG enhancement

ratio is

ηq =4(1 − R)

[(1 − R) + 2ξ]2. (6.9)

Maximizing with respect to R, the largest quantum SFG enhancement is obtained

for R = 1 − 2ξ; for this reflectivity, the enhancement ratio is equal to the inverse of

the round trip power loss, 1/(2ξ).

To study the effect of the cavity on the classical component of SFG, we assume

that the signal and idler spectra are sufficiently broadband that the dipole moment

at the sum frequency is constant over many cavity modes. In this case, g(ω) is a

constant, and the generated SFG spectrum is periodic with resonant peaks separated

from each other by the cavity free spectral range. To obtain the net enhancement of

the classical term, we integrate over a single free spectral range centered at ωq. The

resulting classical SFG enhancement ratio is then

ηc =2

π

∫ ωq+π∆

ωq−π∆

γ

Γ2 + 4 (ω − ωq)2dω

∼= γ

Γ=

1 − R

(1 − R) + 2ξ, (6.10)

where the second equality follows for sufficiently high finesse that tan−1(2π∆/Γ) →π/2. If the cavity is lossless, ηc = 1. Though generation at resonance is enhanced,

generation off-resonance is suppressed so that the integrated classical SFG enhance-

ment ratio is unity.



We now describe a proof of principal experiment where quantum SFG with frequency-

degenerate, time-energy entangled biphotons is enhanced by a factor of twelve. A

schematic of the experiment is shown in Fig. 6.2. Frequency-degenerate biphotons,

with a calculated bandwidth of 32 nm, are generated by spontaneous parametric

down-conversion in a 20-mm-long periodically-poled, magnesium-oxide-doped, stoi-

chiometric lithium tantalate crystal pumped by an 8-W, cw laser at 532 nm. The

pump, signal, and idler photons are polarized along the extraordinary axis of the

crystal. A four-prism setup is used to filter out the strong pump and to provide dis-

persion compensation for the down-converted biphotons. SFG occurs inside a cavity

that is reflecting at 532 nm and transparent at 1064 nm.

Down-converting

crystal

Pump

532nm

Cavity enhanced SFG

Beam

dump

SPCM

R Rin out

Figure 6.2: Schematic of experiment. Details in text.

The SFG crystal is identical to the down-conversion crystal and is anti-reflection

coated at 1064 nm and 532 nm. The confocal cavity consists of two 20-mm radius of

curvature spherical mirrors, one of which is mounted onto a piezo-electric transducer.

The input mirror has a reflectance of Rin > 99.5%, and the output mirror has a

reflectance of Rout = 95.5%. (The measured single-pass crystal loss is ξ ∼ 2%, so

that the optimum reflectivity of the output mirror is Rout = 1−2ξ ∼= 96%.) The SFG

signal is filtered, collected into a multimode fiber and detected with a single-photon-

counting module (Perkin-Elmer SPCM-AQR-16-FC).

The cavity is aligned in three stages. Initial alignment is performed by optimizing

the coupling of a 532-nm-reference beam into the TEM00 cavity mode. Further align-

ment is performed using the SFG signal by scanning the cavity over a free spectral


0

10000

20000

30000

40000

50000

SF

G c

ou

nt

rate

(co

un

ts/s

)q q + 1

Detuning

0

Figure 6.3: SFG count rate versus detuning. Circles are data points from a singlescan. Solid curve is a theoretical fit using Eq. (6.6) (see text for normalization). Thesmall peak at ∆/2 comes from odd-parity transverse modes in the confocal resonator[51].

range at 50 Hz and maximizing the average detected SFG signal. Final adjustments

are made by slowly scanning the cavity and optimizing the detected resonance peaks.

A typical scan of the cavity over a free spectral range is shown in Fig. 6.3.

The resonant behavior of the SFG process is clearly observed, with peak genera-

tion rates on resonance that are significantly higher than the traveling wave rate of

∼ 3200 counts/s. The solid theoretical curve is a plot of Rcav in Eq. (6.6) normalized

to best fit the average of 26 measurements of the resonant SFG rate.

To better quantify the enhancement, we measure the resonantly-enhanced SFG

rate at various down-converted infrared input powers. For each input power, the

cavity is repeatedly scanned over a single free spectral range to obtain 22-26 samples

of the resonant SFG rate. These are averaged and plotted in Fig. 6.4. The large error

bars (standard deviations) are caused by air currents and temperature instability in

the cavity. Also shown is the traveling-wave SFG rate obtained by removing the

out-coupling mirror.


