PHYSICAL REVIEW A VOLUME 43, NUMBER 1 1 JANUARY 1991

Relationship between second-harmonic generationand electric-field-induced second-harmonic generation

R. Bavli and Y. B. Band*Department of Chemistry, Ben Guri-on University, Beer Sheva, Israel 84105

(Received 27 June 1990)

We calculate the electric-field induced second-harmonic-generation (EFISH) susceptibility of ahomogeneously broadened two-level system with permanent dipole moments. This susceptibilityhas contributions from the second-harmonic-generation (SHCs) susceptibility of the same systemwithout the presence of a dc field and the four-wave-mixing (FWM) component of the susceptibility

y, (—2';co, co, 0). The magnitude and phase of the various contributions to EFISH arising from the

SHG and from FWM depend on the frequency of the fundamental optical field, the transition dipolemoment p, I„ the permanent dipole moment of the ground-state level p„, and the diA'erence of theground- and excited-state dipole moments Ap. We present non-rotating-wave-approximation re-sults for both the quasi-steady-state in systems where the dephasing time T, is much shorter thanthe population decay time Tl, so the polarization is in a steady state but the population is not, andfor the complete steady state. Extraction of the SHG susceptibility from the EFISH measurementsis considered.

I. INTRODUCTION

In recent years there has been a growing interest in thediscovery and engineering of materials with largesecond-order optical nonlinearities. Even-order non-linearities vanish in centrosymmetric media, thereforesecond-order nonlinearities are optimally observed in sin-gle crystals lacking inversion symmetry. It is usuallydifficult (and/or expensive) to grow single crystals andtherefore methods for predicing second-order nonlinearoptical properties of single crystals without necessitatingthe growth of single crystals have been developed.Among these methods are the Kurtz powder method, '

the monolayer method, and the electric-field-inducedsecond-harmonic-generation (EFISH) method. Much ofthe experimental work to predict second-harmonic sus-ceptibilities of organic molecules was performed usingEFISH. A major problem with EFISH is that interac-tion of the medium and the static electric field breaks thesymmetry by partially orienting the molecules, therebyallowing contributions from the second-harmonic-generation (SHG) hyperpolarizability /3( —2co;co, co) to theobserved second-harmonic signal. It also adds contribu-tions to the observed second-harmonic intensity fromfour-wave-mixing (FWM) processes that contribute to theobserved second-harmonic signal. Thus the observedEFISH intensity is proportional to the square of themolecular susceptibility

y ( 2co; co, co, 0 ) =y ( 2co; co, co, 0 )

+P( —2co; co, co)SkT '

where y, (—2co; co, co, 0) is the microscopic third-order

molecular susceptibility of the molecules at frequency 2~,/3( —2co;co, co) is the microscopic second-order molecularsusceptibility, r~ is the frequency of the applied funda-

mental optical field, F is the applied dc field, and p„ isthe permanent dipole moment of the ground level. Thecontributions of orientationally induced SHCx, i.e., thesecond term in Eq. (I), and FWM [the first term in Eq.(I)] to the molecular susceptibility at frequency 2co inter-fere in a way that depends on the frequency co. This is be-cause the quantities y, and /3 are complex functions of co.

In the previous experimental work, extraction of theSHG hyperpolarizability /3(

—2co; co, co ) from the mea-sured EFISH third-order molecular susceptibilityy( —2co;co, co, O) was accomplished in several ways. Inwork by Oudar and co-workers ' the third-order hyper-polarizabilities for FWM processes at frequenciesco=m, +co& —

co2 and ~=co, +co&+co, were measured andthe results were corrected for dispersion to estimatey, ( —2co;co, co, O). In another work by Oudar and Le Per-son on the nonlinear optical properties of derivatives ofbenzene, it was assumed that y, does not change muchupon substitution of a functional group, so that the y, ofthe substituted molecules could be approximated by the yof benzene (in benzene y =y, since benzene does nothave a permanent dipole moment). In another work itwas assumed that y, is much smaller than /3p, „/5kT andtherefore can be neglected. Meredith, Van Dusen, andWilliams have tried to extract P and y, by measuring yat various temperatures and plotting it versus 1/T, there-by isolating the temperature-dependent factor in y. Theymeasured a strong unexplained temperature dependenceof y (not 4 +B/T) that rules out the possibility of ex-tracting /3 or y, from the temperature dependence of y.

