Abstract

Radiation-mode coupling is stronger and more efficient in tilted fiber Bragg gratings than in other fiber gratings; it can be used to good advantage in such fields as optical communication and optical sensors. A simplified coupled-mode theory (CMT) approach to the analysis of radiation-mode coupling is proposed for what we believe to be the first time, whose validity and accuracy is demonstrated by comparing its simulation results with that of the complete CMT equations and the volume current method (VCM). With the simplified CMT approach, a theoretical spectral analysis of coupling from core mode to radiation modes in both reflective and transmissive tilted fiber Bragg gratings is presented. The influence of tilt angle on the transmission spectrum characteristics is comprehensively investigated. The different dependences between s-polarized and p-polarized radiation-mode coupling on grating tilt angle are discussed, and an analysis is performed on how to obtain a high-performance in-fiber polarizer that involves a compromise between polarization extinction and insertion loss.

© 2008 Optical Society of America

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References

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  1. K. Feder, P. Westbrook, J. Ging, and P. Reyes, “A compact, low resolution, wavelength monitor applied to Raman pump power monitoring,” in Proceedings of IEEE Conference on Optical Fiber Communications (IEEE, 2003), pp. 42-43.
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    [CrossRef]
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  4. C. Chan, C. Chen, A. Jafari, A. Laronche, D. J. Thomson, and J. Albert, “Optical fiber refractometer using narrowband cladding-mode resonance shifts,” Appl. Opt. 46, 1142-1149 (2007).
    [CrossRef] [PubMed]
  5. X. Chen, K. Zhou, L. Zhang, and I. Bennion, “Optical chemsensor based on etched tilted Bragg grating structures in multimode fiber,” IEEE Photonics Technol. Lett. 17, 864-866 (2005).
    [CrossRef]
  6. C. Zhao, X. Yang, M. S. Demokan, and W. Jin, “Simultaneous temperature and refractive index measurements using a 3 degrees slanted multimode fiber Bragg grating,” J. Lightwave Technol. 24, 879-883 (2006).
    [CrossRef]
  7. C. Caucheteur and P. Mégret, “Demodulation technique for weakly tilted fiber Bragg grating refractometer,” IEEE Photonics Technol. Lett. 17, 2703-2705 (2005).
    [CrossRef]
  8. X. Chen, K. Zhou, L. Zhang, and I. Bennion, “In-fiber twist sensor based on a fiber Bragg grating with 81° tilted structure,” IEEE Photonics Technol. Lett. 18, 2596-2598 (2006).
    [CrossRef]
  9. R. Kashyap, R. Wyatt, and R. J. Campbell, “Wideband gain flattened erbium fibre amplifier using a photosensitive fibre blazed grating,” Electron. Lett. 29, 154-156 (1993).
    [CrossRef]
  10. K. S. Feder, P. S. Westbrook, J. Ging, P. I. Reyes, and G. E. Carver, “In-fiber spectrometer using tilted fiber gratings,” IEEE Photonics Technol. Lett. 15, 933-935 (2003).
    [CrossRef]
  11. P. S. Westbrook, T. A. Strasser, and T. Erdogan, “In-line polarimeter using blazed fiber gratings,” IEEE Photonics Technol. Lett. 12, 1352-1354 (2000).
    [CrossRef]
  12. J. Peupelmann, E. Krause, A. Bandemer, and C. Schaffer, “Fibre-polarimeter based on grating taps,” Electron. Lett. 38, 1248-1250 (2002).
    [CrossRef]
  13. K. Zhou, G. Simpson, X. Chen, L. Zhang, and I. Bennion, “High extinction ratio in-fiber polarizers based on 45° tilted fiber Bragg gratings,” Opt. Lett. 30, 1285-1287 (2005).
    [CrossRef] [PubMed]
  14. S. J. Mihailov, R. B. Walker, T. J. Stocki, and D. C. Johnson, “Fabrication of tilted fiber-grating polarization-dependent loss equalizer,” Electron. Lett. 37, 284-286 (2001).
    [CrossRef]
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    [CrossRef]
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    [CrossRef]
  18. Y. Li, S. Wielandy, P. I. Reyes, and P. S. Westbrook, “Scattering from nonuniform tilted fiber gratings,” Opt. Lett. 29, 1330-1332 (2004).
    [CrossRef] [PubMed]
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    [CrossRef] [PubMed]
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    [CrossRef]
  21. K. S. Lee and T. Erdogan, “Fiber mode conversion with tilted gratings in an optical fiber,” J. Opt. Soc. Am. A 18, 1176-1185 (2000).
    [CrossRef]
  22. K. S. Lee and T. Erdogan, “Fiber mode coupling in transmissive and reflective tilted fiber gratings,” Appl. Opt. 39, 1394-1404 (2000).
    [CrossRef]

