Abstract

The contribution of a periodic stratified structure to the optical guiding properties of a multilayer planar waveguide is analyzed. We present an equivalent-layer method for the periodic stratified structure in a multiplayer waveguide that not only allows the whole periodic stratified structure to be treated as a single layer but offers exact solutions for the propagation constant of the waveguide and the power-confinement factor within the periodic stratified structure in a multilayer waveguide. This method is applied to study the optical guiding characteristics of multiquantum-well waveguides, such as the propagation constant, the field distribution, the power-confinement factor, and the group refractive index. The effective indices and the confinement factors of several practical GaInAsP/InP multiquantum-well lasers are calculated and analyzed.

© 1992 Optical Society of America

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  1. For example, see T. Tamir, Integrated Optics (Springer-Verlag, New York, 1979).
  2. R. Dingle, “Confined carrier quantum states in ultrathin semiconductor heterostructures,” Festkoerperprobleme XV, 21–48 (1975).
    [CrossRef]
  3. N. Holonyak, R. M. Kolbas, R. D. Dupuis, P. D. Dapkus, “Quantum-well heterostructure lasers,” IEEE J. Quantum Electron. QE-16, 170–186 (1980).
    [CrossRef]
  4. T. H. Wood, C. A. Burrus, D. A. B. Miller, D. S. Chemla, T. C. Damen, A. C. Gossard, W. Wiegmann, “High-speed optical modulation with GaAs/GaAlAs quantum wells in a p-i-n diode structure,” Appl. Phys. Lett. 44, 16–18 (1984).
    [CrossRef]
  5. D. A. B. Miller, D. S. Chemla, T. C. Damen, T. H. Wood, C. A. Burrus, A. C. Gossard, W. Wiegmann, “Optical-level shifter and self-linearized optical modulator using a quantum-well self-electro-optic effect device,” Opt. Lett. 9, 567–569 (1984).
    [CrossRef] [PubMed]
  6. D. A. B. Miller, D. S. Chemla, T. C. Damen, A. C. Gossard, W. Wiegmann, T. H. Wood, C. A. Burrus, “Novel hybrid optically bistable switch: the quantum well self-electro-optic effect device,” Appl. Phys. Lett. 45, 13–15 (1984).
    [CrossRef]
  7. Y.-F. Li, J. W. Y. Lit, “General formulas for the guiding properties of a multilayer slab waveguide,” J. Opt. Soc. Am. A 4, 671–677 (1987).
    [CrossRef]
  8. Y.-F. Li, J. W. Y. Lit, “Contribution of low-index layers to mode number in multilayer slab waveguides,” J. Opt. Soc. Am. A 4, 2233–2239 (1987).
    [CrossRef]
  9. Y.-F. Li, J. W. Y. Lit, “Effective thickness, group velocity, power flow, and stored energy in a multilayer dielectric planar waveguide,” J. Opt. Soc. Am. A 7, 617–635 (1990).
    [CrossRef]
  10. J. W. Y. Lit, Y.-F. Li, D. W. Hewak, “Guiding properties of multilayer dielectric planar waveguide,” Can. J. Phys. 66, 914–940 (1988).
    [CrossRef]
  11. Y.-F. Li, J. W. Y. Lit, “Guided even and odd modes in symmetric periodic stratified dielectric waveguides,” J. Opt. Soc. Am. A 5, 1050–1057 (1988).
    [CrossRef]
  12. Y.-F. Li, J. W. Y. Lit, “Generalized dispersion properties of a multilayer dielectric planar waveguide,” J. Opt. Soc. Am. A 9, 121–131 (1992).
    [CrossRef]
  13. M. Abramowitz, I. A. Stegun, Handbook of Mathematical Functions (Dover, New York, 1972).
  14. T. Miyoshi, H. Goto, H. Kimura, “Transmission characteristics of optical waveguides with multiquantum-well structures,” Electron. Lett. 22, 953–954 (1986).
    [CrossRef]
  15. W. Streifer, D. R. Scifres, R. D. Burnham, “Optical analysis of multiple-quantum-well lasers,” Appl. Opt. 18, 3547–3548 (1979).
    [CrossRef] [PubMed]
  16. S. Ohke, T. Umeda, Y. Cho, “Optical waveguides using GaAs–AlxGa1−xAs multiple quantum well,” Opt. Commun. 56, 235–239 (1985).
    [CrossRef]
  17. Y. J. Chen, C. Jagannath, G. M. Cater, E. S. Koteles, S. W. Grown, G. J. Sonek, J. M. Ballantyne, “Optical properties of GaAs/AlGaAs multiple quantum well waveguides,” Super-lattices Microstruct. 3, 287–290 (1987).
    [CrossRef]
  18. The curves in Figs. 3–5 and 11 should be plotted as discontinuous dots because the well number mmust be an integer, and thus the thickness of the well d2cannot be chosen arbitrarily. Since we use the data for these figures from Ref. 14, however, we keep the figures in the same style as theirs.
  19. H. Kogelnik, H. P. Weber, “Rays, stored energy and power flow in dielectric waveguides,”J. Opt. Soc. Am. 64, 174–185 (1974).
    [CrossRef]
  20. H. Ishiguro, T. Kawabata, S. Koike, “InGaAsP multiple quantum well lasers with planar buried heterostructure prepared by metalorganic chemical vapor deposition,” Appl. Phys. Lett. 52, 2099–2101 (1988).
    [CrossRef]
  21. M. Yamada, S. Ogita, T. Miyabo, Y. Nishida, “A theoretical analysis of lasing gain and threshold current in GaAs-AlGaAs SCH lasers,” Trans. IECE Jpn. E-69, 948–955 (1986).
  22. W. T. Tsang, R. A. Logan, J. A. Ditzenberger, “Ultra-low threshold, graded index waveguide separate confinement CW buried heterostructure lasers,” Electron. Lett. 18, 845–846 (1982).
    [CrossRef]
  23. A. Kasukawa, Y. Imajo, T. Makino, “1.3 μm GaInAsP/InP buried heterostructure graded index separate confinement multiple quantum well (BH-GRIN-SC-MQW) lasers entirely grown by metalorganic chemical vapor deposition (MOCVD),” Electron. Lett. 25, 104–105 (1989).
    [CrossRef]

