Non‐Hermitian Broadside Coupled Split Ring Resonators with Directional Sensitivity

Directional radiofrequency (RF) sensing, also known as radio direction finding, is essential in various applications that involve localizing RF sources. However, existing RF sensing technologies face challenges due to their large antenna sizes and complex RF signal processing circuits, which hinder the realization of miniaturized subwavelength direction sensors. In this article, a non‐Hermitian broadside coupled split ring resonators (BC‐SRRs) are presented, which can identify the direction of an incident RF signal with angles ranging from 0° to 180°. The non‐Hermitian Hamiltonian of the system, which can be interpreted by the temporal coupled mode theory, gives rise to asymmetric resonant modes of the BC‐SRRs. The asymmetry allows the direct measurement of RF wave incident angle by probing the resonant strength of the two BC‐SRRs. For the proof of concept, a one‐stage Dickson voltage multiplier is employed to rectify the RF signals and demonstrate angle sensing ability by relating the rectified voltage to the incident angle. The results showcase the potential of the proposed technique using non‐Hermitian BC‐SRRs as a pathway toward subwavelength antenna‐based radio direction finding.


Introduction
Radio direction finding (RDF) is a crucial radio frequency (RF) sensing technique used in various applications, such as telecommunications, [1] radar systems, [2] navigation, [3,4] healthcare, [5] and electronic warfare, [6] among others.Conventional RDF techniques enable the detection of signal angles with respect to a large receiving antenna or an array of antennas, [7,8] which is on the order of the wavelength.In recent years, the development of miniaturized antennas has gained significant attention due to the increasing demand for smaller and lightweight RDF devices that offer high accuracy and resolution. [9,10]By reducing the physical size of the antennas while maintaining or even enhancing their angle-sensing performance, miniaturization techniques enable the integration of RDF capabilities into small form factors, such as handheld devices, unmanned aerial vehicles (UAVs), microrobots, or microsystems.However, achieving miniaturization presents significant challenges.For instance, the high-efficiency electrically small antennas have been critical components in reducing the size of the RF receiving antenna array. [11,12]The mutual coupling between the closely packed elements cannot be eliminated by optimization and may deteriorate the performance of the whole system.[15] Moreover, it is impractical to achieve the complicated signal processing, such as obtaining phase differences between output signals, in the low-power miniaturized systems, or microsystems.
Seeking inspirations from nature, we found that the auditory system of animals provides insights into directional sensing.The large animals can detect and determine the direction of sounds based on the difference in arrival times between their ears, facilitated by the large distance between the ears. [16]In contrast, small animals with short ear-to-ear separations relative to the sound wavelength cannot directly perceive the intensity differences of sound waves between their ears. [17,18]However, certain small creatures, including geckos and flies, exhibit directional hearing through a coherent coupling of acoustic waves between the two closely spaced ears. [19,20]Thus, this natural phenomenon offers a design paradigm to achieve compact directional sensors with a deep subwavelength spacing.Biologically inspired directional sensors, consisting of two coupled components, have been reported in the fields of acoustics, [21] radio frequency, [22] and optics [23] for localizing emitting sources.These sensors greatly reduce system complexity compared to previously reported array structures, promoting their application in microsystems.
The fundamental physics of the coupling-mediated directional sensing lies in a generalized non-Hermitian theory, which offers a framework to study open systems. [24]For instance, non-Hermitian coupling plays a crucial role for introducing a constructive interference among different radiative channels to enhance the scattering. [25]By breaking the system symmetry through direct near-field interaction between resonators, non-Hermitian coupling enables the occurrence of unidirectional reflectionless transparency, [26] exceptional point enhanced sensing, [27] and non-Hermitian skin effects, [28] among others.Although actual coupling coefficients are generally complex matrices, non-Hermitian coupling can be achieved by carefully choosing the resonant modes and coupling factors at a subwavelength scale. [29]Consequently, within a deep subwavelength separation, the coupled resonators can be individually excited from the far field using non-Hermitian coupling, resulting in a significant difference in resonant magnitudes between the two resonators.
