1. Introduction

At macroscopic scales, the law of reflection and Snell’s law provide a complete explanation for common optical phenomena [1]. However, at subwavelength scales, due to conservation of optical spin-orbit angular momentum, light beams may not precisely follow geometric optics principles. When a linearly polarized beam interacts with an optical interface through reflection or refraction, its left-circularly polarized (LCP) and right-circularly polarized (RCP) components will be split along the direction perpendicular to the incidence plane. This phenomenon is known as the photonic spin Hall effect (PSHE) [2–8].

Recently, the PSHE has been widely investigated in various physical systems such as plasmonics [9–13], metamaterials [14–18], image edge detection [19,20], uniaxial crystal [21–26] and even two-dimensional materials [27–32]. Moreover, the spin shifts of the PSHE show high sensitivity to material properties, providing potential applications in optical sensing [33–35] and precise metrology [36–38]. These studies on the PSHE are mostly rooted in the wave nature of light as defined by Maxwell’s Equations.

However, in the quantum theory of light [39], a photon is the quantum of light, representing an uncharged particle. Moreover, photons exhibit some peculiar behaviors; they can exhibit properties of interference and diffraction, similar to waves, and also possess energy and momentum like particles. This behavior is known as the wave-particle duality of light. Therefore, can we reexamine the PSHE from the perspective of particle nature of light?

In this work, we begin with the particle nature of light and rederive the spin shifts of the out-of-plane and in-plane PSHE. We find that the spin shifts of out-of-plane PSHE result from the conservation of spin-orbit angular momentum of one photon and the spin shifts of in-plane PSHE resut from the spread of in-plane wavevector. We also find that the spin shifts of the PSHE do not arise from the overall shift of photons with the same handedness. Instead, they stem from the coherent superposition of photons sharing the same handedness. Their weight shifts contribute to the spin shifts of the PSHE. Moreover, our findings align perfectly with the spin shifts derived from the wave nature of light.

2. Out-of-plane PSHE based on the particle nature of light

To establish a theoretical model for the conservation of spin-orbit angular momentum applicable to all photons, we assume that all the wavevector components of the incident beam are concentrated in a straight line, represented by a single wavevector, which characterizes the entire incident beam. Meanwhile, this wavevector encompasses all spin photons, as shown in Fig. 1(a). Let us consider the partial reflection and refraction of a polarized monochromatic wavevector at an interface between two homogeneous isotropic media with different refractive indices $n^1$ and $n^2$, see Fig. 1(b) and 1(c). Along with the laboratory coordinate system $(x, y, z)$, attached to the interface $z=0$, we define $(x^a, y^a, z^a)$ as local coordinate systems for individual wavevectors, where the superscripts $a={i, r, t}$ label the incident, reflected and refracted wavevectors, respectively. The angles between the propagating direction of wavevector and $z$ axis are denoted as $\theta ^a$.

 

Fig. 1. (a): schematic of the wavevector; (b) and (c): spin shift and IF shift in the process of reflection and refraction for LCP incident wavevector. Note that green arrow shows the direction of spin angular momentum.

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The total angular momentum (TAM) of one photon can be expressed as a sum of the spin angular momentum and orbital angular momentum [3]:

Here, $\boldsymbol {k}^{a}$ and $\boldsymbol {r}^{a}$ are the linear momentum and position vector of one photon, respectively. $\sigma ^{a}=\left \langle e^{a}\left |\hat {\sigma }_{3}\right |e^{a}\right \rangle$ is the mean helicity of one photon. $\left |e^{a}\right \rangle$ is represented in the basis of circular polarizations. $\hat {\sigma }_{3}=\operatorname {diag}(1,-1)$ is the Pauli matrix. Note that we use $\hbar =\mathbf {1}$. Owing to the axial symmetry with respect to z axis, the z component of TAM of one photon is conserved [40]. Equation (1) equals

Here, $\delta ^{a}$ is the spin shift of one photon along the y axis.

