Abstract

An electromagnetic vector-field model for design of optical components based on the finite-difference time-domain method and radiation integrals is presented. Its ability to predict the optical electromagnetic dynamics in structures with complex material distributions is demonstrated. Theoretical and numerical investigations of finite-length surface-relief structures embedded in polymer dielectric waveguiding materials are presented. The importance of several geometric parameter dependencies is indicated as far-field power distributions are rearranged between diffraction orders. The influences of the variation in grating period, modulation depth, length, and profile are investigated.

© 1999 Optical Society of America

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References

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  1. T. K. Gaylord, M. G. Moharam, “Analysis and applications of optical diffraction by gratings,” Proc. IEEE 73, 894–937 (1985).
    [CrossRef]
  2. M. G. Moharam, T. K. Gaylord, “Diffraction analysis of dielectric surface-relief gratings,” J. Opt. Soc. Am. 72, 1385–1392 (1982).
    [CrossRef]
  3. H. Nishihara, S. Ura, T. Suhara, J. Koyama, “An integrated-optic disk pickup device,” J. Lightwave Technol. LT-4, 913–918 (1986).
  4. L. Lading, C. D. Hansen, E. Rasmussen, “Surface light scattering: integrated technology and signal processing,” Appl. Opt. 36, 7593–7600 (1997).
    [CrossRef]
  5. R. W. Ziolkowski, “The incorporation of microscopic material models into the FDTD approach for ultrafast optical pulse simulations,” IEEE Trans. Antennas Propag. 45, 375–391 (1997).
    [CrossRef]
  6. M. S. Mirotznik, D. W. Prather, J. N. Mait, “A hybrid finite element-boundary element method for the analysis of diffractive elements,” J. Mod. Opt. 43, 1309–1321 (1996).
    [CrossRef]
  7. D. W. Prather, M. S. Mirotznik, J. N. Mait, “Boundary integral methods applied to the analysis of diffractive optical elements,” J. Opt. Soc. Am. A 14, 34–43 (1997).
    [CrossRef]
  8. P. G. Dinesen, J. S. Hesthaven, J. P. Lynov, L. Lading, “Pseudo-spectral method for the analysis of diffractive optical elements,” submitted to J. Opt. Soc. Am. A.
  9. M. G. Moharam, D. A. Pommet, E. B. Grann, “Limits of scalar diffraction theory for diffractive phase elements,” J. Opt. Soc. Am. A 11, 1827–1834 (1994).
    [CrossRef]
  10. J. J. H. Wang, Generalized Moment Methods in Electromagnetics: Formulation and Computer Solution of Integral Equations (Wiley, New York, 1991).
  11. J. Jin, The Finite Element Method in Electromagnetics (Wiley, New York, 1993).
  12. B. Yang, D. Gottlieb, J. S. Hesthaven, “Spectral simulations of electromagnetic wave scattering,” J. Comput. Phys. 134, 216–230 (1997).
    [CrossRef]
  13. 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).
  14. A. Taflove, Computational Electrodynamics: The Finite-Difference Time-Domain Method (Artech House, Boston, Mass., 1995).
  15. J. P. Zhang, D. Y. Chu, S. L. Wu, W. G. Bi, R. C. Tiberio, R. M. Joseph, A. Taflove, C. W. Tu, S. T. Ho, “Nanofabrication of 1-D photonic bandgap structures along a photonic wire,” IEEE Photon. Technol. Lett. 8, 491–493 (1996).
    [CrossRef]
  16. H. Y. D. Yang, “Finite difference analysis of 2-D photonic crystals,” IEEE Trans. Microwave Theory Tech. 44, 2688–2695 (1996).
    [CrossRef]
  17. B. Yang, D. Gottlieb, J. S. Hesthaven, “On the use of PML ABC’s in spectral time-domain simulations of electromagnetic scattering,” in 13th Annual Review of Progress in Applied Computational Electromagnetics at the Naval Postgraduate School, A. Kishk, A. Glisson, eds., conference proceedings of the Applied Computational Electromagnetics Society (U. Mississippi Press, Jackson, Miss., 1997), pp. 926–933.
  18. S. Ramo, J. R. Whinnery, T. Van Duzer, Fields and Waves in Communication Electronics, 3rd ed. (Wiley, New York, 1993).
  19. T. Tamir, “Beam and waveguide couplers,” in Integrated Optics, 2nd. ed., T. Tamir, ed. (Springer-Verlag, New York, 1979), pp. 309–327.

