Abstract

The analytic solutions (AS) for the spectral properties of short- and long-period waveguide gratings with the effects of discrete phase shift (PS), Gaussian-apodization (GA) and superstructure are presented in this paper, which are derived from the Fourier mode coupling (FMC) theory proposed recently. The spectral properties include the reflectivity of short-period gratings, and the transmission of long-period gratings. The calculated spectra based on the analytic solutions are achieved and compared with measured cases and that on the transfer matrix (TM) method, in the case of changing grating parameters. The AS-based calculation requires the average time of several milliseconds at common PC, and the AS-based efficiency is improved up to ~6700 times the TM-based one. The comparisons have confirmed that the FMC-based analytic solutions are suitable for the real-time and accurate analyses of some non-uniform waveguide gratings.

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References

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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
  23. B. O. Guan, H. Y. Tam, X. M. Tao, and X. Y. Dong, “Simultaneous Strain and Temperature Measurement Using a Superstructure Fiber Bragg Grating,” IEEE Photon. Technol. Lett. 12(6), 675–677 (2000).
    [CrossRef]

2010

2009

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. Electromag. Res., PIER 93, 385–401 (2009).
[CrossRef]

Q. Ye, R. Huang, Q. Xu, H. Cai, R. Qu, and Z. Fang, “Numerical investigation of ultrashort complex pulse generation based on pulse shaping using a superstructure fiber Bragg grating,” J. Lightwave Technol. 27(13), 2449–2456 (2009), http://www.opticsinfobase.org/jlt/abstract.cfm?URI=jlt-27-13-2449 .
[CrossRef]

2008

C. Wang and J. Yao, “Photonic generation of chirped microwave pulses using superimposed chirped fiber Bragg gratings,” IEEE Photon. Technol. Lett. 20(11), 882–884 (2008).
[CrossRef]

2006

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]

R. J. Espejo, M. Svalgaard, and S. D. Dyer, “Characterizing fiber Bragg grating index profiles to improve the writing process,” IEEE Photon. Technol. Lett. 18(21), 2242–2244 (2006).
[CrossRef]

2005

X. Chen, J. Yao, F. Zeng, and Z. Deng, “Single-longitudinal-mode fiber ring laser employing an equivalent phase-shifted fiber Bragg grating,” IEEE Photon. Technol. Lett. 17(7), 1390–1392 (2005).
[CrossRef]

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]

2002

2000

B. O. Guan, H. Y. Tam, X. M. Tao, and X. Y. Dong, “Simultaneous Strain and Temperature Measurement Using a Superstructure Fiber Bragg Grating,” IEEE Photon. Technol. Lett. 12(6), 675–677 (2000).
[CrossRef]

1997

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

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

1985

1976

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

1962

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]

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.

Cai, H.

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]

Cartaxo, A. V. T.

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. Electromag. Res., PIER 93, 385–401 (2009).
[CrossRef]

Chen, X.

X. Chen, J. Yao, F. Zeng, and Z. Deng, “Single-longitudinal-mode fiber ring laser employing an equivalent phase-shifted fiber Bragg grating,” IEEE Photon. Technol. Lett. 17(7), 1390–1392 (2005).
[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. Electromag. Res., PIER 93, 385–401 (2009).
[CrossRef]

Chiang, K. S.

Deng, Z.

X. Chen, J. Yao, F. Zeng, and Z. Deng, “Single-longitudinal-mode fiber ring laser employing an equivalent phase-shifted fiber Bragg grating,” IEEE Photon. Technol. Lett. 17(7), 1390–1392 (2005).
[CrossRef]

Dong, B.

Dong, X. Y.

B. O. Guan, H. Y. Tam, X. M. Tao, and X. Y. Dong, “Simultaneous Strain and Temperature Measurement Using a Superstructure Fiber Bragg Grating,” IEEE Photon. Technol. Lett. 12(6), 675–677 (2000).
[CrossRef]

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]

Dyer, S. D.

R. J. Espejo, M. Svalgaard, and S. D. Dyer, “Characterizing fiber Bragg grating index profiles to improve the writing process,” IEEE Photon. Technol. Lett. 18(21), 2242–2244 (2006).
[CrossRef]

Erdogan, T.

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

Espejo, R. J.

R. J. Espejo, M. Svalgaard, and S. D. Dyer, “Characterizing fiber Bragg grating index profiles to improve the writing process,” IEEE Photon. Technol. Lett. 18(21), 2242–2244 (2006).
[CrossRef]

Fang, Z.

Guan, B. O.

A. P. Zhang, B. O. Guan, X. M. Tao, and H. Y. Tam, “Mode coupling in superstructure fiber Bragg grating,” IEEE Photon. Technol. Lett. 14(4), 489–491 (2002).
[CrossRef]

B. O. Guan, H. Y. Tam, X. M. Tao, and X. Y. Dong, “Simultaneous Strain and Temperature Measurement Using a Superstructure Fiber Bragg Grating,” IEEE Photon. Technol. Lett. 12(6), 675–677 (2000).
[CrossRef]

Hall, D. G.

