Revisiting physical mechanism of longitudinal photonic spin splitting and Goos-Hänchen shift

The intrinsic connection between the transverse photonic spin Hall effect (PSHE) and the Imbert–Fedorov shift has been well characterized. However, physical insights into the longitudinal photonic spin splitting associated with the Goos-Hänchen (GH) shift remain elusive. This paper aims to expand the theory of the PSHE generation mechanism from the transverse to the longitudinal case by examining the reflection of each spin component from an arbitrarily linearly polarized incident Gaussian beam on the air-dielectric interface. Unlike the transverse case, both spin-maintained and spin-flipped modes exhibit non-zero longitudinal displacements, with the latter being affected by the second-order expansion term of the Fresnel reflection coefficient with respect to the in-plane wave-vector component. Meanwhile, the polarization angle plays a crucial role in determining the longitudinal PSHE since each reflected total spin component is a coherent superposition of these two corresponding modes. Remarkably, the imaginary part of the relative permittivity of the dielectric significantly affects the symmetry of the longitudinal PSHE. Furthermore, the GH shift results from a superposition of individual spin states’ longitudinal displacements, taking into account their energy weights. By incorporating the corresponding extrinsic orbital angular momentum, we explore the generation mechanism of the symmetric/asymmetric longitudinal PSHE. The unified physical framework elucidating the longitudinal photonic spin splitting and GH shift provides a comprehensive understanding of the fundamental origin of the PSHE and beam shifts, paving the way for potential applications in spin-controlled nanophotonics.


Introduction
Nonspecular effects occur when the reflection and refraction of physical light beams at an interface differ from those predicted by geometric optics, which obey Snell's law and the Fresnel equations [1].Two well-known phenomena in this context are the Goos-Hänchen (GH) shift and the Imbert-Fedorov (IF) shift [2][3][4][5].The GH shift represents the total longitudinal displacement of the centroid of the reflected beam when a linearly polarized is incident, whereas the IF shift refers to the total transverse displacement of the centroid of the reflected beam when left or right circularly polarized (LCP or RCP) beam is incident.These shifts arise because the eigenpolarization modes for GH shifts involve P and S linearly polarized waves, which polarize parallel and perpendicular to the incident plane, respectively, while circularly polarized waves are associated with IF shifts [5].Furthermore, the transverse photonic spin Hall effect (PSHE) has been proposed, which amounts to the transverse spin splitting from an arbitrarily linearly polarized incident light [6].The PSHE originates from the conservation of spin angular momentum and extrinsic orbital angular momentum (EOAM) in the spin-orbit interaction [7,8].Afterwards, an analogous effect called the longitudinal PSHE (or longitudinal photonic spin splitting) has been observed, involving the in-plane spin separation [9].Due to their extensive practical applications such as high-precision measurements [10,11] and optical spatial differentiation [12,13], the PSHE and beam shifts have been investigated extensively for various beam types as well as materials [14][15][16][17][18][19][20][21][22][23][24][25][26][27][28].
Various theories have been developed to understand the physical mechanism of the PSHE.However, these theories primarily focused on the transverse shift [5,7,[29][30][31][32].When circularly polarized light is reflected from a generic interface, it produces two modes: a spin-maintained (normal) mode and a spin-flipped (abnormal) mode, which exhibit distinct transverse displacements.These displacements can be interpreted by the geometric phase theory [5,8,28,29,[31][32][33][34].As such, the transverse PSHE results from the splitting between centroids of the LCP and RCP reflected light waves.Remarkably, the generation mechanisms of the transverse and longitudinal PSHE differ due to their distinct physical origins.Specifically, the transverse PSHE arises when the incident beam possesses arbitrary linear polarization, whereas the longitudinal PSHE occurs exclusively when the incident polarization deviates from being parallel or perpendicular to the interface.Despite the discovery of the longitudinal spin splitting, little research has been conducted to gain insights into their physical nature [35].The growing body of research on the longitudinal PSHE [9,[35][36][37] highlights the increasing need for a comprehensive investigation into its physical mechanisms.Furthermore, the inherent physical connection between the GH shift and longitudinal PSHE remains elusive, although it is known that both phenomena originate from the in-plane spread of wave vectors.
In this paper, we extend upon the theory of the PSHE generation mechanism from the transverse to the longitudinal case by studying the reflection of each spin component from an arbitrarily linearly polarized (consisting of LCP and RCP components) incident Gaussian beam on the air-dielectric interface.Different from the transverse case, both spin-maintained and spin-flipped modes exhibit non-zero longitudinal displacements with various energy weights.Specifically, we observe that the second-order expansion term from the in-plane spread of wave vectors can affect the longitudinal spin shift for the spin-flipped mode.Each reflected total spin component from a linear polarization incidence is a coherent superposition between its two corresponding modes, leading to polarization angle-dependent longitudinal spin shifts.Significantly, the relative permittivity of the dielectric plays a great role in determining the symmetry of the longitudinal PSHE.Moreover, the reflected total GH shift is a superposition of individual spin states' longitudinal displacements with considering their energy weights.Through these discoveries, we provide novel physical insights into the underlying mechanism governing the generation of the longitudinal PSHE.These findings establish a unified physical framework, enabling a deeper understanding of the inherent connection between the longitudinal spin separation and GH shift, and offer potential avenues for future applications of spin-controlled nanophotonics.

