## Abstract

We report the manipulation of spin-dependent splitting by geometric Doppler effect based on dielectric metasurfaces. The extrapolation of rotational Doppler effect from temporal to spatial coordinate gives the phase change when the local optical axes of dielectric metasurfaces are rotating in space. Therefore, the continuous variation of local optical axes in a certain direction will introduce a phase gradient in the same direction at the beam cross section. This is additive to the phase gradient appeared when breaking the rotational symmetry of linearly polarized cylindrical vector beams, which leads to the deflections of different spin components of light, i.e., photonic spin Hall effect. Hence, it is possible to manipulate the spin-dependent splitting by introducing the geometric Doppler effect. Theoretically and experimentally, we show that the magnitude and orientation of the spin-dependent splitting are both tunable when changing the spatial rotation rate of local optical axes and incident polarization.

© 2015 Optical Society of America

## 1. Introduction

As an excellent representation of the analogy between optics and particle dynamics, the photonic spin Hall effect (SHE) is the optical counterpart of conventional Hall effect, which originates from the coupling between the spin and orbital degrees of freedom of photons [1–3], and hence theoretically provides an approach for exploring the applications based on the spin and orbital angular momentum of photons. In general, photonic SHE occurs when a constrained beam be reflected or refracted at the interface between different media, which manifests itself as a spin-dependent splitting [1, 2]. However, the magnitude of splitting is so small that we should always implement a precise measurement to observe it [4–6]. Therefore, many research efforts in recent years are dedicated to find a way to steer the photonic SHE and magnify the relevant spin-dependent splitting [7, 8].

Rotational Doppler effect is usually involved in the dynamics with a rotating reference frame, which would lead to the frequency shift of wave in conditions with rotating wave source, propagation medium, or signal receiver [9–11]. However, this effect can also be extrapolated from temporal rotation to the spatial rotation situations [12–14]. As depicted by Bilokh *et al.* [13], the spatial rotation of beam polarization or wave vector may led to the Pancharatnam-Berry (PB) phase [15] or the Rytov-Vladimirskii-Berry (RVB) phase [16,17], respectively. These geometric phases are essentially originated from the spin-orbital coupling. Therefore, the spatial rotational Doppler effect possesses great potential and value for the manipulating of light-matter interaction. A particular incarnation of this effect has developed as the geometric Doppler effect when referring to the medium with spatially rotating patterns [18].

In the present paper, we demonstrate that the dielectric metasurfaces designed based on the geometric Doppler effect can be applied to steer the photonic SHE, where the dielectric metasurfaces are an artificial anisotropic medium with locally varying optical axes and fixed phase retardation. Firstly, we show that the metasurfaces be designed with continuous variation in the azimuthal direction can be used to modulate the vortex phase, i.e., orbital angular momentum (OAM) of light due to the geometric phase change induced by the geometric Doppler effect. And then we demonstrate the manipulation of photonic SHE, which is visualized when we break the rotational symmetry of a cylindrical vector beam (CVB) [19, 20]. It is show that the spin-dependent shift of the photonic SHE can be modulated by changing the metasurfaces with different spatial rotation rates.

## 2. Geometric Doppler effect within metasurfaces

The OAM of light is usually characterized by the topological charge *l*, which may take any integer values (0, ± 1, ± 2,···). The amount of OAM carried by the radiated beam is *lh̄* per photon along the propagation axis, where *h̄* is reduced Planck constant [21,22]. In contrast, the spin angular momentum (SAM) can only be differentiated by the *σ* = ±1 for a single photon with left- or right-circular polarization. As for a light beam with arbitrary elliptic polarization, the SAM will follow *σ̄* ∈ [−1, 1]. In this work, we refer to the total angular momentum of light beam as **J** = *J***u**, where *J* = *l* + *σ̄*, *u* = **k**/*k*. Thus, *J* can take any value in the range [− *j*, *j*] for the vortex beam with arbitrary polarization states.

