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

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. Y. A. Kravtsov, Y. I. Orlov, Geometrical Optics of Inhomogeneous Media (Springer-Verlag, Berlin, 1990).
  2. Y. A. Kravtsov, Y. I. Orlov, Caustics, Catastrophes and Wave Fields, 2nd ed. (Springer-Verlag, Berlin, 1999).
  3. R. C. Hansen, ed. Geometric Theory of Diffraction (IEEE Press Selected Reprint Series, New York, 1981).
  4. M. A. Alonso, G. W. Forbes, “Using rays better. II. Ray families for simple wavefields,” J. Opt. Soc. Am. A 18, 1146–1159 (2001).
    [CrossRef]
  5. M. A. Alonso, G. W. Forbes, “Using rays better. III. Error estimates and illustrative applications in smooth media,” J. Opt. Soc. Am. A18 (to be published).
  6. M. Born, E. Wolf, Principles of Optics, 7th ed. (Cambridge U. Press, Cambridge, UK, 1999). See, for example, the footnote on p. 519.
  7. A. W. Snyder, J. D. Love, Optical Waveguide Theory (Chapman & Hall, London, 1983). See Chap. 35.
  8. G. W. Forbes, 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, K. P. Thompson, eds., Proc. SPIE3482, 22–31 (1998).
    [CrossRef]
  9. M. A. Alonso, G. W. Forbes, “New approach to semiclassical analysis in mechanics,” J. Math. Phys. 40, 1699–1718 (1999).
    [CrossRef]
  10. M. A. Alonso, 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).
    [CrossRef]
  11. G. W. Forbes, M. A. Alonso, “Asymptotic estimation of the optical wave propagator. II. Relative validity,” J. Opt. Soc. Am. A 15, 1341–1354 (1998).
    [CrossRef]
  12. 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, G. W. Forbes, “Semigeometrical estimation of Green’s functions and wave propagators in optics,” J. Opt. Soc. Am. A 14, 1076–1086 (1997).
    [CrossRef]
  13. The issue of the initial phase reference is clarified in Section 5.
  14. 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.
  15. 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.
  16. See Ref. 4.
  17. Provided that the real part of γ is positive and f is not too pathological at infinity, the integrated part vanishes.
  18. 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.
  19. 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 Ra≔[f(ξ)/(X′Ya′)](n), where the superscript denotes n derivatives with respect to ξ, it follows that Ra-Rb=[f(ξ)(Yb′-Ya′)/(X′Ya′Yb′)](n)=(γb-γa)[f(ξ)/(Ya′Yb′)](n). Since G(ξ, z; γa, γb)=A1(ξ, z, γb)-A1(ξ, z, γa), 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 Ra=(X′Ya″+X″Ya′)/(X′Ya′)2 and Ra-Rb=(γb-γa)[(Ya′Yb″+Ya″Yb′)/(Ya′Yb′)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 A1 must diverge. However this is not so, as we can appreciate by writing Eq. (2.14) in a more general form: A1(ξ, z, γ)=F(ξ, z; γa)+G(ξ, z; γa, γ). [Equation (2.14) is just a special case where γa=∞.] 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.
  20. 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 A0, the result of the integral is similarly unchanged by the components added to A0.
  21. See, for example, Eqs. (63)–(66) on page 1.23 of M. Bass, ed., Handbook of Optics (McGraw-Hill, New York, 1995).
  22. See, for example, Chap. 8 of H. Goldstein, Classical Mechanics (Addison Wesley, Reading, Mass., 1980).
  23. 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 L˙=L=pV-H=PX˙+H. Similarly, L′=(∂L/∂x˙)X′=PX′ follows directly from the integrated part that emerges when one is working with integration by parts to derive Eq. (3.2).
    [CrossRef]
  24. 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 ∫|a0(ξ)|2dξ.
  25. See Ref. 4.
  26. The key step in this derivation follows simply upon observing that, by definition, F(ξ, z) is just a˙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 a˙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).
  27. 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 γa=γ and γb=2βγ) must hold for all |β|<1, it is reasonable to expect that this is in fact a necessary condition.
  28. 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.
  29. 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.
  30. 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=ξs(x), the saddle-point method can be used with Eq. (2.15) to derive an expression for the associated asymptotic contribution to Uγ of the form a0[ξs(x)]H[ξs(x), z]X′[ξs(x), z]+O(k-1)exp{ikL[ξs(x), z]}. This expression is also valid for Uγ(0) of Eq. (4.16), and, more generally, the estimate involves a sum of such contributions from each ray through the point of interest.
  31. See, for example, Section 2.3 of Ref. 8.
  32. V. P. Maslov, M. V. Fedoriuk, Semiclassical Aproximation in Quantum Mechanics (Reidel, Boston, 1981).
  33. J. B. Delos, “Semiclassical calculation of quantum mechanical wavefunctions,” Vol. 67 of Advances in Chemical Physics, I. Prigogine, S. A. Rice, eds. (Wiley Interscience, 1986), pp. 161–213.
  34. A description of the application of Maslov’s canonical operator method to optical fields is given in Chap. 6 of Ref. 2.
  35. M. M. Popov, “A new method of computation of wavefields using Gaussian beams,” Wave Motion 4, 85–97 (1982).
    [CrossRef]
  36. A. W. Greynolds, “Propagation of generally astigmatic Gaussian beams along skew ray paths,” in Diffraction Phenomena in Optical Engineering Applications, D. M. Byrne, J. E. Harvey, eds., Proc. SPIE560, 33–50 (1985).
    [CrossRef]
  37. V. M. Babich, M. M. Popov, “Gaussian summation method (review),” Radiophys. Quantum Electron. 39, 1063–1081 (1989).
    [CrossRef]
  38. J. M. Arnold, “Phase-space localization and discrete representations of wave fields,” J. Opt. Soc. Am. A 12, 111–123 (1995).
    [CrossRef]

