Abstract

By using the concept of transfer matrices and Bloch waves, we have derived a set of equations that provide insight into the operation of asymmetric Bragg reflectors that have been demonstrated to be useful in achieving high reflectivities in strained-material systems. These equations will be useful in the design of asymmetric mirrors and can be used to compare the trade-offs between the conventional, symmetric (quarter-wavelength), and asymmetric mirrors.

© 1996 Optical Society of America

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References

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  1. K. Iga, F. Koyama, S. Kinoshita, “Surface emitting semiconductor lasers,” IEEE J. Quantum Electron. 24, 1845–1853 (1988).
    [CrossRef]
  2. J. L. Jewell, K. F. Huang, K. Tai, Y. H. Lee, R. J. Fischer, S. L. McCall, A. Y. Cho, “Vertical cavity single quantum well laser,” Appl. Phys. Lett. 55, 424–426 (1989).
    [CrossRef]
  3. R. S. Geels, S. W. Corzine, J. W. Scott, D. B. Young, L. A. Coldren, “Low threshold planarized vertical-cavity surface-emitting lasers,” IEEE Photon. Tech. Lett. 2, 234–236 (1990).
    [CrossRef]
  4. D. L. Huffaker, J. Shin, D. G. Deppe, “Low threshold half-wave vertical-cavity lasers,” Electron. Lett. 30, 1946–1948 (1994).
    [CrossRef]
  5. R. Kuchibhotla, A. Srinivasan, J. C. Campbell, C. Lei, D. G. Deppe, Y. S. He, B. G. Streetman, “Low-voltage high-gain resonant-cavity avalanche photodiode,” IEEE hoton. Tech. Lett. 3, 354–356 (1991).
    [CrossRef]
  6. 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]
  7. R. Hull, J. C. Bean, F. Cerdeira, A. T. Fiory, J. M. Gibson, “Stability of semiconductor strained-layer superlattices,” Appl. Phys. Lett. 48, 56–58 (1986).
    [CrossRef]
  8. E. D. Palik, ed., Handbook of Optical Constants of Solids (Academic, San Diego, 1991).
  9. J. C. Bean, D. L. Windt, R. Hull, L. J. Peticolas, R. Kuchibhotla, J. C. Campbell, “Design and fabrication of asymmetric strained layer mirrors for optoelectronic applications,” Appl. Phys. Lett. 63, 444–446 (1993).
    [CrossRef]
  10. M. Born, E. Wolf, Principles of Optics (Pergamon, Oxford, England, 1991), pp. 51–69.
  11. P. C. Yeh, A. Yariv, C-S. Hong, “Electromagnetic propagation in periodic stratified media. I. General theory,” J. Opt. Soc. Am. 67, 423–438 (1977).
    [CrossRef]
  12. F. Bloch, “Über die quantenmechanik der elektronen in kristallgittern,” Z. Phys. 52, 555–600 (1928).
  13. F. Abeles, “Sur l’itération des matrices carrées a quatre éléments,” Ann. Phys. 5, 777–782 (1950).

1994 (1)

D. L. Huffaker, J. Shin, D. G. Deppe, “Low threshold half-wave vertical-cavity lasers,” Electron. Lett. 30, 1946–1948 (1994).
[CrossRef]

1993 (1)

J. C. Bean, D. L. Windt, R. Hull, L. J. Peticolas, 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 (2)

R. Kuchibhotla, A. Srinivasan, J. C. Campbell, C. Lei, D. G. Deppe, Y. S. He, B. G. Streetman, “Low-voltage high-gain resonant-cavity avalanche photodiode,” IEEE hoton. Tech. Lett. 3, 354–356 (1991).
[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]

1990 (1)

R. S. Geels, S. W. Corzine, J. W. Scott, D. B. Young, L. A. Coldren, “Low threshold planarized vertical-cavity surface-emitting lasers,” IEEE Photon. Tech. Lett. 2, 234–236 (1990).
[CrossRef]

1989 (1)

J. L. Jewell, K. F. Huang, K. Tai, Y. H. Lee, R. J. Fischer, S. L. McCall, A. Y. Cho, “Vertical cavity single quantum well laser,” Appl. Phys. Lett. 55, 424–426 (1989).
[CrossRef]

1988 (1)

K. Iga, F. Koyama, S. Kinoshita, “Surface emitting semiconductor lasers,” IEEE J. Quantum Electron. 24, 1845–1853 (1988).
[CrossRef]

1986 (1)

R. Hull, J. C. Bean, F. Cerdeira, A. T. Fiory, J. M. Gibson, “Stability of semiconductor strained-layer superlattices,” Appl. Phys. Lett. 48, 56–58 (1986).
[CrossRef]

1977 (1)

1950 (1)

F. Abeles, “Sur l’itération des matrices carrées a quatre éléments,” Ann. Phys. 5, 777–782 (1950).

1928 (1)

F. Bloch, “Über die quantenmechanik der elektronen in kristallgittern,” Z. Phys. 52, 555–600 (1928).

Abeles, F.