Longitudinal spin separation and their total longitudinal (GH) shift of each spin state
It is well known that the transverse displacement originates mainly from the rotation of the local coordinates induced by the out-plane wave vector spread that shares the same incident angle but in different incident planes.Although the in-plane spread of wave vectors also affects the transverse shift values, typically only the first-order expansion term of the Fresnel reflection coefficient with respect to the in-plane wave vector needs to be considered.However, the longitudinal displacement stems from the in-plane spread of wave vectors that still shares the same incident plane albeit with different incident angles.This is the reason why P and S waves serve as the eigenpolarization modes of the incident beam for the longitudinal displacement rather than for the transverse displacement.Owing to the higher sensitivity of the longitudinal shift to variations of the in-plane wave vector, we will retain up to the second-order expansion term when calculating the reflected electric fields to ensure accuracy in the following calculations.
Figure 1(a) schematically depicts a general reflection model of the longitudinal PSHE, where the laboratory coordinate system represented by (x, y, z) is attached to the air-dielectric interface at z = 0, while the local coordinate systems denoted by (x i , y i , z i ) and (x r , y r , z r ) are attached to the geometric paths of the incident and reflected beams, respectively.The relative permittivity of the dielectric is represented by ε = ε r + iε i , with ε r and ε i being its real and imaginary parts, respectively.Considering a paraxial Gaussian beam (with the waist w 0 ) as the incident beam with narrow continuous distributions of wave vector k centered around k 0 = |k|ẑ i = k 0 ẑi (where k 0 = 2π/λ is a wave number with incident wavelength λ in vacuum and ẑi is a unit vector along the incident beam's central propagation direction) and an arbitrary linear polarization described by the Jones vector by |α⟩ = [cos τ sin τ ] T , where the polarization angle τ (τ is the angle between the linear polarization vector and the x i axis and T denotes the transpose operation, the incident angular spectrum can be written as The reflected angular spectrum can be calculated as (see appendix for a detailed derivation): where Û = r P 0 0 r S is the reflective matrix relating the incident and reflected angular spectrum in linear polarization basis, with r P and r S being the corresponding Fresnel reflection coefficients of P and S polarizations, respectively.The Fresnel reflection coefficients can be approximated by Taylor expansion around the central wave vector (k ix = k iy = 0), here considering the second-order expansion terms, as (see appendix for a detailed derivation): where r P,S (θ) is the ordinary Fresnel reflection coefficient of the air-dielectric interface.The second (third) term is the first-(second-) order expansion term of the in-plane spread of wave vectors k ix .Equation (3) can also be formulated in terms of the reflected angular spectrum r S,P k rx , k ry , under the boundary conditions: k rx = −k ix and k ry = k iy .To derive the reflected electric field of each spin component generated by each incident spin field, we perform the basis vector transformations of linear and circular polarizations for the reflective matrix in equation ( 2), resulting in: where ÛC is the reflective matrix connecting the incident and reflected angular spectrum in circular polarization basis, whose main-diagonal components denote the reflection coefficients of the spin-maintained part, and off-diagonal components represent the