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Abstract
A noteworthy challenge in actual wireless sensors is the intrinsic sensing resolution and the sensitivity associated with the response to external perturbation to be measured. To address the issue, we report the realization of enhanced sensitivity in a passive wireless sensing system, consisting of three coupled passive resonators. The input wave is exploited as an effective gain in our open system, thus the ideal parity-time (PT) symmetry can be established, rather than balancing real gain and loss. Then the third-order exceptional points are obtained in ternary PT symmetric systems. With the extrinsic perturbation imposed on any one of resonators, we demonstrate analytically and experimentally that the resonance response of the system follows the cube-root dependence on perturbation. Making use of the effective gain, our results pave a new way, to the best of our knowledge, to realize the ultra-sensitivity of a passive wireless sensing system.
© 2019 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
1. Introduction
Taking the advantage of small size, long lifetime and no physical connection, the passive wireless sensor is essential to many industrial and medical applications [1] including intraocular eye pressure testing [2] in medical sensing, tire pressure monitoring [3] in automotive applications and pressure sensing at high temperature [4]. Recently, inductor–capacitor (LC) passive wireless sensors [5] have been proposed for pressure [6], strain [7], temperature [8], humidity [9], biochemical [10], gas [11] and so on. Typically, the capacitor changes in response to the parameter of interest, resulting in a shift in resonant frequency. The resonant frequency of the sensor is detected by measuring the input return loss (reflection) of an off-resonance readout coil, which is magnetically coupled with the sensor. Although there has been continuous progress in passive wireless sensors with the development of micro-electro-mechanical-system (MEMS) technology [1] in the past three decades, the basics of the readout technique remain essentially unchanged since its invention in 1967 [5]. Improving performance is often hindered by the available levels of Q-factor, the sensing resolution, and particularly the sensitivity related to the spectral shift of resonance in response to variations of quantities to be measured [12–14].
Recently, the physics of non-Hermitian degeneracies, also known as exceptional points in parity-time (PT) systems have been revealed to have great potential for enhancement of sensitivity [15–19]. Exceptional points are the points in parameter space at which the eigenmodes of PT system merge with each other [20,21], namely the eigenvalues and the eigenvectors coalesce simultaneously. A range of extraordinary phenomena in photonics associated with exceptional points have been proposed and demonstrated [22], such as loss-induced transparency [23], power oscillations [24–26], unidirectional invisibility [27–29], PT-symmetric lasers [30–32], and topological chirality [33,34]. Especially, it has been shown that at the exceptional points the eigenvalue bifurcation properties can greatly amplify the extrinsic perturbation effect, namely the sensitivity of resonance frequencies of optical structures to external perturbations can be enhanced [15]. This sensitivity enhancement has been attributed to the nth root topology of the eigenvalue surfaces in parameter space [18]. Exceptional points in PT systems can be tailored by carefully tuning the balanced loss and gain, as well as the coupling between the resonators [35]. The square root behavior of the eigenfrequency splitting dependent on external perturbations has been experimentally observed at the second-order exceptional points with PT optical resonators [19]. Very recently, the cube-root dependence has also been demonstrated at the high-order exceptional points in an optical coupled micro-resonator sensor with PT symmetry [18]. The high-order exceptional points (greater than second order) which further amplify the effect of perturbations can lead to even greater sensitivity.
In this paper, we investigate the third-order exceptional points in a passive wireless sensing system with PT symmetry, and experimentally observe the enhancement of sensitivity (the sensor’s resonance frequencies to external perturbations) in the vicinity of exceptional points in radio frequency regime. Different from the previous experiments in optics where the real physical gain is used to construct PT systems [18,19], here a passive but ideal PT systems (without need of an additional gauge transformation of the states or a mathematical biasing in the diagonal terms of the Hamiltonian) is adopted [36,37], at price of sacrificing the improvement of resolvability from the physical gain mediated ultra-high extrinsic Q-factor [12]. We note that the enhanced wireless sensor telemetry has been proposed with a generalized PT theory, and the cases with and without the physical gain have been compared [12]. We emphasize our work is different. We focus on the sensitivity in a parameter space very close to the exceptional points, which is not discussed in [12] on one hand; on the other hand, the exceptional points we treated are third-order rather than second-order. Our results demonstrate the intriguing dynamics of exceptional points in the ternary system without any active components, and may benefit the development of passive wireless sensor with the ultra-sensitivity.
