Abstract

An analytic study of complete cylindrical focusing of pulses in two dimensions is presented, and compared with the analogous three-dimensional case of focusing over a complete sphere. Such behavior is relevant for understanding the limiting performance of ultrafast, planar photonic and plasmonic devices. A particular spectral distribution is assumed that contains finite energy. Separate ingoing and outgoing pulsed waves are considered, along with the combination that would be generated in free space by an ingoing wave. It is shown that for the two dimensional case, in order to produce a temporally symmetrical pulse at the focus, an asymmetric pulse must be launched. A symmetrical outgoing pulse is generated from a source with asymmetric time behavior, or an anti-symmetric input pulse. These results are very different from the corresponding three-dimensional case, and imply fundamental limitations on the performance of ultrafast, tightly focused, two-dimensional devices.

© 2014 Optical Society of America

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References

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2013 (1)

2012 (2)

T. Tyc, “Gouy phase for full-aperture spherical and cylindrical waves,” Opt. Lett. 37(5), 924–926 (2012).
[Crossref] [PubMed]

J. Lin, J. Dellinger, P. Genevet, B. Cluzel, F. de Fornel, and F. Capasso, “Cosine-Gauss plasmon beam: A localized long-range nondiffracting surface wave,” Phys. Rev. Lett. 109(9), 093904 (2012).
[Crossref] [PubMed]

2011 (3)

Y. G. Ma, S. Sahebdivan, C. K. Ong, T. Tyc, and U. Leonhardt, “Evidence for subwavelength imaging with positive refraction,” New J. Phys. 13(3), 033016 (2011).
[Crossref]

X. Zhang, “Perfect lenses in focus: No drain, no gain,” Nature 480, 42–43 (2011).
[Crossref] [PubMed]

J. C. Gonzalez, P. Benitez, and J. C. Miñano, “Perfect drain for the Maxwell fish eye lens,” New J. Phys. 13(2), 023038 (2011).
[Crossref]

2001 (1)

2000 (2)

S. Feng and H. G. Winful, “Spatiotemporal structure of isodiffracting ultrashort electromagnetic pulses,” Phys. Rev. E Stat. Phys. Plasmas Fluids Relat. Interdiscip. Topics 61(1), 862–873 (2000).
[Crossref] [PubMed]

S. Quabis, R. Dorn, M. Eberler, O. Glockl, and G. Leuchs, “Focusing light to a tighter spot,” Opt. Commun. 179(1-6), 1–7 (2000).
[Crossref]

1999 (2)

S. Feng, H. G. Winful, and R. W. Hellwarth, “Spatiotemporal evolution of focused single-cycle electromagnetic pulses,” Phys. Rev. E Stat. Phys. Plasmas Fluids Relat. Interdiscip. Topics 59(4), 4630–4649 (1999).
[Crossref]

C. F. R. Caron and R. M. Potvliege, “Free-space propagation of ultra-short pulses: space-time couplings in Gaussian pulse beams,” J. Mod. Opt. 46(13), 1881–1891 (1999).
[Crossref]

1998 (4)

G. P. Agrawal, “Spectrum-induced changes in diffraction of pulsed optical beams,” Opt. Commun. 157(1-6), 52–56 (1998).
[Crossref]

M. A. Porras, “Ultrashort pulsed Gaussian light beams,” Phys. Rev. E Stat. Phys. Plasmas Fluids Relat. Interdiscip. Topics 58(1), 1086–1093 (1998).
[Crossref]

A. E. Kaplan, “Diffraction-induced transformation of near-cycle and sub-cycle pulses,” J. Opt. Soc. Am. B 15(3), 951–956 (1998).
[Crossref]

S. Feng, H. G. Winful, and R. W. Hellwarth, “Gouy shift and temporal reshaping of focused single-cycle electromagnetic pulses,” Opt. Lett. 23(5), 385–387 (1998).
[Crossref] [PubMed]

1997 (3)

D. You and P. H. Bucksbaum, “Propagation of half-cycle far infrared pulses,” J. Opt. Soc. Am. B 14(7), 1651–1655 (1997).
[Crossref]

C. J. R. Sheppard and X. Gan, “Free-space propagation of femto-second light pulses,” Opt. Commun. 133(1-6), 1–6 (1997).
[Crossref]

Z. Wang, Z. Zhang, Z. Xu, and Q. Lin, “Space-time profiles of an ultrashort pulsed Gaussian beam,” IEEE J. Quantum Electron. 33(4), 566–573 (1997).
[Crossref]

