Geometric optics limit of Marcuse's coupled power equations Allan W. Snyder and Colin Pask

Institute of Advanced Studies, Department of Applied Mathematics, Australian National University, Canberra 2600, Australia. Received 6 May 1975.

Using coupled mode theory and statistical averaging methods, Marcuse has derived a finite system of coupled power equations for the analysis of optical waveguides.1

An alternative equation has been derived for multimode optical waveguides from elementary plane wave concepts.2

We show here the equivalence of the two results when V » 1.

where ρ is the radius of the core, λ the wavelength in vacu­um, nco, ncl the refractive indexes of the core and cladding, respectively, and θc the complement of the critical angle. This necessitates transforming summations to integra­tions. We assume that V ∞ in what follows.

In the study of mode propagation on optical waveguides of circular cross section, we must often evaluate the sum Q

where F is an arbitrary function and the sum is taken over all modes that can exist for a particular V. Ulm is the transverse eigenvalue within the core, and the longitudinal fields have a Bessel function Jl(Ulmp) dependence at the core-cladding boundary.

The UlmS can be found exactly by solving the eigenvalue equation for the dielectric rod; however, for large V they are approximately

where θ is the inclination of a ray to the axis of the wave­guide. Similarly l can be expressed as

where y expresses the skewness of the ray.3 In Ref. 3 γ = (π/2) - θφ and θ = θz.

The transformation necessary for converting Q to inte­grals is implicit in a previous paper.4 Thus,

where Eq. (3) has been used for U and 7. This equation is valid only if many modes are excited. The limits depend on the specified modes (ray families) to be summed.4 If we sum only the bound modes, as does Marcuse (see Ref. 1), θ1 = 0, θ2 = θc, γ1 = 0, and 72 = π/2; however, if we also in­clude the tunneling leaky modes, (see Ref. 5), θ1 = 0, θ2 = π/2, γ2 = π/2, and γ1 = 0 for 0 ≤ θ ≤ θC and γ1 = cos-1(sinθc/sinθ) for θ0 ≤ θ ≤ π/2.

We now use these results to find the V » 1 limit of Mar­cuse's coupled power equation given as1

868 APPLIED OPTICS / Vol. 15, No. 4 / April 1976

where i is shorthand for l, m, αi is the power loss coeffi­cient, hij the coupled power coefficient, and N indicates the number of modes for a given V. The total power

Application of Eqs. (3) and (4) to the coupled power equation leads to

where P = P(θ,γ), a = a(θ,γ), P t o t = ∫ ∫ P cosθ sinθ cos2γ dydθ. Bound modes, far from cutoff, have Ulm « V, in which case the mode coupling coefficient hlm is indepen­dent1 of l and hence independent of angle 7. It is then ex­pedient to define the quantities I and S

Multiplying Eq. (6) by COS2Y and integrating over the re­gion 0 ≤ γ ≤ π/2 leads to

where we have assumed that θC « 1 and α is independent of the skewness angle 7.

Equation (9) is in exact agreement with the result de­rived from elementary plane wave concepts,2 where I(θ) is the energy distribution contained within a hollow annular core between angles θ and θ + Δθ and S(θ,θ') is expressed as a function of the well-known differential scattering cross section of a scatterer.

When tunneling modes5 are included, hlm depends on l or angle 7, and it is no longer possible to obtain a simple differential equation in I. Instead, Eq. (6) must be solved for P with the lower limit of integration on the 7 integral changed from 0 to cos-1(sinθc/sinθ) for θC ≤ θ ≤ π/2 and the upper limit of integration on the 0 integral changed from θC to π/2.

In summary, Eq. (4), which has been derived from geo­metric optics,2 is identical to Marcuse's1 coupled power equation for bound modes, provided V » 1, θC « 1, a inde­pendent of l or the skewness angle 7, and the modes are far from cutoff.

We thank the Australian Post Office for financial sup­port.

References 1. D. Marcuse, Theory of Dielectric Optical Waveguides (Aca­

demic, New York, 1974), Chapter 5. 2. A. W. Snyder and D. J. Mitchell, Electron. Lett. (1975). 3. A. W. Snyder and D. J. Mitchell, J. Opt. Soc. Am. 64, 599

(1974). 4. C. Pask, A. W. Snyder, and D. J. Mitchell, J. Opt. Soc. Am. 65,

356 (1975). 5. A. W. Snyder and J. D. Love, Opt. Commun. 12, 326 (1974).

April 1976 / Vol. 15, No. 4 / APPLIED OPTICS 869