Abstract

The solution of the multiple-scattering problem for N parallel dielectric cylinders is considered for plane-wave illumination perpendicular to the cylinder axes. We describe a nonlinear programming approach to solve the multiple-scattering matrix for an arbitrary planar array of N parallel dielectric cylinders. To our knowledge, no calculations have been made previously for multiple scattering by more than two parallel dielectric cylinders. Numerical results for four abutting cylinders with end-on illumination demonstrate damping of internal resonance features similar to previously published results for two cylinders. Furthermore, we present numerical examples of scattering from eight unequally spaced, parallel dielectric cylinders with broadside illumination. Because of coupling between the cylinders, the incident energy is spread evenly between the intensity peaks behind the array of cylinders.

© 1992 Optical Society of America

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References

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  1. Lord Rayleigh, “On the electromagnetic theory of light,” Philos. Mag. 12, 81–101 (1881).
    [CrossRef]
  2. J. R. Wait, “Scattering of a plane wave from a circular dielectric cylinder at oblique incidence,” Can. J. Phys. 33, 189–195 (1955).
    [CrossRef]
  3. L. B. Evans, J. C. Chen, S. W. Churchill, “Scattering of electromagnetic radiation by infinitely long, hollow, and coated cylinders,” J. Opt. Soc. Am. 54, 1004–1007 (1964).
    [CrossRef]
  4. W. A. Farone, C. W. Querfeld, “Electromagnetic scattering from radially inhomogeneous infinite cylinders at oblique incidence,” J. Opt. Soc. Am. 56, 476–491 (1966).
    [CrossRef]
  5. D. D. Cooke, M. Kerker, “Light scattering from long thin glass cylinders at oblique incidence,” J. Opt. Soc. Am. 59, 43–49 (1969).
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  6. C. Saekeang, P. L. Chu, “Backscattering of light from optical fibers with arbitrary refractive-index distributions: uniform approximation approach,” J. Opt. Soc. Am. 68, 1298–1305 (1978).
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  7. H. M. Presby, “Refractive index and diameter measurements of unclad optical fibers,” J. Opt. Soc. Am. 64, 280–284 (1974).
    [CrossRef]
  8. V. Twersky, “Multiple scattering of radiation by an arbitrary planar configuration of parallel cylinders,” J. Acoust. Soc. Am. 24, 42–46 (1952).
    [CrossRef]
  9. V. Twersky, “Scattering of waves by two objects,” in Electromagnetic Waves, R. E. Langer, ed. (University of Wisconsin, Madison, Wisc., 1962), pp. 361–389.
  10. G. O. Olaofe, “Scattering by two cylinders,” Radio Sci. 5, 1351–1360 (1970).
    [CrossRef]
  11. J. W. Young, J. C. Bertrand, “Multiple scattering by two cylinders,” J. Acoust. Soc. Am. 58, 1190–1195 (1975).
    [CrossRef]
  12. B. Schlicht, K. F. Wall, R. K. Chang, “Light scattering by two parallel glass fibers,” J. Opt. Soc. Am. A 4, 800–809 (1987).
    [CrossRef]
  13. D. S. Benincasa, P. W. Barber, J.-Z. Zhang, W.-F. Hsieh, R. K. Chang, “Spatial distribution of the internal and near-field intensities of large cylindrical and spherical scatterers,” Appl. Opt. 26, 1348–1356 (1987).
    [CrossRef] [PubMed]
  14. J. F. Owen, R. K. Chang, P. W. Barber, “Internal electric field distributions of a dielectric cylinder at resonant wavelengths,” J. Opt. Soc. Am. 6, 540–542 (1981).
  15. T.-G. Tsuei, P. W. Barber, “Multiple scattering by two parallel dielectric cylinders,” Appl. Opt. 27, 3375–3381 (1988).
    [CrossRef] [PubMed]
  16. H. A. Yousif, S. Kohler, “Scattering by two penetrable cylinders at oblique incidence. I. The analytical solution,” J. Opt. Soc. Am. A 5, 1085–1096 (1988).
    [CrossRef]
  17. H. Machida, J. Nitta, A. Seko, H. Kobayashi, “High-efficiency fiber grating for producing multiple beams of uniform intensity,” Appl. Opt. 23, 330–332 (1984).
    [CrossRef] [PubMed]
  18. R. Magnusson, D. Shin, “Diffraction by periodic arrays of dielectric cylinders,” J. Opt. Soc. Am. A 6, 412–414 (1989).
    [CrossRef]
  19. H. C. van de Hulst, Light Scattering by Small Particles (Wiley, New York, 1957).
  20. M. A. Abramowitz, I. A. Stegun, eds., Handbook of Mathematical Functions (National Bureau of Standards, Washington, D.C., 1964), Chap. 9, Sec. 9.1.79, p. 363.
  21. P. E. Gill, S. J. Hammarling, W. Murray, M. A. Saunders, M. H. Wright, “User’s guide for LSSOL (version 1.0): a fortran package for constrained linear least-squares and convex quadratic programming,” SOL 86-1 (Department of Operations Research, Stanford University, Stanford, Calif., 1986).
  22. P. E. Gill, W. Murray, M. H. Wright, Practical Optimization (Academic, New York, 1981).
  23. International Mathematics and Statistics Libraries, “IMSL edition 10.0” (IMSL, Inc., Houston, Tex., 1988).
  24. M. A. Abushagur, N. George, “Polarization and wavelength effects on the scattering from dielectric cylinders,” Appl. Opt. 27, 3375–3381 (1988).

