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824 J. Opt. Soc. Am. B/Vol. 27, No. 4 /April 2010 Charbonneau-Lefort et al.

Theory and simulation of gain-guidednoncollinear modes in chirped

quasi-phase-matched optical parametric amplifiers

Mathieu Charbonneau-Lefort,1,* Bedros Afeyan,2 and M. M. Fejer1

1E. L. Ginzton Laboratory, Stanford University, Stanford, California 94305, USA2Polymath Research Inc., Pleasanton, California 94566, USA

*Corresponding author: [email protected]

Received October 19, 2009; accepted December 20, 2009;posted January 21, 2010 (Doc. ID 117468); published March 31, 2010

Chirped quasi-phase-matched (QPM) gratings offer essentially constant gain over wide bandwidths, makingthem promising candidates for short-pulse optical parametric amplifiers. However, we discovered that high-gain noncollinear processes can compete with the desired broadband gain of such amplifiers. Here, we inves-tigate these noncollinear gain-guided modes both numerically and analytically, including longitudinal nonuni-formity of the phase-matching profile, lateral localization of the pump beam, and the noncollinear propagationof the interacting waves. © 2010 Optical Society of America

OCIS codes: 140.4480, 190.4970, 190.4410, 320.1590.

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. INTRODUCTIONn this paper, we consider optical parametric amplifica-ion in quadratic nonlinear media. This process is widelysed to amplify optical pulses in high-power ultrafast la-er systems [1–3]. Unlike laser amplifiers, which provideain at a wavelength determined by an atomic transition,ptical parametric amplifiers (OPAs) enable gain at anyesired wavelength and offer large single-pass gains andeduced thermal effects [4]. Recently, OPAs using chirpeduasi-phase-matching (QPM) gratings have been pro-osed as a means to achieve constant gain over a broadandwidth [5–8].QPM is a technique that involves microstructuring the

onlinear optical material in order to effectively phase-atch an interaction at a desired wavelength [9]. In opti-

al materials with ��2� nonlinearity, such as ferroelectrics,t is most commonly implemented by periodically invert-ng the sign of the nonlinear susceptibility. It is wellnown that uniform QPM gratings provide parametricain, which increases exponentially with the gratingength [10]. Their bandwidth is determined by the disper-ive properties of the material. In the case of a nondegen-rate interaction (with different signal and idler wave-engths), the amplification bandwidth is typicallyelatively narrow (on the order of a few nanometers).

Chirped, or nonuniform, QPM gratings offer widerandwidths. However, because each wavelength is ampli-ed only in the vicinity of the perfectly phase-matchedoint (PPMP), the cumulative gain of a chirped QPMrating is always lower than that of a uniform grating ofhe same length. Nevertheless, having constant gain over

wide bandwidth is a major advantage for applicationsuch as femtosecond-pulse amplification.

At present, broadband OPAs are achieved by operatingear degeneracy (with nearly equal signal and idler wave-

engths) [3] or using a noncollinear geometry [11]. The use

0740-3224/10/040824-18/$15.00 © 2

f chirped QPM gratings, investigated previously in [5–8],s an alternative approach. The advantages of this tech-ique are an inherently unrestricted bandwidth (deter-ined simply by the chirp rate and length of the QPM

rating) and a simple collinear geometry.Optical parametric amplification using chirped QPM

ratings are an example of parametric amplification inonuniform phase-matched media. The same phenomenaccur in laser–plasma interactions where they were firsttudied [12,13]. The theoretical model is made up of a pairf coupled differential equations involving only the longi-udinal dimension, called the Rosenbluth model. It wasound that, in linearly inhomogeneous media, the ampli-cation is finite and occurs over a distance that scales as1/2, where L is the scale length of the inhomogeneity. In

he Rosenbluth model, the medium nonuniformity intro-uces dephasing, which limits the amount of amplifica-ion to a small region around the PPMP.

Experiments with chirped QPM gratings [14] revealedhe existence of high-gain parametric processes. They pro-ided evidence that the 1D Rosenbluth model does not ad-quately describe the interaction of real beams in crys-als. The discrepancy between the experimentalbservations and the theoretical predictions triggered ournvestigation of transverse (non-1D) effects, includingoncollinear propagation, diffraction, and lateral localiza-ion of the pump beam, and lead to the discovery thatain-guided modes can exist in spite of the dephasingaused by the axial nonuniformity of the medium.

This paper is a theoretical study of gain-guided modesn nonuniformly phase-matched media. Because of the ap-lication in mind, the subject will be approached from theoint of view of chirped quasi-phase-matched opticalarametric amplifiers (chirped QPM OPAs).The novelty of the model presented here lies in the

ombined treatment of a laterally localized pump and a

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Charbonneau-Lefort et al. Vol. 27, No. 4 /April 2010 /J. Opt. Soc. Am. B 825

onuniform phase-matching profile in a nearly copropa-ating geometry. Pump localization and dephasing areell-known limits, which have been studied separately.he initial-value problem where neither is present haseen solved by Bobroff and Haus [15]. In the presence ofephasing but no pump localization, we recover theosenbluth model [12,13]. The opposite limit, that of a lo-

alized pump with no dephasing, has been solved by Sus-chik [16], who showed that gain-guided modes can exist

n a noncollinear (near-forward) geometry.This paper is organized as follows. We begin by a deri-

ation of the coupled-mode equations describing theropagation of noncollinear waves interacting in a non-niform phase-matching medium with a narrow-widthump beam, including diffraction. This general model ishen investigated with increasing levels of complexity. Af-er reviewing briefly the 1D case in Section 3, we examineoncollinear interactions in uniform media without dif-raction in Section 4. The gain-guided modes that exist inhis case are studied numerically and analytically. The ef-ect of diffraction is included in Section 5. The case of theinearly nonuniform medium is investigated in Sections 6nd 7, first without, then with diffraction. Findings aboutoncollinear gain-guided modes are summarized in Sec-ion 8.

. NORMALIZED COUPLED-MODEQUATIONS. Coupled-Mode Equations for Noncollineararametric Amplificationhe derivation of the equations describing optical para-etric amplification including noncollinear propagation

nd diffraction is very similar to that in one dimension10,18].

Starting from the scalar wave equation, we decomposehe electric field into three waves: pump, signal and idler,t frequencies �0,1,2, satisfying the frequency-matchingondition �0=�1+�2. Each one satisfies a wave equationriven by the nonlinear polarization resulting from theoupling between the waves:

�2Ej −1

c2

�2Ej

�t2 = �0

�2Pj

�t2 , �1�

ith j=0,1,2 for the pump, signal, and idler, respectively.ere, Ej and Pj represent the physical electric fields andolarizations, respectively.Each polarization can be separated into its linear and

onlinear response. The linear part is responsible for theefractive index, through the linear susceptibility ��1�.he nonlinear component, PNL,j, is proportional to theonlinear coefficient of the material.In the case of QPM gratings, the nonlinear coefficient isodulated periodically. Among the multiple Fourier com-

onents contained in the profile, we assume that one ofhem (typically, the fundamental) enables wave-vectoratching between the three waves, thus leading to non-

inear polarizations of significant amplitude. Denoting byj the wave-vectors of the pump, signal, and idler waves,nd by K that of the QPM grating, the wave-vector
g
atching condition is written k0=k1+k2+Kg. We denotey deff the effective nonlinear coefficient for this interac-ion (i.e., the amplitude of the Fourier component Kg).herefore the nonlinear polarizations contributing to sig-ificant coupling are

