Unveiling the photonic spin Hall effect with asymmetric spin-dependent splitting Xinxing Zhou1,3 and Xiaohui Ling2∗ 1 College
of Physics and Information Science, Hunan Normal University, Changsha 410081, China 2 Laboratory for Optics and Optoelectronics, College of Physics and Electronic Engineering, Hengyang Normal University, Hengyang 421002, China 3 [email protected] ∗ [email protected]
Abstract: The photonic spin Hall effect (SHE) manifests itself as the spin-dependent splitting of light beam. Usually, it shows a symmetric spin-dependent splitting, i.e., the left- and right-handed circularly polarized components are equally separated in position and intensity for linear polarization incidence. In this paper, we theoretically propose an asymmetric spin-dependent splitting at an air-glass interface under the illumination of elliptical polarization beam and experimentally demonstrate it with the weak measurement method. The left- and right-handed circularly polarized components show expectedly unequal intensity distributions and unexpectedly different spin-dependent shifts. Remarkably, the asymmetric spin-dependent splitting can be modulated by adjusting the handedness of incident polarization. The inherent physics behind this interesting phenomenon is attributed to the additional spatial Imbert-Fedorov shift. These findings offer us potential methods for developing new spin-based nanophotonic applications. © 2016 Optical Society of America OCIS codes: (260.5430) Polarization; (240.0240) Optics at surfaces; (120.0120) Instrumentation, measurement, and metrology.
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Received 1 Dec 2015; revised 2 Feb 2016; accepted 2 Feb 2016; published 5 Feb 2016 8 Feb 2016 | Vol. 24, No. 3 | DOI:10.1364/OE.24.003025 | OPTICS EXPRESS 3025
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Received 1 Dec 2015; revised 2 Feb 2016; accepted 2 Feb 2016; published 5 Feb 2016 8 Feb 2016 | Vol. 24, No. 3 | DOI:10.1364/OE.24.003025 | OPTICS EXPRESS 3026
1.
Introduction
Recently, similar to the spin Hall effect (SHE) in electronic system [1, 2], the photonic SHE manifesting itself as the spin-dependent splitting of left- and right-handed circularly polarized components of a spatially confined light beam reflecting or refracting at an optical interface has been theoretically and experimentally investigated in different physical systems such as optical physics [3–9], semiconductor physics [10], high-energy physics [11, 12], and plasmonics [13–15]. The photonic SHE is generally believed to be a result of an effective spinorbit coupling which corresponds to geometric Berry phase. There are two types of geometric phases: the spin-redirection Berry phase that is associated with the variations of direction of light beam propagation and the Pancharatnam-Berry phase related to beam’s polarization state evolution [16,17]. The photonic SHE itself can be developed into a useful metrological tool for characterizing the structure parameter variations of different physical systems. For example, we can use the photonic SHE to probe spatial distributions of electron spin states [10], measure the thickness of nanometal film [18], identify graphene layers [19], detect the axion coupling in topological insulators [20], and determine the magneto-optical constant of Fe films [21]. Remarkably, the photonic SHE offers new opportunities for manipulating photons, which provides an additional spin degree of freedom for developing spin-controlled nanophotonic applications [14, 15, 22]. Generally, the spin-dependent splitting in photonic SHE exhibit symmetry properties where the left- and right-handed circularly polarized components are equally separated in position and intensity for linear polarization incidence. In this work, we establish a general propagation model to describe an asymmetric spin-dependent splitting in photonic SHE when an elliptical polarization beam impinges upon an air-glass interface. We find that the left- and right-handed circularly polarized components show distinct intensity distributions and unequal spin accumulations. Importantly, the asymmetric distributions of the spin-dependent splitting can be modulated by adjusting the handedness of incident polarization. Therefore, we can develop a tunable photonic SHE with any desired intensity distributions and spin-dependent splitting for generation and manipulation of spin-polarized photons. These findings offer us potential methods for developing new spin-controlled nanophotonic devices. The rest of the paper is organized as follows. First, we theoretically analyze the asymmetric spin-dependent splitting in photonic SHE under a slightly elliptical polarization beam illumination. Next, we focus our attention on the weak measurements experiment for observing this phenomenon. Under this condition, the weak value includes both of the real and imaginary parts. Finally, a conclusion is given. 2.
Theoretical model
In this part, we first establish a general propagation model to describe the process of elliptical polarization beam reflected at an air-glass interface and theoretically investigate the corresponding asymmetric photonic SHE. Here, the electric field distributions and the spin-dependent shifts of the left- and right-handed circularly polarized components will be discussed. The asymmetric photonic SHE is schematically shown in Fig. 1. The first and second columns show the polarization and intensity distributions of incident light beam with linear and elliptical polarization, respectively. We can see that the electric field intensity distributions are symmetrical for linear polarization light [Figs. 1(b) and 1(e)]. However, for the elliptical polarization beam, the left- and right-handed circularly polarized components indicate asymmetric splitting [Figs. 1(h) and 1(k)]. The last column of Fig. 1 describes the intensity distributions and spindependent splitting of the left- (solid line) and right-handed (dashed line) circularly polarized components of reflected beam. When the incident light beam is linear polarization, the filed intensity (the height of Gaussian components) and spin-dependent shifts (δ± ) of these two spin components show equal values [the same heights and δ+ = δ− , as shown in Figs. 1(c) and 1(f)]. #254802 © 2016 OSA
Received 1 Dec 2015; revised 2 Feb 2016; accepted 2 Feb 2016; published 5 Feb 2016 8 Feb 2016 | Vol. 24, No. 3 | DOI:10.1364/OE.24.003025 | OPTICS EXPRESS 3027
Fig. 1. Schematic of the symmetric and asymmetric photonic SHE at the air-glass interface. The first and second column show the polarization and intensity distributions of incident Gaussian beam. The intensity distributions and spin-dependent shifts of left- and righthanded circularly polarized components of reflected beam are described in the third column. Here, we consider the elliptical polarization beam with its long and short axis along to the horizontal and vertical directions.
