Ultra-sensitive passive wireless sensor exploiting high-order exceptional point for weakly coupling detection

Since the quantum concept of parity-time (PT) symmetry has been introduced into the conventional inductor–capacitor resonance, strategies based on exceptional points (EP) based strategies redefine our understanding of sensitivity limitation. This considerable enhancement of sensitivity originated in exploration of the non-Hermitian physics in photonics, acoustics and electronics, which exhibits a substantial application to the miniaturization of implanted electronic sensors in medicine field. By continuously accessing the EP, the spectral response of reader ∆ω follows a dependency of Δω ∼ κ 2/3 to a weakly coupling rate (|κ| ≈ 0), which may approach the theoretical limit of sensitivity in a second-order EP system. In this paper, we experimentally demonstrate a high-order (higher than second-order) PT symmetric system for weak coupling detection, in which a third-order EP can be employed to fulfill the sensitivity of Δω ∼ κ 1/2. Particularly, we introduce the incoming wave as an effective gain to balance the loss and obtain a pair of purely real eigenfrequencies. There are absence of imaginary parts despite corresponding real parts shifts dramatically by using a neutral resonator, without a broadening of the reflection spectrum so that maintaining a high resolution on the sensitivity. This work may reveal the physical mechanics of a small perturbation at a high-order EP and promote applications in implanted medicine devices.


Introduction
Wireless sensors (WS) can be used to continuously operate in harsh environments or human bodies where wired connections are unable to.The first compact passive WS was proposed by Collins in 1967, which utilized a spiral inductor (L) and a pressure-sensitive capacitor (C) to realize an intraocular pressure sensor implanting in the eye [1].Recently, WS based on LC-resonance have experienced a rapid expansion in the last few decades with many industrial and medical applications, including radio-frequency identification [2], ocular diagnostics [3] and physiological monitoring [4][5][6][7][8][9].Generally, the working mechanism of these WS is based on detecting concomitant resonance frequency shifts caused by the variation of L or C, which will be easily affected by external factors such as pressure [1,10,11], strain [4] and humidity [12].Besides, when a read-out coil is very close to a sensor coil, the change of the read-out coil spectral response Δω induced by the variation of coupling rate κ between them through inductive coupling will also emerge.However, limited by the depth and small dimensions of an implanted sensor, the coupling κ is unable to induce the resonant frequency shifts Δω to a critical value for measuring.A strategy to amplify the response Δω need to be proposed from a weakly coupled version to dramatically enhance the sensitivity of the reader.
In this paper, we experimentally demonstrate ultra-sensitive frequency response in passive WS for the small coupling rate (|κ| ≈ 0) near to a third-order EP (EP3) for weak coupling detection.Inspired by the EP2 scheme, we design a three-resonance system based on kHz inductor-capacitor (LC) circuit that emerges an EP3 with suitable separation between the resonators.Both theoretical and experimental results show that the spectral response of the reader system Δω near the EP3 follows a dependency of Δω ∼ κ 1/2 .Compared to a recently proposed EP2 scheme (figure 1(a)), in that the spectral sensitivity follows Δω ∼ κ 2/3 , EP3 scheme defines more competitive in practice.Using an additional detector coil as a sensor coil weakly coupled with the relay coil (neutral resonator) of the system, purely real eigenfrequencies shifts (red and blue shifts) away from EP3 can be clearly observed simultaneously (figure 1(b)).We also consider a higher-order EP scheme for WS and the theoretical prediction results show an enhancement of sensitivity with increasing the order of the strong coupled resonators in a sensor system.