2001 (1)

1999 (1)

M. A. Alonso, G. W. Forbes, “New approach to semiclassical analysis in mechanics,” J. Math. Phys. 40, 1699–1718 (1999).
[CrossRef]

1998 (2)

1997 (1)

1995 (1)

1991 (1)

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 L˙=L=pV-H=PX˙+H. Similarly, L′=(∂L/∂x˙)X′=PX′ follows directly from the integrated part that emerges when one is working with integration by parts to derive Eq. (3.2).
[CrossRef]

1989 (1)

V. M. Babich, M. M. Popov, “Gaussian summation method (review),” Radiophys. Quantum Electron. 39, 1063–1081 (1989).
[CrossRef]

1982 (1)

M. M. Popov, “A new method of computation of wavefields using Gaussian beams,” Wave Motion 4, 85–97 (1982).
[CrossRef]

Alonso, M. A.

Arnold, J. M.

Babich, V. M.

V. M. Babich, M. M. Popov, “Gaussian summation method (review),” Radiophys. Quantum Electron. 39, 1063–1081 (1989).
[CrossRef]

Born, M.

M. Born, E. Wolf, Principles of Optics, 7th ed. (Cambridge U. Press, Cambridge, UK, 1999). See, for example, the footnote on p. 519.

Delos, J. B.

J. B. Delos, “Semiclassical calculation of quantum mechanical wavefunctions,” Vol. 67 of Advances in Chemical Physics, I. Prigogine, S. A. Rice, eds. (Wiley Interscience, 1986), pp. 161–213.

Fedoriuk, M. V.

V. P. Maslov, M. V. Fedoriuk, Semiclassical Aproximation in Quantum Mechanics (Reidel, Boston, 1981).

Forbes, G. W.

