Abstract

The diffraction characteristics of a volume grating (VG) illuminated by a three-dimensional (3-D) converging–diverging Gaussian beam at conical incidence are investigated by applying 3-D finite-beam (FB) rigorous coupled-wave analysis (RCWA) based on the conventional 3-D RCWA in conjunction with two-dimensional plane-wave decomposition. The Gaussian beam is assumed to have an arbitrary incidence angle, an arbitrary azimuthal angle, and any linear polarization. The two cases with linear polarizations of the central beam of the Gaussian (EK and HK) are investigated. The diffraction efficiencies and the diffracted beam profiles for both unslanted VGs and slanted VGs (designed for substrate-mode optical interconnects) are presented. In general, the diffraction efficiencies of a converging–diverging spherical Gaussian beam diffracted by both unslanted VGs and slanted VGs increase and approach the central-beam results as the refractive-index modulation increases.

© 2005 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. J.-H. Yeh, R. K. Kostuk, “Substrate-mode holograms used in optical interconnects: design issues,” Appl. Opt. 34, 3152–3164 (1995).
    [CrossRef] [PubMed]
  2. J.-H. Yeh, R. K. Kostuk, “Free-space holographic optical interconnects for board-to-board and chip-to-chip interconnects,” Opt. Lett. 21, 1274–1276 (1996).
    [CrossRef] [PubMed]
  3. T. Tanaka, H. Takahashi, Y. Hibino, T. Hashimoto, A. Himeno, Y. Yamada, Y. Tohmori, “Hybrid external cavity lasers composed of spot-size converter integrated LDs and UV written Bragg grating in a planar lightwave circuit on Si,” IEICE Trans. Electron. E83-C, 875–883 (2000).
  4. S. M. Schultz, E. N. Glytsis, T. K. Gaylord, “Design, fabrication, and performance of preferential-order volume grating waveguide couplers,” Appl. Opt. 39, 1223–1232 (2000).
    [CrossRef]
  5. R. A. Villalaz, E. N. Glytsis, T. K. Gaylord, “Volume grating couplers: polarization and loss effect,” Appl. Opt. 41, 5223–5229 (2002).
    [CrossRef] [PubMed]
  6. S.-D. Wu, E. N. Glytsis, “Volume holographic grating couplers: rigorous analysis using the finite-difference frequency-domain method,” Appl. Opt. 43, 1009–1023 (2004).
    [CrossRef] [PubMed]
  7. B. Wang, J. Jiang, G. P. Nordin, “Compact slanted grating couplers,” Opt. Express 12, 3313–3326 (2004).
    [CrossRef] [PubMed]
  8. S. L. Chuang, J. A. Kong, “Wave scattering from a periodic dielectric surface for a general angle of incidence,” Radio Sci. 17, 545–557 (1982).
    [CrossRef]
  9. M. G. Moharam, T. K. Gaylord, “Three-dimensional vector coupled-wave analysis of planar-grating diffraction,” J. Opt. Soc. Am. 73, 1105–1112 (1983).
    [CrossRef]
  10. M. G. Moharam, E. B. Grann, D. A. Pommet, 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).
    [CrossRef]
  11. E. Popov, L. Mashev, “Conical diffraction mounting generalization of a rigorous differential method,” J. Opt. 17, 175–180 (1986).
    [CrossRef]
  12. S. D. Gupta, “Theoretical study of plasma resonance absorption in conical diffraction,” J. Opt. Soc. Am. B 4, 1893–1898 (1987).
    [CrossRef]
  13. R. A. Depine, “Conformal mapping method for finitely conducting diffraction gratings in conical mountings,” Optik (Stuttgart) 81, 95–102 (1989).
  14. M. Gigli, R. A. Depine, “Conical diffraction from uniaxial gratings,” J. Mod. Opt. 42, 1281–1299 (1995).
    [CrossRef]
  15. M. Abe, A. Koshiba, “Three-dimensional diffraction analysis of dielectric surface-relief gratings,” J. Opt. Soc. Am. A 11, 2038–2044 (1994).
    [CrossRef]
  16. P. Cornet, J. Chandezon, C. Faure, “Conical diffraction of a plane-wave by an inclined parallel-plate grating,” J. Opt. Soc. Am. A 14, 437–449 (1997).
    [CrossRef]
