Abstract

We consider the numerical scattering of plane waves by a metallic diffraction grating with a single defect. Besides different diffracted orders, a perturbed scattered field with arbitrary reflection direction is generated by the defect. We transform the diffraction grating into a closed waveguide by introducing a perfectly matched layer. The diffracted field is solved by applying pseudoperiodic boundary conditions on cell boundaries. Then we take two steps to resolve the perturbed scattered field. On the defect cell it is obtained by solving the governing wave equation with absorbing boundary conditions derived by a fast recursive doubling procedure. On the rest of the domain the perturbed scattered field is computed by using the recursive matrix operators efficiently. An optical theorem is employed to evaluate the proposed method.

© 2008 Optical Society of America

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References

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  1. R.Petit, ed., Electromagnetic Theory of Gratings, Vol. 22 of Topics in Current Physics (Springer-Verlag, 1980).
    [CrossRef]
  2. 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]
  3. O. P. Bruno and F. Reitich, “Numerical solution of diffraction problems: A method of variation of boundaries,” J. Opt. Soc. Am. A 10, 1168-1175 (1993).
    [CrossRef]
  4. H. Ammari and G. Bao, “Scattering by a nonhomogeneous object embedded in a perieodic structure,” C. R. Acad. Sci. Paris 330, 333-338 (2000).
  5. H. Ammari and G. Bao, “Maxwell's equations in a perturbed periodic structure,” Adv. Comput. Math. 16, 99-112 (2002).
    [CrossRef]
  6. A. S. Bonnet-Bendhia and K. Ramdani, “Diffraction by an acoustic grating perturbed by a bounded obstacle,” Adv. Comput. Math. 16, 113-138 (2002).
    [CrossRef]
  7. P. Joly, J. R. Li, and S. Fliss, “Exact boundary conditions for periodic waveguides containing a local perturbation,” Comm. Comp. Phys. 1, 945-973 (2006).
  8. K. Hattori and J. Nakayama, “Scattering of TE plane wave from periodic grating with single defect,” IEICE Trans. Electron. E90-C, 312-319 (2007).
    [CrossRef]
  9. J. P. Berenger, “A perfectly matched layer for the absorption of electromagnetic waves,” J. Comput. Phys. 114, 185-200 (1994).
    [CrossRef]
  10. L. Yuan and Y. Y. Lu, “A recursive doubling Dirichlet-to-Neumann map method for periodic waveguides,” J. Lightwave Technol. 25, 3649-3656 (2007).
    [CrossRef]
  11. H. Han, M. Ehrhardt, and C. Zheng, “Numerical simulation of waves in periodic structures,” Comm. Comp. Phys. 5,849-872 (2009).
  12. B. Bao, Z. Chen, and H. Wu, “An adaptive finite element method for diffraction gratings,” J. Opt. Soc. Am. A 22, 1106-1114 (2005).
    [CrossRef]
  13. A. Kirsch, “Diffraction by Periodic Structures,” in Inverse Problems in Mathematical Physics, Lecture Note in Physics 422 (Springer, 1993), pp. 87-102.
    [CrossRef]
  14. S. N. Chandler-Wilde, “The impedance boundary value problem for the Helmholtz equation in a halfplane,” Math. Methods Appl. Sci. 20, 813-840 (1997).
    [CrossRef]
  15. S. N. Chandler-Wilde and A. T. Peplow, “A boundary integral equation formulation for the Helmholtz equation in a locally perturbed half-plane,” Z. Angew. Math. Mech. 85, 79-88 (2005).
    [CrossRef]
  16. M. Duran, I. Muga, and J. C. Nedelec, “The Helmholtz equation in a locally perturbed half-plane with passive boundary,” IMA J. Appl. Math. 71, 853-876 (2006).
    [CrossRef]
  17. S. N. Chandler-Wilde and P. Monk, “Existence, uniqueness, and variational methods for scattering by unbounded rough surfaces,” SIAM J. Math. Anal. 37, 598-618 (2005).
    [CrossRef]
  18. S. N. Chandler-Wilde, P. Monk, and M. Thomas, “The mathematics of scattering by unbounded, rough, inhomogeneous layer,” J. Comput. Appl. Math. 204, 549-559 (2007).
    [CrossRef]
  19. P. McIver, C. M. Linton, and M. McIver, “Construction of trapped modes for wave guides and diffraction gratings,” Proc. R. Soc. London, Ser. A 454, 2593-2616 (1998).
    [CrossRef]
  20. B. P. Belinskii, “Optical theorem for the scattering of waves in an elastic plate,” J. Math. Sci. (N.Y.) 20, 1758-1760 (1998).