reflection coefficients of the spin-flipped is the transformation matrix linking the basic vectors of linear and circular , and V−1 is its corresponding inverse matrix.Therefore, considering only the LCP or RCP component as the incident field, i.e 0 Ẽ|R⟩ i , the spin-maintained and spin-flipped parts of the reflective electric fields of each spin component of the incident electric fields can be calculated by using the above equations and performing an inverse Fourier transform, as where  (6) shows that the reflected fields from the linear polarized incident beam have four individual parts: two from the incident LCP component and the other two from the RCP component, as also illustrated in figure 1(b).
In the first case, we study the longitudinal spin separation of individual reflected spin states induced by an LCP incident beam.By substituting equations (6a) and (6b) into the expression of the centroid of the beam, denoted as ⟨x r ⟩ in equation (A.5) of appendix, we derive the longitudinal spin displacements generated by this LCP incident component for two opposite polarization components, expressed by When the second-order terms are neglected, i.e. ξ = ς = 0, equations (7a) and (7b) degenerate into the first-order term case.Equation (7) shows that the spin shift of each spin component is independent of the polarization angle τ , because calculating the centroid is correlated to the field intensity which is also independent of τ despite the corresponding electric fields associated with τ (as shown in equation ( 6)).In  dx r dy r being their total energy in the cross-section.Equation (8) indicates that the longitudinal spin shifts of both spin-maintained and spin-flipped components contribute to the GH shift values, and are identical to the expression of the GH shift of the circularly polarized (LCP and RCP) incident Gaussian beam presented by the previous studies [5,17,19] when the second-order term is neglected (i.e.ξ = ς = 0) and k 2 0 w 2 0 (|r This also proves that the above-derived formulas are correct. In our numerical simulation, we select the waist w 0 = 20 µm and incident wavelength λ = 632.8nm. Figure 2 shows the dependence of reflected spin shifts of LCP and RCP components on the incident angle θ generated by the LCP incident beam at an air-weakly absorbed dielectric material with ε = 1.8 + 0.18i [14].As shown in figure 2(a), the spin shifts δ |L⟩ |L⟩ of the normal mode are nearly identical, regardless of whether the second-order expansion term of the in-plane spread of wave vectors k ix is considered or not.However, when neglecting the second-order expansion term of k ix , there are obvious differences in the spin shift values δ |L⟩ |R⟩ of the abnormal mode, particularly near the grazing incidence angle (θ = 90 • ), where the maximum difference in shift can reach 14.5 nm.Notably, this is the first study, to the best of our knowledge, to calculate the longitudinal spin shifts while incorporating the second-order expansion term of k ix .This distinction sets it apart from the previous research endeavours [9,[35][36][37].
Figure 3  grazing incidence angle, where the normal mode prevails.That is why the accuracy is adequate when calculating the GH shift only based on the first-order term of the reflection coefficient with respect to k ix .
For an RCP incident case, the longitudinal spin separation of each spin state and their total GH shift are In contrast to the transverse case where the spin shifts of the abnormal mode for incident LCP and RCP beams have equal magnitudes but opposite directions [30], the corresponding results of an RCP incident case are the same as those of an LCP incident case, as shown in equations ( 7)-(9).