As for the illumination of elliptical polarization beam, the asymmetric photonic SHE appears, resulting in the unequal separation in position and intensity of left- and right-handed circularly polarized components [with different heights and δ+ = δ− , as shown in Figs. 1(i) and 1(l)]. Next, we will quantitatively study the reflection process of elliptical polarization beam and the corresponding asymmetric photonic SHE. For simplicity, we choose the long and short axis of the elliptical polarization beam directed along the horizontal (H) and vertical (V) directions, respectively. The elliptical polarization beam can be decomposed into two orthogonal polarization states H and √ V. In the spin basis, the H and √ V polarization states can be expressed as |H = (|+ + |−)/ 2 and |V = i(|− − |+)/ 2. We theoretically analyze the asymmetric photonic SHE by establishing the relationship between the incident and reflected fields. As for the incident light beam with elliptical polarization, the Jones vector can be written as (cos β , eiϕ sin β )T . Here β represents the azimuth angle (the angle between the crystal axis of wave plate and horizontal axis) and ϕ denotes the phase difference between the H and V polarization components. In the present study, we consider an elliptical polarization beam with its long and short axis along to the horizontal and vertical directions. Therefore, the Jones vector can be simplified to (cos β , +i sin β )T or (cos β , −i sin β )T representing the left- or right-
#254802 © 2016 OSA
Received 1 Dec 2015; revised 2 Feb 2016; accepted 2 Feb 2016; published 5 Feb 2016 8 Feb 2016 | Vol. 24, No. 3 | DOI:10.1364/OE.24.003025 | OPTICS EXPRESS 3028
elliptical polarization in the case of ϕ = ±π /2. It is noted that the azimuth angle β (also the ellipticity) mentioned above is a tiny value allowing for a slightly elliptical polarization. We consider the incident source is a monochromatic Gaussian beam whose spectrum is arbitrarily narrow and can be written as 2 + k2 ) w20 (kix w0 iy , (1) |Ei = (eix + iσ eiy ) √ exp − 4 2π where w0 is the beam waist. The polarization operator σ = ±1 stand for left- and right-handed circularly polarized components, respectively. According to the transversality, we can obtain the reflected field of left- and right-handed circularly polarized components in the case of leftelliptical polarization incidence [7] (the case of right-elliptical polarization can be obtained in the similar way), with the long axis being along to the horizontal and vertical directions: cos β H H H = rp√ exp k (+i δ + i δ tan β ) exp(+ η ) |+ |E ry L 2 r p cos β exp kry (−iδ H + iδ H tan β ) exp(−η ) |−, (2) + √ 2 V |E L
rs sin β V V exp kry (+iδ + iδ cot β ) exp(+ϑ ) |+ = √ 2 rs sin β V V exp −kry (−iδ + iδ cot β ) exp(−ϑ ) |−. + √ 2
(3)
Here, the r p and rs denote the Fresnel reflection coefficients for parallel and perpendicular polarizations, respectively. As shown in Eq. (2), we provide that kry δ H 1, kry δ H tan β 1, and η 1. In Eq. (3), we provide that kry δ V 1, kry δ V cot β 1, and ϑ 1. Here, δ H =(1 + rs /r p ) cot θi /k0 , δ V =(1+r p /rs ) cot θi /k0 , η =rs tan β /r p , and ϑ =r p cot β /rs . k0 is the wave number in free space. The terms exp(±ikry δ H ) and exp(±ikry δ V ) stand for the spin-orbit coupling and we find that the shifts induced by these terms are symmetrical, whereas the other electric field components exp(ikry δ H tan β ) and exp(ikry δ V cot β ) are not the spin-orbit coupling terms. However, they can affect the spin-dependent spatial splitting in photonic SHE, which will be discussed in the follwing. From Eqs. (2) and (3), we can obtain the intensity distributions of left- and right-handed circularly polarized components as shown in Figs. 2(a)-2(d). It should be noted that the electric field components exp(±η ) and exp(±ϑ ) cause the asymmetric intensity distributions. For simplicity, we only consider the long axis of incident elliptical polarization beam along to the horizontal direction. Here, for example, the ellipticity are chosen as 0.2◦ and 5◦ . And the incident angles are selected as θi = 30◦ and θi = 45◦ , respectively. From the pictures, we find that the left- and right-handed circularly polarized components show unequal intensity distributions which can be modulated by changing the ellipticity and incident angles. The photonic SHE is described for the left- and right-circularly polarized components undergoing spin-dependent shifts, so the reflected field centroid should be determined. In the present work, we only consider the spin-dependent splitting in the transverse direction. According to Eqs. (2) and (3), we can calculate the field centroid distribution. At any given plane z = const., the transverse displacement of field centroid compared to the geometrical-optics prediction is given by ∂k |E ry |E E|i z E|k ry + . (4) yH,V ± = E E k0 E| E|
#254802 © 2016 OSA
Received 1 Dec 2015; revised 2 Feb 2016; accepted 2 Feb 2016; published 5 Feb 2016 8 Feb 2016 | Vol. 24, No. 3 | DOI:10.1364/OE.24.003025 | OPTICS EXPRESS 3029
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Fig. 2. The intensity distributions (a-d) and initial spin-dependent shifts (e-h) of left- and right-handed circularly polarized components in the asymmetric photonic SHE. The ellipticity are chosen as β = 0.2◦ and β = 5◦ . The intensity is plotted in normalized units. (a) and (b) show the intensity distributions when the incident angle is selected as θi = 30◦ . The long axis of incident elliptical polarization beam is along to the horizontal direction and the handedness of incident polarization is left. (c) and (d) stand for the intensity distributions under the condition of incident angle is selected as θi = 45◦ . Here, the handedness of incident polarization and the direction of long axis are chosen as the same as above. (e) and (f) denote the displacements in the case of the handedness of incident polarizations are left and right, and the long axis of incident elliptical polarization beam is along to the horizontal direction. (g) and (h) describe the shifts under the condition of the long axis of incident elliptical polarization beam along to the vertical direction.
#254802 © 2016 OSA
Received 1 Dec 2015; revised 2 Feb 2016; accepted 2 Feb 2016; published 5 Feb 2016 8 Feb 2016 | Vol. 24, No. 3 | DOI:10.1364/OE.24.003025 | OPTICS EXPRESS 3030
In the above equation, ∂kry = ∂ ∂kry ery and kry = kry ery . The first and second terms in Eq. (4) stand for the spatial and angular shifts, which are independent of and dependent on z, respectively. In fact, after calculating, we find that the photonic SHE shows pure spatial displacements which are independent of z. Evaluating Eq. (4), we can obtain the asymmetric spin-dependent shifts of elliptical polarization beam. Figures 2(e)-2(h) show the initial transverse shifts of left- and right-handed circularly polarized components changing with the ellipticity and handedness of incident polarization. Under this condition, we find that the left- and right-handed circularly polarized components show unequal spin-dependent shifts, which is different from the previous works [4,5,22]. Here, the ellipticity are also chosen as 0.2◦ and 5◦ . First, we consider the long axis of elliptical polarization beam along to the horizontal direction [shown in Figs. 2(e) and 2(f)]. In this case, we find that the left- and right-handed circularly polarized components represent unequal values of spin-dependent shifts which are extremely sensitive to the ellipticity variations. Remarkably, the asymmetric spin-dependent splitting can be modulated by adjusting the handedness of incident polarization. Figures 2(g) and 2(h) denote the condition of the long axis of elliptical polarization beam along to the vertical direction. Here, the transverse displacements of left- and right-handed circularly polarized components change slowly with the variations of ellipticity. However, this asymmetric spin-dependent splitting can also be adjusted when the handedness of incident polarization alters. So, we can develop a tunable photonic SHE with any desired electric field intensity and spin-dependent splitting. We can attribute this asymmetric photonic SHE to the additional spatial Imbert-Fedorov shift [23, 24]. From the Eqs. (2) and (3), as for the left elliptically polarized light impinges at the media interface, we find that there exists a unified spatial displacement induced by the terms exp(ikry δ H tan β ) and exp(ikry δ V cot β ) in addition to the normal symmetric spin-dependent shifts. After calculating Eq. (4), we can describe the final field centroid as follows: H,V H,V yH,V ± = δ± + δIF ,
(5)
where δ±H,V stand for the symmetric spin-dependent shifts of left- and right-handed circularly H,V denotes the unified spatial Imbert-Fedorov shift. For simplicpolarized components, and δIF ity, the specific expressions of these two field centroids are not given. We can see that the normal symmetric spin-dependent displacements of left- and right-handed circularly polarized components will undergo an overall shift (the same sizes and signs), then the asymmetric spindependent shift appears. When the state of the incident light beam changing from left-elliptical polarization to the right one, the left- and right-handed circularly polarized components will also undergo an overall shift but with the opposite directions (as shown in Fig. 2). 3.
