Abstract

Single mode fiber (SMF) birefringence effects have been a limiting factor for a variety of Sagnac applications over longer distance SMF links. In this report, we present a new concept of the SMF birefringence compensation in a Sagnac interferometric setup, based on a novel polarization control system. For the destructive interference, our control system guarantees a perfect compensation of both the SMF birefringence and imperfect propagation times matching of the setup’s components. For the stabilization of the constructive interference, we have applied a fiber stretcher and a simple proportional-integral-derivative (PID) controller. The enclosed experimental data of the setup’s visibility confirm validity of our polarization control system. We have also showed that the SMF birefringence model used in a “plug & play” interferometric setup [19], widely cited in the papers on quantum key distribution [11, 12, 13], cannot be applied in SMF Sagnac interferometric setup. However, the SMF birefringence model based on the Kapron equivalence well describes SMF Sagnac.

© 2009 Optical Society of America

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References

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  1. E. Udd, "Sensing and instrumentation applications of the Sagnac fiber optic interferometer," in Interferometric Fiber Sensing, Proc. SPIE 2341, 52-59 (1994).
    [CrossRef]
  2. J. Zheng, "Differential singlemode fibre frequency-modulated continuous-wave Sagnac gyroscope," Electron. Lett. 41, 727-728 (2005).
    [CrossRef]
  3. B. Culshaw, "The optical fibre Sagnac interferometer: an overview of its principles and applications," Meas. Sci. Technol. 17, 1-16 (2006).
    [CrossRef]
  4. L. R. Jaroszewicz, "Fiber-optic Sagnac Interferometer as real sensor of the physical quantities," in Proceedings of the Symposium on Photonics Technologies (Wroclaw, Poland, 2006), pp. 99-101.
  5. L. R. Jaroszewicz and Z. Krajewski, "Application of fiber-optic Sagnac interferometer for detection of rotational seismic events," Molecular Quantum Acoustics 22, 133-134 (2001).
  6. T. Nishioka, H. Ishizuka, T. Hasegawa, and J. Abe, "Circular type quantum key distribution," IEEE Photon. Technol. Lett. 14, 576-578 (2002).
    [CrossRef]
  7. C. Zhou, G. Wu, L. Ding, and H. Zeng, "Single-photon routing by time-division phase modulation in a Sagnac interferometer," Appl. Phys. Lett. 83, 15-17 (2003).
    [CrossRef]
  8. B. Qi, L. L. Huang, H. K. Lo, and L. Qian, "Quantum key distribution based on a Sagnac loop interferometer and polarization-insensitive phase modulators," in IEEE International Symposium on Information Theory (Institute of Electrical and Electronics Engineers, 2006), pp. 2090-2093.
    [CrossRef]
  9. W. A. de Brito and R. V. Ramos, "Quantum information technology with Sagnac interferometer: interaction-free measurement, quantum key distribution and quantum secret sharing," J. Mod. Opt. 55, 1231-1241 (2008).
    [CrossRef]
  10. J. Bogdanski, J. Ahrens, and M. Bourennane, "Sagnac secret sharing over telecom fiber networks," Opt. Express 17, 1055-1063 (2009).
    [CrossRef] [PubMed]
  11. N. Gisin, G. Ribordy, W. Tittel, and H.  Zbinden, "Quantum cryptography," Rev. Mod. Phys. 74, 145-195 (2002).
    [CrossRef]
  12. D. S. Bethune and W. P. Risk, "Autocompensating quantum cryptography," New J. Phys. 4, 42.1-42.15 (2002).
    [CrossRef]
  13. A. Muller, T. Herzog, B. Huttner, W. Tittel, H. Zbinden, and N. Gisin, "Plug and play systems for quantum cryptography," Appl. Phys. Lett. 70, 793-795 (1997).
    [CrossRef]
  14. A. Kuzin, H. Cerecedo Nunez, and N. Korneev, "Alignment of a birefringent fiber Sagnac interferometer by fiber twist," Opt. Commun. 160, 37-41 (1999).
    [CrossRef]
  15. B. Ibarra-Escamilla, E. A. Kuzin, O. Pottiez, J. W. Haus, F. Gutierrez-Zainos, R. Grajales-Coutin, and P. Zaca-Moran, "Fiber optical loop mirror with a symmetrical coupler and a quarter-wave retarder plate in the loop," Opt. Commun. 242, 191-197 (2004).
    [CrossRef]
  16. D. B. Mortimore, "Fiber loop reflectors," J. Lightwave Technol. 6, 1217-1224 (1988).
    [CrossRef]
  17. F. P. Kapron, N. F. Borrelli, and D. B. Keck, "Birefringence in. dielectric optical waveguides," IEEE J. Quantum Electron. QE-8, 222-230 (1972)
    [CrossRef]
  18. C. Tsao, Optical fibre waveguide analysis (Oxford Science Publ. 1992).
  19. M. Martinelli, "A universal compensator for polarization changes induced by birefringence on a retracing beam," Opt. Commun. 72, 341-344 (1989).
    [CrossRef]

