Abstract

The effect of chromatic dispersion on a photonic link employing polarization modulation is studied analytically and experimentally. For analog polarization modulation, dispersion introduces a frequency-dependent rotation to the orientation of modulation about the center of modulation. As a result, the dispersion-limited bandwidth of the link will depend on receiver design.

© 2006 Optical Society of America

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References

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  1. K. Yonenaga and S. Kuwano, " Dispersion-tolerant optical transmission system using duobinary transmitter and binary receiver," J. Lightwave Technol. 15, 1530- 1537 ( 1997).
    [CrossRef]
  2. A. S. Siddiqui, S. G. Edirisinghe, J. J. Lepley, J. G. Ellison, and S. D. Walker, " Dispersion-tolerant transmission using a duobinary polarization-shift keying transmission scheme," IEEE Photon. Technol. Lett. 14, 158- 160 ( 2002).
    [CrossRef]
  3. S. Benedetto, R. Gaudino, and P. Poggiolini, " Direct detection of optical digital transmission based on polarization shift keying modulation," IEEE J. Sel. Areas Commun. 13, 531- 542 ( 1995).
    [CrossRef]
  4. Y. W. Song, Z. Pan, Y. Arieli, S. M. R. Motaghian, S. A. Havstad, and A. E. Willner, " Enhanced suppression of nonlinearity-induced crosstalk in WDM systems using optical polarization shift keying," presented at the Conference on Lasers and Electro-Optics/Quantum Electronics and Laser Science Conference, Baltimore, Maryland, 1- 6 June 2003, paper CThQ2.
  5. A. L. Campillo, F. Bucholtz, J. L. Dexter, and K. J. Williams, " Crosstalk reduction in wavelength division multiplexed analog links through polarization modulation," presented at the Conference on Lasers and Electro-Optics/International Quantum Electronics Conference, San Francisco, California, 18- 20 May 2004, paper CWQ3.
  6. D. S. Kliger, J. W. Lewis, and C. E. Randall, Polarized Light in Optics and Spectroscopy (Academic, 1990).
  7. J. Wang and K. Petermann, " Small signal analysis for dispersive optical fiber communication systems," J. Lightwave Technol. , 10,96- 100 (1992).
    [CrossRef]
  8. J. D. Bull, N. A. F. Jaeger, H. Kato, M. Fairburn, A. Reid, and P. Ghanipour, " 40 GHz electro-optic polarization modulator for fiber optic communications systems," in Optical Components and Devices, Proc. SPIE 5577, 133- 143 ( 2004).
  9. H. Schmuck, " Comparison of optical millimetre-wave system concepts with regard to chromatic dispersion," IEEE Electron. Lett. 31, 1848- 1849 ( 1995).
    [CrossRef]
  10. J. L. Corral, J. Marti, and J. M. Fuster, " General expressions for IM/DD dispersive analog optical links with external modulation or optical up-conversion in a Mach-Zehnder electrooptical modulator," IEEE Trans. Microwave Theory Technol. 49,1968- 1976 (2001).
    [CrossRef]
  11. U. Gliese, S. Nørskov, and T. N. Nielsen, " Chromatic dispersion in fiber-optic microwave and millimeter-wave links," IEEE Trans. Microwave Theory Tech. 44, 1716- 1724 ( 1996).
    [CrossRef]
  12. A. F. Elrafaie, R. E. Wagner, D. A. Atlas, and D. G. Daut, " Chromatic dispersion limitations in coherent lightwave transmission systems," J. Lightwave Technol. 6, 704- 709 ( 1988).
    [CrossRef]
  13. S. Benedetto and P. Poggiolini, " Theory of polarization shift keying modulation," IEEE Trans. Commun. 40, 708- 720 ( 1992).
    [CrossRef]

2002 (1)

A. S. Siddiqui, S. G. Edirisinghe, J. J. Lepley, J. G. Ellison, and S. D. Walker, " Dispersion-tolerant transmission using a duobinary polarization-shift keying transmission scheme," IEEE Photon. Technol. Lett. 14, 158- 160 ( 2002).
[CrossRef]

2001 (1)

