Abstract

A truly endless polarization stabilization experiment with a tracking speed of 15 krad/s is presented. The high-speed polarization tracking is realized by using calibrated lithium niobate linear retardation waveplates as the polarization transformers combined with a very fast digital controller running on a field-programmable gate array (FPGA).

© 2008 Optical Society of America

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References

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  1. P. M. Krummrich, E.-D. Schmidt, W. Weierhausen, and A. Mattheus, "Field trial results on statistics of fast polarization changes in long haul WDM transmission systems," in Optical Fiber Communication Conference and Exposition and The National Fiber Optic Engineers Conference, Technical Digest (CD) (Optical Society of America 2005), Paper OThT6.
  2. R. Noé, D. Sandel, and V. Mirvoda, "PMD in High-Bit-Rate Transmission and Means for Its Mitigation," J. Sel. Top. Quantum. Electron. 10, 341-355 (2004).
    [CrossRef]
  3. R. Noé, D. Sandel, M. Yoshida-Dierolf, S. Hinz, V. Mirvoda, A. Schöpflin, C. Glingener, E. Gottwald, C. Scheerer, G. Fischer, T. Weyrauch, and W. Haase, "Polarization mode dispersion compensation at 10, 20 and 40 Gb/s with various optical equalizers," J. Lightwave Technol. 17, 1602-1616 (1999).
    [CrossRef]
  4. R. Noé, H. Heidrich, and D. Hoffmann, "Endless Polarization Control Systems for Coherent Optics," J. Lightwave Technol. 6, 1199-1208 (1988).
    [CrossRef]
  5. R. Noé, D. Sandel, S. Hinz, M. Yoshida-Dierolf, V. Mirvoda, G. Feise, H. Herrmann, R. Ricken, W. Sohler, F. Wehrmann, C. Glingener, A. Schöpflin, A. Färbert, and G. Fischer, "Integrated optical LiNbO3 distributed polarization mode dispersion equalizer in 20 Gbit/s transmission system," Electron. Lett. 35, 652-654 (1999).
    [CrossRef]
  6. A. van Haasteren, J. van der Tol, O. van Deventer, and H. Frankena, "Modeling and characterization of an electrooptic polarization controller on LiNbO3," J. Lightwave Technol. 11, 1151-1157 (1993).
    [CrossRef]
  7. JackB.  Kuipers, Quaternions and rotation sequences (Princeton Univ. Press, Princeton, 1999).
  8. N. G. Walker and G. R. Walker, "Polarization control for coherent communications," J. Lightwave Technol.  8, 438-458 (1990).
    [CrossRef]
  9. F. Heismann and M. S. Whalen, "Fast automatic polarization control system," IEEE Photon. Technol. Lett. 4, 503-505 (1992).
    [CrossRef]
  10. C. K. Madsen et al, "Reset-Free Integrated Polarization Controller Using Phase Shifters," J. Sel. Top. Quantum. Electron. 11, 431-438 (2005).
    [CrossRef]
  11. M. Martinelli, P. Martelli, and S. M. Pietralunga, "Polarization Stabilization in Optical Communication Systems," J. Lightwave Technol. 24, 4172-4183 (2006).
    [CrossRef]
  12. P. Martelli et al, "Polarization Stabilizer for Polarization-Division Multiplexed Optical Systems," Proc. ECOC 2007, Berlin, Germany, Paper We6.6.5.
  13. P. Oswald and C. K. Madsen, "Deterministic Analysis of Endless Tuning of Polarization Controllers," J. Lightwave Technol. 24, 2932-2939 (2006).
    [CrossRef]
  14. F. Heismann, "Analysis of a reset-free polarization controller for fast automatic polarization stabilization in fiber-optic transmission systems," J. Lightwave Technol. 12, 690-699 (1994).
    [CrossRef]
  15. A. Hidayat, B. Koch, V. Mirvoda, H. Zhang, S. Bhandare, S.K. Ibrahim, D. Sandel, and R. Noé, "Fast Optical Endless Polarization Tracking with LiNbO3 Component," in Optical Fiber Communication Conference and Exposition and The National Fiber Optic Engineers Conference, Technical Digest (CD) (Optical Society of America 2008), Feb. 24-28, 2008, Paper JWA28, San Diego, CA, USA.

