Abstract

In this paper we investigate the wavelength conversion and regeneration properties of a tunable all-optical signal regenerator (TASR). In the TASR, the wavelength conversion is done by a semiconductor optical amplifier, which is incorporated in an asymmetric Sagnac loop (ASL). We demonstrate both theoretically and experimentally that the ASL regenerates the incident signal’s bit pattern, reduces its noise, increases the extinction ratio (which in many aspects is equivalent to noise reduction) and improves its bit-error rate. We also demonstrate the general behavior of the TASR with a numerical simulation.

© 2005 Optical Society of America

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  1. J. Y. Emery, M. Picq, F. Poingt, F. Gaborit, R. Brenot, M. Renaud, B. Lavigne, and A. Dupas, "Optimised 2-R all-optical regenerator with low polarization sensitivity penalty (<1 dB) for optical networking applications," in Optical Fiber Conference (Optical Society of America, 2000), paper Mb4-1.
  2. W. Idler, K. Daub, G. Laube, M. Schilling, P. Wiedemann, K. Dutting, M. Klenk, E. Lach, and K. Wunstel, "10 Gb/s wavelength conversion with integrated multiquantum-well-based 3-port Mach-Zehnder interferometer," IEEE Photonics Technol. Lett. 8, 1163-1165 (1996).
    [CrossRef]
  3. J. Leuthold, C. H. Joyner, B. Mikkelsen, G. Raybon, J. L. Pleumeekers, B. I. Miller, K. Dreyer, and C. A. Burrus, "100 Gbit/s all-optical wavelength conversion with integrated SOA delayed-interference configuration," Electron. Lett. 36, 1129-1130 (2000).
    [CrossRef]
  4. K. L. Deng, I. Glesk, K. I. Kang, and P. R. Prucnal, "Unbalanced TOAD for optical data clock separation in self-clocked transparent OTDM networks," IEEE Photonics Technol. Lett. 9, 830-832 (1997).
    [CrossRef]
  5. B. C. Wang, L. Xu, V. Baby, I. Glesk, and P. R. Prucnal, "State selection of bistable SOA ring laser for bit level optical memory applications," IEEE Photonics Technol. Lett. 14, 989-991 (2002).
    [CrossRef]
  6. K. I. Kang, T. G. Chang, I. Glesk, and P. R. Prucnal, "Comparison of Sagnac and Mach-Zehnder ultrafast all-optical interferometric switches based on a semiconductor resonant optical nonlinearity," Appl. Opt. 35, 417-426 (1996).
    [CrossRef] [PubMed]
  7. L. Xu, I. Glesk, V. Baby and P. R. Prucnal, "All-optical wavelength conversion using SOA at nearly symmetric position in a fiber-based Sagnac interferometric loop," IEEE Photonics Technol. Lett. 16, 539--541 (2004).
    [CrossRef]

2004 (1)

L. Xu, I. Glesk, V. Baby and P. R. Prucnal, "All-optical wavelength conversion using SOA at nearly symmetric position in a fiber-based Sagnac interferometric loop," IEEE Photonics Technol. Lett. 16, 539--541 (2004).
[CrossRef]

2002 (1)

B. C. Wang, L. Xu, V. Baby, I. Glesk, and P. R. Prucnal, "State selection of bistable SOA ring laser for bit level optical memory applications," IEEE Photonics Technol. Lett. 14, 989-991 (2002).
[CrossRef]

2000 (1)

J. Leuthold, C. H. Joyner, B. Mikkelsen, G. Raybon, J. L. Pleumeekers, B. I. Miller, K. Dreyer, and C. A. Burrus, "100 Gbit/s all-optical wavelength conversion with integrated SOA delayed-interference configuration," Electron. Lett. 36, 1129-1130 (2000).
[CrossRef]

1997 (1)

K. L. Deng, I. Glesk, K. I. Kang, and P. R. Prucnal, "Unbalanced TOAD for optical data clock separation in self-clocked transparent OTDM networks," IEEE Photonics Technol. Lett. 9, 830-832 (1997).
[CrossRef]

1996 (2)

W. Idler, K. Daub, G. Laube, M. Schilling, P. Wiedemann, K. Dutting, M. Klenk, E. Lach, and K. Wunstel, "10 Gb/s wavelength conversion with integrated multiquantum-well-based 3-port Mach-Zehnder interferometer," IEEE Photonics Technol. Lett. 8, 1163-1165 (1996).
[CrossRef]

K. I. Kang, T. G. Chang, I. Glesk, and P. R. Prucnal, "Comparison of Sagnac and Mach-Zehnder ultrafast all-optical interferometric switches based on a semiconductor resonant optical nonlinearity," Appl. Opt. 35, 417-426 (1996).
[CrossRef] [PubMed]

Baby, V.

