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Discrete structure of hybrid modes in nonlinear Kerr media
I March 1996
OPTICS
COMMUNICATIONS ELSEVIER
Optics Communications
124
( 1996) 373-382
Full length article
Discrete structure of hybrid modes in nonlinear Kerr media Jerzy Jasiliski Received 6 June 1995; revised version received 13 September
1995
Abstract In the paper the stationary nonlinear fields containing both TE and TM components of the same propagation constant are considered. The exact numerical solutions of Maxwell’s equations describing these nonlinear hybrid fields are obtained for different mechanisms of Kerr nonlinearity. The mode structure is determined by additional conditions satisfied by fields. The symmetry of the solutions in the unbounded Kerr medium or continuity of the fields at the interface between nonlinear and linear media specifies the discrete (with respect to polarization) system of hybrid modes of any order (modes with many maxima of the field in space). The dependence of the field profile, power flux and polarization of modes on propagation constant, type of nonlinearity, mode number and permittivity of the media is discussed.
1. Introduction The electromagnetic waves transmitted in nonlinear Kerr-type media reveal very interesting properties. Many of their features, giving a chance to control light by light. have been investigated very extensively in different planar systems [ l-l 1] Although such structures can support the pure TE [ 2-61 or pure TM fields [ 2-3.7- 121, the material equations enable the internal coupling between the TE and TM components. A similar effect appears in the linear media due to anisotropy or gyrotropy. but two normal solutions (in uniaxially anisotropic media the normal solutions describe the ordinary and the extraordinary waves) can interactwith each other only at the interface between two media. In Kerr materials, if both polarizations are present. coupling is obtained at every point of the medium. To describe TE-TM coupling several techniques have been used. The TE or TM-dominant approach [ 13-161 assumes a strong field of one polarization that forms a channel in which the linear, weak field of the opposite polarization is guided. Both waves travel with different 00%4018/96/$12.00 0 1996 Elsevier Science B.V. All rights reserved SSD10030.4018(9S)OOS81-I
propagating constants PTE and PTM. Consequently, at the &-AM plane we observe three regions. Between the lines describing the pure TE and pure TM polarizations, we obtain a region of the stationary TE-TM waves. This term means the resulting fields with the intermediate quotient of the TE to TM components. Moreover, the additional TE-TM conversion can be obtained by contact with the linear anisotropic [ 151 or gyrotropic [ 161 material. In [ 171 the TE-TM mode coupling in nonlinear planar waveguides is treated as the four-wave mixing obtained by means of two strong, orthogonally polarized bulk waves falling from outside. Snyder et al. [ 181 described the propagation of the hybrid nonlinear fields (they named the corresponding solutions the spatial solitons of Maxwell’s equations) of arbitrary polarization in Kerr media in the weak-guidance approximation. Vukovic and Dragila [ 19,201 considered the general properties of the hybrid TE-TM waves. They obtained the dispersion relation expressing the propagation constant by means of the boundary values of the electric field and came to the conclusion that the electric field vector at the inter-
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.I. Jusitiski /Optics
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face rotates with the adiabatic increase of power flux, passing through all possible hybrid states of polarization. In the present paper the exact hybrid solutions for different nonlinearity mechanisms are obtained by means of numerical integration of Maxwell’s equations with the help of the Runge-Kutta method. In the unbounded medium the imposition of the symmetry conditions and the conditions at infinity upon the obtained solutions leads to a set of discrete (with respect to polarization) hybrid modes. The same technique applied to fields in the Kerr medium in contact with the linear material (instead of symmetry conditions we should use the boundary conditions across the interface) gives a system of discrete, nonlinear, hybrid surface waves. All obtained solutions describe fields with a certain number (one or more) of maxima in space. Naturally, the hybrid modes can be interpreted as a particular case of stationary TE-TM [ 141 fields with p = /?kiE= PTM. They correspond to the modes of any order and one polarization propagating in a channel generated by the tield of another polarization and vice versa. By now the case of nonlinear coupling with the presence of the higher order modes has not been discussed. The discrete structure of modes means that instead of the continuous spectrum of the TE-TM stationary waves with polarizations lying between the pure TE and pure TM and different PTE and hM [ 141, for any common value of the propagation constant P = PTE = &&I we have a discrete number of solutions with specific polarization. Note that, since the maximum values of the fields in nonlinear media are defined by the nonlinearity coefficient, the power flux of the modes is also discrete. Therefore the conclusions obtained by Vukovic and Dragila [ 201 are not correctthe increase of the power flux results in increasing the propagation constant p of a particular mode. Naturally, since the polarization also depends on /3, we can control one of these quantities by means of the other, but the range of the possible changes is quite limited.
124 (I 996) 373-382
express the magnetic field (H,, H!, Hi) in electric field units and introduce the phase shift ~12 between z and other components [ 15,16,19,20]. In this way the electromagnetic field which is uniform in the y direction and of which all components are characterized by the same propagation constant /3 can be written in the form: E(x, ?‘, Z, t) = (E,(x),
E,.(x), S:(x)
Xexp(iw(Pzlc-t)) H(x, y, Z.‘,I) = -$ Xexp(iw(pzlc-t))
(la) (H,(x),
F,.(x).
Consider a wave of frequency w propagating along the: axis in a planar structure uniform in they direction. The structure consists of a semi-infinite nonlinear Kerr medium in contact with a linear dielectric. Let us
-iH,(x))
.
(lb)
According to [3] and [ 141, for the different mechanisms of nonlinearity the permittivity tensor, elements for the Kerr medium are given by: Ex1
(2b)
where E,_is the linear limit, cy> 0 the nonlinearity coefficient and y= 1 for thermal effects or electrostriction, y= l/3 for electronic distortion and y= -0.5 for molecular orientation. Although all field components in Maxwell’s equations are coupled by the relations (2)) the terms TE and Th4 components shall still be used to classify them. By now only one first integral of Maxwell’s equations with permittivity (2) is known [ 3,14,15,19,20] :
+( 1 + y)(E;+E.;)E;+2yE;E;) 2. The hybrid modes equations in Kerr medium
1
,
(3)
where C is the integration constant with value depending on the boundary’conditions. The first integral (3) enables us to reduce Maxwell’s equations to a system of the three first-order differential equations. Although Eq. (3) is quite complicated, it contains only even powers of the fields, exactly as the material equations
.I. Jasiriski /Optics
Cornrnunicarions
(2) do. Therefore, we can simplify the equations describing hybrid modes by introducing the new variables u, Y, R and Q proportional to the squares of they field components and the squares of their derivatives: 2 CfE’. u=--‘,+Y dE,. tiL ( dx ) ’ EL (4) The coefficients (Yand eL in definitions (4) have been used to normalize the new variables. In a similar way let us introduce the normalized propagation constant 21 and the normalized permittivities e,, e, and eZ: ej=
EJj -,j=x,y,z,. EL
In terms of the above normalized, dimensionless quantities we can easily express thexand z field components: Ef=qbu!(aet). E~=E,Q/(ae~), Hs=&buIa and H_‘=
I24 (1996) 373-382
31s
tions U, v and e,. Eq. (6~) can be rewritten in an alternative form, expressing directly the derivative of e, in terms of U, I, and e,. Differentiating R( ZA,v, e,) given by (7~) and solving the relation (6~) with respect to e_$we arrive at: ei=
&(Te$‘+(b+
5
(b-e.,))u’).
