In a class of high quality (Q-) factor dielectric resonators with low radiative losses, including popular whispering-gallery mode (WGM) resonators with high azimuthal mode numbers, due to high confinement of modal field in dielectric, the Q-factor is limited by the value of inverse dielectric loss tangent of dielectric material. Metal enclosures necessary for device integration only marginally affect the Q-factor while eliminating the residual radiative loss and allowing the optimization of input and output coupling. While very high Q-factors 200 000 are available in sapphire WGM resonators in X-band, at millimeter wave frequencies increasing dielectric loss limits the Q-factor to much smaller values, e.g. ∼50000 and ∼25000 for quasi-TE and quasi-TM modes, correspondingly, at 36 GHz. The use of distributed Bragg reflection (DBR) principle allows to push modal energy outside dielectric while also isolating it from Joule losses in metallic enclosure walls. Very high Q 600 000 > t g δ has been demonstrated in X-band [C. A. Flory and R. C. Taber, IEEE Trans. Ultrason., Ferroelectr., Freq. Control 44, 486–495 (1997).] at the expense of impractically large dimensions. In this work, we report on the assembly and testing of a compact Ka-band sapphire distributed Bragg reflector cavity characterized with Q-factor seven times larger than one predicted by the material’s dielectric loss at the frequency of interest. An intrinsic Q-factor of 200 000 is demonstrated at 36 GHz for the lowest order TM-mode of a sapphire DBR. The resonator has 50 cm 3 volume, smaller than previously demonstrated DBRs.

Dielectric microwave resonators are widely used for creating spectrally pure microwave oscillators.1,2 The figures of merit for the frequency reference resonators are their quality (Q-) factors and volume. The Q-factor defines the close-in phase noise of microwave oscillators utilizing the resonator, while volume impacts the sensitivity of the resonator to acceleration and vibrations, as well as defines the density of the frequency spectrum. The highest Q-factor can be achieved in a large empty resonator, since the photon lifetime in the resonator scales with the resonator size. However, the large volume and mass of the resonator are usually impractical. A dielectric resonator can have a much smaller size and still high Q-factor. The size reduction leading to Q-factor reduction is somewhat compensated by the effect of confinement of the electromagnetic radiation in the low loss material having a large dielectric constant. On the other hand, the dielectric unavoidably attenuates the electromagnetic radiation. The problem is how to create a resonator having the highest possible Q-factor and the smallest possible volume. In this paper, we report on the development of a sapphire distributed Bragg reflector (DBR) resonator operational in the Ka frequency band. For the demonstrated Q-factor, the resonator has the smallest size, to our knowledge, if compared with its predecessors operating at similar frequencies. The compactness and high-Q factor make the resonator attractive for applications in low-noise microwave oscillators, high efficiency frequency discriminators, and many other physical systems.

There are multiple types of high-Q microwave resonators developed over the last century. These include bulk dielectric [e.g., whispering-gallery mode (WGM)] and empty metal cavity resonators. Metal cavities have the lowest Q-factors for the same volume, if compared with the dielectric resonators. The Q-factor value in metal cavities is defined by Joule losses, i.e., by residual surface resistance of metal walls. The Q-factor of the bulk dielectric resonators is ultimately limited by the dielectric loss tangent. Additionally, practical very high Q dielectric devices utilize metal enclosures for compensation of radiative effects and optimized coupling. The enclosures add resistance losses driving the need to increase the overall size of the enclosure to reduce the surface effect. Optimized shielded dielectric WGM resonators remain much smaller than hollow metal cavities of similar Q because of the field confinement in the large dielectric constant cavity host material. For example, the dielectric constant of the popular cavity host material, sapphire, is about 10. The ultimate Q-factor of WGM resonators does not exceed the inverse loss tangent of dielectric material,3,4 since the bulk of modal energy is localized inside the dielectric. Temperature sensitivity of frequency in WGM resonators is also dominated by the temperature dependence of the dielectric constant. The loss of a dielectric usually increases with the increase of frequency, making the problem of creation of high-Q high frequency dielectric resonators rather hard.

