Abstract

We derive a general analytical expression for the width of a femtosecond laser pulse after passing through an angular dispersion device, valid for the plane wave, spherical wave, and Gaussian beam models. This expression is a simple function of two effects: spectral lateral walkoff and group delay dispersion. Plane waves and spherical waves experience no spectral lateral walkoff, as the beams are not explicitly limited in space. The group delay dispersion (GDD) of a Gaussian beam is similar to that of a plane wave at distances much less than the Rayleigh length, and similar to that of the spherical wave at distances far exceeding the Rayleigh length. The GDD of the spherical wave and Gaussian beam approach a constant value at large distances, which is entirely determined by the dispersion parameters of the optical component and the distance between disperser and source point or beam waist. The width of a plane wave pulse always increases in proportion to the propagation distance. The width of a spherical wave or Gaussian beam pulse widens rapidly at first, but soon levels off to nearly constant value.

© 2008 Optical Society of America

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References

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2007

2005

2004

S. Akturk, Xun Gu, Erik Zeek, and Rick Trebino, "Pulse-front tilt caused by spatial and temporal chirp," Opt. Express 12, 4399-4410 (2004).
[CrossRef] [PubMed]

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

K. Osvay, A. P. Kovács, Z. Heiner, G. Kurdi, J. Klebniczki, and M. Csatári, "Angular dispersion and temporal change of femtosecond pulses from misaligned pulse compressors," IEEE J. Sel. Top. Quantum. Electron. 10, 213-220 (2004).
[CrossRef]

2003

2002

K. Varjú, A. P. Kovács, K. Osvay, and G. Kurdi, "Angular dispersion of femtosecond pulses in a Gaussian beam," Opt. Lett. 27, 2034-2036 (2002).
[CrossRef]

K. Varjú, A. P. Kovács, G. Kurdi, and K. Osvay, "High-precision measurement of angular dispersion in a CPA laser," Appl. Phys, B 74, 259-263 (2002).
[CrossRef]

1996

1993

Z. L. Horváth, Z. Benkö, A. P. Kovács, H. A. Hazim, and Z. Bor, "Propagation of femtosecond pulses through lenses, gratings, and slits," Opt. Eng. 32, 2491-2500 (1993).
[CrossRef]

1990

1986

O. E. Martinez, "Grating and prism compressors in the case of finite beam size," J. Opt. Soc. Am. B 3, 929-934 (1986).
[CrossRef]

O. E. Martinez, "Pulse distortions in tilted pulse schemes for ultrashort pulses," Opt. Commun. 59, 229-232 (1986).
[CrossRef]

1984

1969

E. B. Treacy, "Optical pulse compression with diffraction gratings," IEEE J. Quantum Electron. QE-5, 454-458 (1969).
[CrossRef]

Appl. Opt.

Appl. Phys, B

K. Varjú, A. P. Kovács, G. Kurdi, and K. Osvay, "High-precision measurement of angular dispersion in a CPA laser," Appl. Phys, B 74, 259-263 (2002).
[CrossRef]

IEEE J. Quantum Electron.

E. B. Treacy, "Optical pulse compression with diffraction gratings," IEEE J. Quantum Electron. QE-5, 454-458 (1969).
[CrossRef]

IEEE J. Sel. Top. Quantum. Electron.

K. Osvay, A. P. Kovács, Z. Heiner, G. Kurdi, J. Klebniczki, and M. Csatári, "Angular dispersion and temporal change of femtosecond pulses from misaligned pulse compressors," IEEE J. Sel. Top. Quantum. Electron. 10, 213-220 (2004).
[CrossRef]

J. Opt. Soc. Am. A

J. Opt. Soc. Am. B

Opt. Commun.

O. E. Martinez, "Pulse distortions in tilted pulse schemes for ultrashort pulses," Opt. Commun. 59, 229-232 (1986).
[CrossRef]

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

Opt. Eng.

Z. L. Horváth, Z. Benkö, A. P. Kovács, H. A. Hazim, and Z. Bor, "Propagation of femtosecond pulses through lenses, gratings, and slits," Opt. Eng. 32, 2491-2500 (1993).
[CrossRef]

Opt. Express

Opt. Lett.

Other

J. C. Diels and W. Rudolph, Ultrashort Laser Pulse Phenomena (Academic, San Diego, Calif., 1996).

A. E. Siegman, Lasers, (University Science, Mill Valley, CA, 1986).

M. Born and E. Wolf, Principles of Optics, 6th ed. (Pergamon, Oxford, 1980).

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Figures (4)

Fig. 1.
Fig. 1.

