Abstract

We present a rigorous, but mathematically relatively simple and elegant, theory of first-order spatio-temporal distortions, that is, couplings between spatial (or spatial-frequency) and temporal (or frequency) coordinates, of Gaussian pulses and beams. These distortions include pulse-front tilt, spatial dispersion, angular dispersion, and a less well-known distortion that has been called “time vs. angle.” We write pulses in four possible domains, xt, xω, kω, and kt; and we identify the first-order couplings (distortions) in each domain. In addition to the above four “amplitude” couplings, we identify four new spatio-temporal “phase” couplings: “wave-front rotation,” “wave-front-tilt dispersion,” “angular temporal chirp,” and “angular frequency chirp.” While there are eight such couplings in all, only two independent couplings exist and are fundamental in each domain, and we derive simple expressions for each distortion in terms of the others. In addition, because the dimensions and magnitudes of these distortions are unintuitive, we provide normalized, dimensionless definitions for them, which range from -1 to 1. Finally, we discuss the definitions of such quantities as pulse length, bandwidth, angular divergence, and spot size in the presence of spatio-temporal distortions. We show that two separate definitions are required in each case, specifically, “local” and “global” quantities, which can differ significantly in the presence of spatio-temporal distortions.

© 2005 Optical Society of America

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References

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Appl. Opt. (1)

Appl. Phys. B-Lasers and Optics (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 and Optics B74[Suppl], 259-263 (2002).
[CrossRef]

C. Dorrer, E. M. Kosik, and I. A. Walmsley, "Spatio-temporal characterization of ultrashort optical pulses using two-dimensional shearing interferometry," Appl. Phys. B-Lasers and Optics 74 [suppl.], 209-219 (2002).
[CrossRef]

Appl. Phys. B-Lasers and Optics B (1)

I. Z. Kozma, G. Almasi, and J. Hebling, "Geometrical optical modeling of femtosecond setups having angular dispersion," Appl. Phys. B-Lasers and Optics B 76, 257-261 (2003).

IEEE J. Quantum Electron. (1)

D. J. Kane and R. Trebino, "Characterization of Arbitrary Femtosecond Pulses Using Frequency Resolved Optical Gating," IEEE J. Quantum Electron. 29, 571-579 (1993).
[CrossRef]

IEEE J. Quantum. Electron. (4)

O. E. Martinez, "Matrix formalism for pulse compressors," IEEE J. Quantum. Electron. 24, 2530-2536 (1988).
[CrossRef]

O. E. Martinez, "Matrix Formalism for Dispersive Laser Cavities," IEEE J. Quantum. Electron. 25, 296-300 (1989).
[CrossRef]

S. P. Dijaili, A. Dienes, and J. S. Smith, "ABCD Matrices for dispersive pulse propagation," IEEE J. Quantum. Electron. 26, 1158-1164 (1990).
[CrossRef]

A. G. Kostenbauder, "Ray-Pulse Matrices: A Rational Treatment for Dispersive Optical Systems," IEEE J. Quantum. Electron. 26, 1148-1157 (1990).
[CrossRef]

IEEE JSTQE (1)

K. Osvay, A. Kovacs, Z. Heiner, G. Kurdi, J. Klebniczki, and M. Csatari, "Angular Dispersion and Temporal Change of Femtosecond Pulses From Misaligned Pulse Compressors," IEEE JSTQE 10(1), 213-220 (2004).

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

Opt. Commun. (3)

M. A. Larotonda and A. A. Hnilo, "Short laser pulse parameters in a nonlinear medium: different approximations of the ray-pulse matrix," Opt. Commun. 183, 207-213 (2000).
[CrossRef]

O. E. Martinez, "Pulse distortions in tilted pulse schemes for ultrashort pulses," Opt. Commun. 59(3), 229-232 (1986).
[CrossRef]

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

Opt. Engineering (1)

Z. Bor, B. Racz, G. Szabo, M. Hilbert, and H. A. Hazim, "Femtosecond pulse front tilt caused by angular dispersion," Opt. Engineering 32(10), 2501-2503 (1993).
[CrossRef]

Opt. Express (4)

Opt. Lett. (2)

Opt. Quantum Electron. (1)

Q. Lin and S. Wang, "Spatial-temporal coupling in a grating-pair pulse compression system analysed by matrix optics," Opt. Quantum Electron. 27, 785-798 (1995).
[CrossRef]

Opt. Quantum Eng. (1)

J. Hebling, "Derivation of pulse-front tilt casued by angular dispersion," Opt. Quantum Eng. 28, 1759-1763 (1996).
[CrossRef]

Other (4)

M. Born and E. Wolf, Principles of Optics: Electromagnetic Theory of Propagation, Interference and Diffraction of Light (Cambridge Univ Pr, 1999).

