## Abstract

We report on coupling of two whispering gallery mode resonators in the Terahertz frequency range. Due to the long wavelength in the millimeter to submillimeter range, the resonators can be macroscopic allowing for accurate size and shape control. This is necessary to couple specific modes of two or more resonators. Sets of polyethylene (PE) and quartz disk resonators are demonstrated, with medium (loaded) quality (*Q*)-factors of 40-800. Both exhibit coinciding resonance frequency spectra over more than ten times the free spectral range. Loading effects of single resonators are investigated which provide strong *Q*-factor degradation and red-shifts of the resonances in the 0.2% range. By coupling two resonators of the same size, we observe mode splitting, in very good agreement with our numerical calculations.

©2008 Optical Society of America

## 1. Introduction

Whispering gallery mode (WGM) resonators offer a great potential of applications due to their extremely high quality (*Q*)-factors [1, 2, 3, 4]. Lord Rayleigh first studied single WGM resonators in the acoustical domain [5]. By now they have been investigated in almost all frequency domains, the RF electronics [6, 7], W-band [9], the THz frequency range [10], up to the optical domain [1, 2, 3, 4]. CoupledWGM resonators show many new exciting phenomena, such as resonator waveguides [11, 12] and drastic *Q*-factor enhancements [13, 14]. In addition they are a breeding ground for applications, for instance band pass filters or bandstops [3, 7, 8]. For a review of early work see Ref. [15].

Coupling resonators over multiple resonances requires resonators with coinciding resonances. Similar to the formation of bonding and antibonding states for electrons in molecules, coupled identicalWGM resonators show mode splitting [13, 16, 17]. Correspondingly, the split mode with lower frequency (“bonding mode”) shows field enhancement in the area between the disks, whereas the higher frequency mode (“antibonding mode”) has a node there (see inset in Fig. 2). A recent publication [18] shows that these bonding (antibonding) modes are also causing attractive (repulsive) forces between the disks, in accordance with the electronic states in molecules. Due to these similarities, such systems are often addressed as “photonic molecules”. Mode splitting of a specific mode demands coinciding resonances and, thus, size and shape control in the range of *λ*/*Q*. Therefore, in the optical domain, a manufacturing accuracy in the nm range is required for resonators with medium *Q*-factors in the range of 10^{3}. Nevertheless, coupling between size mismatched [19, 20], low *Q* [17], or temperature tuned WGM resonators in a cryostat [21] were demonstrated so far. In these cases, it is very difficult to deliberately design a mode spectrum as required e. g. for frequency specific filter structures. A common procedure to generate identical WGM resonators is to produce many similar WGM resonators and pick two that accidentally provide the required matching mode spectrum [22]. Here as well, producing resonators with defined mode spectrum is extremely difficult. In the Terahertz frequency range (0.1 THz–10 THz, corresponding to wavelengths of 3 mm–30 *µ*m), the manufacturing condition is facilitated by a factor of 1000 due to the larger wavelength. In addition, the resonators are about the same factor larger, allowing for millimeter sizes, and can thus be mechanically post-processed for fine-tuning to attain the required mode spectrum. This allows for designingWGM resonators at will. Similar techniques as in the optical domain, waveguiding and optics can be used. Hence, the Terahertz frequency range offers the possibility to test and verify many theoretical predictions for photonic molecules. Effects on the *Q*-factor and the mode distribution related to shape and size can also be investigated. Due to scale invariance of electromagnetism, the results can directly be used in other wavelength ranges. Because of their large size, Terahertz WGM resonators are easy to handle and also their fundamental modes are accessible. Here we demonstrate WGM resonators of similar shape and identical mode spectrum over tens of modes for sets of polyethylene (PE)-plastic and quartz WGM disk resonators for the THz frequency range. We investigatedTM modes with loaded *Q*-factors up to 800. By coupling two resonators, we observe distance dependent mode splitting. We compare the experimental data to numerical calculations of frequency- and spatial dependence.