2009 (1)

H. Han, M. Ehrhardt, and C. Zheng, “Numerical simulation of waves in periodic structures,” Comm. Comp. Phys. 5,849-872 (2009).

2007 (3)

K. Hattori and J. Nakayama, “Scattering of TE plane wave from periodic grating with single defect,” IEICE Trans. Electron. E90-C, 312-319 (2007).
[CrossRef]

S. N. Chandler-Wilde, P. Monk, and M. Thomas, “The mathematics of scattering by unbounded, rough, inhomogeneous layer,” J. Comput. Appl. Math. 204, 549-559 (2007).
[CrossRef]

L. Yuan and Y. Y. Lu, “A recursive doubling Dirichlet-to-Neumann map method for periodic waveguides,” J. Lightwave Technol. 25, 3649-3656 (2007).
[CrossRef]

2006 (2)

M. Duran, I. Muga, and J. C. Nedelec, “The Helmholtz equation in a locally perturbed half-plane with passive boundary,” IMA J. Appl. Math. 71, 853-876 (2006).
[CrossRef]

P. Joly, J. R. Li, and S. Fliss, “Exact boundary conditions for periodic waveguides containing a local perturbation,” Comm. Comp. Phys. 1, 945-973 (2006).

2005 (3)

S. N. Chandler-Wilde and P. Monk, “Existence, uniqueness, and variational methods for scattering by unbounded rough surfaces,” SIAM J. Math. Anal. 37, 598-618 (2005).
[CrossRef]

S. N. Chandler-Wilde and A. T. Peplow, “A boundary integral equation formulation for the Helmholtz equation in a locally perturbed half-plane,” Z. Angew. Math. Mech. 85, 79-88 (2005).
[CrossRef]

B. Bao, Z. Chen, and H. Wu, “An adaptive finite element method for diffraction gratings,” J. Opt. Soc. Am. A 22, 1106-1114 (2005).
[CrossRef]

2002 (2)

H. Ammari and G. Bao, “Maxwell's equations in a perturbed periodic structure,” Adv. Comput. Math. 16, 99-112 (2002).
[CrossRef]

A. S. Bonnet-Bendhia and K. Ramdani, “Diffraction by an acoustic grating perturbed by a bounded obstacle,” Adv. Comput. Math. 16, 113-138 (2002).
[CrossRef]

2000 (1)

H. Ammari and G. Bao, “Scattering by a nonhomogeneous object embedded in a perieodic structure,” C. R. Acad. Sci. Paris 330, 333-338 (2000).

1998 (2)

P. McIver, C. M. Linton, and M. McIver, “Construction of trapped modes for wave guides and diffraction gratings,” Proc. R. Soc. London, Ser. A 454, 2593-2616 (1998).
[CrossRef]

B. P. Belinskii, “Optical theorem for the scattering of waves in an elastic plate,” J. Math. Sci. (N.Y.) 20, 1758-1760 (1998).

1997 (1)

S. N. Chandler-Wilde, “The impedance boundary value problem for the Helmholtz equation in a halfplane,” Math. Methods Appl. Sci. 20, 813-840 (1997).
[CrossRef]

1996 (1)

1994 (1)

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

1993 (1)

Ammari, H.

H. Ammari and G. Bao, “Maxwell's equations in a perturbed periodic structure,” Adv. Comput. Math. 16, 99-112 (2002).
[CrossRef]

H. Ammari and G. Bao, “Scattering by a nonhomogeneous object embedded in a perieodic structure,” C. R. Acad. Sci. Paris 330, 333-338 (2000).

Bao, B.

Bao, G.

H. Ammari and G. Bao, “Maxwell's equations in a perturbed periodic structure,” Adv. Comput. Math. 16, 99-112 (2002).
[CrossRef]

H. Ammari and G. Bao, “Scattering by a nonhomogeneous object embedded in a perieodic structure,” C. R. Acad. Sci. Paris 330, 333-338 (2000).

Belinskii, B. P.

B. P. Belinskii, “Optical theorem for the scattering of waves in an elastic plate,” J. Math. Sci. (N.Y.) 20, 1758-1760 (1998).