J. L. Corral, J. Marti, and J. M. Fuster, " General expressions for IM/DD dispersive analog optical links with external modulation or optical up-conversion in a Mach-Zehnder electrooptical modulator," IEEE Trans. Microwave Theory Technol. 49,1968- 1976 (2001).
[CrossRef]

1997 (1)

K. Yonenaga and S. Kuwano, " Dispersion-tolerant optical transmission system using duobinary transmitter and binary receiver," J. Lightwave Technol. 15, 1530- 1537 ( 1997).
[CrossRef]

1996 (1)

U. Gliese, S. Nørskov, and T. N. Nielsen, " Chromatic dispersion in fiber-optic microwave and millimeter-wave links," IEEE Trans. Microwave Theory Tech. 44, 1716- 1724 ( 1996).
[CrossRef]

1995 (2)

H. Schmuck, " Comparison of optical millimetre-wave system concepts with regard to chromatic dispersion," IEEE Electron. Lett. 31, 1848- 1849 ( 1995).
[CrossRef]

S. Benedetto, R. Gaudino, and P. Poggiolini, " Direct detection of optical digital transmission based on polarization shift keying modulation," IEEE J. Sel. Areas Commun. 13, 531- 542 ( 1995).
[CrossRef]

1992 (2)

J. Wang and K. Petermann, " Small signal analysis for dispersive optical fiber communication systems," J. Lightwave Technol. , 10,96- 100 (1992).
[CrossRef]

S. Benedetto and P. Poggiolini, " Theory of polarization shift keying modulation," IEEE Trans. Commun. 40, 708- 720 ( 1992).
[CrossRef]

1988 (1)

A. F. Elrafaie, R. E. Wagner, D. A. Atlas, and D. G. Daut, " Chromatic dispersion limitations in coherent lightwave transmission systems," J. Lightwave Technol. 6, 704- 709 ( 1988).
[CrossRef]

Arieli, Y.

Y. W. Song, Z. Pan, Y. Arieli, S. M. R. Motaghian, S. A. Havstad, and A. E. Willner, " Enhanced suppression of nonlinearity-induced crosstalk in WDM systems using optical polarization shift keying," presented at the Conference on Lasers and Electro-Optics/Quantum Electronics and Laser Science Conference, Baltimore, Maryland, 1- 6 June 2003, paper CThQ2.

Atlas, D. A.

A. F. Elrafaie, R. E. Wagner, D. A. Atlas, and D. G. Daut, " Chromatic dispersion limitations in coherent lightwave transmission systems," J. Lightwave Technol. 6, 704- 709 ( 1988).
[CrossRef]

Benedetto, S.

S. Benedetto, R. Gaudino, and P. Poggiolini, " Direct detection of optical digital transmission based on polarization shift keying modulation," IEEE J. Sel. Areas Commun. 13, 531- 542 ( 1995).
[CrossRef]

S. Benedetto and P. Poggiolini, " Theory of polarization shift keying modulation," IEEE Trans. Commun. 40, 708- 720 ( 1992).
[CrossRef]

Bucholtz, F.

A. L. Campillo, F. Bucholtz, J. L. Dexter, and K. J. Williams, " Crosstalk reduction in wavelength division multiplexed analog links through polarization modulation," presented at the Conference on Lasers and Electro-Optics/International Quantum Electronics Conference, San Francisco, California, 18- 20 May 2004, paper CWQ3.

Bull, J. D.

J. D. Bull, N. A. F. Jaeger, H. Kato, M. Fairburn, A. Reid, and P. Ghanipour, " 40 GHz electro-optic polarization modulator for fiber optic communications systems," in Optical Components and Devices, Proc. SPIE 5577, 133- 143 ( 2004).

Campillo, A. L.

A. L. Campillo, F. Bucholtz, J. L. Dexter, and K. J. Williams, " Crosstalk reduction in wavelength division multiplexed analog links through polarization modulation," presented at the Conference on Lasers and Electro-Optics/International Quantum Electronics Conference, San Francisco, California, 18- 20 May 2004, paper CWQ3.

Corral, J. L.

