1. Introduction

Metamaterials (MM) are artificial structures consisting of regular arrays of sub-wavelength sized resonant elements, so-called meta-atoms [1]. The resonant elements are dielectric and plasmonic resonators composed of dielectrics and conductors, such as metals, highly doped semiconductors or superconductors [2, 3]. The term metamaterial originates from the ability to design the material properties - the permittivity and the permeability - beyond those available in nature (e.g. negative or near-zero permeability and permittivity). This is achieved by engineering the resonator geometry and the distance between the individual elements, enabling novel optics with e.g. negative-refraction, hyper-lensing, electromagnetic cloaking [4, 5], and flat optics [6, 7].

A salient feature of the resonant nature of meta-atoms is the local enhancement of the electric field of an impinging electromagnetic wave. Further, the resonator mode volume V_eff of a single plasmonic resonator can be deeply sub-wavelength of the order of ${10}^{-3}\times {\lambda }_0^3$ [8, 9]. These two features enable to achieve an enhanced interaction between the light and entities such as quantum dots, quantum wires and quantum wells, and in the weak coupling regime it can be used to boost nonlinear optical effects for the light generation and sensing (e.g. frequency mixing processes and multi-photon absorption).

Solid state systems are good candidates for building a cavity quantum electrodynamic system with extremely strong coupling since the coupling strength scales with the square root of the number of carriers N involved in an optical transition and the mode volume V_eff of the cavity as Ω ∝ (N/V_eff)^1/2 [10]. This allowed to reach the ultra-strong coupling regime [11–15], where the vacuum Rabi frequency Ω becomes comparable to the bare transition frequency ω₀. Although the ultra-strong coupling can also be reached with the photochromatic molecules [13] and the flux qubits in microwave resonators [14, 15], the present world record was established using cyclotron resonance transitions in the 2-dimensional electron gas coupled to a planar metamaterial, achieving Ω/ω₀ = 0.87 [16]. According to De Liberato [17], even the deep ultra-strong coupling regime, defined as Ω/ω₀ > 1, can be reached offering a transition to Dicke phase [18]. It is worth noting that the resonators used for the ultra-strong coupling typically have quite small quality factor (a ratio of the resonance linewidth and a resonant frequency ω_FWHM/ω₀) Q < 15 due to the combination of radiative and material losses.

In this work, we present modified circular patch resonator designs with eigenmodes that offer Q factors of 40. We discuss their properties and the conditions for their excitation. The eigenmodes are studied using the finite-element electromagnetic simulation and the obtained data is verified experimentally. In addition, we demonstrate a flexible approach of fine-tuning of the eigenmodes’ resonant frequency, a key degree of freedom for optimizing the cQED system performance. This is realized by the post-processing of the resonator structure.

2. Disk patch resonator

So far, in the mid and long wavelength (THz) regime strong light-matter coupling has been demonstrated using resonators in the form of dog-bone [19], patch [20], explicit LC circuit resonators [9, 21] and planar meta-atom [16, 22, 23]. All those resonators exhibit a significant radiation loss since the resonant modes create a large magnetic or electric dipole. To target a high Q factor, the resonator has to feature a low radiation loss. The patch resonator (an arbitrary-shaped metal patch above a metallic ground plane, separated by a dielectric layer) is an efficient antenna. On the other hand, the patch resonator is a unique system that enables a very strong confinement of the electric field in the deep subwavelength space. It features a high field uniformity and the electric field orientation is ideal for the interaction with optical transitions in a semiconductor quantum well. All these features favor the patch resonator as the photonic part of a system providing strong light-matter interaction. Similarly to the patch resonator, the explicit LC resonator [9, 21, 24] has a large radiative loss because the energy is lost via the patch fringe electric field and the magnetic field that is generated by the oscillating current in the conductor loop.

