1972 J. Opt. Soc. Am. B/Vol. 11, No. 10/October 1994
Dynamics of multiple two-wave mixingand fanning in photorefractive materials
Michael Snowbell, Moshe Horowitz, and Baruch Fischer
Department of Electrical Engineering, Advanced Opto-Electronics Research Center,Technion -Israel Institute of Technology, Haifa 32000, Israel
Received May 4, 1993; revised manuscript received December 13, 1993
We investigate the dynamics of multiple two-wave mixing in photorefractive materials. This is the first timeto our knowledge that a study of the dynamics of many pairs of two-wave mixing interactions has taken place.A basic case with three waves undergoing multiple two-wave mixing is studied numerically. A numericalstudy is also performed in which many waves are used to model the time development of the fanning effect.Comparison of this numerical model with a basic experiment involving fanning in the presence of a two-wavemixing interaction is given. A qualitative agreement is exhibited. In addition, it is found that effectivelyreducing the coupling constant of a two-wave mixing interaction by the addition of an incoherent backgroundbeam can, under certain circumstances, significantly reduce the fanning without lowering the signal level.
1. INTRODUCTION
Two-wave mixing has been analyzed extensively in thesteady state.' 2 Studies were also been performed to ana-lyze the dynamic behavior of two-wave mixing, which isof particular importance in applications for which thetime response is of relevance, such as novelty filters.3'4
Most of these studies-1 developed analytical solutionsin the undepleted-pump approximation (where the lossof intensity experienced by the pump is negligible com-pared with the pump intensity itself). However, in mostpractical situations there is significant depletion of thepump. Horowitz et al. 1""12 developed analytical solutionsand a numerical solution for the general case of two-wavemixing that explained many of the observed properties oftwo-wave mixing.
Two-wave mixing applications are inevitably affectedby problems of fanning. The fanning effect is a processin photorefractive crystals by which a single beam oflight generates an asymmetrical fanned-out profile ofscattering.13
1'7 For a BaTiO3 crystal, and when holes
are the charge carriers being excited, scattered beamsfrom an extraordinarily polarized source beam of lightare amplified if they are directed closer to the c axisfrom both sides than the source beam itself. This isillustrated schematically in Fig. 1. In a steady-stateanalysis it was shown'7 that this fanned-out profile oflight is due to noise derived from broad-angled scat-tering from impurities or defects in the crystal, whichare then amplified in two-wave mixing processes. Thisanalysis was based on the propagation of a sum of manywaves in the crystal, where there is a two-wave couplingmechanism between each pair of beams. The analy-sis was performed in the steady state and also includedan analysis of the effect of fanning on the two-wave mix-ing process. Another steady-state analysis of fanning inthe presence of a two-wave mixing process was performedby Ewbank et al.' 8 This analysis phenomenologicallyincluded fanning as a single beam that coupled equally
with both the signal and the pump beams. A similar ap-proach was taken to study the dynamics of a self-pumpedring mirror with fanning.' 9
In the present investigation we extend the dynamictwo-wave mixing analysis of Refs. 11 and 12 to a dynamicmultiple two-wave mixing analysis. This is to the best ofour knowledge the first time that the dynamics of multipletwo-wave mixing has been analyzed. We first examine asingle case of dynamic multiwave mixing in the most ba-sic three-wave case. We further show how, by furtherincreasing the number of waves in this analysis to modelthe scattering profile, one can arrive at a qualitative ap-proximation of the time development of fanning.
2. MULTIPLE TWO-WAVEMIXING: EQUATIONS
The basic photorefractive two-wave mixing equations aregiven by
where the Ai are the electric-field amplitudes, 6i is theangle that each beam makes with the c axis (usually itis included in g, but not here, since the angle is differentfor each beam), a is an absorption term, and g(z, t) isdefined as3,11,20
= -i 2A exp(+iO),
where ni is the first harmonic variation in the indexof refraction caused by the photorefractive effect, A isthe wavelength of the beams, and So is the phase shiftbetween the index of refraction modulation and the inter-ference pattern written by the two beams. The differen-tial equation describing the time development of g can bederived as 3 "'1
Vol. 11, No. 10/October 1994/J. Opt. Soc. Am. B 1973
PR
A
z
Fig. 1. Schematic diagram of fanning for BaTiO3 with holes ascharge carriers. The dotted lines represent the fanning compo-nents. We have ignored fanning components directed above thec axis. These are amplified too (if their angle with the c axis issmaller than the angle of the input wave with the c axis), but lessstrongly than those denoted in the figure. PR, photorefractive.
ag(Z, t) Ai(z, t)A2*(z, t)at I
where yij is the coupling coefficient between the twowaves Ai and Aj and Io is the local average intensity.
If we consider a system of n mutually coherent wavesAi (each at an angle Oi to the c axis) in a photorefractivecrystal, then in general each of these waves will undergoa two-wave mixing interaction with each of the other n -
1 waves. We can thus generalize the two-wave mixingprocess of Eqs. (1) as the system of equations8,1""'7
(cos Oi) az = E gijAj - aAi, (3)
agij ALAJr- at + gj = - Yij I i j. (4)
Equation (3) includes n equations. Equation (4) includesn(n - 1) equations, but, noting that ij = - ji* so thatgij = -gji*, we can eliminate half of these. It shouldalso be noted that Io in Eq. (4) is the sum of all the in-tensities that are present in the same interaction areaas the beams involved in the writing of the grating gij;thus immediately we see that the presence of other wavesin the crystal reduces the grating amplitude. Unless allthe beams involved overlap in exactly the same location,Io must be calculated separately for each interaction sothat it includes only those beams that are physically inthe same location and also increase the total local inten-sity. The parameter and its dependence on the inten-sity is also not fully understood in the literature. It isconvenient to rely on the experimental determination ofr ro(I/Ic)-0 7, 2 2 ,22 where To and I are constants andI = Io is the total local intensity.
