Abstract

The combined method of ray tracing and diffraction (CMRD) is an efficient and accurate technique for computing the scattered field in focal regions of optical systems. Here we extend the CMRD concept so it can be used to compute fields scattered by objects of simple as well as nonsimple shapes. To that end we replace the scattering object by an equivalent, planar phase object; use ray tracing to determine its location, aperture area, amplitude distribution, and phase distribution; and use standard Kirchhoff diffraction theory to compute the field scattered by the equivalent phase object. To illustrate the practical use of the CMRD we apply it to a two-dimensional problem in which a plane or cylindrical wave is normally incident upon a circular cylinder. For this application we determine the range of validity of the CMRD by comparing its results for the scattered field with those obtained by use of an exact eigenfunction expansion.

© 1998 Optical Society of America

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References

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  1. J. J. Bowman, T. B. A. Senior, P. L. E. Uslenghi, Electromagnetic and Acoustic Scattering by Simple Shapes (North-Holland, Amsterdam, 1969).
  2. J. J. Stamnes, B. Spjelkavik, “New method for computing eigenfunctions (Mathieu functions) for scattering by elliptical cylinders,” Pure Appl. Opt. 4, 251–262 (1995).
    [CrossRef]
  3. W. J. Wiscombe, “Improved Mie scattering algorithm,” Appl. Opt. 19, 1505–1509 (1980).
    [CrossRef] [PubMed]
  4. Y. Takano, K. N. Liou, “Solar radiative transfer in cirrus clouds. I. Single-scattering and optical properties of hexagonal ice crystals,” J. Atmos. Sci. 46, 3–19 (1989).
    [CrossRef]
  5. K. Muinonen, “Scattering of light by crystals: a modified Kirchhoff approximation,” Appl. Opt. 28, 3044–3050 (1989).
    [CrossRef] [PubMed]
  6. P. Yang, K. N. Liou, “Light scattering by hexagonal ice crystals: comparison of finite-difference time domain and geometric optics models,” J. Opt. Soc. Am. A 12, 162–176 (1995).
    [CrossRef]
  7. K. Muinonen, K. Lumme, J. Peltoniemi, W. M. Irvine, “Light scattering by randomly oriented crystals,” Appl. Opt. 28, 3051–3060 (1989).
    [CrossRef] [PubMed]
  8. A. Macke, “Scattering of light by polyhedral ice crystals,” Appl. Opt. 32, 2780–2788 (1993).
    [CrossRef] [PubMed]
  9. M. I. Mishchenko, “Light scattering by size-shape distributions of randomly oriented axially symmetric particles of a size comparable to a wavelength,” Appl. Opt. 32, 4652–4666 (1993).
    [CrossRef] [PubMed]
  10. A. Macke, M. I. Mishchenko, K. Muinonen, B. E. Carlson, “Scattering of light by large nonspherical particles: ray-tracing approximation versus T-matrix method,” Opt. Lett. 20, 1934–1936 (1995).
    [CrossRef] [PubMed]
  11. J. J. Stamnes, Waves in Focal Regions (Hilger, Bristol, UK, 1986), Chap. 3.
  12. J. J. Stamnes, H. Heier, “Scalar and electromagnetic diffraction point-spread functions,” Appl. Opt. (to be published).
  13. B. Chen, J. J. Stamnes, “Diffraction effects on laser computed tomography and time-resolved imaging in random media,” in Photon Propagation in Tissues II, B. Chance, D. A. Benaron, G. J. Mueller, eds., Proc. SPIE2925, 89–105 (1996).
    [CrossRef]
  14. B. Chen, J. J. Stamnes, “Reconstruction algorithm for diffraction tomography of photon density waves in random media,” to be submitted to J. Eur. Opt. Soc. A special issue on physical optics and coherence theory, in honor of Emil Wolf on his 75th birthday.
  15. J. J. Stamnes, B. Spjelkavik, H. M. Pedersen, “Evaluation of diffraction integrals using local phase and amplitude approximations,” Opt. Acta 30, 207–222 (1983).
    [CrossRef]
  16. Ref. 11, Chap. 7.
  17. Ref. 11, Chap. 5.
  18. B. Chen, Diffraction Tomography and Its Applications for Optical Imaging in Random Media (University of Bergen, Bergen, Norway, 1996).
  19. A. Ishimaru, Wave Propagation and Scattering in Random Media (Academic, New York, 1978).

1995 (3)

1993 (2)

1989 (3)

1983 (1)

J. J. Stamnes, B. Spjelkavik, H. M. Pedersen, “Evaluation of diffraction integrals using local phase and amplitude approximations,” Opt. Acta 30, 207–222 (1983).
[CrossRef]

1980 (1)

Bowman, J. J.

