Abstract

We present a systematic study of linear propagation of ultrashort laser pulses in media with dispersion, dispersionless media, and vacuum. The applied method of amplitude envelopes makes it possible to estimate the limits of the slowly varying amplitude approximation and to describe an amplitude integrodifferential equation governing propagation of optical pulses in the single-cycle regime in solids. The well-known slowly varying amplitude equation and the amplitude equation for the vacuum case are written in dimensionless form. Three parameters are obtained defining different linear regimes of optical pulse evolution. In contrast to previous studies we demonstrate that in the femtosecond region the nonparaxial terms are not small and can dominate over the transverse Laplacian. The normalized amplitude nonparaxial equations are solved using the method of Fourier transforms. Fundamental solutions with spectral kernels different from those according to Fresnel are found. Exact unidirectional analytical solution of the nonparaxial amplitude equations and the 3D wave equations with initial conditions compatible with Gaussian light bullets are obtained also. One unexpected new result is the relative stability of light bullets (pulses with spherical and spheroidal spatial form) when we compare their transverse enlargement with paraxial diffraction of light beams in air. It is important to emphasize here the case of light disks, i.e., pulses whose longitudinal size is small with respect to the transverse one, which in some partial cases are practically diffractionless over distances of a thousand kilometers. A new formula that calculates the diffraction length of optical pulses is suggested. Finally, propagation of single-cycle pulses in air and vacuum was investigated, and a coronal (semispherical) form of diffraction at short distances was observed.

© 2008 Optical Society of America

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References

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  1. A. Braun, G. Korn, X. Liu, D. Du, J. Squier, and G. Mourou, “Self-channeling of high-peak-power femtosecond laser pulses in air,” Opt. Lett. 20, 73-75 (1997).
    [CrossRef]
  2. C. Ruiz, J. San Romain, C. Mendez, V. Diaz, L. Plaja, I. Arias, and L. Roso, “Observation of spontaneous self-channeling of light in air below the collapse threshold,” Phys. Rev. Lett. 95, 053905 (2005).
    [CrossRef] [PubMed]
  3. J. V. Moloney and M. Kolesik, “Full vectorial, intense ultrashort pulse propagators: derivations and applications,” in Progress in Ultrafast Intense Laser Science II, Vol. 85 (Springer, 2007).
    [CrossRef]
  4. A. Couairon and A. Mysyrowicz, “Femtosecond filamentation in transparent media,” Phys. Rep. 441, 47-189 (2007).
    [CrossRef]
  5. S. L. Chin, S. A. Hosseini, W. Liu, Q. Luo, F. Théberge, N. Aközbek, A. Becker, V. P. Kandidov, O. G. Kosareva, and H. Schoeder, “The propagation of powerful femtosecond laser pulses in optical media: physics, applications, and new challenges,” Can. J. Phys. 83, 863-905 (2005).
    [CrossRef]
  6. Y. Silberbers, “Collapse of optical pulses,” Opt. Lett. 15, 1282-1284 (1990).
    [CrossRef]
  7. T. Brabec and F. Krausz, “Nonlinear optical pulse propagation in the single-cycle regime,” Phys. Rev. Lett. 78, 3282-3285 (1997).
    [CrossRef]
  8. Yu. S. Kivshar and G. P. Agrawal, Optical Solitons: From Fibers to Photonic Crystals (Academic, 2003).
  9. N. N. Akhmediev and A. Ankiewicz, Solitons: Nonlinear Pulses and Beams (Chapman and Hall, 1997).
  10. G. Fibich and G. S. Papanicolaou, “Self-focusing in the presence of small time dispersion and nonparaxiality,” Opt. Lett. 22, 1379-1381 (1997).
    [CrossRef]
  11. K. L. Kovachev, “Limits of application of the slowly varying amplitude approximation,” Bachelor thesis (University of Sofia, 2003).
  12. V. I. Karpman, Nonlinear Waves in Dispersive Media (Nauka, Moscow, 1973).
  13. M. Jain and N. Tzoar, “Nonlinear pulse propagation in optical fibres,” Opt. Lett. 3, 202-204 (1978).
    [CrossRef] [PubMed]
  14. J. V. Moloney and A. C. Newell, Nonlinear Optics (Addison-Wesley, 1991).
  15. D. N. Christodoulides and R. H. Jouseph, “Exact radial dependence of the field in a nonlinear dispersive dielectric fiber: bright pulse solutions,” Opt. Lett. 9, 229-231 (1984).
    [CrossRef] [PubMed]
  16. R. W. Boyd, Nonlinear Optics (Academic, 2003).
  17. L. M. Kovachev, “Optical vortices in dispersive nonlinear Kerr-type media,” Int. J. Math. Math. Sci. 18, 949-967 (2004).
    [CrossRef]
  18. E. M. Dianov, P. V. Mamyshev, A. M. Prokhorov, and V. N. Serkin, Nonlinear Effects in Optical Fibers (Harwood Academic, 1989).
  19. C. Menyuk, “Application of multiple-length methods to the study of optical fiber transmission,” J. Eng. Math. 36, 113-136 (1999).
    [CrossRef]
  20. L. M. Kovachev, L. Pavlov, L. M. Ivanov, and D. Dakova, “Optical filaments and optical bullets in dispersive nonlinear media,” J. Russ. Laser Res. 27, 185-203 (2006).
    [CrossRef]
  21. L. M. Kovachev, “Collapse arrest and self-guiding of femtosecond pulses,” Opt. Express 15, 10318-10323 (2007).
    [CrossRef] [PubMed]

