Abstract

The angular sensitivities of slanted volume gratings (VGs) illuminated by three-dimensional (3-D) converging–diverging spherical Gaussian beams for substrate-mode optical interconnects in microelectronics are analyzed by application of 3-D finite-beam rigorous coupled-wave analysis. Angular misalignments about the z, y, and x axes that correspond to yaw, pitch, and roll misalignments resulting from manufacturing tolerances of chips are investigated. Two cases of linear polarization of the central beam of the Gaussian are considered: EK and HK, where K is the grating vector. From worst-case manufacturing tolerances, the ranges of yaw, pitch, and roll misalignment angles are α = ±1.17°, β = ±3.04°, and γ = ±3.04°, respectively. Based on these ranges of misalignment angles, the decreases of diffraction efficiencies for slanted VGs that are due to both the yaw and the roll misalignments are relatively small. However, the efficiency of substrate-mode optical interconnects achieved by slanted VGs could be reduced by 61.04% for EK polarization and by 58.63% for HK polarization because of the pitch misalignment. Thus the performance of a VG optical interconnect is most sensitive to pitch misalignment.

© 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. Q. Huang, P. R. Ashley, “Holographic Bragg grating input–output couplers for polymer waveguides at an 850-nm wavelength,” Appl. Opt. 36, 1198–1203 (1997).
    [CrossRef] [PubMed]
  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. S.-D. Wu, E. N. Glytsis, T. K. Gaylord, “Optimization of finite-length input volume holographic grating couplers illuminated by finite-width incident beams,” Appl. Opt. 44, 4435–4446 (2005).
    [CrossRef] [PubMed]
  8. E. N. Leith, A. Kozma, J. Upatnieks, J. Marks, N. Massey, “Holographic data storage in three-dimensional media,” Appl. Opt. 5, 1303–1311 (1966).
    [CrossRef] [PubMed]
  9. H. Kogelnik, “Coupled-wave theory for thick hologram gratings,” Bell Syst. Tech. J. 48, 2909–2947 (1969).
    [CrossRef]
  10. A. A. Friesem, J. L. Walker, “Thick absorption recording media in holography,” Appl. Opt. 9, 201–214 (1970).
    [CrossRef] [PubMed]
  11. T. Kubota, “Characteristics of thick hologram grating recorded in absorptive medium,” Opt. Acta 25, 1035–1053 (1987).
    [CrossRef]
  12. M. J. Damzen, Y. Matsumoto, G. J. Grofts, R. P. M. Green, “Bragg-selectivity of a volume gain grating,” Opt. Commun. 123, 182–188 (1996).
    [CrossRef]
  13. R.-S. Chu, J.-A. Kong, “Modal theory of spatially periodic media,” IEEE Trans. Microwave Theory Tech. 25, 18–24 (1977).
    [CrossRef]
  14. M. R. Chatterjee, D. D. Reagan, “Examination of beam propagation in misaligned holographic gratings and comparison with the acousto-optic transfer function model for profiled beams,” Opt. Eng. 38, 1113–1121 (1999).
    [CrossRef]
  15. 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]
  16. J. A. Frantz, R. K. Kostuk, D. A. Waldman, “Model of noise-grating selectivity in volume holographic recording materials by use of Monte Carlo simulations,” J. Opt. Soc. Am. A 21, 378–387 (2004).
    [CrossRef]
  17. 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]
  18. S.-D. Wu, T. K. Gaylord, E. N. Glytsis, Y.-M. Wu, “Three-dimensional converging/diverging Gaussian beam diffraction by a volume grating,” J. Opt. Soc. Am. A (to be published).
  19. ITRS, “International technology roadmap for semiconductors 2003: assembly and packaging,” http://public.itrs.net/ (2003), pp. 14–20.

2005 (1)

2004 (2)

2002 (1)

2000 (1)

1999 (1)

M. R. Chatterjee, D. D. Reagan, “Examination of beam propagation in misaligned holographic gratings and comparison with the acousto-optic transfer function model for profiled beams,” Opt. Eng. 38, 1113–1121 (1999).
[CrossRef]

1997 (1)

1996 (3)

1995 (1)

1987 (1)

T. Kubota, “Characteristics of thick hologram grating recorded in absorptive medium,” Opt. Acta 25, 1035–1053 (1987).
[CrossRef]

1983 (1)

1977 (1)

R.-S. Chu, J.-A. Kong, “Modal theory of spatially periodic media,” IEEE Trans. Microwave Theory Tech. 25, 18–24 (1977).
[CrossRef]

1970 (1)

1969 (1)

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

1966 (1)

Ashley, P. R.

