Enhanced chiroptical responses through coherent perfect absorption in a parity-time symmetric system

Coherent amplification of chiroptical activity from a molecularly-thin optically-active substance has been a long-standing challenge due to the inherently weak nature of chiral responses. Here we report how a coherent perfect absorber (CPA) enabled by an achiral optical system obeying parity-time (PT) symmetry has an enhanced ability to effectively sense molecular chirality of monolayered substances. We demonstrate that such a CPA-based PT-symmetric system enables us in complete darkness to probe a subtle signal change induced by the introduction of a small disturbance, such as adsorbed chiral monolayer, to the unperturbed PT-symmetric system, and allows for absolute measurement and quantitative detection of the magnitude and sign of both real and imaginary parts of the chirality parameter in a background-free environment. Moreover, the CPA-based PT-symmetric system also exhibits three orders of magnitude enhancement in chiroptical responses of molecules, which is consistent with analytical calculations of differential absorption. Optical sensors traditionally struggle to detect inherently weak chiral signals from molecules. Here, coherent perfect absorption in a parity-time symmetric platform is predicted to increase chiral sensitivity by three orders of magnitude.

O wing to its extraordinary properties, parity-time (PT) symmetry has drawn considerable attention in the physics of non-Hermitian systems since Bender et. al. first discovered in 1998 that non-Hermitian Hamiltonians obeying PT symmetry can possess real energy spectra in quantum mechanics 1,2 . In optical systems, PT symmetry can be readily established in a 1D bilayer structure through incorporating gain and loss materials in a judiciously balanced fashion. Such a PTsymmetric photonic system in general can be characterized by a general two-port scattering matrix whose eigenvalues are either unimodular in the PT-symmetric phase or come in inversecomplex-conjugate pairs in PT-broken phase [3][4][5][6] . The transition from the PT-symmetric to the PT-broken phase takes place at socalled exceptional points (EPs) [7][8][9][10] . During the past decade, an enormous amount of theoretical and experimental work on EPs has stimulated various applications, such as unidirectional invisibility 11 , non-reciprocal light transmission and energy transfer 12,13 , single-mode lasing 14,15 , mode switching 16 , electromagnetically induced transparency 17,18 , as well as enhanced sensing sensitivity enabled by eigenfrequency splitting [19][20][21] . In addition to the presence of the symmetry-breaking transition points [22][23][24] , the PT-symmetric system also features a self-dual spectral singularity in the regime of broken-PT symmetry, where a pole and a zero of a scattering matrix appearing in a complex conjugate pair in the complex frequency domain coincide at a specific real frequency, called a coherent perfect absorber-laser (CPAL) point. In general, a real-frequency pole (zero) corresponds to an infinite (infinitesimal) eigenvalue, which is associated with the lasing (coherent perfect absorption (CPA)) eigenstate of the scattering matrix 3 . Thanks to singular characters at the CPAL point, PT-symmetric CPAL systems have been experimentally realized to function as a laser oscillator and as an anti-laser [25][26][27] . Experimental implementations of CPAs have also been demonstrated in different fields, including quantum optics 28 , optical logic gates 29 , and nonlinear physics 30 . Although CPA-related theoretical studies on chiral metamaterials 31,32 and refractive index sensing 33,34 were previously reported, the application of PT-symmetry for the detection of chiral substances remains largely unexplored.
Chirality is a fundamental property of life. Sensing the dominant handedness of chiral substances is crucial to numerous scientific subfields of chemistry, biology, and medicine, and to pharmaceutical, cosmetic and food industries. Detecting the inherently weak chiral responses of molecules remains one of the great challenges in the fields of optics and photonics. The poor detection sensitivities of current optical techniques and the difficulties in analyzing the weak signals of substrate-based sensors do not allow important biological and medical processes to be precisely probed and prevent further understanding and treatment of various diseases. Developing highly sensitive chiroptical spectroscopy methods is important because these methods are used to probe the folding of proteins, the yield of enantioselective chemical synthesis and the purity of processed chemicals (i.e., in the pharmaceutical and chemical industries), and to study fundamental optical and physicochemical interactions that can be developed to separate the enantiomers 35 . Therefore, the photonics community is actively seeking powerful new ways to enhance the chiroptical signals of molecules.
