Abstract

The reconstruction problem for periodic (arbitrary profiled within a period) boundary between two homogeneous media is considered. Our approach to the solution of the inverse problem is based on the Tikhonov regularization technique, which requires successive selection of the boundaries on the basis of multiple solutions of the direct problem of wave diffraction by the candidate boundaries. The analytical numerical C method has been chosen as a simple but rather efficient tool for the direct problem solving. The scheme for numerical tests of algorithms and criteria for reconstruction accuracy have been suggested and verified. Results of numerical experiments that prove the validity of the approach are presented.

© 2014 Optical Society of America

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References

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  1. L. Rayleigh, “On the dynamical theory of gratings,” Proc. R. Soc. Lond. A 79, 399–416 (1907).
    [CrossRef]
  2. R. Petit, ed., Electromagnetic Theory of Gratings (Springer, 1980).
  3. G. Bao, L. Cowsar, and W. Masters, Mathematical Modeling in Optical Science, SIAM Frontiers in Applied Math Series(SIAM: Society for Industrial and Applied Mathematics, 2001).
  4. Y. K. Sirenko and S. Strom, eds., Modern Theory of Gratings. Resonant Scattering: Analysis Techniques and Phenomena (Springer, 2010).
  5. V. Lakshmi, “Remote sensing of soil moisture. Review article,” ISRN Soil Sci. 2013, 424178 (2013).
    [CrossRef]
  6. Y. Sirenko, S. Strom, and N. Yashina, Modeling and Analysis of Transient Processes in Open Resonant Structures (Springer, 2007).
  7. A. Malcolm and D. P. Nicholls, “A boundary perturbation method for recovering interface shapes in layered media,” Inverse Probl. 27, 095009 (2011).
    [CrossRef]
  8. J. Elschner and G. Hu, “An optimization method in inverse elastic scattering for one-dimensional grating profiles,” Commun. Comput. Phys. 12, 1434–1460 (2012).
  9. H. Gross, J. Richter, A. Rathsfeld, and M. Bär, “Investigations on a robust profile model for the reconstruction of 2D periodic absorber lines in scatterometry,” J. Eur. Opt. Soc. Rapid Publ. 5, 10053 (2010).
  10. Y. Altuncu, A. Yapar, and I. Akduman, “Numerical computation of the Green’s function of a layered media with rough interface,” Microw. Opt. Technol. Lett. 49, 1204–1209 (2007).
  11. G. Bruckner and J. Elschner, “The numerical solution of an inverse periodic transmission problem,” Math. Methods Appl. Sci. 28, 757–778 (2005).
  12. J. Elschner and M. Yamamoto, “Uniqueness results for an inverse periodic transmission problem,” Inverse Probl. 20, 1841–1852 (2004).
    [CrossRef]
  13. G. Bao, K. Huang, and G. Schmidt, “Optimal design of nonlinear diffraction gratings,” J. Comput. Phys. 184, 106–121 (2003).
    [CrossRef]
  14. T. Arens and A. Kirsch, “The factorization method in inverse scattering from periodic structures,” Inverse Probl. 19, 1195–1211 (2003).
    [CrossRef]
  15. J. Chandezon, A. Poyedinchuk, Y. Tuchkin, and N. Yashina, “Mathematical modeling of electromagnetic wave scattering by wavy periodic boundary between two media,” Prog. Electromagnet. Res. 38, 130–143 (2002).
  16. A. Tikhonov and Y. Arsenin, Methods for the Solution of Ill-Posed Problems (Wiley, 1977).
  17. P. Melezhik, A. Poyedinchuk, N. Yashina, and G. Granet, “Periodic boundary of metamaterial: eigen regimes and resonant radiation,” J. Opt. A 9, S403–S409 (2007).
    [CrossRef]
  18. A. Sveshnikov, “Radiation principal,” Doklady SSSR 73, 917–920 (1950).
  19. A. Sveshnikov, “Principal of limiting absorption,” Doklady SSSR 80, 345–347 (1951).
  20. A. A. Kirilenko, A. Y. Poyedinchuk, and N. P. Yashina, “Non-destructive control of dielectrics: mathematical models based on analytical regularization,” in Progress in Analysis. Proceeding of the Third International ISSAC Congress (World Scientific, 2001), Vol. II, pp. 1359–1367.
  21. V. Ivanov, V. Vasin, and V. Talanov, Theory of Linear Ill-Posed Problems and Its Applications (Nauka, 1978) (in Russian).
  22. A. Tikhonov, A. Goncharsky, V. Stepanov, and A. Yagola, Numerical Methods for the Solution of Ill-Posed Problems (Kluwer Academic, 1995).
  23. V. Hadson and J. Pim, Application of Functional Analyses for the Operator Theory (Mir, 1987) (in Russian).

