analysis and optimization of optical parametric chirped pulse amplification

Ross et al. Vol. 19, No. 12 /December 2002 /J. Opt. Soc. Am. B 2945

Analysis and optimization of optical parametricchirped pulse amplification

Ian N. Ross and Pavel Matousek

Central Laser Facility, Council for the Central Laboratory of the Research Councils, Rutherford AppletonLaboratory, Chilton, Didcot, Oxfordshire OX11 0QX, United Kingdom

Geoffrey H. C. New

Blackett Laboratory, Imperial College, Prince Consort Road, London SW7 2BW, United Kingdom

Karoly Osvay

Department of Optics and Quantum Electronics, University of Szeged, P.O. Box 406, Szeged 6701, Hungary

Received January 9, 2002; revised manuscript received June 26, 2002

Optical parametric amplification can give particularly high values for gain, gain bandwidth, energy, efficiency,and wave-front quality. In combination with chirped pulse amplification, in a technique we call optical para-metric chirped pulse amplification, it offers the prospect of generating peak powers up to 100 PW and inten-sities greater than 1024 W/cm2 with existing technology. Here we study the technique in detail using bothanalytical and computational techniques, and the limit of validity of the analytical approach is identified. Theeffects of spectral phase, energy extraction, signal pulse chirp, and pump pulse nonuniformity are analyzed,and optimization techniques are proposed and discussed. © 2002 Optical Society of America

OCIS codes: 140.3580, 140.7090, 190.4410, 190.4970.

1. INTRODUCTIONOptical parametric devices play a key role in many laserapplications. They have proved versatile as widely tun-able coherent sources, especially for short pulses sincethey can offer both high gain and high-gain bandwidth.Gain bandwidths of several thousand wave numbers arepossible,1–6 corresponding to transform-limited pulse du-rations down to a few femtoseconds, and amplification ofsuch short pulses has been demonstrated.7,8 In addition,high powers have been generated over a wide range ofwavelengths.9–11 At these extremes, however, it becomesincreasingly important to analyze the optical parametricprocess in detail to understand the limits of operation andto enable optimum performance to be realized experimen-tally. A number of aspects of optical parametric amplifier(OPA) performance have been treated analytically12–15

and computationally, but these do not fully cover the situ-ation under consideration here.

A new technique has been proposed that combines op-tical parametric amplification with chirped pulse amplifi-cation (CPA) in a process we call optical parametricchirped pulse amplification (OPCPA).16–18 The key prin-ciple of OPCPA is to amplify a broad bandwidth linearlychirped signal pulse with an energetic and relativelynarrow-bandwidth pump pulse of approximately the sameduration. If the process is implemented for the appropri-ate geometry in a large nonlinear crystal, it is possible toachieve recompressed signal pulses with powers and in-tensities orders of magnitude larger than values currentlyavailable.

This paper contains a detailed analysis of OPCPA, in-

0740-3224/2002/122945-12$15.00 ©

cluding ways of optimizing the process. We address thefollowing specific issues:

(a) The influence of chirp on the OPA process.(b) The optimization of OPCPA with respect to effi-

ciency and bandwidth, particularly in the presence ofnonflat temporal pulse profiles.

(c) Calculation of the phase imposed on the amplifiedsignal pulse by the OPA process and its influence on thefinal recompressed pulse profile.

(d) The effect of group velocity mismatch.(e) The tolerances to misalignment and input phase ab-

errations on the pump beam.(g) The use of a chirped pump pulse and the possibility

of extending the bandwidth by use of a chirp compensa-tion scheme.

The primary analytical tool is a formulation based onthat of Armstrong et al.,12 which gives useful and instruc-tive results despite its inherent approximations. The va-lidity of this analysis will be assessed, and the resultstested against a computer model with fewer simplifyingassumptions (Section 6). Two model systems of particu-lar interest will be considered. One is a broadband type IOPCPA pumped by the second harmonic of a Nd:glass la-ser at 526 nm, which is the scheme that leads to genera-tion of the highest powers. The second, also a type IOPCPA, uses a frequency-doubled Nd:YAG laser to pumpa b-barium borate (BBO) OPCPA in both a collinear and anoncollinear geometry. This illustrates the high-gainbandwidths achievable at more modest energies.

2002 Optical Society of America

2946 J. Opt. Soc. Am. B/Vol. 19, No. 12 /December 2002 Ross et al.

2. ANALYSISThe analysis of Armstrong et al.12 is based on the plane-wave solution of the coupled-wave equations that assumea slowly varying pulse envelope and a lossless medium.This yields an analytical solution for the intensity andphase of the three propagating waves in the presence ofsignificant pump depletion and/or a phase mismatch.

In this paper we are primarily concerned with the am-plification of chirped pulses in optical parametric amplifi-ers and in particular with geometries that closely matchthe group velocities of the signal and idler pulses sincethis yields the largest gain bandwidth as will be shownlater. For these cases, the analytical solutions are validto first order if a time-dependent phase mismatch is in-cluded to take account of the varying phase mismatchacross the signal and idler spectra. This approach doesnot account for higher-order effects such as group velocitydispersion of the signal and idler pulses, or for the timeslippage between signal and pump pulses. The limit ofvalidity can then be expected to occur when the grouptime difference between pump and signal over the crystallength is a significant fraction of the pulse duration. Anassessment of this limit will be carried out in Section 6 byuse of numerical simulation.

