Phase matching is known to enhance the nonlinear optical response in materials with a non-centrosymmetric crystallographic or electronic structure. In contrast, phase-matched frequency doubling driven by non-centrosymmetric magnetism that induces acentricity in otherwise centrosymmetric structures has not been reported yet. In our study we demonstrate the emergence of magnetically driven second-harmonic generation (SHG) with phase matching in MnWO4. The phase-matched wavelength for SHG can be tuned continuously between 450 nm to 630 nm with the conversion efficiency being determined by the refractive indices and their dispersion. Our findings reveal a new strategy towards magnetism-based conversion-materials and a route for controlling the nonlinear signal yield by acting primarily on the material’s spin degree of freedom rather than employing its electronic or structural properties.
© 2015 Optical Society of America
Optical frequency conversion due to non-linear light-matter interactions plays an important role in technological applications and for advanced materials characterization [1–3]. Certain optical wavelength regions, for instance, are not directly accessible with lasers so that frequency conversion, foremost frequency doubling, “second harmonic generation” (SHG), is required in order to cover these spectral regimes. Particularly high conversion efficiencies are achieved when the incident and the frequency-converted light propagate at the same phase velocity in non-linear materials, being known as optical phase matching. In the case of phase matching the whole illuminated crystal volume can contribute to the non-linear conversion process without back-conversion due to destructive interference . Phase matching has been intensively studied in materials with a non-centrosymmetric crystallographic structure, such as α-BaB2O4 (BBO) , KH2PO4 (KDP) , and BiB3O6 (BIBO) , and ferroelectrics including LiNbO3 [8,9], LiTaO3 , and KTiOPO4 . In contrast, phase matching in connection with magnetism has not yet been reported. The emergence of phase matching in centrosymmetric crystal structures with non-centrosymmetic magnetic order would significantly widen the current pool of conversion materials. Moreover, the magnetic order can be used as a control parameter to tailor the material’s performance and activate or de-activate its optical phase-matched response on demand.
Here we demonstrate magnetically driven SHG combined with optical phase matching in the frustrated magnet MnWO4. The linear optical properties of MnWO4 render phase matching possible, but its centrosymmetric crystal structure prohibits the emergence of electric-dipole-type SHG. Only in the spin-spiral phase, where the magnetic order breaks the inversion symmetry, such electric-dipole-type SHG signals are symmetry-allowed. The SHG response exhibits two spectral maxima associated with electronic intra- and inter-atomic excitations. By performing non-linear optical spectroscopy we show that the central wavelength of phase-matched frequency-doubled light can be continuously tuned within a range of 450 nm to 630 nm and we relate the obtained energy-dependent conversion efficiency to the linear optical properties of MnWO4.
One of the most fundamental nonlinear conversion processes whose efficiency strongly depends on optical phase matching is SHG. The corresponding process of optical frequency doubling in the electric-dipole approximation is described by the equationEq. (1) is symmetry-allowed in all media without a center of inversion (with exception of symmetry 432). The inversion symmetry may be broken due to a non-centrosymmetric arrangement of lattice, orbital, charge, or spin degrees of freedom . The highest conversion efficiency is achieved when the refractive indices are equal, i.e., n(ω) = n(2ω). This is denoted as phase matching.
2. Experimental details
For our study of magnetically enabled phase-matched frequency conversion we investigate the spin-spiral system manganese tungstate (MnWO4). The linear optical properties of this material are well-characterized  and its refractive indices allow for phase-matched optical frequency doubling. MnWO4 crystallizes in the monoclinic space group P2/c (point group 2/m), with the b-axis as twofold axis, as sketched in Fig. 1(a) . Thus, although the condition for collinear phase matching is fulfilled by the refractive indices, phase-matched SHG cannot occur because of the centrosymmetry of the material. Only in the magnetically ordered phase between 7 K and 12 K, where the helical order of Mn2+ spins breaks the inversion symmetry, as depicted in Fig. 1(a) , electric-dipole SHG as described by Eq. (1) becomes allowed and phase-matched frequency doubling can occur. As a byproduct of the magnetic order, the system further develops an improper electric polarization P of about5 nC/cm2, classifying MnWO4 as a spin-driven multiferroic . It is important to note, however, that P is a secondary order parameter that is a consequence and not the origin of the inversion symmetry breaking. Hence, this secondary polarization will be neglected in the following. In Figs. 1(b) and 1(c) we define different characteristic axes that will be used in this work. They refer to the optical indicatrix (X, Y, and Z, with n(X)<n(Y)<n(Z)) of MnWO4, as well as to the configuration we choose for our experiments.
