Abstract

Polarization-insensitive conversion of return-to-zero (RZ) ON-OFF keying (RZ-OOK) to RZ binary phase-shift keying (RZ-BPSK) has been achieved by cross-phase modulation (XPM) in a nonlinear birefringent fiber. This work presents a theoretical analysis of the dependence of format conversion on pump-probe detuning, and the pump state-of-polarization (SOP) that can fluctuate unpredictably in a realistic system. An investigation of the impact of pump polarization fluctuation on receiver sensitivity and receiver optimal threshold for the converted RZ-BPSK probe is also carried out. It was found that although the desired XPM-induced n phase shift can be achieved by launching both the RZ-OOK pump and the probe along the same birefringent axis of the fiber, the phase shift degrades to π/3 if the SOP of the RZ-OOK pump unpredictably switches to the other axis of the fiber, resulting in a large receiver sensitivity penalty fluctuation of 14 dB. By contrast, launching the probe at 45° relative to the birefringent axes can reduce the polarization-dependent receiver sensitivity penalty fluctuation to about 2 dB as the SOP of the RZ-OOK pump is swept over the Poincaré sphere. These conclusions are in good agreement with recently published experimental results.

© 2009 Optical Society of America

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References

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  1. A. H. Gnauck and P. J. Winzer, “Optical phase-shift-keyed transmission,” J. Lightwave Technol.  23, 115–130 (2005).
    [Crossref]
  2. K. Mishina, A. Maruta, S. Mitani, T. Miyahara, K. Ishida, K. Shimizu, T. Hatta, K. Motoshima, and K.-I. Kitayama, “NRZ-OOK-to-RZ-BPSK modulation-format conversion using SOA-MZI wavelength converter,” J. Lightwave Technol.  24, 3751–3758 (2006).
    [Crossref]
  3. W. Astar, A. S. Lenihan, and G. M. Carter, “Performance of DBPSK in a 5 × 10 Gb/s mixed modulation format Raman/EDFA WDM system,” IEEE Photon. Technol. Lett.  17, 2766–2768 (2005).
    [Crossref]
  4. W. Astar and G. M. Carter, “10 Gbit/s RZ-OOK to RZ-BPSK format conversion using SOA and synchronous pulse carver,” Electron. Lett.  44, 369–370 (2008).
    [Crossref]
  5. Cishuo Yan, Yikai Su, Lilin Yi, Lufeng Leng, Xiangqing Tian, Xinyu Vu, and Yue Tian, “All-optical format conversion from NRZ to BPSK using a single saturated SOA,” IEEE Photon. Technol. Lett.  18, 2368–2370 (2006).
    [Crossref]
  6. H. Jiang, He Wen, Liuyan Han, Yili Guo, and Hanyi Zhang, “All-optical NRZ-OOK to BPSK format conversion in an SOA-based nonlinear polarization switch,” IEEE Photon. Technol. Lett.  19, 1985–1987 (2007).
    [Crossref]
  7. K. Mishina, S. Kitagawa, and A. Maruta, “All-optical modulation format conversion from on-off-keying to multiple-level phase-shift-keying based on nonlinearity in optical fiber,” Opt. Express 15, 8444–8453 (2007), http://www.opticsexpress.org/abstract.cfm?uri=OE-15-13-8444.
    [Crossref] [PubMed]
  8. W. Astar, C.-C. Wei, Y.-J. Chen, J. Chen, and G. M. Carter, “Polarization-insensitive, 40 Gb/s wavelength and RZ-OOK-to-RZ-BPSK modulation format conversion by XPM in a highly nonlinear PCF,” Opt. Express 16, 12039–12049 (2008), ttp://www.opticsinfobase.org/abstract.cfm?URI=oe-16-16-12039.
    [Crossref] [PubMed]
  9. S. Kumar, A. Selvarajan, and G. Anand, “Nonlinear propagation of two optical pulses of two different frequencies in birefringent fibers,” J. Opt. Soc. Am. B  11, 810–817 (1994).
    [Crossref]
  10. G. P. Agrawal, Nonlinear Fiber Optics (Academic Press, San Diego, CA, 2001), Chap. 7.
  11. D. Marcuse, “Derivation of analytical expressions for the bit-error probability in lightwave systems with optical amplifiers,” J. Lightwave Technol.  8, 1819–1826 (1990).
    [Crossref]
  12. E. W. Weisstein, “Noncentral Chi-Squared Distribution.” From MathWorld--A Wolfram Web Resource. http://mathworld.wolfram.com/NoncentralChi-SquaredDistribution.html.
  13. S. Stein, “Unified analysis of certain coherent and noncoherent binary communications systems,” IEEE. Trans. Info. Theory 10, 43–51 (1964).
    [Crossref]
  14. K.-P. Ho, Phase-Modulated Optical Communication System (Springer, 2005), Appendix 3.A.