In this paper we use the formalism developed by uspreviously' '" to compute the various components of thenonlinear optical susceptibility of homogeneouslybroadened systems and apply it to SHG and EFISH. Wecalculate the nonlinear susceptibilities for SHG and forEFISH in a two-level system with permanent dipole mo-ments. We then try to correlate the results of EFISH

43 507 1991 The American Physical Society

508 R. BAVLI AND Y. B. BAND 43

with those of SHG in an attempt to learn about SHGfrom EFISH experiments of randomly oriented sampleswith the orientation of the molecules (a) frozen or (b) sus-ceptible to orientation by the dc field. This work also re-lies upon the results of previous studies of two-level sys-tems with permanent dipole moments. Multiphoton ab-sorption calculations in two-level systems with per-manent dipole moments were reported by Meath and co-workers. ' ' Two-photon absorption of two-level sys-tems with permanent dipole moments was discussed byScharf and Band, ' and two-photon absorption and Ra-man properties of such molecules were investigated byBavli, Heller, and Band. ' ' Reference 18 also computedthree-wave mixing in such systems.

We assume for simplicity that all three dipole momentsp„, pbb, and p, b are parallel and calculate y„„where z isthe polarization axis of both the fundamental optical fieldand the direction of the applied dc field. In Sec. II wepresent the formalism and show that y, has nine contri-butions to lowest order in perturbation theory. Some ofthese contributions are proportional to p,b(h)M), whereAp is the difference between the permanent dipole mo-ments of the ground and excited states Ap—=p„—pbb,while others are proportional to p,b. Since the per-manent dipole moments change upon substitution offunctional groups, using y of benzene as y, of substitutedbenzene might introduce substantial errors. Also, since

y, has nine complex contributions whose dependence onco are different, the correction for dispersion is complicat-ed (even for a two-level system). In Sec. III we calculateP and use the results of Sec. II to calculate ~y, ~

and ~y~

for different dipole moments, Ap, p„, and p,b. We thencalculate the ratio

~ y, ~ /~ y ~

as a function of the frequency~ and these dipole moments, and use our results to sug-gest a simple method of extracting )33 from experimentalresults. We also show that in cases where the dephasingtime T2 is much shorter than the population decay timeT, (a common case in solutions), the nonlinear mediumresponds to short and long pulses in different ways. Sec-tion IV contains a discussion and conclusion.

II. FORMALISM

The state of the optically active two-level medium atany time is given by its density matrix

+B(x t)e2i[n(2~)kx —cut]+C. C. +F (5)

where F is a static electric field and A and 8 are theslowly varying vector amplitudes of the field at frequen-cies co and 2', respectively. The development of the den-sity matrix with time is given by its commutator with theHamiltonian

p= —(i /fi)[II, p] I p, —

where (6)

H=H o—gEp,

I is the decay tensor and q is the Lorentz-Lorentzcorrection factor which is given by q(co) = [n (co)+2]/3.In what follows we include the local field correction fac-tors in the field amplitudes. For simplicity, we now dropthe vector index of the fields and the dipole moments,thereby treating only the case where the transition andpermanent dipole moments are parallel to the fields andto themselves. More general cases can be treated in asimilar manner. ' By substituting Eqs. (2), (4), and (5)into Eq. (6) we obtain the derivatives of the four matrixelements of p~ paa~ pbb~ pab~ and pba The interaction ofthe optical fields with the nonlinear medium causes thedevelopment of oscillating terms in the density matrixand therefore we may use the following Fourier expan-sion for each one of the elements

(m) im [n (co)kx —cot]pij (7)

Some of these Fourier components appear in the opticalpolarization at frequency 2', which is obtained by substi-tuting Eqs. (2)—(5) into the density-matrix equations andequating terms with identical time dependencies. Makingthe slowly varying envelope approximation of the waveEq. (3) (i.e., assuming ~V E~ && ~2inkVE~ and((),E~ && ~2i co(3,E)() we obtain

2

B+ 8 =1Np' P(2)

(3t n (2co) ()x ll (2')

where we have used p,"=(((, ;. The electric field in Eq. (3)is taken in the form

E(x t) A( t) i [n(co)kx c—ot]

