Abstract

A new ray-based method is extended to include the modeling of optical interfaces. The essential idea is that the wave field and its derivatives are always expressed as a superposition of ray contributions of flexible width. Interfaces can be analyzed in this way by introducing a family of surfaces that smoothly connects them. Even though the ray-to-wave link may appear to be obscured at caustics, the standard Fresnel coefficients (for plane waves at flat interfaces between homogeneous media) are shown to be universally applicable on a ray-by-ray basis. Thus, in the interaction at the interface, the surface’s curvature and any gradients in the refractive indices influence only the higher asymptotic corrections. Further, this method finally gives access to such corrections.

© 2001 Optical Society of America

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References

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  1. G. W. Forbes, M. A. Alonso, “Using rays better. I. Theory for smoothly varying media,” J. Opt. Soc. Am. A 18, 1132–1145 (2001).
    [CrossRef]
  2. M. Born, E. Wolf, Principles of Optics, 7th ed. (Cambridge U. Press, 1999). See Sec. 1.5.
  3. Y. A. Kravtsov, Y. A. Orlov, Geometrical Optics of Inhomogeneous Media (Springer-Verlag, Berlin, 1990). See Section 2.5.
  4. A detailed extension to three dimensions is beyond the scope of this paper, but many of the necessary ideas are contained in Ref. 1 and are demonstrated in the examples given by M. A. Alonso, G. W. Forbes, “Using rays better. II. Ray families for simple wave fields,” J. Opt. Soc. Am. A 18, 1146–1159 (2001).
    [CrossRef]
  5. It is possible, of course, to consider other families which smoothly connect one interface to the next in an optical system or where the xcoordinate may also scale with ζ. However, Eq. (2.3) is sufficient for the present purposes.
  6. G. W. Forbes, M. A. Alonso, “What on earth is a ray and how can we use them best?” in Proceedings of the International Optical Design Conference 1998, L. R. Gardner, K. P. Thompson, eds., Proc. SPIE3482, 22–31 (1998).
    [CrossRef]
  7. Since f has the dimensions of length, it would be safer in general to write f(x)≡f0,where f0is a constant, and then z=ζf0.However, the special case given in the body (which follows when f0is taken to be one unit of length—either mm or whatever) is more convenient for making the connection to the earlier results.
  8. The Legendre transformation of the square root of a quadratic is straightforward. For L(x˙)=(a+2bx˙+cx˙2)1/2,it follows that p=(b+cx˙)/L(x˙),which can be solved to find x˙={p[(ac-b2)/(c-p2)]1/2-b}/c.The end result then takes the form H(p)≔px-L=-bp/c-[(a-b2/c)(1-p2/c)]1/2, and this gives an easy path to Eq. (3.4).
  9. An alternative interpretation for pfollows directly upon observing that (p,0)=κN+nT/|T| for κ=-n(f+ζf′x˙)/|T|.That is, pis the transverse displacement (i.e., zero z component) between the surface normal and a point that sits n length units along the ray tangent from the surface.This gives the useful connection that pis exactly the ray-direction variable used by G. W. Forbes, B. D. Stone, “Restricted characteristic functions for general optical configurations,” J. Opt. Soc. Am. A 10, 1263–1269 (1993). See Fig. 1 of that work. The results derived here for modeling interfaces can therefore be expected to couple naturally to the characteristic functions treated in this reference.
  10. This generalization is discussed at Eq. (4.16) of the reference cited in Note 4.
  11. The connection for Pis least obvious but follows directly from the geometric interpretations given in Section 3.
  12. Notice from the geometric interpretation that C(ξ, ζ)vanishes only when a ray becomes tangent to the surface z=ζf(x),and this is analogous to rays turning back in Ref. 1. Such cases have been excluded consistently in this series of papers.
  13. Because γ is now a constant, there is no need for the open overdots on Ythat were used in Ref. 1. That is, Y˙=Y˚when γ is independent of ζ.
  14. Given Eq. (3.10) and the geometric interpretation of Cthat was presented in the third paragraph of Section 3, the factors in Eq. (5.9) can be seen to correspond to the familiar factors found when a uniform plane wave is incident on a flat interface. Even though this is a scalar field, the results match those of an electromagnetic field when the electric field is normal to the plane of incidence; see T⊥and R⊥of Eqs. (20) and (21) of Subsection 1.5.2 of Ref. 2.
  15. Such matters are treated for the standard models by Y. A. Kravtsov et al., “Theory and applications of complex rays,” in Progress in Optics XXXIX, E. Wolf, ed. (Elsevier, Amsterdam, 1999), pp. 1–62.