2013

V. Lakshmi, “Remote sensing of soil moisture. Review article,” ISRN Soil Sci. 2013, 424178 (2013).
[CrossRef]

2012

J. Elschner and G. Hu, “An optimization method in inverse elastic scattering for one-dimensional grating profiles,” Commun. Comput. Phys. 12, 1434–1460 (2012).

2011

A. Malcolm and D. P. Nicholls, “A boundary perturbation method for recovering interface shapes in layered media,” Inverse Probl. 27, 095009 (2011).
[CrossRef]

2010

H. Gross, J. Richter, A. Rathsfeld, and M. Bär, “Investigations on a robust profile model for the reconstruction of 2D periodic absorber lines in scatterometry,” J. Eur. Opt. Soc. Rapid Publ. 5, 10053 (2010).

2007

Y. Altuncu, A. Yapar, and I. Akduman, “Numerical computation of the Green’s function of a layered media with rough interface,” Microw. Opt. Technol. Lett. 49, 1204–1209 (2007).

P. Melezhik, A. Poyedinchuk, N. Yashina, and G. Granet, “Periodic boundary of metamaterial: eigen regimes and resonant radiation,” J. Opt. A 9, S403–S409 (2007).
[CrossRef]

2005

G. Bruckner and J. Elschner, “The numerical solution of an inverse periodic transmission problem,” Math. Methods Appl. Sci. 28, 757–778 (2005).

2004

J. Elschner and M. Yamamoto, “Uniqueness results for an inverse periodic transmission problem,” Inverse Probl. 20, 1841–1852 (2004).
[CrossRef]

2003

G. Bao, K. Huang, and G. Schmidt, “Optimal design of nonlinear diffraction gratings,” J. Comput. Phys. 184, 106–121 (2003).
[CrossRef]

T. Arens and A. Kirsch, “The factorization method in inverse scattering from periodic structures,” Inverse Probl. 19, 1195–1211 (2003).
[CrossRef]

2002

J. Chandezon, A. Poyedinchuk, Y. Tuchkin, and N. Yashina, “Mathematical modeling of electromagnetic wave scattering by wavy periodic boundary between two media,” Prog. Electromagnet. Res. 38, 130–143 (2002).

1951

A. Sveshnikov, “Principal of limiting absorption,” Doklady SSSR 80, 345–347 (1951).

1950

A. Sveshnikov, “Radiation principal,” Doklady SSSR 73, 917–920 (1950).

1907

L. Rayleigh, “On the dynamical theory of gratings,” Proc. R. Soc. Lond. A 79, 399–416 (1907).
[CrossRef]

Akduman, I.

Y. Altuncu, A. Yapar, and I. Akduman, “Numerical computation of the Green’s function of a layered media with rough interface,” Microw. Opt. Technol. Lett. 49, 1204–1209 (2007).

Altuncu, Y.

Y. Altuncu, A. Yapar, and I. Akduman, “Numerical computation of the Green’s function of a layered media with rough interface,” Microw. Opt. Technol. Lett. 49, 1204–1209 (2007).

Arens, T.

T. Arens and A. Kirsch, “The factorization method in inverse scattering from periodic structures,” Inverse Probl. 19, 1195–1211 (2003).
[CrossRef]

Arsenin, Y.

