Abstract

In proximity side-writing fiber Bragg gratings (FBGs) using a phase shifted phase mask, the phase shift in the phase mask is split into two half-amplitude phase shifts. The effects of the phase-shift split on FBGs is investigated. A FBG with a split π-phase shift is modeled as two cascaded and tightly coupled grating-based Fabry–Perot filters. A compact expression of its reflectivity spectrum is obtained showing spectral asymmetry errors and a shift of the transmission peak. Our numerical results are in good agreement with previous experimental results. These effects decrease with increasing FBG length and become negligible for gratings more than a few millimeters long.

© 2006 Optical Society of America

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References

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  1. R. Kashyap, Fiber Bragg Gratings (Academic, 1999), Chap. 6.
  2. F. Bakhti and P. Sansonetti, "Wide bandwidth, low loss and highly rejective doubly phase-shifted uv-written fibre bandpass filter," Electron. Lett. 32, 581-582 (1996).
    [CrossRef]
  3. R. Zengerle and O. Leminger, "Phase-shifted Bragg grating filters with improved transmission characteristics," J. Lightwave Technol. 13, 2354-2358 (1995).
    [CrossRef]
  4. L. Wei and J. W. Y. Lit, "Phase-shifted Bragg grating filters with symmetrical structures," J. Lightwave Technol. 15, 1405-1410 (1997).
    [CrossRef]
  5. H. Li, Y. Sheng, Y. Li, and J. E. Rothenberg, "Phased-only sampled fiber Bragg gratings for high-channel-count chromatic dispersion of compensation," J. Lightwave Technol. 21, 2074-2083 (2003).
    [CrossRef]
  6. J. E. Rothenberg, H. Li, Y. Li, J. Popelek, Y. Sheng, Y. Wang, P. B. Wilcox, and J. Zweiback, "Dammann fiber Bragg gratings and phase-only sampling for high channel counts," IEEE Photon. Technol. Lett. l4, 1309-1311 (2002).
    [CrossRef]
  7. L. Poladian, B. Ashton, and W. Padden, "Interactive design and fabrication of complex FBGs," in Optical Fiber Communication Conference (OFC), Vol. 86 of OSA Trends in Optics and Photonics Series (Optical Society of America, 2003), pp. 378-379.
  8. S. J. Mihailov, C. W. Smelser, D. Grobnic, R. Walker, P. Lu, H. Ding, and J. Unruh, "Fiber Bragg gratings made with a phase mask and 800-nm femtosecond radiation," Opt. Lett. 28, 995-997 (2003).
    [CrossRef] [PubMed]
  9. R. Kashyap, P. F. Mckee, and D. Armes, "UV written reflection grating structures in photosensitive optical fibers using phase-shifted phase masks," Electron. Lett. 30, 1977-1978 (1994).
    [CrossRef]
  10. Y. Sheng, J. E. Rothenberg, H. Li, Y. Wang, and J. Zweiback, "Split of phase shifts in a phase mask for fiber Bragg gratings," IEEE Photon. Technol. Lett. 16, 1316-1318 (2004).
    [CrossRef]
  11. Y. Sheng and L. Sun, "Near-field diffraction of irregular phase gratings with multiple phase-shifts," Opt. Express 13, 6111-6116 (2005).
    [CrossRef] [PubMed]
  12. T. Erdogan, "Fiber Grating spectra," J. Lightwave Technol. 15, 1277-1294 (1997).
    [CrossRef]
  13. S. Legoubin, M. Douay, P. Bernage, P. May, S. Boj, and E. Delevaque, "Free spectral range variations of grating-based Fabry-Perot filters photowritten in optical fibers," J. Opt. Soc. Am. A 12, 1687-1694 (1995).
    [CrossRef]
  14. J. A. R. Williams, X. Liu, L. A. Everall, I. Bennion, J. S. Aitchison, S. Thorns, and R. M. De La Rue, "The effects of phase steps in e-beam written phase masks used for fiber grating fabrication by near field holography," in European Conference on Optical Communication (ECOC, 1997), pp. 187-190.

