Abstract

In this paper the analytic models (AMs) of the spectral responses of fiber-grating-based interferometers are derived from the Fourier mode coupling (FMC) theory proposed recently. The interferometers include Fabry-Perot cavity, Mach-Zehnder and Michelson interferometers, which are constructed by uniform fiber Bragg gratings and long-period fiber gratings, and also by Gaussian-apodized ones. The calculated spectra based on the analytic models are achieved, and compared with the measured cases and those on the transfer matrix (TM) method. The calculations and comparisons have confirmed that the AM-based spectrum is in excellent agreement with the TM-based one and the measured case, of which the efficiency is improved up to ~2990 times that of the TM method for non-uniform-grating-based in-fiber interferometers.

© 2012 OSA

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References

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  1. Y. Bai, Q. Liu, K. P. Lor, and K. S. Chiang, “Widely tunable long-period waveguide grating couplers,” Opt. Express 14(26), 12644–12654 (2006), http://www.opticsinfobase.org/abstract.cfm?URI=oe-14-26-12644 .
    [Crossref] [PubMed]
  2. S. K. Liaw, L. Dou, and A. S. Xu, “Fiber-bragg-grating-based dispersion-compensated and gain-flattened raman fiber Amplifier,” Opt. Express 15(19), 12356–12361 (2007), http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-15-19-12356 .
    [Crossref] [PubMed]
  3. K. P. Koo, M. LeBlanc, T. E. Tsai, and S. T. Vohra, “Fiber-chirped grating Fabry-Perot sensor with multiple-wavelength-addressable free-spectral ranges,” IEEE Photon. Technol. Lett. 10(7), 1006–1008 (1998).
    [Crossref]
  4. X. J. Gu, “Wavelength-division multiplexing isolation fiber filter and light source using cascaded long-period fiber gratings,” Opt. Lett. 23(7), 509–510 (1998), http://www.opticsinfobase.org/ol/abstract.cfm?URI=ol-23-7-509 .
    [Crossref] [PubMed]
  5. P. L. Swart, “Long-period grating Michelson refractometric sensor,” Meas. Sci. Technol. 15(8), 1576–1580 (2004).
    [Crossref]
  6. G. D. Marshall, R. J. Williams, N. Jovanovic, M. J. Steel, and M. J. Withford, “Point-by-point written fiber-Bragg gratings and their application in complex grating designs,” Opt. Express 18(19), 19844–19859 (2010), http://www.opticsinfobase.org/abstract.cfm?URI=oe-18-19-19844 .
    [Crossref] [PubMed]
  7. T. Erdogan, “Fiber grating spectra,” J. Lightwave Technol. 15(8), 1277–1294 (1997).
    [Crossref]
  8. M. Yamada and K. Sakuda, “Analysis of almost-periodic distributed feedback slab waveguides via a fundamental matrix approach,” Appl. Opt. 26(16), 3474–3478 (1987).
    [Crossref] [PubMed]
  9. L. A. Weller-Brophy and D. G. Hall, “Analysis of waveguide gratings: application of Rouard’s method,” J. Opt. Soc. Am. A 2(6), 863–871 (1985).
    [Crossref]
  10. J. J. Liau, N. H. Sun, S. C. Lin, R. Y. Ro, J. S. Chiang, C. L. Pan, and H. W. Chang, “A new look at numerical analysis of uniform fiber Bragg gratings using coupled mode theory,” Prog. Electromagn. Res. 93, 385–401 (2009).
    [Crossref]
  11. H. Kogelnik, “Filter response of nonuniform almost-periodic structure,” Bell Syst. Tech. J. 55, 109–126 (1976).
  12. E. Mazzetto, C. G. Someda, J. A. Acebron, and R. Spigler, “The fractional Fourier transform in the analysis and synthesis of fiber Bragg gratings,” Opt. Quantum Electron. 37(8), 755–787 (2005).
    [Crossref]
  13. H. V. Baghdasaryan and T. M. Knyazyan, “Modeling of linearly chirped fiber Bragg gratings by the method of single expression,” Opt. Quantum Electron. 34(5-6), 481–492 (2002).
    [Crossref]
  14. E. Peral and J. Capmany, “Generalized Bloch wave analysis for fiber and waveguide gratings,” J. Lightwave Technol. 15(8), 1295–1302 (1997).
    [Crossref]
  15. A. Bouzid and M. A. G. Abushagur, “Scattering analysis of slanted fiber gratings,” Appl. Opt. 36(3), 558–562 (1997).
    [Crossref] [PubMed]
  16. L. Poladian, “Graphical and WKB analysis of nonuniform Bragg gratings,” Phys. Rev. E Stat. Phys. Plasmas Fluids Relat. Interdiscip. Topics 48(6), 4758–4767 (1993).
    [Crossref] [PubMed]
  17. X. K. Zeng and Y. J. Rao, “Theory of Fourier mode coupling for fiber Bragg gratings,” Acta Phys. Sin. 59, 8597–8606 (2010).
  18. X. K. Zeng and Y. J. Rao, “Theory of Fourier mode coupling for long-period fiber gratings,” Acta Phys. Sin. 59, 8607–8614 (2010).
  19. X. K. Zeng, “Application of Fourier mode coupling theory to real-time analyses of nonuniform Bragg gratings,” IEEE Photon. Technol. Lett. 23(13), 854–856 (2011).
    [Crossref]
  20. X. K. Zeng and K. Liang, “Analytic solutions for spectral properties of superstructure, Gaussian-apodized and phase shift gratings with short- or long-period,” Opt. Express 19(23), 22797–22808 (2011), http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-19-23-22797 .
    [Crossref] [PubMed]
  21. J. A. Armstrong, N. Bloembergen, J. Ducuing, and P. S. Pershan, “Interactions between light waves in a nonlinear dielectric,” Phys. Rev. 127(6), 1918–1939 (1962).
    [Crossref]