Theoretical and experimental results
First, let us consider a sensing scheme based on a third-order PT symmetric reader system consisting of three coupled resonators (transmitter, relay and receiver coils), as shown in figure 1(a).With respect to an input port with forward wave s 1+ = S 1+ e −iωt and backward wave s 1− = −s 1+ + √ 2γa 1 connected to the transmitter coil, the coupled mode equations are given by [54,62,63]: where |a 1 | 2 , |a 2 | 2 and |a 3 | 2 correspond to the energy stored in transmitter, relay and receiver coils, respectively.ω 0 is the resonant frequency of the resonant coil.μ is normalized coupling rate between the nearest-neighboring coils.γ signifies the coupling rate from transmitter/receiver to port 1/2 (input/output port) in vector network analyzer (VNA).By putting zero backward wave s 1− = 0 into equation ( 1), an equivalent eigenfrequency problem in solving the perfect absorption (zero reflection) states can be described as follows [40,54,63]: where the effective Hamiltonian (H) is By solving |ωI − H| = 0, where I is an identical matrix, the corresponding eigenfrequencies are as follows: Equation (4) indicates that all three eigenfrequencies merge at ω 0 , providing the critical value condition of μ = √ 2γ/2 is satisfied, thus EP3 is obtained.When a detector coil (exotic resonator)is coupled with the relay coil of this system in EP3 condition, equation (1) can become: For analogical reasoning, we can predict that a spectral response of the reader system near EPn will follow the relationship ∼κ 2/(n+1) .The higher the order n, the higher the sensitivity of EPn scheme.
where |a x | 2 represents the energy stored in detector coil.The eigenfrequencies around the EP3 are derived from equation ( 4) as follows: The above eigenfrequencies can potentially mask the hypersensitive resonance shift, corresponding to a deep reflection (R) dip for reflection spectrums obtained by R = |s 1− /s 1+ |.
We also study an analogous non-Hermitian sensing scheme of reader system in EP2 condition (μ = γ), as is shown in figure 1(b).Similarly, the dependence of the purely real eigenfrequency on κ around this EP2 is Figures 1(c) and (d) plot the theoretical values of complex eigenfrequencies as a function of κ for the parameter setting μ = 1, respectively.It is clear that the third-order PT symmetric system demonstrates giant frequency splitting near EP3 (red and pink) compared with that with EP2 (orange).For comparison, we also present these results on the same figure.The Newton-Puiseux series expansion of equation ( 6) can be found as follows [28,54]: where O stands for higher-order term.Consequently, the dependence of the frequency shift is expected to approximately follow Based on equation ( 9), the sensitivity of this reader system can be expressed as Δω/κ = 2 1/4 (μ/κ) 1/2 .For μ = 1, this sensitivity becomes 2 1/4 /κ 1/2 , thus Δω/κ > 1 can be obtained when κ is very small, which means enhanced sensitivity.Besides, we consider a model where the coupling exists between the detector and both of the resonators in the EP3 system simultaneously, and the coupling between the detector coil and any coil in the EP3 system is supposed to κ.Using methods in reference [28,54], we can still derive equation ( 9), namely a spectral response of the reader near EP3 following ∼κ 1/2 .Under this circumstance, this EP3 WS still works normally.Similarly, the approximate result of equation ( 7) can be derived as follows: We find a good agreement between analytical (deep color solid line) and approximate (light color dashed line) solutions.The observed linear slope of 1/2 (yellow) and 2/3 (grey) in the corresponding logarithmic plot shown in figure 2 affirm the behavior of square-root and three-second power, respectively.Compared with equations ( 9) and (10), for μ = 1, we can also observe Δω/Δω = 1/(2 1/12 κ 1/6 ) > 1 with small κ.It means the frequency response of EP3 is more significant than that of EP2.Similarly, we can derive a spectral response of the reader near EP4 and EP5 following ∼κ 2/(4+1) (brillant blue) and ∼κ 2/(5+1) (light green), respectively.For analogical reasoning, we can predict that a spectral response of the reader near EPn will follow ∼κ 2/(n+1) .The use of higher order EP will further amplify the frequency response to coupling, leading to even greater sensitivity.