To solve these equations, we extend the numericalmethod of Ref. 11 used for the two-wave mixing case. Inthis method each of the variables Ai and gij are expandedin a Taylor series in the time domain about any specifictime to. This allows us to simplify this system of partial
equations in both time and position into a system of equa-tions depending only on position. We use only the firstthree terms in the expansion (assuming that the changein the variables with time is well behaved enough thatthe first three terms of the expansion are an accurate ap-proximation for a short enough change of time At), so wecan write
AiO(z, to + At) = Y Aik (z, to)(At)k,k=O
2gij0(z, to + At) = _ gijk(Z, to)(At)k.
k=o
(5)
(6)
To determine the other coefficients, we substitute Eqs. (5)and (6) into Eqs. (3) and (4). By collecting terms with thesame power of At and ignoring terms with powers of Athigher than (At)2 , we arrive at the following system ofequations that are dependent on position z alone (for atime t + At): gijo(z, to) and Ai,(z, to) are given from theformer time interval, and
MA~ = -1 ij - I j A A. *-1-1 * I*2gij2 (c =-ij 1 - (AioAjl + AiAjo
(Cos 0i) aA = E (gijo)Aj, + gijAjo) - aAij,
(7)
(8)
(9)
(cos 0i) a i = E (gijoAj2 + gij1Aj + gij2Ajo) - aAi2 -az joi
(10)
Once these equations are solved by standard numericaltechniques, then the Taylor coefficients can be used tocalculate Aik and gijk for time to + At. These values canthen be used for the next iteration.
In the remainder of this paper we use the same materialparameters for all calculations of yij for any particularcase of wave mixing. As was the case for all measure-ments in the laboratory, we used beams at the argon lineof 514.5 nm, extraordinarily polarized light, and a BaTiO3crystal with a length of 0.7 cm. The expression for yijcan be derived (with the assumptions that the waves havethe same frequency and that no external electric field isapplied, from standard expressions for the index of refrac-tion modulation and space-charge fields," 4'8 1"'1
7 and withthe assumption (valid for BaTiO3) that r4,2 dominates allthe electro-optic tensor elements),
ii = 2'7rr42n 04k 1T kg os( - 2)Yi leAe 1 + (kg! ko)2 CS0 2
X cos2(0 + 02).(1 + 02),w kgth 2 w sinu 2 g
where kg, the grating wave number, is given as
kg = ine si[: 21
(11)
ko = (NAe 2)
Here NA is the density of traps in the material, kB is theBoltzmann constant, T is the temperature (in kelvins) and
Snowbell et al.
1974 J. Opt. Soc. Am. B/Vol. 11, No. 10/October 1994
no and ne are the ordinary and the extraordinary indexesof refraction.
The relevant material parameters for the BaTiO3 crys-tal are r42 = 1640 x 10-12 V/m, n = 2.488, and ne =
2.424. The dc permittivity along the direction of the grat-ing is given by e = eo(ec cos2 ij + Ea sin2
35j), where38 j is the angle between the wave vector and the c axisand is given as ij = 90° - [(0 + 6j)/2]. For BaTiO3 ,e = 4300 and ec = 106. Room temperature is assumed,and we chose a typical doping concentration of NA =
2 x 1016 cm-3 , which of course may vary from crystal tocrystal. For the time constant we take I = 1 and mea-sure increments in units of so. The pump is always takento be of intensity I, = 1.
The wave-mixing analysis that we have used has beenone dimensional and does not take into account vari-ations in the beam profile in the transverse direction,essentially assuming plane waves of infinite extent.However, since finite waves overlap only for a finitelength, we must somehow take into account the width ofthe beams to calculate the interaction length. For sim-plicity we assume all the beams to be of the same widthw. The effective interaction length in the c-axis directionbetween any two pairs of waves is given by (see Fig. 2)
csl + 02)Co 2 )
linter =W *( - 62) (12)
The physical dimensions of the crystal add an additionallimit on the interaction length as well. Thus each pairof waves i, j has its own yij as given and its own interac-tion range (Zbegin, Zend), which depends on the interactionlength and the point of intersection of the two waves.
3. NUMERICAL SOLUTION FORTHREE WAVES
To illustrate some of the basic concepts involved in mul-tiple two-wave mixing, we consider the basic case of mul-tiple two-wave mixing with three waves. We used theexample illustrated in Fig. 3, with the three beams hav-ing incident intensities of I1(0) = 1, I2(0) = 1 x 10-2,and 13(0) = 1 X 10-4, with each beam having a width of0.5 mm, which corresponds to complete overlapping of allbeams in the crystal. This geometrical configuration re-sults in the following yijl pairs: 21
1 = 5.39, Y3l1 = 5.24,and Y321 = 3.87.
There are a number of factors that are involved in thetime dependence of each of these beams, and all thesefactors interact with one another. For example, it wasdemonstrated" that in two-wave mixing the buildup timeis longer for a lower signal-to-pump-ratio (SPR) signalthan for a high-SPR signal. Part of the reason for thisis that the higher depletion of the pump that occurs inhigher-SPR signals leaves less pump to be drawn from,and hence the pump quickly reaches the point at whichit has little energy to transfer to the signal. However,with a number of waves interacting with a particularpump at the same time, depletion of the pump will occurfrom many beams, which will affect the dynamics of thebuildup of all the beams. Hence the dynamics of multipletwo-wave mixing cannot be understood by simple isolation
of the various two-wave mixing interactions but must beseen as a combination of the various factors interactingwith one another.
