Abstract

We develop a computational method for calculating the spatial profile of the electromagnetic field after scattering by an array of waveguides. Our formalism is very general and includes chromatic dependence, the influence of the array arrangement, and other effects such as the effect of stress. Our calculations of the amplitude and phase of scattered light provide valuable information about the features of the waveguides. These results can be applied to different areas of study, such as biological waveguides and fiber sensing.

© 2003 Optical Society of America

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  1. A. W. Snyder, “Excitation and scattering of modes on a dielectric of optical fiber,” IEEE Trans. Microwave Theory Tech. MIT-17, 1138–1144 (1969).
    [CrossRef]
  2. R. F. Álvarez-Estrada and M. L. Calvo, “Electromagnetic scattering by an infinite inhomogeneous dielectric cylinder: new Green’s function and integral equations,” J. Math. Phys. 21, 389–394 (1980).
    [CrossRef]
  3. S. K. Sharma and D. J. Somerford, “Scattering of light in the Eikonal approximation,” in Progress in Optics, Vol. XXXIX, E. Wolf, ed. (Elsevier, North-Holland, Amsterdam, 1992), Chap. III.
  4. M. L. Calvo and A. Durán, “Freedholm’s method for multiple scattering of electromagnetic waves by fixed obstacles,” Il Nuovo Cimento 45B, 68–76 (1978).
    [CrossRef]
  5. P. Cheben and M. L. Calvo, “Multiple scattering of classical electromagnetic waves by volume gratings: completion of Fujiwara’s solution,” J. Mod. Opt. 46, 181–198 (1999).
    [CrossRef]
  6. M. L. Calvo and R. F. Álvarez-Estrada, “Coupling of dielectric waveguides,” in Max Born Centenary Conference, M. J. Colles and D. Swift, eds., Proc. SPIE 369, 401–406 (1982).
    [CrossRef]
  7. T. B. Smith, “Multiple scattering in the cornea,” J. Mod. Opt. 35, 93–101 (1988).
    [CrossRef]
  8. Q. Zhou and R. W. Knighton, “Light scattering and form birefringence of parallel cylindrical arrays that represent cellular organelles of the retinal nerve fiber layer,” Appl. Opt. 36, 2273–2285 (1997).
    [CrossRef] [PubMed]
  9. T. R. Wolinski, “Polarimetric optical fibers and sensors,” in Progress in Optics, Vol. XL, E. Wolf, ed. (Elsevier, Amsterdam, 2000), Chap. I.
  10. A. Snyder and J. D. Love, Optical Waveguide Theory, (Chapman & Hall, London, 1983), Chap. 30.
  11. R. F. Alvarez-Estrada and M. L. Calvo, “Single-mode anisotropic cylindrical dielectric waveguides,” Opt. Acta 30, 481–503 (1983).
    [CrossRef]
  12. A. M. Benoit, K. Naoun, V. Louis-Dorr, L. Mala, and A. Raspiller, “Linear dichroism of the retinal nerve fiber layer expressed with Mueller matrices,” Appl. Opt. 40, 565–569 (2001).
    [CrossRef]
  13. W. H. Press, B. P. Flannery, S. A. Teukolsky, and W. T. Vetterling, Numerical Recipes in C: the Art of Scientific Computing (Cambridge University, New York, 1988).
  14. M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions (Dover, New York, 1965), p. 360, formula 9.1.8.
  15. R. Barakat, “Application of the sampling theorem to optical diffraction theory,” J. Opt. Soc. Am. 54, 921–930 (1964).
    [CrossRef]
  16. D. R. Williams and N. J. Coletta, “Cone spacing and the visual resolution limit,” J. Opt. Soc. Am. A 4, 1514–1523 (1987).
    [CrossRef] [PubMed]
  17. J. W. Goodman, Introduction to Fourier Optics, 2nd ed. (McGraw-Hill, New York, 1996), Chap. 2, Sec. 2.4.
  18. V. Lakshminarayanan and J. M. Enoch, “Biological Waveguides” in Handbook of Optics Vol. III, 2nd ed., M. Bass, ed. (McGraw-Hill, New York, 2001), Chap. 9.
  19. R. Dändliker, “Rotational effects of polarization in optical fibers,” in Anisotropic and Non-Linear Optical Waveguides, C. G. Someda and G. Stegman, eds. (Elsevier, Amsterdam, 1992), p. 42, Sec. 2.2.
  20. L. D. Landau and E. M. Lifshitz, Electrodynamics of Continuous Media (Pergamon, Oxford, 1960), Chap. XI, Sec. 81.
  21. S. Choi, M. Kono, and J. M. Enoch, “Evidence for Transient Strains at the Optics Disc and Nerve in Myopia: I. Stiles-Crawford Effect Studies Performed Over Time,” presented at the Annual Meeting of Association for Research in Vision and Ophthalmology, Fort Lauderdale, Florida, May 5–10, 2002.
  22. J. Limeres, M. L. Calvo, V. Lakshminarayanan, and J. M. Enoch, “Stress sensor based on light scattering by an array of birefringent optical waveguides,” in International Commission for Optics XIX: Optics for the Quality of Life, A. Consortini and G. C. Righini, eds., Proc. SPIE 4829, 881–882 (2002).