Experimental observation and discussion
The asymmetric photonic SHE induced spin-dependent splitting are too small to be detected directly. However, the signal enhancement technique known as the weak measurements [25–30] can resolve this problem. The weak measurements based on preselection and postselection states has attracted a lot of attention and holds great promise for precision metrology such as beam deflection measurement [31], measuring small optical phase shift [32,33], direct detection of the quantum wavefunction [34], observing the average trajectories of single photons [35], and full characterization of polarization states of light [36]. It should be noted that the weak measurements method involved in the present work shows some adjustments based on the previous weak measurements, which will be discussed in the following. There exists an amplified factor so-called weak value playing a great role in weak measurements. The weak value establishes the relationship between the observable and the shifts in
#254802 © 2016 OSA
Received 1 Dec 2015; revised 2 Feb 2016; accepted 2 Feb 2016; published 5 Feb 2016 8 Feb 2016 | Vol. 24, No. 3 | DOI:10.1364/OE.24.003025 | OPTICS EXPRESS 3031
Fig. 3. (a) Schematic of the preselection and postselection states on Bloch sphere. (b) and (c) show the results of | f |iRe(Aw )| and | f |iIm(Aw )| as a function of the the ellipticity β and amplification angle Δ. Both of the functions are plotted in normalized units.
measurement pointer’s mean position and mean momentum Aw =
ˆ f |A|i . f |i
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|i and | f stand for the preselection and postselection states. As shown in Fig. 3(a), the Bloch sphere can be introduced to describe the weak measurements process [37]. We consider an elliptical polarization beam incident and the long and short axis are along to the horizontal and vertical directions. Therefore the preselection state of the system can be written as:
Θ+β Θ+β |+ + eiΦ sin |−, (7) |i = cos 2 2 where Θ and Φ denote the qubit state on the surface of the Bloch sphere. And the postselection state of the system is chosen as:
Θ Θ −i(Φ+Δ) |+ − e |−. (8) cos | f = sin 2 2 Here, the angle Δ show the amplification angle which will be discussed in the following. For practical purposes, we set Θ = π /2. Using Eq. (6) together with Eq. (7) and (8), we can calculate the real and imaginary parts of the weak value: Re(Aw ) =
#254802 © 2016 OSA
sin β , cos β cos Δ − 1
(9)
Received 1 Dec 2015; revised 2 Feb 2016; accepted 2 Feb 2016; published 5 Feb 2016 8 Feb 2016 | Vol. 24, No. 3 | DOI:10.1364/OE.24.003025 | OPTICS EXPRESS 3032
Fig. 4. Experimental setup for measuring the asymmetric photonic SHE and the corresponding weak measurements process. (a) shows the experimental setup. The sample is a BK7 glass. The light source is a 21mW linearly polarized He-Ne laser at 632.8nm (Thorlabs HNL210L-EC). L1 and L2, lenses with effective focal length 50mm and 250mm, respectively. HWP, half-wave plate (for adjusting the intensity). QWP, quarter-wave plate for generating elliptical polarization beam. P1 and P2, Glan Laser polarizers. CCD, chargecoupled device (Coherent LaserCam HR). The inset: The incident beam is preselected in the left- or right-elliptical polarization state by P1 and QWP. The optical axis of P1 make the angles β or −β with horizontal axis. (b) denotes the weak measurements process with linear polarization (the first row) and elliptical polarization (the second row) beam.
Im(Aw ) =
1 . cot Δ − csc Δ sec β
(10)
It can be seen that the real and imaginary parts of weak value are all nonzero when we introduce the tiny angles β and Δ. This is different from the previous work [5] in which the angle β = 0 and the weak value is a pure imaginary number. In order to obtain the large output, we need to maximize the weak value including both of the real and imaginary parts. Figures 3(b) and 3(c) show the products of | f |iRe(Aw )| and | f |iIm(Aw )|. We can see that both of the ellipticity β and the amplification angle Δ should be chosen as the tiny values for obtaining a large weak value. For simplicity, in the following, we replace the symbols Re(Aw ) and Im(Aw ) with ARe and AIm . There also exists another amplification mechanism called propagation amplification [29] that produces the amplified factor F accompanying by the imaginary part of weak value, which leads to the possibility of even larger enhancements following the beam free evolution. Finally, we
#254802 © 2016 OSA
Received 1 Dec 2015; revised 2 Feb 2016; accepted 2 Feb 2016; published 5 Feb 2016 8 Feb 2016 | Vol. 24, No. 3 | DOI:10.1364/OE.24.003025 | OPTICS EXPRESS 3033
can obtain the amplified shifts [from Eq. (5)] by the weak value and propagation amplifications: yw = (ARe + FAIm )|δ±H,V |,
(11)
so it can be detected directly by our experimental setup. It should be noted that, in this case, we can not amplify the spatial Imbert-Fedorov shift due to the fact that the observable is only the σˆ 3 which is corresponding to the spin projection along the central propagation direction. The weak value amplification process can convert the position displacements caused by the photonic SHE into a momentum shift, and it also converts the momentum shifts into a position shift. However, in our work, the contribution of the real part is much smaller than that in imaginary part due to two reasons: the tiny ellipticity and the beam free evolution. Therefore, the imaginary part plays a key role in the final amplification and we can rewrite the Eq. (11) as follows: yw ≈ FAIm |δ±H,V |.
(12)
This is the important result for the following experimental measurement. We can measure the amplified shifts by reading out the difference between two states |V ± Δ or |H ± Δ (the AIm shows opposite signs). Our experimental setup shown in Fig. 4(a) is similar to that in [5, 8, 18]. A He-Ne laser is used to generate linear polarization Gauss beam which firstly impinges onto the HWP. This HWP can adjust the polarization of incident light beam, which is used to control the light intensity preventing the charge-coupled device (CCD) from saturation. Then, the light beam passes through a short focal length lens (L1) and is preselected as a slightly elliptical polarization state by P1 and QWP, which is slightly different from the previous work [5]. Here, the P1 is chosen as |H or |V . When the beam reaches the sample (BK7 glass) interface, the asymmetric photonic SHE takes place allowing for the reflected beam separated into two unequal spin components. As the reflected beam splits by a fraction of the wavelength, the left- and right-handed 400
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Fig. 5. The amplified (a, b) and initial (c, d) spin-dependent shifts of asymmetric photonic SHE in the case of elliptical polarization beam with its long and short axis along to the horizontal (the left column) and vertical directions (the right column). Here, the ellipticity is chosen as β = 0.2◦ and the handedness of incident polarization is left. The lines indicate the theoretical value. The diamonds, squares, and triangles show the experimental results.
