Abstract

The total-field–scattered-field formulation of the finite-difference time-domain method (FDTD) is used to analyze the diffraction of finite incident beams by finite-number-of-periods holographic and surface-relief gratings. Both second-order and fourth-order FDTD formulations are used with various averaging schemes to treat permittivity discontinuities and a comparative study is made with alternative numerical methods. The diffraction efficiencies for gratings of several periods and various beam sizes, for both TE and TM polarization cases, are calculated and the FDTD results are compared with the finite- difference frequency-domain (FDFD) method results in the case of holographic gratings, and with the boundary element method results in the case of surface-relief gratings. Furthermore, the convergence of the FDTD results to the rigorous coupled-wave analysis results is investigated as the number of grating periods and the incident beam size increase.

© 2008 Optical Society of America

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  1. Feature issue on “Diffractive optics applications,” Appl. Opt. 34, 2399-2559 (1995).
  2. S. Sinzinger and J. Janns, “Integrated micro-optical imaging system with a high interconnection capacity fabricated in planar optics,” Appl. Opt. 36, 4729-4735 (1997).
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  4. R. T. Chen, L. Lin, C. Choi, Y. J. Liu, B. Bihari, L. Wu, S. Tang, R. Wickman, B.Picor, M. K. Hibbs-Brenner, J. Bristow, and Y. S. Liu, “Fully embedded board-level guided-wave optoelectronic interconnects,” Proc. IEEE 88, 780-793 (2000).
    [CrossRef]
  5. J. M. Bendickson, E. N. Glytsis, and T. K. Gaylord, “Focusing diffractive cylindrical mirrors: rigorous evaluation of various design methods,” J. Opt. Soc. Am. A 18, 1487-1494 (2001).
    [CrossRef]
  6. S. M. Schultz, E. N. Glytsis, and T. K. Gaylord, “Design of a high-efficiency volume grating couplers for line focusing,” Appl. Opt. 37, 2278-2287 (1998).
  7. S. M. Schultz, E. N. Glytsis, and T. K. Gaylord, “Volume grating preferential-order focusing waveguide coupler,” Opt. Lett. 24, 1708-1710 (1999).
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  9. T. K. Gaylord and M. G. Moharam, “Analysis and applications of optical diffraction by gratings,” Proc. IEEE 73, 894-937(1985).
  10. M. G. Moharam, E. B. Grann, D. A. Pommet, and T. K. Gaylord, “Formulation for stable and efficient implementation of the rigorous coupled-wave analysis of binary gratings,” J. Opt. Soc. Am. A 12, 1068-1076 (1995).
  11. M. G. Moharam, E. B. Grann, D. A. Pommet, and T. K. Gaylord, “Stable implementation of the rigorous coupled-wave analysis for surface-relief gratings: enhanced trasmittance matrix approach,” J. Opt. Soc. Am. A 12, 1077-1086 (1995).
  12. E. E. Kriezis, P. K. Pandelakis, and A. G. Papagiannakis, “Diffraction of a Gaussian beam from a periodic planar screen,” J. Opt. Soc. Am. A 11, 630-636 (1994).
  13. J. M. Bendickson, E. N. Glytsis, and T. K. Gaylord, “Guided-mode resonant subwavelength gratings: effects of finite beams and finite gratings ,” J. Opt. Soc. Am. A 18, 1912-1928 (2001).
  14. Y.-L. Kok, “General solution to the multiple-metallic-grooves scattering problem: the fast-polarization case,” Appl. Opt. 32, 2573-2581 (1993).
  15. O. Mata-Mendez and J. Sumaya-Martinez, “Scattering of TE-polarized waves by a finite-grating: giant resonant enhancement of the electric field within the grooves,” J. Opt. Soc. Am. A 14, 2203-2211 (1997).
  16. G. Pelosi, G. Manara, and G. Toso, “Heuristic diffraction coefficient for plane-wave scattering from edges in periodic planar surfaces,” J. Opt. Soc. Am. A 13, 1689-1697 (1996).
