Abstract

The exact solution for scattering from cylindrical objects is compared with the following weak-scattering models: the first Born approximation, the Rytov approximation, the straight-ray model of geometric optics, and ray tracing. Computer calculations of the various approximate and the exact virtual fields at the center of a transparent cylinder are compared in order to find the practical limitations of the various approximations with respect to cylinder radii and indices of refraction. It is found that the Rytov approximation introduces a phase error that increases linearly with the cylinder radius. The straight-ray model yields better results than the first Rytov approximation for all cylinder radii and real indices of refraction, even at the edges. The first Born approximation is best for cylinders with radii less than a wavelength.

© 1995 Optical Society of America

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References

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  1. M. A. Fiddy, “Inversion of optical scattered field data,” J. Phys. D 19, 301–317 (1986).
    [Crossref]
  2. V. I. Tatarski, Wave Propagation in a Turbulent Medium (Dover, New York, 1961).
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    [Crossref] [PubMed]
  4. A. J. Devaney, “Reconstructive tomography with diffracting wavefields,” Inverse Probl. 2, 161–183 (1986).
    [Crossref]
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    [Crossref]
  7. H. C. van de Hulst, Light Scattering by Small Particles (Wiley, New York, 1957).
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    [Crossref]
  10. T. C. Wedberg, J. J. Stamnes, “Analytical and numerical examination of the qualitative imaging properties of optical diffraction tomography,” J. Mod. Opt. (to be published).
  11. A. J. Devaney, “A computer simulation study of diffraction tomography,” IEEE Trans. BioMed. Eng. 30, 377–386 (1983).
    [Crossref] [PubMed]
  12. A. J. Devaney, “Structure determination from intensity measurements in scattering experiments,” Phys. Rev. Lett. 62, 2385–2388 (1989).
    [Crossref] [PubMed]
  13. T. C. Wedberg, W. Wedberg, “Tomographic reconstruction of the cross-sectional refractive index distribution in semi-transparent birefringent fibers,” J. Microsc. 117, 53–67 (1995).
    [Crossref]
  14. T. C. Wedberg, J. J. Stamnes, W. Singer, “Comparison of the filtered backpropagation and the backprojection algorithms for quantitative tomography,” Appl. Opt. 34, 6575–6581 (1995).
    [Crossref] [PubMed]
  15. L.-J. Gelius, J. J. Stamnes, “Computer experiments in acoustical diffraction tomography,” in Inverse Problems in Optics, E. R. Pike, ed., Proc. Soc. Photo-Opt. Instrum. Eng. 808, 209–217 (1987).
  16. L.-J. Gelius, J. J. Stamnes, “Diffraction tomography: potentials and problems,” in Scattering in Volumes and Surfaces, M. Nieto-Vesperinas, J. C. Daintry, eds. (Elsevier, New York, 1990), pp. 91–109.
  17. J. J. Stamnes, B. Spjelkavik, “New method for computing the eigenfunctions (Mathieu functions) for scattering by elliptical cylinders,” Pure Appl. Opt. 4, 251–262 (1995).
    [Crossref]
  18. J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1968).
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    [Crossref]
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    [Crossref]
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    [Crossref]
  28. N. Sponheim, I. Johansen, A. Devaney, “Initial testing of a clinical ultrasound mammograph,” in Acoustical Imaging, M. Lee, G. Wade, eds. (Plenum, New York, 1990), Vol. 18, pp. 401–411.
    [Crossref]
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    [Crossref]
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  32. H. Nareid, H. M. Pedersen, “Modeling of the Lippman color process,” J. Opt. Soc. Am. A 8, 257–265 (1991).
    [Crossref]

1995 (3)

T. C. Wedberg, W. Wedberg, “Tomographic reconstruction of the cross-sectional refractive index distribution in semi-transparent birefringent fibers,” J. Microsc. 117, 53–67 (1995).
[Crossref]

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

T. C. Wedberg, J. J. Stamnes, W. Singer, “Comparison of the filtered backpropagation and the backprojection algorithms for quantitative tomography,” Appl. Opt. 34, 6575–6581 (1995).
[Crossref] [PubMed]

1992 (1)

M. H. Maleki, A. J. Devaney, A. Schatzberg, “Tomographic reconstruction from optical scattered intensities,” J. Opt. Soc. Am. 9, 1356–1363 (1992).
[Crossref]

1991 (2)