We show in Figs. 4, 5, and 6 the time dependence ofbeams 1, 2, and 3, respectively. In this case beam 1(Fig. 6) loses energy to both beam 2 and beam 3 con-currently, and hence it experiences immediate depletion.At first beam 2 (Fig. 5) experiences a strong gain at theexpense of beam 1 and draws most of the energy frombeam 1, with little energy going into beam 3 despite thefact that 21 Y31. This is because beam 3 is initiallyweaker than beam 2, and the index of refraction gratingthat beam 3 initially writes with beam 1 is not as strongas that written by beam 2 with beam 1. In addition, thegrating between beams 2 and 3 is weak (their intensi-ties are weak compared with the background). However,after beam 2 reaches a peak intensity it becomes strongenough to write stronger gratings with beam 3 and be-gins to couple some of the energy it acquired from beam 1to beam 3. This strengthening of beam 3 allows it tothen establish a stronger grating with beam 1. Beam 3(Fig. 4) at first rises slowly. But then, as beam 2 grows,beam 3 will also begin to receive energy from beam 2 and
Fig. 2. Effective interaction length for two beams.
A PR
A 2
A 3
0 =8
02 =6 0
03 =4
Fig. 3. Configuration for dynamic multiwave mixing, threewaves. PR, photorefractive.
Snowbell et al.
I
Vol. 11, No. 10/October 1994/J. Opt. Soc. Am. B 1975
n A
0.4
0.35k
0.31
.>0.25
c 0.2
0.15k
0.1k
0.05
v1 2 4 6 ZS 1U 12 14 10 1 2Utime (units of TO)
Fig. 4. Time development of beam 3, multiple two-wave mixing,three-wave case.
a
._
0
0
0
0
0
0
0
0
0
I
.9 -
.8 ,
.7
.6-
.5-
..3
.2
.1-
(1 I 6_ - _. . ._U 2 4 6 1 IU 12 14 1 tI L2U
time (units of x)
Fig. 5. Time development of beam 2, multiple two-wave mixing,three-wave case.
I
0.9
0.8
0.7k
0.6
0.5k
0.4
0.3 .
0.2k
0.1A
"U) 2 4 6 8 10 12 14 16 18 20time (units ofTo)
Fig. 6. Time development of beam 1, multiple two-wave mixing,three-wave case.
finally reach steady state, having received over 40% ofthe total energy.
An additional understanding of this multiwave-mixingprocess can be acquired by examining the time develop-ment of the grating amplitudes. In Fig. 7 the three grat-ing amplitudes at the end of the crystal (z = 4,) are plottedas a function of time. From this plot we can see that thegrating written between beams 1 and 2 (1,2) increases
in amplitude from the turn-on time and then falls steeplyas beam 3 begins to gain in energy at the expense of bothbeam 1 and beam 2.
The grating written between beams 1 and 3 (g13) un-dergoes a more complex development. At first the grat-ing strengthens at a slow rate as beam 1 begins to coupleinto beam 3. At the same time most of beam 1's energy iscoupled into beam 2, thus limiting the coupling that beam1 can have with beam 3. At approximately t = 2 (in unitsof To) beam 1 becomes almost completely depleted by beam2 (refer to Figs. 5 and 6). This weakening of beam 1 thusalso weakens the grating that it can write with beam 3,and hence g 3 experiences a drop at approximately t = 2.At the same time beam 2, having been greatly strength-ened, begins to couple its energy into beam 3. Beam 3 isthus strengthened to the point that it almost equals beam2. This increase in size allows beam 3 to write strongergratings with beam 1, so that by the time steady stateis approached the strength of the grating between beam1 and beam 3 almost equals that of the grating betweenbeam 1 and beam 2. The grating written between beam 2and beam 3 (g23) grows monotonically, as there is no rivalprocess to weaken this grating: as beam 2 strengthens,it can then write a stronger grating with beam 3, whichthen itself grows, and a maximum grating amplitude isreached as the beam energies approach the steady state.Since beam 1 is almost completely depleted at z = 1, andbeams 2 and 3 are strongest at this point, beams 2 and 3write the strongest grating.
It should be realized that this plot of the grating ampli-tude (Fig. 7) is valid only for the crystal output. If onewere to record the gratings amplitudes at different placesin the crystal, the form would be different because as thevarious intensities change along the length of the crys-tal and hence the coupling strength between the beamsvaries along the length of the crystal. For example, inFig. 8 we plot the grating amplitudes at z = ,/2. At thispoint in space only half of the interaction region has beentraversed by the beams, and the full extent of the wave-mixing processes is not developed compared with whatoccurs at the end of the interaction region. Hence thepump beam is not so depleted at this point in space, andit couples strongly into beam 2, and hence gl,2 dominates.Also, the coupling between beams 2 and 3 has only be-
250F
200
00.5
00
150
50 /
U .I
0 2 4 6 8 10 12 14 16 18 20
time
Fig. 7. Time development of grating amplitudes at z = l1, mul-tiple two-wave mixing, three-wave case.
923
13 --
unto l
I 1 ' A . - - .- , .- ' .' A .- .^
3W,
Snowbell et a.
crZ
I
1976 J. Opt. Soc. Am. B/Vol. 11, No. 10/October 1994
-
co._S
tb
400
912
350
300
250
20 4
150
00 2 4 6 8 10 12 14 16 18 20
timeFig. 8. Time development of grating amplitudes at z = ,/2,multiple two-wave mixing, three-wave case.
gun to occur at this point in the crystal, and hence g23remains small.
The change in dynamics from two-wave mixing withtwo waves to that with three waves thus adds a great dealof complexity. Certainly, because the number of gratingsis given by (n2 - n)/2, this complexity increases manyfoldas the number of waves introduced further increases.
4. NUMERICAL SOLUTION FORTWO-WAVE MIXING WITH FANNINGAn accurate description of fanning would involve a largenumber of fanning elements to more fully approximatethe continuum of fanning elements. However, becauseof the number of equations involved (of the order of n2
owing to the gij terms), using a large number of fanningelements could become numerically prohibitive. Thus weused a smaller number of fanning elements for practi-cal purposes. We modeled one particular configuration,shown in Fig. 9. In this configuration a pump beam isshone into the crystal by itself at time t = 0. This beamis allowed to develop its own fanning, and then at timet = 10 (in units of rO) a signal beam is added to the sys-tem. Of interest here is how the fanning from the pumpbeam develops, how it is affected by the insertion of thesignal beam, and how the fanning from the signal beamthen develops.
