Vol. 9, No. 8/August 1992/J. Opt. Soc. Am. B 1499

Investigation of photorefractive self-pumpedphase-conjugate mirrors in the presence of loss and

high modulation depth

James E. Millerd and Elsa M. Garmire

Center for Laser Studies, University of Southern California, University Park, DRB 17,Los Angeles, California 90089-112

Marvin B. Klein

Hughes Research Laboratories, 3011 Malibu Canyon Road, Malibu, California 90265

Received November 19, 1991; revised manuscript received February 7, 1992

Previous theoretical models for linear and ring self-pumped phase-conjugate mirrors are reviewed and modifiedto include the effects of linear absorption. The nonlinear photorefractive response at high modulation depths,present when large electric fields are applied in semiconductor and sillenite materials, is included in the coupledwave equations for the ring mirror, and the equations are solved numerically. Results of the analysis show thatphase-conjugate reflectivities can be severely limited because of both linear absorption and large-signal effects.The calculations are compared with actual measurements of ring mirrors by using InP and Bil2TiO2o.

INTRODUCTION

The photorefractive self-pumped phase-conjugate mirrorwas first demonstrated by White et al.' in 1982. A pho-torefractive crystal was placed inside a linear resonator,and a phase conjugate of the input beam was generated.The process uses a four-wave mixing geometry in whichthe pump beams are derived from amplified scattering ofthe incoming signal beam. Feinberg,2 Cronin-Golombet al.3 , Chang and Hellwarth,4 and others have since intro-duced alternative geometries. Under ideal conditionsphase-conjugate reflectivities can approach 100%.5 WithBaTiO3, ref lectivities were measured of the order of 60%.6Self-pumped mirrors have been demonstrated with semi-conductor and sillenite materials. However, the measuredreflectivities for these materials are surprisingly low (nolarger than 11%, Ref. 7), even though gain coefficients arecomparable with those in BaTiO3 (as large as 35 cm'). 7 '10

The relatively poor performance of the semiconductorsand sillenite is due to the combination of linear absorption(typically higher than in BaTiO3) as well as the nonlinearphotorefractive response present when large electric fieldsare applied at high modulation depths. Wolffer et al.1 ex-amined the effects of absorption and nonlinear photore-fractive responses on the double-pumped phase-conjugatemirror. We present here an investigation of the individualand combined effects on self-pumped phase conjugators.

thresholds for oscillation and because external electricfields are difficult to apply in other geometries. It is usualto make the single-grating approximation, which amountsto considering only the gratings written by the interfer-ence of beams 1 with 4 and 3 with 2. This approximationis quite good when applied fields are used, since the othergratings are either at too high a spatial frequency or in thewrong direction to experience enhancement. The basiccoupled wave equations for a four-wave interaction in aphotorefractive material under this approximation are5

dA 1 m a- 2 A4 - -A l ,dr 2 2

dA 2* m ay 2A?* + 2A2*,dr 2 2

dA3 m a-= +y2A 2 + -A 3 ,dr 2 2

dA,* m adr 2 2

(la)

(lb)

(ic)

(1d)

where

ii7rno3reffEscV = exp(i'lsc),

mA cos 0

m = 2(AlA 4 * + A2*A3)/IO.

(2)

(3)

THEORETICAL BACKGROUND

The theory for self-pumped phase conjugation using pho-torefractive four-wave mixing has been developed byCronin-Golomb et al.5 The geometry and the designationof the beams for the ring and the linear self-pumpedphase-conjugate mirrors are shown in Fig. 1. These tworesonators were considered because they offer the lowest

Ai is the electric-field amplitude coefficient of the ithbeam, the asterisk denotes the complex conjugate, no is thebackground index of refraction, reff is the effective electro-optic coefficient, A is the free-space wavelength, is thehalf-angle between beams, & is the magnitude of the fun-damental Fourier component of the space-charge field, cD(is the phase of the space-charge field with respect to theincident intensity pattern, a is the linear absorption coef-

0740-3224/92/081499-08$05.00 ( 1992 Optical Society of America

Millerd et al.

1500 J. Opt. Soc. Am. B/Vol. 9, No. 8/August 1992

ficient, and Io = I + 2 + 3 + 14 is the combined beamintensity. Notice that the standard two-wave mixinggain coefficient F and y are related by Re(y) = F/2.We assume here that D8C = T/2 , so that may bereplaced by r/2 (since y is pure real). This condition ismet when symmetric applied ac fields are used or when dcfields are used with enhancement techniques such as thetemperature-intensity resonance.'2 In the dc case, how-ever, it may not be possible to maintain exactly a /2phase shift over the entire length of the sample, becausethe intensity distribution caused by absorption will makeit difficult to maintain the resonance everywhere.

