The measurement of three-photon quantum correlation requires one to perform efficient thirdorder nonlinear interactions, like third harmonic generation (THG) or triple photon generation (TPG). Both of these interactions were recently performed with success by using phasematching in KTiOPO₄ (KTP): a THG conversion efficiency of 2.5% was obtained [1], and the first experiment of TPG, stimulated by two photons, was demonstrated, leading to 3.34×10¹³ triple photons/pulse [2]. Despite these good results, KTP is probably not the best material for the purpose: owing to the absence of inversion center, the cubic interactions can be polluted by quadratic cascading processes, with contributions of about 10% for the THG and 0.5 % for the TPG mentioned above [1, 3], and the magnitude of the third-order electric susceptibility χ⁽³⁾ is not so high : χ₁₁=2.32×10^-21 m²/V² and χ₂₂=1.96×10^-21 m²/V² at 1064 nm measured by z-scan experiments [4]; χ₂₄=1.46×10^-21 m²/V² at 539 nm and χ₁₆=0.80×10^-21 m²/V² at 491 nm determined by THG [5]. The identification of a better phase-matched crystal with a higher cubic nonlinearity is then an open issue for the achievement of quantum experiments.

A good candidate seems to be TiO₂ rutile. It belongs to the centrosymmetric tetragonal crystal class 4/mmm and to the positive uniaxial optical class, i.e. n_e>n_o where no and ne denote the principal ordinary and extraordinary refractive indices. The χ⁽³⁾ third order electric susceptibility tensor of rutile has the four following independent elements under Kleinman symmetry [6]:

The indices x, y and z refer to the dielectric frame where the z-axis is oriented along the quaternary axis. The χ_ij correspond to the contracted notation according to the standard convention. The third order effective coefficient magnitude ranges between 13.0×10^-21m²/V² and 16.0×10^-21m²/V² according to nonlinear index measurements performed by degenerate four-wave mixing (DFWM) or by nearly degenerate three-wave mixing (TWM) methods respectively [7, 8], and according to non phase-matched THG [9]. Furthermore rutile exhibits phase-matching: the THG ω(e)+ω(e)+ω(e)→3ω(o), where ω(o) and ω(e) denote the circular frequencies of the ordinary and extraordinary waves respectively, was investigated in a (110) oriented plate, which lead to the determination of the phase-matching angle corresponding to a fundamental wavelength of λ_ω=1900 nm [10].

The present work deals with a quantitative investigation of collinear phase-matched third order parametric interactions in a TiO₂ rutile single crystal involving two ordinary waves (o) and two extraordinary waves (e) : the THG [ω(e)+ω(e)+ω(o)→3ω(o)] and the TPG [ω₄(o)→ω₁(e)+ω₂(e)+ω₃(o)]. Collinear phase-matching corresponds to Δk=0 where the mismatch parameter Δk, is defined as:

with (ω_a, ω_b, ω_c, ω_d)=(3ω, ω, ω, ω) for THG and (ω_a, ω_b, ω_c, ω_d)=(ω₄, ω₁, ω₂, ω₃) for TPG; k_o,e(ω_i)=[ω_i/c] n_o,e(ω_i) are the wave vectors where no,e correspond to the ordinary and extraordinary refractive indices in the considered direction of propagation. The associated effective coefficient is given by [6]:

The indices i, j, k and l refer to the dielectric frame (x, y, z). $e_{\alpha }^{\beta }$ , with α=(i, j, k, l) and β=(o,e), is the α Cartesian coordinate of the unit vector of the ordinary (o) or extraordinary (e) electric field. χ ijkl refers to the third order nonlinear coefficients given by (1).

We performed THG measurements on a 1mm-thick (100) rutile crystal provided by MTI Corporation. A Continuum Panther OPO pumped by a tripled Nd:YAG laser (10-Hz repetition-rate, 5-ns half-width at 1/e²) and tuneable from 400 nm to 2400 nm, is used as the fundamental beam for the THG experiments. The beam is focused with a 10-mm-focal length in the rutile plate. The optimization of the THG conversion efficiency is made by adjusting the polarization of the fundamental wave with an achromatic half wave plate, and by translating the crystal with respect to the fundamental beam waist. A collecting lens that is placed behind the crystal allows the emergent beams to be focused on a prism. The third harmonic (TH) beam is separated from the non converted fundamental by the prism and a long-wavelength rejection filter. The TH power is measured with a silicon detector and the corresponding wavelength is determined by using a Chromex 250SM monochromator. The THG fundamental wavelength, corresponding to the maximum of the THG conversion efficiency, is measured at ${\mathrm{\lambda }}_{\mathrm{\omega }}^{\text{PM}}$=1839.6 nm. Calculations made from available dispersion equations of rutile lead to smaller values of ${\mathrm{\lambda }}_{\mathrm{\omega }}^{\text{PM}}$, which ranges between 1752 nm and 1794 nm [11–13]. We refined the dispersion equations of the ordinary and extraordinary principal refractive indices of rutile by taking into account our phase-matching measurement. We found the following equations:

The measured spectral acceptance (full width of the tuning curve at 0.405 of the maximum) is L.Δλ_ω=0.47nm.cm, as shown in Fig. 1 where the normalized TH intensity I(ξ)/I(ξ^PM) is plotted as a function of the fundamental wavelength ξ=λ_ω from either side of the phasematching fundamental wavelength ξ^PM=${\mathrm{\lambda }}_{\mathrm{\omega }}^{\text{PM}}$=1839.6 nm.