2005

2004

Y. Sheng, J. E. Rothenberg, H. Li, Y. Wang, and J. Zweiback, "Split of phase shifts in a phase mask for fiber Bragg gratings," IEEE Photon. Technol. Lett. 16, 1316-1318 (2004).
[CrossRef]

2003

2002

J. E. Rothenberg, H. Li, Y. Li, J. Popelek, Y. Sheng, Y. Wang, P. B. Wilcox, and J. Zweiback, "Dammann fiber Bragg gratings and phase-only sampling for high channel counts," IEEE Photon. Technol. Lett. l4, 1309-1311 (2002).
[CrossRef]

1997

L. Wei and J. W. Y. Lit, "Phase-shifted Bragg grating filters with symmetrical structures," J. Lightwave Technol. 15, 1405-1410 (1997).
[CrossRef]

T. Erdogan, "Fiber Grating spectra," J. Lightwave Technol. 15, 1277-1294 (1997).
[CrossRef]

1996

F. Bakhti and P. Sansonetti, "Wide bandwidth, low loss and highly rejective doubly phase-shifted uv-written fibre bandpass filter," Electron. Lett. 32, 581-582 (1996).
[CrossRef]

1995

R. Zengerle and O. Leminger, "Phase-shifted Bragg grating filters with improved transmission characteristics," J. Lightwave Technol. 13, 2354-2358 (1995).
[CrossRef]

S. Legoubin, M. Douay, P. Bernage, P. May, S. Boj, and E. Delevaque, "Free spectral range variations of grating-based Fabry-Perot filters photowritten in optical fibers," J. Opt. Soc. Am. A 12, 1687-1694 (1995).
[CrossRef]

1994

R. Kashyap, P. F. Mckee, and D. Armes, "UV written reflection grating structures in photosensitive optical fibers using phase-shifted phase masks," Electron. Lett. 30, 1977-1978 (1994).
[CrossRef]

Aitchison, J. S.

J. A. R. Williams, X. Liu, L. A. Everall, I. Bennion, J. S. Aitchison, S. Thorns, and R. M. De La Rue, "The effects of phase steps in e-beam written phase masks used for fiber grating fabrication by near field holography," in European Conference on Optical Communication (ECOC, 1997), pp. 187-190.

Armes, D.

R. Kashyap, P. F. Mckee, and D. Armes, "UV written reflection grating structures in photosensitive optical fibers using phase-shifted phase masks," Electron. Lett. 30, 1977-1978 (1994).
[CrossRef]

Ashton, B.

L. Poladian, B. Ashton, and W. Padden, "Interactive design and fabrication of complex FBGs," in Optical Fiber Communication Conference (OFC), Vol. 86 of OSA Trends in Optics and Photonics Series (Optical Society of America, 2003), pp. 378-379.

Bakhti, F.

F. Bakhti and P. Sansonetti, "Wide bandwidth, low loss and highly rejective doubly phase-shifted uv-written fibre bandpass filter," Electron. Lett. 32, 581-582 (1996).
[CrossRef]

Bennion, I.

J. A. R. Williams, X. Liu, L. A. Everall, I. Bennion, J. S. Aitchison, S. Thorns, and R. M. De La Rue, "The effects of phase steps in e-beam written phase masks used for fiber grating fabrication by near field holography," in European Conference on Optical Communication (ECOC, 1997), pp. 187-190.

Bernage, P.

Boj, S.

De La Rue, R. M.

J. A. R. Williams, X. Liu, L. A. Everall, I. Bennion, J. S. Aitchison, S. Thorns, and R. M. De La Rue, "The effects of phase steps in e-beam written phase masks used for fiber grating fabrication by near field holography," in European Conference on Optical Communication (ECOC, 1997), pp. 187-190.

Delevaque, E.

Ding, H.

Douay, M.

Erdogan, T.

T. Erdogan, "Fiber Grating spectra," J. Lightwave Technol. 15, 1277-1294 (1997).
[CrossRef]

Everall, L. A.

J. A. R. Williams, X. Liu, L. A. Everall, I. Bennion, J. S. Aitchison, S. Thorns, and R. M. De La Rue, "The effects of phase steps in e-beam written phase masks used for fiber grating fabrication by near field holography," in European Conference on Optical Communication (ECOC, 1997), pp. 187-190.

Grobnic, D.

Kashyap, R.

R. Kashyap, P. F. Mckee, and D. Armes, "UV written reflection grating structures in photosensitive optical fibers using phase-shifted phase masks," Electron. Lett. 30, 1977-1978 (1994).
[CrossRef]

R. Kashyap, Fiber Bragg Gratings (Academic, 1999), Chap. 6.

Legoubin, S.