  17. K. Hirayama, E. N. Glytsis, T. K. Gaylord, and D. W. Wilson, “Rigorous electromagnetic analysis of diffractive cylindrical lenses,” J. Opt. Soc. Am. A 13, 2219-2231 (1996).
  18. K. Hirayama, E. N. Glytsis, and T. K. Gaylord, “Rigorous electromagnetic analysis of diffraction by finite-number-of-periods gratings,” J. Opt. Soc. Am. A 14, 907-917 (1997).
  19. O. Mata-Mendez and J. Sumaya-Martinez, “Diffraction of Gaussian and Hermite-Gaussian beams by finite gratings,” J. Opt. Soc. Am. A 18, 537-545 (2001).
    [CrossRef]
  20. S. D. Wu and E. N. Glytsis, “Finite-number-of-periods holographic gratings with finite-width incident beams: Analysis using the finite-difference frequency domain method,” J. Opt. Soc. Am. A 19, 2018-2029 (2002).
    [CrossRef]
  21. B. Wang, J. Jiang, and G. P. Nordin, “Compact slanted grating couplers,” Opt. Express 12, 3313-3326 (2004).
    [CrossRef]
  22. B. Wang, J. Jiang, and G. P. Nordin, “Systematic design process for slanted grating couplers,” Appl. Opt. 45, 6223-6226(2006).
  23. S. Banerjee, J. B. Cole, and T. Yatagai, “Calculation of diffraction characteristics of subwavelength conducting gratings using a high accuracy nonstandard finite-difference time-domain method,” Opt. Rev. 12, 274-280 (2005).
    [CrossRef]
  24. A. Taflove and S. C. Hagness, Computational Electrodynamics: The Finite-Difference Time-Domain Method, 3rd ed. (Artech House, 2005), Ch. 3-5 and 7.
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    [CrossRef]
  27. N. V. Kantartzis and T. D. Tsiboukis, Higher Order FDTD Schemes for Waveguides and Antenna Structures, Ch. 2 (Morgan and Claypool, 2006).
  28. T. T. Zygiridis and T. D. Tsiboukis, “Low-dispersion algorithms based on the higher order (2.4) FDTD method,” IEEE Trans. Microwave Theory Tech. 52, 1321-1327 (2004).
    [CrossRef]
  29. A. Yefet and P. G. Petropoulos, “A staggered fourth-order accurate explicit finite difference scheme for the time-domain Maxwell's equations,” J. Comput. Phys. 168, 286-315 (2001).
  30. D. T. Prescott and N. V. Shuley, “Reducing solution time in monochromatic FDTD waveguide simulations,” IEEE Trans. Microwave Theory Tech. 42, 1582-1584 (1994).
    [CrossRef]
  31. L. Gurel and U. Oguz, “Signal-processing techniques to reduce the sinusoidal steady-state error in the FDTD method,” IEEE Trans. Antennas Propagat. 48, 585-593 (2000).
  32. T. Hirono, Y. Shibata, W. W.Lui, S. Seki, and Y. Yoshikuni, “The second-order condition for the dielectric interface orthogonal to the Yee-lattice axis in the FDTD scheme,” IEEE Microwave Guided Lett. 10, 359-361 (2000).
  33. U. Anderson, “Time domain methods for the Maxwell equations,” Ph.D. dissertation, Royal Institute of Technology, Sweden (2001).
  34. K. P. Hwang and A. C. Cangellaris, “Effective permittivities for second-order accurate FDTD equations at dielectric interfaces,” IEEE Microwave Wireless Comp. Lett. 11, 158-160(2001).
  35. E. Kashdan and E. Turkel, “High-order accurate modeling of electromagnetic wave propagation across media--grid conforming bodies,” J. Comput. Phys. 218, 816-835 (2006).
    [CrossRef]
  36. P. Lalanne and G. M. Morris, “Highly improved convergence of the coupled-wave method for TM polarization,” J. Opt. Soc. Am. A 13, 779-784 (1996).
  37. L. Li, “Use of Fourier series in the analysis of discontinuous periodic structures,” J. Opt. Soc. Am. A 13, 1870-1876 (1996).
  38. P. G. Petropoulos, “Phase error control for the FD-TD methods of second and fourth order accuracy,” IEEE Trans. Antennas Propagat. 42, 859-862 (1994).
  39. S. D. Wu and E. N. Glytsis, “Volume holographic grating couplers: rigorous analysis by use of the finite-difference frequency-domain method,” Appl. Opt. 43, 1009-1023 (2004).
    [CrossRef]