We have modeled the fanning beams by taking 30beams that are (angularly) equally spaced between thepump beam and the c axis. (We neglected fanning abovethe c axis.) Since fanning beams are derived from thesource beam itself, we take the intensity of the fanningbeams to be proportional to the incident intensity of thepump beam itself with proportionality constant a. Sincethe fanning results from amplified scattering off of defectsin the crystal, we assume a Rayleigh-type scattering,' 7
so the angular profile of the fanning seed intensity isgiven as
If(z = 0, 0) = aIp COS2 (0 - p), (13)
where If (0, ) is the intensity of a particular fanning ele-ment associated with angle 0 with respect to the c axisevaluated at position z = 0 and 0, is the angle of the pump
beam (ofintensityIp) with respect to the c axis. As an ap-proximation we considered the fanning elements to havetheir center of interaction with the source beam in the cen-ter of the crystal. This would give the fanning elementsthe maximum interaction area within the crystal, whichwould best take into account the fact that fanning occursover the entire length of the crystal. We took a = 10-4.
It should be noted that if one uses more fanning beamsin the simulation then this factor can be reduced. Bytaking only 30 fanning beams, we are ignoring countlessother fanning beams, and hence we must compensate forthis approximation by using a higher fanning proportion-ality constant. All the beams, including the pump andsignal beams, were assumed to have the same beam widthw = 1 mm. The SPR was taken to be 1:10. For eachpair of beams yij was calculated [Eq. (11)] as well as theinteraction range for each pair of waves [Eq. (12)]. Theyij were then scaled by a constant factor so that yl be-tween the pump and the signal beams would be 5.1, whichis in the range found experimentally in BaTiO3 crystals.
The results from the simulation are shown in the fig-ures below. Figure 10 shows the time development of thepump. From time t = 0 to t = 10 we can see that thepump wave is strongly depleted by the fanning beams.At time t = 10, when the signal beam is introduced, thepump is further depleted by the signal, which is much
AI
A 2
PRcrystal 01 = 11.5
02 = 8.0
<Of< 11.50
z
Fig. 9. Configuration used in model of dynamic two-wave mix-ing with fanning. PR, photorefractive.
0.9
0.8 -
0.7-
> 0.6 -
c 0.5 -' 0.4 -
0.3-
0.2-
0.1
0) 4 6 8 l0 12 14 16 1'820
time
Fig. 10. Time development of pump beam with fanning, signalon at t = 10.
Snowbell et al.
Vol. 11, No. 10/October 1994/J. Opt. Soc. Am. B 1977
Z'C
0.9
0.8
0.7
0.6
0.5
0.4-
0.3
0.2
0.1
0~~~~~~~~ 1 U L 4 0 8 IU
time
IL I4 10 18 LU
Fig. 11. Time development of signal beam, signal on at t = 10.
stronger than any of the fanning elements. Figure 11shows the time development of the signal beam. Afterbeing turned on at time t = 10, the signal experiencesa quick gain. This gain is mostly at the expense of thefanning, as most of the energy was already depleted fromthe pump to the fanning beams and now is redirected fromthe pump to the signal beam. However, after reaching apeak, the signal begins to trail off as it experiences itsown fanning. Rather than showing the time-dependentbehavior of all the fanning elements individually, we plotonly two samples, one located between the pump and thesignal at an angle of 9.3° and one located between thesignal and the c axis at an angle of 5.2°.
It can be seen from Fig. 12 that the fanning elementat an angle of 9.3° receives a strong gain from the pumpbeam and then begins lose its energy, which it passes onto other fanning elements into which it couples. It thenreaches an essentially steady-state value. When the sig-nal is turned on, this fanning element is strongly depleted.Part of this effect is due to the stronger coupling of thepump to the signal, which draws the pump energy awayfrom the fanning wave. In addition, since this fanningwave has a positive coupling constant to the signal beam,it also couples its own energy into the signal, thus effec-tively lowering its intensity to almost zero. The fanningelement at angle 5.2°, whose time dependence is shown inFig. 13, begins by receiving energy from the pump beamand also from other fanning waves situated between itand the pump beam. Since the beam at 5.20 is closerto the c axis, the coupling constants between it and theother beams even closer to the c axis are relatively weak{because of the sin[(61 + 2)/2] term in the definition of yjin Eq. (11)} and hence this beam is not significantly de-pleted into other fanning beams as was the beam at 9.3°.When the signal beam is turned on at t = 10, the fanningwave at 5.2° begins to lose energy as the coupling fromthe pump to the fanning elements is overtaken by thecoupling from the pump to the signal. However, the sig-nal beam itself experiences fanning, and since its fanningis directed toward the c axis, the fanning element at 5.20will receive energy from the signal. It should be notedthat the fanning that the signal experiences is greaterthan it would if it were in the crystal by itself (withoutthe pump). The pump establishes stronger fanning ele-ments than the signal alone would be able to because the
pump is stronger than the signal to begin with; the sig-nal can then couple into these more-established fanningelements and lose a larger share of its energy in that way.
To picture the angular profile of the fanning dynami-cally, Fig. 14 shows the plot of the angular profile of thefanning as a function of time. We point out salient fea-tures of this plot by isolating certain time cross sections inFigs. 15 and 16. Figure 15 plots the angular spectrum ofthe fanning at a number of times before the signal entersthe system at t = 10. This plot shows that as the timeadvances the fanning intensity increases until it reachesa steady value. In addition the center of mass of thefanning profile shifts toward the c axis. This can be ex-plained in light of the beginning part of the time depen-dence of the fanning element at 9.3°, shown in Fig. 15.Certain fanning elements immediately begin to receivea strong gain from the pump owing to large couplingconstants that the pump has with them. Once thesefanning elements grow, they then couple to neighboringfanning elements toward which they have positive cou-pling constants. This is thus a gradual process of trans-ferring beam energies toward the c axis. This transferdoes not continue all the way to the c axis, as the fanningelements closer to the c axis have weaker coupling con-stants, and the finite crystal length does not allow enoughenergy transfer to occur to these elements. Rather, asteady-state profile develops.
timeFig. 12. Time development of fanning beam at 0 = 9.3.
timeFig. 13. Time development of fanning beam at 0 = 5.2°.
