We present results for a photonic microwave resonator designed and fabricated at 17.4GHz with a record high Quality factor (Q = 26,400) at room temperature over a mode volume smaller than one cubic wavelength. The cavity is uniquely designed to have its electric field concentrated in air, which allows for efficient coupling to free space and facilitates interactions with gaseous atomic systems and fluids.
©2011 Optical Society of America
Photonic crystal cavities with high Quality factors (Qs) are capable of extending the lifetime of cavity photons confined within mode volumes (V) of the order of a cubic-wavelength, and hence can greatly enhance the interaction between electromagnetic fields and matter. Recently, high Q/V photonic crystal cavities have been extensively investigated with various geometries at optical frequencies, acting as a powerful platform for studying cavity quantum electrodynamics (cQED)  and developing a myriad of nano-photonic devices. Yet few photonic crystal cavities have been reported at microwave frequencies [2–7].
A high Q/V microwave cavity is demanded for a number of applications. In atomic physics, a single Rydberg atom coupled to a single photon is the fundamental system for cavity QED studies, especially when one reaches the interesting strong coupling regime where energy can be coherently exchanged between the two. For strong coupling to occur, the atom-photon coupling rate g (inversely proportional to square root of V) must exceed both the photon leakage rate (inversely proportional to Q) and the atom decoherence rate. Since the frequency difference of Rydberg states is in the microwave region , a high Q/V microwave cavity is crucial to achieve this type of coupling. Strong coupling in Rydberg atoms has been previously observed in large-scale Fabry-Perot cavities and superconducting circuits [8,9]. Photonic crystal cavities, in contrast, offer an alternative platform that is more compact and can be operated at room temperature.
Secondly, given the cavity’s intrinsic field enhancement factor, there are also significant advantages in practical applications such as tunable microwave filters and antennas, where the efficiency of a radiator can be significantly improved by coupling with the cavity mode ; fluorescence microscopy for biological systems, where a high Q/V microwave cavity can be used to control the temperature and drive the biological reactions ; resonance-enhanced microwave detectors ; particle accelerators [5,6]; and refractive index sensors.
In this work, we have designed and experimentally demonstrated an all-dielectric photonic crystal microwave cavity. The cavity can be fabricated with a variety of materials using conventional machining techniques. We report a cavity mode at 17.4GHz with a very small mode volume limited to about one cubic-wavelength and a record-high Q-factor of 26,400 at room temperature. Besides the high Q/V factor, the defect mode has a TM-polarized electric field concentrated in the air region, as opposed to most photonic crystal cavities designs with TE-polarized electric field concentrated in the host dielectric. These features are crucial for effectively coupling microwaves to matter in a number of applications. Moreover, when a center-fed antenna is placed inside the cavity acting as a radiating dipole, we observe a strong signature of the cavity’s Purcell enhancement factor. In summary, we believe this type of device offers great promise for studying cavity QED phenomena as well as for enabling novel applications at microwave frequencies.
2. Cavity design
The structure we propose starts from a so-called rectangular dielectric rod waveguide with a periodic array of circular holes, as shown in Fig. 1(a) [12–16]. We choose a refractive index of 3.1 for the rod, which corresponds to the refractive index of alumina (Al2O3) at microwave frequencies. The rectangular waveguide has an aspect ratio (thickness/width) of 2:1, and supports a fundamental TM-polarized mode with its major component aligned along the y-axis [Fig. 1(c)]. From our previous work , introducing a suitable periodic array of holes in the bulk materials of this high aspect-ratio waveguide results in a relatively wide bandgap for TM-polarized modes. Here we select the thickness to be w = 4.72mm, depth h = 9.43mm, and hole periodicity a = 4.15mm. The radius of the holes is chosen as r = 1.25mm.
Next, we introduce a defect region into the cavity by gradually increasing the periodicity (hole-to-hole distance) and hole diameter for each segment starting from a pair of outer holes and symmetrically moving towards the center. When the feature size of a segment is enlarged, the band-gap is red-shifted, resulting in a graded photonic band, as shown in Fig. 1(b). This allows confining an air-band mode in the defect region: the air-band mode is coupled to the evanescent Bloch modes within the band-gaps at each end, effectively trapping it between a pair of Bragg mirrors.
A three-dimensional finite-difference time-domain (FDTD) code was used to calculate the resonant frequency and the Q factor of the cavity mode . On each side of the cavity we introduce 10 segments with progressive tapering of the hole diameter and spacing, in order to adiabatically couple the cavity mode to the evanescent Bloch mode. This adiabatic process is tuned to suppress the scattering loss resulting from the effective index mismatch between the two modes [18,19]. Without material losses, the cavity mode has an ultra-high Q factor of 2,500,000, limited by scattering alone (Q = Q sc). The cavity mode profiles are shown in Fig. 1(d), where the Ey component is plotted at the mirror plane of y-axis and x-axis, respectively. The tapered section’s parameters are tailored to localize an air-band mode resonating at 17.4GHz, which is closer to the air band-edge. At this frequency, the photonic bandgap has maximized confinement along the z-axis, which results in a small cavity mode volume. As expected from the general features of air-band modes, the electric field concentrates in the air-region. However, the maximum of the electric field density [ε|E| 2] is not in the air region, so we define the effective cavity mode volume as,
where E max, air is the electric field maximum in the air region, located at the center of the cavity. For our cavity we found an effective mode volume of 0.99λ 3.
The cavity mode is normally excited from an external waveguide port. Our device can be modeled accurately with temporal coupled-mode theory [20,21]. The amount of energy stored (U) is proportional to the launched power (P0).
