PTX-symmetric metasurfaces for sensing applications

Abstract

In this paper, we introduce an ultra-sensitive optical sensing platform based on the parity-time-reciprocal scaling (PTX)-symmetric non-Hermitian metasurfaces, which leverage exotic singularities, such as the exceptional point (EP) and the coherent perfect absorber-laser (CPAL) point, to significantly enhance the sensitivity and detectability of photonic sensors. We theoretically studied scattering properties and physical limitations of the PTX-symmetric metasurface sensing systems with an asymmetric, unbalanced gain-loss profile. The PTX-symmetric metasurfaces can exhibit similar scattering properties as their PT-symmetric counterparts at singular points, while achieving a higher sensitivity and a larger modulation depth, possible with the reciprocal-scaling factor (i.e., X transformation). Specifically, with the optimal reciprocal-scaling factor or near-zero phase offset, the proposed PTX-symmetric metasurface sensors operating around the EP or CPAL point may achieve an over 100 dB modulation depth, thus paving a promising route toward the detection of small-scale perturbations caused by, for example, molecular, gaseous, and biochemical surface adsorbates.

Key words： parity-time symmetry; exceptional point (EP); laser oscillator; coherent perfect absorber; electromagnetic sensor; radio frequency (RF) and microwave sensing; optical sensing

1 Introduction

The past decades have witnessed the evolution of optical and photonic micro/nano-sensors and the significant contributions they have made for a variety of applications, such as biomedical analysis, non-invasive detection, and environmental monitoring, to name a few [1–8]. With necessity of detecting miniature disturbances, it is desirable to explore and investigate photonic sensors with an extremely high sensitivity and an ultrasmall detection limit. So far, numerous sensing systems have been evaluated toward this goal, such as fiber-optic sensors [9,10], nanowire-based sensors [11,12], tactile sensors based on contact electrification [13]. Along different lines, exceedingly-sensitive sensors with unusual points have recently attracted intense attention [14–21]. Such unusual points are observed in optical and photonic systems described by non-Hermitian Hamiltonians. Possibly the best-known example would be the exceptional points (EPs) typically observed in parity-time (PT)-symmetric systems. Interest in PT-symmetric systems was originally triggered by developments in quantum mechanics [22], showing that a certain class of non-Hermitian Hamiltonians can exhibit wholly real energy spectra. In electromagnetics, PT-symmetric structures, which involve balanced gain and loss components, coalesce two or more eigenvalues at the exceptional points. The indicated EPs have been proven their potential of being utilized for ultra-sensitive sensing systems, thanks to dramatic shifts of eigenfrequencies with respect to external perturbations [14–18,23,24]. Additionally, the coherent perfect absorber-laser (CPAL) point has been found as a self-dual singular point in the PT-symmetric system with special relevance for sensing purposes [19–21,25]. At the CPAL point, the eigenvalues of a PT-symmetric system will go separate ways, reaching out for zero (CPA state) and infinity (lasing state) individually. Motivated by this CPAL action, we have recently demonstrated a CPAL-locked sensing system in electromagnetic domain, initially kept at CPA state by achieving the designed ratio of complex amplitude between two monochromatic incoming waves [20,21]. A slight disturbance may break the specific condition of initial state, leading a mode switch from CPA to lasing, which results in an unprecedented sensitivity surpassing that of traditional sensors based on Fabry-Perot cavities [20,21].

