Abstract

This paper extends the area of application of the Fourier modal method (FMM) from periodic structures to aperiodic ones, in particular for plane-wave illumination at arbitrary angles. This is achieved by placing perfectly matched layers at the lateral sides of the computational domain and reformulating the governing equations in terms of a contrast field that does not contain the incoming field. As a result of the reformulation, the homogeneous system of second-order ordinary differential equations from the original FMM becomes non-homogeneous. Its solution is derived analytically and used in the established FMM framework. The technique is demonstrated on a simple problem of planar scattering of TE-polarized light by a single rectangular line.

© 2010 Optical Society of America

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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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2008 (1)

S. Rafler, P. Götz, M. Petschow, T. Schuster, K. Frenner, and W. Osten, “Investigation of methods to set up the normal vector field for the differential method,” Proc. SPIE 6995, 69960Y (2008).

2007 (4)

2006 (1)

R. J. Swift and S. A. Wirkus, A Course in Ordinary Differential Equations (CRC Press, 2006).

2005 (2)

A. Taflove and S. C. Hagness, Computational Electrodynamics: The Finite-Difference Time-Domain Method, 3rd ed. (Artech House Publishers, 2005).

J. P. Hugonin and P. Lalanne, “Perfectly matched layers as nonlinear coordinate transforms: a generalized formalization,” J. Opt. Soc. Am. A 22, 1844–1849 (2005).
[CrossRef]

2001 (2)

2000 (2)

1999 (1)

1997 (1)

W. C. Chew, J. M. Jin, and E. Michielssen, “Complex coordinate stretching as a generalized absorbing boundary condition,” Microwave Opt. Technol. Lett. 15, 363–369 (1997).
[CrossRef]

1996 (4)

1995 (4)

1994 (2)

W. C. Chew and W. H. Weedon, “A 3D perfectly matched medium from modified Maxwell’s equations with stretched coordinates,” Microwave Opt. Technol. Lett. 7, 599–604 (1994).
[CrossRef]

J. Berenger, “A perfectly matched layer for the absorption of electromagnetic waves,” J. Comput. Phys. 114, 185–200 (1994).
[CrossRef]

1990 (1)

K. A. Michalski and D. Zheng, “Electromagnetic scattering and radiation by surfaces of arbitrary shape in layered media. I—Theory. II—Implementation and results for contiguous half-spaces,” IEEE Trans. Antennas Propag. 38, 335–352 (1990).
[CrossRef]

1982 (1)

1981 (1)

1980 (1)

R. Petit, Electromagnetic Theory of Gratings(Topics in Applied Physics) (Springer, 1980).
[CrossRef]

1949 (1)

A. Sommerfeld, Partial Differential Equations in Physics(Pure and Applied Mathematics: A Series of Monographs and Textbooks, Vol. 1) (Academic Press, 1949).

Bai, B.

Baida, F.

P. Lalanne, M. Besbes, J. P. Hugonin, S. van Haver, O. T. A. Janssen, A. M. Nugrowati, M. Xu, S. F. Pereira, H. Urbach, A. S. van de Nes, P. Bienstman, G. Granet, A. Moreau, S. Helfert, M. Sukharev, T. Seideman, F. Baida, B. Guizal, and D. van Labeke, “Numerical analysis of a slit-groove diffraction problem,” J. Eur. Opt. Soc. Rapid Publ. 207022, (2007).
[CrossRef]

Berenger, J.

J. Berenger, “A perfectly matched layer for the absorption of electromagnetic waves,” J. Comput. Phys. 114, 185–200 (1994).
[CrossRef]

Besbes, M.

P. Lalanne, M. Besbes, J. P. Hugonin, S. van Haver, O. T. A. Janssen, A. M. Nugrowati, M. Xu, S. F. Pereira, H. Urbach, A. S. van de Nes, P. Bienstman, G. Granet, A. Moreau, S. Helfert, M. Sukharev, T. Seideman, F. Baida, B. Guizal, and D. van Labeke, “Numerical analysis of a slit-groove diffraction problem,” J. Eur. Opt. Soc. Rapid Publ. 207022, (2007).
[CrossRef]

Bienstman, P.

