Abstract

A second-order correlation coefficient is defined to estimate the severity of second-order spatiotemporal couplings. The correlation coefficient is scale invariant, normalized, dimensionless, etc. Using this correlation coefficient, the spatial chirp of ultrashort pulsed Gaussian beams is studied. It can be shown that the spatial chirp of ultrashort pulsed Gaussian beams depends on the pulse width, spectrum shape, and local frequency shift characteristics, and different types of ultrashort pulsed Gaussian beams exhibit different second-order spatiotemporal couplings characteristics.

© 2009 Optical Society of America

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References

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    [CrossRef]
  4. Z. Wang, Z. Zhang, Z. Xu, and Q. Lin, “Space-time profiles of an ultrashort pulsed gaussian beam,” IEEE J. Quantum Electron. 33, 566-573 (1997).
    [CrossRef]
  5. M. A. Porras, “Pulse correction to monochromatic light-beam propagation,” Opt. Lett. 26, 44-46 (2001).
    [CrossRef]
  6. M. A. Porras, “Diffraction effects in few-cycle optical pulses,” Phys. Rev. E 65, 026606 (2002).
    [CrossRef]
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    [CrossRef] [PubMed]
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    [CrossRef] [PubMed]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef] [PubMed]

2008 (1)

2007 (1)

2005 (1)

2004 (2)

2003 (2)

2002 (4)

K. Varju, A. P. Kovacs, and K. Osvay, “Angular dispersion of femtosecond pulses in a Gaussian beam,” Opt. Lett. 27, 2034-2036 (2002).
[CrossRef]

K. Varju, A. P. Kovacs, G. Kurdi, and K. Osvay, “High-precision measurement of angular dispersion in a CPA laser,” Appl. Phys. B: Lasers Opt. B74, 259-263 (2002).
[CrossRef]

C. Dorrer, E. M. Kosik, and I. A. Walmsley, “Spatiotemporal characterization of the electric field of ultrashort optical pulses using two-dimensional shearing interferometry,” Appl. Phys. B: Lasers Opt. 74, 209-217 (2002).
[CrossRef]

M. A. Porras, “Diffraction effects in few-cycle optical pulses,” Phys. Rev. E 65, 026606 (2002).
[CrossRef]

2001 (1)

1999 (1)

G. P. Agrawal, “Far-field diffraction of pulsed optical beams in dispersive media,” Opt. Commun. 167, 15-22 (1999).
[CrossRef]

1998 (1)

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

1997 (1)

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

1992 (1)

1986 (1)

Agrawal, G. P.

S. A. Ponomarenko and G. P. Agrawal, “Phase-space quality factor for ultrashort pulsed beams,” Opt. Lett. 33, 767-769 (2008).
[CrossRef] [PubMed]

G. P. Agrawal, “Far-field diffraction of pulsed optical beams in dispersive media,” Opt. Commun. 167, 15-22 (1999).
[CrossRef]

Akturk, S.

Apostol, T. M.

T. M. Apostol, Mathematical Analysis (Addison-Wesley, 1974).

Diels, J. C.

J. C. Diels and W. Rudolph, Ultrashort Laser Pulse Phenomena, 2nd ed. (Academic, 2006).

Dorrer, C.

C. Dorrer, E. M. Kosik, and I. A. Walmsley, “Spatiotemporal characterization of the electric field of ultrashort optical pulses using two-dimensional shearing interferometry,” Appl. Phys. B: Lasers Opt. 74, 209-217 (2002).
[CrossRef]

Gabolde, P.

Gu, X.

Judkins, J. B.

Kimmel, M.

Kosik, E. M.

C. Dorrer, E. M. Kosik, and I. A. Walmsley, “Spatiotemporal characterization of the electric field of ultrashort optical pulses using two-dimensional shearing interferometry,” Appl. Phys. B: Lasers Opt. 74, 209-217 (2002).
[CrossRef]

Kovacs, A. P.

K. Varju, A. P. Kovacs, G. Kurdi, and K. Osvay, “High-precision measurement of angular dispersion in a CPA laser,” Appl. Phys. B: Lasers Opt. B74, 259-263 (2002).
[CrossRef]

K. Varju, A. P. Kovacs, and K. Osvay, “Angular dispersion of femtosecond pulses in a Gaussian beam,” Opt. Lett. 27, 2034-2036 (2002).
[CrossRef]

Kurdi, G.

K. Varju, A. P. Kovacs, G. Kurdi, and K. Osvay, “High-precision measurement of angular dispersion in a CPA laser,” Appl. Phys. B: Lasers Opt. B74, 259-263 (2002).
[CrossRef]

Lee, D.

