Modified Ray Tracing in a Dielectric Rod Colin Pask

Department of Applied Mathematics, Institute of Ad­vanced Studies, Australian National University, Can­berra, ACT 2600 Australia. Received 13 May 1974. In this Letter we describe a simple approximate method

for calculating the power carried by trapped modes along an ideal, circular, dielectric waveguide or optical fiber that is uniformly illuminated over one end by a coherent beam. The electromagnetic theory for this system has been studied in detail,1-3 and we refer the reader to Ref. 1 for mathematical formulation and to Refs. 2 and 3 for ap­plication of the theory and numerical results.

The important waveguide parameter involved is the di-mensionless frequency

where ρ is the fiber radius, n1 and n2 are the refractive in­dexes of the fiber and its surrounding, respectively, and λ is the wavelength of light in vacuum. The number N of trapped modes increases as V increases, being 12, 54, and 210 for V = 5, 10, and 20, respectively, and approaching N = V2/2 for large V. As V and N increase, the analysis based on trapped modes becomes progressively more cum­bersome. For this reason the applicability of the simpler methods of geometric optics is of great interest. The value of V is a measure of the importance of wave con­cepts, and electromagnetic theory results in the V ∞ limit should be equivalent to geometric optics.2,4,5

We seek to modify the treatment of classical ray tracing (CRT) as developed by Potter6 so it may be used for small values of V, i.e., V < 20. Straightforward application of CRT gives V independent results that are poor approxi­mations to the exact electromagnetic results for low V values. Further, the sensitivity to beam angle of inci­dence is also poorly described. A succinct summary of these points is obtained by examining Fig. 5 of Ref. 2.

Our new scheme, modified ray tracing (MRT), is de­rived from CRT by introducing the following two modifi­cations:

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(1) At the end of the fiber, the rays forming the coher­ent beam gain an angular spread due to diffraction ef­fects. To account for this we use the power distribution given by diffraction theory for a circular aperture7 to weight the rays over all angles rather than assume a par­allel bundle of rays as in CRT.

(2) Let θ be the angle that a ray makes with the fiber axis, and let θν = c o s - 1 (n2 /n1) be the complement of the usual critical angle. According to geometric optics,6 ray's trapped within the fiber have θ ≤ θc, or they are skew in such a way that they still make angles with the local nor­mal at the point of reflection that are greater than the critical angle even though θ > θc. This second set of rays has recently been shown4,5 to consist of leaky rays that attenuate along the fiber, and thus they correspond to the radiation or continuous spectrum field rather than to trapped mode or discrete spectrum fields. To calculate trapped mode power we integrate over rays with θ ≤ θν.

We now use modified ray tracing (MRT) as specified above to calculate the power P of trapped modes propa­gating along the fiber. The parameters to be considered are V[Eq.(l)] and

Table I. Percentage Errors in Modified and Classical Ray Tracing as Defined by Eq. (5) for Various V [Eq. (1)] and α [Eq.

(3)] and √δ [Eq. (2)] = 0.1

which we take to be small as is the case in practice, and

where θt is the angle that the incident beam makes with the fiber axis on entering the fiber (see Fig. 1 of Ref. 2).

For on-axis light, a = 0, and √δ small the MRT answer can be found analytically and

where J is the usual Bessel function. age error by

We define percent-

and similarly for ECRT, where P E M is the power calculated using electromagnetic theory. Using Eq. (4), EMRT ≤ 1.7% for 8 < V < 20, and EMRT ≤ 3.5% for 8 < V < 3. Classical ray tracing gives P C R T (α = 0) = 1, and |E C R T | increases from 2.8% at V = 20 to 20% at V = 3. Equation (4) therefore represents a considerable improvement over the classical geometric optics result.

If V were fixed and a varied, 0 < a < 1, electromagnet­ic theory shows P decreasing as a increases, while classical ray tracing gives PCRT = 1 (see Fig. 5 of Ref. 2). Modi­fied ray tracing gives results that are similar to P E M matching the behavior as both a changes and curves with different V values are considered. Table I shows percent­age errors [Eq. (5)J for various V and a and √δ = 0.1. ECRT is always negative, and | E C R T | increases monotoni-cally as a increases. The superiority of the modified method is quite obvious.

If we fix a, the P E M VS V curves show rapid variations whenever new modes can propagate.3 MRT gives the cor­rect general trend but does not reproduce the effects of mode cutoff. As an example, Fig. 1 shows P vs V for 2 < V < 7 and a = 0.6. The MRT results are for √δ = 0.1 and 0.3 and indicate the usefulness of the method for a wide range of practical \ /δ values.

We conclude that MRT provides a useful approximation

Fig. 1. Summed power of trapped modes propagating along a di­electric rod illuminated uniformly at one end by a coherent beam with incidence angle a [Eq. (3)] = 0.6 vs dimensionless frequency V [Eq. (1)]. Curves compare classical ray tracing (CRT) and modified ray tracing (MRT) approximations with the exact elec­

tromagnetic theory result for two values of √δ [Eq. (2)].

to the full electromagnetic theory results for total trapped mode power. It must be pointed out, however, that for larger angles of incidence, i.e. a > 1, the total power in the fiber is strongly aided by the leaky rays that may per­sist for considerable distances.4-5

The author gratefully acknowledges the help and advice given by Allan Snyder.

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References 1. A. W. Snyder, IEEE Trans. Microwave Theory Tech. MTT-17,

1138(1969). 2. A. W. Snyder, C. Pask, and D. J. Mitchell, J. Opt. Soc. Am.

63,59(1973). 3. C. Pask and A. W. Snyder, Appl. Opt. 13, 1889 (1974). 4. A. W. Snyder, D. J. Mitchell, and C. Pask, J. Opt. Soc. Am.

64,608(1974). 5. C. Pask and A. W. Snyder, Opto-Electron. 6, 297 (1974). 6. R. J. Potter, J. Opt. Soc. Am. 51, 1079 (1961). 7. J. D. Jackson, Classical Electrodynamics (Wiley, New York,

1967), Sec. 9.8.

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