Rotational Doppler effect usually helps to interpret the scattering in a rotating reference fames, which gives the phase change in the frame rotating with some instantaneous angular velocity Ω [13],

**J**and

**J**

_{0}denote the total angular momenta of emitted and scattered waves, respectively. For more general cases, this deduction can also be extrapolated to the spatial evolution (in coordinate

*ζ*) through the substitution:

We applied the dielectric metasurfaces to show the geometric Doppler effect in this work. Metasurfaces is a kind of artificial two-dimensional material with various of functionalities like vector beam generators [23–25], phase plates [26–28], and concentrators [29, 30], which represent now the most promising route to planar photonics [31]. By rationally designing the geometry of local miniatures of metasurfaces, it is able to access novel optical devices and explore the unknown regimes which are previously constrained by the electro-magnetic responds of natural materials. More recently, metasurfaces based on the transparent dielectric material is rapidly developed because of their amazing and unique properties in propagation and polarization manipulations [32]. The dielectric metasurfaces applied in this work can be viewed as an anisotropic medium with constant phase retardation *π*, and the local optical axis distribution of these metasurfaces can be depicted as

*θ*is the angle of local optical axis;

*ϕ*and

*r*are the azimuthal and radial coordinates, respectively;

*α*

_{0}denotes the initial angle of optical axis;

*q*is the parameter characterizing the spatial rotation rate of optical axes in the azimuthal direction.

In considering these metasurfaces under geometric Doppler effect, the azimuthal angle *ϕ* can be taken as the coordinate *ζ* in the preceding deduction. Firstly, the rotation rate of reference frame along the coordinate *ζ* can be obtained from the Eq. (3),

*z*axis, is left- or right-circularly polarized:

**J**

_{0}= ±

**e**

*(*

_{z}**e**, the unit vector). After that, due to the fixed phase retardation

*π*, the angular momentum of passing beam would be inverted as shown in Fig. 1(a):

**J**= ∓

**e**

*. Thus the phase change involved in this process equals*

_{z}*ϕ*. This calculation intuitively demonstrates the overall function of the metasurfaces, which is different from the Jones calculus way [33, 34]. By Jones calculus, only the phase change in each particular position is acquired.

This geometric phase can be used to modify the phase front of beams. When the metasurfaces are illuminated by a circularly polarized planar beam, the phase front of light would be inverted into a vortex with 2*q* segments of spirals within the pitch of a wavelength, namely, with topological charge ±2*q*. For a circularly polarized vortex beam, the topological charge would be modulated according to the form *l* ±2*q*, where the sign depends on the chirality of incident circular polarization. Figure 1(a) shows the polarization evolution when a left-circularly polarized beam (*σ* = 1) passes through the metasurface (*q* = 1). The beam with *l* = 0 order of OAM is transformed to the *l* = 2 order with the opposite circular polarization (*σ* = −1). For simplicity, this evolution can also be elucidated intuitively by the hybrid-order Poincaré sphere as shown in Fig. 1(b), which simultaneously describe the evolution of phase and polarization of light beams [35]. In general, the phase changes induced by the geometric Doppler effect provide a convenient scheme for OAM manipulation. Figures 1(c) and 1(d) are the examples of phase addition and subtraction, respectively. For a metasurface with *q* = 1/2, the vortex beam with topological charge *l* = 4 would be converted to the vortex with topological charge *l* = 5 or *l* = 3, depending on the spin (*σ* = ±1) of incoming beam.

As the geometric phase induced by the geometric Doppler effect is spin-dependent in nature, it can also be applied to steer the photonic SHE. Originally, both the RVB and PB phases can lead to the photonic SHE, which contribute a spin-dependent shift in real space or momentum space by introduce the spin-dependent phase discontinuity or phase gradient in the beam cross section [36, 37]. Essentially, the geometric phase induced by the geometric Doppler effect in our scheme is the PB phase, as the polarization evolution included, which can be imposed to the phases that cause the photonic SHE. Therefore, it is possible to steer the photonic SHE by designing the metasurfaces based on the geometric Doppler effect.

An intrinsic photonic SHE has been reported in our past work when the rotational symmetry of linearly polarized CVB is broken [38]. Similar to the geometric photonic SHE observed in an oblique plane, which is not originated from the light-matter interaction [39]. And that it is a spin-dependent angular shift in real space, of which the magnitude of splitting is proportional to the beam propagation. The linearly polarized CVB can be viewed as the equal superposition of two opposite circularly polarized vortex components. When the rotational symmetry of this beam is broken, the opposite defective vortex phases of the two spin components will give rise to a spin-dependent splitting. In this work we applied dielectric metasurfaces with rotational asymmetry structure to steer the photonic SHE, as an example in Fig. 2(b). This incomplete structure can break the rotational symmetry of CVB, thus lead to the spin-dependent splitting of photonic SHE. Besides, the spatially rotating optical axes in the azimuthal direction will impose a spin-dependent PB phase on the original vortex phase of the spin components.