DBR principle allows to create a metal-enclosed dielectric resonator with Q-factor exceeding one of a hollow metal cavity, or a fully dielectric structure of the same volume. It should be considered the best solution to overcome inverse loss tangent limit for the dielectric resonator, since it allows confinement of the bulk of microwave field energy outside the dielectric material. It also effectively shields the modal field from the metal enclosure of the resonator. Proper design of the resonator ensures that only a single high-Q DBR mode is supported by the structure having a volume exceeding the microwave wavelength. Multiple kinds of DBRs were demonstrated and reported so far. Some of the achievements are summarized in Table I. The main challenge here is to develop a structure that allows achieving a very high Q-factor for a high frequency resonator while keeping the resonator size as small as possible. While very high values have been obtained in X-band, dimensions were very large and coupling not well optimized for practical RF applications.

TABLE I.

Values of frequency and Q-factor of various demonstrated experimentally sapphire DBRs. The values of the internal volume of the cavity enclosure are listed, when provided. As the microwave loss typically increases inversely proportional to frequency, for better comparison we include in the table the product Q × f for each of the DBR embodiments. Note the highest value of Q × f per unit of cavity volume achieved in this work.

f, GHzQV, cm3Q × f, GHzRef.
19 5.3 × 105 105 1 × 107 5  
9/13.2 6.5/4.5 × 105 1,280 5.85 × 106 6  
7 × 105 … 6.3 × 106 7  
13.4 2 × 105 334 2.68 × 106 8  
9.8 2 × 105 325 1.96 × 106 9  
9.4 1.2 × 105 961 1.13 × 106 10  
36 2 × 105 50 7.2 × 106 This work 
f, GHzQV, cm3Q × f, GHzRef.
19 5.3 × 105 105 1 × 107 5  
9/13.2 6.5/4.5 × 105 1,280 5.85 × 106 6  
7 × 105 … 6.3 × 106 7  
13.4 2 × 105 334 2.68 × 106 8  
9.8 2 × 105 325 1.96 × 106 9  
9.4 1.2 × 105 961 1.13 × 106 10  
36 2 × 105 50 7.2 × 106 This work 

We must mention here alternative methods suggested to overcome the inverse loss tangent limit in resonator quality factor. The authors of Ref. 11 suggest application of photonic bandgap (PBG) principle and discuss the potential of a very high Q of 500 000. They also demonstrate a 30 GHz prototype with modest Q photonic crystal defect mode; however, compact embodiment with Q higher than inverse loss tangent remains to be seen. Methods of proposed tuning of DBR modes have been recently suggested in Ref. 10.

In this work, we report on the development of a 36 GHz DBR resonator with an intrinsic Q-factor of 2 × 10 5, which is seven times better than the Q-factor limited by the material loss. The latter was qualified using a WGM resonator operated at the same frequency and created with the same material. While the general configuration idea of the resonator was taken from Ref. 6, the final shape and periodicity of the sapphire layers were optimized numerically to increase the lifetime of the fundamental DBR mode. As the result of our study, the demonstrated resonator has very small overall size for the achieved Q-factor and microwave frequency. As can be seen from Table I, our work would represent the best result in terms of possible figure of merit Q × f per unit of resonator volume. In addition to optimization of design for quality factor, we suggest and implement critical coupling solutions important for applications such as low phase noise oscillators.

The paper is organized as follows. Section II describes the design of the Ka-band sapphire DBR and the results of the resonator modeling using ANSYS software. Section III outlines the experimental setup. Section IV presents general physical characteristics of the assembled DBR measured in our experiments. Section V describes calibration of the material attenuation using a WGM resonator made of the same material. Section VI concludes the paper.