J. L. Corral, J. Marti, and J. M. Fuster, " General expressions for IM/DD dispersive analog optical links with external modulation or optical up-conversion in a Mach-Zehnder electrooptical modulator," IEEE Trans. Microwave Theory Technol. 49,1968- 1976 (2001).
[CrossRef]

Daut, D. G.

A. F. Elrafaie, R. E. Wagner, D. A. Atlas, and D. G. Daut, " Chromatic dispersion limitations in coherent lightwave transmission systems," J. Lightwave Technol. 6, 704- 709 ( 1988).
[CrossRef]

Dexter, J. L.

A. L. Campillo, F. Bucholtz, J. L. Dexter, and K. J. Williams, " Crosstalk reduction in wavelength division multiplexed analog links through polarization modulation," presented at the Conference on Lasers and Electro-Optics/International Quantum Electronics Conference, San Francisco, California, 18- 20 May 2004, paper CWQ3.

Edirisinghe, S. G.

A. S. Siddiqui, S. G. Edirisinghe, J. J. Lepley, J. G. Ellison, and S. D. Walker, " Dispersion-tolerant transmission using a duobinary polarization-shift keying transmission scheme," IEEE Photon. Technol. Lett. 14, 158- 160 ( 2002).
[CrossRef]

Ellison, J. G.

A. S. Siddiqui, S. G. Edirisinghe, J. J. Lepley, J. G. Ellison, and S. D. Walker, " Dispersion-tolerant transmission using a duobinary polarization-shift keying transmission scheme," IEEE Photon. Technol. Lett. 14, 158- 160 ( 2002).
[CrossRef]

Elrafaie, A. F.

A. F. Elrafaie, R. E. Wagner, D. A. Atlas, and D. G. Daut, " Chromatic dispersion limitations in coherent lightwave transmission systems," J. Lightwave Technol. 6, 704- 709 ( 1988).
[CrossRef]

Fairburn, M.

J. D. Bull, N. A. F. Jaeger, H. Kato, M. Fairburn, A. Reid, and P. Ghanipour, " 40 GHz electro-optic polarization modulator for fiber optic communications systems," in Optical Components and Devices, Proc. SPIE 5577, 133- 143 ( 2004).

Fuster, J. M.

J. L. Corral, J. Marti, and J. M. Fuster, " General expressions for IM/DD dispersive analog optical links with external modulation or optical up-conversion in a Mach-Zehnder electrooptical modulator," IEEE Trans. Microwave Theory Technol. 49,1968- 1976 (2001).
[CrossRef]

Gaudino, R.

S. Benedetto, R. Gaudino, and P. Poggiolini, " Direct detection of optical digital transmission based on polarization shift keying modulation," IEEE J. Sel. Areas Commun. 13, 531- 542 ( 1995).
[CrossRef]

Ghanipour, P.

J. D. Bull, N. A. F. Jaeger, H. Kato, M. Fairburn, A. Reid, and P. Ghanipour, " 40 GHz electro-optic polarization modulator for fiber optic communications systems," in Optical Components and Devices, Proc. SPIE 5577, 133- 143 ( 2004).

Gliese, U.

U. Gliese, S. Nørskov, and T. N. Nielsen, " Chromatic dispersion in fiber-optic microwave and millimeter-wave links," IEEE Trans. Microwave Theory Tech. 44, 1716- 1724 ( 1996).
[CrossRef]

Havstad, S. A.

Y. W. Song, Z. Pan, Y. Arieli, S. M. R. Motaghian, S. A. Havstad, and A. E. Willner, " Enhanced suppression of nonlinearity-induced crosstalk in WDM systems using optical polarization shift keying," presented at the Conference on Lasers and Electro-Optics/Quantum Electronics and Laser Science Conference, Baltimore, Maryland, 1- 6 June 2003, paper CThQ2.

Jaeger, N. A. F.

J. D. Bull, N. A. F. Jaeger, H. Kato, M. Fairburn, A. Reid, and P. Ghanipour, " 40 GHz electro-optic polarization modulator for fiber optic communications systems," in Optical Components and Devices, Proc. SPIE 5577, 133- 143 ( 2004).

Kato, H.