In spite of these advantages in traditional PT-symmetric systems, practical implementations for the ultrasensitive PT sensor may face challenges in achieving an exact impedance profile with identical gain/loss pair. To mitigate these difficulties, we here introduce a thorough research on the generalized sensing platform based on the parity-time-reciprocal scaling (PTX)-symmetric metasurfaces operating in the proximity of EP or CPAL point. We will show that an ultrahigh sensitivity and an ultrawide detection range can still be enabled by the reciprocal-scaling operation, first demonstrated in electronic systems [17,26] and later optical metasurface systems [19]. Moreover, the PTX-symmetric system can have an eigenspectrum identical to that of the standard PT-symmetric system, while allowing unequal gain and loss, and even observation of EP in a fully-passive system [17,19]. More importantly, the scaling operation offers an additional degree of freedom in sensor design, which delicately controls the sensitivity, detection range, and modulation depth. Figure 1(a) shows the two-port transmission line network (TLN) model that can describe the proposed PTX-symmetric metasurfaces (which could be realized in different spectra with different techniques). The model consists of a pair of active and passive metasurfaces with surface conductances of -G/k and kG, where k is the reciprocal-scaling factor provided by the X operator. The negative surface conductance may be realized with an active metasurface formed by meta-atoms with photoexcited gain [19,27], as schematically shown in Fig. 1(b). When operating in the vicinity of EP, the metasurface sensing system shows a unique characteristic of unidirectional reflectionless transparency [27,28], which can be disturbed by minute perturbations and hence offers possibility for sensing applications. As for the CPAL point, the proposed sensing system employs lasing state as initial state, of which the sensing functionality is realized by the disappearance of lasing state when an ultrasmall perturbation is applied. Unlike the traditional PT sensors utilizing eigenfrequency shifts near the exceptional point, the proposed PTX sensors exploits a monochromatic sensing scheme, such that the system is rather insusceptible to the environmental noises, such as phase noise or flicker noise, thereby providing higher signal-to-noise ratio. As shown in the following, besides an ultrahigh sensitivity, the sensing limitation of the EP- or CPAL-based PTX-symmetric sensor could be pushed toward infinitesimal by applying the optimum scaling factor or phase offset, consequently providing unprecedentedly excellent sensing performance. Moreover, the introduction of reciprocal-scaling factor k not only allows the adjustment of sensitivity and working range, but also releases the restrictions on the required phase offset in transmission lines (which must be precisely controlled for high sensitivity in standard PT-symmetric sensing systems [20]). Compared with the CPAL sensor in Ref. [20] with tight constraints on the complex amplitude ratio between two incident waves, the proposed single port excited scheme also simplifies experimental requirements, thereby offering a preeminent robustness and practical feasibility in optical and photonic sensing.

Fig.1 (a) Equivalent transmission line model for the PTX-symmetric sensing system and (b) its practical realization at optical frequencies, where the positive and negative elements can be realized by a passive metal sheet and an optical-pumped active metasurface respectively. In this scheme, two metasurfaces with conductances of \mathit{kG} and -G/k (G=\left|-G\right|=\gamma Y_{\mathrm{0}}) are separated by an air gap (which is equivalent to a transmission line of characteristic admittance Y₀ and electrical length x). The system is perturbed by a variable admittance \delta Y, which could be a reactive contribution or a conductive one

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2 Concept of EP and CPAL actions in PTX-symmetric metasurface systems

Figures 1(a) and 1(b) present the schematics and one possible geometry to realize the PTX-symmetric metasurface system, comprising a scaled gain-loss pair (i.e., -G\to -G/k, G\to kG) spatially located in two sides of a transmission line with a characteristic admittance Y_{\mathrm{0}} and an electrical length of x={\beta }_{\mathrm{0}}d=\pi /\mathrm{2}+\delta x, where {\beta }_{\mathrm{0}}, d and \delta x severally denote the propagation constant, physical length and phase offset of the transmission line segment. In the optical realm, the active and passive metasurfaces, representing the gain and loss elements of the TLN model in Fig. 1(a), must be delicately designed to achieve desired surface impedance required by PTX-symmetric conditions. In practice, the passive metasurface can be simply realized with a resistive metal sheet, while an active one can be built using a planar array of photopumped graphene nanoribbons [27]. With a certain value of k, an active metasurface with surface conductance of -G/k can be accomplished by utilizing a suitable photodoping intensity and proper dimension properties of the graphene-nanoribbon array [27]. Moreover, the characteristic impedances of the port 1 (left/loss side) and port 2 (right/gain side) are also scaled by the dimensionless scaling factor k and 1/k, respectively. Applying this scaling rule in optics domain, the dielectric permittivity of the host substrates should be provided by k^{\mathrm{2}}{\epsilon }_{\mathrm{0}} and {\epsilon }_{\mathrm{0}}/k^{\mathrm{2}}, where {\epsilon }_{\mathrm{0}} is the permittivity of free space (Fig. 1(b)). We should note that to meet this criterion, the permittivity {\epsilon }_{\mathrm{eff}} of the host medium is possible to be less than 1 or near zero realized by metamaterials or plasmas at optical frequencies [19]. In this TLN (Fig. 1), it is noticed that the typical PT-symmetric system could be regarded as a degenerate case of the PTX-scenario with k=\mathrm{1}. As a result, we can likewise use scattering matrix S to describe input and output waves in the two-port PTX-symmetric system with the form of |{\psi }_{\mathrm{o}\mathrm{u}\mathrm{t}}\rangle =S|{\psi }_{\mathrm{i}\mathrm{n}}\rangle, where |{\psi }_{\mathrm{i}\mathrm{n}}\rangle ={({\psi }_f^{-}\hspace{0.17em},\hspace{0.17em}\hspace{0.17em}{\psi }_b^{+})}^{\mathrm{T}}, |{\psi }_{\mathrm{o}\mathrm{u}\mathrm{t}}\rangle ={({\psi }_f^{+}\hspace{0.17em},\hspace{0.17em}\hspace{0.17em}{\psi }_b^{-})}^{\mathrm{T}}, and S=\left(\begin{array}{ll}t& r^{+}\\ r^{-}& t\end{array}\right). Here, {\psi }_f^{\pm } and {\psi }_b^{\pm } sequentially refer to incoming and outgoing waves in the left (-) and right (+) ports. For the PTX-symmetric system in Fig. 1, the scattering coefficients in S can be derived using the transfer-matrix method and given by (see Supplementary Material for the details)