(6~‘) Although for the pure TE and pure TM waves the above equation can be solved analytically, giving the permittivity profile (for the TM case only in quadrature [ 8]), it is so complicated that for the hybrid case its usefulness is quite limited. Therefore for the numerical purpose we shall still apply Eq. (6~). Nevertheless Eq. (6~‘) enables us to treat the permittivity as a function of E; and H; only. Such a function e,( u, u) is a solution of the partial differential equation:
s_ au
l+y -2
e: 2bu + e.:)
’
uR 2
I/’ = 4uR ,
(6a)
u”=4uQ,
(6b)
R’= - (e,.-b)u’
,
(6~)
where prime is used to denote differentiation with res ect to the dimensionless x-coordinate [= w P E~X/C. The dielectric tensor components e? and eZ as well as the functions R and Q can be expressed in terms of U. v and e_,: e,.=l+
!+,-,)-
(ly)‘+ 4Y
.!I+? 2y
ef’
(7a) (7b)
V
(7c)
which means that Eqs. (6)-(7) give a system of three coupled nonlinear differential equations for three func-
b+y: (b-e,)
UQ
(6~“)
and the initial condition e,( 0, 0) = 1. The equations (6)-(7) express the ~1field components II and u, their derivatives R and Q, and permittivity components e,, e, and e, as functions of the transverse normalized coordinate 5. Note that the pure TE and TM cases are covered by these equations. For the pure TE fields we have u =O, Q = 0 which gives e,=e,=l +(l+y)u/2. e,.=l fu and R= (b - 1 )u - ~‘12. Eq. (6a) has the analytical solution U( 5) = 2ti/cosh’( a,$) with 0’ = b - I 121. Similarly, in the pure TM case we should write u = 0 and R = 0. After introducing (7d) into (6~‘) we obtain a differential equation with a solution for the thermal nonlinearity mechanism y = 1 that expressed e,( 5) as a quadrature, exactly the same as published by Leung [ 81 or Joseph and Christodoulides [ 91.
3. The solutions in an unbounded
medium
The solutions of the hybrid mode equations (6)) (7) depend on certain constants of integration. Their values depend on additional conditions satisfied by the fields. In this section we shall consider such conditions in the
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uniform, unbounded Kerr medium. Note that the geomctry of the system does not distinguish the positive and negative direction of the x axis and, moreover, all equations depend only on the squares of the fields E< and Hf. These two facts cause the functions E_:(x) and H:(x) to be symmetric or antisymmetric with respect to a certain point. From symmetry follows the existence of a certain constant of integration. Two such integration constants can be the position xOEof the point of symmetry of E,f and position xOMof the point of symmetry of H;’ (another possible explanation of the appearance of these integration constants is based on the observation that ?I is a cyclic coordinate in both wave equations). Because the hybrid field as a whole must also reveal symmetry, we conclude that xoE = xoM. The symmetry of the hybrid solution in the unbounded medium means that all four combinations of the functions E,.(x) and H,(x), namely {E,, H,.), (E,, -H,}, { -E,., H, ) and { -E,., -H,.}, are exactly equivalent. Two other integration constants (one of them is the constant C appearing in the function R (7~) and the first integral (3)) can be obtained by specifying the behavior of the TE and TM components for x + + 00. If both fields EJ and Hy and both the derivatives dE,l dr and dH,/dr vanish at infinity then both these constants will equal zero [ 3,14,19,20]. Therefore, in an unbounded Kerr medium all constants of integration are set, except for .~~~n = xoM=x<,, which is arbitrary. At this point, note that the maximum values of the field components, Eymaxand H,,,,, in the nonlinear medium are determined by the mode and system parameters (for the pure TE case we have E,,~,,=(~(~‘-E~)/(Y)“~ [ 21, while for y= 1 and pure TM case the maximum field is expressed by the maximum permittivity H,,,,;,, = l,,,,,,, ( ( E,,,, - eL) I( P’a) ) ’‘2, where G,,, =(/3(9/3’--8~,_)“~+3/3~)/(4~~) [8]).Consequently, the quotient of the TM to TE component is not arbitrary, as it is in the linear case. Let us take the polarization of the field defined by means of the normalized quantities u and V: U= -
II
11 +
1’
(8)
as a measure of this quotient (the polarization (T lies between 0 for the pure TE modes and 1 for the pure TM case). Therefore, the nonlinear hybrid modes in an unbounded Kerr medium are characterized by specific
polarization. The polarization changes with x, depends on the propagation constant /3 and, naturally, on the nonlinearity mechanism (described by -y). A given hybrid mode (that is a solution of a given type) may appear for only one value of the polarization, several different polarizations or none. The solutions of the hybrid equations (6)-( 7) can be obtained only numerically. In this paper these solutions have been found using the Runge-Kutta method; the particular functions have been calculated at 4 X IO“ to 3 X 10’ points. Starting from a certain point tini we arrive at a whole family of solutions depending on the assumed (Tini= a( ~i”i). AS the initial point bni we take the point of symmetry to discussed above or any point lying sufficiently far away from the field maxima ( .$+ + “) . The values of the fields near tini (and even the approximate, analytical expressions describing the fields) can be obtained by introducing the appropriate Taylor series (or asymptotic expansions for c-+ +E) into Eqs. (6)-( 7). But in the process of the numerical integration starting from the point of symmetry we obtain a solution that, in general, does not vanish at infinity. Similarly, starting from infinity, we arrive at a solution not satisfying the condition of symmetry. Therefore we must select among all obtained solutions only those which satisfy the above requirements. In this way we arrive at a set of solutions characterized by discrete values of (Tini. This does not mean that the all remaining solutions are unphysical. Some of them represent fields that can appear in systems without symmetry, for instance in multilayered media. Having functions describing field profiles, we can calculate the power flow per unit width carried by the hybrid nonlinear modes. This power flow can also be expressed in terms of the normalized variables: (9)
(the coefficient before the integral gives the power flow in SI units W/m). Naturally, the above formula establishes the relation P(p), so any quantity depending on /3 we can interpret as depending on power. The obtained hybrid numerical solutions can be divided into a few classes. The first class describes guided modes of any order, that is solutions exhibiting a certain number of maxima and minima in the field distribution and vanishing at infinity. Although the
J. Jasiriski / Optics Communications
Electrome d>stort,on -1
0 x
Molecular
ar,entat,an 1
2
[ml
Fig. I, The field distributions of three second-order hybrid modes obtained for different types of Kerr nonlinearity. h = 0.5 145 @m, ~,,=2.3716. a=6.4 p.m’/V’and p2= 1.065.~,_.
0 1.0
1.05
1.1
1.15
1.2
P’/s
Fig. 2. The powerflowof the firstand the second-orderhybrid modes against the propagation constant. A=0.5145 km, l,=2.3716 and 1x=6.4 pm’/V’.
classification of a particular solution as corresponding to the n-th order mode is sometimes ambiguous (because the number of extremes in E, and Hy field distributions can be different and, on the other hand, in certain solutions this number varies with fi), this type of solution satisfies all conditions necessary to represent the physical fields in the unbounded medium. In Fig. 1 we show three second-order solutions obtained for the same normalized propagation constant B = 1.065, but for three different mechanisms of nonlinearity. All existing hybrid solutions of the first and the second order (including the pure TE and TM modes) are given in Fig. 2, which illustrates the power flow P as a function of b. Note a few characteristic
377
I24 (I 996) 373-382
features of the hybrid fields. First, the amplitudes (and power flows) are largest for modes obtained for molecular orientation and smallest in those for thermal effects (this observation is valid for solutions of any order). In other words, the capacity of the nonlinear medium (where the term capacity means ability to accumulate energy) for molecular orientation is greater than that for electronic distortion, which, in turn, is greater than the capacity for thermal effects. This property can be explained as a consequence of different weight coefficients for E$ E_tand E: in the permittivity tensor element eu- (we can see from (9) that the power flow P increases with decreasing values of E,). Second, the power flow carried by the first-order hybrid modes is greater than power flows of the pure TM mode (and, of course, of the pure TE mode)-that is, the capacity of the medium for both types of fields (TE and TM components) together, is greater than for only one. In Fig. 3 we see a certain third-order mode in the medium with molecular orientation having an even greater amplitude than the pure TM mode! Therefore, addition of the TE (or TM) component to the existing TM (or TE) field always results in increasing the transmitted power and sometimes (only for molecular orientation) in increasing the amplitude of the existing component. Third, odd-order hybrid modes have both field components Ey and H? symmetric with respect to lo, while for even-order modes we have two types of symmetrysymmetric E, and antisymmetric Hy or antisymmetric Ey and symmetric H,.. The number of existing solutions depends on the type of symmetry, the nature of the
1.0
1.02
I .04
I .06
1.06
1.1
Fig. 3. Polarization and relative maximum values of the field components for the third-order hybrid mode versus propagation constant. A=0.5145 pm, l,_=2.3716, a=6.4 p,m’/V’ and y= I (molecular orientational mechanism of nonlinearity).
