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

1. Introduction

Diffraction gratings have been the subject of numerous studies due to the applications that have been developed in many different areas such as holography, spectroscopy or integrated optics. Different mathematical methods have been applied in order to solve the propagation of electromagnetic waves in periodically dielectric materials such as are the rigorous modal approach [1], [2], integral equation methods [3], coupled wave and rigorous coupled wave theory [4], [5], [6], [7], [8], the complex Poynting theorem [9], [10], or the sparse Hill matrix method [11]. Recently, there have been renewed the interest in the analysis of one-dimensional periodic structures due to their applicability as 1D photonic crystals [12],[13], [14], [15]. These systems are periodically structured electromagnetic media characterized by photonic band gaps. Propagation of the electromagnetic waves in a one-dimensional photonic crystal has been investigated for applications like pulse compression [16] and spatial beam shaping [17]. In this paper we present a recursive method based on the Fixed Point Theorem (FPT) for solve the 1D Helmholtz equation in materials that present a continuous (real or complex) dielectric constant. Rigorous expressions for electric field, transmission and reflection coefficients as combinations of Mathieu functions are also reported for the particular case of a sinusoidal non absorbing dielectric grating. Good agreement between exact solutions and results based on the FPT method are obtained.

2. Numerical method

Consider a dielectric medium (see Fig. 1) of length L with a one-dimensional permittivity characterized by a continuous complex function εr(z).

 

Fig. 1. Physical system and coordinates used.

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We wish to calculate the electric field inside the material and the transmission and reflection efficiencies for light incident upon this material at normal incidence (propagation is assumed in the direction and electric field polarization in the direction). We also assume that the material is linear and isotropic and the incident plane wave has a harmonic time dependence of exp(iωt). The amplitude of the electric field Ex , is described by the Helmholtz equation.:

2Ex(z)z2+(2πλ)2εr(z)Ex(z)=0

where λ is the free-space wavelength of the incident light and εr(z) is the relative complex permittivity of the material. In order to solve the Helmholtz equation, we transform the second order differential equation into a system of linear first order differential equations by introducing the change of variables:

y1(z)=Ex(z)andy2(z)=y1(z)z

Taking into account 2, the Eq. (1) is equivalent to the system of first order differential equations:

(y1(z)y2(z))=(01εr(z)0)·(y1(z)y2(z))

where yr(z) = ∂yi (z)/εr(z)=(2πλ)2εr(z). It is easy to show that the solution of the system of differential Eqs. (3) is equivalent to solving the integral equation:

y1(z)=e1+0zy2(τ)y2(z)=e20zεr(τ)y1(τ)}z[0,L]

where e 1 and e 2 are the complex constants given by the initial conditions. In order to solve this problem we are going to introduce the notation and the theorem that will be used in the simulations.

Theorem Let be εr : [0, L] → ℂ a continuous application and let be e 1, e 2 ∈ ℂ. Then, the integral equation given by:

y1(z)=e1+0zy2(τ)y2(z)=e20zεr(τ)y1(τ)}z[0,L]

has a unique solution in [0,L]. Let be f1(0), f2(0) :[0, L] → ℂ continuous applications with fi(0) = ei , i ∈ {1,2}. Let be the function succession:

(f1(n+1)(z)f2(n+1)(z))=(f1(n)(0)+0zf2(n)(τ)f2(n)(0)0zεr(τ)f1(n)(τ))

Then the limn+(f1(n)(z)f2(n)(z))=(f1(z)f2(z))2,z[0,L] and also (f1(z)f2(z)) is the solution to the integral equation previously described by (4).

Using this theorem whose demonstration is based on the Fixed Point Theorem [18], the Helmholtz Eq. (1) can be solved, obtaining the numeric limit of the functions succession 6. As a result of the theorem, we can obtain an analytical function that fits the solution in the interval [0,L].