## 2. Theory

Maxwell’s equations govern all classical electrodynamics. In the case of resonators they reduce to the Helmholtz equation. For cylindrical geometries, the Helmholtz equation is known to separate into vertical, radial and angular components. As the experimental cavities are thin with respect to the wavelength, they can be modelled as a two dimensional disk. Their finite vertical extent leads to a frequency dependent shift of the resonances, which can be described by a frequency dependent effective index of refraction. The discrete resonances of the disk’s horizontal plane can be identified by two quantum numbers (*j*,*m*), where *j* is the radial and *m* the angular momentum quantum number. A connection to ray optics can be found via *m*=sin *χ*/(*nkR*), where *χ* is the angle of incidence, *n* the refractive index and *kR* the dimensionless wavenumber [23]. These quantum numbers are commonly noted with the two different polarizations as TM_{m, j} and TE_{m, j}. For coupled disks, the Helmholtz equation does not separate in the horizontal plane and there are no simple quantum numbers, although in the weak coupling regime one often associates the coupled system with the unperturbed quantum numbers. We find general solutions in the horizontal plane via a combination of a boundary matching multipole method [24, 25, 26] and the root searching scheme of the scattering quantization [23, 27]. Details will be presented elsewhere [28]. We expand the field around the origin of the disks with Bessel functions, relate the fields with the appropriate addition theorem, impose smoothness of the field at the boundary and obey the Sommerfeld boundary condition at infinity (no incoming fields). Discretizing the expansion and the boundary, the resulting matrix equation is reformulated in a generalized Eigenvalue problem and solved following the idea of Türeci *et al.* [23, 27]. Contrary to solutions of the scattering formulation, solutions obeying the Sommerfeld boundary condition are known to be discrete in the complex wavenumber *k*∈𝓒. The imaginary part and real part of the wavenumber can be associated with the quality factor according to *Q _{ideal}*=-2Re[

*k*]/Im[

*k*]. Material specific losses are taken into account by calculating

*Q*=(1/

*Q*+1/

_{abs}*Q*)

_{ideal}^{-1}, where

*Q*is the absorption limited quality factor.

_{abs}## 3. Experimental setup

Two temperature-stabilized telecom DFB laser diodes at wavelengths around 1545 nm are frequency offset by the desired THz frequency and heterodyned to get the optical beat signal for the photomixing process. A subsequent Er-doped fiber amplifier provides the required optical power. The lasers drive ballistic transport enhanced n-i-pn-i-p In(Al)GaAs photomixers (PM) [29, 30] that convert the optical beat signal into a continuous wave THz beam at room temperature. By temperature tuning the wavelength of one telecom diode, the THz frequency can easily be chosen. The THz linewidth of the system is about 3 MHz and the temporal frequency stability can be estimated from an RF measurement to about 50 MHz, limiting the maximum *Q*-factors to be investigated to about 1000 at 0.2 THz. A silicon lens mounted on the back side of the photomixer allows for efficient outcoupling into free space and pre-collimation of the THz beam. A schematic of the setup is depicted in Fig. 1 (a). A wire grid polarizer (P) selects the TM polarization. Subsequent parabolic mirrors focus the beam on a thin, rectangular teflon waveguide, an analogon to a tapered polarization maintaining fiber. In order to improve the coupling efficiency, a round horn, adapted to the numerical aperture of 0.45 of the parabolic mirror, is mounted to the waveguide. The cross section of the waveguide is 1.0mm×1.4mm at a refractive index of *n*=1.44. The small axis is used to couple from the waveguide to the disks. At a frequency of 0.3 THz the optical thickness to wavelength ratio of the small axis is thus 1.44, comparable to tapered waveguides in the optical domain. In the THz frequency range, the large extent of the evanescent field in the tens of *µ*m-range allows for simple access without active position stabilization of the disks. At the resonator coupling position, the waveguide is bent (maximum radius of curvature ROC≈ 5 mm) to further enhance the evanescent field and the coupling efficiency [31]. The resonators are mounted on micrometer stages to allow for scanning the distance between the resonators and the waveguide. Throughout the experiment, we measure the signal transmitted through the waveguide, *T*=*I _{T}*/

*I*

_{0}by a Golay cell detector (D), where

*I*

_{0}is the signal without resonators attached.

## 4. Experiment

We fabricated two types of WGM resonators: polyethylene (PE300)-plastic (*n*=1.5±0.03) disks of 6 mm diameter and 1.1 mm thickness and quartz (*n*=2.0±0.03) disks with rounded edges (ROC=0.5 mm) of 3.62 mm diameter and 0.98 mm thickness (see insets of Fig. 2 and 3). Both types were produced with a lathe. Post-processing using emery paper was necessary for both types to match the mode spectra of two resonators over many resonances. First, we measure the influence of the waveguide to a single disk. The disk was moved towards the waveguide and the transmission through the waveguide was measured over a broad frequency range with more than 10 high Q resonances. Figure 1 (b) shows the effect of the waveguide on the resonator. Both the *Q*-factor and the peak frequency are affected. The loaded *Q*-factors (inset of Fig. 1 (b)) are much smaller than the nearly unloaded ones, in agreement with theory [32, 33]. As the waveguide has a higher dielectric constant in contrast to air, the waveguide red shifts the peaks of the loaded cavity slightly. The maximum red shift, Δ*f _{load}*=0.5 GHz, is small compared to the linewidth of the loaded cavity. Hence, there is sufficient mode overlap of a waveguide loaded cavity with the second cavity.

