ChapterPDF Available

Space-Time Analogies in Optics

December 2011
Progress in Optics 56:1-80

December 2011
56:1-80

DOI:10.1016/B978-0-444-53886-4.00001-0

Authors:

Victor Torres Company

Chalmers University of Technology

Jesús Lancis

Universitat Jaume I

Pedro Andrés

University of Valencia

The so-called space-time analogy constitutes a source of inspiration to understand, engineer, and implement new systems for ultrafast optical signal processing based on concepts borrowed from the well-established field of Fourier Optics. In this review, we start by describing in a comprehensive manner the most basic notions of this analogy and discuss some recent developments with state-of-the-art technology, including the silicon-chip-based time lens and ultra-dispersive Raman devices, among others. Apart from the applications in optical communications, special emphasis is paid on the collateral benefits that the "ultra" appellative brings in fields as diverse as optical frequency comb generation, arbitrary waveform generation, optical coherence tomography, sensors, imaging, or quantum information processing.

Different schemes for implementing a temporal lens: (a) electro-optic phase modulation (EOPM); (b) cross-phase modulation (XPM) in a nonlinear medium; (c) sum frequency generation (SFG); and (d) four-wave mixing (FWM). The solid lines mean amplitude and the dashed ones phase. The star represents complex conjugation.

…

Space-Time analogy transfer rules Space Time

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An optical Fourier transformer is a device that implements physically the Fourier transform of an input signal. In (a), the idea is represented in the spatial domain, where an input beam is focused by a lens. The intensity distribution at the focal plane is the modulus square of the Fourier transform of the transversal profile of the input beam. In (b), we have the temporal counterpart situation, where the output pulse becomes a replica in intensity of the input optical spectrum after traversing an ultrafast optical Fourier transformer.

…

Scheme for implementing an RF-AWG using the temporal far-field phenomenon and a Fourier-transform pulse shaper as reconfigurable spectrum synthesizer. The optical spectrum of a broadband input pulse is synthesized in a user-defined fashion with a Fourier-transform pulse shaper. After the synthesis, the pulse is launched into a GDD medium that satisfies the far-field condition, so that the output intensity is a scaled replica of the engineered spectrum. The light is detected with a high-speed photodiode, so electrical impulses with user-defined current variation can be easily created.

…

+15

Example of RF-AWG using the scheme of Figure 6. (a) synthesized optical spectrum; (b) achieved waveform after frequency-to-time mapping conversion. The frequency-to-time mapping operation is evident in this figure; (c) corresponding RF spectrum. We observe that the frequency content of the synthesized electrical signal is indeed very broad (spanning several gigahertz). (After Lin, McKinney, & Weiner, 2005). c [2005] IEEE.

…

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CHAPTER 1

Space-Time Analogies

in Optics

V´

ıctor Torres-Companya,b,c

, Jes ´

us Lancisc

and Pedro Andr ´

esd

aDepartment of Electrical and Computer Engineering,

McGill University, H3A 2A7 Montreal, QC, Canada

bCurrent address: Department of Electrical and Computer

Engineering, Purdue University, IN-47907, USA

cDepartament de F´ısica, Universitat Jaume I, 12007

Castell´o de la Plana, Spain

dDepartamento de ´

Optica, Universitat de Val`encia, 46100

Burjassot, Spain

Contents 1. Introduction 2

2. Ultrashort Light Pulse Propagation in Dispersive

Homogeneous Media 3

3. First-Order Approximation: Space-Time Analogy 5

4. Elements and their Implementations 6

4.1. Temporal ABCD Matrices 6

4.2. Spectral Dual Formalism 7

4.3. Basic Photonic Components 8

4.4. Some ABCD Matrix Properties 17

5. Coherent Ultra-High-Speed Optical Systems

and their Applications 18

5.1. Tunable Delays 18

5.2. Real-Time Fourier Transformation 20

5.3. Time-to-Frequency Converters 27

5.4. Temporal Imaging Systems 32

5.5. Spectral Imaging System 35

5.6. Ultrafast Fourier Processing Systems 37

5.7. Joint Transform Correlator 41

5.8. Temporal Talbot Effect 45

5.9. Temporal Array Illuminators 49

6. Temporal Optics in the Noncoherent Regime 52

Progress in Optics, Volume 56, c

ISSN 0079-6638, DOI: 10.1016/B978-0-444-53886-4.00001-0.

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2Space-Time Analogies in Optics

6.1. Scalar Coherence Theory for Nonstationary

Partially Coherent Wave ﬁelds 53

6.2. Space-Time Analogy for Partially Coherent

Wave ﬁelds 55

6.3. Propagation of Nonstationary Partially

Coherent Pulses 56

6.4. Temporal van Cittert-Zernike Theorem 57

6.5. Temporal Lau Effect 62

7. Temporal Optics in the Two-Photon Regime 65

7.1. Quantum Two-Photon Correlation Functions

and Two-Photon States 66

7.2. Tailoring Two-Photon States in Spontaneous

Parametric Down-Conversion 67

7.3. Propagation of Ultrafast Two-Photon Wave

Packets 70

7.4. Two-Photon Temporal Far-Field

Phenomenon 72

7.5. Two-Photon Temporal Imaging 73

7.6. Some Remarks on Ultrafast Two-Photon

Processors 74

8. Conclusions 75

Acknowledgments 76

References 76

1. INTRODUCTION

In the last decades, the generation of pulsed beams with pulse durations in

the order of pico- and femtosecond has constituted an important topic for

the physics and engineering communities, where researchers ﬁnd them-

selves continuously pushing the limits to satisfy quite radical premises

such as ultrafast, ultrabroad or ultrasmall. The characteristics of this kind

of optical radiation, that is, broadband spectrum, enormous temporal res-

olution, high peak but low average power, potentially high repetition

rate, and high spatial coherence make it an indispensable tool to develop

many applications in different ﬁelds of science and technology (Fermann,

Galvanauskas, & Sucha,2003).

At a systems level, the so-called space-time analogy constitutes a source

of inspiration to design and implement new schemes for processing

these ultrafast optical signals based on concepts borrowed from the well-

established ﬁeld of Fourier optics (Goodman,1996), leading to what is

now popularly termed as Temporal Optics. The key relies on noting the

mathematical similarity between the equations that govern the parax-

ial diffraction of one-dimensional monochromatic light beams and those

describing the distortion of plane-wave pulses in a ﬁrst-order dispersive

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Ultrashort Light Pulse Propagation in Dispersive Homogeneous Media 3

medium (Akhmanov, Sukhorukov, & Chirkin,1969;Treacy,1969). Thanks

to the continuous advances in the optoelectronics and optical commu-

nication industries, the analogy can be extended to include other ele-

ments such as imaginglenses (Kolner,1994a) or prisms (van Howe &

Xu,2006). These optoelectronic tools have paved an avenue for creating

innovative temporal processing systems, bringing the “ultra” appella-

tive to ﬁelds as diverse as optical interconnects, optical communications,

microwave photonics, biophotonics, or quantum information processing,

among others.

In this review, we shall provide the most basic notions for the under-

standing of the space-time analogy, including the fundamental elements

and their possible implementations with state-of-the-art technology. We

provide a comprehensive approach, based on a temporal matrix formal-

ism (Nakazawa et al.,1998), to describe the characteristics of some of the

most widespread system processing architectures. Special emphasis shall

be paid on their applications in the above mentioned ﬁelds, highlighting

not only their innovative character but offering a comparative study with

respect to other, more conventional, solutions. Finally, we will review the

extension of the space-time analogy to the noncoherent (Lancis, Torres-

Company et al.,2005) and even to the nonclassical (quantum) regime (Tsang

& Psaltis,2006;Torres-Company, Lajunen et al.,2008), where this anal-

ogy still offers some potential to design – and sometimes interpret – new

physical phenomena with the aid of the notions of classical Fourier optics.

2. ULTRASHORT LIGHT PULSE PROPAGATION IN

DISPERSIVE HOMOGENEOUS MEDIA

In this section, we introduce the fundamental equations describing the

linear distortion of an ultrashort light pulse in a dispersive homogeneous

medium. Such a basic physical problem constitutes the cornerstone of

ultrafast optical signal processing.

Let us assume a scalar optical ﬁeld, described by its analytic signal

U(r,t0), propagating linearly in a waveguide with translational symmetry

through the z-direction zand ﬁlled with a homogeneous lossless disper-

sive medium. The propagation constant is β(ω0)=n(ω0)ω0/c, where c is

the speed of light in vacuum and n(ω0)the frequency-dependent refractive

index. We can then write

U(r,t0)=A(x,y)ψ(z,t0)exp[−i(ω0t0−β0z)]. (1)

where ψ(z,t0)is the pulse envelope, which modulates the monochromatic

carrier wave of angular frequency ω0; the constant β0=β(ω0); and A(x,y)

is the transversal spatial distribution of the mode, evaluated at ω0. Once

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4Space-Time Analogies in Optics

A(x,y)is calculated, we only need to know ψ(z,t0)to determine U(r,t0)

at every zposition. The evolution of ψ(z,t0)is then described by a wave

equation.

Since the functional form of β(ω0)is usually unknown, it is very useful

to perform a Taylor expansion (Agrawal,2007)

β(ω0)=β0+β1(ω0−ω0)+β2

2! (ω0−ω0)2+β3

3! (ω0−ω0)3+ · ·· (2)

where βn=dnβ(ω0)/dω0n|ω0=ω0is the nth-order dispersion coefﬁcient of

the waveguide, with n=0, 1, 2, 3, . . . From now on, we will express the

temporal variations of the pulse in a reference framework moving at the

group velocity of the wave packet, t=t0−β1z.

The wave equation describing the envelope distortion is of second

order in z. In order to reduce this equation to ﬁrst order, the slowly

varying envelope approximation (SVEA) is usually invoked in the mul-

ticycle regime (Agrawal,2007). This approximation requires that the

function ψ(z,t)does not change signiﬁcantly through a distance com-

pared with the carrier wavelength, and the pulse duration to be much

larger than the carrier oscillation period. Mathematically, it is translated

into |∂ψ(z,t)/∂z|β0|ψ (z,t)|and |∂ ψ (z,t)/∂ t|ω0|ψ (z,t)|. Both inequali-

ties are guaranteed whenever the optical frequency bandwidth, 1ω, is

much less than the carrier frequency, 1ω ω0.

Within the SVEA, we have

i∂ψ(z,t)

∂z=Hψ(z,t), (3)

where the (Hamiltonian operator) H= −Pn=2inβn

∂n

∂tn. Alternatively,

Equation (3) can be rewritten in the frequency domain by Fourier trans-

formation,1

i∂˜

ψ(z,ω)

∂z=˜

H˜

ψ(z,ω), (4)

where ˜

ψ(z,ω) is the Fourier transform of ψ(z,t)and

H= −X

n=2

βn

n!ωn. (5)

1In the following, the angular frequencies are referred at the baseband. They are referred to the optical

frequencies, ω0, by a shift ω0=ω+ω0.

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First-Order Approximation: Space-Time Analogy 5

This equation can be easily integrated as

ψ(z,ω) =exp "iX

n=2

βnz

n!ωn#˜

ψ(z=0, ω). (6)

Therefore, the dispersive medium acts as a phase-only spectral ﬁlter.

Sometimes, it is useful to write the dispersive terms in the more compact

form 8n=βnz. When n=2, β2is called the group velocity dispersion

(GVD) coefﬁcient, and 82the group delay dispersion (GDD) param-

eter. Analogously, when n=3, β3is the third-order dispersion (TOD)

coefﬁcient.

3. FIRST-ORDER APPROXIMATION: SPACE-TIME ANALOGY

We are particularly interested in the case in which only the ﬁrst term of

the operator contributes,

i∂ψ(z,t)

∂z=β2

∂2ψ(z,t)

∂t2. (7)

Since it implies a second-order expansion in Equation (2), the medium is

said to be parabolic. This ﬁrst-order approximation is physically plausi-

ble whenever 1ω 3|β2/β3|. As an example, for standard single-mode

ﬁber (SMF) and a waveform centered in the telecommunication wave-

length (λ0=1.5 µm), the ﬁber coefﬁcients are β2= −21.7 ps2/km and β3=

0.1 ps3/km. The signal should then have an optical bandwidth shorter

than ∼100 THz.

Equation (7) is a Schr¨

odinger-like equation for a free particle, ubiqui-

tous in Physics scenarios. In particular, this equation is mathematically

identical to that describing the one-dimensional (1D) scalar diffraction of a

paraxial monochromatic beam propagating in the z-direction (Goodman,

1996)

i∂Ue(z,x)

∂z= − 1

2k0

∂2Ue(z,x)

∂x2, (8)

where Ue(z,x)denotes the transversal proﬁle of the 1D beam and k0the

wave number.

The mathematical similarity between the Equations (7) and (8) is what

we know as the space-time analogy, mentioned in the Introduction. Table 1

summarizes the transfer rules connecting both domains. This connection

between diffraction and dispersion was found independently by two dif-

ferent groups at the end of the sixties (Akhmanov, Sukhorukov, & Chirkin,

1969;Treacy,1969). On the other hand, Papoulis had pointed out a formal

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6Space-Time Analogies in Optics

TABLE 1 Space-Time analogy transfer rules

Space Time

Description Variable/Parameter Variable/Parameter Description

Position x t Proper time

Spatial frequency 2πuω(Baseband) angular

frequency

Wave number−11/k0−β2GVD coefﬁcient

Paraxial propagation

factor (distance z)

exp[−i2π2zu2/k0] exp[i82ω2/2] First-order dispersion

(GDD parameter 82)

Spatial lens factor

(focal length f)

exp[−ik0x2/(2f)] exp[iKt2/2] Time lens factor

(chirping rate K)

similarity between diffraction and chirp radar, including the equivalent of

spatial lenses (Papoulis,1968a). Later on, we found in the literature some

theoretical research work performed by Saleh and Irshid (1982) about an

extension into the temporal domain of the Collet-Wolf equivalent theo-

rem regarding spatially partially coherent light (Mandel & Wolf,1995). In

the same decade, it is worth mentioning the temporal equivalents of the

Talbot effect proposed by Jannson and Jannson (1981) and spatial Fourier

transformation (Jannson,1983). However, it was not until the pioneering

work of Kolner and Nazarathy (1989) who, inspired on developing pulse

compression techniques based on electro-optic phase modulators (Kolner,

1988), developed a formal treatment of this analogy to include what we

know today as “time lenses”. It became then evident that a huge avenue

of temporal equivalent systems for ultrafast signal processing was fea-

sible to build, given the instrumentation available at that time (see e.g.,

Lohman & Mendlovic,1992;Kolner,1994a;Godil, Auld, & Bloom,1994;

Papoulis,1994;Mendlovic, Melamed, & Ozaktas,1995).

4. ELEMENTS AND THEIR IMPLEMENTATIONS

4.1. Temporal ABCD Matrices

In the previous section, we have introduced the formalism of the space-

time analogy and advanced that it can be extended to include other

photonic components apart from ﬁrst-order dispersive media, such as

time lenses. In this section, we provide a uniﬁed formalism to describe

the linear distortion of the pulse envelope in a system composed by con-

catenating different elements that are susceptible to be described within

the framework of this analogy: the so-called “Gaussian” systems. Each

element in a Gaussian system is mathematically characterized by a uni-

tary 2 ×2 matrix. The whole system is quantiﬁed by a matrix calculated

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Elements and their Implementations 7

by the multiplication, in the right order, of each of the elements that com-

pose it. This formalism is well known in spatial ﬁrst-order Fourier optics,

which works in the paraxial regime (Siegman,1985;Collins,1970;Palma &

Bagini,1997), and has been adapted into the temporal domain (Dijaili,

Dienes, & Smith,1990;Nakazawa et al.,1998;Mookherjea & Yariv,2001).

Concretely, the action of any linear system on the input complex

envelope of a short light pulse, ψin(t), can be characterized as a linear

superposition

ψout(t)=Zψin (t0)K(t,t0)dt0, (9)

where the system is described by the kernel K(t,t0)and ψout(t)denotes

the output complex envelope. In the case in which the linear system is

Gaussian, the Kernel takes the following form

K(t,t0)=



qi

2πBexp h−i

2BAt02+Dt2−2tt0iif B6= 0

Aexp h−iCt2

2Aiδ(t0−t/A)if B=0

. (10)

Here the constants A,B,C, and Daccount for the system’s matrix coefﬁ-

cients. The reason why these systems are called Gaussian is because of the

quadratic dependence in the exponential term.

4.2. Spectral Dual Formalism

In temporal optics, there are some situations in which the Fourier trans-

form of the envelope, ˜

ψ(ω), may be the physical magnitude of interest,

rather than ψ(t). Of course, they are connected each other by a Fourier

transform relation and both carry the same quantity of information. Since

any linear system in time is linear in frequency too, it is useful to pro-

vide a similar analysis of Equations (9) and (10) in the dual space. By dual

we mean that these devices behave identically from a mathematical point

of view, but their action is performed in the Fourier domain (Papoulis,

1968b). Thus, we can write

ψout(ω) =Z˜

ψin(ω0)˜

K(ω,ω0)dω0, (11)

where the Kernel in the frequency domain forms a Fourier transform pair

with the Kernel in time, that is,

K(ω,ω0)=ZZ K(t0,t00)exp[i(ωt0−ω0t00)]dt0dt00. (12)

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8Space-Time Analogies in Optics

For the particular case in which the system is Gaussian, it becomes Gaus-

sian in the spectral domain too and therefore a similar structural form of

Equation (10) holds,

K(ω,ω0)=



qi

2πBωexp h−i

2BωAωω02+Dωω2−2ωω0iif Bω6= 0

Aωexp h−iCωω2

2Aωiδ(ω0−ω/Aω)if Bω=0

. (13)

Of course, the dual coefﬁcients are related to the matrix parameters A,B,

C, and Din the time domain. By inserting Equation (10) into (12), we easily

obtain

AωBω

CωDω=A B

C D−1

=D−C

−B A . (14)

Equations (12) and (14) imply that the action of a Gaussian system in the

spectral domain is mathematically identical to the action of a Gaussian

system in the temporal domain whose matrix elements are provided by

the inverse matrix. The implications of this statement will become clearer

in the following sections.

4.3. Basic Photonic Components

Within the previous matrix formalism, we now proceed to describe the

mathematical structure of some devices that will probe very useful in tem-

poral optics applications and describe brieﬂy their implementation with

current technology.

4.3.1. Group Delay Dispersion (GDD) Circuit

As previously advanced, a GDD circuit is an element designed to

introduce a quadratic phase factor in the spectral domain, ˜

ψout(ω) =

exp[i82ω2/2] ˜

ψin(ω), where 82is the GDD coefﬁcient. The corresponding

matrix is

A B

C D=182

0 1 . (15)

These photonic components, as highlighted in Table 1, constitute the tem-

poral equivalent of the paraxial diffraction. However, we must note that

while diffraction only takes place for positive wave numbers in tempo-

ral optics, the GDD parameter can be positive or negative, depending

on the material component, waveguide structure, and the signal’s carrier

frequency. This subtleness certainly opens exciting new possibilities for

ultrafast signal processing.

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Elements and their Implementations 9

4.3.1.1. Possible Implementations

There exist many different photonic devices that can act as a GDD circuit.

Table 2 summarizes some of the most commonly used in the telecommu-

nications band. The most popular solution is the single-mode ﬁber. In this

case, the dispersion comes from the frequency dependence of the refrac-

tive index of the material, and the GDD parameter is 82=β2z, where β2

is the GVD parameter. Therefore, the GDD can be easily controlled with

the ﬁber length. Alternatively, dispersion compensating ﬁbers (DCFs) can

be used. Thanks to the special core structure design, with shorter length

higher dispersion can be achieved. Both SMF and DCF are especially

attractive devices to achieve, when needed, high dispersion amounts.

Besides, it becomes particularly interesting that Raman ampliﬁcation can

be implemented together with dispersion in the same ﬁber link.

Fiber solutions however present technical difﬁculties owing to the rel-

atively high nonlinear parameter, which makes it difﬁcult to handle them

for dispersion management using powerful broadband pulses. In this

case, alternatives such as grating pairs or prisms can be more suitable

candidates. In other situations, it is interesting to have more compact

dispersive circuits and/or capable to operate in broader or just differ-

ent spectral windows. For this aim, different proposals including chirped

Bragg gratings inscribed in silica glass or silicon-on-insulator (SOI) have

received increasing attention in the last years. In this case, the GVD

coefﬁcient can be engineered with a proper choice of the waveguide

dimensions and structure. The low nonlinearity and compactness of these

devices make them highly suitable for several optical communication

applications. Other alternatives such as photonic crystal ﬁbers (PCFs)

are very attractive candidates for engineering the GDD amount over a

much broader spectral range, but the nonconventional core size, together

with the difﬁculty to fabricate them, imposes signiﬁcant constraints that

prevent them to use in optical communication systems, where other

dispersion compensator devices easier to manage are readily accessible.

4.3.2. Temporal Lens

A temporal lens consists of any element that imprints a temporal quad-

ratic phase factor on the envelope of a light pulse without altering the

intensity proﬁle, that is, ψout(t)=exp[iKt2/2]ψin(t). This lensing device

performs the same operation as a conventional lens does it on a spatial

light beam, that is, it modiﬁes quadratically the phase of the input wave

front. Taking into account Equations (9) and (10), we can conclude that the

matrix of a temporal lens has the form

A B

C D=1 0

−K1. (16)

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TABLE 2 Several examples of photonic devices acting as GDD circuits in the telecommunications band (λ=1.5 µm). For the grating pair,

the formula parameters mean bthe spacing between gratings, dthe period and γthe incidence angle. For the linearly chirped ﬁber gratings

(LCFGs), ngis the group index, 3lthe grating period, and Lthe ﬁber length. Typical achievable numerical values are shown as example.

