Abstract

A method based in the application of Fixed Point Theorem (FPT) techniques to the solution of the 1D wave equation at normal incidence for materials that present a continuous (real or complex) dielectric constant is presented. As an example, the method is applied for the calculation of the electric field, reflection and transmission spectra in volume holographic gratings. It is shown that the solution obtained using this method agrees with the exact Mathieu solutions also obtained in this paper for volume holographic reflection gratings.

© 2005 Optical Society of America

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References

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  1. T. Tamir, H. Wang, and A. Oliner, �??Wave propagation in sinusoidally stratified dielectric media,�?? IEEE Trans. Microwave Theory MTT-12 323, 141 (1964).
  2. D. Yeh, K. F. Casey, and Z. A. Kaprilean, �??Transverse magnetic wave propagation in sinusoidally stratified dielectric media,�?? IEEE Trans. Microwave Theory MTT-13 13, 297 (1965).
    [CrossRef]
  3. D. Maystre, �??A new integral theory for dielectric coated gratings,�?? J. Opt. Soc. Am. 68, 490 (1978).
    [CrossRef]
  4. H. Kogelnik, �??Coupled wave theory for thick hologram gratings,�?? Bell. Sys. Tech. J. 48(9), 2909 (1969).
  5. R. Magnusson and T. K. Gaylord, �??Use of dynamic theory to describe experimental results from volume holography,�?? J. Appl. Phys. 47(1), 190 (1976).
    [CrossRef]
  6. R. Magnusson and T. K. Gaylord, �??Analysis of multiwave diffraction of thick holograms,�?? J. Opt. Soc. Am. 67, 1165 (1977).
    [CrossRef]
  7. M. G. Moharam and T. K. Gaylord, �??Rigorous coupled-wave analysis of planar-grating diffraction,�?? J. Opt. Soc. Am. 71, 37 (1981).
    [CrossRef]
  8. M. G. Moharam and T. K. Gaylord, �??Three-dimensional vector coupled-wave analysis of planar-grating,�?? J. Opt. Soc. Am 73, 1105 (1983).
    [CrossRef]
  9. J. M. Jarem and P. P. Banerjee, �??Application of the complex Poynting theorem to diffration gratings,�?? J. Opt. Soc. Am. 16(5), 1097 (1999).
    [CrossRef]
  10. J. M. Jarem and P. Banerjee, "Computational methods for electromagnetic and optical systems" (Marcell Dekker, 2000).
  11. D. W. Diehl and N. George, �??Analysis of multitone holographic interference filters by use of a sparse Hill matrix method,�?? Appl. Opt. 433, 88 (2004).
    [CrossRef]
  12. L. Carretero, M. Ulibarrena, S. Blaya, and A. Fimia, �??One-dimensional photonic crystals with an amplitude-modulated dielectric constant in the unit cell,�?? Appl. Opt 43(14), 2895 (2004).
    [CrossRef] [PubMed]
  13. G. V. Morozov, D. Sprung, and J. Martorell, �??Semicassical coupled-wave theory and its applications to TE waves in one-dimensional photonic crystals,�?? Phys. Rev. E 69, 016,612 (2004).
    [CrossRef]
  14. C. Sibila, M. Centini, K. Sakoda, J. Haus, and M. Bertolotti, �??Coherent emission in one-dimensional photonic band gap materials,�?? J. Opt. A:Pure Appl. Opt 7 (2005).
  15. V. A. Trofimov, E. B. Tereshin, and M. V. Fedotov, �??Localization of the femtosecond pulse train energy in a one-dimensional nonlinear photonic crystal,�?? Opt. Quantum Electron. 74, 66 (2004).
  16. A. V. Andreev, A. V. Balakin, I. A. Ozheredov, A. Shkurinov, P. Masselin, G. Mouret, and D. Boucher, �??Compression of femtosecond laser pulses in thin one-dimensional photonic crystals,�?? Phys. Rev. E 6(1), 016 (2000).
  17. I. V. Shadirvov, A. A. Sukhorukov, and Y. S. Kivshar, �??Beam shapin bya a periodic structure with negative refraction,�?? Appl. Phys. Lett 82 (2003).
  18. G. Jameson, "Topology and normed spaces" (Chapman and Hall, 1974).
  19. C. B. Burckhardt, �??Diffraction of a plane at a sinusoidally stratified dielectric grating,�?? J. Opt. Soc. Am 56, 1502 (1966).
    [CrossRef]
  20. M. G. Moharam and T. K. Gaylord, �??Diffraction analysis of dielectric surface-relief gratings,�?? J. Opt. Soc. Am 72, 1385 (1982).
    [CrossRef]
  21. I. Wolfram Research, "Mathematica" (Wolfram Research, Inc., Champaign, Illinois, 2004).

