Abstract

A matrix method which relates the field and its derivative is presented for the study of wave propagation in any type of one-dimensional media. The transfer matrix is obtained from the canonical solutions of Helmholtz equations at normal incidence. The method is applied to different optical systems like a Fabry-Perot cavity formed by uniform fiber Bragg gratings, periodic dielectric structures and different quasi-periodic structures based on Fibonacci and Thue-Morse sequences of layers with constant and variable refractive index.

© 2006 Optical Society of America

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References

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  1. M. Born and E. Wolf, Principles of Optics (Macmillan, London, 1964).
  2. J. Lekner and M. Dorf, "Matrix methods for the calculation of reflection amplitudes," J. Opt. Soc. Am. A 4, 2092 (1987).
    [CrossRef]
  3. J. Lekner, "Light in periodically stratified media," J. Opt. Soc. Am. A 11, 2892 (1994).
    [CrossRef]
  4. M. Kohmoto, B. Sutherland, and K. Iguchi, "Localization in Optics: Quasiperiodic Media," Phys. Rev. Lett. 58, 2436 (1987).
    [CrossRef] [PubMed]
  5. W. Gellermann, M. Kohmoto, B. Sutherland, and P. Taylor, "Localization of light waves in Fibonacci dielectric multilayers," Phys. Rev. Lett. 72, 633 (1994).
    [CrossRef] [PubMed]
  6. E. Macia, "Exploiting quasiperiodic order in design of optical devices," Phys. Rev. B 63, 205,421 (2001).
    [CrossRef]
  7. M. Yamada and K. Sakuda, "Analysis of almost-periodic distributed feedback slab waveguides via fundamental matrix approach," Applied Optics 26, 3474 - 3478 (1987).
    [CrossRef] [PubMed]
  8. T. Erdogan, "Fiber grating spectra," J. Lightwave Technol. 15(8), 1277 (1997).
    [CrossRef]
  9. L. Carretero, M. Perez-Molina, S. Blaya, R. Madrigal, P. Acebal, and A. Fimia, "Application of the fixed point theorem for the solution of the 1D wave equation: comparison with exact Mathieu solutions," Opt. Express 13, 9078 (2005).
    [CrossRef] [PubMed]
  10. Y. O. Barmenkov, D. Zalvidea, S. Torres-Peiro, J. Cruz, and M. Andres, "Effective length of short Fabry-Perot cavity formed by uniform fiber Bragg gratings," Opt. Express 14, 6394 (2006).
    [CrossRef] [PubMed]

2006

2005

2001

E. Macia, "Exploiting quasiperiodic order in design of optical devices," Phys. Rev. B 63, 205,421 (2001).
[CrossRef]

1997

T. Erdogan, "Fiber grating spectra," J. Lightwave Technol. 15(8), 1277 (1997).
[CrossRef]

1994

J. Lekner, "Light in periodically stratified media," J. Opt. Soc. Am. A 11, 2892 (1994).
[CrossRef]

W. Gellermann, M. Kohmoto, B. Sutherland, and P. Taylor, "Localization of light waves in Fibonacci dielectric multilayers," Phys. Rev. Lett. 72, 633 (1994).
[CrossRef] [PubMed]

1987

M. Kohmoto, B. Sutherland, and K. Iguchi, "Localization in Optics: Quasiperiodic Media," Phys. Rev. Lett. 58, 2436 (1987).
[CrossRef] [PubMed]

M. Yamada and K. Sakuda, "Analysis of almost-periodic distributed feedback slab waveguides via fundamental matrix approach," Applied Optics 26, 3474 - 3478 (1987).
[CrossRef] [PubMed]

J. Lekner and M. Dorf, "Matrix methods for the calculation of reflection amplitudes," J. Opt. Soc. Am. A 4, 2092 (1987).
[CrossRef]

Acebal, P.

Andres, M.

Barmenkov, Y. O.

Blaya, S.

Carretero, L.

Cruz, J.

Dorf, M.

Erdogan, T.

T. Erdogan, "Fiber grating spectra," J. Lightwave Technol. 15(8), 1277 (1997).
[CrossRef]

Fimia, A.

Gellermann, W.

W. Gellermann, M. Kohmoto, B. Sutherland, and P. Taylor, "Localization of light waves in Fibonacci dielectric multilayers," Phys. Rev. Lett. 72, 633 (1994).
[CrossRef] [PubMed]

Iguchi, K.