In this article, we proposed a subwavelength RF directional sensor comprising broadside coupled split ring resonators (BC-SRRs) capable of detecting the incident angle of electromagnetic (EM) waves.Using the temporal coupled mode theory (TCMT), we theoretically demonstrated that the interference of scattered waves from the BC-SRRs was close to non-Hermitian coupling at a critical separation (< 0.1 0 ) between the two BC-SRRs.The giant difference of transmission coefficients in resonant frequency originated from this non-Hermitian coupling, which helped to improve the angle sensing ability.Experimental validation confirms that the BC-SRRs in conjunction with the voltage multiplier exhibit sensitive response for incident angle of RF signal.By simply calculating the difference between two output voltages, the incident angle of RF signal can be obtained directly without the need for complicated signal processing circuitry.This work provides an insightful understanding of non-Hermitian coupled resonance phenomena, and presents a practical and effective solution for directional RF sensing solution using the coupled subwavelength resonators.

Design of the Non-Hermitian Coupled System
The proposed system is described in Figure 1a, in which the transmitter is considered as a resonator a 1 , and the detector comprises of a 2 and a 3 , respectively.Within the framework of tempo-ral coupled mode theory (TCMT), the response of the system may be modeled by in which a i is the oscillation strength of each resonator (i denotes the number of the resonator, and i = 1, 2, or 3), H 0 is the usual Hamiltonian for the independent resonators, H i represents the coupling between resonators, and Κ is the coupling matrix between the resonator and each port.The Hamiltonians and coupling may be expressed by in which,  0i ,  ni , and  ri represent the resonance frequency, intrinsic loss, and radiative loss of each resonator, respectively, k im (i, m = 1, 2, 3, and i ≠ m) is the coupling strength between resonators, k ii is the coupling strength between the resonator to the corresponding port.In the case of directional RF sensing, the excitation is only fed through Port 1, i.e., s 1+ ≠ 0, while s 2+ , s 3+ = 0. Due to the reciprocity of this linear and time-invariant RF antenna system, the couplings between the ports inside the same finite region are equal, and thus k im is equal to k mi . [30]The resonator a 1 emanates RF energy and excites resonance in a 2 and a 3 through k 21 and k 31 .The coupling strength between a 2 and a 3 may be calculated from the mutual coupling (including inductive coupling and capacitive coupling) factor μ by k 23 = k 32 = μ 0 /2.By solving the equation in the frequency domain, we may obtain the oscillation strength of each resonator with various frequency detuning, i.e.,  = (- 0 )/ 0 .When there is a phase difference between k 21 and k 31 , the coupling Hamiltonian becomes non-Hermitian, leading to a response with the breaking symmetry.
When separation between the a 2 and a 3 is on a deep subwavelength scale, the magnitudes of k 21 and k 31 are expected to be equal. [23]However, due to the distinct locations and configurations of the two coupled resonators, a phase difference can arise between k 21 and k 31 , resulting in distinctive oscillation strengths through the coupling, as shown in Figure 1b.When the coupling between resonators 2 and 3 are weak (e.g., μ = 0.03), resonant responses of a 1 and a 2 are identical when there is no phase difference between k 21 and k 31 , i.e., Δ =  (k 21 )−  (k 31 ) = 0.As Δ increases, signifying an out-of-phase relationship between the resonant responses of a 2 and a 3 , non-trivial distinctions between the resonance behavior emerge.At specific frequencies, one resonator is with high oscillation strength (in a bright mode) while the other is with minimum oscillation strength (in a dark mode).Notably, when k 21 and k 31 are orthogonal (i.e., Δ = 90°), one resonator exhibits strong resonance while the other resonator is diminished at the original resonant frequency ( 0 ).
