Low-power, single-interface guided waves mediated byhigh-power nonlinear guided waves: TE case
T. P. Shen
Rockwell International Science Center, 1049 Camino dos Rios, Thousand Oaks, California 91360
A. A. Maradudin
Department of Physics, University of California at Irvine, Irvine, California 92717
G. I. Stegeman
Optical Sciences Center, University of Arizona, Tucson, Arizona 85721
Received August 11, 1987; accepted February 3, 1988
The dispersion relation for a low-power (probe) TE-polarized wave guided by an interface between a linear andnonlinear medium was derived and investigated numerically as a function of the power of a high-power (pump), TE-polarized nonlinear wave guided by the same interface. The birefringence induced by a Kerr-type nonlinearityleads to a difference between the dispersion relations for pump and probe beams propagating at right angles to eachanother.
INTRODUCTION
A new class of waves guided by single interfaces'- 3 involvingat least one medium with an intensity-dependent refractiveindex has been the subject of intense theoretical investiga-tion over the last few years. These waves are characterizedby optically induced changes in the refractive index (An =n2I, where I is the local intensity) that are larger than theindex discontinuities across the interface between the linearand nonlinear media. The transverse field distributionpeaks in the nonlinear medium near the boundary, and -thewave effectively writes its own waveguide by producing alocalized high-index region similar to that found in integrat-ed-optics waveguides. Therefore it is reasonable to assumethat low-power integrated-optics modes can also be guidedby the effective waveguide produced by a high-power beam.In this paper we investigate the dispersion relations of suchlow-power modes.
To date, only the nonlinear, guided-wave dispersion rela-tions for the case when only one distinct wave is present havebeen analyzed. The most general case, that of two high-power guided waves, possibly of orthogonal polarization anddifferent propagation directions, is analytically intractableand requires numerical techniques to make any progresstoward the coupled dispersion relations. However, if one ofthe two beams is strong (high power) and one is weak (suffi-ciently low enough power so that it does not affect the refrac-tive-index distribution), it is possible to use the well-knownanalytical solutions for the high-power wave in order todefine the index distribution for the low-power wave. Thelow-power-wave dispersion relation can then be evaluatedeither by the WKB method or by the analytical solution ofthe weak-wave dispersion relation.
In this paper we consider an interface between a linear anda nonlinear dielectric upon which a high-power TE-polar-
ized pump wave is propagating. We derive and solve nu-merically the dispersion relations for a second, weak (probe)TE-polarized wave that propagates at right angles to thestrong pump beam. Two models for the nonlinearity aretreated. For the first, the refractive-index distribution isisotropic in planes parallel to the surface, typical of an ab-sorptive nonlinearity. Because the index distribution isidentical to that seen by the high-power wave, the dispersionrelations are expected to be the same as for the high-powerwave itself. This will provide a check of the formalism.The second is a Kerr-type nonlinearity, which leads to bire-fringence in the plane of the surface because of the well-known weak-wave-retardation effect. For this case, theweak-wave dispersion relations should differ from those ofthe high-power beam that writes the index distribution.
THEORY
The geometry considered here (see Fig. 1) consists of a linearmedium occupying the region X3 > 0 and a nonlinear medi-um occupying the region X3 < 0. The goal of the calculationis to determine the dispersion relations for a low-intensityelectromagnetic wave in the presence of a high-intensityelectromagnetic wave propagating in a direction orthogonalto the weak wave. By the terms weak and low-intensitywaves we specifically mean that these waves do not perturbthe refractive-index distribution. It is well known thatabove a threshold power a linear surface polariton can prop-agate along the interface X3 = 0. For a local nonlinearity ofthe form n = no + n2I, the index distribution will be shown tohave a Pschl-Teller profile in the nonlinear medium, andthis structure provides, under appropriate conditions, theguiding conditions for the probe wave, which by itself is notsupported by the interface. For simplicity, we shall restrict
Fig. 1. The guided-wave geometry and beam propagation direc-tions considered here.
ourselves to monochromatic, s-polarized (TE) pump andprobe waves.
The total electric field of the s-polarized strong and weakwaves, which propagate perpendicular to each other, can beexpressed as
where a 2(kw) = k2- O(W)W2 /C2 and a 2(qw) = q2- Eo(W)W2 /c2
are both positive for guided waves, which are localized nearthe interface X3 = 0, and EO(w) is the dielectric constant of thelinear medium, which is assumed to be isotropic. In Eqs. (5)E0 and E0 are determined by the boundary conditions at X3 =0, and E0 also depends on the power level of the strong beam.