Snowbell et al.
1978 J. Opt. Soc. Am. B/Vol. 11, No. 10/October 1994
0a
Fig. 14. Time development of angular profile of fanning beam.
._
I Ic axis pump
angle (deg)
Fig. 15. Angular profile of fanning beam at times before thesignal is turned on.
2::.OC.
pumpangle (deg)
Fig. 16. Angular profilesignal is turned on.
of fanning beam at times after the
Figure 16 shows the plot of the angular spectrum offanning after t = 10 when the signal enters the crystalat 8°. It can be seen that the introduction of the signalbeam causes a fast immediate depletion of the overallfanning as the pump establishes a stronger grating withthe strong signal. In addition, its profile is shifted. Thefanning elements between the signal and the pump are
depleted strongly compared with the fanning elementsbetween the signal and the c axis. This occurs because,not only does the signal draw energy from the pump awayfrom fanning elements between the signal and the pump,but also these fanning elements couple their own energyinto the signal. However, the fanning elements betweenthe signal and the c axis are positioned to receive energyfrom the signal. Hence, when the signal is turned on,they are not completely depleted. Also, as the signalbeam grows, it begins to couple more of its own energyinto these fanning elements, and hence the fanning grows(with a similar shift in its profile) again, although not toits previous value.
The main drawback of this simulation is the number ofwaves used to represent the fanning. Fanning is essen-tially the combination of thousands of fanning elements.Individually, each of them is small, but the combination ofall of them can yield significant depletion of source beams.In our model, in which only 30 beams are used to modelthe fanning, we had to resort to increasing the magnitudeof each of the elements manyfold in order that the 30 mod-eled fanning beams could combine to give the strengthof thousands of actual fanning waves. This changes thedynamics of the system somewhat, since the developmentof a single strong beam is not the same as the sum ofmany weaker beams. Thus, until we extend the numeri-cal model to include many more fanning waves, we expectexperimental results to differ from the numerical simula-tion. Nonetheless, the simulation can give a qualitativefeel of the dynamics of fanning.
5. EXPERIMENTAL RESULTS
To verify the qualitative agreement of experiment withthe numerical simulations, an argon laser at 514.5 nmwith an 6talon was used as the source of both the pumpbeam and the signal beam, with the experimental con-figuration being the same as in Fig. 9. The time de-velopment of the pump beam and the signal beam wasmeasured, and the angular profile of the fanning was mea-sured with a CCD camera. A line profile of the fanningon the CCD was used to measure the fanning intensityand thus does not represent the entire fanning intensitybut corresponds to the line alone. This fits our numeri-cal analysis. The signal- and the pump-beam intensitiesare removed from the plot of the fanning intensity so thatthe fanning profile alone can be displayed. (Usually thesignal and the pump beam saturated the CCD detectors).
The time development of the pump beam is shown inFig. 17. The pump experienced a depletion that is dueto its fanning and then a further depletion that is dueto the presence of the signal turned on at t = 57 s. Thedepletion of the pump beam that is due to the fanningis weaker than in the numerical solution. This may bebecause of an overestimation of the fanning-beam intensi-ties in the model or of the yI for the fanning waves in themodel. Another possible factor is a dc internal field inthe crystal. " 2 The depletion of the pump by the signalis also weaker than in the simulation. The time devel-opment of the signal turned on at t = 57 s is shown inFig. 18. The depletion of the signal that is due to signalfanning is similar to that in the numerical simulation.
The time development of the fanning profile is shown
Snowbell et al.
Vol. 11, No. 10/October 1994/J. Opt. Soc. Am. B 1979
1.2
i9 1 -
xc 0.8 C.
.3 0.60.4
0.2-
°0 20 40 60 80 100 120
time (sec)
Fig. 17. Experimental time dependence of pump beam; signalon at t = 57 s.
fanning directly from the signal. This profile indicatesthat the fanning that results after the signal is added isdue to separate fanning from the signal, and little is fromthe pump directly. In the numerical model, however, weused strong fanning elements for the pump beam; the sig-nal beam did not have any independent fanning elementsof its own, but was free to interact with the fanning beamsfrom the signal. Since in the numerical model the indi-vidual fanning beams are stronger than in reality, the sig-nal was able only partially to reduce the transfer of energyfrom the pump to the fanning beams and then shift thisfanning profile somewhat toward the c axis. The pro-file that resulted in the model was thus a combinationof fanning from the pump and the signal and was notmoved toward the c axis as much as in reality. This con-trasts with the experiment, in which the fanning profilethat builds up is almost exclusively derived from the sig-nal. In both the numerical and the experimental resultsa strong reduction of fanning situated between the pumpand the signal beams is observed.
C.
time (sec)
dependence of signal beam; signal
in Figs. 19 (before the signal beam is added) and 20 (afterthe signal is added). In these plots we see that, althoughthere is some qualitative agreement with the numericalresults, there are some important differences. In Fig. 19the profile is shown to grow with time as expected, butthe shift toward the c axis that the fanning profile experi-ences is less than in the numerical simulations. This isbecause of the aforementioned limitation of the numberand size of fanning elements in the simulation. Since itwas necessary to increase the size of each fanning elementto compensate for the smaller number of fanning waves inthe simulation, the interaction between these artificiallystrengthened fanning beams could be stronger in thesimulation. This allowed a transfer of energy towardsthe c axis. In the experiment, however, the fanningbeams are much weaker, and the interaction between thepump and the fanning beams dominates the interactionsamong the fanning waves themselves, and hence less of ashift toward the c axis occurs.
