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THEORY OF NONLINEAR WAVEGUIDES : STATIONARY AND PROPAGATING WAVES
J. Moloney
To cite this version:
J. Moloney. THEORY OF NONLINEAR WAVEGUIDES : STATIONARY AND PROP- AGATING WAVES. Journal de Physique Colloques, 1988, 49 (C2), pp.C2-277-C2-282.
�10.1051/jphyscol:1988265�. �jpa-00227682�
THEORY OF NONLINEAR WAVEGUIDES : STATIONARY AND PROPAGATING WAVES
J.V. MOLONEY
Physics Department, Heriot-Watt University, Riccarton, GB-Edinburgh EH14 4AS, Scotland, Great-Britain
Resum6 - On donne un apercu sur les techniques d'obtention des solutions d'ondes guidee non lineaires et stationaires, et d70ndes de surface dans les ondes guidees dielectriques, ainsi que determination de leur proprietes d'instabilit6 et l'etude des caractbristiques de propagation.
Abstract
-
Techniques for obtaining stationary nonlinear guided and surface wave solutions in dielectric waveg- uides, determining their stability properties and investigating global propagation characteristics will be overviewed 1 - INTRODUCTION.Theoretical techniques for enumerating stationary TE and TM nonlinear guided (NGW) and nonlinear surface (NSW) wave solutions for various permutations of nonlinear dielectric layered geometries will be briefly overviewed. Emphasis will be placed on discussing the stability of such stationary waves t o propagation along the guiding structure and it will be shown that these nonlinear waves represent a small subclass of special solutions to a much more general propaga- tion problem, whose solutions are related to spatial soliton/solitary wave solutions of a modified nonlinear Schrodinger equation. It is therefore natural to try t o understand the nature of these more general propagating nonlinear waves and identify the familiar stationary NSW and NGW solutions with equilibria (stable and unstable) of the latter. Most stability studies to date, with two important exceptions [1,2], have involved direct numerical integration of this modified NLS equation using the appropriate nonlinear stationary wave solution as initial data. The analytic stability technique [l]
for stationary TE NGWs is applicable t o layered media with arbitrary nonlinear optical dielectric responses. Application of this technique to nonlinear waveguides with both saturable and defocussing nonlinear substrate and cladding layers is discussed in the article by Etrich et al. in the present volume. The analytic stability result for NSWs at optical interfaces [2] follows from a global self-focussed channel propagation theory. This latter equivalent particle theory is discussed in more detail in another article in this book by Aceves et al. Recent developements in the study of diffusion effects and instabilities in the 3D problem will be referred to briefly. The discussion here will be confined to transverse electric (TE) nonlinear waves propagating in layered linear/nonlinear dielectric media with positive (self-focussing) nonlinear optical responses. An excellent up to date exposition on theory and experiments in nonlinear waveguides and a t optical interfaces in general is contained in the recent "Special Issue on Nonlinear Waveguides" published by the Journal of the Optical Society of America [3].
Fig. 1
-
(a) Single interface and (b) thin-film waveguide geometries of interest. The film width is '2d' and the material parameters are defined in the text.The general picture that emerges from a study of nonlinear guided and surface waves is that there exist two distinct types of nonlinear wave. One emerges in a thin-film waveguide from the low power linear guided T E or TM wave and exists due to the linear guiding properties of the system. This wave is stable to propagation at low power and exhibits apower dependent effective guide index
P
which can be exploited in feedback (optical bistabiity) or directional coupling geometries. This type of nonlinear wave cannot be supported at a single interface between nonlinear dielectrics. With increasing power the "local effective refractive index" n f ( f d, IE12) = ni2+
aiIE(f d, z)I2 a t each interface can become equal, causing theNGW
to lose its guiding properties. Here ni, ai refer respectively to the linear refractive index and nonlinear coefficient in the i-th dielectric layer; the linear film interface is a t z = f d [ see Figure I]. This is the physical me~hanism that causes a symmetricN G W
in a thin-film planar waveguide with a linear guiding film and with bounding nonlinear substrate andArticle published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:1988265
C2-278 JOURNAL DE PHYSIQUE
cladding layers to lose its guiding properties and become unstable [I]. A bifurcation to a new class of asymmetric solutions occurs at this point and these eventually degenerate with increasing power into nonlinear surface waves localized at each interface in the guide. The symmetric NGW loses stability at the bifurcation point. The newly emerging asymmetric nonlinear stationary waves (NSWs) evolve into perturbed spatial solitons of the modified NLS equation and their shapes are determined by the material constants of the nonlinear medium in which the NSW peak resides. The propagating self-focussed channels are spatial soliton/solitary wave solutions of the same equation and are closely related to the latter NSWs. These spatial solitons are localized a s a consequence of a balance between linear diffractive spreading and nonlinear self-focussing. They do not therefore rely on discontinuities oi gradations in the material properties for their existance. The latter act as trapping sites for such waves. Using soliton perturbation techniques, the general propagation characteristics of these NSWs can be reduced to the much simpler Newtonian dynamical problem of an equivalent particle moving in an equivalent potential [2].