Equations 1(a) and 1(b) may be decoupled from Eqs. 1(c)and 1(d) by assuming that the loss is negligible and by us-ing the well-known relations

12 + 3 - I1 - 4 = A, A1A2 + A 3A 4 = C/2,

A1A3* + A2A 4* = D/2, (4)

constant C and as functions of coupling strength, FL, andmirror reflectivity. The reflectivity is plotted as a func-tion of coupling strength for various values of exter-nal mirror reflectivity in Fig. 2, which agree with thepreviously published calculations of Cronin-Golombet al.5 The threshold condition of the ring mirror may befound by taking the limit of Eq. (5b) as C goes to zero (i.e.,as reflectivity goes to zero). This calculation gives theexpression

rL= 2(M +1) InM + (8)

In the limit of perfect external mirrors, M = 1, thethreshold coupling strength is 2.

The equations may be solved for the case of the linearmirror by observing the following boundary conditions:

A122(0) = Ml, A122(L) = 1/M2. (9)

Ml

Aiwhere C, D and A are constants. The first relation maybe regarded as conservation of intensity or flux with re-spect to z. The last two relations have been interpretedas the results of reciprocity.13 Differential expressionscan be found for A34 (defined as A3 divided by A4) and forA12 (defined as Al divided by A2). With some rearrange-ment we arrive at

[FL 2+ C2)1/2l= -C*A 3 4 (0) + A + (A2 + C2)1I21

exp,- + I - C*A 4 (0) + A _ (A2+ C2 )1/2

[-C*A3 4(L) + A - (A2+ C2)1/2

X LC*A3 4(L) + A + ( 2 + C2)12J(5a)

- F (A2 + C2)1/2]exp 2Io-

47

0 L

I ,Z

3-01

> 2~ M

(a) Linear

0

2

L

1 M1

M2

- r-C*A12(0) - A + (A2 + C2)12 1- C*A12(0) _ A - (A2

+ C2)1/21

r-C*A12(W _ A _ ( 2+ C2 )1/2

L C*A12L - A + (A2 + C2)1/2](5b)

If enough is known about the boundary conditions, theequations can be solved. Notice that A34(0) is the phase-conjugate reflectivity for the linear mirror and that A12 (L)is the reflectivity for the ring mirror.

2

(b) RingFig. 1. Geometry and beam designations for ring and linear self-pumped phase-conjugate mirrors.

1 0'0SOLUTION FOR LOSSLESS MATERIALS

In the case of the ring mirror, the boundary conditions are

I4(0)/I2(0) = M I(0)/I3(0) = M, (6)

where M = M1M2 is the product of the intensity ref lectivi-ties of the two external mirrors. . Since the A3 wave isgenerated entirely inside the crystal, A34(L) = 0, whileA2(L) is the known amplitude of the input beam. Twoadditional relationships are needed to solve the equations:

M -1A= M

M + 1

CA12(L = 22 L (7)

These additional relationships can be confirmed by substi-tuting the boundary conditions into Eqs. (4). At thispoint the roots of Eqs. (5) may be found in terms of the

80I80

. _

0a)0)a:

60

40

20

00 1 2 3 4 5 6 7 8

Coupling strength FLFig. 2. Theoretical reflectivity of the lossless ring mirror as afunction of coupling strength for several values of external mirrorreflectivity M.

Millerd et al.

Vol. 9, No. 8/August 1992/J. Opt. Soc. Am. B 1501

1 00

80

6 00~~~~~~~~~~~~00

0)

20 * 11=0.1~~~~~~~~0.20

0 2 4 6 8 1 0Coupling strength FL

Fig. 3. Effects of absorption on the ring mirror reflectivity.The parameter = a/F, the relative amount of absorption;M = 1. The points are illustrative examples plotted using theapproximation ofEq. (11) for two cases.

The threshold condition is found in a similar manner to be

FL = in M, (10)

which has a threshold of zero for perfect external mirrors.The reflectivity as a function of coupling strength forM = 1 is shown as the top curve in Fig. 4 below.

