1. Introduction

TiO₂ in its rutile phase has been identified as a promising crystal for third-harmonic generation (THG: ω + ω + ω → 3ω) as well as triple photon generation (TPG: 3ω → ω + ω + ω) in the framework of fundamental quantum optics and quantum information [1–4]. Rutile TiO₂ belongs to the tetragonal crystal class 4/mmm with lattice parameters a = 4.5937Å and c = 2.9581Å [5]. It is centrosymmetric so that it does not exhibit any cascading quadratic process that would pollute the TPG. Its optical class is positive uniaxial, which means that n_o < n_e where n_o and n_e are the ordinary and extraordinary linear indices respectively. It is transparent from 430 nm to 6200 nm [6]. The birefringence of rutile is high, with a magnitude of about 0.25, considering the strong discrepancy between the different published data of n_o and n_e [2,6–9]. Among its four independent elements of χ⁽³⁾ under Kleinman symmetry approximation, i.e. χ₁₁, χ₁₆, χ₁₈ and χ₃₃, only two have been determined for certain: |χ₁₆ (613.2 nm)| = 5.0 × 10⁻²¹ m²V⁻² from THG [2] and χ₁₁ (532 nm) = - 6.15 × 10⁻²⁰ m²V⁻² from coherent Raman ellipsometry [10]. Other results previously published based on nonlinear index n₂ measurements do not specify which coefficients were involved so that they only give estimates of the magnitude of |χ⁽³⁾|: 2.05 × 10⁻²¹ m²V⁻² at 1064 nm from nearly degenerate four-wave mixing experiments [11]; 5.6 × 10⁻²⁰ m²V⁻² or 3.08 × 10⁻¹⁹ m²V⁻² at 800 nm from non-phase-matched THG in thin films [12,13]. Note also that calculations using the bond-orbital theory at a wavelength of 1 µm give a magnitude ranging between 3.4 × 10⁻²¹ and 6.8 × 10⁻²⁰ m²V⁻² [14]. In the present paper, we performed phase-matched THG experiments that allowed us to determine accurate Sellmeier equations, the magnitude of χ₁₈, as well as the relative sign between χ₁₆ and χ₁₈.

2. Theory

There are three possible polarization schemes enabling phase-matching for THG [14]. In rutile TiO₂, one of the best situation to achieve a maximal conversion efficiency is to propagate the beams in the xz plane of the dielectric frame (x, y, z) with the following configuration of polarization: ${\omega }^{(o)}+{\omega }^{(e)}+{\omega }^{(e)}\to 3{\omega }^{(o)}$, so-called type II, where ${\omega }^{(o)}$ and ${\omega }^{(e)}$ are the circular frequencies of the ordinary and extraordinary polarized waves respectively [15]. The corresponding phase-matching relationship is then:

The effective coefficient ${\chi }_{eff}^{\left(3\right)}\left({\theta }_{AP}\right)$ corresponding to such a conversion in the xz plane of the crystal is given by [14]:

The THG energy conversion efficiency in the considered phase-matching direction is defined as η = ξ_3ω / ξ_ω, where ξ_ω and ξ_3ω are the incident fundamental and generated third-harmonic energies respectively. According to our experiments, η can be calculated in the undepleted pump approximation with Gaussian beams, which gives [16]:

3. Experimental setup

In order to carry out the THG experiments, we use the cylinder method we previously used for the study of RTP and TiO₂ [2,3,17]. We cut and polished TiO₂ as a cylinder with a radius of 4 mm and a thickness of 5 mm, the y-axis of the crystal being its revolution axis oriented by X-rays with an accuracy of 0.1°. Then the cylinder was stuck on a goniometric head as depicted in Fig. 1(a) .

 

Fig. 1 (a) Picture of the oriented 4-mm radius TiO₂ cylinder with the y-axis as rotation axis; (b) Schematic top view of the THG experimental setup where ω and 3ω are the fundamental and third harmonic circular frequencies. HWP is a Half-Wave Plate.

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The cylinder is placed at the center of a rotation stage with an angular resolution of 0.5°. Using this experimental setup and rotating the cylinder around the y-axis make possible the access to any direction of propagation over the whole xz plane of TiO₂ while keeping the perpendicular incidence [2,18]. The phase-matching direction corresponding to a given fundamental wavelength is detected when the associated THG conversion efficiency is maximal. An angular accuracy of about ± 0.3° is then accessible by measuring the four symmetrical phase-matching directions in the plane of the cylinder. Note that the cylindrical geometry is more suited than the slab geometry for that purpose since the refraction inside the crystal remains normal to the surface for any orientation of the cylinder.

The fundamental beam is generated using two tunable parametric sources. A first device emits a beam tunable from 400 nm up to 2500 nm from a Topas Light Conversion optical parametric generator (OPG) pumped by the third harmonic of a 10 Hz Leopard Continuum Nd:YAG laser of 15-ps pulse duration at 1/e². The second source is a DFG Light Conversion generator that mixes by difference frequency generation the idler beam of the previous OPG and the Nd:YAG beam at 1064 nm in order to generate between 2 µm and 12 µm. Different half-wave plates (HWP) suitable for the considered spectral range are placed just before the rutile crystal to adjust the polarization of the incident tunable fundamental beam at 30° of the z-axis so as to achieve the considered THG phase-matching polarization configuration. Focalizing and collecting spherical lenses (75mm focal lengths) from each sides of the crystal form an afocal system. The incidence lens is placed so that it focuses at the focal plane of the first half of the cylinder, allowing a parallel propagation of the fundamental beam within the crystal [17]. The residual fundamental beam at the exit of the crystal is removed using filters. Fundamental beam power is measured using an Ophir PE10 energy-meter, while third-harmonic was detected using a Silicium Hamamatsu C2719 photodiode calibrated in energy prior to the experiments.

