Frequency conversion in KTP crystal and its isomorphs ( invited )

We report the results of an analysis of the functional capabilities of the KTP crystal and its isomorphs for nonlinear-optical frequency conversion of all types of interaction in the transparency range of the crystal. The possibility of implementing angle-, wavelength(frequency-) and temperature-noncritical phase matching is shown.


Introduction
Since the first publication of the data on the synthesis of the KTP crystal (potassium titanyl phosphate, KTiOPO4) [1] and of the results of measuring its characteristics, it became evident that the crystal would take its rightful place for frequency conversion tasks and has fully justified these hopes [2--5].
For a certain but rather wide range of tasks, these crystals have no alternative.They have a high effective nonlinearity coefficient, rather large values of all the phase-matching widths and of the thermal conductivity coefficient, good optical quality, small absorption and linear expansion coefficients, as well as non-hygroscopicity.Besides they are inexpensive in manufacture.
Not very high value of the damage threshold determines the field of the most effective applications of these crystals, which includes generation of harmonics and parametric frequency conver-sion in the near-IR range.In these crystals, noncritical processes were realized for all parameters, i.e. angles, wavelength, and temperature.Moreover, the possibility of producing periodically and non-periodically poled structures in them at record high values of the nonlinear susceptibility coefficient d33 allowed them to find wide application for the problems of frequency conversion of low-intensity radiation in the crystal transparency range [22--25].
In addition to frequency conversion, these crystals are used as modulators and Q-switches [26--27].Work is underway to design fibers and waveguide structures [28--37], photonic structures [38--40] from these media.Also, these crystals are very promising for the generation of THz radiation [41--45].
To date, a large number of reviews on these crystals have been published.It is impossible to enumerate all the problems on the generation of radiation at different wavelengths in the KTP crystal and its isomorphs, which were obtained experimentally.But, nevertheless, not all their capabilities are fully defined.In this paper, we present the results of an analysis of the functional capabilities of the KTP crystal and its isomorphs for all frequency conversion tasks including generation of harmonics and sum and difference frequencies, as well as parametric generation in the range of their transparency (0.4--5.0 μm).
The KTP crystal and its isomorphs belong to mm2 point-group symmetry, with the mutual orientation of the axes XYZ ¬ abc.A common property of these crystals is that the signs of the nonlinear susceptibility tensor coefficients dij are identical (in contrast to the crystals of point group 3m), and their values differ insignificantly.This leads to the fact that the distributions of the effective nonlinearity coefficients have practically the same form.Figure 1 shows the distributions of the effective nonlinearity coefficients deff in the KTP crystal for two types of interaction, ssf and fsf = sff.The lines of white color show the phase-matching directions for the second harmonic generation (SHG), i.e. ssf (SHG at 1 = 2= 3.4 μm) and fsf = sff (SHG at 1 = 2 = 1.064 μm).For a large number of applications, a cut of the crystal is selected on the phase-matching curve, for which deff has a maximum value.For the particular cases of ssf type shown in Fig. 1, this value of deff is 0.65 pm/V at  = 42 and  = 49.7, and for the cases of fsf = sff type we have deff = 3.42 pm/V at  = 23.5 and  = 90.The maximum value deff takes place for the second type of phase matching, i.e. sff = fsf, which most widely used in practice.
Let us consider the functional possibilities of frequency conversion for all possible pro- cesses and types of phase matching in the crystal transparency range.