Figure 6.4: SFG count rate versus infrared input power. Circles are resonant SFGrates. Squares are traveling-wave SFG rates. Curves are theoretical fits (see text).The SFG enhancement ratio is 12.


The solid curves in Fig. 6.4 are theoretical fits to the cavity data using Eq. (6.6)

and to the traveling-wave data using Eq. (6.8). Both are scaled to minimize mean-

square error. This scaling includes the effects of SPCM detection efficiency, crystal

nonlinearity, periodic poling errors, and transverse mode overlap. Based on the fits,

the enhancement ratio is 12 and is about a factor of two less than the theoretical

prediction using Eq. (6.9). The discrepancy may be due to imperfect mode matching

or cavity instability.

We note that, in our experiment, the classical terms containing g(ω) in Eqs. (6.6)

and (6.8) are about three orders of magnitude smaller than the quantum terms con-

taining f(ω). Because a long summing crystal is used, the phase matching factor

Φ(ω1, ω2) greatly restricts the bandwidth of the generated dipole moment, suppress-

ing classical SFG. The dashed line in Fig. 6.4 is tangent to the cavity theory curve

at zero input power. The small deviation from linearity of the theory curve comes

entirely from the quantum term in Eq. (6.6) (see Appendix B).


6.6 Summary of Resonant Enhancement

It seems likely that, by using crystals with lower bulk absorption loss, higher-quality

AR coatings, or perhaps by using a microtoroidal crystal cavity [52], much larger

enhancement factors can be obtained. With sufficient enhancement, measurement of

the biphoton wavefunction amplitude may be possible even for single-cycle biphotons

that require extremely thin summing crystals for correlation. Furthermore, enhance-

ment may generate large enough signals to allow a homodyning technique where the

output of the resonant sum generator is mixed with the original pumping laser, and

the relative path length of the signal and idler beams is varied. One thereby obtains

the complete biphoton waveform, including phase information [44].

In summary we have shown how the quantum nature of sum frequency generation

with time-energy entangled photons allows resonant enhancement. Experimentally

we obtain a 12-fold enhancement of the near-monochromatic sum frequency radiation

that is generated by summing signal and idler radiation with a bandwidth of about

32 nm.

Chapter 7

Conclusion

The unique properties of time-energy entangled photons have made possible new

quantum optical effects that rely on temporal and spectral correlations that are not

bounded by Heisenberg uncertainty. Nonlocal dispersion cancellation was one of the

earliest of such effects and served as motivation for nonlocal modulation as demon-

strated in Chapter 4. It also allows control over the temporal biphoton wavefunc-

tion by acting on just one of the photons, as used in Chapter 5 to compress the

chirped biphoton. Dispersion compensation is a major challenge in classical ultrafast

optics—where even air can broaden a femtosecond pulse—and is equally difficult with

biphotons. Combined with much lower signal levels typical in the quantum optical

regime, the task of generating a femtosecond biphoton is especially daunting; nonlocal

dispersion cancellation simplifies this task.

Nonlocal modulation, the time-frequency analog to nonlocal dispersion cancella-

tion, allows one to control the spectral wavefunction of the biphoton through phase

modulation in a single channel. While it is a new effect that has not yet developed a

compelling application in quantum optics, usage in quantum cryptography is a dis-

tinct possibility. Because phase modulation is a dynamic process and easily varied at

the discretion of the experimentalist (in contrast to dispersion), we envision proto-

cols in which two parties randomly choose from a set of modulation waveforms and

build a secret key based on the frequency correlation between the entangled photons.

With currently-available commercial phase modulators and microwave electronics, it

51

CHAPTER 7. CONCLUSION 52

is possible to modulate at frequencies up to about 40 GHz. The frequency correlation

setup described in Chapter 4 consumes most of an optical table and has a resolution

of about 9 GHz, which allows us to resolve the nonlocal modulation effects. Future

advances in modulator technology will allow more freedom in the choice of waveforms

and will ease the technical requirements for the correlation setup.