1992 (1)

1990 (1)

1989 (1)

A. Kasukawa, Y. Imajo, T. Makino, “1.3 μm GaInAsP/InP buried heterostructure graded index separate confinement multiple quantum well (BH-GRIN-SC-MQW) lasers entirely grown by metalorganic chemical vapor deposition (MOCVD),” Electron. Lett. 25, 104–105 (1989).
[CrossRef]

1988 (3)

H. Ishiguro, T. Kawabata, S. Koike, “InGaAsP multiple quantum well lasers with planar buried heterostructure prepared by metalorganic chemical vapor deposition,” Appl. Phys. Lett. 52, 2099–2101 (1988).
[CrossRef]

J. W. Y. Lit, Y.-F. Li, D. W. Hewak, “Guiding properties of multilayer dielectric planar waveguide,” Can. J. Phys. 66, 914–940 (1988).
[CrossRef]

Y.-F. Li, J. W. Y. Lit, “Guided even and odd modes in symmetric periodic stratified dielectric waveguides,” J. Opt. Soc. Am. A 5, 1050–1057 (1988).
[CrossRef]

1987 (3)

Y.-F. Li, J. W. Y. Lit, “General formulas for the guiding properties of a multilayer slab waveguide,” J. Opt. Soc. Am. A 4, 671–677 (1987).
[CrossRef]

Y.-F. Li, J. W. Y. Lit, “Contribution of low-index layers to mode number in multilayer slab waveguides,” J. Opt. Soc. Am. A 4, 2233–2239 (1987).
[CrossRef]

Y. J. Chen, C. Jagannath, G. M. Cater, E. S. Koteles, S. W. Grown, G. J. Sonek, J. M. Ballantyne, “Optical properties of GaAs/AlGaAs multiple quantum well waveguides,” Super-lattices Microstruct. 3, 287–290 (1987).
[CrossRef]

1986 (2)

T. Miyoshi, H. Goto, H. Kimura, “Transmission characteristics of optical waveguides with multiquantum-well structures,” Electron. Lett. 22, 953–954 (1986).
[CrossRef]

M. Yamada, S. Ogita, T. Miyabo, Y. Nishida, “A theoretical analysis of lasing gain and threshold current in GaAs-AlGaAs SCH lasers,” Trans. IECE Jpn. E-69, 948–955 (1986).

1985 (1)

S. Ohke, T. Umeda, Y. Cho, “Optical waveguides using GaAs–AlxGa1−xAs multiple quantum well,” Opt. Commun. 56, 235–239 (1985).
[CrossRef]

1984 (3)

T. H. Wood, C. A. Burrus, D. A. B. Miller, D. S. Chemla, T. C. Damen, A. C. Gossard, W. Wiegmann, “High-speed optical modulation with GaAs/GaAlAs quantum wells in a p-i-n diode structure,” Appl. Phys. Lett. 44, 16–18 (1984).
[CrossRef]

D. A. B. Miller, D. S. Chemla, T. C. Damen, T. H. Wood, C. A. Burrus, A. C. Gossard, W. Wiegmann, “Optical-level shifter and self-linearized optical modulator using a quantum-well self-electro-optic effect device,” Opt. Lett. 9, 567–569 (1984).
[CrossRef] [PubMed]

D. A. B. Miller, D. S. Chemla, T. C. Damen, A. C. Gossard, W. Wiegmann, T. H. Wood, C. A. Burrus, “Novel hybrid optically bistable switch: the quantum well self-electro-optic effect device,” Appl. Phys. Lett. 45, 13–15 (1984).
[CrossRef]

1982 (1)

W. T. Tsang, R. A. Logan, J. A. Ditzenberger, “Ultra-low threshold, graded index waveguide separate confinement CW buried heterostructure lasers,” Electron. Lett. 18, 845–846 (1982).
[CrossRef]

1980 (1)

N. Holonyak, R. M. Kolbas, R. D. Dupuis, P. D. Dapkus, “Quantum-well heterostructure lasers,” IEEE J. Quantum Electron. QE-16, 170–186 (1980).
[CrossRef]

1979 (1)

1975 (1)

R. Dingle, “Confined carrier quantum states in ultrathin semiconductor heterostructures,” Festkoerperprobleme XV, 21–48 (1975).
[CrossRef]

1974 (1)

Abramowitz, M.

M. Abramowitz, I. A. Stegun, Handbook of Mathematical Functions (Dover, New York, 1972).

Ballantyne, J. M.

Y. J. Chen, C. Jagannath, G. M. Cater, E. S. Koteles, S. W. Grown, G. J. Sonek, J. M. Ballantyne, “Optical properties of GaAs/AlGaAs multiple quantum well waveguides,” Super-lattices Microstruct. 3, 287–290 (1987).
[CrossRef]

Burnham, R. D.

Burrus, C. A.