2006 (2)

M. Martinelli, P. Martelli, and S. M. Pietralunga, "Polarization Stabilization in Optical Communication Systems," J. Lightwave Technol. 24, 4172-4183 (2006).
[CrossRef]

P. Oswald and C. K. Madsen, "Deterministic Analysis of Endless Tuning of Polarization Controllers," J. Lightwave Technol. 24, 2932-2939 (2006).
[CrossRef]

2005 (1)

C. K. Madsen et al, "Reset-Free Integrated Polarization Controller Using Phase Shifters," J. Sel. Top. Quantum. Electron. 11, 431-438 (2005).
[CrossRef]

2004 (1)

R. Noé, D. Sandel, and V. Mirvoda, "PMD in High-Bit-Rate Transmission and Means for Its Mitigation," J. Sel. Top. Quantum. Electron. 10, 341-355 (2004).
[CrossRef]

1999 (2)

R. Noé, D. Sandel, M. Yoshida-Dierolf, S. Hinz, V. Mirvoda, A. Schöpflin, C. Glingener, E. Gottwald, C. Scheerer, G. Fischer, T. Weyrauch, and W. Haase, "Polarization mode dispersion compensation at 10, 20 and 40 Gb/s with various optical equalizers," J. Lightwave Technol. 17, 1602-1616 (1999).
[CrossRef]

R. Noé, D. Sandel, S. Hinz, M. Yoshida-Dierolf, V. Mirvoda, G. Feise, H. Herrmann, R. Ricken, W. Sohler, F. Wehrmann, C. Glingener, A. Schöpflin, A. Färbert, and G. Fischer, "Integrated optical LiNbO3 distributed polarization mode dispersion equalizer in 20 Gbit/s transmission system," Electron. Lett. 35, 652-654 (1999).
[CrossRef]

1994 (1)

F. Heismann, "Analysis of a reset-free polarization controller for fast automatic polarization stabilization in fiber-optic transmission systems," J. Lightwave Technol. 12, 690-699 (1994).
[CrossRef]

1993 (1)

A. van Haasteren, J. van der Tol, O. van Deventer, and H. Frankena, "Modeling and characterization of an electrooptic polarization controller on LiNbO3," J. Lightwave Technol. 11, 1151-1157 (1993).
[CrossRef]

1992 (1)

F. Heismann and M. S. Whalen, "Fast automatic polarization control system," IEEE Photon. Technol. Lett. 4, 503-505 (1992).
[CrossRef]

1990 (1)

N. G. Walker and G. R. Walker, "Polarization control for coherent communications," J. Lightwave Technol.  8, 438-458 (1990).
[CrossRef]

1988 (1)

R. Noé, H. Heidrich, and D. Hoffmann, "Endless Polarization Control Systems for Coherent Optics," J. Lightwave Technol. 6, 1199-1208 (1988).
[CrossRef]

Färbert, A.

R. Noé, D. Sandel, S. Hinz, M. Yoshida-Dierolf, V. Mirvoda, G. Feise, H. Herrmann, R. Ricken, W. Sohler, F. Wehrmann, C. Glingener, A. Schöpflin, A. Färbert, and G. Fischer, "Integrated optical LiNbO3 distributed polarization mode dispersion equalizer in 20 Gbit/s transmission system," Electron. Lett. 35, 652-654 (1999).
[CrossRef]

Feise, G.

R. Noé, D. Sandel, S. Hinz, M. Yoshida-Dierolf, V. Mirvoda, G. Feise, H. Herrmann, R. Ricken, W. Sohler, F. Wehrmann, C. Glingener, A. Schöpflin, A. Färbert, and G. Fischer, "Integrated optical LiNbO3 distributed polarization mode dispersion equalizer in 20 Gbit/s transmission system," Electron. Lett. 35, 652-654 (1999).
[CrossRef]

Fischer, G.

R. Noé, D. Sandel, S. Hinz, M. Yoshida-Dierolf, V. Mirvoda, G. Feise, H. Herrmann, R. Ricken, W. Sohler, F. Wehrmann, C. Glingener, A. Schöpflin, A. Färbert, and G. Fischer, "Integrated optical LiNbO3 distributed polarization mode dispersion equalizer in 20 Gbit/s transmission system," Electron. Lett. 35, 652-654 (1999).
[CrossRef]

R. Noé, D. Sandel, M. Yoshida-Dierolf, S. Hinz, V. Mirvoda, A. Schöpflin, C. Glingener, E. Gottwald, C. Scheerer, G. Fischer, T. Weyrauch, and W. Haase, "Polarization mode dispersion compensation at 10, 20 and 40 Gb/s with various optical equalizers," J. Lightwave Technol. 17, 1602-1616 (1999).
[CrossRef]

Frankena, H.

