Abstract

A model with two roughness levels for the diffraction of a plane wave by a metallic grating with periodic imperfections is presented. The grating surface is the sum of a reference profile and a perturbation profile. First, the diffraction by the reference grating is treated. At this stage the Chandezon method is used. This method leads to the resolution of eigenvalue systems. Each eigensolution defines an elementary wave function that characterizes a propagating or an evanescent wave. Second, the periodic errors are taken into account and a Rayleigh hypothesis is expressed: Everywhere in space the diffracted fields can be written as a linear combination of reference wave functions. The boundary conditions on the perturbed grating allow the diffraction amplitudes to be determined and therefore lead to the energetic magnitudes (efficiencies). The domain of analytical validity of this hypothesis is not defined. In fact, this method is considered to be an approximation. The proposed numerical study leads to some utilization rules. With a plane as the reference surface, the electromagnetic fields are given by classical Rayleigh expansions. Here the reference profile is a grating, hence the term generalized Rayleigh expansion.

© 1998 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. M. Breidne, D. Maystre, “Variational theory of diffraction gratings and its application to the study of ghosts,” J. Opt. Soc. Am. 72, 499–506 (1982).
    [CrossRef]
  2. N. A. Finkelstein, C. H. Brawley, R. J. Meltzer, “The reduction of ghosts on diffraction spectra,” J. Opt. Soc. Am. 42, 121–126 (1952).
    [CrossRef]
  3. R. Dusséaux, “Etude de la diffraction d’une onde plane par un réseau—équations de Maxwell covariantes et méthodes de perturbation,” Ph.D. dissertation (Université Blaise Pascal, Clermont-Ferrand, France, 1993).
  4. R. Dusséaux, C. Faure, J. Chandezon, F. Molinet, “New perturbation theory of diffraction gratings and its application to the study of ghosts,” J. Opt. Soc. Am. A 12, 1271–1282 (1995).
    [CrossRef]
  5. E. J. Post, Format Structure of Electromagnetic (North-Holland, Amsterdam, 1962).
  6. J. Chandezon, “Les équations de Maxwell sous forme covariante—application à l’étude de la propagation dans les guides périodiques et à la diffraction par les réseaux,” Ph.D. dissertation (Université Blaise Pascal, Clermont-Ferrand, France, 1979).
  7. J. Chandezon, D. Maystre, G. Raoult, “A new theoretical method for diffraction gratings and its numerical application,” J. Opt. (Paris) 11, 235–241 (1980).
    [CrossRef]
  8. J. M. Elson, R. H. Ritchie, “Photon interaction at a rough metal surface,” Phys. Rev. B 4, 4129–4138 (1971).
    [CrossRef]
  9. J. Chandezon, M. T. Dupuis, G. Cornet, D. Maystre, “Multicoated gratings: a differential formalism applicable in the entire optical region,” J. Opt. Soc. Am. 72, 839–846 (1982).
    [CrossRef]
  10. L. Li, “Multilayer-coated diffraction gratings: differential method of Chandezon et al. revisited,” J. Opt. Soc. Am. A 11, 2816–2828 (1994).
    [CrossRef]
  11. T. W. Preist, N. P. Cotter, J. R. Sambles, “Periodic multilayer gratings of arbitrary shape,” J. Opt. Soc. Am. A 12, 1740–1748 (1995).
    [CrossRef]
  12. G. Granet, “Analysis of diffraction by crossed gratings using a non-orthogonal coordinate system,” Pure Appl. Opt. 4, 777–793 (1995).
    [CrossRef]
  13. L. Li, G. Granet, J. P. Plumey, J. Chandezon, “Some topics in extending the C method to multilayer-coated gratings of different profiles,” Pure Appl. Opt. 5, 141–156 (1996).
    [CrossRef]
  14. J. P. Plumey, G. Granet, J. Chandezon, “Differential covariant formalism for solving Maxwell’s equations in curvilinear coordinates: oblique scattering from lossy periodic surfaces,” IEEE Trans. Antennas Propag. 43, 835–842 (1995).
    [CrossRef]
  15. Lord Rayleigh, “On the dynamical theory of gratings,” Proc. R. Soc. London, Ser. A 79, 399–416 (1907).
    [CrossRef]
  16. Lord Rayleigh, The Theory of Sound (Dover, New York, 1945), Vol. 2, pp. 89–96.
  17. R. Petit, ed., Electromagnetic Theory of Gratings (Springer-Verlag, Heidelberg, 1980).
  18. R. Petit, M. Cadilhac, “Sur la diffraction d’une onde plane par un réseau infiniment conducteur,” C. R. Acad. Sci., Ser. B 262, 468–471 (1966).
  19. R. F. Millar, “On the Rayleigh assumption in scattering by a periodic surface,” Proc. Cambridge Philos. Soc. 69, 773–791 (1971).
    [CrossRef]
  20. R. F. Millar, “The Rayleigh hypothesis and a related least-squares solution to scattering problems for periodic surfaces and other scatterers,” Radio Sci. 8, 785–796 (1973).
    [CrossRef]
  21. M. Nevière, M. Cadilhac, “Sur une nouvelle formulation du problème de la diffraction d’une onde plane par un réseau infiniment conducteur: cas général,” Opt. Commun. 3, 379–383 (1971).
    [CrossRef]
  22. P. M. van den Berg, J. T. Fokkema, “Rayleigh hypothesis in the theory of reflection by a grating,” J. Opt. Soc. Am. 69, 27–31 (1979).
    [CrossRef]
  23. P. M. van den Berg, “Reflection by a grating: Rayleigh methods,” J. Opt. Soc. Am. 71, 1224–1229 (1981).
    [CrossRef]
  24. J. P. Hugonin, R. Petit, M. Cadilhac, “Plane-wave expansions used to describe the field diffracted by a grating,” J. Opt. Soc. Am. 71, 593–598 (1981).
    [CrossRef]
  25. J. M. Soto-Crespo, M. Nieto Vesperinas, A. T. Friberg, “Scattering from slightly rough random surfaces: a detailed study on the validity of the small perturbation method,” J. Opt. Soc. Am. A 7, 1185–1201 (1990).
    [CrossRef]
  26. H. Raether, Surface Plasmons on Smooth and Rough Surfaces and on Gratings (Springer-Verlag, Berlin, 1988), Chap. 6 and Appendix 4.

