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Journal of the Optical Society of America A

Journal of the Optical Society of America A

| OPTICS, IMAGE SCIENCE, AND VISION

  • Vol. 18, Iss. 6 — Jun. 1, 2001
  • pp: 1357–1370

Using rays better. III. Error estimates and illustrative applications in smooth media

M. A. Alonso and G. W. Forbes  »View Author Affiliations


JOSA A, Vol. 18, Issue 6, pp. 1357-1370 (2001)
http://dx.doi.org/10.1364/JOSAA.18.001357


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Abstract

A new method for computing ray-based approximations to optical wave fields is demonstrated through simple examples involving wave propagation in free space and in a gradient-index waveguide. The analytic solutions that exist for these cases make it easy to compare the new estimates with exact results. A particularly simple RMS error estimate is developed here, and corrections to the basic field estimate are also discussed and tested. A key step for any ray-based method is the choice of a family of rays to be associated with the initial wave field. We show that, for maximal accuracy, not only must the initial field be considered in choosing the rays, but so too must the medium that is to carry the wave.

© 2001 Optical Society of America

OCIS Codes
(000.3860) General : Mathematical methods in physics
(080.2710) Geometric optics : Inhomogeneous optical media
(080.2720) Geometric optics : Mathematical methods (general)
(350.7420) Other areas of optics : Waves

Citation
M. A. Alonso and G. W. Forbes, "Using rays better. III. Error estimates and illustrative applications in smooth media," J. Opt. Soc. Am. A 18, 1357-1370 (2001)
http://www.opticsinfobase.org/josaa/abstract.cfm?URI=josaa-18-6-1357


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References

  1. G. W. Forbes and M. A. Alonso, “Using rays better. I. Theory for smoothly varying media,” J. Opt. Soc. Am. A 18, 1132–1145 (2001).
  2. G. W. Forbes and M. A. Alonso, “Asymptotic estimation of the optical wave propagator. II. relative validity,” J. Opt. Soc. Am. A 15, 1341–1354 (1998).
  3. M. A. Alonso and G. W. Forbes, “Using rays better. II. Ray families for simple wave fields,” J. Opt. Soc. Am. A 18, 1146–1159 (2001).
  4. If γ is changed after the integration, the change in θγ can be found by using the fact that round(θγ/π), where round(t) denotes the closest integer to t, is independent of γ. It follows that, if γ is changed from γa to γb (where γb may be complex) after the integration of Eq. (2.9), the new value of θγ can be found by using θθγb=π round(θγa/π)+arctan{[Im(γb)X+P]/[Re(γb)X]}, where |arctan(t)| ≥ π/2 for all t.
  5. To verify this, consider the γ¯ derivative of the right-hand side of Eq. (3.2). By using Eqs. (2.1) as well as Eqs. (3.33.43.53.6), it can be shown that this derivative vanishes identically.
  6. This definition follows upon averaging the relative error with a weight that is proportional to the squared modulus of the field. In the resulting expression the normalization factor is then the integral of |U|2, but this is replaced by |Uγ(j)|2 in Eq. (3.7). Regardless, εγ(j) can be interpreted as an RMS relative error.
  7. With Eq. (2.5), the integral of |Uγ(0)(x, z)|2 over all x can be carried out in closed form, leaving |Uγ(0)(x, z)|2dx=12 kπγ a0(ξ)a0*(τ)Y(ξ, z)Y′*(τ, z)H(ξ, z)H(τ, z)1/2×exp−kγ4 [X(ξ, z)−X(τ, z)]2−k4γ [P(ξ, z)−P(τ, z)]2×exp−ik2 [X(ξ, z)−X(τ, z)][P(ξ, z)+P(τ, z)]+ik[L(ξ, z)−L(τ, z)]dτdξ. Notice that this integral is most significant when ξ ≈ τ. The expression given in Eq. (3.9) results from approximating the integral over τ here by using the form of the saddle-point method given in an appendix in Ref. 3. [The saddle point is at τ=ξ, where Ω2=|Y(ξ, z)|2/2γ ≠ 0].
  8. L. Mandel and E. Wolf, Optical Coherence and Quantum Optics, 1st ed. (Cambridge U. Press, New York, 1995), pp. 109–127.
  9. There is an important exception, however, where the phase-space curve can coincide with a track of the medium. Consider, for example, a plane wave traveling in the z direction in free space. The ray family is described by Eqs. (5.13), although a0 is now a constant. From Eq. (5.11) we see that a1 vanishes identically in this case for any γ, and the estimate in Eq. (2.5) actually gives the exact solution. More generally, when the phase-space curve is chosen to coincide with a track of the medium and a0 is chosen such that the relative weight associated with any segment of the track is conserved under propagation, Eq. (2.5) gives an accurate estimate of an eigenstate or mode of the medium (provided that γ is chosen appropriately). Of course, there is an extra restriction for closed tracks, which follows from requiring the agreement of the phase of the integrand in Eq. (2.5) at the (arbitrary) limits of integration: the phase-space area enclosed by the track must equal (m+1/2)λ, where m is an integer. This is consistent with the well-known quantization condition for the modes of a waveguide. The use of this method for solving eigenstate problems, where tunneling effects can be present, is the subject of further research.
  10. Y. A. Kravtsov and Y. I. Orlov, Caustics, Catastrophes and Wave Fields, 2nd ed. (Springer-Verlag, Berlin, 1999). See Chap. 6.
  11. This effect is not accounted for in Ref. 2, where the objective was to compare the errors of a variety of methods by using nothing more than the ray information. Note that this analysis considered a special case of the new method that was specific to the estimation of wave propagators for which the initial field is a delta function. The initial phase-space curve was then just a vertical line with constant A0 and with P ≡ ξ.

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