8/3/2019 JOSAB 4-8-1235 Amplitude and Phase of a Pulsed SH Signal

1/6

Vol. 4, No. 8/August 1987/J. Opt. Soc. Am. B 1235

Amplitude and phase of a pulsed second-harmonic signaljamal T. Manassah

Photonic Engineering Center, Department of Electrical Engineering, City CollegeofNew York,New York,New York 10031Received December 15, 1986; accepted April 8, 1987

The equations for the amplitude and phase of the second-harmonic signal generated in a nonlinear quadraticmedium by an exciting ultrashort primary pulse are derived. These equations are approximately solved for the caseof negligibledispersion effects. Explicit expressions for the shape and phase functions of the second-harmonicpulse are obtained.

1. INTRODUCTIONSince the seminal theoretical and experimental work of the1960's,1,2which heralded the era of nonlinear optics, second-harmonic generation has been the standard technique toexplore regions of the electromagnetic spectrum that areprimary-source sterile. The shape3 of the second-harmonicpulse generated in a quadratic nonlinear medium by thepropagation of a primary pulse can be approximated, inmost applications, by folding the primary pulse shape onitself. More recently, in applications with ultrashort com-plex pulses4 ,5 or very long samples, 6 a more precise determi-nation of the pulse shape and the chirp rate is needed. Inthis paper, we derive the coupled partial differential equa-tions relating the amplitude and the phase of the primaryand second-harmonic pulses under general conditions. Wesolve these equations for the case of negligible dispersioneffects.In Section 2, the coupled differential equations for theamplitude and phase of the primary and secondary wavesare obtained. In Section 3, approximate solutions to thesedifferential equations in the case of negligible dispersion arederived. In Section 4, experimental consequences to theabove solutions are explored. In Appendix A the detailedderivation of the coupled differential equations, using themethod of multiple scales, is given.2. NONLINEAR WAVE EQUATIONThe propagation of a monochromatic electromagnetic wavein a linear medium is described by

a2 R(z, w)+ k 2((c,)E(z, L ) = 0. (1)Denoting k(coo) = ko, Maxwell's Eq. (1) can be described forany Xnear co to second order in the difference (co o) bya2 E + [ + 2koko'(- ) + (k0'2 + kok 1 )(c-C O) 2 ]fE = 0,

(2)where

ko= dkdw ww = d2kdo 2

For a pulse with center frequency o, the electric field thenobeys the equation 7az, + exp(-icoot)[ko2 + 2kcko'(i )

- (h0 2 + kokos) at21 exp(ico0t)E(z, t) = 0, (3)where con Fourier space has been replaced by [i(0/0t)] n thetime domain. Next, defining the dimensionless variables

E= AE, T t/T, Z = Z/CT,W c=coT, KO (cr)ko, (4)

where A is the maximum pulse amplitude and T is its width,Eq. (3) then reduces to_A r02E F 21.0

c2 2 2 + exp(-iWoT)[Ko + 2KoKo' i+ KK 011) T2] exp(iWoT)} 0,

(5)where

I dK I d 2K cK d=Ww0 = ck , K 0" = - - /.dW ~~dWw 0 1r (6)In a quadratic nonlinear medium, the nonlinear polariza-tion2 is given by

(7)NO) =-dijkE(1)E(k).The source term for the Maxwell equation is

0P (i) I 2 EJEk.NL2 - 0 A 2dl aTThe primary and second-harmonic components' contribu-tions to the electric field are written as

E = /2[A(Z, T)exp(iKZ - iW1T)+ B(Z, T)exp(iK 2Z - iW2 T) + c. c.], (9)

where W2 = 2W1 and K 2 = 2K + a.The equations describing the propagation of the pulse in

(8)

0740-3224/87/081235-06$02.00 1987 Optical Society of America

Jamal T. Manassah

8/3/2019 JOSAB 4-8-1235 Amplitude and Phase of a Pulsed SH Signal

2/6

1236 J. Opt. Soc. Am. B/Vol. 4, No. 8/August 1987the nonlinear medium are obtained by replacing the zero onthe right-hand side of Eq. (5) by expression (8).Denoting

= c Ad, (10)and using the method of multiple scales8 (see Appendix A fora detailed derivation and comparison with other methods),then the equations for A and B in terms of the Z and Tvariables are given by

aA aA i , 2Ad + K1 dT + - K" lT 2 = ieIA*B exp(io-Z)and

O9 i K2 2B 2-d~z K2/ dd +_ 2 K d 2= i'A 2 exp(-iaZ),aZ aT 2 aT 2where

e2 WV2

In the instance of exact phase-matching conditions, i.e.0 and K = K2' = K', the above equations can reducequadratically coupled Schr6dinger-like system of equatSpecifically,define the comovingvariable U such thatTK' V=Z;

then Eqs. (11) and (12) reduce toaA i Kj" 2AdA + - I1A =ie'A*B,aV 2 12 au 2-B -2" B= ie'A2.aV 2 2 aU 2

(11)

(12)

B = b exp(io),the equations of motion reduce to

O9a-d = -E'ab sin y,a

a aV = e'ab cosy,dV =e'a sin ,

b # = e'a2cos-y,where.

y = - 2a + aV.