A. van Haasteren, J. van der Tol, O. van Deventer, and H. Frankena, "Modeling and characterization of an electrooptic polarization controller on LiNbO3," J. Lightwave Technol. 11, 1151-1157 (1993).
[CrossRef]

Glingener, C.

R. Noé, D. Sandel, M. Yoshida-Dierolf, S. Hinz, V. Mirvoda, A. Schöpflin, C. Glingener, E. Gottwald, C. Scheerer, G. Fischer, T. Weyrauch, and W. Haase, "Polarization mode dispersion compensation at 10, 20 and 40 Gb/s with various optical equalizers," J. Lightwave Technol. 17, 1602-1616 (1999).
[CrossRef]

R. Noé, D. Sandel, S. Hinz, M. Yoshida-Dierolf, V. Mirvoda, G. Feise, H. Herrmann, R. Ricken, W. Sohler, F. Wehrmann, C. Glingener, A. Schöpflin, A. Färbert, and G. Fischer, "Integrated optical LiNbO3 distributed polarization mode dispersion equalizer in 20 Gbit/s transmission system," Electron. Lett. 35, 652-654 (1999).
[CrossRef]

Gottwald, E.

Haase, W.

Heidrich, H.

R. Noé, H. Heidrich, and D. Hoffmann, "Endless Polarization Control Systems for Coherent Optics," J. Lightwave Technol. 6, 1199-1208 (1988).
[CrossRef]

Heismann, F.

F. Heismann, "Analysis of a reset-free polarization controller for fast automatic polarization stabilization in fiber-optic transmission systems," J. Lightwave Technol. 12, 690-699 (1994).
[CrossRef]

F. Heismann and M. S. Whalen, "Fast automatic polarization control system," IEEE Photon. Technol. Lett. 4, 503-505 (1992).
[CrossRef]

Herrmann, H.

R. Noé, D. Sandel, S. Hinz, M. Yoshida-Dierolf, V. Mirvoda, G. Feise, H. Herrmann, R. Ricken, W. Sohler, F. Wehrmann, C. Glingener, A. Schöpflin, A. Färbert, and G. Fischer, "Integrated optical LiNbO3 distributed polarization mode dispersion equalizer in 20 Gbit/s transmission system," Electron. Lett. 35, 652-654 (1999).
[CrossRef]

Hinz, S.

R. Noé, D. Sandel, S. Hinz, M. Yoshida-Dierolf, V. Mirvoda, G. Feise, H. Herrmann, R. Ricken, W. Sohler, F. Wehrmann, C. Glingener, A. Schöpflin, A. Färbert, and G. Fischer, "Integrated optical LiNbO3 distributed polarization mode dispersion equalizer in 20 Gbit/s transmission system," Electron. Lett. 35, 652-654 (1999).
[CrossRef]

R. Noé, D. Sandel, M. Yoshida-Dierolf, S. Hinz, V. Mirvoda, A. Schöpflin, C. Glingener, E. Gottwald, C. Scheerer, G. Fischer, T. Weyrauch, and W. Haase, "Polarization mode dispersion compensation at 10, 20 and 40 Gb/s with various optical equalizers," J. Lightwave Technol. 17, 1602-1616 (1999).
[CrossRef]

Hoffmann, D.

R. Noé, H. Heidrich, and D. Hoffmann, "Endless Polarization Control Systems for Coherent Optics," J. Lightwave Technol. 6, 1199-1208 (1988).
[CrossRef]

Madsen, C. K.

P. Oswald and C. K. Madsen, "Deterministic Analysis of Endless Tuning of Polarization Controllers," J. Lightwave Technol. 24, 2932-2939 (2006).
[CrossRef]

C. K. Madsen et al, "Reset-Free Integrated Polarization Controller Using Phase Shifters," J. Sel. Top. Quantum. Electron. 11, 431-438 (2005).
[CrossRef]

Martelli, P.

Martinelli, M.

Mirvoda, V.