Paa Pabp(t) =

Pba Pbb(2)

and its effect on the electric field is given by the waveequation

XMC (2') ~ 2

n (2')where the polarization at 2m is given by

P(2) ( (2)+ (2)) g D(2)/2Pab Pba Pab I

(8)

E(x, t) —c E(x, t) = —X)Lloc 2 P(x, t),Bt Bx Bt

(3)

Paa Pab

Pba Pbb

where the microscopic polarization vector P is given byP=Tr(pp) and X is the density of the optically activemolecures. The effective dipole moment vector matrix )M

is given by

Here D' '=p,', ' —pb

' is the population difference of theground and excited state oscillating at frequency men.Using Eqs. (7) and (8) we obtain the nonlinear susceptibil-ity

)(2")=y( )/N =[@Dc/n (2')][P' '/2] . (10)

Figure 1 shows all lowest-order paths in the develop-ment of terms contributing to the optical polarization atfrequency 2', P( '. The first index in the superscripts ofthe terms in Fig. 1 is the same index (m ) as in Eq. (7), and

43 RELATIONSHIP BETWEEN SECOND-HARMONIC GENERATION. . . 509

the second index is the order in field amplitudes (Aand/or F) and therefore the order of perturbation theoryof the term. The Appendix lists the steady-state values ofthese contributions to y obtained by substituting expan-sion (7) into the density-matrix equations and making the

I

steady-state approximation. As is evident from Fig. 1,there are 11 paths through which third-order contribu-tions to P' ' may develop. The magnitudes of these termsin the nonlinear optical susceptibility at frequency 2' areobtained by orientationally averaging:

Ip, b I f cos (8)exp[Fp„cos(8)/kT]dcos(8)(P4b) =

f exp [Fp„cos(8) /k T]d cos( 8)0

similarly

II .b I'I &P I'Bab e@

I y.bI'

5

and

Ip.bl'i~pl f cos'(8)exp[ Fp..cos(8)/kT]dcos(8) Ip.bl'Ial I Ip..l(p'.,&p) =f exp[Fp„cos(8)/kT]dcos(8) skT (13)

where the 11 C s resulting from the 11 contributionsshown in Fig. 1 are presented in the Appendix. All 11

D(o,o) AJ ~+

Ah, P (2,2) FP~ (2,3) P~:P~' F'

AP~. D(22) FPm . D(23) P

A+P (2,2) FQP (2,3) P~: P+'

AP~ D(2,2) FPab p(2,3) Pab.F' F P

Ah P (2,2)F

FPab . D(1,2)

+

FhP (1,2)

AP :P+'Ab, P (2 3): P+'

A P~ D(2'3) h, P

where we assumed Fp„/k T « 1 so the exponentialcould be expanded to lowest order. The resulting opticalpolarization can be expressed in the form

I'"'=( A'F)LII

D' 2co~b

5(co,b co +I —2icoI )—

x+C, ,

contributions have identical dependence on field ampli-tudes ( A F) and therefore include the same product oflocal field factors L—:g (co)g(2')g(0). It is important tonote that the C s, and therefore the susceptibilities, de-pend only upon the magnitude of the dipole moments,Ip, b I, lp„l, and lb@; their sign is of no consequence tothese susceptibilities. The sum of contributions 3—11 inFig. 1 accounts for the temperature-independent part ofy' ', y, . This contribution can be observed even in a ran-domly oriented solid medium in which the molecularorientation cannot be modified by application of a staticelectric field, as well as in liquid media. The sum of thecontributions 1 and 2 in Fig. 1 is the temperature-dependent part of g' '. It cannot be observed in a rota-tionally "frozen" medium. This temperature-dependentcontribution top' 'is proportional top' '.

In cases where the dephasing time is much smallerthan the excited-state decay time, T2 &(T, , the develop-ment of oscillating terms in the diagonal elements of thedensity matrix is much slower than the development inthe ofF-'diagonal elements. Therefore all contributionscontaining oscillating diagonal elements will not be ob-served when pulses of duration comparable to or shorterthan T& are used. The fast response of the medium con-sists of contributions 1, 3, 7, and 10 in Fig. 1, since thesepaths contain no oscillating diagonal elements. SucheA'ects were discussed in detail elsewhere. '

(0 p) FPBb p(0 1)D +

Ah, P (1 2): P+'

P. p(, )

APF'

AkP (2,3) 10III. RESULTS AND DISCUSSION

AP~ AP~ (2,3)

FICi. l. All lowest-order paths in the development of termscontributing to the optical polarization at frequency 2', P' '.The first index of the superscripts denotes the order of frequen-cy oscillation, mco, and the second index denotes the order infield amplitudes ( A and/or F) and therefore the order of pertur-bation theory of the term.

