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Charbonneau-Lefort et al. Vol. 25, No. 4 /April 2008 /J. Opt. Soc. Am. B 463

Optical parametric amplifiers using chirped quasi-phase-matching gratings I: practical design

formulas

Mathieu Charbonneau-Lefort,1,* Bedros Afeyan,2 and M. M. Fejer2

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

*Corresponding author: [email protected]

Received August 20, 2007; revised January 15, 2008; accepted January 17, 2008;posted January 24, 2008 (Doc. ID 86421); published March 6, 2008

Optical parametric amplifiers using chirped quasi-phase-matching (QPM) gratings offer the possibility of en-gineering the gain and group delay spectra. We give practical formulas for the design of such amplifiers. Weconsider linearly chirped QPM gratings providing constant gain over a broad bandwidth, sinusoidally modu-lated profiles for selective frequency amplification and a pair of QPM gratings working in tandem to ensureconstant gain and constant group delay at the same time across the spectrum. The analysis is carried out inthe frequency domain using Wentzel–Kramers–Brillouin analysis. © 2008 Optical Society of America

OCIS codes: 140.4480, 190.4970, 190.4410, 190.7110, 320.1590, 320.5540.

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. INTRODUCTIONeveral scientific and technological applications requirehort optical pulses with large peak power [1]. A numberf practical mode-locked femtosecond oscillators are avail-ble, but scaling from nanojoule pulse energies to milli-oules and above generally requires optical amplification2]. The chirped-pulse amplification technique [3], whichvoids the undesired effects associated with large peakower by first stretching the optical pulse and then re-ompressing it after the amplification stage, made ampli-cation possible over 9 orders of magnitude while en-bling high-pulse energies.Solid-state optical amplifiers have played a major role

n the development of high-power laser systems [1,2].owever a limitation of solid-state laser amplifiers lies in

he fact that their gain spectrum is determined by a par-icular atomic transition. This determines the operatingavelength and limits the amplification bandwidth.Optical parametric amplifiers (OPAs) provide a solu-

ion to this problem. OPAs allow a wider choice of pumpnd signal wavelengths. Their large single-pass gainvoids the need for complex multipass or regenerativeonfigurations. Moreover, parametric amplifiers do notely on optical absorption as a pumping mechanism, thusliminating the problem of thermal lensing common toigh average-power systems. Their use as femtosecondulse amplifiers [4–6] and their use in chirped-pulse am-lification systems [7] has been successfully demon-trated.

There are two techniques commonly used to broadenhe amplification bandwidth of OPAs. The first one is toperate at degeneracy, where the signal and idler wave-engths are equal. In this case the group velocities of theaves are matched, which leads to a broad amplificationandwidth. Chirped-pulse amplification systems usinghis technique are described in [8–10]. A drawback of op-

0740-3224/08/040463-18/$15.00 © 2

rating at degeneracy is that the wavelength requiredrom the pump laser is fixed once the center of the gainpectrum is chosen, precluding the use of convenientump lasers in some applications. Another disadvantages that the bandwidth, although larger than in a nonde-enerate case, remains limited by the dispersion of theaterial.The second bandwidth-broadening technique involves a

oncollinear geometry [11]. The input signal beam is dis-ersed angularly using diffraction gratings in order to en-ure simultaneously phase matching and group-velocityatching between the signal and idler waves. Amplifica-

ion of pulses as short as 5 fs was achieved using thisethod [12,13].A less complicated approach, collinear in nature, is to

se chirped quasi-phase-matching (QPM) gratings in or-er to build broadband OPAs. This is the technique ex-lored in this paper.The QPM technique consists of a periodic reversal of

he sign of the nonlinear coefficient in order to compen-ate for the accumulated phase mismatch [14]. Since theeriod of the sign reversal is a controlled parameter, QPMevices operate equally well over a range of wavelengths.PM allows more freedom over the choice of polarizationf the waves than birefringent phase matching, enablinghe use of the largest component of the nonlinear suscep-ibility tensor. For these reasons, QPM gratings have be-ome a widely used alternative to traditional birefringenthase matching. In particular, femtosecond-pulse amplifi-rs based on QPM materials have been successfully dem-nstrated [15–17].

In addition to the benefits mentioned above, QPM of-ers a distinct advantage over conventional nonlinearrystals. It allows the engineering of nonuniform phase-atching profiles, which can be used to obtain desirable

nd highly tunable gain and phase spectra. For instance,

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464 J. Opt. Soc. Am. B/Vol. 25, No. 4 /April 2008 Charbonneau-Lefort et al.

hirped QPM gratings have been used to broaden theecond-harmonic acceptance bandwidth [18] and to ma-ipulate the phase of the generated wave in order tochieve pulse compression [19–23]. Previous work in theontext of OPA includes the calculation of the single-passain [24], the use of aperiodic QPM gratings in opticalarametric oscillators [25], and the proposal of a tandem-rating design for the simultaneous control of the gainnd group delay [26].This paper is an investigation of the properties of

hirped QPM gratings used as broadband OPAs. The easi-st model to describe OPAs is the Rosenbluth model [27],hich is presented in Section 2. In Section 3, we give ex-ressions for the amplitude and phase response of anylow-varying QPM grating profile and explore a variety ofesigns in Section 4. We look at the linear phase-atching profile in depth. It is the most important in

ractice because it allows one to achieve an essentiallyonstant gain over a wide bandwidth. We also considerpodization techniques to reduce the gain and phaseipple. We then explore the effect of a sinusoidal modula-ion of the grating profile, which provides selective ampli-cation at certain frequencies. Finally, we describe aandem-grating design that achieves constant gain andonstant group delay across the spectrum. We also give ahort discussion of parametric amplification in the low-ain regime, showing the connection with the work donen the context of the second-harmonic generation (SHG)nd the difference-frequency generation (DFG) [22,23].In this paper, the analysis is carried out in the fre-

uency domain. A companion paper [28] investigates thevolution of the amplified pulses in time.

. OPTICAL PARAMETRIC AMPLIFICATIONN NONUNIFORM QUASI-PHASE-ATCHING GRATINGSdiagram illustrating OPA in nonuniform QPM gratings

s shown in Fig. 1. We consider three copropagatingaves, the pump, the signal, and the idler, with frequen-

ies �p, �s, and �i, respectively. For their coupling to beost efficient, these three waves must satisfy the

requency-matching condition �p=�s+�i. We denote by��s−�s0 the frequency shift of the signal wave withespect to a nominal frequency �s0 and consider the pumprequency fixed for the present discussion. Due to the dis-ersive properties of the material, the three wavenum-ers will not necessarily be matched [29]. We definek��kp−ks−ki to be the intrinsic k-vector mismatch.

n ��2� materials, perfect wave vector matching also maxi-izes the parametric gain.The contribution of the QPM grating cancels most of

he intrinsic wave vector mismatch. Here we considerather general grating profiles. The only requirement thate impose on the grating period ��z� is that it be a piece-ise slow-varying function of position. We define Kg�z�

Fig. 1. Illustration of OPA in a chirped QPM grating.

