Abstract

A new formalism for equivalent refractive indices in multi-quantum-well (MQW) waveguides with arbitrarily shaped base periods, which uses the transfer matrix technique in the thin-film approximation, has been developed for TM and TE polarizations. It clearly illustrates the intrinsic birefringence feature and can be used to produce the dispersion relation of MQW waveguides in a way that is more straightforward than in previous methods.

© 1991 Optical Society of America

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References

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  1. S. M. Rytov, “Electromagnetic properties of a finely stratified medium,” Sov. Phys. JETP 2, 466–475 (1956).
  2. S. Ohke, T. Umeda, Y. Cho, “Optical waveguides using GaAs–Alx Ga1−x As multiquantum well,” Opt. Commun. 56, 235–239 (1985).
    [CrossRef]
  3. S. Ohke, T. Umeda, Y. Cho, “TM-mode propagation and form birefringence in a GaAs–AlGaAs multiple quantum well optical waveguide,” Opt. Commun. 70, 92–96 (1989).
    [CrossRef]
  4. I. M. Skinner, R. Shail, B. L. Weiss, “Modal propagation within MQW waveguides,” IEEE J. Quantum Electron. 25, 6–11 (1989).
    [CrossRef]
  5. P.-f. Yuh, K. L. Wang, “Formalism of the Kronig–Penney model for superlattices of variable basis,” Phys. Rev. B 38, 13307–13315 (1988).
    [CrossRef]
  6. J. P. van der Ziel, A. C. Gossard, “Optical birefringence of ultra-thin Alx Ga1−x As–GaAs multilayer heterostructures,” J. Appl. Phys. 49, 2919–2921 (1978).
    [CrossRef]

1989 (2)

S. Ohke, T. Umeda, Y. Cho, “TM-mode propagation and form birefringence in a GaAs–AlGaAs multiple quantum well optical waveguide,” Opt. Commun. 70, 92–96 (1989).
[CrossRef]

I. M. Skinner, R. Shail, B. L. Weiss, “Modal propagation within MQW waveguides,” IEEE J. Quantum Electron. 25, 6–11 (1989).
[CrossRef]

1988 (1)

P.-f. Yuh, K. L. Wang, “Formalism of the Kronig–Penney model for superlattices of variable basis,” Phys. Rev. B 38, 13307–13315 (1988).
[CrossRef]

1985 (1)

S. Ohke, T. Umeda, Y. Cho, “Optical waveguides using GaAs–Alx Ga1−x As multiquantum well,” Opt. Commun. 56, 235–239 (1985).
[CrossRef]

1978 (1)

J. P. van der Ziel, A. C. Gossard, “Optical birefringence of ultra-thin Alx Ga1−x As–GaAs multilayer heterostructures,” J. Appl. Phys. 49, 2919–2921 (1978).
[CrossRef]

1956 (1)

S. M. Rytov, “Electromagnetic properties of a finely stratified medium,” Sov. Phys. JETP 2, 466–475 (1956).

Cho, Y.

S. Ohke, T. Umeda, Y. Cho, “TM-mode propagation and form birefringence in a GaAs–AlGaAs multiple quantum well optical waveguide,” Opt. Commun. 70, 92–96 (1989).
[CrossRef]

S. Ohke, T. Umeda, Y. Cho, “Optical waveguides using GaAs–Alx Ga1−x As multiquantum well,” Opt. Commun. 56, 235–239 (1985).
[CrossRef]

Gossard, A. C.

J. P. van der Ziel, A. C. Gossard, “Optical birefringence of ultra-thin Alx Ga1−x As–GaAs multilayer heterostructures,” J. Appl. Phys. 49, 2919–2921 (1978).
[CrossRef]

Ohke, S.

S. Ohke, T. Umeda, Y. Cho, “TM-mode propagation and form birefringence in a GaAs–AlGaAs multiple quantum well optical waveguide,” Opt. Commun. 70, 92–96 (1989).
[CrossRef]

S. Ohke, T. Umeda, Y. Cho, “Optical waveguides using GaAs–Alx Ga1−x As multiquantum well,” Opt. Commun. 56, 235–239 (1985).
[CrossRef]

Rytov, S. M.

