Abstract

A modified coupled-mode (CM) model is proposed for the optical behavior of thermally chirped Bragg gratings. The model accounts for the axial gradient in the modulation wavenumber, which has been ignored in the classical CM model. The model is used to characterize the optical behavior of a polymethyl methacrylate-based polymer Bragg grating subjected to nonisothermal conditions. The validity of the proposed method is verified by comparing the results of the modified CM model with those obtained from the exact numerical solution.

© 2010 Optical Society of America

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References

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  2. C. S. Hong, J. B. Shellan, A. C. Livanos, A. Yariv, and A. Katzir, “Broad-band grating filters for thin-film optical waveguides,” Appl. Phys. Lett. 31, 276-278 (1977).
    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
  10. G. D. Peng and P. L. Chu, “Polymer optical fiber photosensitivities and highly tunable fiber gratings,” Fiber Integr. Opt. 19, 277-293 (2000).
    [CrossRef]
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    [CrossRef]
  12. I. C. M. Littler, T. Grujic, and B. J. Eggleton, “Photothermal effects in fiber Bragg gratings,” Appl. Opt. 45, 4679-4685(2006).
    [CrossRef]
  13. K. J. Kim, A. Bar-Cohen, and B. Han, “Thermo-optical modeling of an intrinsically heated polymer fiber Bragg grating,” Appl. Opt. 46, 4357-4370 (2007).
    [CrossRef]
  14. S. J. Orfanidis, Electromagnetic Waves and Antennas, an online book, www.ece.rutgers.edu/~orfanidi/ewa (2004).

2007

2006

2002

H. Y. Liu, G. D. Peng, and P. L. Chu, “Thermal stability of gratings in PMMA and CYTOP polymer fibers,” Opt. Commun. 204, 151-156 (2002).
[CrossRef]

2000

G. D. Peng and P. L. Chu, “Polymer optical fiber photosensitivities and highly tunable fiber gratings,” Fiber Integr. Opt. 19, 277-293 (2000).
[CrossRef]

1997

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

1994

1993

L. Poladian, “Graphical and WKB analysis of nonuniform Bragg gratings,” Phys. Rev. E 48, 4758-4767 (1993).
[CrossRef]

1987

1985

1979

1977

A. C. Livanos, A. Katzir, A. Yariv, and C. S. Hong, “Chirped grating demultiplexers in dielectric waveguides,” Appl. Phys. Lett. 30, 519-521 (1977).
[CrossRef]

C. S. Hong, J. B. Shellan, A. C. Livanos, A. Yariv, and A. Katzir, “Broad-band grating filters for thin-film optical waveguides,” Appl. Phys. Lett. 31, 276-278 (1977).
[CrossRef]

1976

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

Bar-Cohen, A.

Chu, P. L.

H. Y. Liu, G. D. Peng, and P. L. Chu, “Thermal stability of gratings in PMMA and CYTOP polymer fibers,” Opt. Commun. 204, 151-156 (2002).
[CrossRef]

G. D. Peng and P. L. Chu, “Polymer optical fiber photosensitivities and highly tunable fiber gratings,” Fiber Integr. Opt. 19, 277-293 (2000).
[CrossRef]

de Sterke, C. M.

Eggleton, B. J.

Erdogan, T.

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

Fukuzawa, T.

Grujic, T.

Hall, D. G.

Han, B.

Hong, C. S.

C. S. Hong, J. B. Shellan, A. C. Livanos, A. Yariv, and A. Katzir, “Broad-band grating filters for thin-film optical waveguides,” Appl. Phys. Lett. 31, 276-278 (1977).
[CrossRef]

A. C. Livanos, A. Katzir, A. Yariv, and C. S. Hong, “Chirped grating demultiplexers in dielectric waveguides,” Appl. Phys. Lett. 30, 519-521 (1977).
[CrossRef]

Katzir, A.

A. C. Livanos, A. Katzir, A. Yariv, and C. S. Hong, “Chirped grating demultiplexers in dielectric waveguides,” Appl. Phys. Lett. 30, 519-521 (1977).
[CrossRef]

C. S. Hong, J. B. Shellan, A. C. Livanos, A. Yariv, and A. Katzir, “Broad-band grating filters for thin-film optical waveguides,” Appl. Phys. Lett. 31, 276-278 (1977).
[CrossRef]

Kim, K. J.