A. Tikhonov and Y. Arsenin, Methods for the Solution of Ill-Posed Problems (Wiley, 1977).

Bao, G.

G. Bao, K. Huang, and G. Schmidt, “Optimal design of nonlinear diffraction gratings,” J. Comput. Phys. 184, 106–121 (2003).
[CrossRef]

G. Bao, L. Cowsar, and W. Masters, Mathematical Modeling in Optical Science, SIAM Frontiers in Applied Math Series(SIAM: Society for Industrial and Applied Mathematics, 2001).

Bär, M.

H. Gross, J. Richter, A. Rathsfeld, and M. Bär, “Investigations on a robust profile model for the reconstruction of 2D periodic absorber lines in scatterometry,” J. Eur. Opt. Soc. Rapid Publ. 5, 10053 (2010).

Bruckner, G.

G. Bruckner and J. Elschner, “The numerical solution of an inverse periodic transmission problem,” Math. Methods Appl. Sci. 28, 757–778 (2005).

Chandezon, J.

J. Chandezon, A. Poyedinchuk, Y. Tuchkin, and N. Yashina, “Mathematical modeling of electromagnetic wave scattering by wavy periodic boundary between two media,” Prog. Electromagnet. Res. 38, 130–143 (2002).

Cowsar, L.

G. Bao, L. Cowsar, and W. Masters, Mathematical Modeling in Optical Science, SIAM Frontiers in Applied Math Series(SIAM: Society for Industrial and Applied Mathematics, 2001).

Elschner, J.

J. Elschner and G. Hu, “An optimization method in inverse elastic scattering for one-dimensional grating profiles,” Commun. Comput. Phys. 12, 1434–1460 (2012).

G. Bruckner and J. Elschner, “The numerical solution of an inverse periodic transmission problem,” Math. Methods Appl. Sci. 28, 757–778 (2005).

J. Elschner and M. Yamamoto, “Uniqueness results for an inverse periodic transmission problem,” Inverse Probl. 20, 1841–1852 (2004).
[CrossRef]

Goncharsky, A.

A. Tikhonov, A. Goncharsky, V. Stepanov, and A. Yagola, Numerical Methods for the Solution of Ill-Posed Problems (Kluwer Academic, 1995).

Granet, G.

P. Melezhik, A. Poyedinchuk, N. Yashina, and G. Granet, “Periodic boundary of metamaterial: eigen regimes and resonant radiation,” J. Opt. A 9, S403–S409 (2007).
[CrossRef]

Gross, H.

H. Gross, J. Richter, A. Rathsfeld, and M. Bär, “Investigations on a robust profile model for the reconstruction of 2D periodic absorber lines in scatterometry,” J. Eur. Opt. Soc. Rapid Publ. 5, 10053 (2010).

Hadson, V.

V. Hadson and J. Pim, Application of Functional Analyses for the Operator Theory (Mir, 1987) (in Russian).

Hu, G.

J. Elschner and G. Hu, “An optimization method in inverse elastic scattering for one-dimensional grating profiles,” Commun. Comput. Phys. 12, 1434–1460 (2012).

Huang, K.

G. Bao, K. Huang, and G. Schmidt, “Optimal design of nonlinear diffraction gratings,” J. Comput. Phys. 184, 106–121 (2003).
[CrossRef]

Ivanov, V.

V. Ivanov, V. Vasin, and V. Talanov, Theory of Linear Ill-Posed Problems and Its Applications (Nauka, 1978) (in Russian).

Kirilenko, A. A.

A. A. Kirilenko, A. Y. Poyedinchuk, and N. P. Yashina, “Non-destructive control of dielectrics: mathematical models based on analytical regularization,” in Progress in Analysis. Proceeding of the Third International ISSAC Congress (World Scientific, 2001), Vol. II, pp. 1359–1367.

Kirsch, A.

T. Arens and A. Kirsch, “The factorization method in inverse scattering from periodic structures,” Inverse Probl. 19, 1195–1211 (2003).
[CrossRef]

Lakshmi, V.