A. Solution for IntensityThe solution for the output intensity of an OPA was givenby Eq. 6.9 of Ref. 12. With some manipulation and sub-stitution, this equation can be rewritten in terms of mea-surable parameters as

2gz 5 6E0

f~z ! df

$ p~1 2 f !~ f 1 gs2!~ f 1 g i

2! 2 @gsg i cos u~0 !Ap 1 fDk/2g#2%1/2, (1)

2gz 5 6E0

f df

@ p~1 2 f !~ f 1 gs2!f 2 ~Dk/2g !2f 2#1/2 .

(2)

Amplification continues until, at propagation distance z5 zA , the fractional pump depletion f rises to a maxi-mum value at which the denominator of Eq. (2) vanishes,which is determined by

fmax2 2 F1 2 gs

2 21

p S Dk

2g D 2G fmax 2 gs2 5 0. (3)

In the case of perfect phase matching (Dk 5 0), fmax5 1, and the pump is fully depleted; the pump-to-signalconversion efficiency is then 100% times vs /vp . With anincrease in phase mismatch, the conversion efficiency isreduced. As will become clear later from the solution forthe phase, sin u changes sign at this asymptotic point and,for z . za , the process reverses symmetrically to transferenergy back into the pump from the signal and the idler.

Equation (2) has the form of a Jacobi elliptical integral,its solution being written as the elliptical function sn.Since energy conservation requires that the energy lost bythe pump be divided between signal and idler in propor-tion to their frequencies, we can write the signal intensityas

Is 5 fIp~0 !vs

vp1 Is~0 !. (4)

For practical applications it is useful to estimate the pulse

where

g 5 4pdeffF Ip~0 !

2e0npnsniclsl iG1/2

;

f 5 1 2 Ip /Ip(0) is the fractional depletion of the pumpbeam;

gs2 5

vp

vs

Is~0 !

Ip~0 !, g i

2 5vp

v i

Ii~0 !

Ip~0 !

are the input photon intensity ratios; p 5 Ip(0)/@Ip(0)1 Is(0) 1 Ii(0)# is the pump to total input intensity ra-tio; u(t) 5 fp(t) 2 fs(t) 2 f i(t) is the OPA phase; andDk is the crystal phase mismatch per unit length.

Equation (1) is applicable to any three-wave mixingprocess. The (1) and (2) sign option in front of the inte-gral corresponds to positive and negative values of sin u,which, respectively, deliver amplification and attenuationof the signal. If there is no input idler intensity @Ii(0)5 0#, the initial idler phase w i(0) self-adjusts to ensuremaximum initial signal gain, in which case u(0)5 2p/2. In this case, Eq. (1) reduces to

energy (W) by integration of the intensity over space andtime, yielding for the signal beam

Ws 5 EE IsdAdt. (5)

Note that, as it stands, the analysis ignores group velocityand pulse chirping effects. However, to take account ofthe wavelength dependence of Dk a simple extension toaccommodate chirped pulses can be effected by assuminga time-dependent value for Dk for evaluation of Eq. (2);the phase mismatch is of course usually zero at one wave-length only. Group velocity effects in the OPA will resultin group velocity dispersion within each pulse, as well asa temporal slippage among the pump, the signal, and theidler pulses. If the beams are also chirped, this will ingeneral be accompanied by a relative wavelength slippagebetween beams. More exact computer simulations to bedescribed below show that the above solutions are validprovided (as noted earlier) the time slippage is small com-pared to the pump pulse duration. The cases under con-sideration satisfy this requirement.

B. Solution for PhaseWith the increasing interest in shorter and shorterpulses, careful attention must be paid to the analysis of


the spectral phase that is reflected in the temporal phaseof amplified chirped pulses. We obtained the phases in athree-wave mixing process by solving the imaginary partsof the coupled-wave equations that yield

dfs

dz5 2K

vs2

ks

rpr i

rscos u,

df i

dz5 2K

v i2

ki

rprs

r icos u,

dfp

dz5 2K

vp2

kp

rsr i

rpcos u, (6)

where K 5 (2p/c2)xeff , xeff is the effective nonlinear sus-ceptibility, and the amplitude of each wave has been writ-ten in the form r exp(if ). Equations (6) are readily com-bined and integrated to give12

cos u 5 2S G 1dk

2Avpgup

2D Y upusui , (7)

where up2 5 Ip /vpI0 , Ip 5 (c2kp)/(8pvp)rp

2 and analo-gous relations apply for the signal and the idler, I0 5 Ip1 Is 1 Ii , and G is a constant of integration determinedby the initial conditions.

Using Eq. (7) to eliminate u from Eqs. (6) yields

dfs

dz5

vs~AvpgGI0 1 DkIp/2vp!

Is,

df i

dz5

v i~AvpgGI0 1 DkIp/2vp!

Ii,

dfp

dz5

vp3/2gGI0

Ip1

Dk

2. (8)

These equations apply to the general case of three-wavemixing and can be integrated by use of the solutions of theintensity equations. For the case with which we are pri-marily concerned, an OPA in which the zero initial idler(Ii(0) 5 0), the solution for G is AvpgGI05 2DkIp(0)/2vp . Substituting for G into Eqs. (8) andmaking use of the Manley–Rowe relations yield

dfs

dz5 2

Dk

2F1 2

gs2

~ f 1 gs2!

G ,

df i

dz5 2

Dk

2,

dfp

dz5 2

Dk

2

f

1 2 f. (9)

The initial phases of pump and signal are determined bythe input beams and, as noted earlier, the input phase ofthe idler adjusts itself to maximize the signal gain. Byinspection of the coupled-wave equation for the signal, itcan be seen that this occurs at sin u 5 21 or f i(0)5 fp(0) 2 fs(0) 2 p/2.