Berenger, J. P.

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

Bonnet-Bendhia, A. S.

A. S. Bonnet-Bendhia and K. Ramdani, “Diffraction by an acoustic grating perturbed by a bounded obstacle,” Adv. Comput. Math. 16, 113-138 (2002).
[CrossRef]

Bruno, O. P.

Chandler-Wilde, S. N.

S. N. Chandler-Wilde, P. Monk, and M. Thomas, “The mathematics of scattering by unbounded, rough, inhomogeneous layer,” J. Comput. Appl. Math. 204, 549-559 (2007).
[CrossRef]

S. N. Chandler-Wilde and P. Monk, “Existence, uniqueness, and variational methods for scattering by unbounded rough surfaces,” SIAM J. Math. Anal. 37, 598-618 (2005).
[CrossRef]

S. N. Chandler-Wilde and A. T. Peplow, “A boundary integral equation formulation for the Helmholtz equation in a locally perturbed half-plane,” Z. Angew. Math. Mech. 85, 79-88 (2005).
[CrossRef]

S. N. Chandler-Wilde, “The impedance boundary value problem for the Helmholtz equation in a halfplane,” Math. Methods Appl. Sci. 20, 813-840 (1997).
[CrossRef]

Chen, Z.

Duran, M.

M. Duran, I. Muga, and J. C. Nedelec, “The Helmholtz equation in a locally perturbed half-plane with passive boundary,” IMA J. Appl. Math. 71, 853-876 (2006).
[CrossRef]

Ehrhardt, M.

H. Han, M. Ehrhardt, and C. Zheng, “Numerical simulation of waves in periodic structures,” Comm. Comp. Phys. 5,849-872 (2009).

Fliss, S.

P. Joly, J. R. Li, and S. Fliss, “Exact boundary conditions for periodic waveguides containing a local perturbation,” Comm. Comp. Phys. 1, 945-973 (2006).

Han, H.

H. Han, M. Ehrhardt, and C. Zheng, “Numerical simulation of waves in periodic structures,” Comm. Comp. Phys. 5,849-872 (2009).

Hattori, K.

K. Hattori and J. Nakayama, “Scattering of TE plane wave from periodic grating with single defect,” IEICE Trans. Electron. E90-C, 312-319 (2007).
[CrossRef]

Joly, P.

P. Joly, J. R. Li, and S. Fliss, “Exact boundary conditions for periodic waveguides containing a local perturbation,” Comm. Comp. Phys. 1, 945-973 (2006).

Kirsch, A.

A. Kirsch, “Diffraction by Periodic Structures,” in Inverse Problems in Mathematical Physics, Lecture Note in Physics 422 (Springer, 1993), pp. 87-102.
[CrossRef]

Li, J. R.

P. Joly, J. R. Li, and S. Fliss, “Exact boundary conditions for periodic waveguides containing a local perturbation,” Comm. Comp. Phys. 1, 945-973 (2006).

Li, L.

Linton, C. M.

P. McIver, C. M. Linton, and M. McIver, “Construction of trapped modes for wave guides and diffraction gratings,” Proc. R. Soc. London, Ser. A 454, 2593-2616 (1998).
[CrossRef]

Lu, Y. Y.

McIver, M.

P. McIver, C. M. Linton, and M. McIver, “Construction of trapped modes for wave guides and diffraction gratings,” Proc. R. Soc. London, Ser. A 454, 2593-2616 (1998).
[CrossRef]

McIver, P.

P. McIver, C. M. Linton, and M. McIver, “Construction of trapped modes for wave guides and diffraction gratings,” Proc. R. Soc. London, Ser. A 454, 2593-2616 (1998).
[CrossRef]

Monk, P.

S. N. Chandler-Wilde, P. Monk, and M. Thomas, “The mathematics of scattering by unbounded, rough, inhomogeneous layer,” J. Comput. Appl. Math. 204, 549-559 (2007).
[CrossRef]

S. N. Chandler-Wilde and P. Monk, “Existence, uniqueness, and variational methods for scattering by unbounded rough surfaces,” SIAM J. Math. Anal. 37, 598-618 (2005).
[CrossRef]

Muga, I.

M. Duran, I. Muga, and J. C. Nedelec, “The Helmholtz equation in a locally perturbed half-plane with passive boundary,” IMA J. Appl. Math. 71, 853-876 (2006).
[CrossRef]

Nakayama, J.