L. Xu, I. Glesk, V. Baby and P. R. Prucnal, "All-optical wavelength conversion using SOA at nearly symmetric position in a fiber-based Sagnac interferometric loop," IEEE Photonics Technol. Lett. 16, 539--541 (2004).
[CrossRef]

B. C. Wang, L. Xu, V. Baby, I. Glesk, and P. R. Prucnal, "State selection of bistable SOA ring laser for bit level optical memory applications," IEEE Photonics Technol. Lett. 14, 989-991 (2002).
[CrossRef]

Brenot, R.

J. Y. Emery, M. Picq, F. Poingt, F. Gaborit, R. Brenot, M. Renaud, B. Lavigne, and A. Dupas, "Optimised 2-R all-optical regenerator with low polarization sensitivity penalty (<1 dB) for optical networking applications," in Optical Fiber Conference (Optical Society of America, 2000), paper Mb4-1.

Burrus, C. A.

J. Leuthold, C. H. Joyner, B. Mikkelsen, G. Raybon, J. L. Pleumeekers, B. I. Miller, K. Dreyer, and C. A. Burrus, "100 Gbit/s all-optical wavelength conversion with integrated SOA delayed-interference configuration," Electron. Lett. 36, 1129-1130 (2000).
[CrossRef]

Chang, T. G.

Daub, K.

W. Idler, K. Daub, G. Laube, M. Schilling, P. Wiedemann, K. Dutting, M. Klenk, E. Lach, and K. Wunstel, "10 Gb/s wavelength conversion with integrated multiquantum-well-based 3-port Mach-Zehnder interferometer," IEEE Photonics Technol. Lett. 8, 1163-1165 (1996).
[CrossRef]

Deng, K. L.

K. L. Deng, I. Glesk, K. I. Kang, and P. R. Prucnal, "Unbalanced TOAD for optical data clock separation in self-clocked transparent OTDM networks," IEEE Photonics Technol. Lett. 9, 830-832 (1997).
[CrossRef]

Dreyer, K.

J. Leuthold, C. H. Joyner, B. Mikkelsen, G. Raybon, J. L. Pleumeekers, B. I. Miller, K. Dreyer, and C. A. Burrus, "100 Gbit/s all-optical wavelength conversion with integrated SOA delayed-interference configuration," Electron. Lett. 36, 1129-1130 (2000).
[CrossRef]

Dupas, A.

J. Y. Emery, M. Picq, F. Poingt, F. Gaborit, R. Brenot, M. Renaud, B. Lavigne, and A. Dupas, "Optimised 2-R all-optical regenerator with low polarization sensitivity penalty (<1 dB) for optical networking applications," in Optical Fiber Conference (Optical Society of America, 2000), paper Mb4-1.

Dutting, K.

W. Idler, K. Daub, G. Laube, M. Schilling, P. Wiedemann, K. Dutting, M. Klenk, E. Lach, and K. Wunstel, "10 Gb/s wavelength conversion with integrated multiquantum-well-based 3-port Mach-Zehnder interferometer," IEEE Photonics Technol. Lett. 8, 1163-1165 (1996).
[CrossRef]

Emery, J. Y.

J. Y. Emery, M. Picq, F. Poingt, F. Gaborit, R. Brenot, M. Renaud, B. Lavigne, and A. Dupas, "Optimised 2-R all-optical regenerator with low polarization sensitivity penalty (<1 dB) for optical networking applications," in Optical Fiber Conference (Optical Society of America, 2000), paper Mb4-1.

Gaborit, F.

J. Y. Emery, M. Picq, F. Poingt, F. Gaborit, R. Brenot, M. Renaud, B. Lavigne, and A. Dupas, "Optimised 2-R all-optical regenerator with low polarization sensitivity penalty (<1 dB) for optical networking applications," in Optical Fiber Conference (Optical Society of America, 2000), paper Mb4-1.

Glesk, I.