Kogelnik, H.

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

Littler, I. C. M.

Liu, H. Y.

H. Y. Liu, G. D. Peng, and P. L. Chu, “Thermal stability of gratings in PMMA and CYTOP polymer fibers,” Opt. Commun. 204, 151-156 (2002).
[CrossRef]

Livanos, A. C.

C. S. Hong, J. B. Shellan, A. C. Livanos, A. Yariv, and A. Katzir, “Broad-band grating filters for thin-film optical waveguides,” Appl. Phys. Lett. 31, 276-278 (1977).
[CrossRef]

A. C. Livanos, A. Katzir, A. Yariv, and C. S. Hong, “Chirped grating demultiplexers in dielectric waveguides,” Appl. Phys. Lett. 30, 519-521 (1977).
[CrossRef]

Nakamura, M.

Orfanidis, S. J.

S. J. Orfanidis, Electromagnetic Waves and Antennas, an online book, www.ece.rutgers.edu/~orfanidi/ewa (2004).

Peng, G. D.

H. Y. Liu, G. D. Peng, and P. L. Chu, “Thermal stability of gratings in PMMA and CYTOP polymer fibers,” Opt. Commun. 204, 151-156 (2002).
[CrossRef]

G. D. Peng and P. L. Chu, “Polymer optical fiber photosensitivities and highly tunable fiber gratings,” Fiber Integr. Opt. 19, 277-293 (2000).
[CrossRef]

Poladian, L.

J. E. Sipe, L. Poladian, and C. M. de Sterke, “Propagation through non-uniform grating structures,” J. Opt. Soc. Am. A 11, 1307-1320 (1994).
[CrossRef]

L. Poladian, “Graphical and WKB analysis of nonuniform Bragg gratings,” Phys. Rev. E 48, 4758-4767 (1993).
[CrossRef]

Sakuda, K.

Shellan, J. B.

C. S. Hong, J. B. Shellan, A. C. Livanos, A. Yariv, and A. Katzir, “Broad-band grating filters for thin-film optical waveguides,” Appl. Phys. Lett. 31, 276-278 (1977).
[CrossRef]

Sipe, J. E.

Weller-Brophy, L. A.

Yamada, M.

Yariv, A.

A. C. Livanos, A. Katzir, A. Yariv, and C. S. Hong, “Chirped grating demultiplexers in dielectric waveguides,” Appl. Phys. Lett. 30, 519-521 (1977).
[CrossRef]

C. S. Hong, J. B. Shellan, A. C. Livanos, A. Yariv, and A. Katzir, “Broad-band grating filters for thin-film optical waveguides,” Appl. Phys. Lett. 31, 276-278 (1977).
[CrossRef]

Appl. Opt.

Appl. Phys. Lett.

C. S. Hong, J. B. Shellan, A. C. Livanos, A. Yariv, and A. Katzir, “Broad-band grating filters for thin-film optical waveguides,” Appl. Phys. Lett. 31, 276-278 (1977).
[CrossRef]

A. C. Livanos, A. Katzir, A. Yariv, and C. S. Hong, “Chirped grating demultiplexers in dielectric waveguides,” Appl. Phys. Lett. 30, 519-521 (1977).
[CrossRef]

Bell Syst. Tech. J.

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

Fiber Integr. Opt.

G. D. Peng and P. L. Chu, “Polymer optical fiber photosensitivities and highly tunable fiber gratings,” Fiber Integr. Opt. 19, 277-293 (2000).
[CrossRef]

J. Lightwave Technol.

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

J. Opt. Soc. Am. A

Opt. Commun.

H. Y. Liu, G. D. Peng, and P. L. Chu, “Thermal stability of gratings in PMMA and CYTOP polymer fibers,” Opt. Commun. 204, 151-156 (2002).
[CrossRef]

Opt. Lett.

Phys. Rev.

