Abstract

We present the study of thermal and optical properties of 15.5 pairs of GaAlAsSb/AlAsSb layers deposited upon a GaSb substrate by photothermal deflection spectroscopy. This stacking of layers constitutes a distributed Bragg mirror.

© 2002 Optical Society of America

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  1. C. L. Felix, W. W. Bewley, I. Vurgaftman, J. R. Meyer, L. Goldberg, D. H. Chow, E. Selvig, “Midinfrared vertical-cavity surface-emitting laser,” Appl. Phys. Lett. 71, 3483–3485 (1997).
    [CrossRef]
  2. I. Vurgaftman, J. R. Meyer, L. R. Ram-Mohan, “Mid-IR vertical-cavity surface-emitting lasers,” IEEE J. Quantum Electron. 34, 147–156 (1998).
    [CrossRef]
  3. J. Koeth, R. Dietrich, A. Forchel, “GaSb vertical-cavity surface-emitting lasers for the 1.5 µm rang,” Appl. Phys. Lett. 72, 1638–1640 (1999).
    [CrossRef]
  4. A. N. Baranov, N. Bertru, Y. Cuminal, G. Goissier, Y. Rouillard, J. C. Nicolas, P. Grech, A. F. Joullie, C. L. Alibert, “Mid-infrared GaSb-InAs-based multiple quantum well lasers,” in In-Plane Seminconductor Lasers: from Ultraviolet to Mid-Infrared II, H. K. Choi, P. S. Lory, eds., Proc. SPIE3284, 247–257 (1998).
  5. J. Geske, V. Jayaraman, T. Goodwin, M. Culick, M. MacDougal, T. Goodnough, D. Welch, J. E. Bowers, “2.5-Gb/s transmission over 50 km with a 1.3-µm vertical-cavity surface-emitting laser,” IEEE Photon Technol. Lett. 12, 1707–1709 (2000).
    [CrossRef]
  6. O. Blum, I. J. Fritz, L. R. Dawson, T. J. Drummond, “Digital alloy AlAsSb/AlGaAsSb distributed Bragg reflectors lattice matched to InP for 1.3–1.55 µm wavelength range,” Electron. Lett. 31, 1247–1248 (1995).
    [CrossRef]
  7. A. Khol, J. C. Harmand, J. L. Oudar, E. V. K. Rao, R. Kuszelewicz, E. L. Deplon, “AlGaAsSb/AlAsSb microcavity designed for 1.55 µm and grown by molecular beam epitaxy,” Electron. Lett. 33, 708–710 (1997).
    [CrossRef]
  8. F. Genty, G. Almuneau, N. Bertru, L. Chusseau, P. Grech, D. Cot, J. Jacquet, “Molecular beam epitaxy growth and characterizations of AlGaAsSb/AlAsSb Bragg reflectors on InP,” J. Cryst. Growth 183, 15–22 (1998).
    [CrossRef]
  9. G. Almuneau, E. Hall, S. Nakagawa, J. K. Kim, D. Lofgreen, C. Luo, D. R. Clarke, J. H. English, L. A. Coldren, “Molecular beam epitaxial growth of monolithic 1.55 µm vertical cavity surface emitting lasers with AlGaAsSb/AlAsSb Bragg mirrors,” J. Vac. Sci. Technol. B 18, 1601–1604 (2000).
    [CrossRef]
  10. S. Callard, A. Gagnaire, M. P. Besland, J. Joseph, “Adapted wavelength method from in situ ellipsometry,” Thin Solid Films 479, 313–314 (1998).
  11. A. C. Boccara, D. Fournier, J. Badoz, “Thermo-optical spectroscopy: detection by the mirage effect,” Appl. Phys. Lett. 36, 130–132 (1980).
    [CrossRef]
  12. J. C. Murphy, L. C. Aamodt, “Photothermal spectroscopy using optical beam probing: mirage effect,” J. Appl. Phys. 9, 4580–4588 (1980).
    [CrossRef]
  13. N. Yacoubi, B. Girault, J. Fesquet, “Determination of absorption coefficient and thermal conductivity of GaAlAs/GaAs heterostructure using a photothermal method,” Appl. Opt. 25, 4622–4625 (1986).
    [CrossRef]
  14. N. Yacoubi, A. Hafaiedh, A. Joullié, “Determination of the optical and thermal properties of semiconductors with the photothermal method,” Appl. Opt. 33, 7171–7174 (1994).
    [CrossRef] [PubMed]
  15. N. Yacoubi, C. Alibert, “Determination of very thin semiconductor layer thickness by a photothermal method,” J. Appl. Phys. 69, 8310–8312 (1991).
    [CrossRef]
  16. F. Saadallah, N. Yacoubi, A. Hafaiedh, “Determination of thermal properties of semiconductors using the photothermal method in the many thin layer cases,” Opt. Mater. 6, 35–39 (1996).
    [CrossRef]
  17. S. Adachi, “Lattice thermal resistivity of III–V compound alloys,” J. Appl. Phys. 54, 1844–1848 (1983).
    [CrossRef]
  18. C. Alibert, M. Skouri, A. Joullié, M. Benoua, S. Sadiq, “Refractive indices of AlSb and GaSb-lattice-matched AlxGa1–xAsySb1–y in the transparent wavelength region,” J. Appl. Phys. 69, 3208–3211 (1991).
    [CrossRef]
  19. M. Muñoz, K. Wei, F. H. Pollak, J. L. Freeouf, G. W. Charache, “Spectral ellipsometry of GaSb: experiment and modeling,” Phys. Rev. B 60, 8105–8110 (1999).
    [CrossRef]
  20. M. Guden, J. Piprek, “Material parameters of quaternary III–V semiconductors for multilayer mirrors at 1.55 µm wavelength,” Modell. Simul. Mater. Sci. Eng. 4, 349–357 (1996).
    [CrossRef]
  21. R. Ferrini, M. Patrini, S. Franchi, “Optical functions from 0.02 to 6 eV of AlxGa1–xSb/GaSb epitaxial layers,” J. Appl. Phys. 84, 4517–4524 (1998).
    [CrossRef]
  22. J. Tauc, R. Grigorovici, A. Vancu, “Optical properties and electronic structure of amorphous germanium,” Phys. Status Solidi 15, 627–637 (1966).
    [CrossRef]
  23. S. Adachi, “Band gaps and refractive indices of AlGaAsSb, GaInAsSb, and InPAsSb:key properties for a variety of the 2–4-µm optoelectronic device applications,” J. Appl. Phys. 61, 4869–4876 (1987).
    [CrossRef]

2000

J. Geske, V. Jayaraman, T. Goodwin, M. Culick, M. MacDougal, T. Goodnough, D. Welch, J. E. Bowers, “2.5-Gb/s transmission over 50 km with a 1.3-µm vertical-cavity surface-emitting laser,” IEEE Photon Technol. Lett. 12, 1707–1709 (2000).
[CrossRef]