2007 (2)

E. Chehura, S. W. James, and R. P. Tatam, “Temperature and strain discrimination using a single tilted fibre Bragg grating,” Opt. Commun. 3, 1-4 (2007).

C. Chan, C. Chen, A. Jafari, A. Laronche, D. J. Thomson, and J. Albert, “Optical fiber refractometer using narrowband cladding-mode resonance shifts,” Appl. Opt. 46, 1142-1149 (2007).
[CrossRef] [PubMed]

2006 (2)

C. Zhao, X. Yang, M. S. Demokan, and W. Jin, “Simultaneous temperature and refractive index measurements using a 3 degrees slanted multimode fiber Bragg grating,” J. Lightwave Technol. 24, 879-883 (2006).
[CrossRef]

X. Chen, K. Zhou, L. Zhang, and I. Bennion, “In-fiber twist sensor based on a fiber Bragg grating with 81° tilted structure,” IEEE Photonics Technol. Lett. 18, 2596-2598 (2006).
[CrossRef]

2005 (4)

C. Caucheteur and P. Mégret, “Demodulation technique for weakly tilted fiber Bragg grating refractometer,” IEEE Photonics Technol. Lett. 17, 2703-2705 (2005).
[CrossRef]

X. Chen, K. Zhou, L. Zhang, and I. Bennion, “Optical chemsensor based on etched tilted Bragg grating structures in multimode fiber,” IEEE Photonics Technol. Lett. 17, 864-866 (2005).
[CrossRef]

K. Zhou, G. Simpson, X. Chen, L. Zhang, and I. Bennion, “High extinction ratio in-fiber polarizers based on 45° tilted fiber Bragg gratings,” Opt. Lett. 30, 1285-1287 (2005).
[CrossRef] [PubMed]

R. B. Walker, S. J. Mihailov, P. Lu, and D. Grobnic, “Shaping the radiation field of tilted fiber Bragg gratings,” J. Opt. Soc. Am. B 22, 962-975 (2005).
[CrossRef]

2004 (1)

2003 (1)

K. S. Feder, P. S. Westbrook, J. Ging, P. I. Reyes, and G. E. Carver, “In-fiber spectrometer using tilted fiber gratings,” IEEE Photonics Technol. Lett. 15, 933-935 (2003).
[CrossRef]

2002 (1)

J. Peupelmann, E. Krause, A. Bandemer, and C. Schaffer, “Fibre-polarimeter based on grating taps,” Electron. Lett. 38, 1248-1250 (2002).
[CrossRef]

2001 (3)

2000 (4)

1996 (1)

1995 (1)

1993 (1)

R. Kashyap, R. Wyatt, and R. J. Campbell, “Wideband gain flattened erbium fibre amplifier using a photosensitive fibre blazed grating,” Electron. Lett. 29, 154-156 (1993).
[CrossRef]

Albert, J.

Bandemer, A.

J. Peupelmann, E. Krause, A. Bandemer, and C. Schaffer, “Fibre-polarimeter based on grating taps,” Electron. Lett. 38, 1248-1250 (2002).
[CrossRef]

Bennion, I.