  17. M. Ohki, H. Tateno, S. Kozaki, “T-matrix analysis of the electromagnetic wave diffraction from a metallic Fourier grating for an arbitrary incidence and polarization,” Int. J. Electron. 85, 787–796 (1998).
    [CrossRef]
  18. M. G. Moharam, T. K. Gaylord, R. Magnusson, “Bragg diffraction of finite beams by thick gratings,” J. Opt. Soc. Am. 70, 300–304 (1980).
    [CrossRef]
  19. G. Notni, R. Kowarschik, “Simultaneous diffraction of two finite waves at a nonuniform mixed dynamic transmission grating,” J. Opt. Soc. Am. A 7, 1475–1482 (1990).
    [CrossRef]
  20. P. Boffi, J. Osmond, D. Piccinin, M. C. Ubaldi, M. Martinelli, “Diffraction of optical communication Gaussian beams by volume gratings: comparison of simulations and experimental results,” Appl. Opt. 43, 3854–3865 (2004).
    [CrossRef] [PubMed]
  21. R.-S. Chu, T. Tamir, “Bragg diffraction of Gaussian beams by periodically modulated media,” J. Opt. Soc. Am. 66, 220–226 (1976).
    [CrossRef]
  22. R.-S. Chu, T. Tamir, “Diffraction of Gaussian beams by periodically modulated media for incidence close to a Bragg angle,” J. Opt. Soc. Am. 66, 1438–1440 (1976).
    [CrossRef]
  23. B. Benlarbi, P. St. J. Russell, L. Solymar, “Diffraction of a Gaussian beam incident upon a thick phase grating,” Int. J. Electron. 52, 209–216 (1982).
    [CrossRef]
  24. B. Benlarbi, P. St. J. Russell, L. Solymar, “Bragg diffraction of Gaussian beams by thick gratings: numerical evaluations by plane-wave decomposition,” Appl. Phys. B: Photophys. Laser Chem. 28, 383–390 (1982).
    [CrossRef]
  25. Em. E. Kriezis, P. K. Pandelakis, A. G. Papagiannakis, “Diffraction of a Gaussian beam from a periodic planar screen,” J. Opt. Soc. Am. A 11, 630–636 (1994).
    [CrossRef]
  26. M. D. McNeill, T.-C. Poon, “Gaussian-beam profile shaping by acousto-optic Bragg diffraction,” Appl. Opt. 33, 4508–4514 (1994).
    [CrossRef] [PubMed]
  27. D. C. Skigin, R. A. Depine, “Model theory for diffraction from a dielectric aperture with arbitrarily shaped corrugations,” Opt. Commun. 149, 117–126 (1998).
    [CrossRef]
  28. O. Mata-Mendez, F. Chavez-Rivas, “Diffraction of a Gaussian and Hermite Gaussian beams by finite gratings,” J. Opt. Soc. Am. A 18, 537–545 (2001).
    [CrossRef]
  29. J. Sumaya-Martines, O. Mata-Mendez, F. Chavez-Rivas, “Rigorous theory of the diffraction of Gaussian beams by finite gratings: TE polarizations,” J. Opt. Soc. Am. A 20, 827–835 (2003).
    [CrossRef]
  30. A. E. Siegman, “Bragg diffraction of a Gaussian beam by a crossed-Gaussian volume grating,” J. Opt. Soc. Am. 67, 545–550 (1977).
    [CrossRef]
  31. A. Y. Hamad, J. P. Wicksted, “Volume grating produced by intersecting Gaussian beams in an absorbing medium: a Bragg diffraction model,” Opt. Commun. 138, 354–364 (1997).
    [CrossRef]
  32. M. G. Moharam, T. K. Gaylord, R. Magnusson, “Diffraction characteristics of three-dimensional crossed-beam volume gratings,” J. Opt. Soc. Am. 70, 437–442 (1980).
    [CrossRef]
  33. M. R. Wang, “Analysis and observation of finite beam Bragg diffraction by a thick planar phase grating,” Appl. Opt. 35, 582–592 (1996).
    [CrossRef] [PubMed]
  34. G. D. Landry, T. A. Maldonado, “Gaussian beam transmission and reflection from a general anisotropic multilayer structure,” Appl. Opt. 35, 5870–5879 (1996).
    [CrossRef] [PubMed]
  35. H. Kogelnik, “Coupled-wave theory for thick hologram gratings,” Bell Syst. Tech. J. 48, 2909–2947 (1969).
    [CrossRef]
  36. S.-D. Wu, E. N. Glytsis, “Holographic grating formation in photopolymers: analysis and experimental results based on a nonlocal diffusion model and rigorous coupled-wave analysis,” J. Opt. Soc. Am. B 20, 1177–1188 (2003).
    [CrossRef]
  37. S.-D. Wu, E. N. Glytsis, “Characteristics of DuPont photopolymers for slanted holographic grating formations,” J. Opt. Soc. Am. B 21, 1722–1731 (2004).
    [CrossRef]

