adiabatic processes in frequency conversion2.1. the nonlinear coupled wave equations in linear...

Laser Photonics Rev. 8, No. 3, 333–367 (2014) / DOI 10.1002/lpor.201300107

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Abstract Adiabatic evolution, an important dynamical processin a variety of classical and quantum systems providing a ro-bust way of steering a system into a desired state [1–5], wasintroduced only recently to frequency conversion [6–9]. Adia-batic frequency conversion allowed the achievement of efficientscalable broadband frequency conversion [8,9] and was appliedsuccessfully to the conversion of ultrashort pulses, demonstrat-ing near-100% efficiency for ultrabroadband spectrum [10–16].The underlying analogy between undepleted pump nonlinearprocesses and coherently excited quantum systems was ex-tended in the past few years to multi-level quantum systems,demonstrating new concepts in frequency conversion, such ascomplete frequency conversion through an absorption band[17–20]. Additionally, the undepleted pump restriction was re-moved, enabling the exploration of adiabatic processes in thefully nonlinear dynamics regime of nonlinear optics [21–26]. In

Conv

ersi

on E

ffici

ency

Conversion axisω ω

Δk<0 Δk=0 Δk>0

this article, the basic concept of adiabatic frequency conversionis introduced, and recent advances in ultrashort physics, multi-process systems, and the fully nonlinear dynamics regime arereviewed.

Adiabatic processes in frequency conversion

Haim Suchowski1,∗,∗∗, Gil Porat2,∗∗, and Ady Arie2

1. Introduction

Adiabatic processes in a dynamical system occur whenan external perturbation of the system varies very slowlycompared to its internal dynamics, allowing the system thetime to adapt to the external changes [1]. Mathematically,it means that for the entire dynamical evolution, the systemremains at one of the system’s eigenmodes. These pro-cesses were investigated in many subfields in physics andengineering, ranging from adiabatic evolution in nuclearmagnetic resonance, coherently excited quantum atomicsystems, optical switching, waveguide arrays and recentlyeven in quantum computation [1–5]. With the use of ultra-short lasers with controllable shapes, adiabatic processeshave gained practical importance for coherent manipulationof atoms and molecules, providing a robust way of steeringa quantum system into desired states. Only in recent years, itwas understood that adiabatic dynamical processes can playa significant role also in optical frequency conversion, sug-gesting alternative schemes for efficient conversion [6–9].

Using standard frequency conversion techniques, onecould achieve either complete frequency conversion fornarrowband spectrum or inefficient conversion for broadbandwidth [28]. The introduction of adiabatic frequencyconversion resolved the bandwidth-efficiency trade-off, andachieved efficient scalable broadband frequency conversion[6–9]. Adiabatic dynamics in frequency conversion wasfirst theoretically suggested for second harmonic genera-

1 NSF Nano-scale Science and Engineering Center (NSEC), 3112 Etcheverry Hall, University of California,, Berkeley, CA 94720, USA2 Department of Physical Electronics, Fleischman Faculty of Engineering, Tel-Aviv University,, Tel-Aviv 69978, Israel∗∗These authors contributed equally to this work.∗Corresponding author: e-mail: [email protected]

tion (SHG), predicting robust conversion of fundumentallight source to its second harmonic [6, 7]. The analogybetween coherently excited multi-level quantum systemsand electromagnetic waves coupled by an undepleted pumpwave in a nonlinear crystal was introduced in Ref. [8], sug-gesting and experimentally realizing the concept of RapidAdiabatic Passage (RAP) in frequency conversion. In a setof experiments, a robust broad bandwidth conversion withvery high efficiency for sum frequency generation (SFG)from the near-IR into the visible has been performed. Itwas confirmed that the conversion process is insensitive tosmall changes in parameters that affect the phase mismatchsuch as crystal temperature, interaction length, angle of in-cidence, and input wavelength [9]. The method was appliedsuccessfully to the up-conversion and down-conversion ofultrashort pulses, where conversion of Ti:S oscillator pulseswith near-100% efficiency for ultrabroadband spectrum hasbeen obtained [10–12,16], allowing the generation of high-energy, multi-octave-spanning IR pulsed sources.

The concept of adiabatic evolution has been extendedbeyond RAP mechanism by introducing analogous schemesof adiabatic dynamics of coherently excited multi-levelquantum systems into frequency conversion, predictingand demonstrating new and unique phenomena. Two suchnovel schemes are adiabatic elimination mechanisms andthe introduction of a scheme analogous to Stimulated Ra-man Adiabatic Passage (STIRAP) from three level atomicdynamics, providing complete frequency conversion

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334 H. Suchowski, G. Porat, and A. Arie: Adiabatic processes in frequency conversion

through a highly absorptive frequency band [17,18]. In thecase of adiabatic elimination, it was shown that in additionto the material dispersion, phase matching also dependson the pump intensities, in analogy to the Stark shift inatomic systems. Another method directly extends the ba-sic approach, facilitating efficient broadband multi-processfrequency conversion between very far or near frequencies[19, 20].

Recently, the restriction of the undepleted pump as-sumption in the analysis, which linearizes the dynamics,was removed, thereby allowing the exploration of adiabaticprocesses in the fully nonlinear dynamics regime of non-linear optics [21–26], in which all the interacting wavesmay be depleted or amplified. Experimental results havealready been obtained for optical parametric amplification(OPA) [13–15], and the analysis is expected to significantlyexpand the use of the method to other nonlinear processessuch as SHG and four wave mixing (FWM).

Here we outline the background of the adiabatic mech-anism and review recent advances in the field. Our goal inthis paper is to allow the AMO (atomic, molecular, optics)and the NLO (nonlinear optics) researchers to share a com-mon language and understanding in the use of adiabaticprocesses for frequency conversion. This review is orga-nized as follow: in Section 2, we start with a short overviewof the fundamentals of frequency conversion using nonlin-ear optics. We continue by presenting the analogy betweenthe dynamics of sum and difference frequency generation(SFG and DFG) processes to the known dynamics of twolevel atomic systems and spin–1/2 from nuclear magneticresonance. In Section 3, we introduce the concept of adia-batic frequency conversion, and present the Landau-Zenercriterion in the context of nonlinear optics, as an analyticaltool of measuring adiabaticity (and thus conversion effi-ciency) of the process. Also, we present, the eigenvaluesdiagram, which as the dressed-state picture from atomicphysics, provides an intuitive understanding of the dynam-ics. In Section 4, we discuss the mechanism using ultrashortpulses, and review recent works on the achievement of oc-tave spanning mid-IR sources. Section 5 is devoted to theextension of adiabatic frequency conversion into multi-stepprocesses in nonlinear optics, such as adiabatic eliminatedsystem and STIRAP. In Section 6, several recent studies onadiabatic processes in the fully nonlinear dynamics regimewill be reviewed. We summarize and discuss future outlookin Section 7.

2. Fundamentals of frequency conversion

Nonlinear optics is the study of phenomena occurring whenthe optical properties of a material system are modified bythe presence of light. The modification depends nonlin-early on the strength of the optical field, and as a con-sequence, new optical frequencies can be generated in aprocess known as frequency conversion. In this Section, webriefly develop the essential dynamical equations of wavesmixing in nonlinear optics, discuss the requirements for fre-quency conversion and present the physics behind the quasi

phase matching (QPM) method as a way to manipulate theconversion process. Further reading on those topics can befound in any textbook of nonlinear optics [28, 29].

2.1. The nonlinear coupled wave equations

In linear optics, the induced polarization, P(t), of a mate-rial system depends linearly upon the applied electric fieldstrength, E(t) = A(z)ei(kz−ωt), in a manner that follows therelationship: P(t) = ε0χ

(1) E(t), where ε0 is the electricalpermittivity of vacuum and χ (1) is known as the linearsusceptibility. In nonlinear optics, the optical response canoften be described by expressing the polarization P(t) as apower series in the field strength E(t) as:

P(t) = ε0

[χ (1) E(t) + χ (2) E2(t) + χ (3) E3(t)

+ · · · + χ (N ) E N (t)]

= ε0χ(1) E(t) + P N L (t). (1)

The quantities χ (i) are known as the i-th order non-linear optical susceptibilities, which are tensors of ranki. In general, the nonlinear susceptibilities depend on thefrequencies of the applied fields, but under our present as-sumption of instantaneous response and the fact that thosefrequencies are far from the material’s resonances, we cantake them to be constants.

The general dynamical equation, which is written as-suming the scalar approximation and employing the slowlyvarying envelope approximation (assuming slow variationof the envelope over a wavelength scale) can be derived forthe propagation of the electromagnetic fields,

∂ A j (z, t)

∂z+ 1

vg

∂ A j (z, t)

∂t+ iβ

2

∂2 A j (z, t)

∂t2

= i2πω

ncP N L

j (z, t), (2)

where, 1vg

= ∂k∂ω

and β = ∂2k∂ω2 are the inverse of the group ve-

locity and group velocity dispersion (GVD), respectively.This derivation, which takes into account the main dis-persion properties of the waves inside the materials, isimportant for the propagation of ultrashort pulses (broadspectral bandwidth). Also, we note that the intensity thatis associated with the wave at frequency ωj in MKS the

system, is given by I j = 2n(

∈0μ0

) 12 ∣∣A j

∣∣2, where ∈0 =

8.85 × 10−12 F/m, μ0 = 4π × 10−7 H/m. The full deriva-tion and related quantities can be found elsewhere [28].

In the important case of monochromatic and quasi-monochromatic lasers, one can neglect the group velocityand GVD terms and obtain the known propagation equationin a nonlinear medium:

∂ A j (z, t)

∂z= i

2πω

ncP N L

j (z, t). (3)

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We will start by analyzing frequency conversion usingthree wave mixing (TWM) process, where the nonlinearpolarization contains only the second-order nonlinear termof the induced polarization. The indices j = 1,2,3 repre-sent signal, pump, and idler waves, respectively. In theseprocesses, light of two frequencies is mixed in a nonlin-ear crystal, resulting in the generation of a third color atthe sum or difference frequency. These three-wave mixingprocesses, also known as SFG and DFG, are typically verysensitive to the incoming frequencies. This is due to thephase mismatch between the waves, which is the lack ofoptical momentum conservation, originated from the dis-persion of the waves in the medium. In more rigorous form,the phase mismatch parameter is defined as �k = k1 +k2 − k3, where the wavevector is ki = n(ωi)·ωi

c and n(ωi )is the refraction index of the material involved. Due tothe fact that in all materials n(ωi ) is frequency depen-dent, a phase matching condition is usually not satisfied.Also, one can calculate the total accumulated phase asφ(z) = ∫

�k(z′) dz′, where in the general case the phase

mismatch may vary over the propagation length.By substituting explicitly the interacting waves into the

dynamical equations presented in Eq. (3), three couplednonlinear wave equations are obtained:

d A1(z)

dz= i

χ (2)ω21

k1c2A∗

2 A3e−i�kz, (4a)

d A2(z)

dz= i

χ (2)ω22

k2c2A∗

1 A3e−i�kz, (4b)

d A3(z)

dz= i

χ (2)ω23

k3c2A1 A2ei�kz . (4c)

2.2. Phase matching and quasi phase-matching

As explained in the previous section, the phase mismatch�k between the waves causes the conversion process to beinefficient. Here, we would like to elaborate more on thiscrucial parameter, and to describe techniques for phase-matching compensation. We begin by defining the coher-ence length, lc ≡ π/ |�k|. In the absence of any phasematching mechanism, waves generated at locations sep-arated by lc along the crystal are out of phase with eachother (i.e. their phases are π -shifted with respect to eachother). As a result, the direction of energy flow betweenthe waves is reversed with propagation of distance lc, pre-venting any buildup of the generated wave and causing theconversion to be inefficient. Note that lc is inversely propor-tional to the phase mismatch, meaning that large |�k| valueleads to a shorter distance over which conversion can effec-tively be performed. When a phase matching technique isbeing used, the phase-mismatch between the waves couldbe compensated, so the interaction length over which thedirection of energy flow is maintained is no longer limitedby the phase-mismatch, but rather by the depletion of theinteracting waves.

Compensating phase mismatch can be done inseveral ways. The two most widely used meth-ods to facilitate control over the phase-mismatchare birefringence phase-matching [30] and QPM[28, 31–33]. In the former, birefringent nonlinear crystalsare used to mix different polarizations [28], which pose alimitation on the tensor elements of χ (2) that can be utilized.The geometry required for birefringence phase-matchingmay also introduce unwanted walk-off between the inter-acting beams. On the other hand, the QPM method allows asimple and robust way of manipulating the phase mismatchparameter, which makes it the most attractive in terms ofexperimental realization. In the following section, we ex-plain the QPM technique in detail. Further information canbe found elsewhere [32, 33].

In a periodically poled crystal, the sign of the nonlinearsusceptibility is inverted with each successive domain, (i.e.,the sign of χ (2) is modulated along a ferroelectric crystal bymeans of electric field poling [34]) with each domain widthequal to the coherence length of the nonlinear process lc. Tounderstand how this modulation facilitates phase-matching,let us consider a periodic modulation with period ,

χ (2)(z) = χ(2)i jksign [cos (2π z/)] , (5)

where χ(2)i jk is the relevant χ (2) tensor element. This periodic

function can be written in the form of a Fourier series:

χ (2)(z) = χ(2)i jk

∞∑m=−∞

2

mπsin

(mπ

2

)exp

(i2πm

z

)

(6)

Let us assume that for all the terms in the series, exceptthe first one (m = 1), the phase mismatch is very large,hence we can ignore their contribution to the nonlinearprocess. Substituting only the first order term into Eq. (6)yields

d A1(z)

dz= i

2χ (2)ω21

πk1c2A∗

2 A3e−i(�k−2π/)z (7a)

d A2(z)

dz= i

2χ (2)ω22

πk2c2A∗

1 A3e−i(�k−2π/)z (7b)

d A3(z)

dz= i

2χ (2)ω23

πk3c2A∗

1 A2ei(�k−2π/)z (7c)

As seen in Eqs. (7), the effective phase mis-match parameter is the summation of the dispersionphase mismatch and an artificial phase-mismatch,i.e., �k + �k(z) = ksignal + kpump − kidler + �k(z),where �k(z) = −2π/ and(z) is the local polingperiod. The modulation period can thus be chosen such that�k − 2π/ = 0 and so phase-matching will be obtainedin the first-order Fourier approximation. The magnitude ofthe coupling coefficients for this case will be determinedby the amplitude of this Fourier component, namely 2/π .This modification stems from the fact that the contributionto the coupling from all the other terms in the Fourier series

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Figure 1 (a) Non-phase matched interac-tion (b) Quasi phased-matched interaction(c) The generated SFG intensity for the non-phase matched chase. (d) The generatedSFG intensity for QPM interaction.

is negligible, since they are characterized by a rapidlyoscillating exponential term which is averaged to zero byintegration over the interaction length. In this manner, anysingle process involving any triplet of frequencies, with anycombination of polarizations, can be quasi-phase-matched,in principle (i.e. ignoring technological limitations). Italso allows the use of any of the tensor elements of χ (2),and in particular large diagonal terms that cannot be usedin the case of birefringent phase matching. Furthermore,walk-off is eliminated by operating along principle axes ofthe crystal. The QPM mechanism is illustrated in Fig. 1.

More generally, by tuning the spatial structure of thedomains, this technique allows to design almost any de-sired function of the phase mismatch parameter. Usually,for an aperiodic design it is reasonable to expand �k(z)in a power series: �k(z) = �k0 + ∂�k

∂z z + 12

∂2�k∂z2 z2 +

· · · + 1N !

∂ N �k∂zN zN . For such arbitrary binary modulation, the

Fourier series is replaced with a Fourier transform, and thesame procedure can be repeated.

2.3. Analogy of SFG/DFG processes with twolevel systems

The three nonlinear coupled equations, shown in Eq. (4),can be simplified assuming that one of the incoming wave(termed pump in our analysis) is much stronger than theother two. In this very special simplification, which isknown as the undepleted pump approximation, the pumpamplitude, A2, can be considered to be constant along thepropagation, resulting in two linear coupled equation ratherthan the three nonlinear ones [28]:

d A1(z)

dz= iκ A3e−i�k z (8a)

d A3(z)

dz= iκ∗ A1ei�k z (8b)

Here again, �k = k1 + k2 − k3 is the phase mismatch,z is the position along the propagation axis, κ = χ (2)ω1ω3√

k1k3c2A2

is the coupling coefficient. The normalized signal andidler amplitudes are A1 ≡

√k1

ω1 A2A1 and A3 ≡

√k3

ω3 A2A3. These

equations are termed linear in the sense that they describelinear dynamics (dictated by parameters independent of sys-tem state), as opposed to Eq. (4), which describe a nonlineardynamics that depends on the state of the system.

The coupled wave equations of SFG and DFG processesin the undepleted pump approximation share their dynami-cal behavior with other two states systems, such as nuclearmagnetic resonance (NMR), polarization optics, the dy-namics of two coupled pendulums, and the interaction ofcoherent light with a two-level atom, due to the fact thatit possess SU(2) dynamical symmetry [35]. Some of thoseanalogous systems are presented in Fig. 2, where the evo-lution parameter is either the time parameter (spin 1/2 andtwo level atom) or the propagation parameter as in polar-ized light in single mode optical fiber and in SFG or DFGprocesses. The dynamics in any SU(2) system is dictated bytwo parameters – the coupling between the modes or states(which is sometime referred to as the off-diagonal term inthe Hamiltonian of a dynamical system) and the phase mis-match between the modes (which is the diagonal term in theHamiltonian). These are written explicitly below each dy-namical system shown in Fig. 2. In frequency conversion,those are the two z-dependent parameters, �k(z) and κ(z),the phase-mismatch and coupling coefficient respectively[8].