1996 (1)

M. M. Wefers and K. A. Nelson, “Space-time profiles of shaped ultrafast optical waveforms,” IEEE J. Quantum Electron. 32(1), 161–172 (1996).
[Crossref]

1995 (1)

A. M. Shaarawi, I. M. Besieris, and R. W. Ziolkowski, “The propagating and evanescent field components of localized wave solutions,” Opt. Commun. 116(1-3), 183–192 (1995).
[Crossref]

1994 (1)

E. Heyman and T. Melamed, “Certain considerations in aperture synthesis of ultrawideband/short-pulse radiation,” IEEE Trans. Antenn. Propag. 42(4), 518–525 (1994).
[Crossref]

1992 (3)

J. Y. Lu and J. F. Greenleaf, “Nondiffracting X waves-exact solutions to free-space scalar wave equation and their finite aperture realizations,” IEEE Trans. Ultrason. Ferroelectr. Freq. Control 39(1), 19–31 (1992).
[Crossref] [PubMed]

S. Hell and E. H. K. Stelzer, “Fundamental improvement of resolution with a 4Pi-confocal fluorescence microscope using two-photon excitation,” Opt. Commun. 93(5-6), 277–282 (1992).
[Crossref]

R. W. Ziolkowski and J. B. Judkins, “Propagation characteristics of ultrawide-bandwidth pulsed Gaussian beams,” J. Opt. Soc. Am. A 9(11), 2021–2030 (1992).
[Crossref]

1989 (2)

E. Heyman and L. B. Felson, “Complex-source pulsed-beam fields,” J. Opt. Soc. Am. A 6(6), 806–817 (1989).
[Crossref]

R. W. Ziolkowski, “Localized transmission of electromagnetic energy,” Phys. Rev. A 39(4), 2005–2033 (1989).
[Crossref] [PubMed]

1987 (1)

1985 (1)

I. P. Christov, “Propagation of femtosecond light pulses,” Opt. Commun. 53(6), 364–366 (1985).
[Crossref]

1890 (1)

L. G. Gouy, “Sur une propriété nouvelle des ondes lumineuses,” Comptes Rendus de l'Académie des Sciences, Paris 110, 1251–1253 (1890).

Agrawal, G. P.

G. P. Agrawal, “Spectrum-induced changes in diffraction of pulsed optical beams,” Opt. Commun. 157(1-6), 52–56 (1998).
[Crossref]

Benitez, P.

J. C. Gonzalez, P. Benitez, and J. C. Miñano, “Perfect drain for the Maxwell fish eye lens,” New J. Phys. 13(2), 023038 (2011).
[Crossref]

Besieris, I. M.

A. M. Shaarawi, I. M. Besieris, and R. W. Ziolkowski, “The propagating and evanescent field components of localized wave solutions,” Opt. Commun. 116(1-3), 183–192 (1995).
[Crossref]

Bucksbaum, P. H.

Capasso, F.

J. Lin, J. Dellinger, P. Genevet, B. Cluzel, F. de Fornel, and F. Capasso, “Cosine-Gauss plasmon beam: A localized long-range nondiffracting surface wave,” Phys. Rev. Lett. 109(9), 093904 (2012).
[Crossref] [PubMed]

Caron, C. F. R.

C. F. R. Caron and R. M. Potvliege, “Free-space propagation of ultra-short pulses: space-time couplings in Gaussian pulse beams,” J. Mod. Opt. 46(13), 1881–1891 (1999).
[Crossref]

Chaumet, P. C.

Christov, I. P.

I. P. Christov, “Propagation of femtosecond light pulses,” Opt. Commun. 53(6), 364–366 (1985).
[Crossref]

Cluzel, B.

J. Lin, J. Dellinger, P. Genevet, B. Cluzel, F. de Fornel, and F. Capasso, “Cosine-Gauss plasmon beam: A localized long-range nondiffracting surface wave,” Phys. Rev. Lett. 109(9), 093904 (2012).
[Crossref] [PubMed]

de Fornel, F.

J. Lin, J. Dellinger, P. Genevet, B. Cluzel, F. de Fornel, and F. Capasso, “Cosine-Gauss plasmon beam: A localized long-range nondiffracting surface wave,” Phys. Rev. Lett. 109(9), 093904 (2012).
[Crossref] [PubMed]

Dellinger, J.