M. A. Alonso, G. W. Forbes, “Using rays better. II. Ray families for simple wavefields,” J. Opt. Soc. Am. A 18, 1146–1159 (2001).
[CrossRef]

M. A. Alonso, G. W. Forbes, “New approach to semiclassical analysis in mechanics,” J. Math. Phys. 40, 1699–1718 (1999).
[CrossRef]

M. A. Alonso, 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).
[CrossRef]

G. W. Forbes, M. A. Alonso, “Asymptotic estimation of the optical wave propagator. II. Relative validity,” J. Opt. Soc. Am. A 15, 1341–1354 (1998).
[CrossRef]

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, G. W. Forbes, “Semigeometrical estimation of Green’s functions and wave propagators in optics,” J. Opt. Soc. Am. A 14, 1076–1086 (1997).
[CrossRef]

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 L˙=L=pV-H=PX˙+H. Similarly, L′=(∂L/∂x˙)X′=PX′ follows directly from the integrated part that emerges when one is working with integration by parts to derive Eq. (3.2).
[CrossRef]

M. A. Alonso, G. W. Forbes, “Using rays better. III. Error estimates and illustrative applications in smooth media,” J. Opt. Soc. Am. A18 (to be published).

G. W. Forbes, 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, K. P. Thompson, eds., Proc. SPIE3482, 22–31 (1998).
[CrossRef]

Goldstein, H.

See, for example, Chap. 8 of H. Goldstein, Classical Mechanics (Addison Wesley, Reading, Mass., 1980).

Greynolds, A. W.

A. W. Greynolds, “Propagation of generally astigmatic Gaussian beams along skew ray paths,” in Diffraction Phenomena in Optical Engineering Applications, D. M. Byrne, J. E. Harvey, eds., Proc. SPIE560, 33–50 (1985).
[CrossRef]

Kravtsov, Y. A.

Y. A. Kravtsov, Y. I. Orlov, Geometrical Optics of Inhomogeneous Media (Springer-Verlag, Berlin, 1990).

Y. A. Kravtsov, Y. I. Orlov, Caustics, Catastrophes and Wave Fields, 2nd ed. (Springer-Verlag, Berlin, 1999).

Love, J. D.

A. W. Snyder, J. D. Love, Optical Waveguide Theory (Chapman & Hall, London, 1983). See Chap. 35.

Maslov, V. P.

V. P. Maslov, M. V. Fedoriuk, Semiclassical Aproximation in Quantum Mechanics (Reidel, Boston, 1981).

Orlov, Y. I.

Y. A. Kravtsov, Y. I. Orlov, Caustics, Catastrophes and Wave Fields, 2nd ed. (Springer-Verlag, Berlin, 1999).

Y. A. Kravtsov, Y. I. Orlov, Geometrical Optics of Inhomogeneous Media (Springer-Verlag, Berlin, 1990).

Popov, M. M.

V. M. Babich, M. M. Popov, “Gaussian summation method (review),” Radiophys. Quantum Electron. 39, 1063–1081 (1989).
[CrossRef]

M. M. Popov, “A new method of computation of wavefields using Gaussian beams,” Wave Motion 4, 85–97 (1982).
[CrossRef]

Snyder, A. W.

A. W. Snyder, J. D. Love, Optical Waveguide Theory (Chapman & Hall, London, 1983). See Chap. 35.

Wolf, E.

M. Born, E. Wolf, Principles of Optics, 7th ed. (Cambridge U. Press, Cambridge, UK, 1999). See, for example, the footnote on p. 519.

Am. J. Phys. (1)

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 L˙=L=pV-H=PX˙+H. Similarly, L′=(∂L/∂x˙)X′=PX′ follows directly from the integrated part that emerges when one is working with integration by parts to derive Eq. (3.2).
[CrossRef]

J. Math. Phys. (1)

M. A. Alonso, G. W. Forbes, “New approach to semiclassical analysis in mechanics,” J. Math. Phys. 40, 1699–1718 (1999).
[CrossRef]

J. Opt. Soc. Am. A (5)

Radiophys. Quantum Electron. (1)

V. M. Babich, M. M. Popov, “Gaussian summation method (review),” Radiophys. Quantum Electron. 39, 1063–1081 (1989).
[CrossRef]

Wave Motion (1)

M. M. Popov, “A new method of computation of wavefields using Gaussian beams,” Wave Motion 4, 85–97 (1982).
[CrossRef]

Other (29)

A. W. Greynolds, “Propagation of generally astigmatic Gaussian beams along skew ray paths,” in Diffraction Phenomena in Optical Engineering Applications, D. M. Byrne, J. E. Harvey, eds., Proc. SPIE560, 33–50 (1985).
[CrossRef]

M. A. Alonso, G. W. Forbes, “Using rays better. III. Error estimates and illustrative applications in smooth media,” J. Opt. Soc. Am. A18 (to be published).