To obtain the spin shifts of out-of-plane PSHE, we need to acquire the number of photons of LCP state $(|+\rangle )$ and RCP state $(|-\rangle )$, and spin shift of one photon along the y axis. In the following discussion, we will present their weight shifts result in the Imbert-Fedorov (IF) shifts and spin shifts of the PSHE.

To simplify matters, let us begin by considering the case of the incident monochromatic wavevector in an LCP state, expressed as $\left |\psi ^{i}\right \rangle =|+\rangle$, see Figs. 1(b) and 1(c). Upon reflection and refraction at an interface between two homogeneous isotropic media [1], the initial state of system evolves to

Afterwards, we discuss the change of number of photons in the reflection and refraction process for an LCP incident wavevector. Note that TAM can be expressed as $N^{i} \boldsymbol {j}^{i}$, where $N^{i}$ is the total number of photons in the incident wavevector. Based on the energy conservation law, we have $N^{i}=\sum _{m=r,t}N^{m}$, $N^{r}$ and $N^{t}$ are the number of photons in the reflected and refracted wavevectors, respectively. According to the Fresnel law [1], we have ${N}^{m}=N^{i} M\left (m_{++}^{2}+m_{+-}^{2}\right )$, where $M=n^{m} \cos \theta ^{m} / n^{i} \cos \theta ^{i}$. Note that $n^r=n^i=n^1$, $n^t=n^2$ and $\theta ^r=-\theta ^i$. The number of photons of LCP component $(N_{++}^{m})$ can be obtained from Eq. (3) with $N_{++}^{m}=N^{m} \eta _{++}^{m}$, where

Here, $\langle +\left |m_{++}^{*}\right |\psi ^{m}\rangle$ is the projection of polarization state $|\psi ^{m}\rangle$ on LCP component $m_{++}|+\rangle$. $m_{++}^{*}$ is the complex conjugate of $m_{++}$. $\langle \psi ^m|\psi ^m\rangle$ is the normalization coefficient of polarization state. Hence, $\eta _{++}^{m}$ represents the probability of being converted into LCP component for an LCP incident wavevector.

Similarly, the number of photons of RCP component $(N_{+-}^{m})$ can also be obtained from Eq. (3) with $N_{+-}^{m}=N^{m} \eta _{+-}^{m}$, where

Here, $\langle -\left |m_{+-}^{*}\right |\psi ^{m}\rangle$ is the projection of polarization state $|\psi ^{m}\rangle$ on RCP component $m_{+-}|-\rangle$. $m_{+-}^{*}$ is the complex conjugate of $m_{+-}$. Hence, $\eta _{+-}^{m}$ represents the probability of being converted into RCP component for an LCP incident wavevector. From Eqs. (3)–(5), when an LCP photon interacts with an optical interface, there is a possibility that it can either remain as an LCP photon (spin-maintained photon) or transform into a RCP photon (spin-flipped photon), each with a specific probability. The transition probabilities are decided by Eqs. (4) and (5), respectively.

After obtaining the number of photons for the LCP and RCP states, our next step is to determine the spin shift of one photon in the process of reflection and refraction. It can be given by Eq. (2):

Here, $\sigma ^m=\sigma ^r=-1$ in the reflection process; $\sigma ^m=\sigma ^t=1$ in the refraction process; $\beta ^{m}=\cos \theta ^{m} / \cos \theta ^{i}$; and $\delta _{++}^{m}$ and $\delta _{+-}^{m}$ are the spin shifts of spin-maintained photon and spin-flipped photon, respectively.

By combining Eqs. (4) (6), the transverse shift of barycenter of all photons can be obtained with

Note that $\delta _{+IF}^{m}$ is commonly referred to as the IF shift [41,42]. From Eq. (7), the IF shift can be seen as the weighted superposition of the spin shifts.

