Abstract

Previous numerical work is extended by deriving simple analytic expressions for the impedance of periodic layers over a wide frequency range within the reflection stop band (not just the center Bragg frequency) for an arbitrary number of periods in the structure, for arbitrary layer thicknesses (not just quarter-wave layers), for sizable absorption, and for arbitrary sizes of the refractive index differences. When the number of periods in the structure is infinite, exact expressions for impedance, which are valid for all frequencies in the reflection stop band, are derived. For a finite number of periods in the structure, it is shown that the asymptotic approach of the impedance toward its value for an infinite structure has a decaying exponential dependence. It is shown that the characteristic number of periods in this decaying exponential dependence is determined by the condition number of the transverse field matrix. Simple analytic expressions for the phase shift throughout the reflection stop band are derived, as well as simple analytic expressions to show that a small fractional error in the VCSEL cavity mode frequency can still result from a large fractional error in the cavity thickness if the layers in the Bragg mirror have a small refractive index difference. These simple analytic expressions are useful for design.

© 2007 Optical Society of America

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References

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  1. A. Thelen, Design of Optical Interference Coatings (McGraw-Hill, 1989).
  2. M. A. Matin, T. M. Benson, L. R. Chen, and P. W. E. Smith, "Analysis of distributed Bragg reflectors using thin-film optics," Microwave Opt. Technol. Lett. 21, 11-15 (1999).
    [CrossRef]
  3. P. Yeh, A. Yariv, and C.-S. Hong, "Electromagenetic propagation in periodic stratified media. I. General theory," J. Opt. Soc. Am. 67, 423-438 (1977).
    [CrossRef]
  4. D. L. MacFarlane and E. M. Dowling, "Z-domain techniques in the analysis of Fabry-Perot etalons and multilayer structures," J. Opt. Soc. Am. A 11, 236-245 (1994).
    [CrossRef]
  5. S. W. Corzine, R. H. Yan, and L. A. Coldren, "A tanh substitution technique for the analysis of abrupt and graded interface multilayer dielectric stacks," IEEE J. Quantum Electron. 27, 2086-2090 (1991).
    [CrossRef]
  6. M. A. Parent, S. S. Murtaza, and J. C. Campbell, "Analytic solution for the peak reflectivity of an asymmetric mirror," Appl. Opt. 36, 4265-4268 (1997).
    [CrossRef] [PubMed]
  7. S. S. Murtaza, M. A. Parent, J. C. Bean, and J. C. Campbell, "Theory of reflectivity of an asymmetric mirror," Appl. Opt. 35, 2054-2059 (1996).
    [CrossRef] [PubMed]
  8. D. I. Babic and S. W. Corzine, "Analytic expressions for the reflection delay, penetration depth, and absorptance of quarter-wave dielectric mirrors," IEEE J. Quantum Electron. 28, 514-524 (1992).
    [CrossRef]
  9. D. I. Babic, Y. Chung, N. Dagli, and J. E. Bowers, "Modal reflection of quarter-wave mirrors in vertical-cavity lasers," IEEE J. Quantum Electron. 29, 1950-1962 (1993).
    [CrossRef]
  10. F. Koyama, Y. Suematsu, S. Arai, and T. Tawee, "1.5-1.6 μm GaInAsP/InP dynamic-single-mode (DSM) lasers with distributed Bragg reflector," IEEE J. Quantum Electron. 19, 1042-1051 (1983).
    [CrossRef]
  11. N. Matuschek, F. Kaertner, and U. Keller, "Exact coupled-mode theories for multilayer interference coating with arbitrary strong index modulation," IEEE J. Quantum Electron. 33, 295-302 (1997).
    [CrossRef]

1999 (1)

M. A. Matin, T. M. Benson, L. R. Chen, and P. W. E. Smith, "Analysis of distributed Bragg reflectors using thin-film optics," Microwave Opt. Technol. Lett. 21, 11-15 (1999).
[CrossRef]

1997 (2)

