202 J. Opt. Soc. Am. B/Vol. 5, No. 2/February 1988

Low-power (10.6-gim) laser-beam amplificationby thermal-grating-mediated degenerate four-wave mixing

in a nematic liquid-crystal film

I. C. Khoo, P. Y. Yan, G. M. Finn, T. H. Liu, and R. R. Michael

Department of Electrical Engineering, The Pennsylvania State University, University Park, Pennsylvania 16802

Received March 2, 1987; accepted September 28, 1987We present a detailed theoretical analysis of degenerate four-wave mixing by a laser-induced thermal grating in anematic liquid-crystal film. In particular, we show that the coupling of the (strong) pump beam to the first-orderdiffracted beam can give rise to substantial amplification of a (weak) probe beam. Experimental verification of thiseffect with a CO 2 laser beam is also made. A probe-beam gain of greater than 20 can easily be observed in a 120-Amfilm with a pump intensity of the order of a few watts per square centimeter.

1. INTRODUCTION

Degenerate optical wave mixing in highly nonlinear materi-als has been the subject of intensive research in the past fewyears. Various useful and novel processes have been experi-mentally demonstrated and are beginning to find applica-tion.' One of the phenomena associated with optical wavemixing is the amplification of one beam by another.'- 9

These two beams form an intensity grating on the mediumthat in turn creates a refractive-index grating through theunderlying nonlinear mechanism. In the case of photore-fractive materials,3 -5 the intensity grating is phase shiftedwith respect to the resultant index grating, thereby creatingan energy-coupling mechanism between the two beams. Ina Kerr-like medium, the phase shift is zero, and amplifica-tion of one beam by the other is possible, provided that thetwo input beams are appropriately frequency shifted.6 -8

We recently showed that it is indeed possible to haveamplification of a weak beam by a strong beam of the samefrequency, even in a Kerr medium, provided that the mixingconditions favor the generation of the diffracted beam.7 Inthat study, the nonlinearity is due to director axis reorienta-tion, and the laser-induced refractive-index change (An) isproportional to the square modulus of the optical electricfield, i.e., An = n2 JEI2, where n2 is the nonlinear coefficientand E is the optical electric field. In this paper we extendthe study to another class of nonlinearity, namely, thermalnonlinearity that is induced by cw lasers. Because of heatdiffusion within the liquid crystals and to the cell walls, thelaser-induced refractive-index change An is, in general, anonlocal function of the optical field, i.e., An is not simplydependent on the square modulus of the optical intensity ata particular point; for a given applied optical field, An mustbe solved subject to the boundary conditions. This changein refractive index, in turn, will affect the optical field distri-bution in the medium, and a correct solution of the tempera-ture distribution, the refractive index, and the optical fieldwithin the nonlinear medium can be obtained only by self-consistent calculations, subject to the boundary conditions.Nevertheless, in Section 2, by identifying the various wavemixing terms in the basic coupled equations, we show that

the basic mechanism for probe-beam amplification is similarto the four-wave mixing effect observed in Kerr media, i.e.,the weak probe beam is amplified through the diffractionsfrom the pump beam from a grating formed by the pumpbeam and the diffracted beam; cf. Fig. 1. For quantitativedetermination of the effect, a detailed numerical calculationis needed; it is presented in Section 2. Some preliminaryexperimental results are also presented.

We have conducted a beam-amplification experiment thatuses a CO 2 laser for several reasons. One is to demonstrateexplicitly that the extraordinarily large optical nonlinearityof liquid crystals can be applied to this spectral region, wherefew room-temperature, highly nonlinear materials exist.Second, because of the rather long wavelength of the CO2laser (compared with the usual -0.5-Am wavelength of visi-ble light), optical wave mixing involving large grating con-stants (>200 ,lm) presents no practical problems. Largegrating constants permit efficient thermal-grating-mediat-ed wave mixing in liquid crystals. There are other equallyimportant reasons for extending the success of optical wavemixing effects in the visible and near-infrared regimes to thefar-infrared regime. One is the possibility of constructing aring oscillator and a self-pumped phase conjugator in theinfrared (10.6-,um) region.

