Photorefractive two-wave mixing in the presence of high-speed optical phase modulation

$Page 1: Photorefractive two-wave mixing in the presence of high-speed optical phase modulation$
Photorefractive two-wave mixing in thepresence of high-speed optical phase modulation

Christopher T. Field and Frederic M. Davidson

Equations that describe the steady-state dependence of the coherent-coupling properties of photorefrac-tively induced refractive-index gratings on high-speed periodic biphase, sinusoidal, and triangular phasemodulation impressed on one of the input optical beams are found and solved for both depleted and

undepleted pump conditions. The period of the phase modulation wave form was kept short comparedwith the grating-formation time but did not cause significant spectral broadening. The results obtainedwere verified with data obtained from measurements of two-wave mixing in the photorefractive material,InP:Fe.

Key words: Photorefractive effect, homodyne optical detection, InP:Fe photorefractive properties.

1. Introduction

Two-wave mixing in photorefractive materials hasbeen shown to effectively, coherently combine anunmodulated optical pump beam with an amplitudeor phase-modulated optical signal beam.1-8 Con-sequently photorefractive materials have been pro-posed for use as beam combiners in coherent opticalcommunication receivers2 3 and in holographic meth-ods of vibration analysis.48 In photorefractive mate-rials, two mutually coherent optical beams form anoptical interference pattern inside the material. Aninternal space-charge field is established because ofmigration of photoexcited charge carriers into thedarker regions of the interference pattern where theybecome trapped. The space-charge field modulatesthe refractive index of the photorefractive materialvia the linear electro-optic effect. The volume indexof refraction grating that is formed diffracts subse-quent light entering the material and is responsiblefor the coherent combining properties of the material.

There are some important differences betweenconventional and photorefractive beam combiners.The conventional optical beam splitter with fixedpower transmission and reflection coefficients E and1 - e gives rise to a combined optical-field intensitythat is determined by the optical intensity levels andthe instantaneous difference between the phases of

The authors are with the Department of Electrical and Com-puter Engineering, The Johns Hopkins University, Baltimore,Maryland 21218.

Received 24 July 1992.0003-6935/93/275285-14$06.00/0.e 1993 Optical Society of America.

the two input optical fields. The photorefractivebeam combiner relies on a dynamically formed refrac-tive-index diffraction grating with properties thatdepend on the intensity levels of the two interferingbeams. The spatial position of the refractive-indexmaxima is determined by the average value of thephase difference between the two input (signal andpump) optical beams. The magnitude of the space-charge field depends on the modulation depth of theoptical interference pattern averaged over the grating-formation time. Because phase modulation will de-crease this value relative to the unmodulated situa-tion, the space-charge field and the resultant coherentbeam coupling will be reduced.

In this paper the effects of high-speed temporalphase modulation on the signal beam are included ina simple and direct way, beginning with the differen-tial equation that describes the time evolution of theinternal space-charge field. Here high speed is takento mean phase variations on a time scale that is shortcompared with the grating-formation time, Tg, butsuch that the spectral broadening AX/X is still lessthan 10-4. The wave equation is then solved for thetwo optical fields that emerge from the photorefrac-tive medium. The dependence of the steady-statecoherent-coupling properties of the grating on thedetailed nature of the phase modulation impressed onthe signal beam is found. The wave equation for theoptical field is solved in steady state without use ofthe undepleted pump approximation. The predicteddependence of the energy coupling on the detailednature of the signal beam phase modulation waveform was verified experimentally by use of the photore-fractive material InP:Fe. The solutions obtained are

20 September 1993 / Vol. 32, No. 27 / APPLIED OPTICS 5285

shown to reduce to those obtained in the undepletedpump approximation9 10 and those obtained in theholographic solution given in Refs. 4 and 5.

The equations that describe the steady-state effectsof phase modulation on the two-wave energy-cou-pling properties of the refractive-index grating aredeveloped in Section 2. Section 3 shows how theseequations reduce to the expressions developed in theundepleted pump approximation. Section 4 presentsresults for several specific phase modulation formats.Section 5 presents experimental results that verifythe theoretically predicted behavior. Although sev-eral approximations are made in Section 2, and theresults do not describe transient behavior, the experi-mental results presented in Section 5 verify theaccuracy of the equations obtained.

2. Theory

Two plane-wave optical fields interact in a photorefrac-tive medium as shown in Fig. 1. As in Ref. 9, theoptical beams, which are assumed to be linearlypolarized in they direction, are described as

charge electric field. It evolves temporally as9

aEsc'l'(z, t) Escl)(z t)at Trg

G Al(z, t)A2*(z, t)T

g IO(Z) (3)

where Tg, the grating response time, and G depend onmaterial parameter values of the photorefractivematerial. The precise form of G is not importanthere because it will cancel out of the steady-stateequations that couple the envelopes Al(z, t) and A2(z,t). The total intensity is given by Io(z) = IA,(z) 12 +

TA2 (Z) 12 eThe field envelopes evolve as9

aAl(z, ) = - G Esc()(z, tA 2(Z, t) -

aA2(z, 2G°* Es(l)*(z, t)Al(z, t)

Ot

2 cos 0 A,(z, t),

(4a)

a2 cos o A2 (z, t),

(4b)

E,(r, t) = Al(z, t)exp[j(k. r - owt)] + cc., (la)

E 2 (r, t) A2 (z, t)exp[j(k 2 r - oct)] + c.c., (lb)

where A,2(z, t) are the complex amplitudes of thetwo propagating optical fields, k1,2 are their propaga-tion vectors, is the spatial position, o is theoptical-field-carrier angular frequency, and c.c. de-notes complex conjugate. According to standardtreatments of the photorefractive effect"",2"3 theinternal electric field is of the form

E(x, z, t) = E + /2[Es.cl)(z, t)exp(jKgx) + c.c.], (2)

where E represents an externally applied electricfield and Kg = (kl)x - (k2), is the grating vector. Thequantity E84(')(z, t) is the amplitude of the fundamen-tal spatial frequency component of the internal space-

Photorefractive Optical Beam Combining

Pump

Signal

Signal + coherentlydiffracted pump

= fringe pattern maxima- = refractive index maxima

Fig. 1. Plane-wave interaction in a photorefractive material.

where FO and aL are the exponential photorefractivepower gain and absorption coefficients in inversecentimeters and 0 is the angle between each beam andthe z axis. The following standard approximationswere made in the derivation of Eqs. (4a) and (4b).The envelope functions Al and A2 vary slowly with zand t compared with distance scale kz-1 and time scaleoC-', dark conductivity in the photorefractive mate-rial is negligible compared with the photoexcitedcharge carriers, the intensity ratio, I(0)/12(0), is largeso that the space-charge field is accurately repre-sented by just its fundamental spatial frequencycomponent, Esc(')(z, t), and only light that satisfies theBragg condition is considered.

In Ref. 9, Eqs. (3), (4a), and (4b) were solved forA2(z, t) in the undepleted pump approximation, thatis, Al(z, t) independent of z and t. Here the steady-state solutions A(z, t >> g) and A2(z, t >> Tg) areobtained that yield the steady-state coherent beamcombining properties of the grating in the presence ofhigh-speed phase modulation. Other discussions ofthe space-charge field response to step changes in theinput phase or amplitude may be found in Refs.12-16.

