transient second-harmonic generation: influence of effective group-velocity dispersion

756 J. Opt. Soc. Am. B/Vol. 5, No. 4/April 1988

Transient second-harmonic generation: influence ofeffective group-velocity dispersion

N. C. Kothari

Laboratoire d'Optique Quantique du Centre National de la Recherche Scientifique, Ecole Polytechnique, 91128Palaiseau Cedex, France

X. Carlotti

Centre de Math6matiques Appliqu6es, Ecole Polytechnique, 91128 Palaiseau Cedex, France

Received July 13, 1987; accepted November 16, 1987

Second-harmonic generation using high-intensity ultrashort pulses is studied numerically. In a highly dispersivemedium and for ultrashort (subpicosecond to femtosecond) fundamental pulses the effective group-velocity disper-sion (GVD) as well as the group-velocity mismatch should be taken into account. The numerical results indicatethat in this case the conversion efficiency is reduced, compared with the case when only the group-velocity mismatchis taken into account, and for somewhat larger values of effective GVD parameters it exhibits an oscillatorybehavior. Fundamental and harmonic pulse shapes inside the medium are also calculated. The cases with theeffective GVD taken into account show that inside the medium both of the pulses split into two or more pulses.

INTRODUCTION

Second-harmonic generation (SHG), being conceptuallyand experimentally the simplest nonlinear optical effect,was the first to be discovered.' It has now been observed inhundreds of crystals. In the first experiments the amount ofharmonic light generated was quite small. With the adventof phase-matching techniques,2 the conversion ratios haveincreased, and 100% harmonic conversion has been report-ed.3 The detailed theoretical analysis for SHG has beengiven by Armstrong et al.

4

In most practical situations a pulsed fundamental lightsource is used to generate second-harmonic (SH) light. Thecw or so-called quasi-stationary approximation can still beapplied if the light pulses are not too short. This approxi-mation applies down to the subnanosecond range but cannotadequately describe the main features of the interactionwhen picosecond giant pulses are used. An important re-cent development in pulsed lasers is the generation of pulsesdown to the femtosecond range5 ; the nonlinear-optical ef-fects resulting from using such short pulses are then essen-tially of a nonstationary character.

In the nonstationary case, the dispersion of a mediumbecomes an important factor in determining the transientcharacteristics of nonlinear wave phenomena. The wave-vector dispersion (h/lca) gives rise to a group-velocity mis-match6 in the creation of a harmonic wave owing to the twodifferent group velocities of the fundamental and SH pulses.If the group velocity u2 of the harmonic pulse is less than ulof the fundamental pulse, the harmonic pulse lags behindand quickly stops growing. Equivalently, one interpretsthis by saying that for a broadband fundamental in a disper-sive medium, the phase-matching condition cannot beachieved for a complete spectrum.7' 8 Thus the SH conver-sion efficiency may be considerably reduced when short

pulses are used if the group-velocity mismatch between thetwo waves is large.

Comly and Garmire7 have calculated the SH pulse shapesand widths after neglecting the fundamental laser powerdepletion due to the creation of the SH. They have foundthat in long dispersive crystals the SH pulses become muchbroader than the incident fundamental pulses, and the pow-er-conversion ratios saturate at low values. Garmire andYariv9 have shown that the dispersion of the crystals doesnot greatly alter the characteristics of the fundamental lightpulses. Ducuing and Flytzanis10 have analytically obtainedthe same results for a Gaussian fundamental incident pulseby solving the time-dependent SH amplitude equationalone.

