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Abstract
Here, joint spatial weak measurements with higher-order Laguerre-Gaussian (LG) point states are investigated experimentally. From the intensity patterns of the final LG point states, two dimensional position operators 〈X〉 and 〈Y〉 as well as high-order position operators 〈XY〉, 〈X2 − Y2〉, 〈X3〉, and 〈Y3〉 are extracted simultaneously, from which both the complex weak values and joint weak values of two non-commuting observables can be obtained. The enhancement of joint weak values by the postselection state are analyzed. The simple relationship between the expectation values of position operators and the azimuthal and radial indexes of LG modes allows us to identify the mode indexes directly. A simple and robust scheme based on an optical window is demonstrated to monitor the LG mode indexes. These findings deepen the understanding of the weak measurement and provide an alternate and effective method for LG mode index monitoring.
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1. Introduction
Since originally proposed by Aharonov, Albert and Vaidman in 1988, weak value measurement (WVM) has attracted significant attention owing to its unique properties such as weak value amplification and non-collapse system state [1]. The weak value amplification can enhance the small change of system parameters, promising wider-range applications in precise metrology [2–6]. Various parameters have been detected by WVM method such as temperature [7], frequency [8], magnetic field [9], tiny displacements [10] and deflection angles [11] of light beam. The non-collapse system state indicates that WVM can provide information without greatly disturbing the evolution of the measured system, differing distinctly from the conventional quantum measurement, where the measured system is irrecoverably collapsed into one of the eigenstates of the observable [12–15]. The WVM enables us to access information that cannot be obtained by the conventional quantum measurements. Thus, the WVM has been used to resolve Hardy’s Paradox [16]; to realize Leggett-Garg inequality violations [17]; to characterize wave functions [12,18,19]; and to measure the geometric phase from quantum measurement back-action [20,21]. Comparing to the fundamental Gaussian point state in WVM, it is predicted that non-separable pointer state such as orbital angular momentum states can provide additional information. The non-separable pointer state can help extract weak values of higher-order moments of single-particle operators [22] and joint weak values [23–25]. To our knowledge, the joint weak values with non-separable pointer states have not been demonstrated experimentally.
Here, we perform joint spatial WVM with inseparable Laguerre-Gaussian (LG) point states at an Au film coated on a prism. Both the real and imaginary parts of the weak values and joint weak values of the Goos-Hänchen (GH) and Imbert-Fedorov (IF) operators are measured. The excitation of surface plasmon resonance leads to large joint weak values, which are further enhanced by postselection. The azimuthal and radial indexes of LG beams can be obtained simultaneously by extracting the high-order position operator Ŷ3 together with two dimensional position operators $\hat{X}$ and Ŷ from the final LG point states. Comparing to reported methods for identification of azimuthal and radial indexes, the proposed method based on joint spatial WVM has the advantages of collimation-free, high precision, low cost, suitable for LG mode monitoring [26–30].
2. Theory and model
To begin with, the GH and IF operators in the reflection process are in form of [31]
respectively. χp,s=-i∂lnrp,s/∂θ is the first derivation of reflection coefficients rp,s with respect to incident angle θ. γp=-iM/rp, γs = iM/rs with M = [rp+rs]cotθ. The GH and IF operators are complex values when the reflection coefficients are complex. The product of GH and IF operators areAfter the interaction between the system and probe states, we postselect the system in state $|{{\psi_f}} \rangle = \cos \psi |H \rangle + \sin \psi |V \rangle$ with ψ being the polarization angle, resulting in the point state in form of
The final point state can be recorded by a CCD camera, from which one can extract the expectation values of position operators ($\left\langle X \right\rangle$ and $\left\langle Y \right\rangle$), joint position operators $\left\langle {XY} \right\rangle$ and $\left\langle {[{X^2} - {Y^2}]/2} \right\rangle$, and higher-order position operators ($\left\langle {{X^3}} \right\rangle$ and $\left\langle {{Y^3}} \right\rangle$). From these expectation values, various information about the weak interaction process can be obtained.
