Scaling rules for nonlinear thin film opticalwaveguides
Marie Fontaine
A mode power measure is applied to characterize nonlinear thin film optical waveguides in an approachanalogous to that of Chelkowski and Chrostowski. Together with the normalized film thickness and the
asymmetry coefficient, it allows us to get a concise overview of the waveguiding properties at a given power.
For self-focusing film, we discuss the design conditions under which the degree of asymmetry significantly
affects the waveguiding properties.
1. Introduction
Planar optical waveguides in which one of the mediaexhibits an intensity-dependent refractive index havereceived increasing attention in the last few years1 -l0
due to their potential use in optical communication.A basic problem in the design stage is to solve the
Maxwell equations for nonlinear slab waveguides inwhich the nonlinear, usually Kerr-type, material isused as either the guiding layer or the cladding. Fornonlinear film waveguides bounded by two linear me-dia, several forms of dispersion equations have beensolved for particular geometries. 4 5
Our goal here is to present a parameterization modelfor nonlinear film waveguides bounded by two linearmedia to obtain a universal description of differentgeometries for this type of waveguide. This generalapproach, initiated by Kogelnik and RamaswamylI forlinear waveguides, permits a simple determination ofthe various design possibilities. More recently, Chel-kowski and Chrostowski'0 extended the approach tooptical waveguides bounded by one nonlinear medium.By defining a new power-dependent parameter, theywere able to get a concise overview of the waveguidingproperties at a given power. The approach we havetaken in this paper is similar to that of Chelkowski andChrostowski,10 since it provides universal plots fromwhich the effective index and thickness of a waveguidecan be determined for a given power. We limit our-selves here to the case of nonlinear self-focusing film (a> 0) and TE mode propagation. With only slight
The author is with University of Quebec at Hull, Laboratory ofOptoelectronics, P.O. Box 1250 Station B, Hull, Quebec J8X 3X7,
modification, however, the present analysis could alsobe applied to a waveguide with defocusing nonlinearity(a < O).
In Sec. II, we present the dispersion equation for theTE modes in a nonlinear film waveguide in scaledvariables. Two general situations are considered: (1)where the linear part of the film index is greater thanthe linear index of the substrate, and (2) where it issmaller. In Sec. III, we show that the newly definedparameter bi proposed by Chelkowski and Chros-towski1 0 is also useful for investigating the waveguideproperties of nonlinear thin film waveguides. Ourresults are summarized in Sec. IV.
All the computations for this work have been per-formed with the Mathematica software.
11. Dispersion Equation for an Optically Nonlinear Film in
Scaled Variables
The geometry of the nonlinear waveguide underconsideration is shown in Fig. 1. The waveguide con-sists of a thin, optically nonlinear, dielectric film sand-wiched between semi-infinite linear dielectrics. Theproperties of such a waveguide depend on the parame-ters nj, nf, ns, d, X and a = nfeeon2,I. For linear wave-guides in which a = 0, Kogelnik and Ramaswamy1lhave shown that the guiding properties of differentgeometries of waveguide can be universally describedin terms of two independent parameters: the normal-ized film thickness,
V= 2r d (f2 2)1/2 (1)
and the asymmetry coefficient,
(n2 - n 2)
(n2 _ n 2) *
The boundary conditions at the interfaces yield a com-
model uses only three independent variables, V, a, andbl, to obtain the dispersion curves V(b,a,b) and thepower plots P(Va,b). For the nonlinear self-focusingcase, two different nonlinear structures must be con-sidered; the first being when nf > n and the secondwhen nf < n,. We begin with the case where nf > ns,since the unified description outlined here is directlypatterned on the universal description of linear wave-
Region 3 guides proposed by Kogelnik and Ramaswamy.11
A. The Case Where nf > ns
The characteristic curves V(b,a,bj) of the waveguideRegion 2 shown in Fig. 1 are obtained by matching the boundary
conditions of the fields at the interfaces z = 0 and z = d.In linear regions 1 and 3, the fields are:
c
X
Semi-infinite 2linear medium: n
s
E = E0 exp(klz),
E3 = Eb exp[k3(d - z)],
(5)
(6)
Region 1 where,
k-2= k2(N2 n)
Fig. 1. Nonlinear waveguide bounded by linear dielectrics.
pact dispersion equation, the solution to which deter-mines the normalized modal index,
(N2- n2)
(nf - n,)
where N = fl/k, being the propagation constant of thewaveguide and k the wave number of light at free-spacewavelength X. The guiding properties of any geometryof slab linear waveguide are then summarized in theaniversal dispersion curve V(b,a,M) with differentmodes M.