1989 (1)

1988 (3)

1987 (2)

1984 (1)

1981 (1)

J. F. Owen, R. K. Chang, P. W. Barber, “Internal electric field distributions of a dielectric cylinder at resonant wavelengths,” J. Opt. Soc. Am. 6, 540–542 (1981).

1978 (1)

1975 (1)

J. W. Young, J. C. Bertrand, “Multiple scattering by two cylinders,” J. Acoust. Soc. Am. 58, 1190–1195 (1975).
[CrossRef]

1974 (1)

1970 (1)

G. O. Olaofe, “Scattering by two cylinders,” Radio Sci. 5, 1351–1360 (1970).
[CrossRef]

1969 (1)

1966 (1)

1964 (1)

1955 (1)

J. R. Wait, “Scattering of a plane wave from a circular dielectric cylinder at oblique incidence,” Can. J. Phys. 33, 189–195 (1955).
[CrossRef]

1952 (1)

V. Twersky, “Multiple scattering of radiation by an arbitrary planar configuration of parallel cylinders,” J. Acoust. Soc. Am. 24, 42–46 (1952).
[CrossRef]

1881 (1)

Lord Rayleigh, “On the electromagnetic theory of light,” Philos. Mag. 12, 81–101 (1881).
[CrossRef]

Abushagur, M. A.

Barber, P. W.

Benincasa, D. S.

Bertrand, J. C.

J. W. Young, J. C. Bertrand, “Multiple scattering by two cylinders,” J. Acoust. Soc. Am. 58, 1190–1195 (1975).
[CrossRef]

Chang, R. K.

Chen, J. C.

Chu, P. L.

Churchill, S. W.

Cooke, D. D.

Evans, L. B.

Farone, W. A.

George, N.

Gill, P. E.

P. E. Gill, W. Murray, M. H. Wright, Practical Optimization (Academic, New York, 1981).

P. E. Gill, S. J. Hammarling, W. Murray, M. A. Saunders, M. H. Wright, “User’s guide for LSSOL (version 1.0): a fortran package for constrained linear least-squares and convex quadratic programming,” SOL 86-1 (Department of Operations Research, Stanford University, Stanford, Calif., 1986).

Hammarling, S. J.

P. E. Gill, S. J. Hammarling, W. Murray, M. A. Saunders, M. H. Wright, “User’s guide for LSSOL (version 1.0): a fortran package for constrained linear least-squares and convex quadratic programming,” SOL 86-1 (Department of Operations Research, Stanford University, Stanford, Calif., 1986).

Hsieh, W.-F.

Kerker, M.

Kobayashi, H.

Kohler, S.

Machida, H.

Magnusson, R.

Murray, W.

P. E. Gill, S. J. Hammarling, W. Murray, M. A. Saunders, M. H. Wright, “User’s guide for LSSOL (version 1.0): a fortran package for constrained linear least-squares and convex quadratic programming,” SOL 86-1 (Department of Operations Research, Stanford University, Stanford, Calif., 1986).

P. E. Gill, W. Murray, M. H. Wright, Practical Optimization (Academic, New York, 1981).

Nitta, J.

Olaofe, G. O.

G. O. Olaofe, “Scattering by two cylinders,” Radio Sci. 5, 1351–1360 (1970).
[CrossRef]

Owen, J. F.

J. F. Owen, R. K. Chang, P. W. Barber, “Internal electric field distributions of a dielectric cylinder at resonant wavelengths,” J. Opt. Soc. Am. 6, 540–542 (1981).

Presby, H. M.

Querfeld, C. W.