3. Reflection gratings: Comparison of FPT method and rigorous Mathieu solution

In this section we are going to apply the FPT method to a lossless volume holographic reflection grating with a variation of the relative permittivity described by Eq. (7):

ε'r=ε0+ε1Cos(2πzΛ)

where ε 0 is the average relative permittivity and ε 1 the modulation of permittivity. Introducing Eq. (7) in the Helmholtz Eq. (1), the differential equation obtained is:

2Ex(z)z2+(2πλ)2(ε0+ε1Cos(2πzΛ))Ex(z)=0

This is a well known equation that has been solved by different authors. In particular, Kogelnik [4] solved it using coupled wave theory, and rigorous methods [6, 7, 8, 10, 19, 20] have been also applied. For this particular case, Eq. (8) is a Mathieu differential equation [2], [1], so the exact solution can be obtained as a linear combination of Mathieu functions:

Ex(z)=c1mc(a,b,πzΛ)+c2ms(a,b,πzΛ)

where c 1 and c 2 are the constant of integration and mc and ms the even and odd Mathieu functions [21] with characteristic value a0=4ε0Λ2λ2 and parameter b=2ε1Λ2λ2 respectively. If a monochromatic beam of unit amplitude is incident upon the material of thickness L, the electric field in each of the regions will be:

exp(i2πnzλ)+ρexp(i2πnzλ)ifz0
Ex(z)if0zL
τexp(i2πn(zLλ))ifzL

where ρ and τ are the amplitudes of the reflected and transmitted waves and n is the refraction index of the material, which is that assumed to be the same for the region inside and outside the grating boundaries. Assuming that the electric field must be continuous across the interfaces at (0, L), the boundary conditions will be:

1+ρ=Ex(0)
τ=Ex(L)
(Ex(z)z)z=0=2πni(ρ1)λ
(Ex(z)z)z=L=2πniτλ

In order to simplify the expressions we introduce the notation:

ζL=πLΛ,p=2inΛ,αs0=msab0,αsL=msabζL
αc0=mcab0,αcL=mcabζL
βs0=(Λπms(a,b,πzΛ)z)z=0,βsL=(Λπms(a,b,πzΛ)z)z=L
βc0=(Λπmc(a,b,πzΛ)z)z=0,βcL=(Λπmc(a,b,πzΛ)z)z=L

Taking into account the Eqs. (9) and (13)–(14)–(15)–(16) the amplitudes ρ and τ will be:

τ=2(βcLαsLαcLβsL)λβcL(pαs0λβs0)αcLp(pαs0+λβs0)+(pαc0+λβc0)(pαsL+λβsL)
ρ=λβcL(pαs0+λβs0)αcLp(pαs0+λβs0)(pαc0+λβc0)(pαsL+λβsL)λβcL(pαs0λβs0)αcLp(pαs0+λβs0)+(pαc0+λβc0)(pαsL+λβsL)
 

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.

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Figure 2 shows a typical transmission spectra obtained from Eq. (18) for a reflection holographic grating with parameters ε 0 = 2.3716, ε 1 = 0.154 and thickness L = 10 μm, Λ = 0.205 nm, as can be seen a partial band gap centered at 633 nm is obtained.

Applying the FPT method, the functions succession for this particular problem will be given by

(f1(n+1)(z)f2(n+1)(z))=((ρ+1)+0zf2(n)2πni(ρ1)λ(2πλ)20z(ε0+ε1Cos(2πτΛ))f1(n)(τ))

being f1(0) (z) = ρ + 1 and f2(0)(z)=2πni(ρ1)λ.

Figure 3 shows the local power in E-field ∣Ex (z,λ)∣2 obtained by using the Eq. (20) for wavelengths from 400 to 700 nm. In the range of 400 to 500 nm, the movie of Fig. 3 shows that ∣Ex (0,400 - 500) ∣2 is approximately 1, so there are not significant effects of the material dielectric constant for these wavelengths, being ∣τ2 = 100%. In the range of 500 to 606 nm, the number of modes decreases from 13 to 2 as the amplitude increases from 0.4 to 2. When the wavelengths are in the range of 607 to 617 nm in the first side lobe of the transmission curve (see Fig. 2), the number of modes changes from 2 to 1 and the amplitude is higher than 4 for λ = 617 nm inside the material. When the wavelengths are in the principal minimum region (λ =618-650) only a part of the mode is observed and as can be seen, the local power ∣Ex (0,618 - 650) ∣2 > ∣Ex (L,618 - 650)∣2 due to the multiple reflections inside the periodic dielectric material, with the result that, for example ∣τ(633)∣2 = 2.7 %, and the error with regard to the exact value obtained from Eq. (18) of 0.8 %. For values of λ = 650-700 nm, the local power behaviour is symmetrical to that obtained in the previously described region of λ = 618-568 nm.