2006 (2)

B. Wang, J. Jiang, and G. P. Nordin, “Systematic design process for slanted grating couplers,” Appl. Opt. 45, 6223-6226(2006).

E. Kashdan and E. Turkel, “High-order accurate modeling of electromagnetic wave propagation across media--grid conforming bodies,” J. Comput. Phys. 218, 816-835 (2006).
[CrossRef]

2005 (1)

S. Banerjee, J. B. Cole, and T. Yatagai, “Calculation of diffraction characteristics of subwavelength conducting gratings using a high accuracy nonstandard finite-difference time-domain method,” Opt. Rev. 12, 274-280 (2005).
[CrossRef]

2004 (3)

2002 (1)

2001 (5)

O. Mata-Mendez and J. Sumaya-Martinez, “Diffraction of Gaussian and Hermite-Gaussian beams by finite gratings,” J. Opt. Soc. Am. A 18, 537-545 (2001).
[CrossRef]

A. Yefet and P. G. Petropoulos, “A staggered fourth-order accurate explicit finite difference scheme for the time-domain Maxwell's equations,” J. Comput. Phys. 168, 286-315 (2001).

K. P. Hwang and A. C. Cangellaris, “Effective permittivities for second-order accurate FDTD equations at dielectric interfaces,” IEEE Microwave Wireless Comp. Lett. 11, 158-160(2001).

J. M. Bendickson, E. N. Glytsis, and T. K. Gaylord, “Focusing diffractive cylindrical mirrors: rigorous evaluation of various design methods,” J. Opt. Soc. Am. A 18, 1487-1494 (2001).
[CrossRef]

J. M. Bendickson, E. N. Glytsis, and T. K. Gaylord, “Guided-mode resonant subwavelength gratings: effects of finite beams and finite gratings ,” J. Opt. Soc. Am. A 18, 1912-1928 (2001).

2000 (4)

S. M. Schultz, E. N. Glytsis, and T. K. Gaylord, “Design, fabrication, and performance of preferential-order volume grating waveguide couplers,” Appl. Opt. 39, 1223-1232 (2000).

R. T. Chen, L. Lin, C. Choi, Y. J. Liu, B. Bihari, L. Wu, S. Tang, R. Wickman, B.Picor, M. K. Hibbs-Brenner, J. Bristow, and Y. S. Liu, “Fully embedded board-level guided-wave optoelectronic interconnects,” Proc. IEEE 88, 780-793 (2000).
[CrossRef]

L. Gurel and U. Oguz, “Signal-processing techniques to reduce the sinusoidal steady-state error in the FDTD method,” IEEE Trans. Antennas Propagat. 48, 585-593 (2000).

T. Hirono, Y. Shibata, W. W.Lui, S. Seki, and Y. Yoshikuni, “The second-order condition for the dielectric interface orthogonal to the Yee-lattice axis in the FDTD scheme,” IEEE Microwave Guided Lett. 10, 359-361 (2000).

1999 (1)

1998 (2)

1997 (3)

1996 (5)

1995 (3)

1994 (3)

E. E. Kriezis, P. K. Pandelakis, and A. G. Papagiannakis, “Diffraction of a Gaussian beam from a periodic planar screen,” J. Opt. Soc. Am. A 11, 630-636 (1994).

P. G. Petropoulos, “Phase error control for the FD-TD methods of second and fourth order accuracy,” IEEE Trans. Antennas Propagat. 42, 859-862 (1994).