PNL,0 = 2�0deffeiKg·rE1E2, �2�

PNL,1 = 2�0deffe−iKg·rE0E2

* , �3�

PNL,2 = 2�0deffe−iKg·rE0E1

* . �4�

We extract the fast carrier phases:

Ej = Ejei�kj·r−�jt�, �5�

PNL,j = PNL,jei�kj·r−�jt�, �6�

here Ej�r� and PNL,j�r� are the field envelopes. Substi-uting into the wave equation, and using the dispersionelation �j=ckj /nj, where nj= �1+�j

�1��1/2 is the refractivendex, we obtain

�2Ej + 2ikj · �Ej = − �0�j2PNL,j. �7�

e assume that the wave vectors kj form small anglesith respect to the axis of propagation (z axis). Then weake the slowly varying envelope approximation, drop-

ing the second derivative of the envelope with respect to. Considering only one transverse dimension, x, the wavequation describing noncollinear propagation includingiffraction is

�Ej

�z+

kjx

kjz

�Ej

�x−

i

2kjz

�2Ej

�x2 = i�0�j

2

2kjzPNL,j. �8�

e assume that the pump wave propagates along the zxis, remains undepleted, is focused with a long f-numberens, and can be approximated as being longitudinally in-ariant, with the transverse profile A0�x�:

E0 = E0A0�x�, �9�

here E0 is the peak electric field of the pump beam. Weormalize the signal and idler fields with respect to theirhoton numbers:

E1,2 =� �1,2

n1,2 cos �1,2A1,2, �10�

here �1 and �2 are the angles of propagation of the sig-al and idler waves with respect to the z axis, defined byan �1=k1x /k1z and tan �2=k2x /k2z, respectively. Thesengles are related to each other by the requirement ofoncollinear phase-matching, developed below.Finally, we obtain the following coupled-mode equa-

ions for the spatial evolution of the signal and idleraves:

�A1

�z+ tan �1

�A1

�x−

i

2k1 cos �1

�2A1

�x2 = i�0A0�x�A2*ei�·r,

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�A2*

�z+ tan �2

�A2*

�x+

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2k2 cos �2

�2A2*

�x2 = − i�0A0*�x�A1e−i�·r,

�12�

here

�0 =� �1�2

n1n2 cos �1 cos �2

deff

cE0 �13�

s the coupling coefficient, and �=k0−k1−k2−Kg is theollinear wave-vector mismatch (also labeled by �k in theptics literature). In this work we assume that both theump and grating wave vectors lie along the z axis, andllow the signal and idler wave vectors to propagate non-ollinearly while preserving phase-matching (see Subsec-ion 2.B below). The phase mismatch appearing in Eqs.11) and (12) is then simply ��z. In the case of a non-niform QPM grating, the accumulated phase mismatch,z, must be replaced by �z�=�z��z��dz�.

. Noncollinear Phase-Matchingn the derivation of the coupled-mode equations, Eqs. (11)nd (12), it was assumed that the carrier wave vectors ofhe three interacting waves satisfy the phase-matchingondition k0−k1−k2−Kg=0. In a uniform QPM grating,his condition can be satisfied throughout the crystal by aroper choice of angles. (Throughout this discussion, wessume that the wavelengths of the pump, signal, anddler waves are fixed and unequal.) In a nonuniform grat-ng, where Kg is a function of position, a given set of waveectors k0,1,2 can only satisfy the matching condition at aingle location, called the perfectly phase-matched pointPPMP), zpm. The vectorial phase-matching condition,alid at that location, can be rewritten as

k1 sin �1 + k2 sin �2 = 0, �14�

k1 cos �1 + k2 cos �2 = kp − Kg�zpm�. �15�

he transverse component yields a relation between theignal and idler noncollinear angles:

sin �2 = −k1

k2sin �1. �16�

n the limit of small angles, Eq. (16) becomes �2��k1 /k2��1. The ratio between the phase-matching angles

s determined by the ratio of the wave numbers. In thease of degenerate interactions �k1=k2�, �1 and �2 must bequal.

In a chirped QPM grating, the phase-matching anglesre a function of position. From Eq. (15) in the small-ngle approximation, we obtain the angle �1 at the PPMP:

�1�zpm� ��− 2�k0 − k1 − k2 − Kg�zpm��

k1�1 + k1/k2�. �17�

way from zpm, the wave vector mismatch ��z�=k0−k1zk2z−Kg�z� is nonzero. Assuming a linear QPM gratingrofile, it can be written as

��z� = ��z − zpm�, �18�

here �� is the dephasing rate, or chirp rate of the QPMrating ��=−dKg /dz�. The incidence angle ensuringhase-matching at any location along the grating is

�1,pm�z� =��12�zpm� −

2��z − zpm�

k1�1 + k1/k2�. �19�

hus, in a linearly varying phase-matching medium thehase-matching angle varies as the square root of posi-ion.

. Normalized Equationse return to the coupled-mode equations, Eqs. (11) and

12). We normalize the transverse dimensions with re-pect to the half-width of the pump beam, w0:

x �x

w0. �20�

ach wave overlaps with the pump over a length0/tan �1,2. The geometric mean of these signal and idler

verlap lengths defines the pump width traversal length:

Lpwt �w0

�tan �1 tan �2

. �21�

ince the carrier wave vectors are chosen to obey theave-vector matching condition, �1 and �2 are related byq. (16). Therefore, in the small-angle limit,

Lpwt �w0

�1�k12

, �22�

here we have introduced

k12 �k1

k2. �23�

e use this pump-width traversal length to normalize theongitudinal dimension:

z �z

Lpwt. �24�

he normalized coupled-mode equations are

�A1

�z+ 1

�A1

�x− i�1

�2A1

�x2= i�1/2A0�x�A2

*ei�z�, �25�

�A2*

�z+ 2

�A2*

�x+ i�2

�2A2*

�x2= − i�1/2A0

*�x�A1e−i�z�, �26�

ith

1 =� tan �1

tan �2 �

1

�k12

, �27�

= −� tan �2� − �k , �28�
2 tan �1 12

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�j =Lpwt

2kj cos �jw02 �

Lpwt

Ldiff,j, �29�

� = ��0Lpwt�2. �30�

he parameters �j are proportional to the ratio of theump-width traversal length to the diffraction lengthsdiff,j=2kjw0

2. Note that the parameters 2 and �2 arexed once 1 and �1 have been specified, since 12=−1nd �2= k12�1.In normalized units, a linearly chirped grating profile

s given by

��z� = ��z − zpm�, �31�

here

�� Lpwt2 �32�

s the normalized chirp rate and zpm is the PPMP. The ac-umulated phase mismatch is then

�z� =��

2��z − zpm�2 − �z0 − zpm�2�, �33�

here z0 is the position of the input plane.The normalized units introduced in this section are

onvenient for the study of noncollinear narrow trans-erse pump-width interactions. This necessitates the in-orporation of the angle into the normalization. In prac-ice, it will be useful to plot the growth rate K of theignal and idler waves as a function of angle. In normal-zed units the growth rate is K=LpwtK. The physicalrowth rate can be extracted, and its dependence on anglehown, by plotting K /��=K /�0 as a function of 1/��

1/�0Lpwt�tan �1�k12/�0w0.