Appl. Opt (1)

L. Carretero, M. Ulibarrena, S. Blaya, and A. Fimia, �??One-dimensional photonic crystals with an amplitude-modulated dielectric constant in the unit cell,�?? Appl. Opt 43(14), 2895 (2004).
[CrossRef] [PubMed]

Appl. Opt. (1)

D. W. Diehl and N. George, �??Analysis of multitone holographic interference filters by use of a sparse Hill matrix method,�?? Appl. Opt. 433, 88 (2004).
[CrossRef]

Appl. Phys. Lett (1)

I. V. Shadirvov, A. A. Sukhorukov, and Y. S. Kivshar, �??Beam shapin bya a periodic structure with negative refraction,�?? Appl. Phys. Lett 82 (2003).

Bell. Sys. Tech. J. (1)

H. Kogelnik, �??Coupled wave theory for thick hologram gratings,�?? Bell. Sys. Tech. J. 48(9), 2909 (1969).

IEEE Trans. Microwave Theory (2)

T. Tamir, H. Wang, and A. Oliner, �??Wave propagation in sinusoidally stratified dielectric media,�?? IEEE Trans. Microwave Theory MTT-12 323, 141 (1964).

D. Yeh, K. F. Casey, and Z. A. Kaprilean, �??Transverse magnetic wave propagation in sinusoidally stratified dielectric media,�?? IEEE Trans. Microwave Theory MTT-13 13, 297 (1965).
[CrossRef]

J. Appl. Phys. (1)

R. Magnusson and T. K. Gaylord, �??Use of dynamic theory to describe experimental results from volume holography,�?? J. Appl. Phys. 47(1), 190 (1976).
[CrossRef]

J. Opt. A:Pure Appl. Opt (1)

C. Sibila, M. Centini, K. Sakoda, J. Haus, and M. Bertolotti, �??Coherent emission in one-dimensional photonic band gap materials,�?? J. Opt. A:Pure Appl. Opt 7 (2005).

J. Opt. Soc. Am (3)

M. G. Moharam and T. K. Gaylord, �??Three-dimensional vector coupled-wave analysis of planar-grating,�?? J. Opt. Soc. Am 73, 1105 (1983).
[CrossRef]

C. B. Burckhardt, �??Diffraction of a plane at a sinusoidally stratified dielectric grating,�?? J. Opt. Soc. Am 56, 1502 (1966).
[CrossRef]

M. G. Moharam and T. K. Gaylord, �??Diffraction analysis of dielectric surface-relief gratings,�?? J. Opt. Soc. Am 72, 1385 (1982).
[CrossRef]

J. Opt. Soc. Am. (4)

R. Magnusson and T. K. Gaylord, �??Analysis of multiwave diffraction of thick holograms,�?? J. Opt. Soc. Am. 67, 1165 (1977).
[CrossRef]

D. Maystre, �??A new integral theory for dielectric coated gratings,�?? J. Opt. Soc. Am. 68, 490 (1978).
[CrossRef]

J. M. Jarem and P. P. Banerjee, �??Application of the complex Poynting theorem to diffration gratings,�?? J. Opt. Soc. Am. 16(5), 1097 (1999).
[CrossRef]

M. G. Moharam and T. K. Gaylord, �??Rigorous coupled-wave analysis of planar-grating diffraction,�?? J. Opt. Soc. Am. 71, 37 (1981).
[CrossRef]

Opt. Quantum Electron. (1)

V. A. Trofimov, E. B. Tereshin, and M. V. Fedotov, �??Localization of the femtosecond pulse train energy in a one-dimensional nonlinear photonic crystal,�?? Opt. Quantum Electron. 74, 66 (2004).

Phys. Rev. E (2)

A. V. Andreev, A. V. Balakin, I. A. Ozheredov, A. Shkurinov, P. Masselin, G. Mouret, and D. Boucher, �??Compression of femtosecond laser pulses in thin one-dimensional photonic crystals,�?? Phys. Rev. E 6(1), 016 (2000).

G. V. Morozov, D. Sprung, and J. Martorell, �??Semicassical coupled-wave theory and its applications to TE waves in one-dimensional photonic crystals,�?? Phys. Rev. E 69, 016,612 (2004).
[CrossRef]

Other (3)

J. M. Jarem and P. Banerjee, "Computational methods for electromagnetic and optical systems" (Marcell Dekker, 2000).

I. Wolfram Research, "Mathematica" (Wolfram Research, Inc., Champaign, Illinois, 2004).

G. Jameson, "Topology and normed spaces" (Chapman and Hall, 1974).

Supplementary Material (2)

» Media 1: GIF (517 KB)     
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Figures (5)

Fig. 1.
Fig. 1.