N. Matuschek, F. Kaertner, and U. Keller, "Exact coupled-mode theories for multilayer interference coating with arbitrary strong index modulation," IEEE J. Quantum Electron. 33, 295-302 (1997).
[CrossRef]

M. A. Parent, S. S. Murtaza, and J. C. Campbell, "Analytic solution for the peak reflectivity of an asymmetric mirror," Appl. Opt. 36, 4265-4268 (1997).
[CrossRef] [PubMed]

1996 (1)

1994 (1)

1993 (1)

D. I. Babic, Y. Chung, N. Dagli, and J. E. Bowers, "Modal reflection of quarter-wave mirrors in vertical-cavity lasers," IEEE J. Quantum Electron. 29, 1950-1962 (1993).
[CrossRef]

1992 (1)

D. I. Babic and S. W. Corzine, "Analytic expressions for the reflection delay, penetration depth, and absorptance of quarter-wave dielectric mirrors," IEEE J. Quantum Electron. 28, 514-524 (1992).
[CrossRef]

1991 (1)

S. W. Corzine, R. H. Yan, and L. A. Coldren, "A tanh substitution technique for the analysis of abrupt and graded interface multilayer dielectric stacks," IEEE J. Quantum Electron. 27, 2086-2090 (1991).
[CrossRef]

1983 (1)

F. Koyama, Y. Suematsu, S. Arai, and T. Tawee, "1.5-1.6 μm GaInAsP/InP dynamic-single-mode (DSM) lasers with distributed Bragg reflector," IEEE J. Quantum Electron. 19, 1042-1051 (1983).
[CrossRef]

1977 (1)

Arai, S.

F. Koyama, Y. Suematsu, S. Arai, and T. Tawee, "1.5-1.6 μm GaInAsP/InP dynamic-single-mode (DSM) lasers with distributed Bragg reflector," IEEE J. Quantum Electron. 19, 1042-1051 (1983).
[CrossRef]

Babic, D. I.

D. I. Babic, Y. Chung, N. Dagli, and J. E. Bowers, "Modal reflection of quarter-wave mirrors in vertical-cavity lasers," IEEE J. Quantum Electron. 29, 1950-1962 (1993).
[CrossRef]

D. I. Babic and S. W. Corzine, "Analytic expressions for the reflection delay, penetration depth, and absorptance of quarter-wave dielectric mirrors," IEEE J. Quantum Electron. 28, 514-524 (1992).
[CrossRef]

Bean, J. C.

Benson, T. M.

M. A. Matin, T. M. Benson, L. R. Chen, and P. W. E. Smith, "Analysis of distributed Bragg reflectors using thin-film optics," Microwave Opt. Technol. Lett. 21, 11-15 (1999).
[CrossRef]

Bowers, J. E.

D. I. Babic, Y. Chung, N. Dagli, and J. E. Bowers, "Modal reflection of quarter-wave mirrors in vertical-cavity lasers," IEEE J. Quantum Electron. 29, 1950-1962 (1993).
[CrossRef]

Campbell, J. C.

Chen, L. R.

M. A. Matin, T. M. Benson, L. R. Chen, and P. W. E. Smith, "Analysis of distributed Bragg reflectors using thin-film optics," Microwave Opt. Technol. Lett. 21, 11-15 (1999).
[CrossRef]

Chung, Y.

D. I. Babic, Y. Chung, N. Dagli, and J. E. Bowers, "Modal reflection of quarter-wave mirrors in vertical-cavity lasers," IEEE J. Quantum Electron. 29, 1950-1962 (1993).
[CrossRef]

Coldren, L. A.

S. W. Corzine, R. H. Yan, and L. A. Coldren, "A tanh substitution technique for the analysis of abrupt and graded interface multilayer dielectric stacks," IEEE J. Quantum Electron. 27, 2086-2090 (1991).
[CrossRef]

Corzine, S. W.

