Abstract

This paper deals with the problem of electromagnetic wave scattering from rough surfaces. By means of a generalized Floquet modal representation an analytic solution can be found that is valid for any Fourier-expandable surface pattern, with no limitation on the corrugation amplitude. By means of the special relativistic frame-hopping method, the motionless solution is generalized to the case of uniform translational relative motion between the surface and the observer. Plane-wave simplification techniques are employed to minimize the algebraic complexity of the field covariance transformations. A detailed signal analysis of the electromagnetic scattered field is performed in both the frequency and the time domains.

© 2006 Optical Society of America

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References

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  1. J. Van Bladel, Relativity and Engineering (Springer-Verlag, 1984).
    [CrossRef]
  2. D. Censor, "Scattering in velocity-dependent systems," Radio Sci. 7, 331-337 (1972).
    [CrossRef]
  3. D. Censor, "The mathematical elements of relativistic free-space scattering," J. Electromagn. Waves Appl. 19, 907-923 (2005).
    [CrossRef]
  4. P. De Cupis, G. Gerosa, and G. Schettini, "Electromagnetic scattering by uniformly moving bodies," J. Electromagn. Waves Appl. 14, 1037-1062 (2000).
    [CrossRef]
  5. P. De Cupis, P. Burghignoli, G. Gerosa, and M. Marziale, "Electromagnetic wave scattering by a perfectly conducting wedge in uniform translational motion," J. Electromagn. Waves Appl. Vol. 16, 345-364 (2002).
    [CrossRef]
  6. P. De Cupis, D. Anatriello, and G. Gerosa, "An asymptotically exact technique for electromagnetic scattering by uniformly moving objects with arbitrary geometry," XXVIIth Union Radio-Scientifique Internationale General Assembly, Maastricht (the Netherlands), August 17-24, 2002. PDF available at http://www.die.uniromal.it/personale/gerosa/bibliografia/2002URSIlowbarp509.pdf.
  7. G. Valenzuela, "Scattering of electromagnetic waves from a slightly rough surface moving with uniform velocity," Radio Sci. 3, 1154-1157 (1968).
  8. A. Tzikas, D. Chrissoulidis, and E. Kriezis, "Relativistic bistatic scattering by a uniformly moving random rough surface," IEEE Trans. Microwave Theory Tech. 34, 1046-1052 (1986).
  9. P. De Cupis, "Relativistic scattering by moving rough surfaces," VIth Società Italiana di Matematica Applicata e Industriale (SIMAI) National Congress, Domus de Maria (CA), Italy, May 27-31, 2002. PDF available at http://www.die.uniromal.it/personale/gerosa/bibliografia/2002SIMAI.pdf.
  10. D. P. Chrissoulidis, L. P. Ivrissimtzis, and E. E. Kriezis, "Relativistic scattering by a random rough Kirchhoff surface in uniform motion," Radio Sci. 22, 803-814 (1986).
    [CrossRef]
  11. S. L. Chuang and J. A. Kong, "Scattering of waves from periodic surfaces," Proc. IEEE 69, 1132-1144 (1981).
    [CrossRef]
  12. S. Savaidis, P. Frangos, D. L. Jaggard, and K. Hizanidis, "Scattering from fractally corrugated surfaces: an exact approach," Opt. Lett. 20, 2357-2359 (1995).
    [CrossRef] [PubMed]
  13. S. Savaidis, P. Frangos, D. L. Jaggard, and K. Hizanidis, "Scattering from fractally corrugated surfaces with use of the extended boundary method," J. Opt. Soc. Am. A 14, 475-485 (1997).
    [CrossRef]
  14. P. De Cupis, "Relativistic scattering by moving rough surfaces," Opt. Lett. 28, 849-850 (2003).
    [CrossRef] [PubMed]
  15. P. De Cupis and G. Gerosa, "Relativistic scattering by moving rough surfaces," in Electromagnetics in a Complex World, I.Pinto, ed. (Springer-Verlag, 2004). PDF draft http://www.die.uniromal.it/personale/gerosa/bibliografia/2003lowbarbenevento.pdf.
  16. For instance, in the case of a periodical pattern with roughness parameters given by Eq. 13, different indices Q not = L epsilon Z^M should redundantly assign the same Floquet mode (i.e., K'_Q = K'_L) if Sigma^M _m=1(qm−lm)jm=0.
  17. D. Censor, "The Doppler effect—now you see it, now you don't," J. Math. Phys. 25, 309-316 (1984).
    [CrossRef]
  18. Within the proposed relativistic extension of the EBC methodology, the influence on the Doppler spectrum profile of the shape of the diffractive corrugation is summarized through the following theoretical chain: The surface geometric features are directly represented by the Fourier parameters fm,Xim, varphim, m=1-M, given by Eq. . Their values determine the algebraic linear system obtainable from Eq. , whose solutions furnish the amplitudes {CP′,DP′} of the various Floquet modes in the co-moving frame Σ′. Then, after covariance transformation to the laboratory frame Σ, the amplitude coefficients of any given Floquet mode, i.e., {CP,DP}, set the magnitude level of its associated Doppler-shifted tune; see Eq. . Also the exponential term exp[i(kp∙r)], whose magnitude is not = 1 for an evanescent Floquet mode, is directly dependent on K′={xim}m=1−M, as one can infer from Eqs. .
  19. Y. Ben Shimol and D. Censor, "Contribution to the problem of near-zone inverse Doppler effect," Radio Sci. 33, 463-474 (1998).
    [CrossRef]
  20. D. Censor, "Theory of the Doppler effect: fact, fiction and approximation," Radio Sci. 19, 1027-1040 (1984).
    [CrossRef]
  21. Main aliases are shifted by Deltax′=l2π/xi_1, l epsilon Z−{0}, where xi_1 is the slowest spatial "tune" used in Eq. , and are in general dissimilar with respect to the actual peak, since the Riemann approximate summation is in general not periodical, i.e., usually xi_m not = mxi_1, m=2,3...,M in Eq. .
  22. J. A. Kong, Electromagnetic Wave Theory (Wiley, 1986), Chap. 6.3.