2009

2008

W. A. de Brito and R. V. Ramos, "Quantum information technology with Sagnac interferometer: interaction-free measurement, quantum key distribution and quantum secret sharing," J. Mod. Opt. 55, 1231-1241 (2008).
[CrossRef]

2006

B. Culshaw, "The optical fibre Sagnac interferometer: an overview of its principles and applications," Meas. Sci. Technol. 17, 1-16 (2006).
[CrossRef]

2005

J. Zheng, "Differential singlemode fibre frequency-modulated continuous-wave Sagnac gyroscope," Electron. Lett. 41, 727-728 (2005).
[CrossRef]

2004

B. Ibarra-Escamilla, E. A. Kuzin, O. Pottiez, J. W. Haus, F. Gutierrez-Zainos, R. Grajales-Coutin, and P. Zaca-Moran, "Fiber optical loop mirror with a symmetrical coupler and a quarter-wave retarder plate in the loop," Opt. Commun. 242, 191-197 (2004).
[CrossRef]

2003

C. Zhou, G. Wu, L. Ding, and H. Zeng, "Single-photon routing by time-division phase modulation in a Sagnac interferometer," Appl. Phys. Lett. 83, 15-17 (2003).
[CrossRef]

2002

T. Nishioka, H. Ishizuka, T. Hasegawa, and J. Abe, "Circular type quantum key distribution," IEEE Photon. Technol. Lett. 14, 576-578 (2002).
[CrossRef]

N. Gisin, G. Ribordy, W. Tittel, and H.  Zbinden, "Quantum cryptography," Rev. Mod. Phys. 74, 145-195 (2002).
[CrossRef]

2001

L. R. Jaroszewicz and Z. Krajewski, "Application of fiber-optic Sagnac interferometer for detection of rotational seismic events," Molecular Quantum Acoustics 22, 133-134 (2001).

1999

A. Kuzin, H. Cerecedo Nunez, and N. Korneev, "Alignment of a birefringent fiber Sagnac interferometer by fiber twist," Opt. Commun. 160, 37-41 (1999).
[CrossRef]

1997

A. Muller, T. Herzog, B. Huttner, W. Tittel, H. Zbinden, and N. Gisin, "Plug and play systems for quantum cryptography," Appl. Phys. Lett. 70, 793-795 (1997).
[CrossRef]

1994

E. Udd, "Sensing and instrumentation applications of the Sagnac fiber optic interferometer," in Interferometric Fiber Sensing, Proc. SPIE 2341, 52-59 (1994).
[CrossRef]

1989

M. Martinelli, "A universal compensator for polarization changes induced by birefringence on a retracing beam," Opt. Commun. 72, 341-344 (1989).
[CrossRef]

1988

D. B. Mortimore, "Fiber loop reflectors," J. Lightwave Technol. 6, 1217-1224 (1988).
[CrossRef]

1972

F. P. Kapron, N. F. Borrelli, and D. B. Keck, "Birefringence in. dielectric optical waveguides," IEEE J. Quantum Electron. QE-8, 222-230 (1972)
[CrossRef]

Abe, J.

