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Observation of photonic spin Hall effect (SHE) near the phase singularity at dielectric metasurfaces is presented. The structured metasurface works as a space-variant Pancharatnam-Berry phase element and produces a vortex beam with phase singularity. The dynamical vortex phase is introduced to eliminate or enhance the phase singularity, thus realizing the manipulation of spin-dependent Pancharatnam-Berry phase. The spin-orbit coupling near the singularity of the Pancharatnam-Berry phase leads to the observation of the photonic SHE which manifests itself as spin-dependent splitting. The underlying mechanism is significantly different from previously reported cases. It thereby provides an alternative way to manipulate the spin states of photons.
© 2015 Optical Society of America
1. Introduction
Photonic spin Hall effect (SHE) is generally believed to be a result of topological spin-orbit interaction which describes the coupling between the spin and the trajectory of light beam propagation [1–3]. The spin-orbit interaction corresponds to two types of geometric phases: the Rytov-Vladimirskii-Berry phase and the Pancharatnam-Berry phase. The former is related to the evolution of the propagation direction of light and the latter to the manipulation with the polarization state of light [4, 5]. When a linearly polarized light beam impinges obliquely upon an interface, the spin-dependent splitting in the real position space occurs, which is associated with the Rytov-Vladimirskii-Berry phase [6–10]. While for the light beam propagating in inhomogeneous anisotropic media, the generated spin-dependent splitting in the momentum space is related to the Pancharatnam-Berry phase [11,12]. In these two cases, the magnitude of spin-dependent splitting are both determined by the gradient of geometric phase.
In this work, we present an observation of photonic SHE near singularity of geometric phase. The phase singularity is strongly related to the vortex beam with optical phase undetermined at the zero point of intensity. A dielectric metasurface works as a space-variant Pancharatnam-Berry phase element and produces a vortex beam with phase singularity. The input beam with dynamical vortex phase generated by the spatial light modulator is introduced to eliminate or enhance the phase singularity, and thus realizing the manipulation of spin-dependent Pancharatnam-Berry phase. The spin-orbit coupling near the singularity of the Pancharatnam-Berry phase leads to the observation of photonic SHE.
2. Phase singularity in Pancharatnam-Berry geometric phase
Optical metasurfaces comprise a class of optical metamaterials with a reduced dimensionality that demonstrate exceptional abilities for controlling light [13]. A dielectric-based metasurface can be fabricated by femtosecond laser writing of self-assembled nanostructures in silica glass. The local optical axes at each point are oriented parallel and perpendicular to the subwavelength grooves [14]. This leads to build an artificial uniaxial crystal with locally varying optical axes and homogeneous phase retardation. In particular, the direction of optical axis can be specified by the following expression:
where (r, φ) is the polar coordinate representation, α0 is a constant angle denoting the initial orientation for φ = 0, and q is a constant describing the spatial rotation rate of the metasurface structure. The nanostructures in the metasurface are smaller than the input wavelength, so only the zero order is a propagating one, and all other orders are evanescent.For the sake of simplicity while neglecting the loss, the Jones matrix describing a uniaxial crystal with fast axis in the x-direction, can be represented as [15]
where δ is the retardation phase at the working wavelength λ. For our metasurface, the optical axis orientation is space-variant, and its Jones matrix can be described by a coordinate-dependent matrix [16]: Here, the position-dependent Jones matrix T(r, φ) can be written as andWe first consider the case where the metasurface is normally illuminated by a circularly polarized vortex beam with spin angular momentum (SAM) σ±h̄ and orbital angular momentum (OAM) lh̄ [17, 18], where σ+ = +1 for the left-handed circular polarization and σ− = −1 for the right-handed circular one, respectively. Its Jones vector of input beam is then given by Ein(r, φ) = E0(r, φ) · (1, σ±i)T exp(ilφ). The output beam Eout (r, φ) = T(r, φ)Ein(r, φ) can be written as
We find that the output field can be regarded as a superposition of two parts: One has the same SAM and OAM as the input, and the other having a reversed SAM and a modified OAM given by (l + 2σ±q)h̄. The amplitudes of these two components of the output beam depend on the phase retardation δ, i.e., cosδ/2 and sinδ/2, respectively. It means that the input beam only partially converts its spin angular momentum to orbital part [19].The output field acquires a nonuniform phase retardation which depends on the incident spin states and the metasurface geometry. Based on our derivation, the second term of Eq. (6) indicates the generation of a component which is polarized orthogonally to the input wave, and its additional phase equals to twice of the local optical axis orientation 2σα±. Similarly, the additional phase with an opposite topological charge should be generated if the input polarization handedness is inverted. This particular approach for optical wavefront shaping is purely geometric in nature, and thereby the additional phase factor is the so-called Pancharatnam-Berry geometric phase [16]. The wavefront-shaping devices based on this principle have been realized and can be referred to as Pancharatnam-Berry phase optical elements [20]. In addition, the phase factor exp(ilφ) [see Eq. (6)] in the output field is spin-independent, which is purely dynamic in nature.