T. H. Wood, C. A. Burrus, D. A. B. Miller, D. S. Chemla, T. C. Damen, A. C. Gossard, W. Wiegmann, “High-speed optical modulation with GaAs/GaAlAs quantum wells in a p-i-n diode structure,” Appl. Phys. Lett. 44, 16–18 (1984).
[CrossRef]

D. A. B. Miller, D. S. Chemla, T. C. Damen, T. H. Wood, C. A. Burrus, A. C. Gossard, W. Wiegmann, “Optical-level shifter and self-linearized optical modulator using a quantum-well self-electro-optic effect device,” Opt. Lett. 9, 567–569 (1984).
[CrossRef] [PubMed]

D. A. B. Miller, D. S. Chemla, T. C. Damen, A. C. Gossard, W. Wiegmann, T. H. Wood, C. A. Burrus, “Novel hybrid optically bistable switch: the quantum well self-electro-optic effect device,” Appl. Phys. Lett. 45, 13–15 (1984).
[CrossRef]

Cater, G. M.

Y. J. Chen, C. Jagannath, G. M. Cater, E. S. Koteles, S. W. Grown, G. J. Sonek, J. M. Ballantyne, “Optical properties of GaAs/AlGaAs multiple quantum well waveguides,” Super-lattices Microstruct. 3, 287–290 (1987).
[CrossRef]

Chemla, D. S.

D. A. B. Miller, D. S. Chemla, T. C. Damen, A. C. Gossard, W. Wiegmann, T. H. Wood, C. A. Burrus, “Novel hybrid optically bistable switch: the quantum well self-electro-optic effect device,” Appl. Phys. Lett. 45, 13–15 (1984).
[CrossRef]

D. A. B. Miller, D. S. Chemla, T. C. Damen, T. H. Wood, C. A. Burrus, A. C. Gossard, W. Wiegmann, “Optical-level shifter and self-linearized optical modulator using a quantum-well self-electro-optic effect device,” Opt. Lett. 9, 567–569 (1984).
[CrossRef] [PubMed]

T. H. Wood, C. A. Burrus, D. A. B. Miller, D. S. Chemla, T. C. Damen, A. C. Gossard, W. Wiegmann, “High-speed optical modulation with GaAs/GaAlAs quantum wells in a p-i-n diode structure,” Appl. Phys. Lett. 44, 16–18 (1984).
[CrossRef]

Chen, Y. J.

Y. J. Chen, C. Jagannath, G. M. Cater, E. S. Koteles, S. W. Grown, G. J. Sonek, J. M. Ballantyne, “Optical properties of GaAs/AlGaAs multiple quantum well waveguides,” Super-lattices Microstruct. 3, 287–290 (1987).
[CrossRef]

Cho, Y.

S. Ohke, T. Umeda, Y. Cho, “Optical waveguides using GaAs–AlxGa1−xAs multiple quantum well,” Opt. Commun. 56, 235–239 (1985).
[CrossRef]

Damen, T. C.

T. H. Wood, C. A. Burrus, D. A. B. Miller, D. S. Chemla, T. C. Damen, A. C. Gossard, W. Wiegmann, “High-speed optical modulation with GaAs/GaAlAs quantum wells in a p-i-n diode structure,” Appl. Phys. Lett. 44, 16–18 (1984).
[CrossRef]

D. A. B. Miller, D. S. Chemla, T. C. Damen, T. H. Wood, C. A. Burrus, A. C. Gossard, W. Wiegmann, “Optical-level shifter and self-linearized optical modulator using a quantum-well self-electro-optic effect device,” Opt. Lett. 9, 567–569 (1984).
[CrossRef] [PubMed]

D. A. B. Miller, D. S. Chemla, T. C. Damen, A. C. Gossard, W. Wiegmann, T. H. Wood, C. A. Burrus, “Novel hybrid optically bistable switch: the quantum well self-electro-optic effect device,” Appl. Phys. Lett. 45, 13–15 (1984).
[CrossRef]

Dapkus, P. D.

N. Holonyak, R. M. Kolbas, R. D. Dupuis, P. D. Dapkus, “Quantum-well heterostructure lasers,” IEEE J. Quantum Electron. QE-16, 170–186 (1980).
[CrossRef]

Dingle, R.

R. Dingle, “Confined carrier quantum states in ultrathin semiconductor heterostructures,” Festkoerperprobleme XV, 21–48 (1975).
[CrossRef]

Ditzenberger, J. A.

W. T. Tsang, R. A. Logan, J. A. Ditzenberger, “Ultra-low threshold, graded index waveguide separate confinement CW buried heterostructure lasers,” Electron. Lett. 18, 845–846 (1982).
[CrossRef]

Dupuis, R. D.

N. Holonyak, R. M. Kolbas, R. D. Dupuis, P. D. Dapkus, “Quantum-well heterostructure lasers,” IEEE J. Quantum Electron. QE-16, 170–186 (1980).
[CrossRef]

Gossard, A. C.

D. A. B. Miller, D. S. Chemla, T. C. Damen, T. H. Wood, C. A. Burrus, A. C. Gossard, W. Wiegmann, “Optical-level shifter and self-linearized optical modulator using a quantum-well self-electro-optic effect device,” Opt. Lett. 9, 567–569 (1984).
[CrossRef] [PubMed]

T. H. Wood, C. A. Burrus, D. A. B. Miller, D. S. Chemla, T. C. Damen, A. C. Gossard, W. Wiegmann, “High-speed optical modulation with GaAs/GaAlAs quantum wells in a p-i-n diode structure,” Appl. Phys. Lett. 44, 16–18 (1984).
[CrossRef]

D. A. B. Miller, D. S. Chemla, T. C. Damen, A. C. Gossard, W. Wiegmann, T. H. Wood, C. A. Burrus, “Novel hybrid optically bistable switch: the quantum well self-electro-optic effect device,” Appl. Phys. Lett. 45, 13–15 (1984).
[CrossRef]

Goto, H.