L. Xu, I. Glesk, V. Baby and P. R. Prucnal, "All-optical wavelength conversion using SOA at nearly symmetric position in a fiber-based Sagnac interferometric loop," IEEE Photonics Technol. Lett. 16, 539--541 (2004).
[CrossRef]

B. C. Wang, L. Xu, V. Baby, I. Glesk, and P. R. Prucnal, "State selection of bistable SOA ring laser for bit level optical memory applications," IEEE Photonics Technol. Lett. 14, 989-991 (2002).
[CrossRef]

K. L. Deng, I. Glesk, K. I. Kang, and P. R. Prucnal, "Unbalanced TOAD for optical data clock separation in self-clocked transparent OTDM networks," IEEE Photonics Technol. Lett. 9, 830-832 (1997).
[CrossRef]

K. I. Kang, T. G. Chang, I. Glesk, and P. R. Prucnal, "Comparison of Sagnac and Mach-Zehnder ultrafast all-optical interferometric switches based on a semiconductor resonant optical nonlinearity," Appl. Opt. 35, 417-426 (1996).
[CrossRef] [PubMed]

Idler, W.

W. Idler, K. Daub, G. Laube, M. Schilling, P. Wiedemann, K. Dutting, M. Klenk, E. Lach, and K. Wunstel, "10 Gb/s wavelength conversion with integrated multiquantum-well-based 3-port Mach-Zehnder interferometer," IEEE Photonics Technol. Lett. 8, 1163-1165 (1996).
[CrossRef]

Joyner, C. H.

J. Leuthold, C. H. Joyner, B. Mikkelsen, G. Raybon, J. L. Pleumeekers, B. I. Miller, K. Dreyer, and C. A. Burrus, "100 Gbit/s all-optical wavelength conversion with integrated SOA delayed-interference configuration," Electron. Lett. 36, 1129-1130 (2000).
[CrossRef]

Kang, K. I.

K. L. Deng, I. Glesk, K. I. Kang, and P. R. Prucnal, "Unbalanced TOAD for optical data clock separation in self-clocked transparent OTDM networks," IEEE Photonics Technol. Lett. 9, 830-832 (1997).
[CrossRef]

K. I. Kang, T. G. Chang, I. Glesk, and P. R. Prucnal, "Comparison of Sagnac and Mach-Zehnder ultrafast all-optical interferometric switches based on a semiconductor resonant optical nonlinearity," Appl. Opt. 35, 417-426 (1996).
[CrossRef] [PubMed]

Klenk, M.

W. Idler, K. Daub, G. Laube, M. Schilling, P. Wiedemann, K. Dutting, M. Klenk, E. Lach, and K. Wunstel, "10 Gb/s wavelength conversion with integrated multiquantum-well-based 3-port Mach-Zehnder interferometer," IEEE Photonics Technol. Lett. 8, 1163-1165 (1996).
[CrossRef]

Lach, E.

W. Idler, K. Daub, G. Laube, M. Schilling, P. Wiedemann, K. Dutting, M. Klenk, E. Lach, and K. Wunstel, "10 Gb/s wavelength conversion with integrated multiquantum-well-based 3-port Mach-Zehnder interferometer," IEEE Photonics Technol. Lett. 8, 1163-1165 (1996).
[CrossRef]

Laube, G.

W. Idler, K. Daub, G. Laube, M. Schilling, P. Wiedemann, K. Dutting, M. Klenk, E. Lach, and K. Wunstel, "10 Gb/s wavelength conversion with integrated multiquantum-well-based 3-port Mach-Zehnder interferometer," IEEE Photonics Technol. Lett. 8, 1163-1165 (1996).
[CrossRef]

Lavigne, B.

J. Y. Emery, M. Picq, F. Poingt, F. Gaborit, R. Brenot, M. Renaud, B. Lavigne, and A. Dupas, "Optimised 2-R all-optical regenerator with low polarization sensitivity penalty (<1 dB) for optical networking applications," in Optical Fiber Conference (Optical Society of America, 2000), paper Mb4-1.

Leuthold, J.

J. Leuthold, C. H. Joyner, B. Mikkelsen, G. Raybon, J. L. Pleumeekers, B. I. Miller, K. Dreyer, and C. A. Burrus, "100 Gbit/s all-optical wavelength conversion with integrated SOA delayed-interference configuration," Electron. Lett. 36, 1129-1130 (2000).
[CrossRef]

Mikkelsen, B.

J. Leuthold, C. H. Joyner, B. Mikkelsen, G. Raybon, J. L. Pleumeekers, B. I. Miller, K. Dreyer, and C. A. Burrus, "100 Gbit/s all-optical wavelength conversion with integrated SOA delayed-interference configuration," Electron. Lett. 36, 1129-1130 (2000).
[CrossRef]

Miller, B. I.