2001 (2)

1993 (1)

Alonso, M. A.

Born, M.

M. Born, E. Wolf, Principles of Optics, 7th ed. (Cambridge U. Press, 1999). See Sec. 1.5.

Forbes, G. W.

Kravtsov, Y. A.

Such matters are treated for the standard models by Y. A. Kravtsov et al., “Theory and applications of complex rays,” in Progress in Optics XXXIX, E. Wolf, ed. (Elsevier, Amsterdam, 1999), pp. 1–62.

Y. A. Kravtsov, Y. A. Orlov, Geometrical Optics of Inhomogeneous Media (Springer-Verlag, Berlin, 1990). See Section 2.5.

Orlov, Y. A.

Y. A. Kravtsov, Y. A. Orlov, Geometrical Optics of Inhomogeneous Media (Springer-Verlag, Berlin, 1990). See Section 2.5.

Stone, B. D.

Wolf, E.

M. Born, E. Wolf, Principles of Optics, 7th ed. (Cambridge U. Press, 1999). See Sec. 1.5.

J. Opt. Soc. Am. A (3)

Other (12)

It is possible, of course, to consider other families which smoothly connect one interface to the next in an optical system or where the xcoordinate may also scale with ζ. However, Eq. (2.3) is sufficient for the present purposes.

G. W. Forbes, M. A. Alonso, “What on earth is a ray and how can we use them best?” in Proceedings of the International Optical Design Conference 1998, L. R. Gardner, K. P. Thompson, eds., Proc. SPIE3482, 22–31 (1998).
[CrossRef]

Since f has the dimensions of length, it would be safer in general to write f(x)≡f0,where f0is a constant, and then z=ζf0.However, the special case given in the body (which follows when f0is taken to be one unit of length—either mm or whatever) is more convenient for making the connection to the earlier results.

The Legendre transformation of the square root of a quadratic is straightforward. For L(x˙)=(a+2bx˙+cx˙2)1/2,it follows that p=(b+cx˙)/L(x˙),which can be solved to find x˙={p[(ac-b2)/(c-p2)]1/2-b}/c.The end result then takes the form H(p)≔px-L=-bp/c-[(a-b2/c)(1-p2/c)]1/2, and this gives an easy path to Eq. (3.4).

This generalization is discussed at Eq. (4.16) of the reference cited in Note 4.

The connection for Pis least obvious but follows directly from the geometric interpretations given in Section 3.

Notice from the geometric interpretation that C(ξ, ζ)vanishes only when a ray becomes tangent to the surface z=ζf(x),and this is analogous to rays turning back in Ref. 1. Such cases have been excluded consistently in this series of papers.

Because γ is now a constant, there is no need for the open overdots on Ythat were used in Ref. 1. That is, Y˙=Y˚when γ is independent of ζ.

Given Eq. (3.10) and the geometric interpretation of Cthat was presented in the third paragraph of Section 3, the factors in Eq. (5.9) can be seen to correspond to the familiar factors found when a uniform plane wave is incident on a flat interface. Even though this is a scalar field, the results match those of an electromagnetic field when the electric field is normal to the plane of incidence; see T⊥and R⊥of Eqs. (20) and (21) of Subsection 1.5.2 of Ref. 2.

Such matters are treated for the standard models by Y. A. Kravtsov et al., “Theory and applications of complex rays,” in Progress in Optics XXXIX, E. Wolf, ed. (Elsevier, Amsterdam, 1999), pp. 1–62.

