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Wavelength tuning of conventional mirror-based optical parametric oscillators (OPOs) exhibits parabolically-shaped tuning curves (type-0 and type-I phase matching) or tuning branches that cross each other with a finite slope (type-II phase matching). We predict and experimentally prove that whispering gallery OPOs based on type-0 phase matching show both tuning behaviors, depending on whether the mode numbers of the generated waves coincide or differ. We investigate the wavelength tuning of optical parametric oscillation in a millimeter-sized radially-poled lithium niobate disk pumped at 1 μm wavelength generating signal and idler waves between 1.7 and 2.6 μm wavelength. Our experimental findings excellently coincide with the theoretical predictions. The investigated whispering gallery optical parametric oscillator combines the employment of the highest nonlinear-optical coefficient of the material with a controlled type-II-like wavelength tuning and with the possibility of self-phase locking.
© 2016 Optical Society of America
1. Introduction
Continuous-wave optical parametric oscillators (OPOs) convert a monochromatic pump wave at the wavelength λp into two waves (signal and idler) at λs,i such that 1/λp = 1/λs + 1/λi. The tuning range of the generated waves is limited by the transparency of the nonlinear-optical material only [1]. Thus, they are ideally suited for wide-range high-resolution spectroscopy [2]. Furthermore, they are employed as sources of non-classical light [3]. Regarding the phase-matching concept, OPOs can be divided up into three types: Type-0 phase matching (all interacting waves have equal polarization), type-I phase matching (signal and idler waves have equal polarizations being perpendicular to the one of the pump wave) and type-II phase matching (the polarizations of signal and idler waves differ). These schemes have different pros and cons.
In type-0 OPOs based on materials such as lithium niobate, lithium tantalate or potassium titanyl phosphate, the highest nonlinear-optical coefficient d333 is employed leading to low oscillation thresholds. Since type-0 as well as type-I OPOs provide equal polarizations for the generated waves, self-phase locking can be achieved at wavelength degeneracy (λs = λi = 2λp) [4, 5]. Then, the OPO acts a phase-coherent optical frequency-by-2-divider which is useful for optical frequency metrology [6]. However, the wavelength tuning behavior of type-0 and type-I OPOs has a parabolic shape with an infinite slope at wavelength degeneracy. Consequently, a controlled tuning is hampered. To circumvent this drawback, one cavity mirror of the OPO can be replaced by a movable transversely-chirped narrowband Bragg grating [7].
In contrast to this, type-II OPOs show tuning branches that cross each other with a finite slope because the refractive indices of the generated waves differ at wavelength degeneracy. This strongly simplifies a controlled wavelength tuning in this spectral region. However, the highest nonlinear-optical coefficient of the abovementioned materials cannot be employed here. Furthermore, self-phase locking can be achieved only with additional effort, e.g. by adding an intracavity quarter-wave plate [8].
In this article, we show that whispering gallery optical parametric oscillators (WGR-OPOs) enable to combine the advantages of all abovementioned phase-matching configurations without the need of additional optical elements. WGR-OPOs do not require any cavity mirrors since the circulating light is guided by total internal reflection [9,10]. In contrast to their conventional mirror-based counterparts, spheroidally-shaped whispering gallery resonators are multimode cavities, i.e. the generated waves might have unequal transverse mode structures and consequently unequal effective refractive indices, even if their frequencies and their polarizations coincide. Consequently, a type-0 or type-I WGR-OPO might show a wavelength-tuning behavior similar to that of a type-II OPO (branches crossing each other at wavelength degeneracy with a finite slope). Although wavelength tuning of WGR-OPOs has been studied before [10–13], this has not been observed yet. Here, we experimentally verify the pseudo-type-II tuning behavior in a type-0 WGR-OPO made of radially-poled lithium niobate. This device combines the employment of the highest nonlinear-optical coefficient with a controlled wavelength-tuning around the wavelength degeneracy and with the possibility of self-phase locking.
2. Simulation of the tuning behavior
The three interacting waves propagate in a spheroidally-shaped whispering gallery resonator with the two radii R, r and a quasi-phase-matching structure with M periods along the circumference. Several conditions for the respective mode numbers as well as for the frequencies have to be fulfilled [14]:
Here, mp,s,i and pp,s,i denote the azimuthal and polar mode numbers. Furthermore, we have the frequencies νp,s,i as well as the detuning δ̂ from the respective resonances at νres,s,i normalized to the linewidths Δνs,i. The resonance frequencies νres,s,i are given by the dispersion relation [15] and are a function of the mode numbers m, p, q, of the resonator radii R, r and of the bulk refractive index n. The temperature dependence of R and r due to thermal expansion [16] is considered as well as the temperature and frequency dependence of n [17].In order to simulate the tuning behavior, we fix the mode numbers pp,s,i considering the second and third relation of (1) and qp,s,i as well as the pump frequency νp. We require that the latter corresponds to a resonance frequency νres,p. Knowing the resonator radii, we calculate the corresponding resonance temperature Tres by numerical evaluation of the dispersion relation.