The cavity design shown in Figs. 1(a) and 1(c) is similar to the original design of the lower frequency DBR cavities.5,6 The resonator geometry is the intersection of two cylindrical shells and four planar disk bodies of sapphire inserted into a metal cylinder enclosure. Component geometry is finalized via numerical simulations using ANSYS HFSS software.

FIG. 1.

Design (a) and implementation (b) of two layer distributed sapphire Bragg reflector structure for 36 GHz resonator, along with designed critically coupled housing (c) and prototype brass housing with circular coupling apertures (d), top lid removed.

FIG. 1.

Design (a) and implementation (b) of two layer distributed sapphire Bragg reflector structure for 36 GHz resonator, along with designed critically coupled housing (c) and prototype brass housing with circular coupling apertures (d), top lid removed.

Close modal

The brass enclosure of our prototype cavity (25.5 mm internal diameter and 28 mm height) had attenuation corresponding to Q 0 = 7000 of hollow cavity in the K a band, frequency of 36 GHz. Circular grooves cut in the metal enclosure’s top and bottom caps, which are used to mate with cylinder cavity and align the sapphire cylinders, do not change the properties of the field distribution significantly. The expected Q-factor of the infinite DBR structure is 3 × 10 5. The estimation is based on an assumption that approximately 10% of modal energy is confined in the material with loss tangent t g δ = 3.3 × 10 5 (Ka-band sapphire loss tangent component perpendicular to the crystal axis; data extrapolated to 36 GHz from Ref. 3). One has to select the number of sapphire layers capable of isolating the DBR mode from metal walls enough to ensure that their resistive loss influence on the quality factor is negligible. To quantify this, we calculated numerically the Q factor of hypothetical DBR cavity with “lossless” sapphire and obtained unloaded Q factors of 2.0 × 10 6 and 8.3 × 10 6, in two layer and three layer 36 GHz structures correspondingly. This indicates that a two layer structure is enough to not dampen the unloaded quality factor of the DBR mode with real sapphire by more than 15% due to Joule losses. Assuming this level of field suppression, one expects Q 300 000.

The sapphire structure of our DBR resonator [Fig. 1(a)] has been constructed with four disks of diameter 25.4 mm and thickness 0.71 mm and ten cylindrical shells with external diameters 19.43 and 13.1 mm, a thickness of 0.69 mm, and heights 8.33, 4.18, and 4.15 mm, respectively, from the middle toward the end caps. All components have been fabricated out of HEMEX grade sapphire (from Crystal Systems Inc.) and oriented with their symmetry axis parallel to crystal axis to within 0.5 °. The group of four sapphire disks forms the Bragg reflector cavity in the vertical direction and isolates the DBR mode from Joule loss contribution of end caps, similar to the function of cylindrical sapphire shells for isolating the resistive losses in metal cylinder. Further elimination of Joule losses may have been achieved using three layer structure but that would come with nearly twofold increase in overall resonator internal volume, from 14.2 to 27.5 cm 3. Alternatively, silver or gold plating can be used for reduction of surface resistance.

We have simulated the full structure using ANSYS software, varying the parameters and arriving at the optimum values listed above, and predict an unloaded Q factor of 297 700 for a two-layer sapphire structure in the brass enclosure. We have also verified the sensitivity of design to dimensional variations of cavity structure, and also to misalignment of components during assembly, and have found that 0.1mm variations and displacements do not reduce the resulting Q-factor of the DBR mode by more than 10%. If special measures are not taken to selectively couple to the ultimate DBR mode, mode spectrum of resonator will be rather dense, consisting of metal box modes perturbed by sapphire layers, with unloaded Q-factors similar to the empty cavity ones (7000), whispering-gallery modes (WGMs) with Q-factors of 30 000–60 000, and Bragg grating (DBR) mode with Q of 200 000–300 000. All these modes were observed in the follow-on experiment, as is shown below.