T. Miyoshi, H. Goto, H. Kimura, “Transmission characteristics of optical waveguides with multiquantum-well structures,” Electron. Lett. 22, 953–954 (1986).
[CrossRef]

Grown, S. W.

Y. J. Chen, C. Jagannath, G. M. Cater, E. S. Koteles, S. W. Grown, G. J. Sonek, J. M. Ballantyne, “Optical properties of GaAs/AlGaAs multiple quantum well waveguides,” Super-lattices Microstruct. 3, 287–290 (1987).
[CrossRef]

Hewak, D. W.

J. W. Y. Lit, Y.-F. Li, D. W. Hewak, “Guiding properties of multilayer dielectric planar waveguide,” Can. J. Phys. 66, 914–940 (1988).
[CrossRef]

Holonyak, N.

N. Holonyak, R. M. Kolbas, R. D. Dupuis, P. D. Dapkus, “Quantum-well heterostructure lasers,” IEEE J. Quantum Electron. QE-16, 170–186 (1980).
[CrossRef]

Imajo, Y.

A. Kasukawa, Y. Imajo, T. Makino, “1.3 μm GaInAsP/InP buried heterostructure graded index separate confinement multiple quantum well (BH-GRIN-SC-MQW) lasers entirely grown by metalorganic chemical vapor deposition (MOCVD),” Electron. Lett. 25, 104–105 (1989).
[CrossRef]

Ishiguro, H.

H. Ishiguro, T. Kawabata, S. Koike, “InGaAsP multiple quantum well lasers with planar buried heterostructure prepared by metalorganic chemical vapor deposition,” Appl. Phys. Lett. 52, 2099–2101 (1988).
[CrossRef]

Jagannath, C.

Y. J. Chen, C. Jagannath, G. M. Cater, E. S. Koteles, S. W. Grown, G. J. Sonek, J. M. Ballantyne, “Optical properties of GaAs/AlGaAs multiple quantum well waveguides,” Super-lattices Microstruct. 3, 287–290 (1987).
[CrossRef]

Kasukawa, A.

A. Kasukawa, Y. Imajo, T. Makino, “1.3 μm GaInAsP/InP buried heterostructure graded index separate confinement multiple quantum well (BH-GRIN-SC-MQW) lasers entirely grown by metalorganic chemical vapor deposition (MOCVD),” Electron. Lett. 25, 104–105 (1989).
[CrossRef]

Kawabata, T.

H. Ishiguro, T. Kawabata, S. Koike, “InGaAsP multiple quantum well lasers with planar buried heterostructure prepared by metalorganic chemical vapor deposition,” Appl. Phys. Lett. 52, 2099–2101 (1988).
[CrossRef]

Kimura, H.

T. Miyoshi, H. Goto, H. Kimura, “Transmission characteristics of optical waveguides with multiquantum-well structures,” Electron. Lett. 22, 953–954 (1986).
[CrossRef]

Kogelnik, H.

Koike, S.

H. Ishiguro, T. Kawabata, S. Koike, “InGaAsP multiple quantum well lasers with planar buried heterostructure prepared by metalorganic chemical vapor deposition,” Appl. Phys. Lett. 52, 2099–2101 (1988).
[CrossRef]

Kolbas, R. M.

N. Holonyak, R. M. Kolbas, R. D. Dupuis, P. D. Dapkus, “Quantum-well heterostructure lasers,” IEEE J. Quantum Electron. QE-16, 170–186 (1980).
[CrossRef]

Koteles, E. S.

Y. J. Chen, C. Jagannath, G. M. Cater, E. S. Koteles, S. W. Grown, G. J. Sonek, J. M. Ballantyne, “Optical properties of GaAs/AlGaAs multiple quantum well waveguides,” Super-lattices Microstruct. 3, 287–290 (1987).
[CrossRef]

Li, Y.-F.

Lit, J. W. Y.

Logan, R. A.

W. T. Tsang, R. A. Logan, J. A. Ditzenberger, “Ultra-low threshold, graded index waveguide separate confinement CW buried heterostructure lasers,” Electron. Lett. 18, 845–846 (1982).
[CrossRef]

Makino, T.

A. Kasukawa, Y. Imajo, T. Makino, “1.3 μm GaInAsP/InP buried heterostructure graded index separate confinement multiple quantum well (BH-GRIN-SC-MQW) lasers entirely grown by metalorganic chemical vapor deposition (MOCVD),” Electron. Lett. 25, 104–105 (1989).
[CrossRef]

Miller, D. A. B.

D. A. B. Miller, D. S. Chemla, T. C. Damen, T. H. Wood, C. A. Burrus, A. C. Gossard, W. Wiegmann, “Optical-level shifter and self-linearized optical modulator using a quantum-well self-electro-optic effect device,” Opt. Lett. 9, 567–569 (1984).
[CrossRef] [PubMed]

D. A. B. Miller, D. S. Chemla, T. C. Damen, A. C. Gossard, W. Wiegmann, T. H. Wood, C. A. Burrus, “Novel hybrid optically bistable switch: the quantum well self-electro-optic effect device,” Appl. Phys. Lett. 45, 13–15 (1984).
[CrossRef]

T. H. Wood, C. A. Burrus, D. A. B. Miller, D. S. Chemla, T. C. Damen, A. C. Gossard, W. Wiegmann, “High-speed optical modulation with GaAs/GaAlAs quantum wells in a p-i-n diode structure,” Appl. Phys. Lett. 44, 16–18 (1984).
[CrossRef]

Miyabo, T.