M. Born, E. Wolf, Principles of Optics, 7th ed. (Cambridge U. Press, 1999). See Sec. 1.5.

Y. A. Kravtsov, Y. A. Orlov, Geometrical Optics of Inhomogeneous Media (Springer-Verlag, Berlin, 1990). See Section 2.5.

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Figures (2)

Fig. 1
Fig. 1

(a) A free-space wave field for a ray family that contains a caustic. The wave amplitude is given by the shading, and the phase contours are shown as black curves. Geometric length is marked out with dots on a representative ray. Notice that, under propagation from left to right, the wave fronts drift away from the dots and end up with an offset of λ/4. Further, near the caustic, the rays are not normal to the wave fronts and the ray spacing no longer gives the wave amplitude. (As discussed in Ref. 3, the product of the square of the amplitude, the refractive index, and the perpendicular spacing between neighboring rays is effectively constant away from caustics.) It therefore appears that the ray–wave connection is profoundly disrupted at a caustic. A natural and long-standing question then is whether a ray-based analysis is applicable when an interface is encountered near a caustic. For example, the same wave field is shown in (b), but now with a flat interface at the center. The interference between the incident and the reflected components is evident to the left of the interface (where the index steps by a factor of 1.5). The reflected rays are shown as dashed lines, but it is not clear that the transmitted and the reflected rays will generally retain a useful connection to the wave field: What happens to the caustic phase shift, and can the Fresnel coefficients be used when the interface is curved or the media are inhomogeneous? Such matters motivate the developments presented below.

Fig. 2
Fig. 2

A ray approaching a refractive interface at the surface described by z=f(x). The family of surfaces of the form z=ζf(x), where 0<ζ<1 are shown in gray. Together with the lines of constant x, these surfaces provide a non-Cartesian coordinate system that is effective for characterizing both the rays and the wave field in the analysis of the field’s interaction with the interface.

Equations (66)

