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A type of phase matching for difference frequency generation with Bessel-type pump beams is proposed. In this geometry, the phase matching is achieved in a cone around the laser path by properly controlling the beam profile. An experimental case that 1.5THz generation with ~2μm lasers pumped bulk GaAs crystal is considered. Calculations of the energy conversion characteristics are performed based on a semi-analytical model. The results indicate that this configuration could relax the phase matching condition in a wide range of nonlinear crystals and pump wavelengths.
© 2016 Optical Society of America
1. Introduction
Difference frequency generation (DFG) between two laser wavelengths is a proven approach for tunable, monochromatic and coherent mid-infrared and terahertz (THz) wave generation, which has been applied in the fields of high-resolution spectroscopy [1,2 ]. Zinc-blende semiconductors (e.g. ZnTe, GaP and GaAs) are a kind of nonlinear materials well suited for the infrared and THz sources, due to such advantages: well-developed growing technique, good optical quality, relatively large second-order nonlinear coefficient, low absorption, and high damage threshold. In a DFG process, phase-matching (PM) condition greatly determines the efficiency of nonlinear frequency conversion. However, the traditional birefringent PM does not work in these isotropic crystals. The PM geometries commonly used in such materials include: bulk crystal based noncollinear PM [3], periodically-inverted structure based quasi-PM (QPM) [4], waveguide based modal PM [5], as well as recently proposed front-tilting PM [6]. The noncollinear PM requires the critical control of the angle between two input beams. In the case of QPM, the fabrication of periodically-inverted structure is difficult. Unlike the ferroelectric materials (e.g. LiNbO3), the QPM zinc-blende crystals are mainly in three types: diffusion-bonded, orientation-patterned and optical-contacted. For the waveguide based DFG, the coupling efficiency of the pump into the nonlinear core is relatively low, which limits the output power. Thus, efficient PM configuration in bulk zinc-blende crystals is necessary to be explored.
Recently, Bessel laser beams (also called “diffraction-free beam”) have attracted considerable interest [7,8 ], owing to the merits: tunable longitudinal wavenumber and unique transverse mode distribution. It was first applied in second harmonic generation (SHG) by Wulle and Herminghaus [9], which provided a novel approach for PM in nonlinear frequency conversion. Later, Bessel beam pumped optical parametric oscillation and third harmonic generation were studied [10,11 ]. In the presented paper, we go further and propose a PM geometry for DFG based on Bessel pump beams, which is especially suitable for isotropic mediums.
2. Analysis of the phase-matching condition
Considering the lowest order of Bessel beam:
where J 0 is the zeroth-order Bessel function of the first-kind, κ and β are the radial and longitudinal wavenumbers, respectively. The beam contains a series of wavevectors k (k 2 = κ 2 + β 2), which lie on a cone. Two types of PM for SHG have already been studied comprehensively in [12] [Figs. 1(a) and 1(b) ]. For the DFG process, there are also two kinds of typical PM configurations [Figs. 1(c) and 1(d)]. We assume that the two input wavelengths have the same beam profile (κ). In type-I, each pair of interactive photons is with opposite radial wavevectors. A conic phase front of the difference-frequency wave is generated. On the contrary, two photons with identical radial wavevectors contribute to a plane wave in the longitudinal direction (type-II). At given conditions (pump wavelengths and nonlinear medium), the longitudinal propagation constant β can be varied continuously by appropriately focusing the input Bessel beams, which allows us to achieve the PMs above.According to the momentum conservation, the required transverse wavenumbers of pump beam for two types PM are derived:
In the case of 1.5THz generation with GaAs, the phase-matched beam radius r I,II (radius of the bright central spot: 2.405/κ I,II) as functions of input wavelength λ1 (λ1 < λ2) are plotted in Fig. 2(a) . The refractive indices of GaAs at pump and THz frequencies are given by [13]. The wavenumber mismatch in collinear geometry Δk = k T + k 2 – k 1 is also presented. At given Δk, type-II PM requires a κ II two orders larger than that of type-I κ I (much smaller transverse mode). For example, the pump wavelength of λ1 = 1.98μm corresponds to the two types PM radius of r I = 135μm and r II = 0.73μm. Since the spot size r II is even smaller than the wavelength λ1, type-II PM is commonly unachievable. Thus, type-I geometry is worth to be investigated from the experimental point of view, compared with type-II.
Fig. 2 Wavenumber mismatch Δk (dashed and dotted), required Bessel beam radius for type-I (solid) and type-II PM (dotted) versus the pump wavelength at 1.5THz generation (a) and the oblique angle of conic THz wavevector θ in type-I at different frequencies versus the pump wavelength (b).