General features of frequency conversion
The method of analysis of the functional possibilities of the KTP crystal and its isomorphs proposed in [46,47] uses the form of presentation for the crystal figure-of-merit from the wavelengths 1 and 2 for uniaxial [46] and biaxial [47] crystals.Hereafter, the relation 1  2 > 3 is adopted.For all the values of the wavelengths 1 and 2, the value of 3 (1/ 3=1/ 1 + 1/ 2) is uniquely determined, the plots of the dependences for which are given for all the results presented below.For each pair of wavelengths 1 and 2, a cone of phase-matching directions was calculated.Along these directions there was defined one for which deff has a maximum value.It was used to calculate FOMD(1, 2), each value of which on the distributions presented below in Figs. 2, 4, 6--10, 12--17 has its color from the right-hand palette.Here, the parameter FOMD(1, 2) corresponds to the maximum value deff on the phasematching curve, unlike the other FOM parameter defined below in Section 3. In all the figures of the FOM(1, 2) distributions the maximum values are showed.The following data were used for the crystal parameters: KTP [48,69], RTP [49], RTA [50], KTA [51], and CTA [52,53].There is one peculiarity here.All this group of crystals is grown by different technologies [54--61], in different regimes and with different composition of the initial charge.This leads to the fact that the crystals have different refractive indices.As a result, the phase-matching angles can differ by a few degrees.The data [48--53, 69] used in the calculations most closely correspond to the crystals supplied by the majority of manufacturers.Below, we will show the difference between the results for FOM(1, 2) using various optical and thermo-optical parameters of the KTP crystal.
It is known (see, e.g., [62,63]) that the coefficients of the nonlinear susceptibility tensor dijk are characterized by dispersion.But due to the lack of complete data for all crystals, dispersion was not taken into account in the calculations.We used typical values [53] in the crystal transparency range.The variation in the values of dijk in this range does not change the general character of the distributions.
Figure 2 shows the FOMD(1, 2) distributions for the wavelengths 1 and 2 for all types of interaction for the KTP crystal in its transparency range (the boundaries of the range are shown by dashed lines).For the used ratio of wavelengths i, the results appear below the diagonal of the graph.It is easy to see that for ssf-type interactions the distribution is symmetric with respect to the diagonal.For sff and fsf types, the results are mutually complementary with respect to the diagonal.
Almost throughout the crystal transparency range, phase matching is realized for the first and second types of interaction.The boundary of the FOMD(1, 2) distribution determines combinations of wavelengths at which angular noncritical phase matching takes place.This is most fully obvious for the sff type of interaction in Fig. 2. For all crystals of the KTP group, phase matching with a change in wavelength appears and disappears along the y axis [64].In this case, it is noncritical in angles  and .In all the figures, a combination of wavelengths for which phase matching exists along the x axis is shown by the white line.For SHG, this is realized at 1 = 2 = 1.078 μm and 1 = 2 = 3.18 μm.This is also angular noncritical phase matching.For type-II phase matching along the x axis, the coefficient deff has a maximum value.Thus, at all combinations of wavelengths with phase matching along the x axis, the maximum conversion efficiency can be obtained.
For this group of crystals, phase matching along the z axis is absent.In the KTP crystal, the maximum value of the wavelength for the sum frequency generation is possible with type-II phase matching for SHG at 1 = 2 = 3.308 μm, whereas the minimum value of the wavelength for SHG is observed at 1 = 2 = 0.994 μm.The minimum value of the wavelength with ssf-and fsf-type interaction can be obtained by sum frequency generation (SFG) at the boundary of the transparency range.
The character of the FOMD(1, 2) distribution for the ssf type in the main part of the wavelength region of the transparency range is determined by the fact that the terms with different elements of the tensor dij with opposite signs contribute to the nonlinear polarizability of the medium.For a value of 1 at the boundary of the transparency range, a large variance for the angle of the optical axis Vz(), a large difference Vz(1) -Vz(2), leads to an increase in the values of deff.But even in this region the maximum value of FOMD(1, 2) for the ssf type is less than that for fsf and sff types, the region of phase matching for ssf type being maximal.
The presented results allow us to determine the possible tuning range of optical parametric oscillators.For a given value of 3, the phase matching region shows the tuning range for 1 and 2.This can all be achieved at a maximum value of deff.It can be seen from the results of Fig. 2 that the largest tuning range can be obtained by changing the phase-matching angle in the xz plane.
The maximum pump wavelength for KTP is 1.7 μm.The largest tuning range can be obtained for 3 = 0.8--1.2μm.In this case, the wavelength range is 1 = 1.1--4.5 μm.This can all be achieved at a maximum value of deff in the xy plane, since the value of FOMD(1, 2) is determined for these values.
The method of analysis proposed in Refs [46,47] allows us to determine combinations of wavelengths at which the regime of frequency-noncritical phase matching (FNCPM) is realized.The condition dk/d = 0 corresponds to it.Figure 3  on the phase-matching curve [65], but at a smaller value of deff.
The dash-dotted line in Fig. 4 shows the combinations of the wavelengths for the FNCPM regime in the yz plane, which occurs in the KTP crystal and its isomorphs.The FNCPM regime is also possible when the frequency of ultrashort pulses is converted into a field of quasi-continuous wave (quasi-CW) radiation.Figure 6 shows the special case of the FOMD(1, 2) distribution for sum frequency generation for type-II phase matching with broadband radiation at 1 = 2.4 μm and quasi-CW radiation at 2 = 1.75 μm.In this case, the spectral width of phase matching with respect to 1 is 170 nmcm 1/2 .This possibility follows from the fact that in the case when the tangent to the isolines of the FOMD(1, 2) distribution is parallel to the axis, the value of deff does not change in a wide range of the wavelengths.Taking into account the results of Fig. 2, we find that in a wide range of the wavelengths, the phase-matching angle preserves its value.For the KTP crystal, for example, in the xz plane, this is the angle phm.The character of the FOMD(1, 2) distribution with a minimal value in the central region (Fig. 6) shows that the FNCPM is possible in a wide range of the wavelengths.Also possible is the FNCPM regime with a different ratio of the spectral widths of two wavelengths λ1 and λ2.
In general, the character of the distributions for all these crystals is similar to that for KTP.As in the case of KTP, for the ssf type, phase matching exists almost everywhere in the crystal transparency range.But the value of deff for it is significantly less than that for fsf and sff types.For the fsf type, phase matching is realized in most of the crystal transparency range.In the case of the CTA crystal, in the vicinity of the x axis the rate of change in FOMD(1, 2) in the complete wavelengths range is much less than that for other crystals.This corresponds to the fact that the spectral width of phase matching for CTA is larger.At a wavelength of 1.548 μm, the spectral width in CTA is 4.3 nmcm, whereas the spectral width in KTP at 1.076 μm is 0.6 nmcm.In all crystals, the FNCPM regime can be obtained both for the generation of harmonics and sum and difference frequencies.