37s
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nonlinearity mechanism and, ofcourse, the mode number. We never obtain modes with both fields antisymmetric. Moreover. for the isotropic mechanism of nonlinearity ( y= I ) we have only even-order hybrid modes. The multiple solutions are characterized by difI’erent types of symmetry-one second-order hybrid mode obtained for molecular orientation has symmetric H, and antisymmetric E,. (Fig. I), two others exhibit the opposite symmetry. The higher-order modes have much more complex properties. In Fig. 3 we can observe two modes, characterized by two different polarizations, in a medium with molecular orientational nonlinearity that exists in the limited regime of the propagation constant. For increasing p both these modes close into each other and disappear at a certain PlllilY.The very interesting property of these two modes is degeneracy-both of them carry the same energy (with an accuracy of 0. I %). As we can conclude, the mode structure for molecular orientation is more complicated than the structure for other nonlinearity mechanisms. The second class of solutions describes fields periodic in tge .Xdirection. Such fields satisfy the symmetry conditions but do not vanish at infinity, so they can appear only in the multilayered systems, where the Kerr medium occupies a spatially bounded region. A specific periodic solution corresponding to the constant value of the nonlinear permittivity across the whole nonlinear medium of thermal type (e, = e! = e, = const) we are even able to describe analytically for any value of the integration constant C: L[= zl,,sin’R&
R=
1’ =
Q =
P,,COS’R5;
d2’z4,,cos2f25,
fl%,,sin’@
.
(10)
where LJ’=e-_, u,,=(2b-e)(e-1)/b, L’~)= e’(e-1)/h and C=(e-1)(3e+l-4b)/2 (e>b, e=(4bI )/3 for C=O). The constant value of the permittivity across the nonlinear medium means that the changes of the dielectric functions induced by two components E, and E: (both fields depend on the same function sin 0{) and the changes induced by E, (described by cos n[) balance each other. The special case of the above trigonometric solution applied to a TM nonlinear waveguide has been discussed in [21] and [ 221. There are many different types of periodic solutions-three or them are depicted in the phase plane E,-H, (Fig. 4).
124 (1996) 373-382
_0,4
Thermal effects Electronic distortion ~~~.l~,,~~~.,,,,,,/,,,,, -0.2 -0.1
--
E,
..
-.-.
.___’
Molecular orientation ,.,,,,,,.,,,,,,,,,,,,.,.i 0.0 0.1 0.2
[v/clml
Fig. 4. The phase trajectories corresponding to three periodic solutions and a fragment of the open phase trajectory (chaotic solution). All solutions were obtained for the material and held parameter values the same as for Fig. 1.
The third class of solutions corresponds to open trajectories in phase plane (Fig. 4). Such fields to not vanish at infinity nor exhibit any periodicity in space. Their field distribution depends very much on the initial values, so two solutions very close to each other in one region can differ very significantly away from this region. Therefore, we can interpret such solutions as spatial chaos. Since the energy carried by spatially chaotic fields in the unbounded medium is infinite, we should treat them as unphysical, although certain solutions from this class can describe physical nonlinear fields occurring in multilayered systems. The presented picture reveals certain properties of the mode structure of the hybrid fields. The appearance of the discrete guided modes, which is a consequence of the symmetry. can easily be understood if we consider the values of the fields at the common extremes. Since e, can be treated as function of 14 and 11(e.,( 14, ~1) is in principle the solution of the differential equation (6~“) ), then solving the set of equations R( ~1,I!, e.,( u, u)) =0 and Q(u, u, e_,(u, ~1)) =0 defining extreme values of the fields, we shall have discrete solutions. This is in accordance with the interpretation of the hybrid modes as a particular case of the mixed TE-TM stationary waves [ 3,141 (because for a given PTE we observe a continuous spectrum of the stationary fields differing with polarizations and &,, then by putting PTM = &, discrete polarizations result) but, unfortunately, it does not agree with the picture described by Vukovic and Dragila [ 19,201. We do not pass through
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/Optics
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all possible states of polarization by a continuous increase of the transmitted power for a fixed &for the case considered by them ( y= 1 and coupling between the first-order modes), no hybrid solution is obtained. Naturally, since both polarization and power depend on the propagation constant, we can observe the behavior reported in Ref. [ 201 for higher-order hybrid modes or other nonlinearity mechanism, but the change of polarization with power has a quite limited range (Fig. 3).
4. The nonlinear hybrid surface waves
Let us now assume that a linear medium with permittivity E, occupies the half-space x
(lla) H. H -2 ZE,, E,. E._ ‘
(1 lb)
Of the two above equations, only ( 1lb) involves the dielectric constant of the medium, cc. Eq. ( 1 la) does not contain lc, so it describes a quite general rule of simultaneous decay (or increase) of all field components in the linear medium. That is why this relation can be named a condition of spatial synchronization. Since the four field components that appear in ( 11) are continuous across the interface x = 0, these equations are also satisfied by the nonlinear field at the boundary. Therefore we can say that the hybrid fields in a nonlinear dielectric in contact with a semi-infinite linear medium must satisfy the condition of spatial synchronization ( I la) and one boundary condition ( 11b). Both these relations can be expressed in terms of the normalized quantities:
R u
124 (1996) 373-382
+e, f;
=b,
319
(12a)
( 12b) where eF= EJ+. Moreover, since the equations ( 12) are also satisfied by exponentially growing fields. we should take into account one more condition-for the physically acceptable solution describing the nonlinear hybrid surface waves in the vicinity of the interface all field components must diminish toward the linear medium. In the considered system there is no more symmetry, so the solutions of Eqs. (6)-( 7) and ( 12) do not show any symmetry either. Using the classification introduced in the previous Section, the nonlinear hybrid surface waves belong to spatially chaotic solutions (which means that there is no direct correspondence between the solutions in the unbounded and the bounded Kerr medium). Since the lack of symmetry causes xOE# xOM,we have two free constants of integration. In spite of the unbounded medium case, where one integration constant remained arbitrary, now Eqs. (12) specify both of them. But the positions of the extremes of the TE and TM components determines the boundary polarization as. Therefore, the nonlinear hybrid surface waves also exhibit a discrete spectrum of polarization. The continuity conditions in the form ( 12a)-( 12b) describe a certain interesting aspect of the hybrid nonlinear fields. In the pure TE or TM case. the field maximum can be situated at any distance from the boundary-its position depends on the relation between p. eL and E, only. But for the hybrid fields two more continuity conditions have to be satisfied, however only one more free parameter appears: the amplitude of the second field component in the linear medium. Therefore, not every point of the field can appear at the boundary. The condition of spatial synchronization ( 12a) defines the set of such points, while the boundary condition ( 12b) specifies the corresponding values of boundary polarization. Like in the unbounded Kerr medium, the hybrid surface solutions that vanish for x -+ + cc possess a certain number of maxima and minima in the nonlinear medium. This number defines the order of the solution. For a specific mechanism of nonlinearity the number of possible solutions changes with mode order, the
.I. Jmitiski
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hermal
/ Optics Communicutions
effects
Fig. 5. The power flow as a function of the propagation constant for the first and second order hybrid nonlinear surface waves. A = OS 145 pm, E, =2.3716. E,= i.OS.q,and cu=6.4 p,m’/V’
I”“,,“,
Linear
‘,‘_111-1,1,,.
substrate
1
Kerr medium
i
~> ’
-0.2 -2
-1
--. 1
0
x
Thermal effects Electronic distortion
I
:
\_,’ _““““‘~““““_““““““““““““““’
2
3
124 (1996) 373-382
and TM components very close to each other. Such a relation between fields means that both of them together generate certain regions of increased permittivity (partially overlapping in space) in which they both are focused. This model of propagation is characteristic for hybrid modes in the unbounded Kerr medium. The presence of the boundary shifts the positions of the held maxima and their values quite slightly. On the other hand, for the second solution presented in Fig. 6, obtained for thermal effects, the changes are very significant: the TM component practically adjoins the boundary, while the maximum of the TE component lies at a certain distance. In other words, the boundary polarization approaches unity, while the polarization at infinity is close to zero.Therefore we can say that the propagating mode produces two parallel waveguides. separated in space: one generated by a strong TE component in which a weak TM component is transmitted and the other generated by a strong TM component and carrying a weak TE field (Fig. 7). This model of propagation is similar to the picture introduced in the TE and TM-dominant approach [ 13-161, but since we have two separate channels and the same propagating constant for both TE and TM components, a more detailed comparison is difficult. The effect of generation of the separate channels by the fields needs lC to be close to E,_and p’ close to E,. It can also appear for other mechanisms of nonlinearity and for higher mode orders, but it is not so spectacular as for the secondorder mode in thermal media. In Fig. 7 we see a certain
[ml
Fig. 6. The field distributions of the second-order nonlinear hybrid surface waves. p’= 1.06S~et.. h=0.5145 pm. 6, =2.3716, e, = i.OS~e, and cu=6.4 pm’/V’.