(17b)

(18a)(18b)(18c)

(18d)

(18e)The above equations admit of two quasi-conservation laws,specifically,

(13), =to aions.

a 2 + b2 = 2(U (19)and

ba2cos ya b2= G(U); (20)the functional form of the functions F(U) and G(U) can be(14) obtained from the initial conditions, i.e., V = 0. For the caseof most common interest b(U, 0) = 0, and consequently G(U)= 0 and F(U) is the shape function ofthe incomingprimarypulse. If we parameterize a and b as

(15a)

(15b)This author is not aware of any general solution to Eqs. (11)and (12). The method of inverse scattering9 seems promis-ing for tackling Eqs. (15a) and (15b). In this paper weconcentrate only on approximate solutions to Eqs. (11) and(12), in the instance that dispersion effects (Ki' terms) arecompletely neglected and the a dependence is neglected inall but the oscillatory terms.

a2(U, V) = t(U, V)F 2(U),b2(U, V) = [1- (U, V)]F 2(U),

the equation for (U, V) is then( 9 )2= 4E'2F2(U)[2(1 ) _ __2_ __-)2

(21)(22)

(23)If we denote the roots of the right-hand side by P1,2, 3 suchthat Pl< 2 < 3, then 3 = 1, and P,and {2 are the roots ofthequadratic equation

! + - S = 0,where

3. APPROXIMATE SOLUTIONS TO THEDISPERSIONLESS CASENext we solve Eqs. (11) and (12) in the case of Kj' = K2" = 0and Kj' = K2' in all except the oscillatory terms (i.e., theexponent terms on the right-hand side). The solutions arereminiscent of the cw treatment of Bloembergen.2 Underthe above approximations, Eqs. (11) and (12) reduce to

A~j7=EIA*B exp(io-V) (16a)and

(24)'24,E,2F2(UEquation (23) can directly be integrated, with the initialcondition P(U, 0) = 1, to give 0

-= (3-3 2)sn2(KV, ), (25a)where

K = e'F(U)(3 - 1)1 (25b)and

aB =iE'A2 exp(-irV).0V (16b)Denoting the polar representation of the complex ampli-tudes A and B by

A = a exp(ia), (17a)

3)- 2 1/2Xt3-1 (25c)In Figs. 1 and 2, b2(0, V) is plotted for different s. Maximumconversion is obtained for perfect phase matching. In theinstance of perfect phase matching, i.e., a = 0, thenl

Jamal T. Manassah

8/3/2019 JOSAB 4-8-1235 Amplitude and Phase of a Pulsed SH Signal

3/6

Vol. 4, No. 8/August 1987/J. Opt. Soc. Am. B 1237

b2 (0,v)

/

1 2Fig. 1. Variation of the second-harmonicmaximum (amplitude)2 as function of the distance traveled in the dispersionless quadratic medium.Dashed line, s = 0 (perfect phase matching); solid line, s = 1.

0.6

0.31

0

2Bmax

20 40 SFig. 2. Variation of the second-harmonicmaximum (amplitude)2 withs, a parameter proportional to the angle between the primary beam andthe phase-matching direction. (E'V = 1.)

1.0

0.5

0

- L__

Jamal T. Manassah

8/3/2019 JOSAB 4-8-1235 Amplitude and Phase of a Pulsed SH Signal

4/6

1238 J. Opt. Soc. Am. B/Vol. 4, No. 8/August 1987

1.0

0.5

0

b

0.5 1.0 TFig. 3. The second-harmonic pulse normalized shape as functionof the parameter p. The primary pulse is Gaussian. (T = t/r).Dashed line, p = 0.1; solid line, p = 10.