#254802 © 2016 OSA
Received 1 Dec 2015; revised 2 Feb 2016; accepted 2 Feb 2016; published 5 Feb 2016 8 Feb 2016 | Vol. 24, No. 3 | DOI:10.1364/OE.24.003025 | OPTICS EXPRESS 3034
circularly polarized components interfere destructively at the second polarizer (P2) [as shown in Fig. 4(b)]. From Eq. (12), we can measure the amplified shifts by discriminating the difference between the displacements for rotating the P2 as two states | f = |V ± Δ or |H ± Δ. Here, the Δ is a small angle that we called amplification angle. This process can be seen as the postselection. Using this method, we can deduce the initial symmetrical spin-dependent shifts δ±H,V . By considering the additional spatial Imbert-Fedorov shift in Eq. (5), we can finally obtain the asymmetric spin-dependent displacements yH,V ± . In our weak measurement experiment, we choose the amplification angle Δ = 0.8◦ . After passing through the polarizer P2, we use L2 to collimate the beam and make the beam shifts insensitive to the distance between L2 and the CCD, which will improve the measurement precision. Finally, a CCD is used to measure the amplified shift after L2. In the case of elliptical polarization beam with its long axis along to the horizontal direction, we measure the amplified displacements of asymmetric photonic SHE on the BK7 glass every 0.5◦ from 52◦ to 60◦ [as shown in Fig. 5(a)]. After obtaining the amplified shifts, we can deduce the initial displacements [Fig. 5(c)]. In our experiment, we choose the ellipticity as β = 0.2◦ and the incident beam is fixed to left-elliptical polarization. As for the left-handed circularly polarized component of reflected light beam, the spin-dependent shift first increases with the incident angle. After reaching the peak value at the incident angle about θi = 56.3◦ , the shift decreases rapidly and then gets the negative maximum value. As for the right-handed circularly polarized component, the spin-dependent shift shows opposite trend. It first decreases with the incident angle and reaches the negative peak value. We should note that, under this condition, the spin-dependent shifts of left- and right-handed circularly polarized components in photonic SHE represent asymmetrical values which is different from the previous symmetrical photonic SHE. We measure the shifts of the asymmetric photonic SHE every 5◦ from 30◦ to 85◦ under the condition of elliptical polarization beam with its long axis along to the vertical direction [Figs. 5(b) and 5(d)]. The ellipticity is also chosen as β = 0.2◦ . Limited by the large holders of experimental equipments, the spin-dependent shift at small incident angles can not be measured. Here, the ellipticity and the polarization of incident beam are chosen as the same as the above condition (long axis along to horizontal direction). The spin-dependent shifts of left- and right-handed circularly polarized components change slowly with the incident angles. Both of them exhibit a peak value but with opposite signs. We note that, in this condition, the degree of asymmetry is smaller than the above case. In fact, in the case of elliptical polarization beam with its long axis along to the horizontal direction, there exists a relatively large spatial Imbert-Fedorov shift [38]. 4.
Conclusions
In conclusion, we have examined the asymmetric photonic SHE when an elliptical polarization beam reflected at a glass interface. It was found that the left- and right-handed circularly polarized components in photonic SHE represent distinct intensity distributions and the corresponding spin-dependent shifts show unequal values. We can attribute this asymmetric photonic SHE to the additional spatial Imbert-Fedorov shift which is corresponding to the overall transverse displacement of light beam. We have also found that the asymmetric distributions of the spin-dependent splitting can be modulated by adjusting the handedness of incident polarization. The weak measurement technique was used to measure this asymmetric photonic SHE, and the experimental results are in good agreement with the theoretical calculations. These findings provide an additional degree of freedom for generation and manipulation of spin-polarized photons and thereby open the possibility of developing new spin-controlled nanophotonic devices.
#254802 © 2016 OSA
Received 1 Dec 2015; revised 2 Feb 2016; accepted 2 Feb 2016; published 5 Feb 2016 8 Feb 2016 | Vol. 24, No. 3 | DOI:10.1364/OE.24.003025 | OPTICS EXPRESS 3035
Acknowledgments This research was partially supported by the National Natural Science Foundation of China (Grant No. 11447010) and the Natural Science Foundation of Hunan Province (Grant No. 2015JJ3026).
#254802 © 2016 OSA
Received 1 Dec 2015; revised 2 Feb 2016; accepted 2 Feb 2016; published 5 Feb 2016 8 Feb 2016 | Vol. 24, No. 3 | DOI:10.1364/OE.24.003025 | OPTICS EXPRESS 3036
of Physics and Information Science, Hunan Normal University, Changsha 410081, China 2 Laboratory for Optics and Optoelectronics, College of Physics and Electronic Engineering, Hengyang Normal University, Hengyang 421002, China 3 [email protected] ∗ [email protected]
Abstract: The photonic spin Hall effect (SHE) manifests itself as the spin-dependent splitting of light beam. Usually, it shows a symmetric spin-dependent splitting, i.e., the left- and right-handed circularly polarized components are equally separated in position and intensity for linear polarization incidence. In this paper, we theoretically propose an asymmetric spin-dependent splitting at an air-glass interface under the illumination of elliptical polarization beam and experimentally demonstrate it with the weak measurement method. The left- and right-handed circularly polarized components show expectedly unequal intensity distributions and unexpectedly different spin-dependent shifts. Remarkably, the asymmetric spin-dependent splitting can be modulated by adjusting the handedness of incident polarization. The inherent physics behind this interesting phenomenon is attributed to the additional spatial Imbert-Fedorov shift. These findings offer us potential methods for developing new spin-based nanophotonic applications. © 2016 Optical Society of America OCIS codes: (260.5430) Polarization; (240.0240) Optics at surfaces; (120.0120) Instrumentation, measurement, and metrology.