1996

L. Li, G. Granet, J. P. Plumey, J. Chandezon, “Some topics in extending the C method to multilayer-coated gratings of different profiles,” Pure Appl. Opt. 5, 141–156 (1996).
[CrossRef]

1995

J. P. Plumey, G. Granet, J. Chandezon, “Differential covariant formalism for solving Maxwell’s equations in curvilinear coordinates: oblique scattering from lossy periodic surfaces,” IEEE Trans. Antennas Propag. 43, 835–842 (1995).
[CrossRef]

G. Granet, “Analysis of diffraction by crossed gratings using a non-orthogonal coordinate system,” Pure Appl. Opt. 4, 777–793 (1995).
[CrossRef]

R. Dusséaux, C. Faure, J. Chandezon, F. Molinet, “New perturbation theory of diffraction gratings and its application to the study of ghosts,” J. Opt. Soc. Am. A 12, 1271–1282 (1995).
[CrossRef]

T. W. Preist, N. P. Cotter, J. R. Sambles, “Periodic multilayer gratings of arbitrary shape,” J. Opt. Soc. Am. A 12, 1740–1748 (1995).
[CrossRef]

1994

1990

1982

1981

1980

J. Chandezon, D. Maystre, G. Raoult, “A new theoretical method for diffraction gratings and its numerical application,” J. Opt. (Paris) 11, 235–241 (1980).
[CrossRef]

1979

1973

R. F. Millar, “The Rayleigh hypothesis and a related least-squares solution to scattering problems for periodic surfaces and other scatterers,” Radio Sci. 8, 785–796 (1973).
[CrossRef]

1971

M. Nevière, M. Cadilhac, “Sur une nouvelle formulation du problème de la diffraction d’une onde plane par un réseau infiniment conducteur: cas général,” Opt. Commun. 3, 379–383 (1971).
[CrossRef]

J. M. Elson, R. H. Ritchie, “Photon interaction at a rough metal surface,” Phys. Rev. B 4, 4129–4138 (1971).
[CrossRef]

R. F. Millar, “On the Rayleigh assumption in scattering by a periodic surface,” Proc. Cambridge Philos. Soc. 69, 773–791 (1971).
[CrossRef]

1966

R. Petit, M. Cadilhac, “Sur la diffraction d’une onde plane par un réseau infiniment conducteur,” C. R. Acad. Sci., Ser. B 262, 468–471 (1966).