P. Lalanne, M. Besbes, J. P. Hugonin, S. van Haver, O. T. A. Janssen, A. M. Nugrowati, M. Xu, S. F. Pereira, H. Urbach, A. S. van de Nes, P. Bienstman, G. Granet, A. Moreau, S. Helfert, M. Sukharev, T. Seideman, F. Baida, B. Guizal, and D. van Labeke, “Numerical analysis of a slit-groove diffraction problem,” J. Eur. Opt. Soc. Rapid Publ. 207022, (2007).
[CrossRef]

Cao, Q.

Chew, W. C.

W. C. Chew, J. M. Jin, and E. Michielssen, “Complex coordinate stretching as a generalized absorbing boundary condition,” Microwave Opt. Technol. Lett. 15, 363–369 (1997).
[CrossRef]

W. C. Chew and W. H. Weedon, “A 3D perfectly matched medium from modified Maxwell’s equations with stretched coordinates,” Microwave Opt. Technol. Lett. 7, 599–604 (1994).
[CrossRef]

Edee, K.

Frenner, K.

S. Rafler, P. Götz, M. Petschow, T. Schuster, K. Frenner, and W. Osten, “Investigation of methods to set up the normal vector field for the differential method,” Proc. SPIE 6995, 69960Y (2008).

Gaylord, T. K.

Götz, P.

S. Rafler, P. Götz, M. Petschow, T. Schuster, K. Frenner, and W. Osten, “Investigation of methods to set up the normal vector field for the differential method,” Proc. SPIE 6995, 69960Y (2008).

Granet, G.

Grann, E. B.

Guizal, B.

P. Lalanne, M. Besbes, J. P. Hugonin, S. van Haver, O. T. A. Janssen, A. M. Nugrowati, M. Xu, S. F. Pereira, H. Urbach, A. S. van de Nes, P. Bienstman, G. Granet, A. Moreau, S. Helfert, M. Sukharev, T. Seideman, F. Baida, B. Guizal, and D. van Labeke, “Numerical analysis of a slit-groove diffraction problem,” J. Eur. Opt. Soc. Rapid Publ. 207022, (2007).
[CrossRef]

G. Granet and B. Guizal, “Efficient implementation of the coupled-wave method for metallic lamellar gratings in TM polarization,” J. Opt. Soc. Am. A 13, 1019–1023 (1996).
[CrossRef]

Hagness, S. C.

A. Taflove and S. C. Hagness, Computational Electrodynamics: The Finite-Difference Time-Domain Method, 3rd ed. (Artech House Publishers, 2005).

Helfert, S.

P. Lalanne, M. Besbes, J. P. Hugonin, S. van Haver, O. T. A. Janssen, A. M. Nugrowati, M. Xu, S. F. Pereira, H. Urbach, A. S. van de Nes, P. Bienstman, G. Granet, A. Moreau, S. Helfert, M. Sukharev, T. Seideman, F. Baida, B. Guizal, and D. van Labeke, “Numerical analysis of a slit-groove diffraction problem,” J. Eur. Opt. Soc. Rapid Publ. 207022, (2007).
[CrossRef]

Hugonin, J. P.

P. Lalanne, M. Besbes, J. P. Hugonin, S. van Haver, O. T. A. Janssen, A. M. Nugrowati, M. Xu, S. F. Pereira, H. Urbach, A. S. van de Nes, P. Bienstman, G. Granet, A. Moreau, S. Helfert, M. Sukharev, T. Seideman, F. Baida, B. Guizal, and D. van Labeke, “Numerical analysis of a slit-groove diffraction problem,” J. Eur. Opt. Soc. Rapid Publ. 207022, (2007).
[CrossRef]

J. P. Hugonin and P. Lalanne, “Perfectly matched layers as nonlinear coordinate transforms: a generalized formalization,” J. Opt. Soc. Am. A 22, 1844–1849 (2005).
[CrossRef]

Hugonin, J.-P.

