Abstract

We give the second- and third-order approximations at large times of cylindrical focus wave modes. As an application, we analyze the scattering of focus wave modes from a circular cylinder.

© 1999 Optical Society of America

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References

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  1. J. N. Brittingham, “Focus wave modes in Maxwell’s equations,” J. Appl. Phys. 54, 78–82 (1983).
    [CrossRef]
  2. A. D. Kiselev, “Modulated Gaussian beams,” Radiophys. Quantum Electron. 26, 1014–1023 (1983).
    [CrossRef]
  3. R. Courant, D. Hilbert, Methods of Mathematical Physics, Vol. 2 (Interscience, New York, 1962), pp. 760–764.
  4. P. M. Morse, H. Feschbach, Methods of Theoretical Physics (Mc-Graw Hill, New York, 1953), Vol. 2, pp. 1376–1379.
  5. D. Palmer, R. Donnelly, R. MacIsaac, “Spherical scattering of superpositions of localized waves,” Phys. Rev. E 48, 1410–1417 (1993).
    [CrossRef]
  6. R. Ziolkowski, J. Besieris, A. Shaarawi, “Localized wave representations of acoustic and electromagnetic radiation,” Proc. IEEE 79, 1378–1385 (1991).
    [CrossRef]
  7. R. Donnelly, R. Ziolkowski, “A method for constructing localized waves,” Proc. R. Soc. London 437, 637–692 (1992).
  8. R. Ziolkowski, J. Judkins, “Propagation characteristics of ultrawide-bandwith Gaussian beams,” J. Opt. Soc. Am. A 9, 2021–2030 (1992).
    [CrossRef]
  9. P. Hillion, “Approximations of the scalar focus wave modes,” J. Opt. Soc. Am. A 9, 137–142 (1992).
    [CrossRef]
  10. V. Borisov, A. Utkins, “On formation of focus wave modes,” J. Phys. A 26, 406–411 (1993).
  11. V. Borisov, A. Utkins, “Focus wave modes in conducting media,” Can. J. Phys. 72, 725–730 (1994).
    [CrossRef]
  12. R. Ziolkowski, K. Lewis, B. Cook, “Evidence of localized wave transmission,” Phys. Rev. Lett. 62, 147–150 (1989).
    [CrossRef] [PubMed]

1994

V. Borisov, A. Utkins, “Focus wave modes in conducting media,” Can. J. Phys. 72, 725–730 (1994).
[CrossRef]

1993

V. Borisov, A. Utkins, “On formation of focus wave modes,” J. Phys. A 26, 406–411 (1993).

D. Palmer, R. Donnelly, R. MacIsaac, “Spherical scattering of superpositions of localized waves,” Phys. Rev. E 48, 1410–1417 (1993).
[CrossRef]

1992

1991

R. Ziolkowski, J. Besieris, A. Shaarawi, “Localized wave representations of acoustic and electromagnetic radiation,” Proc. IEEE 79, 1378–1385 (1991).
[CrossRef]

1989

R. Ziolkowski, K. Lewis, B. Cook, “Evidence of localized wave transmission,” Phys. Rev. Lett. 62, 147–150 (1989).
[CrossRef] [PubMed]

1983

J. N. Brittingham, “Focus wave modes in Maxwell’s equations,” J. Appl. Phys. 54, 78–82 (1983).
[CrossRef]

A. D. Kiselev, “Modulated Gaussian beams,” Radiophys. Quantum Electron. 26, 1014–1023 (1983).
[CrossRef]

Besieris, J.

R. Ziolkowski, J. Besieris, A. Shaarawi, “Localized wave representations of acoustic and electromagnetic radiation,” Proc. IEEE 79, 1378–1385 (1991).
[CrossRef]

Borisov, V.

V. Borisov, A. Utkins, “Focus wave modes in conducting media,” Can. J. Phys. 72, 725–730 (1994).
[CrossRef]

V. Borisov, A. Utkins, “On formation of focus wave modes,” J. Phys. A 26, 406–411 (1993).

Brittingham, J. N.

J. N. Brittingham, “Focus wave modes in Maxwell’s equations,” J. Appl. Phys. 54, 78–82 (1983).
[CrossRef]

Cook, B.

R. Ziolkowski, K. Lewis, B. Cook, “Evidence of localized wave transmission,” Phys. Rev. Lett. 62, 147–150 (1989).
[CrossRef] [PubMed]

Courant, R.

R. Courant, D. Hilbert, Methods of Mathematical Physics, Vol. 2 (Interscience, New York, 1962), pp. 760–764.

Donnelly, R.

D. Palmer, R. Donnelly, R. MacIsaac, “Spherical scattering of superpositions of localized waves,” Phys. Rev. E 48, 1410–1417 (1993).
[CrossRef]

R. Donnelly, R. Ziolkowski, “A method for constructing localized waves,” Proc. R. Soc. London 437, 637–692 (1992).

Feschbach, H.