In Fig. 20 the differences between the modeled and ex-perimental behavior are more pronounced. In the experi-mental behavior it is clear that the fanning peak thatcorresponds to the fanning of the pump is almost com-pletely reduced to zero soon after the signal is added,and then a separate peak develops, corresponding to the
140
120
100
80
60
40
200I
cx0I.
c-axis
2 4 6 8 10 121
pumpangle inside crystal (deg)
Fig. 19. Fanning profile at various times before the addition ofthe signal beam. (Units of intensity are that of the CCD andare arbitrary.)
Isignal
angle inside crystal (deg)
Fig. 20. Fanning profile at various times before the additionof signal beam. t = 0 corresponds to the addition of the sig-nal beam. (Units of intensity are that of the CCD and arearbitrary.)
E
._
Pc5
Fig. 18. Experimental timeon at t = 57 s.
t=0s
> ~~~~~~~~~~~t=209's
Snowbell et al.
1980 J. Opt. Soc. Am. B/Vol. 11, No. 10/October 1994
Fanning in a two-wave mixing application adds an ele-ment of noise to the system, and hence it is desirableto reduce the fanning intensity somehow without signifi-cantly changing the desired signal gain, i.e., to increasethe signal-to-noise ratio. Rajbenbach et al.
2 3 determinedthat rotating the crystal gave the desired effect. This isunderstood in terms of the rise-time dependence on thesignal-to-pump ratio." If we consider the buildup timefor a low-SPR signal to be t and that of a high-SPR sig-nal to be th, then th << t If we rotate the crystal ata period t that lies between these two values, then thedesired effect takes place. For a low-SPR signal turningthe crystal is too fast for the grating to establish itself be-cause of the large rise time of low-SPR signals. However,for a high-SPR signal, whose rise time is much smaller,this movement of the crystal will not be fast enough tocompletely disturb the grating formation. Since fanningis a result of scattering off of impurities, it is in the realmof small SPR signals and hence will have a long builduptime. The actual signal that enters the crystal to be am-plified will be of a much higher SPR than the fanningelements and hence will have a faster buildup time. Inthis way rotating the crystal at an appropriate rate caneffectively increase the signal-to-noise ratio. While thismethod is effective, it does require moving parts, however,which might not be desirable in some applications.
We propose here another method of effectively decreas-ing fanning without significantly reducing the signal,which is mechanically simpler but which works only whenthe signal is strong enough to induce strong depletion ofthe pump beam. This method involves effectively chang-ing the coupling constant yl by adding another incoherentbackground beam into the interaction region. While thismethod is similar to using a weaker crystal, it also al-lows one to continuously tune 7l from a maximum value.Similar ideas have been recently presented by Bogodaevet al.19
For a basic understanding we consider the steady-stateexpression for the signal intensity which is given as17
I2(z) = 12(0)exp(-2az) I 1(0)exp(-z) + ( (14)
where I2(z) is the signal intensity at position z, I,(z) isthe pump intensity at position z, z = 0 corresponds tothe input face of the crystal, and r = 2 Re(y). Usingthis expression, we evaluate the sensitivity of the outputsignal intensity [I2(1)] to changes in yl (for real yl andwith a = 0) as
SI2(V -_ a ln[12(1)] _ aI2(l) yla ln(yl) a (yl) 2(l)
2I1(0)exp(-2yl)yl . (15)Ii(0)exp(-2y1) + I2(0)
We consider two extreme cases. In the first I2(0) <<I,(0)exp(-2y1). Physically this is the case in which thesignal beam is small and in which, despite the signalgain, the pump remains essentially nondepleted. In thiscase the sensitivity to changes in yl is given by 2y1.On the opposite extreme, I2(0) >> I,(0)exp(-2yz). This
corresponds to the case in which the signal depletes thepump completely, and in this case the sensitivity is
271 I,(0)exp(-2y1)I2(0)
where the small fractional factor markedly reduces thesensitivity. Physically this means that, when a signalstrongly saturates a pump beam, then a small decreasein y will not significantly affect the pump saturation.Thus, for example, we consider a decrease of couplingfrom yl = 4 to yl = 2. If we have an incident pumpintensity of I,(0) = 1, then for an input signal of intensity12(0) = 10-6 the output signal intensity I2(1) will decreaseby more than 54 times. However, if the input signalis of intensity I2(0) = 10-1, then the change is muchsmaller-only a 17% decrease. Thus fanning elementswill decrease much more significantly with a decrease inyi then will strong signals. In this example, where thesignal is 105 times as large as the fanning and is 1/10times as large as the pump, then there will be an increasein the signal-to-noise ratio by more than 45 times. Thusan operation that reduces the coupling constant actuallyimproves the signal-to-noise ratio. However, it should benoted that this method is limited to cases in which theinput signal intensity is high enough to cause a strongdepletion of the signal.
Changing yl is typically done by changing the orienta-tion of the beams with respect to one another and withrespect to the crystal axis, essentially changing the ef-fective electro-optic tensor reff. (Other angle-dependentparameters change at the same time.) While this is pos-sible, it requires the reconfiguration of the system, whichis not always desirable. Although this method of con-trolling yl allows for easy control over the yl between thesignal and the pump, the y1's involved in fanning are notnecessarily changed by the same proportions. By exam-ining the equation for the grating amplitude [Eq. (2)], wesee that in steady state g is proportional to y/Io and thatthis dependence of g on y is the only term including y inthe two-wave mixing equations. Thus increasing Io (bya laser beam which is mutually incoherent to the mixingbeams) while keeping y constant is equivalent to reducingy while keeping Io constant. Therefore we add anotherterm to the total intensity so that
IO = I + 2 + Ib, (16)
where Ib is the background intensity. Of course, if thisbackground intensity beam is in any way mutually co-herent with either the pump or the signal, then it willalso couple with both waves and change the entire dy-namics. To avoid this the background intensity shouldbe completely incoherent with the interacting beams, andit could be of a different wavelength. This introductionof a background beam is much easier and more practicalthan reconfiguring the system, as the background sourceis only to be turned on and off without any realignment.In addition changing the effective y by this method canbe performed in a continuous manner by simply varyingthe intensity of the background beam. Thus a systemusing this method of effective y reduction should be con-figured once for the maximum possible y, and then thebackground intensity could be added to lower y as desired.