S. M. Rytov, “Electromagnetic properties of a finely stratified medium,” Sov. Phys. JETP 2, 466–475 (1956).

Shail, R.

I. M. Skinner, R. Shail, B. L. Weiss, “Modal propagation within MQW waveguides,” IEEE J. Quantum Electron. 25, 6–11 (1989).
[CrossRef]

Skinner, I. M.

I. M. Skinner, R. Shail, B. L. Weiss, “Modal propagation within MQW waveguides,” IEEE J. Quantum Electron. 25, 6–11 (1989).
[CrossRef]

Umeda, T.

S. Ohke, T. Umeda, Y. Cho, “TM-mode propagation and form birefringence in a GaAs–AlGaAs multiple quantum well optical waveguide,” Opt. Commun. 70, 92–96 (1989).
[CrossRef]

S. Ohke, T. Umeda, Y. Cho, “Optical waveguides using GaAs–Alx Ga1−x As multiquantum well,” Opt. Commun. 56, 235–239 (1985).
[CrossRef]

van der Ziel, J. P.

J. P. van der Ziel, A. C. Gossard, “Optical birefringence of ultra-thin Alx Ga1−x As–GaAs multilayer heterostructures,” J. Appl. Phys. 49, 2919–2921 (1978).
[CrossRef]

Wang, K. L.

P.-f. Yuh, K. L. Wang, “Formalism of the Kronig–Penney model for superlattices of variable basis,” Phys. Rev. B 38, 13307–13315 (1988).
[CrossRef]

Weiss, B. L.

I. M. Skinner, R. Shail, B. L. Weiss, “Modal propagation within MQW waveguides,” IEEE J. Quantum Electron. 25, 6–11 (1989).
[CrossRef]

Yuh, P.-f.

P.-f. Yuh, K. L. Wang, “Formalism of the Kronig–Penney model for superlattices of variable basis,” Phys. Rev. B 38, 13307–13315 (1988).
[CrossRef]

IEEE J. Quantum Electron. (1)

I. M. Skinner, R. Shail, B. L. Weiss, “Modal propagation within MQW waveguides,” IEEE J. Quantum Electron. 25, 6–11 (1989).
[CrossRef]

J. Appl. Phys. (1)

J. P. van der Ziel, A. C. Gossard, “Optical birefringence of ultra-thin Alx Ga1−x As–GaAs multilayer heterostructures,” J. Appl. Phys. 49, 2919–2921 (1978).
[CrossRef]

Opt. Commun. (2)

S. Ohke, T. Umeda, Y. Cho, “Optical waveguides using GaAs–Alx Ga1−x As multiquantum well,” Opt. Commun. 56, 235–239 (1985).
[CrossRef]

S. Ohke, T. Umeda, Y. Cho, “TM-mode propagation and form birefringence in a GaAs–AlGaAs multiple quantum well optical waveguide,” Opt. Commun. 70, 92–96 (1989).
[CrossRef]

Phys. Rev. B (1)

P.-f. Yuh, K. L. Wang, “Formalism of the Kronig–Penney model for superlattices of variable basis,” Phys. Rev. B 38, 13307–13315 (1988).
[CrossRef]

Sov. Phys. JETP (1)

S. M. Rytov, “Electromagnetic properties of a finely stratified medium,” Sov. Phys. JETP 2, 466–475 (1956).

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Figures (1)

Fig. 1
Fig. 1

Refractive-index profile of the MQW waveguide with an arbitrarily shaped base period.