It is known that the circularly polarized input light reverses its handedness of circular polarization and acquires a constant phase factor when passing through a homogeneous half wave-plate. The metasurface possessing inhomogeneous anisotropic axes, however, can result in the occurrence of the spin-orbit interaction [21]. As expected, the metasurface with phase retardation δ = π not only inverts the handedness of circular polarization completely, but also endows the output wave with a spin-dependent geometric phase [22, 23]. Hence, it is possible to realize the evolution of vector beam with any desired polarization distribution in high-order Poincaré sphere [24–26]. Note that the cylindrical vector beam with vortex can also be obtained when the metasurface with phase retardation δ = π/2 is illuminated by the circular polarized light [27–29].
We now consider the case that the metasurface illuminated by a linearly polarized vortex wave whose Jones vector is then given by Ein(r, φ) = E0(r, φ)(cosθ, sinθ)T exp(ilφ). The output beam Eout (r, φ) = T(r, φ)Ein(r, φ) can be obtained as
Here, we have assumed δ = π. For the right-handed circular polarization, the spin-orbit coupling of light in the metasurface leads to the appearance of the helical phase with topological charge l + 2q. Similarly, for the left-handed one, the vortex beam with topological charge l − 2q is obtained (see Fig. 1 for example). This interesting phenomenon can be explained as the appearance of the Pancharatnam-Berry phase when the beam passes through the metasurface. The input beam with vortex phase exp(ilφ) generated by the SLM is dynamical in nature, and can therefore eliminate the phase singularity of one spin component (l − 2q = 0 or l + 2q = 0), while enhances the phase singularity of the other. The spin-orbit interaction near the singularity of the Pancharatnam-Berry phase leads to the observation of spin-dependent splitting of light, i.e., the photonic SHE. By switching the signs of the topological charges of the input beam, the spin-dependent splitting pattern can be manipulated effectively.
Fig. 1 Schematic illustration of the generation of spin-dependent phase singularity. The input beams possess the same orbital angular momenta and opposite spin angular momenta. (a) The phase singularity is eliminated for a incident beam with σ = +1 and l = +1. (b) The phase singularity is enhanced for a incident beam with σ = −1 and l = +1. The spin-dependent splitting in output beam presents when the input beam with linear polarization.
3. Observation of spin-dependent splitting
Figure 2 shows the schematic illustration of the experimental setup. In this experiment, we use a Gaussian beam with fundamental mode as the input beam, which is produced by a He-Ne laser working at the wavelength λ = 632.8nm. The laser beam is converted to linear polarization state by a Glan laser polarizer (GLP1), and then transformed into a vortex-bearing beam using a reflective phase-only spatial light modulator (SLM) (Holoeye Pluto-Vis). A quarter-wave plate (QWP1) with its optical axis ±45° inclined to horizontal direction converts the horizontal polarizations to two orthogonal circular polarizations. The metasurface (MS) is used to transform the LHC (RHC) polarization beam into RHC (LHC) polarization with an additional helical phase. As shown in Fig. 2, we can add another Glan laser polarizer (GLP2) and another quarter waveplate (QWP2) behind the metasurface to analyze the polarization of the output beam. Insets (a) and (b) in Fig. 2 present two schematic pictures of the metasurface with q = 1/2 and q = 1, respectively. In our scheme, the dielectric metasurface is fabricated by femtosecond laser writing of self-assembled nanostructures in a silica glass (Altechna). In principle, the metasurfaces can be regarded as a two dimensional dielectric nanograting, and the photonic SHE can thereby be detected in far field with high transmission efficiency.
Fig. 2 Experimental setup for observation of photonic SHE near phase singularity. The He-Ne laser inputs a linearly polarized Gaussian beam, and the Glan laser polarizer (GLP1) ensures the polarization direction to be horizontal. A linearly polarized vortex beam is produced by the phase-only spatial light modulator (SLM) with nonpolarized beam splitter (BS), and then passes through the structured metasurface (MS). The quarter waveplate (QWP1) with its optical axis direction 45° inclined to the horizontal direction can convert the linearly polarization vortex into a circular polarization one. A polarizer (GLP2) combined with a quarter waveplate is the typical setup for measuring the Stokes parameter. The Insets (a) and (b): Schematic pictures of metasurfaces with q = +1/2 and q = +1, respectively.