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[k-22+n2(x, z)]U(x, z)=0,
n2(x, z)=NI(x, z),z<f(x)NII(x, z),z>f(x),
U(x, ζ)U[x, ζf(x)].
f2(x)2Ux2-2ζf(x)f(x)2Uxζ
+{1+[ζf(x)]2}2Uζ2-ζ[f(x)f(x)-2 f2(x)]Uζ
+k2f2(x)n2[x, ζf(x)]U(x, ζ)=0.
OPL=Lf[x(ζ), x˙(ζ), ζ]dζ,
Lf(x, x˙, ζ)n[x, ζf(x)]{[f(x)+ζf(x)x˙]2+x˙2}1/2.
p Lfx˙=n[x, ζ, f(x)]ζf(x)(x)+ζf(x)x˙]+x˙[f(x)+ζf(x)x˙]2+x˙2
Hf(x, p, ζ)=-f(x)1+[ζf(x)]2[ζf(x)p+C(x, p, ζ)],
C(x, p, ζ)  n[x, ζf(x)](1+[ζf(x)]2-{p/n[x, ζf(x)]}2)1/2.
x˙=Hfp(x, p, ζ),
p˙=-Hfx(x, p, ζ),
ddζ{Hf[x(ζ), p(ζ), ζ]}=Hfζ[x(ζ), p(ζ), ζ].
{1+[ζf(x)]2}Hf2+2ζf(x)f(x)pHf
-f2(x){n2[x, ζf(x)]-p2}=0,
x˙A=-fB,
p˙A=ζHf[Bf+pf2]-ff(n2-p2)-12f2ddx{n2[x, ζf(x)]},
ddζ{Hf[x(ζ), p(ζ), ζ]}A=12f3n2z[x, ζf(x)]-fHfB,
A [1+(ζf)2]Hf+ζffp,
B  ζfHf+pf.
H(ξ, ζ) -Hf[X(ξ, ζ), P(ξ, ζ), ζ],
C(ξ, ζ)C[X(ξ, ζ), P(ξ, ζ), ζ],
-CY˙f+(γf+iζHf)(Pf-ζHf)
+iζHPf2+i(n2-P2)ff
+i2n2x[X, ζf(X)]+n2z[X, ζf(X)]ζff2=0.
L(ξ, ζ)=P(ξ, ζ)X(ξ, ζ),
L˙(ξ, ζ)=H(ξ, ζ)+P(ξ, ζ)X˙(ξ, ζ),
H(ξ, ζ)=X(ξ, ζ)P˙(ξ, ζ)-X˙(ξ, ζ)P(ξ, ζ),
Uγ(x, ζ)k2π A(ξ, ζ, γ)Y(ξ, ζ)g(x, ζ;γ, ξ)dξ,
g(x, ζ; γ, ξ)exp{-(kγ/2)[x-X(ξ, ζ)]2}×exp(ik{L(ξ, ζ)+[x-X(ξ, ζ)]P(ξ, ζ)}).
[E0+E1/k+E2/k2+O(k-3)]gdξ=0,
E1=i2RS+RS+2 fCYddζ(AC)=0,
A(ξ, ζ, γ)=a(ξ, ζ, γ)C(ξ, ζ)=1C(ξ, ζ)j=0aj(ξ, ζ, γ)(ik)j.
Uγ(0)(x, ζ)k2π a0(ξ)Y(ξ, ζ)C(ξ, ζ) g(x, ζ;γ, ξ)dξ,
XR(ξ, 1)XT(ξ, 1)XI(ξ, 1),
PR(ξ, 1)PT(ξ, 1)PI(ξ, 1),
LR(ξ, 1)LT(ξ, 1)LI(ξ, 1).
UγS(x, ζ)=k2π aS(ξ, ζ, γ)×YS(ξ, ζ)|CS(ξ, ζ)| gS(x, ζ;γ, ξ)dξ,
wI(ξ, 1, γ)+wR(ξ, 1, γ)-wT(ξ, 1, γ)0.
1ikUγSζ=k2π×HSwS-ik[w˙S-(wSY˙S/YS)]+O(k-2)gSdξ.
wS(ξ, ζ, γ)=j=0wjS(ξ, ζ, γ)(ik)j=YS(ξ, ζ)|CS(ξ, ζ)| j=0ajS(ξ, ζ, γ)(ik)j,
vS(ξ, ζ, γ)
 HSw0S-ik
×[HSw1S-(w0S/YS)Y˙S-12w0S(Y˙S/YS+C˙S/CS)]
+O(k-2),
vI(ξ, 1, γ)+vR(ξ, 1, γ)-vT(ξ, 1, γ)0.
w0I(ξ, 1, γ)+w0R(ξ, 1, γ)-w0T(γ, 1, γ)0
[w0I(ξ, 1, γ)-w0R(ξ, 1, γ)]CI(ξ, 1)
-w0T(ξ, 1, γ)CT(ξ, 1)0.
w0T(ξ, 1, γ)=2CI(ξ, 1)CI(ξ, 1)+CT(ξ, 1)w0I(ξ, 1, γ)
w0R(ξ, 1, γ)=CI(ξ, 1)-CT(ξ, 1)CI(ξ, 1)+CT(ξ, 1)w0I(ξ, 1, γ).
a0T(ξ, 1, γ)=2[CI(ξ, 1)CT(ξ, 1)]1/2CI(ξ, 1)+CT(ξ, 1)a0I(ξ, 1, γ)τ(ξ)a0I(ξ, 1, γ),
a0R(ξ, 1, γ)=CI(ξ, 1, γ)-CT(ξ, 1)CI(ξ, 1)+CT(ξ, 1)a0I(ξ, 1, γ)ρ(ξ)a0I(ξ, 1, γ).
w1I(ξ, 1, γ)+w1R(ξ, 1, γ)-w1T(ξ, 1, γ)0,
[w1I(ξ, 1, γ)-w1R(ξ, 1, γ)]CI(ξ, 1)
-w1T(ξ, 1, γ)CT(ξ, 1)Qγ(ξ),
DγS(ξ) 1+f2[X(ξ, 1)]f[X(ξ, 1)]
×[(w0S/YS)Y˙S+12w0S(Y˙S/YS+C˙S/CS)].
a1T(ξ, 1, γ)=τ(ξ)a1I(ξ, 1, γ)-CT(ξ, 1)Y(ξ, 1) Qγ(ξ)CI(ξ, 1)+CT(ξ, 1),
a1R(ξ, 1, γ)=ρ(ξ)a1I(ξ, 1, γ)-CI(ξ, 1)Y(ξ, 1) Qγ(ξ)CI(ξ, 1)+CT(ξ, 1).
Uγ(0)(x, z)=k2π T(ξ)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)+Y¨S-YS¨+2Y˙S˙-12NS]/4HY,
a˙1={(FS)+MS-(1+ζ2f2)YS¨+ZS˙+2[Y˙-iζfPf+ζf(γf+2iζHf)+ζf2(ζY˙-iP)]S˙+BS}/4f2CY.
H(p)px-L=-bp/c-[(a-b2/c)(1-p2/c)]1/2,
(p,0)=κN+nT/|T|forκ=-n(f+ζfx˙)/|T|.

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