We now present results of calculations of P( —2';co, co),y( —2';co, co, O), and y, (

—2';co, co, O) versus frequency.The product of all local field factors, L, is taken as unitysince these factors depend upon the dispersion of themedium that may include bulk contributions of the re-fractive index and are therefore sample specific.

Figure 2 shows the sum of the fast and slow contribu-tions to p( —2';co, co) versus co/co, b for the case p, b =Ap.The slow contribution (path 2 in Fig. 1) is very weak so

510 R. BAVLI AND Y. B. BAND 43

TABLE I. Parameters of the medium.

Parameters

Transition dipole momentGround-state dipole momentPermanent dipole momentResonance frequencyDecay time'Dephasing timePopulation densityRefractive index

Symbol

PaaI"aa

ApCOQUE

TlT, ( =(1.0X 10-'T, )

a.u.

3.03.03.00.1

2.74X 10'2.74X 10'

2.96 X 10 Bohr1.33

7.62 X 10 " esu cm7.62X10 " esucm7.62 X 10 ' esu cm4.13 X 10' sec

66.2 nsec6.62 psec

2.00 X 10' cm1.33

'Calculated.Chosen because of lack of data.

that the fast response (path 1 in Fig. 1) and the overall(i.e., slow) response are similar. Since both contributionsto I' have the same dependence on dipole moments, theshape of the P curve is independent of both p,& and bp,and its magnitude is proportional to p,bhp. Clearly, thenonlinear polarization has maxima at the one- and thetwo-photon resonances. This is due to the fact that theperturbative expressions (see Appendix) for the denomi-nator of the dominant contribution to /3 is a product oftwo factors (co,& co i—I )(—co,&

—2' —iI ). The real partof f' is much larger than the imaginary part of P awayfrom the resonance peaks. Between the one- and thetwo-photon resonances, the Re(P) is of opposite signcompared to that outside this region. Of course, Re(P)vanishes very close to these resonances. ' ' Note thatthe ordinate of Fig. 2 is a log scale, and that the peaks of~y' '~ at the resonances are very much larger than awayfrom the resonances. As I increases these peaks becomeless pronounced and wider.

As shown in Sec. II, the phase and magnitude of eachone of the contributions to y has its own dependence onthe frequency of the fundamental field. Since the magni-tudes of these contributions are proportional to differentproducts of dipole moments, the interference betweenthem is affected by the relative values of p,b, p„, and Ap.Figures 3(a)—3(d) show the frequency dependence of thetemperature-independent contribution arising fromFWM processes (dashed curves) and the total ~y' '~ (solidcurves). Each of these figures describes a case withdifferent magnitudes of the transition and permanent di-pole moments (the parameters of the medium are given inTable I, and the four cases are specified in Table II).

lO"

lp-20

lp-21

10

]0-23

lp-24

lO"

lp-20

lO21

10

10

10

10'

lp-19

lQ 23

lp-25

RandoI I I I I I I

P,b=~P =P~

'~

rI ~ I

Randomly orientedsolid medium

I I I II

~ 1 I I

0.5 1.5

Ran domlliquid m

I s s s s s s ' ~

= h, p. = lop.„

I I I I I

rientedcolum

I I II

~ I I I I I ~

0.5 1.5

=(c) Randomly orientedliquid medium

= lpbp=p,

Randomly onentedsolid medium

I I I I I I I I II

t I

0.5I I I I I I '

II I I I I I I I I

1.5

I I l I I I I I ~ 11 I l l I I l I I I I I I I I I

1p-l3

10-15

I I I I I I I I I I I I I! I I I I I I I I I I I I I I I

p, =hp.

lO"

lO-"

lp-21

102

lp-25

I I I

== (cI)I I I I I I I I I I I I I I 1 I I

Randomly orientedliquid medium

'~

I I I I i I I

lpg, b——hP. = P,

IIE

rII

10 "

10-"I I I I I I I I I

I

I I I I I I I I I

I

I I I I I I I I I

0.5 1(0'(0b

1.5

FIG. 2. Absolute value of the second-order susceptibility forSHG, y( —2', co, co), vs co/coQQ.

