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We report the experimental determination of the ordinary and extraordinary refractive index of 8 mol% Mg-doped congruent lithium tantalate (MCLT). Refractive index measurements cover a spectral range from 450nm to 1550nm and temperatures varying from 22°C to 200°C. Experimental data are fitted to a temperature-dependent dispersion relation that has not been previously used with this material family. Based on this relation, various optical properties of MCLT are calculated, including thermo-optic coefficient, group velocity dispersion, phase matching curve and temporal walk-off. In an additional quasi-phase-matching second-harmonic-generation experiment it is shown that the proposed dispersion relation may be used to predict grating period with remarkable nanometer-scale accuracy.
©2011 Optical Society of America
1. Introduction
Ferroelectric nonlinear crystals exhibit high nonlinear coefficients and may be domain-engineered, thus are widely used for quasi-phase-matched (QPM) frequency conversion applications. Over the last decades, lithium niobate (LiNbO3, LN) has dominated ferroelectric QPM technologies. However, this material suffers from photorefractive damage, an effect that becomes particularly significant with high power visible radiation. In recent years, ferroelectric lithium tantalate (LiTaO3, LT) has emerged as an alternative solution to LN, mainly due to its wider transparency range towards the ultraviolet. Several authors have reported efficient nonlinear interactions in both near-stoichiometric (SLT) and congruent (CLT) lithium tantalate, including second-harmonic-generation, sum and difference frequency mixing, as well as parametric oscillation and amplification [1–17].
Mg-doped LN and LT crystals are often preferred over the undoped material. Mg-doping, in combination with operation at elevated temperatures, suppresses photorefractive damage by increasing material conductivity. Contrary to the case of Mg-doped LN, which is an extensively investigated material system, linear and nonlinear optical properties of Mg-doped LT are relatively understudied.
Refractive index (along with its wavelength and temperature dependence) is the most important optical constant of a nonlinear medium that is required for the design of nonlinear converters. Typically, semi-empirical Sellmeier equations are employed to model this dependence. There exist several reports on Sellmeier coefficients for both SLT and CLT undoped crystals [18–25]. In recent works, Sellmeier relations for 1.0 mol% and 0.5 mol% Mg-doped SLT were also presented [26,27].
In the present article, we report on experimental measurements of the ordinary and extraordinary refractive index dispersion of 8 mol% Mg-doped CLT (thereafter, referred to as MCLT). Our measurements include five laser wavelengths in the visible to near-infrared spectral range and temperatures varying from 22°C to 200°C. Our study differs from previous works in that (a) it exploits a doping level that is significantly heavier and (b) focuses on doped LT crystals grown form congruent (rather than stoichiometric) melt. It is worth noting that MCLT samples with this particular doping level have exhibited higher surface damage threshold compared to MCLT samples of lower doping concentration, as well as compared to undoped SLN and 5 mol% MCLN samples. At the same time, relatively large (3-inch in diameter) 8 mol% MCLT substrates are now commercially available and may be fabricated by use of Czochralski method easier than, for example, SLT substrates.
Experimental refractive index data are consequently fitted to a dispersion relation, which resembles Schott formula often used with glasses, but is properly modified to account for temperature dependence. By use of this theoretical fit, linear optical properties of MCLT are calculated and compared with the undoped crystal. Such properties include thermo-optic coefficient and group-velocity dispersion. In a further experimental step, second-harmonic-generation (SHG) in MCLT is studied. More specific, the QPM grating period is determined as a function of fundamental wavelength and temperature. Experimental grating period values are then compared with theoretical predictions based on the dispersion relation of this work. We establish that our model reproduces experimental grating period values with a high precision that reaches few nanometers. Finally, the new dispersion relation is employed to calculate phase-matching curve and temporal walk-off effect.
2. Wavelength and temperature dispersion of refractive index
A prism-coupling refractometry technique is used for refractive index measurement [28]. The sample under investigation is brought into contact with a reference prism of known index np. Monochromatic laser radiation is directed upon the base of the reference prism and then reflected onto a photodetector. The angle of light incidence (which may be controlled via a rotary table) eventually reaches a critical value at which, frustrated total internal reflection at the prism base leads to abrupt decrease in photodetector reading. This critical angle θc directly relates to the unknown index nx of the sample via:
Our measuring system is equipped with five different lasers emitting radiation at 450.0nm, 532.0nm, 632.8nm, 964.0nm and 1551.0nm. Heating elements are used to simultaneously vary the prism and sample temperature from room conditions up to 200°C. Proper polarizing elements enable switching laser output from TE to TM mode, thus providing easy access to the determination of both ordinary (no) and extraordinary (ne) sample index. The experimental setup allows temperature control with an accuracy of ~0.1°C and refractive index measurement with an accuracy (resolution) reaching at least third (fourth) decimal place. Experimental refractive index data corresponding to five different wavelengths in a nearly two-octave spanning spectral region and ten different temperatures from 22°C to 200°C are presented in Table 1 .