2011 (2)

2010 (3)

G. D. Marshall, R. J. Williams, N. Jovanovic, M. J. Steel, and M. J. Withford, “Point-by-point written fiber-Bragg gratings and their application in complex grating designs,” Opt. Express 18(19), 19844–19859 (2010), http://www.opticsinfobase.org/abstract.cfm?URI=oe-18-19-19844 .
[Crossref] [PubMed]

X. K. Zeng and Y. J. Rao, “Theory of Fourier mode coupling for fiber Bragg gratings,” Acta Phys. Sin. 59, 8597–8606 (2010).

X. K. Zeng and Y. J. Rao, “Theory of Fourier mode coupling for long-period fiber gratings,” Acta Phys. Sin. 59, 8607–8614 (2010).

2009 (1)

J. J. Liau, N. H. Sun, S. C. Lin, R. Y. Ro, J. S. Chiang, C. L. Pan, and H. W. Chang, “A new look at numerical analysis of uniform fiber Bragg gratings using coupled mode theory,” Prog. Electromagn. Res. 93, 385–401 (2009).
[Crossref]

2007 (1)

2006 (1)

2005 (1)

E. Mazzetto, C. G. Someda, J. A. Acebron, and R. Spigler, “The fractional Fourier transform in the analysis and synthesis of fiber Bragg gratings,” Opt. Quantum Electron. 37(8), 755–787 (2005).
[Crossref]

2004 (1)

P. L. Swart, “Long-period grating Michelson refractometric sensor,” Meas. Sci. Technol. 15(8), 1576–1580 (2004).
[Crossref]

2002 (1)

H. V. Baghdasaryan and T. M. Knyazyan, “Modeling of linearly chirped fiber Bragg gratings by the method of single expression,” Opt. Quantum Electron. 34(5-6), 481–492 (2002).
[Crossref]

1998 (2)

K. P. Koo, M. LeBlanc, T. E. Tsai, and S. T. Vohra, “Fiber-chirped grating Fabry-Perot sensor with multiple-wavelength-addressable free-spectral ranges,” IEEE Photon. Technol. Lett. 10(7), 1006–1008 (1998).
[Crossref]

X. J. Gu, “Wavelength-division multiplexing isolation fiber filter and light source using cascaded long-period fiber gratings,” Opt. Lett. 23(7), 509–510 (1998), http://www.opticsinfobase.org/ol/abstract.cfm?URI=ol-23-7-509 .
[Crossref] [PubMed]

1997 (3)

T. Erdogan, “Fiber grating spectra,” J. Lightwave Technol. 15(8), 1277–1294 (1997).
[Crossref]

E. Peral and J. Capmany, “Generalized Bloch wave analysis for fiber and waveguide gratings,” J. Lightwave Technol. 15(8), 1295–1302 (1997).
[Crossref]

