Light waves at the interface of linear andphotorefractive media
Ron Daisy and Baruch Fischer
Department of Electrical Engineering, Advanced Opto Electronics Center, Technion-Israel Institute of Technology, Haifa 32000, Israel
Received March 30, 1993; revised manuscript received August 16, 1993
We study the behavior of light waves at an interface between linear and photorefractive media, where theincident wave propagates from the linear side. We find bistable behavior of the reflection coefficient as afunction of the incident angle at incident wave directions in the vicinity of the critical angle for total internalreflection. The analysis is based on approximations in the photorefractive-process equations, for which ananalytical plane-wave solution for the boundary problem is found. In this regime an exponentially decayingevanescent field induces a uniform change of the refractive index. The limitation of this solution is alsostudied by a numerical analysis.
The large optical nonlinearities of photorefractive ma-terials have motivated much research and a variety ofapplications in past years. Most of these studies usedbulk photorefractive crystals. However, the obviousattractiveness of nonbulky materials produced severalrecent studies of photorefractive waveguides,1' 2 fibers,3 4
and surfaces.5 In a recent investigation 5 we studiedthe reflection properties of nonlinear photorefractive andlinear interfaces. It was shown that the nonlinear two-wave mixing of the incident and the reflected waves cancause bistability. Previously, Kaplan 6 first presentedstudies of an interface between linear and nonlinearKerr media in which the incident wave propagates fromthe linear side (opposite to the direction used in Ref. 5).He also found bistability for the reflection coefficient as afunction of the incident intensity or angle. In his case thefeedback mechanism, which is necessary for bistability, isprovided by index-of-refraction perturbations induced bythe transmitted wave. Experiments carried out later 7 8
showed quasi-bistable behavior.In this paper we study the configuration in which
the incident wave propagates from the linear side, butthe Kerr nonlinearity is replaced by the photorefractiveeffect. We then have a new interface problem, since thephotorefractive effect is nonlocal and depends on light-intensity variations in the material and not on the localfield and its intensity. We present a simplified solutionfor the induced index-of-refraction change and for theinvolved light waves at the boundary and obtain a simplesolution for the reflectivity, which is found to possessbistability.
A scheme of the linear-photorefractive interface withthe relevant waves is given in Fig. 1. We first give aqualitative description of the processes. Let us assumethat the refractive index of the linear material is slightlygreater than the photorefractive one. When an incidentplane wave propagates with an angle larger than the criti-cal angle, a total internal reflection state is not reached,and the transmitted wave will also be a plane wave. Inthis situation there are no intensity variations in thephotorefractive material. Hence it responds in a linear
manner, and the solution for the boundary problem isknown. However, when the incident angle is smallerthan the critical angle, an evanescent wave is formed onthe photorefractive side, and its intensity variations leadto index-of-refraction changes in the photorefractive mate-rial. This provides a feedback mechanism that can causebistability, as in the linear-Kerr interface. In the steadystate the photorefractive effect is almost insensitive to thetotal intensity in the material. Therefore the reflectioncoefficient does not depend on the incident intensity as inthe Kerr case.
Before starting the analysis on the photorefractive sideit is important to note that we assume in this paperthat the two media that form the interface differ only inrefractive-index properties. There is no interface, how-ever, for the charge carriers that produce the space-chargefield, and they can pass freely through the interface; i.e.,the interface is optical and not electronic. This situationcan be obtained experimentally, for example, by use of theinterface of two domains in the same ferroelectric crystal.The other situation, with a boundary for the carriers, andits effect on the space-charge field is another interestingproblem that we will consider at another time.
For the photorefractive process we bring a simplifiedpresentation and solution of the equations that couplethe light and the material in a specific crystal parameterregime. In this way we are able to obtain a simpletreatment of the waves at the interface. The materialequations for the photorefractive process were given byKukhtarev et al.9 In steady state the space-charge fieldESC and the mobile-carrier concentration ne are given by
Esc= kBT Vnee n.
ne =D - ND+ (g + ,YRND+
(1)
(2)
where I is the light intensity, k is the Boltzmann con-stant, T is the temperature in kelvins, e is the charge ofthe electron, ND is the total trap concentration, ND+ isthe ionized trap concentration, YR is the recombination
Fig. 1. Schematic of the linear-photorefractive (PR) interfaceand the waves.