J. D. Bull, N. A. F. Jaeger, H. Kato, M. Fairburn, A. Reid, and P. Ghanipour, " 40 GHz electro-optic polarization modulator for fiber optic communications systems," in Optical Components and Devices, Proc. SPIE 5577, 133- 143 ( 2004).

Kliger, D. S.

D. S. Kliger, J. W. Lewis, and C. E. Randall, Polarized Light in Optics and Spectroscopy (Academic, 1990).

Kuwano, S.

K. Yonenaga and S. Kuwano, " Dispersion-tolerant optical transmission system using duobinary transmitter and binary receiver," J. Lightwave Technol. 15, 1530- 1537 ( 1997).
[CrossRef]

Lepley, J. J.

A. S. Siddiqui, S. G. Edirisinghe, J. J. Lepley, J. G. Ellison, and S. D. Walker, " Dispersion-tolerant transmission using a duobinary polarization-shift keying transmission scheme," IEEE Photon. Technol. Lett. 14, 158- 160 ( 2002).
[CrossRef]

Lewis, J. W.

D. S. Kliger, J. W. Lewis, and C. E. Randall, Polarized Light in Optics and Spectroscopy (Academic, 1990).

Marti, J.

J. L. Corral, J. Marti, and J. M. Fuster, " General expressions for IM/DD dispersive analog optical links with external modulation or optical up-conversion in a Mach-Zehnder electrooptical modulator," IEEE Trans. Microwave Theory Technol. 49,1968- 1976 (2001).
[CrossRef]

Motaghian, S. M. R.

Y. W. Song, Z. Pan, Y. Arieli, S. M. R. Motaghian, S. A. Havstad, and A. E. Willner, " Enhanced suppression of nonlinearity-induced crosstalk in WDM systems using optical polarization shift keying," presented at the Conference on Lasers and Electro-Optics/Quantum Electronics and Laser Science Conference, Baltimore, Maryland, 1- 6 June 2003, paper CThQ2.

Nielsen, T. N.

U. Gliese, S. Nørskov, and T. N. Nielsen, " Chromatic dispersion in fiber-optic microwave and millimeter-wave links," IEEE Trans. Microwave Theory Tech. 44, 1716- 1724 ( 1996).
[CrossRef]

Nørskov, S.

U. Gliese, S. Nørskov, and T. N. Nielsen, " Chromatic dispersion in fiber-optic microwave and millimeter-wave links," IEEE Trans. Microwave Theory Tech. 44, 1716- 1724 ( 1996).
[CrossRef]

Pan, Z.

Y. W. Song, Z. Pan, Y. Arieli, S. M. R. Motaghian, S. A. Havstad, and A. E. Willner, " Enhanced suppression of nonlinearity-induced crosstalk in WDM systems using optical polarization shift keying," presented at the Conference on Lasers and Electro-Optics/Quantum Electronics and Laser Science Conference, Baltimore, Maryland, 1- 6 June 2003, paper CThQ2.

Petermann, K.

J. Wang and K. Petermann, " Small signal analysis for dispersive optical fiber communication systems," J. Lightwave Technol. , 10,96- 100 (1992).
[CrossRef]

Poggiolini, P.

S. Benedetto, R. Gaudino, and P. Poggiolini, " Direct detection of optical digital transmission based on polarization shift keying modulation," IEEE J. Sel. Areas Commun. 13, 531- 542 ( 1995).
[CrossRef]

S. Benedetto and P. Poggiolini, " Theory of polarization shift keying modulation," IEEE Trans. Commun. 40, 708- 720 ( 1992).
[CrossRef]

Randall, C. E.

D. S. Kliger, J. W. Lewis, and C. E. Randall, Polarized Light in Optics and Spectroscopy (Academic, 1990).

Reid, A.

J. D. Bull, N. A. F. Jaeger, H. Kato, M. Fairburn, A. Reid, and P. Ghanipour, " 40 GHz electro-optic polarization modulator for fiber optic communications systems," in Optical Components and Devices, Proc. SPIE 5577, 133- 143 ( 2004).

Schmuck, H.

H. Schmuck, " Comparison of optical millimetre-wave system concepts with regard to chromatic dispersion," IEEE Electron. Lett. 31, 1848- 1849 ( 1995).
[CrossRef]

Siddiqui, A. S.