Janssen, O. T. A.

P. Lalanne, M. Besbes, J. P. Hugonin, S. van Haver, O. T. A. Janssen, A. M. Nugrowati, M. Xu, S. F. Pereira, H. Urbach, A. S. van de Nes, P. Bienstman, G. Granet, A. Moreau, S. Helfert, M. Sukharev, T. Seideman, F. Baida, B. Guizal, and D. van Labeke, “Numerical analysis of a slit-groove diffraction problem,” J. Eur. Opt. Soc. Rapid Publ. 207022, (2007).
[CrossRef]

Jin, J. M.

W. C. Chew, J. M. Jin, and E. Michielssen, “Complex coordinate stretching as a generalized absorbing boundary condition,” Microwave Opt. Technol. Lett. 15, 363–369 (1997).
[CrossRef]

Kerwien, N.

Lalanne, P.

Li, L.

Michalski, K. A.

K. A. Michalski and D. Zheng, “Electromagnetic scattering and radiation by surfaces of arbitrary shape in layered media. I—Theory. II—Implementation and results for contiguous half-spaces,” IEEE Trans. Antennas Propag. 38, 335–352 (1990).
[CrossRef]

Michielssen, E.

W. C. Chew, J. M. Jin, and E. Michielssen, “Complex coordinate stretching as a generalized absorbing boundary condition,” Microwave Opt. Technol. Lett. 15, 363–369 (1997).
[CrossRef]

Moharam, M. G.

Moreau, A.

P. Lalanne, M. Besbes, J. P. Hugonin, S. van Haver, O. T. A. Janssen, A. M. Nugrowati, M. Xu, S. F. Pereira, H. Urbach, A. S. van de Nes, P. Bienstman, G. Granet, A. Moreau, S. Helfert, M. Sukharev, T. Seideman, F. Baida, B. Guizal, and D. van Labeke, “Numerical analysis of a slit-groove diffraction problem,” J. Eur. Opt. Soc. Rapid Publ. 207022, (2007).
[CrossRef]

Morf, R. H.

Morris, G. M.

Nevière, M.

Noponen, E.

J. Saarinen, E. Noponen, and J. P. Turunen, “Guided-mode resonance filters of finite aperture,” Opt. Eng. 34, 2560–2566 (1995).
[CrossRef]

Nugrowati, A. M.

P. Lalanne, M. Besbes, J. P. Hugonin, S. van Haver, O. T. A. Janssen, A. M. Nugrowati, M. Xu, S. F. Pereira, H. Urbach, A. S. van de Nes, P. Bienstman, G. Granet, A. Moreau, S. Helfert, M. Sukharev, T. Seideman, F. Baida, B. Guizal, and D. van Labeke, “Numerical analysis of a slit-groove diffraction problem,” J. Eur. Opt. Soc. Rapid Publ. 207022, (2007).
[CrossRef]

Osten, W.

S. Rafler, P. Götz, M. Petschow, T. Schuster, K. Frenner, and W. Osten, “Investigation of methods to set up the normal vector field for the differential method,” Proc. SPIE 6995, 69960Y (2008).

T. Schuster, J. Ruoff, N. Kerwien, S. Rafler, and W. Osten, “Normal vector method for convergence improvement using the RCWA for crossed gratings,” J. Opt. Soc. Am. A 24, 2880–2890 (2007).
[CrossRef]

Pereira, S. F.

P. Lalanne, M. Besbes, J. P. Hugonin, S. van Haver, O. T. A. Janssen, A. M. Nugrowati, M. Xu, S. F. Pereira, H. Urbach, A. S. van de Nes, P. Bienstman, G. Granet, A. Moreau, S. Helfert, M. Sukharev, T. Seideman, F. Baida, B. Guizal, and D. van Labeke, “Numerical analysis of a slit-groove diffraction problem,” J. Eur. Opt. Soc. Rapid Publ. 207022, (2007).
[CrossRef]

Petit, R.