Though the dynamics is dictated only by those two pa-rameters, it can be solved analytically only for limited cases[35, 36]. One such solvable example, in frequency conver-sion, is when the phase mismatch is constant [28]. In thiscase, full energy transfer from signal to idler (SFG process)or vice versa (DFG process) is achieved only in the case ofperfect phase matching (�k(z) = 0), and only when κz =nπ is satisfied with odd n. This is known in the literature ofNMR and atomic physics as complete Rabi flopping. Other


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Figure 2 A formal equivalence can be shown between a number of diverse physical systems, such as polarization modes in a fiber withdifference �β(z) in propoagation constants and coupling κ(z), spin 1/2 case excited by a magnetic pulse at a radial frequency �R(t),which is detuned by �f(t) from the resonance frequency, coherently excited two level atom with a coupling coefficient of Veg(t) = μεp(t)

�,

(where εp(t) is the pump’s electromagnetic field and μ is the electronic dipole moment), also termed as the optical Rabi frequency,detuned by �(t) from the resonance frequency, and the SFG process with an undepleted pump.

Figure 3 Constant coupling dynamics (a) Two dimensional map of the conversion efficiency as a function of propagation length andthe input wavelength. (b) Geometrical visualization of the SFG dynamics on a SFG Bloch Sphere. Two trajectories are plotted: perfectphase matching (blue, torque vector points to the equator) which can result in efficient conversion, and a constant nonzero phasemismatch (orange, torque vector points to a point in the south hemisphere), always resulting in an inefficient conversion process.Upper inset shows the projection of the trajectory onto the w axis, which is the conversion efficiency. Lower inset shows a QPM crystalwith a constant phase matching along the propagation.

values of constant phase mismatch result in an inefficientfrequency conversion. In Fig. 3a we show numerical simu-lations of the expected conversion efficiency as a function ofpropagation length (horizontal axis) and signal wavelength(vertical axis), where we choose the perfect phase matchingfor λ = 1550 nm. As seen, only for that wavelength, com-plete conversion could occur, and only in a discrete numberof location along the propagation length.

Methods for approximate solutions, such as perturba-tion theory, are also available. In the weak coupling limit,the dynamics can be solved fully in the Fourier domain[37], but due to its perturbative nature, it will be limited tolow conversion efficiency. In the general case of a complexvalued dynamics of SU(2) symmetry, where the phase mis-match parameter varies along the propagation, there isn’t aknown analytical solution, a statement which is also true inNMR and in light-matter interaction in two level systems[35, 36]. For those cases which cannot be solved analyti-

cally, it is particularly convenient to use a geometrical rep-resentation, which gives a physical intuition to the evolutionof the dynamics (the conversion along the propagation inthe case of frequency conversion), without solving or fullysimulating the dynamical process.

2.4. Geometrical representation of SU(2)dynamics

For the geometrical representation, the geometrical formu-lation approach presented by F. Bloch for spin 1/2 systems[38], R. Feynman et al. for atomic systems [39] and byH. Poincare in polarization optics [40], were adopted towrite the dynamics of SFG/DFG system as a real three di-mensional vector. Such representation allows the geomet-rical visualization on a sphere, known as Bloch sphere. Infrequency conversion, the state vector, ρSFG = (U, V, W )




is defined as follows

USFG = A∗3 A1 + A∗

1 A3 (9a)

VSFG = i(

A∗3 A1 − A∗

1 A3)

(9b)

WSFG = |A3|2 − |A1|2 (9c)

This vector represents the relation between the signaland idler fields along the crystal. In particular, the WSFG

component gives information about the conversion effi-ciency. The South pole ρSFG = (0, 0, −1) corresponds tozero conversion (A3 = 0), while the north pole ρSFG =(0, 0, 1) corresponds to full conversion. In between, theconversion efficiency can be evaluated by η = (WSFG+1)

2 . Theloss-free evolution equations can be written as a single vec-tor precession equation:

d ρSFG

dz= g × ρSFG (10)

where the torque vector g = (Re{κ}, lm{κ},�k) representsthe coupling between the fields. Figure 3b presents the caseof constant phase matching case, where the perfect phasematched solution (i.e. �k = 0) for full conversion [28], hasthe same dynamical trajectory on the Bloch sphere surfaceas on-resonant interaction in atomic physic [35]. This re-sults in oscillatory dynamics between the two modes (“Rabioscillations” in atomic or spin 1/2 systems). An odd π -pulse in optical resonance is analogous to full transfer ofenergy from ω1 to ω3 (ω3 to ω1), i.e. the SFG state vec-tor is rotated from the south pole (north pole) to the northpole (south pole). Any non-zero phase mismatch will leadto a dynamics similar to a detuned resonance interaction,which exhibits faster oscillations and lower conversion ef-ficiencies, as shown in the upper inset of Fig. 3b. The ef-ficiency value of the constant phase-mismatch dynamics isη = κ2

g2 sin2(gz) [28].

3. Rapid adiabatic passage in frequencyconversion

The analogy of the dynamics in nonlinear optics media andthe rich framework of discrete level dynamics in atomicsystems, spin 1/2 system and other SU(2) dynamical sys-tems open exciting new possibilities. Here we consider oneunique scheme, the RAP mechanism, which was imple-mented recently in the realm of frequency conversion [8].Adiabatic processes in general, and RAP mechanism inparticular, occur when an external perturbation of the sys-tem varies very slowly compared to the internal dynamicsof the system, allowing the system time to adapt to thechanges. Mathematically, it means that the system remainsin one of its eigenmode for the entire dynamical evolution.In frequency conversion, we will show that by changingthe coupling or the phase-mismatch between the interac-tion waves in an adiabatic manner, the system can evolvefrom an initial mode (in which all the energy is at ω1 for

instance) and end in another mode (ω3), while remaining inthe same eigen-mode along the entire propagation, allow-ing nearly complete frequency conversion for ultrabroadoptical bandwidth.

Here, we layout the mathematical and physical back-ground for the adiabatic process in two level dynamics, andanalyze the robustness of the adiabatic SFG scheme, boththeoretically and experimentally. We introduce the Landau-Zener conversion efficiency formula, which is used to es-timate the efficiency of the adiabatic conversion process.The robustness of the adiabatic method is discussed as well,showing efficient conversion for wide range of frequenciesand temperatures, up to two orders of magnitude larger thanstandard phase matching. We also present the eigenvaluesdiagram that gives an intuitive illustration of the diabaticand adiabatic evolution in such systems, which will thenbe used in the following sections to analyze more complexmulti-step processes.

3.1. Mathematical layout

Let us write the z-dependent dynamics of the SFG processas presented in Eqs. (8) in the rotating frame using matrixform:

i∂

∂z

(A1A3

)=

(0 χ (2)ω2

1k1c2 A∗

2ei�kz

χ (2)ω23

k3c2 A2e−i�kz 0

)(A1A3

)

→ i∂

∂z

(A1

A3

)=

(−�k(z) κ

κ∗ �k(z)

)(A1

A3

)(11)

In that frame, the states with amplitudes A1 =√k1

ω1 A2A1e−i�kz and A3 =

√k3

ω3 A2A3e−i�kz are commonly

called the bare states in the terminology that is taken fromquantum literature [27]. In order to retrieve the adiabatic cri-teria, it is preferable to move to the adiabatic basis, whichis the basis of the system’s eigenvectors. This will be doneby the following transformation [27, 41]:(

B1B3

)=

(cos(θ ) sin(θ )

− sin(θ ) cos(θ )

) (A1

A3

), (12)

where,

tan(θ ) = κ(z)

�k(z)

The new states, with amplitudes Bi(z), are known asthe adiabatic states or the dressed states of the system inthe quantum literature, and they obey the following set ofcoupled equations:

i∂

∂z

(B1B3

)=

(−ε(z) −i θ (z)i θ (z) ε(z)

)(B1B3

)(13)

where

ε(z) =√

κ2(z) + �k2(z)


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Figure 4 Adiabatic conversion scheme of SFG. (a) Conversion efficiency for the adiabatic aperiodic design along the propagation axis(horizontal axis) and different input wavelength (vertical axis). (b) The adiabatic following trajectory. Upper inset shows the projectionof the trajectory onto the w axis. In this trajectory, the phase matching condition is fulfilled only at z = 1 cm, and the effective length ofthe conversion is 3 mm. The lower inset shows that continuous adiabatic variation of the phase mismatch parameter is required.

The condition for adiabatic evolution is obtained whenthe off diagonal parts of the dynamics in the adiabatic frameare much smaller than the diagonal ones. It results in thefollowing inequality:∣∣∣∣dθ

dz

∣∣∣∣ � ε(z) →∣∣∣∣dκ

dz�k − κ

d�k

dz

∣∣∣∣ � (κ2 + �k2

) 32

(14)

If the coupling coefficient is constant along the propa-gation, the adiabatic criterion reduces to:

∣∣∣∣d�k

dz

∣∣∣∣ �(κ2 + �k2

) 32

|κ| (15)

From the adiabatic inequality, we see that in order to ac-complish adiabatic passage, the sweep rate of the phase mis-match parameter along the propagation should vary slowlywith respect to the square of the coupling term. The phasemismatch parameter, �k(z), also should be very large inthe entrance and exit of the crystal (positive or negative),i.e. |�k| � κ , and �k(z = 0) < 0, �k(z = L) > 0 (orvice versa). This requirement makes the bare states and theadiabatic states coincide at the beginning and end of theinteraction (see Eq. 12). If the rate of variation is not slowenough, or the coupling coefficient is not large enough, thisinequality will not be satisfied and the conversion efficiencywill be poor. In any practical realization, where the crystallength is finite, the adiabaticity condition, corresponding toa conversion efficiency of 100%, can only be asymptoti-cally reached. The exact variation of �k(z) function alongthe propagation is not so critical as long as the adiabaticitycondition is satisfied. Due to such variation in the nonlin-

ear crystal, many wavelengths can meet this criterion, thusallowing a unique opportunity for efficient broadband con-version devices. In Fig. 4a, we plot a numerical simulationof the conversion efficiency of such process as a functionof propagation axis (horizontal axis) and seed wavelength(vertical axis) in the case of the adiabatic design. As seen, atthe end facet of the crystal all the wavelengths in that opti-cal range were converted efficiently. The conversion of eachfrequency component occurs in a different location alongthe nonlinear crystal, correlating with the location of �k(z)≈ 0 for the interacting waves involved. The geometricaltrajectory of the adiabatic evolution for seed wavelength ofλ = 1550 nm can be viewed on the surface of the Blochsphere as shown in Fig. 4b.

It should be noted that the adiabatic criterion could beobtained also by tuning adiabatically the coupling coeffi-cient of the dynamics, κ(z), as can learned from Eq. (14).With QPM, this can be done by varying the duty cycle of thenonlinear modulation pattern, which has a direct influenceon the coupling coefficient.

3.2. The dressed state picture - Visualizationusing Eigenvalue diagrams

The eigenvalues of the dynamical system can be viewed ina graphical visualization, known as the eigenvalue diagram[20], or the dressed state picture in the quantum literature[27]. In the diagram, it is common to plot two types ofcurves: the eigenvalues in the absence of coupling, i.e. theeigenvalues of the bare states of the system, and the eigen-values when the coupling term is involved, which are theeigenvalues of the dressed states. Such diagram providesvaluable information on the dynamics of the system and




Figure 5 Adiabatic evolution of SFG withundepleted pump. Numerically obtained in-tensities along an adiabatically chirped crys-tal: (a) increasing period (c) decreasing pe-riod. The adiabatic interactions in (a) and(c) follow the paths marked with arrows in theeigenvalues plots in (b) and (d), respectively.

in particular gives a physical intuitive understanding if thedynamics is abrupt or adiabatic.

In order to describe the eigenvalues diagram for SFG,we start with the case where there is no coupling, i.e. κ =0, so the coupling matrix in the right hand side of Eq. (11)is diagonal. Therefore, if only one of the interacting waveshas nonzero amplitude, the system will be in an eigenstate.Clearly, in this case, the two interacting waves are eigen-states of the system, i.e. they are the bare states. Each wavethen evolves as A j (z) = A j (0) exp(ikbare

j z), where kbarej is

its eigenvalue. Mathematically, kbarej are the eigenvalues of

the coupling matrix in the right hand side of Eq. (11). (Notethat kbare

j are not the same as the wavenumbers kj, which arethe carrier spatial frequencies of the waves with the slowcomplex envelopes Aj of Eq. (11)). In Fig. 5b, we haveplotted the eigenvalues of the bare states against the localQPM modulation period as black solid lines. Due to ourchoice of the reference frame, the eigenvalue of A3, kbare

3 ,is identically zero, and the eigenvalue of the A1 wave, kbare

1 ,is inversely proportional to the QPM period. At the cross-ing point of these two lines we have kbare

1 = kbare3 , meaning

that at this point the total phase difference between the twowaves is zero. This is the phase-matching point, i.e. it cor-responds to the QPM modulation period that phase matchesthe SFG process. When coupling is present (due to the pres-ence of the pump laser), the eigenstates of the system arethe dressed states. They are different from the bare states,and correspondingly have different eigenvalues. We haveplotted these eigenvalues as dashed blue lines in Fig. 5b.These lines bend such that the crossing is avoided, wheregreater pump intensity results in greater bending. Far fromthe crossing the dashed and solid lines overlap. The phys-ical interpretation gained from this diagram is that owingto the coupling, the situation where only one of the twoamplitudes is nonzero is no longer an eigenstate of the sys-tem near the crossing. Therefore, if such a state is obtained

near the crossing, energy will be transferred from one waveto another as light propagates along the crystal. This cor-responds to the explanation given above, on the effect ofphase mismatch on the coupling between the two interact-ing waves. Note that far from the crossing the dashed andsolid lines overlap, meaning that when the phase-mismatchis large, the eigestates in the presence or absence of cou-pling are the same (there is no difference between the barestates and the dressed states). Therefore, in this situation, thetwo waves are the eigenstates of the system, regardless ofthe pump. The corresponding physical interpretation is thatwhen the phase-mismatch is large, energy transfer betweenthe waves will be insignificant, regardless of the presenceof the coupling via the pump laser, as we expect.

The usefulness of the eigenvalues plot in describing adi-abatic interactions will be given in the following numericalexample. We consider the case where the QPM modula-tion period is chirped adiabatically, from = 14.3 μm to = 15.7 μm. This modulation causes the phase-mismatchto start with a very large positive value, gradually decreasealong the crystal, and end with a very large negative value.Graphically this would correspond to a motion from left toright on the eigenvalues plot as light propagates along thecrystal. At the beginning of the interaction = 14.3 μm,and energy is present only at the ω1 frequency, so thesystem is “on” the solid black line of kbare

1 at the top leftof the eigenvalues plot of Fig. 5b. This point overlaps witha dashed blue line since it is far from phase-matching. Ac-cording to the adiabatic theorem, as the phase-mismatchchanges due to the adiabatically varying QPM modulationperiod, the system will stay in the same eigenstate. Thismeans that the system will remain on the same dashed blueline, going from left to right, as illustrated with arrows inFig. 5b. At the end of the interaction, the dashed blue linethat was followed overlaps with the solid black line thatcorresponds to kbare

3 , i.e. the state at which there is energy


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only at ω3, with none at ω1. The eigenvalues plot represen-tation thus predicts that if adiabatic interaction is achieved,all of the energy will be transferred from ω1 to ω3, which isindeed the result obtained numerically, seen in Fig. 5a (inthis figure, the output intensity exceeds the input intensitydue to contribution from the pump). In the same manner, ifthe QPM modulation period will be chirped in the oppositedirection, i.e. from = 15.7 μm to = 14.3 μm in thisnumerical example, the system will start on the kbare

1 solidblack line at the bottom right, and follow the overlappingblue dashed line from right to left as illustrated in Fig. 5d.Once again at the end of the process the blue dashed lineoverlaps with the solid black line of kbare

3 , so all of theenergy gets transferred to ω3, as seen in the numerical sim-ulation of Fig. 5c. In this example the same behavior isobtained for an up-chirped or down-chirped crystals, butlater we will describe a different example in which thegenerated signal depends on the direction of the chirp.

The robustness of adiabatic conversion is also mani-fested graphically in the eigenvalues diagram. A changein the parameters of the interaction, e.g. input wavelength,will cause a horizontal shift of the diagram. As long asthe crossing-point of the solid black lines (the bare stateseigenvalues) is within the QPM period chirp range, theabove description holds, so the interaction will be effi-cient. Since the crossing-point is the phase-matching point,this graphical phenomenon exactly corresponds to the rig-orous description of adiabatic interaction given earlierSection 3.2.