J. J. Bowman, T. B. A. Senior, P. L. E. Uslenghi, Electromagnetic and Acoustic Scattering by Simple Shapes (North-Holland, Amsterdam, 1969).

Carlson, B. E.

Chen, B.

B. Chen, J. J. Stamnes, “Diffraction effects on laser computed tomography and time-resolved imaging in random media,” in Photon Propagation in Tissues II, B. Chance, D. A. Benaron, G. J. Mueller, eds., Proc. SPIE2925, 89–105 (1996).
[CrossRef]

B. Chen, J. J. Stamnes, “Reconstruction algorithm for diffraction tomography of photon density waves in random media,” to be submitted to J. Eur. Opt. Soc. A special issue on physical optics and coherence theory, in honor of Emil Wolf on his 75th birthday.

B. Chen, Diffraction Tomography and Its Applications for Optical Imaging in Random Media (University of Bergen, Bergen, Norway, 1996).

Heier, H.

J. J. Stamnes, H. Heier, “Scalar and electromagnetic diffraction point-spread functions,” Appl. Opt. (to be published).

Irvine, W. M.

Ishimaru, A.

A. Ishimaru, Wave Propagation and Scattering in Random Media (Academic, New York, 1978).

Liou, K. N.

P. Yang, K. N. Liou, “Light scattering by hexagonal ice crystals: comparison of finite-difference time domain and geometric optics models,” J. Opt. Soc. Am. A 12, 162–176 (1995).
[CrossRef]

Y. Takano, K. N. Liou, “Solar radiative transfer in cirrus clouds. I. Single-scattering and optical properties of hexagonal ice crystals,” J. Atmos. Sci. 46, 3–19 (1989).
[CrossRef]

Lumme, K.

Macke, A.

Mishchenko, M. I.

Muinonen, K.

Pedersen, H. M.

J. J. Stamnes, B. Spjelkavik, H. M. Pedersen, “Evaluation of diffraction integrals using local phase and amplitude approximations,” Opt. Acta 30, 207–222 (1983).
[CrossRef]

Peltoniemi, J.

Senior, T. B. A.

J. J. Bowman, T. B. A. Senior, P. L. E. Uslenghi, Electromagnetic and Acoustic Scattering by Simple Shapes (North-Holland, Amsterdam, 1969).

Spjelkavik, B.

J. J. Stamnes, B. Spjelkavik, “New method for computing eigenfunctions (Mathieu functions) for scattering by elliptical cylinders,” Pure Appl. Opt. 4, 251–262 (1995).
[CrossRef]

J. J. Stamnes, B. Spjelkavik, H. M. Pedersen, “Evaluation of diffraction integrals using local phase and amplitude approximations,” Opt. Acta 30, 207–222 (1983).
[CrossRef]

Stamnes, J. J.

J. J. Stamnes, B. Spjelkavik, “New method for computing eigenfunctions (Mathieu functions) for scattering by elliptical cylinders,” Pure Appl. Opt. 4, 251–262 (1995).
[CrossRef]

J. J. Stamnes, B. Spjelkavik, H. M. Pedersen, “Evaluation of diffraction integrals using local phase and amplitude approximations,” Opt. Acta 30, 207–222 (1983).
[CrossRef]

B. Chen, J. J. Stamnes, “Reconstruction algorithm for diffraction tomography of photon density waves in random media,” to be submitted to J. Eur. Opt. Soc. A special issue on physical optics and coherence theory, in honor of Emil Wolf on his 75th birthday.

B. Chen, J. J. Stamnes, “Diffraction effects on laser computed tomography and time-resolved imaging in random media,” in Photon Propagation in Tissues II, B. Chance, D. A. Benaron, G. J. Mueller, eds., Proc. SPIE2925, 89–105 (1996).
[CrossRef]

J. J. Stamnes, Waves in Focal Regions (Hilger, Bristol, UK, 1986), Chap. 3.

J. J. Stamnes, H. Heier, “Scalar and electromagnetic diffraction point-spread functions,” Appl. Opt. (to be published).

Takano, Y.

Y. Takano, K. N. Liou, “Solar radiative transfer in cirrus clouds. I. Single-scattering and optical properties of hexagonal ice crystals,” J. Atmos. Sci. 46, 3–19 (1989).
[CrossRef]

Uslenghi, P. L. E.

J. J. Bowman, T. B. A. Senior, P. L. E. Uslenghi, Electromagnetic and Acoustic Scattering by Simple Shapes (North-Holland, Amsterdam, 1969).

Wiscombe, W. J.