Owing to the spatially inseparable of LG point states, the expectation values of position operators, i.e., the GH and IF shifts, are in form of [32,33]
The expectation values of second-order position operators are
3. Experimental setup
To measure the joint weak values of the GH and IF operators experimentally, a 532 nm laser (SQLE-LS-532-2000TA, Leoptics) emits a light beam with a diameter of 2.5 mm, as shown in Fig. 1(a). The beam is expanded by a beam expander into 10 mm, slightly larger than the height of the active region (15.36 mm×8.64 mm) of spatial light modulator (SLM, Holoeye PLUTO-2-NIR-011). The expanded light beam passes through a polarizer and then is sent to the SLM, where a phase mask (see inset in Fig. 1(a)) encoding with the amplitude and phase of LG mode [34] is loaded to create the desired LG modes. After passing through a Glan polarizer to preselect the incident polarization, the light beam is focused into a glass prism coated with Au film of 40 nm thickness. The reflected beam is post-selected by another polarizer, and then imaged by a combination of focal lens and a CCD camera (8051C-USB, Thorlabs, pixel size: 5.5 µm×5.5 µm). Figure 1(b,c) gives the recorded post-selected intensity patterns for LG beams with ℓ=±2. The patterns are deformed from their initially standard LG modes. There are outside diffracted rings in the LG modes generated by SLM, which is clearly shown in Fig. 1(b,c) and the insets in Fig. 4(a). These outside rings will affect the calculation of expectation values of position operators, which can be removed simply in image processing. To calculate the expectation values of position operators, the origin of the x- and y-axes should be determined firstly. To this end, we record an undeformed intensity pattern by rotating the Glan polarizers. The coordinate origin can be obtained from this undeformed symmetric pattern.
Fig. 1. (a) Experimental setup for the measurement of joint weak value of GH and If observables. The inset shows the light beam in Au/prism structure and the phase mask loaded on SLM. (b, c) The CCD recorded intensity patterns for incident LG modes of ℓ=±2 and p = 0.
4. Results and discussion
The Au film is deposited directly on the prism. The reflectivity of Au/prism changing with incident angle is measured and shown in Fig. 2. The theoretical and experimental results are in good agreement. The reflectivity of Au/prism has a dip at θ=48.1°, corresponding to the surface plasmonic resonance angle. The LG beams with ℓ=±2, p = 0, w0 = 20 µm are illuminating sequentially on the Au/prism with an incident angle of 52°. The reflected beam is post-selected by a polarizer. From the imaged intensity patterns, we extract the expectation values for the first-order position operators $\left\langle X \right\rangle$ and $\left\langle Y \right\rangle$ and for the second-order position operators $\left\langle {XY} \right\rangle$ and $\left\langle {({X^2} - {Y^2})/2} \right\rangle$. The results are shown in Fig. 3(a1-b2). For the cases of ℓ=±2, $\left\langle Y \right\rangle$ are opposite in sign, while $\left\langle X \right\rangle$ has small difference. $\left\langle X \right\rangle$ and $\left\langle Y \right\rangle$ vanish identically when postselction angle ψ is 90°, i.e., the preselection and postselection states are orthogonal. At ψ=89.4° and 90.6°, $\left\langle X \right\rangle$ reach their peaks for both cases of ℓ=±2. The peak positions of $\left\langle Y \right\rangle$ for ℓ=±2 have small relative displacements. The real and imaginary parts of weak values GH and IF operators can be obtained directly from the measured $\left\langle X \right\rangle$ and $\left\langle Y \right\rangle$ according to Eq. (5a) and Eq. (5b). The experimental results and theoretical predictions are in good agreement, as shown in Fig. 3(c1). The postselection angle has no influence on the both the real and imaginary parts of ${\left\langle {\hat{G}} \right\rangle _w}$. However, the the real and imaginary parts of ${\left\langle {\hat{F}} \right\rangle _w}$ will be amplified strongly when the postselection angle ψ approaches to 90°. Two middle points for both $\textrm{Re} [{\left\langle {\hat{F}} \right\rangle _w}]$ and ${\mathop{\textrm{Im}}\nolimits} [{\left\langle {\hat{F}} \right\rangle _w}]$ are not shown in Fig. 3(c1), as the scale of the longitudinal axis is too large. Since the measurement errors will also be amplified by the postselection angle, the errors increase when ψ approaches to 90°.
