Abstract

A recently proposed ray-based method for wave propagation is used to provide a meaningful criterion for the validity of rays in wave theory. This method assigns a Gaussian contribution to each ray in order to estimate the field. Such contributions are inherently flexible. By means of a simple example, it is shown that superior field estimates can result when the contributions are no longer forced to evolve like parabasal beamlets.

© 2002 Optical Society of America

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References

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  1. M. Born and E. Wolf, Principles of Optics. Electromagnetic Theory of Propagation, Interference and Diffraction of Light (Cambridge, Cambridge, 7th edition, 1999), pp. 116-129.
  2. Yu. A. Kravtsov andYu. I. Orlov, Geometrical Optics of Inhomogeneous Media (Springer, Berlin, 1990).
    [CrossRef]
  3. Yu. A. Kravtsov andYu. I. Orlov, Caustics, Catastrophes and Wave Fields (Springer, Berlin, 2nd Edition, 1999).
    [CrossRef]
  4. 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).
    [CrossRef]
  5. M.A. Alonso and G.W. Forbes, �??Using rays better. II. Ray families to match prescribed wave fields,�?? J. Opt. Soc. Am. A 18, 1146-1159 (2001).
    [CrossRef]
  6. 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).
    [CrossRef]
  7. G.W. Forbes, �??Using rays better. IV. Refraction and reflection,�?? J. Opt. Soc. Am. A 18, 2557-2564 (2001).
    [CrossRef]
  8. E.J. Heller, �??Frozen Gaussians: A very simple semiclassical approximation,�?? J. Chem. Phys. 75, 2923-2931 (1981).
    [CrossRef]
  9. M.M. Popov, �??A new method of computation of wave fields using Gaussian beams,�?? Wave Motion 4, 85-97 (1982).
    [CrossRef]
  10. V.M. Babich and M.M. Popov, �??Gaussian summation method (review),�?? Izvestiya Vysshikh Zavedenii, Radiofizika 32, 1447-1466 (1989).
  11. M.J. Bastiaans, �??The expansion of an optical signal into a discrete set of Gaussian beams,�?? Optik 57, 95-102 (1980).
  12. A.N. Norris, �??Complex point-source representation of real point sources and the Gaussian beam summation method,�?? J. Opt. Soc. Am. A 12, 2005-2010 (1986).
    [CrossRef]
  13. P.D. Einziger, S. Raz, and M. Shapira, �??Gabor representation and aperture theory,�?? J. Opt. Soc. Am. A 3, 508-522 (1986).
    [CrossRef]
  14. P.D. Einziger and S. Raz, �??Beam-series representation and the parabolic approximation: the frequency domain,�?? J. Opt. Soc. Am. A 5, 1883-1892 (1988).
    [CrossRef]
  15. B.Z. Steinberg, E. Heyman, and L.B. Felsen, �??Phase-space beam summation for time-harmonic radiation from large apertures,�?? J. Opt. Soc. Am. A 8, 41-59 (1991).
    [CrossRef]
  16. J.M. Arnold, �??Phase-space localization and discrete representation of wave fields,�?? J. Opt. Soc. Am. A 12, 111-123 (1995).
    [CrossRef]

Izvestiya Vysshikh Zavedenii, Radiofizik (1)

V.M. Babich and M.M. Popov, �??Gaussian summation method (review),�?? Izvestiya Vysshikh Zavedenii, Radiofizika 32, 1447-1466 (1989).

J. Chem. Phys. (1)

E.J. Heller, �??Frozen Gaussians: A very simple semiclassical approximation,�?? J. Chem. Phys. 75, 2923-2931 (1981).
[CrossRef]

J. Opt. Soc. Am. (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).
[CrossRef]

J. Opt. Soc. Am. A (8)

Optik (1)

M.J. Bastiaans, �??The expansion of an optical signal into a discrete set of Gaussian beams,�?? Optik 57, 95-102 (1980).

Wave Motion (1)

M.M. Popov, �??A new method of computation of wave fields using Gaussian beams,�?? Wave Motion 4, 85-97 (1982).
[CrossRef]

Other (3)

M. Born and E. Wolf, Principles of Optics. Electromagnetic Theory of Propagation, Interference and Diffraction of Light (Cambridge, Cambridge, 7th edition, 1999), pp. 116-129.