/ = (/2) V + H(U). (31)To determine the function H(U), use Eq. (24) at V = 0,which gives specifically

cos[-2a(U, 0) + H(U)] = 0, (32)i.e., [H(U)/2]differs from a(U,0) by aconstant. Notice thata(U, V) can be obtained directly by combining Eqs. (20)-(25) and (31). Consequently, the phase of the second har-monic, in our approximation, is uniquely determined fromthe phase ofthe incomingprimary pulse.The experimental range whereby the dispersion effectscan be neglected can be obtained by imposing the physicalcondition that the distance for efficient second-harmonicconversion (i.e., the distance at which max b - 1) be smallerthan the distance over which the pulse width will noticablyincrease. This condition imposes the followinginequalityover the problem's parameters:

k0"/(1i0_Adw2)

8/3/2019 JOSAB 4-8-1235 Amplitude and Phase of a Pulsed SH Signal

5/6

Vol. 4, No. 8/August 1987/J. Opt. Soc. Am. B 1239

The spectral distributions of the outgoing second-harmonicpulse are similarly given by

Wanti-Stokes - W Max(Oa/hU) (37a)and

2tokes -W2 Min(81/aU)- (37b)But from Eq. (32) we know that

ad 2[aU ) (38)

5. CONCLUSIONIn this paper I have started a study of the shape and phasefunctions of the second-harmonic pulse. So far I have suc-ceeded in solving only the case of essentially no dispersion.Dispersion effects should be incorporated into those in-stances when the sample is long (such as in a fiber) or insearching for solitary-wave solutions. Nevertheless, thelimited solutions obtained were shown to have some inter-esting experimental consequences on second-harmonic su-percontinuum generation and on the measurement of theparameters of an ultrashort pulse.

Consequently,Wanti-Stokes= 2Wanti-Stokes, (39a)

p/Stokes 2pStokes (39b)This result is hardly surprising, in the sense that the nonlin-ear quadratic medium with no dispersion scales similarlyeach Fourier component of the incoming pulse.B. Double Interferometric TechniqueRecently, we proposed an interferometric techniques forcharacterizing a chirped ultrafast pulse. We proved thatthe parameter X,

1 + g2X= f2 (40a)where

f = ()- WI, (40b)g = (b2r2), (40c)and b2 is the chirped rate, can be determined through aninterferometric measurement. This meant that either f or gor, equivalently, b or r would have to be determined byanother technique. The double interferometric technique'4seeks to find a certain transformation that, when applied tothe pulse, permits a new set of parameters (b', r') to beobtained, following which a corresponding X' is measured bythe interferometric technique. Through a knowledgeof thetransformation law of (b,T) and from the experimental mea-surements of Xand X', the pulse parameters b and rcan thenbe separately deduced. Any linear optical system would

givebut a trivial transform of X, and the scheme cannot beimplemented. However, if the transformation is taken asthe one corresponding to second-harmonic conversion, anonlinear transformation on (b, r) is achievable. Specifical-ly, for '/V >> , the transformation laws are12 = T1 (41a)

b2 = 2bl2 . (41b)Consequently,giventhe experimentally measured Xland X2,the parameters f1 and g, are deduced as

4 (X1 - X2) (42a)2 4X2- Xl (42b)

4(X1- X2)

APPENDIX AIn this appendix, we summarize the derivation of Eqs. (11)and (12)by the method of multiple scales. The functionaldependence of E on Z, T, and E n the solution of Maxwell'sequations is not disjoint. For example, to first order in E,Edepends on the combinations ET and eZ as well as on theindividual T, Z, and E. Carrying the perturbation to highorders, E depends additionally on E2 T, 2Z, e3T, e3Z, etc.Hence it is convenient to write E(Z, T; E)as

E(Z, T; e) = E (ZO T(, Z1, T,, Z2, T2 ... *;e), (Al)where the new scaledvariables Z,, T,, Z2, T2, etc. are definedas

TO= T, T, =ET, T2 = E2 T...,Zo = Z Z = EZ, Z2 = 2Z.... (A2)

T, and Zn represent different time and distance scales.Next we seek a uniform expansion solution to E in the form

E = E"' (TO,ZO,T,, Z1,T2, Z2 ... )(A3)

In the new variables the time and space derivatives can beexpressed asadT a + e a + e2 a +a a a ___

-Z azo aZ aZ2a2 a2 a2 + e(2 a + d2-T +O E a~oaT, aToaT2 aT 2 ]

a2 a2 + a2 +2e(2 a2 + a2az2 azo2 azoaz, aZ4aZ2 az 2 /

(A4)

In what follows,we solve he equations through the follow-ing ansatz:E) = 1/{A(Zl, ,, Z2, T2 )exp[i(KIZ0 - W1T0)]+ B(Z1, Tp, Z 2, T2 )exp[i(K2Z0 - W2TO)]+ c. c.},E(1) = 0,

Jamal T. Manassah

+EE(1)(T,), (, T,, Z1, T21 Z2 + - -')'

8/3/2019 JOSAB 4-8-1235 Amplitude and Phase of a Pulsed SH Signal

6/6

1240 J. Opt. Soc. Am. B/Vol. 4, No. 8/August 1987E(2) = 12C(Zl, T1, Z2, T2)exp[i(KZo - W1TO)]

+ D(Z 1 , T1 , Z2 , T 2 )exp[i(K 2 ZO - W 2 TO)] + c. C.}

Equating terms of the same order in e, the equationsand B are given byOA aA+K'1 =0,0Z1 1 T