M. Kohmoto, B. Sutherland, and K. Iguchi, "Localization in Optics: Quasiperiodic Media," Phys. Rev. Lett. 58, 2436 (1987).
[CrossRef] [PubMed]

Kohmoto, M.

W. Gellermann, M. Kohmoto, B. Sutherland, and P. Taylor, "Localization of light waves in Fibonacci dielectric multilayers," Phys. Rev. Lett. 72, 633 (1994).
[CrossRef] [PubMed]

M. Kohmoto, B. Sutherland, and K. Iguchi, "Localization in Optics: Quasiperiodic Media," Phys. Rev. Lett. 58, 2436 (1987).
[CrossRef] [PubMed]

Lekner, J.

Macia, E.

E. Macia, "Exploiting quasiperiodic order in design of optical devices," Phys. Rev. B 63, 205,421 (2001).
[CrossRef]

Madrigal, R.

Perez-Molina, M.

Sakuda, K.

M. Yamada and K. Sakuda, "Analysis of almost-periodic distributed feedback slab waveguides via fundamental matrix approach," Applied Optics 26, 3474 - 3478 (1987).
[CrossRef] [PubMed]

Sutherland, B.

W. Gellermann, M. Kohmoto, B. Sutherland, and P. Taylor, "Localization of light waves in Fibonacci dielectric multilayers," Phys. Rev. Lett. 72, 633 (1994).
[CrossRef] [PubMed]

M. Kohmoto, B. Sutherland, and K. Iguchi, "Localization in Optics: Quasiperiodic Media," Phys. Rev. Lett. 58, 2436 (1987).
[CrossRef] [PubMed]

Taylor, P.

W. Gellermann, M. Kohmoto, B. Sutherland, and P. Taylor, "Localization of light waves in Fibonacci dielectric multilayers," Phys. Rev. Lett. 72, 633 (1994).
[CrossRef] [PubMed]

Torres-Peiro, S.

Yamada, M.

M. Yamada and K. Sakuda, "Analysis of almost-periodic distributed feedback slab waveguides via fundamental matrix approach," Applied Optics 26, 3474 - 3478 (1987).
[CrossRef] [PubMed]

Zalvidea, D.

Applied Optics

M. Yamada and K. Sakuda, "Analysis of almost-periodic distributed feedback slab waveguides via fundamental matrix approach," Applied Optics 26, 3474 - 3478 (1987).
[CrossRef] [PubMed]

J. Lightwave Technol.

T. Erdogan, "Fiber grating spectra," J. Lightwave Technol. 15(8), 1277 (1997).
[CrossRef]

J. Opt. Soc. Am. A

Opt. Express

Phys. Rev. B

E. Macia, "Exploiting quasiperiodic order in design of optical devices," Phys. Rev. B 63, 205,421 (2001).
[CrossRef]

Phys. Rev. Lett.

M. Kohmoto, B. Sutherland, and K. Iguchi, "Localization in Optics: Quasiperiodic Media," Phys. Rev. Lett. 58, 2436 (1987).
[CrossRef] [PubMed]

W. Gellermann, M. Kohmoto, B. Sutherland, and P. Taylor, "Localization of light waves in Fibonacci dielectric multilayers," Phys. Rev. Lett. 72, 633 (1994).
[CrossRef] [PubMed]

Other

M. Born and E. Wolf, Principles of Optics (Macmillan, London, 1964).

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Figures (3)

Fig. 1.
Fig. 1.

Transmission curve vs. λ(nm) for (A) P 16(1) gray lines, P 16(2) black lines,(B) TM 5(1) gray lines, TM 5(2) black lines, (C) F 8(1) gray lines, F 8(2) black color

Fig. 2.
Fig. 2.

Transmittance vs. λ(µm) of a Fabry-Perot fiber cavity formed by two equal uniform 4-cm FBG separated by 5 cm with R=99 % (dotted lines, εm =0.0001), R=95 % (black lines εm =0.000075) and R=66 % (grey lines, εm =0.00004)

Fig. 3.
Fig. 3.