J. Leuthold, C. H. Joyner, B. Mikkelsen, G. Raybon, J. L. Pleumeekers, B. I. Miller, K. Dreyer, and C. A. Burrus, "100 Gbit/s all-optical wavelength conversion with integrated SOA delayed-interference configuration," Electron. Lett. 36, 1129-1130 (2000).
[CrossRef]

Picq, M.

J. Y. Emery, M. Picq, F. Poingt, F. Gaborit, R. Brenot, M. Renaud, B. Lavigne, and A. Dupas, "Optimised 2-R all-optical regenerator with low polarization sensitivity penalty (<1 dB) for optical networking applications," in Optical Fiber Conference (Optical Society of America, 2000), paper Mb4-1.

Pleumeekers, J. L.

J. Leuthold, C. H. Joyner, B. Mikkelsen, G. Raybon, J. L. Pleumeekers, B. I. Miller, K. Dreyer, and C. A. Burrus, "100 Gbit/s all-optical wavelength conversion with integrated SOA delayed-interference configuration," Electron. Lett. 36, 1129-1130 (2000).
[CrossRef]

Poingt, F.

J. Y. Emery, M. Picq, F. Poingt, F. Gaborit, R. Brenot, M. Renaud, B. Lavigne, and A. Dupas, "Optimised 2-R all-optical regenerator with low polarization sensitivity penalty (<1 dB) for optical networking applications," in Optical Fiber Conference (Optical Society of America, 2000), paper Mb4-1.

Prucnal, P. R.

L. Xu, I. Glesk, V. Baby and P. R. Prucnal, "All-optical wavelength conversion using SOA at nearly symmetric position in a fiber-based Sagnac interferometric loop," IEEE Photonics Technol. Lett. 16, 539--541 (2004).
[CrossRef]

B. C. Wang, L. Xu, V. Baby, I. Glesk, and P. R. Prucnal, "State selection of bistable SOA ring laser for bit level optical memory applications," IEEE Photonics Technol. Lett. 14, 989-991 (2002).
[CrossRef]

K. L. Deng, I. Glesk, K. I. Kang, and P. R. Prucnal, "Unbalanced TOAD for optical data clock separation in self-clocked transparent OTDM networks," IEEE Photonics Technol. Lett. 9, 830-832 (1997).
[CrossRef]

K. I. Kang, T. G. Chang, I. Glesk, and P. R. Prucnal, "Comparison of Sagnac and Mach-Zehnder ultrafast all-optical interferometric switches based on a semiconductor resonant optical nonlinearity," Appl. Opt. 35, 417-426 (1996).
[CrossRef] [PubMed]

Raybon, G.

J. Leuthold, C. H. Joyner, B. Mikkelsen, G. Raybon, J. L. Pleumeekers, B. I. Miller, K. Dreyer, and C. A. Burrus, "100 Gbit/s all-optical wavelength conversion with integrated SOA delayed-interference configuration," Electron. Lett. 36, 1129-1130 (2000).
[CrossRef]

Renaud, M.

J. Y. Emery, M. Picq, F. Poingt, F. Gaborit, R. Brenot, M. Renaud, B. Lavigne, and A. Dupas, "Optimised 2-R all-optical regenerator with low polarization sensitivity penalty (<1 dB) for optical networking applications," in Optical Fiber Conference (Optical Society of America, 2000), paper Mb4-1.

Schilling, M.

W. Idler, K. Daub, G. Laube, M. Schilling, P. Wiedemann, K. Dutting, M. Klenk, E. Lach, and K. Wunstel, "10 Gb/s wavelength conversion with integrated multiquantum-well-based 3-port Mach-Zehnder interferometer," IEEE Photonics Technol. Lett. 8, 1163-1165 (1996).
[CrossRef]

Wang, B. C.

B. C. Wang, L. Xu, V. Baby, I. Glesk, and P. R. Prucnal, "State selection of bistable SOA ring laser for bit level optical memory applications," IEEE Photonics Technol. Lett. 14, 989-991 (2002).
[CrossRef]

Wiedemann, P.

W. Idler, K. Daub, G. Laube, M. Schilling, P. Wiedemann, K. Dutting, M. Klenk, E. Lach, and K. Wunstel, "10 Gb/s wavelength conversion with integrated multiquantum-well-based 3-port Mach-Zehnder interferometer," IEEE Photonics Technol. Lett. 8, 1163-1165 (1996).
[CrossRef]

Wunstel, K.