R. Noé, D. Sandel, and V. Mirvoda, "PMD in High-Bit-Rate Transmission and Means for Its Mitigation," J. Sel. Top. Quantum. Electron. 10, 341-355 (2004).
[CrossRef]

R. Noé, D. Sandel, S. Hinz, M. Yoshida-Dierolf, V. Mirvoda, G. Feise, H. Herrmann, R. Ricken, W. Sohler, F. Wehrmann, C. Glingener, A. Schöpflin, A. Färbert, and G. Fischer, "Integrated optical LiNbO3 distributed polarization mode dispersion equalizer in 20 Gbit/s transmission system," Electron. Lett. 35, 652-654 (1999).
[CrossRef]

R. Noé, D. Sandel, M. Yoshida-Dierolf, S. Hinz, V. Mirvoda, A. Schöpflin, C. Glingener, E. Gottwald, C. Scheerer, G. Fischer, T. Weyrauch, and W. Haase, "Polarization mode dispersion compensation at 10, 20 and 40 Gb/s with various optical equalizers," J. Lightwave Technol. 17, 1602-1616 (1999).
[CrossRef]

Noé, R.

R. Noé, D. Sandel, and V. Mirvoda, "PMD in High-Bit-Rate Transmission and Means for Its Mitigation," J. Sel. Top. Quantum. Electron. 10, 341-355 (2004).
[CrossRef]

R. Noé, D. Sandel, S. Hinz, M. Yoshida-Dierolf, V. Mirvoda, G. Feise, H. Herrmann, R. Ricken, W. Sohler, F. Wehrmann, C. Glingener, A. Schöpflin, A. Färbert, and G. Fischer, "Integrated optical LiNbO3 distributed polarization mode dispersion equalizer in 20 Gbit/s transmission system," Electron. Lett. 35, 652-654 (1999).
[CrossRef]

R. Noé, D. Sandel, M. Yoshida-Dierolf, S. Hinz, V. Mirvoda, A. Schöpflin, C. Glingener, E. Gottwald, C. Scheerer, G. Fischer, T. Weyrauch, and W. Haase, "Polarization mode dispersion compensation at 10, 20 and 40 Gb/s with various optical equalizers," J. Lightwave Technol. 17, 1602-1616 (1999).
[CrossRef]

R. Noé, H. Heidrich, and D. Hoffmann, "Endless Polarization Control Systems for Coherent Optics," J. Lightwave Technol. 6, 1199-1208 (1988).
[CrossRef]

Oswald, P.

P. Oswald and C. K. Madsen, "Deterministic Analysis of Endless Tuning of Polarization Controllers," J. Lightwave Technol. 24, 2932-2939 (2006).
[CrossRef]

Pietralunga, S. M.

Ricken, R.

R. Noé, D. Sandel, S. Hinz, M. Yoshida-Dierolf, V. Mirvoda, G. Feise, H. Herrmann, R. Ricken, W. Sohler, F. Wehrmann, C. Glingener, A. Schöpflin, A. Färbert, and G. Fischer, "Integrated optical LiNbO3 distributed polarization mode dispersion equalizer in 20 Gbit/s transmission system," Electron. Lett. 35, 652-654 (1999).
[CrossRef]

Sandel, D.

R. Noé, D. Sandel, and V. Mirvoda, "PMD in High-Bit-Rate Transmission and Means for Its Mitigation," J. Sel. Top. Quantum. Electron. 10, 341-355 (2004).
[CrossRef]

R. Noé, D. Sandel, S. Hinz, M. Yoshida-Dierolf, V. Mirvoda, G. Feise, H. Herrmann, R. Ricken, W. Sohler, F. Wehrmann, C. Glingener, A. Schöpflin, A. Färbert, and G. Fischer, "Integrated optical LiNbO3 distributed polarization mode dispersion equalizer in 20 Gbit/s transmission system," Electron. Lett. 35, 652-654 (1999).
[CrossRef]

R. Noé, D. Sandel, M. Yoshida-Dierolf, S. Hinz, V. Mirvoda, A. Schöpflin, C. Glingener, E. Gottwald, C. Scheerer, G. Fischer, T. Weyrauch, and W. Haase, "Polarization mode dispersion compensation at 10, 20 and 40 Gb/s with various optical equalizers," J. Lightwave Technol. 17, 1602-1616 (1999).
[CrossRef]

Scheerer, C.

Schöpflin, A.