Akhamanov et al.6 and Bamberger et al.11 have also taken

into account the fundamental pulse's power depletion due tothe growth of a SH pulse, thus allowing for an arbitraryconversion ratio of the fundamental into the SH power.The analytical results6 using an incident Lorentzian funda-mental pulse shape show that the fundamental pulse widthdecreases and hence the SH efficiency also decreases whenthe group-velocity mismatch between the fundamental andSH pulses increases. Moreover, under certain conditionsthe SH pulse width becomes shorter6 than the fundamentalone. In Ref. 11 it was mathematically proved that the con-version efficiency asymptotically approaches unity for a suf-ficiently long interaction length, as in the cw case.4

In all the previous calculations only the dispersion of thewave vectors was taken into account. In the present work,we also take into account the effective group-velocity disper-sion (GVD) of the two waves. This may become importantwhen high-intensity ultrashort (subpicosecond to femtose-cond) pulses are used to generate a SH inside a nonlinearmedium. We have studied this problem numerically andobtained the results described below.

0740-3224/88/040756-09$02.00 © 1988 Optical Society of America

N. C. Kothari and X. Carlotti

Vol. 5, No. 4/April 1988/J. Opt. Soc. Am. B 757

THEORY

In this section we formulate the theory for transient SHGand obtain the set of two coupled time-dependent differen-tial equations for the field amplitudes of the fundamentaland SH waves. The macroscopic mechanism underlyingSHG is quite simple: a laser wave of frequency w (funda-mental) propagating in a nonlinear medium induces a sourcepolarization at frequency 2w, which radiates and creates thewave of frequency 2w (SH). We shall assume that the fun-damental and SH beams are represented by linearly polar-ized plane waves. Let Al,2 (r, z, t), kl,2, and W1,2 be, respec-tively, the complex amplitudes, wave vectors, and frequen-cies of the two waves. The total electric field E, given by

E = Al(r, z, t)exp[i(klz - lt)] + A2(r, z, t)

X exp[i(k 2z - 2 t)], (1)

satisfies Maxwell's equation

In Eqs. (5) we have introduced the following normalizedquantities: q, 2 = so, 2/A, where A is a reference electric fieldamplitude; t = zIL, where L designates a reference distance;and r = (t - k'z)/T, where k' is the inverse of the referencevelocity and T is the initial fundamental pulse width. Fur-thermore,

k = Ok,, 2W W1,2

denotes the inverse of the group velocities of the two waves,

- aw2|1,2

denotes the GVD of the two waves, and c is the velocity oflight in vacuum. We choose the reference distance L andthe inverse k' of reference velocity as given by L = 2T/(k2 -kl) and k' = (k + k2)/2. To simplify the notation in Eqs. (5),we define the following parameters:

V2E 1 2DL 1 O PNLe c2 0t2 E c2 0t2 (2)

For phase-matched SHG, we have (02 = 2 and k2 = 2k1.The total polarization vector P has been written as P = PL +

PNL, where the nonlinear polarization PNL for two frequen-cies wl and cw2 may be given by

PNL(wl) = EO[2x (2)(P2Sl* exp[i(k 2 - kl)z

+ X( (1,1I2 + 2IP212)01 exp(iklz)]exp(-iwlt),

(3a)

PNL(W2) = E0 [X( 2) 12 exp(2iklz)

+ X(Nk((P212 + 2ke,112) ,o2 exp(ik2 z)]exp(-iw2 t).

(3b)

In Eqs. (3) eo is the free-space permittivity, x(2) and X(3) are,respectively, the second- and third-order nonlinear suscepti-bilities of the medium, and the field amplitudes A1,2 areexpressed in the form

A1, 2(r, z, t) = R(r)so1,2(z, t). (4)

In Eqs. (3), we have taken R(r) = 1 by assuming that thebirefringence angle is small enough that the transverse vari-ation of the fields can be neglected.' 0 Substitution of Eqs.(1), (3), and (4) into Eq. (2) yields, after the slowly varyingenvelope approximation is applied and the coefficients ofexp[i(kl,2z - W1, 2 t)] are separated out, the following twocoupled differential equations, written in the normalizedforms, for the field amplitudes p01,2 inside the medium:

o- a + k -k' a O~ T Or/

(k,) 2- (k')2 + k 1k' 02 q,

2kT 2 Or2

+ k 2 2X q2ql* + 12 X()(jq 1

2 + 21q 212)ql = 0,

i(-1 + k - k'O a' aO T Or/

(5a)

