Vol. 7, No. 3/March 1990/J. Opt. Soc. Am. B 335

Theory of phase conjugation in frequency doubling

Lee M. Frantz

TRW Electronics & Defense Sector, One Space Park, Redondo Beach, California 90278

Received April 10, 1989; accepted November 8, 1989

It is shown that, in frequency doubling, the phase aberrations that are due to the crystal's surface unevenness can beremoved by phase conjugation, leaving a residue that is insignificant in most practical cases.

1. INTRODUCTION

Ordinarily, frequency doubling leads to beam-quality degra-dation. For conversion efficiency, it is desirable to reducetemperature variation in the crystal; this can be done bysegmenting the crystal longitudinally and flowing a coolingmedium between segments. Residual unevenness of themany crystal surfaces, however, then becomes a source ofpoorer beam quality. Here it is shown theoretically that, ifthe beam is phase conjugated and passed back through thecrystal, the surface-induced phase aberrations are removedeven while the doubling process proceeds, provided thatcertain conditions are satisfied. Experimentally, the re-moval of the aberrations in this way has already been dem-onstrated.'

To see why this effect occurs, we first note that, if there isan amplifier between the crystal and the phase-conjugatemirror, then the aberrations induced by it on the fundamen-tal wave are nullified before any further conversion takesplace. Furthermore, the surface-induced aberrations areprogressively removed from the fundamental as it returns inconjugated form through those surfaces that originallycaused them, so they are not transmitted to any part of theharmonic that is generated after their removal.

But what of those aberrations that are still in the funda-mental wave when an element of harmonic wave is generat-ed? And what is the effect of the remaining surfaces on thiselement? We shall see that the aberrations transmittedfrom the fundamental are nearly canceled on the harmonicby the unevenness of the remaining surfaces, in much thesame way that they are canceled from the conjugate of thefundamental itself. Thus we shall conclude that if the fun-damental is incident on the frequency-doubling crystal as anearly plane wave, the harmonic will emerge as a nearlyplane wave.

assume, for the moment, that the incident fundamental is aplane wave,

EF(p, - urn) = E = const.

Throughout the analysis, we shall treat it as monochromatic.In practice, the peak-to-valley unevenness um can be helddown to the order of 600 A (Ref. 2); this is small enough thatthe effect of the passage of the wave to z = 0 can be describedby a phase increment calculated from geometric optics. Ig-noring reflection, which is irrelevant to this argument, wehave

EF(P, 0) = E0 exp[iOF(P)], (1)

where

OF(P) = kO[um - u(p)] + nkou(p) (2)

and where ko is the free-space wave number of the funda-mental (for this argument we also ignore the presence of thecoolant) and n is its refractive index in the crystal. Thereturning fundamental, having reflected from the phase-conjugate mirror, is proportional to the conjugate of Eq. (1);if the proportionality factor is ignored, it is

EF*(p, 0) = E0 exp (ikF) . (3)

The theory of harmonic generation shows that the spatialgrowth rate of the harmonic wave is proportional to thesquare of the amplitude of the generating wave,3 which inthis case is the complex conjugate of the fundamental:

(4)d 0EH 0)12,

( ) = A [EF* (p, )2

where A is a constant. From Eq. (4) we see that the differen-tial element of harmonic wave created in the differentialelement of length dz at z = o is

2. THEORETICAL BASIS

To understand the physics underlying this assertion, consid-er the first crystal segment encountered by the fundamentalwave. In Fig. 1 the hatched region represents the segment,and the curve is its left-hand surface. The propagationdirection is along the z axis, and the origin is set so that the xand y axes just touch that element of surface that extendsthe farthest to the right. At any location p in the x-y plane,the magnitude of the surface's distance from z = 0 is denotedby u(p), and the maximum distance is um = max[u(p)]. We

dEH(P, 0) = A [EF*(p, 0)]2dz (5)

or, using Eq. (3),

dEH(p, 0) = AEO exp[-i2bF(p)]dz.