W. Idler, K. Daub, G. Laube, M. Schilling, P. Wiedemann, K. Dutting, M. Klenk, E. Lach, and K. Wunstel, "10 Gb/s wavelength conversion with integrated multiquantum-well-based 3-port Mach-Zehnder interferometer," IEEE Photonics Technol. Lett. 8, 1163-1165 (1996).
[CrossRef]

Xu, L.

L. Xu, I. Glesk, V. Baby and P. R. Prucnal, "All-optical wavelength conversion using SOA at nearly symmetric position in a fiber-based Sagnac interferometric loop," IEEE Photonics Technol. Lett. 16, 539--541 (2004).
[CrossRef]

B. C. Wang, L. Xu, V. Baby, I. Glesk, and P. R. Prucnal, "State selection of bistable SOA ring laser for bit level optical memory applications," IEEE Photonics Technol. Lett. 14, 989-991 (2002).
[CrossRef]

Appl. Opt. (1)

Electron. Lett. (1)

J. Leuthold, C. H. Joyner, B. Mikkelsen, G. Raybon, J. L. Pleumeekers, B. I. Miller, K. Dreyer, and C. A. Burrus, "100 Gbit/s all-optical wavelength conversion with integrated SOA delayed-interference configuration," Electron. Lett. 36, 1129-1130 (2000).
[CrossRef]

IEEE Photonics Technol. Lett. (4)

K. L. Deng, I. Glesk, K. I. Kang, and P. R. Prucnal, "Unbalanced TOAD for optical data clock separation in self-clocked transparent OTDM networks," IEEE Photonics Technol. Lett. 9, 830-832 (1997).
[CrossRef]

B. C. Wang, L. Xu, V. Baby, I. Glesk, and P. R. Prucnal, "State selection of bistable SOA ring laser for bit level optical memory applications," IEEE Photonics Technol. Lett. 14, 989-991 (2002).
[CrossRef]

L. Xu, I. Glesk, V. Baby and P. R. Prucnal, "All-optical wavelength conversion using SOA at nearly symmetric position in a fiber-based Sagnac interferometric loop," IEEE Photonics Technol. Lett. 16, 539--541 (2004).
[CrossRef]

W. Idler, K. Daub, G. Laube, M. Schilling, P. Wiedemann, K. Dutting, M. Klenk, E. Lach, and K. Wunstel, "10 Gb/s wavelength conversion with integrated multiquantum-well-based 3-port Mach-Zehnder interferometer," IEEE Photonics Technol. Lett. 8, 1163-1165 (1996).
[CrossRef]

Other (1)

J. Y. Emery, M. Picq, F. Poingt, F. Gaborit, R. Brenot, M. Renaud, B. Lavigne, and A. Dupas, "Optimised 2-R all-optical regenerator with low polarization sensitivity penalty (<1 dB) for optical networking applications," in Optical Fiber Conference (Optical Society of America, 2000), paper Mb4-1.

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Figures (15)

Fig. 1
Fig. 1

System schematic. P in is the incident data transcribed on the original wavelength λ 1 , P cw is the cw beam with the converted wavelength λ 2 , the PC terms are polarization controllers, C1 and C2 are couplers, F is a spectral filter, and ϕ cw R and ϕ cw L are the phases of the clockwise- and counterclockwise-propagating beams, respectively.

Fig. 2
Fig. 2

An ideal presentation of the phases in the system. The upper panel displays the input signal’s power. The central panel represents the phases of the beams that exist in the SOA. Δ T is the time lag between the two beams, and β is the relative phase between them (in this case, β is negative). In the lower panel the two phases ϕ cw R and ϕ cw L are presented.

Fig. 3
Fig. 3

Schematic presentation of the coupler operation. In the upper panel the two phases ϕ cw R and ϕ cw L are presented again for comparison purposes. In the central panel the phase difference ϕ cw R ϕ cw L is plotted, whereas the lower panel represents the coupler exit.

Fig. 4
Fig. 4

Illustration of the inversion algorithm. By changing the relative phase between the two propagating beams from β = ϕ 1 Δ T p dc to β = ϕ 1 Δ T p dc , an inverted pattern appears.

Fig. 5
Fig. 5

Calculation results of the ASL as a regenerator for a time delay of 14 ps. The SNR increases slightly, but the bit shape remains almost intact.