G. Almuneau, E. Hall, S. Nakagawa, J. K. Kim, D. Lofgreen, C. Luo, D. R. Clarke, J. H. English, L. A. Coldren, “Molecular beam epitaxial growth of monolithic 1.55 µm vertical cavity surface emitting lasers with AlGaAsSb/AlAsSb Bragg mirrors,” J. Vac. Sci. Technol. B 18, 1601–1604 (2000).
[CrossRef]

1999

J. Koeth, R. Dietrich, A. Forchel, “GaSb vertical-cavity surface-emitting lasers for the 1.5 µm rang,” Appl. Phys. Lett. 72, 1638–1640 (1999).
[CrossRef]

M. Muñoz, K. Wei, F. H. Pollak, J. L. Freeouf, G. W. Charache, “Spectral ellipsometry of GaSb: experiment and modeling,” Phys. Rev. B 60, 8105–8110 (1999).
[CrossRef]

1998

F. Genty, G. Almuneau, N. Bertru, L. Chusseau, P. Grech, D. Cot, J. Jacquet, “Molecular beam epitaxy growth and characterizations of AlGaAsSb/AlAsSb Bragg reflectors on InP,” J. Cryst. Growth 183, 15–22 (1998).
[CrossRef]

S. Callard, A. Gagnaire, M. P. Besland, J. Joseph, “Adapted wavelength method from in situ ellipsometry,” Thin Solid Films 479, 313–314 (1998).

I. Vurgaftman, J. R. Meyer, L. R. Ram-Mohan, “Mid-IR vertical-cavity surface-emitting lasers,” IEEE J. Quantum Electron. 34, 147–156 (1998).
[CrossRef]

R. Ferrini, M. Patrini, S. Franchi, “Optical functions from 0.02 to 6 eV of AlxGa1–xSb/GaSb epitaxial layers,” J. Appl. Phys. 84, 4517–4524 (1998).
[CrossRef]

1997

C. L. Felix, W. W. Bewley, I. Vurgaftman, J. R. Meyer, L. Goldberg, D. H. Chow, E. Selvig, “Midinfrared vertical-cavity surface-emitting laser,” Appl. Phys. Lett. 71, 3483–3485 (1997).
[CrossRef]

A. Khol, J. C. Harmand, J. L. Oudar, E. V. K. Rao, R. Kuszelewicz, E. L. Deplon, “AlGaAsSb/AlAsSb microcavity designed for 1.55 µm and grown by molecular beam epitaxy,” Electron. Lett. 33, 708–710 (1997).
[CrossRef]

1996

M. Guden, J. Piprek, “Material parameters of quaternary III–V semiconductors for multilayer mirrors at 1.55 µm wavelength,” Modell. Simul. Mater. Sci. Eng. 4, 349–357 (1996).
[CrossRef]

F. Saadallah, N. Yacoubi, A. Hafaiedh, “Determination of thermal properties of semiconductors using the photothermal method in the many thin layer cases,” Opt. Mater. 6, 35–39 (1996).
[CrossRef]

1995

O. Blum, I. J. Fritz, L. R. Dawson, T. J. Drummond, “Digital alloy AlAsSb/AlGaAsSb distributed Bragg reflectors lattice matched to InP for 1.3–1.55 µm wavelength range,” Electron. Lett. 31, 1247–1248 (1995).
[CrossRef]

1994

1991

C. Alibert, M. Skouri, A. Joullié, M. Benoua, S. Sadiq, “Refractive indices of AlSb and GaSb-lattice-matched AlxGa1–xAsySb1–y in the transparent wavelength region,” J. Appl. Phys. 69, 3208–3211 (1991).
[CrossRef]

N. Yacoubi, C. Alibert, “Determination of very thin semiconductor layer thickness by a photothermal method,” J. Appl. Phys. 69, 8310–8312 (1991).
[CrossRef]

1987

S. Adachi, “Band gaps and refractive indices of AlGaAsSb, GaInAsSb, and InPAsSb:key properties for a variety of the 2–4-µm optoelectronic device applications,” J. Appl. Phys. 61, 4869–4876 (1987).
[CrossRef]

1986

1983

S. Adachi, “Lattice thermal resistivity of III–V compound alloys,” J. Appl. Phys. 54, 1844–1848 (1983).
[CrossRef]

1980

A. C. Boccara, D. Fournier, J. Badoz, “Thermo-optical spectroscopy: detection by the mirage effect,” Appl. Phys. Lett. 36, 130–132 (1980).
[CrossRef]

J. C. Murphy, L. C. Aamodt, “Photothermal spectroscopy using optical beam probing: mirage effect,” J. Appl. Phys. 9, 4580–4588 (1980).
[CrossRef]

1966

J. Tauc, R. Grigorovici, A. Vancu, “Optical properties and electronic structure of amorphous germanium,” Phys. Status Solidi 15, 627–637 (1966).
[CrossRef]

Aamodt, L. C.

J. C. Murphy, L. C. Aamodt, “Photothermal spectroscopy using optical beam probing: mirage effect,” J. Appl. Phys. 9, 4580–4588 (1980).
[CrossRef]

Adachi, S.

S. Adachi, “Band gaps and refractive indices of AlGaAsSb, GaInAsSb, and InPAsSb:key properties for a variety of the 2–4-µm optoelectronic device applications,” J. Appl. Phys. 61, 4869–4876 (1987).
[CrossRef]

S. Adachi, “Lattice thermal resistivity of III–V compound alloys,” J. Appl. Phys. 54, 1844–1848 (1983).
[CrossRef]

Alibert, C.

N. Yacoubi, C. Alibert, “Determination of very thin semiconductor layer thickness by a photothermal method,” J. Appl. Phys. 69, 8310–8312 (1991).
[CrossRef]

C. Alibert, M. Skouri, A. Joullié, M. Benoua, S. Sadiq, “Refractive indices of AlSb and GaSb-lattice-matched AlxGa1–xAsySb1–y in the transparent wavelength region,” J. Appl. Phys. 69, 3208–3211 (1991).
[CrossRef]

Alibert, C. L.

A. N. Baranov, N. Bertru, Y. Cuminal, G. Goissier, Y. Rouillard, J. C. Nicolas, P. Grech, A. F. Joullie, C. L. Alibert, “Mid-infrared GaSb-InAs-based multiple quantum well lasers,” in In-Plane Seminconductor Lasers: from Ultraviolet to Mid-Infrared II, H. K. Choi, P. S. Lory, eds., Proc. SPIE3284, 247–257 (1998).

Almuneau, G.

G. Almuneau, E. Hall, S. Nakagawa, J. K. Kim, D. Lofgreen, C. Luo, D. R. Clarke, J. H. English, L. A. Coldren, “Molecular beam epitaxial growth of monolithic 1.55 µm vertical cavity surface emitting lasers with AlGaAsSb/AlAsSb Bragg mirrors,” J. Vac. Sci. Technol. B 18, 1601–1604 (2000).
[CrossRef]

F. Genty, G. Almuneau, N. Bertru, L. Chusseau, P. Grech, D. Cot, J. Jacquet, “Molecular beam epitaxy growth and characterizations of AlGaAsSb/AlAsSb Bragg reflectors on InP,” J. Cryst. Growth 183, 15–22 (1998).
[CrossRef]

Badoz, J.