2004

2003

2002

2001

2000

T. Tanaka, H. Takahashi, Y. Hibino, T. Hashimoto, A. Himeno, Y. Yamada, Y. Tohmori, “Hybrid external cavity lasers composed of spot-size converter integrated LDs and UV written Bragg grating in a planar lightwave circuit on Si,” IEICE Trans. Electron. E83-C, 875–883 (2000).

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

1998

M. Ohki, H. Tateno, S. Kozaki, “T-matrix analysis of the electromagnetic wave diffraction from a metallic Fourier grating for an arbitrary incidence and polarization,” Int. J. Electron. 85, 787–796 (1998).
[CrossRef]

D. C. Skigin, R. A. Depine, “Model theory for diffraction from a dielectric aperture with arbitrarily shaped corrugations,” Opt. Commun. 149, 117–126 (1998).
[CrossRef]

1997

A. Y. Hamad, J. P. Wicksted, “Volume grating produced by intersecting Gaussian beams in an absorbing medium: a Bragg diffraction model,” Opt. Commun. 138, 354–364 (1997).
[CrossRef]

P. Cornet, J. Chandezon, C. Faure, “Conical diffraction of a plane-wave by an inclined parallel-plate grating,” J. Opt. Soc. Am. A 14, 437–449 (1997).
[CrossRef]

1996

1995

1994

1990

1989

R. A. Depine, “Conformal mapping method for finitely conducting diffraction gratings in conical mountings,” Optik (Stuttgart) 81, 95–102 (1989).

1987

1986

E. Popov, L. Mashev, “Conical diffraction mounting generalization of a rigorous differential method,” J. Opt. 17, 175–180 (1986).
[CrossRef]

1983

1982

S. L. Chuang, J. A. Kong, “Wave scattering from a periodic dielectric surface for a general angle of incidence,” Radio Sci. 17, 545–557 (1982).
[CrossRef]

B. Benlarbi, P. St. J. Russell, L. Solymar, “Diffraction of a Gaussian beam incident upon a thick phase grating,” Int. J. Electron. 52, 209–216 (1982).
[CrossRef]

B. Benlarbi, P. St. J. Russell, L. Solymar, “Bragg diffraction of Gaussian beams by thick gratings: numerical evaluations by plane-wave decomposition,” Appl. Phys. B: Photophys. Laser Chem. 28, 383–390 (1982).
[CrossRef]

1980

1977

1976

1969

H. Kogelnik, “Coupled-wave theory for thick hologram gratings,” Bell Syst. Tech. J. 48, 2909–2947 (1969).
[CrossRef]

Abe, M.