A. Bouzid and M. A. G. Abushagur, “Scattering analysis of slanted fiber gratings,” Appl. Opt. 36(3), 558–562 (1997).
[Crossref] [PubMed]

1993 (1)

L. Poladian, “Graphical and WKB analysis of nonuniform Bragg gratings,” Phys. Rev. E Stat. Phys. Plasmas Fluids Relat. Interdiscip. Topics 48(6), 4758–4767 (1993).
[Crossref] [PubMed]

1987 (1)

1985 (1)

1976 (1)

H. Kogelnik, “Filter response of nonuniform almost-periodic structure,” Bell Syst. Tech. J. 55, 109–126 (1976).

1962 (1)

J. A. Armstrong, N. Bloembergen, J. Ducuing, and P. S. Pershan, “Interactions between light waves in a nonlinear dielectric,” Phys. Rev. 127(6), 1918–1939 (1962).
[Crossref]

Abushagur, M. A. G.

Acebron, J. A.

E. Mazzetto, C. G. Someda, J. A. Acebron, and R. Spigler, “The fractional Fourier transform in the analysis and synthesis of fiber Bragg gratings,” Opt. Quantum Electron. 37(8), 755–787 (2005).
[Crossref]

Armstrong, J. A.

J. A. Armstrong, N. Bloembergen, J. Ducuing, and P. S. Pershan, “Interactions between light waves in a nonlinear dielectric,” Phys. Rev. 127(6), 1918–1939 (1962).
[Crossref]

Baghdasaryan, H. V.

H. V. Baghdasaryan and T. M. Knyazyan, “Modeling of linearly chirped fiber Bragg gratings by the method of single expression,” Opt. Quantum Electron. 34(5-6), 481–492 (2002).
[Crossref]

Bai, Y.

Bloembergen, N.

J. A. Armstrong, N. Bloembergen, J. Ducuing, and P. S. Pershan, “Interactions between light waves in a nonlinear dielectric,” Phys. Rev. 127(6), 1918–1939 (1962).
[Crossref]

Bouzid, A.

Capmany, J.

E. Peral and J. Capmany, “Generalized Bloch wave analysis for fiber and waveguide gratings,” J. Lightwave Technol. 15(8), 1295–1302 (1997).
[Crossref]

Chang, H. W.

J. J. Liau, N. H. Sun, S. C. Lin, R. Y. Ro, J. S. Chiang, C. L. Pan, and H. W. Chang, “A new look at numerical analysis of uniform fiber Bragg gratings using coupled mode theory,” Prog. Electromagn. Res. 93, 385–401 (2009).
[Crossref]

Chiang, J. S.

J. J. Liau, N. H. Sun, S. C. Lin, R. Y. Ro, J. S. Chiang, C. L. Pan, and H. W. Chang, “A new look at numerical analysis of uniform fiber Bragg gratings using coupled mode theory,” Prog. Electromagn. Res. 93, 385–401 (2009).
[Crossref]

Chiang, K. S.

Dou, L.

Ducuing, J.

J. A. Armstrong, N. Bloembergen, J. Ducuing, and P. S. Pershan, “Interactions between light waves in a nonlinear dielectric,” Phys. Rev. 127(6), 1918–1939 (1962).
[Crossref]

Erdogan, T.

T. Erdogan, “Fiber grating spectra,” J. Lightwave Technol. 15(8), 1277–1294 (1997).
[Crossref]

Gu, X. J.

Hall, D. G.

Jovanovic, N.

Knyazyan, T. M.

H. V. Baghdasaryan and T. M. Knyazyan, “Modeling of linearly chirped fiber Bragg gratings by the method of single expression,” Opt. Quantum Electron. 34(5-6), 481–492 (2002).
[Crossref]

Kogelnik, H.

H. Kogelnik, “Filter response of nonuniform almost-periodic structure,” Bell Syst. Tech. J. 55, 109–126 (1976).

Koo, K. P.

K. P. Koo, M. LeBlanc, T. E. Tsai, and S. T. Vohra, “Fiber-chirped grating Fabry-Perot sensor with multiple-wavelength-addressable free-spectral ranges,” IEEE Photon. Technol. Lett. 10(7), 1006–1008 (1998).
[Crossref]

LeBlanc, M.

