Abstract

A generalization of type 3 ultrashort pulses (also known as pulse beams or isodiffracting pulses) is introduced. The Bessel beam form of this generalized beam consists of pulses that propagate in free space, without spreading, with a velocity that can be less than that of light. A model spectral distribution that is zero outside a finite range is investigated.

© 2002 Optical Society of America

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References

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  1. E. Heyman, T. Melamed, “Certain considerations in aperture synthesis of ultrawideband/short-pulse radiation,” IEEE Trans. Antennas Propag. 42, 518–525 (1994).
    [Crossref]
  2. C. J. R. Sheppard, X. Gan, “Free-space propagation of femto-second light pulses,” Opt. Commun. 133, 1–6 (1997).
    [Crossref]
  3. Z. Wang, Z. Zhang, Z. Xu, Q. Lin, “Space-time profiles of an ultrashort pulsed beam,” IEEE J. Quantum Electron. 33, 566–573 (1997).
    [Crossref]
  4. M. A. Porras, “Ultrashort pulsed Gaussian light beams,” Phys. Rev. E 58, 1086–1093 (1998).
    [Crossref]
  5. C. F. R. Caron, R. M. Potvliege, “Free-space propagation of ultrashort pulses: space-time couplings in Gaussian pulse beams,” J. Mod. Opt. 45, 1881–1892 (1999).
    [Crossref]
  6. S. Feng, H. G. Winful, “Spatiotemporal structure of isodiffracting ultrashort electromagnetic pulses,” Phys. Rev. E 61, 862–873 (2000).
    [Crossref]
  7. J. N. Brittingham, “Focus wave modes in homogeneous Maxwell’s equations: transverse electric modes,” J. Appl. Phys. 54, 1179–1189 (1983).
    [Crossref]
  8. K. Reivelt, P. Saari, “Optical generation of focused wave modes,” J. Opt. Soc. Am. A 17, 1785–1790 (2000).
    [Crossref]
  9. C. J. R. Sheppard, “Bessel pulse beams and focus wave modes,” J. Opt. Soc. Am. A 18, 2594–2600 (2001).
    [Crossref]
  10. C. J. R. Sheppard, M. D. Sharma, “Spatial frequency content of focused ultra-short pulsed beams,” J. Opt. A Pure Appl. (to be published).
  11. C. W. McCutchen, “Generalized aperture and the three-dimensional diffraction image,” J. Opt. Soc. Am. 54, 240–244 (1964).
    [Crossref]
  12. J. Lu, J. F. Greenleaf, “Nondiffracting X waves—exact solutions to free-space scalar wave equations and their finite aperture realizations,” IEEE Trans. Ultrason. Ferroelectr. Freq. Control 39, 19–31 (1992).
    [Crossref]
  13. P. A. Bélanger, “Packetlike solutions of the homogeneous wave equation,” J. Opt. Soc. Am. A 1, 723–724 (1984).
    [Crossref]
  14. R. W. Ziolkowski, I. M. Besieris, A. M. Shaarawi, “Aperture realizations of exact solutions to homogeneous-wave equations,” J. Opt. Soc. Am. A 10, 75–87 (1993).
    [Crossref]

2001 (1)

2000 (2)

S. Feng, H. G. Winful, “Spatiotemporal structure of isodiffracting ultrashort electromagnetic pulses,” Phys. Rev. E 61, 862–873 (2000).
[Crossref]

K. Reivelt, P. Saari, “Optical generation of focused wave modes,” J. Opt. Soc. Am. A 17, 1785–1790 (2000).
[Crossref]

1999 (1)

C. F. R. Caron, R. M. Potvliege, “Free-space propagation of ultrashort pulses: space-time couplings in Gaussian pulse beams,” J. Mod. Opt. 45, 1881–1892 (1999).
[Crossref]

1998 (1)

M. A. Porras, “Ultrashort pulsed Gaussian light beams,” Phys. Rev. E 58, 1086–1093 (1998).
[Crossref]

1997 (2)

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

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

1994 (1)

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

1993 (1)

1992 (1)

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

1984 (1)

1983 (1)

J. N. Brittingham, “Focus wave modes in homogeneous Maxwell’s equations: transverse electric modes,” J. Appl. Phys. 54, 1179–1189 (1983).
[Crossref]

1964 (1)

Bélanger, P. A.