EFFECT OF ABSORPTIONIn the steps leading to Eqs. (5) it was assumed that theloss coefficient was zero. This is clearly inappropriate inthe case of band-edge enhancement,' since its existencerequires moderate loss. In our InP crystal operating at970 nm the one-way transmission loss over a 2-mm crys-tal length is of the order of 50%. To evaluate the effectsof loss on the self-pumped phase-conjugate mirror, it isnecessary, in general, to solve the coupled equations nu-merically. We have found it convenient to introduce a ma-terial figure of merit -q, defined by

7 = a/r, (11)

where a and r are the loss and the gain coefficients, re-spectively. Introduction of -q permits mirror reflectivityto be calculated as a function of sample length for a givenmaterial parameter. Numerical solutions of the coupledequations can be obtained in a straightforward manner inthe case of the ring mirror because the boundary condi-tions at z = L are well defined. They are given by 2 =1 (we are free to select input intensity); 3 = 0; I =RI 2 , where R is the unknown reflectivity; I4 = I2 [Mexp(-2aL) - R] from energy conservation. By guessingR and propagating the beams to z = 0, we may check theboundary conditions given by Eqs. (6). The process is it-erated until a solution is found. Computing time is gov-erned by the integration step size, Az, and the desiredaccuracy.

Results for the ring mirror are shown in Fig. 3 for differ-ent values of -q. It can be seen that for a given value of qthere is an optimum length of sample such that any fur-ther increase will reduce performance because of the over-all increase in loss.

It was found that under certain conditions the effects ofabsorption may be approximated by lumping the absorp-

tion losses into the external mirror reflectivity. Theroots of Eqs. (5) may be solved for with the following sub-stitution for the boundary conditions:

M = M1M2 exp(-21FL). (12)

This approximation is valid provided that the couplingstrength is well above threshold and that the photorefrac-tive gain is not sensitive to intensity. Physically what oc-curs is that for large coupling strengths the reflectivity islimited by the loss and becomes equal to the effectivetransmission calculated with Eq. (12). Comparisons withthe numerical solutions show that this approximation isquite good at all coupling strengths for -q < 0.1 (see datapoints in Fig. 3). For larger -q values this approximationbegins to underestimate reflectivity at coupling strengthsnear threshold.

Investigation of the coupling strength necessary forthreshold in the ring mirror reveals that for - > 0.48there is no length of sample for which oscillation may beachieved. This agrees with the results of Wolffer et al."for the double-pumped phase conjugator that oscillationrequires F > 2a. The similarities between the twoconfigurations were previously identified by Cronin-Golomb.'5 Unlike for laser resonators, simply having netgain, F - a > 0, is not enough to guarantee oscillation.Because of the requirement F > 2a, the ratio a/F (or F/a)provides a good figure of merit to determine a material'sability to achieve threshold in a ring self-pumped phase-conjugate mirror geometry.

The effect of loss on the linear mirror was modeled withan approximation similar to Eq. (12). Here the effectivereflectivity for each mirror was given by the product of itsreflectivity, and the loss that accrued in one pass throughthe sample. The results are plotted in Fig. 4. Again, anoptimum length of sample is evident. By comparingFigs. 2 and 4 it can be seen that for the same -q value thering mirror has a higher maximum reflectivity, althoughthe linear mirror still has a lower threshold. The reasonthat loss has a more severe effect on the linear mirror maybe that the effective pump beams must make multiplepasses in the linear resonator, whereas only two passesare made in the case of the ring. The approximationbreaks down near threshold, as mentioned above, and sothe curves in this region are omitted.

100

80

10

:> 60

= 40a)C

20

00 2 4 6

Coupling strength FL8 10

Fig. 4. Effects of absorption on the linear self-pumped phase-conjugate mirror reflectivity; M = 1.

Millerd et al.

1502 J. Opt. Soc. Am. B/Vol. 9, No. 8/August 1992

100

80

:0

-a:CE5

60

40

20

00 2 4 6

Coupling strength rL8 10

Fig. 5. Reflectivity of the ring mirror for various degrees oflarge-signal effects. The empirical fitting parameter, af, indi-cates the magnitude of large-signal effects; M = 1.