. REVIEW OF THE 1D MODELhe 1D model corresponds to the case of collinear geom-try ��1=�2=0� in the absence diffraction. The coupled-ode equations in this case are

dA1

dz= i�0A2

*ei�z�, �34�

dA2*

dz= − i�0A1e−i�z�, �35�

ith

�z� =z0

z

��z��dz�. �36�

. Uniform Mediumet us first consider the simple case of a uniform medium

n which the interaction is perfectly phase-matched overhe entire length of the device (i.e., =0). Assuming thatnly the signal wave is present initially (i.e., adopting theoundary conditions A1�z0�=1, A2�z0�=0), the solutionsre

A1�z� = cosh��0z�, �37�

A2�z� = sinh��0z�. �38�

he waves grow exponentially in space at a growth rate of0. In the large-gain regime, the overall gain is G

12 exp �0L, where L is the length of the grating. It will be

seful for subsequent discussion to define the uniform-edium gain length,

Lg =1

�0. �39�

e note that, using this definition, the normalized gainarameter � defined in Eq. (30) can be rewritten as ��Lpwt /Lg�2.

. Nonuniform Medium (Rosenbluth Model)he case of two coupled plane waves in 1D, interactingollinearly in an axially nonuniformly phase-matchedrystal, is called the Rosenbluth model [12]. We considernly the linear profile, given by

��z� = ��z − zpm�. �40�

ere, �� is the dephasing rate and zpm is the location ofhe PPMP. Combining Eqs. (34) and (35), and using thehange of variables Aj=ajei/2, we obtain

d2

dz2aj + ��z�

2 2

+i��

2− �0

2�aj = 0. �41�

pproximate solutions can be obtained using WKB analy-is [7,12]. Alternatively, exact solutions exist in the case ofhe linear profile in terms of parabolic cylinder functions8,13]. In any case, solutions of Eq. (41) are oscillatoryhen ��z�� 2�0 and exponential when ��z��2�0. Am-lification is restricted to the vicinity of zpm, where the so-utions have exponential character. The dephasing lengthefines the limits of the amplification region, beyondhich the amplification stops due to excessive dephasing.

n the case of the linear profile given by Eq. (40), theength of the amplification region is given by

Ldeph =2�0

��. �42�

he amplification region extends from zpm−Ldeph to zpmLdeph. The gain is essentially given by �� /2�2

�02�1/2dz, integrated over the amplification region (i.e.,

etween the turning points). In the case of the linear pro-le, the amplification of the signal amplitude is given byhe Rosenbluth gain formula [12]:

G = exp��02

�� . �43�

t is useful to define the Rosenbluth gain parameter:

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hen the Rosenbluth gain formula for the intensity can beewritten as G=exp�2��R�. The dephasing length, Eq.42), can be expressed as

Ldeph =2��R

��. �45�

or fixed amplification (i.e., fixed �R), the interactionength scales as 1/��1/2.

The main feature of the nonuniform phase-matchingodel is that the amplification is finite and localized

round the phase-matched point. This behavior must beontrasted with that in a uniform medium, where theaves grow exponentially with the length of the medium.

. GAIN-GUIDED NONCOLLINEAR MODESN UNIFORM MEDIAe begin our study of transverse effects by considering

oncollinear propagation in the presence of a laterally lo-alized pump beam. We will look at various cases of in-reasing complexity. In this section we consider the casef a uniform medium without diffraction. We derive theroperties of gain-guided noncollinear modes discoveredy Sushchik [16].The equations under consideration in this section are

qs. (25) and (26), with �1=�2=0 (no diffraction) and ��0 (no chirp):

�A1

�z+ 1

�A1

�x= i�1/2A0�x�A2

* , �46�

�A2*

�z+ 2

�A2*

�x= − i�1/2A0

*�x�A1. �47�

e also recall that 2=−1/1.

. Numerical Simulationsn this section, we solve Eqs. (46) and (47) numerically us-ng the operator splitting technique. We use an input athe signal wave only. The coupling coefficient � is turnedn adiabatically to suppress long-lasting transient effects.e consider a gaussian pump transverse profile:

A0�x� = exp�− x2�. �48�

. Detailed Example: 1=1, �=4e start by exploring the case �=4, 1=1 (and, therefore,

2=−1) in detail.Figure 1(a) shows 2D plots of the amplitude of the sig-

al and idler beams on a logarithmic scale. The signalnd idler waves are amplified along the direction of theump beam.Figure 1(b) shows the peak amplitude along the propa-

ation direction. The waves grow exponentially (increas-ng linearly on a logarithmic scale). The signal and idler
xperience the same growth rate. The transverse beam c
rofiles at the end of the simulation �z=15� are shown inig. 1. The signal wave is purely real, while the idlerave is purely imaginary.In order to examine the beam shape in more detail, we

ake transverse cuts at various positions along the propa-ation direction. These are shown in Fig. 2. Clearly, afterome distance, the shape of the amplified wave is inde-endent of position. The amplified waves are gain-guidedodes localized inside the pump beam, as predicted byushchik [16]. In the present case, the signal and idlerodes are the mirror image of each other because we are

ig. 1. Numerical results for a gain parameter �=4 and the de-enerate case 1=1, 2=−1 (i.e. k1=k2): (a) 2-D plots of the am-litude of the signal and idler beams, A1 and A2 (logarithmiccale); (b) Peak amplitude; (c) Mode profile at position z=15.

onsidering the degenerate case 1=1, 2=−1.

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. Growth Raten the example shown above we saw that the signal anddler grow exponentially at the same rate. In this subsec-ion we study how the growth rate depends on � and 1.

Figure 3(a) shows how the growth rate depends on theain parameter �, with the degeneracy parameter 1=1xed. There is a threshold below which localized growingodes do not exist. This threshold depends on the pump

ig. 2. Signal and idler mode shape at various positions alonghe direction of propagation.

ig. 3. Growth rate, K, as a function of (a) the gain parameter �n the degenerate case 1=1, 2=−1, and (b) the asymmetry pa-ameter for �=2 and �=4.
1
rofile. Although very interesting, the behavior close tohreshold will not be explored in this paper, other than toote that the threshold lies close to �=1, which is the con-ition where the gain length and pump-width traverseength are the same size—a regime where variational

ethods are more aptly suited [19].The effect of the degeneracy parameter 1 is shown in

ig. 3(b). The growth is maximum in the degenerate case1=1, 2=−1, and decreases away from degeneracy. Natu-ally, the growth rate remains the same when the roles ofhe signal and idler are interchanged, namely, upon theransformations 1→1/1 and 2→1/2.

. Conclusion from Numerical Investigationshe insight provided by the numerical simulations is that

he beam shape is invariant in z. The signal and idleraves are separable in x and z and can be described by

A1,2�z, x� = eKz�1,2�x�, �49�

here �1,2�x� is the mode shape. This form of the solutionill be used to carry out the analysis in the following sec-

ion.

. Analysis

. Equation for Bound Statesquations (46) and (47) can be combined following tech-iques and notation given in [19] to give

L1L2A1 − ��A0�x��2A1 − 2

d ln A0

dxL1A1 = 0, �50�

ith L1,2=� /�z+1,2� /�x. For now there is no need topecify the shape of the pump beam. The only require-ent is that it be of half-width 1. We also recall that

12=−1.We make the assumption that the solution A1�z , x� is

eparable in z and x. This assumption is motivated by theumerical simulations, which indicated that the beamrofile is invariant along the direction of propagation.Taking the Fourier transform in z of Eq. (50) (or,

quivalently, assuming exponential solutions in z) gives

d2A1

dx2− �d ln A0

dx− i�1 + 2�kz�dA1

dx+ �kz

2 + ��A0�x��2

− i2kz

d ln A0

dx �A1 = 0, �51�

here kz is the transform variable. The mode shape A1 isow a function of x only. We eliminate the first-order de-ivative by the substitution

A1�x� = A01/2�x�e−i/2�1+2�kzxa1�x�. �52�

he mode amplitude a1�x� obeys the second-order ordi-ary differential equation in standard form

d2a1

dx2+ Q�x�a1 = 0 �53�

ith

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Q�x� = ��A0�x��2 −1

4�d ln A0

dx− i�1 − 2�kz�2

+1

2

d2 ln A0

dx2.