P. M. Morse, H. Feschbach, Methods of Theoretical Physics (Mc-Graw Hill, New York, 1953), Vol. 2, pp. 1376–1379.

Hilbert, D.

R. Courant, D. Hilbert, Methods of Mathematical Physics, Vol. 2 (Interscience, New York, 1962), pp. 760–764.

Hillion, P.

Judkins, J.

Kiselev, A. D.

A. D. Kiselev, “Modulated Gaussian beams,” Radiophys. Quantum Electron. 26, 1014–1023 (1983).
[CrossRef]

Lewis, K.

R. Ziolkowski, K. Lewis, B. Cook, “Evidence of localized wave transmission,” Phys. Rev. Lett. 62, 147–150 (1989).
[CrossRef] [PubMed]

MacIsaac, R.

D. Palmer, R. Donnelly, R. MacIsaac, “Spherical scattering of superpositions of localized waves,” Phys. Rev. E 48, 1410–1417 (1993).
[CrossRef]

Morse, P. M.

P. M. Morse, H. Feschbach, Methods of Theoretical Physics (Mc-Graw Hill, New York, 1953), Vol. 2, pp. 1376–1379.

Palmer, D.

D. Palmer, R. Donnelly, R. MacIsaac, “Spherical scattering of superpositions of localized waves,” Phys. Rev. E 48, 1410–1417 (1993).
[CrossRef]

Shaarawi, A.

R. Ziolkowski, J. Besieris, A. Shaarawi, “Localized wave representations of acoustic and electromagnetic radiation,” Proc. IEEE 79, 1378–1385 (1991).
[CrossRef]

Utkins, A.

V. Borisov, A. Utkins, “Focus wave modes in conducting media,” Can. J. Phys. 72, 725–730 (1994).
[CrossRef]

V. Borisov, A. Utkins, “On formation of focus wave modes,” J. Phys. A 26, 406–411 (1993).

Ziolkowski, R.

R. Ziolkowski, J. Judkins, “Propagation characteristics of ultrawide-bandwith Gaussian beams,” J. Opt. Soc. Am. A 9, 2021–2030 (1992).
[CrossRef]

R. Donnelly, R. Ziolkowski, “A method for constructing localized waves,” Proc. R. Soc. London 437, 637–692 (1992).

R. Ziolkowski, J. Besieris, A. Shaarawi, “Localized wave representations of acoustic and electromagnetic radiation,” Proc. IEEE 79, 1378–1385 (1991).
[CrossRef]

R. Ziolkowski, K. Lewis, B. Cook, “Evidence of localized wave transmission,” Phys. Rev. Lett. 62, 147–150 (1989).
[CrossRef] [PubMed]

Can. J. Phys.

V. Borisov, A. Utkins, “Focus wave modes in conducting media,” Can. J. Phys. 72, 725–730 (1994).
[CrossRef]

J. Appl. Phys.

J. N. Brittingham, “Focus wave modes in Maxwell’s equations,” J. Appl. Phys. 54, 78–82 (1983).
[CrossRef]

J. Opt. Soc. Am. A

J. Phys. A

V. Borisov, A. Utkins, “On formation of focus wave modes,” J. Phys. A 26, 406–411 (1993).

Phys. Rev. E

D. Palmer, R. Donnelly, R. MacIsaac, “Spherical scattering of superpositions of localized waves,” Phys. Rev. E 48, 1410–1417 (1993).
[CrossRef]

Phys. Rev. Lett.

R. Ziolkowski, K. Lewis, B. Cook, “Evidence of localized wave transmission,” Phys. Rev. Lett. 62, 147–150 (1989).
[CrossRef] [PubMed]

Proc. IEEE

R. Ziolkowski, J. Besieris, A. Shaarawi, “Localized wave representations of acoustic and electromagnetic radiation,” Proc. IEEE 79, 1378–1385 (1991).
[CrossRef]

Proc. R. Soc. London

R. Donnelly, R. Ziolkowski, “A method for constructing localized waves,” Proc. R. Soc. London 437, 637–692 (1992).

Radiophys. Quantum Electron.

A. D. Kiselev, “Modulated Gaussian beams,” Radiophys. Quantum Electron. 26, 1014–1023 (1983).
[CrossRef]

Other

R. Courant, D. Hilbert, Methods of Mathematical Physics, Vol. 2 (Interscience, New York, 1962), pp. 760–764.

P. M. Morse, H. Feschbach, Methods of Theoretical Physics (Mc-Graw Hill, New York, 1953), Vol. 2, pp. 1376–1379.

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Equations (41)

