Abstract

Prism light guides are hollow dielectric tubes that use prismatic facets to guide light by means of total internal reflection. An unresolved problem has been to determine the magnitude of loss caused by diffraction in prism light guides. Neither experimental measurement nor an analytical solution has yet been achieved, so we attacked the problem numerically, in two steps. First, we found a way to represent such a transitionally invariant three-dimensional system as an equivalent two-dimensional problem. Second, we employed the finite-difference time-domain algorithm, with periodic boundary conditions, to yield a computation problem of manageable size. We found that the diffraction-induced transmissivity of a prism light guide wall is of the order of the wavelength divided by the prism size—a result that has encouraging practical implications.

© 1998 Optical Society of America

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References

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  1. L. A. Whitehead, “Simplified ray tracing in cylindrical systems,” Appl. Opt. 21, 3536–3538 (1982).
    [CrossRef] [PubMed]
  2. L. A. Whitehead, R. A. Nodwell, F. L. Curzon, “New efficient light guide for interior illumination,” Appl. Opt. 21, 2755–2757 (1982).
    [CrossRef] [PubMed]
  3. L. A. Whitehead, “Prism light guide having surfaces which are octature,” U.S. Patent4,260,220 (7April1981).
  4. L. A. Whitehead “Transport and distribution of light energy for illuminating engineering applications,” Ph.D. dissertation (University of British Columbia, Vancouver, Canada, 1989).
  5. J. D. Jackson, Classical Electrodynamics (Wiley, New York, 1962), p. 618.
  6. K. S. Yee, “Numerical solution of initial boundary value problems involving Maxwell’s equations in isotropic media,” IEEE Trans. Antennas Propag. AP-14, 302–307 (1966).
  7. A. Taflove, K. R. Umashankar, “The finite-difference time-domain method for numerical modeling of electromagnetic wave interactions with arbitrary structures,” in Progress in Electromagnetic Research 2: Finite Element and Finite Difference Methods in Electromagnetic Scattering, M. A. Morgan, ed. (Elsevier, New York, 1990), pp. 288–373.
  8. A. Taflove, M. E. Brodwin, “Numerical solution of steady-state electromagnetic scattering problems using the time-dependent Maxwell’s equations,” IEEE Trans. Microwave Theory Tech. MTT-23, 623–630 (1975).
    [CrossRef]
  9. C.-S. Joo, J.-W. Ra, S.-Y. Shin, “Scattering by right angle dielectric wedge,” IEEE Trans. Antennas Propag. AP-32, 61–69 (1984).

1984 (1)

C.-S. Joo, J.-W. Ra, S.-Y. Shin, “Scattering by right angle dielectric wedge,” IEEE Trans. Antennas Propag. AP-32, 61–69 (1984).

1982 (2)

1975 (1)

A. Taflove, M. E. Brodwin, “Numerical solution of steady-state electromagnetic scattering problems using the time-dependent Maxwell’s equations,” IEEE Trans. Microwave Theory Tech. MTT-23, 623–630 (1975).
[CrossRef]

1966 (1)

K. S. Yee, “Numerical solution of initial boundary value problems involving Maxwell’s equations in isotropic media,” IEEE Trans. Antennas Propag. AP-14, 302–307 (1966).

Brodwin, M. E.

A. Taflove, M. E. Brodwin, “Numerical solution of steady-state electromagnetic scattering problems using the time-dependent Maxwell’s equations,” IEEE Trans. Microwave Theory Tech. MTT-23, 623–630 (1975).
[CrossRef]

Curzon, F. L.

Jackson, J. D.

J. D. Jackson, Classical Electrodynamics (Wiley, New York, 1962), p. 618.

Joo, C.-S.

C.-S. Joo, J.-W. Ra, S.-Y. Shin, “Scattering by right angle dielectric wedge,” IEEE Trans. Antennas Propag. AP-32, 61–69 (1984).

Nodwell, R. A.

Ra, J.-W.

C.-S. Joo, J.-W. Ra, S.-Y. Shin, “Scattering by right angle dielectric wedge,” IEEE Trans. Antennas Propag. AP-32, 61–69 (1984).

Shin, S.-Y.

C.-S. Joo, J.-W. Ra, S.-Y. Shin, “Scattering by right angle dielectric wedge,” IEEE Trans. Antennas Propag. AP-32, 61–69 (1984).

Taflove, A.

A. Taflove, M. E. Brodwin, “Numerical solution of steady-state electromagnetic scattering problems using the time-dependent Maxwell’s equations,” IEEE Trans. Microwave Theory Tech. MTT-23, 623–630 (1975).
[CrossRef]

A. Taflove, K. R. Umashankar, “The finite-difference time-domain method for numerical modeling of electromagnetic wave interactions with arbitrary structures,” in Progress in Electromagnetic Research 2: Finite Element and Finite Difference Methods in Electromagnetic Scattering, M. A. Morgan, ed. (Elsevier, New York, 1990), pp. 288–373.

Umashankar, K. R.

