Electromagnetic study of a split-ring resonator metamaterial with Cold-Electron Bolometers

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This document is not formatted, has not undergone copyediting or typesetting, and may contain errors, unsubstantiated scientific claims or preliminary data. Preprint Title Electromagnetic study of a split-ring resonator metamaterial with Cold-Electron Bolometers Authors Ekaterina A. Matrozova, Alexander V. Chiginev, Leonid S. Revin and Andrey L. Pankratov Publication Date 06 Okt. 2025 Article Type Full Research Paper ORCID® iDs Ekaterina A. Matrozova - https://orcid.org/0000-0003-1013-1365; Alexander V. Chiginev - https://orcid.org/0000-0002-6676-9141; Andrey L. Pankratov - https://orcid.org/0000-0003-2661-2745 Electromagnetic study of a split-ring resonator metamaterial with1 Cold-Electron Bolometers2 Ekaterina A. Matrozova1, Alexander V. Chiginev1,2, Leonid S. Revin1,2 and Andrey L.3 Pankratov1,2∗14 Address: 1Nizhny Novgorod State Technical University n.a. R.E. Alekseev, Minin Street, 24,5 Nizhny Novgorod, 603155, Russia6 2Institute for Physics of Microstructures of the Russian Academy of Sciences, Akademicheskaya7 Street, 7, Nizhny Novgorod, 603950, Russia8 Email: Andrey L. Pankratov1,2 - [email protected] ∗ Corresponding author10 Abstract11 We present an electromagnetic study of a metamaterial receiver based on split-ring resonators with12 integrated cold-electron bolometers. We suggest a modified antenna design that allows one to sig-13 nificantly increase the absorbed power and the bandwidth. The trade-off between the bandwidth14 expansion due to miniaturization and the reduction in absorption efficiency determined by the Airy15 spot size of the coupling lens is investigated. To solve this issue, a simultaneous miniaturization of16 the size of the entire structure with an increase in the number of array elements is proposed. The17 design with a 37-element array demonstrates an increase in power absorption by a factor of 1.418 compared to the original 19-element single-ring array, as well as an increase in operating band-19 width from 160 to 820 GHz.20 Keywords21 Metamaterial, split-ring resonator, cold electron bolometer22 1 Introduction23 Highly sensitive receivers with broadband antennas are of significant interest for advanced spec-24 troscopic applications and various radioastronomy tasks [1-5]. In particular, broadband receiving25 systems are required for use with a Fourier Transform Spectrometer based on the Martin-Paplett in-26 terferometer that is planned to be used in future missions, such as BISOU (Balloon Interferometer27 for Spectral Observations of the Universe) [3,4] and Millimetron [2,5]. The use of Cold-Electron28 Bolometers (CEBs) is particularly advantageous for such systems, enabling operation in a wide29 frequency range from GHz to X-ray [6-8] due to a normal-metal absorber. CEBs offer several ad-30 vantages over other types of receiver, such as Transition Edge Sensors (TESs) [9-11]. These ad-31 vantages include their micrometer scale size, which facilitates direct integration into antenna slots32 without the need for microwave feed lines (e.g., microstrip or coplanar lines), thus simplifying the33 design and preventing signal degradation at higher frequencies [12]. Furthermore, the natural elec-34 tron cooling mechanism in CEBs [13-15] is highly suitable for operation with cryogenic systems35 such as3He sorption fridges. Perhaps most critically, CEBs demonstrate exceptional immunity to36 cosmic rays [16], a paramount requirement for balloon and space missions.37 Our group has recently designed, fabricated and characterized a metamaterial receiver with inte-38 grated CEBs, operating in a broad frequency range [17]. In that work, each element represented a39 ring antenna with two embedded CEBs connected parallel in DC, whereas the antennas in the array40 were connected in series. In the present work, we propose and numerically investigate a new de-41 sign of a CEB metamaterial