A. C. Boccara, D. Fournier, J. Badoz, “Thermo-optical spectroscopy: detection by the mirage effect,” Appl. Phys. Lett. 36, 130–132 (1980).
[CrossRef]

Baranov, A. N.

A. N. Baranov, N. Bertru, Y. Cuminal, G. Goissier, Y. Rouillard, J. C. Nicolas, P. Grech, A. F. Joullie, C. L. Alibert, “Mid-infrared GaSb-InAs-based multiple quantum well lasers,” in In-Plane Seminconductor Lasers: from Ultraviolet to Mid-Infrared II, H. K. Choi, P. S. Lory, eds., Proc. SPIE3284, 247–257 (1998).

Benoua, M.

C. Alibert, M. Skouri, A. Joullié, M. Benoua, S. Sadiq, “Refractive indices of AlSb and GaSb-lattice-matched AlxGa1–xAsySb1–y in the transparent wavelength region,” J. Appl. Phys. 69, 3208–3211 (1991).
[CrossRef]

Bertru, N.

F. Genty, G. Almuneau, N. Bertru, L. Chusseau, P. Grech, D. Cot, J. Jacquet, “Molecular beam epitaxy growth and characterizations of AlGaAsSb/AlAsSb Bragg reflectors on InP,” J. Cryst. Growth 183, 15–22 (1998).
[CrossRef]

A. N. Baranov, N. Bertru, Y. Cuminal, G. Goissier, Y. Rouillard, J. C. Nicolas, P. Grech, A. F. Joullie, C. L. Alibert, “Mid-infrared GaSb-InAs-based multiple quantum well lasers,” in In-Plane Seminconductor Lasers: from Ultraviolet to Mid-Infrared II, H. K. Choi, P. S. Lory, eds., Proc. SPIE3284, 247–257 (1998).

Besland, M. P.

S. Callard, A. Gagnaire, M. P. Besland, J. Joseph, “Adapted wavelength method from in situ ellipsometry,” Thin Solid Films 479, 313–314 (1998).

Bewley, W. W.

C. L. Felix, W. W. Bewley, I. Vurgaftman, J. R. Meyer, L. Goldberg, D. H. Chow, E. Selvig, “Midinfrared vertical-cavity surface-emitting laser,” Appl. Phys. Lett. 71, 3483–3485 (1997).
[CrossRef]

Blum, O.

O. Blum, I. J. Fritz, L. R. Dawson, T. J. Drummond, “Digital alloy AlAsSb/AlGaAsSb distributed Bragg reflectors lattice matched to InP for 1.3–1.55 µm wavelength range,” Electron. Lett. 31, 1247–1248 (1995).
[CrossRef]

Boccara, A. C.

A. C. Boccara, D. Fournier, J. Badoz, “Thermo-optical spectroscopy: detection by the mirage effect,” Appl. Phys. Lett. 36, 130–132 (1980).
[CrossRef]

Bowers, J. E.

J. Geske, V. Jayaraman, T. Goodwin, M. Culick, M. MacDougal, T. Goodnough, D. Welch, J. E. Bowers, “2.5-Gb/s transmission over 50 km with a 1.3-µm vertical-cavity surface-emitting laser,” IEEE Photon Technol. Lett. 12, 1707–1709 (2000).
[CrossRef]

Callard, S.

S. Callard, A. Gagnaire, M. P. Besland, J. Joseph, “Adapted wavelength method from in situ ellipsometry,” Thin Solid Films 479, 313–314 (1998).

Charache, G. W.

M. Muñoz, K. Wei, F. H. Pollak, J. L. Freeouf, G. W. Charache, “Spectral ellipsometry of GaSb: experiment and modeling,” Phys. Rev. B 60, 8105–8110 (1999).
[CrossRef]

Chow, D. H.

C. L. Felix, W. W. Bewley, I. Vurgaftman, J. R. Meyer, L. Goldberg, D. H. Chow, E. Selvig, “Midinfrared vertical-cavity surface-emitting laser,” Appl. Phys. Lett. 71, 3483–3485 (1997).
[CrossRef]

Chusseau, L.

F. Genty, G. Almuneau, N. Bertru, L. Chusseau, P. Grech, D. Cot, J. Jacquet, “Molecular beam epitaxy growth and characterizations of AlGaAsSb/AlAsSb Bragg reflectors on InP,” J. Cryst. Growth 183, 15–22 (1998).
[CrossRef]

Clarke, D. R.

G. Almuneau, E. Hall, S. Nakagawa, J. K. Kim, D. Lofgreen, C. Luo, D. R. Clarke, J. H. English, L. A. Coldren, “Molecular beam epitaxial growth of monolithic 1.55 µm vertical cavity surface emitting lasers with AlGaAsSb/AlAsSb Bragg mirrors,” J. Vac. Sci. Technol. B 18, 1601–1604 (2000).
[CrossRef]

Coldren, L. A.

G. Almuneau, E. Hall, S. Nakagawa, J. K. Kim, D. Lofgreen, C. Luo, D. R. Clarke, J. H. English, L. A. Coldren, “Molecular beam epitaxial growth of monolithic 1.55 µm vertical cavity surface emitting lasers with AlGaAsSb/AlAsSb Bragg mirrors,” J. Vac. Sci. Technol. B 18, 1601–1604 (2000).
[CrossRef]

Cot, D.

F. Genty, G. Almuneau, N. Bertru, L. Chusseau, P. Grech, D. Cot, J. Jacquet, “Molecular beam epitaxy growth and characterizations of AlGaAsSb/AlAsSb Bragg reflectors on InP,” J. Cryst. Growth 183, 15–22 (1998).
[CrossRef]

Culick, M.

J. Geske, V. Jayaraman, T. Goodwin, M. Culick, M. MacDougal, T. Goodnough, D. Welch, J. E. Bowers, “2.5-Gb/s transmission over 50 km with a 1.3-µm vertical-cavity surface-emitting laser,” IEEE Photon Technol. Lett. 12, 1707–1709 (2000).
[CrossRef]

Cuminal, Y.

A. N. Baranov, N. Bertru, Y. Cuminal, G. Goissier, Y. Rouillard, J. C. Nicolas, P. Grech, A. F. Joullie, C. L. Alibert, “Mid-infrared GaSb-InAs-based multiple quantum well lasers,” in In-Plane Seminconductor Lasers: from Ultraviolet to Mid-Infrared II, H. K. Choi, P. S. Lory, eds., Proc. SPIE3284, 247–257 (1998).

Dawson, L. R.

O. Blum, I. J. Fritz, L. R. Dawson, T. J. Drummond, “Digital alloy AlAsSb/AlGaAsSb distributed Bragg reflectors lattice matched to InP for 1.3–1.55 µm wavelength range,” Electron. Lett. 31, 1247–1248 (1995).
[CrossRef]

Deplon, E. L.