Fig. 3. (a1-b2) The expectation values of first- (a1, b1) and second-order (a2, b2) spatial displacements changing with the postselection angle. The complex weak values (c1) and joint weak values (c2) for the GH and IF operators.
When ℓ=±2, p = 0, the expectation values of second-order position operators are reduced into
Therefore, both the weak values and joint weak values of the GH and IF operators can be obtained simultaneously by extracting the first- and second-order position operators from the intensity patterns of final point states. Additionally, the expectation values of higher-order position operators ($\left\langle {{X^3}} \right\rangle$ and $\left\langle {{Y^3}} \right\rangle$) can also be extracted. The higher-order position operators can be given by
In the following, we focus on the joint spatial weak measurement at air-glass interface. As shown by Fig. 4(a), LG beams are reflected a BK7 window of 5 mm thickness to ensure the full separation of beams reflected at the front and back interfaces. The reflected beam from the front interface is collected and passes through a Soleil-Babinet (SB) compensator and a GL polarizer sequentially. The SB compensator provides a phase delay of δ between horizontal and vertical polarizations. Thus, the postselection state is $|{{\psi_f}} \rangle = \cos \psi |H \rangle + \sin \psi \exp (i\delta )|V \rangle$. The GH and IF operators are thus ${\left\langle {\hat{G}} \right\rangle _w} = {\chi _p}$ and ${\left\langle {\hat{F}} \right\rangle _w} = {\gamma _p}\tan \psi \exp (i\delta )$, respectively. The weak values GH operator is independent of the postselection. However, the weak values IF operator can be flexibly controlled by the polarization angle and phase delay of the postselection state.
Fig. 4. (a) Experimental setup for the monitoring of azimuthal and radial indexes of LG beams with joint weak measurement technique. The inset shows the transmitted unaffected LG beams. (b-e) The GH and IF shifts changing with polarization angle (b, c) and phase delay (d, e) for ℓ=±4 and p = 2, respectively.
Figure 4(b) and (c) show the the GH and IF shifts changing with the polarization angle ψ for ℓ=±4, p = 2, when the phase delay and incident angle are δ=π/4 and θ=20°, respectively. The GH and IF shifts are in subwavelength scale without postselection. With proper postselection polarization angle, the GH and IF shifts can be up to 8.4 and 33 µm, respectively. The measurement errors of $\left\langle Y \right\rangle$ in Fig. 4(b,c) increase for the rapid change region (ψ is around 90°). This may be caused by the impurity of the generated LG modes (see insets in Fig. 4(a)).
By fixed the postseclection angle at 89.8°, the GH shifts vanish for both cases of ℓ=±4 when δ=0, while the vanishing IF shifts appear at δ=90°. The GH shifts are nearly identical for the cases of ℓ=±4 and they are much smaller than IF shifts, as shown by Fig. 4(c) and (d). The IF shifts however are opposite in sign but nearly identical in amplitude, indicating that the terms containing ${\left\langle {\hat{G}} \right\rangle _w}$ in Eq. (5a) and Eq. (5b) have mere influence on the GH and IF shifts.