2008 (2)

2007 (2)

H. Jiang, He Wen, Liuyan Han, Yili Guo, and Hanyi Zhang, “All-optical NRZ-OOK to BPSK format conversion in an SOA-based nonlinear polarization switch,” IEEE Photon. Technol. Lett.  19, 1985–1987 (2007).
[Crossref]

K. Mishina, S. Kitagawa, and A. Maruta, “All-optical modulation format conversion from on-off-keying to multiple-level phase-shift-keying based on nonlinearity in optical fiber,” Opt. Express 15, 8444–8453 (2007), http://www.opticsexpress.org/abstract.cfm?uri=OE-15-13-8444.
[Crossref] [PubMed]

2006 (2)

Cishuo Yan, Yikai Su, Lilin Yi, Lufeng Leng, Xiangqing Tian, Xinyu Vu, and Yue Tian, “All-optical format conversion from NRZ to BPSK using a single saturated SOA,” IEEE Photon. Technol. Lett.  18, 2368–2370 (2006).
[Crossref]

K. Mishina, A. Maruta, S. Mitani, T. Miyahara, K. Ishida, K. Shimizu, T. Hatta, K. Motoshima, and K.-I. Kitayama, “NRZ-OOK-to-RZ-BPSK modulation-format conversion using SOA-MZI wavelength converter,” J. Lightwave Technol.  24, 3751–3758 (2006).
[Crossref]

2005 (2)

W. Astar, A. S. Lenihan, and G. M. Carter, “Performance of DBPSK in a 5 × 10 Gb/s mixed modulation format Raman/EDFA WDM system,” IEEE Photon. Technol. Lett.  17, 2766–2768 (2005).
[Crossref]

A. H. Gnauck and P. J. Winzer, “Optical phase-shift-keyed transmission,” J. Lightwave Technol.  23, 115–130 (2005).
[Crossref]

1994 (1)

S. Kumar, A. Selvarajan, and G. Anand, “Nonlinear propagation of two optical pulses of two different frequencies in birefringent fibers,” J. Opt. Soc. Am. B  11, 810–817 (1994).
[Crossref]

1990 (1)

D. Marcuse, “Derivation of analytical expressions for the bit-error probability in lightwave systems with optical amplifiers,” J. Lightwave Technol.  8, 1819–1826 (1990).
[Crossref]

1964 (1)

S. Stein, “Unified analysis of certain coherent and noncoherent binary communications systems,” IEEE. Trans. Info. Theory 10, 43–51 (1964).
[Crossref]

Agrawal, G. P.

G. P. Agrawal, Nonlinear Fiber Optics (Academic Press, San Diego, CA, 2001), Chap. 7.

Anand, G.

S. Kumar, A. Selvarajan, and G. Anand, “Nonlinear propagation of two optical pulses of two different frequencies in birefringent fibers,” J. Opt. Soc. Am. B  11, 810–817 (1994).
[Crossref]

Astar, W.

W. Astar and G. M. Carter, “10 Gbit/s RZ-OOK to RZ-BPSK format conversion using SOA and synchronous pulse carver,” Electron. Lett.  44, 369–370 (2008).
[Crossref]

W. Astar, C.-C. Wei, Y.-J. Chen, J. Chen, and G. M. Carter, “Polarization-insensitive, 40 Gb/s wavelength and RZ-OOK-to-RZ-BPSK modulation format conversion by XPM in a highly nonlinear PCF,” Opt. Express 16, 12039–12049 (2008), ttp://www.opticsinfobase.org/abstract.cfm?URI=oe-16-16-12039.
[Crossref] [PubMed]

W. Astar, A. S. Lenihan, and G. M. Carter, “Performance of DBPSK in a 5 × 10 Gb/s mixed modulation format Raman/EDFA WDM system,” IEEE Photon. Technol. Lett.  17, 2766–2768 (2005).
[Crossref]

Carter, G. M.