1952

1907

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

Brawley, C. H.

Breidne, M.

Cadilhac, M.

J. P. Hugonin, R. Petit, M. Cadilhac, “Plane-wave expansions used to describe the field diffracted by a grating,” J. Opt. Soc. Am. 71, 593–598 (1981).
[CrossRef]

M. Nevière, M. Cadilhac, “Sur une nouvelle formulation du problème de la diffraction d’une onde plane par un réseau infiniment conducteur: cas général,” Opt. Commun. 3, 379–383 (1971).
[CrossRef]

R. Petit, M. Cadilhac, “Sur la diffraction d’une onde plane par un réseau infiniment conducteur,” C. R. Acad. Sci., Ser. B 262, 468–471 (1966).

Chandezon, J.

L. Li, G. Granet, J. P. Plumey, J. Chandezon, “Some topics in extending the C method to multilayer-coated gratings of different profiles,” Pure Appl. Opt. 5, 141–156 (1996).
[CrossRef]

J. P. Plumey, G. Granet, J. Chandezon, “Differential covariant formalism for solving Maxwell’s equations in curvilinear coordinates: oblique scattering from lossy periodic surfaces,” IEEE Trans. Antennas Propag. 43, 835–842 (1995).
[CrossRef]

R. Dusséaux, C. Faure, J. Chandezon, F. Molinet, “New perturbation theory of diffraction gratings and its application to the study of ghosts,” J. Opt. Soc. Am. A 12, 1271–1282 (1995).
[CrossRef]

J. Chandezon, M. T. Dupuis, G. Cornet, D. Maystre, “Multicoated gratings: a differential formalism applicable in the entire optical region,” J. Opt. Soc. Am. 72, 839–846 (1982).
[CrossRef]

J. Chandezon, D. Maystre, G. Raoult, “A new theoretical method for diffraction gratings and its numerical application,” J. Opt. (Paris) 11, 235–241 (1980).
[CrossRef]

J. Chandezon, “Les équations de Maxwell sous forme covariante—application à l’étude de la propagation dans les guides périodiques et à la diffraction par les réseaux,” Ph.D. dissertation (Université Blaise Pascal, Clermont-Ferrand, France, 1979).

Cornet, G.

Cotter, N. P.

Dupuis, M. T.

Dusséaux, R.

R. Dusséaux, C. Faure, J. Chandezon, F. Molinet, “New perturbation theory of diffraction gratings and its application to the study of ghosts,” J. Opt. Soc. Am. A 12, 1271–1282 (1995).
[CrossRef]

R. Dusséaux, “Etude de la diffraction d’une onde plane par un réseau—équations de Maxwell covariantes et méthodes de perturbation,” Ph.D. dissertation (Université Blaise Pascal, Clermont-Ferrand, France, 1993).

Elson, J. M.

J. M. Elson, R. H. Ritchie, “Photon interaction at a rough metal surface,” Phys. Rev. B 4, 4129–4138 (1971).
[CrossRef]

Faure, C.

Finkelstein, N. A.

Fokkema, J. T.

Friberg, A. T.

Granet, G.

L. Li, G. Granet, J. P. Plumey, J. Chandezon, “Some topics in extending the C method to multilayer-coated gratings of different profiles,” Pure Appl. Opt. 5, 141–156 (1996).
[CrossRef]

G. Granet, “Analysis of diffraction by crossed gratings using a non-orthogonal coordinate system,” Pure Appl. Opt. 4, 777–793 (1995).
[CrossRef]

J. P. Plumey, G. Granet, J. Chandezon, “Differential covariant formalism for solving Maxwell’s equations in curvilinear coordinates: oblique scattering from lossy periodic surfaces,” IEEE Trans. Antennas Propag. 43, 835–842 (1995).
[CrossRef]

Hugonin, J. P.

Li, L.