Figure 2 illustrates the variation of the normalized number of photons and transverse shift in relation to the incident angle. From Fig. 2(a), it is evident that prior to the Brewster angle, the reflected wavevector mainly consists of spin-flipped photons. However, subsequent to the Brewster angle, the reflected wavevector is gradually comprised of spin-maintained photons. From Fig. 2(b), the spin shift of spin-maintained photons gradually decreases with the increase of the incident angle. It is worth noting that the spin shift of spin-flipped photons is always zero. This is because the z component of spin angular momentum is continuous in the process of reflection, see Fig. 1(b). Combining Figs. 2(a) and 2(b), as the number of spin-maintained photons increase, the IF shift gradually aligns with the spin shift of spin-maintained photons, see the inset of Fig. 2(b). From Fig. 2(c), there are hardly any spin-flipped photons in the process of refraction. Moreover, the spin-maintained photons also gradually decrease to zero. From Fig. 2(d), the spin shift of spin-flipped photons gradually decreases with the increase of the incident angle. However, the IF shift always aligns with the spin shift of spin-maintained photons due to the extremely low spin-flipped photons, see the inset of Fig. 2(d).

Next, let us examine the scenario of the incident monochromatic wavevector in a RCP state, represented as $|\psi \rangle =|-\rangle$, see Fig. 3(a) and 3(b). Upon reflection and refraction at an interface between two homogeneous isotropic media, the polarization state of system evolves to

From Eq. (8), $\left |\psi ^{m}\right \rangle$ is still a linear superposition of $|+\rangle$ and $|-\rangle$. $m_{-+}$ $(m_{--})$ determines the conversion of the incident wavevector of a RCP state into LCP (RCP) component.

 

Fig. 2. (a) and (c): normalized number of photons in the process of reflection and refraction; (b) and (d): transverse shift of one photon and all photons in the process of reflection and refraction. The incident wavelength is chosen as 632.8 nm, and the refractive indices of medium 1 and medium 2 are 1 and 1.5, respectively.

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Fig. 3. Schematic of spin shift and IF shift in the process of reflection (a) and refraction (b) for RCP incident wavevector.

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According to the Fresnel law [1], we have ${N}^{m}=N^{i} M\left (m_{-+}^{2}+m_{--}^{2}\right )$. The number of photons of LCP component $(N_{-+}^{m})$ can be obtained from Eq. (8) with $N_{-+}^{m}=N^{m} \eta _{-+}^{m}$, where

Here, $\langle +\left |m_{-+}^{*}\right |\psi ^{m}\rangle$ is the projection of polarization state $|\psi ^{m}\rangle$ on LCP component $m_{-+}|+\rangle$. $m_{-+}^{*}$ is the complex conjugate of $m_{-+}$. Hence, $\eta _{-+}^{m}$ represents the probability of being converted into LCP component for a RCP incident wavevector.

Similarly, the number of photons of RCP component $(N_{--}^{m})$ can also be obtained from Eq. (8) with $N_{--}^{m}=N^{m} \eta _{--}^{m}$, where

Here, $\langle -\left |m_{--}^{*}\right |\psi ^{m}\rangle$ is the projection of polarization state $|\psi ^{m}\rangle$ on RCP component $m_{--}|-\rangle$. $m_{--}^{*}$ is the complex conjugate of $m_{--}$. Hence, $\eta _{--}^{m}$ represents the probability of being converted into RCP component for a RCP incident wavevector. Based on Eqs. (8)–(10), when a RCP photon encounters an optical interface, it has the potential to transform into an LCP photon (spin-flipped photon), or remain as a RCP photon (spin-maintained photon), each with a specific probability. The transition probabilities are determined by Eqs. (9) and (10), respectively.

The spin shift of one photon can also be given by Eq. (2):

By combining Eqs. (9)–(11), the IF shift can be obtained with

From Eq. (12), the IF shift can also be regarded as the weighted superposition of the spin shifts.