The solution is found in three steps. First thespace-charge field at z and t is found in terms of theunknown envelopes at z and previous time t' < t.Next the envelopes at z, t are found in terms of theknown input envelopes and unknown space-chargefield at z' < z. Finally the two solutions are madeself-consistent under steady-state conditions.

The solution to Eq. (3) is given, formally, by

Es LM(z, t) = G ftexp[(t - t')/g]

x A(z, t')A 2*(z, t')dt', (5)

5286 APPLIED OPTICS / Vol. 32, No. 27 / 20 September 1993

where it is assumed that the optical fields first enterthe material at t = 0. The space-charge field at timet is then given by the convolution of the low-passcausal impulse response hLp(t) = exp(-t/Tg)/Tg, withthe input s(z, t) = Aj(z, t)A 2*(z, t). The Fouriertransform of Es,(')(z, t) may be written as

GE5(')(z, o)) = I(Z) HLP(d)SAA 2*(Z, )

G 1

IO(Z) 1 + jiYg SA1A2*(Z, (), (6)

where HLp(w) = 1/(1 + jwrg) = 7[exp(-t/Tg)/Tg] andSA1A2*(Z, o) is the Fourier transform of Aj(z, t)A2*(z, t).From Eq. (6), it is clear that only the low-frequencycomponents (i.e. O << Tg-1 ) of SA A2*(Z, o) stronglyinfluence the form of E8,(')(z, w), whereas the high-frequency components (i.e., o >> Tg

1 ) of SA1A2*(Z, o)have no effect because 1/(1 + joTg) 0 for w >> Tg

Frequency components in the transition region,(w)Tg 1, are characterized by phase shifts betweenthe space-charge field and the input and are notdescribed by this model. Equation (6) is simply afrequency domain expression of the well-known time-averaging properties of the photorefractive effect.'7-20

In this paper the convolution (5) will be approxi-mated by the time average of s(z, t) over the timeinterval [t - Tg, t]. Consider the impulse responsedenoted by hAv(t) = Tg-' 0 < t < Tg, and zerootherwise. The Fourier transform of hAV(t),HAV(w) = exp( -JoTg/2) sinc(oTg/2), may be substi-tuted for HLp(w) in Eq. (6) without much error as longas the input energy at low frequencies wOTg << 1 ismuch larger than the input energy in the transitionregion. These conditions will be explored in moredetail later. Under this assumption, the photorefrac-tively induced internal space-charge field is wellapproximated as

Ec(1)(z = I 0 _- (Al(z, t)A 2 *(z, t))Tg (7)

where ( ) denotes a time average over the intervalt - g to gt. The argument of the complex quantity(Al(z, t)A2*(z, t)), determines the average locationalong the x axis of the interference pattern maxima.Although this theory is intended to describe two-wavecoupling in the presence of high-speed phase modula-tion, the phase of the time average and hence thelocation of the space-charge field maxima may changeslowly, for example because of air currents in theoptical paths traversed by the two interfering beams.However, the theory presented here does not describethe dynamics of such drifts, which are assumed tooccur slowly on time scale Tg. Although the locationof the space-charge field maxima may slowly change,the space-charge field magnitude, and hence themagnitude of the time average (Al(z, t)A2*(z, t))T ,will be assumed to be stationary, i.e., independent oftime t for all planes z in the medium. However, forclarity of the following development, the notation willcontinue to show dependence on time t.

Equation (7) represents only a formal approximatesolution for the space-charge field. Below it will beshown that (Al(z, t)A2*(z, t)),g depends only on tempo-ral variations of the phases of the two input beams atthe entrance plane z = 0. The steady-state two-wavecoupling properties of the grating, as characterized bya grating-rotation angle, (Z),

2,3 depend only on

I (A(z, t)A2*(z, t)),g I, which is assumed to be timeindependent.

Now that the space-charge field has been formallyexpressed in terms of the envelopes, the envelopefunctions will be found by solving the coupled Eqs.(4a) and (4b), which can be accomplished using vectornotation. Let the two-component column vectorA(z, t) be defined as

A(z, t) = |Aj t)] (8)

The coupled wave Eqs. (4a) and (4b) can be written asa single-matrix differential equation as23

aA(z, t) F0 a( = 21o(z) T(z, t)A(z, t) -A(z, t),

where Jow2cos 0

where

(9)

T(z, t) = (Ai*(z, t)A2(z, t))Tg

(10)

In the absence of phase modulation of the inputbeams, maximum energy coupling occurs when theinternal space-charge field maxima form 7r/2 radiansthat are out of phase with respect to the interference-pattern maxima. The optical envelope phases do notdepend on z under these conditions." In our nota-tion, 1o is real for this situation. When phase modu-lation is present, the corresponding condition is a rr/2shift between the time-averaged position of the fringepattern maxima and the space-charge field maxima.Under these conditions, (f(z, t), defined as the argu-ment of (Al(z, t)A2*(z, t)), is independent of z, asshown in the Appendix, and will hereafter be denotedof (t). Recall that af (t) determines the location alongthe x axis of the interference-pattern maxima andmay be a slowly changing function of time.

Next define the unitary transform, U(t), by

U(t) = [exp[jf(t)/2]

The quantity T'(z) = Ut(t)T(z, t)U(t), has only off-diagonal elements consisting of the real, time-independent quantity I (A,(z, t)A 2*(z, t))g 1:

[

T'(z) = I (Al*(Z, t)A2(Z, *,))

- I (A1(z, t)A 2 *(Z, *))TI

(12)


-/A\.c3.l(Z, t)A2*(Z, 0,,

0

0

exp[_jl�f(0/21 ' (11)

The optical absorption term in Eq. (9) can beeliminated through the transformation A'(z, t) =exp(az/2 cos O)Ut(t)A(z, t). Equation (9) can then bewritten as

aA'(z, t) r0az 2IO(Z) T'(z)A'(z, t).

(13)

The solution to Eq. (13) is given by A'(z, t) = A'(0, t)exp[B(z)], where B(z) = (Fo/2)flozIO-(z')T'(z')dz'.The grating-rotation angle, (z), is defined as

1Oz I (A 1 z, t)A 2 *(Z', t)) I5(z) = - l- jd I'z- o r ) (14)

which we assume is independent of t given theassumption of the stationarity of IAl(z', t) A 2*(z', t))Tgfor all planes z. It follows that B(z) is given by

B(z) [ 0 -8(z) (15)5(z) 0 (5

Because, for n an integer, B 2 n = )n62n(z)1, where I

is the identity matrix and

O -1B2n+ = (-1)n82n+l(z) +1 0j , (16)

the power series expansion for exp[B(z)] separatesinto two sums that are easily evaluated to yield23

[ cos (z) -sin (z)]

A'(z, t) = sin (z) cos (z) A(0, t).(17)

After transforming A'(z, t) back to A(z, t), the individ-ual amplitude functions are then given by

Al(z, t) = A1(0, t)cos (z) - A2 (0, t)

x exp[j(f(t)]sin 8(z)lexp(-az/2 cos 0),

(18a)

A2 (z, t) = fA1 (0, t)exp[-j>f(t)]sin 8(z)

+ A2 (0, t)cos 8(z)}exp(-az/2 cos 0). (18b)

Equations (18a) and (18b) are the solution to thesecond part of the problem. They relate the enve-lopes at z to the input envelopes at z = 0 and to anintegral involving the magnitude of the unknownspace-charge field. Each propagating beam is com-posed of two terms, an undiffracted part with a phasegiven by the beam's input phase and a diffracted partwith a phase shifted by f (t).