V. Lakshmi, “Remote sensing of soil moisture. Review article,” ISRN Soil Sci. 2013, 424178 (2013).
[CrossRef]

Malcolm, A.

A. Malcolm and D. P. Nicholls, “A boundary perturbation method for recovering interface shapes in layered media,” Inverse Probl. 27, 095009 (2011).
[CrossRef]

Masters, W.

G. Bao, L. Cowsar, and W. Masters, Mathematical Modeling in Optical Science, SIAM Frontiers in Applied Math Series(SIAM: Society for Industrial and Applied Mathematics, 2001).

Melezhik, P.

P. Melezhik, A. Poyedinchuk, N. Yashina, and G. Granet, “Periodic boundary of metamaterial: eigen regimes and resonant radiation,” J. Opt. A 9, S403–S409 (2007).
[CrossRef]

Nicholls, D. P.

A. Malcolm and D. P. Nicholls, “A boundary perturbation method for recovering interface shapes in layered media,” Inverse Probl. 27, 095009 (2011).
[CrossRef]

Pim, J.

V. Hadson and J. Pim, Application of Functional Analyses for the Operator Theory (Mir, 1987) (in Russian).

Poyedinchuk, A.

P. Melezhik, A. Poyedinchuk, N. Yashina, and G. Granet, “Periodic boundary of metamaterial: eigen regimes and resonant radiation,” J. Opt. A 9, S403–S409 (2007).
[CrossRef]

J. Chandezon, A. Poyedinchuk, Y. Tuchkin, and N. Yashina, “Mathematical modeling of electromagnetic wave scattering by wavy periodic boundary between two media,” Prog. Electromagnet. Res. 38, 130–143 (2002).

Poyedinchuk, A. Y.

A. A. Kirilenko, A. Y. Poyedinchuk, and N. P. Yashina, “Non-destructive control of dielectrics: mathematical models based on analytical regularization,” in Progress in Analysis. Proceeding of the Third International ISSAC Congress (World Scientific, 2001), Vol. II, pp. 1359–1367.

Rathsfeld, A.

H. Gross, J. Richter, A. Rathsfeld, and M. Bär, “Investigations on a robust profile model for the reconstruction of 2D periodic absorber lines in scatterometry,” J. Eur. Opt. Soc. Rapid Publ. 5, 10053 (2010).

Rayleigh, L.

L. Rayleigh, “On the dynamical theory of gratings,” Proc. R. Soc. Lond. A 79, 399–416 (1907).
[CrossRef]

Richter, J.

H. Gross, J. Richter, A. Rathsfeld, and M. Bär, “Investigations on a robust profile model for the reconstruction of 2D periodic absorber lines in scatterometry,” J. Eur. Opt. Soc. Rapid Publ. 5, 10053 (2010).

Schmidt, G.

G. Bao, K. Huang, and G. Schmidt, “Optimal design of nonlinear diffraction gratings,” J. Comput. Phys. 184, 106–121 (2003).
[CrossRef]

Sirenko, Y.

Y. Sirenko, S. Strom, and N. Yashina, Modeling and Analysis of Transient Processes in Open Resonant Structures (Springer, 2007).

Stepanov, V.

A. Tikhonov, A. Goncharsky, V. Stepanov, and A. Yagola, Numerical Methods for the Solution of Ill-Posed Problems (Kluwer Academic, 1995).

Strom, S.

Y. Sirenko, S. Strom, and N. Yashina, Modeling and Analysis of Transient Processes in Open Resonant Structures (Springer, 2007).

Sveshnikov, A.

A. Sveshnikov, “Principal of limiting absorption,” Doklady SSSR 80, 345–347 (1951).

A. Sveshnikov, “Radiation principal,” Doklady SSSR 73, 917–920 (1950).

Talanov, V.

V. Ivanov, V. Vasin, and V. Talanov, Theory of Linear Ill-Posed Problems and Its Applications (Nauka, 1978) (in Russian).

Tikhonov, A.