A. Taflove, K. R. Umashankar, “The finite-difference time-domain method for numerical modeling of electromagnetic wave interactions with arbitrary structures,” in Progress in Electromagnetic Research 2: Finite Element and Finite Difference Methods in Electromagnetic Scattering, M. A. Morgan, ed. (Elsevier, New York, 1990), pp. 288–373.

Whitehead, L. A.

L. A. Whitehead, “Simplified ray tracing in cylindrical systems,” Appl. Opt. 21, 3536–3538 (1982).
[CrossRef] [PubMed]

L. A. Whitehead, R. A. Nodwell, F. L. Curzon, “New efficient light guide for interior illumination,” Appl. Opt. 21, 2755–2757 (1982).
[CrossRef] [PubMed]

L. A. Whitehead “Transport and distribution of light energy for illuminating engineering applications,” Ph.D. dissertation (University of British Columbia, Vancouver, Canada, 1989).

L. A. Whitehead, “Prism light guide having surfaces which are octature,” U.S. Patent4,260,220 (7April1981).

Yee, K. S.

K. S. Yee, “Numerical solution of initial boundary value problems involving Maxwell’s equations in isotropic media,” IEEE Trans. Antennas Propag. AP-14, 302–307 (1966).

Appl. Opt. (2)

IEEE Trans. Antennas Propag. (2)

C.-S. Joo, J.-W. Ra, S.-Y. Shin, “Scattering by right angle dielectric wedge,” IEEE Trans. Antennas Propag. AP-32, 61–69 (1984).

K. S. Yee, “Numerical solution of initial boundary value problems involving Maxwell’s equations in isotropic media,” IEEE Trans. Antennas Propag. AP-14, 302–307 (1966).

IEEE Trans. Microwave Theory Tech. (1)

A. Taflove, M. E. Brodwin, “Numerical solution of steady-state electromagnetic scattering problems using the time-dependent Maxwell’s equations,” IEEE Trans. Microwave Theory Tech. MTT-23, 623–630 (1975).
[CrossRef]

Other (4)

A. Taflove, K. R. Umashankar, “The finite-difference time-domain method for numerical modeling of electromagnetic wave interactions with arbitrary structures,” in Progress in Electromagnetic Research 2: Finite Element and Finite Difference Methods in Electromagnetic Scattering, M. A. Morgan, ed. (Elsevier, New York, 1990), pp. 288–373.

L. A. Whitehead, “Prism light guide having surfaces which are octature,” U.S. Patent4,260,220 (7April1981).

L. A. Whitehead “Transport and distribution of light energy for illuminating engineering applications,” Ph.D. dissertation (University of British Columbia, Vancouver, Canada, 1989).

J. D. Jackson, Classical Electrodynamics (Wiley, New York, 1962), p. 618.

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Figures (7)

Fig. 1
Fig. 1

Prism light guide.

Fig. 2
Fig. 2

Characteristic parameters of a prism light guide.

Fig. 3
Fig. 3

Schematic diagram of the model setup.

Fig. 4
Fig. 4

Thin-film interference pattern in reflected light.

Fig. 5
Fig. 5

x component of the electric field in full vector FDTD model of the prism light guide. (a) Propagation is in plane; TIR does not occur at this angle. (b) Propagation has an out-of-plane component; TIR does occur.

Fig. 6
Fig. 6

Propagation of the evanescent wave.

Fig. 7
Fig. 7

Diffraction losses of the prism light guide.

Equations (58)