D. I. Babic and S. W. Corzine, "Analytic expressions for the reflection delay, penetration depth, and absorptance of quarter-wave dielectric mirrors," IEEE J. Quantum Electron. 28, 514-524 (1992).
[CrossRef]

S. W. Corzine, R. H. Yan, and L. A. Coldren, "A tanh substitution technique for the analysis of abrupt and graded interface multilayer dielectric stacks," IEEE J. Quantum Electron. 27, 2086-2090 (1991).
[CrossRef]

Dagli, N.

D. I. Babic, Y. Chung, N. Dagli, and J. E. Bowers, "Modal reflection of quarter-wave mirrors in vertical-cavity lasers," IEEE J. Quantum Electron. 29, 1950-1962 (1993).
[CrossRef]

Dowling, E. M.

Hong, C.-S.

Kaertner, F.

N. Matuschek, F. Kaertner, and U. Keller, "Exact coupled-mode theories for multilayer interference coating with arbitrary strong index modulation," IEEE J. Quantum Electron. 33, 295-302 (1997).
[CrossRef]

Keller, U.

N. Matuschek, F. Kaertner, and U. Keller, "Exact coupled-mode theories for multilayer interference coating with arbitrary strong index modulation," IEEE J. Quantum Electron. 33, 295-302 (1997).
[CrossRef]

Koyama, F.

F. Koyama, Y. Suematsu, S. Arai, and T. Tawee, "1.5-1.6 μm GaInAsP/InP dynamic-single-mode (DSM) lasers with distributed Bragg reflector," IEEE J. Quantum Electron. 19, 1042-1051 (1983).
[CrossRef]

MacFarlane, D. L.

Matin, M. A.

M. A. Matin, T. M. Benson, L. R. Chen, and P. W. E. Smith, "Analysis of distributed Bragg reflectors using thin-film optics," Microwave Opt. Technol. Lett. 21, 11-15 (1999).
[CrossRef]

Matuschek, N.

N. Matuschek, F. Kaertner, and U. Keller, "Exact coupled-mode theories for multilayer interference coating with arbitrary strong index modulation," IEEE J. Quantum Electron. 33, 295-302 (1997).
[CrossRef]

Murtaza, S. S.

Parent, M. A.

Smith, P. W. E.

M. A. Matin, T. M. Benson, L. R. Chen, and P. W. E. Smith, "Analysis of distributed Bragg reflectors using thin-film optics," Microwave Opt. Technol. Lett. 21, 11-15 (1999).
[CrossRef]

Suematsu, Y.

F. Koyama, Y. Suematsu, S. Arai, and T. Tawee, "1.5-1.6 μm GaInAsP/InP dynamic-single-mode (DSM) lasers with distributed Bragg reflector," IEEE J. Quantum Electron. 19, 1042-1051 (1983).
[CrossRef]

Tawee, T.

F. Koyama, Y. Suematsu, S. Arai, and T. Tawee, "1.5-1.6 μm GaInAsP/InP dynamic-single-mode (DSM) lasers with distributed Bragg reflector," IEEE J. Quantum Electron. 19, 1042-1051 (1983).
[CrossRef]

Thelen, A.

A. Thelen, Design of Optical Interference Coatings (McGraw-Hill, 1989).

Yan, R. H.

S. W. Corzine, R. H. Yan, and L. A. Coldren, "A tanh substitution technique for the analysis of abrupt and graded interface multilayer dielectric stacks," IEEE J. Quantum Electron. 27, 2086-2090 (1991).
[CrossRef]

Yariv, A.

Yeh, P.

Appl. Opt. (2)

IEEE J. Quantum Electron. (5)

D. I. Babic and S. W. Corzine, "Analytic expressions for the reflection delay, penetration depth, and absorptance of quarter-wave dielectric mirrors," IEEE J. Quantum Electron. 28, 514-524 (1992).
[CrossRef]

D. I. Babic, Y. Chung, N. Dagli, and J. E. Bowers, "Modal reflection of quarter-wave mirrors in vertical-cavity lasers," IEEE J. Quantum Electron. 29, 1950-1962 (1993).
[CrossRef]