In Eq. (2) Qw denotes the quality factor responsible for the field energy leakage from the cavity section via the dielectric waveguide. The total cavity quality factor Q, including material losses, can be written as
Therefore, for given Qsc and Qm, it is important to tune Qw by fixing the number of Bragg mirror pairs on each side of the resonator so as to achieve optimal critical coupling conditions by satisfying Eq. (4).
3. Fabrication and measurement
The resonator was fabricated with ultra-high purity alumina from Coorstek (AD998 PlasmaPure). This material has a relatively high refractive index of 3.1 at microwave frequencies and very small dielectric losses. Dielectric losses are important as they ultimately limit the quality factor of the resonator. The material that we use has a nominal loss factor of tan(δ) = 2.5 × 10−5 at 6GHz, which results in Qm = 44,000. For our device, we put 14 Bragg mirror pairs at each side, which corresponds to Qw = 210,000. Therefore, the total Q factor is expected to be 36,000. Fabrication was done with standard computer-numerical-control (CNC) milling techniques, with a nominal positional accuracy of ~10 μm for each hole.
The fabricated structure is shown in Fig. 2(a) , and the setup used for our two-port transmittance measurements in Fig. 2(b). The device was connected to a network analyzer via conventional WR62 microwave waveguides placed at each end. Because of the geometrical mismatch between the metallic waveguide (15.8mm × 7.9mm) and our dielectric waveguide resonator (4.72mm × 9.43mm), we designed slant sections at both ends of the resonator to allow partial insertion into the WR62 waveguide and thereby facilitate coupling [Fig. 2(c)]. However, the coupling efficiency of the experimental setup is still limited by the coupling between the metallic waveguide and the dielectric waveguide. In Fig. 3(a) , we show the amplitude spectrum from the transmission measurement. It demonstrates a large bandgap ranging from 15.4GHz to 17.7GHz, which is consistent with theory.
The transmission coefficient of the cavity mode can be generally fitted using a Fano model [22,23]. Unlike most photonic crystal cavity experiments at optical frequencies, here both amplitude and phase of the transmittance can be obtained. We plot the resulting spectra in Fig. 3(b), where the cavity resonance is clearly seen at 17.405GHz, in good agreement with the design value. We obtain a full-width half-maximum linewidth ΔωFWHM of 0.659MHz, corresponding to a Q factor of 26,400, which is slightly smaller than the expected Q value of 36,000. The observed discrepancy of Q factor can be attributed to higher dielectric losses in the bulk material at this frequency, to fabrication tolerances on the holes position and size and to resonant losses at the transition between waveguide and dielectric. In general, a higher Q factor can be obtained with less lossy materials or at cryogenic temperatures.
Next, we insert a small antenna inside the cavity in order to study the interaction between the cavity mode and a radiating dipole. The antenna is constructed by stripping both the outer conductor and the dielectric core of a coaxial cable (Megaphase Corp., model ClearPath-A06), exposing the inner conductor for a total length of 9 mm. The relatively small diameter (0.305 mm) of the inner conductor was chosen to minimize any perturbation of the cavity mode.
In Fig. 4(a) , we plot the reflectance spectrum measured with a network analyzer connected to the above mentioned center-fed antenna, when this is placed in free space (red) and in the center hole of cavity (blue). A clearly visible reflectance dip with a 2.2dB depth at ~17.43GHz is observed, which we interpret as a convincing signature of the Purcell effect of the cavity. When the antenna is coupled to the cavity mode, the radiation rate is enhanced, and thus the load impedance is modified. This results in a variation of the reflectance at the analyzer port, due to the impedance mismatch between the antenna load and the cable. From Fig. 4(a), we also note that there is a change in the resonant frequency of the system, from 17.40GHz to 17.43GHz. This is due to the small but non-negligible perturbation of the metallic antenna, which blue-shifts the resonance. The free-space background in Fig. 4(a) (red-solid curve) arises from the small reflectance between the network analyzer/cable connection, and the cable/antenna connection. This background prevents us from accurately quantifying the load impedance variation and subsequently extracting a numerical value for the Purcell enhancement factor and the perturbed Q factor.
Next, we scanned the antenna in z-direction along the length of the dielectric rod to probe the electric field distribution of the cavity mode. The height of the antenna and the distance between the antenna and the photonic crystal (0.5 mm) were kept constant during the measurements. This is analogous to near-field scanning optical microscopy (NSOM) [24,25]. In Fig. 4(b), we plot the measured reflectance depth as a function of the z-position (blue-solid). In comparison, we also plot the electric field amplitude (black-dash). It can be seen that the two curves follow the same z-dependence, which verifies that the S11 dip we observe is due to coupling to the cavity mode.
In summary, a high-Q microwave resonator with a mode volume smaller than one cubic wavelength has been designed and fabricated. A record high Quality factor (Q = 26,400) for photonic crystal cavities at microwave frequencies has been measured at room temperature. In addition to its ultra-high Q/V factor, the cavity is uniquely designed to have its electric field concentrated in air. Coupling to this cavity mode is enhanced by the Purcell factor.
The photonic crystal cavity that we demonstrated has unique advantages when compared to other types of microwave cavities such as sapphire ring resonators. First of all, it supports a mode confined in the air region. Moreover, due to the integrated dielectric waveguides better coupling efficiencies can be realized. Finally, it has a very small mode volume on order of one cubic wavelength. We believe that this device is well suited for conducting microwave cavity QED experiments and for developing a variety of novel microwave devices.
Y. Zhang would like to dedicate this paper to Mughees Khan and his wife. The authors would also like to acknowledge the generous support from Schlumberger-Doll Research Center at Cambridge, Massachusetts.
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