1. Introduction

Laguerre-Gaussian (LG) beams are solutions of the scalar Helmholtz equation in the paraxial approximation. These solutions form a complete basis set of orthogonal functions, where the azimuthal index $l$ and the radial index $p$ represent an orbital angular momentum (OAM) of $l\hbar$ the beam carries and the number of radial modes in the intensity distribution, respectively [1–3]. These two indices provide LG beams with high capacity and rich design that are applied in both classical and quantum fields, such as micromanipulation [4,5], optical communications [6–11], quantum information science [12–15], gravitational wave detector [16] and so on.

One of the crucial issues for applying LG beams is to achieve the precise determination of the indices of a unknown LG beam. There are quite a number of different methods that are able to implement the determination of the azimuthal index $l$, such as interfering the measured LG beam with a uniform plane wave or its mirror image [17,18], diffracting the beam by a special mask [19–23], mode conversion [24,25] and geometric transformation [26,27]. In contrast, only a few of methods are applicable to the two indices of LG beams simultaneously, such as interference [17,18] and mode conversion [24,25]. Furthermore, it is the problem that almost all these methods destroy the intensity distribution of the measured beams or cause collapse of the measured states.

Weak measurement offers a feasible way to solve the problem and figure out the two indicese concurrently. Proposed by Aharonov et al. [28] as an extension to the standard von Neumann model of quantum measurement, weak measurement is characterized by the pre- and post-selected states of the measured system. Through this approach, little information about the measured system from a single outcome can be extracted without causing state collapse. This feature makes weak measurement an ideal tool for examining the fundamentals of quantum physics, such as measuring the profile of the wave function [29], engaging the signal amplification [30], affording a resolution of the Hardy paradox [31,32], and directly measuring the density matrix of a quantum system [33]. Recently, some theoretical researches extend the application of this method to measure the indices of LG beams [34,35]. Based on the theoretical study [35] and our previous work [36], we have firstly succeeded in the performing the simultaneous measurement of the azimuthal and radial indices of LG beams by weak measurement experimentally with the final intensity distributions approximating to their origin.

2. Theoretical principle

In the position space, a Laguerre-Gaussian beam which can be deemed as a LG state is expressed as

If we choose polarization states for $\hat {A}$ system, the preselected state can then be expressed as $\vert i \rangle =B_H\vert H\rangle +B_V\vert V\rangle$, and $\hat {A}$ can be any of the pauli operators. For $\hat {A}=\sigma _{z}=\vert H\rangle \langle H\vert -\vert V\rangle \langle V\vert$, the initial state becomes $\vert \varphi \rangle _f=B_H\vert H\rangle \vert \psi _{l,p}(x-\gamma ,y)\rangle +B_V\vert V\rangle \vert \psi _{l,p}(x+\gamma ,y)\rangle$ after weak interaction. Obviously, it indicates that horizontal polarization component of the state takes the initial shift $-\gamma$ followed by the shift $\gamma$ of the vertical one after weak interaction. This can be achieved easily by a birefringent crystal or a polarizing Sagnac interferometer. On the other hand, for $\hat {A}=\sigma _{x}$ and $\hat {A}=\sigma _{y}$ the states $\vert \varphi \rangle _H$ will become more complicated and present in the forms of the superimposing states of $\vert \psi _{l,p}(x-\gamma ,y)\rangle$ and $\vert \psi _{l,p}(x+\gamma ,y)\rangle$. Those suggest that more devices and operations are needed. To avoid this, $\hat {A}=\sigma _{z}$ was chosen in our experiment.