2007 (2)

A. Couairon and A. Mysyrowicz, “Femtosecond filamentation in transparent media,” Phys. Rep. 441, 47-189 (2007).
[CrossRef]

L. M. Kovachev, “Collapse arrest and self-guiding of femtosecond pulses,” Opt. Express 15, 10318-10323 (2007).
[CrossRef] [PubMed]

2006 (1)

L. M. Kovachev, L. Pavlov, L. M. Ivanov, and D. Dakova, “Optical filaments and optical bullets in dispersive nonlinear media,” J. Russ. Laser Res. 27, 185-203 (2006).
[CrossRef]

2005 (2)

C. Ruiz, J. San Romain, C. Mendez, V. Diaz, L. Plaja, I. Arias, and L. Roso, “Observation of spontaneous self-channeling of light in air below the collapse threshold,” Phys. Rev. Lett. 95, 053905 (2005).
[CrossRef] [PubMed]

S. L. Chin, S. A. Hosseini, W. Liu, Q. Luo, F. Théberge, N. Aközbek, A. Becker, V. P. Kandidov, O. G. Kosareva, and H. Schoeder, “The propagation of powerful femtosecond laser pulses in optical media: physics, applications, and new challenges,” Can. J. Phys. 83, 863-905 (2005).
[CrossRef]

2004 (1)

L. M. Kovachev, “Optical vortices in dispersive nonlinear Kerr-type media,” Int. J. Math. Math. Sci. 18, 949-967 (2004).
[CrossRef]

1999 (1)

C. Menyuk, “Application of multiple-length methods to the study of optical fiber transmission,” J. Eng. Math. 36, 113-136 (1999).
[CrossRef]

1997 (3)

1990 (1)

1984 (1)

1978 (1)

Agrawal, G. P.

Yu. S. Kivshar and G. P. Agrawal, Optical Solitons: From Fibers to Photonic Crystals (Academic, 2003).

Akhmediev, N. N.

N. N. Akhmediev and A. Ankiewicz, Solitons: Nonlinear Pulses and Beams (Chapman and Hall, 1997).

Aközbek, N.

S. L. Chin, S. A. Hosseini, W. Liu, Q. Luo, F. Théberge, N. Aközbek, A. Becker, V. P. Kandidov, O. G. Kosareva, and H. Schoeder, “The propagation of powerful femtosecond laser pulses in optical media: physics, applications, and new challenges,” Can. J. Phys. 83, 863-905 (2005).
[CrossRef]

Ankiewicz, A.

N. N. Akhmediev and A. Ankiewicz, Solitons: Nonlinear Pulses and Beams (Chapman and Hall, 1997).

Arias, I.

C. Ruiz, J. San Romain, C. Mendez, V. Diaz, L. Plaja, I. Arias, and L. Roso, “Observation of spontaneous self-channeling of light in air below the collapse threshold,” Phys. Rev. Lett. 95, 053905 (2005).
[CrossRef] [PubMed]

Becker, A.

S. L. Chin, S. A. Hosseini, W. Liu, Q. Luo, F. Théberge, N. Aközbek, A. Becker, V. P. Kandidov, O. G. Kosareva, and H. Schoeder, “The propagation of powerful femtosecond laser pulses in optical media: physics, applications, and new challenges,” Can. J. Phys. 83, 863-905 (2005).
[CrossRef]

Boyd, R. W.