A. Khol, J. C. Harmand, J. L. Oudar, E. V. K. Rao, R. Kuszelewicz, E. L. Deplon, “AlGaAsSb/AlAsSb microcavity designed for 1.55 µm and grown by molecular beam epitaxy,” Electron. Lett. 33, 708–710 (1997).
[CrossRef]

Dietrich, R.

J. Koeth, R. Dietrich, A. Forchel, “GaSb vertical-cavity surface-emitting lasers for the 1.5 µm rang,” Appl. Phys. Lett. 72, 1638–1640 (1999).
[CrossRef]

Drummond, T. J.

O. Blum, I. J. Fritz, L. R. Dawson, T. J. Drummond, “Digital alloy AlAsSb/AlGaAsSb distributed Bragg reflectors lattice matched to InP for 1.3–1.55 µm wavelength range,” Electron. Lett. 31, 1247–1248 (1995).
[CrossRef]

English, J. H.

G. Almuneau, E. Hall, S. Nakagawa, J. K. Kim, D. Lofgreen, C. Luo, D. R. Clarke, J. H. English, L. A. Coldren, “Molecular beam epitaxial growth of monolithic 1.55 µm vertical cavity surface emitting lasers with AlGaAsSb/AlAsSb Bragg mirrors,” J. Vac. Sci. Technol. B 18, 1601–1604 (2000).
[CrossRef]

Felix, C. L.

C. L. Felix, W. W. Bewley, I. Vurgaftman, J. R. Meyer, L. Goldberg, D. H. Chow, E. Selvig, “Midinfrared vertical-cavity surface-emitting laser,” Appl. Phys. Lett. 71, 3483–3485 (1997).
[CrossRef]

Ferrini, R.

R. Ferrini, M. Patrini, S. Franchi, “Optical functions from 0.02 to 6 eV of AlxGa1–xSb/GaSb epitaxial layers,” J. Appl. Phys. 84, 4517–4524 (1998).
[CrossRef]

Fesquet, J.

Forchel, A.

J. Koeth, R. Dietrich, A. Forchel, “GaSb vertical-cavity surface-emitting lasers for the 1.5 µm rang,” Appl. Phys. Lett. 72, 1638–1640 (1999).
[CrossRef]

Fournier, D.

A. C. Boccara, D. Fournier, J. Badoz, “Thermo-optical spectroscopy: detection by the mirage effect,” Appl. Phys. Lett. 36, 130–132 (1980).
[CrossRef]

Franchi, S.

R. Ferrini, M. Patrini, S. Franchi, “Optical functions from 0.02 to 6 eV of AlxGa1–xSb/GaSb epitaxial layers,” J. Appl. Phys. 84, 4517–4524 (1998).
[CrossRef]

Freeouf, J. L.

M. Muñoz, K. Wei, F. H. Pollak, J. L. Freeouf, G. W. Charache, “Spectral ellipsometry of GaSb: experiment and modeling,” Phys. Rev. B 60, 8105–8110 (1999).
[CrossRef]

Fritz, I. J.

O. Blum, I. J. Fritz, L. R. Dawson, T. J. Drummond, “Digital alloy AlAsSb/AlGaAsSb distributed Bragg reflectors lattice matched to InP for 1.3–1.55 µm wavelength range,” Electron. Lett. 31, 1247–1248 (1995).
[CrossRef]

Gagnaire, A.

S. Callard, A. Gagnaire, M. P. Besland, J. Joseph, “Adapted wavelength method from in situ ellipsometry,” Thin Solid Films 479, 313–314 (1998).

Genty, F.

F. Genty, G. Almuneau, N. Bertru, L. Chusseau, P. Grech, D. Cot, J. Jacquet, “Molecular beam epitaxy growth and characterizations of AlGaAsSb/AlAsSb Bragg reflectors on InP,” J. Cryst. Growth 183, 15–22 (1998).
[CrossRef]

Geske, J.

J. Geske, V. Jayaraman, T. Goodwin, M. Culick, M. MacDougal, T. Goodnough, D. Welch, J. E. Bowers, “2.5-Gb/s transmission over 50 km with a 1.3-µm vertical-cavity surface-emitting laser,” IEEE Photon Technol. Lett. 12, 1707–1709 (2000).
[CrossRef]

Girault, B.

Goissier, G.

A. N. Baranov, N. Bertru, Y. Cuminal, G. Goissier, Y. Rouillard, J. C. Nicolas, P. Grech, A. F. Joullie, C. L. Alibert, “Mid-infrared GaSb-InAs-based multiple quantum well lasers,” in In-Plane Seminconductor Lasers: from Ultraviolet to Mid-Infrared II, H. K. Choi, P. S. Lory, eds., Proc. SPIE3284, 247–257 (1998).

Goldberg, L.

C. L. Felix, W. W. Bewley, I. Vurgaftman, J. R. Meyer, L. Goldberg, D. H. Chow, E. Selvig, “Midinfrared vertical-cavity surface-emitting laser,” Appl. Phys. Lett. 71, 3483–3485 (1997).
[CrossRef]

Goodnough, T.

J. Geske, V. Jayaraman, T. Goodwin, M. Culick, M. MacDougal, T. Goodnough, D. Welch, J. E. Bowers, “2.5-Gb/s transmission over 50 km with a 1.3-µm vertical-cavity surface-emitting laser,” IEEE Photon Technol. Lett. 12, 1707–1709 (2000).
[CrossRef]

Goodwin, T.

J. Geske, V. Jayaraman, T. Goodwin, M. Culick, M. MacDougal, T. Goodnough, D. Welch, J. E. Bowers, “2.5-Gb/s transmission over 50 km with a 1.3-µm vertical-cavity surface-emitting laser,” IEEE Photon Technol. Lett. 12, 1707–1709 (2000).
[CrossRef]

Grech, P.

F. Genty, G. Almuneau, N. Bertru, L. Chusseau, P. Grech, D. Cot, J. Jacquet, “Molecular beam epitaxy growth and characterizations of AlGaAsSb/AlAsSb Bragg reflectors on InP,” J. Cryst. Growth 183, 15–22 (1998).
[CrossRef]

A. N. Baranov, N. Bertru, Y. Cuminal, G. Goissier, Y. Rouillard, J. C. Nicolas, P. Grech, A. F. Joullie, C. L. Alibert, “Mid-infrared GaSb-InAs-based multiple quantum well lasers,” in In-Plane Seminconductor Lasers: from Ultraviolet to Mid-Infrared II, H. K. Choi, P. S. Lory, eds., Proc. SPIE3284, 247–257 (1998).

Grigorovici, R.

J. Tauc, R. Grigorovici, A. Vancu, “Optical properties and electronic structure of amorphous germanium,” Phys. Status Solidi 15, 627–637 (1966).
[CrossRef]

Guden, M.