X. Chen, K. Zhou, L. Zhang, and I. Bennion, “In-fiber twist sensor based on a fiber Bragg grating with 81° tilted structure,” IEEE Photonics Technol. Lett. 18, 2596-2598 (2006).
[CrossRef]

X. Chen, K. Zhou, L. Zhang, and I. Bennion, “Optical chemsensor based on etched tilted Bragg grating structures in multimode fiber,” IEEE Photonics Technol. Lett. 17, 864-866 (2005).
[CrossRef]

K. Zhou, G. Simpson, X. Chen, L. Zhang, and I. Bennion, “High extinction ratio in-fiber polarizers based on 45° tilted fiber Bragg gratings,” Opt. Lett. 30, 1285-1287 (2005).
[CrossRef] [PubMed]

Campbell, R. J.

R. Kashyap, R. Wyatt, and R. J. Campbell, “Wideband gain flattened erbium fibre amplifier using a photosensitive fibre blazed grating,” Electron. Lett. 29, 154-156 (1993).
[CrossRef]

Carver, G. E.

K. S. Feder, P. S. Westbrook, J. Ging, P. I. Reyes, and G. E. Carver, “In-fiber spectrometer using tilted fiber gratings,” IEEE Photonics Technol. Lett. 15, 933-935 (2003).
[CrossRef]

Caucheteur, C.

C. Caucheteur and P. Mégret, “Demodulation technique for weakly tilted fiber Bragg grating refractometer,” IEEE Photonics Technol. Lett. 17, 2703-2705 (2005).
[CrossRef]

Chan, C.

Chehura, E.

E. Chehura, S. W. James, and R. P. Tatam, “Temperature and strain discrimination using a single tilted fibre Bragg grating,” Opt. Commun. 3, 1-4 (2007).

Chen, C.

Chen, X.

X. Chen, K. Zhou, L. Zhang, and I. Bennion, “In-fiber twist sensor based on a fiber Bragg grating with 81° tilted structure,” IEEE Photonics Technol. Lett. 18, 2596-2598 (2006).
[CrossRef]

X. Chen, K. Zhou, L. Zhang, and I. Bennion, “Optical chemsensor based on etched tilted Bragg grating structures in multimode fiber,” IEEE Photonics Technol. Lett. 17, 864-866 (2005).
[CrossRef]

K. Zhou, G. Simpson, X. Chen, L. Zhang, and I. Bennion, “High extinction ratio in-fiber polarizers based on 45° tilted fiber Bragg gratings,” Opt. Lett. 30, 1285-1287 (2005).
[CrossRef] [PubMed]

de Sterke, C. M.

Demokan, M. S.

Erdogan, T.

Feder, K.

K. Feder, P. Westbrook, J. Ging, and P. Reyes, “A compact, low resolution, wavelength monitor applied to Raman pump power monitoring,” in Proceedings of IEEE Conference on Optical Fiber Communications (IEEE, 2003), pp. 42-43.

Feder, K. S.

K. S. Feder, P. S. Westbrook, J. Ging, P. I. Reyes, and G. E. Carver, “In-fiber spectrometer using tilted fiber gratings,” IEEE Photonics Technol. Lett. 15, 933-935 (2003).
[CrossRef]

Froggatt, M.

Ging, J.

K. S. Feder, P. S. Westbrook, J. Ging, P. I. Reyes, and G. E. Carver, “In-fiber spectrometer using tilted fiber gratings,” IEEE Photonics Technol. Lett. 15, 933-935 (2003).
[CrossRef]

K. Feder, P. Westbrook, J. Ging, and P. Reyes, “A compact, low resolution, wavelength monitor applied to Raman pump power monitoring,” in Proceedings of IEEE Conference on Optical Fiber Communications (IEEE, 2003), pp. 42-43.

Grobnic, D.

Jafari, A.

James, S. W.

