Abstract

The current contribution is concerned with obtaining the relativistic two-dimensional (three-dimensional in relativity jargon) Green’s function of a time-harmonic line current that is embedded in a moving dielectric–magnetic medium with a planar discontinuity. By applying a plane-wave (PW) spectral representation for the relativistic electromagnetic Green’s function of a dielectric–magnetic medium that is moving in a uniform velocity, the exact reflected and transmitted (refracted) fields are obtained in the form of a spectral integral over PWs in the so-called laboratory and comoving frames. We investigate these spectral representations, as well as their asymptotic evaluations, and discuss the associated relativistic wave phenomena of direct reflected/transmitted rays and relativistic head waves (lateral waves).

© 2012 Optical Society of America

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References

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  1. C. Tai, “The dyadic Green’s function for a moving isotropic medium,” IEEE Trans. Antennas Propag. 13, 322–323 (1965).
    [CrossRef]
  2. J. V. Bladel, Relativity and Engineering (Springer, 1984).
  3. T. Mackay and A. Lakhtakia, “Negative phase velocity in a uniformly moving, homogeneous, isotropic, dielectric-magnetic medium,” J. Phys. A 37, 5697–5711 (2004).
    [CrossRef]
  4. T. Mackay, A. Lakhtakia, and S. Setiawan, “Positive-, negative-, and orthogonal-phase-velocity propagation of electromagnetic plane waves in a simply moving medium,” Optik 118, 195–202(2007).
    [CrossRef]
  5. A. Lakhtakia and W. Weiglhofer, “On electromagnetic fields in a linear medium with gyrotropic-like magnetoelectric properties,” Microwave Opt. Technol. Lett. 15, 168–170 (1997).
    [CrossRef]
  6. A. Lakhtakia and T. Mackay, “Simple derivation of dyadic Green functions of a simply moving, isotropic, dielectric-magnetic medium,” Microwave Opt. Technol. Lett. 48, 1073–1074 (2006).
    [CrossRef]
  7. M. Idemen and A. Alkumru, “Relativistic scattering of a plane-wave by a uniformly moving half-plane,” IEEE Trans. Antennas Propag. 54, 3429–3440 (2006).
    [CrossRef]
  8. P. De Cupis, G. Gerosa, and G. Schettini, “Gaussian beam diffraction by uniformly moving targets,” Atti della Fondazione Giorgio Ronchi 56, 799–811 (2001).
  9. P. D. Cupis, “Radiation by a moving wire-antenna in the presence of plane interface,” J. Electromagn. Waves Appl. 14, 1119–1132 (2000).
    [CrossRef]
  10. W. Arrighetti, P. De Cupis, and G. Gerosa, “Electromagnetic radiation from moving fractal sources: A plane-wave spectral approach,” Prog. Electromagn. Res. 58, 1–19 (2006).
    [CrossRef]
  11. P. D. Cupis, “Electromagnetic wave scattering from moving surfaces with high-amplitude corrugated pattern,” J. Opt. Soc. Am. A 23, 2538–2550 (2006).
    [CrossRef]
  12. T. Danov and T. Melamed, “Spectral analysis of relativistic dyadic Green’s function of a moving dielectric-magnetic medium,” IEEE Trans. Antennas Propag. 59, 2973–2979 (2011).
    [CrossRef]
  13. J. Kong, Theory of Electromagnetic Waves (Wiley, 1975).
  14. M. Idemen and A. Alkumru, “Influence of motion on the edge-diffraction,” Prog. Electromagn. Res. B 6, 153–168 (2008).
    [CrossRef]
  15. L. Felsen and N. Marcuvitz, Radiation and Scattering of Waves (IEEE, 1994), Chap. 4.2.
  16. L. Felsen and N. Marcuvitz, Radiation and Scattering of Waves (IEEE, 1994), Chap. 4.8.