2� /��z� as the associated grating wavenumber. Theverall wavenumber mismatch, �, is the difference be-ween the intrinsic mismatch and the potentially spa-ially nonuniform QPM grating:

��z,�� = �k�� − Kg�z�. �1�

he chirp rate ��z� is the rate of change of the wavenum-er mismatch in the axial direction:

��z� =��z,��

�z= −

dKg�z�

dz. �2�

We define the perfect phase-match point (PPMP) to bet the position zpm where ��zpm,��=0. The usual treat-ent of OPAs makes use of the slow-varying envelope ap-

roximation [29,30]. The frequency-domain signal anddler envelope functions Es,i�z , ±�� are obtained from theourier-domain representations of the electric fields

˜s,i�z ,�s,i� by extracting their fast carrier phases, namely,

˜s,i�z ,�s,i�=Es,i�z , ±��exp�iks,i��s,i�z�. The wavenumberss,i��s,i� introduced here are frequency-dependent; theyccount for material dispersion. [The conventional enve-ope description uses constant k-vectors for the envelopes,.g., E�z ,�s,i�=E�z , ±��exp�iks,iz�. The relationship be-ween these two envelopes is discussed in more detail in22].] We normalize the optical fields to make them pro-ortional to the photon fluxes by introducing As,i�ns,i /�s,i�1/2Es,i, where ns,i are the refractive indices at

he respective frequencies of the two waves. We treat theump as an undepleted, monochromatic plane wave. Theesulting steady-state rate equations for the spatial evo-ution of a pair of signal and idler frequency componentsre

dAs

dz= i��z,��Ai

*ei�z,��, �3�

dAi*

dz= − i��z,��Ase

−i�z,��. �4�

he coupling coefficient is ��z ,��s�i /nsni�1/2�deff�z� /c� �Ep�, where deff is the amplitudef the effective nonlinear coefficient of the QPM gratingi.e., the amplitude of the Fourier coefficient of the spa-ially modulated structure), �Ep� is the magnitude of theump wave electric field, and c is the speed of light inacuum. We allow the coupling coefficient � to vary slowlyith position and frequency. The phase mismatch accu-ulated between the three waves is

�z,�� =�z0

z

��z�,��dz�, �5�

here z0 is the position of the input plane of the grating.inally, we will in general allow both signal and idleraves to be incident on the grating, with amplitudess�z0 ,��=As0 and Ai�z0 ,−��=Ai0, respectively.With the change of variables As,i=as,i�

1/2ei/2, we com-ine the coupled-mode Eqs. (3) and (4) to eliminate one ofhe fields, leading to a second-order linear differentialquation in standard form [31]:

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Q�z� = ��

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��

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nd where prime denotes differentiation with respect to z.The solutions to Eq. (6) will have an oscillatory charac-

er when Re�Q�0 and will be exponentials when Re�Q�0. The two regimes are separated by the turning points,

iven by the condition 1/2 �� (here we assume that� /� is small and can be neglected). Located on either sidef the PPMP, the turning points define the limits of themplification region. A typical grating profile ��z ,�� ishown in Fig. 2, together with the location of the PPMPnd the two turning points, which define the extent of themplification region.

. PRACTICAL FORMULAS FOR OPTICALARAMETRIC AMPLIFICATIONESIGN. Wentzel–Kramers–Brillouin Solutione use the WKB formalism and notation developed byeading [32]. In this notation, the end points of the phase

ntegrals as well as the order of dominance of the solu-ions are easily conveyed. We set the phase reference levelt one of the complex turning points, where the function

vanishes. Two such turning points exist, ztp1 and ztp2,ocated in the complex plane on the left- and right-handides of the PPMP, respectively (see Appendix B, Fig. 11).he general WKB solutions using the left-hand-side turn-

ng point ztp1 as the phase reference level are written as

�ztp1,z� � Q−1/4�z�exp�i�ztp1

z

Q1/2�z��dz� , �8�

�z,ztp1� � Q−1/4�z�exp�i�z

ztp1

Q1/2�z��dz� . �9�

linear combination of these two expressions defines ap-roximate solutions that are valid away from the turningoints. The “complex WKB method” consists of finding theinear combination of the general solutions that satisfy

ig. 2. Nonuniform grating profile showing the PPMP, the turn-ng points, and the nature of the solutions in each region.

he boundary conditions in a given region and then in ex-ending this solution to the entire complex plane by find-ng the coefficients that ensure continuity (asymptoti-ally) between adjacent regions [32–34]. The details of thealculation are shown in Appendix B and C; here we onlytate the result. The signal at the end of the grating zL is

As�zL� ��zL�

��z0� �1/2

ei�zL�/2�C+ − iC−�i�ztp2,ztp1��zL,ztp2�

− i�ztp2,zL��, �10�

ith

C+ =1

�ztp1,z0��1 +�2�z0�

�2�z0�As0 −��z0�

��z0�Ai0

* � , �11�

C− =1

�z0,ztp1��−�2�z0�

�2�z0�As0 +

��z0�

��z0�Ai0

* � , �12�

�ztp2,ztp1� � exp�i�ztp2

ztp1

Q1/2�z�dz . �13�

s mentioned before, z0 is the position of the input planend ztp1 and ztp2 are the turning points such that�ztp1,2�=0. The frequency argument has been sup-ressed to simplify the notation.The factor C+− iC− together with the phase term ei/2

epresent the contribution of the portion of grating lo-ated before the amplification region where the wavesropagate, accumulate a relative delay and are combinedefore entering the amplification region. The factor�ztp2 ,ztp1� is the contribution from the amplification re-ion where the waves grow with little phase accumula-ion. Propagation over the remaining portion of the grat-ng is given by �zL ,ztp2�− i�ztp2 ,zL�.

. Design Formulas for a Single PPMPe can simplify Eq. (10) further by retaining only itsost significant contributions. An adequate approxima-

ion (derived in Appendix F) is given by

As R�As0 + iAi0* ei��z0,zpm��eg�ztp1,ztp2�, �14�

ith R= ��zL� /��z0��1/2. The phase integral is

��z0,zpm� � �z0

zpm

��z�dz, �15�

nd the gain integral is

g�ztp1,ztp2� � �ztp1

ztp2

��2 − �2/4�1/2dz, �16�

here the integration is carried out between the twourning points ztp1 and ztp2.

Equation (14) is the central result of this paper. It ishe basis of the design procedure explained below. It wille used repeatedly in several examples.In a majority of cases we can linearize the grating pro-

le around the PPMP. The amplification factor G=eg thenecomes

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��zpm�� , �17�

here the chirp rate and the coupling coefficient arevaluated at the PPMP. This is the Rosenbluth amplifica-ion formula [27], an important result first obtained in theontext of laser–plasma interactions. For polynomial pro-les the gain is given in terms of beta functions [24]. Inarticular, a case that we will be using later is the cubicrofile �=��z−zpm�3, for which

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ith �−11 �1−u6�1/2du= 1

3 � 16 , 3

2 �1.82.

. Design Formulas for Multiple PPMPshen a profile contains multiple PPMPs, the output of

ach monotonic segment becomes the input of the next.his situation is illustrated in Fig. 3.A subtlety arises because the phase mismatch at the in-

ut of a given segment is not zero but results from thehase accumulation over the previous segments. Thisroblem can be solved by absorbing the accumulatedhase into the fields, i.e., letting As,i

�j� =Bs,i�j�ei�j�/2, where the

uperscripts refer to the segment number, and �j�

�z0

�1�z0

�j�

��z�dz is the phase mismatch accumulated beforeegment j. Then the new fields obey the coupled-modeqs. (3) and (4) with initial conditions Bs,i0

�j� =As,i0�j� e−i�j�/2,

nd we can use the results derived above. Consequently,he contribution of a segment j can be written

As�j� = R�j��As0

�j� + iAi0�j�*ei��z0

�1�,zpm�j� ��eg�ztp1

�j� ,ztp2�j� �, �19�

Ai�j� = R�j��Ai0

�j� + iAs0�j�*ei��z0

�1�,zpm�j� ��eg�ztp1

�j� ,ztp2�j� �. �20�

n the case of periodic profiles, As0�j�=As

�j−1�, Ai0�j�=Ai

�j−1�.owever, when multiple gratings are used in a cascaded

onfiguration this is not necessarily the case. A typical ex-mple is the tandem configuration discussed in Subsec-ion 4.E.

. ANALYSIS OF VARIOUS GRATINGROFILEShis section illustrates the engineering of chirped grat-

ngs. We first review the well-known case of a uniform

Fig. 3. Grating profile with multiple PPMPs.