F. Koyama, Y. Suematsu, S. Arai, and T. Tawee, "1.5-1.6 μm GaInAsP/InP dynamic-single-mode (DSM) lasers with distributed Bragg reflector," IEEE J. Quantum Electron. 19, 1042-1051 (1983).
[CrossRef]

N. Matuschek, F. Kaertner, and U. Keller, "Exact coupled-mode theories for multilayer interference coating with arbitrary strong index modulation," IEEE J. Quantum Electron. 33, 295-302 (1997).
[CrossRef]

S. W. Corzine, R. H. Yan, and L. A. Coldren, "A tanh substitution technique for the analysis of abrupt and graded interface multilayer dielectric stacks," IEEE J. Quantum Electron. 27, 2086-2090 (1991).
[CrossRef]

J. Opt. Soc. Am. (1)

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

Microwave Opt. Technol. Lett. (1)

M. A. Matin, T. M. Benson, L. R. Chen, and P. W. E. Smith, "Analysis of distributed Bragg reflectors using thin-film optics," Microwave Opt. Technol. Lett. 21, 11-15 (1999).
[CrossRef]

Other (1)

A. Thelen, Design of Optical Interference Coatings (McGraw-Hill, 1989).

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Figures (4)

Fig. 1
Fig. 1

Example of a periodic layered structure. Layers having characteristic impedance Z A and thickness L A are alternated with layers having characteristic impedance Z G and thickness L G on top of a substrate of characteristic impedance Z S . The impedance of N periods of the layered structure (with the layers of impedance Z A on top) is denoted as Z BIG ( N ) . The impedance of N 1 2 periods of the layered structure (with the layers of impedance Z G on top) is denoted as Z SMALL ( N ) . The layer from which the radiation is incident has characteristic impedance Z INC . The reflection coefficient of the entire structure, as seen from the layer of impedance Z INC , is denoted Γ BIG ( N ) .

Fig. 2
Fig. 2

Calculations using our approximation in Eq. (8) (solid curves in the figure) and exact results using transverse field matrices (filled circles) for radiation incident upon an AlAs / GaAs DBR from free space. Here we assume n A = 2.97 (for AlAs), n G = 3.65 (for GaAs), an operating wavelength of 873   nm , a Bragg center wavelength of 883   nm , a GaAs absorption coefficient of 3930 cm 1 , and no absorption in the AlAs. (A) Square of the reflection coefficient, | Γ BIG ( N ) | 2 . (B) Fractional error, 1− | Γ BIG ( N ) | 2 | Γ BIG ( N = ) | 2 . An exact result for | Γ BIG ( N ) | 2 as N was obtained using Eq. (32). Note that, as N , the error approaches zero in an exponentially decaying manner, as predicted by Eq. (8).

Fig. 3
Fig. 3

Reflection coefficient in the entire stop band as a function of wave vector ( K A Z or K G Z ) for a beam incident from free space upon a lossless AlAs∕GaAs infinite periodic layered structure. Here, the wave vector is normalized so that K A Z L A = K G Z L G = 90 ° represents quarter-wavelengths. We assume that n A = 2.91 (for AlAs) and n G = 3.25 (for GaAs). In the reflection stop band of an infinite periodic structure, the magnitude of the reflection coefficient is unity, and thus only the phase of the reflection coefficient is plotted. The filled circles represent numerical multiplication of transverse field matrices for N = 1000 . The solid curve represents our exact result in Eq. (32). Note that our exact result in Eq. (32) is in excellent agreement with the transverse field matrix method for all frequencies in the reflection stop band. The dashed curve represents our approximate result, Eq. (34), which we used in the approximate recursion relation, Eq. (8).

Fig. 4
Fig. 4

Electric field amplitude in a cavity of thickness d.