In a similar way, Chelkowski and Chrostowski' 0
have shown that the guiding properties of a slab wave-guide with nonlinear substrate and linear core mediumcan be described by three independent parameters V,a, and b which determine the parameter b via thedispersion equation. They were able to achieve a uni-versal description of different geometries of this typeof nonlinear waveguide by presenting universal disper-sion curves to be used simultaneously with powercurves for fixed values of V, a, and b. In this case,parameters V, a, and b are also defined by Eqs. (1), (2),and (3), respectively, and the new parameter b isdefined as follows:
, aE2(0) (4)2(nf-n5)
Chelkowski and Chrostowski'0 show that the vari-able b is proportional to the mode power P for smallpowers. Their results also establish that the defini-tion of a power parameter Po is useful since valuesdetermine the order of magnitude of power for whichnonlinear effects can be observed.
In the following, we offer a similar universal descrip-tion of different geometries of nonlinear thin film opti-cal waveguides bounded by two linear media. Our
(7)
k2= k 2(N2 - n2). (8)
The parameter N is defined as previously by fl/k. Theparameters Eo and Eb are the values of the field ampli-tude at the interfaces z = 0 and z = d, respectively.
A first integration of the differential equation for theguiding layer field yields a differential equation forwhich the solution is a Jacobian elliptic function. 2'l 3
There are, in fact, twelve Jacobi functions and thechoice of a particular one depends upon the sign of theconstant of integration C2,
C2 = k 2E 2 [(nf2 - 2) + '/2aEo],
or
C2 k2E2 [(n2 - n2) + '/2 aE2I
(9)
(10)
generated by the first integration of the differentialequation for the guiding layer.
Since in Sec. II.A we are considering only self-focus-ing nonlinearity (a > 0) and the case where nf > n, thisconstant of integration is always positive and the solu-tion of the field in the guiding layer is given by the cnJacobian function:
E = P cn[q(z + z)Im]. (11)
The identification of the differential equation forthe guiding layer field as the differential equation forthe cn Jacobi function defines the parameters q, p, andm in terms of known physical constants. In terms ofparameters a, b, and b defined by Eqs. (2)-(4), we canshow that:
q = k(nf2- n') 2[(b - 1)2 + 4b,(1 + b,)]/4,
p = (E 0 ) 1[(b - 1)2 + 4b,(1 + bl)]"/2 + (b -1)1/2,
We observe that 0 < m < 1 for all positive values of band all values of b, greater than 0 (the self-focusingcase), so that the choice of the cn Jacobi function isjustified. The parameter zo in Eq. (11) is a secondconstant of integration and the mathematical proper-ties of the cn Jacobi function permit us to write thefield function Eq. (11) in terms of two other Jacobifunctions, sn and dn, without using the parameter zo.
We now consider the dispersion equation generatedby the boundary conditions of the field functions Eqs.(5), (6), and (11) at the interfaces z = 0 and z = d.Following Boardman and Egan,4 we write the disper-sion equation in mathematical form without using theinverse Jacobi functions. After some algebraic ma-nipulations, we can write this equation in terms of onlyfour parameters, V, a, b, and bI defined, respectively,by Eqs. (1), (2), (3), and (4):
E, E, b)b + a) /2a
cn P X ) (b -1)2+4b,(1 + b,) (15)cn(X~dlm) - (- )2] [1 _ ( b )2]} (5
According to the definition of p and m in Eqs. (13) and(14), the parameters m and Eo/p are only functions of band bi. From Eqs. (1) and (12), we can also see that qdis only a function of V, b, and bj:
qd = V[(b - 1)2 + 4b,(1 + b,)]114.
b
0 1 2 3 4 5 6 7 8
V
Fig. 2. Universal dispersion curves for a symmetrical (a = 0) non-linear thin film waveguide in the case where nf > n,. The curves arelabeled with values of b,. The dotted line corresponds to the linear
thin film waveguide where bi = 0.
