A millimeter-sized, monolithic whispering gallery resonator made of a lithium tetraborate, , crystal was employed for doubly resonant second-harmonic generation with a continuous-wave laser source at 490 nm. An intrinsic quality factor of was observed at the pump wavelength. A conversion efficiency of 2.2% was attained with 5.9 mW of mode-matched pump power. In the lithium tetraborate resonator, it is feasible to achieve phase-matching of second-harmonic generation for pump wavelengths between 486 and 506 nm.
© 2015 Optical Society of America
A compact implementation of continuous-wave (CW) laser sources in the middle ultraviolet is based on efficient and resonant second-harmonic generation (SHG) of diode laser light [1 –6]. For this purpose, crystals such as -barium borate, (BBO), cesium lithium borate, (CLBO), and—more recently—potassium fluoroboratoberyllate, (KBBF), are used among others as nonlinear materials. Borate crystals are suitable for these applications, as their transparency range frequently reaches far into the ultraviolet. Lithium tetraborate, (LB4), however, received only little attention as a crystal for CW frequency doubling due to its small nonlinear-optical coefficient of . An efficiency of 0.08% at 10 W pump power was observed in a single-pass experiment . The material was, however, intensively studied in the pulsed regime .
The conversion efficiency of CW SHG can dramatically be increased, using resonant enhancement of the optical fields. A remarkable reduction of the required pump power for efficient frequency conversion was recently observed in whispering gallery resonators (WGRs) made of lithium niobate [10,11]. This geometry was successfully transferred to BBO , and harmonic generation was demonstrated therein with the help of cyclic phase-matching .
Combining LB4 with the WGR geometry offers a way to efficiently use the crystal’s small nonlinear coefficient. Frequency conversion may already reach efficiencies in the percent regime using laser sources with milliwatts of output powers. Moreover, LB4 can be noncritically phase-matched for SHG of 488 nm light . There, the wavelength of the harmonic light still lies well within the transparency window of the crystal, which makes LB4 even more preferable than BBO. In particular, the phase-matching wavelength is close to the blue Argon-ion lasing transition, around which diverse applications have been established. With this motivation, we investigate LB4 in the WGR geometry as a compact source of ultraviolet light around 244 nm.
The article is structured as follows. A figure of merit for SHG—the characteristic pump power—is established and estimated for an LB4 WGR. On this basis, SHG is investigated.
The material LB4 exhibits Type-I noncritical phase-matching for SHG of pump light at wavelength . The -cut configuration is used for investigating this process. There, the symmetry axis of the WGR coincides with the optic axis of the crystal. In a -cut WGR, second-harmonic generation reaches a maximum conversion efficiency at the characteristic pump power . Assuming critical coupling  of both light fields, we arrive at an estimate for this power to be
To assess the characteristic pump power in LB4, the two quality factors and the effective volume still have to be determined. The quality factors can be estimated from the resonance linewidth of the WGR made from LB4. The effective volume can be calculated from the electric field distribution of the whispering gallery modes . There, only those modes, that fulfill the phase-matching condition , are of interest and have to be selected accordingly.
At first, the optical quality factor at the pump wavelength of 490.4 nm was estimated. For this, a WGR was fabricated from a bulk LB4 single crystal. A cylindrical crystal preform was mechanically machined to a spheroid, and its equator was lapped to further reduce the surface roughness. The symmetry axis of this spheroidal WGR was carefully aligned with the optic axis of the crystal during the machining. The two radii of the spheroid were measured to be 1.15 and 0.16 mm (see Fig. 1). The WGR was placed into a holder at with a temperature stabilization on a millikelvin scale. The same holder supported a sapphire prism. The optic axis of the prism coincided with the symmetry axis of the resonator. The spacing between the WGR and the sapphire prism was controlled with a piezoelectric actuator with nanometer precision.
For the optical analysis of the LB4 resonator, CW laser light at 980.8 nm with a linewidth of 75 kHz was frequency doubled to 490.4 nm with a less than twofold linewidth. The laser frequency was tunable without mode-hops in the range of 5 GHz and with a tuning rate of 1.5 Hz. The frequency scan was calibrated with a Fabry–Perot cavity with 1.5 GHz free spectral range, which monitored the output of the infrared laser.
As sketched in Fig. 1, the visible laser light was focused with a lens onto the back of a sapphire prism, with the focal point at the back surface and closest to the resonator. The polarization of light was linear and in ordinary direction with respect to the optic axis of LB4. The angle of incidence was chosen to match the critical angle for total internal reflection between the prism and resonator. This provided the possibility for evanescent coupling of the light to the WGR. The out-coupled pump light was monitored with a photodetector.
The transmission spectrum of the LB4 resonator was recorded, while scanning the laser frequency. A normalized transmission measurement of the resonator at 490.4 nm and 3.2 mW optical power at critical coupling is presented in Fig. 2. The intrinsic quality factor was estimated from the linewidth measurement to be . From the intrinsic quality factor, one can derive the optical extinction coefficient of the resonator. The extinction coefficient at 490 nm is . This value is comparable with absorption data on LB4 . This indicates, that surface scattering of light in the WGR plays a negligible role. No laser source was available at 244 nm. Thus, the absorption coefficient from  at 266 nm was taken to estimate an intrinsic quality factor of for the harmonic wavelength.
In a second step, the phase-matched whispering gallery modes have to be selected, and the corresponding effective volumes are calculated. For phase-matching , the dispersion relation for WGRs is taken as a basis . For this calculation, knowledge of the refractive index is necessary.