Here we present a chiral sensing mechanism based on the unusual and distinctive eigenbehaviors of the CPA offered by PT-symmetry. In contrast to extracting a faint AC signal from an intense DC signal, typically existing in homodyne measurements 36,37 , the PT-symmetric system operating at the CPA mode reveals a self-sustaining null intensity on its output channels with an extraordinarily large extinction depth, which enables us in complete darkness to probe a subtle change in weak chiroptical signals induced by the binding of chiral molecules to the surfaces of the unperturbed PT-symmetric system. Just like you observe a minimum transmitted power or intensity, I t → 0, as incident light passes through two crossed polarizers commonly used in polarization measurement. If a perturbed object is placed between the crossed polarizers, any polarization effects present in the object, such as birefringence, will cause an increase in transmitted intensity, I t + ΔI t . Distinct from the crossed polarizers, the sensitivity of such a CPA-based PT-symmetric system to a small perturbation is in principle infinitely large nearby the CPAL point. In this work, we demonstrate that the chiroptical responses are substantially enhanced at the CPAL point and highly sensitive to molecular chirality, compared to those in the absence of the PT-symmetric substrate, with a potential for the detection of the chirality of monolayers. Besides, the structurally achiral PT-symmetric substrate itself does not contribute to undesired background signals so that absolute measurements of the chirality of molecules become feasible. Our work represents a characteristic example of applying non-Hermitian systems to boost chiroptical responses and could lead to a promising solution for the detection of extremely faint chiroptical signals without the need for complicated fabrication of nanoengineered structures.

Results and discussion
Eigenbehaviors of PT-symmetric system S PT . Since chiral molecules only manifest their handedness during their interaction with the chiral state of light (such as circularly-polarized (CP) light), the vectorial electromagnetic waves are represented in terms of the unit circular basis vectors (e ± ) and then expressed as in the form of circular Jones vectors (Fig. 1a). The solution to the eigenvalue problem of S PT yields two degenerate pairs of eigenvalues, s 1 = s 2 and s 3 = s 4 , and four corresponding eigenstates,  Fig. 1b, the eigenvalue spectrum of S PT , the absolute value of the eigenvalues of S PT as a function of wavelength λ, clearly illustrates two distinct regimes of the PT phase separated by a phase transition point or an EP at which all four eigenvalues coalesce with two degenerate pairs of eigenstates, ψ 1 = ψ 3 and ψ 2 = ψ 4 . In the PT-symmetric phase, in spite of s 1 = s 2 ≠ s 3 = s 4 , the eigenvalues are unimodular, |s i | = 1, i = {1,2,3,4}, implying that there is neither net amplification nor net attenuation between output and input states. Their eigenstates that thePT operator maps one eigenstate into the oppositely handed eigenstate associated with the same eigenvalue. Although each eigenstate is not itself PT-symmetric, a linear combination of two PT-nonsymmetric eigenstates associated with the same eigenvalue is PT-symmetric (i.e.,PT ψ 1 þ ψ 2 À Á / ψ 1 þ ψ 2 and PT ψ 3 þ ψ 4 À Á / ψ 3 þ ψ 4 ). In the PT-broken phase, despite s 1 = s 2 ≠ s 3 = s 4 and |s 1 | = |s 2 | > 1 > |s 3 | = |s 4 |, the eigenvalues come in pairs with reciprocal moduli, s i s Ã j ¼ 1; i; j È É ¼ 1 or f 2; 3 or 4g; 3 or 4; 1 or 2 f g , meaning that there exists amplification and attenuation between output and input states. Their eigenstates satisfyPTψ i ¼ ψ j ; i; j È É ¼ 1; 4 f g; 2; 3 f g; 3; 2 f g; 4; 1 f g, signifying that thePT operator transforms one eigenstate into the oppositely handed eigenstate associated with different eigenvalue. Similarly, the sum of two PT-nonsymmetric eigenstates with the same eigenvalue fulfillsPT which are not PT-symmetric. Besides, the eigenvalue spectrum of S PT in the PT-broken phase exhibits a self-dual spectral singularity at a CPAL point whose eigenvalues approach zero and infinity associated with the CPA and lasing eigenstates, respectively (Supplementary Notes 3 and 4 for details of s i and ψ i at the EP and the CPAL wavelength).