Figure 1 shows the geometry of the laser-nematic interac-tion. A strong pump beam (ko) intersects with a muchweaker probe beam (k,), forming a thermal index grating inthe liquid crystals. The first-order diffraction (k2) occurson the side of the pump beam. (Because of the weakness ofthe probe beam, the diffraction on the probe-beam side isnegligible.) The physics of the multibeam coupling andheat diffusion in this system is described by the Maxwellwave equation and the diffusion equation, respectively:

2 -2 472 p'NLc7E - 2 at c2

( W)

and, at steady state,

d2T _ 2 T caIE 2

-D ax 2-Dl aX2 47r

(1)

(2)

0740-3224/88/020202-05$02.00 © 1988 Optical Society of America

Khoo et al.

Vol. 5, No. 2/February 1988/J. Opt. Soc. Am. B 203

LIQUID CRYSTALZn Se WINDOW

PROBE

hPUMP

=- \Z2 DIFFRACTED

Fig. 1. Schematic of the amplification of a weak probe beam (E1 ,kl) by a strong pump beam (E0, ko) and the simultaneous generationof the principal diffracted beam (E2). The laser is polarized in adirection perpendicular to the plane of the paper.

In Eqs. (1) and (2), E is the optical electric field, pNL thenonlinear polarization, T the temperature, a the absorptionloss coefficient, and D1 and D11 the thermal diffusion con-stants for directions perpendicular and parallel, respective-ly, to the director axis.

In the plane-wave approximation, the pump (Eo), probe(El), and diffracted beams (E2 ) may be expressed as follows:

Eo(z) = E0(z)exp(ik 0 r) = E0 exp(ikz),

E1(z) = El(z)exp(ik, - r) = exp(ikz cos 0 + ikx sin 0),

E2 (z) = e2 (z)exp(ik2 r) = 2 exp(ikz cos 0 - ikx sin 0), (3)

where k = n/c. Note that ko is along the direction.The generated polarization pNL is given by

pNL = an (E + E1 + E2 )AT,4i7r OTOT

(4)

where To is the initial equilibrium temperature and AT is thetemperature rise.

Consider the IE12 term on the right-hand side of Eq. (2).Using Eq. (3), we have

IE12 = [1E012 + IE112 + IE21

2 + EoE,*

+ E1 E2 * + EOE2 * + c.c.]

= [lE012 + E112 + le212 + e0el* exp(ikx sin 6) + E0E2*

X exp(ikx sin 0) + E1e2* exp(2ikx sin 0) + c.c], (5)

where, for small 0, we have approximated 1 - cos 0 0, i.e.,we include effects coming from the first order in 0. Further-more, in view of the fact that el and e2 are much smaller thaneo (usually EO is about 102 times el or E2), we have

IEl 2= lE012 + le112 + E2

12 + (Eoel* + e2Eo*)exp(-iIBoX)

+ (E 0* + EOE2 *)exp(iIlox), (6)

where 03 - k sin 0.For the geometry depicted in Fig. 1, the temperature dis-

tribution may be assumed to be independent of the y dimen-sion for laser-beam sizes that are much larger than the grat-ing constant and the film thickness. (This is true in prac-tice.) The solution for the temperature rise AT may thus beexpressed in the form

AT(x, z) = f0(z) + f1(z)exp(-i~ox) + f2(z)exp(if30x). (7)

Substituting Eqs. (6) and (7) into Eq. (2) gives

= o(lel 2 + lE1l2 + le21

2), (8)

DL, 02f 1- DI1fl' = -&(EoEl* + EoE2 *), (9)

and

Df 02 f2 -Dljf21 = (el eo* + EoE2 *), (10)

where -a = ca/4ir and double primes mean a2/az2.In standard differential-equation form, Eqs. (8)-(10) be-

come

to" = - (1eO12 + E112 + E2

12),

+ (EOEl* + E2 e0*),

and

f2- - Dj 2 - (ElEO* + E0 E2*)

