Abstract

The wavelength response of a waveguide volume grating coupler (WVGC) is analyzed for coupling light from a slab waveguide into the superstrate. A leaky-mode approach is used in conjunction with rigorous coupled-wave analysis. A quantitative theoretical study of the effect of index modulation, waveguide index, and grating thickness on the wavelength bandpass of a WVGC is also presented. The FWHM wavelength bandpasses found for high-efficiency couplers range from 173 to 525 nm. The various Bragg conditions that can be used in designing a WVGC are also presented and compared. The use of the propagation constant of the mode being outcoupled as the incident wave vector in the Bragg condition is shown to produce the highest coupling efficiency.

© 2004 Optical Society of America

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  1. S. M. Schultz, E. N. Glytsis, T. K. Gaylord, “Design of a high-efficiency volume grating coupler for line focusing,” Appl. Opt. 37, 2278–2287 (1998).
    [CrossRef]
  2. S. M. Schultz, E. N. Glytsis, T. K. Gaylord, “Volume grating preferential-order focusing waveguide coupler,” Opt. Lett. 24, 1708–1710 (1999).
    [CrossRef]
  3. S. M. Schultz, E. N. Glytsis, T. K. Gaylord, “Design, fabrication, and performance of preferential-order volume grating waveguide couplers,” Appl. Opt. 39, 1223–1231 (2000).
    [CrossRef]
  4. R. A. Villalaz, E. N. Glytsis, T. K. Gaylord, “Volume grating couplers: polarization and loss effects,” Appl. Opt. 41, 5223–5229 (2002).
    [CrossRef] [PubMed]
  5. F. Koyama, S. Kinoshita, K. Iga, “Room-temperature continuous wave lasing characteristics of a GaAs vertical cavity surface-emitting laser,” Appl. Phys. Lett. 55, 221–222 (1989).
    [CrossRef]
  6. J. W. Scott, R. S. Geels, S. W. Corzine, L. A. Coldren, “Modeling temperature effects and spatial hole burning to optimize vertical-cavity surface-emitting laser performance,” IEEE J. Quantum Electron. 29, 1295–1308 (1993).
    [CrossRef]
  7. S. Rapp, J. Piprek, K. Streubel, J. André, J. Wallin, “Temperature sensitivity of 1.54-μm vertical-cavity lasers with an InP-based Bragg reflector,” IEEE J. Quantum Electron. 33, 1839–1845 (1997).
    [CrossRef]
  8. H. Kogelnik, “Coupled wave theory for thick hologram gratings,” Bell Syst. Tech. J. 48, 2909–2947 (1969).
    [CrossRef]
  9. T. Stone, N. George, “Wavelength performance of holographic optical elements,” Appl. Opt. 24, 3797–3810 (1985).
    [CrossRef] [PubMed]
  10. T. K. Gaylord, M. G. Moharam, “Analysis and applications of optical diffraction by gratings,” Proc. IEEE 73, 894–937 (1985).
    [CrossRef]
  11. M. G. Moharam, T. K. Gaylord, “Rigorous coupled-wave analysis of planar grating diffraction,” J. Opt. Soc. Amer. 71, 811–818 (1981).
    [CrossRef]
  12. 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. Amer. A 12, 1068–1076 (1995).
    [CrossRef]
  13. S. D. Wu, 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] [PubMed]

2004

2002

2000

1999

1998

1997

S. Rapp, J. Piprek, K. Streubel, J. André, J. Wallin, “Temperature sensitivity of 1.54-μm vertical-cavity lasers with an InP-based Bragg reflector,” IEEE J. Quantum Electron. 33, 1839–1845 (1997).
[CrossRef]

1995

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. Amer. A 12, 1068–1076 (1995).
[CrossRef]

1993

J. W. Scott, R. S. Geels, S. W. Corzine, L. A. Coldren, “Modeling temperature effects and spatial hole burning to optimize vertical-cavity surface-emitting laser performance,” IEEE J. Quantum Electron. 29, 1295–1308 (1993).
[CrossRef]

1989

F. Koyama, S. Kinoshita, K. Iga, “Room-temperature continuous wave lasing characteristics of a GaAs vertical cavity surface-emitting laser,” Appl. Phys. Lett. 55, 221–222 (1989).
[CrossRef]

1985

T. Stone, N. George, “Wavelength performance of holographic optical elements,” Appl. Opt. 24, 3797–3810 (1985).
[CrossRef] [PubMed]

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

1981

M. G. Moharam, T. K. Gaylord, “Rigorous coupled-wave analysis of planar grating diffraction,” J. Opt. Soc. Amer. 71, 811–818 (1981).
[CrossRef]

1969

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

André, J.