(k)2

- (k')2 + k2 k; 0

2q2

2k2T2 ar2

+ A22 X(2)q 2 + A2

22 X(3)(jq 2 + 2lq,12 )q2 = 0.

(k'1)2 - (k')2 + klk'lL

2kT 2

(k) 2- (k') 2 + k2k2

2k 2 T2

(kl )2- (k') 2 + k k;

klT(k 2 - k',)

(k) 2 -(k')2 + k2k2

k2T(k 2- ki)

Awl 2 AW2= 1 2X ML = 22 X(2)L,

2klc2 2k 2C2

A__ 2-2 ( 1 A2W22

X3

2klc2 2 2k 2c2

(6)

(7)

(8)

(9)

In terms of these parameters, Eqs. (5) can be written asfollows:

Oq, Oq, a2 Oq, *

______ + ia _ izyq 2q* - v(lq12 + 21q212)ql = 0,O~ Or Or2

(10a)

0q2 0q2 0q2 . 2 1 1___ + + I3 - iyq 2 - i2v(1q 212 + 21q )q2 0-

O a Or (r2(l0b)

a and 3 are the effective GVD parameters of the medium forthe two waves. Equations (10) in the steady-state (cw orquasi-stationary) condition with v = 0 were solved analyti-cally in Ref. 4, and the nonstationary case with a = / v = 0was solved analytically in Ref. 6 and numerically in Ref. 11.It should be noted here that a and / are in general nonzeroeven when k%,2 = 0. In deriving Eqs. (10) we have neglectedthe absorption of the waves inside the medium.

Let us now estimate the orders of the parameters a, /, 'y,and v. Generally, the influence of the effective GVD param-eters (dispersion spreading) is not significantl2 in typicalnonlinear crystals with pulse durations as great as 10-3 sec,as in the transparency region the typical value of 02k/0co2 is ofthe order of 10-25 sec2 m-1. For the fundamental wave-length X, = 1.064 ,um with a pulse duration T = 10-12 sec,typical values of the other parameters for the crystal KDPare taken as followsl2 : k2 - k - 2.7 X 10-11 sec/m, k,,-10-25 sec2 /m, n - 1.5, X (2) - 1.26 X 10-12 m/V, and X(3)10-22 m2/V2. The values of k and k2 are not known inde-


(5b)


pendently. We take k- 5 X 10-9 sec/m. For the fieldamplitude A - 5 X 106 V/m, we get a - 3 X 10-3, ,- 4 X 10-3,,y - 1.8, and v - 3.6 X 10-4. Thus for picosecond pulses wecan neglect a, /, and v in comparison with y. However, for T-3 X 1014 sec and A 108 V/, we get a - 0.1, - 0.13, -y1.1, and v - 4.4 X 10-3, and thus a, , and y are of the sameorder, whereas can still be neglected. This rough estimateof the parameters indicates that for pulse durations T 30fsec and for the field amplitudes A > 108 V/m with some-what larger values of kl,2, the effective GVD should be takeninto account. The higher-order nonlinear contributions canstill be neglected. The slowly varying envelope approxima-tion, which requires that X << T/kj,2, may be still assumed tobe valid for the pulse widths considered above.

We have performed the numerical computations with a =0-0.5, = -0.5-0.5, y = 2, and v = 0. The slightly largervalue of the coupling coefficient y is chosen in order toreduce the computing time, and somewhat larger values of aand are chosen in order to produce pronounced effects ofthese parameters. Furthermore, it may be noted from Eqs.(10) with v = 0 that -ql* and -q2* also satisfy the sameequations if a and :3 are replaced by -a and -3, respectively.This means that, while calculating the intensity profiles ofthe two pulses and the conversion efficiency, we can fix thesign of a (say, positive) and choose positive as well as nega-tive values for . Then the results with a < 0 are alreadyincluded in the above combinations.