H. Nareid, H. M. Pedersen, “Modeling of the Lippman color process,” J. Opt. Soc. Am. A 8, 257–265 (1991).
[Crossref]

N. Sponheim, L. Gelius, I. Johansen, J. Stamnes, “Quantitative results in ultrasonic tomography using line sources and curved detector arrays,” IEEE Ultrason. Ferroelectrics Frequency Control 38, 370–379 (1991).
[Crossref]

1989 (1)

A. J. Devaney, “Structure determination from intensity measurements in scattering experiments,” Phys. Rev. Lett. 62, 2385–2388 (1989).
[Crossref] [PubMed]

1987 (1)

1986 (2)

A. J. Devaney, “Reconstructive tomography with diffracting wavefields,” Inverse Probl. 2, 161–183 (1986).
[Crossref]

M. A. Fiddy, “Inversion of optical scattered field data,” J. Phys. D 19, 301–317 (1986).
[Crossref]

1985 (1)

1983 (2)

A. J. Devaney, “A computer simulation study of diffraction tomography,” IEEE Trans. BioMed. Eng. 30, 377–386 (1983).
[Crossref] [PubMed]

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]

1982 (1)

A. J. Devaney, “A filtered backpropagation algorithm for diffraction tomography,” Ultrasonic Imaging 4, 336–350 (1982).
[Crossref] [PubMed]

1981 (1)

1970 (1)

M. I. Sancer, A. D. Varvatsis, “A comparison of the Born and Rytov methods,” Proc. IEEE 58, 140–141 (1970).
[Crossref]

1969 (2)

1968 (2)

Abramowitz, M.

M. Abramowitz, I. A. Stegun, Handbook of Mathematical Functions (Dover, New York, 1964).

Born, M.

M. Born, E. Wolf, Principles of Optics (Pergamon, New York, 1981).

Devaney, A.

N. Sponheim, I. Johansen, A. Devaney, “Initial testing of a clinical ultrasound mammograph,” in Acoustical Imaging, M. Lee, G. Wade, eds. (Plenum, New York, 1990), Vol. 18, pp. 401–411.
[Crossref]

Devaney, A. J.

M. H. Maleki, A. J. Devaney, A. Schatzberg, “Tomographic reconstruction from optical scattered intensities,” J. Opt. Soc. Am. 9, 1356–1363 (1992).
[Crossref]

A. J. Devaney, “Structure determination from intensity measurements in scattering experiments,” Phys. Rev. Lett. 62, 2385–2388 (1989).
[Crossref] [PubMed]

A. J. Devaney, “Reconstructive tomography with diffracting wavefields,” Inverse Probl. 2, 161–183 (1986).
[Crossref]

A. J. Devaney, “A computer simulation study of diffraction tomography,” IEEE Trans. BioMed. Eng. 30, 377–386 (1983).
[Crossref] [PubMed]

A. J. Devaney, “A filtered backpropagation algorithm for diffraction tomography,” Ultrasonic Imaging 4, 336–350 (1982).
[Crossref] [PubMed]

Feshbach, H.

P. M. Morse, H. Feshbach, Methods of Theoretical Physics (McGraw-Hill, New York, 1953).

Fiddy, M. A.

M. A. Fiddy, “Inversion of optical scattered field data,” J. Phys. D 19, 301–317 (1986).
[Crossref]

Fried, D. L.

Gelius, L.

N. Sponheim, L. Gelius, I. Johansen, J. Stamnes, “Quantitative results in ultrasonic tomography using line sources and curved detector arrays,” IEEE Ultrason. Ferroelectrics Frequency Control 38, 370–379 (1991).
[Crossref]

Gelius, L.-J.

L.-J. Gelius, J. J. Stamnes, “Computer experiments in acoustical diffraction tomography,” in Inverse Problems in Optics, E. R. Pike, ed., Proc. Soc. Photo-Opt. Instrum. Eng. 808, 209–217 (1987).

L.-J. Gelius, J. J. Stamnes, “Diffraction tomography: potentials and problems,” in Scattering in Volumes and Surfaces, M. Nieto-Vesperinas, J. C. Daintry, eds. (Elsevier, New York, 1990), pp. 91–109.

Goodman, J. W.

J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1968).

Gradstein, I. S.

I. S. Gradstein, I. M. Ryshik, Tables of Series, Products and Integrals (Springer-Verlag, Berlin, 1981).

Johansen, I.

