## Abstract

We experimentally investigated third harmonic generation in TiO_{2} rutile single crystal, including phase-matching and cubic nonlinearity. We refined the dispersion equations of rutile and we demonstrated that this crystal allows angular non critical phase-matching at useful wavelengths, with a figure of merit 7.5 times that of KTiOPO_{4}. The measured cubic non linear coefficient and the corresponding Miller coefficients are : χ_{16}=5.0×10^{-21} m^{2}/V^{2} at 613.2 nm and Δ_{16}=3.5×10^{-24} m^{2}/V^{2}. These results are used to predict the phase-matching conditions and the efficiency of triple photon generation in rutile.

©2006 Optical Society of America

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 KTiOPO_{4} (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 3.34×10^{13} 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 : χ_{11}=2.32×10^{-21} m^{2}/V^{2} and χ_{22}=1.96×10^{-21} m^{2}/V^{2} at 1064 nm measured by z-scan experiments [4]; χ_{24}=1.46×10^{-21} m^{2}/V^{2} at 539 nm and χ_{16}=0.80×10^{-21} m^{2}/V^{2} 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 TiO_{2} 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]:

$${\chi}_{\mathrm{xxzz}}={\chi}_{\mathrm{xzxz}}={\chi}_{\mathrm{xzzx}}={\chi}_{\mathrm{yyzz}}={\chi}_{\mathrm{yzyz}}={\chi}_{\mathrm{yzzy}}={\chi}_{\mathrm{zyyz}}={\chi}_{\mathrm{zyzy}}={\chi}_{\mathrm{zzyy}}={\chi}_{\mathrm{zxxz}}={\chi}_{\mathrm{zxzx}}={\chi}_{\mathrm{zzxx}}\left(\equiv {\chi}_{16}\right)$$

$${\chi}_{\mathrm{xxyy}}={\chi}_{\mathrm{xyxy}}={\chi}_{\mathrm{xyyx}}={\chi}_{\mathrm{yxxy}}={\chi}_{\mathrm{yxyx}}={\chi}_{\mathrm{yyxx}}\left(\equiv {\chi}_{18}\right)$$

$${\chi}_{\mathrm{zzzz}}\left(\equiv {\chi}_{33}\right)$$

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^{-21}m^{2}/V^{2} and 16.0×10^{-21}m^{2}/V^{2} 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 TiO_{2} 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 [ω_{4}(o)→ω_{1}(e)+ω_{2}(e)+ω_{3}(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})=(ω_{4}, ω_{1}, ω_{2}, ω_{3}) 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/e^{2}) 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:

$${n}_{e}^{2}\left(\lambda \right)=7.07355+\frac{0.29834}{{\lambda}^{2}-0.07957}$$

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.

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.

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 TiO_{2} 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/cm^{2}. For both crystals, η_{THG} versus I_{ω} exhibit a quadratic behavior as expected by the theory.

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 χ_{16} (3ω) for TiO_{2} and χ_{24} (3ω) for KTP according to (1) and (3) for the considered phase-matching type and direction of propagation. The corresponding refractive indices are (n_{1}, n_{2}, n_{3}, n_{4})≡(n_{e}, n_{e}, n_{o}, n_{o}) for TiO_{2} and (n_{1}, n_{2}, n_{3}, n_{4})≡(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 TiO_{2} 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 χ_{24} of KTP [5], we find that χ_{16}=5.0×10^{-21} m^{2}/V^{2} for TiO_{2} 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 χ_{16} at any wavelength can be done by considering the cubic Miller coefficient Δ_{16}, which is a non dispersive parameter given by [14]:

We find Δ_{16}=3.5×10^{-24} m^{2}/V^{2} from Eq. (7) and the magnitude of χ_{16} measured at λ_{3ω}=613.2nm.

The present measurements of F_{OM} and dispersion equations show us that TiO_{2} 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/cm^{2} 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 (${\mathrm{\lambda}}_{1}^{\mathrm{e}}$,${\mathrm{\lambda}}_{2}^{\mathrm{e}}$).

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 TiO_{2} 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 TiO_{2} compared with that of KTP, it is possible to expect more than 10^{14} triple photons per pulse for few tenths of GW/cm^{2} and a crystal length of 10 mm, which is suited to correlation measurements [15].

## Acknowledgments

The authors whish to thank Martin M. Fejer from Stanford University for stimulating discussions.

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