In addition, the eigenstates of S PT display very intriguing and unique properties in different PT phases. The matrix configuration of S PT implies two independent responses at normal incidence: for input pair of a + and d − , the output state |Ψ O 〉 is a RCP reflected wave b − and a LCP transmitted wave c + , while in response to the oppositely handed input pair of a − and d + , |Ψ O 〉 is a LCP reflected wave b + and a RCP transmitted wave c − . With the incoming CP waves [a +,i , d −,i , a −,i , d +,i ] extracted from each eigenstate ψ T i , i = {1,2,3,4} associated with the same eigenvalue s i for different wavelength λ, we obtain the following significant relationships: a þ;1 =d À;1 ¼ a À;2 =d þ;2 ¼ A p e iϕ p and a þ;3 =d À;3 ¼ a À;4 =d þ;4 ¼ A q e iϕ q , where A and ϕ, respectively, represent the relative amplitude and phase of the two counterpropagating oppositely handed CP incident eigenwaves of the same wavelength, with the subscript p (q) denoting the associated eigenvalues s 1 = s 2 (s 3 = s 4 ). The relationships among the relative amplitude A, phase ϕ and wavelength λ in different PT phase domains are illustrated in Fig. 1c, d, respectively, and summarized in Table 1 (Supplementary Note 5). Figure 1d specifies that the phase difference between the two counter-propagating oppositely handed CP incident beams of the same wavelength is exactly π/2 for m = 0 in the PT-broken phase. By fixing their relative phase at π/2, we can achieve coherent switching between the same-handed pair of absorbing and amplifying eigenstates, that is, (ψ 3 ↔ ψ 1 ) and (ψ 4 ↔ ψ 2 ), by simply varying the intensity ratio of the two CP incident beams, as indicated in Fig. 1c. The intensity ratios and phase differences of two independent pairs of [a + , d − ] and [a − , d + ] beams can be experimentally tuned through adjustments of two separate sets of neutral density filters and optical delay lines. We can locate the wavelength of the CPAL point, λ CPAL , by sweeping the wavelength of the coherent incident beams while observing a minimum of output coefficient (Θ) defined as Once the minimum Θ is reached, the settings of the neutral density filters and the optical delay lines remain unchanged, in which the input state |Ψ I 〉 is locked to the CPA eigenstates ψ CPA i À Á , called the CPA-locked PT-symmetric system. The switching from the CPA to lasing eigenstates can be readily accomplished by rotating the neutral density filters until Θ goes to maximum. Figure 1e shows how Θ varies with the wavelength of the coherent incident beams for the lasing-locked, CPA-locked, and eigen S PT systems. Significant linewidth narrowing of the CPA mode at λ CPAL for CPA-locked S PT can be attributed to the use of either CPA eigenstates (ψ CPA  lasing-locked responses (the red solid curves) no matter whether the S PT system is locked or not.
Chiroptical responses of CPA-locked S C . Now let us consider that a one-nanometer-thick layer of a chiral substance adheres to both surfaces of the PT-symmetric structure (layer 2 and 5 in Fig. 2) as the substrate is immersed in a solution containing chiral molecules. The adsorbed chiral layer has an average refractive index n c ¼ 1:33 þ 10 À4 i and a variable chirality parameter κ. The relation between the output and input states of the PT-symmetric system in the presence of a chiral substance is given by where |Ψ I 〉 can be expressed as a coherent superposition of the CPA eigenstates of S PT ; jΨ . Please refer to Supplementary Notes 6, 7, and 11 for the details of calculations of optical rotation ϑ and ellipticity angle χ. As shown in Fig. 2, we demonstrate the capability to detect chiroptical signals from molecularly thin chiral layers bound to the exterior surfaces of the PT-symmetric structure. For one-nanometer-thick layers of chiral substance with κ = ±10 −5 and ±10 −5 i (typical values for the chirality parameter of chiral molecules) 38,39 , ϑ (arising from circular birefringence(CB)) and χ (originating from circular dichroism (CD)) are as large as 10 mdeg near λ CPAL and highly sensitive to the sign of κ. Furthermore, it is feasible to achieve the absolute chirality measurements since such a PT-symmetric system composed of a gain-loss bilayer does not make any contributions to the total chiroptical signals owing to its structurally achiral nature (Supplementary Note 8). As a comparison, the reference chiroptical signals (ϑ Ref and χ Ref ) contributed