GDD 82Bandwidth 1ω

Parameter Typical value (ps2) Parameter Range (THz) Tuning Ref.

SMF β2z∼ −2,500 ∼β2/(10β3)∼100 No [a]

DCF β2z∼ −2,500 ∼β2/(10β3)∼100 No [b]

Grating pair −λb

2πc2λ

d21

1−λ

d−sin γ2!3/2−0,29 −c

5λ0

1−λ

d−sin γ2!

1+λ

dsin γ−sin2γ∼12 Yes [c]

LCFGs λ2ng3l/(πc2)∼[−2,500 2,500] c3lL/λ2∼2.8 (C-band) Yes [d]

Integrated gratings n/a ∼[−9 9] n/a ∼0.5−1.5 No [e]

[a]Agrawal [2007]

[b]Gr ¨

uner-Nielsen, Wandel, Kristensen, et al. [2005]

[c]Weiner [2009]

[d]Venghaus [2006]

[e]Tan, Ikeda, Saperstein, et al. [2008]

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Elements and their Implementations 11

The coefﬁcient Kis a real number called chirping rate. This parameter

is closely related to the concept of the focal length (Kolner,1994b), and it

accounts for the quantity of spectral broadening that a pulse acquires after

time lensing. The higher the chirping rate the higher the broadening is,

regardless of the sign of K.

4.3.2.1. Possible Implementations

Electro-Optic Phase Modulation The most straightforward way to

imprint a desired quadratic phase modulation on a short light pulse

is using an electro-optic phase modulator (EOPM). An EOPM is usu-

ally built with a nonlinear crystal having a high electro-optic coefﬁcient,

such as LiNbO3. By applying an electrical radio-frequency (RF) signal

on the crystal, the optical ﬁeld propagating through acquires a phase

proportional to the applied voltage. If the electric ﬁeld is just com-

posed by a single-tone RF waveform and assuming the optical pulse

is synchronized with the maximum of the RF signal, we get ψout(t)=

exp[i1θ cos(2πfrt)]ψin(t). Here, fris the frequency of the electrical wave-

form and 1θ is a real constant, called the modulation index, propor-

tional to the peak-to-peak voltage of the applied waveform, V0,1θ =

πV0/(2Vπ). The coefﬁcient Vπestablishes the voltage needed to acquire

aπ-phase shift and depends on the crystal material and waveguide struc-

tural parameters. For standard LiNbO3waveguide in travelling-wave

conﬁguration, it is possible to achieve typical values around Vπ∼5V.

To relate these parameters with the time-lens chirping rate, we further

require the temporal pulse width to be shorter than the period. In this case,

we can approximate ψout(t)=exp[i1θ ] exp[−i1θ (2πfrt)2/2]ψin (t). Then,

by direct inspection we get K= −4π21θf2

In radar literature, the capabilities of quadratic phase modulation to

increase the bandwidth of a temporal signal were very well known

(Klauder et al.,1960), even the analogy of chirping with spatial lenses was

reported as early as in the sixties (Papoulis,1968a). In optics, it was also

known that by chirping a light pulse (Giordmaine, Duguay, & Hansen,

1968) or a continuous-wave laser (Bjorkholom, Turner, & Pearson,1975)

one could enhance its optical spectrum and subsequently obtain a shorter

pulse by properly compensating for the instantaneous frequency chirp

with an adequate passive dispersive medium (Treacy,1968). The merit

of Kolner’s works on time lenses (Kolner,1988;Kolner & Nazarathy,

1989) was that of unifying these discoveries and putting them in the more

general framework of the space-time analogy, opening a door to new,

more sophisticated, ultrashort pulse processing schemes such as temporal

imaging (Bennett, Scott, & Kolner,1994).

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12 Space-Time Analogies in Optics

The advantages of using the EOPM time lens include wavelength trans-

parency, high-speed operation, and easiness of integration with optical

communication components. On the other hand, a simple Taylor expan-

sion analysis on the above approximation concludes that the pulses over

which the EOPM time lens operate must satisfy σ0(√πfr)−1, where σ0

is a measure of the pulse duration. This restricts the EOPM time lens to

operate over low duty cycle (typically <30%) pulse trains (Kolner,1994a).

Longer pulses get affected by the higher order (nonquadratic) terms in

phase modulation. In the context of the space-time analogy, this devia-

tion from the ideal quadratic operation can be understood as the temporal

equivalent of a lens’ spatial aberrations, a subject that has been extensively

studied in the literature (Bennett & Kolner,2001;Marinho & Bernardo,

2006).

To extend the time window and chirping rate, alternative solutions

involving nonlinear phenomena have been reported, which we explain

brieﬂy in the following paragraphs.

Cross-Phase Modulation (XPM) An elegant all-optical alternative for

time lensing consists of using a highly nonlinear ﬁber. The idea is that two

equally polarized optical ﬁelds are nonlinearly coupled due to the Kerr

effect (Agrawal,2007), which is a phenomenon based on the third-order

nonlinear susceptibility, χ(3)

, and therefore takes place in all dielectric

materials. In the simplest case where dispersion, losses, and walk-off are

ignored, the output pump envelope, ψp, out(t), and signal’s, ψs,out(t), are

analytically related to their input complex envelopes

ψs,out(t)=ψs,in (t)exp[iγL(|ψs,in(t)|2+2|ψp,in (t)|2)], (17a)

ψp,out(t)=ψp,in (t)exp[iγL(|ψp,in(t)|2+2|ψs,in (t)|2)]. (17b)

Here, γis the nonlinear Kerr coefﬁcient and Lthe length of the nonlinear

medium. We are assuming the envelope square has dimensions of power.

We shall focus on the particular case in which the signal is weak, so that

the ﬁrst term in the exponential (self-phase modulation), in Eq. (17a), can

be neglected when compared to the other one (cross-phase modulation).

In this case, ψs,out(t)=ψs,in (t)exp[i2γL|ψp,in(t)|2], so that the pump ﬁeld

modiﬁes the instantaneous phase of the signal, regardless of the shape of

the input pulse. For the particular case in which the pump ﬁeld is Gaus-

sian, ψs,out(t)=ψs,in (t)exp[i2γLP0exp(−t2/T2

0)], where P0is related to the

pump’s peak power and T0is a parameter related to its duration. If the

temporal width of the signal is much shorter than the temporal width

of the pump’s, ψs,out(t)≈ψs,in (t)exp[i2γLP0] exp[−i2γLP0t2/T2

0]. In this

case, by direct inspection we get K= −4γLP0/T2

0. The temporal window

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Elements and their Implementations 13

over which the phase modulation remains approximately quadratic

equals the pump pulse duration T0. The main advantage of using XPM

as a medium for time lensing is the fact that, for some materials, the Kerr

coefﬁcient, γ, can be extremely large.

XPM as a mechanism for time lensing was ﬁrstly reported by Moura-

dian et al. (2000), who used a pump pulse with Gaussian intensity proﬁle.

However, one can easily realize that by using preshaped parabolic-like

pump pulses (Finot et al.,2009), the temporal aberrations are substan-

tially removed. This XPM time lens with a parabolic-pump conﬁguration

has been recently reported, achieving a nearly aberration-free operation

(Hirooka & Nakazawa,2008;Ng et al.,2008).

Sum Frequency Generation (SFG) If one relaxes the wavelength

transparency operation of the time lens, parametric phenomena can be

used to achieve the desired time lens operation. Sum-frequency genera-

tion, a second-order nonlinear phenomenon, was indeed exploited in the

ﬁrst conﬁgurations for temporal imaging (Bennett, Scott, & Kolner,1994;

Bennett & Kolner,1999). The idea consists of mixing the pulse over which

we want to imprint the chirping, ψs,in(t), with a pump pulse that has

been previously dispersed in a GDD circuit, ψp,in(t). Because of the dis-

persion, and assuming a pump pulse with Gaussian proﬁle, we can write

(Agrawal,2007)

ψp,in(t)∝exp "−t2

2T2

z#exp "−iCzt2

2#. (18)

Here, the temporal duration and chirping rate of the pump pulse are con-

trolled by the GDD amount of dispersion, that is, Tz=T0[1 +(82/T2

0)2]1/2

and Cz=82/(T4

0+82

2).T0is the initial pump pulse duration. After the

predistored pump pulse is mixed with the signal pulse, we look at the

sum-frequency component, which is isolated using a spatial or frequency

ﬁlter, depending on the chosen conﬁguration for SFG. Assuming ideal

phase-matching conditions and ignoring pump depletion and walk-off in

the nonlinear medium, the complex envelope for the up-converted ﬁeld is

ψs,out(t)∝ψp,in (t)ψs,in(t)=exp "−t2

2T2

z#exp "−iCzt2

2#ψs,in(t). (19)

By direct inspection, we conclude that the chirping rate for a time lens

implemented with SFG is Cz, over a time window Tz. However, the tem-

poral duration of a parametric time lens and the chirping rate cannot be

tuned independently. For example, at 82=T2

0, the chirping rate achieves

its maximum possible value, but the temporal window is only Tz=√2T0.

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14 Space-Time Analogies in Optics

On the other hand, it is very important to note that in this parametric

case, the chirping rate coefﬁcient does not depend on the nonlinear crys-

tal parameters but on the pump pulse duration and dispersion amount. If

one is able to balance the trade-off between the chirping rate and temporal

window, there are beneﬁts of using parametric processes for time lensing.

The ﬁrst conﬁgurations demonstrated for SFG-based time lensing

involved the use of beta barium borate, leading to phase matching over

a huge bandwidth, albeit with a rather bulky setup (Bennett & Kolner,

1999). Recent advances in quasi-phase matching techniques in waveg-

uides (Yang et al.,2004) have enabled the generation of more compact

devices to produce SFG-based time lenses (Bennett et al.,2007).

Despite these advances, residual second-harmonic generation from the

pump pulse and the frequency shift of the chirped waveform make it

difﬁcult to launch the up-converted signal into a subsequent optical sys-

tem for further processing that operates in the same spectral window. In

the next paragraphs, we review a recent alternative that overcomes these

difﬁculties, though still making use of a parametric nonlinear process.

Four-Wave Mixing (FWM) The conﬁguration is very similar to the one

used for SFG, since it relies on mixing a predistorted pump pulse with the

signal over which we desire to imprint the chirp. Instead of using SFG,

which depends on the second-order nonlinear susceptibility and therefore

is only possible in certain materials, the idea is to exploit the four-wave

mixing effect, which is due to the third-order susceptibility and therefore

takes place in all dielectric materials (Salem et al.,2008). After conversion

in the nonlinear medium and assuming ideal phase-matching conditions,

the envelope of the idler wave becomes

ψs,out(t)∝ψ2

p,in(t)ψ ∗

s,in(t)=exp "−t2

z#exp h−iCzt2iψ∗

s,in(t). (20)

Now the chirping rate is 2Czand the time lens operates in an effective

time window Tz/√2, thus leading to the same ﬁgure of merit as with

SFG. However, the beneﬁts of FWM rely on the fact that the converted

wavelength is very near to that of the signal, so it becomes easier to per-

form further processing. Even more, with today’s technology, FWM can

be easily achieved in highly nonlinear ﬁbers (Agrawal,2007;Saruwatari,

2000), nanophotonic wires built in silica (Foster, Turner et al.,2008), silicon

(Osgood et al.,2009), highly nonlinear chalcogenide waveguides (Pelusi

et al.,2008) or even doped silica glass (Ferrera et al.,2008), and all of them

are becoming more used for advanced optical communication applica-

tions. It is important to note that, unlike with the previous conﬁgurations,

the FWM time lens also conjugates the input complex ﬁeld. This change

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Elements and their Implementations 15

must be taken into account as it usually leads to different dispersion com-

pensation rules for performing optical signal processing applications like,

for example, temporal imaging (Kuzucu et al.,2009).

As shall become evident in the following sections, the possibility to

realize the time lens using FWM has permitted to implement for the ﬁrst

time many schemes proposed in the framework of temporal optics for

ultrafast signal processing, with applications in modern optical commu-

nication systems. As with other parametric processes, we note that once

the phase-matching conditions are achieved, the chirping rate parameter

only depends on the settings of the pump pulse. However, a word of cau-

tion must be brought up here. Although a broad tuning range of chirping

rates can be achieved with parametric time lenses, the fact that the pump

pulse needs to be predistorted imposes a fundamental limit on the maxi-

mum speed at which devices based on these time lenses can work. This is

the reason why most of the reported schemes implement the processing

at a relatively low speed (megahertz rate). As the optical communication

ﬁeld demands for higher operation speeds, further research work should

be done to upgrade the time lens operation.

Table 3 summarizes the main characteristics of the different devices

proposed here to work as time lenses. Figure 1 represents schematically

the possibilities for implementing the earlier mentioned time lenses.

4.3.3. Temporal Gaussian Modulator

This device ﬁlters an input signal in the temporal domain, so that the

output pulse envelope is given by ψout(t)=exp[−02t2/2]ψin (t), where

0−1is a real constant related to the gate duration. This device can be

TABLE 3 Several physical principles upon which time lenses can be constructed.

Some typically achievable rates and time apertures are shown as an example, as well

as the deﬁnition of a ﬁgure of merit (FOM) aimed to account for the time aperture and

the power of the chirping lens.

Chirping KTemporal window T

Typical

Parameter value (ps−2)Parameter Range (ps) FOM (|K|T2)

EOPM −4π21θ2f2

r∼–0.5 (3fr)−1∼30 41θ2

XPM −4γLP0/T2

0∼–5 T0∼100 4γLP0

SFG 82/82

2+T4

0∼–0.1 T0

1+ 82

0!2



1/2

∼3 1

FWM 282/82

2+T4

0∼–0.2 T0

√2

1+ 82

0!2



1/2

∼2 1

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“Wolf 04-ch01-001-080-9780444538864” — 2011/11/18 — page 16 — #16

16 Space-Time Analogies in Optics

Input pulse Output pulse

XPM

Input pulse

Probe pulse

Output pulse

SFG

Probe pulse

Output pulse

FWM

Input pulse

Probe pulse

Output pulse

GDD circuit

(a) (c)

(

)(

)

EOPM

FIGURE 1 Different schemes for implementing a temporal lens: (a) electro-optic

phase modulation (EOPM); (b) cross-phase modulation (XPM) in a nonlinear medium;

mean amplitude and the dashed ones phase. The star represents complex

conjugation.

understood as a temporal lens with an imaginary chirping rate. Thus, the

matrix associated is

A B

C D=1 0

−i021. (21)

This action can be readily achieved with an active amplitude modulator

providing Gaussian-like modulation, such as an electro-optic modula-

tor (EOM) in a push–pull conﬁguration or an electroabsorption modulator

(EAM). For these cases, assuming that an external clock signal with fre-

quency frdrives the devices, the typically achieved values for the pulse

gate are around 1/(3fr)for EOMs and 1/(4fr)for EAMs.

4.3.4. Spectral Gaussian Modulator

This device ﬁlters spectrally the complex ﬁeld envelope as ˜

ψout(ω) =

exp[−12ω2/2] ˜

ψin(ω), where 1is a real parameter related with the spectral

width. The corresponding matrix is

A B

C D=1i12

0 1 . (22)

Spectral Gaussian ﬁlters can be conveniently fabricated using ﬁber Bragg

grating devices, microring resonators, Fabry-Perot cavities, followed by a

long etcetera. With optical communication instrumentation, today we can

ﬁnd suitable ﬁlters satisfying quite demanding constraints.

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Elements and their Implementations 17

4.4. Some ABCD Matrix Properties

The aim of this section is to enumerate some of the most important char-

acteristics of the temporal ABCD matrix systems, which will provide

very useful to understand the properties of the processing systems to be

described in the next section. These characteristics are adapted from the

analysis performed in the spatial domain (Siegman,1985;Collins,1970;

Palma & Bagini,1997).

1. The determinant of an ABCD matrix satisﬁes AD −BC =1.

2. Any Gaussian system having a matrix coefﬁcient B=0 conjugates

the input and output, meaning that the output intensity becomes a

scaled replica of the input, with a scaling factor being provided by the

matrix coefﬁcient A, that is, Iout(t)=1

|A|Iin(t/A). These are the so-called

temporal imaging systems (TISs).

3. A Gaussian system having C=0 behaves as a temporal afocal system.

4. Combining 2 and 3, and given the spectral formalism developed in

Section 4.2, with Equation (14) in particular, we conclude that any

temporal imaging system is a spectral afocal system and vice versa.

5. Conversely to 4, any afocal system (C=0)behaves as a spectral imag-

ing system (Bω=0), meaning that the output spectrum of a signal

that has traversed a temporal afocal system constitutes a scaled replica

of the input’s optical spectrum, with a scaling factor given by D(or

Aω), that is, Sout(ω) =1

|D|Sin(ω/D). These systems are called spectral

imaging systems (Torres-Company, Lancis, & Andr ´

es,2007a).

6. Any Gaussian system having A=0 is an optical Fourier transformer,

meaning that the output intensity pulse becomes a scaled replica of the

optical spectrum of the input pulse. In this case, the scaling factor is

provided by the matrix coefﬁcient B, that is, Iout(t)=1

|B|Sin(ω/B).

7. Following the dual analysis, a system having Aω=0 (D=0)consti-

tutes a time-to-frequency transformer, where the optical spectrum of

the output signal becomes a scaled replica of the input intensity proﬁle,

with a scaling factor provided by the matrix coefﬁcient Bω(i.e., −C). In

mathematical terms, Sout(ω) =1

|C|Iin(−t/C).

8. Gaussian systems having B=0 and C=0 perform simultaneously a

temporal and spectral imaging of the input signal, which is only possi-

ble if both the amplitude and phase are scaled. These are called scaling

systems and perform a temporal image of the input’s complex ﬁeld.

The temporal scaling factor is provided by the matrix coefﬁcient A.

Therefore, ψout(t)=1

|A|1/2ψin(t/A).

9. Any Gaussian system with Adifferent from zero can be described

with the combination of three equivalent systems: a temporal lens

with equivalent chirping coefﬁcient Keq = −C/A, a scaling system with

equivalent scaling factor meq =A, and a GDD circuit with equivalent

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“Wolf 04-ch01-001-080-9780444538864” — 2011/11/18 — page 18 — #18

18 Space-Time Analogies in Optics

GDD coefﬁcient 82eq =B/A. This can be easily seen by the fact that any

2×2 matrix can be rewritten as (see, e.g., Lancis et al.,2004)

A B

C D=1 0

C/A1A0

0 1/A1B/A

0 1 . (23)

5. COHERENT ULTRA-HIGH-SPEED OPTICAL SYSTEMS

AND THEIR APPLICATIONS

In the last section, we reviewed some of the most widely used pho-

tonic components in the framework of the space-time analogy. The aim

of this section is to provide a comprehensive review of the systems built

using those photonic components and pointing out the unique capabilities

which make them so attractive in a rich variety of ﬁelds.

5.1. Tunable Delays

Many applications today require an optical delay line providing an accu-

rate and programmable temporal delay with very ﬁne resolution. The

most widespread solution relies on using precise mechanical translation,

where the timing of a signal is controlled by varying its free-space path

length. In the framework of the space-time analogy, we can envision that

introducing a controllable delay on a short light pulse is fully equiva-

lent to introduce a controllable lateral shift on a 1D beam (van Howe &

Xu,2005a). The latter scheme can be accomplished by a sequence of two

prisms. The ﬁrst prism tilts an incoming light beam which, after prop-

agating in free space a certain distance, is redirected towards the same

direction as it had in the beginning. There appears a lateral shift with

respect to the input beam, which is proportional to the travelled distance

and the angle tilt introduced by the prisms. The counterpart temporal sit-

uation is a widespread technique and it involves three main operations.

First, a wavelength conversion scheme changes the carrier frequency of

the incoming light pulse whose temporal delay is to be controlled. This

element plays the role of a “temporal prism” since a change in the carrier

frequency of a pulse is equivalent to a change in the propagating wave

number. The second element is a GDD circuit, in complete analogy with

the action of diffraction. And ﬁnally, a wavelength reconverter shifts back

the frequency of the pulse to its original value, in the same way the last

prism leads to a beam whose propagation direction is parallel to that of

the input beam. This general arrangement is known in the literature as the

“wavelength conversion and dispersion” scheme and it has been widely

used to achieve fractional delays exceeding 1000 (Sharping et al.,2005) at

high speed and in a format transparent operation (Nuccio et al.,2010).

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“Wolf 04-ch01-001-080-9780444538864” — 2011/11/18 — page 19 — #19

Coherent Ultra-High-Speed Optical Systems and their Applications 19

Frequency

shifter

Time

GDD circuit Φ2

Δω

FIGURE 2 Scheme for understanding the role of the frequency shift in an optical

delay line. Two pulses with identical complex envelopes but with different frequency

carrier suffer the same distortion in a GDD circuit but undergo a delay proportional to

the difference between their frequencies and the GDD amount.