M. Guden, J. Piprek, “Material parameters of quaternary III–V semiconductors for multilayer mirrors at 1.55 µm wavelength,” Modell. Simul. Mater. Sci. Eng. 4, 349–357 (1996).
[CrossRef]

Hafaiedh, A.

F. Saadallah, N. Yacoubi, A. Hafaiedh, “Determination of thermal properties of semiconductors using the photothermal method in the many thin layer cases,” Opt. Mater. 6, 35–39 (1996).
[CrossRef]

N. Yacoubi, A. Hafaiedh, A. Joullié, “Determination of the optical and thermal properties of semiconductors with the photothermal method,” Appl. Opt. 33, 7171–7174 (1994).
[CrossRef] [PubMed]

Hall, E.

G. Almuneau, E. Hall, S. Nakagawa, J. K. Kim, D. Lofgreen, C. Luo, D. R. Clarke, J. H. English, L. A. Coldren, “Molecular beam epitaxial growth of monolithic 1.55 µm vertical cavity surface emitting lasers with AlGaAsSb/AlAsSb Bragg mirrors,” J. Vac. Sci. Technol. B 18, 1601–1604 (2000).
[CrossRef]

Harmand, J. C.

A. Khol, J. C. Harmand, J. L. Oudar, E. V. K. Rao, R. Kuszelewicz, E. L. Deplon, “AlGaAsSb/AlAsSb microcavity designed for 1.55 µm and grown by molecular beam epitaxy,” Electron. Lett. 33, 708–710 (1997).
[CrossRef]

Jacquet, J.

F. Genty, G. Almuneau, N. Bertru, L. Chusseau, P. Grech, D. Cot, J. Jacquet, “Molecular beam epitaxy growth and characterizations of AlGaAsSb/AlAsSb Bragg reflectors on InP,” J. Cryst. Growth 183, 15–22 (1998).
[CrossRef]

Jayaraman, V.

J. Geske, V. Jayaraman, T. Goodwin, M. Culick, M. MacDougal, T. Goodnough, D. Welch, J. E. Bowers, “2.5-Gb/s transmission over 50 km with a 1.3-µm vertical-cavity surface-emitting laser,” IEEE Photon Technol. Lett. 12, 1707–1709 (2000).
[CrossRef]

Joseph, J.

S. Callard, A. Gagnaire, M. P. Besland, J. Joseph, “Adapted wavelength method from in situ ellipsometry,” Thin Solid Films 479, 313–314 (1998).

Joullie, A. F.

A. N. Baranov, N. Bertru, Y. Cuminal, G. Goissier, Y. Rouillard, J. C. Nicolas, P. Grech, A. F. Joullie, C. L. Alibert, “Mid-infrared GaSb-InAs-based multiple quantum well lasers,” in In-Plane Seminconductor Lasers: from Ultraviolet to Mid-Infrared II, H. K. Choi, P. S. Lory, eds., Proc. SPIE3284, 247–257 (1998).

Joullié, A.

N. Yacoubi, A. Hafaiedh, A. Joullié, “Determination of the optical and thermal properties of semiconductors with the photothermal method,” Appl. Opt. 33, 7171–7174 (1994).
[CrossRef] [PubMed]

C. Alibert, M. Skouri, A. Joullié, M. Benoua, S. Sadiq, “Refractive indices of AlSb and GaSb-lattice-matched AlxGa1–xAsySb1–y in the transparent wavelength region,” J. Appl. Phys. 69, 3208–3211 (1991).
[CrossRef]

Khol, A.

A. Khol, J. C. Harmand, J. L. Oudar, E. V. K. Rao, R. Kuszelewicz, E. L. Deplon, “AlGaAsSb/AlAsSb microcavity designed for 1.55 µm and grown by molecular beam epitaxy,” Electron. Lett. 33, 708–710 (1997).
[CrossRef]

Kim, J. K.

G. Almuneau, E. Hall, S. Nakagawa, J. K. Kim, D. Lofgreen, C. Luo, D. R. Clarke, J. H. English, L. A. Coldren, “Molecular beam epitaxial growth of monolithic 1.55 µm vertical cavity surface emitting lasers with AlGaAsSb/AlAsSb Bragg mirrors,” J. Vac. Sci. Technol. B 18, 1601–1604 (2000).
[CrossRef]

Koeth, J.

J. Koeth, R. Dietrich, A. Forchel, “GaSb vertical-cavity surface-emitting lasers for the 1.5 µm rang,” Appl. Phys. Lett. 72, 1638–1640 (1999).
[CrossRef]

Kuszelewicz, R.

A. Khol, J. C. Harmand, J. L. Oudar, E. V. K. Rao, R. Kuszelewicz, E. L. Deplon, “AlGaAsSb/AlAsSb microcavity designed for 1.55 µm and grown by molecular beam epitaxy,” Electron. Lett. 33, 708–710 (1997).
[CrossRef]

Lofgreen, D.

G. Almuneau, E. Hall, S. Nakagawa, J. K. Kim, D. Lofgreen, C. Luo, D. R. Clarke, J. H. English, L. A. Coldren, “Molecular beam epitaxial growth of monolithic 1.55 µm vertical cavity surface emitting lasers with AlGaAsSb/AlAsSb Bragg mirrors,” J. Vac. Sci. Technol. B 18, 1601–1604 (2000).
[CrossRef]

Luo, C.

G. Almuneau, E. Hall, S. Nakagawa, J. K. Kim, D. Lofgreen, C. Luo, D. R. Clarke, J. H. English, L. A. Coldren, “Molecular beam epitaxial growth of monolithic 1.55 µm vertical cavity surface emitting lasers with AlGaAsSb/AlAsSb Bragg mirrors,” J. Vac. Sci. Technol. B 18, 1601–1604 (2000).
[CrossRef]

MacDougal, M.

J. Geske, V. Jayaraman, T. Goodwin, M. Culick, M. MacDougal, T. Goodnough, D. Welch, J. E. Bowers, “2.5-Gb/s transmission over 50 km with a 1.3-µm vertical-cavity surface-emitting laser,” IEEE Photon Technol. Lett. 12, 1707–1709 (2000).
[CrossRef]

Meyer, J. R.

I. Vurgaftman, J. R. Meyer, L. R. Ram-Mohan, “Mid-IR vertical-cavity surface-emitting lasers,” IEEE J. Quantum Electron. 34, 147–156 (1998).
[CrossRef]

C. L. Felix, W. W. Bewley, I. Vurgaftman, J. R. Meyer, L. Goldberg, D. H. Chow, E. Selvig, “Midinfrared vertical-cavity surface-emitting laser,” Appl. Phys. Lett. 71, 3483–3485 (1997).
[CrossRef]

Muñoz, M.

M. Muñoz, K. Wei, F. H. Pollak, J. L. Freeouf, G. W. Charache, “Spectral ellipsometry of GaSb: experiment and modeling,” Phys. Rev. B 60, 8105–8110 (1999).
[CrossRef]

Murphy, J. C.

