Time evolution of Symmetry-forbidden Raman lines activated by photorefractivity

Transmission Raman spectroscopy experiments were performed on iron doped congruent lithium niobate within two –in principle equivalent- configurations, namely Y(ZX)Y and Y(XZ)Y. While the former respects the Raman selection rules, the other configuration gives a time dependent spectrum that, after a transient time of several minutes, finally results in a mixture of expected and forbidden modes. This breaking of Raman selection rules is caused by the spontaneous conversion of a part of the ordinarily polarized pump beam into an extraordinarily polarized beam by photorefractive anisotropic self-scattering. A numerical modelling of the phenomenon is developed and fairly reproduces the time dependence of conversion energy.

www.nature.com/scientificreports www.nature.com/scientificreports/ that one of the two equivalent configurations produces, after a transient time of the order of few minutes, a completely different Raman  modes. This can be interpreted as the partly conversion of ordinary polarization into the extraordinary polarization. We attempt a quantitative modelling of this phenomenon and in particular of its temporal dependence.

Experimental Details and Results
The Fe:LN crystal sample studied here was grown by the Czochralski method from a congruent melt with concentration of . wt 0 03 % Fe. Iron ions were added to the melt in the form of Fe O 2 3 oxide. The sample was prepared in an optical-grade parallelepiped with a thickness of . mm 4 7 along the beam propagation direction and treated at °1050 C for 5 hours in oxidizing atmosphere to improve the transparency. The optical absorption of the sample was measured with a JascoV670 spectrophotometer to be α = .
− cm 0 75 1 . Transmission Raman scattering measurements were performed with a laser source at nm 532 and a fixed power of mW 18 . The TEM 00 laser mode was collimated with the help of a set of lenses to create a waist radius and transmitted through the sample.
The incident beam is propagating along the crystal Y axis, and can be polarized along the Z or X axis by using a λ/2 waveplate. An iris diaphragm is placed behind the sample to define the total angular acceptance of the light scattered within the sample. The transmitted beam is passed through a polarizer crossed with respect to the incident polarization, collected by an appropriate optics and measured by a spectrometer HORIBA Jobin Yvon iHR320. An edge filter cutting at − cm 200 1 was utilized to remove the strong excitation line. The spectral resolution of the spectrometer is around − cm 2 1 . The configurations in trans Raman scattering Y(ZX)Y and Y(XZ)Y were successively recorded and analyzed. The far-field image of the transmitted beam was recorded as well within both configurations. For this, the image of the emerging light was projected on a screen at a distance of about 35 cm behind the rear face of the sample. Both the Raman and far-field pattern measurements were performed as function of time under a constant pump power. For comparison of Raman data, the same sample was also measured in the usual backscattering Y ZX Y ( ) and Y XZ Y ( ) configurations using a HORIBA Jobin Yvon Aramis spectrometer with a spectral resolution of about − cm 1 1