The chirp and compress technique demonstrated in Chapter 5 generated a bipho-

ton with a temporal width of 130 fs. This width is a factor of five improvement over

a non-chirped crystal of the same length, but it is still two orders of magnitude wider

than the limit of a single-cycle biphoton. To reach the single-cycle limit requires high-

order dispersion control, which is a difficult task even in classical ultrafast optics. In

the classical realm, advanced correlation techniques such as frequency-resolved opti-

cal gating (FROG) are typically used to characterize short pulses and determine how

to best compress them (often involving prisms, gratings, spatial light modulators, or

chirped mirrors). With our SFG correlation technique, the count rate is prohibitively

low, with correlation scans taking hours to complete. Extension to the single-cycle

domain will require extremely thin summing crystals that reduce the efficiency fur-

ther. The combination of complex, high-order dispersion control and weak correlation

signal poses a significant barrier in the path towards the shortest biphoton. We antic-

ipate that resonant enhancement will become an integral part of ultrafast biphoton

correlators to counteract the reduced efficiency in thin crystals. Also, it may be

possible to emulate the chirped biphoton wavefunction by shaping classical femtosec-

ond pulses, which may then be used with techniques such as FROG to fine-tune the

dispersion compensation.

Chapter 6 introduced a resonant-enhancement technique to improve the efficiency

of biphoton SFG. By using a macroscopic cavity with commercially-available mirrors,

we observed a factor of 12 enhancement in quantum SFG efficiency with broadband

biphotons. In our experiment, we did not stabilize the cavity; the generated SFG rate

was too low to be used directly in a locking circuit. While we believe it is possible

to lock a macroscopic cavity using an external laser, a better technique may be to

use a monolithic cavity such as a microtoroid resonator. Such a cavity, if sufficiently

small, may be stabilized through temperature control. This stabilization is necessary

CHAPTER 7. CONCLUSION 53

for resonant enhancement to be used in an SFG correlator for biphotons.

It is our hope that these new effects raise interest in time-energy entanglement

and lead to new applications in quantum optics.

Appendix A

Extended Theory of Down

Conversion

This chapter provides additional equations useful for modeling biphoton generation

by spontaneous parametric down conversion. In particular, we give the solutions to

the coupled generation equations [Eqs. (2.4)] for the case of high parametric gain. We

also present an iterative integration technique that allows one to numerically calculate

ABCD coefficients for arbitrary quasi-phase-matched crystals, including the effects of

periodic poling errors and nonlinear chirping profiles.

A.1 High-Gain Parametric Down Conversion

The ABCD coefficients in Eqs. (2.7) are valid only in the low-gain regime, where

[bpκ(ω)L]2 < 1. In the high-gain regime, we may not assume that b†i (ωi, z) = b†i (ωi, 0)

and b†s(ω, z) = b†s(ω, 0) in the right-hand sides of Eqs. (2.4); the equations remain

coupled and must be solved simultaneously. (Note that we continue to assume that

the pump field is not depleted.) With s(ω) =√

b2pκ2(ω) − ∆k2(ω)/4, the resulting

ABCD coefficients are

54

APPENDIX A. EXTENDED THEORY OF DOWN CONVERSION 55

A1(ω) = exp

i

[

∆k(ω)

2+ ks(ω)

]

L

×

cosh[s(ω)L] − i∆k(ω)

2s(ω)sinh[s(ω)L]

,

B1(ω) =ibpκ(ω)

s(ω)exp

i

[

∆k(ω)

2+ ks(ω)

]

L

sinh[s(ω)L],

C1(ω) =−ibpκ(ω)

s∗(ω)exp

−i[

∆k(ω)

2+ ki(ωi)

]

L

sinh[s∗(ω)L],

D1(ω) = exp

−i[

∆k(ω)

2+ ki(ωi)

]

L

×

cosh[s∗(ω)L] +i∆k(ω)

2s∗(ω)sinh[s∗(ω)L]

. (A.1)

Figure A.1: Photo of SLT crystal with 8-W pumping. This pump power is just belowthe damage threshold of our crystal. It is unlikely that the gain threshold can bereached with bulk crystals and cw pumping.

For the experiments in this thesis, the maximum pump power used for down

conversion was about 8 W. The slight nonlinearity in the quantum SFG rate versus


input power in Fig. 6.4 can be explained using the high-gain equations above. As the

gain threshold is approached in down conversion, even the quantum term begins to

scale quadratically in input power. The photo in Fig. A.1 shows our stoichiometric

lithium tantalate crystal pumped by 8 W from a 532-nm laser.