In order to efficiently handle the radiative loss, we focus on a structure supporting a weakly-radiative mode, i.e. a mode that has a very small electric dipole moment and a well wrapped magnetic field with zero net magnetic dipole. We start with a standard picture of a planar resonator - split ring - featuring a capacitor C (gap between conductors) connected to an inductor L (conductor loop), see Fig. 1(a). The resonator mode of interest (LC mode) is characterized by a periodic exchange of energy between the capacitor and the inductor. As already mentioned, the energy in such a resonator is radiatively lost via the electric and magnetic fields that are generated by the split ring resonator. In order to get the electric and the magnetic fields confined, we choose a circularly symmetric patch structure featuring two embedded capacitors -a disk-shaped capacitor (C₁) in the centre and a ring-shaped one (C₂), concentric with the inner capacitor see Fig. 1(b). These two capacitors are separated by a region with holes drilled into the dielectric layer and are connected by a ground plane on the bottom and a top conductor exhibiting an inductance (see Fig. 1(c) and Fig. 1(d) for the cross-section and the resonator layout, respectively). The electrical equivalent circuit of such a disk patch resonator (DP) consists of two capacitors connected via an inductor. The basic LC resonant mode is thus described as a periodic exchange of the electric field energy in the two capacitors and the magnetic field energy in the inductive part of the resonator. The electric field of the mode exhibits a rotational symmetry with electric field concentrated in the capacitors regions. Such mode we label as axisymmetric mode and it is known as TM₀₁₀ standing wave eigenmode in the double-metal waveguide [25, 26].

 

Fig. 1 (a) Sketch of a standard split ring (LC) resonator and its lumped elements model; (b) Sketch of a modified disk path resonator and its lumped elements model; (c,d) Cross-section and top view of the resonator showings a LC resonator consisting of two capacitors C₁ and C₂ connected via top and bottom conductor forming a loop with inductance L; (e) Frequency of the LC resonant mode of disk patch resonator as a function of geometrical parameters of the conductor obtained by 2D eigenmode simulation (colored symbols) and from the LC model (black solid lines).

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To understand the fundamental nature of the structure we carried out 2D rotational symmetric eigenmodes simulations as well as an analysis of a lumped element LC model. For the comparison of this model with the simulations, we keep the capacitance of both capacitors constant C₁ = C₂ = const, while the radial spacing and the connecting conductor dimensions are varied. In the lumped element LC model, the bridge-like shaped top conductor and the flat bottom conductor form an inductive loop that has rotational symmetry around the z-axis with the radius r_toroid. Such a toroidal inductor has an inductance proportional to the enclosed cross-sectional area A = w_loop h_loop divided by the inner toroid radius r_i,loop, where w_loop is the width and h_loop the height of the bridge-shaped top-conductor. In our model the eigenfrequency ν for fixed capacitances C₁ = C₂ = C is thus given by

2.1. Resonator eigenmodes

For a simple and reproducible fabrication of the modified patch resonator we further focused on structures with h_loop = 0 µm. Such a resonator design is extensively simulated by 3D full wave simulations in order to map all eigenmodes with frequencies up to 3.2 THz. The eigenmodes for the disk patch resonator with holes are shown in Fig. 2(a). For the mode labeling we have adapted the nomenclature from Ref. [26]. Modes with an antisymmetric electric field distribution are labeled as ’odd’ modes o_mn, where m and n indicate the order of the angular and the radial mode order, respectively. Modes with a symmetric electric field distribution are called ’even’ modes e_mn. The eigenmodes of the DP resonator are plotted in Fig. 2(a) (right). Their electric field distribution without tuning resembles the eigenmodes of the DP resonators without holes that are discussed in detail in Ref. [25, 26]. Here, we just point to the fact that except of the axisymmetric mode (e₀₂), all the other modes (o₁₁, e₂₁, o₃₁, o₁₂) are degenerate with two orthogonal electric field distributions. The modes o₁₁, e₂₁, o₃₁ are in fact the lowest order whispering gallery modes of a disk resonator known from the semiconductor lasers technology.

 

Fig. 2 Electric field distribution of the eigenmodes up to 3.2 THz of (a) the disk patch resonator (DP), (b) the split disk patch resonator (SDP) and (c) the double split disk patch resonator (dSDP) with a radius of 21 µm and a dielectric layer (GaAs) thickness of 4 µm. The eigenfrequencies of individual modes are listed in Table 1. The mode notation e/o_mn reflects orders of angular m and radial n modes according to Ref. [26]. Subscript ρ is assigned to modes derived from the axisymmetric mode e₀₂; axis related subscripts refer to orientation of the axis of symmetry for the mode pattern.