A. S. Siddiqui, S. G. Edirisinghe, J. J. Lepley, J. G. Ellison, and S. D. Walker, " Dispersion-tolerant transmission using a duobinary polarization-shift keying transmission scheme," IEEE Photon. Technol. Lett. 14, 158- 160 ( 2002).
[CrossRef]

Song, Y. W.

Y. W. Song, Z. Pan, Y. Arieli, S. M. R. Motaghian, S. A. Havstad, and A. E. Willner, " Enhanced suppression of nonlinearity-induced crosstalk in WDM systems using optical polarization shift keying," presented at the Conference on Lasers and Electro-Optics/Quantum Electronics and Laser Science Conference, Baltimore, Maryland, 1- 6 June 2003, paper CThQ2.

Wagner, R. E.

A. F. Elrafaie, R. E. Wagner, D. A. Atlas, and D. G. Daut, " Chromatic dispersion limitations in coherent lightwave transmission systems," J. Lightwave Technol. 6, 704- 709 ( 1988).
[CrossRef]

Walker, S. D.

A. S. Siddiqui, S. G. Edirisinghe, J. J. Lepley, J. G. Ellison, and S. D. Walker, " Dispersion-tolerant transmission using a duobinary polarization-shift keying transmission scheme," IEEE Photon. Technol. Lett. 14, 158- 160 ( 2002).
[CrossRef]

Wang, J.

J. Wang and K. Petermann, " Small signal analysis for dispersive optical fiber communication systems," J. Lightwave Technol. , 10,96- 100 (1992).
[CrossRef]

Williams, K. J.

A. L. Campillo, F. Bucholtz, J. L. Dexter, and K. J. Williams, " Crosstalk reduction in wavelength division multiplexed analog links through polarization modulation," presented at the Conference on Lasers and Electro-Optics/International Quantum Electronics Conference, San Francisco, California, 18- 20 May 2004, paper CWQ3.

Willner, A. E.

Y. W. Song, Z. Pan, Y. Arieli, S. M. R. Motaghian, S. A. Havstad, and A. E. Willner, " Enhanced suppression of nonlinearity-induced crosstalk in WDM systems using optical polarization shift keying," presented at the Conference on Lasers and Electro-Optics/Quantum Electronics and Laser Science Conference, Baltimore, Maryland, 1- 6 June 2003, paper CThQ2.

Yonenaga, K.

K. Yonenaga and S. Kuwano, " Dispersion-tolerant optical transmission system using duobinary transmitter and binary receiver," J. Lightwave Technol. 15, 1530- 1537 ( 1997).
[CrossRef]

IEEE Electron. Lett. (1)

H. Schmuck, " Comparison of optical millimetre-wave system concepts with regard to chromatic dispersion," IEEE Electron. Lett. 31, 1848- 1849 ( 1995).
[CrossRef]

IEEE J. Sel. Areas Commun. (1)

S. Benedetto, R. Gaudino, and P. Poggiolini, " Direct detection of optical digital transmission based on polarization shift keying modulation," IEEE J. Sel. Areas Commun. 13, 531- 542 ( 1995).
[CrossRef]

IEEE Photon. Technol. Lett. (1)

A. S. Siddiqui, S. G. Edirisinghe, J. J. Lepley, J. G. Ellison, and S. D. Walker, " Dispersion-tolerant transmission using a duobinary polarization-shift keying transmission scheme," IEEE Photon. Technol. Lett. 14, 158- 160 ( 2002).
[CrossRef]

IEEE Trans. Commun. (1)

S. Benedetto and P. Poggiolini, " Theory of polarization shift keying modulation," IEEE Trans. Commun. 40, 708- 720 ( 1992).
[CrossRef]

IEEE Trans. Microwave Theory Tech. (1)

U. Gliese, S. Nørskov, and T. N. Nielsen, " Chromatic dispersion in fiber-optic microwave and millimeter-wave links," IEEE Trans. Microwave Theory Tech. 44, 1716- 1724 ( 1996).
[CrossRef]

IEEE Trans. Microwave Theory Technol. (1)