Hao, J.

He, S.

Huang, R.

Jovanovic, N.

Kogelnik, H.

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

Liang, S.

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. Electromag. Res., PIER 93, 385–401 (2009).
[CrossRef]

Lin, B.

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. Electromag. Res., PIER 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. Electromag. Res., PIER 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]

Qu, R.

Rao, Y. J.

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).

Rebola, J. L.

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. Electromag. Res., PIER 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. Electromag. Res., PIER 93, 385–401 (2009).
[CrossRef]

Svalgaard, M.

R. J. Espejo, M. Svalgaard, and S. D. Dyer, “Characterizing fiber Bragg grating index profiles to improve the writing process,” IEEE Photon. Technol. Lett. 18(21), 2242–2244 (2006).
[CrossRef]

Tam, H. Y.

A. P. Zhang, B. O. Guan, X. M. Tao, and H. Y. Tam, “Mode coupling in superstructure fiber Bragg grating,” IEEE Photon. Technol. Lett. 14(4), 489–491 (2002).
[CrossRef]

B. O. Guan, H. Y. Tam, X. M. Tao, and X. Y. Dong, “Simultaneous Strain and Temperature Measurement Using a Superstructure Fiber Bragg Grating,” IEEE Photon. Technol. Lett. 12(6), 675–677 (2000).
[CrossRef]

Tang, D.

Tang, Y.

Tao, X. M.

A. P. Zhang, B. O. Guan, X. M. Tao, and H. Y. Tam, “Mode coupling in superstructure fiber Bragg grating,” IEEE Photon. Technol. Lett. 14(4), 489–491 (2002).
[CrossRef]

B. O. Guan, H. Y. Tam, X. M. Tao, and X. Y. Dong, “Simultaneous Strain and Temperature Measurement Using a Superstructure Fiber Bragg Grating,” IEEE Photon. Technol. Lett. 12(6), 675–677 (2000).
[CrossRef]

Tjin, S. C.

Wang, C.

C. Wang and J. Yao, “Photonic generation of chirped microwave pulses using superimposed chirped fiber Bragg gratings,” IEEE Photon. Technol. Lett. 20(11), 882–884 (2008).
[CrossRef]

Wang, Z.

Weller-Brophy, L. A.

Westergren, U.

Williams, R. J.

Withford, M. J.

Wosinski, L.

Xu, Q.

Yamada, M.

Yao, J.

C. Wang and J. Yao, “Photonic generation of chirped microwave pulses using superimposed chirped fiber Bragg gratings,” IEEE Photon. Technol. Lett. 20(11), 882–884 (2008).
[CrossRef]

X. Chen, J. Yao, F. Zeng, and Z. Deng, “Single-longitudinal-mode fiber ring laser employing an equivalent phase-shifted fiber Bragg grating,” IEEE Photon. Technol. Lett. 17(7), 1390–1392 (2005).
[CrossRef]

Ye, Q.

Zeng, F.

X. Chen, J. Yao, F. Zeng, and Z. Deng, “Single-longitudinal-mode fiber ring laser employing an equivalent phase-shifted fiber Bragg grating,” IEEE Photon. Technol. Lett. 17(7), 1390–1392 (2005).
[CrossRef]

Zeng, X. K.

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).

Zhang, A. P.

A. P. Zhang, B. O. Guan, X. M. Tao, and H. Y. Tam, “Mode coupling in superstructure fiber Bragg grating,” IEEE Photon. Technol. Lett. 14(4), 489–491 (2002).
[CrossRef]

Zhang, H.

Acta. Phys. Sin.

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.

Bell Syst. Tech. J.

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

IEEE Photon. Technol. Lett.

R. J. Espejo, M. Svalgaard, and S. D. Dyer, “Characterizing fiber Bragg grating index profiles to improve the writing process,” IEEE Photon. Technol. Lett. 18(21), 2242–2244 (2006).
[CrossRef]

X. Chen, J. Yao, F. Zeng, and Z. Deng, “Single-longitudinal-mode fiber ring laser employing an equivalent phase-shifted fiber Bragg grating,” IEEE Photon. Technol. Lett. 17(7), 1390–1392 (2005).
[CrossRef]

C. Wang and J. Yao, “Photonic generation of chirped microwave pulses using superimposed chirped fiber Bragg gratings,” IEEE Photon. Technol. Lett. 20(11), 882–884 (2008).
[CrossRef]

A. P. Zhang, B. O. Guan, X. M. Tao, and H. Y. Tam, “Mode coupling in superstructure fiber Bragg grating,” IEEE Photon. Technol. Lett. 14(4), 489–491 (2002).
[CrossRef]

B. O. Guan, H. Y. Tam, X. M. Tao, and X. Y. Dong, “Simultaneous Strain and Temperature Measurement Using a Superstructure Fiber Bragg Grating,” IEEE Photon. Technol. Lett. 12(6), 675–677 (2000).
[CrossRef]

J. Lightwave Technol.

J. Opt. Soc. Am. A

Opt. Express

Opt. Lett.