To demonstrate the non-Hermitian topology, we begin by examining the eigen values of the Hamiltonian (H i ) for a system composed of three resonators.As shown in Figure 1c, the real part of the complex eigen value varies along with the coupling factor (μ) and phase difference (Δ).The middle plane corresponds to the mode mainly dominated by a 1 , while the upper and lower planes are mainly governed by the coupling of a 2 and a 3 .The modes converge to a third-order exceptional point (EP3) at (μ = 0.035, Δ = 45°), revealed by the coalesce of the real and imaginary parts of the eigen values, [24,[31][32][33] as shown in Figure S1 (Supporting Information).Moreover, we evaluate the transmission coefficients (S 21 and S 31 ) with the TCMT model.The relationship between the EP3 and transmission coefficients has been derived in Note S2 (Supporting Information).In the case of  = 0, |S 21 | and |S 31 | varied differently with the changes of μ and Δ, as shown in Figure 1d.|S 31 | approaches 0 while |S 21 | remains a high level when the phase difference is set to Δ = 90°, and the coupling factor is μ = 0.05.This behavior indicates a significant suppression of transmission to resonator a 3 , while resonator a 2 maintains strong transmission characteristics.
The contrast between the bright and dark resonant strengths for a specific Δ is strongly influenced by the coupling factor in the case of the orthogonally coupled resonators, as shown in Figure 1e,f.The results demonstrate frequency splitting occurring in both resonators under strongly coupled conditions (when μ is >0.06).As the coupling factor decreases, |S 31  A commonly used approach to implement the coupled system is through the utilization of broadside coupled split ring resonators (BC-SRRs). [34,35]Each BC-SRR consists of two square rings with a slit, separated by a dielectric substrate, as shown in Figure 2a.The two identical square rings are placed in parallel, and their slits are rotated 180°with respect to each other.When the magnetic field component of the incident EM wave crosses through the SRR along the z-axis, the surface currents will be excited in the rings, and electric field enhancement appears at the slits (Figure S1, Supporting Information).The electromagnetic properties of the BC-SRR, such as resonant frequency and reflection coefficient, are related to its geometric sizes.More detailed information regarding the geometry of the BC-SRR can be found in the Figure S2 (Supporting Information).
The relative position of in a pair of BC-SRRs can be described by the relative angle () and the separation (d).The coupling effects within BC-SRRs arise from capacitive and inductive coupling, resulting in the mutual coupling factor (μ) and the mode splitting.The calculation of the coupling factor involves analyzing the charge and current distributions in the SRRs.The phase difference between the SRRs is related to the relative rotation angle, as shown in Figure 2b.Specially, the phase difference is 90°f or  = 0°, while it is 0°for  = 180°, which aligns with the expected phase response of loop antennas based on theoretical models. [30]umerical simulations were conducted to investigate the effects of separation (d) on the transmission coefficients in BC-SRRs (Figure S3, Supporting Information).The phase difference (Δ) of two SRRs is set to be 90°in the  = 0°configuration.The calculated results based on simulations of varied d are shown in the colormaps in Figure 2c (S 21 ) and Figure 2d (S 31 ), respectively.These transmission spectra exhibit significant different variations for the d ranging from 7 to 16 mm.As the separation (d) increases, the coupling factor (μ) decreases.When d is <11 mm, the two BC-SRRs are strongly coupled, and the nearfield interaction between Port 2 and Port 3 will be enhanced.In the strong coupling regime, the eigenmodes of the BC-SRR split to a pair of hybrid electromagnetic transmission resonance peaks. [36]The electromechanically induced transparency appears in both RF resonators, resulting in frequency splitting.
At a critical coupling separation (d = 11 mm, which is ≈0.1 of the resonating wavelength), the S 31 response reaches a minimum of ≈−75 dB at the resonant frequency, while the S 21 has a magnitude of ≈−25 dB.When d exceeds 11 mm, the two BC-SRRs are weakly coupled, frequency splitting diminishes in both S 21 and S 31 .Thus, the proposed TCMT (Figure 1e,f) can be employed to explain the influence of separation (d) on the transmission coefficient of the non-Hermitian BC-SRRs.Additionally, a critical separation can be chosen to achieve the maximum difference in transmission coefficients (|ΔT| = |S 21 -S 31 |).