Longitudinal PSHE of refelction
Similar to the transverse case [30], the longitudinal PSHE produced from an arbitrary linear polarization |α⟩ = [cos τ sin τ ] T can be calculated by considering that each spin component of the reflected fields consists of a pair with the same handedness from different incident spin states.Consequently, the expression for this process can be represented as  When the incident beam is the parallel (τ = 0) or orthogonal (τ = π /2) polarization, equations (10a) and (10b) become and where the longitudinal spin splitting no longer occurs but the same longitudinal displacement of LCP and RCP components exists, which are the total longitudinal (GH) shifts for the entire reflected beam.Therefore equations (10a) and (10b) are consistent with the expression of reflected GH shifts produced by the incident parallel and orthogonal polarization beams [5,17,19].It should be noted that despite the angle of polarization with τ = 0 or τ = π /2, the transverse spin splitting always occurs.However, to obtain the longitudinal non-zero spin splitting for the linearly polarized incident beam, the polarization angle τ cannot be either 0 or π /2.The dependence of the longitudinal PSHE on the incident angle θ and the polarization angle τ is demonstrated in figures 4(a) and (b), respectively.It can be observed from figure 4(b) that when τ = 0 or τ = π /2, the longitudinal spin splitting no longer occurs.Figures 4(c) and (d) present the Stokes parameters S 3 of the reflected fields with the polarization angle τ = 0 and τ = π /4, respectively.It is seen that there is a transverse spin separation but no longitudinal spin splitting when τ = 0 (as shown in figure 4(c)) while both transverse and longitudinal spin shifts exist when τ = π /4 (as shown in figure 4(d)).
Additionally, it can also be observed from figure 4 that the longitudinal spin splitting exhibits an asymmetry, i.e. |δ It is not difficult to find that the relative permittivity of the dielectric (ε = ε r + iε i ) is a key factor for the symmetry of the longitudinal PSHE.When the relative permittivity is a purely real value (i.e.ε i = 0), equations (10a) and (10b) can be simplified as indicating the symmetric longitudinal splitting.Figure 5 illustrates the impact of the imaginary part ε i of the relative permittivity on the symmetry of the longitudinal PSHE, with the real part set as ε r = 0.18 [14].As shown in figures 5(a)-(c), the asymmetric longitudinal spin splitting occurs at ε i ̸ = 0, while the symmetric splitting is observed at ε i = 0.This observation highlights that manipulating the relative permittivity of the dielectric provides flexible control over the symmetry of the longitudinal PSHE.Furthermore, we study the total energy (I This is the physical mechanism by which an incident paraxial beam with an arbitrarily linear polarization can generate longitudinal photonic spin Hall shifts upon reflection.