Snowbell et al.
Vol. 11, No. 10/October 1994/J. Opt. Soc. Am. B 1981
Aside from its ease of configuration, this effective y re-duction method is an improvement in terms of the gratingdynamics as well. Changing the effective y by adding abackground intensity allows the interacting beams to re-main in the same location, which leaves the refractive-index grating in place, only varying its amplitude. Onthe other hand, y adjustment by realignment would intro-duce an entirely new grating to be built up in the mate-rial, while the old grating would still exist. Thus steadystate can be reached much more easily by this method.Another important advantage of this method is that ifthere is more than one wave-mixing interaction going onin the crystal (including fanning processes) then the in-troduction of the background intensity reduces y for allthe interactions by the same amount (as long as E Ii andIb remain constant).
Thus for fanning reduction in two-wave mixing appli-cations this method of effective y reduction can be highlyuseful if the signal causes a significant amount of pumpdepletion (to the point of saturation) but the fanning isalso strong. A background intensity is shone into thecrystal so that y is effectively reduced. If this is doneby the right amount, then the signal will still depletethe pump appreciably and will not lose most of its gain.However, this same reduction will cause the weakertwo-wave mixing interactions (i.e., the fanning) to becomesignificantly weakened. This weakening of the fanningcan also permit more of the pump energy to be transferredto the signal beam, thus giving the signal the possibilityof more gain. It should be noted that this method of uni-form y reduction is identical to using a crystal with aweaker electro-optic constant, which seems to be counter-productive. However, as is apparent from this section,for some applications a weaker crystal could give approxi-mately the same gain to stronger signals while having sig-nificantly less fanning. Thus using a strong crystal andadding a variable background intensity would allow one tooptimize the parameters for the best signal-to-noise ratiofor a particular application.
This basic type of noise reduction has been used in adifferent sense to increase the double-phase-conjugate re-flectivity in certain passive self-pumped phase-conjugatemirrors. James and Eason2 4 reported that in certaingeometries adding an ordinarily polarized component oflight to the input beam actually improved the phase-conjugate reflectivity in BaTiO3 . They attributed this toa reduction of parasitic gratings caused by the presenceof the ordinarily polarized light. This simple analysis ex-plains their result. The ordinarily polarized componentis not involved in the primary wave-mixing processes be-cause of the much lower electro-optic tensor coefficientsin BaTiO3 for ordinarily polarized light. Thus the in-troduction of this component is equivalent to adding anincoherent background intensity. Thus this backgroundintensity, the amount of which was controlled by chang-ing the polarization of the input beam, was able to re-duce y in the crystal, thereby strongly reducing weakerparasitic gratings in the phase-conjugate mirror, and atthe same time it did not significantly reduce the desiredwave-mixing effects.
The effects of a background incoherent beam in the nu-merical model is shown in Fig. 21, in which we plot thetime development of the signal when different background
intensities are added. (The other parameters remain thesame as in Section 4.) It can be seen that the introduc-tion of the background beam in this model reduces thefall-off that the signal undergoes because of fanning, thusimproving the signal-to-noise ratio. Also, the signal in-tensity actually reaches a higher peak value than in theno-background case despite the effective reduction in y.
The configuration of Fig. 9 was set up in the laboratorywith an argon-ion laser source at 514 nm, with a signal-to-pump ratio of approximately 1:10. An mutually inco-herent beam of ordinary polarization was added into theinteraction region. (Since the effective electro-optic ten-sor is much smaller for ordinary beams in BaTiO3, thisbeam would experience minimum fanning and its inten-sity profile in the crystal would remain constant.) Thewidth of this beam was chosen to be wider than the wave-mixing beams to ensure that the background intensitywas present in the entire interaction region. Figure 22is a plot of the time dependence of the signal for vari-ous background intensities. (The background intensitieshave been scaled to account for the beam width differ-ences. The beam widths are difficult to measure, andthe scaling factor is only approximate.)
The experiment demonstrates the same effect andqualitatively the same behavior as the numerical sim-ulation. Without a background intensity, the signalbeam experiences a drop of almost half owing to the
1.2
1 Ib=0.5
0,8 b025
Q 0.6 b0
0.4
0.2
4 1 141 18 0
time
Fig. 21. Time development of signal beam with added back-ground intensities (b): numerical simulation.
8
= 3~~~~~~~~~l=5
7 - .. .......... . . .. ... ......M~~~~~~
i~ 2-
0( ) 0 2 0:06 0 8 i 1L2 14 1.6 18 2
time (sec)
Fig. 22. Time development of signal beam with added back-ground intensities (b): experimental results.
Snowbell et al.
1982 J. Opt. Soc. Am. B/Vol. 11, No. 10/October 1994
fanning that it develops. As the background intensityis increased, this drop becomes smaller, until at a par-ticular value of the background intensity (6.0 mW in thefigure) the drop is not discernible. It is also observedwith the eye that the fanning profile is weakened almostcompletely. If the intensity is increased much beyondthat amount (16.5 mW in the figure), then the effective yreduction is high enough to affect the signal buildup aswell, so that the intensity of the signal does not reach thesame peak value. The experimental results, however,do not show the actual increase in peak intensity withthe addition of the background noise as predicted in thenumerical simulation. As explained above, because thesmall fanning waves in reality are not 30 but thousandsin number and because the value of a (the fanning pro-portionality constant) was chosen arbitrarily to be 104,quantitative differences in the modeled behavior and thephysical behavior are to be expected. Nonetheless, aqualitative agreement is seen in the findings presentedhere. A model that includes many more fanning ele-ments is expected to be in better quantitative agreementwith experimental findings.