Figure 3(a) shows the schematics of the third-order PT symmetric WS system, corresponding sample photo is shown in figure 3(b).All coils are arranged coaxially towards.The transmitter, the relay, the receiver and the detector coils are equidistant resonant coils (LC tanks) composed of wire-wound inductors and film capacitors.They are made of identical Litz wire whose width is about 2.36 mm (0.078 mm × 400) and fixed on the polymethyl methacrylate hollow box.It has each side length of 12 cm, with thickness is 3.06 cm and empty center section outer diameter of 6 cm.The film capacitors have fixed values of about 4.7 nF, and they are loaded on each resonant coil in parallel with the adjustable capacitors ranging from 10-100 pF.By carefully tuning the adjustable capacitors, the three resonant coils have nearly the same resonance frequency at f 0 = ω 0 /2π = 113.3kHz.
Besides, the source and the load coils are non-resonant coil, made of identical Litz wire whose width is about 2.82 mm (0.07 mm × 714) and fixed on transmitter and receiver coils, respectively.They are connected to port 1 and port 2 of the VNA (Keysight E5071C), respectively.A harmonic wave at a frequency ω is coupled to the transmitter coil at a rate γ, transferred across relay coil before delivered to the load coil at the same rate γ.The transmitter-relay resonant coupling rate as well as the relay-receiver resonant coupling rate is μ.The following fitting parameters are set as γ = 8.00 kHz and μ = 5.65 kHz to satisfy EP3 condition of γ = √ 2μ.The relay coil is subjected to the coupling rate of κ come from one identically resonant coil.Particularly, we can consider the coupling rate κ to be zero with the reader system in EP3 condition.Figure 3(c) plot the calculated (dots) and experimental (lines) reflection coefficient versus frequency with different coupling rate κ for the proposed third-order PT symmetry.When the value of κ increases from zero to 0.1 kHz, the separation distance d between the reader system and the detector (center to center) reduces from about 25 cm to 18.2 cm.It is indeed that there is nearly zero reflection at f 0 at EP3 (κ = 0) with about −60 dB signal power.Below −60 dB, the noise and residual crosstalk contributions will be more significant relative to other sources of measurement errors.With the increasing of κ, the same magnitude of blue and red shifts of the frequency resonance (reflection dip) are observed simultaneously, corresponding to purely real parts of eigenfrequencies in figure 1(d), which coincides with the theoretical predictions.
We also present the results for the frequency shifts near EP3 and EP2 on the same figure, in which we have normalized Δf by selecting the ratio of Δf/Δf max .In this case, the comparison of frequency response plot between EP3 and EP2 scheme is significantly clear.As can be observed from figure 4, The observed linear slopes of 1/2 (red) and 2/3 (grey) in the logarithmic plot affirm square-root and three-second power behaviors of calculated (lines) and experimental (dots) results, respectively.Inset shows experimental (dots) and fitting (line) results of coupling κ as a function of coupling distance d, where the fitting result is κ = 1.86 exp(−0.16d).Using EP3, we can observe an enhanced sensitivity of at least 8 times with the frequency response to coupling, even up to about 15 times.EP3 offers a clear advantage as measured by larger splitting than that of EP2.The frequency response enhancement based on EP3 scheme reveals at least 1.8 times compared to that of EP2 scheme, even up to about 2.2 times.
For analogical reasoning, under ideal circumstance, when t independent resonators are coupled with the system in EP3 condition with coupling rate κ, the effective coupling rate is κ = √ tκ.Using methods in references [28,54], we can predict that a generalization of spectral response of the reader near EP3 will follow (2t) 1/4 (μκ) 1/2 .The application of such techniques to draw near the theoretical limits of sensitivity for weak coupling remains an important direction in the long term.

