TECHNICAL NOTES

Bound modes in anisotropic multilayer thin film waveguides Ian Hodgkinson and Dominique Endelema

Ian Hodgkinson is with University of Otago, Dunedin, New Zealand; D. Endelema is with Ecole Nationale Super-ieure de Physique de Marseille, CNRS Laboratoire d'Op-tique des Surfaces et des Couches Minces, Domaine Uni-versitaire de St. Jerome, F-13397 Marseille CEDEX 13, France. Received 6 March 1990. 0003-6935/90/304424-03$02.00/0. © 1990 Optical Society of America.

We discuss the construction of field transfer matrices and derive expressions for the modal condition and elliptically polarized fields of anisotropic multilayer waveguides.

In the 4 × 4 matrix method for describing the propagation of light in anisotropic dielectric stratified media, optical fields are represented by column vectors, and matrices are used to transform the fields from one plane to another and from one representation to another.1–3 A vector based on tangential components of the total fields E and H,

where T indicates the transpose and the axes are defined in Fig. 1, allows direct passage through an interface. For trans­lation within an anisotropic layer it is convenient to use the natural basis provided by the four (j = 1,4) linearly polarized traveling waves

where α. and β have the general definitions4 β = n sinθ and α = n cosθ.

At any point in the anisotropic layer, the complex ampli­tudes of the optical fields can be expressed as weighted sums of the real amplitudes of the basis fields,

and the column vector a formed by the complex weighting factors αj provides a second representation of the fields.

Next, consider the matrix F that is constructed using the traveling waves of the basis as columns,

As can be seen by inspection, F has the property of trans­forming traveling wave fields to total fields, while its inverse F – 1 transforms total fields to traveling wave fields:

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Fig. 1. Coordinate axes for the anisotropic multilayer waveguide, the basis of the field vector a within a layer, and the basis of a in the

substrate.

Both F and F – 1 transform field representations at the same point. Two additional matrices, A and M, transform the fields from one plane to another:

In the special case when the two planes are the interfaces of a single layer, the matrix A is required to account for the phase differences of the basis traveling wave fields at the two locations, and hence it has the form

where φ(j) = ka(j)d and d is the physical thickness of the layer. Thus the matrix M, constructed as

has the required property of transforming total fields to total fields across a layer. Useful physical insight into the trans­formation processes can be obtained by considering the transform M as the sequence of three operations; for an individual layer in Fig. 1, F – 1 decomposes total fields at the right interface into traveling wave fields just inside the layer, Ad transforms the phases to correspond to the traveling waves just inside the left interface, and F transforms back to total fields at the left interface.

Finally, consider the system matrix

where M is the stack matrix formed as the product M1M2M3, . . ., and the substrate matrix

has been constructed using a TM+,TE+,TM–,TE– basis for the isotropic substrate normalized so that the l's are in specific locations; the γ terms defined by4 γs = ns cosθs/z0 and

γs = Z0 cosθs/ns are positive real or positive imaginary. The cover matrix Fc has a similar form, and by inspection its inverse is found to be

With this particular choice of notation, the column vectors in the matrix equation

represent the traveling wave electric fields at the cover and the substrate:

Thus, whereas the stack matrix M transforms tangential components of the total fields from the substrate to cover interfaces, the system matrix A transforms traveling wave electric fields from just inside the substrate to just inside the cover.

Reflection and transmission coefficients from a stack of anisotropic layers follow from the solutions to the matrix equation ac = Aas with E–

sy = 0 and E–sy = 0:

For example, with E+cz = 0, the reflection coefficient rpp =

E–cy/E+

cy is obtained in the form

The special field requirement for a bound mode in an aniso­tropic thin film multilayer waveguide is just outward-going evanescent traveling waves in the cover and the substrate. Thus, using ac = Aas with outward-going waves and with A the system matrix defined in Eq. (8), we see that the four equations implied by

must be satisfied for a mode to occur. The modal condition,

follows from the requirement of a nontrivial solution to the first two equations. Clearly, the expression a11a22 — α12α21 in the modal condition is the denominator of the reflection and transmission coefficients obtainable from Eq. (13), and hence the modes of an anisotropic waveguide occur at poles of the reflection and transmission coefficients,5 just as for an isotropic guide.6 An alternative modal condition can be derived by solving the equations mc = Mms where mT

c =

(Ecz,—γcEcz,Hcz,—γ′cHcz) and ms

T = (Eszs,γsEszHsz,γ′sHsz),

with the solution

Relative values of the amplitudes of the bound mode tangen­tial electric fields at the cover and substrate are obtained by solving Eq. (15). In terms of unit vectors along the y- and z-axes, the solutions are

The tangential electric fields at the cover and the substrate are elliptically polarized with the axes of the ellipse, y' and z' say, rotated (in a positive sense y to z) by an angle ε and with axial ratio Ey′/Ez′ given by the general equations

For the cover, Ey = E–cy etc. Modal field profiles can be

traced throughout the guide and its bounding media by operating on the field vector at the substrate

by the M matrix for the intervening material to give the elliptically polarized fields

Also, at a point in the guide the fields can be written as sums of the basis fields weighted by the αj terms and for displace­ments x from that point, within the same layer, a field profile such as Ey varies as Σ4

j=1 aj E (j)

y exp[ika(j)x]. Thus the field profiles are in general anharmonic in x but harmonic with distance y and time t because of the implied common factors exp(ikβy) and exp(–iωt). Finally, the time averaged power flow per unit area in the plane of the guide can be calculated point by point as the fields are traced. For example, in the y-direction,

and the x-components of the fields that are necessary for the calculation are just the weighted sums for Ex and Hx given by Eq. (2).

Ian Hodgkinson would like to thank Direction des Re-cherches et Etudes Techniques for financial support re­ceived during his stay in Laboratoire UA1120 de Marseille.

References 1. S. Teitler and B. W. Henvis, "Refraction in Anisotropic Stratified

Media," J. Opt. Soc. Am. 60, 830–834 (1966). 2. D. W. Berreman, "Optics in Stratified and Anisotropic Media," J.

Opt. Soc. Am. 62, 502-510 (1972). 3. K. Eidner, "Light Propagation in Stratified Anisotropic Media:

Orthogonality and Symmetry Properties of 4 × 4 Matrix Forma­lisms," J. Opt. Soc. Am. A 6, 1657-1660 (1989).

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4. J. T. Chilwell and I. J. Hodgkinson, "Thin-Films Field-Transfer Matrix Theory of Planar Multilayer Waveguides and Reflection from Prism-Loaded Waveguides," J. Opt. Soc. Am. A 1, 742-753 (1984).

5. P. Yeh, "Electromagnetic Propagation in Birefringent Layered Media," J. Opt. Soc. Am. 69, 742-756 (1979).

6. R. Ulrich, "Theory of the Prism-Film Coupler by Plane-Wave Analysis," J. Opt. Soc. Am. 60, 1337-1350 (1970).

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