Abstract

We present an analytical derivation of the phase change of the reflection coefficient of a distributed-Bragg-reflector mirror when the thickness of a layer in the mirror changes. These phase changes are additive, as has been verified by an exact numerical calculation using transmission matrices. Such a phase change will affect the lasing wavelength of vertical-cavity surface-emitting lasers and the center wavelength of various integrated Bragg devices.

© 1990 Optical Society of America

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References

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  1. F. Koyama, S. Kinoshita, K. Iga, Appl. Phys. Lett. 55, 221 (1989).
    [Crossref]
  2. D. Botez, L. M. Zinkiewicz, T. J. Roth, L. J. Mawst, G. Peterson, IEEE Photon. Tech. Lett. 1, 205 (1989).
    [Crossref]
  3. P. L. Gourley, T. M. Brennan, B. E. Hammons, S. W. Corzine, R. S. Geels, R. H. Yan, J. W. Scott, L. A. Coldren, Appl. Phys. Lett. 54, 1209 (1989).
    [Crossref]
  4. R. H. Yan, R. J. Simes, L. A. Coldren, Appl. Phys. Lett. 55, 1946 (1989).
    [Crossref]
  5. S. Wang, in Semiconductors and Semimetals, W. T. Tsang, ed. (Academic, New York, 1985), Vol. 22, Part E, pp. 52–64.
    [Crossref]
  6. J.-P. Weber, K. Malloy, S. Wang, IEEE Photon. Tech. Lett. 2, 162 (1990).
    [Crossref]
  7. G. Björk, O. Nilsson, IEEE J. Lightwave Technol. 5, 140 (1987).
    [Crossref]
  8. S. Adachi, J. Appl. Phys. 58, R1 (1985).
    [Crossref]

1990 (1)

J.-P. Weber, K. Malloy, S. Wang, IEEE Photon. Tech. Lett. 2, 162 (1990).
[Crossref]

1989 (4)

F. Koyama, S. Kinoshita, K. Iga, Appl. Phys. Lett. 55, 221 (1989).
[Crossref]

D. Botez, L. M. Zinkiewicz, T. J. Roth, L. J. Mawst, G. Peterson, IEEE Photon. Tech. Lett. 1, 205 (1989).
[Crossref]

P. L. Gourley, T. M. Brennan, B. E. Hammons, S. W. Corzine, R. S. Geels, R. H. Yan, J. W. Scott, L. A. Coldren, Appl. Phys. Lett. 54, 1209 (1989).
[Crossref]

R. H. Yan, R. J. Simes, L. A. Coldren, Appl. Phys. Lett. 55, 1946 (1989).
[Crossref]

1987 (1)

G. Björk, O. Nilsson, IEEE J. Lightwave Technol. 5, 140 (1987).
[Crossref]

1985 (1)

S. Adachi, J. Appl. Phys. 58, R1 (1985).
[Crossref]

Adachi, S.

S. Adachi, J. Appl. Phys. 58, R1 (1985).
[Crossref]

Björk, G.

G. Björk, O. Nilsson, IEEE J. Lightwave Technol. 5, 140 (1987).
[Crossref]

Botez, D.

D. Botez, L. M. Zinkiewicz, T. J. Roth, L. J. Mawst, G. Peterson, IEEE Photon. Tech. Lett. 1, 205 (1989).
[Crossref]

Brennan, T. M.

P. L. Gourley, T. M. Brennan, B. E. Hammons, S. W. Corzine, R. S. Geels, R. H. Yan, J. W. Scott, L. A. Coldren, Appl. Phys. Lett. 54, 1209 (1989).
[Crossref]

Coldren, L. A.

P. L. Gourley, T. M. Brennan, B. E. Hammons, S. W. Corzine, R. S. Geels, R. H. Yan, J. W. Scott, L. A. Coldren, Appl. Phys. Lett. 54, 1209 (1989).
[Crossref]

R. H. Yan, R. J. Simes, L. A. Coldren, Appl. Phys. Lett. 55, 1946 (1989).
[Crossref]

Corzine, S. W.

P. L. Gourley, T. M. Brennan, B. E. Hammons, S. W. Corzine, R. S. Geels, R. H. Yan, J. W. Scott, L. A. Coldren, Appl. Phys. Lett. 54, 1209 (1989).
[Crossref]

Geels, R. S.