2011 (1)

T. Danov and T. Melamed, “Spectral analysis of relativistic dyadic Green’s function of a moving dielectric-magnetic medium,” IEEE Trans. Antennas Propag. 59, 2973–2979 (2011).
[CrossRef]

2008 (1)

M. Idemen and A. Alkumru, “Influence of motion on the edge-diffraction,” Prog. Electromagn. Res. B 6, 153–168 (2008).
[CrossRef]

2007 (1)

T. Mackay, A. Lakhtakia, and S. Setiawan, “Positive-, negative-, and orthogonal-phase-velocity propagation of electromagnetic plane waves in a simply moving medium,” Optik 118, 195–202(2007).
[CrossRef]

2006 (4)

A. Lakhtakia and T. Mackay, “Simple derivation of dyadic Green functions of a simply moving, isotropic, dielectric-magnetic medium,” Microwave Opt. Technol. Lett. 48, 1073–1074 (2006).
[CrossRef]

M. Idemen and A. Alkumru, “Relativistic scattering of a plane-wave by a uniformly moving half-plane,” IEEE Trans. Antennas Propag. 54, 3429–3440 (2006).
[CrossRef]

P. D. Cupis, “Electromagnetic wave scattering from moving surfaces with high-amplitude corrugated pattern,” J. Opt. Soc. Am. A 23, 2538–2550 (2006).
[CrossRef]

W. Arrighetti, P. De Cupis, and G. Gerosa, “Electromagnetic radiation from moving fractal sources: A plane-wave spectral approach,” Prog. Electromagn. Res. 58, 1–19 (2006).
[CrossRef]

2004 (1)

T. Mackay and A. Lakhtakia, “Negative phase velocity in a uniformly moving, homogeneous, isotropic, dielectric-magnetic medium,” J. Phys. A 37, 5697–5711 (2004).
[CrossRef]

2001 (1)

P. De Cupis, G. Gerosa, and G. Schettini, “Gaussian beam diffraction by uniformly moving targets,” Atti della Fondazione Giorgio Ronchi 56, 799–811 (2001).

2000 (1)

P. D. Cupis, “Radiation by a moving wire-antenna in the presence of plane interface,” J. Electromagn. Waves Appl. 14, 1119–1132 (2000).
[CrossRef]

1997 (1)

A. Lakhtakia and W. Weiglhofer, “On electromagnetic fields in a linear medium with gyrotropic-like magnetoelectric properties,” Microwave Opt. Technol. Lett. 15, 168–170 (1997).
[CrossRef]

1965 (1)

C. Tai, “The dyadic Green’s function for a moving isotropic medium,” IEEE Trans. Antennas Propag. 13, 322–323 (1965).
[CrossRef]

Alkumru, A.

M. Idemen and A. Alkumru, “Influence of motion on the edge-diffraction,” Prog. Electromagn. Res. B 6, 153–168 (2008).
[CrossRef]

M. Idemen and A. Alkumru, “Relativistic scattering of a plane-wave by a uniformly moving half-plane,” IEEE Trans. Antennas Propag. 54, 3429–3440 (2006).
[CrossRef]

Arrighetti, W.

W. Arrighetti, P. De Cupis, and G. Gerosa, “Electromagnetic radiation from moving fractal sources: A plane-wave spectral approach,” Prog. Electromagn. Res. 58, 1–19 (2006).
[CrossRef]

Bladel, J. V.

J. V. Bladel, Relativity and Engineering (Springer, 1984).

Cupis, P. D.

P. D. Cupis, “Electromagnetic wave scattering from moving surfaces with high-amplitude corrugated pattern,” J. Opt. Soc. Am. A 23, 2538–2550 (2006).
[CrossRef]

P. D. Cupis, “Radiation by a moving wire-antenna in the presence of plane interface,” J. Electromagn. Waves Appl. 14, 1119–1132 (2000).
[CrossRef]

Danov, T.

T. Danov and T. Melamed, “Spectral analysis of relativistic dyadic Green’s function of a moving dielectric-magnetic medium,” IEEE Trans. Antennas Propag. 59, 2973–2979 (2011).
[CrossRef]

De Cupis, P.

W. Arrighetti, P. De Cupis, and G. Gerosa, “Electromagnetic radiation from moving fractal sources: A plane-wave spectral approach,” Prog. Electromagn. Res. 58, 1–19 (2006).
[CrossRef]

P. De Cupis, G. Gerosa, and G. Schettini, “Gaussian beam diffraction by uniformly moving targets,” Atti della Fondazione Giorgio Ronchi 56, 799–811 (2001).

Felsen, L.

L. Felsen and N. Marcuvitz, Radiation and Scattering of Waves (IEEE, 1994), Chap. 4.8.

L. Felsen and N. Marcuvitz, Radiation and Scattering of Waves (IEEE, 1994), Chap. 4.2.

Gerosa, G.

