Abstract

We theoretically analyze the diffraction of light by gratings that are photogenerated in Fabry–Perot microcavities. The coupled-wave theory of volume gratings is combined with appropriate boundary conditions to yield expressions for the diffraction efficiency. Multiple round trips within the cavity are seen to increase the effective grating thickness and therefore the efficiency. Numerical calculations specific to GaAs microcavities show that the diffraction efficiency can be enhanced by more than 2 orders of magnitude at the resonant wavelengths.

© 1995 Optical Society of America

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  1. A. M. Glass, D. D. Nolte, D. H. Olson, G. E. Doran, D. S. Chemla, and W. H. Knox, "Resonant photodiffractive four-wave mixing in semi-insulating As/AlGaAs quantum wells," Opt. Lett. 15, 264–266 (1990).
    [CrossRef] [PubMed]
  2. D. D. Nolte, D. H. Olson, G. E. Doran, W. H. Knox, and A. M. Glass, "Resonant photodiffractive effect in semi-insulating multiple quantum wells," J. Opt. Soc. Am. B 7, 2217–2225 (1990).
    [CrossRef]
  3. N. T. Pelekanos, H. Haas, N. Magnea, H. Mariette, and A. Waliela, "Room-temperature exciton absorption engineering in II-VI quantum wells," Appl. Phys. Lett. 61, 3154–3156 (1992).
    [CrossRef]
  4. J. T. Sheridan, "Stacked volume holographic gratings: part I, transmission gratings in series," Optik 95, 73–80 (1993).
  5. F. Lin, H. Chou, E. Strzelecki, and J. B. Shellan, "Multiplexed holographic Fabry–Perot étalons," Appl. Opt. 31, 2478–2484 (1992).
    [CrossRef] [PubMed]
  6. W. Wang, "Reflection and transmission properties of holographic mirrors and holographic Fabry–Perot filters. I. Holographic mirrors with monochromatic light," Appl. Opt. 33, 2560–2566 (1994).
    [CrossRef] [PubMed]
  7. W. D. Cornish and L. Young, "Influence of multiple internal reflections and thermal expansion on the effective diffraction efficiency of holograms stored in lithium niobate," J. Appl. Phys. 46, 1252–1254 (1975).
    [CrossRef]
  8. J. Ctyroky, "Coupled-mode theory of Bragg diffraction in the presence of multiple internal reflections," Opt. Commun. 16, 259–261 (1994).
    [CrossRef]
  9. F. Vasey, J. M. Stauffer, Y. Oppliger, and F. K. Reinhart, "Characterization of an AlGaAs rib waveguide using a grating in a Fabry–Perot étalon configuration," Appl. Opt. 30, 3897–3906 (1991).
    [CrossRef] [PubMed]
  10. W. H. Steier, G. T. Kavounas, R. T. Sahara, and J. Kumar, "Enhanced optical light deflection using cavity resonance," Appl. Opt. 27, 1603–1606 (1988).
    [CrossRef] [PubMed]
  11. J. Krawczak, R. Dean, E. J. Torok, and G. Nelson, "Diffraction efficiency gain, blazing, and apodizing of a symmetric square reflection grating in an étalon," Opt. Lett. 15, 1264–1266 (1990).
    [CrossRef] [PubMed]
  12. D. D. Nolte, "Holographic phase grating in a multiple-quantum-well asymmetric Fabry–Perot modulator," in Conference on Lasers and Electro-Optics, Vol. 8 of 1994 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1994), CTuD4.
  13. H. Kogelnik, "Coupled wave theory for thick hologram gratings," Bell Syst. Tech. J. 48, 2909–2946 (1969).
  14. L. Banyai and S. W. Koch, "A simple theory for the effects of plasma screening on the optical spectra of highly excited semiconductors," Z. Phys. B 63, 283–291 (1986).
    [CrossRef]
  15. H. Haug and S. W. Koch, Quantum Theory of the Optical and Electronical Properties of Semiconductors, 1st ed. (World Scientific, New York, 1988), Chap. 13.

