The behavior of a bounded beam of light in reflection differs from that exhibited by plane waves, the latter being ruled by geometrical optics. For finite-diameter light beams, diffractive corrections occur that shift the beam in directions parallel and perpendicular to the plane of incidence. The parallel shift is known as the Goos–Hänchen (GH) [1] shift and the perpendicular one is the Imbert–Fedorov (IF) shift [2,3]. These deviations from geometrical optics predictions can be either spatial or angular [4]. These effects have been extensively studied not only for total internal reflection (TIR), which is the context wherein the GH and IF effects were originally addressed, but also in partial dielectric reflection and transmission.

Among the various experimental techniques for the observation of optical beam shifts, weak measurements have proven to be very successful. This approach was used for the first time by Hosten and Kwiat [5] for the observation of the spin-Hall effect of light (SHEL) [6,7]. This phenomenon is a spin-dependent displacement, perpendicular to the plane of incidence, for photons transmitted through an interface. It can be seen as a photonic version of the spin-Hall effect in electronic systems. Weak measurements allowed for others results, like the observation of SHEL in optical reflections [8], as well as in plasmonic systems [9]. A spin-dependent beam shift in the plane of incidence was reported as well [10]. Recently, we observed the GH shift via a weak measurement approach [11]. Although weak measurements were introduced in quantum mechanics, theoretical analysis proved that, in the context of optical beam shifts, they can be described by classical optics [12–16].

Here, we report the weak measurement of the spatial IF effect. The effect is observed in TIR, i.e., the simplest case in which the IF shift can be investigated. The IF effect does not depend only on the spin of the photon, but it also requires a more complete description in terms of the polarization properties of the incident light.

Our experimental setup shows a strict analogy with the original proposal of Aharonov, Albert, and Vaidman (AAV) [17,18], which describes the weak measurement of the spin of the electron. In their paper, an electron Gaussian beam is prepared in a preselected spin state. A tiny magnetic field gradient, acting as the weak measurement apparatus, splits the electron beam in two parts, and a Stern–Gerlach apparatus postselects the final spin state. AAV noticed how weak measurements can be used as an amplification scheme for the observation of the tiny magnetic field gradient. We measure the weak value of the polarization of a light beam. We preselect the polarization of a Gaussian beam that undergoes TIR. The IF shift acts as the weak measuring effect, and an analyzer postselects the final polarization state. As a consequence of this, the displacement of the beam centroid on a position-sensitive detector following the analyzer is a faithful amplification of the IF effect.

Even if the IF effect is completely classical, we measure it here with an experimental scheme derived from quantum mechanics. We can give a complete classical description of both the IF effect and the measurement approach, or we can alternatively describe the IF effect using a quantum mechanical description. This last approach has the merit to furnish a very good physical insight and to simplify the mathematical treatment necessary for the analysis of our experimental data. Owing to the one-to-one correspondence between the paraxial wave equation and the Schrödinger equation, the electric field of a paraxial beam can be described in a way formally equivalent to the wave function of a nonrelativistic quantum-mechanical particle with spin $1/2$. Let ${\mathit{e}}_1=(1,0)$ and ${\mathit{e}}_2=(0,1)$ be two unit vectors that span the transverse plane perpendicular to the beam propagation axis $z$ (Fig. 1). The polarization of a light beam can be described as a two-component spin, i.e., $〈A|\equiv a_1{\mathit{e}}_1+a_2{\mathit{e}}_2$, where ${|a_1|}^2+{|a_2|}^2=1$. Using the bracket notation, $〈p|=(1,0)$ is the $p$-polarization and $〈s|=(0,1)$ is the $s$-polarization. If we limit ourselves (without loss of generality) to the TIR case, then the quantum operator describing the IF shift is given by the $2\times 2$ matrix [15]

 

Fig. 1. Setup for the weak measurement of the Imbert–Fedorov shift. L1 and L2, lenses with focal length 20 and 30 cm, respectively; P1 and P2, Glan–Thompson polarizers; QWP, quarter-wave plate; HWP, half-wave plate; and QD, quadrant detector. P1 preselects the polarization state and the polarization analyzer composed of QWP, HWP, and P2 postselects the final polarization state.