R. Petit, Electromagnetic Theory of Gratings(Topics in Applied Physics) (Springer, 1980).
[CrossRef]

Petschow, M.

S. Rafler, P. Götz, M. Petschow, T. Schuster, K. Frenner, and W. Osten, “Investigation of methods to set up the normal vector field for the differential method,” Proc. SPIE 6995, 69960Y (2008).

Plumey, J.-P.

Pommet, D. A.

Popov, E.

Rafler, S.

S. Rafler, P. Götz, M. Petschow, T. Schuster, K. Frenner, and W. Osten, “Investigation of methods to set up the normal vector field for the differential method,” Proc. SPIE 6995, 69960Y (2008).

T. Schuster, J. Ruoff, N. Kerwien, S. Rafler, and W. Osten, “Normal vector method for convergence improvement using the RCWA for crossed gratings,” J. Opt. Soc. Am. A 24, 2880–2890 (2007).
[CrossRef]

Ruoff, J.

Saarinen, J.

J. Saarinen, E. Noponen, and J. P. Turunen, “Guided-mode resonance filters of finite aperture,” Opt. Eng. 34, 2560–2566 (1995).
[CrossRef]

Schuster, T.

S. Rafler, P. Götz, M. Petschow, T. Schuster, K. Frenner, and W. Osten, “Investigation of methods to set up the normal vector field for the differential method,” Proc. SPIE 6995, 69960Y (2008).

T. Schuster, J. Ruoff, N. Kerwien, S. Rafler, and W. Osten, “Normal vector method for convergence improvement using the RCWA for crossed gratings,” J. Opt. Soc. Am. A 24, 2880–2890 (2007).
[CrossRef]

Seideman, T.

P. Lalanne, M. Besbes, J. P. Hugonin, S. van Haver, O. T. A. Janssen, A. M. Nugrowati, M. Xu, S. F. Pereira, H. Urbach, A. S. van de Nes, P. Bienstman, G. Granet, A. Moreau, S. Helfert, M. Sukharev, T. Seideman, F. Baida, B. Guizal, and D. van Labeke, “Numerical analysis of a slit-groove diffraction problem,” J. Eur. Opt. Soc. Rapid Publ. 207022, (2007).
[CrossRef]

Silberstein, E.

Sommerfeld, A.

A. Sommerfeld, Partial Differential Equations in Physics(Pure and Applied Mathematics: A Series of Monographs and Textbooks, Vol. 1) (Academic Press, 1949).

Sukharev, M.

P. Lalanne, M. Besbes, J. P. Hugonin, S. van Haver, O. T. A. Janssen, A. M. Nugrowati, M. Xu, S. F. Pereira, H. Urbach, A. S. van de Nes, P. Bienstman, G. Granet, A. Moreau, S. Helfert, M. Sukharev, T. Seideman, F. Baida, B. Guizal, and D. van Labeke, “Numerical analysis of a slit-groove diffraction problem,” J. Eur. Opt. Soc. Rapid Publ. 207022, (2007).
[CrossRef]

Swift, R. J.

R. J. Swift and S. A. Wirkus, A Course in Ordinary Differential Equations (CRC Press, 2006).

Taflove, A.

A. Taflove and S. C. Hagness, Computational Electrodynamics: The Finite-Difference Time-Domain Method, 3rd ed. (Artech House Publishers, 2005).

Turunen, J.

Turunen, J. P.

J. Saarinen, E. Noponen, and J. P. Turunen, “Guided-mode resonance filters of finite aperture,” Opt. Eng. 34, 2560–2566 (1995).
[CrossRef]

Urbach, H.

P. Lalanne, M. Besbes, J. P. Hugonin, S. van Haver, O. T. A. Janssen, A. M. Nugrowati, M. Xu, S. F. Pereira, H. Urbach, A. S. van de Nes, P. Bienstman, G. Granet, A. Moreau, S. Helfert, M. Sukharev, T. Seideman, F. Baida, B. Guizal, and D. van Labeke, “Numerical analysis of a slit-groove diffraction problem,” J. Eur. Opt. Soc. Rapid Publ. 207022, (2007).
[CrossRef]

van de Nes, A. S.