Leminger, O.

R. Zengerle and O. Leminger, "Phase-shifted Bragg grating filters with improved transmission characteristics," J. Lightwave Technol. 13, 2354-2358 (1995).
[CrossRef]

Li, H.

Y. Sheng, J. E. Rothenberg, H. Li, Y. Wang, and J. Zweiback, "Split of phase shifts in a phase mask for fiber Bragg gratings," IEEE Photon. Technol. Lett. 16, 1316-1318 (2004).
[CrossRef]

H. Li, Y. Sheng, Y. Li, and J. E. Rothenberg, "Phased-only sampled fiber Bragg gratings for high-channel-count chromatic dispersion of compensation," J. Lightwave Technol. 21, 2074-2083 (2003).
[CrossRef]

J. E. Rothenberg, H. Li, Y. Li, J. Popelek, Y. Sheng, Y. Wang, P. B. Wilcox, and J. Zweiback, "Dammann fiber Bragg gratings and phase-only sampling for high channel counts," IEEE Photon. Technol. Lett. l4, 1309-1311 (2002).
[CrossRef]

Li, Y.

H. Li, Y. Sheng, Y. Li, and J. E. Rothenberg, "Phased-only sampled fiber Bragg gratings for high-channel-count chromatic dispersion of compensation," J. Lightwave Technol. 21, 2074-2083 (2003).
[CrossRef]

J. E. Rothenberg, H. Li, Y. Li, J. Popelek, Y. Sheng, Y. Wang, P. B. Wilcox, and J. Zweiback, "Dammann fiber Bragg gratings and phase-only sampling for high channel counts," IEEE Photon. Technol. Lett. l4, 1309-1311 (2002).
[CrossRef]

Lit, J. W. Y.

L. Wei and J. W. Y. Lit, "Phase-shifted Bragg grating filters with symmetrical structures," J. Lightwave Technol. 15, 1405-1410 (1997).
[CrossRef]

Liu, X.

J. A. R. Williams, X. Liu, L. A. Everall, I. Bennion, J. S. Aitchison, S. Thorns, and R. M. De La Rue, "The effects of phase steps in e-beam written phase masks used for fiber grating fabrication by near field holography," in European Conference on Optical Communication (ECOC, 1997), pp. 187-190.

Lu, P.

May, P.

Mckee, P. F.

R. Kashyap, P. F. Mckee, and D. Armes, "UV written reflection grating structures in photosensitive optical fibers using phase-shifted phase masks," Electron. Lett. 30, 1977-1978 (1994).
[CrossRef]

Mihailov, S. J.

Padden, W.

L. Poladian, B. Ashton, and W. Padden, "Interactive design and fabrication of complex FBGs," in Optical Fiber Communication Conference (OFC), Vol. 86 of OSA Trends in Optics and Photonics Series (Optical Society of America, 2003), pp. 378-379.

Poladian, L.

L. Poladian, B. Ashton, and W. Padden, "Interactive design and fabrication of complex FBGs," in Optical Fiber Communication Conference (OFC), Vol. 86 of OSA Trends in Optics and Photonics Series (Optical Society of America, 2003), pp. 378-379.

Popelek, J.

J. E. Rothenberg, H. Li, Y. Li, J. Popelek, Y. Sheng, Y. Wang, P. B. Wilcox, and J. Zweiback, "Dammann fiber Bragg gratings and phase-only sampling for high channel counts," IEEE Photon. Technol. Lett. l4, 1309-1311 (2002).
[CrossRef]

Rothenberg, J. E.

Y. Sheng, J. E. Rothenberg, H. Li, Y. Wang, and J. Zweiback, "Split of phase shifts in a phase mask for fiber Bragg gratings," IEEE Photon. Technol. Lett. 16, 1316-1318 (2004).
[CrossRef]

H. Li, Y. Sheng, Y. Li, and J. E. Rothenberg, "Phased-only sampled fiber Bragg gratings for high-channel-count chromatic dispersion of compensation," J. Lightwave Technol. 21, 2074-2083 (2003).
[CrossRef]

J. E. Rothenberg, H. Li, Y. Li, J. Popelek, Y. Sheng, Y. Wang, P. B. Wilcox, and J. Zweiback, "Dammann fiber Bragg gratings and phase-only sampling for high channel counts," IEEE Photon. Technol. Lett. l4, 1309-1311 (2002).
[CrossRef]

Sansonetti, P.