J. Lin, J. Dellinger, P. Genevet, B. Cluzel, F. de Fornel, and F. Capasso, “Cosine-Gauss plasmon beam: A localized long-range nondiffracting surface wave,” Phys. Rev. Lett. 109(9), 093904 (2012).
[Crossref] [PubMed]

Dorn, R.

S. Quabis, R. Dorn, M. Eberler, O. Glockl, and G. Leuchs, “Focusing light to a tighter spot,” Opt. Commun. 179(1-6), 1–7 (2000).
[Crossref]

Eberler, M.

S. Quabis, R. Dorn, M. Eberler, O. Glockl, and G. Leuchs, “Focusing light to a tighter spot,” Opt. Commun. 179(1-6), 1–7 (2000).
[Crossref]

Felson, L. B.

Feng, S.

S. Feng and H. G. Winful, “Spatiotemporal structure of isodiffracting ultrashort electromagnetic pulses,” Phys. Rev. E Stat. Phys. Plasmas Fluids Relat. Interdiscip. Topics 61(1), 862–873 (2000).
[Crossref] [PubMed]

S. Feng, H. G. Winful, and R. W. Hellwarth, “Spatiotemporal evolution of focused single-cycle electromagnetic pulses,” Phys. Rev. E Stat. Phys. Plasmas Fluids Relat. Interdiscip. Topics 59(4), 4630–4649 (1999).
[Crossref]

S. Feng, H. G. Winful, and R. W. Hellwarth, “Gouy shift and temporal reshaping of focused single-cycle electromagnetic pulses,” Opt. Lett. 23(5), 385–387 (1998).
[Crossref] [PubMed]

Gan, X.

C. J. R. Sheppard and X. Gan, “Free-space propagation of femto-second light pulses,” Opt. Commun. 133(1-6), 1–6 (1997).
[Crossref]

Genevet, P.

J. Lin, J. Dellinger, P. Genevet, B. Cluzel, F. de Fornel, and F. Capasso, “Cosine-Gauss plasmon beam: A localized long-range nondiffracting surface wave,” Phys. Rev. Lett. 109(9), 093904 (2012).
[Crossref] [PubMed]

Glockl, O.

S. Quabis, R. Dorn, M. Eberler, O. Glockl, and G. Leuchs, “Focusing light to a tighter spot,” Opt. Commun. 179(1-6), 1–7 (2000).
[Crossref]

Gonzalez, J. C.

J. C. Gonzalez, P. Benitez, and J. C. Miñano, “Perfect drain for the Maxwell fish eye lens,” New J. Phys. 13(2), 023038 (2011).
[Crossref]

Gouy, L. G.

L. G. Gouy, “Sur une propriété nouvelle des ondes lumineuses,” Comptes Rendus de l'Académie des Sciences, Paris 110, 1251–1253 (1890).

Greenleaf, J. F.

J. Y. Lu and J. F. Greenleaf, “Nondiffracting X waves-exact solutions to free-space scalar wave equation and their finite aperture realizations,” IEEE Trans. Ultrason. Ferroelectr. Freq. Control 39(1), 19–31 (1992).
[Crossref] [PubMed]

Hell, S.

S. Hell and E. H. K. Stelzer, “Fundamental improvement of resolution with a 4Pi-confocal fluorescence microscope using two-photon excitation,” Opt. Commun. 93(5-6), 277–282 (1992).
[Crossref]

Hellwarth, R. W.

S. Feng, H. G. Winful, and R. W. Hellwarth, “Spatiotemporal evolution of focused single-cycle electromagnetic pulses,” Phys. Rev. E Stat. Phys. Plasmas Fluids Relat. Interdiscip. Topics 59(4), 4630–4649 (1999).
[Crossref]

S. Feng, H. G. Winful, and R. W. Hellwarth, “Gouy shift and temporal reshaping of focused single-cycle electromagnetic pulses,” Opt. Lett. 23(5), 385–387 (1998).
[Crossref] [PubMed]

Heyman, E.

E. Heyman and T. Melamed, “Certain considerations in aperture synthesis of ultrawideband/short-pulse radiation,” IEEE Trans. Antenn. Propag. 42(4), 518–525 (1994).
[Crossref]

E. Heyman and L. B. Felson, “Complex-source pulsed-beam fields,” J. Opt. Soc. Am. A 6(6), 806–817 (1989).
[Crossref]

Judkins, J. B.

Kaplan, A. E.

Leonhardt, U.