To understand mathematically the temporal delay that can be achieved

from the previous discussion, we make use of the system sketched in

Figure 2. A short light pulse is split into two. The copy travelling through

the upper arm is shifted in frequency by a quantity 1ω, whereas the

copy travelling through the lower arm remains at the same ﬁxed car-

rier frequency. We now consider both pulses travelling through identical

GDD circuits. Taking into account the Equations (9),(10), and (15), the

corresponding output pulse envelopes transform as

ψdown(t)=exp −it2

282!Zψin(t0)exp −it02

282!exp itt0

82dt0, (24)

for the wavelength-preserved pulse and

ψup(t)=exp −it2

282!Zψin(t0)exp −it02

282!exp it0t

82−1ωdt0,

(25)

for the wavelength converted one. By comparing Equations (24) and (25),

we conclude that, after ignoring an irrelevant constant phase factor,

ψup(t)=exp(−i1ωt)ψdown (t−T). This result indicates that by shifting

the carrier frequency, a short light pulse gets delayed after propagating

in a GDD circuit. The delay, T, is measured with respect to the unshifted

version and its magnitude is given by

T=821ω. (26)

Note that this time delay only depends on the GDD parameter and the

frequency shift. The aim of using a second temporal prism is to shift the

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20 Space-Time Analogies in Optics

carrier frequency back to the original value, so that

ψup(t)=ψdown (t−T). (27)

This equation indicates that by introducing a wavelength change and let-

ting the pulse spread in a GDD medium, a controllable time delay can

be easily achieved. Unfortunately, the envelope gets distorted owing to

GDD, and for long delays (large amounts of dispersion) it needs to be

compensated for (van Howe & Xu,2005b).

To derive the above equations, we have assumed that the dispersive

medium introduces quadratic dispersion only (i.e., third- and higher dis-

persion terms can be ignored) and that the temporal prisms do not change

the envelope of the signal. These are very restrictive assumptions and,

in practice, are the main limiting factors that prevent to achieve signiﬁ-

cant delays in the optical domain. The tuning capabilities will be satisﬁed

whenever the time prism introduces a controllable frequency shift or

when the dispersive medium introduces a tunable GDD amount. The ﬁrst

feature can be achieved easily with an EOPM. Using a similar argument

as the one described in Section 4.3.2.1, an EOPM driven by an electrical

sinusoidal signal can introduce the required linear temporal phase if the

input optical pulse is properly aligned and is short enough in duration

(van Howe & Xu,2005a). Although this procedure introduces a highly

accurate and electrically controllable frequency shift, its magnitude is only

of few gigahertz and therefore achieving larger delays becomes challeng-

ing with this time prism implementation. An alternative procedure is to

introduce wavelength conversion making use, for example, of the FWM in

either highly nonlinear ﬁber (Sharping et al.,2005), a silicon nanowaveg-

uide (Okawachi et al.,2008), or SFG in a periodically poled lithium niobate

waveguide (Langrock et al.,2006;Wang et al.,2007). With these devices,

tunable delays of 0.8 ns (Sharping et al.,2005), 243 ns (Okawachi et al.,

2008), and 44 ns (Wang et al.,2007) have been reported for 10 Gb/s non-

return to zero (NRZ) signals. Distortion owing to GDD can be effectively

precompensated in these schemes.

It is worth mentioning that by cascading several wavelength converter

devices, tunable delays in the microsecond range have been recently

reported (Alic et al.,2010;Dai et al.,2010), making this technique more

attractive than other alternatives including dispersive resonances or slow

light, where dispersion management becomes a signiﬁcant issue.

5.2. Real-Time Fourier Transformation

Real-time Fourier transformers (RTFTs) are optical devices capable to

modify the light intensity proﬁle of an ultrashort light pulse so that the

achieved shape becomes a scaled version of the input coherent optical

spectrum. Two main approaches are proposed for this aim.

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“Wolf 04-ch01-001-080-9780444538864” — 2011/11/18 — page 21 — #21

Coherent Ultra-High-Speed Optical Systems and their Applications 21

5.2.1. Optical Fourier Transformation Technique

This system was originally proposed by Jannson (1983) and corresponds

to the temporal counterpart of the classic Fourier transformer constituted

by a spatial lens and diffraction until the focal length (Goodman,1996), as

shown in Figure 3.

Making use of the ABCD matrix approach, one can calculate the ABCD

matrix corresponding to the situation in Figure 3(b). Imposing A=0

requires having K82=1, that is, the chirp rate introduced by the time lens

must be completely compensated for. In this situation, the output intensity

proﬁle resembles the shape of the input optical spectrum, with a scaling

Lens

Focal plane

(a)

(b)

GDD circuit Output intensity

Time

Input intensity

Temporal lens

Time

FIGURE 3 An optical Fourier transformer is a device that implements physically the

Fourier transform of an input signal. In (a), the idea is represented in the spatial

domain, where an input beam is focused by a lens. The intensity distribution at the

focal plane is the modulus square of the Fourier transform of the transversal proﬁle of

the input beam. In (b), we have the temporal counterpart situation, where the output

pulse becomes a replica in intensity of the input optical spectrum after traversing an

ultrafast optical Fourier transformer.

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“Wolf 04-ch01-001-080-9780444538864” — 2011/11/18 — page 22 — #22

22 Space-Time Analogies in Optics

factor provided by the GDD amount. This system can be used to compress

and reshape ultrashort pulses, provided a suitable spectral shaping device

is introduced before the optical Fourier transformation technique. In this

case, the temporal width can be tuned by a proper selection of the tempo-

ral chirp and the GDD parameter, and even made to occupy less than one

bit period in a stream of pulses. It is important to note that this system does

not constitute a time-to-frequency converter, that is, the output spectrum

of the pulse is not, in general, a scaled replica of the input intensity pro-

ﬁle because the element Dof the resulting matrix is not zero. The physical

reason behind is that the output optical envelope appears multiplied by a

quadratic phase factor in time.

However, a simple modiﬁcation of this optical Fourier transformer

can be done to achieve time-to-frequency conversion as well (Lohman &

Mendlovic,1992). It involves the concatenation of a GDD circuit with

identical GDD parameter, sandwiching the time lens. In this case, the

Fourier transformation rule remains the same, K82=1, but the chirp-

ing induced by the dispersion in the ﬁrst GDD circuit compensates for

the chirp acquired through the time lens, so that the pulse at the tem-

poral focus becomes transform limited. It is easy to show that in this

case D=0, meaning that this system also performs time-to-frequency

conversion.

In any case, the most important characteristic of an optical Fourier

transformer (OFT) is the fact that the output intensity pulse is a scaled ver-

sion of the optical spectrum regardless of the input spectral phase. This

feature has provided the key to compensate for the spectral phase dis-

tortions in ultra-high-speed optical time-division multiplexing (OTDM)

linear systems (Hirooka & Nakazawa,2006). The main limitation is that

the spectral phase distortions must not lead to a pulse with a temporal

structure larger than the temporal aperture of the OFT’s time lens. At that

work, 160 Gb/s RZ differential phase-shift keying (DPSK) signals were

demultiplexed to 40 Gb/s and transmitted through a DCF-compensated

1000 km ﬁber link with the OFT technique. The OFT results show that

broadening due to remaining GDD and part of the TOD can be fully

compensated for, as well as reducing timing jitter (note that all these

effects imply spectral phase distortion). This regeneration property led to

a 2-dB improvement in bit error rate performance at 10−9, very close to

the spontaneous emission limit.2Further achievements could be obtained

if the temporal aberrations in the EOPM-based time lens were fully

compensated for.

2We remind the reader that the usual protocol in optical communication systems for considering “error

free” transmission is having a BER <10−9without applying forward error correction algorithms.

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“Wolf 04-ch01-001-080-9780444538864” — 2011/11/18 — page 23 — #23

Coherent Ultra-High-Speed Optical Systems and their Applications 23

5.2.2. Temporal Far-Field Phenomenon

In order to achieve an RTFT, the temporal lens is not really required. In the

spatial domain, it is well known that the diffraction pattern achieved at a

certain distance away from an input object, usually called the far-ﬁeld dis-

tance, coalesces with the (modulus square of the) Fourier transform of the

object. In the temporal domain, this effect is well known (Fetterman et al.,

1979;Tong, Chan, & Tsang,1997), but the interpretation as a far-ﬁeld or

Fraunhofer phenomenon was realized later (Muriel, Aza˜

na, & Carballar,

1999;Aza˜

na et al.,1999), an effect usually coined as frequency-to-time

mapping. The far-ﬁeld or Fraunhofer condition translates to (Aza˜

na &

Muriel,2000)

82σ2

0/(4π), (28)

where σ0is a measure of the input pulse width. When this condition is

satisﬁed, the output intensity shape becomes a scaled replica of the input

optical spectrum too. As in the previous case, the scaling factor is also

given by the GDD coefﬁcient. In general, the exact amount of dispersion

to reach the far-ﬁeld depends on the speciﬁc input waveform, and Equa-

tion (28) is just a simple estimation. As sketched in Figure 4, once the

temporal far-ﬁeld condition is practically satisﬁed, by adding larger dis-

persion amounts, the output intensity proﬁle does not change its shape

but only its scale. This means that this speciﬁc RTFT brings the advantage

Input intensity

Time

GDD circuit

Output intensites

FIGURE 4 Schematic representation of the frequency-to-time mapping in a GDD

circuit or temporal far-ﬁeld phenomenon. The intensity proﬁle of a pulse at the output

of a GDD medium that satisﬁes the far-ﬁeld condition becomes a scaled replica of the

input optical spectrum. Once the far-ﬁeld condition is reached, by adding larger

dispersion amounts, the proﬁle of the pulse does not change, only its scale becomes

larger.

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“Wolf 04-ch01-001-080-9780444538864” — 2011/11/18 — page 24 — #24

24 Space-Time Analogies in Optics

of resolution tuning simply by adding larger GDD amounts, since the

optical spectral resolution, δω, is provided by δω =δt/82, where δtis the

temporal resolution of the acquisition unit (assuming that the ﬁnest spec-

tral feature is adequetely mapped into the time domain). Thus, this system

is highly interesting for measuring spectra of optical pulses. The advan-

tage of using a GDD circuit and a high-speed photodiode (followed by

a high-speed digitizer) instead of a simple OSA relies on the fact that,

assuming after the photodiode there is a real-time digitizer, the optical

spectra can be measured in a pulse-by-pulse basis. In other words, the

acquisition speed is given by the repetition rate of the pulse train used.

However, given that the output stretched pulses must not suffer tempo-

ral overlapping, the repetition rate of the pulse train cannot be arbitrarily

increased and indeed should satisfy f< σ0/(4|82|). Assuming transform-

limited pulses, σ01ω =4, where 1ω is the spectral bandwidth of the input

pulse, and combining the above inequality with the resolution formula,

we get 1ω/δω < (fδt)−1, indicating that there is a trade-off between the

maximum acquisition rate and the spectral resolution of the system. The

number of spectral features (the optical bandwidth divided by the spectral

resolution) is bounded by a quantity depending on the temporal resolu-

tion and repetition rate of the pulse train, regardless of the GDD amount.

Nevertheless, for typical subpicosecond optical pulses and high-speed

photodetectors, it is possible to acquire optical spectra with subgigahertz

spectral resolutions at megahertz acquisition rates. This is in contrast

to the much longer acquisition speeds (kilohertz rate in a best-case sce-

nario) in commercial OSAs offering similar resolutions, where mechanical

and/or electronic artifacts limit the acquisition speeds.

The frequency-to-time mapping technique has been used in many dif-

ferent areas. For instance, real-time spectral measurements were initially

applied in absorption spectroscopy (Kelkar et al.,1999), with dispersive

ﬁber as the GDD medium. More recently, researchers have incorporated

Raman ampliﬁcation in the dispersive medium, thus compensating for

the energy spreading of the pulse as it stretches (Solli, Chou, & Jalali,

2008). Using this ampliﬁed Fourier transformation modality, real-time and

quasi real-time absorption spectroscopies have been reported at subgiga-

hertz optical resolution and few megahertz acquisition rate (Chou, Solli, &

Jalali,2008).

This technique has also impacted the ﬁeld of radio-frequency arbitrary

waveform generation (RF-AWG). Here, the aim is to tailor the intensity

proﬁle of a short light pulse to generate electrical signals on demand

after proper opto-electronic conversion with a high-speed photodiode

(Capmany & Novak,2007). Using conventional pulse shaping techniques

(Weiner,2000), it is difﬁcult to achieve long temporal waveforms owing to

the relatively high spectral resolution of typical pulse shapers. In other

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“Wolf 04-ch01-001-080-9780444538864” — 2011/11/18 — page 25 — #25

Coherent Ultra-High-Speed Optical Systems and their Applications 25

words, while it is relatively straightforward to achieve high-frequency

electrical signals using pulse shaping, for example, in the millimeter

wave (around 60 GHz and beyond; see McKinney, Leaird, & Weiner,

2002) or terahertz regime (∼0.1–3 THz, see Liu, Park, & Weiner,1996), it

becomes challenging to achieve the submillimeter wave regime. However,

once the temporal far ﬁeld is achieved, the waveform does not change

by adding more dispersion, and it becomes relatively easy to achieve

submillimeter-wave electrical signals with the frequency-to-time mapping

method (Chou, Han, & Jalali,2003). The concept for RF-AWG is illustrated

in Figure 5. The scheme consists on synthesizing the optical spectrum of

a coherent broadband signal with a Fourier-transform pulse shaper and

later transferring the designed spectral shape into the electrical domain by

stretching the optical pulse in a GDD circuit and subsequently detecting

it. This technique has been used to generate high-speed electrical sig-

nals (in particular ultra-wideband, UWB) with a temporal proﬁle meeting

very stringent regulations for wireless communications (Lin, McKinney, &

Weiner,2005). An experimental example is illustrated in Figure 6, show-

ing the synthesized spectrum and the corresponding achieved electrical

signal after stretching. Due to the high frequency content (spanning the

fs laser

Electrical signal

Pulse shaper

Reflective LCOS

display

Lens

Grating

Collimator

Circulator Mirror

Synthesized

Dispersive element O/E

conversion

Time

FIGURE 5 Scheme for implementing an RF-AWG using the temporal far-ﬁeld

phenomenon and a Fourier-transform pulse shaper as reconﬁgurable spectrum

synthesizer. The optical spectrum of a broadband input pulse is synthesized in a

user-deﬁned fashion with a Fourier-transform pulse shaper. After the synthesis, the

pulse is launched into a GDD medium that satisﬁes the far-ﬁeld condition, so that the

output intensity is a scaled replica of the engineered spectrum. The light is detected

with a high-speed photodiode, so electrical impulses with user-deﬁned current

variation can be easily created.

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26 Space-Time Analogies in Optics

−2−1012

Times (ns)

(a)

(b)

(c)

1530 1540 1550 1560 1570

Wavelength (nm)

1.25

(dBm)

12 34 567 8 910

Frequency (GHz)

2.5 5.0

−50

−100

FIGURE 6 Example of RF-AWG using the scheme of Figure 6. (a) synthesized optical

spectrum; (b) achieved waveform after frequency-to-time mapping conversion. The

frequency-to-time mapping operation is evident in this ﬁgure; (c) corresponding RF

spectrum. We observe that the frequency content of the synthesized electrical signal

is indeed very broad (spanning several gigahertz). (After Lin, McKinney, & Weiner,

2005). c

[2005] IEEE.

range 0–10 GHz), this kind of electrical waveforms is extremely difﬁ-

cult to generate using purely electrical approaches. Given the high level

of complexity in the pulse shapes that can be achieved with this tech-

nique, interesting applications have been reported, like compensation for

antenna dispersion (McKinney & Weiner,2006) or distortion in broadband

electrical ampliﬁers (Bortnik et al.,2006).

Finally, this frequency-to-time mapping technique also offers interest-

ing possibilities in the ﬁeld of imaging. For instance, in Fourier-domain

optical coherence tomography (FD-OCT), the depth information of a

layered object is encoded in a spectral interferogram which is usually

recorded using a regular OSA. Replacing the OSA with the GDD circuit

and a high-speed digitizer permits to obtain the depth information of

layered tissues at a very high speed (typically megahertz rate) using the

same optical source. The use of the frequency-to-time mapping method for

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Coherent Ultra-High-Speed Optical Systems and their Applications 27

FD-OCT was ﬁrstly reported by Moon and Kim (2006), using a dispersive

DCF as GDD circuit, leading to an axial rate of 5 MHz and 5-µm reso-

lution at 1.5-µm illumination wavelength. However, the signal-to-noise

ratio was still low to achieve a high-quality OCT image for more demand-

ing applications (to give an example, at a different wavelength, retinal

images require sensitivities as high as 100 dB). Subsequent works used

a linearly chirped ﬁber Bragg grating as GDD element (Park et al.,2007;

Saperstein et al.,2007) and Raman-ampliﬁed dispersive Fourier transfor-

mation (Goda, Solli, & Jalali,2008), leading to better SNRs, but still far

from the values offered by other advanced high-speed techniques involv-

ing Fourier-domain mode-locked lasers, which have recently reported

high sensitivity and megahertz acquisition rates in OCT with similar axial

resolutions (Wieser et al.,2010).

Nevertheless, this frequency-to-time technique still shows potential as

a high-speed version of an imaging modality called spectrally encoded

endoscopy (Tearney, Shishkov, & Bouma,2002). This imaging modality

uses a scheme identical to the Fourier pulse shaper depicted in Figure 5.

The idea is to replace the SLM at the focal plane of the pulse shaper by the

object whose image is to be measured. The reﬂectance of the object is thus

encoded in the optical spectrum of the broadband light source used and it

can be easily measured with an OSA and decoded ofﬂine. Using 2D spec-

tral dispersers (Xiao & Weiner,2004) and the frequency-to-time mapping

technique, the reﬂectance of a moving object can be acquired (Goda, Tsia,

& Jalali,2009). This device can capture 2D spatial proﬁles of objects mov-

ing at a speed as fast as the repetition rate of the mode-locked laser used.

Experimental results of 2D dynamic imaging (a metallic particle travelling

through a tube) are shown in Figure 7.

5.3. Time-to-Frequency Converters

A time-to-frequency converter is a device operating over an incoming

ultrashort pulse in such a way that the shape of the output spectrum

is a scaled version of the input temporal intensity (Kauffman et al.,

1994;Mouradian et al.,2000;Aza˜

na,2003a). This device constitutes a

viable alternative to measure temporal intensity waveforms with a simple

OSA. It corresponds to the dual conﬁgurations presented in the previ-

ous section. Similarly, there are two different approaches to achieve this

operation.

5.3.1. GDD Circuit with Time Lens

A time-to-frequency converter has a corresponding matrix whose spectral

element satisﬁes Aω=0. The simplest device to achieve this operation

is presented in Figure 8. It is easy to show that the time lens and GDD

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“Wolf 04-ch01-001-080-9780444538864” — 2011/11/18 — page 28 — #28

28 Space-Time Analogies in Optics

Metal microspheres

(a)

(c)

(b)

Water

Flow

20 µm

t=7.335 µs

t=7.172 µs

t=7.009 µs

t=0.163 µs

t=2.934 µs

t=5.705 µs

t=8.476 µs

FIGURE 7 Two-dimensional image of a moving particle acquired with the ultrafast

camera implemented with the concepts of frequency-to-time mapping. The key relies

on encoding the image information in the frequency domain. Later, the spectrum is

mapped into the temporal domain so that the object information can be acquired

with a single high-speed photodiode and subsequent digital signal processing. The

frame at which the images are acquired is equal to the pulse repetition rate and can

be as high as a few megahertz, thus opening the possibility to acquire 2D images at a

very high speed. (Image courtesy of K. Goda. After Goda, Solli, & Jalali,2008).

parameter must satisfy the same condition as an RTFT, K82=1. However,

note that the elements are disposed in the reversed order. This conﬁgura-

tion was ﬁrst proposed by Kauffman et al. (1994), where the time lens

was implemented with an EOPM. More recently, using the silicon time

lens, Foster, Salem et al. (2008) have shown an on-chip time-to-frequency

transformer for measuring the intensity of optical pulses with an unprece-

dented 0.2-ps resolution in a time span larger than 100 ps, which can also

operate in a single shot. Figure 9 shows a pulse featuring a short temporal

structure in a large span captured with this device.

The main limitation of this technique is the relatively long refresh time

of the OSA, which prevents its use as a real-time optical oscilloscope for

telecommunication signals typically operating at Gb/s rate. To overcome

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Coherent Ultra-High-Speed Optical Systems and their Applications 29

Time

GDD circuit

Temporal lens

Output ES

Frequency

Input intensity

FIGURE 8 Scheme of a time-to-frequency converter. This device encodes the

temporal intensity distribution of an input light pulse into the optical spectrum at the

output. Therefore, after proper calibration, a measurement in the spectral domain

with a regular OSA reveals the temporal intensity distribution of the input signal.

10 ps

10 ps 10 ps

FIGURE 9 Measurement of a complex pulse featuring ultrafast variations on a large

scale. Left part is the pulse measured with the chip-based time-to-frequency converter

and right part is the measurement done with an optical ultrafast cross-correlator.

(Adapted with permission from Macmillan Publishers Ltd. After Foster, Salem et al.,

2008). c

[2008] NPG.

this limitation, several alternative solutions have been reported recently,

enabling not only real-time operation but also amplitude and phase

retrieval information, albeit at a lower time-bandwidth product (Fontaine

et al.,2010).

