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Pulse-front tilt caused by spatial and temporal chirp

Selcuk Akturk, Xun Gu, Erik Zeek, and Rick Trebino

Open Access Open Access

Abstract

Pulse-front tilt in an ultrashort laser pulse is generally considered to be a direct consequence of, and equivalent to, angular dispersion. We show, however, that, while this is true for certain types of pulse fields, simultaneous temporal chirp and spatial chirp also yield pulse-front tilt, even in the absence of angular dispersion. We verify this effect experimentally using GRENOUILLE.

©2004 Optical Society of America

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Figures (4)
Fig. 1.
Fig. 1. Two sources of pulse-front tilt. Left: The well-known angular dispersion. Right: The combination of spatial and temporal chirp.

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Fig. 2.
Fig. 2. Apparatus to introduce constant spatial chirp, variable temporal chirp and no angular dispersion.

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Fig. 3.
Fig. 3. GRENOUILLE traces of a beam that has a constant spatial chirp and variable temporal chirp. Note that the amount of shift of the center (a measure of pulse-front tilt) increases with increasing temporal chirp.

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Fig. 4.
Fig. 4. Experimental measurements (plus-sign symbols) of pulse-front tilt for different amounts of GDD. The red line shows the linear fit.

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Equations (56)

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E ( x , z , t ) = E xz ( x , z ) E t ( t − px )
E ˜ ̂ ( k x , k z , ω ) = E ˜ ̂ k x k z ( k x − pω , k z ) E ˜ ̂ ω ( ω )
E ( x , ω ) = E ( ω ) exp [ − i k ( x − ζω ) 2 2 q ] exp ( − i k 0 βωx )
q ( z ) = ( z + d ) + i π w 2 λ = ( z + d ) + i k 0 w 2 2
E ( ω ) = E 0 exp ( − ω 2 τ 0 2 4 ) exp ( − i φ ( 2 ) 2 ω 2 )
q ( z ) ≈ q 0 ( d ) = i k w 2 2
E ( x , ω ) = E 0 exp ( − ω 2 τ 0 2 4 ) exp [ − ( x − ζω ) 2 w 2 ] exp ( − i φ ( 2 ) 2 ω 2 ) exp ( − i k 0 βωx )
υ = ζ ζ 2 + w 2 τ 0 2 4
E ( x , ω ) = E 0 exp [ − ( x w′ ) 2 ] exp ( − ( τ′ ) 2 4 ( ω − υx ) 2 ) exp ( − i φ ( 2 ) 2 ω 2 ) exp ( − i k 0 βωx )
E ( x , ω ) = E 0 exp [ − ( x w′ ) 2 ] exp [ − i ( k 0 β + φ ( 2 ) 2 ) υ x 2 ]
× exp [ − ( τ′ ) 2 4 ( ω − υx ) 2 ] exp [ − i φ ( 2 ) 2 ( ω − υx ) 2 ]
× exp [ − i ( k 0 β + φ ( 2 ) υ ) x ( ω − υx ) ]
E 0 ( x , t ) = f ( x ) exp [ − ( t − t 0 ) 2 τ 2 ] exp { i [ ϕ ( 1 ) ( t − t 0 ) + ϕ ( 2 ) 2 ( t − t 0 ) 2 ] }
f ( x ) = 1 π [ ( τ′ ) 2 + i 2 φ 2 ] − 1 / 2 E 0 exp [ − ( x w′ ) 2 ] exp ( i φ ( 2 ) 2 υ 2 x 2 )
t 0 = ( k 0 β + φ ( 2 ) υ ) x
τ = [ ( τ′ ) 2 + 4 ( φ ( 2 ) ) 2 ( τ′ ) 2 ] 1 / 2 = [ τ 0 2 + 4 ζ 2 w 2 + 4 ( φ ( 2 ) ) 2 τ 0 2 + 4 ζ 2 w 2 ] 1 / 2
ϕ ( 1 ) = υ x