Temperature-noncritical processes of frequency conversion
The above results in the form of FOMD(1, 2) distributions allow us to determine combinations of the wavelengths for which deff has a maximum value and for which angle-and frequency-noncritical phase matching takes place.It is also possible to implement temperaturenoncritical phase matching (TNCPM) by determining the value of FOMT(1, 2) on the phasematching cone along the directions for which dk/dT = 0.This regime of frequency conversion in the KTP crystal has been repeatedly obtained by various authors [66--73].As in the case of angle-and frequency-noncritical phase matching, the first-order derivative with respect to temperature dk/dT = 0 determines the TNCPM direction.The temperature width is determined by derivatives of a higher order.
It is important that the TNCPM direction is not strictly fixed in the crystal.It has dispersion as well as phase-matching and optical axis directions.To analyze the feasibility of the TNCPM regime and its dispersion, it was proposed [73] to determine the directions (cone) of temperature-noncritical interactions (TNCIs) independently of the phase-matching condition for which k(, ) = 0.These are the directions along which dk(, )/dT = 0, no matter if phase matching takes place or not.The intersection of the phase-matching and TNCI cones determines the direction of TNCPM, since in this direction k(, ) and dk(, )/dT are simultaneously equal to zero.With changing the radiation wavelength, both cones (phase matching and TNCI) change, which leads to a change in the TNCPM direction.This shows that this regime takes place in a finite range of wavelengths for a given frequency conversion process.For the sff-type interactions, the TNCPM regime can be obtained in the wavelength range corresponds to the temperature-noncritical phase matching (wavelength region with TNCPM).
A comparison of Fig. 2 and Fig. 11 for the KTP crystal shows that the values of FOM(1, 2) are different for the same combinations of the wavelengths.When these values are equal for the KTP crystal, the TNCPM direction lies in the main plane, where deff has a maximum value.
In the case of SHG, this takes place for fsf = sff type phase matching at a wavelength of 3.18 μm (Fig. 11).Also it is possible at different combinations of 1 and 2.With FOMD(1, 2) differs from FOMT(1, 2), the direction of TNCPM has the most common orientation: 90 >  > 0 и 90 >  > 0.In this case, deff will be less than the maximum possible value for the selected combination of wavelengths.
The FOMT(1, 2) distributions, similar to those in Fig. 12, are presented for KTA (Fig. 13), RTP (Fig. 14), RTA (Fig. 15), and CTA (Fig. 16) crystals.One can see from these figures that only in the KTP and RTP crystals there are directions in the crystal transparency range along which TNCPM is realized.For KTA and RTA crystals, the TNCPM region is much smaller than the phase-matching region.For the CTA crystal, no TNCPM is realized at any combination of wavelengths 1 and 2.In analyzing the results of Figs.12--16, it is necessary to pay attention to one peculiarity.
For example, more than 10 papers have been published for the KTP crystal in which the Sellmeier equations ni() are given for the principal values of the refractive indices, and the data are lesser extent, from the values of ni().As noted above, the following data were used to calculate the FOMT(1, 2) distributions for the KTP crystal (Fig. 12): ni() [48], dni()/dT [69].The give a fairly good agreement with the results of calculations and the experimental data for phasematching angles, mainly in the visible and near-IR ranges.Also, a good agreement was obtained for the temperature widths of phase-matching.A comparison of the experimental results with the TNCPM [70] was carried out using the data for dni()/dT from [69].As a result, a good agreement was obtained.
Later, more precise measurements of the parameters were made for the KTP crystal [48].
The obtained data for ni() are in very good agreement with the results of calculations for the phase-matching angles in the crystal transparency range.The data for dni()/dT in [48] give good agreement for the temperature-critical phase matching in the visible --near-IR range.But in the crystal transparency range of the KTP crystal, the FOMT(1, 2) distributions (see Fig. 17) considerably differ from the results of Fig. 12.The ranges of wavelengths within which TNCPM is present also differ.Comparison of the results in Fig. 12 and Fig. 17   essary to measure the temperature derivatives for refractive indices of the second and higher orders to determine the temperature widths of phase matching [72,73].Based on this, at present, the reliability of the above results for RTP, KTA, RTA, and CTA crystals cannot be guaranteed (Figs.13--16).Much less research was carried out for these crystals than for the KTP crystal.
Without pretending to rigorous determination of the results (see Figs. 12--16) at this stage, it can be formally noted that in the largest wavelength region, the TNCPM regime takes place for phosphate crystals (KTP and RTP).In a much smaller region, the TNCPM is realized for crystals containing arsenic (RTA and KTA).The presence of cesium in the crystal together with arsenic (CTA) leads to the fact that the TNCPM regime is absent in the crystal transparency.This is confirmed by the results of [52], in which the temperature width of phase matching did not exceed 11.1 Сcm for different frequency conversion processes in the range 0.532--2.02μm.All this requires an appropriate analysis.

