1790 J. Opt. Soc. Am. B/Vol. 4, No. 11/November 1987

Phase-matched frequency tripling and phase conjugation inisotropic materials

Goran Manneberg

Department of Physics II, Royal Institute of Technology, S-100 44 Stockholm, Sweden

Received January 5, 1987; accepted June 17, 1987

A new phase-matching scheme for noncollinearly phase-matched fifth-order difference-frequency generation(FODGE), yielding the third harmonic, is described. The coupled-wave equations are solved for some special cases,and the phase-conjugating properties of FODGE are discussed. Different causes of phase mismatch are alsodiscussed.

INTRODUCTION

Noncollinear phase matching of third-order difference-fre-quency generation was originally described by Auston' andwas subsequently treated by many others2 4; see Figs. 1 and 2for the phase-matching schemes. The efficiency of theseschemes is, however, hampered by the inherent short inter-action length.

Another noncollinear phase-matching scheme for third-order difference-frequency generation was used for phaseconjugation, with frequency conversion (see Fig. 3) in poly-chromatic phase conjugation (PPC), 5 in which a longer in-teraction length is achieved through creation of a cavity forthe pump field.

The fifth-order nonlinear susceptibility was utilized byDucloy6 to achieve phase conjugation with frequency con-version, combined with a wide field of view. In the samepaper he also mentions some possible methods for enhancingthe fifth-order nonlinear susceptibility.

In all these cases, however, the wavelength of the generat-ed field can never be shorter than half of the pump wave-length.

BASIC IDEA

The primary aim of this paper is to present an experimentwith which the fifth-order nonlinear susceptibility can bemeasured in nonabsorbing bulk materials. No such mea-surements have, to my knowledge, been performed, andhence no estimations of necessary power levels, etc. are pre-sented in this paper. There are, however, indications thatmaterials with enhanced fifth-order susceptibilities7 (metal-colloid suspensions and semiconductor-doped glasses) arebeing developed, and thus a growing demand exists for selec-tive probing experiments.

In this paper we propose a noncollinear phase-matchingscheme for degenerate fifth-order difference-frequency gen-eration (FODGE), i.e., the created beam is at the third-harmonic frequency of the fundamental, W3 = 4w, - wl.

The phase-matching scheme of FODGE can be seen inFig. 4. The experimental difficulties would be prohibitive ifthe assumed overlapping of four fields (k1, k2a, k2b, k4 ) couldnot be simplified as in Fig. 5, where the process involves two

parts of a folded laser cavity with mirrors that are 100%reflecting for the fundamental wavelength but, in the idealcase, are totally transparent for the third harmonic. Thetwo pump fields E2a and E2b are thus created by reflectingthe beam an angle 20.

The cavity is preferably limited by cylindrical mirrorswith the cylinder axis in the plane of the paper in Fig. 5.This is to maximize the flatness of the wave front in thecavity plane and to maximize the energy concentration inthe perpendicular direction.

If the fields in the interaction area can be approximated asplane waves, they can be expressed as follows:

A1 = E1 exp[i(wlt -klz)],

A2a = E2a expli[wt - kl(cos )x - kl(sin 0)z]}

A2b = E2b expli[wlt + kl(cos 0)x - kl(sin 0)z]l,

A4 = E4 exp[i(wlt + klz)],

A5a = E5a expli[wt - kl(cos O)x + kl(sin 0)z]b

A5b = E5b expli[wlt + kl(cos O)x + kl(sin O)z]}.

(1)

(2)

(3)

(4)

(5)

(6)

The full, expanded expression for the fifth-order polariza-tion would now contain 125 = 248,832 (the sum of six fieldexpressions, with complex conjugates, to the fifth power)terms, among which there are several classes that can bephase matched.

First, it can be seen that there are three pairs of antiparal-lel fields (1 with 4, 2a with 5b, and 2b with 5a). One photoncan be taken from each of these fields to create pump fieldsfor phase conjugation with degenerate four-wave mixing.All these fields generate photons of the fundamental wave-length into the modes already prevalent in the cavity. Thus,in the steady state, they do not change the physical situationin any respect other than in maintaining the balance be-tween the pump fields.

