Abstract

We discuss the wavelength for maximum reflectivity of a Bragg reflector that has both index and absorption modulation. We show there that the blue shift away from the Bragg wavelength can be no larger than that given by the wavelength of maximum coupling, which we derive analytically. For gratings with fewer layers, the blue shift will be smaller than this maximum value. In typical semiconductor Bragg reflectors, this shift may be as much as 32 nm.

© 1992 Optical Society of America

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References

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  1. P. L. Gourley, T. J. Drummond, “Single crystal, epitaxial multilayers of AlAs, GaAs, and AlxGa1−xAs for use as optical interferometric elements,” Appl. Phys. Lett. 49, 489–491 (1986).
    [CrossRef]
  2. M. Ogura, T. Hata, T. Yao, “Distributed feedback surface emitting laser diode with multilayered heterostructure,” Jpn. J. Appl. Phys. 23, L512–L514 (1984).
    [CrossRef]
  3. J. L. Jewell, A. Scherer, S. L. McCall, A. C. Gossard, J. H. English, “GaAs-ALAs monolithic microresonator arrays,” Appl. Phys. Lett. 51, 94–96 (1987).
    [CrossRef]
  4. B. G. Kim, E. Garmire, S. G. Hummel, P. D. Dapkus, “Nonlinear Bragg reflector based on saturable absorption,” Appl. Phys. Lett. 54, 1095–1097 (1989).
    [CrossRef]
  5. H. Kogelnik, C. V. Shank, “Coupled-wave theory of distributed feedback lasers,” J. Appl. Phys. 43, 2327–2335 (1972).
    [CrossRef]
  6. W. Streifer, D. R. Scifres, R. D. Burnham, “Coupled wave analysis of DFB and DBR lasers,” IEEE J. Quantum Electron. QE-13, 134–141 (1977).
    [CrossRef]
  7. B. G. Kim, E. Garmire, “Comparison between the matrix method and the coupled wave method in analyzing Bragg reflector structures,” J. Opt. Soc. Am. A. 9, 132–136 (1992).
    [CrossRef]
  8. A. Yariv, P. Yeh, Optical Waves in Crystals (Wiley, New York, 1984).
  9. H. C. Casey, M. B. Panish, Heterostructure Lasers Part A: Fundamental Principles (Academic, New York, 1978).

1992 (1)

B. G. Kim, E. Garmire, “Comparison between the matrix method and the coupled wave method in analyzing Bragg reflector structures,” J. Opt. Soc. Am. A. 9, 132–136 (1992).
[CrossRef]

1989 (1)

B. G. Kim, E. Garmire, S. G. Hummel, P. D. Dapkus, “Nonlinear Bragg reflector based on saturable absorption,” Appl. Phys. Lett. 54, 1095–1097 (1989).
[CrossRef]

1987 (1)

J. L. Jewell, A. Scherer, S. L. McCall, A. C. Gossard, J. H. English, “GaAs-ALAs monolithic microresonator arrays,” Appl. Phys. Lett. 51, 94–96 (1987).
[CrossRef]

1986 (1)

P. L. Gourley, T. J. Drummond, “Single crystal, epitaxial multilayers of AlAs, GaAs, and AlxGa1−xAs for use as optical interferometric elements,” Appl. Phys. Lett. 49, 489–491 (1986).
[CrossRef]

1984 (1)

M. Ogura, T. Hata, T. Yao, “Distributed feedback surface emitting laser diode with multilayered heterostructure,” Jpn. J. Appl. Phys. 23, L512–L514 (1984).
[CrossRef]

1977 (1)

W. Streifer, D. R. Scifres, R. D. Burnham, “Coupled wave analysis of DFB and DBR lasers,” IEEE J. Quantum Electron. QE-13, 134–141 (1977).
[CrossRef]

1972 (1)

H. Kogelnik, C. V. Shank, “Coupled-wave theory of distributed feedback lasers,” J. Appl. Phys. 43, 2327–2335 (1972).
[CrossRef]

Burnham, R. D.

W. Streifer, D. R. Scifres, R. D. Burnham, “Coupled wave analysis of DFB and DBR lasers,” IEEE J. Quantum Electron. QE-13, 134–141 (1977).
[CrossRef]

Casey, H. C.

H. C. Casey, M. B. Panish, Heterostructure Lasers Part A: Fundamental Principles (Academic, New York, 1978).

Dapkus, P. D.

