Measuring the Topological Charge of Orbital Angular Momentum Beams by Utilizing Weak Measurement Principle

According to the principle of weak measurement, when coupling the orbital angular momentum (OAM) state with a well-defined pre-selected and post-selected system of a weak measurement process, there will be an indirect coupling between position and topological charge (TC) of OAM state. Based on this we propose an experiment scheme and experimentally measure the TC of OAM beams from −14 to 14 according to the weak measurement principle. After the experiment the intrinsic OAM of the beams changed very little. Weak measurement, Topological Charge, OAM beams.

where l is the TC, l ≠ 0 and sgn(⋅) is the sign function. Following the scheme of weak measurement, the initial state |ϕ〉 i can be prepared as |ϕ〉 i = |i〉 ⊗ |ψ l 〉, in which |i〉 is the preselected state. By considering the von Neumann measurement, the interaction Hamiltonian can be described as γ = ⊗ˆĤ A P x , where γ is the coupling constant, Â is an observable of the preselected state and P x is the momentum observable of the unknown OAM state. Consequently, the unitary transformation is = γ − ⊗ U e i A P xˆˆ. After this unitary transformation, system Â is post-selected to the state |f〉 and the unknown OAM state is projected to |x, y〉 basis. Then the final state become is satisfied, a simplified result can be obtained as On the other hand, according to Eq. (28) of ref. 52 the final state is the superposition of two OAM states with same TC which separate from initial position with the same distances of γ along the opposite direction. Meanwhile, the change of the intrinsic OAM which equal to the theoretical erro is very little after the measurement because of the weak interaction 53,54 . Figure 1 shows the experimental setup of the weak measurement. Light from a He-Ne laser passes through a half-wave plate (HWP) and a polarizing beam splitter (PBS). The HWP is used to adjust the polarizing ratios and the PBS filters the desired polarization. The beam is then expanded by two lenses, L1 and L2. The expanded beam is vertically illuminated on a spatial light modulator (SLM) with the resolution of 20 μm per pixel to generate OAM states. The beam then passes through two HWPs with a quarter-wave plate (QWP) inserted in-between to prepare a known polarization state as the preselected state |i〉. At this point, the www.nature.com/scientificreports www.nature.com/scientificreports/ state preparation is completed. To achieve the weak measurement operation, a polarizing Sagnac interferometer is employed in which one of the mirrors, M3, is connected with a piezo-transmitter (PZT). A single polarizing beam splitter (PBS) is used as the entry and exit gates of the device. When entering the interferometer, the incident beam is split into different polarization components |H〉 and |V〉, which traverse the interferometer in opposite directions. Without the PZT, the two components would combine again when they exit the PBS. Because of the existence of the PZT, a tiny rotation can be imposed on M3 and the two components can be separated slightly in different directions at the exit. This executes exactly the operation Û we mentioned above. Meanwhile the lenses, L3 and L4, constitute the 4f system which images the distribution of both H and V components after the beam is reflected by the SLM to the position of the charge-coupled device (CCD) camera. After the weak measurement, a HWP and a PBS are used to post-select the state |f〉. Finally, the intensity pattern is recorded by a CCD camera.

Experiment and result.
In the setup, the observable of the weak value Â corresponds exactly to Â = |H〉〈H| − |V〉〈V|. The pre-selected state is prepared as |i〉 = a exp(−i2θ)|H〉 + b exp(iφ) exp(i2θ)|V〉 and the post-selected state is |f〉 = cos(2η)|H〉 + sin(2η)|V〉. The values of the parameters involved in our experiment are listed in Table 1. Figure 2 shows the experimental and the numerically simulated light intensity distribution of the final states. Table 2 lists the OAM of experimental and simulated beams over a range l = −14 to l = +14. where l represent the TC originally generated, l m is the experimentally measured one and l n is the result of numerical calculating.

Discussion
From Fig. 1 and Table 2, it is clearly observed that the absolute value of experimental result is a little larger than the simulated one and the accuracy of the simulated value obviously decline more quickly than the experimental one with the increase of the |l|. This is caused by an experimental treatment. To avoid over exposure, we always adjust the parameters of the CCD to control the maximal incident light intensity. As a result, the intensity distribution recorded I m is the product of the actual distribution I and a scaling coefficient μ. For the CCD has a response threshold so that the intensity below the value will not be recorded, the scale coefficient of the intensity will cause some intensity loss. Suppose that the actual light intensity is I(x, y) = ϕ(x, y) * ϕ(x, y) and the response threshold is ξI max , and then, the intensity we record in the picture (I m ) becomes where O(ξI(x, y)) represents the effect of the faint intensity that is not recorded by the CCD. Since the second term of Eq. (5) is much smaller than the first term, we can ignore it. It is then obvious that l m is a little larger than l n . For the Eq. (3) is derived by neglecting the higher terms of 2 2 γ σ , the result of l n is smaller than the value initially generated l. In this sense, Eq. (5) gives some compensation and the accuracy of the experimental results decrease more slowly. This means the restriction γ σ l 1 2 2 2  in ref. 52 can be relaxed and the difficulty of the experiment can be reduced substantially. In addition, all the theories involved are based on the fact that l is an integer, so that it is reasonable to round l m . In this situation, our experiment is perfectly consistent with the true value.
It is important to collimate the waist radius σ and the coupling constant γ for they are the pivotal parameters to determine the theoretical error of the scheme. To collimate the beam profile we have combined two aspects. Theoretically, as we employ a 4f system, the waist radius we used must be as same as the one loaded on the SLM. Experimentally, we determine the waist radius by the following steps: (a) Record a full spot when the HWP of the post-select system is at 0° (or 45°) and calculate its centroid, P H (or P V ). (b) Choose a line passing through P H (or P V ) and retrieve the intensity along this line. The distance between the two peaks is the measured diameter of the spot in the direction of the line, named D 1 . (c) Choose another 3 lines which also passing through P H (or P V ) and calculate D2 to D4. The average diameter is D D /4 (d) According to the relationship σ = || D l / 2 , the waist radius for each picture can be determined. During our experiment, all the error of σ between the theoretical and experimental results are less than 6%, so we still use the value of waist radius loaded on the SLM.
The shift of the two beams, |H〉 and |V〉, exiting from the polarizing Sagnac interferometer is 2γ. It is also necessary to determine the origin and the plus direction of the shift. To solve those problems, we can find the centroid of H beam, P H , when the HWP of the post-select system is at 0°. And we can determine the centroid of V beam, P V , when the HWP is at 45°. Then the distance between P H and P V is 2γ, the direction from P V to P H is the plus direction of x, the midpoint of P V and P H is the origin of the shift and the legnth of P V P H /2 is γ.   Table 2. The TC values of experimental and numerical results. l g , l m and l are as the same as Fig. 2.