J. L. Corral, J. Marti, and J. M. Fuster, " General expressions for IM/DD dispersive analog optical links with external modulation or optical up-conversion in a Mach-Zehnder electrooptical modulator," IEEE Trans. Microwave Theory Technol. 49,1968- 1976 (2001).
[CrossRef]

J. Lightwave Technol. (3)

A. F. Elrafaie, R. E. Wagner, D. A. Atlas, and D. G. Daut, " Chromatic dispersion limitations in coherent lightwave transmission systems," J. Lightwave Technol. 6, 704- 709 ( 1988).
[CrossRef]

K. Yonenaga and S. Kuwano, " Dispersion-tolerant optical transmission system using duobinary transmitter and binary receiver," J. Lightwave Technol. 15, 1530- 1537 ( 1997).
[CrossRef]

J. Wang and K. Petermann, " Small signal analysis for dispersive optical fiber communication systems," J. Lightwave Technol. , 10,96- 100 (1992).
[CrossRef]

Other (4)

J. D. Bull, N. A. F. Jaeger, H. Kato, M. Fairburn, A. Reid, and P. Ghanipour, " 40 GHz electro-optic polarization modulator for fiber optic communications systems," in Optical Components and Devices, Proc. SPIE 5577, 133- 143 ( 2004).

Y. W. Song, Z. Pan, Y. Arieli, S. M. R. Motaghian, S. A. Havstad, and A. E. Willner, " Enhanced suppression of nonlinearity-induced crosstalk in WDM systems using optical polarization shift keying," presented at the Conference on Lasers and Electro-Optics/Quantum Electronics and Laser Science Conference, Baltimore, Maryland, 1- 6 June 2003, paper CThQ2.

A. L. Campillo, F. Bucholtz, J. L. Dexter, and K. J. Williams, " Crosstalk reduction in wavelength division multiplexed analog links through polarization modulation," presented at the Conference on Lasers and Electro-Optics/International Quantum Electronics Conference, San Francisco, California, 18- 20 May 2004, paper CWQ3.

D. S. Kliger, J. W. Lewis, and C. E. Randall, Polarized Light in Optics and Spectroscopy (Academic, 1990).

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Figures (6)

Fig. 1
Fig. 1

Effect of dispersion on PolM. For an initial modulation along the great circle on the S 1, S 2 plane, dispersion produces a polarization-dependent rotation of orientation of the modulation about the axis of dc polarization. RCP, right-circular polarization; LCP, left-circular polarization.

Fig. 2
Fig. 2

Experimental setup used to confirm polarization relation of carrier and sidebands predicted by Eq. (4). DFB, distributed feedback laser.

Fig. 3
Fig. 3

Optical spectrum of a 20-GHz PolM signal transmitted through a polarizer (a) parallel to the dc polarization and (b) rotated 90° relative to the dc polarization. The optical spectrum when the polarizer is removed is shown as a dotted curve in both (a) and (b).

Fig. 4
Fig. 4

Experimental set-up. The polarization of the light from a DFB laser is modulated with a JGKB 40-GHz TE–TM mode converter polarization modulator. The PolM signal is transmitted along a 25-km single mode fiber. The S 1 and S 3 components of the modulated signal are detected by two polarization controllers, a polarizing beam splitter, and a photodetector.

Fig. 5
Fig. 5

Rotation of PolM before the polarizing element of receiver. (a) A manual polarization controller rotates the orientation of modulation relative to the polarization of the carrier, (b) an inline polarization manager, consisting of a polarimeter, a polarization controller, and a feedback system, sets the polarization of the carrier.

Fig. 6
Fig. 6

S 1 and S 3 responses of the PolM signal described by Eq. (1) transmitted through 25 km of single mode fiber. The plot shows the measured and calculated transmission responses for S 3 (data as black circles and calculation as solid curve) and S 1 (data as gray triangles and calculation as dashed gray curve).