propagation constant and the permittivity of the linear medium. But the number of parameters is greater than in the unbounded Kerr case, so the behavior of the solutions is a bit more complicated. Although for l, close to lL.we obtain a quite similar structure of modes (at least for the lowest orders; see Fig. 5). the field distributions can differ very much from the fields of corresponding solutions in the unbounded medium. In Fig. 6 we see such distributions for two second-order surface waves. One of them, obtained for the electronic distortional nonlinearity, is very similar to the mode in the unbounded medium, illustrated in Fig. 1. It describes the hybrid fields with the maxima of the TE
x [ml Fig. 7. The permittivity distributions inside the nonlinear medium generated by two nonlinear hybrid surface modes. Both modes correspond to the field and material parameter values as for Fig. 6.
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fourth-order molecular orientational surface mode, in which we can observe four such channels-two of them, generated by strong TM fields, are situated between two others induced by a TE component. Note that the TM peaks are higher that the TE ones (for the secondorder thermal case the TE and TM peaks have almost equal heights) and that the separation is not so perfect as in the thermal case. The higher-orders surface modes or modes obtained for larger permittivity of the linear substrate can exhibit a greater difference with respect to the mode structure in the unbounded nonlinear dielectric. For instance, for E, = 1.05 eL.we get only one third-order surface mode for molecular orientational nonlinearity, while for electronic distortional there are three modes-in the unbounded medium we have the opposite situation. Moreover, two degenerate third-order modes for molecular orientation have disappeared, but not the possibility of degeneracy-the three third-order hybrid surface modes for electronic distortion mentioned above carry the same energy. Similarly, the other features of the structure remain-the energetic capacity of the medium for the hybrid field is greater than for the pure TE or TM boundary solution and for the thermal nonlinearity we can obtain only even-order modes.
5. Summary
and conclusions
In the unbounded nonlinear Kerr medium and in the vicinity of the boundary between the nonlinear and linear media, hybrid fields can exist for any nonlinearity mechanism. These hybrid fields possess TE and TM components coupled through the material equations and propagate as an entity with the same propagation constant. Since the physical fields must vanish at infinity and satisfy symmetry conditions in the unbounded medium (or continuity conditions at the interface between the two media), the hybrid solutions of Maxwell’s equations can appear only for certain values of polarization. The solutions that do not satisfy one or both of the above conditions also exist-some of them can, at the phase plane E,.-I$, be represented by closed trajectories, the others by open curves (Fig. 4). The fields corresponding to the closed trajectories are periodic in space and one of these periodic hybrid solution can even be expressed analytically (Eqs. (10)). On the other hand. the fields represented by open phase
124 (1996) 373-382
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trajectories describe spatial chaos. Such non-decaying solutions can appear in multilayered structures. The physical hybrid solutions in the Kerr region may have more than one field maximum (Figs. I,6 and 7). That is why they can be interpreted as n-th order guided or surface modes. For certain orders we obtain more than one solution-each of them corresponds to a different polarization. The very interesting fact is that certain multiple modes are degenerate-two or more solutions carry the same energy (within an accuracy of 0.1%). The degenerate modes appear both in the unbounded medium and near the boundary with the linear dielectric and are solutions of at least third order. The energy carried by hybrid waves always exceeds the power flow of the pure TE and pure TM modes. Since the carried energy depends on the propagation constant as does polarization, we can change the polarization of the field by means of the power flow. The most complex hybrid mode structure is obtained for the molecular orientational mechanism of nonlinearity, the simplest for the thermal or electrostrictive mechanism. In this case only modes of even order exist. This property also exists for the nonlinear hybrid surface waves, at least when the dielectric constants of the media are close to each other (Figs. 2 and 5). The electronic distortional or molecular orientational nonlinearity enables existence of the hybrid modes of any order. In the unbounded medium the model of propagation dominates in which TE and TM components together create a certain number of overlapping waveguides inside which they are transmitted. For the hybrid surface waves we can obtain another situation. When the propagation constant is close to its minimum value, certain modes induce parallel.channels more isolated in space (Fig. 7) in which the TE and TM components propagate separately. This phenomenon, similar to the effect occurring in the description of the stationary TETM waves in the TE and TM-dominant approach, is most apparent for the second-order mode and for the thermal nonlinearity case. Comparing the obtained results with the properties of the stationary TE-TM fields discussed by Boardman and Twardowski [ 141, we can say that a few of their statements are too general, while a few others are too limited. The regime of TE-TM solutions obtained, which makes the appearance of the hybrid fields with PTE = &,,, impossible, concerns only the thermal (elec-
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trostrictivc) nonlinearity type and only the coupling hetwecn the first-order TE and the first-order TM modes (this particular case was considered by them as 3n example). It is in perfect agreement with the results ohtaincd in Section 3 and Section 4 of the present paper-the odd-order hybrid fields that vanish at infinity do not exist for y= 1. But in the region of the nonstationary solutions (one of them, obtained for y= 1, is the trigonometric analytical solution ( 10) ), certain specific hybrid fields appear. They exhibit more than one maximum in space and vanish for x+ +m, i.e. they are the stationary higher-order TE-TM modes. The other types of Kerr nonlinearity extend boundaries of the stationary TE-TM regime, so this region contains the first-order hybrid solution with PTE = &,. Therefore, the regime of all possible stationary solutions of the coupled Maxwell equations is wider than Boardman and Twardowski reported. Although the problem of stability of the obtained hybrid solutions is difficult to examine exactly, discrete modes of a few lowest orders in the unbounded medium and at the boundary between the two media should be stable at the branches where dP/d/3> 0. Qualitative analysis is based on the comparison of the energy of close-by solutions. Since in the sufficiently close vicinity of any discrete mode (maybe except for certain higher-order of multiple solutions) only solutions that do not vanish at infinity appear, a given hybrid guided or surface mode carries the smallest power of them all, so it ought to be stable.
I24 (I 996) 373-382
]2] G.I. Stegeman, C.T. Seaton. J. Ariyasu. R.F. Wallis and A.A. Maradudin, J. Appl. Phys. 158 ( 1985) 2543. 131 A. Boardman, P. Egan, F. Lrderer, U. Langbein and D. Mihalache. in: Modem Problems in Condensed Matter Physics. eds. V.M. Agranovich. A.A. Maradudin, H.E. Ponath and G.I. Stegeman (North-Holland, Amsterdam. 1987) pp. 160-192. 14 ] A.D. Boardman and P. Egan, IEEE J. Quant. Electron., QE-22 (1986) 319. ]5] W. Chen and A.A. Maradudin, J. Opt. Sot. Am. B 5 (1988) 529. [6] A.D. Boardman, M.M. Shabat and R.F. Wallis, J. Phys. D 24 (1991) 1702. [ 7 1F. Lederer. U. Langbein and H.-E. Ponath. Appl. Phys. B 3 1 (1983) 187. 181 K.M. Leung. Phys. Rev. B 32 (1985) 5093. [ 91 R.1. Joseph and D.N. Christodoulides. Optics Lett. I? ( 1987) 826. [IO] A.D. Boardman, A.A. Maradudin. G.I. Stegeman. T. Twardowski and E.M. Wright, Phys. Rev. A 35 ( 1987) 1159. 1I1 1Mihalache. G.1. Stegeman, C.T. Seaton, E.M. Wright, R. Zanoni, A.D. Boardman and T. Twardowski. Optics Lett. 12, 3 (1987) 187. [ 121J. Jasidski and K. Gniadek,Opt. Quantum Electron. 26 ( 1994) 865. [ 13) A.D. Boardman and T. Twardowski, J. Opt. Sot. Am. B 5 (1988) 523. [ 141 A.D. Boardman and T. Twardowski, Phys. Rev. A 39 ( 1989) 2481. 151 J.P. Torres, L. Tomer, F. Lederer. D. Mihalache. D. Baboiu and M. Ciumac, J. Opt. Sot. Am. B 11 ( 1994) 983. 161 N.N. Akhmediev. A.K. Zvezdin. V.I. Korneev and N.V. Mitskevich. Sov. Phys. Solid State 30 ( 1988) 1337. 171 E. Weiner--Raczka, W. Leutheuser and F. Lederer, Optics Comm. 81 (1991) 162. 181 A.W. Snyder, D.J. Mitchell and Y. Chen. Optics Lett. 19 (1994) s24.