K. Hattori and J. Nakayama, “Scattering of TE plane wave from periodic grating with single defect,” IEICE Trans. Electron. E90-C, 312-319 (2007).
[CrossRef]

Nedelec, J. C.

M. Duran, I. Muga, and J. C. Nedelec, “The Helmholtz equation in a locally perturbed half-plane with passive boundary,” IMA J. Appl. Math. 71, 853-876 (2006).
[CrossRef]

Peplow, A. T.

S. N. Chandler-Wilde and A. T. Peplow, “A boundary integral equation formulation for the Helmholtz equation in a locally perturbed half-plane,” Z. Angew. Math. Mech. 85, 79-88 (2005).
[CrossRef]

Ramdani, K.

A. S. Bonnet-Bendhia and K. Ramdani, “Diffraction by an acoustic grating perturbed by a bounded obstacle,” Adv. Comput. Math. 16, 113-138 (2002).
[CrossRef]

Reitich, F.

Thomas, M.

S. N. Chandler-Wilde, P. Monk, and M. Thomas, “The mathematics of scattering by unbounded, rough, inhomogeneous layer,” J. Comput. Appl. Math. 204, 549-559 (2007).
[CrossRef]

Wu, H.

Yuan, L.

Zheng, C.

H. Han, M. Ehrhardt, and C. Zheng, “Numerical simulation of waves in periodic structures,” Comm. Comp. Phys. 5,849-872 (2009).

Adv. Comput. Math. (2)

H. Ammari and G. Bao, “Maxwell's equations in a perturbed periodic structure,” Adv. Comput. Math. 16, 99-112 (2002).
[CrossRef]

A. S. Bonnet-Bendhia and K. Ramdani, “Diffraction by an acoustic grating perturbed by a bounded obstacle,” Adv. Comput. Math. 16, 113-138 (2002).
[CrossRef]

C. R. Acad. Sci. Paris (1)

H. Ammari and G. Bao, “Scattering by a nonhomogeneous object embedded in a perieodic structure,” C. R. Acad. Sci. Paris 330, 333-338 (2000).

Comm. Comp. Phys. (2)

P. Joly, J. R. Li, and S. Fliss, “Exact boundary conditions for periodic waveguides containing a local perturbation,” Comm. Comp. Phys. 1, 945-973 (2006).

H. Han, M. Ehrhardt, and C. Zheng, “Numerical simulation of waves in periodic structures,” Comm. Comp. Phys. 5,849-872 (2009).

IEICE Trans. Electron. (1)

K. Hattori and J. Nakayama, “Scattering of TE plane wave from periodic grating with single defect,” IEICE Trans. Electron. E90-C, 312-319 (2007).
[CrossRef]

IMA J. Appl. Math. (1)

M. Duran, I. Muga, and J. C. Nedelec, “The Helmholtz equation in a locally perturbed half-plane with passive boundary,” IMA J. Appl. Math. 71, 853-876 (2006).
[CrossRef]

J. Comput. Appl. Math. (1)

S. N. Chandler-Wilde, P. Monk, and M. Thomas, “The mathematics of scattering by unbounded, rough, inhomogeneous layer,” J. Comput. Appl. Math. 204, 549-559 (2007).
[CrossRef]

J. Comput. Phys. (1)

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

J. Lightwave Technol. (1)

J. Math. Sci. (N.Y.) (1)

B. P. Belinskii, “Optical theorem for the scattering of waves in an elastic plate,” J. Math. Sci. (N.Y.) 20, 1758-1760 (1998).

J. Opt. Soc. Am. A (3)

Math. Methods Appl. Sci. (1)

S. N. Chandler-Wilde, “The impedance boundary value problem for the Helmholtz equation in a halfplane,” Math. Methods Appl. Sci. 20, 813-840 (1997).
[CrossRef]

Proc. R. Soc. London, Ser. A (1)

P. McIver, C. M. Linton, and M. McIver, “Construction of trapped modes for wave guides and diffraction gratings,” Proc. R. Soc. London, Ser. A 454, 2593-2616 (1998).
[CrossRef]

SIAM J. Math. Anal. (1)

S. N. Chandler-Wilde and P. Monk, “Existence, uniqueness, and variational methods for scattering by unbounded rough surfaces,” SIAM J. Math. Anal. 37, 598-618 (2005).
[CrossRef]