A. Tikhonov, A. Goncharsky, V. Stepanov, and A. Yagola, Numerical Methods for the Solution of Ill-Posed Problems (Kluwer Academic, 1995).

A. Tikhonov and Y. Arsenin, Methods for the Solution of Ill-Posed Problems (Wiley, 1977).

Tuchkin, Y.

J. Chandezon, A. Poyedinchuk, Y. Tuchkin, and N. Yashina, “Mathematical modeling of electromagnetic wave scattering by wavy periodic boundary between two media,” Prog. Electromagnet. Res. 38, 130–143 (2002).

Vasin, V.

V. Ivanov, V. Vasin, and V. Talanov, Theory of Linear Ill-Posed Problems and Its Applications (Nauka, 1978) (in Russian).

Yagola, A.

A. Tikhonov, A. Goncharsky, V. Stepanov, and A. Yagola, Numerical Methods for the Solution of Ill-Posed Problems (Kluwer Academic, 1995).

Yamamoto, M.

J. Elschner and M. Yamamoto, “Uniqueness results for an inverse periodic transmission problem,” Inverse Probl. 20, 1841–1852 (2004).
[CrossRef]

Yapar, A.

Y. Altuncu, A. Yapar, and I. Akduman, “Numerical computation of the Green’s function of a layered media with rough interface,” Microw. Opt. Technol. Lett. 49, 1204–1209 (2007).

Yashina, N.

P. Melezhik, A. Poyedinchuk, N. Yashina, and G. Granet, “Periodic boundary of metamaterial: eigen regimes and resonant radiation,” J. Opt. A 9, S403–S409 (2007).
[CrossRef]

J. Chandezon, A. Poyedinchuk, Y. Tuchkin, and N. Yashina, “Mathematical modeling of electromagnetic wave scattering by wavy periodic boundary between two media,” Prog. Electromagnet. Res. 38, 130–143 (2002).

Y. Sirenko, S. Strom, and N. Yashina, Modeling and Analysis of Transient Processes in Open Resonant Structures (Springer, 2007).

Yashina, N. P.

A. A. Kirilenko, A. Y. Poyedinchuk, and N. P. Yashina, “Non-destructive control of dielectrics: mathematical models based on analytical regularization,” in Progress in Analysis. Proceeding of the Third International ISSAC Congress (World Scientific, 2001), Vol. II, pp. 1359–1367.

Commun. Comput. Phys.

J. Elschner and G. Hu, “An optimization method in inverse elastic scattering for one-dimensional grating profiles,” Commun. Comput. Phys. 12, 1434–1460 (2012).

Doklady SSSR

A. Sveshnikov, “Radiation principal,” Doklady SSSR 73, 917–920 (1950).

A. Sveshnikov, “Principal of limiting absorption,” Doklady SSSR 80, 345–347 (1951).

Inverse Probl.

A. Malcolm and D. P. Nicholls, “A boundary perturbation method for recovering interface shapes in layered media,” Inverse Probl. 27, 095009 (2011).
[CrossRef]

J. Elschner and M. Yamamoto, “Uniqueness results for an inverse periodic transmission problem,” Inverse Probl. 20, 1841–1852 (2004).
[CrossRef]

T. Arens and A. Kirsch, “The factorization method in inverse scattering from periodic structures,” Inverse Probl. 19, 1195–1211 (2003).
[CrossRef]

ISRN Soil Sci.

V. Lakshmi, “Remote sensing of soil moisture. Review article,” ISRN Soil Sci. 2013, 424178 (2013).
[CrossRef]

J. Comput. Phys.

G. Bao, K. Huang, and G. Schmidt, “Optimal design of nonlinear diffraction gratings,” J. Comput. Phys. 184, 106–121 (2003).
[CrossRef]

J. Eur. Opt. Soc. Rapid Publ.

H. Gross, J. Richter, A. Rathsfeld, and M. Bär, “Investigations on a robust profile model for the reconstruction of 2D periodic absorber lines in scatterometry,” J. Eur. Opt. Soc. Rapid Publ. 5, 10053 (2010).