NUMERICAL SOLUTIONS

Computational SchemeWe have solved Eqs. (10) with v = 0 numerically by using theFourier-transform-based beam-propagation technique.' 3

The numerical procedure may be explained as follows: thedistance is divided into N (=100, 200, ... , 500) equalsegments of length A4 = 0.1, and for each of these segmentsthree steps are followed. In the first step, the nonlinearequations

dq, = iyql*q2 dq2 *2

d = yq

qj( = 0, T) I exp I, (13a)chr \ch 2r

q2 (Q = 0, T) = 0. (13b)In our calculations, we have taken 0 = 0, 7r/4, 7r/2, r. Weshall describe our results separately for the two cases 0 = 0and 0 0 0. The calculations are performed to obtain theintensity profiles of the two pulses and the conversion effi-ciency given by

J q2 (Q, )12 d,

n() = - (14)

J Iqj(, r)12dr + JIq 2(, )I'd

The Case with 0 = 0In Fig. 1 we show plots of 1(t) versus for three differentapproximations: the stationary case, 4 the nonstationarycase with wave-vector dispersion only,6 "'1 and the nonsta-tionary case with both wave-vector dispersion and the effec-tive GVD taken into account. As is well known,4 in thestationary case the efficiency is given by (7) =tanh2 (qjo-y4), which asymptotically approaches unity as 4 -a. The quantity qlo is the initial value of ql at = Owhich istaken equal to unity. The initial value of q2 is assumed to bezero. When the wave-vector dispersion is taken into consid-eration, the efficiency is reduced compared with that for theprevious case. However, it still approaches unity asymptot-ically. This was proved mathematically in Ref. 11. In thecase when a, /3, or both are nonzero but small, the efficiencyis reduced further. It may asymptotically approach somevalue -. different from unity. For instance, when thegroup-velocity mismatch is absent (i.e., when k = k) and

1.0

0.8(11)

are solved for a step length At/2, using the fourth-orderRunge-Kutta method. In the second step, the linear equa-tions

aq, aq, . 02q,

Ot aT - 0T2

aq2 aq2 . q2

04 Or hi ar2

0.6

'1(12a)

0.4

(12b)

are solved for a step length A4 by means of the fast Fouriertransform (FFT). The third step is the repetition of thefirst, again over the step length A4/2. For each of thesesteps, the initial conditions are the result of the precedingstep. The scheme has a global second-order accuracy in tv4.In our calculations, we have checked the stability of thenumerical scheme by increasing the number of segments fora given distance and/or the number of points in the FFT.

Numerical ResultsEquations (10) with v = 0 were solved by using the numericalscheme described above for the initial conditions

0.2

00 40 80 120 160 200

4 x 10Fig. 1. Plots of efficiency (4) versus for (i) the stationary case,(ii) the nonstationary case with only wave-vector dispersion, and(iii) the nonstationary case with both wave-vector dispersion andthe effective GVD taken into account. In case (iii) a = 0.1 and = 0;the efficiency is further reduced compared with that for the case (ii).


Vol. 5, No. 4/April 1988/J. Opt. Soc. Am. B 759

0.

0.4 T 0.41

CF

0 200 0 60 00

D~~~~~~~~~~0 200 400 ]600 800 1000

4°6080.E _ _ ; X

T x 32/jr

Fig. 2. Intensity profiles of the fundamental (labeled A) and SH (labeled B) waves for the nonstationary cases (ii) and (iii) of Fig. 1. When theeffective GVD is taken into account in case (iii), both the pulses have split into two.

when the effective GVD parameter : of the SH wave is zero,the coupled equations for qi and q2 exhibit a soliton solutionfor which -. = 2/5 (see Appendix A).

