Whispering-gallery resonators have become extensively used frequency-synthesizers. Their high quality factors, combined with small modal volumes, mechanical robustness, and compact dimensions, make them especially attractive in the field of nonlinear optics [1–3]. Because they highly enhance the pump intensity and therefore allow efficient frequency conversion already at low input powers [4], nonlinear light-generation covering the ultraviolet down to the mid-infrared spectral range has been demonstrated using different materials [5,6]. This impressively demonstrates their versatility. In contrast to this broad accessible wavelength range, the fine-tuning capabilities as they are needed for most spectroscopic applications are still very limited [7,8]. This constraint results from a nonlinear relation between tuning efficiency and wavelength—often temperature tuning is deployed. When used as nonlinear frequency converters, the resonators usually are doubly (for second-harmonic generation) or triply (for optical parametric oscillation) resonant, while the interacting waves can span more than one octave in wavelength. An extreme case is the generation of harmonics using frequency combs—requiring near-zero dispersion over a large wavelength range. Unfortunately, when tuned with a dispersive tuning mechanism, the cavity resonances shift asynchronously. Because energy conservation dictates the frequencies of the interacting waves, the generated light may no longer be resonant with the cavity resonance. Besides temperature tuning, various other tuning mechanisms have been employed: electro-optic tuning with an external electric field offers fast tuning, but internal compensation charges compensate the effect in lithium niobate, hindering DC operation [9,10]; additionally, not every crystal features a strong electro-optic effect. Also, ferroelectric poling structures might be affected by externally applied electric fields. A further method was demonstrated using an external dielectric material in close proximity to the resonator [8]. By changing the distance, the resonances can be shifted. This has the drawback of additional external components and limited tuning range. Also, compression or stretching of the resonator with an external piezo actuator [11–14] has been tried and tested. Because this method changes the geometry, the wavelength dispersion is small. But, again, the use of external components contradicts the principle of a microresonator and impedes an easy integration. A promising method to address the aforementioned issues is to change the geometry of the resonator with an internal actuator. Here, we present a method to place a standard piezo actuator inside a crystalline whispering-gallery resonator ring to achieve a fast and widely tunable device suitable in particular to tuning nonlinear optical processes.

To manufacture the tunable resonator, we start with a $z$-cut 5%-MgO-doped congruent lithium niobate wafer. A femtosecond (fs) laser emitting at 388 nm wavelength with a 2 kHz repetition rate and approximately 300 mW average output power is used to cut out a resonator blank, as shown in Fig. 1(a). Here, an inner circle with a nominal diameter of 1.3 mm radius is removed. This creates the void that will accommodate the piezo actuator. Subsequently, the ring is placed around the piezo actuator and fixated with an adhesive. The piezo disk features a taper to match the tilted sidewalls created by the laser cutting. This way, the clearance between the crystal and the piezo actuator, which will be filled with the adhesive, is kept at a minimum. In the next process step, the outer rim of the crystal is shaped to get its final resonator geometry. Again, the fs laser is used. It is focused in grazing incidence at the rim of the spinning resonator, as shown in Fig. 1(b). Successively, the focal spot ablates the crystal until the resonator has a major radius of 1.5 mm and a minor radius of approximately 500 μm. After this process, the crystal has less than 200 μm thickness in radial direction, leaving only a small sheet that can be strained significantly without cracking. To create an optical-grade surface quality, a subsequent chemical-mechanical polishing step using a colloidal ${\mathrm{SiO}}_2$ suspension is conducted. This allows us to achieve an intrinsic linewidth of approximately 10 MHz at 1 μm wavelength. The final device is depicted in Figs. 1(c) and 1(d). The piezo actuator (PI Ceramic GmbH type PIC 255) used here is polarized along the resonator symmetry axis and silver coated on the top and bottom surfaces. For better handling, the back electrode (ground) is soldered to a brass post. When a positive voltage is applied to the top electrode, the actuator elongates along the symmetry axis but, because of the transverse piezo effect, it also compresses in the radial direction. Hereby, the strain $s_1$ in radial direction is given by $s_1=d_{31}{E}_3$, where $E_3$ is the electric field in axial direction and $d_{31}$ is the direct transverse piezoelectric-coefficient, which has a nominal value of $d_{31}\approx -180\mathrm{pC}/\mathrm{N}$ according to the manufacturer. In axial direction, the strain consequently adds up to $s_3=d_{33}{E}_3$ with $d_{33}\approx 400\mathrm{pC}/\mathrm{N}$. To demonstrate the versatility of this geometric tuning, we employ two crystals with different quasi phase-matching patterns. One radial pattern with 746 domain lines allows efficient second-harmonic generation of pump laser light in the 1 μm region. The second crystal features a linear poling pattern with 28.5 μm period length, enabling optical parametric oscillation.