D. T. Prescott and N. V. Shuley, “Reducing solution time in monochromatic FDTD waveguide simulations,” IEEE Trans. Microwave Theory Tech. 42, 1582-1584 (1994).
[CrossRef]

1993 (1)

1985 (1)

T. K. Gaylord and M. G. Moharam, “Analysis and applications of optical diffraction by gratings,” Proc. IEEE 73, 894-937(1985).

Appl. Opt. (8)

Feature issue on “Diffractive optics applications,” Appl. Opt. 34, 2399-2559 (1995).

S. Sinzinger and J. Janns, “Integrated micro-optical imaging system with a high interconnection capacity fabricated in planar optics,” Appl. Opt. 36, 4729-4735 (1997).

A. C. Walker, T.-Y. Yang, J. Gourlay, J. A. B. Danes, M. G. Forbes, S. M. Prince, D. A. Baillie, D. T. Neilson, R. Williams, L. C. Wilkinson, G. R. Smith, M. P. Y. Desmulliez, G. S. Buller, M. R. Taghizadeh, A. Waddie, I. Underwood, C. R. Stanley, F. Pottier, B. Vögele, and W. Sibbett, “Optoelectronic systems based on InGaAs-complementary-metal-oxide-semiconductor smart-pixel arrays and free-space optical interconnects,” Appl. Opt. 37, 2822-2830 (1998).

S. M. Schultz, E. N. Glytsis, and T. K. Gaylord, “Design of a high-efficiency volume grating couplers for line focusing,” Appl. Opt. 37, 2278-2287 (1998).

S. M. Schultz, E. N. Glytsis, and T. K. Gaylord, “Design, fabrication, and performance of preferential-order volume grating waveguide couplers,” Appl. Opt. 39, 1223-1232 (2000).

Y.-L. Kok, “General solution to the multiple-metallic-grooves scattering problem: the fast-polarization case,” Appl. Opt. 32, 2573-2581 (1993).

B. Wang, J. Jiang, and G. P. Nordin, “Systematic design process for slanted grating couplers,” Appl. Opt. 45, 6223-6226(2006).

S. D. Wu and E. N. Glytsis, “Volume holographic grating couplers: rigorous analysis by use of the finite-difference frequency-domain method,” Appl. Opt. 43, 1009-1023 (2004).
[CrossRef]

Guided-mode resonant subwavelength gratings: effects of finite beams and finite gratings (1)

J. M. Bendickson, E. N. Glytsis, and T. K. Gaylord, “Guided-mode resonant subwavelength gratings: effects of finite beams and finite gratings ,” J. Opt. Soc. Am. A 18, 1912-1928 (2001).

IEEE Microwave Guided Lett. (1)

T. Hirono, Y. Shibata, W. W.Lui, S. Seki, and Y. Yoshikuni, “The second-order condition for the dielectric interface orthogonal to the Yee-lattice axis in the FDTD scheme,” IEEE Microwave Guided Lett. 10, 359-361 (2000).

IEEE Microwave Wireless Comp. Lett. (1)

K. P. Hwang and A. C. Cangellaris, “Effective permittivities for second-order accurate FDTD equations at dielectric interfaces,” IEEE Microwave Wireless Comp. Lett. 11, 158-160(2001).

IEEE Trans. Antennas Propag. (1)

S. D. Gedney, “An anisotropic perfectly matched layer-absorbing medium for the truncation of FDTD lattices,” IEEE Trans. Antennas Propag. 44, 1630-1639 (1996).
[CrossRef]

IEEE Trans. Antennas Propagat. (2)

L. Gurel and U. Oguz, “Signal-processing techniques to reduce the sinusoidal steady-state error in the FDTD method,” IEEE Trans. Antennas Propagat. 48, 585-593 (2000).

P. G. Petropoulos, “Phase error control for the FD-TD methods of second and fourth order accuracy,” IEEE Trans. Antennas Propagat. 42, 859-862 (1994).