E. Chehura, S. W. James, and R. P. Tatam, “Temperature and strain discrimination using a single tilted fibre Bragg grating,” Opt. Commun. 3, 1-4 (2007).

Jin, W.

Johnson, D. C.

S. J. Mihailov, R. B. Walker, T. J. Stocki, and D. C. Johnson, “Fabrication of tilted fiber-grating polarization-dependent loss equalizer,” Electron. Lett. 37, 284-286 (2001).
[CrossRef]

Kashyap, R.

R. Kashyap, R. Wyatt, and R. J. Campbell, “Wideband gain flattened erbium fibre amplifier using a photosensitive fibre blazed grating,” Electron. Lett. 29, 154-156 (1993).
[CrossRef]

Krause, E.

J. Peupelmann, E. Krause, A. Bandemer, and C. Schaffer, “Fibre-polarimeter based on grating taps,” Electron. Lett. 38, 1248-1250 (2002).
[CrossRef]

Laronche, A.

Lee, K. S.

Li, Y.

Li, Y. F.

Lu, P.

Mégret, P.

C. Caucheteur and P. Mégret, “Demodulation technique for weakly tilted fiber Bragg grating refractometer,” IEEE Photonics Technol. Lett. 17, 2703-2705 (2005).
[CrossRef]

Mihailov, S. J.

R. B. Walker, S. J. Mihailov, P. Lu, and D. Grobnic, “Shaping the radiation field of tilted fiber Bragg gratings,” J. Opt. Soc. Am. B 22, 962-975 (2005).
[CrossRef]

S. J. Mihailov, R. B. Walker, T. J. Stocki, and D. C. Johnson, “Fabrication of tilted fiber-grating polarization-dependent loss equalizer,” Electron. Lett. 37, 284-286 (2001).
[CrossRef]

Parker, R.

Peupelmann, J.

J. Peupelmann, E. Krause, A. Bandemer, and C. Schaffer, “Fibre-polarimeter based on grating taps,” Electron. Lett. 38, 1248-1250 (2002).
[CrossRef]

Reyes, P.

K. Feder, P. Westbrook, J. Ging, and P. Reyes, “A compact, low resolution, wavelength monitor applied to Raman pump power monitoring,” in Proceedings of IEEE Conference on Optical Fiber Communications (IEEE, 2003), pp. 42-43.

Reyes, P. I.

Y. Li, S. Wielandy, P. I. Reyes, and P. S. Westbrook, “Scattering from nonuniform tilted fiber gratings,” Opt. Lett. 29, 1330-1332 (2004).
[CrossRef] [PubMed]

K. S. Feder, P. S. Westbrook, J. Ging, P. I. Reyes, and G. E. Carver, “In-fiber spectrometer using tilted fiber gratings,” IEEE Photonics Technol. Lett. 15, 933-935 (2003).
[CrossRef]

Riziotis, C.

Schaffer, C.

J. Peupelmann, E. Krause, A. Bandemer, and C. Schaffer, “Fibre-polarimeter based on grating taps,” Electron. Lett. 38, 1248-1250 (2002).
[CrossRef]

Simpson, G.

Sipe, J. E.

Stocki, T. J.

S. J. Mihailov, R. B. Walker, T. J. Stocki, and D. C. Johnson, “Fabrication of tilted fiber-grating polarization-dependent loss equalizer,” Electron. Lett. 37, 284-286 (2001).
[CrossRef]

Strasser, T. A.

P. S. Westbrook, T. A. Strasser, and T. Erdogan, “In-line polarimeter using blazed fiber gratings,” IEEE Photonics Technol. Lett. 12, 1352-1354 (2000).
[CrossRef]

Tatam, R. P.

E. Chehura, S. W. James, and R. P. Tatam, “Temperature and strain discrimination using a single tilted fibre Bragg grating,” Opt. Commun. 3, 1-4 (2007).

Thomson, D. J.

Walker, R. B.