1994 (2)

1993 (1)

J. T. Sheridan, "Stacked volume holographic gratings: part I, transmission gratings in series," Optik 95, 73–80 (1993).

1992 (2)

F. Lin, H. Chou, E. Strzelecki, and J. B. Shellan, "Multiplexed holographic Fabry–Perot étalons," Appl. Opt. 31, 2478–2484 (1992).
[CrossRef] [PubMed]

N. T. Pelekanos, H. Haas, N. Magnea, H. Mariette, and A. Waliela, "Room-temperature exciton absorption engineering in II-VI quantum wells," Appl. Phys. Lett. 61, 3154–3156 (1992).
[CrossRef]

1991 (1)

1990 (3)

1988 (1)

1986 (1)

L. Banyai and S. W. Koch, "A simple theory for the effects of plasma screening on the optical spectra of highly excited semiconductors," Z. Phys. B 63, 283–291 (1986).
[CrossRef]

1975 (1)

W. D. Cornish and L. Young, "Influence of multiple internal reflections and thermal expansion on the effective diffraction efficiency of holograms stored in lithium niobate," J. Appl. Phys. 46, 1252–1254 (1975).
[CrossRef]

1969 (1)

H. Kogelnik, "Coupled wave theory for thick hologram gratings," Bell Syst. Tech. J. 48, 2909–2946 (1969).

Banyai, L.

L. Banyai and S. W. Koch, "A simple theory for the effects of plasma screening on the optical spectra of highly excited semiconductors," Z. Phys. B 63, 283–291 (1986).
[CrossRef]

Chemla, D. S.

Chou, H.

Cornish, W. D.

W. D. Cornish and L. Young, "Influence of multiple internal reflections and thermal expansion on the effective diffraction efficiency of holograms stored in lithium niobate," J. Appl. Phys. 46, 1252–1254 (1975).
[CrossRef]

Ctyroky, J.

J. Ctyroky, "Coupled-mode theory of Bragg diffraction in the presence of multiple internal reflections," Opt. Commun. 16, 259–261 (1994).
[CrossRef]

Dean, R.

Doran, G. E.

Glass, A. M.

Haas, H.

N. T. Pelekanos, H. Haas, N. Magnea, H. Mariette, and A. Waliela, "Room-temperature exciton absorption engineering in II-VI quantum wells," Appl. Phys. Lett. 61, 3154–3156 (1992).
[CrossRef]

Haug, H.

H. Haug and S. W. Koch, Quantum Theory of the Optical and Electronical Properties of Semiconductors, 1st ed. (World Scientific, New York, 1988), Chap. 13.

Kavounas, G. T.

Knox, W. H.

Koch, S. W.

L. Banyai and S. W. Koch, "A simple theory for the effects of plasma screening on the optical spectra of highly excited semiconductors," Z. Phys. B 63, 283–291 (1986).
[CrossRef]

H. Haug and S. W. Koch, Quantum Theory of the Optical and Electronical Properties of Semiconductors, 1st ed. (World Scientific, New York, 1988), Chap. 13.

Kogelnik, H.

H. Kogelnik, "Coupled wave theory for thick hologram gratings," Bell Syst. Tech. J. 48, 2909–2946 (1969).

Krawczak, J.

Kumar, J.

Lin, F.

Magnea, N.

N. T. Pelekanos, H. Haas, N. Magnea, H. Mariette, and A. Waliela, "Room-temperature exciton absorption engineering in II-VI quantum wells," Appl. Phys. Lett. 61, 3154–3156 (1992).
[CrossRef]

Mariette, H.

N. T. Pelekanos, H. Haas, N. Magnea, H. Mariette, and A. Waliela, "Room-temperature exciton absorption engineering in II-VI quantum wells," Appl. Phys. Lett. 61, 3154–3156 (1992).
[CrossRef]

Nelson, G.

Nolte, D. D.

Olson, D. H.

Oppliger, Y.

Pelekanos, N. T.

N. T. Pelekanos, H. Haas, N. Magnea, H. Mariette, and A. Waliela, "Room-temperature exciton absorption engineering in II-VI quantum wells," Appl. Phys. Lett. 61, 3154–3156 (1992).
[CrossRef]

Reinhart, F. K.

Sahara, R. T.

Shellan, J. B.

Sheridan, J. T.

J. T. Sheridan, "Stacked volume holographic gratings: part I, transmission gratings in series," Optik 95, 73–80 (1993).

Stauffer, J. M.

Steier, W. H.

Strzelecki, E.

Torok, E. J.

Vasey, F.

Waliela, A.