P. Lalanne, M. Besbes, J. P. Hugonin, S. van Haver, O. T. A. Janssen, A. M. Nugrowati, M. Xu, S. F. Pereira, H. Urbach, A. S. van de Nes, P. Bienstman, G. Granet, A. Moreau, S. Helfert, M. Sukharev, T. Seideman, F. Baida, B. Guizal, and D. van Labeke, “Numerical analysis of a slit-groove diffraction problem,” J. Eur. Opt. Soc. Rapid Publ. 207022, (2007).
[CrossRef]

van Haver, S.

P. Lalanne, M. Besbes, J. P. Hugonin, S. van Haver, O. T. A. Janssen, A. M. Nugrowati, M. Xu, S. F. Pereira, H. Urbach, A. S. van de Nes, P. Bienstman, G. Granet, A. Moreau, S. Helfert, M. Sukharev, T. Seideman, F. Baida, B. Guizal, and D. van Labeke, “Numerical analysis of a slit-groove diffraction problem,” J. Eur. Opt. Soc. Rapid Publ. 207022, (2007).
[CrossRef]

van Labeke, D.

P. Lalanne, M. Besbes, J. P. Hugonin, S. van Haver, O. T. A. Janssen, A. M. Nugrowati, M. Xu, S. F. Pereira, H. Urbach, A. S. van de Nes, P. Bienstman, G. Granet, A. Moreau, S. Helfert, M. Sukharev, T. Seideman, F. Baida, B. Guizal, and D. van Labeke, “Numerical analysis of a slit-groove diffraction problem,” J. Eur. Opt. Soc. Rapid Publ. 207022, (2007).
[CrossRef]

Weedon, W. H.

W. C. Chew and W. H. Weedon, “A 3D perfectly matched medium from modified Maxwell’s equations with stretched coordinates,” Microwave Opt. Technol. Lett. 7, 599–604 (1994).
[CrossRef]

Wirkus, S. A.

R. J. Swift and S. A. Wirkus, A Course in Ordinary Differential Equations (CRC Press, 2006).

Xu, M.

P. Lalanne, M. Besbes, J. P. Hugonin, S. van Haver, O. T. A. Janssen, A. M. Nugrowati, M. Xu, S. F. Pereira, H. Urbach, A. S. van de Nes, P. Bienstman, G. Granet, A. Moreau, S. Helfert, M. Sukharev, T. Seideman, F. Baida, B. Guizal, and D. van Labeke, “Numerical analysis of a slit-groove diffraction problem,” J. Eur. Opt. Soc. Rapid Publ. 207022, (2007).
[CrossRef]

Zheng, D.

K. A. Michalski and D. Zheng, “Electromagnetic scattering and radiation by surfaces of arbitrary shape in layered media. I—Theory. II—Implementation and results for contiguous half-spaces,” IEEE Trans. Antennas Propag. 38, 335–352 (1990).
[CrossRef]

IEEE Trans. Antennas Propag. (1)

K. A. Michalski and D. Zheng, “Electromagnetic scattering and radiation by surfaces of arbitrary shape in layered media. I—Theory. II—Implementation and results for contiguous half-spaces,” IEEE Trans. Antennas Propag. 38, 335–352 (1990).
[CrossRef]

J. Comput. Phys. (1)

J. Berenger, “A perfectly matched layer for the absorption of electromagnetic waves,” J. Comput. Phys. 114, 185–200 (1994).
[CrossRef]

J. Eur. Opt. Soc. Rapid Publ. (1)

P. Lalanne, M. Besbes, J. P. Hugonin, S. van Haver, O. T. A. Janssen, A. M. Nugrowati, M. Xu, S. F. Pereira, H. Urbach, A. S. van de Nes, P. Bienstman, G. Granet, A. Moreau, S. Helfert, M. Sukharev, T. Seideman, F. Baida, B. Guizal, and D. van Labeke, “Numerical analysis of a slit-groove diffraction problem,” J. Eur. Opt. Soc. Rapid Publ. 207022, (2007).
[CrossRef]