Y. G. Ma, S. Sahebdivan, C. K. Ong, T. Tyc, and U. Leonhardt, “Evidence for subwavelength imaging with positive refraction,” New J. Phys. 13(3), 033016 (2011).
[Crossref]

Leuchs, G.

Lin, J.

J. Lin, J. Dellinger, P. Genevet, B. Cluzel, F. de Fornel, and F. Capasso, “Cosine-Gauss plasmon beam: A localized long-range nondiffracting surface wave,” Phys. Rev. Lett. 109(9), 093904 (2012).
[Crossref] [PubMed]

Lin, Q.

Z. Wang, Z. Zhang, Z. Xu, and Q. Lin, “Space-time profiles of an ultrashort pulsed Gaussian beam,” IEEE J. Quantum Electron. 33(4), 566–573 (1997).
[Crossref]

Lu, J. Y.

J. Y. Lu and J. F. Greenleaf, “Nondiffracting X waves-exact solutions to free-space scalar wave equation and their finite aperture realizations,” IEEE Trans. Ultrason. Ferroelectr. Freq. Control 39(1), 19–31 (1992).
[Crossref] [PubMed]

Ma, Y. G.

Y. G. Ma, S. Sahebdivan, C. K. Ong, T. Tyc, and U. Leonhardt, “Evidence for subwavelength imaging with positive refraction,” New J. Phys. 13(3), 033016 (2011).
[Crossref]

Matthews, H. J.

Melamed, T.

E. Heyman and T. Melamed, “Certain considerations in aperture synthesis of ultrawideband/short-pulse radiation,” IEEE Trans. Antenn. Propag. 42(4), 518–525 (1994).
[Crossref]

Miñano, J. C.

J. C. Gonzalez, P. Benitez, and J. C. Miñano, “Perfect drain for the Maxwell fish eye lens,” New J. Phys. 13(2), 023038 (2011).
[Crossref]

Nelson, K. A.

M. M. Wefers and K. A. Nelson, “Space-time profiles of shaped ultrafast optical waveforms,” IEEE J. Quantum Electron. 32(1), 161–172 (1996).
[Crossref]

Ong, C. K.

Y. G. Ma, S. Sahebdivan, C. K. Ong, T. Tyc, and U. Leonhardt, “Evidence for subwavelength imaging with positive refraction,” New J. Phys. 13(3), 033016 (2011).
[Crossref]

Porras, M. A.

M. A. Porras, “Ultrashort pulsed Gaussian light beams,” Phys. Rev. E Stat. Phys. Plasmas Fluids Relat. Interdiscip. Topics 58(1), 1086–1093 (1998).
[Crossref]

Potvliege, R. M.

C. F. R. Caron and R. M. Potvliege, “Free-space propagation of ultra-short pulses: space-time couplings in Gaussian pulse beams,” J. Mod. Opt. 46(13), 1881–1891 (1999).
[Crossref]

Quabis, S.

S. Quabis, R. Dorn, M. Eberler, O. Glockl, and G. Leuchs, “Focusing light to a tighter spot,” Opt. Commun. 179(1-6), 1–7 (2000).
[Crossref]

Saari, P.

Sahebdivan, S.

Y. G. Ma, S. Sahebdivan, C. K. Ong, T. Tyc, and U. Leonhardt, “Evidence for subwavelength imaging with positive refraction,” New J. Phys. 13(3), 033016 (2011).
[Crossref]

Sentenac, A.

Shaarawi, A. M.

A. M. Shaarawi, I. M. Besieris, and R. W. Ziolkowski, “The propagating and evanescent field components of localized wave solutions,” Opt. Commun. 116(1-3), 183–192 (1995).
[Crossref]

Sheppard, C. J. R.

C. J. R. Sheppard and X. Gan, “Free-space propagation of femto-second light pulses,” Opt. Commun. 133(1-6), 1–6 (1997).
[Crossref]

C. J. R. Sheppard and H. J. Matthews, “Imaging in high aperture optical systems,” J. Opt. Soc. Am. A 4(8), 1354–1360 (1987).
[Crossref]

Stelzer, E. H. K.

S. Hell and E. H. K. Stelzer, “Fundamental improvement of resolution with a 4Pi-confocal fluorescence microscope using two-photon excitation,” Opt. Commun. 93(5-6), 277–282 (1992).
[Crossref]

Tyc, T.

T. Tyc, “Gouy phase for full-aperture spherical and cylindrical waves,” Opt. Lett. 37(5), 924–926 (2012).
[Crossref] [PubMed]

Y. G. Ma, S. Sahebdivan, C. K. Ong, T. Tyc, and U. Leonhardt, “Evidence for subwavelength imaging with positive refraction,” New J. Phys. 13(3), 033016 (2011).
[Crossref]

Wang, Z.

