Chirality-enabled topological phase transitions in parity-time symmetric systems

Photonic spin Hall effect (PSHE) in chiral PT-symmetric systems exhibits many exotic features, but the underlying physical mechanism has not been well elucidated. Here, through rigorous calculations based on full-wave theory, we reveal the physical mechanism of the exotic PSHE and identify a chirality-enabled topological phase transition. When circularly polarized light is incident on a chiral PT-symmetric system, the transmitted beam contains two components: a spin-flipped abnormal mode that acquires a geometric phase (exhibiting a vortex or a spin-Hall shift), and a spin-maintained normal mode that does not exhibit such a phase. If the phase difference between the cross-polarized Fresnel coefficients cannot be ignored, it results in a chirality-enabled phase and intensity distribution in the abnormal mode, which induces an exotic PSHE. Consequently, as the incident angle increases, a chirality-induced topological phase transition occurs, namely the transition from the vortex generation to the exotic PSHE. Finally, we confirm that the asymmetric and periodic PSHE in the chiral slab is also related to the phase difference between the cross-polarized Fresnel coefficients. These concepts and findings also provide an opportunity for unifying the phenomena of topological phase transitions in various spin-orbit photonic systems.


Introduction
As the counterpart of angular momentum in mechanical systems, light can also have angular momenta, including spin angular momentum (SAM) and orbital angular momentum (OAM) [1,2].The SAM is related to the polarization of light.A circularly polarized (CP) light with left-or right-handedness carries a SAM of ±h per photon.Meanwhile, there are two types of OAM, namely, intrinsic OAM and extrinsic OAM, the former and the latter being dictated by the vortex nature of the wavefront and the spin-dependent transverse displacement, respectively [3,4].The conversion or coupling between the SAM and OAM of light is called the optical spin-orbit interaction (SOI) [1,5].The SOIs are widely present in optical systems with interface reflection and refraction, such as optical slab, anisotropic medium, strong focusing, particle scattering, and evanescent wave [6][7][8][9][10].The conversion from SAM to intrinsic OAM results in an optical vortex with a hole (singularity, with zero light intensity) at its center, which can be regarded as a topological state [11][12][13][14].The conversion from SAM to extrinsic OAM manifests the generation of photonic spin Hall effect (PSHE, i.e. the spin-dependent transverse displacement of the centroid of the beam) without a hole represents another type of topological state [4,6,15,16].Therefore, changing the relative weights of these two OAM contributions makes a transition between these two types of SOIs, which can be regarded as a topological phase transition (TPT) [17][18][19][20][21][22][23].
Numerous methods have been proposed to manipulate the TPT [20,22,24].A common category of these methods involves modulating the properties of the light beam, such as the incident angle, angular momentum, and spatial structure.The underlying mechanism is associated with the momentum-dependent Pancharatnam-Berry (PB) phase [17-20, 22, 23].As the incident angle increases, it induces a TPT related to this PB phase [17].Consequently, manipulating the TPT enables the modulation of the PSHE through the properties of the light beam.Furthermore, the PSHE can also be modulated by the properties of the medium [25,26], and chirality is an effective degree of freedom for regulating PSHE [27][28][29][30].Although natural materials usually exhibit weak chirality [31], an increase in the chirality parameter is possible since chiral metamaterials can dramatically enhance the electromagnetic coupling and have greater chirality [32,33], thereby more effectively regulating the PSHE [27,28,34].Additionally, non-Hermitian optical systems, by adjusting their geometric and material parameters, can also effectively manipulate the PSHE [35][36][37][38][39][40].Particularly, systems that exhibit parity-time (PT) symmetry serve as powerful tools for manipulating PSHE [35,36,39,41,42].Moreover, systems with PT symmetry that incorporate chiral metamaterials [43][44][45][46], known as chiral PT-symmetric systems, can offer more methods for manipulating the PSHE and TPT.The chiral PT-symmetric systems, as an example, can be formed using realistic metamaterials consisting of two metallic crosses twisted by 30 • with respect to each other and embedded in a dielectric host of low refractive index [33,43].However, the PSHE and TPT in chiral PT-symmetric systems exhibit anomalous behaviors, such as an asymmetric PSHE that is influenced by the sign of the imaginary component of chirality [44,45].The underlying physics of the exotic PSHE and TPT in chiral PT-symmetric systems is not yet fully understood.