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, 566-573 (1997).
[CrossRef]

Martinez, O. E.

O'Shea, P.

Osvay, K.

K. Varju, A. P. Kovacs, and K. Osvay, “Angular dispersion of femtosecond pulses in a Gaussian beam,” Opt. Lett. 27, 2034-2036 (2002).
[CrossRef]

K. Varju, A. P. Kovacs, G. Kurdi, and K. Osvay, “High-precision measurement of angular dispersion in a CPA laser,” Appl. Phys. B: Lasers Opt. B74, 259-263 (2002).
[CrossRef]

Ponomarenko, S. A.

Porras, M. A.

M. A. Porras, “Diffraction effects in few-cycle optical pulses,” Phys. Rev. E 65, 026606 (2002).
[CrossRef]

M. A. Porras, “Pulse correction to monochromatic light-beam propagation,” Opt. Lett. 26, 44-46 (2001).
[CrossRef]

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

Rudolph, W.

J. C. Diels and W. Rudolph, Ultrashort Laser Pulse Phenomena, 2nd ed. (Academic, 2006).

Svelto, O.

O. Svelto, Principles of Lasers, 4th ed. (Plenum, 1998).

Trebino, R.

Varju, K.

K. Varju, A. P. Kovacs, and K. Osvay, “Angular dispersion of femtosecond pulses in a Gaussian beam,” Opt. Lett. 27, 2034-2036 (2002).
[CrossRef]

K. Varju, A. P. Kovacs, G. Kurdi, and K. Osvay, “High-precision measurement of angular dispersion in a CPA laser,” Appl. Phys. B: Lasers Opt. B74, 259-263 (2002).
[CrossRef]

Walmsley, I. A.

C. Dorrer, E. M. Kosik, and I. A. Walmsley, “Spatiotemporal characterization of the electric field of ultrashort optical pulses using two-dimensional shearing interferometry,” Appl. Phys. B: Lasers Opt. 74, 209-217 (2002).
[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, 566-573 (1997).
[CrossRef]

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, 566-573 (1997).
[CrossRef]

Zeek, E.

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, 566-573 (1997).
[CrossRef]

Ziolkowski, R. W.

Appl. Phys. B: Lasers Opt. (2)

K. Varju, A. P. Kovacs, G. Kurdi, and K. Osvay, “High-precision measurement of angular dispersion in a CPA laser,” Appl. Phys. B: Lasers Opt. B74, 259-263 (2002).
[CrossRef]

C. Dorrer, E. M. Kosik, and I. A. Walmsley, “Spatiotemporal characterization of the electric field of ultrashort optical pulses using two-dimensional shearing interferometry,” Appl. Phys. B: Lasers Opt. 74, 209-217 (2002).
[CrossRef]

IEEE J. Quantum Electron. (1)

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

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

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

Opt. Commun. (2)

X. Gu, S. Akturk, and R. Trebino, “Spatial chirp in ultrafast optics,” Opt. Commun. 242, 599-604 (2004).
[CrossRef]

G. P. Agrawal, “Far-field diffraction of pulsed optical beams in dispersive media,” Opt. Commun. 167, 15-22 (1999).
[CrossRef]

Opt. Express (5)

Opt. Lett. (3)

Phys. Rev. E (2)

M. A. Porras, “Diffraction effects in few-cycle optical pulses,” Phys. Rev. E 65, 026606 (2002).
[CrossRef]

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

Other (3)

J. C. Diels and W. Rudolph, Ultrashort Laser Pulse Phenomena, 2nd ed. (Academic, 2006).

T. M. Apostol, Mathematical Analysis (Addison-Wesley, 1974).

O. Svelto, Principles of Lasers, 4th ed. (Plenum, 1998).

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Figures (6)

Fig. 1
Fig. 1

Variation of relative frequency shift with axial distance z along several caustic surfaces. The values are calculated for a few-cycle Gaussian pulse of Gaussian envelope, for which Δ ω r m s ω 0 = 0.15 .

Fig. 2
Fig. 2

Variation of second-order correlation coefficient with axial distance z. The source is E ( x , t ) = exp ( t 2 b 2 ) exp ( i ω 0 t ) exp ( x 2 s 2 ) with ω 0 = 3.2 fs 1 , (period T 0 = 1.96 fs ), b = 1.67 fs (FWHM of intensity Δ t = 2 In 2 b = T 0 ), and s = 20 μ m ( L ω 0 = 2.1 mm ) .