N. Sponheim, L. Gelius, I. Johansen, J. Stamnes, “Quantitative results in ultrasonic tomography using line sources and curved detector arrays,” IEEE Ultrason. Ferroelectrics Frequency Control 38, 370–379 (1991).
[Crossref]

N. Sponheim, I. Johansen, A. Devaney, “Initial testing of a clinical ultrasound mammograph,” in Acoustical Imaging, M. Lee, G. Wade, eds. (Plenum, New York, 1990), Vol. 18, pp. 401–411.
[Crossref]

Keller, J. B.

Lira, I. H.

Maleki, M. H.

M. H. Maleki, A. J. Devaney, A. Schatzberg, “Tomographic reconstruction from optical scattered intensities,” J. Opt. Soc. Am. 9, 1356–1363 (1992).
[Crossref]

Morse, P. M.

P. M. Morse, H. Feshbach, Methods of Theoretical Physics (McGraw-Hill, New York, 1953).

Nareid, H.

Oristaglio, M. L.

Pedersen, H. M.

H. Nareid, H. M. Pedersen, “Modeling of the Lippman color process,” J. Opt. Soc. Am. A 8, 257–265 (1991).
[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]

Ryshik, I. M.

I. S. Gradstein, I. M. Ryshik, Tables of Series, Products and Integrals (Springer-Verlag, Berlin, 1981).

Sancer, M. I.

M. I. Sancer, A. D. Varvatsis, “A comparison of the Born and Rytov methods,” Proc. IEEE 58, 140–141 (1970).
[Crossref]

Schatzberg, A.

M. H. Maleki, A. J. Devaney, A. Schatzberg, “Tomographic reconstruction from optical scattered intensities,” J. Opt. Soc. Am. 9, 1356–1363 (1992).
[Crossref]

Sherman, G. C.

Sherwell, J. R.

Singer, W.

Spjelkavik, B.

J. J. Stamnes, B. Spjelkavik, “New method for computing the 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]

Sponheim, N.

N. Sponheim, L. Gelius, I. Johansen, J. Stamnes, “Quantitative results in ultrasonic tomography using line sources and curved detector arrays,” IEEE Ultrason. Ferroelectrics Frequency Control 38, 370–379 (1991).
[Crossref]

N. Sponheim, I. Johansen, A. Devaney, “Initial testing of a clinical ultrasound mammograph,” in Acoustical Imaging, M. Lee, G. Wade, eds. (Plenum, New York, 1990), Vol. 18, pp. 401–411.
[Crossref]

Stamnes, J.

N. Sponheim, L. Gelius, I. Johansen, J. Stamnes, “Quantitative results in ultrasonic tomography using line sources and curved detector arrays,” IEEE Ultrason. Ferroelectrics Frequency Control 38, 370–379 (1991).
[Crossref]

J. Stamnes, “Focusing of two-dimensional waves,” J. Opt. Soc. Am. 71, 15–31 (1981).
[Crossref]

Stamnes, J. J.

T. C. Wedberg, J. J. Stamnes, W. Singer, “Comparison of the filtered backpropagation and the backprojection algorithms for quantitative tomography,” Appl. Opt. 34, 6575–6581 (1995).
[Crossref] [PubMed]

J. J. Stamnes, B. Spjelkavik, “New method for computing the 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]

J. J. Stamnes, Waves in Focal Regions (Hilger, London, 1986), Sec. 7.2.

L.-J. Gelius, J. J. Stamnes, “Diffraction tomography: potentials and problems,” in Scattering in Volumes and Surfaces, M. Nieto-Vesperinas, J. C. Daintry, eds. (Elsevier, New York, 1990), pp. 91–109.

L.-J. Gelius, J. J. Stamnes, “Computer experiments in acoustical diffraction tomography,” in Inverse Problems in Optics, E. R. Pike, ed., Proc. Soc. Photo-Opt. Instrum. Eng. 808, 209–217 (1987).

T. C. Wedberg, J. J. Stamnes, “Analytical and numerical examination of the qualitative imaging properties of optical diffraction tomography,” J. Mod. Opt. (to be published).

Stegun, I. A.

M. Abramowitz, I. A. Stegun, Handbook of Mathematical Functions (Dover, New York, 1964).

Tatarski, V. I.

V. I. Tatarski, Wave Propagation in a Turbulent Medium (Dover, New York, 1961).

van de Hulst, H. C.

H. C. van de Hulst, Light Scattering by Small Particles (Wiley, New York, 1957).

Varvatsis, A. D.

M. I. Sancer, A. D. Varvatsis, “A comparison of the Born and Rytov methods,” Proc. IEEE 58, 140–141 (1970).
[Crossref]

Vest, C. M.