On leaving the crystal through the surface, the harmonicwave element acquires an additional phase factor, just as thefundamental did on entering it:

dEH(p, - um) = CEH(p, 0)exp[ikH(p)]

= AE02 expli[OH(p) - 2OF(p)Ildz. (6)

0740-3224/90/030335-05$02.00 © 1990 Optical Society of America

Lee M. Frantz

336 J. Opt. Soc. Am. B/Vol. 7, No. 3/March 1990

X

Z= Um z=

Fig. 1. Uneven surface of a crystal segment. The y axis is normalto the plane of the figure. The phase-conjugate mirror is to theright of all the crystal segments, and the incident fundamental wavegoes from left to right.

The phase distribution 0H(P) is of exactly the same form asthat given for OF(P) by Eq. (2), except that ko is replaced bythe free-space wave number of the harmonic wave and n isreplaced by the refractive index of the harmonic. We as-sume now that the fundamental is incident at precisely thephase-matching angle, so that the two refractive indices areidentical. We also note that the harmonic's wave number is2kO, so that

PH(P) = 2F(P).

Inserting this into Eq. (6), we find that

dEH(p, - U.) = AEo'dz,

that is, the emerging harmonic element is a plane wave. Theaberrations passed on to it by the fundamental have beenremoved at the surface in the same way that the conjugate ofthe fundamental has its aberrations removed.

If the element of harmonic wave were generated on thefirst pass of the fundamental through the crystal, instead ofon the return pass, it would have the form

dEH(p, 0) = A[EF(p, 0)]2dz.

After propagating through the rest of the crystal, picking upphase aberrations, being phase conjugated, then returningthrough the crystal and thereby having all these phase aber-rations removed, it would arrive at z = 0 in precisely the formshown in Eq. (5). The argument would then follow as be-fore.

It should be noted that the same type of argument holdsfor harmonic conversion of any order.

3. GENERAL CASE

The above argument applies to an element of a harmonicwave generated at the first surface of the first crystal seg-ment. Now we consider the more general case of generationanywhere in any segment; we show that for frequency dou-bling the same conclusion can be drawn, provided, however,that certain conditions are satisfied. The generalization forhigher-order harmonic conversion will not be consideredhere.

Let the phase-conjugated fundamental wave, on its returnpass, be denoted by EFC. It has the form

EFC(P, Z) = f(p, z)EF* (P, Z), (7)

where f(p, z) describes the effect on EF* of gain in the ampli-fier and loss in the crystal. The factor f may depend on pbecause of a nonuniform transverse gain distribution or be-cause of gain saturation. The latter effect is likely to be thebigger, since the fundamental beam intensity must fall offstrongly from its central value to avoid edge diffraction.But f is real, because it describes only an intensity effect.We shall assume that the scale size of the p variations in f issufficiently large, and the propagation distance sufficientlysmall, that f is not altered by diffraction; that is, we shallignore any changes in f with propagation. As before, wetreat the wave as monochromatic. We make the approxima-tion, therefore, of ignoring whatever dependence f may haveon the frequencies within the actual frequency spread of thewave.

If EFC is known at z = zg inside one of the crystal segments,then at the segment surface, z = ZN < Zg, it can be obtainedfrom Fresnel propagation theory as 4

EFC(P, ZN) = 2 kiN J EFC(pl, zg)exp[i (p1 - P)2 dpl, (8)

where 6 N = Zg - ZN and k is the wave number. This relation-ship can be mathematically inverted to give

EFC(p Z,) = -k f EFC(Pl, ZN)expF -i (P1 -P) dpl.2 7r 3 N J[ 2 0 N ~P2dl

(9)

If at some z this wave generates an element of harmonicwave, it will be given, as in Eq. (5), by

dEH(p, z) = A[EFC(p, z)]2dz.