We have optimized the dimensions of the resonator so that the fundamental DBR mode of the resonator had the frequency of 36 GHz. The field distribution of the DBR mode of the cavity evaluated with ANSYS software is shown in Fig. 2(a). The confinement is good enough to achieve the desirable Q-factor. Importantly, the amount of field confined in the sapphire structure was less than 15%. The WGMs have a smaller mode volume concentrated in sapphire rings, and most of the electric field energy confined in the dielectric material. This is the main reason not to select WGM for achieving the highest possible Q-factor in the structure and utilize input–output structure that is isolated from WGMs and decoupled from box modes.

FIG. 2.

Calculated electric field distribution of eigenmode (a) and optimized excitation (b) of DBR mode with a waveguide coupler [see Fig. 1(c)]. The color code represents 25x reduction of the field from red to blue.

FIG. 2.

Calculated electric field distribution of eigenmode (a) and optimized excitation (b) of DBR mode with a waveguide coupler [see Fig. 1(c)]. The color code represents 25x reduction of the field from red to blue.

Close modal

Both localized coaxial inductive probes and optimized pinholes from adjacent hollow waveguides are not suitable to decouple from the WGMs that have alternating azimuthal maxima. We developed the azimuthally expanded constructive mini-horns with optimized opening into WR-28 waveguides [Fig. 1(c)]. As was found in Ref. 6, the cylindrical DBR structure supports only one “super-Q” DBR mode polarization, with an electrical field perpendicular the cylinder axis. Hence, the WR-28 waveguides are oriented with their wider side along the cylinder axis.

Azimutally expanded opening into cylindrical cavity is intended to suppress coupling to WGMs, and variation of the opening into the waveguide, as well as the length of taper allows to adjust coupling strength to DBR mode and achieve undercoupled, overcoupled, and critically coupled regimes. Figure 2(b) illustrates the electrical field in critically coupled regime at DBR resonance. Results are represented for simplified, single dielectric layer Bragg structure, since three-dimensional resonance field simulation in a very high Q microwave cavity is still beyond the capacity of lab computers. The DBR mode effectively amounts to a standing wave bouncing between two Bragg mirrors, in Fig. 2 represented by combination of a single dielectric layer and metallic cylinder. Azimuthal expansion of two coupling port constructive horns leads to effective averaging out the excitation for high mode number WGMs.