Equations (63)

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Z SMALL ( N + 1 ) = Z G [ ( Z BIG ( N ) Z G ) + i   tan   K G Z L G 1 + i ( Z BIG ( N ) Z G ) tan   K G Z L G ] ,
Z BIG ( N + 1 ) = Z A [ ( Z SMALL ( N + 1 ) Z A ) + i   tan   K A Z L A 1 + i ( Z SMALL ( N + 1 ) Z A ) tan   K A Z L A ] ,
Z A,TE = μ A ω K A Z , Z A,TM = K A Z ϵ A ω
Z SMALL ( N + 1 ) Z G [ 1 i   tan   K G Z L G + Z G Z BIG ( N ) ] .
Z BIG ( N + 1 ) Z A [ 1 1 i   tan   K A Z L A + Z SMALL ( N + 1 ) Z A ] .
Z A Z BIG ( N + 1 ) = [ 1 i   tan   K A Z L A + Z G Z A 1 i   tan   K G Z L G ] + ( Z G Z A ) 2 Z A Z BIG ( N ) ,
Z SMALL ( N + 1 ) Z A = Z G Z A [ 1 i   tan   K G Z L G + Z G Z A 1 i   tan   K A Z L A ] + ( Z G Z A ) 2 Z SMALL ( N ) Z A .
1 Z BIG ( N + 1 ) = 1 Z BIG,MC [ 1 C N ] + C N Z BIG ( 1 ) ,
Z SMALL ( N + 1 ) = Z SMALL,MC [ 1 C N ] + Z SMALL ( 1 ) C N ,
C = ( Z G Z A ) 2 [ 1 { 1 i   tan   K G Z L G } 2 ] [ 1 { 1 i   tan   K A Z L A } 2 ] ,
( Z G Z A ) 2 , for   | Z A | > | Z G | ,
1 Z BIG,MC = 1 ( Z A 2 Z G 2 ) [ Z A i   tan   K A Z L A + Z G i   tan   K G Z L G ] ,
Z SMALL,MC = Z A 2 Z G 2 ( Z A 2 Z G 2 ) [ 1 Z G i   tan   K G Z L G + 1 Z A i   tan   K A Z L A ] .
| ( Z BIG ( N ) Z G ) tan   K G Z L G | 1 ,
| tan   K A Z L A | | Z SMALL ( N + 1 ) Z A | .
Γ BIG | Γ BIG | exp ( i Γ BIG )
= [ 1 Z INC Z BIG 1 + Z INC Z BIG ] .
Γ BIG 2 = arctan [ Z INC { Z BIG } ] ,
Γ BIG ( N = ) 2 = arctan [ Z INC ( Z A 2 Z G 2 ) { Z A tan   K A Z L A + Z G tan   K G Z L G } ] .
K A Z L A = K G Z L G ,
Z BIG ( N = ) = Z BIG,MC = [ Z A Z G ] i   tan   K A Z L A ,
Z SMALL ( N = ) = Z SMALL,MC = Z A Z G [ Z A Z G ] i   tan   K A Z L A ,
Γ BIG ( N = ) 2 = arctan [ Z INC ( Z A Z G )   tan   K A Z L A ] .
Z BIG ( N + 1 ) Z BIG ( N ) = N Z , 1 P D Z , 1 P ,
N Z , 1 P = [ cos   K G Z L G ] [ cos   K A Z L A ] + Z A Z BIG ( N ) [ cos   K G Z L G ] [ i   sin   K A Z L A ] + Z G Z BIG ( N ) [ i   sin   K G Z L G ] [ cos   K A Z L A ] + Z A Z G [ i   sin   K G Z L G ] [ i   sin   K A Z L A ] ,
D Z , 1 P = [ cos   K G Z L G ] [ cos   K A Z L A ] + Z BIG ( N ) Z A [ cos   K G Z L G ] [ i   sin   K A Z L A ] + Z BIG ( N ) Z G [ i   sin   K G Z L G ] [ cos   K A Z L A ] + Z G Z A [ i   sin   K G Z L G ] [ i   sin   K A Z L A ] .
