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A new tuning method for tuning whispering gallery modes (WGMs) in a cryogenic environment is presented. Within a home-made exchange gas cryostat the applicability of pressure tuning in microbubbles at liquid nitrogen (LN) temperature is shown. The general thermal shift and tuning behavior of borosilicate microbubbles is theoretically analyzed and compared to experimental data. We show that stress/strain tuning using compressed gas is widely unaffected by system temperature.
©2013 Optical Society of America
1. Introduction
During the last few decades optical microresonators have proven to be extremely versatile systems. The combination of simple manufacturing techniques and ultra high quality factors makes microsphere resonators attractive tools for experimental research [1]. In experiments where the foremost interest is the efficient coupling of an emitter to a single resonator mode, the question arises, how to match a single resonance frequency of the microresonator to the emission line of an applied photon emitter. Although these coupled systems can be analyzed experimentally at room temperature, cryogenic conditions are preferred due to the highly suppressed interaction with the environment, e.g. the reduced decoherence rate of coupled emitters. Under these conditions also the thermal bistability of a resonance can be suppressed and the intrinsic Kerr non-linearity becomes dominant [2]. Attempts to couple the narrow zero-phonon line of color centers in diamond to WGMs have been reported by several groups [3–5], but a required reliable tuning method could not be realized so far. In a cryostat, changing the circumference of a WGM resonator by temperature tuning through thermal expansion is not appropriate over long ranges [6,7]. Another method is based on mechanical strain, for example compressing or stretching the microsphere via a piezoelectric actuator [8].
An alternative to such well known methods is offered by a quite new type of microresonator, the so called microbubble resonator. These are normally produced from thin-walled or tapered micropipettes which are locally heated to the softening point by an arc discharge or a CO2 laser beam and then expanded by either compressive mechanical strain or by applying an internal pressure [9,10]. Internal pressure can also be used after production for reversibly changing the effective size of the microresonator. With such a tuning scheme resonance shifts in the order of a free spectral range (FSR) have recently been shown [11]. Tuning the microresonator using internal aerostatic pressure gives a negligible thermal load on the system which makes it ideally suited for cryogenic applications. Nevertheless, when applying this kind of resonator in a cryogenic environment the precursory resonance shift by cooling down the system has to be fully taken into account. In this paper we will demonstrate the ability to apply pressure tuning to microresonators under cryogenic environments and analyze the overall tuning properties of that system, ranging from room temperature to liquid nitrogen (LN) temperature.
2. Experimental setup
Microbubbles for this experiment were made by heating the tip of sealed glass capillaries in the focus of two counter propagating CO2 laser beams. During heating, the air in the capillary was pressurized. When the glass softens the internal air pressure pushes out the capillary walls and forms a microbubble. The walls of bubbles were typically in a range between 1 and 2 μm [11]. To excite WGMs, laser light at 770 nm was coupled into the thin spherical shell of the microbubble via evanescent waves on a tapered optical fiber. The laser frequency was ramped ± 25 GHz with a repetition rate of 5 Hz. The WGMs were observed as Lorentzian shaped dips in the laser power transmitted by the fiber taper. The WGM spectra were then recorded using a photodiode and digital oscilloscope.
Figure 1 shows a picture of a microbubble and the schematic of the experimental setup. The set up consists of a small metal box which was held inside a plastic container. On the wall of the metal box was a feedthrough for gases and vacuum. Inside the metal box the gas feedthrough was connected to a plastic tube for holding the microbubble. After the bubble was inserted the plastic tube was sealed by UV curing glue. Also inside the metal box was a U-shaped plastic mount which held the tapered optical fiber. The fiber mount was connected to a three-dimensional stage by a post fed through the lid of the metal box. The lid of the metal box was made from clear plastic so that the position of the taper could be monitored using a microscope and a camera.
Fig. 1 (a) Microbubble under a microscope. (b) Schematic of the cryogenic setup with laser control, camera, digital storage oscilloscope (DSO), and gas supply system.
To cool the microbubble the gap between the inner metal box and plastic container was filled with LN. During filling and subsequent cooling, the inside of the metal box was flooded with nitrogen gas (N2) to prevent icing on the taper and the microbubble. The temperature of the gas inside the metal box was measured using a thermocouple positioned next to the microbubble. After sufficient cooling, it was possible to stop the flow of N2 and the optical transmission of the fiber remained constant for the rest of the cool-down process. The temperature inside the box settled at 86 K and remained at this temperature for some minutes as the liquid boiled off but the temperature could be maintained by adding more LN. When the LN boiled off to a certain level, the box started to heat up very slowly. It reached room temperature after evaporating the entire LN but without any icing of the optical components. In this way it was possible to measure the shift of the WGMs continuously from the cold to the hot state without the need of stabilizing a specific temperature in between. Also this direction for the shift offers better stability for the measurement due to the fact that there are no mechanical disturbances associated with boiling or refilling the LN.