Z. Angew. Math. Mech. (1)

S. N. Chandler-Wilde and A. T. Peplow, “A boundary integral equation formulation for the Helmholtz equation in a locally perturbed half-plane,” Z. Angew. Math. Mech. 85, 79-88 (2005).
[CrossRef]

Other (2)

A. Kirsch, “Diffraction by Periodic Structures,” in Inverse Problems in Mathematical Physics, Lecture Note in Physics 422 (Springer, 1993), pp. 87-102.
[CrossRef]

R.Petit, ed., Electromagnetic Theory of Gratings, Vol. 22 of Topics in Current Physics (Springer-Verlag, 1980).
[CrossRef]

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Figures (10)

Fig. 1
Fig. 1

Scattering of TE plane wave from a periodic grating with a defect. The surface is a periodic array of rectangular grooves and has a defect where a groove is not formed. The incident wave is u i ( x , y ) with angle θ and the scattered field is u s ( x , y ) with angle ϕ. The period of the surface and the width and depth of groove are denoted by L, w, and d, respectively. The unbounded domain above the grating surface is truncated by a PML layer between y r and y p .

Fig. 2
Fig. 2

Ω is divided into unit cells such that Ω j = + C j . For the cell C j , C j = Σ + j Σ j Γ j Γ j + 1 . C 0 is a defect cell.

Fig. 3
Fig. 3

Relative diffraction power versus the angle of incidence θ for depth d = 0.1 λ with period L = 2 λ and width w = 1.3 λ , where λ is the wavelength. Power of incident wave is normalized to 1. [ × 5 ] and [ × 20 ] mean that values are multiplied by 5 and 20, respectively.

Fig. 4
Fig. 4

Relative diffraction power versus the angle of incidence θ for depth d = 0.542 λ with period L = 2 λ and width w = 1.3 λ , where λ is the wavelength. Power of incident wave is normalized to 1.

Fig. 5
Fig. 5

Total scattering cross section versus the angle of incidence θ for widths w = 0.7 λ , 1.0 λ , 1.3 λ with period L = 2 λ , depth d = 0.1 λ ; λ is the wavelength.

Fig. 6
Fig. 6

Total scattering cross section versus the angle of incidence θ for widths w = 0.7 λ , 1.0 λ , 1.3 λ with period L = 2 λ , depth d = 0.542 λ ; λ is the wavelength.

Fig. 7
Fig. 7

Relative errors Err opt versus the angle of incidence θ for widths w = 0.7 λ , 1.0 λ , 1.3 λ with period L = 2 λ , depth d = 0.1 λ ; λ is the wavelength.

Fig. 8
Fig. 8

Relative errors Err opt versus the angle of incidence θ for widths w = 0.7 λ , 1.0 λ , 1.3 λ with period L = 2 λ , depth d = 0.542 λ ; λ is the wavelength.

Fig. 9
Fig. 9

Differential scattering cross section σ ( ϕ θ ) for d = 0.1 λ , 0.271 λ , 0.542 λ . The period is L = 2 λ , and w = 1.3 λ . The angle of incidence of the plane wave is θ = 60 ° .

Fig. 10
Fig. 10

Differential scattering cross section σ ( ϕ θ ) for d = 0.1 λ , 0.271 λ , 0.542 λ . The period is L = 2 λ , and d = 0.1 λ . The angle of incidence of the plane wave is θ = 60 ° .

Equations (53)

Equations on this page are rendered with MathJax. Learn more.