W. Arrighetti, P. De Cupis, and G. Gerosa, “Electromagnetic radiation from moving fractal sources: A plane-wave spectral approach,” Prog. Electromagn. Res. 58, 1–19 (2006).
[CrossRef]

P. De Cupis, G. Gerosa, and G. Schettini, “Gaussian beam diffraction by uniformly moving targets,” Atti della Fondazione Giorgio Ronchi 56, 799–811 (2001).

Idemen, M.

M. Idemen and A. Alkumru, “Influence of motion on the edge-diffraction,” Prog. Electromagn. Res. B 6, 153–168 (2008).
[CrossRef]

M. Idemen and A. Alkumru, “Relativistic scattering of a plane-wave by a uniformly moving half-plane,” IEEE Trans. Antennas Propag. 54, 3429–3440 (2006).
[CrossRef]

Kong, J.

J. Kong, Theory of Electromagnetic Waves (Wiley, 1975).

Lakhtakia, A.

T. Mackay, A. Lakhtakia, and S. Setiawan, “Positive-, negative-, and orthogonal-phase-velocity propagation of electromagnetic plane waves in a simply moving medium,” Optik 118, 195–202(2007).
[CrossRef]

A. Lakhtakia and T. Mackay, “Simple derivation of dyadic Green functions of a simply moving, isotropic, dielectric-magnetic medium,” Microwave Opt. Technol. Lett. 48, 1073–1074 (2006).
[CrossRef]

T. Mackay and A. Lakhtakia, “Negative phase velocity in a uniformly moving, homogeneous, isotropic, dielectric-magnetic medium,” J. Phys. A 37, 5697–5711 (2004).
[CrossRef]

A. Lakhtakia and W. Weiglhofer, “On electromagnetic fields in a linear medium with gyrotropic-like magnetoelectric properties,” Microwave Opt. Technol. Lett. 15, 168–170 (1997).
[CrossRef]

Mackay, T.

T. Mackay, A. Lakhtakia, and S. Setiawan, “Positive-, negative-, and orthogonal-phase-velocity propagation of electromagnetic plane waves in a simply moving medium,” Optik 118, 195–202(2007).
[CrossRef]

A. Lakhtakia and T. Mackay, “Simple derivation of dyadic Green functions of a simply moving, isotropic, dielectric-magnetic medium,” Microwave Opt. Technol. Lett. 48, 1073–1074 (2006).
[CrossRef]

T. Mackay and A. Lakhtakia, “Negative phase velocity in a uniformly moving, homogeneous, isotropic, dielectric-magnetic medium,” J. Phys. A 37, 5697–5711 (2004).
[CrossRef]

Marcuvitz, N.

L. Felsen and N. Marcuvitz, Radiation and Scattering of Waves (IEEE, 1994), Chap. 4.8.

L. Felsen and N. Marcuvitz, Radiation and Scattering of Waves (IEEE, 1994), Chap. 4.2.

Melamed, T.

T. Danov and T. Melamed, “Spectral analysis of relativistic dyadic Green’s function of a moving dielectric-magnetic medium,” IEEE Trans. Antennas Propag. 59, 2973–2979 (2011).
[CrossRef]

Schettini, G.

P. De Cupis, G. Gerosa, and G. Schettini, “Gaussian beam diffraction by uniformly moving targets,” Atti della Fondazione Giorgio Ronchi 56, 799–811 (2001).

Setiawan, S.

T. Mackay, A. Lakhtakia, and S. Setiawan, “Positive-, negative-, and orthogonal-phase-velocity propagation of electromagnetic plane waves in a simply moving medium,” Optik 118, 195–202(2007).
[CrossRef]

Tai, C.

C. Tai, “The dyadic Green’s function for a moving isotropic medium,” IEEE Trans. Antennas Propag. 13, 322–323 (1965).
[CrossRef]

Weiglhofer, W.

A. Lakhtakia and W. Weiglhofer, “On electromagnetic fields in a linear medium with gyrotropic-like magnetoelectric properties,” Microwave Opt. Technol. Lett. 15, 168–170 (1997).
[CrossRef]

Atti della Fondazione Giorgio Ronchi (1)

P. De Cupis, G. Gerosa, and G. Schettini, “Gaussian beam diffraction by uniformly moving targets,” Atti della Fondazione Giorgio Ronchi 56, 799–811 (2001).