Chatterjee, M. R.

M. R. Chatterjee, D. D. Reagan, “Examination of beam propagation in misaligned holographic gratings and comparison with the acousto-optic transfer function model for profiled beams,” Opt. Eng. 38, 1113–1121 (1999).
[CrossRef]

Chu, R.-S.

R.-S. Chu, J.-A. Kong, “Modal theory of spatially periodic media,” IEEE Trans. Microwave Theory Tech. 25, 18–24 (1977).
[CrossRef]

Damzen, M. J.

M. J. Damzen, Y. Matsumoto, G. J. Grofts, R. P. M. Green, “Bragg-selectivity of a volume gain grating,” Opt. Commun. 123, 182–188 (1996).
[CrossRef]

Frantz, J. A.

Friesem, A. A.

Gaylord, T. K.

Glytsis, E. N.

Green, R. P. M.

M. J. Damzen, Y. Matsumoto, G. J. Grofts, R. P. M. Green, “Bragg-selectivity of a volume gain grating,” Opt. Commun. 123, 182–188 (1996).
[CrossRef]

Grofts, G. J.

M. J. Damzen, Y. Matsumoto, G. J. Grofts, R. P. M. Green, “Bragg-selectivity of a volume gain grating,” Opt. Commun. 123, 182–188 (1996).
[CrossRef]

Huang, Q.

Kogelnik, H.

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

Kong, J.-A.

R.-S. Chu, J.-A. Kong, “Modal theory of spatially periodic media,” IEEE Trans. Microwave Theory Tech. 25, 18–24 (1977).
[CrossRef]

Kostuk, R. K.

Kozma, A.

Kubota, T.

T. Kubota, “Characteristics of thick hologram grating recorded in absorptive medium,” Opt. Acta 25, 1035–1053 (1987).
[CrossRef]

Leith, E. N.

Marks, J.

Massey, N.

Matsumoto, Y.

M. J. Damzen, Y. Matsumoto, G. J. Grofts, R. P. M. Green, “Bragg-selectivity of a volume gain grating,” Opt. Commun. 123, 182–188 (1996).
[CrossRef]

Moharam, M. G.

Reagan, D. D.

M. R. Chatterjee, D. D. Reagan, “Examination of beam propagation in misaligned holographic gratings and comparison with the acousto-optic transfer function model for profiled beams,” Opt. Eng. 38, 1113–1121 (1999).
[CrossRef]

Schultz, S. M.

Upatnieks, J.

Villalaz, R. A.

Waldman, D. A.

Walker, J. L.

Wang, M. R.

Wu, S.-D.

Wu, Y.-M.

S.-D. Wu, T. K. Gaylord, E. N. Glytsis, Y.-M. Wu, “Three-dimensional converging/diverging Gaussian beam diffraction by a volume grating,” J. Opt. Soc. Am. A (to be published).

Yeh, J.-H.

Appl. Opt. (9)

Q. Huang, P. R. Ashley, “Holographic Bragg grating input–output couplers for polymer waveguides at an 850-nm wavelength,” Appl. Opt. 36, 1198–1203 (1997).
[CrossRef] [PubMed]

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]

R. A. Villalaz, E. N. Glytsis, T. K. Gaylord, “Volume grating couplers: polarization and loss effect,” Appl. Opt. 41, 5223–5229 (2002).
[CrossRef] [PubMed]

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]

S.-D. Wu, E. N. Glytsis, T. K. Gaylord, “Optimization of finite-length input volume holographic grating couplers illuminated by finite-width incident beams,” Appl. Opt. 44, 4435–4446 (2005).
[CrossRef] [PubMed]

E. N. Leith, A. Kozma, J. Upatnieks, J. Marks, N. Massey, “Holographic data storage in three-dimensional media,” Appl. Opt. 5, 1303–1311 (1966).
[CrossRef] [PubMed]