161
'
I PC
c
i o l'5 $0 2'5 3'0 3'5 4'0 4'5 i 0 015 2 0 2.5
P
P
Fig. 2
-
Power versus effective index characteristics (a) for a single interface separating two self-focussing nonlinear dielectric media and (b) for a symmetric thin-iilm waveguide with a linear film and identical self-focussing cladding and substrate layers. The TEo branch of stationary solutions is given for two guide widths; (2dlXz.4 (left characteristic) and .8).2 - BACKGROUND THEORY
The general beam propaga1;ion problem is described by the following nonlinear partial differential equation for the slowly varying envelope of the electric field F(x, z),
where p = 2nocos$i is the effective index of the propagating wave with $ j the incident angle and n(x, F F * ) the intensity dependent refractive index. The specific form of n(x, F F * ) will be irrelevant for most of the ensuing discussion although we will specialize to the case of self-focussing Kerr media when discussing the equivalent particle theory. In this latter case n2(x, F F * ) = nf
+
uilF1'.
In any event the refractive index will be assumed to change discontinously across each interface. Equation (1) has discontinous linear and nonlinear coefficients in general and we refer to it as a modified Nonlinear Schrijdinger equation. Solutions to equation (1) which are bounded in space (F, F, --+ 0 as x -+ f co) will be of interest. There are essentially two fundamental types of bounded stationary solution, one which appears as a consequence of the discontinuity in the coefficients (discontinuity in material properties) of the above equation and the second which arises from a balance between nonlinear self-focussing and linear diffraction. The first type of solution emerges from the linear T E , guided wave at low incident powers whereas the second appears above a critical power threshold P,. We identify this second type of stationary solution as a perturbed spatial soliton. When well separated from the interface and residing in a nonlinear medium, this latter type of wave represents a propagating self-focussed channel and is a soliton solution of the NLS'equation whose shape depends on the material properties of the medium in which it resides.Stationary (2-independent) solutions to equation (1) are obtained by making the ansatz F(x, z ) = G'(x)eiBZ in (1) yielding the second order ordinary differential equations for the NGW or NSW field shape,
These equations must be supplemented by the boundary conditions on G(+) at each interface; namely, continuity of G(r) and it's first derivative G,(r) across each interface. The boundary conditions yield a general nonlinear dispersion relation which determines the value of
p
( in the present contextp
is a free parameter rather than a prescribed propagation constant) and other free parameters in the appropriate problem. Considerable effort has gone into determining stationary NSW and NGW profiles from eqns(2) for a variety of permutations and combinations of linearlnonlinear dielectric layerssolutions have been obtained using first integral methods and phase portrait analysis[5].
Figure 2a shows the total power of the stationary NSW at a single interface separating two self-focussing Kerr media plotted against the effective index P. In Figure 2b we show corresponding plots for TEo waves in a symmetric thin- film waveguide with a linear guiding film and two identical bounding self-focussing Kerr media. The single interface characteristic represents stationary NSWs which are perturbed spatial solitons of equation (I). Figure 2b contains both types of solution referred to above. The branch which emerges from zero power (S-branch) represents the conventional linear guided TEo mode. As we move to higher power dong this characteristic the effective waveguide index /3 for the TEo wave shows a significant power dependence. For relatively low powers we can extend conventional coupled mode analysis to study optical bistability in a feedback geometry, harmonic mixing or directional coupling between adjacent waveguides. As the power increases to the critical point /3, we enter the strongly nonlinear region and coupled mode analysis does not work. Beyond PC the branch of symmetric solutions still exists but now it is unstable. The new branch of solutions which emerges at PC is referred to as the surface polariton branch (A-branch) and for the case shown here is doubly degenerate. This degeneracy is broken in an asymmetric thin-film waveguide. Solutions on this branch show behavior similar to those on the single interface characteristic in Figure 2a. Power versus
P
characteristics for higher order TE, modes show similiar features undergoing a bifurcation at increased power to asymmetric waves.I 8 r (a) A
X X
Fig. 3
-
(a) Stationary NSW shapes representing points on the power versus f3 characteristic of Figure 2a. The G(-) wave lies in the region (DE) and G(+) in the region (AB). (b) Equivalent particle potential U ( 5 ) a t a fked incident power.The local equilibria represent unstable (G(-1) and stable (G(+)) stationary NSWs.