A.2 Iterative Technique for Computing ABCDs

The ABCD coefficients of Eqs. (2.7) and (A.1) are valid for bulk crystals and quasi-

phase-matched crystals in which the poling period is constant throughout the length

of the crystal (replacing d with dQPM = 2πd). In realistic quasi-phase-matched crys-

tals, the positions of the domain reversals deviate from their ideal positions with some

statistical variance, resulting in reduced generation efficiency [53]. While these simple

equations still work—with a reduced scale factor that accounts for poling errors—it

may be helpful to calculate exact ABCD coefficients in crystal models that include

poling errors. Also, the ABCD coefficients for chirped crystals from Ref. [11] assume

an ideal linear k-vector chirp, where the poling frequency varies linearly over the crys-

tal length. To allow for nonlinear chirping profiles, or to calculate ABCD coefficients

for chirped crystals in the high-gain regime, we may use an iterative technique in

which each domain of the quasi-phase-matched crystal is analyzed separately, using

the output fields of the previous domain as its inputs.

!m(")!m-1(") !m+1(")

zm zm+1 zm+2zm-1

Lm

Figure A.2: Quasi-phase-matched crystal domains. The domain widths Lm = zm+1−zm are arbitrary.

In a quasi-phase-matched crystal, the sign of the nonlinear coefficient alternates

between positive and negative at specific locations in the crystal that restore phase


matching when the input and output fields slip out of phase. This reversal of the

sign creates a sequence of domains, each of which may be viewed as an independent

bulk crystal. In this technique, we define the positions of each domain edge (starting

with z1 = 0) and the nonlinear coefficients between them, as shown in Fig. A.2. For

the case of a periodically-poled crystal with poling period Λ, zm = (m − 1)Λ/2 and

κm(ω) = (−1)(m−1)κ1(ω).

To begin the technique, we first calculate the ABCD coefficients at the output of

the first domain (m = 1) whose input and output edges are z1 = 0 and z2 = L1, where

L1 is the length of the domain. The field operators at the output of this domain are

given by Eqs. (2.6):

as(ω, L1) = A1(ω)as(ω, 0) +B1(ω)a†i(ωi, 0),

a†i(ωi, L1) = C1(ω)as(ω, 0) +D1(ω)a†i(ωi, 0), (A.2)

where we have replaced L with L1 and κ(ω) (inside the ABCD coefficients) with κ1(ω).

The ABCD coefficients may be taken from Eqs. (2.7) for low gain or Eqs. (A.1) for

high gain. For the next domain, having length L2 and coupling factor κ2(ω), we solve

the coupled equations by taking the output of the previous domain as the input. The

signal and idler operators at the output of the second domain are therefore

as(ω, L1 + L2) = A2(ω)as(ω, L1) +B2(ω)a†i(ωi, L1),

a†i(ωi, L1 + L2) = C2(ω)as(ω, L1) +D2(ω)a†i(ωi, L1). (A.3)

Because the operators on the right-hand side of Eqs. (A.3) are not evaluated at

z = 0, they do not represent the vacuum field, and Eqs. (A.1) cannot be used to

calculate A2(ω) and B2(ω). Instead, we use ABCD coefficients that take the field

operators from position zm to position zm+1:


Am(ω) = exp

i

[

∆k(ω)

2+ ks(ω)

]

Lm

×

cosh[sm(ω)Lm] − i∆k(ω)

2sm(ω)sinh[sm(ω)Lm]

,

Bm(ω) =ibpκm(ω)

sm(ω)exp

[

i

kp(ωp)zm +

[

∆k(ω)

2+ ks(ω)

]

Lm

]

sinh[sm(ω)Lm],

Cm(ω) =−ibpκm(ω)

s∗m(ω)exp

[

−i

kp(ωp)zm +

[

∆k(ω)

2+ ki(ωi)

]

Lm

]

sinh[s∗m(ω)Lm],

Dm(ω) = exp

−i[

∆k(ω)

2+ ki(ωi)

]

Lm

×

cosh[s∗m(ω)Lm] +i∆k(ω)

2s∗m(ω)sinh[s∗m(ω)Lm]

. (A.4)