In Fig. 2 we show the intensity profiles of the two waves forboth nonstationary cases of Fig. 1 at a distance t = 20. Thecase with the effective GVD taken into account shows thatinside the medium both pulses have split into two..

In Fig. 3 we show the plots of efficiency for three differentcases when a, A, or both are nonzero but small. The efficien-cy in each case (see also the effective GVD case in Fig. 1)develops a kind of oscillatory behavior. This is seen moreclearly for somewhat larger values of a, 1, or both (see Fig. 4).Figure 5 shows the intensity profiles of the two waves for thedifferent cases of Fig. 3 at a distance t = 20 inside themedium. Pulse shapes of both the waves in all the casesshow the splitting into two or more peaks. Again this ismore pronounced for somewhat larger values of a, A, or both(see Fig. 6).

The Case with 0 #d 0In Fig. 7 we show the plots of efficiency for a = 1 = 0 and 0 =7r/4, 7r/ 2 , 7r. The oscillatory behavior in all the three cases issomewhat similar to the cases for which 0 = 0, a = 0, and A is

o [see Fig. 4(ii)]. Figure 8 shows the intensity profiles of thetwo waves, which exhibit splittings similar to those in theprevious cases.

Figure 9 shows the efficiency plots for the case when a id0, = 0, and 0 = 7r/4, 7r/2, 7r. The behavior in all the cases is

qualitatively similar to those in Figs. 4(i)-4(iv) for a 5 0.However, an important difference seems to be that in Fig. 9the initial oscillatory behavior tends to smooth out asymp-

'1

0 40 80 120 160 200

x x10

Fig. 3. Plots of efficiency 1(t) versus t for three different cases: (i)a = 0, 3 = 0.1, (ii) a = 0.1, a = -0.1, (iii) a = = 0.1. For nonzerovalues of either a or 13, efficiency plots in all the cases indicateoscillatory behavior.



1. 0

0 . 5

1.

0.5

I I 1. 0

0.5

1.0

0.5

0

20 40 60 80 100 120 140 IO 180 200

20 40 60 80 100 120 140 160 180 200

X 10Fig. 4. Plots of efficiency 1() versus t for four different cases: (i) a = 0.5, 9 = 0; (ii) a = 0, fi = 0.5; (iii) a = = 0.5; (iv) a = 0.5, f =-0.5. Thecurves labeled A are the plots of 1 - m7. The oscillatory behavior of the efficiency in all the cases is highly pronounced for the somewhat largervalues of a and al chosen for these plots.

11)

TG' \4<

- -

I


Vol. 5, No. 4/April 1988/J. Opt. Soc. Am. B

0.

(i

11Ti1

IH, a

0 .50

0 .2 5

200 400 60 0 0 200 400 600 00 1000

.4 0

0 .2 0

- 200 * o 600 O00 low

Fig. 5. Intensity profiles of thehave split into two.

fundamental (labeled A)

r x 32/I

and the SH (labeled B) waves for the three cases of Fig. 3. In all the cases, the pulses

0.241 (iii)

1

I

I 1- A1 \

2.; 40 EO 8 00 l0 140 160 180 200 220 04-

0.2

0.1

(iv) 1

Ia,

CD

11

F

L 61 0 ' 60 b5 "I 1 ' 0 140 160 180 20; '22 240

x 4/2

Fig. 6. Intensity profiles of the fundamental (labeled A) and the SH (labeled B) waves for the last two cases [(iii) and (iv)] of Fig. 4. In thesetwo cases, splitting of the pulses into two or more pulses is seen for somewhat larger values of the effective GVD parameters.

totically for large t. Figure 10 shows the case when a = 0and 3, 0 # 0. The plots show an identical behavior to thosein Fig. 7. Thus the qualitative similarities of the plots inFigs. 4(ii), 7, and 10 indicate that when a = 0, the nonzerovalues of /3, 0, or both produce similar results. Also, the

qualitative similarities of the plots in Figs. 4(i)-4(iv) and 9indicate that when a 0, the results are similar in all thecases for which /' and 6 are either zero or nonzero (the casewith all a, 3, and 6 nonzero is not shown).