�54�

quation (53) has the familiar form of a Schrödingerquation describing a wave function in a potential well. Inur case, the “well” is defined by the pump beam profilequared and the eigenvalue is kz.

In the case of a top-hat pump profile, Eq. (53) can beolved exactly: the solutions are exponentials, and boundtates can be found by matching the solution and its de-ivative at the edges of the pump. More generally, theound states can be found using WKB analysis. Enforcing

¯
ecay of the solution as x→ ±� leads to a quantization T
It

wma

t�=w

Tc�toTw

=r

ondition that determines the eigenvalue. The gain-uided modes found this way are the Sushchik modes16].

. Growth Rate for a Top-Hat Pump Profileet us consider the top-hat pump profile:

�A0�2 = �1, �x� � 1

0, �x� 1...� �55�

e introduce the growth rate parameter K=−ikz. Theoundary conditions are A1=0 at x=−1 and at x=+�since the signal wave can only propagate to the right).
he solutions of Eq. (51) satisfying these conditions are
A1�x� � �0, x � − 1

e1/2�1+2�Kx sin�� −1

4�1 − 2�2K2�x + 1�� , − 1 � x � 1

e−Kx/1, x 1�. �56�

e define K� 12 �1−2�K in order to simplify the notation.

nforcing continuity of the solution and its derivative at=1 determines the growth rate, which satisfies the fol-

owing transcendental equation:

cot 2�� − K2 = −K

�� − K2. �57�

e introduce

u = 2�� − K2. �58�

he quantization condition can be rewritten

tan�u −�

2 =2�� − �u/2�2

u. �59�

he graphical solution of this equation is shown in Fig. 4.s � is increased from 0, a new mode appears every time�1/2= �n+1/2��, for n=0,1,2, . . .. Therefore the thresholdf mode n is

ig. 4. Graphical solution of the eigenvalue condition tan�u� /2�=2��− �u /2�2 /u, Eq. (59), for �=10.

�th,n = ��n +1

2 �

2�2

. �60�

n the infinite pump-strength limit, i.e., �→�, the solu-ions are un= �n+1��, or

Kn =2

1 + 1/1

�� − �n + 1�2��/2�2, �61�

here we have made use of the fact that 2=−1/1. Allodes grow at a rate that asymptotes to 2�1/2 / �1+1/1�

s �→�.Let us convert these results into physical units. The

hreshold condition for the fundamental mode �n=0� isth,0= �� /4�2. We recall that �= ��0Lpwt�2, with Lpwtw0 / �tan �1 tan �2�1/2. The threshold condition can beritten

�0

��tan �1 tan �2�� 2w0 =

�

2. �62�

he term �0 / �tan �1 tan �2�1/2�� is the projection of theoupling coefficient onto the x direction. The factor 2w0L� is the full width of the pump beam. Therefore, the

hreshold condition simply states that ��L�=� /2; inther words, the transverse gain must be larger than � /2.his is identical to the instability threshold of a backwardave oscillator [17].The growth rate in physical units is given by K

K /Lpwt. In the strong pump limit (large �), the growthate of mode n is

K =2�0 �1 −

�n + 1�2�2

. �63�
n1 + 1/1 4�

3FaTOt

W

It

wT

saaag

Tfifbns

Ip

Wtg

T

W�E

a

F

wTcts(

p

i

Taa

4Tpnff

F�qglnt

F�ntn


. Growth Rate for More General Pump Profilesor more general pump profiles, the mode shape and thessociated growth rate can be found using WKB analysis.his method is described, for instance, in Bender andrszag [20] in the context of the bound states of a poten-

ial well.Inside the pump (i.e., between the turning points), theKB solutions of Eq. (53) are

a1 � exp ± ix

�Qdx. �64�

mposing the boundary conditions a1→0 at x→ ±� leadso the Bohr–Sommerfeld quantization condition [20,21]:

x1

x2

�Qdx = �n +1

2 �, �65�

here x1 and x2 are the turning points (zeros of Q�x�).his condition determines the eigenvalues kz.The quantization condition obtained from WKB analy-

is can be compared to the exact calculation in the case oftop-hat pump profile. In this case, the turning points

re located at the edges of the pump beam (x1=−1, x2=1)nd Q=�+ �1−2�2kz

2 /4. The quantization condition (65)ives the growth rate

Kn =2

1 + 1/1�� − �n +

1

2 2��

2 2

. �66�

his value tends to the exact solution, Eq. (61), in the in-nite pump-strength limit. (The discrepancy is due to theact that the WKB method is valid for highly excitedound states. Except for the parabolic profile, it shouldot be expected to give the exact solution for the groundtate [19].)

Let us now return to the WKB quantization condition.n order to proceed further analytically, we assume aarabolic pump profile:

�A0�2 � 1 − x2. �67�

e also use the approximation �d /dx�ln A0=−x. Applica-ion of the quantization condition, Eq. (65), gives the ei-envalues

kz = ±2i

1 − 2�1 +

1

4�� −

1

2−�� +

1

4�2n + 1�.

�68�

he growth rate of these modes is given by Kn= �Im�kz��.As noted above, the calculation of the growth rate usingKB analysis is accurate in the infinite pump-strength

�→�� limit. In this case, the growth rate obtained fromq. (68) becomes

Kn �2

1 + 1/1

�� − �2n + 1��, �69�

nd the threshold is

�th,n = �1 + 2n�2. �70�

igure 5 compares this expression for the growth rate

ith the values obtained from the numerical simulations.he infinite pump-strength expression is the one thataptures best the behavior close to threshold [19]. We notehat the infinite pump-strength approximation corre-ponds to dropping the terms involving �d /dx�ln A0 in Eq.50).

Figure 6 shows plots of the growth rate as a function of1. Except for a constant offset, Eq. (69) captures the de-endence on 1 accurately.The growth rate given above was expressed in normal-

zed units. In physical units, we have

Kn �2�0

1 + 1/1�1 −

2n + 1

��. �71�

he growth rate in the step-function and parabolic pumppproximations can both be found in [16], by Sushchiknd Freidman.

. Mode Shapehe growth rate was calculated assuming a parabolicump profile. For a more general pump profile A0�x�, theumerical value of the growth rate will be different. Theundamental mode �n=0� has the following approximateorm between the turning points:

ig. 5. Normalized growth rate, K, of the fundamental moden=0� vs �. This plot shows a comparison between the WKBuantization condition evaluated numerically in the case of aaussian, top-hat (“flat”), and parabolic pump profiles; the ana-ytical formula obtained assuming a parabolic pump in the infi-ite pump-strength limit, Eq. (69); and the numerical simula-ions using a gaussian pump.

ig. 6. Normalized growth rate, K, of the fundamental moden=0� vs 1, comparing the analytical formula, Eq. (69) with theumerical simulations. Note that the growth rate for 1 is equalo that for 1/1, corresponding to the exchange of the roles of sig-al and idler.

ft

Ppp

w

TdOt

5IDSnc

Aob

w

otfa


a1�x� � sin�x1

x

�Q�x��dx� + �� , �72�

or some phase shift �. Outside of the pump, away fromhe turning points, the WKB solutions are

a1 � exp ±x

�− Qdx � A0±1/2e�i/2�1−2�kzx, �x� � 1.