constant of an empty trap and a free electron, § = s(hv)is the optical ionization cross section divided by the singlephoton energy, and 6 is the thermal ionization rate.The mobile carriers are assumed to be electrons butcan also be holes. We precede the calculations with thefollowing assumptions: The total trap concentration ismuch larger than the ionized trap concentration (ND >>ND'), and the change in the concentration of the ionizedtrap in comparison with its dark concentration Pdark isvery small, i.e., Pdark >> AND', where ND' = Pdark +
AND'. Note that in the regime that is usually describedin the literature the second assumption does not hold.Then AND+ can be of the same order of magnitude asPdark, and the induced change in the concentration of theionized sites ND+ = Pdark + AND+ is significant and influ-ences the recombination. The number of ionizable sites,however, is very large as in our case. Our assumptionsfit the case of low-intensity illumination with small gra-dients, which well describes our interface problem nearthe critical angle (with a strongly penetrating evanescentwave). Thus we can approximate the prefactor in Eq. (2),
ND - ND+ ND
YRND+ YRPdark
and obtain for the space-charge field
E kBT VIsce I + 8 (3)
Because of the electro-optic effect, this electric field in-duces index-of-refraction changes that are given by
n(I) = n)1 + a I (4)
where a = (1/2)ni2(kBT/e)reff; rff is an effective electro-optic constant along the direction of the space-charge field(the surface normal) that depends on the crystal cut, thesurface orientation, the wave polarization, and the crystalparameters; and nj is the dark linear refractive index ofthe material.
Another assumption simplifies Eq. (4) so that an ana-lytical solution can be obtained in the wave analysisbelow. We assume that the rate of the thermal ionizationis much smaller than the optical ionization, i.e., I >>P3. This assumption can be justified under certain situ-ations, since ,3 is a small constant at low temperatures.However, because the evanescent field decays as it pen-etrates the material, at some length (say, 1) from theinterface the intensity is so small that §I(l) /3, and
our assumption does not hold. Nevertheless, for lowenough temperatures the intensity I(1) is so small that itsinfluence on the solution is weak. Then the refractiveindex is given by
n(I) = n(1 + aI . (5)
A similar expression was used in a previous study byFeinberg.10 In this simple form, which is valid onlyfor the specified regime, the light-induced change of therefractive index is constant in space for the exponentialsolution of the wave equation in a homogeneous medium.Therefore the perturbation in the refractive index doesnot change the homogeneity of the nonlinear medium, andthe exponential solution remains a proper solution of thenonlinear wave equation. This is an important point forthe analysis below.
We now turn to the wave analysis on both sides ofthe interface and apply the boundary conditions. On thelinear side we assume that the incident and the reflectedwaves are ordinary-polarized plane waves;
Ej = (/2)[Aj exp(-ikj r) + c.c.], j=1,2, (6)
where the complex amplitudes Aj are constants and theamplitude Al (the amplitude of the incident wave) isknown. The k vector components satisfy
kj 2 +kj 2= k02 o2 j = 1, 2, (7)
where no is the refractive index of the linear material andko is the vacuum wave number.
The nonlinear wave equation in the photorefractive sideis
V2 E3 + k 2n 2 (I)E3 = 0, (8)
where n(I) is given by Eq. (5). As argued above, weexpect a plane-wave solution with a possible decayingamplitude:
E3 = Y [A3(x)exp(-ik3,z) + c.c.],2
A3(x) = A3,0 exp(-ax). (9)
Then the intensity is I = [1/(2?7)]f A3,o12 exp[-2 Re(a)x],where q is the wave impedance of the material, Y7-120,wnl. This gives, for Eq. (5),
n(I) = nj[1 - 2a Re(a)]. (10)
As expected, the medium remains homogeneous. Substi-tuting Eq. (10) into the wave equation gives the relationbetween a and k3,:
a 2-ka3 2 22R1 - 2a Re(a)]2 =-kon (11)
Having determined the wave solution on both sides ofthe interface, we apply the boundary conditions (conti-nuity of the tangential electric and magnetic fields) andobtain
(12a)
(12b)
k+z = k2z = k3z,
Al + A2 = A3,
-iaA 3 = klAl + k2xA2 . (12c)
R. Daisy and B. Fischer
Vol. 11, No. 6/June 1994/J. Opt. Soc. Am. B 1061
From these equations one can deduce that the incidentangle equals the reflected angle, and one can obtain thereflectivity (determined by A2/Al):
kono sin 01 + iakono sin 01 - ia
(13)
Plots of the solutions for a, and a as functions ofthe incident angle for the linear-linear interface (a = 0)for negative (a < 0) and positive (a > 0) nonlinearitiesare shown in Figs. 2, 3, and 4. It can be seen that fora < 0 (as for the linear case, a = 0) for each incident
To complete the solution it is necessary to find a as afunction of the incident angle (01). This can be done byusing k3, = k = kono cos 01 [Eq. (12a)] and substitutingthe result into Eq. (11):
a2 - k 2no2 cos2 01[1 - 2a Re(a)]2 = -ko2 n .