Here, through rigorous calculations, we reveal the underlying physics of the exotic PSHE in chiral PT-symmetric systems and identify a chirality-enabled TPT.We discovered that the anomalous PSHE in the chiral PT-symmetric system is caused by the phase difference between the cross-polarized Fresnel coefficients that cannot be ignored.The PT-symmetric condition suppresses the cross-polarized Fresnel coefficients caused by the real part of the chirality parameters, leading to an asymmetrical PSHE mainly dependent on the imaginary part of these parameters.Furthermore, the phase difference induces a chirality-enabled TPT in chiral PT-symmetric systems as the incident angle increases.We have confirmed that such a TPT widely exists in chirality-related systems, causing asymmetrical or periodic PSHE.These findings provide an opportunity to unify the TPT phenomena in various spin-orbit photonics systems.

Theoretical model
A chiral medium is characterized by magnetoelectric coupling (i.e. an applied electric field results not only in electric polarization but also in magnetic polarization; same for an applied magnetic field).This results in a different index of refraction for left-handed circularly polarized (LCP) waves and right-handed circularly polarized (RCP) waves, and circularly polarized waves are the eigensolutions of the wave equation in such media [46].The chiral constitutive relations are written as [47] where ε, µ, κ refer to the relative permittivity, permeability and the chirality parameter, respectively (ε 0 and µ 0 are the vacuum permittivity, permeability, respectively, while c represents the speed of light in a vacuum.).As the chiral constitutive relations are incorporated into Maxwell's equations [47][48][49][50][51][52], the conditions of the chiral PT-symmetric system are as follows [53]: In PT-symmetric systems, it is necessary to maintain a balance between loss and gain.Therefore, at least two components are required in the model [52,54].To facilitate theoretical simplification and analysis of physical mechanisms, we consider only the chiral PT-symmetric system with two layers, as shown in figure 1. Certain designs of chiral metamaterials have demonstrated extremely high optical activity and strong circular dichroism, from microwave frequencies up to the visible range [53].The parameters of chiral metamaterials that satisfy the chiral PT-symmetric system are given in the literature [43,45,47,54].Here, we adopt the parameters: ε (z) = 4 + i0.6, µ (z) = 1.5 + i0.15, with |κ re | < 0.5 and |κ im | < 0.5.Applying the boundary conditions of the continuity of the tangential components of the electric and magnetic fields at the three interfaces, z = −d, z = 0, and z = d, the fields inside and outside the system can be calculated through the solution of a 12 × 12 linear system of equations [47].More details about the model of the chiral PT-symmetric system can be found in the literature [47,54].The transmission coefficients of the circular polarizations can be obtained via the calculated fields and the Fresnel-Jones matrix F t cp can be written as [47]: The subscripts + and-denote RCP and LCP, respectively.The s and p denote the transverse electric and magnetic waves, respectively.The reflection coefficients for the circular polarizations can be obtained in the same way.
Since a plane wave theory cannot fully describe the nature of the light beam, a full-wave theory is required to obtain more information about light beam propagation in the chiral PT-symmetric system.As shown in figure 1, the coordinate systems (x, y, z) and (x a , y a , z a ) are the laboratory and local coordinate systems, respectively, where y and y a point in the same direction.Here, the superscripts a = {i, t, r} denote the incident, transmitted and reflected light, respectively.It is assumed that U a ± (k a ) dictates the transverse patterns of the ath beam in k space in the CP basis.At a transverse reference plane z a , the light beams can be written as where 2 being position vector, wave vector, transverse wave vector component, and longitudinal wave vector component.It has In previous work [17], the full-wave theory has been established to connect the transmitted and incident Gaussian beam, i.e.
where the 2 × 2 matrix M t cp can be written as where P t, i is the projection operation connecting the arbitrary plane wave and the central wave in the CP bases.More details of P t, i can be found in [17,55].