1997 (4)

L. Lading, C. D. Hansen, E. Rasmussen, “Surface light scattering: integrated technology and signal processing,” Appl. Opt. 36, 7593–7600 (1997).
[CrossRef]

R. W. Ziolkowski, “The incorporation of microscopic material models into the FDTD approach for ultrafast optical pulse simulations,” IEEE Trans. Antennas Propag. 45, 375–391 (1997).
[CrossRef]

D. W. Prather, M. S. Mirotznik, J. N. Mait, “Boundary integral methods applied to the analysis of diffractive optical elements,” J. Opt. Soc. Am. A 14, 34–43 (1997).
[CrossRef]

B. Yang, D. Gottlieb, J. S. Hesthaven, “Spectral simulations of electromagnetic wave scattering,” J. Comput. Phys. 134, 216–230 (1997).
[CrossRef]

1996 (3)

M. S. Mirotznik, D. W. Prather, J. N. Mait, “A hybrid finite element-boundary element method for the analysis of diffractive elements,” J. Mod. Opt. 43, 1309–1321 (1996).
[CrossRef]

J. P. Zhang, D. Y. Chu, S. L. Wu, W. G. Bi, R. C. Tiberio, R. M. Joseph, A. Taflove, C. W. Tu, S. T. Ho, “Nanofabrication of 1-D photonic bandgap structures along a photonic wire,” IEEE Photon. Technol. Lett. 8, 491–493 (1996).
[CrossRef]

H. Y. D. Yang, “Finite difference analysis of 2-D photonic crystals,” IEEE Trans. Microwave Theory Tech. 44, 2688–2695 (1996).
[CrossRef]

1994 (1)

1986 (1)

H. Nishihara, S. Ura, T. Suhara, J. Koyama, “An integrated-optic disk pickup device,” J. Lightwave Technol. LT-4, 913–918 (1986).

1985 (1)

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

1982 (1)

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).

Bi, W. G.

J. P. Zhang, D. Y. Chu, S. L. Wu, W. G. Bi, R. C. Tiberio, R. M. Joseph, A. Taflove, C. W. Tu, S. T. Ho, “Nanofabrication of 1-D photonic bandgap structures along a photonic wire,” IEEE Photon. Technol. Lett. 8, 491–493 (1996).
[CrossRef]

Chu, D. Y.

J. P. Zhang, D. Y. Chu, S. L. Wu, W. G. Bi, R. C. Tiberio, R. M. Joseph, A. Taflove, C. W. Tu, S. T. Ho, “Nanofabrication of 1-D photonic bandgap structures along a photonic wire,” IEEE Photon. Technol. Lett. 8, 491–493 (1996).
[CrossRef]

Dinesen, P. G.

P. G. Dinesen, J. S. Hesthaven, J. P. Lynov, L. Lading, “Pseudo-spectral method for the analysis of diffractive optical elements,” submitted to J. Opt. Soc. Am. A.

Gaylord, T. K.

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

M. G. Moharam, T. K. Gaylord, “Diffraction analysis of dielectric surface-relief gratings,” J. Opt. Soc. Am. 72, 1385–1392 (1982).
[CrossRef]

Gottlieb, D.

B. Yang, D. Gottlieb, J. S. Hesthaven, “Spectral simulations of electromagnetic wave scattering,” J. Comput. Phys. 134, 216–230 (1997).
[CrossRef]

B. Yang, D. Gottlieb, J. S. Hesthaven, “On the use of PML ABC’s in spectral time-domain simulations of electromagnetic scattering,” in 13th Annual Review of Progress in Applied Computational Electromagnetics at the Naval Postgraduate School, A. Kishk, A. Glisson, eds., conference proceedings of the Applied Computational Electromagnetics Society (U. Mississippi Press, Jackson, Miss., 1997), pp. 926–933.