Opt. Quantum Electron.

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]

Phys. Rev.

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

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. Electromag. Res., PIER

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. Electromag. Res., PIER 93, 385–401 (2009).
[CrossRef]

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Figures (8)

Fig. 1
Fig. 1

Schematic diagram of the mode couplings in (a) short- and (b) long-period waveguide gratings.

Fig. 2
Fig. 2

Schematic diagram of the refractive index modulation of conventional SBG.

Fig. 3
Fig. 3

Calculated reflectivities of a Bragg GA-grating, according to Eq. (13) (solid lines labeled 1,2 and 3) and the TM method (dotted lines labeled 1', 2' and 3′), in the case of changing length (a), amplitude (b) and Gaussian-apodized coefficient (c).

Fig. 4
Fig. 4

Calculated bar-transmissions of a long-period GA-grating, according to Eq. (14) (solid lines labeled 1,2 and 3) and the TM method (dotted lines labeled 1', 2' and 3′).

Fig. 5
Fig. 5

Calculated and measured reflectivities of an SBG. (a) and (b) are based on Eq. (17) and the TM method, respectively. (c) is the measured one adapted from Ref [22].

Fig. 6
Fig. 6

Calculated and measured bar-transmissions of an SBG. (a) is the calculated one based on Eq. (18). (b) is the measured one adapted from Ref [23].

Fig. 7
Fig. 7

Calculated reflectivities of a PSBG with ϕ = 0 (a), π/2 (b), π (c) and 3π/2 (d), according to the analytic solution Eq. (23) (solid lines) and the TM method (dotted lines).

Fig. 8
Fig. 8

Calculated bar-transmissions of a LP-PSG with shifted phase ϕ = 0 (a), π/2 (b), π (c) and 3π/2 (d), according to Eq. (24) (solid lines) and the TM method (dotted lines).

Equations (24)

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,
B s (0) B s (L) d B s (z) B m (z) =±j m 0 L k s Δn(z) e j( β m ± β s )z dz ,
B m 2 (z)= B m 2 (L)+ B s 2 (z),
B m 2 (z)= B m 2 (0) B s 2 (z).
0 L Δn(z) e j2π v s z dz = γ s ( v s )+j η s ( v s )=H e jφ ,
R= sinh 2 ( k s η s )+ sin 2 ( k s γ s ) cosh 2 ( k s η s )+ sin 2 ( k s γ s ) ,
T= cos 2 [ k s η s ( ν s ) ] sinh 2 [ k s γ s ( ν s ) ].
R= tanh 2 ( k s H ),
T= cos 2 ( k s H ).
Δ n G (z)=δn[ 1+ e α z 2 L 2 sin( 2π Λ z ) ] , L 2 z L 2 ,
η s = Lδn 2α { πα 2 e π 2 L 2 σ 2 α + e α 4 [1cos(πLσ)] },
R G = tanh 2 { k s Lδn 2α { πα 2 e π 2 L 2 σ B 2 α + e α 4 [1cos(πL σ B )]} },
T G = cos 2 { k s Lδn 2α { πα 2 e π 2 L 2 σ F 2 α + e α 4 [1cos(πL σ F )]} },
Δ n SB (z)={ δ 0 δ n cos( 2πz/Λ ) , qP<zqP+a 0,qP+a<z(q+1)P,q=0,1,2..,N1 ,
H SB = Naδ π i=4 P/a 4 P/a sinc( πai P )sinc[ πNP(σ i P ) ] ,
R SB = tanh 2 { k s Na δ n π i=4 P/a 4 P/a sinc( πai P )sinc[ πNP( σ b i P ) ] }.
T SB = s cos 2 { π 1 k s Na δ 0 sinc( πa/P )sinc[ πNP( σ sc 1/P) ] } cosh 2 { k m Na δ n π i=4 P/a 4 P/a sinc( πai P )sinc[ πNP( σ b i P ) ] } .
λ p (g)= 2( n m + δ 0 )Λ 1+gΛ/P = λ 0 1+gΛ/P ,
Δ λ 0 Λ/P,
Δ n P (z)={ δn[ 1+cos( 2πz/Λ ) ], 0z<l δn[ 1+cos( 2πz/Λ+φ ) ],lzL ,
H 2 = 1 16 δ n 2 { { lsinc( πσl )( Ll )sinc[ πσ( Ll ) ] } 2 +4l( Ll )sinc( πσl )sinc[ πσ( Ll ) ] cos 2 [ 0.5( ϕπσL ) ] }.
R p = tanh 2 [ k s Lδn 4 sinc( πL σ B 2 )cos( ϕπL σ B 2 ) ],
T p = cos 2 [ k s Lδn 4 sinc( πL σ F 2 )cos( ϕπL σ F 2 ) ],

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