Yu. A. Kravtsov andYu. I. Orlov, Geometrical Optics of Inhomogeneous Media (Springer, Berlin, 1990).
[CrossRef]

Yu. A. Kravtsov andYu. I. Orlov, Caustics, Catastrophes and Wave Fields (Springer, Berlin, 2nd Edition, 1999).
[CrossRef]

Supplementary Material (3)

» Media 1: MOV (155 KB)     
» Media 2: MOV (1435 KB)     
» Media 3: MOV (1761 KB)     

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Figures (6)

Fig. 1.
Fig. 1.

Intensity profile (grayscale) and rays (red lines) for the waveguide mode in Eq. (12) with m = 8. This illustration is applicable even if its aspect ratio is changed to make the ray angles either more or less extreme.

Fig. 2.
Fig. 2.

(156 KB) Movie illustrating insensitivity in the construction of even a low-order waveguide mode, namely m = 8. (a) The deformation of the scaled phase space curve caused by changing γ by a factor of 4. (The value of γ is indicated by the blue dot.) (b) The corresponding SAFE estimate (black curve), as well as the associated error (red curve) magnified by a factor of 1000. These results are valid regardless of the number of propagating modes supported by the waveguide, i.e regardless of M.

Fig. 3.
Fig. 3.

(1.44 MB) Movie showing (a) the deterioration of the sums of Gaussian beamlets for the m = 8 waveguide mode, and (b) the evolution of the effective phase space areas occupied by a set of these beamlets. The animation covers propagation from z = 0 out to z = 7.5 (where the rays are periodic in z with period 2π.). The sums in (a) include 16 (red), 64 (green), and 256 (blue) beamlets, and are compared with the exact profile (black line). By contrast, SAFE propagates this mode exactly, and 16 rays then give sufficient accuracy for many purposes.

Fig. 4.
Fig. 4.

Amplitude (green curve) and phase (red curve) of the ratio between an estimate made up of a continuous superposition of Gaussian beamlets and the corresponding exact result given in Eq. (12) with m = 8. Notice that the phase discrepancy goes to -π/4, and the slope of the amplitude curve for high z goes approximately to -0.53.

Fig. 5.
Fig. 5.

(1.76 MB) Movie showing the intensity of the field excited when the mode in the waveguide considered in Sec. 3 is incident upon an interface with a homogeneous medium. The axes are scaled to units of X 0. The red rays are incident from the left, the blue rays are reflected by the interface, and the green rays are transmitted. Additional transmitted rays (in orange) have been added in the lower right corner to clarify the ray’s caustic structure. The initial frame has q = 0.71, and the final frame has q = 1.0, which means that the index is matched initially at the edge of the mode, but ends up matched at the center. The interference between the incident and reflected fields reveals interesting features of their relative phase and strength (in keeping with the properties of the familiar Fresnel reflection coefficient).

Fig. 6.
Fig. 6.

Plots of the rms relative errors at the interface itself for the transmitted (green) and reflected (orange) field estimates associated with Fig. 5. Notice that the error climbs rapidly as q approaches a value that represents the onset of internal reflection in the waveguide. This is due to the presence, in this case, of rays normal to the z axis, which represent a problem for SAFE. Since the problematic rays travel in the x direction, the error peaks disappear as we move away from the interface.

Equations (29)