(1)\begin{array}{l}t={\displaystyle \frac{\mathrm{i}\mathrm{2}k\csc x}{({\mathit{\gamma }}^{\mathrm{2}}-\mathrm{2})k+\mathrm{i(1+}{k}^{\mathrm{2}}+\mathit{\gamma }(k^{\mathrm{2}}-\mathrm{1))cot}x}},\hfill \\ \hfill \\ r^{+}=-{\displaystyle \frac{(\mathit{\gamma }+\mathrm{2})\mathit{\gamma }k+\mathrm{i}(\mathit{\gamma }+\mathrm{1})(k^{\mathrm{2}}-1)\cot x}{({\mathit{\gamma }}^{\mathrm{2}}-\mathrm{2})k+\mathrm{i(1+}{k}^{\mathrm{2}}+\mathit{\gamma }(k^{\mathrm{2}}-\mathrm{1))cot}x}},\hfill \\ \hfill \\ r^{-}=-{\displaystyle \frac{(\mathit{\gamma }-\mathrm{2})\mathit{\gamma }k+\mathrm{i}(\mathit{\gamma }-\mathrm{1})(k^{\mathrm{2}}-1)\cot x}{({\mathit{\gamma }}^{\mathrm{2}}-\mathrm{2})k+\mathrm{i(1+}{k}^{\mathrm{2}}+\mathit{\gamma }(k^{\mathrm{2}}-\mathrm{1))cot}x}},\hfill \end{array}

where \gamma indicates the conductance ratio of the gain/loss component to the characteristic admittance of the transmission line (\gamma =G/Y_0). From Eq. (1), it can be seen apparently that the scaling factor k could modify the scattering properties in terms of reflection (r) and transmission (t) coefficients. In addition, we also observe from Eq. (1) that when x=\pi /\mathrm{2} (i.e., \delta x=\mathrm{0}), all scattering coefficients in S and the correlative eigenvalues {\lambda }_{\pm } become independent of scaling coefficient k, implying that the unbalanced PTX-symmetric system shares exactly the same scattering properties as its PT-symmetric counterpart with balanced gain and loss. Similarly, the evolution of eigenvalues (\left|{\lambda }_{\pm }\right|) of S matrix could be used to characterize the phase transition in this PTX-symmetric system. Figure 2 reports the eigenvalues as a function of \gamma and k for the PTX-symmetric system with \delta x={\mathrm{10}}^{-\mathrm{3}}(\pi /\mathrm{2}), in which the exact and broken symmetry phases are divided by the exceptional point (\gamma =\mathrm{2}). More specifically, when , the PTX-symmetric system is in a broken phase where the eigenvalues are non-unimodular (i.e., {\lambda }_{+}={({\lambda }_{-}^{\ast })}^{-\mathrm{1}}); when \gamma >\mathrm{2}, the PTX-symmetry system enters the exact phase and the eigenvalues become nondegenerate and unimodular (i.e., \left|{\lambda }_{\pm }\right|=\mathrm{1}). In addition, it is observed that a CPAL point induced by self-dual spectral singularity occurs in the broken symmetry phase (\gamma =\sqrt{\mathrm{2}}), where the PTX-symmetric electromagnetic system possesses zero and infinity eigenvalues simultaneously, representing the CPA and lasing states respectively.