F. Bakhti and P. Sansonetti, "Wide bandwidth, low loss and highly rejective doubly phase-shifted uv-written fibre bandpass filter," Electron. Lett. 32, 581-582 (1996).
[CrossRef]

Sheng, Y.

Y. Sheng and L. Sun, "Near-field diffraction of irregular phase gratings with multiple phase-shifts," Opt. Express 13, 6111-6116 (2005).
[CrossRef] [PubMed]

Y. Sheng, J. E. Rothenberg, H. Li, Y. Wang, and J. Zweiback, "Split of phase shifts in a phase mask for fiber Bragg gratings," IEEE Photon. Technol. Lett. 16, 1316-1318 (2004).
[CrossRef]

H. Li, Y. Sheng, Y. Li, and J. E. Rothenberg, "Phased-only sampled fiber Bragg gratings for high-channel-count chromatic dispersion of compensation," J. Lightwave Technol. 21, 2074-2083 (2003).
[CrossRef]

J. E. Rothenberg, H. Li, Y. Li, J. Popelek, Y. Sheng, Y. Wang, P. B. Wilcox, and J. Zweiback, "Dammann fiber Bragg gratings and phase-only sampling for high channel counts," IEEE Photon. Technol. Lett. l4, 1309-1311 (2002).
[CrossRef]

Smelser, C. W.

Sun, L.

Thorns, S.

J. A. R. Williams, X. Liu, L. A. Everall, I. Bennion, J. S. Aitchison, S. Thorns, and R. M. De La Rue, "The effects of phase steps in e-beam written phase masks used for fiber grating fabrication by near field holography," in European Conference on Optical Communication (ECOC, 1997), pp. 187-190.

Unruh, J.

Walker, R.

Wang, Y.

Y. Sheng, J. E. Rothenberg, H. Li, Y. Wang, and J. Zweiback, "Split of phase shifts in a phase mask for fiber Bragg gratings," IEEE Photon. Technol. Lett. 16, 1316-1318 (2004).
[CrossRef]

J. E. Rothenberg, H. Li, Y. Li, J. Popelek, Y. Sheng, Y. Wang, P. B. Wilcox, and J. Zweiback, "Dammann fiber Bragg gratings and phase-only sampling for high channel counts," IEEE Photon. Technol. Lett. l4, 1309-1311 (2002).
[CrossRef]

Wei, L.

L. Wei and J. W. Y. Lit, "Phase-shifted Bragg grating filters with symmetrical structures," J. Lightwave Technol. 15, 1405-1410 (1997).
[CrossRef]

Wilcox, P. B.

J. E. Rothenberg, H. Li, Y. Li, J. Popelek, Y. Sheng, Y. Wang, P. B. Wilcox, and J. Zweiback, "Dammann fiber Bragg gratings and phase-only sampling for high channel counts," IEEE Photon. Technol. Lett. l4, 1309-1311 (2002).
[CrossRef]

Williams, J. A. R.

J. A. R. Williams, X. Liu, L. A. Everall, I. Bennion, J. S. Aitchison, S. Thorns, and R. M. De La Rue, "The effects of phase steps in e-beam written phase masks used for fiber grating fabrication by near field holography," in European Conference on Optical Communication (ECOC, 1997), pp. 187-190.

Zengerle, R.

R. Zengerle and O. Leminger, "Phase-shifted Bragg grating filters with improved transmission characteristics," J. Lightwave Technol. 13, 2354-2358 (1995).
[CrossRef]

Zweiback, J.

Y. Sheng, J. E. Rothenberg, H. Li, Y. Wang, and J. Zweiback, "Split of phase shifts in a phase mask for fiber Bragg gratings," IEEE Photon. Technol. Lett. 16, 1316-1318 (2004).
[CrossRef]

J. E. Rothenberg, H. Li, Y. Li, J. Popelek, Y. Sheng, Y. Wang, P. B. Wilcox, and J. Zweiback, "Dammann fiber Bragg gratings and phase-only sampling for high channel counts," IEEE Photon. Technol. Lett. l4, 1309-1311 (2002).
[CrossRef]

Electron. Lett.