From Eqs. (4) and (7), we get

p0NL = kaTaTo(Eofo + elf, + E2f2)exp(ikz),

PNL = 2n (tan' (Eoo + Eof2)exp(ikz + ikdo),47r \aTIT0 (Jo 0 2 -fr" ik{3),

p2NL 2n tan' (o + E-flexpt ikz - ~)

(11)

(12)

(13)

(14)

(15)

(16)

Substituting these polarization terms into the correspond-ing Maxwell equations for EO, El, and 2 yields

8E0 .2nw2 taOn\dz = - ( EA + Elf + E) I Eo'az c2k (T)T 0( +2

ael . 2nco2 adn c Eldz = .2n (On\- (Ef + N -E

a c2k (aT)TO'" J 2"

aE i2nw2 t(an (E~fo + Eof) --0E2,c2 k k~ JTo E 2

(17)

(18)

(19)

where we have included the effect of the absorption loss inthe last terms on the right-hand sides of Eqs. (17)-(19).

The coupled equations for the temperature distribution[Eqs. (11)-(13)] and the Maxwell equations for the pump,probe, and diffracted beams [Eqs. (17)-(19)] form the basisfor our theoretical and experimental studies of the variouswave mixing processes (such as diffraction and beam ampli-fication) that are mediated by the thermal grating. In thispaper we shall limit our discussion to the possibility ofprobe-beam amplification by the pump-diffracted-beamcoupling and to some experimental confirmation.

2. DISCUSSION

The special problem associated with thermal nonlinearity isits nonlocal diffusive nature; the equilibrium temperaturerise (and therefore the refractive-index change) at a point (x,z) is dependent on the boundary conditions imposed at vari-ous other points. Since f2 involves e2, and since the magni-tude of E2, the diffracted beam, grows from 0 at z = 0 to somefinite value at z = d, f2 is therefore an asymmetric function inreference to the midplane z = d/2. This asymmetry will bereflected in the grating term f2eo, which is responsible for theamplification of the probe beam (as we will see presently).

Khoo et al.

204 J. Opt. Soc. Am. B/Vol. 5, No. 2/February 1988

This asymmetry, in addition to the great variations of theprobe beam (it can increase in magnitude by severalfoldfrom z = 0 to z = d) and the pump beam (owing to losses),renders analytical solutions of these equations impossible.On the other hand, numerical iterative techniques can beemployed to solve the problem from which various new in-sights into the multiwave coupling can be gained.

In analogy to similar wave mixing effects in a Kerr medi-um10 11 the optical intensity grating terms on the right-handsides of Eqs. (11)-(13) may be classified into two distincttypes. One arises from the interference between the pumpand the probe beams (e.g., foe,* and eleo*), which can give riseto probe-beam gain only if the resulting refractive index isappropriately phase shifted to the intensity grating. Theother arises from the interference between the pump and thediffracted beams (E2* and e2eo*), which can give rise toprobe-beam gain without the phase-shift requirement.This observation is borne out in the numerical solutions ofEqs. (17)-(19). Without accounting for the pump-diffract-ed-beam coupling, the pump- and probe-beam intensitiesare found to be monotonic decreasing functions of distance,and the main effect of the pump-probe interference is thegeneration of the diffracted beam. The formalism devel-oped here, however, permits several detailed insights intothe problem and the generalization of this diffraction effectto phase conjugation (which involves another strong beam).Optical phase conjugation with gain using thermal effectshas been observed in several studies12 with high-power nano-second lasers. It is important to point out that the funda-mental mechanism for gain in those four-wave mixing pro-cesses is different from the one under study here, which thephase-conjugated signal arises from four-wave mixing in-volving a retroreflected pump beam; i.e., the conjugated(reflected) probe beam and the amplification of the forwardprobe beam is due to the diffraction of the retroreflectedpump beam from the grating set up by the pump and theprobe beams. The effect of the diffracted beam was ignoredin the previous studies.