Rayleigh, Lord

Lord Rayleigh, “On the electromagnetic theory of light,” Philos. Mag. 12, 81–101 (1881).
[CrossRef]

Saekeang, C.

Saunders, M. A.

P. E. Gill, S. J. Hammarling, W. Murray, M. A. Saunders, M. H. Wright, “User’s guide for LSSOL (version 1.0): a fortran package for constrained linear least-squares and convex quadratic programming,” SOL 86-1 (Department of Operations Research, Stanford University, Stanford, Calif., 1986).

Schlicht, B.

Seko, A.

Shin, D.

Tsuei, T.-G.

Twersky, V.

V. Twersky, “Multiple scattering of radiation by an arbitrary planar configuration of parallel cylinders,” J. Acoust. Soc. Am. 24, 42–46 (1952).
[CrossRef]

V. Twersky, “Scattering of waves by two objects,” in Electromagnetic Waves, R. E. Langer, ed. (University of Wisconsin, Madison, Wisc., 1962), pp. 361–389.

van de Hulst, H. C.

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

Wait, J. R.

J. R. Wait, “Scattering of a plane wave from a circular dielectric cylinder at oblique incidence,” Can. J. Phys. 33, 189–195 (1955).
[CrossRef]

Wall, K. F.

Wright, M. H.

P. E. Gill, W. Murray, M. H. Wright, Practical Optimization (Academic, New York, 1981).

P. E. Gill, S. J. Hammarling, W. Murray, M. A. Saunders, M. H. Wright, “User’s guide for LSSOL (version 1.0): a fortran package for constrained linear least-squares and convex quadratic programming,” SOL 86-1 (Department of Operations Research, Stanford University, Stanford, Calif., 1986).

Young, J. W.

J. W. Young, J. C. Bertrand, “Multiple scattering by two cylinders,” J. Acoust. Soc. Am. 58, 1190–1195 (1975).
[CrossRef]

Yousif, H. A.

Zhang, J.-Z.

Appl. Opt. (4)

Can. J. Phys. (1)

J. R. Wait, “Scattering of a plane wave from a circular dielectric cylinder at oblique incidence,” Can. J. Phys. 33, 189–195 (1955).
[CrossRef]

J. Acoust. Soc. Am. (2)

V. Twersky, “Multiple scattering of radiation by an arbitrary planar configuration of parallel cylinders,” J. Acoust. Soc. Am. 24, 42–46 (1952).
[CrossRef]

J. W. Young, J. C. Bertrand, “Multiple scattering by two cylinders,” J. Acoust. Soc. Am. 58, 1190–1195 (1975).
[CrossRef]

J. Opt. Soc. Am. (6)

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

Philos. Mag. (1)

Lord Rayleigh, “On the electromagnetic theory of light,” Philos. Mag. 12, 81–101 (1881).
[CrossRef]

Radio Sci. (1)

G. O. Olaofe, “Scattering by two cylinders,” Radio Sci. 5, 1351–1360 (1970).
[CrossRef]

Other (6)

V. Twersky, “Scattering of waves by two objects,” in Electromagnetic Waves, R. E. Langer, ed. (University of Wisconsin, Madison, Wisc., 1962), pp. 361–389.

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

M. A. Abramowitz, I. A. Stegun, eds., Handbook of Mathematical Functions (National Bureau of Standards, Washington, D.C., 1964), Chap. 9, Sec. 9.1.79, p. 363.

P. E. Gill, S. J. Hammarling, W. Murray, M. A. Saunders, M. H. Wright, “User’s guide for LSSOL (version 1.0): a fortran package for constrained linear least-squares and convex quadratic programming,” SOL 86-1 (Department of Operations Research, Stanford University, Stanford, Calif., 1986).

P. E. Gill, W. Murray, M. H. Wright, Practical Optimization (Academic, New York, 1981).

International Mathematics and Statistics Libraries, “IMSL edition 10.0” (IMSL, Inc., Houston, Tex., 1988).

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Figures (13)

Fig. 1
Fig. 1

Diffractive optical bar code that consists of transparent bars on an opaque background.

Fig. 2
Fig. 2

Modulated ribbon grating that could also be used as a diffractive optical bar code.

Fig. 3
Fig. 3

Geometry of the problem. The axis of each cylinder is parallel to the z axis, which points out of the plane of the page.

Fig. 4
Fig. 4

Calculated intensity along the line of centers C0C3 of four abutting infinite cylinders with ka in resonance with a natural mode for end-on illumination and TM polarization (ka = 45.329, m = 1.530, and δ = 2ka = 90.658). The location of each cylinder is shown by a circle below the kx axis.