J. Opt. A

P. Melezhik, A. Poyedinchuk, N. Yashina, and G. Granet, “Periodic boundary of metamaterial: eigen regimes and resonant radiation,” J. Opt. A 9, S403–S409 (2007).
[CrossRef]

Math. Methods Appl. Sci.

G. Bruckner and J. Elschner, “The numerical solution of an inverse periodic transmission problem,” Math. Methods Appl. Sci. 28, 757–778 (2005).

Microw. Opt. Technol. Lett.

Y. Altuncu, A. Yapar, and I. Akduman, “Numerical computation of the Green’s function of a layered media with rough interface,” Microw. Opt. Technol. Lett. 49, 1204–1209 (2007).

Proc. R. Soc. Lond. A

L. Rayleigh, “On the dynamical theory of gratings,” Proc. R. Soc. Lond. A 79, 399–416 (1907).
[CrossRef]

Prog. Electromagnet. Res.

J. Chandezon, A. Poyedinchuk, Y. Tuchkin, and N. Yashina, “Mathematical modeling of electromagnetic wave scattering by wavy periodic boundary between two media,” Prog. Electromagnet. Res. 38, 130–143 (2002).

Other

A. Tikhonov and Y. Arsenin, Methods for the Solution of Ill-Posed Problems (Wiley, 1977).

R. Petit, ed., Electromagnetic Theory of Gratings (Springer, 1980).

G. Bao, L. Cowsar, and W. Masters, Mathematical Modeling in Optical Science, SIAM Frontiers in Applied Math Series(SIAM: Society for Industrial and Applied Mathematics, 2001).

Y. K. Sirenko and S. Strom, eds., Modern Theory of Gratings. Resonant Scattering: Analysis Techniques and Phenomena (Springer, 2010).

Y. Sirenko, S. Strom, and N. Yashina, Modeling and Analysis of Transient Processes in Open Resonant Structures (Springer, 2007).

A. A. Kirilenko, A. Y. Poyedinchuk, and N. P. Yashina, “Non-destructive control of dielectrics: mathematical models based on analytical regularization,” in Progress in Analysis. Proceeding of the Third International ISSAC Congress (World Scientific, 2001), Vol. II, pp. 1359–1367.

V. Ivanov, V. Vasin, and V. Talanov, Theory of Linear Ill-Posed Problems and Its Applications (Nauka, 1978) (in Russian).

A. Tikhonov, A. Goncharsky, V. Stepanov, and A. Yagola, Numerical Methods for the Solution of Ill-Posed Problems (Kluwer Academic, 1995).

V. Hadson and J. Pim, Application of Functional Analyses for the Operator Theory (Mir, 1987) (in Russian).

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Figures (6)

Fig. 1.
Fig. 1.

Geometry of the problem: infinite periodic grating of arbitrary profile.

Fig. 2.
Fig. 2.

Behavior of lg(χN) as function of truncation number N for profile a(y)=A0sin2(πy/d), A0=1.0, ε2=2.25, κ=0.6, φ=0. Solid line corresponds to C method, dashed line to problem after regularization.

Fig. 3.
Fig. 3.

Dependence of δ=δ(α) on α; point αm=104 is a position of minimum of δ(α), points αl=101αm and αr=10αm, a(y)=A0sin2(πy/d), A0=1.0, ε2=2.25, κ=0.6.

Fig. 4.
Fig. 4.

Minimization problem solution versus parameter: (a) α=αl=101αm; (b) α=αr=10αm; (c) α=αm. Solid line: exact profile a(y)=A0[0.375+0.25sin(2πy/d)+0.125cos(4πy/d)]; crosses: reconstructed profile; dotted line: absolute value of deviation between the exact and reconstructed profile functions A01(|a(y)ar(y)|), N=6, 0.5d/λ2, The hundredfold residual function 100A01(|a(y)ar(y)|) computed for P=6 and P=36 is presented in the bottom of the fragment (c).

Fig. 5.
Fig. 5.