2. Theoretical and experimental results
The theoretical model of our ternary PT system is schematically shown in Figs. 1(a) and 1(b). The model is composed of linearly arranged three passive resonators with the same resonance frequency ${\omega _0}$ and the same intrinsic loss $\Gamma $. The two side resonators (denoted by the red and the green sphere) are subjected to the same amount of radiative loss $\gamma$ to the external two channels, respectively. One of them (here the red one) is stimulated by an input wave from its only external channel at the frequency $\omega$. The intermediate resonator can not couple to any external channel. Consider the symmetric nearest-neighbor coupling $\kappa$ only. The dynamic equations for the three resonance modes can be described by the coupled-mode theory [38–39] as
Fig. 1. (a-b) Schemes and principle illustration of the ternary PT system with extrinsic perturbation $\varepsilon$ imposed on (a) grey and (b) green resonators, respectively. The effective gain, neutral and loss resonators with the same frequency ${\omega _0}$ are shown in red, grey and green spheres, respectively, and $\Gamma $ is the intrinsic loss of each resonator. The green and the red resonators are subjected to the same radiative loss $\gamma$. Coupling strength between the neighboring resonators is denoted by $\kappa$. (c-d) Calculated reflection spectra for the two cases (a) and (b), with different perturbations. Purple, mazarine, skyblue, cyan and claybank lines represent perturbations ${\varepsilon _0} = 0$, ${\varepsilon _0} = 0.001$, ${\varepsilon _0} = 0.01$ and ${\varepsilon _0} = 0.1$, respectively. (e-f) Calculated real (Re) and imaginary (Im) parts of the eigenfrequencies for the two cases (a) and (b) as a function of ${\varepsilon _0}$, respectively. The skyblue, reseda and orange lines in (e)–(f) describe three eigenfrequencies ${\omega _n}$ and ${\omega ^{\prime}_n}$ (n = 1, 2, 3) for the cases of (a)-(b), respectively.
To comprehend how a small resonance shift in the neutral or the loss resonators affects the detectable reflection spectrum, we assume a small value of perturbation $\varepsilon$. Then Eq. (1) becomes as follows:
Consider the following normalized parameters: $\Gamma = 0$, $\gamma = {\gamma _0} = \sqrt 2$, $\kappa = {\kappa _0} = 1$. Figures 1(c) and 1(d) show the reflection spectra for the ternary systems, and the normalized perturbation strength ${\varepsilon _0}$ (${\varepsilon _0} = \varepsilon /\kappa$) is varied from 0 to 0.4 (0, 0.001, 0.01, 0.1 and 0.4).
Here a very deep reflection dip is found for each spectrum, which corresponds to the eigenmode whose imaginary part of eigenfrequency is minimum. We note that the eigenfrequency needs to be real in order to have perfect absorption occur. Therefore, the non-zero perturbations will lead to the formation of multiple eigenmodes, corresponding to multiple poles on the reflection spectra. Among them the modes whose complex eigenfrequencies has the smallest imaginary part can be used for sensing, because it corresponds to a very deep reflection dip. The complex eigenfrequencies as a function of perturbation are plotted in Figs. 1(e) and 1(f), where the sensing modes form the blue branch (stars are added to guide the eye). It is noted the dip frequencies are blue and red shifted in two cases, which can be explained by the opposite evolution direction of the real parts of the eigenfrequencies of the sensing modes.
Using methods in [18], our calculations show that for the perturbations on the neutral resonator, the real part shift of the eigenfrequency of the sensing mode (frequencies of reflection dip) is approximately Re(${\omega _1}$)=${\omega _0} + {2^{1/3}}{\kappa ^{2/3}}{\varepsilon ^{1/3}}$, and the imaginary part is always zero. For the perturbations on the loss resonator, the real parts shift is approximately Re(${\omega ^{\prime}_1}$)=${\omega _0} - {\kappa ^{2/3}}{\varepsilon ^{1/3}}$, and the corresponding imaginary part is Im(${\omega ^{\prime}_1}$)=$\sqrt 2 /3{\kappa ^{1/3}}{\varepsilon ^{2/3}}$. As are shown in Figs. 2(a) and 2(b), when ${\varepsilon _0}$ acts on the neutral resonator, the dip frequency shifts $|\Delta \omega |$ ($\Delta \omega = {\omega _{dip}} - {\omega _0}$) are completely consistent with Re(${\omega _1}$). But with the case of ${\varepsilon _0} > 0.08$ in the loss resonator, the shifts $|\Delta \omega ^{\prime}|$ ($\Delta \omega ^{\prime} = {\omega ^{\prime}_{dip}} - {\omega _0}$) are deviated clearly from Re(${\omega ^{\prime}_1}$) for Im(${\omega ^{\prime}_1}$) increasing. The observed linear slope of 1/3 (blue and green solid lines) clearly demonstrates a cube-root behavior on a logarithmic scale, as shown in Fig. 2(b), confirming the existence of third-order exceptional point. However, it is noted that the coefficients are different. Perturbations on the neutral resonator offer higher sensitivity than those on the loss resonator. This difference may provide another point of view to think about anisotropic EPs. [43] When ${\varepsilon _0}$ is large (${\varepsilon _0}$>0.08), the trends of the response deviate obviously from the cube-root attributed to high-order clearly terms [44].