K. P. Koo, M. LeBlanc, T. E. Tsai, and S. T. Vohra, “Fiber-chirped grating Fabry-Perot sensor with multiple-wavelength-addressable free-spectral ranges,” IEEE Photon. Technol. Lett. 10(7), 1006–1008 (1998).
[Crossref]

Liang, K.

Liau, J. J.

J. J. Liau, N. H. Sun, S. C. Lin, R. Y. Ro, J. S. Chiang, C. L. Pan, and H. W. Chang, “A new look at numerical analysis of uniform fiber Bragg gratings using coupled mode theory,” Prog. Electromagn. Res. 93, 385–401 (2009).
[Crossref]

Liaw, S. K.

Lin, S. C.

J. J. Liau, N. H. Sun, S. C. Lin, R. Y. Ro, J. S. Chiang, C. L. Pan, and H. W. Chang, “A new look at numerical analysis of uniform fiber Bragg gratings using coupled mode theory,” Prog. Electromagn. Res. 93, 385–401 (2009).
[Crossref]

Liu, Q.

Lor, K. P.

Marshall, G. D.

Mazzetto, E.

E. Mazzetto, C. G. Someda, J. A. Acebron, and R. Spigler, “The fractional Fourier transform in the analysis and synthesis of fiber Bragg gratings,” Opt. Quantum Electron. 37(8), 755–787 (2005).
[Crossref]

Pan, C. L.

J. J. Liau, N. H. Sun, S. C. Lin, R. Y. Ro, J. S. Chiang, C. L. Pan, and H. W. Chang, “A new look at numerical analysis of uniform fiber Bragg gratings using coupled mode theory,” Prog. Electromagn. Res. 93, 385–401 (2009).
[Crossref]

Peral, E.

E. Peral and J. Capmany, “Generalized Bloch wave analysis for fiber and waveguide gratings,” J. Lightwave Technol. 15(8), 1295–1302 (1997).
[Crossref]

Pershan, P. S.

J. A. Armstrong, N. Bloembergen, J. Ducuing, and P. S. Pershan, “Interactions between light waves in a nonlinear dielectric,” Phys. Rev. 127(6), 1918–1939 (1962).
[Crossref]

Poladian, L.

L. Poladian, “Graphical and WKB analysis of nonuniform Bragg gratings,” Phys. Rev. E Stat. Phys. Plasmas Fluids Relat. Interdiscip. Topics 48(6), 4758–4767 (1993).
[Crossref] [PubMed]

Rao, Y. J.

X. K. Zeng and Y. J. Rao, “Theory of Fourier mode coupling for fiber Bragg gratings,” Acta Phys. Sin. 59, 8597–8606 (2010).

X. K. Zeng and Y. J. Rao, “Theory of Fourier mode coupling for long-period fiber gratings,” Acta Phys. Sin. 59, 8607–8614 (2010).

Ro, R. Y.

J. J. Liau, N. H. Sun, S. C. Lin, R. Y. Ro, J. S. Chiang, C. L. Pan, and H. W. Chang, “A new look at numerical analysis of uniform fiber Bragg gratings using coupled mode theory,” Prog. Electromagn. Res. 93, 385–401 (2009).
[Crossref]

Sakuda, K.

Someda, C. G.

E. Mazzetto, C. G. Someda, J. A. Acebron, and R. Spigler, “The fractional Fourier transform in the analysis and synthesis of fiber Bragg gratings,” Opt. Quantum Electron. 37(8), 755–787 (2005).
[Crossref]

Spigler, R.

E. Mazzetto, C. G. Someda, J. A. Acebron, and R. Spigler, “The fractional Fourier transform in the analysis and synthesis of fiber Bragg gratings,” Opt. Quantum Electron. 37(8), 755–787 (2005).
[Crossref]

Steel, M. J.

Sun, N. H.

J. J. Liau, N. H. Sun, S. C. Lin, R. Y. Ro, J. S. Chiang, C. L. Pan, and H. W. Chang, “A new look at numerical analysis of uniform fiber Bragg gratings using coupled mode theory,” Prog. Electromagn. Res. 93, 385–401 (2009).
[Crossref]

Swart, P. L.