3.3. Adiabaticity measurement – theLandau-Zener criterion

A known theorem from quantum physics literature, whichprovides valuable insight to the adiabatic dynamics, is theLandau-Zener theorem [42, 43]. This theorem, which canestimate the probability of electron transitions in two-levelsystems, gives a measurement of the adiabaticity, or moreaccurately the amount of diabaticity (non-adiabatic cor-rections) of the interaction. This theorem was adopted forfrequency conversion [8]. For example, the conversion effi-ciency of the adiabatic SFG process with undepleted pumpas described above, where the phase mismatch is slowlyscanned through the phase-matching point, is

ηL Z (z → ∞) = 1 − e− 2π |κ|2|d�k/dz| . (16)

This expression, which gives the signal-to-idler conver-sion efficiency of the SFG process, is analytical when thevariation of the phase mismatch is linear, but only approx-imate otherwise. The efficiency depends exponentially onan adiabatic parameter, defined as α ≡ |d�k/dz|

2π |κ|2 , i.e. the ra-tio between the sweep rate of the phase mismatch, d�k/dz,and the square of the coupling coefficient, κ2. Mathemati-cally it represents the ratio between the left hand side and

the right hand side of Eq. (15), for �k = 0. Adiabaticpropagation is obtained when α � 1, which is the asymp-totic case where the conversion efficiency reaches unity.This can be achieved either by changing the sweep rateslowly at a given pump intensity, or by applying strongpump for a given sweep rate. A nice comparison of thethree different dynamical regimes, α � 1, α ∼ 1, α � 1,for diabatic, semi-daibatic and adiabatic trajectories can befurther viewed in Ref. [9].

Due to the importance of Eq. (16), we present it usingmore practical parameters:

ηL Z (z → ∞) = 1 − exp

(− 2 · 104 · π2

(χ (2)

)2I2

n1n2n3λ1λ3ε0c |d�k/dz|

).

(17)

Here, c = 3 · 108 m/sec, ε0 = 8.85 · 10−12 F/m, λ1 andλ3 are measured in nm, I2 is measured in MW/cm2, χ (2)

is measured in pm/V and |d�k/dz| is measured in m−2.In the adiabatic limit, the conversion process occurs in alocalized region along the crystal. An estimation of thischaracteristic length can be found by adopting the Landau-Zener inversion time from quantum literature [44, 45] tothe realm of frequency conversion. In nonlinear optics itis Ladiabatic = κ

|d�k/dz| , which gives a linear relation to thecoupling coefficient term (i.e. the pump field), and inverselyproportional to the sweep rate. Outside the adiabatic regime,the effective length does not hold [44,45]. This length is par-ticularly important for frequency conversion of ultrashortpulses, as will be discussed in the next section.

The bandwidth of the conversion is determined roughlyby the wavelengths that meet perfect phase matching alongthe adiabatic propagation in the nonlinear crystal. The avail-able bandwidth is proportional to the length of the nonlinearcrystal, thus it is scalable by a proper design. The efficiencyof each wavelength can be easily determined by Eq. (17),the Landau-Zener formula in frequency conversion. Thescalable bandwidth can be shown in Fig. 4a, where eachseed wavelength meets, in an adiabatic way, perfect phasematching at a different location along the nonlinear crystal,allowing full conversion of more than 100 nm bandwidth.A precise definition of the bandwidth in such processes isdetailed in Section 6, where we review the rigorous analysisof the fully nonlinear dynamics, i.e. without assuming anundepleted pump.

To summarize this section, in order to transfer energyfrom A1(z) to A3(z) the phase mismatch parameter, �k(z),should be very large compared to the coupling coefficientκ , and has to change adiabatically from a large negative(positive) value to a large positive (negative) value alongthe crystal. With the adiabatic solution, one can obtain ro-bust efficient broad bandwidth conversion. Alternatively,the adiabatic constraint can be achieved also by manip-ulation the coupling coefficient through the interaction,κ(z), by varying the duty cycle of the nonlinear modulationpattern.




Figure 6 Conversion efficiency as afunction of (a) input wavelength and (b)crystal temperature, using an adiabati-cally chirped KTP crystal at a pump in-tensity of 60 MW/cm2 (c). A good cor-respondence between the experimen-tal results (shown in solid red) and thesimulation of the design (dashed-dottedblue) is shown. The low efficiency around1485nm is associated with a manufactur-ing defect. Reprinted figure with permis-sion from Ref. [8].

3.4. Experimental verification of adiabaticfrequency conversion

The verification of the adiabatic dynamics was done inseveral sets of experiments in using quasi-CW (tunablemonochromatic) lasers and using ultrashort-pulse sources[8–12]. In the current section, the most significant experi-mental results of the adiabatic dynamics in SFG/DFG in theundepleted pump approximation will be presented. It willinclude bandwidth scalability, the robustness of the methodand also verification of the Landau-Zener prediction of theadiabatic dynamics.

The adiabatic frequency conversion scheme was real-ized experimentally for the first time using a KTP crystaland the chirp modulation of the nonlinear coefficient wasachieved by electric field poling. This modulation patternwas designed to satisfy the constraints posed by Eq. (15)[8]. By slowly changing the poling periodicity along thepropagation direction, highly efficient signal-to-idler con-version over a bandwidth of 140nm and for 100◦C crystaltemperature variation was achieved, as shown in Figs. 6aand b, respectively [8,9]. The adiabatic conversion schemewas also shown to be robust to variations in the param-eters of the crystal as well as those of the light. Thoseinclude the input wavelength, crystal temperature, crystallength and the angle of incidence. In particular, the processis robust for high enough pump intensities that facilitateconversion efficiency approaching unity, thus allowing anefficient conversion of Gaussian beam profiles. In addition,color tunability of the conversion band through temperaturecontrol was demonstrated [9]. A detailed study of the prop-erties, robustness and tunability of the conversion processis presented in Ref. [9].

The conversion efficiency as a function of the pumppeak intensity for a fixed signal wavelength was exam-ined as well, where a maximal efficiency of 74% ± 3%(limited by the available pump intensity) for a narrowbandseed source was achieved [8]. The dependency with pumpintensity variation had a very good correspondence withthe numerical simulation [8]. Higher efficiency, near 100%conversion was recently demonstrated [12]. The experi-ment used an adiabatic DFG technique realized with a KTPcrystal, demonstrated the principle of complete Landau-Zener adiabatic transfer in nonlinear optical wave mixing[9]. As can be seen in Fig. 7, the matching between the

experimental measurements and the best fit Landau-Zenercalculation predicted using Eq. (17) is clearly seen. It isimportant to note that unlike the case of a phase-matchedcrystal, where the conversion efficiency oscillates betweenunity and zero upon increase in the pump intensity, the con-version efficiency showed to remain nearly unity also forpump intensities exceeding 1 GW/cm2. This fact is of par-ticular importance, because it allows a uniform conversionacross a Gaussian beam (or other beam profiles), wherethe intensity can vary in the transverse directions, whichin conventional frequency conversion devices affects theconversion efficiency dramatically [46].

This implementation of the adiabatic dynamics, by pol-ing a quasi-phase matched crystal, is only one possiblerealization of such a structure. The same effect (yet withnarrower bandwidth) can be obtained by inducing a tem-perature gradient across a nonlinear crystal.

4. Adiabatic ultrashort pulse frequencyconversion

For frequency conversion of monochromatic or quasi-monochromatic laser beams, one can omit the influence ofthe waves’ group velocities, but this is not the case for ultra-fast pulses (femtosecond and picosecond), where the highorder dispersion properties of the nonlinear medium mustalso be considered. The methods described in the previoussection, which are usually only efficient for narrow spectra,are not suitable for efficiently converting broadband signals.For ultrashort pulses, dispersion causes the three interactingwaves to travel at different group velocities, thereby limit-ing the effective nonlinear interaction region. Also, it leadsto temporal stretching during propagation, which results inlower pulse peak intensity and therefore lower conversionefficiency.

Several possible methods were suggested in order todeal with the conversion of a broadband source. In bire-fringent phase-matched crystals, the common solution toachieving broadband conversion is to use very short crys-tals - less than 1 mm in length. The short propagation lengthlessens the effect of dispersion; however, it also consider-ably limits the achievable conversion efficiency. Anotherapproach for coping with these challenges is to use ran-dom crystal structures that circumvent the need for phase


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Figure 7 Verification of Landau-Zener for-mulation in frequency conversion. Mea-sured PbSe photodiode response plottedversus peak pump intensity. Error bars in-dicate the relative measurement error. Solidline: exponential fit of the experimental data.Inset: Predicted conversion efficiency ver-sus idler wavelength for several pump in-tensities based on numerical simulations.Reprinted figure with permission fromRef. [12].

matching completely [47]. Researchers demonstrated thisin a disordered polycrystalline material that consisted ofrandomly distributed and oriented single crystal domains.However, the conversion efficiencies that have been demon-strated to date have been low. The efficiency of the methodalso depends on the ability to manipulate the average sizeof the domain structures - a fact that carries implicationsfor the applicability of this scheme to certain wavelengthregimes.

Many types of QPM structures have been implementedfor frequency conversion with broader bandwidths [10,32].Multi-periodic modulation patterns can be arranged in se-quence along the length of the crystal, leading to phase-matching of several processes simultaneously. This is aneffective method for converting a number of narrow band-width light sources, but not for a single broadband lightsource, because each distinct modulation period providesoptimal phase matching only for a specific wavelength. Thesame is true for QPM modulation with short-range order(ordered unit cells with random variation, e.g. rotation ortranslation) [48,49], though this provides better efficiency-bandwidth tradeoff. In a chirped QPM crystal, the modu-lation period is varied continuously as a function of posi-tion along the propagation axis. Each frequency componentin the pulse is phase-matched only at a particular positionin the crystal, however; the smooth sweep of the modula-tion period ensures that a broad phase-matching bandwidthcan be supported after propagation over the entire crys-tal length. In addition, the spatial dependence of the con-version process means that the total group delay disper-sion acquired by each distinct frequency component canbe modulated, resulting in broadband frequency conver-sion with simultaneous temporal pulse shaping or pulsecompression [50].

Here, the adiabatic conversion method is generalized toinclude efficient broadband ultrafast pulse conversion. Thisis done by introducing the concept of chirp pulse conversionas part of the adiabatic scheme. The concept, which wasshown to be scalable and could be applied both for SFG andDFG conversion, allows the generation of intense ultrafastsources over a wide wavelength range covering the visiblethrough to the mid-infrared.

4.1. The concept of chirp pulse conversion

Group Velocity Mismatch (GVM) causes a temporalwalkoff between the incoming signal pulse and the gener-ated sum frequency pulse. This effect is typically character-ized by the quasi-static interaction length, L QS = τ/GVM ,which is propagation length over which the pulses still havesignificant overlap. Here τ is the incoming signal pulse du-ration. For the case of wave mixing with a narrowbandpump, in which the pump does not put any limitation fromthe group dispersion point of view, the GVM is defined asthe difference between the inverse group velocities of thepulses, i.e. 1

vg1− 1

vg3.

To avoid the problem of GVM, a requirement that usesthe length parameter, Ladiabatic = κ

|d�k/dz| [8], as the ef-fective length over which the frequency conversion processtakes place within the nonlinear crystal. As seen, this lengthis dependent on the material parameters of the crystal, thefield strength of the pump and the sweep rate of the phasemismatch along the propagation axis in the crystal. To ob-tain a tolerable walkoff between the pulses over the adia-batic interaction region, the following constraint should besatisfied: L QS � Ladiabatic. This sets a limit for the mini-mum temporal width of the seed pulse, therefore the seedpulse should be stretched to a duration τMIN satisfying:

τM I N ≥ L QS · GVM �∣∣∣∣ κ

d�k/dz

(1

vg1− 1

vg3

)∣∣∣∣ . (18)

Typically, a pulse length in the 1−10 ps range will suf-fice for conversion over the entire visible and near infraredspectral range. GVM also leads to some temporal spread-ing within the crystal, since one of the pulses lags behindthe other one, resulting in frequency conversion along alonger temporal interval. The converted pulse should be re-compressed to a transform-limited pulse after exiting fromthe nonlinear crystal.

The second order dispersion term, known as the GVDis defined by β = d2ω

dk2 , and has characteristic length of

LGVD = τ 2

β. GVD leads to temporal spreading of the pulses




Figure 8 Conceptual implementation apparatus of adiabatic frequency conversion of ultrafast pulses. The stretched seed pulse froman ultrashort pulse or a broadband source (red line) and a strong pump beam (pink thick line) are mixed in the adiabatically chirpednonlinear crystal. Depending on the adiabatic crystal design, the energy in the seed beam can be efficiently converted to either thesum frequency or difference frequency beam (blue line).

propagating along the crystal, and LGVD is the propagationlength over which this spreading is of the same order as theinput pulse width. Each of the interacting wavelengths hasa different GVD, and the limit is set by the shortest GVDcharacteristic length, which typically determined by theshortest wavelength. The constraint is LGVD � Ladiabatic,i.e. it requires the interaction to effectively take place over alength for which GVD is negligible. For typical parametervalues, this constraint is very easy to satisfy. Several nu-merical examples can be found in Ref. [11]. Often, higherorder dispersion effects can be neglected as well [8, 11].Another inherent phenomenon which exists in wavelengthconversion through chirped poling structures is the addi-tion of linear chirp, positive or negative, when propagatingalong the crystal [32, 37]. This chirp, too, can be removedat the output using a compressor.

A general apparatus for the adiabatic ultrashort fre-quency conversion scheme using chirp pulse conversionmethod is shown in Fig. 8. As seen, the same setup canbe used for both the SFG and DFG implementations. Thestrong pump pulses are electronically synchronized in timeto the seed pulses (oscillator or amplifier). The seed pulsespectrum does not pose an inherent limitation - it can varyin wavelength regimes and can have a scalable bandwidthas long as the adiabaticity criterion is preserved. In Ref. [11]the seed was generated by a multipass amplifier, centerednear 790 nm with approximately 30 nm FWHM bandwidthwith transform limited pulse duration of 30 fs, whereas inRef. [12], the seed pulse was generated by an oscillator,centered near 740 nm with approximately 100 nm FWHMbandwidth with transform limited pulse duration of 10 fs.The seed pulse energies could vary as well (as long as un-depleted pump approximation holds). In Ref. [11] it was oforder 0.5–1.5 μJ, while in Ref. [12] it was 50-100 nJ. Priorto combination with the strong pump, the seed pulse shouldbe either stretched (using glass or SF10 for example) toa duration of 1–10 ps, or alternatively the uncompressedoutput of the Ti:S amplifier can be used. This satisfies therequirement set by Eq. (18) and reduces unwanted SHG ofthe seed that would lower the efficiency of the conversionprocess.

The first experimental proof of this concept was pre-sented in Ref. [11], where an excellent conversion of thespectral shape and bandwidth of the seed to both the vis-ible and mid-IR wavelength regions were observed. Forthe SFG pulse, the spectrum is centered near 450 nm andits bandwidth is sufficient for producing 30 fs pulses ifcompressed to the transform limit. As for the DFG pulse,the spectrum is centered near 3150 nm, where 20 nm ofthe FWHM seed bandwidth was converted to the mid-IR.Also, both the DFG and SFG efficiency curves are largelyflat across the seed bandwidth (falling off only in the tailsof the spectrum), indicating the robustness of the adiabaticfrequency conversion method. The conversion efficienciesreach a peak of 11% for SFG and 50% for DFG, in goodagreement with the calculated efficiencies for the nonlinearcrystals and pump intensities used in the experiment.

For both SFG and DFG, the efficiency was limitedin large part due to the available pump power, but muchlarger efficiencies, even approaching 100% in principle,are achievable by a sufficient increase in the pump inten-sity. Such efficiencies were demonstrated recently in DFGconversion of a near-IR broadband pulse to the mid-IRusing a nonlinearly chirped QPM grating in KTP [12]. Examining the spectra, shown in Fig. 9, effi-cient broadband conversion of Ti:S frequencies (630-720 nm) to the 1550–2450 nm range was observed (therange between 2450–2800 nm was not detected due to thelimited sensitivity of the InGaAs spectrometer). In Fig. 10,the time domain traces of the seed pulses with- (dashedgrey) and without- (solid black) an overlap of signal andpump pulses in the nonlinearly chirped crystal are pre-sented. As seen, when overlap occurs between the 85 MHztrain pulses of the seed and the 1kHz train pulses of thepump, a roughly 90% depletion of the signal in the rangewas measured.

A recent experimental observation, reported in Ref. [16]has shown the first demonstration of an octave-spanningmid-IR source of approximately 1 μJ energy by adiabaticfrequency conversion of near-IR optical parametric chirpedpulse amplifier (OPCPA) pulses. A spectrum spanning from2 to 5 μm, spanning an octave from 2.1 to 4.2 μm at −15 dB


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Figure 9 (a) Ti:S oscillator spectrum that entered the crystal.(b) Measured DFG idler spectrum (solid) and expected spec-trum (dashed) based on simulated conversion of the signal spec-trum in (a) with pump wavelength of 1047 nm and 2.5 GW/cm2

pump intensity. The shaded regions indicate the signal and idlerbands chosen for conversion in the nonlinearly chirped crystal.Reprinted figure with permission from Ref. [12].

Figure 10 Oscilloscope traces of Si photodiode response tothe 674nm to 686nm band of the Ti:S signal pulses transmittedthrough the nonlinearly chirped crystal with (dashed line) >20 psdelay or (dark solid line) no delay between the pump and sig-nal pulses. The depletion of the synchronized pulse at 30 ns is∼90%. Reprinted figure with permission from Ref. [12].

of peak, was obtained by near full photon number conver-sion of a 3-μJ OPCPA pulse with broad bandwidth of 670–900 nm mixed with a narrowband 1047 nm pulse. This wasachieved by an adiabatic DFG device with a pump inten-sity of 13.2 GW/cm2. A uniform spectral power depletionof the near-IR spectrum was observed, indicating nearlycomplete photon number conversion from signal to idlerand corresponding to a signal to idler energy conversionratio of ∼1/4. Figure 11 shows the normalized measuredpower spectrum (solid red curve) alongside the normalized“expected” power spectral density (dashed black curve) cal-culated by assuming 100% conversion of the OPCPA powerspectrum (top right inset) to the mid-IR via adiabatic DFG.As seen, a direct transfer of the near-IR spectral amplitudeprofile to the mid-IR is obtained, spanning more than anoctave (2.05–4.6 μm).