. Only E TO
[ ] modes are expected in all these four configurations according to Raman selection rules 18 and since the Raman tensor is symmetric, all configurations are in principle fully equivalent.
The transmitted Raman spectra are reported in Fig. 1 after different time intervals, and compared with corresponding back-scattered spectra, used as references. Whereas only the lines corresponding to E TO [ ] phonons are detected at any time as expected in the Y ZX Y ( ) configuration, the spectra in the Y XZ Y ( ) configuration show a large evolution with time. Just at the beginning of illumination, the spectrum in the transmission geometry is very sample. At the top: backscattering Raman spectra independent of time are reported for comparison. We can note that the lines are any time, broader in the transmission spectra than in the back-scattering. This is due to the geometry and to the lower resolution of the spectrometer.
www.nature.com/scientificreports www.nature.com/scientificreports/ similar to the corresponding spectrum in back-scattering, but rapidly new lines appear and increase in intensity with time. After one minute of illumination, the spectrum completely differs from expected E TO [ ]. The unexpected lines cannot be attributed to the direct effect of the incorporation of iron into the lattice. First, any compositionally-induced modification of the Raman profile is expected to be static and cannot possibly explain a time-evolution on the scale of the minutes such as the one here reported. Moreover, Fe doping in the concentration used in this work is expected only to induce a slight change in the position and width of the Raman line 26 . These new lines in fact arise from another mode symmetry and are clearly assigned as due to A TO [ ] 1 phonons 18 . It is to be mentioned that A TO [ ] 1 modes are polarized along the extraordinary Z axis and normal to E TO [ ] modes. Generally, the detection of unexpected lines by symmetry and selection rules can arise from a small disorientation of the crystal axes with respect of lab references axes, and/or non-perfect quality of polarizer or analyzer leading to a leakage of the polarization. These reasons are fully excluded here owing to the time dependence and magnitude of the observed feature. Furthermore, the comparison between both configurations within transmission geometry discards any source of error coming from optical set up. The contrast with the back-scattering data in which only expected E TO [ ] lines are detected is remarkable since both geometries are in principle equivalent. The far-field pattern of the transmitted beam recorded with the pump beam polarized along X and the analyzer along Z, i.e. in the Y XZ Y ( ) geometry at different times are presented in Fig. 2. One can notice the beam pattern evolution, showing that after some time a new light field extraordinarily polarized is produced along two lobes symmetrically placed with respect to the pump spot. This explains the activation of A TO [ ] 1 Raman lines, the intensity of which increases with time: this part of the Raman spectrum is produced by the newly generated extraordinary beams which corresponds to a Y ZZ Y ( ) scattering geometry. The lobes are most pronounced in the XY plane, with a maximum at an angle of about 5 degrees with respect to the pump beam. By changing the iris diaphragm aperture placed behind the crystal it is possible to verify that the contribution of the A TO [ ] 1 Raman lines in the detected spectrum comes essentially from the scattered intensity emitted in the direction of the lobes maxima.
In contrast, the beam pattern for the Y ZX Y ( ) configuration remains unchanged (Fig. 2). In this last case, the transmitted beam between crossed polarizers is co-linear with the incident beam (along x) and the polarization is not changed. Therefore, the activation of forbidden lines occurs solely in the Y XZ Y ( ) geometry, and its time dependence is connected to the build-up of the extraordinary scattered beam, as the ordinarily polarized beam propagates through the PR sample.
The origin of the activation of A TO [ ] 1 (denoted hereafter  Note that the intensity of the ordinary and extraordinary components have a different amplitude: this is due to a number of factors, including the different magnitude of the Raman tensor elements, the different optical path followed by the two beam and possible different efficiencies in the collection optics. To evaluate quantitatively the phenomenon it is therefore convenient to recast it in terms of the conversion efficiency, where P 2 is the power of the extraordinary beam after the sample and + P P 1 2 is the total beam power. If absorption can be neglected, as in our case, this latter term is equal to the power of the pump beam before the sample. Assuming that the pump beam is coupled to one single extraordinary wave propagating along the maximum intensity direction (i.e. at 5° from the ordinary beam), by energy conservation we can renormalize the signal of the forbidden Raman line so that the fraction of power gained by the extraordinary beam is equal to the one lost by the pump beam. In this way we get rid of most of the systematic errors in intensity measurements due to unknown conversion factors between the Raman signals and the measured beam powers, and to the fact that a fraction of the extraordinary signal is lost because of the limited angular acceptance of the detection system. It should be noted that measuring the ratio of the Raman lines represents a very convenient experimental approach to access the ordinary-to extraordinary conversion. This is more robust against stray light and misalignments than a basic polarization measurement, provided that the angular direction of the scattered light collected by the spectrometer is somehow defined by appropriate collection optics.