T. Nishioka, H. Ishizuka, T. Hasegawa, and J. Abe, "Circular type quantum key distribution," IEEE Photon. Technol. Lett. 14, 576-578 (2002).
[CrossRef]

Ahrens, J.

Bogdanski, J.

Borrelli, N. F.

F. P. Kapron, N. F. Borrelli, and D. B. Keck, "Birefringence in. dielectric optical waveguides," IEEE J. Quantum Electron. QE-8, 222-230 (1972)
[CrossRef]

Bourennane, M.

Cerecedo Nunez, H.

A. Kuzin, H. Cerecedo Nunez, and N. Korneev, "Alignment of a birefringent fiber Sagnac interferometer by fiber twist," Opt. Commun. 160, 37-41 (1999).
[CrossRef]

Culshaw, B.

B. Culshaw, "The optical fibre Sagnac interferometer: an overview of its principles and applications," Meas. Sci. Technol. 17, 1-16 (2006).
[CrossRef]

de Brito, W. A.

W. A. de Brito and R. V. Ramos, "Quantum information technology with Sagnac interferometer: interaction-free measurement, quantum key distribution and quantum secret sharing," J. Mod. Opt. 55, 1231-1241 (2008).
[CrossRef]

Ding, L.

C. Zhou, G. Wu, L. Ding, and H. Zeng, "Single-photon routing by time-division phase modulation in a Sagnac interferometer," Appl. Phys. Lett. 83, 15-17 (2003).
[CrossRef]

Gisin, N.

N. Gisin, G. Ribordy, W. Tittel, and H.  Zbinden, "Quantum cryptography," Rev. Mod. Phys. 74, 145-195 (2002).
[CrossRef]

A. Muller, T. Herzog, B. Huttner, W. Tittel, H. Zbinden, and N. Gisin, "Plug and play systems for quantum cryptography," Appl. Phys. Lett. 70, 793-795 (1997).
[CrossRef]

Grajales-Coutin, R.

B. Ibarra-Escamilla, E. A. Kuzin, O. Pottiez, J. W. Haus, F. Gutierrez-Zainos, R. Grajales-Coutin, and P. Zaca-Moran, "Fiber optical loop mirror with a symmetrical coupler and a quarter-wave retarder plate in the loop," Opt. Commun. 242, 191-197 (2004).
[CrossRef]

Gutierrez-Zainos, F.

B. Ibarra-Escamilla, E. A. Kuzin, O. Pottiez, J. W. Haus, F. Gutierrez-Zainos, R. Grajales-Coutin, and P. Zaca-Moran, "Fiber optical loop mirror with a symmetrical coupler and a quarter-wave retarder plate in the loop," Opt. Commun. 242, 191-197 (2004).
[CrossRef]

Hasegawa, T.

T. Nishioka, H. Ishizuka, T. Hasegawa, and J. Abe, "Circular type quantum key distribution," IEEE Photon. Technol. Lett. 14, 576-578 (2002).
[CrossRef]

Haus, J. W.

B. Ibarra-Escamilla, E. A. Kuzin, O. Pottiez, J. W. Haus, F. Gutierrez-Zainos, R. Grajales-Coutin, and P. Zaca-Moran, "Fiber optical loop mirror with a symmetrical coupler and a quarter-wave retarder plate in the loop," Opt. Commun. 242, 191-197 (2004).
[CrossRef]

Herzog, T.

A. Muller, T. Herzog, B. Huttner, W. Tittel, H. Zbinden, and N. Gisin, "Plug and play systems for quantum cryptography," Appl. Phys. Lett. 70, 793-795 (1997).
[CrossRef]

Huttner, B.