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ψa(t, x, z)=Aga(t, x, z)exp{ik[(ct+Z)-X2(a+ct+Z)-1]},
Z=x cos θ+z sin θ,X=x sin θ-z cos θ,ga(t, x, z)=(a+ct-Z)-1/2,
ψa(t, r, ϕ)=Aga(t, r, ϕ)exp{ik{ct+r cos(θ-ϕ)-r2 sin2(θ-ϕ)[a+ct-r cos(θ-ϕ)]-1}},
ga(t, r, ϕ)=[a+ct-r cos(θ-ϕ)]-1/2.
β=ct/r,μ=ct/a.
ψ0=A[ct-r cos(θ-ϕ)]-1/2 exp{ik(c2t2-r2)×[ct-r cos(θ-ϕ)]-1}.
ψ0=A(ct)-1/2[1+(2β)-1 cos(θ-ϕ)]×exp{ikct[1+β-1 cos(θ-ϕ)]}+0(β-2)=A(t)[1+(2β)-1 cos(θ-ϕ)]×exp{ikr cos(θ-ϕ)}+0(β-2),
cos(θ-ϕ)exp{ikr cos(θ-ϕ)}
=-ir-1k exp{ikr cos(θ-ϕ)}.
ψ0=A(t)[1-i(2βr)-1k]×exp{ikr cos(θ-ϕ)}+0(β-2).
exp{ikr cos(θ-ϕ)}=-+inJn(kr)exp{in(θ-ϕ)},
ψ0=A(t)in[Jn(kr)-i(2β)-1Jn(kr)]×exp{in(θ-ϕ)}+0(β-2).
g0=(ct)-1/2[1+(2β)-1 cos(θ-ϕ)+(8β2/3)-1 cos2(θ-ϕ)]+0(β-3),
exp{ }=exp(ikct)exp{ikr cos(θ-ϕ)-ikctβ-2 sin2(θ-ϕ)}+0(β-3),
=exp(ikct)exp{ikr cos(θ-ϕ)}×[1-ikctβ-2 sin2(θ-ϕ)]+0(β-3).
ψ0=A(t)B0(β)exp{ikr cos(θ-ϕ)}+0(β-3),
B0(β)=1-ikctβ-2+(2β)-1 cos(θ-ϕ)+β-2(ikct+3/8)cos2(θ-ϕ).
cos2(θ-ϕ)exp{ikr cos(θ-ϕ)}
=-r-2k2 exp{ikr cos(θ-ϕ)},
B0(β)=u0+v0r-1k+w0r-2k2,
u0=1-ikctβ-2,v0=-i(2β)-1,
w0=-β-2(ikct+3/8).
ψ0=A(t)nin[u0Jn(kr)+v0Jn(kr)+w0Jn(kr)]×exp{in cos(θ-ϕ)}+0(β-3).
ψa=Aga exp{ik[c2t2-r2+act+ar cos(θ-ϕ)]×[a+ct-r cos(θ-ϕ)]-1}.
ψa=A(t)[1-(2µ)-1-(2β)-1 cos(θ-ϕ)]×exp{ikr cos(θ-ϕ)}+0(-2).
ψa=A(t)nin{[1-(2µ)-1]Jn(kr)-i(2β)-1Jn(kr)}×exp{in(θ-ϕ)}+0(-2).
exp{ }=exp{ikct+ikr cos(θ-ϕ)-ikctβ-2×sin2(θ-ϕ)}+0(-3).
ga=1-(2µ)-1-(2β)-1 cos(θ-ϕ)+3/8[μ-2+β-2×cos2(θ-ϕ)]-(4µβ/3)-1 cos(θ-ϕ)+0(-3).
ψa=A(t)(ua+var-1k+war-2k2)×exp{ikr cos(θ-ϕ)}+0(-3),
ua=1-(2µ)-1+3µ-2/8-ikctβ-2,
va=-i(2β)-1(1-3/2µ),wa=-β2(ikct+3/8).
ψa=A(t)nin[uaJn(kr)+vaJn(kr)+waJn(kr)]×exp{in(θ-ϕ)}+0(-3).
ψ0=A(t)n=0εnin[Jn(kr)-i(2β)-1Jn(kr)]×cos nϕ+0(β-2).
χ0=A(ct-r cos ϕ)-1/2×F0[ik(c2t2-r2)(ct-r cos ϕ)-1].
χ0=A(ct)-1/2[1-i(2βr)-1k]×F0(ikct+ikr cos ϕ)+0(β-2).
F0(ikct+ikr cos ϕ)=exp(ikct)n=0bnϵnin×Hn(kr)cos nϕ,
χ0=A(t)n=0bnϵnin[Hn(kr)-i(2β)-1Hn(kr)]×cos nϕ+0(β-2).
bn[Hn(kr0)-i(2β0)-1Hn(kr0)]
=Jn(kr0)-i(2β0)-1Jn(kr0),
bn=Jn(kr0)Hn-1(kr0){1-i(2β0)-1[HnHn-1-JnJn-1]×(kr0)}+0(β0-2),
(xΩ)2+(zΩ)2-c-2(tΩ)2=0.

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