2002 (1)

J. Limeres, M. L. Calvo, V. Lakshminarayanan, and J. M. Enoch, “Stress sensor based on light scattering by an array of birefringent optical waveguides,” in International Commission for Optics XIX: Optics for the Quality of Life, A. Consortini and G. C. Righini, eds., Proc. SPIE 4829, 881–882 (2002).

2001 (1)

1999 (1)

P. Cheben and M. L. Calvo, “Multiple scattering of classical electromagnetic waves by volume gratings: completion of Fujiwara’s solution,” J. Mod. Opt. 46, 181–198 (1999).
[CrossRef]

1997 (1)

1988 (1)

T. B. Smith, “Multiple scattering in the cornea,” J. Mod. Opt. 35, 93–101 (1988).
[CrossRef]

1987 (1)

1983 (1)

R. F. Alvarez-Estrada and M. L. Calvo, “Single-mode anisotropic cylindrical dielectric waveguides,” Opt. Acta 30, 481–503 (1983).
[CrossRef]

1982 (1)

M. L. Calvo and R. F. Álvarez-Estrada, “Coupling of dielectric waveguides,” in Max Born Centenary Conference, M. J. Colles and D. Swift, eds., Proc. SPIE 369, 401–406 (1982).
[CrossRef]

1980 (1)

R. F. Álvarez-Estrada and M. L. Calvo, “Electromagnetic scattering by an infinite inhomogeneous dielectric cylinder: new Green’s function and integral equations,” J. Math. Phys. 21, 389–394 (1980).
[CrossRef]

1978 (1)

M. L. Calvo and A. Durán, “Freedholm’s method for multiple scattering of electromagnetic waves by fixed obstacles,” Il Nuovo Cimento 45B, 68–76 (1978).
[CrossRef]

1969 (1)

A. W. Snyder, “Excitation and scattering of modes on a dielectric of optical fiber,” IEEE Trans. Microwave Theory Tech. MIT-17, 1138–1144 (1969).
[CrossRef]

1964 (1)

R. Barakat, “Application of the sampling theorem to optical diffraction theory,” J. Opt. Soc. Am. 54, 921–930 (1964).
[CrossRef]

Alvarez-Estrada, R. F.

R. F. Alvarez-Estrada and M. L. Calvo, “Single-mode anisotropic cylindrical dielectric waveguides,” Opt. Acta 30, 481–503 (1983).
[CrossRef]

Álvarez-Estrada, R. F.

M. L. Calvo and R. F. Álvarez-Estrada, “Coupling of dielectric waveguides,” in Max Born Centenary Conference, M. J. Colles and D. Swift, eds., Proc. SPIE 369, 401–406 (1982).
[CrossRef]

R. F. Álvarez-Estrada and M. L. Calvo, “Electromagnetic scattering by an infinite inhomogeneous dielectric cylinder: new Green’s function and integral equations,” J. Math. Phys. 21, 389–394 (1980).
[CrossRef]

Barakat, R.

R. Barakat, “Application of the sampling theorem to optical diffraction theory,” J. Opt. Soc. Am. 54, 921–930 (1964).
[CrossRef]

Benoit, A. M.