2005

D. Censor, "The mathematical elements of relativistic free-space scattering," J. Electromagn. Waves Appl. 19, 907-923 (2005).
[CrossRef]

2003

2002

P. De Cupis, P. Burghignoli, G. Gerosa, and M. Marziale, "Electromagnetic wave scattering by a perfectly conducting wedge in uniform translational motion," J. Electromagn. Waves Appl. Vol. 16, 345-364 (2002).
[CrossRef]

2000

P. De Cupis, G. Gerosa, and G. Schettini, "Electromagnetic scattering by uniformly moving bodies," J. Electromagn. Waves Appl. 14, 1037-1062 (2000).
[CrossRef]

1998

Y. Ben Shimol and D. Censor, "Contribution to the problem of near-zone inverse Doppler effect," Radio Sci. 33, 463-474 (1998).
[CrossRef]

1997

1995

1986

A. Tzikas, D. Chrissoulidis, and E. Kriezis, "Relativistic bistatic scattering by a uniformly moving random rough surface," IEEE Trans. Microwave Theory Tech. 34, 1046-1052 (1986).

D. P. Chrissoulidis, L. P. Ivrissimtzis, and E. E. Kriezis, "Relativistic scattering by a random rough Kirchhoff surface in uniform motion," Radio Sci. 22, 803-814 (1986).
[CrossRef]

1984

D. Censor, "The Doppler effect—now you see it, now you don't," J. Math. Phys. 25, 309-316 (1984).
[CrossRef]

D. Censor, "Theory of the Doppler effect: fact, fiction and approximation," Radio Sci. 19, 1027-1040 (1984).
[CrossRef]

1981

S. L. Chuang and J. A. Kong, "Scattering of waves from periodic surfaces," Proc. IEEE 69, 1132-1144 (1981).
[CrossRef]

1972

D. Censor, "Scattering in velocity-dependent systems," Radio Sci. 7, 331-337 (1972).
[CrossRef]

1968

G. Valenzuela, "Scattering of electromagnetic waves from a slightly rough surface moving with uniform velocity," Radio Sci. 3, 1154-1157 (1968).

Anatriello, D.

P. De Cupis, D. Anatriello, and G. Gerosa, "An asymptotically exact technique for electromagnetic scattering by uniformly moving objects with arbitrary geometry," XXVIIth Union Radio-Scientifique Internationale General Assembly, Maastricht (the Netherlands), August 17-24, 2002. PDF available at http://www.die.uniromal.it/personale/gerosa/bibliografia/2002URSIlowbarp509.pdf.

Ben Shimol, Y.

Y. Ben Shimol and D. Censor, "Contribution to the problem of near-zone inverse Doppler effect," Radio Sci. 33, 463-474 (1998).
[CrossRef]

Burghignoli, P.

P. De Cupis, P. Burghignoli, G. Gerosa, and M. Marziale, "Electromagnetic wave scattering by a perfectly conducting wedge in uniform translational motion," J. Electromagn. Waves Appl. Vol. 16, 345-364 (2002).
[CrossRef]

Censor, D.