W. Astar and G. M. Carter, “10 Gbit/s RZ-OOK to RZ-BPSK format conversion using SOA and synchronous pulse carver,” Electron. Lett.  44, 369–370 (2008).
[Crossref]

W. Astar, C.-C. Wei, Y.-J. Chen, J. Chen, and G. M. Carter, “Polarization-insensitive, 40 Gb/s wavelength and RZ-OOK-to-RZ-BPSK modulation format conversion by XPM in a highly nonlinear PCF,” Opt. Express 16, 12039–12049 (2008), ttp://www.opticsinfobase.org/abstract.cfm?URI=oe-16-16-12039.
[Crossref] [PubMed]

W. Astar, A. S. Lenihan, and G. M. Carter, “Performance of DBPSK in a 5 × 10 Gb/s mixed modulation format Raman/EDFA WDM system,” IEEE Photon. Technol. Lett.  17, 2766–2768 (2005).
[Crossref]

Chen, J.

Chen, Y.-J.

Gnauck, A. H.

A. H. Gnauck and P. J. Winzer, “Optical phase-shift-keyed transmission,” J. Lightwave Technol.  23, 115–130 (2005).
[Crossref]

Guo, Yili

H. Jiang, He Wen, Liuyan Han, Yili Guo, and Hanyi Zhang, “All-optical NRZ-OOK to BPSK format conversion in an SOA-based nonlinear polarization switch,” IEEE Photon. Technol. Lett.  19, 1985–1987 (2007).
[Crossref]

Han, Liuyan

H. Jiang, He Wen, Liuyan Han, Yili Guo, and Hanyi Zhang, “All-optical NRZ-OOK to BPSK format conversion in an SOA-based nonlinear polarization switch,” IEEE Photon. Technol. Lett.  19, 1985–1987 (2007).
[Crossref]

Hatta, T.

K. Mishina, A. Maruta, S. Mitani, T. Miyahara, K. Ishida, K. Shimizu, T. Hatta, K. Motoshima, and K.-I. Kitayama, “NRZ-OOK-to-RZ-BPSK modulation-format conversion using SOA-MZI wavelength converter,” J. Lightwave Technol.  24, 3751–3758 (2006).
[Crossref]

Ho, K.-P.

K.-P. Ho, Phase-Modulated Optical Communication System (Springer, 2005), Appendix 3.A.

Ishida, K.

K. Mishina, A. Maruta, S. Mitani, T. Miyahara, K. Ishida, K. Shimizu, T. Hatta, K. Motoshima, and K.-I. Kitayama, “NRZ-OOK-to-RZ-BPSK modulation-format conversion using SOA-MZI wavelength converter,” J. Lightwave Technol.  24, 3751–3758 (2006).
[Crossref]

Jiang, H.

H. Jiang, He Wen, Liuyan Han, Yili Guo, and Hanyi Zhang, “All-optical NRZ-OOK to BPSK format conversion in an SOA-based nonlinear polarization switch,” IEEE Photon. Technol. Lett.  19, 1985–1987 (2007).
[Crossref]

Kitagawa, S.

Kitayama, K.-I.

K. Mishina, A. Maruta, S. Mitani, T. Miyahara, K. Ishida, K. Shimizu, T. Hatta, K. Motoshima, and K.-I. Kitayama, “NRZ-OOK-to-RZ-BPSK modulation-format conversion using SOA-MZI wavelength converter,” J. Lightwave Technol.  24, 3751–3758 (2006).
[Crossref]

Kumar, S.

S. Kumar, A. Selvarajan, and G. Anand, “Nonlinear propagation of two optical pulses of two different frequencies in birefringent fibers,” J. Opt. Soc. Am. B  11, 810–817 (1994).
[Crossref]

Leng, Lufeng

Cishuo Yan, Yikai Su, Lilin Yi, Lufeng Leng, Xiangqing Tian, Xinyu Vu, and Yue Tian, “All-optical format conversion from NRZ to BPSK using a single saturated SOA,” IEEE Photon. Technol. Lett.  18, 2368–2370 (2006).
[Crossref]

Lenihan, A. S.