Grann, E. B.

Hansen, C. D.

Hesthaven, J. S.

B. Yang, D. Gottlieb, J. S. Hesthaven, “Spectral simulations of electromagnetic wave scattering,” J. Comput. Phys. 134, 216–230 (1997).
[CrossRef]

P. G. Dinesen, J. S. Hesthaven, J. P. Lynov, L. Lading, “Pseudo-spectral method for the analysis of diffractive optical elements,” submitted to J. Opt. Soc. Am. A.

B. Yang, D. Gottlieb, J. S. Hesthaven, “On the use of PML ABC’s in spectral time-domain simulations of electromagnetic scattering,” in 13th Annual Review of Progress in Applied Computational Electromagnetics at the Naval Postgraduate School, A. Kishk, A. Glisson, eds., conference proceedings of the Applied Computational Electromagnetics Society (U. Mississippi Press, Jackson, Miss., 1997), pp. 926–933.

Ho, S. T.

J. P. Zhang, D. Y. Chu, S. L. Wu, W. G. Bi, R. C. Tiberio, R. M. Joseph, A. Taflove, C. W. Tu, S. T. Ho, “Nanofabrication of 1-D photonic bandgap structures along a photonic wire,” IEEE Photon. Technol. Lett. 8, 491–493 (1996).
[CrossRef]

Jin, J.

J. Jin, The Finite Element Method in Electromagnetics (Wiley, New York, 1993).

Joseph, R. M.

J. P. Zhang, D. Y. Chu, S. L. Wu, W. G. Bi, R. C. Tiberio, R. M. Joseph, A. Taflove, C. W. Tu, S. T. Ho, “Nanofabrication of 1-D photonic bandgap structures along a photonic wire,” IEEE Photon. Technol. Lett. 8, 491–493 (1996).
[CrossRef]

Koyama, J.

H. Nishihara, S. Ura, T. Suhara, J. Koyama, “An integrated-optic disk pickup device,” J. Lightwave Technol. LT-4, 913–918 (1986).

Lading, L.

L. Lading, C. D. Hansen, E. Rasmussen, “Surface light scattering: integrated technology and signal processing,” Appl. Opt. 36, 7593–7600 (1997).
[CrossRef]

P. G. Dinesen, J. S. Hesthaven, J. P. Lynov, L. Lading, “Pseudo-spectral method for the analysis of diffractive optical elements,” submitted to J. Opt. Soc. Am. A.

Lynov, J. P.

P. G. Dinesen, J. S. Hesthaven, J. P. Lynov, L. Lading, “Pseudo-spectral method for the analysis of diffractive optical elements,” submitted to J. Opt. Soc. Am. A.

Mait, J. N.

D. W. Prather, M. S. Mirotznik, J. N. Mait, “Boundary integral methods applied to the analysis of diffractive optical elements,” J. Opt. Soc. Am. A 14, 34–43 (1997).
[CrossRef]

M. S. Mirotznik, D. W. Prather, J. N. Mait, “A hybrid finite element-boundary element method for the analysis of diffractive elements,” J. Mod. Opt. 43, 1309–1321 (1996).
[CrossRef]

Mirotznik, M. S.

D. W. Prather, M. S. Mirotznik, J. N. Mait, “Boundary integral methods applied to the analysis of diffractive optical elements,” J. Opt. Soc. Am. A 14, 34–43 (1997).
[CrossRef]

M. S. Mirotznik, D. W. Prather, J. N. Mait, “A hybrid finite element-boundary element method for the analysis of diffractive elements,” J. Mod. Opt. 43, 1309–1321 (1996).
[CrossRef]

Moharam, M. G.

Nishihara, H.