(16)
From Eqs. (9) and (10), we can show that Ebip = Eb/p(b,b1 ,a):
(Eb\ [(1 + a)2 + 4b,(1 + b,)]"12-(1 + a) 1/2
\P/ [(b_1)2 + 4b,(1 + b)]1/2 +(b -1)(17)
We can then write the dispersion equation in the nor-malized form Eq. (15). This form indicates that thenormalized modal index b depends on only three inde-pendent parameters: V, a, and bI. Simple numericalmethods can be used to solve Eq. (15) if we choose V asthe unknown parameter, since V appears only to theleft side of the equation.
The plot of the dispersion Eq. (15) for a symmetricalwaveguide (a = 0) is presented in Fig. 2. This showsthe dependence of the guide index b on the normalizedfrequency V for various values of bi. The dotted linecorresponds to the value of bI equal to 0. This curvereproduces the results obtained by Kogelnik and Ra-maswamyll for the linear film waveguide. In this case,the maximum value of b is 1.0 and all the solutionscorrespond to guided modes since fl/k < nf. For thenonlinear case, solutions exist for which b > 1 (fl/k >nf). Following the classification proposed by Board-man and Egan,4 these solutions correspond to surfacemodes. As shown in Fig. 3, these waves do not decaytoward the center as we would expect but are describedby periodic waves where the intensity of the field ismaximum in the guiding layer. These surface wavesare typical of highly nonlinear waveguide and requirematerials with large self-focusing nonlinearities par-ticularly for structure waveguides with nf > n,. Figure
E
6)
-1 0 2
Z.
Fig. 3. Electric field configuration at V = 7.0 for a symmetricalnonlinear waveguide and a symmetrical linear waveguide. (1) b, =0.01 and b = 1.265; (2) bl = 0.01 and b = 1.117; (3) bl = 0 and b = 0.87.
4, an enlargement of Fig. 2, shows the variation of thenormalized guide index for only the guided modes (b <1) for very small values of the nonlinearity coefficientb1. In terms of applicability, as long as no new materi-als with large self-focusing nonlinearities appear, 14 thedispersion curves of Fig. 4 are the most useful.
It is clear from these universal dispersion curves thatfor a fixed value of V, there is only a certain range ofvalues of bi within which physical solutions for the TEo
Fig. 4. Universal dispersion curves for normalized guided index b <1 in the case where nf> n,(a = 0). The curves are labeled with valuesof b, < 0.10. The dotted line corresponds to the linear thin film
waveguide where b = 0.
2.0
3.0 1.0 0.5 0.1 005 0.01
1.5
b 1.0-
0.0~~~~~~~~~~~~~00
0 1 2 3 4 5 6 7 8
VFig. 5. Universal dispersion curves for an asymmetrical nonlinearthin film waveguide in the case where n > n. The asymmetrycoefficient a = 10. The curves are labeled with values of b. Thedotted line corresponds to the linear thin film waveguide where b =
0.
mode can be found. In the next section, we will seethat, at a fixed value of V, a given value of the totalnormalized power carried by the waveguide deter-mines the value of the parameter b.
The plot of the dispersion equation (15) for an asym-metrical waveguide (a = 10) is shown in Fig. 5. Acomparison of Figs. 2 and 5 shows that when V > 1, theguide index b depends very slightly on the asymmetryparameter a, particularly for small values of b, butwhen V < 1, its dependence on the asymmetry coeffi-cient is more important. Figure 6 shows that the
b 0.001 l 11.15 1.20 1.25 1.30
0.4.
0.2.
0.041.0 1.5 2.0 2.5 3.0
VFig. 6. Universal dispersion curves for normalized guided index b <1 in the case where nf > na = 10). The curves are labeled withvalues of b < 0.10. The dotted line corresponds to the linear thinfilm waveguide where b = 0. In the left upper corner, we show thevariation with b of the cutoff normalized frequency of the TEO
mode.
cutoff value of the normalized frequency V for eachTEO mode decreases even with small values of b.