The fundamental beam waist radius within the TiO₂ crystal is equal to w_ω = 120 ± 11 µm, which leads to a Rayleigh length z_Rω = πn_ωw_ω²/λ equal to 30 ± 9 mm for the ordinary polarization (along the y-axis), and to 33 ± 9 mm for the extraordinary polarization. This ensures us that the beams remain parallel in the cylinder since its diameter of 8 mm is smaller than the Rayleigh length. This is a crucial point to perform accurate measurements of the phase-matching angles.

4. Phase-matching measurements and determination of the Sellmeier equations

Figure 2 gives the phase-matching angles measured in the TiO₂ cylinder using the experimental setup described in section 3 with respect to the fundamental and down-converted wavelengths. It appears that rutile can be phase-matched for fundamental wavelengths ranging between 1836 nm and 4449 nm.

 

Fig. 2 Type II THG phase-matching curve of rutile TiO₂. Black squares are experimental data from the cylinder experiment and the solid red line corresponds to their numerical interpolation. Color dashed lines are phase-matching curves calculated from data of [2,6–8].

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For comparison, Fig. 2 also exhibits in dashed lines phase-matching curves calculated from the dispersion equations of the refractive indices of TiO₂ previously published [2,6–8]. It clearly appears that the discrepancy between these calculated curves and the experimental data increases as a function of wavelength leading to strong divergences above 2000 nm. This feature can be well understood knowing that these curves were calculated from refractive indices that had been measured over the visible and near infrared range up to 1500 nm. Then, it is possible to refine those equations by interpolating the phase-matching angles we obtained experimentally since they are widely distributed over the whole transparency range of the crystal. The corresponding interpolating curve is shown as a solid red line in Fig. 2; it is based on a Sellmeier form [19] for the expressions of the principal refractive indices n_o and n_e expressed as following where the wavelength $\lambda$ is expressed in nanometers:

The wavelength validity domain of these equations corresponds to the experimental measurement range, i.e. from 612 nm to 4449 nm.

Equations (7) and (8) are plotted in Fig. 3 . They are compared with previous works including measurements on prisms [6–9] and calculations [8]. Note that Sellmeier Eqs. (7) and (8) are in perfect agreement with the refractive index measurements previously published, which is an additional validation of our equations. We can also see the strong discrepancy between the curves of the present work and those previously published. Figure 3 gives an example, from [7], where it clearly appears a splitting between the curves from 1500 nm, this wavelength corresponding to the upper value of the range over which were measured the refractive indices [7].

 

Fig. 3 Ordinary n_o and extraordinary n_e principal refractive indices of TiO₂ as a function of wavelength from present work in solid lines, and from [6–9] in symbols (experiments) and dashed lines (calculations).

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5. Determination of the third order nonlinear coefficients

The energy conversion efficiency of type II THG in the TiO₂ cylinder is measured at different phase-matching angles using two different values of the energy of the incident fundamental beam, i.e. 16 µJ and 1.9 µJ as shown in Fig. 4 , which corresponds to 7.4 MW/cm² and 0.9 MW/cm² respectively. The experimental data corresponding to these two sets of measurements are coherent one to each other, which attests to the good reproducibility. The conversion efficiency can reach 1.3% which is the higher value ever reported for this crystal. From these experimental data we were able to determine the magnitude of χ₁₈ and its relative sign with χ₁₆ since these two coefficients are involved in the effective coefficient according to Eq. (3). We took as a reference the absolute value |χ₁₆(613.2nm)| = 5.0 × 10⁻²¹ m²V⁻² previously determined by type II phase-matched THG experiments at θ = 90° which does not involve χ₁₈ [2,3]. Numerical computation of the conversion efficiencies using Eqs. (3)–(6) and the absolute value of χ₁₆ have been done in the two hypotheses of identical and opposite signs for χ₁₆ and χ₁₈. The two corresponding curves relative to the effective coefficient are shown in the insert of Fig. 4 together with calculations arising from the conversion efficiency measurements. It clearly appears that χ₁₆ and χ₁₈ are of opposite signs, the Miller coefficient for χ₁₈ being [20]:

 

Fig. 4 Type II THG conversion efficiency of rutile TiO₂ as a function of the phase-matching angle for fundamental energies ξ_ω = 16µJ (black dots) and 1.9µJ (blue triangles). In the insert is shown the magnitude of the effective coefficient χ_eff⁽³⁾ where the black squares correspond to values determined from conversion efficiencies experimental data and Eq. (4); the dashed red and solid blue lines are calculations considering χ₁₆ and χ₁₈ with same and opposite signs respectively.

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As an example, Eq. (9) leads to |χ₁₈(616.7nm)| = 9.7 × 10⁻²⁰ m²V⁻², that is close to the absolute magnitude of χ₁₁ [10], and significantly larger than χ₁₆ [2].

6. Conclusion

To the best of our knowledge, we performed the most exhaustive study of THG phase-matching properties of rutile TiO₂ up to now. We showed that this crystal enables phase-matching for fundamental wavelengths ranging from 1836 nm and 4449 nm with an energy conversion efficiency that can reach more than 1%. We also refined the Sellmeier equations and determined the absolute magnitude of χ₁₈. Furthermore we showed that the sign of this coefficient is different than that of χ₁₆. Since THG (ω + ω + ω → 3ω) is the exact reverse process of TPG (3ω → ω + ω + ω), the phase-matching properties are exactly the same. Thus these results can be directly used for further quantum TPG experiments.