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Table 1. Summary of the eigenfrequencies in THz of the modes shown in Fig. 2. We utilize the mode notation from [26]. The additional letters and the prime denote modes that are generated by breaking the symmetry. y and xy describe the orientation of the modes and the relation to the incidence E_x and E_xy of the light that can excite those modes. ρ states the relation to the axisymmetric mode e₀₂.

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To break the rotational symmetry of the DP resonator, and thereby lifting the modes’ degeneracy, we introduce a gap that splits the circular patch into two halves. Such a resonator is labeled as split disk patch (SDP) resonator. Figure 2(b) shows a sketch of the resonator and the electric field distribution of its lowest eigenmodes. As expected, due to the splitting, the degeneracy is lifted and the originally degenerate modes are split into two modes with a frequency difference that depends on the size of the gap. One of each mode pair resembles the eigenmodes of the resonator without a slit, see the top row of Fig. 2(b). The modes in the bottom row of Fig. 2(b) differentiate themselves from the modes in the top row by a π phase shift in the mode fields of the two patch halves. That enables a selective excitation of the modes depending on the angle of incidence and the polarization of the incoming plane wave for the SDP resonator. Consequently, we expect that the properties of the modes with a π phase shift differ considerably from the original modes of a DP resonator, especially in term of the coupling to free space.

Finally, we consider the disk patch resonator with two orthogonal slits that split the original resonator into four equal sections. Such a resonator, labeled as double split disk patch (dSDP) resonator, together with the electric field distribution of several lowest order eigenmodes, are shown in Fig. 2(c). The additional slit allows to eliminate certain modes of the original DP completely, namely, the modes o_(2n+1)1, o_1(2n+2),… for n = 0,1,2,…. The fields and currents in the individual four sections of the dSDP can again oscillate with a zero or π phase shift. This leads to the appearance of new modes, shown in the bottom row of the electric field mode maps. Remarkably, the axisymmetric mode e₀₂ is present in all resonators and its associated antisymmetric counterpart o_12,ρ appears in the SDP and dSDP resonators due to the slits. In the following, we will analyze these two modes in more detail.

2.2. Excitation of the eigenmodes

A quantitative description of the selection rules for the excitation of metal-dielectric-metal structures by an electromagnetic wave is discussed in [27]. The authors compared the charges, the current densities and the fields that are induced at the boundaries of a quadratic patch resonator by an incident plane wave, with those that are generated by the resonator eigenmodes at the very same boundaries. If there is a partial overlap, the mode can be excited. The same approach we have applied to our disk patch resonators.

The selection rules for the excitation of odd modes are easy to understand, as the modes manifest themselves as an oscillating electric dipole on the metallic patch, with positive and negative charges at the opposite edges. Additionally, with similar considerations, it can be shown that each odd mode of a double-metal resonator has a net magnetic dipole moment. In contrast, the even modes feature a symmetric charge distribution, which cannot be achieved by a normal incident plane wave. Such modes exhibit a lower radiative loss and consequently a higher Q factor compared to the odd modes. To excite these modes with a field distribution symmetric or antisymmetric along the x- and y- axes, an oblique incidence of light is required, as illustrated in the schematic in Fig. 1(c). For the rotational symmetric disk patch resonator without a slit (DP), there is no structural breaking of the rotational symmetry. Consequently, the odd cavity modes can be excited with an arbitrary polarization and angle of incidence. Only the polarization of the incident wave projected onto the xy-plane determines the orientation of the induced electric net dipole moment of the corresponding excited odd eigenmode. The even modes of the DP resonator, which have an even number of angular nodes (e₂₁, e₀₂, e₄₁), can be excited by oblique incident light with an arbitrary polarization. The axisymmetric mode e₀₂ with no angular node can only be excited by p-polarized waves. In the following, we consider only p-polarized waves.