The next task is to find (z). The integral equa-tion definition of (z) in Eq. (14) can be converted tothe differential equation

with (z = 0) = 0. From Eqs. (18a) and (18b),A1(z, t)A2*(z, t) is given by

A1 (z, t)A2 *(z, t)

= exp(-az/cos O)exp[jf(t)]{[I1 (0) - 2(0)]x cos (z)sin 8(z) + A1(0, t)A2*(0, t)exp[ -jf(t)]

x cos2 a(z) - Aj*(0, t)A2 (0, t)exp[j(f(t)]sin 2 a(z)},

(20)

where I and I2(0) are the time-independent intensi-ties of the input beams. Substitution of Eq. (20) intoEq. (19) and use of Io(z) = Io exp(-az/cos 0) yields

- I [II(0) - 12 (0)]cos b(z)sin (z)

+ (A1(, t)A2*(0, t)), exp[-j(f(t)]cos 2 (z)

- (A1*(0, t)A 2 (0, t))Texp[j-f (t)]sin2 5(z)] I, (21)

where the time-average notation of the input intensi-ties has been suppressed. Define the ratio of theinput intensities by mO = I(0)/I2(0). Note from thedefinition of f(t) that the coefficients of cos2 andsin2 are both real. Next define the real quantity Kby

I (A=(0, t)A 2 *(0, t))g 12

11(0)I2(0)(22)

K is time independent because I (A1(0, t)A2*(0, t))g I hasbeen assumed to be stationary. The K is a measureof the reduction in the magnitude of the space-chargefield at z = 0 because of the presence of the phasemodulation and is always less than or equal to one.Below, K will be related to the frequency spectrum ofthe phase modulation. With these definitions Eq.(21) becomes

dB rodz = 2Io 2 (0)cos 2 (z)

X {[mO - 1]tan 5(z) + K^Mh[1 - tan 2 (z)] 1. (23)

Note that I2(0)/Io = 1/(1 + mO). For the situation ofinterest here, the quanity within the brackets ispositive, and the absolute value signs can be dropped.With the change of variable x = tan (z) Eq. (23) isseparable, and from Equation (2.172) in Ref. 21 hasthe solution

tan (z)

2M[exp(Fz/2) - 1]

[A - (O - 1)]exp(Fz/2) + [s + (mO - 1)]

(24)

where F and A are given by

dB= 2I( ) 1 (A(z, t)A2*(z, t))Tg , (19) Ir= 1 A = I4(K 2 - )mO]1/2

(mO + 1)21


(25)

A = (O-1) 2 + 4moK2. (26)

Equations (24) and (25) contain all of the relevantdetails of the coherent beam coupling. As expected,the exponential gain is reduced by the phasemodulation, becoming proportional to K when mO = 1.However, Eq. (3) assumes large beam ratios (mO >> 1)and for these ratios F F o. This is because (A)1/2 is

bounded by mO - 1 for K = 0 and by mO + 1 for K = 1

and can generally be approximated by mO - 1 + 2 K2 .

Consequently Eq. (24) can be approximated as

t zMK[exp(Fz/2) - 1]

K2[exp(Fz/2) + 1] + mO - 1

Now we examine the conditions under which(A,(z, t)A2*(z, t)), may be substituted for the convolu-tion (5). The input envelopes, A,(0, t) and A2(0, t)may be written

A1(0, t) = [I,(0)]/ 2exp[j4,(t)], (28a)

A 2(0, t) = [I 2 (0)]1/2 exp[j+ 2 (t)], (28b)

where [I,(0)]1/2 and [12(0)]1/2 are the time-independentenvelope magnitudes at the input plane z = 0 and, 1(t) and P2(t) represent the instantaneous phases ofthe optical beams at (x = 0, z = 0) at time t. Letthe signal phase +2(t) be expressed as the sum of acarrier phase 42,(t) and an externally impressed phasemodulation tm(t) and assume that +,(t) = (1jjt)The signal spectrum at the input plane, SA A2 (0, o),may be expressed in terms of the convoiution ofthe carrier phase difference spectrum, SA,,((w) =-flexp[j[(,(t) - +2,(t)]1), and the modulation spec-trum, Sm(') = 7(expfj[4m(t)]}), as

SA1A2*(O, ) = [1(0)I2(0)]/ 2SA,,((0) * S(W). (29)

For -m(t) a deterministic, repetitive signal of period T,S m(0) is a line spectrum that may be written in termsof the continuous variable o) as

= 1 COlD(W - | ')(30)n=- Trwhere BD(-) is the Dirac delta function.

For perfect carrier phase tracking of the two lightbeams, k, (t) - I2 (t) = A~c is a constant, SA,> (03) =

exp(jA4)8D(3) and SAjA2 *(0, w) = [IM(0)I2 0)]1/2exp(jA(\#c)Sm(Jo). More generally, if there is sometime-varying carrier phase error, SA (w) will havesome finite width. Under these conditions

SA 1A2 *(O, o) = [I(0)12(0)]1/2 SA ()

X > CD( - -o 0)d'.n= -x

(31)

The spectrum of the space-charge field, E,,(')(z, w),involves the product of Eq. (31) with either HLp(w) orHAv(o)), which both vanish at wrrg >> 1. As long asthe phase modulation period, T, is short comparedwith Tg, and the spectrum SA,(o) vanishes well beforethe transition region wrrg 1 is reached, the twoproducts become indistinguishable as both 1/(1 + jwTg)

and exp(-j0)Tg/2)sinc((oTg/2) have approximately equalmagnitudes and phases over the range of frequenciesfor which E(')(z, 0) is appreciable. Under theseassumptions, only the n = 0 term in Eq. (31) contrib-utes and the convolution (5) is proportional to coexp[jA+0(t)], where by definition, co = (exp[-j4im(t)])Tand A4(t) is approximately constant over the timeinterval [t - Tg, t]. For these repetitive modulations,the time average over one period is the same as theaverage over the time Tg provided T << Tg.

Next we evaluate K and (f(t) in terms of the inputoptical frequency spectra for close carrier phase track-ing, which occurs when A~(t) varies slowly on timescale T From Eq. (22), K2 can be expressed as K2 =

I (expjrJO(t) - m(t)]) 12. Because Al)(t) is approxi-mately constant on time scale T

g, it does not contrib-ute to the magnitude of the time average. ThereforeK

2 may be written as

K2 = j(exp[im(t)]),gI 2 = Co 12, (32)

where the equality of the time average over Tg and the

phase modulation period, T, has been used. Fromthe definition of (3f(t), and the form of the inputenvelope functions, Eqs. (28a) and (28b), 4f(t) may berewritten in terms of A4(t) and (m(t) as

4f(t) = IWO(t) - "av, (33)

where ikau = -arg(exp[-jfm(t)]),. Again, noting thatfor the periodic phase modulation with T << Tg, timeaverages over Tg and T are identical, it is also true that

,av = -arg c0 . The real number K may be written as

K = CO exp(jau), (34)

which will be useful in resolving a sign ambiguity in K,

which can arise under certain phase modulationconditions.