R. W. Boyd, Nonlinear Optics (Academic, 2003).

Brabec, T.

T. Brabec and F. Krausz, “Nonlinear optical pulse propagation in the single-cycle regime,” Phys. Rev. Lett. 78, 3282-3285 (1997).
[CrossRef]

Braun, A.

Chin, S. L.

S. L. Chin, S. A. Hosseini, W. Liu, Q. Luo, F. Théberge, N. Aközbek, A. Becker, V. P. Kandidov, O. G. Kosareva, and H. Schoeder, “The propagation of powerful femtosecond laser pulses in optical media: physics, applications, and new challenges,” Can. J. Phys. 83, 863-905 (2005).
[CrossRef]

Christodoulides, D. N.

Couairon, A.

A. Couairon and A. Mysyrowicz, “Femtosecond filamentation in transparent media,” Phys. Rep. 441, 47-189 (2007).
[CrossRef]

Dakova, D.

L. M. Kovachev, L. Pavlov, L. M. Ivanov, and D. Dakova, “Optical filaments and optical bullets in dispersive nonlinear media,” J. Russ. Laser Res. 27, 185-203 (2006).
[CrossRef]

Dianov, E. M.

E. M. Dianov, P. V. Mamyshev, A. M. Prokhorov, and V. N. Serkin, Nonlinear Effects in Optical Fibers (Harwood Academic, 1989).

Diaz, V.

C. Ruiz, J. San Romain, C. Mendez, V. Diaz, L. Plaja, I. Arias, and L. Roso, “Observation of spontaneous self-channeling of light in air below the collapse threshold,” Phys. Rev. Lett. 95, 053905 (2005).
[CrossRef] [PubMed]

Du, D.

Fibich, G.

Hosseini, S. A.

S. L. Chin, S. A. Hosseini, W. Liu, Q. Luo, F. Théberge, N. Aközbek, A. Becker, V. P. Kandidov, O. G. Kosareva, and H. Schoeder, “The propagation of powerful femtosecond laser pulses in optical media: physics, applications, and new challenges,” Can. J. Phys. 83, 863-905 (2005).
[CrossRef]

Ivanov, L. M.

L. M. Kovachev, L. Pavlov, L. M. Ivanov, and D. Dakova, “Optical filaments and optical bullets in dispersive nonlinear media,” J. Russ. Laser Res. 27, 185-203 (2006).
[CrossRef]

Jain, M.

Jouseph, R. H.

Kandidov, V. P.

S. L. Chin, S. A. Hosseini, W. Liu, Q. Luo, F. Théberge, N. Aközbek, A. Becker, V. P. Kandidov, O. G. Kosareva, and H. Schoeder, “The propagation of powerful femtosecond laser pulses in optical media: physics, applications, and new challenges,” Can. J. Phys. 83, 863-905 (2005).
[CrossRef]

Karpman, V. I.

V. I. Karpman, Nonlinear Waves in Dispersive Media (Nauka, Moscow, 1973).

Kivshar, Yu. S.

Yu. S. Kivshar and G. P. Agrawal, Optical Solitons: From Fibers to Photonic Crystals (Academic, 2003).

Kolesik, M.

J. V. Moloney and M. Kolesik, “Full vectorial, intense ultrashort pulse propagators: derivations and applications,” in Progress in Ultrafast Intense Laser Science II, Vol. 85 (Springer, 2007).
[CrossRef]

Korn, G.

Kosareva, O. G.

S. L. Chin, S. A. Hosseini, W. Liu, Q. Luo, F. Théberge, N. Aközbek, A. Becker, V. P. Kandidov, O. G. Kosareva, and H. Schoeder, “The propagation of powerful femtosecond laser pulses in optical media: physics, applications, and new challenges,” Can. J. Phys. 83, 863-905 (2005).
[CrossRef]

Kovachev, K. L.

K. L. Kovachev, “Limits of application of the slowly varying amplitude approximation,” Bachelor thesis (University of Sofia, 2003).

Kovachev, L. M.

L. M. Kovachev, “Collapse arrest and self-guiding of femtosecond pulses,” Opt. Express 15, 10318-10323 (2007).
[CrossRef] [PubMed]

L. M. Kovachev, L. Pavlov, L. M. Ivanov, and D. Dakova, “Optical filaments and optical bullets in dispersive nonlinear media,” J. Russ. Laser Res. 27, 185-203 (2006).
[CrossRef]

L. M. Kovachev, “Optical vortices in dispersive nonlinear Kerr-type media,” Int. J. Math. Math. Sci. 18, 949-967 (2004).
[CrossRef]

Krausz, F.