N. T. Pelekanos, H. Haas, N. Magnea, H. Mariette, and A. Waliela, "Room-temperature exciton absorption engineering in II-VI quantum wells," Appl. Phys. Lett. 61, 3154–3156 (1992).
[CrossRef]

Wang, W.

Young, L.

W. D. Cornish and L. Young, "Influence of multiple internal reflections and thermal expansion on the effective diffraction efficiency of holograms stored in lithium niobate," J. Appl. Phys. 46, 1252–1254 (1975).
[CrossRef]

Appl. Opt. (4)

Appl. Phys. Lett. (1)

N. T. Pelekanos, H. Haas, N. Magnea, H. Mariette, and A. Waliela, "Room-temperature exciton absorption engineering in II-VI quantum wells," Appl. Phys. Lett. 61, 3154–3156 (1992).
[CrossRef]

Bell Syst. Tech. J. (1)

H. Kogelnik, "Coupled wave theory for thick hologram gratings," Bell Syst. Tech. J. 48, 2909–2946 (1969).

J. Appl. Phys. (1)

W. D. Cornish and L. Young, "Influence of multiple internal reflections and thermal expansion on the effective diffraction efficiency of holograms stored in lithium niobate," J. Appl. Phys. 46, 1252–1254 (1975).
[CrossRef]

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

Opt. Commun. (1)

J. Ctyroky, "Coupled-mode theory of Bragg diffraction in the presence of multiple internal reflections," Opt. Commun. 16, 259–261 (1994).
[CrossRef]

Opt. Lett. (2)

Optik (1)

J. T. Sheridan, "Stacked volume holographic gratings: part I, transmission gratings in series," Optik 95, 73–80 (1993).

Z. Phys. B (1)

L. Banyai and S. W. Koch, "A simple theory for the effects of plasma screening on the optical spectra of highly excited semiconductors," Z. Phys. B 63, 283–291 (1986).
[CrossRef]

Other (2)

H. Haug and S. W. Koch, Quantum Theory of the Optical and Electronical Properties of Semiconductors, 1st ed. (World Scientific, New York, 1988), Chap. 13.

D. D. Nolte, "Holographic phase grating in a multiple-quantum-well asymmetric Fabry–Perot modulator," in Conference on Lasers and Electro-Optics, Vol. 8 of 1994 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1994), CTuD4.

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Figures (5)

Fig. 1
Fig. 1

(a) Absorption of bulk GaAs at room temperature in the presence of an electron-hole plasma with density ρ, (b) index modulation deduced from the absorption by the Kramers–Kronig transformation.

Fig. 2
Fig. 2

Cavity configuration. The different wave vectors are the following: q1 for the transmitted beam, q2 for the forward diffracted beam, q1′ for the reflected beam, and q2′ for the backward diffracted beam. G is the grating wave vector. The different angles are the following: θ0 for the incident beam, θ1 for the transmitted beam in the cavity, and θ2 for the diffracted beam in the cavity.

Fig. 3
Fig. 3

Efficiency of the backward diffracted beam as a function of the incident-beam wavelength: parameter is the angle of incidence θ0; the front and the back reflectivities are RF = 0.9 and RB = 0.995, respectively; the cavity thickness is L = 1 μm; the index modulation is Δn = 0.003; the grating spacing is Λ = 684 nm.

Fig. 4
Fig. 4

Efficiency of the backward diffracted beam as a function of the index modulation under the Bragg incidence θ0 = 40° at 685 nm. The different curves correspond to different front mirror reflectivities RF = 0.9, 0.7, 0.5, 0.3, 0.1, and 0; RB = 0.995; the cavity thickness is L = 1 μm; the grating spacing is Λ = 684 nm. The dotted line corresponds to the index variation Δn = 0.003 of Fig. 3.

Fig. 5
Fig. 5

Backward diffraction efficiency at the Bragg incidence for different grating spacings ranging from 628 to 698 nm. Δn = 0.003, RF = 0.7, RB = 0.995, incidence angle θ0 = 40°, L = 1 μm.

