Abstract

Asymmetric Bragg reflectors have been shown to optimize mirror performance in strained-layer material systems. Although the theory behind the reflectivity of symmetric mirrors has been well studied, little is known about asymmetric mirror designs. We present a closed-form solution for the peak reflectivity of an asymmetric mirror that results from applying a tanh substitution. This elegant technique has been shown to yield a markedly simplified calculation of the peak reflectivity of a symmetric mirror. These analytic expressions will be useful in mirror design by providing a straightforward way to compare the trade-offs between asymmetric and symmetric mirror designs.

© 1997 Optical Society of America

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References

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  1. S. S. Murtaza, M. A. Parent, J. C. Bean, J. C. Campbell, “Theory of reflectivity of an asymmetric mirror,” Appl. Opt. 35, 2054–2059 (1996).
    [CrossRef]
  2. K. Iga, K. Koyama, S. Kinoshita, “Surface emitting semi-conductor lasers,” IEEE J. Quantum Electron. 24, 1945–1953 (1988).
    [CrossRef]
  3. K. Kishino, M. S. Unlu, J. I. Chyi, J. Reed, L. Arsenault, H. Morkoc, “Resonant cavity-enhanced photodetectors,” IEEE J. Quantum Electron. 27, 2025–2034 (1991).
    [CrossRef]
  4. J. C. Bean, L. J. Peticolas, R. Hull, D. L. Windt, R. Kuchibhotla, J. C. Campbell, “Design and fabrication of asymmetric strained layer mirrors for optoelectronic applications,” Appl. Phys. Lett. 63, 444–446 (1993).
    [CrossRef]
  5. S. S. Murtaza, J. C. Campbell, J. C. Bean, L. J. Peticolas, “High reflectivity GeSi/Si asymmetric Bragg reflector at 0.8 µm,” Electron. Lett. 30, 315–316 (1994).
    [CrossRef]
  6. S. W. Corzine, R. H. Yan, 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]
  7. M. Born, E. Wolf, Principles of Optics (Pergamon, Oxford, England, 1991).

1996

1994

S. S. Murtaza, J. C. Campbell, J. C. Bean, L. J. Peticolas, “High reflectivity GeSi/Si asymmetric Bragg reflector at 0.8 µm,” Electron. Lett. 30, 315–316 (1994).
[CrossRef]

1993

J. C. Bean, L. J. Peticolas, R. Hull, D. L. Windt, R. Kuchibhotla, J. C. Campbell, “Design and fabrication of asymmetric strained layer mirrors for optoelectronic applications,” Appl. Phys. Lett. 63, 444–446 (1993).
[CrossRef]

1991

K. Kishino, M. S. Unlu, J. I. Chyi, J. Reed, L. Arsenault, H. Morkoc, “Resonant cavity-enhanced photodetectors,” IEEE J. Quantum Electron. 27, 2025–2034 (1991).
[CrossRef]

S. W. Corzine, R. H. Yan, 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]

1988

K. Iga, K. Koyama, S. Kinoshita, “Surface emitting semi-conductor lasers,” IEEE J. Quantum Electron. 24, 1945–1953 (1988).
[CrossRef]

Arsenault, L.

K. Kishino, M. S. Unlu, J. I. Chyi, J. Reed, L. Arsenault, H. Morkoc, “Resonant cavity-enhanced photodetectors,” IEEE J. Quantum Electron. 27, 2025–2034 (1991).
[CrossRef]

Bean, J. C.

S. S. Murtaza, M. A. Parent, J. C. Bean, J. C. Campbell, “Theory of reflectivity of an asymmetric mirror,” Appl. Opt. 35, 2054–2059 (1996).
[CrossRef]

S. S. Murtaza, J. C. Campbell, J. C. Bean, L. J. Peticolas, “High reflectivity GeSi/Si asymmetric Bragg reflector at 0.8 µm,” Electron. Lett. 30, 315–316 (1994).
[CrossRef]

J. C. Bean, L. J. Peticolas, R. Hull, D. L. Windt, R. Kuchibhotla, J. C. Campbell, “Design and fabrication of asymmetric strained layer mirrors for optoelectronic applications,” Appl. Phys. Lett. 63, 444–446 (1993).
[CrossRef]

Born, M.

M. Born, E. Wolf, Principles of Optics (Pergamon, Oxford, England, 1991).

Campbell, J. C.