L. Poladian, “Graphical and WKB analysis of nonuniform Bragg gratings,” Phys. Rev. E 48, 4758-4767 (1993).
[CrossRef]

Other

S. J. Orfanidis, Electromagnetic Waves and Antennas, an online book, www.ece.rutgers.edu/~orfanidi/ewa (2004).

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Figures (4)

Fig. 1
Fig. 1

Contribution of thermo-optic effects to spectral variation (shift from Bragg wavelength) in reflected power for a 10 mm glass grating. A 10 mm glass grating under an exponential temperature profile of Δ T = 85 ° 3 ° C is considered. Calculations are separated for the sole effects of k, B, B & B , and the combined effects models CM (k and B) and CMT (k and B & B ). Power is normalized to the incident light of unity.

Fig. 2
Fig. 2

Contribution of thermo-optic effects to spectral variation (shift from Bragg wavelength) in reflected power for a 10 mm PMMA grating, under an exponential temperature profile of Δ T = 35 ° 1 ° C . The individual effects of k, B, B & B , and the combined effects models CM (k and B) and CMT (k and B & B ) are shown. Power is normalized to the incident light of unity.

Fig. 3
Fig. 3

Normalized backward wave power for a thermally chirped 10 mm PMMA grating (at Δ λ B = 0 and 1.0 nm ) under a linear temperature profile of Δ T = 5 ° 0 ° C (start to end of grating). Calculations are according to exact numerical model, CM, and CMT. Power is normalized to the incident light of 100 units.

Fig. 4
Fig. 4

Effect of temperature nonuniformity on spectral variation (shift from Bragg wavelength) in reflected power for a 10 mm PMMA grating. Power is normalized to the incident light of unity.

Tables (1)

Tables Icon

Table 1 Physical Parameters for Glass and PMMA (from Kim [13]) at Standard Room Temperature

Equations (16)

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[ F j R j ] = 1 τ j [ 1 ρ j * ρ j 1 ] [ e i k j Δ z 0 0 e i k j Δ z ] [ F j + 1 R j + 1 ] ,
τ j = 2 n j n j + n j + 1 , ρ j = n j n j + 1 n j + n j + 1 ,
n ( z ) = n 0 ( z ) + δ n e i B ( z ) z .
d [ F R ] = A [ F R ] ,
A = ( 1 1 τ e i k Δ z ρ * τ e i k Δ z ρ τ e i k Δ z 1 1 τ e i k Δ z ) .
ρ j = n j n j + 1 n j + n j + 1 n Δ z 2 n + n Δ z , τ j = 2 n j n j + n j + 1 = 2 n 2 n + n Δ z , e i k Δ z 1 + i k Δ z ,
A = ( 1 2 n n i k 1 2 n n * 1 2 n n 1 2 n n + i k ) Δ z .
n ( z ) n ( z ) = n 0 ( z ) + i [ B ( z ) + z B ( z ) ] δ n e i B ( z ) z n 0 ( z ) + δ n e i B ( z ) z i δ n n 0 ( z ) [ B ( z ) + z B ( z ) ] e i B ( z ) z ,
A i ( k ( z ) 1 2 δ n n 0 ( z ) [ B ( z ) + z B ( z ) ] e i B ( z ) z 1 2 δ n n 0 ( z ) [ B ( z ) + z B ( z ) ] e i B ( z ) z k ( z ) ) Δ z .
d d z [ f r ] = a [ f r ] ,
a = i ( 1 2 [ B ( z ) + z B ( z ) ] + k ( z ) 1 2 δ n n 0 ( z ) [ B ( z ) + z B ( z ) ] 1 2 δ n n 0 ( z ) [ B ( z ) + z B ( z ) ] 1 2 [ B ( z ) + z B ( z ) ] k ( z ) ) .
B ( z ) = 2 π Λ ( z ) = 2 π Λ 0 [ 1 + α Δ T ( z ) ] ,
k ( z ) = 2 π λ 0 n ( z ) 2 π λ 0 [ n 0 + n T Δ T ( z ) ] ,
[ F N R N ] = [ 1 0 ] .
[ f j r j ] = e a j Δ z [ f j + 1 r j + 1 ] ,
[ f N r N ] = [ 1 0 ] .

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