 

Fig. 1. (a) Femtosecond laser cuts out a ring from a lithium niobate (LN) chip. (b) After the LN ring is glued onto a piezo actuator, the fs laser shapes the outer rim of the crystal. (c) Final tunable resonator. The piezo actuator is contacted on its top and bottom surface and contracts or expands radially. (d) Photograph of the resonator with the coupling prism. Like a swimming ring on a human, the elastic whispering-gallery resonator ring is fitted on a piezoelectric element.

Download Full Size | PDF

The measurement setup is illustrated in Fig. 2 and comprises a tunable diode laser emitting at around 1040 nm wavelength, which is used to pump the whispering-gallery resonator with approximately 20 mW mode-matched power. It is fiber-coupled, and the polarization is set parallel to the crystal $z$ axis. A GRIN lens and a rutile prism are used to couple the light into the resonator while the gap between the prism and the resonator can be tuned. A temperature controller and a housing around the setup ensure good temperature stability. The transmitted and generated light are collimated and separated with a dichroic mirror reflecting the pump wave. A beam splitter reflects a part of the generated light onto a photodiode. This signal is used to stabilize the pump laser via a servo-loop (PID control) onto the side-of-fringe of the resonance of the generated light. This fulfills several purposes simultaneously. First, this ensures that the pump laser wavelength tracks the cavity resonance while the cavity is tuned. Second, it ensures a constant power of the generated light. This is especially beneficial for spectroscopic applications because no reference detector is necessary. Third, the pump spectrum is—particularly when strongly overcoupled—a superposition of several transversal cavity modes, which can be distorted by the nonlinear conversion [15]. In contrast, the spectrum at the generated wavelength typically appears much cleaner because only modes that allow a phase-matched process are excited. This allows easy and robust locking. To analyze the fine-tuning capabilities, two methods are employed. In case of second-harmonic generation, the wavelength-dependent transmission through an iodine-vapor reference-cell (Thorlabs GC19100-I) is recorded. At the second-harmonic wavelength, iodine offers a dense and well-known spectrum. In the case of optical parametric oscillation, a wavemeter (HighFinesse WS-7 IR-II) directly records the signal-light wavelength.

 

Fig. 2. Schematic of the measurement setup. Abbreviations: tunable external-cavity diode laser (ECDL); polarization controller (PC); lens (L1); dichroic mirror (DM); photodiode (D1-3); beam splitter (BS); optical parametric oscillation (OPO); and second-harmonic generation (SHG).

Download Full Size | PDF

As described earlier, crucial for mode-hop-free tuning in a doubly or triply resonant system is an equal relative resonance tuning efficiency $\mathrm{\Delta }\nu /\nu$ of all interacting waves. This ensures that the actual frequencies of the light still match their respective cavity resonances during tuning. To determine the wavelength-dependent tuning efficiency, we assume a simple eigenfrequency relation of the resonator $\nu n=m{c}_0/(2\pi R)$, where $\nu$ is the resonance frequency, $n$ is the bulk refractive index of the host material, $m$ is the mode index (number of wavelength periods per cavity roundtrip), $c_0$ is the speed of light, and $R$ is the major resonator radius. Consideration of these eigenfrequencies is sufficient because we found that higher terms of the eigenfrequency relation, as presented in Ref. [16], make no significant impact. Furthermore, we take into account that the refractive index changes with frequency, $\mathrm{d}n/\mathrm{d}\nu$, while the resonator tunes, and that a strained crystal shows an elasto-optic effect, changing the refractive index as the radius changes, $\mathrm{d}n/\mathrm{d}R$, and the crystal is strained. This gives us the following ansatz:

Starting from that, it is straightforward to show (see Supplement 1) that the tuning efficiency is approximately given by

 

Fig. 3. Calculated tuning efficiency (orange) of the piezo actuator normalized with the relative geometric change of the resonator. The individual contributions of the refractive index dispersion (blue) and the elasto-optic effect (green) are dotted. The refractive index was obtained from [18].