Finally, by integrating Eqs. (9) we can write the equa-tions for the phase of the three waves as

fs 5 fs~0 ! 2Dkz

21

Dkgs2

2E dz

f 1 gs2 ,

f i 5 fp~0 ! 2 fs~0 ! 2p

22

Dkz

2,

fp 5 fp~0 ! 2Dk

2E fdz

1 2 f. (10)

Inspection of these equations allows one to make the fol-lowing statements about the phase relationships in anOPA:

(a) The phase of the amplified signal is independent ofthe initial phase of the pump and consequently it is pos-sible to maintain the optical quality of the signal while weuse, for example, a pump with both spatial aberrationsand having temporal phase variations that result from achirp.

(b) Phase changes resulting from amplification of thesignal and idler occur only at wavelengths for which thereis a phase mismatch (Dk Þ 0).

(c) The phase of the idler is particularly simple [seeEqs. (10)], depending only on the initial pump and signalphases and the phase mismatch term Dkz/2.

(d) A good approximation to the phase of the signal canbe obtained if the input signal intensity is small comparedto the input pump intensity. In this case y2 ! f andthere is only a small contribution to the integral part ofthe equation for fs from the region of significant pumpdepletion. After some manipulation, we can then use thelow depletion solution for the signal phase18:

fs 5 fs~0 ! 2Dkz

21 tan21H Dk

2@ g2 2 ~Dk/2!2#1/2

3 tanh@ g2 2 ~Dk/2!2#1/2zJ . (11)

(e) The direction of energy flow in an optical parametricprocess is determined by the phase term u. When thereis no input idler field, the initial idler phase self-adjusts tou(0) 5 2p/2 and energy is transferred from pump to sig-nal and idler. By combining Eqs. (3) and (7), we canshow that, at maximum depletion, cos u 5 1 or u 5 0.With further propagation, u becomes positive and the di-rection of energy flow is reversed.

3. NONCOLLINEAR GEOMETRYIt is important to consider the case of noncollineargeometry,19 which leads to the highest gain bandwidthsfor an OPA. The following simple analysis enables us tounderstand and optimize this geometry.

A. Gain Bandwidth on the Signal BeamWe confine our attention to the important case of anarrow-bandwidth pump beam and a signal beam that isnot angularly dispersed. The phase-matching vector dia-gram for the general noncollinear case is shown in Fig. 1.The variation of the phase-matching angle j (defined asthe angle between pump beam and c axis) with the non-collinear angle a can be determined from the cosine rule

2kpks~1 2 cos a! 5 ki2 2 ~kp 2 ks!

2, (12)

where k 5 2pnv/c, with appropriate subscripts forpump, signal, and idler beams; vi 5 vp 2 vs by conserva-tion of energy; and the variation with j is contained


within the extraordinary wave vectors. For type I phasematching, the pump beam is the extraordinary wave andthe variation with j is determined from

np22 2 npo

22 5 ~npe22 2 npo

22!sin2 j, (13)

where npo and npe are the ordinary and extraordinary in-dices for the pump beam. An example is given as curve Ain Fig. 2, which shows the variation of a with j for BBOwith a pump at 532 nm and a signal at 800 nm.

A variation in the signal wave vector Dks that is due toa change in signal wavelength will, in general, lead to aphase mismatch over the length of the crystal. However,we can seek to achieve maximum bandwidth by requiringthat the variation of the phase mismatch with wave-length is zero to first order. By differentiating Eq. (12)and using the fact that dvi /dvs 5 21 and dki /dks5 2ngi /ngs (where ng refers to the group index), thebroad bandwidth condition can be written as

kp~1 2 cos a! 5 kp 2 ks 2 kingi /ngs . (14)

Combining Eqs. (12) and (14) leads to the requirementthat

cos b 5 @ngi /ngs#. (15)

As pointed out by Wilhelm et al.,20 this simple result isjust the condition for group velocity matching of the sig-nal and idler beams in the signal beam direction. It is

Fig. 1. Phase-matching k-vector triangle for noncollinear opti-cal parametric amplification.

Fig. 2. A, Variation of the phase-matching angle with the non-collinear angle for a BBO OPA with a 532-nm pump and an800-nm signal beam. B, Condition for maximum gain band-width. C, Condition for maximum angular tolerance for thepump beam.

also the condition for individual signal and idler wave-lengths within chirped pulses to remain in step over longdistances.

It is also useful to write Eq. (15) in terms of the fixedangle between pump and signal:

sin a 5 ki /kp sin~cos21 ngi /ngs!. (16)

Curve B in Fig. 2 shows the variation of the optimum non-collinear angle with phase-matching angle (which is re-flected in the value of kp) for type I BBO pumped at 532nm. The intersection of curves A and B at a 5 2.3 degindicates that maximum bandwidth should be achieved atthis point. Note also that this geometry results in an an-gularly dispersed idler wave.

B. Spectral Bandwidth Tolerance on the Pump BeamThe condition for maximum pump bandwidth can be cal-culated from Eqs. (12) and (13) and is represented byequations that are similar to Eqs. (15) and (16), namely,

cos g 5 ngi /ngp , sin a 5 nils sin g/nsl i . (17)

For the BBO example, the optimum values are g5 10.8 deg and a 5 5.4 deg. This value for a is sub-stantially different from that for the maximum signalbandwidth, but this does not pose a problem for theOPCPA schemes proposed, all of which use a narrow-bandwidth pump beam.