Fig. 2. (a) Calculated frequency shifts of reflection dips compared with ${\omega _0}$ as a function of perturbation strength ${\varepsilon _0}$. (b) The results from (a) on a logarithmic coordinate. The blue and green solid lines follow the lines of ${y_1} = {2^{1/3}}\varepsilon _0^{1/3}$ and ${y_2} = \varepsilon _0^{1/3}$, respectively, showing a slope of 1/3. The orange and black solid lines are Re(${\omega _1}$) and Re(${\omega ^{\prime}_1}$) from Figs. 1(e)–(f), respectively. The orange and black marks denote the dip frequency shift due to perturbation operated on the grey and green spheres, respectively.
To make evidence of the effect of perturbation, we carry out the experiments with resonant coils, to simulation a passive wireless sensing system. Figures 3(a) and 3(b) show the schematics of the ternary system with perturbations acted on neutral and loss resonators, respectively.
Fig. 3. (a)-(b) Schematics of PT symmetric passive wireless sensing system based on three resonant coils (transmitter, relay and receiver coils), where Γ is the intrinsic loss rate of the resonant coil, and $\varepsilon$ is the extrinsic perturbation imposed on the relay (a) and the receiver (b) coils, respectively. (c) Photograph of our sample.
Figure 3(c) gives a photograph of our experimental setup. The transmitter, the relay and the receiver 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 0.95 mm (0.1mm×50) and fixed on the polymethyl methacrylate (PMMA) hollow box. It has each side length of a = 12 cm, with thickness is b = 3.1 cm and empty center section side length of c = 7.6 cm. The film capacitors have fixed values of about 1 nF, and they are loaded on each resonant coil in parallel with the adjustable capacitors. By carefully tuning the adjustable capacitors, the three resonant coils have nearly the same resonance frequencies at ${f_0} = {\omega _0}/2\pi = 144.5$ kHz, which regime is widely used in industrial electronics. In addition, the small perturbation $\varepsilon$ can be introduced by changing the adjustable capacitors simulating the variation in capacitance versus parameters of interest in an actual sensor. In addition, source and load coils, which are made of Litz wire whose width is about 2.87 mm (0.1mm×500), are identical off-resonant coils, for simplicity. They are mirrored on both sides (fixed on the prepared transmitter and receiver coils), and connected to port 1 and port 2 of the vector network analyzer (Keysight E5071C), respectively.
We first investigate the effect of perturbations for a conventional passive wireless sensor using single LC tank. By properly arranging a non-resonant and a resonant coil (here we use the source and the transmitter coils in experiments), we measure the reflection of this conventional sensing system with different perturbations, as shown in Fig. 4(a). The gray dashed lines are corresponding calculations considering intrinsic loss $\Gamma = 0.3$ kHz, which agree with the measurements well. Figure 4(b) presents the extracted resonant frequency as a function of different capacitance values, which is in agreement with the theoretical calculated resonance frequencies from the following equation:
where L is inductance provided by wound coils with $L = 113.58$ mH, and ${C_0}$ is fixed capacitance value with ${C_0} = 1059.1$ pF. Here the adjustable capacitor ${C_1}$ is the tuned from 4.5 to 9 pF, which leads to the small perturbations $\varepsilon$ about from 0 to 0.4 kHz acted on the resonant frequency.Fig. 4. (a) Reflection spectra of a single passive resonant coil with different perturbations. (b) Resonance frequency shifts of the passive resonant coil compared with f0 as a function of capacitance C1. (c-d) Reflection spectra with different perturbations imposed on (c) relay and (d) receiver coils, respectively. The solid and dashed lines denote experimental data and theoretical results, respectively. (e-f) Calculated real parts of the eigenfrequencies. The skyblue and orange lines in (e) donate describe three eigenfrequencies ${f_n}$ and ${f^{\prime}_n}$ (n = 1, 2, 3) for the cases of Figs. 3(a)–(b), respectively.
Then we investigate the ternary system with the measured perturbation strength. The experimental reflection spectrum with imposed perturbations $\varepsilon$ on the relay and the receiver coils are shown in Figs. 4(c) and 4(d), respectively. It is indeed that there is nearly zero reflection at ${f_0}$ at the exceptional point ($\varepsilon = 0$). Corresponding calculations (gray dashed lines) which are consistent with measurements very well are also given, with the following fitting parameters: $\gamma$ = 8.1 kHz, $\kappa$ = 5.727 kHz and $\Gamma $ = 0.3 kHz. Obviously, the critical value condition of $\gamma = \sqrt {2\kappa }$ is satisfied. Hence the third-order exceptional point is realized in this ternary system with the distance between each of two resonant coils about 8.4 cm experimentally. With the increasing of perturbation, blue and red shifts of the resonance (reflection dip) are observed respectively in Figs. 4(c) and 4(d) corresponding to real prats of eigenfrequencies form the blue branch (stars are added to guide the eye) in Figs. 4(e)-(f), respectively, which coincides with the theoretical predictions. From sensing point of view, the opposite shift direction of these two cases may provide another degree of freedom to sensing the position of the tagged objects.