PM grating. Then, we examine the linear profile, whichssentially yields constant gain over a large bandwidth,nd we also examine ways of eliminating the ripple affect-ng the spectrum. Then, as an example of nonuniformhirp rate we discuss a sinusoidal modulation superposednto a linear ramp providing enhanced amplification atelected frequencies. Finally, we examine a tandem de-ign for simultaneous gain and group delay control.

. Uniform Profilen the case of a uniform QPM grating, there are no turn-ng points since the wave vector mismatch � is constant,nd as a consequence the nature of the solutions (i.e., realr complex exponentials) remains unchanged throughouthe medium. The WKB analysis developed herein doesot apply in this simple case.When the wave vector mismatch � and the coupling co-

fficient � are constant, the coupled Eqs. (3) and (4) can beolved exactly [29,30]:

As�zL� = As0ei�L/2�cosh �L −i�

2�sinh �L�

+ i�

�Ai0

* ei�L/2 sinh �L, �21�

Ai�zL� = Ai0ei�L/2�cosh �L −i�

2�sinh �L�

+ i�

�As0

* ei�L/2 sinh �L, �22�

here L=zL−z0 is the length of the grating and

� = �2 − ��

22

�23�

s the growth rate in the presence of constant wave vectorismatch. With input at the signal wave only and in the

arge-gain regime ��L�1�, the amplitudes of the wavesre approximately

�As,i�zL�� As0

�

2�e�L. �24�

he peak gain, achieved when �=0, is

Guniform =1

2e�L. �25�

he gain increases exponentially with the length of theevice. The FWHM bandwidth is reached when the waveector mismatch is equal to

�FWMH = ±2

L �2L2 − �ln 2�2. �26�

he amplification bandwidth takes a simple form if weeglect group-velocity dispersion at the signal and idleravelengths and assume off-degenerate operation. In this

ase, the intrinsic wave vector mismatch is

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k�� = kp − ks − ki

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here ks0 and ki0 are the signal and idler wave vectors athe nominal frequencies and vs,i=�� /�k��s0,i0

are the groupelocities. The grating wave vector is equal to Kg=kpks0−ki0 (by definition of the nominal frequencies), so

hat the overall mismatch, given by Eq. (1), is simply

�� = � 1

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vi��. �28�

e define the group-velocity mismatch parameter, �v, as

1

�v�

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vs−

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ubstituting into Eq. (26), we find the following expres-ion for the FWHM bandwidth:

��uniform =4��v�

L �2L2 − �ln 2�2 4��v��, �30�

here the approximation is valid in the large-gain re-ime. Therefore the bandwidth of a uniform QPM gratingn the high-gain limit is essentially independent of therating length. It depends only on the strength of the cou-ling coefficient and on the dispersive properties of theaterial.

. Linear Profile for Broadband Amplification

. Application of the Design Formulahe most basic chirped QPM grating profile is the linearhirp. We consider an input signal wave only, as this ishe most common situation in practice and let the idlerevelop from the interaction. For simplicity, we neglectroup-velocity dispersion at the signal and idler wave-engths so that the phase mismatch varies linearly withrequency. (The extension to higher-order dispersion istraightforward if somewhat more tedious.) The totalave vector mismatch, Eq. (1), is given by

��z,�� = ��z − zpm0� −��

�v, �31�

here �� is the constant chirp rate and zpm0 is the posi-ion of the PPMP at the nominal frequency. As the inputrequency is varied, the PPMP is shifted linearly with fre-uency according to

zpm = zpm0 +��

��v. �32�

he mismatch then takes the simple form

� = ��z − zpm�. �33�

he two turning points are located at a distance given by� / �� on each side of the PPMP; therefore the length ofhe amplification region is

Lg =4�

��. �34�

he present approximate treatment is valid for gratingsor which L�Lg. If this condition is satisfied, then theephasing effects due to the chirp of the grating dominatehe behavior of the device. Conversely, if L�Lg, then therating is essentially uniform.

The limits of the amplification spectrum are reachedhen the frequency shift results in one of the turningoints being at the edge of the grating. Therefore the am-lification bandwidth is

��chirped = ��v��L − Lg�. �35�

s expected, the bandwidth is proportional to the productf the chirp rate and the grating length, i.e., to the rangef grating k-vectors in the device. The bandwidth takeshis simple form when the group-velocity dispersion of theaterial can be neglected. Otherwise, higher dispersive

rders have to be included in Eq. (31), and the bandwidthay have to be calculated directly from the dispersion re-

ation, as discussed in [26].The application of Eq. (14) for each wave gives

As = As0e��2/��, �36�

Ai = iAs0* e��2/��e−i��zpm − z0�2/2. �37�

oth waves experience the constant amplitude gain giveny the Rosenbluth gain formula:

GRosenbluth = e��2��/��. �38�

owever, they differ in their phases. While the signal ex-eriences negligible phase shift, the idler accumulates auadratic phase corresponding to a time delay (with re-pect to a reference traveling at the idler velocity) of

�i =zpm − z0

�v. �39�

ince this delay is linear itself in the input frequencythrough its dependence on zpm��], the idler experiencesroup delay dispersion, the magnitude of which dependsn the chirp rate. Alternatively, the idler group delay withespect to the corresponding signal wave Fourier compo-ent is

�i−s = −zL − zpm

�v. �40�

It is important to keep in mind that these phases rep-esent the contribution from the grating only; the disper-ive properties of the material must be considered sepa-ately. They are accounted for by the carrier phase ks,iz� t, which must be added to the envelopes in order to
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ecover the Fourier representation of the fields. Materialispersion is in fact buried in the wave vectors ks,i�s,in��s,i� /c.

. Comparison with Wentzel–Kramers–Brillouin Solutionhe approximate design formulas lead to simple expres-ions for the gain, bandwidth, and phase. It is natural toxpect that this simplicity comes at the price of a loss ofccuracy. The accuracy of our expressions can be verifiedy comparing it with the fuller WKB and numerical solu-ions.

The WKB solution, Eq. (10), is valid for general gratingrofiles. In Appendix E, we obtain an explicit expressionn the special case of a linear profile [see Eq. (E7)]. Thepatial evolution of the waves is plotted in Fig. 12 alongith the numerical solution. The underlying assumptionsehind the WKB solution are also discussed there.Figure 4 compares the gain spectra obtained from the

xplicit WKB solution, Eq. (E7), with the numerical solu-ion, together with the simplified result obtained in Sub-ection 4.B.1, Eqs. (36) and (37). Figure 5 shows the phasepectra. The signal has a small phase drift that is not cap-ured by the simplified expressions. However, the idleras a large quadratic spectral phase predicted by Eq. (37).inally, Fig. 6 shows the relative group delay, �, with re-pect to the input signal. As expected, the idler experi-nces a linear group delay, reflecting the linear relation-hip between the position of the PPMP and frequency.

A striking feature of those plots is the significant ripplen the gain, phase, and group delay spectra. While theKB solution in Eq. (10) recovers this ripple correctly,

he small-scale features of the amplification are lost whenaking the simplifications leading to the design formula

n Eq. (14). The origins of the ripple and the ways to re-uce it will be discussed in Section 5.Figure 6 indicates that the magnitude of the group de-

ay ripple is typically of the order of 10% of the delay ac-umulated between the signal and idler, and that it de-reases with an increasing chirp rate or increasingength. For femtosecond pulse amplification, the pulseistortion resulting from this group delay ripple is oftennacceptable.