D. Censor, "The mathematical elements of relativistic free-space scattering," J. Electromagn. Waves Appl. 19, 907-923 (2005).
[CrossRef]

Y. Ben Shimol and D. Censor, "Contribution to the problem of near-zone inverse Doppler effect," Radio Sci. 33, 463-474 (1998).
[CrossRef]

D. Censor, "Theory of the Doppler effect: fact, fiction and approximation," Radio Sci. 19, 1027-1040 (1984).
[CrossRef]

D. Censor, "The Doppler effect—now you see it, now you don't," J. Math. Phys. 25, 309-316 (1984).
[CrossRef]

D. Censor, "Scattering in velocity-dependent systems," Radio Sci. 7, 331-337 (1972).
[CrossRef]

Chrissoulidis, D.

A. Tzikas, D. Chrissoulidis, and E. Kriezis, "Relativistic bistatic scattering by a uniformly moving random rough surface," IEEE Trans. Microwave Theory Tech. 34, 1046-1052 (1986).

Chrissoulidis, D. P.

D. P. Chrissoulidis, L. P. Ivrissimtzis, and E. E. Kriezis, "Relativistic scattering by a random rough Kirchhoff surface in uniform motion," Radio Sci. 22, 803-814 (1986).
[CrossRef]

Chuang, S. L.

S. L. Chuang and J. A. Kong, "Scattering of waves from periodic surfaces," Proc. IEEE 69, 1132-1144 (1981).
[CrossRef]

De Cupis, P.

P. De Cupis, "Relativistic scattering by moving rough surfaces," Opt. Lett. 28, 849-850 (2003).
[CrossRef] [PubMed]

P. De Cupis, P. Burghignoli, G. Gerosa, and M. Marziale, "Electromagnetic wave scattering by a perfectly conducting wedge in uniform translational motion," J. Electromagn. Waves Appl. Vol. 16, 345-364 (2002).
[CrossRef]

P. De Cupis, G. Gerosa, and G. Schettini, "Electromagnetic scattering by uniformly moving bodies," J. Electromagn. Waves Appl. 14, 1037-1062 (2000).
[CrossRef]

P. De Cupis and G. Gerosa, "Relativistic scattering by moving rough surfaces," in Electromagnetics in a Complex World, I.Pinto, ed. (Springer-Verlag, 2004). PDF draft http://www.die.uniromal.it/personale/gerosa/bibliografia/2003lowbarbenevento.pdf.

P. De Cupis, D. Anatriello, and G. Gerosa, "An asymptotically exact technique for electromagnetic scattering by uniformly moving objects with arbitrary geometry," XXVIIth Union Radio-Scientifique Internationale General Assembly, Maastricht (the Netherlands), August 17-24, 2002. PDF available at http://www.die.uniromal.it/personale/gerosa/bibliografia/2002URSIlowbarp509.pdf.

P. De Cupis, "Relativistic scattering by moving rough surfaces," VIth Società Italiana di Matematica Applicata e Industriale (SIMAI) National Congress, Domus de Maria (CA), Italy, May 27-31, 2002. PDF available at http://www.die.uniromal.it/personale/gerosa/bibliografia/2002SIMAI.pdf.

Frangos, P.

Gerosa, G.

P. De Cupis, P. Burghignoli, G. Gerosa, and M. Marziale, "Electromagnetic wave scattering by a perfectly conducting wedge in uniform translational motion," J. Electromagn. Waves Appl. Vol. 16, 345-364 (2002).
[CrossRef]

P. De Cupis, G. Gerosa, and G. Schettini, "Electromagnetic scattering by uniformly moving bodies," J. Electromagn. Waves Appl. 14, 1037-1062 (2000).
[CrossRef]

P. De Cupis and G. Gerosa, "Relativistic scattering by moving rough surfaces," in Electromagnetics in a Complex World, I.Pinto, ed. (Springer-Verlag, 2004). PDF draft http://www.die.uniromal.it/personale/gerosa/bibliografia/2003lowbarbenevento.pdf.

P. De Cupis, D. Anatriello, and G. Gerosa, "An asymptotically exact technique for electromagnetic scattering by uniformly moving objects with arbitrary geometry," XXVIIth Union Radio-Scientifique Internationale General Assembly, Maastricht (the Netherlands), August 17-24, 2002. PDF available at http://www.die.uniromal.it/personale/gerosa/bibliografia/2002URSIlowbarp509.pdf.

Hizanidis, K.

Ivrissimtzis, L. P.

D. P. Chrissoulidis, L. P. Ivrissimtzis, and E. E. Kriezis, "Relativistic scattering by a random rough Kirchhoff surface in uniform motion," Radio Sci. 22, 803-814 (1986).
[CrossRef]

Jaggard, D. L.

Kong, J. A.