L. Li, G. Granet, J. P. Plumey, J. Chandezon, “Some topics in extending the C method to multilayer-coated gratings of different profiles,” Pure Appl. Opt. 5, 141–156 (1996).
[CrossRef]

L. Li, “Multilayer-coated diffraction gratings: differential method of Chandezon et al. revisited,” J. Opt. Soc. Am. A 11, 2816–2828 (1994).
[CrossRef]

Maystre, D.

Meltzer, R. J.

Millar, R. F.

R. F. Millar, “The Rayleigh hypothesis and a related least-squares solution to scattering problems for periodic surfaces and other scatterers,” Radio Sci. 8, 785–796 (1973).
[CrossRef]

R. F. Millar, “On the Rayleigh assumption in scattering by a periodic surface,” Proc. Cambridge Philos. Soc. 69, 773–791 (1971).
[CrossRef]

Molinet, F.

Nevière, M.

M. Nevière, M. Cadilhac, “Sur une nouvelle formulation du problème de la diffraction d’une onde plane par un réseau infiniment conducteur: cas général,” Opt. Commun. 3, 379–383 (1971).
[CrossRef]

Nieto Vesperinas, M.

Petit, R.

J. P. Hugonin, R. Petit, M. Cadilhac, “Plane-wave expansions used to describe the field diffracted by a grating,” J. Opt. Soc. Am. 71, 593–598 (1981).
[CrossRef]

R. Petit, M. Cadilhac, “Sur la diffraction d’une onde plane par un réseau infiniment conducteur,” C. R. Acad. Sci., Ser. B 262, 468–471 (1966).

Plumey, J. P.

L. Li, G. Granet, J. P. Plumey, J. Chandezon, “Some topics in extending the C method to multilayer-coated gratings of different profiles,” Pure Appl. Opt. 5, 141–156 (1996).
[CrossRef]

J. P. Plumey, G. Granet, J. Chandezon, “Differential covariant formalism for solving Maxwell’s equations in curvilinear coordinates: oblique scattering from lossy periodic surfaces,” IEEE Trans. Antennas Propag. 43, 835–842 (1995).
[CrossRef]

Post, E. J.

E. J. Post, Format Structure of Electromagnetic (North-Holland, Amsterdam, 1962).

Preist, T. W.

Raether, H.

H. Raether, Surface Plasmons on Smooth and Rough Surfaces and on Gratings (Springer-Verlag, Berlin, 1988), Chap. 6 and Appendix 4.

Raoult, G.

J. Chandezon, D. Maystre, G. Raoult, “A new theoretical method for diffraction gratings and its numerical application,” J. Opt. (Paris) 11, 235–241 (1980).
[CrossRef]

Rayleigh, Lord

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

Lord Rayleigh, The Theory of Sound (Dover, New York, 1945), Vol. 2, pp. 89–96.

Ritchie, R. H.

J. M. Elson, R. H. Ritchie, “Photon interaction at a rough metal surface,” Phys. Rev. B 4, 4129–4138 (1971).
[CrossRef]

Sambles, J. R.

Soto-Crespo, J. M.

van den Berg, P. M.

C. R. Acad. Sci., Ser. B

R. Petit, M. Cadilhac, “Sur la diffraction d’une onde plane par un réseau infiniment conducteur,” C. R. Acad. Sci., Ser. B 262, 468–471 (1966).

IEEE Trans. Antennas Propag.

J. P. Plumey, G. Granet, J. Chandezon, “Differential covariant formalism for solving Maxwell’s equations in curvilinear coordinates: oblique scattering from lossy periodic surfaces,” IEEE Trans. Antennas Propag. 43, 835–842 (1995).
[CrossRef]

J. Opt. (Paris)

J. Chandezon, D. Maystre, G. Raoult, “A new theoretical method for diffraction gratings and its numerical application,” J. Opt. (Paris) 11, 235–241 (1980).
[CrossRef]

J. Opt. Soc. Am.

J. Opt. Soc. Am. A

Opt. Commun.

M. Nevière, M. Cadilhac, “Sur une nouvelle formulation du problème de la diffraction d’une onde plane par un réseau infiniment conducteur: cas général,” Opt. Commun. 3, 379–383 (1971).
[CrossRef]

Phys. Rev. B

J. M. Elson, R. H. Ritchie, “Photon interaction at a rough metal surface,” Phys. Rev. B 4, 4129–4138 (1971).
[CrossRef]

Proc. Cambridge Philos. Soc.