C. Angular Tolerance on the Pump BeamSome degree of angular tolerance (dz) on the pump waveis desirable to allow one to use a spatially imperfect beamor a narrow beam that necessarily has a significant angu-lar content. Simulations21 of the effect of finite beam sizeenable a minimum acceptable angular tolerance to bespecified. We calculated the angular tolerance by differ-entiating the k vector Eq. (12) with respect to a small an-gular error (dz) on the pump beam. For a type I processthis yields

df 5L

kiF ~kp 2 ks cos a!

kp

np

dnp

dj1 kpks sin aGdz,

(18)

where df is the phase error that results from the angularerror dz. For the simplest case of coplanar geometrywith the signal beam between the pump beam and the caxis, an in-plane angular increment dz of the pump beamimplies an equal change in both the phase-matchingangle (dz) and the noncollinear angle (da).

Maximum tolerance is achieved by setting dw to zero inEq. (18). After some rearrangement and with the help ofEq. (13), this yields

sin a 51

2 S npls

nslp2 cos a Dnp

2~npe22 2 npo

22!sin 2j.

(19)

Curve C in Fig. 2 shows the relationship between the non-collinear and the phase-matching angles for the BBO ex-ample. The coincidence between curves A and C indi-cates that the maximum angular tolerance occurs at a5 1.6 deg. Since the two coincidence points in Fig. 2 are


quite close, BBO can be operated with both high band-width and high angular tolerance.

For the same coplanar geometry but for an out-of-planeangular increment on the pump beam, j and a are af-fected only to second order, regardless of the values of jand a. The tolerance is consequently also large in thisplane. For noncoplanar geometry, curve C in Fig. 2 canbe adjusted to bring the coincidence points together, toachieve maximum bandwidth and optimal angular toler-ance at the same time. However, this occurs only for anangular variation of the pump beam in a one plane; in theorthogonal plane the tolerance will be reduced.

D. Broad Bandwidth by Use of Chirp CompensationA large gain bandwidth can be generated in a differentway. Previously suggested for harmonic generation,22–24

this can be referred to as a chirp-compensation techniquesince it requires the imposition and control of a chirp onthe pump pulse to compensate for the chirp on the signal.The operation of this scheme follows from the phase-matching Eq. (12), which can be solved for fixed values ofthe noncollinear and phase-matching angles to give thepump wavelength as a function of the signal wavelengthto maintain phase matching. The result for our BBO ex-ample is plotted in Fig. 3(a). The slope of this curve,shown in Fig. 3(b), gives the associated pump-to-signalchirp ratio. Note that the value is always less than unity,so the signal bandwidth is larger than that of the pumpand leads to a shorter compressed pulse. For the caseshown in Fig. 3, an amplified and recompressed 10-fs sig-nal pulse at 800 nm is possible by use of a chirped 532-nmpump pulse with a bandwidth that corresponds to 60 fs.

Fig. 3. Chirp compensation scheme for a collinear BBO forwavelengths close to 532 nm for the pump and 800 nm for thesignal: (a) the required variation of pump wavelength with sig-nal wavelength to maintain phase matching and (b) the ratio ofpump-to-signal pulse chirp necessary to maintain phase match-ing.

4. CALCULATED EXAMPLES ANDOPTIMIZATIONA. Top-Hat Profile BeamsFor beams of uniform intensity in space and time, the in-tensity and phase of the amplified signal can be calcu-lated by use of Eqs. (2) and (10). We can model chirpedpulses by making the phase mismatch time dependentand by using the material dispersion equations. We il-lustrate this procedure for an OPCPA that consists of atype I BBO crystal operating near degeneracy andpumped by a narrow-bandwidth line at 532 nm (secondharmonic of a Nd:YAG laser). Since the OPA can deliverhigh stage gain in comparison with the low gain of a con-ventional amplifier, the intensity and interaction lengthwere chosen to give a parametric gain of 106.

Figure 4 shows the calculated signal and pump inten-sities for zero phase mismatch as a function of distance inthe crystal up to and beyond the region of maximumpump depletion. The cyclic nature of the process is atonce apparent and indicates that 100% depletion of thepump is possible; however, this can only be achieved forbeams of uniform intensity and at specific values of crys-tal length and beam intensity. The figure also shows theevolution of signal and pump at wavelengths at whichthere is a significant phase mismatch (DkL 5 3p at z5 za). The maximum pump depletion is now only ;50%and requires a greater length of crystal to develop.

Figure 5(a) illustrates the development in the crystal ofthe calculated output signal energy and the spectralbandwidth in the region of maximum pump depletion. Itis apparent that maximum efficiency and spectral band-width do not occur at the same point. This turns out tobe true in almost all cases of interest, and further analy-sis is needed if the system is required to give the best bal-ance of efficiency and bandwidth. As a qualitative esti-mate of optimal balance, we can take the maximumproduct of the two, since this gives a measure of the maxi-mum peak power of the recompressed pulse in a CPA sys-tem. Figure 5(b) illustrates the different development ofthe efficiency and efficiency–bandwidth product throughthe region of maximum depletion. For a proper quanti-tative assessment, the shape of the spectrum should alsobe taken into consideration.

The output spectrum with greatest efficiency–bandwidth product is displayed in Fig. 6(a) together with

Fig. 4. Energy conversion in an OPA from pump (upper curves)to signal (lower curves) for a wavelength with perfect phasematching and a wavelength with a phase mismatch of 3p at z5 za . Solution is for a plane wave with uniform intensity inspace and time.