Figure 4 also shows that the normalized number of photons and transverse shift vary with the incident angle for the incident RCP wavevector. Comparing Fig. 2 and Fig. 4, it is evident to find that $N_{++}^{m}=N_{--}^{m}$, $N_{+-}^{m}=N_{-+}^{m}$, $\delta _{++}^{m}=-\delta _{--}^{m}$, $\delta _{+-}^{m}=-\delta _{-+}^{m}$ and $\delta _{+IF}^{m}=-\delta _{-IF}^{m}$. These results indicate that the number of photons generated from incident LCP and RCP states are identical, while the transverse shifts are completely opposite. The scenario of incident RCP state can be discussed in the same manner as that of incident LCP state.

 

Fig. 4. (a) and (c): normalized number of photons in the process of reflection and refraction; (b) and (d): transverse shift of one photon and all photons in the process of reflection and refraction.

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According to the above analysis, for the LCP incident wavevector, after reflection and refraction, it simultaneously contains both the LCP and RCP components; for the RCP incident wavevector, after reflection and refraction, it also simultaneously contains both the LCP and RCP components. Therefore, for the incident wavevector in the horizontal polarization state, which can be viewed as a superposition of LCP and RCP states, after reflection and refraction, its LCP component necessarily originates partly from the incident LCP component and partly from the incident RCP component. The same applies to the RCP component. Next, we will specifically discuss this process.

The horizontal input polarization state $|\psi ^i\rangle$ can be defined as:

Here, $|\psi _+^{m}\rangle$ and $|\psi _{-}^{m}\rangle$ are the LCP and RCP states after reflection and refraction, respectively.

From Eq. (15a), $m_{++}|+\rangle$ derived from LCP and $m_{-+}|+\rangle$ derived from RCP are coherently superposed and form a new coherent state $|\psi _+^{m}\rangle$. According to the Fresnel law, the number of photons of LCP component $N_+^{m}=N^{i} A^{2} M\left |m_{++}+m_{-+}\right |^{2}$. Combined with the Eq. (15a), $N_+^{m}$ is partly derived from LCP component of horizontal input polarization, and partly from RCP component of horizontal input polarization. Therefore, we can obtain that

 

Fig. 5. Schematic of the spin shifts of out-of-plane PSHE in the process of reflection (a) and refraction (b) for a horizontal input polarization.

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Fig. 6. The reverse of direction of the spin angular momentum in the process of reflection.

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Similarly, from Eq. (15b), $m_{+-}|-\rangle$ derived from LCP and $m_{--}|-\rangle$ derived from RCP are also coherently superposed and form a new coherent state $|\psi _{-}^{m}\rangle$. According to the Fresnel law, the number of photons of RCP component $N_{-}^{m}=N^{i} A^{2} M\left |m_{-+}+m_{--}\right |^{2}$. Combined with the Eq. (15b), $N_{-}^{m}$ is partly derived from LCP component of horizontal input polarization, and partly from RCP component of horizontal input polarization. Therefore, we can obtain that

Here, $\langle -|m_{+-}^{*}|\psi _{-}^{m}\rangle (\langle -|m_{-+}^{*}|\psi _{-}^{m}\rangle )$ is the projection of coherent state $|\psi _{-}^{m}\rangle$ on RCP $m_{+-}|-\rangle (m_{--}|-\rangle )$; $\gamma _{+-}^{m} (\gamma _{--}^{m})$ is the coherence factor converted into RCP component for LCP (RCP) component of horizontal input polarization; and $\langle \psi _{-}^{m}|\psi _{-}^{m}\rangle$ is the normalization coefficient of polarization state. Note that $\gamma _{--}^{m}<0$ when $m_{+-}>0$ and $m_{--}<0$, which means that $N_{--}^{m}<0$. It can still be regarded as the reversal of direction of the spin angular momentum. In addition, it is easy to verify $N^{i}=\sum _{m=r, t}\left (N_+^{m}+N_{-}^{m}\right )$ by the Fresnel law.

