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Quasi-phase-matching for third harmonic generation in noble gases employing ultrasound

U. K. Sapaev, I. Babushkin, and J. Herrmann

Open Access Open Access

Abstract

We study a novel method of quasi-phase-matching for third harmonic generation in a gas cell using the periodic modulation of the gas pressure and thus of the third order nonlinear coefficient in the axial direction created by an ultrasound wave. Using a comprehensive numerical model we describe the quasi-phase matched third harmonic generation of UV (at 266 nm) and VUV pulses (at 133 nm) by using pump pulses at 800 nm and 400 nm, respectively, with pulse energy in the range from 3 mJ to 1 J. In addition, using chirped pump pulses, the generation of sub-20-fs VUV pulses without the necessity for an external chirp compensation is predicted.

© 2012 Optical Society of America

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Figures (8)
Fig. 1
Fig. 1 QPM for THG using an ultrasound wave. (a) - The basic scheme: the pump with frequency ω is sent into a tube where an ultrasound wave is exited to achieve the QPM. At the output, third harmonic 3ω is observed. (b) - Dependence of the ultrasound frequency Ωs on the pump wavelength λ for THG QPM in argon at 1 atm. (c) - The absorption rate of the ultrasound wave in argon at the normal conditions in dependence on its frequency Ωs.

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Fig. 2
Fig. 2 THG for a 0.7 ps pump pulse at 800 nm with 3 mJ energy. In (a) the conversion efficiency η3ω calculated analytically by Eq. (8) (red-dashed) and numerically (red-solid) in dependence on the propagation distance z (the result in the absence of ultrasound is shown by the green curve); in (b) the evolution of the peak intensity of the pump (blue) and the TH (red); in (c) the spatial intensity profile of the TH at the output and in (d) the change of the radii of the pump (red) and the TH (blue) are presented.

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Fig. 3
Fig. 3 THG for a 1 ps pump pulse at 800 nm with 1 J energy. The description of (a)–(d) and the curves are analogous as Fig. 2.

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Fig. 4
Fig. 4 THG for a 1.4 ps pump pulse at 400 nm with 3 mJ energy. In (a) the conversion efficiency calculated analytically by Eq. (8) (dashed) and numerically (solid); in (b) the evolution of the peak intensity of the fundamental (blue) and the TH (red); in (c) the maximum electron density and in (d) the change of radii of the fundamental (red) and the TH (blue) are presented.

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Fig. 5
Fig. 5 THG for a 1 ps pump pulse at 400 nm with 0.1 J energy. In (a) the conversion efficiency calculated analytically by Eq. (8) (dashed) and numerically (solid); in (b) the evolution of the peak intensity of the fundamental (blue) and the TH (red); in (c) the spatial intensity profile of the TH at the output are presented.

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Fig. 6
Fig. 6 THG for a negatively chirped pump with the energy of 3 mJ. (a) The THG efficiency η3ω in dependence on z; (b) Peak intensity of the fundamental (blue) and the TH (red) pulses versus z; (c) Spatial profile of the TH at the output; (d) Change of radii of the fundamental (red) and TH (blue) with z.

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Fig. 7
Fig. 7 Evolution of the pulse duration (a) as well as the spectral (b) and temporal (c) shapes of the TH pulse at the output for the parameters of Fig. 6. Blue and red curves in (a) refer to the fundamental and the TH pulses, respectively. In (c) blue curve represents the phase of the TH.

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Fig. 8
Fig. 8 THG for a negatively chirped pump with energy of 100 mJ using. (a) THG efficiency in dependence on z; (b) Peak intensity of the fundamental (blue) and the TH (red); (c) Spatial profile of the TH at the output; (d) Temporal shape of the TH at the output.

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Equations (8)

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β s = Ω s 2 8 π 2 ρ 0 c s 3 ( 4 3 η + ( γ − 1 ) κ C p ) ,
∂ A ω ∂ z + M ^ ω A ω = i γ ω ( z ) { A ω ( | A ω | 2 + 2 | A 3 ω | 2 − Γ ω ) + A ω * 2 A 3 ω e i Δ k z } ,
∂ A 3 ω ∂ z + M ^ 3 ω A 3 ω = i γ 3 ω ( z ) { A 3 ω ( | A 3 ω | 2 + 2 | A ω | 2 − Γ 3 ω ) + A ω 3 e − i Δ k z / 3 } ,
∂ ρ ∂ t = ( ρ o ( z ) − ρ ) ( σ K ω | A ω | 2 K ω + σ K 3 ω | A 3 ω | 2 K 3 ω ) + ρ U i ( σ ω ( z ) | A ω | 2 + σ 3 ω ( z ) | A 3 ω | 2 ) .
M ^ m = − i ( k m ( z ) − k m o ) + ν m ( z ) ∂ ∂ t + i g m ( z ) 2 ∂ 2 ∂ 2 t − i 2 k m ( z ) [ ∂ 2 ∂ x 2 + ∂ 2 ∂ y 2 ] ,
Γ m = σ m ( z ) 2 ( 1 + i ω m o τ c ( z ) ) + β K m 2 ( ρ o ( z ) − ρ ) | A m | 2 K m − 2 ,
K S = Δ k + 3 γ ω | A ω o | 2 − 2 γ 3 ω | A ω o | 2
I 3 ω ( z ) ≈ c ε 0 ( γ 3 ω p z ) 2 12 | A ω o | 6 .
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