Second, terms of type Ei12IEj12Ea describe a polarization

at the same frequency and with the same k vector as the fieldEh¢. This polarization is therefore a fifth-order correspon-dence to the optical Kerr effect on the field Ek; thus it doesnot couple energy between the fields but can be considered

0740-3224/87/111790-04$02.00 © 1987 Optical Society of America

Goran Manneberg

Vol. 4, No. 11/November 1987/J. Opt. Soc. Am. B 1791

There is, however, one class of terms at the third-harmon-ic frequency in the nonlinear polarization that can be phasematched; that is, Eqs. (1)-(4) can be combined to yield thefollowing nonlinear polarization:

Pnl = x5AlSA2aA2bA4*. (7)

Fig. 1. Phase-matching scheme for third-order difference-fre-quency generation, as proposed and demonstrated by Auston. 1

This Fourier component of the polarization can give rise to afield

A 3 = E 3 expi[3wt - k1(3 + 2 sin 0)z] (8)

for which the phase-matching condition can be written as

k3w = k(3 + 2 sin 0), (9)

which can be expressed in the linear refractive indices atfundamental frequency n, and third-harmonic frequency n3:

sin 0 = 1.5(n 3 - nl)/nl.Fig. 2. Phase-matching scheme used for BOXCARS proposed byEckbreth. 2

k-

-kl ~~~~ 2b

Fig. 3. Phase matching in the third-order process: PPC. 5

'kp p k3

Fig. 4. Phase-matching scheme used in this paper: k, k 2a, kb,and k4 represent beams at the fundamental frequency, whereas k3represents a beam at the third-harmonic frequency. The differencein length between 3k1 and k 3 caused by chromatic dispersion iscompensated for by adjustment of the angle 0.

R=1 ER R=1MDIUM Ir' R31=0

R =1R=1 R 3 0

Fig. 5. Suggestion for experimental setup for FODGE. A lasercavity without outcoupling mirrors for the fundamental wavelength(fields E1 , E 2, and E4 ) is folded as depicted. The generation of thethird harmonic (E3) is the only outcoupling mechanism.

merely a change in the index of refraction that is included inthe coupling coefficients of the coupled-wave equations.

Third, there is also a preponderance of terms at the funda-mental and third-harmonic frequencies that are not phasematched.

Finally, there are the fifth-harmonic frequency terms thatfor phase matching require that nw, = n5w, which is unrealiz-able in most materials.

(10)

Such a phase-matching angle 0 can be found when the dis-persion in the medium is not so strong.

Equations (1), (2), (5), and (6) will further yield, as asimilar solution, a frequency-tripled field propagating in theopposite direction. There is no term in the fifth-order po-larization that will couple energy directly between this fieldand the aforementioned E3. Also, if the cavity end mirrorsare 100% transmissive for the third harmonic, the forward-and backward-traveling third-harmonic fields will not affecteach other and, hence, can be treated separately.

From a photon point of view, this process can be seen as anannihilation of two photons from beam 1 and one photoneach from beams 2a and 2b, respectively, as the generation ofone photon in both the third-harmonic beam and beam 4(for proof see Appendix A). The reverse process is, ofcourse, also compatible with the phase-matching condition,but, as is seen from the solution of the coupled-wave equa-tions below, this does not, in the limit of no pump depletion,take place if there is no input E 3.

COUPLED-WAVE EQUATIONS

If all beams were approximated as plane and slowly varyingin amplitude, the system of coupled-wave equations can beexpressed as

___ + n1 aE1 = -2 El*E2a*E2 b*E3E 4,6Z c bt n 1 c

(la)

5E2a 5E2a . nl E2,-cos 0 + sin +-

5x 3z c 3t

=- - 5El*2E2*E3E4, (lib)nlc

6E2 b 6E 2b . , E2 b- cos +-sin 0 +-

5x 5z c bt

= - n1cEE2a*E3E4,

5E3 n3 _E3 ____5 - + - = -°35El E2aE2bE4*15Z ±C; t n3 C

6E4 n, E 4 iolx55z c bt n1c El 2 E2UE2 bEa*.

(llc)

(lid)

(lie)

Gbran Manneberg

1792 J. Opt. Soc. Am. B/Vol. 4, No. 11/November 1987

The simplest approximation in which they can be solved iswhen all the pump beams (1, 2a, 2b, and 4) are cw andundepleted, in which case the only necessary boundary con-dition can be written as

E3(z = 0) = E30.