To observe the spin-dependent intensity distribution of the output beams, we first normally illuminate the metasurface with a circularly polarized vortex beam. We measure the intensity pattern and the Stokes parameters when the incident beam is circularly polarized. The CCD (Coherent LaserCam HR) camera is used to record the transmitted intensity distribution, and the quarter waveplate (QWP2) combined with a polarizer (GLP2) acts as a polarization analyzer. By rotating the GLP to two angles (±45°) and holding QWP2 in the x direction, we can obtain the intensity distributions on the CCD. As the Stokes parameter S3 is defined as [30] S3 = (I+45° − I−45°)/(I+45° + I−45°), where I±45° represents the recorded intensity when the transmission axis of the GLP2 is set as ±45°. The experimental results are plotted in Fig. 3. We find that the intensity distributions depend on the signs of spin polarization and topological charges.
Fig. 3 Intensity distribution (first and third rows) and spin distribution (S3 parameters, second and forth rows) of output beam when the input beam is circularly polarized. The spin-dependent distribution can be modulated by the topological charges l and circular polarization σ of input beams (columns). The metasurface structures are chosen with q = +1/2 and q = +1 (rows).
We now consider that the metasurface is illuminated by a linearly polarized beam. Figure 4 shows the measured intensity distributions of the Stokes parameters when the input beam exhibits different topological charges. According to the measured intensity distributions by rotating the polarization analyzer GLP2, the Stokes parameter S3 of the output beams are sketched. In Fig. 4, we can clearly observe a typical spin-dependent splitting pattern when the input beam with topological charge l = ±1 and l = ±2, respectively. We find that the dynamical phase generating by the SLM can be introduced to eliminate or enhance the phase singularity, and thus realize the manipulation of the splitting pattern in photonic SHE. Due to the phase with singularity is rotation symmetry, one of spin component present a donut intensity distribution. The other component, however, presents a Gaussian distribution at the center of donut due to its singularity is eliminated by the dynamical phase. Therefore, we can realize a tunable photonic SHE by modulating the incident dynamic phase.
Fig. 4 The output beams present a spin-dependent splitting near the phase singularity when the input beam is linearly polarized. The pattern of spin-dependent splitting can be modulated by the topological charges of input beams. (a) and (b) are l = +1 and l = −1, respectively. The structure parameter of metasurface is chosen as q = +1/2. (c) and (d) are l = +2 and l = −2, respectively. The structure parameter of metasurface is chosen as q = +1.
Note that the photonic SHE at a plasmonic metasurface has also been reported recently [31]. The spin-dependent splitting in that case is perpendicular to designed phase gradient, and is thereby attributed to the Rytov-Vladimirskii-Berry phase. Another very important result shows that the spin degeneracy removal occurs the metasurface structure presents an inversion asymmetry [32]. The spin-dependent splitting in that situation is parallel to the designed phase gradient, and is thereby related to the Pancharatnam-Berry phase. In our scheme, obviously different from the previously reported photonic SHE, the spin-dependent splitting is founded near the phase singularity of Pancharatnam-Berry phase. Furthermore, the all dielectric metasurface applied in our case can dramatically enhance the efficiency of transmission and spin-dependent splitting can be easily observed in far filed. Both the dynamic phase and the geometric phase are introduced in our experiment to eliminate or enhance the phase singularity. The dynamic phase generated by the spatial light modulator is introduced to switch the spin-dependent splitting by controlling the sign of its topological charges. The interesting phenomena reported here are also significantly different from the geometric photonic SHE in which the light-matter interaction is not necessary [33–35].
4. Conclusions
In conclusion, we have demonstrated that the spin-dependent splitting can be observed near the phase singularity when a structured metasurface is illuminated by a linearly polarized vortex beam. The metasurface works as a space-variant Pancharatnam-Berry phase element and produce vortex phase with singularity. The dynamical vortex phase generated by the spatial light modulator is introduced to eliminate or enhance the phase singularity, and thus realize the manipulation of the spin-dependent splitting of light. For these reasons, the photonic SHE relating to the phase gradient near the phase singularity is attributed to the combined contributions of dynamic phase and geometric Pancharatnam-Berry phase. Meanwhile, the underlying mechanism of spin-orbit coupling is significantly different from the previously reported photonic SHE, and thereby provides an alternative way to manipulate the spin states of photons.
Acknowledgments
This research was partially supported by the National Natural Science Foundation of China (Grants Nos. 11274106 and 11474089).
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