10

Randomly orientedsolid medium

I I [ I t[

I I I

0.5 ~COb

1

I Ill I t I I

1.5

FIG. 3. Frequency dependence of the temperature-independent contribution of Iy' '~ arising from FWM processes~y',"~ (dashed curves) and the total ~y"'~ (solid curves) for thefour different values of the dipole moments described in Table I.

43 RELATIONSHIP BETWEEN SECOND-HARMONIC GENERATION. . . 511

TABLE II. Cases with diferent magnitudes of the transition and permanent dipole moments.

Dipolemoments

Pab

PaaAp

Case 1

Equal moments

3.03.03.0

Case 2Small p„

3.00.33.0

Case 3Small Ap

3.03.00.3

Case 4Small p.b

0.33.03.0

Clearly, ~y' '~ has a maximum at the one-photon and thetwo-photon resonances. As in the g' ' case, the real partof g' ' is dominant over the imaginary part away fromthe resonances. There are nine contributions to y', ' andits detailed shape depend upon the interferences amongthose nine complex functions of co. All the FWM contri-butions (~g', ' dashed curves) have a minimum betweenthe resonances. The position and depth of the minimumdepend upon the ratio p,b/Ap. Note that the FWM con-tribution in Figs. 2(a) and 2(b) are identical; there is nocontribution from p„. The total third-order susceptibili-ty at frequency 2' is obtained by addition ofP( —2co;co, co)p„/5kT to the FWM contribution rsee Eq.(I)]. Therefore y' ' depends upon p„as well as p, b andAp.

Figure 4 show the ratio ~y, ~/~y ~for the four cases de-

scribed in Table II. Figure 4(a) plots the total susceptibil-ities which will be observed using pulses of duration

much longer than T, . Figure 4(b) shows the fast contri-butions to the susceptibilities which will be observed us-ing pulses of duration shorter than T, (but much longerthan Tz). As already mentioned, this latter quantity doesnot depend upon the oscillating diagonal elements of thedensity matrix. Note that both these ratios have maximaand minima whose frequency and magnitude dependupon the relative values of the dipole moments. If ap-proximate information about the transition frequency co,b

and the relative values of p,b, p„, and Ap is available, itis possible to estimate frequencies where y, is negligiblecompared to y. For these frequencies y is nearly propor-tional to P. The second-order susceptibility /3 can be ob-tained by measuring y at these frequencies and thencorrecting the resultant value of 13 for dispersion (correc-tions for dispersion in order to obtain P at a particularfrequency from the value of P at a diff'erent frequency aremuch simpler than corrections of y, ).

10 I I I I I ! I I I l I I I I I I I I I I I I I I I I I

(a) IV. CONCLUSION

10

0.1

0.01

0.001small

I I I I I I I I I

i

I I I I I I I I

i

I I I I I I

0.5

&nbI I I

1.5

100 I I I I I I I I I t I I I I I I I I I I I I I I I I I

0C0

0V

r5

0.01

104

0

I/small h, p.

I I I I I I

i

I I I

0.5I I I I

[

I I I I I I I

03/Ib

1.5

FIG. 4. ~y, ~/~y ~vs co/co, z for the four different values of the

dipole moments described in Table I. (a) Total susceptibilitiesobserved using pulses of duration much longer than T&. {b) Fastcontributions to the susceptibilities observed using pulses ofduration shorter than T& (but much longer than T2 ).