R. B. Walker, S. J. Mihailov, P. Lu, and D. Grobnic, “Shaping the radiation field of tilted fiber Bragg gratings,” J. Opt. Soc. Am. B 22, 962-975 (2005).
[CrossRef]

S. J. Mihailov, R. B. Walker, T. J. Stocki, and D. C. Johnson, “Fabrication of tilted fiber-grating polarization-dependent loss equalizer,” Electron. Lett. 37, 284-286 (2001).
[CrossRef]

Westbrook, P.

K. Feder, P. Westbrook, J. Ging, and P. Reyes, “A compact, low resolution, wavelength monitor applied to Raman pump power monitoring,” in Proceedings of IEEE Conference on Optical Fiber Communications (IEEE, 2003), pp. 42-43.

Westbrook, P. S.

Y. Li, S. Wielandy, P. I. Reyes, and P. S. Westbrook, “Scattering from nonuniform tilted fiber gratings,” Opt. Lett. 29, 1330-1332 (2004).
[CrossRef] [PubMed]

K. S. Feder, P. S. Westbrook, J. Ging, P. I. Reyes, and G. E. Carver, “In-fiber spectrometer using tilted fiber gratings,” IEEE Photonics Technol. Lett. 15, 933-935 (2003).
[CrossRef]

P. S. Westbrook, T. A. Strasser, and T. Erdogan, “In-line polarimeter using blazed fiber gratings,” IEEE Photonics Technol. Lett. 12, 1352-1354 (2000).
[CrossRef]

Wielandy, S.

Wyatt, R.

R. Kashyap, R. Wyatt, and R. J. Campbell, “Wideband gain flattened erbium fibre amplifier using a photosensitive fibre blazed grating,” Electron. Lett. 29, 154-156 (1993).
[CrossRef]

Yang, X.

Zervas, M. N.

Zhang, L.

X. Chen, K. Zhou, L. Zhang, and I. Bennion, “In-fiber twist sensor based on a fiber Bragg grating with 81° tilted structure,” IEEE Photonics Technol. Lett. 18, 2596-2598 (2006).
[CrossRef]

X. Chen, K. Zhou, L. Zhang, and I. Bennion, “Optical chemsensor based on etched tilted Bragg grating structures in multimode fiber,” IEEE Photonics Technol. Lett. 17, 864-866 (2005).
[CrossRef]

K. Zhou, G. Simpson, X. Chen, L. Zhang, and I. Bennion, “High extinction ratio in-fiber polarizers based on 45° tilted fiber Bragg gratings,” Opt. Lett. 30, 1285-1287 (2005).
[CrossRef] [PubMed]

Zhao, C.

Zhou, K.

X. Chen, K. Zhou, L. Zhang, and I. Bennion, “In-fiber twist sensor based on a fiber Bragg grating with 81° tilted structure,” IEEE Photonics Technol. Lett. 18, 2596-2598 (2006).
[CrossRef]

X. Chen, K. Zhou, L. Zhang, and I. Bennion, “Optical chemsensor based on etched tilted Bragg grating structures in multimode fiber,” IEEE Photonics Technol. Lett. 17, 864-866 (2005).
[CrossRef]

K. Zhou, G. Simpson, X. Chen, L. Zhang, and I. Bennion, “High extinction ratio in-fiber polarizers based on 45° tilted fiber Bragg gratings,” Opt. Lett. 30, 1285-1287 (2005).
[CrossRef] [PubMed]

Appl. Opt. (2)

Electron. Lett. (3)

R. Kashyap, R. Wyatt, and R. J. Campbell, “Wideband gain flattened erbium fibre amplifier using a photosensitive fibre blazed grating,” Electron. Lett. 29, 154-156 (1993).
[CrossRef]

J. Peupelmann, E. Krause, A. Bandemer, and C. Schaffer, “Fibre-polarimeter based on grating taps,” Electron. Lett. 38, 1248-1250 (2002).
[CrossRef]