receiver based on double split-ring resonators (SRRs) to increase both42 the magnitude of the absorbed signal and the working bandwidth. We consider various geometrical43 modifications of this design and perform a comparative analysis.44 Design and Simulation Approach45 In our previous work [17], a metamaterial comprising 19 ring antennas enabled the reception of46 external electromagnetic signals in the broad band from 150 to 550 GHz, as well as in the band47 from 900 to 1300 GHz. To further enhance signal absorption, we propose replacing simple ring48 2 Figure 1: Schematic layout of the investigated metamaterial arrays: a) 19-element array of single- ring antennas; b) 19-element array of split-ring resonators; c) 37-element array of miniaturized SRRs. The Inset: a single unit cell with two embedded CEBs represented as an RC circuit. antennas with SRRs [18-21]. The SRR is a well-established magnetic metamaterial element whose49 resonant properties are governed by its internal inductance and capacitance, allowing for a strong50 magnetic response and associated current loops at the designed resonance frequency.51 The simulated receiving structure is placed on a 500𝜇m-thick silicon substrate. A 4-mm-diameter52 silicon hyperhemispherical lens is placed on the back side of the substrate to efficiently couple the53 incident radiation into the planar structure. The external signal is incident from a waveguide port54 located behind the Si lens, simulating a realistic excitation source.55 The signal is received by an array of the proposed ring resonators. Two Cold-Electron Bolometers56 3 are embedded into the outer ring of each SRR element. In the simulation, each CEB is modeled57 as an RC circuit (see inset in Fig. 1), where𝑅𝑎𝑏𝑠 = 75Ω represents the resistance of the CEB’s58 normal-metal absorber, and𝐶𝑆𝐼 𝑁 = 20 fF is the capacitance of the two SIN junctions of the CEB59 connected in series. The total absorbed power is calculated as the sum of the powers absorbed in60 these discrete ports representing the CEBs.61 The design of the previously studied metamaterial with CEBs and single-ring antennas is shown62 in Fig. 1a. To increase the absorbed power and the working frequency band, we propose and ana-63 lyze a new design based on SRR (Fig. 1b and c). The geometric parameters of the structures are as64 follows:65 • Single ring: outer ring diameter𝑑𝑒𝑥𝑡 = 80𝜇m, inner ring diameter𝑑𝑖𝑛𝑡 = 70𝜇m. The lattice66 constant (period) of the metamaterial array is𝑃 = 86𝜇m. The total size of the structure is67 424 𝜇m.68 • SRR, large scale: the outer ring has an external diameter of𝑑𝑒𝑥𝑡, 1 = 80𝜇m and an internal69 diameter of𝑑𝑖𝑛𝑡, 1 = 70𝜇m. The inner ring has an external diameter of𝑑𝑒𝑥𝑡, 2 = 40𝜇m and an70 internal diameter of𝑑𝑖𝑛𝑡, 2 = 30𝜇m. The period of the metamaterial array is𝑃 = 86𝜇m. The71 total size of the structure is 424𝜇m.72 • SRR, small scale: A scaled-down version with𝑑𝑒𝑥𝑡, 1 = 40𝜇m, 𝑑𝑖𝑛𝑡, 1 = 35𝜇m; 𝑑𝑒𝑥𝑡, 2 = 2073 𝜇m, 𝑑𝑖𝑛𝑡, 2 = 15𝜇m. The lattice period for this dense array is𝑃 = 43𝜇m. The total size of the74 structure is reduced to 298𝜇m.75 This scaling of the SRR geometry is intended to shift the central frequency of the metamaterial to a76 higher value while maintaining the increasing absorption of the double-ring design.77 The transition from a single-ring antenna to a double split-ring resonator design, while keeping78 the number of elements to be constant, resulted in a significant improvement in performance. The79 addition of the inner ring, which increases the total capacitance of the resonant element, leads to80 a slight reduction of the central frequency [19]. More importantly, it yielded a 1.5-fold increase in81 the total absorbed power.82 4 0 5 001 0001 5002 0000,00 ,10 ,20 ,3Absorbed powerF requency (GHz) a) Single-ring antennas b) SRRsE xperimental response (a.u.) Figure 2: Amplitude-frequency characteristics of the metamaterial receiver: a) 19-element array of single-ring antennas with a lattice period of𝑃 = 86𝜇m (red curve); b) 19-element array of SRRs