The original phase gradient contributed by the opposite vortex phases of the two circular polarized components of CVB can be deduced as

where*m*denotes the order of CVB. The phase gradient is additive to that introduced by the geometric Doppler effect. Therefore, the shift of wave vector

**k**can be deduced as

**k**) is in the direction of polarization evolving, which is orthogonal to the propagation orientation

*z*, the Δ

**k**would cause a splitting in real space: where

*k*= 2

*π/λ*,

*λ*is the wavelength of incident light. This formula of splitting is obtained in small-angle approximation, which demonstrates a spatial shift proportional to the absolute value of

*m*± 2

*q*.

According to these analyses, The metasurfaces designed based on the geometric Doppler effect will introduce a spin-dependent geometric phase to the passing beam. When a circularly polarized beam is incident upon the metasurfaces with spatially rotating optical axes, the additional phase will modify the phase front of beam. Therefore, a metasurface with rotational symmetry can be used to genetrate vortex beams or to modulate the OAM of beams. Furthermore, as the geometric phase is spin-dependent in nature, it can also be applied to steer the spin-dependent splitting of photonic SHE. The generated phase will superpose with the original phase which lead to the spin-dependent splitting, hence, it will change the magnitude and orientation of splitting. Meaningfully, the geometric Doppler effect do not need to differentiate the effects according to the propagation or polarization variation, which would provide a broad prospects for the designing of planar photonics.

## 3. Modulation of beam phase front

The metasurfaces produced by femtosecond laser writing of space-variant nano-grooves in a fused silica sample (Altechna R&D) are applied in our experiments to verify the conceives about geometric Doppler effect. This technique artificially creates an inhomogeneous form birefringence in the isotropic sample, and the local optical axis directions (fast and slow axes) are perpendicular and parallel to the grooves, respectively [40–42]. As the groove spacing is much smaller than the operation wavelength (632.8nm), the fabricated metasurfaces can be viewed as birefringent waveplates with homogeneous phase retardation and locally varying optical axis direction. Moreover, the transparent silica glass would offer the metasurfaces a relative high transmission ratio, which makes the observation in optical far field reliable. In this work, the phase retardation of our metasurfaces are fixed at *π*, and the optical axis directions are constructed according to Eq. (3) (where, *α*_{0} = 0). Three metasurfaces with different spatial rotation rates Ω in the azimuthal direction are applied in our experiments, which correspond to *q* = 1/2, 1, and 3/2. As an example, the distribution of local optical axes in the metasurface corresponding to *q* = 1/2 are depicted in the insets of Figs. 1(c) and 1(d).

To verify the phase front modulation with metasurfaces base on the geometric Doppler effect, the interference experiments are used to analysis the topological charge of transmitted beams [43]. The interference patterns would show fork dislocations as a vortex beam superposed with a Gaussian reference beam, and that the number of dislocations would equal to the topological charge of the vortex beam. Moreover, the direction of interference fringes is in the charge of the handedness of the vortex. Figure 3 shows the interference patterns of vortexes generated by the metasurfaces [Figs. 3(a)–3(c), respectively] and by a phase-only spatial light modulator (SLM, Holoeye Pluto-Vis) [Figs. 3(d)–3(f) corresponding to *l* = 1, 2, and 3]. As the SLM is a most commonly used device for generating the vortex beams, by comparison, Fig. 3 shows that the metasurfaces are high-quality vortex beam generators. Additionally, the metasurfaces with spatially rotating optical axes in azimuthal direction (Ω = *q*) would generate a vortex beam with topological charge *l* = ±2*q*, when a circularly polarized beam passes through the metasurfaces.

As what we have discussed above, the phase front of beam can also be added or be subtracted by the geometric phases generated from the geometric Doppler effect. When a vortex beam passes through the metasurfaces, its topological charge would be modulated according to the form *l* ± 2*q*. Therefore, we managed to verify the topological charge of output beams generated after vortexes passing through the metasurfaces. Figure 4 is the interference patterns get from the superposition with a Gaussian reference beam. It is found that the OAM of incident light is added or subtracted when we change the handedness of incident vortexes. Therefore, the geometric Doppler effect provides a convenient approach for the phase front modulation.