Z BIG ( N + 1 ) = Z BIG ( N ) .
N Z , 1 P = D Z , 1 P .
a Z B I G 2 ( N = ) + b Z BIG ( N = ) + c = 0 ,
a = 1 Z A [ cos   K G Z L G ] [ i   sin   K A Z L A ] + 1 Z G [ i   sin   K G Z L G ] × [ cos   K A Z L A ] ,
b = [ Z G Z A Z A Z G ] [ i   sin   K G Z L G ] [ i   sin   K A Z L A ] ,
c = { Z A [ cos   K G Z L G ] [ i   sin   K A Z L A ] + Z G [ i   sin   K G Z L G ] × [ cos   K A Z L A ] } .
Z BIG ( N = ) = Z BIG,MC 2 + ( Z BIG,MC 2 ) 2 + Z CHAR 2 ,
Z SMALL ( N = ) = Z BIG,MC 2 ( Z BIG,MC 2 ) 2 + Z CHAR 2 ,
b a = Z BIG,MC = [ Z A 2 Z G 2 ] [ Z G i   tan   K G Z L G + Z A i   tan   K A Z L A ] ,
c a = Z CHAR 2 = [ Z A Z G ] 2 [ 1 Z G i   tan   K G Z L G + 1 Z A i   tan   K A Z L A ] [ Z G i   tan   K G Z L G + Z A i   tan   K A Z L A ] .
( Z BIG,MC 2 ) 2 + Z CHAR 2 < 0.
K A Z L A = K G Z L G
Z BIG,MC = [ Z A Z G ] i   tan   K A Z L A ,
Z CHAR 2 = [ Z A Z G ]
tan 2 K A Z L A > 4 Z A Z G [ Z A Z G ] 2 .
k z d 2 ϕ M = m π ,
ϕ M = Γ BIG ( N = ) 2 = arctan [ Z CAV [ Z A Z G ] tan   K A Z L A ] ,
2 π λ z d = m π + 2 ϕ M ,
d λ z = m 2 + ϕ M π .
K A Z L A = K G Z L G = π 2
λ z = λ M C + Δ λ .
d [ λ M C + Δ λ ] = m 2 + ϕ M π .
tan   K A Z L A = tan   K G Z L G = tan ( π 2 λ M C [ λ M C + Δ λ ] ) tan ( π 2 [ 1 Δ λ λ M C ] ) 1 cos ( π 2 [ 1 Δ λ λ M C ] ) 1 sin ( π 2 Δ λ λ M C ) ,
ϕ M = arctan [ Z CAV [ Z A Z G ] tan   K A Z L A ] arctan [ Z CAV [ Z A Z G ]   sin ( π 2 Δ λ λ M C ) ] arctan [ Z CAV [ Z A Z G ] { π 2 Δ λ λ M C } ] [ Z CAV [ Z A Z G ] { π 2 Δ λ λ M C } ] .
d m 2 λ M C .
Δ d = m 2 Δ λ + ϕ M π λ M G ,
Δ d λ M C = [ m + { Z CAV Z A Z G } ] [ 1 2 Δ λ λ M C ] .
Δ d d = [ 1 + 1 m { Z CAV Z A Z G } ] [ Δ λ λ M C ] ,
Δ λ λ M C = Δ d d [ 1 + 1 m { Z CAV Z A Z G } ] 1 .
Δ λ λ M C = Δ d d [ 1 + 1 m { n A n G n A } ] 1 .
Δ λ λ M C = Δ d d [ 1 5.28 ] ,
Δ d d = 5 % implies Δ λ λ M C = 0.95 % .
u n + 1 = A + B u n
u n = [ A 1 B ] [ 1 B n 1 ] + B n 1 u 1 , for   n 1 ,
u n = 1 u 1 .
u N = [ A 1 B ] [ 1 B N 1 ] + B N 1 u 1 ,
u N + 1 = A + B u N = A + B { [ A 1 B ] [ 1 B N 1 ] + B N 1 u 1 } = A + [ A B 1 B ] [ 1 B N 1 ] + B N u 1 = A { [ 1 B ] + B [ 1 B N 1 ] 1 B } + B N u 1 = [ A 1 B ] [ 1 B N ] + B N u 1 .

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