The time-to-frequency conversion can be understood as the dual equiv-

alent of the previous optical Fourier transformation. In the same way that

an optical Fourier transformer is insensitive to the input spectral phase,

the time-to-frequency conversion performs its operation regardless of the

input temporal phase proﬁle. This ensures that the device measures the

intensity distribution in such a way that the signal’s inherent random phase

ﬂuctuations do not affect the measurement (Fern´

andez-Pousa,2006).

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“Wolf 04-ch01-001-080-9780444538864” — 2011/11/18 — page 30 — #30

30 Space-Time Analogies in Optics

Input spectrum

Output spectrum

Frequency

Temporal lens

FIGURE 10 Simpliﬁed time-to-frequency converter with a single time lens with high

chirping rate. This device performs the same operation as the device in Figure 9. After

introducing a huge chirp, a temporal lens maps the intensity proﬁle of an input light

pulse into the optical spectrum. Once the spectral far-ﬁeld condition is achieved, after

introducing higher chirping rates, the output spectrum distribution does not change in

shape, only in scale.

5.3.2. Spectral Far Field

By recalling that a time lens corresponds to the dual device of a GDD

circuit, it is straightforward to note that a similar far-ﬁeld regime to that

reported in Section 5.2.2 also exists in the spectral domain (Aza˜

na,2003a).

This provides a route to achieve a simpliﬁed version of the previous time-

to-frequency converters by removing the input GDD parameter, as shown

in Figure 10, if the input pulse satisﬁes

K1ω2/(4π ), (29)

where 1ω is the spectral bandwidth of the pulse. Analogously to the

temporal far-ﬁeld phenomenon, once the above limit (also called spectral

Fraunhofer limit) is satisﬁed, larger chirping amounts do not change the

spectral shape but only its scale.

The ﬁrst experimental demonstrations of this phenomenon made use

of a pre-distorted pump pulse in order to achieve a high chirping rate.

The idea was to use this time-to-frequency conversion to map the inten-

sity proﬁle from a high-speed electrical signal (even a terahertz beam) into

the optical spectrum of the predistorted pulse (Jiang & Zhang,1998;Cop-

pinger, Bhushan, & Jalali,1999). By measuring the optical spectrum with

a regular OSA, one then gets access to the highly varying temporal infor-

mation of the electrical pulse. The same concept can be used for analog-to-

digital conversion, where the optical spectrum can be demultiplexed and

parallelized with lower speed detectors (Bhushan et al.,1999).

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Coherent Ultra-High-Speed Optical Systems and their Applications 31

More recently, in a different context, it has been recognized that this

simpliﬁed time-to-frequency conversion can be used for generating opti-

cal frequency combs with a spectral proﬁle with enhanced ﬂatness when

the time lens is implemented with an EOPM (Torres-Company, Lancis, &

Andr´

es,2008). Optical frequency combs generated by sinusoidal phase

modulation are widely used for pulse shaping in the ﬁeld of optical com-

munications (e.g., Miyamoto et al.,2006;Fontaine et al.,2007;Jiang et al.,

2007) because the spacing between the comb lines can be easily controlled

with an external high-speed clock signal (typically few gigahertz), and be

large enough so that the comb spectral lines are resolved and manipu-

lated with state-of-the-art pulse shapers (Jiang et al.,2005). For quite some

time, it became challenging to build an optical frequency comb with truly

ﬂat spectral proﬁle prior to the shaping stage. In recent years, the works

of Fujiwara et al. (2003) and Yamamoto et al. (2007) reported two archi-

tectures where an outstanding ﬂatness was easily achieved. The process

involved the generation of a ﬂat-top pulse train followed by an EOPM.

Figure 11 shows an example of the ﬂat frequency comb achieved by

Yamamoto et al. (2007). The work of Torres-Company, Lancis, and Andr ´

(2008) pointed out that the physical phenomenon behind this was the

simpliﬁed time-to-frequency conversion, where the ﬂat intensity proﬁle is

mapped into the spectral domain, thanks to the action of the EOPM, which

acted as a time lens working in the spectral Fraunhofer regime. Because

the time lens operated in a pulse-by-pulse basis, the spectral envelope of

the signal (and thus that of the frequency comb) became ﬂat. That work

explicitly pointed out that by correcting the time lens temporal aberrations

in the EOPM a better ﬂatness than reported could be achieved. In addition,

thanks to the spectral Fraunhofer-ﬁeld properties, it became clear that a

higher chirping leads to broader frequency combs without altering the

desired spectral ﬂatness.

−20

−40

−60

−80

−1.0 −0.5

−120

−140

−160

0 0.5 1.0

61 carrier light

Experiment Calculation

RIN (dB/Hz)

Relative optical frequency (THz)

Power (dBm)

RIN

FIGURE 11 Ultra-ﬂat frequency comb achieved by time-to-frequency conversion of a

ﬂat-top intensity pulse using an EOPM temporal lens, based on the principles of the

simpliﬁed time-to-frequency converter of Figure 10. (After Yamamoto et al., 2007).

[2007] IEEE.

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32 Space-Time Analogies in Optics

5.4. Temporal Imaging Systems

These devices provide a scaled replica in intensity of an input short light

pulse. Three different schemes are considered. When designed for mag-

nifying, the output pulse duration becomes large enough to overcome

the limited temporal resolution of current fast photodetectors, acting as

a temporal microscope for ultrashort events (Bennett & Kolner,1999).

Alternatively, TISs can also be applied for pulse compression while main-

taining the initial intensity proﬁle (Bennett & Kolner,2000). Due to the

possibility of engineering the magniﬁcation factor to be negative, TISs

have also been proposed for the creation of anticorrelated two-photon

light in spontaneous parametric down-conversion (Tsang & Psaltis,2006),

for compensating for the TOD effect in WDM systems (Kumar,2007), or

achieving phase conjugation (Kuzucu et al.,2009). Here, three different

schemes are reviewed that can act as a TIS.

5.4.1. Conventional Temporal Imaging

The basic TIS conﬁguration is sketched in Figure 12, where two ﬁrst-order

dispersive elements are used (Kolner & Nazarathy,1989). The spatial

counterpart is also illustrated for comparison. The temporal imaging con-

dition links the chirp of the temporal lens and the GDD parameters. By

calculating the ABCD matrix corresponding to the Gaussian system of

Figure 12 and imposing B=0, we obtain

821 +1

822 =K. (30)

GDD circuit

Output intensity

Time

GDD circuit

Input intensity

Temporal lens

FIGURE 12 Schematic representation of a conventional temporal imaging system.

This system scales the temporal intensity distribution of an input pulse, so that it can

be larger (like a temporal microscope), shorter (pulse compressor), or even reversed.

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Coherent Ultra-High-Speed Optical Systems and their Applications 33

The magnifying factor, m, is given by the matrix coefﬁcient A, which

under the above constraint results in m=822/821. Thus, with this scheme

and selecting properly the GDD parameters, it is possible to engineer the

scaling factor of the TIS.

In the earlier demonstrations of TISs, magniﬁcation factors of ∼12×

(Bennett, Scott, & Kolner,1994) and ∼24×compression (Godil, Auld, &

Bloom,1994) were achieved using SFG and EOPM as time lenses, respec-

tively. With today’s time lenses implemented using FWM in silicon chips,

a temporal magniﬁcation factor >500 has been recently reported by Salem

et al. (2009), enabling high-speed optical sampling using a lower rate

digital sampling oscilloscope.

5.4.2. Time-Stretching Technique

As a remark, the scheme displayed in Figure 12 can be understood as the

concatenation of two different subsystems, a time-to-frequency converter

(ﬁrst GDD medium and part of the time lens) and a frequency-to-time

converter (consisting of the remaining part of the time lens and the sec-

ond GDD medium). This two-fold concatenation is property inherent to

all TISs.

On the other hand, from Sections 5.2 and 5.3, we have learned that

simpliﬁed conﬁgurations for time-to-frequency and frequency-to-time

mapping exist. Then, as ﬁrstly reported by Caputti in the context of elec-

tronic systems (Caputi,1971), a simpliﬁed TIS for time magniﬁcation can

also be built. This is known in the literature as the time-stretching tech-

nique and the ﬁrst implementation in the optical domain was reported by

Bhushan, Coppinger, and Jalali (1998). The operational principle is illus-

trated in Figure 13. Here, one makes use of the chirp of a predistorted

Gaussian pulse in a highly dispersive GDD medium as time lens in order

to induce the spectral Fraunhofer regime on an electrical signal (Cop-

pinger, Bhushan, & Jalali,1999) whose temporal image is to be acquired.

After this time-to-frequency conversion, the optical spectrum is mapped

into the temporal domain by a subsequent frequency-to-time operation in

a highly dispersive material (Tong, Chan, & Tsang,1997). Then, the scaling

factor for the magniﬁed electrical signal becomes m=822/821 (Bhushan,

Coppinger, & Jalali,1998). Because the second dispersive medium must

be highly dispersive to achieve the frequency-to-time mapping opera-

tion, the typical magniﬁcation factors achievable with this technique are

very high. It is also interesting to note that a similar time-stretching tech-

nique has been recently reported. The main difference is the substitution

of the ﬁrst time-to-frequency converter introduced by the dispersed pulse

with a deeply phase-modulated EOPM (Aza˜

na, Berger et al.,2005b). Some

authors have pointed out an analogy with the spatial phenomenon of

shadow casting (Aza˜

na,2003b;Aza˜

na, Lugo et al.,2005).

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34 Space-Time Analogies in Optics

GDD circuit Φ21

GDD circuit Φ22

E/O device

Input

pulse

Chirped pulse

Electrical pulse to image

Stretched output pulse (temporal image)

1. Time-to-frequency conversion

2. Frequency-to-time conversion

FIGURE 13 Scheme for temporal imaging based on the time-stretching technique.

Here the signal to be imaged is the electrical impulse. The operation involves a

two-step process. First the electrical signal is mapped into the spectral domain of an

optical pulse, thanks to the chirping it acquires through propagation in a GDD circuit.

Later, the spectrum of the modiﬁed pulse is mapped to the temporal domain, thus

achieving a large magniﬁcation factor.

This time-stretching technique has become particularly useful for mag-

niﬁcation of high-speed electrical signals. Combined with high-speed

electrical analog-to-digital conversion, this method upgrades the effec-

tive sampling rate by mtimes. Because the magniﬁcation factor increases

proportionally with the GDD parameter, it is interesting to get the high-

est possible value for 822. Making use of Raman ampliﬁcation in a GDD

circuit based on a DCF, a magniﬁcation factor of 250 has been recently

reported, leading to an effective sampling rate of 10 Tsample/s using a

state-of-the-art electronic digitizer (Chou et al.,2007).

5.4.3. General Temporal Imaging Phenomenon

Apart from the two previously mentioned TISs, there is another, less

known, TIS that is capable of performing an image of the temporal inten-

sity distribution of a virtual Fresnel plane (Aza˜

na & Chen,2003;Lancis,

Caraquitena et al.,2005). The idea can be understood with the help of

Figure 14. The distortion of a short light pulse with envelope ψin(t)

propagating in a GDD circuit is given by the Fresnel integral

ψun(t)=exp −it2

282!Zψin(t0)exp −it02

282!exp itt0

82dt0. (31)

If we consider the distortion of a chirped light pulse with envelope

exp(iKt2/2)ψin(t)through a different GDD circuit with coefﬁcient 82,eq,

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“Wolf 04-ch01-001-080-9780444538864” — 2011/11/18 — page 35 — #35

Coherent Ultra-High-Speed Optical Systems and their Applications 35

Input pulse

Time lens

Output pulse

“virtual temporal image”

Time

Linearly distorted

pulse

Time

GDD circuit Φ2

GDD circuit Φ2,eq

FIGURE 14 Scheme to understand the general temporal imaging phenomenon. The

distortion of a chirped light pulse can be always explained as a scaled replica of the

unchirped version.

the corresponding Fresnel integral would be

ψchirped(t)=exp −it2

282,eq !Zψin(t0)exp iKt02

×exp −it02

282,eq !exp itt0

82,eq dt0. (32)

We note that for 82,eq =82/(1+K82), the intensity proﬁle becomes a

scaled replica, with scaling factor 1 +K82, that would be achieved at 82

for an unchirped pulse. In other words, the distortion of a chirped light

pulse through a GDD circuit with GDD coefﬁcient 82,eq can always be

interpreted as the temporal image of the unchirped version propagat-

ing through a GDD medium satisfying the imaging condition 82,eq =

82/(1+K82). This generalized temporal imaging law has been used to

compress electrical signals, thus increasing their electrical bandwidth a

factor given by 1 +K82, which is easily controlled by the chirp and GDD

coefﬁcient (Aza˜

na et al.,2004;Torres-Company et al.,2006).

5.5. Spectral Imaging System

In the previous section, we have established the rules to achieve temporal

imaging, that is, a system that provides a scaled replica in intensity of an

input light pulse. However, because the scaling is done for the intensity, a

TIS does not necessarily scale the optical spectrum of the input signal. If

one desires to scale the optical spectrum of an input light pulse, regardless

of the intensity proﬁle, an alternative procedure would be to use a spectral

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36 Space-Time Analogies in Optics

GDD circuit

Temporal lens

Output ES

Frequency

Input ES

Temporal lens

FIGURE 15 Scheme of a spectral imaging system. This device scales the optical

spectrum of an input light pulse, so that it can be larger (spectral microscope), shorter

(spectral compressor), or even reversed in sign.

imaging system (SIS), which is the dual of a TIS (Torres-Company, Lan-

cis, & Andr´

es,2007a). Following the ABCD matrix analysis exposed in

Section 4.4, an SIS is a Gaussian system with a matrix coefﬁcient C=0.

The most basic implementation is illustrated in Figure 15, which consists

of two temporal lenses sandwiching a GDD circuit. If we invoke the SIS

condition, the chirp rates and GDD parameter must satisfy the relation

(Torres-Company, Lancis, & Andr ´

es,2007a)

K1+1

K2=82, (33)

This can be easily understood if we remind that a time lens constitutes the

dual element of a GDD circuit. As a consequence, an SIS can be consid-

ered as the concatenation of a frequency-to-time mapping (i.e., the optical

Fourier transformer composed by the ﬁrst time lens and part of the GDD

circuit) followed by a time-to-frequency converter (composed by part of

the GDD circuit and the second time lens). Consequently, the spectral

scaling factor would be provided by 1 −K282.

Combined with a high-resolution OSA, this processing system can

be used as a spectral telescope for spectroscopy applications. The ﬁrst

implementation of this original proposal involved the use of two sili-

con time lenses (Okawachi et al.,2009), achieving a magniﬁcation factor

of 105×, leading to an effective 1-GHz optical resolution. These results

appear in Figure 16. We must note however that in this SIS, the time

aperture of the ﬁrst time lens will limit the shortest discernible spec-

tral feature size, regardless of the resolution of the OSA employed. This

means that by using an OSA with better resolution it might not be possi-

ble to observe low-frequency spectral features if these do not fall into the

time window imposed by the time lenses of the system. This constitutes

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Coherent Ultra-High-Speed Optical Systems and their Applications 37

1.0

0.8

0.6

0.4

0.2

1536.1 1536.14 1536.18 1536.1 1536.14 1536.18

Transmission

1.0

0.8

0.6

0.4

0.2

1540 1544

Wavelength (nm)

Transmission

Wavelength (nm)

1548 1540 1544 1548

FIGURE 16 Experimental results of the measurement of the absorption spectra of

C2H2using a conventional OSA (top row) and a spectral imaging system (bottom row)

implemented using FWM time lenses. A 105×spectral magniﬁcation is observed.

From left to right there is a change in pressure of the gas under study. (Adapted from

permission of OSA. After Okawachi et al.,2009). c

[2009] OSA.

another example of how important is to minimize the temporal aberration

effects in temporal optics applications.

5.6. Ultrafast Fourier Processing Systems

In optical information processing, a Fourier ﬁltering system is a basic com-

ponent for manipulating the spectrum of an optical signal. For ultrafast

optical pulses, the most widespread solution to do so in a reconﬁgurable

manner is probably the Fourier transform pulse shaper (Weiner,2000).

Brieﬂy, the frequency components of a short light pulse are angularly dis-

persed using, typically, a grating placed at the back focal plane of a spatial

lens. At the focus of the lens, a spatial light modulator (SLM) manipulates

the amplitude and/or phase of the complex spectrum of the signal. Later,

the ﬁltered spectrum is recombined through a reversed operation either in

a reﬂective conﬁguration or by traveling through identical lens and grat-

ing. The ultimate goal of this device is the possibility for reconﬁguring

the synthesized waveform, due to capability to reprogram the mask at the

SLM. With some modiﬁcations, this device can manipulate other degrees

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38 Space-Time Analogies in Optics

of freedom, such as the beam proﬁle or polarization (Feurer, Vaughan,

& Nelson,2003;Ninck et al.,2007;Esumi, Kabir, & Kannari,2009). The

simultaneous high bandwidth and high resolution obtained with these

devices have allowed for the highest complexity in pulse shaping today

(Jiang et al.,2007;Supradeepa et al.,2008). Some applications, however,

require the use of more compact and rapidly reconﬁgurable pulse shap-

ing processors (Geisler et al.,2009). For these applications, the space-time

analogy offers a wide range of possibilities by direct inspection into the

counterpart spatial Fourier processors.

In general terms, the Fourier processors inspired by the space-time

analogy concepts involve a three-step process, which in terms of the

temporal optics could be explained as follows. First, a frequency-to-time

converter performs physically the Fourier transform of the signal which

we want to ﬁlter. Second, the optical spectrum of the signal is ﬁltered in

real time with an optical modulator. The point is that this active device

manipulates (in amplitude and/or phase) the spectral components of the

signal that are displayed in time. Finally a time-to-frequency conversion

reverses the operation of the ﬁrst step and brings back the ﬁltered optical

spectrum of the input signal to the temporal domain. We can distinguish

between two different systems capable of implementing this basic Fourier

processing operation.

5.6.1. 4-f Temporal Processing Systems

The scheme of this architecture is illustrated in Figure 17. An optical

Fourier transformer performs the required frequency-to-time mapping

K1K2

Φ21 Φ21

Φ22 Φ22

Input pulse

Time lens

Output pulse

GDD

circuit

GDD

circuit

Time lens

GDD

circuit

GDD

circuit

Chirp free

optical Fourier transformer 1

Chirp free

optical Fourier transformer 2

E/O

device

Electrical pulse

(filter mask)

Time Time

FIGURE 17 Scheme of a temporal 4f Fourier processing system. This device ﬁlters an

input short light pulse. The idea involves a two-step process. First, an optical Fourier

transformer performs the Fourier transform of the input signal. The spectral

components of the pulse are manipulated in real time with an electro-optical device

driven by an arbitrary electrical signal. In this way, the spectrum of the pulse can be

manipulated in a user-deﬁned fashion. After that, a second optical Fourier transformer

undoes the operation of the ﬁrst system, so that the output pulse is a ﬁltered scaled

version of the input, as predicted by Equation (34).

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Coherent Ultra-High-Speed Optical Systems and their Applications 39

operation. We note that the introduction of the ﬁrst GDD circuit to achieve

optical Fourier transformation without the inherent temporal quadratic

phase factor, that is, after frequency-to-time mapping, the pulse is trans-

form limited. The optical Fourier transformation rule dictates 821K1=1.

Later, the spectrum of the pulse is manipulated directly in the time domain

using an active modulator with complex transfer function m(t). And

ﬁnally, a time-to-frequency converter undoes the operation implemented

in the ﬁrst step, for which we must require 822K2=1. Consequently, the

ﬁltered spectral complex envelope at the output of the system, ˜

ψout(ω), is

given by

ψout(ω) =˜

ψin(mωω)m(−822 ω). (34)

Here, ˜

ψin(ω) is the input complex spectral envelope and the spectral scal-

ing factor mω= −K1/K2. Therefore, one can scale and ﬁlter independently

the complex spectrum of the signal with the proper selection of the sys-

tem parameters. We note that in the particular case in which mω=1, the

system behaves mathematically as a pulse shaper device, in which the

complex transfer function of the modulator plays the role of the system’s

transfer function. In this conﬁguration, the spectral resolution of the pulse

shaping system is given by δω =δt/822, where δtis the temporal resolu-

tion of the active modulator. Probably due to the complexity of the system,

no demonstration of temporal ﬁltering with active modulation has been

reported so far.

An interesting modiﬁcation of the previous device appears when there

is no active modulation, that is, m(t)=1. In this case, the system behaves

as a temporal telescope and ˜

ψout(ω) =˜

ψin(mωω), indicating that scaling

in amplitude and phase becomes possible. Although the theoretical pro-

posal of temporal telescopes was reported in the early years of temporal

imaging, it was not until very recently that the ﬁrst experimental imple-

mentation has been reported, making use of two time lenses based on

FWM in silicon nano waveguides (Foster et al.,2009).