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Received 1 Dec 2015; revised 2 Feb 2016; accepted 2 Feb 2016; published 5 Feb 2016 8 Feb 2016 | Vol. 24, No. 3 | DOI:10.1364/OE.24.003025 | OPTICS EXPRESS 3025
10. J.-M. M´enard, A. E. Mattacchione, M. Betz, and H. M. van Driel, “Imaging the spin Hall effect of light inside semiconductors via absorption,” Opt. Lett. 34(15), 2312–2314 (2009). 11. P. Gosselin, A. B´erard, and H. Mohrbach, “Spin Hall effect of photons in a static gravitational field,” Phys. Rev. D 75(8), 084035 (2007). 12. C. A. Dartora, G. G. Cabrera, K. Z. Nobrega, V. F. Montagner, M. H. K. Matielli, F. K. R. de Campos, and H. T. S. Filho, “Lagrangian-Hamiltonian formulation of paraxial optics and applications: Study of gauge symmetries and the optical spin Hall effect,” Phys. Rev. A 83(1), 012110 (2011) 13. Y. Gorodetski, K. Y. Bliokh, B. Stein, C. Genet, N. Shitrit, V. Kleiner, E. Hasman, and T. W. Ebbesen, “Weak measurements of light chirality with a plasmonic slit,” Phys. Rev. Lett. 109(1), 013901 (2012). 14. N. Shitrit, I. Yulevich, E. Maguid, D. Ozeri, D. Veksler, V. Kleiner, and E. Hasman, “Spin-Optical metamaterial route to Spin-Controlled photonics,” Science 340(6133), 724–726 (2013). 15. X. Yin, Z. Ye, J. Rho, Y. Wang, and X. Zhang, “Photonic spin Hall effect at metasurfaces,” Science 339(6126), 1405–1407 (2013). 16. K. Y. Bliokh, A. Niv, V. Kleiner, and E. Hasman, “Geometrodynamics of spinning light,” Nature Photon. 2, 748–753 (2008). 17. K. Y. Bliokh, Y. Gorodetski, V. Kleiner, and E. Hasman, “Coriolis effect in optics: Unified geometric phase and Spin-Hall effect,” Phys. Rev. Lett. 101(3), 030404 (2008). 18. X. Zhou, Z. Xiao, H. Luo, and S. Wen, “Experimental observation of the spin Hall effect of light on a nanometal film via weak measurements,” Phys. Rev. A 85(4), 043809 (2012). 19. X. Zhou, X. Ling, H. Luo, and S. Wen, “Identifying graphene layers via spin Hall effect of light,” Appl. Phys. Lett. 101(25), 251602 (2012). 20. X. Zhou, J. Zhang, X. Ling, S. Chen, H. Luo, and S. Wen, “Photonic spin Hall effect in topological insulators,” Phys. Rev. A 88(5), 053840 (2013). 21. X. Qiu, X. Zhou, D. Hu, J. Du, F. Gao, Z. Zhang, and H. Luo, “Determination of magneto-optical constant of Fe films with weak measurements,” Appl. Phys. Lett. 105(13), 131111 (2014). 22. X. Ling, X. Zhou, X. Yi, W. Shu, Y. Liu, S. Chen, H. Luo, S. Wen, and D. Fan, “Giant photonic spin Hall effect in momentum space in a structured metamaterial with spatially varying birefringence,” Light Sci. Appl. 4, e290 (2015). 23. F. I. Fedorov, “Theory of total reflection,” Dokl. Akad. Nauk SSSR 105, 465–468 (1955). 24. C. Imbert, “Calculation and experimental proof of the transverse shift induced by total internal reflection of a circularly polarized beam,” Phys. Rev. D 5(4), 787–796 (1972). 25. Y. Aharonov, D. Z. Albert, and L. Vaidman, “How the result of a measurement of a component of the spin of a spin -1/2 particle can turn out to be 100,” Phys. Rev. Lett. 60(14), 1351–1354 (1988). 26. N. W. M. Ritchie, J. G. Story, and R. G. Hulet, “Realization of measurement of a weak value,” Phys. Rev. Lett. 66, 1107 (1991). 27. Y. Susa, Y. Shikano, and A. Hosoya, “Optimal probe wave function of weak-value amplification,” Phys. Rev. A 85(5), 052110 (2012) 28. J. Dressel, M. Malik, F. M. Miatto, A. N. Jordan, and R. W. Boyd, “Colloquium: Understanding quantum weak values: Basics and applications,” Rev. Mod. Phys. 86(1), 307–316 (2014). 29. X. Zhou, X. Li, H. Luo, and S. Wen, “Optimal preselection and postselection in weak measurements for observing photonic spin Hall effect,” Appl. Phys. Lett. 104(5), 051130 (2014). 30. A. Di Lorenzo, “Weak values and weak coupling maximizing the output of weak measurements,” Ann. Phys. 345, 178 (2014). 31. P. B. Dixon, D. J. Starling, A. N. Jordan, and J. C. Howell, “Ultrasensitive beam deflection measurement via interferometric weak value amplification,” Phys. Rev. Lett. 102(17), 173601 (2009). 32. N. Brunner and C. Simon, “Measuring small longitudinal phase shifts: Weak measurements or standard interferometry,” Phys. Rev. Lett. 105(1), 010405 (2010). 33. X.-Y. Xu, Y. Kedem, K. Sun, L. Vaidman, C.-F. Li, and G.-C. Guo, “Phase estimation with weak measurement using a white light source,” Phys. Rev. Lett. 111(3), 033604 (2013). 34. J. S. Lundeen, B. Sutherland, A. Patel, C. Stewart, and C. Bamber, “Direct measurement of the quantum wavefunction,” Nature 474, 188–191 (2011). 35. S. Kocsis, B. Braverman, S. Ravets, M. J. Stevens, R. P. Mirin, L. K. Shalm, and A. M. Steinberg, “Observing the average trajectories of single photons in a Two-Slit interferometer,” Science 332(6034), 1170–1173 (2011). 36. J. Z. Salvail, M. Agnew, A. S. Johnson, E. Bolduc, J. Leach and R. W. Boyd, “Full characterization of polarization states of light via direct measurement,” Nature Photon. 7, 316–321 (2013). 37. A. N. Jordan, J. Mart´ınez-Rinc´on, and J. C. Howell, “Technical advantages for weak-value amplification: When less is more,” Phys. Rev. X 4(1), 011031 (2014). 38. J. B. G¨otte and M. R. Dennis, “Limits to superweak amplification of beam shifts,” Opt. Lett. 38(13), 2295 (2013).
#254802 © 2016 OSA
Received 1 Dec 2015; revised 2 Feb 2016; accepted 2 Feb 2016; published 5 Feb 2016 8 Feb 2016 | Vol. 24, No. 3 | DOI:10.1364/OE.24.003025 | OPTICS EXPRESS 3026
1.
Introduction
Recently, similar to the spin Hall effect (SHE) in electronic system [1, 2], the photonic SHE manifesting itself as the spin-dependent splitting of left- and right-handed circularly polarized components of a spatially confined light beam reflecting or refracting at an optical interface has been theoretically and experimentally investigated in different physical systems such as optical physics [3–9], semiconductor physics [10], high-energy physics [11, 12], and plasmonics [13–15]. The photonic SHE is generally believed to be a result of an effective spinorbit coupling which corresponds to geometric Berry phase. There are two types of geometric phases: the spin-redirection Berry phase that is associated with the variations of direction of light beam propagation and the Pancharatnam-Berry phase related to beam’s polarization state evolution [16,17]. The photonic SHE itself can be developed into a useful metrological tool for characterizing the structure parameter variations of different physical systems. For example, we can use the photonic SHE to probe spatial distributions of electron spin states [10], measure the thickness of nanometal film [18], identify graphene layers [19], detect the axion coupling in topological insulators [20], and determine the magneto-optical constant of Fe films [21]. Remarkably, the photonic SHE offers new opportunities for manipulating photons, which provides an additional spin degree of freedom for developing spin-controlled nanophotonic applications [14, 15, 22]. Generally, the spin-dependent splitting in photonic SHE exhibit symmetry properties where the left- and right-handed circularly polarized components are equally separated in position and intensity for linear polarization incidence. In this work, we establish a general propagation model to describe an asymmetric spin-dependent splitting in photonic SHE when an elliptical polarization beam impinges upon an air-glass interface. We find that the left- and right-handed circularly polarized components show distinct intensity distributions and unequal spin accumulations. Importantly, the asymmetric distributions of the spin-dependent splitting can be modulated by adjusting the handedness of incident polarization. Therefore, we can develop a tunable photonic SHE with any desired intensity distributions and spin-dependent splitting for generation and manipulation of spin-polarized photons. These findings offer us potential methods for developing new spin-controlled nanophotonic devices. The rest of the paper is organized as follows. First, we theoretically analyze the asymmetric spin-dependent splitting in photonic SHE under a slightly elliptical polarization beam illumination. Next, we focus our attention on the weak measurements experiment for observing this phenomenon. Under this condition, the weak value includes both of the real and imaginary parts. Finally, a conclusion is given. 2.