J.-H. Yeh, R. K. Kostuk, “Substrate-mode holograms used in optical interconnects: design issues,” Appl. Opt. 34, 3152–3164 (1995).
[CrossRef] [PubMed]

A. A. Friesem, J. L. Walker, “Thick absorption recording media in holography,” Appl. Opt. 9, 201–214 (1970).
[CrossRef] [PubMed]

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]

Bell Syst. Tech. J. (1)

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

IEEE Trans. Microwave Theory Tech. (1)

R.-S. Chu, J.-A. Kong, “Modal theory of spatially periodic media,” IEEE Trans. Microwave Theory Tech. 25, 18–24 (1977).
[CrossRef]

J. Opt. Soc. Am. (1)

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

Opt. Acta (1)

T. Kubota, “Characteristics of thick hologram grating recorded in absorptive medium,” Opt. Acta 25, 1035–1053 (1987).
[CrossRef]

Opt. Commun. (1)

M. J. Damzen, Y. Matsumoto, G. J. Grofts, R. P. M. Green, “Bragg-selectivity of a volume gain grating,” Opt. Commun. 123, 182–188 (1996).
[CrossRef]

Opt. Eng. (1)

M. R. Chatterjee, D. D. Reagan, “Examination of beam propagation in misaligned holographic gratings and comparison with the acousto-optic transfer function model for profiled beams,” Opt. Eng. 38, 1113–1121 (1999).
[CrossRef]

Opt. Lett. (1)

Other (2)

S.-D. Wu, T. K. Gaylord, E. N. Glytsis, Y.-M. Wu, “Three-dimensional converging/diverging Gaussian beam diffraction by a volume grating,” J. Opt. Soc. Am. A (to be published).

ITRS, “International technology roadmap for semiconductors 2003: assembly and packaging,” http://public.itrs.net/ (2003), pp. 14–20.

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 (8)

Fig. 1
Fig. 1

Physical implementation of a substrate-mode optical interconnect utilizing a VG to couple an optical signal emitted by a single-mode laser mounted on a printed wiring board into a substrate for the board-to-chip interconnection.

Fig. 2
Fig. 2

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

Fig. 3
Fig. 3

Configurations of a VG at (a) the Bragg condition, (b) angular misalignment about the z axis (corresponding to yaw misalignment) by α, (c) angular misalignment about the y axis (corresponding to pitch misalignment) by β, and (d) angular misalignment about the x axis (corresponding to roll misalignment) by γ.

Fig. 4
Fig. 4

Diffraction efficiencies of −1st forward-diffracted order DE−1T of a slanted VG as a function of yaw misalignment angle α for both central-beam EK polarization and central-beam HK polarization.

Fig. 5
Fig. 5

Diffraction efficiencies of −1st forward-diffracted order DE−1T of a slanted VG for the central subbeam, the ±kx 1/e subbeams, and the ±ky 1/e subbeams of the 3-D converging–diverging spherical Gaussian beam as a function of yaw misalignment angle α for central-beam EK polarization.

Fig. 6
Fig. 6

Diffraction efficiencies of −1st forward-diffracted order DE−1T of a slanted VG as a function of pitch misalignment angle β for both central-beam EK polarization and central-beam HK polarization.

Fig. 7
Fig. 7

Diffraction efficiencies of −1st forward-diffracted order DE−1T of a slanted VG as a function of roll misalignment angle γ for both central-beam EK polarization and central-beam HK polarization.

Fig. 8
Fig. 8

Diffraction efficiencies of the −1st forward-diffracted order DE−1T of a slanted VG for the central subbeam, the ±kx 1/e subbeams, and the ±ky 1/e subbeams of the 3-D converging–diverging spherical Gaussian beam as a function of roll misalignment angle γ for central-beam EK polarization.

Tables (1)

Tables Icon

Table 1 Typical Tolerance for Each Component in a GSI Chip

Equations (3)

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

ɛ = ɛ 0 + p = 1 ɛ p c cos ( p K · r ) + p = 1 ɛ p s sin ( p K · r ) ,
E inc = exp { - [ ( x w 0 x ) 2 + ( y w 0 y ) 2 ] } e ^ ,
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 cos ϕ cos θ + sin Ψ E sin ϕ ) y ^ + ( cos Ψ E sin θ ) z ^ .

Metrics