7. SUMMARY
In this paper we have presented a numerical techniquefor the modeling of the dynamic behavior of multiple two-wave mixing. We have presented the numerical resultsin two situations. In the first we have presented the so-lution of a basic three-wave case and analyzed the time-dependent behavior of all three waves and of the gratingamplitude. In the second we used a much larger num-ber of waves that we took to model the fanning effect.We studied how fanning develops from a pump beam andhow the introduction of a signal beam affects the fanningprofile as a function of time. Experimental results wereshown to be in qualitative agreement, but some featuresdiffered. This is attributed to the limited number of fan-ning beams that could be used in the numerical model,which necessitated an increase in the intensity of eachof the fanning beams. This approximation in the modelchanged the dynamics of the system somewhat, which ac-counts for much of the observed differences between thenumerical simulation and the experimental results.
This is the first time to our knowledge that dynamicmultiple two-wave mixing has been analyzed. Thisanalysis gives a greater understanding of the time de-velopment of the fanning effect. As a possible extensionof this research one could model the fanning effect infour-wave mixing processes. This could be of particu-lar importance for self-pumped phase-conjugate mirrors,where the oscillation is built up from fanning elements.This type of modeling would allow one to get a better un-derstanding of the development of the phase conjugates,since the grating responsible for the phase conjugate hasto overcome the other fanning gratings to become thedominant grating.
We have also proposed and demonstrated experimen-tally a method for an improvement of the signal-to-noiseratio by inserting an incoherent background beam. This
background beam, which reduces the coupling constantyl, is shown to affect weaker signal beams much morethan ones that are strong enough to deplete the pumpstrongly. Thus fanning beams, which consist of a contin-uum of amplified beams that were originally weak, will bereduced much more strongly than will a stronger signalbeam. The background intensity can be varied continu-ously to achieve an optimal signal-to-noise ratio.
ACKNOWLEDGMENTS
We thank the American Technion Society, New YorkMetro region, and especially Norman Seiden, Mel Dubin,and Steve Shapiro for the support of our research. Thiswork is a part of the M.S. thesis of Michael Snowbell atthe Electrical Engineering Department, Technion-IsraelInstitute of Technology, Haifa, Israel, submitted in De-cember 1992.
REFERENCES
1. N. V. Kukhtarev, V. Markov, S. Odoulov, M. Soskin, and V.Vinetski, Ferroelectrics, 22, 946, 961 (1979).
2. D. L. Staebler and J. J. Amodei, J. Appl. Phys. 43, 1042(1972).
3. D. Z. Anderson and J. Feinberg, IEEE J. Quantum Electron.25, 635 (1989).
4. M. Cronin-Golomb, A. M. Biernacki, C. Lin, and H. Kong,Opt. Lett. 12, 1029 (1987).
5. L. Solymar and J. M. Heaton, Opt. Commun. 51,76 (1984).6. J. M. Heaton and L. Solymar, Opt. Acta 32, 397 (1985).7. J. Goltz, F. Laeri, and T. Tshudi, Optik 71, 163 (1985).8. M. Cronin-Golomb, in Photorefractive Materials, Effects, and
Devices, Vol. 17 of 1987 OSA Technical Digest Series (Opti-cal Society of America, Washington, D.C., 1987), p. 142.
9. J. Goltz and T. Tshudi, Opt. Commun. 67, 414 (1988).10. D. R. Erbschloe and T. Wilson, Opt. Commun. 72, 135 (1989).11. M. Horowitz, D. Kligler, and B. Fischer, J. Opt. Soc. Am.
B 8, 2204 (1991); M. Horowitz, R. Daisy, and B. Fischer,J. Opt. Soc. Am. B 9, 1685 (1992).
12. M. Horowitz, "Filtering behavior of wave mixing in photore-fractive media," M.S. thesis (Technion-Israel Institute ofTechnology, Haifa, Israel, 1991).
13. V. V. Voronov, I. R. Dorosh, Y. S. Kuzffiinov, and N. V.Tkachenko, Sov. J. Quantum Electron. 10, 1346 (1981).
14. J. Feinberg, J. Opt. Soc. Am. 72, 46 (1982).15. 0. V. Kandikova, V. V. Lemanov, and B. V. Sukharev, Sov.
Phys. Solid State 28, 424 (1986).16. G. Zhang, Q. X. Li, P. P. Ho, Z. K. Wu, and R. R. Alfano,
Appl. Opt. 25, 2955 (1986).17. M. Segev, Y. Ophir, and B. Fischer, Opt. Commun. 77, 265
(1990).18. M. D. Ewbank, F. R. Vachss, and R. A. Vazquez, in Photo-
refractive Materials, Effects and Devices, Vol. 14 of 1991OSA Technical Digest Series (Optical Society of America,Washington, D.C., 1991), p. 43.
19. N. V. Bogodaev, A. A. Zuzulya, L. I. Ivleva, A. S. Korshunov,A. V. Mamaev, and N. M. Polozkov, Sov. J. Quantum Elec-tron. 22, 410 (1992).
20. J. 0. White, S.-K. Kwong, M. Cronin-Golomb, B. Fischer, andA. Yariv, in Photorefractive Materials and their ApplicationsII, P. Guinter and J.-P. Huignard, eds. (Springer-Verlag,Berlin, 1989), p. 101.
21. S. Ducharme and J. Feinberg, J. Appl. Phys. 56, 838 (1984).22. D. Mahgerefteh and J. Feinberg, Opt. Lett. 13,111 (1988).23. H. Rajbenbach, A. Delboulb6, and J.-P. Huignard, Opt. Lett.
14, 1275 (1989).24. S. W. James and R. W. Eason, Opt. Lett. 16,633 (1991).