By substituting Eqs. (A.2) into Eqs. (A.3), we may define a set of ABCD coeffi-

cients that take the operators from z = 0 to z = z3 = L1 + L2:

as(ω, L1 + L2) = A2(ω)as(ω, 0) + B2(ω)a†i(ωi, 0),

a†i(ωi, L1 + L2) = C2(ω)as(ω, 0) + D2(ω)a†i(ωi, 0), (A.5)

where A2(ω) = A1(ω)A2(ω) + C1(ω)B2(ω), B2(ω) = B1(ω)A2(ω) + D1(ω)B2(ω),

C2(ω) = A1(ω)C2(ω) + C1(ω)D2(ω), and D2(ω) = B1(ω)C2(ω) +D1(ω)D2(ω). Since

the coefficients X2(ω) take the field from z = 0 to z = z3, the two domains they tra-

verse may be viewed as a single equivalent domain whose output may be substituted

into a set of equations like Eqs. (A.3) to calculate the field at z = z4. This process

may be iterated, each time including the next domain by substitution, to compute

the field operators at the output of the crystal. The operators Xm(ω) (representing

m domains from z = 0 to z = zm+1) thus obey the recursion relations


Am(ω) = Am−1(ω)Am(ω) + Cm−1(ω)Bm(ω),

Bm(ω) = Bm−1(ω)Am(ω) + Dm−1(ω)Bm(ω),

Cm(ω) = Am−1(ω)Cm(ω) + Cm−1(ω)Dm(ω),

Dm(ω) = Bm−1(ω)Cm(ω) + Dm−1(ω)Dm(ω), (A.6)

and the initial conditions X1(ω) = X1(ω). Equations (A.6) and (A.4) can calculate

the output field operators for arbitrary quasi-phase-matched crystals, where zm, Lm,

and κm are the input-edge position, length, and coupling factor, respectively, for the

mth domain.

Appendix B

Theory of SFG with Biphotons

When the signal and idler photons are focused into a nonlinear crystal of length Ls,

they may recombine to generate photons at the sum frequency. Following the strategy

of Chapter 2, the equation describing the SFG field may be written as

∂bm(ωm, z)

∂z=

∫ ∞

−∞iκs(ωm, ω)bs(ω, z)bi(ωm − ω, z) exp [−i∆k(ωm, ω)z] dω, (B.1)

where the subscript m denotes the Fourier component of the SFG field at frequency

ωm, and ∆km(ωm, ω) = km(ωm) − [ks(ω) + ki(ωm − ω)]. For quasi-phase-matched

crystals, ∆k(ωm, ω) also includes a constant term K0 which specifies the grating k-

vector. The coupling factor κs(ωm, ω) is the same as in Eq. (2.5) but with the pump

frequency replaced by ωm:

κs(ωm, ω) = ǫ0d

[

8~ωmωωiη(ωm)η(ω)η(ωi)AmAsAi

(AmAs + AmAi + AsAi)2

]1

2

. (B.2)

The low depletion assumption is made in the summing process so that bj(ω, z) =

bj(ω, z0), where z0 is the position of the input edge of the crystal. Noting that

aj(ω, z) = bj(ω, z) exp[ikj(ω)z], the general form of the solution to Eq. (B.1) is

am(ωm, z0 + Ls) =

∫ ∞

−∞E1(ωm, ω)as(ω, z0)ai(ωm − ω, z0)dω, (B.3)

60

APPENDIX B. THEORY OF SFG WITH BIPHOTONS 61

where the coefficient E1(ωm, ω) is

E1(ωm, ω) = iκs(ωm, ω)Ls exp

−i[

∆k(ωm, ω)

2+ km(ωm)

]

Ls

×sinc

[

∆k(ωm, ω)Ls

2

]

. (B.4)

The time-domain operator for the sum field is given by the inverse Fourier trans-

form of the frequency-domain operator:

am(t, z0 + Ls) =

∫ ∞

−∞am(ωm, z0 + Ls) exp(−iωmt)dωm. (B.5)

The sum photon rate can be found by evaluating the expectation value a†mam

against the vacuum state:

Rm(t) = 〈a†m(t, z0 + Ls)am(t, z0 + Ls)〉

=1

4π2

∣

∣

∣

∣

∫ ∞

−∞A(ω)C∗(ω)E1(ωp, ω)dω

∣

∣

∣

∣

2

+1

4π2

∫ ∞

−∞

∫ ∞

−∞|B(ω)|2 |C(ωp − ωm + ω)|2 |E1(ωm, ω)|2 dωdωm.