�73�

utting everything together, we obtain the following ex-ressions for the fundamental growing modes with ap-
roximate shapes:
A1�z, x� � �A0�x�eK�z−x/2� x � − 1

A01/2�x�eK�z+�1+2�x� sin�

x1

x

�Q�x�dx� + �� − 1 � x � 1

eK�z−x/1� x � 1

,� , �74�

lfii[eskd

i

w

wweapottwp

ith

K �2

1 + 1/1

�� − ��. �75�

he fact that A0 appears in the expression for x�−1 in-icates that the signal mode is almost zero in that region.n the other side, where x�1, the mode is constant along

he characteristics �= z− x /1.

. GAIN-GUIDED NONCOLLINEAR MODESN UNIFORM MEDIA, INCLUDINGIFFRACTIONtill considering the case of a uniform medium ��=0�, weow include the effect of diffraction ��1,2�0�. Theoupled-mode equations describing this case are

�A1

�z+ 1

�A1

�x− i�1

�2A1

�x2= i�1/2A0�x�A2

* , �76�

�A2*

�z+ 2

�A2*

�x+ i�2

�2A2*

�x= − i�1/2A0

*�x�A1. �77�

s usual, these equations can be combined to eliminatene of the waves. Ignoring the derivatives of the pumpeam profile, we obtain

L1L2A1 − ��A0�x��2A1 = 0, �78�

here L1,2=� /�z+1,2� /�x� i�1,2�2 /�x2.Equation (78) is a partial differential equation of sec-

nd order in z and fourth order in x. The method used inhe previous section (4), which consists in Fourier trans-orming in z and then finding bound states using WKBnalysis in x, cannot be employed in this case.

An alternative solution technique is to solve the prob-em in the Fourier domain. This approach was used in theeld of plasma physics, where the interaction is described

n x-space by typically complicated differential operators22]. Assuming a parabolic pump profile, Eq. (78) has co-fficients that are at most quadratic, which translates to aecond-order PDE in the Fourier domain. Working in-space allows a reduction from 4 to 2 of the order of theifferential equation.We consider a parabolic pump profile: �A0�2=1− x2. Tak-

ng Fourier transforms in z and x, Eq. (78) becomes

d2A1

dkx2

+ Q�kx�A1 = 0, �79�

ith the k-space “potential”

Q�kx� = 1 +1

��kz + 1kx + �1kx

2��kz + 2kx − �2kx2�, �80�

here kz and kx are the transform variables associatedith z and x, respectively. Here, kz plays the role of the

igenvalue and kx is the independent variable. Since Q isquartic potential, there are four turning points. Ap-

roximate solutions for the ground-state eigenvalues arebtained by solving Q�kx�=0 [this amounts to droppinghe factor of � /2 in the right-hand-side term in the quan-ization condition, Eq. (65)]. Then the most unstableaves (i.e., the eigenvalues kz with the largest imaginaryart) correspond to

U(w

cvvtti

atm+letlDittte

lsgcodr

arbz

6IWn�

Fg

ATFftfo

1Lfh

esTmt

btdTe

2

Fn(


kx* = �

0

− �1 − 2

�1 + �2 � . �81�

sing the definitions of the normalized parameters, Eqs.21) and (27)–(29), and the phase-matching condition (16),e can rewrite kx

* as

kx* = �0

− 2w0k1 sin �1� . �82�

The physical wave vector is recovered by adding thearrier wave vector k1, which was extracted when the en-elope functions were introduced in Eq. (5). The trans-erse component of the signal wave vector correspondingo maximum amplification, k1x

* =k1 sin �1+ kx* /w0, is found

o be k1x* = ±k1 sin �1. The angle of maximum amplification

s �1* = ±�1.

Let us comment on the significance of this result. In thebsence of diffraction, the signal wave is amplified alonghe incidence angle �1. When diffraction is included in aore realistic model, light is allowed to propagate in thex and −x directions and, as a consequence, two noncol-

inear gain-guided modes exist that are mirror images ofach other, with k-vectors along the +�1 and −�1 direc-ions. They correspond to the two ways in which noncol-inear phase-matching can be realized in two dimensions.ue to the symmetry of the problem, both modes have

dentical growth rates. Although the mode correspondingo +�1 is the only one initially excited by the input beam,he second mode can be seeded by diffraction or by spon-aneous emission, so that, in general, both modes will co-xist.

In three dimensions, assuming a pump beam with cy-indrical symmetry, the phase-matching angle �1 de-cribes a cone. There are, therefore, an infinity of gain-uided modes that have the same angle and,onsequently, the same growth rate. Because of the 3D ge-metry, the exact value of the growth rate will be slightlyifferent than in 2D, but the physics of Sushchik modesemains the same.

As long as diffraction remains small, the growth ratend mode shape of Sushchik modes obtained in Section 4emain approximately valid. The perturbation introducedy diffraction can be calculated by evaluating the quanti-ation condition and expanding the potential Q around kx

*.

. GAIN-GUIDED NONCOLLINEAR MODESN NONUNIFORM MEDIAe now turn our attention to the nonuniform medium, ig-

oring diffraction. We consider Eqs. (25) and (26) with1=�2=0:

�A1

�z+ 1

�A1

�x= i�1/2A0�x�A2

*ei�z�, �83�

�A2*

�z+ 2

�A2*

�x= − i�1/2A0

*�x�A1e−i�z�. �84�

or a linearly varying medium, the phase mismatch isiven by Eq. (33), repeated here:

�z� =��

2��z − zpm�2 − �z0 − zpm�2�. �85�

. Numerical Simulationshe numerical simulations are carried out as follows.irst, the waves interact in a uniform segment with per-

ect phase matching. Once gain-guided modes are excited,he nonuniformity is turned on adiabatically and the ef-ect of dephasing on the growth rate and mode shape isbserved.

. Noncollinear Modes in Nonuniform Mediaet us begin our numerical investigation of the nonuni-

orm medium by two examples of completely different be-avior.Figure 7 shows the evolution of the beams for param-

ters 1=1, �=3, and ��=6. The amplification ceases asoon as the nonuniformity is turned on, around z�13.his behavior is consistent with the 1D Rosenbluthodel, according to which the amplification is limited to

he phase-matching region.For different gain and chirp rate parameters, the insta-

ility can behave completely differently. Figure 8 showshe case 1=1, �=4, ��=4. The waves grow even whenephasing is present, although at a reduced growth rate.his shows that gain-guided noncollinear modes can alsoxist in nonuniform media.

. Detailed Example: 1=1, �=4, ��=4

ig. 7. Example of a case where amplification is suppressed in aon-uniform medium: (a) 2-D plots of the beam amplitudes andb) peak amplitudes for =1, �=3 and � =6.
1 �

Wogft

baps−atp8f

att

tfa

3Fgr�s�

dt

udFog1

B

1To(

wL

Fept

Fp�


e have seen in the previous section (6.A.1) that the setf parameters 1=1, �=4, and ��=4 corresponds to a re-ime where gain-guided noncollinear modes in nonuni-orm media can exist. Let us explore this case in more de-ail.