ϕ ( 2 ) = ϕ ( 2 ) ( τ′ ) 2 4 + ( φ ( 2 ) ) 2 = φ ( 2 ) 1 4 ( τ 0 2 + 4 ζ 2 σ 2 ) 2 + ( φ ( 2 ) ) 2
p ≡ d t 0 d x
tan ψ = pc
p = p AD + p SC + TC
p AD = k 0 β
p SC + TC = φ ( 2 ) υ
E ˜ ( x , z , ω ) = E xz ( x , z ) E ˜ ω ( ω ) exp ( − i pxω )
E ˜ ( x , z , ω ) = E xz ( x − ζω , z ) E ˜ ω ( ω ) exp ( − i pxω )
E ˜ ( x , z , ω ) = E xz ( x , z ) E ˜ ω ( ω − υx ) exp ( − i pxω )
E ( x , ω , z ) = i λz ∫ − ∞ ∞ E ( x ′ , ω , z = 0 ) exp [ − i π λz ( x − x′ ) 2 ] d x′
E ( x , ω , z = 0 ) = E ( ω , z = 0 ) exp [ − i k 0 ( x − ζ 0 ω ) 2 2 q 0 ] exp ( − i k 0 βωx )
= E 0 exp ( − ω 2 τ 0 2 4 ) exp ( − i φ 0 ( 2 ) 2 ω 2 ) exp [ − i k 0 ( x − ζ 0 ω ) 2 2 q 0 ] exp ( − i k 0 βωx )
E ( x , ω , z ) = i k 0 2 πz E 0 exp ( − ω 2 τ 0 2 4 ) exp ( − i φ 0 ( 2 ) 2 ω 2 )
× ∫ − ∞ ∞ exp [ − i k 0 ( x ′ ) 2 2 q ( 0 ) ] exp [ − i k 0 βω ( x′ + ζ 0 ω ) ] exp [ − i k 0 2 z ( x′ + ζ 0 ω − x ) 2 ] d x′
= [ i k 0 2 πz q ( 0 ) q ( z ) ] 1 / 2 E 0 exp ( − ω 2 τ 0 2 4 ) exp ( − i φ 0 ( 2 ) 2 ω 2 )
× exp { i k 0 z 2 q ( 0 ) q ( 0 ) ( x − ζ 0 ω z − βω ) 2 − i k 0 [ β ζ 0 ω 2 + ( x − ζ 0 ω ) 2 2 z ] }
q ( 0 ) q ( z ) = d + i k 0 w 2 2 z + d + i k 0 w 2 2 = 1 + i 2 z k 0 w 2
E ( x , ω , z ) = ( i k 0 2 πz ) 1 / 2 ( 1 + i 2 z k 0 w 2 ) 1 / 2 E 0 exp ( − ω 0 τ 0 2 4 ) exp [ − i 2 ( φ 0 ( 2 ) − k 0 β 2 z ) ω 2 ]
× exp { − [ x − ( ζ 0 + βz ) ω ] 2 w 2 } exp ( − i k 0 βωx )
ζ ( z ) = ζ 0 + βz
φ ( 2 ) ( z ) = φ 0 ( 2 ) − k 0 β 2 z
υ ( z ) = ζ 0 + βz ( ζ 0 + βz ) 2 + w 2 τ 0 2 4
p = k 0 β + ( φ 0 ( 2 ) − k 0 β 2 z ) υ ( z )
K = [ A B 0 E C D 0 F G H 1 I 0 0 0 1 ] = [ 1 L 0 2 πζ 0 1 0 0 0 − 2 πζ / λ 0 1 2 π φ ( 2 ) 0 0 0 1 ]
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 ) ]
∣ E ( x , t ) ∣ ∝ exp [ − ( t − px ) 2 τ 2 ]
τ = [ π λ 0 Im { ( Q − 1 ) 22 } ] − 1 2
p = π τ 2 2 λ 0 Im { ( Q − 1 ) 12 − ( Q − 1 ) 21 } = Im { ( Q − 1 ) 12 − ( Q − 1 ) 21 } 2 Im { ( Q − 1 ) 22 }
( Q in − 1 ) 11 = 1 q
( Q in − 1 ) 22 = i λ 0 π τ 0 2
Q in = [ q 0 0 − i π τ 0 2 λ 0 ]
Q out = { [ A 0 G 1 ] Q in + [ B E / λ 0 H I / λ 0 ] } · { [ C 0 0 1 ] Q in + [ D F / λ 0 0 1 ] } − 1
Q out = [ q + L 2 πζ λ 0 − 2 πζ λ 0 2 πφ ( 2 ) λ 0 − i π τ 0 2 λ 0 ]
Q out − 1 = [ λ 0 ( φ ( 2 ) − i 2 τ 0 2 ) φ ( 2 ) ( L + q ) λ 0 + 2 π ζ 2 − i 2 ( L + q ) λ 0 τ 0 2 λ 0 ζ φ ( 2 ) ( L + q ) λ 0 + 2 π ζ 2 − i 2 ( L + q ) λ 0 τ 0 2 λ 0 ζ φ ( 2 ) ( L + q ) λ 0 + 2 π ζ 2 − i 2 ( L + q ) λ 0 τ 0 2 1 2 π ( L + q ) λ 0 2 φ ( 2 ) ( L + q ) λ 0 + 2 π ζ 2 − i 2 ( L + q ) λ 0 τ 0 2 ]
q ( L ) = L + q ≈ i π w 2 λ 0
Q out − 1 = [ λ 0 π 2 φ ( 2 ) − i τ 0 2 4 ζ 2 + w 2 τ 0 2 + i 2 φ ( 2 ) w 2 − 2 λ 0 π ζ 4 ζ 2 + w 2 τ 0 2 + i 2 φ ( 2 ) w 2 2 λ 0 π ζ 4 ζ 2 + w 2 τ 0 2 + i 2 φ ( 2 ) w 2 i λ 0 π w 2 4 ζ 2 + w 2 τ 0 2 + i 2 φ ( 2 ) w 2 ]
τ = [ τ 0 2 + 4 ζ 2 w 2 + 4 ( φ ( 2 ) ) 2 τ 0 2 + 4 ζ 2 w 2 ] 1 / 2
p = φ ( 2 ) ζ ζ 2 + 1 4 w 2 τ 0 2
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