June 15, 2014 / Vol. 39, No. 12 / OPTICS LETTERS

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Measuring spin Hall effect of light by cross-polarization intensity ratio Bo Wang,1,2 Yan Li,1,2,* Meng-Meng Pan,1 Jin-Li Ren,1,2 Yun-Feng Xiao,1,2 Hong Yang,1 and Qihuang Gong1,2 1

State Key Laboratory for Mesoscopic Physics, Department of Physics, Peking University, Beijing 100871, China 2 Collaborative Innovation Center of Quantum Matter, Beijing 100871, China *Corresponding author: [email protected] Received March 21, 2014; revised April 22, 2014; accepted April 28, 2014; posted April 30, 2014 (Doc. ID 208684); published June 4, 2014

We propose and realize a simple technique to measure the tiny spin Hall effect of light from the ratio of the minimum and the maximum intensities along two cross-polarization directions, without the requirement of a positionsensitive detector in the conventional weak measurement. Furthermore, the weak intensity ratio is dramatically amplified by purposely choosing the intensity along the direction close to that of the minimum instead of the maximum along the perpendicular polarization direction, which is verified by the experimental results. In principle, this method also can be modified for measurement of the high extinction ratio of a polarizer. © 2014 Optical Society of America OCIS codes: (240.3695) Linear and nonlinear light scattering from surfaces; (260.5430) Polarization. http://dx.doi.org/10.1364/OL.39.003425

When plane waves reflect from and transmit through a plane interface, they obey Snell’s law and Fresnel equations [1]. However, anomaly phenomena occur when the incident waves are spatially confined polarized beams, among which the Goos–Hänchen (GH) [2–4] and Imbert–Fedorov (IF) [5] effects are most important. The GH effect is a beam displacement in the plane of incidence. While the IF effect, or in the spin basis, known as spin Hall effect of light (SHEL) [6], is a spin-dependent displacement perpendicularly to the plane of incidence. The SHEL originates from geometric phase or spin-orbit interaction and can be explained by the total angular momentum conservation [7]. The geometric SHEL, one kind of beam displacement that occurs without the existence of optical interface and merely depends on the observation direction, also arouses the interest of researchers [8,9]. In the recent experiment [10], the authors have directly measured this intriguing phenomenon and analyzed the distinction of result caused by using real-world polarizers. The SHEL is usually a subwavelength phenomenon, which is much smaller than the beam waist; thus it hardly can be detected directly. The extensively used approach is the application of weak measurement realized by Hosten and Kwiat [11], which has attained sensitivity of displacements of 1 Å and boosted the SHEL investigation at a variety of interfaces [12–18]. The weak measurement is originally a quantum concept [19], but it also can be understood using classical interaction of wavepackage and polarizations in this case [20]. The analogy between classical beam shifts and the quantum perspective of weak measurement have been clearly discussed in [21,22]. To greatly amplify the displacement, the spatial shift is converted into angular shift, resulting in enhanced weak values, accompanied with a propagation-enhancement factor [12–18]. This method has achieved great success in ultrasmall displacement measurements, but it relies on the long propagation distance and leads to large beam spot sizes on the beam position displacement detectors, such as quadrant detectors (QD) [4], position-sensitive detectors (PSD) [3,11,12,15,17,18] or, in 0146-9592/14/123425-04$15.00/0