[ 191 S. Vukovic and R. Dragila, Optics Lett. 14 ( 1989) 529. References 1I 1C.T. Seaton. Xu Mai. G.1. Stegeman and H.G. Winfui. Optical Engin. 24 ( 1985) 593.
]20] S. Vukovic and R. Dragila, Optics Lett. 1.5 ( 1990) 168. I21 ] J. Jasiriski, Optica Applicata, Vol. XXIV ( 1994) 287. [22] J. Jasiriski, “Balanced modes approximation for TM fields in Kerr thin films”, J. Opt. Sot. Am. B. in print.
OPTICS
COMMUNICATIONS ELSEVIER
Optics Communications
124
( 1996) 373-382
Full length article
Discrete structure of hybrid modes in nonlinear Kerr media Jerzy Jasiliski Received 6 June 1995; revised version received 13 September
1995
Abstract In the paper the stationary nonlinear fields containing both TE and TM components of the same propagation constant are considered. The exact numerical solutions of Maxwell’s equations describing these nonlinear hybrid fields are obtained for different mechanisms of Kerr nonlinearity. The mode structure is determined by additional conditions satisfied by fields. The symmetry of the solutions in the unbounded Kerr medium or continuity of the fields at the interface between nonlinear and linear media specifies the discrete (with respect to polarization) system of hybrid modes of any order (modes with many maxima of the field in space). The dependence of the field profile, power flux and polarization of modes on propagation constant, type of nonlinearity, mode number and permittivity of the media is discussed.
1. Introduction The electromagnetic waves transmitted in nonlinear Kerr-type media reveal very interesting properties. Many of their features, giving a chance to control light by light. have been investigated very extensively in different planar systems [ l-l 1] Although such structures can support the pure TE [ 2-61 or pure TM fields [ 2-3.7- 121, the material equations enable the internal coupling between the TE and TM components. A similar effect appears in the linear media due to anisotropy or gyrotropy. but two normal solutions (in uniaxially anisotropic media the normal solutions describe the ordinary and the extraordinary waves) can interactwith each other only at the interface between two media. In Kerr materials, if both polarizations are present. coupling is obtained at every point of the medium. To describe TE-TM coupling several techniques have been used. The TE or TM-dominant approach [ 13-161 assumes a strong field of one polarization that forms a channel in which the linear, weak field of the opposite polarization is guided. Both waves travel with different 00%4018/96/$12.00 0 1996 Elsevier Science B.V. All rights reserved SSD10030.4018(9S)OOS81-I
propagating constants PTE and PTM. Consequently, at the &-AM plane we observe three regions. Between the lines describing the pure TE and pure TM polarizations, we obtain a region of the stationary TE-TM waves. This term means the resulting fields with the intermediate quotient of the TE to TM components. Moreover, the additional TE-TM conversion can be obtained by contact with the linear anisotropic [ 151 or gyrotropic [ 161 material. In [ 171 the TE-TM mode coupling in nonlinear planar waveguides is treated as the four-wave mixing obtained by means of two strong, orthogonally polarized bulk waves falling from outside. Snyder et al. [ 181 described the propagation of the hybrid nonlinear fields (they named the corresponding solutions the spatial solitons of Maxwell’s equations) of arbitrary polarization in Kerr media in the weak-guidance approximation. Vukovic and Dragila [ 19,201 considered the general properties of the hybrid TE-TM waves. They obtained the dispersion relation expressing the propagation constant by means of the boundary values of the electric field and came to the conclusion that the electric field vector at the inter-
37.4
.I. Jusitiski /Optics
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face rotates with the adiabatic increase of power flux, passing through all possible hybrid states of polarization. In the present paper the exact hybrid solutions for different nonlinearity mechanisms are obtained by means of numerical integration of Maxwell’s equations with the help of the Runge-Kutta method. In the unbounded medium the imposition of the symmetry conditions and the conditions at infinity upon the obtained solutions leads to a set of discrete (with respect to polarization) hybrid modes. The same technique applied to fields in the Kerr medium in contact with the linear material (instead of symmetry conditions we should use the boundary conditions across the interface) gives a system of discrete, nonlinear, hybrid surface waves. All obtained solutions describe fields with a certain number (one or more) of maxima in space. Naturally, the hybrid modes can be interpreted as a particular case of stationary TE-TM [ 141 fields with p = /?kiE= PTM. They correspond to the modes of any order and one polarization propagating in a channel generated by the tield of another polarization and vice versa. By now the case of nonlinear coupling with the presence of the higher order modes has not been discussed. The discrete structure of modes means that instead of the continuous spectrum of the TE-TM stationary waves with polarizations lying between the pure TE and pure TM and different PTE and hM [ 141, for any common value of the propagation constant P = PTE = &&I we have a discrete number of solutions with specific polarization. Note that, since the maximum values of the fields in nonlinear media are defined by the nonlinearity coefficient, the power flux of the modes is also discrete. Therefore the conclusions obtained by Vukovic and Dragila [ 201 are not correctthe increase of the power flux results in increasing the propagation constant p of a particular mode. Naturally, since the polarization also depends on /3, we can control one of these quantities by means of the other, but the range of the possible changes is quite limited.
124 (I 996) 373-382
express the magnetic field (H,, H!, Hi) in electric field units and introduce the phase shift ~12 between z and other components [ 15,16,19,20]. In this way the electromagnetic field which is uniform in the y direction and of which all components are characterized by the same propagation constant /3 can be written in the form: E(x, ?‘, Z, t) = (E,(x),
E,.(x), S:(x)
Xexp(iw(Pzlc-t)) H(x, y, Z.‘,I) = -$ Xexp(iw(pzlc-t))
(la) (H,(x),
F,.(x).
Consider a wave of frequency w propagating along the: axis in a planar structure uniform in they direction. The structure consists of a semi-infinite nonlinear Kerr medium in contact with a linear dielectric. Let us
-iH,(x))
.
(lb)
According to [3] and [ 141, for the different mechanisms of nonlinearity the permittivity tensor, elements for the Kerr medium are given by: Ex1
(2b)
where E,_is the linear limit, cy> 0 the nonlinearity coefficient and y= 1 for thermal effects or electrostriction, y= l/3 for electronic distortion and y= -0.5 for molecular orientation. Although all field components in Maxwell’s equations are coupled by the relations (2)) the terms TE and Th4 components shall still be used to classify them. By now only one first integral of Maxwell’s equations with permittivity (2) is known [ 3,14,15,19,20] :
+( 1 + y)(E;+E.;)E;+2yE;E;) 2. The hybrid modes equations in Kerr medium
1
,
(3)
where C is the integration constant with value depending on the boundary’conditions. The first integral (3) enables us to reduce Maxwell’s equations to a system of the three first-order differential equations. Although Eq. (3) is quite complicated, it contains only even powers of the fields, exactly as the material equations
.I. Jasiriski /Optics
Cornrnunicarions
(2) do. Therefore, we can simplify the equations describing hybrid modes by introducing the new variables u, Y, R and Q proportional to the squares of they field components and the squares of their derivatives: 2 CfE’. u=--‘,+Y dE,. tiL ( dx ) ’ EL (4) The coefficients (Yand eL in definitions (4) have been used to normalize the new variables. In a similar way let us introduce the normalized propagation constant 21 and the normalized permittivities e,, e, and eZ: ej=
EJj -,j=x,y,z,. EL
In terms of the above normalized, dimensionless quantities we can easily express thexand z field components: Ef=qbu!(aet). E~=E,Q/(ae~), Hs=&buIa and H_‘=
I24 (1996) 373-382
31s
tions U, v and e,. Eq. (6~) can be rewritten in an alternative form, expressing directly the derivative of e, in terms of U, I, and e,. Differentiating R( ZA,v, e,) given by (7~) and solving the relation (6~) with respect to e_$we arrive at: ei=
&(Te$‘+(b+
5
(b-e.,))u’).