Then, we determine combinations ms, mi that fulfill the first relation of (1). Knowing the temperature Tres, we calculate the resonance frequencies νres,s,i for the generated waves. Close to the resonances, optical parametric oscillation can occur. Finally, we can easily determine the frequencies νs,i. By setting the linewidths Δνs,i, we are left with the two conditions (2) and (3) where only the frequencies νs,i are unknown. Thus, for every value of Tres, we find sets (νs, νi, δ̂). This is done for several mode numbers mp, i.e. for several temperature values Tres in order to achieve the temperature tuning of νs,i.
Figure 1 shows the results of the simulations for parameters that are close to the ones in our experiment. The resonator is made of stoichiometric lithium niobate doped with 1.3 % MgO having R = 1.03 mm, r = 0.13 mm (both at 22 °C) and M = 225. The pump frequency is νp = 288.205 THz (λp = 1040.2 nm). Furthermore, it is assumed that all waves are polarized along the optic axis, i.e. type-0 phase matching. We find optical parametric oscillation for the mode-number combinations [(pp = 0, qp = 4), (ps = 1, qs = 3), (pi = 1, qi = 2)] and [(0, 5), (1, 3),(1, 3)]. If the mode numbers of signal and idler coincide, we observe the typical parabolically-shaped type-0 tuning behavior. In contrast to this, unequal mode numbers lead to tuning branches that cross each other with a finite slope at wavelength degeneracy as it is known for type-II phase matching.
Fig. 1 Simulated tuning branches of type-0 WGR-OPO with R = 1.03 mm and r = 0.13 mm and M = 225 pumped at λp = 1040.2 nm wavelength for different mode-number combinations. a) Unequal radial mode numbers for the generated waves. b) Equal radial mode numbers for the generated waves. The polar mode numbers are pp = 0 and ps = pi = 1 in both cases.
We restrict our simulation to solutions with |δ̂| < 2 assuming linewidths of 10 MHz for signal and idler waves. The detuning-induced oscillation-threshold variation is then limited to a factor of five, because Pth ∝ (1 + δ̂2) [14]. Since typical pump thresholds Pth are considerably below 1 mW in WGR-OPOs, we can expect to observe parametric oscillation experimentally at every temperature and consequently to verify the two different tuning behaviors. Furthermore, Fig. 1 shows that the number of solutions, i.e. the number of possible parametric oscillation processes at the wavelength degeneracy strongly depends on the shape of the tuning branches. For qs ≠ qi, we find only a few processes, whereas for qs = qi there are a lot of them. This indicates that a controlled tuning around wavelength degeneracy is hampered for the parabolically-shaped curves.
3. Experimental setup and identification of pump modes
The experimental setup used to analyze the tuning behavior of the type-0 WGR-OPO and identify the mode triplets is shown in Fig. 2(a). It comprises an external-cavity diode laser (ECDL) with a wavelength of 1040.2 nm, mode-hop-free tuning over more than 25 GHz and a linewidth of 100 kHz. The laser light is evanescently coupled from a single-mode fiber into the radially-poled WGR via a rutile prism. A Fabry-Pérot interferometer (FPI) is used to measure the frequency sweep of the ECDL. Additionally, a fiber polarization controller ensures extraordinary polarization. The resonator is made of z-cut stoichiometric lithium niobate doped with 1.3 % MgO. In order to achieve type-0 phase matching a radial domain structure with M = 225 is employed. More details about the fabrication of radially-poled resonators can be found in [11]. The major and minor radii are R = 1.03 ± 0.01 mm and r = 0.13 ± 0.01 mm, as measured using a white light interferometer. The distance between resonator and prism is adjusted by a piezo actuator and the resonator is temperature controlled with millikelvin stability using a proportional-integral-derivative controller and a housing around the setup. A low-pass filter (LP) separates the pump light from the generated light. The two light beams are detected on photodiodes (D) and the wavelengths of the generated signal and idler light are monitored on an optical spectrum analyzer (OSA) which is sensitive up to 2.55 μm with a wavelength accuracy of about 6 nm.