In addition, numerical simulations were also conducted to study the EM responses of the BC-SRRs-based RF directional sensor for different incident angles.The simulated EM spectra of S 21 and S 31 exhibit a nearly symmetric characteristic with reverse changes corresponding to the incident angle (), as shown in Figure 3a and Figure 3b.To quantify the difference between S 21 and S 31 , the parameter ΔT was calculated, as shown in Figure 3c for  = 0°, 30°, 60°, 90°, 120°, 150°, and 180°.As  varies from 0°t o 180°, ΔT decreases from the largest positive value to the smallest negative value at the operating frequency.When  is set to be 90°, ΔT remains below 2 dB in the range from 2.3 to 2.5 GHz.In Figure 3d, the simulated electric field distribution of the BC-SRRs is presented for different incident angles.The bright white color indicates strong enhancement of the electric field confined at the slit of SRRs.At an incidence angle of  = 0°, the electric field at Port 2 is stronger than that at Port 3, resulting in a higher value of S 21 compared to S 31 .As  increases, the electric field strength at Port 2 (Port 3) change from strong (weak) to weak (strong).At  = 90°, the electric field at Port 2 and Port 3 becomes comparable, leading to similar values of S 21 and S 31 .Consequently, the transmission coefficient of the two ports shows a strong angular dependence, which holds promise for directional RF sensing applications.

Experimental Demonstration
To assess the transmission coefficient difference (ΔT) between the transmitter antenna and BC-SRRs, we set up an experimental configuration as depicted in Figure 4a.The setup using a commercial RF antenna (transmitter) connected to the output port of a vector network analyzer (VNA, Agilent E8361A) via a co-axial cable.With the VNA, we had the flexibility to adjust the   operating frequency and power of the RF signals.The fabrication of the BC-SRRs was accomplished by using the standard printed circuit board (PCB) process, and the SMA connectors were employed for interconnection at the output ports (Port 2 and Port 3).The detailed geometric parameters for the BC-SRRs are included in the Figure S2 and Table S1 (Supporting Information).For a comprehensive analysis, we measured both the reflection and transmission coefficients of the fabricated BC-SRRs, as shown in Figures S4 and S5 (Supporting Information).It should be noted that the spectra of the two BC-SRRS are slightly different due to manufacturing deviations.
The critical coupling of two BC-SRRs was experimentally determined by measuring the transmission coefficients for varied separation distances.The separation (d) of the two BC-SRRs was determined by the different lengths of nylon hexagonal columns (See Note S9, Supporting Information).The BC-SRRs were placed at a 10 cm from the transmitter antenna.At a separation distance (d) of 10 mm, the transmission coefficient difference |ΔT| reaches 26.25 dB at a resonant frequency of 2.16 GHz, as shown in Figure 4b.The simulation results (dashed lines) and experimental results (symbols) for S 21 and S 31 exhibit a high degree of agreement.The slight deviation of the resonant frequency and transmission coefficient mainly related to the fabrication error.To further investigate the influence of the separation distance, we characterize the transmission coefficients with varied d.The measured S 21 and S 31 curves for five different separation distances are shown in Figure S6 (Supporting Information).Figure 4c demonstrates the relationship between ΔT and d.According to the experimental results, when the separation is small (e.g., 5 mm) or large (e.g., 20 mm), ΔT is below 3 dB at the resonant frequency with limited angle sensitivity.Therefore, an optimal separation distance (10 mm) should be chosen to achieve a large ΔT.