Transmittance vs. λ(µm ) of a double Fabry-Perot fiber cavity formed by two equal uniform 4-cm FBG separated by 5 cm with R=99 % (εm =0.0001)

Equations (25)

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ξ = L z
2 E x ( ξ ) ξ 2 + ( 2 π λ ) 2 ε ( ξ ) E x ( ξ ) = 0
E c 1 ( 0 ) = 1 , E c 1 ( 0 ) = 0 , E c 2 ( 0 ) = 0 and E c 2 ( 0 ) = 1
E c ( ξ ) = c 1 E c 1 ( ξ ) + c 2 E c 2 ( ξ )
M = ( E c 1 ( L ) E c 2 ( L ) E c 1 ξ | ξ = L E c 2 ξ | ξ = L )
M T = i = 1 N M N i + 1
ρ = 2 π i ε 1 ( λ M T ( 11 ) 2 π i ε 3 M T ( 12 ) ) + λ ( λ M T ( 21 ) 2 π i ε 3 M T ( 22 ) ) 2 π i ε 1 ( λ M T ( 11 ) 2 π i ε 3 M T ( 12 ) ) λ ( λ M T ( 21 ) 2 π i ε 3 M T ( 22 ) )
τ = 4 λ π i ε 1 2 π i ε 1 ( λ M T ( 11 ) 2 π i ε 3 M T ( 12 ) ) λ ( λ M T ( 21 ) 2 π i ε 3 M T ( 22 ) )
E c ( ξ ) = c 1 cos ( 2 π ξ n λ ) + c 2 sin ( 2 π ξ n λ )
M y ( λ , n , L ) = ( cos ( 2 π L n λ ) λ 2 π n sin ( 2 π L n λ ) 2 π n λ sin ( 2 π L n λ ) cos ( 2 π L n λ ) )
ε ( ξ ) = ε 1 exp ( B ξ )
E c ( ξ ) = c 1 J 0 ( a 1 exp ( B ξ ) ) + c 2 Y 0 ( a 1 exp ( B ξ ) )
M g a ( λ , L , ε 1 , ε 1 ) = ( π a 1 2 ( J 1 ( a 1 ) . Y 0 ( a 3 ) J 0 ( a 3 ) . Y 1 ( a 1 ) ) λ a 3 4 ε 3 ( J 0 ( a 3 ) . Y 0 ( a 1 ) J 0 ( a 1 ) . Y 0 ( a 3 ) ) π 2 ε 3 a 1 λ ( J 1 ( a 3 ) . Y 1 ( a 1 ) J 1 ( a 1 ) . Y 1 ( a 3 ) ) π a 3 2 ( J 1 ( a 3 ) . Y 0 ( a 1 ) J 0 ( a 1 ) . Y 1 ( a 3 ) ) )
ε ( ξ ) = ε 0 + ε m cos ( 2 π ξ Λ )
L = π L Λ , ζ 1 = λ 0 2 λ 2 and ζ 2 = 2 ε m Λ 2 λ 2
E c ( ξ ) = c 1 mc ( ζ 1 , ζ 2 , π ξ Λ ) + c 2 ms ( ζ 1 , ζ 2 , π ξ Λ )
M g ( λ , ε 0 , ε m , Λ , L ) = 1 p ( β c 0 . α s L α c L . β s 0 Λ π ( α c 0 . α s 0 α c 0 . α s L ) π Λ ( β c L . β s 0 + β c 0 . β s L ) β c L . α s 0 α c 0 . β s L )
α q w = mq ( ζ 1 , ζ 2 , w ) and β q w = [ Λ π mq ( ζ 1 , ζ 2 , π ξ Λ ) ξ ] ξ = w
P n ( i ) = ( H . L i ) n , i = 1 , 2
F n ( i ) = F n 1 ( i ) . F n 2 ( i ) with F 0 = H , F 1 = L i , i = 1 , 2
T M n ( i ) = T M n 1 ( i ) . T M n 1 ( i )
T M n ( i ) = T M n 1 ( i ) . T M n 1 ( i )
T M 0 = H and T M 0 = L i , i = 1 , 2
M F P ( λ , 2.085 , ε m , d 1 , d 2 ) = M g ( λ , 2.085 , ε m , Λ , d 1 ) . M y ( λ , 1.444 , d 2 ) . M g ( λ , 2.085 , ε m , Λ , d 1 )
M F P 2 ( λ , 2.085 , ε m , d 1 , d 2 ) = M F P ( λ , 2.085 , ε m , Λ , d 1 , d 2 ) . M y ( λ , 1.444 , d 2 ) . M F P ( λ , 2.085 , ε m , Λ , d 1 , d 2 )

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