The frequency and dipole moment dependence of P, y,and y, for a homogeneously broadened two-level systemwith permanent dipole moments have been analyticallydeveloped. There are a total of 11 contributions to y and9 contributions to y, . These quantities were numericallyevaluated as functions of frequency and ~p, ~ ~, ~IM„~, and~bp~. Both y and y, have peaks at the one- and two-photon resonances. The real parts of y and y, are dom-inant over their imaginary parts away from the reso-nances. We have shown that there are frequency regionswhere y, is negligible but there are also regions where

y, &&y. Since y, has contributions with various dipolemoment dependencies, the ratio

~ y, ~ /~ y ~depends on the

relative magnitudes of the dipole moments. The second-order susceptibility P can be obtained by measuring y atthe frequencies for which the ratio ~y, ~/~y~ is minimum,and then correcting the resultant value of 13 for disper-sion. Corrections for dispersion in order to obtain /3 at aparticular frequency from the value of )33 at a different fre-quency are much simpler than corrections of y, . The ra-tio ~y, ~/~y~ may be significantly diff'erent for short andlong pulses. These differences have been properly ac-counted for within the present theory. Hopefully, experi-mental data can be analyzed in terms of the presenttheory in order to extract accurate values of I33. However,great care must be exercised in doing so, because realmultilevel systems may not be well approximated by atwo-level system in which coupling to other levels is ab-sent. It remains to generalize the present work for a sys-tem in which two manifolds of levels that are each in

512 R. BAVLI AND Y. B. BAND 43

thermal equilibrium are modeled as an effective homo-geneously broadened two-level system.

ACKNOWLEDGMENTS

needed for contributions to y( —2co;co, co, 0):

p(2 2) 3m 2l r ~ ~p(1 1)2 4 2+r2 4,

(A12)

This work was supported in part by a grant from theU.S.-Israel Binational Science Foundation.

APPENDIX

The off-diagonal contributions to the optical polariza-tions are products of the transition dipole moment p, b

and off-diagonal density-matrix frequency components inthe form

p(i,j) (i,j )+ (i,j)+ =Pba Pab (A 1)

p(t, g) (&,g) (&,g)=Pab Pba (A2)

Now we use these definitions and Eqs. (7)—(11) to calcu-late the steady-state values of the contributions to the op-tical polarization at frequency 2' as shown in Fig. I:

pb', "=p,,bAD' '/(co, b

—co —iI ),AD]0, 0]/(~ b+~+l r),

P""=]]c AD' '[(co +co+i I )

+(co,b co i I )—']—,

(A3)

(A4)

(A5)

The expressions for frequency components of diagonaldensity-matrix elements always involve differences be-tween off-diagonal elements of the form

( 2 2 ) gab AP

(2co+iy )

p, b—A P+ (path 2) .(co+ i r ) (] ] )

2co+l p co~b(A13)

Similarly, we calculate contributions to y( —2co; co, co, 0),(2, 2)

(2, 3 )ApFPha

Pba'(co b 2co —l I

(2, 3)Pab

(bp, ) FA(co~b 2co l I )

(co,b+2co+i I )

COab +CO+ i I2~ab

(A14)

(b p)2F A co~b—co —i r

(co, +2co+iI )(A15)

The contribution of path 3 to y( —2co;co, co, 0) is given bythe sum of these two equations,

co,b +Sco —3I + 10i cur

(co,b—4co + I —4icol")

In path 2 of Fig. 1, the diagonal contribution toI3( —2co;co, co) is given by

P""= —[(co+iI )/co, ]P"" . (A6) (path 3) . (A16)

In order to calculate higher-order terms in frequency andfield amplitudes, it is useful to use the relations

pb'."= [(co.b+co+ ]r )/2co. b )P'+'",

p.","=[(~.b —~—l r)/2~. b ]P'+ ",(A7)

(Ag)

to obtain the following expressions for the oscillating off-diagonal density-matrix elements:

For path 4, we can use Eq. (A13) to obtain

(2, 3) ab ~ab~+ 2 2 2co b 4' + I 4icoI

2(co+i —I )FA p,bP]+ "(path 4) .

(~.', —4~'+ r' —4i~r)(2'+ iy)

(2,2)Pba

apw p""co b 2co /I

For path 5, we can use Eq. (A12) to obtain

(A17)

ApAco b +2co+ l I

M CO /Iab p(1 1)

2 +

QP g Crab +CO+ / I( co~b

—2co l I 2co~b

(1,1)I Pab'

Pab'N „+2'+/ I

(A9)

(A10)

FP'2'2'(2 3) Pab

2'+ l P

( —3co 2i I )b.pp, b AFP'+ "—(path 5) .