by the pure chiral layers of the same thickness in the absence of the PTsymmetric substrate under the identical illumination are as small as 9 μdeg (Supplementary Note 9), thus achieving three orders of magnitude enhancement in the chiroptical responses using the CPA-locked PT-symmetric detection scheme. Specifically, ϑ and χ are related to the difference in refractive index and absorption of LCP and RCP light propagating in a homogeneous isotropic chiral medium, respectively. Therefore, it is unsurprised that ϑ scales linearly with the real part of κ (Re(κ)) while χ is proportional to the imaginary part of κ (Im(κ)) 40,41 . However, those extremely weak chiroptical signals (~8 μdeg) shown in the insets of Fig. 2 counterintuitively imply that Re(κ) can produce χ and Im(κ) can yield ϑ. Our results are in agreement with recent observations that CD or χ signals depend on both Re(κ) and Im(κ) 39,42,43 . In order to investigate the origin of such unexpected contributions, we examine plots of the κ-dependence of ϑ and χ measured at λ CPAL for different values of Im(n c ), where n c is the average refractive index of the chiral layers, as shown in Fig. 3. For Im(n c ) = 0, which excludes the nonchiral loss term from contributing to the total signal, the chiroptical signals follow the expected behavior, ϑ ∝ Re(κ) and χ ∝ Im(κ). For Im(n c ) ≠ 0, however, Re(κ) and Im(κ) contribute to a different extent to ϑ and χ. The slopes of χ(Re(κ)) and ϑ(Im(κ)) in the insets of Fig. 3 scale monotonically with Im(n c ), but the change in Im(n c ) has no effect on both ϑ(Re(κ)) and χ(Im(κ)). On the whole, Re(κ) and Im(κ) are dominantly responsible to ϑ and χ, respectively, regardless of Im(n c ) as obviously indicated by overlaid dashed and dotted lines shown in Fig. 3. Consequently, the appearance of the extremely small chiroptical signals shown in the insets of Fig. 2 can be attributed to the consequence of Im(n c ) ≠ 0, and their relative contributions to the overall chiroptical signals are negligibly small and can be ignored (Supplementary Note 10 for purely real n c case). Figure 4a, b show the κ-dependence of ϑ and χ measured at λ CPAL for different chiral layer thicknesses (L c ). The linear dependence between enhanced ϑ (χ) and the real (imaginary) part of κ indicates that the magnitude and sign of the real (imaginary) part of κ can be quantitatively determined through measurements of far-field chiroptical signals for a given chiral layer thickness, which allows for unambiguous discrimination of the chirality of an unknown sample. As shown in Fig. 4c, d, the sensitivity, expressed as the change in ϑ (χ) with respect to the change in the real and imaginary part of κ, is nearly unaltered for L c less than 10 nm, even though there is a small variation in the real part of n c , namely 1.33 + δn c , which may be caused by fluctuations in temperature and humidity. As L c gets thicker, the influence of δn c on the effective refractive index of the overall PT-symmetric system becomes prominent, and the contribution of the refractive index perturbation to the system's sensitivity might need to be taken into consideration. Once again, for those relatively weak chiroptical shown in the insets of Fig. 4, their emergence can be attributed to the consequence of the presence of Im(n c ) and their relative contributions to the total chiroptical signals are negligible for thinner L c . The CPA-locked PT-symmetric detection scheme allows for quantitative detection of the chirality parameter κ of the adsorbed chiral substance even though there exists a nonchiral loss term in its refractive index. We have to emphasize that our analytical model is valid only when the unperturbed CPA-locked S PT is exposed to a small perturbation (such as a small κ and δn c with a comparable thin chiral layer). As the external perturbation becomes larger, the perturbed system S C will not be able to hold the eigenstates of the unperturbed system S PT . Moreover, even though κ is assumed independent of wavelength over a specified wavelength range, it is still possible to probe the chiroptical activity of an optically active sample as long as the sample to be detected has a finite-valued κ nearby λ CPAL . This gives us a great opportunity to explore vibrational optical activity outside the ultraviolet wavelength region in which the optical rotatory dispersion and electronic CD measurements are usually carried out 44 .