P. L. Swart, “Long-period grating Michelson refractometric sensor,” Meas. Sci. Technol. 15(8), 1576–1580 (2004).
[Crossref]

Tsai, T. E.

K. P. Koo, M. LeBlanc, T. E. Tsai, and S. T. Vohra, “Fiber-chirped grating Fabry-Perot sensor with multiple-wavelength-addressable free-spectral ranges,” IEEE Photon. Technol. Lett. 10(7), 1006–1008 (1998).
[Crossref]

Vohra, S. T.

K. P. Koo, M. LeBlanc, T. E. Tsai, and S. T. Vohra, “Fiber-chirped grating Fabry-Perot sensor with multiple-wavelength-addressable free-spectral ranges,” IEEE Photon. Technol. Lett. 10(7), 1006–1008 (1998).
[Crossref]

Weller-Brophy, L. A.

Williams, R. J.

Withford, M. J.

Xu, A. S.

Yamada, M.

Zeng, X. K.

X. K. Zeng, “Application of Fourier mode coupling theory to real-time analyses of nonuniform Bragg gratings,” IEEE Photon. Technol. Lett. 23(13), 854–856 (2011).
[Crossref]

X. K. Zeng and K. Liang, “Analytic solutions for spectral properties of superstructure, Gaussian-apodized and phase shift gratings with short- or long-period,” Opt. Express 19(23), 22797–22808 (2011), http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-19-23-22797 .
[Crossref] [PubMed]

X. K. Zeng and Y. J. Rao, “Theory of Fourier mode coupling for long-period fiber gratings,” Acta Phys. Sin. 59, 8607–8614 (2010).

X. K. Zeng and Y. J. Rao, “Theory of Fourier mode coupling for fiber Bragg gratings,” Acta Phys. Sin. 59, 8597–8606 (2010).

Acta Phys. Sin. (2)

X. K. Zeng and Y. J. Rao, “Theory of Fourier mode coupling for fiber Bragg gratings,” Acta Phys. Sin. 59, 8597–8606 (2010).

X. K. Zeng and Y. J. Rao, “Theory of Fourier mode coupling for long-period fiber gratings,” Acta Phys. Sin. 59, 8607–8614 (2010).

Appl. Opt. (2)

Bell Syst. Tech. J. (1)

H. Kogelnik, “Filter response of nonuniform almost-periodic structure,” Bell Syst. Tech. J. 55, 109–126 (1976).

IEEE Photon. Technol. Lett. (2)

K. P. Koo, M. LeBlanc, T. E. Tsai, and S. T. Vohra, “Fiber-chirped grating Fabry-Perot sensor with multiple-wavelength-addressable free-spectral ranges,” IEEE Photon. Technol. Lett. 10(7), 1006–1008 (1998).
[Crossref]

X. K. Zeng, “Application of Fourier mode coupling theory to real-time analyses of nonuniform Bragg gratings,” IEEE Photon. Technol. Lett. 23(13), 854–856 (2011).
[Crossref]

J. Lightwave Technol. (2)

T. Erdogan, “Fiber grating spectra,” J. Lightwave Technol. 15(8), 1277–1294 (1997).
[Crossref]

E. Peral and J. Capmany, “Generalized Bloch wave analysis for fiber and waveguide gratings,” J. Lightwave Technol. 15(8), 1295–1302 (1997).
[Crossref]

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

Meas. Sci. Technol. (1)

P. L. Swart, “Long-period grating Michelson refractometric sensor,” Meas. Sci. Technol. 15(8), 1576–1580 (2004).
[Crossref]

Opt. Express (4)

Opt. Lett. (1)

Opt. Quantum Electron. (2)

E. Mazzetto, C. G. Someda, J. A. Acebron, and R. Spigler, “The fractional Fourier transform in the analysis and synthesis of fiber Bragg gratings,” Opt. Quantum Electron. 37(8), 755–787 (2005).
[Crossref]

H. V. Baghdasaryan and T. M. Knyazyan, “Modeling of linearly chirped fiber Bragg gratings by the method of single expression,” Opt. Quantum Electron. 34(5-6), 481–492 (2002).
[Crossref]

Phys. Rev. (1)

J. A. Armstrong, N. Bloembergen, J. Ducuing, and P. S. Pershan, “Interactions between light waves in a nonlinear dielectric,” Phys. Rev. 127(6), 1918–1939 (1962).
[Crossref]