Simulation
To analyze the dependence of the forbidden lines intensity upon time we realized a simulation code based on previous works by Wilson 27 and Fluck 28 . Essentially, the coupling phenomenon is enabled by a transfer of power between the ordinary (pump) and the extraordinary (generated) power mediated by the following occurrences: (i) a nonzero component of the photo-galvanic tensor which, upon the simultaneous presence of two orthogonally polarized beams with differing wavevectors, produces a spatially modulated current and in turn a space charge field (ii) a nonzero component of the electro-optic tensor, which transforms the spatially modulated field into a refractive index grating diffracting the pump beam into the generated one. Note that the refractive index term (in Voigt notation) responsible for the coupling is nondiagonal, so that the ordinary beam is diffracted into an extraordinary one. The phenomenon is therefore self-starting: any weak extraordinary seed with sufficient intensity 29 initially present into the sample produces the modulated photogalvanic current, which in turns amplifies the extraordinary beam itself and so on, until a saturation regime is attained. The problem is to describe the spatio-temporal evolution of the probe beam and of the PR -generated beam as a function of the position and the time.
Let us consider the experimental situation sketched in Fig. 4. The Raman probe at 532 nm is described by a Gaussian function characterized by the waist and by the power. Since our typical sample thickness is much smaller than the Rayleigh range of our transmitted probe beam (several cm), the intensity of the pump beam can be described along a set of parallel lines. As shown in Fig. 2, the generated extraordinary beam has an intensity that depends on the direction with respect to the primary beam. In the following, for sake of simplicity, we will consider that the extraordinary beams are emitted only in one specific direction, i.e. in the plane xy at an angle theta with the primary beam corresponding to the direction along which the scattered intensity is maximized. In this way we can build the grid reported in Fig. 4 where the evolution of the two beams can be computed numerically as a function of the position on each node and for any time interval.
The power transfer between the two polarization states inside a photorefractive crystal is described by the following coupled differential equations 27 :  I  x  C  I I  I  I  I   I  x  C  I I  I  I  I  2 (1) where I 1(2) are the intensities and x 1(2) the propagation directions of the two beams, α is the absorption coefficient and C 12 is a coupling coefficient. The factor 2 visible in the second line is due to the fact that the pump power is depleted by the two symmetrically displaced extraordinary beams. The coupling coefficient C 12 depends on the position and on time, according to the following equations: = Γ C x y z t f x y z t D x y z t S x y z t ( , , , ) ( , , , ) ( , , , ) ( , , , ) In Eq. (2) the coefficient Γ is the photorefractive gain which is material constant with the dimensions of m −1 . Photorefractivity theory gives 27 :  www.nature.com/scientificreports www.nature.com/scientificreports/ should be noted that this factor show a quick increase for θ between 0 and 4 degrees, and then remains almost constant in a wide angular range (in agreement with our data in Fig. 2). In the following we set therefore θ =°5 .
In Eq. (3) τ 0 is the time constant of the PR process. Equations (3 and 4) govern the space and time evolution of the coupling constant, as they are dependent on the beam intensities. Here we make substantial use of a "local" approximation which states that, for slowly varying intensities, the photorefractive equations for uniform illumination remain valid provided that intensity-dependent quantities are replaced by their local value 28 . Thus, the term τ + I I 1 2 0 in Eq. (3) has the dimensions of s −1 and corresponds to the "local" Maxwell relaxation time of the system. If the beam intensities were constant, this term would remain equal everywhere and the solution to Eq.
(3) would be the familiar saturating exponential function. Equation (4) describes the effect of the material background conductivity. It plays a role of fundamental importance since without it, for sufficiently long times, all the points of the sample would reach the saturation gain coefficient Γ. Instead, for local intensities + I I 1 2 much smaller than the I dark value, the coupling coefficient remains small and proportional to +  I I I / 1 dark 1 1 . Finally, Eq. (5) confines the possibility of a coupling only inside the sample. In summary, the time evolution of the extraordinary beam intensity is described by three free parameters: E sat , τ 0 and I dark , all the other parameters being known or independently measurable.
We wrote a dedicated simulation software that solves Eqs (1-6) on the discrete grid reported in Fig. 4. The initial conditions are given by the input pump beam and by a uniform seed, assumed to have everywhere the arbitrary initial intensity of − W m 10 /  where D 1(2) are the propagation steps in the two beams directions, as shown in Fig. 4. The procedure is then iterated for a sufficiently long time, until the system reaches the saturation. The powers of the transmitted (pump) beam 1 and of the converted beam 2, P 1 and P 2 respectively, are calculated by integrating the beam intensities I 1 and I 2 on the output face of the crystal for each time step. We can finally compute the conversion efficiency by calculating as a function of time η = + P P P 2 1 2 . The simulation parameters E sat , τ 0 and I dark best reproducing the experimental curve were determined for our sample by a trial-and error procedure. It is important to stress that this monotonic saturation curve has a quite complicated behavior: initially the conversion efficiency is small and increases slowly, then it speeds up and finally reaches a stationary value. The corresponding results are shown in Fig. 2  , where it appears clearly that our simulation code is able to reproduce satisfactorily the full dynamics of the spectrum evolution. The first parameter, E sat determines the amplitude of the forbidden Raman peaks at the end of the temporal evolution, while τ 0 describes the duration of the transient. The parameter I dark affects both the amplitude and the evolution speed of the curve, so it is partially interdependent on the other two parameters. However, the first part of the curve between 0 and 1000 s is only weakly affected by the choice of I dark . This fact allows finding a unique combination of parameters providing a satisfactory agreement.
The incertitude reported in the simulation parameters concerns only the random errors and is estimated taking into account the mentioned partial dependency among them, as well as the fact that the data to be fitted are obtained through the renormalization procedure described in Section 2. In particular, it is especially important to determine accurately the depletion of power from pump beam 1 as this sets the saturation value of the curve in Fig. 3b and thus the final value of E sat . A systematic error source is due to the fact that in our simulation for simplicity the extraordinary beam is treated as a uni-directional plane wave directed at 5° from the ordinary one. A more accurate treatment would consist in taking into account the angular dependence of the scattered beams and in calculating the evolution of the extraordinarily polarized intensity as an average over the angular range accepted by the detector. However, as discussed above, in the range between 4° to about 20°, the angular dependence of the gain coefficient (Eq. 6) is weak, while between 0 and 4 degrees this quantity goes quickly to zero, so that the signals coming from this range are contributing in a limited amount to the final curve. The overall relative error in performing our simplified treatment is thus expected to be in the range of 10-20% depending on the parameters.   8 reflects the conversion efficiency as a function of time. This time dependence is fairly well reproduced by simulations based upon the model earlier proposed by Wilson et al. These calculations provide an estimation of the parameters characterizing the photorefractive effect and were discussed in terms of sources of random and systematic errors. We stress that the photorefractive parameters E sat , τ 0 and I dark can be directly related with the help of standard photorefractivity theory (see e.g. ref. 6 ) to sample characteristics, such as [Fe] concentration and reduction degree. Our study combining experimental Raman measurements and simulations therefore opens a new way to investigate photorefractive photonics by using transmitted Raman spectroscopy.