For a horizontal input polarization, the transverse shifts of barycenter of all photons of same handedness (i.e., the spin shifts of out-of-plane PSHE) can be obtained with

Here, $\delta _+^{m}$ and $\delta _{-}^{m}$ are the spin shifts of LCP and RCP components, respectively. From Eq. (20), each spin component is a result of the coherent superposition of all photons with the same handedness. Their weight shifts result in the overall shift of the corresponding spin component. The conclusion is consistent with the findings of Ren $et$ $al$. [43] in 2015. Note that the IF shift refers to the overall transverse shift of barycenter of all photons (without distinguishing between LCP and RCP photons). The spin shift, on the other hand, pertains to the transverse shift of barycenter of all LCP photons or all RCP photons (distinguishing between LCP and RCP photons).

Figure 7 shows that normalized number of photons and spin shift change with the incident angle. From Figs. 7(a) and 7(c), we can find that $N_{++}^{m}=N_{--}^{m}$ and $N_{+-}^{m}=N_{-+}^{m}$. From Figs. 7(b) and 7(d), we can obtain that $\delta _+^{m}=-\delta _{-}^{m}$, showing that the spin shifts are equal in magnitude and opposite in direction. Based on the previous analysis, we know that during the reflection process, the shift of spin-flipped photons remains zero. The primary cause of inducing IF shift is the spin-maintained photons, see Figs. 2(b) and 4(b). From the inset of Fig. 7(a), before the Brewster angle, the proportion of spin-maintained photons is extremely low in the total number of photons. However, it is the primary cause of inducing spin shift of the PSHE. Moreover, note that the spin-maintained photons are less than zero before the Brewster angle, indicating a reversal in the direction of spin angular momentum. This also leads to a negative spin shift for LCP component and a positive spin shift for RCP component, which is exactly opposite to the case of one photon reflection, see Figs. 2(b), 4(b) and 7(b). Combining Figs. 2(d), 4(d), 7(c) and 7(d), we can find that there is no phenomenon of spin angular momentum reversal during the refraction process.

 

Fig. 7. (a) and (c): normalized number of photons in the process of reflection and refraction; (b) and (d): spin shift in the process of reflection and refraction.

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Finally, we calculate the spin shifts of the out-of-plane PSHE for an arbitrary input polarization. We define the input polarization state $\left |\psi ^{i}\right \rangle$ as:

Upon reflection and refraction at an interface, the initial state of system evolves to

Here,

From Eq. (23a), $m_{++}\left (1+\alpha _{c}\right )|+\rangle$ derived from LCP and $m_{-+}\left (1-\alpha _{c}\right )|+\rangle$ derived from RCP are coherently superposed and form a new coherent state $\left |\psi _+^{m}\right \rangle$. From Eq. (23b), $m_{+-}\left (1+\alpha _{c}\right )|-\rangle$ derived from LCP and $m_{--}\left (1-\alpha _{c}\right )|-\rangle$ derived from RCP are coherently superposed and form a new coherent state $\left |\psi _{-}^{m}\right \rangle$. According to the Fresnel law, the number of photons of LCP and RCP components can be obtained with

Combined with Eq. (23), both $N_+^{m}$ and $N_{-}^{m}$ are partly derived from LCP component of an arbitrary input polarization, and partly derived from RCP component of an arbitrary input polarization. Therefore, we can obtain that

Here, $N_{++}^{m} (N_{-+}^{m})$ is the number of photons of LCP component derived from LCP (RCP) component of an arbitrary input polarization, and $N_{+-}^{m} (N_{--}^{m})$ is the number of photons of RCP component derived from LCP (RCP) component of an arbitrary input polarization. By analyzing Eq. (23), we can obtain $N_{++}^{m}=N_+^{m} \operatorname {Re}\left [\gamma _{++}^{m}\right ]$, $N_{-+}^{m}=N_+^{m} \operatorname {Re}\left [\gamma _{-+}^{m}\right ]$, $N_{+-}^{m}=N_{-}^{m} \operatorname {Re}\left [\gamma _{+-}^{m}\right ]$ and $N_{--}^{m}=N_{-}^{m} \operatorname {Re}\left [\gamma _{--}^{m}\right ]$, where