P. L. Gourley, T. M. Brennan, B. E. Hammons, S. W. Corzine, R. S. Geels, R. H. Yan, J. W. Scott, L. A. Coldren, Appl. Phys. Lett. 54, 1209 (1989).
[Crossref]

Gourley, P. L.

P. L. Gourley, T. M. Brennan, B. E. Hammons, S. W. Corzine, R. S. Geels, R. H. Yan, J. W. Scott, L. A. Coldren, Appl. Phys. Lett. 54, 1209 (1989).
[Crossref]

Hammons, B. E.

P. L. Gourley, T. M. Brennan, B. E. Hammons, S. W. Corzine, R. S. Geels, R. H. Yan, J. W. Scott, L. A. Coldren, Appl. Phys. Lett. 54, 1209 (1989).
[Crossref]

Iga, K.

F. Koyama, S. Kinoshita, K. Iga, Appl. Phys. Lett. 55, 221 (1989).
[Crossref]

Kinoshita, S.

F. Koyama, S. Kinoshita, K. Iga, Appl. Phys. Lett. 55, 221 (1989).
[Crossref]

Koyama, F.

F. Koyama, S. Kinoshita, K. Iga, Appl. Phys. Lett. 55, 221 (1989).
[Crossref]

Malloy, K.

J.-P. Weber, K. Malloy, S. Wang, IEEE Photon. Tech. Lett. 2, 162 (1990).
[Crossref]

Mawst, L. J.

D. Botez, L. M. Zinkiewicz, T. J. Roth, L. J. Mawst, G. Peterson, IEEE Photon. Tech. Lett. 1, 205 (1989).
[Crossref]

Nilsson, O.

G. Björk, O. Nilsson, IEEE J. Lightwave Technol. 5, 140 (1987).
[Crossref]

Peterson, G.

D. Botez, L. M. Zinkiewicz, T. J. Roth, L. J. Mawst, G. Peterson, IEEE Photon. Tech. Lett. 1, 205 (1989).
[Crossref]

Roth, T. J.

D. Botez, L. M. Zinkiewicz, T. J. Roth, L. J. Mawst, G. Peterson, IEEE Photon. Tech. Lett. 1, 205 (1989).
[Crossref]

Scott, J. W.

P. L. Gourley, T. M. Brennan, B. E. Hammons, S. W. Corzine, R. S. Geels, R. H. Yan, J. W. Scott, L. A. Coldren, Appl. Phys. Lett. 54, 1209 (1989).
[Crossref]

Simes, R. J.

R. H. Yan, R. J. Simes, L. A. Coldren, Appl. Phys. Lett. 55, 1946 (1989).
[Crossref]

Wang, S.

J.-P. Weber, K. Malloy, S. Wang, IEEE Photon. Tech. Lett. 2, 162 (1990).
[Crossref]

S. Wang, in Semiconductors and Semimetals, W. T. Tsang, ed. (Academic, New York, 1985), Vol. 22, Part E, pp. 52–64.
[Crossref]

Weber, J.-P.

J.-P. Weber, K. Malloy, S. Wang, IEEE Photon. Tech. Lett. 2, 162 (1990).
[Crossref]

Yan, R. H.

R. H. Yan, R. J. Simes, L. A. Coldren, Appl. Phys. Lett. 55, 1946 (1989).
[Crossref]

P. L. Gourley, T. M. Brennan, B. E. Hammons, S. W. Corzine, R. S. Geels, R. H. Yan, J. W. Scott, L. A. Coldren, Appl. Phys. Lett. 54, 1209 (1989).
[Crossref]

Zinkiewicz, L. M.

D. Botez, L. M. Zinkiewicz, T. J. Roth, L. J. Mawst, G. Peterson, IEEE Photon. Tech. Lett. 1, 205 (1989).
[Crossref]

Appl. Phys. Lett. (3)

F. Koyama, S. Kinoshita, K. Iga, Appl. Phys. Lett. 55, 221 (1989).
[Crossref]

P. L. Gourley, T. M. Brennan, B. E. Hammons, S. W. Corzine, R. S. Geels, R. H. Yan, J. W. Scott, L. A. Coldren, Appl. Phys. Lett. 54, 1209 (1989).
[Crossref]

R. H. Yan, R. J. Simes, L. A. Coldren, Appl. Phys. Lett. 55, 1946 (1989).
[Crossref]

IEEE J. Lightwave Technol. (1)