B. G. Kim, E. Garmire, S. G. Hummel, P. D. Dapkus, “Nonlinear Bragg reflector based on saturable absorption,” Appl. Phys. Lett. 54, 1095–1097 (1989).
[CrossRef]

Drummond, T. J.

P. L. Gourley, T. J. Drummond, “Single crystal, epitaxial multilayers of AlAs, GaAs, and AlxGa1−xAs for use as optical interferometric elements,” Appl. Phys. Lett. 49, 489–491 (1986).
[CrossRef]

English, J. H.

J. L. Jewell, A. Scherer, S. L. McCall, A. C. Gossard, J. H. English, “GaAs-ALAs monolithic microresonator arrays,” Appl. Phys. Lett. 51, 94–96 (1987).
[CrossRef]

Garmire, E.

B. G. Kim, E. Garmire, “Comparison between the matrix method and the coupled wave method in analyzing Bragg reflector structures,” J. Opt. Soc. Am. A. 9, 132–136 (1992).
[CrossRef]

B. G. Kim, E. Garmire, S. G. Hummel, P. D. Dapkus, “Nonlinear Bragg reflector based on saturable absorption,” Appl. Phys. Lett. 54, 1095–1097 (1989).
[CrossRef]

Gossard, A. C.

J. L. Jewell, A. Scherer, S. L. McCall, A. C. Gossard, J. H. English, “GaAs-ALAs monolithic microresonator arrays,” Appl. Phys. Lett. 51, 94–96 (1987).
[CrossRef]

Gourley, P. L.

P. L. Gourley, T. J. Drummond, “Single crystal, epitaxial multilayers of AlAs, GaAs, and AlxGa1−xAs for use as optical interferometric elements,” Appl. Phys. Lett. 49, 489–491 (1986).
[CrossRef]

Hata, T.

M. Ogura, T. Hata, T. Yao, “Distributed feedback surface emitting laser diode with multilayered heterostructure,” Jpn. J. Appl. Phys. 23, L512–L514 (1984).
[CrossRef]

Hummel, S. G.

B. G. Kim, E. Garmire, S. G. Hummel, P. D. Dapkus, “Nonlinear Bragg reflector based on saturable absorption,” Appl. Phys. Lett. 54, 1095–1097 (1989).
[CrossRef]

Jewell, J. L.

J. L. Jewell, A. Scherer, S. L. McCall, A. C. Gossard, J. H. English, “GaAs-ALAs monolithic microresonator arrays,” Appl. Phys. Lett. 51, 94–96 (1987).
[CrossRef]

Kim, B. G.

B. G. Kim, E. Garmire, “Comparison between the matrix method and the coupled wave method in analyzing Bragg reflector structures,” J. Opt. Soc. Am. A. 9, 132–136 (1992).
[CrossRef]

B. G. Kim, E. Garmire, S. G. Hummel, P. D. Dapkus, “Nonlinear Bragg reflector based on saturable absorption,” Appl. Phys. Lett. 54, 1095–1097 (1989).
[CrossRef]

Kogelnik, H.

H. Kogelnik, C. V. Shank, “Coupled-wave theory of distributed feedback lasers,” J. Appl. Phys. 43, 2327–2335 (1972).
[CrossRef]

McCall, S. L.

J. L. Jewell, A. Scherer, S. L. McCall, A. C. Gossard, J. H. English, “GaAs-ALAs monolithic microresonator arrays,” Appl. Phys. Lett. 51, 94–96 (1987).
[CrossRef]

Ogura, M.

M. Ogura, T. Hata, T. Yao, “Distributed feedback surface emitting laser diode with multilayered heterostructure,” Jpn. J. Appl. Phys. 23, L512–L514 (1984).
[CrossRef]

Panish, M. B.

H. C. Casey, M. B. Panish, Heterostructure Lasers Part A: Fundamental Principles (Academic, New York, 1978).

Scherer, A.

J. L. Jewell, A. Scherer, S. L. McCall, A. C. Gossard, J. H. English, “GaAs-ALAs monolithic microresonator arrays,” Appl. Phys. Lett. 51, 94–96 (1987).
[CrossRef]

Scifres, D. R.

W. Streifer, D. R. Scifres, R. D. Burnham, “Coupled wave analysis of DFB and DBR lasers,” IEEE J. Quantum Electron. QE-13, 134–141 (1977).
[CrossRef]

Shank, C. V.

H. Kogelnik, C. V. Shank, “Coupled-wave theory of distributed feedback lasers,” J. Appl. Phys. 43, 2327–2335 (1972).
[CrossRef]

Streifer, W.