Total GH shift reflected from the linearly polarized beam
In the transverse case, P and S waves do not act as the eigenpolarization modes.Consequently, the total transverse shift (i.e. the IF displacement) of the centroid of the entire reflected beam from the linearly polarized incident beam is always zero.However, there is the transverse spin separation for each handedness, i.e. the transverse PSHE.In the longitudinal case, an incident beam of an arbitrary linear polarization can induce a total longitudinal shift, which is indeed the total GH shift of the entire reflected beam.According to the above analyses, the total GH shift generated by an arbitrary linear incident polarization beam (|α⟩ = [cos τ sin τ ] T ) can be calculated as In fact, equation ( 13) is precisely the expression of the reflected GH shift for an arbitrary linear incident polarization state given in [5,17,19].
Especially, when τ = π /4, equation ( 13) becomes It can be observed that equation ( 14) is consistent with equation ( 8) when the second-order term is neglected, indicating that the GH displacements are equivalent for linear polarization of τ = π /4 and either left or right circular polarization.It demonstrates again that the P and S polarizations represent the eigenpolarization modes for GH shifts.This differs substantially from the transverse case, where the IF shift arises during circular polarization but disappears during any arbitrary linear polarization.

Conclusion
In conclusion, we have extended the theory of the PSHE generation mechanism from the transverse to the longitudinal case.Our results reveal that each incident spin state can generate two different reflective spin components: one with the same handedness as incidence (referred to as the normal mode) and another with the opposite handedness as incidence (known as the abnormal mode).Both modes have non-zero longitudinal displacements with different energy weights.We have also observed that the second-order expansion term of the in-plane wave vector affects the longitudinal spin shift of the abnormal mode near the grazing incidence angle.Under an arbitrary linear polarization incidence, each reflected total spin component is a coherent superposition between its corresponding normal and abnormal modes, leading to polarization-angle-dependent longitudinal spin shifts.In particular, we have discovered that the longitudinal PSHE can occur when the incidence is neither parallelly (τ ̸ = 0) nor orthogonally (τ ̸ = π /2) polarized.A significant finding is that the symmetry of the longitudinal spin splitting can be controlled by manipulating the relative permittivity of the dielectric.Specifically, the symmetrical longitudinal PSHE occurs when the permittivity is real, while the asymmetrical PSHE exists when it becomes complex.Moreover, regardless of whether the incident beam is circularly or linearly polarized, the reflected total GH shift is a superposition of individual spin states' longitudinal shifts, taking into account their energy weights.These outcomes significantly advance our comprehension of the physical mechanisms underlying the PSHE and beam shift in both transverse and longitudinal directions.This study establishes a unified physical theory encompassing transverse and longitudinal PSHE as well as beam shifts, which holds considerable promise for future advancements in spin-controlled nanophotonics.

Appendix. The calculation of the longitudinal centroid of the beam through the reflection
The relation between reflected and incident angular spectrum can be written as: where |P⟩ and |S⟩ refer to the field vector parallel or orthogonal to the incidence plane, r P (k ix , k iy ) and r S (k ix , k iy ) are the corresponding Fresnel reflection coefficients of P and S polarizations of the air-dielectric interface, which can be represented as follows [16]: (A.2) For simplicity, the rotation of the incident plane is ignored as we focus solely on calculating the longitudinal shift.To ensure accurate calculations, we consider up to the second-order expansion terms when characterizing the Fresnel reflection coefficients using a Taylor expansion around the central wave vector (k ix = k iy = 0), yielding Note that the boundary conditions are k rx = −k ix and k ry = k iy .By substituting equation (A.5) into equation (A.1) and performing the corresponding basis vector transformations as well as the inverse Fourier transform, the reflected electric field E(x r , y r ) can be obtained.Consequently, the corresponding longitudinal centroid ⟨x r ⟩ (in the x r direction) of the reflection can be calculated as (z r = 0)

Figure 1 .
Figure 1.(a) Schematic diagram of the reflected longitudinal PSHE for an arbitrarily linearly polarized incident beam (|α⟩ = [cos τ sin τ ] T ) on the air-dielectric interface (with the relative permittivity ε = εr + iε i ), where the purple arrows represent the geometric paths of the incident and reflected beams, while the blue and red arrows denote the LCP and RCP components of the reflected beam, δ |α⟩ |L⟩ and δ |α⟩ |R⟩ are longitudinal spin shifts of the LCP and RCP components, and θ is the incident angle.(b) Four separate components in real reflection from an arbitrarily linearly polarized incident beams with the spin-maintained mode (E |L⟩ |L⟩ and E |R⟩ |R⟩ ) and spin-flipped mode (E |L⟩ |R⟩ and E |R⟩ |L⟩ ), where the superscript (subscript) |L⟩ or |R⟩ of E |σ⟩ |σ⟩ represents LCP or RCP component of the incident (reflected) beam.
∂θ 2 ; the superscript (subscript) |L⟩ or |R⟩of E |σ⟩ |σ⟩ represents the LCP or RCP component in the incident (reflected) fields.For example, E |L⟩ |L⟩ (x r , y r ) indicates the reflected LCP component induced by the incident LCP component (the spin-maintained term) while E |L⟩ |R⟩ (x r , y r ) denotes the reflected RCP component induced by the incident LCP component (the spin-flipped term).Equation