F. Bakhti and P. Sansonetti, "Wide bandwidth, low loss and highly rejective doubly phase-shifted uv-written fibre bandpass filter," Electron. Lett. 32, 581-582 (1996).
[CrossRef]

R. Kashyap, P. F. Mckee, and D. Armes, "UV written reflection grating structures in photosensitive optical fibers using phase-shifted phase masks," Electron. Lett. 30, 1977-1978 (1994).
[CrossRef]

IEEE Photon. Technol. Lett.

Y. Sheng, J. E. Rothenberg, H. Li, Y. Wang, and J. Zweiback, "Split of phase shifts in a phase mask for fiber Bragg gratings," IEEE Photon. Technol. Lett. 16, 1316-1318 (2004).
[CrossRef]

J. E. Rothenberg, H. Li, Y. Li, J. Popelek, Y. Sheng, Y. Wang, P. B. Wilcox, and J. Zweiback, "Dammann fiber Bragg gratings and phase-only sampling for high channel counts," IEEE Photon. Technol. Lett. l4, 1309-1311 (2002).
[CrossRef]

J. Lightwave Technol.

R. Zengerle and O. Leminger, "Phase-shifted Bragg grating filters with improved transmission characteristics," J. Lightwave Technol. 13, 2354-2358 (1995).
[CrossRef]

L. Wei and J. W. Y. Lit, "Phase-shifted Bragg grating filters with symmetrical structures," J. Lightwave Technol. 15, 1405-1410 (1997).
[CrossRef]

H. Li, Y. Sheng, Y. Li, and J. E. Rothenberg, "Phased-only sampled fiber Bragg gratings for high-channel-count chromatic dispersion of compensation," J. Lightwave Technol. 21, 2074-2083 (2003).
[CrossRef]

T. Erdogan, "Fiber Grating spectra," J. Lightwave Technol. 15, 1277-1294 (1997).
[CrossRef]

J. Opt. Soc. Am. A

Opt. Express

Opt. Lett.

Other

R. Kashyap, Fiber Bragg Gratings (Academic, 1999), Chap. 6.

L. Poladian, B. Ashton, and W. Padden, "Interactive design and fabrication of complex FBGs," in Optical Fiber Communication Conference (OFC), Vol. 86 of OSA Trends in Optics and Photonics Series (Optical Society of America, 2003), pp. 378-379.

J. A. R. Williams, X. Liu, L. A. Everall, I. Bennion, J. S. Aitchison, S. Thorns, and R. M. De La Rue, "The effects of phase steps in e-beam written phase masks used for fiber grating fabrication by near field holography," in European Conference on Optical Communication (ECOC, 1997), pp. 187-190.

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Figures (6)

Fig. 1
Fig. 1

Fiber Bragg grating with a split phase shift.

Fig. 2
Fig. 2

Split-phase-shifted grating model as two cascaded FP resonators with wavelength-dependent reflectors and cavity lengths.

Fig. 3
Fig. 3

Reflection spectrum computed with Eq. (12) of a FBG with a single split π-phase shift.

Fig. 4
Fig. 4

Asymmetry ratio R max 2 R max 1 as a function of the grating length for different values of Δ n eff and L 2 = 25 μ m .

Fig. 5
Fig. 5

Numerically calculated transmission peak shift for L 2 = 25 μ m , Δ n eff = 10 4 and 10 3 , and λ 0 = 1550 nm .

Fig. 6
Fig. 6

Effect of the phase-shift split: bandwidth ratio as a function of grating length for a given separation L 2 = 25 μ m and Δ n eff = 10 4 and 10 3 .

Equations (23)