From Eqs. (10) it may be seen that for a = = v = 0 the

r.

0 2 5

(i i)

111

~1

11

. _L . ,7la

'iii')a

1..

j!

IIAI

. - .. , .p -.TI

0.16

04

cr

0.08

26.

. .. . . 1 l1 . - . .I .

_.. -as b n J °> Be '�u | '4 " | a ^ '

I s _ I,, . I . or ! | nt. |-& No | r J - | Bum H -B H v.-

Ad H U u - v' G son *

N. C. Kothari and X. Carlotti 761

c n

To

' I r I

r!u -1. In


0-

(i)

-0 - 0

I0 -s 0!' 2 20 6-2

0 20 V. 6t 8; 100 120 140 160' 18G,

I .0

0 5

2,

(ii)

1T

II

/T - - ---

7

]I0 E el I0C 120 140 S6C 18'

X 10

Fig. 7. Plots of efficiency n(t) versus t for a = / = O and (i) O = ir/4, (ii) 0 = 7r/2, and (iii) 0 = 7r. The curves labeled A are the plots of 1 - . Theoscillatory behavior in all the three cases is somewhat similar to the cases for which 0 = 0, a = 0, and fl # 0 [see Fig. 4(ii)].

0.5

c,0 .25

00 100 0o0 300 - 400

Fig. 8. Intensity profiles of the fundamental (labeledsplitting as in the previous cases shown in Fig. (6).

0.6

0.4

0.2

.........................I...... - 1

,,,,,,,. -1. .1M .... ' . . . 0 '500 0 oo 200 300 -

A) and the SH (labeled B) waves for cases (i) and (ii) of Fig. 7.qoo - b

Both the pulses show

I

0 0 5

(i)

IO

W~~~~~~ I

i' (ii)

7'a

.... . . . . . I .. . . . . . . . . . . . . . . . . . .I I I I r l I I

.... .... .... ,......... . . . . .......................

. * s ,H,,


T'!

p C,

I

I

N. C. Kothari and X. Carlotti Vol. 5, No. 4/April 1988/J. Opt. Soc. Am. B 763

I(i)

0.

0, 20 40 60 80 100 120 140 160 18 200

1

0 sF

Of

) 21: SD 60 80 00 120 140 16C, 18.

1 .0

o s

200

rI

VSA(i ii)

j

0 I-D or 60 80 lDD ]20 190 l6G I8C 200)21 40 60 80 100 120 140 16 180I 200

x 10

Fig. 9. Plots of efficiency 1(t) versus t for a = 0.1, A = 0, and (i) O = 7r/4, (ii) 0 = 1r/2, and (iii) 0 = 7r. The curves labeled A are the plots of 1 -.The efficiency behavior in all the cases is qualitatively similar to those in Figs. 4(i)-4(iv). However, the initial oscillatory behavior tends tosmooth out asymptotically for large t.

r (i)

L - -I

i 1I

20 40 60 80 100 120 140 16., 180 200

1.0

05

jf

I .0 . . ...

(ii)

�0�

I 20 40 60 80 100 120 140 160 180

0.

200

0 'i 6 so O Z 4 6 o

5--H

0 20 40 602 00 10 10 6 10 0

X 10

Fig. 10. Same as in Fig. 9 except that a = 0, i = 0.1. The plots show an identical behavior to those in Fig. 7.

amplitude profiles q1(Q, r) and q2Q, -) inside the mediumremain real if the initial profiles qj(0, -) and q2(0, r) are real(i.e., 0 = 0). If the initial profiles are complex (O 5'4 0), theefficiency plots exhibit oscillations and the intensity profilesshow splittings. Similar results are obtained when the ini-tial amplitude profiles are real and a, f, or both are nonzero.This means that initial real amplitude profiles do not remainreal inside the medium when the effective GVD of the wavesis taken into account. The effective GVD broadens thepulses and introduces the chirp, as a result of which at largedistances thepulses do not separate from each other and a

strong interaction still remains between them. This causesthe oscillations into the efficiency plots, as part of the energyis transferred alternately from the fundamental to SH andback.

