On the use of structured light in nonlinear optics studies of the symmetry group of a crystal

We put forward a technique that allows to extract information about the symmetry group to which certain nonlinear crystals belong using a single illuminating beam. It provides such information by considering the outcome of a nonlinear optics process characterized by the electric nonlinear susceptibility tensor, whose structure is dictated by such symmetry group. As an example, we consider the process of spontaneous parametric down-conversion, when it is pumped with a special type of Bessel beam. The observation of the spatial angular dependence of the lower-frequency generated light provides direct information about the symmetry group of the crystal. We should stress that the choice of the appropriate illumination is of paramount importance for unveiling the sought-after information.

where E i (i = x, y, z) are the components of the electric field, χ ( ) ij 1 is the linear susceptibility tensor that determines the linear polarization response of the material, and χ ( ) ijk 2 , χ ( ) ijkl 3 … are nonlinear susceptibility tensors of different ranks responsible for the nonlinear polarization generated in the medium.
Here we show two main things. Firstly, that it is possible to perform complementary symmetry studies of nonlinear materials by using nonlinear optics processes besides second harmonic generation (SHG), such as spontaneous parametric down-conversion (SPDC). In SPDC, an intense pump beam with frequency ω p interacts with the atoms or molecules of a second order nonlinear crystal with nonlinear coefficient χ ( ) ijk 2 . In the process, a flow of paired photons is generated, the signal and idler, with central frequencies ω s and ω i , such that ω s + ω i = ω p .
Secondly, that instead of illuminating the nonlinear crystal with paraxial Gaussian beams that propagate along a myriad of different directions, as it is usually done in similar cases 8 , it is more advantageous to choose as illuminating beam a non-paraxial optical beam. The use of a laser source with fast switching of its pointing direction might be technically cumbersome, and can severely limit the applicability of the method due to the need to control mechanical noise. However, the use of an appropriately designed non-paraxial beam allows to unveil critical information on the crystal symmetry group in a experiment using a single structured pump beam.

SPDC using structured pump beams designed for crystallographic studies
Under standard conditions 11 , most SPDC configurations consist of a Gaussian pump beam that impinges normally onto the surface of the nonlinear crystal, which is endowed with a second order nonlinear electric susceptibility χ ( ) ijk 2 . The cut angle of the crystal is chosen to guarantee the fulfillment of the phase matching conditions, which are equivalent to the conservation of energy and momentum of the photons involved in the SPDC process. In most cases, the pump beam is linearly polarized. Under these circumstances, the distribution of wave vectors of the signal and idler photons, for given frequencies of the resulting photon pairs, is restricted to a single cone (in the case of type I SPDC) or to two cones (in the case of type II SPDC) with axes symmetrically arranged relative to the pump beam 12 . These conical distributions are observed for different crystal symmetries, so they do not bear useful information about the point group to which the crystal belongs. This is because in the configuration considered, the nonlinear process addresses only certain elements of the nonlinear susceptibility tensor.
We show that a different configuration can lead to a spatial distribution of the photon pairs that directly reflects the symmetry group of the crystal. For the sake of simplicity, we will consider the case of an uniaxial birefringent crystal with its optic axis parallel to the normal of its surface. The pump beam that illuminates the crystal is a vectorial Bessel mode 13,14 , prepared to guarantee a vectorial extraordinary character inside the nonlinear media. This can be done by choosing a transverse magnetic (TM) mode with its main propagation direction coincident with the axis of the nonlinear crystal.
A Bessel beam is formed by the superposition of plane waves with wave vectors confined in a cone and with a circular cylindrical symmetry on the angular spectrum. In this way, the cylindrical symmetry of the photon pairs produced in the SPDC process is directly broken by the intrinsic symmetry of the crystal, which manifests on the spatial distribution of the resulting photon pairs. The geometry of the set-up is illustrated in Fig. 1. The axicon angle ϕ a of the incident vectorial Bessel mode is the parameter to be optimized for both the fulfilment of Note that choosing a Bessel beam as a pump of the nonlinear process is equivalent to observing the crystal structure simultaneously for many different angles. In this configuration, SPDC (or any other nonlinear optical process) is sensible to all the components of the nonlinear optical susceptibility tensor, including that along the main direction of propagation of the pump beam; that is due to the fact that TM modes acquire a significant component of its electric field along their main direction of propagation as they depart from the paraxial limit 14 .
In this paper we consider vectorial monochromatic beams, i.e., beams with an electric field  ω where ω is the angular frequency of the beam, r designates the spatial location, t is time, E m is the vectorial spatially-varying amplitude of the beam and m is related to the orbital angular momentum of the beam.