3. Experimental scheme

The experimental setup of the weak measurement is schematically shown in Fig. 1. A Guaussian beam at wavelength 633 nm emitted from the He-Ne laser passes through a half-wave plate (HWP) and a polarizing beam splitter (PBS) to adjust the light intensity and purify the polarization. Then the filtered beam was coupled into a single-mode fiber to choose a little spot which can be deemed as plane wave. The spot is then expanded by two lenses, L1 and L2, and vertically illuminated on the spatial light modulator (SLM) with the resolution of 20 $\mu$m per pixel to generate the desired LG states. The LG beam is loaded the preselected polarization state $\vert i \rangle$ by a quarter-wave plate (QWP) sandwiched by two HWPs. So, the initial state preparation is completed. A polarizing Sagnac interferometer with a piezo-transmitter (PZT) connected to M3 is used to achieve the weak measurement operation of $\hat {A}=\sigma _{z}$. The entry and exit gates of the interferometer is a PBS. When entering the interferometer, the beam is divided into horizontal polarization component $\vert H \rangle$ and vertical polarization component $\vert V \rangle$, which traverse the interferometer in opposite directions. Without the PZT, the two components would combine again when they exit the PBS. However, with the PZT a tiny rotation can be imposed on M3 to separate the two components slightly in different directions at the exit. This is the weak interaction that was chosen above. Meanwhile the 4f system constituted by L3 and L4 images the distribution of the beam after reflected from the SLM to the position of the charge-coupled device (CCD) camera. The post-selection of $\vert f \rangle$ is operated by the last combination of a HWP and a PBS. Finally, the intensity pattern is recorded by the CCD camera.

 

Fig. 1. A sket2ch of the experimental setup.

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In our experiment, the pre-selected state is $\vert i \rangle =a\exp (-i2\theta )\vert H \rangle +b\exp (i\phi )\exp (i2\theta )\vert V \rangle$ and the post-selected state is $\vert f \rangle =\cos (2\eta )\vert H \rangle +\sin (2\eta )\vert V \rangle$. The values of the parameters involved in our experiment are listed in Table 1.

Table 1. The values of experimental parameters

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Since the final state is the superposition of two LG states with different phases and amplitudes that are separated from each other with distance of $2\gamma$ for $\hat {A}=\sigma _{z}$ [28,29], it is very important to determine the origin and the direction of their translation. To solve this, three steps are followed. Firstly, we recorded the centroid of $\vert H \rangle$, $P_{H}$ at the HWP4 of $0^\circ$ and the centroid of $\vert V \rangle$, $P_{V}$, at the HWP4 of $45^\circ$. The midpoint of $P_{H}$ and $P_{V}$, $(x_{0i}, y_{0i})$ then chosen as the initialized origin and the direction from $P_{H}$ to $P_{V}$, $(\theta _{0i})$, as the initialized orientation. Secondly, an appropriate calculating range around $(x_{0i}, y_{0i})$ was performed to deem the position of the origin and the direction of the translation, $(x_{0}, y_{0}, \theta _{0})$. By these two steps, three parameters will be gained. The third step is to record the three intensity pattern by the CCD with a large range of mode indices spanning, the final results of $x_{0}, y_{0}, \theta _{0}$ we achieved which satisfy $min(\Delta )$ within the calculating range according to Eqs. (2) and (3), where $\Delta =( \vert l_{1}- l_{1m}\vert +\vert l_{2}- l_{2m}\vert +\vert l_{3}- l_{3m}\vert +\vert p_{1}- p_{1m}\vert +\vert p_{2}- p_{2m}+\vert p_{3}- p_{3m}\vert$). The subscript $m$ denotes the result calculated by Eqs. (2) and (3) for the intensity pattern and the indices without $m$ are those generated originally. In our experiment, we chose $l_{1}=-3$, $p_{1}=1$, $l_{2}=6$, $p_{2}=3$, $l_{3}=-6$, $p_{3}=8$ respectively.

4. Result and error analysis

Figure 2 shows the experimental and the numerical simulated patterns of the light intensity distribution of LG beams in the final states. The experimental and numerical results about the mode indices are shown in Fig. 3 with $l$ spanning from $l=-6$ to $l=+6$ and $l\neq 0$ while $p$ change from $p=0$ to $p=8$. The error analysis and statistics are shown in Fig. 3, Fig. 4 and Table 2. Where the subscript $n$ represents the indices obtained by numerically calculating. $m$ implies that these indices are measured experimentally and those parameters without subscripts are originally generated data.

 

Fig. 2. Intensity distributions of the final states for LG beams with different mode indices. The top row is the intensity distribution in experiment while the bottom row is the simulative ones. The cyan lines are the (x,y) coordinate lines and the orange crosses denote centroids of the intensity distribution.