G. Björk, O. Nilsson, IEEE J. Lightwave Technol. 5, 140 (1987).
[Crossref]

IEEE Photon. Tech. Lett. (2)

J.-P. Weber, K. Malloy, S. Wang, IEEE Photon. Tech. Lett. 2, 162 (1990).
[Crossref]

D. Botez, L. M. Zinkiewicz, T. J. Roth, L. J. Mawst, G. Peterson, IEEE Photon. Tech. Lett. 1, 205 (1989).
[Crossref]

J. Appl. Phys. (1)

S. Adachi, J. Appl. Phys. 58, R1 (1985).
[Crossref]

Other (1)

S. Wang, in Semiconductors and Semimetals, W. T. Tsang, ed. (Academic, New York, 1985), Vol. 22, Part E, pp. 52–64.
[Crossref]

Cited By

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Figures (3)

Fig. 1
Fig. 1

Model of the DBR mirror.

Fig. 2
Fig. 2

Absolute change of the phase of the reflection coefficient of the mirror due to one more monolayer as a function of position in the mirror. The steps are exact calculations, and the curves are analytical approximations.

Fig. 3
Fig. 3

Phase change as a function of δ/κ for one more monolayer in the Al0.05Ga0.95As layer of pair 23. The solid curve is the exact numerical calculation, the dashed curve is from Eq. (8), and the dashed–dotted curve is the approximation from relation (9).

Equations (16)

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E ( z , λ ) = A 0 [ 1 + s ( λ ) exp ( j 2 K B z ) ] exp ( Γ z ) + B 0 [ 1 + s ( λ ) exp ( j 2 K B z ) ] exp ( Γ z ) = A 0 ( 1 + s f ) exp ( Γ z ) + B 0 ( 1 + s b ) exp ( Γ z ) ,
δ = K B β ¯ ( λ ) ,
( G + j δ eff ) 2 = ( g + ) 2 + κ 2 ,
Γ = G + j δ eff j K B ,
s = G + g + j ( δ + δ eff ) ,
R 0 = s f 0 , T 0 = [ 1 + s f 0 ] , R 1 = s b 0 [ 1 + s f 0 1 + s b 0 ] , T 1 = 1 s f 0 s b 0 [ 1 + s b 0 ] , R 2 = ( P A 2 1 ) s fA 1 P A 2 s fA s bA [ 1 + s bA 1 + s fA ] , P A = exp ( j β t t ) , D f = exp ( Γ L ) [ 1 + s fA 1 + s f 0 ] , D b = exp ( Γ L ) [ 1 + s b 0 1 + s bA ] ,
R T = r exp ( ) = R 0 + R 2 T 0 T 1 D f D b 1 R 1 R 2 D f D b .
Δ ϕ = ϕ ( t ) ϕ ( 0 ) = arg [ R T ( t ) R T ( 0 ) ] .
R T ( t ) R T ( 0 ) = 1 P A 2 s 2 + ( P A 2 1 ) exp ( 2 GL ) 1 P A 2 s 2 + s 2 ( P A 2 1 ) exp ( 2 GL ) ,
s = exp ( )
cos ( ψ ) = δ / κ
sin ( ψ ) = [ 1 ( δ / κ ) 2 ] 1 / 2 .
Δ ϕ = arctan { sin [ 2 ( ψ + β t t ) ] exp ( 2 GL ) sin ( 2 β t t ) 1 cos [ 2 ( ψ + β t t ) ] + exp ( 2 GL ) [ cos ( 2 β t ) 1 ] } arctan ( sin [ 2 ( ψ + β t t ) + exp ( 2 GL ) { sin ( 2 ψ ) sin [ 2 ( ψ + β t t ) ] } 1 cos [ 2 ( ψ + β t t ) ] + exp ( 2 GL ) { cos [ 2 ( ψ + β t t ) ] cos ( 2 ψ ) } ) .
Δ ϕ = arctan [ 2 β t t exp ( 2 GL ) ] 2 β t t exp ( 2 GL ) .
R T ( t ) [ 1 exp ( 2 GL ) ] exp ( ) + exp ( 2 GL ) exp [ j ( ϕ 2 β t t ) ] [ 1 j 2 β t t exp ( 2 GL ) ] exp ( ) ,
Δ ϕ = arg [ R T ( t ) R T ( 0 ) ] arctan [ 2 β t t exp ( 2 GL ) ] ,

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