with 𝑃 = 86𝜇m (blue curve). The dashed black curve shows the experimentally measured bolome- ter response. The amplitude-frequency characteristics (AFC) for the simulated single-ring and SRR designs83 are presented in Fig. 2. For the single-ring array, the absorbed power in the first resonance maxi-84 mum reached a value of 0.18 (normalized units, with 0.5 maximal total power) at half maximum85 (FWHM) spanning from 100 to 545 GHz (Fig. 2, red curve). In contrast, the SRR array demon-86 strated a higher absorbed power of 0.27 within a bandwidth of 105 to 440 GHz (Fig. 2, blue curve).87 As an experimental reference for our simulations, Fig. 2 also shows the frequency response mea-88 sured for a fabricated sample consisting of a 19-element single-ring metamaterial (black dashed89 curve). This sample had the design described in [17] and was characterized using the same exper-90 imental setup described there. This setup employs a YBaCuO Josephson junction oscillator as a91 broadband source, with the signal delivered to the sample via an oversized waveguide. Therefore,92 the measured frequency response is the combined frequency response of the entire path (oscilla-93 tor, waveguide-feeder, lens and the CEB metamaterial itself), with "fingers" due to the used log-94 periodic antenna of the Josephson oscillator, which was not fully matched to the antenna. Despite95 this convolution, the experimental data clearly confirm the calculated dual-band behavior of the96 metamaterial, showing two broad peaks centered at approximately 350 GHz and 1100 GHz. This97 agreement validates our simulation model.98 5 0 5 001 0001 5002 0000,000 ,150 ,300 5 001 0001 5002 0000,000 ,150 ,301 9-element array of the SRRsA bsorbed power a) Period = 86 µm b) Period = 68,8 µm c) Period = 51,6 µm d) Period = 34,4 µm A bsorbed powerF requency (GHz) a) Period = 86 µm b) Period = 68,8 µm c) Period = 51,6 µm d) Period = 34,4 µm 1 9-element array of the Ring Antennas Figure 3: The upper plot: AFC of the 19 single-ring antenna metamaterial for different geometric scaling factors: a) black curve: outer ring diameter𝑑𝑜𝑢𝑡 = 80𝜇m, inner ring diameter𝑑𝑖𝑛 = 70𝜇m, period 𝑃 = 86𝜇m; b) red curve:𝑑𝑜𝑢𝑡 = 64𝜇m, 𝑑𝑖𝑛 = 56𝜇m, 𝑃 = 68.8𝜇m; c) blue curve:𝑑𝑜𝑢𝑡 = 48 𝜇m, 𝑑𝑖𝑛 = 42𝜇m, 𝑃 = 51.6𝜇m; d) purple curve:𝑑𝑜𝑢𝑡 = 32𝜇m, 𝑑𝑖𝑛 = 28𝜇m, 𝑃 = 34.4𝜇m. The bottom plot: AFC of the 19 SRR-based metamaterial for different geometric scaling factors. The design parameters and scaling factors (0%, 20%, 40%, 60%) correspond to the upper plot. The amplitude-frequency characteristics (AFC) of the single-ring and SRR metamaterials with99 various scaling factors are presented in Fig. 3. The optimal number and size of the resonators are100 governed by the requirement to fill the Airy spot of the silicon lens. If the total array size is smaller101 than the Airy spot, a portion of the incident signal will not interact with the metamaterial, instead102 scattering into the surrounding space. Our simulations confirm this principle: a reduction in the103 SRR dimensions and the array period by 20% led to a broadening of the absorption bandwidth and104 a small shift of the first resonance maximum towards higher frequencies. A further reduction of105 dimensions by 40%resulted in an even wider bandwidth; however, the peak absorbed power began106 to decrease, indicating that the array size was becoming insufficient relative to the Airy spot. A107 drastic 60%size reduction caused a severe deterioration of absorption.108 To achieve the widest possible bandwidth using SRRs, our results shown in Fig. 3 suggest prior-109 itizing somewhat smaller unit cell sizes. Simply scaling down a fixed 19-element array leads to110 less efficient signal reception due to the array becoming smaller than the Airy spot. As an efficient111 alternative, we propose to halve the SRR dimensions and array period while simultaneously in-112 6 creasing the number of elements from 19 to 37 (Fig. 1c). This approach successfully increased113 the absorbed power to 0.25, which