W. Streifer, D. R. Scifres, R. D. Burnham, “Coupled wave analysis of DFB and DBR lasers,” IEEE J. Quantum Electron. QE-13, 134–141 (1977).
[CrossRef]

Yao, T.

M. Ogura, T. Hata, T. Yao, “Distributed feedback surface emitting laser diode with multilayered heterostructure,” Jpn. J. Appl. Phys. 23, L512–L514 (1984).
[CrossRef]

Yariv, A.

A. Yariv, P. Yeh, Optical Waves in Crystals (Wiley, New York, 1984).

Yeh, P.

A. Yariv, P. Yeh, Optical Waves in Crystals (Wiley, New York, 1984).

Appl. Phys. Lett. (3)

P. L. Gourley, T. J. Drummond, “Single crystal, epitaxial multilayers of AlAs, GaAs, and AlxGa1−xAs for use as optical interferometric elements,” Appl. Phys. Lett. 49, 489–491 (1986).
[CrossRef]

J. L. Jewell, A. Scherer, S. L. McCall, A. C. Gossard, J. H. English, “GaAs-ALAs monolithic microresonator arrays,” Appl. Phys. Lett. 51, 94–96 (1987).
[CrossRef]

B. G. Kim, E. Garmire, S. G. Hummel, P. D. Dapkus, “Nonlinear Bragg reflector based on saturable absorption,” Appl. Phys. Lett. 54, 1095–1097 (1989).
[CrossRef]

IEEE J. Quantum Electron. (1)

W. Streifer, D. R. Scifres, R. D. Burnham, “Coupled wave analysis of DFB and DBR lasers,” IEEE J. Quantum Electron. QE-13, 134–141 (1977).
[CrossRef]

J. Appl. Phys. (1)

H. Kogelnik, C. V. Shank, “Coupled-wave theory of distributed feedback lasers,” J. Appl. Phys. 43, 2327–2335 (1972).
[CrossRef]

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

B. G. Kim, E. Garmire, “Comparison between the matrix method and the coupled wave method in analyzing Bragg reflector structures,” J. Opt. Soc. Am. A. 9, 132–136 (1992).
[CrossRef]

Jpn. J. Appl. Phys. (1)

M. Ogura, T. Hata, T. Yao, “Distributed feedback surface emitting laser diode with multilayered heterostructure,” Jpn. J. Appl. Phys. 23, L512–L514 (1984).
[CrossRef]

Other (2)

A. Yariv, P. Yeh, Optical Waves in Crystals (Wiley, New York, 1984).

H. C. Casey, M. B. Panish, Heterostructure Lasers Part A: Fundamental Principles (Academic, New York, 1978).

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Figures (3)

Fig. 1
Fig. 1

Geometry of a Bragg reflector that includes both index and absorption modulation.

Fig. 2
Fig. 2

Normalized wavelength shift of the reflectivity peak as a function of L for α = 1, 0.1, 0.01 (μm−1) in the case of K = 1.331 1μm −1 (n h = 3.6, n L = 3), and λ B = 0.9 μm.

Fig. 3
Fig. 3

Normalized wavelength shift of the reflectivity peak as a function of L for α = 1, 0.1, 0.01 (μm−1) in the case of K = 0.444 μm−1 (n h = 3.2, n L = 3), and λ B = 0.9 μm.

Equations (8)

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r = E b ( 0 ) E f ( 0 ) = i ( K - i F ) sinh ( s L ) s cosh ( s L ) + i ( δ - i a α / 2 ) sinh ( s L ) ,
s = { [ ( K 2 - F 2 ) - δ 2 + ( a α / 2 ) 2 ] + i ( δ a α - 2 K F ) } 1 / 2 .
V = ( π / Λ ) ± i s .
( B δ - D ) [ 4 D δ 2 + ( B 3 - 4 A B ) δ - B 2 D ] = 0 ,
δ m = D B = sin a π a π K .
Δ λ C = - λ B ( n h 2 - n L 2 ) sin 2 ( a π ) 2 a π 2 n av 2 + ( n h 2 - n L 2 ) sin 2 ( a π ) .
Δ λ C λ B = ( 2 + Δ n / n L ) 4 ( Δ n / n L ) sin 2 a π ( 2 + Δ n / n L ) 4 ( Δ n / n L ) sin 2 a π + 8 π 2 ( 1 + Δ n / n L ) 2 .
Δ λ C λ B = - 2 ( Δ n / n L ) π 2 .

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