As mentioned above, the dielectric structure of the designed resonator was fabricated using custom made sapphire components. The structure has been assembled on a specially prepared rotary alignment rig, allowing coaxiality of components’ placement to within 0.03 mm margin. All components have been consecutively added and attached to each other using a minimal amount of epoxy. With the total estimated amount of epoxy less than 2 mm 3, dielectric constant ε = 3.3, microwave loss tangent t g δ < 1 × 10 3, and due to its location in areas of the relatively small electrical field of DBR mode, loss contribution of epoxy was negligible. While the four disks forming the vertical Bragg structure were optically polished and planarized to optical class parallelity at the level of λ / 4, surface finish of cylindrical shells was “as grinded,” with residual rms roughness of 10  μm, and thickness variation (non-concentricity of inner and outer cylindrical surfaces) reaching 50  μm, both due to limitations of fabrication process and tools. In a free standing DBR structure, surface roughness, non-concentricity, and assembly misalignment should result in additional scattering and radiation loss. In enclosed structure with metallic walls, the inhomogeneities result in perturbation of modal frequencies and intermodal coupling, the latter effect significant only in the case of their frequency degeneracy. To illustrate the importance of enclosure for achievement of high Q in a compact device, we tested the assembled bare DBR structure using external horn emitters, as shown in Fig. 3(a). The measurements were performed using an Agilent E8364C network analyzer. The quality factor of standalone sapphire assembly, inferred from the 3 dB bandwidth of the transmission resonance in Fig. 3(b), was rather low around 480. The measured Q-factor of the DBR mode was somewhat lower than the calculated value of 575 we had obtained by numerical simulation, but is not drastically different, and the discrepancy can be attributed to additional scattering loss produced by non-concentricity and geometrical imperfections. To support the highest possible Q, all-dielectric structure would need a larger number of layers, optimized to allow critical radiative coupling—a bulky and complex option that is better replaced with our optimized metal-dielectric construction in which Bragg mirrors are formed by two layers of dielectric and reflective metal wall. Not only this technique reduces the overall size of the resonator, but it also facilitates compact critical coupling. Placing the dielectric structure into metal housing drastically increased the Q-factor of Bragg mode. For evaluation of its intrinsic (unloaded) quality factor Q 0, and optimization to near critical regime, the sapphire assembly shown in Fig. 3(a) was inserted into the metal housing [Fig. 4(a)] having originally two small circular apertures of diameter 3.175 mm at opposite sides. Those two sides had exactly symmetric flats suitable for either direct attachment of WR28 waveguide flange or joint assembly with intermediately placed optimized horn. That single waveguide was then connected to a vector network analyzer, and S-parameter tested. Figure 5 represents the result of S 11 measurement with opposite aperture blocked by attached brass blank (a) and open (b). In both cases, the depth of S 11 dip of DBR modes did not exceed 1.3 dB in log scale, so this regime remained low-coupled. Single aperture bandwidth value (a) δ f 1 = 249 kHz already indicates that Q 0 > 145 000. Assuming an equal effect of two identical coupling apertures on opposite sides of enclosure, and assumption that small holes additively contribute to energy losses of the cavity, we can use results of two measurements to extrapolate and evaluate Q 0 as a value at which both holes are closed. This method, known since early microwave studies, results in estimated unloaded bandwidth δ f 0 = 2 δ f 1 δ f 2 = 187 kHz and corresponds to intrinsic Q 0 = 1.9 × 10 5.

FIG. 3.

Test setup (a) and result of quality factor measurement (b) of free standing sapphire Bragg reflector structure. Lorentzian fit of transmission peak near 36 GHz yields the estimated bandwidth of 74.8 MHz that corresponds to Q = 481. Background transmission and fringes are caused by direct horn to horn transmission and residual interference in waveguides.

FIG. 3.

Test setup (a) and result of quality factor measurement (b) of free standing sapphire Bragg reflector structure. Lorentzian fit of transmission peak near 36 GHz yields the estimated bandwidth of 74.8 MHz that corresponds to Q = 481. Background transmission and fringes are caused by direct horn to horn transmission and residual interference in waveguides.

Close modal
FIG. 4.

Structure and implementation of test housing [(a)–(c)] with interchangeable circular aperture used for verification of critical coupling regime. (d) Results of S 11 parameter by network analyzer measurement illustrate 17 dB extinction at DBR mode, and absence of high contrast modes within 500 MHz vicinity, important for oscillator applications.

FIG. 4.

Structure and implementation of test housing [(a)–(c)] with interchangeable circular aperture used for verification of critical coupling regime. (d) Results of S 11 parameter by network analyzer measurement illustrate 17 dB extinction at DBR mode, and absence of high contrast modes within 500 MHz vicinity, important for oscillator applications.

Close modal
FIG. 5.

The S 11 parameter measurement results (in linear magnitude format) of DBR mode in housing with one (a) and two (b) identical opposite apertures of diameter 3.175 mm. Lorentzian fit yields bandwidth of δ f 1 = 249 kHz and δ f 2 = 311 kHz correspondingly. Extrapolated unloaded bandwidth of the DBR 2 δ f 1 δ f 2 = 187 kHz corresponds to intrinsic Q 0 = 1.9 × 10 5.

FIG. 5.

The S 11 parameter measurement results (in linear magnitude format) of DBR mode in housing with one (a) and two (b) identical opposite apertures of diameter 3.175 mm. Lorentzian fit yields bandwidth of δ f 1 = 249 kHz and δ f 2 = 311 kHz correspondingly. Extrapolated unloaded bandwidth of the DBR 2 δ f 1 δ f 2 = 187 kHz corresponds to intrinsic Q 0 = 1.9 × 10 5.