The transverse patterns of an incident Gaussian beam of a CP wave can be written as where w 0 = 50λ is the beam waist.The transverse patterns U t ± k t and electric field E t R t ⊥ of the transmitted beam are expressed as below Therefore, the normal mode can be written as: ⊥ and the abnormal mode can be written as: Finally, with the E-field distribution of the reflected beam fully known, we can then use the following formula to calculate the beam shifts [56].Longitudinal and transverse shifts are defined as δ x± and δ y± , respectively For the reflected beam, similar results for longitudinal and transverse shifts can be achieved through the same methodology.

The mechanism of anomalous PSHE in parity-time symmetric systems
In the chiral PT-symmetric systems, as chirality parameter κ im = 0 (A-chiral PT-symmetric system, κ = κ re ) or κ re = 0 (B-chiral PT-symmetric system, κ = κ im ), it is found that the exotic PSHE exhibits both longitudinal and transverse shifts that are symmetrically distributed with respect to κ re and distributed with respect to κ im [45].
As shown in figure 1, under CP incidence, the transmitted beam contains two components: a spin-flipped abnormal mode acquiring geometric phase (exhibiting spin-Hall shift effect) and a spin-maintained normal mode without such phase [56].Under linearly polarized (LP) incidence, however, the spin component of the transmitted beam must be the sum of normal and abnormal components of transmitted beams corresponding to the CP incidence of different helicities.Therefore, whether it is CP or LP incidence, PSHE will be greatly affected by abnormal modes.To comprehend the exotic PSHE, the intensity patterns ( E t In the A-chiral PT-symmetric system with κ re = ±0.1 (figure 2(a)), all intensity patterns exhibit x-axis symmetry.It is observed that the direction of the shift in the weight center of the beam remains unchanged, and the magnitude of the shift is the same when the sign of the chiral parameter is reversed.This implies that the magnitude of the photonic spin Hall shift is invariant regardless of whether the chiral parameter is positive or negative.However, in the B-chiral PT-symmetric system with κ im = ±0.1 (figure 2(b)), it is difficult to find the symmetry of the intensity patterns.Altering the sign of the chiral parameter leads to a variation in the magnitude of the shift of the beam's weight center, indicating that the magnitude of the photonic spin Hall shift is influenced by the sign of the chiral parameter.
Comparing the phase patterns of M t −+ in figures 2(a) and (b), it can be observed that the phase patterns in figure 2 According to equations ( 1) and ( 4), the terms of M t −+ can be divided into two parts, co-polarized (related to t ss and t pp ) and cross-polarized (related to t ps and t sp ) components, by assigning one of the components a zero value.The phase patterns of M t −+ and its components are shown in figure 2(c).The phase patterns of M t −+ with two singularities for κ re of ±0.1 differ from that with one singularity for κ re of zero, as shown in figure 2(c)-[Total].Interestingly, the phase patterns of co-polarized component ([Co.]) for κ re of 0, ±0.1 are similar and have only one singularity.However, the phase patterns of cross-polarized components ([Cross]) for κ re of ±0.1 are different and have two singularities.This suggests that the deviations in the phase pattern are primarily caused by the cross-polarized components.Moreover, there is only one singularity in the phase patterns of the terms containing t ps ( As shown in figure 3(a), in the A-chiral PT-symmetric system, the cross-polarized Fresnel coefficients are always several orders of magnitude lower than those of the co-polarized ones.Therefore, the cross-polarized components can be neglected in the A-chiral PT-symmetric system.However, in the B-chiral PT-symmetric system, the cross-polarized Fresnel coefficients should not be ignored when |κ im | is larger than 0.01.As shown in figure 3(b), the cross-polarized Fresnel coefficients remain at the same level as the co-polarized ones when |κ im | exceeds 0.08.