Spatially dependent differential absorption. To further understand the underlying mechanism of how the CPA-locked S C enhances chiroptical signals, especially CD, we first investigate the optical absorption of two independent pairs of internal ½c j þ ; d j À and ½c j À ; d j þ beams in response to their respective input states, , as schematically illustrated in Fig. 5. Based upon the time-averaged power densities evaluated Table 1 Characteristics of eigenstates of a 4 × 4 S PT in different phases.

Relative amplitude A
Relative phase ϕ ψ T i at a given λ in the PT-unbroken phase A p = A q = 1 ϕ p + ϕ q = (2m + 1)π ψ T i at exceptional point (EP) A p = A q = 1 ϕ p = ϕ q = (4m + 1) π/2 ψ T i at a given λ in the PT-broken phase A p A q = 1 ϕ p = ϕ q = (4m + 1) π/2 Each eigenstate ψ T i only consists of two counter-propagating circularly-polarized incident waves of opposite handedness, namely either The relationships between different ψ T i of the same eigenvalue s i at a given wavelength λ are a þ;1 =d À;1 ¼ a À;2 =d þ;2 ¼ A p e iϕ p and a þ;3 =d À;3 ¼ a À;4 =d þ;4 ¼ A q e iϕ q , where A and ϕ are the relative amplitude and phase with the subscript p (q) denoting the associated eigenvalues s1 = s2 (s3 = s4) and m is an integer.
at the initial position z 2 and at a relative position z' (0 ≤ z' ≤ L j ) inside the layer j with a length of L j , j = {2,3,4,5}, we are capable of examining how the differential absorption evolves internally as the internal ½c    same position, thus resulting in Δα(z) = 0, irrespective of the presence of the balanced gain and loss in the CPA-locked PTsymmetric system. In the presence of chiral absorption (Im(κ) ≠ 0), we observe chirality-dependent Δα(z) and dissymmetry in the spatial profiles of Δα(z) with respect to the center of PTsymmetric structure, with ripples caused by the differential interference of ½c j À ; d j þ and ½c j þ ; d j À beams. In fact, the differential transmitted intensity in layer 6 ðjc 6 þ j 2 À jc 6 À j 2 Þ that scales with the CD signal (Supplementary equation (23)) is proportional to the differential absorption at the position z 6 , Δα(z 6 ). As shown in Fig. 5a, the signs of the chirality-dependent Δα(z 6 ), denoted by blue downward and red upward arrows corresponding respectively to Im(κ) >0 and Im(κ) <0, are in accord with those of the calculated χ shown in Fig. 2. The magnitudes of the chiralitydependent Δα(z 6 ) get considerably increased by a factor of up tõ 10 3 compared to those of the reference differential absorption (Δα(z 6 ) Ref ) from the pure chiral layers of the same thickness in the absence of the PT-symmetric substrate under the identical illumination shown in Fig. 5b, which are consistent with the aforementioned enhancement of three orders of magnitude in the chiroptical responses ( Fig. 2 and Supplementary Fig. S2).
An ultrasensitive monochromatic chiroptical sensor. We now consider the responses of the CPA-locked S C with L c = 50 nm to complex-valued κ. Figure 6 illustrates the polarization ellipses of the time-dependent transmitted beams (c + e + + c − e − ) at λ CPAL for four possible combinations of the complex-valued κ. The results show that a positive value of Re(κ) contributes to a negative ϑ, while a positive value of Im(κ) leads to a negative χ. For purely real n c = 1.33 with the specified values of κ given in Fig. 6, we obtain symmetric distributions of (ϑ, χ) = (±0.532°, ±0.532°), whereas (ϑ,χ) for n c = 1.33 + 10 −4 i are equal to (0.509°, 0.553°), (−0.553°, 0.509°), (−0.509°, −0.553°), and (0.553°, −0.509°). This asymmetry is because there exists a lossy term in the nonchiral part of the sample's refractive index (n ± ) and the differences between the results of two different cases of n c are all less than ±5%. Potentially being an ultrasensitive monochromatic chiroptical sensor, the CPA-locked S C reveals the enhanced chiroptical signals of~±0.5°merely from two 50-nm-thick chiral layers adhering to the PT-symmetric surfaces, which are well above the sensitivity range of the commercial CB and CD instruments that commonly have an accuracy of 1 mdeg. In general, the chiroptical enhancement at λ CPAL is mediated by the extinction depth of the CPA mode which in principle can be extremely high through fine-tuning the refractive indices of the gain and loss slabs to exactly match the CPA eigensolution 3 . However, the overall performance of the PT-symmetric system is practically limited by noises, such as photon shot noise level nearby λ CPAL as well as flicker noise and amplitude/phase noise from the detectors and laser sources used in the measurements 34,46 .