Fig. 5. Axial variation of output signal beam parameters for aBBO OPA with a spatially and temporally uniform pump beam at532 nm and a collinear 800-nm signal beam uniform in space andtime and having large input spectral bandwidth: (a) comparisonof the energy transfer efficiency into the signal and the signalspectral bandwidth and (b) comparison of the energy transfer ef-ficiency and the efficiency–bandwidth product.

Fig. 6. (a) Amplified signal intensity and phase for the maxi-mum efficiency–bandwidth product of Fig. 5. The crystal lengthis 1.5 mm and the pump intensity is 120 GW/cm2. (b) Fouriertransform of this spectrum, indicating the optimum recom-pressed pulse profile. Phase terms up to the cubic are assumedto be compensated.

the signal phase calculated from Eq. (10). Figure 6(b)shows the resulting recompressed pulse profile calculatedby taking the Fourier transform of the amplitude andphase of Fig. 6(a) and assuming that the compressor is setto compensate for linear and quadratic chirps. Althoughthis represents an idealized case, the potential of theOPCPA scheme is demonstrated in these calculations,which give an amplified bandwidth of 3140 cm21 and a re-compressed pulse duration of 8.6 fs.

1. Gain Spectral Narrowing and Optimization in thePresence of Limiting ProcessesThe effects demonstrated above result from a kind of gainspectral narrowing in the OPA. However, there is an im-portant difference between an OPA and a conventionalamplifier in this respect. The OPA gain spectral narrow-ing is primarily a consequence of the phase mismatch(DkL), which is a function only of the crystal length and,unlike a conventional amplifier, has only a small depen-dence on the gain. The OPA gain also depends on thecrystal length but, because it is a function of gAIpL, italso depends on the pump intensity. This makes it pos-sible to achieve a large bandwidth and a high gain in theOPA by use of a high pump intensity and a short crystal.We illustrate this property in Fig. 7 by plotting the gainbandwidth against gain for both constant pump intensityand constant crystal length.

Since maximum pump intensity is desirable to maxi-mize the gain bandwidth, one must take care not to ex-ceed the limits on intensity resulting from either nonlin-ear processes or damage. Competing nonlinear processestypically set a limit on the intensity–length product

E0

L

Idz < A, (20)

where A is a constant determined by the process and thematerial parameters. The process in question might beself-focusing or self-phase modulation for which there is alimit on the so-called B integral, or it might be two-photonabsorption or Raman generation. For BBO at the param-eters of interest here, the B integral is plotted in Fig. 8together with a line defining a gain of 106. The maxi-mum intensity and hence also bandwidth is determinedby the crossing point of the two graphs and it can be seenthat the B integral is the key parameter up to approxi-mately 10 ps. Optical damage is also represented in Fig.8 by a limit on the intensity, which is pulse length depen-dent. For pulses longer than 10 ps, the damage thresh-old requires that the pump intensity be reduced and thecrystal length increased, with a consequent reduction in

Fig. 7. Variation of OPA gain bandwidth with gain at either con-stant intensity or constant crystal length for BBO at degeneracy.


the gain bandwidth. For 100-ps pulses, for example, theintensity must be reduced by a factor of approximately 6,leading to a reduction in gain bandwidth by more than afactor of 2.

2. Maximum Bandwidth OptionsAmong materials commonly used for OPAs, BBO offersone of the highest bandwidths for type I operation at de-generacy, and this is why we have used it as an example.In a particular material, however, there are several op-tions for achieving large OPA bandwidth. For OPCPAwith a narrow-bandwidth pump and no angular disper-sion on the signal, the principal options are collinear op-eration at degeneracy or noncollinear operation awayfrom degeneracy. In general, the latter possibility givesthe larger gain bandwidth for BBO, and the amplifiedspectral intensity and phase for this case are plotted inFig. 9 for the optimum signal wavelength and a gain of

Fig. 8. Operational limits for pump intensity versus crystallength of a BBO OPA as determined by the required gain, thenonlinear B integral limit, and the damage threshold.

Fig. 9. Optimum performance of a BBO OPA of 1.5-mm lengthin noncollinear geometry for a 532-nm, 120-GW/cm2 pump beamand 800-nm signal beam: (a) amplified spectral intensity andphase, and (b) pulse profile as determined by the Fourier trans-form of the spectral amplitude and phase after compensation forphase terms up to the quartic.

106; the Fourier transform showing the recompressedpulse profile is also displayed. The gain bandwidth isnow 4830 cm21 and the recompressed pulse duration 5.7fs. As noted in Subsection 3.D, an additional option forbroad bandwidth is the chirp-compensation scheme.

3. Effect of the Optical Parametric Amplifier PhaseThe importance of the spectral phase generated in anOPA when the signal beam is chirped is demonstrated inFig. 10. This takes the example of Fig. 9 and calculatesthe recompressed (and transform-limited) temporal pro-file of the pulse (a) when the calculated phase term is ig-nored; (b) when the full phase is taken into account, as-suming the quadratic phase term is compensated by thecompressor; and (c) if the compressor compensates forphase terms up to the quartic. Comparison of Figs. 10(a)and 10(b) shows that the cubic phase term is associatedwith significant distortion, whereas comparison of Figs.10(a) and 10(c) demonstrates that terms higher than thequartic are not significant in this case.