The question is why the real part of the chirality parameter generates ignorable cross-polarized Fresnel coefficients in the A-chiral PT-symmetric system.As we know, the real part of the chirality parameter describes the rotation of the polarization, whereas the imaginary part one describes the circular dichroism [46].The disappearance of cross-polarized Fresnel coefficients implies that the effect of polarization rotation appears to be suppressed in the A-chiral PT-symmetric systems.Reconsidering the PT-symmetric condition on chirality in the A-chiral PT-symmetric system, κ (z) = −κ * (−z), i.e. κ (z) = κ re and κ (−z) = −κ re .This indicates that the real part of chirality in the gain and loss materials is opposite, meaning that the rotations of polarization have opposite directions.Since the beam paths in the gain and loss materials are equal, the effect of the real part of the chirality (the polarization rotations) on the cross-polarized Fresnel coefficients is completely suppressed.Therefore, the cross-polarized Fresnel coefficients in the A-chiral PT-symmetric system can be ignored, resulting in the magnitude of the photon spin Hall shift being the same regardless of whether the sign of the chiral parameter is positive or negative.
It is interesting that although t ps and t sp are almost the same, their phase difference at κ re ̸ = 0 is π .Under the paraxial-wave approximation, the abnormal model can be written as: In figure 3(c), for κ im > 0, Arg t ps ≈ Arg t sp + π and t ps ≈ t sp .Therefore, for example, the cross terms of M t −+ , e 2iϕ k i t ps + t sp , can be written as it sp e 2iϕ k + e 2i(ϕ k +π /2+o(δ)) .Here, o (δ) represents an arbitrarily small value.For κ im < 0, a similar result can be obtained, i.e. it ps e 2iϕ k + e 2i(ϕ k +π /2+o(δ)) .This indicates that the phase difference (π ) between t ps and t sp will result in the phase difference (π /2) between t ps and t sp , leading to double singularities in [Cross].That is to say, the two singularities in [Cross] are the result of interference between the t ps term and the t sp term.
It can be summarized that the combined effect of PT symmetry and chirality parameters on the cross-polarized Fresnel coefficients leads to the exotic PSHE.When the phase difference between the cross-polarized Fresnel coefficients cannot be ignored, it introduces a new phase in the abnormal model, resulting in the exotic PSHE.The PT symmetry condition allows the cross-polarized Fresnel coefficients generated by the real part of the chiral parameter to be negligible.Therefore, the asymmetric PSHE is solely related to the imaginary part of the chiral parameters.This is the reason why the asymmetric PSHE is observed in the B-chiral PT-symmetric system.

Chirality-enabled topological phase transitions in parity-time symmetric systems
As mentioned above, when the cross-polarized Fresnel coefficient cannot be ignored, it introduces a new phase in the phase pattern.As the incidence angle increases, during the transition from vortex generation to PSHE, the question arises whether a new type of TPT exists.To verify the generality of this phenomenon, we consider a chiral PT-symmetric system with κ re = κ im = ±0.2 to clarify this issue.