Undoubtedly, any slight relative changes on the refractive index and thickness of the gain-loss bilayer, mainly caused by imperfections during the sensor manufacturing process, will significantly affect the chiroptical responses (ϑ S C and χ S C ). The former n 3 ≠n Ã 4 can be alleviated with the bilayer structure made by low refractive index materials, for instance, Re[n] = 1.5, and considerably improved by allowing refractive index and dispersion accurate to the fourth decimal place or better. As for the latter L 3 ≠ L 4 , the thin slab of micrometer size imposes a constraint that requires a thickness precision of sub-nm, which can be achieved by atomic layer deposition given that the dielectric material is chosen as the gain-loss material. With these requirements, one can have a decent CPA mode, while maintaining at least 50% of maximal chiroptical signals of the ideally PT-symmetric system.

Conclusions.
In summary, we have demonstrated that the CPAlocked PT-symmetric system is capable of, in a highly sensitive manner, boosting chirality-dependent signals from molecularly thin chiral layers bound to the surfaces of the PT-symmetric substrate. The structurally achiral PT-symmetric system itself does not make any contributions to the total chiroptical signals, which enables absolute chirality measurements without requiring the background subtraction and the substrate removal for acquiring a reference measurement. A 1000-fold enhancement in the chiroptical responses offered by the CPA-locked PT-symmetric system makes it feasible to sense the molecular chirality of monolayered substances with existing research-grade instruments, which was previously unattainable. The linear dependence between enhanced ϑ (χ) signals and Re(κ) (Im(κ)) allows for quantitative and unambiguous discrimination of the chirality of an unknown sample even though there exists a lossy term in the nonchiral part of the sample's refractive index. Furthermore, we develop an analytical model to provide an insightful interpretation of physical mechanisms behind chiroptical enhancements supported by the CPA-locked PT-symmetric system. The CPAlocked PT-symmetric detection scheme has not been studied before and may be regarded as a promising chiral-sensitive platform for detections of extremely small chiral biomolecules, monolayers of chiral substances, and self-assembled chiral layers with high sensitivity. Our study explores the fundamental properties of non-Hermitian chiral photonics and paves the way for ultrasensitive chiroptical sensors and their applications.

Methods
Transfer matrix in circular polarization representation. Transfer matrix formalism is used to analyze the propagation of electromagnetic waves through a PTsymmetric system in the absence and presence of chiral layers present in mediums 2 and 5, respectively. Because there are two possible circular polarizations (lefthanded circular polarization, LCP (+), and right-handed circular polarization, RCP (−)) for the forward-and back-propagating waves on each of two sides of the system, the PT-symmetric system can be described by a 4 × 4 transfer matrix M which relates a 4 × 1 circular Jones vector on the one side to the other 4 × 1 circular Jones vector on the opposite side, as illustrated by the green arrows in medium 1 and 6 of Fig. 1a.
where T ij is a 4 × 4 interface transition matrix from medium i toward medium j, which relates the circular Jones vector a þ ; a À ; b þ ; b À Â Ã T i on the left-hand side to the other c þ ; c À ; d þ ; d À Â Ã T j on the right-hand side of the interface, with the superscript T indicating transpose, P j is a 4 × 4 propagation matrix that accounts for the phase contributed by forward-and backward-CP waves propagating inside the medium j with a thickness of L j , and M is a 4 × 4 global transfer matrix obtained by successive multiplication of individual T ij and P j matrices. The number of rows of the transfer matrix corresponds to four independent boundary conditions at the interface, i.e., the continuity of tangential electric and magnetic fields. The transfer matrix method explicitly takes into consideration the effect of multiple reflection and transmission of light at all interfaces on the overall system responses.