As seen in Fig. 1(c), oblique wavevectors k T generated via type-I PM form a conic THz wave-front. The angle between k T and longitudinal direction θ at different THz frequencies is shown in Fig. 2(b). Although the conic wave-front looks like the Cherenkov PM, there are distinctions between these two configurations. Cherenkov PM is automatically satisfied when a tightly focused laser beam propagates in a superluminal material (e.g. Ti:sapphire laser pumped LiNbO3 and fiber laser pumped GaAs) [14]. General pump beams (e.g. Gaussian beam) consist of a continuum of radial wavenumbers, which contribute to the transverse components of Cherenkov wavevector as long as the beam is focused into a size in the order of THz wavelength. Cherenkov geometry is not perfect PM, because of the transverse phase mismatch originated from the beam size [15]. On the other hand, tightly focusing causes severe divergence of pump beam, decreasing the effective radiation length. These factors restrict the conversion efficiency of Cherenkov-type DFG to a rather low level [16]. For the case of Bessel beam pumped type-I PM, the radial Fourier component of pump beam is a single value κ, which should strictly satisfy the condition Eq. (2). The drawbacks above no longer exist, due to the perfect vector PM as well as the non-diffraction properties of Bessel beams. This proposed type-I geometry is a promising PM approach for nonlinear frequency down-conversion, especially for the circumstances where traditional PMs do not apply.
3. Dynamic of energy conversion
In this section, we perform calculations to investigate the characteristic of this energy conversion process, quantitatively. An experimental case that bulk GaAs crystal pumped by Tm3+-doped fiber laser is considered. The general expression of the interactive electric fields can be written as:
Here, the subscripts m ( = 1, 2 and T) denote the input two wavelengths and the generated difference-frequency-wave. The slowly varying complex amplitudes A m obey the coupled-wave equations with transverse effects taken into account:
where ∇⊥ 2 is the transverse Laplacian, n eff ( = βc/ω) is the effective refractive indices. Effective nonlinear coefficient d eff is determined by the crystallographic orientation and polarizations of input beams, which reaches the maximum (4/3)1/2 d 14 when two input wavelengths are parallel polarized along the <1-11> direction of GaAs crystal. Four terms on the right hand-side of each equation describe the diffraction, noncollinear walk-off, absorption, and nonlinear interaction (gain or depletion), respectively. According to the vector PM condition [Fig. 1(c)], wavevector components of three waves satisfy such relationships: κ T = 2κ I, and β T = β 1 – β 2.Equations (5) can be solved numerically with the split-step method [17]. Parameters used in the calculation are as follows: input laser wavelengths of λ1 = 1.98μm and λ2 = 2.00μm (free from the two photon absorption in GaAs), the corresponding difference frequency of ω T/2π = 1.5THz, intensity at the beam center of 300MW/cm2 (dual-wavelengths), type-I PM required beam radius of r I = 135μm, and nonlinear coefficient of d 14 = 94pm/V [18]. Spatial distribution of THz intensity generated via type-I PM DFG is presented in Fig. 3 . THz wave spreads obviously during the course of interaction, which coincides with the qualitative analysis (conic THz wave-front). The dashed line labels the direction of wavevector k T.
Fig. 3 Variation of THz intensity in radial and longitudinal directions during the process of Bessel beam pumped type-I PM DFG.
The variations of total powers of three waves along the longitudinal direction are plotted in Fig. 4(a) . Strictly speaking, the total power of ideal Bessel beams should be infinite. Here, we integrate the pump intensities over a beam radius of 4mm (initial power of 675.1kW at each wavelength). At the length z = 40mm, the THz power reaches 536.8W and the corresponding conversion efficiency is 0.8‰. It can be seen that the proposed type-I method greatly improves the PM efficiency, by comparison with the collinear non-PM and QPM geometries [Fig. 4(b)].
Fig. 4 (a) Dynamic of energy conversion among the three waves: high-frequency pump λ1 (dashed), low-frequency pump λ2 (dotted) and THz wave (solid curve), (b) comparison between THz powers generated by type-I PM (solid), collinear non-PM (dotted) and QPM DFG (dashed curve) under the same pump condition.
4. Conclusion
In conclusion, we presented a theoretical study on a novel PM geometry for DFG by utilizing Bessel-type pump beams with proper transverse mode. The mechanism and PM condition were discussed in detail, which were also compared with the Cherenkov configuration. The energy conversion dynamic was investigated, based on the split-step method. Our results demonstrated that the proposed configuration could achieve efficient phase-matched THz DFG in bulk zinc-blende crystals without the periodically-inverted or waveguide structure (difficult for fabrication). It would become a promising approach for the DFG under circumstances where traditional PM methods were difficult, owing to the well-developed techniques of fiber laser and Bessel beam generation.
Acknowledgment
This work is supported by the National Basic Research Program of China (973) (Grant No. 2014CB339802), the National Natural Science Foundation of China (Grant Nos. 61505089, 61275102 and 61471257), China Postdoctoral Science Foundation funded project (No. 2015M581291), and the Doctoral Fund of Ministry of Education (No. 20130032110051). The author P. Liu thanks Prof. Koji Suizu (Chiba University) for recommending the split-step method.
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