The solution will then be that

LE3(Z) = E30 - i f Q3El2E22E4*dz,

(12)

(13)

with

W3x(5

° 2n 3 c(14)

where I have approximated the z dependence for E2a and E2b

as equal. It can be seen from Eq. (13) that, if there is nofrequency-tripled input (E30 = 0), energy is always trans-ferred to beam E3. This situation is somewhat altered in themore realistic case when the cavity end mirrors have a finitereflectivity for 3w. If this reflectivity is large, the lasercavity must be doubly resonant for the generation to beefficient.

BSE

1\ I / / X5 E,'�i

iI

IN E,/1'

I

DIELEC TRICMIRROR

_E3_1 --- __ -1

Fig. 6. Suggestion for experimental setup to study the phase-con-jugating property of FODGE. A high-power beam, entering fromthe left, is divided by the beam splitter (BS) to create the two pumpfields El and E2. The weak probe beam E 4 enters from the right,and the conjugate beam E 3 at third-harmonic frequency leaves onthe right.

and no conjugate input

E3(z = 0) = . (15b)

The remaining equations (ld) and (lie) can be written as

6E 3_R = -iK 3E*, (16a

6E4*R- =KlE3'

PHASE CONJUGATION WITH FREQUENCYCONVERSION

In the experimental configuration mentioned above, El andE4 have antiparallel k vectors because of the laser cavitycondition, and, as can be seen from Eq. (13), E3 has theconjugate phase of E4. However, this phase-conjugationproperty is of little consequence for the physical situation, asit essentially means only that E3 and El will have the sametransverse phase distribution.

If, on the other hand, the experimental setup is changedaccording to Fig. 6, the process no longer takes place inside alaser cavity, and the situation is changed in many respects.

First, the fields E5a and E5b are eliminated, which will alsoeliminate two of the pairs of fields mentioned in Section 2 asbeing capable of pumping phase conjugation with degener-ate four-wave mixing (DFWM).

Second, if the field E4 is taken to be several orders ofmagnitude weaker than El, this pair is also not capable ofpumping DFWM. Hence the pairs of beams creating thepump fields for wavelength degenerate phase conjugation nolonger exist, and complications resulting from this can bedisregarded.

Third, in this situation the field E can be made up from.several plane-wave components, as is the case in an image-carrying beam. This means that E3 can, in a real sense, beregarded as the phase conjugate of E4. In FODGE, however,the conjugate E3 is at the third-harmonic frequency of theprobe E4. Phase conjugating with such large wavelengthchanges can be used for image conversion, as discussed inRef. 8.

Mathematically, this problem can be solved in the approx-imation of undepleted cw pumps (El, E2a,2b), which meansthat the only necessary boundary conditions are that therebe a finite probe input

E 4(z = L) = E40

Solving these equations in a manner inspiredPepper 9 yields a conjugate field

(16b)

by Yariv and

E3 (z) = -i - E siKtEz) (17)n3 cos(VK1K3L)'

with

wx(5)

=, El E2 (18)

The conjugate intensity can then be expressed as

13(L) = 314(L)tan( KK 3L), (19)

where it is clear, from inspection, that a singularity can beexpected when the argument of the trigonometric function islarger than 7r/2. This corresponds to the case of oscillationin which both beams E3 and E4 are started from noise.

At oscillation, the approximation of undepleted pumps is,of course, not valid, and because of that it is not possible withthis method to give any approximation of the conversionefficiency.

PHASE MISMATCH

One of the reasons for phase mismatch is inaccurate adjust-ment of the phase-matching angle 0. It can be seen fromdifferentiating the phase-matching condition [Eq. (9)] that asmall change in 0, AO will yield a change in the k vector of thenonlinear polarization

Ak = 2k cos OAO, (20)

from which it can be seen that phase mismatch is a first-order phenomenon in 0.