Close modal

After that, we kept the opposite aperture closed, increased the front opening in the metal enclosure, and inserted the layer with optimized constructive horn [shown in Figs. 4(a)4(c)]. To simplify machining, we have replaced the rectangular horn structure shown in Fig. 1(c), with the conical one, having an equal length and opening areas on the side of WR28 waveguide and DBR cavity. Test results shown in Fig. 4(d) indicate similar efficiency of conical mini-horn structure for achievement of critical coupling and suppression of unwanted modes: S 11 = 17 dB extinction at DBR mode near 36 GHz and absence of adjacent modes within 500 MHz vicinity, important for oscillator and discriminator applications. WGMs at these frequencies, or quasi-TM type with azimuthal number ∼24 that must have free spectral range of ∼1.5GHz, are not clearly identifiable in the spectrum and thus effectively isolated by this coupler.

The Q-factor of a resonator mode depends on the loading as well as the intrinsic attenuation of the mode. We introduce the amplitude loss coefficients γ c and γ 0 to quantify these parameters, correspondingly. To characterize the intrinsic quality factor experimentally, we tuned up the two waveguide couplers in identical ways and noticed that the mode frequency spectrum has full width at the half maximum (FWHM) given by δ f 1 = 2 ( γ c + γ 0 ) when a single coupler is involved and δ f 2 = 2 ( 2 γ c + γ 0 ) when two couplers are involved. Finding difference 2 δ f 1 δ f 2 = 2 γ 0, we can evaluate Q-factor of the mode of frequency ν 0 as Q = ν 0 / ( 2 γ 0 ).

The result of the measurement is presented in Fig. 5. As one can see, the signal to noise was high, and no averaging was involved. The accuracy of the bandwidth measurement was at 1% level. The resultant Q-factor reached nearly Q 0 = 2 × 10 5. This is more than an order of magnitude more than the Q-factor of the metal cavity enclosure and the standing alone sapphire structure.

The constructed resonator package is versatile enough to enable the measurements of its unloaded, critical as well as loaded characteristics. We used different apertures embedded in the coupling waveguide to modify the coupling. The network analyzer was used to detect phase characteristics of the assembly. A few measurement results are presented in Fig. 6. The resonator clearly demonstrated all the coupling regimes and the noise associated with the measurements in the vicinity of the critical regime confirm the steep slope of the phase characteristic when γ 0 γ c.

FIG. 6.

Reflected phase response of DBR mode in undercoupled (a), near-critical (b), and overcoupled (c) regime. With increased coupling, phase swing across resonance increases to full 2 π. Maximum slope is achieved at near critical coupling. Increased noise at exact resonance is a consequence of strong signal extinction in the critical coupling regime.

FIG. 6.

Reflected phase response of DBR mode in undercoupled (a), near-critical (b), and overcoupled (c) regime. With increased coupling, phase swing across resonance increases to full 2 π. Maximum slope is achieved at near critical coupling. Increased noise at exact resonance is a consequence of strong signal extinction in the critical coupling regime.

Close modal
The measurements of the phase characteristics of the coupled resonator correspond well enough to the simplest theory of a single port cavity. The amplitude transfer function of the cavity is
H ( f ) = γ 0 γ c i ( f ν 0 ) γ 0 + γ c i ( f ν 0 ) .
(1)
While the absolute value of the transfer function defines the result of the power spectrum transmission of the cavity S 11 = | H ( f ) | 2, arg ( H ( f ) ) defines the phase characteristics of the cavity. In the case of the nearly critically coupled system, the frequency slope of the dependence d [ arg ( H ( f ) ) ] / d f goes to infinity. Figure 6(b) with noise appearing near zero detuning also illustrates how critically coupled resonator acts as a discriminator of frequency noise of microwave source, which is, in this case, the tunable oscillator part of the network analyzer.