M. Yamada, S. Ogita, T. Miyabo, Y. Nishida, “A theoretical analysis of lasing gain and threshold current in GaAs-AlGaAs SCH lasers,” Trans. IECE Jpn. E-69, 948–955 (1986).

Miyoshi, T.

T. Miyoshi, H. Goto, H. Kimura, “Transmission characteristics of optical waveguides with multiquantum-well structures,” Electron. Lett. 22, 953–954 (1986).
[CrossRef]

Nishida, Y.

M. Yamada, S. Ogita, T. Miyabo, Y. Nishida, “A theoretical analysis of lasing gain and threshold current in GaAs-AlGaAs SCH lasers,” Trans. IECE Jpn. E-69, 948–955 (1986).

Ogita, S.

M. Yamada, S. Ogita, T. Miyabo, Y. Nishida, “A theoretical analysis of lasing gain and threshold current in GaAs-AlGaAs SCH lasers,” Trans. IECE Jpn. E-69, 948–955 (1986).

Ohke, S.

S. Ohke, T. Umeda, Y. Cho, “Optical waveguides using GaAs–AlxGa1−xAs multiple quantum well,” Opt. Commun. 56, 235–239 (1985).
[CrossRef]

Scifres, D. R.

Sonek, G. J.

Y. J. Chen, C. Jagannath, G. M. Cater, E. S. Koteles, S. W. Grown, G. J. Sonek, J. M. Ballantyne, “Optical properties of GaAs/AlGaAs multiple quantum well waveguides,” Super-lattices Microstruct. 3, 287–290 (1987).
[CrossRef]

Stegun, I. A.

M. Abramowitz, I. A. Stegun, Handbook of Mathematical Functions (Dover, New York, 1972).

Streifer, W.

Tamir, T.

For example, see T. Tamir, Integrated Optics (Springer-Verlag, New York, 1979).

Tsang, W. T.

W. T. Tsang, R. A. Logan, J. A. Ditzenberger, “Ultra-low threshold, graded index waveguide separate confinement CW buried heterostructure lasers,” Electron. Lett. 18, 845–846 (1982).
[CrossRef]

Umeda, T.

S. Ohke, T. Umeda, Y. Cho, “Optical waveguides using GaAs–AlxGa1−xAs multiple quantum well,” Opt. Commun. 56, 235–239 (1985).
[CrossRef]

Weber, H. P.

Wiegmann, W.

T. H. Wood, C. A. Burrus, D. A. B. Miller, D. S. Chemla, T. C. Damen, A. C. Gossard, W. Wiegmann, “High-speed optical modulation with GaAs/GaAlAs quantum wells in a p-i-n diode structure,” Appl. Phys. Lett. 44, 16–18 (1984).
[CrossRef]

D. A. B. Miller, D. S. Chemla, T. C. Damen, T. H. Wood, C. A. Burrus, A. C. Gossard, W. Wiegmann, “Optical-level shifter and self-linearized optical modulator using a quantum-well self-electro-optic effect device,” Opt. Lett. 9, 567–569 (1984).
[CrossRef] [PubMed]

D. A. B. Miller, D. S. Chemla, T. C. Damen, A. C. Gossard, W. Wiegmann, T. H. Wood, C. A. Burrus, “Novel hybrid optically bistable switch: the quantum well self-electro-optic effect device,” Appl. Phys. Lett. 45, 13–15 (1984).
[CrossRef]

Wood, T. H.

D. A. B. Miller, D. S. Chemla, T. C. Damen, A. C. Gossard, W. Wiegmann, T. H. Wood, C. A. Burrus, “Novel hybrid optically bistable switch: the quantum well self-electro-optic effect device,” Appl. Phys. Lett. 45, 13–15 (1984).
[CrossRef]

D. A. B. Miller, D. S. Chemla, T. C. Damen, T. H. Wood, C. A. Burrus, A. C. Gossard, W. Wiegmann, “Optical-level shifter and self-linearized optical modulator using a quantum-well self-electro-optic effect device,” Opt. Lett. 9, 567–569 (1984).
[CrossRef] [PubMed]

T. H. Wood, C. A. Burrus, D. A. B. Miller, D. S. Chemla, T. C. Damen, A. C. Gossard, W. Wiegmann, “High-speed optical modulation with GaAs/GaAlAs quantum wells in a p-i-n diode structure,” Appl. Phys. Lett. 44, 16–18 (1984).
[CrossRef]

Yamada, M.

M. Yamada, S. Ogita, T. Miyabo, Y. Nishida, “A theoretical analysis of lasing gain and threshold current in GaAs-AlGaAs SCH lasers,” Trans. IECE Jpn. E-69, 948–955 (1986).