Here, $\left \langle +\left |m_{++}^{*}\left (1+\alpha _{c}^{*}\right )\right | \psi _+^{m}\right \rangle \left (\left \langle +\left |m_{-+}^{*}\left (1-\alpha _{c}^{*}\right )\right | \psi _+^{m}\right \rangle \right )$ is the projection of coherent state $\psi _+^{m}$ on LCP $m_{++}\left (1+\alpha _{c}\right )|+\rangle \left (m_{-+}\left (1-\alpha _{c}\right )|+\rangle \right )$; $\left \langle -\left |m_{+-}^{*}\left (1+\alpha _{c}^{*}\right )\right | \psi _{-}^{m}\right \rangle \left (\left \langle -\left |m_{--}^{*}\left (1-\alpha _{c}^{*}\right )\right | \psi _{-}^{m}\right \rangle \right )$ is the projection of coherent state $\psi _{-}^{m}$ on RCP $m_{+-}\left (1+\alpha _{c}\right )|-\rangle \left (m_{--}\left (1-\alpha _{c}\right )|-\rangle \right )$; $\gamma _{++}^{m}\left (\gamma _{-+}^{m}\right )$ is the coherence factor converted into LCP component for LCP (RCP) component of an arbitrary input polarization; $\gamma _{+-}^{m}\left (\gamma _{--}^{m}\right )$ is the coherence factor converted into RCP component for LCP (RCP) component of an arbitrary input polarization; both $\left \langle \psi _+^{m} \mid \psi _+^{m}\right \rangle$ and $\left \langle \psi _{-}^{m} \mid \psi _{-}^{m}\right \rangle$ are the normalization coefficients of polarization state. Note that these coherence factors can be expressed as complex numbers due to $\alpha _c$ being a complex number. It can be demonstrated that, for the out-of-plane PSHE, the real part of the coherence factor corresponds to transverse shift, while the imaginary part corresponds to angular shift. Therefore, when calculating the transverse shift, we exclusively consider the real component of the coherence factors.

By the coherent superposition of all photons with the same handedness, the spin shifts of the out-of-plane PSHE can be obtained with

Equation (27) is the general expression of spin shifts for an arbitrary input polarization. Let $\alpha _{c}=0$, Eq. (27) can be simplified as Eq. (20). Moreover, from Eq. (27), the spin shifts of the out-of-plane PSHE exhibits asymmetric distribution with $\alpha _{c}$ being a general value [44]. In order to confirm reliability of our results, we will reexamine the spin shifts of the out-of-plane PSHE from the perspective of wave nature of light.

3. Out-of-plane PSHE based on the wave nature of light

The coordinate frames are shown in Fig. 1. We consider the incident one-dimensional Gaussian beam with an arbitrary polarization state, and its angular spectrum can be expressed as

Here, $H=\sqrt {\omega _{0} / \sqrt {2 \pi }} / \sqrt {2\left (1+\left |\alpha _{c}\right |^{2}\right )}$ is the normalized coefficient; $\Phi \left (k_{y}^{i}\right )=\exp \left [-\left (\omega _{0} k_{y}^{i}\right )^{2}/4\right ]$ is the Gaussian distribution; $\omega _{0}$ is the half width of the beam waist; $k_{y}^{i}$ is the transverse wave vector along $y^i$ direction; and $\hat {e}_+^{i}$ and $\hat {e}_{-}^{i}$ are the unit vectors in the spin basis set in the local coordinate system. Note that the definition of polarization state here is similar to Eq. (21).