Figure 2 .
Figure 2. The dependence of the reflected spin displacements of (a) LCP and (b) RCP components from an LCP incident beam on the incident angle θ, with considering the second-order (solid lines) and first-order (dashed lines) expansion terms of in-plane wave vector spread k ix .Here, w0 = 20 µm, λ = 632.8nm, and ε = 1.8 + 0.18i.
corresponding energy weight for the spin-maintained and spin-flipped modes, with I |σ⟩ |σ⟩ = ˜|E |σ⟩ |σ⟩ (x r , y r )| 2 (a) illustrates the total longitudinal GH shift resulting from the incident LCP component, which is a combination of the spin shifts δ |L⟩ |L⟩ and δ |L⟩ |R⟩ , arising from reflected LCP and RCP components by considering their energy weights shown in figure 3(b).By comparing figures 2(a), (b) and 3(a), it can be observed that the GH shift trend is akin to the spin shift δ |L⟩ |R⟩ of the abnormal mode at small angles, whereas it resembles that of the spin displacement δ |L⟩ |L⟩ of the normal mode at larger angles.Initially, the abnormal mode predominates.However, as the incidence angle increases, the normal mode becomes more prominent, as indicated in figure 3(b).Remarkably, unlike the longitudinal spin shift δ |L⟩ |R⟩ of the abnormal mode, the GH shift values calculated with and without considering the second-order terms of k ix are entirely consistent.This consistency arises because the energy weight w |L⟩ |R⟩ of the abnormal mode is substantially low near the

Figure 3 .
Figure 3. (a) The total longitudinal (GH) shift for an LCP incident beam considering the second-order (solid lines) and first-order (dashed lines) expansion terms of k ix .(b) The energy weights of the corresponding reflected LCP (solid lines) and RCP (dashed lines) components.Other parameters are the same as those in figure 2.

Figure 4 .
Figure 4.The longitudinal PSHE of reflected LCP (solid lines) and RCP (dashed lines) components as functions of (a) the incident angle θ and (b) the polarization angle τ .(c) and (d) The Stokes parameter S3 of the reflected fields from different polarization angles τ = 0 and τ = π /4, respectively.Other parameters are the same as those in figure 2.
|α⟩ |L⟩ and I |α⟩ |R⟩ ) in the cross-section and the EOAM in the y r direction (L ry |α⟩ |L⟩ and L ry |α⟩ |R⟩ ), which are involved in the spin-orbit interaction of light and correspond to the longitudinal spin splitting of the reflected LCP and RCP components.These quantities are expressed as I |α⟩ |σ⟩ = ˜|E |α⟩ |σ⟩ (x r , y r )| 2 dx r dy r and L ry |α⟩ |σ⟩ = −δ |α⟩ |σ⟩ p rz |α⟩ |σ⟩ , respectively, where p rz |α⟩ |σ⟩ is the linear momentum z r direction.Figures 5(d1)-(d3) show that when the relative permittivity of the dielectric is complex (i.e.ε i ̸ = 0), the total energies (I |α⟩ |L⟩ and I |α⟩ |R⟩ ) of reflected LCP and RCP fields are different, and their EOAMs in the y r direction exhibit different directions and magnitudes (i.e.|L ry |α⟩ |L⟩ | ̸ = |L ry |α⟩ |R⟩ |), leading to the asymmetric longitudinal spin splitting.On the other hand, when the relative permittivity of the dielectric is real (i.e.ε i = 0), the total energies of both LCP and RCP fields are consistent (I |α⟩ |L⟩ = I |α⟩ |R⟩ ) after reflection.Their EOAMs in the y r direction have opposite directions but the same value (L ry |α⟩ |L⟩ = −L ry |α⟩ |R⟩ ), resulting in symmetric longitudinal spin separation, as shown in figures 5(e1)-(e3).As a summary for this section, we demonstrate that the existence of the phase difference exp (2iτ ) between incident opposite spin states induced by the polarization angle (τ ̸ = 0 ∩ τ ̸ = π /2) results in the coherent superposition of two same reflected spin parts originating from opposite incident spin states (E |L⟩ |L⟩ , E |L⟩ |R⟩ , E |R⟩ |L⟩ , E |R⟩ |R⟩ ).This redistribution of these four reflected components (E |α⟩ |L⟩ = E |L⟩ |L⟩ + E |R⟩ |L⟩ and E |α⟩ |R⟩ = E |L⟩ |R⟩ + E |R⟩ |R⟩ ) leads to their different EOAM values (L ry |α⟩ |L⟩ and L ry |α⟩ |R⟩ ) along the y r -direction due to the effect of in-plane spread wave vectors k ix , resulting in the symmetric/asymmetric longitudinal spin splitting.