IEEE Trans. Microwave Theory Tech. (2)

D. T. Prescott and N. V. Shuley, “Reducing solution time in monochromatic FDTD waveguide simulations,” IEEE Trans. Microwave Theory Tech. 42, 1582-1584 (1994).
[CrossRef]

T. T. Zygiridis and T. D. Tsiboukis, “Low-dispersion algorithms based on the higher order (2.4) FDTD method,” IEEE Trans. Microwave Theory Tech. 52, 1321-1327 (2004).
[CrossRef]

J. Comput. Phys. (2)

A. Yefet and P. G. Petropoulos, “A staggered fourth-order accurate explicit finite difference scheme for the time-domain Maxwell's equations,” J. Comput. Phys. 168, 286-315 (2001).

E. Kashdan and E. Turkel, “High-order accurate modeling of electromagnetic wave propagation across media--grid conforming bodies,” J. Comput. Phys. 218, 816-835 (2006).
[CrossRef]

J. Opt. Soc. Am. A (12)

P. Lalanne and G. M. Morris, “Highly improved convergence of the coupled-wave method for TM polarization,” J. Opt. Soc. Am. A 13, 779-784 (1996).

L. Li, “Use of Fourier series in the analysis of discontinuous periodic structures,” J. Opt. Soc. Am. A 13, 1870-1876 (1996).

J. M. Bendickson, E. N. Glytsis, and T. K. Gaylord, “Focusing diffractive cylindrical mirrors: rigorous evaluation of various design methods,” J. Opt. Soc. Am. A 18, 1487-1494 (2001).
[CrossRef]

M. G. Moharam, E. B. Grann, D. A. Pommet, and T. K. Gaylord, “Formulation for stable and efficient implementation of the rigorous coupled-wave analysis of binary gratings,” J. Opt. Soc. Am. A 12, 1068-1076 (1995).

M. G. Moharam, E. B. Grann, D. A. Pommet, and T. K. Gaylord, “Stable implementation of the rigorous coupled-wave analysis for surface-relief gratings: enhanced trasmittance matrix approach,” J. Opt. Soc. Am. A 12, 1077-1086 (1995).

E. E. Kriezis, P. K. Pandelakis, and A. G. Papagiannakis, “Diffraction of a Gaussian beam from a periodic planar screen,” J. Opt. Soc. Am. A 11, 630-636 (1994).

O. Mata-Mendez and J. Sumaya-Martinez, “Scattering of TE-polarized waves by a finite-grating: giant resonant enhancement of the electric field within the grooves,” J. Opt. Soc. Am. A 14, 2203-2211 (1997).

G. Pelosi, G. Manara, and G. Toso, “Heuristic diffraction coefficient for plane-wave scattering from edges in periodic planar surfaces,” J. Opt. Soc. Am. A 13, 1689-1697 (1996).

K. Hirayama, E. N. Glytsis, T. K. Gaylord, and D. W. Wilson, “Rigorous electromagnetic analysis of diffractive cylindrical lenses,” J. Opt. Soc. Am. A 13, 2219-2231 (1996).

K. Hirayama, E. N. Glytsis, and T. K. Gaylord, “Rigorous electromagnetic analysis of diffraction by finite-number-of-periods gratings,” J. Opt. Soc. Am. A 14, 907-917 (1997).

O. Mata-Mendez and J. Sumaya-Martinez, “Diffraction of Gaussian and Hermite-Gaussian beams by finite gratings,” J. Opt. Soc. Am. A 18, 537-545 (2001).
[CrossRef]

S. D. Wu and E. N. Glytsis, “Finite-number-of-periods holographic gratings with finite-width incident beams: Analysis using the finite-difference frequency domain method,” J. Opt. Soc. Am. A 19, 2018-2029 (2002).
[CrossRef]

Opt. Express (1)

Opt. Lett. (1)

Opt. Rev. (1)

S. Banerjee, J. B. Cole, and T. Yatagai, “Calculation of diffraction characteristics of subwavelength conducting gratings using a high accuracy nonstandard finite-difference time-domain method,” Opt. Rev. 12, 274-280 (2005).
[CrossRef]

Proc. IEEE (2)

T. K. Gaylord and M. G. Moharam, “Analysis and applications of optical diffraction by gratings,” Proc. IEEE 73, 894-937(1985).