Fig. 3
Fig. 3

Variation of second-order correlation coefficient with pulse width Δ t . z = 10 L ω 0 , and the other parameters are the same as in Fig. 2.

Fig. 4
Fig. 4

Pulse spectra as functions of ω ω * for five values of q, and ω 1 ω * = 10 , ω * = 0.32 fs 1 .

Fig. 5
Fig. 5

Variation of relative frequency shift with axial distance z along several caustic surfaces. The values are calculated for a pulsed Gaussian beam E ̂ ω ( x ) = p ̂ ω exp ( x 2 s 2 ) for which p ̂ ω is described by Eq. (29), q = 1 , s 1 = 20 μ m , ω 1 ω * = 10 , ω * = 0.32 fs 1 .

Fig. 6
Fig. 6

Second-order correlation coefficient varies with the value of q, and other parameters are the same as in Fig. 5.

Equations (28)

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ω 0 ( x , z ) = 0 d ω ω I ( x , ω ) 0 d ω I ( x , ω ) ,
ω ¯ = d x 0 d ω ω I ( x , ω ) d x 0 d ω I ( x , ω ) ,
ω 0 ( x , z ) x m ,
ρ x ω = d x d ω I ( x , ω ) x ω Δ x Δ ω ,
Δ x = [ x 2 I ( x , ω ) d x d ω ] 1 2 ,
Δ ω = [ ω 2 I ( x , ω ) d x d ω ] 1 2 ,
σ x ω = d x 0 I ( x , ω ) x 2 ( ω ω ¯ ) d ω Δ x Δ ω ,
Δ x = [ d x 0 x 4 I ( x , ω ) d ω ] 1 2 ,
Δ ω = [ d x 0 ( ω ω ¯ ) 2 I ( x , ω ) d ω ] 1 2 ,
1 < σ x ω < 1 .
E ̂ ω ( x ) = p ̂ ω exp ( x 2 s 2 ) ,
a ω ( x , z ) = s s ω ( z ) exp [ x 2 s ω 2 ( z ) ] ,
s ω ( z ) = s ( 1 + z 2 L ω 2 ) 1 2 ,
I ( x , ω ) = | p ̂ ω | 2 | a ω ( x , z ) | 2 .
ω 0 = 0 d ω ω | p ̂ ω | 2 0 d ω | p ̂ ω | 2 .
ω ¯ = ω 0 = ω 0 ( x , z ) ,
E ( x , t ) = p ( t ) exp ( x 2 s 2 ) ,
ω 0 ( x , z ) = ω 0 { 1 + 2 1 + L ω 0 2 z 2 [ 1 2 x 2 s ω 0 2 ( z ) ] ( Δ ω rms ω 0 ) 2 } ,
σ x ω = 3 3 0 | p ̂ ω | 2 ( ω ω ¯ ) M 1 2 d ω [ 0 | p ̂ ω | 2 M 3 2 d ω 0 | p ̂ ω | 2 ( ω ω ¯ ) 2 M 1 2 d ω ] 1 2 = F ( p ̂ ω , z ) ,
M = 1 + z 2 L ω 2 ,
σ x ω = 3 3 0 | p ̂ ω | 2 ( ω ω ¯ ) d ω [ 0 | p ̂ ω | 2 d ω 0 | p ̂ ω | 2 ( ω ω ¯ ) 2 d ω ] 1 2 = J ( p ̂ ω ) .
σ x ω = 3 3 0 | p ̂ ω | 2 ( ω ω ¯ ) ω 1 d ω [ 0 | p ̂ ω | 2 ω 3 d ω 0 | p ̂ ω | 2 ( ω ω ¯ ) 2 ω d ω ] 1 2 = Q ( p ̂ ω ) .
s ( ω ) = s 1 ( ω 1 ω ) 1 2 ,
L ω = ω s 2 2 c = ω 1 s 1 2 2 c L 0 .
a ω ( x , z ) = 1 ( 1 + z 2 L 0 2 ) 1 2 exp [ x 2 ω 2 L 0 c ( 1 + z 2 L 0 2 ) ] .
p ̂ ω { ( ω ω * ) q e α ω if ω ω * 0 if ω ω * } ,
ω 0 ( x , z ) = ω 0 2 x 2 ω 0 s ω 0 2 ( z ) ( Δ ω rms ) 2 .
σ x ω = 3 3 0 | p ̂ ω | 2 ( ω ω ¯ ) ω 3 2 d ω [ 0 | p ̂ ω | 2 ω 5 2 d ω 0 | p ̂ ω | 2 ( ω ω ¯ ) 2 ω 1 2 d ω ] 1 2 = K ( p ̂ ω ) .

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