LARGE-SIGNAL EFFECTS

Large-signal effects, the reduction in the apparent gaincoefficient with an increasing modulation index, havebeen reported in both Bi12SiO20 and GaAs when appliedfields and moving gratings are used. 6 "7 We have mea-sured similar effects in InP:Fe (Ref. 10) and Bi12TiO2(BTO) (Ref. 18; see next section). These large-signaleffects are due to a nonlinear response of the space-chargefield with the modulation index [defined by Eq. (3)]. Thenonlinearity is believed to be caused by the clamping ofthe space-charge field magnitude to the applied field andhas been investigated by several authors.'9 -22 No analytictheory has been developed to model this dependence forthe dc-field case; however, Refregier et al.' 6 introduced asimple empirical formula to model this behavior:

Im(E..) - f(m) = - [1 - exp(-afm)],af

(13)

where af is a fitting parameter. We have substituted f (m)for m in Eqs. (1) because of the function's ability to fit awide range of data. Again a numerical solution was neces-sary. Results for the lossless ring mirror are shown inFig. 5 for several values of af. The main effect of af is todecrease the rate at which reflectivity rises with couplingstrength; the reflectivity still can grow large if a longenough sample is used.

With the addition of absorption, however, the reflectiv-ity peaks at an optimum coupling strength (sample length).Figures 6 and 7 show the effect of both loss and large-signal effects on the ring mirror for several values of af.Although both physical phenomena reduce performance,loss appears to be the greatest problem, because it limitsthe length of sample that can be used.

EXPERIMENTS IN InP:Fe

We recently demonstrated large two-beam coupling gainsin InP:Fe by combining band-edge resonant nonlinearitieswith a second resonant enhancement involving tempera-ture stabilization.' 4 Net gain coefficients larger than25 cm-' were measured, making semiconductor gainscomparable with those in ferroelectric materials. In or-der to evaluate the performance of InP:Fe using these en-hancements as a self-pumped phase-conjugate mirror, it

was first necessary to determine the two-beam couplingperformance. Beam-coupling experiments were per-formed using an argon-pumped titanium-sapphire lasertuned near 970 nm. Two silver-paste electrodes werepainted on the sample and connected to copper electrodes,which in turn were each mounted on thermoelectric cool-ers. The temperature was stabilized within 1MC by usinga feedback loop that monitored thermocouples mountednear the edge of the electrodes. The directions of thecrystal and the applied dc field were oriented so that thecontributions of Pockels and Franz-Keldysh effects wereadditive.2 3 The samples were antireflection coated,R < 0.5%, to eliminate spurious gratings from Fresnel re-flections. The electrode spacing was 2 mm, and the beamdiameter was expanded to 5 mm to avoid field screeningeffects caused by nonuniform illumination in the trans-verse direction.

The effective gain coefficient measured as a function ofincident intensity for two different sample lengths at afixed temperature is shown in Fig. 8. The gain coefficientwas calculated with F = (ln yo)/L, where L is the length ofthe crystal and yo is the ratio of the transmitted signalbeam with and without the pump beam on. High beamintensity ratios were used ( = 1.6 X 106, the ratio ofpump to signal intensity) to avoid pump depletion andlarge-signal effects. The two samples were cut fromneighboring positions in the same boule. We believe that

50

40

180:= 3

= 20a)

1 0

00 2 4 6 8 10

Coupling strength FLFig. 6. Combination of large-signal effects and linear absorptionon the reflectivity of the ring mirror for the case of af = 4 andM = 1.

30

25

:,ao

C

20

1 5

1 0

5

00 2 4 6 8 10

Coupling strength rLFig. 7. Combination of large-signal effects and linear absorptionon the reflectivity of the ring mirror for the case of af = 6 andM = 1.

Millerd et al.

Vol. 9, No. 8/August 1992/J. Opt. Soc. Am. B 1503

32

7E0

0)0)._?

.2000

28

24

20

16

12 1 ... '.. '.. '.. ... I.. I.... ''l'''' 2e''l|s I10 15 20 25 30 35 40 45 50

Incident intensity mW/cm2

Fig. 8. Gain coefficient versus incident intensity for two-wavemixing in two InP samples maintained at constant temperature.T = 20'C; applied field Eo = 10 kV/cm; ratio of pump to signalintensity /3 = 106; grating period A = 7.7 ,.m; A = 8 m;A = 970 nm; a = 5 cm-'.

