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

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  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 (1)

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

2003 (1)

2002 (1)

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 (1)

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

1998 (1)

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

1997 (1)

1995 (1)

1986 (2)

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 (2)

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]

1981 (1)

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

1972 (1)

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

1968 (1)

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, 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.

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.

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, 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 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.

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. (1)

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. (3)

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. (1)

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 (1)

Opt. Lett. (2)

Proc. IEEE (1)

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

Radio Sci. (5)

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]

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

Other (8)

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.

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. .

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.

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.

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 (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 } .

Metrics