Another cause of phase mismatch, especially in the casewhen the process takes place outside the laser cavity as in

G6ran Manneberg

I4 E,I

(15a)

Vol. 4, No. 11/November 1987/J. Opt. Soc. Am. B 1793

Fig. 6, is the nonperfect antiparallelism if k, and k4. If weassume k4 to have a small transverse component:

k4= (-kl cos 'k, 0, k, sin ), (21)

the square of the k vector of the polarization correspondingto Eq. (7) will be

kp2 = 12[(2 + cos 0 + 2 sin 0)2 + sin2 0], (22)

which should be compared with the square of the correctlength of the k vector at 3w1 given by Eq. (9):

k 3 = k1 (3 + 2 sin 0). (23)

Subtracting Eq. (23) from Eq. (22) gives

A(k2) = 4k1 (cos 0 - 1)(1 + sin 0),

from which

2n1Ak = (1 + sin0)023n 3

for small 0.It can thus be seen that phase mismatch is a second-order

phenomenon in , which will enhance the possibilities forphase conjugation of even more-complicated fields thanGaussian fields.8

APPLICATIONS

The most natural application of FODGE is, of course, intra-cavity third-harmonic generation with high conversion effi-ciency. An experimentally more interesting application forFODGE is as a probe experiment for measuring X5 withhigher selectivity than can be done by performing third-order experiments, such as phase conjugation with DFWMand measuring the (small) intensity-dependent part of X3

and identifying that as mainly X5.Finally, FODGE can be used for phase conjugation with

large wavelength shifts.

SUMMARY

A new phase-matching scheme was presented for degenerateFODGE, yielding the third-harmonic frequency. The en-tire system of coupled-wave equations was formulated andsolved for some special cases. Further, the phase-conjugat-ing property of the process was recognized and discussed.Some possible applications were also outlined.

APPENDIX A

In the steady state and the plane-wave approximation, thecoupled-wave equations can be written as

6E,_ = -2iQ 1 Ei*E2 *2 E3E 4, (Al)6Z

5E2_ = -2iQ 1 /sin 0E1*2E2*E3E4 , (A2)

3Z5E3 22__Z = Q3EJ E 2 E4 *, (A3)

(24)

By multiplying Eq. (Al) with El* and taking its complexconjugate,

MIE,12E = iQlEi2E 2

2 E 3 *E 4 * (A6)

is obtained. In a similar fashion Eq. (A3) can be multipliedwith E3 * to yield

=IE3I12 2- 2AA- Q3 1E2 E3*E4*. (A7)

By dividing Eqs. (A6) and (A7) with the photon energies ofthe fields E(hw 1) and E3(hw3) and multiplying these fields

(25) with their respective refractive indices, the following rela-tion is obtained:

--z h + = 0) o6Z hwl hw3)

which can be expressed as

2m, + m3 const.,

(A8)

(A9)

where mi is the number of photons in field Ei.By combining the other equations as described in the

above paragraph, it is easily proved that the only two possi-ble processes in steady state are that two photons each aretaken from fields E2 and E2, respectively, to yield one photoneach at the third-harmonic frequency in field E 3 and at thefundamental frequency in field E4 , or the reverse.

ACKNOWLEDGMENTS

I gratefully acknowledge valuable discussions with K. Bie-dermann, B. Jaskorzynska, and M. Breidne.

REFERENCES

1. D. H. Auston, "Nonlinear spectroscopy of picosecond pulses,"Opt. Commun. 3, 272 (1972).

2. A. Eckbreth, "BOXCARS-crossed beams phase matched CARSgeneration in gases," Appl. Phys. Lett. 32, 421 (1978).

3. R. A. Wood, R. G. Harrison, S. R. Butcher, and C. R. Pidgeon,"Noncollinearly phase matched four wave mixing of CO2 andNH3 laser radiation in germanium," Opt. Commun. 26, 463(1978).

4. A. Penzkoefer, J. Kraus, and J. Sperka, "Noncollinearly phasematched four photon frequency mixing in water," Opt. Commun.37, 437 (1981).

5. G. Manneberg, "Polychromatic phase conjugation with noncol-linearly phase matched difference frequency generation," J. Opt.Soc. Am. B 3, 849 (1986).

6. M. Ducloy, "Optical phase conjugation with frequency upconver-sion via high order nondegenerate multiwave mixing," Appl.Phys. Lett. 46, 1020 (1985).

7. C. Flytzanis, presented at the European Conference on Optics,Optical Systems and Applications, Florence, Italy, 1986.

8. G. Manneberg, "Image formation in polychromatic phase conju-gation," J. Opt. Soc. Am. A 3, 2033 (1986).

9. A. Yariv and D. M. Pepper, "Amplified reflection, phase conjuga-tion and oscillation in degenerate four wave mixing," Opt. Lett. 1,16 (1977).

(A4)E4 = iQ1E12 E2

2E3*,

where

Q 2nic(A5)

Gbran Manneberg