Figure 2 reveals the crystallographic directions along which collinear phase matching can occur in MnWO4 at room temperature. The so-called Hobden plot in Fig. 2 is calculated based on the tabulated room-temperature refractive indices . The plot is derived from the cones formed by the directions along which phase matching, i.e., n(ω) = n(2ω), is obtained for a given frequency ω. These cones are stereographically projeced onto the XZ equatorial plane of the optical indicatrix . Directions for phase matching of type I (ss → f, s = slow wave, f = fast wave) with parallel polarization of the two fundamental light fields Ej and Ek are marked in red, while blue marks indicate the phase matching directions of type II (sf → f) with perpendicular polarization of the two fundamental waves. The Hobden plot reveals the strong dependence of the phase matching conditions on the orientation of the wave vector k(ω) of the fundamental waves, their polarization, and their photon energy.
In order to analyze the polarization dependence, we perform SHG measurements in a transmission setup described elsewhere . As light source we use an optical parametric oscillator (OPO) pumped by the frequency-tripled emission of a Nd:YAG laser, at a pulse length of 5 ns, which allows to tune the photon energy ħω of the incident light from 0.5 eV to 1.5 eV. The experiments are performed within the spin-spiral phase of MnWO4 at 7 K, i.e., at the temperature at which the magnetically induced SHG signal reaches its maximum [18,19]. The incident light is linearly polarized so that a rotation of the sample around its vertical baxis continuously changes the polarization direction of D(ω) from a to c. This setting allows us to measure the SHG intensity and conversion efficiency related to Pb(2ω) as function of the direction and polarization of the incident light field. A detailed illustration of the relationship between sample frame, crystallographic (a, b, c) and optical (X, Y, Z) coordinate system is depicted in Fig. 1(c): The surface normal of the sample is chosen to be parallel to the magnetic easy axis, measy, which lies in the ac-plane at an angle ∠(a,measy) = + 33.9° . The principal X-axis of the index ellipsoid n(ω,φ,θ) is parallel to the two-fold rotation axis b || X, while the Y and Z axes deviate by about −20° from the crystallographic a and c axes . These relations are important when calculating the internal direction k(ω) of the fundamental wave E(ω,φ) as function of the sample rotation by φ’ around the b || X direction .
Due to the optical absorption of MnWO4 with its maximum at about 2.20 eV, anomalous dispersion of the refractive indices is to be expected around this energy. This may lead to the occurrence of anomalous-dispersion phase-matched SHG, which was, as a general possibility for phase matching, already discussed in the early days of nonlinear optics [21,22]. Unfortunately, the prism method used for the determination of precise refractive indices of MnWO4  did not allow to cover the wavelength range below 850 nm (above 1.46 eV) because of the increasing absorption. So, precise refractive indices for the calculation of SHG phase matching conditions are not available and have to be estimated as discussed below.
3. Phase-matched SHG in MnWO4
Here, we begin our discussion with the investigation of the SHG response of MnWO4, which occurs in the polar magnetic phase between 7 K and 12 K only, as shown in Fig. 3(b). For normal incidence (i.e. k(ω) parallel to measy) the peak is centred at 2.32 eV and its temperature-dependence closely follows the behaviour of the magnetic order parameter with maximum intensity at 7 K .
The dependence of the SHG signal for different propagation directions of the incident light field E(ω) is presented in Fig. 3(a). The offset of the curves is proportional to the internal propagation direction of the light field which is parameterized by the angle φ = ∠(a, k(ω)) between the crystallographic axis a and k(ω). The angle φ can be calculated from φ’ and the refractive indices, see Figs. 1(b) and 1(c). Note that while the sample is rotated by angles φ’ between−40° and + 80°, the angle for the internal propagation direction of the fundamental wave φ varies between 7° and 51°. Peak energy and line width (as obtained by Lorentz fit) are indicated by the bar at the bottom of each SHG peak with the line colour reflecting the central wavelength.