. Comparison with Uniform Gratingsrom the point of view of applications, the clear advan-age of linearly chirped QPM gratings lies in their arbi-rarily wide amplification bandwidth. For a fixed gratingength, the bandwidth is essentially proportional to thehirp rate ��. Naturally, an increase of bandwidth comest the expense of a reduction of gain since the Rosenbluthactor is proportional to 1/��. The trade-off between gainnd bandwidth is expressed by the fact that theogarithmic-gain-bandwidth product is a quantity inde-endent of the chirp rate:

ln GRosenbluth � ��chirped ��v��2L. �41�

In the case of a uniform grating, the peak gain and themplification bandwidth are given by Eqs. (25) and (30),espectively. The logarithmic-gain-bandwidth product inhis case is approximately

ln Guniform � ��uniform 4��v��2L. �42�

he logarithmic-gain-bandwidth products of uniform andhirped gratings are essentially the same (except for aactor of 4 instead of �).

The advantages of chirped gratings over uniform grat-ngs are to enable (in principle) arbitrarily large band-idths and to allow shaping of the gain and group delay

pectra. However, this increase of bandwidth is accompa-ied by a reduction of the gain, according to Eq. (41).

ig. 4. Gain spectrum of a linear profile comparing the numeri-al solution with the WKB solution and the Rosenbluth gain fac-or for various grating lengths. The numerical values used are2/��=2 and (a) ��1/2L=20, (b) ��1/2L=30, and (c) ��1/2L=40.

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F(u


. Concrete Design Exampleo give an idea of typical experimental values, let us con-ider an OPA consisting of a chirped QPM grating de-igned to offer a (power) gain of 50 dB over a bandwidth of00 nm�1550 nm. The nonlinear crystal is made of peri-dically poled lithium niobate (PPLN) and has a length ofcm. We assume that the OPA is pumped by a Nd:YAG

aser �1064 nm�. The numerical values of the various ex-erimental parameters involved are listed in Table 1. Thehirp rate required is ��=4.13�105 m−2. To obtain the de-ired gain, the coupling coefficient must be �=870 m−1.he required pump intensity to achieve this is38 MW/cm2. In terms of normalized quantities, the gainarameter is �2 /��=1.8, and the length is ��1/2L=32. Thisituation is very similar to the one illustrated in Fig. 4(b).

ig. 5. Phase spectrum of a linear profile comparing the numeri36) and (37), for various grating lengths. Plots (a)–(c) correspondsed are �2 /��=2 and (a), (d) ��1/2L=20; (b), (e) ��1/2L=30; and (c

By comparison, the pump intensity required to achievehe same gain in a uniform grating of equal length is.6 MW/cm2. However, in this case the bandwidth is6 nm only. The 1D model studied here assumes that the

ump, signal, and idler are plane waves. In a free-spacexperiment, however, the light pulses are localized inpace and time. A reasonable approximation (neglectingiffraction and dispersion) is to average the gain over thentire pulse:

G3D =�� exp��2�x,y,t�

�� dxdydt, �43�

here now the coupling coefficient has a transverse andemporal profile related to the intensity of the pump

tion with the WKB solution and the simplified expressions, Eqs.signal, plots, (d)–(f) correspond to the idler. The numerical values�1/2L=40.

cal soluto the), (f) �

pp3(aOwf

5Atliiftd

vtbd

ssag

taelQgtt

Fdts�


ulse, �2�x ,y , t��Ip�x ,y , t�. If we assume that the pumpulses are Gaussian in space and time with a spot size of00 �m (1/e2 intensity diameter) and a duration of 1 ns1/e2 intensity full duration) but the same peak intensitys the plane wave example just given, then the gain of thePA is 35 dB instead of 50 dB. Diffraction, inevitablehen gain narrowing occurs, will decrease this value even

urther.

. Comparison with Noncollinear Phase Matchings mentioned in Section 1, a common way of broadening

he amplification bandwidth of OPAs is to use a noncol-inear geometry [11]. This technique consists of introduc-ng an angle between the signal and the pump beams andn spatially dispersing the input signal, so that at eachrequency component the signal group velocity is equal tohe projection of the idler group velocity along the signalirection. This approach effectively cancels the group-

ig. 6. Group delay spectrum of a linear profile normalized withelays are relative to reference waves traveling at the signal andion with the WKB solution and the simplified expressions, Eqs. (ignal, plots (d)–(f) correspond to the idler. The numerical values�1/2L=40.

elocity mismatch between the two waves. Consequently,he bandwidth is limited not by group-velocity dispersionut by higher dispersive orders in a manner similar to aegenerate OPA.The drawback of noncollinear phase matching is that

patial walk-off limits the gain length of the amplifier. As-uming a pump beam of a width 2w0 and a noncollinearngle � between the pump and signal beams, the effectiveain length is approximately Leff2w0 / tan �.

By comparison, chirped QPM gratings do not attempto correct for group-velocity mismatch. Instead, they offerbroad range of phase-matching periods. Since they op-

rate in a collinear geometry, spatial walk-off does notimit the gain length. Other major advantages of chirpedPM OPAs are the possibility of engineering the gain androup delay spectra. Their drawback is that, unlike withhe noncollinear phase-matching technique, there exist arade-off between gain and bandwidth.

ct to the delay between the waves � / �1/vs−1/vi �L=� ��v � /L. Thevelocities, respectively. These plots compare the numerical solu-d (37), for various grating lengths. Plots (a)–(c) correspond to theare �2 /��=2 and (a), (d) ��1/2L=20; (b), (e) ��1/2L=30; and (c), (f)

respeidler

36) anused

fdaOwbTpwwtClg=oLb5q(it

tcOt

CTplbt

wttp

dt(ti

ttsotcst

pQs

Ttfgtbctggl=

pagpl

watndg

ioum

MCBPGPO

QCN

ECGGP


Since the performance of each approach depends on dif-erent physical quantities (e.g., high-order dispersive or-ers in the case of noncollinear phase matching and thevailable grating length in the case of chirped QPMPAs), it is impossible to assert in an absolute mannerhich approach is best. We can work out an exampleased on the experimental conditions reported in [35].he authors report a noncollinear OPA in periodicallyoled lithium tantalate. They use a pump laser at 785 nmith a spot size of 180 �m to achieve gain over a band-idth ranging from 1.1 to 1.6 �m. Material dispersion at

hose wavelengths imposed a noncollinear angle �2°.onsequently, spatial walk-off limits the effective gain

ength to Leff5.2 mm. We will assume that the powerain is given as in the case of a uniform medium by Gexp 2�Leff. Then, in order to achieve, for instance, a gainf 40 dB, the required coupling coefficient is �893 m−1.et us now calculate the gain obtained by the same pumpeam in a chirped QPM OPA assuming a grating length ofcm. Neglecting high-order dispersion, the chirp rate re-uired to achieve the desired bandwidth, given by Eq.35), is ��8.8�105 m−2. The Rosenbluth gain parameters then �2 /��1.65, corresponding to a power amplifica-ion of 45 dB.

In the example given above, the performances of thewo techniques are similar. Chirped QPM OPAs would be-ome advantageous if the grating length were increased.n the other hand, noncollinear OPAs would perform bet-

er if the material were less dispersive.

. Tapered Profiles for Ripple Reductionhe gain, phase, and group delay ripple affecting the am-lification spectrum of chirped gratings can be relativelyarge. Subsection 4.C explores ways of reducing the rippley tapering the magnitude of the coupling coefficient orhe grating profile.

Table 1. Numerical Values for the OPA Design

Specifications

aterial LiNbO3

enter signal wavelength 1550 nmandwidth 100 nmump wavelength 1064 nmrating length, L 5 cmower gain 50 dBperating temperature 150°C

QPM Grating

PM grating period range 28.3–31.2 �mhirp rate, �� 4.13�105 m−2

ormalized length, ��L 32

Pump Intensity

ffective nonlinear coefficient, deff 2 /��27 pm/Voupling coefficient, � 870 m−1

ain parameter, �2 /�� 1.83ain length, Lg 8.4 mmump intensity 438 MW/cm2

The ripple is due to the fact that some amount of idlerave is generated before reaching the gain region. When

hey reach the phase-matching region, contributions fromhe signal and idler are superposed, and their relativehase affects the magnitude of the gain.This phenomenon can be understood using the WKB

escription. The solution is constructed by the superposi-ion of the two elementary WKB solutions given by Eqs.8) and (9). These positive and negative complex exponen-ials interfere, causing small oscillations in the signal anddler amplitudes.