3. Temperature tuning properties of microbubbles
The well known equation for the relative wavelength shift λ of a WGM is dependent on the radius a and the refractive index n and is used here under the assumption that it is still valid in the case of microbubbles [11]. By using the thermal expansion coefficient and the thermo-optic coefficient the temperature (T) induced resonance shift of the WGM can be rewritten as [6]
Please note that only the material parameters have to be taken into account and no direct dependency on the absolute size of the resonator exists. From elasticity theory there is also no difference between a full sphere and a microbubble resonator consisting of just a thin spherical silica shell [12]. As long as the wall thickness of the resonator is able to fully support the relevant mode structures the theory is completely independent on that parameter.To analyze the temperature dependency of the mode shift it should be taken into account that the coefficients α and β as well as n are not constant, but also depend on the respective temperature range. For the refractive index this change is normally quite small. The overall resonance shift by cooling down the resonator from room to LN temperature is then rewritten as
Here n0 is fixed to the refractive index at room temperature, considering that its change with temperature is neglectable in the last quotient. For borosilicate capillaries (Schott Duran) the thermal expansion coefficient at room temperature and the refractive index for a wavelength of 770 nm are given directly by the manufacturer (α = 3.3∙10−6 K−1, n0 = 1.4712). The temperature dependency of the thermal expansion coefficient can be found in the literature [13]. For the range starting from room temperature down to LN temperature this behavior can be fitted by a simple quadratic function, see also Fig. 3(a). The room temperature value and temperature dependency of the thermo-optic coefficient β(T) is unknown for this type of industrial glass. However, this function can be deduced by looking at different optical borosilicate glasses offered from Schott. Normally for this kind of glass α and β are of the same order and so we simply set β(T) = α(T) for a rough estimate. At ambient temperatures around 300 K the differential shift rate (DSR) from Eq. (2) is around −2.2 GHz/K. For the lowest temperature reachable with our setup the estimation gives a DSR of −1.0 GHz/K based on Eq. (2). In Fig. 2 the theoretical tuning curves (dotted red lines) based on our estimation can be seen. In Fig. 2(a) the absolute shift of the resonator modes is shown, whereas in Fig. 2(b) the DSR is presented. In total an absolute shift of nearly −350 GHz is expected. It follows from theory that, as in the case of ordinary silica microspheres, the temperature tuning properties of microbubbles are strongly decreased with lower temperatures.Fig. 2 (a) Absolute and (b) differential shift curves for the WGMs resonance positions. The dotted lines are based on theoretical assumptions while the straight lines are experimental data.
The experimentally measured temperature shift rates of the WGM resonances in a microbubble are also presented in Fig. 2 (solid blue lines). While the system slowly heats up from LN to room temperature, the modes become red shifted and move out of the 50 GHz laser scan window while new modes continuously appear from the blue side. The tuning properties of the different modes in microbubbles slightly depend on their interaction with the environment, but nearly all WGMs shift at the same rate. Due to the different linear expansion coefficients of the materials used in the box the coupling was lost during the heat up process and instant recoupling was required. All the scan data during the heat up was saved on a PC and the differential shift rates of the modes were analyzed afterwards. One mode was traced for as long as possible and then we changed to another mode with the exact same shift rate, i.e. modes with identical tuning properties, on the blue side of the laser scan window. The timing resolution of the setup allows for 0.5 K discrimination. The stepwise difference values were then plotted together to give a total accumulated thermal mode shift of nearly −325 GHz, see the solid blue line in Fig. 2(a). The resulting experimental differential values are plotted in Fig. 2(b) (solid blue line). The experimental data show good agreement with the theoretical curves and validates our previous approximation of β(T).
The results in Fig. 2 were used for calculating a temperature dependent curve for the material property β(T), see Fig. 3(b) . For the thermo-optic coefficient at room temperature a value of β295K = (4.48 ± 0.19)∙10−6 K−1 was determined. In the LN temperature range this value goes down to β90K = (0.05 ± 0.01)∙10−6 K−1. The error of these two values is based on the fitting accuracies of the reference α(T) and the assumption that the measured temperature shift curve is in complete agreement of the tuning behavior from our theory. Lacking information about the thermo-optic coefficient of pure borosilicate glass at room temperature these results cannot be fully proven. However, given the good agreement between the theoretical estimation and the measured data these values seem reasonable.
Fig. 3 (a) Temperature dependency of the linear expansion coefficient α(T) in borosilicate glass. The presented data points are directly taken from ref [13]. The solid line is a quadratic fit to these values within the relevant temperature range. (b) Temperature dependency of the corresponding thermo-optic coefficient β(T) gained from the experimental data. Also here a quadratic fit to the data is given for the examined temperature range.