+K 2 - = 0,Z 1and

2iK 0Z +2iKK'd -(K1 +KIK )d 2(9Z2 '0T2 aT12= -A*B(W 2 - WI)2

X exp[i(K 2 - 2K1 )ZO i(W 2 -2W)Zo]2iK28 2iKK'2 ddT-K 2 + K2K"2 )2iK220 AT2T, A 2 W)= - -(2W)2X exp[-i(K 2 - 2K,)Zo + i(W 2 - 2W,)Zo]. (A9)From Eq. (A6) one obtains

d2A +(dZ,2

2B+0Z1 2

02A 0 (0A a Az2A = d (dA= d _K'T 1aZ12 Z Z aZ1 adl - K'2 0 A

T1 2Similarly, from Eq. (A7) one obtains

dZ1 2 2 dT 1 2 (All)Using the chain rule for differentiation [Eqs. (A4)]

01 B 11BB _ 1 / B ,0B0Z K T2 E \OZ 2/ TJ k +Z 2 aT,C2 Z aro)

(A12)but

dB B =0 (A13)

because there is no Z and To dependence in the envelope.Also, by using Eq. (A6), Eq. (A8) reduces to1 aA a~A 02A(2iK1 d + 2iK1K'l d-KIK- I

X exp[i(K 1 - W1 T)]= -A*B(W 2 - W1)2 exp[i(K2 - K)Z - i(W2 - Wl)71,

and Eq. (A9)reduces to(A5) e2(2iK29B + 2iK2K 2 aB - K2K 2

for A X exp[i(K 2Z - 2T)]

(A6) A2

=- 2 (2W 2exp[i(2KZ - 2W1T)]. (A15)(A7) From Eqs. (A14)and (A15),Eqs. (11)and (12)ofthe text canbe trivially obtained.The curious reader may wonder why the slowly varyingapproximation (SVA) was not used to obtain these equa-tions. Well, although the left-hand sides of Eqs. (11) and(12) can be thus obtained, if the SVA were applied consis-tently to the right-hand sides (i.e., nonlinear terms), first-derivative terms ofthe nonlinearity wouldappear. Apriori,there is no reason to neglect these terms. Using the multi-(A8) ple-scales technique, it is straightforward to see that these

terms would not appear, except to order 3. (Many otherterms also contribute to this order of on the left-handsides.) Consequently, to order e2, these derivative terms ofthe nonlinearity can be neglected without further assump-tion.REFERENCES AND NOTES

1. P. A. Franken, A. E. Hill, C. W. Peters, and G. Weinreich,"Generation of optical harmonics," Phys. Rev. Lett. 7, 118(1961).2. N. Bloembergen, Nonlinear Optics (Benjamin, New York,1965).3. See, for example, E. P. Ippen and C. V. Shank, "Techniques formeasurement," in Ultrashort Light Pulses, S. L. Shapiro, ed.(Springer-Verlag, New York, 1984).4. J. T. Manassah, M. A. Mustafa, R. R. Alfano, and P. P. Ho,"Induced supercontinuum and steepening of an ultrafast laserpulse," Phys. Lett. 113A, 242 (1985).5. J. T. Manassah, "Interferometric characterization of chirpedultrafast pulses," Appl. Opt. 25, 2480 (1986).6. J. T. Manassah, R. R. Alfano, and S. A. Ahmed, "Second har-monic generation in birefringent optical fibers," Phys. Lett.115A, 135 (1986).7. N. Tzoar and M. Jain, "Propagation of nonlinear optical pulsesin fibers," in Fiber Optics, B. Bendow and S. Mitra, eds. (Ple-num, New York, 1979).8. See, for example, A. H. Nayfeh, Introduction to PerturbationTechniques (Wiley, New York, 1981).9. See, for example, F. Calogero, and A. Degasperis, SpecialTransform and Solitons (North-Holland, Amsterdam, 1982).10. sn is a Jacobi elliptic function. See, for example, I. Gradshteynand I. Ryzhik, Table of Integrals, Series and Products (Aca-demic, New York, 1980), Secs. 8.11-8.15.11. sn(a, 1) = tanh a.12. J. T. Manassah, M. A. Mustafa, R. R. Alfano, and P. P. Ho,"Spectral extent and pulse shape of the supercontinuum forultrashort laser pulse," IEEE J. Quantum Electron. QE-22, 197(1986).13. J. T. Manassah, "Interferometric characterization of chirpedultrafast pulses," Appl. Opt. 25, 2480 (1986).14. J. T. Manassah, "Direct and second harmonics interferometricdetermination of chirped pulses parameters," submitted toAppl. Opt.

Jamal T. Manassah

I