5.6.2. Simpliﬁed Conﬁguration

If we recall that the frequency-to-time mapping operation can be achieved

using a highly dispersive GDD circuit (see Section 5.2.2), then an ultra-

fast optical ﬁlter in the time domain can be implemented following a

conﬁguration like the one presented in Figure 18 (see Haner & Warren,

1989; Weiner & Heritage, 1990; Saperstein & Fainman,2008). The short

light pulse to be ﬁltered is stretched in time through propagation in the

GDD circuit, so that the spectral components are decomposed in the

time domain. Then, a temporal modulator modiﬁes the amplitude and or

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40 Space-Time Analogies in Optics

Input pulse Output pulse

Electrical pulse

(filter mask)

Time Time

E/O device

GDD

circuit

Φ21

GDD

circuit

Φ22 = − Φ21

FIGURE 18 Scheme of a simpliﬁed temporal processing system. The idea is very

similar to that depicted in Figure 17. An optical pulse is stretched in time so that the

optical frequencies are displayed in the temporal domain. These optical frequencies

are manipulated in real-time by an active modulator. Then a second GDD circuit

has a reversed dispersion to undo the operation of the ﬁrst stretcher element.

phase of these physical spectral components. Later, a second GDD circuit

with dispersion parameter designed to compensate for the ﬁrst one (and

possibly the GDD introduced by the modulator) is inserted, with the aim

to bring back the synthesized spectral proﬁle to the temporal domain. The

main goal of this conﬁguration is that the temporal resolution of the whole

shaping device is determined by the duration of the pulse to be shaped

and not by the temporal response of the modulator.

In mathematical terms, the output complex envelope becomes

ψout(t)=exp −it2

2822 !ZZ ψin(t00)m(t0)exp −it02

21

822 +1

821 !

×exp −it002

2821 !exp it0t00

821 +t

822 dt0dt00 (35)

By assuming that the temporal width of the input pulse is much shorter

than the temporal width of the modulator, invoking the far-ﬁeld condi-

tion, and demanding the GDD parameters to compensate for each other,

we can then approximate the previous equation to

ψout(t)≈exp −it2

2822 !Zψin(t0)˜

mt−t0

822 dt0, (36)

where ˜

m(ω) is the Fourier transform of m(t).

The previous equation indicates that the modulation response of the

modulator manipulates the optical spectrum of the input signal. The spec-

tral resolution of the system is provided by δω =δt/822, where again δt

is the ﬁnest temporal resolution of the active modulator. By introducing

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Coherent Ultra-High-Speed Optical Systems and their Applications 41

larger dispersion amounts, the system provides better resolution. It is

instructive to determine the minimum spectral resolution this system is

capable to achieve, given an input pulse train operating at a certain rep-

etition rate f. The minimum spectral resolution is given by the maximum

GDD parameter, 822,max =σ0/(4f). This maximum dispersion amount is

ﬁxed by the condition that two neighbor pulses must not overlap after

frequency-to-time mapping conversion, that is, the system should operate

in a pulse-by-pulse basis. Thus, δωmin =4fδt/σ0, indicating that there is

a tradeoff between the complexity of the ﬁlter and the maximum speed.

In particular, in order to shape pulses in a line-by-line basis, a tempo-

ral modulator as fast as the initial pulse duration should be used. Other

details including the deviation from ideal quadratic phase modulation in

the GDD circuits have been studied by Chi and Yao (2007).

Despite its limitations, this basic Fourier processor has been used

for several applications, where compactness at relatively high speed

rates is required. The ﬁrst implementation involved the use of a hybrid

ﬁber/grating compressor as GDD circuits and an electro-optic modu-

lator as the active ﬁltering device (Haner & Warren, 1989). Using an

EOPM driven by a sinusoidal signal as ﬁlter, this system can be used to

compensate for the TOD effects of short light pulses in ﬁber transmission

links (Pelusi, Matsui, & Suzuki,1999). The use of high-speed electro-optic

amplitude and phase modulators has triggered applications in the genera-

tion of high-speed electrical signals (Saperstein et al.,2005;Aza˜

na, Berger

et al.,2005a), generation of single-sideband modulation (Chou et al.,2009),

controllable time delays (Xin et al.,2010), and optical pulse shaping based

on binary phase coding (Thomas et al.,2009,2010). Some experimental

examples obtained with this last conﬁguration are shown in Figure 19.

5.7. Joint Transform Correlator

A widely extended system for processing optical information in the spa-

tial domain is the joint transform correlator (JTC), ﬁrstly reported by

Goodman (1996). As sketched in Figure 20(a), this system enables the cor-

relation of two arbitrary objects, a fundamental operation for matched

ﬁltering, pattern recognition, and encryption. The implementation of the

JTC involves a two-step process. First, we require performing the optical

Fourier transformation of the objects to correlate, which are spatially sep-

arated at the input plane. The interference pattern (or interferogram) is

recorded in intensity. The next step is to calculate the Fourier transform

of the interference pattern which contains a term proportional to the cor-

relation of the objects. We remark that in the ﬁrst step the interferogram

is actually recorded in intensity prior to calculate its Fourier transform

in the second step. Thus, if one wishes to implement an in-line JTC, an

intensity-to-ﬁeld conversion element is required.

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42 Space-Time Analogies in Optics

Normalized intensity

1.0

0.8

0.6

0.4

0.2

0.0

–20 –15 –10 –5 0

Time (ps)

51015

Normalized intensity

1.0

0.8

0.6

0.4

0.2

0.0

–15 –10 –5 0

Time (ps)

5101520

20 –20

1.6 ns

FIGURE 19 Examples of synthesized pulses using the simpliﬁed conﬁguration

of Figure 28, with an electro-optic phase modulator as the active modulator.

The electrical signal consists of binary square pulses with amplitude leading to π

phase changes (shown in the inset of the ﬁgure). The key is that different codes can

be programmed for neighbour pulses therefore allowing for an ultrafast transition

from one synthesized pulse to the next one. (Image courtesy of A. Malacarne and

J. Aza ˜

na. After Thomas et al.,2010). c

[2010] IEEE.

In the temporal domain, there are few theoretical proposals to achieve

the correlation of two arbitrary signals (Lohman & Mendlovic,1992;Shab-

tay, Mendlovic, & Zalevsky,2000;Geraghty et al.,2009), but the temporal

equivalent of the JTC sketched in Figure 20(a) has been only recently

reported (Torres-Company & Chen,2009). The conﬁguration is shown in

Figure 20(b). Here, two pulses with envelopes ψa(t)and ψb(t)are tem-

porally separated by a quantity T, playing the role of the spatial objects

to be correlated. The idea is to perform the frequency-to-time mapping

operation of the sequence of two pulses using a highly dispersive GDD

medium. We note that to mimic strictly the scheme in Figure 20(a), the

frequency-to-time mapping operation should be achieved by the ultra-

fast optical Fourier transformer of Section 5.2.1. However, for the pur-

poses of a JTC, the far-ﬁeld phenomenon of Section 5.2.2 can be used

instead of frequency-to-time mapping. For achieving the correlation of

the pulses, one needs to perform the Fourier transformation of the inten-

sity interferogram, that is, measuring the radio-frequency spectrum of the

interferogram (Torres-Company & Chen,2009).

In mathematical terms, after dispersion in the GDD circuit with disper-

sion parameter 82, the intensity proﬁle becomes a scaled replica of the

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“Wolf 04-ch01-001-080-9780444538864” — 2011/11/18 — page 43 — #43

Coherent Ultra-High-Speed Optical Systems and their Applications 43

input spectrum,

Iout(t)= | ˜

ψa(ω)|2+ | ˜

ψb(ω)|2+˜

ψ∗

a(ω) ˜

ψb(ω) exp(iωT)+c.c.|ω=t/82, (37)

where ˜

ψa(ω) and ˜

ψb(ω) are the Fourier transform of ψa(t)and ψb(t),

respectively. To achieve a temporal JTC, the key is to calculate the Fourier

transform of the temporal signal given by Equation (37),

Iout(f)=dc(f)+X(f+1f)+X∗(−f−1f). (38)

Δx

Lens

Computer

(a)

(b)

Correlation

peak

Interferogram

as object

CCD

camera

Fourier transformation Fourier transformation

Object

λ1

Pump

Dispersive

medium

CW laser

HNLF

OSA

TimeTime

Frequency

Frequency-to-time mapping

(optical) RF analyzer

FIGURE 20 Scheme of a joint transform correlator in: (a) spatial domain. An optical

Fourier transformer followed by a camera records the interference pattern between

the objects. Then, by digital or optical means, we perform the Fourier transform of

the interference, consisting of three terms, one of them containing the information

of the cross-correlation between the objects. In (b), we have the same idea presented

in the temporal domain. Two pulses for which we desire to measure their

cross-correlation are stretched in a GDD medium. The intensity proﬁle is a scaled

replica of their spectral interferogram. Performing the Fourier transform of the

interferogram is identical to calculate the RF spectrum of this intensity proﬁle, which

is done by a nonlinear optics method in the ﬁgure.

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“Wolf 04-ch01-001-080-9780444538864” — 2011/11/18 — page 44 — #44

44 Space-Time Analogies in Optics

The above equation implies that the spectrum is constituted by three

terms: dc(f)∝R[|˜

ψa(ω)|2+ | ˜

ψb(ω)|2] exp(i2πfω82)dω, centered exactly

at the zero-frequency component and the cross-correlation terms X(f)∝

Rψ∗

a(t)ψb(t−2πf82)dt, centered at

1f=T

2π82

. (39)

In principle, if there is no spectral overlapping between the dc and cross-

correlation terms (for which a proper choice of delay Tand dispersion

parameter 82are required), the X(f)can be isolated.

−20

−40 −20 20 400

Frequency offset (GHz)

(b) (d)

−40

−60

0 500

Matched case

2.9 dB

Time (ps)

(a) (c)

Time (ps)

1000

Input sequence

Power JTC (dBm)

0 500

Unmatched case

1000

Input sequence

−20

−40 −20 20 400

Frequency offset (GHz)

−40

−60

7.2 dB

Power JTC (dBm)

FIGURE 21 Experimental results for a temporal joint-transform correlator obtained

using the setup of Figure 20(b). This system can be used to identify whether two

pulses in a sequence are identical or not. Two different cases are considered and

plotted numerically, matched (left column), when the signals are identical, and

unmatched (right column), when the signals to correlate are different. Figures (b) and

(d) show the corresponding RF spectra of the Fourier transform of the sequence,

measured with the all-optical technique in Figure 20(b). We observe that the

unmatched case leads to a decreasing in power with respect to the zero-frequency

component.

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“Wolf 04-ch01-001-080-9780444538864” — 2011/11/18 — page 45 — #45

Coherent Ultra-High-Speed Optical Systems and their Applications 45

The recognition of ultrafast sequences is a topic of increasing rele-

vance in optical communications. A possible way to achieve this in an

all-optical and in-line manner is by using an all-optical radio-frequency

spectrum analyzer based on XPM in a highly nonlinear medium (Dorrer

& Maywar,2004;Pelusi et al.,2009). In Figure 21, we show experimental

measurements of the RF spectrum of two signals multiplexed in time

when they are the same (matched case) and when they are not (unmatched

case). By measuring the peak height relative to the zero-frequency compo-

nent in the RF spectrum of the interferogram, one can assess the similarity

of the two signals, thus opening the possibility to perform optical recog-

nition of ultrafast pulses. The advantage of using this system is that with

a suitable choice of the GDD parameter, the pulses to be correlated may

come from a sequence going at an ultra-high-speed rate (i.e., Tcould be in

the subnanosecond range).

5.8. Temporal Talbot Effect

Throughout the previous sections, we have studied the distortion of a

pulse envelope in a GDD circuit without specifying its shape. In the par-

ticular case in which the signal is a train of ultrashort light pulses, the

temporal spreading and interference between adjacent pulses may lead to

unexpected results. In this section, we focus our attention in the distortion

of periodic signals in GDD media and review an interesting phenomenon

which bares an analogy with a well-known physical effect in Fourier

optics: the Talbot effect (for a review see Patorski,1989).

Let us consider an input pulse train at repetition rate f, with periodic

envelope

ψin(t)=∞

n=−∞

cnexp(−i2πnft), (40)

where, of course,

cn=f

−1

ψin(t)exp(i2πnft)dt. (41)

We now propagate the above signal into a GDD circuit. By substituting

the matrix elements from Equation (15) to Equations (10) and (9) and

considering Equation (40), we get

ψout(t)=∞

n=−∞

cnexp(i2π282n2f2)exp(−i2πnft). (42)

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“Wolf 04-ch01-001-080-9780444538864” — 2011/11/18 — page 46 — #46

46 Space-Time Analogies in Optics

This equation indicates that the output complex envelope is always a

periodic signal. And the corresponding light intensity proﬁle

Iout(t)=∞

m=−∞

Cm(82)exp(−i2πmft), (43)

Is, of course, periodic too. With

Cm(82)=exp i2πm282

82T∞

n=−∞

c∗

ncn+mexp i4πmn 82

82T, (44)

where we are deﬁning the Talbot GDD parameter as

82T=1

πf2. (45)

If we now restrict our study to the speciﬁc GDD parameters that satisfy

82=P

Q82T, (46)

where Pand Qare coprime integer numbers, then, the output complex

ﬁeld envelope can be written as (Berger et al.,2004)

ψout(t)=

Q−1

L=0

G(L,Q,P)ψin t−L

fQ. (47)

with the complex coefﬁcients given by the summation

G(L,Q,P)=1

Q−1

q=0

exp −i2πq

Q(L−qP). (48)

These formulae are the basic results that allow us to study the temporal

Talbot effect or self-imaging phenomenon. This effect appears whenever

the input light sequence is recovered after propagation in a GDD circuit.

If the repetition rate is maintained, we discuss about integer Talbot effect,

whereas if the repetition rate is increased (while preserving the individual

pulse shapes), we discuss about fractional Talbot effect.

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Coherent Ultra-High-Speed Optical Systems and their Applications 47

5.8.1. Integer Temporal Self-Imaging

As sketched in Figure 22, the integer temporal Talbot effect or temporal

self-imaging phenomenon leads to the regeneration of an input periodic

pulse sequence propagating through a GDD circuit. The output pulse train

is an exact replica in ﬁeld of the input one at those GDD amounts for which

Q=1 or in intensity for Q=2. This corresponds to the temporal counter-

part of the well-known phenomenon in the spatial domain. This effect was

discovered by Talbot in 1836, when he realized that the diffracted ﬁeld of

a grating replicated itself after certain propagation distances. The tempo-

ral version of this phenomenon was proposed by Jannson and Jannson

(1981) and veriﬁed a decade later by Andrekson (1993) although with-

out pointing out the connection with the Talbot effect. Figure 23 shows an

Input train

GDD circuit

Output train

Time

FIGURE 22 Schematic representation of the integer temporal Talbot effect. An input

periodic pulse train is launched into a GDD medium that satisﬁes the Talbot condition.

The output is an identical replica in intensity of the input, with the same period.

Intensity (a.u.)

Input sequence Temporal self-image

Time (ps)

(a) (b)

100 200 0

Time (ps)

100 200

FIGURE 23 Example of the temporal self-imaging or integer Talbot phenomenon.

(a) A periodic input sequence is launched into a GDD medium that satisﬁes the

integer Talbot condition. (b) The output sequence is a periodic sequence identical to

the input.

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“Wolf 04-ch01-001-080-9780444538864” — 2011/11/18 — page 48 — #48

48 Space-Time Analogies in Optics

example of a 12.5-GHz repetition rate pulse train at the input and prop-

agated through a GDD circuit satisfying the integer Talbot condition. We

observe how the pulse train is completely recovered at the output.

Giving a further step, the temporal Talbot effect has been proposed in

telecommunication systems as a simple mechanism to reduce the timing

jitter of periodic pulse trains (Fern´

andez-Pousa et al.,2004), as well as a

clock recovery device (Pudo, Depa, & Chen,2007). These conﬁgurations

have also been adapted to the temporal domain from their correspond-

ing spatial counterpart phenomena (Maheswari, Takai, Asakura,1991;

Kalestynski & Smolinska,1978). More recently, the temporal Talbot effect

has been used to compute factorial number (Bigourd et al.,2008). As an

example of the capabilities of the temporal Talbot effect for self-restoring

an imperfect sequence, with potential applications for clock recovery,

Figure 24 shows how an input sequence consisting of ones and zeros

(observed by an eye diagram with a zero base line) is transformed after

propagation into a sequence of only ones (an eye diagram without zero

base line).

5.8.2. Fractional Temporal Self-Imaging

This is the case when Qand Pare coprime integer numbers. Here, the

output intensity is constituted by pulses with identical intensity shape

but the repetition rate has increased by a factor of Q/2, as can be derived

from Equation (47). In the spatial domain, replicas in intensity of the input

grating spatial distribution can be achieved at fractional distances of the

fundamental one with a reduced period. In the temporal domain, this is

schematically represented in Figure 25. This phenomenon allows us to

achieve multiplication of the repetition rate of an input optical pulse train

just by dispersing it into a GDD circuit. This effect was ﬁrstly observed by

Shake et al. (1998) and Arahira et al. (1998), where an SMF was used as

GDD circuit. However, the interpretation with the fractional Talbot effect

was provided later (Aza˜

na et al.,1999), where the use of an LCFG as

GDD circuit was suggested. Such an experiment was later performed by

Longhi et al. (2000). It is worth mentioning that the Talbot effect can also

take place in higher order dispersion elements (Duchesne, Morandotti, &

Aza˜

na,2007).

The achievement of ultra-high repetition rate pulse sequences is an

important requisite in optical time-division multiplexing (OTDM) sys-

tems. What makes the fractional Talbot effect so interesting is its simplicity.

However, it is important to note two main limitations which make it difﬁ-

cult to use as an effective mechanism for OTDM applications: the high

sensitivity to slight mismatches of the fractional Talbot condition and

the fact that after multiplication, not every pulse has the same phase.

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“Wolf 04-ch01-001-080-9780444538864” — 2011/11/18 — page 49 — #49

Coherent Ultra-High-Speed Optical Systems and their Applications 49

(a)

(b)

100 ps

2.5 mW

FIGURE 24 Experimental demonstration of the temporal Talbot effect for restoring an

imperfect sequence. (a) Eye diagram of the input signal. The zero base line indicates

that the sequence is constituted by 1s and 0s, that is, it is not a perfect periodic

signal. (b) Output after travelling a GDD medium satisfying the integer Talbot

condition. The zero base line has disappeared, indicating that the output signal is

conformed by 1s only, that is, it is a periodic sequence. (After Pudo, Depa, & Chen,

2007). c

[2007] IEEE.

To overcome this last problem, Atkins and Fischer (2003) demonstrated a

method that involves the cross-gain modulation effect in a semiconductor

optical ampliﬁer to induce intensity-to-ﬁeld conversion.

5.9. Temporal Array Illuminators

The electro-optic pulse generation method is a technique that produces

high-repetition-rate pulse trains while preserving throughput (Kobayashi

et al.,1988;Murata et al.,2000;Komukai, Yamamoto, & Kawanishi,2005).

As sketched in Figure 26, the basic system consists of a continuous-wave

narrowband laser modulated in phase by a periodic microwave signal.

Pulsed light is achieved by temporal spreading in a GDD circuit, which

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“Wolf 04-ch01-001-080-9780444538864” — 2011/11/18 — page 50 — #50

50 Space-Time Analogies in Optics

Time

Input train at f

Output train at 2

GDD circuit

Φ2=Φ

2T/4

FIGURE 25 Schematic representation of the fractional temporal Talbot effect. An

input periodic sequence can have its repetition rate multiplied by Q/2 times after

launching the train onto a GDD circuit that satisﬁes the fractional condition given by

Equation (46).

Time

CW laser

EOPM

GDD circuit

Output intensity

RF signal

FIGURE 26 Schematic representation of the electro-optic pulse generation method.

A continuous wave signal is modulated periodically in phase with a sinusoidal signal.

The modulated signal is dispersed in a GDD circuit to induce phase-to-intensity

conversion and thus achieving a high-repetition-rate pulse train.

converts the input phase modulation into intensity modulation. Electro-

optic phase modulation is commonly related to sinusoidal proﬁles only.

This is because commercially available single-tone RF modulators can

produce signal bandwidths >80 GHz, whereas arbitrary RF-waveform

generators have a limited maximum bandwidth ∼10 GHz. Pulse gener-

ation by sinusoidal phase modulation is usually understood by means of

the so-called bunching parameter (Kobayashi et al.,1988). This approach

establishes the required GDD amount to focus the parabolic region of

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“Wolf 04-ch01-001-080-9780444538864” — 2011/11/18 — page 51 — #51

Coherent Ultra-High-Speed Optical Systems and their Applications 51

the sinusoidal proﬁle. Then, it is usually accepted that the pulsed light

is optimally compressed if 82K=1, where Kis the chirping rate of the

approximation to a parabolic proﬁle. However, this compression rule can-

not explain the rich panoply of achievable pulse proﬁles at different GDD

parameters. On the other hand, the space–time analogy provides some

tools to analyze the problem of ultrashort light pulse generation based

on propagation of periodically phase-modulated light (Torres-Company,

Lancis, & Andr´

es,2006a). The key relies on noticing the equivalence with

the spatial diffraction of phase structures. Here, the so-called Fresnel array

illuminators use pure phase diffraction gratings for producing a two-

dimensional (2D) array of bright spots with equal amounts of light by

free-space propagation at a ﬁnite distance. In this direction, an important

amount of research work has been devoted to understand the properties

of light diffracted by periodic phase structures and, above all, to opti-

mize their design in order to achieve a set of high-contrast two-level light

intensity dots by diffraction (see, for example, Winthrop & Worthington,

1965;Guigay,1971;Patorski & Parfjanowicz,1981;Leger & Swanson,1990;

Arriz´

on & Ojeda-Casta˜

neda,1992).