Benlarbi, B.

B. Benlarbi, P. St. J. Russell, L. Solymar, “Diffraction of a Gaussian beam incident upon a thick phase grating,” Int. J. Electron. 52, 209–216 (1982).
[CrossRef]

B. Benlarbi, P. St. J. Russell, L. Solymar, “Bragg diffraction of Gaussian beams by thick gratings: numerical evaluations by plane-wave decomposition,” Appl. Phys. B: Photophys. Laser Chem. 28, 383–390 (1982).
[CrossRef]

Boffi, P.

Chandezon, J.

Chavez-Rivas, F.

Chu, R.-S.

Chuang, S. L.

S. L. Chuang, J. A. Kong, “Wave scattering from a periodic dielectric surface for a general angle of incidence,” Radio Sci. 17, 545–557 (1982).
[CrossRef]

Cornet, P.

Depine, R. A.

D. C. Skigin, R. A. Depine, “Model theory for diffraction from a dielectric aperture with arbitrarily shaped corrugations,” Opt. Commun. 149, 117–126 (1998).
[CrossRef]

M. Gigli, R. A. Depine, “Conical diffraction from uniaxial gratings,” J. Mod. Opt. 42, 1281–1299 (1995).
[CrossRef]

R. A. Depine, “Conformal mapping method for finitely conducting diffraction gratings in conical mountings,” Optik (Stuttgart) 81, 95–102 (1989).

Faure, C.

Gaylord, T. K.

Gigli, M.

M. Gigli, R. A. Depine, “Conical diffraction from uniaxial gratings,” J. Mod. Opt. 42, 1281–1299 (1995).
[CrossRef]

Glytsis, E. N.

Grann, E. B.

Gupta, S. D.

Hamad, A. Y.

A. Y. Hamad, J. P. Wicksted, “Volume grating produced by intersecting Gaussian beams in an absorbing medium: a Bragg diffraction model,” Opt. Commun. 138, 354–364 (1997).
[CrossRef]

Hashimoto, T.

T. Tanaka, H. Takahashi, Y. Hibino, T. Hashimoto, A. Himeno, Y. Yamada, Y. Tohmori, “Hybrid external cavity lasers composed of spot-size converter integrated LDs and UV written Bragg grating in a planar lightwave circuit on Si,” IEICE Trans. Electron. E83-C, 875–883 (2000).

Hibino, Y.

T. Tanaka, H. Takahashi, Y. Hibino, T. Hashimoto, A. Himeno, Y. Yamada, Y. Tohmori, “Hybrid external cavity lasers composed of spot-size converter integrated LDs and UV written Bragg grating in a planar lightwave circuit on Si,” IEICE Trans. Electron. E83-C, 875–883 (2000).

Himeno, A.

T. Tanaka, H. Takahashi, Y. Hibino, T. Hashimoto, A. Himeno, Y. Yamada, Y. Tohmori, “Hybrid external cavity lasers composed of spot-size converter integrated LDs and UV written Bragg grating in a planar lightwave circuit on Si,” IEICE Trans. Electron. E83-C, 875–883 (2000).

Jiang, J.

Kogelnik, H.

H. Kogelnik, “Coupled-wave theory for thick hologram gratings,” Bell Syst. Tech. J. 48, 2909–2947 (1969).
[CrossRef]

Kong, J. A.

S. L. Chuang, J. A. Kong, “Wave scattering from a periodic dielectric surface for a general angle of incidence,” Radio Sci. 17, 545–557 (1982).
[CrossRef]

Koshiba, A.

Kostuk, R. K.

Kowarschik, R.

Kozaki, S.