T. Brabec and F. Krausz, “Nonlinear optical pulse propagation in the single-cycle regime,” Phys. Rev. Lett. 78, 3282-3285 (1997).
[CrossRef]

Liu, W.

S. L. Chin, S. A. Hosseini, W. Liu, Q. Luo, F. Théberge, N. Aközbek, A. Becker, V. P. Kandidov, O. G. Kosareva, and H. Schoeder, “The propagation of powerful femtosecond laser pulses in optical media: physics, applications, and new challenges,” Can. J. Phys. 83, 863-905 (2005).
[CrossRef]

Liu, X.

Luo, Q.

S. L. Chin, S. A. Hosseini, W. Liu, Q. Luo, F. Théberge, N. Aközbek, A. Becker, V. P. Kandidov, O. G. Kosareva, and H. Schoeder, “The propagation of powerful femtosecond laser pulses in optical media: physics, applications, and new challenges,” Can. J. Phys. 83, 863-905 (2005).
[CrossRef]

Mamyshev, P. V.

E. M. Dianov, P. V. Mamyshev, A. M. Prokhorov, and V. N. Serkin, Nonlinear Effects in Optical Fibers (Harwood Academic, 1989).

Mendez, C.

C. Ruiz, J. San Romain, C. Mendez, V. Diaz, L. Plaja, I. Arias, and L. Roso, “Observation of spontaneous self-channeling of light in air below the collapse threshold,” Phys. Rev. Lett. 95, 053905 (2005).
[CrossRef] [PubMed]

Menyuk, C.

C. Menyuk, “Application of multiple-length methods to the study of optical fiber transmission,” J. Eng. Math. 36, 113-136 (1999).
[CrossRef]

Moloney, J. V.

J. V. Moloney and A. C. Newell, Nonlinear Optics (Addison-Wesley, 1991).

J. V. Moloney and M. Kolesik, “Full vectorial, intense ultrashort pulse propagators: derivations and applications,” in Progress in Ultrafast Intense Laser Science II, Vol. 85 (Springer, 2007).
[CrossRef]

Mourou, G.

Mysyrowicz, A.

A. Couairon and A. Mysyrowicz, “Femtosecond filamentation in transparent media,” Phys. Rep. 441, 47-189 (2007).
[CrossRef]

Newell, A. C.

J. V. Moloney and A. C. Newell, Nonlinear Optics (Addison-Wesley, 1991).

Papanicolaou, G. S.

Pavlov, L.

L. M. Kovachev, L. Pavlov, L. M. Ivanov, and D. Dakova, “Optical filaments and optical bullets in dispersive nonlinear media,” J. Russ. Laser Res. 27, 185-203 (2006).
[CrossRef]

Plaja, L.

C. Ruiz, J. San Romain, C. Mendez, V. Diaz, L. Plaja, I. Arias, and L. Roso, “Observation of spontaneous self-channeling of light in air below the collapse threshold,” Phys. Rev. Lett. 95, 053905 (2005).
[CrossRef] [PubMed]

Prokhorov, A. M.

E. M. Dianov, P. V. Mamyshev, A. M. Prokhorov, and V. N. Serkin, Nonlinear Effects in Optical Fibers (Harwood Academic, 1989).

Roso, L.

C. Ruiz, J. San Romain, C. Mendez, V. Diaz, L. Plaja, I. Arias, and L. Roso, “Observation of spontaneous self-channeling of light in air below the collapse threshold,” Phys. Rev. Lett. 95, 053905 (2005).
[CrossRef] [PubMed]

Ruiz, C.

C. Ruiz, J. San Romain, C. Mendez, V. Diaz, L. Plaja, I. Arias, and L. Roso, “Observation of spontaneous self-channeling of light in air below the collapse threshold,” Phys. Rev. Lett. 95, 053905 (2005).
[CrossRef] [PubMed]

San Romain, J.

C. Ruiz, J. San Romain, C. Mendez, V. Diaz, L. Plaja, I. Arias, and L. Roso, “Observation of spontaneous self-channeling of light in air below the collapse threshold,” Phys. Rev. Lett. 95, 053905 (2005).
[CrossRef] [PubMed]

Schoeder, H.