Fig.2 Evolution of eigenvalues as a function of \gamma and k for the PTX system in Fig. 1; here, \delta x={\mathrm{10}}^{-\mathrm{3}}(\pi /\mathrm{2}) and \delta Y=\mathrm{0}. We should note that if x=\pi /\mathrm{2} (or \delta x=\mathrm{0}), the eigenvalues are unaffected by scaling factor k. The system can be divided into the exact symmetry phase (\gamma >\mathrm{2}) and the broken PTX-symmetry phase (), with a discontinuous phase transition at the exceptional point (\gamma =\mathrm{2}). In the broken phase, two eigenvalues approach infinity (lasing state) and zero (CPA state) at the CPAL point (\gamma =\sqrt{\mathrm{2}})

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3 EP-based PTX-symmetric metasurface sensing systems

To characterize scattering properties of the PTX-symmetric metasurfaces, we introduce the output coefficient Θ defined as the total output power divided by the input power, which could be written as

(2)\mathrm{\Theta }={\displaystyle \frac{k|{\psi }_b^{-}{|}^{\mathrm{2}}+|{\psi }_f^{+}{|}^{\mathrm{2}}/k}{k|{\psi }_f^{-}{|}^{\mathrm{2}}+|{\psi }_b^{+}{|}^{\mathrm{2}}/k}}.

For the system operating at the EP (\gamma =\mathrm{2}), a distinctive property of unidirectional reflectionless may be obtained (i.e., \left|r^{-}\right|=\mathrm{0} according to Eq. (1)), and its sensing functionality may be realized by drastically breaking this reflection response with a tiny perturbation. On that account, we consider here the EP-locked sensing scenario with left (−) port (port 1) excitation and only the reflected wave {\psi }_b^{-} is taken into account for the output factor in Eq. (2), i.e., {\psi }_f^{-}=\mathrm{1},{\psi }_b^{+}={\psi }_f^{+}=\mathrm{0}. By applying a small-scale perturbation \nu =\delta {\mathit{Y/Y}}_{\mathrm{0}} to the loss side (which could be either resistive or reactive), the initial state of unidirectional reflectionless may be disturbed and the associated output coefficient can be approximately expressed as (see Supplementary Material for the detailed derivations of output coefficients)

(3)\mathrm{\Theta }(\nu )\approx {\displaystyle \frac{{\nu }^{\mathrm{2}}}{\mathrm{4}{k}^{\mathrm{2}}}}+\mathrm{O}({\nu }^{\mathrm{3}}).

From Eq. (3), we find that the slope of the output coefficient versus {\nu }^{\mathrm{2}}, which is a measurement of sensitivity, is governed by the dimensionless parameter \mathrm{1/(4}{k}^{\mathrm{2}}) and unrelated to the phase offset \delta x. In addition, the output factor \mathrm{\Theta } is limited with upper and lower bounds of 1 and {\displaystyle \frac{{(k^{\mathrm{2}}-\mathrm{1})}^{\mathrm{2}}\delta x^{\mathrm{2}}}{\mathrm{4}{k}^{\mathrm{2}}}} respectively, from which it is noted that the initial or minimum value of \mathrm{\Theta } could approach zero when k is set to 1 or \delta x=\mathrm{0}. Under this specific circumstance, the sensing limitation (i.e., lower detection bound) may be remarkably infinitesimal and the sensing system accordingly achieves its maximal working range. Figure 3(a) plots the output coefficient under the effect of a minuscule disturbance \nu with different phase offset \delta x (k is kept as constant 2), where the approximate output factor (dashed line) obtained from Eq. (3) matches the accurate value (solid lines) truly well in the operation range. From Fig. 3(a), it is evidently observed that such unidirectional reflectionless property at the EP, indeed, enables photonic sensors with an extremely high sensitivity. Additionally, its sensing limitation and working range could be modulated through the adjustment of phase offset \delta x. The contours of \mathrm{\Theta } as a function of \nu and \delta x are shown in Fig. 3(b), which further illustrate that the lower bound of \mathrm{\Theta } or detection limit could be minimized by decreasing \delta x. Particularly, a large modulation depth up to 70 dB may be obtained with a phase offset \delta x\sim {\mathrm{10}}^{-\mathrm{4}}(\pi /\mathrm{2}).