R. Noé, D. Sandel, M. Yoshida-Dierolf, S. Hinz, V. Mirvoda, A. Schöpflin, C. Glingener, E. Gottwald, C. Scheerer, G. Fischer, T. Weyrauch, and W. Haase, "Polarization mode dispersion compensation at 10, 20 and 40 Gb/s with various optical equalizers," J. Lightwave Technol. 17, 1602-1616 (1999).
[CrossRef]

R. Noé, D. Sandel, S. Hinz, M. Yoshida-Dierolf, V. Mirvoda, G. Feise, H. Herrmann, R. Ricken, W. Sohler, F. Wehrmann, C. Glingener, A. Schöpflin, A. Färbert, and G. Fischer, "Integrated optical LiNbO3 distributed polarization mode dispersion equalizer in 20 Gbit/s transmission system," Electron. Lett. 35, 652-654 (1999).
[CrossRef]

Sohler, W.

R. Noé, D. Sandel, S. Hinz, M. Yoshida-Dierolf, V. Mirvoda, G. Feise, H. Herrmann, R. Ricken, W. Sohler, F. Wehrmann, C. Glingener, A. Schöpflin, A. Färbert, and G. Fischer, "Integrated optical LiNbO3 distributed polarization mode dispersion equalizer in 20 Gbit/s transmission system," Electron. Lett. 35, 652-654 (1999).
[CrossRef]

van der Tol, J.

A. van Haasteren, J. van der Tol, O. van Deventer, and H. Frankena, "Modeling and characterization of an electrooptic polarization controller on LiNbO3," J. Lightwave Technol. 11, 1151-1157 (1993).
[CrossRef]

van Deventer, O.

A. van Haasteren, J. van der Tol, O. van Deventer, and H. Frankena, "Modeling and characterization of an electrooptic polarization controller on LiNbO3," J. Lightwave Technol. 11, 1151-1157 (1993).
[CrossRef]

van Haasteren, A.

A. van Haasteren, J. van der Tol, O. van Deventer, and H. Frankena, "Modeling and characterization of an electrooptic polarization controller on LiNbO3," J. Lightwave Technol. 11, 1151-1157 (1993).
[CrossRef]

Walker, G. R.

N. G. Walker and G. R. Walker, "Polarization control for coherent communications," J. Lightwave Technol.  8, 438-458 (1990).
[CrossRef]

Walker, N. G.

N. G. Walker and G. R. Walker, "Polarization control for coherent communications," J. Lightwave Technol.  8, 438-458 (1990).
[CrossRef]

Wehrmann, F.

R. Noé, D. Sandel, S. Hinz, M. Yoshida-Dierolf, V. Mirvoda, G. Feise, H. Herrmann, R. Ricken, W. Sohler, F. Wehrmann, C. Glingener, A. Schöpflin, A. Färbert, and G. Fischer, "Integrated optical LiNbO3 distributed polarization mode dispersion equalizer in 20 Gbit/s transmission system," Electron. Lett. 35, 652-654 (1999).
[CrossRef]

Weyrauch, T.

Whalen, M. S.

F. Heismann and M. S. Whalen, "Fast automatic polarization control system," IEEE Photon. Technol. Lett. 4, 503-505 (1992).
[CrossRef]

Yoshida-Dierolf, M.

R. Noé, D. Sandel, S. Hinz, M. Yoshida-Dierolf, V. Mirvoda, G. Feise, H. Herrmann, R. Ricken, W. Sohler, F. Wehrmann, C. Glingener, A. Schöpflin, A. Färbert, and G. Fischer, "Integrated optical LiNbO3 distributed polarization mode dispersion equalizer in 20 Gbit/s transmission system," Electron. Lett. 35, 652-654 (1999).
[CrossRef]

R. Noé, D. Sandel, M. Yoshida-Dierolf, S. Hinz, V. Mirvoda, A. Schöpflin, C. Glingener, E. Gottwald, C. Scheerer, G. Fischer, T. Weyrauch, and W. Haase, "Polarization mode dispersion compensation at 10, 20 and 40 Gb/s with various optical equalizers," J. Lightwave Technol. 17, 1602-1616 (1999).
[CrossRef]

Electron. Lett. (1)

R. Noé, D. Sandel, S. Hinz, M. Yoshida-Dierolf, V. Mirvoda, G. Feise, H. Herrmann, R. Ricken, W. Sohler, F. Wehrmann, C. Glingener, A. Schöpflin, A. Färbert, and G. Fischer, "Integrated optical LiNbO3 distributed polarization mode dispersion equalizer in 20 Gbit/s transmission system," Electron. Lett. 35, 652-654 (1999).
[CrossRef]

IEEE Photon. Technol. Lett. (1)