In order to compress a stretched pulse, one needs toaccurately characterize the spectral phase of the convertedpulse (which is predicted to be smooth also for an adia-

batic chirp QPM design). Spectral phase characterizationof such a broadband spectral range is not an easy task.Compressing ultra-broad bandwidths (single cycle pulses),which is the second challenge, requires the application ofvarious methodologies, such as the use anomalous disper-sion materials and pulse shaping techniques. Encourag-ing experimental results that have been achieved by Heeseet al. [13–15] with adiabatic OPA are discussed in Section6 of this review.

The scheme described here is expected to have broadapplicability as a post-amplification method for near-IR tomid-IR conversion and potentially allowing single-cyclepulsed sources. It is immediately suited for the seedingof a degenerate OPCPA with a narrowband 1 μm pumpand chirped 2 μm signal, as shown in Refs. [51, 52]. Theamplified pulses can be subsequently compressed to pro-vide a high-energy, few-cycle source. In such a scheme, ahigher pulse-rate pump laser would be desirable. For exam-ple, bulk or thin-disk amplifiers can provide ∼0.01–1 mJ,ps pulses at a 100 kHz to few-MHz repetition rate oper-ating at 10 to 100 W of average power. Used as the DFGpump, they could provide tens of mW of DFG power andmany Watts of few-cycle pulses when further amplified bythe same pump in an OPA. Using the scalability of thedesign, the adiabatic DFG technique could potentially beused to generate multiple-octave-spanning spectra (i.e. hav-ing bandwidth that contains wavelengths their harmonics),and allowing the generation of the shortest pulse mid IRsource possible.

5. Multi-process adiabatic interaction withundepleted pump

In this section we extend the analogy presented earlierbetween SFG/DFG and the dynamics of two level sys-tems. We will first present how two frequency-cascadedand spatially-simultaneous SFG/DFG processes is analo-gous to the dynamics of a three level system. This type ofprocess combination results in an effective FWM process(with two undepleted input fields), so some of the samegoals can also be realized by third-order nonlinearity pro-cesses, e.g. in optical fibers where tapering can be used tocontrol phase-matching conditions. Further on, we will dis-cuss how to extend this analogous mechanism to multilevelquantum system.

Frequency conversion processes can be cascaded inorder to reach frequencies that are difficult to generatewith a single process [53]. For example, a generated sum-frequency wave at ω2 = ω1 + ωp1 can be further combinedwith another frequency in another SFG process, which pro-duces ω3 = ω2 + ωp2 = ω1 + ωp1 + ωp2 . In this manner,available lasers can be used to reach a frequency far re-moved from the input frequency ω1, than what is possiblewith the same lasers and a single process (Fig. 12a). Sim-ilarly, two DFG processes can be utilized (see Fig. 12b).Another example is cascading SFG and DFG, i.e. ω3 =ω2 − ωp2 = ω1 + ωp1 − ωp2 . In this case, available lasers




Figure 11 Depletion of the near-IR OPCPA pulse (left) and generation of the octave-spanning mid-IR ADFG pulse (right, “Measured”).Intensities (left) refer to the pump wave. The “Expected” curve (right), depicts the mid-IR spectrum corresponding to 100% conversionfrom the near-IR. Reprinted figure with permission from Ref. [16].

Figure 12 The four cases of SFG/DFG cascade: (a) two SFG processes (b) two DFG processes (c) SFG followed by DFG (d) DFGfollowed by SFG. Reprinted figure with permission from Ref. [18].

can be used to reach a frequency closer to the input ω1 thana single process can provide, as depicted in Fig. 12c. In thesame manner a DFG processes can be followed by SFG, asdisplayed in Fig. 12d. Note that the last two cases canyield either up-conversion or down-conversion, depend-ing on the difference between the two pump frequenciesωp1 − ωp2 . Furthermore, not all five frequencies involvedhave to be distinct. For example, a single pump can beused to pump two SFG or DFG processes, or even performas one of the inputs. In the most degenerate case, whereωp1 = ωp2 = ω1, the intermediate and output frequenciesare the second harmonic (SH) and third harmonics (TH),ω2 = 2ω1 and ω3 = 3ω1, respectively. Another interestingcase is ωp1 = ω1 and ωp2 = ω2 = 2ω1, where fourth har-monic generation (FHG) is obtained, i.e. ω3 = 4ω1. Allthese cascaded processes are especially appealing as meansto perform frequency shifting and all-optical switching inoptical communication systems.

A straightforward approach to cascading would be toperform the second conversion process in a spatially sep-arate zone, either in another crystal or in the same crys-tal. These processes are frequency cascaded and spatially-cascaded. In the context of adiabatic processes, an opti-cal parametric oscillator (OPO) has been cascaded withadiabatic SFG, and was experimentally demonstrated toprovide efficient and tunable up-conversion of a fixed-frequency laser [54]. Unfortunately, the length of a sin-gle QPM modulated crystal is technologically limited. Thisputs constraints on the efficiency achievable by combiningdifferently modulated segments in a single crystal. Further-more, using two crystals greatly increases the complexity ofphysical apparatus, making it less feasible for applications.A more appealing approach is to perform the two pro-cesses simultaneously, making them frequency-cascadedand spatially-simultaneous. This introduces two difficul-ties: (i) the dynamics of simultaneous processes is more


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complicated (ii) simultaneous phase-matching control forboth processes is required. Saltiel et al. [53] outlined threeways of dealing with these issues: (i) phase-matching onlyone of the two processes (ii) phase-matching both pro-cesses (iii) compensating for the sum of the two phase-mismatches. In the first case, the lack of phase-matchingfor one of the two processes obviously limits conversionefficiency. Simultaneous phase-matching of two processesrequires some trade-off between their efficiencies; how-ever it can still be highly beneficial. Details about suchmethods can be found in Refs. [55, 56]. Due to the com-plex dynamics involved, compensation for the sum of thetwo phase-mismatches was applied so far to THG, FHGand/or the case where the input frequency wave is neg-ligibly depleted [53, 57–59]. Recently another approachto frequency-cascaded and spatially-simultaneous TWM,based on adiabatic processes, has begun to be implemented[17–20]. An adiabatic method was presented, which gen-eralizes the aforementioned compensation of the sum ofthe phase-mismatches, yielding novel phenomena, such asa vanishingly small intermediate frequency ω2 and inten-sity dependent phase-matching [17]. In another work [18],QPM modulation was utilized to control the efficiencies ofthe two simultaneous processes, such that adiabatic evolu-tion is maintained, as well as simultaneous phase-matchingof both processes. In this manner a highly efficient fre-quency conversion from input ω1 to output ω3 is obtained,while keeping the intensity at the intermediate frequencyω2 negligible. A. Rangelov and N. Vitanov [19] discussedan adiabatic method for efficient, broadband and robust up-conversion using two SFG processes pumped by the samelaser. This approach was later generalized to the case oftwo distinct pumps and any of the four cases displayedin Fig. 12, i.e. up-conversion or down-conversion to eithernear or far frequencies [20]. Furthermore, it was shown thatsuch adiabatic interaction involves five distinct frequencies,so it has a very rich and interesting dynamics, with uniquefeatures that are demonstrated experimentally. The remain-der of this section will review each of these cases in moredetail.

5.1. Frequency conversion through opaquemedia using adiabatic methods

It is commonly assumed that in order for frequency con-version to be efficient, the nonlinear medium has to betransparent at all of the interacting frequencies. This sec-tion discusses two adiabatic methods that circumvent thisrequirement for the case of two frequency-cascaded andspatially-simultaneous SFG/DFG processes: they enableefficient conversion from the input to the output of thecascade, even in the presence of strong absorption at theintermediate frequency.

Porat et al. [17] experimentally demonstrated a SFG-SFG cascade where the intermediate frequency power wasnegligible throughout the entire interaction, using a methodknown in atomic physics as adiabatic elimination [27]. The

basic idea was to cause one process to transfer energy fromthe input to the intermediate frequency at the same rate atwhich a second process transfers energy from the interme-diate frequency to the output frequency. By keeping thisrate fast, accumulation of energy at the intermediate fre-quency is avoided, while keeping a net transfer of energyfrom the input to the output frequencies of the cascade. Thefact that the energy transfer rate of each process dependson pump intensity leads to an intensity-dependent effectivephase-matching condition.

For simplicity, in the analysis, both SFG processes aretaken to be pumped by the same undepleted pump, thusω1 + ωp = ω2 and ω2 + ωp = ω3, where ω1, ω2 and ωp

are the input, intermediate and pump frequencies, respec-tively. Each of the two processes is taken to have a verylarge phase mismatch, while their sum is rather small,i.e. |�k1| L � 1, |�k2| L � 1 and |�k1 + �k2| L � 1,where �kj is the phase mismatch of the j-th process andL is the length of the nonlinear crystal. It was shown thatunder these conditions, the intermediate frequency waveperforms fast oscillations with low amplitude. In fact, therate of these oscillations increases with

∣∣�k j

∣∣ and withpump intensity. The system’s dynamics can be separatedinto two length scales: on the short length scale, only thelow amplitude fast oscillations take place. These can then beeliminated from the overall dynamics, and so the interme-diate wave is said to be adiabatically eliminated. The slow,long scale dynamics system is composed only of the inputω1 and output ω3, coupled by two pump photons, whichis quite similar to a case of FWM process. This effectiveprocess has an effective phase mismatch that can be writtenas

�kef f = �kTP + δkStark(Ip), (19)

where �kTP = �k1 + �k2 has been defined as the two-photon phase-mismatch, and the phase matching conditionis �kef f = 0. Notably, in addition to the two-photon mis-match term considered previously [53], an additional, pumpintensity dependent term, δkStark(Ip), should also be takeninto account. The physical origin of this term is in thedependence of the rate of oscillation of the intermediatewave amplitude on pump intensity. In other words, in orderto match the rate of ω1 ↔ ω2 transition to the ω2 ↔ ω3transition, the pump intensity should be taken into consid-eration in addition to the phase mismatch that results fromdispersion. This effect is the analogous mechanism of theStark effect in NMR and atomic physics [17].

The separation of slow and fast dynamics, low am-plitude of the intermediate wave, as well as intensity-dependent phase matching condition, were demonstratedby numerical simulation and analytical calculation. In thecase under consideration, of two SFG processes, the input,intermediate and output wavelengths were 3010 nm, 786nm and 452 nm, respectively. Both SFG processes werepumped by the same 1064 nm laser. In order to satisfy theabove phase-matching requirements, the waves were prop-agated through a KTP crystal with a = 8.6 μm QPMperiod, kept at a temperature of 125 oC, and the pump




Figure 13 Adiabatic elimination theory: (a) Numerically simulated intensities of the 3010nm input (red dashed line), 452 nm outputwave (blue dash-dotted line) and 786 nm intermediate wave (green solid line). Inset: close-up view of the intermediate wave over34.4 μm. Shading indicates poling. (b) Analytically calculated photon conversion efficiency vs. the two-photon phase-mismatch.The difference between the peak efficiencies phase-mismatches is the Stark shift. Inset: Experimentally measured (blue dots) andanalytically calculated (green line) output power vs. pump power in an adiabatic elimination experiment. Reprinted figure with permissionfrom Ref. [17].

intensity was Ip = 10 GW/cm2. In Fig. 13a the 3010nminput and 452nm output are clearly seen to exchange en-ergy with each other, which takes place on a much longerscale than the fast and low amplitude oscillations of theintermediate wave, shown in the inset (shading indicatesQPM modulation). Furthermore, introducing a 10 cm−1

absorption at the intermediate wavelength decreased themaximum 452nm intensity by just 0.5%, showing that suchscheme is indeed robust against strong absorption of theintermediate wave. Figure 13b shows the photon conver-sion efficiency from the input to the output for two cases:low pump intensity of 1 GW/cm2 and high pump intensityof 12.4 GW/cm2. The results are plotted against the twophoton phase mismatch �kTP. For the case of low pumpintensity, maximum intensity is obtained near �kTP = 0,since in this case the Stark term in Eq. (19) is negligible.However, for high pump intensity, the efficiency maximumshifts to �kTP = −δkStark , showing that phase-matchingindeed depends on pump intensity.

The predictions of this theory were also verified ex-perimentally, for the interaction of the same wavelengthsas in the simulation. The inset of Fig. 13b depicts the ex-perimentally measured output (452 nm) power as a func-tion of the pump power. The experimental results (bluedots) fit very well on the analytical calculation (greencurve) for the output wave. Furthermore, in the exper-iment, the intermediate wave power was so low that itwas below the noise level, so it can only be boundedfrom above. At most, it was 733 times lower than thepeak output power, in correspondence with the theoreticalprediction.

Cascaded frequency conversion with negligible energyat the intermediate wave can also be obtained via anotheradiabatic technique analogous to the atomic STIRAP pro-cess [18]. In the atomic version of this method, two delayedpulses facilitate transfer of atomic population through adark state [27]. The nonlinear analog makes use of two

SFG/DFG processes, where their coupling strengths arevaried along the interaction. In this case �k of Eq. (14)is kept constant while a slow variation of κ satisfies theadiabaticity condition. This variation can be chosen suchthat efficient conversion from the input to the output fre-quency of the overall cascade takes place without having asignificant amount of energy at the intermediate frequency.An important difference between this method and that ofadiabatic elimination is that in this case both processes arephase-matched throughout the entire interaction, while inadiabatic elimination each of the two processes experiencesa very large phase mismatch parameter throughtout the en-tire interaction. As a result, better efficiency is obtained forgiven pumps intensities and interaction length.

In the case of a single SFG/DFG process, it was ex-plained that an adiabatically evolving system remains al-ways in one of its instantaneous (or local) eigenstates, whichchange due to a slowly varying parameter. When two pro-cesses are involved, the evolving system can be kept in thesame eigenstate by adiabatically varying the ratio betweenthe coupling strengths of the two processes. This variationis performed in the counter-intuitive order: the interactionbegins with strong coupling between the intermediate andoutput waves and weak coupling between the input and in-termediate waves. As the waves propagate along the crystal,the ratio between these two coupling strengths is slowly var-ied, achieving at the end of the interaction a strong couplingbetween the input and intermediate waves while the cou-pling between the intermediate and output waves is weak.The variation of the coupling ratio is manifested in a varia-tion of the eigenstate that the system follows. Consequently,all of the energy is transferred from the input to the outputwithout going through the intermediate frequency.

The fact that the intermediate wave does not containsignificant amount of energy enables to transfer energyfrom the input frequency to the output frequency ofthe cascade even in the presence of strong absorption


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Figure 14 STIRAP-analog numerical simu-lation of intensities of the input (dashed blueline), output (dash-dotted red line) and in-termediate (solid green line) waves alongthe nonlinear medium, with high intermedi-ate wave absorption and assuming arbitrarycontrol of coupling strengths. The inset showsthe intermediate wave intensity on a smallerscale. Reprinted figure with permission fromRef. [18].

at the intermediate frequency. Such dynamics wasdemonstrated by numerical simulation, where the twoprocesses were the SFG process (λ1 = 546.4 nm) +(λp1 = 900 nm

) → (λ2 = 340 nm) and the DFG process(λ2 = 340 nm) − (

λp2 = 2266 nm) → (λ3 = 400 nm).

The nonlinear medium was taken to be KTP, where theabsorption coefficient of the intermediate wavelength isα2 = 229.9 cm−1, i.e. KTP is practically opaque at thiswavelength. The results are displayed in Fig. 14, in whichfull energy transfer from the input to the output is seen.Furthermore, the inset shows that the intermediate waveintensity is about 4 orders of magnitude lower than theinput and output waves intensities, as expected. Note thatin this case energy only flows in one direction, from theinput to the output, i.e. there is no back-conversion as inthe adiabatic elimination method. It worth noting here thatthe numerical simulation takes into account the resonantenhancement of the nonlinear susceptibility associatedwith the high absorption at the intermediate frequency, andalso two photons absorption, which is responsible for theslight decline of the output intensity towards the end of theinteraction [18].

Several schemes were already suggested for controllingthe coupling strengths of the two processes, where the firstwas by focusing the two pumps at different positions alongthe crystal, thereby causing their intensities to decay alongthe crystal (due to diffraction) and leading to the requiredvariation of the coupling strengths. The second scheme toachieve the same goal is a QPM based scheme that was an-alyzed in detail and demonstrated by numerical simulations[18].

Both adiabatic elimination and the STIRAP-analog canbe applied to any of the four combinations of SFG and DFGprocesses depicted in Fig. 12. Their robustness against ab-sorption of the intermediate wave provides access to newspectral regions when using near infra-red solid state laser

sources, as it enables cascaded frequency conversion goingthrough an absorptive spectral range. These methods alsoprovide an efficient way to perform conversion betweenoptically near frequencies, that would otherwise requirenon-existing pump lasers operating at far IR frequencieswhere nonlinear media is generally opaque. In additionto frequency conversion between near and far frequen-cies, these schemes can be used to study the electronicstructure of nonlinear crystals. When the intermediate fre-quency is in the crystal’s UV absorption band, it can beused as a probe into this spectral region, which is otherwisedifficult to study due to the strong absorption. Electronicresonances will be manifested mainly because of theircorresponding effect on dispersion and phase-matchingconditions.