M. Born, E. Wolf, Principles of Optics, 7th ed. (Cambridge U. Press, Cambridge, UK, 1999). See, for example, the footnote on p. 519.

A. W. Snyder, J. D. Love, Optical Waveguide Theory (Chapman & Hall, London, 1983). See Chap. 35.

G. W. Forbes, 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, K. P. Thompson, eds., Proc. SPIE3482, 22–31 (1998).
[CrossRef]

Y. A. Kravtsov, Y. I. Orlov, Geometrical Optics of Inhomogeneous Media (Springer-Verlag, Berlin, 1990).

Y. A. Kravtsov, Y. I. Orlov, Caustics, Catastrophes and Wave Fields, 2nd ed. (Springer-Verlag, Berlin, 1999).

R. C. Hansen, ed. Geometric Theory of Diffraction (IEEE Press Selected Reprint Series, New York, 1981).

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 ∫|a0(ξ)|2dξ.

See Ref. 4.

The key step in this derivation follows simply upon observing that, by definition, F(ξ, z) is just a˙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 a˙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 γa=γ and γb=2βγ) 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=ξs(x), the saddle-point method can be used with Eq. (2.15) to derive an expression for the associated asymptotic contribution to Uγ of the form a0[ξs(x)]H[ξs(x), z]X′[ξs(x), z]+O(k-1)exp{ikL[ξs(x), z]}. This expression is also valid for Uγ(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, 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, 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.

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 Ra≔[f(ξ)/(X′Ya′)](n), where the superscript denotes n derivatives with respect to ξ, it follows that Ra-Rb=[f(ξ)(Yb′-Ya′)/(X′Ya′Yb′)](n)=(γb-γa)[f(ξ)/(Ya′Yb′)](n). Since G(ξ, z; γa, γb)=A1(ξ, z, γb)-A1(ξ, z, γa), 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 Ra=(X′Ya″+X″Ya′)/(X′Ya′)2 and Ra-Rb=(γb-γa)[(Ya′Yb″+Ya″Yb′)/(Ya′Yb′)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 A1 must diverge. However this is not so, as we can appreciate by writing Eq. (2.14) in a more general form: A1(ξ, z, γ)=F(ξ, z; γa)+G(ξ, z; γa, γ). [Equation (2.14) is just a special case where γa=∞.] 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 A0, the result of the integral is similarly unchanged by the components added to A0.

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).

Cited By

OSA participates in CrossRef's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

Alert me when this article is cited.


Figures (3)

Fig. 1
Fig. 1

Plot of U(x, z) in free space for a Gaussian beam. The gray levels correspond to the field amplitudes and the dotted curves correspond to wave fronts (i.e., phase contours separated by 2π). The superimposed black lines represent a ray family that, away from the waist, clearly matches this wave in terms of both relative phase and amplitude. Notice that the geometric phase is marked out uniformly with white ticks along a couple of representative rays and that near the caustic the wave fronts advance more rapidly and end up one quarter of a cycle out of step with the geometric phase, as is well known. Further, the rays are not normal to the wave fronts near the caustic. Not only does the phase go astray in this region, but the amplitude can no longer be estimated simply in terms of the ray spacing. (The squared modulus of the field is generally inversely proportional to the product of the local refractive index and the transverse spacing between the rays.) That is, in the vicinity of the caustic, the connection between the ray family and the wave field appears to be lost.

Fig. 2
Fig. 2

Two neighboring rays in an inhomogeneous medium, labeled ξ and ξ+δξ. It follows by similar triangles that Sδξ=(P/n)Xδξ and, from the definition of L(ξ, z), that Lδξ=nSδξ. Eliminating S from these two relations gives L=PX.