R. F. Millar, “On the Rayleigh assumption in scattering by a periodic surface,” Proc. Cambridge Philos. Soc. 69, 773–791 (1971).
[CrossRef]

Proc. R. Soc. London, Ser. A

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

Pure Appl. Opt.

G. Granet, “Analysis of diffraction by crossed gratings using a non-orthogonal coordinate system,” Pure Appl. Opt. 4, 777–793 (1995).
[CrossRef]

L. Li, G. Granet, J. P. Plumey, J. Chandezon, “Some topics in extending the C method to multilayer-coated gratings of different profiles,” Pure Appl. Opt. 5, 141–156 (1996).
[CrossRef]

Radio Sci.

R. F. Millar, “The Rayleigh hypothesis and a related least-squares solution to scattering problems for periodic surfaces and other scatterers,” Radio Sci. 8, 785–796 (1973).
[CrossRef]

Other

R. Dusséaux, “Etude de la diffraction d’une onde plane par un réseau—équations de Maxwell covariantes et méthodes de perturbation,” Ph.D. dissertation (Université Blaise Pascal, Clermont-Ferrand, France, 1993).

E. J. Post, Format Structure of Electromagnetic (North-Holland, Amsterdam, 1962).

J. Chandezon, “Les équations de Maxwell sous forme covariante—application à l’étude de la propagation dans les guides périodiques et à la diffraction par les réseaux,” Ph.D. dissertation (Université Blaise Pascal, Clermont-Ferrand, France, 1979).

Lord Rayleigh, The Theory of Sound (Dover, New York, 1945), Vol. 2, pp. 89–96.

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

H. Raether, Surface Plasmons on Smooth and Rough Surfaces and on Gratings (Springer-Verlag, Berlin, 1988), Chap. 6 and Appendix 4.

Cited By

OSA participates in CrossRef's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

Alert me when this article is cited.


Figures (14)

Fig. 1
Fig. 1

Metallic grating illuminated by a plane wave under incidence θ.

Fig. 2
Fig. 2

Reference surface and perturbation profile orientation of the field components at the coordinate surfaces u0=0 and u0=a1(x).

Fig. 3
Fig. 3

Accuracy ΔS(M) in the sum of E diffraction efficiencies for the metallic grating referenced in Subsection 5.A with h1=0.02λ. CM denotes the C method: GRE, the generalized Rayleigh expansion; wl, wavelength.

Fig. 4
Fig. 4

Accuracy ΔS(M) in the sum of H diffraction efficiencies obtained by the generalized Rayleigh expansion for the metallic grating referenced in Subsection 5.A with hl=0.02λ.

Fig. 5
Fig. 5

Accuracies Δξ-1(1) and Δξ0(1) in the negative first-order and zero-order H diffraction efficiencies obtained by the generalized Rayleigh expansion for the metallic grating referenced in Subsection 5.A with h1=0.04λ.

Fig. 6
Fig. 6

Same as in Fig. 5, except that h1=0.12λ.

Fig. 7
Fig. 7

Reflected efficiencies ξ-1(1) and ξ0(1) versus groove depth obtained by the C method and the generalized Rayleigh expansion for the sinusoidal grating referenced in Subsection 5.C with E polarization.

Fig. 8
Fig. 8

Same as in Fig. 7, but the sinusoidal grating is H polarized.

Fig. 9
Fig. 9

Reflected efficiencies ξ-1(1) and ξ0(1) versus groove depth obtained by the C method and the generalized Rayleigh expansion for the second grating referenced in Subsection 5.C with E polarization.

Fig. 10
Fig. 10

Same as in Fig. 9, but the grating is H polarized.

Fig. 11
Fig. 11

Accuracy ΔS(M) in the sum of E diffraction efficiencies obtained by the generalized Rayleigh expansion for the metallic grating referenced in Subsection 5.D with an incidence angle of 20°, 50°, and 80°.

Fig. 12
Fig. 12

Same as in Fig. 11, but the grating is H polarized.

Fig. 13
Fig. 13

Sum of E diffraction efficiences versus incidence angle for the grating referenced in Subsection 5.D. The sum of efficiencies of the reference grating is obtained by the C method; the sum of efficiencies of the real grating and the sum of supplementary efficiencies obtained by the generalized Rayleigh expansion.