According to the transversality, the reflected field and the refracted field can be obtained with [5,45]

Here, the terms $\exp \left (-i k_{y}^{m} \delta _{++}^{m}\right )$, $\exp \left (-i k_{y}^{m} \delta _{-+}^{m}\right )$, $\exp \left (-i k_{y}^{m} \delta _{+-}^{m}\right )$ and $\exp \left (-i k_{y}^{m} \delta _{--}^{m}\right )$ indicate the spin-orbit coupling of one photon; $k_{y}^{m}$ is the transverse wave vectors along $y^m$ direction. A detailed derivation of Eq. (29) can be seen in Appendix B.

By combining Eqs. (28) and (29), the LCP component of the reflected and refracted angular spectrum can be expressed as

Similarly, the RCP component of the reflected and refracted angular spectrum can be expressed as

4. In-plane PSHE based on the particle nature of light

Due to the close correlation between the in-plane PSHE and in-plane wavevector [46], we need to expand the incident wavevector along the in-plane direction. To simplify the discussion of the in-plane PSHE, we assume that all spin photons are concentrated along the direction of the in-plane wavevector expansion (slightly deviating from the central wave vector of the incident wave), as shown in Fig. 8(a). In this case, the Fresnel coefficients also need to be expanded along the direction of the in-plane wavevector, i.e.,

Note that $m_{++\left (\theta ^{i}\right )}$, $m_{+-\left (\theta ^{i}\right )}$, $m_{-+\left (\theta ^{i}\right )}$ and $m_{--\left (\theta ^{i}\right )}$ are the Fresenl coefficients corresponding to the central wave vector and $k_{x}^{i}$ is the transverse wave vector along $x^i$ direction.

Let us first consider the incident wavevector slightly deviating from the central wavevector, which is an LCP state expressed as $\left |\psi ^{i}\right \rangle =|+\rangle$, see Figs. 8(b) and 8(c). Upon reflection and refraction at an interface, the polarization state of system evolves to

Here, $\Delta _{++}^m=\sigma ^m\beta ^m\left (\partial \ln m_{++}/\partial \theta ^i\right )/k_0$; $\Delta _{+-}^m=\sigma ^m\beta ^m\left (\partial \ln m_{+-}/\partial \theta ^i\right )/k_0$; for the reflection process, $\sigma ^m=\sigma ^r=-1$ and $\beta ^m=\beta ^r=1$; and for the refraction process, $\sigma ^{m}=\sigma ^{t}=1$ and $\beta ^m=\beta ^t=\cos \theta ^t/\cos \theta ^i$. Note that we use the boundary conditions that $k_x^i=-k_x^r$ and $k_x^i=\left (\cos \theta ^t/\cos \theta ^i\right )k_x^t$. From Eq. (34), for the LCP incident wavevector slightly deviating from the central wavevector, after reflection and refraction, it simultaneously induces the LCP spin angular shift $\Delta _{++}^m$ and RCP spin angular shift $\Delta _{+-}^m$, as illustrated in Figs. 8(b) and 8(c). Note that the spin angular shift in Eq. (34) corresponds to the in-plane spin shift during the propagation Raleigh distance. Additionally, the change in the number of photons in this process can be determined by Eqs. (3)–(5).

 

Fig. 8. (a): schematic of the wavevector; (b) and (c): in-plane spin anglar shift in the process of reflection and refraction for LCP incident wavevector.

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Next, let us consider the incident wavevector slightly deviating from the central wavevector, which is an RCP state expressed as $\left |\psi ^{i}\right \rangle =|-\rangle$, see Figs. 9(a) and 9(b). Upon reflection and refraction at an interface, the polarization state of system evolves to

Here, $\Delta _{-+}^m=\sigma ^m\beta ^m\left (\partial \ln m_{-+}/\partial \theta ^i\right )/k_0$ and $\Delta _{--}^m=\sigma ^m\beta ^m\left (\partial \ln m_{--}/\partial \theta ^i\right )/k_0$. From Eq. (35), for the RCP incident wavevector slightly deviating from the central wavevector, after reflection and refraction, it also simultaneously induces the LCP spin angular shift $\Delta _{-+}^m$ and RCP spin angular shift $\Delta _{--}^m$, as illustrated in Figs. 9(a) and 9(b). The change in the number of photons in this process can be determined by Eqs. (8)–(10).