Fig.3 (a) Output coefficient of the PTX-symmetric EP sensor in Fig. 1 under small perturbations \nu =\delta {\mathrm{Y/Y}}_{\mathrm{0}}; here, \gamma =\mathrm{2} and k=\mathrm{2}. Dashed line is the approximate results obtained from Eq. (3). We note that \mathrm{\Theta }(\nu ) is not affected by the sign of \delta x. (b) Contours of output coefficient as a function of \nu and \delta x. The lower bound of output coefficient (or detection limit) can be minimized as {(\delta x)}^{\mathrm{2}} decreases

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Next, the influence of scaling factor k on output intensity is also studied and the results are depicted in Figs. 4(a) and 4(b); here \delta x={\mathrm{10}}^{-\mathrm{3}}(\pi /\mathrm{2}). As expected, the scaling factor k has the potential for tuning sensitivity and working range. To be more precise, sensitivity which appears as the slope of \Theta ({\nu }^{\mathrm{2}}) can be improved by reducing k, and a scaling factor k near 1 could help the PTX-symmetric EP sensor achieve an initial condition of near-zero output coefficient \mathrm{\Theta }, thus realizing an ultrasmall sensing limitation and the maximum working range. Also, the optimum condition k = 1 releases the restrictions on the phase offset, so as to offer the EP-locked sensing system excellent tolerance to errors in the assembly and fabrication of transmission lines.

Fig.4 (a) Output coefficient \mathrm{\Theta }(\nu ) of the PTX-symmetric EP sensor with different scaling factor k; here, \gamma =\mathrm{2} and \delta x={\mathrm{10}}^{-\mathrm{3}}(\pi /\mathrm{2}). (b) Contours of output intensity as a function of \nu and k. The slope of \mathrm{\Theta }(\nu ), or sensitivity, and working range can be tuned by modifying scaling factor. When k=\mathrm{1}, the lower detection limit is approaching zero, leading to infinitesimal sensing limitation and the topmost operation range

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4 CPAL-based PTX-symmetric metasurface sensing systems

In this section, we study the effect of perturbations on the output coefficient of the PTX-symmetric metasurface system locked to the CPAL point. We note that under the CPAL condition, i.e., x=\pi /\mathrm{2} and \gamma =\sqrt{\mathrm{2}}, the lasing mode could always be obtained with single port excitation (i.e., {\psi }_f^{-}=\mathrm{1},{\psi }_b^{+}=\mathrm{0} for port 1 incidence while {\psi }_f^{-}=\mathrm{0},{\psi }_b^{+}=\mathrm{1} for wave coming from right side, i.e., port 2) [19]. It is worthwhile mentioning that the proposed single port excited PTX-symmetric CPAL sensing system has hardly any restrictions on input signals which are necessarily required for achieving the initial CPA mode in Ref. [20]. However, it could still accomplish a superior sensitivity owing to the rapid amplitude dropping of the output coefficient at design frequency when applying ultrasmall-scale perturbations. Let us first analyze the performance of the CPAL-locked PTX-symmetric sensor with left-incident wave (loss side) excited by port 1. Considering a small resistive or reactive perturbation \nu =\delta {\mathit{Y/Y}}_{\mathrm{0}} () applied on the loss component and solely detecting the reflected wave {\psi }_b^{-}, the output factor could be approximated as (see Supplementary Material for the detailed derivations of output coefficients)

(4)\mathrm{\Theta }(\nu )\approx {\displaystyle \frac{\mathrm{4}{k}^{\mathrm{2}}}{{\nu }^{\mathrm{2}}}}+\mathrm{O}({\nu }^{\mathrm{3}}),

which accurately predicts the output coefficient affected by small values of \nu, as shown in Fig. 5(a). It is evidently seen from Eq. (4) that sensitivity, or the slope of \mathrm{\Theta }({\nu }^{-\mathrm{2}}), is dominated by the factor \mathrm{4}{k}^{\mathrm{2}} and rather independent of the phase offset \delta x, revealing that sensitivity of the CPAL-locked sensor can be modified by tuning the scaling factor k. In point of fact, the suggested PTX-symmetric CPAL sensing system offers eminent sensitivity, which is also tunable and may be ameliorated by choosing large scaling coefficient k appropriately (Fig. 6).