S. L. Chin, S. A. Hosseini, W. Liu, Q. Luo, F. Théberge, N. Aközbek, A. Becker, V. P. Kandidov, O. G. Kosareva, and H. Schoeder, “The propagation of powerful femtosecond laser pulses in optical media: physics, applications, and new challenges,” Can. J. Phys. 83, 863-905 (2005).
[CrossRef]

Serkin, V. N.

E. M. Dianov, P. V. Mamyshev, A. M. Prokhorov, and V. N. Serkin, Nonlinear Effects in Optical Fibers (Harwood Academic, 1989).

Silberbers, Y.

Squier, J.

Théberge, F.

S. L. Chin, S. A. Hosseini, W. Liu, Q. Luo, F. Théberge, N. Aközbek, A. Becker, V. P. Kandidov, O. G. Kosareva, and H. Schoeder, “The propagation of powerful femtosecond laser pulses in optical media: physics, applications, and new challenges,” Can. J. Phys. 83, 863-905 (2005).
[CrossRef]

Tzoar, N.

Can. J. Phys. (1)

S. L. Chin, S. A. Hosseini, W. Liu, Q. Luo, F. Théberge, N. Aközbek, A. Becker, V. P. Kandidov, O. G. Kosareva, and H. Schoeder, “The propagation of powerful femtosecond laser pulses in optical media: physics, applications, and new challenges,” Can. J. Phys. 83, 863-905 (2005).
[CrossRef]

Int. J. Math. Math. Sci. (1)

L. M. Kovachev, “Optical vortices in dispersive nonlinear Kerr-type media,” Int. J. Math. Math. Sci. 18, 949-967 (2004).
[CrossRef]

J. Eng. Math. (1)

C. Menyuk, “Application of multiple-length methods to the study of optical fiber transmission,” J. Eng. Math. 36, 113-136 (1999).
[CrossRef]

J. Russ. Laser Res. (1)

L. M. Kovachev, L. Pavlov, L. M. Ivanov, and D. Dakova, “Optical filaments and optical bullets in dispersive nonlinear media,” J. Russ. Laser Res. 27, 185-203 (2006).
[CrossRef]

Opt. Express (1)

Opt. Lett. (5)

Phys. Rep. (1)

A. Couairon and A. Mysyrowicz, “Femtosecond filamentation in transparent media,” Phys. Rep. 441, 47-189 (2007).
[CrossRef]

Phys. Rev. Lett. (2)

C. Ruiz, J. San Romain, C. Mendez, V. Diaz, L. Plaja, I. Arias, and L. Roso, “Observation of spontaneous self-channeling of light in air below the collapse threshold,” Phys. Rev. Lett. 95, 053905 (2005).
[CrossRef] [PubMed]

T. Brabec and F. Krausz, “Nonlinear optical pulse propagation in the single-cycle regime,” Phys. Rev. Lett. 78, 3282-3285 (1997).
[CrossRef]

Other (8)

Yu. S. Kivshar and G. P. Agrawal, Optical Solitons: From Fibers to Photonic Crystals (Academic, 2003).

N. N. Akhmediev and A. Ankiewicz, Solitons: Nonlinear Pulses and Beams (Chapman and Hall, 1997).

K. L. Kovachev, “Limits of application of the slowly varying amplitude approximation,” Bachelor thesis (University of Sofia, 2003).

V. I. Karpman, Nonlinear Waves in Dispersive Media (Nauka, Moscow, 1973).

J. V. Moloney and A. C. Newell, Nonlinear Optics (Addison-Wesley, 1991).

R. W. Boyd, Nonlinear Optics (Academic, 2003).

J. V. Moloney and M. Kolesik, “Full vectorial, intense ultrashort pulse propagators: derivations and applications,” in Progress in Ultrafast Intense Laser Science II, Vol. 85 (Springer, 2007).
[CrossRef]

E. M. Dianov, P. V. Mamyshev, A. M. Prokhorov, and V. N. Serkin, Nonlinear Effects in Optical Fibers (Harwood Academic, 1989).

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Figures (8)

Fig. 1
Fig. 1

Gaussian LB propagates in air with 330 fs time duration, 100 μ m initial waist, on carrying frequency 800 nm , and governed by linear SVEA (30) in the Galilean frame under initial condition A x ( x , y , z , t = 0 ) = exp ( x 2 + y 2 + z 2 2 ) , α = 785.0 , δ = 1.0 . The dimensionless dispersion parameter β 1 = 2.1 × 10 5 is very small, and the dynamics is identical with pulse evolution governed by VLAE (32). The surfaces A ( x , y , z = 0 , t = 0 , t = 785 2 , t = 785 ) 2 are plotted. The transverse size (the spot) grows by the factor 2 over normalized time–distance t = z = 785 . For the laser source selected in the paper this corresponds to a real distance z diff pulse = α δ 2 z diff beam 62 m .