(6~‘) Although for the pure TE and pure TM waves the above equation can be solved analytically, giving the permittivity profile (for the TM case only in quadrature [ 8]), it is so complicated that for the hybrid case its usefulness is quite limited. Therefore for the numerical purpose we shall still apply Eq. (6~). Nevertheless Eq. (6~‘) enables us to treat the permittivity as a function of E; and H; only. Such a function e,( u, u) is a solution of the partial differential equation:
s_ au
l+y -2
e: 2bu + e.:)
’
uR 2
I/’ = 4uR ,
(6a)
u”=4uQ,
(6b)
R’= - (e,.-b)u’
,
(6~)
where prime is used to denote differentiation with res ect to the dimensionless x-coordinate [= w P E~X/C. The dielectric tensor components e? and eZ as well as the functions R and Q can be expressed in terms of U. v and e_,: e,.=l+
!+,-,)-
(ly)‘+ 4Y
.!I+? 2y
ef’
(7a) (7b)
V
(7c)
which means that Eqs. (6)-(7) give a system of three coupled nonlinear differential equations for three func-
b+y: (b-e,)
UQ
(6~“)
and the initial condition e,( 0, 0) = 1. The equations (6)-(7) express the ~1field components II and u, their derivatives R and Q, and permittivity components e,, e, and e, as functions of the transverse normalized coordinate 5. Note that the pure TE and TM cases are covered by these equations. For the pure TE fields we have u =O, Q = 0 which gives e,=e,=l +(l+y)u/2. e,.=l fu and R= (b - 1 )u - ~‘12. Eq. (6a) has the analytical solution U( 5) = 2ti/cosh’( a,$) with 0’ = b - I 121. Similarly, in the pure TM case we should write u = 0 and R = 0. After introducing (7d) into (6~‘) we obtain a differential equation with a solution for the thermal nonlinearity mechanism y = 1 that expressed e,( 5) as a quadrature, exactly the same as published by Leung [ 81 or Joseph and Christodoulides [ 91.
3. The solutions in an unbounded
medium
The solutions of the hybrid mode equations (6)) (7) depend on certain constants of integration. Their values depend on additional conditions satisfied by the fields. In this section we shall consider such conditions in the
J. Jusin’ski /Optics Commur~ications 124 (1996) 373-382
376
uniform, unbounded Kerr medium. Note that the geomctry of the system does not distinguish the positive and negative direction of the x axis and, moreover, all equations depend only on the squares of the fields E< and Hf. These two facts cause the functions E_:(x) and H:(x) to be symmetric or antisymmetric with respect to a certain point. From symmetry follows the existence of a certain constant of integration. Two such integration constants can be the position xOEof the point of symmetry of E,f and position xOMof the point of symmetry of H;’ (another possible explanation of the appearance of these integration constants is based on the observation that ?I is a cyclic coordinate in both wave equations). Because the hybrid field as a whole must also reveal symmetry, we conclude that xoE = xoM. The symmetry of the hybrid solution in the unbounded medium means that all four combinations of the functions E,.(x) and H,(x), namely {E,, H,.), (E,, -H,}, { -E,., H, ) and { -E,., -H,.}, are exactly equivalent. Two other integration constants (one of them is the constant C appearing in the function R (7~) and the first integral (3)) can be obtained by specifying the behavior of the TE and TM components for x + + 00. If both fields EJ and Hy and both the derivatives dE,l dr and dH,/dr vanish at infinity then both these constants will equal zero [ 3,14,19,20]. Therefore, in an unbounded Kerr medium all constants of integration are set, except for .~~~n = xoM=x<,, which is arbitrary. At this point, note that the maximum values of the field components, Eymaxand H,,,,, in the nonlinear medium are determined by the mode and system parameters (for the pure TE case we have E,,~,,=(~(~‘-E~)/(Y)“~ [ 21, while for y= 1 and pure TM case the maximum field is expressed by the maximum permittivity H,,,,;,, = l,,,,,,, ( ( E,,,, - eL) I( P’a) ) ’‘2, where G,,, =(/3(9/3’--8~,_)“~+3/3~)/(4~~) [8]).Consequently, the quotient of the TM to TE component is not arbitrary, as it is in the linear case. Let us take the polarization of the field defined by means of the normalized quantities u and V: U= -
II
11 +
1’
(8)
as a measure of this quotient (the polarization (T lies between 0 for the pure TE modes and 1 for the pure TM case). Therefore, the nonlinear hybrid modes in an unbounded Kerr medium are characterized by specific
polarization. The polarization changes with x, depends on the propagation constant /3 and, naturally, on the nonlinearity mechanism (described by -y). A given hybrid mode (that is a solution of a given type) may appear for only one value of the polarization, several different polarizations or none. The solutions of the hybrid equations (6)-( 7) can be obtained only numerically. In this paper these solutions have been found using the Runge-Kutta method; the particular functions have been calculated at 4 X IO“ to 3 X 10’ points. Starting from a certain point tini we arrive at a whole family of solutions depending on the assumed (Tini= a( ~i”i). AS the initial point bni we take the point of symmetry to discussed above or any point lying sufficiently far away from the field maxima ( .$+ + “) . The values of the fields near tini (and even the approximate, analytical expressions describing the fields) can be obtained by introducing the appropriate Taylor series (or asymptotic expansions for c-+ +E) into Eqs. (6)-( 7). But in the process of the numerical integration starting from the point of symmetry we obtain a solution that, in general, does not vanish at infinity. Similarly, starting from infinity, we arrive at a solution not satisfying the condition of symmetry. Therefore we must select among all obtained solutions only those which satisfy the above requirements. In this way we arrive at a set of solutions characterized by discrete values of (Tini. This does not mean that the all remaining solutions are unphysical. Some of them represent fields that can appear in systems without symmetry, for instance in multilayered media. Having functions describing field profiles, we can calculate the power flow per unit width carried by the hybrid nonlinear modes. This power flow can also be expressed in terms of the normalized variables: (9)
(the coefficient before the integral gives the power flow in SI units W/m). Naturally, the above formula establishes the relation P(p), so any quantity depending on /3 we can interpret as depending on power. The obtained hybrid numerical solutions can be divided into a few classes. The first class describes guided modes of any order, that is solutions exhibiting a certain number of maxima and minima in the field distribution and vanishing at infinity. Although the
J. Jasiriski / Optics Communications
Electrome d>stort,on -1
0 x
Molecular
ar,entat,an 1
2
[ml
Fig. I, The field distributions of three second-order hybrid modes obtained for different types of Kerr nonlinearity. h = 0.5 145 @m, ~,,=2.3716. a=6.4 p.m’/V’and p2= 1.065.~,_.