W. Astar, A. S. Lenihan, and G. M. Carter, “Performance of DBPSK in a 5 × 10 Gb/s mixed modulation format Raman/EDFA WDM system,” IEEE Photon. Technol. Lett.  17, 2766–2768 (2005).
[Crossref]

Marcuse, D.

D. Marcuse, “Derivation of analytical expressions for the bit-error probability in lightwave systems with optical amplifiers,” J. Lightwave Technol.  8, 1819–1826 (1990).
[Crossref]

Maruta, A.

K. Mishina, S. Kitagawa, and A. Maruta, “All-optical modulation format conversion from on-off-keying to multiple-level phase-shift-keying based on nonlinearity in optical fiber,” Opt. Express 15, 8444–8453 (2007), http://www.opticsexpress.org/abstract.cfm?uri=OE-15-13-8444.
[Crossref] [PubMed]

K. Mishina, A. Maruta, S. Mitani, T. Miyahara, K. Ishida, K. Shimizu, T. Hatta, K. Motoshima, and K.-I. Kitayama, “NRZ-OOK-to-RZ-BPSK modulation-format conversion using SOA-MZI wavelength converter,” J. Lightwave Technol.  24, 3751–3758 (2006).
[Crossref]

Mishina, K.

K. Mishina, S. Kitagawa, and A. Maruta, “All-optical modulation format conversion from on-off-keying to multiple-level phase-shift-keying based on nonlinearity in optical fiber,” Opt. Express 15, 8444–8453 (2007), http://www.opticsexpress.org/abstract.cfm?uri=OE-15-13-8444.
[Crossref] [PubMed]

K. Mishina, A. Maruta, S. Mitani, T. Miyahara, K. Ishida, K. Shimizu, T. Hatta, K. Motoshima, and K.-I. Kitayama, “NRZ-OOK-to-RZ-BPSK modulation-format conversion using SOA-MZI wavelength converter,” J. Lightwave Technol.  24, 3751–3758 (2006).
[Crossref]

Mitani, S.

K. Mishina, A. Maruta, S. Mitani, T. Miyahara, K. Ishida, K. Shimizu, T. Hatta, K. Motoshima, and K.-I. Kitayama, “NRZ-OOK-to-RZ-BPSK modulation-format conversion using SOA-MZI wavelength converter,” J. Lightwave Technol.  24, 3751–3758 (2006).
[Crossref]

Miyahara, T.

K. Mishina, A. Maruta, S. Mitani, T. Miyahara, K. Ishida, K. Shimizu, T. Hatta, K. Motoshima, and K.-I. Kitayama, “NRZ-OOK-to-RZ-BPSK modulation-format conversion using SOA-MZI wavelength converter,” J. Lightwave Technol.  24, 3751–3758 (2006).
[Crossref]

Motoshima, K.

K. Mishina, A. Maruta, S. Mitani, T. Miyahara, K. Ishida, K. Shimizu, T. Hatta, K. Motoshima, and K.-I. Kitayama, “NRZ-OOK-to-RZ-BPSK modulation-format conversion using SOA-MZI wavelength converter,” J. Lightwave Technol.  24, 3751–3758 (2006).
[Crossref]

Selvarajan, A.

S. Kumar, A. Selvarajan, and G. Anand, “Nonlinear propagation of two optical pulses of two different frequencies in birefringent fibers,” J. Opt. Soc. Am. B  11, 810–817 (1994).
[Crossref]

Shimizu, K.

K. Mishina, A. Maruta, S. Mitani, T. Miyahara, K. Ishida, K. Shimizu, T. Hatta, K. Motoshima, and K.-I. Kitayama, “NRZ-OOK-to-RZ-BPSK modulation-format conversion using SOA-MZI wavelength converter,” J. Lightwave Technol.  24, 3751–3758 (2006).
[Crossref]

Stein, S.