Applications of the resonators in oscillators call for high frequency stability of the device. The stability is usually impacted by ambient thermal variations. The demonstrated device has low thermal sensitivity, smaller than that of the full body dielectric resonator, making it attractive for practical applications.

Environmental sensitivity of a resonator assembly is defined by the thermal sensitivity of its components. We have a sapphire structure in a brass enclosure. Thermal expansion coefficients of brass and sapphire are 1.7 × 10 5 K 1 and 4.5 × 10 6 K 1 at 25 °C, respectively. The 36 GHz thermo-refractive coefficient of sapphire (component perpendicular to crystal axis) at 25 °C is ( 1 / ε ) ( d ε / d T ) = 1.8 × 10 5 K 1.12 Therefore, it is expected that the thermal sensitivity of the frequency of a bulk sapphire resonator is rather high and depends primarily on its thermo-refractive coefficient.

It was shown, though, that DBR resonators have much smaller thermal sensitivity. Indeed, DBRs have a few times less temperature sensitivity than WGM resonators without using temperature compensating techniques. The best thermal sensitivity achieved so far is about 1.2 × 10 5 K 1 at 22 °C.9 We have measured thermal sensitivity of our resonator and observed similar performance for the Ka band resonator (Fig. 7). The thermal sensitivity is less than both the thermal expansion coefficient of the copper enclosure and the thermo-refractive coefficient of the sapphire. Apparently, the limitation is primarily related to the thermal expansion of the sapphire. The sapphire structure determines the shape of the mode and the other thermal effects impact the frequency due to interaction with the mode residual field.

FIG. 7.

Measurement of temperature coefficient of frequency averaged between 24 and 34 °C, T C f = 428 kHz / °C, which corresponds to ( 1 / f ) ( d f / d T ) = 1.19 × 10 5 K 1, about five times smaller than reported room temperature sensitivity of bulk sapphire resonators with whispering-gallery modes.

FIG. 7.

Measurement of temperature coefficient of frequency averaged between 24 and 34 °C, T C f = 428 kHz / °C, which corresponds to ( 1 / f ) ( d f / d T ) = 1.19 × 10 5 K 1, about five times smaller than reported room temperature sensitivity of bulk sapphire resonators with whispering-gallery modes.

Close modal

To perform the measurement shown in Fig. 7, we placed the assembled sapphire resonator [Fig. 4(d)] into a convection oven able to vary temperature in a 30 °C range around room temperature with 0.1 degree accuracy. We have additionally monitored the temperature of resonator assembly using film thermistor taped to the metallic lid of enclosure. Each data point in Fig. 7 was taken after enough time was allowed for resonator to thermalize, and frequency was read out from the position of the DBR mode S 11 dip observed with a network analyzer. We limited our test within the range 24– 34 °C, which was enough to determine the temperature coefficient of frequency. Frequency of the DBR mode was tracked using a network analyzer. As expected, the demonstrated thermal dependence of the frequency dependence was linear. The calculated thermal coefficient was ( 1 / f ) ( d f / d T ) = 1.2 × 10 5 K 1.

Quality factor of a sapphire resonator depends on the material quality as well as the fabrication quality of the resonator parts. To validate the high purity low loss grade of the material utilized in the DBR structure, we measured its attenuation using the WGM technique.3 