4. DetuningAn increase in the efficiency and bandwidth for an OPAclose to degeneracy might be expected for a slight detun-

Fig. 10. Effects of the optical parametric phase on the calcu-lated pulse profile: (a) assuming constant spectral phase, (b)taking account of all phase terms of Fig. 9, (c) taking account ofphase terms only higher than the quartic.


ing of the crystal such that it is phase matched for twowavelengths one on each side of the exactly degeneratevalue. A similar technique has been used in sum-frequency generation.25 Figure 11 plots comparable sig-nal spectra with and without this detuning. We obtainedan efficiency and bandwidth increase of 13% correspond-ing to a 13% increase in the power of the recompressedpulse.

B. Real Beam ProfilesWe consider the case of an OPCPA with flat-top spatialdistributions for both pump and signal beams and Gauss-ian distributions for both temporal profiles. This corre-sponds to the best situation that can usually be achievedin practice, since it is possible to approximate quiteclosely to a flat-top spatial distribution but not to a flat-top temporal profile. Because this is a chirped pulse am-plifier with a close-to-linear chirp the temporal distribu-tion relates directly to the spectral distribution (apartfrom a constant factor). For the BBO example of Fig. 6with equal pump and signal pulse durations, Fig. 12shows curves for the input and output signals when thecrystal length (or pump intensity) is set for either maxi-mum energy conversion efficiency or maximum product ofefficiency and bandwidth. As shown in Fig. 13, weachieved only a slight improvement in performance by de-tuning the crystal in this case. Further performance op-timization can be achieved when the pulse durations ofthe input pump and signal beams are different. Figure14 shows the variation with pump pulse duration of thebest efficiency–bandwidth product and points to an opti-mum value of 1.75 for the ratio of the pulse durations.At this value, the efficiency–bandwidth product is 19%higher than the value for equal-pulse durations.

1. Optimized Beam ProfilingIt has been demonstrated26,27 that control of either orboth the spatial and the temporal beam profiles can beused to maximize efficiency and bandwidth in an OPA andthat there are compensating pairs of profiles for pumpand signal that give optimum performance. The simplestsuch pair is a flat-top profile for each but if, as is usuallythe case, the pump beam temporal profile is unavoidablybell shaped, the compensating signal beam profile shouldbe shaped to look like an inversion of the pump beam pro-

Fig. 11. Calculated increase in the gain spectral bandwidth forthe collinear degenerate BBO OPA arising from an angularchange to introduce a phase mismatch at the central wavelength.

file. There are two ways that an approximation to thisprofile can be achieved in practice:

(a) Since the signal pulse in an OPCPA is stronglychirped, with the temporal and spectral profiles closelyrelated, the application of spectral filtering, possibly inthe spectral plane of the pulse stretcher, can be used tocontrol the temporal profile.

(b) Some OPCPA systems could require the use of twoor more OPA stages in series. Since, however, it is onlythe final power amplifier stage that needs to be optimized

Fig. 12. Performance of a collinear degenerate BBO OPA withGaussian-shaped pulses of equal duration for the pump and thesignal. The upper curve is the input signal spectrum (magnifiedgreatly) and the lower curves show the output amplified signalfor (i) maximum efficiency and (ii) maximum efficiency–bandwidth product.

Fig. 13. Performance of a collinear degenerate BBO OPA withGaussian-shaped pulses of equal duration for the pump and thesignal. The upper curve is the input signal spectrum and thelower curves show the output amplified signal for (i) maximumefficiency–bandwidth product with additional crystal detuningand (ii) maximum efficiency–bandwidth product without detun-ing.

Fig. 14. Variation of the normalized efficiency–bandwidth prod-uct with the ratio of pump-to-signal pulse duration for the ex-ample of Fig. 12.


for maximum efficiency and bandwidth, the earlier ampli-fiers can be adjusted to create an optimum signal profilefor the final stage. This can be approximated by drivingthe earlier stages into heavy saturation.

Figure 15 shows the optimized design for a petawattOPCPA scheme that uses a large Nd:glass laser to providethe pump beam. The calculated temporal profile gener-ated by the preamplifiers is close to the required shapeand leads to an output signal beam with maximum effi-ciency and spectral bandwidth, and an intensity profilethat can be compressed to a short pulse with good fidelity.

2. Stability and ReproducibilityIt is important for most laser systems to have both goodstability and insensitivity to variations of the input pumpbeam. However, since an OPA is a high-gain amplifierwith an unsaturated gain that depends on the exponen-tial of the pump intensity, OPCPA systems are potentially

Fig. 15. Optimized design for a petawatt OPCPA systempumped by a Nd:glass laser. Note the inverted profile of the sig-nal from the second amplifier to enable optimum performance ofthe final power amplifier.

Fig. 16. Variation of the calculated recompressed power (nor-malized) of the petawatt OPCPA of Fig. 15 with the pump power,showing increased stability with increased saturation of the sys-tem.

sensitive to pump fluctuations.28 To minimize this sensi-tivity, one should ensure that all OPA stages are stronglysaturated by adjusting the levels of depletion in the se-quence of amplifiers. For the petawatt OPCPA systempumped by the Nd:glass laser of Fig. 15, the peak powerof the recompressed pulse is displayed as a function of thepump beam intensity in Fig. 16. This demonstrates that,at the design operating point (Ip /Ipo 5 1), a stability re-quirement of, for example, 10% can be maintained in thepresence of both shot-to-shot and profile variations in thepump beam of 15%. It can also be seen that improvedstability could be achieved if it were possible to increasethe pump beam intensity, although this would reduce theconversion efficiency and exceed the acceptable intensitylimit of the crystal.