S. Stein, “Unified analysis of certain coherent and noncoherent binary communications systems,” IEEE. Trans. Info. Theory 10, 43–51 (1964).
[Crossref]

Su, Yikai

Cishuo Yan, Yikai Su, Lilin Yi, Lufeng Leng, Xiangqing Tian, Xinyu Vu, and Yue Tian, “All-optical format conversion from NRZ to BPSK using a single saturated SOA,” IEEE Photon. Technol. Lett.  18, 2368–2370 (2006).
[Crossref]

Tian, Xiangqing

Cishuo Yan, Yikai Su, Lilin Yi, Lufeng Leng, Xiangqing Tian, Xinyu Vu, and Yue Tian, “All-optical format conversion from NRZ to BPSK using a single saturated SOA,” IEEE Photon. Technol. Lett.  18, 2368–2370 (2006).
[Crossref]

Tian, Yue

Cishuo Yan, Yikai Su, Lilin Yi, Lufeng Leng, Xiangqing Tian, Xinyu Vu, and Yue Tian, “All-optical format conversion from NRZ to BPSK using a single saturated SOA,” IEEE Photon. Technol. Lett.  18, 2368–2370 (2006).
[Crossref]

Vu, Xinyu

Cishuo Yan, Yikai Su, Lilin Yi, Lufeng Leng, Xiangqing Tian, Xinyu Vu, and Yue Tian, “All-optical format conversion from NRZ to BPSK using a single saturated SOA,” IEEE Photon. Technol. Lett.  18, 2368–2370 (2006).
[Crossref]

Wei, C.-C.

Weisstein, E. W.

E. W. Weisstein, “Noncentral Chi-Squared Distribution.” From MathWorld--A Wolfram Web Resource. http://mathworld.wolfram.com/NoncentralChi-SquaredDistribution.html.

Wen, He

H. Jiang, He Wen, Liuyan Han, Yili Guo, and Hanyi Zhang, “All-optical NRZ-OOK to BPSK format conversion in an SOA-based nonlinear polarization switch,” IEEE Photon. Technol. Lett.  19, 1985–1987 (2007).
[Crossref]

Winzer, P. J.

A. H. Gnauck and P. J. Winzer, “Optical phase-shift-keyed transmission,” J. Lightwave Technol.  23, 115–130 (2005).
[Crossref]

Yan, Cishuo

Cishuo Yan, Yikai Su, Lilin Yi, Lufeng Leng, Xiangqing Tian, Xinyu Vu, and Yue Tian, “All-optical format conversion from NRZ to BPSK using a single saturated SOA,” IEEE Photon. Technol. Lett.  18, 2368–2370 (2006).
[Crossref]

Yi, Lilin

Cishuo Yan, Yikai Su, Lilin Yi, Lufeng Leng, Xiangqing Tian, Xinyu Vu, and Yue Tian, “All-optical format conversion from NRZ to BPSK using a single saturated SOA,” IEEE Photon. Technol. Lett.  18, 2368–2370 (2006).
[Crossref]

Zhang, Hanyi

H. Jiang, He Wen, Liuyan Han, Yili Guo, and Hanyi Zhang, “All-optical NRZ-OOK to BPSK format conversion in an SOA-based nonlinear polarization switch,” IEEE Photon. Technol. Lett.  19, 1985–1987 (2007).
[Crossref]

Electron. Lett (1)

W. Astar and G. M. Carter, “10 Gbit/s RZ-OOK to RZ-BPSK format conversion using SOA and synchronous pulse carver,” Electron. Lett.  44, 369–370 (2008).
[Crossref]

From MathWorld--A Wolfram Web Resource (1)

E. W. Weisstein, “Noncentral Chi-Squared Distribution.” From MathWorld--A Wolfram Web Resource. http://mathworld.wolfram.com/NoncentralChi-SquaredDistribution.html.