Theoretical model
In this part, we first establish a general propagation model to describe the process of elliptical polarization beam reflected at an air-glass interface and theoretically investigate the corresponding asymmetric photonic SHE. Here, the electric field distributions and the spin-dependent shifts of the left- and right-handed circularly polarized components will be discussed. The asymmetric photonic SHE is schematically shown in Fig. 1. The first and second columns show the polarization and intensity distributions of incident light beam with linear and elliptical polarization, respectively. We can see that the electric field intensity distributions are symmetrical for linear polarization light [Figs. 1(b) and 1(e)]. However, for the elliptical polarization beam, the left- and right-handed circularly polarized components indicate asymmetric splitting [Figs. 1(h) and 1(k)]. The last column of Fig. 1 describes the intensity distributions and spindependent splitting of the left- (solid line) and right-handed (dashed line) circularly polarized components of reflected beam. When the incident light beam is linear polarization, the filed intensity (the height of Gaussian components) and spin-dependent shifts (δ± ) of these two spin components show equal values [the same heights and δ+ = δ− , as shown in Figs. 1(c) and 1(f)]. #254802 © 2016 OSA
Received 1 Dec 2015; revised 2 Feb 2016; accepted 2 Feb 2016; published 5 Feb 2016 8 Feb 2016 | Vol. 24, No. 3 | DOI:10.1364/OE.24.003025 | OPTICS EXPRESS 3027
Fig. 1. Schematic of the symmetric and asymmetric photonic SHE at the air-glass interface. The first and second column show the polarization and intensity distributions of incident Gaussian beam. The intensity distributions and spin-dependent shifts of left- and righthanded circularly polarized components of reflected beam are described in the third column. Here, we consider the elliptical polarization beam with its long and short axis along to the horizontal and vertical directions.
As for the illumination of elliptical polarization beam, the asymmetric photonic SHE appears, resulting in the unequal separation in position and intensity of left- and right-handed circularly polarized components [with different heights and δ+ = δ− , as shown in Figs. 1(i) and 1(l)]. Next, we will quantitatively study the reflection process of elliptical polarization beam and the corresponding asymmetric photonic SHE. For simplicity, we choose the long and short axis of the elliptical polarization beam directed along the horizontal (H) and vertical (V) directions, respectively. The elliptical polarization beam can be decomposed into two orthogonal polarization states H and √ V. In the spin basis, the H and √ V polarization states can be expressed as |H = (|+ + |−)/ 2 and |V = i(|− − |+)/ 2. We theoretically analyze the asymmetric photonic SHE by establishing the relationship between the incident and reflected fields. As for the incident light beam with elliptical polarization, the Jones vector can be written as (cos β , eiϕ sin β )T . Here β represents the azimuth angle (the angle between the crystal axis of wave plate and horizontal axis) and ϕ denotes the phase difference between the H and V polarization components. In the present study, we consider an elliptical polarization beam with its long and short axis along to the horizontal and vertical directions. Therefore, the Jones vector can be simplified to (cos β , +i sin β )T or (cos β , −i sin β )T representing the left- or right-
#254802 © 2016 OSA
Received 1 Dec 2015; revised 2 Feb 2016; accepted 2 Feb 2016; published 5 Feb 2016 8 Feb 2016 | Vol. 24, No. 3 | DOI:10.1364/OE.24.003025 | OPTICS EXPRESS 3028
elliptical polarization in the case of ϕ = ±π /2. It is noted that the azimuth angle β (also the ellipticity) mentioned above is a tiny value allowing for a slightly elliptical polarization. We consider the incident source is a monochromatic Gaussian beam whose spectrum is arbitrarily narrow and can be written as 2 + k2 ) w20 (kix w0 iy , (1) |Ei = (eix + iσ eiy ) √ exp − 4 2π where w0 is the beam waist. The polarization operator σ = ±1 stand for left- and right-handed circularly polarized components, respectively. According to the transversality, we can obtain the reflected field of left- and right-handed circularly polarized components in the case of leftelliptical polarization incidence [7] (the case of right-elliptical polarization can be obtained in the similar way), with the long axis being along to the horizontal and vertical directions: cos β H H H = rp√ exp k (+i δ + i δ tan β ) exp(+ η ) |+ |E ry L 2 r p cos β exp kry (−iδ H + iδ H tan β ) exp(−η ) |−, (2) + √ 2 V |E L
rs sin β V V exp kry (+iδ + iδ cot β ) exp(+ϑ ) |+ = √ 2 rs sin β V V exp −kry (−iδ + iδ cot β ) exp(−ϑ ) |−. + √ 2
(3)
Here, the r p and rs denote the Fresnel reflection coefficients for parallel and perpendicular polarizations, respectively. As shown in Eq. (2), we provide that kry δ H 1, kry δ H tan β 1, and η 1. In Eq. (3), we provide that kry δ V 1, kry δ V cot β 1, and ϑ 1. Here, δ H =(1 + rs /r p ) cot θi /k0 , δ V =(1+r p /rs ) cot θi /k0 , η =rs tan β /r p , and ϑ =r p cot β /rs . k0 is the wave number in free space. The terms exp(±ikry δ H ) and exp(±ikry δ V ) stand for the spin-orbit coupling and we find that the shifts induced by these terms are symmetrical, whereas the other electric field components exp(ikry δ H tan β ) and exp(ikry δ V cot β ) are not the spin-orbit coupling terms. However, they can affect the spin-dependent spatial splitting in photonic SHE, which will be discussed in the follwing. From Eqs. (2) and (3), we can obtain the intensity distributions of left- and right-handed circularly polarized components as shown in Figs. 2(a)-2(d). It should be noted that the electric field components exp(±η ) and exp(±ϑ ) cause the asymmetric intensity distributions. For simplicity, we only consider the long axis of incident elliptical polarization beam along to the horizontal direction. Here, for example, the ellipticity are chosen as 0.2◦ and 5◦ . And the incident angles are selected as θi = 30◦ and θi = 45◦ , respectively. From the pictures, we find that the left- and right-handed circularly polarized components show unequal intensity distributions which can be modulated by changing the ellipticity and incident angles. The photonic SHE is described for the left- and right-circularly polarized components undergoing spin-dependent shifts, so the reflected field centroid should be determined. In the present work, we only consider the spin-dependent splitting in the transverse direction. According to Eqs. (2) and (3), we can calculate the field centroid distribution. At any given plane z = const., the transverse displacement of field centroid compared to the geometrical-optics prediction is given by ∂k |E ry |E E|i z E|k ry + . (4) yH,V ± = E E k0 E| E|
#254802 © 2016 OSA
Received 1 Dec 2015; revised 2 Feb 2016; accepted 2 Feb 2016; published 5 Feb 2016 8 Feb 2016 | Vol. 24, No. 3 | DOI:10.1364/OE.24.003025 | OPTICS EXPRESS 3029
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Fig. 2. The intensity distributions (a-d) and initial spin-dependent shifts (e-h) of left- and right-handed circularly polarized components in the asymmetric photonic SHE. The ellipticity are chosen as β = 0.2◦ and β = 5◦ . The intensity is plotted in normalized units. (a) and (b) show the intensity distributions when the incident angle is selected as θi = 30◦ . The long axis of incident elliptical polarization beam is along to the horizontal direction and the handedness of incident polarization is left. (c) and (d) stand for the intensity distributions under the condition of incident angle is selected as θi = 45◦ . Here, the handedness of incident polarization and the direction of long axis are chosen as the same as above. (e) and (f) denote the displacements in the case of the handedness of incident polarizations are left and right, and the long axis of incident elliptical polarization beam is along to the horizontal direction. (g) and (h) describe the shifts under the condition of the long axis of incident elliptical polarization beam along to the vertical direction.