Z. Wang, Z. Zhang, Z. Xu, and Q. Lin, “Space-time profiles of an ultrashort pulsed Gaussian beam,” IEEE J. Quantum Electron. 33(4), 566–573 (1997).
[Crossref]

Wefers, M. M.

M. M. Wefers and K. A. Nelson, “Space-time profiles of shaped ultrafast optical waveforms,” IEEE J. Quantum Electron. 32(1), 161–172 (1996).
[Crossref]

Winful, H. G.

S. Feng and H. G. Winful, “Spatiotemporal structure of isodiffracting ultrashort electromagnetic pulses,” Phys. Rev. E Stat. Phys. Plasmas Fluids Relat. Interdiscip. Topics 61(1), 862–873 (2000).
[Crossref] [PubMed]

S. Feng, H. G. Winful, and R. W. Hellwarth, “Spatiotemporal evolution of focused single-cycle electromagnetic pulses,” Phys. Rev. E Stat. Phys. Plasmas Fluids Relat. Interdiscip. Topics 59(4), 4630–4649 (1999).
[Crossref]

S. Feng, H. G. Winful, and R. W. Hellwarth, “Gouy shift and temporal reshaping of focused single-cycle electromagnetic pulses,” Opt. Lett. 23(5), 385–387 (1998).
[Crossref] [PubMed]

Xu, Z.

Z. Wang, Z. Zhang, Z. Xu, and Q. Lin, “Space-time profiles of an ultrashort pulsed Gaussian beam,” IEEE J. Quantum Electron. 33(4), 566–573 (1997).
[Crossref]

You, D.

Zhang, X.

X. Zhang, “Perfect lenses in focus: No drain, no gain,” Nature 480, 42–43 (2011).
[Crossref] [PubMed]

Zhang, Z.

Z. Wang, Z. Zhang, Z. Xu, and Q. Lin, “Space-time profiles of an ultrashort pulsed Gaussian beam,” IEEE J. Quantum Electron. 33(4), 566–573 (1997).
[Crossref]

Ziolkowski, R. W.

A. M. Shaarawi, I. M. Besieris, and R. W. Ziolkowski, “The propagating and evanescent field components of localized wave solutions,” Opt. Commun. 116(1-3), 183–192 (1995).
[Crossref]

R. W. Ziolkowski and J. B. Judkins, “Propagation characteristics of ultrawide-bandwidth pulsed Gaussian beams,” J. Opt. Soc. Am. A 9(11), 2021–2030 (1992).
[Crossref]

R. W. Ziolkowski, “Localized transmission of electromagnetic energy,” Phys. Rev. A 39(4), 2005–2033 (1989).
[Crossref] [PubMed]

Comptes Rendus de l'Académie des Sciences, Paris (1)

L. G. Gouy, “Sur une propriété nouvelle des ondes lumineuses,” Comptes Rendus de l'Académie des Sciences, Paris 110, 1251–1253 (1890).

IEEE J. Quantum Electron. (2)

M. M. Wefers and K. A. Nelson, “Space-time profiles of shaped ultrafast optical waveforms,” IEEE J. Quantum Electron. 32(1), 161–172 (1996).
[Crossref]

Z. Wang, Z. Zhang, Z. Xu, and Q. Lin, “Space-time profiles of an ultrashort pulsed Gaussian beam,” IEEE J. Quantum Electron. 33(4), 566–573 (1997).
[Crossref]

IEEE Trans. Antenn. Propag. (1)

E. Heyman and T. Melamed, “Certain considerations in aperture synthesis of ultrawideband/short-pulse radiation,” IEEE Trans. Antenn. Propag. 42(4), 518–525 (1994).
[Crossref]

IEEE Trans. Ultrason. Ferroelectr. Freq. Control (1)

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Supplementary Material (2)

» Media 1: MOV (999 KB)     
» Media 2: MOV (627 KB)     

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Figures (8)