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F 11 = cosh ( γ L ) i σ ̂ γ sinh ( γ L ) ,
F 21 = i κ γ sin ( γ L ) ,
R ( λ ) = r ( λ ) 2 = F 21 F 22 2 ,
T ( λ ) = τ ( λ ) 2 = 1 F 22 2 .
F = [ F 11 exp ( i ϕ 1 ) F 12 F 21 F 22 exp ( i ϕ 1 ) ] = [ exp ( i ϕ 1 2 ) 0 0 exp ( i ϕ 1 2 ) ] [ F 11 F 12 F 21 F 22 ] × [ exp ( i ϕ 1 2 ) 0 0 exp ( i ϕ 1 2 ) ] .
R = F sin 2 ϕ 1 + F sin 2 ϕ ,
F = F ( 1 ) F ps F ( 2 ) F ps F ( 3 ) ,
F ps = [ exp ( i ϕ ps 2 ) 0 0 exp ( i ϕ ps 2 ) ] ,
F = [ e 1 0 0 e 1 * ] [ F 11 ( 1 ) F 21 ( 1 ) F 21 ( 1 ) F 22 ( 1 ) ] [ e C 0 0 e C * ] × [ F 11 ( 2 ) F 21 ( 2 ) F 21 ( 2 ) F 22 ( 2 ) ] [ e C 0 0 e C * ] [ F 11 ( 1 ) F 21 ( 1 ) F 21 ( 1 ) F 22 ( 1 ) ] [ e 1 0 0 e 1 * ]
d 1 ( λ ) = ϕ 1 λ 4 π n eff ,
d 2 ( λ ) = ε 2 + ( ϕ 1 + ϕ 2 ) λ 4 π n eff .
R = F ( sin ϕ ϕ M ) 2 T 2 + F ( sin ϕ ϕ M ) 2 ,
T = T 2 T 2 + F ( sin ϕ ϕ M ) 2 .
F ( sin ϕ ϕ M ) 2 = F [ sin 2 ϕ + ϕ M 2 2 sin ( ϕ ) ϕ M ] ,
F = [ f 11 f 12 f 21 f 22 ] ,
f 11 = e 1 2 ( e C 2 F 11 ( 1 ) 2 F 11 ( 2 ) 2 F 11 ( 1 ) F 21 ( 1 ) F 21 ( 2 ) + e C * 2 F 21 ( 1 ) 2 F 11 ( 2 ) ) ,
f 21 = F 21 ( 2 ) ( F 11 ( 1 ) 2 + F 21 ( 1 ) 2 ) + F 11 ( 1 ) F 11 ( 2 ) f 21 ( 1 ) ( e C 2 + e C * 2 ) ,
R = F 21 ( 2 ) 2 ( F 11 ( 1 ) 2 + F 21 ( 1 ) 2 ) 2 + 4 F 11 ( 1 ) 2 F 21 ( 1 ) 2 F 11 ( 2 ) 2 cos 2 ( ϕ C ) 4 F 11 ( 1 ) F 11 ( 2 ) F 21 ( 1 ) F 21 ( 2 ) cos ( ϕ C ) ( F 11 ( 1 ) 2 + F 21 ( 1 ) 2 ) F 11 ( 2 ) 2 ( F 11 ( 1 ) 4 + F 12 ( 1 ) 4 ) + 2 F 11 ( 1 ) 2 F 21 ( 1 ) 2 [ F 11 ( 2 ) 2 cos ( 2 ϕ C ) + 2 F 21 ( 2 ) 2 ] 4 F 11 ( 1 ) F 11 ( 2 ) F 21 ( 1 ) F 21 ( 2 ) cos ( ϕ C ) ( F 11 ( 1 ) 2 + F 21 ( 1 ) 2 ) .
R = 2 R 1 [ 1 + cos ( 2 ϕ C ) ] + R 2 ( 1 + R 1 ) 2 4 R 1 R 2 ( 1 + R 1 ) cos ( ϕ C ) 1 + R 1 2 + 4 R 1 R 2 + 2 R 1 cos ( 2 ϕ C ) 4 R 1 R 2 ( 1 + R 1 ) cos ( ϕ C ) .
R = 2 R 1 ( 2 sin 2 ϕ 4 ϕ M sin ϕ + 2 ϕ M 2 ) 1 + R 1 2 + 4 R 1 R 2 + 2 R 1 ( 2 sin 2 ϕ 1 ) 8 ϕ M R 1 sin ϕ = 4 R 1 ( sin ϕ ϕ M ) 2 ( 1 R 1 ) 2 + 4 R 1 R 2 + 4 R 1 ( sin 2 ϕ 2 ϕ M sin ϕ ) .
R = F ( sin ϕ ϕ M ) 2 1 + F ( R 2 ϕ M 2 ) + F ( sin ϕ ϕ M ) 2 .
R = F ( sin ϕ ϕ M ) 2 T 2 + F ( sin ϕ ϕ M ) 2 .
R = F sin 2 ϕ 1 + F sin 2 ϕ .

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