Inclusion of the diffracted beam and the numerous grat-ings that can be formed among the pump, the probe, and thediffracted beams leads to numerous interesting theoreticalconclusions. In this paper, we concentrate on some of thesalient aspects of this pump-probe-diffracted-beam cou-pling pertaining to the probe-beam amplification processand the nonlocal nature of the thermal effect on the intensi-ty distribution within the nonlocal media.

Figure 2a shows a plot of the diffracted-beam intensity asa function of the normalized distance zid, for the parametersIo = 80 W/cm2, I, = 0.8 W/cm 2 (i.e., a pump/probe ratio of100/1), a grating constant A = V/[2 sin(0/2)] 300 ,m, d =100 ,um, a = 80 cm'1, and dn/dT = dno/dT 10-3 K-1. Itshows the expected growth of the diffracted beam with dis-tance. Owing to loss (from a), the dependence on z/d showssigns of saturation for z = d. For the same parameters, Fig.2b shows that the corresponding pump-beam intensity de-cays monotonically by a considerable amount. Figure 2cshows a rather interesting result, i.e., that the probe-beamintensity can be amplified by more than 200% in traversingthis high-loss, but also highly nonlinear, medium. Evenhigher amplification can be obtained at a higher pump in-tensity, and/or at a larger grating constant, or by using high-er nonlinearity. (For a liquid crystal, higher thermal non-

linearity can be obtained either by bringing the sample tem-perature closer to the nematic - isotropic phase-transitiontemperature or by using dnll/dT rather than dnI/dTat a given temperature.) Our detailed theoretical study hasgeneralized this theory to include the interplay among vari-ous parameters and physical characteristics, such as intensi-ty, losses, grating constant, thickness, phase mismatch, tem-perature, presence of other diffracted beams, frequency dif-ference between the incident beams, anisotropies (As, D11 -

DI), and pump-probe beams, and will be reported else-where.

0.08

n

LU 0.06

Z

:Ld 0.04 -

c

LLJ

U< 0.02a1

C]

00

0.34

z 0.32

Lu

:2 E 0.30

L0.

:20.2 8D

0.26 0

7

U)Z

6Z:2

5mLu

004

0.2 0.4 0.6 0.8 1.0z/da

0.2 0.4 0.6 0.8 1.0

z/db

0.2 0.4

z/dC

Fig. 2. a, Plot of the diffracted-beam intensity dependence on thereduced distance z/d. b, Plot of the pump-beam intensity as afunction of propagation distance into the sample. c, Plot of theprobe-beam intensity as a function of the distance z into the sample,showing gain at the exit plane.

Khoo et al.

Vol. 5, No. 2/February 1988/J. Opt. Soc. Am. B 205

In Section 3 we present what we deem to be more importantinformation, i.e., some experimental confirmation.

3. EXPERIMENT

Figure 3 is a schematic of the experimental setup. A TEM0oCO2 laser beam (Advanced Kinetic Power-Laser 50) operat-ing in a cw or pulsed (msec) mode is split into a strong pumpand a weak probe beam (the pump/probe intensity ratio is60) and is overlapped on the sample at an (external) crossingangle of 2.30 (corresponding to a grating constant inside theliquid crystal of about 260 ,um). The beam diameter at thesample is 4 mm. The pump power is varied from 1.7 to 3.3W, and the power of the transmitted probe beam is moni-tored. The geometry of interaction of the laser polarizationand propagation vectors with the liquid-crystal axis is shownin Fig. 1. The polarization is perpendicular to h, the direc-tor axis of the nematic liquid-crystal film. The thermalindex gradient is therefore given by dno/dT, where no is therefractive index for ordinary rays.13 The film is fabricatedusing ZnSe plates coated with a surfactant for homeotropicalignment. The liquid crystal used is pentylcyanobiphenyl(PCB), and the film is 120 gm thick, maintained at a roomtemperature of 220 C. PCB was shown in a previous study 3

to have good transmission charcteristics at 10.6 /Am and doesabsorb appreciably at 10.6 /im (a = 80 cm-').