From a physical point of view, the ripple is caused byhe “hard” edges of the grating where the interaction isurned on and off abruptly. Therefore, ripple reductionchemes should aim at making the transition into and outf the interaction region as smooth as possible. There arewo ways of accomplishing this. One way is to turn on theoupling coefficient adiabatically; the other consists oftarting from a completely mismatched interaction that ishen brought progressively into phase matching.

Tapering of the coupling coefficient ��z� can be accom-lished, for example, by varying the duty cycle of thePM grating or by omitting domain reversals [36]. For in-

tance, let us consider the profile:

��z�

�max= a + b � tanh� z − z0 − l1

w1 � tanh�L − z + z0 − l2

w2 .

�44�

he various constants can be chosen to achieve satisfac-ory ripple reduction. The Rosenbluth gain factor is nowrequency-dependent with �=��zpm�� and zpm��iven by Eq. (32). The reduction of the gain at the edges ofhe spectrum causes a narrowing of the amplificationandwidth. Figure 7 shows the tapering function and theorresponding amplification and group delay spectra. Forhis particular bandwidth and gain, the amplitude of theain ripple could be reduced from �100% of the averageain (peak-to-peak variation) to 2% with the parameters1= l2=w1=w2=0.04�L and a and b chosen so that ��z0��zL�=0 and max ��z�=�max.Let us now turn our attention to the tapering of the

hase mismatch profile of the grating, ��z�. As an ex-mple, we consider a profile that is linear for most of therating, but becomes large at the ends. This is accom-lished for instance by adding a large, odd power to theinear grating profile:

� = �0��z − zpm0� + �� z − zpm0

L/2 �

− �1/vs − 1/vi��, �45�

here � is the amplitude of the departure from linearitynd � is a large, odd integer. The Rosenbluth amplifica-ion factor is now dependent on frequency through theonuniform chirp rate ��=��zpm��. The gain is re-uced at the edges of the spectrum corresponding to re-ions of large chirp rate.

Figure 8 shows the grating profile and the correspond-ng amplification and group delay spectra. The amplitudef the ripple could be kept below 5% of the average gainsing the parameters � /�0�

1/2=100 and �=21. These nu-erical results show that tapering of the coupling coeffi-

F(r�

Fdaand the length is �� L=20. The gain and group delay spectra without apodization were shown in Figs. 4 and 6, top case.


ig. 8. Ripple reduction using tapering of the QPM profile: (a) grating profile, (b) gain spectrum, (c) signal group-delay spectrums andd) idler group delay spectrum. The group delays are defined with respect to reference waves traveling at the signal and idler velocities,espectively. The grating profile is given by Eq. (45) with � /�0�

1/2=100 and �=21. The gain parameter is �2 /�0�=2 and the length is1/2L=20.

ig. 7. Ripple reduction using tapering of the coupling coefficient: (a) coupling coefficient profile, (b) gain spectrum, (c) signal groupelay spectrum, and (d) idler group delay spectrum. The group delays are defined with respect to reference waves traveling at the signalnd idler velocities, respectively. The tapering profile is given by Eq. (44) with l1= l2=w1=w2=0.04�L. The gain parameter is �2 /��=2

1/2

�

cr

sp2Qpcp3

DAAcwattt

lggs

womofdz�ilbvi

s=tebe

SoTazs3

EGTlispt

tcobaHed

cfiwioo

Fn(t


ient seems more effective at eliminating the group delayipple than tapering of the phase profile of the grating.

As a concrete example, we consider the design de-cribed in Subsection 4.B.4. To obtain a 100 nm wide am-lification bandwidth, the QPM period must range from8.3 to 31.2 �m over a length of 5 cm. Tapering of thePM profile can be done by keeping �90% of the centralortion of the grating unchanged and by progressively in-reasing the chirp rate at the two ends so that the gratingeriod reaches, for example, 25.3 �m at one end and3.2 �m at the other.

. Sinusoidal Profile for Selective Frequencymplificationccording to the design formulas, the amplification in thease of a monotonic profile with no input idler is simply eg,here g is the gain integral given in Eq. (16). Thus themplification depends predominantly on the local proper-ies of the grating in the vicinity of the PPMP. This openshe possibility of engineering the amplification spectrumhrough careful design of the grating profile.

By way of illustration, we consider a sinusoidal modu-ation superposed onto a linear profile and show that itives rise to an amplification spectrum with enhancedain around certain frequencies only. Such a profile is de-cribed by

� = �0��z − zpm0� − � sin�k��z − zpm0�� − �1/vs − 1/vi��,

�46�

here � and k� are the amplitude and spatial frequencyf the modulation, respectively. If the amplitude of theodulation is small (i.e., �k��0�), then the linearization

f the profile is valid everywhere and the gain is obtainedrom the Rosenbluth formula using the frequency-ependent chirp rate ��=�0�−�k� cos k��zpm−zpm0

�, wherepm is now the solution of the transcendental equation�zpm�=0. However, as the amplitude of the modulationncreases the chirp rate becomes close to zero at certainocations inside the grating. When �k�=�0� the profile cane approximated by a cubic at those locations where ��anishes, and in these cases we can use the gain formulan Eq. (18), which in our particular case becomes

ig. 9. Sinusoidal profile for selective frequency amplification:umerical values with the Rosenbluth amplification formula. The

marked by the dots on the plot), while the linear approximationhe maxima. The numerical parameters in this example are �2 /�
0
G = exp�4.17� �4

�0�k�21/3� . �47�

The amplification spectrum of this sinusoidal profile ishown in Fig. 9 for parameters �2 /�0�=2, �0�

1/2L=40, k�

2� / �L /3�, and �=�0� /k�. The minimum channel spacinghat can be achieved with a sinusoidal profile is roughlyqual to the bandwidth of a uniform QPM grating giveny Eq. (30). It is typically of the order of a few nanom-ters.

Let us take as an example the design introduced inubsection 4.B.4. We keep the same range of QPM peri-ds, namely, 28.3–31.2 �m, and grating length �5 cm�.he profile described in Fig. 9 is achieved by introducingsinusoidal modulation such that the chirp rate reaches

ero at three locations (0.83, 2.5, and 4.17 cm), corre-ponding to QPM periods of approximately 28.8, 29.8, and0.8 �m.

. Tandem Gratings for Simultaneously Flat Gain androup Delay Spectrahe examples presented above illustrate how one can tai-

or the amplification spectrum through careful engineer-ng of the grating profile. Little has been said about de-igning the phase response. Nevertheless, control of thehase spectrum is often critical, especially for applica-ions in ultrafast optics.

Recently, we proposed the use of a pair of gratings in aandem configuration in order to achieve simultaneousontrol of the gain and group delay spectra [26]. This ideaf using the idler wave in a cascaded geometry has alsoeen used to reduce the amplified spontaneous emissionnd improve the pulse contrast in high-gain OPAs [10].ere is an analysis of this design using the simple design

xpressions. We will see that the equations governing theesign procedure follow in a straightforward manner.The tandem configuration is shown in Fig. 10. Its prin-

iple of operation is the following. The signal to be ampli-ed is incident on the first grating. After the first grating,e block the signal wave so that only the pump and the

dler are incident on the second grating. The output idlerf the first grating is used as the input signal of the sec-nd. The “idler” generated by the second grating then has

M grating profile and (b) amplification spectrum, comparing theapproximation to the grating profile gives the peak amplificationed curve) is valid away from the peaks but not in the vicinity of�1/2L=40, k =2� / �L /3�, and �=�� /k .