S. S. Murtaza, M. A. Parent, J. C. Bean, J. C. Campbell, “Theory of reflectivity of an asymmetric mirror,” Appl. Opt. 35, 2054–2059 (1996).
[CrossRef]

S. S. Murtaza, J. C. Campbell, J. C. Bean, L. J. Peticolas, “High reflectivity GeSi/Si asymmetric Bragg reflector at 0.8 µm,” Electron. Lett. 30, 315–316 (1994).
[CrossRef]

J. C. Bean, L. J. Peticolas, R. Hull, D. L. Windt, R. Kuchibhotla, J. C. Campbell, “Design and fabrication of asymmetric strained layer mirrors for optoelectronic applications,” Appl. Phys. Lett. 63, 444–446 (1993).
[CrossRef]

Chyi, J. I.

K. Kishino, M. S. Unlu, J. I. Chyi, J. Reed, L. Arsenault, H. Morkoc, “Resonant cavity-enhanced photodetectors,” IEEE J. Quantum Electron. 27, 2025–2034 (1991).
[CrossRef]

Coldren, L. A.

S. W. Corzine, R. H. Yan, 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.

S. W. Corzine, R. H. Yan, 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]

Hull, R.

J. C. Bean, L. J. Peticolas, R. Hull, D. L. Windt, R. Kuchibhotla, J. C. Campbell, “Design and fabrication of asymmetric strained layer mirrors for optoelectronic applications,” Appl. Phys. Lett. 63, 444–446 (1993).
[CrossRef]

Iga, K.

K. Iga, K. Koyama, S. Kinoshita, “Surface emitting semi-conductor lasers,” IEEE J. Quantum Electron. 24, 1945–1953 (1988).
[CrossRef]

Kinoshita, S.

K. Iga, K. Koyama, S. Kinoshita, “Surface emitting semi-conductor lasers,” IEEE J. Quantum Electron. 24, 1945–1953 (1988).
[CrossRef]

Kishino, K.

K. Kishino, M. S. Unlu, J. I. Chyi, J. Reed, L. Arsenault, H. Morkoc, “Resonant cavity-enhanced photodetectors,” IEEE J. Quantum Electron. 27, 2025–2034 (1991).
[CrossRef]

Koyama, K.

K. Iga, K. Koyama, S. Kinoshita, “Surface emitting semi-conductor lasers,” IEEE J. Quantum Electron. 24, 1945–1953 (1988).
[CrossRef]

Kuchibhotla, R.

J. C. Bean, L. J. Peticolas, R. Hull, D. L. Windt, R. Kuchibhotla, J. C. Campbell, “Design and fabrication of asymmetric strained layer mirrors for optoelectronic applications,” Appl. Phys. Lett. 63, 444–446 (1993).
[CrossRef]

Morkoc, H.

K. Kishino, M. S. Unlu, J. I. Chyi, J. Reed, L. Arsenault, H. Morkoc, “Resonant cavity-enhanced photodetectors,” IEEE J. Quantum Electron. 27, 2025–2034 (1991).
[CrossRef]

Murtaza, S. S.

S. S. Murtaza, M. A. Parent, J. C. Bean, J. C. Campbell, “Theory of reflectivity of an asymmetric mirror,” Appl. Opt. 35, 2054–2059 (1996).
[CrossRef]

S. S. Murtaza, J. C. Campbell, J. C. Bean, L. J. Peticolas, “High reflectivity GeSi/Si asymmetric Bragg reflector at 0.8 µm,” Electron. Lett. 30, 315–316 (1994).
[CrossRef]

Parent, M. A.

Peticolas, L. J.

S. S. Murtaza, J. C. Campbell, J. C. Bean, L. J. Peticolas, “High reflectivity GeSi/Si asymmetric Bragg reflector at 0.8 µm,” Electron. Lett. 30, 315–316 (1994).
[CrossRef]

J. C. Bean, L. J. Peticolas, R. Hull, D. L. Windt, R. Kuchibhotla, J. C. Campbell, “Design and fabrication of asymmetric strained layer mirrors for optoelectronic applications,” Appl. Phys. Lett. 63, 444–446 (1993).
[CrossRef]

Reed, J.

K. Kishino, M. S. Unlu, J. I. Chyi, J. Reed, L. Arsenault, H. Morkoc, “Resonant cavity-enhanced photodetectors,” IEEE J. Quantum Electron. 27, 2025–2034 (1991).
[CrossRef]

Unlu, M. S.

K. Kishino, M. S. Unlu, J. I. Chyi, J. Reed, L. Arsenault, H. Morkoc, “Resonant cavity-enhanced photodetectors,” IEEE J. Quantum Electron. 27, 2025–2034 (1991).
[CrossRef]

Windt, D. L.