The experimental setup shown in Figure 4a was also employed to measure transmission coefficients (S 21 and S 31 ) of different incident angles.The BC-SRRs with a separation distance of 10 mm were mounted on the center of a rotation stage, while maintaining a fixed distance (D) between the transmitter antenna and BC-SRRs.The incident angle  is controlled using the rotation stage.To enable the extraction of DC voltage from the RF signal, we introduced a single-stage Dickson voltage multiplier to the output port of the BC-SRRs, as shown in Figure 6a.For the efficient rectification, we utilized Schottky diodes (SMS7630-061) due to their low threshold voltage (0.34 V).The capacitances of C 1 and C 2 are 100 nF and 1 nF, respectively.The rectified DC voltages from the two BC-SRRs were then measured using a digital oscilloscope (Tektronix TBS2000B).The incident RF signal power was set to 8 dBm at 2.16 GHz.Then, we examined the behavior of the rectified voltages with respect to the incident angle .As  increases, the rectified voltage of BC-SRR1 (V 1 ) decreases while that of BC-SRR2 (V 2 ) increases, as shown in Figure 6b.The maximum output voltage of each BC-SRR is up to be ≈250 mV.At  = 90°, both BC-SRRs yield similar DC outputs, measuring ≈60 mV.The slight deviation between the measurement results of the two BC-SRRs are attributed to inherent fabrication inconsistencies and assembly error, including difference in size parameters between the two BC-SRRs and misalignment of two PCB antennas.By reducing machining and assembly errors, the performance of this RF directional sensor can be further optimized.
To demonstrate angle sensing, the rectified voltages from two BC-SRRs of different incident angles were normalized to the peak value (V max ).Next, we calculated the difference between the two normalized voltages, i.e., ΔV = (V 1 -V 2 )/V max , as plotted in Figure 6c.Note that the normalized voltage difference drops as the incident angle  increases.The good agreement proves that the proposed angle sensing method is robust despite the discrepancy observed in Figure 6b.The relationship between ΔV and  is not ideally linear, and the sensing sensitivity defined by |Δ(ΔV)/Δ| throughout all the angles is not constant.The sensitivity is higher than 1.33 (mV/°) over the entire angles except the angle range of 135°to 157.5°.The angle sensing sensitivity can be further improved by increasing the rectified voltages and using low noise amplifier.

Conclusion
In summary, we have developed a direction-sensing scheme of RF sources by employing the BC-SRRs.By analyzing the coupling behavior of this non-Hermitian coupled system (Δ = 90°) based on the TCMT theory, we uncovered distinct spectral responses of the two resonators when they were close to critical coupling (d = 10 mm).The measured transmission coefficient difference |ΔT| exhibited a remarkable maximum value of 26.25 dB at the resonant frequency.Furthermore, the ΔT gradually decreased from 27.7 to −26.1 dB as the incident angle () increased.This implies that the proposed coupled system can effectively sense the angle of an incident RF signal within a range of 0°to 180°, which has the potential applications including microrobot positioning and automatic navigation, robotic arm controlling, etc.
Our design provides a promising platform for realizing compact RF directional sensor, which has a deep subwavelength side length (< 0.1  0 ), surpassing previously reported RF directional sensing systems.Each BC-SRR can directly convert the RF signal into DC voltage through the integration of rectifying circuits.Our experimental demonstration has showcased the multifunctionality of the BC-SRRs-based microsystems, encompassing directional sensing and RF energy harvesting capabilities.In future design, we envision to enhance the output voltage of BC-SRRs by increasing the maximum power of transmitting antenna and incorporating additional stages in the rectifier circuits.Thus, a higher output voltage can be obtained under the condition of the same incident angle of RF signal, and the sensitivity of the RF directional sensor can be improved.Our demonstration paves the way for a battery-less and high-performance RF-powering microsystem with exceptional directional sensing capabilities.Furthermore, by leveraging appropriate materials for the spacer in each BC-SRR, such as high permittivity and low-loss dielectrics, it is plausible to shrink the size of this platform while maintaining its functionality.By optimizing the RF directional sensor structure and constructing array topology, the detection in three dimensions will be realized, thereby expanding its potential for various applications such as micro-robotics. [37,38]

Experimental Section
Simulation: CST Studio Suite was employed to numerically model the response of the BC-SRRs detector.In the simulation, the transmitter antenna was an electrical dipole antenna, which was modeled as a single thin perfect electric conducting (PEC) cylinder with a gap.The BC-SRRs are copper split ring resonators separated by FR4 substrate.The conductivity of copper is 5.8×10 5 S cm −1 , and the relative dielectric constant of FR4 is 4.3 + j0.11.Geometry of the structure may be found in the Supporting Information.Perfect match layers were employed at the bounds of the model.
