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Broadband sum frequency mixing using noncollinear angularly dispersed geometry for indirect phase control of sub-20-femtosecond UV pulses

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Abstract

We report on a novel scheme for generating a broad spectrum in the UV region. This scheme enables us to control the phase of the UV pulse through a frequency-mixing process in a nonlinear crystal. For group velocity matching, it is essential that a monochromatic beam should be sum-frequency mixed with an angularly dispersed beam having a broad spectrum in noncollinear geometry. We found analytically unique solutions for a noncollinear angle, for an angular dispersion of the broadband input beam, and for an angle of the beam from the optical axis in a nonlinear crystal, with the condition that there is no angular dispersion in the output beam. Based on the analysis of this scheme, we obtained UV pulses with a sufficiently broad spectrum for obtaining a sub-20-fs pulsewidths in the experiment. The improvement of conversion efficiency and compensation of chirp are also discussed.

©2003 Optical Society of America

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Figures (7)

Fig. 1.
Fig. 1. (a) Phase matching (wave-vector matching) in NOPA. (b) Noncollinear angularly dispersed wave-vector matching in sum-frequency mixing.
Fig. 2.
Fig. 2. Definitions of angles. An angle of κ b and that of κ c from κ a0 are defined as α ab and α ac , respectively. A sign of the angle is plus when the direction of the angle is the same as that of the angle of κ a0 vector from z-(optical) axis. This figure shows these angles at the center angular frequencies of ω b0 and ω c0 . Angularly dispersed wave-vector components in beam B are schematically shown as three arrows with different directions and lengths. The wave-vector of the monochromatic beam, κ a0 , and that of the generated beam at the center angular frequency, κ c0 , are far from the optical axis with the angles θ a0 and θ c0 , respectively. The direction of the wave vector of the generated beam, κ c , generally depends on the angular frequency, ω c , although in this figure we depict the ends of the three arrows for κ c c ) as being aligned.
Fig. 3.
Fig. 3. Wave-vector mismatches to the wavelength of Ti:sapphire laser, Δκ b )L, in collinear (brown dotted curve) and noncollinear angularly dispersed geometries (red dot-dashed curve and blue solid curve) for Type I (a) and for Type II (b) SFM in a BBO crystal. Wavelength of the monochromatic beam is set to be 532 nm. The acceptable power spectra as the functions of the wavelength of the Ti:sapphire laser, |η(λ b )|2, are also shown below the wave-vector mismatches. The slight tilt of the crystal from the complete WVMC (blue solid curves) broadened the bandwidth in spite of a central dip.
Fig. 4.
Fig. 4. Experimental setup for NADG for GVMC.
Fig. 5.
Fig. 5. Spectra of the input Ti:sapphire laser pulse (a) and the UV pulse (b) with respect to wavelengths of λ b and λ c .
Fig. 6.
Fig. 6. Power spectra of the input Ti:sapphire laser pulse (red, dotted curve) and UV pulse (blue, solid curve). Bottom and top axes correspond to the optical frequency of UV, νUV=c c , and the Ti:sapphire laser, νTiS=c b , respectively. A brown, dotdashed curve is the spectrum of the Ti:sapphire laser filtered with an acceptable power spectrum.
Fig. 7.
Fig. 7. Energy of UV pulse (solid circles) and pulse shape of the second harmonic of the Nd:YAG laser.

Equations (32)

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E ˜ 3 ( ω ) d ω E ˜ 1 ( ω ) E ˜ 2 ( ω ω ) .
E ˜ 1 ( ω ) A 1 δ ( ω ω 1 0 ) ,
E ˜ 3 ( ω ) E ˜ 2 ( ω ω 1 0 ) ,
Δ k k c ( ω c ) { k a ( ω a ) + k b ( ω b ) } ,
ω c = ω a + ω b .
Δ k = k c ( ω c 0 ) { k a ( ω a 0 ) + k b ( ω b 0 ) }
+ { d k c d ω c | 0 d k b d ω b | 0 } Δ ω + O ( Δ ω 2 ) .
0 = k c ( ω c 0 ) { k a ( ω a 0 ) + k b ( ω b 0 ) } ,
0 = d k c d ω c | 0 d k b d ω b | 0 ,
k a ( ω a ) = k a ( ω a 0 ) = k a 0 e a 0 ,
k b ( ω b ) = k b ( ω b ) cos { α ab ( ω b ) } e a 0 + k b ( ω b ) sin { α ab ( ω b ) } e b 0 ,
k c ( ω c ) = k c ( ω c ) cos { α ac ( ω c ) } e a 0 + k c ( ω c ) sin { α ac ( ω c ) } e b 0 ,
k c 0 cos ( α ac 0 ) { k a 0 + k b 0 cos ( α ab 0 ) } = 0 ,
k c 0 sin ( α ac 0 ) k b 0 sin ( α ab 0 ) = 0 ,
d k b d ω b | 0 cos ( α ab 0 ) k b 0 sin ( α ab 0 ) d α ab d ω b | 0
{ d k c d ω c | 0 cos ( α ac 0 ) k c 0 sin ( α ac 0 ) d α ac d ω c | 0 } = 0 ,
d k b d ω b | 0 sin ( α ab 0 ) + k b 0 cos ( α ab 0 ) d α ab d ω b | 0
{ d k c d ω c | 0 sin ( α ac 0 ) + k c 0 cos ( α ac 0 ) d α ac d ω c | 0 } = 0 ,
d α ac d ω c | 0 0 ,
υ g 0 b cos ( α ab 0 α ac 0 ) = υ g 0 c ,
υ g 0 b = ( d k b d ω b | 0 ) −1 ,
υ g 0 c = ( d k c d ω c | 0 ) −1 .
d α ab d ω b | 0 gvm = tan ( α ab 0 gvm α ac 0 gvm ) 1 k b 0 d k b d ω b | 0 gvm ,
sin θ c 0 = { [ n o ( ω c 0 ) n ¯ ab 0 ( α ab 0 ) ] 2 1 [ n o ( ω c 0 ) n e ( ω c 0 ) ] 2 1 } 1 2 ,
n ¯ ab 0 ( α ab 0 )
{ [ ω bc 0 n o ( ω b 0 ) sin α ab 0 ] 2 + [ ω ac 0 n o ( ω a 0 ) + ω bc 0 n o ( ω b 0 ) cos α ab 0 ] 2 } 1 2 ,
η ( ω a 0 , ω b ) L −1 0 L d ζ e i Δ k ( ω a 0 , ω b ) ζ
= e i Δ k ( ω a 0 , ω b ) L 2 sinc { Δ k ( ω a 0 , ω b ) L 2 } ,
d eff I = [ d 11 cos ( 3 φ ) d 22 sin ( 3 φ ) ] cos θ c 0 + d 31 sin θ c 0
d 11 cos ( 3 φ ) cos θ c 0 ,
d eff II = d 11 cos θ a 0 cos θ c 0 sin ( 3 φ ) + d 22 cos θ a 0 cos θ c 0 cos ( 3 φ )
d 11 cos θ a 0 cos θ c 0 sin ( 3 φ ) ,
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