For notational simplicity, we drop the differential sym-bols, absorb the dz into A, and rewrite this equation as

EH(P, z) = A[EFC(p, Z)]2. (10)

If this harmonic element is generated at z = zg, then at thesegment surface it is given, in analogy to Eq. (8), by

EH(P, ZN) = k f |LEH(Pl, z)exp[ ,, (P1 P) dpl.-IriSNJ [ NJ

Now, into the integral we insert the expression [Eq. (10)]for EH(P, zg). Then we substitute Eq. (9) for EFC(p, g),reverse the orders of integration, identify a formal integralexpression for the Dirac delta function, and integrate over it.The result, after a simple transformation of integration vari-ables, is

AC [ X 1/2 1EH(PZN) f JEFC [P + 2r) {, ZNJ

X EFCP -( 2) t, ZN] exp(-i )dt,

where A is the wavelength. Next we introduce Eq. (7). Be-cause we are ignoring changes in f with propagation, wemake the approximation

X6N 1/2 ]

Lee M. Frantz

Vol. 7, No. 3/March 1990/J. Opt. Soc. Am. B 337

takefout ofthe integral, and absorb it into the factor A. Fornotational simplicity, we ignore the new p dependence of A,since it will always be assumed unaltered by propagation; wealso ignore loss in the crystal and, therefore, drop the new ZN

dependence of A. The result is

EH(P, ZN)= . EF* + (2) e ZN]

X EF*[ - (XN 1/2

, ZNexP(-i2)d.

The integration variable t is dimensionless. Because of thefactor exp(-i%2), which oscillates extremely rapidly for >>1, we expect that there will be no significant contribution tothe integral for t >> 1.

Therefore, in the argument of EF*, the ratio of the magni-tude of the term (N/27r) 1/2t to that of the term p is of theorder of (X6N/27rp2)1/2. 5N is a propagation distance, and p isa transverse dimension of the wave, so the ratio XbN/p 2 is,roughly speaking, the ratio of the diffraction spread XeN/ tothe beam width p. As is discussed in more detail below, weanticipate that the dimensions of a phase-conjugation fre-quency-doubling experiment can be such that diffractionwill be insignificant. Intuitively, then, we expect that the t-dependent term in the argument of Ep* can be dropped togood approximation. If it can be dropped, the t integrationthen simply provides a numerical factor in front of [EF*(p)]2,and the resulting equation is effectively equivalent to Eq.(5). The principal conclusion of Section 1 then follows fordifferential elements of the second harmonic generated any-where in any crystal segment, namely, that phase conjuga-tion removes all the phase aberrations from the second har-monic. Thus it appears intuitively that, if the experimentaldimensions are such that the fractional diffraction spread ofthe fundamental wave is small, phase conjugation shouldresult in an aberration-free second harmonic. We now showmathematically that this is indeed the case. We expand EF*in a power series of {x and ty, keeping only terms up to secondorder. The integrals can now be done exactly, giving

where QFN(P) is understood to be of the same form4 as givenin Eq. (2) for 'kF(P), UmN is defined as UmN = max[uN(p)], andUN(p) is generally different for each of the surfaces. Similar-ly, the harmonic wave gains a phase factor in passing out ofthe crystal segment,

(14)EH(P ZN - UmN) = EH(P, ZN)exp[ioHN(P)],

where, as at the first surface,

'OHN(P) = 20FN(P).

Using Eqs. (13)-(15) in Eq. (11), we obtain

EH(P, ZN - UmN) = A[EF* (p, ZN - UmN)]2

(1 + EN).

Comparing this equation with Eq. (10), and recalling thatEq. (10) describes the harmonic element at the point ofcreation, we see that, to within the error term N, the har-monic element has exactly the form that it would have if ithad been generated at ZN - UmN. Repeating this process totake the harmonic through the intersegment gap and thenjust through the next surface at z = ZN-1 yields

EH(p, ZN_1 - U.,vj,) = A[EF*(p, ZN-1 Um,N-1)I

X (1 + N)(l + N-)-

In doing this, we have ignored the effect of propagation onEN, since we shall be looking only for an estimate of the error.Now, repeating the process through all the remaining seg-ments, we finally obtain

N

EH(p, Z1 - Ur) = A[Eo* (p)]2 71 (1+Ej)j=1

(16)

where EO(p) is the fundamental wave incident upon the firstsegment and zi is the location of the first surface. Equation(12) holds for e, except that for j #M N we have bj = Zj+I - Zj.