(co,b—4co +I 4l'coI )(2co+iy)—

The off-diagonal density-matrix term which contributesto ]33(

—2co;co, co) (path 1) is given by

(A 1 8)

~ b+2M r +3/~rm, b

—4' + I —4/car

(path 1) . (All)

(1,1)(1 2) PabF+ — ~ah+ l r= —VbF, . ~+'

~+/p co +ly'co(A19)

For calculation of the contribution of path 6 we calculatethe diagonal component,

Similarly, we calculate the value of the following term and obtain

43 RELATIONSHIP BETWEEN SECOND-HARMONIC GENERATION. . . 513

(2, 3) ~ab Pah (1,2)P+ —2

CO b 4CO + I 4iCOI

Using Eqs. (A28} and (A29), in a fashion similar to theprocedure in Eqs. (A 1)—(A8), we calculate the relations

2(—co+iI )FAp, ,bp'+"(path 6) .

(co,b 4—co + I 4—icol )(co+ i y')

(A20)

P' "=2P,bFD' ' '[~,b Z(~',b+r')],p(0, 1)=—((re }p(0,1)

p,"."=[(~., +ir}y2~.,]p"",

(A30)

(A31)

(A32)

In order to calculate the contribution of path 7 the fol-lowing off-diagonal elements are required:

(1,2)Pba'

F (1 1)

CO b CO /I

b,pF COab +CO+ / ICO b CO /I 2CO b

(A21)

(1,1)ApFpab'b + + I

ApFCO b +CO+/I

CO CO / Iab p(1 1)+

2COab(A22}

(2 3)=Pba'

COab—2CO —l I

(Ap) FA (co,b+co+il ) P"", (A23)2co,b (co,b 2co —i I )—(co,b co i r—) —+

(1,2)~p A pat

b +2co+/ I

(b,p) FA(co, b—co —iI ) P(+'" . (A24)

2co,b(co,b+2co+iI )(co,b+co+iI )

Using Eqs. (A23) and (A24) we obtain

(~'.„+S~' —3r'+ 8i ~r)(Si.)'F~P(,( "(~'., 4~'+—r' 4i~r—)(~.', ~'+ r—' 2i~r—)

(path 7) . (A25)

p."b"= [(co., i r—)/2co. b ]P'+ " (A33)

(1,2)Pba

ap A p'b'."CO b CO /I

ap, a (~., +ir)p", "2co b(co b co / I )

(A34)

(0, 1)ApAP b'

Pab'CO b+CO+/I

bed(co„—ir)P'+"2~.„(~.b+~+ i r )

( co 2il )b—pAP—( ')p(1,2)

CO CO + I 2lCOI

(A35)

(A36)

The contribution of path 9 to y( —2co;co, co, 0) is obtainedusing Eq. (A36),

pab AP

2CO+ l P

( co 2i—l )b,pp,—b A P'+"(path 9) .

(co,b—co +I 2icoI —)(2co+iy)

(A37)

Equations (A34) and (A35) are also used to calculate theoff-diagonal contribution of path 10,

U»ng Eqs. (A30)—(A33) we calculate the second-orderoff-diagonal terms,

Using Eqs. (A21) and (A22) we find

P(, ) 2 gpFP(, )

COab CO +I 2lCOI(A26)

(1,2)ApApb '

Pba'COab 2CO / I

and from Eq. (A26) we obtain

p, b AP

2CO+ / P2( co i I )b,p—,p, b

—AFP(+ "(path 8) .

(co b—co +I" —2icor)(2co+iy)

(A27)

(bp) FA (co,b+il ) P( '), (A38)2co,b(co, b

—2co —iI )(co,b—co —i I )

(1,2)hpAP, '

Pab'CO +2CO+ / I

(bp) FA(co, b iI )—P '+' ), (A39)

2co,b(co,b+2co+ i r )(co,b+ co+ i r )

pb,"=p,bFD' '/(co, b i I ), —

+(r)(A28)

(A29)

In order to calculate the contributions of the paths withF interacting with the medium first, we calculate thevalues of the following zero-frequency first-order (in thefield amplitudes) off-diagonal elements:

By adding Eqs. (A38} and (A39) we find

(„) (M.'b+2~2 —3r'+6i mr }(3bP )'P(")(co,b

—4co + I —4icoI )(co,b —co + I —2icoI )

(path 10) . (A40)

514 R. BAVLI AND Y. B.BAND 43

Using Eq. (A31) we obtain

D""=l., As"' "y(~+i y)= —[tI ico,b(co+ ty)]p, &

AP'+ ",from which the contribution of path 11 is obtained,

(A41)

co,b +See —3I" + 10icoI

(co,b—4co +1 —4icol )

2Q)~b + E I 67gbIPat I

(2co +iyco)(co, b—4co +I 4—i col )

(A44c)

(&,3)— 2Q)ab 3Pgb (&,2)P+' =co b 4m +I 4icoI

—l'21 ( A p ) P'+"(path 11) .