For the nonlinear medium, its nonlinear polarization maybe expanded in higher powers of the electric field, i.e., sec-ond, third, etc. If the nonlinear medium possesses a centerof inversion, the first nonvanishing nonlinear polarization isthe third order in electric fields. This third-order nonlinearpolarization has two contributions, one at frequency 3w, i.e.,third-harmonic generation, and the other at frequency w. Inmost materials the third-harmonic generation process is notphase matched and is therefore small and can be neglectedhere. Then the third-order nonlinear polarization at thefrequency w is given by
P 2 (3 (r, co) = {aIE2 (r, W 12 + b[1E 3 (r, 4)124ir
+ IE,(r, )2] 1E2(r, )
+ d [E32 (r, w) + E 1
2 (r, w)]E2*(r, c),
P 3(3)(r, ) = faIE3(r, W 12 + b[1E1(r, 4)124ir
+ E2(r, )12] 1E3 (r, 4)
+ I [EJ2(r, ) + E22(r, w]E3*(r, ),
(6a)
(6b)
(6c)
E(r,co) = E(1)(r, ) + E(2)(r, w),
where
E(')(r, ) = x2 E(kwlx3)exp(jkxi),
E(2)(r, w) = xE(qwlx 3)exp(iqx 2 ).
The total electric field obeys the wave equation
2
V[V E(r, 4] -VE(r, w4 = 2D(r, 4),c
(1) where E(r, w) is given by Eqs. (2) and cubic symmetry isassumed for the nonlinear medium whose third-order non-linear coefficients a, b, and d may be frequency dependent.Now we can write the electric displacement vector in the
(2a) region X3 < 0 as
(2b)
(3)
where the electric displacement vector D(r, ) is given by
Di(r, c) = EijM(Ej(r, c) + 47r PNL(r, O. (4)
In Eq. (4), eij(w) is the local linear dielectric tensor andPiNL(r, ) is the nonlinear polarization, which vanishes inthe linear medium.
For the linear medium (X3 > 0), the electric-field ampli-tudes in Eqs. (2) for surface polaritons are given by
where E(U) is the linear part of the dielectric constant of thenonlinear medium, which is henceforth assumed to be iso-tropic. Then the wave equation in the nonlinear region isgiven by
- l >
Shen et al.
(5b)
Vol. 5, No. 7/July 1988/J. Opt. Soc. Am. B 1393
[ d - a2(kw)lE(kwIx3) + 2 [a1E(kwlx3 ) 2
dX32 c2
+ bIE(qlx 3)12]E(qcolx3)
+ 2 dE2 (qxIX3)E*(kWIx3)exp(2jqx2 - 2jkxl) = O, (8a)
where a 2(ko) = k2 - E(W)c2 /c2 and a2(qw) = q2 - E(w)w,2 /c2 areboth positive. If we neglect the non-phase-matching termsand only keep the lowest-order nonlinearities, we obtain thefollowing decoupled equations:
The solution of Eq. (9a) is well known' and is given for a >0 by
/2\1/2 a(kw)E(kwIX3) = V -- (a cosh a(ko)(x3 -xe) (10)
where k can be expressed as O/3kc and k is the so-calledeffective index of the pump wave and xc < 0 is determined bythe boundary conditions at X3 = 0 and the pump-beam pow-er level. Once we have solved Eq. (9a), we can cast Eq. (9b)in the form
d__ a2(kw)
[dX32 -a(qo)] E(qlX3) a cosh2 a(ko)(x 3 - )
X E(qlx 3) = (11)
which is exactly the same form as the one-dimensionalSchr6dinger equation with the modified P6schl-Teller-typepotential hole,4 if b > 0. The solution to Eq. (11) is given by
E(qcolx3) = A cosh ac(k)(x 3 - xI)
X F[-y, , 1/2; -sinh 2 a(ko)(x 3 - )]
+ B coshX a(ko)(x 3 - xc)sinh a(ko)(x 3 - xI)
X F[y + /2, + 1/2, /2; -sinh2 a(ko)(x3 -X],
(12)
where X(X - 1) = 2b/a, with > 1, y = 1/2(X -a), = 1/2( +a), a = a(q)/a(ko), and F is the hypergeometric function.In Eq. (12) A and B are arbitrary constants that will bedetermined by the boundary condition as X3 - -o and at x3=0.
For X3 - , the electric field must vanish. We use the
asymptotic behavior of the solution in Eq. (12) for largenegative values of the argument to obtain one relation be-tween A and B, given by
= ( a) ( X+a)B 2r(+ X+ a),, (2 X- a) (13)
where r(argument) is the r function. The boundary condi-tions at X3 = 0 require that
Equation (15) is the dispersion relation for the weak probewave in a modified Poschl-Teller-type waveguide, which isproduced by the pump wave. The allowed values for q inEq. (15) define the probe guided wave. Again one can definethe effective index /q for the secondary wave with q qW/C.
NUMERICAL RESULTS AND DISCUSSIONS
The variation in the effective indices with the guided-wavepower for the strong pump beam was calculated numericallyfrom the usual dispersion relations'-3 for the pump wave andfrom Eq. (15) for the weak beam. We introduced Euler'stransformation5 in order to facilitate series convergence ofthe hypergeometric functions in the expression for E(qwlx3)in Eq. (12). For these numerical examples we assumed awavelength (X = 27rc/co) of 0.515 ,um, and an intensity (I)-
1.71
1.69
1.67 F
1.65
1.63 F
1.61 F
1.59
1.57
1.551_20 40 60 80 100 120 140
P(mW/mm)
160
Fig. 2. The effective index Ok of the s-polarized pump wave versusits guided-wave power for ns = 1.55, n2s = 10-9 m2/W, and n, = 1.56,1.58, 1.60.
n,= 1.55n2 = 10-9 m2 /W
n = 1.60
Shen et al.