A. Muller, T. Herzog, B. Huttner, W. Tittel, H. Zbinden, and N. Gisin, "Plug and play systems for quantum cryptography," Appl. Phys. Lett. 70, 793-795 (1997).
[CrossRef]

Ibarra-Escamilla, B.

B. Ibarra-Escamilla, E. A. Kuzin, O. Pottiez, J. W. Haus, F. Gutierrez-Zainos, R. Grajales-Coutin, and P. Zaca-Moran, "Fiber optical loop mirror with a symmetrical coupler and a quarter-wave retarder plate in the loop," Opt. Commun. 242, 191-197 (2004).
[CrossRef]

Ishizuka, H.

T. Nishioka, H. Ishizuka, T. Hasegawa, and J. Abe, "Circular type quantum key distribution," IEEE Photon. Technol. Lett. 14, 576-578 (2002).
[CrossRef]

Jaroszewicz, L. R.

L. R. Jaroszewicz and Z. Krajewski, "Application of fiber-optic Sagnac interferometer for detection of rotational seismic events," Molecular Quantum Acoustics 22, 133-134 (2001).

Kapron, F. P.

F. P. Kapron, N. F. Borrelli, and D. B. Keck, "Birefringence in. dielectric optical waveguides," IEEE J. Quantum Electron. QE-8, 222-230 (1972)
[CrossRef]

Keck, D. B.

F. P. Kapron, N. F. Borrelli, and D. B. Keck, "Birefringence in. dielectric optical waveguides," IEEE J. Quantum Electron. QE-8, 222-230 (1972)
[CrossRef]

Krajewski, Z.

L. R. Jaroszewicz and Z. Krajewski, "Application of fiber-optic Sagnac interferometer for detection of rotational seismic events," Molecular Quantum Acoustics 22, 133-134 (2001).

Kuzin, A.

A. Kuzin, H. Cerecedo Nunez, and N. Korneev, "Alignment of a birefringent fiber Sagnac interferometer by fiber twist," Opt. Commun. 160, 37-41 (1999).
[CrossRef]

Kuzin, E. A.

B. Ibarra-Escamilla, E. A. Kuzin, O. Pottiez, J. W. Haus, F. Gutierrez-Zainos, R. Grajales-Coutin, and P. Zaca-Moran, "Fiber optical loop mirror with a symmetrical coupler and a quarter-wave retarder plate in the loop," Opt. Commun. 242, 191-197 (2004).
[CrossRef]

Martinelli, M.

M. Martinelli, "A universal compensator for polarization changes induced by birefringence on a retracing beam," Opt. Commun. 72, 341-344 (1989).
[CrossRef]

Mortimore, D. B.

D. B. Mortimore, "Fiber loop reflectors," J. Lightwave Technol. 6, 1217-1224 (1988).
[CrossRef]

Muller, A.

A. Muller, T. Herzog, B. Huttner, W. Tittel, H. Zbinden, and N. Gisin, "Plug and play systems for quantum cryptography," Appl. Phys. Lett. 70, 793-795 (1997).
[CrossRef]

Nishioka, T.

T. Nishioka, H. Ishizuka, T. Hasegawa, and J. Abe, "Circular type quantum key distribution," IEEE Photon. Technol. Lett. 14, 576-578 (2002).
[CrossRef]

Pottiez, O.

B. Ibarra-Escamilla, E. A. Kuzin, O. Pottiez, J. W. Haus, F. Gutierrez-Zainos, R. Grajales-Coutin, and P. Zaca-Moran, "Fiber optical loop mirror with a symmetrical coupler and a quarter-wave retarder plate in the loop," Opt. Commun. 242, 191-197 (2004).
[CrossRef]

Ramos, R. V.

W. A. de Brito and R. V. Ramos, "Quantum information technology with Sagnac interferometer: interaction-free measurement, quantum key distribution and quantum secret sharing," J. Mod. Opt. 55, 1231-1241 (2008).
[CrossRef]

Ribordy, G.

N. Gisin, G. Ribordy, W. Tittel, and H.  Zbinden, "Quantum cryptography," Rev. Mod. Phys. 74, 145-195 (2002).
[CrossRef]

Tittel, W.