H. Nishihara, S. Ura, T. Suhara, J. Koyama, “An integrated-optic disk pickup device,” J. Lightwave Technol. LT-4, 913–918 (1986).

Pommet, D. A.

Prather, D. W.

D. W. Prather, M. S. Mirotznik, J. N. Mait, “Boundary integral methods applied to the analysis of diffractive optical elements,” J. Opt. Soc. Am. A 14, 34–43 (1997).
[CrossRef]

M. S. Mirotznik, D. W. Prather, J. N. Mait, “A hybrid finite element-boundary element method for the analysis of diffractive elements,” J. Mod. Opt. 43, 1309–1321 (1996).
[CrossRef]

Ramo, S.

S. Ramo, J. R. Whinnery, T. Van Duzer, Fields and Waves in Communication Electronics, 3rd ed. (Wiley, New York, 1993).

Rasmussen, E.

Suhara, T.

H. Nishihara, S. Ura, T. Suhara, J. Koyama, “An integrated-optic disk pickup device,” J. Lightwave Technol. LT-4, 913–918 (1986).

Taflove, A.

J. P. Zhang, D. Y. Chu, S. L. Wu, W. G. Bi, R. C. Tiberio, R. M. Joseph, A. Taflove, C. W. Tu, S. T. Ho, “Nanofabrication of 1-D photonic bandgap structures along a photonic wire,” IEEE Photon. Technol. Lett. 8, 491–493 (1996).
[CrossRef]

A. Taflove, Computational Electrodynamics: The Finite-Difference Time-Domain Method (Artech House, Boston, Mass., 1995).

Tamir, T.

T. Tamir, “Beam and waveguide couplers,” in Integrated Optics, 2nd. ed., T. Tamir, ed. (Springer-Verlag, New York, 1979), pp. 309–327.

Tiberio, R. C.

J. P. Zhang, D. Y. Chu, S. L. Wu, W. G. Bi, R. C. Tiberio, R. M. Joseph, A. Taflove, C. W. Tu, S. T. Ho, “Nanofabrication of 1-D photonic bandgap structures along a photonic wire,” IEEE Photon. Technol. Lett. 8, 491–493 (1996).
[CrossRef]

Tu, C. W.

J. P. Zhang, D. Y. Chu, S. L. Wu, W. G. Bi, R. C. Tiberio, R. M. Joseph, A. Taflove, C. W. Tu, S. T. Ho, “Nanofabrication of 1-D photonic bandgap structures along a photonic wire,” IEEE Photon. Technol. Lett. 8, 491–493 (1996).
[CrossRef]

Ura, S.

H. Nishihara, S. Ura, T. Suhara, J. Koyama, “An integrated-optic disk pickup device,” J. Lightwave Technol. LT-4, 913–918 (1986).

Van Duzer, T.

S. Ramo, J. R. Whinnery, T. Van Duzer, Fields and Waves in Communication Electronics, 3rd ed. (Wiley, New York, 1993).

Wang, J. J. H.

J. J. H. Wang, Generalized Moment Methods in Electromagnetics: Formulation and Computer Solution of Integral Equations (Wiley, New York, 1991).

Whinnery, J. R.

S. Ramo, J. R. Whinnery, T. Van Duzer, Fields and Waves in Communication Electronics, 3rd ed. (Wiley, New York, 1993).

Wu, S. L.

J. P. Zhang, D. Y. Chu, S. L. Wu, W. G. Bi, R. C. Tiberio, R. M. Joseph, A. Taflove, C. W. Tu, S. T. Ho, “Nanofabrication of 1-D photonic bandgap structures along a photonic wire,” IEEE Photon. Technol. Lett. 8, 491–493 (1996).
[CrossRef]

Yang, B.

B. Yang, D. Gottlieb, J. S. Hesthaven, “Spectral simulations of electromagnetic wave scattering,” J. Comput. Phys. 134, 216–230 (1997).
[CrossRef]

B. Yang, D. Gottlieb, J. S. Hesthaven, “On the use of PML ABC’s in spectral time-domain simulations of electromagnetic scattering,” in 13th Annual Review of Progress in Applied Computational Electromagnetics at the Naval Postgraduate School, A. Kishk, A. Glisson, eds., conference proceedings of the Applied Computational Electromagnetics Society (U. Mississippi Press, Jackson, Miss., 1997), pp. 926–933.