When translated into the time domain, we note that the analysis car-

ried out in the previous section is still valid even when the input sequence

is phase-only modulated (as long as it is periodic in time). However,

thanks to the phase-only restriction, some analytical formulae previously

reported in the ﬁeld of spatial Fresnel array illuminators can be derived

for particular GDD amounts (Torres-Company et al.,2006). For example,

for P=1 and Q=4, the summation in Equation (48) can be performed

analytically (Guigay,1971), and the corresponding intensity proﬁle can be

written as

Iout(t)=1−sin[V(t−1/(2f)) −V(t)], (49)

where V(t)is the periodic phase proﬁle. This result has provided the

key to explain the a priori counterintuitive ﬁnding reported by Komukai,

Yamamoto, and Kawanishi (2005), who achieved a ﬂat-top-pulse proﬁle

from a sinusoidally phase-modulated CW laser at a GDD amount different

from the bunching condition. With a careful insight in their experimental

settings, we note that they selected a GDD parameter at the one-quarter

Talbot condition. By substituting V(t)=1θ sin(2πft)in Equation (49) we

achieve

Iout(t)=1+sin[21θ sin(2πft)]. (50)

This output pulse proﬁle is schematically represented in Figure 27 for

1θ =π/4 and f=40 GHz. This waveform ﬁts perfectly to the experimen-

tal results previously reported. This constitutes another example of how

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“Wolf 04-ch01-001-080-9780444538864” — 2011/11/18 — page 52 — #52

52 Space-Time Analogies in Optics

20 40 60 80 1000

Time (ps)

Intensity (a.u.)

FIGURE 27 Example of a ﬂat-top pulse generated using the temporal array

illumination theory. A sinusoidally phase-modulated signal is launched into a GDD

circuit satisfying a fractional temporal Talbot dispersion. The intensity proﬁle

achieved in this case can be analytically expressed using Equations (49) and (50).

the space-time analogy can be useful to interpret ultrafast phenomena

from a completely different perspective.

6. TEMPORAL OPTICS IN THE NONCOHERENT REGIME

The previous section focused on ultrafast signal processing applications

operating on fully coherent signals. However, realistic pulses are subject

to random ﬂuctuations, either of fundamental nature (due to the quan-

tum origin of the radiation) or technical (like the spontaneous emission

radiation in ampliﬁer media or thermal or mechanical variations). Opti-

cal coherence theory takes into account the random variations of the

electric ﬁeld based on the statistical theory of stochastic signals, through

the so-called correlation functions. The modern formulation of optical

coherence theory was developed in the 1950s (Mandel & Wolf,1995).

Since then, the research has been mainly focused on the case of sta-

tistically stationary light ﬁelds, for which the averaged light intensity

remains constant in time. This model is suitable to describe the coher-

ence properties of thermal-like light, such as that emitted by diode (LED)

or ampliﬁed spontaneous emission (ASE) sources. However, it fails to

deal with the coherence properties of pulsed radiation, where the average

intensity does not remain constant in time. For this aim, optical coher-

ence theory has been recently extended to the nonstationary case (see, e.g.,

Paakonen et al.,2002;Lajunen, Vahimaa, & Tervo,2005;Torres-Company,

Lajunen, & Friberg,2007). In this section, we provide a brief summary

of the basics of the scalar optical coherence theory applied to nonstation-

ary radiation and extend the concepts of the space-time analogy beyond

the bounds of coherent processing. This will be done highlighting the

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“Wolf 04-ch01-001-080-9780444538864” — 2011/11/18 — page 53 — #53

Temporal Optics in the Noncoherent Regime 53

stochastic nature of light not only from a pernicious point of view, but

also pointing out some possible technical advantages in ultrafast signal

processing applications.

6.1. Scalar Coherence Theory for Nonstationary Partially

Coherent Wave ﬁelds

Let us consider an ensemble of stochastic complex scalar ﬁelds, {U(t)},

whose spatial structure is assumed to be a plane-wave. In this case, the

meaningful physical magnitude becomes the mutual coherence function,

0(t1,t2)=U∗(t1)U(t2), (51)

which contains the ﬁrst-order statistical information of the complex ﬁeld

associated with the light source. The angle brackets denote statistical

ensemble average. From the above function, one can recover important

physical quantities like the average intensity

I(t)=0(t,t), (52)

or the complex degree of coherence

γ (t1,t2)=0(t1,t2)

√I(t1)I(t2), (53)

which determines the strength of the ﬁeld correlations at the two par-

ticular instant times t1and t2. An optical pulse is said to be ﬁrst-order

coherent if |γ (t1,t2)| = 1, ∀t1,t2, and incoherent if |γ (t1,t2)| = 0, ∀t1,t2.

Intermediate cases deal with partially coherent radiation.

A similar analysis can be done in the spectral domain. In this case, we

study the ﬁrst-order correlations among the frequency components of the

optical ﬁeld through the cross-spectral density function

W(ω1,ω2)=D˜

U∗(ω1)˜

U(ω2)E, (54)

where ˜

U(ω) denotes the Fourier transform of U(t). Important physi-

cal magnitudes can also be obtained from this analysis, like the optical

spectrum

S(ω) =W(ω,ω), (55)

or the complex degree of spectral coherence

µ(ω1,ω2)=W(ω1,ω2)

√S(ω1)S(ω2). (56)

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54 Space-Time Analogies in Optics

Time and frequency representations contain the same quantity of infor-

mation and are linked each other through the so-called generalized

Wiener-Khintchine theorem

0(t1,t2)=ZZ W(ω1,ω2)exp [i(ω1t1−ω2t2)]dω1dω2, (57)

and, conversely,

W(ω1,ω2)=1

4π2ZZ 0(t1,t2)exp [−i(ω1t1−ω2t2)]dt1dt2. (58)

In the limit of a stationary light ﬁeld, we have 0(t1,t2)=0s(t2−t1)and,

accordingly,

W(ω1,ω2)=Ss(ω1)δ(ω2−ω1), (59)

indicating that a stationary light ﬁeld has completely uncorrelated fre-

quencies. Here, according to the Wiener-Khintchine theorem, Ss(ω) forms

a Fourier transform pair with 0s(τ ).

6.2. Space-Time Analogy for Partially Coherent Wave ﬁelds

In the noncoherent regime, the meaningful physical quantity becomes

the mutual coherence function, rather than the individual complex

ﬁeld realizations. Our aim in this section is to get the set of equations

describing the distortion of the mutual coherence function in a dispersive

medium with GVD coefﬁcient β2. To extend the problem studied in

Section 2 to the partially coherent regime, we must remind that the

complex ﬁeld U(z,t)is now a complex random function. We assume the

randomness can be fully accounted for in the complex envelope ψ(z,t).

Then, the corresponding mutual coherence function is 0(z1,z2;t1,t2)=

hU∗(z1,t1)U(z2,t2)i=hψ∗(z1,t1)ψ (z2,t2)iexp{−i[ω0τ−β0(z2−z1)]} =

0e(z1,z2;t1,t2)exp{−i[ω0τ−β0(z2−z1)]}, where τ=t2−t1. Since each of

the individual realizations of the envelope satisﬁes Equation (7), we easily

ﬁnd that the mutual coherence function associated with the envelope,

0e(z1,z2;t1,t2), must satisfy the following pair of equations

"∂2

∂t2

j−2i(−1)j1

β2

∂

∂zj#0e(z1,z2;t1,t2)=0, (60)

with j=(1, 2). The key is to note that these equations are mathemati-

cally identical to those describing the free-space propagation of a spatially

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“Wolf 04-ch01-001-080-9780444538864” — 2011/11/18 — page 55 — #55

Temporal Optics in the Noncoherent Regime 55

partially coherent light beam at frequency ω, under the paraxial approxi-

mation (Lancis, Torres-Company et al.,2005)

"∂2

∂x2

j+2ik(−1)j∂

∂zj#Ws,e (r1,r2;ω)=0, (61)

where r=(x,z),Ws,e(r1,r2;ω) is the spatial envelope part of the cross-

spectral density function deﬁned in the space-frequency domain, and

k=ω/cis the free-space wave number. We note that the transfer rules

highlighted in Table 1 remain valid in the partially coherent regime. The

goal of this formal extension is that it paves the way to design new ultra-

fast signal processors based on the concepts of Fourier optics incoherent

processing.

6.3. Propagation of Nonstationary Partially Coherent Pulses

To study the propagation of the mutual coherence function through tem-

poral Gaussian systems, one can simply take the results from Section 4.1

and evaluate the corresponding propagated mutual coherence function by

calculating the ensemble average on Equation (9) and assuming the enve-

lope is a complex random function. Taking into account the deﬁnition of

the input and output mutual coherence functions, it is straightforward

to get

0e,out (t1,t2)=ZZ K∗(t1,t0

1)K(t2,t0

2)0e,in t0

1,t0

2dt0

1dt0

2. (62)

Alternatively, one can study the evolution of the cross-spectral density

function3

Wb,out (ω1,ω2)=ZZ ˜

K∗(ω1,ω00

1)˜

K(ω2,ω00

2)Wb,in ω00

1,ω00

2dω00

1dω00

2. (63)

The key of the above analysis relies on noting that the kernel operators are

exactly the same as those used for the fully coherent case.

Although a rich variety of noncoherent processing systems can be par-

ticularized from the previous analysis, we shall concentrate in a couple of

interesting arrangements. We proceed to propagate an arbitrary mutual

coherence function inside a GDD circuit. For this aim, we take the corre-

sponding matrix coefﬁcients from Equation (15), particularize the kernel

3The subindex bindicates that the cross-spectral density function is referred at the baseband, that is,

Wb(ω1,ω2)=W(ω0

1−ω0,ω0

2−ω0).

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“Wolf 04-ch01-001-080-9780444538864” — 2011/11/18 — page 56 — #56

56 Space-Time Analogies in Optics

of Equation (10), and insert them into Equation (62) to get

0e,out (t1,t2)=I0exp "it2

1−t2

282#ZZ exp "it02

1−t02

282#

×exp −it0

1t1−t0

2t2

820e,in t0

1,t0

2dt0

1dt0

2, (64)

where I0is an irrelevant constant. We provide a detailed analysis for the

case in which the input mutual coherence function corresponds to a source

like the one reported by Lajunen et al. (2003). It is a spectrally incoherent

stationary light source (e.g., ASE radiation) that is externally modulated

by a fast modulator, as sketched in Figure 6.1. The optical spectral band-

width of the source ﬁxes the coherence time of the light (by an inverse

relation) and the temporal duration of the modulator ﬁxes the duration of

the partially coherent signal (Lajunen et al.,2003). By inserting the corre-

sponding expression of the mutual coherence function for this light source

into Equation (64) and doing some straightforward algebra, the averaged

intensity proﬁle at the output of the GDD circuit is given by

Iout(t)=Sb(t/82)⊗Icoh(t), (65)

where irrelevant constant factors have been dropped for clarity. Here,

Sb(ω) is the spectral density of the source centered at the baseband and

Icoh(t)is the intensity that would be achieved if the spectral source was a

perfect Dirac’s delta, that is,

Icoh(t)=Zexp i82

2ω2˜

M(ω) exp(−iωt)dω

2, (66)

where ˜

M(ω) is the Fourier transform of the external complex modulation

m(t). This general result is known as the extension of the Collett–Wolf

equivalent theorem to the Fresnel region, and it was reported ﬁrst in the

spatial domain by Saleh (1979) and formulated recently in the temporal

domain by Chantada, Fern´

andez-Pousa, and G´

omez-Reino (2006). The rel-

evance of the previous result is that by a proper optical spectrum synthesis

of the light source one can either shape an output intensity proﬁle or ﬁlter

the signal that would be achieved in the coherent regime.

6.4. Temporal van Cittert-Zernike Theorem

Let us come back to Equation (64). In this section, we aim to focus on the

equivalent far-zone expression for the output mutual coherence function,

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Temporal Optics in the Noncoherent Regime 57

which is achieved when the quadratic factor inside the integrand can be

removed. Following the analysis performed by Gori (2005) in the spatial

domain and having in mind the space-time analogy transfer rules, we eas-

ily ﬁnd that the far-zone is reached in partially coherent temporal optics

whenever

|82|σ

4πmin[2σ,tc], (67)

where σis a measure of the temporal intensity width at the input; min[a,b]

means the smaller of the real numbers a,b; and tcis a measure of the coher-

ence time at the input. Once the above inequality is satisﬁed, the output

mutual coherence function becomes

0e,out (t1,t2)=I0exp "it2

1−t2

282#Wb,in(t1/82,t2/82), (68)

that is, a scaled replica of the cross-spectral density function at the input,

in close resemblance to the results presented in Section 5.2.2.

Now, we consider again the same optical source used to achieve Equa-

tion (65). In particular, we further assume that the coherence time is much

smaller than the temporal duration of the modulator, so that the mutual

coherence function behaves as a quasi-homogeneous light pulse (Lajunen,

Friberg, & Ostlund,2006),

0e,in (t1,t2)=Iin t1+t2

2γe,in (t2−t1), (69)

where Iin(t)and γe,in(t2−t1)denote the averaged intensity and the enve-

lope of the complex degree of coherence of the input partially coherent

light pulse, respectively. Substituting these into Equation (68) and after

some straightforward algebra, one can easily ﬁnd that the output aver-

aged temporal intensity is given by the Fourier transform of the input

complex degree of coherence and vice versa, the output complex degree

of coherence is given by the Fourier transform of the input averaged

intensity (Lajunen, Friberg, & Ostlund,2006). This double Fourier trans-

formation rule is what is known as the generalized temporal van Cittert-

Zernike theorem, which was ﬁrstly proposed and experimentally veriﬁed

by Dorrer (2004).

6.4.1. Ultrafast Spectrometer

The temporal van Citter-Zernike may have practical applications. From

the previous section, we have learned that if we consider the source from

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58 Space-Time Analogies in Optics

Spectrally incoherent light

Deterministic

temporal modulator

ASE light

Partiall coherent

nonstationary pulses

FIGURE 28 Practical scheme to generate partially coherent pulses. A broadband

stationary light source is externally modulated by a deterministic modulator. The

modiﬁed mutual coherence function corresponds to an ensemble of non-stationary

light pulses. The coherence time of these pulses is ﬁxed by the coherence time of the

stationary light, while the spectral correlations are ﬁxed by the temporal gate of

the deterministic modulator. (After Lajunen et al.,2003).

Figure 28 having a coherence time much narrower than the temporal

duration of the temporal modulator, the complex degree of coherence of

the input light source determines the output intensity proﬁle, leading to

(Dorrer,2004)

Iout(t)≈Sin (t/82), (70)

which can be alternatively interpreted as an approximation of Equa-

tion (65). The important point here is that the averaged output intensity is

a scaled replica of the input’s spectral density. This can be considered as an

extension of the frequency-to-time mapping operation from Section 5.2.2

to the incoherent regime (Torres-Company, Lancis, & Andr´

es,2007b).

Figure 29 represents an experimental veriﬁcation of this frequency-to-time

mapping operation. As spectrally incoherent source we used the ASE from

the emission spectrum of an Erbium-doped ﬁber ampliﬁer. The spectral

density is measured with an OSA and sketched in Figure 29(a). Typical

coherence times of EDFAs are around 100 fs. The deterministic modulator

was a 10 Gb/s LiNbO3electro-optic modulator (EOM) biased at quadra-

ture and driven with a short electrical Gaussian impulse. The resulting

pulse has a Gaussian shape with 88 ps full-width at half maximum

(FWHM) duration. In this way, the quasi-homogeneous requirement is

satisﬁed. Finally, the modulated light is launched into an SMF of 1.44-km

length. Note that this GDD amount is much larger than the minimum

required by Equation (67), assuming β2= −21 ps2/km. The output inten-

sity shape is measured again with the same sampling scope and the result

is shown in Figure 29(b). The frequency-to-time mapping predicted by

Equation (70) is thus achieved.

The relevance of this mapping is very similar to that previously

exposed in the coherent regime (see Section 5.2.2). There we learned that

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Temporal Optics in the Noncoherent Regime 59

1520 1530 1540

Wavelength (nm)

(a) (b)

Time (ns)

Optical spectrum (a.u.)

Optical intensity (a.u.)

1550 1560 1570 0 0.5 1.0 1.5 2.0

FIGURE 29 Experimental veriﬁcation of the incoherent frequency-to-time mapping.

The optical spectrum of an incoherent light source can be mapped into the time

domain by modulation and subsequent dispersion in a GDD medium, in a similar

manner to the coherent effect explained in Section 5.2.2. After statistical averaging,

the output intensity proﬁle becomes a scaled replica of the optical spectrum. Once the

mapping is achieved, adding more dispersion does not change the intensity proﬁle,

only its scale.

the coherent frequency-to-time conversion is allowed for capturing opti-

cal spectra in a high-speed manner using broadband coherent pulses,

with potential applications in absorption spectroscopy, imaging, arbitrary

RF waveform generation, sensing, and reﬂectometry. In the noncoherent

regime, Equation (70) establishes that high-speed spectral measurements

can be done too using an incoherent light source instead.

However, a word of caution must be brought up here. Unlike in the

coherent regime, where the spectral measurements can be done in sin-

gle shot (i.e., the mapping takes place for a single individual pulse), in

the incoherent regime we note that Equation (70) is valid only for aver-

age quantities. In other words, both the spectral density of the input light

source and the acquired intensity are constructed upon an ensemble aver-

age operation. In practical terms, this means that one needs a statistically

signiﬁcant amount of data (i.e., individual intensity shots) to construct

the quantity to be measured, which in turn reduces the operation speed.

In general, the SNR of the acquired spectra asymptotically increases as

√N(Dorrer,2009), while the spectrum acquisition speed gets reduced

by N, thus indicating a trade-off between SNR and acquisition speed in

incoherent frequency-to-time mapping. As an example, for an SNR of 20

dB and an initial clock rate of 1 GHz (i.e., the active modulator repeti-

tion rate), it would be possible to achieve optical spectra varying at a

rate of 100 kHz, which is nevertheless quite competitive with the per-

formance acquired with the ultrafast coherent spectrometer presented at

Section 5.2.2.

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60 Space-Time Analogies in Optics

6.4.2. Arbitrary RF-Waveform Generator

A possible application of the generalization of the frequency-to-time map-

ping operation to the incoherent case lies in the ﬁeld of arbitrary RF

waveform generation. Let us consider a scheme like the one presented in

Figure 30. A spectrally incoherent stationary light source is manipulated

in amplitude with a reconﬁgurable ﬁlter. Similar to the previous section,

the modiﬁed optical source is actively modulated with an electro-optic

modulator providing optical pulses with a temporal duration larger than

the inverse of the spectral resolution of the shaper in order to satisfy the

condition of quasi-homogeneous light pulse. After this, the modulated

light signal is launched into a GDD medium satisfying condition (67).

Then, as predicted by Equation (70), the synthesized spectral density is

transferred to the averaged temporal proﬁle. This pulse is ﬁnally mapped

into the electrical domain by using a high-speed photodiode (Torres-

Company, Lancis, & Andr´

es,2006b). A possible implementation of this

scheme can be done with the setup of Figure 30, where the reconﬁg-

urable spectral ﬁlter is achieved with a Fourier transform pulse shaper

(Torres-Company et al.,2008a). Two examples of the synthesized spec-

tra and the achieved intensity proﬁles acquired with a sampling scope

are displayed in Figure 31, when using a short length of ﬁber for the

frequency-to-time mapping operation. The ﬁber length is the minimum

FIGURE 30 Schematic representation of arbitrary RF waveform generation based on

incoherent pulse shaping. In the optical implementation a reﬂective Fourier-transform

pulse shaper is used as reconﬁgurable ﬁlter.

Electrical signal

Applied mask

Dispersive element

O/E

conversion

Time

Deterministic

modulator

Spectrally

incoherent

source

Pulse shaper

Reflective LCOS

display

Lens

Grating

Collimator

Circulator Mirror

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“Wolf 04-ch01-001-080-9780444538864” — 2011/11/18 — page 61 — #61

Temporal Optics in the Noncoherent Regime 61

1530 1532

(a)

1534

ES linear scale (a.u.)

λ (nm)

1536 1538 1540

(g)

(c)(b)

(

)

500 ps 500 ps

500 ps

02468

f (GHz)

(d)

RF spectrum (10dB/div)

10 12 14

(

)

RF spectrum (10dB/div)

02468

(GHz)

10 12 14

(

)

ES linear scale (a.u.)

1525 1530 1535

λ (nm)

1540

FIGURE 31 Two examples (one per row) of high-bandwidth electrical signals

synthesized with the incoherent technique of Figure 30. The synthesized optical

spectra with a Fourier-transform pulse shaper are shown in (a) and (e). The achieved

intensity proﬁles are shown in (b) and (f) without performing average. A reduced SNR

is apparent from the ﬁgure. For 4×average, the waveform ﬁdelity increases as seen

from (c) and (g). The corresponding electrical RF spectra are displayed in (d) and (h),

reﬂecting the high-bandwidth content of the synthesized signals.

to guarantee the far-zone operation while at the same time maximizes the

achievable RF bandwidth. The corresponding RF spectra synthesized with

this method are also displayed in the ﬁgure, leading to a relatively broad

frequency content (∼0–10 GHz). As mentioned in the previous section,

we also note the reduced signal-to-noise ratio (SNR) for the synthesized

waveforms when no statistical averaging is performed on the signal. This

indicates that this technique is more suitable for RF-AWG applications not

requiring single-shot high waveform ﬁdelity, like spectral measurements

of broadband optoelectronic equipment (Torres-Company et al.,2008b).

