1. INTRODUCTION

Optical vortices with azimuthal phase carry orbital angular momentum (OAM) of per photon, where is the topological charge, is the reduced Planck’s constant, and is the azimuthal angle [13" target="_self" style="display: inline;">–3]. Vortex beams with OAM have found applications in many fields, such as quantum information processing [4,5], free-space communications [6,7], super-resolution microscopy [8], optical micromanipulations [9,10], and astronomical sources detection [11]. Accordingly, it is important to measure the topological charge (or OAM) of a vortex beam in many applications [11–16" target="_self" style="display: inline;">–16]. Until now, numerous methods have been developed for measuring the topological charge of vortex beams. For instance, we can use a spatial light modulator and monofiber to convert vortex beams to Gaussian beams by holographic techniques, while the efficiency is low [17]. The widely used method is to observe the interference pattern by interfering the wavefront of the vortex beam with its own mirror image or a reference wavefront, such as the Mach–Zehnder interferometer [18] and double-slit interference [19]. In 2008, Berkhout and Beijersbergen [20] presented a multipoint interferometer (MI) to detect the OAM of vortex beams by observing the interference patterns. It is shown that this method can find applications in measuring the OAM of light from astronomical sources. Later, a ring-sampled multipoint plate was developed for measuring the OAM of vortex beams based on the retrieving algorithm [21]. Also, we can measure the OAM by optical transformation, namely, converting the vortex beam with the spiral phase into a beam with a transverse phase gradient [22]. Besides the symmetrical structures, some asymmetric structures were also developed for measuring OAM, such as a screen with two nonparallel air slits [23], dynamic angular double slits [24], and a gradually changing period diffraction element [25]. More recently, several new devices, such as a translated single slit [26], single stationary lens [27], and gray-scale algorithm [28], have been introduced to determine the OAM of vortex beams. Furthermore, some methods for measuring fractional topological charge of vortex beams have been proposed as well [2931" target="_self" style="display: inline;">–31]. Among the aforementioned methods, the multipoint interferometer (MI) is one of the most robust and simple. We found that such a simple MI can also be used to manipulate the OAM spectrum [32]. However, it is shown that the interference patterns, generated from the MI, will repeat periodically once the value of topological charge becomes larger than half-value of the number of pinholes. In other words, this method is only available for vortex beams with low topological charge [33].

In this work, we propose an improved multipoint interferometer (IMI), which is formed by uniformly distributed circular apertures instead of point-like pinholes, where the radius of aperture () is not negligible, as shown in Fig. 1, to solve such problems. The results demonstrate that the far-field interference patterns using such a method will not repeat any more for vortex beams with any topological charge. Therefore, the IMI can be used to measure the vortex beams with high topological charge.

Fig. 1. Schematic of the experimental setup for detecting the topological charge of optical vortex with an IMI.

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2. THEORY AND METHODS

2.1 A. Theory of IMI

The schematic for measuring the topological charge of an optical vortex by an IMI is illustrated in Fig. 1. When a vortex beam illuminates on the IMI, we can measure its OAM by detecting the interference patterns in the far field.

We start with the proposed method theoretically. The complex amplitude of the optical vortex beam in the source plane, namely, just after passing through the IMI, can be written as o(x,y)=∑n=1Ncirc[[x−xn)2+(y−yn)2/r0]Apl,(1)where is the transmittance function of the aperture, and are the central coordinates of the th aperture. The complex amplitude of Laguerre–Gaussian (LG) beams with topological charge is given by Apl(r,ϕ)∝rLpll(2r2w2)exp(−r2w2)exp(−ilϕ),(2)where is the waist size of the beam, is the associated Laguerre polynomial, and and are the radial and azimuthal mode index, respectively.

Considering the Fraunhofer limit, the intensity pattern in the detector plane can be written as $$\begin{gathered} Ipl∝|FT{o(x,y)}|2 \\ =|∑n=0N−1FT{Apl(r,ϕ,z)}⊗{r0J1(2πr0fx2+fy2)fx2+fy2×exp[−ikaf(xcosan+ysinan)]}|2, \end{gathered}$$(3)where denotes the Fourier transform operator, and is the convolution operator. is the Bessel function of the first order, and are given by and , and is the azimuthal coordinate of the th aperture.

Provided the radius of each aperture is small enough and can be neglected, the far-field intensity pattern can be reduced as [20] Ipl∝|∑n=0N−1exp(−ilan)exp[−ikaf(xcosan+ysinan)]|2.(4)

To demonstrate the power of our method, the simulated intensity patterns of vortex beams with different OAM diffracted by IMI with six circular apertures are shown in Fig. 2(a).

Fig. 2. Far-field intensity patterns for vortex beams with different topological charges diffracted by (a) IMI and (b) MI with N=6. (a) and (b) are calculated by Eqs. (3) and (4), respectively.

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As a comparison, the corresponding results for MI are shown in Fig. 2(b) as well. From Fig. 2(a), we can see that, for IMI, all the intensity patterns differ significantly, even when the topological charge is larger than the number of apertures. Therefore, the OAM of the vortex beams can be measured by detecting the interference patterns directly. In Fig. 2(b), the intensity patterns repeat periodically from , viz., the patterns are the same for The intensity patterns can be inferred to repeat periodically when the topological charges satisfy the equation , or , where , is an arbitrary integer, and is the number of the pinholes in MI. For example, using MI, the interference pattern of the vortex beam with is the same as those with and . Figure 1 shows that the MI with six pinholes does not work for vortex beams with , while the IMI can still work for vortex beams with high topological charge.