 

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.

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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.

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Maximum errors (shown in Fig. 4) were obtained for the wavelength of 633 nm, with the relative error function (e) for the local power being defined as the difference in E-field between the exact Mathieu solution and that obtained with the FPT method. As can be observed in Fig. 4 the maximum errors are lower than 4%.

In order to analyze whether the periodicity of the ∣Ex (z, λ)∣2 is the same as the dielectric constant we studied the Lissajaus figures (see Fig. 5) obtained from the plot of E = ∣Ex (z, λ) ∣2 vs εr (z). As can be seen none of the wavelengths shows a periodicity equal to the dielectric constant inside the material, because the number of Lissajaus figures that can be observed for each wavelength is equal to L/Λ, so the distribution of electric field has no direct relation to the spatial variation of the dielectric constant within the medium [1].

 

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.

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4. Conclusions

We have reported a mathematical method based on the Fixed Point Theorem (FPT) that can be used to study 1D wave equations for materials that present a continuous (real or complex) dielectric constant. It was applied to the study of reflection gratings at normal incidence obtaining the exact electric field, transmission and reflection coefficients were obtained as a combination of Mathieu functions. Good agreement was found between the exact solution and that obtained with the FPT.

Acknowledgments

This work has received financial support from the Comision Interministerial de Ciencia y Tecnologia (CICYT) of Spain (Project No. MAT2004-04643-C03-03) and the Generalitat Valenciana (Project No. GVA04A-642).

References and links

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 ).

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 ).

2005 (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 ).

2004 (4)

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 ).

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]

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]

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]

2003 (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 ).

2000 (1)

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 ).

1999 (1)

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]

1983 (1)

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

1982 (1)

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

1981 (1)

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

1978 (1)

1977 (1)

1976 (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]

1969 (1)

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

1966 (1)

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

1965 (1)

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]

1964 (1)

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

Andreev, A. V.

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 ).

Balakin, A. V.

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 ).

Banerjee, P.

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

Banerjee, P. P.

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]

Bertolotti, M.

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 ).

Blaya, S.

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]

Boucher, D.

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 ).

Burckhardt, C. B.

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

Carretero, L.

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]

Casey, K. F.

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]

Centini, M.

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 ).

Diehl, D. W.

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]

Fedotov, M. V.

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 ).

Fimia, A.

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]

Gaylord, T. K.

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

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

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

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

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]

George, N.

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]

Haus, J.

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 ).

Jameson, G.

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

Jarem, J. M.

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]

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

Kaprilean, Z. A.

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]

Kivshar, Y. S.

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

Kogelnik, H.

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

Magnusson, R.

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

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]

Martorell, J.

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]

Masselin, P.

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 ).

Maystre, D.

Moharam, M. G.

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

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

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

Morozov, G. V.

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]

Mouret, G.

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 ).

Oliner, A.

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

Ozheredov, I. A.

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 ).

Sakoda, K.

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 ).

Shadirvov, I. V.

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

Shkurinov, A.

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 ).

Sibila, C.

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 ).

Sprung, D.

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]

Sukhorukov, A. A.

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

Tamir, T.

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

Tereshin, E. B.

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 ).

Trofimov, V. A.

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 ).

Ulibarrena, M.

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]

Wang, H.

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

Yeh, D.

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]

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 MTT-12 (1)

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

IEEE Trans. Microwave Theory MTT-13 (1)

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)

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]

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

J. Opt. Soc. Am. (4)

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]

D. Maystre , “ A new integral theory for dielectric coated gratings ,” J. Opt. Soc. Am.   68 , 490 ( 1978 ).
[Crossref]

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

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

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

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

Supplementary Material (2)

» Media 1: GIF (517 KB)     
» Media 2: GIF (976 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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