J. C. Bean, L. J. Peticolas, R. Hull, D. L. Windt, R. Kuchibhotla, J. C. Campbell, “Design and fabrication of asymmetric strained layer mirrors for optoelectronic applications,” Appl. Phys. Lett. 63, 444–446 (1993).
[CrossRef]

Wolf, E.

M. Born, E. Wolf, Principles of Optics (Pergamon, Oxford, England, 1991).

Yan, R. H.

S. W. Corzine, R. H. Yan, 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]

Appl. Opt.

Appl. Phys. Lett.

J. C. Bean, L. J. Peticolas, R. Hull, D. L. Windt, R. Kuchibhotla, J. C. Campbell, “Design and fabrication of asymmetric strained layer mirrors for optoelectronic applications,” Appl. Phys. Lett. 63, 444–446 (1993).
[CrossRef]

Electron. Lett.

S. S. Murtaza, J. C. Campbell, J. C. Bean, L. J. Peticolas, “High reflectivity GeSi/Si asymmetric Bragg reflector at 0.8 µm,” Electron. Lett. 30, 315–316 (1994).
[CrossRef]

IEEE J. Quantum Electron.

S. W. Corzine, R. H. Yan, 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]

K. Iga, K. Koyama, S. Kinoshita, “Surface emitting semi-conductor lasers,” IEEE J. Quantum Electron. 24, 1945–1953 (1988).
[CrossRef]

K. Kishino, M. S. Unlu, J. I. Chyi, J. Reed, L. Arsenault, H. Morkoc, “Resonant cavity-enhanced photodetectors,” IEEE J. Quantum Electron. 27, 2025–2034 (1991).
[CrossRef]

Other

M. Born, E. Wolf, Principles of Optics (Pergamon, Oxford, England, 1991).

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Figures (4)

Fig. 1
Fig. 1

Schematic of an N layer symmetric mirror. The r i are single interface reflectivities, the Γ i are the net reflectivities from the ith interface, and ϕ is the phase difference the wave undergoes in traveling twice through the layer.

Fig. 2
Fig. 2

Comparison of standard symmetric with novel asymmetric mirror designs. The duty cycle D is the fraction of the half-wavelength thickness taken up by the first layer in the pair.

Fig. 3
Fig. 3

Terminology used in finding the peak reflectivity of an M period mirror. Here, r 0, r mj , and r bj are single interface reflectivities: r m from the middle interface in the period and r b from the bottom interface. R j is the reflectivity of the jth period. The S j and s 0 are the transformed period and surface interface reflectivities, respectively. The Γ j are the net coefficients of reflection from the jth period interface.

Fig. 4
Fig. 4

Comparison of reflected power predicted by our analytic model (solid curves) with computer simulation data (data points) for different asymmetric mirrors. (a) Two-period (lower curve) and five-period (upper curve) mirrors where the layers have indices of refraction that alternate between 3.0 and 2.0, bordered by air and a substrate of the high-index material. (b) A two-period mirror on a 4.0 index substrate, with layers having indices of refraction of, in order, 2.5, 4.0, 2.0, 3.0, bordered by air.

Equations (23)

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ri=1-nlinhi1+nlinhi,
Γi=ri+Γi-1 exp-iϕ1+riΓi-1 exp-iϕ.
Γi=ri+Γi-11+riΓi-1.
tanhX+Y=tanhX+tanhY1+tanhXtanhY,
ri=tanhsi=1-exp-2si1+exp-2si.
Γ1=tanhs0+s1, Γ2=tanhs0+s1+s2,
ΓN=tanhi=0Nsi.
si=-12 lnnlinhi.
ΓN=1-b1+b,
b=i=0Nnlinhi.
Rj=rbj-rmj exp-iϕ1-rbjrmj exp-iϕ.
ϕ1=2πD layer with thickness Dλ/2,  ϕ2=2π1-Dlayer with thickness 1-Dλ/2,
ϕ2=-2πD.
ΓM=1=r0+R11+r0R1.
r0=tanhs0,  Rj=tanhSj,
ΓM=1=tanhs0+S1.
ΓM=2=R2+Γ11+R2Γ1=tanhs0+S1+S2.
ΓM=tanhs0+j=1MSj.
Rj=tanhSj=tanhsm+sb.
Sj=2si.
R=r+r exp-iϕ1+r2 exp-iϕ,
ΓM=tanhs0+MS,
S=tanh-1R=-ln21-R1+R.

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