Figure 2 shows the transmission of the PE disks scanned over 12 high *Q* resonances in the Terahertz frequency range. For clarity, the individual graphs are vertically shifted. Numerical results for the outermost modes (*j*=1) of the resonator are also included. By comparing the mode spectrum to theory, the modes can be identified as TM_{13,1} to TM_{25,1}. Due to the low mode numbers investigated here, the effective mode radius is in the range of 0.8–0.9×*r _{disk}*. With a 1.0 mm waveguide it is therefore difficult to reach the point of critical coupling of such low order modes, as the modes are buried inside the disk. Thus, for most resonances, touching disks and touching the waveguide achieved best coupling with the transmission through the waveguide lowered by up to 92%. The theoretical free spectral range agrees very well with the experiment. The lower modes are affected by the PE mount of the disk and, thus, are blue shifted, due to the higher effective index of refraction. The experimental

*Q*-factors are about 2.2 times smaller over the whole frequency range than predicted by the calculation. Here, loading has already been taken into account which reduces the

*Q*-factor according to Fig. 1 (b) by a factor of approximately three. The absorption limited

*Q*-factor for PE, can be calculated to

*Q*=4

_{abs}*n*/

*ε*″≈3300, using values for the imaginary part of the dielectric constant,

*ε*″≈1.7·10

^{-3}, from ref. [34]. Note that this is only an estimate as there exist different kinds of high density PE plastics. At higher frequencies, some higher order/inner (

*j*>1) modes become visible but are much less excited than the outermost modes. The coupled disk spectrum shows mode splitting (see also Fig. 4) as expected. However, there is a small asymmetry between the bonding and the antibonding mode with respect to the

*Q*-factor. Most resonances show antibonding modes with a higher

*Q*-factor. A similar effect was found in the microwave range in ref. [12] and was described as an interplay between mode splitting and absorption. Also a remaining difference in shape of the two disks or cavity loading could have similar effects.

The good agreement of the resonance positions for both cavities over such a large spectrum ensures that several modes outside the investigated frequency range will also coincide. Evanescent waveguide coupling excites only the well confined resonances with *m*≫*j*. With a different coupling scheme, e.g. prism coupling, coupling to the other 76 lower *Q* resonances should also be possible.

Similar results are obtained for quartz WGM resonators. The resonance spectra are depicted in Fig. 3. In contrast to plastic resonators, the surface of the quartz disks can be cured by heat after the processing to improve the *Q*-factor. However, for the medium *Q*-factors investigated here, curing was not necessary as surface roughness does not play a significant role. The loaded *Q*-factors of the single disks range between 80 and 800 for the quartz resonators and are about three times higher than those of the PE disks owing to their higher index of refraction and smaller absorption according to ref. [34]. *Q _{abs}* is calculated to about 26000 for quartz (

*ε*″≈3·10

^{-4}) and decreases slightly with increasing frequency. For quartz, however, there is a second loss mechanism due to birefringence which was neglected in the calculation. Therefore, a reduced

*Q*≈5000 for the random oriented quartz was assumed to fit the trend of the experimental

_{abs}*Q*-factor. Assuming that the loading reduces the

*Q*by a factor three similar to the PE disks, the experimental

*Q*-factors are 3.7 times smaller than predicted by theory.

To determine the strength of the mode splitting, the second disk is moved away from the first and the splitting vs. distance of the two PE disks is recorded (Fig. 4). Similar to a hydrogen molecule, the coupling gets weaker with increased distance as the mode overlap decreases, decreasing the difference of the split peak frequencies, Δ*f _{s}*. The experiment is compared to numerical results of the splitting determined by solving the Helmholtz equation. The experimental touching point was fitted to the theory resulting in a small offset of 80

*µ*m (180

*µ*m) for PE (quartz), as a result of small misalignment and, in the case of the quartz resonators, due to the curved surface of the disks. Note that the mode frequency splitting, Δ

*f*is always much larger than the shift induced by coupling to the waveguide, Δ

_{s}*f*(c. f. Fig. 1 (b)). The latter should be almost common for both split modes. Therefore, the theoretical splitting agrees very well with the experimental data as depicted in Fig. 4. sThe inset of Fig. 4 shows the splitting vs. frequency for nearly touching disks. As the extent of the evanescent field decreases with increasing frequency, the coupling and the mode splitting decreases. At large distances, the modes in the cavities are unperturbed and can be described with TM

_{load}*. Good agreement with numerical calculations is achieved as depicted in Fig. 4.*

_{m,j}## 5. Conclusion

In conclusion, we have demonstrated the feasibility to fabricate quartz and polyethylene whispering gallery mode resonators for the THz frequency range with coinciding mode spectra over more than ten times the free spectral range. We point out the advantage of large size and the possibility of post-processing ofWGMresonators in the Terahertz frequency range. This allows for designing, manufacturing and examining photonic molecules at will. We demonstrated this by fabricating two cavities with matched resonance spectra and coupling them to photonic molecules. The achieved mode splitting is in excellent agreement with theoretical calculations.

## Acknowledgments

Authors Sascha Preu and Harald Schwefel contributed equally to this paper. The authors acknowledge the DFG for financial support.

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