Fig. 3
Fig. 3

Plot of the elemental ray contribution g[x, z; γ(z), ξ] for a ray (shown as a black curve) that is propagating in a waveguide-like medium where the index is highest along x=0. Again, the gray levels give the field amplitude and the dotted curves are equally spaced phase contours that are now shown in gray so that their form can be seen even when the field amplitude is relatively small. In any given vertical slice, the amplitude has a perfectly Gaussian profile with a width set by the local value of γ(z). The intersections of the phase contours with such a slice are uniformly spaced with a separation determined by the direction of the ray at that slice. When k is increased, the phase contours move closer together and the Gaussian shroud tightens on the ray. It turns out that, as suggested in this figure, the Gaussian width can vary almost arbitrarily with z so that the ray contribution can be either localized or distributed. Surprisingly, the result of a continuous superposition of these elemental contributions is insensitive to such changes, and this is a central result that underlies our new method.

Equations (97)

Equations on this page are rendered with MathJax. Learn more.

[k-22+n2(x, z)]U(x, z)=0,
g(x, z;γ, ξ)=exp{-(kγ/2)[x-X(ξ, z)]2}×exp(ik{L(ξ, z)+[x-X(ξ, z)]P(ξ, z)}),
P(ξ, z)=n[X(ξ, z), z]X˙(ξ, z)/[1+X˙2(ξ, z)]1/2.
L(ξ, z)=P(ξ, z)X(ξ, z).
Uγ(x, z)k2π w[ξ, z, γ(z)]g[x, z;γ(z), ξ]dξ,
gξ=k[x-X(ξ, z)][γX(ξ, z)+iP(ξ, z)]g(x, z;γ, ξ).
Y(ξ, z)γ(z)X(ξ, z)+iP(ξ, z).
Fm(x, z, γ)f(ξ)[x-X(ξ, z)]mg(x, z;γ, ξ)dξ,
F1(x, z, γ)=-1k f(ξ)Y(ξ, z)g(x, z;γ, ξ)dξ,
F2=fXkY+1k2 ξ 1Y fYgdξ,
F3=-1k2 3fYXY+2fYXY+O(k-3)gdξ,
F4=3k2 fX2Y2+O(k-3)gdξ,
Uγγ=k2π wγ-k2[x-X(ξ, z)]2w(ξ, z, γ)×g(x, z;γ, ξ)dξ.
Uγγ=k2π wγ-wX2Y-12k ξ 1Y wYgdξ.
w(ξ, z, γ)=Y(ξ, z)A(ξ, z, γ)=Y(ξ, z) j=0 Aj(ξ, z, γ)(ik)j.
Ajγ=i2Y ξ 1Y ξ Aj-1Y.