S. L. Chuang and J. A. Kong, "Scattering of waves from periodic surfaces," Proc. IEEE 69, 1132-1144 (1981).
[CrossRef]

J. A. Kong, Electromagnetic Wave Theory (Wiley, 1986), Chap. 6.3.

Kriezis, E.

A. Tzikas, D. Chrissoulidis, and E. Kriezis, "Relativistic bistatic scattering by a uniformly moving random rough surface," IEEE Trans. Microwave Theory Tech. 34, 1046-1052 (1986).

Kriezis, E. E.

D. P. Chrissoulidis, L. P. Ivrissimtzis, and E. E. Kriezis, "Relativistic scattering by a random rough Kirchhoff surface in uniform motion," Radio Sci. 22, 803-814 (1986).
[CrossRef]

Marziale, M.

P. De Cupis, P. Burghignoli, G. Gerosa, and M. Marziale, "Electromagnetic wave scattering by a perfectly conducting wedge in uniform translational motion," J. Electromagn. Waves Appl. Vol. 16, 345-364 (2002).
[CrossRef]

Savaidis, S.

Schettini, G.

P. De Cupis, G. Gerosa, and G. Schettini, "Electromagnetic scattering by uniformly moving bodies," J. Electromagn. Waves Appl. 14, 1037-1062 (2000).
[CrossRef]

Tzikas, A.

A. Tzikas, D. Chrissoulidis, and E. Kriezis, "Relativistic bistatic scattering by a uniformly moving random rough surface," IEEE Trans. Microwave Theory Tech. 34, 1046-1052 (1986).

Valenzuela, G.

G. Valenzuela, "Scattering of electromagnetic waves from a slightly rough surface moving with uniform velocity," Radio Sci. 3, 1154-1157 (1968).

Van Bladel, J.

J. Van Bladel, Relativity and Engineering (Springer-Verlag, 1984).
[CrossRef]

IEEE Trans. Microwave Theory Tech.

A. Tzikas, D. Chrissoulidis, and E. Kriezis, "Relativistic bistatic scattering by a uniformly moving random rough surface," IEEE Trans. Microwave Theory Tech. 34, 1046-1052 (1986).

J. Electromagn. Waves Appl.

D. Censor, "The mathematical elements of relativistic free-space scattering," J. Electromagn. Waves Appl. 19, 907-923 (2005).
[CrossRef]

P. De Cupis, G. Gerosa, and G. Schettini, "Electromagnetic scattering by uniformly moving bodies," J. Electromagn. Waves Appl. 14, 1037-1062 (2000).
[CrossRef]

P. De Cupis, P. Burghignoli, G. Gerosa, and M. Marziale, "Electromagnetic wave scattering by a perfectly conducting wedge in uniform translational motion," J. Electromagn. Waves Appl. Vol. 16, 345-364 (2002).
[CrossRef]

J. Math. Phys.

D. Censor, "The Doppler effect—now you see it, now you don't," J. Math. Phys. 25, 309-316 (1984).
[CrossRef]

J. Opt. Soc. Am. A

Opt. Lett.

Proc. IEEE

S. L. Chuang and J. A. Kong, "Scattering of waves from periodic surfaces," Proc. IEEE 69, 1132-1144 (1981).
[CrossRef]

Radio Sci.

G. Valenzuela, "Scattering of electromagnetic waves from a slightly rough surface moving with uniform velocity," Radio Sci. 3, 1154-1157 (1968).

D. P. Chrissoulidis, L. P. Ivrissimtzis, and E. E. Kriezis, "Relativistic scattering by a random rough Kirchhoff surface in uniform motion," Radio Sci. 22, 803-814 (1986).
[CrossRef]

Y. Ben Shimol and D. Censor, "Contribution to the problem of near-zone inverse Doppler effect," Radio Sci. 33, 463-474 (1998).
[CrossRef]

D. Censor, "Theory of the Doppler effect: fact, fiction and approximation," Radio Sci. 19, 1027-1040 (1984).
[CrossRef]

D. Censor, "Scattering in velocity-dependent systems," Radio Sci. 7, 331-337 (1972).
[CrossRef]

Other

J. Van Bladel, Relativity and Engineering (Springer-Verlag, 1984).
[CrossRef]

P. De Cupis, "Relativistic scattering by moving rough surfaces," VIth Società Italiana di Matematica Applicata e Industriale (SIMAI) National Congress, Domus de Maria (CA), Italy, May 27-31, 2002. PDF available at http://www.die.uniromal.it/personale/gerosa/bibliografia/2002SIMAI.pdf.

P. De Cupis, D. Anatriello, and G. Gerosa, "An asymptotically exact technique for electromagnetic scattering by uniformly moving objects with arbitrary geometry," XXVIIth Union Radio-Scientifique Internationale General Assembly, Maastricht (the Netherlands), August 17-24, 2002. PDF available at http://www.die.uniromal.it/personale/gerosa/bibliografia/2002URSIlowbarp509.pdf.