For the SDP resonator with a slit, the excitation conditions become more complex. Each odd eigenmode of the resonator has a distinct polarization direction, which is determined by the slit orientation. If we define the slit in the x-direction, the odd modes o₁₁, o₃₁, o₁₂ feature the electric dipoles in x-direction, while the mode dipoles associated with the other odd modes o_12,y, o_12,ρ are oriented in y-direction. The former modes are excited by a p-polarized light for the incidence in the xz-plane (E_x orientation) and the latter modes are excited by p-polarized light in the yz-plane (E_y orientation). For even modes, in general, there is a plane of incidence for p-polarized light to fulfill the excitation condition. For example, the mode e₂₁ can be excited with p-polarized light in the xz-plane (E_x) or yz-plane of incidence (E_y).

The dSDP resonator modes are again degenerate because the structure is invariant to a π/2 rotation around the z-axis (two modes except of the even modes). For the eigenmodes of the dSDP resonator that also exist in the SDP resonator, namely those that are excited by E_y incidence, the same selection rules are valid. All the other modes, which are generated due to the second slit, are either symmetric or antisymmetric to the (±1, 1, 0) plane or antisymmetric to both slits. Consequently, the odd modes are orientated with an azimuthal angle φ_I of ±45° ((±1, 1, 0) plane) and the plane of incidence has to be parallel to these directions. Thus, as an example, modes that are symmetric around the (−1, 1, 0) plane are excited by a p-polarized light that is incident in the very same plane, denoted as E_xy. The axisymmetric mode e₀₂ has no electric net dipole moment and the magnetic field is enclosed within the plasmonic patch resonator. Hence, it matches our requirement of a weak-interacting meta-atom properly, see Fig. 3(d). Finally, the broken symmetry generates the related antisymmetric mode o_12,ρ whose fields and current in the first patch resonator half oscillate with a π phase shift relative to the second half. As a result, this mode has a net dipole moment that enables the mode to couple to normally incident free space wave.

 

Fig. 3 Tuning the split disk resonator by partially removing the dielectrics. 3D schematic sketch of (a) the initial unetched case; (b) the intermediate state after anisotropic etching of the dielectric and (c) the final state (removed top electrode on one side is for the sake of clarity to emphasise the top electrode undercut). (d) and (e) Electric displacement field D_z and the magnitude of magnetic field |B| of the axisymmetric eigenmodes e₀₂ for the initial unetched case and the final case, respectively.

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2.3. Tuning of the circular patch resonators

Since the tuning of a cavity mode to an intersubband transition (ISBT) is an essential step to achieve optimal light-matter coupling conditions, we have investigated the flexibility of the SDP and dSDP resonators with respect to post-process tuning. Such a fabrication technique has been demonstrated for planar resonators by etching the ambient material [22]. We have modeled the tuning process in order to estimate the achievable tuning range and the resonance linewidth. The simulation model is based on a finite element method for 2D rotational symmetric and 3D simulations.

The etching process can be divided into two subprocesses, namely the removal of the dielectrics around the patch resonator using an anisotropic etching procedure (aniso) and the partial removal of the dielectrics underneath the metallic patches employing a more isotropic etching process (iso). The structural changes of the resonator structures due to the two etching processes are shown in Figs. 3(a)–3(c). Without loss of generality, we have chosen the axisymmetric mode e₀₂ featuring a high Q factor with a simple mode field shape to demonstrate the effect of the etching. As expected, both etching sub-processes induce a blue-shift of the resonance frequency, see Fig. 4. The removal of the dielectric around the resonator is accompanied by a redistribution of the electric field at the edge of the resonator, as it was reported for planar resonators [22]. In addition, the etching through the holes leads to the reconfiguration of the mode’s electric field distribution and can be used to increase the separation of the electric displacement field and the magnetic field, see Fig. 3(d) and Fig. 3(e). Considering the size of the structure, the model predicts a shift of the resonance frequency by a factor of 0.30, a reasonably high value to adjust the resonator to a given material’s optical transition.

 

Fig. 4 (left) Tuning of the resonance frequency for etching the dielectric anisotropically down to the ground layer and subsequently for etching isotropically underneath the top layer. (right) Electric and the magnetic field distribution for various etch steps.