N. Gisin, G. Ribordy, W. Tittel, and H.  Zbinden, "Quantum cryptography," Rev. Mod. Phys. 74, 145-195 (2002).
[CrossRef]

A. Muller, T. Herzog, B. Huttner, W. Tittel, H. Zbinden, and N. Gisin, "Plug and play systems for quantum cryptography," Appl. Phys. Lett. 70, 793-795 (1997).
[CrossRef]

Udd, E.

E. Udd, "Sensing and instrumentation applications of the Sagnac fiber optic interferometer," in Interferometric Fiber Sensing, Proc. SPIE 2341, 52-59 (1994).
[CrossRef]

Wu, G.

C. Zhou, G. Wu, L. Ding, and H. Zeng, "Single-photon routing by time-division phase modulation in a Sagnac interferometer," Appl. Phys. Lett. 83, 15-17 (2003).
[CrossRef]

Zaca-Moran, P.

B. Ibarra-Escamilla, E. A. Kuzin, O. Pottiez, J. W. Haus, F. Gutierrez-Zainos, R. Grajales-Coutin, and P. Zaca-Moran, "Fiber optical loop mirror with a symmetrical coupler and a quarter-wave retarder plate in the loop," Opt. Commun. 242, 191-197 (2004).
[CrossRef]

Zbinden, H

N. Gisin, G. Ribordy, W. Tittel, and H.  Zbinden, "Quantum cryptography," Rev. Mod. Phys. 74, 145-195 (2002).
[CrossRef]

Zbinden, H.

A. Muller, T. Herzog, B. Huttner, W. Tittel, H. Zbinden, and N. Gisin, "Plug and play systems for quantum cryptography," Appl. Phys. Lett. 70, 793-795 (1997).
[CrossRef]

Zeng, H.

C. Zhou, G. Wu, L. Ding, and H. Zeng, "Single-photon routing by time-division phase modulation in a Sagnac interferometer," Appl. Phys. Lett. 83, 15-17 (2003).
[CrossRef]

Zheng, J.

J. Zheng, "Differential singlemode fibre frequency-modulated continuous-wave Sagnac gyroscope," Electron. Lett. 41, 727-728 (2005).
[CrossRef]

Zhou, C.

C. Zhou, G. Wu, L. Ding, and H. Zeng, "Single-photon routing by time-division phase modulation in a Sagnac interferometer," Appl. Phys. Lett. 83, 15-17 (2003).
[CrossRef]

Appl. Phys. Lett.

C. Zhou, G. Wu, L. Ding, and H. Zeng, "Single-photon routing by time-division phase modulation in a Sagnac interferometer," Appl. Phys. Lett. 83, 15-17 (2003).
[CrossRef]

A. Muller, T. Herzog, B. Huttner, W. Tittel, H. Zbinden, and N. Gisin, "Plug and play systems for quantum cryptography," Appl. Phys. Lett. 70, 793-795 (1997).
[CrossRef]

Electron. Lett.

J. Zheng, "Differential singlemode fibre frequency-modulated continuous-wave Sagnac gyroscope," Electron. Lett. 41, 727-728 (2005).
[CrossRef]

IEEE J. Quantum Electron.

F. P. Kapron, N. F. Borrelli, and D. B. Keck, "Birefringence in. dielectric optical waveguides," IEEE J. Quantum Electron. QE-8, 222-230 (1972)
[CrossRef]

IEEE Photon. Technol. Lett.

T. Nishioka, H. Ishizuka, T. Hasegawa, and J. Abe, "Circular type quantum key distribution," IEEE Photon. Technol. Lett. 14, 576-578 (2002).
[CrossRef]

J. Lightwave Technol.

D. B. Mortimore, "Fiber loop reflectors," J. Lightwave Technol. 6, 1217-1224 (1988).
[CrossRef]

J. Mod. Opt.

W. A. de Brito and R. V. Ramos, "Quantum information technology with Sagnac interferometer: interaction-free measurement, quantum key distribution and quantum secret sharing," J. Mod. Opt. 55, 1231-1241 (2008).
[CrossRef]

Meas. Sci. Technol.