M. Ohki, H. Tateno, S. Kozaki, “T-matrix analysis of the electromagnetic wave diffraction from a metallic Fourier grating for an arbitrary incidence and polarization,” Int. J. Electron. 85, 787–796 (1998).
[CrossRef]

Kriezis, Em. E.

Landry, G. D.

Magnusson, R.

Maldonado, T. A.

Martinelli, M.

Mashev, L.

E. Popov, L. Mashev, “Conical diffraction mounting generalization of a rigorous differential method,” J. Opt. 17, 175–180 (1986).
[CrossRef]

Mata-Mendez, O.

McNeill, M. D.

Moharam, M. G.

Nordin, G. P.

Notni, G.

Ohki, M.

M. Ohki, H. Tateno, S. Kozaki, “T-matrix analysis of the electromagnetic wave diffraction from a metallic Fourier grating for an arbitrary incidence and polarization,” Int. J. Electron. 85, 787–796 (1998).
[CrossRef]

Osmond, J.

Pandelakis, P. K.

Papagiannakis, A. G.

Piccinin, D.

Pommet, D. A.

Poon, T.-C.

Popov, E.

E. Popov, L. Mashev, “Conical diffraction mounting generalization of a rigorous differential method,” J. Opt. 17, 175–180 (1986).
[CrossRef]

Russell, P. St. J.

B. Benlarbi, P. St. J. Russell, L. Solymar, “Bragg diffraction of Gaussian beams by thick gratings: numerical evaluations by plane-wave decomposition,” Appl. Phys. B: Photophys. Laser Chem. 28, 383–390 (1982).
[CrossRef]

B. Benlarbi, P. St. J. Russell, L. Solymar, “Diffraction of a Gaussian beam incident upon a thick phase grating,” Int. J. Electron. 52, 209–216 (1982).
[CrossRef]

Schultz, S. M.

Siegman, A. E.

Skigin, D. C.

D. C. Skigin, R. A. Depine, “Model theory for diffraction from a dielectric aperture with arbitrarily shaped corrugations,” Opt. Commun. 149, 117–126 (1998).
[CrossRef]

Solymar, L.

B. Benlarbi, P. St. J. Russell, L. Solymar, “Diffraction of a Gaussian beam incident upon a thick phase grating,” Int. J. Electron. 52, 209–216 (1982).
[CrossRef]

B. Benlarbi, P. St. J. Russell, L. Solymar, “Bragg diffraction of Gaussian beams by thick gratings: numerical evaluations by plane-wave decomposition,” Appl. Phys. B: Photophys. Laser Chem. 28, 383–390 (1982).
[CrossRef]

Sumaya-Martines, J.

Takahashi, H.

T. Tanaka, H. Takahashi, Y. Hibino, T. Hashimoto, A. Himeno, Y. Yamada, Y. Tohmori, “Hybrid external cavity lasers composed of spot-size converter integrated LDs and UV written Bragg grating in a planar lightwave circuit on Si,” IEICE Trans. Electron. E83-C, 875–883 (2000).

Tamir, T.

Tanaka, T.

T. Tanaka, H. Takahashi, Y. Hibino, T. Hashimoto, A. Himeno, Y. Yamada, Y. Tohmori, “Hybrid external cavity lasers composed of spot-size converter integrated LDs and UV written Bragg grating in a planar lightwave circuit on Si,” IEICE Trans. Electron. E83-C, 875–883 (2000).

Tateno, H.

M. Ohki, H. Tateno, S. Kozaki, “T-matrix analysis of the electromagnetic wave diffraction from a metallic Fourier grating for an arbitrary incidence and polarization,” Int. J. Electron. 85, 787–796 (1998).
[CrossRef]

Tohmori, Y.

T. Tanaka, H. Takahashi, Y. Hibino, T. Hashimoto, A. Himeno, Y. Yamada, Y. Tohmori, “Hybrid external cavity lasers composed of spot-size converter integrated LDs and UV written Bragg grating in a planar lightwave circuit on Si,” IEICE Trans. Electron. E83-C, 875–883 (2000).