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3. Measurements and discussion

We have fabricated the disk patch resonators SDP and dSDP using processing techniques employed for quantum cascade laser fabrication [29]. A 4 µm thick undoped GaAs layer grown by molecular beam epitaxy on an undoped GaAs substrate was covered by a 900 nm thick Au layer and wafer bonded to a host substrate (undoped GaAs) also covered with a 900 nm thick Au layer. Subsequently, the original GaAs substrate has been removed by lapping and wet-chemical etching to leave the epitaxial GaAs layer on the host substrate. The final device was formed by a standard photolithography to pattern the 200 nm thick Au top electrodes of the resonators. The fabricated double-metal disk patch resonators with one and two slits (SDP and dSDP) have a disk radius of 21 µm with holes of 3 µm in diameter and a gap size of 3.6 µm. They are arranged in a square array with a pitch of 65 µm.

The samples with an array of disk patch resonators were measured in a reflection geometry with an angle of incidence of 45^◦ using a standard THz time-domain spectroscopy setup in the frequency range between 0.2 THz and 3.5 THz and with a frequency resolution of ~50 GHz. The SDP resonators are excited by broadband THz pulses with a p-polarization in two different planes of incidence. Namely, in the plane parallel (xz-plane or E_x) and orthogonal to the slit (yz-plane or E_y), see Fig. 5(a). The dSDP resonators were excited with THz pulses with p-polarization in a plane of incidence orthogonal to one of the slits (E_y) and with an azimuthal angle of 45° with respect to the slits (E_xy), see Fig. 5(b).

 

Fig. 5 SEM micrograph of (a) a SDP resonator, and (b) a dSDP resonator. The THz light pulses are incident at 45° with p-polarization. Each structure is measured for two different orientations as indicated in the images.

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The measured THz reflectance spectra for SDP and dSDP resonator arrays are shown in Figs. 6(a) and 6(b) and Figs. 7(a) and 7(b), respectively. The spectra of the resonators have been measured before any etching of GaAs and after gradual stepwise removal of this dielectric in several consecutive steps. Each observed reflectance spectrum features a series of local minima, indicating several resonances of the investigated structures. These resonances (resonator eigenmodes) are characterized by their frequency and linewidth, from which the Q factor is calculated. The resonances are labeled with the names of associated excited eigenmodes and the arrows in the figures indicate the tuning range of the individual modes.

 

Fig. 6 Comparison of the measured SDP reflectance spectra (top row) with 3D simulations (bottom row). In the experiment the resonators were excited at 45° incidence in (a) with the electric field parallel to the slit (E_x), and (b) orthogonal to the slit (E_y), see Fig. 5(a). The spectra were measured for the initial unetched case and after stepwise anisotropic (aniso) or isotropic (iso) dry etching. (c), (d) In the 3D simulations the resonators are excited for the same angle of incidence and the same configuration of the dielectric environment as in the experiment.

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Fig. 7 Comparison of the measured dSDP reflectance spectra (top row) with 3D simulations (bottom row). In the experiment the resonators were excited at 45° incidence in (a) with the electric field orthogonal the one of slits (E_y), and (b) at azimuthal angle of 45° to slits (E_xy), see Fig. 5(b). The reflectance spectra were measured for the initial unetched case and after two etch steps of anisotropic etching (aniso). (c), (d) In the 3D simulations the resonators are excited for the same angle of incidence and the same configuration of the dielectric environment as in the experiment.

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In the case of the SDP resonators, Fig 6(a) and 6(b) show two different sets of resonances in the reflectance spectra obtained for the plane of incidence parallel (E_x) and perpendicular to the slit (E_y), respectively. As mentioned earlier, the orientation of the polarization determines the charge and the current distribution induced at the boundaries and allows to selectively excite particular eigenmodes. In this way, we identify the modes o₁₁, e₂₁ and e₀₂ that are excited with E_x incident THz light and the modes e₂₁, o_12,y, e₀₂ and o_12,ρ that are excited in E_y incidence. Noteworthy is the SDP resonance at 1.7 THz (no etching) in Figs. 6(a) and 6(b), which exhibits different apparent linewidths for E_x and E_y excitations. While at E_x incidence only the even mode e₂₁ is excited, at E_y incidence also the odd mode o_12,y is excited, but these two modes cannot be resolved spectrally due to the limited frequency resolution of the measurement.