B. Culshaw, "The optical fibre Sagnac interferometer: an overview of its principles and applications," Meas. Sci. Technol. 17, 1-16 (2006).
[CrossRef]

Molecular Quantum Acoustics

L. R. Jaroszewicz and Z. Krajewski, "Application of fiber-optic Sagnac interferometer for detection of rotational seismic events," Molecular Quantum Acoustics 22, 133-134 (2001).

Opt. Commun.

A. Kuzin, H. Cerecedo Nunez, and N. Korneev, "Alignment of a birefringent fiber Sagnac interferometer by fiber twist," Opt. Commun. 160, 37-41 (1999).
[CrossRef]

B. Ibarra-Escamilla, E. A. Kuzin, O. Pottiez, J. W. Haus, F. Gutierrez-Zainos, R. Grajales-Coutin, and P. Zaca-Moran, "Fiber optical loop mirror with a symmetrical coupler and a quarter-wave retarder plate in the loop," Opt. Commun. 242, 191-197 (2004).
[CrossRef]

M. Martinelli, "A universal compensator for polarization changes induced by birefringence on a retracing beam," Opt. Commun. 72, 341-344 (1989).
[CrossRef]

Opt. Express

Proc. SPIE

E. Udd, "Sensing and instrumentation applications of the Sagnac fiber optic interferometer," in Interferometric Fiber Sensing, Proc. SPIE 2341, 52-59 (1994).
[CrossRef]

Rev. Mod. Phys.

N. Gisin, G. Ribordy, W. Tittel, and H.  Zbinden, "Quantum cryptography," Rev. Mod. Phys. 74, 145-195 (2002).
[CrossRef]

Other

D. S. Bethune and W. P. Risk, "Autocompensating quantum cryptography," New J. Phys. 4, 42.1-42.15 (2002).
[CrossRef]

B. Qi, L. L. Huang, H. K. Lo, and L. Qian, "Quantum key distribution based on a Sagnac loop interferometer and polarization-insensitive phase modulators," in IEEE International Symposium on Information Theory (Institute of Electrical and Electronics Engineers, 2006), pp. 2090-2093.
[CrossRef]

L. R. Jaroszewicz, "Fiber-optic Sagnac Interferometer as real sensor of the physical quantities," in Proceedings of the Symposium on Photonics Technologies (Wroclaw, Poland, 2006), pp. 99-101.

C. Tsao, Optical fibre waveguide analysis (Oxford Science Publ. 1992).

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Figures (4)

Fig. 1.
Fig. 1.

SMF Sagnac interferometer without attenuation losses [16]. Ein 1 , Eout 1 , Eout 4 , Eclk 2 , Ecnt 2 , Eclk 3 , and Ecnt 3 denote the electric field amplitudes. The CPL coupling ratio k = 0.5 is assumed to be the same for both Ex and Ey , i.e. kx = ky .

Fig. 2.
Fig. 2.

“Plug & play” interferometric setup. 1 denotes the input, T transmitted output, and R reflected output of the polarization beam splitter, while 1 denotes the input, T the transmitted output in direction 1 → T, and R transmitted output in direction T→R of the circulator. The long arm includes the delay line DL, which causes an additional signal delay compared to the short arm, which does not have such a device.

Fig. 3.
Fig. 3.

SMF birefringence compensation in Sagnac interferometer. All the components inside the boxes, marked with dot dashed lines, are polarization maintaining. Couplers CPL1, CPL2, and CPL3 coupling ratios are k=0.5. The delay line’s (DL) length is matched to the stretcher’s (STR) length. The SMF denotes single mode fiber. For the circulator, 1 denotes the input, T the transmitted output in direction 1 → T, and R transmitted output in direction T → R. For the polarization beam splitters PBS1 and PBS2, with the outputs aligned to the slow (horizontal) axis, 1 denotes the input, T transmitted output, and R reflected output. The signal transmitted from the output R into the input 1 shifts its polarization (from the horizontal to the vertical), while there is no such shift for the signal transmitted from the output T into the input 1. DET1, DET2, and DET3 denote photo detectors.