J. Opt. Soc. Am. (2)

J. Opt. Soc. Am. A (14)

G. Granet, “Reformulation of the lamellar grating problem through the concept of adaptive spatial resolution,” J. Opt. Soc. Am. A 16, 2510–2516 (1999).
[CrossRef]

L. Li, “Formulation and comparison of two recursive matrix algorithms for modeling layered diffraction gratings,” J. Opt. Soc. Am. A 13, 1024–1035 (1996).
[CrossRef]

R. H. Morf, “Exponentially convergent and numerically efficient solution of Maxwell’s equations for lamellar gratings,” J. Opt. Soc. Am. A 12, 1043–1056 (1995).
[CrossRef]

K. Edee, G. Granet, and J.-P. Plumey, “Complex coordinate implementation in the curvilinear coordinate method: application to plane-wave diffraction by nonperiodic rough surfaces,” J. Opt. Soc. Am. A 24, 1097–1102 (2007).
[CrossRef]

M. G. Moharam, E. B. Grann, D. A. Pommet, and T. K. Gaylord, “Formulation for stable and efficient implementation of the rigorous coupled-wave analysis of binary gratings,” J. Opt. Soc. Am. A 12, 1068–1076 (1995).
[CrossRef]

M. G. Moharam, D. A. Pommet, E. B. Grann, and T. K. Gaylord, “Stable implementation of the rigorous coupled-wave analysis for surface-relief gratings: enhanced transmittance matrix approach,” J. Opt. Soc. Am. A 12, 1077–1086 (1995).
[CrossRef]

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Figures (9)

Fig. 1
Fig. 1

Geometry of the periodic model problem and division into layers.

Fig. 2
Fig. 2

Imaginary part β ( x ) of the transformation (10).

Fig. 3
Fig. 3

Problems P 1 (top) and P 2 (bottom) have equal solutions on Ω 0 (for an ideal non-reflecting PML).

Fig. 4
Fig. 4

Incident field on the complex contour x ̃ for z = 0 .

Fig. 5
Fig. 5

Permittivities involved in the source term of Eq. (23).

Fig. 6
Fig. 6

Background problem.

Fig. 7
Fig. 7

Source term in the contrast-field equation.

Fig. 8
Fig. 8

Contrast field computed with aFMM-CFF.

Fig. 9
Fig. 9

Logarithmic plots log 10 e 1 (left) and log 10 e 2 (right) of the errors defined in Eq. (50).

Equations (65)