Ubaldi, M. C.

Villalaz, R. A.

Wang, B.

Wang, M. R.

Wicksted, J. P.

A. Y. Hamad, J. P. Wicksted, “Volume grating produced by intersecting Gaussian beams in an absorbing medium: a Bragg diffraction model,” Opt. Commun. 138, 354–364 (1997).
[CrossRef]

Wu, S.-D.

Yamada, Y.

T. Tanaka, H. Takahashi, Y. Hibino, T. Hashimoto, A. Himeno, Y. Yamada, Y. Tohmori, “Hybrid external cavity lasers composed of spot-size converter integrated LDs and UV written Bragg grating in a planar lightwave circuit on Si,” IEICE Trans. Electron. E83-C, 875–883 (2000).

Yeh, J.-H.

Appl. Opt.

Appl. Phys. B: Photophys. Laser Chem.

B. Benlarbi, P. St. J. Russell, L. Solymar, “Bragg diffraction of Gaussian beams by thick gratings: numerical evaluations by plane-wave decomposition,” Appl. Phys. B: Photophys. Laser Chem. 28, 383–390 (1982).
[CrossRef]

Bell Syst. Tech. J.

H. Kogelnik, “Coupled-wave theory for thick hologram gratings,” Bell Syst. Tech. J. 48, 2909–2947 (1969).
[CrossRef]

IEICE Trans. Electron.

T. Tanaka, H. Takahashi, Y. Hibino, T. Hashimoto, A. Himeno, Y. Yamada, Y. Tohmori, “Hybrid external cavity lasers composed of spot-size converter integrated LDs and UV written Bragg grating in a planar lightwave circuit on Si,” IEICE Trans. Electron. E83-C, 875–883 (2000).

Int. J. Electron.

M. Ohki, H. Tateno, S. Kozaki, “T-matrix analysis of the electromagnetic wave diffraction from a metallic Fourier grating for an arbitrary incidence and polarization,” Int. J. Electron. 85, 787–796 (1998).
[CrossRef]

B. Benlarbi, P. St. J. Russell, L. Solymar, “Diffraction of a Gaussian beam incident upon a thick phase grating,” Int. J. Electron. 52, 209–216 (1982).
[CrossRef]

J. Mod. Opt.

M. Gigli, R. A. Depine, “Conical diffraction from uniaxial gratings,” J. Mod. Opt. 42, 1281–1299 (1995).
[CrossRef]

J. Opt.

E. Popov, L. Mashev, “Conical diffraction mounting generalization of a rigorous differential method,” J. Opt. 17, 175–180 (1986).
[CrossRef]

J. Opt. Soc. Am.

J. Opt. Soc. Am. A

J. Opt. Soc. Am. B

Opt. Commun.

D. C. Skigin, R. A. Depine, “Model theory for diffraction from a dielectric aperture with arbitrarily shaped corrugations,” Opt. Commun. 149, 117–126 (1998).
[CrossRef]

A. Y. Hamad, J. P. Wicksted, “Volume grating produced by intersecting Gaussian beams in an absorbing medium: a Bragg diffraction model,” Opt. Commun. 138, 354–364 (1997).
[CrossRef]

Opt. Express

Opt. Lett.

Optik (Stuttgart)

R. A. Depine, “Conformal mapping method for finitely conducting diffraction gratings in conical mountings,” Optik (Stuttgart) 81, 95–102 (1989).

Radio Sci.

S. L. Chuang, J. A. Kong, “Wave scattering from a periodic dielectric surface for a general angle of incidence,” Radio Sci. 17, 545–557 (1982).
[CrossRef]

Cited By

OSA participates in CrossRef's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

Alert me when this article is cited.