Besieris, I. M.

Brittingham, J. N.

J. N. Brittingham, “Focus wave modes in homogeneous Maxwell’s equations: transverse electric modes,” J. Appl. Phys. 54, 1179–1189 (1983).
[Crossref]

Caron, C. F. R.

C. F. R. Caron, R. M. Potvliege, “Free-space propagation of ultrashort pulses: space-time couplings in Gaussian pulse beams,” J. Mod. Opt. 45, 1881–1892 (1999).
[Crossref]

Feng, S.

S. Feng, H. G. Winful, “Spatiotemporal structure of isodiffracting ultrashort electromagnetic pulses,” Phys. Rev. E 61, 862–873 (2000).
[Crossref]

Gan, X.

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

Greenleaf, J. F.

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

Heyman, E.

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

Lin, Q.

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

Lu, J.

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

McCutchen, C. W.

Melamed, T.

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

Porras, M. A.

M. A. Porras, “Ultrashort pulsed Gaussian light beams,” Phys. Rev. E 58, 1086–1093 (1998).
[Crossref]

Potvliege, R. M.

C. F. R. Caron, R. M. Potvliege, “Free-space propagation of ultrashort pulses: space-time couplings in Gaussian pulse beams,” J. Mod. Opt. 45, 1881–1892 (1999).
[Crossref]

Reivelt, K.

Saari, P.

Shaarawi, A. M.

Sharma, M. D.

C. J. R. Sheppard, M. D. Sharma, “Spatial frequency content of focused ultra-short pulsed beams,” J. Opt. A Pure Appl. (to be published).

Sheppard, C. J. R.

C. J. R. Sheppard, “Bessel pulse beams and focus wave modes,” J. Opt. Soc. Am. A 18, 2594–2600 (2001).
[Crossref]

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

C. J. R. Sheppard, M. D. Sharma, “Spatial frequency content of focused ultra-short pulsed beams,” J. Opt. A Pure Appl. (to be published).

Wang, Z.

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

Winful, H. G.

S. Feng, H. G. Winful, “Spatiotemporal structure of isodiffracting ultrashort electromagnetic pulses,” Phys. Rev. E 61, 862–873 (2000).
[Crossref]

Xu, Z.

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

Zhang, Z.

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

Ziolkowski, R. W.

IEEE J. Quantum Electron. (1)

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

IEEE Trans. Antennas Propag. (1)

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

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

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

J. Appl. Phys. (1)

J. N. Brittingham, “Focus wave modes in homogeneous Maxwell’s equations: transverse electric modes,” J. Appl. Phys. 54, 1179–1189 (1983).
[Crossref]

J. Mod. Opt. (1)

C. F. R. Caron, R. M. Potvliege, “Free-space propagation of ultrashort pulses: space-time couplings in Gaussian pulse beams,” J. Mod. Opt. 45, 1881–1892 (1999).
[Crossref]

J. Opt. Soc. Am. (1)

J. Opt. Soc. Am. A (4)

Opt. Commun. (1)

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

Phys. Rev. E (2)

M. A. Porras, “Ultrashort pulsed Gaussian light beams,” Phys. Rev. E 58, 1086–1093 (1998).
[Crossref]

S. Feng, H. G. Winful, “Spatiotemporal structure of isodiffracting ultrashort electromagnetic pulses,” Phys. Rev. E 61, 862–873 (2000).
[Crossref]

Other (1)

C. J. R. Sheppard, M. D. Sharma, “Spatial frequency content of focused ultra-short pulsed beams,” J. Opt. A Pure Appl. (to be published).

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Figures (4)

Fig. 1
Fig. 1

Three-dimensional pupil function for generalized pulse beams. For β=1, the pupil function is a paraboloid of revolution. For β less than unity, the pupil function is a prolate spheroid. For β greater than unity, the pupil function is a hyperboloid of two sheets.

Fig. 2
Fig. 2

Model spectral distribution (a) for different values of the parameter n, for a constant value of k1z0=0, and (b) for different values of the parameter k1z0, for a constant value of n=3.