F. Heismann and M. S. Whalen, "Fast automatic polarization control system," IEEE Photon. Technol. Lett. 4, 503-505 (1992).
[CrossRef]

J. Lightwave Technol. (3)

J. Sel. Top. Quantum. Electron. (1)

C. K. Madsen et al, "Reset-Free Integrated Polarization Controller Using Phase Shifters," J. Sel. Top. Quantum. Electron. 11, 431-438 (2005).
[CrossRef]

J. Lightwave Technol. (4)

A. van Haasteren, J. van der Tol, O. van Deventer, and H. Frankena, "Modeling and characterization of an electrooptic polarization controller on LiNbO3," J. Lightwave Technol. 11, 1151-1157 (1993).
[CrossRef]

R. Noé, H. Heidrich, and D. Hoffmann, "Endless Polarization Control Systems for Coherent Optics," J. Lightwave Technol. 6, 1199-1208 (1988).
[CrossRef]

N. G. Walker and G. R. Walker, "Polarization control for coherent communications," J. Lightwave Technol.  8, 438-458 (1990).
[CrossRef]

P. Oswald and C. K. Madsen, "Deterministic Analysis of Endless Tuning of Polarization Controllers," J. Lightwave Technol. 24, 2932-2939 (2006).
[CrossRef]

J. Sel. Top. Quantum. Electron. (1)

R. Noé, D. Sandel, and V. Mirvoda, "PMD in High-Bit-Rate Transmission and Means for Its Mitigation," J. Sel. Top. Quantum. Electron. 10, 341-355 (2004).
[CrossRef]

Other (4)

P. M. Krummrich, E.-D. Schmidt, W. Weierhausen, and A. Mattheus, "Field trial results on statistics of fast polarization changes in long haul WDM transmission systems," in Optical Fiber Communication Conference and Exposition and The National Fiber Optic Engineers Conference, Technical Digest (CD) (Optical Society of America 2005), Paper OThT6.

JackB.  Kuipers, Quaternions and rotation sequences (Princeton Univ. Press, Princeton, 1999).

P. Martelli et al, "Polarization Stabilizer for Polarization-Division Multiplexed Optical Systems," Proc. ECOC 2007, Berlin, Germany, Paper We6.6.5.

A. Hidayat, B. Koch, V. Mirvoda, H. Zhang, S. Bhandare, S.K. Ibrahim, D. Sandel, and R. Noé, "Fast Optical Endless Polarization Tracking with LiNbO3 Component," in Optical Fiber Communication Conference and Exposition and The National Fiber Optic Engineers Conference, Technical Digest (CD) (Optical Society of America 2008), Feb. 24-28, 2008, Paper JWA28, San Diego, CA, USA.

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Figures (6)

Fig. 1.
Fig. 1.

Structure of an x-cut z-propagation lithium niobate retarder (left) and its polarization transformation on the Poincaré sphere (right).

Fig. 2.
Fig. 2.

Characterization result of a lithium-niobate retarder.

Fig. 3.
Fig. 3.

Calibrated retarder voltages for different retardation and eigenmode orientation.

Fig. 4.
Fig. 4.

Endless polarization control experiment setup (left), and block diagram of control algorithm (right).

Fig. 5.
Fig. 5.

Complementary distribution function 1-F(RIE) of relative intensity error (RIE) for polarization tracking of different HWP rotation rates.

Fig. 6.
Fig. 6.

Tracking error for different polarization changes.

Equations (8)

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

φ ~ E x 2 + E y 2 , tan ( ψ π 2 ) ~ E x E y .
E x = κ x ( V 1 + V 2 V 0 , x )
E y = κ y ( V 1 + V 2 V 0 , y )
φ = π V π E x 2 + E y 2
L ( V 1 , V 2 ) = cos π V π κ x 2 ( V 1 + V 2 V 0 , x ) 2 + κ y 2 ( V 1 + V 2 V 0 , y ) 2 +
( i κ x ( V 1 + V 2 V 0 , x ) κ x 2 ( V 1 + V 2 V 0 , x ) 2 + κ y 2 ( V 1 + V 2 V 0 , y ) 2 +
j κ y ( V 1 + V 2 V 0 , y ) κ x 2 ( V 1 + V 2 V 0 , x ) 2 + κ y 2 ( V 1 + V 2 V 0 , y ) 2 )
sin π V π κ x 2 ( V 1 + V 2 V 0 , x ) 2 + κ y 2 ( V 1 + V 2 V 0 , y ) 2

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