Table 1. Experimental Ordinary and Extraordinary Refractive Index Data for MCLT
By use of a Levenberg–Marquardt nonlinear algorithm, the experimental data were fitted to an 8-term temperature-dependent dispersion relation of the form:
The temperature-dependent parameter F(T) follows a commonly used empirical form with a given offset at the initial room temperature of 22°C:so that when T = 22°C Eq. (1) reduces to:Temperature-independent Eq. (2) is identical to the Schott function (that is, Cauchy’s dispersion relation with an added second-order wavelength term) which is commonly used with optical glasses. On the contrary and to the best of our knowledge, the generalized temperature-dependent Eq. (1) has never been used before. During the fitting process, we initially accounted for the five experimental data points corresponding to room temperature in order to compute the “wavelength-sensitive” A1 to A5 terms. Having determined these values, we then used all available experimental data points to compute the “temperature-sensitive” B1 to B3 terms. By substituting wavelength in μm and temperature in °C, the values of the 8 coefficients for the ordinary and extraordinary index of MCLT are presented in Table 2 .
Table 2. Dispersion Relation Coefficients for the Ordinary and Extraordinary Refractive Index of MCLT
It is worth noting that, since our refractive index measurement apparatus uses five laser wavelengths, experimental data may only be fitted to dispersion relations with five (or less) “wavelength-sensitive” coefficients. Apart from Eq. (1), other dispersion equations satisfying this requirement were also tested, including the forms of references [20,25]. Among all alternative dispersion relations that were investigated, Eq. (1) provided best fitting quality.
Figures 1(a) and 1(b) present fitted no and ne values, respectively, along with the corresponding experimental data. Throughout the entire temperature range of our measurement, average difference between calculated and measured values for both no and ne is found to be ~0.0004. The discrepancy between fit and measurement becomes larger at higher temperatures. This observation indicates that further improvement of fitting quality may be possible via inclusion of higher-order temperature terms in Eq. (1). However, an average inaccuracy of ~0.0004 already compares well with experimental error, thus inclusion of additional temperature coefficients is considered unnecessary. Figure 1(b) also presents (dashed lines) the extraordinary index for undoped CLT based on the Sellmeier equations of Bruner et al [21]. It is observed that both doped and undoped crystals exhibit nearly identical indices near ~120 °C. However, Mg-doping results in a significantly decrease (increase) in ne towards low (high) temperatures, respectively.
Fig. 1 Fitted (solid lines) and experimental (open circles) refractive index dispersion as a function of temperature and five different wavelengths for the ordinary (a) and extraordinary (b) optic axis. For comparison, refractive index dispersion for undoped CLT based on the fit of Ref [21]. is also shown (dashed lines) in plot (b).
The new dispersion relation may also be employed for the calculation of various temperature-dependent linear optical effects in MCLT. In this study, our interest is limited to the extraordinary principal axis which provides access to the largest nonlinear coefficient d33 and thus is nearly exclusively utilized in QPM devices. Figure 2(a) presents calculated thermo-optic coefficient for MCLT (dispersion fit of this work), as well as for undoped CLT crystal (dispersion fit of Ref [21].), as a function of temperature and for five different wavelengths. It is observed that Mg-doping leads to significant increase of thermo-optic coefficient by a factor of approximately 5. In both MCLT and CLT crystals, thermo-optic coefficient increases with increasing temperature and decreasing wavelength. Figure 2(b) shows group-velocity dispersion (GVD) calculations as a function of wavelength for two indicative temperatures of 22°C and 200°C. Again, calculation is repeated for both MCLT (dispersion fit of this work) and undoped CLT material (dispersion fit of Ref [21].). It is observed that Mg doping and temperature variation has negligible influence on GVD value, which however decreases rapidly with increasing wavelength. More specific, GVD is found to be in the order of 1000fs2/mm near 450nm, reducing to ~100 fs2/mm near 1550nm.
Fig. 2 (a) Caclulated thermo-optic coefficient as a function of temperature for MCLT (solid lines) and undoped CLT (dashed lines). Calculations are carried out for five indicative wavelengths (450nm, 550nm, 650nm, 1050nm and 1550nm). (b) Calculated group-velocity dispersion as a function of wavelength for MCLT (solid lines) and undoped CLT (dashed lines). Calculations are repeated for two indicative temperatures of 22°C and 200°C. Calculated values for undoped CLT are based on dispersion relation of ref [21].
3. Second-harmonic generation investigations
Second-harmonic generation experiment in MCLT is also carried out. This study provides an evaluation of the precision of our dispersion relation. For this purpose, four 1cm-long MCLT samples were domain inverted via electric field poling [29] with corresponding experimental grating periods (Λexp) of 7.86μm, 7.92μm, 7.97μm and 8.02μm, respectively. The z-cut samples were predesigned for first-order SHG process with all waves polarized along extraordinary axis (eee). This geometry corresponds to the most common QPM interaction that exploits the largest nonlinear tensor element d33. A commercial Titanium:sapphire CW laser (Millenia 3900S, Spectra-Physics Lasers Inc., USA) was used as source of fundamental radiation. The pump source is tunable between 675nm and 1100nm with a wavelength resolution of 0.01nm. For each one of the available grating periods and three different temperatures (28°C, 55°C, and 100°C) the corresponding phase-matching fundamental wavelength λF is determined. The results of this experiment are presented in Table 3 .
