TE waves in circular nonlinear optical antiwaveguide

B.V.Gisin

Indexing terms: Kerr medium, Nonlinear waveguiding, Spatial solitons

Abstract: Eigenvalues and eigenfunctions of TE,, modes in a two-layer circular nonlinear optical waveguide are numerically investigated for various refractive index differences between the core and cladding and various values of the core radius. It is shown that for a negative difference bctween the refractive indices of the corc and thc cladding, there are possible regions which may be interpreted as stability regions for TE modes. In these regions the core diameter is small (of the order of light wavelength) and the longitudinal magnetic field is comparable with the transverse one.

1 Introduction

It is known that under certain conditions a light beam in a homogeneous Kerr medium can produce its own dielectric waveguide [I]. Such waveguides can also arise in nonlinear optical step or graded index structures. The problem of nonlinear waves and their stability in such structures has been the topic of research since the 1980s [2-41. In [5] it was shown that in two-layered nonlinear optical antiwaveguides stability regions for self-trapping optical beams may exist. A nonlinear optical antiwaveguide (NOA) is a ’reverse optical waveguide’ where the refractive index of the core is less than that of the cladding. In the stability region a con- tinuum of eigenmodes exists and these modes may con- tinuously transform from one into another in certain energy intervals. Note that we use here the concept of the mode as a specific solution that satisfies the appro- priate boundary conditions. The stability regions may exist in cylindrical and planar structures [6]. They are characterised by a threshold value, i.e. the incident mode has to exceed some minimum power level in order to propagate, and by the small core cross-section. Note that such stability regions were found only in NOAs. Clues for stability regions were also obtained by employing an energy approach [7]. NOAs may be used in applications for the transmission or switching of optical energy, harmonic generation and parametric mixing, as well as laser active mediums and bi- or multistable devices.

Usually, plane-polarised waves of nonlinear waveguides and NOAs are considered, However, the

0 IEE, 1995 IEE Proteednzgs online no. 19952365 Paper first received 26th June 1995 and in final revised form 6th October 1995 The author IS with the Faculty of Engineering, Tel-Aviv University, Tel-Aviv 69978, Israel

electric field of these waves always have a component alon3 the propagation direction, due to the condition divD = 0, where d i s the dielectric displacement. The longitudinal component is usually small and can be neglected in the nonlinear part of the dielectric con- stant. Decreasing the optical beam diameter results in an increase in the longitudinal component. When it becames comparable with the transverse component then the description of plane polarised waves turns into a complicated multidimensional problem. In contrast, TE modes with circular symmetry allow simple investi- gation as their symmetry does not change when the optical beam diameter is decreased. We can therefore study peculiarities of these modes for extremely small core radii and for large differences between the refrac- tive indices of the core and the cladding when the weakly guiding approximation [8] is not correct. Such cylindrical symmetrical modes correspond to the TEol modes of an ordinary waveguide. The modes of the lin- ear waveguide may be continuously transformed into modes of the nonlinear waveguide and into modes of the NOA by continuously varying the difference in refractive indices of the core and cladding and by a continuous change in the light amplitude.

In this paper we extend a previous study [6] and per- form an eigenvalue analysis of the TE modes of two- layered nonlinear waveguides and NOAs and find sta- bility regions for these modes.

2 Formulation

The electric E” and magnetic I? field of the cylindrical symmetrical TE modes is assumed to be of the follow- ing symmetry

Y r

E, = -E( r ) cos CP

E, = - -E(r) cos CP

E, = O

H, = - H ( r ) COS

(1) X

r

X

r Y H - - -H(r)cos@

H, = H , ( r ) sin @ where x, y are the transverse coordinates, z is the longi- tudinal coordinate, t is the time, y2 = x2 + y2, @ = cot - pz, w is the frequency, p is the propagation constant. We assume that material of the core and the cladding has a Kerr-type nonlinearity+ i.e.,+the electric displace- ment into this material is D = E[& + 4/3~2(Ei + E;)], where the coefficient 413 before is introduced for convenience. The linear dielectric constant is different for the core E = E, and for the cladding E = E ~ , while the optical Kerr constant is assumed invariable.