At an input pump power of 1.7 W [which is reduced to 0.8W on the liquid crystal because of the 50% reflection loss atthe air-(uncoated) ZnSe window interface], an increase ofthe transmitted probe beam of about 10% is observed. Theprobe-beam gain is observed to be rather nonlinear withrespect to the pump power (cf Fig. 4), obeying close to an I2pump dependence. At a pump power of 3 W (intensity ofthe order of 25 W/cm 2), a gain of 40% is observed. Themagnitude of these probe gains for the parameters used inthe experiment is in good agreement with the numericalsolutions (solid line in Fig. 4). The numerical solution isobtained using the parameters a = 80 cm-, thickness d =120 m, 0 = 2.3°, and dno/dT = 1 X 10-3 K-1.

The basic mechanism responsible for the probe-beam gainis the scattering of the much stronger pump beam from thepump-diffracted-beam grating in the direction of the probe.Obviously, at an equal pump-probe-beam ratio, the probegain will be zero. A higher pump-probe-beam intensityratio will lead to a higher probe-beam amplification. Thiswas verified experimentally. Figure 5 shows the experimen-tally observed probe-beam gain as a function of the pump-to-probe-beam ratio for the following parameters: pumppower, 1.8 W; ai = 2.60; laser-beam size, 0.3 cm2; initialsample temperature, 29 0C. It shows that the probe gainsaturates at a value of about 70% for a ratio >100. Theexperimentally observed dependence is in agreement withthe theoretical prediction (Fig. 6). The thermal grating,owing to diffusion, is also highly sensitive to the gratingconstant. Smaller grating constants can sustain a smallertemperature difference and therefore a smaller refractive-index modulation. Using the same parameters as thoseused for Fig. 4, we observe that the gain drops from 40% at 0= 2.3° (i.e., A 260 ,um) to a vanishing value at 0 > 100. Thisrequirement of a large grating constant for appreciable am-plification, incidentally, is (in practice) easily met by usinginfrared lasers (such as CO 2 and CO lasers). Although it is

CO, LASER ATTENUATOR

BS

Fig. 3. Schematic of the experimental setup for CO 2 probe-laseramplification. The ratio of the pump-beam to the probe-beamintensity is 60 to 1. The He-Ne laser is for probing the grating.Diffractions from the He-Ne laser are easily visible with the eye.BS, beam splitter; LC, liquid crystal.

60 -

50

0o 40

0zd 30

Lii

M0IW 200L

I 0

G

THEORY

0 0.5 1.0 1.5 2.0 2.5 3.0 3.5

PUMP-BEAM POWER (Watts)

Fig. 4. Plot of the observed probe-beam gain as a function of thepump-beam power, showing nonlinear dependence on the pump-beam intensity. The solid line is a theoretical fit, with a 80 cm-1,0 = 2.30, d = 120 um, and dno/dT = 1 X 10-3 K-1.

no -__

0

3 IT

() 25 50 75 100 125 150 175

PUMP/PROBE RATIO

Fig. 5. Experimentally observed probe-beam gain as a function ofpump-probe-beam ratio for the experimental parameters stated inthe text. The curve shows the theoretically predicted dependence.

Khoo et al.

206 J. Opt. Soc. Am. B/Vol. 5, No. 2/February 1988

0no

Lii

0T

3.0 h 20 mV/Div

GAIN =30%

2.0 -

1.0 _

I A J AWITH PUMP

4.0

LJ

i 3.00~

LLim 2.00

1.0

WITHOUT PUMP

0.2 V/Div bGAIN =2000%PUMP POWER = 4.6 Watts

WITH PUMP WITHOUT PUMP

Fig. 6. a, Oscilloscope traces of probe-beam power with and with-out pump for a pump power of 1 W. The laser pulses used are of 40-msec duration. b, Same as a but for a pump power of 4.6 W. A gainof 20 is observed.

true that, in general, a larger grating constant will produce ahigher probe-beam gain, the choice of the input probe beam(much weaker than the pump beam, of course) is important.In general, one should choose it to be as small as possible, sothat only one diffracted order is observed. If the probebeam, in combination with the high index modulation ob-tained at a large grating constant, is strong enough to pro-duce many high-order diffractions, we have observed thatthe probe-beam gain will be quenched at a low value, as aresult of the loss through these multiorder diffractions.