Equations (22)

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n 2 ( x ) = n 0 + Δ n sin ( 2 π x Λ ) ,
[ E 1 ( x , 0 + ) E 2 ( x , 0 + ) ] - r 21 [ E 1 ( x , 0 + ) E 2 ( x , 0 + ) ] = ( t 12 0 ) .
[ E 1 ( x , L - ) E 2 ( x , L - ) ] = M ( L ) exp ( i q 1 L cos θ 1 ) [ E 1 ( x , 0 + ) E 2 ( x , 0 + ) ] .
[ E 1 ( x , L - ) E 2 ( x , L - ) ] = r 23 [ E 1 ( x , L - ) E 2 ( x , L - ) ] .
[ E 1 ( x , 0 + ) E 2 ( x , 0 + ) ] = M ( L ) exp ( i n 0 q 0 L cos θ 1 ) [ E 1 ( x , L - ) E 2 ( x , L - ) ] .
[ 1 - r 21 r 23 M 2 ( L ) exp ( i 2 L q 0 n 0 cos θ 1 ) ] [ E 1 ( x , 0 + ) E 2 ( x , 0 + ) ] = ( t 12 0 ) .
E 1 ( x , 0 + ) = t 12 [ 1 - r 21 r 23 M 23 ( 2 L ) exp ( i 2 δ ) ] Δ ,
E 2 ( x , 0 + ) = t 12 r 21 r 23 M 21 ( 2 L ) exp ( i 2 δ ) Δ ,
Δ = [ 1 - r 21 r 23 M 11 ( 2 L ) exp ( i 2 δ ) ] × [ 1 - r 21 r 23 M 22 ( 2 L ) exp ( i 2 δ ) ] - r 21 2 r 23 2 M 12 ( 2 L ) M 21 ( 2 L ) exp ( i 4 δ ) .
M = [ 1 0 0 1 ] ,
E R ( x , 0 ) = r 12 + t 21 E 1 ( x , 0 + ) = r 12 + t 21 [ E 1 ( x , 0 + ) - t 12 r 21 ] = r 21 - t 12 t 21 r 21 + t 21 r 21 E 1 ( x , 0 + ) ,
E R ( x , 0 ) = - 1 ( R F ) 1 / 2 + 1 - R F ( R F ) 1 / 2 × [ 1 - ( R F R B ) 1 / 2 M 22 ( 2 L ) exp ( i 2 δ ) ] Δ .
E BD ( x , 0 ) = t 21 E 2 ( x , 0 + ) = t 21 r 21 E 2 ( x , 0 + ) = t 21 r 21 t 12 r 21 r 23 M 21 ( 2 L ) exp ( i 2 δ ) Δ E BD ( x , 0 ) = ( 1 - R F ) ( R B ) 1 / 2 M 21 ( 2 L ) exp ( i 2 δ ) / Δ .
E FD ( x , L ) = t 23 E 2 ( x , L - ) = t 23 exp ( i δ ) [ M 21 ( L ) E 1 ( x , 0 + ) + M 22 ( L ) E 2 ( x , 0 + ) ] .
E T ( x , L ) = t 23 E 1 ( x , L - ) = t 23 exp ( i δ ) [ M 11 ( L ) E 1 ( x , 0 + ) + M 12 ( L ) E 2 ( x , 0 + ) ] .
M ( L ) = 1 ( γ 1 - γ 2 ) [ - [ γ 2 exp ( γ 1 L ) - γ 1 exp ( γ 2 L ) ] i K [ exp ( γ 2 L ) - exp ( γ 1 L ) ] cos θ 1 i k [ exp ( γ 2 L ) - exp ( γ 1 L ) ] cos θ 1 [ γ 1 exp ( γ 1 L ) - γ 2 exp ( γ 2 L ) ] ] ,
γ i = - i 2 V cos θ 1 ± i 2 cos θ 1 ( V 2 + 4 K 2 ) 1 / 2
V = ( n 0 2 q 0 2 - q 2 2 ) 2 n 0 q 0 .
E R ( x , 0 ) 2 + E BD ( x , 0 ) 2 + E FD ( x , L ) 2 + E T ( x , L ) 2 = 1.
cos θ 1 = N λ 0 2 n 0 L 1 ,             N = 1 , 2 , 3 , .
tan θ 2 = G - q 1 sin θ 1 q 1 cos θ 1 = 2 π - Λ q 1 sin θ 1 Λ q 1 cos θ 1 .
Δ θ D = - 2 n 2 sin ( θ 2 ) cos 2 ( θ 2 ) cos ( θ D ) Δ Λ Λ = 180 π tan θ D [ 1 - sin 2 ( θ 0 ) n 0 2 ] Δ Λ Λ             ( Δ θ D in degrees ) .

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