2.2 B. Physical Explanation

Now, let’s discuss why the intensity patterns can repeat for different topological charges using MI. Suppose we use two vortex beams to illuminate the MI successively, satisfying . Then, the far-field complex amplitude of the beam passing through the th pinhole can be written as $$\begin{gathered} Enl1=exp(−il1an)exp[−ikaf(xcosan+ysinan)] \\ =exp[−i(l2+Nm)an]exp[−ikaf(xcosan+ysinan)] \\ =exp(−il2an)exp(−i2mnπ)exp[−ikaf(xcosan+ysinan)] \\ =Enl2. \end{gathered}$$(5)

This is why the intensity patterns for and are the same using the MI. For , the situation is similar, and the expression for the far-field complex amplitude can be written as $$\begin{gathered} Enl1=exp(−il1an)exp[−ikaf(xcosan+ysinan)] \\ =exp[−i(Nm−l2)an]exp[−ikaf(xcosan+ysinan)] \\ =exp[−i(−l2)an]exp(−i2mnπ)⁢exp[−ikaf(xcosan+ysinan)] \\ =Enl2*. \end{gathered}$$(6)

Here, * denotes the complex conjugate. Equation (6) shows that, if one uses MI, the interference patterns for and are the same when .

To illustrate the new insight gained from our improved method, we visually analyze the two different interfering processes diffracted by MI and IMI, using the local phase information for vortex beams with and 7 in Fig. 3. Figures 3(d₁) and 3(d₂) show that the local phase information obtained by MI is nearly the same, due to the tiny size of the pinhole, and consequently causes the identical interference patterns in Figs. 3(e₁) and 3(e₂). However, the local phase information obtained by the apertures in IMI [Fig. 3(g₂)] is obviously different from that in MI [Fig. 3(g₁)], due to the bigger size of the aperture and because the apertures are closer to the center of the beam axis, where the phase changes quicker than in the outer region. Accordingly, the distinguishable interference patterns can be observed [see Figs. 3(h₁) and 3(h₂)]. It is clear from Fig. 3 that, although the structure of our IMI is similar to MI, the IMI can get more phase information and can distinguish any vortex beams with different OAM.

Fig. 3. Distinctions between the two methods with (c) MI and (f) IMI. (a) and (b) The phase distributions of vortex beams with l=1 and l=7, respectively. (d₁) and (d₂) are the corresponding local phase distributions obtained by MI for l=1 and l=7, respectively. (e₁) and (e₂) are the corresponding far-field intensity patterns, respectively. (g₁), (g₂), (h₁), and (h₂) are the same as (d₁), (d₂), (e₁), and (e₂) but for IMI.

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3. EXPERIMENTAL RESULTS

To prove the feasibility of our proposed method, we measure the interference patterns diffracted by IMI experimentally. The experimental setup used to generate and measure the OAM of optical vortices is shown in Fig. 4. The measured vortex beams possessing OAM are generated by the spiral phase plate [34,35] displayed on a phase-only liquid crystal spatial light modulator (, Holoeye PLUTO VIS), illuminated by a helium-neon laser with the wavelength of 632.8 nm that emits a Gaussian beam with after collimated and broadened by a telescope system. Meanwhile, the IMI, simulated by the intensity-only SLM (, Holoeye LC-R 1080), is placed in front of the Fourier lens with the focal length of . We placed Fresnel lens in an geometry system, imaging the far-field diffraction pattern of the plate. The far-field diffraction intensity pattern was recorded by a digital camera with the pixel size of 3.45 μm. We used SLMs to generate vortex beams () and to create the desired IMI (), respectively. The polarization state of the beam produced by the helium-neon laser is controlled by the polarizer, collimated and expanded by the lens, and sent to the phase-only . A diaphragm allowed selecting the desired diffraction order, which was generated by spiral phase plate displayed on . The desired plate created by the is in the front focal plane of the Fourier-transforming lens. By using this optical system, the far-field diffraction patterns of the plate were imaged onto the CCD camera. We used an analyzer to ensure appropriate state incidence on the CCD camera.

Fig. 4. Schematic overview of the setup for measuring OAM of the vortex beams.

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The far-field diffraction intensity patterns recorded in our experiment, for different topological charges, are shown in Fig. 5, where it is shown that the experimental results agree well with the simulations shown in Fig. 2. Figure 5 also shows that an IMI with larger apertures enables us to characterize high topological charge from the diffracted intensity patterns.

Fig. 5. Experimental results of the far-field intensity patterns corresponding to Fig. 2(a).

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4. CONCLUSIONS

In conclusion, we demonstrated an approach to measure the OAM of vortex beams by distinguishing the far-field intensity patterns with an IMI. This robust method is as simple as MI; accordingly, it can be used to measure the OAM of light from astronomical sources as well. The only difference between IMI and MI is the size of pinholes. The bigger pinholes in IMI enable us to measure the high topological charge of vortex beams. Furthermore, we want to claim that our method is available for measuring the OAM of Bessel vortex beams as well, and the robust IMI is also applicable to other scalar vortex waves with larger topological charge, such as electron vortex beams, neutron vortex beams, and X-ray vortex beams. The IMI will be useful for many applications based on OAM [6,7,36].