(a) QPcubic(dash�=2, �
0 � 0 �

tpct

dt(

wtgtio

dsstc

Satwd

caPolttd

5IWrpg[attiFrg

attew

Iatgp

Ti�tbdPt

6Idmiotf

Wrircwpad


he same frequency as the original signal. By choosing theosition of the PPMPs in the two gratings and their localhirp rates, we can control the gain and group delay spec-ra at the same time.

Let us now describe the output of the tandem-gratingesign. We use the expressions describing the amplifica-ion in the presence of multiple PPMPs, Eqs. (19) and20), with Ai0=0. The output of the first grating is

As�1� = R�1�As0eg�1�

, �48�

Ai�1� = iR�1�As0

* eg�1�ei��z0

�1�,zpm�1� �, �49�

here g�1� denotes the gain integral around the PPMP ofhe first grating [Eq. (16)]. For a locally linear chirp rate,�1�=��zpm

�1� � /��zpm�1� �. Before entering the second grating,

he signal is filtered out. The inputs to the second grat-ngs are therefore As0

�2�=0, Ai0�2�=Ai

�1�. A second applicationf Eq. (19) then gives

As�2� = As0

* R�1�R�2�eg�1�+g�2�ei��zpm

�1� ,zpm�2� �. �50�

This expression contains all the information needed toesign grating profiles with the desired gain and phasepectra. First, the total logarithmic gain is equal to theum of the individual gains. Using the Rosenbluth fac-ors, we obtain the total logarithmic gain in terms of thehirp rates:

ln G�� =��2�zpm

�1� �

��zpm�1� ��

+��2�zpm

�2� �

��zpm�2� ��

. �51�

econd, the accumulated phase corresponds to the delayccumulated by the propagation at the idler velocity be-ween the two PPMPs. Differentiating the total phaseith respect to frequency, we can express the total groupelay in terms of the positions of the PPMPs:

�� =zpm

�1� − z0�1�

vs+

zL�1� − zpm

�1�

vi+

zpm�2� − z0

�2�

vi+

zL�2� − zpm

�2�

vs.

�52�

The group delay for a particular spectral componentorresponds to traveling from the input to the first PPMPt the signal group velocity, from the first to the secondPMPs at the idler group velocity, and then from the sec-nd PPMP to the end of the crystal at the signal group ve-ocity. By choosing the lengths of the two crystals and therajectory of the two PPMPs as a function of frequency,here are enough degrees of freedom to flatten the groupelay over a broad spectrum [26].

Fig. 10. Tandem configuration.

. OPTICAL PARAMETRIC AMPLIFICATIONN THE LOW-GAIN LIMIThen the gain is low, we can assume that the signal wave

emains unamplified and that only the idler grows. Thisrocess is called DFG. DFG and SHG in chirped QPMratings have been studied in detail by Imeshev et al.22,23]. They have shown that the spectrum of the gener-ted wave is related to the spectrum of the input by aransfer function that is given by the Fourier transform ofhe grating profile. In the case of linearly chirped grat-ngs, this transfer function can be expressed in terms ofresnel integrals. Its spectrum is broad and flat withapid oscillations, similar to the gain spectrum of chirped-rating OPAs.

We can recover these results by considering parametricmplification, Eqs. (3) and (4), in the low-gain limit. Inhis case, the signal amplitude is constant, and the evolu-ion of the idler is described by a single first-order differ-ntial equation that can be integrated in a straightfor-ard manner:

Ai�z� = i�As*�

z0

z

ei�z��dz�. �53�

n the case of linearly chirped gratings, �z� is quadratic,nd the solution can be expressed in terms of error func-ions or Fresnel integrals [22,23]. Alternatively, the inte-ral can be approximately evaluated using the stationaryhase method [31]:

Ai�z� iei�/4� 2�

��As

*e−�i/2��z0 − zpm�2. �54�

he gain, which was given by the Rosenbluth gain factorn the case of parametric amplification, is now equal to 2� /��1. Nevertheless, both regimes are similar inwo aspects: their gain is essentially constant over a wideandwidth, and the idler wave experiences group-velocityispersion through the frequency dependence of thePMP. Although their mathematical descriptions differ,

he physics of OPA and DFG are very similar.

. CONCLUSIONn this paper, we presented a general procedure for theesign of OPAs using nonuniform QPM gratings. Theodel used was 1D and assumed an undepleted, time-

ndependent pump wave. We considered slow-varying buttherwise general profiles for the coupling coefficient andhe wave vector mismatch. We treated the problem in therequency domain.

We solved for the wave evolution using the complexKB method. These expressions were then simplified to

eveal the main features of the amplification process. Wellustrated the use of the design formula by studying a va-iety of QPM grating profiles. We considered the canoni-al linear profile, which provides gain over wide band-idths; tapered profiles for ripple reduction; sinusoidalrofiles for selective frequency amplification; and finally,tandem-grating design for engineered gain and group

elay spectra (such as flat profiles for both).

plcaap

f2t

AGWSWffiPaoirtc�

c

wb

w

IbaSf�b

T=u

I

Tbtbaw

wsaimpeiepT

ACWItptSpW

sotr

Msyd

1FsoWlh


We also discussed the similarities existing betweenarametric amplification in chirped QPM gratings and itsow-gain limit DFG. The analysis presented here dealt ex-lusively with the spectral properties of the amplifiers. In

companion paper [28], we present the time-domainnalysis, which will lead to a discussion of the temporalroperties of the amplified pulses.The 1D model presented here ignored transverse ef-

ects such as diffraction and noncollinear interactions. AD model including those effects will be described in fu-ure publications.

PPENDIX A: DERIVATION OF THEENERALENTZEL–KRAMERS–BRILLOUIN

OLUTIONSe consider profiles for which ��z� is a smooth, monotonic

unction of position. The smoothness condition is requiredor the validity of the WKB method. Monotonicity is alsomportant because it ensures that there exists only onePMP at any given input frequency. We can define a char-cteristic chirp rate, �0�, which sets the scale of ��z�. [Inther words, we decompose the wavenumber mismatchnto a linear part, �0��z−zpm0�, and another function thatepresents the departure from linearity.] We assume thathis chirp rate is positive. The solution for the negativehirp rate can then be obtained using the substitutions�→−��, As→As

*, and Ai→−Ai*.

We normalize the position axis using the characteristichirp rate �0�:

� = �0��z − zpm�, �A1�

here zpm is the PPMP satisfying ��zpm�=0. Then Eq. (6)ecomes

d2as

d�2 + Q��as = 0, �A2�

here

Q�� = � �

2−

i

4

��

�2

− � +i��

2+

1

4��

��

. �A3�

n this expression, �=� /�0�1/2 is the normalized wavenum-

er mismatch, �=�2 /�0� is the Rosenbluth gain exponent,nd the primes denote differentiation with respect to �.ince the profiles � and � are, by assumption, smooth

unctions of position, we neglect small terms such as �� or��2 from now on. The solution is also subject to theoundary conditions:

as��0� = �−1/2��0�As0, �A4�

das��0�

d�= �−1/2��0��i�1/2��0�Ai0

* −i

2��0�As0 −

1

4

��0�

��0�As0� .

�A5�

hese come from the input conditions As�z0�=As0, Ai�z0�Ai0, where z0 is the position of the input plane in realnits.