#254802 © 2016 OSA
Received 1 Dec 2015; revised 2 Feb 2016; accepted 2 Feb 2016; published 5 Feb 2016 8 Feb 2016 | Vol. 24, No. 3 | DOI:10.1364/OE.24.003025 | OPTICS EXPRESS 3030
In the above equation, ∂kry = ∂ ∂kry ery and kry = kry ery . The first and second terms in Eq. (4) stand for the spatial and angular shifts, which are independent of and dependent on z, respectively. In fact, after calculating, we find that the photonic SHE shows pure spatial displacements which are independent of z. Evaluating Eq. (4), we can obtain the asymmetric spin-dependent shifts of elliptical polarization beam. Figures 2(e)-2(h) show the initial transverse shifts of left- and right-handed circularly polarized components changing with the ellipticity and handedness of incident polarization. Under this condition, we find that the left- and right-handed circularly polarized components show unequal spin-dependent shifts, which is different from the previous works [4,5,22]. Here, the ellipticity are also chosen as 0.2◦ and 5◦ . First, we consider the long axis of elliptical polarization beam along to the horizontal direction [shown in Figs. 2(e) and 2(f)]. In this case, we find that the left- and right-handed circularly polarized components represent unequal values of spin-dependent shifts which are extremely sensitive to the ellipticity variations. Remarkably, the asymmetric spin-dependent splitting can be modulated by adjusting the handedness of incident polarization. Figures 2(g) and 2(h) denote the condition of the long axis of elliptical polarization beam along to the vertical direction. Here, the transverse displacements of left- and right-handed circularly polarized components change slowly with the variations of ellipticity. However, this asymmetric spin-dependent splitting can also be adjusted when the handedness of incident polarization alters. So, we can develop a tunable photonic SHE with any desired electric field intensity and spin-dependent splitting. We can attribute this asymmetric photonic SHE to the additional spatial Imbert-Fedorov shift [23, 24]. From the Eqs. (2) and (3), as for the left elliptically polarized light impinges at the media interface, we find that there exists a unified spatial displacement induced by the terms exp(ikry δ H tan β ) and exp(ikry δ V cot β ) in addition to the normal symmetric spin-dependent shifts. After calculating Eq. (4), we can describe the final field centroid as follows: H,V H,V yH,V ± = δ± + δIF ,
(5)
where δ±H,V stand for the symmetric spin-dependent shifts of left- and right-handed circularly H,V denotes the unified spatial Imbert-Fedorov shift. For simplicpolarized components, and δIF ity, the specific expressions of these two field centroids are not given. We can see that the normal symmetric spin-dependent displacements of left- and right-handed circularly polarized components will undergo an overall shift (the same sizes and signs), then the asymmetric spindependent shift appears. When the state of the incident light beam changing from left-elliptical polarization to the right one, the left- and right-handed circularly polarized components will also undergo an overall shift but with the opposite directions (as shown in Fig. 2). 3.
Experimental observation and discussion
The asymmetric photonic SHE induced spin-dependent splitting are too small to be detected directly. However, the signal enhancement technique known as the weak measurements [25–30] can resolve this problem. The weak measurements based on preselection and postselection states has attracted a lot of attention and holds great promise for precision metrology such as beam deflection measurement [31], measuring small optical phase shift [32,33], direct detection of the quantum wavefunction [34], observing the average trajectories of single photons [35], and full characterization of polarization states of light [36]. It should be noted that the weak measurements method involved in the present work shows some adjustments based on the previous weak measurements, which will be discussed in the following. There exists an amplified factor so-called weak value playing a great role in weak measurements. The weak value establishes the relationship between the observable and the shifts in
#254802 © 2016 OSA
Received 1 Dec 2015; revised 2 Feb 2016; accepted 2 Feb 2016; published 5 Feb 2016 8 Feb 2016 | Vol. 24, No. 3 | DOI:10.1364/OE.24.003025 | OPTICS EXPRESS 3031
Fig. 3. (a) Schematic of the preselection and postselection states on Bloch sphere. (b) and (c) show the results of | f |iRe(Aw )| and | f |iIm(Aw )| as a function of the the ellipticity β and amplification angle Δ. Both of the functions are plotted in normalized units.
measurement pointer’s mean position and mean momentum Aw =
ˆ f |A|i . f |i
(6)
|i and | f stand for the preselection and postselection states. As shown in Fig. 3(a), the Bloch sphere can be introduced to describe the weak measurements process [37]. We consider an elliptical polarization beam incident and the long and short axis are along to the horizontal and vertical directions. Therefore the preselection state of the system can be written as:
Θ+β Θ+β |+ + eiΦ sin |−, (7) |i = cos 2 2 where Θ and Φ denote the qubit state on the surface of the Bloch sphere. And the postselection state of the system is chosen as:
Θ Θ −i(Φ+Δ) |+ − e |−. (8) cos | f = sin 2 2 Here, the angle Δ show the amplification angle which will be discussed in the following. For practical purposes, we set Θ = π /2. Using Eq. (6) together with Eq. (7) and (8), we can calculate the real and imaginary parts of the weak value: Re(Aw ) =
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sin β , cos β cos Δ − 1
(9)
Received 1 Dec 2015; revised 2 Feb 2016; accepted 2 Feb 2016; published 5 Feb 2016 8 Feb 2016 | Vol. 24, No. 3 | DOI:10.1364/OE.24.003025 | OPTICS EXPRESS 3032
Fig. 4. Experimental setup for measuring the asymmetric photonic SHE and the corresponding weak measurements process. (a) shows the experimental setup. The sample is a BK7 glass. The light source is a 21mW linearly polarized He-Ne laser at 632.8nm (Thorlabs HNL210L-EC). L1 and L2, lenses with effective focal length 50mm and 250mm, respectively. HWP, half-wave plate (for adjusting the intensity). QWP, quarter-wave plate for generating elliptical polarization beam. P1 and P2, Glan Laser polarizers. CCD, chargecoupled device (Coherent LaserCam HR). The inset: The incident beam is preselected in the left- or right-elliptical polarization state by P1 and QWP. The optical axis of P1 make the angles β or −β with horizontal axis. (b) denotes the weak measurements process with linear polarization (the first row) and elliptical polarization (the second row) beam.