is by a factor of 1.4 higher than for the single-ring array, while114 also achieving an ultra-wide receiving band from 160 to 820 GHz (Fig. 4, black line). If the 37-115 element array structure occupies the same area as the original single-ring structure, larger absorp-116 tion efficiency at the first peak can be achieved (Fig. 4, red line), but the working bandwidth will117 be narrower than for the structure with smaller rings. Thus, by selecting the overall structure size,118 a compromise can be found between the maximum absorption efficiency and the widest receiving119 bandwidth.120 0 5 001 0001 5002 0000,00 ,10 ,20 ,33 7-element array of SRRsA bsorbed PowerF requency (GHz) a) Period = 43 µm b) Period = 62 µm c) Period = 86 µm Figure 4: The amplitude-frequency characteristics of the 37-element array of SRR-based metama- terial for different periods of the lattice. It is important to note that the choice of the number of receiving antennas should be in a proper121 balance. Although a larger array can better fill the Airy spot, it also increases the total number of122 bolometers. This, in turn, increases the differential resistance of the structure at the operating point123 and increases the current noise contribution of the readout amplifier [17,30]. Furthermore, a larger124 number of elements increases the fabrication complexity. Crucially, nearly doubling the number of125 elements (from 19 to 37) does not produce a proportional increase in the absorbed power (Fig. 5).126 Figure 5 shows the AFC of the SRR metamaterial with a different number of elements. For the127 large-scale design (period𝑃 = 86𝜇m, rings: 𝑑𝑜𝑢𝑡, 1/𝑑𝑖𝑛,1 = 80/70𝜇m, 𝑑𝑜𝑢𝑡, 2/𝑑𝑖𝑛,2 = 40/30𝜇m),128 doubling the number of elements increased the absorbed power by only about 7%, with a minor in-129 7 0 5 001 0001 5002 0000,00 ,10 ,20 ,3Absorbed powerF requency (GHz) a) 19 SRR, Period = 43 µm b) 19 SRR, Period = 86 µm c) 37 SRR, Period = 43 µm d) 37 SRR, Period = 86 µm Figure 5: Dependence of the absorbed power on the number of elements in the SRR array. crease in bandwidth. The same doubling for the miniaturized design (𝑃 = 43𝜇m, rings: 𝑑𝑜𝑢𝑡, 1/𝑑𝑖𝑛,1130 = 40/35𝜇m, 𝑑𝑜𝑢𝑡, 2/𝑑𝑖𝑛,2 = 20/15𝜇m) is more efficient, leading to 17% increase in power. This131 higher efficiency is directly linked to the Airy spot coverage: adding elements to the smaller array132 more effectively increases its total area towards the optimal size. For the already-large array, new133 elements are added at the periphery or outside the most intense part of the Airy spot, which do not134 actually help.135 Discussion136 Solving the problem of broadband high-sensitivity reception for terahertz applications naturally137 entails comparing the metamaterial-based approach presented here with traditional broadband138 antenna solutions such as the log-periodic [22-24] or spiral antennas [25,26]. These antennas are139 indeed a well-established technology, providing wideband frequency response and high detec-140 tion/radiation efficiency. However, their widespread use is subject to a fundamental limitation: the141 active receiving element is typically a single detector unit located at the antenna’s feed point. This142 configuration can become a bottleneck when detecting ultra-low power signals in the presence of143 high background radiation, as the single detector must handle the entire power load, potentially144 limiting the dynamic range and complicating the optimization of noise-equivalent power (NEP).145 8 There have been proposals to integrate multiple sensing elements directly into the structure of a146 log-periodic antenna [27-29]. While promising, such designs face significant challenges in imple-147 mentation. The complex geometry of the antenna makes it difficult to integrate a large number of148 detectors and to design complex series-parallel electrical networks necessary for optimal power dis-149 tribution and impedance matching. In contrast, the metamaterial approach offers a fundamentally150 more flexible