Equations (26)

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cos K Λ = ½ Tr M ( Λ ) ,
K = ( k 2 n e 2 - β 2 ) 1 / 2 ,
M j = [ cos α j h ( f j / α j ) sin α j h ( α j / f j ) sin α j h cos α j h ] ,
f = { 1 TE modes n 2 ( x j ) TM modes ,
α j = [ k 2 n 2 ( x j ) - β 2 ] 1 / 2
M j = [ 1 - ½ ( α j h ) 2 f j h ( α j 2 / f j ) h 1 - ½ ( α j h ) 2 ] = [ 1 0 0 1 ] + f j h [ 0 1 0 0 ] - α j 2 f j h [ 0 0 0 1 ] - 1 2 ( α j h ) 2 [ 1 0 0 1 ] .
M j = E + ( f j h ) A - ( α j 2 f h h ) B - 1 2 ( α j h ) 2 E ,
M ( Λ ) = M l M l - 1 M l - 2 M 3 M 2 M 1 .
Tr ( M + N ) = Tr ( M ) + Tr ( N ) , Tr ( M N ) = Tr ( N M ) .
Tr M ( Λ ) = Tr ( E l ) + j ( f j h ) Tr ( E l - 1 A ) - ( a j 2 f j h ) Tr ( E l - 1 B ) - 1 2 j ( α j h ) 2 Tr ( E l ) + 1 2 s i ( f s h ) ( f i h ) Tr ( A E s - i - 1 A E l - s + i - 1 ) + 1 2 s i ( α s 2 f s h ) ( α i 2 f i h ) Tr ( B E s - i - 1 B E l - s + i - 1 ) - s i ( f s h ) ( α i 2 f i h ) Tr ( A E s - i - 1 B E l - s + i - 1 ) ( i , j , s = 1 , 2 , 3 , , l ) ,
Tr ( E l - 1 A ) = Tr ( A ) = 0 , Tr ( E l - 1 B ) = Tr ( B ) = 0 , Tr ( A E s - i - 1 A E l - s + i - 1 ) = Tr ( A 2 ) = 0 , Tr ( B E s - i - 1 B E l - s + i - 1 ) = Tr ( B 2 ) = 0 , Tr ( E l ) = Tr ( E ) = 2 , Tr ( A E s - i - 1 B E l - s + i - 1 ) = Tr ( A B ) = 1.
Tr M ( Λ ) = 2 - s i ( f s h ) ( α i 2 f i h ) - j ( α j h ) 2 .
cos K Λ = 1 - ½ ( K Λ ) 2 .
( k 2 n e 2 - β 2 ) Λ 2 = s i ( f s h ) ( α i 2 f i h ) + j ( α j h ) 2 .
s i ( f s h ) ( α i 2 f i ) + j ( α j h ) 2 = 0 Λ d x 0 Λ [ k 2 n 2 ( x ) - β 2 ] d x = k 2 Λ 0 Λ n 2 ( x ) d x - β 2 Λ 2 .
( n e TE ) 2 = 1 Λ 0 Λ n 2 ( x ) d x .
( n e TM ) 2 = 1 Λ 0 Λ n 2 ( x ) d x - [ 1 Λ 2 0 Λ n 2 ( x ) d x 0 Λ d x n 2 ( x ) - 1 ] β 2 / k 2 .
n 2 ( x ) = { n 1 2 0 x d 1 n 2 2 d 1 x d 1 + d 2 = Λ .
( n e TE ) 2 = n 1 2 d 1 + n 2 2 d 2 d 1 + d 2 ,
( n e TE ) 2 = n 1 2 d 1 + n 2 2 d 2 d 1 + d 2 - d 1 d 2 ( d 1 + d 2 ) 2 ( n 1 - n 2 ) 2 n 1 2 n 2 2 β 2 / k 2 .
( n e TM ) 2 = n 1 2 n 2 2 ( d 1 + d 2 ) n 1 2 d 2 + n 2 2 d 1 .
n e TM n e TM ,
K f c ( p f 0 + q f 3 ) = ( K 2 f e 2 - p q f 0 f 3 ) tan K w ,
p = ( β 2 - k 2 n 0 2 ) 1 / 2 , q = ( β 2 - k 2 n 3 2 ) 1 / 2 ,
f i = { 1 TE modes n i 2 TM modes             ( i = 0 , 3 , e ) .
n e TM < n 0 < n e TE ,

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