Apparently, both adiabatic elimination and theSTIRAP-analog serve the same purpose. However, theyhave different advantages and disadvantages. Adiabaticelimination requires only periodic QPM modulation, whichis easy to implement. Furthermore, it has the special featureof intensity dependent phase-matching that can be used foroptical switching. However, adiabatic elimination requiresrelatively high pump intensity or long interaction lengths inorder to provide high conversion efficiency. The STIRAP-analog yields significantly higher conversion efficiency forgiven pump intensities and interaction lengths due to itsuse of phase-matched processes. On the down side, controlof the coupling strengths is technologically challenging.Therefore, the decision between adiabatic elimination andthe STIRAP-analog depends on the specific application.

Finally, we would like to mention that in Section 6, wediscuss another scheme suggested by Longhi [60], whichprovides THG via cascaded SHG and SFG, with low secondharmonic power, in the regime of nonlinear dynamics (i.e.with pump depletion). It should also be stressed that whilethe methods presented in this section merit some special




properties, they do not possess the robustness and broadbandwidth of the single adiabatic process, as discussed inSection 3.1. We shall present methods for robust multipleprocess conversion in the next section.

5.2. Cascading adiabatic processes for efficient,robust and broadband frequency conversion

As mentioned above, the phase matching requirements ofmultiple processes pose a significant challenge when broad-band frequency conversion is desired. Rangelov and Vi-tanov [19] and Porat and Arie [20] have shown that cas-caded adiabatic interactions can facilitate high conversionefficiency for the overall cascade, maintaining its robust-ness and bandwidth scalability as in the single adiabaticprocess case (unlike the two methods presented in the pre-vious section).

Rangelov and Vitanov [19] studied the case of twospatially-simultaneous and frequency-cascaded SFG pro-cesses, where both processes are pumped by the samepump: ω1 + ωp = ω2 and ω2 + ωp = ω3. Here, ωp is thefrequency of the pump and ω1 is the input wave frequency.These waves were taken to propagate along a chirped QPMcrystal, where the chirp rate and pump intensity are suchthat the adiabatic condition of Eq. (15) is satisfied for bothprocesses. In order to understand this interaction’s dynam-ics, it is convenient to use an eigenvalue diagram, which isplotted in Fig. 15. Each of the three solid black lines in theplot represents the eigenvalue of one of the three interact-ing waves, which are the eigenstates of the system in theabsence of the pump (the bare states). The kbare

1 ↔ kbare2

and kbare2 ↔ kbare

3 crossing points are the phase matchingpoints of the two processes. The kbare

1 ↔ kbare3 crossing

point corresponds to phase matching of an effective five-wave mixing process, which is in fact a three-photon adi-abatic elimination processes (see Section 5.1). Since theseinherently involve large phase-mismatches for each of theparticipating processes, they have a negligible effect onthe adiabatic interaction considered here. The dashed lightblue lines represent the eigenvalues of the eigenstates of thesystem when the pump is present (the dressed states).

When the modulation periodicity increases along thepropagation (left to right motion on the eigenvalues dia-gram) the two processes are phase-matched in the sameorder of the frequency cascaded (i.e. first ω1 + ωp = ω2and then ω2 + ωp = ω3). This is called the intuitive order.As for the single process case described in Section 3.1, thesystem point follows the eigenstate it started with. In thiscase, it is the one that is marked by arrows at the top of theplot. Initially (left hand side of the diagram) this eigenstatecorresponds to all energy being at ω1, which is the initialstate of the system. At the end of the interaction, the sameeigenstate corresponds to all energy being at ω3, which isthe desired result. A numerical simulation of the interactionthat shows the normalized intensities of the three interactingwaves, is presented in the top inset of Fig. 15. Full conver-sion from ω1 to ω3 is clearly evident. For a decreasing mod-

Figure 15 Eigenvalues diagram for two cascaded SFG pro-cesses. The top and bottom insets depict the numerically simu-lated normalized intensities along the propagation for the intuitiveand counter-intuitive order, respectively.

ulation periodicity ω2 + ωp = ω3 is phase-matched beforeω1 + ωp = ω2, which is termed the counter-intuitive order.In this case the system follows the eigenstate marked witharrows at the bottom of Fig. 15, from right to left. The bot-tom inset of Fig. 15 depicts the corresponding numericalresult, showing once again full conversion from ω1 to ω3.Note that for the counter-intuitive order there is less energyin the intermediate wave than in the case of intuitive order.This phenomenon also has a graphical manifestation in theeigenvalues plot. Graphically, if the eigenstate being fol-lowed is closer to the solid black line of kbare

2 , then moreenergy will be transferred through ω2 on the way from ω1to ω3. Indeed, the eigenstate followed in the intuitive casetraverses closer to the kbare

2 line than the eigenstate followedin the counter-intuitive case.

For undepleted pump, the interaction was found to bevery robust [19], i.e. high efficiency is provided for a widerange of parameters that influence the phase matching con-ditions (e.g. crystal temperature and angle of incidence) asthe case of a single adiabatic process. Specifically, highefficiency can be provided for a wide range of input fre-quencies, resulting in broadband conversion.

Simulations were also conducted taking pump depletioninto consideration, with pump intensity that is comparableto the input wave intensity. The interaction remain effi-cient even in the case of highly depleted pump. However,parasitic processes (e.g. SHG of the pump or input wave)become significant, making the interaction generally lessrobust. Porat et al. found the physical mechanism behindthis phenomenon, as part of a general analysis for adia-batic interaction with depleted pump [20]. We will elab-orate more on this issue within Section 6 of the currentreview.

A more general case, studied both theoretically andexperimentally by Porat and Arie where SFG was followed


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Figure 16 Eigenvalues plot for two SFG-DFG cascades pumpedby two distinct pumps. The insets show the nonlinear processeson the frequency axis. The input wavelength is (a) λ1 = 1542nm(b) λ1 = 1509 nm.

by DFG and the two processes were pumped by two distinctpumps [20]. Here the two processes are ω1 + ωp1 = ω2α

and ω2α + ωp2 = ω3α , where ω1, ω2α , ω3α , ωp1 and ωp2 arethe input, intermediate, output, first pump and second pumpfrequencies, respectively. If the two pumps frequencies arevery close spectrally, the same chirped QPM modulationcan also support another SFG-DFG cascade, in which theorder of the pumps in reversed: ω1 + ωp2 = ω2b and ω2b −ωp1 = ω3b. All four processes are illustrated in the insetsof Fig. 16a and b. The same figures show the eigenvaluesdiagram for λp1 = 1047.5 nm and λp2 = 1064.5 nm, for λ1= 1542 nm in Fig. 16a and λ1 = 1509 nm in Fig. 16b.

In this interaction, the KTP crystal has chirped QPMmodulation, with linear period variation from 14.38 μmto 14.92 μm. This chirp range is bound between the twodashed orange lines in Fig. 16. For each of the two casesillustrated in Fig. 16, the chirp range includes only twocrossings that correspond to the phase matching points oftwo cascaded processes (the third crossing corresponds toan adiabatic elimination process, as before). The processescorresponding to these crossings will have a significanteffect on the interaction, while those with crossings out-side the chirp range will be negligible, since their phase-mismatches are always large. Clearly, Fig. 16a and b depict

the same diagram shifted horizontally with respect to eachother, in correspondence to the explanation given earlier inSection 3.1. For other values of the input wavelength λ1,the eigenvalues plot will shift horizontally to a differentextent. For each input wavelength, the crossings that residewithin the chirp range will determine the interaction. Wethus expect different conversion to take place for differentinput wavelengths λ1. Furthermore, due to the structure ofthe eigenvalues plot, different processes will take place de-pending on whether the QPM modulation period increases(left to right motion on the eigenvalues diagram) or de-creases (right to left motion on the eigenvalues diagram)along the propagation. This is contrast to all of the adia-batic processes detailed so far, in which the same conversiontakes place regardless of the period chirp direction (thoughsometimes through a different adiabatic path, as in Fig. 15).

The adiabatic paths followed for the case of λ1 = 1542nm and λ1 = 1509 nm are marked with arrows in Fig. 16aand b, respectively. The direction of the arrows indicatesincreasing (left to right) or decreasing (right to left) QPMmodulation period. Note that, in correspondence to the ini-tial condition of the interaction, the system always startson the kbare

1 solid black line. In Fig. 16a we see that λ1= 1542 nm and an increasing period would result in ef-ficient conversion to ω2α (i.e. via the SFG process ω1 +ωp1 = ω2α). For the same input wavelength, a decreasingperiod would yield conversion to ω3α (the latter SFG pro-cess would be cascaded with the DFG process ω2α + ωp2= ω3α). Similarly, for λ1 = 1509 nm, Fig. 16b shows adi-abatic conversion to ω2b (by virtue of ω1 + ωp2 = ω2b)for decreasing period and adiabatic conversion to ω3b (thelatter cascaded with ω2b − ωp1 = ω3b). The eigenvaluesplot therefore predicts that the conversion will depend bothon chirp direction as well as input wavelength.

These phenomena can be seen in Fig. 17a and b, whichshow the numerically simulated output intensities of thedominant conversion process, as a function of input wave-length, for increasing and decreasing QPM period, respec-tively. From these two figures the dependence of the dom-inant conversion process on both the input wavelength andthe chirp direction is clear. Altogether, four different con-version processes can be performed, depending on inputwavelength and chirp direction. Furthermore, each of theseprocesses can be performed with high efficiency over a widerange of input wavelength.

These results were also verified experimentally. The ex-perimental results are displayed in Fig. 17c and Fig. 17d,which show the measured output power vs. input wave-length for increasing and decreasing QPM period, respec-tively. The experimental conversion dependence on inputwavelength and chirp direction, as well as the broad conver-sion bandwidth, correspond very well with the theoreticalpredictions and simulation results. The lower conversionefficiency in the experiment as compared to the simulationresults from technical limitations.

Interestingly, in the experiment, for a certain range of in-put wavelengths, the product of the two-process conversionwas itself converted. The reason was that the acceptancebandwidth of the two-process conversion was greater than




Figure 17 Numerical simulation of domi-nant output intensity vs. input wavelengthfor (a) increasing QPM period (b) decreas-ing QPM period. Experimental results of thesame for (c) increasing QPM period (d) de-creasing QPM period. Reprinted figure withpermission from Ref. [20].

Figure 18 Optical spectrum analyzer mea-surement showing multiple adiabatic con-versions. The adiabatic SFG-DFG cascadehad a conversion bandwidth so wide that itcontained its own output: the input λ1 wasconverted to λ3b, which in turn was con-verted to λ5b by the same adiabatic SFG-DFG conversion process. Reprinted figurewith permission from Ref. [20].

the difference between the frequencies of the two pumplasers, so the product was also in the acceptance bandwidthof the same two-process conversion. An example of sucha case is seen in Fig. 18, where the optical power againstwavelength exhibits multiple adiabatic conversions for in-creasing QPM period. The measurement shows that theinput ω1 was converted to ω3b. The same frequency thenparticipated in the SFG process ω3b + ωp2 = ω4b, followedby the DFG process ω4b + ωp1 = ω5b. In fact, in addi-tion to this, another SFG process took place: ω5b + ωp1= ω6α , since λ5b = 1559.8 nm was inside the acceptancebandwidth of this process (see Fig. 17a). Altogether, a fiveadiabatic process cascade was obtained.

These multiple frequency-cascaded processes suggestan interesting application: an all-optically tunable fre-quency comb. The pump frequencies should be chosensuch that the difference between them is much smaller thanthe two-process conversion bandwidth. There will then bemany cascades of conversion, since each cascade-productwill still be within the acceptance bandwidth of the sameprocess that generated it. The spectral spacing betweenthese products would be equal to the spectral spacing be-tween the two pumps. In this manner a frequency comb canbe obtained, where the central frequency and teeth spacingcan be all-optically tuned by tuning the frequencies of theinput and pumps. High efficiency for many cascades can

be obtained by placing the chirped crystal in a resonator(see Fig. 19), or using CW lasers with a fiber-coupled QPMwaveguide [32] in a fiber loop.

6. Adiabatic dynamics in the fully nonlinearthree wave mixing

The fully nonlinear regime of frequency mixing concernsthe general case where the undepleted pump approxima-tion does not necessarily hold, meaning that the interact-ing waves are allowed to have comparable intensity. Asseen in previous sections, the undepleted pump assump-tion linearizes the dynamics and makes it isomorphousto the linear Schrodinger equation of quantum mechan-ics. Such linearization process allows the use of quantummechanical adiabatic theorem [1], and thus was essentialfor the for the research work presented in previous sections[8,9,11,12,18–20]. Frequency conversion in the fully non-linear regime is very appealing, since there is no require-ment for a strong pump laser whose power is mostly unusedas in the undepleted pump regime. It is, however, generallydifficult to obtain efficient and robust conversion in thefully nonlinear regime, due to the more complex dynam-ics. Specifically for adiabatic interactions, the isomorphismwith the linear Schrodinger equation breaks down and the


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Figure 19 Schematics of frequency comb generation by cascaded adiabatic conversions, facilitated by a chirped crystal in a resonator.The input frequency ω1 gets converted to another frequency that is within the conversion bandwidth of the same adiabatic SFG-DFGprocess, which in turn gets converted itself, and so on. The spectral spacing between all of these frequencies is equal and opticallytunable.

quantum mechanical adiabatic theorem does not apply anymore. In this section, we will review the recent researchworks that have been performed in the fully nonlinear inter-action regime. We will focus here on pump-depleted TWMprocesses and adiabatic dynamics.

Adiabatic interaction with nonlinear dynamics was firstconsidered by Baranova et al. [6, 7] for the special caseof SHG. In Ref. [6], an analytical investigation of theeigenstates of the SHG system lead to the hypothesis thatadiabatic interaction is possible under some adiabaticitycondition. Ref. [7] presented numerical simulations thatsupport the hypothesis. Phillips et al. [21, 61] and Heeseet al. [13–15] conducted extensive research into adiabaticOPA, both theoretically and experimentally. In these worksthe notion of adiabatic following of an eigenstate was ex-plained in detail. In a recent publication, a geometricalrepresentation of adiabatic OPA was introduced, which canalso be applied to other adiabatic conversion processes,and a heuristic adiabaticity condition was put forth, basedon this geometrical representation. Furthermore, it wasshown how apodization (meaning, reducing spectral rip-ples) can be used to reduce gain oscillations across theconversion bandwidth, which is crucial for broadband ul-trashort applications. Phillips et al. also analyzed adiabaticOPO [22], which was found to have an inherent instabil-ity. Another interesting perspective on adiabatic OPA wasprovided by Yaakobi et al. [23], who found analytical ex-pressions for the waves evolution and adiabaticity condi-tion for this case, where the pump input intensity is muchhigher than the input signal intensity [23]. The analysiswas based on autoresonance process: it was shown that thepump-perturbed nonlinear system gets captured into a res-onant state, and stays phase-locked with the perturbationdespite its nonlinearity, as long as the phase-matching con-ditions vary slowly enough. The autoresonance approachwas also applied to adiabatic SHG in a more recent pub-lication [24]. In this context we also mention the work ofRichards [62], who presented a theoretical study of SHGin tapered optical fibers, based on modal phase match-ing. By sweeping the phase mismatch across the phase-matched condition during the propagation, he had demon-strated that this SHG scheme is both efficient and robustwith respect to nonlinear phase modulation. Though theadiabaticity mentioned in this work is that of the taper (i.e.the eigenmodes of the fiber follow the taper adiabatically)and not of the nonlinear interaction, the phenomena that

Figure 20 Phase portrait for the motion of a pendulum. Thephase space axes, q and p, are the angle of deviation from thevertical and the angular momentum, respectively (see inset). Thependulum drawings illustrate the corresponding motions.

he has observed is exactly the outcome of the adiabaticconstraint.

A general, comprehensive physical model of adia-batic frequency conversion in the fully nonlinear dynamicsregime, which applies to all three wave mixing processes,was presented recently by Porat and Arie [20]. Using adia-batic invariance theorem from classical mechanics [63,64],it was shown that the nonlinear system can follow a sta-tionary state that changes with adiabatically varying phasematching conditions. An adiabaticity condition was also de-rived analytically. These analytical expressions match thosefound by Yaakobi et al. [23,24] in the special cases consid-ered by them. Furthermore, the linear dynamics case, ob-tained with the fixed pump approximation, was also shownto be a special case described by the general theorem [20].

In the following section, we choose to start by reviewingin detail the general physical model of adiabatic frequencyconversion based on classical mechanics adiabatic invari-ance [20]. Next we review the autoresonant approach thatwas applied to the special cases of OPA and SHG. Theheuristic geometrical approach of Phillips et al. [61] andthe adiabatic OPA experiments of Heese et al. [13–15] willbe discussed as well. This will be followed by an analysis ofadiabatic OPO and its inherent instability [22]. Finally, thefindings of Rangelov et al. [19] and Longhi [60], regarding




THG via adiabatic interaction of cascaded processes, will bereviewed.

6.1. Classical mechanics adiabatic invariancetheorem

Classical mechanics also boasts an adiabatic theorem[20,21], called the adiabatic invariance theorem. This theo-rem is valid for any conservative system that evolves period-ically or quasi-periodically, with either linear or nonlineardynamics, as long as its dynamics can be cast in the formof a canonical Hamiltonian structure, with a Hamiltonianthat is smooth (meaning that the first and second derivativesof the Hamiltonian with respect to generalized coordinatesand momenta are always finite).