(B.6)

The first term in Eq. (B.6) represents monochromatic SFG photons generated by

the combination of signal and idler photons from a single entangled pair (quantum

SFG). The second term represents broadband SFG photons generated when signal

and idler photons from different pairs recombine (classical SFG).

If the summing crystal is very thin, it is usually a good approximation to replace

E1(ωm, ω) with a constant, e.g. |E1(ωm, ω)|2 = η1. We may define signal and idler

power spectral densities Ss(ω) and Si(ωi) such that the respective photon rates are

Rs = 〈a†s(t, z0)as(t, z0)〉 =∫ ∞−∞ Ss(ω)dω and Ri = 〈a†i(t, z0)ai(t, z0)〉 =

∫ ∞−∞ Si(ωi)dωi.

[These expectation values are similar to that which produces Eq. (2.8).] The densities

APPENDIX B. THEORY OF SFG WITH BIPHOTONS 62

are Ss(ω) = 12π|B(ω)|2 and Si(ωi) = 1

2π|C(ω)|2, where ωi = ωp − ω. With the thin-

crystal assumption, the first term of Eq. (B.6) becomes

R(1)m (t) =

η1

4π2

∣

∣

∣

∣

∫ ∞

−∞A(ω)C∗(ω)dω

∣

∣

∣

∣

2

. (B.7)

The second term of Eq. (B.6) becomes

R(2)m (t) = η1

∫ ∞

−∞(Ss ∗ Si)(Ω)dΩ

= η1RsRi, (B.8)

where (Ss ∗ Si)(Ω) is the convolution of the signal and idler power spectral densities.

If there is no loss in the system, the rate of signal and idler photons are equal, and

the classical term [Eq. (B.8)] is η1R2pair.

Sum frequency generation with biphotons is therefore the sum of two terms: one

which is monochromatic and can be shown to scale linearly with input power, and

one which is broadband—whose spectrum is given by the convolution of the signal

and idler spectra—and scales quadratically with input power.

Appendix C

Molecular Modulation in a Hollow

Fiber

The following section describes work unrelated to entangled photons but which served

as experimental training during the first year of my PhD. Molecular modulation

was pioneered by Steve Harris and Alexei Sokolov in 1998 [54]. By the time the

following work was published, nearly a decade later in 2006, many students wrote

PhD dissertations about molecular modulation. With the help of Sunil Goda, I

transferred the technology to a hollow fiber implementation described below.

C.1 Molecular Modulation

The technique of molecular modulation employs two driving lasers whose frequency

difference is close, but not equal, to that of a Raman transition. Because they are

detuned from the resonance, these lasers adiabatically drive the molecular transition

and thereby modulate the electronic polarizability. To first order, the incident laser

beams are phase modulated and generate a comb of sidebands that have Bessel func-

tion spectral amplitudes [54, 55]. By using variations of the original technique termed

as multiplicative and additive, hundreds of sidebands with a spectral range of four

octaves may be generated [56, 57]. It has been shown that the phases of these side-

bands may be modified so as to produce synthesized waveforms and optical pulses as

63

APPENDIX C. MOLECULAR MODULATION IN A HOLLOW FIBER 64

short as a single cycle [58, 59].

In order to obtain the necessary coherence on the molecular transition, driving

lasers must have power densities of several GW/cm2. For a free-space interaction

length of 25 cm and a Q-switched pulse length of 15 ns, the necessary driving laser

pulse energy is about 100 mJ.

C.2 Overview of Molecular Modulation in a

Hollow Fiber

This appendix reports the demonstration of molecular modulation using a hollow

deuterium-filled optical fiber. We drive the fundamental vibrational transition of

deuterium using laser pulse energies of several mJ incident on a 22.5-cm-long, 200-

µm-diameter fused-silica fiber. We generate twelve sidebands with wavelengths from

1.56 µm to 254 nm. As compared to generation in free space with the same beam

waist, this fiber extends the interaction length by a factor of six, thereby allowing

generation with millijoules of incident pulse energy.

There is now extensive literature demonstrating the use of hollow optical fibers

for the enhancement of nonlinear optical processes. Following the demonstration by

Miles and colleagues in 1974 [60], these fibers now play an important role in areas such

as high-energy-laser pulse compression [61, 62] and high order harmonic generation

[63, 64]. The use of hollow fibers in the generation of ultrashort pulses has been

studied in great detail [65]. As it bears on this work, Sokolov and colleagues have

described the use of a hollow fiber to enhance molecular modulation in deuterium [66].