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Fig. 3. The results of the mode indices from experiment and simulation. The top row is the results for $l$ where these pink transparent bars are the numerical ones and the cyan bars are the experimental ones. The bottom row is the results for $p$ where these blue transparent bars are the numerical ones and the green bars are the experimental ones. The left column is the situation for $l>0$ while the right column is for $l<0$.

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Fig. 4. Error analysis. The top row is the result for $l$. $erl_{k}(l)$ satisfy $erl_{k}(l)=\sum _{p=0}^8[l_k(l,p)-l]/9$, where $k=m,n$, meaning the average error about $l$ with $l$ fixed and $p$ changed. $erl_{k}(p)$ satisfy $erl_{k}(p)=\sum _{l=-6,l\neq 0}^6[l_k(l,p)-l]/12$, where $k=m,n$, meaning the average error about $l$ with $p$ fixed and $l$ changed. The bottom row is the result for $p$. $erp_{k}(l)$ satisfy $erp_{k}(l)=\sum _{p=0}^8[p_k(l,p)-p]/9$, where $k=m,n$, meaning the average error about $p$ with $l$ fixed and $p$ changed. $erp_{k}(p)$ satisfy $erp_{k}(p)=\sum _{l=-6,l\neq 0}^6[p_k(l,p)-p]/12$, where $k=m,n$, meaning the average error about $p$ with $p$ fixed and $l$ changed.

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Table 2. Error statistics

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From Fig. 2, the experimental patterns with small mode indices are in good agreement with their numerical counterpart. However, for the large mode indices, such as $l=-5$ $p=7$ and $l=6$ $p=8$, the experimental distribution on the right edge is obviously brighter than their left one. This is caused by the size of light spots modulated by the SLM. The spots with small indices are very tiny so that the gratings needed in the SLM are small. In this small area, the intensity of modulated light is almost the same. As a result, the experimental patterns are consistent with the numerical ones. At the same time, the area of grating needed in the SLM is large with high mode indices. When the modulating region is on the fringe of the incident light, where the intensity is weak, the final distribution on this area is dim. The final distribution is bright when the modulating region is in the center area of the incident light. The uneven distribution can be deemed as the superposition of a plane wave and the background noise.

With the increase of modulating area, the measuring errors caused by background noise rise up remarkably. However, based on the optimization of the translation origin and the direction we developed above, this error can be partially offset. As a result, the measuring error for small spots is obviously lager than the simulated one, as proven clearly in Fig. 2 , Fig. 3, Fig. 4 and Table 2. Especially, the compensation effect can be seen distinctly in Fig. 4, where the tendency of the average error curve for the measurement is not in line with the numerical one and there are some sharp changes for the medium size indices during experiment causing the error increase of small indices but the error decrease of large indices in the experiment. On the other hand, the measuring error of $p$ is much higher than the simulated one in Fig. 1 and Fig. 4. This can be explained by Eq. (3) in which $p$ is related to $\bar {x^3}/\bar {x}$. With the existence of background noise, the error of $\bar {x^3}$ is much bigger than the one of $\bar {x}$ so that the measuring error of $p$ is much higher than its simulating one. Finally, the average of the absolute measurement error about $l$ is 0.2104 and the one of $p$ is 0.2016 in Table 2. Furthermore, since the theoretical derivation of the relationship as shown in Eq. (2) (3) is based on the fact that $l$ and $p$ are all integers, it is reasonable to round off the experimental results to get the initial values of the indices. In this situation, our experiment is perfectly consistent with the generated values while $l$ and $p$ span across a large range. Meanwhile, Fig. 2 shows that the final intensity distribution is approximate to their initial one which means the disturbances of the LG beams during the experiment are small.

5. Conclusion

In conclusion, as an extension to the standard von Neumann model of quantum measurement, there are no states suffering from collapse after weak measurement and the information of the measured system can be extracted. When coupling the LG state with a well-defined pre- and post-selected system in a weak measurement process, an indirect coupling between $\bar {x}$, $\bar {x^3}$, $\bar {y}$ and $l$, $p$ exists. Based on this, we have proposed an effective scheme to experimentally measure the mode indices of LG beams by weak measurement. We have managed to measure the mode indices of LG beams spanning from $l=-6$ to $l=+6$, $p=0$ to $l=+8$ accurately with the final intensity distribution approximating to their initial one.