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High-order pulse front tilt caused by high-order angular dispersion

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Abstract

We have found general expressions relating the high-order pulse front tilt and the high-order angular dispersion in an ultrashort pulse, for the first time to our knowledge. The general formulae based on Fermat’s principle are applicable for any ultrashort pulse with angular dispersion in the limit of geometrical optics. By virtue of these formulae, we can calculate the high-order pulse front tilt in the sub-20-fs UV pulse generated in a novel scheme of sum-frequency mixing in a nonlinear crystal accompanied by angular dispersion. It is also demonstrated how the high-order angular dispersion can be eliminated in the calculation.

©2003 Optical Society of America

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Supplementary Material (4)

Media 1: GIF (641 KB)     
Media 2: GIF (488 KB)     
Media 3: GIF (2313 KB)     
Media 4: GIF (1401 KB)     

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Figures (8)

Fig. 1.
Fig. 1. Schematic figure of two rays passing through a diffractive device. The argument of the pulse front tilt in this section is general for any diffractive devices although we show a prism as an example in this figure.
Fig. 2.
Fig. 2. Wave-vector matching in sum-frequency mixing with a noncollinear angularly dispersed geometry.
Fig. 3.
Fig. 3. Spectrum of the generated DUV with the GDD-matched SFM (light blue hatched area on bottom graph) obtained in the experiment, and calculated output angle (blue curve on top graph). An acceptable power spectrum is also shown as the red hatched area in the bottom graph.
Fig. 4.
Fig. 4. Residual angular dispersions at the zeroth (angle), first, and second orders. Dotted curves in each graph correspond to those compensated with an inverse angular disperser consisting of a one-to-one telescope and a fused silica prism. Solid curves are the ones corrected by adding a grating to the inverse angular disperser and replacing the one-to-one telescope with another one having a magnification factor of 2.4/1.5.
Fig. 5.
Fig. 5. (2.1 MB) Change of the pulse front tilt caused by the residual high-order angular dispersions compensated with the inverse angular disperser of a prism. The color bar at the top left indicates the incident angle relative to the prism.
Fig. 6.
Fig. 6. (1.6 MB) Change of the pulse front tilt caused by the residual high-order angular dispersions compensated with the hybrid compensation of a prism and a grating. The color bar at the top left indicates the relative incident angle to the prism. The incident angle to the grating is fixed in the calculation.
Fig. 7.
Fig. 7. (2.2 MB) Change of the interferogram corresponding the change of the pulse front tilt in Fig. 5.
Fig. 8.
Fig. 8. (1.6 MB) Change of the interferogram corresponding the change of the pulse front tilt in Fig. 6.

Equations (13)

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tan γ = λ 0 | 0 ,
ϕ A ( ω 0 ) = ϕ B ( ω 0 ) ,
ϕ B ( ω 0 + Δ ω ) ϕ A ( ω 0 + Δ ω )
= ( ω 0 + Δ ω c ) sin θ 0 cos Δ ω ( ω 0 + Δ ω c ) sin ( θ 0 + Δ θ )
= ( ω 0 + Δ ω c ) x 0 sin Δ θ ,
d ϕ B d ω | 0 d ϕ A d ω | 0 = x 0 c ω 0 d θ d ω | 0
d 2 ϕ B d ω 2 | 0 d 2 ϕ A d ω 2 | 0 = x 0 c ( ω 0 d 2 θ d ω 2 | 0 + 2 d θ d ω | 0 )
d 3 ϕ B d ω 3 | 0 d 3 ϕ A d ω 3 | 0 = x 0 c [ ω 0 { d 3 θ d ω 3 | 0 ( d θ d ω | 0 ) 3 } + 3 d 2 θ d ω 2 | 0 ]
τ B ( ω 0 ) τ A ( ω 0 ) = x 0 c λ 0 d θ d λ | 0 ,
d 2 ϕ B d ω 2 | 0 d 2 ϕ A d ω 2 | 0 = x 0 2 π c 2 λ 0 3 d 2 θ d λ 2 | 0 ,
d 3 ϕ B d ω 3 | 0 d 3 ϕ A d ω 3 | 0 = x 0 4 π 2 c 3 λ 0 2 { 3 λ 0 2 d 2 θ d λ 2 | 0 + λ 0 3 d 3 θ d λ 3 | 0 + λ 0 3 ( d θ d λ | 0 ) 3 } .
θ c = θ a 0 + α ac ( ω c ) ,
α ac ( ω c ) = arctan { k b ( ω b ) sin α ( ω b ) k a 0 ( ω a 0 ) + k b ( ω b ) cos α ( ω b ) } ,
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