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2 x 2 E y ( x , z ) + 2 z 2 E y ( x , z ) + k 0 2 ε ( x , z ) E y ( x , z ) = 0 ,
E y inc ( x , z ) = exp ( i ( k x 0 x + k z 0 z ) ) ,
E y inc ( 0 , z ) = E y inc ( Λ , z ) exp ( i k x 0 Λ ) .
2 x 2 E y , l ( x , z ) + 2 z 2 E y , l ( x , z ) + ε l ( x ) E y , l ( x , z ) = 0 .
E y , l ( x , z ) = n = s n , l ( z ) exp ( i k x n x ) ,
ε l ( x ) = n = ε ̂ n , l exp ( i 2 π n Λ x ) ,
k x n = k 0 n 1 sin θ n 2 π Λ , n Z .
k x n 2 s n , l ( z ) + d 2 d z 2 s n , l ( z ) + m = N N ε ̂ n m , l s m , l ( z ) = 0 ,
n = N , , N ,
d 2 d z 2 s l ( z ) = k 0 2 A l s l ( z ) , with A l = K x 2 E l ,
s l ( z ) = s l + ( z ) + s l ( z ) = W l ( exp ( Q l ( z h l 1 ) ) c l + + exp ( Q l ( z h l ) ) c l ) ,
s 1 + ( h 1 ) = d 0 exp ( i k z 0 h 1 ) ,
s 3 ( h 2 ) = 0 .
x ̃ = x + i β ( x ) , x R .
exp ( i ( k x 0 x ̃ + k z 0 z ) ) = exp ( i ( k x 0 x + k z 0 z ) ) exp ( k x 0 β ( x ) ) .
E ( x ) { 1 } E ̃ ( x ̃ ) { 2 } E ̃ ( x ) , with x R , x ̃ C .
x ̃ = f ( x ) = x + i β ( x )
x ̃ = d x d x ̃ x = 1 f ( x ) x .
1 f ( x ) x ( 1 f ( x ) x E ̃ y , l ( x , z ) ) + 2 z 2 E ̃ y , l ( x , z ) + k 0 2 ε l ( x ) E ̃ y , l ( x , z ) = 0 .
d 2 d z 2 s l ( z ) = k 0 2 A l s l ( z ) , A l = ( F K x ) 2 E l ,
E ̃ y = E ̃ y inc + E ̃ y s .
E y inc ( x , z ) = exp ( i ( k x 0 x + k z 0 z ) ) .
E ̃ y inc ( x ̃ , z ) = exp ( i ( k x 0 x ̃ + k z 0 z ) ) = exp ( i ( k x 0 x + k z 0 z ) ) exp ( k x 0 β ( x ) ) .
E ̃ y inc ( f ( 0 ) , z ) E ̃ y inc ( f ( Λ ) , z ) exp ( i k x 0 Λ ) .
1 f ( x ) x ( 1 f ( x ) x E ̃ y ) + 2 z 2 E ̃ y + k 0 2 ε ( x , z ) E ̃ y = 0 .
E ̃ = E ̃ c + E ̃ b ,
1 f ( x ) x ( 1 f ( x ) x E ̃ y b ) + 2 z 2 E ̃ y b + k 0 2 ε b ( x , z ) E ̃ y b = 0 .
1 f ( x ) x ( 1 f ( x ) x E ̃ y c ) + 2 z 2 E ̃ y c + k 0 2 ε ( x , z ) E ̃ y c = k 0 2 ( ε ( x , z ) ε b ( x , z ) ) E ̃ y b .
2 x 2 E y b + 2 z 2 E y b + k 0 2 ε b ( x , z ) E y b = 0 .
E y , 2 b = E y inc + E y r = exp ( q 2 z ) exp ( i k x 0 x ) + r exp ( q 2 z ) exp ( i k x 0 x ) .
E y , 3 b = E y t = t exp ( q 3 ( z h ) ) exp ( i k x 0 x ) ,
E y inc ( x , h ) + E y r ( x , h ) = E y t ( x , h ) ,
z E y inc ( x , h ) + z E y r ( x , h ) = z E y t ( x , h ) .
r b 1 + b = t ,
r q 2 b 1 q 2 b = t q 3 .
r = q 2 q 3 q 2 + q 3 b 2 , t = 2 q 2 q 2 + q 3 b .
1 f ( x ) x ( 1 f ( x ) x E ̃ y , l c ) + 2 z 2 E ̃ y , l c k 0 2 ε l ( x ) E ̃ y , l c = 0 , l = 1 , 3 .