Yang, H. Y. D.

H. Y. D. Yang, “Finite difference analysis of 2-D photonic crystals,” IEEE Trans. Microwave Theory Tech. 44, 2688–2695 (1996).
[CrossRef]

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).

Zhang, J. P.

J. P. Zhang, D. Y. Chu, S. L. Wu, W. G. Bi, R. C. Tiberio, R. M. Joseph, A. Taflove, C. W. Tu, S. T. Ho, “Nanofabrication of 1-D photonic bandgap structures along a photonic wire,” IEEE Photon. Technol. Lett. 8, 491–493 (1996).
[CrossRef]

Ziolkowski, R. W.

R. W. Ziolkowski, “The incorporation of microscopic material models into the FDTD approach for ultrafast optical pulse simulations,” IEEE Trans. Antennas Propag. 45, 375–391 (1997).
[CrossRef]

Appl. Opt. (1)

IEEE Photon. Technol. Lett. (1)

J. P. Zhang, D. Y. Chu, S. L. Wu, W. G. Bi, R. C. Tiberio, R. M. Joseph, A. Taflove, C. W. Tu, S. T. Ho, “Nanofabrication of 1-D photonic bandgap structures along a photonic wire,” IEEE Photon. Technol. Lett. 8, 491–493 (1996).
[CrossRef]

IEEE Trans. Antennas Propag. (2)

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).

R. W. Ziolkowski, “The incorporation of microscopic material models into the FDTD approach for ultrafast optical pulse simulations,” IEEE Trans. Antennas Propag. 45, 375–391 (1997).
[CrossRef]

IEEE Trans. Microwave Theory Tech. (1)

H. Y. D. Yang, “Finite difference analysis of 2-D photonic crystals,” IEEE Trans. Microwave Theory Tech. 44, 2688–2695 (1996).
[CrossRef]

J. Comput. Phys. (1)

B. Yang, D. Gottlieb, J. S. Hesthaven, “Spectral simulations of electromagnetic wave scattering,” J. Comput. Phys. 134, 216–230 (1997).
[CrossRef]

J. Lightwave Technol. (1)

H. Nishihara, S. Ura, T. Suhara, J. Koyama, “An integrated-optic disk pickup device,” J. Lightwave Technol. LT-4, 913–918 (1986).

J. Mod. Opt. (1)

M. S. Mirotznik, D. W. Prather, J. N. Mait, “A hybrid finite element-boundary element method for the analysis of diffractive elements,” J. Mod. Opt. 43, 1309–1321 (1996).
[CrossRef]

J. Opt. Soc. Am. (1)

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

Proc. IEEE (1)

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

Other (7)

J. J. H. Wang, Generalized Moment Methods in Electromagnetics: Formulation and Computer Solution of Integral Equations (Wiley, New York, 1991).

J. Jin, The Finite Element Method in Electromagnetics (Wiley, New York, 1993).

P. G. Dinesen, J. S. Hesthaven, J. P. Lynov, L. Lading, “Pseudo-spectral method for the analysis of diffractive optical elements,” submitted to J. Opt. Soc. Am. A.

A. Taflove, Computational Electrodynamics: The Finite-Difference Time-Domain Method (Artech House, Boston, Mass., 1995).

B. Yang, D. Gottlieb, J. S. Hesthaven, “On the use of PML ABC’s in spectral time-domain simulations of electromagnetic scattering,” in 13th Annual Review of Progress in Applied Computational Electromagnetics at the Naval Postgraduate School, A. Kishk, A. Glisson, eds., conference proceedings of the Applied Computational Electromagnetics Society (U. Mississippi Press, Jackson, Miss., 1997), pp. 926–933.