S. J. Mihailov, R. B. Walker, T. J. Stocki, and D. C. Johnson, “Fabrication of tilted fiber-grating polarization-dependent loss equalizer,” Electron. Lett. 37, 284-286 (2001).
[CrossRef]

IEEE Photonics Technol. Lett. (5)

K. S. Feder, P. S. Westbrook, J. Ging, P. I. Reyes, and G. E. Carver, “In-fiber spectrometer using tilted fiber gratings,” IEEE Photonics Technol. Lett. 15, 933-935 (2003).
[CrossRef]

P. S. Westbrook, T. A. Strasser, and T. Erdogan, “In-line polarimeter using blazed fiber gratings,” IEEE Photonics Technol. Lett. 12, 1352-1354 (2000).
[CrossRef]

C. Caucheteur and P. Mégret, “Demodulation technique for weakly tilted fiber Bragg grating refractometer,” IEEE Photonics Technol. Lett. 17, 2703-2705 (2005).
[CrossRef]

X. Chen, K. Zhou, L. Zhang, and I. Bennion, “In-fiber twist sensor based on a fiber Bragg grating with 81° tilted structure,” IEEE Photonics Technol. Lett. 18, 2596-2598 (2006).
[CrossRef]

X. Chen, K. Zhou, L. Zhang, and I. Bennion, “Optical chemsensor based on etched tilted Bragg grating structures in multimode fiber,” IEEE Photonics Technol. Lett. 17, 864-866 (2005).
[CrossRef]

J. Lightwave Technol. (4)

J. Opt. Soc. Am. A (2)

J. Opt. Soc. Am. B (1)

Opt. Commun. (1)

E. Chehura, S. W. James, and R. P. Tatam, “Temperature and strain discrimination using a single tilted fibre Bragg grating,” Opt. Commun. 3, 1-4 (2007).

Opt. Lett. (3)

Other (1)

K. Feder, P. Westbrook, J. Ging, and P. Reyes, “A compact, low resolution, wavelength monitor applied to Raman pump power monitoring,” in Proceedings of IEEE Conference on Optical Fiber Communications (IEEE, 2003), pp. 42-43.

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Figures (9)

Fig. 1
Fig. 1

Configuration of tilted fiber Bragg gratings.

Fig. 2
Fig. 2

Phase-matching conditions in TFBGs.

Fig. 3
Fig. 3

Comparison of calculated transmission spectra from the complete CMT approach (solid curve) and the simplified function expression of this paper for gratings (dashed curve) with tilt angles of 1°, 5°, and 30°.

Fig. 4
Fig. 4

Comparison of calculated transmission spectra from the simplified function expression of this paper and the VCM method for a 55°-tilted grating.

Fig. 5
Fig. 5

Comparison of calculated transmission spectra for 55°-tilted gratings with different cladding refractive indices.

Fig. 6
Fig. 6

Calculated transmission spectra of TFBGs with different tilt angles for both s-polarized and p-polarized light.

Fig. 7
Fig. 7

(a) Minimum transmission versus tilt angle for both s and p polarization. (b) Minimum transmission wavelength and the wavelength difference between the two polarization states versus tilt angle.

Fig. 8
Fig. 8

Polarization extinction as the ratio between the minimum transmission of p-polarized light and s-polarized light versus wavelength for gratings with seven different tilt angles.

Fig. 9
Fig. 9

Polarization extinction versus modulation amplitude for five gratings with different angles.