Appl. Opt. (1)

Appl. Phys. Lett. (3)

T. H. Wood, C. A. Burrus, D. A. B. Miller, D. S. Chemla, T. C. Damen, A. C. Gossard, W. Wiegmann, “High-speed optical modulation with GaAs/GaAlAs quantum wells in a p-i-n diode structure,” Appl. Phys. Lett. 44, 16–18 (1984).
[CrossRef]

D. A. B. Miller, D. S. Chemla, T. C. Damen, A. C. Gossard, W. Wiegmann, T. H. Wood, C. A. Burrus, “Novel hybrid optically bistable switch: the quantum well self-electro-optic effect device,” Appl. Phys. Lett. 45, 13–15 (1984).
[CrossRef]

H. Ishiguro, T. Kawabata, S. Koike, “InGaAsP multiple quantum well lasers with planar buried heterostructure prepared by metalorganic chemical vapor deposition,” Appl. Phys. Lett. 52, 2099–2101 (1988).
[CrossRef]

Can. J. Phys. (1)

J. W. Y. Lit, Y.-F. Li, D. W. Hewak, “Guiding properties of multilayer dielectric planar waveguide,” Can. J. Phys. 66, 914–940 (1988).
[CrossRef]

Electron. Lett. (3)

T. Miyoshi, H. Goto, H. Kimura, “Transmission characteristics of optical waveguides with multiquantum-well structures,” Electron. Lett. 22, 953–954 (1986).
[CrossRef]

W. T. Tsang, R. A. Logan, J. A. Ditzenberger, “Ultra-low threshold, graded index waveguide separate confinement CW buried heterostructure lasers,” Electron. Lett. 18, 845–846 (1982).
[CrossRef]

A. Kasukawa, Y. Imajo, T. Makino, “1.3 μm GaInAsP/InP buried heterostructure graded index separate confinement multiple quantum well (BH-GRIN-SC-MQW) lasers entirely grown by metalorganic chemical vapor deposition (MOCVD),” Electron. Lett. 25, 104–105 (1989).
[CrossRef]

Festkoerperprobleme (1)

R. Dingle, “Confined carrier quantum states in ultrathin semiconductor heterostructures,” Festkoerperprobleme XV, 21–48 (1975).
[CrossRef]

IEEE J. Quantum Electron. (1)

N. Holonyak, R. M. Kolbas, R. D. Dupuis, P. D. Dapkus, “Quantum-well heterostructure lasers,” IEEE J. Quantum Electron. QE-16, 170–186 (1980).
[CrossRef]

J. Opt. Soc. Am. (1)

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Opt. Commun. (1)

S. Ohke, T. Umeda, Y. Cho, “Optical waveguides using GaAs–AlxGa1−xAs multiple quantum well,” Opt. Commun. 56, 235–239 (1985).
[CrossRef]

Opt. Lett. (1)

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Y. J. Chen, C. Jagannath, G. M. Cater, E. S. Koteles, S. W. Grown, G. J. Sonek, J. M. Ballantyne, “Optical properties of GaAs/AlGaAs multiple quantum well waveguides,” Super-lattices Microstruct. 3, 287–290 (1987).
[CrossRef]

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M. Yamada, S. Ogita, T. Miyabo, Y. Nishida, “A theoretical analysis of lasing gain and threshold current in GaAs-AlGaAs SCH lasers,” Trans. IECE Jpn. E-69, 948–955 (1986).

Other (3)

For example, see T. Tamir, Integrated Optics (Springer-Verlag, New York, 1979).

The curves in Figs. 3–5 and 11 should be plotted as discontinuous dots because the well number mmust be an integer, and thus the thickness of the well d2cannot be chosen arbitrarily. Since we use the data for these figures from Ref. 14, however, we keep the figures in the same style as theirs.

M. Abramowitz, I. A. Stegun, Handbook of Mathematical Functions (Dover, New York, 1972).

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Figures (12)

Fig. 1
Fig. 1

Refractive-index profile of a PS structure in a multilayer waveguide. The two layers a and b are the layers bordering the PS structure and are not necessarily the outermost layers.

Fig. 2
Fig. 2

EL of the PS structure. dq is the equivalent thickness, nq is the equivalent refractive index, and ψq is the half-shift at the boundary x = 0 for the EL.

Fig. 3
Fig. 3

Effective index N as a function of the well thickness d2 or the well number m. The other data are d1 = d2, na = ns = n1 = 3.36, n2 = 3.62, λ = 875 nm, and dt = 440.7 nm.

Fig. 4
Fig. 4

Refractive index of the EL of the MQW structure as a function of the well thickness d2 or the well number m. The other data are the same as in Fig. 3.

Fig. 5
Fig. 5

Equivalent thickness dq as a function of d2 or m. The total thickness dt of the MQW structure is included as a reference. The other data are the same as in Fig. 3.

Fig. 6
Fig. 6

Field distribution of the fundamental TE0 mode for various values of m. The values of N used to calculate the field distributions are taken from Fig. 3.

Fig. 7
Fig. 7

Effective guide thickness and effective layer thickness of each layer in the equivalent three-layer waveguide as functions of the well thickness d2 or the well number m. The equivalent thickness dq is included as a reference. The other data are the same as in Fig. 3.

Fig. 8
Fig. 8

Fraction of power confined in the MQW structure of the waveguide as a function of the well thickness d2 or the well number m. The other data are the same as in Fig. 3.

Fig. 9
Fig. 9

Product of the transit time τ and the speed of light c in vacuum for each layer in the equivalent three-layer waveguide as a function of the well thickness d2 or the well number m. The other data are the same as in Fig. 3.

Fig. 10
Fig. 10

Fraction of energy stored in the MQW structure of the waveguide as a function of the well thickness d2 or the well number m. The other data are the same as in Fig. 3.

Fig. 11
Fig. 11

Group index Ng and effective index N of the MQW waveguide as functions of the well thickness d2 or the well number m. The other data are the same as in Fig. 3.