Phys. Rev. E Stat. Phys. Plasmas Fluids Relat. Interdiscip. Topics (1)

L. Poladian, “Graphical and WKB analysis of nonuniform Bragg gratings,” Phys. Rev. E Stat. Phys. Plasmas Fluids Relat. Interdiscip. Topics 48(6), 4758–4767 (1993).
[Crossref] [PubMed]

Prog. Electromagn. Res. (1)

J. J. Liau, N. H. Sun, S. C. Lin, R. Y. Ro, J. S. Chiang, C. L. Pan, and H. W. Chang, “A new look at numerical analysis of uniform fiber Bragg gratings using coupled mode theory,” Prog. Electromagn. Res. 93, 385–401 (2009).
[Crossref]

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Figures (4)

Fig. 1
Fig. 1

Schematic diagram of the mode couplings in the fiber-grating-based FP cavity (a), MZ interferometer (b) and Michelson interferometer (c).

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Fig. 2
Fig. 2

Spectral responses of a FP cavity formed by two uniform FBGs. (a) and (b) are the calculated reflectivities and transmissions, respectively, according to Eq. (12) (solid lines) and the TM method (dotted lines), and (c) is the measured transmission.

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Fig. 3
Fig. 3

Bar-transmissions of a MZ interferometer formed by two uniform LPFGs. (a) is the calculated bar-transmissions based on Eq. (13) (solid line) and the TM method (dotted line), and (b) is the measured one.

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Fig. 4
Fig. 4

Calculated spectra of the interferometers formed by GA-gratings, according to Eqs. (15) and (16) (solid lines), and the TM method (dotted lines). (a) is the reflectivities of a FP cavity constructed by two GA-FBGs, and (b) is the bar-transmissions of a MZ or Michelson interferometer by GA-LPFGs.

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Equations (16)

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

d B s (z) dz =±j ∑ m k s B m (z)Δn(z) e −j( β m ± β s )z ,
k s = ε 0 ω n 0 2 ∬ A∞ E m ( r,ϕ ) E s * ( r,ϕ ) dA,
Δn(z)=Δ n 0 ( z+ P+L 2 )+Δ n 0 ( z− P+L 2 ),
∫ B s ( z 0 ) B s ( z 1 ) d B s (z) B m (z) =±2j k s cos[ π v s ( P+L ) ] ∫ −L/2 L/2 Δ n 0 (z) e −j2π ν s z dz ,
B m 2 (z)= B m 2 ( z 1 )+ B s 2 (z),
B m 2 (z)= B m 2 ( z 0 )− B s 2 (z).
2cos[ π v s ( P+L ) ] ∫ −L/2 L/2 Δ n 0 (z) e −j2π v s z dz = γ s ( v s )+j η s ( v s ),
R= sinh 2 [ k s η s ( v B ) ]+ sin 2 [ k s γ s ( v B ) ] cosh 2 [ k s η s ( v B ) ]+ sin 2 [ k s γ s ( v B ) ] ,
T= cos 2 [ k s η s ( ν L ) ]− sinh 2 [ k s γ s ( ν L ) ],
Δ n 0u (z)= δ n [ 1+sin( 2π Λ z ) ],− L 2 ≤z≤ L 2 ,
η su = L δ n 2 cos[ π v s ( P+L ) ]sinc[ πL( v s − 1 Λ ) ],
R Fu = tanh 2 { k s L δ n 2 cos[ π v B ( P+L ) ]sinc( πL σ B ) },
T Mu = cos 2 { k s L δ n 2 cos[ π v L ( P+L ) ]sinc( πL σ L ) },
Δ n 0G (z)= δ n [ 1+exp( −α z 2 L 2 )sin( 2π Λ z ) ],− L 2 ≤z≤ L 2 ,
R FG = tanh 2 { k s L δ n 2α cos[ π v B ( P+L ) ]{ πα e − π 2 L 2 σ B 2 α +2 e −α 4 [ 1−cos( πL σ B ) ] } },
T MG = cos 2 { k s L δ n 2α cos[ π v L ( P+L ) ]{ πα e − π 2 L 2 σ L 2 α +2 e −α 4 [ 1−cos( πL σ L ) ] } }.

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