Fig. 5
Fig. 5

As in Fig. 4 but with TE polarization (ka = 45.726, m = 1.530, and δ = 2ka = 91.452).

Fig. 6
Fig. 6

As in Fig. 4 but with TM polarization (ka = 45.239, m = 1.530, and δ = 101.99).

Fig. 7
Fig. 7

As in Fig. 4 but with TE polarization (ka = 45.726, m = 1.530, and δ = 102.88).

Fig. 8
Fig. 8

As in Fig. 4 but with TM polarization (ka = 45.329, m = 1.530, and δ = 150.00).

Fig. 9
Fig. 9

As in Fig. 4 but with TE polarization (ka = 45.726, m = 1.530, and δ = 151.31).

Fig. 10
Fig. 10

Near-field intensity distribution over the xy plane of eight unequally spaced, parallel dielectric cylinders for a = 2.52 μm with ka = 25 and m = 1.530 for a TM-polarized wave incident from the left. The internal intensity of each cylinder is set equal to zero.

Fig. 11
Fig. 11

As in Fig. 10 but for a TE-polarized wave.

Fig. 12
Fig. 12

As in Fig. 10 except that the isolated-cylinder coefficients were used instead of the multiple-scattering coefficients.

Fig. 13
Fig. 13

Near-field intensity distribution over the xy plane on the shadow side of a single fiber from an optical-fiber sheet for a = 12.5 μm with ka = 124 and m = 1.530 for a TM-polarized wave incident from the left. The internal intensity of the fiber is set equal to zero.

Equations (25)

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

u 0 inc = - i exp ( - ω t ) k exp ( - i ρ 0 cos γ 0 ) ,
v 0 inc = i exp ( - i ω t ) k exp ( i ρ 0 cos γ 0 ) ,
exp ( i ρ cos γ ) = n = - i n J n ( ρ ) exp ( - i n γ )
u p inc = - exp ( - i ω t ) k exp ( i δ p cos β ) × n = - i n + 1 J n ( ρ p ) exp ( - i n γ p ) ,
v p inc = exp ( - i ω t ) k exp ( i δ p cos β ) × n = - i n + 1 J n ( ρ p ) exp ( - i n γ p ) .
u p s = exp ( - i ω t ) k n = - i n + 1 b n p H n ( ρ p ) exp ( - i n γ p ) ,
v p s = - exp ( - i ω t ) k n = - i n + 1 a n p H n ( ρ p ) exp ( - i n γ p ) ,
u p trans = exp ( - i ω t ) k n = - i n + 1 d n p J n ( m ρ p ) exp ( - i n γ p ) .
v p trans = - exp ( - i ω t ) k n = - i n + 1 c n p J n ( m ρ p ) exp ( - i n γ p ) ,
exp ( - i n θ q ) H n ( ρ q ) = { l = - ( - 1 ) l H n + l ( δ p q ) J l ( ρ p ) exp ( i l θ p ) , q < p , ( - 1 ) n l = - H n + l ( δ p q ) J l ( ρ p ) exp ( i l θ p ) , q > p
b n p = b n [ exp ( i δ p cos β ) + i n + 1 exp ( i n β ) q p B - n p q ] ,
B n p q = { l = - ( - 1 ) l i l + 1 b n q exp ( - i l β ) H n + l ( δ p q ) p < q , ( - 1 ) n l = - i l + 1 b n q exp ( - i l β ) H n + l ( δ p q ) , p > q
b n = m J n ( k a ) J n ( m k a ) - J n ( k a ) J n ( m k a ) m H n ( k a ) J n ( m k a ) - H n ( k a ) J n ( m k a )
a n p = a n [ exp ( i δ p cos β ) + i n + 1 exp ( i n β ) q p B - n p q ] ,
a n = J n ( k a ) J n ( m k a ) - m J n ( k a ) J n ( m k a ) H n ( k a ) J n ( m k a ) - m H n ( k a ) J n ( m k a )
L = F + C L ,
L = [ b - M 0 , , b 0 0 , , b M 0 , , b - M N - 1 , , b 0 N - 1 , , b M N - 1 ] T ,
F = [ b - M 0 , , b 0 0 , , b M 0 , , b - M N - 1 , , b 0 N - 1 , , b M N - 1 ] T ,
L = ( I - C ) - 1 F .
( I - C ) - 1 = I + C + C 2 +
L = F + C F + C 2 F + .
err = 1 2 F - ( I - C ) L 2 .
minimize x R n G ( x )
1 { x D x } u ,
G ( x ) = ½ b - A x 2

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