Next, we need to examine and estimate the E/s to showthat, under conditions easily attainable in the laboratory,they may be ignored. In Eq. (12) we approximate EF* by

(15)

EH(p, ZN) = A[EF*(p, ZN)] 2(1 + EN),

where

EN [ 1 V EF*(PI ZN)l - 1N=lEF* PX ZN)V1 EF (P, ZN)

Xv I2EF* (PI ZN)) 1X8

4r

(11)EF*(p, ZN) = EO*(p)exp[ i0N(p)], (17)

whereN

'PN(P) = OFj(P)

j=1

(12)

(18)

is the sum of the phase disturbances imposed at each of theaberrating surfaces seen by the fundamental on its initialtrip to the Nth segment and each Fj has the form5

where VI is the transverse component of the gradient. Thegoal now is to show that N is small enough to be neglectedrelative to unity; the intuitive reason for expecting it to be sois similar to that discussed just above. Both (V1EF*/EF*)2and V_2EF*/EF* are of the order of /p2 , where p is thetransverse beam dimension. Therefore we have eN_ XbN/p 2,the fractional diffraction spread of the beam in propagatinga distance N, which we expect to be small.

Now, exactly as at the first surface, the fundamental wavehad previously acquired a phase factor that was due to thesurface figure,

EF(P, ZN) = EF(P, ZN - U.N)exp[i0FN(p)], (13)

'kFj(p) = ko[Umj - uj(p)] + nkouj(p), (19)

as in Eq. (2). Intuitively, this appears to be a fairly accurateapproximation for this system. The number of segmentswill probably be no more than 10, with probably no morethan 10 cm between segments and a 1-cm segment thickness,for a total distance of the order of 100 cm. The wavelength Xis -1 lim, and the transverse scale size of a surface undula-tion is of the same order as the aperture diameter D, which isof the order of centimeters (Ref. 2); therefore the propaga-tion distance for the imposed phase aberrations to be strong-ly affected by diffraction is approximately D2/X = 104 cm,which is 100 times the actual propagation distance. Fur-

Lee M. Frantz

338 J. Opt. Soc. Am. B/Vol. 7, No. 3/March 1990

thermore, the surface gradients,2 du/dp 10-5, are so smallthat there will be little refraction.

If we write Eo in terms of its magnitude and phase,

E = AO exp(io 0),

and insert this, along with Eq. (17), into the expression [Eq.(12)] for EN, we find that

FN = [ VA\2 -1 VL2AO + iVL2AO + iV2 I -AN O 10 AO~LO~~' ~W 4ir

We have kept only the linear term in the product expansion,because we are interested only in cases in which the error issmall compared to unity. We denote the crystal segmentlength by a and the intersegment gap length by b. Toestimate the maximum error, we take N + 1 to be the totalnumber of segment surfaces. We also take AN = Zg - ZN a,so that

a, j odd6j =-, Zj+1-Zj{,, j even

(20) and

We would like to determine under what conditions thiserror term EN is small compared to unity. In its original form[Eq. (12)], it could be seen to be of the order of XbN/p 2, wherep is the transverse beam width, so it could be interpretedphysically as being proportional to the fractional diffractionspread of the fundamental wave. It is now of the samegeneral form but is broken up into several contributingparts. The terms involving AO are evidently the fractionaldiffraction spread that is due to the shape of the incidentfundamental's intensity distribution. Similarly, the terminvolving 00 describes the diffractive effect of the incidentfundamental's phase distribution. The last term, involvingq1N, describes the effect of the phase aberrations induced bythe crystal segments' surface undulations.

We would like to estimate V 124N by taking the average

over a statistical ensemble of surfaces, but because the aver-age (v±2 uj) vanishes, so also does the average (± 2

,pN).

Therefore we use the square root oN of the variance as ameasure. Combining Eqs. (18) and (19), and treating the ujas independent but identically distributed stochastic vari-ables, we get

YN2 = ((V 21,pN) 2) = Nk(n 2- 1) 2 ((VI 2 U)2 ), (21)

where u is any of the uj's. To estimate this, we think of theslow surface undulations as part of a product of sine waves,u(p) = w(sin 27rx/D)(sin 27ry/D), where D is the aperturediameter, and find for the maximum value of the transverseLaplacian

IV±2 UImax = 2w(27r/D)2.