A plane wave passes through an angular dispersion device, represented by a vertical line passing through point O. After the dispersion, each spectral component propagates in a different direction. We may freely define the z-axis as the propagation direction of the central frequency component (ω 0).Some other frequency ω will then propagate along the zω direction, where the angle between the two directions is denoted ε.The angle ε′ between their two wave fronts is equivalent to ε. We also define the distances PO ¯ = d and OQ ¯ = z .

Fig. 2.
Fig. 2.

A spherical wave passes through an angular dispersion component. The angle between spectral components ω 0 and ω is ε. The angle between their wave fronts ε′, however, is less than ε. We also have PO ¯ = d , OQ ¯ = z . d′=d″=d/α 2. P represents the source of the spherical wave.

Fig. 3.
Fig. 3.

GDD of the plane wave, spherical wave, and Gaussian beam as a function of distance.

Fig. 4.
Fig. 4.

Pulse width as a function of distance for the plane wave, spherical wave, and Gaussian beam after passing through an angular dispersion component.

Equations (34)

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τ P = τ 0 [ 1 + ( 4 ln 2 ) 2 ( k β 2 z ) 2 τ 0 4 ] 1 2 ,
τ S = τ 0 [ 1 + ( 4 ln 2 ) 2 ( d d + α 2 z k β 2 z ) 2 τ 0 4 ] 1 2 ,
τ G = τ 0 [ 1 + 2 ln 2 z R 2 ( d + α 2 z ) 2 + z R 2 ( 2 α β z w 0 τ 0 ) 2 + ( 4 ln 2 ) 2 [ ( d + α 2 z ) d + z R 2 ( d + α 2 z ) 2 + z R 2 k β 2 z ] 2 [ 1 + 2 ln 2 z R 2 ( d + α 2 z ) 2 + z R 2 ( 2 α β z w 0 τ 0 ) 2 ] τ 0 4 ] 1 2 ,
τ = τ 0 [ ( 1 + U ) + V 2 ( 1 + U ) ] 1 2 ,
U P = 0 .
V P = 4 ln 2 · k β 2 z τ 0 2 .
U S = 0 .
V S = 4 ln 2 · d d + α 2 z k β 2 z τ 0 2 .
U G = 2 ln 2 z R 2 ( d + α 2 z ) 2 + z R 2 ( 2 α β z w 0 τ 0 ) 2 .
V G = 4 ln 2 · ( d + α 2 z ) d + z R 2 ( d + α 2 z ) 2 + z R 2 k β 2 z τ 0 2 .
Δ θ ( γ , ω ) = α Δ γ + β Δ ω = θ γ Δ γ + θ ω Δ ω .
A i ( x 1 , ω ) = a ( ω ) a ( x 1 ) .
a ( ω ) = exp [ τ 0 2 8 ln 2 ω 2 ]
A d ( x 2 , ω ) = exp ( ik β ω x 2 ) A i ( α x 2 , ω ) ,
A z ( x 3 , z , ω ) = i λ z + A d ( x 2 , 0 , ω ) exp [ i π λ z ( x 3 x 2 ) 2 ] dx 2 .
A z ( x , z , t ) = + A z ( x 3 , z , ω ) exp ( i ω t ) d ω .
A ( x ω , z ω , ω ) = exp [ ρ ] exp [ i ϕ ] .
z ω = z cos ε + x sin ε ,
x ω = z sin ε + x cos ε .
z ω 0
x ω ' z d ε d ω
z ω z ( d ε d ω ) 2
x ω z ( d 2 ε d ω 2 )
β = θ ω = d ε d ω
A ( x ω , z ω , ω ) = exp [ ik ( d + z ω ) ] .
ρ P = 0
ϕ P = k β 2 z
A ( x ω , z ω , ω ) = 1 x ω 2 + ( d α 2 + z ω ) 2 exp [ ik x ω 2 + ( d α 2 + z ω ) 2 ] .
ρ S = 0
ϕ S = d d + α 2 z k β 2 z .
A ( x ω , z ω , ω ) = 1 w ( z ω + d α 2 ) exp [ x ω 2 w 2 ( z ω + d α 2 ) ]
× exp [ ik ( x ω 2 2 R ( z ω + d α 2 ) + z ω + d α 2 ) + i tan 1 ( z ω + d α 2 z R ) ]
ρ G = 2 z R 2 ( d + α 2 z ) 2 + z R 2 ( α β z w 0 ) 2
ϕ G = ( d + α 2 z ) d + z R 2 ( d + α 2 z ) 2 + z R 2 k β 2 z

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