(co,b 4co—+1 4icol—)(co+iy)

C=(2co+iy)(co, b

—co +I —2icol )

(2co+iy)(co, b—4co +I —4icol ) 2

(A44cl)

(A44e)

(A42)

Since the contributions of paths 1—8 to y( —2co;co, co, O)are a11 proportional to P'+'", we express the remainingcontributions (paths 9—11) in terms of P'+" instead ofP'+'" using the relation

—2( + '1")

(2co+i y )(co,b—4co + I —4icol )

(A44g)

co b+5cu —3I +8coI

(co,b—co +1 —2icoI')(co, b

—4co +1 4i col—)

co,b—co +1 —Zicol )P' '"=

Q) b+I(A43)

Using the expressions for the contributions of the 11terms to y( —2co;co, co, O), Eq. (A43) and Eqs. (15)—(18), weobtain the complex C&'s of Eq. (18):

x fbpf',co+2ir 1

Ig(2co+iy)(co,q+I ) 2

co, +2' —3I +6icoI

(~'.,—4~'+ r' —4t ~i )(~'.„+r')

(A44h)

(A441)

, +2 —I +3' 1 fhpf fp„f67 b co + 4 coI c77+7 b

——I ~i I IP..I,co+E I 1

( 2co + t y )cocoa t,

(A44a)

(A44b)

i2&c—o,b—co +I —2icol )

(co, 4co + I 4i —col )(co„+—I )(co+i y )

X Pabf

where co =—kT/A.

(A441c)

*Electronic address: band@bguvms.~S. K. Kurtz and T. T. Perry, J. Appl. Phys. 39, 3798 (1968).G. Berkovic and Y. R. Shen, J. Opt. Soc. Am. B 4, 945 (1987).G. Mayer, C. R. Acad. Sci. Paris B 267, 54 (1968).

4Nonlinear Optical Properties of Organic Molecules and Crystals,edited by D. S. Chemla and J. Zyss (Academic, New York,1987).

5J. L. Oudar, D. S. Chemla, and Batifol, J. Chem. Phys. 67, 1626(1977).

J. L. Oudar, J. Chem. Phys. 67, 446 (1977).7J. L. Oudar and H. Le Person, Opt. Commun. 15, 258 (1975).8M. Barzoukas, D. Josse, P. Fremaux, J. Zyss, J. F. Nicoud, and

J. O. Morley, J. Opt. Soc. Am. B 4, 977 (1987).G. R. Meredith, J. G. VanDusen, and D. J. Williams, Macro-

molecules 15, 1385 (1982).' Y. B.Band and R. Bavli, Phys. Rev. A 36, 3203 {1987).

R. Bavli, D. F. Heller, and Y. B. Band, J. Chem. Phys. 91,6714 (1989).

' M. A. Kmetic, R. A. Thuraisingham, and W. J. Meath, Phys.Rev. A 33, 1688 {1986).

~3W. J. Meath and E. A. Power, Mol. Phys. 51, 585 (1984).~4M. A. Kmetic and W. J. Meath, Phys. Lett. 108A, 340 (1985).'~G. F. Thomas and W. J. Meath, Mol. Phys. 46, 743 (1982).~W. J. Meath and E. A. Power, J. Phys. B 17, 763 (1984).7B. E. Scharf and Y. B. Band, Chem. Phys. Lett. 144, 165

(1988).R. Bavli, D. F. Heller, and Y. B. Band, Phys. Rev. A 41, 3960(1990).

' Y. B. Band, R. Bavli, and D. F. Heller, Chem. Phys. Lett. 156,405 (1989).R. Bavli and Y. B. Band, Phys. Rev. A 36, 5200 (1987).