Fig. 2 a) Experimental setup for analyzing the transmission spectrum of the pump light (blue) and optical parametric oscillation (signal light: orange, idler light: red) in a radially-poled WGR. b) Identification of the equatorial pump modes qp = 1 − 5 in the transmission spectrum. The theoretically predicted positions are indicated above.
Our simulations above show that the two different tuning behaviors are found at the radial pump mode numbers qp = 4 and 5. Consequently, we need to identify these modes first. In order to do that, we compare the transmission spectrum of the resonator with the theoretical prediction by the dispersion relation [15]. This technique has been successfully employed for spheres and for spheroidally-shaped WGRs before [18, 19]. The measured transmission spectrum as well as the simulated one are displayed in Fig. 2. One can clearly identify the equatorial (pp = 0) whispering gallery modes with qp = 1 − 5.
4. Comparison of the measured tuning behavior with the simulated one
In accordance with the simulations, measurements with the equatorial pump modes qp = 4 and 5 are performed in the overcoupling regime between 20 and 90 °C in steps of 2 °C. Figure 3 shows the experimental results. At qp = 4, we observe tuning branches crossing each other with a finite slope. At qp = 5, a parabolically-shaped tuning branch is found. Thus, we have experimentally verified the different tuning behaviors predicted above (comp. Fig. 1). A closer look reveals that the experimentally determined curves are shifted in temperature by about 40 °C with respect to the simulated ones. We achieve a better accordance by changing the major and minor radii to R = 1.026 mm and r = 0.155 mm (at 22 °C). The slight discrepancy of the radii may be explained by the fact that the Sellmeier equation employed here does not perfectly describe the refractive indices of our crystal. For qp = 4, the simulation predicts that two parametric processes are phase-matched simultaneously. However, the outer parts (signal wavelengths below 1.8 μm and idler wavelengths above 2.5 μm) are suppressed due to a higher pump power threshold. These processes have a lower mode overlap because the wavelengths strongly differ. Furthermore, the quality factor decreases significantly above 2.5 μm wavelength [20].
Fig. 3 Measured and calculated tuning branches for two different mode combinations [(pp, qp), (ps, qs), (pi, qi)]. (a) [(0, 4), (1, 3), (1, 2)] and (b) [(0, 5), (1, 3), (1, 3)]. Idler wavelengths above 2.55 μm are derived from the signal wavelengths and energy conservation. The electric field distributions of the corresponding pump, signal and idler modes are shown (not to scale).
The good accordance of the calculated and measured tuning branches allows us to identify the mode combinations (electric field distributions in Fig. 3). We assume ps = pi = 1 because it has the lowest pump power threshold due to the largest mode overlap.
Additional tuning branches due to imperfections of the poling structure, as observed in a previous publication [10], have not been measured in our experiments. Presumably, our off-center parameter δR/R, estimated to be 0.008 from microscopic investigation of the used resonator, has been sufficiently small and the quality of the poling structure high enough [11] to have one dominating Fourier coefficient of the structure.
Once the WGR is calibrated by determining R and r for the computer simulations, the position of other tuning branches can also be predicted. We verified this for four pump modes. We have been able to record the same tuning branches after several months and infer that a single calibration using an OSA is sufficient to characterize the tuning behavior of this WGR-OPO.
5. Conclusion
In this work, we have predicted and experimentally verified that the tuning behavior of type-0 WGR-OPOs strongly depends on the mode-number combination of the generated waves. Equal mode numbers lead to the known parabolically-shaped tuning behavior, while unequal mode numbers yield to tuning branches that cross each other with a finite slope at wavelength degeneracy. The good accordance of the measured tuning behavior with the simulated one has enabled us to unambiguously identify the radial mode numbers of signal and idler light. Our work shows that type-0 WGR-OPOs allow to combine the advantages of different phase matching schemes, i.e. employment of the highest nonlinear-optical coefficient, controlled tuning around wavelength degeneracy and the possibility of self-phase locking.
Acknowledgments
We gratefully acknowledge financial support from the German Federal Ministry of Education and Research (funding program Photonics Research Germany, 13N13648) and the Deutsche Forschungsgemeinschaft. S-K. Meisenheimer acknowledges funding by the Deutsche Telekom Stiftung. The article processing charge was funded by the German Research Foundation (DFG) and the Albert Ludwigs University Freiburg in the funding program Open Access Publishing.
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