Using this in Eq. (21) as an upper-limit estimate then gives

aN= FNko(n - 1)87r 2w1D2. (22)

Returning now to Eq. (16) and expanding the product oferror terms,

N

I=i(1+e) = 1 +etot+...,

where, with the aid of Eqs. (20) and (22), we have

N

OEtt = E j=1

=E(!VA 0 - - 1 0 S= [(AO V AO) -A V2AO +iv 2ko]4

j.1

+ 21n jJ=1

6J = zg-z = Ma + (M-1)b,

where M isDefining

the number of crystal segments (N = 2M - 1).

MRM = 2k-1,

k=1

M-1

SM= k=1

we obtain from Eq. (23) the final error expression,

F/i .2 v 2A i2~ ixEtot = [ AO vA) -A V A0+ iv0o 4

X [Ma + (M - 1)b] + 4Dw(n ) (RMa + SMb)- (24)

When this error Etot is small compared to unity, diffractionspreading is insignificant. Equation (16) is then equivalentto Eq. (5), and, as stated above, it follows that phase conjuga-tion will remove the phase aberrations from differential ele-ments of the second harmonic generated anywhere in anycrystal segment. The terms involving Ao and 00 representthe fractional diffraction spread that is due to the incidentfundamental wave's intensity and phase distributions. Thelast term represents the fractional spread due to the surface-induced aberrations.

To illustrate conditions under which the error is small, wetake the following possible design values, which are readilyachievable in the laboratory: the crystal segment length is a= 0.4 cm, the intercrystal gap length is b = 7.4 cm, thenumber of segments is M = 10, the wavelength is Xo = 1.06 X10-4 cm (Nd:glass), the crystal diameter is D = 1 cm, thesurface undulation amplitude iS2 W = 3 X 10-6 cm, and therefractive index is n = 1.49 (KDP). For a beam with aGaussian intensity distribution, I = Io exp(-p 2 /u-2), the am-plitude is Ao = 1I exp(-p 2/2a 2 ); we then have

(A v 2~o) - 1 2A0 =2.

Using the above design values, we find that for the term inEq. (24) involving this quantity to be small compared tounity, the requirement is that a > 0.09 cm. To avoid edgediffraction effects, we might impose the condition Pmax > 3a= 0.27 cm, which would lead to the result Imin/IO = 1.2 X 10-4.Both of these conditions can then be satisfied with the cho-sen crystal diameter of D = 1 cm. For the term involving

23) vL2 00, we need IvL20oI << 1.7 X 103 cm-2, a requirementeasily satisfied by a nearly plane wave. Finally, putting

Lee M. Frantz

Vol. 7, No. 3/March 1990/J. Opt. Soc. Am. B 339

numbers into the last term, we find that [47r2 w(n - 1)/D2n](RMa + SMa) = 8.3 X 10-3, so this is already smallcompared to unity for the design values.

4. CONCLUSIONS

Provided that certain reasonable requirements are imposedon the design values of a frequency-doubling system, thephase aberrations caused by the surface figure of the crystalsegments can be removed by phase conjugation.

ACKNOWLEDGMENT

The author thanks W. W. Simmons for valuable discussions.

REFERENCES AND NOTES

1. C. Hoefer, H. Injeyan, B. Zukowski, L. Frantz, and J. Brock,"Phase conjugated harmonic conversion, in Digest of Conferenceon Lasers and Electro-Optics (Optical Society of America,Washington, D.C., 1988), paper TUA5.

2. P. E. Coyle, ed., "Laser program annual report-1976," UCRL-50021-76 (Lawrence Livermore Laboratory, Livermore, Calif.,1976), p. 2-261.

3. See for example, A. Yariv, Quantum Electronics, 2nd ed. (Wiley,New York, 1975), p. 421.

4. The subscript N has been added to indicate that we are dealingwith the Nth surface.

5. With the proper definition of U, Eq. (19) applies at right as wellas left segment edges.

Lee M. Frantz