Fig. 14
Fig. 14

Same as in Fig. 13, but the grating is H polarized.

Tables (5)

Tables Icon

Table 1 Accuracy in Efficiencies versus Grating Period with the C Methoda

Tables Icon

Table 2 Spectra of Elementary Wave Functions Associated with n0 Incidences

Tables Icon

Table 3 Accuracy in Efficiencies versus Grating Period with the Generalized Rayleigh Expansiona

Tables Icon

Table 4 Norm Values σc and σd for the Gratings Studied in Subsection 5.A

Tables Icon

Table 5 Truncation Order Range ΔM Ensuring the Exploitation of Diffraction Efficiencies for the Metallic Grating Referenced in Subsection 5.D

Equations (78)

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

Ez,i(1)(x, y)orZ(1)Hz,i(1)(x, y)=Fi(1)(x, y)=exp(-jα0x+jβ0y),
Z(1)=μ001/2,α0=k(1) sin θ,
β0=k(1) cos θ,k(1)=2πλ.
-1<sin θn(j)=1ν(j)sin θ+n λD<1,
ifIm(ν(j))=0and(ν(1), ν(2))=(1, ν).
Lψ(x, u)=1jkψ(x, u)u,L=L11L12L21L22,
ψ(x, u)=F(x, u)G(x, u),
L11L12L21L22=d(x)jk.x-c(x)-1k2xc(x) .x+k.2,1jk[d(x).]x,
c(x)=11+a˙(x)2,d(x)=a˙(x)c(x),a˙(x)=da(x)dx.
Lϕ(x)=rϕ(x),ϕ(x)=f(x)g(x).
F(x, u)=Ez(x, u),G(x, u)=ZHx(x, u),
ZHu(x, u)=Gu(x, u)=c(x)jkF(x, u)x+d(x)G(x, u),
Ex(x, u)=Eu(x, u)=Hz(x, u)=0;
F(x, u)=ZHz(x, u),G(x, u)=-Ex(x, u),
Eu(x, u)=Gu(x, u)=-c(x)jkF(x, u)x-d(x)G(x, u),
Hx(x, u)=Hu(x, u)=Ez(x, u)=0.
Gu(x, u)=ZHu(x, u)=Z{-a˙(x)Hx(x, u)+[1+a˙2(x)]Hu(x, u)}=1jkF(x, u)u;
Gu(x, u)=Eu(x, u)=-a˙(x)Ex(x, u)+[1+a˙2(x)]Eu(x, u)=-1jkF(x, u)u.
ϕ(x)=m=-m=+ϕm exp(-jαmx),ϕm=fmgm,
f(x)=m=-m=+fm exp(-jαmx),
g(x)=m=-m=+gm exp(-jαmx),
αm=k(1) sin θ+mK,K=2πD.
[L(j)]ϕ(j)=r(j)ϕ(j),[L(j)]=[L11(j)][L12(j)][L21(j)][L22(j)],
ϕ(j)=f(j)g(j),j=1, 2,
[L11(j)]mq=-αqk(j)dm-q,[L12(j)]mq=-cm-q,
[L21(j)]mq=αmαq(k(j))2cm-q-δmq,
[L22(j)]mq=-αmk(j)dm-q,
ψn(j)(x, u)=ϕn(j)(x)exp(jk(j)rn(j)u),j=1, 2,
ϕn(j)(x)=m=-Mm=+Mϕmn(j) exp(-jαmx),
n[1, 4M+2].
ψ(j)(x, u)=n=12M+1Cn(j)ψn(j)(x, u),
ψt(1)(x, u)=ψ(1)(x, u)+ψi(1)(x, u).
ξn(j)=ν(j) cos θn(j)cos θ|Cn(j)|2if Im(ν(j))=0,
S=j=12nξn(j).