Fig. 1
Fig. 1 (a) The real part of the amplitude of a source/sink 3D pulse, given by the instantaneous amplitude for δ=0 , (b) an outgoing 3D pulse, δ=π/2 . The light cone is shown in red, with the direction of propagation indicated with arrows.
Fig. 2
Fig. 2 3D source/sink pulses for different values of the parameter δ . (a) Amplitude at the focus, normalized to unity at t=0 . (b) Output pulse amplitude (× k 1 r) at k 1 r=50 . A symmetrical pulse at the focus produces an antisymmetric output pulse in the far field.
Fig. 3
Fig. 3 3D pulsed source for different values of the parameter δ . (a) Amplitude at the source, normalized to unity at t=0 . (b) Output pulse amplitude (× k 1 r) .
Fig. 4
Fig. 4 The amplitude of different 2D source/sink pulses generated by different launched input pulse shapes: (a) launched by an input pulse with a wake, δ=0 , (b) an input pulse with a slow rise, δ=π/2 , (c) symmetric input pulse, anti-symmetric output pulse, δ=π/4 . The light ‘cone’ is shown in red, with the direction of propagation indicated with arrows.
Fig. 5
Fig. 5 (a) The function f(x) . (b) The instantaneous amplitude in the far field for different values of δ , as given by Eq. (14).
Fig. 6
Fig. 6 2D source/sink pulses for different values of the parameter δ . (a) Time variation of the input pulse amplitude at k 1 r=50 , multiplied by k 1 r = 50 . (b) Time variation of the amplitude at the source/sink, normalized to unity at t=0 . (c) Time variation of the output pulse amplitude at k 1 r=50 , multiplied by k 1 r = 50 . See Media 1: a movie showing how the amplitude variation (× k 1 r ) in space changes with time.
Fig. 7
Fig. 7 A 2D outgoing pulse generated by a time-symmetric source, δ=π/2 . The light ‘cone’ is shown in red, with the direction of propagation indicated with an arrow. The output pulse decays slowly above the light ‘cone’.
Fig. 8
Fig. 8 The amplitude of a 2D pulsed source for different values of the parameter δ . (a) Time variation of the amplitude at the source, normalized to unity at t=0 . (b) Time variation of the output pulse amplitude at k 1 r=50 , multiplied by k 1 r = 50 . See Media 2: a movie showing how the amplitude variation (× k 1 r ) in space changes with time.

Equations (14)

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f(k)dk= 1 k 1 exp( k k 1 )dk,k>0 =0,k<0,
U(r,t)=A 0 f(k) exp(ikr) r exp(ikct)dk = A r 1 [ 1+i k 1 c( t r c ) ] ,
U(r,t)= 1 2i k 1 r { 1 [ 1+i k 1 c( t r c ) ] 1 [ 1+i k 1 c( t+ r c ) ] } = 1 k 1 2 r 2 + (1+i k 1 ct) 2 .
U(0,t)= 1 ( 1+i k 1 ct ) 2 ,
U(r,0)= 1 1+ k 1 2 r 2 ,
U(ρ,t)=A 0 f(k) H 0 (1) (kρ)exp(ikct)dk = A k 1 2 ρ 2 + (1+i k 1 ct) 2 { 1 2i π ln[ 1 k 1 ρ + ict ρ + 1 ρ ρ 2 + ( 1 k 1 +ict ) 2 ] },
U(ρ,t)= A k 1 2 ρ 2 + (1+i k 1 ct) 2 { 1+ 2i π ln[ 1 k 1 ρ + ict ρ + 1 ρ ρ 2 + ( 1 k 1 +ict ) 2 ] }.
U(ρ,t)= 1 k 1 2 ρ 2 + (1+i k 1 ct) 2 .
I(ρ,t)= 1+ k 1 2 ( ρ 2 c 2 t 2 ) 2[ 1+2 k 1 2 ( ρ 2 + c 2 t 2 )+ k 1 4 ( ρ 2 c 2 t 2 ) 2 ] + 1 2 1+2 k 1 2 ( ρ 2 + c 2 t 2 )+ k 1 4 ( ρ 2 c 2 t 2 ) 2 .
t =tρ/c
I(ρ, t )= 1 4 k 1 ρ [ 1 1+ k 1 2 c 2 t 2 k 1 c t (1+ k 1 2 c 2 t 2 ) ].
f(x)= 1 1+ x 2 + x (1+ x 2 ) ,
U(0,t)= 1 k 1 ct 1+ ( k 1 ct) 2 .
U(ρ, t )= cos(δ+ π 4 ) ( 1+ k 1 2 c 2 t 2 +1 ) 1/2 sgn( t )sin(δ+ π 4 ) ( 1+ k 1 2 c 2 t 2 1 ) 1/2 4 k 1 ρ(1+ k 1 2 c 2 t 2 ) .

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