Im(Aw ) =
1 . cot Δ − csc Δ sec β
(10)
It can be seen that the real and imaginary parts of weak value are all nonzero when we introduce the tiny angles β and Δ. This is different from the previous work [5] in which the angle β = 0 and the weak value is a pure imaginary number. In order to obtain the large output, we need to maximize the weak value including both of the real and imaginary parts. Figures 3(b) and 3(c) show the products of | f |iRe(Aw )| and | f |iIm(Aw )|. We can see that both of the ellipticity β and the amplification angle Δ should be chosen as the tiny values for obtaining a large weak value. For simplicity, in the following, we replace the symbols Re(Aw ) and Im(Aw ) with ARe and AIm . There also exists another amplification mechanism called propagation amplification [29] that produces the amplified factor F accompanying by the imaginary part of weak value, which leads to the possibility of even larger enhancements following the beam free evolution. Finally, we
#254802 © 2016 OSA
Received 1 Dec 2015; revised 2 Feb 2016; accepted 2 Feb 2016; published 5 Feb 2016 8 Feb 2016 | Vol. 24, No. 3 | DOI:10.1364/OE.24.003025 | OPTICS EXPRESS 3033
can obtain the amplified shifts [from Eq. (5)] by the weak value and propagation amplifications: yw = (ARe + FAIm )|δ±H,V |,
(11)
so it can be detected directly by our experimental setup. It should be noted that, in this case, we can not amplify the spatial Imbert-Fedorov shift due to the fact that the observable is only the σˆ 3 which is corresponding to the spin projection along the central propagation direction. The weak value amplification process can convert the position displacements caused by the photonic SHE into a momentum shift, and it also converts the momentum shifts into a position shift. However, in our work, the contribution of the real part is much smaller than that in imaginary part due to two reasons: the tiny ellipticity and the beam free evolution. Therefore, the imaginary part plays a key role in the final amplification and we can rewrite the Eq. (11) as follows: yw ≈ FAIm |δ±H,V |.
(12)
This is the important result for the following experimental measurement. We can measure the amplified shifts by reading out the difference between two states |V ± Δ or |H ± Δ (the AIm shows opposite signs). Our experimental setup shown in Fig. 4(a) is similar to that in [5, 8, 18]. A He-Ne laser is used to generate linear polarization Gauss beam which firstly impinges onto the HWP. This HWP can adjust the polarization of incident light beam, which is used to control the light intensity preventing the charge-coupled device (CCD) from saturation. Then, the light beam passes through a short focal length lens (L1) and is preselected as a slightly elliptical polarization state by P1 and QWP, which is slightly different from the previous work [5]. Here, the P1 is chosen as |H or |V . When the beam reaches the sample (BK7 glass) interface, the asymmetric photonic SHE takes place allowing for the reflected beam separated into two unequal spin components. As the reflected beam splits by a fraction of the wavelength, the left- and right-handed 400
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Fig. 5. The amplified (a, b) and initial (c, d) spin-dependent shifts of asymmetric photonic SHE in the case of elliptical polarization beam with its long and short axis along to the horizontal (the left column) and vertical directions (the right column). Here, the ellipticity is chosen as β = 0.2◦ and the handedness of incident polarization is left. The lines indicate the theoretical value. The diamonds, squares, and triangles show the experimental results.
#254802 © 2016 OSA
Received 1 Dec 2015; revised 2 Feb 2016; accepted 2 Feb 2016; published 5 Feb 2016 8 Feb 2016 | Vol. 24, No. 3 | DOI:10.1364/OE.24.003025 | OPTICS EXPRESS 3034
circularly polarized components interfere destructively at the second polarizer (P2) [as shown in Fig. 4(b)]. From Eq. (12), we can measure the amplified shifts by discriminating the difference between the displacements for rotating the P2 as two states | f = |V ± Δ or |H ± Δ. Here, the Δ is a small angle that we called amplification angle. This process can be seen as the postselection. Using this method, we can deduce the initial symmetrical spin-dependent shifts δ±H,V . By considering the additional spatial Imbert-Fedorov shift in Eq. (5), we can finally obtain the asymmetric spin-dependent displacements yH,V ± . In our weak measurement experiment, we choose the amplification angle Δ = 0.8◦ . After passing through the polarizer P2, we use L2 to collimate the beam and make the beam shifts insensitive to the distance between L2 and the CCD, which will improve the measurement precision. Finally, a CCD is used to measure the amplified shift after L2. In the case of elliptical polarization beam with its long axis along to the horizontal direction, we measure the amplified displacements of asymmetric photonic SHE on the BK7 glass every 0.5◦ from 52◦ to 60◦ [as shown in Fig. 5(a)]. After obtaining the amplified shifts, we can deduce the initial displacements [Fig. 5(c)]. In our experiment, we choose the ellipticity as β = 0.2◦ and the incident beam is fixed to left-elliptical polarization. As for the left-handed circularly polarized component of reflected light beam, the spin-dependent shift first increases with the incident angle. After reaching the peak value at the incident angle about θi = 56.3◦ , the shift decreases rapidly and then gets the negative maximum value. As for the right-handed circularly polarized component, the spin-dependent shift shows opposite trend. It first decreases with the incident angle and reaches the negative peak value. We should note that, under this condition, the spin-dependent shifts of left- and right-handed circularly polarized components in photonic SHE represent asymmetrical values which is different from the previous symmetrical photonic SHE. We measure the shifts of the asymmetric photonic SHE every 5◦ from 30◦ to 85◦ under the condition of elliptical polarization beam with its long axis along to the vertical direction [Figs. 5(b) and 5(d)]. The ellipticity is also chosen as β = 0.2◦ . Limited by the large holders of experimental equipments, the spin-dependent shift at small incident angles can not be measured. Here, the ellipticity and the polarization of incident beam are chosen as the same as the above condition (long axis along to horizontal direction). The spin-dependent shifts of left- and right-handed circularly polarized components change slowly with the incident angles. Both of them exhibit a peak value but with opposite signs. We note that, in this condition, the degree of asymmetry is smaller than the above case. In fact, in the case of elliptical polarization beam with its long axis along to the horizontal direction, there exists a relatively large spatial Imbert-Fedorov shift [38]. 4.
Conclusions
In conclusion, we have examined the asymmetric photonic SHE when an elliptical polarization beam reflected at a glass interface. It was found that the left- and right-handed circularly polarized components in photonic SHE represent distinct intensity distributions and the corresponding spin-dependent shifts show unequal values. We can attribute this asymmetric photonic SHE to the additional spatial Imbert-Fedorov shift which is corresponding to the overall transverse displacement of light beam. We have also found that the asymmetric distributions of the spin-dependent splitting can be modulated by adjusting the handedness of incident polarization. The weak measurement technique was used to measure this asymmetric photonic SHE, and the experimental results are in good agreement with the theoretical calculations. These findings provide an additional degree of freedom for generation and manipulation of spin-polarized photons and thereby open the possibility of developing new spin-controlled nanophotonic devices.
#254802 © 2016 OSA
Received 1 Dec 2015; revised 2 Feb 2016; accepted 2 Feb 2016; published 5 Feb 2016 8 Feb 2016 | Vol. 24, No. 3 | DOI:10.1364/OE.24.003025 | OPTICS EXPRESS 3035
Acknowledgments This research was partially supported by the National Natural Science Foundation of China (Grant No. 11447010) and the Natural Science Foundation of Hunan Province (Grant No. 2015JJ3026).
#254802 © 2016 OSA
Received 1 Dec 2015; revised 2 Feb 2016; accepted 2 Feb 2016; published 5 Feb 2016 8 Feb 2016 | Vol. 24, No. 3 | DOI:10.1364/OE.24.003025 | OPTICS EXPRESS 3036