IEEE Trans. Antennas Propag. (3)

C. Tai, “The dyadic Green’s function for a moving isotropic medium,” IEEE Trans. Antennas Propag. 13, 322–323 (1965).
[CrossRef]

T. Danov and T. Melamed, “Spectral analysis of relativistic dyadic Green’s function of a moving dielectric-magnetic medium,” IEEE Trans. Antennas Propag. 59, 2973–2979 (2011).
[CrossRef]

M. Idemen and A. Alkumru, “Relativistic scattering of a plane-wave by a uniformly moving half-plane,” IEEE Trans. Antennas Propag. 54, 3429–3440 (2006).
[CrossRef]

J. Electromagn. Waves Appl. (1)

P. D. Cupis, “Radiation by a moving wire-antenna in the presence of plane interface,” J. Electromagn. Waves Appl. 14, 1119–1132 (2000).
[CrossRef]

J. Opt. Soc. Am. A (1)

J. Phys. A (1)

T. Mackay and A. Lakhtakia, “Negative phase velocity in a uniformly moving, homogeneous, isotropic, dielectric-magnetic medium,” J. Phys. A 37, 5697–5711 (2004).
[CrossRef]

Microwave Opt. Technol. Lett. (2)

A. Lakhtakia and W. Weiglhofer, “On electromagnetic fields in a linear medium with gyrotropic-like magnetoelectric properties,” Microwave Opt. Technol. Lett. 15, 168–170 (1997).
[CrossRef]

A. Lakhtakia and T. Mackay, “Simple derivation of dyadic Green functions of a simply moving, isotropic, dielectric-magnetic medium,” Microwave Opt. Technol. Lett. 48, 1073–1074 (2006).
[CrossRef]

Optik (1)

T. Mackay, A. Lakhtakia, and S. Setiawan, “Positive-, negative-, and orthogonal-phase-velocity propagation of electromagnetic plane waves in a simply moving medium,” Optik 118, 195–202(2007).
[CrossRef]

Prog. Electromagn. Res. (1)

W. Arrighetti, P. De Cupis, and G. Gerosa, “Electromagnetic radiation from moving fractal sources: A plane-wave spectral approach,” Prog. Electromagn. Res. 58, 1–19 (2006).
[CrossRef]

Prog. Electromagn. Res. B (1)

M. Idemen and A. Alkumru, “Influence of motion on the edge-diffraction,” Prog. Electromagn. Res. B 6, 153–168 (2008).
[CrossRef]

Other (4)

L. Felsen and N. Marcuvitz, Radiation and Scattering of Waves (IEEE, 1994), Chap. 4.2.

L. Felsen and N. Marcuvitz, Radiation and Scattering of Waves (IEEE, 1994), Chap. 4.8.

J. Kong, Theory of Electromagnetic Waves (Wiley, 1975).

J. V. Bladel, Relativity and Engineering (Springer, 1984).

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Figures (7)

Fig. 1.
Fig. 1.

Physical configuration of the relativistic Green’s function of a line current that is embedded in a dielectric–magnetic medium with a planar discontinuity. The medium is uniformly moving in the z direction with a velocity of v=vz^ in the laboratory frame and the discontinuity is located in z=z0 at t=0. The medium is an isotropic dielectric–magnetic in the comoving frame with ε1,2 and μ1,2 denoting its permittivity and permeability on either side of the discontinuity.

Fig. 2.
Fig. 2.

Integration contour of the PW spectral representation in Eq. (14). The upper Riemann sheet is define by Rek¯z10.

Fig. 3.
Fig. 3.

Integration contour in the spectral φ˜ plane.

Fig. 4.
Fig. 4.

Geometrical optics interpretation of the ray field forms: (a) the incident field in Eq. (50) and (b) the reflected field in Eq. (58).

Fig. 5.
Fig. 5.

Integration contour in φ˜ plane for the reflected field (a) under and (b) over critical angle incidence.

Fig. 6.
Fig. 6.

Geometrical optics interpretation of the lateral (head) wave in Eq. (63).

Fig. 7.
Fig. 7.

Geometrical optics interpretation of asymptotic transferred field in Eq. (80).

