1. Introduction

Despite the considerable demand in terms of applications [1], the current state of the art THz sources are still far from ideal. Photoconductive dipole antennas, for example, are very popular for generating and detecting broadband THz pulses, but they require the external excitation of femtosecond lasers. For Continuous-Wave (CW) sources, optical heterodyne mixing (photo-mixing) is still limited to output powers in the 100 nW range below 1 THz, quickly decreasing to few nW at about 2 THz [2]. THz Quantum Cascade Lasers (QCLs) can emit more than 50 mW (peak) in the pulsed regime, but they are poorly tunable and only operate at cryogenic temperatures [3].

The fabrication of a THz source that is, at the same time, compact, providing high output power, and operating at room temperature is still an unsolved challenge. In this context, nonlinear THz sources constitute a promising approach, but they are normally affected by two main drawbacks: 1) the required excitation of external (high-power) pump lasers; and/or 2) small conversion efficiencies.

The first problem can be circumvented by integrating laser pump sources and nonlinear medium. Although adding more constraints both in terms of design and technological processing, this provides considerable improvements in terms of ease of use for any applications. This approach is exemplified by the THz source based on Difference Frequency Generation (DFG) in dual-wavelength mid-IR QCLs reported in Refs. [4,5] and operating in pulsed regime up to room temperature.

The problem of small conversion efficiencies can be tackled by exploiting optical microcavities in order to boost the nonlinear phenomenon [6]. In this respect, Whispering Gallery Mode (WGM) resonators are promising candidates, because of their ultra-high quality factors. In particular, recent technological progress has allowed the fabrication of AlGaAs microdisks with Q factors exceeding 10⁵ [7, 8]. These resonators could therefore be used for efficient frequency conversion sources based on DFG [9, 10].

High Q factors are desirable since they translate into long storage times and therefore into long interaction times between the modes involved in the nonlinear generation. However, they come at a price: the higher the Q factor, the smaller the width of the resonance. Therefore, the nonlinear process becomes extremely sensitive to even small changes of physical parameters and/or minimal fabrication imperfections.

It is clear that some degree of fabrication tolerance is highly desirable, and thus one needs to employ a well-controlled tuning parameter to compensate for any deviations with respect to the nominal design. Temperature is a classic example: by changing the temperature, the refractive index of the materials making up the cavity can be adjusted until the modes are brought into phase matching.

In this letter we numerically study an alternative and powerful tuning approach based on defining a hole in an AlGaAs WGM THz nonlinear microcavity source similar to the one we discussed in Ref. [9]. Interestingly, this opens the way for a second tuning parameter, provided by the height of a metallic tip positioned in the hollow part of the ring.

The paper is organized as follows: in Section 2, we present our design guidelines and briefly summarize the Coupled Mode Theory (CMT) results pertinent to a DFG process in a WGM cavity. In Section 3, we report the results of our analysis on ring resonators for THz generation by DFG. We show that ring-like cavities are advantageous over pillar-like structures not only in terms of tuning capabilities but also in terms of nonlinear conversion efficiencies.

2. Cavity design

As shown in Fig. 1(a), the vertical structure of our cavity is composed of AlGaAs layers placed between two gold mirrors. The central Al_0.34Ga_0.66As slab sandwiched between Al_0.8Ga_0.2As spacers provides dielectric guiding for two near-IR pump modes with wavelengths around 1.3 μm. The vertical confinement of the THz mode is made possible by the metallic layers, which create a double-metal waveguide similar to the one used for THz QCLs [11]. Please note that both the Al molar fraction and the pump wavelengths are chosen so to minimize the effect of two photon absorption.

 

Fig. 1 Sketch of the nonlinear source studied in text (a). Schematic of the DFG process with pump fields injected via a tapered waveguide (b).

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By slightly changing the vertical design and by burying quantum dots in the cavity central layer, it should be possible to implement an active THz source. For the following however, we will limit our discussion to a passive structure.

A schematic of the DFG process is presented in Fig. 1(b): the circularly symmetric cavity is side-coupled to a tapered waveguide in order to excite the relevant pump WGMs. Due to the material nonlinearity, a third mode at the difference frequency is generated and radiated into free space.

Based on standard CMT [12], we have already described THz DFG in WGM cavities in Ref. [9]. Neglecting the pump depletion, the nonlinear efficiency of a passive emitter can be written as:

According to Eq. (1), η is maximized when the critical coupling condition is obtained for the two pump fields, i. e. when $Q_i^{\mathit{cpl}}=Q_i^{\mathit{int}}$ (i = 1,2). This can be achieved by an appropriate choice of 1) the waveguide width; and 2) the waveguide to resonator distance.

3. Modeling and discussion

In Fig. 2 we report the calculated THz frequency (ν₃, solid black line) of a WGM with m₃ = 2 versus R_int for a ring cavity with R_ext = 17.48 μm - see Fig. 1(b) for the definition of internal (R_int) and external (R_ext) radii. The results were obtained with a fully-vectorial finite-difference frequency-domain code developed following the guidelines of Ref. [13].

 

Fig. 2 THz WGM frequency ν₃ (solid black curve) and near-IR mode spacing Δν (dash-dotted blue curves) versus internal radius of the ring. The black dotted line corresponds to the THz resonance frequency for a structure without hole. The inset is a zoomed-in view around R_int = 15.7μm.