Table 3. Experimental SHG Phase-matching Wavelength and Grating Period at Three Different Temperatures, along with Calculated Period Values. Last Column Presents Absolute Value of Difference Between Experimental and Theoretical Grating Period.
Consequently, corresponding theoretical grating period values (Λth) were computed for each phase-matching fundamental wavelength and temperature, via:
The last equation accounts for thermal expansion effect by inclusion of the thermal expansion coefficient a(T) which, according to ref [30]. may be approximated by:
During theoretical calculation of grating period the result of our dispersion model ne(λ,T) is required as input to Eq. (3). Theoretically computed grating periods Λth are also presented in Table 3.
The absolute value of the difference between Λexp and Λth is as small as ≤5nm at 28°C and increases to just ≤16nm (≤25nm) as temperature increases to 55°C (100°C), respectively. Accounting for the fact that Λexp is in the range of ~8μm, these values indicate that our dispersion relation is able to predict grating period with an error as small as (not exceeding) ~0.02% (~0.3%), respectively. The observed small increase of error at elevated temperatures may be attributed to the accumulation of several effects, ranging from compromised performance of the dispersion model to misevaluation of thermal expansion effect. It is worth noting that alternative dispersion relations that were also used for fitting refractive index data resulted in grating period prediction errors exceeding ~150nm and reaching up to ~500nm.
To gain further insight on the significance of accurate grating period prediction, one has to account for grating period acceptance ΔΛ. Parameter ΔΛ defines the range over which Λ may be shifted before nonlinear conversion efficiency falls below 50% of its maximum value. For first-order QPM SHG and within the limits of plane-wave approximation, grating-period acceptance is given by [31]:
where L is the interaction length, Λ is the grating period that corresponds to ideal QPM configuration (that is, maximum conversion efficiency), and Δk is the phase-mismatch parameter, which in this case is equal to Accounting for experimental grating period values in the range of ~8μm, grating acceptance normalized with respect to interaction length is readily found to be in the range of ΔΛ∙L ≈ 6nm∙cm (this value increases/decreases with increasing/decreasing fundamental wavelength, respectively). Since in many QPM devices optimum crystal length is in the ~1cm range, this estimate of ΔΛ∙L ≈ 6nm∙cm suggests that nanometer-scale grating period accuracy is indeed required for a variety of practical applications.Finally, based on the proposed dispersion relation, grating period scaling for first-order QPM SHG as a function of fundamental wavelength and two different temperatures of 22°C and 200°C is calculated and presented in Fig. 3(a) . This phase-matching curve is produced via Eq. (3) and thus accounts for thermal expansion effect. Grating period reduces from ~20μm to ~5μm as fundamental wavelength tunes from 1.55μm to 0.9μm, respectively. By varying the temperature from 22°C to 200°C, grating period shifts by several hundreds of nm. Figure 3(b) presents theoretical calculation of temporal walk-off between fundamental and second-harmonic radiation, as a function of fundamental wavelength and for two different temperatures of 22°C and 200°C. Temporal walk-off effect, which in the spectral-domain manifests itself via spectral acceptance considerations [31], is an important parameter in the design of ultrashort pulses wavelength converters. It is found that temporal walk-off decreases from ~1.1ps/mm to ~0.26ps/mm as fundamental wavelength increases from 0.9nm to 1.55nm. These values are nearly identical to temporal walk-off for undoped CLT crystal (calculated by use of dispersion relation of ref [21].) and only slightly smaller than walk-off values for standard commercial Mg-doped and undoped LN crystals. Finally, temporal walk-off in MCLT crystal is observed to only moderately increase with increasing temperature.
Fig. 3 (a) Calculated SHG grating period versus fundamental wavelength for two temperatures of 22°C and 200°C. (b) Calculated temporal walk-off between fundamental and second-harmonic radiation versus fundamental wavelength for two temperatures of 22°C and 200°C.
4. Conclusions
We have presented ordinary and extraordinary refractive index measurements for 8 mol% Mg-doped congruent lithium tantalate. Based on experimental data, temperature and wavelength dispersion relation was determined for the spectral region between 450nm and 1550nm and a temperature range from 25°C to 200°C. Fitting quality was found to be comparable to experimental measurements error. Second-harmonic generation measurements established that the proposed dispersion relation is able to predict quasi-phase-matching grating period with nanometer-scale accuracy. Finally, the present dispersion model was employed to theoretically calculate various optical properties of this material, including thermo-optic coefficient, group-velocity dispersion, phase-matching curve and temporal walk-off.
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