case of large SHEL, charge-coupled devices (CCD) [13,14,16] or a camera [23]. Recently, a different method based on interference also has been reported [24]. Compared with weak measurement, it measures the SHEL across the entire exit pupil with a relatively complex experimental setup. In this Letter, we reveal that the SHEL also can be regarded as an interface eigenstate polarization conversion process, and its spatial spin displacements can be simply calculated by the intensity ratio of the minimum and the maximum along two cross-polarization directions, without requirement of the detailed intensity distribution. The intensity ratio can be amplified 3 to 4 orders by purposely choosing a weak intensity instead of the maximum, which is verified by the experimental results. In the expression derivation and experimental measurement, we have taken into consideration the extinction ratio of the polarizers. In principle, this technique can be modified for measurement of the high extinction ratio of a polarizer. Now we consider a linearly polarized beam reflected from a planar interface, as shown in Fig. 1. The z axis of the laboratory coordinate xyz is perpendicular to the interface, while xi yi zi  and xr yr zr  are coordinates attached to the center of the incident and reflected beams, respectively. The conversion relations between these coordinates have been expressed in [18]. In momentum space, each plane wave is decomposed along the center coordinates of the incident and reflected coordinates, respectively. Thus we have k⃗ i  kxi xˆ i  ky yˆ  kˆzi and k⃗ r  kxr xˆ r  ky yˆ  kˆzr , with k  2π∕λ  1∕λ̵ and kxi;r ;y ≪ k in paraxial approximation condition [11]. For P (electric field parallel to the plane-of-incidence) or S (electric field perpendicular to the plane-ofincidence) polarized incident beam, its polarization direction is represented as Pˆ i or Sˆ i , respectively. The SHEL after reflection is derived as the out-of-plane spin displacement of the beam and expressed as δp  −λ̵ r s  r p  cot θ∕r p or δs  −λ̵ r s  r p  cot θ∕r s [12], where θ is the angle of incidence, and r p and r s are reflection coefficients that are determined by Fresnel © 2014 Optical Society of America

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equations [1]. These values are for the left-handed spin component ji of the beam, while the displacements of the right-handed spin component j−i are −δp and −δs . The essential step for acquirement of the spin displacements is the polarization basis transformation from the interface eigenstates (Pˆ i and Sˆ i ) to the spin states (ji and j−i) [11]. That is to say, we may calculate the SHEL displacements in the interface eigenstate basis as confirmed as follows. In fact, SHEL can be regarded as the conversion between p and s polarizations, which generate cross-polarization terms, and the relationship between SHEL and cross polarization has been briefly discussed [25]. The special case of cross polarization at Brewster’s angle also has been previously studied [26]. In other words, when a pure p or s beam is incident, the reflected beam is almost p or s polarized but with a slight converted p or s polarization and can be expressed as   kx ∂ ln r p ˆ Pˆ i → r p Pˆ r − r P r  ky δp Sˆ r ; k ∂θ   kx ∂ ln r s ˆ Sˆ i → r s Sˆ r − r S r − ky δs Pˆ r : k ∂θ

(1)

Here, the subscript r (r will be omitted later on) represents the polarization component of the reflected beam. It can be seen from Eq. (1) that the amount of the converted polarization is directly related to the SHEL displacement (δp or δs ). More important, the SHEL displacement can be measured by intensity ratio of the converted polarization to its own polarization as demonstrated as follows. We first consider an s-polarized Gaussian beam E⃗ i xi ; y  Sˆ i exp−x2i  y2 ∕w20  or expressed in the momentum space as E⃗ i kxi ;ky   Sˆ i exp−w20 k2xi  k2y ∕4. The reflected electric field distribution is   w2 ⃗E r kx ; ky   r s Sˆ − kxr ∂ ln r s Sˆ − ky δs Pˆ × e− 40 k2xr k2y  ; r k ∂θ where its propagation phase is neglected since only the intensity is considered. The electric field along the direction with an azimuth angle α from Pˆ is ˆ E⃗ r kxr ; ky  · cos αPˆ  sin αS    kxr ∂ ln r s s sin α − ky δ cos α  rs 1 − k ∂θ ×e

w2

− 40 k2xr k2y 

:

I s P 1    I s S 1  ∂ ln rs 2 ∂θ

The corresponding intensity is ZZ ˆ 2 dkx dky : jE⃗ r kxr ; ky  · cos αPˆ  sin αSj I s α  r kxr ;ky

(3) From Eqs. (2) and (3), when α  0 (P state) the intensity I s P is minimum, while α  π∕2 (S state), the intensity I s S is maximum since the reflected beam is almost s polarized. The ratio of the two intensities is

(4)

w0

Usually, λ̵ ≪w0 , and Eq. (4) can be simplified into: I s P jδs j2  2 : I s S w0

(5)