In the non-chiral PT-symmetric system (that is, κ re = κ im = 0), the intensity patterns ( E t −+ 2 ) of the transmitted abnormal model vary with incident angles and are the same as those driven by TPT related to the momentum-dependent PB phase (as shown in figure 4, the second row) [17].When the chirality parameter is non-zero, the intensity pattern is also donut-shaped at normal incidence and dot-shaped at an angle of incidence of 5 • , which is consistent with the previous result.However, there is a significant difference in the intensity patterns of the chiral PT-symmetric systems when the incident angles are between 0 and 5 • (in figure 4, the first and third rows).The evolution of the intensity pattern in the chiral PT-symmetric system differs from that in the non-chiral system.It implies that there is a chirality-related transition between the two types of SOIs.At normal incidence, all phase patterns are similar and have only one singularity, as shown in figure 5(a).This is the reason why the intensity patterns in figure 4 appear as a doughnut shape for normal incidence [17].At oblique incidence, in a non-chiral PT-symmetric system, the phase patterns of M t −+ (Φ 0 ) are similar to the momentum-dependent PB phase patterns (Φ PB ) and have only one phase singularity, which gradually disappears with the increase of incident angle.However, the evolution of phase patterns of M t −+ for κ re of ±0.2 (Φ ±0.2 ) are different from Φ PB and Φ 0 .As the incident angle reaches 0.2 • , Φ PB and Φ 0 have one singularity, while the Φ 0.2 and Φ −0.2 have two singularities.At the incident angle of 1 • , the phase singularity disappears in the Φ PB and Φ 0 patterns, but phase singularity also appears in Φ 0.2 and Φ −0.2 .This means that the phase pattern of M t −+ in the non-chiral PT-symmetric system is mainly determined by the momentum-dependent PB phase, and that in the chiral PT-symmetric system is determined by the combination of the momentum-dependent PB phase and the new phase.At the incident angle of 5 • , the phase singularity disappears in all phase patterns, which results in the intensity patterns in figure 4 appearing as the dot shape (i.e.Gaussian profile) at the angle of incidence of 5 • [17].
In addition, for the incident angle of 5 • , although there are no phase singularities in both Φ 0.2 and Φ −0.2 patterns, their phase distributions are different.Namely, for larger incident angles, the influence of topological phase transitions driven by chirality does not vanish.The different signs of chirality lead to different phase distributions, which in turn result in different phase gradients.Due to the correlation between PSHE enhancement and phase gradient at large incident angles, the enhancement of PSHE varies with the signs of chirality.Similar results have been observed in reflected light at large angles of incidence, for example, at 71 • as reported in the literature [45].When k im = 0.01 and k im = −0.01,δ + = 10.8λ and δ + = 1.64λ, respectively.It should be noted that, for linearly polarized incident light, it is necessary to consider the coherent effects of different spin components, such as, M a ++ + M a +− or M a −+ + M a −− .Moreover, the distributions of light intensity and phase become complex at large angles of incidence.Therefore, the reflected and transmitted light at large angles of incidence cannot be simplistically divided into cross-polarized and co-polarized components.Here, to more clearly reveal the topological phase transitions in the chiral PT-symmetric system, we primarily focus on the SOIs at small angles of incidence.
Vortex mode decomposition offers an alternative perspective for understanding the SOIs and is considered an effective method for describing the TPT phenomenon.The abnormal mode (E t −+ ) can be decomposed into a series of Laguerre-Gaussian (LG) modes.More specifically, C m p denotes the normalized weight coefficient of the mth-order azimuthal LG mode with radial index p [18,57]: where E LG m p stands for the electric field distribution of the LG beam.Since the mode of the high-order radial index component (p > 0) can be ignored in the present system, only the normalized weight coefficient of the p = 0 component (C m ) is shown in figures 5(b)-(e).At normal incidence, only a single vortex mode with a topological charge of m = 2 can be found, as shown in figure 5(b).At oblique incidence, new vortex modes (0, 1) appear, competing and superposing with m = 2 vortex mode, as shown in figures 5(c)-(e).It is observed that the normalized weight coefficient of m = 2 vortex mode for κ re of ±0.2 is always greater than that for κ re of 0, which is consistent with what is found in the evolution of the singularity of the phase pattern of the M t −+ .Therefore, it is further confirmed that the chirality can induce a TPT, i.e. chirality-enabled TPT.