A1(ξ, z, γ)=F(ξ, z)-i2 A0XY+A024 51XY-XY+XY(XY)2,
Uγ(x, z)=k2π A[ξ, z, γ(z)]Y(ξ, z)×g[x, z; γ(z), ξ]dξ,
Uγ(0)(x, z)k2π A0(ξ, z)Y(ξ, z)×g[x, z; γ(z), ξ]dξ.
Uγ(1)(x, z)k2π {A0(ξ, z)+A1[ξ, z,γ(z)]/(ik)}×Y(ξ, z)g[x, z; γ(z), ξ]dξ,
G(ξ)Ya(ξ, z)g(x, z; γa, ξ)dξ
=[G(ξ)+O(k-1)]Yb(ξ, z)g(x, z;γb, ξ)dξ,
Uγb(0)(x, z)-Uγa(0)(x, z)=k2π {[A1(ξ, z, γa)-A1(ξ, z, γb)]/(ik)+O(k-2)}×Ya(ξ, z)g(x, z;γa; ξ)dξ.
Uγb(0)(x, z)=k2π {A0(ξ, z)-G(ξ, z; γa, γb)/(ik)+O(k-2)}Ya(ξ, z)g(x, z; γa, ξ)dξ,
G(ξ, z; γa, γb)i2(γb-γa)A0YaYb+A024 51YaYb-YaYb+YaYb(YaYb)2.
|G(ξ, z; γa, γb)||kA0(ξ, z)|.
A1(ξ, z, γb)=A1(ξ, z, γa)+G(ξ, z; γa, γb),
OPL=L[x(z), x˙(z), z]dz,
Lx[x(z), x˙(z), z]-ddz Lx˙[x(z), x˙(z), z]=0.
pLx˙=n(x, z)x˙/1+x˙2,
x˙=V(x, p, z)=p/[n(x, z)2-p2]1/2.
H(x, p, z)pV(x, p, z)-L[x,V(x, p, z), z]
=-[n(x, z)2-p2]1/2.
Hx(x, p, z)=-Lx[x,V(x, p, z), z],
Hp(x, p, z)=V(x, p, z),
Hz(x, p, z)=-Lz[x,V(x, p, z), z].
x˙=Hp(x, p, z)=p/[n2(x, z)-p2]1/2,
p˙=-Hx(x, p, z)=12 n2x[n2(x, z)-p2]1/2.
ddz H[x(z), p(z), z]=Hxx˙+Hpp˙+Hz=Hz[x(z), p(z), z].
H(ξ, z)-H[X(ξ, z), P(ξ, z), z]={n2[X(ξ, z), z]-P2(ξ, z)}1/2,
X˙(ξ, z)=P(ξ, z)H(ξ, z),
P˙(ξ, z)=12H(ξ, z) n2x[X(ξ, z), z].
H˙(ξ, z)=12H(ξ, z) n2z[X(ξ, z), z].
L˙(ξ, z)=n2[X(ξ, z), z]H(ξ, z)=H(ξ, z)+P(ξ, z)X˙(ξ, z).
H(ξ, z)=X(ξ, z)P˙(ξ, z)-X˙(ξ, z)P(ξ, z).
1k2 2gx2=-{[P(ξ, z)+iγΔ(x, ξ, z)]2+γ/k}g(x, z; γ, ξ),
1k2 2Uγx2=k2π {-P2(ξ, z)A[ξ, z, γ(z)]Y(ξ, z)+B1/k+B2/k2}g[x, z; γ(z), ξ]dξ,
B1=iγ(z){2P(ξ, z)S[ξ, z, γ(z)]+P(ξ, z)S[ξ, z, γ(z)]},
B2=γ(z)2 ξ{S[ξ, z, γ(z)]/Y(ξ, z)}.
1k2 2Uγz2=k2π [-H2AY+C1/k+C2/k2+O(k-3)]gdξ,
C1=-i{2HY˚S+[(HY˚)-H˙Y]S-2HA˙Y},
Y2C2=Y(Y˚2-iγ˙H)S-[YY2+Y˚(Y˚Y-3Y˚Y)-iγ˙HY]S-[12 YY2-(34Y˚2+Y˚Y˚)Y+Y˚Y˚Y]S+Y[(A¨Y-2Y˚A˙-Y˚A˙)Y+Y˚YA˙].
n2(x, z)=N0+N1Δ+12N2Δ2+13!N3Δ3+14!N4Δ4+O(Δ5),
Nj(ξ, z)jn2xj[X(ξ, z), z].
n2Uγ=k2π [N0AY+D1/k+D2/k2+O(k-3)]gdξ,
D1=-12(2N1S+N2XS),
D2=12[(N2S/Y)+NS],
N[3N2+N3Y4(X/Y4)]/(12Y).