Main aliases are shifted by Deltax′=l2π/xi_1, l epsilon Z−{0}, where xi_1 is the slowest spatial "tune" used in Eq. , and are in general dissimilar with respect to the actual peak, since the Riemann approximate summation is in general not periodical, i.e., usually xi_m not = mxi_1, m=2,3...,M in Eq. .

J. A. Kong, Electromagnetic Wave Theory (Wiley, 1986), Chap. 6.3.

P. De Cupis and G. Gerosa, "Relativistic scattering by moving rough surfaces," in Electromagnetics in a Complex World, I.Pinto, ed. (Springer-Verlag, 2004). PDF draft http://www.die.uniromal.it/personale/gerosa/bibliografia/2003lowbarbenevento.pdf.

For instance, in the case of a periodical pattern with roughness parameters given by Eq. 13, different indices Q not = L epsilon Z^M should redundantly assign the same Floquet mode (i.e., K'_Q = K'_L) if Sigma^M _m=1(qm−lm)jm=0.

Within the proposed relativistic extension of the EBC methodology, the influence on the Doppler spectrum profile of the shape of the diffractive corrugation is summarized through the following theoretical chain: The surface geometric features are directly represented by the Fourier parameters fm,Xim, varphim, m=1-M, given by Eq. . Their values determine the algebraic linear system obtainable from Eq. , whose solutions furnish the amplitudes {CP′,DP′} of the various Floquet modes in the co-moving frame Σ′. Then, after covariance transformation to the laboratory frame Σ, the amplitude coefficients of any given Floquet mode, i.e., {CP,DP}, set the magnitude level of its associated Doppler-shifted tune; see Eq. . Also the exponential term exp[i(kp∙r)], whose magnitude is not = 1 for an evanescent Floquet mode, is directly dependent on K′={xim}m=1−M, as one can infer from Eqs. .

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Figures (10)

Fig. 1
Fig. 1

Geometry of the problem in the rest frame Σ of the diffractive surface; f ( x ) is a 10th-order approximation of a rectangular pulse of height h and width W .

Fig. 2
Fig. 2

Normalized Doppler spectra s ̃ P ( r ) = S P ( r ) max Q G M 0 S Q ( r ) at observation point r = 2 λ z ̂ versus normalized frequency shift ϖ P = Re [ ν ¯ P ] β , P G M 0 . Single rectangular pulse with h = W = λ . Circular polarization, A = B ; incidence angle, θ = π 2 . Relative velocity angles, ψ = π 2 (solid curve) and ψ = π 2 (dashed curve); counter-parting intervals Δ τ = ( , t 0 ] and Δ τ = [ t 0 , ) , respectively, t 0 = λ ( 2 γ 1 ) γ β c . (a) Low-speed case, β = 10 6 . (b) High-speed case, β = 0.2 .

Fig. 3
Fig. 3

Same as Fig. 2 except β = 10 7 , θ = π 4 . (a) ψ = 7 12 π , i.e., χ = π 3 , Δ τ = [ t 0 , ) . (b) ψ = 5 12 π , i.e., χ = 2 3 π , Δ τ = ( , t 0 ] . t 0 = λ ( γ β c sin ψ ) [ 1 + 2 ( γ 1 ) sin ψ 2 ] .

Fig. 4
Fig. 4

Same as Fig. 3 except β = 0.3 .

Fig. 5
Fig. 5

Same as Fig. 2 except for elliptical polarization ( A B = 2 ) ; θ = π 2 ; scattering pulse with isosceles triangular shape, W = λ , h = λ ; ψ = 0 ; Δ τ = ( , ) . (a) β = 10 7 , (b) β = 0.3 .

Fig. 6
Fig. 6

Normalized amplitude modulation Υ ̃ η ( t ) = Υ η ( t ) Υ η ( t min ) relevant to η = E π ( 1 3 ) ( x ̂ + y ̂ + z ̂ ) versus normalized time t ̃ = β c t λ , in the same conditions as Fig. 2 except for r = 1.25 λ z ̂ , θ = π 2 , ψ = 0 , Δ τ = λ β c [ 4 , 4 ] . (a) β = 10 7 , (b) β = 0.3 .

Fig. 7
Fig. 7

Same as Fig. 6 except for θ = 5 π 4 , η = H π ( 1 3 ) ( x ̂ y ̂ + z ̂ ) , and B A = 2 . Dashed curves ψ = 2 π 3 with Δ τ = [ 4 λ β c , 0 ] ; solid curves ψ = 5 π 3 with Δ τ = [ 0 , 4 λ β c ] . (a) β = 10 7 , (b) β = 0.3 .