4. Pressure tuning properties of microbubbles at low temperature
For the estimation of the pressure tuning properties of the WGM resonances at LN temperature the microbubble resonator is approximated as a thin spherical glass shell. Based on the solution of the elasticity and the Maxwell-Neumann equations in [11] the pressure tuning abilities of microbubbles are derived as
Here p0 and pi are the externally and internally applied uniform pressures, a and b denote the outer and inner radius of the silica shell, G is the shear modulus, and C the elasto-optic coefficient of the material. It can be shown that the tuning properties described by Eq. (3) are nearly independent of the small change in the refractive index which appears by cooling down the system from room to LN temperature. Therefore the refractive index of the stress-free material at room temperature, n0, can be used also at LN temperature. Figure 4 shows an example of WGM shift in a microbubble when the internal pressure is increased from 1 bar to 1.5 bar. For this specific microbubble the tuning rate is −5 GHz/bar.Fig. 4 Example of mode shift in a microbubble due to internal pressure. It can be seen, that all of the modes shift nearly by the same frequency. A residual is based on slightly different tuning conditions for the various mode orders in the microbubble.
When applying an internal pressure at a specific temperature the total resonance shift in a microbubble is, by neglecting the constant term depending on p0, directly proportional to the geometric parameter χ = b3/(a3-b3). Therefore the shift is also proportional to the ratio between the whole enclosed volume and the volume of the spherical shell alone. This ratio is independent of the relative size changes due to cooling or heating. Thus different pressure tuning properties at different temperatures can only arise from the two temperature dependent material parameters G and C. At room temperature these values are given by the manufacturer as G = 26.6∙109 Pa and C = 4∙10−12 m2/N. For a comparable borosilicate glass made by Corning (Pyrex) the temperature dependency of the elastic moduli were found to behave positively [14]. Following from this reference, G should slightly increase with lower temperatures, but not differ by more than 2-3 percent from its room temperature value. For the temperature dependency of C it was not possible to find any reference stating this behavior. For simplicity a constant value for C is used over the full temperature range. Under these assumptions we estimate the overall cryogenic pressure tuning properties of microbubbles to be more or less of the same order as they are under room temperature conditions.
Tuning of the WGMs can be achieved by pressurizing or evacuating the gas inside the bubble. The shift rate of any bubble can be estimated from the bubble dimensions and the material properties [11]. For the borosilicate glass used here the shift rate at room temperature is simply −1.1 GHz/bar times the geometrical parameter χ as introduced earlier. Based on this expression and taking the bubble dimensions (a = 108 µm, b = 106.8 µm) into account, the bubble used here is expected to have a room temperature tuning rate of −14 GHz/bar. For pressure tuning in this experiment compressed helium gas was used inside the microbubble to prevent icing effects on the inside of the microbubble. For testing the room temperature tuning properties the internal helium gas pressure was increased from 0 bar to 4 bar and the corresponding positions of the WGMs were recorded. The WGMs showed an identical tuning behavior, only a few modes fractionally differed due to their higher environmental sensitivity. The experiment was then repeated after cooling down the whole setup and recoupling the resonator to the fiber taper coupler, the result of these measurements is plotted in Fig. 5 .
Fig. 5 Pressure tuning curves of a microbubble at LN and room temperature. The observed room temperature tuning rate for this microbubble corresponds well with the estimated value from its geometrical parameter χ.
The pressure tuning measurements were repeated several times, hence the error bars in Fig. 5. The measured room temperature tuning rate of the microbubble used was nearly −15 GHz/bar and corresponds with the theoretical value estimated from the geometrical parameter χ or by directly solving of Eq. (3). Within the error range of these measurements the two curves appear to have slightly different slopes. The tuning properties of the microbubbles at LN temperature seem to be a bit lower compared to the room temperature values. This is in good agreement with our theoretical estimation about the temperature dependent pressure tuning properties. By using the Pyrex values for the elastic properties it is also possible to calculate the corresponding LN temperature value of the elasto-optic coefficient C. From our measurement it is about 10% lower as the referenced room temperature value and turns out to be C90K = (3.55 ± 0.08)∙10−12 m2/N based on our measured statistics and the fitting accuracy of the Pyrex values.
6. Summary and outlook
In conclusion, we have successfully shown WGM tuning using the internal aerostatic pressure in a microbubble resonator at cryogenic temperatures. Compared to the room temperature values, the influence of the mechanical properties is proven to be negligible at least down to LN temperature and a reliable shift of the resonance positions is still possible under these conditions. There is no external change to the environment so this method allows for applications where a fast and flexible active tuning method is needed at low temperatures. From the mechanical point of view this technique should be applicable also at LH temperature. A widely tunable microresonator operating at cryogenic temperature is highly desirable for studying solid-state model systems in cavity quantum electrodynamics or to exploit non-linear effects for confined light at low temperatures.
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