1000

.C0)

U,C

0)C

100

10

10° 102 104 106Beam ratio ,B (at entrance face)

Fig. 9. Two-wave mixing gain as a function of input-beam ratio:total intensity is 32 mW/cm2; T = 20'C; E = 10 kV/cm;A = 970 nm; L = 4 mm. The circles are experimental results;the bold curve is a theoretical fit with rFat = 18.5 cm-' anda = 3.7; the thin curve is the falloff expected from pump deple-tion alone.

the smaller gain coefficient measured in the longer samplearises primarily from the inability to maintain the opti-mum intensity over the entire length of the absorbingsample. Thus the measured (average) gain coefficient isless than optimum.

It should be noted that the introduction of the back-ward-propagating beam in the ring mirror configurationwill alter the intensity profile and therefore change thegain profile. In general this modification will tend to in-crease the uniformity of illumination and to raise the av-erage gain coefficient. The extent to which the averagegain is altered depends on the strength of the redirectedsignal. We use the average gain coefficient measured inthe two-wave mixing experiments for the ring mirror cal-culations and point out that, in the case of the 4-mmsample, the redirected signal is only of the order of 10% ofthe incident beam intensity. For shorter sample lengthsthe correction will be more significant.

Large-signal effects were examined by measuring theintensity amplification, y0, as a function of beam ratio(see Fig. 9). The combined intensity of the two beamswas held constant to maintain the temperature-intensityresonance. The thin curve in Fig. 9 shows the falloff in

amplification expected from pump depletion under thelinear approximation. Large-signal effects are requiredfor explanation of the more drastic reduction of gain withdecreased beam ratio. The data are fitted by using thenonlinear model of relation (13). To account for bothpump depletion and large-signal effects, we integrated thecoupled wave equations numerically. The equations aregiven by Eqs. (1) with A3 as the signal-beam amplitude, A2as the pump-beam amplitude, and Al = A4 = 0. Wefound, for a grating spacing of 7.7 /-m, that af is approxi-mately 3.7 when the saturated gain, rsat (measured in thelimit of small m), is 18.5 cm-' (see the bold curve inFig. 9). This fitting procedure was repeated for differentgrating spacings, and the values obtained for af and Fsatare plotted in Fig. 10. Note that af and Fsat follow thesame trend, being large at comparable grating spacings.From numerical calculations for BTO with ac fields it hasbeen found empirically that there is a linear relationshipbetween the normalized space-charge field, Es/m, givenby the standard linear photorefractive theory and af.Fsat is equivalent to the gain coefficient given by the stan-dard linear theory and is proportional to E,,/m. There-fore these numerical calculations agree qualitatively withour measurements here that the values of Fsat and af arerelated to each other.

5

4.5

a)

MaE

CZCL

4

3.5

3

2.5

2

0 5 10 15 20 25Grating spacing A, gim

20

18 2)-

::00

14 C_L

12 ,,'

10

830

Fig. 10. Grating spacing dependence of the saturated gain coef-ficient, rsat, and the magnitude of the large-signal effects, af.T = 20'C; E0 = 10 kV/cm; A = 970 nm; L = 4 mm. Curves areguides for the eye.

`0

0)0)CE

0.1

0.01

0.0014.5 5 5.5 6 6.5 7 7.5 8

Coupling strength tL

Fig. 11. Theoretical and measured reflectivity in a ring mirrorusing InP near the band edge. T = 20'C; A = 970 nm;L = 4 mm; A = 7.7 ,um. The coupling strength was changed byreducing the applied voltage incrementally from 10 to 8.25 kV/cm.

i%�L4m�m1

Millerd et al.

1

1504 J. Opt. Soc. Am. B/Vol. 9, No. 8/August 1992

0.35

0.3 T=200C

0.25

0.2

0) 0.15

CE 0.1

0.05

010 15 20 25 30 35 40

Incident intensity mW/cm2

Fig. 12. Measured reflectivity from the ring mirror as a functionof input intensity. Same parameters as in Fig. 11; Eo = 10 kV/cm.

5

x0:.

0)a)

.5

4

3

2

1

00 5 10 15

Grating spacing gim20 25

Fig. 13. Calculated reflectivity for the ring mirror as a functionof grating spacing for data from Fig. 10. The square is the maxi-mum measured reflectivity, shown also in Fig. 11. The curve isdrawn as a guide to the eye; deviations come from data.

1 0

-0~~~~~~

CD ~ ~ ~ ~~

1

2.5 3 3.5 4 4.5 5 5.5Coupling strength FL

Fig. 14. Theoretical and measured reflectivity in a ring mirrorusing BTO with 60-Hz ac fields. Eo = 10 kV/cm; A = 633 nm;L = 1.37 mm; A = 5.5 Am. Coupling strength was changed byreducing the applied voltage from 10 to 7.2 kV/cm.