Conclusions
The paper presents the results showing the functional capabilities of the KTP crystal and its isomorphs for nonlinear-optical frequency conversion in the range of their transparency for all
shows the wavelength dependence of the phase-matching angle phm and the coefficient deff in the xz plane for the SHG for the type-II interaction in the KTP crystal.One can see that these dependences exhibit a consistent variation of these parameters.In this case, the FNCPM regime can be determined by the equality d/d = 0. Consequently, the minimum value of FOMD(1, 2) for the KTP crystal on the straight line representing SHG (Fig.2) corresponds to the FNCPM regime.Similarly, the combination of the wavelengths 1 and 2 for FNCPM can be determined for all the frequency conversion processes, i.e. generation of the third (THG), fourth (FoHG), fifth (FiHG) harmonics, and SFG (in Fig.4they are indicated by the red line).In the FNCPM regime, the minimum values of FOMD(1, 2) along the straight line will correspond to the above frequency conversion processes.For fsf and sff interaction types in Fig.4, the dashed lines show the combination of 1 and 2 of the FNCPM regime.It should also be noted that FNCPM takes place for the combinations of the wavelengths 1 and 2 on all these lines, which are tangent to the isolines of the FOMD(1, 2) distributions.The FNCPM regime is realized accurately for the given ratio of the wavelengths.In addition, it can also be obtained in the vicinity of these values of 1 and 2

Fig. 3 .
Fig.3.Distribution of phase matching angle  and deff coefficient versus wavelength for SHG in KTP crystal with (sff)=(fsf) type of interaction.

Figure 11
Figure 11 shows the angular dependences for phase matching and TNCI of ssf and fsf interaction types for SHG in the KTP crystal at different wavelengths.For the ssf-type interaction, the TNCPM regime is initially obtained at a wavelength of 1 = 2 = 0.747 μm in the xy plane ( = 64,  = 90).As the wavelength of the radiation increases, the direction of TNCPM changes, and the values of the angles  and  change.At a wavelength of 1 = 2 = 1.064 μm, the TNCPM regime takes place at  = 48 and  = 43, and at 1 = 2 = 3.48 μm it occurs in the xz plane ( = 0,  = 54).Thus, for SHG with the ssf type of interaction, the TNCPM regime can be obtained in the range from 0.774 to 3.48 μm with a change in the direction from the xy plane to the xz plane.The results of Fig.1.ademonstrate that in the principal planes xy, yz and xz of the crystal (up to the optical axis), deff = 0, and the results of Figs.11a and 11c are of no practical value.The maximum conversion efficiency for the ssf type with TNCPM can be obtained at 1 = 2 of about 3.25 μm.
raises the problem of refinement of the data on dni()/dT in the KTP crystal transparency range.At the same time, it is neca) ssf b) fsf c) sff Fig.17 . FOMT(1, 2) distribution for KTP crystal for all types of interactions with data on dni/dT from[48].