Figures (10)

Fig. 1
Fig. 1

Geometry of a planar VG illuminated by a converging–diverging spherical Gaussian beam with wave vector k at an arbitrary incidence angle θ, at an arbitrary azimuthal angle ϕ, and with an arbitrary linear polarization (specified by the polarization angle Ψ E ). The VG has period Λ, slant angle ϕ g , and thickness d. The refractive indices of the incident region, the grating, and the substrate are n I , n g , and n s , respectively.

Fig. 2
Fig. 2

Configuration of a converging–diverging spherical Gaussian beam at an arbitrary incidence angle θ and at zero azimuthal angle ϕ = 0 ° on the x z plane. w 0 x is the beam radius of the incident beam at the beam waist along the x direction. The two cases with linear polarizations of the central beam of the Gaussian of E K and H K corresponding to the polarization angles Ψ E = 90 ° and Ψ E = 0 ° , respectively, are presented.

Fig. 3
Fig. 3

Diffraction efficiencies of the 1 st forward-diffracted order of an unslanted VG for (a) a converging–diverging spherical Gaussian beam and (b) its corresponding five major subbeams as a function of the refractive-index modulation for the central-beam E K polarization case.

Fig. 4
Fig. 4

(a) 3-D beam profile at z = 10 μ m and (b) its corresponding angular spectrum of the 1 st forward-diffracted order of an unslanted VG for the central-beam E K polarization case with grating Δ n 1 = 0.02 .

Fig. 5
Fig. 5

Diffraction efficiencies of the 1 st forward-diffracted order of an unslanted VG for (a) a converging–diverging spherical Gaussian beam and (b) its corresponding five major subbeams as a function of the refractive-index modulation for central-beam H K polarization case.

Fig. 6
Fig. 6

(a) 3-D beam profile at z = 10 μ m and (b) its corresponding angular spectrum of the 1 th forward-diffracted order of an unslanted VG for the central-beam H K polarization case with grating Δ n 1 = 0.02 .

Fig. 7
Fig. 7

Diffraction efficiencies of the 1 st forward-diffracted order of a slanted VG designed to support a substrate-mode optical interconnect for (a) a converging–diverging spherical Gaussian beam and (b) its corresponding five major subbeams as a function of the refractive-index modulation for the central-beam E K polarization case.

Fig. 8
Fig. 8

(a) 3-D beam profile at z = 10 μ m and (b) its corresponding angular spectrum of the 1 st forward-diffracted order of a slanted VG designed to support a substrate-mode optical interconnect for the central-beam E K polarization case with grating Δ n 1 = 0.02 .

Fig. 9
Fig. 9

Diffraction efficiencies of the 1 st forward-diffracted order of a slanted VG designed to support a substrate-mode optical interconnect for (a) a converging–diverging spherical Gaussian beam and (b) its corresponding five major subbeams as a function of the refractive-index modulation for the central-beam H K polarization case.

Fig. 10
Fig. 10

(a) 3-D beam profile at z = 10 μ m and (b) its corresponding angular spectrum of the 1 st forward-diffracted order of a slanted VG designed to support a substrate-mode optical interconnect for the central-beam H K polarization case with grating Δ n 1 = 0.02 .

Equations (26)