Fig. 3
Fig. 3

Mean of the spectral distribution as a function of k1z0, for different values of the parameter n.

Fig. 4
Fig. 4

Variance of the spectral distribution as a function of k1z0, for different values of the parameter n.

Equations (46)

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U(r, z)=J0(kr sin θ)exp(ikz cos θ)exp(-ikct),
U(r, z, t)=-+f(k)J0(kr sin θ)exp(ikz cos θ)×exp(-ikct)dk,
U(r, z, t)=-f(k)J0(kr sin θ)exp-ikz1β-cos θ×exp-ikct-zβcdk,
k1β-cos θ=kc,
U(r, z, t)=exp(-ikcz)-f(k)J0(kr sin θ)×exp-ikct-zβcdk.
t=t-zβc,
kmin=kc/2.
k<kc.
kmin=βkc1+β
kmax=βkc1-β.
e=(1-β2)-1/2.
U(r, z, t)=exp(-ikcz)kminkmaxf(k)J0k(1+β)β-kc×kc-k(1-β)β1/2rexp(-ikct)dk.
f(k)=1π1/2k1(k1z0)n2nΓ(n+12)In(k1z0)1-k-k0k12n-1/2×exp[-(k-k0)z0],
(k0+k1)>k>(k0-k1),
k1-k0k1+k0f(k)exp(-ikct)dk
=exp(-ik0ct)
×k1z0k1(ct-iz0)nJn(k1(ct-iz0))In(k1z0),
k¯=k0-k1In+1(k1z0)In(k1z0),
σ2=k121-n+122k1z0In+1(k1z0)In(k1z0)-In+1(k1z0)In(k1z0)2.
k12/[2(n+1)].
U(r, z, t)=exp(-ikcz)π1/2k1(k1z0)n2nΓ(n+12)In(k1z0)×k0-k1k0+k11-k-k0k12n-1/2×exp[-(k-k0)z0]×J0k(1+β)β-kc×kc-k(1-β)β1/2r2×exp(-ikct)dk.
k0+k1=kmax=β1-β kc
k0-k1=kmin=kα
k1=kc2β2(1-cos α)(1-β)(1-β cos α)
k0=kc2β(2-β(1+cos α))(1-β)(1-β cos α).
kmaxkα=1+2β1-βsin2α2;
p=k-k0k1,
U(r, z, t)=exp[-i(k0ct+kcz)]π1/2(k1z0)n2nΓ(n+12)In(k1z0)×-11(1-p2)n-1/2J0×(v[(1-p)(p+a)]1/2)×exp[-ik1p(ct-iz0)]dp,
v=k0r sin2(α/2)(1-β2)1/2(1-β)+β sin2(α/2)
a=(3β-1)sin2(α/2)+2(1-β)(1+β)sin2(α/2).
U(0, z, t)=exp-ik0ct-zvcp k1z0k1(ct-iz0)n×Jn(k1(ct-iz0))In(k1z0),
vcp=βc1-β(kc/k0).
kck0=2(1-β)(1-β cos α)β(2-β(1+cos α)),
vcp=2-β(1+cos α)1+(1-2β)cos α c.
k1z0k1(ct-iz0)nJn(k1(ct-iz0))In(k1z0)
=1+i(k0-k¯)ct-(σct)22+. . . .
ϕt=-k0c+k1c In+1(k1z0)In(k1z0).
T=2σc,
N=k¯πσ.
k0+k1=kmax=β1-β kc,
k0-k1=kmin=β1+β kc,
k1=β21-β2 kc,
k0=β1-β2 kc,
U(r, z, t)=exp[-ik0(ct-βz)]π1/2(k1z0)n2nΓ(n+12)In(k1z0)×0πsin2n γJ0(k0r(1-β2)1/2sin γ)×exp[-ik1(ct-iz0)cos γ]dγ.
U(0, z, t)=exp[-ik0(ct-βz)]k1z0k1(ct-iz0)n×Jn(k1(ct-iz0))In(k1z0).
U(r, z, t)=exp[-ik0(ct-βz)]×sin[k0((1-β2)r2+β2c2t2)1/2]k0((1-β2)r2+β2c2t2)1/2.

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