1394 J. Opt. Soc. Am. B/Vol. 5, No. 7/July 1988
1.7'
1.69 F
1.67 .
1.65 _
0q 1.63 F
1.61 _
1.59 _
1.57
1.55 0
1.73
1.71 -
1.69 F-
1.67 F
1.65
1.63 F
1.61 -
1.59 k
.57'*0
1.73
1.71 -
1.69 _
1.67 _
1.65-
1.63 -
1.61 _
1.59 20 40 60 8) 100
P(mW/mm)(a)
20 40 60P(mW/mm)
(b)
1.57
80 oo
dependent refractive index in, = [E(W)]1/2 + n 2 I) for the sub-
strate of [e(w)]1/2 = 1.55, and n2 j=47r/c[a/e(w)]j is 10-9 m2/W.
For the linear cladding, the following values were used: n =[fo(W)]1/2 = 1.56, 1.58, 1.60.
In Fig. 2 we plot the effective index Ok for the s-polarizedpump wave versus its guided-wave power for comparisonwith Fig. 3. In Fig. 3 we plot the effective index 1 q of the s-polarized weak waves versus the guided-wave power of thestrong wave for two choices of b/a, which are 1 and 2/3. Thecase b/a = 2/3 corresponds to a classical Kerr-law nonlinear-
20 40 60 80 100PlmW/min
(C)
120
Fig. 3. The effective index q of the s-polarized probe wave versusthe guided-wave power of the primary wave for n, = 1.55 and n2s =
10-9 m2/W: (a) n, = 1.56, (b) n, = 1.58, (c) n, = 1.60 for b/a = 1 and
b/a = 2/3.
ity, for example, an electronic nonlinearity that producesbirefringence. The case b/a = 1 describes, for example, athermal nonlinearity without spatial diffusion.
We first examine the case b = a. For all choices of clad-ding refractive index, q is equal to Ok to within the numeri-cal accuracy. This is illustrated in Fig. 3 for three specificcases. This is a direct consequence of Eqs. (9) and (15).Given the refractive-index distribution created by the strongbeam [aE(kwlx3)], the nonlinear wave equations for thestrong and weak beam in Eqs. (9) are identical. Because the
Shen et al.
TE + TEnc = 1.56
b/a= 1
2/3
nc= 1.60
= 1
-b/a = 2/3
TE + TEnc = 1.58
b/a = 1
b/a = 2/3
I 1 1ffi 1
Il I |
- - - -
I
Vol. 5, No. 7/July 1988/J. Opt. Soc. Am. B 1395
boundary conditions [Eq. (15)] are also identical for thepump and probe beams, the equivalence of 1
3k and flq for a
given value of pump power (which creates the specific indexdistribution) is necessary. The fact that this is verifiednumerically is a check on our numerics.
The situation is different for the case b/a = 2/3. The twononlinear wave equations are governed by the same spatiallyvarying refractive-index distribution but with differentmagnitudes. Because the probe beam encounters a smallerindex change than the pump beam for a given pump-beampower, 3
k > 13q, as shown in Fig. 3. Note that the minimum
pump-beam power required for both the pump and probebeams to exist is the same, within numerical accuracy, forthe two cases. However, because A
3k # f3q for b/a = 2/3, the
field distributions are different for the pump and probebeams at every pump-beam power.
One desirable application for a strong-weak-beam inter-action is to produce all-optical modulation of the weak beamby the strong beam, or vice versa. The simplest way toachieve this is to use a beam configuration in which theminimum pump-beam power required for the existence of aweak-beam guided wave is larger than the minimum strongbeam power required for the pump beam itself. This, unfor-
tunately, is not the case found here for copolarized waves.However, it is well known from integrated optics that a TMmode can be cutoff for film thicknesses that do support a TEwave, which suggests that a strong-weak-wave interactionusing mixed (TE pump, TM probe) waves might be usefulfor light-by-light modulation. This case is currently underinvestigation.
ACKNOWLEDGEMENT
This research was supported in part by the U.S. Army Re-search Office (DAAG29-84-K-0026).
REFERENCES
1. A. E. Kaplan, Sov. Phys. JETP 45, 896 (1977).2. A. A. Maradudin, in Optical and Acoustic Waves in Solids-
Modern Topics, M. Borissov, ed. (World Scientific, Singapore,1983), p. 72.
3. G. I. Stegeman, C. T. Seaton, J. Ariyasu, R. F. Wallis, and A. A.Maradudin, J. Appl. Phys. 58, 2453 (1985).
4. S. Fl6gge, Practical Quantum Mechanics I (Springer-Verlag,New York, 1976), p. 89.
5. P. M. Morse and H. Feshbach, Methods of Theoretical Physics(McGraw-Hill, New York, 1953), p. 587.