Fig. 2
Fig. 2

Transverse intensity distribution of 33 fs pulse with initial waist 100 μ m on carrying frequency 800 nm (LD) governed by the same SVEA (30) in Galilean coordinates and initial conditions A x ( x , y , z , t = 0 ) = exp ( x 2 + y 2 + z 2 2 ) , α = 78.5 , δ = 100 , β 1 = 2.1 × 10 5 . The surfaces A ( x , y , z = 0 , t = 0 , t = 7850 2 , t = 7850 ) 2 are presented. The LD enlarges its transverse size by the factor 2 over the normalized time–distance t = z = 7850 . This corresponds to 7850 diffraction lengths of a laser beam or for the selected laser source, z diff pulse = α δ 2 z diff beam 621 m .

Fig. 3
Fig. 3

Initial Gaussian shape of the solution (52) of VLAE (31) when t = 0 . The surface A ( x , y = 0 , z , t = 0 ) is plotted.

Fig. 4
Fig. 4

Deformation of the Gaussian LB with 330 fs time duration obtained from exact solution (52) of VLAE (31). The surface A ( x , y = 0 , z , t = 785 ) is plotted. The waist grows by the factor 2 over normalized time–distance t = z = 785 , while the amplitude decreases with A 0 = 1 2 , as can be expected for one pulse diffraction length. It is noted in the text that the pulse diffraction length is 785 times greater than those of a beam. Compare with Fig. 1.

Fig. 5
Fig. 5

Shaping of Gaussian pulse obtained from exact solution (56) of VLAE (32) in Galilean coordinates. The surface A ( x , y = 0 , z , t = 785 ) is plotted. The spot grows by the factor 2 over the same normalized time t = 785 , while the pulse remains in initial position z = 0 , as can be expected from Galilean invariancy. Compare with Fig. 4.

Fig. 6
Fig. 6

Propagation of LB with two cycles in air on normalized time–distance t = z = 3 π . The evolution is governed by fundamental solution (41) of the amplitude equation (29) calculated by the FFT technique. The initial LB admits two optical periods only α = 2 and the small dispersion parameter β 1 = 2.1 × 10 5 . We plot the x z plane of the pulse.

Fig. 7
Fig. 7

Shaping of LB amplitude with two optical periods governed by the solution of normalized wave equation (22) ( c = 1 ) under the same initial conditions as Fig. 6 and normalized wave vector: k 0 = 2 , E = E x x , E x ( x , y , z , t = 0 ) = exp ( i k 0 z ) exp [ ( x 2 + y 2 + z 2 ) 2 ] . Surface x z is presented. We point out here the equal dynamics of LB with few optical periods in the framework of both Eqs. (29, 22).

Fig. 8
Fig. 8

Coronal (semispherical) deformation of LB with two optical periods governed by exact solution (52) of the VLAE (31), on the same distance z = 3 π and under the same initial conditions as Fig. 6: α = 2 , A = A x x , A x ( x , y , z , t = 0 ) = exp [ ( x 2 + y 2 + z 2 ) 2 ] . Surface x z is plotted. The dynamics is identical with the numerical solutions using the FFT method for wave equation (22) and SVEA (29) (Compare with Fig. 6 and Fig. 7). The equality among solutions of SVEA (29), wave equation (22), and VLAE (31) was expected as the dispersion parameter in air is negligible ( β 1 1 ) .

Equations (59)

