## Asymptotic estimation of the optical wave propagator. I. Derivation of a new method

JOSA A, Vol. 15, Issue 5, pp. 1329-1340 (1998)

http://dx.doi.org/10.1364/JOSAA.15.001329

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### Abstract

A new method is proposed for modeling wave propagation in two-dimensional, smoothly varying inhomogeneous media. This method is valid in the limit of small wavelength and constructs the wave propagator by combining contributions from all relevant rays. The form of each contribution was reported in J. Opt. Soc. Am. A 14, 1279 (1997). The resulting estimate for the total field takes a strikingly simple form, which associates a Gaussian field with each ray. This form is asymptotically correct and free of the usual difficulties associated with caustics.

© 1998 Optical Society of America

**OCIS Codes**

(350.5500) Other areas of optics : Propagation

(350.7420) Other areas of optics : Waves

**History**

Original Manuscript: June 11, 1997

Revised Manuscript: November 18, 1997

Manuscript Accepted: October 6, 1997

Published: May 1, 1998

**Citation**

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)

http://www.opticsinfobase.org/josaa/abstract.cfm?URI=josaa-15-5-1329

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### References

- The assumption that the field is forward propagating is essential to the definition of the propagator. A discussion of forward-propagating fields is given in M. A. Alonso, G. W. Forbes, “Semigeometrical estimation of Green’s functions and wave propagators in optics,” J. Opt. Soc. Am. A 14, 1076–1086 (1997). [CrossRef]
- M. A. Alonso, G. W. Forbes, “Uniform asymptotic expansions for wave propagators via fractional transformations,” J. Opt. Soc. Am. A 14, 1279–1292 (1997). [CrossRef]
- A. W. Conway, J. L. Synge, eds., The Mathematical Papers of Sir William Rowan Hamilton (Cambridge U. Press, Cambridge, 1931), Vol. 1.
- H. A. Buchdahl, An Introduction to Hamiltonian Optics (Dover, New York, 1993), p. 8.
- M. Born, E. Wolf, Principles of Optics, 6th ed. (Pergamon, Oxford, 1989), p. 133.
- A. Walther, “Lenses, wave optics and eikonal functions,” J. Opt. Soc. Am. 59, 1325–1333 (1969). [CrossRef]
- A. Walther, The Ray and Wave Theory of Lenses (Cambridge U. Press, Cambridge, 1995), pp. 169–187.
- J. H. Van Vleck, “The correspondence principle in the statistical interpretation of quantum mechanics,” Proc. Natl. Acad. Sci. USA 14, 178–188 (1928). [CrossRef] [PubMed]
- Y. A. Kravtsov, Y. A. Orlov, Caustics, Catastrophes and Wave Fields (Springer-Verlag, Berlin, 1993), p. 18.
- The definition of the mixed characteristic varies from one author to the next. The one used here is consistent with the one given by R. Luneburg, Mathematical Theory of Optics (U. California Press, Berkeley, Calif., 1964), pp. 111–115.
- R. Courant, D. Hilbert, Methods of Mathematical Physics (Interscience, New York, 1962), Vol. II, pp. 32–39.
- An analogous expression for three-dimensional propagation is given in Refs. 6 and 7. A rigorous derivation of a more general result is given in Eq. (5.21) of Ref. 1.
- A momentum caustic occurs when the rays in the bundle are locally parallel. Momentum caustics carry a sign: They are positive when the bundle goes from being divergent to convergent and negative when the converse is true. The net sum of momentum caustics is given by the difference between the number of positive ones and the number of negative ones.
- V. P. Maslov, M. V. Fedoriuk, Semiclassical Approximation in Quantum Mechanics (Reidel, Boston, Mass., 1981).
- J. B. Delos, “Semiclassical calculation of quantum mechanical wavefunctions,” in Advances in Chemical Physics, Vol. 65 of Advances in Chemical Physics, I. Prigogine, S. A. Rice, eds. (Wiley/Interscience, New York, 1986), pp. 161–213.
- See Chap. 6 of Ref. 9.
- M. A. Alonso, G. W. Forbes, “Fractional Legendre transformation,” J. Phys. A 28, 5509–5527 (1995). [CrossRef]
- E. U. Condon, “Immersion of the Fourier transform in a continuous group of functional transformations,” Proc. Natl. Acad. Sci. USA 23, 158–164 (1937). [CrossRef] [PubMed]
- For the purposes of the asymptotics considered here, a factor of k is included in the exponent of the kernel of the FrFT (with the corresponding factor of k in the amplitude for normalization). Comprehensive bibliographies on this subject are given in A. W. Lohmann, D. Mendlovic, Z. Zalevsky, eds., The Fractional Fourier Transform–Status Report 1995 (Tel Aviv U. Press, Tel Aviv, 1995); “Fractional Fourier transform and its applications in optics and signal processing—a bibliography,” http://www.ee.bilkent.edu.tr/∼haldun/ffbiblio/ffbiblio.html .
- See Eq. (5.21) of Ref. 1. In that reference the initial representation was also allowed to be fractional, but the initial parameter ϕ is set to zero here. The fractional propagator and characteristic, referred to here as Kθ(x, y; x′, ρ′) and Vθ(x, y; x′, ρ′), correspond, respectively, to Q(x, ρ, 0;x′, ρ′, θ) and F(x, ρ, 0; x′, ρ′, θ) of Ref. 1.
- Let y′ and p′ be the parametric position and momentum at x′ of a ray. The relative change of y′ as x′ increases is given by the ratio of the components of the normal vector, that is, ∂y′/∂x′=αy′/αx′=p′/[n2(x′, y′)-p′2]1/2. Notice that for a vertical phase-space curve segment, y′ is constant. Because ∂y′/∂x′ is evidently a monotonically increasing function of p′, the points with higher momentum have greater displacement as x′ increases, and this corresponds to a clockwise rotation through the vertical in phase space.
- A Gaussian of different width or even a different window function could be chosen here. However, it turns out that this particular choice greatly simplifies the results that follow.
- Given that, from Eq. (4.3), ϑ is the angle from the y′ axis to the local tangent of the phase-space curve, Eq. (3.10) states that |ρs′-ρc′|max=2|R|.
- This theorem states that phase space area (or étendue) is conserved during propagation; i.e., ∂y′/∂y ∂p′/∂p-∂p′/∂y ∂y′/∂p=1. See, for example, W. T. Welford, R. Winston, High Collection Nonimaging Optics (Academic, San Diego, Calif., 1989), pp. 225–226.
- M. Alonso, G. Forbes, “Asymptotic estimation of the optical wave propagator. II. Relative validity,” J. Opt. Soc. Am. A 15, 1341–1354 (1998). [CrossRef]
- A. Erdélyi, Asymptotic Expansions (Dover, New York, 1956), p. 51.

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