Equations (40)

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

E = M [ E x E y ] = [ exp [ i δ ( t ) 2 ] 0 0 exp [ i δ ( t ) 2 ] ] A 0 2 [ 1 1 ]
= A 0 2 [ exp [ i δ ( t ) 2 ] exp [ i δ ( t ) 2 ] ] ,
S = A 0 2 ( 1 , 0 , cos [ δ ( t ) ] , sin [ δ ( t ) ] ) .
E = [ exp [ i δ ( t ) ] 0 0 1 ] 1 2 [ 1 1 ] = 1 2 [ exp [ i δ ( t ) ] 1 ] ,
E   = [ E x ( t ) E y ( t ) ]
= P 0 [ cos [ ω 0 t m sin ( Ω RF t ) ] cos [ ω 0 t + m sin ( Ω RF t ) ] ]
= P 0 2 [ exp [ i m sin ( Ω RF t ) ] exp [ i m sin ( Ω RF t ) ] ] exp ( i ω 0 t ) + P 0 2 [ exp [ i m sin ( Ω RF t ) ] exp [ i m sin ( Ω RF t ) ] ] exp ( i ω 0 t ) ,
E   =   [ E x ( t ) E y ( t ) ] = P 0 { J 0 ( m ) cos ( ω 0 t + ϕ 0 ) [ 1 1 ] J 1 ( m ) cos [ ( ω 0 + Ω RF ) t + ϕ 1 ] [ 1 1 ] + J 1 ( m ) cos [ ( ω 0 Ω RF ) t + ϕ 1 ] [ 1 1 ] + J 2 ( m ) cos [ ( ω 0 + 2 Ω RF ) t + ϕ 2 ] [ 1 1 ] + J 2 ( m ) cos [ ( ω 0 2 Ω RF ) t + ϕ 2 ] [ 1 1 ] + + } ,
ϕ n ω 0 τ ( ω 0 ) + π D c L [ ( n Ω RF ) 2 ω 0     2 1 ] ,
ϕ n ϕ 0 + n 2 θ ,
ϕ 0 ω 0 τ ( ω 0 ) π D c L ,
θ = π D c L [ Ω RF 2 ω 0 2 ] ,
E  =   P 0 ( n = J 2 n ( m ) cos [ ( ω 0 + 2 n Ω RF ) t + ( 2 n ) 2 θ ] [ 1 1 ] + n = J 2 n + 1 ( m ) cos { [ ω 0 + ( 2 n + 1 ) Ω RF ] t + ( 2 n + 1 ) 2 θ } [ 1 1 ] ) .
E  =   P 0 ( n = J 2 n ( m ) cos [ ( ω 0 + 2 n Ω RF ) t ] [ 1 1 ] ( 1 ) p n = J 2 n + 1 ( m ) × sin { [ ω 0 + ( 2 n + 1 ) Ω RF ] t } [ 1 1 ] ) ,
E   =   P 0 2 [ cos ( m sin Ω RF t ) sin ( m sin Ω RF t ) cos ( m sin Ω RF t ) + sin ( m sin Ω RF t ) ] ×  exp ( i ω 0 t ) + c .c . ( p   odd ) ,
E   =   P 0 2 [ cos ( m sin Ω RF t ) + sin ( m sin Ω RF t ) cos ( m sin Ω RF t ) sin ( m sin Ω RF t ) ] ×  exp ( i ω 0 t ) + c .c . ( p   even ) ,
E   =   A 0 [ cos [ π 4 + m sin ( Ω RF t ) ] sin [ π 4 + m sin ( Ω RF t ) ] ] ( p   odd ) ,
E   =   A 0 [ sin [ π 4 + m sin ( Ω RF t ) ] cos [ π 4 + m sin ( Ω RF t ) ] ] ( p   even ) .
E = P 0 ( n = J 2 n ( m ) cos [ ( ω 0 + 2 n Ω RF ) t ] [ 1 1 ] + ( 1 ) p n = J 2 n + 1 ( m ) × cos { [ ω 0 + ( 2 n + 1 ) Ω RF ] t } [ 1 1 ] ) ,
S 1 = | E 0 ° | 2 | E 90 ° | 2 ,