S. Rapp, J. Piprek, K. Streubel, J. André, J. Wallin, “Temperature sensitivity of 1.54-μm vertical-cavity lasers with an InP-based Bragg reflector,” IEEE J. Quantum Electron. 33, 1839–1845 (1997).
[CrossRef]

Coldren, L. A.

J. W. Scott, R. S. Geels, S. W. Corzine, L. A. Coldren, “Modeling temperature effects and spatial hole burning to optimize vertical-cavity surface-emitting laser performance,” IEEE J. Quantum Electron. 29, 1295–1308 (1993).
[CrossRef]

Corzine, S. W.

J. W. Scott, R. S. Geels, S. W. Corzine, L. A. Coldren, “Modeling temperature effects and spatial hole burning to optimize vertical-cavity surface-emitting laser performance,” IEEE J. Quantum Electron. 29, 1295–1308 (1993).
[CrossRef]

Gaylord, T. K.

R. A. Villalaz, E. N. Glytsis, T. K. Gaylord, “Volume grating couplers: polarization and loss effects,” Appl. Opt. 41, 5223–5229 (2002).
[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, 1223–1231 (2000).
[CrossRef]

S. M. Schultz, E. N. Glytsis, T. K. Gaylord, “Volume grating preferential-order focusing waveguide coupler,” Opt. Lett. 24, 1708–1710 (1999).
[CrossRef]

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

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. Amer. A 12, 1068–1076 (1995).
[CrossRef]

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, “Rigorous coupled-wave analysis of planar grating diffraction,” J. Opt. Soc. Amer. 71, 811–818 (1981).
[CrossRef]

Geels, R. S.

J. W. Scott, R. S. Geels, S. W. Corzine, L. A. Coldren, “Modeling temperature effects and spatial hole burning to optimize vertical-cavity surface-emitting laser performance,” IEEE J. Quantum Electron. 29, 1295–1308 (1993).
[CrossRef]

George, N.

Glytsis, E. N.

Grann, E. B.

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. Amer. A 12, 1068–1076 (1995).
[CrossRef]

Iga, K.

F. Koyama, S. Kinoshita, K. Iga, “Room-temperature continuous wave lasing characteristics of a GaAs vertical cavity surface-emitting laser,” Appl. Phys. Lett. 55, 221–222 (1989).
[CrossRef]

Kinoshita, S.

F. Koyama, S. Kinoshita, K. Iga, “Room-temperature continuous wave lasing characteristics of a GaAs vertical cavity surface-emitting laser,” Appl. Phys. Lett. 55, 221–222 (1989).
[CrossRef]

Kogelnik, H.

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

Koyama, F.

F. Koyama, S. Kinoshita, K. Iga, “Room-temperature continuous wave lasing characteristics of a GaAs vertical cavity surface-emitting laser,” Appl. Phys. Lett. 55, 221–222 (1989).
[CrossRef]

Moharam, M. G.

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. Amer. A 12, 1068–1076 (1995).
[CrossRef]

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, “Rigorous coupled-wave analysis of planar grating diffraction,” J. Opt. Soc. Amer. 71, 811–818 (1981).
[CrossRef]

Piprek, J.

S. Rapp, J. Piprek, K. Streubel, J. André, J. Wallin, “Temperature sensitivity of 1.54-μm vertical-cavity lasers with an InP-based Bragg reflector,” IEEE J. Quantum Electron. 33, 1839–1845 (1997).
[CrossRef]

Pommet, D. A.

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. Amer. A 12, 1068–1076 (1995).
[CrossRef]

Rapp, S.

S. Rapp, J. Piprek, K. Streubel, J. André, J. Wallin, “Temperature sensitivity of 1.54-μm vertical-cavity lasers with an InP-based Bragg reflector,” IEEE J. Quantum Electron. 33, 1839–1845 (1997).
[CrossRef]

Schultz, S. M.

Scott, J. W.

J. W. Scott, R. S. Geels, S. W. Corzine, L. A. Coldren, “Modeling temperature effects and spatial hole burning to optimize vertical-cavity surface-emitting laser performance,” IEEE J. Quantum Electron. 29, 1295–1308 (1993).
[CrossRef]

Stone, T.

Streubel, K.

S. Rapp, J. Piprek, K. Streubel, J. André, J. Wallin, “Temperature sensitivity of 1.54-μm vertical-cavity lasers with an InP-based Bragg reflector,” IEEE J. Quantum Electron. 33, 1839–1845 (1997).
[CrossRef]

Villalaz, R. A.

Wallin, J.