Equation (5) reveals that the SHEL displacement δs can be measured by the intensity ratio of two orthogonal polarization states, which is exactly the result of the cross-polarization conversion. For p-polarized incidence, the corresponding relationship is I p S∕I p P  jδp j2 ∕w20 . The I s P or I p S is very weak for a common SHEL with tiny displacement δs or δp as shown in Eq. (5), so its accurate measurement is crucial for determination of the SHEL. Therefore the correction must be taken into account for a real polarizer that cannot generate or transmit an ideally linearly polarized beam. When we use a perfect polarizer P1 to generate an s-polarized incident beam, we actually obtain a state Sˆ i  Δ1 expiϕ1 Pˆ i . Δ1 is not zero because of the finite extinction ratio ρ1  Δ21 , and ϕ1 is the phase difference generated by the polarizer P1. For a typical Glan-polarizer, ρ ≈ 10−6 ∼ 10−5 . With such a polarized incident beam, the momentum space polarization distribution of the reflected one is 

 kxr ∂r s ˆ ky S  cot θr p  r s Pˆ k ∂θ k   kxr ∂r p ˆ ky iϕ1 ˆ ˆ P − cot θr p  r s S : rpP −  Δ1 e k ∂θ k

r s Sˆ −

To measure the intensities of certain polarizations, another polarizer, P2, with the extinction ratio of ρ2  Δ22 is also needed. When the transmission axis of P2 inclines to Pˆ with an angle α, the transmitted wave ampliˆ  Δ2 expiϕ2  tude is projected to cos αPˆ  sin αS ˆ ˆ cos αS − sin αP. For a Gaussian wave-package, we can obtain the total intensity after P2 in case of α  0 and α  π∕2, respectively. Since Δ21;2 is usually very small, we ignore its higher orders to obtain the intensity ratio: ̵

(2)

jδs j2 : w20 2

λ̵ 2

2

r 2p  wλ 2 ∂r p ∕∂θ2 jδs j2 I s P  ρ2  ρ1 2 λ̵ 20  2 : I s S w0 r s  2 ∂r s ∕∂θ2

(6)

w0

In principle, we can derive the SHEL displacement by measuring the intensity ratio I s P∕I s S, according to Eq. (6). However, the ratio is too small to ensure high resolution. In experiments, we realize an amplified value by purposely orienting P2 near perpendicular to P1 with a small angle α. Specifically, in the case of s-polarized incidence, we measure I s α; α ≪ 1 rather than I s S; α  π∕2. From Eq. (3), we have I s α  I s Pcos2 α I s Ssin2 α, which leads to an amplified intensity ratio:

June 15, 2014 / Vol. 39, No. 12 / OPTICS LETTERS

Fig. 1.

Schematic of the SHEL at an air–glass interface.

I s P I s α : cot2 α  I s S I s α∕cos2 α − I s P

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Fig. 3. Experimental setup: P1 and P2, Glan-polarizers; L1 and L2, lenses with 25 and 125 mm focal lengths, respectively; prism, a right-angle prism with an angle of 45 deg. He–Ne laser at 632.8 nm, 18 mW.

Using I s α and I s P, we obtain a dramatically enhanced cot2 αjδs j2 ∕w20 for a small α. In our experiment, the typical value of α is 1 deg, so the amplification factor cot2 α ∼ 103 . Another advantage of this approach is that I s α is much closer to I s P than I s S to considerably alleviate the requirement of high dynamic range of the detector. The sign of δs is easily determined by direct observation. The electric distribution of the beam after P2 is calculated from Eq. (3) and expressed as Eα; kxr ; ky  ∝ exp−δs ky ∕α exp−w20 k2xr  k2y ∕4, with α ≪ 1, and the ˆ polarization state transmitting through P2 is cos αP ˆ When α  0, i.e., P2 perpendicular to P1, the sin αS. intensity distribution at the far field after P2 has a double-peak profile as shown in Fig. 2(a), which is exactly the result of SHEL. When 0 < α ≪ 1, the intensity distributions at the far field after P2 for the negative and positive are shown in Figs. 2(b) and 2(c), respectively.