F. Abeles, “Sur l’itération des matrices carrées a quatre éléments,” Ann. Phys. 5, 777–782 (1950).

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.

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

R. Hull, J. C. Bean, F. Cerdeira, A. T. Fiory, J. M. Gibson, “Stability of semiconductor strained-layer superlattices,” Appl. Phys. Lett. 48, 56–58 (1986).
[CrossRef]

Bloch, F.

F. Bloch, “Über die quantenmechanik der elektronen in kristallgittern,” Z. Phys. 52, 555–600 (1928).

Born, M.

M. Born, E. Wolf, Principles of Optics (Pergamon, Oxford, England, 1991), pp. 51–69.

Campbell, J. C.

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

R. Kuchibhotla, A. Srinivasan, J. C. Campbell, C. Lei, D. G. Deppe, Y. S. He, B. G. Streetman, “Low-voltage high-gain resonant-cavity avalanche photodiode,” IEEE hoton. Tech. Lett. 3, 354–356 (1991).
[CrossRef]

Cerdeira, F.

R. Hull, J. C. Bean, F. Cerdeira, A. T. Fiory, J. M. Gibson, “Stability of semiconductor strained-layer superlattices,” Appl. Phys. Lett. 48, 56–58 (1986).
[CrossRef]

Cho, A. Y.

J. L. Jewell, K. F. Huang, K. Tai, Y. H. Lee, R. J. Fischer, S. L. McCall, A. Y. Cho, “Vertical cavity single quantum well laser,” Appl. Phys. Lett. 55, 424–426 (1989).
[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.

R. S. Geels, S. W. Corzine, J. W. Scott, D. B. Young, L. A. Coldren, “Low threshold planarized vertical-cavity surface-emitting lasers,” IEEE Photon. Tech. Lett. 2, 234–236 (1990).
[CrossRef]

Corzine, S. W.

R. S. Geels, S. W. Corzine, J. W. Scott, D. B. Young, L. A. Coldren, “Low threshold planarized vertical-cavity surface-emitting lasers,” IEEE Photon. Tech. Lett. 2, 234–236 (1990).
[CrossRef]

Deppe, D. G.

D. L. Huffaker, J. Shin, D. G. Deppe, “Low threshold half-wave vertical-cavity lasers,” Electron. Lett. 30, 1946–1948 (1994).
[CrossRef]

R. Kuchibhotla, A. Srinivasan, J. C. Campbell, C. Lei, D. G. Deppe, Y. S. He, B. G. Streetman, “Low-voltage high-gain resonant-cavity avalanche photodiode,” IEEE hoton. Tech. Lett. 3, 354–356 (1991).
[CrossRef]

Fiory, A. T.

R. Hull, J. C. Bean, F. Cerdeira, A. T. Fiory, J. M. Gibson, “Stability of semiconductor strained-layer superlattices,” Appl. Phys. Lett. 48, 56–58 (1986).
[CrossRef]

Fischer, R. J.

J. L. Jewell, K. F. Huang, K. Tai, Y. H. Lee, R. J. Fischer, S. L. McCall, A. Y. Cho, “Vertical cavity single quantum well laser,” Appl. Phys. Lett. 55, 424–426 (1989).
[CrossRef]

Geels, R. S.

R. S. Geels, S. W. Corzine, J. W. Scott, D. B. Young, L. A. Coldren, “Low threshold planarized vertical-cavity surface-emitting lasers,” IEEE Photon. Tech. Lett. 2, 234–236 (1990).
[CrossRef]

Gibson, J. M.

R. Hull, J. C. Bean, F. Cerdeira, A. T. Fiory, J. M. Gibson, “Stability of semiconductor strained-layer superlattices,” Appl. Phys. Lett. 48, 56–58 (1986).
[CrossRef]

He, Y. S.

R. Kuchibhotla, A. Srinivasan, J. C. Campbell, C. Lei, D. G. Deppe, Y. S. He, B. G. Streetman, “Low-voltage high-gain resonant-cavity avalanche photodiode,” IEEE hoton. Tech. Lett. 3, 354–356 (1991).
[CrossRef]

Hong, C-S.