Wedberg, T. C.

T. C. Wedberg, J. J. Stamnes, W. Singer, “Comparison of the filtered backpropagation and the backprojection algorithms for quantitative tomography,” Appl. Opt. 34, 6575–6581 (1995).
[Crossref] [PubMed]

T. C. Wedberg, W. Wedberg, “Tomographic reconstruction of the cross-sectional refractive index distribution in semi-transparent birefringent fibers,” J. Microsc. 117, 53–67 (1995).
[Crossref]

T. C. Wedberg, J. J. Stamnes, “Analytical and numerical examination of the qualitative imaging properties of optical diffraction tomography,” J. Mod. Opt. (to be published).

Wedberg, W.

T. C. Wedberg, W. Wedberg, “Tomographic reconstruction of the cross-sectional refractive index distribution in semi-transparent birefringent fibers,” J. Microsc. 117, 53–67 (1995).
[Crossref]

Wolf, E.

Appl. Opt. (2)

IEEE Trans. BioMed. Eng. (1)

A. J. Devaney, “A computer simulation study of diffraction tomography,” IEEE Trans. BioMed. Eng. 30, 377–386 (1983).
[Crossref] [PubMed]

IEEE Ultrason. Ferroelectrics Frequency Control (1)

N. Sponheim, L. Gelius, I. Johansen, J. Stamnes, “Quantitative results in ultrasonic tomography using line sources and curved detector arrays,” IEEE Ultrason. Ferroelectrics Frequency Control 38, 370–379 (1991).
[Crossref]

Inverse Probl. (1)

A. J. Devaney, “Reconstructive tomography with diffracting wavefields,” Inverse Probl. 2, 161–183 (1986).
[Crossref]

J. Microsc. (1)

T. C. Wedberg, W. Wedberg, “Tomographic reconstruction of the cross-sectional refractive index distribution in semi-transparent birefringent fibers,” J. Microsc. 117, 53–67 (1995).
[Crossref]

J. Opt. Soc. Am. (6)

J. Opt. Soc. Am. A (2)

J. Phys. D (1)

M. A. Fiddy, “Inversion of optical scattered field data,” J. Phys. D 19, 301–317 (1986).
[Crossref]

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]

Phys. Rev. Lett. (1)

A. J. Devaney, “Structure determination from intensity measurements in scattering experiments,” Phys. Rev. Lett. 62, 2385–2388 (1989).
[Crossref] [PubMed]

Proc. IEEE (1)

M. I. Sancer, A. D. Varvatsis, “A comparison of the Born and Rytov methods,” Proc. IEEE 58, 140–141 (1970).
[Crossref]

Pure Appl. Opt. (1)

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

Ultrasonic Imaging (1)

A. J. Devaney, “A filtered backpropagation algorithm for diffraction tomography,” Ultrasonic Imaging 4, 336–350 (1982).
[Crossref] [PubMed]

Other (12)

J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1968).

L.-J. Gelius, J. J. Stamnes, “Computer experiments in acoustical diffraction tomography,” in Inverse Problems in Optics, E. R. Pike, ed., Proc. Soc. Photo-Opt. Instrum. Eng. 808, 209–217 (1987).

L.-J. Gelius, J. J. Stamnes, “Diffraction tomography: potentials and problems,” in Scattering in Volumes and Surfaces, M. Nieto-Vesperinas, J. C. Daintry, eds. (Elsevier, New York, 1990), pp. 91–109.

V. I. Tatarski, Wave Propagation in a Turbulent Medium (Dover, New York, 1961).

T. C. Wedberg, J. J. Stamnes, “Analytical and numerical examination of the qualitative imaging properties of optical diffraction tomography,” J. Mod. Opt. (to be published).

H. C. van de Hulst, Light Scattering by Small Particles (Wiley, New York, 1957).

P. M. Morse, H. Feshbach, Methods of Theoretical Physics (McGraw-Hill, New York, 1953).

N. Sponheim, I. Johansen, A. Devaney, “Initial testing of a clinical ultrasound mammograph,” in Acoustical Imaging, M. Lee, G. Wade, eds. (Plenum, New York, 1990), Vol. 18, pp. 401–411.
[Crossref]

J. J. Stamnes, Waves in Focal Regions (Hilger, London, 1986), Sec. 7.2.

I. S. Gradstein, I. M. Ryshik, Tables of Series, Products and Integrals (Springer-Verlag, Berlin, 1981).

M. Born, E. Wolf, Principles of Optics (Pergamon, New York, 1981).

M. Abramowitz, I. A. Stegun, Handbook of Mathematical Functions (Dover, New York, 1964).

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Figures (11)

Fig. 1
Fig. 1

Illustration of the scattering geometry.