Equation (A2) is in a form suitable for WKB analysis.ts two linearly independent WKB solutions are [31–34]

Q−1/4 exp�±i��

�Q��1/2d�� . �A6�

he conditions of validity of the WKB approximation wille discussed below. As observed in Section 2 and illus-rated in Fig. 2, the solutions are of an exponential typeetween the turning points, where Re�Q��0 (region II)nd of an oscillatory type outside the interaction regionhere Re�Q�0 (regions I and III).Approximations to the signal and idler waves can be

ritten as linear combinations of these elementary WKBolutions. As we will see, different linear combinationsre required in each of regions I–III. One way of calculat-ng the correct coefficients is to use connection formulas to

atch the WKB solutions on either side of each turningoint [31]. Another, possibly more elegant, approach is tomploy the complex WKB method [32,34], which consistsn extending the solution to the entire complex plane andnforcing continuity asymptotically far from the turningoints. In this paper we will take the second approach.he procedure for doing so is detailed in Appendix B.

PPENDIX B: APPLICATION OF THEOMPLEXENTZEL–KRAMERS–BRILLOUIN METHOD

n this section we apply the complex WKB method to ex-end each elementary solution to the entire complexlane. We establish the Stokes diagram corresponding tohe function Q and the use of the usual rules for crossingtokes and anti-Stokes lines. (Complete accounts of theserocedures are given by Heading [32], Budden [34], andhite [37].)To uniquely define the WKB solutions we have to

pecify the lower bound of integration. It is typical to takene of the two complex turning points �1 and �2, which arehe values of � for which Q=0. Following Heading [32], weepresent the WKB solutions by the notation

��1,�� Q−1/4 exp�i��1

�

�Q��1/2d�� , �B1�

��,�1� � Q−1/4 exp�i��

�1

�Q��1/2d�� . �B2�

oreover, and still following Heading, we will use theubscripts d or s according to whether a solution is as-mptotically dominant (exponentially growing) or sub-ominant (exponentially decaying) as �� →�.The Stokes diagram of the function Q is shown in Fig.

1. We assume that the turning points are well separated.rom each turning point emerges a branch cut, which wepecify to be away from the real axis: three Stokes linesn which Re�Q1/2d��=0, where the magnitude of the twoKB solutions are most different; and three anti-Stokes

ines on which Im�Q1/2d��=0, where the two solutionsave equal magnitude. We number the various regions

fte�c

s

Tcl

Tr

Am

Wp

w

I

F1l

Tph

�l

Iwsttgt

AWSWIrtliupg

T�

FSz


rom 1 through 8 as shown in Fig. 11. Our goal is to de-ermine how each global WKB solution becomes a differ-nt linear combination of ��1 ,�� and �� ,�1� or ��2 ,�� and� ,�2� when � moves from one region to another in theomplex plane.

Let us start from ��1 ,�� in region 8 �Re��→−� �. Thisolution being dominant in this sector, we write

region 8: ��1,��d. �B3�

o go to region 7 we cross a Stokes line in the counter-lockwise direction, so we must add the subdominant so-ution multiplied by the Stokes coefficient i:

region 7: ��1,��d + i��,�1�s. �B4�

o get to region 6, we cross an anti-Stokes line and theoles of dominance and subdominance are interchanged

region 6: ��1,��s + i��,�1�d. �B5�

gain, as we cross the Stokes line to go to region 5, weust add the subdominant solution:

region 5: ��1,��s + i��,�1�d + i��1,��s� = i��,�1�d. �B6�

e can match regions 5 and 3 by connecting turningoints �1 and �2:

region 3: i��2,�1��,�2�s, �B7�

here we have introduced the coefficient

��2,�1� � exp�i��2

�1

�Q��1/2d�� . �B8�

n region 2, this solution becomes dominant:

region 2: i��2,�1��,�2�d. �B9�

inally, as we cross the Stokes line between regions 2 andin the clockwise direction, we add the subdominant so-

ution multiplied by −i:

ig. 11. Stokes diagram of the function Q defined by Eq. (A3).olid curve, anti-Stokes curves; dashed curves, Stokes curves;ig–zag, branch cut.

region 1: i��2,�1��,�2�d − i��2,��s�. �B10�

hus we obtain the continuation in the complex plane. Inarticular, we have a representation of the asymptotic be-avior of this solution over the entire real axis.Now let us consider the second solution. We start from

� ,�1� in region 8. Proceeding as before, we have the fol-owing connections:

region 8: ��,�1�s, �B11�

region 7: ��,�1�s, �B12�

region 6: ��,�1�d, �B13�

region 5: ��,�1�d + i��1,��s, �B14�

region 3: ��2,�1��,�2�s + i��1,�2��2,��d ��2,�1��,�2�s,

�B15�

region 2: ��2,�1��,�2�d, �B16�

region 1: ��2,�1��,�2�d − i��2,��s�. �B17�

n going from regions 5 to 3, we have dropped the termith the coefficient ��1 ,�2� because it is exponentially

mall compared to ��2 ,�1�. (This follows the assumptionhat the turning points are well separated.) We note thathe two global solutions, which can be represented in re-ion 8 by ��1 ,�� and �� ,�1�, only differ in region 1 by a fac-or of i.

PPENDIX C:ENTZEL–KRAMERS–BRILLOUIN

OLUTIONS FOR THE SIGNAL AND IDLERAVES

n Appendix B we have given the proper asymptotic rep-esentations for two global, linearly independent solu-ions. Now we are left with the task of determining theinear combinations that satisfy the boundary conditionsn Eqs. (A4) and (A5). Then each solution can be contin-ed over the entire real axis using the results from Ap-endix B. We obtain the following solutions in each re-ion:

asI Cs

+��1,�� + Cs−��,�1�, � � − �tp, �C1�

asII �iCs

+ + Cs−��,�1�, − �tp � � � �tp, �C2�

asIII �iCs

+ + Cs−��2,�1��,�2��1 − i

��2,��

��,�2��, � � �tp.

�C3�

he coefficients Cs± are given approximately (provided

0�−�tp) by

Cs+

1

��1,�0��1/2��0��1 +��0�

��0�2As0 −�1/2��0�

��0�Ai0

* � ,

�C4�

Tt

mgpgtwaT�tr

AWSA(

taWatcadvg

cutCp�tse

bsalta

ATLfQg

se

F

S

Fs

Tp

w


Cs−

1

��0,�1��1/2��0��−��0�

�2��0�As0 +

�1/2��0�

��0�Ai0

* . �C5�

he corresponding coefficients for the idler wave are ob-ained by interchanging the role of the signal and idler.

We recover the frequency-domain envelope functions byultiplying by �1/2 exp�i /2�, resulting in the expression

iven in the text, Eq. (10). Even when only one wave isresent initially (e.g., Ai0=0), the coefficients Cs

± are ineneral both nonzero. The solution is then the superposi-ion of a positive and a negative complex exponential,hich interferes and causes the ripple observed on themplification spectrum (as discussed in Subsection 4.C).hese oscillations can be suppressed by letting either��0�→0, ��L�→0 or ��0�→�, ��L�→� (where �L referso the output plane). This is accomplished by the ripple-eduction techniques presented in Subsection 4.C.

PPENDIX D: VALIDITY OF THEENTZEL–KRAMERS–BRILLOUIN

OLUTIONSfew key assumptions have been made to obtain Eqs.

C1)–(C3). They will be examined in this section.We have assumed when obtaining the continuation of

he solution in the complex plane that the turning pointsre well separated. This requirement is necessary for theKB solutions to be valid between the turning points. It

lso allowed us to drop the exponentially decaying solu-ion when connecting between the turning points. In thease of a linear grating for which the magnitude of themplification factor is ��2 ,�1� � =exp��, neglecting theecaying exponential is valid provided ��1/�. Thisalue can be considered as the threshold of significantain.