Calvo, M. L.

J. Limeres, M. L. Calvo, V. Lakshminarayanan, and J. M. Enoch, “Stress sensor based on light scattering by an array of birefringent optical waveguides,” in International Commission for Optics XIX: Optics for the Quality of Life, A. Consortini and G. C. Righini, eds., Proc. SPIE 4829, 881–882 (2002).

P. Cheben and M. L. Calvo, “Multiple scattering of classical electromagnetic waves by volume gratings: completion of Fujiwara’s solution,” J. Mod. Opt. 46, 181–198 (1999).
[CrossRef]

R. F. Alvarez-Estrada and M. L. Calvo, “Single-mode anisotropic cylindrical dielectric waveguides,” Opt. Acta 30, 481–503 (1983).
[CrossRef]

M. L. Calvo and R. F. Álvarez-Estrada, “Coupling of dielectric waveguides,” in Max Born Centenary Conference, M. J. Colles and D. Swift, eds., Proc. SPIE 369, 401–406 (1982).
[CrossRef]

R. F. Álvarez-Estrada and M. L. Calvo, “Electromagnetic scattering by an infinite inhomogeneous dielectric cylinder: new Green’s function and integral equations,” J. Math. Phys. 21, 389–394 (1980).
[CrossRef]

M. L. Calvo and A. Durán, “Freedholm’s method for multiple scattering of electromagnetic waves by fixed obstacles,” Il Nuovo Cimento 45B, 68–76 (1978).
[CrossRef]

Cheben, P.

P. Cheben and M. L. Calvo, “Multiple scattering of classical electromagnetic waves by volume gratings: completion of Fujiwara’s solution,” J. Mod. Opt. 46, 181–198 (1999).
[CrossRef]

Coletta, N. J.

Durán, A.

M. L. Calvo and A. Durán, “Freedholm’s method for multiple scattering of electromagnetic waves by fixed obstacles,” Il Nuovo Cimento 45B, 68–76 (1978).
[CrossRef]

Enoch, J. M.

J. Limeres, M. L. Calvo, V. Lakshminarayanan, and J. M. Enoch, “Stress sensor based on light scattering by an array of birefringent optical waveguides,” in International Commission for Optics XIX: Optics for the Quality of Life, A. Consortini and G. C. Righini, eds., Proc. SPIE 4829, 881–882 (2002).

Knighton, R. W.

Lakshminarayanan, V.

J. Limeres, M. L. Calvo, V. Lakshminarayanan, and J. M. Enoch, “Stress sensor based on light scattering by an array of birefringent optical waveguides,” in International Commission for Optics XIX: Optics for the Quality of Life, A. Consortini and G. C. Righini, eds., Proc. SPIE 4829, 881–882 (2002).

Limeres, J.

J. Limeres, M. L. Calvo, V. Lakshminarayanan, and J. M. Enoch, “Stress sensor based on light scattering by an array of birefringent optical waveguides,” in International Commission for Optics XIX: Optics for the Quality of Life, A. Consortini and G. C. Righini, eds., Proc. SPIE 4829, 881–882 (2002).

Louis-Dorr, V.

Mala, L.

Naoun, K.

Raspiller, A.

Smith, T. B.

T. B. Smith, “Multiple scattering in the cornea,” J. Mod. Opt. 35, 93–101 (1988).
[CrossRef]

Snyder, A. W.

A. W. Snyder, “Excitation and scattering of modes on a dielectric of optical fiber,” IEEE Trans. Microwave Theory Tech. MIT-17, 1138–1144 (1969).
[CrossRef]

Williams, D. R.

Zhou, Q.