IEEE Photon. Technol. Lett (3)

W. Astar, A. S. Lenihan, and G. M. Carter, “Performance of DBPSK in a 5 × 10 Gb/s mixed modulation format Raman/EDFA WDM system,” IEEE Photon. Technol. Lett.  17, 2766–2768 (2005).
[Crossref]

Cishuo Yan, Yikai Su, Lilin Yi, Lufeng Leng, Xiangqing Tian, Xinyu Vu, and Yue Tian, “All-optical format conversion from NRZ to BPSK using a single saturated SOA,” IEEE Photon. Technol. Lett.  18, 2368–2370 (2006).
[Crossref]

H. Jiang, He Wen, Liuyan Han, Yili Guo, and Hanyi Zhang, “All-optical NRZ-OOK to BPSK format conversion in an SOA-based nonlinear polarization switch,” IEEE Photon. Technol. Lett.  19, 1985–1987 (2007).
[Crossref]

IEEE. Trans. Info. Theory (1)

S. Stein, “Unified analysis of certain coherent and noncoherent binary communications systems,” IEEE. Trans. Info. Theory 10, 43–51 (1964).
[Crossref]

J. Lightwave Technol (3)

D. Marcuse, “Derivation of analytical expressions for the bit-error probability in lightwave systems with optical amplifiers,” J. Lightwave Technol.  8, 1819–1826 (1990).
[Crossref]

A. H. Gnauck and P. J. Winzer, “Optical phase-shift-keyed transmission,” J. Lightwave Technol.  23, 115–130 (2005).
[Crossref]

K. Mishina, A. Maruta, S. Mitani, T. Miyahara, K. Ishida, K. Shimizu, T. Hatta, K. Motoshima, and K.-I. Kitayama, “NRZ-OOK-to-RZ-BPSK modulation-format conversion using SOA-MZI wavelength converter,” J. Lightwave Technol.  24, 3751–3758 (2006).
[Crossref]

J. Opt. Soc. Am (1)

S. Kumar, A. Selvarajan, and G. Anand, “Nonlinear propagation of two optical pulses of two different frequencies in birefringent fibers,” J. Opt. Soc. Am. B  11, 810–817 (1994).
[Crossref]

Opt. Express (2)

Other (2)

G. P. Agrawal, Nonlinear Fiber Optics (Academic Press, San Diego, CA, 2001), Chap. 7.

K.-P. Ho, Phase-Modulated Optical Communication System (Springer, 2005), Appendix 3.A.

Supplementary Material (2)

» Media 1: AVI (704 KB)     
» Media 2: AVI (1726 KB)     

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Figures (5)

Fig. 1.
Fig. 1.

RZ-OOK-to-RZ-BPSK format conversion in a birefringent fiber.

Fig. 2.
Fig. 2.

With Φ = π , (a) (Media 1)the effective phase shift with L|∆K|≫2π as a function of ψ 2 , and (b) (Media 2) the effective phase shift for the best scenario (ψ 2 = π/4) as a function of L|∆K|.

Fig. 3.
Fig. 3.

Maximum and minimum effective phase shifts as functions of Φ (Eq. (12)).

Fig. 4.
Fig. 4.

(a) The optimal thresholds, and (b) the sensitivities as functions of ϕeff .

Fig. 5.
Fig. 5.

(a) The optimal thresholds, and (b) the sensitivity penalty as functions of Φ.

Equations (29)