Extensions
In contrast to the results in the nonchiral slab system, both asymmetrical PSHE and periodical PSHE have been found in the chiral interface or chiral slab systems with only the real parts of the chirality parameter [27][28][29][30].Therefore, what is the difference in the origin of this exotic PSHE?To explore this, we consider a thin slab system characterized by real chirality parameters, i.e. κ = κ re .For simplicity, we also assume that there is neither loss nor gain, i.e. ε = 4, µ = 1.5.
In the chiral slab system, both the transverse (δ y± ) and longitudinal (δ x± ) shifts are asymmetrically distributed as the incident angle increases, as shown in figure 6(a).Unlike the results in the nonchiral slab system, the amplitudes of δ y± are not equal, and the same is true for δ x± , except when all amplitudes are zero.In figure 6(b, upper one), it appears that amplitudes of δ y± are equal within a certain range of κ re , but in fact, they are only very close but not exactly equal.In figure 6(b, lower one), the amplitudes of δ x± are not equal, unless all of them are zero.This indicates that the transverse shifts are asymmetrical with respect to the handedness of light.Additionally, the amplitudes of δ x± and δ y± oscillate periodically with κ re , and the period of these oscillations period is approximately 0.24.In figure 6(b), the longitudinal shifts exhibit asymmetric variations depending on the sign of chirality and the spin components.In the previous report, the mechanism of the longitudinal photonic spin splitting is related to the Goos-Hänchen shift and the imaginary part of the relative permittivity of the dielectric significantly affects the symmetry of the longitudinal PSHE [58].However, here, the permittivity and permeability do not have an imaginary part, and the chirality, which has only a real part, can also control the symmetry of the longitudinal PSHE.This further demonstrates that there is an alternative mechanism through which chirality regulates the PSHE.
In figure 6(c), the lines of the Fresnel coefficients overlap because t sp ≈ t ps and |t ss | ≈ t pp .The co-polarized and cross-polarized Fresnel coefficients exhibit periodic oscillations.The period of the amplitude of these coefficients with respect to κ re is approximately 0.24.Obviously, within most of the κ re range, the co-polarized and cross-polarized coefficients vary in the same order.Therefore, the cross-polarized coefficients cannot be ignored.Consistent with the above results, the phase difference between t sp and t ps is approximately π, and the phase of t ss is close to that of t pp , as shown in figure 6(d).The period of the phases of the Fresnel coefficients with respect to κ re is also about 0.24.This indicates that the periodic PSHE is related to the periodic variation in the Fresnel coefficients, which is caused by the rotation of polarization.
With the increase in incident angle, the phase patterns (Arg M t −+ and Arg M t +− ) in the chiral slab system with κ re of 0.01 are shown in figure 7(a).The number of phase singularities first increases from one to two, and then the phase singularity disappears, which is consistent with that in the chiral PT-symmetric system.It is also found that the evolution of Arg M t −+ and Arg M t +− patterns are different.For example, at the incident angle of 0.5 • , the Arg M t −+ pattern still exhibits two singularities, whereas the Arg M t +− pattern shows only one.This difference in pattern evolution is the reason why asymmetric PSHE, with respect to the handedness of light, can occur in the chiral slab system.With the increase in κ re , the intensity patterns ( E t −+ 2 ) and phase patterns (Arg M t −+ ) in the chiral slab system are shown in figure 7(b).It can be found that the number of phase singularities changes periodically (figure 7(b), first row).The period of the number of phase singularities with respect to κ re is about 0.24, which is similar to that of PSHE.The intensity patterns at κ re values of 0.241 and 0.483 are similar to these at κ re = 0. Consequently, the intensity patterns exhibit a period in κ re that is similar to that observed in PSHE (figure 7(b), second row).These results validate that the asymmetric and periodic PSHE in chiral slabs are related to the differences between cross-polarized Fresnel coefficients, which cannot be ignored.It also indicates that chirality-enabled TPT widely exists in chirality-related systems.