When rotating the sample, a blue shift is observed for decreasing φ whereas the increase of φ leads to a red shift. The phase-matched SHG signal covers a wide spectral range from red to blue. Including the cases of k(ω) || c and k(ω) || a  the wavelength range spans from 450 nm to 630 nm. The shift in SHG central energy with φ is almost linear with a slope of 11.8 meV/degree. These experimental results are in qualitative agreement with the pronounced polarization and wavelength dependence of phase-matched SHG reflected by the Hobden plot in Fig. 2.
Note that the dashed line in Fig. 3 compares the calculated phase matching angles expected at room temperature with the measured values in the magnetically ordered phase at 7 K. Despite the significant temperature difference, calculation and experiment reveal the same trend, indicating that the refractive indices, and thus the phase matching condition, do not drastically change toward low temperature. The increasing deviation between experiment and theory toward higher photon energy can be attributed to the resonant increase of the refractive indices near the charge-transfer gap of MnWO4 . Extrapolating this increase from the measured low-temperature values under the assumption of no fundamental changes of the dispersion by the increasing absorption is a rough approximation. Consequently, the agreement between the measured and extrapolated phase matching is not perfect. Nevertheless the reproduction of the basic trend is surprisingly good. In any case, Fig. 3 demonstrates that the wavelength of the magnetically driven phase-matched SHG response in MnWO4 can readily be tuned, covering almost the entire visible spectral range.
The optical absorption of MnWO4 has a pronounced influence on the SHG conversion efficiency. This relation is depicted in Fig. 4(a), where we present the linear absorption coefficient α(2ω) (solid line) of the SHG wave together with the SHG intensity at ωc (open circles) as extracted from the data in Fig. 3. The data reveals that the SHG signal is roughly proportional to the linear absorption at 2ω for energies below about 2.50 eV. For higher energies the SHG signal is strongly suppressed.
This experimentally observed spectral dependence of the SHG conversion efficiency can be explained based on the electronic structure of MnWO4. Peak-like features of the envelope relate to transitions between crystal-field-split energy levels of the Mn2+ ions (electronic configuration d5) at 2.20 eV and 2.65 eV as indicated by the black arrows in Fig. 4. In addition, the onset of charge-transfer excitations above 2.50 eV [19,23] leads to a re-absorption of the frequency doubled light, explaining the drastic decrease observed in the SHG yield toward higher energy. Besides codetermining the spectral dependence of the SHG conversion efficiency, the linear absorption properties of MnWO4 also have an impact on the line width of the SHG peaks. The latter is presented in Fig. 4(b) and is consistent with the solution of the non-linear wave equation .
In summary, we observed a frequency-doubled signal in the spin-spiral phase of MnWO4, where the long-range magnetic order breaks the spatial inversion symmetry and gives rise to SHG which is greatly enhanced by optical phase matching. We showed that by crystal rotation the phase-matching wavelength of the SHG light can be tuned continuously between 450 nm and 630 nm, with the linear optical dispersion and absorption codetermining the conversion efficiency. Within the investigated spectral range we found two maxima in the phase-matched SHG response originating from intra- and inter-atomic excitations, respectively. Our findings demonstrate the feasibility of magnetic insulators with non-centrosymmetric spin order as nonlinear medium for phase-matched light conversion. In the present case the magnetically driven symmetry lowering arises at a technologically unfeasible temperature. Successful and ongoing attemps to obtain spin-driven acentricity under abient conditions are continuously improving these odds [25,26]. The application of materials with non-centrosymmetic spin textures widens the pool of suitable conversion-materials and expands the currently accessible parameter space. In addition, with magnetism being responsible for the acentricity and, hence, emergence of phase-matched SHG in otherwise centrosymmetric media (or new phased-matched SHG components in non-centrosymmetric systems), we identified a handle for controlling the conversion efficiency of nonlinear optical processes by acting primarily on the material’s spin degree of freedom rather than electric or structural properties.
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