Fig. 8
Fig. 8

Normalized frequency modulations ʊ ̃ η = ʊ η ( β ω ) relevant to η = E π z ̂ versus normalized time t ̃ = β c t λ for various values of β. (a) Parallel relative motion, ψ = 0 ; normal incidence, θ = π 2 ; Δ τ = λ β c [ 4 , 4 ] . (b) Synopsis of complementary experiments: ψ = π 4 with Δ τ = [ 4 λ β c , 0 ] and ψ = 5 π 4 with Δ τ = [ 0 , 4 λ β c ] ; oblique incidence θ = π 12 . All other parameters are the same as for Fig. 6.

Fig. 9
Fig. 9

The same as Fig. 5 for the periodical triangular surface T per ( x ) . (a) β = 10 7 , (b) β = 0.3 .

Fig. 10
Fig. 10

Instantaneous modulations. (a) Normalized amplitude Υ ̃ η ( t ) = Υ η ( t ) Υ η ( 0 ) . (b) Normalized frequency shift ʊ ̃ η = ʊ η ( β ω ) versus normalized time t ̂ = t ω Λ ( 2 π λ ) for β = 10 6 (solid curves) and β = 0.4 (dashed curves). η = H π z ̂ . Other parameters are the same as for Fig. 9.

Tables (1)

Tables Icon

Table 1 Inaccuracy Comparison between the EBC Method (Exemplar Implementations a, b, c, and d) and the OP, MM, and RR Techniques a

Equations (56)