S. Ramo, J. R. Whinnery, T. Van Duzer, Fields and Waves in Communication Electronics, 3rd ed. (Wiley, New York, 1993).

T. Tamir, “Beam and waveguide couplers,” in Integrated Optics, 2nd. ed., T. Tamir, ed. (Springer-Verlag, New York, 1979), pp. 309–327.

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Figures (13)

Fig. 1
Fig. 1

FDTD two-dimensional computational grid surrounding a binary diffractive structure embedded in a dielectric waveguiding material. The substrate, the film layer, and air have refractive indices n 3 (n3), n 2 (n2), and n 1 (n1), respectively. d is the thickness of film layer in the region outside the grating region. The (x, y, z) coordinate system is used to define the geometric surface and in connection with the transformation surface Ω, where we have secondary auxiliary electromagnetic sources. These are needed for the solution of the exact radiation integrals. A transparent nonphysical rectangular box is used to verify energy conservation. Pd, Ptr, Prd, and Prb are the time-averaged power intensity flows for the upward-diffracted, transmitted, downward-reflected, and backward-reflected power, respectively.

Fig. 2
Fig. 2

Main diffraction angle (ν = -1) in the far-field region, shown as a function of the grating-period-to-free-space wavelength ratio Λ/λ for different grating profiles. The amplitude-to-film-layer thickness ratio A/d is 0.3 for the finite structures (10λ long). The refractive indices are n 1 = 1, n 2 = 1.58, and n 3 = 1.47. The film-layer to free-space wavelength ratio d/λ is 0.65.

Fig. 3
Fig. 3

Normalized far-field power radiation pattern as a function of the far-field observation angle θ for surface-relief gratings with sine profiles of different lengths in a two-layered dielectric slab waveguide. The grating-period-to-free-space wavelength ratio Λ/λ is 1. The grating-amplitude-to-film-layer thickness ratio A/d is 0.3. The refractive indices are n 1 = 1 for free space, n 2 = 1.58 for the film layer, and n 3 = 1.47 for the semi-infinite substrate. The film-layer-thickness-to-free-space wavelength ratio d/λ is 0.65.

Fig. 4
Fig. 4

Normalized far-field power radiation pattern as a function of the far-field observation angle θ for surface-relief gratings with sine profiles of different grating-period-to-free-space wavelength ratios Λ/λ in a two-layered dielectric slab waveguide. The grating-amplitude-to-film-layer thickness ratio A/d is 0.307. The refractive indices are n 1 = 1 for free space, n 2 = 1.58 for the film layer, and n 3 = 1.47 for the semi-infinite substrate. The film-layer-thickness-to-free-space wavelength ratio d/λ is 0.65.

Fig. 5
Fig. 5

Normalized far-field power radiation pattern as a function of the far-field observation angle θ for surface-relief gratings with binary profiles of different grating-period-to-free-space wavelength ratios Λ/λ in a two-layered dielectric slab waveguide. The grating-amplitude-to-film-layer thickness ratio A/d is 0.307. The refractive indices are n 1 = 1 for free space, n 2 = 1.58 for the film layer, and n 3 = 1.47 for the semi-infinite substrate. The film-layer-thickness-to-free-space wavelength ratio d/λ is 0.65.

Fig. 6
Fig. 6

Normalized far-field power radiation pattern as a function of the far-field observation angle θ for surface-relief gratings with positively blazed profiles of different grating-period-to-free-space wavelength ratios Λ/λ in a two-layered dielectric slab waveguide. The grating-amplitude-to-film-layer thickness ratio A/d is 0.307. The refractive indices are n 1 = 1 for free space, n 2 = 1.58 for the film layer, and n 3 = 1.47 for the semi-infinite substrate. The film-layer-thickness-to-free-space wavelength ratio d/λ is 0.65.