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The black dotted line represents the frequency of the mode for a cavity with no hole (R_int = 0) and radius R = R_ext = 17.48 μm. On the same plot we report, as blue dash-dotted lines, the frequency difference between pump modes Δν = ν₁ − ν₂ for a few different pairs of azimuthal numbers (m₁, m₂). Each pair and the THz WGM satisfy the phase-matching condition (Δm = 2). However, only the intersections between the dash-dotted curves and the solid black curve represent frequency triplets that strictly fulfill energy conservation. For example, if we consider a pillar with no hole (black-dotted line in Fig. 2), then the energy is strongly not conserved and DFG is inhibited.

By adjusting R_int we can fine tune the THz eigenfrequency - while leaving the pump modes unperturbed - so to enforce the energy conservation, and this can be done for different WGMs triplets. Therefore the central hole can be effectively used as a tuning parameter, with the possibility of implementing it - even after processing and testing the sample - via a Focused Ion Beam (FIB) session. While any treatments applied to the external semiconductor-air boundary would be detrimental for the quality factors of the pump WGMs involved in the DFG, the same does not hold for the internal radius. This is due to the fact that the pump WGMs have their maximum intensity at the periphery of the pillar, while they have negligible intensity close to the pillar axis.

Coming back to Fig. 2, if we restrict our attention to the pump modes (m₁ = 249, m₂ = 253), we find that the energy is strictly conserved for an internal radius R_int = 15.72 μm (gray circle in the inset). Figure 3 shows the calculated THz output power provided by a ring with such internal radius and for pump quality factors Q_p = 510⁵, i. e. the current state of the art for AlGaAs cylindrical cavities. To obtain this figure, we assumed that both pump modes are critically coupled to the ring. Output powers of few hundred nW could be reached with input powers (inside the waveguide) ranging from 5 to 20 mW.

 

Fig. 3 THz output power (in nW) versus waveguide input power (in mW) calculated for a ring-like structure with internal radius R_int = 15.72 μm and external radius R_ext = 17.48 μm. Both pump modes are assumed to be critically coupled to the cavity.

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An additional benefit of the microring with respect to a structure without hole is the increase of the overlap integral between the three modes. This is due to the fact that the hole allows to squeeze the THz mode into a smaller volume, and it typically results in an efficiency increase of about 50%.

The exploitation of ring-like resonators provides more tuning options. In Fig. 4 we show the effect of a metallic tip placed in the hollow part of the above ring structure. In particular, the top panel shows the shift of the THz frequency due to a gold tip with varying radius (R_T) and height fixed at h_T = h/2, h being the height of the resonator (see inset). A shift close to 100 GHz (around 3% of the WGM original frequency) can be obtained by exploiting large tips (i.e. for R_T > 0.8R_int). On the other hand, the bottom panel of Fig. 4 shows the shift of the THz frequency due to a gold tip with radius R_T = 0.9R_int and as a function of the relative tip height h_T/h. As we move the tip closer to the bottom mirror, the frequency of the THz WGM shifts up to 300 GHz (around 9% of the WGM original frequency).

 

Fig. 4 Frequency shift of the THz WGM versus: the radius of a metallic tip placed at half pillar height (top panel); the height of a tip with fixed radius R_T = 0.9R_int (bottom panel). Inset: sectional view of the tip and AlGaAs microring (the dashed line represents the axis of cylindrical symmetry).

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The same principle can also be used with a non-metallic tip. For example, in Fig. 5, we show the shift of the THz mode resonance frequency versus h_T/h, for a GaAs tip with radius R_T = 0.76R_int. As expected, the shift is negative and decreases in modulus if we move the tip further away. Please note that, in this case, the tip radius must be smaller than 0.8R_int to avoid the leakage of the WGM.

 

Fig. 5 Frequency shift of the THz WGM versus the height of a GaAs tip with fixed radius R_T = 0.76R_int. The solid black line is a guide to the eye.

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It is important to stress that the pump modes are localized close to the external ring periphery, and therefore they are not affected by the presence of the tip. This allows to have a powerful tuning mechanism for the DFG process.

4. Conclusion

In this letter we studied a passive THz source based on DFG between WGMs in semiconductor microcavities. We showed that ring-like resonators not only can be effectively used to introduce a tuning parameter (the pillar internal radius) that selectively acts on the THz WGM, but they also allow to increase the nonlinear efficiency of the DFG process. In particular, changing the ring internal radius allows to modify the frequency of the THz WGM (which is delocalized on the whole pillar volume) while not perturbing the frequencies of the pump modes (which are localized close to the periphery of the cavity). In this way, starting with a triplet satisfying the phase matching condition, we can exploit R_int to enforce the energy conservation.

Additionally, a controlled displacement of a non-absorbing tip in the hollow part of the ring, allows us an even finer tuning of the THz WGM frequency. The interest in this case is due to the fact that this fine tuning could also be performed while the source is operating. In the case of an active device, this would allow to compensate for e.g. temperature variations due to current injection. A piezo controller moving the tip could therefore be exploited to optimize the power emitted by the source. Moreover, the tip itself could be shaped so to act as a microantenna and increase the directivity of the extracted THz field.

Finally, the cavity considered here lends itself to pump laser integration by burying quantum dots in the central slab. In recent works the lasing action of WGMs sustained by electrically driven AlGaAs pillars has been demonstrated both at 10 K [14] and, more recently, at room temperature [15], a promising step towards the fabrication of the first electrically-pumped CW nonlinear THz emitter at room temperature.