Δ u + k 2 u = 0 ,
u i = e i α x i β ( α ) y , α = k cos θ , β ( ζ ) = k 2 ζ 2 + , ζ R .
c y ( c y u s ) + x 2 u s + k 2 u s = 0 , y r < y < y p ,
c y ( c y u s ) + x 2 u s + k 2 u s = 0 , ( x , y ) Ω j = + C j ,
y u s = 0 , ( x , y ) Σ + j = + Σ j + ,
u s = u i , ( x , y ) Σ j = + Σ j ,
f j s = ( x + i k ) u s Γ j , g j s = ( x i k ) u s Γ j .
g j s = A j f j s + B j g j + 1 s + E j ( u i ) , f j + 1 s = C j f j s + D j g j + 1 s + F j ( u i ) ,
j Z .
E j ( u i ) = γ j E , F j ( u i ) = γ j F , j 0 ,
g j s = A f j s + B g j + 1 s + γ j E , f j + 1 s = C f j s + D g j + 1 s + γ j F , j 0 ,
g 0 s = A 0 f 0 s + B 0 g 1 s + E 0 ( u i ) , f 1 s = C 0 f 0 s + D 0 g 1 s + F 0 ( u i ) .
f j d = ( x + i k ) u d Γ j , g j d = ( x i k ) u d Γ j
g j d = A f j d + B g j + 1 d + γ j E , f j + 1 d = C f j d + D g j + 1 d + γ j F , j Z .
f j d = γ j f , g j d = γ j g , j Z .
g = A f + γ B g + E , γ f = C f + γ D g + F ,
f = [ γ I C γ D ( I γ B ) 1 A ] 1 [ γ D ( I γ B ) 1 E + F ] ,
g = [ I γ B γ A ( γ I C ) 1 D ] 1 [ A ( γ I C ) 1 F + E ] .
u s = u d + u p .
f j p = ( x + i k ) u p Γ j , g j p = ( x i k ) u p Γ j ,
g j p = A f j p + B g j + 1 p , f j + 1 p = C f j p + D g j + 1 p , j 0 .
f 2 N p = 0 , g 1 + 2 N p = 0 ,
f 0 p = D N g 0 p , g 1 p = A N f 1 p ,
f j s = f j d + f j p , g j s = g j d + g j p ,
g 0 p = A 0 f 0 p + B 0 g 1 p + E 0 p ,
f 1 p = C 0 f 0 p + D 0 g 1 p + F 0 p ,
E 0 p = E 0 ( u i ) + A 0 f 0 d + B 0 g 1 d g 0 d ,
F 0 p = F 0 ( u i ) + C 0 f 0 d + D 0 g 1 d f 1 d .
g 0 p = [ I A 0 D N B 0 A N ( I D 0 A N ) 1 C 0 D N ] 1 × [ B 0 A N ( I D 0 A N ) 1 F 0 p + E 0 p ] ,
f 1 p = [ I D 0 A N C 0 D N ( I A 0 D N ) 1 B 0 A N ] 1 × [ C 0 D N ( I A 0 D N ) 1 E 0 p + F 0 p ] .
g j p = A m f j p + B m g j + 2 m p , f j + 2 m p = C m f j p + D m g j + 2 m p .
g j p = A m ( C m f j 2 m p + D m g j p ) + B m g j + 2 m p ,
f j p = C m f j 2 m p + D m ( A m f j p + B m g j + 2 m p ) ,
g j p = A m * f j 2 m p + B m * g j + 2 m p , f j p = C m * f j 2 m p + D m * g j + 2 m p ,
A m * = ( I A m D m ) 1 A m C m , B m * = ( I A m D m ) 1 B m ,
C m * = ( I D m A m ) 1 C m , D m * = ( I D m A m ) 1 D m B m .
g j 2 m p = A m f j 2 m p + B m ( A m * f j 2 m p + B m * g j + 2 m p ) ,
f j + 2 m p = C m ( C m * f j 2 m p + D m * g j + 2 m p ) + D m g j + 2 m p ,
A m + 1 = A m + B m A m * , B m + 1 = B m B m * ,
C m + 1 = C m C m * , D m + 1 = D m + C m D m * .
u s , j y = y r = V l j f j s + V r j g j + 1 s + W j ( u i ) .
W j ( u i ) = γ j W , j 0 .
u d = e i α x m = + A m e i m k L x + i β ( α + m k L ) y ,
u = u i + u d = e i α x i β ( α ) y + e i α x m = + A m e i m k L x + i β ( α + m k L ) y .
β ( α ) = m = + Re [ β ( α + m k L ) ] A m 2 .
R m = Re [ β ( α + m k L ) ] A m 2 β ( α ) .
u p = e i α x + a ( s ) e i s x + i β ( α + s ) y d s ,
P c = Φ s ,
P c = 2 k m = + Re [ β ( α + m k L ) ] Re [ a ( m k L ) A m * ] ,
Φ s = 1 k + Re [ β ( α + s ) ] a ( s ) 2 d s .
Φ s = L 2 π 0 π σ ( ϕ θ ) d ϕ ,
σ ( ϕ θ ) = 2 π k sin 2 ϕ a ( k cos ϕ α ) 2 L .
Err opt = Φ s P c Φ s .

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