A brief discussion of classical mechanics adiabatic in-variance theorem is put forth here. The interested readercan find the full mathematical analysis in classical mechan-ics textbooks [20, 21]. Consider a system with one degreeof motion. The dynamics of the system is described by acanonical Hamiltonian structure, i.e. two coupled differen-tial equations for the canonical coordinate and momentum,q and p, respectively:

dq

dt= ∂ H

∂p(20a)

dp

dt= −∂ H

∂q(20b)

The system is assumed to evolve periodically, i.e. q andp are both periodic functions of time, with the same period-icity. Therefore, for a given set of fixed system parameters,the motion of the system point in the (q, p) phase-space de-scribes a closed or open curve that is repeated indefinitely.A simple example would be the pendulum, where q andp are the pendulum’s angle of deflection from the verti-cal and angular momentum, respectively. The pendulum’sphase portrait, where each point corresponds to a state ofthe pendulum, is illustrated in Fig. 20. Solid black linesrepresent trajectories of the system point for different ini-tial conditions. Since the Hamiltonian is a constant of themotion, the phase-space trajectory curve (either closed oropen) is a contour line, also known as iso-line, of the Hamil-tonian. Therefore the phase portrait is the contour plot ofa given Hamiltonian. Furthermore, if the Hamiltonian de-pends on a parameter, then changing this parameter willchange the phase portrait. The extrema of the Hamiltonian,in which ∂ H/∂q = 0 and ∂ H/∂p = 0, are stationary states(i.e. eigenstates) of the system, as evident from Eqs. (20).The corresponding points in phase space are called fixedpoints.

For the example of a pendulum, a fixed point is foundat q = 0 and p = 0, the state where the pendulum sitsstill at the vertical. For a certain range of p, for which thependulum performs back and forth oscillations, the phaseportrait would consist of ellipses centered about the fixedpoint. For values of p beyond this range, the pendulumcircles around, so the phase portrait is made of open curves.

All of these cases are shown in Fig. 20. Generally, a fixedpoint that is enclosed within a trajectory curve is called anelliptic fixed point.

Suppose that the system point is orbiting an elliptic fixedpoint. Furthermore, the Hamiltonian is taken to dependon a parameter, which is changed slowly. The adiabaticinvariance theorem states that if the rate of this change issufficiently low, then the system point stays near the ellipticfixed point throughout the change process. Specifically, fora system point orbiting at frequency ω with correspondingperiod τ = 2π /ω, the rate of change needs to satisfy

τdω

dt� ω (21)

i.e. the change in ω in one period of the motion is requiredto be much smaller than ω itself. For example, the parame-ter can be changed such that the elliptic fixed point movesin phase-space. The adiabatic invariance theorem then pre-dicts that the system point will continue to orbit the sameelliptic fixed point as it moves in phase-space. It is herethat the significance of this theorem is revealed: if a systemparameter is varied slowly enough, the state of the system(represented by the position of the system point in phase-space) can be controlled. The system can thus be broughtinto a physical state, at the end of the change process, whichis very different from the one it started with.

The adiabatic invariance theorem has been applied toquantum systems with nonlinear dynamics, by representingthe Schrodinger equation in a canonical Hamiltonian for-mulation [65–67]. The approach was recently implementedin Ref. [25] in order to apply adiabatic invariance to fullynonlinear TWM. In the following it will be explained indetail.

6.2. Canonical representation

The first step towards applying the adiabatic invari-ance theorem to fully nonlinear TWM is to find anappropriate canonical Hamiltonian representation. Westart by defining the complex amplitudes qj by A j ≡−√

κ j q j exp[−i

∫ z0 �k(z′)dz′], j = 1,2,3, where κ j =

χ (2)ω2j

k j c2 are the coupling coefficients from Eq. (7). This defini-

tion makes |q j |2 proportional to the photon flux at frequencyωj. The TWM system is known to have three constants ofmotion, namely the Manley-Rowe relations,

K1 = |q1|2 + |q3|2 (22a)

K2 = |q1|2 − |q2|2 (22b)

K3 = |q2|2 + |q3|2 (22c)

Remembering that |q j |2 is proportional to the photonflux at ωj, the physical meaning of these constants is easyto see: every time a ω1 photon is annihilated, a ω2 photonis also annihilated, and a ω3 photon is created. Similarly,annihilation of a ω3 photon is accompanied by creation of aω1 photon and a ω2 photon. Therefore, the sum of photons


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Figure 21 (a) Phase portrait with P2 = 0.3P3,i.e. 30% photon excess in ω1 over ω2, and nor-malized phase-mismatch �� = 0.6

√P3. (+)

and (−) mark the positions, in phase space, ofthe stationary states. The arrows indicate thedirection of their motion with increasing phase-mismatch. (b) Normalized canonical momen-tum vs. normalized phase-mismatch, for the twostationary states, where P2 = 0.3P3. These arethe vertical positions of the (+) and (−) station-ary states in phase space for various valuesof the phase mismatch. Reprinted figure withpermission from Ref. [25].

at ω1 and ω3 (K1) is constant, and so is the sum of photonsat ω2 and ω3 (K3) as well as the difference between thenumber of photons at ω1 and ω2 (K2). In fact, only two ofthese three constants are independent, since K3 = K1 − K2.As the Manley-Rowe relations are constants of the motion,the system is invariant under the phase-transformations thatthey generate.

Now we are in a position to define the real generalizedcoordinates Qj and momenta Pj, which will be used for thecanonical Hamiltonian representation:

Q1 =[− arg(q1) − arg(q2) + arg(q3)

]8

Q2 =[− arg(q1) + arg(q2)

]4

(23a)

Q3 =[− arg(q1) − arg(q2) − arg(q3)

]8

P1 = |q1|2 + |q2|2 − 2 |q3|2

P2 = |q1|2 − |q2|2 (23b)

P3 = |q1|2 + |q2|2 + 2 |q3|2

The reason that these parameters provide a useful rep-resentation will now be explained. From their definitions,P2 = K2 and P3 = K1 + K3, so P2 and P3 are constantsof the motion. Note P2 represents the photon flux excessat ω1 over ω2, and P3 represents the overall photon fluxbalance between the three waves. Their corresponding gen-eralized coordinates, Q2 and Q3, are the phases that remaininvariant under the transformations generated by P2 andP3, respectively. As a result, the interaction can be com-pletely described by the dynamics of Q1 and P1, where Q1is proportional to the phase difference between the two lowfrequencies and the high frequency, and P1 represents theexcess of photon flux in the two low frequency waves overthe high frequency wave. P1 is thus directly related to theconversion efficiency.

We further define the scaled propagation length ξ =z√

κ1κ2κ3 and the parameter �� = �k/√

κ1κ2κ3, whichdescribes the relative strength of the phase-mismatch com-pared to the nonlinearity. With these definitions, the canon-

ical Hamiltonian equations are given by

d Q1

dξ= ∂ H

∂ P1(24a)

d P1

dξ= − ∂ H

∂ Q1(24b)

where the Hamiltonian is

H = 1

8

[√(P1 + 2P2 + P3) (P1 − 2P2 + P3) (−P1 + P3)

cos (8Q1) − �� (P1 + 3P3)]

(25)

The phase portrait of the system is determined by theconstants of motion P2 and P3 (i.e. by the Manley-Rowerelations) and the phase-mismatch ��. In other words,the constraints on the distribution of photons between thethree waves, expressed with the Manley-Rowe constants orequivalently with P2 and P3, determine the interaction forgiven initial conditions and the phase-mismatch parameter��. Figure 21a shows the (Q1, P1) phase portrait, whereP2 = 0.3P3 and �� = 0.6

√P3. Note that P1 is bounded

between 2 |P2| − P3 and P3. This reflects the boundaries ofup-conversion and down-conversion due to depletion (e.g.the up-conversion process can continue only until one ofthe low frequency waves is depleted). The phase portraitcontains two elliptic fixed points, labeled “+” and “−” (thephase portrait is periodic in Q1 so the “+” fixed point ap-pears on both sides of Fig. 21a). The arrows in this figureindicate the direction of motion of the fixed points withincreasing phase-mismatch. This motion is purely vertical,so each elliptic fixed point can have different values of P1depending on the phase-mismatch, i.e. it will correspondto a different conversion efficiency for different values ofthe phase-mismatch. Additionally, each point will alwayshave the same value of Q1, i.e. Q−

1 = 0 and Q+1 = π/8.

Therefore, at an eigenstate, the phase difference betweenthe low frequencies and the high frequency is either 0 or π .This point will serve to elucidate the connection to autores-onance process in the next section.

Figure 21b depicts the vertical position of the two sta-tionary points in phase space, as a function of the phase-mismatch. The important property to note here is that thesepositions span the entire range of values allowed for P1. This




Figure 22 Illustration of adia-batic criteria and evolution fora case with equal number ofphotons at ω1 and ω2. The topand bottom rows correspond tofast and slow chirp rate, respec-tively. In both rows the phase-mismatch is swept over thesame range. The dashed linescorrespond to a stationary state.These were calculated analyti-cally, as was the nonlinear adia-batic condition rnl in (c) and (f).Only the bottom row, in whichrnl � 1, satisfies the adiabaticcondition. Reprinted figure withpermission from Ref. [25].

means that adiabatic variation of the phase mismatch canfacilitate evolution over the entire range of P1. Specifically,the system can start with no energy in the high frequency ω3(top of the phase space, where P1 = P3) and end up with themaximum possible energy in ω3, bounded only by deple-tion (bottom of the phase space, where P1 = 2 |P2| − P3).The efficient conversion in the opposite direction, fromhigh to low frequency where ω3 is completely depleted,corresponds to motion from bottom to top. Interestingly,if P2 = 0, then each of the two low frequencies has thesame photon flux, which means that they deplete simul-taneously. In this case, adiabatic up-conversion would re-sult in complete transfer of their initial energy to the highfrequency.

6.3. Adiabatic criteria and evolution

In the following, we summarize the conditions requiredfor adiabatic evolution, which were presented fully inRef. [25]. Adiabatic following can be obtained when thesystem is prepared to be near a stationary state with P±

1(see Fig. 21b), i.e. such that

∣∣P1 − P±1

∣∣ � P3, and the rateof change of the scaled phase mismatch �� is small enoughto satisfy

rnl =∣∣∣∣∣d

(P±

1 /P3)

d��

d��

dξ

∣∣∣∣∣ v−1 � 1 (26)

where v =√

∂2 H∂ Q2

1

∣∣∣Q±1 ,P±

1

∂2 H∂ P2

1

∣∣∣Q±1 ,P±

1is the frequency in

which the system point orbits a fixed point in phase space.Adiabaticity can thus be more closely satisfied when theoverall intensity is larger (which increases the overall pho-ton flux P3) and when the rate of change of the phase-mismatch is smaller. For given Manley-Rowe constants (orequivalently P2 and P3), the frequency v is generally fasterfor greater phase-mismatch magnitude |��|. The conditionis thus more stringent close to phase-matching. If addition-

ally �� varies monotonically, changing signs from begin-ning to end, and |��| � √

P3 at the beginning and end ofthe interaction, the system will evolve adiabatically fromP1 = P3 to P1 = 2 |P2| − P3 or vice versa.

An example illustrating adiabatic criteria and evolutionfor a case with the same number of photons in ω1 and ω2(i.e. P2 = 0) is depicted in Fig. 22. In this figure, the dashedcurves correspond to the minus stationary state, calculatedanalytically. Each row of panels in Fig. 22 corresponds toa different case. In both cases the system started in the mi-nus state and the phase-mismatch was swept over the samerange (from −10

√P3 to 10

√P3). However, in each case the

phase-mismatch chirp rate was different, (the interactionlength was different). In the first case, shown in Figs. 22a-c, the normalized interaction length was �ξ

√P3 = 1, and

the system does not follow the stationary state. Correspond-ingly, the adiabatic condition is not satisfied, as rnl reachesa value much greater than 1. In the second case, displayedin Figs. 22d-f, �ξ

√P3 = 100, the stationary state is very

closely followed, and rnl � 1 throughout the entire inter-action. Subfigures c and f, which depict rnl, were calculatedusing Eq. (26).

6.4. Robustness and bandwidth

Adiabatic TWM processes have numerically been shownto be robust against changes in various parameters, suchas wavelength and temperature [8, 9, 11, 12, 19, 20] (alsosee Section 3.1), which are manifested in changes in thephase-mismatch. This robustness stems from the fact thatthe phase-mismatch is swept along a large range of val-ues, so a wide range of physical conditions can result inphase-matching conditions within the range that satisfiesthe requirements for adiabatic evolution.

The conversion bandwidth is defined as the full widthat half maximum of the conversion efficiency. For up-conversion, the conversion efficiency can be expressed


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Figure 23 Adiabatic SFG simulation. Thepump wavelength was λ2 = 1064 nm andthe sum-frequency wave was taken to startwith no energy. The nonlinear medium was aMgO:LiNbO3 crystal with a poling period lin-early chirped from 11.52 μm to 11.79 μm.(a) Intensities of the three waves along the crys-tal for input wavelength λ1 = 1550 nm, and to-tal input intensity 200 MW/cm2. (b) Conversionefficiency vs. λ1 for different total input inten-sities, which are indicated in units of MW/cm2.Reprinted figure with permission from Ref. [25].

as [25]

η ≡ P3

2 (|P2| − P3)

(P1

P3− 1

)(27)

while for down-conversion it is simply 1 − η. Notethat, since P2 and P3 are constant for a given interac-tion, the conversion efficiency dynamics is determinedentirely by P1. Under this definition, η(P1 = P3) = 0and, η (P1 = |P2| − P3) = 1. The full width at half max-imum of η was estimated to be ��BW = ��(�ξ/2) −��(−�ξ/2). The estimated bandwidth is therefore inde-pendent of the intensities of the interacting waves, as itdepends only on the phase-mismatch chirp range. This es-timate was shown to be within 3% of numerical results[25].

Figure 23 shows numerical simulation results of an ex-ample of an adiabatic broadband process in the fully non-linear regime. The intensities of the three waves along thecrystal when the input wavelength is λ1 = 1550 nm and thetotal input intensity (i.e. at both λ1 and λ2) is 200 MW/cm2

are displayed in Fig. 23a. Clearly, energy is efficientlytransferred from the low frequencies to the high frequency(η = 0.93). Figure 23b depicts the conversion efficiencyas a function of input wavelength, for several total inputintensities. For low intensity of 2 MW/cm2, the conver-sion efficiency is low (η ≤ 0.11) but constant over a 54.2nm bandwidth. This is an example of non-adiabatic inter-action, where the adiabatic condition of Eq. (26) is notsatisfied due to the low intensity (low P3). For total inputintensity of 200 MW/cm2, the conversion is very efficient(reaching η = 0.97) over a wide 55.5 nm bandwidth. Thishigh and broadband efficiency is the hallmark of adiabaticinteraction. Interestingly, for higher total input intensity of20, 000 MW/cm2, the efficiency oscillates across the λ1 tun-ing range. The reason for this is that the high intensity (i.e.high P3) invalidates the requirement that |��| � P3 at thebeginning and end of the interaction. As a result, the sys-tem point orbits the stationary point from a large distance,which in turn results in large oscillations of the conversionefficiency with changing conditions (such as changing λ1).This demonstrates how, due to the nonlinearity of the dy-namics, a given adiabatic design will perform optimally notonly above a certain input intensity, but also below someintensity upper bound. In order to support adiabatic interac-tion for higher intensities, a larger chirp range is required,so that |��| � P3 at the beginning and end of the interac-

tion. Finally, we note that numerical simulations for SHG,DFG and OPA were shown to yield similar results [25].

Several earlier works had already provided insightsinto fully nonlinear adiabatic TWM. To the best of ourknowledge, Baranova et al. [6, 7] were the first to consideradiabatic SHG with nonlinear dynamics. Explicitly admit-ting that the linear adiabatic theory does not apply, theauthors hypothesized that adiabatic interaction could stilltake place, based on similarities between the eigenstatesand eigenvalues of the SHG system and those of linearsystems with two coupled modes. The same authors thenwent forth to show correspondence between the system’seigenstates and numerical solutions with a slowly chang-ing phase-mismatch. Other concepts, while contained in thedescription given above, shed light on special cases of thenonlinear dynamics as will be shown in the following.