J. C. Murphy, L. C. Aamodt, “Photothermal spectroscopy using optical beam probing: mirage effect,” J. Appl. Phys. 9, 4580–4588 (1980).
[CrossRef]

Nakagawa, S.

G. Almuneau, E. Hall, S. Nakagawa, J. K. Kim, D. Lofgreen, C. Luo, D. R. Clarke, J. H. English, L. A. Coldren, “Molecular beam epitaxial growth of monolithic 1.55 µm vertical cavity surface emitting lasers with AlGaAsSb/AlAsSb Bragg mirrors,” J. Vac. Sci. Technol. B 18, 1601–1604 (2000).
[CrossRef]

Nicolas, J. C.

A. N. Baranov, N. Bertru, Y. Cuminal, G. Goissier, Y. Rouillard, J. C. Nicolas, P. Grech, A. F. Joullie, C. L. Alibert, “Mid-infrared GaSb-InAs-based multiple quantum well lasers,” in In-Plane Seminconductor Lasers: from Ultraviolet to Mid-Infrared II, H. K. Choi, P. S. Lory, eds., Proc. SPIE3284, 247–257 (1998).

Oudar, J. L.

A. Khol, J. C. Harmand, J. L. Oudar, E. V. K. Rao, R. Kuszelewicz, E. L. Deplon, “AlGaAsSb/AlAsSb microcavity designed for 1.55 µm and grown by molecular beam epitaxy,” Electron. Lett. 33, 708–710 (1997).
[CrossRef]

Patrini, M.

R. Ferrini, M. Patrini, S. Franchi, “Optical functions from 0.02 to 6 eV of AlxGa1–xSb/GaSb epitaxial layers,” J. Appl. Phys. 84, 4517–4524 (1998).
[CrossRef]

Piprek, J.

M. Guden, J. Piprek, “Material parameters of quaternary III–V semiconductors for multilayer mirrors at 1.55 µm wavelength,” Modell. Simul. Mater. Sci. Eng. 4, 349–357 (1996).
[CrossRef]

Pollak, F. H.

M. Muñoz, K. Wei, F. H. Pollak, J. L. Freeouf, G. W. Charache, “Spectral ellipsometry of GaSb: experiment and modeling,” Phys. Rev. B 60, 8105–8110 (1999).
[CrossRef]

Ram-Mohan, L. R.

I. Vurgaftman, J. R. Meyer, L. R. Ram-Mohan, “Mid-IR vertical-cavity surface-emitting lasers,” IEEE J. Quantum Electron. 34, 147–156 (1998).
[CrossRef]

Rao, E. V. K.

A. Khol, J. C. Harmand, J. L. Oudar, E. V. K. Rao, R. Kuszelewicz, E. L. Deplon, “AlGaAsSb/AlAsSb microcavity designed for 1.55 µm and grown by molecular beam epitaxy,” Electron. Lett. 33, 708–710 (1997).
[CrossRef]

Rouillard, Y.

A. N. Baranov, N. Bertru, Y. Cuminal, G. Goissier, Y. Rouillard, J. C. Nicolas, P. Grech, A. F. Joullie, C. L. Alibert, “Mid-infrared GaSb-InAs-based multiple quantum well lasers,” in In-Plane Seminconductor Lasers: from Ultraviolet to Mid-Infrared II, H. K. Choi, P. S. Lory, eds., Proc. SPIE3284, 247–257 (1998).

Saadallah, F.

F. Saadallah, N. Yacoubi, A. Hafaiedh, “Determination of thermal properties of semiconductors using the photothermal method in the many thin layer cases,” Opt. Mater. 6, 35–39 (1996).
[CrossRef]

Sadiq, S.

C. Alibert, M. Skouri, A. Joullié, M. Benoua, S. Sadiq, “Refractive indices of AlSb and GaSb-lattice-matched AlxGa1–xAsySb1–y in the transparent wavelength region,” J. Appl. Phys. 69, 3208–3211 (1991).
[CrossRef]

Selvig, E.

C. L. Felix, W. W. Bewley, I. Vurgaftman, J. R. Meyer, L. Goldberg, D. H. Chow, E. Selvig, “Midinfrared vertical-cavity surface-emitting laser,” Appl. Phys. Lett. 71, 3483–3485 (1997).
[CrossRef]

Skouri, M.

C. Alibert, M. Skouri, A. Joullié, M. Benoua, S. Sadiq, “Refractive indices of AlSb and GaSb-lattice-matched AlxGa1–xAsySb1–y in the transparent wavelength region,” J. Appl. Phys. 69, 3208–3211 (1991).
[CrossRef]

Tauc, J.

J. Tauc, R. Grigorovici, A. Vancu, “Optical properties and electronic structure of amorphous germanium,” Phys. Status Solidi 15, 627–637 (1966).
[CrossRef]

Vancu, A.

J. Tauc, R. Grigorovici, A. Vancu, “Optical properties and electronic structure of amorphous germanium,” Phys. Status Solidi 15, 627–637 (1966).
[CrossRef]

Vurgaftman, I.

I. Vurgaftman, J. R. Meyer, L. R. Ram-Mohan, “Mid-IR vertical-cavity surface-emitting lasers,” IEEE J. Quantum Electron. 34, 147–156 (1998).
[CrossRef]

C. L. Felix, W. W. Bewley, I. Vurgaftman, J. R. Meyer, L. Goldberg, D. H. Chow, E. Selvig, “Midinfrared vertical-cavity surface-emitting laser,” Appl. Phys. Lett. 71, 3483–3485 (1997).
[CrossRef]

Wei, K.

M. Muñoz, K. Wei, F. H. Pollak, J. L. Freeouf, G. W. Charache, “Spectral ellipsometry of GaSb: experiment and modeling,” Phys. Rev. B 60, 8105–8110 (1999).
[CrossRef]

Welch, D.

J. Geske, V. Jayaraman, T. Goodwin, M. Culick, M. MacDougal, T. Goodnough, D. Welch, J. E. Bowers, “2.5-Gb/s transmission over 50 km with a 1.3-µm vertical-cavity surface-emitting laser,” IEEE Photon Technol. Lett. 12, 1707–1709 (2000).
[CrossRef]

Yacoubi, N.

F. Saadallah, N. Yacoubi, A. Hafaiedh, “Determination of thermal properties of semiconductors using the photothermal method in the many thin layer cases,” Opt. Mater. 6, 35–39 (1996).
[CrossRef]

N. Yacoubi, A. Hafaiedh, A. Joullié, “Determination of the optical and thermal properties of semiconductors with the photothermal method,” Appl. Opt. 33, 7171–7174 (1994).
[CrossRef] [PubMed]

N. Yacoubi, C. Alibert, “Determination of very thin semiconductor layer thickness by a photothermal method,” J. Appl. Phys. 69, 8310–8312 (1991).
[CrossRef]

N. Yacoubi, B. Girault, J. Fesquet, “Determination of absorption coefficient and thermal conductivity of GaAlAs/GaAs heterostructure using a photothermal method,” Appl. Opt. 25, 4622–4625 (1986).
[CrossRef]

Appl. Opt.