Fig.5 (a) Output coefficient of the PTX-symmetric CPAL sensor in Fig. 1 under tiny perturbations \nu =\delta {\mathit{Y/Y}}_{\mathrm{0}}; here, \gamma =\sqrt{\mathrm{2}} and k=\mathrm{1}. Dashed line is the approximate results obtained from Eq. (4). (b) Contours of output coefficient as a function of \nu and \delta x. The upper bound (or maximum value) of the output coefficient can be greatly enhanced as {(\delta x)}^{\mathrm{2}} decreases

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Fig.6 (a) Output coefficient \mathrm{\Theta }(\nu ) of the PTX-symmetric CPAL sensor with different scaling factor k; here, \gamma =\sqrt{\mathrm{2}} and \delta x={\mathrm{10}}^{-\mathrm{3}}\mathrm{(\pi }/2). (b) Contours of output intensity as a function of \nu and k. When k=\sqrt{\mathrm{2}}-\mathrm{1}, \max ({\mathrm{\Theta }}_{\nu }) is pushed toward infinity, conforming to near-zero sensing limitation

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In addition, the detection limit and working range of the proposed PTX system could be described by physical bounds of the output intensity. For this port 1 excited PTX-symmetric CPAL metasurface sensor, the initial or maximum value of output intensity could be expressed by \max ({\mathrm{\Theta }}_{\nu })={\displaystyle \frac{\mathrm{4}(\mathrm{17}-\mathrm{12}\sqrt{\mathrm{2}})k^{\mathrm{2}}}{{(-\mathrm{3}+\mathrm{2}\sqrt{\mathrm{2}}+k^{\mathrm{2}})}^{\mathrm{2}}\delta x^{\mathrm{2}}}}. If the perturbation is fairly massive, the output coefficient will converge to its lower limitation: \min ({\mathrm{\Theta }}_{\nu })=\mathrm{1}. Interestingly, the denominator of \max ({\mathrm{\Theta }}_{\nu }) becomes zero when \delta x=\mathrm{0} or k=\sqrt{\mathrm{2}}-\mathrm{1}, showing that the initial (maximum) value of output coefficient may theoretically achieve infinity. As a result, the sensing limitation, directly linked with \max ({\mathrm{\Theta }}_{\nu }), may be significantly enhanced and even pushed toward infinitesimal with a tiny \delta x or a scaling factor near \sqrt{\mathrm{2}}-\mathrm{1}. Figure 5 compares the metasurface sensing systems based on different phase offset \delta x (here k=\mathrm{1}), in response to a small perturbation \nu, which proves that the reduction of phase offset from \delta x\sim {\mathrm{10}}^{-\mathrm{2}}(\pi /\mathrm{2}) to \delta x\sim {\mathrm{10}}^{-\mathrm{4}}(\pi /\mathrm{2}) could result in a significant enhancement of sensing limitation and modulation depth. The output coefficients as a function of k and \nu are depicted in Fig. 6 with a phase offset \delta x={\mathrm{10}}^{-\mathrm{3}}(\pi /\mathrm{2}), illustrating that the scaling factor k offers the possibility of adjustable sensing limitation and tunable working range. It should be noted that, although k=\sqrt{\mathrm{2}}-\mathrm{1} is not capable of bringing the largest sensitivity, it can still be regarded as the optimal scaling factor since it offers decent sensitivity with the lowest detection limit and the maximum working range. In practice, the phase offset may be minimized by utilizing advanced complimentary metal-oxide-semiconductor (CMOS) photonics and integrated optics technologies. On the other hand, the optimal value of k could be readily realized, for instance, through careful design of the active/passive metasurface with desired surface conductance values.

Here, we also fully investigate the cases where the PTX-symmetric CPAL sensing system (a) is excited with left-incident wave (port 1) and both reflection and transmission are detected for output intensity; (b) is excited with right-incident wave (port 2) and only reflected wave is considered; (c) is excited with right-incident wave (port 2) and both reflection and transmission are counted for output coefficient. For all three cases, the output factors \mathrm{\Theta } under perturbation \nu =\delta {\mathrm{Y/Y}}_{\mathrm{0}} are listed in Table 1, where we perceive the similarities between these circumstances and the aforementioned sensing scheme (i.e., left-port excitation and single reflection detection); these include adjustable sensitivity and tunable working range provided by k, and near-zero sensing limitation accomplished by a minimized \delta x or a scaling factor k=\sqrt{\mathrm{2}}-\mathrm{1}. Noticeably, the lower bound of output intensity in case (a) is 1 whereas those in cases (b) and (c) are \mathrm{17+12}\sqrt{2}, showing that sensing systems with incident wave coming from the loss side possess broader working range and larger modulation depth in contrast to the gain-side excitation scenarios. Since two-port detection requires more complicated experimentation platform without producing preferable performance, the single port excited sensing system with left-incident wave and reflection detection is concluded as the most distinguished sensing scheme for its manageable feature and marvelous sensing performance.