Equations (102)

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y=y,z=γ(zβct),t=γ(tβz/c),
y=y,z=γ(z+βct),t=γ(t+βz/c),
β=v/c,γ=1/1β2.
E(r,t)=V̲̲·[E+v×B],B(r,t)=V̲̲·[Bv×E/c2],D(r,t)=V̲̲·[D+v×H/c2],H(r,t)=V̲̲·[Hv×D],
V̲̲=diag{γ,γ,1}.
J(r,t)=I0δ(z)δ(y)exp(jωt)x^,
D=ϵ1E,B=μ1H,z<z0,D=ϵ2E,B=μ2H,z>z0,
D=ϵ1α̲̲·E+c1mz^×H,B=μ1α̲̲·Hc1mz^×E,
m=βn1211n12β2,n1=cϵ1μ1,
α̲̲=diag{α,α,1},α=1β21n12β2.
z¯=αz,k¯1=ωαϵ1μ1.
Ei(r,t)=12πdkyE˜i(r,t;ky),
E˜i(r,t;ky)=I0ωμ1α2k¯z1x^exp(jΨi),Ψi=ωtkyyk¯z1z¯+kmz,
k¯z1=k¯12ky2,Rek¯z10,Imk¯z10.
Hi(r,t)=12πdkyH˜i(r,t;ky),
H˜i(r,t;ky)=I02k¯z1(k¯z1y^+αkyz^)exp(jΨi),
Di(r,t)=12πdkyD˜i(r,t;ky),Bi(r,t)=12πdkyB˜i(r,t;ky),
D˜i(r,t;ky)=I02k¯z1(ωμ1ε1α3/2mk¯z1c)x^exp(jΨi),B˜i(r,t;ky)=μ1αI02k¯z1[(αk¯z1mωc)y^kyz^]exp(jΨi).
Ei(r,t)=12πdkyE˜i(r,t;ky),Hi(r,t)=12πdkyH˜i(r,t;ky),
E˜i(r,t;ky)=E˜0x^exp(jΨi),H˜i(r,t;ky)=1η1ωωE˜0[γα(k¯z1k¯1n1β)y^kyαk¯1z^]exp(jΨi).
E˜0=I0ωμ1α3/22k¯z1γ(1n1βk¯z1/k¯1),
Ψi(r,t;ky)=ωtkyykz1z,
ω=ωγα(1n1βk¯z1/k¯1),kz1=αγ(k¯z1n1βk¯1)
z¯=αγz,t¯=αγt,
Ψi(r,t;ky)=ω¯t¯kyyk¯z1z¯,
ω¯ω/αγ=ωα(1n1βk¯z1/k¯1),k¯z1kz1αγ=(k¯z1n1βk¯1).
Er(r,t)=12πdkyE˜r(r,t;ky),Hr(r,t)=12πdkyH˜r(r,t;ky),
E˜r(r,t;ky)=E˜0Γ(ky)x^exp(jΨr),H˜r(r,t;ky)=1η1Γ(ky)E˜0γ1k¯1ωω¯(kz1y^+kyz^)exp(jΨr).
Ψr(r,t;ky)=ω¯t¯kyyk¯z1(2z¯0z¯),
Γ(ky)=μ2k¯z1μ1k¯z2μ2k¯z1+μ1k¯z2,
k¯z2=ω¯2ε2μ2ky2/αγ2,
Et(r,t)=12πdkyE˜t(r,t;ky),Ht(r,t)=12πdkyH˜t(r,t;ky),
E˜t(r,t;ky)=E˜0T(ky)x^exp(jΨt),H˜t(r,t;ky)=1ωμ2E˜0T(ky)(kz2y^kyz^)exp(jΨt),
Ψt(r,t;ky)=ω¯t¯kyyk¯z2(z¯z¯0)k¯z1z¯0,
T(ky)=1+Γ(ky).
sinφt=kyω¯αγε2μ2.
Er(r,t)=12πdkyE˜r(r,t;ky),Hr(r,t)=12πdkyH˜r(r,t;ky),
E˜r(r,t;ky)=x^E˜0Γ(ky)(1n1βαωk¯z1ω¯k¯1)γexp(jΨr),H˜r(r,t;ky)=E˜0Γ(ky)η1(γωαk¯z1ω¯n1βk¯1ω¯k¯1ωn1βαk¯z1y^+ωω¯kyγ1k¯1z^)exp(jΨr).
Ψr(r,t;ky)=αγ2(ω¯βck¯z1)tkyyαγ2[ω¯βc1z+k¯z1(2z0z)].