0 1.0
1.05
1.1
1.15
1.2
P’/s
Fig. 2. The powerflowof the firstand the second-orderhybrid modes against the propagation constant. A=0.5145 km, l,=2.3716 and 1x=6.4 pm’/V’.
classification of a particular solution as corresponding to the n-th order mode is sometimes ambiguous (because the number of extremes in E, and Hy field distributions can be different and, on the other hand, in certain solutions this number varies with fi), this type of solution satisfies all conditions necessary to represent the physical fields in the unbounded medium. In Fig. 1 we show three second-order solutions obtained for the same normalized propagation constant B = 1.065, but for three different mechanisms of nonlinearity. All existing hybrid solutions of the first and the second order (including the pure TE and TM modes) are given in Fig. 2, which illustrates the power flow P as a function of b. Note a few characteristic
377
I24 (I 996) 373-382
features of the hybrid fields. First, the amplitudes (and power flows) are largest for modes obtained for molecular orientation and smallest in those for thermal effects (this observation is valid for solutions of any order). In other words, the capacity of the nonlinear medium (where the term capacity means ability to accumulate energy) for molecular orientation is greater than that for electronic distortion, which, in turn, is greater than the capacity for thermal effects. This property can be explained as a consequence of different weight coefficients for E$ E_tand E: in the permittivity tensor element eu- (we can see from (9) that the power flow P increases with decreasing values of E,). Second, the power flow carried by the first-order hybrid modes is greater than power flows of the pure TM mode (and, of course, of the pure TE mode)-that is, the capacity of the medium for both types of fields (TE and TM components) together, is greater than for only one. In Fig. 3 we see a certain third-order mode in the medium with molecular orientation having an even greater amplitude than the pure TM mode! Therefore, addition of the TE (or TM) component to the existing TM (or TE) field always results in increasing the transmitted power and sometimes (only for molecular orientation) in increasing the amplitude of the existing component. Third, odd-order hybrid modes have both field components Ey and H? symmetric with respect to lo, while for even-order modes we have two types of symmetrysymmetric E, and antisymmetric Hy or antisymmetric Ey and symmetric H,.. The number of existing solutions depends on the type of symmetry, the nature of the
1.0
1.02
I .04
I .06
1.06
1.1
Fig. 3. Polarization and relative maximum values of the field components for the third-order hybrid mode versus propagation constant. A=0.5145 pm, l,_=2.3716, a=6.4 p,m’/V’ and y= I (molecular orientational mechanism of nonlinearity).
37s
J. Jmiriski /Optics
Comtnunicutims
nonlinearity mechanism and, ofcourse, the mode number. We never obtain modes with both fields antisymmetric. Moreover. for the isotropic mechanism of nonlinearity ( y= I ) we have only even-order hybrid modes. The multiple solutions are characterized by difI’erent types of symmetry-one second-order hybrid mode obtained for molecular orientation has symmetric H, and antisymmetric E,. (Fig. I), two others exhibit the opposite symmetry. The higher-order modes have much more complex properties. In Fig. 3 we can observe two modes, characterized by two different polarizations, in a medium with molecular orientational nonlinearity that exists in the limited regime of the propagation constant. For increasing p both these modes close into each other and disappear at a certain PlllilY.The very interesting property of these two modes is degeneracy-both of them carry the same energy (with an accuracy of 0. I %). As we can conclude, the mode structure for molecular orientation is more complicated than the structure for other nonlinearity mechanisms. The second class of solutions describes fields periodic in tge .Xdirection. Such fields satisfy the symmetry conditions but do not vanish at infinity, so they can appear only in the multilayered systems, where the Kerr medium occupies a spatially bounded region. A specific periodic solution corresponding to the constant value of the nonlinear permittivity across the whole nonlinear medium of thermal type (e, = e! = e, = const) we are even able to describe analytically for any value of the integration constant C: L[= zl,,sin’R&
R=
1’ =
Q =
P,,COS’R5;
d2’z4,,cos2f25,
fl%,,sin’@
.
(10)
where LJ’=e-_, u,,=(2b-e)(e-1)/b, L’~)= e’(e-1)/h and C=(e-1)(3e+l-4b)/2 (e>b, e=(4bI )/3 for C=O). The constant value of the permittivity across the nonlinear medium means that the changes of the dielectric functions induced by two components E, and E: (both fields depend on the same function sin 0{) and the changes induced by E, (described by cos n[) balance each other. The special case of the above trigonometric solution applied to a TM nonlinear waveguide has been discussed in [21] and [ 221. There are many different types of periodic solutions-three or them are depicted in the phase plane E,-H, (Fig. 4).
124 (1996) 373-382
_0,4
Thermal effects Electronic distortion ~~~.l~,,~~~.,,,,,,/,,,,, -0.2 -0.1
--
E,
..
-.-.
.___’
Molecular orientation ,.,,,,,,.,,,,,,,,,,,,.,.i 0.0 0.1 0.2
[v/clml
Fig. 4. The phase trajectories corresponding to three periodic solutions and a fragment of the open phase trajectory (chaotic solution). All solutions were obtained for the material and held parameter values the same as for Fig. 1.
The third class of solutions corresponds to open trajectories in phase plane (Fig. 4). Such fields to not vanish at infinity nor exhibit any periodicity in space. Their field distribution depends very much on the initial values, so two solutions very close to each other in one region can differ very significantly away from this region. Therefore, we can interpret such solutions as spatial chaos. Since the energy carried by spatially chaotic fields in the unbounded medium is infinite, we should treat them as unphysical, although certain solutions from this class can describe physical nonlinear fields occurring in multilayered systems. The presented picture reveals certain properties of the mode structure of the hybrid fields. The appearance of the discrete guided modes, which is a consequence of the symmetry. can easily be understood if we consider the values of the fields at the common extremes. Since e, can be treated as function of 14 and 11(e.,( 14, ~1) is in principle the solution of the differential equation (6~“) ), then solving the set of equations R( ~1,I!, e.,( u, u)) =0 and Q(u, u, e_,(u, ~1)) =0 defining extreme values of the fields, we shall have discrete solutions. This is in accordance with the interpretation of the hybrid modes as a particular case of the mixed TE-TM stationary waves [ 3,141 (because for a given PTE we observe a continuous spectrum of the stationary fields differing with polarizations and &,, then by putting PTM = &, discrete polarizations result) but, unfortunately, it does not agree with the picture described by Vukovic and Dragila [ 19,201. We do not pass through
J. Jasihki
/Optics
Communications
all possible states of polarization by a continuous increase of the transmitted power for a fixed &for the case considered by them ( y= 1 and coupling between the first-order modes), no hybrid solution is obtained. Naturally, since both polarization and power depend on the propagation constant, we can observe the behavior reported in Ref. [ 201 for higher-order hybrid modes or other nonlinearity mechanism, but the change of polarization with power has a quite limited range (Fig. 3).
4. The nonlinear hybrid surface waves
Let us now assume that a linear medium with permittivity E, occupies the half-space x
(lla) H. H -2 ZE,, E,. E._ ‘
(1 lb)
Of the two above equations, only ( 1lb) involves the dielectric constant of the medium, cc. Eq. ( 1 la) does not contain lc, so it describes a quite general rule of simultaneous decay (or increase) of all field components in the linear medium. That is why this relation can be named a condition of spatial synchronization. Since the four field components that appear in ( 11) are continuous across the interface x = 0, these equations are also satisfied by the nonlinear field at the boundary. Therefore we can say that the hybrid fields in a nonlinear dielectric in contact with a semi-infinite linear medium must satisfy the condition of spatial synchronization ( I la) and one boundary condition ( 11b). Both these relations can be expressed in terms of the normalized quantities:
R u
124 (1996) 373-382
+e, f;
=b,
319
(12a)
( 12b) where eF= EJ+. Moreover, since the equations ( 12) are also satisfied by exponentially growing fields. we should take into account one more condition-for the physically acceptable solution describing the nonlinear hybrid surface waves in the vicinity of the interface all field components must diminish toward the linear medium. In the considered system there is no more symmetry, so the solutions of Eqs. (6)-( 7) and ( 12) do not show any symmetry either. Using the classification introduced in the previous Section, the nonlinear hybrid surface waves belong to spatially chaotic solutions (which means that there is no direct correspondence between the solutions in the unbounded and the bounded Kerr medium). Since the lack of symmetry causes xOE# xOM,we have two free constants of integration. In spite of the unbounded medium case, where one integration constant remained arbitrary, now Eqs. (12) specify both of them. But the positions of the extremes of the TE and TM components determines the boundary polarization as. Therefore, the nonlinear hybrid surface waves also exhibit a discrete spectrum of polarization. The continuity conditions in the form ( 12a)-( 12b) describe a certain interesting aspect of the hybrid nonlinear fields. In the pure TE or TM case. the field maximum can be situated at any distance from the boundary-its position depends on the relation between p. eL and E, only. But for the hybrid fields two more continuity conditions have to be satisfied, however only one more free parameter appears: the amplitude of the second field component in the linear medium. Therefore, not every point of the field can appear at the boundary. The condition of spatial synchronization ( 12a) defines the set of such points, while the boundary condition ( 12b) specifies the corresponding values of boundary polarization. Like in the unbounded Kerr medium, the hybrid surface solutions that vanish for x -+ + cc possess a certain number of maxima and minima in the nonlinear medium. This number defines the order of the solution. For a specific mechanism of nonlinearity the number of possible solutions changes with mode order, the
.I. Jmitiski
Ml
hermal
/ Optics Communicutions
effects
Fig. 5. The power flow as a function of the propagation constant for the first and second order hybrid nonlinear surface waves. A = OS 145 pm, E, =2.3716. E,= i.OS.q,and cu=6.4 p,m’/V’
I”“,,“,
Linear
‘,‘_111-1,1,,.