Tables (2)

Tables Icon

Table 1 Transmission Ratio from Reflective and Transmissive Formulas for a 45°-Tilted Grating

Tables Icon

Table 2 Modulation Amplitude, Loss of s-Polarized Light, and Loss of p-Polarized Light for Gratings with Tilt Angle of 45°, 50°, and 55° To Achieve 20 dB Polarization Extinction

Equations (20)

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E t ( x , y , z , t ) = j [ A j ( z ) exp ( i β j z ) + B j ( z ) exp ( i β j z ) ] e j t ( x , y ) exp ( i ω t ) ,
β ¯ 1 d A μ d z = i σ { ν g μ ν + + A ν ( z ) exp [ i ( β ν β μ ) z ] + ν g μ ν + B ν ( z ) exp [ i ( β ν + β μ ) z ] } + i κ exp ( i K z ) { ν f μ ν + + A ν ( z ) exp [ i ( β ν β μ ) z ] + ν f μ ν + B ν ( z ) exp [ i ( β ν + β μ ) z ] } + i κ exp ( i K z ) { ν h μ ν + + A ν ( z ) exp [ i ( β ν β μ ) z ] + ν h μ ν + B ν ( z ) exp [ i ( β ν + β μ ) z ] }
β ¯ 1 d B μ d z = i σ { ν g μ ν + A ν ( z ) exp [ i ( β ν + β μ ) z ] + ν g μ ν B ν ( z ) exp [ i ( β ν + β μ ) z ] } i κ exp ( i K z ) { ν f μ ν + A ν ( z ) exp [ i ( β ν + β μ ) z ] + ν f μ ν B ν ( z ) exp [ i ( β ν + β μ ) z ] } i κ exp ( i K z ) { ν h μ ν + A ν ( z ) exp [ i ( β ν + β μ ) z ] + ν h μ ν B ν ( z ) exp [ i ( β ν + β μ ) z ] } .
β ¯ 1 d A 01 d z = i σ g 01 , 01 + , + A 01 ( z ) + i κ ( z ) f 01 , 01 + , B 01 ( z ) exp [ i ( K 2 β 01 ) z ] + i κ { ν f 01 , ν + , B ν ( z ) exp [ i ( K β 01 β ν ) z ] } .
β ¯ 1 d A 01 d z = i κ { ν f 01 , ν + , B ν ( z ) exp [ i ( K β 01 β ν ) z ] }
β ¯ 1 d B ν d z = i κ h ν , 01 , + A 01 ( z ) exp [ i ( K β 01 β ν ) z ] .
β ¯ 1 d u d z = i δ u ( z ) + i κ ν f 01 , ν + , B ν ( z ) exp [ i ( K 2 β ν ) z ] ,
β ¯ 1 d B ν d z = i κ h ν , 01 , + u ( z ) exp [ i ( K 2 β ν ) z ] ,
u ( z ) u ( z 0 ) exp [ i β ¯ δ ( z z 0 ) ] .
B ν ( z ) = β ¯ κ h ν , 01 , + u ( z ) exp [ i ( K 2 β ν ) z ] β ν ( K β 01 ) .
β ¯ 1 d u d z = i κ ν f 01 , ν + , B ν ( z ) exp [ i ( K 2 β ν ) z ]
d u d z = i κ 2 Ω u ( z ) ,
Ω = i β ¯ ν f 01 , ν + , 2 β ν ( K β 01 ) .
Ω = ν ( β ν π ρ β ¯ f 01 , ν + , 2 ) β ν = K β 01 = ν Ω ν , ν = 0 , 1 , 2 , .
β ¯ 1 d A 01 d z = i σ g 01 , 01 + , + A 01 ( z ) + i κ ( z ) f 01 , 01 + , B 01 ( z ) exp [ i ( K 2 β 01 ) z ] + i κ { ν f 01 , ν + , + B ν ( z ) exp [ i ( K + β ν β 01 ) z ] } .
β ¯ 1 d A 01 d z = i κ { ν f 01 , ν + , + B ν ( z ) exp [ i ( K + β ν β 01 ) z ] } .
β ¯ 1 d B ν d z = i κ h ν , 01 + , + A 01 ( z ) exp [ i ( K + β ν β 01 ) z ] .
u ( z ) = exp ( i κ 2 Ω z ) ,
Ω = ν Ω ν β ν ,
{ β ν = K β 01 , θ 45 ° ( reflective gratings ) β ν = β 01 K , θ 45 ° ( transmissive gratings ) } .

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