Et(r,t)=12πdkyE˜t(r,t;ky),Ht(r,t)=12πdkyH˜t(r,t;ky),
E˜t(r,t;ky)=E˜0T(ky)γ(1+cβk¯z2ω¯)x^exp(jΨt),H˜t(r,t;ky)=1η2E˜0T(ky)[γ(n2β+ck¯z2n2ω¯)y^ckyn2ωz^]exp(jΨt).
Ψt(r,t;ky)=αγ2(ω¯+cβk¯z2)tkyyαγ2[(ω¯βc1+k¯z2)z+(k¯z1k¯z2)z0].
ky=k¯1sinφ˜.
Ricosφi=z¯+vt¯,Risinφi=y.
Ei(r,t)=x^Ii(r,t)exp(jΨ0i),Ψ0i=γαωt+βn1k¯1z¯.
Ii(r,t)=12πCdφ˜fi(φ˜)exp[jΨi(r,t;φ˜)]
Ψi(r,t;φ˜)=k¯1Ricos(φ˜φi),fi(φ˜)=I0μ1αω/2.
ω(φ˜)=ωγα(1n1βcosφ˜).
Ii(r,t)ISDPi=12π|φ˜2Ψi(φ˜si)|fi(φ˜si)exp[jΨi(φ˜si)±jπ/4],
ISDPi=αμ1ω(φi)I0exp(jk¯1Rijπ/4)8πk¯1Ri,
Hi(r,t)1η1ISDPiexp(jΨ0i)h^i(φi),h^i(φi)=y^(cosφin1β)+z^sinφi/(γα)(1βn1cosφi),
Rrcosφr=2z¯0z¯+vt¯,Rrsinφr=y.
Er(r,t)=x^Ir(r,t)exp(jΨ0r),Ψ0r=γαωtβn1k¯1(z¯2z¯0).
Ir(r,t)=12πCdφ˜fr(φ˜)exp[jΨr(r,t;φ˜)],
Ψr(r,t;φ˜)=k¯1Rrcos(φ˜φr),fr(φ˜)=Γ(φ˜)fi(φ˜),
Γ(φ˜)=μ2(cosφ˜n1β)μ1n212(1n1βcosφ˜)2sin2φ˜/(αγ2)μ2(cosφ˜n1β)+μ1n212(1n1βcosφ˜)2sin2φ˜/(αγ2),
cosφ˜b1,2=n21n2βαγ2±(n21n2βαγ2)2(n22β2αγ2+1)(n212αγ21)n22β2αγ2+1.
IrISDPr=Γ(φr)αμ1ω(φr)I0exp(jk¯1Rrjπ/4)8πk¯1Rr,
Hr(r,t)1η1ISDPrexp(jΨ0r)h^r(φr),h^r(φr)=y^(cosφrn1β)+z^sinφr/(γα)(1βn1cosφr).
Er(r,t)=ESDPr(r,t)+Ebr(r,t),
Γ(φ˜)1+j2μ12sinφ˜b1,2φ˜φ˜b1,2γα(cosφ˜b1,2n1β)(1n1βcosφ˜b1,2).
Ib=Pbdφ˜(a+bφ˜φ˜b)exp[Ωq(φ˜)]bπ[Ωq(φ˜b)]3/2exp[Ωq(φ˜b)],
Ebr(r,t)x^Eb1rDEb2rEb3rexp(jαγωt),
Eb1r=I0αμ1ω(φr)8πk¯1L1exp(jΨb1),
Ψb1=k¯1L1(π/4)+βn1k¯1z¯0,L1=(z¯0vt¯)/cosθc.
sinθc=n21
D=2jsin2θck¯1L1αγ(cosθc)3/2cosθcn1β1n1βcosθc,
Eb2r=exp(jk¯2L2)(k¯2L2)3/2,L2=yL1sinθc+(z¯z¯0)tanθc,
Eb3r=exp(jk¯1L3)exp[jβn1k¯1(z¯0z¯)],L3=(z¯0z¯)/cosθc.
Hbr(r,t)1η1Eb1rDEb2rEb3rexp(jαγωt)h^r(φ˜b1,2),h^r(φ˜b1,2)=y^(cosφ˜b1,2n1β)+z^sinφ˜b1,2/(γα)(1βn1cosφ˜b1,2).
R1tcosφ1t=z¯0+vt¯,R2tcosφ2t=z¯z¯0,
tanφ2t=(n21αγ2)1sinφ1tn2(1n1βcosφ1t)βtanφ1t(1n1βcosφ1t)2sin2φ1t/n212αγ2.
y=R1tsinφ1t+R2tsinφ2t.