Fig. 6
Fig. 6

Same as Fig. 5 but for a delay time of 40 ps. In this case the SNR increases significantly at the expense of a small pattern deformation.

Fig. 7
Fig. 7

Simulation results. The upper panel stands for the incident signal; the central panel stands for the cross-gain modulation signal that exits the SOA; and the lower panel is the simulated TASR output.

Fig. 8
Fig. 8

The phases Φ cw R and Φ cw L that correspond to the simulation presented in Fig. 7.

Fig. 9
Fig. 9

Same SOA as in Fig. 8 but with a 43 : 57 coupler instead of a 50 : 50 one. In this case the dc part is totally reduced, but the spikes increase.

Fig. 10
Fig. 10

Same parameters as in Fig. 8 with a focus on the seventh bit (around 0.7 ns).

Fig. 11
Fig. 11

Experimental results of the input and output signals of the TASR.

Fig. 12
Fig. 12

BER measurements of the incident data signal (dashed line) and the output signal (solid line) for a high (27 dB) OSNR signal.

Fig. 13
Fig. 13

BER measurements of the incident data signal (dashed line) and the output signal (solid line) for a low (18 dB) OSNR signal.

Fig. 14
Fig. 14

Schematic representation of the SOA in a cross-gain–phase operation.

Fig. 15
Fig. 15

Physical realization of a device (by a PC) that adds a different phase for the two beams copropagating and counterpropagating

Equations (27)

Equations on this page are rendered with MathJax. Learn more.

ϕ out ( τ ) = ϕ in + 1 2 α C [ τ e p dc + τ p ac ( ξ ) d ξ ] ,
E SOA ( t + Δ T 2 ) E SOA ( t Δ T 2 ) Δ T d E SOA ( t ) d t .
ϕ cw L ( t ) = ϕ 0 + ϕ 1 t p ac ( ξ ) d ξ ,
ϕ cw R ( t ) = ϕ 0 + ϕ 1 t + Δ T p ac ( ξ ) d ξ + β .
P out ( t ) exp [ i ϕ cw R ( t ) ] exp [ i ϕ cw L ( t ) ] 2 = 4 sin 2 [ ϕ 1 2 t t + Δ T p ac ( ξ ) d ξ + β 2 ] .
P out ( t ) 4 sin 2 [ ( ϕ 1 Δ T p ac ( t ) + β ) 2 ] .
P out ( t ) { ϕ 1 Δ T [ p ac ( t ) + p dc ] } 2 = [ ϕ 1 Δ T P in ( t ) ] 2 .
P out ( t ) { ϕ 1 Δ T [ p ac ( t ) p dc ] } 2 [ ϕ 1 Δ T P in ¯ ( t ) ] 2 ,
P z = ( g α int ) P ,
ϕ z = 1 2 α g ,
g τ = g 0 g τ c g P E s ,
P out ( τ ) = P 0 exp [ h ( τ ) ] ,
ϕ out ( τ ) = ϕ in ( τ ) 1 2 α h ( τ ) ,
d h d τ = g 0 L h τ c P in ( τ ) E s [ exp ( h ) 1 ] .
d δ h d τ + δ h τ e = p ( τ ) C ,
δ h = C exp ( τ τ e ) τ p ( ξ ) exp ( ξ τ e ) d ξ .
δ h C [ τ e p dc + τ p ac ( ξ ) d ξ ] ,
ϕ out ( τ ) = ϕ in + 1 2 α C [ τ e p dc + τ p ac ( ξ ) d ξ ] .
M ( a b b * a * ) ,
M = [ cos θ exp ( i φ ) sin θ exp i μ sin θ exp ( i μ ) cos θ exp ( i φ ) ] .
M V = W ,
M T F V = W exp ( i β ) ,
A ( M T ) * M T F = [ cos 2 θ sin 2 θ exp ( 2 i μ ) sin 2 θ cos μ exp ( i φ ) sin 2 θ cos μ exp ( i φ ) cos 2 θ + sin 2 θ exp ( 2 i μ ) ] ,
sin β = sin 2 θ sin 2 μ .
M = [ 0 exp ( i μ ) exp ( i μ ) 0 ] ,
A ( M T ) * M T F = [ exp ( 2 i μ ) 0 0 exp ( 2 i μ ) ] .
Λ 1 = exp ( 2 i μ ) V 1 = ( 1 0 ) , Λ 2 = exp ( 2 i μ ) V 2 = ( 0 1 ) ,

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