Appl. Opt. (2)

IEEE Trans. Microwave Theory Tech. (1)

A. W. Snyder, “Excitation and scattering of modes on a dielectric of optical fiber,” IEEE Trans. Microwave Theory Tech. MIT-17, 1138–1144 (1969).
[CrossRef]

Il Nuovo Cimento (1)

M. L. Calvo and A. Durán, “Freedholm’s method for multiple scattering of electromagnetic waves by fixed obstacles,” Il Nuovo Cimento 45B, 68–76 (1978).
[CrossRef]

J. Math. Phys. (1)

R. F. Álvarez-Estrada and M. L. Calvo, “Electromagnetic scattering by an infinite inhomogeneous dielectric cylinder: new Green’s function and integral equations,” J. Math. Phys. 21, 389–394 (1980).
[CrossRef]

J. Mod. Opt. (2)

P. Cheben and M. L. Calvo, “Multiple scattering of classical electromagnetic waves by volume gratings: completion of Fujiwara’s solution,” J. Mod. Opt. 46, 181–198 (1999).
[CrossRef]

T. B. Smith, “Multiple scattering in the cornea,” J. Mod. Opt. 35, 93–101 (1988).
[CrossRef]

J. Opt. Soc. Am. (1)

R. Barakat, “Application of the sampling theorem to optical diffraction theory,” J. Opt. Soc. Am. 54, 921–930 (1964).
[CrossRef]

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

Opt. Acta (1)

R. F. Alvarez-Estrada and M. L. Calvo, “Single-mode anisotropic cylindrical dielectric waveguides,” Opt. Acta 30, 481–503 (1983).
[CrossRef]

Proc. SPIE (2)

M. L. Calvo and R. F. Álvarez-Estrada, “Coupling of dielectric waveguides,” in Max Born Centenary Conference, M. J. Colles and D. Swift, eds., Proc. SPIE 369, 401–406 (1982).
[CrossRef]

J. Limeres, M. L. Calvo, V. Lakshminarayanan, and J. M. Enoch, “Stress sensor based on light scattering by an array of birefringent optical waveguides,” in International Commission for Optics XIX: Optics for the Quality of Life, A. Consortini and G. C. Righini, eds., Proc. SPIE 4829, 881–882 (2002).

Other (10)

W. H. Press, B. P. Flannery, S. A. Teukolsky, and W. T. Vetterling, Numerical Recipes in C: the Art of Scientific Computing (Cambridge University, New York, 1988).

M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions (Dover, New York, 1965), p. 360, formula 9.1.8.

J. W. Goodman, Introduction to Fourier Optics, 2nd ed. (McGraw-Hill, New York, 1996), Chap. 2, Sec. 2.4.

V. Lakshminarayanan and J. M. Enoch, “Biological Waveguides” in Handbook of Optics Vol. III, 2nd ed., M. Bass, ed. (McGraw-Hill, New York, 2001), Chap. 9.

R. Dändliker, “Rotational effects of polarization in optical fibers,” in Anisotropic and Non-Linear Optical Waveguides, C. G. Someda and G. Stegman, eds. (Elsevier, Amsterdam, 1992), p. 42, Sec. 2.2.

L. D. Landau and E. M. Lifshitz, Electrodynamics of Continuous Media (Pergamon, Oxford, 1960), Chap. XI, Sec. 81.

S. Choi, M. Kono, and J. M. Enoch, “Evidence for Transient Strains at the Optics Disc and Nerve in Myopia: I. Stiles-Crawford Effect Studies Performed Over Time,” presented at the Annual Meeting of Association for Research in Vision and Ophthalmology, Fort Lauderdale, Florida, May 5–10, 2002.

S. K. Sharma and D. J. Somerford, “Scattering of light in the Eikonal approximation,” in Progress in Optics, Vol. XXXIX, E. Wolf, ed. (Elsevier, North-Holland, Amsterdam, 1992), Chap. III.

T. R. Wolinski, “Polarimetric optical fibers and sensors,” in Progress in Optics, Vol. XL, E. Wolf, ed. (Elsevier, Amsterdam, 2000), Chap. I.

A. Snyder and J. D. Love, Optical Waveguide Theory, (Chapman & Hall, London, 1983), Chap. 30.

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Figures (9)

Fig. 1
Fig. 1

N coupled waveguides; configuration for each single fiber as in Fig. 2. daverage distance between two adjacent (parallel) fibers. The impact plane XY defines the plane where the amplitude of the light scattered by the fiber array is determined (see text for details).