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θ θ max   =   sin - 1 3 - 2 2 n 2 - 1 1 / 2 ,
E x 1 ,   y ,   z ,   t = E x 2 ,   y ,   z ,   t ,
H x 1 ,   y ,   z ,   t = H x 2 ,   y ,   z ,   t .
K x δ = 2 m π ,     m = ± 0 ,   1 ,   2 ,   ,
E x ,   y ,   z ,   0 = E 0 exp - c 1 y - y 0 2 exp - i K · r E ,
H = ε / μ | E | H ˆ ,
E x ,   y ,   z ,   0 = Re E x ,   y ,   t exp - iK z z ,
H x ,   y ,   z ,   0 = Re H x ,   y ,   t exp - iK z z .
ε z = μ z = 0 ,
E x ,   y ,   z ,   t = Re E x ,   y ,   t exp - iK z z ,
H x ,   y ,   z ,   t = Re H x ,   y ,   t exp - iK z z .
H t = - 1 μ   × E - ρ μ H ,
E t = 1 ε   × H - σ ε .
H x t = 1 μ 0 E y z - E z y - ρ H x ,
H y t = 1 μ 0 E z x - E x z - ρ H y ,
H z t = 1 μ 0 E x y - E y x - ρ H z ,
E x t = 1 ε 0 H z y - H y z - σ E x ,
E y t = 1 ε H x z - H z x - σ E y ,
E z t = 1 ε H y x - H x y - σ E z .
H x t = 1 μ 0 - ik z E y - E z y ,
H y t = 1 μ 0 E z x + ik z E x ,
H z t = 1 μ 0 E x y - E y x ,
E x t = 1 ε H z y + ik z H y - σ E x ,
E y t = 1 ε - ik z H x - H z x - σ E y ,
E z t = 1 ε H y x - H x y - σ E z .
i ,   j = i Δ x ,   j Δ y ,
F i ,   j ,   n = F i Δ x ,   j Δ y ,   n Δ t .
Δ x = Δ y = Δ .
E x 2 i + 1 ,   2 j ,   2 n + 2 = E x 2 i + 1 ,   2 j ,   2 n + 1 ε Δ t Δ H z 2 i + 1 ,   2 j + 1 ,   2 n + 1 - H z 2 i + 1 ,   2 j - 1 ,   2 n + 1 + Δ t iK z H y 2 i + 1 ,   2 j ,   2 n + 1 - Δ t σ E x 2 i + 1 ,   2 j ,   2 n + 1 ,
E y 2 i ,   2 j + 2 ,   2 n + 2 = E y 2 i ,   2 j + 1 ,   2 n - 1 ε Δ t Δ H z 2 i + 1 ,   2 j + 1 ,   2 n + 1 - H z 2 i - 1 ,   2 j + 1 ,   2 n + 1 + Δ t iK z H x 2 i ,   2 j + 1 ,   2 n + 1 - Δ t σ E y 2 i ,   2 j + 1 ,   2 n + 1 ,
E z 2 i ,   2 j ,   2 n + 2 = E y 2 i ,   2 j ,   2 n + 1 ε Δ t Δ H y 2 i + 1 ,   2 j ,   2 n + 1 - H z 2 i - 1 ,   2 j ,   2 n + 1 - Δ t Δ H x ) 2 i ,   2 j   + 1 ,   2 n + 1 - H z 2 i ,   2 j - 1 ,   2 n + 1 - Δ t σ E z 2 i ,   2 j ,   2 n + 1 ,
H x 2 i ,   2 j + 1 ,   2 n + 1 = H x 2 i ,   2 j + 1 ,   2 n - 1 - Δ t Δ E z 2 i ,   2 j + 2 ,   2 n - E z 2 i ,   2 j ,   2 n - Δ t iK z E y 2 i ,   2 j + 1 ,   2 n ] ,
H y 2 i + 1 ,   2 j ,   2 n + 1 = H x 2 i + 1 ,   2 j ,   2 n - 1 + Δ t Δ E z 2 i + 2 ,   2 j ,   2 n - E z 2 i ,   2 j ,   2 n - Δ t iK z E x 2 i + 1 ,   2 j ,   2 n ,
H z 2 i + 1 ,   2 j + 1 ,   2 n + 1 = H z 2 i + 1 ,   2 j + 1 ,   2 n - 1 - Δ t Δ E y 2 i + 2 ,   2 j + 1 ,   2 n - E y 2 i ,   2 j + 1 ,   2 n - E x 2 i + 1 ,   2 j + 2 ,   2 n - E x 2 i + 1 ,   2 j ,   2 n ,
H x i ,   j ,   n     i = 0 ,   2 ,   4 , , j = 1 ,   3 ,   5 , , n = 1 ,   3 ,   5 , ;
H y i ,   j ,   n     i = 1 ,   3 ,   5 , , j = 0 ,   2 ,   4 , , n = 1 ,   3 ,   5 , ;
H z i ,   j ,   n     i = 1 ,   3 ,   5 , , j = 1 ,   3 ,   5 , , n = 1 ,   3 ,   5 , ;
E x i ,   j ,   n     i = 1 ,   3 ,   5 ,   , j = 0 ,   2 ,   4 , , n = 0 ,   2 ,   4 , ;
E y i ,   j ,   n     i = 0 ,   2 ,   4 , , j = 1 ,   3 ,   5 , , n = 0 ,   2 ,   4 , ;
E z i ,   j ,   n     i = 0 ,   2 ,   4 , , j = 0 ,   2 ,   4 , , n = 0 ,   2 ,   4 , .
K x = | K |   sin   θ   cos   ϕ ,
K y = | K |   sin   θ   sin   ϕ ,
K z = | K |   cos   θ ,
E ˆ = A E ˆ 1 + B E ˆ 2 ,
A 2 + B 2 = 1 .
E ˆ 1 = K ˆ × k ˆ ,
E ˆ 2 = K ˆ × E ˆ 1 ,
E ˆ = A   sin   ϕ + B   cos   θ   cos   ϕ ı ˆ + - A   cos   ϕ + B   cos   θ   sin   ϕ J ˆ - B   sin   θ k ˆ ,
H ˆ = K ˆ × E ˆ .
Δ t 1 c max 1 Δ x 2 + 1 Δ y 2 1 / 2 = Δ 2 c max ,
L = I t I t + I r .
I t = t x 0 x N S y y = y t ,
I r = - t x 0 x N S y y = y r ,
S y = c 0 2 ε 0 E z H x - E x H z
θ B = tan - 1 n 2 / n 1 ,
I r / I i max = 4 r 2 1 + r 2 2 ,
r = r 2 + r 2 ,
2 n 2 b   cos θ = m + 0.5 λ 0 ,     m = 0 ,   1 ,   2 , .

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