d 2 d z 2 s ̃ l ( z ) = k 0 2 A l s ̃ l ( z ) , l = 1 , 3.
1 f ( x ) x ( 1 f ( x ) x E ̃ y , 2 c ) + 2 z 2 E ̃ y , 2 c + k 0 2 ε 2 ( x ) E ̃ y , 2 c = k 0 2 ( ε 2 ( x ) ε 2 b ) E y , 2 b .
E ̃ y , 2 c ( x , z ) = n = s ̃ 2 , n ( z ) exp ( i k x n x ) ,
E y , 2 b ( x , z ) = exp ( q 2 z ) exp ( i k x 0 x ) + r exp ( q 2 z ) exp ( i k x 0 x ) ,
ε 2 ( x ) = n = ε ̂ 2 , n exp ( i 2 π n Λ x ) ,
1 f ( x ) = n = f ̂ n exp ( i 2 π n Λ x ) .
n = N N m = N N ( f ̂ n m k x m r = N N f ̂ m r k x r s ̃ 2 , r ( z ) exp ( i k x n x ) ) + n = N N d 2 d z 2 s ̃ 2 , n ( z ) exp ( i k x n x ) + k 0 2 n = N N m = N N ε ̂ 2 , n m s ̃ 2 , m ( z ) exp ( i k x n x ) = k 0 2 n = N N m = N N ( ε ̂ 2 , n m ε 2 b δ n m ) ( exp ( q 2 z ) + r exp ( q 2 z ) ) δ n exp ( i k x n x ) ,
m = N N ( f ̂ n m k x m r = N N f ̂ m r k x r s ̃ 2 , r ( z ) ) + d 2 d z 2 s ̃ 2 , n ( z ) + k 0 2 m = N N ε ̂ 2 , n m s ̃ 2 , m ( z ) = k 0 2 m = N N ( ε ̂ 2 , n m ε 2 b δ n m ) δ n ( exp ( q 2 z ) + r exp ( q 2 z ) ) , n = N , , N .
d 2 d z 2 s ̃ 2 ( z ) = k 0 2 A 2 s ̃ 2 ( z ) + k 0 2 ( ε 2 b I E 2 ) d 0 ( exp ( q 2 z ) + r exp ( q 2 z ) ) ,
A 2 = ( F K x ) 2 E 2 .
s ̃ 2 = s ̃ 2 , hom + s ̃ 2 , part .
s ̃ 2 , part ( z ) = p ( exp ( q 2 z ) + r exp ( q 2 z ) ) ,
( k 0 2 A 2 q 2 2 I ) p = k 0 2 ( ε 2 b I E 2 ) d 0 .
s ̃ 2 ( z ) = W 2 ( exp ( k 0 Q 2 z ) c 2 + + exp ( k 0 Q 2 ( z h ) ) c 2 ) + p ( exp ( q 2 z ) + r exp ( q 2 z ) ) .
s ̃ 1 ( 0 ) = s ̃ 2 ( 0 ) ,
1 k 0 d d z s ̃ 1 ( 0 ) = 1 k 0 d d z s ̃ 2 ( 0 ) ,
s ̃ 2 ( h ) = s ̃ 3 ( h ) ,
1 k 0 d d z s ̃ 2 ( h ) = 1 k 0 d d z s ̃ 3 ( h ) .
s ̃ 1 + ( 0 ) = 0 ,
s ̃ 3 ( h ) = 0 ,
[ W 1 W 1 Q 1 ] c 1 = [ W 2 W 2 X 2 W 2 Q 2 W 2 Q 2 X 2 ] [ c 2 + c 2 ] + [ s ̃ 2 , part ( 0 ) k 0 1 s ̃ 2 , part ( 0 ) ] ,
[ W 2 X 2 W 2 W 2 Q 2 X 2 W 2 Q 2 ] [ c 2 + c 2 ] + [ s ̃ 2 , part ( h ) k 0 1 s ̃ 2 , part ( h ) ] = [ W 3 W 3 Q 3 ] c 3 + ,
s ̃ 2 , part ( z ) = p ( exp ( q 2 z ) + r exp ( q 2 z ) ) ,
s ̃ 2 , part ( z ) = p ( q 2 exp ( q 2 z ) + r q 2 exp ( q 2 z ) ) .
x ̃ = f ( x ) = { x + i σ 0 | x x l | ( p + 1 ) ( p + 1 ) , 0 x x l x , x l < x < x r , x i σ 0 | x x r | ( p + 1 ) ( p + 1 ) , x r x Λ }
d d x f ( x ) = { 1 + i σ 0 | x x l | p , 0 x x l 1 , x l < x < x r . 1 i σ 0 | x x r | p , x r x Λ }
e 1 ( N , Λ ) = E y c ( N , Λ ) E ref 2 ,
e 2 ( N , σ 0 ) = E ̃ y c ( N , σ 0 ) E ref 2 ,

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