Conclusion
The underlying physics of the SOIs for the exotic PSHE in the chiral PT-symmetric system is revealed, and a chirality-enabled TPT is identified based on rigorous calculations.We found that the exotic PSHE is caused by the combined influences of PT symmetry and chirality parameters on the cross-polarized Fresnel coefficients in the chiral PT-symmetric system.If the phase difference between the cross-polarized Fresnel coefficients cannot be ignored, it introduces a chirality-enabled phase and intensity distribution.As the incident angle increases, this phase difference results in a chirality-enabled TPT, thereby causing exotic PSHE.It is confirmed that the asymmetric and periodical PSHE in chiral slabs are also related to the chirality-enabled TPT, which widely exists in the chirality-related system.We note that, based on the similar methods and concepts in this paper, other TPTs may exist depending on the properties of media in the system where SOIs are manipulated by the anisotropy, including anisotropic liquid crystal [12], graphene [59], and anisotropic metasurfaces [7], etc.Therefore, these concepts and findings provide an opportunity for unifying the TPT phenomena in various spin-orbit photonics systems.
phase patterns (Arg M t −+ and Arg[M t +− ]) of the abnormal mode in the transmitted beams are calculated and displayed in figures 2(a) and (b).
(a) only have one singularity, whereas all phase patterns in figure 2(b) have double singularities.Additionally, all phase patterns in figure 2(a) exhibit vertical symmetry, whereas symmetry is difficult to discern in the phase patterns of figure 2(b).

Figure 2 .
Figure 2. The intensity and phase patterns of the abnormal modes in the beams transmitted through the chiral PT-symmetric system at an incident angle of 0.2 • and a thickness of d = 1.04λ.(a) κ = κre (A-chiral PT-symmetric system) and (b), (c) κ = κ im (B-chiral PT-symmetric system).The [Total] indicates the phase patterns of M t −+ .The [Co.] and [Cross] denote the phase patterns of the co-polarized (including tpp and tss) and cross-polarized (including tps and tsp) components of M t −+ , respectively.The [tps] and [tsp] denote the phase patterns of the terms containing only tps and tsp, respectively.Note: since tps and tps tend to zero when κ im equals to zero, the corresponding [Cross], [tps] and [tsp] cannot be determined and are labeled an 'N' .The other parameters match those in figure 1.

Figure 3 .
Figure 3. Fresnel coefficients at an incident angle of 0.2 • for thickness d = 1.04λ.The magnitude of the Fresnel coefficients for (a) A-chiral PT-symmetric system and (b) B-chiral PT-symmetric system.(c)The phase of Fresnel coefficients for B-chiral PT-symmetric system.Overlapping lines due to |tsp| ≈ |tps|, |tss| ≈ |tpp| and Arg [tss ]≈ Arg[ tpp].Other relevant parameters as in figure 1.

Figure 5 .
Figure 5.The phase patterns and vortex mode components of the abnormal model of the beams transmitted through the chiral PT-symmetric system with thickness d = 1.04λ.(a) In the first, second, third, and fourth rows, there are patterns of Pancharatnam-Berry phase (Φ PB), the phase of M t −+ for κre of 0 (Φ 0), 0.2(Φ 0.2) and −0.2 (Φ −0.2 ), respectively.The vortex mode components of E t −+ at incident angles of (b) 0, (c) 0.2, (d) 0.5, and (e) 1 • .The relevant other parameters are the same as in figure 4.

Figure 7 .
Figure 7.The phase and intensity patterns in the chiral slab system.(a) The evolution of phase (Arg[M t −+ ] and Arg[M t +− ]) patterns vary with incident angle for κre of 0.01.(b) The evolution of phase (Arg[M t −+ ]) and intensity (|E t −+ | 2 ) patterns vary with κre at incident angle of 0.2 • .The relevant other parameters are the same as in figure 6.
[t ps ]) or t sp ([t sp ]).The [t ps ], when rotated by about 90 • around the singularity, coincides with the [t sp ].This indicates that the double singularities in [Cross] are due to the phase difference between [t ps ] and [t sp ].