Fig. 2
Fig. 2

Thin-object model: (a) weakly scattering cylinder, (b) equivalent thin object with transmission t(x) = exp[ik(n − 1)d(x)]. The same geometric ray is plotted in (a) and (b).

Fig. 3
Fig. 3

Refraction through a cylinder. The ray appears to come from the virtual point C in the plane z = 0.

Fig. 4
Fig. 4

Phase (a) and amplitude deviation (b) of the approximate and the exact virtual fields for a cylinder with radius 500λ and index of refraction n = 1.001 at the plane z = 0: dashed–dotted curve, first Born approximation; thick line, overlap of the Rytov, straight-ray, ray-tracing, and exact virtual fields. The aperture’s half-angle θ0 is 10°.

Fig. 5
Fig. 5

Phase difference (a) and relative amplitude (b) of the approximate fields compared with the exact fields at the center of cylinders with λ ≤ a ≤ 72λ and refractive index n = 1.001: dashed–dotted curve, first Born approximation; dashed curve, Rytov approximation; solid curve, geometric-optics models. The aperture’s half angle θ0 is 45°.

Fig. 6
Fig. 6

Phase difference (a) and relative amplitude (b) as in Fig. 5 for the Rytov approximation and the geometric-optics models but for large cylinders with radii 50λ ≤ a ≤ 500λ, n = 1.001, and θ0 = 10°.

Fig. 7
Fig. 7

Phase difference (a) and relative amplitude (b) of the first Born and the Rytov fields compared with the geometric-optics fields at the center of cylinders with λ ≤ a ≤ 72λ and refractive index n = 1.001. The large-radius test functions are indicated: open circles, first Born approximation; pluses, Rytov approximation; solid lines and curves, test functions. The aperture’s half-angle θ0 is 45°.

Fig. 8
Fig. 8

Phase difference (a) and relative amplitude (b) as in Fig. 7 but for large cylinders with radii 50λ ≤ a ≤ 500λ, n = 1.001, and θ0 = 10°.

Fig. 9
Fig. 9

Phase difference (a) and relative-amplitude deviation (b) of the approximate compared with the exact virtual total fields for a cylinder with radius 5λ and refractive index n = 1.001 at the plane z = 0: dashed–dotted curve, first Born approximation; dashed curve, Rytov approximation; curve, straight-ray model; solid curve, ray tracing. The aperture’s half-angle θ0 is 45°.

Fig. 10
Fig. 10

Phase difference (a) and relative-amplitude deviation (b) as in Fig. 9 but with a = 5λ, n = 1.001, and θ0 = 10°.

Fig. 11
Fig. 11

Phase difference (a) and relative-amplitude deviation (b) as in Fig. 9 but with a = 250λ, n = 1.001, and θ0 = 10°.

Equations (43)