 

Fig. 1. Normalized generated third harmonic intensity as a function of the fundamental wavelength around phase-matching. The squares are the experimental data and the line is the calculation from the refined dispersion equations of the principal refractive indices.

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As shown in Fig. 1, there is a good agreement between the measurement and the calculation from the dispersion Eq. (4) and the following general expression :

where L is the crystal length, ξ is the dispersive parameter, and Δk is the mismatch parameter given by Eq. (2).

The fundamental wavelength being kept maintained at the phase-matching value ${\mathrm{\lambda }}_{\mathrm{\omega }}^{\text{PM}}$, the rutile plate is then rotated on it-self around the y-axis of the dielectric frame (x, y, z) in order to determine the θ angular acceptance associated with the phase-matching direction [100], i.e. the x-axis with the spherical coordinates θ=90° and ϕ=0°: the measured angular acceptance is equal to L.Δθ=0.42°.cm as shown in Fig. 2; it is close to the theoretical value calculated with relation (5), where ξ=θ and ξ^PM=90°, and with the dispersion Eq. (4). Note that L.Δϕ is infinite because rutile is an uniaxial crystal.

 

Fig. 2. Normalized generated third harmonic intensity as a function of the direction of propagation : θ is the angle between the z-axis and the x-axis of the dielectric frame (x,y,z). The squares are the experimental data and the line is the calculation from the refined dispersion equations of the principal refractive indices.

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The very good agreement between the measured and calculated angular and spectral acceptances indicates that the refined dispersion equations are valid over the concerned wavelength range on one hand, and that the whole crystal length L corresponds to the effective interaction length on the other hand. This last point is of prime importance for the determination of the effective coefficient from frequency conversion measurements.

The THG [ω(e)+ω(e)+ω(o)→3ω(o)] in rutile was compared with a THG of same type in a 1 mm-thick (100) KTP crystal phase-matched at λ_ω=1618 nm. In the two cases, there is not spatial walk-off because the waves propagate along a principal axis of the dielectric frame. The THG energy conversion efficiencies, η_THG=u_3ω/u_ω, of KTP and TiO₂ are measured by using the same experimental setup; they are given in Fig. 3 as a function of the incident fundamental intensity, I_ω. The measured efficiencies are weak because the considered fundamental intensities are very low, below 50 MW/cm². For both crystals, η_THG versus I_ω exhibit a quadratic behavior as expected by the theory.

 

Fig. 3. Third harmonic energy conversion efficiency as a function of the fundamental intensity in a 1-mm-long TiO₂ crystal and a 1-mm-long KTP crystal phase-matched along the x-axis. The squares are the experimental data and the lines correspond to the calculation.

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The exact calculation of η_THG under the plane-wave limit and the undepleted pump approximation, in the case of temporally and spatially Gaussian beams that are phase-matched in a direction without spatial and temporal walk-off gives:

The different T_a coefficients correspond to the Fresnel transmissions of the interacting waves, L is the crystal length, and F_OM is the figure of merit expressed as:

χ_eff is the THG effective coefficient that reduces to χ₁₆ (3ω) for TiO₂ and χ₂₄ (3ω) for KTP according to (1) and (3) for the considered phase-matching type and direction of propagation. The corresponding refractive indices are (n₁, n₂, n₃, n₄)≡(n_e, n_e, n_o, n_o) for TiO₂ and (n₁, n₂, n₃, n₄)≡(n_z, n_z, n_y, n_y) for KTP.

Equation (5) can be applied to our experimental conditions since the Rayleigh length of the focused beam is larger than the crystal length, the walk-off angle is nil, and the group velocity dispersion is negligible in nanosecond regime. FOM of TiO₂ is found to be 7.5 larger than that of KTP according to Fig. 3 and Eq. (5). Then from Eq. (6) and the magnitude of χ₂₄ of KTP [5], we find that χ₁₆=5.0×10^-21 m²/V² for TiO₂ at λ_3ω=613.2 nm. This value is smaller than those previously determined by nonlinear index measurements [7–9], which can be explained in part by the fact that the involved coefficients of the χ⁽³⁾ tensor are different. The calculation of χ₁₆ at any wavelength can be done by considering the cubic Miller coefficient Δ₁₆, which is a non dispersive parameter given by [14]:

We find Δ₁₆=3.5×10^-24 m²/V² from Eq. (7) and the magnitude of χ₁₆ measured at λ_3ω=613.2nm.