The experiment setup is similar to that in [12,15,17,18] as shown in Fig. 3. An He–Ne laser generates a 632.8 nm laser Gaussian beam, with an output power of 18 mW. L1 and L2 are lens; P1 and P2 are Glan-polarizers. The SHEL occurs at the air–glass interface. The detector PSD can output the total intensity as well as the position of the barycenter of beams as done in the conventional weak measurement [12,15,17,18]. In our experiment, only the intensity is needed. For s polarization, we measure the minimum intensity I s P and a weak intensity I s α with α  1° and cot2 α  3.3 × 103 . Similar steps are taken in the p-polarization measurement. We have measured nine points every 5 deg for the angle of incidence from 30 to 70 deg. The extinction ratio of P2, the beam waist w0 , and jδs j are derived from Eq. (8). The conventional weak measurements of SHEL [11–18] amplify the shift of centroid of beams. Two linear polarizers are used to prepare the pre and postselection of states and convert the spatial shifts into angular deviations. The resulting imaginary weak values also depend on the propagation distance of the beam, resulting in large beam spot sizes on the detectors such as PSD. Our method, however, utilizing the cross-polarization effect of SHEL, measures the beam intensities only, which is independent of the propagation distance. Although at the loss of the propagation enhancement, it is easy to carry out with sufficient amplification to derive the reflected SHEL at an air–glass interface. The experimental results (dots) are shown in Fig. 4 for both S and P states, which perfectly agree with the

Fig. 2. Schematics of the intensity distribution after P2. (a) P1 perpendicular to P2, i.e., α  0. (b) P2 is rotated by a small angle α  1° ; in case of δs < 0, the lower part of the beam become brighter. (c) α  1° ; in case of δs > 0, the upper part of the beam become brighter.

Fig. 4. SHEL displacements of left-handed component, as a function of the incident angle θ, in the case of (a) s polarization and (b) p polarization. The dots are experimental results, and solid curves are theoretical prediction.

(7)

Combining Eqs. (6) and (7), the SHEL is successfully related to I s α and I s P. Usually, ρ1 ≈ ρ2 . In the case of air–glass interface, where the r 2p is smaller than r 2s by one order for most incident angles, we have   I s α jδs j2 ≈ ρ  cot2 α: 2 w20 I s α∕cos2 α − I s P

(8)

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prediction (solid curves) of the SHEL at the air–glass interface, which have been confirmed by the conventional weak measurement [12]. The derived extinction ratio ρ2 ∼ 10−5 , or 50 dB, and the focused beam waist w0 ≈ 16 μm. Both of them are in accordance with actual parameters. These results verify that the SHEL can be accurately determined by measuring the intensity ratio only, without the requirement of the detailed intensity distribution. Moreover, the intensity ratio can be amplified by 3 orders of magnitude, and it provides a potential method for measuring intensity-ratio-related values such as the extinction ratio of a polarizer. To summarize, we demonstrate that the SHEL measurement can be realized by using only the ratio of beam intensities along two polarization directions, utilizing the cross-polarization effect. Because only intensity rather than its distribution in the conventional weak measurement is required, the detector can be simpler. Choosing the minimum and a weak one instead of the maximum, we successfully enhance the intensity ratio 1000 times to derive accurate SHEL displacements. In the meantime, this approach considerably alleviates the requirement of high dynamic range of the detector. It is apparent that this method can be modified for measurement of high extinction ratio of polarizers. This work was supported by the National Basic Research Program of China under grant no. 2013CB921904 and the National Natural Science Foundation of China under grant nos. 11074013, 11023003, 11121091, and 11134001. References 1. M. Born and E. Wolf, Principles of Optics, 7th ed. (Cambridge University, 2003). 2. F. Goos and H. Hanchen, Ann. Phys. 436, 333 (1947). 3. H. Gilles, S. Girard, and J. Hamel, Opt. Lett. 27, 1421 (2002). 4. M. Merano, A. Aiello, G. W. ’t Hooft, M. P. van Exter, E. R. Eliel, and J. P. Woerdman, Opt. Express 15, 15928 (2007).

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Measuring spin Hall effect of light by cross-polarization intensity ratio.

We propose and realize a simple technique to measure the tiny spin Hall effect of light from the ratio of the minimum and the maximum intensities alon...
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