Tab.1 Performance of PTX-symmetric CPAL sensor

excitation /detection port	approximate \mathrm{\Theta }(\nu )	upper bound	lower bound
port 1 /port 1	{\displaystyle \frac{\mathrm{4}{k}^{\mathrm{2}}}{{\nu }^{\mathrm{2}}}}	{\displaystyle \frac{\mathrm{4}(\mathrm{17}-\mathrm{12}\sqrt{\mathrm{2}})k^{\mathrm{2}}}{{(-\mathrm{3}+\mathrm{2}\sqrt{\mathrm{2}}+k^{\mathrm{2}})}^{\mathrm{2}}\delta x^{\mathrm{2}}}}	1
port 1 /port 1&2	{\displaystyle \frac{\mathrm{4}(\mathrm{3}+\mathrm{2}\sqrt{\mathrm{2}}+k^{\mathrm{2}})}{{\nu }^{\mathrm{2}}}}	{\displaystyle \frac{\mathrm{4}(\mathrm{17}-\mathrm{12}\sqrt{\mathrm{2}})(\mathrm{3}+\mathrm{2}\sqrt{\mathrm{2}}+k^{\mathrm{2}})}{{(-\mathrm{3}+\mathrm{2}\sqrt{\mathrm{2}}+k^{\mathrm{2}})}^{\mathrm{2}}\delta x^{\mathrm{2}}}}	1
port 2 /port 2	{\displaystyle \frac{\mathrm{4}(\mathrm{17}+\mathrm{12}\sqrt{\mathrm{2}})k^{\mathrm{2}}}{{\nu }^{\mathrm{2}}}}	{\displaystyle \frac{\mathrm{4}{k}^{\mathrm{2}}}{{(-\mathrm{3}+\mathrm{2}\sqrt{\mathrm{2}}+k^{\mathrm{2}})}^{\mathrm{2}}\delta x^{\mathrm{2}}}}	\mathrm{17}+\mathrm{12}\sqrt{\mathrm{2}}
port 2 /port 1&2	{\displaystyle \frac{\mathrm{4}{k}^{\mathrm{2}}(\mathrm{17}+\mathrm{12}\sqrt{\mathrm{2}}+(\mathrm{3}+\mathrm{2}\sqrt{\mathrm{2}})k^{\mathrm{2}})}{{\nu }^{\mathrm{2}}}}	{\displaystyle \frac{\mathrm{4}{k}^{\mathrm{2}}+\mathrm{4}(\mathrm{3}-\mathrm{2}\sqrt{\mathrm{2}})k^{\mathrm{4}}}{{(-\mathrm{3}+\mathrm{2}\sqrt{\mathrm{2}}+k^{\mathrm{2}})}^{\mathrm{2}}\delta x^{\mathrm{2}}}}	\mathrm{17}+\mathrm{12}\sqrt{\mathrm{2}}

5 Conclusions

We have thoroughly analyzed the monochromatic PTX-symmetric metasurfaces for sensing applications, which may take advantages of drastic changes in optical scattering properties nearby the EP and CPAL point. We have theoretically shown that the output coefficient of the proposed sensing scheme is extremely sensitive to external perturbations. Further, our results have shown that the phase offset \delta x related to the separation distance between two metasurfaces is responsible for the sensing limitation and detection range, while the reciprocal-scaling factor k associated with the X transformation determines the sensitivity of the system. Through a systematic study, we have found out that the optimal scaling factor (k=\mathrm{1} for the EP-based sensor and k=\sqrt{\mathrm{2}}-\mathrm{1} for the CPAL-based sensor) may achieve the maximum detection range (for both lower and upper bounds). The proposed PTX-symmetric metasurface sensors with advantages of low profile and ultrahigh sensitivity may open a new avenue of the next-generation high-performance optical and photonic sensors in different spectral ranges.

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