Using rays better. I. Theory for smoothly varying media
JOSA A, Vol. 18, Issue 5, pp. 1132-1145 (2001)
http://dx.doi.org/10.1364/JOSAA.18.001132
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Abstract
We present a method for computing ray-based approximations to optical fields that not only offers unprecedented accuracy but is also accompanied by accessible error estimates. The basic elements of propagation through smooth media, refraction and reflection at interfaces, and diffraction by obstacles give the foundations for the new framework, and the first of these is treated here. The key in each case is that the wave field and any relevant derivatives are expressed consistently as a superposition of delocalized ray contributions. In this way, the mysteries surrounding the sometimes perplexing tenaciousness of ray-based estimates are clearly resolved. Further, an essential degree of freedom in this approach offers an attractive resolution of part of the apparent conflict of particle/wave duality.
© 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
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)
http://www.opticsinfobase.org/josaa/abstract.cfm?URI=josaa-18-5-1132
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References
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- R. C. Hansen, ed. Geometric Theory of Diffraction (IEEE Press Selected Reprint Series, New York, 1981).
- M. A. Alonso and G. W. Forbes, “Using rays better. II. Ray families for simple wavefields,” J. Opt. Soc. Am. A 18, 1146–1159 (2001).
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- M. Born and E. Wolf, Principles of Optics, 7th ed. (Cambridge U. Press, Cambridge, UK, 1999). See, for example, the footnote on p. 519.
- A. W. Snyder and J. D. Love, Optical Waveguide Theory (Chapman & Hall, London, 1983). See Chap. 35.
- G. W. Forbes and M. A. Alonso, “What on earth is a ray and how can we use them best?” in International Optical Design Conference 1998, L. R. Gardner and K. P. Thompson, eds., Proc. SPIE 3482, 22–31 (1998).
- M. A. Alonso and G. W. Forbes, “New approach to semiclassical analysis in mechanics,” J. Math. Phys. 40, 1699–1718 (1999).
- M. A. Alonso and G. W. Forbes, “Asymptotic estimation of the optical wave propagator. I. Derivation of a new method,” J. Opt. Soc. Am. A 15, 1329–1340 (1998).
- 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).
- This claim is easy to establish for cases where n is independent of z, but it is nontrivial more generally. These issues are discussed in more detail in Secs. 4–6 of M. A. Alonso and G. W. Forbes, “Semigeometrical estimation of Green’s functions and wave propagators in optics,” J. Opt. Soc. Am. A 14, 1076–1086 (1997).
- The issue of the initial phase reference is clarified in Section 5.
- This form is precisely what emerged in Ref. 10 from a relatively cumbersome approach based on fractional Fourier transformation. Its analog enters more naturally in Ref. 9 [see Eq. (7.2) there], where the GWFT is used throughout.
- In Refs. 9 and 10, the ray contribution width was taken to be constant. However, this leads to a paradox concerning error estimates and the variation of γ. We now clarify this matter by explicitly allowing γ to vary from the outset and, in Section 5, discuss this paradox and its resolution. What is more, we allow γ to also depend on ξ in the second paper in this series; see Ref. 4.
- See Ref. 4.
- Provided that the real part of γ is positive and f is not too pathological at infinity, the integrated part vanishes.
- Because Eq. (2.11) is an identity, all higher-order γ derivatives are also suppressed by one asymptotic order owing to the first relation in Eq. (2.12). Equation (2.18) then follows.
- The derivation of Eq. (2.14) can be performed easily by using computer algebra: with Y replaced by γX+iP, expanding the derivatives of Eq. (2.13) is straightforward, and, after integrating with respect to γ, the final result is then simplified by replacing all occurrences of P with i(γX− Y). However, the transition to Eq. (2.21) is not as straightforward to automate. Fortunately there is an easy manual shortcut: First notice that with R_{a} ≔ [f(ξ)/(X^{′}Y_{a}^{′})]^{(n)}, where the superscript denotes n derivatives with respect to ξ, it follows that R_{a}−R_{b}=[f(ξ)(Y_{b}^{′}− Y_{a}^{′})/(X^{′}Y_{a}^{′}Y_{b}^{′})]^{(n)}=(γ_{b}−γ_{a})[f(ξ)/(Y_{a}^{′}Y_{b}^{′})]^{(n)}.Since G(ξ, z; γ_{a}, γ_{b})=A_{1}(ξ, z, γ_{b})−A_{1}(ξ, z, γ_{a}), this result gives the first and second terms of Eq. (2.21) directly from Eq. (2.14). The third term follows by considering thecase f(ξ)≡−1 and n=1, so R_{a}=(X^{′}Y_{a}^{″}+X^{″}Y_{a}^{′})/(X^{′}Y_{a}^{′})^{2} and R_{a}−R_{b}=(γ_{b}−γ_{a})[(Y_{a}^{′}Y_{b}^{″}+Y_{a}^{″}Y_{b}^{′})/(Y_{a}^{′}Y_{b}^{′})^{2}].If this is derived the long way (i.e., by replacing all Y’s with γX+iP, etc.) then it is easy to see that the same result holds where the double primes are replaced by any number of primes. This gives the third component of Eq. (2.21). Notice also that Eq. (2.14) appears to signal a catastrophe at caustics; i.e., wherever X^{′} vanishes it seems that A_{1} must diverge. However this is not so, as we can appreciate by writing Eq. (2.14) in a more general form: A_{1}(ξ, z, γ)= F(ξ, z; γ_{a})+G(ξ, z; γ_{a}, γ). [Equation (2.14) is just a special case where γ_{a}=∞.] It follows from this more general form that there will be problems at caustics only for infinite γ. This will become clearer in Section 5.