A formula for the refractive index as a function of wavelength and temperature is developed in the following. The refractive index of LB4 at 25°C was available from a Sellmeier equation in . With knowledge on the thermo-optic coefficients , the temperature-dependent refractive index is given as21] for several wavelengths from 436 to 644 nm. These data showed that the thermo-optic coefficient changes linearly with temperature and the slope is the same for all wavelengths studied. Therefore, an average slope was estimated from a linear fit. The wavelength dependence of the thermo-optic coefficient was also qualitatively similar for all measured temperatures within a range of . A cubic fit averaged over this temperature range together with the slope from the linear fit resulted in 1. Strictly speaking, this refractive index formula can only be applied for wavelengths between 436 and 644 nm and in a temperature range from to . Nonetheless, the formula was taken as a qualitative tool for phase-matching calculations in the ultraviolet. Near room temperature, the calculations should also be quantitatively reasonable for the ultraviolet.
Now, it is possible to calculate the wavelengths and temperatures, where phase-matching is fulfilled in the present LB4 resonator. Phase-matching was checked for whispering gallery modes with polar and radial mode numbers within and and temperatures from 0°C to 50°C. These modes are the most dominant modes in a typical WGR spectrum . These modes can be addressed within the tuning range of the laser light. Then, the effective volumes were calculated for all phase-matched mode combinations.
Phase-matching for the fundamental mode combination with and is achieved at and 13°C with a minimum effective volume of . For a pump wavelength between 486 and 506 nm, phase-matching with effective volumes less than a hundred times the one of the fundamental mode were found. This gives a large flexibility in the choice of the pump wavelength. Exemplarily, the simulation for the experimentally available wavelength of 490.4 nm is presented in Fig. 3. Various conversion channels are available with quite different effective volumes. The minimal effective volume in this situation (for , , and , ) is 3.5 times the one for the fundamental mode. In this simulation, ranges from 0.5 to 35 mW. This pump power is well within optical powers available from diode lasers. In experiment, critical coupling can be arranged for the pump light. Simultaneously, the harmonic light is likely to be undercoupled instead of critically coupled, as its quality factor and its coupling strength is smaller, compared to that of the pump mode. This will only weakly affect . Even the conversion efficiency is still expected to be on the order of 5%.
On this basis, SHG was studied experimentally in the LB4 resonator. The WGR was pumped with horizontally polarized CW light at 490.4 nm at a temperature of . The pump frequency was scanned across a few GHz. The ultraviolet second-harmonic light was monitored with a second photodetector behind a color filter with a 30 dB pump light suppression. Additionally, the pump and harmonic light emerged from the prism under different angles, as sketched in Fig. 1. This spatial separation further increased the pump light suppression in the ultraviolet light detection.
For the SHG measurement, the whispering gallery mode presented in Fig. 2 was used. At critical coupling and 3.2 mW pump power, this resonance showed a transmission of 0.7. Thus, the spatial overlap  of this mode to the external pump beam was 30%, and the mode-matched pump power was 0.95 mW. Phase-matching was found for this mode by temperature tuning. Within a range of 5°C only a few other less efficient conversion channels could be observed with other pump modes. Up to 4.8 μW of optical power was observed on the second photodetector. The light impinging on this detector was measured to be extraordinarily polarized. Additionally, the spectral density of the emitted light was recorded with an optical spectrum analyzer. The result is presented in Fig. 4 (inset). It shows the fundamental wavelength and a second peak exactly at the harmonic wavelength. This verifies a Type-I SHG process.
The conversion efficiency in critical coupling was 0.5% at a mode-matched pump power of 0.95 mW. This scales well with the estimate of , where about 5% conversion efficiency is expected. Nevertheless, the specific conversion channel cannot be identified on the basis of this estimate.
Finally, the second-harmonic power was measured as a function of the pump power utilizing still the same mode. The measurement was performed in the overcoupled regime with a pump linewidth of 43 MHz. Within a temperature window of about 10 mK the second-harmonic signal was pronounced. The oven temperature was adjusted to the maximum of this emission within a stability of a few millikelvin. The result is shown in Fig. 4.
The second-harmonic power grows quadratically with pump power until around 15 mW, where the absorption of light at 490.4 nm starts to significantly contribute to the thermal load of the LB4 resonator. To maintain phase-matching, the oven temperature was gradually reduced by 0.02°C while increasing the pump power up to 19.7 mW. The slope efficiency in this process was 0.36% per milliwatt mode-matched pump power, with a maximum efficiency of 2.2% at 5.9 mW mode-matched pump power.
An active locking technique  might help to reduce the effect of the thermal instability caused by pump light absorption. And identification of the whispering gallery modes might be possible with a mode identification method from . With this information, it would be possible to employ the modes in the WGR with the smallest effective volume. Locking and mode selection bear the potential to significantly increase the efficiency. Simulations indicate that convenient conversion channels can be found for a pump wavelength range of twenty nanometers.
In conclusion, second-harmonic generation in LB4 was demonstrated with conversion efficiencies exceeding 2.2% using a continuous-wave pump laser. This value represents a more than twenty-times improvement compared to data of previous experiments , and this at a thousand times lower pump power. The whispering gallery geometry unfolded its full potential in the regime of nonlinear optics, when using crystals with small nonlinear coefficients. Active stabilization combined with mode-hop-free-tuning  will pave the way for WGR-based CW light sources in the ultraviolet and visible frequency regime based on efficient frequency conversion, and with several milliwatts of output power.
The authors thank M. Leidinger, R. Wolf, and Ch. Werner for valuable discussions, and the Deutsche Forschungsgemeinschaft for financial support.
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