Fig. 12
Fig. 12

Refractive-index profile of a BH–GRIN–SC–MQW waveguide. Here n1 = 3.2022, n2 = 3.3219, n3 = 3.1781, n4 = 3.1433, n5 = 3.1081, and n6 = 3.0903. The thickness of each layer within the MQW structure is 0.01 μm, and the thickness of each layer outside the MQW structure is 0.03 μm.

Tables (2)

Tables Icon

Table 1 Values Calculated for the TE0 Mode for the MQW Laser Geometry Sketched in Fig. 1a

Tables Icon

Table 2 Values Calculated for the TE0 Mode at the Wavelength in Vacuum λ = 1.3 μm for the BH–GRIN–SC–MQW Laser Geometry Sketched in Fig. 12

Equations (79)

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n ( x ) = { n a ( t a - 1 < x < t a = 0 ) n 2 ( 0 < x < d 2 ) n 1 ( d 2 < x < Λ ) n b ( d t < x < t b ) ,
n ( x + Λ ) = n ( x )             [ 0 < x < ( m - 2 ) Λ + d 2 ] .
Λ = d 1 + d 2 .
d t = ( m - 1 ) d 1 + m d 2 .
p i 2 = k 2 ( N 2 - n i 2 ) ,
h i = j p i ,
ζ r s = η r s p s p r ,
η r s = { 1 TE modes ( n r / n s ) 2 TM modes .
tanh ψ 2 i - 1 = X 2 U i - 1 - X 1 U i - 2 Y 2 U i - 1 - Y 1 U i - 2 .
tanh ψ 2 i = Z 1 U i - 1 + ζ 2 a tanh ( p a d a - ψ a ) U i - 2 Z 2 U i - 1 - U i - 2 .
X 1 = sinh ( p 1 d 1 ) - ζ 1 a tanh ( p a d a - ψ a ) cosh ( p 1 d 1 ) ,
X 2 = - ζ 12 sinh ( p 2 d 2 ) - ζ 1 a tanh ( p a d a - ψ a ) cosh ( p 2 d 2 ) ,
Y 1 = cosh ( p 1 d 1 ) - ζ 1 a tanh ( p a d a - ψ a ) sinh ( p 1 d 1 ) ,
Y 2 = cosh ( p 2 d 2 ) + ζ 2 a tanh ( p a d a - ψ a ) sinh ( p 2 d 2 ) ,
Z 1 = - cosh ( p 1 d 1 ) sinh ( p 2 d 2 ) - ζ 21 sinh ( p 1 d 1 ) cosh ( p 2 d 2 ) - ζ 2 a tanh ( p a d a - ψ a ) [ cosh ( p 1 d 1 ) cosh ( p 2 d 2 ) + ζ 21 sinh ( p 1 d 1 ) sinh ( p 2 d 2 ) ] ,
Z 2 = cosh ( p 1 d 1 ) cosh ( p 2 d 2 ) + ζ 12 sinh ( p 1 d 1 ) sinh ( p 2 d 2 ) + ζ 1 a tanh ( p a d a - ψ a ) [ sinh ( p 1 d 1 ) cosh ( p 2 d 2 ) + ζ 21 cosh ( p 1 d 1 ) sinh ( p 2 d 2 ) ] .
U i = sin ( i + 1 ) K Λ / sin K Λ ,
cos K Λ = cosh ( p 1 d 1 ) cosh ( p 2 d 2 ) + ½ ( ζ 12 + ζ 21 ) × sinh ( p 1 d 1 ) sinh ( p 2 d 2 ) .
tanh ψ b = tanh ψ 2 m - 1 = ζ b 1 X 2 U m - 1 - X 1 U m - 2 Y 2 U m - 1 - Y 1 U m - 2 ,
tanh ψ b = - ζ b a tanh ( p a d a - ψ a ) + ζ a 1 ( R m - 1 / T m - 1 ) 1 + ζ 1 a tanh ( p a d a - ψ a ) ( S m - 1 / T m - 1 ) ,
R i = sinh ( p 1 d 1 ) U i - 1 + ζ 12 sinh ( p 2 d 2 ) U i ,
S i = sinh ( p 1 d 1 ) U i - 1 + ζ 21 sinh ( p 2 d 2 ) U i ,
T i = - cosh ( p 1 d 1 ) U i - 1 + cosh ( p 2 d 2 ) U i ,
F a ( x ) = F a cosh [ p a ( x - t a - 1 ) - ψ a ] cosh ψ a             ( t a - 1 < x < t a = 0 ) ;
F 2 i - 1 ( x ) = F 2 i - 1 cosh [ p 1 ( x - i Λ + d 1 ) - ψ 2 i - 1 ] cosh ψ 2 i - 1 ( i = 1 , 2 , , m - 1 ;             i Λ - d 1 < x < i Λ ) ;
F 2 i ( x ) = F 2 i cosh [ p 2 ( x - i Λ ) - ψ 2 i ] cosh ψ 2 i ( i = 0 , 1 , , m - 1 ;             i Λ < x < i Λ + d 2 ) ;
F b ( x ) = F b cosh [ p b ( x - d t ) - ψ b ] cosh ψ b             ( d t < x < t b ) .
F 2 i - 1 = F 0 ( Y 2 U i - 1 - Y 1 U i - 2 ) ,
F 2 i = F 0 ( Z 2 U i - 1 - U i - 2 ) ,
F b = F a cosh ( p a d a - ψ a ) cosh ψ a ( Y 2 U m - 1 - Y 1 U m - 2 ) .