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

ϵ = ϵ 0 + p = 1 ϵ p c cos ( p K r ) + p = 1 ϵ p s sin ( p K r ) ,
E inc = [ w 0 x w x ( z ) ] 1 2 [ w 0 y w y ( z ) ] 1 2 exp { [ ( x w x ( z ) ) 2 + ( y w y ( z ) ) 2 ] } exp { j 1 2 k [ x 2 R x ( z ) + y 2 R y ( z ) ] } × exp { j 1 2 [ tan 1 ( z z 0 x ) + tan 1 ( z z 0 y ) ] } exp ( j k z ) e ̂ = E inc e ̂ ,
w u ( z ) = w 0 u [ 1 + ( z z 0 u ) 2 ] 1 2 ,
R u ( z ) = z [ 1 + ( z 0 u z ) 2 ] ,
z 0 u = π n I λ 0 w 0 u 2 .
e ̂ = e x x ̂ + e y y ̂ + e z z ̂ = ( cos Ψ E cos ϕ cos θ sin Ψ E sin ϕ ) x ̂ + ( cos Ψ E sin ϕ cos θ + sin Ψ E cos ϕ ) y ̂ + ( cos Ψ E sin θ ) z ̂ .
E inc ( x , y , z = 0 ) = m x = M x 2 M x 2 1 m y = M y 2 M y 2 1 F ( k x , m x , k y , m y ) exp [ j ( k x , m x x + k y , m y y ) ] ,
F ( k x , m x , k y , m y ) = 1 M x M y n x = M x 2 M x 2 1 n y = M y 2 M y 2 1 E inc ( x n , y n , z = 0 ) exp [ j ( k x , m x x n + k y , m y y n ) ] ,
θ m x , m y = cos 1 ( k z , m x , m y k 0 n I ) ,
ϕ m x , m y = tan 1 ( k y , m y k x , m x ) .
Ψ E , m x , m y = cos 1 ( e ̂ TM e ̂ m x , m y ) ( e ̂ TM × e ̂ m x , m y ) k e ̂ TM × e ̂ m x , m y k ,
e ̂ m x , m y = ( k ̂ × e ̂ ) × k ̂ m x , m y ( k ̂ × e ̂ ) × k ̂ m x , m y ,
E R ( x , y , z ) = i m x = M x 2 M x 2 1 m y = M x 2 M y 2 1 F ( k x , m x , k y , m y ) R m x , m y , i exp [ j ( k x , m x , i x + k y , m y , i y + k z , m x , m y , i R z ) ] ,
E T ( x , y , z ) = i m x = M x 2 M x 2 1 m y = M y 2 M y 2 1 F ( k x , m x , k y , m y ) T m x , m y , i exp { j [ k x , m x , i x + k y , m y , i y + k z , m x , m y , i T ( z d ) ] } ,
k z , m x , m y , i R = { ( k 0 2 n I 2 k x , m x , i 2 k y , m y , i 2 ) 1 2 for propagation waves + j ( k x , m x , i 2 + k y , m y , i 2 k 0 2 n I 2 ) 1 2 for evanescent waves } ,
k z , m x , m y , i T = { + ( k 0 2 n s 2 k x , m x , i 2 k y , m y , i 2 ) 1 2 for propagation waves j ( k x , m x , i 2 + k y , m y , i 2 k 0 2 n s 2 ) 1 2 for evanescent . waves } .
DE i l = P i l P inc = m x m y F ( k x , m x , k y , m y ) 2 Re ( k z , m x , m y * ) DE m x , m y , i l m x m y F ( k x , m x , k y , m y ) 2 Re ( k z , m x , m y * ) , ( l = R , T ) ,
Δ n 1 p = λ 0 C R C S d ( r ̂ E s ̂ E ) ,
E inc = exp { [ ( x w 0 x ) 2 + ( y w 0 y ) 2 ] } ,
F inc ( k x , k y ) = w 0 x w 0 y 4 π exp { [ ( w 0 x 2 k x ) 2 + ( w 0 y 2 k y ) 2 ] } ,
k = k 0 n I z ̂ ,
k = ± 2 w 0 x x ̂ + [ k 0 2 n I 2 ( 2 w 0 x ) 2 ] 1 2 z ̂ ,
k = ± 2 w 0 y y ̂ + [ k 0 2 n I 2 ( 2 w 0 y ) 2 ] 1 2 z ̂ .
k = R ̿ θ R ̿ ϕ k ,
R ̿ θ = [ cos θ 0 sin θ 0 1 0 sin θ 0 cos θ ] ,
R ̿ ϕ = [ cos ϕ sin ϕ 0 sin ϕ cos ϕ 0 0 0 1 ] .

Metrics