The instantaneous phase difference A4(t) =arg{exp[-fjlf(t)]A(0, t)A2*(0, t)} will appear in thetime-dependent photorefractively coupled intensi-ties. Substituting for -f(t) and the phases of A1(0, t)and A2(0, t), A\#(t) may be written as A,(t) =-A [1(t) - -)aj + A4)c(t) - -(t), which reduces to

A (t) = av - t. (35)

The time average, (exp[jA4(t)]),, may be evaluatedusing Eqs. (34) and (35). Eq. (34) may be written asK = (exp(j(tav exp[-j( m(t)]) . Splitting the timeaverage into real and imaginary parts and substitut-ing Al\(t), we get K = (COS At+(t))Tg + j(sin A4(t)Xg.


Equating the real and imaginary parts, we get

K = (COS A((t)),

0 = (sin A4+(t)XTg.

(36a)

(36b)

These will be useful later.Finally the optical intensities that leave the photore-

fractive material, I1,2 (d, t) = IA1,2(d, t) 12 are given by

Ij(d, t) = exp(-ad/cos O){I(0)cos 28(d) + I2(0)sin28(d)- 2[1(0)1 2(0)]'/2 cos 8(d)sin 8(d)cos[A4(t)]j,

(37a)

I2 (d, t) = exp(-cxd/cos 0)I(0)sin 22(d) + 12(0)cos2 8(d)

MOK2 [exp(Fd/2) - 1]2

K2 exp(Fd)[(mo + 1)] + [(mO - 1)2 + K

2 (3mo 1)]

(39b)

For K = 1, A = (O + 1)2, and these reduce to theexpression for diffraction efficiency given by Kukh-tarev et al.'7 after accounting for the fact that ourdefinition of mO is the inverse of the one used in Ref.17. For no phase modulation, K = 1, A+(t) = 0, andthe intensity expressions (37a) and (37b) become

Ij(d) = exp(-ad/cos )I(0) M0) + (0)e()'

(40a)

+ 2[I1(0)12(0)]'/2sin (d )cos (d )cos[A+(t)]}, I(0) + 2(0)(37b) 12(d) = exp(-xd/cos 0)12(0) I,(0)exp(-rd) + I2(0)

where A(4(t) was defined aboveNow the sign ambiguity in K may be resolved.

First notice that Eq. (37b) may be written in terms oftan 8(d) and the beam ratio, mO, as

I 2 (d, t) = exp(- d/cos 0)12(0)cos2 B(d){mo tan2 5(d) + 1

+ 2Fim tan (d)cos[A4(t)]}. (38)

From the definitions of tan (d), Eq. (24), and A4(t),Eq. (35), the time-varying term in Eq. (38) is propor-tional to K cos[ka - m(t)]. For some phase modula-tion formats, c is a real number that may be positiveor negative. When c is positive, then from Eq. (34),K is positive, av = 0, and the time-varying term in Eq.(38) is proportional to K cos[-+m(t)]. When c isnegative, there are two choices. Either k may beselected as positive and av = r or K may be selected asnegative and av = 0. The former choice, which isconsistent with the standard definition of the magni-tude and phase of a complex number, leads to atime-varying term proportional to K cos[Tr - m(t)] =- K cos[4m(t)]. The latter choice leads directly to KCOS[-im(t)] = - I Kcos[(4m(t)]. The two choices areequivalent and lead to an inversion of the timedependence of the intensity wave form for negative c0.This is discussed in more detail in the experimentalsection (Section 5), where plots of the temporal waveform I 2(d, t) show the inversion.

The output intensities can be written in terms ofthe grating-diffraction efficiency, q, which is definedas the fraction of incident pump intensity diffractedinto the signal beam direction.22 From Eq. (37b), thediffraction efficiency is given by = exp(-ad/cos 0)sin2 (d). If we use trigonometry identity, sin2 maybe written as

sin2 8(d)

2mOK2 [exp(d/2) - 1]2

exp(Fd)[A - (O - 1)rA] + [A + (O -1)V]

(39a)

(40b)

which are equivalent to the expressions given byKukhtarev et al. for the stationary photorefractivelycoupled output intensities.23

Photorefractive beam combiners have propertiesthat are significantly different from conventionalbeam splitters when combiners are used to detectphase-modulated optical signals. From Eqs. (37a)and (40) and the definition of A(t), the outputintensities, I 2(d, t) do not depend at all on thedifference between the carrier phases of the two inputbeams, but only on the difference between 4m(t) and4av. This implies the output will always have acosine response and cannot be forced to have a sineresponse by adding a carrier phase offset to eitherbeam. Phrased differently, unlike the case of aconventional beam splitter, the beams cannot bemixed in quadrature. In addition, although notdescribed here, the grating compensates for slowchanges in the input beam carrier phases providedsuch changes are much slower than the grating-response time. If the grating did not track the slowphase variations, then it would be possible to offsetthe carrier phase of one beam and mix the slow phasevariations in quadrature. This tracking ability isborne out in experiments with m(t) = 0 in which aircurrents produce no noticeable effect on the outputintensities of photorefractively coupled beams,whereas the output intensity of beams coupled by aconventional beam splitter underwent several transi-tions between bright and dark per minute.

3. Undepleted Pump

To compare our results with previous work for anundepleted pump by Cronin-Golomb,9"10 we neglectabsorption and assume mO >> exp(Fz), which impliesthat is small, cos (z) 1, and

sin 8(z) tan 8(z) K[exp(Fz/2) - 1]

(41)


and from Eq. (18b),

A2 (z, t) = exp[-f 41(t)A,(0, t) K[exp(Fz/2) - 1] + A2(0, t)

(42)

The definition of K, Eq. (22), may be written as K2=

I (A(0, t)A2*)exp[-j.f(t)] 12/1,I2. Notice that K and(A,(0, t)A2*)exp[-jif(t)] are both real. By assump-tion, the magnitude of A1(0, t) is constant. Assumethat the phase of A,(O, t) changes so slowly that it maybe considered constant during a time interval Tg long.Then A1(0, t) may be brought out of the time average,K exp[-j>f(t)] may be written as A1*(0, t)(A2(0, t)),g/(I1I2)1/2 and Eq. (42) becomes

A2 (z, t) = (A2 (0, t))T[exp(Fz/2) - 1] + A2(0, t)- (43)

Cronin-Golomb's result,9 10 after correcting a typo-graphical error, is

A2 *(z, t) = A2 *(0, O)K(z, t)

+ [A2*(Ot') + dA2*(O, t)]K(Z,t-t

where

K(z, t) = exp( t/Tg)Jo[2(- 2o)t]/' (45)

and J is the zero-order Bessel function of the firstkind. The second term in the integral of Eq. (44) isconverted to one involving A2*(0, t') by integrating byparts. As before, for t >> Tg and high-speed phasemodulation, A2 *(0, t') can be replaced by its timeaverage, which is assumed to be time independent forlarge t and can be removed from the integral. ThenEq. (44) becomes

A2*(z, t) = A2 *(0, t) + (A2 *(0, t))Tg

x [1 | K(z, t - t')dt' - t d(z dt t() dt'

(46)

The second integral evaluates to - 1, whereas the firstcan be approximated by

which, after taking the complex conjugate, matchesEq. (43) if F = Fo, which is true for mO >> 1.