In free space, the electric field of a transverse-magnetic (TM) Bessel beam of order m, which is an exact solution of Maxwell's equations, can be written 13 as a sum of plane-waves with equal amplitude and wavevector confined in a cone around the z-axis, the main direction of propagation of the beam, with axicon angle ϕ a , i.e., is the polarization of each plane-wave with wavevector k; this vector can be written as a sum of a longitudinal vector with amplitude ω ϕ = ( / ) k c cos z a , and a transverse wavevector cos sin a q q ; φ q is the polar angle around the z-axis of each wavevector k, it can take any value between 0 and 2π, with zero giving the orientation of the x-axis. Notice that even though the polarization of each k-wave is perpendicular to k 15 , the superposition of all k-waves yields a beam with a non-zero field component along the main direction of propagation ( ) z . For paraxial beams (ϕ a small) and m = 0, one obtains the so-called radial modes 13,16 .
In experiments, one never generates such an ideal beam. However, one can produce vectorial Bessel beams close to the one described by Eq. (2) by superposing scalar Bessel modes with the proper topological charge m and polarization 17 . Scalar Bessel modes were generated for the first time by illuminating, with a quasi-plane wave, an annular slit placed in the back focal plane of a lens 18 , later an alternative to this method was found on the usage of a lens with a conical surface -called an axicon-with cone angle ϕ α 19 ; other options for the generation of Bessel beams are implemented using a space light modulator, or a structured diffraction plate. In most circumstances, the generated beam is designed to be close to the paraxial limit, i. e. ϕ  1 a , even though there is no fundamental limitation on achieving higher values of ϕ a . Under experimental conditions, the wavevectors k are no longer confined to a cone of angle ϕ a , with transverse wavevector q, but instead the wavevectors spread narrowly around a central value, q + Δ q, or equivalently, where V is the volume of interaction and χ ( ) 2 is the nonlinear tensor that characterizes the nonlinear electric response of the material. Let us consider as example a type I (eoo) SPDC process in an uniaxial crystal 21 . A configuration that will give us information about the symmetry of the crystal is the one where the pump beam propagates inside the nonlinear crystal along the optic axis ( = )ĉ z . The signal and idler waves propagate as ordinary waves. For the case of an intense classical extraordinary pump beam that generates ordinary signal and idler photons, the pump beam, and the electric field operators for signal and idler photons write 11   , respectively. Since in most situations the nonlinear interaction is weak, we can obtain an accurate quantum description by calculating the first-order solution of the Schrödinger equation, i.e., The interaction Hamiltonian is effectively zero when the classical beams that pump the nonlinear process are zero, so that, the time of integration can be extended to t = ∞. The quantum state of the down-converted photons at the output face of the nonlinear crystal, neglecting for the sake of simplicity the contribution from the vacuum term, can be written as It is important to stress out the crucial role that the tensorial character of χ ( ) 2 plays in the SPDC configuration considered. Most experiments that make use of SPDC, due to the paraxial character of the pump, can be described using an effective nonlinear index 21  Let us illustrate these results and how they help to determine the symmetry group of the crystal. In Fig. 2, we exemplify the angular dependence of the flux rate of signal photons (singles detections as given by Eq. (11) Fig. 2(a). As | | q p increases the down-converted photons are emitted in a wider region in the transverse wavevector space, and the symmetry of the crystal becomes more evident; so that the crystallographic structure is directly observable for | | q p values above a crystal dependent threshold. In Fig. 2, that threshold has already been exceeded at q p = 0.15 μm −1 for GaSe, and crystallographic traces are evident in its angular spectra. GaSe is a layered crystal that belongs to the space group D 3h ; an individual layer consists of four planes of Se − Ga − Ga − Se with the Ga − Ga bond normal to the layer plane arranged on a hexagonal lattice, and the Se anions located in the eclipsed conformation when viewed along the optic axis 9 ; these leads to the hexagonal shape of the corresponding maximum flux rate regions. A key point concerning the determination of the symmetry group of a crystal comes from the fact that structurally characteristic patterns on the flux rates R s (p) arise from observations with a common moderate axicon angle as illustrated in Fig. 2. As a consequence a direct comparison between numerical simulations for moderate -though, in general, out of the paraxial regime− | | q p values and the observed results would yield the desired identification of the symmetry group. This comparison can be automatized in actual experimental set ups. Properties such as the radius of the concentric rings, the values of | | q p at which the visibility of the symmetry of the crystal is significant, and the maximum value of R s , could be used to determine some characteristics of the linear susceptibility tensor and the second-order nonlinear susceptibility tensors. For instance, KDP (Fig. 2(b)), CsH 2 AsO 4 (Fig. 2(c)) and DKDP (Fig. 2(i)) belong to the same point group, 42m, but they have slightly different values of ε o , ε e and the relevant components of χ ( ) lmn 2 . This manifests in general common properties between the angular spectrum of those crystals, but there are still measurable differences of the relative values of R s for each one of the crystals, for small values of p. These differences are mainly due to a higher sensitivity of CsH 2 AsO 4 to the electric field component along the z-axis.