Raman generation in hollow fibers has been reported using sub-millijoule, ultrashort

pulses to drive Raman transitions in the transient regime [67, 68, 69].

C.3 Experimental Description

Our experimental setup is shown in Fig. C.1. The hollow optical fiber (Polymicro

Technologies TSPE200350) rests on a brass V-groove mount inside a 2.54-cm outer


diameter, 58.7-cm-long thin-wall stainless steel tube. The V-groove keeps the fiber

straight which prevents optical loss and excitation into high-order modes. Steel shims

placed on top of the groove prevent the fiber from bending vertically but allow it

to slide as the brass contracts from cooling. An insulated steel cooling chamber

surrounds the center portion of the tube where the fiber is located. The chamber is

filled with liquid nitrogen for cooling. The tube is filled with 200-torr deuterium gas

which surrounds and fills the hollow fiber. Since the brass mount is in contact with

both the tube and the fiber, it serves as a thermal pathway to cool the fiber and the

deuterium gas inside.

FUSED SILICA HOLLOW FIBER

(22.5 cm, 200 m I.D.)

LIQUID NITROGEN-COOLED GAS CHAMBER

D2

FUSED SILICA WINDOWS

f = 25 cm

D21064 nm

807 nm

1.56 m 254 nm

f = 20 cm

V-GROOVE MOUNT

Figure C.1: Experimental setup. Drive lasers are combined on a dichroic beamsplitter.UV-grade fused-silica lenses focus the drive lasers into the laser-cleaved hollow fiberand focus the output light onto a pyroelectric detector (not shown). The gas chamberis filled with 200-torr deuterium.

We use two solid-state lasers that are nearly two-photon resonant with the funda-

mental vibrational transition in deuterium. The 1064-nm laser is an injection-seeded,

Q-switched Quanta-Ray GCR-290 Nd:YAG laser with a 12-ns pulse width and a calcu-

lated linewidth of 37 MHz. The 807-nm laser is a custom injection-seeded Ti:sapphire

laser with a 16-ns, nearly-Gaussian pulse width. The lasers are combined on a beam

splitter and aligned through a 25-cm lens into the fiber. The alignment path is set by

a backward-coupled HeNe laser exciting the fundamental EH11 mode of the hollow

fiber. The focused spot sizes are 150 µm for the 1064-nm laser and 120 µm for the

807-nm laser. Each laser is attenuated to provide incident pulse energies of 2.8 mJ at

the input of the fiber. The two-photon detuning from the vibrational Raman tran-

sition at 2994 cm−1 is varied by adjusting the frequency of the 807-nm laser. With


the detuning set at 230 MHz below resonance we generate twelve sidebands from

1.56 µm to 254 nm. A 20-cm lens is used at the output of the chamber to refocus the

diffracting sidebands through a Pellin-Broca prism and onto a pyroelectric detector.

C.4 Conversion Efficiency

Figure C.2 shows the generated energies in each sideband with the fiber present

and with it removed. The two-photon detuning is optimized to produce the widest

bandwidth in both cases. The optimal detunings are 230 MHz below resonance in

the hollow fiber and 100 MHz below resonance in free space. With the hollow fiber in

place, eleven sidebands have energies above 0.1 µJ. The remaining 254-nm sideband

is visible when focused onto a fluorescing card. Fig. C.2 shows a strong increase

in generation efficiency from the hollow-fiber guiding. Generation into the fourth

anti-Stokes sideband is 48 µJ in the hollow fiber compared to 20 nJ in free space.

It should be noted that the total energy in all sidebands drops by almost half

when the fiber is present. The lower total output energy is primarily due to coupling

and propagation losses associated with the hollow fiber. Measuring the fiber input

and output energies for the two drive lasers independently yields a total energy trans-

mittance of 62%. When the drive lasers propagate together, the total output energy

including all sidebands is 51% of the incident energy. We believe that coupling of

generated sidebands into higher-order spatial modes and significant propagation loss

at 1.56 µm (21% for an ideal hollow fiber with our dimensions) are responsible for

the additional loss.