6.5. Temporal Lau Effect

The temporal Talbot effect explained in Section 5.8 can also be extended

to the noncoherent regime. The spatial noncoherent Talbot effect is known

in the Fourier optics literature as the Lau effect (Jahns & Lohmann,1979).

This effect consists of overlapping several temporal self-images coming

from different light sources. The temporal counterpart of this effect was

ﬁrstly reported by Zalvidea, Duchowicz, and Sicre (2004), who proposed

the use of different mode-locked pulse trains as different light sources.

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62 Space-Time Analogies in Optics

An alternative implementation is by Lancis et al. (2006), who proposed

the use of several CW light sources that are externally modulated by a

common electro-optic modulator. This proposal has been recently veriﬁed

experimentally by Torres-Company, Fern ´

andez-Pousa, and Chen (2009),

highlighting the potential beneﬁts of the temporal Lau effect for ultrafast

photonic applications.

Such an implementation can be described as follows. In the one-

dimensional (1D) spatial case, sketched in Figure 32(a), a quasi-

monochromatic spatially incoherent source is imaged onto a 1D grating

(G1) thus producing an array of incoherent spots, equidistant in the

transversal direction to the propagation. Since G1 is located at the focal

plane of a lens, each of the spots produces a tilted plane wave that

illuminates a second grating (G2). By diffracting a distance corresponding

to an integer Talbot plane, each of the plane waves produces a self-image

shifted in the transversal coordinate. Owing to the incoherent nature of

the illuminating points, the Talbot images are summed in intensity. If

the distance between consecutive spots satisﬁes a certain relation, the

intensities overlap. Far from being a simple curiosity, this phenomenon

(a) (b)

GDD circuit

Modulator

Laser 0

Laser 1

Laser 2

Laser L−1

Input pulse

train

ω0

Δω

φ2

ω1ω2ωL−1

1 st self-image

G1 f

Incoherent

point sources

Lens

1st

Self-image

(incoherent

superposition)

ω1

ω2

ωL−1

FIGURE 32 Schematic representation of the Lau effect in (a) spatial domain. Here, an

input array of equidistant spatially incoherent emitters (G1) at the back focal plane of

a lens emits rays in different directions illuminating a grating (G2). Each emitter

produces a self-image of the G2 after propagating a Talbot distance. If the distance

between emitters satisﬁes a certain criterion, each of the self-images will overlap in

intensity. (b) Shows the corresponding scheme in the temporal domain. An array of

CW lasers equally spaced in frequency plays the role of the incoherent spots at the

focal plane in (a). The lasers are modulated externally by a temporal periodic function

emulating the role of G2. Each of these lasers generates a temporal self-image. When

the spectral distance between lasers satisﬁes a certain criterion, the superposition of

the temporal self-images becomes in intensity.

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Temporal Optics in the Noncoherent Regime 63

has been used for spatial signal processing applications (see, e.g., Patorski,

1989). Following Lancis et al. (2006), we now consider the temporal analog

sketched in Figure 32(b). An array of LCW lasers with equally spaced

frequencies plays the role of the tilted plane waves, with 1ω being the

spectral separation. The monochromatic carriers are modulated externally

by a repetitive sequence with repetition rate f, playing the role of G2. Then

the light is launched into a GDD circuit that satisﬁes the integer temporal

self-imaging condition [Equation (46) with Q=2]. In view of the analysis

carried out at Section 5.1, we know that consecutive carriers undergo a

relative shift provided by Equation (26). Thus, if

f T =m, (71)

where mis an integer number, there will be an overlapping in the out-

put temporal envelopes. This overlapping can be in intensity as long as

the photodetector bandwidth is shorter than the possible beat between

the highest and lowest modulation sidebands of any pair of consecutive

lasers, thus leading to

Iout(t)=

L−1

n=0

PnIin(t−nT), (72)

where Iin(t)is the temporal intensity proﬁle provided by the external

modulator and Pnis the relative power of the nth carrier.

An illustrative example of this effect is shown in Figure 33. For sim-

plicity, only two carriers are used. The corresponding input and output

0 100

ω1

ω2

ω1

ω1and ω2

ω2

200

Time (ps)

(a) (b) (c)

0 100 200

Time (ps)

0 100 200

Time (ps)

Input intensity (a.u.)

Temporal

self-images (a.u.)

Intensity out (a.u.)

FIGURE 33 Experimental results for the temporal Lau effect. (a) Input sequence for

each of the corresponding frequencies. (b) Achieved ﬁrst temporal self-image for the

corresponding carriers. We observe how the frequencies have been properly selected

so that the self-images are perfectly synchronized in time. (c) Intensity superposition

of the self-images.

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64 Space-Time Analogies in Optics

0 100 200

Time (ps)

Intensity out (a.u.)

FIGURE 34 A small detuning from the frequency spacing in the temporal Lau effect

can lead to completely different shapes by intensity superposition, opening the way

to alternative electrical pulse shaping mechanisms.

intensity proﬁles are shown for each frequency. We observe that each

of the sources generates a perfect self-image. The separation between

the light sources is such that it ensures intensity superposition, as illus-

trated in Figure 33. This way to achieve “incoherence” guarantees that

the generated electrical signal is free of phase-induced noise, thus offer-

ing nice features for the generation of arbitrary RF electrical impulses.

Indeed, thanks to the dispersion-wavelength controllable delay, together

with the fact that the individual lasers’ power can be easily controlled,

this temporal Lau effect offers a tremendous potential for arbitrary RF-

waveform generation (Torres-Company, Fern ´

andez-Pousa, & Chen,2009).

As an example, Figure 34 shows how a slight detuning from the perfect

self-imaging superposition condition can lead to a high-speed pulse train

consisting of triangle-like pulses.

7. TEMPORAL OPTICS IN THE TWO-PHOTON REGIME

The previous sections dealt with light sources whose properties can be

described using the laws of classical physics. However, some recent exper-

iments show that there are other properties and effects which are better

understood within the framework of a quantum theory (Mandel & Wolf,

1995). A particularly relevant example of this is the entanglement between

particles.

In optics, the most versatile method for producing entangled photon

pairs (or two-photon states) has been spontaneous parametric down-

conversion (SPDC), where an intense pump beam impinges onto a

second-order nonlinear medium, eventually producing a pair of lower

frequency photons in a nonseparable state (Shih,2003). These photon

pairs may be entangled in polarization, momentum, space, frequency, and

time. There exist alternative schemes for producing entangled photons, for

example through electromagnetically induced transparency, where two

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Temporal Optics in the Two-Photon Regime 65

pump beams interact with a cloud of cold atoms with Raman-type transi-

tions (Balic et al.,2005). More compact solutions have been reported in the

last few years, including spontaneous four-wave mixing in microstruc-

tured ﬁbers (Alibart et al.,2006) or silicon nano waveguides (Sharping

et al.,2006), and two-photon emission from semiconductor quantum wells

(Hayat, Ginzburg, & Orenstein,2008).

Two-photon states are a key resource for quantum information process-

ing applications, including cryptography (Ekert et al.,1992), teleportation

(Bouwmeester et al.,1997), or metrology (Giovannetti, Lloyd, & Maccone,

2001), just to name a few.

The aim of this chapter is to establish some counterpart ultrafast sig-

nal processors for broadband two-photon light emitted from SPDC. The

similarity with the previous schemes and their relevance with quantum

information applications will be pointed out.

7.1. Quantum Two-Photon Correlation Functions

and Two-Photon States

For light in an arbitrary quantum state, the probability of observing a pho-

ton at a time t1and another at t2is proportional to the intensity correlation

function (Glauber,1963)

G(2)

12 (t1,t2)=D9|ˆ

E−(t1)ˆ

E−(t2)ˆ

E+(t2)ˆ

E+(t1)|9E, (73)

where ˆ

E−(t)and ˆ

E+(t)denote the negative- and positive-frequency com-

ponents of the optical ﬁeld operator at time instant t, and |9irepresents

the vector state.

For the particular case of light in a two-photon state, the right-

hand side of Equation (73) factorizes as (Shih,2003)h9|ˆ

E−(t1)ˆ

E−(t2)

E+(t2)ˆ

E+(t1)|9i=h9|ˆ

E−(t1)ˆ

E−(t2)|0ih0|ˆ

E+(t1)ˆ

E+(t2)|9i, with |0ibeing

the vacuum state. Therefore, we can write as

G(2)

12 (t1,t2)=|912(t1,t2)|2, (74)

with

912(t1,t2)=D0|ˆ

E+(t1)ˆ

E+(t2)|9E. (75)

Then, 912(t1,t2)can be interpreted as the probability amplitude distribu-

tion of the two-photon state or two-photon wave packet.

The above discussion has a dual interpretation in the frequency

domain. In this case, the meaningful quantity is the frequency-domain

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66 Space-Time Analogies in Optics

two-photon probability amplitude, ˜

912(ω1,ω2), whose modulus square

determines the probability of detecting a photon at frequency ω1and

the other at frequency ω2. This function forms with 912(t1,t2)a Fourier

transform pair, that is,

912(t1,t2)=ZZ ˜

912(ω1,ω2)exp [−i(ω1t1+ω2t2)]dω1dω2, (76)

and

912(ω1,ω2)=1

4π2ZZ 912(t1,t2)exp [i(ω1t1+ω2t2)]dt1dt2. (77)

Obviously, when we ﬁx the two-photon probability amplitude in the time

domain, the properties in the dual (frequency) space are automatically

settled.

7.2. Tailoring Two-Photon States in Spontaneous Parametric

Down-Conversion

In this section, because of its high versatility and easy tuning, we concen-

trate in the particular method of SPDC for generating two-photon states.

As mentioned before, in SPDC, a high-power pump beam impinges onto

a nonlinear crystal, eventually producing two photons at lower frequency,

usually referred for historical reasons as signal and idler (Shih,2003).

Energy and momentum are conserved in this process. The two-photon

probability amplitude can be written as 912(t1,t2)=D0|ˆ

E+

s(t1)ˆ

E+

i(t2)|9E,

where ˆ

E+

s,i(t)is the positive-frequency electric-ﬁeld operator at time t

for the signal and idler photon, respectively. We are concentrating in

the temporal distribution only because we consider a single-mode spa-

tial structure (which could be achieved by imaging the down-converted

beams onto a waveguide with translational symmetry, like, for example,

a single-mode ﬁber). But we remark that a similar analysis can be car-

ried out in the spatial domain. The structure of the two-photon state, |9i,

depends on the particular arrangement used for the SPDC process, which

includes the spatiotemporal distribution of the pump beam, the material

intrinsic and synthesized properties, and the optical system to bring the

pump source onto the crystal. The interpretation in the spectral domain is

straightforward. The spectral two-photon probability amplitude becomes

912(ω1,ω2)= h0|ˆ

E+

s(ω1)ˆ

E+

i(ω2)|9i, where ˆ

E+

s,i(ω) is the positive-frequency

ﬁeld operator for the signal and idler photons, respectively. From Equa-

tions (76) and (77), it is clear that 912(t1,t2)and ˜

912(ω1,ω2)still form a

Fourier transform pair.

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Temporal Optics in the Two-Photon Regime 67

In the same way that the synthesis of the electric ﬁeld through pulse

shaping techniques constitutes the workhorse for numerous applications

based on signal processing, in quantum information processing based on

two-photon states, the important magnitude to be tailored is the two-

photon wave packet, 912(t1,t2). Certainly, one could write another review

paper about the current techniques that permit to synthesize this mag-

nitude. We will just brieﬂy mention, for example, the use of appropriate

materials and wavelengths (Kuzucu et al.,2005;Mosley et al.,2008); intro-

ducing angular dispersion into the pump and down-converted beams

(Hendrych, Micuda, & Torres,2007); inserting phase (Pe’er et al.,2005)

and/or amplitude (Zah, Halder, & Feurer,2008) spectral ﬁlters in sig-

nal and idler beams; using chirped quasi-phase-matched materials (Nasr

et al.,2008); the so-called spatial-to-spectral mapping technique (Valencia

et al.,2007); and the combination of angular dispersion and noncollinear

geometries (Shi et al.,2008).

We will not go into the details on how to achieve them, but rather

we shall concentrate on the remarkable implications derived from the

912(t1,t2)structures that lead to unique photon correlations. For exam-

ple, when 912(t1,t2)∝912 (t2−t1), the signal and idler photons are said

to be temporally correlated. This can be better understood by looking into

Figure 35, which depicts the two-photon probability amplitude in both

time and frequency domains. Once the signal photon “clicks” its cor-

responding detector at time instant t1, the idler photon will then click

the other within a very narrow temporal uncertainty given by the two-

photon coherence time [which is the temporal width of the antidiagonal

of 912(t1,t2)]. This temporal structure is the natural shape appearing in

SPDC when pumped by a powerful beam with a very narrow line width

(see, e.g., Hong & Mandel,1985;Shih et al.,1994;Gatti et al.,2003).

Based on the Fourier properties, temporal correlation implies frequency

anticorrelation, that is, ˜

912(ω1,ω2)∝8(ω1)δ(ω1+ω2), indicating that the

sum of the frequencies in the photon pair gives back the frequency of

the pump beam, which is consistent with energy conservation in the

SPDC process [see Figure 35(a)]. Here, the function 8(ω) is the two-

photon spectrum (often called biphoton spectrum) an forms a Fourier

transform pair with 912(τ =t2−t1)in the variable τ. Thus, the inverse

of its spectral width determines the coherence time of the second-order

intensity correlation. On the other hand, the width of the diagonal of

912(ω1,ω2)in the (ω1,ω2)plane is related with the pump line width

and therefore approaches a Dirac delta. An unequivocal signature of the

quantum character of these photon correlations is that the width of the

temporal correlation and the width of the frequency anticorrelation are

not bounded by a Fourier relation (Khan & Howell,2006). For example,

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68 Space-Time Analogies in Optics

Time t1 (a.u.)

(a)

Time t2 (a.u.)

−1

Frequency ω2 (a.u.)

−1

Frequency ω1 (a.u.)

(b)

−1

Time t1 (a.u.)

Time t2 (a.u.)

(c)

−1

Frequency ω2 (a.u.)

−1

Time t1 (a.u.)

Time t2 (a.u.)

(e)

−1

Frequency ω2 (a.u.)

−1

Frequency ω1 (a.u.)

(d)

−11

Frequency ω1 (a.u.)

(f)

−1

FIGURE 35 Schematic representation of different two-photon correlations. Upper

row shows the two-photon wavepacket, 912(t1,t2), and lower row shows the

corresponding frequency-domain representation, ˜

912(ω1,ω2), for: (a) (b) temporally

correlated biphoton; (c) (d) frequency-correlated biphoton; and (e) (f) uncorrelated

biphoton.

having strong and simultaneous frequency anticorrelation and temporal

correlation may lead to important implications in clock synchronization

(Valencia, Scarcelli, & Shih,2004) and cryptography (Khan & Howell,

2006).

Another interesting shape of the two-photon probability amplitude is

the dual of the previous case. Here, the signal and idler photons are

correlated in frequency instead, meaning that ˜

912(ω1,ω2)∝8(ω1−ω2).

Now, if we were able to detect the signal photon at frequency ω1, then

the idler would have the same frequency within an uncertainty given

by the width of the antidiagonal of ˜

912(ω1,ω2)in the (ω1,ω2)plane, as

can be understood from Figure 35(b). Based on the Fourier relations, fre-

quency correlation leads to temporal anticorrelation, that is,912(t1,t2)∝

0(t1)δ(t1+t2). Here, the temporal function 0(t)forms a Fourier trans-

form pair with 8(1ω =ω1−ω2)in the variable 1ω. The most remarkable

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Temporal Optics in the Two-Photon Regime 69

feature of these two-photon states is that the temporal anticorrelation

width is independent of the biphoton function 8(1ω =ω1−ω2). These

two-photon states are a valuable resource for timing and positioning

metrology because the precision of the measurements can be enhanced

with respect to the standard quantum limit by a factor of √2 (Giovannetti,

Lloyd, & Maccone,2001). A possible scheme for achieving frequency cor-

related photon pairs from SPDC involves the use a broadband pump pulse

with proper engineering of the phase-matching conditions (Price, U’ Ren,

& Walmsley,2001). The ﬁrst demonstration of this technique was reported

by Kuzucu et al. (2005).

Finally, another increasingly interesting shape of the probability ampli-

tude is the corresponding to a separable state, for which 912(t1,t2)=

f(t1)g(t2). Obviously, separability in the time domain also implies sepa-

rability in the spectral domain. This means that the two photons in the

pair are fully uncorrelated. This kind of light source is very interest-

ing for the implementation of single-photon sources based on a concept

known as heralding, where the idea relies on detecting one photon,

thereby revealing the presence of its twin. Poorly speaking, if the shape

of ˜

912(ω1,ω2)is the closest to a circle in the (ω1,ω2)plane, the two

photons would have the same spectral shape, thus becoming spectrally

indistinguishable. Besides, the separability in the two-photon probability

amplitude ensures that each photon is in a single spectral mode (ensuring

a high purity). The typical approach to achieve this shape was by narrow-

band ﬁltering (Ou, Rhee, & Wang,1999). However, having a separable

two-photon probability amplitude with broad bandwidth (i.e., directly

achieved without ﬁltering) remained a challenge for quite a long time. The

generation of ultrabroad pure heralded single photons constitutes a major

step towards the implementation of some quantum information protocols.

It is known that these appreciated wave packets can be achieved by a care-

ful manipulation of the phase-matching conditions in type-II SPDC when

pumped with a broadband pulse (Price, U’ Ren, & Walmsley,2001). The

ﬁrst experimental report of these states was recently presented by Mosley

et al. (2008).

7.3. Propagation of Ultrafast Two-Photon Wave Packets

From the previous section, we have learned that there are several impor-

tant applications that rely on a particular shape of the two-photon wave

packet, 912(t1,t2), and that this can be adequately tailored with a careful

engineering of the SPDC process. In this section, we show how the two-

photon wave packet propagates, after its generation, through the sort of

processing systems that we have studied in Section 5. We shall ﬁnd that

they can assist us to manipulate further the engineered two-photon states.

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70 Space-Time Analogies in Optics

Ki(t, t′)

G(2)(t1, t2)

Ks(t, t′)

Nonlinear

crystal

Signal

Idler

Pump

FIGURE 36 General scheme for two-photon propagation through linear systems. The

two-photon pair is generated through interaction in a nonlinear crystal. Signal and

idler are physically separated and travel different linear systems, which are

mathematically described by the corresponding kernel. Later the photons are

detected in coincidence with ideally ultrafast detectors, leading to the measurement

of the Glauber intensity correlation function.

Let us consider a scheme like the one presented in Figure 36. Each of

the photons from a pair in a properly tailored SPDC process is launched

into a different linear processing system. To study the propagation of

the two-photon probability amplitude, 912(t1,t2)= h0|ˆ

E+

s(t1)ˆ

E+

i(t2)|9i,

we just have to assess the change of the electric-ﬁeld operator, ˆ

E+

s,i(t). Giv-

ing a further step, we can rewrite this electric-ﬁeld operator as ˆ

E+

s,i(t)=

ψs,i(t)exp[−i(ω0s,i t−β0s,iz)], where ω0s,i is the central frequency of the sig-

nal (idler) photon and β0s,i is the corresponding phase factor. Analogously

to what we did in Section 4.1, we just have to study the evolution of the

envelope operator. Since the systems are linear, the transformation of the

complex envelope operators is described by

ψs,out(t)=Zˆ

ψs,in(t0)Ks(t,t0)dt0, (78a)

ψi,out(t)=Zˆ

ψi,in(t0)Ki(t,t0)dt0, (78b)

which is an obvious quantum generalization of Equation (9). Here, the ker-

nel Ks(i)(t,t0)corresponds to the system through which the signal (idler)

photon propagates. An important consequence of these equations, is that

the system operates individually on the electric ﬁeld of the corresponding

photon. The implications of this statement will be clear later. In any case,

the measurable quantity is the joint-probability detection, which is pro-

vided by the modulus square of the propagated two-photon probability

amplitude, 912,out(t1,t2). From the previous equations, it is easy to ﬁnd

that this magnitude evolves as

912e,out(t1,t2)=ZZ Ks(t1,t0

1)Ki(t2,t0

2)912e,in(t0

1,t0

2)dt0

1dt0

2, (79)

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Temporal Optics in the Two-Photon Regime 71

where the subscript e refers to the envelope part of 912(t1,t2). Of course,

alternatively, the two-photon wave packet can be tracked in the spectral

domain,

912b,out (ω1,ω2)=ZZ ˜

Ks(ω1,ω0

1)˜

Ki(ω2,ω0

2)˜

912b,in ω0

1,ω0

2dω0

1dω0

2, (80)

where the subscript b refers to the baseband two-dimensional function

912(ω1,ω2).

7.4. Two-Photon Temporal Far-Field Phenomenon

This section deals with the extension of the technique presented in

Section 5.2.2 to the two-photon regime. For simplicity, we shall consider

the case of spectrally anticorrelated photons that naturally arise from

CW-pumped SPDC. We assume a non-collinear geometry, so that after

generation each photon from the pair is launched onto a different GDD

medium. The output two-photon probability amplitude can be calcu-

lated using Equation (79), considering the kernel for the corresponding

GDD medium and reminding that the generated photons are frequency

anticorrelated,

912e,out(t1,t2)=Zexp i82eff

2ω28(ω) exp[i(t2−t1)ω]dω, (81)

where 8(ω) is the biphoton spectrum and the GDD parameters are

grouped into the coefﬁcient 82eff =82s +82i, with 82s(i) being the GDD

parameter of the dispersive circuit that the signal (idler) photon traverses.