Note that for the undepleted pump, the diffractionefficiency becomes

N = K2Tqo = K2[exp(rd/2) - 112

(49)mO

where 'r0 is the diffraction efficiency without modula-tion.

4. Particular Modulation Formats

This section derives K and the output intensity forseveral particular phase modulation formats.Throughout this section the input fields are given byEqs. (28a) and (28b). The 1(0, t) becomes 4, and(2(0, t) becomes (12c + (1m(t), where 4+, and (2, are thepump and signal beam carrier phases, respectively,which vary so slowly that they can be consideredconstant, and 1)m(t) is the phase modulation im-pressed on the signal. The input intensities I1(0)and I2(0) will be denoted by I, and I2, respectively.Also it will be assumed that the period of the modula-tion is much shorter than the grating-response timeTg. Thus, as described in Section 2, co = (exp[fim(t)])Tand A-+(t) = av - m(t), where c)av is the phase of theaverage of the modulation, ')av = arg(exp[jkm(t)])T.

The first modulation format considered is biphasemodulation, in which the signal beam phase is repeti-tively switched between the two phases T, and T2over a time interval T << Tg. This modulation mightbe used in a phase-shift-keyed optical communicationsystem. The signal envelope is

A2(0, t)

= FI2 exp{j[4N + m(t)1}( \I2 exp[j(46, + 4')] nT < t < nT + t,

; I;eXp[j('k2c + q2)] nT + t, < t < (n + 1)T'(50)

where t, is the time the signal has phase T, and n isan integer. Because co = (t, exp ji + (T - t1)expj4" 2 )/T, K and 'kav are given by

K = [T2 - 2Tt, + 2t, + 2(Tt, - t12)cos(P - P2)31/,

(51)

and

K(z, t - t')dt' = rg exp(Foz/2), (47)

where Equations (6.614) and (8.467) in Ref. 21 wereused. So finally

A2 *(z, t) = (A2 *(0, t))Jexp(loz/2) - 1] + A2*(0, t),

(48)

[t, sin 'P, + (T - t)sin T2 4~av = tan |t, cos l', + (T - t)cos T 2

(52)

When T = 2t,, the photorefractively coupled outputintensity is constant regardless of 4', and *2, because4a. = (4i + AI2)/2 and A4(t) = +(T1 - T2)/2. WhenT • 2 t1, the output intensity has two distinct levels,with the higher value of I 2(d, t) corresponding to themore common signal beam phase.


In holographic vibrations studies, the phase ofA2(0, t) is modulated sinusoidally with amplitude Tat angular frequency 2/T. Therefore 4m(t) =T sin(2irt/T) and

A2(0, t) = vFI exp[j((2c + T sin 2rrt/T)];

c0 is given by

(53)

0=-T | Jexp(jP sin 27rt/T)dt = J0 (P), (54)

where J0 is the zero-order Bessel function of the firstkind. As discussed in Section 2, c0 is real and may bepositive or negative. Here we define av = 0 regard-less of the sign of c and let K = c so that A4v(t) =

)m(t) For low grating-diffraction efficiency, theoutput light intensity may be written in terms of thediffraction efficiency in the absence of modulation oand the modulation amplitude P by use of Eqs. (49)and (54) as

I2 (d, t) = ['9oJ 02(p) + 2 + 2 (IlI2'ri0)1/2

x Jo(T)cos(P sin flt]exp(-az), (55)

where 1 - o has been approximated as unity. Thisis the expression used in Ref. 24 but is inconsistentwith the coupling expression used in Ref. 5.

The last phase modulation format considered is onein which the signal phase is linearly ramped from -Pto +' and back to -P during a time interval T. Forthis format

A2(0, t) = VI2expjj[12 + km(t)]}

F2 exp{j[2 + P + 4(t - nT)/T]}(n - /2)T < t < nT

= I +expfj[' 2 + - 4(t - nT)/T]}' (56)nT < t < (n + /2)T

where, as before, 2 is the signal beam input intensityand n is an integer. The c is given by

J1 oCO = - f&+ T/2

+T/2

exp[j(P + 4Pt/T)]dt

exp[j(P - 4Pt/T)]dt]sin(P)

= T =sinc(P).

Again we choose a = 0 so that K = co and A(t) =+m(t) regardless of the sign of c0.

5. Experiment

The apparatus shown in Fig. 2 was used to measurethe effects of signal beam phase modulation on pho-torefractive two-wave coupling. The laser used was

* To Scope

Filters InP:Fe

Optca Pas M To Scope

BS / RC PD

Local Osc light MI

Fig. 2. The experimental setup used to measure photorefractivebeam coupling with phase modulation. BS, beam splitter; M,mirror; ND, neutral density.

a Lightwave Electronics model 122 Nd:YAG unidirec-tional nonplanar ring oscillator at = 1.06 m with60-mW output optical power and an optical linewidthspecified at less than 5 kHz. Approximately 20% ofthe laser light was diverted for the signal beam, whichwas phase modulated by a Crystal Technology inte-grated optic traveling-wave phase modulator. Thefrequency response of the phase modulator extendedfrom dc to well above 1 GHz. The optical signalbeam was guided into and out of the phase modulatorby Newport F-SY single-mode optical fibers. New-port F-1015 fiber couplers were used to couple thelight into and out of the optical fibers. The polariza-tion state of the light entering and exiting the phasemodulator was controlled by fiber loops approxi-mately 50 mm in diameter.2 5 The modulatorconversion gain of 6.2 V/T rad was verified bymeasurements with a conventional Mach-Zehnderinterferometer. The photodiode used was an RCAC30617E InGaAs p-i-n that had a quantum efficiencyof approximately 60% at = 1.06 m. The outputphotocurrent was converted to a voltage by a ComLin-ear CLC415 current feedback op-amp in a transimped-ance configuration with a 1-kfl feedback resistor andwas further amplified 31 times. The frequency re-sponse of the photodetection circuit extended from dcto a 3-dB point at approximately 40 MHz. The noisecharacteristics of the amplifiers were not importantbecause the Tektronix 11402 digital oscilloscope thatwas used averaged 16 or 32 sweeps before the datawas processed.

A Microwave Logic BERT660Tx bit error rate testset transmitter provided a signal source at 7 Mbitsper second for biphase modulation [Eq. (50)]. TheBERT660Tx is designed for testing binary communi-cation systems and can generate a fixed 16-bit binarypattern (modulation period, T = 2.3 sec.). Thequantity K was varied from 0.875 to 0 in steps of 0.125by setting from 1 to 8 of the 16 bits generated by theBERT660Tx to one while the rest were set to zero.In addition, the BERT660Tx can insert errors intothe data stream by inverting a single bit at a program-mable rate from once every 103 to 107 bits. The Kvalues of 0.99998 and 0.998 were generated by settingthe transmitted word to all zeros and inverting a bitevery 105 or 103 bits, respectively. The BERT66OTxoutput, suitably amplified by a Comlinear CLC166 dcto a 150-MHz amplifier, was used to drive the phasemodulator. A 330-pF capacitor across the CLC166input slowed the pulse rise and fall time to about 19


ns to reduce pulse overshoot. A Tektronix FG504function generator provided sinusoidal [Eq. (53)] andtriangular [Eq. (56)] modulation at 500 kHz (modula-tion period, T = 2 jis). The FG504 output was ampli-fied by an Aventek amplifier, which could produce upto a 20-V peak-to-peak drive signal into the 50-finput impedance of the phase modulator to produce aphase modulation range of approximately + 1.7 I rad.