Huang, K. F.

J. L. Jewell, K. F. Huang, K. Tai, Y. H. Lee, R. J. Fischer, S. L. McCall, A. Y. Cho, “Vertical cavity single quantum well laser,” Appl. Phys. Lett. 55, 424–426 (1989).
[CrossRef]

Huffaker, D. L.

D. L. Huffaker, J. Shin, D. G. Deppe, “Low threshold half-wave vertical-cavity lasers,” Electron. Lett. 30, 1946–1948 (1994).
[CrossRef]

Hull, R.

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

R. Hull, J. C. Bean, F. Cerdeira, A. T. Fiory, J. M. Gibson, “Stability of semiconductor strained-layer superlattices,” Appl. Phys. Lett. 48, 56–58 (1986).
[CrossRef]

Iga, K.

K. Iga, F. Koyama, S. Kinoshita, “Surface emitting semiconductor lasers,” IEEE J. Quantum Electron. 24, 1845–1853 (1988).
[CrossRef]

Jewell, J. L.

J. L. Jewell, K. F. Huang, K. Tai, Y. H. Lee, R. J. Fischer, S. L. McCall, A. Y. Cho, “Vertical cavity single quantum well laser,” Appl. Phys. Lett. 55, 424–426 (1989).
[CrossRef]

Kinoshita, S.

K. Iga, F. Koyama, S. Kinoshita, “Surface emitting semiconductor lasers,” IEEE J. Quantum Electron. 24, 1845–1853 (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, F.

K. Iga, F. Koyama, S. Kinoshita, “Surface emitting semiconductor lasers,” IEEE J. Quantum Electron. 24, 1845–1853 (1988).
[CrossRef]

Kuchibhotla, R.

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

R. Kuchibhotla, A. Srinivasan, J. C. Campbell, C. Lei, D. G. Deppe, Y. S. He, B. G. Streetman, “Low-voltage high-gain resonant-cavity avalanche photodiode,” IEEE hoton. Tech. Lett. 3, 354–356 (1991).
[CrossRef]

Lee, Y. H.

J. L. Jewell, K. F. Huang, K. Tai, Y. H. Lee, R. J. Fischer, S. L. McCall, A. Y. Cho, “Vertical cavity single quantum well laser,” Appl. Phys. Lett. 55, 424–426 (1989).
[CrossRef]

Lei, C.

R. Kuchibhotla, A. Srinivasan, J. C. Campbell, C. Lei, D. G. Deppe, Y. S. He, B. G. Streetman, “Low-voltage high-gain resonant-cavity avalanche photodiode,” IEEE hoton. Tech. Lett. 3, 354–356 (1991).
[CrossRef]

McCall, S. L.

J. L. Jewell, K. F. Huang, K. Tai, Y. H. Lee, R. J. Fischer, S. L. McCall, A. Y. Cho, “Vertical cavity single quantum well laser,” Appl. Phys. Lett. 55, 424–426 (1989).
[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]

Peticolas, L. J.

J. C. Bean, D. L. Windt, R. Hull, L. J. Peticolas, 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]

Scott, J. W.

R. S. Geels, S. W. Corzine, J. W. Scott, D. B. Young, L. A. Coldren, “Low threshold planarized vertical-cavity surface-emitting lasers,” IEEE Photon. Tech. Lett. 2, 234–236 (1990).
[CrossRef]

Shin, J.

D. L. Huffaker, J. Shin, D. G. Deppe, “Low threshold half-wave vertical-cavity lasers,” Electron. Lett. 30, 1946–1948 (1994).
[CrossRef]

Srinivasan, A.

R. Kuchibhotla, A. Srinivasan, J. C. Campbell, C. Lei, D. G. Deppe, Y. S. He, B. G. Streetman, “Low-voltage high-gain resonant-cavity avalanche photodiode,” IEEE hoton. Tech. Lett. 3, 354–356 (1991).
[CrossRef]

Streetman, B. G.

R. Kuchibhotla, A. Srinivasan, J. C. Campbell, C. Lei, D. G. Deppe, Y. S. He, B. G. Streetman, “Low-voltage high-gain resonant-cavity avalanche photodiode,” IEEE hoton. Tech. Lett. 3, 354–356 (1991).
[CrossRef]

Tai, K.

J. L. Jewell, K. F. Huang, K. Tai, Y. H. Lee, R. J. Fischer, S. L. McCall, A. Y. Cho, “Vertical cavity single quantum well laser,” Appl. Phys. Lett. 55, 424–426 (1989).
[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, D. L. Windt, R. Hull, L. J. Peticolas, 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), pp. 51–69.