The absorption coefficient at 970 nm was 5 cm-', whichgives = 0.27 (at the 7.7-Am grating spacing). Clearly,based on absorption alone, we can see from Fig. 3 that ourreflectivity is reduced to less than 15% (L = 4 mm). The

addition of large-signal effects, demonstrated in Fig. 6,further reduces the reflectivity to less than 1l%. Calcula-tions for the 1-mm sample show that it should have a re-flectivity of approximately 0.1%.

To compare experimental results with the theoreticalanalysis, we constructed a ring self-pumped phase-conjugate mirror and measured its performance as a func-tion of coupling strength. The coupling strength wasvaried by changing the applied field. The measured re-flectivity for the ring mirror with the 4-mm sample isshown in Fig. 11 as the circles, and the calculated reflec-tivity is shown as the curve. Figure 12 shows the reflec-tivity as the intensity is tuned through the resonance.The turn-on time of the mirror, (defined here withR = Rol[ - exp(-t/T)], where Ro is the maximum reflec-tivity of the mirror) was measured to be 100 ms for aninput intensity of 23 mW/cm2. The external mirror re-flectivity M = 86%. The relationship between appliedvoltage and coupling strength was measured separately ina two-beam coupling experiment and was used to obtainthe scaling of coupling strength for Fig. 11. Although afis a function of the applied field (af is related to Fsat asdiscussed above), we have assumed af to be a constant(af = 3.7 as measured at 10 kV/cm) in order to simplify ourcalculations. From the measurements of Fig. 10 wewould expect af to vary between 3 and 3.7 over the experi-mental range, which results in a 20% overestimate of af atlow field values. With this approximation the agreementis quite good, especially for predicting the relative magni-tude of the reflectivity. Without including absorptionand large-signal effects in the calculations one would pre-dict reflectivities of the order of 80%. Clearly these areimportant corrections to the theory.

As is shown in Fig. 13, the calculated reflectivity as afunction of grating spacing follows the amount of en-hancement, Lsat. The trend is the same as that which onewould expect without including large-signal effects. Thisis not surprising, since the relative gain at any given beamratio also follows the shape of the curve of Fig. 10, al-though the absolute magnitude is a function of beam ratio.

EXPERIMENTS IN Bi12TiO20

We have also confirmed our model using BTO with appliedac square-wave fields at 633 nm. BTO is from the sillen-ite family of crystals, symmetry class 23, and is similar toBi,2 SiO2 0 and Bi12 GeO but has a higher electro-optic coef-ficient (5.7 pm/V) and lower optical activity (6 0/mm at633 nm).24 The sample used here was grown at HughesResearch Laboratory. Details concerning our recent pho-torefractive measurements in this material are discussedin Ref. 18.

The saturated gain coefficient rsat and af were measuredin a two-wave mixing experiment to be 35.5 cm-' and 6,respectively, with an applied field strength of 10 kV/cm(60 Hz) in a 1.37-mm-long sample. The absorption coeffi-cient was 1 cm-', giving a value for -q of 0.028. Althoughaf is larger in BTO than in InP, qj is smaller by an order ofmagnitude. This difference translates to roughly a factorof 20 increase in self-pumped phase-conjugate mirrorreflectivity.

Figure 14 shows the measured and calculated ref lectiv-ity as a function of coupling strength for a 1.37-mmsample. Again the coupling strength was varied bychanging the applied field. The effect of optical activityis neglected in the equations, although for small interac-tion lengths this should not introduce a large error.

' ' ' I I . . . . I . I I I| I I* II r I -T

.... I .... I . .. -a l - ,

0.... 11.11.11.111-1 ...

Millerd et al.

Vol. 9, No. 8/August 1992/J. Opt. Soc. Am. B 1505

DISCUSSION

Clearly it is advantageous to reduce both -q and af whilemaintaining the same coupling strength. The three mostimportant material requirements for self-pumped phase-conjugate mirror operation are, in order of importance,(1) small q (i.e., a large ratio of rsat to a), (2) small af, and(3) the ability to maintain high rsar-over any length of crys-tal. The first criterion is a strong function of operatingwavelength as well as the magnitude of applied fields. Ifthe applied fields can be increased without bound, -q willcontinue to decrease. From a realistic standpoint there issome question as to how practical it is to use field valuesmuch larger than 10 kV/cm because of signal generationand dielectric breakdown at surfaces. Certainly crystalparameters such as doping can play an important role inthe minimization of -q through the interplay of trap den-sity, conductivity, and loss. The second factor, a, ap-pears to be strongly tied to rsat, as discussed above, so thatit is difficult to change them independently. The finalcriterion is influenced by factors such as having to main-tain a temperature-intensity resonance or such as walk-off caused by optical activity. These issues will beconsidered further in terms of each material.