B. The Case Where f < n
In this case, since nf < ns, the parameters V, a, b, andbj in the dispersion equation must be defined, respec-tively, as follows:
V = 2r (n-2 _ n2)-1 2
X
(n' - n2)a 2 _ 2)
(n2 _ ni)
b -aE2 (0)
2(nn2)
(18)
(19)
(20)
(21)
The fields in the linear regions are also taken as Eqs.(5) and (6) but for the nonlinear guiding layer, depend-ing on the sign of C2 in Eq. (9), different Jacobi func-tions must be considered. From Eq. (9), we verify thatC2 is positive if b > 1 and negative if b < 1. Conse-quently, depending on the value of b, different Jacobifunctions must be considered:1. If b > 1, the dispersion relation is identical to Eq.(15) and parameters q, p, and m now are:
Fig. 8. Universal dispersion curves for an asymmetrical (a = 10)nonlinear thin film waveguide in the case where nf < n,. The curvesare labeled with values of bI. The solid lines refer to TEO modes and
the dotted lines to TE, modes.
The function Eb/p(b,bI,a) is also defined by Eq. (25).As discussed previously, when nf < ns, the only casewhere b, can be <1 is when a < 1.
Plots of the dispersion equation for the situationwhen a = 0 and a = 10 while nf < n are shown in Figs. 7and 8, respectively. When nf < ns, the solutions areinduced by the effect of nonlinearity so that no solu-tion exists for bI = 0. For a fixed value of V, there isonly a certain range of values of bI within which physi-cal solutions for the TE mode can be found. We willsee in the next section that this range always deter-mines the power threshold below which no waves couldbe propagated.
Figure 8 reproduces results very close to those ofBoardman and Egan,4 since their parameters ns, nf,
and n, correspond to the situation where a = 9.67. Acomparison of Figs. 7 and 8 shows that the dispersioncurves are different and it is clear that the guidingproperties of the nonlinear waveguide depend stronglyon the asymmetry coefficient when nf < ns. Thissituation also obtains when nf > n but only for valuesof V< 1.
Ill. Relation Between Parameter b, and Power Carriedby the TEO Mode
In this section, we establish the normalized powercurves P/Po = P/Po(Vabj) to show that these curvescan be used simultaneously with the previous universaldispersion curves to fully investigate waveguide prop-erties of the nonlinear film at a given power P.
The guided wave power per unit length along x isgiven by
P = 1 Nceo | E 2(z)dz.2 J_
We use a parameter Po similar to the parameter Podefined by Chelkowski and Chrostowski' 0 to see howthe normalized power flow P/PO varies with bI. As
2.0
(23)
(24)
We observe that 0 < m < 1 for all positive values of band all values of b > 1, so that the choice of the cnJacobi function in Eq. (15) is justified.
There are two different functions Eb/p(b,bI,a) to beconsidered depending on the value of the asymmetrycoefficient to be sure that Eb is real:
If a > 1, we have to choose the minus sign and if a < 1,the plus sign. We observe from Eq. (25) that when a <1, physical solutions exist for all values of b > 0 butwhen a > 1, the value of bI must be >1 to ensure thatEbip is a real function. This condition does not applyin the case where nf > n since physical solutions al-ways exist for bI at values <1 regardless of the value ofthe asymmetry coefficient a.2. If bi < 1, we can see from Eq. (24) that m > 1 and it isnecessary to use Jacobi's real transformations to bringm within the range (0,1). This results in the substitu-tion of dn(q'dlm') in place of cn(qdim) in the dispersionEq. (15), the new parameters q' and m' being definedby:
Fig. 7. Universal dispersion curves for a symmetrical (a 0) non-linear thin film waveguide in the case where nf < n.. The curves arelabeled with values of bl. The solid lines refer to TEO modes and the
We have defined the parameter Po following Chel-kowski and Chrostowski.10 Our parameter P0 differsonly by a factor of 2 from their parameter. We canconsider PIPo as only a function of V, a, and bj since theparameter b = b(Va,bj) is found from dispersion rela-tion Eq. (15).
The normalized power flow P/Po as a function of bjfor the symmetrical waveguide is shown in Fig. 9. Weobserve that P/PO is proportional to b. For smallvalues of bi (<0.5), P/Po is nearly a linear function of b1.However, when b < 0.2, the field is confined in theguiding layer only for values of V> 1.5. In this case, P/Po is nearly independent of V.