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P lin = t ( δ ( τ t ) + 4 π χ ( 1 ) ( τ t ) ) E ( τ , r ) d τ = t ϵ ( τ t ) E ( τ , x , y , z ) d τ ,
× E = 1 c B t ,
× H = 1 c D t ,
D = 0 ,
B = H = 0 ,
B = H , D = P lin ,
( E ) Δ E = 1 c 2 2 D t 2 ,
Δ E = 1 c 2 2 D t 2 .
E ( r , t ) = + E ̂ ( r , ω ) exp ( i ω t ) d ω ,
Δ E = 1 c 2 2 t 2 ( t ϵ ( τ t ) E ̂ ( r , ω ) exp ( i ω τ ) d ω d τ ) .
k 2 = ω 2 ϵ ̂ ( ω ) c 2 ,
Δ E = k 2 ( ω ) E ̂ ( r , ω ) exp ( i ω t ) d ω .
E ( x , y , z , t ) = A ( x , y , z , t ) exp [ i ( k 0 z ω 0 t ) ] ,
A ( r , t ) = + A ̂ ( r , ω ω 0 ) exp [ i ( ω ω 0 ) t ] d ω ,
E ̂ ( r , ω ) exp ( i ω t ) = exp [ i ( k 0 z ω 0 t ) ] A ̂ ( r , ω ω 0 ) exp [ i ( ω ω 0 ) t ] .
Δ A ( r , t ) + 2 i k 0 A ( r , t ) z k 0 2 A ( r , t ) = k 2 ( ω ) A ̂ ( r , ω ω 0 ) exp [ i ( ω ω 0 ) t ] d ω .
Δ A ̂ ( r , ω ω 0 ) + 2 i k 0 A ̂ ( r , ω ω 0 ) z + [ k 2 ( ω ) k 0 2 ( ω 0 ) ] A ̂ ( r , ω ω 0 ) = 0 .
k 2 ( ω ) = ω 2 ϵ 0 ̂ ( ω ) c 2 = k 2 ( ω 0 ) + [ k 2 ( ω 0 ) ] ω 0 ( ω ω 0 ) + 1 2 [ k 2 ( ω 0 ) ] ω 0 2 ( ω ω 0 ) 2 + .
Δ A + 2 i k 0 A z + 2 i k 0 k A t = ( k 0 k + k 2 ) 2 A t 2 ,
v ( ω 0 ) = 1 k = c ϵ ( ω 0 ) + ω 0 2 1 ϵ ϵ ω ;
i ( A t + v A z ) = v 2 k 0 Δ A v 2 ( k + 1 k 0 v 2 ) 2 A t 2 .
Δ E = 1 c 2 2 E t 2 .
E ( x , y , z , t ) = V ( x , y , z , t ) exp [ i ( k 0 z ω 0 t ) ] ,
i ( V t + c V z ) = c 2 k 0 Δ V 1 2 k 0 c 2 V t 2 .
i A t = v 2 k 0 Δ A v 3 k 0 2 2 A z 2 v 2 ( k + 1 k 0 v 2 ) ( 2 A t 2 2 v 2 A t z ) ,
i V t = c 2 k 0 Δ V 1 2 k 0 c 2 V t 2 + 1 k 0 2 V t z .
A = A 0 A , V = V 0 V , x = r x , y = r y , z = z 0 z ,
t = t 0 t , z = z 0 z , t = t 0 t .
α = k 0 z 0 , δ 2 = r 2 z 0 2 , β = z dif z disp .
2 i α δ 2 ( A t + A z ) = Δ A + δ 2 ( 2 A z 2 2 A t 2 ) β 2 A t 2 .
2 i α δ 2 A t = Δ A β 2 A z 2 ( β + δ 2 ) ( 2 A t 2 2 2 A t z ) ,
2 i α δ 2 ( V t + V z ) = Δ V + δ 2 ( 2 V z 2 2 V t 2 ) .
2 i α δ 2 V t = Δ V δ 2 ( 2 V t 2 2 V t z ) .
2 i α δ 2 A L t = [ k x 2 + k y 2 + δ 2 ( k z 2 2 α k z ) ] A L δ 2 ( β 1 + 1 ) 2 A L t 2 ,
2 i δ 2 [ α ( β 1 + 1 ) k z ] A G t = ( k x 2 + k y 2 δ 2 β 1 k z 2 ) A G δ 2 ( β 1 + 1 ) 2 A G t 2 ,
2 i α δ 2 B L t = [ k x 2 + k y 2 + δ 2 ( k z 2 2 α k z ) ] B L δ 2 2 B L t 2 ,
2 i δ 2 ( α k z ) B G t = ( k x 2 + k y 2 ) B G δ 2 2 B G t 2 .
A L = A L ( k x , k y , k z , t = 0 ) exp { i [ α β 1 + 1 ± α 2 ( β 1 + 1 ) 2 + k ̂ 2 δ 2 ( β 1 + 1 ) ] t } .