Et(r,t)=x^exp[jΨ0i(r,t)]It(r,t),It(r,t)=12πCdφ˜ft(φ˜)exp[jΨt(r,t;φ˜)],
ft(φ˜)=T(φ˜)fi(φ˜),
Ψt(r,t;φ˜)=k¯1[R1tcos(φ˜φ1t)+R2t(sinφ2tsinφ˜+n21cosφ2t(1n1βcosφ˜)2sin2φ˜/n212αγ2)].
T(φ˜)=2μ2(cosφ˜n1β)μ2(cosφ˜n1β)+μ1n21(1n1βcosφ˜)2sin2φ˜/n212αγ2.
ytanφ˜s1(z¯0+vt¯)tanφ˜s2(z¯z¯0)=0,
tanφ˜s2=(n21αγ2)1sinφ˜s1n2(1n1βcosφ˜s1)βtanφ˜s1(1n1βcosφ˜s1)2sin2φ˜s1/n212αγ2.
Et(r,t)x^E1tE2texp(jωγαt),
E1t=αμ1ω(φ1t)I0exp(jk¯1R1t+jβn1k¯1z¯0jπ/4)8πk¯1R1t
E2t=T(φ1t)exp(jk¯2R2t)1+R2t/ρt,
ρt=R1tsinφ2tcos2φ1t[n12[1β2(n12n22)]sinφ1tn2βtanφ1tn12[1β2(n12n22)]n2β/cos3φ1t+n12tan2φ2t].
Ht(r,t)1η2E1tE2texp(jωγαt)h^t(φi),h^t(φi)=y^cosφ2t+z^sinφ1t/(n21γα)(1n1βcosφ1t).
Ei(r,t)x^Exi(r)exp(jωt),Exi(r)=I0ωμ1α1/2exp[jk¯1Rijπ/4]8πk¯1Riexp(jkmz),Hi(r,t)1η1Exi(r)exp(jωt)(y^cosφi+z^αsinφi),
φ˜sr=cos1[γ2α[2(z0+vt)z(1+β2)]y2+γ4α[2(z0+vt)z(1+β2)]2].
Er(r,t)x^Exr(r,t)exp(jΨr),
Ψr=γ2α(1+β2n12)ωtβγ2αk¯1[z(1+n12)/n12n1z0],Exr(r,t)=ωμ1γ2α3/2Γ(φr)I0exp(jk¯1Rrjπ4)8πk¯1Rr,
Rr=y2+γ4α[2(z0+vt)z(1+β2)]2,cosφr=γ2α[2(z0+vt)z(1+β2)]/Rr,Γ(φr)=(12βn1cosφr+β2n12)Γ(φr).
Hr(r,t)1η1Exrexp(jΨr)h^r(φr),h^r(φr)=y^[cosφr(1+β2n12)2βn1]+z^sinφr/γ2α12βn1cosφr+β2n12.
Er(r,t)=ESDPr(r,t)+Ebr(r,t),
L1=αγ2(z0+β2zvt)/cosθc,L3=αγ2(z0z+vt)/cosθc,L2=RrsinφrL1sinθc+αγ2(zz0vt)tanθc.
Ebr(r,t)x^Eb1rDEb2rEb3rexp[jαγ2ω(tβz/c)],
Eb1r=I0γ2α3/2μ1ω(φr)8πk¯1L1(12βn1cosφr+β2n12)exp(jΨb1),
Ψb1=k¯1L1π/4+βn1k¯1αγ2z0.
D=2jsinθck¯1L1αγ(cosθc)3/2cosθcn1β1n1βcosθc,
Eb2r=exp(jk¯2L2)(k¯2L2)3/2,
Eb3r=exp(jk¯1L3)exp[jβn1k¯1αγ2(z0z+vt)].
Ext(r,t)x^E1t(r,t)E2t(r,t)exp[jαγ2ω(tβz/c)],
E1t=I0μ1γ2α3/2ω(φ1t)exp(jk¯1R1t+jk¯1n1βαγ2z0jπ/4)8πk¯1R1t,E2t=T(φ1t)exp(jk¯2R2t)1+R2t/ρt,
cosφ1t=αγ2(z0+vtβ2z)/R1t,cosφ2t=αγ2(zz0vt)/R2t,
R1t=yP2+αγ4(z0+vtβ2z)2,R2t=(yyP)2+αγ4(zz0vt)2,

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