The photorefractive crystal used was lightly Fedoped InP crystal of dimensions 1cm x 1 cm x 0.8 cmobtained from Crystacomm. The refractive-indexgrating spacing used was typically 10-11 jum. The(110) crystal faces were antireflection coated for X =1.06 jim, and a dc electric field of 4.0 or 6.8 kV/cmwas applied in the [001] direction. The absorptioncoefficient, which was not corrected for reflectionlosses, was measured to be a = 0.6 cm- 1 , which wassomewhat smaller than reported elsewhere in photore-fractive experiments on this material.26 The signaland pump light incident on the crystal were S polar-ized. The optical pump beam fully illuminated thecrystal. The signal beam diameter was between 3and 4 mm for the measurements made at 4.0 kV/cmand was approximately 0.5 mm for the measurementsmade at 6.8 kV/cm.

InP:Fe is an unusual photorefractive material inthat both thermally generated electrons and opticallygenerated holes contribute to the space-charge field.26

The phase shift between the interference pattern andthe space-charge field depends on both generationrates. In an externally applied dc electric field, thephase shift between the optical intensity maxima andthe internal space-charge field maxima is I/2 onlywhen the optical hole-generation rate is exactly bal-anced by the electron thermal-generation rate.Therefore stabilization of the crystal temperaturewas required. Sapphire disks 25 mm in diameterand 2 mm thick provided electrical isolation butthermal contact between the top and bottom (110)crystal faces and a water-cooled mount was nominallymaintained at 23 'C and 19 C when the crystal biaswas 4.0 kV/cm and 6.8 kV/cm, respectively. Theoptical pump power was adjusted for maximum steady-state photorefractive two-wave mixing gain and wastypically 50 mW/cm2. Because the crystal absorbedapproximately 40% of the light, the optical intensitywas not optimal throughout the crystal. As theintensity decreased, the changing phase shift betweenthe intensity maxima of the fringe pattern and thespace-charge field resulted in a continuum of phasesfor the diffracted pump beam and reduced beammixing efficiency. This caused the peak-to-peak in-tensity change of the photorefractively coupled signalbeam to be less than predicted. The low value of Foreported here is largely due to the low iron concentra-tion, the low external bias voltages used, and thenonoptimal phase shift between the grating andintensity maxima in some regions of the photorefrac-tive material.

Figures 3(a), 3(b), and 3(c) show the time evolutionof the photorefractively coupled output intensity in

0A(t) = 0

sm(t) =

A

m(t) = + tm(t) = 0

O'm(t) = - I

B

5bm(t) = 0

7

I

C

.......

.. .. .

Fig. 3. Phase modulation wave form (top of each graph) andoutput intensity wave forms for three phase modulation formats:(A) biphase modulation (O and Ti), (B) sinusoidal modulation(middle, IP = 44 and bottom, P = 2200), and (C) triangular modu-lation (middle, T = 44° and bottom, =220). Top trace of eachgraph is the phase modulation wave form, and the other traces arethe photorefractively coupled signal intensity. All traces 500ns/div. horizontally and 20 mV/div. vertically. The bottom line ofeach graph is zero intensity. Signal without photorefractivecoupling was 24 mV. Crystal bias was 4 kV/cm. Mount temper-ature was approximately 23 'C.


I .

-. 1 .a

..... .... ..

.... ....... ... ........ .......... .. .... .... . ........ . ...... .. ....

... .. .. . .... ....... ...................

...... . ........

...................

............... . ...........

.

........ ........ .....

..........................

..................................

................... ...................

..........

......................................................................................... ..........

.. ..I .......... - . .. . ..........!......... ............. .....

. ~. ... I.. ... ._

.

...... ..... ...... .....

.............................. ..........

..........

...... .....

the signal beam direction for each of the phasemodulation formats discussed in Section 4. The toptrace of each is the phase modulation signal, whereasthe other traces are proportional to the photorefrac-tively coupled output intensity in the signal beamdirection. In Figure 3(a) the signal beam phase iseither 0 or r rad. The correspondence between thephase modulation and the output intensity is clear.When the phase is zero, the signal beam outputintensity is maximum, and when the phase is Tr, theoutput intensity abruptly decreases. The time evolu-tion of the output intensity in the signal beamdirection for sinusoidal and triangular modulationare more complex as shown in Figures 3(b) and 3(c).The middle and bottom traces show the outputintensity in the signal beam direction when the phasemodulation amplitude, ', is 44° and 220, respec-tively.

A small asymmetry in the center wave form isevident in Figures 3(b) and 3(c). Equation (37b)predicts that the intensity for '4(t) = P should be thesame as the intensity for (m(t) = -T. The differencein the measured minimum intensity corresponds to a0.1- rad difference in 4m(t) at the positive and negativelimits. This difference can be accounted for if thephase shift between the average fringe system inten-sity maxima and the space-charge field is 870 and not90° as assumed. The amount of asymmetry dependsstrongly on the pump intensity because of the proper-ties peculiar to InP:Fe26 and also seems to dependslightly on K.

Because the time evolution of the photorefractivelycoupled output intensity is complex, good ways tocharacterize the phase modulation effects on photore-fractive two-wave coupling are the maximum, mini-mum, and time-averaged values of the photorefrac-tively coupled output intensity. From Eq. (36a),(cos A(t)) = K, where the time average may becomputed over an integral number of periods or aninterval much longer than T. Therefore, from Eq.(38), the normalized time-average output intensity inthe signal beam direction is

(12(d, t))I2(0)exp( -ad)

= cos2 (d)[1 + mO tan 2 8(d) + 2KfiMi tan 8(d)] (58)

for any phase modulation format. This is the pho-torefractively coupled signal intensity that would berecorded by any slow detector, such as photographicfilm, a video camera, or a slow optical power meter.The maximum and minimum output intensities inthe signal beam direction are not so simply expressed.

If the signal beam modulation is such that m(t) hasonly the two values 0 and IT, then the normalizedmaximum and minimum photorefractively coupledoutput optical intensities in the signal beam directionare given by

Imax(min) = cos 2 8(d)[1 ± /54 tan 8(d)]2I2(0)exp( -ad) -" (59)

where K is given by Eq. (51) and may be expressed asK = 1 - 2t,/T.

If 4'm(t) is sinusoidally [Eq. (53) and Fig. 3(b)] ortriangularly [Eq. (56) and Fig. 3(c)] modulated withamplitude 4, then (T) is a continuous function oftime taking on values between -P and +'. Theextreme values of cos m(t) are unity when m(t) = 0and cos P, where P, = P for 0 < T < r and P = rfor T 2 r. The extremums of the normalizedphotorefractively coupled output signal intensity aretherefore given by

Imax(min)

I2(0)exp( -ad)

Tcos2 8(d)[1 + MO tan 2 (d) + 2im tan 8(d)]

cos 2 8(d)(1 + mO tan 2 8(d) + 2Vmi tan 8(d)cos P.)'