A. E. Siegman, Lasers (Univ Science Books, 1986).

I. S. Gradshteyn and I. M. Ryzhik, Table of integrals, series and products (Academic Press, 1994).

R. Trebino, Frequency-Resolved Optical Gating (Kluwer Academic Publishers, Boston, 2002).
[CrossRef]

Supplementary Material (2)

» Media 1: AVI (2320 KB)     
» Media 2: AVI (2096 KB)     

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Figures (4)

Fig. 1.
Fig. 1.

The effect of the WFR. The movie shows the wave fronts in (x,t) domain as a function of time and position (file size 2.26 MB).

Fig. 2.
Fig. 2.

The effect of the WFD. The movie shows the wave fronts in (x,ω) as a function of frequency and position (file size 2.06 MB).

Fig. 3.
Fig. 3.

Intensity profile of a pulse expressed in the four different domains. This pulse simultaneously has all four spatio-temporal amplitude couplings: PFT, SPC, AGD and TVA, as can be seen from the tilted images.

Fig. 4.
Fig. 4.

Intensity profile of a pulse expressed in the four different domains. This pulse has significant PFT and SPC but very small AGD and TVA, as can be seen from the tilt of the traces.

Tables (3)

Tables Icon

Table 1. Summary of the relations of spatio-temporal distortions in all four domains.

Tables Icon

Table 2 The correlation coefficients of the pulse field shown in Figure 3.

Tables Icon

Table 3 The correlation coefficients of the pulse field shown in Fig. 4.

Equations (86)