Similarly, several resonances are present in the reflectance spectra of the dSDP resonators in Fig. 7(a) and Fig. 7(b) observed for the THz light incidence in the plane perpendicular to a slit (φ_I = 90° or E_y) and at 45° to a slit (φ_I = 45° or E_xy), respectively (see Fig. 5(b)). The observed resonances have been identified as the modes o_12,y, e₀₂ and o_12,ρ for the case of E_y incidence and the modes o_11,xy, e₀₂, o_12,ρ,_xy and e_22,ρ for E_x incidence. The axisymmetric modes e₀₂ is excited in both geometries as expected. Finally, the geometry of the resonator (dSDP) does not support the modes o₁₁, o₃₁ and o₁₂ of SDP resonator that are indeed missing in the reflectance spectrum.

In order to confirm the assignment of the individual modes, we have compared the experimental reflectance spectra for the SDP and the dSDP resonators to the spectra that were obtained by the 3D simulations done for the same geometry as the experiment and are shown in Figs. 6(c) and 6(d) and Figs. 7(c) and 7(d), respectively. The simulated reflectance spectra are very similar to the experimental data in term of position of the resonances. The linewidth of the resonances differs because only the radiative loss and the material losses in the metal patch were included into the simulation.

The etching of the dielectric results in a blue-shift of the resonance frequency of each mode as demonstrated in the reflectance spectra in Figs. 6(a) and 6(b) and Figs. 7(a) and 7(b). Such observed behavior is in accordance with our simulations results shown in Figs. 6(c) and 6(d) and Figs. 7(c) and 7(d) and they are done for the similar detailed geometry as featured by the etched samples (i.e. partial removal of surrounding dielectrics). To summarize the tendencies observed in the tuning process, the measured resonance frequencies (connected colored symbols) and the associated quality factors Q are plotted in Fig. 8 as a function of the etching depth of the resonators. The modes’ eigenfrequencies are blue shifted up to 0.3 – 0.5 THz by etching through the 4 µm thick GaAs layer and by undercutting the top metal patch. For the SDP resonators we observe also an increase of the modes’ frequency separation when the isotropic etching is applied. It is explained by the strong sensitivity of the modes to a selective removal of the dielectric from the patch volume. As a result, the modes e₂₁ and o_12,y, which are unresolved in the original unetched SDP resonator for the E_y excitation, get resolved after the etching.

 

Fig. 8 The eigenfrequencies (a,b) and the Q factors (c,d) of the SDP and dSDP resonator as a function of the etching depth in the dielectric extracted from the measured reflectance spectra in Figs. 6(a,b) and Figs. 7(a,b). The color of symbols assigned to the individual eigenmode indicates the orientation of incident electric field (red - E_x, blue - E_y, green -45°/E_xy).

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The etching has only a weak effect on the Q factor of the resonator eigenmodes as shown in Figs. 8(c) and 8(d). Generally, according to expectations, the even modes have higher Q factors than the odd modes as the in-plane electric dipole of the even modes is nearly zero. The odd mode o₁₁ has a Q factor of 12−18, while the even modes typically have a value between 30 and 40. The highest Q factor of at least ∼40 is observed for the modes e₀₂, e₂₁ and e_22,ρ. This value is already at the limit of the measurement setup frequency resolution.

4. Conclusion

We have studied modified disk patch resonators in order to provide an improvement of the light-matter interaction for the semiconductor heterostructure based quantum systems operating at terahertz frequencies. To obtain control over the creation and elimination of the resonator eigenmodes, we have introduced a ring chain of holes in the top metal of the disk patch that enables unique control over the electric displacement field distribution of the resonator radial modes. In addition, to further expand mode control, we introduced slits to the standard disk structure providing a symmetry breaking of the resonator and leading to appearance of new modes that can be selective addressed. Such modified disk patch resonators feature more dense modal spectrum than the standard disk patch with odd modes having Q factors (ω_FWHM/ω₀) of 12 – 18, while the even modes have Q factors between 30 and 40. The disk patch tuning capability, a technical requirement for the optimization of light-matter interaction, was addressed by structure post processing using a reactive-ion etching technique. The combination of isotropic and anisotropic etching processes allowed us to blue-shift the resonator eigenfrequencies up to 50 % without a significant modification of the modes electric field profile and deterioration of their Q factor. The designed resonators might be good candidates for cavity based quantum electrodynamic systems with intersubband transitons in nanostructures.