(60)

For sinusoidal and triangular modulation, c0 isequal to J0 (T) or sinc(T), respectively, as stated inSection 4. Both functions are positive for P betweenzero and their first zero (2.405 and r, respectively).The maximum photorefractively coupled output opti-cal intensity in the signal direction then correspondsto Pm(t) = 0. This is evident in the middle traces(T = 44 = 0.766 rad) in Figs. 3(b) and 3(c), where themaximum values of the curve correspond to fm(t) = 0.The minimum intensity values correspond to theextreme values of the phase modulation (top trace).For values of T between the first and second zero(5.520 or 2), co is negative, and the sign of thetime-varying part of the output signal beam intensitywave form is inverted as discussed below Eq. (38).The minimum photorefractively coupled intensitiesthen occur at m(t) = 0. The wave-form inversion isevident in the bottom traces of Figs. 3(b) and 3(c),(P = 220 = 3.84 rad).

The effects of high-speed 4m(t) = 0 or Tr modulationon the photorefractive coupling are shown in Figs. 4and 5. Figure 4 plots the measured values of thenormalized maximum and minimum output opticalintensities in the signal beam direction as a functionof K for two values of beam ratio, mO = 180 and mO =30. The solid and dashed curves represent the ex-pected behavior, computed from Eq. (59), with tan8(d) computed from Eqs. (24) and (25), and theidentity cos2 = (1 + tan2 b)-. The InP:Fe wassubjected to a 4-kV/cm externally applied electricfield, which gave a static (K = 1) two-wave mixinggain of exp(rod) = 7.3. The minimum signal beamoutput intensity should vanish at ( 0)/ 2 tan 8(d) = 1.For the two values of mO used, from Eq. (24), thisoccurs at K = [exp(Fd/2) - 1]-1 = 0.60, in goodagreement with the measured data. Figure 5 showsresults obtained at m = 550 with a 6.8-kV/cmexternally applied electric field, which yielded a statictwo-wave mixing gain of exp(Fd) = 32. The mini-mum photorefractive coupled output signal beamintensity should vanish at K = 0.21, again in goodagreement with the measured data.


48 0 Measured maximum m0=1 80° Measured minimum mo=1 80

U) + Measured maximum mo= 30+ Measured minimum mo= 30

6 - Predicted maximum mo=1 801 6 - Predicted minimum m0=1 80

0.4z - -- Predicted maximum mo= 30 -V

-- Predicted minimum m= 3 00

C0) 2

~0in

E o.o 0.2 0.4 0.6 0.8 1.0

o KappazFig. 4. Maximum and minimum normalized output signal beamintensity versus K when the phase was 0 or ir. m = 180 and30. The external crystal voltage was 4.0 kV/cm. Mount temper-ature was approximately 23 C.

Figures 6 and 7 are similar to Figures 4 and 5 andshow the photorefractive coupling response in thepresence of sinusoidal phase modulation, describedby Eq. (53) or Eq. (60). For this type of phasemodulation, K = 0 = tan 8(d) if T = 2.405 or 5.520.Under this condition, the maximum and minimumsignal beam output intensities, given by Eq. (60),coincide as shown in the figures. The solid anddashed curves are the theoretically expected valuesgiven by Eq. (60). The response shown in Fig. 6 wasobtained with a 4-kV/cm externally applied electricfield, which yielded a value of exp(Fod) = 7.3 for bothmo = 180 and mo = 30. The response shown in Fig.7 was obtained with a 6.8-kV/cm field, which yieldedexp(Fod) = 32. The experimentally observed valuesof T for the first and second zero crossings of K were

8

2

o Measuredo Measured+ Measured+ Measured

- Predicted- Predicted--- Predicted- -- Predicted

maximum m0=1 80minimum m0=1 80maximum mo= 30minimum mo= 30maximum mo=1 80minimum mo= 180maximum mo= 30minimum mo= 30

0 O 1 2 3 4 5 60z Psi (radians)

Fig. 6. Maximum and minimum normalized output signal beamintensity versus 'e for sinusoidal phase modulation. m = 180and 30. The external crystal voltage was 4.0 kV/cm. Mounttemperature was approximately 23 C.

observed to occur within 0.1 rad and 0.26 rad (or 5%)of their expected values of 2.405 and 5.520 rad,respectively, for Fig. 6. At the higher two-wavemixing gain used in Fig. 7, the differences were 0.04rad and 0.21 rad, respectively, which are within 4% ofthe expected values.

Figures 8 and 9 represent the results obtained withtriangular phase modulation and are otherwise thesame as Figs. 6 and 7. The zero crossings of K areexpected to occur at P = 3.14 and 6.28 rad. Themeasured values of the first zero crossings are all 3.2rad, which is in good agreement with the theoreticalvalue of Ir. The phase modulation could not reachthe second zero, but the extrapolated values are 6.3

x Measured n+ Measured ao Measured n

- Predicted r------ Predicted a

Predicted r

- 4-- ' __0.2

Fig. 5. Maximum, minimuput signal beam intensity vmo = 550. The external cr;temperature was approximat

.4 40-40in

C x Measured maximummaximum : - + Measured averageaverage - Measured minimumminimum 4 30 - Predicted maximummaximum --- Predicted averageaverage , / 4- ---- Predicted minimumninimum .~0

0/X z U)7 m 10 0) 1

- 0 -~~~~~~~~~~

0.4 0.6 0.8 1.0 E 1 2 3 4 5 6

Kappa 0 Psi (radians)

m, and time-average normalized out- Fig. 7. Maximum, minimum, and time-average normalized signalersus K when the phase was 0 or r. beam output intensity versus T for sinusoidal phase modulation.ystal voltage was 6.8 kV/cm. Mount mo = 550. The crystal voltage was 6.8 kV/cm. Mount tempera-;ely 19 'C. ture was approximately 19 C.


±J 40U,Ce)as4-

CL4-' 30*

4-J

0E 20aU

C) 1 00_

N

0E_ lo

0)z

0.0

-

.+' 8 0 Measured maximum m0=1 80C 0 Measured minimum mo=1 80a) s~s + Measured maximum mo= 30C + Measured minimum mo= 304~. 6 * N- Predicted maximum mo=1 800. ~ +Q- Predicted minimum mo= 1804- \ , --- Predicted maximum mo= 30o 4 . --- Predicted minimum mo= 30

E 4- 'co

N_ .

E 0 1 2 3 4 5 6L_o Psi (radians)Z

Fig. 8. Maximum and minimum normalized output signal beamintensity versus t for triangular phase modulation. mo = 180and 30. The external crystal voltage was 4.0 kV/cm. Mounttemperature was approximately 23 C.

8

2

+ Measured maximum+ Measured minimum

- Predicted maximum- Predicted minimum

++

+

+

+ ,

E 0.0 0.2 0.4 0.6 0.8 1.0

o Kappa

Fig. 10. Maximum and minimum normalized output signal beamintensity versus K when the phase was 0 or r. m = 16. Theexternal crystal voltage was 4.0 kV/cm. Mount temperature wasapproximately 23 °C.

and 5.9 rad. The solid and dashed curves are, asbefore, the maximum and minimum values of signalbeam output intensity as given by Eq. (60).