DISCUSSION AND CONCLUSION

We have studied SHG numerically, using the high-intensityultrashort pulses of fundamental waves. The quasi-station-ary approximation cannot be applied when ultrashort (sub-picosecond to femtosecond) pulses are used to generate SH

(ii)

1. ,I-(

VI

i~

4k -

.0 l

_

oC

'L

C,r,

C S


waves; the problem is essentially of a nonstationary charac-ter, and the dispersion of a medium becomes an importantfactor in determining the transient characteristics of thegenerated SH pulses.

When the wave-vector dispersion

ak|

a Wi(j = 1, 2)

is taken into account, the SH conversion efficiency reducescompared with that in the quasi-stationary case, owing to agroup-velocity mismatch between the two waves. The effi-ciency, however, still approaches unity asymptotically forlarge interaction lengths, as in the quasi-stationary case. Ifthe initial fundamental pulse amplitude is not real (0 i 0),the conversion efficiency develops an oscillatory behaviorand the intensity profiles of the two waves split into two ormore pulses. This is because of the presence of a chirp in theinitial fundamental pulse, which broadens both of the pulsesinside the medium, as a result of which even at large dis-tances the pulses do not separate from each other and astrong interaction still remains between them. This causesthe oscillations in the efficiency plots, as part of the energykeeps flowing from the fundamental to the SH pulses andvice versa.

When effective GVD is also included in the calculations,two apparently distinct cases emerge, corresponding towhether the effective GVD of the fundamental wave, a, iszero or nonzero. In either case, the conversion efficiencydevelops an oscillatory behavior, and the intensity profilesof the two waves split into two or more pulses. This is trueirrespective of whether initial fundamental amplitude pro-file is real ( = 0) or complex (0 # 0).

The effective GVD essentially introduces the chirp intothe waves. In the case when nonlinearity is absent this chirpproduces the oscillations into the pulse profile. The pulseamplitude also oscillates with the distance t. In the nonlin-ear medium, the nonlinearity interacts with this oscillatorybehavior of the pulses. Owing to this interplay between thenonlinearity and dispersion, the pulse profiles split into twoor more pulses, and the energy in the pulses exhibits modu-latory behavior with respect to the distance traveled insidethe medium. As a result, the SH efficiency q exhibits oscil-lations.

APPENDIX A

If we suppose that k = k = k' in Eqs. (5a) and (5b) (nogroup-velocity mismatch), and in the absence of any third-order effect, Eqs. (Oa) and (lOb) become

+ ia - iyq 2 q1 * 0, (Al)

al Ž+i/3 =q 2 i 2 (A2)

where a and /3 are now given by

(A3)2T

/3= k;L.2T

(A4)

For the case when /3 is zero, the system of Eqs. (Al) and (A2)admits, for an arbitrary Q such that aQ > 0, the stationarysolution (ql, q2) given by

ql(r,~ =2 1 ex(fl,ch[T(Q/a)1/21 1

'Y ch [T(Q/a)1/2]

(A5)

(A6)

and the conversion efficiency, calculated from Eq. (14), isequal to the constant

x

1 +x

where

ch-4tdt

J ch-2tdt 3

In the present case 7 = 2/5.

ACKNOWLEDGMENTS

We are grateful to C. Flytzanis, who introduced us to thesubject, and to A. Bamberger and D. Jennev6 for the helpthat they provided us in the mathematical and numericalstudy of the problem.

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transient second-harmonic generation: influence of effective group-velocity dispersion

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