The performance of an OPCPA system could also be ex-pected to be sensitive to errors in synchronization be-tween the pump and the signal beams. One solution toeliminate these errors is to derive the pump and signalfrom the same oscillator. If this is not feasible there willbe a timing jitter, which, for the cases discussed here, cor-responds to the accuracy of locking a Pockels cell opticalgate in the pump laser to the signal oscillator. This canbe typically 50–100 ps. As an example, for the petawattsystem of Fig. 15 with a jitter of 10% (65 ps) of the pumppulse duration, the resulting maximum decrease in re-compressed pulse power was calculated to be 8%.

5. MAXIMUM POWER OPTICALPARAMETRIC CHIRPED PULSEAMPLIFICATION DESIGNSThe design of Fig. 15 was chosen to provide a comparisonbetween a Nd:glass laser-pumped OPCPA system and aconventional Nd:glass CPA system giving the presentmaximum achieved power of approximately 1 PW. Sucha comparison favors OPCPA since it requires a smallerNd:glass laser and much smaller gratings and gratingseparation in the compressor. However this design doesnot make full use of the capacity of available OPA crystalsand gratings since both can be obtained in larger sizes.Further advantage can be obtained by use of the beam-combining ability of the OPA, which allows a single signalbeam to extract energy from a number of pump beams.Consequently, it is interesting to estimate the maximumpower that can be generated by an OPCPA system by useof current technology.

We propose a scheme based on a multibeam Nd:glasspump laser, since this system has seen the greatest devel-opment and can supply the maximum pump energy at aconvenient wavelength and pulse duration. A KDP crys-tal is used as the main power OPA because only this crys-tal can be grown to the size required to handle the highpump energy available; it is already used as a second-harmonic generator for the glass laser. As in conven-tional CPA systems, the limiting factor in the OPCPA de-sign is the energy capacity of the compression grating.For this exercise we assume a 100 cm 3 100 cm squaregrating with a groove density of 750 lines/mm and an in-cident angle of 31 deg (diffraction angle of 16 deg). Thesevalues were chosen to give a large beam geometry, whichis feasible without the appearance of unwanted diffracted


orders and with a broad cutoff spectral bandwidth (6.5times the FWHM). Although this lower groove densityhas not previously been used with large holographic com-pressor gratings, the production process will be the sameand should achieve at least equal efficiency and equiva-lent bandwidth to gratings used currently at 1200lines/mm (for Ti:sapphire CPA lasers with similar band-width) and 1500 lines/mm (for Nd:glass lasers requiringthe largest sizes). The maximum grating area we can il-luminate with a square beam input at the incident angleis 8600 cm2 and, at a maximum fluence on the secondgrating of 0.3 J/cm2, the maximum energy capacity is 2.6kJ. If the grating efficiency is 90% the maximum energyon target is 2.3 kJ. The grating separation for this sys-tem is 3.3 m, which is much more manageable and cost-effective than the 13 m required for current 1-PW sys-tems.

Figure 17 presents a schematic of this high-power de-sign and includes the results of a simulation of its perfor-mance. Current glass laser technology29 can provide anenergy of approximately 3.4 kJ per beam in 1 ns at apump wavelength of 526 nm and in a square 34 cm3 34 cm beam with flat profiles in both space and time.One beam drives a three-stage OPCPA to amplify a 1-nJsignal pulse that has been stretched from 20 fs to 400 ps.Preamplification is carried out by use of lithium triborate(LBO) OPAs since this material has higher gain, gainbandwidth, and damage threshold and is more tolerant ofangular errors and pump beam aberrations. An outputsignal energy of 1.4 kJ is anticipated from the third stage.Two subsequent KDP booster amplifiers, each pumped bya second and third beam from the glass laser, enable fur-ther amplification of the signal up to 4.45 kJ, which ex-ceeds the capacity of the compressor. Limiting the out-put of the compressor to its maximum of 2.3 kJ, wecalculated the peak power by taking the Fourier trans-form of the predicted output spectral amplitude andphase distribution and assuming that phase terms up tothe cubic can be compensated. A value of 22 fs was ob-tained, finally giving an estimated power for this schemeof 105 PW. (If gratings with a fluence limit of 0.5 J/cm2

were available then all the OPCPA output pulse energy

Fig. 17. Design for an ultrahigh-power multipump OPCPA sys-tem.

could be utilized to yield a peak power of ;160 PW, as indicated in Fig. 17).

The fluence in the OPAs is kept below 3 J/cm2 for KDPand below 5 J/cm2 for LBO, and with a total path in LBOand KDP of 29 and 57 mm, respectively, the effective over-all B integral on the signal beam is estimated to be lessthan 1.

The angular tolerance on the pump beams requiresthem to have a divergence no less than 0.1 mrad, which ismore than 30 times the diffraction limit. In addition theOPCPA system could be placed in vacuum to minimizebeam distortion on the amplified signal, and, with theimplementation if needed of an adaptive optic, an outputbeam quality close to the diffraction limit can be expected.Focusing this beam to a focal spot size of say 3 mm wouldthen provide intensities in excess of 1024 W/cm2. Sincethe signal and idler pulses are at the same wavelength, itmight appear that the schemes shown in Figs. 15 and 17operate at degeneracy. However, it is important to stressthat in both cases the signal and the idler are noncol-linear with an angular separation of approximately 1 deg,so all the problems associated with degenerate operationare avoided.