Fig. 2
Fig. 2

Model of an individual fiber; the wave vector of the incoming radiation β is defined at the Z direction (parallel to the optic axis of the waveguide). According to the properties of birefringent media, the electric displacement vector is defined at the XY plane. The magnetic field is plane-polarized at the XY plane. In assuming this model, the propagation modes are considered with propagation constant β, along with the light scattering. This is the configuration considered in the present model [see Eqs. (2) and (3)].

Fig. 3
Fig. 3

Integration region: (a) Circular section of a single fiber Ω=πR2, where R is the radius of the fiber, ρ is the point of observation, and ρ is the radial variable of integration. In this case, |ρ|>R, so |ρ-ρ|>0. (b) In this case, |ρ|<R. Then, at |ρ-ρ|=0 the integrand shows a singularity; see text for details.

Fig. 4
Fig. 4

Packing arrangements: (a) The section of the array fits a hexagon formed by seven waveguides. (b) The section of the array fits a square formed by nine waveguides.

Fig. 5
Fig. 5

A model for mechano-optical effect. A weak external force F is applied and defined in some arbitrary XY plane. The conditions of incidence of light are similar to those of Figs. 1 and 2. Note that here the optic axis may have a certain arbitrary angle with respect to the Z axis (original optic axis prior to force application).

Fig. 6
Fig. 6

Numerical analysis of Eq. (6) for the profile of the total field scattered by a single waveguide. The intensity of the scattered light is a function of the position. We consider three values for the ratio of the fiber radius to the wavelength of the incoming radiation: (a) R/λ=0.67, (b) R/λ=1.33, and (c) R/λ=2.67.

Fig. 7
Fig. 7

Representation of the phase of the complex scattering-field amplitude as a function of the spatial coordinates X and Y.

Fig. 8
Fig. 8

Representation of the total intensity of the scattered light for different arrangements of the fibers forming the bunch (see Fig. 4): (a) seven fibers in a hexagonal lattice packing, (b) nine fibers arranged in a square lattice.

Fig. 9
Fig. 9

Forward-scattering intensity versus the strength of the applied force. The solid curve corresponds to X-polarized light, the dashed curve to Y-polarized light.

Equations (22)

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ε(p)=ε11ε120ε21ε22000ε33.
hin(ρ)=h0(ρ)exp(iβz).
hscatt(ρ)=hi(ρ)exp(iβz).
hi(ρ)=h(0)(ρ)+Ωid2ρ[(1/4i)H0(1)(K|ρ-ρ|)P^1(ρ)+P^2(ρ,ρ)]hi(ρ),
P^1(ρ)=-K2ε22-100ε11-1,
P^2(ρ,ρ)=14iε33-(ε22-ε33) 2H0(1)(K|ρ-ρ|)x2(ε22-ε33) 2H0(1)(K|ρ-ρ|)xy-(ε33-ε11) 2H0(1)(K|ρ-ρ|)yx(ε33-ε11) 2H0(1)(K|ρ-ρ|)y2.
hi(ρ)=h(0)(ρ)+Ωid2ρ14i H0(1)(K|ρ-ρ|)P^1(ρ)+P^2(ρ,ρ)h(0)(ρ).
P^1=-K2ε11-100ε11-1,
P^2(ρ, ρ)=14iε33 (ε33-ε11)2H0(1)x2-2H0(1)xy-2H0(1)yx2H0(1)y2.
hi(ρ)=h(0)-14iP^1I1h(0)+P^2I^2h(0),
I1=Ωd2ρH0(1)(K|ρ-ρ|),
I^2=14iε33 (ε33-ε11)I2xx-I2xy-I2yxI2yy.
I2αβ=Ωd2ρ2H0(1)xαxβ (K|ρ-ρ|).
htot=π4mnhi(nd/2, md/2) ×J1{2π/d[(x-nd/2)2+(y-md/2)2]1/2}2π/d[(x-nd/2)2+(y-md/2)2]1/2,
ε˜=εeδik+a1uik+a2ullδik,
u11=|F|πRi2,
u22=-3u11,
u33=0.
ε˜=ε˜11000ε˜22000ε˜33,
ε˜11=εe+a1u11+a2u22+a2u33,
ε˜22=εe+a2u11+a1u22+a2u33,
ε˜33=εe+a1u11+a2u22+a1u33.

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