R. T. Chen, L. Lin, C. Choi, Y. J. Liu, B. Bihari, L. Wu, S. Tang, R. Wickman, B.Picor, M. K. Hibbs-Brenner, J. Bristow, and Y. S. Liu, “Fully embedded board-level guided-wave optoelectronic interconnects,” Proc. IEEE 88, 780-793 (2000).
[CrossRef]

Other (4)

A. Taflove and S. C. Hagness, Computational Electrodynamics: The Finite-Difference Time-Domain Method, 3rd ed. (Artech House, 2005), Ch. 3-5 and 7.

A. Taflove, Editor, Advances in Computational Electrodynamics: The Finite-Difference Time-Domain Method (Artech House, 1998), Ch. 2.

N. V. Kantartzis and T. D. Tsiboukis, Higher Order FDTD Schemes for Waveguides and Antenna Structures, Ch. 2 (Morgan and Claypool, 2006).

U. Anderson, “Time domain methods for the Maxwell equations,” Ph.D. dissertation, Royal Institute of Technology, Sweden (2001).

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Figures (15)

Fig. 1
Fig. 1

Geometric configuration of the diffractive structure and the complete ( actual + UPML ) computational domain for FDTD analysis. The surrounding UPML region is used to eliminate any waves reflected back into the actual computational region. The TF–SF planar boundary as well as the incident-beam excitation plane also are shown.

Fig. 2
Fig. 2

Grating types used in this paper: (a) a general slanted volume holographic grating, (b) a two-level surface-relief grating, (c) an eight-level asymmetric surface-relief grating.

Fig. 3
Fig. 3

Beam profile of the finite-incident-beam of flat width W and total width ( 2 D W ) .

Fig. 4
Fig. 4

Electric field intensity for a L = 5 Λ unslanted volume holographic grating ( Λ = 2.5 μm , d = 8 μm , ε 0 = 2.25 , and ε 1 = 0.006 ) at steady state for TE incident polarization. The steady -state was obtained after 6750 time steps. Regions 1, 2, 3, and 4 along with the grating region are delineated in order to compare with the general Fig. 1.

Fig. 5
Fig. 5

± 1 forward diffraction efficiency DE ± 1 f for the unslanted holographic grating as a function of the grating length for TE incident polarization. Δ x = Δ y = Δ = λ min / 25 stands for higher cell density, i.e., 25 cells per minimum wavelength, while all other cases were for 20 cells per minimum wavelength.

Fig. 6
Fig. 6

± 1 forward diffraction efficiency DE ± 1 f for the unslanted holographic grating as a function of the number of grating length for TM incident polarization. Δ x = Δ y = Δ = λ min / 25 stands for higher cell density, i.e., 25 cells per minimum wavelength, while all other cases were for 20 cells per minimum wavelength.

Fig. 7
Fig. 7

Difference (in percent) of the ± 1 forward diffraction efficiency DE f ± 1 of the unslanted holographic grating (of L = 15 Λ ), calculated by the FDTD method for TE and TM polarization, with respect to the DE f ± 1 calculated by the FDTD(2,4) method for 40 cells per minimum wavelength.

Fig. 8
Fig. 8

1 forward diffraction efficiency DE 1 f for the slanted holographic grating as a function of the grating length for TE incident polarization.

Fig. 9
Fig. 9

1 forward diffraction efficiency DE 1 f for the slanted holographic grating as a function of the grating length for TM incident polarization.

Fig. 10
Fig. 10

± 1 forward diffraction efficiency DE ± 1 f for the two-level surface-relief grating as a function of the grating length for TE incident polarization.