Yang, P.

Appl. Opt. (5)

J. Atmos. Sci. (1)

Y. Takano, K. N. Liou, “Solar radiative transfer in cirrus clouds. I. Single-scattering and optical properties of hexagonal ice crystals,” J. Atmos. Sci. 46, 3–19 (1989).
[CrossRef]

J. Opt. Soc. Am. A (1)

Opt. Acta (1)

J. J. Stamnes, B. Spjelkavik, H. M. Pedersen, “Evaluation of diffraction integrals using local phase and amplitude approximations,” Opt. Acta 30, 207–222 (1983).
[CrossRef]

Opt. Lett. (1)

Pure Appl. Opt. (1)

J. J. Stamnes, B. Spjelkavik, “New method for computing eigenfunctions (Mathieu functions) for scattering by elliptical cylinders,” Pure Appl. Opt. 4, 251–262 (1995).
[CrossRef]

Other (9)

J. J. Bowman, T. B. A. Senior, P. L. E. Uslenghi, Electromagnetic and Acoustic Scattering by Simple Shapes (North-Holland, Amsterdam, 1969).

Ref. 11, Chap. 7.

Ref. 11, Chap. 5.

B. Chen, Diffraction Tomography and Its Applications for Optical Imaging in Random Media (University of Bergen, Bergen, Norway, 1996).

A. Ishimaru, Wave Propagation and Scattering in Random Media (Academic, New York, 1978).

J. J. Stamnes, Waves in Focal Regions (Hilger, Bristol, UK, 1986), Chap. 3.

J. J. Stamnes, H. Heier, “Scalar and electromagnetic diffraction point-spread functions,” Appl. Opt. (to be published).

B. Chen, J. J. Stamnes, “Diffraction effects on laser computed tomography and time-resolved imaging in random media,” in Photon Propagation in Tissues II, B. Chance, D. A. Benaron, G. J. Mueller, eds., Proc. SPIE2925, 89–105 (1996).
[CrossRef]

B. Chen, J. J. Stamnes, “Reconstruction algorithm for diffraction tomography of photon density waves in random media,” to be submitted to J. Eur. Opt. Soc. A special issue on physical optics and coherence theory, in honor of Emil Wolf on his 75th birthday.

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Figures (10)

Fig. 1
Fig. 1

Scattering geometry. The incident wave u i , generated by the source S, is scattered by the scatterer SC, and the total field is observed at P. The refractive indices of the scatterer and the surrounding background medium are n 2 = n 2′ + in 2′ and n 1, respectively.

Fig. 2
Fig. 2

Equivalent diffraction geometry. The incident wave u i , generated by the source S, is scattered by an EPO, which occupies an area A in the plane z = b.

Fig. 3
Fig. 3

Geometry of the CMRD for the case in which a plane wave is normally incident upon a circular cylinder and n 2> n 1.

Fig. 4
Fig. 4

Geometry of the second EPO of the CMRD for the case in which a plane wave is normally incident upon a circular cylinder and n 2 > n 1.

Fig. 5
Fig. 5

Geometry of the CMRD for the case in which a plane wave is normally incident upon a circular cylinder and n 2< n 1.

Fig. 6
Fig. 6

Geometry of the third EPO of the CMRD for the case in which a plane wave is normally incident upon a circular cylinder and n 2 < n 1.

Fig. 7
Fig. 7

Geometry of the CMRD for the case in which a cylindrical wave is normally incident upon a circular cylinder and n 2 > n 1.

Fig. 8
Fig. 8

Geometry of the CMRD for the case in which a cylindrical wave is normally incident upon a circular cylinder and n 2 < n 1.

Fig. 9
Fig. 9

Comparison of CMRD results and exact results for scattered fields from a circular cylinder of radius a = 5λ. The incident field was a plane wave of wavelength λ = 0.6328 μm, which was normally incident upon the cylinder axis. The observation distance was 500λ. The refractive index of the background medium was n 1 = 1.00, and that of the cylinder was n 2. There were 500 sampling points, and the sampling interval was 1.5λ. (a) n 2 = 1.05. (b) n 2 = 1.10. (c) n 2 = 1.33.

Fig. 10
Fig. 10

Comparison of CMRD results and exact results along the z axis (cf. Fig. 3) for the total field from a cylinder of radius a = 5λ, which was incident in the z direction normally to the cylinder axis. The incident field was a plane wave of wavelength λ = 0.6328 μm. The refractive index of the background medium was n 1 = 1.00, and that of the cylinder was n 2 = 1.33.