For a given pump-probe-beam ratio and grating constant,the probe gain can be increased (>100%) by simply increas-ing the pump intensity. However, because of absorption,the operating temperature of the liquid crystal under cwlaser illumination will rise, eventually exceeding the transi-tion temperature. We observe that at an input cw powerexceeding 6 W the sample becomes isotropic (the Tc of PCBis 35.3°C). This ambient overheating problem can be elimi-nated by simply pulsing the input laser. Typically, we findthat laser pulses of tens of milliseconds' duration are idealfor these thermal wave mixing processes (similar to our pre-vious work" with a visible laser).

With a pulsed laser, it is possible to utilize the muchhigher thermal index near Tc without overheating the sam-ple. Figure 6 shows oscilloscope traces of the probe-beam

pulse amplitude with and without the pump beam, when thesample is maintained at 320C, i.e., close to Tc, where dn/dTis large. The pulse duration is 40 msec, chosen because thethermal-grating decay time is of this order. In Fig. 6a, thepump power is 1 W (40 mJ in 40 msec), and a gain of 30% isobserved. Figure 6b shows the large gain (20 times) ob-served at an input pump power of 4.6 W. This large gain isdue to the large dn/dT near T, as well as to more efficientbuildup of the thermal refractive-index grating by pulsedlasers.

In conclusion, we have demonstrated, for the first time toour knowledge, how a probe beam can be amplified by apump beam in a diffusive medium involving a thermal grat-ing. Also, we have experimentally demonstrated the effectand how it may be applied to CO2 lasers of moderate power.Since the thermal-index gradient for the extraordinary raydne/dT is larger, and also, since these thermal gradientsincrease greatly near T,, considerably lower power and/orhigher gain of the probe beam can be achieved. By usingthis high-gain wave mixing effect, processes such as imageamplification, ring oscillation, and self-pumped phase con-jugation are clearly feasible, and they are currently understudy.

ACKNOWLEDGMENT

This research is supported by National Science Foundationgrant ECS8415387.

REFERENCES

1. See, for example, R. J. F. Reintjes, Nonlinear Optical Paramet-ric Processes in Liquid and Gases (Academic, New York, 1984).

2. R. Y. Chiao, P. L. Kelley, and E. Garmire, Phys. Rev. Lett. 17,1158 (1966).

3. J. P. Huignard, H. Rajbenbach, Ph. Refr6gier, and L. Solymar,Opt. Eng. 24, 586 (1985), and references therein.

4. S. J. Stepanov, V. V. Kulikov, and M. P. Petrov, Opt. Commun.44, 19 (1982).

5. V. L. Vinetskii and N. V. Kukhtarev, Sov. J. Quantum Electron.8, 231 (1978).

6. P. Yeh, J. Opt. Soc. Am. B 3, 747 (1986).7. S. Chang and T. Sato, Appl. Opt. 25, 1635 (1986).8. Y. Silberg and I. Bar-Joseph, J. Opt. Soc. Am. B 1, 622 (1984).9. E. S. Bliss, D. R. Speck, J. F. Holzrichter, J. H. Erkkila, and A. J.

Glass, Appl. Phys. Lett. 25, 448 (1974).10. I. C. Khoo and S. L. Zhuang, Appl. Phys. Lett. 37, 3 (1980); I. C.

Khoo and T. H. Liu, IEEE J. Quantum Electron. 23, 171 (1987).11. I. C. Khoo and S. Shepard, J. Appl. Phys. 54, 5491 (1983).12. See, for example, H. Hoffman, J. Opt. Soc. Am B 3, 253 (1986),

and references therein.13. I. C. Khoo and R. Normandin, IEEE J. Quantum Electron. QE-

21, 329 (1985); see also Ref. 11.14. J. G. Pasko, J. Tracy, and W. Elser, Proc. Soc. Photo-Opt.

Instrum. Eng. 202, 82 (1979).

Khoo et al.