S 2 = | E 45 ° | 2 | E 45 ° | 2 ,
S 3 = | E RCP | 2 | E LCP | 2 .
E 0 ° = P 0 ( n = J 2 n ( m ) cos [ ( ω 0 + 2 n Ω RF ) t + ( 2 n ) 2 θ ] n = J 2 n + 1 ( m ) cos { [ ω 0 + ( 2 n + 1 ) Ω RF ] t + ( 2 n + 1 ) 2 θ } ) ,
E 90 ° = P 0 ( n = J 2 n ( m ) cos [ ( ω 0 + 2 n Ω RF ) t + ( 2 n ) 2 θ ] + n = J 2 n + 1 ( m ) cos { [ ω 0 + ( 2 n + 1 ) Ω RF ] t + ( 2 n + 1 ) 2 θ } ) ,
S 1 = 4 P 0 { n = J 2 n ( m ) × cos [ ( ω 0 + 2 n Ω RF ) t + ( 2 n ) 2 θ ] } × ( l = J 2 l + 1 ( m ) cos { [ ω 0 + ( 2 l + 1 ) Ω RF ] t + ( 2 l + 1 ) 2 θ } ) ,
E 45 ° = 2 P 0 n = J 2 n ( m ) cos [ ( ω 0 + 2 n Ω RF ) t + ( 2 n ) 2 θ ] ,
E 45 ° = 2 P 0 n = J 2 n + 1 ( m ) cos { [ ω 0 + ( 2 n + 1 ) Ω RF ] t + ( 2 n + 1 ) 2 θ } ,
S 2 = 2 P 0 [ { n = J 2 n ( m ) cos [ ( ω 0 + 2 n Ω RF ) t + ( 2 n ) 2 θ ] } 2 ( n = J 2 n + 1 ( m ) cos { [ ω 0 + ( 2 n + 1 ) Ω RF ] t + ( 2 n + 1 ) 2 θ } ) 2 ] .
E LCP = P 0 ( n = J 2 n ( m ) cos [ ( ω 0 + 2 n Ω RF ) t + ( 2 n ) 2 θ ] n = J 2 n + 1 ( m ) sin { [ ω 0 + ( 2 n + 1 ) Ω RF ] t + ( 2 n + 1 ) 2 θ } ) ,
E RCP = P 0 ( n = J 2 n ( m ) cos [ ( ω 0 + 2 n Ω RF ) t + ( 2 n ) 2 θ ] + n = J 2 n + 1 ( m ) sin { [ ω 0 + ( 2 n + 1 ) Ω RF ] t + ( 2 n + 1 ) 2 θ } ) ,
S 3 = 4 P 0 { n = J 2 n ( m ) cos [ ( ω 0 + 2 n Ω RF ) t + ( 2 n ) 2 θ ] } × ( l = J 2 l + 1 ( m ) sin { [ ω 0 + ( 2 l + 1 ) Ω RF ] t + ( 2 l + 1 ) 2 θ } ) .
S 1 4 P 0 J 0 ( m ) J 1 ( m ) sin ( Ω RF t ) sin ( θ ) ,
S 2 P 0 [ J 0     2 ( m ) 4 J 1     2 ( m ) + 4 J 1     2 ( m ) × cos ( 2 Ω RF t ) ]
P 0 J 0     2 ( m ) ,
S 3 4 P 0 J 0 ( m ) J 1 ( m ) sin ( Ω RF t ) cos ( θ ) .
I 1 1 2 P 0 + 2 P 0 J 0 ( m ) J 1 ( m ) sin ( Ω RF t ) sin ( θ ) ,
I 2 1 2 P 0 2 P 0 J 0 ( m ) J 1 ( m ) sin ( Ω RF t ) sin ( θ ) ,
I 1 1 2 P 0 + 2 P 0 J 0 ( m ) J 1 ( m ) sin ( Ω RF t ) cos ( θ ) ,
I 2 1 2 P 0 2 P 0 J 0 ( m ) J 1 ( m ) sin ( Ω RF t ) cos ( θ ) .
S corrected = [ cos ( θ ) 0 sin ( θ ) 0 1 0 sin ( θ ) 0 cos ( θ ) ] [ S 1 S 2 S 3 ] = [ S 1 cos θ S 3 sin θ S 2 S 1 sin θ + S 3 cos θ ] ,

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