Proposal of high quality walk-off compensated sum frequency generation of ultra-short pulses
Introduction
Frequency up-conversion processes of ultra-short pulses have many potential applications in spectroscopy, optical communications and optical biology, etc. The sum frequency generation (SFG) is a basic frequency mixing step to generate ultraviolet femtosecond pulses. For a low power pumping at a seed-pulse level, a problem often occurs when one simultaneously requires conversion efficiency and beam quality. The group-velocity mismatch (GVM) between interacting waves places a severe restriction on the conversion process, which reduces efficiency and incurs pulse width broadening [1]. A classical method is to use spectral angular dispersion [or achromatic phase-matching (APM)] by dispersive elements (grating, prism or grism pairs) [2], [3], [4], [5] to increase spectral bandwidth. This technique is easily applied in second harmonic generation (SHG), however becomes quite complicated when it is used in other three-wave interacting processes, where commonly a noncollinear phase-matching (PM) is taken [6]. Generally, a trade-off must be taken among spatial walk-off, conversion efficiency, diffraction loss [5], accompanied by a complex spectral technique of precise chirp compensation through dispersive element pairs. Another effective approach is to compensate for GVM by designing specific optics. Since the typical APM and noncollinear PM approaches are difficult to be applied in a multi-pass structure to increase conversion efficiency, some group-velocity compensating methods are in expectation. Smith et al. [7] had presented an idea of temporal walk-off compensation through a multi-crystal structure, which was implemented later by use of five programmed β-Barium Borate (BBO) segments [8]. Fülöp et al. proposed a two-pass scheme but through a prism pair using a single nonlinear crystal [9]. Recently, Sapaev and Assanto designed a scheme by frequency-selective mirrors to compensate for GVM in picosecond SHG [10], which we think is able to be applied to femtosecond frequency conversion via a special design of light paths. A similar concept of “quasi-group-velocity matching” was used by Huang et al. [11], who used bent fibers to compensate for GVM in a quasi-phase matched nonlinear waveguide. The GVM compensation method is found to be an effective approach to enhance the PM bandwidth and increase conversion efficiency without degrading beam’s spatial-temporal quality. However to our knowledge, these compensation methods have not applied to other nondegenerate three-wave interactions except SHG. In this paper, we present two schemes to realize walk-off compensated sum frequency-third harmonic generation (SF-THG) of Ti:sapphire femtosecond laser, as will boost their applications in photonics and spectroscopy.
A general SF-THG is realized by a cascaded process, in which a SHG process ω + ω → 2ω is accompanied by a SFG ω1 + ω2 → ω3 (ω + 2ω → 3ω), where the angular frequencies are ω1 = ω, ω2 = 2ω and ω3 = 3ω, which correspond to the wavelengths λ1, λ2, λ3. Here we only consider a SF-THG process in a negative nonlinear crystal, such as BBO, which is one of most common cases for ultraviolet femtosedond pulse generation [12], [13]. For a concise representation, we introduce the undepleted-pump approximation to sketch out the proposal. Under the undepleted-pump approximation for SF-THG of either type I (ooe) or type II (oee), wave evolution follows the equations [7] (modified)where, diffraction, group-velocity dispersion (GVD) and absorption are neglected here (GVD will be reconsidered later); z, the coordinate in the propagating direction with time t; x, one transverse coordinate as well as y (omitted here); Δk = k1 + k2 − k3, the known wave vector mismatch at center wavelengths; σ, the coupling strength proportional to the second order effective susceptibility; ρj and uj, the tangent values of spatial walk-off angles (ρ1 = 0) and the group-velocities of the pulses with ωj (j = 1, 2, 3) respectively. The pulse amplitudes A1((x,z),t − L/u1), A2((x,z),t − L/u2 + Δt′) (A1,2 are traveling pulse waves), A3(x,z,t) are connected to real fields bywith the unit vectors , crystal length L and a predelay time Δt′. Eq. (1c) has a simple solution, i.e.,withwhere Δt = L/u3 and (j, l = 1, 2, 3), the inverse GVM. , in proportion to the third harmonic (TH) amplitude generated at the source spatial-temporal point (x − ρ3(L − z), z, t − Δt), is described at the observation point (x, L, t) on the exit surface of the crystal. This relationship can be understood by the principle of light rays’ propagation. Since thin nonlinear crystals are commonly used for frequency conversion of ultra-short pulses, the spatial walk-off can be neglected in many cases. Thus, the temporal walk-off is our main concern in this paper. We can see, the generation of the TH pulses of ω3 only happens at the superposition time of the first harmonic (FH) pulses of ω1 and the second harmonic (SH) pulses of ω2, which are related to Δu12, L and pulse widths.
Section snippets
Multi-crystal design
A previous SHG stage for blue ultra-short pulse generation can be simply made by a thin BBO crystal. A seed-pulse level pumping is able to produce an efficiency over 40%. Higher efficiencies are obtainable by multi-crystal configuration [7], where a kind of GVM compensating crystal with large birefringence should be used. This GVM compensating principle will be used in the following SF-THG scheme. Fig. 1 shows the first sketch of our designs, where a multi-crystal array, each of which is
Frequency-selective design
Fig. 3 demonstrates a frequency-selective scheme. The SHG stage can be designed as the same as before. The mechanism in the SFG part is optical path rectifications by a frequency-division method in air. First, the TH pulses will be turned aside by one dichroic plate or harmonic separator and then be totally reflected by a precise positioning mirror (generally the accuracy is at an interference level) which will advance them with respect to the FH pulses, next, the SH pulses will be reflected
Discussions
The output amplitude of a TH pulse in the two extra-cavity schemes will formally be a superposition of N amplitude phasors, i.e.,where, N arrays (passes) are considered; m denotes the mth TH pulse generated in the mth array (pass); δm represent the interference factors among the TH pulses by imperfect interference. Actually, they will bring a practical requirement, i.e., the compensation for phase difference must be taken to guarantee a coherence of the generated TH
Conclusions
In summary, we have demonstrated two compensating schemes designed for a cascaded SF-THG of the typical Ti:sapphire femtosecond laser, which we believe are viable. The mechanisms and discussions are presented by schematic figures. The multi-crystal scheme can be used for other ultra-short pulse lasers at different wavelengths with specially designed polarization rotating elements. However without perfect AR-coatings, the Fresnel losses by so many elements are commonly severe. Practical
Acknowledgements
The authors thank the Education Department of Heilongjiang province for a support project for key young scholars under Grant No. 1151G013.
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