Figure 8 shows plots of the amplitude and phase of theeams. The 2D plot (a) indicates that the logarithm of themplitude is constant along the noncollinear direction ofropagation (i.e., along the characteristics defined by thetraight lines z− x /1 in the case of the signal and zx /2 in the case of the idler). Figure 8(b) shows the peakmplitude along the direction of propagation. The reduc-ion in the growth rate (around z=13) corresponds to theosition where the nonuniformity is turned on. Figure(c) shows the phase accumulation caused by the nonuni-ormity. The signal and idler phases at peak amplitude

ig. 8. Example of a case where noncollinear gain-guided modesxist in a non-uniform medium even though phase mismatch isresent: (a) 2-D plots of the beam amplitudes, (b) peak ampli-udes and (c) peak phases for 1=1, �=4 and ��=4.

re essentially equal to �z� /2. Finally, Fig. 9 showsransverse cuts of the mode shape and phase at the end ofhe simulation.

It is insightful to look at the k-space representation ofhe waves. Figure 10 shows 2D plots of the Fourier trans-orm in x. The field is shifted in kx-space, consistent withphase accumulation in x, which increases along z.

. Growth Rateigure 11 shows the growth rate, K, as a function of theain parameter �, for various values of the dephasingate. An increasing dephasing rate translates to a higher-threshold and a lower growth rate. The same figure alsohows the growth rate as a function of the dephasing rate�, for different values of �.It was mentioned in Subsection 2.C that the depen-

ence of growth rate on angle is best understood by plot-ing K /��=K /�0 (where K is the growth rate in physical

nits) as a function of 1/��=tan �1�k12/w0�0. This is

one in Fig. 12 for various values of the ratio �� /�=1/�R.rom this plot, it is easy to extract the range of anglesver which gain-guided modes exist, and the maximumrowth rate as a function of the chirp rate, shown in Fig.3.

. Nonuniform Medium: Analytical Results

. Equation for Bound Stateshe equation describing the evolution of the signal wave,btained from combining the coupled equations (83) and84), is

L1L2A1 − ��A0�x��2A1 − �i��z� + 2

d ln A0

dx L1A1 = 0,

�86�

here ��z�=d /dz is the phase-matching profile and1,2=� /�z+1,2� /�x.

ig. 9. Transverse cuts at the end of the simulation �z=25�. Thearameters used for this simulation are 1=1, 2=−1, �=4 and=4.
�

4rWts=

bsdlbsu

wp

AcTd

Fe

Fa

Fn=�

FmbTtgr


In the discussion of the uniform medium (Subsection.B), it was argued that it is valid to obtain the growthate and mode shape in the infinite pump-strength limit.e therefore drop the term involving �d /dx�ln A0 in order

o simplify the expressions. We will also limit our discus-ion to the linear phase-matching profile, given by ��z��z.The solution in the case of a uniform medium was

ased on the separation of variables. However, numericalimulations showed that, in the case of a nonuniform me-ium, this assumption is not valid. The phase, in particu-ar, is a nonseparable function of x and z and was found toe constant along the characteristics z− x /1,2. It is pos-ible to remove any dependence on z from the equation bysing the following change of variables:

ig. 10. 2-D plots of the Fourier transforms in x. The param-ters used for this simulation are 1=1, 2=−1, �=4 and ��=4.

ig. 11. Growth rate K as a function of (a) the gain parameter �,nd (b) chirp rate � , for the degenerate case =1.
� 1
A1�z, x� = B1�z, x�exp�i1

1 − 2�z − x/1�� , �87�

ith being the phase mismatch given by Eq. (85). Ap-lying operators L1 and L2, we get

L1L2A1 = L1L2B1 + i��z − x/1�L1B1. �88�

fter substitution into Eq. (86), we find that the coeffi-ient of L1B1 involves ��z− x /1�− ��z�=−��x /1=2��x.herefore, in the special case of a linear profile the depen-ence on z disappears from the equation:

ig. 12. Normalized growth rate, K /��=K /�0, as a function oformalized angle, 1/��=tan �1 /�0w0, for the degenerate case 11, 2=−1 and various values of the Rosenbluth gain parameter� /�=1/�R=�� /�0

2.

ig. 13. (a) Angle range for which gain-guided noncollinearodes exist, as a function of �� /�=1/�R, where �R is the Rosen-

luth gain parameter, and for the degenerate case 1=1, 2=−1.he solid lines show the maximum and minimum boundaries ofhis region, and the dashed line shows the angle of maximumrowth rate. (b) Maximum growth rate as a function of the chirpate.

Nz

m

W

T

w

Stt

2Tcacu

Ttp

I

wd

sptsmt

hgii

CTgcto

blltr

Tgl

twewilaaelo

tw

FltRg


L1L2B1 − ��A0�x��2B1 + i2��xL1B1 = 0. �89�

ow we can look for a solution that is separable in x and.

The rest of the derivation is similar to that of a uniformedium. We take the Fourier transform in z:

d2B1

dx2+ i��1 + 2�kz + ��x�

dB1

dx+ �kz

2 + ��A0�x��2 − i2��x�B1

= 0. �90�

e eliminate the first-order derivative using

B1�x� = b1�x�exp�−i

2��1 + 2�kzx +��

2x2�� . �91�

hen b1�x� satisfies a Shrödinger-like equation,

d2b1

dx2+ Q�x�b1�x� = 0, �92�

ith the “potential”

Q�x� = ��A0�x��2 +1

4��1 − 2�kz + ��x�2 −

i

2��. �93�

ince the curvature of the term ��2x2 is positive whilehat of ��A0�2 is negative, the dephasing rate decreaseshe “strength” of the potential.

. Growth Ratehe calculation of the eigenvalues kz is similar to thatarried out in the case of the uniform medium. We assume

parabolic pump profile �A0�2=1− x2. The quantizationondition, �x1

x2Q1/2dx= �n+1/2��, gives rise to the eigenval-es

kz =2i

1 + 1/1�1 −

��2

4�� −

i��

2− �2n + 1��1 −

��2

4�.

�94�

he imaginary part of the kz gives the growth rate. Forhe fundamental mode, the growth rate in the infiniteump-strength limit is approximately

K =2

1 + 1/1�1 −

��2

4�� − ��1 −

��2

4�. �95�

n physical units, K=K /Lpwt, so we obtain

K =2�0

1 + 1/1�1 −

�

4�R2�1 −

1

��1 −

�

4�R2 , �96�

here �R=�2 / ��=�02 /�� is the Rosenbluth gain parameter

efined in Eq. (44).As discussed in Subsection 2.C, a useful way of repre-

enting the angular dependence of the growth rate is tolot K /�=K /�0 as a function of tan �1 /�0w0, for a fixed ra-io of �� /�=�� /�0

2=1/�R. This plot is shown in Fig. 14. Ithould be compared with the growth rate obtained nu-erically (see Fig. 12). Although the actual values differ,

he trends are similar. One aspect in which they disagree,

owever, is that the analytical formula suggests that theain of chirped gratings can exceed that of uniform grat-ngs at large angles. Numerical results (Fig. 12) show thatt is not the case.

. Physical Interpretationhe most important feature of Fig. 14 is the fact thatain-guided modes are allowed for sufficiently large non-ollinear angles, in spite of the dephasing introduced byhe nonuniformity of the medium. This is a consequencef the lateral localization of the pump beam.