Fig. 4.
Fig. 4.

Sagnac visibility measurement result for the total SMF fiber loop length of 50 km. The measurements were carried out during one hour at the 1550 nm laser wavelength with pulse repetition rate of 2 MHz in the burst mode with the burst duty cycle of 20 %. The laser pulse level was set to a classical (non-quantum) signal level. The measurement density is one result/sec since the DAQ card’s reading period of the photo detector DET3 was set to 1 sec.

Equations (42)

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

J sys = R ( Ω ) · e iR / 2 0 0 e iR / 2 · R ( Φ ) ,
R ( Θ ) = cos Θ sin Θ sin Θ cos Θ .
J sys = P Q * Q P * ,
P = cos ( R / 2 ) cos ( Φ + Ω ) i sin ( R / 2 ) cos ( Φ Ω ) ,
Q = cos ( R / 2 ) sin ( Φ + Ω ) + i sin ( R / 2 ) sin ( Φ Ω ) ,
J smf clk = P Q * Q P * ,
J smf cnt = ( J smf clk ) T = P Q Q * P * .
E 2 clk = 2 2 E 1 in ,
E 3 cnt = i 2 2 E 1 in ,
E 1 in = E 1 x in E 1 y in .
E 3 clk = J smf clk E 2 clk ,
E 2 cnt = J smf cnt E 3 cnt .
E 1 out = i 2 2 E 3 clk + 2 2 E 2 cnt ,
E 4 out = 2 2 E 3 clk + i 2 2 E 2 cnt .
J f = P Q Q P * ,
P = cos ( R / 2 ) i sin ( R / 2 ) cos ( 2 Φ ) ,
Q = i sin ( R / 2 ) sin ( 2 Φ ) ,
P * = cos ( R / 2 ) + i sin ( R / 2 ) cos ( 2 Φ ) .
J b = P Q Q P * .
J sys = J b J FM J f ,
J FM = 0 1 1 o .
J sys = P Q Q P * 0 1 1 0 P Q Q P * = 0 1 1 0
J = E x e i δ x E y e i δ y .
E a clk = 2 2 E cir clk ,
E d cnt = i 2 2 E cir clk ,
E cir clk = E cir clk E cir y clk = E cir x clk 0 ,
J int 1 clk = 2 2 1 0 i e i δ 1 0 .
E b clk = J int 1 clk E a clk = 2 2 J int 1 clk E cir clk .
E c clk = J smf clk E b clk = 2 2 J smf clk J int 1 clk E cir clk .
J int 2 clk = 2 2 i e i δ 2 0 0 .
E d clk = J int 2 clk E c clk = 2 2 J int 2 clk J smf clk J int 1 clk E cir clk .
E a cnt = i 2 2 J int 2 cnt J smf cnt J int 2 cnt E cir clk ,
J int 1 cnt = 2 2 1 i e i δ 1 0 0 ,
J int 2 cnt = 2 2 1 0 e i δ 2 0 .
E DET 1 = 2 2 E d clk + 1 2 2 E a cnt ,
E DET 2 = i 2 2 E d clk + 2 2 E a cnt ,
E DET 1 = 0 0 ,
E DET 2 = E D 2 x 0 ,
E D 2 x = 1 2 E cir x clk { sin R 2 [ ( e i δ 1 e i δ 2 ) sin ( Φ Ω )
i ( e i ( δ 1 + δ 2 ) 1 ) cos ( Φ Ω ) ]
cos R 2 [ ( e i ( δ 1 δ 2 ) + 1 ) cos ( Φ Ω )
i ( e i δ 1 + e i δ 2 ) sin ( Φ + Ω ) ] } .

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