To perform the measurement, we took bare cylindrical layers used in the Bragg resonator Fig. 8 assembly and used a single disk or a stack of a few disks as a standalone WGM cylindrical sapphire resonator. The sapphire blanks were nicely polished and cleaned, so the estimated distance between the layers did not exceed 5  μm, which is negligible for the Ka-band frequency. Tunable coupling to the quasi-TM whispering-gallery modes was achieved with dielectric (acrylic) waveguide in appropriate horizontal orientation, electrical field parallel to the plane of the disk. Broad span S 21 measurement using network analyzer allowed easy identification of quasi-TM WG modes with consecutive azimuthal numbers in 26–30 range and evaluation of their Q-factors from the Lorentzian fits [Fig. 8(b)]. The resonator was free standing, with no enclosure utilized. The enclosure is not required, since radiative loss limitation of Q-factor for the WGM of these orders is higher than 10 9, while the expected Q limit by dielectric loss tangent is below 5 × 10 4 for both polarizations. By increasing the distance between dielectric waveguide and resonator, we were able to reduce the coupling from critical to very low coupling regime and measure the value of intrinsic Q-factor that is very close to inverse loss tangent for chosen polarization, perpendicular to crystal axis for the DBR. While a single disk of thickness 0.72 mm was not enough to support the quasi-TM with loss tangent-limited Q, stack of two disks showed numbers close to those projected from the data obtained in earlier studies. Stack of three disks with a total thickness of 2.16 mm increased the Q to approach the limit by dielectric loss tangent extrapolated to be t g δ = 3.3 × 10 5. Further addition of thickness did not raise the Q significantly. Estimated Q-factor of quasi-TM modes did not vary measurably (beyond 10% uncertainty) for three consecutive modes around 36 GHz and confirmed that the material we utilized had properties close to those of high-quality HEMEX standard sapphire. This result also emphasizes the fact that DBR structure allows improving the Q-factor by the factor of 7, compared with the bulk WGM resonator made of the same material. Let us note here in the end of the section that stacking dielectric disks for the purpose of creating compound dielectric WGM resonators has been demonstrated with different materials in Ref. 13.

FIG. 8.

Sapphire dielectric loss qualification by quality factor of whispering-gallery mode (WGM) in sapphire disk (a). Lorentzian bandwidth 1.39 MHz in the unloaded regime corresponds to Q 0 = 26 × 10 3 of TE mode, in agreement with reported data for high-grade sapphire.

FIG. 8.

Sapphire dielectric loss qualification by quality factor of whispering-gallery mode (WGM) in sapphire disk (a). Lorentzian bandwidth 1.39 MHz in the unloaded regime corresponds to Q 0 = 26 × 10 3 of TE mode, in agreement with reported data for high-grade sapphire.

Close modal

We have designed, built, and characterized a high-Q factor, compact distributed Bragg reflector-based resonator in the Ka frequency band. This first realization of DBR resonator in millimeter wave band overcomes the inverse loss tangent limit typical of dielectric resonators with whispering-gallery modes. As shown in Table I in Sec. I, our work represents the best result in terms of merit parameter Q × f per unit of resonator volume, 7.2 × 10 6 GHz per 50 cm 3, compared to previous demonstrations at lower microwave frequencies. Resonator has a reduced temperature coefficient of frequency compared to bulk dielectric resonators with whispering-gallery modes. In addition, we have designed and implemented critical coupling solution important for crucial target applications such as low phase noise oscillators.

The reported research here was carried out at the Jet Propulsion Laboratory, California Institute of Technology, under a contract with the National Aeronautics and Space Administration (No. 80NM0018D0004).

The authors have no conflicts to disclose.

Vladimir Iltchenko: Conceptualization (lead); Data curation (supporting); Formal analysis (lead); Funding acquisition (lead); Investigation (equal); Methodology (lead); Supervision (lead); Writing – original draft (lead); Writing – review & editing (equal). Rabi Wang: Conceptualization (lead); Formal analysis (lead); Investigation (equal); Methodology (lead); Writing – original draft (lead); Writing – review & editing (equal). Michael Toennies: Data curation (equal); Formal analysis (lead); Investigation (equal); Software (lead); Writing – review & editing (equal). Andrey Matsko: Conceptualization (supporting); Data curation (supporting); Formal analysis (supporting); Funding acquisition (lead); Investigation (equal); Software (lead); Supervision (lead); Writing – review & editing (equal).

The data that support the findings of this study are available from the corresponding author upon reasonable request.

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