substrate
1
Kerr medium
i
~> ’
-0.2 -2
-1
--. 1
0
x
Thermal effects Electronic distortion
I
:
\_,’ _““““‘~““““_““““““““““““““’
2
3
124 (1996) 373-382
and TM components very close to each other. Such a relation between fields means that both of them together generate certain regions of increased permittivity (partially overlapping in space) in which they both are focused. This model of propagation is characteristic for hybrid modes in the unbounded Kerr medium. The presence of the boundary shifts the positions of the held maxima and their values quite slightly. On the other hand, for the second solution presented in Fig. 6, obtained for thermal effects, the changes are very significant: the TM component practically adjoins the boundary, while the maximum of the TE component lies at a certain distance. In other words, the boundary polarization approaches unity, while the polarization at infinity is close to zero.Therefore we can say that the propagating mode produces two parallel waveguides. separated in space: one generated by a strong TE component in which a weak TM component is transmitted and the other generated by a strong TM component and carrying a weak TE field (Fig. 7). This model of propagation is similar to the picture introduced in the TE and TM-dominant approach [ 13-161, but since we have two separate channels and the same propagating constant for both TE and TM components, a more detailed comparison is difficult. The effect of generation of the separate channels by the fields needs lC to be close to E,_and p’ close to E,. It can also appear for other mechanisms of nonlinearity and for higher mode orders, but it is not so spectacular as for the secondorder mode in thermal media. In Fig. 7 we see a certain
[ml
Fig. 6. The field distributions of the second-order nonlinear hybrid surface waves. p’= 1.06S~et.. h=0.5145 pm. 6, =2.3716, e, = i.OS~e, and cu=6.4 pm’/V’.
propagation constant and the permittivity of the linear medium. But the number of parameters is greater than in the unbounded Kerr case, so the behavior of the solutions is a bit more complicated. Although for l, close to lL.we obtain a quite similar structure of modes (at least for the lowest orders; see Fig. 5). the field distributions can differ very much from the fields of corresponding solutions in the unbounded medium. In Fig. 6 we see such distributions for two second-order surface waves. One of them, obtained for the electronic distortional nonlinearity, is very similar to the mode in the unbounded medium, illustrated in Fig. 1. It describes the hybrid fields with the maxima of the TE
x [ml Fig. 7. The permittivity distributions inside the nonlinear medium generated by two nonlinear hybrid surface modes. Both modes correspond to the field and material parameter values as for Fig. 6.
J. Jusiriski/Optics
Communications
fourth-order molecular orientational surface mode, in which we can observe four such channels-two of them, generated by strong TM fields, are situated between two others induced by a TE component. Note that the TM peaks are higher that the TE ones (for the secondorder thermal case the TE and TM peaks have almost equal heights) and that the separation is not so perfect as in the thermal case. The higher-orders surface modes or modes obtained for larger permittivity of the linear substrate can exhibit a greater difference with respect to the mode structure in the unbounded nonlinear dielectric. For instance, for E, = 1.05 eL.we get only one third-order surface mode for molecular orientational nonlinearity, while for electronic distortional there are three modes-in the unbounded medium we have the opposite situation. Moreover, two degenerate third-order modes for molecular orientation have disappeared, but not the possibility of degeneracy-the three third-order hybrid surface modes for electronic distortion mentioned above carry the same energy. Similarly, the other features of the structure remain-the energetic capacity of the medium for the hybrid field is greater than for the pure TE or TM boundary solution and for the thermal nonlinearity we can obtain only even-order modes.
5. Summary
and conclusions
In the unbounded nonlinear Kerr medium and in the vicinity of the boundary between the nonlinear and linear media, hybrid fields can exist for any nonlinearity mechanism. These hybrid fields possess TE and TM components coupled through the material equations and propagate as an entity with the same propagation constant. Since the physical fields must vanish at infinity and satisfy symmetry conditions in the unbounded medium (or continuity conditions at the interface between the two media), the hybrid solutions of Maxwell’s equations can appear only for certain values of polarization. The solutions that do not satisfy one or both of the above conditions also exist-some of them can, at the phase plane E,.-I$, be represented by closed trajectories, the others by open curves (Fig. 4). The fields corresponding to the closed trajectories are periodic in space and one of these periodic hybrid solution can even be expressed analytically (Eqs. (10)). On the other hand. the fields represented by open phase
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trajectories describe spatial chaos. Such non-decaying solutions can appear in multilayered structures. The physical hybrid solutions in the Kerr region may have more than one field maximum (Figs. I,6 and 7). That is why they can be interpreted as n-th order guided or surface modes. For certain orders we obtain more than one solution-each of them corresponds to a different polarization. The very interesting fact is that certain multiple modes are degenerate-two or more solutions carry the same energy (within an accuracy of 0.1%). The degenerate modes appear both in the unbounded medium and near the boundary with the linear dielectric and are solutions of at least third order. The energy carried by hybrid waves always exceeds the power flow of the pure TE and pure TM modes. Since the carried energy depends on the propagation constant as does polarization, we can change the polarization of the field by means of the power flow. The most complex hybrid mode structure is obtained for the molecular orientational mechanism of nonlinearity, the simplest for the thermal or electrostrictive mechanism. In this case only modes of even order exist. This property also exists for the nonlinear hybrid surface waves, at least when the dielectric constants of the media are close to each other (Figs. 2 and 5). The electronic distortional or molecular orientational nonlinearity enables existence of the hybrid modes of any order. In the unbounded medium the model of propagation dominates in which TE and TM components together create a certain number of overlapping waveguides inside which they are transmitted. For the hybrid surface waves we can obtain another situation. When the propagation constant is close to its minimum value, certain modes induce parallel.channels more isolated in space (Fig. 7) in which the TE and TM components propagate separately. This phenomenon, similar to the effect occurring in the description of the stationary TETM waves in the TE and TM-dominant approach, is most apparent for the second-order mode and for the thermal nonlinearity case. Comparing the obtained results with the properties of the stationary TE-TM fields discussed by Boardman and Twardowski [ 141, we can say that a few of their statements are too general, while a few others are too limited. The regime of TE-TM solutions obtained, which makes the appearance of the hybrid fields with PTE = &,,, impossible, concerns only the thermal (elec-
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trostrictivc) nonlinearity type and only the coupling hetwecn the first-order TE and the first-order TM modes (this particular case was considered by them as 3n example). It is in perfect agreement with the results ohtaincd in Section 3 and Section 4 of the present paper-the odd-order hybrid fields that vanish at infinity do not exist for y= 1. But in the region of the nonstationary solutions (one of them, obtained for y= 1, is the trigonometric analytical solution ( 10) ), certain specific hybrid fields appear. They exhibit more than one maximum in space and vanish for x+ +m, i.e. they are the stationary higher-order TE-TM modes. The other types of Kerr nonlinearity extend boundaries of the stationary TE-TM regime, so this region contains the first-order hybrid solution with PTE = &,. Therefore, the regime of all possible stationary solutions of the coupled Maxwell equations is wider than Boardman and Twardowski reported. Although the problem of stability of the obtained hybrid solutions is difficult to examine exactly, discrete modes of a few lowest orders in the unbounded medium and at the boundary between the two media should be stable at the branches where dP/d/3> 0. Qualitative analysis is based on the comparison of the energy of close-by solutions. Since in the sufficiently close vicinity of any discrete mode (maybe except for certain higher-order of multiple solutions) only solutions that do not vanish at infinity appear, a given hybrid guided or surface mode carries the smallest power of them all, so it ought to be stable.
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