A G = A G ( k x , k y , k z , t = 0 ) × exp { i [ α ( β 1 + 1 ) k z β 1 + 1 ± ( α ( β 1 + 1 ) k z ) 2 ( β 1 + 1 ) 2 + k x 2 + k y 2 δ 2 β 1 k z 2 δ 2 ( β 1 + 1 ) ] t } .
B L = B L ( k x , k y , k z , t = 0 ) exp [ i ( α ± α 2 + k ̂ 2 δ 2 ) t ] .
B G = B G ( k x , k y , k z , t = 0 ) exp { i [ ( α k z ) ± α 2 + k ̂ 2 δ 2 ] t } .
A ( x , y , z , t ) = F 1 [ A L ( k x , k y , k z , t = 0 ) ] F 1 ( exp { i [ α β 1 + 1 ± α 2 ( β 1 + 1 ) 2 + k ̂ 2 δ 2 ( β 1 + 1 ) ] t } ) .
A ( x , y , z , t ) = F 1 [ A G ( k x , k y , k z , t = 0 ) ] F 1 [ exp ( i { α ( β 1 + 1 ) k z β 1 + 1 ± [ α ( β 1 + 1 ) k z ] 2 ( β 1 + 1 ) 2 + k x 2 + k y 2 δ 2 β 1 k z 2 δ 2 ( β 1 + 1 ) } t ) ] .
V ( x , y , z , t ) = F 1 [ B L ( k x , k y , k z , t = 0 ) ] F 1 { exp [ i ( α ± α 2 + k ̂ 2 δ 2 ) t ] } .
V ( x , y , z , t ) = F 1 [ B G ( k x , k y , k z , t = 0 ) ] F 1 ( exp { i [ ( α k z ) ± α 2 + k ̂ 2 δ 2 ] t } ) .
z diff pulse = α δ 2 z diff beam = k 0 2 r 4 z 0 .
A = A x x , α = 785 , δ 2 = r 2 z 0 2 = 1 , β 1 = 2.1 × 10 5 ;
A x ( x , y , z , t = 0 ) = exp ( x 2 + y 2 + z 2 2 ) .
A = A x x , α = 78.5 , δ 2 = r 2 z 0 2 = 100 ,
A x ( x , y , z , t = 0 ) = exp ( x 2 + y 2 + z 2 2 ) .
A ( x , y , z , t ) = 1 ( 2 π ) 3 exp [ ( k x 2 + k y 2 + k z 2 ) 2 ] × exp { i [ α + k x 2 + k y 2 + ( k z α ) 2 ] t } × exp [ i ( x k x ) ] exp [ i ( y k y ) ] exp [ i ( z k z ) ] d k x d k y d k z .
A ( x , y , z , t ) = 1 ( 2 π ) 3 exp [ α 2 2 + i α ( t z ) ] exp [ ( k x 2 + k y 2 + k ̂ z 2 ) 2 ] exp ( i k x 2 + k y 2 + k ̂ 2 ) t × exp [ i ( x k x ) ] exp [ i ( y k y ) ] exp { i [ ( z i α ) k ̂ z ] } d k x d k y d k ̂ z .
r ̂ = x 2 + y 2 + ( z i α ) 2 = r 2 2 i α z α 2 ,
A ( x , y , z , t ) = 1 ( 2 π ) 3 exp [ α 2 2 + i α ( t z ) ] 1 r ̂ k r exp ( k r 2 2 ) × exp ( i k r t ) sin ( r ̂ k r ) d k r .
A x ( x , y , z , t ) = i 2 r ̂ exp [ α 2 2 + i α ( t z ) ] × { i ( t + r ̂ ) exp [ 1 2 ( i t + i r ̂ ) 2 ] erfc [ i 2 2 ( t + r ̂ ) ] i ( t r ̂ ) exp [ 1 2 ( i t i r ̂ ) 2 ] erfc [ i 2 2 ( t r ̂ ) ] } .
E ( x , y , z , t ) = x A x ( x , y , z , t ) exp [ i ( k 0 z ω 0 t ) ] .
A ( x , y , z , t ) = 1 2 π 3 exp [ α 2 2 + i α ( t z ) ] 1 r ̃ 0 k r exp ( k r 2 2 ) × exp ( i k r t ) sin ( r ̃ k r ) d k r ,
r ̃ = x 2 + y 2 + ( z + t i α ) 2 .
A x ( x , y , z , t ) = i 2 r ̃ exp [ α 2 2 + i α ( t z ) ] × { i ( t + r ̃ ) exp [ 1 2 ( i t + i r ̃ ) 2 ] erfc [ i 2 2 ( t + r ̃ ) ] i ( t r ̃ ) exp [ 1 2 ( i t i r ̃ ) 2 ] erfc [ i 2 2 ( t r ̃ ) ] } .

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