The coincidence detections, given by Eq. (10) provide complementary useful information about the symmetry group of the medium under investigation. A direct calculation shows 22 that for ω | | /  c q p p , the detection of a signal photon with transverse wavevector p is accompanied by the detection of an idler photon that is confined to a ring with radius ε ε ∼ | | / q q p o e around q = − p. As q p increases, the symmetry of the crystal becomes more visible in the angular distribution of idler photons, the flux rate becomes inhomogeneous along the ring with a structure that depends on the symmetry of the crystal, and which is compatible with the conditional generation of stationary Bessel photons, i. e. photons with an angular spectra resulting from superpositions of m and − m Bessel modes. The conservation of optical angular momentum restricts the possible values of m i and m s depending on which components of the nonlinear tensor χ ( ) 2 are different from zero 22 , a property that is directly related to the point symmetry group of the crystal. In Fig. 3, this effect is illustrated for various types of crystals. In the GaSe case (Fig. 3(a)), the conditional angular spectrum is a superposition of m i = ± 1 Bessel modes. This agrees with the option m s = 0 and m i = ± 1, expected 22  2 ; the resulting maximal conditional spectrum shown in Fig. 3(b) is a superposition of m i = ± 1

Discussion
Summarizing, we have shown that, under the adequate experimental set up, the correlations of the photons emitted in a SPDC process, contain crystallographic information that can be accessed by measurements of the SPDC angular spectra and the conditional angular spectra. It is expected that a complete characterization of the twin photons correlations as a function of the properties of the structured pump beam could be used to obtain precise measurements of the first and second order electric susceptibility tensors. The results shown in this manuscript refer to the particular case of a beam impinging into an uniaxial crystal with the optic axis parallel to the normal of the crystal surface, other orientations of the relevant axes deserve further analysis. Preliminary numerical results show that the angular spectra loses gradually the direct visualization traces of the symmetry group as the orientation of the optic axis departs from the normal of the crystal surface. This makes the identification of the symmetry group a more computational demanding issue, that would be always based in the comparison between the observed angular spectra and the expected spectra from numerical simulations. Such a comparison would yield information not just on the symmetry group but also on the specific values of the electric susceptibility tensors. A similar treatment could be used for biaxial crystals.
It is also important to mention that the general procedure described in this work has facets that require a quantum analysis but also shares many features with its classical analogs. Phase matching conditions have clearly a classical interpretation; meanwhile, the conditional angular spectrum is by definition a quantum observable.
In general SPDC is a weak effect as noticed from the flux rates expected from moderate power of the pump beam. Recent advances in weak signal detection, e.g. the so called intensified cameras, make feasible the measurement of the flux rates shown in last section. Nevertheless, the effective pump power can be greatly enhanced using pulsed Bessel pump beams 25 yielding, as a consequence, great enhancement of the down converted photons flux rates. For the theoretical study of SPDC with ultrashort pump pulses, the formalism described in this work can be extended to incorporate effects of the spectral density α(ω P ) of the pump beam. Basically the coupling factor g, Eq. (8), becomes a frequency dependent term α(ω P )g that must be integrated over the involved frequency range. That kind of approach to SPDC has been studied experimentally in the context of Gaussian beams for which other interesting effects have been reported 26 .
Notice that the benefits arising from structured pump beams for nonlinear crystallography should not be restricted to the parametric down conversion process. Second harmonic generation is the most widely used nonlinear mechanism for crystallographic analyzes 7,8,27 . In fact, the adequate selection of the electromagnetic waves involved in SHG surmounts the intrinsic limitation of any nonlinear electromagnetic process induced by the bulk electric susceptibility χ ( ) ijk 2 : the necessity of dealing with noncentrosymmetric materials so that inversion symmetry is guaranteed. The key point is to induce combined electric dipole and magnetic dipole transitions 28,29 . Enhancement of this otherwise rare effect requires both the proper selection of the frequencies involved, so that the nonlinear process is resonant in both the incoming and outgoing light waves, and the usage of pulsed pump beams to achieve high intensities (typically pulses of tens of mJ in a pair of nanoseconds). External magnetic fields can also be used to manipulate the ferromagnetic domains. An outcome of such studies is the characterization of the magnetic point group of a material 27 . Nonlinear harmonic generation crystallography with pulsed Bessel beams could be an alternative to rotational anisotropy measurements, since using pulsed Bessel modes as a pump is equivalent to simultaneous measurements along different directions in a single shot. This idea deserves further studies which should have in mind tailoring of all the properties at hand of the conical superposition of plane waves depending on the susceptibility to be measured. They include: its spectral content α(ω), its angular spectra β(φ q ) (taken here as φ e im q , Eq. (2)), and its polarization.