Bi-isotropic constitutive relations and eigenmodes in chiral medium. Chiral medium, present in layers 2 and 5, is a bi-isotropic medium, in which complex electric and magnetic flux densities (D and B) and fields (E and H) are related by the biisotropic constitutive relations which can be written in the time-harmonic convention e −iωt as D = ε 0 ε r E + i(κ/c) H and B = μ 0 μ r H − i(κ/c) E, where ε 0 and μ 0 are the permittivity and permeability of free space, ε r and μ r are the relative permittivity and permeability of the chiral layer, c is the speed of light in free space, and κ is the chirality parameter of the chiral substance. κ measures the degree of the handedness of the material and a change in the sign of κ means taking the mirror image of the material 47 . The eigenmodes propagating along the z axis in the chiral region can be found by inserting the constitutive relations into time-harmonic Maxwell equations, ∇ × E = iωB and ∇ × H = −iωD, in a matrix formulation, whereby the matrix is represented in terms of its eigenvalues k ± = k 0 n ± = k 0 (n j ± κ j ) & = −k 0 (n j ± κ j ), the wavenumbers of the medium for the LCP (+) and RCP (−) waves propagating along the positive and negative z axis, and its associated eigenvectors , unit column vectors that play the role of the circular basis for the LCP (+) and RCP (−) electric field, where n j and κ j denote the average refractive index and the chirality parameter of the medium j, respectively. The relation between magnetic and electric unit vectors is given by h ± = ∓ie ± . Therefore, the allowed solutions of the plane waves for the chiral medium are four distinct ± h ± e Àik ± z with the complex electric-field amplitude E) propagating along the z axis with different phase velocities (v ± = c/(n j ± κ j )), where η ± = η 0 /n j is the impedance parameters of the medium j with η 0 being the characteristic impedance of free space. It should be noted that the impedance parameters do not depend on the chirality parameter while the wavenumbers do. The aforementioned equations hold for the fields in nonchiral medium (i.e., layer 1, 3, 4, and 6) with κ = 0.
where r and t are complex amplitude reflection and transmission coefficients, the superscript L and R indicate the incoming CP waves on the left-or right-handed sides of the system, and the first and second subscripts denote the handedness of the scattered and incident CP waves, respectively. Unlike the transfer matrix, each matrix element of the scatting matrix has its physical meaning. In the case of S PT , eight of offdiagonal elements are zero, r L ÀÀ ¼ t R Àþ ¼ t L þÀ ¼ r R þþ ¼ 0 and r L þþ ¼ t R þÀ ¼ t L Àþ ¼ r R ÀÀ ¼ 0, because of the conditions of continuity at the interfaces, and the rest of off-diagonal elements are nonzero and equal, t R ÀÀ ¼ t R þþ ¼ t L þþ ¼ t L ÀÀ t, owing to the conservation of energy and time-reversal symmetry. Four diagonal elements can be reduced to two, r L Àþ ¼ r L þÀ r L and r R þÀ ¼ r R Àþ r R , irrespective of the handedness of the incident waves. It is apparent that S PT is a non-Hermitian matrix, S PT ≠ S PT y , where the superscript dagger indicates the Hermitian adjoint. Because of respecting PT symmetry, S PT can be characterized by the invariance σ x S PT À Á À1 h i Ã σ x ¼ S PT and satisfies the relation PTS PTPT ¼ S PT À Á À1 , where the asterisk denotes complex conjugate,P is the parity operator (P σ x , the x component of 4 × 4 Pauli matrices) andT is the time-reversal operator (T K, the complex conjugation operator) 48,49 . The resultant S PT matrix is verified by COMSOL Multiphysics (Supplementary Note 14).

Data availability
The data that support the findings of this study are available from the corresponding author upon reasonable request.

Code availability
The code that supports the findings of this study is available from the corresponding author upon reasonable request.
Received: 29 November 2021; Accepted: 10 March 2022; Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made. The images or other third party material in this article are included in the article's Creative Commons license, unless indicated otherwise in a credit line to the material. If material is not included in the article's Creative Commons license and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this license, visit http://creativecommons.org/ licenses/by/4.0/.