6.5. Autoresonance process

An interesting view point on adiabatic interaction was pro-vided by Yaakobi et al. [23, 24] by applying the concept ofautoresonance [68, 69]. Autoresonance is a nonlinear pro-cess in which nonlinearly interacting waves can phase lock(hence stay in resonance) owing to an adiabatic change inthe system parameters, thereby enabling efficient transferof energy between these waves. In fact, the first publi-cation on autoresonant optical frequency conversion wasapplied to OPA by FWM [26], however a similar rea-soning was used for TWM OPA and SHG, where in allcases the phase mismatch is swept from a very large neg-ative (positive) to a very large positive (negative) value.In all of these cases, a key requirement is that the interactionbegins in conditions that satisfy undepleted pump approxi-mation (for FWM there were two undepleted pumps). It wasthen shown that with an undepleted pump, and using theManley-Rowe relations, the entire dynamics of the interac-tion can be described as the dynamics of Z = |B1| exp(iφ),where B1 is a normalized complex amplitude of the signaland φ is the difference between the phases of the unde-pleted pump(s) and the phases of the weak signal and idler(in SHG the signal and idler are degenerate), e.g. for TWMOPA:

φ = arg(B1) + arg(B2) − arg(B3) (28)




Figure 24 Numerical simulation of autoresonant OPA. (a) Photon fluxes of the interacting waves and (b) The relative phase definedby Eq. (28) along the propagation. The bottom row of panes shows illustrations of the corresponding phase-space picture, where thestationary state being followed is marked with a magenta star. Reprinted Fig. 24a and Fig. 24b with permission from Ref. [23].

where B2 and B3 are the normalized complex amplitudesof the idler and pump, respectively. Subsequently, it wasfound that, for a linearly varying phase mismatch, Z isreal and inversely proportional to the chirp rate near thephase-matching point (for FWM, nonlinear phase modu-lation compensates the dispersion phase-mismatch at thispoint). As a result, as the propagating waves approach thephase-matching point, the manifestation of autoresonanceis that the phase φ locks onto either 0 or π for increasing ordecreasing phase-mismatch, respectively. Next, under theassumption that φ is either 0 or π , the undepleted pumpdynamics equations predict that the signal and idler willhave the same number of photons after a finite propagationlength, in the limit of an asymptotically small chirp rate.The physical interpretation is simply that the pump(s) gen-erated an equal number of signal and idler photons, and thatthis number is much larger than the initial number of signalphotons, after some finite propagation length. Finally, thefull nonlinear dynamics is solved under the conditions ofφ = 0 and identical signal and idler photon flux. This solu-tion leads to a full transfer of energy from the pump(s) tothe signal and idler.

A numerical simulation of autoresonant OPA with pos-itive chirp rate is shown in Fig. 24. The normalized photonfluxes of the interacting waves are shown in Fig. 24a, andthe evolution of the phase φ in Fig. 24b. In correspondenceto the above explanation, in the first part of the interac-tion the photon fluxes are effectively constant, while the

phase φ oscillates and gradually approaches 0. When φ

is finally very close to 0, energy starts getting transferredfrom the pump to the signal and idler, where they havethe same photon flux. This process continues until the fulldepletion of the pump, where the phase φ remains near zerovalue all along the propagation.

Yaakobi et al. also provide stability analysis for eachof the three cases that were studied [24, 26]. In simpleterms, they found that adiabatic interaction will take placeas long as the chirp rate is much smaller than the oscillationfrequency of φ during the locking stage (see Fig. 24b). Thisfrequency is itself proportional to the chirp rate and to theamplitudes of the interacting waves.

The autoresonant perspective is isomorphous to a spe-cial case of the general theory that was presented in theprevious section, i.e. OPA and SHG with an initially weaksignal. This is illustrated for the case of adiabatic OPA inthe bottom row of panels in Fig. 24. First, the locking stagein the autoresonance process is the same as the initial stageof the general theory, where the stationary point is nearthe bottom of the phase portrait (leftmost bottom pane inFig. 24) while the phase-mismatch is very large. At thisstage, the system point is moving along an open curve (thecurve that passes through the center of the star). As thephase-mismatch decreases in magnitude, this curve bendsto orbit a stationary point. From this point the system pointorbits this stationary point, which is marked with a magentastar in the four rightmost bottom panes in Fig. 24. This fixed


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point moves vertically up in phase space, along the line ofQ1 = π/8, which is equivalent to φ = 0. More generally, asexplained above and can be seen in Fig. 21, the stationarypoints move vertically in phase-space with increasing ordecreasing phase-mismatch (i.e. positive or negative chirprate), along the Q1 = 0 and Q1 = π/8 vertical lines. Since,by their definitions in Eqs. (23) and (28), Q1 = (π − φ)/8,following a stationary point with Q1 = 0 (Q1 = π/8) isequivalent to maintaining φ = π (φ = 0).

Furthermore, the case of identical signal and idler pho-ton flux was solved analytically in the context of the generaltheory, yielding the same results as those obtained with au-toresonance. Finally, the frequency of the oscillations ofφ during the autoresonance locking stage is the same asthe frequency of the system point around the fixed point inthe general theory phase space. Hence, the autoresonancestability condition is in fact a special case of the generaladiabaticity condition given by Eq. (26).

6.6. Adiabatic OPA and geometricalvisualization of general TWM

Due to its significance to ultrashort laser physics, adiabaticOPA received special attention by researchers. A parametricinvestigation of adiabatic OPA was carried out by Phillipset al. [21]. In this publication the authors have already laidout, in general terms, the main features of the general the-ory, namely that the system follows an eigenstate due tothe slowly varying phase-mismatch. Following a series ofexperimental works by Heese et al. [13–15], Phillips et al.published a detailed analysis of apodized adiabatic OPA[61], taking a heuristic approach based on the geometricalrepresentation of Ref. [70]. This theory will now be ex-plained, and then the experimental results of Ref. [13–15]will be reviewed. Here a QPM modulation with 50% dutycycle will be assumed, and adiabaticity will be obtainedby chirping. We note that Phillips et al. [61] consideredother schemes, but found them to be less useful due totechnological fabrication limitations or inherent physicalreasons.

We begin by describing the geometrical representationof Ref. [70]. Three real variables, X, Y and Z, are definedaccording to

X + iY = γ ′q1q2q∗3 (29a)

Z = γ ′3 |q3|2 (29b)

where γ ′m = nmγ 2

m/

[ωm

∑3j=1

n j γ2j |q j (0)|2

ω j

]and γ ′ =√

γ ′1γ

′2γ

′3 are normalization constants. The phase of X +

iY is the phase difference between the two low frequenciesand the high frequency, and Z is proportional to the pumpphoton flux. The geometrical representation is constructedby considering X, Y and Z as components of a vector W ina three dimensional space, i.e. W = [X, Y, Z ]T .

Due to the constraints represented by the Manley-Rowerelations, this vector is restricted to a surface defined by theimplicit equation φ = 0, where

ϕ = X2 + Y 2 − γ ′1γ

′2

γ ′3

Z(Z − γ ′

3 K1) (

Z − γ ′3 K3

)(30)

It can be shown that this conserved surface is alwaysclosed and convex [70]. For a constant phase-mismatch, theevolution of the system is geometrically represented as amotion of the vector W along a closed curve. This curve isformed by the intersection between the φ = 0 surface anda plane, which is entirely determined by the Manley-Roweconstants and the phase-mismatch. Additionally, each suchcurve is circling a point on the Y = 0 line that correspondsto an eigenstate. An example of such evolution is shown inFig. 25. Figure 25a depicts the φ = 0 surface, where theblue curve is the trajectory traced by the motion of W inthe direction indicated by the blue arrow. The black arrowexits the surface at the point corresponding to the eigenstatethat the trajectory curve circles. Figure 25b displays thecorresponding photon flux of the OPA signal (ω2) and pump(ω3).

Phillips et al. [61] found that the eigenstates correspondto points along Y = 0 on the surface (i.e. in the X-Z plane),so an ideal adiabatic OPA interaction would have the vectorW following the trajectory from the top to the bottom ofthe surface, along Y = 0. Correspondingly, an adiabaticitycondition was formulated based on two heuristic require-ments: (i) the angle of deviation of W from the Y = 0 line,in the direction perpendicular to the X-Z plane, is requiredto be much smaller than the angle W subtends in the X-Zplane; (ii) the rate of variation of Z along the crystal is re-quired to match that of the Z component of the eigenstate(the latter varies due to the variation of the phase-mismatchalong the crystal). This results in an intuitive geometricalrequirement, written as∣∣∣∣dθ‖

d�

∣∣∣∣ � R⊥R‖

(31)

where θ‖ = tan−1[(

∂ϕ

∂z

)/(∂ϕ

∂x

)]Y= 0

is the angle sub-

tended on the X-Z plane by a vector pointing at theeigenstate point that the local trajectory is circling. � =∫ z

0

√4γ ′′2 + �k2

1 (z′)dz′ is a scaled propagation coordinate,

γ ′′ = 2χ (2)

πc

√ω1ω2ω3

∑j

n j γ2j

ω j|q j (0)|2

n1n2n3is a constant, and R‖

and R⊥ are the local radius of curvature of the φ = 0 sur-face, at the point described by W, in the X-Z plane and thedirection perpendicular to it, respectively. In loose terms,the left hand side of this inequality describes the rate ofmotion of the eigenstate point, on the surface, due to thevarying phase-mismatch. In order to understand the mean-ing of the right hand side, recall that W is constrained tomove on the φ = 0 surface. Therefore, for a given phase-mismatch, the local curvature of the surface determines therate of change of W. The right hand side thus describes the




Figure 25 Geometrical representation of OPA and its phase-space analog. (a) The surface represents all physically allowed systemstates, i.e. allowed positions of the state vector W. For the given phase-mismatch, W evolves along the blue curve in the directionindicated by the blue arrow. This curve circles a point that represents an eigenstate. The black arrow exits the surface at this point.(b) Corresponding periodic evolution of the signal and pump along the crystal. (c) Corresponding phase-space diagram, where thesystem follows the blue contour line in the direction indicated by the blue arrow, and the stationary point that it circles is marked by themagenta star. Reprinted Fig. 25a and Fig. 25b with permission from Ref. [61].

local ratio between the rate of motion of W in the X-Z planeand in the perpendicular direction, due to evolution of Walong the local trajectory curve (i.e. the curve that corre-sponds to the local phase-mismatch). Large R⊥/R‖ meansthat W is moving faster along the X-Z plane than in the per-pendicular direction. As long as this ratio is much greaterthan the angular rate of motion of the eigenstate point, Wwill stay near the Y = 0 line, so the adiabaticity conditionwill be satisfied.

The relation between this heuristic geometric approachand the general adiabatic theory will now be explained.We note first that, for a constant phase-mismatch, the tra-jectory followed along the φ = 0 surface in the geomet-rical representation is analogous to a contour line in thephase space representation (see Fig. 25c). They are bothcompletely defined by the Manley-Rowe constants and thephase-mismatch, and both circle a point that corresponds toan eigenstate. More precisely, horizontal (in the X-Y plane)and vertical (along the Z axis) motion of W on the φ = 0surface correspond to horizontal and vertical motion of thesystem point in phase space, respectively. The reason is thatX + iY ∝ |q1q2q3| exp(−i8Q1) and Z ∝ |q3|2. HorizontalW motion thus means that the phase between the waves ischanging while their amplitudes stay constant. This corre-sponds to varying Q1, i.e. the horizontal phase-space coor-dinate, and constant P1 = |q1|2 + |q2|2 − 2 |q3|2, which isthe vertical phase-space coordinate. Similarly, vertical Wmotion corresponds to constant Q1 and varying P1, i.e. ver-tical motion in phase-space. Note that due to the definitionof the coordinate systems in each of these representations,the direction of motion is reversed both horizontally andvertically. Furthermore, the eigenstate line on the surface,Y = 0, corresponds to Q1 = 0 or Q1 = π/8 in phase space,i.e. the Q1 values of the eigenstates in the general theory.Consequently, in terms of the general theory, the adiabatic-ity requirement of Eq. (31) means the rate of vertical motionof the stationary point in phase space, due to the changingphase-mismatch, has to be much smaller than the ratio be-tween the vertical and horizontal local rates of motion of

the system point evolving along the local contour line. As aresult, the system point will move primarily vertically alongthe Q1 = 0 or Q1 = π/8 line. Apparently, this requirementis more strict than the adiabaticity condition formulated inthe context of the general theory (Eq. (26)). The formerrequires the system point to be near the stationary pointthroughout the entire evolution, while the latter only limitsits rate of motion with respect to the frequency with whichthe system point is orbiting it. However, since in the generaltheory the system point is required to be near a stationarypoint at the start of the interaction, these two conditionsqualitatively amount to the same thing.

An experimental demonstration of adiabatic OPA in thenonlinear dynamics regime was conducted by Heese et al.[14]. The chirped QPM structure used in this experimentis illustrated in Fig. 26a, which shows the poling periodalong the crystal. Figure 26b displays the experimentallymeasured gain spectrum. The result was obtained by tuningan ultrashort mid-IR source across the 3 − 4 μm range. Theinput pulse had a 72 fs transform limit, and was stretched to2.5 ps before conversion in order to avoid group dispersioneffects. The 10 dB gain bandwidth was more than 800 nmwide around 3.5 μm, peaking at 45 dB. When the output ofthis adiabatic OPA was let into an additional, identical OPAstage, the gain increased by 14 dB. Before compression, theconverted pulse had 1.5 μJ of energy. This pulse was thencompressed to 75 fs duration, although it suffered from aprepulse with a peak intensity of about 20% as comparedto the main pulse. Overcoming this temporal distortion isthe motivation behind the work on apodization, discussedbelow.

6.7. Apodization procedure

For frequency conversion of ultrashort pulses, the use ofan apodization procedure in the adiabatic design is veryimportant. It reduces the spectral ripples that appear in theconversion efficiency when a constant-chirp QPM mod-ulation is used (as seen in Fig. 26b), and also yields a


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Figure 26 Adiabatic OPA experiment in the nonlinear dynamics regime. (a) Chirped poling period along the 7.4 mm long MgO:LiNbO3

crystal used in the experiment. The top inset illustrates the corresponding binary QPM modulation. (b) Experimentally measured gainspectrum. The 10dB bandwidth is greater than 800 nm. Reprinted figure with permission from Ref. [14].

smoother spectral phase [13, 14, 61]. Both of these phe-nomena, i.e. the spectral ripple and complex spectral phasestructure, produce a lower bound on the pulse duration ofthe compressed converted pulses and even facilitate pre-pulses. Apodization offers a way to mitigate these effects,with the aim of producing shorter converted pulses. Suchan apodization procedure has been demonstrated both theo-retically and experimentally [13,14,61]. While apodizationwas investigated in the context of non-adiabatic interactionsin Ref. [61], as well as in other publications (e.g. [71–75]),the discussion here is restricted to adiabatic interaction.

Before addressing the apodization procedure, we wouldlike to explain the reason for the spectral ripples, as laid outin Ref. [61]. All of the approaches to adiabatic interactiontheories discussed so far require the interaction to beginand end with very large phase-mismatch, and at the sametime keep a low chirp rate. For a finite interaction length,a trade-off is required between these two demands. There-fore, as the waves enter or leave the nonlinear medium,the phase-mismatch is finite. This constitutes an abrupt,highly non-adiabatic change in system parameters, essen-tially a discrete switch-on or switch-off of the couplingbetween the frequencies. Consequently, the system doesnot transit adiabatically between free-space eigenstates andcoupled-system eigenstates. Rather, a superposition of sucheigenstates is excited. The spectral ripples are a manifes-tation of interference between these eigenstates, as well asthe complex phase-structure imparted on converted broad-band ultrashort pulses. This can also be described in termsof the general theory: due to the abrupt change, the sys-tem point does not start very near to the stationary point. Ithenceforth orbits the stationary point from a non-negligible,frequency-dependent distance. Each frequency componentthus performs a different trajectory across phase-space, re-sulting in spectral ripple.

In the context of adiabatic interaction, apodization tech-niques provide a gradual (in fact, adiabatic) switch-on andswitch-off of the coupling, at the beginning and end of theinteraction. The idea is to take advantage of the fact that farfrom phase-matching, the adiabatic criterion is relaxed. In-deed, as was noted above, as the phase-mismatch increases,the phase-space orbiting frequency, v, of the system pointaround the fixed point, also increases. By considering a

constant chirp-rate, the methods introduced so far had lim-ited the chirp-rate to that which satisfies adiabaticity nearphase-matching. For a given adiabatically chriped QPMmodulation, Phillips et al. [61] suggested adding fast chirprate segments at both ends. The main design criterion forthese apodization segments is the conversion bandwidth in-tended to be converted by the adiabatic interaction. The fastchirp begins at points where the phase-mismatch is large forconversion of any frequency in this range, at both ends ofthe QPM modulation pattern. The chirp rate in the apodiza-tion segments still satisfies the adiabatic condition, howeverthis condition is now very much relaxed so the chirp ratecan be greatly increased with position away from the edgesof the original constant-chirp design. A numerical methodfor finding an optimal form of the apodization segments,for a desired degree of adiabaticity, is detailed in Ref. [61].

Figure 27 displays a numerical example of apodizationof adiabatic OPA. This OPA is designed to amplify an in-put wavelength in the range 1450–1650 nm with a 1064nm pump. The apodized grating vector, Kg(z) = 2π/(z),where (z) is the local modulation period, is depicted inFig. 27b. The two fast-chirped apodization segments areclearly visible at both ends of the central segment. The factthat their chirp rate is so much faster than that of the cen-teral segment, while still adiabatic, illustrates how relaxedthe adiabaticity condition is in these regions. In Fig. 27a thenumerically calculated pump depletion, corresponding tothe conversion efficiency, is shown for both the unapodizedand apodized case vs. the input wavelength. In the absenceof apodization, the efficiency has as much as 10% fluctu-ation across the desired wavelength conversion range. Theapodized conversion efficiency is near 100% throughout thesame range, with a less than 1% ripple.