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In our setup, the IF effect takes place when the incident Gaussian beam is totally reflected by a 45°–90°–45° BK7 prism. A lens (L1) focuses the light from a collimated single-mode fiber-coupled 826 nm laser diode to a $1/e^2$ intensity spot size $w_0=60\mathrm{\mu m}$. The beam is $p$-polarized by means of a Glan–Thompson polarizing prism P1. After reflection, an analyzer composed of a quarter-wave plate (QWP), a half-wave plate (HWP), and a Polarizer P2, which is used to postselect the desired polarization states. A lens L2 (focal length $f=30\mathrm{cm}$) collimates the beam emerging from the analyzer and directs it to a quadrant detector (QD). The output signal from the QD is read by a voltmeter. In our setup, the propagation distance $z=f=22·z_0$. With such a $z/z_0$ ratio, the beam propagation distance is large enough to disregard in Eqs. (5) and (6) the addends that do not contain it, being them smaller than our experimental error. The polarizer P2 is exactly orthogonal to the polarizer P1 so that it transmits $s$-polarized light. We first set the QWP and the HWP to minimize the light transmitted through P2. In this condition, the transmitted beam is no longer a Gaussian beam, but has a double peak intensity profile [Fig. 2(b)] along the $y$ direction [7,19,20]. We observe it by replacing the QD with a CCD camera. The two peaks are separated by a distance $\sqrt{2}·w$, where $w$ is the beam waist expected for the input Gaussian beam on the CCD.

 

Fig. 2. Beam profile of the light beam transmitted through the polarization analyzer. (a) Beam profile for when the weak measurements settings is still Gaussian. (b) Beam profile for minimum analyzer transmission.

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Our measurement procedure runs as follows. We perform two different measurements. In the first one we rotate the HWP at an angle ($+ϵ/2$) (Fig. 1 shows the positive rotation direction), and then at an angle ($-ϵ/2$), and we measure the relative positions of the beam centroid for each case with respect to the other. With these settings, the Jones matrices describing the action of the QWP ($Q_1$) and of the HWP ($H$) are:

 

Fig. 3. Spatial IF beam shift for a 45° linearly polarized incident Gaussian beam. Shown is the observed polarization-differential shift (dots), where the solid line represents the theoretical prediction. The $y$ axis on the right represents the amplified data with an amplification factor of 445.

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The experimental data (dots) reported in the graph are our measurements divided by this amplification factor. The line in the graph correspond to the spatial IF shifts of a $+45°$ linearly polarized beam with respect to a $-45°$ linearly polarized beam in TIR. The agreement in between theory and experiment is excellent. This is the most important experimental result of this Letter, since it proves that weak measurements allow us to also observe the linearly polarized dependent part of the IF shift.

The second measurement scheme runs as follows. We rotate the HWP and the QWP at angles of $ϵ/2$ and $ϵ$, respectively, then measure the relative positions of the beam centroid in this case with respect those measured when the HWP and QWP are rotated, respectively, at $-ϵ/2$ and $-ϵ$. In this case, the Jones matrix for the QWP is

 

Fig. 4. Spatial IF beam shift for a $\pm$ circularly polarized incident Gaussian beam. Shown is the observed polarization-differential shift (dots). The solid line represents the theoretical prediction. The $y$ axis on the right represents the amplified data with an amplification factor of 445.

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The range of values of $ϵ$ for which the weak measurement works is quite large. Even for the smallest value of $ϵ$ that we use in our measurements, the profile of the beam transmitted through P2 is still Gaussian [Fig. 2(a)]. In Fig. 5, we report a series of weak measurements for different $ϵ$ taken with postselected final states $(\mp ϵ,1)$ at an angle of incidence of 42° and at the fixed $z/z_0=22$. As expected, there is a linear dependence on $1/ϵ$. We see a departure from linearity at $1/ϵ=6.4$ where, due to a small amplification factor, the noise becomes larger than our signal. We experimentally observed that the use of the lens L2 helps to achieve a better linear dependence on $1/ϵ$. We wonder if the discrepancy observed in Fig. 4 cannot be partially related to this.

 

Fig. 5. Experimental study of the weak value amplification scheme as a function of $ϵ$. A faithful amplification of the optical beam shift is observed as long as the amplified signal scales linearly with $1/ϵ$.

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In conclusion, we have observed the IF shift using weak measurements in close analogy with the original AAV proposal. Even if we observe here a spatial shift, the imaginary part of the weak value is amplified by the propagation distance. We have shown that our approach allows for measuring both the IF effect that it is diagonal in the circular $\pm$ polarization basis, that the one diagonal in the $\pm 45°$ linear polarization basis.

Our Letter demonstrates the advantages of weak measurement as an amplification scheme. While measuring spatial beam shifts of tens or hundreds of nanometers is, in general, a challenging task, measuring amplified light beam shifts of tens or hundreds of microns is quite easy. Weak value measurement techniques, such as the one described here, amplify the signal but not the technical noise (thermal, electrical, vibrational, etc.) that confounds measurements at the nanometer scale. The experimental approach, presented here, discriminates in between two different physical contributions to the IF effect, showing a remarkable sensitivity of the weak amplification scheme in postselection. Weak measurement is finding more and more applications, as reported in recent research [21–24]. We think it would be very interesting to extend our results to applications that use beam shifts as a sensitive meter [25], such as in bio-sensing, or to beam shifts of matter waves [26] that are intrinsically quantum.