Δξn(j)(M)=-log10ξn(j)(M+1)-ξn(j)(M)ξn(j)(M),
ΔS(M)=-log10S(M+1)-S(M)S(M).
ϕn(j)(x)=fn(j)(x)gn(j)(x)=m=-n0Mm=+n0Mϕmn(j) exp(-jαmx)=m=-n0Mm=+n0Mfmn(j)gmn(j)exp(-jαmx),
αm=k(1) sin θ+mK,K=2πD.
ϕn(j)(x)=fn(j)(x)gn(j)(x)=p=0n0-1ϕp,n(j)(x)=p=0n0-1fp,n(j)(x)gp,n(j)(x),j=1, 2,
ϕp,n(j)(x)=q=-Mq=Mpϕq(p,n)(j) exp(-jαn0q+px),
ϕq(p,n)(j)=fq(p,n)(j)gq(p,n)(j),M0=MandMp0=M-1,
αn0q+p=k(1)sin θ+pλD+qK0,K0=2πD0.
kx,p(1)=k(1)sin θ+p λD,p[0, n0-1].
(rp,n(j))2=(cos θp,n(j))2=1-kx,p(1)+nK0k(j)2=1-sin θ+(nn0+p)λ/Dν(j)2,
Im(ν(j))=0andj=1, 2.
ifm=n0q+p,fmn(j)=fq(p,n)(j),gmn(j)=gq(p,n)(j),elsefmn(j)=gmn(j)=0;ifp=0,q[-M, M]andn=n;ifp[1, n0-1],q[-M, M-1]andn=n+2Mp+1.
ψ(j)(x, u0)=F(j)(x, u0)G(j)(x, u0)=n=12Mn0+1Cn(j)ψn(j)(x, u0),
u0=y-a0(x),
ψn(j)(x, u0)=Fn(j)(x, u0)Gn(j)(x, u0)=ϕn(j)(x)exp(jk(j)rn(j)u0),
ϕn(j)(x)=fn(j)(x)gn(j)(x).
n=12Mn0+1Cn(1)fn(1)(x)exp[jk(1)rn(1)a1(x)]-νfn=12Mn0+1Cn(2)fn(2)(x)exp[jk(2)rn(2)a1(x)]
=-fi(1)(x)exp[jkri(1)a1(x)].
c(x)G(j)(x, u0)=-1jk(j)F(j)(x, u0)u+1jk(j)d(x) F(j)(x, u0)x,
u0=u+a1(x),
F(j)(x, u0)=n=12Mn0+1Cn(j)Fn(j)(x, u0).
n=12Mn0+1Cn(1)-rn(1)hn(1)(x)+d(x)jk(1)hn(1)(x)x-νgn=12Mn0+1Cn(2)-rn(2)hn(2)(x)+d(x)jk(2)hn(2)(x)(x)x
=ri(1)hi(1)(x)-d(x)jk(1)hi(1)(x)x,
hn(j)(x)=fn(j)(x)exp[jk(j)rn(j)a1(x)]=m=-n0Mm=+n0Mhmn(j) exp(-jαmx).
[H(1)]-νf[H(2)]-[H(1)][R(1)]+[L11(1)][H(1)]νg([H(2)][R(2)]-[L11(2)][H(2)])C(1)C(2)=Hi(1)ri(1)Hi(1)-[L11(1)]Hi(1),
(C(j))n=Cn(j),[R(j)]mn=δmnrn(j),
(Hi(1))m=hmi(1),[H(j)]mn=hmn(j),j=1, 2,
hmn(j)=q=-n0M+n0Mfqn(j)Vm-q,n(j),
Vm,n(j)=1D0D exp[jk(j)rn(j)a1(x)]exp(jmKx)dx.
F(j)(x, y)=n=-n=+Cn(j)Fn(j)(x, y)
fory>max[a1(x)],ifj=1
fory<min[a1(x)]ifj=2.
Fn(j)(x, y)=exp(-jαnx)exp(jk(j)rn(j)y),
αn=k(1) sin θ+nK1,K1=2πD1,
rn(j)=(-1)j1-1ν(j)2sin θ+n λD121/2,
Im(rn(j))0.
σc=12π1D0D|c˙0(x)-c˙(x)|2dx1/2,
σd=12π1D0D|d˙0(x)-d˙(x)|2dx1/2.
σc<0.1176,σd<0.2776.
a0(x)=0.2688λ cos2πxD0-10°+0.0672λ cos4πxD0+20°,λD0=1,
a1(x)=0.06λ cos2πxD1-45°+0.02λ cos4πxD1-45°,λD1=4.
n0-1AdditionalIncidences:
sin θ+p λ/D,p0
S=nξn(1)

Metrics