Fig. 11
Fig. 11

± 1 forward diffraction efficiency DE ± 1 f for the two-level surface-relief grating as a function of the grating length for TM incident polarization.

Fig. 12
Fig. 12

Difference (in percent) of the ± 1 forward diffraction efficiency DE f ± 1 of the two-level surface-relief grating ( L = 15 Λ ) calculated by the FDTD method with respect to the DE f ± 1 calculated by the BEM method for the two-level surface-relief grating for TE and TM polarization.

Fig. 13
Fig. 13

Electric field intensity for the eight-level asymmetric surface-relief grating ( L = 15 Λ ) at steady state for TE incident polarization. The steady state was obtained after 2568 time steps.

Fig. 14
Fig. 14

+ 1 forward diffraction efficiency DE ± 1 f for the eight-level surface-relief grating as a function of the grating length for TE incident polarization. The FDTD(2,2) and FDTD(2,4) results are almost identical in the scale of the figure. Δ x , Δ y Δ = λ min / 30 stands for higher cell density, i.e., 30 cells per minimum wavelength, while all other cases were for 20 cells per minimum wavelength.

Fig. 15
Fig. 15

+ 1 forward diffraction efficiency DE ± 1 f for the eight-level surface-relief grating as a function of the grating length for TM incident polarization. Δ x , Δ y Δ = λ min / 30 stands for higher cell density, i.e., 30 cells per minimum wavelength, while all other cases were for 20 cells per minimum wavelength.

Tables (2)

Tables Icon

Table 1 Forward Diffraction Efficiencies DE ± 1 f for an Eight-Level Surface-Relief Grating (TE Polarization Case)

Tables Icon

Table 2 Forward Diffraction Efficiencies DE ± 1 f for an Eight-Level Surface-Relief Grating (TM Polarization Case)

Equations (28)