## 4. Manipulating the spin-dependent splitting

To observe photonic SHE and realize the steering of spin-dependent splitting, we firstly set up an a Sagnac interferometer to generate the linearly polarized CVB, as shown in Fig. 2, which is based on polarization beam splitter (PBS), SLM, and Mirror. A linearly polarized Gaussian beam output from a He-Ne laser is fixed at vertical polarization by the first polarizer (P). Then it passes through a beam splitter (BS) and a quarter-wave plate (QWP) with 45° optical axis orientation to be converted to a circularly polarized beam. The following PBS would split the circularly polarized beam into a horizontal (transmission) and a vertical (reflection) components with equal intensities. Subsequently, the horizontal polarized transmission component is transformed into a vortex beam with vertical polarization after passing through the SLM and the half-wave plate (HWP) with its optical axis 45° inclined to the horizontal direction. In contrast, the vertical polarized component is converted to a vortex beam with horizontal polarization after the modulation of HWP and SLM. An inserted Dove prism makes sure that the sign of the dynamic phase applied by the SLM are opposite for the two components. Finally, both components return back to the QWP which changes them into opposite circular polarizations. After superposition, the two orthogonal circular polarizations with just opposite dynamic phase would produce the linearly polarized CVB as what we wanted. The two cascaded HWPs are used to adjust the polarization state of the linearly polarized CVB.

We realized the steering of spin-dependent splitting occurring in three rotational asymmetrical metasurfaces with spatial rotation rates *q* = 1/2, 1, and 3/2. The three metasurfaes are designed with a 60° fan-shape to break the rotational symmetry of linearly polarized CVB, as represented by the inset (b) of Fig. 2. The combination of QWP, P, and charge couple device (CCD, Coherent LaserCam HR) is applied to measure the *S*_{3} distributions as shown in Fig. 2. which is used to characterize the spin (circular polarization) of light [44]. Figure 5 shows the *S*_{3} distributions observed when linearly polarized CVBs pass through the metasurfaces, red and blue represent the left- and right-circular polarizations, respectively. It is shown that the spin-dependent splitting is in variation with the spatial rotation rates *q* of metasurfaces and with the order *m* of CVBs. The affections of parameters *q* and *m* are respectively demonstrated by the columns and rows in the Fig. 5, which evidence that the magnitude of splitting is proportional to the absolute value of *m* − 2*q* as Eq. (8) predicted. Moreover, the orientation of splitting is inverted when the sign of *m* − 2*q* is reversed.

A more intuitive way for the observation of photonic SHE is to measure the intensity splitting directly, which is usually a challenge as the very small magnitude of splitting caused by the photonic SHE [4,8]. The splitting of spin components of light in the interface between two different media is the effect of RVB phase, because the alteration of light propagation in reflection or refraction [16,17]. It is an extreme tiny effect of light, the technique dubbed weak measurement is used to detect it [4–6]. However, the photonic SHE in our experiments is originated from the PB phase due to the polarization transformation. Thus, replacing a spatial transverse shift in the position space, it is a angular splitting which can be amplified by propagation in our experiments. Choosing a proper observing distance behind the metasufaces enabled us to record the intensity splitting of light. The intensity patterns in a same observing distance are demonstrated in Fig. 6. We chose the different combinations of *m* and *q* to exhibit this splitting. The same as the *S*_{3} splitting demonstrated in Fig. 5, it is shown that a larger absolute value of *m* − 2*q* promises a bigger intensity splitting pattern.

## 5. Discussions and conclusions

In comparing with our past photonic spin Hall effect relevant works [33, 34], this work developed the Doppler effect to explain the relation between the structure patterns and the photonic spin Hall effect. Although the same structures were applied in our past works, the physical mechanisms and main concerns are totally different for them. The first reported work Ref. [33] presented a novel form of photonic spin Hall effect near the phase singularity. And the second work Ref. [33] gave the precondition to observing the photonic spin Hall effect, which is supposed to break the rotational symmetry of the metasurfaces. However, in this work we found the solution to manipulate the spin-dependent splitting. The magnitude of splitting is directly related the spatially rotating structures by the geometric Doppler effect.

In conclusion, we realized the manipulation of spin-dependent splitting by the geometric Doppler effect. The metasurfaces with spatially rotating optical axes will impose a phase change on the passing beam, which is donated by the geometric Doppler effect. By constructing metasurfaces with rotating optical axes in azimuth and making this structure with rotational symmetry, the produced phase change forms a vortex front. Thus the metasurfaces can be applied to modulate the orbital angular momentum of passing beams. Furthermore, the additional phase induced by the geometric Doppler effect will alter the spin-dependent phase gradient which appears when rotational symmetry of a linearly polarized cylindrical vector beam is broken, where the phase gradient gives rise to the spin-dependent splitting of passing beam (i.e., photonic spin Hall effect). Accordingly, the magnitude of spin-dependent splitting of the photonic spin Hall effect is tunable by replacing the metasurfaces with different spatial rotation rates. The geometric Doppler effect in metasurfaces possessing spatially rotating configuration may facilitate the conceiving of novel planar photonics devices.

## Acknowledgments

This research was partially supported by the National Natural Science Foundation of China (Grants Nos. 11274106 and 11474089).

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