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

[ E i ζ H i ] r , t = Re [ E i ζ H i ] r exp ( i ω t ) = Re [ y ̂ y ̂ × κ ̂ i y ̂ × κ ̂ i y ̂ ] [ A B ] × exp [ i ( κ i r ω t ) ] ;
r = [ I ͇ + ( γ 1 ) v ̂ v ̂ ] r γ β c t v ̂ , t = γ ( t β c 1 v ̂ r ) ,
[ E i ζ H i ] r , t = Re [ E i ζ H i ] r exp ( i ω t ) = Re [ y ̂ y ̂ × κ ̂ i y ̂ × κ ̂ i y ̂ ] [ A B ] exp [ i ( κ i r ω t ) ] ,
{ ω , A , B } = γ [ 1 β cos ( θ ψ ) ] { ω , A , B } ,
κ i = ω c 1 κ ̂ i = ω c 1 ( x ̂ cos θ + z ̂ sin θ ) ,
{ cos θ sin θ } = ( 1 + { cos 2 ψ sin 2 ψ } ( γ 1 ) ) { cos θ sin θ } + ( γ 1 ) cos ψ sin ψ { sin θ cos θ } γ β { cos ψ sin ψ } γ [ 1 β cos ( θ ψ ) ] .
z = f ( x ) = m = 1 M f m sin ( ξ m x + φ m ) .
f ̃ ( x ) = 2 0 + F ̆ ( ξ ) sin { ξ x + arg [ F ̆ ( ξ ) ] + π 2 } d ξ ,
F ̆ ( ξ ) = 1 2 π + f ̃ ( x ) exp ( i ξ x ) d x ;
ξ m = ξ ̃ m , f m = 2 F ̆ ( ξ ̃ m ) δ ξ m , φ m = arg [ F ̆ ( ξ ̃ m ) ] + π 2 , m = 1 , , M .
f ̃ ( x ) = 2 j = 1 F ̆ j sin { j 2 π Λ x + arg [ F ̆ j ] + π 2 } ,
F ̆ j = 1 Λ Λ 2 Λ 2 f ̃ ( x ) exp ( i j 2 π Λ x ) d x .
f m = 2 F ̆ j m , ξ m = j m 2 π Λ , φ m = arg [ F ̆ j m ] + π 2 , m = 1 , , M ,
[ E s ζ H s ] r , t = P Z M [ E s P ζ H s P ] r , t = Re P Z M [ y ̂ y ̂ × κ ̂ P y ̂ × κ ̂ P y ̂ ] [ C P D P ] × exp [ i ( κ P r ω t ) ] ,
κ P = x ̂ κ x P + z ̂ κ z P , κ x P = κ i x + P K , κ z P = + ( ω c 1 ) 2 ( κ x P ) 2 ,
κ ̂ P = κ P ω c 1 = x ̂ cos ϑ P + z ̂ sin ϑ P ; cos ϑ P = cos θ + P K ( ω c 1 ) , sin ϑ P = + 1 cos 2 ϑ P ;
[ E s ζ H s ] r , t = lim G M Z M P G M [ E s P ζ H s P ] r , t = Re lim G M Z M P G M [ y ̂ y ̂ × κ ̂ P y ̂ × κ ̂ P y ̂ ] [ C P D P ] exp [ i ( κ P r Ω P t ) ] ,
{ Ω P , C P , D P } = γ [ 1 + β cos ( ϑ P ψ ) ] { ω , C P , D P } ,
κ P = Ω P c 1 κ ̂ P = Ω P c 1 ( x ̂ cos ϑ P + z ̂ sin ϑ P ) ,
{ cos ϑ P sin ϑ P } = ( 1 + { cos 2 ψ sin 2 ψ } ( γ 1 ) ) { cos ϑ P sin ϑ P } + ( γ 1 ) cos ψ sin ψ { sin ϑ P cos ϑ P } + γ β { cos ψ sin ψ } γ [ 1 + β cos ( ϑ P ψ ) ] ,
Γ + = { ( r , t ) x ( γ 1 ) cos ψ sin ψ + z [ 1 + ( γ 1 ) sin 2 ψ ] c t γ β sin ψ = z > z max } .
[ E π ζ H π ] r , t = Re [ E ¯ π ζ H ¯ π ] r , t ; ω exp ( i ω t ) ,
[ E ¯ π ζ H ¯ π ] r , t ; ω P G M 0 [ y ̂ y ̂ × κ ̂ P y ̂ × κ ̂ P y ̂ ] [ C P D P ] exp [ i κ P r ] exp [ i ν ¯ P t ] ,
ν ¯ P = { γ 2 [ 1 + β cos ( ϑ P ψ ) ] [ 1 β cos ( θ ψ ) ] 1 } ω ,
S ( r ) = { ( S P ( r ) , ν P ) } P G M 0 = { ( C P 2 + D P 2 exp [ i ( κ P r ) ] , Re [ ν ¯ P ] ) } P G M 0 .
δ S u = 1 g M 1 P G M 0 S P S P u S P , δ Υ η u = 1 Δ τ t Δ τ Υ η Υ η u Υ η d t ,
δ ʊ η u = 1 Δ τ t Δ τ ʊ η ʊ η u ω d t ,
{ E i y H i y } r + S { g ( r , r 0 ) S 0 E y r 0 H y ( r 0 ) S 0 g ( r , r 0 ) } w ̂ 0 d S 0 = { E y H y } r u [ z f ( x ) ] ,
g ( r , r 0 ) = i 4 H 0 ( 1 ) ( κ r r 0 ) = i 4 π 1 k z exp { i [ k x ( x x 0 ) + k z z z 0 ] } d k x ,
d S 0 w ̂ 0 S 0 g ( r , r 0 ) = d x 0 [ D f ( x 0 ) x 0 + z 0 ] g ( r , r 0 ) = d x 0 4 π D f ( x 0 ) k x + sgn ( z z 0 ) k z k z × exp { i [ k x ( x x 0 ) + k z z z 0 ] } d k x ,
d S 0 { w ̂ 0 S 0 E y r 0 ζ H y ( r 0 ) } = d x 0 { κ 1 } exp [ i κ i x x 0 ] Q Z M { c Q d Q } × exp [ i Q K x 0 ] ,