A physical interpretation of this result can be obtainedy expressing the growth rate in terms of characteristicengths. The gain, dephasing, and pump-width traversalengths were defined in Eqs. (39), (42), and (21), respec-ively. In terms of these characteristic lengths, the growthate from Eq. (96) becomes

K

�0=

2

1 + 1/1�1 − � Lpwt

Ldeph 2�1 −

Lg

Lpwt�1 − � Lpwt

Ldeph 2

.

�97�

he growth rate is real if Lpwt�Ldepth. In other words,ain-guided modes exist if the pump-width traversalength is shorter than the dephasing length.

Let us look at the various regimes successively. If LpwtLdeph, the pump-width traversal length is longer than

he dephasing length (i.e., the angle is small) and theaves remain in the interaction region sufficiently long toxperience dephasing. In this case, the growth of theaves is suppressed. If the pump-width traversal length

s shorter than the dephasing length (i.e., if the noncol-inear angle is large enough), the waves leave the inter-ction region before experiencing significant dephasingnd, as a consequence, gain-guided modes can exist. How-ver, if the angle is increased further and becomes tooarge, the gain is low and the instability is below thresh-ld. Figure 15 illustrates these three cases.

In the absence of lateral pump localization, �w0→��,he pump-width traversal length becomes infinite (theaves never escape the interaction region). In this case,

ig. 14. Angular dependence of the growth rate obtained ana-ytically, Eq. (96). This plot shows K /�=K /�0 as a function ofan �1 /�0w0, for fixed ratios of �� /�=�� /�0

2=1/�R (where �R is theosenbluth gain parameter). This plot was obtained for the de-enerate case k12=1, i.e. 1=1, 2=−1.

tLa

SccSmwwtmSmiwwib

7IDTne

aicm

ntdm

ADnotr�atl

m1sTo

BWpswp

c

wWl

FnnWctg

Ff=


he absolute instability cannot exist in nonuniform media.ateral pump localization is crucial to the existence of anbsolute instability.The results of this section bridge the gap between the

ushchik modes and the Rosenbluth model. Each caseorresponds to a different limit of the problem: lateral lo-alization of the pump but uniform medium in the case ofushchik’s model, and nonlocalized pump but nonuniformedium in the case of Rosenbluth’s. In the present paper,e address the case where both are present. We find thathen the collinear angle is small enough, the amplifica-

ion stops because of dephasing, as in the Rosenbluthodel. However, when the angle is large enough,ushchik-like modes can exist in spite of the nonunifor-ity. In other words, the instability, which is convective

n the case of the Rosenbluth model, can become absolutehen the pump is confined laterally. A similar conclusionas reached in the case of axial localization of the pump

n earlier work in the plasma physics literature forackward-wave oscillators [23–25].

. GAIN-GUIDED NONCOLLINEAR MODESN NONUNIFORM MEDIA, INCLUDINGIFFRACTIONhe equations describing the noncollinear interaction inonuniform media, including diffraction, were given byxpressions (25) and (26).

We saw in Section 5 that two gain-guided modes, whichre mirror images of each other, exist when diffraction isncluded in the model, corresponding to the two inter-hangeable ways of achieving phase-matching in two di-ensions. Not surprisingly, this remains true when chirp

ig. 15. Physical picture of gain-guided noncollinear modes inon-uniform media. (a) When Lpwt Ldeph, gain-guided modes doot exist because they are suppressed by the dephasing. (b)hen Lpwt�Ldeph, gain-guided modes exist because the waves es-

ape the pump before experiencing dephasing. (c) However, whenhe angle is too large, gain-guided modes do not exist because theain is too low.

onuniformity is introduced. However, positive and nega-ive dephasing rates behave differently in the presence ofiffraction. This was discovered numerically and will beotivated analytically in this section.

. Numerical Simulationsiffraction breaks the degeneracy between positive andegative chirp rates. Figure 16 shows the peak amplitudef a noncollinear gain-guided mode as a function of posi-ion, obtained with equal but opposite chirp rates. The pa-ameters in this example are 1=0.7, 2=−1.4, �2=2�1,=5, and ��= ±3. In the case of positive chirp rate, themplification length is finite. When the chirp rate is nega-ive, the growth rate is reduced, but the amplificationength is infinite just as in the absence of diffraction.

As mentioned previously, two noncollinear gain-guidedodes are supported when diffraction is included. Figure

7 shows the signal and idler beam amplitudes in realpace and in k-space, in the case of negative chirp rate.he two branches correspond to noncollinear modes withpposite transverse wave vector.

. Analysishen both medium nonuniformity and diffraction are

resent, it is no longer possible to obtain a solution that iseparable in x and z. A more careful look at this problemill be published elsewhere; here we simply show whyositive and negative chirp rates behave differently.The two coupled-mode equations (25) and (26), once

ombined, become

L1L2A1 − ��A0�x��2A1 − i��z�L1A1 = 0, �98�

ith L1,2=� /�z+1,2� /�x� i�1,2�2 /�x2 and ��z�= ��z− zpm�.e simplify the differential operators by solving the prob-

em in the Fourier domain, assuming a slow-varying

ig. 16. Effect of diffraction on noncollinear gain-guided modes,or positive and negative chirp rates. The parameters are 10.7, 2=−1.4, �=5. The chirp rate is (a) ��=3 and (b) ��=−3.

�b

w

AQw

T=

Tm(Fpds

it

Fg2t�

Ft


-profile and a parabolic pump beam. Equation (98) thenecomes

d2A1

dkx2

+ Q�kx�A1 = 0, �99�

ith the k-space “potential”

Q�kx� = 1 +1

��kz + 1kx + �1kx

2��kz + 2kx − �2kx2� − ��z��kz

+ 1kx + �1kx2��. �100�

s before, we find approximate eigenvalues by solvingˆ �kx�=0. The kx values associated with the most unstableaves are

ig. 17. (a) Beam amplitude and (b) their spectral content inhe case of negative chirp rate. The parameters in this case are1=0.7, 2=−1.4, �=10 and ��=−2.

kx* = −

1

2�1 − 2

�1 + �2 ±�1

4�1 − 2

�1 + �2 2

−��z�

�1 + �2.

�101�

he corresponding physical transverse wave vectors, k1x*

k1 sin �1+ kx* /w0, are

k1x* = ± k1��1

2 −2��z�

1 + k1/k2. �102�

his is precisely the value ensuring noncollinear phase-atching, k1x

* � ±k1�1,pm�z�, with �1,pm�z� given by Eq.19). The two gain-guided modes observed numerically inig. 17 correspond to two symmetric noncollinearlyhase-matched interactions. The phase of these modes isescribed by wave vectors that vary along z in order toatisfy the phase-matching condition locally.

This explains why the amplification behaves differentlyn the case of positive and negative chirp rate. If ��0,hen noncollinear phase-matching is possible for all zzpm; once excited at the PPMP, the modes can grow until

ig. 18. Amplification of noncollinear modes in a non-uniformain medium seeded collinearly, in the presence of diffraction.-D plots of the (a) amplitudes of the fields and (b) their Fourierransforms in x. The parameters are 1=2=0, �=4, ��=−2 and

=� =0.01.
1 2

t�wEg

wecnmm

CGWcaiSl

ptamtncdecd

ttc

gwp=ra

bgcnd

ps=r2ft

wnR

tit

FgTtrPs=

FeT


hey reach the end of the medium. On the other hand, if� 0, then phase-matching is possible only until theaves reach the collinear PPMP, where �1=0 [or k1x

* =0 inq. (102)]. Therefore, in the case of positive chirp rate, theain length of noncollinear gain-guided modes is finite.