S. Rapp, J. Piprek, K. Streubel, J. André, J. Wallin, “Temperature sensitivity of 1.54-μm vertical-cavity lasers with an InP-based Bragg reflector,” IEEE J. Quantum Electron. 33, 1839–1845 (1997).
[CrossRef]

Wu, S. D.

Appl. Opt.

Appl. Phys. Lett.

F. Koyama, S. Kinoshita, K. Iga, “Room-temperature continuous wave lasing characteristics of a GaAs vertical cavity surface-emitting laser,” Appl. Phys. Lett. 55, 221–222 (1989).
[CrossRef]

Bell Syst. Tech. J.

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

IEEE J. Quantum Electron.

J. W. Scott, R. S. Geels, S. W. Corzine, L. A. Coldren, “Modeling temperature effects and spatial hole burning to optimize vertical-cavity surface-emitting laser performance,” IEEE J. Quantum Electron. 29, 1295–1308 (1993).
[CrossRef]

S. Rapp, J. Piprek, K. Streubel, J. André, J. Wallin, “Temperature sensitivity of 1.54-μm vertical-cavity lasers with an InP-based Bragg reflector,” IEEE J. Quantum Electron. 33, 1839–1845 (1997).
[CrossRef]

J. Opt. Soc. Amer.

M. G. Moharam, T. K. Gaylord, “Rigorous coupled-wave analysis of planar grating diffraction,” J. Opt. Soc. Amer. 71, 811–818 (1981).
[CrossRef]

J. Opt. Soc. Amer. A

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. Amer. A 12, 1068–1076 (1995).
[CrossRef]

Opt. Lett.

Proc. IEEE

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

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Figures (11)

Fig. 1
Fig. 1

Diagrams of the two WVGC configurations discussed in this paper: (a) volume grating in the cover layer and (b) volume grating in the waveguide. The grating vector K as well as the period Λ and slant angle ϕ are shown. The outcoupling angle is θ c . The thickness of the grating layer is t g , and the thickness of the waveguide layer is t w .

Fig. 2
Fig. 2

Coupling efficiency versus wavelength for TE- and TM-polarized light in the “VG in the cover layer” structure, with normal outcoupling for the design wavelength λ o = 1 μm.

Fig. 3
Fig. 3

Coupling efficiency versus wavelength for TE- and TM-polarized light in the “VG in the cover layer” structure, with outcoupling at 45° for the design wavelength λ o = 1 μm.

Fig. 4
Fig. 4

Coupling efficiency versus wavelength for TE- and TM-polarized light in the “VG in the waveguide” structure, with normal outcoupling for the design wavelength λ o = 1 μm.

Fig. 5
Fig. 5

Coupling efficiency versus wavelength for TE- and TM-polarized light in the “VG in the waveguide” structure, with outcoupling at 45° for the design wavelength λ o = 1 μm.

Fig. 6
Fig. 6

Coupling efficiency versus wavelength for TE-polarized light in the “VG in the cover layer” structure, with normal outcoupling for the design wavelength λ o = 1 μm. The structure with Δn = 0.06 has a slightly narrower FWHM bandpass than the original structure with Δn = 0.02.

Fig. 7
Fig. 7

Coupling efficiency versus wavelength for TE-polarized light in the “VG in the waveguide” structure, with normal outcoupling for the design wavelength λ o = 1 μm. The structure with Δn = 0.06 has a slightly narrower FWHM bandpass than the original structure with Δn = 0.02.

Fig. 8
Fig. 8

Coupling efficiency versus wavelength for TE-polarized light in the “VG in the cover layer” structure, with normal outcoupling for the design wavelength λ o = 1 μm. The structure with n wg = 1.55 has a significantly narrower FWHM bandpass than the original structure with n wg = 1.56.

Fig. 9
Fig. 9

Coupling efficiency versus wavelength for TE-polarized light in the “VG in the waveguide” structure, with normal outcoupling for the design wavelength λ o = 1 μm. The structures with t wg = 3 μm and t wg = 6 μm have significantly narrower FWHM bandpass than the original structure with t wg = 1.8 μm.

Fig. 10
Fig. 10

Three options for the Bragg condition for a guided incident wave.

Fig. 11
Fig. 11

Coupling coefficient versus wavelength for TE-polarized light in the “VG in the cover layer” structure, with normal outcoupling for the design wavelength λ o = 1 μm. The Bragg condition determined by use of the propagation constant β results in a higher coupling coefficient at the design wavelength.

Equations (4)

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CE,i=η,i1-exp-2αL,
θc=sin-1β-Kxkonsup.
Kz=kong2-β-Kx2]1/2,
Kz=kong2-kinc,x-Kx2]1/2±kinc,z,

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