1 ± 1 2 { E y ζ H y } r = { E i y ζ H i y } r 1 4 π { i κ 1 } × k x = ( 1 k z exp [ i ( k x x ± k z z ) ] Q Z M { c Q d Q } { I Q ± ( k x ) k x L Q ± ( k x ) k z I Q ± ( k x ) } ) d k x ,
{ I Q ± ( k x ) L Q ± ( k x ) } = { 1 D f ( x 0 ) } exp { i [ ( k x κ i x Q K ) x 0 ± k z f ( x 0 ) ] } d x 0 ,
exp { i [ k z f ( x 0 ) ] } = exp { i [ k z m = 1 M f m sin ( ξ m x 0 + φ m ) ] } = m = 1 M exp { i [ k z f m sin ( ξ m x 0 + φ m ) ] } = m = 1 M p m = ( 1 ) p m J p m ( k z f m ) exp [ i p m ( ξ m x 0 + φ m ) ] = P Z M ( 1 ) χ ( P ) exp [ i ( P K x 0 + P Φ ) ] Λ P ( k x ) .
Λ P ( k x ) = m = 1 M J p m [ ( κ 2 k x 2 ) 1 2 f m ] , χ ( P ) = m = 1 M p m , Φ = { φ m } m = 1 M .
L Q ± ( k x ) = ± [ R Q ± ( k x ) + κ i x + Q K k z I Q ± ( k x ) ] ,
R Q ± ( k x ) = i exp ( i k x x 0 ) k z d d x 0 exp { i [ ( κ i x + Q K ) x 0 k z f ( x 0 ) ] } d x 0 .
I Q ± ( k x ) = P Z M ( 1 ) χ ( P ) exp ( i P Φ ) m = 1 M J p m ( k z f m ) exp { i [ k x κ i x ( Q + P ) K ] x 0 } d x 0 = P Z M ( 1 ) χ ( P ) exp ( i P Φ ) 2 π δ { k x [ κ i x + ( Q + P ) K ] } Λ P ( k x ) ,
R Q ± ( k x ) = i P Z M ( 1 ) χ ( P ) exp ( i P Φ ) 1 k z Λ P ( k x ) exp ( i k x x 0 ) d d x 0 exp ( i { [ κ i x + ( Q + P ) K ] x 0 } ) d x 0 = i P Z M ( 1 ) χ ( P ) exp ( i P Φ ) i [ κ i x + ( Q + P ) K ] k z exp ( i { k x [ κ i x + ( Q + P ) K ] } x 0 ) d x 0 Λ P ( k x ) = P Z M ( 1 ) χ ( P ) exp ( i P Φ ) [ κ i x + ( Q + P ) K ] k z 2 π δ { k x [ κ i x + ( Q + P ) K ] } Λ P ( k x ) ,
1 ± 1 2 { E y ζ H y } r = { E i y ζ H i y } r 1 2 { i κ 1 } P Z M Q Z M { c Q d Q } 1 κ z ( Q + P ) exp [ i ( κ x ( Q + P ) x ± κ z ( Q + P ) z ) ] ( 1 ) χ ( P ) exp ( i P Φ ) { 1 ± ( κ x ( Q + P ) κ z ( Q + P ) + κ x Q κ z ( Q + P ) ) κ x ( Q + P ) κ z ( Q + P ) } Λ P ( κ x ( Q + P ) ) = { E i y ζ H i y } r 1 2 P Z M Q Z M { c Q d Q } exp [ i ( κ x P x ± κ z P z ) ] × ( 1 ) χ ( P Q ) exp [ i ( P Q ) Φ ] { i κ κ z P ( κ 2 κ x P κ x Q ) κ z P 2 } Λ P Q ( κ x P ) ,
{ C P D P } = 1 2 Q Z M { c Q d Q } ( 1 ) χ ( P Q ) { i κ κ x P ( κ 2 κ z P κ z Q ) ( κ z P ) 2 } × exp [ i ( P Q ) Φ ] Λ P Q ( κ x P ) ,
[ E i ζ H i ] r , t = Re P Z M [ y ̂ y ̂ × κ ̂ P y ̂ × κ ̂ P y ̂ ] [ A P B P ] exp [ i ( κ P r ω t ) ] ,
{ A P B P } = Q Z M { c Q A P Q d Q B P Q } ,
{ A P Q B P Q } = 1 2 { i κ κ z P ( κ 2 κ z P κ z Q ) ( κ z P ) 2 } exp [ i ( P Q ) Φ ] Λ P Q ( κ x P ) .
k = [ I ͇ + ( γ 1 ) v ̂ v ̂ ] k + γ β c 1 Ω v ̂ ;
Ω = γ [ 1 + β v ̂ k ( Ω c 1 ) ] Ω .
Ω = γ [ 1 + β cos ( ϑ ψ ) ] Ω ,
{ cos ϑ sin ϑ } = ( 1 + { cos 2 ψ sin 2 ψ } ( γ 1 ) ) { cos 2 ϑ sin 2 ϑ } + ( γ 1 ) cos ψ sin ψ { sin ϑ cos ϑ } + γ β { cos ψ sin ψ } γ [ 1 + β cos ( ϑ ψ ) ] .
{ e , h } = { e TE , h TE } + { e TM , h TM } ,
{ e , h } = { e TE , h TE } + { e TM , h TM } ,
{ e TE , ζ h TE } = A { y ̂ , ( z ̂ cos ϑ x ̂ sin ϑ ) } ,
{ e TE , ζ h TE } = A { y ̂ , ( z ̂ cos ϑ x ̂ sin ϑ ) } ,
{ e TM , ζ h TM } = B { ( x ̂ sin ϑ z ̂ cos ϑ ) , y ̂ } ;
{ e TM , ζ h TM } = B { ( x ̂ sin ϑ z ̂ cos ϑ ) , y ̂ } .
{ e ζ h } = [ γ ( I ͇ v ̂ v ̂ ) + v ̂ v ̂ ] { e ζ h } γ β v ̂ × { ζ h e } .
{ A , B } = γ [ 1 + β cos ( ϑ ψ ) ] { A , B } .

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