Download Full Size | PDF

Now, we focus on the results generated with the resonator supporting second-harmonic generation. The recorded pressure and Doppler-broadened transmission spectrum of the iodine cell and the predicted spectrum [19] are shown in Fig. 4 and agree well. A mode-hop-free tuning is evident and corresponds to almost 28 GHz at second-harmonic wavelength or 14 GHz at pump wavelength, while the piezo voltage has been changed by approximately 500 V. Since piezo actuators tend to have a nonlinear strain voltage dependence as well as a hysteresis, the frequency axis has been linearized with the help of a Fabry–Perot interferometer tracking the detuning of the pump laser. To achieve this wide tuning range, the resonator was strongly overcoupled, leading to an increased resonator linewidth of about 100 MHz and, consequently, a wider tolerance window to compensate the dispersion of the tuning efficiency. It is worth mentioning that the linewidth of the generated light is mostly defined by the pump laser, which is less than 1 MHz. Thus, it is considerably smaller than the resonance linewidth and suitable for high-resolution spectroscopy [7]. Furthermore, the lock point was chosen at around 1% of the maximum possible output to ensure that locking is possible, although the conversion efficiency drops because of resonance mismatch. The measured tuning efficiency at 1 μm wavelength slightly exceeds the predicted tuning efficiency of 24 GHz per 1000 V. Because our simplified model neglects complex stress and strain patterns, this can be considered a good agreement. However, the conversion efficiency drops by a factor of 100, measured when the pump laser is not locked to a cavity resonance, resembling the predicted wavelength-dependent behavior.

 

Fig. 4. Measured transmission spectrum of the iodine gas cell (top) compared with the predicted absorption lines (bottom) at an absolute frequency of 576.30 THz [19].

Download Full Size | PDF

Next, the resonator supporting optical parametric oscillation is used. The signal-wavelength tuning while the piezo voltage changes is shown in Fig. 5. Again, with a strongly overcoupled resonator, mode-hop-free tuning of approximately 4.5 GHz was possible. In contrast to the second-harmonic generation process, the maximum achievable output power stays almost constant while the piezo voltage is changed. This is in agreement with the predicted dispersion of the tuning efficiency. Because the pump, signal, and idler resonances tune with almost the same rate, as can be seen from the weak dispersion of the tuning rate (Fig. 3) in the range 1–2 μm, all three waves stay close to the resonance center, which allows efficient light conversion. But another phenomenon becomes evident: although the same pump resonance is excited, the signal wave shows a pronounced tendency to make frequency hops spaced by one free spectral range. This indicates that the longitudinal signal mode index $m_{\mathrm{s}}$ changes by 1. To understand this behavior, we need to take a look at simulated radius-tuning branches, as presented in Fig. 6. In this simulation, we assume that the pump laser tracks one specific cavity resonance. Furthermore, the pump, signal, and idler waves are assumed to be transversal fundamental modes. Then, possible combinations of signal and idler modes with different longitudinal mode indices offering a resonance mismatch of less than 10 MHz—a conservative assumption for a strongly overcoupled resonator—are searched. Especially close to the point of degeneracy (left display), the same pump mode allows phase-matched processes with many combinations of signal and idler resonances simultaneously. Therefore, the actual process is arbitrary. It is likely that processes further away from the point of degeneracy (right display) show improved frequency-hop-free tuning behavior. Unfortunately, the available linear poling pattern had a nonideal period length, only supporting near-degenerate processes. Another aspect is the long-term stability. To measure this, we applied a voltage jump of $\mathrm{\Delta }{U}_{\mathrm{piezo}}=50\mathrm{V}$ and recorded the frequency fluctuation for more than 1.5 h. The actual frequency stays within $\pm 100\mathrm{MHz}$; the remaining weak fluctuations are most likely caused by temperature changes of the resonator. It is important to mention that no directional drift or decay was observed. Again, this is a strong argument that no significant electro-optic influence caused by piezo charges is active since such charges are generated just once and are always compensated on a minute-to-hour time scale [20].

 

Fig. 5. Measured signal-light wavelength while changing the piezo voltage.

Download Full Size | PDF

 

Fig. 6. Simulated tuning branches for a WGR with $R=1.5\mathrm{mm}$, 1.04 μm pump wavelength, and a quasi-phase matching pattern (QPM) with the indicated periods. The resonance mismatch is color coded, while a mismatch of 10 MHz is considered tolerable.

Download Full Size | PDF

The presented method focused on bulk whispering-gallery resonators. Despite this, application in batch-processed chip-integrated resonators seems feasible as etching processes to create the void for the piezo actuator are well established. Also, the used laser cutting process is fast and might be adapted.

To summarize, an innovative geometric tuning approach of a whispering-gallery resonator introduces new and fast tuning capability. The lack of external components keeps the dimensions small—which is a major feature of whispering-gallery resonators. Using lithium niobate as a host material, the relative frequency tuning efficiency is nearly wavelength independent when operated at wavelengths between 0.5 and 1 μm. This allows mode-hop-free tuning of nonlinear processes and paves the way for using such light sources in high-resolution spectroscopic applications. Because this approach does not depend on specific material properties, it offers promising prospects as a universal tuning scheme in many materials and applications.