Yariv, A.

Yeh, P. C.

Young, D. B.

R. S. Geels, S. W. Corzine, J. W. Scott, D. B. Young, L. A. Coldren, “Low threshold planarized vertical-cavity surface-emitting lasers,” IEEE Photon. Tech. Lett. 2, 234–236 (1990).
[CrossRef]

Ann. Phys. (1)

F. Abeles, “Sur l’itération des matrices carrées a quatre éléments,” Ann. Phys. 5, 777–782 (1950).

Appl. Phys. Lett. (3)

J. L. Jewell, K. F. Huang, K. Tai, Y. H. Lee, R. J. Fischer, S. L. McCall, A. Y. Cho, “Vertical cavity single quantum well laser,” Appl. Phys. Lett. 55, 424–426 (1989).
[CrossRef]

R. Hull, J. C. Bean, F. Cerdeira, A. T. Fiory, J. M. Gibson, “Stability of semiconductor strained-layer superlattices,” Appl. Phys. Lett. 48, 56–58 (1986).
[CrossRef]

J. C. Bean, D. L. Windt, R. Hull, L. J. Peticolas, 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. (1)

D. L. Huffaker, J. Shin, D. G. Deppe, “Low threshold half-wave vertical-cavity lasers,” Electron. Lett. 30, 1946–1948 (1994).
[CrossRef]

IEEE hoton. Tech. Lett. (1)

R. Kuchibhotla, A. Srinivasan, J. C. Campbell, C. Lei, D. G. Deppe, Y. S. He, B. G. Streetman, “Low-voltage high-gain resonant-cavity avalanche photodiode,” IEEE hoton. Tech. Lett. 3, 354–356 (1991).
[CrossRef]

IEEE J. Quantum Electron. (2)

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]

K. Iga, F. Koyama, S. Kinoshita, “Surface emitting semiconductor lasers,” IEEE J. Quantum Electron. 24, 1845–1853 (1988).
[CrossRef]

IEEE Photon. Tech. Lett. (1)

R. S. Geels, S. W. Corzine, J. W. Scott, D. B. Young, L. A. Coldren, “Low threshold planarized vertical-cavity surface-emitting lasers,” IEEE Photon. Tech. Lett. 2, 234–236 (1990).
[CrossRef]

J. Opt. Soc. Am. (1)

Z. Phys. (1)

F. Bloch, “Über die quantenmechanik der elektronen in kristallgittern,” Z. Phys. 52, 555–600 (1928).

Other (2)

M. Born, E. Wolf, Principles of Optics (Pergamon, Oxford, England, 1991), pp. 51–69.

E. D. Palik, ed., Handbook of Optical Constants of Solids (Academic, San Diego, 1991).

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Figures (6)

Fig. 1
Fig. 1

(a) Schematic diagram and (b) the reflectivity spectrum of a strained-layer GeSi/Si asymmetric mirror.

Fig. 2
Fig. 2

Terminology used in analyzing an M-period Bragg reflector.

Fig. 3
Fig. 3

Calculated reflectivity of a buried single-period Bragg reflector as a function of the duty cycle, which is defined as the fraction of the quarter-wavelength thickness taken up in the strained layer. The plots correspond to three different refractive-index steps.

Fig. 4
Fig. 4

Calculated peak reflectivity of a symmetric (quarter-wavelength) Bragg reflector as a function of the number of periods.

Fig. 5
Fig. 5

Reflectivity of an asymmetric mirror expressed as a fraction of the peak reflectivity of a symmetric (quarter-wavelength) mirror as a function of the duty cycle of the asymmetric mirror. A duty cycle of 0.5 corresponds to the conventional, symmetric mirrors.

Fig. 6
Fig. 6

It is possible to recover peak reflectivity from an asymmetric mirror that is equal to that of a symmetric mirror. This figure shows a plot of the number of periods of an asymmetric mirror required for it to equal the reflectivity of a corresponding 20-period symmetric mirror.