Fig. 7
Fig. 7

Normalized far-field power radiation pattern as a function of the far-field observation angle θ for surface-relief gratings with negatively blazed profiles of different grating-period-to-free-space wavelength ratios Λ/λ in a two-layered dielectric slab waveguide. The grating-amplitude-to-film-layer thickness ratio A/d is 0.307. The refractive indices are n 1 = 1 for free space, n 2 = 1.58 for the film layer, and n 3 = 1.47 for the semi-infinite substrate. The film-layer-thickness-to-free-space wavelength ratio d/λ is 0.65.

Fig. 8
Fig. 8

Upward-diffracted, transmitted, and downward-reflected and backward-reflected power through the sides of the fictitious box for the finite periodic grating with a sine profile as a function of the amplitude-to-film-layer thickness ratio. The grating-period-to-free-space wavelength ratio Λ/λ is 1. The grating length is 10λ.

Fig. 9
Fig. 9

Upward-diffracted, transmitted, and downward-reflected and backward-reflected power through the sides of the fictitious box for the finite periodic grating with binary profile as a function of the amplitude to film-layer thickness ratio. The grating-period-to-free-space wavelength ratio Λ/λ is 1. The grating length is 10λ.

Fig. 10
Fig. 10

Upward-diffracted, transmitted, and downward-reflected and backward-reflected power through the sides of the fictitious box for the finite periodic grating with positively blazed profile as a function of the amplitude-to-film-layer thickness ratio. The grating-period-to-free-space wavelength ratio Λ/λ is 1. The grating length is 10λ.

Fig. 11
Fig. 11

Upward-diffracted, transmitted, and downward-reflected and backward-reflected power through the sides of the fictitious box for the finite periodic grating with negatively blazed profile as a function of the amplitude-to-film-layer thickness ratio. The grating-period-to-free-space wavelength ratio Λ/λ is 1. The grating length is 10λ.

Fig. 12
Fig. 12

Normalized far-field power radiation pattern as a function of the far-field observation angle θ for surface-relief gratings with sine profiles of different amplitude-to-film-layer thickness ratios A/d in a two-layered dielectric slab waveguide. The grating-period-to-free-space wavelength ratio Λ/λ is 1. The refractive indices are n 1 = 1 for free space, n 2 = 1.58 for the film layer, and n 3 = 1.47 for the semi-infinite substrate. The film-layer-thickness-to-free-space wavelength ratio d/λ is 0.65.

Fig. 13
Fig. 13

Normalized far-field power radiation pattern as a function of the far-field observation angle θ for surface-relief gratings with positively blazed profiles of different amplitude-to-film-layer thickness ratios A/d in a two-layered dielectric slab waveguide. The grating-period to free-space wavelength ratio Λ/λ is 1. The refractive indices are n 1 = 1 for free space, n 2 = 1.58 for the film layer, and n 3 = 1.47 for the semi-infinite substrate. The film-layer-thickness-to-free-space wavelength ratio d/λ is 0.65.

Equations (27)

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

J=nˆ×H
M=-nˆ×E.
Pn=12Ω nˆ·ReE×H*dΩ.
gr, rΩ=exp-jk|r-rΩ||r-rΩ|,
Ir=½|ReEr×H*r|,
Er=ΩMrΩ×rgr, rΩ-1jω0 r×JrΩ×rgr, rΩdΩ,
Hr=Ω-JrΩ×rgr, rΩ-1jωμ0 r×MrΩ×rgr, rΩdΩ.
Pθ=-½ ReEϕHθ*-EθHϕ*
Eyt=1nx, z2Hxz-Hzx,
Hxt=Eyz,
Hzt=-Eyx,
Es=Etotal-Ei,
tanhd=h p+qh2-pq,
Eyt=1nx, z2Hxz-Hzx-Qz+μQx,
Hxt=Eyz-2Hx-Pz,
Hzt=-Eyx-2μHz-μPx,
Pzt=Hx,
Pxt=μHz,
Qzt=-Qz-Hx,
Qxt=-μQx-Hz,
x=0
x=C|x-xi|n
2xt±np0c2t2±p2cn2z2A=0,
2zt±np0c2t2±p2cn2x2A=0.
ΔtΔc2,
Δλ10,
sin θ=1n1neff+νΛ,

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