293 IEE Proc.-Optoelectron., Vol. 142, No. 6, December 1995

Positive and negative values of represent self-focus- ing and self-defocusing material. In case of the NOAs confined solutions do not exist for c2 < 0. Thus, we limit ourselves to the analysis of self-focusing materials with >+O. Neglecting the third-order processes we can write D as follows:

Dx,y = Ex,,(€ + €aE2) (3) Note that the identity d i d = 0 follows from the defini- tions in eqn. 1 and the boundary condition at the core- cladding interface corresponds to the continuity of E and H,.

Substituting eqns. 1 and 2 into the Maxwell’s equa- tions and using the new variables

we obtain the normalised equation for TE modes d 1 -f + -f = g dP P d

dP -g = (1 - V ) f - f 3

(5)

where

P > P c

nc = 4~~ and no = dcO is the refractive index of the core and the cladding respectively. c is the speed of light.

Multiplying the first equation from eqn. 5 by pdp and integrating from zero to infinity it may be straight- forwardly stated that g must have at least one node. We restrict ourselves to one-node solutions, i.e. the TEol modes. These solutions also have to be bounded, square integrable and continuous at the core-cladding interface.

For small f? so that f3 is negligible, eqn. 5 corre- sponds to ordinary linear waveguides. It is known that for the linear waveguides desirable solutions exist at positive discrete values A = A,(p,) > 0, where m is the number of nodes. Corresponding one-node solutions of the nonlinear eqn. 8 exist for any value of A , provided

Continuous and bounded solutions of eqn. 5 have near p = 0 the asymptotic behaviour g(p) = go + g2p2 + ..., f(p) = fip, + f3p3 + _ _ _ _ All the coefficient g, and f, in these expansions depend on go = g(0). go varies continu- ously with A for a given value of p, and obtains single values for any A provided AI > A > 0. However, go may be multivalued for A < 0, i.e. when n, < no. We define the discrete values of go, as the eigenvalues of eqn. 5. These eigenvalues are found from the require- ment that at infinityf(p) + 0 and therefore the asymp- totic behaviour of f ( p ) is given by the Macdonald’s function [9].

Regions of stability are determined when part of an eigenvalue curve go versus A may be approximated by a straight line parallel to the go axis. A heuristic expla- nation for this assumption is given in [6]. However, using an energy approach [7], this assumption has been partially justified. The regions of stability is associated

A < Al.

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with a continuum of eigenfunctions. In these regions, eigenfunctions may continuously transform from one into another in certain power intervals.

A Eigenvalue curves for various p r Fig. 1

Solid line represents the curve with region stability. For pc 2 2.5 the eigenvalue curve breaks into two separate branches

4 r

pc P Fig.2 and various A

Examples of antiwaveguide eigenfunctions gjp), f(p) for pc = 2.6

~ A = -0.5 A = -2.5 A = -2.5

- - - -

3 Numerical results

Results of numerical analysis of curves of eigenvalues (i.e. curves go versus A) are shown in Fig. 1. All the curves start on the A axis at a point corresponding to eigenvalues of ordinary linear waveguide A , (p,). In the nonlinear waveguide domain, i.e. for A I > A > 0 the curves change monotonously with A and intersect at the point A = 0 corresponding to homogeneous medium. In the antiwaveguide domain, i.e. €or A < 0 the curves break into two separate branches if pc 2 2.5. The antiwaveguide eigenfunctions for p, = 2.6 are shown in Fig. 2, as an illustration. Note that at the core-cladding interface g(p), f(p) and f’(p) are continu- ous. A jump of g’(p) at the point (p,) equals Af(p,). For pc = 2.45 the eigenvalue curve has a maximal sec- tion which may be approximated by a straight line par- allel to the go axis, approximately between the values go = 1.1 and go = 2. The evolution of the eigenfunctions at this section is shown in Fig. 3. The eigenfunctions g(p) andf(p) for a certain eigenvalue go are pictured in Figs. 2 and 3. Note that the longitudinal magnetic field is confined, in general, to the antiwaveguide core which has a small cross-section. We can estimate, for exam- ple, the core diameter of fused silica antiwaveguide in

IEE Proc.-Optoelectron., Vol. 142, No. 6, December 1995