Equations (42)

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E δ , m ( x , t ) = { F δ , m exp [ i k δ ( x - m T ) ] + B δ , m exp [ - i k δ ( x - m T ) ] } exp ( - i ω t ) ,
k δ = 2 π λ 0 n δ = ω c n δ , δ = 1 , 2 ,
E δ , m = ( F δ , m B δ , m ) ,
F 1 , m - 1 + B 1 , m - 1 = exp ( - i k 2 T ) F 2 , m + exp ( i k 2 T ) B 2 , m ,
i k 1 ( F 1 , m - 1 - B 1 , m - 1 ) = i k 2 [ exp ( - i k 2 T ) F 2 , m - exp ( - i k 2 T 1 ) F 2 , m - exp ( i k 2 T ) B 2 , m ] ,
exp ( - i k 2 t 1 ) F 2 , m + exp ( i k 2 t 1 ) B 2 , m = exp ( - i k 1 t 1 ) F 1 , m - exp ( i k 1 t 1 ) B 1 , m ,
i k 2 [ exp ( - i k 2 t 1 ) F 2 , m - exp ( i k 2 t 1 ) B 2 , m ] = i k 1 [ exp ( - i k 1 t 1 ) F 1 , m - exp ( i k 1 t 1 ) B 1 , m ] ,
( F 1 , m B 1 , m ) = [ τ 11 τ 12 τ 21 τ 22 ] ( F 1 , m - 1 B 1 , m - 1 ) ,
E 1 , m = τ E 1 , m - 1 ,
τ 11 = exp ( i k 1 t 1 ) ( cos k 2 t 2 + i η sin k 2 t 2 ) ,
τ 12 = exp ( i k 1 t 1 ) ( i Δ sin k 2 t 2 ) ,
τ 21 = exp ( - i k 1 t 1 ) ( - i Δ sin k 2 t 2 ) ,
τ 22 = exp ( - i k 1 t 1 ) ( cos k 2 t 2 - i η sin k 2 t 2 ) ,
η = 1 2 ( k 2 k 1 + k 1 k 2 ) = 1 2 ( n 2 n 1 + n 1 n 1 ) ,
Δ = 1 2 ( k 2 k 1 - k 1 k 2 ) = 1 2 ( n 2 n 1 - n 1 n 2 ) .
E κ ( x , t ) = E κ ( x ) exp ( i κ x ) exp ( - i ω t ) ,
E κ ( x + T , t ) = E κ ( x , t ) .
( F 1 , m B 1 , m ) = exp ( i κ T ) ( F 1 , m - 1 B 1 , m - 1 ) .
κ = 1 T cos - 1 ( τ 11 + τ 22 2 ) .
r M = ( B 1 , 0 F 1 , 0 ) B 1 , M = 0 .
( F 1 , 0 B 1 , 0 ) = [ τ 11 τ 12 τ 21 τ 22 ] - M ( F 1 , M B 1 , M ) ,
( F 1 , 0 B 1 , 0 ) = [ τ 22 - τ 12 - τ 21 τ 11 ] M ( F 1 , M B 1 , M ) .
( F 1 , 0 B 1 , 0 ) = [ τ 22 U M - 1 - U M - 2 - τ 12 U M - 1 - τ 21 U M - 1 τ 11 U M - 1 - U M - 2 ] ( F 1 , M B 1 , M ) ,
U M = sin [ ( M + 1 ) κ T ] sin ( M κ T ) .
r M = - τ 21 U M - 1 τ 22 U M - 1 - U M - 2 .
R M = r M 2 = τ 21 2 τ 21 2 + [ sin ( κ T ) sin ( M κ T ) ] 2 .
R 1 = r 1 2 = τ 21 2 τ 21 2 + 1 .
τ 21 2 = R 1 1 - R 1 .
r 1 = ( B 1 , 0 F 1 , 0 ) B 1 , 1 = 0 .
r 1 = - τ 21 τ 11 = i Δ sin ( k 2 t 2 ) cos ( k 2 t 2 ) - i η sin ( k 2 t 2 ) .
R 1 = r 1 2 = Δ 2 sin 2 ( k 2 t 2 ) cos 2 ( k 2 t 2 ) + η 2 sin 2 ( k 2 t 2 ) .
R 1 = Δ 2 η 2 + cot 2 ( k 2 t 2 ) .
sin ( 2 k 2 t 2 ) = 0 ,
t 2 = n π 2 k 2 = n λ 0 4 n 2 ,
R 1 , max = Δ 2 η 2 .
R 1 = Δ 2 η 2 + cot 2 ( D π ) .
R M , max = Δ 2 Δ 2 + 1 M 2 [ ( η 2 - Δ 2 ) + cot 2 ( D π ) ] .
R M , max = Δ 2 Δ 2 + csc 2 ( D π ) M 2 .
R M , max QW = Δ 2 Δ 2 + 1 M 2 .
R M , max D < 0.5 R M , max QW = M 2 Δ 2 + 1 M 2 Δ 2 + csc 2 ( D π ) .
N 2 = M 2 [ 1 + cot 2 ( D π ) ] ,
N D < 0.5 = M QW csc ( D π ) .

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