;-1
QC)C)C)xI Sf
d
(14)
With a a, + iai (where ai and ar are real numbers),the last equation gives
Here a can be either a real number (ai = 0) or a purelyimaginary number (a, = 0), which corresponds to a total-internal-reflection regime or to a transmission regime,respectively. The solution for a, = 0 is the well-knownlinear solution. In this case ai is given from Eq. (15b) by
ai = ±kono[(n1 /no)2 - COS2 01]112,
where only the positive sign is relevant for us.solution is valid for angles
01 > c = cos-'(nl/no),
01 [deg.]
Fig. 2. Roots of ar and ai as a function of the incident angle01 for the linear-linear interface: a = 0 cm, no = 2.4, n =2.399, and A = 0.5 um.
(16)
The I
C)CD'-4
(17) x~
where Ec is the critical angle for the transition from atransmission regime to a total-internal-reflection regime.The solution for a, is obtained by substitution of ai = 0into Eq. (15b), which gives
(1 + 4a2 ko2 n 12 )a, 2 - 4ako2n 12a,.
+ k 2(n12 - n 2 cos2 )1) = 0. (18)
By definition a, must be a real number. Therefore
cos2 01 > 1 +C1/*0 cos2 ( cNL). (19)1+4a2ko2n, 2
Thus the nonlinearity shifts the critical angle from itsvalue in the linear case (,) to larger values. This is thefirst indication for a bistable solution. The solution ofEq. (18) [for incident angles that obey condition (19)] isgiven by
Fig. 3. Roots of ar and ai as a function of the incident angle01 for a negative nonlinearity: a = -4 x 10-8 cm, no = 2.4,ni = 2.399, and A = 0.5 ,im.
20
15-
-II 10-
C)CD .--
X
(20)
A physically meaningful solution requires that ar mustbe a positive number. We find below that, for a positivenonlinearity (a > 0) when 01 is between 0, and 0CNL,the two solutions of ar are both positive, but only one isstable, as we discuss below.
E), E3N.L.-5- c
-10-
-15
-2071.5 1.6 1.7 1.8 1.9 2 2.1 2 2 2 3 2.4 2.5
81 [deg.]
Fig. 4. Roots of ar and ai as a function of the incident angle01 for a positive nonlinearity: a = 4 x 10- 8 cm, no = 2.4, ni =2.399, and A = 0.5 Am.
__Lar
ar
(3c s~~~~~~-X-_ t
_._r. ~~~~~~~~~~~~
ar
zu-
15-
110-
5-
I
R. Daisy and B. Fischer
1062 J. Opt. Soc. Am. B/Vol. 11, No. 6/June 1994
angle there is only one solution (positive ar and a).Therefore there is no bistable behavior for a < 0. This isas expected, since in this case the nonlinear perturbationin the refractive index is positive, and it acts as a negativefeedback; i.e., when the system enters the regime of totalinternal reflection, the nonlinear perturbation tends tobring it back to a transmission regime. Figure 5 showsthe dependence of the reflectivity as a function of theincident angle for negative nonlinearity; no bistabilityis seen. The situation, however, is different for a >
0, as is seen in Fig. 4. In Fig. 4, for incident anglesbetween E), and 0,', there are two positive solutionsfor ar. It can be shown by simple arguments that thelower solution is not stable. Small perturbations in theincident angle will cause the solution to evolve into one ofthe other stable solutions. Here (for a > 0) the nonlinearperturbation in the refractive index is negative, and it actsas a positive feedback. Thus at incident angles between0C and E),NL one stable positive solution remains fora,. (ai = 0) that corresponds to total internal reflection,and there is another solution, ar = 0 and ai 0 0, thatcorresponds to a transmission state. The dependence ofthe reflectivity on the incident angle with the bistability isshown in Fig. 6 for various positive values of the nonlinearparameter a. The width of the hysteresis loop increaseswith the nonlinear parameter a.