The present measurements of F_OM and dispersion equations show us that TiO₂ rutile is a very promising crystal for cubic frequency conversion when compared with the previous performances of KTP [1, 2]. Firstly, rutile should lead to a phase-matched THG [1839.6 nm (e)+1839.6 nm (e)+1839.6 nm (o)→613.2 nm (o)] conversion efficiency of about 30% for a fundamental intensity of few GW/cm² and a crystal length of 10 mm. Such high conversion efficiency should be enough to perform statistical properties measurements of both fundamental and TH beams. Note that this expected efficiency would be very close to that obtained with cascading quadratic processes in KTP: [λ_ω+λ_ω→λ_2ω] coupled with [λ_ω+λ_2ω→λ_3ω]. Secondly, the calculation from Eq. (2), by setting Δk=0, and the dispersion Eq. (4) indicates interesting situations of phase-matching for the TPG [${\mathrm{\lambda }}_4^{\mathrm{o}}$→${\mathrm{\lambda }}_1^{\mathrm{e}}$+${\mathrm{\lambda }}_2^{\mathrm{e}}$+${\mathrm{\lambda }}_3^{\mathrm{o}}$], pumped at ${\mathrm{\lambda }}_4^{\mathrm{o}}$=532nm, as shown in Fig. 4 in the space of coordinates (${\mathrm{\lambda }}_1^{\mathrm{e}}$,${\mathrm{\lambda }}_2^{\mathrm{e}}$).

 

Fig. 4. Calculated phase-matching curves of TPG [${\mathrm{\lambda }}_4^{\mathrm{o}}$→${\mathrm{\lambda }}_1^{\mathrm{e}}$+${\mathrm{\lambda }}_2^{\mathrm{e}}$+${\mathrm{\lambda }}_3^{\mathrm{o}}$] in TiO₂ rutile. The indices o and e refer to the ordinary and extraordinary polarizations respectively. The three dotted lines correspond to the situations where two wavelengths are equal. Point D corresponds to the degeneracy in wavelength, i.e. ${\mathrm{\lambda }}_1^{\mathrm{e}}$=${\mathrm{\lambda }}_2^{\mathrm{e}}$=${\mathrm{\lambda }}_3^{\mathrm{o}}$. Each continuous line refers to phasematching directions of equal θ angles. The points A and B are defined by the intersections between the phase-matching curves at θ=90° and the locations where ${\mathrm{\lambda }}_1^{\mathrm{e}}$=${\mathrm{\lambda }}_3^{\mathrm{o}}$ and ${\mathrm{\lambda }}_2^{\mathrm{e}}$=${\mathrm{\lambda }}_3^{\mathrm{o}}$.

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The two phase-matching areas are symmetrical according to the diagonal, where ${\mathrm{\lambda }}_1^{\mathrm{e}}$=${\mathrm{\lambda }}_2^{\mathrm{e}}$, since these two wavelengths can be permuted due to the identity of their polarization states. Note that the value of ${\mathrm{\lambda }}_3^{\mathrm{o}}$ corresponding to a given set (${\mathrm{\lambda }}_1^{\mathrm{e}}$, ${\mathrm{\lambda }}_2^{\mathrm{e}}$) is easily obtained by using photon-energy conservation, i.e. (${\mathrm{\lambda }}_3^{\mathrm{o}}$)^-1=(${\mathrm{\lambda }}_4^{\mathrm{o}}$)^-1-(${\mathrm{\lambda }}_1^{\mathrm{e}}$)^-1-(${\mathrm{\lambda }}_2^{\mathrm{e}}$)^-1. The two phase-matching areas are formed by a framework of curves, each of them corresponding to a given phasematching angle. The locations where ${\mathrm{\lambda }}_3^{\mathrm{o}}$=${\mathrm{\lambda }}_1^{\mathrm{e}}$ and ${\mathrm{\lambda }}_3^{\mathrm{o}}$=${\mathrm{\lambda }}_2^{\mathrm{e}}$ given in Fig. 4 allow us to identify two interesting situations that are marked out by the two symmetrical points A and B: they correspond to the equivalent triple states [${\mathrm{\lambda }}_1^{\mathrm{e}}$=2940 nm, ${\mathrm{\lambda }}_2^{\mathrm{e}}$=830 nm, ${\mathrm{\lambda }}_3^{\mathrm{o}}$=2940 nm], and [${\mathrm{\lambda }}_1^{\mathrm{e}}$=830 nm, ${\mathrm{\lambda }}_2^{\mathrm{e}}$=2940 nm, ${\mathrm{\lambda }}_3^{\mathrm{o}}$=2940 nm], respectively. These schemes are relevant from the experimental point of view because the spatial walk-off is nil and they facilitate the necessary stimulation of the TPG, by two photons, as previously demonstrated in the case of KTP [2]: the double injection in TiO₂ can be done at a single wavelength, i.e. at 2940 nm, in the ordinary and extraordinary polarizations states. In this situation and according to the magnitude of F_OM of TiO₂ compared with that of KTP, it is possible to expect more than 10¹⁴ triple photons per pulse for few tenths of GW/cm² and a crystal length of 10 mm, which is suited to correlation measurements [15].