Measurement: The S-parameters of the coupled system were characterized using a vector network analyzer (VNA, Agilent E8361A).A commercial RF antenna was connected to one port of the VNA, and the BC-SRRs were connected to the other port of the VNA via co-axial cables.The S-parameters were measured for varied separation distances between the BC-SRRs without rectifiers.After characterization of the S-parameters, the BC-SRRs were connected to a one-stage Dickson voltage multiplier, and the rectified voltage was measured using an oscilloscope (Tektronix TBS2204B).The source frequency was fixed at 2.16 GHz and the output power was 8 dBm.

Figure 1 .
Figure 1.The operation principle of the non-Hermitian coupled resonators for RF directional detector.a) The schematic of the non-Hermitian coupled system design, featuring a transmitter a 1 and two coupled receivers, i.e., a 2 and a 3 .b) Theoretical oscillation strength |a 2 | and |a 3 | as a function of the phase difference between k 21 and k 31 (Δ) with the coupling factor (μ) set to 0.03.c) Real part of the eigenvalue of the Hamiltonian matrix.d) Theoretical transmission coefficients (|S 21 | and |S 31 |) for varied μ and Δ.(e) and (f) are the |S 21 | and |S 31 | spectra for varied coupling factors.e,f) are transmission coefficients |S 21 | and |S 31 | as a function of the coupling factor with a Δ of 90°.

Figure 2 .
Figure 2. Simulation of the broadside coupled split ring resonators (BC-SRRs) detector.a) Schematic of the coupled system, including the transmitter antenna and BC-SRRs detector.b) Implementation of the phase difference (Δ) using the rotation of BC-SRRs ().c) S 21 and d) S 31 for varied separations between the BC-SRRs obtained by the numerical simulation.

Figure 4 .
Figure 4. Characterization of the critical coupling coefficient.a) Configuration of the measurement setup consisting of the transmitter antenna and BC-SRRs in an anechoic chamber.b) Simulated (dashed lines) and measured (symbols) results of S 21 and S 31 for d = 10 mm.c) Measured transmission coefficient difference (ΔT) under resonance frequencies for varied d.The average length of the error bars is 3 dB.
Figure 5a-e show the measured transmission coefficient (S 21 and S 31 ) for  = 0°, 45°, 90°, 135°, and 180°.As the incident angle increases, the S 21 decreases from −40.2 to −64.3 dB, while the S 31 increases from −66.8 to −37.4 dB at the resonant frequency.At  = 90°, the curves of S 21 and S 31 coincide within the frequency band of interest (gray area), and ΔT is ≈0 dB. Figure 5f illustrates the transmission coefficient difference (ΔT) as a function of the incident angle .The measured ΔT at 2.16 GHz decreases from 27.7 to −26.1 dB as  increases from 0°to 180°.The experimental results agree well with the numerical results.Therefore, the incident angle () of RF signal can be obtained by calculating ΔT.

Figure 6 .
Figure 6.Demonstration of direction sensing using the rectified voltage.a) Schematic of the BC-SRRs with a single-stage Dickson voltage multiplier.b) Measured rectified voltages of the two BC-SRRs for different incident angles (Δ = 22.5°).c) Normalized voltage difference for different incident angles.