Typical stationary NSW field shapes are plotted in Figure 3a representing solutions on the branch DE (G(-)) and AB (G(+)) in Figure 2a. These solutions are easily found to be,
and
where the free parameters (xo, xl) are determined from the boundary conditions at the interface (x = 0) and are given
with p 2 = A/((1 - a)q?),+ = ql/qO,A = n i - nf and a = cuo/al. Notice that these hyberbolic secant shapes are reminiscent of stationary soliton amplitude profiles with the parameters (xo,xl) giving the positions of the respective soliton peaks. One can view the stationary NSW as being made up of 'pieces' of two distinct soliton shapes, one being defined by the parameters of the left hand medium and the other by the parameters of the right hand medium. This idea is expolited in [2] and the article in this volume by Aceves et al. in developing the equivalent particle prescription.
The NGW shapes for the thin-flm waveguide contain the same soliton shapes for those pieces which lie in the bounding
C2-280 JOURNAL
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PHYSIQUEnonlinear media and differ from the NSWs in that the piece of the solution lying in the linear film is a cosine or hyperbolic cosine function (for TEo waves) whichever one satisfies the boundary conditions a t both interfaces for a given P. These solutions and their dispersion relations have been derived in references [4] and will not be reproduced here.
3
-
STABILITYThe question of stability is central to all problems on nonlinear guided or surface waves. Two type of stability question may be posed. One, which we call stability t o form, tells us whether the 1D (z-independent) stationary NGW or NSW solution to eqns(2) will retain its shape on propagating along the guide. The other stability question that needs to be addressed is whether the assumption of an infinite plane wavefront in the third (y-) dimension is valid in general. This 3D modulational instability has been investigated numerically very recently [6] for the case of a Kerr nonlinearity and it has been shown that the NSW solution which is stable in 2D is unstable t o a modulational growth on the plane wave envelope in the third space dimension. The question of stability to form is a complicated one and a limited number of analytic instability results exist t o date [I]. Most stability studies for this case have been restricted to numerical simulation in a restricted parameter window. The only existing analytic instability prediction for the stationary NGWs on the S-branch of Figure 2a (and its asymmetric counterpart) is a result by Jones et al. [I] presented at the last Optical Bistability meeting in Tucson in 1985. There a prescription was given for determining the stability of an NGW by tracking a tangent vector around the underlying composite phase portrait of the stationary solution. The article by Etrich et al.
in this volume applies this stabiity scheme t o a nonlinear waveguide with a more general saturable self-focussing and self-defocussing nonlinearity and confirms the instability predictions with explicit numerical beam propagation. Stability of the stationary NSWs follows directly from the equivalent particle theory (see Figure 3b). The earliest stability analysis of NSWs was numerical [7] for the special case of a linear/nonlinear interface. The equivalent particle theory confirms these earlier results. Numerical stability studies of various NGWs, including higher order TE, waves, has been carried out by a number of authors 181. The general conclusion that may be drawn from these works is that many of the NSWs and NGWs are unstable to propagation. The region CDE in Figure 2a and the dashed portions on the curves in Figure 2b represent unstable stationary waves. The question then is whether instabilty represents an undesirable situation from the point of view of designing nonlinear devices or whether it can be exploited directly.
Fig. 4 - Breakup of the incident self-focussed channel into three individual channels after crossing the interface. The lowest amplitude channel propagates to the left and disperses into radiation
A brief comment should be made on the 3D modulational instability. This type of instability was originally investigated by Zakharov and Rubenchik [9] in the context of a 1D soliton embedded in two transverse space dimensions. The waveguide problem differs in that the NGW waves are not solitons, especially the waves on the S-branch in which most of the energy can lie in the linear film. The waves on the surface polariton branch and the NSWs however are perturbed spatial solitons when localized predominantly in the nonlinear medium. We find [6] that the waves with most of their energy in the linear film are stable to a modulational instability whereas those waves with most of their energy localized in a ID spatial soliton are unstable to a modulational instability along the flat wavefront in the third dimension. This instability can be inhibited in a channel waveguide where the width in the third dimension is short enough t o exclude spatial modes from the unstable band in k-space. We will assume that this condition is satisfied.
One consequence of instability of an NGW is that the wave undergoes a profound change of shape and a significant portion of the energy is carried away from the interface as a self-focussed channel [I]. The remainder of the energy remains trapped in the guiding film and behaves as a low amplitude, almost linear, oscillating guided wave. In the case of a surface wave, instability is less dramatic with the wave retaining it's sech shape and propagating away as a self-focussed channel with a small loss of energy as radiation. The self-focussed channel in both cases is a spatial soliton of an NLS equation whose coefficients are dependent on the material constants in which it propagates. This soliton evolves in both cases from different initial data. In designing a device based on this phenomenon one would like to be able to exercise some degree of control over the angle which the self-focussed channel subtends t o the interface. It would also be desirable to retain most of the energy in the self-focussed channel or at least be able to estimate the partitioning of energy between multiple channels. A switching action based on incrementing the input power beyond the critical point in a thin-film waveguide with a single nonlinear bounding medium has recently been proposed [lo]. There is little control over the propagating self-focussed channel angle in this mode of operation, however, as it will be sensitive to critical background fluctuations. The difficulty lies in the waveguide geometry in that the two distinct types of solution discussed above exist.