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

d A mp dz = ( A mp 2 + 2 A np 2 + 2 3 A mq 2 + 2 3 A nq 2 ) A mp + j γ 3 A mp * A mq 2 e j 2 Δ β mpq z
+ j 2 γ 3 A np * A nq A mq e j ( Δ β mpq + Δ β npq ) z + j 2 γ 3 A np A nq * A mq e j ( Δ β mpq Δ β npq ) z ,
d A 1 p dz = ( A 1 p 2 + 2 3 A 1 q 2 ) A 1 p ,
d A 2 p dz = ( 2 A 1 p 2 + 2 3 A 1 q 2 ) A 2 p + j 2 γ 3 A 1 A 1 q * A 2 q e j ( Δ β 1 pq Δ β 2 pq ) z .
A 1 x ( z ) = A 1 x ( 0 ) exp [ j γ P 1 3 ( 2 + cos 2 ψ 1 ) z ] ,
A 1 y ( z ) = A 1 y ( 0 ) exp [ j γ P 1 3 ( 2 + sin 2 ψ 1 ) z ] .
A 2 x ( L ) = A 2 x ( 0 ) [ cos ( kL 2 ) + j μ x sin ( kL 2 ) ] exp [ j ( 3 γ P 1 2 j γ P 1 3 sin 2 ψ 1 j ΔK 2 ) L ] ,
A 2 y ( L ) = A 2 y ( 0 ) [ cos ( kL 2 ) + j μ y sin ( kL 2 ) ] exp [ j ( 3 γ P 1 2 j γ P 1 3 cos 2 ψ 1 + j ΔK 2 ) L ] ,
k = 4 9 γ 2 P 1 2 sin 2 ( 2 ψ 1 ) + [ γ P 1 cos ( 2 ψ 1 ) + ΔK ] 2 ,
μ x = 1 k [ γ P 1 cos ( 2 ψ 1 ) + ΔK + 2 γ P 1 3 tan ψ 2 sin ( 2 ψ 1 ) e j Δ θ ] ,
μ y = 1 k [ γ P 1 cos ( 2 ψ 1 ) ΔK + 2 γ P 1 3 cos ψ 2 sin ( 2 ψ 1 ) e j Δ θ ] ,
ϕ eff = cos 1 ( A 2 x ( 0 ) A 2 x ( L ) cos ϕ x + A 2 y ( 0 ) A 2 y ( L ) cos ϕ y A 2 x ( 0 ) 2 + A 2 y ( 0 ) 2 ) ,
ϕ eff = cos 1 ( cos 2 ψ 2 cos ( Φ ( 1 + 2 cos 2 ψ 1 ) 3 ) + sin 2 ψ 2 cos ( Φ ( 1 + 2 sin 2 ψ 1 ) 3 ) ) ,
ΔK = 1.02 × 10 6 × ( 1 λ 1 1 λ 2 ) + 7.04 × 10 2 × In λ 1 λ 2 ,
i DB ( t ) = p = x , y E p ( t ) + E p ( t T ) 2 + n p ( t ) + n p ( t T ) 2 2 ,
i AMI ( t ) = p = x , y E p ( t ) E p ( t T ) 2 + n p ( t ) n p ( t T ) 2 2 .
ξ = p = x , y E p ( t ) ± E p ( t T ) 2 2 σ n 2 .
p v ( v ) = 4 ρ s ρ s v ξ e 2 ρ s v ξ 2 I 1 ( 2 ξ ρ s v )
p V DB in ( v ) = 2 ρ s v e 2 ρ s ( 1 + v ) I 1 ( 4 ρ s v ) ,
p V AMI in ( v ) = 4 ρ s 2 v e 2 ρ s v ,
p V DB out ( v ) = 2 ρ s v sec ( ϕ eff 2 ) e 2 ρ s [ cos 2 ( ϕ eff 2 ) + v ] I 1 ( 4 ρ s v cos ( ϕ eff 2 ) ) ,
p V AMI out ( v ) = 2 ρ s v csc ( ϕ eff 2 ) e 2 ρ s [ sin 2 ( ϕ eff 2 ) + v ] I 1 ( 4 ρ s v sin ( ϕ eff 2 ) ) ,
P e in = 1 Q 1 ( 2 ρ s , 2 ρ s h ) + 1 2 e ρ s + 2 ρ s h Q 1 ( 2 ρ s , 2 2 ρ s h )
1 8 e 2 ρ s 2 ρ s h h I 1 ( 4 ρ s h ) + 1 8 e 2 ρ s 2 ρ s h n = 1 n ( 2 h ) n I n ( 4 ρ s h ) ,
P e out = e ρ s ( 1 + 2 h ) n = 0 ( a n 4 a n 1 a n 2 + 8 m = 0 n 1 a m ) × cot n 1 ( ϕ eff 2 ) I n 1 ( ρ s sin ( ϕ eff 2 ) ) ,
= e ρ s [ 1 + cos 2 ( ϕ eff 2 ) + 2 h ] m = 0 ( a ˜ n + 4 m = 0 n 1 ( 2 n m 1 ) a ˜ m m = 1 n 1 m = 0 m 1 a ˜ m )
× ( cos ( ϕ eff 2 ) 2 h ) n 1 I n 1 ( 4 ρ s cos ( ϕ eff 2 ) ) ,
a m = 1 8 δ 0 , m + 1 8 l = 1 m ( 1 ) n l l ! ( m 1 l 1 ) ( 4 ρ s h ) l ,
a ¯ m = 1 8 δ 0 , m + 1 8 l = 1 m 1 l ! ( m 1 l 1 ) ( ρ s sin 2 ( ϕ eff 2 ) ) l ,

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