In the above calculations we neglected the thermalionization with respect to the optical ionization (sI >>A3). We argued that in the regime of total internal reflec-tion this condition is not satisfied far from the interface,where the light intensity and the ionization rate are low.Therefore we test the equations with the thermal ioniza-tion component and solve the problem numerically. Asubstitution of Eq. (4) into the wave equation, Eq. (8),givesA3/' -k3zA3 + ko2ni2
X 1 + a Re(AIA3* + A3A 3 *) lA3 = 0, (21)2 IA312/2 n7+/ j
where A3' and A3 " are the first and the second deriva-tives of A 3 and A3* is the complex conjugate of A 3 .Equation (21) requires two initial conditions for thefield amplitude and its derivative. One must determinewhat the two required initial conditions are and atwhich location they should be applied. The field andits derivative at the interface depend on the discon-tinuity there, which itself depends on the solution ofthe field at the photorefractive side. However, farfrom the interface in the nonlinear material the ther-mal ionization smears the effect; hence the solutionthere can be taken as the linear solution. At thisstage we want only to find the field at the photo-refractive side and not to solve the whole boundaryproblem. Therefore we give the initial condition deepinside the material and solve the equation from insidethe material toward the interface.
The solution of Eq. (21) is shown in Fig. 7 for a >
0 and for various thermal ionization rates, It can beseen that there are two distinct regions of exponentialsolutions, each giving a constant refractive-index changeand a homogeneous region. The first, near the interface,corresponds to region where the approximate solution
derived above is valid, and the second is the region inwhich the light intensity becomes weak such that thethermal ionization smears the photorefractive effect. We
1.1-
1-
0.9-
0.8-
0.7-
0.6-
0.5-
0.4-
0.3
0.2-
0.1.
T5 1.6 1.7 1.8 1.9 2 2.1 2'2 2'3 2'4 2.5
0 1 [deg.]
Fig. 5. Reflectivity as a function of the incident angle for anegative nonlinearity: a = -4 X 10-8 cm, no = 2.4 (correspondsto Fig. 3), with no bistability.
0.
0.
0.
0.
-0.
I-0.
0.
0.
0.
1.5
1.6 1.*.
- *7) -
6
5
0 - . .-
EGi [deg.]
Fig. 6. Bistable reflectivity as a function of the incident anglefor various positive nonlinearities: a = 2 X 10-8 cm (dashedline), a = 3 X 10-8 cm (solid line), a = 4 X 10-5 cm (dotted line),and for ni = 2.399, no = 2.4, and A = 0.5 ttm.
1000
100=
m~ 10
0.01-0 5 10 15 20 25 30 35 40 45 50
x [m]
Fig. 7. Field in the nonlinear material near the inter-face for a = 1 10-8 cm, s = I X 10-19 cm 2 , ni =2.399, no = 2.4, A = 0.5 ptm, 01 = 1.60, and for various thermalionization rates: /3 = 0.01 s-1 (solid curve), /3 = 0.001 s-(dotted curve), /3 = 0.0001 s-1 (heavy solid curve), and ,/ = 0 s-1(dashed line).
n |IJI
R. Daisy and B. Fischer
1.8 1.9 2 2.1 2.2 2.6 2.4 Z. DI-
1.6 1.7
Vol. 11, No. 6/June 1994/J. Opt. Soc. Am. B 1063
-5E-05
-0.0001-
a -0.00015
-0.0002
0 5 10 15 20 25 30 35 40 45 50
x [/-'m]
Fig. 8. Induced refractive-index change in the nonlinear ma-terial in the vicinity of the interface for a = 1 10-8 cm,s = 1 x 10-1 9 cm 2 , nl = 2.399, no = 2.4, A = 0.5 /im, 01 = 1.6°,and for various thermal ionization rates: /3 = 0.01 s (solidcurve), /3 = 0.001 s1 (dotted curve), /3 = 0.0001 s1 (heavy solidcurve), and /3 = 0 s1 (dashed-dotted line).
can see that the larger the thermal ionization rate A,the smaller the first region. The nonlinear perturbationon the refractive index is shown in Fig. 8 for a > 0and for various thermal ionization rates /l. We can seethat, as the temperature increases, the width of theinduced homogeneous layer of the refractive index de-creases. For high enough temperatures (or 3) this layercan be strongly depleted, and the nonlinear refractive-index perturbation vanishes. From these calculations
it is possible to determine the conditions at which theprevious approximate solution is valid. For a given ther-mal ionization rate and incident intensity, one can solveEq. (21). The approximate solution is valid when thewidth of the induced homogeneous layer is greater thanthe penetration depth (a,-').
In conclusion, we have solved the boundary problembetween linear and photorefractive materials in whichthe incident plane wave propagates from the linear side.We have shown that the reflectivity can be bistable forpositive nonlinear coefficients. We have also given a nu-merical solution for the field in which thermal ionizationis accounted for, and we have verified the validity of theapproximate solution.
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