The band-edge photorefractive effect, investigated herein InP, makes it possible to increase rsat without changingaf. This is because an additional index change is createdwithout increasing the magnitude of the space-chargefield.23 Unfortunately, however, operation near the bandedge is accompanied by an increase in absorption. Al-though the net gain, r - a, increases as the wavelengthapproaches the band edge, the ratio F/a decreases. Be-cause of this effect -q becomes larger and reduces themaximum performance of the ring mirror. RecentlyWolffer et al.25 showed that the performance of a double-pumped phase-conjugate mirror used with InP improvesas the operating wavelength is moved away from the bandedge, in agreement with the analysis given here. Bylsmaet al.7 demonstrated a ring self-pumped phase-conjugatemirror using InP and ac fields at a wavelength of 1.3 Amwith a maximum reflectivity of 11%. The gain coeffi-cient was reported to be 2.5 cm-' for their 17-mm crystal.However, the absorption was only 0.15 cm-', so that- = 0.06. If we assume that their large-signal effects areroughly the same as in our crystal, the calculated reflec-tivity for this system is 10% (see Fig. 6), in agreementwith published results. It is clear that even though thegain coefficient decreases as wavelength increases, the ab-sorption decreases faster; thus -, decreases and the self-pumped phase-conjugate mirror performance improves.

As mentioned earlier, in an absorbing crystal there is anoptimum length of sample that will maximize reflectivity.Calculation of this length for InP is complicated by theproblem that the temperature-intensity resonance ischanging over the length of the sample. (This is where acfields have an advantage, because the resonance can bemaintained over any length of crystal.) In the best case,when resonance can be held over the entire length (per-haps by a temperature gradient 2 ), we would predict amaximum reflectivity of 3% for a 2.1-mm sample (forFsat = 32 cm-', as measured in the 1-mm sample, andaf = 3.7).

Calculations of the optimum length of sample for BTO

show that, if there were no optical activity, a maximumreflectivity of 31% might be achieved for a length of4.4 mm. In practice optical activity will slightly degradeperformance and increase the optimum sample length.The magnitude of the correction can be estimated by not-ing that, if the incident polarization is optimal (i.e., polar-ized vertical at the middle of the sample), then the angle ofrotation at either end is 130. Circularly polarized lighteliminates problems of optical activity; however, the effec-tive gain coefficient is reduced. With the effective gaincoefficient approximated as 1/2, its value for linearly po-larized light, calculations show a maximum reflectivity of13% at a sample length of 6.4 mm.

The only remaining aspect that can be optimized iscrystal composition. As with any photorefractive mate-rial, the maximum gain coefficient is regulated by thecombination of trap density and diffusion length. In gen-eral gain coefficients can be increased by raising the trapdensity; however, this modification could also increase ab-sorption. The exact effects of doping and heat treatmenton material performance are not clear and remain a pos-sible subject of further study.

Our model has provided several physical insights.First, the presence of absorption and nonlinearities at ahigh modulation index, which appear to be present insemiconductor and sillenite materials when electric fieldsare applied, have a serious effect on the performance ofself-pumped phase-conjugate mirrors, explaining the lowreflectivities that have been reported in InP, GaAs, andBTO. Second, the ratio of the absorption to the gain coef-ficient is a good figure of merit for determining the per-formance of self-pumped phase-conjugate mirrors. In thepresence of loss the ring mirror offers higher reflectivitythan the linear mirror, although the linear still has alower threshold. Finally, oscillation for the ring mirrorrequires that F > 2a.

ACKNOWLEDGMENTS

The authors thank Rahul Asthana for providing antire-flection coatings for the InP, Daniel Camberos for labora-tory assistance, and Afshin Partovi for many usefuldiscussions. This work was funded in part by the De-fense Advanced Research Project Agency through the Na-tional Center for Integrated Photonic Technology and inpart by the National Science Foundation.

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