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E ( x , t ) = exp { i π λ 0 ( x t ) T Q 1 ( x t ) } =
exp [ i π λ 0 ( ( Q 1 ) 11 x 2 + ( Q 1 ) 12 xt ( Q 1 ) 21 xt ( Q 1 ) 22 t 2 ) ]
K = [ x out x in x out θ in 0 x out v in x out x in θ out θ in 0 θ out v in t out x in t out θ in 1 t out v in 0 0 0 1 ] = [ A B 0 E C D 0 F G H 1 I 0 0 0 1 ]
Q out = { [ A 0 G 1 ] Q in + [ B E λ 0 H I λ 0 ] } · { [ C 0 0 0 ] Q in + [ D F λ 0 0 1 ] } 1
E ( x , t ) exp { Q ˜ xx x 2 + 2 Q ˜ xt xt Q ˜ tt t 2 }
Q in = [ Q 11 Q 12 Q 12 Q 22 ] = i λ 0 π [ Q ˜ xx Q ˜ xt Q ˜ xt Q ˜ tt ] 1
Q ˜ xx = i π λ 0 R ( z ) 1 w 2 ( z )
Q ˜ tt = + 1 τ 2
E ( x , t ) exp { Q ˜ xx x 2 + 2 Q ˜ xt xt Q ˜ tt t 2 }
Re { Q ˜ xx } beam spot size ( BSS )
Im { Q ˜ xx } wave front curvature ( WFC )
Re { Q ˜ tt } temporal pulse width ( TPW )
Im { Q ˜ tt } temporal chirp ( TCH )
Re { Q ˜ xt } pulse front tilt ( PFT )
Im { Q ˜ xt } wave front rotation ( WER )
E ( x , ω ) = 1 2 π E ( x , t ) e iωt dt
exp ( p 2 x 2 ± qx ) dx = exp ( q 2 4 p ) π p
E ( x , ω ) exp { R xx x 2 + 2 R R ωω ω 2 }
R xx = Q ˜ xx + Q ˜ xt Q ˜ tt
R = i 2 Q ˜ xt Q ˜ tt
R ωω = 1 4 Q ˜ tt
Re { R xx } beam spot size ( BSS )
Im { R xx } wave front curvature ( WFC )
Re { R ωω } band width ( BDW )
Im { R ωω } frequency chirp ( FCH )
Re { R } spatial chirp ( SPC )
Im { R } wave front tilt dispersion ( WFD )
E ( k , ω ) exp { S kk k 2 + 2 S S ωω ω 2 }
S kk = 1 4 R xx
S = i 2 R R xx
S ωω = R ωω + R 2 R xx
[ Q ˜ ] = [ Q ˜ xx Q ˜ xt Q ˜ xt Q ˜ tt ]
[ S ] = [ S kk S S S ωω ]
[ S ] = 1 4 [ Q ˜ T ] 1
Re { S kk } angular div ergence ( ADV )
Im { S kk } angular phase front curvature ( APC )
Re { S ωω } band width ( BDW )
Im { S ωω } frequency chirp ( FCH )
Re { S } angular dispersion ( AGD )
Im { S } angular spectral chirp ( ASC )
E ( k , t ) exp { P kk k 2 + 2 P kt kt P tt t 2 }
P kk = S kk + S 2 S ωω
P kt = i 2 S S ωω
P tt = 1 4 S ωω
[ P ] = 1 4 [ R T ] 1
[ R ] = [ R xx R R R ωω ] [ P ] = [ P kk P kt P kt P tt ]
Re { P kk } angular div ergence ( ADV )
Im { P kk } angular phase front curvature ( APC )
Re { P tt } temporal pulse width ( TPW )
Im { P tt } temporal chirp ( TCH )
Re { P kt } time vs . angle ( TVA )
Im { P kt } angular temporal chirp ( ATC )
Δ x = [ x 2 I ( x , t ) dx ( xI ( x , t ) dx ) 2 I ( x , t ) dx ] 1 2
Δ x L ( t ) = [ x 2 I ( x , t ) dx ( xI ( x , t ) dx ) 2 I ( x , t ) dx ] 1 2
Δ x G = [ x 2 I ( x , t ) dxdt I ( x , t ) dxdt ] 1 2
Δ x L = 1 2 [ 1 Q ˜ xx R ] 1 2
Δ x G = 1 2 [ Q ˜ tt R Q ˜ xx R Q ˜ tt R + Q ˜ xt R 2 ] 1 2
Δ t L = 1 2 [ 1 Q ˜ tt R ] 1 2
Δ t G = 1 2 [ Q ˜ xx R Q ˜ xx R Q ˜ tt R + Q ˜ xt R 2 ] 1 2
E ( x , ω ) exp [ ( x x 0 ω ω 4 Δ x L ) 2 ω 2 4 Δ ω G 2 ]
E ( x , ω ) exp [ R xx R x 2 + 2 R R R ωω R ω 2 ]
= exp [ R xx R ( x + R R R xx R ω ) 2 ( R R 2 R xx R + R ωω R ) ω 2 ]
Δ t L = 1 2 [ 1 R xx R ] = w 2
x 0 ω = R R R xx R
Δ ω G = 1 2 ( R xx R R xx R 2 + R ωω R R xx R ) 1 2
E ( x , ω ) = exp [ x 2 4 Δ x G 2 ( ω ω 0 x x 4 Δ ω L ) 2 ]
= exp [ ( R R 2 R ωω R + R xx R ) x 2 R ωω R ( ω R R R ωω R x ) 2 ]
Δ x G = 1 2 ( R ωω R R R 2 + R ωω R R xx R ) 1 2
ω 0 x = R R R ωω R
Δ ω L = 1 2 [ 1 R ωω R ] 1 2
ω 0 x = x 0 ω ( x 0 ω ) 2 + Δ x L 2 Δ ω G 2
ρ ∫∫ xωI ( x , ω ) dxdω ∫∫ I ( x , ω ) dxdω 1 Δ x G Δ ω G
ρ = R R R xx R R ωω R
Δ x L ( ω ) = Δ x G 1 ρ 2
Δ ω L ( x ) = Δ ω G 1 ρ 2
Q ˜ xt R = 1 2 R ωω 2 [ R ωω R R I + R ωω I R R ]
PET = 2 WFD + 2 FCH × FRG
PET = AGD + 2 FCH × FRG
SPD = 2 ASC 2 AGD × APC
AGD = 2 ATC + 2 TVA × TCH
TVA = 2 WFR + 2 PFT × WFC
Q ˜ xt = i 2 FRG + Q ˜ tt PFT
WFR = FRG 2 + TCH × PET
WFD = AGD 2 + SPD × WFC
ASC = TVA 2 + AGD × FCH
ATC = PFT 2 + TVA × APC

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