The solution of Eqs. (98), describing the evolution of theaves with nonuniform phase-matching and in the pres-nce of diffraction, will be the object of a separate publi-ation, in which we will show how, when diffraction isegligible, the solution goes to that of the Rosenbluthodel when �� 0 and becomes gain-guided Sushchikodes when ��0.

. Parasitic Amplification in Negatively Chirped QPMratingsithout diffraction, noncollinear gain-guided modes in

hirped QPM gratings can only be excited if the incidencengle is large enough. At small angles, the amplifications adequately described by the Rosenbluth model (seeubsection 3.B). Consequently, an amplifier seeded col-

inearly will see the desired collinear gain.However, when diffraction is present, even collinear in-

ut beams can seed noncollinear gain-guided modes. Inhe presence of focused pump beams, the amplified signalnd idler waves amplified according to the Rosenbluthodel have a finite lateral extent and contain a range of

ransverse wave vectors. If the beam waist is sufficientlyarrow, large-angle spatial frequency components can ex-ite noncollinear gain-guided modes, which eventuallyominate the output. This effect is detrimental to the op-ration of amplifiers designed to take advantage of theollinear process, and it can limit the applicability of suchesigns. Thus, it is important to understand the competi-

ig. 19. Amplification of noncollinear modes in a non-uniformain medium seeded collinearly, in the presence of diffraction.he amplification initially ceases when the nonuniformity is

urned on, as predicted by the Rosenbluth model, but eventuallyesumes as noncollinear gain-guided modes are amplified. (a)eak amplitude and (b) trajectory of the peak amplitude in kxpace. The parameters are 1=2=0, �=4, ��=−2 and �1=�20.01.

ion between these two processes in more detail, so thathe possibility of designs evading the noncollinear pro-esses can be evaluated.

This mechanism of parasitic amplification was investi-ated numerically. At near-collinear angles, the pump-idth traversal length, Lpwt, becomes very large and thearameters �1, �2, �, and �� are large compared to 1 and2. Therefore, in these numerical simulations we set 12=0 in order to use reasonable values for the other pa-ameters. We also limit our investigation to the degener-te case �1=�2.Figures 18 and 19 show a collinear input beam, which,

ecause of diffraction, acts as a seed for noncollinear gain-uided modes. Shortly after the PPMP, the amplificationeases, as predicted by the Rosenbluth model. However,oncollinear gain-guided modes quickly emerge andominate.We call threshold length the distance required for the

arasitic noncollinear modes to emerge. Figure 20(a)hows a plot of the threshold length multiplied by ��1�2, as a function of the dephasing rate. The curves cor-esponding to different values of � are similar. Figure0(b) shows the dependence of the threshold length on �or �� and � fixed. From the numerical results, we deducehat the threshold length Lth is well approximated by

Lth �1

8��

� 2

, �103�

here �=�1=�2. As �→0, Lth→�, and growing modes doot appear, and we recover the results from the simpleosenbluth model.The threshold length given in Eq. (103) is expressed in

erms of the dimensionless parameters of the model. Us-ng the normalized variables defined in Subsection 2.C,he threshold length in physical units is

ig. 20. (a) Product of threshold length and diffraction param-ter, �=�1=�2, as a function of the dephasing rate, for �=4. (b)hreshold length Lth vs �, for fixed �=�1=�2=0.01 and ��=−2.

welde

8GLp-panTdiwatw

napa

FmwptfiTpi

9Tmpapa

tmgtaaws

Qfitgts

ATS0tiss(S

R

Fut(pmpcl(


Lth,physical =k1w0

2

4�R2 =

Ldiff,1

8�R2 , �104�

here, as usual, �R=�02 /�� is the Rosenbluth gain param-

ter defined in Eq. (44) and Ldiff,1=2k1w02 is the diffraction

ength. This expression, developed numerically, is valid ategeneracy (i.e., for �1=�2). Further work is required toxtend these results to the nondegenerate case.

. SUMMARY OF NONCOLLINEAR GAIN-UIDED MODES

et us now summarize our investigation of noncollineararametric interactions in the presence of a laterallylocalized pump beam. Various cases of increasing com-lexity have been considered. The simplest case is that ofuniform medium, neglecting diffraction. Gain-guided

oncollinear modes, or Sushchik modes, exist in this case.heir growth rate is largest in the collinear direction andecreases with increasing angle. If diffraction is takennto account, these modes behave in essentially the sameay, except that two degenerate modes propagating atngles ±�1 coexist. (The second of these modes is effec-ively dropped when the second derivative associatedith diffraction is omitted.)

ig. 21. Summary of noncollinear gain-guided modes in non-niform phase-matched media. The parabolas describe the spa-ial dependence of the phase-matching angle �1,pm�z� given by Eq.19). A signal wave with incidence angle �1 is phase-matched atosition zpm. If this angle lies in the proper range, a gain-guidedode with growth rate K��1� can be excited. If the chirp rate is

ositive, the gain length is limited to the range over which non-ollinear phase-matching is possible (i.e. from zpm to zpm0, the col-inear PPMP). If it is negative, the mode can grow indefinitelyi.e. until it reaches the limit of the medium).

When nonuniformity is included but diffraction is ig-ored, noncollinear gain-guided modes survive only if thengle is large enough. For smaller angles, dephasing sup-resses the existence of gain-guided modes, and the inter-ction is well described by the 1D Rosenbluth model.Including diffraction modifies this picture in two ways.

irst, the two gain-guided modes corresponding to phase-atching angles ±�1,pm�z� are excited simultaneously. Theave vectors of these modes vary with position in order toreserve the phase-matching condition locally. Second,he gain length in the case of positive chirp rate becomesnite, while that of negative chirp rates remains infinite.his is due to the possibility of realizing noncollinearhase-matching in each case. A diagram illustrating thesedeas is shown in Fig. 21.

. CONCLUSIONhis paper is an investigation of noncollinear gain-guidedodes existing in chirped quasi-phase-matched optical

arametric amplifiers (chirped QPM OPAs). These modesrise when transverse effects, namely, noncollinearropagation and lateral localization of the pump beam,re taken into account.Since they can be amplified over the entire length of

he device, the gain of these noncollinear gain-guidedodes is often much larger than the desired collinear

ain. Seeded by spontaneous emission, they can give riseo an intense parametric fluorescence, which may becomeconcern in the design of chirped QPM OPAs. Even if the

mplifier is seeded collinearly, these modes can be excitedhen the pump beam is focused, which can lead to para-

itic amplification in negatively chirped QPM gratings.In order to preserve the desirable features of chirped

PM OPAs—namely, a controllable gain and phase pro-le over a wide bandwidth—it is necessary to suppresshe growth of gain-guided noncollinear modes. Designuidelines to that effect have been formulated [14]. Addi-ional promising approaches include engineering the la-er beam profiles.

CKNOWLEDGMENTShis work was sponsored by the U. S. Air Force Office ofcientific Research, under AFOSR grants F49620-02-1-240 and F49620-01-1-0428. M. C.-L. acknowledges addi-ional support from the Natural Sciences and Engineer-ng Research Council of Canada. The work of B. A. wasupported by the Laser-Plasma branch of the Naval Re-earch Laboratory (NRL) and by a Department of EnergyDOE) National Nuclear Security Administration (NNSA)tewardship Science Academic Alliances (SSAA) grant.

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