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

ϵ = ϵ 0 ε ( x , y ) = ϵ 0 [ ε 0 + p = 1 ε p cos ( p K · r ) ] ,
g ( y ) = { 1 , 0 | y | W 2 , cos 2 [ | y | W 2 2 ( D W ) π ] , W 2 | y | D W 2 , 0 , D W 2 | y | .
U inc = z ^ g ( y ) Re { exp ( j k · r ) exp ( j ω t ) } ,
s ¯ = [ s y s z s x 1 0 0 0 s x s z s y 1 0 0 0 s x s y s z 1 ] ,
H y ˜ x H x ˜ y = j ω κ x D z ˜ + σ x ε D z ˜ ,
E z ˜ y = j ω κ y B x ˜ + σ y ε B x ˜ ,
E z ˜ x = j ω κ z B y ˜ + σ z ε B y ˜ .
D z i 1 / 2 , j + 1 / 2 n + 1 / 2 = 2 ε κ x Δ t σ x 2 ε κ x + Δ t σ x D z i 1 / 2 , j + 1 / 2 n 1 / 2 + 2 ε Δ t 2 ε κ x + Δ t σ x ( H y n x H x n y ) | i 1 / 2 , j + 1 / 2 ,
B x i 1 / 2 , j + 1 n + 1 = 2 ε κ y Δ t σ y 2 ε κ y + Δ t σ y B x i 1 / 2 , j + 1 n + 2 ε Δ t 2 ε κ y + Δ t σ y ( E z y ) | i 1 / 2 , j + 1 ,
B y i , j + 1 / 2 n + 1 = 2 ε κ z Δ t σ z 2 ε κ z + Δ t σ z B y i , j + 1 / 2 n + 2 ε Δ t 2 ε κ z + Δ t σ z ( E z x ) | i , j + 1 / 2 .
f w | l n 1 Δ w s = 1 M / 2 C s [ f | l + ( 2 s 1 ) / 2 n f | l ( 2 s 1 ) / 2 n ] , ( w = x , y ) ,
C s = ( 1 ) s + 1 ( M 1 ) !! 2 2 M 2 ( 1 2 M + s 1 ) ! ( 1 2 M s ) ! ( 2 s 1 ) 2 ,
f w | l n [ C w 1 ( f l + / 2 n f l 1 / 2 n ) + C w 2 ( f l + 3 / 2 n f l 3 / 2 n ) ] Δ w ,
f t | l n A f l n + 1 / 2 f l n 1 / 2 Δ t ,
[ 1 J 0 ( k Δ w ) J 0 ( k Δ w ) J 0 ( 2 k Δ w ) J 0 ( k Δ w ) J 0 ( 2 k Δ w ) 1 J 0 ( 3 k Δ w ) ] [ C w 1 C w 2 ] = [ J 1 ( k Δ w 2 ) J 1 ( 3 k Δ w 2 ) ] k Δ w ,
H y | i 0 1 / 2 , j + 1 / 2 = 315 128 H y | i 0 1 , j + 1 / 2 105 32 H y | i 0 2 , j + 1 / 2 + 189 64 H y | i 0 3 , j + 1 / 2 45 32 H y | i 0 4 , j + 1 / 2 + 35 128 H y | i 0 5 , j + 1 / 2 ,
H y | i 0 1 / 2 , j + 1 / 2 x = ( 1126 315 H y | i 0 1 / 2 , j + 1 / 2 + 315 64 H y | i 0 , j + 1 / 2 35 16 H y | i 0 + 1 , j + 1 / 2 + 189 160 H y | i 0 + 2 , j + 1 / 2 45 112 H y | i 0 + 3 , j + 1 / 2 + 35 576 H y | i 0 + 4 , j + 1 / 2 ) / Δ x .
E z | i 0 1 / 2 , j + 1 / 2 n + 1 / 2 = { E z | i 0 1 / 2 , j + 1 / 2 n } Δ t ε 0 Δ x H y , inc | i 0 1 , j + 1 / 2 n .
E z | i 0 1 / 2 , j + 1 / 2 n + 1 / 2 = { E z | i 0 1 / 2 , j + 1 / 2 n + 1 / 2 } Δ t ε 0 Δ x C 1 H y , inc | i 0 1 , j + 1 / 2 n Δ t ε 0 Δ x C 2 H y , inc | i 0 2 , j + 1 / 2 n ,
E z | i 0 + 1 / 2 , j + 1 / 2 n + 1 / 2 = { E z | i 0 + 1 / 2 , j + 1 / 2 n + 1 / 2 } Δ t ε 0 Δ x C 2 H y , inc | i 0 1 , j + 1 / 2 n ,
E z | i 0 3 / 2 , j + 1 / 2 n + 1 / 2 = { E z | i 0 3 / 2 , j + 1 / 2 n + 1 / 2 } Δ t ε 0 Δ x C 2 H y , inc | i 0 , j + 1 / 2 n ,
( E z | i , j n ) s s = A | i , j cos ( ω n Δ t + ϕ | i , j ) .
f i j ( A | i j , ϕ | i j ) = n | E z | i , j n ( E z | i , j n ) s s | 2
F i ( k y m ) = q = 0 N 1 U ˜ ( x = x i , q Δ y ) exp [ j k y m ( q Δ y ) ] ,
P i , p T E = Δ y 2 M m = k 1 , y ( p + 1 / 2 ) K y k 1 , y ( p 1 / 2 ) K y | F i ( k y m ) | 2 Re ( k i , x m * η i * k i * ) ,
P i , p T M = Δ y 2 M m = k 1 , y ( p + 1 / 2 ) K y k 1 , y ( p 1 / 2 ) K y | F i ( k y m ) | 2 Re ( k i , x m η i k i ) ,
ε ave = { ε 1 + ε 2 2 , E | | S 2 ε 1 ε 2 ε 1 + ε 2 , E S .
ε ˜ ( x ) = 1 2 ( ε 2 + ε 1 ) + 45 16 ( ε 2 ε 1 ) ( x δ ) 25 2 ( ε 2 ε 1 ) ( x δ ) 3 + 21 ( ε 2 ε 1 ) ( x δ ) 5 .

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