Appl. Phys. Lett.

C. L. Felix, W. W. Bewley, I. Vurgaftman, J. R. Meyer, L. Goldberg, D. H. Chow, E. Selvig, “Midinfrared vertical-cavity surface-emitting laser,” Appl. Phys. Lett. 71, 3483–3485 (1997).
[CrossRef]

J. Koeth, R. Dietrich, A. Forchel, “GaSb vertical-cavity surface-emitting lasers for the 1.5 µm rang,” Appl. Phys. Lett. 72, 1638–1640 (1999).
[CrossRef]

A. C. Boccara, D. Fournier, J. Badoz, “Thermo-optical spectroscopy: detection by the mirage effect,” Appl. Phys. Lett. 36, 130–132 (1980).
[CrossRef]

Electron. Lett.

O. Blum, I. J. Fritz, L. R. Dawson, T. J. Drummond, “Digital alloy AlAsSb/AlGaAsSb distributed Bragg reflectors lattice matched to InP for 1.3–1.55 µm wavelength range,” Electron. Lett. 31, 1247–1248 (1995).
[CrossRef]

A. Khol, J. C. Harmand, J. L. Oudar, E. V. K. Rao, R. Kuszelewicz, E. L. Deplon, “AlGaAsSb/AlAsSb microcavity designed for 1.55 µm and grown by molecular beam epitaxy,” Electron. Lett. 33, 708–710 (1997).
[CrossRef]

IEEE J. Quantum Electron.

I. Vurgaftman, J. R. Meyer, L. R. Ram-Mohan, “Mid-IR vertical-cavity surface-emitting lasers,” IEEE J. Quantum Electron. 34, 147–156 (1998).
[CrossRef]

IEEE Photon Technol. Lett.

J. Geske, V. Jayaraman, T. Goodwin, M. Culick, M. MacDougal, T. Goodnough, D. Welch, J. E. Bowers, “2.5-Gb/s transmission over 50 km with a 1.3-µm vertical-cavity surface-emitting laser,” IEEE Photon Technol. Lett. 12, 1707–1709 (2000).
[CrossRef]

J. Appl. Phys.

S. Adachi, “Lattice thermal resistivity of III–V compound alloys,” J. Appl. Phys. 54, 1844–1848 (1983).
[CrossRef]

C. Alibert, M. Skouri, A. Joullié, M. Benoua, S. Sadiq, “Refractive indices of AlSb and GaSb-lattice-matched AlxGa1–xAsySb1–y in the transparent wavelength region,” J. Appl. Phys. 69, 3208–3211 (1991).
[CrossRef]

J. C. Murphy, L. C. Aamodt, “Photothermal spectroscopy using optical beam probing: mirage effect,” J. Appl. Phys. 9, 4580–4588 (1980).
[CrossRef]

N. Yacoubi, C. Alibert, “Determination of very thin semiconductor layer thickness by a photothermal method,” J. Appl. Phys. 69, 8310–8312 (1991).
[CrossRef]

R. Ferrini, M. Patrini, S. Franchi, “Optical functions from 0.02 to 6 eV of AlxGa1–xSb/GaSb epitaxial layers,” J. Appl. Phys. 84, 4517–4524 (1998).
[CrossRef]

S. Adachi, “Band gaps and refractive indices of AlGaAsSb, GaInAsSb, and InPAsSb:key properties for a variety of the 2–4-µm optoelectronic device applications,” J. Appl. Phys. 61, 4869–4876 (1987).
[CrossRef]

J. Cryst. Growth

F. Genty, G. Almuneau, N. Bertru, L. Chusseau, P. Grech, D. Cot, J. Jacquet, “Molecular beam epitaxy growth and characterizations of AlGaAsSb/AlAsSb Bragg reflectors on InP,” J. Cryst. Growth 183, 15–22 (1998).
[CrossRef]

J. Vac. Sci. Technol. B

G. Almuneau, E. Hall, S. Nakagawa, J. K. Kim, D. Lofgreen, C. Luo, D. R. Clarke, J. H. English, L. A. Coldren, “Molecular beam epitaxial growth of monolithic 1.55 µm vertical cavity surface emitting lasers with AlGaAsSb/AlAsSb Bragg mirrors,” J. Vac. Sci. Technol. B 18, 1601–1604 (2000).
[CrossRef]

Modell. Simul. Mater. Sci. Eng.

M. Guden, J. Piprek, “Material parameters of quaternary III–V semiconductors for multilayer mirrors at 1.55 µm wavelength,” Modell. Simul. Mater. Sci. Eng. 4, 349–357 (1996).
[CrossRef]

Opt. Mater.

F. Saadallah, N. Yacoubi, A. Hafaiedh, “Determination of thermal properties of semiconductors using the photothermal method in the many thin layer cases,” Opt. Mater. 6, 35–39 (1996).
[CrossRef]

Phys. Rev. B

M. Muñoz, K. Wei, F. H. Pollak, J. L. Freeouf, G. W. Charache, “Spectral ellipsometry of GaSb: experiment and modeling,” Phys. Rev. B 60, 8105–8110 (1999).
[CrossRef]

Phys. Status Solidi

J. Tauc, R. Grigorovici, A. Vancu, “Optical properties and electronic structure of amorphous germanium,” Phys. Status Solidi 15, 627–637 (1966).
[CrossRef]

Thin Solid Films

S. Callard, A. Gagnaire, M. P. Besland, J. Joseph, “Adapted wavelength method from in situ ellipsometry,” Thin Solid Films 479, 313–314 (1998).

Other

A. N. Baranov, N. Bertru, Y. Cuminal, G. Goissier, Y. Rouillard, J. C. Nicolas, P. Grech, A. F. Joullie, C. L. Alibert, “Mid-infrared GaSb-InAs-based multiple quantum well lasers,” in In-Plane Seminconductor Lasers: from Ultraviolet to Mid-Infrared II, H. K. Choi, P. S. Lory, eds., Proc. SPIE3284, 247–257 (1998).

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Figures (12)

Fig. 1
Fig. 1

Schema of a sample made up of n layers on a substrate. E ij is the electric field in each medium.

Fig. 2
Fig. 2

Normalized amplitude of the PDS signal obtained for a Bragg mirror of 31 alternate layers of GaAlAsSb and AlAsSb (symbols) and amplitude calculated with the theoretical interference model (solid curve) versus the wavelength of heating light.

Fig. 3
Fig. 3

PDS experimental phase obtained for a Bragg mirror of 31 alternate layers of GaAlAsSb and AlAsSb (symbols) and the best theoretical phase calculated with the theoretical interference model (solid curve) versus wavelength.

Fig. 4
Fig. 4

Normalized amplitude calculated with the theoretical model versus wavelength. The interference phenomenon was omitted.

Fig. 5
Fig. 5

Theoretical phase calculated with the theoretical model versus wavelength. The interference phenomenon was omitted.