Physical system and coordinates used.

Fig. 2.
Fig. 2.

Transmission spectra ∣τ2 as a function of λ. Parameters used ε 0 = 2.3716, ε 1 = 0.154 and thickness L = 10 μm, Λ = 0.205 nm.

Fig. 3.
Fig. 3.

(529 KB) Movie of E = ∣Ex (z)∣2 as a function of z for λ = 633 nm to λ = 700. Parameters used ε 0 = 2.3716, ε 1 = 0.154 and thickness L = 10 μm, Λ = 0.205 nm.

Fig. 4.
Fig. 4.

Error e(%) for ∣Ex (z)∣2 as a function of z for λ = 633 nm, showing the absolute value of the difference between the exact Mathieu solutions and those obtained with FPT. Parameters used ε 0 = 2.3716, ε 1 = 0.154 and thickness L = 10 μm, Λ = 0.205 nm.

Fig. 5.
Fig. 5.

(599 KB) Movie of ∣Ex2 vs ε(z) for λ 633 nm to λ = 700. Parameters used ε 0 = 2.3716, ε 1 =0.154 and thickness L = 10 μm, Λ = 0.205 nm.

Equations (23)

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2 E x ( z ) z 2 + ( 2 π λ ) 2 ε r ( z ) E x ( z ) = 0
y 1 ( z ) = E x ( z ) and y 2 ( z ) = y 1 ( z ) z
( y 1 ( z ) y 2 ( z ) ) = ( 0 1 ε r ( z ) 0 ) · ( y 1 ( z ) y 2 ( z ) )
y 1 ( z ) = e 1 + 0 z y 2 ( τ ) y 2 ( z ) = e 2 0 z ε r ( τ ) y 1 ( τ ) } z [ 0 , L ]
y 1 ( z ) = e 1 + 0 z y 2 ( τ ) y 2 ( z ) = e 2 0 z ε r ( τ ) y 1 ( τ ) } z [ 0 , L ]
( f 1 ( n + 1 ) ( z ) f 2 ( n + 1 ) ( z ) ) = ( f 1 ( n ) ( 0 ) + 0 z f 2 ( n ) ( τ ) f 2 ( n ) ( 0 ) 0 z ε r ( τ ) f 1 ( n ) ( τ ) )
ε ' r = ε 0 + ε 1 Cos ( 2 πz Λ )
2 E x ( z ) z 2 + ( 2 π λ ) 2 ( ε 0 + ε 1 Cos ( 2 πz Λ ) ) E x ( z ) = 0
E x ( z ) = c 1 mc ( a , b , πz Λ ) + c 2 ms ( a , b , πz Λ )
exp ( i 2 πnz λ ) + ρ exp ( i 2 πnz λ ) if z 0
E x ( z ) if 0 z L
τ exp ( i 2 πn ( z L λ ) ) if z L
1 + ρ = E x ( 0 )
τ = E x ( L )
( E x ( z ) z ) z = 0 = 2 πni ( ρ 1 ) λ
( E x ( z ) z ) z = L = 2 πniτ λ
ζ L = πL Λ , p = 2 in Λ , α s 0 = ms a b 0 , α sL = ms a b ζ L
α c 0 = mc a b 0 , α cL = mc a b ζ L
β s 0 = ( Λ π ms ( a , b , πz Λ ) z ) z = 0 , β sL = ( Λ π ms ( a , b , πz Λ ) z ) z = L
β c 0 = ( Λ π mc ( a , b , πz Λ ) z ) z = 0 , β cL = ( Λ π mc ( a , b , πz Λ ) z ) z = L
τ = 2 ( β cL α sL α cL β sL ) λ β cL ( p α s 0 λ β s 0 ) α cL p ( p α s 0 + λ β s 0 ) + ( p α c 0 + λ β c 0 ) ( p α sL + λ β sL )
ρ = λ β cL ( p α s 0 + λ β s 0 ) α cL p ( p α s 0 + λ β s 0 ) ( p α c 0 + λ β c 0 ) ( p α sL + λ β sL ) λ β cL ( p α s 0 λ β s 0 ) α cL p ( p α s 0 + λ β s 0 ) + ( p α c 0 + λ β c 0 ) ( p α sL + λ β sL )
( f 1 ( n + 1 ) ( z ) f 2 ( n + 1 ) ( z ) ) = ( ( ρ + 1 ) + 0 z f 2 ( n ) 2 πni ( ρ 1 ) λ ( 2 π λ ) 2 0 z ( ε 0 + ε 1 Cos ( 2 πτ Λ ) ) f 1 ( n ) ( τ ) )

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