In order to quantitatively reveal the effects of perturbations on the resonance shift of the ternary system, we recall the absolute values of the resonance shifts in Figs. 5(a)-(b). As shown in Fig. 5(a), When $\varepsilon$ acts on the neutral resonator, the dip frequency shifts $|\Delta f|$ ($\Delta f = {f_{dip}} - {f_0}$) are consistent with Re(${f_1} - {f_0}$). Nevertheless the dip frequency shifts $|\Delta f^{\prime}|$ ($\Delta f^{\prime} = {f^{\prime}_{dip}} - {f_0}$) are deviated a bit from Re(${f^{\prime}_1} - {f_0}$) with ε acted on the loss resonator. With ε<0.03 kHz, there will be a bit deviation from the calculations in our experiments due to the existence of intrinsic loss in resonant coil. The measured resonance shifts reveal cube-root behavior as a function of $\varepsilon$, which is identified by plotting the experimental data on a logarithmic scale in Fig. 5(b), from which we directly conclude a slope of 1/3 in comparison with the grey solid line. Please be noted that in Fig. 2(b), the perturbations ranges from 0 to $0.4\kappa$ so the deviation is clearly seen due to high order terms. However, in Fig. 5(b), we only experimented with very small perturbations ($\varepsilon < 0.1\kappa$). So there are almost no deviations of experimental results shown in Fig. 5(b).
Fig. 5. (a) Experimental frequency shifts of reflection dips compared with ${f_0}$ of the PT symmetry wireless sensor system as a function of perturbations strength $\varepsilon$. (b) The results from (a) on a logarithmic coordinate. The blue and green solid lines show a slope of 1/3, follow the lines of ${y_3} = {2^{1/3}}{\kappa ^{2/3}}{\varepsilon ^{1/3}}$ and ${y_4} = {2^{1/3}}{\varepsilon ^{1/3}}$, respectively. The orange and black solid lines are Re(${f_1}$) and Re(${f_1}^\prime$) from Figs. 4(e)–(f), respectively. (c) Experimental sensitivity enhancement factor as a function of the strength of perturbations. Orange and black marks denote perturbation imposed on relay and receiver coils, respectively.
The sensitivity enhancement factor $\Delta f|/\varepsilon$ and $|\Delta f^{\prime}|/\varepsilon$ of the ternary PT symmetry system is finally plotted in Fig. 5(c), respectively. In this case, the sensitivity is amplified at least 20 times when the detuning is tiny ($\varepsilon < 0.05$kHz). When $\varepsilon$ is up to 0.4 kHz, the sensitivity enhancement factor is approximate 10. We note that, the sensitivity enhancement factors are different comparing two cases with a very small perturbations applied to the neutral and the loss resonators, respectively. Case 1 with perturbations on the neutral resonator shows higher enhancement factor. In addition, the dips in reflection spectra are also much deeper in case 1, as shown in Figs. 1(c)–(d) and Figs. 4(c)–4(d). So it is a preferred solution to use the neutral resonant cavity as the sensing coil.
3. Conclusion
In conclusion, we have demonstrated theoretically and experimentally an ideal PT symmetry by balancing an effective gain caused by incident wave and radiative loss in a coupled-resonator system. Thus the high-order exceptional points in ternary PT symmetric systems is achieved with purely passive elements. Subsequently, the associated frequency response following the cube-root dependence on small perturbation is clearly observed with the extrinsic perturbation imposed on any one of resonators. Nevertheless, with the identical perturbation added in the neutral and loss resonators together, a deep reflection dip is observed not so clear as the case with perturbation in either the neutral site or the loss site. Besides, corresponding sensitivity is inferior to that with perturbation imposed on the neutral resonator solely. Our results may be useful for realizing the ultra-sensitivity in a passive wireless sensing system benefiting from the third-order exceptional points physics. One possible application is for foreign object detection in wireless power transfer application [38]. Despite the experiments carried out in the radio frequency region but the conclusions do not accept limit of frequency since the parity-time-symmetric systems can be constructed easily in optical or other frequency bands.
Funding
National Key Research Program of China (2016YFA0301101); National Natural Science Foundation of China (91850206, 61621001, 11674247, 11974261); Shanghai Science and Technology Committee (18JC1410900, 18ZR1442900); Fundamental Research Funds for the Central Universities.
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