We have also performed experiments with five different gas pressures ranging from

157 torr to 375 torr, and we find that 200 torr is the optimal pressure. We use the

generated energy in the fourth anti-Stokes sideband as a metric for optimizing gas

pressure.

Figure C.3 compares generation efficiency in a hollow fiber to an optimized exper-

iment in free space where large interaction lengths and intensities are achieved with

loose focusing and high energy. Similar to previous free-space experiments [57], we

loosely focus the same two drive lasers into a 75-cm long chamber filled with 50-torr


−3 −2 −1 0 1 2 3 4 5 6 7 8 910

−2

10−1

100

101

102

103

104

Anti−Stokes Order

Ou

tpu

t E

ne

rgy (

µJ)

With Hollow Fiber

Without Hollow Fiber

Figure C.2: Energy of generated sidebands at the output of the gas cell vs. anti-Stokes order, with and without a hollow fiber. The input drive energies are 2.8 mJfor each drive laser. The 1064-nm and 807-nm drive wavelengths are defined as -1and 0 orders, respectively.

deuterium gas. The drive energies are 65 mJ for the 1064-nm laser and 56 mJ for the

807-nm laser. The focused spot sizes are 460 µm for the 1064-nm laser and 400 µm

for the 807-nm laser with confocal parameters of about 30 cm. To directly compare

the low-energy hollow fiber and high-energy free-space experiments, we define con-

version efficiency for a particular sideband as the percentage of total incident energy

converted into that sideband. By this definition, fiber losses due to coupling and

propagation reduce generation efficiency. Despite fiber-related losses, and at 20 times

lower incident energy, the generation efficiency in the hollow fiber is roughly equal to

that in free space.


−3 −2 −1 0 1 2 3 4 5 6 7 8 90.001

0.01

0.1

1

10

100

Anti−Stokes Order

Co

nve

rsio

n E

ffic

ien

cy (

%)

(a)

(b)

Hollow Fiber; Low Energy

Free Space; High Energy

Figure C.3: Percentage of total incident energy converted to sideband energy. Gener-ation is shown (a) in a hollow fiber with 5.8-mJ total incident energy and (b) in freespace with 120-mJ total incident energy and a 30-cm confocal parameter.

C.5 Spatial Modes of the Sidebands

Figure C.4 shows a photograph of sidebands generated by a laser-cleaved hollow fiber.

The beams are nearly collimated by a 25-cm fused silica lens placed 20 cm from the

fiber output and are separated by a fused silica prism. This figure is representa-

tive of the best spatial mode quality we have achieved. Spatial mode quality varies

unpredictably with different fibers, although laser-cleaved fibers generally perform

better than hand-cleaved fibers. We note that higher-order sidebands have somewhat

degraded spatial modes.


Figure C.4: Spatial modes of generated sidebands projected onto a white screen.Sideband wavelengths from left to right are: 807 nm (faint), 650 nm, 544 nm, 488 nm,410 nm, 365 nm, 330 nm, and 300 nm. The 650 nm, 544 nm, 488 nm, and 410 nmbeams have been attenuated by a factor of 10 with a neutral density filter.

C.6 Summary of Molecular Modulation in a

Hollow Fiber

Raman generation in a hollow fiber is robust. A laser-cut fiber without end-face

defects is operative for many hours at 5.6 mJ of incident energy. There is little

degradation of spatial mode or generation efficiency. When examined with a 20X

microscope objective, there is very slight burn damage to the polyimide coating on

the input end of the fiber. Hand-cleaved fibers show more significant damage. We

find that the quality of the spatial mode of the backward-coupled HeNe laser is a

good indicator of hollow fiber performance. Fibers that cannot support a clean EH11

HeNe mode show more damage than those that do.

The generation efficiency in a hollow optical fiber will improve at higher input

energies and longer fiber lengths. At present we find that the end-face burning of

the fiber limits our input energy to about 6 mJ. We anticipate experiments aimed at

improving the coupling efficiency and increasing the fiber length. Significant increases

in length may be possible. It is likely that hollow fibers may allow an increase in the

repetition rate of broadband light sources that are based on molecular modulation

from the present-day 10 Hz to the kHz range. For applications at higher power,

tapered fibers may allow the input energy to be raised significantly.

In addition to this work, I also worked with Sunil Goda and Miro Shverdin to

demonstrate coherent control of the molecular modulation process using genetic al-

gorithms and a spatial light modulator [70].

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