Finally measurements at the low value of mo = 16were made for biphase ( or Tr) and sinusoidal phasemodulation in an attempt to explore further theaccuracy of the theoretical description of the photore-fractive coupling at even higher levels of pump beamdepletion. Here the measure of pump depletion hasbeen taken as the fraction of energy lost in thetransmitted pump beam resulting from coherentlydiffracted light into the signal beam in the absence ofphase modulation (K = 1) on the input signal beam.From Eq. (37a), this may be expressed as

.4- 40u)C) x Measured maximum

4..I + Measured averageR +x a Measured minimum

4 30 ' Predicted maximumCL ------ Predicted average

- --- Predicted minimumoE 20

X

C0\

o~~~~~~oi

0 2 3 4 5 6

o Psi (radians)

Fig. 9. Maximum, minimum, and time-average normalized out-put signal beam intensity versus T for triangular phase modula-tion. m = 550. The external crystal voltage was 6.8 kV/cm.Mount temperature was approximately 19 C.

Pump depletion = 1 - cos2 8(d) E - t )]2 (61)

From Eqs. (24) and (25), the values of exp(Fod) = 7.3and mo = 180, 30, and 16 yield pump depletions of 3%,17%, and 27% respectively. The results for 27%pump depletions are shown in Figs. 10 and 11. Forbiphase modulation, the value of K for which theminimum signal beam intensity vanishes is predictedto be K = 0.67 and is close to the experimentallyobserved value as shown in Fig. 10. Similarly, forsinusoidal modulation, the values of T for which K =0 are within 5% of the theoretically expected valuesgiven above. The measured data points, however, donot fall as close to the theoretically expected values

U) 8Inas4-

,j 6-C0. P

0E 4m

C)M

can 2-._))2(VN

0z

+ Measured+ Measured

- Predicted- Predicted

maximumminimummaximumminimum

b4\"+

k\ \

\ \

\+' Y + -"4' -,c tt> + + + (

1 2 3 4 5 6

Psi (radians)Fig. 11. Maximum and minimum normalized output signal beamintensity versus 'I for sinusoidal phase modulation. m = 16.The external crystal voltage was 4.0 kV/cm. Mount temperaturewas approximately 23 C.


\

based on Eqs. (59) and (60) as in Figs. 4-9, althoughthe data points are still within 20% of these values.This is not surprising, given the very complicatednature of the photorefractive coupling that occurs inthis photorefractive material. At large values of mo,I2(0) <« I(O) and pump depletion is negligible. Thismeans the entire interaction region within the crystalexperiences exponentially attenuated pump light mod-ulated only slightly by the interference pattern causedby the presence of the much weaker signal beam.At low values of mo, a larger fraction of the totaloptical intensity inside the material is due to thesignal beam, and the deeper intensity modulationdepth causes a larger imbalance in the local opticalhole generation-electron thermal ionization rateswithin the material. This in turn causes departuresfrom the assumed Tr/2 phase shift between intensitymaxima and the photorefractively induced index ofrefraction grating maxima within the material. Asshown in Figs. 3(b) and 3(c), a departure of only a fewdegrees can cause noticeable asymmetries in thepredicted signal beam intensity wave forms.

6. Conclusions

We have presented a simple theoretical model describ-ing phase modulation effects on steady-state photore-fractive two-wave mixing. Pump depletion effectsare included. The photorefractive coupling has beenexpressed in terms of a grating-rotation angle, whichis reduced by phase modulation as given in Eqs. (24)and (25). The model assumes plane-wave interac-tion and a Tr/2 phase shift between the intensitymaxima and space-charge field maxima.

This research showed that phase modulation hassome effect on the exponential gain r, which cannotbe predicted with the undepleted pump approximation.Also it showed that the phase difference, A(t), appear-ing in the photorefractively coupled intensities, Eq.(37), is given by the difference between the instanta-neous modulation phase, 4m(t), and the average 4 ~av =

arg(expj(m(t)),g, when the phase shift between theindex of refraction grating and the average fringesystem is Ir/2 rad. This implies that two opticalbeams cannot be mixed in quadrature, which hasimplications for phase locking two lasers by use ofphotorefractive materials as a phase detector.

We have measured the photorefractive two-wavecoupling in InP:Fe for three modulation formats,biphase, sinusoidal, and triangular modulation.Despite the simplicity of the theoretical treatment, itaccurately predicted the maximum, minimum, andtime-average photorefractively coupled output inten-sities. We attribute most of the differences betweenthe theoretical and measured performance to theintensity-dependent phase shift between the inten-sity and photorefractively induced index of refractionmaxima that occurs in InP:Fe.

The theory will be useful in predicting the perfor-mance of phase-shift-keyed coherent optical commu-nication systems that use photorefractive beam com-biners. It will also be useful in analyzing holographic

vibration studies that use photorefractive crystals asthe holographic storage medium.

Appendix A

Here we show that if the grating is shifted r/2 radfrom the fringe pattern, i.e., 1O is real, then the phaseof (Al(z, t)A2*(z, t), is independent of the z position.To show this, write (Al(z, t)A2 *(z, t))ig in terms of itsmagnitude and phase as

(A1(Z, t)A 2 *(Z, t))g = I (A(Z, t)A 2 *(z, t))Tg I exp[jf(z, t)].

(Al)

Take the derivative of (Al) with respect to z to get

az A2 *(Z, t) + A(z, t) aZ )

d I (A1 (z, t)A 2 *(z, t)), I

azexp[j(f(z, t)]

+ i I (A1 (z, t)A 2 *(z, t)),g I exp[j4y(z, t)] ay(z t) (A2)

Now the derivatives on the left-hand side can beexpressed in terms of the space-charge field, whichcan in turn be expressed in terms of the local fieldintensities as follows. Use Eq. (4a) and (4b) to elimi-nate the derivatives on the left-hand side, yielding

(|- GEC(')(z, t)A2 (z, t) - 2 coA(z, t)1A2*(Z, t)

+ Aj(z, t) Esc(l)(z, t)Ai*(z, t) - 2 cosOA2*(Z, t)|

a | (A 1(z, t)A 2*(z, t))Tg I

az exp[j4f(zt)]

+j I (A1 (z, t)A 2*(z, t))Tg I exp[j-f (z, t)] az (A3)

Substitute for the space-charge field using Eq. (7) torewrite the equation as

(I1(Z, t) - I2 (Z, t))g 2I0 (Z) - COS ot IA1(z, t)A 2 *(Z, t))g I

a I (Al(z, t)A 2 *(z, t)% IX exp[j4f(zt)] = az exp[j4f(z, t)]

+j az (A1 (Z, t)A 2*(Z, t))g I exp[jf(z, t)]. (A4)

Equating the real and imaginary parts of Eq. (A4)gives a(f(z, t)/az = 0 so that the argument of(Ai(z, t)A2 *(z, t))T is independent of z and is denotedby 4f (t) throughout the paper.

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Photorefractive two-wave mixing in the presence of high-speed optical phase modulation