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

U i = exp ( i k z ) = m = - J M ( k r ) exp [ i m ( θ + π 2 ) ] ,
U = U i + U s ,             r a ,
U s = m = - A m H m ( 1 ) ( k r ) i m exp ( i m θ ) ,
A m = - ( 1 / 2 ) ( 1 + C m ) ,
C m = exp { - 2 i arg [ n J m ( k n a ) H m ( 1 ) ( k a ) - J m ( k n a ) H m ( 1 ) ( k a ) ] } .
U s ( x , z ) = 1 π i - ( π / 2 ) - i + ( π / 2 ) exp [ i k ( x sin θ + z cos θ ) ] T ( θ ) d θ ,
T ( θ ) = m = - A m exp ( i m θ )
U s ( x , z ) = 1 π - π / 2 π / 2 exp [ i k ( x sin θ + z cos θ ) ] T ( θ ) d θ .
U s ( x , 0 ) = 1 π - π / 2 π / 2 exp ( i k x sin θ ) T ( θ ) d θ .
{ 2 + k 2 [ 1 - O ( r ) ] } U ( r ) = 0 ,
2 = 2 x 2 + 2 z 2 ,
O ( r ) = n ( r ) 2 - 1 = { n 2 - 1 , r a , 0 , r > a ,
U ( r ) = U i ( r ) + k 2 - π π 0 O ( r ) G ( r - r ) U ( r ) r d r d ϕ ,
G ( r - r ) = - i 4 H 0 ( 1 ) ( k r - r ) .
U B ( r ) = U i ( r ) - i k 2 4 - π π 0 O ( r ) H 0 ( 1 ) ( r - r ) U i ( r ) r d r d ϕ .
H 0 ( 1 ) ( k r - r ) = m = - H m ( 1 ) ( k r ) J m ( k r ) cos [ m ( ϕ - θ ) ] ,
U B s ( r ) = ( - i k 2 4 ) m = - n = - i n H m ( 1 ) ( k r ) × 0 a O ( r ) J n ( k r ) J m ( k r ) r d r × - π π cos [ m ( ϕ - θ ) ] cos ( n θ ) d θ .
0 a J n ( k r ) 2 r d r = ( a ) 2 2 [ J n 2 ( k a ) - J n - 1 ( k a ) J n + 1 ( k a ) ] .
U B s = m = 0 ɛ m i m B m H m ( 1 ) ( k r ) cos ( m ϕ ) ,
B m = i π 4 ( n 2 - 1 ) ( k a ) 2 [ J m 2 ( k a ) - J m - 1 ( k a ) J m + 1 ( k a ) ] .
T B ( θ ) = m = - B m exp ( i m θ ) ,
U B s ( x , 0 ) = 1 π - π / 2 π / 2 exp [ i k x sin ( θ ) ] T B ( θ ) d θ ,
U ( r ) = exp [ i Ψ ( r ) ] = exp { i [ Ψ i ( r ) + Ψ s ( r ) ] } = U i ( r ) exp [ i Ψ s ( r ) ] ,
Ψ s ( r ) = 1 U i ( r ) - π π 0 k 2 O ( r ) G ( r - r ) U i ( r ) × r d r d ϕ = U B s ( r ) U i ( r ) ,
U ( x , 0 ) = exp [ i k ( n - 1 ) d ( x ) ] ,
d ( x ) = { 2 ( a 2 - x 2 ) 1 / 2 , x a 0 , x > a .
U ( x , 0 ) = { exp [ i k a ( 2 n cos { arcsin [ α ( x ) ] n } - cos { 2 arcsin [ α ( x ) ] n - α ( x ) } cos { 2 arcsin [ α ( x ) ] n - 2 α ( x ) } - cos [ α ( x ) ] ) ] , x a 1 , x > a
x = a sin [ α ( x ) ] cos { 2 n arcsin [ α ( x ) ] - 2 α ( x ) } .
U ( x , 0 ) = 1 + U s ( x , 0 ) = 1 + 2 π 0 θ 0 exp [ i k x sin ( θ ) ] T ( θ ) d θ , U B ( x , 0 ) = 1 + U B s ( x , 0 ) = 1 + 2 π 0 θ 0 exp [ i k x sin ( θ ) ] T B ( θ ) d θ .
Ψ ( x , a ) = k a + Ψ s ( x , a ) = k a + exp ( - i k a ) 2 π 0 θ 1 exp [ i k x sin ( θ ) ] T B ( θ ) d θ .
δ x = λ 2 sin θ 0 ,
δ x = λ z N .
U apprx U exact = U rel exp ( i Δ Ψ ) .
U B ( z ) = exp ( i k z ) + i k ( n 2 - 1 ) 2 × - a a exp [ i k ( z + z - z ) ] d z = [ 1 + i k a ( n 2 - 1 ) ] exp ( i k z )
U B ( 0 ) = 1 + i k a ( n 2 - 1 )
U R ( z ) = exp { i k z + [ U B ( z ) - exp ( i k z ) ] exp ( - i k z ) } = exp { i k [ z + a ( n 2 - 1 ) ] }
U R ( 0 ) = exp [ i k a ( n 2 - 1 ) ]
U G ( z ) = exp { i k [ 2 a ( n - 1 ) + z ] }
U G ( 0 ) = exp { i k [ 2 a ( n - 1 ) ] }
Δ Ψ R G = Ψ R - Ψ G = k a [ ( n 2 - 1 ) - 2 ( n - 1 ) ] = k a ( n - 1 ) 2 .
Δ Ψ B G = Ψ B - Ψ G = 2 k a ( 1 - n ) + arctan [ k a ( n 2 - 1 ) ] .
U B ( 0 ) U G ( 0 ) = [ 1 + ( k a ) 2 ( n 2 - 1 ) 2 ] 1 / 2 ,
U R ( 0 ) U G ( 0 ) = 1.

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