 

Fig. 9. Schematic of in-plane spin angular shift in the process of reflection (a) and refraction (b) for RCP incident wavevector.

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For any input polarization state, it is represented by Eq. (21), and the change in the number of photons is determined by Eqs. (22)–(26). It is worth noting that the coherence factors in Eq. (26) are complex numbers. It can be demonstrated that, for the in-plane PSHE, the real part corresponds to angular shift, and the imaginary part corresponds to transverse shift. Therefore, when calculating the transverse shift, we exclusively consider the imaginary component of the coherence factors.

By the coherent superposition of all photons with the same handedness, the spin shifts of the in-plane PSHE can be obtained with

Here, $\mathbb {N}_{++}^m=N_+^{m} \operatorname {Im}\left [\gamma _{++}^{m}\right ]$, $\mathbb {N}_{-+}^m=N_+^{m} \operatorname {Im}\left [\gamma _{-+}^{m}\right ]$, $\mathbb {N}_{+-}^m=N_{-}^{m} \operatorname {Im}\left [\gamma _{+-}^{m}\right ]$ and $\mathbb {N}_{--}^m=N_{-}^{m} \operatorname {Im}\left [\gamma _{--}^{m}\right ]$. Note that the imaginary part of the coherence factor transforms photons with angular shift entirely into transverse shift. Eq. (36) is the general expression of in-plane spin shifts for an arbitrary input polarization. Just like the out-of-plane PSHE, we can also reexamine the in-plane PSHE based on the wave nature of light.

5. In-plane PSHE based on the wave nature of light

We still consider the incident one-dimensional Gaussian beam with an arbitrary polarization state, and its angular spectrum can be expressed as

Here, $H=\sqrt {\omega _{0} / \sqrt {2 \pi }} / \sqrt {2\left (1+\left |\alpha _{c}\right |^{2}\right )}$ is the normalized coefficient; $\Phi \left (k_{x}^{i}\right )=\exp \left [-\left (\omega _{0} k_{x}^{i}\right )^{2}/4\right ]$ is the Gaussian distribution; and $k_{x}^{i}$ is the transverse wave vector along $x^i$ direction.

Based on the Eqs. (34) and (36), the reflected field and the refracted field can be obtained with

Here, the terms $\exp \left (k_{x}^{m}\Delta _{++}^{m}\right )$, $\exp \left (k_{x}^{m}\Delta _{-+}^{m}\right )$, $\exp \left (k_{x}^{m}\Delta _{+-}^{m}\right )$ and $\exp \left (k_{x}^{m}\Delta _{--}^{m}\right )$ indicate that the spin angular shifts induced by the spread of in-plane wavevector. The spin shifts of the in-plane PSHE can be obtained with

In addition, it is worth noting that the model based on the particle nature of light can also be used to explain the enhanced spin shifts of out-of-plane and in-plane PSHE. However, this requires transforming the current single-wavevector model into a model of Gaussian beam composed of infinitely many wavevectors. For detailed discussions, refer to the Supplement 1.

6. Conclusion

In this work, we have recalculated the spin shifts of the out-of-plane and in-plane PSHE for any given input polarization by the coherent superposition of all photons of same handedness. When dealing with a circularly polarized incident wavevector, both reflected and refracted wavevectors simultaneously generate the spin-flipped and spin-maintained photons. Viewing light from a particle perspective, we have discovered that the spin shifts of out-of-plane PSHE can be accounted for by the conservation of spin-orbital angular momentum of one photon and that the spin shifts of in-plane can be accounted for the spread of in-plane wavevector. Furthermore, the spin shifts of the PSHE based on the particle nature of light align with the spin shifts based on the wave nature of light. Our research ensures the unity of the wave-particle duality of light in the PSHE.