tanh ψ q = - ζ q a tanh ( p a d a - ψ a ) ,
tanh ψ b = - ζ b q tanh ( p q d q - ψ q ) .
tanh ψ b = - ζ b a tanh ( p a d a - ψ a ) + ζ a q tanh ( p q d q ) 1 + ζ q a tanh ( p a d a - ψ a ) tanh ( p q d q ) .
tanh ( p q d q ) = ζ q 1 R m - 1 / T m - 1
= ζ 1 q S m - 1 / T m - 1 .
ζ 1 q 2 = R m - 1 / S m - 1
η 1 q 2 p q 2 p 1 2 = R m - 1 / S m - 1 .
n q 2 = N 2 - ( N 2 - n 1 2 ) R m - 1 / S m - 1 .
n q 2 = n 1 2 { [ n 1 4 + 4 N 2 ( N 2 - n 1 2 ) R m - 1 / S m - 1 ] 1 / 2 - n 1 2 } 2 ( N 2 - n 1 2 ) R m - 1 / S m - 1 .
F b = F a cosh ( p a d a - ψ a ) cosh ψ a cosh ( p q d q - ψ q ) cosh ψ q .
cosh ( p q d q ) = T m - 1 ,
sinh ( p q d q ) = ζ 1 q S m - 1 .
d q = cosh - 1 T m - 1 p q ,
tan ( h q d q ) = h q ( η q a p a + η q s p s ) h q 2 - η q a η q s ( p a + p s ) ,
n ¯ = m d 2 n 2 + ( m - 1 ) d 1 n 1 d t .
n ¯ = ( n 1 2 d 1 + n 2 2 d 2 d 1 + d 2 ) 1 / 2
n ¯ = ( d 1 + d 2 d 1 / n 1 2 + d 2 / n 2 2 ) 1 / 2
d t = ( 2 m - 1 ) d 2 .
F a ( x ) = F 0 exp ( p a x )             ( - < x < 0 ) ,
F s ( x ) = F b exp [ - p s ( x - d t ) ]             ( d t < x < + ) .
F 2 i - 1 = F 0 ( Y 2 U i - 1 - Y 1 U i - 2 ) ,
F 2 i = F 0 ( Z 2 U i - 1 - U i - 2 ) .
tanh ψ 2 i - 1 = X 2 U i - 1 - X 1 U i - 2 Y 2 U i - 1 - Y 1 U i - 2 ,
tanh ψ 2 i = Z 1 U i - 1 + ζ 2 a U i - 2 Z 2 U i - 1 - U i - 2 .
X 1 = sinh ( p 1 d 1 ) - ζ 1 a cosh ( p 1 d 1 ) ,
X 2 = - ζ 12 sinh ( p 2 d 2 ) - ζ 1 a cosh ( p 2 d 2 ) ,
Y 1 = cosh ( p 1 d 1 ) - ζ 1 a sinh ( p 1 d 1 ) ,
Y 2 = cosh ( p 2 d 2 ) + ζ 2 a sinh ( p 2 d 2 ) ,
Z 1 = - cosh ( p 1 d 1 ) sinh ( p 2 d 2 ) - ζ 21 sinh ( p 1 d 1 ) cosh ( p 2 d 2 ) - ζ 2 a [ cosh ( p 1 d 1 ) cosh ( p 2 d 2 ) + ζ 21 sinh ( p 1 d 1 ) sinh ( p 2 d 2 ) ] ,
Z 2 = cosh ( p 1 d 1 ) cosh ( p 2 d 2 ) + ζ 12 sinh ( p 1 d 1 ) sinh ( p 2 d 2 ) + ζ 1 a [ sinh ( p 1 d 1 ) cosh ( p 2 d 2 ) + ζ 21 cosh ( p 1 d 1 ) sinh ( p 2 d 2 ) ] .
d e a = η q a h q 2 p a ( h q 2 + η q a 2 p a 2 ) ,
d e q = d q + η q a p a h q 2 + η q a 2 p a 2 + η q s p s h q 2 + η q s 2 p s 2 ,
d e s = η q s h q 2 p s ( h q 2 + η q s 2 p s 2 ) ,
d e = d q + η q a h q 2 + p a 2 p a ( h q 2 + η q a 2 p a 2 ) + η q s h q 2 + p s 2 p s ( h q 2 + η q s 2 p s 2 ) .
q i , j = { 1 for TE modes n i 2 n j 2 [ ( N / n i ) 2 + ( N / n j ) 2 - 1 ] for TM modes ,
d e a = η q a q q a h q 2 p a ( h q 2 + p a 2 ) ,
d e q = d q + η q a q q a p a h q 2 + p a 2 + η q s q q s p s h q 2 + p s 2 ,
d e s = η q s q q s h q 2 p s ( h q 2 + p s 2 ) .
d e = d q + η q a q q a 1 p a + η q s q q s 1 p s .
Γ i = P i P = d e i d e             ( i = a , q , s ) ,
τ = k c h q [ n q 2 d q + N 2 ( η q a q q a 1 p a + η q s q q s 1 p s ) ]
= k c h q [ N 2 d e + ( n q 2 - N 2 ) d q ] .
τ a = k c h q N 2 d e a ,
τ q = k c h q [ n q 2 d e q - ( N 2 - n a 2 ) d e a - ( N 2 - n s 2 ) d e s ] ,
τ s = k c h q N 2 d e s .
ϒ i = W i W = τ i τ             ( i = a , q , s ) ,
N g = 1 N n i 2 Γ i ,
N g = N + n q 2 - N 2 N d q d e .
Γ w = Γ m q w m d 2 d t ,

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