The above design demonstrates the feasibility ofOPCPA for the generation of powers far in excess of cur-rently achieved values. Higher powers than currentlyavailable could also be anticipated by use of a similar CPAscheme but with large aperture Ti:sapphire amplifiers in-stead of OPAs, and it is therefore interesting to comparethe two. They are both pumped by the second harmonicof the Nd:glass laser, they have similar efficiencies ofpump energy extraction, similar gain bandwidths for thishigh-power application, and similar required aperturesfor a given energy. As a storage medium, Ti:sapphire isadvantageous since, for the same output energy, thepump pulse can be of longer duration, leading to a moreefficient and consequently a smaller pump laser. TheOPCPA however is the cleaner system both spectrally andspatially since the material path length is smaller and theabsence of energy deposition within the OPA crystalsminimizes the thermal aberrations. The scaling tohigher powers also currently favors the use of OPA crys-tals that can be grown to larger aperture and at lowercost.

6. NUMERICAL MODELINGThe numerical model that accounts for group velocity ef-fects is based on a solution of the standard OPA coupled-wave equations for the slowly varying complex field enve-lopes for pump, signal, and idler pulses Ep , Es , and Ei ,namely,

]Es

]z5 2iS vs

nscDdeffEpEi* exp~2iDkz !,

]Ei

]z5 2iS v i

nicDdeffEpEs* exp~2iDkz !,

]Ep

]z5 2iS vp

npc DdeffEsEi exp~iDkz !. (21)


A split-step technique is adopted in which the three dif-ferential equations are solved in the space–time domain,whereas spectral effects are implemented in the fre-quency domain. Our normal practice is to use the valueof Dk at the center frequencies vp , vs , and v i in the dif-ferential equations, and to apply phase adjustments inthe frequency domain by use of refractive-index valuesderived from a Sellmeier equation; identical results areobtained if the entire mismatch is applied in the fre-quency domain. Either way, the procedure takes full ac-count of dispersion and group velocity effects to all orders.

We generated the initial signal pulse by applying a lin-ear chirp to a transform-limited wideband parent pulse;the effectiveness of the OPCPA process could be measuredby observing how well the parent pulse was recreated af-ter amplification and subsequent compression. In Fig.18 we compare the analytical results of Fig. 12 with theresults of the full-scale numerical analysis. A 4.8-pspump pulse with a peak intensity of 10 GW/cm2 was usedto amplify an 11.2-fs parent signal pulse stretched to aduration equal to that of the pump pulse. The mediumwas BBO, and the peak unsaturated gain was 106, whichis equal to the value used for Fig. 12. The group time de-lay difference between the pump and the signal over the5-mm propagation distance was 430 fs and, to compensatefor this, the signal was delayed with respect to the pumpby 200 fs. The agreement between the two sets of resultsis clearly good. However, discrepancies begin to be ap-

Fig. 18. Performance of a collinear degenerate BBO opticalparametric amplifier with Gaussian-shaped pulses of equal du-ration for the pump and the signal. The output signal profilewas calculated by use of (i) the analytical solution and (ii) nu-merical modeling.

Fig. 19. Output signal temporal profile calculated by use of nu-merical modeling for pulse lengths in the BBO OPA of (i) 4.8, (ii)1.2, and (iii) 0.6 ps. The group transit time difference in thecrystal was 0.43 ps.

parent for shorter pulse durations. Figure 19 shows thecomputed amplified signal spectra for pulse durations be-tween 4.8 ps and 600 fs and indicates that the analyticaltreatment that neglects the group velocity mismatch ef-fects, presented in Section 2, is valid for stretched pulsedurations greater than approximately 3 times the grouptime dispersion in the crystal.

7. CONCLUSIONSAn analytical treatment has been presented for the opera-tion of optical parametric amplifiers with particular ref-erence to their use as chirped pulse amplifiers. The pri-mary aim is to use the analysis in the design andoptimization of OPCPA systems for the generation of ul-trashort, ultrahigh power pulses. The formulation pro-vides the basis by which it is possible to maximize the en-ergy and bandwidth for both collinear and noncollinearoptical parametric amplifiers and to minimize the sensi-tivity to pump beam aberrations or alignment. An ex-ample of a noncollinear OPCPA by use of BBO gives am-plified and recompressed pulses of less than 6-fs durationfor a gain of 106. The effects of the nonlinear refractiveindex and the damage threshold are also considered andare shown to place a limit on the peak usable pump in-tensity and fluence, respectively. Further use of the ana-lytical results showed that the potential increase in am-plified bandwidth resulting from a detuning of the crystalis small. However a much more promising technique forincreasing gain bandwidth is to use a chirped pump pulsewith this chirp appropriately matched to that of the sig-nal.

Two examples of ultrahigh power designs have beenpresented. The first was designed to generate a petawattpower and illustrates the optimization of a multiamplifierOPCPA system. The second uses a beam-combining tech-nique to make use of several beams from a pump laserand points to the possibility of generating powers in ex-cess of 100 PW with current technology.

Finally we examined the validity of the analytical ap-proach with its simplifying assumptions by comparing itspredictions with those calculated by computer simulation.For the selected example, that of a BBO OPCPA underheavy depletion of the pump and with a strongly chirpedsignal beam, good agreement was achieved for a pumpand chirped signal pulse duration in the amplifier, whichis more than several times the pump-to-signal group timedispersion in the BBO crystal. This indicates that thedynamics of operation of the amplifier is changed onlyslightly as a result of a chirp on the signal beam.

ACKNOWLEDGMENTK. Osvay acknowledges the support of the Hungarian Na-tional Science Foundation under grants T33018 and TS040759.

The e-mail address for I. N. Ross is [email protected].


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analysis and optimization of optical parametric chirped pulse amplification

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