We note the striking similarity of this equation with the one that governs

the distortion of an ultrashort light pulse. Thanks to this similarity that we

can adapt some of the learned notions in the previous sections to this new

exciting ﬁeld. For example, it is clear the two-photon probability ampli-

tude will spread by increasing 82eff. If this parameter keeps increasing to

the point that the inequality [which is just an extension of Equation (28)]

82eq t2

c/(4π) (82)

is satisﬁed, with tcbeing the biphoton coherence time, the output wave

packet becomes

912e,out (t1,t2)=exp "−i(t2−t1)2

282eff #8t2−t1

82eff , (83)

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72 Space-Time Analogies in Optics

800

600

400

200

025 30 35 40 45 50

t (ns)

Number of counts

FIGURE 37 Experimental demonstration of the two-photon temporal far-ﬁeld

phenomenon. The measured two-photon correlation function in a CW-pumped type-II

SPDC becomes a scaled replica of the expected biphoton spectrum. The

frequency-to-time mapping occurs thanks to the distortion of in the shape introduced

by the GDD circuits (SMF in the experiments). (Reprinted with permission from

Valencia et al., 2002). c

[2002] APS.

which is a scaled replica of the biphoton spectrum. Here, the scaling factor

is provided by the effective coefﬁcient 82eff. This frequency-to-time map-

ping was ﬁrst discovered by Valencia et al. (2002), and it can be used to

measure the biphoton spectrum without the need of scanning monochro-

mators, with a spectral resolution given by δω =δt/82eff, where δtis the

temporal resolution of the photon coincidence detectors (again ensuring

that the minimum spectral feature is properly mapped into the spectral

domain). An example of this technique is shown in Figure 37, where the

two-photon coincidence detection is plotted together with the expected

biphoton spectrum calculated from the phase-matching conditions for the

particular SPDC arrangement.

While we have focused on the case of CW-pumped SPDC, the technique

can be easily applied for other schemes with different two-photon probabil-

ity amplitudes. Then, if the joint-probability detection is measured in the

whole (t1,t2)plane and not only in the line (t2−t1), it becomes possible

to build a spectrogram that calculates the joint-probability detection at

two frequencies using the same equipment for calculating the probability

detection in time (Avenhaus et al.,2010).

The challenge in this technique relies on overcoming the timing jit-

ter limited resolution in state-of-the-art photon detectors and introducing

high dispersion with minimal photon losses in order to get a spectral

resolution competitive with more conventional spectral devices.

7.5. Two-Photon Temporal Imaging

Recalling Figure 36, we have already commented that each linear tempo-

ral system acts individually on the envelope of the electric-ﬁeld operator

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Temporal Optics in the Two-Photon Regime 73

of the corresponding photon. Then, we can use the notions exposed in

Section 5 for Gaussian temporal systems. In particular, let us consider

a temporal imaging system introduced in the arm of the signal pho-

ton, while the remaining idler beam unaltered. The transformation of the

complex envelope operator will be (Tsang & Psaltis,2006)

ψs,out(t)=exp h−iK0t2/2iˆ

ψs,in(t/m), (84a)

ψi,out(t)=ˆ

ψi,in(t0). (84b)

Thus, apart from an irrelevant quadratic phase factor with chirping K0,

the output complex envelope operator becomes a scaled replica of the

input one. The scaling factor is the same as the one provided by a classical

temporal imaging system, m=822/821 . In other words, it is possible to

change the scaling of the photon correlations in one of the variables (t1or

t2)while leaving unaffected the other one. As pointed out by Tsang and

Psaltis (2006), by using a temporal imaging system with m= −1, one can,

in principle, achieve a temporally anticorrelated biphoton starting from a

temporally correlated two-photon wave packet.

7.6. Some Remarks on Ultrafast Two-Photon Processors

It is not difﬁcult to think about different alternative conﬁgurations to tune

the correlations of the photons generated from SPDC using the ultrafast

processing schemes that we have studied in Section 5. For example, with

a proper engineering of the spectral scaling factor, the spectral imaging

system from Section 5.5 could be analogously used to create a spec-

trally anticorrelated biphoton from a spectrally correlated one and vice

versa. Or the 4f ultrafast Fourier processing systems could be used to

process in-line the temporal and spectral structure of more complicated

two-photon wave packets.

However, no operation of time lensing at a single-photon scale has

been reported yet. At a low-photon level, implementing time lens conﬁg-

urations based on nonlinear optics constitutes a major challenge, owing

to the intrinsic low-power regime operation. The EOPM time lens con-

stitutes in this case the most reliable alternative. In this direction, there

exist some ultrafast Fourier processing schemes based on the use of an

EOPM (Kolchin et al.,2008;Sensarn, Yin, & Harris,2009), but they are

not strictly operating as a time lens. Nevertheless, the impressive results

achieved in these experiments indicate that there is huge avenue to adapt

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74 Space-Time Analogies in Optics

the notions that we have acquired in classical temporal optics to process

temporal two-photon wave packets.

8. CONCLUSIONS

The space-time analogy establishes a mathematical connection between

the equations that govern the paraxial diffraction of a light beam and

those describing the linear distortion of a short light pulse in a dispersive

medium. Far from being a simple curiosity, this analogy has permitted

us to identify and design new photonic devices and systems, as well as

reinterpret physical phenomena, bringing enhanced capabilities for pro-

cessing ultrafast optical pulses. The key of this success relies on the fact

that we can borrow well-established concepts from the ﬁeld of classical

Fourier optics and adapt them to the particular problem or situation in

the temporal domain.

We have also seen that while most of the research efforts have been

devoted to explore the coherent temporal counterpart situations, the anal-

ogy is not at all restricted to coherent signals. Actually, in the last years,

we have seen increasing efforts to perform incoherent processing of short

pulses. In this direction, there are important challenges associated with

the fact that incomplete coherence necessarily implies dealing with the

stochastic nature of light. From a practical perspective, this means that

one has to balance the advantages of having incoherent sources with the

extra noise associated with the system’s noise performance.

Finally, we have also observed that the space-time analogy can be

formally extended to the ﬁeld of quantum optics. In particular, owing

to the ultrafast nature of the temporal correlations of entangled two-

photon states, this kind of light source can be manipulated using simi-

lar arrangements to those used for ultrafast coherent signal processors.

In this direction, some processing schemes have already been experi-

mentally realized, while some exciting proposals are still waiting to be

achieved.

The continuous and impressive advances in the fabrication of new

nanophotonic devices, having ultra-high nonlinearities with low peak

power and offering the possibility for dispersion engineering; ultrafast

lasers, with ever more compact and robust schemes, delivering ultrashort

pulses with wavelength tuning and high stability; ultra-high-speed digi-

tal acquisition systems, with a continuous scaling in electronic bandwidth

and real-time performance; and ﬁnally quantum light sources with inte-

gration capabilities and high brightness, permit us to conclude that there

is still plenty of room to keep extending and performing new processing

systems conceived within the concepts of this fruitful analogy.

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References 75

ACKNOWLEDGMENTS

We are very grateful to many people with whom we have commented

the ideas exposed in this review and have directly or indirectly partici-

pated in some of the experimental results provided. Lawrence R. Chen

(McGill University, Canada) deserves special gratitude, as the experimen-

tal results presented in this review were taken in his ﬁber optics laboratory

(unless otherwise stated). We thank Jos´

e Aza˜

na (Institut National de la

Recherche Scientiﬁque, INRS, Canada) and Alejandra Valencia (Institute

of Photonic Sciences, ICFO, Spain) for the careful reading and insightful

comments on the contents of the chapter. Hanna Lajunen, Enrique Sil-

vestre (Universitat de Valencia, Spain), Ari T. Friberg (Royal Institute of

Technology, KTH, Sweden), Juan P. Torres, (Institute of Photonic Sciences,

ICFO, Spain), J. Ojeda-Casta˜

neda (University of Guanajuato, Mexico) and

Carlos R. Fern´

andez-Pousa (University of Elche, Spain) are also gratefully

acknowledged for sharing comments about part of the contents exposed

in this review. Thanks also to Keisuke Goda (University of California,

Los Angeles, USA) for providing us with the material of Figure 7 and

to Antonio Malacarne and Jos´

e Aza˜

na (Institut National de la Recherche

Scientiﬁque, INRS, Canada) for providing Figure 19. We also thank the

principal investigators for each of the works explicitly highlighted in

ﬁgures for granting us permission to use their results.

Victor Torres acknowledges the Spanish Ministry of Science (MICINN)

and the Spanish Foundation for Science and Technology (FECYT) for

a postdoctoral fellowship. This work has been partially funded by

the National Science and Engineering Research Council (Canada) and

MICINN, Spain (under projects FIS2010-15746 and Consolider program

CSD2007-00013).

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On the role of chirping in pulsed single photon spectroscopy

Preprint

Full-text available

May 2024

Spatiotemporal optical vortices in atmospheric turbulence

Article

Full-text available

Oct 2023
WAVE RANDOM COMPLEX

We investigate the behaviors of spatiotemporal optical vortex (STOV) beams propagating through atmospheric turbulence. Starting with the extended Huygens-Fresnel integral, we derive an expression for the mutual coherence function (MCF) of a STOV beam, of any vortex order, propagating through turbulence of arbitrary strength. We specialize our general MCF and compare it to existing results from the literature. In addition, we analyze, in detail, first-and second-order STOV beams, focusing on how turbulence degrades both the beams' shapes and vortices. Lastly, we simulate the propagation of a second-order STOV beam through atmospheric turbulence and compare the results to predictions from our theoretical MCF. We find the results to be in good agreement thereby validating our analysis.

Twisted spatiotemporal optical vortex beams in dispersive media

Article

Full-text available

May 2024
OPT COMMUN

Milo Hyde

We derive a closed-form expression for the mutual coherence function (MCF) of a twisted spatiotemporal optical vortex (STOV) beam after propagating a distance z in a linear dispersive medium. A twisted STOV beam is a partially coherent optical field that possesses a coherent STOV and a stochastic twist coupling its space and time dimensions. These beams belong to a special class of space-time-coupled light fields that carry transverse (to the direction of propagation) orbital angular momentum, making them potentially useful in numerous applications including quantum optics, optical manipulation, and optical communications. After presenting the derivation, we validate our new general MCF by showing that it simplifies to the MCFs for two example beams from the literature. Lastly, we present and analyze space-time beam profiles and complex arguments (phases) of the MCF at multiple z and differing amounts of material dispersion, field correlation or coherence, and beam twist to gain insight into how twisted STOV beams evolve in dispersive media. We conclude with a brief summary.

Ultrafast electro-optic Time-Frequency Fractional Fourier Imaging at the Single-Photon Level

Article

Full-text available

Mar 2024
OPT EXPRESS

The Fractional Fourier Transform (FRT) corresponds to an arbitrary-angle rotation in the phase space, e.g., the time-frequency (TF) space, and generalizes the fundamentally important Fourier Transform. FRT applications range from classical signal processing (e.g., time-correlated noise optimal filtering) to emerging quantum technologies (e.g., super-resolution TF sensing) which rely on or benefit from coherent low-noise TF operations. Here a versatile low-noise single-photon-compatible implementation of the FRT is presented. Optical TF FRT can be synthesized as a series of a spectral disperser, a time-lens, and another spectral disperser. Relying on the state-of-the-art electro-optic modulators (EOM) for the time-lens, our method avoids added noise inherent to the alternatives based on non-linear optical interactions (such as wave-mixing, cross-phase modulation, or parametric processes). Precise control of the EOM-driving radio-frequency signal enables fast all-electronic control of the FRT angle. In the experiment, we demonstrate FRT angles of up to 1.63 rad for pairs of coherent temporally separated 11.5 ps-wide pulses in the near-infrared (800 nm). We observe a good agreement between the simulated and measured output spectra in the bright-light and single-photon-level regimes, and for a range of pulse separations (20 ps to 26.7 ps). Furthermore, a tradeoff is established between the maximal FRT angle and optical bandwidth, with the current setup accommodating up to 248 GHz of bandwidth. With the ongoing progress in EOM on-chip integration, we envisage excellent scalability and vast applications in all-optical TF processing both in the classical and quantum regimes.

Spatiotemporal diffractive deep neural networks

Article

Full-text available

Jan 2024
OPT EXPRESS

A spatiotemporal diffractive deep neural network (STD2NN) is proposed for spatiotemporal signal processing. The STD2NN is formed by gratings, which convert the signal from the frequency domain to the spatial domain, and multiple layers consisting of spatial lenses and space light modulators (SLMs), which conduct spatiotemporal phase modulation. An all-optical backpropagation (BP) algorithm for SLM phase tuning is proposed, with the gradient of the loss function computed by the inner product of the forward propagating optical field and the backward propagating conjugated error field. As a proof of concept, a spatiotemporal word “OPTICA” is generated by the STD2NN. Afterwards, a spatiotemporal optical vortex (STOV) beam multiplexer based on the STD2NN is demonstrated, which converts the spatially separated Gaussian beams into the STOV wave-packets with different topological charges. Both cases illustrate the capability of the proposed STD2NN to generate and process the spatiotemporal signals.

Erecting time telescope for photonic quantum networks

Article

Full-text available

Oct 2023
OPT EXPRESS

A time lens allows one to stretch or compress optical waveforms in time, similar to the conventional lens in space. However, a single-time-lens imaging system always imparts a residual temporal chirp on the image, which may be detrimental for quantum networks, where the temporal image interacts with other fields. We show that a two-time-lens imaging system satisfying the telescopic condition, a time telescope, is necessary and sufficient for creating a chirpless image. We develop a general theory of a time telescope, find the conditions for loss minimization, and show how an erecting time telescope creating a real image of a temporal object can be constructed. We consider several applications of such a telescope to making indistinguishable the photons generated by spontaneous parametric downconversion or single emitters such as quantum dots.

Preface of Topical Papers on Time Stretch Spectroscopy and Applications −Space-Time Duality in Ultrafast Optical Signal Processing−「タイムストレッチ分光法とその応用－超高速光信号処理における時空間双対性－」解説小特集号によせて

Article

Jan 2021

Fumihiko KANNARI

Advances in Fourier optics and information processing technology have facilitated a variety of new research areas in telecommunications, biology, quantum information science, spectroscopy, and atomic and molecular physics. The concept of space-time duality plays an important role as a system design tool for ultrafast optical processing and characterization. In particular, time-lens-based temporal imaging systems enable additional features such as waveform expansion and compression that are particularly useful for ultrafast optical signal processing. The time-equivalent optical setup of a spatial 2-f system, which acts as a Fourier processor (time-to-frequency and frequency-to-time conversion)，can be used as a basic component of more complex temporal imaging systems. This special issue focuses on timestretch spectroscopy, which uses frequency chirp ultrafast laser pulses to capture ultrafast phenomena in real time.

Continuous High-Rate Photonically Enabled Compressed Sensing (CHiRP-CS)

Chapter

Mar 2024

Mark Aaron Foster

Due to the ultrahigh bandwidth capacity of photonics, high-speed optical imaging, and sensing technologies can capture and manipulate extremely fast and complex signals. However, the ultimate performance of these sensing systems is entirely constrained by the information throughput of digitization and processing electronics. This digitization bottleneck can be alleviated using the principles of compressed sensing to implement data compression in the optical domain prior to digitization. Continuous High-Rate Photonically Enabled Compressed Sensing (CHiRP-CS) is a flexible photonic architecture to implement such high-speed compressed sensing in continuously recording scenarios. To do so CHiRP-CS leverages high-speed optical signal processing in robust fiber optic hardware based on the principles of the space-time duality of electromagnetic waves. This chapter first provides a brief introduction to the background for CHiRP-CS including a description of compressed sensing and the space-time duality of electromagnetic waves. Subsequently, the working principles of CHiRP-CS are described and several example applications are highlighted.

Fresnel time lens empowers quantum networks

Article

Aug 2023

John M. Donohue

The Lagrangian structure, the Euler equation, and second Newton’s law of ultrafast nonlinear optics

Article

Jul 2023
OPT COMMUN

A.M. Zheltikov

On Fresnel Diffraction by One-dimensional Periodic Objects, with Application to Structure Determination of Phase Objects

Article

Full-text available

Nov 2010

Jean-Pierre Guigay

In the case of a periodic object, the wave amplitude in every point of a Fresnel diffraction pattern may be obtained from the values of the wave amplitude at a finite number of points in the object plane. This results is applied to the case of pure phase objects. A simple condition on the object structure is derived, which would allow the object phase distribution to be determined directly from a measured intensity profile for a certain defocusing distance.

Ultrafast Optics

Article

Oct 2008

Andrew M. Weiner

Half Title Wiley Series Page Title Copyright Dedication Contents Preface

Self-restoration of the autoidolon of defective periodic objects

Article

Jan 1978
Opt Acta

Temporal imaging with a time lens

Article

Jun 1989

We extend the well-known analogy between the problems of paraxial diffraction in space and dispersion in time to optical pulse compression and propose a time-domain analog to spatial imaging that allows for the distortionless expansion or compression of optical power waveforms. We call this new concept temporal imaging and derive equivalent expressions for the focal length and the f-number of a time lens and the magnification of an imaging system. It should now become possible, with a temporal microscope, to expand ultrafast optical phenomena to a time scale that is accessible to conventional high-speed photodiodes.

Theory of the partially-coherent temporal Talbot effect

Article

Oct 2006
OPT COMMUN

The behaviour of the temporal Talbot effect under carrier partial coherence is analysed. This effect finds application as a passive method for multiplying the repetition rate of the intensity of pulse trains in first-order dispersive media. The study is based on the representation of the average intensity of partially coherent modulated light after first order dispersion as a coherence-dependent low pass filter acting on the intensity distribution obtained with a monochromatic carrier. The resulting coherence limitations are more restrictive in Talbot devices with high indices, due to the necessity of highly coherent carriers to produce a faithful reconstruction of trains by multiple pulse-to-pulse interference. Coherence limitations for architectures composed of externally modulated cw sources are analysed. Generalizations of the theory to several spectral lines, and its relation with the Collett–Wolf theorem are also reported.

Weiner, A. M. Femtosecond pulse shaping using spatial light modulators. Rev. Sci. Instrum. 71, 1929-1960

Article

May 2000

Andrew M. Weiner

We review the field of femtosecond pulse shaping, in which Fourier synthesis methods are used to generate nearly arbitrarily shaped ultrafast optical wave forms according to user specification. An emphasis is placed on programmable pulse shaping methods based on the use of spatial light modulators. After outlining the fundamental principles of pulse shaping, we then present a detailed discussion of pulse shaping using several different types of spatial light modulators. Finally, new research directions in pulse shaping, and applications of pulse shaping to optical communications, biomedical optical imaging, high power laser amplifiers, quantum control, and laser-electron beam interactions are reviewed. (C) 2000 American Institute of Physics. [S0034-6748(00)02005-0].

Real-time optical reflectometry enabled by amplified dispersive Fourier transformation

Article

Jul 2008
APPL PHYS LETT

The axial scan rate of optical frequency-domain reflectometry and optical coherence tomography can be increased to megahertz frequencies by dispersive Fourier transformation. However, the fundamental connection between dispersion and loss creates a trade-off between detection sensitivity and acquisition speed. Here we circumvent this predicament by using distributed Raman postamplification of the reflection from the sample. The Raman amplification enables measurement of weak signals, which are otherwise buried in detector noise. It extends the depth range without sacrificing the acquisition speed. Single-shot imaging with improved sensitivity at an axial scan rate of 36.6 MHz is demonstrated.

Compression of picosecond light pulses

Article

Oct 1968
PHYS LETT A

E. B. Treacy

The ultrashort pulses generated by a Nd:glass laser have frequency swept carriers and can therefore be compressed by a pair of gratings to a length much closer to the reciprocal of the bandwidth. The author acknowledges stimulating discussions with M.J. Brienza and A.J. DeMaria, and the technical assistance of R. Bodurtha.

Dual Optical Systems

Article

May 1968
J Opt Soc Am

A. PAPOULIS

To every optical system S consisting of thin lenses illuminated by narrow beams, there corresponds a dual system Su with the following property: If the complex amplitudes of the fields on the input and output planes of S and Su are a(x,y), au(x,y) and b(x,y), bu(x,y), respectively, and au(x,y) equals the Fourier transform of a(x,y), then bu(x,y) equals the Fourier transform of b(x,y). This dualism can be used to simplify the analysis and synthesis of certain idealized systems.

General temporal self-imaging phenomena

Article

Jul 2003
J OPT SOC AM B

Using the space–time duality theory, we introduce and analyze theoretically the temporal equivalent of the spatial self-imaging phenomena under point-source illumination (spherical wave-front illumination). This temporal effect can be produced by passing a given periodic optical pulse sequence through a time lens followed by a dispersive medium. The time lens (quadratic phase modulator) implements the time-domain equivalent of spherical wave-front illumination. Based on this temporal effect, we demonstrate that a system composed of a time lens followed by a dispersive medium can be configured to operate over a periodic optical pulse sequence of finite duration (i) as a conventional temporal imaging system, providing a distortionless temporal compression or expansion of the original pulse sequence, or (ii) as an advanced temporal imaging system, combining the capabilities of a conventional imaging system with those of the fractional temporal self-imaging effect, i.e., multiplication of the original pulse-repetition rate.

Space-Time Analogies in Optics

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