paradigm. A periodic array of resonators, such as our SRR-based design, inherently151 functions as a multi-absorber system. This architecture allows for the precise engineering of the de-152 tector network: the number of CEBs, their individual connection (series or parallel), and the overall153 array configuration to achieve an optimal balance between power load, responsivity, and total noise154 [17,30].155 This capability is particularly critical for applications like cosmic microwave background polarime-156 try or high-resolution spectroscopy, where the detector must operate photon-noise-limited under a157 specific background power load. For CEBs, we have previously demonstrated that the optimal con-158 figuration for minimizing the total NEP with a given readout amplifier involves a specific series-159 parallel combination of bolometers. The metamaterial platform is ideal for implementing such an160 optimized multi-absorber receiver. By adapting the array geometry and the electrical connection161 scheme between CEBs, one can precisely control the power absorbed per bolometer and the re-162 sulting differential resistance, thereby achieving photon-noise-limited performance across a wide163 bandwidth. This level of design control is considerably more challenging to realize within the con-164 strained geometry of a single-feed log-periodic antenna.165 Conclusions166 In this work, we have presented a comprehensive electromagnetic study on the design and opti-167 mization of a metamaterial receiver based on split-ring resonators integrated with cold-electron168 bolometers. The transition from a conventional single-ring antenna design to a double SRR config-169 uration has been demonstrated to be a highly efficient strategy to enhance the receiver performance.170 This design improvement resulted in a substantial 1.5-fold increase in the absorbed power, confirm-171 9 ing the theoretical advantage of SRRs in providing a stronger magnetic resonance and greater field172 concentration within the capacitive gaps where the CEBs are located.173 Our investigation of the scaling of the metamaterial array revealed a critical design trade-off. While174 reducing the dimensions of the SRR unit cells effectively broadens the operational bandwidth, it175 also reduces the total absorbed power if the array’s physical size becomes smaller than the Airy176 spot of the coupling lens. We successfully resolved this issue by implementing a strategy of simul-177 taneous miniaturization and increasing the array density. By halving the SRR dimensions and lat-178 tice period while nearly doubling the number of elements (from 19 to 37), we achieved an optimal179 compromise. The resulting receiver exhibits both enhanced absorption (by a factor of 1.4 larger180 than the original single-ring design) and an ultra-wide bandwidth spanning from 160 to 820 GHz.181 Furthermore, we quantified the non-linear relationship between the number of array elements and182 the absorbed power, showing that the benefit of adding elements is significantly higher for a minia-183 turized array that initially underfills the Airy spot. This provides a crucial practical guideline for184 designing efficient multi-absorber receivers, balancing performance gains against the increased185 technological complexity and noise considerations associated with a larger number of bolometers.186 This work solidifies the position of CEB-based SRR metamaterials as a highly promising platform187 for constructing ultra-broadband, high-sensitivity receivers essential for next-generation spectro-188 scopic and radioastronomical applications, particularly in demanding space and balloon-borne189 environments. Future work will focus on the experimental fabrication and characterization of the190 proposed miniaturized 37-element SRR array to validate these simulation results.191 Funding192 The work is supported by Russian Science Foundation Grant No. 21-79-20227.193 References194 1. 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