In order to identify the radial and azimuthal indexes of LG beam, we extract the higher-order position operator $\left\langle {{Y^3}} \right\rangle$. $\left\langle {{Y^3}} \right\rangle$ relies on weak values GH and IF operators. When ${\left\langle {\hat{G}} \right\rangle _w} < < {\left\langle {\hat{F}} \right\rangle _w}$, $\left\langle {{Y^3}} \right\rangle = a(2p + |\ell |+ 1)\;\textrm{Re} [{\left\langle {\hat{F}} \right\rangle _w}]\} /{k_0}$ with $a = 3w_0^2/4$. Figure 5 shows the the experimental results and theoretical predictions for $\left\langle X \right\rangle$, $\left\langle Y \right\rangle$, and $\left\langle {{Y^3}} \right\rangle$ respectively when δ=π/4. The $\left\langle X \right\rangle$ almost changes linearly with the azimuthal index ℓ. The $\left\langle {{Y^3}} \right\rangle$ are almost linearly proportional to p and |ℓ|. The deflection from the linear relationships should be attributed to the small dependence of ${\left\langle {\hat{F}} \right\rangle _w}$ on p and |ℓ|. This dependence also leads to the dependence of $\left\langle Y \right\rangle$ on p and |ℓ|.
Fig. 5. (a-c) The GH (a) and IF (b) shifts and $\left\langle {{Y^3}} \right\rangle /a$ (c) for different input azimuthal and radial indexes. (d, e) Experimental measured azimuthal (d) and radial (e) indexes.
When the phase delay is δ=π/4, the real and imaginary parts of ${\left\langle {\hat{F}} \right\rangle _w}$ are identical. Thus, the ${\left\langle {\hat{F}} \right\rangle _w}$ terms will be cancelled out in the calculation of radial and azimuthal indexes:
According to these simple relationships, we get the radial and azimuthal indexes from the experimental data directly. As shown by Fig. 5(d), the measured azimuthal indexes ℓ’ are in good agreement with the incident index ℓ for all radial indexes with the standard deviations smaller than 0.18. The measured values of radial indexes are around the input indexes, as shown by Fig. 5(e). The standard deviations are 0.24, 0.17, 0.08, 0.16 for p = 0, 1, 2, and 3, respectively. Therefore, both the azimuthal and radial indexes of LG modes can be determined by the joint spatial weak measurement simultaneously. To increase the measurement accuracy, higher-quality LG modes should be generated [35] and the CCD camera with smaller pixel size should be used.It is worth to point out that, only partial of incident light beam is used for the measurement with much incident light transmitting through the BK7 window. The transmitted beams keep initial mode structure with a transmissivity of 87%. The transmissivity can be increased further by lowering the incident angle. According to Eqs. (5a), (5b), and (8b), the simple relationships between the indexes of LG beams with the expectation values of position operators (Eqs. (9a) and (9b)) hold when $\ell\;{\mathop{\textrm{Im}}\nolimits} [{\left\langle {\hat{G}} \right\rangle _w}] < <\;\textrm{Re} [{\left\langle {\hat{F}} \right\rangle _w}]$. The weak value of GH and IF operators can be flexibly tailored by the polarization angle of the postselection polarizer, thus high precise and high mode capacity can be obtained in the identification of the azimuthal and radial indexes of LG beams.
5. Conclusions
In conclusions, a novel joint spatial WVM platform has been demonstrated based on inseparable LG point states. The inseparable LG states mix the real and imaginary parts of the weak values and joint weak values for two non-commuting operators, allowing us to obtain the complex weak values and joint weak values of the GH and IF operators simultaneously from the final point state. Accompanied with the expectation values of high-order position operator Ŷ3, the azimuthal and radial indexes of LG modes can be determined directly and precisely. These findings deepen the understanding of weak measurements, facilitating the detection LG mode index. The present method might identify the fractional topological charge, since a LG beam carrying fractional topological charge can be considered as a superposition of many LG modes with different integer topological charges [36,37].
Funding
National Natural Science Foundation of China; Natural Science Foundation of Guangdong Province; Jinan Outstanding Young Scholar Support Program; Research Fund of Guangdong-Hong Kong-Macao Joint Laboratory for Intelligent Micro-Nano Optoelectronic Technology.
Disclosures
The authors declare no conflicts of interest.
Data availability
Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.
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