Second, we have assumed that neither turning point islose to the edges of the grating. This allowed us to definenambiguously the three regions I–III. The assumptionhat �0�−�tp was used to obtain the coefficients C+ and−. Similarly, when obtaining expressions involving theosition of the output of the grating, �L, we assume thatL��tp. Consequently, the description given here ceaseso be valid at the edges of the spectrum because our re-ults cannot be applied if the gain region reaches thedges of the grating.

To summarize, the WKB formalism developed here isest suited when (i) the gain is large, (ii) the gain region ishort compared to the grating length, and (iii) the PPMPsre far from the edges of the grating. In the case of theinear profile described in Subsection 4.B, these condi-ions are satisfied when �=�2 / �� 1/�, Lg=4� / �� L,nd ��BW.

PPENDIX E: EXPLICIT EXPRESSIONS FORHE LINEAR PROFILEinear grating profiles are given simply by �=�. There-

ore, assuming a constant coupling coefficient, we have¯ =�2 /4−�+ i /2. The complex turning points are theniven by � ,� = �2��− i /2�1/2.
1 2
In this case the integrals appearing in the WKB expres-ions can be evaluated exactly; approximate and simplerxpressions are useful, however. First, for �0�−�tp:

�a

�0

Q1/2d�� = �� − i/2�� 0

2 � − i/2 �0

2

4�� − i/2�− 1

+ ln� ��0�

2 � − i/2+ �0

2

4�� − i/2�− 1��

−�0

2

4+ �� − i/2�ln� ��0�

�1/2 +�

2. �E1�

or −�tp��tp:

�a

�

Q1/2d�� i ��1 −i

4�� +

i��

2+

�

4. �E2�

imilarly, for �L��tp:

�b

�L

Q1/2d�� L

2

4− �� − i/2�ln� �L

�1/2 −�

2. �E3�

inally, the integral between the two turning points isimply

�a

b

Q1/2d�� = i�� − i/2�. �E4�

he solutions for the linear profile can then be written ap-roximately as

AsI�� 1 + �2��0��As0 + ��0�Ai0

* �exp�i� ln� �

�0�

��1 − ��Ai0* + ��0�As0

As0 + ��0�Ai0*

�exp� i

2��2 − �0

2� − 2i� ln� �

�0�� , �E5�

AsII��

1

2exp��

2+ �1/2� − i� �

4�1/2 +�2

4+ � ln ��0�

−�

2� � ��1 + �2��0��As0 − ��0�Ai0* − �Ai0

* − ��0�As0�

�exp�− i��0��, �E6�

AsIII�� exp�� + i� ln� �

�0�

�F��F*��0�As0 − F��0�e−i��0�Ai0* �, �E7�

ith

F�� = 1 − ��exp�i��, �E8�

�� =�2

2+ 2� ln �� − � +

�

2, �E9�

Tu

tacatst

rra�c

ASTasm

a

cattopacctomh

T

A

mW

Fo


�� =�1/2

��. �E10�

he expressions for the signal wave are obtained, assual, by interchanging the signal and idler subscripts.The spatial evolution of the signal and idler magni-

udes for typical parameter values is plotted in Fig. 12long with the numerical solution. The solutions are os-illatory before and after the gain region where the inter-ction is phase mismatched while they grow exponen-ially in the phase-matched region. As expected, the WKBolution is not a very good approximation in the vicinity ofhe turning points.

Finally, it is interesting to point out that the ripple-eduction schemes described in Subsection 4.C (namely,eduction of � or increase of � at the ends of the grating)mount to reducing the values of ��0� and ��L�. Setting��0�=��L�=0 into Eqs. (E5)–(E7) leads to major simplifi-ations:

AsI�� As0ei� ln��/�0�, �E11�

AsII��

1

2�As0 + iAi0

* e−i��02/2−��

�exp��

2+ �1/2� − i� �

4�1/2 +�2

4−

�

2� ,

�E12�

AsIII�� e��As0 + iAi0

* e−i��02/2−��ei� ln��/�0�. �E13�

PPENDIX F: DERIVATION OF THEIMPLIFIED DESIGN FORMULAhe WKB solutions, Eqs. (C1)–(C3), are quite complicateds they involve several integrals of the function Q1/2. Toimplify these expressions, the integrals can be approxi-ately evaluated for general (smooth) grating profiles.Let us outline the evaluation procedure using as an ex-

mple ��2

�LQ1/2d�. We neglect the variation of the coupling

ig. 12. Comparison between the WKB solution and the numerif the grating: (a) signal, (b) idler. The insets give details on the

oefficient, therefore we can approximate Q from Eq. (A3)s Q �2 /4−�+ i�� /2. The integral can be separated intowo parts. The first one corresponds to the vicinity of theurning point, where ��z� is approximately linear; the sec-nd one corresponds to the portion away from the turningoint, where � is large. Thus the two integration rangesre from �2 to �, and from � to �L, respectively, with � lo-ated on the real axis somewhere between Re��2� and �L,lose enough to �2 so that � is linear but far enough sohat the behavior of Q is dominated by �2 /4. The integralver the linear range is given by Eq. (E3). Retaining theost significant real and imaginary contributions, weave

��2

�

Q1/2d� �2

4+

i

2ln� �

�1/2 . �F1�

he second integral can be approximated by

��

�L

Q1/2d� 1

2��

�L

�d� +i

2ln� ��L�

� . �F2�

dding both contributions gives

��2

�L

Q1/2d� 1

2�0

�L

�d� +i

2ln� ��L�

�1/2 . �F3�

The integral ��1

�0Q1/2d� can be evaluated in a similaranner. Using these approximations, the elementaryKB solutions can be written as

��1,�0� 21/2�−1/4��0�ei/2��0

0 ��d�, �F4�

��0,�1� 21/2�1/4��0�

��0��e−i/2��0

0 ��d�, �F5�

��2,�L� 21/2�1/4��0�

��L�ei/2�

0�L��d�, �F6�

��L,�2� 21/2�−1/4��0�e−i/2�0�L��d�. �F7�

ution for �=1, As0=1, Ai0=0, L=20, and �pm located at the centerudes near the input.

cal solamplit

S

w−iyi

Tt

w

ATt1aewbpma

R

1

1

1

1

1

1

1

1

1

1

2

2

2

2

2

2

2


imilarly, the amplification factor can be approximated by

��2,�1� = ei��2

�1Q1/2d� − ie��1*

�2*�� − �2/4�1/2d�, �F8�

here �1,2* are the real-valued turning points of �

�2�� /4. These approximations can then be substitutednto the expressions for the signal and idler, Eq. (C3), toield the following formula valid for general smooth grat-ng profiles:

AsIII ��1 −

i�1/2��0�

��0�ei��0

0 ��d�As0 + iei��0

0 ��d�

��1 +i�1/2��0�

��0�e−i��0

0 ��d�Ai0* �

�e��1*

�2*�� − �2/4�1/2d��1 −

i�1/2��L�

��L�ei�

0�L��d� . �F9�

o simplify the formulas further, we neglect the oscilla-ory terms. This step yields

As �As0 + iAi0* ei��0

0 �d��e��1*

�2*�� − �2/4�1/2d�, �F10�

hich is the design formula, Eq. (14).

CKNOWLEDGMENTShis work was sponsored by the Air Force Office of Scien-ific Research (AFOSR) under AFOSR grants F49620-02--0240 and F49620-01-1-0428. M. Charbonneau-Lefortcknowledges additional support from the Natural Sci-nces and Engineering Research Council of Canada. Theork of B. Afeyan was supported by the Laser-Plasmaranch of the Naval Research Laboratory and by a De-artment of Energy (DOE) National Nuclear Security Ad-inistration (NNSA) Stewardship Science Academic Alli-

nces (SSAA) grant.

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Documents

Optical parametric ampliﬁers using chirped quasi- …nlo.stanford.edu/system/files/mcl_josab_2008_1.pdfchirped QPM gratings used as broadband OPAs. The easi-est model to describe