This difficulty can be overcome at a single interface separating two nonlinear dielectric media where the self-focussed channels on each side of the interface are both spatial solitons whose shapes are determined from the constants of the material in which they reside.
Fig. 5 - (a) Power or (b) angle adjustable spatial scanner showing trajectories of the self-focussed channel peaks as the power in (a) is decremented at a fixed incident angle or the angle is decremented in (b) at fixed incident power.
An equivalent particle theory developed by Aceves et al.[2] allows one to quantify the propagation characteristics of self-focussed channels in nonlinear dielectric media separated by one or more interfaces. The main results of this theory are stated in another article in the present volume. Here we highlight some of the applications of the theory which are relevant to the present discussion. Consider a self-focussed channel incident from the left at an angle
Gi
to the interface separating two nonlinear dielectric media (see Figure la). Let the initial channel F(x, 0) have the shapewhere u o = 2 n ~ s i n $ ~ is the soliton velocity (proportioval to the angle of incidence) and 2110 is the soliton amplitude
p9;'
is the soliton width). Notice that this solution has the same structure as the stationary NSWs G(+) and G(-) given above.
Now however we allow the self-focussed channel amplitude, peak location and velocity to vary. This basic wavepacket allows us to define an equivalent particle whose location and velocity are determined from Newton's equations of motion (see [2] and the article by Aceves et al. in this volume). The general shape F(x, 0) allows us sufficient latitude to allow for reshaping if the channel crosses the interface. Using this shape we can now transform equation (1) to the much more suggestive form,
iA, +A,,
+
2 1 ~ ) ~ ~ = V(x)A, where G F ( x , r ) = &A(I, r)ei9gZ, 2pr = z and the perturbation potentialC2-282 JOURNAL
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PHYSIQUEThe lefthand side of this equation is now the nonlinear Schrbdinger equation and the s m d perturbation V(x) on the right reflects the discontinuity in material properties across the interface. This equation can be further reduced to Newton's equations of motion for the location 2 = p-'
ST,"
AA'dr and velocity (angle) v = ip-' S:;(AA:-
A8A,)dx of the self-focussed channel where p =J:+m" I A ~ ~ ~ X .
A typical equivalent particle potential U ( F ) is graphed in Figure 3b. The shape of this potential changes as the incident power is varied. The local equilibria of the potential represent the unstable (G(-1) and stable ( ~ ( f )) stationary NSWs in Figure 3a.The equivalent particle theory provides quantitative information on beam reflection, transmission and trapping at non- linear optical interfaces. Its modular structure allows one to treat multiple interfaces by ~ a t c h i n g together single interface equivalent particle potentials of the type shown in Figure 3b. Accurate estimates are given on the amount of radiatian generated due to interaction with an interface. The phenomenon of beam breakup into multiple self-focussed channels is illustrated in Figure 4. The prediction of the theory is that the incident beam power is such that it will break up into three channels, two transmitted and one low amplitude one reflected and dispersing away. This is an example of where the incident self-focussed channel is a bound state of three solitons (N=3 soliton) which is split-up into three individual solitons due the action of the finite perturbation V(x) of the interface. We end by providing an example of where the theory allows us to design a spatial scanning element by either incident power ( ~ i & r e 5a) or angle (Figure 5b) adjustment. These trajectories of the self-focussed channel peak are numerically generated by solving equation (1) and can read off from the phase portraits of the appropriate equivalent potential.
5 - CONCLUSIONS - The above discussion suggests that it is more natural to view a stationary NGW and NSW power versus
p
characteristic curve as the locus of extrema of an appropriate equivalent potential. This point of view is supported in a numerical study of nonstationary NGWs in a thin-film waveguide by Etrich et al. in this volume. The global picture that then emerges makes it much easier to envisage future device designs based on propagationg self-focussed channels.Another important question that has been recently addressed is the effect of diffusion of the excitation on the stationary NSW characteristic curves Ill] and on propagation effects [12]. The main conclusion of these works is that the critical power for the existence on NSWs increases roughly linear with diffusion length and that it is possible to trap a surface wave at an interface separating a linear and a nonlinear dielectric in the case where no diffusion is allowed across the interface.
ACKNOWLEDGMENT
-
Support for the work described herein has been provided by grants from the U.S. Air Force AFOSR F4962086C0130 and the U.K. Science and Engineering Research Council SERC GR/D/84726. The author also acknowledges a collaborative travel research grant from NATO (RG.0005/86).REFERENCES
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