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[ 2 + k 2 n 2 ( x , z ) ] U ( x , z ) = 0 ,
X z ( ξ ; z ) = P ( ξ ; z ) H ( ξ ; z ) , P z ( ξ ; z ) = 1 2 H ( ξ ; z ) n 2 x [ X ( ξ ; z ) , z ] ,
H ( ξ ; z ) = n 2 [ X ( ξ ; z ) , z ] P 2 ( ξ ; z ) > 0 .
L z ( ξ ; z ) = n 2 [ X ( ξ ; z ) , z ] H ( ξ ; z ) , L ξ ( ξ ; z ) = P ( ξ ; z ) X ξ ( ξ ; z ) .
U γ ( x , z ) = w ( ξ , γ ; z ) g ( ξ , γ ; x , z ) d ξ .
g ( ξ , γ ; x , z ) = k 2 π exp ( k γ 2 [ x X ( ξ ; z ) ] 2 + ik { L ( ξ ; z ) + [ x X ( ξ ; z ) ] P ( ξ ; z ) } ) .
z [ H w ( ξ ; z ) ] = w 2 H 5 2 ( P 2 n 2 x i γ n 2 ) ,
γ z ( ξ ; z ) = i H 3 [ γ 2 n 2 + i γ P n 2 x 1 4 ( n 2 x ) 2 + H 2 2 2 n 2 x 2 ] ,
w ( ξ , γ ; z ) = [ γ X ( ξ ; z ) + i P ( ξ ; z ) H ( ξ ; z ) ] 1 2 [ a 0 ( ξ ) + O ( k 1 ) ] ,
U γ ( 0 ) γ = O ( k 1 ) U γ ( 0 ) .
n 2 ( x ) = n 0 2 ( 1 ν 2 x 2 ) ,
U m ( x , z ) = H m ( k ¯ vx ) exp ( k ¯ ν 2 x 2 2 ) exp [ i k ¯ vz ( 1 2 m + 1 k ¯ ) 1 2 ] ,
X ( ξ ; 0 ) = X 0 cos ξ , P ( ξ ; 0 ) = P 0 sin ξ ,
X ( ξ ; z ) = X 0 cos ( ξ n 0 H 0 ν z ) , P ( ξ ; z ) = P 0 sin ( ξ n 0 H 0 ν z ) ,
L ( ξ ; z ) = X 0 P 0 2 [ sin ( ξ n 0 H 0 vz ) cos ( ξ n 0 H 0 vz ) ( ξ n 0 H 0 vz ) ] + H 0 z .
Γ τ ( ξ ; τ ) = i { 1 Γ 2 ( ξ ; τ ) + ρ 2 [ cos ( ξ τ ) + i sin ( ξ τ ) Γ ( ξ ; τ ) ] 2 } ,
Γ ( ξ ; τ ) = Ω ( τ ) cos ( ξ τ ) i sin ( ξ τ ) cos ( ξ τ ) i Ω ( τ ) sin ( ξ τ ) , Ω ( τ ) = 1 + i ρ 2 τ .
w ( ξ ; H 0 τ n 0 ν ) = c i ρ [ cos ( ξ τ ) + i Γ ( ξ ; τ ) sin ( ξ τ ) ]
= c i ρ cos ( ξ τ ) i Ω ( τ ) sin ( ξ τ ) .
a 0 t ( ξ ) = t ( ξ ) a 0 i ( ξ ) = t ( ξ ) c , a 0 r ( ξ ) = r ( ξ ) a 0 i ( ξ ) = r ( ξ ) c .
t ( ξ ) = 2 H 0 H 1 ( ξ ) H 0 + H 1 ( ξ ) , r ( ξ ) = H 0 H 1 ( ξ ) H 0 + H 1 ( ξ ) .
H 1 ( ξ ) = n 1 2 P 2 ( z 1 ; ξ ) = H 0 q 2 ( 1 + ρ 2 ) ρ 2 sin 2 ( ξ n 0 H 0 ν z 1 ) ,
w ( ξ , γ ; z ) = [ γ X ( ξ ; z ) + i P ( ξ ; z ) H ( ξ ; z ) ] 1 2 [ a 0 ( ξ ) + a 1 ( ξ , γ ; z ) ik + O ( k 2 ) ] .
a 1 t ( ξ , γ 0 ; z 1 ) = h a 1 r ( ξ , γ 0 ; z 1 ) = c n 0 ν 4 H 0 2 t 2 h 5 2 ( ih exp [ 2 i ( ξ n 0 H 0 ν z 1 ) ]
+ { t h 3 2 + i ( r 2 ) sin ( ξ n 0 H 0 ν z 1 ) exp [ i ( ξ n 0 H 0 ν z 1 ) ] } h )
ε γ ( z ) = [ U ( x , z ) U γ ( 0 ) ( x , z ) 2 dx U ( x , z ) 2 dx ] 1 2 = 1 k [ H 1 2 ( ξ ) a 1 ( ξ , γ ; z ) 2 d ξ H 1 2 ( ξ ) a 0 ( ξ ) 2 d ξ ] 1 2 + O ( k 2 ) .
T ε γ t ( z 1 ) = R ε γ r ( z 1 ) = ρ 2 4 2 π ( 2 m + 1 ) [ t 2 h 3 ( ih exp [ i ( ξ n 0 H 0 ν z 1 ) ]
+ { t h 3 2 exp [ i ( ξ n 0 H 0 ν z 1 ) ] + i ( r 2 ) sin ( ξ n 0 H 0 ν z 1 ) } h ) | 2 d ξ ] 1 2 ,
T = 1 2 π h 1 t 2 d ξ , R = 1 2 π r 2 d ξ .

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