The merits of apodization of adiabatic OPA were alsodemonstrated experimentally [13, 15]. Figure 28 shows re-sults of an experiment that compared an unapodized andan apodized QPM crystal. The poling period along the un-apodized and apodized crystals are displayed in Figs. 28aand b, respectively. Their corresponding compressed con-verted pulse in Figs. 28c and d. Here, the apodized segmentsfollow a hyperbolic tangent (tanh) function, which is simi-lar to the heuristic optimum found by the algorithm of Ref.[61], yet easier to fabricate. In the unapodized case, the




Figure 27 Numerical demonstration of apodized adiabatic OPA.(a) Pump depletion, corresponding to conversion efficiency, ofapodized (blue) and unapodized (green) QPM modulation, vs.input wavelength. This device is designed for amplification of in-put wavelengths in the range 1450-1650 nm input. (b) ApodizedQPM modulation grating vector, Kg (z) = 2π /∧(z), where ∧(z) isthe local modulation period. This modulation is entirely adiabatic,where the two fast-chirp apodization segments at the ends takeadvantage of relaxed adiabatic conditions for wavelengths out-side the conversion range. Reprinted figure with permission fromRef. [61].

compressed converted pulse suffers from a very significantprepulse distortion and a complex temporal phase. An ad-ditional feature can also be seen around 200 fs in Fig. 28c.Contrarily, in the apodized case, the compressed convertedpulse has no significant prepulse or other distortion fea-tures. Its temporal phase is also well behaved, nearly flatacross the temporal range that contains almost all of thepulse’s energy.

In a recent experiment, using two cascaded apodizedadiabatic OPAs, pulses of 75 fs duration and 7 μJ energywere generated from a 1.56 μm ultrafast input and 1064nmpumps (the generated radiation spectrum was around 3.4μm) [15]. In this experiment, the generated pulse durationwas limited by the bandwidth of the input seed. Interest-ingly, the input pulses were shaped using a SLM pulse

shaper before entering the OPA, and the generated mid-IRradiation was compressed simply by passing through bulksilicon. This scheme takes advantage of mature technologyin the 1.5 μm range for dispersion control, which is moredifficult in the mid-IR, thus facilitating both short pulse du-ration and high efficiency (mid-IR pulse shapers have lowtransmission efficiency).

6.8. Adiabatic OPO

Phillips et al. also investigated apodized adiabatic OPO[22], i.e. the case where the apodized adiabatic OPA crystalis placed inside a resonator and pumped with a single fre-quency. In OPOs, weak signal and idler are generated firstby spontaneous parametric down conversion. Either the sig-nal, or both the signal and idler resonate in the optical cavityand so they get further amplified in following trips throughthe cavity. The gain saturates as the pump gets depleted, soeventually the OPO settles into some operating point, wherethe gain exactly compensates for the round-trip loss (for atextbook analysis of OPO see Ref. [28]). As was shown inprevious sections for other interactions, adiabatic interac-tions lead to robust one-way conversion. These are attractiveproperties for OPO, which generally suffers from signal-to-pump back-conversion and non-uniform gain across thepump’s spatio-temporal profile.

In their research, they have devised a clever way tofind the oscillation threshold and operating point of theadiabatic OPO where only the signal is resonated in thecavity, assuming low cavity losses. First, the signal gainnear threshold was approximated by assuming undepletedpump,

Gs ≈ exp(2πp) (32)

where p = χ (2)ωsωi√ks ki c2

∣∣Ap

∣∣2/(d�k/dz). Oscillation thresh-

old is thus gives by p,th = ln(R−1

s

)/(2π ), where Rs =

1 − as and aS is the round-trip signal power loss. Next,for operation above threshold, pump depletion is esti-mated using Eq. (32) assuming constant signal powerthroughout the crystal, i.e. ηp ≈ exp(−2πs) where s =χ (2)ωpωi√

kpki c2|As |2 /(d�k/dz). The last assumption is reason-

able for the working point, in which signal gain and lossare balanced against each other. Finally, the (low) roundtrip signal loss is equated to the signal gain in order to findthe operating point. Using the last two equations this leadsto

1 − exp(−2πs)

2πs= 1

N(33)

where N is the ratio of pump intensity to the threshold pumpintensity. An implicit equation for the signal intensity at theoperating point has thus been obtained. The conversion ef-ficiency obtained from this equation, as a function of N, isplotted in Fig. 29, along with the result of a full numer-ical simulation. Clearly there is excellent correspondence


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Figure 28 Experimental demonstra-tion of apodized adiabatic OPA. Pol-ing period along the QPM crystal (a)with no apodization (b) with apodiza-tion. Normalized intensity (solid blueline) and spectral phase (dashedred line) of the corresponding com-pressed and converted pulse for the(c) unapodized (d) apodized case.Reprinted figure with permission fromRef. [13].

Figure 29 Adiabatic OPO conversion efficiency as a function ofthe ratio of pump intensity to threshold pump intensity. Reprintedfigure with permission from Ref. [22].

between the two, both showing how efficiency increasesmonotonically with increasing N, reaching nearly 100% atN ∼ 6.

At this point we can conclude that, for a Gaussian beampulsed pump, high conversion efficiency can be obtained ifN � 1 across most of the pump’s spatio-temporal profile.This condition can be satisfied by using a signal cavitymode somewhat larger than the pump beam and a cavity

lifetime comparable to or longer than the duration of thepump pulse.

A more elaborate analysis revealed that adiabatic OPOsuffers from modulation instability. So far, CW pump, sig-nal and idler were considered, so the instability was effec-tively ignored. However, if they are allowed to have side-bands (as in common real physical systems), and quantumnoise at all frequencies is taken into account, the instabil-ity is manifested. The modulation instability comes fromOPO interaction between sidebands and carrier waves, e.g.a signal sideband and an idler sideband with the same shiftfrom their carriers can have a significant interaction with thepump carrier. This broadband sideband gain is made pos-sible by the wide range of phase-mismatch compensationprovided by the chirped crystal. In some cases, parasitic op-tical rectification may further exasperate this phenomenon(see Ref. [22] for details). An example of such a case canbe seen in Fig. 30, that shows the sideband gain G vs. side-band detuning frequency, for various values of N and usingthe steady state solution found using Eq. (32). The distinctpeak at 1.38 THz is related to a wave at the same frequencygenerated by optical rectification. Furthermore, it’s clearthat significant sideband gain is present at a wide range ofother sideband frequencies in the THz range.

These results indicate that the OPO will not operatein a single mode due to modulation instability, and indeedthis was demonstrated by numerical simulation [22]. Fig-ure 31 shows an example of such a case, where the signalspectrum essentially fills the OPO acceptance bandwidth




Figure 30 Sideband gain vs. sideband detuning for variousratios of the pump intensity to the pump threshold intensity.Reprinted figure with permission from Ref. [22].

due to the modulation instability. Phillips et al. discuss anddemonstrate methods to overcome the modulation instabil-ity, suggesting an intra-cavity etalon or narrowband signalseeding [22].

6.9. Adiabatic third harmonic generation

Generating the third harmonic of a fundamental wave (FW)by cascading SHG (ω + ω → 2ω) and SFG (ω + 2ω → 3ω)has attracted much attention in the field of frequency con-version [53]. Two distinct adiabatic schemes for efficientTHG in such a scheme will now be put forth.

First, we consider the work of Longhi [60], which actu-ally utilized an idea very similar to the one used in theSTIRAP-analog presented above. However, Longhi wasable to apply it to the nonlinear dynamics regime, by virtueof an analogy of the SHG-SFG cascade with a three-level

Figure 31 Adiabatic OPO signal spectrum on a frequency scalethat is normalized to the OPO acceptance bandwidth. The centralfrequency is the signal frequency for which the OPO process isphase-matched at the center of the chirped grating. Reprintedfigure with permission from Ref. [22].

atomic system, involving both a two-photon transition anda one-photon transition [76]. In essence, this nonlinear sys-tem has an eigenstate that corresponds to all energy beingin the fundamental frequency when the SH-TH coupling ismuch stronger than the FW-SH coupling. Upon reversal ofthis ratio, the same eigenstate corresponds to all energy be-ing in the TH. Additionally, for any ratio of these couplingstrengths, this eigenstate never has any energy in the SHwave. An adiabatic conversion of the FW to the TH couldbe facilitated by varying this coupling ratio such that thesystem follows this eigenstate along the interaction. Notethat this would result in a counter-intuitive coupling order,i.e. the SH-TH coupling strength starts out higher than theFW-SH coupling strength.

A numerical example of this method is shown inFig. 32a, which depicts the normalized power of each fre-quency. This power is plotted along a QPM waveguide thatis structured to provide the FW-SH coupling and SH-THcoupling κ0 and σ 0, respectively, shown in Fig. 32b. Thelatter also shows the adiabaticity parameter r, where theadiabaticity condition is r � 1. High THG efficiency isclearly obtained, however the SH reaches as much as 20%of the input power along the way. The reason is that in thiscase of nonlinear dynamics it is more difficult to satisfy theadiabatic condition (as was shown analytically by Pu et al.[76] for the atomic case). This can also be seen in Fig. 32b,which shows that the adiabaticity condition r � 1 is notsatisfied throughout a large part of the interaction.

Rangelov et al. also performed numerical simulationsof adiabatic THG via SHG and SFG cascading [19]. In thisscheme, a QPM crystal is chirped in the counter-intuitive or-der, i.e. first the SFG process is phase-matched and then theSHG phase-matching takes place. Numerical simulationsindicate that this results in efficient THG with negligibleenergy being present in the SH throughout the entire in-teraction (see Fig. 33 for an example). Robustness againstvariation in the input wavelength was also demonstratednumerically.

Figure 32 Nonlinear STIRAP THG. (a) Numerically simulatedpower of the three interacting waves along the propagation axis.(b) FW-SH and SH-TH coupling strengths κ0 and σ 0, respectively,and the adiabaticity parameter r. The adiabaticity condition isr � 1. Reprinted figure with permission from Ref. [60].


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Figure 33 Numerically simulated normalized intensities of adia-batic THG via SHG-SFG cascade along a chirped QPM crystal.Reprinted figure with permission from Ref. [19].

In this section, we have tried to unite several interestingand important research works that have been reported in thelast few years, under the framework of the general theory ofthe fully nonlinear adiabatic TWM process. We discussedin detail the approaches taken by Yaakobi, Philips, Heese,and their coworkers, regarding the special cases of OPAand SHG. Each approach provides further insight into thephysics of adiabatic interaction and the design of adiabaticfrequency converters. We also reviewed the work of Longhiand Rangelov et al., regarding adiabatic THG via an adi-abatic SHG-SFG cascade. This special case is an encour-aging result regarding a possible extension of the generaladiabatic TWM theory to multiple processes. Finally, wenote that all of the processes reviewed in this section arehighly efficient and highly nonlinear, indicating that theycan be used in applications such as optical switching orpulse cleaning, where strong nonlinearity is an advantage.

7. Summary and Outlook

In this paper, we have reviewed the research on adiabaticprocesses in frequency conversion. Our aim was to coverthis new emerging sub-field to researchers from a varietyof research areas. For those who work in nonlinear op-tics, we hope that this review will expand their knowledgein frequency conversion, showing that efficient conversioncan occur not only when perfect phase matching is sat-isfied for the entire propagation, and also to demonstratethat the dynamical solutions in frequency conversion havemuch more to offer. For those who work in atomic physics,NMR and related fields, we hope that this review will al-low a smooth entrance to a new research field with somefamiliar concepts, visualization tools, and physical intu-ition, and will show them that the linear evolution discussedin their own fields are only the starting point for a muchricher dynamical evolution that includes the fully nonlineardynamics.

Starting with introducing the propagation of coherentlight in a nonlinear crystal in geometrical control terminol-ogy, sharing equivalent dynamical mechanisms from otheranalogous SU(2) systems, that were utilized to proposedthe RAP mechanism in the field of frequency conversion,we further gave a detailed description of the adiabatic evo-

lution. We have discussed robustness, tunability and scala-bility of the method, and presented the Landau-Zener con-version efficiency formula, which is used to estimate theefficiency of the adiabatic conversion process. Also, wehave showed several visualization tools that were borrowedfrom other two level systems - the ‘Bloch/Poincare’ sphere(and later on the general TWM surface), the ‘dressed state’picture – both introduce a physical intuitive general pre-sentation of the dynamical trajectories in the realm of fre-quency conversion and the distinction between adiabaticand non-adiabatic evolution processes.

We have shown recent progress on the applicability ofthe scheme to efficiently convert ultrashort pulses to otheroptical regimes, and its scalability to reach full octave con-version, spanning from the near-IR to the mid infrared. Bythat it offers the potential to produce the first single-cyclemid-IR pulse in a single efficient conversion step, in con-trast to existing mid-IR ultrafast pulse generation schemesthat normally involve multiple frequency conversion steps,making it highly relevant to established compact and intenselaser systems. The adiabatic method is expected to becomea powerful seeding source for coherent wavelength multi-plexers based on OPAs, and are of particular importancein nonlinear spectroscopy and high harmonic generationprocesses, where higher cut-off frequency can be obtainedwith mid-IR ultrashort lasers [77, 78].

The basic adiabatic dynamics scheme was extended tomulti-step frequency conversion, as well as to the rigor-ous analysis of adiabatic evolution in the fully nonlineardynamical regime. Adiabatic TWM with fully nonlineardynamics was put on a firm physical basis, detailing theconditions for obtaining adiabatic evolution. Just as the adi-abatic TWM in the linear dynamics regime was developedfrom an analogy with linear quantum systems, the gen-eral method reviewed here also follows, in general terms,an analysis of nonlinear adiabatic evolution performed fornonlinear quantum systems. Furthermore, the nonlinearadiabatic condition was determined, and an estimation ofthe bandwidth of adiabatic TWM processes was derivedand shown to be consistent with numerical results. We havereviewed the recent work of Porat et al. [20] , Yaakobi et al.[23] and Phillps et al. [61], where the different theoreticalapproaches were compared by discussing their similarities.Numerical simulations were shown, that demonstrate fullynonlinear adiabatic frequency conversion in several config-urations attainable with current technology, suggesting thatadiabatic SFG, SHG, DFG and OPA can be efficient overa wide band of input frequencies. Additionally, recent ex-periments successfully demonstrating adiabatic OPA withultrashort pulses were reviewed [13–15]. The apodizationprocedure used in these experiments, which relies on adi-abatic interaction as explained by Phillips et al. [54], wasalso discussed in the context of ultrashort pulse conversion.

In conclusion, adiabatic processes in frequency conver-sion and the analogy of the dynamical evolution in nonlin-ear optical media with the rich framework of finite leveldynamics open enormous possibilities. We believe that weare only at the initial stage of both theoretical and exper-imental perspectives. From the theoretical point of view,




the adiabatic theory in the fully nonlinear regime can befurther developed to more accurately account for the spec-tral phase imparted on broadband ultrashort pulses. Thiscould practically lead to shorter converted pulses. Addi-tionally, this theory can be extended to include multi-stepprocesses or higher order nonlinear processes, such as four-wave mixing. From the experimental perspective, adiabaticschemes can be utilized to efficiently convert broadbandfluorescence signals, which will be highly relevant to spec-troscopy of incoherent signals commonly utilized in ma-terial science and molecular spectroscopy. Adiabatic fre-quency conversion may be useful in quantum optics appli-cations, enabling to shift single and bi-photons into desiredspectral ranges. The inherent advantage of the adiabaticconversion scheme, of preserving the signal complex am-plitude modulation during conversion, allows the transferof signal phase to the mid-IR, thus offering fidelity andconvenience for applications requiring phase coherence inthe frequency converted field, allowing the generation ofshaped ultrafast pulses in wavelength regimes where directshaping is difficult or inaccessible using currently avail-able pulse shaping technology. Also, many new theoreticalpredictions in the multi-processes and the fully nonlinearanalysis, which were discussed in Sections 5 and 6, arewaiting for an experimental validation, and for suggestionof new applications.

Received: 22 July 2013, Revised: 1 September 2013,Accepted: 20 September 2013

Published online: 21 October 2013

Key words: Nonlinear optics, adiabatic dynamics, frequencyconversion, three wave mixing, ultrashort conversion.

Haim Suchowski is currently a postdoc-toral fellow at the University of California,Berkeley. He earned his M.A. and Ph.D inPhysics at the Weizmann Institute of Sci-ence (2011). He holds a B.A. in Physics(2004) and a B.Sc. in Electrical Engineer-ing (2004) from Tel Aviv University. He re-ceived the postdoctoral Fulbright fellow-

ship for 2012. His Ph.D research dealt with quantum con-trol of atoms and molecules with ultra-short laser pulses andanalogous schemed in nonlinear optics, whereas his currentresearch deals with nonlinear and ultrashort interaction withnanostructures and metamaterials.

Gil Porat received the B.Sc. degree inelectrical engineering from Tel Aviv Uni-versity, Israel, in 2007. He then pursueda direct track Ph.D. in electrical engineer-ing, also in Tel Aviv University, and histhesis is under final review. He is currentlya post-doctoral fellow at the WeizmannInstitute of Science, Israel.

Ady Arie received his Ph.D. degree inEngineering from Tel-Aviv University in1992. Between 1991 and 1993 he wasa Wolfson and Fulbright postdoctoralscholar at Ginzton Laboratory, StanfordUniversity, U.S.A. In 1993 he joined theDept. of Physical Electronics at Tel-AvivUniversity and in 2006 he became a Pro-fessor of Electrical Engineering. In theyears 2001-2005 he also served as a

co-founder and chief technology officer of ZettaLight Inc., astartup company in the field of optical wavelength manage-ment. His research in the last years is in the areas of non-linear optics – in particular periodic and quasi-periodic non-linear photonic crystals, nonlinear beam shaping and control– as well as electron microscopy and plasmonics. He is theco-author of more than 130 publications in peer-reviewed sci-entific journals and a Fellow of the Optical Society of America.

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adiabatic processes in frequency conversion2.1. the nonlinear coupled wave equations in linear...

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