Conclusion
In conclusion, we have demonstrated ultra-sensitive sensing in passive WS system for weakly coupled detection in the condition of EP3.With regard to our theoretical models in third-order PT symmetry, when an external resonator is coupled to the neutral resonator, the reflection dip matches the purely real eigenfrequencies (red or blue shifts) of the system, not broadening the reflection spectrum and causing large noise.The frequency response is further amplified as Δω ∼ κ 1/2 near EP3, better performance than that with Δω ∼ κ 2/3 near EP2.The designed kHz LC system with balanced gain and loss verified our predictions experimentally and demonstrated that the eigenfrequency bifurcation close to EP3 amplifies more 1.8 times in the sensitivity than that using EP2.Further, the theoretical prediction results show considerable improvement of the sensitive property in a higher-order EP as the spectral response follows a dependency of Δω ∼ κ 2/(n+1) .We envision that such new higher-order PT symmetric systems will enable a superior sensing capability, which can also be extended to other microwave, millimeter wave and terahertz wireless systems.

Figure 1 .
Figure 1.Comparison of sensing between an EP2 scheme (a), (c) and an EP3 scheme (b), (d).Schematics diagrams of the third-order (a) and second-order (b) PT-symmetric reader systems consisting of transmitter, relay and receiver coils, which are analogous to electronic molecules with gain, neutral and loss properties, respectively.The inductive coupling between the nearest-neighboring coils is μ(s), where s is distance between the two coils.VNA inputs forward waves to transmitter through port 1 with coupling γ, so as the receiver and port 2 in VNA.(a) A detector coil is coupled weakly (|κ| ≈ 0) with transmitter and receiver coils of this system in EP2 (μ = γ) condition simultaneously.(b) A detector coil (yellow) is coupled weakly (|κ| ≈ 0) with the relay coil of this system in EP3 condition (μ = √ 2γ/2).Parameters to be detected can be obtained by monitoring the reflection coefficient of these reader systems.Analytical solutions for the real (top) and imaginary (bottom) parts of the eigenfrequencies near EP2 (c) and EP3 (d), respectively.It is clear that frequency response in reflection spectra and purely real eigenfrequencies splitting with EP3 scheme (red and green) is larger than that of EP2 (orange).

Figure 2 .
Figure 2. Comparison of sensitivities of reader systems.Analytical (deep color solid line) and approximate (light color dashed line) results for frequency response as a function of coupling κ near EP2, EP3, EP4 and EP5 on a logarithmic scale, with the slope of 2/3, 1/2, 2/5 and 1/3, respectively.Approximate results represent the corresponding series expansion truncated to the first order.For analogical reasoning, we can predict that a spectral response of the reader system near EPn will follow the relationship ∼κ 2/(n+1) .The higher the order n, the higher the sensitivity of EPn scheme.

Figure 3 .
Figure 3. (a) A schematic diagram of a wireless-sensor system based on third-order PT symmetry for implementing at EP3.It consists of transmitter, relay and receiver coils.A source coil is coupled to a transmitter coil at a rate γ, transferred across relay coil before being delivered to the load coil at the same rate γ.And the transmitter-relay as well as relay-receiver resonant coupling rate is μ.For γ = √ 2μ, all three eigenfrequencies coalesce at the resonant frequency and the system exhibits EP3.A resonant detector coil is coupled with the relay coil of the system in EP3 condition, with the coupling rate κ.(b) Photograph of our sample.(c) Measured (blue dots) and calculated (green lines) reflection spectra as a function of frequency for the proposed wireless sensor under different values of κ (or separation distance d) in the weak-coupling region.Note that a clear frequency splitting emerges with smaller κ (or larger d).

Figure 4 .
Figure 4. (a) Plots of calculated (solid line) and measured (dots) normalized frequency response Δf as a function of coupling rate κ near EP3 (red) and EP2 (grey), respectively.The corresponding separation distance d is shown in blue coordinate axis.The solid lines show the square-root and three-second power behavior, respectively.Inset shows experimental (dots) and fitting (line) results of coupling κ as a function of coupling distance d, where the fitting result is κ = 1.86 exp(−0.16d).