[(N0-P2-H2)AY+E1/k+E2/k2+O(k-3)]gdξ=0,
Ej=Bj+Cj+Dj,j=1, 2.
E1=i[2RS+RS-iSY˚(H+X˙P-XP˙)+(2HA˙+H˙A)Y]=0,
E1=i4HY ddz(AH)=0.
aj(ξ, z, γ)Aj(ξ, z, γ)H(ξ, z).
Uγ(0)(x, z)=k2π a0(ξ)Y(ξ, z)H(ξ, z)×exp{-12kγ(z)[x-X(ξ, z)]2}×exp(ik{L(ξ, z)+[x-X(ξ, z)]P(ξ, z)})dξ.
a˙1=[(FS)+GS+12KS]/4HY,
F=[2iHγ˙-N2-2(γ2+Y˚2)]/(2Y),
G=[HYY-Y˚Y˚-H˙Y˚Y-iγ˙HH]/(HY),
K=Y-2Y˚1/2(H˙Y˚1/2/H)-2Y˚1/4(Y˚Y˚3/4/Y)+YH1/2 z(H˙/H3/2)-N.
L(ξ, z)=L(ξ, z0)+z0z n2[X(ξ, z), z]{n2[X(ξ, z), z]-P2(ξ, z)}1/2 dz.
a˙1=ddz D[ξ, z, γ(z)]+F(ξ, z),
a1[ξ, z, γ(z)]=a1[ξ, z0, γ(z0)]+D[ξ, z, γ(z)]-D[ξ, z0, γ(z0)]+z0zF(ξ, z)dz.
Dγ=iH2Y ξ SY.
D(ξ, z, γb)-D(ξ, z, γa)=H(ξ, z)G(ξ, z; γa, γb)=a1(ξ, z, γb)-a1(ξ, z, γa).
0=D(ξ, z0, γ¯)-D(ξ, z, γ¯)+z0z ddz D(ξ, z, γ¯)dz,
a1(ξ, z, γ¯)=a1(ξ, z0, γ¯)+z0za¯˙1(ξ, z)dz.
maxallξγA0Y2A0,γYA0Y3A0,γYY3,γY2Y4k
Y(ξ, z)Y(ξ, z)/γ.
maxallξˆ{|Aˆ0/Aˆ0|,|YˆAˆ0/Aˆ0|,|Yˆ|,|Yˆ2|}k.
maxallξˆ{|Aˆ0/Aˆ0|,|Aˆ0/Aˆ0|2,|αˆ|,αˆ2}k.
Y(ξ, z)=γreX(ξ, z)+i[P(ξ, z)+γimX(ξ, z)].
tan[θγ(ξ, z)]=P(ξ, z)/[γX(ξ, z)],
θ˙γ=γ(XP˙-X˙P)γ2X2+P2=-γ(γ2X2+P2)H P2-12X2N2+1H2(PP-12XN1)2,
Y=(γ2X2+P2)1/4 exp(iθγ/2).
θγb=πround(θγa/π)+arctan{[Im(γb)X+P]/[Re(γb)X]},
Uγ(0)(x, z)=r a0(ξr)[H(ξr, z)|X(ξr, z)|]1/2×expikL(ξr, z)-i π2M(ξr, z).
f˜(p)k2πf(x)exp(-ikpx)dx,
f(x)=k2πf˜(p)exp(ikpx)dp.
U˜γ(p, z)=ik2πa[ξ, z, γ(z)]×P(ξ, z)/γ(z)-iX(ξ, z)H(ξ, z)1/2×exp-k [p-P(ξ, z)]22γ(z)exp(ik{L¯(ξ, z)-[p-P(ξ, z)]X(ξ, z)})dξ,
U˜γ0(0)(p, z)=r a0(ξr)[H(ξr, z)|P(ξr, z)|]1/2×expikL¯(ξr, z)+i π4-i π2M˜(ξr, z),
UG(0)(x, z)k2π a0(ξ)1H(ξ, z) [G·X(ξ, z)+iP(ξ, z)](ξ)1/2×exp-k2[x-X(ξ, z)]·G·[x-X(ξ, z)]×exp(ik{L(ξ, z)+[x-X(ξ, z)]·P(ξ, z)})d2ξ,
(Q)(ξ)Qxξ1 Qyξ2-Qxξ2 Qyξ1.
G(x, z)=B(z)exp{-kγ(z)[x-X(z)]2/2}(ik{L(z)+[x-X(z)]P(z)}),
γ˙=-i(γ2+12N2+Y˚2)/H,
b˙=-i(γ+Y˚X˙)b/2H,
Ra-Rb=[f(ξ)(Yb-Ya)/(XYaYb)](n)=(γb-γa)[f(ξ)/(YaYb)](n).
Ra-Rb=(γb-γa)[(YaYb+YaYb)/(YaYb)2].
a0[ξs(x)]H[ξs(x), z]X[ξs(x), z]+O(k-1)exp{ikL[ξs(x), z]}.

Metrics