To show how the simultaneous use of the dispersioncurves and the normalized power plots enables us toobtain the optical characteristics of a nonlinear filmwaveguide with n > n,, we consider the theoretical
P
P0
1.0
0.00.0 0.1 0.2 0.3 0.4 0.5
Fig. 9. Relationship between parameter b and the normalizedpower flow P/PO in the TEo mode when n > n, for a series of values ofV: V= 0.8, A V= 1.0, V = 1.5, V= 2.0. The asymmetry
coefficient is 0.
waveguide structure proposed by Stegeman.15 It con-sists of a GaAs film with a thickness of 0.035 Am (nf =3.6 at X = 0.82 Am, n2j = 10-11 m 2/W) grown onto asapphire (ns = n = 1.77). In terms of our universalparameters, V 0.8 and Po N X 2650 mW/mm.Stegeman 5 has shown that for a 0 .035 -am film thick-ness, a minimum power of -150 mW/mm is needed forthe TEO mode to propagate over a 22 -Am distance. Inthis case, according to our results, this order of magni-tude of power would correspond to values of b > 0.003.As shown in Fig. 4 at V 0.8, the normalized guideindex b must be considered as a function of the totalpower carried by the waveguide for values of b > 0.01.For the particular waveguide described above, thiscorresponds to power per unit length >500 mW/mm.
The relations between b, and P/Po for an asymmetri-cal (a = 10) system are displayed in Figs. 10 (V < 1) and11 (V > 1). If we compare these figures, we observethat P/PO is highly nonlinear with b where V < 1 butwhen V > 1, the power is proportional to bj. When V S1, the minimum of the P/PO(Va,b) curves, as a func-tion of bi, corresponds to the minimum value of b, fromwhich the field begins to be confined in the nonlinearfilm. This nonlinearity in the power curves can beexplained by an interplay in the power distributedbetween the substrate and the film regions. For V >1.5, our calculations have shown that the normalizedpower flow P/PO is a linear function of b that variesslightly with the effective thickness V.
Summarizing, for systems where n > ns, the normal-ized dispersion curves show that the properties of thewaveguide do not vary very much with asymmetrywhen V> 1.5. For this range of Vand for small valuesof b, the total normalized power flow may be consid-ered, in a first approximation, as a linear function of b,where PIPO bF(Va). However, when V < 1, thedegree of asymmetry significantly affects the guiding
P
P
0.0 1.0 2.0 3.0 4.0
bFig. 10. Relationship between parameter b and the normalizedpower flow P/Po in the TEo mode when nf> n and V 1. The
Fig. 11. Relationship between parameter bj and the normalizedpower flow P/PO in the TEO mode when nf > n, and V > 1. The
asymmetry coefficient is 10.
properties and the total normalized power is highlynonlinear with bj.
B. The Case Where nf < n,
With V, a, b defined by Eqs. (18), (19), and (21),respectively, the relation P/PO is similar to the previouscase if b, > 1:
P/O = b (1 - a) [(1 - a)2 + 4b,(b, -1)1/2
b12 ° v 2V(b + a)1/2
2b, fqd F cn(u) + f,(bbj)sn(u)dn(U) 2+ -f'fbbl) 2IU du, (30)
1 /2 ° L dn-(u) + f2(b,bj)sn(u) J
where
fl(b,b1) = [(b + 1)2 + 4b,(b, -1)1/4
f2(b,bl) = 2 ,[(b + 1)2 + 4b,(b, -1)]1/2
PO =( (n-nf2)d.a
In Eq. (30) we choose the plus sign if a > 1 and theminus sign if a < 1.
For symmetrical waveguides (a = 0), the relationsbetween bj and P/PO for fixed values of V are displayedin Figs. 12 and 13 for the TEo and TE, modes, respec-tively. For all values of V, the normalized power goesthrough a minimum and, subsequently, increases rap-idly with b1. This means that for any fixed value of V,no waves are propagated until a certain power thresh-old is reached. This result is different from the casewhere nf > n since, when nf < ns, confinement of thefield is induced only by nonlinearity. For the TEO orTE, modes, we also observe that the threshold powerincreases as V decreases.
p
P0
3.0
Fig. 12. Relationship between parameter b, and the normalizedpower flow P/Po in the TEO mode when nf < ns and V < 1. The
asymmetry coefficient is 0.
p
P0
1.0 1.2 1.4 1.6 1.8 2.0 2.2
Fig. 13. Relationship between parameter b and the normalizedpower flow P/Po in the TE, mode when nf < nS, and V > 3. The
asymmetry coefficient is 0.