Fig. 6
Fig. 6

Experimental normalized amplitude versus wavelength (open symbols) compared with theoretical curves obtained when we considered n 1 and n 2 constants equal to their values at λ0 = 1.45 µm (dashes), when we considered n 2 constant and varied n 1 to fit the experimental curve (filled symbols when we varied n 1 and n 2 until we achieved the best with the experimental curve (solid curve).

Fig. 7
Fig. 7

Refractive index n 1 of the GAA1AsSb layer versus wavelength.

Fig. 8
Fig. 8

Refractive index n 2 of the AlAsSb layer versus wavelength.

Fig. 9
Fig. 9

Effect of the average thermal conductivity K m on phase variation Δϕ. The three curves represent the theoretical phase calculated without interference for three values of k m .

Fig. 10
Fig. 10

Experimental phase versus wavelength (open symbols) is smoothed (solid curve) for comparison with the theoretical phase in Fig. 5.

Fig. 11
Fig. 11

Absorption coefficient α1 of the GaAlAsSb layer versus wavelength.

Fig. 12
Fig. 12

PDS normalized amplitude (open symbols) compared with the reflectivity spectrum (solid curve).

Equations (24)

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

Ip=Ipxexpiωt,
2Tbx2-σb2Tb=0 in the backing, 2Tsx2-σs2Ts=-12ksdIsdx in the substrate, 2Tpx2-σp2Tp=-12kpdIpdx in medium p, 2Tfx2-σf2Tf=0 in the fluid,
Tfx=T0 exp-σfx-ln, Tsx=U expσsx+V exp-σsx-Z00 expαsx-Z01 exp-αsx-Z02 cosβsx+l-Z03 sinβsx+l, Tpx=Up expσpx-lp+Vp exp(-σpx-lp-Zp0 expαpx-lp-Zp1 exp-αpx-lp-Zp2 cosβpx-Zp3 sinβpx, Tbx=W expσbx+l,
Tplp=Tp+1lp, Ts-l=Tb-l, Tnln=Tfln, Tpxlp=Tp+1xlp, Tsx-l=Tbx-l, Tnxln=Tfxln.
T0=1Bnp=0n2n-pZp0Yp0+Zp1Yp1+Zp2Yp2+Zp3Yp3,
A1=12.22127, B1=0.17826, C1=1.162×10-6 for GaAlAsSb with χ2=8×10-4, A2=8.678, B2=0.622, C2=0.837×10-6 for AlAsSb with χ2=0.015.
E0=0 exp-imfxexp-iω0t,E0=0 expimfxexp-iω0t,E1p=1p exp-impxexp-iω0t,E2p=2p expimpxexp-iω0t, p=s, 1  n,E1b=1b exp-imbxexp-iω0t,
E1x=τf10 expil1m1-mfexp-im1x+ρ1s exp(im1x.
I1=I0|τf1|2 expαf-α1l1expα1x+|ρ1s|2 exp-α1x+2Reρ1scosβ1x-Imρ1ssinβ1x.
E2x=τf20 expi12m2-mf)exp-im2x+ρ21 exp-2im211expim2x.
E1x=τf2τ210 expil1m1-m2expil2m2-mf×exp-im1x+ρ1s expim1x.
I2=I0|τf2|2 expαf-α2l2expα2x+|ρ21|2 exp2α2l1exp-α2x+2 expα2l1Reρ21cos(β2x-l1-Imρ21sinβ2x-l1
I1=I0|τf2|2|τ21|2 expαf-α1l2expα2-α1l1×expα1x+|ρ1s|2 exp-α1x+2Reρ1scosβ1x-Imρ1ssinβ1x.
Epx=0j=p+1n+1τjj-1 expilj-1mj-1-mj×exp-impx+ρpp-1 exp-2implp-1expimpx.
Ip=I0 exp-αflnj=p+1n+1|τjj-1|2 expαjhj-1×expαpx+|ρpp-1|2 exp2αplp-1exp-αpx+2 expαplp-1Reρpp-1cosβpx-1p-1-IMρpp-1sinβp(x-1p-1).
ρ1s=ns+ρsb exp2imsl1+nsρsb exp2imsl, τ1s=t1s1+nsρsb exp2imsl.
τjj-1=tjj-11+rjj-1ρj-1j exp2imj-1lj-1, ρjj-1=rjj-1+ρj-1j-2 exp2imj-1lj-11+rij-1ρj-1j-2 exp2imj-1lj-1,
rsb=ns-nbns+nb, tsb=2nsns+nb, rij-1=nj-nj-1nj+nj-1, tjj-1=2njnj+nj-1
T0=1B12Z00Y00+Z01Y01+Z02Y02+Z03Y03+Z10Y10+Z11Y11+Z12Y12+Z13Y13.
T0=1B222Z00Y00+Z01Y01+Z02Y02+Z03Y03+2Z10Y10+Z11Y11+Z12Y12+Z13Y13+Z20Y20+Z21Y21+Z22Y22+Z23Y23.
T0=1Bnp=0n2n-pZp0Yp0+Zp1Yp1+Zp2Yp2+Zp3Yp3.
Zp0=I0αp exp-αflnj=p+1n+1τjj-12 expαjhj-12kpαp2-σp2,Zp1=I0-αpρpp-12 exp2αp1p-1exp-αflnj=p+1n+1τjj-12 expαjhj-12kpαp2-σp2,Zp2=I0βpImρpp-1expαp1p-1exp-αflnj=p+1n+1τjj-12 expαjhj-1kpσp2+βp2Zp3=I0βpReρpp-1expαp1p-1exp-αflnj=p+1n+1τjj-12 expαjhj-1kpσp2+βp2,Y00=-L1+rsM1+2cs1-rsCsb, Y01=-L1-rsM1+2ds1+rsCsb,Y02=2-L1 cosβs1-tsM1 sinβs1, Y02=-L1 sinβs1+tsM1 cosβs1-2tsCsb.
Yj0=2cjlj-Cjj-1rjMj-Lj+1+rjMj+1,Yj1=2djlj+Cjj-1rjMj-Lj+1-rjMj+1,Yj2=2Lj-Lj+1 cosβjhj-tjMj+1 sinβjhj,Yj3=-2Cjj-1tjMj-Lj+1 sinβjhj+tjMj+1 cosβjhj,Bn=-anPn1-Cfn+bnQn1+Cfn.
rp=αpσp, tp=βpσp,ap=1bp=exp-σplp, cp=1dp=exp-αplp,Cfn=kfσfknσn, Csb=ksσskbσb, Cpp-1=kpσpkp-1σp-1,P0=Csb-1, Q0=-Csb+1,Pp=ap-1Pp-11+Cpp-1+bp-1Qp-11-Cpp-1Qp=ap-1Pp-11-Cpp-1+bp-1Qp-11+Cpp-1,Lp=-ap-1Pp-1+bp-1Qp-1,Mp=ap-1Pp-1-bp-1Qp-1).

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