- This condition can be justified by using the saddle-point method on Eq. (2.20) to show that, when the value of the factor in braces is effectively determined by just A_{0}, the result of the integral is similarly unchanged by the components added to A_{0}.
- See, for example, Eqs. (63)–(66) on page 1.23 of M. Bass, ed., Handbook of Optics (McGraw-Hill, New York, 1995).
- See, for example, Chap. 8 of H. Goldstein, Classical Mechanics (Addison Wesley, Reading, Mass., 1980).
- If L is identified as the classical action, Eq. (3.12) may appear to be a surprise: the time derivative of the action is normally just the Hamiltonian [see, e.g., Eqs. (32)–(33) of G. W. Forbes, “On variational problems in parametric form,” Am. J. Phys. 59, 1130–1140 (1991)]. The difference here is that the derivative in Eq. (3.12) is taken along a fixed ray, whereas x is held fixed in the usual result. In fact, Eq. (3.12) follows simply from the fact that L(ξ, z) is just the OPL up to the plane of given z for the ray ξ: On account of Eq. (3.1), and then Eq. (3.5) followed by Eqs. (3.4) and (3.9), it can be seen that L̇=L=pV−H=PẊ+ H. Similarly, L^{′}=(∂L/∂ẋ)X^{′}=PX^{′} follows directly from the integrated part that emerges when one is working with integration by parts to derive Eq. (3.2).
- The link to the transport equation of Ref. 1 can be appreciated more directly from the result given later in Eq. (6.1). Alternatively, since the power flux is proportional to Im(U^{*}∇U), the method used in Ref. 5 to derive its Eq. (3.9) can be used to show that the integrated power flux across any plane of fixed z is asymptotically just ∫|a_{0}(ξ)|^{2}dξ.
- See Ref. 4.
- The key step in this derivation follows simply upon observing that, by definition, F(ξ, z) is just ȧ_{1}− (d/dz)D[ξ, z, γ(z)] and the original form of the term inside the integrand of Eq. (5.7) is F(ξ, z)+∂D/∂z(ξ, z, γ¯). Since ∂D/∂z(ξ, z, γ¯)=(d/dz)D(ξ, z, γ¯), it is evident that F(ξ, z)+∂D/∂z(ξ, z, γ¯) is precisely ȧ_{1} with γ(z) replaced by γ¯. Notice also that the advantage of using Eq. (5.7) in place of Eq. (5.3) is that the form of D that follows from the analog of Eq. (2.14) has apparent problems at caustics that are avoided by using G of Eq. (2.21).
- Cancellation between the separate terms is the only way that condition (5.8) may not be a necessary condition. However, because condition (2.22) (with γ_{a}=γ and γ_{b}= 2^{β}γ) must hold for all |β|<1, it is reasonable to expect that this is in fact a necessary condition.
- Following the discussion in Section 6, it will become clear that this lower bound prevents catastrophes in the Fourier transform of the field estimate. This can be appreciated by observing that x and p have simply exchanged roles in Eq. (6.4) and that the ray contribution width in Fourier space is now linked to the inverse of γ.
- The integer multiple of π in Eq. (5.16) [hence of π/2 in condition (5.15)] becomes the Maslov index that appears in Eq. (6.1) and is familiar from semiclassical mechanics. It can therefore also be determined by counting caustics (i.e., the number of sign changes in X^{′}), but Eq. (5.14) is more explicit and straightforward. Recall that an example of such a π/2 phase shift at a caustic is illustrated in Fig. 1.
- From the discussion in Section 2, it follows that the exponent of the integrand in Eq. (4.16) is stationary only where X(ξ, z)=x, and, if one of the solutions of this relation is written as ξ=ξ_{s}(x), the saddle-point method can be used with Eq. (2.15) to derive an expression for the associated asymptotic contribution to U_{γ} of the form a_{0}[ξ_{s}(x)]H[ξ_{s}(x), z]X^{′}[ξ_{s}(x), z]+O(k^{−1})exp{ikL[ξ_{s}(x), z]}. This expression is also valid for U_{γ}^{(0)} of Eq. (4.16), and, more generally, the estimate involves a sum of such contributions from each ray through the point of interest.
- See, for example, Section 2.3 of Ref. 8.
- V. P. Maslov and M. V. Fedoriuk, Semiclassical Aproximation in Quantum Mechanics (Reidel, Boston, 1981).
- J. B. Delos, “Semiclassical calculation of quantum mechanical wavefunctions,” Vol. 67 of Advances in Chemical Physics, I. Prigogine and S. A. Rice, eds. (Wiley Interscience, 1986), pp. 161–213.
- A description of the application of Maslov’s canonical operator method to optical fields is given in Chap. 6 of Ref. 2.
- M. M. Popov, “A new method of computation of wavefields using Gaussian beams,” Wave Motion 4, 85–97 (1982).
- A. W. Greynolds, “Propagation of generally astigmatic Gaussian beams along skew ray paths,” in Diffraction Phenomena in Optical Engineering Applications, D. M. Byrne and J. E. Harvey, eds., Proc. SPIE 560, 33–50 (1985).
- V. M. Babich and M. M. Popov, “Gaussian summation method (review),” Radiophys. Quantum Electron. 39, 1063–1081 (1989).
- J. M. Arnold, “Phase-space localization and discrete representations of wave fields,” J. Opt. Soc. Am. A 12, 111–123 (1995).
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