Equations (111)

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u f x ,   y ,   z = b = u i x ,   y ,   z = b x ,   y A u i x ,   y ,   z = b a x ,   y exp i ϕ x ,   y x ,   y A ,
u x ,   y ,   z =   - +   u f x ,   y ,   z = b × h x - x ,   y - y ,   z - b d x d y ,
h x - x ,   y - y ,   z - b = 2   z   G x - x ,   y - y ,   z - b .
G x - x ,   y - y ,   z - b = - 1 4 π exp ikn 1 R R ,
R = x - x 2 + y - y 2 + z - b 2 1 / 2 .
u x ,   y ,   z =   A c   u i x ,   y ,   z = b × h x - x ,   y - y ,   z - b d x d y +   A   u i x ,   y ,   z = b a x ,   y exp × i ϕ x ,   y h x - x ,   y - y ,   z - b × d x d y ,
u x ,   y ,   z = u i x ,   y ,   z + u sc x ,   y ,   z ,
u sc x ,   y ,   z =   A   u i x ,   y ,   z = b × a x ,   y exp i ϕ x ,   y - 1 h × x - x ,   y - y ,   z - b d x d y .
h x - x ,   y - y ,   z - b = 1 i λ 1 z R 1 + i k 1 R exp ik 1 R R ,
u i x ,   y ,   z = b = exp ik 1 s x i x + s y i y + s z i b ,
u sc x ,   y ,   z = b = 1 i λ 1     A a x ,   y exp i ϕ x ,   y - 1 × z R 2 1 + i k 1 R exp ik 1 R + s x i x + s y i y + s z i b d x d y .
u i x ,   y ,   z = b = exp ik 1 R s R s ,
R s = x s - x 2 + y s - y 2 + z s - b 2 1 / 2 ,
u sc x ,   y ,   z = b = 1 i λ 1     A a x ,   y exp i ϕ x ,   y - 1 × z R 2 1 + i k 1 R 1 R s × exp ik 1 R + R s d x d y .
I =   A   g x ,   y exp if x ,   y d x d y ,
g x ,   y = 1 i λ 1 a x ,   y exp i ϕ x ,   y - 1 z R 2 1 + i k 1 R × 1 plane   incident   wave 1 / R s spherical   incident   wave ,
f x ,   y = k 1 R + s x i x + s y i y + s z i b plane   incident   wave k 1 R + R s spherical   incident   wave .
u sc x ,   y ,   z = j = 1 2     A   g j x ,   y exp if j x ,   y d x d y ,
g 1 x ,   y = - 1 i λ 1 z R 2 1 + i k 1 R × 1 plane   incident   wave 1 / R s spherical   incident   wave ,
g 2 x ,   y = - g 1 x ,   y a x ,   y ,
f 1 x ,   y = f x ,   y ,
f 2 x ,   y = f 1 x ,   y + ϕ x ,   y .
u x ,   z = A   u f x ,   z = b h x - x ,   z - b d x ,
u f x ,   z = b = u i x ,   z = b   x A u i x ,   z = b a x exp i ϕ x   x A .
h x - x ,   z - b = 2   z   G x - x ,   z - b ,
G x - x ,   z - b = - i 4   H 0 1 k 1 R ,
R = x - x 2 + z - b 2 1 / 2 .
u x ,   z = u i x ,   z + u sc x ,   z ,
u sc x ,   z = A   u i x ,   z = b a x exp i ϕ x - 1 × h x - x ,   z - b d x .
h x - x ,   z - b = 1 i λ 1 z R 3 / 2 exp ik 1 R .
u i x ,   z = b = exp ik 1 s x i x + s z i b ,
u sc x ,   z = 1 i λ 1 A a x exp i ϕ x - 1 × z R 3 / 2 exp ik 1 R + s x i x + s z i b d x .
u i x ,   z = b = exp ik 1 R s R s ,
R s = x s - x 2 + z s - b 2 1 / 2 ,
u sc x ,   z = 1 i λ 1 A a x exp i ϕ x - 1 × z R s 3 / 2 1 R s 1 / 2 exp ik 1 R + R s d x .
b = a   tan π 2 - β 0 = a n 2 n 1 2 - 1 1 / 2 ,
β 0 = sin - 1 n 1 / n 2 .
γ = 2 α - β ,
β = sin - 1 n 1 sin   α n 2 .
α max - sin - 1 n 1 sin   α max n 2 = π / 4 .
x α = a sin α - γ + 2   cos   β   cos α - β tan   γ - cos   α   tan   γ - b   tan   γ .
