Abstract

In this communication, certain comments are made on the X-shaped field generated by a really superluminal (point) charge treated in Subsection 2.D in the paper by Zamboni-Rached et al., “Cherenkov radiation versus X-shaped localized waves,” [J. Opt. Soc. Am. A 27, 928 (2010)].

© 2012 Optical Society of America

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References

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  1. M. Zamboni-Rached, E. Recami, and I. M. Besieris, “Cherenkov radiation versus X-shaped localized waves,” J. Opt. Soc. Am. A 27, 928–934 (2010).
    [CrossRef]
  2. I. S. Gradshteyn and I. M. Ryzhik, Tables of Integrals, Series, and Products (Academic, 1965), p. 731, formulas 6.671.7 and 6.671.12.
  3. I. M. Besieris, Bradley Department of Electrical and Computer Engineering, Virginia Polytechnic Institute and State University, Blacksburg, Virginia, USA, 2011 (personal communication).
  4. J. A. Stratton, Electromagnetic Theory (McGraw-Hill, 1941), Chap. 1.

2010 (1)

Besieris, I. M.

M. Zamboni-Rached, E. Recami, and I. M. Besieris, “Cherenkov radiation versus X-shaped localized waves,” J. Opt. Soc. Am. A 27, 928–934 (2010).
[CrossRef]

I. M. Besieris, Bradley Department of Electrical and Computer Engineering, Virginia Polytechnic Institute and State University, Blacksburg, Virginia, USA, 2011 (personal communication).

Gradshteyn, I. S.

I. S. Gradshteyn and I. M. Ryzhik, Tables of Integrals, Series, and Products (Academic, 1965), p. 731, formulas 6.671.7 and 6.671.12.

Recami, E.

Ryzhik, I. M.

I. S. Gradshteyn and I. M. Ryzhik, Tables of Integrals, Series, and Products (Academic, 1965), p. 731, formulas 6.671.7 and 6.671.12.

Stratton, J. A.

J. A. Stratton, Electromagnetic Theory (McGraw-Hill, 1941), Chap. 1.

Zamboni-Rached, M.

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

Other (3)

I. S. Gradshteyn and I. M. Ryzhik, Tables of Integrals, Series, and Products (Academic, 1965), p. 731, formulas 6.671.7 and 6.671.12.

I. M. Besieris, Bradley Department of Electrical and Computer Engineering, Virginia Polytechnic Institute and State University, Blacksburg, Virginia, USA, 2011 (personal communication).

J. A. Stratton, Electromagnetic Theory (McGraw-Hill, 1941), Chap. 1.

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Equations (33)

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(2ρ2+1ρρ+2z21c22t2)ϕ(ρ,z,t)=4πδ(ρ)2πρδ(zvt).
ϕ(ρ,z,t)=vγ2π0dηηJ0(ηρ)dωexp[iω(zvt)/v](ω2η2v2γ2),
ϕ(ρ,z,t)=0forζ>0orz>vt.
ϕ(ρ,z,t)=Fψ(ρ,ζ),
0dω¯N0(ω¯ρ)cos(ω¯γζ)=γ1(ζ2ρ2γ2)1/2for0<ργ1<|ζ|=0for|ζ|<ργ1<,
x=x,y=y,z=γs(vczct),t=γsc(z+vt),
x=x,y=y,z=γs(vczct),t=γsc(zvt).
(γsv/cγscγs/cγsv/c)(γsv/cγscγs/cγsv/c)=(1001).
x2+y2+z2c2t2=0.
x2+y2+z2c2t2=0,
×E=BtJm,
·B=ρm,
×H=Dt+Je,
·D=ρe.
EH,BD,JmJeandρmρe
·Ja+ρat=0a=m,e.
Jmxx+ρmt=0.
x=x,t=γs(czvct).
(Jmx)x+γscρmz+γsvcρmt=0.
Jex=Jmx,Jey=0,Jez=γscρm,ρe=γsvcρm.
Jez=γscρmandρe=γsvcρm.
ρe=4πε0γsvcδ(x)δ(y)δ[γsvc(zc2vt)],
Jez=4πε0γscδ(x)δ(y)δ[γsvc(zc2vt)],
x=x,y=y,z=γs(z+vt),t=γs(vc2zt),
x=x,y=y,z=γs(zvt),t=γs(vc2zt).
x2+y2+c2t2z2=0.
ϕ(ρ,z,t)=iγ0dηJ0(ηρ)[exp(iηγζ)exp(iηγζ)]forζ<0.
J0(ηρ)=12πππdθexp(iηρcosθ)
ϕ(ρ,z,t)=iγ2πππdθ(I1I2)forζ<0,
I1=0dηexp[iη(ρcosθγ|ζ|)],
I2=0dηexp[iη(ρcosθ+γ|ζ|)].
I1,2=0dηexp{η[αi(ρcosθγ|ζ|)]}=[αi(ρcosθγ|ζ|)]1.
ϕL(ρ,z,t)=2γ(γ2|ζ|2ρ2)1/2forγ|ζ|>ρ=0forγ|ζ|<ρ,

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