For asymmetrical waveguides (a = 10), the relationsbetween bi and P/Po are displayed in Fig. 14 and 15 forthe TEo and TE, modes, respectively. As previously,P/Po is highly nonlinear with bj. For situations wherenf < ns, computations of power plots at different valuesof asymmetry show that these curves differ significant-ly with asymmetry as is the case for the dispersioncurves.
In the following, we analyze the optical properties ofa planar waveguide where the nonlinear film mediumis a liquid crystal MBBA, characterized by a very largenonlinearity (ni = 2.460, d = 6 + 1 ptm, n2jI = 10- i 2!
W). We suppose that the film is bounded by twoidentical linear media (glass with n2 = 2.465) and thatthe wavelength is 0.488 ,gm (Ar+ laser). This configu-ration has been used recently16 to study the tunneling
Fig. 14. Relationship between parameterpower flow P/PO in the TEo mode when n
asymmetry coefficient
4.0.
pV 3.0.
0
2.0-
i 1.25 1.50 1.75 2.00
Fig. 15. Relationship between parameterpower flow PIPo in the TE, mode when n
asymmetry coefficient i
of reflected TE plane waves througia function of the input power. configuration as a planar wavepi
lations, without absorption, we assume indirectly acomplete transmission T = 1, since intensity of thefield at the interfaces z = 0 and z = d is the same forsymmetrical waveguides (a = 0). A more realisticmodel would have to include effects of absorption andalso saturation effects in the nonlinearity of the mate-rial. Universal parameterization of these effects is atopic for further study.
IV. Summary
/ Our analysis shows that the number of independentparameters determining the optical characteristics ofnonlinear thin film waveguides can be reduced to threeby the introduction of appropriately normalized pa-rameters and associated scaling rules. These parame-ters are the normalized film thickness V, the asymme-try coefficient a and the parameter b related to the
6 1.8 2.0 normalized power flow in the waveguide. By referringsimultaneously to the universal dispersion curves andthe normalized power curves P/Po, we get a concise
b, and the normalized overview of the characteristics of nonlinear film wave-f < n and V> 1. The guides for various designs: the dispersion curves indi-s 10. cate the range of possible values of bj that will yield a
physical solution at a fixed value of V and a, and thenormalized power curves indicates the total power car-ried by the waveguides for these values of bj.
For self-focusing film, when nf > n'i and V > 1.5, ourcomputations have shown that the characteristics ofthe nonlinear thin film waveguides vary slightly withthe asymmetry coefficient. For small values of b, P/
V = 3.0 P0 is nearly a linear function of bj. However, when V <1, even small variations in the asymmetry coefficientsignificantly affect the guiding properties and P/P ishighly nonlinear with b. The minimum of the P/P0(Vab) curves, as a function of b, corresponds tothe minimum value of bj from which the field begins tobe confined in the nonlinear film.
When nf < ns, the characteristics of nonlinear wave-guides vary significantly with the asymmetry coeffi-cient for all values of the effective thickness V. In this
2.25 2.50 2.75 case, P/PO(Va,b1) is not a linear function of b yet theminimum value of P/PO informs us on the minimumvalue of bj from which the field begins to be confined inthe nonlinear film. For cases where a > 1, physical
b, and the normalized solutions exist only for values of b > 1 and normalizedf < ns. and V > 3. The dispersion curves for different values of asymmetrys 10. coefficient may be obtained by solving Eq. (15). By
selecting values of V, b, a and b from these curves andusing formula Eq. (30) we can compute the relation of
a nonlinear film as bi to the power. If a < 1 and b < 1, only a slightIf we consider this modification is needed in Eqs. (15) and (30) as dis-uidA the striutu1re cussed in the text.
would be described by the universal parameters a =0, V 5.0 and PO N X 12 mW/mm. As shown in Figs.7 and 13, our numerical model predicts a TE1 surfacemode if the total power per unit length carried by thewaveguide is greater than 37 mW/mm. A straightfor-ward comparison with experimental results of Peschelet al.16 is not possible since our theoretical resultscorrespond to modes obtained by solving the nonlinearHelmholtz equation without absorption. In our calcu-
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