f x = k n 1 SA + n 2 AB - n 1 BB ,
SA = a 1 - cos   α ,
AB = 2 a   cos   β ,
BB = AB   cos α - β - a   cos   α - b cos   γ .
ϕ x = f x - kn 1 a + b .
a x = e - σ s a 0 x T x ,
d x = a   cos   α   d α .
d x = d x α d α d α ,
a 0 x = | d x d x | = | a   cos   α d x α / d α | ,
T x = t A x t B x ,
t A = 2 n 1 cos   α n 1 cos   α + n 2 cos   β ,     t B = 2 n 2 cos   β n 1 cos   α + n 2 cos   β ,
T x = t A t B = sin   2 α   sin   2 β sin 2 α + β .
b 2 = - a   cos π / 4 ,
w 2 = a ,
x α = a sin   α + 1 2 tan 2 α - cos   α   tan 2 α .
ϕ 2 x = kn 1 AB - BB ,
AB = b 2 - a   cos   α ,
BB = b 2 - a   cos   α / cos 2 α .
a 2 x = a 2 0 x | R x | ,
a 2 0 x = | d x d x | = | a   cos   α d x α / d α | ,
R x = n 1 cos   α - n 2 cos   β n 1 cos   α + n 2 cos   β .
α 0 = sin - 1 n 2 / n 1 .
b = - a   cos   α 0 .
w = 1 / 2 C 0 C 0 = a   sin   α 0 .
ϕ x = f x - kn 1 a - b ,
f x = k n 1 SA + n 2 AB - n 1 BB ,
x = a   sin   α + AB   sin β - α - BB   sin   γ ,
AB = 2 a   cos   β ,
BB = AB   cos β - α - a   cos   α + b cos   γ .
b 3 = - a cos   α 0 ,
w 3 = a .
x α = a sin   α 1 - cos   α cos   α 0 .
ϕ 3 x = kn 1 AB - BB ,
AB = a 1 - cos   α ,
BB = b 3 - a   cos   α / sin   α .
a 3 x = a 3 0 x | R t x | .
a 3 x = a 3 0 x | R t x | = a 3 0 x = | d x d x | = | a   cos   α d x α / d α | .
b = a 2   cos   β 0 sin θ 0 + β 0 - cos   β 0 × cos   γ 0 / sin   β 0 - sin   θ 0 ,
θ 0 = sin - 1 a / | z s | .
β 0 = sin - 1 n 1 / n 2 ,
γ 0 = π - θ 0 - 2 β 0 ,
w = 1 / 2 C 0 C 0 = | z s | + b tan   θ 0 .
ϕ x = f x - kn 1 SB ,
SB = x 2 + | z s | + b 2 1 / 2 ,
f x = k n 1 SA + n 2 AB - n 1 BB ,
x = a sin α - γ + 2   cos   β   cos α - θ - β tan   γ - cos α - θ tan   γ - b   tan   γ ,
SA = a   sin α - θ sin   θ ,
AB = 2 a   cos   β ,
BB = 2 a   cos   β   cos α - θ - β - a   cos α - θ - b cos   γ .
α 0 = sin - 1 n 2 / n 1 .
θ 0 = sin - 1 a   sin   α 0 | z s | ,
b = - a   cos α 0 - θ 0 ,
w = ½ C 0 C 0 = | z s | + b tan   θ 0 .
ϕ x = f x - kn 1 SB ,
SB = x 2 + | z s | + b 2 1 / 2 ,
f x = k n 1 SA + n 2 AB - n 1 BB ,
x = a   sin α - θ + AB   sin β + θ - α - BB   sin   γ ,
SA = a   sin α - θ sin   θ ,
AB = 2 a   cos   β ,
BB = 2 a   cos   β   cos α - θ - β - a   cos α - θ - b cos   γ .
u sc , M = n = 0 A n M cos n ϕ + A n M sin n ϕ H n 1 k 1 r ,
r = x 2 + z 2 1 / 2 ,     ϕ = tan - 1 x / z ,
A n M = q n M A on / D n ,     A n M = q n M A on / D n ,
A on = k 2 k 1   J n k 1 a J n k 2 a - J n k 1 a J n k 2 a ,
D n = J n k 2 a H n 1 k 1 a - k 2 k 1   J n k 2 a × H n 1 k 1 a ,
a n = n H n 1 k 1 r 0 cos n ϕ 0 ,     a n = n H n 1 k 1 r 0 sin n ϕ 0 ,
b n = n - i n cos n ϕ 0 ,     b n = n - i n sin n ϕ 0 ,
u = u i , M + u sc , M ,
u GO x = 0 ,   z = exp ik n 1 z + 2 a n 2 - n 1 t 12 t 21 ,
A 2 R 12 R 21 A 1 n 1 - n 2 n 1 + n 2 2 A 1 ,

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