Reconstructed profiles of the functions a2(y) (a) and a4(y) (b) for P=36, N=6, ε2=2.25, 0.5d/λ2, φ=0, A0=1.0. Residual functions are shown at the bottoms of the fragments [in (a) increased hundredfold].

Fig. 6.
Fig. 6.

Numerical testing of the profile reconstruction algorithm for a1(y)=A0[0.5+(π3y/6d)((2y/d)1)(1(y/d))], ε2=2.25, A0=1.0, φ=0, 0.5d/λ2. Residual functions for P=16 (dotted line) and P=36 (solid line); N=16, are increased hundredfold and shown at the bottom of the picture.

Equations (38)

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

rotE=ikμH;rotH=ikεE,
(Δ+k2εiμi)U(y,z)=0,
Ex=U(x,y);Ey=0;Ez=0;Hx=0;Hy=1ikμiUz;Hz=1ikμiUy.
U(y+d,z)=ei2πΦU(y,z),
|Φ|1/2.
Ui(y,z)=Aei(kyy+kzz),
ky2+kz2=k2εμ,
Ut(y,z)={Ui(y,z)+Us(y,z),z>ha(y);Us(y,z),z<ha(y).
Ut(+)(p)=Ut()(p),Ut(+)(p)np=γUt()(p)nppL,
γ={μ1/μ2E-polarizationε1/ε2H-polarization.
Us(y,z)={n=RneiΦn2πdy+Γn12πdz,z>h;n=TneiΦn2πdyΓn22πdz,z<0,,
Φn=Φ+n,Γnj=κj2Φn2,κj=κ2εjμj,j=1,2,κ=d/λ.
a=n=anei2πndy,y[0,d].
H(A,λ)=Rc(A,λ),
Φα(A)=m=1Mn=N(λm)N(λm)|Rn(A,λm)Rnc(A,λm)|2+αn=QQ|an|2(1+|n|2k),
Fx=B
F=F1F2G1γG2,x=RT,B=B1B2.
Fp=Fmnpm=1,n=,Gp=Gmnpm=1,n=;p=1,2
Fmn1=Lmn(ρn1A0);Fmn2=L¯mn(ρn2A0).
Ln(v)=1d0dexp{iva(y)i2πndy}dy;L¯n(v)=1d0dexp{iv(1a(y))i2πndy}dy;
Gmn1=ρn1Fmn1+iΦnA0s=+amsFms1(ms);
Gmn2=ρn1Fmn2+iΦnA0s=+amsFms2(ms).
an=1d0da(y)exp{i2πndy}dy;
B1=(Bn1)n=;Bn1=Ln(κ2cos(φ)A0);
B2=(Bn2)n=;Bn2={κ2cos(φ)B01;n=0(ntg(φ)κ2cos(φ))Bn1;n0.
Fmn1k=1P1ei(mn)yk1e(mn)22A0(|n|+1)a¨(yk1)2πA0(|n|+1)a¨(yk1);Fmn2k=1P2ei(mn)yk2e(mn)22A0(|n|+1)|a¨(yk2)|2πA0(|n|+1)|a¨(yk2)|,
a(yk1)=0;a˙(yk1)=0;a¨(yk1)>0;
a(yk2)=1;a˙(yk2)=0;a¨(yk2)<0.
Gmn1im|m|nFmn1;Gmn2im|m|nFmn2.
FNxN=BN
(I00T2)(F1F2G1γG2)(T100T1)
{F1T1y1F21y2=B1;T2G1T1y1T2G2T1y2=T2B2;
{c1y1c2y2+A11y1A12y2=B1;c1y1+c2γy2+A21y1A22y2=T2B2,
c1=k=1P112πA0a¨(yk1),c2=k=1P212πA0|a¨(yk2)|.
(c1c2c1c2γ),
y+Hy=b.
δ(α)=(n(1+|n|2)|anaαnr|2n(1+|n|2)|an|2)1/2
Δα=(n|RmnRmncα|2n|Rmn|2)1/2

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