1. Introduction

Vortices are a unique system which exhibits remarkable behavior in various branches of physics such as Bose condensates, superconductors, superfluids, fluid flow, and optics [1,2]. Vortices are characterized by a topological charge (TC) whose signal is associated with the direction of circulation. In particular, in optics, this circulation or orbital angular momentum (OAM) is related to the optical phase profile of the optical field in a plane orthogonal to the light’s propagation direction. Such fields possess a well-defined net OAM value and can be written in polar coordinates as $E(r,\phi )=E_0(r)\exp (im\phi ),$ where $m$ is an integer that we will call net TC. Note that such fields usually possess an optical axis centered phase singularity. The wave front of this field is composed of m intertwined helical surfaces that result in an annular intensity cross-section, with a handedness given by the sign of$m$.

Optical lattice formation is currently a very active research area. It has been studied in various media such as nonlinear media [3–5], Bose-Einstein condensates [6], and periodic photonic structures [2]. It has also been studied in free space through the interference of three plane waves [7–9]. Recently, optical intensity lattice formation associated with the apertures using light possessing OAM has been observed using, for example, a equilateral triangle [10] and a multipoint interferometer [11–13].

In this paper, we study the optical intensity lattice formation through the diffraction of light possessing OAM by a square aperture. In contrast to the work in ref [10], in which a well-shaped hexagonal truncated intensity lattice is always generated for any value of OAM, we find here that a perfect truncated square intensity lattice is formed only for even values of the TC. For odd values of the TC the lattice is not very well formed. We study both cases theoretically and experimentally, as well as the influence of fractional topological charges.

2. Theoretical results

We determine the Fraunhofer diffraction pattern in the far field region of a beam carrying OAM scattered using a square aperture. If we are interested only in the relative intensities at a fixed plane placed at the position$z=z_0$, the diffracted field $E_d$ is given by the integral [14]:

In this integral, the far field distribution $E_d(k_{\perp })$ is obtained from the Fourier transform of the product of the function describing the square aperture $\tau (r_{\perp })$ and the incident field $E_i(r_{\perp })$. Note that the transverse wavevector $k_{\perp }$ can be associated with the coordinate system of the far field region playing the role of reciprocal space.

The integral shown in Eq. (1) was numerically evaluated using high-order LG beams as the initial condition for the electric field. For LG beams with$p=0$, the Fraunhofer diffraction patterns for different values of the TC ranging from 1 to 12 are shown in Fig. 1 . We can observer that a square intensity lattice forms as $m$increases but only for even values of the TC with $m=2n$ and $n\in \mathbb{Z}$. For odd values of$m$, the maxima are not well defined. It can be seen that there is a relation between the number of lateral spots N and the TC, namely $m=2N-2,$ but only for even values of m.

 

Fig. 1 Diffraction patterns corresponding to the numerical results of Eq. (1).

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To understand the formation of the intensity lattices, we analyze the phase diagrams in the Fourier plane depicted in Fig. 2(a) and 2(b) for m = 5 and m = 6, respectively. Note that the phase has a uniform distribution, mainly in the central region, only in Fig. 2(b). Phase jumps, which form a square shape, are clearly observed in Fig. 2(b). In contrast, in some parts in the center of Fig. 2(a) (in which m has an odd value), the phase distribution is sufficiently smooth that it is impossible to clearly identify the phase jumps. Similar behavior is observed for all phase diagrams for the various values of m, in accordance with the amount and the parity of OAM.

 

Fig. 2 Phase patterns corresponding to the diffraction patterns in Fig. 1, for m = 5 (a) and m = 6 (b). The red dashed square was used only to highlight the center of the pattern.

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3. Experimental setup

The setup used to perform the experiment is depicted in Fig. 3(a) . An Nd:YAG laser operating at 532 nm illuminates a pixilated computer hologram written with a Hamamatsu model X10468-01 spatial light modulator (SLM) to produce high-order LG modes.

 

Fig. 3 (a) Experimental setup; (b) A beam inside of the square aperture (top) and the phase diagram (bottom) for m = 3. In the figure F is a density neutral filter; f_i are lenses; SLM is the spatial light modulator; and SF is a spatial filter.

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Because the LG beam radius increases with m, we used square masks with sides varying from 1.8 mm to 3.3 mm for different LG beams. These masks can be superimposed over the LG hologram in the SLM. The effect is similar to an LG beam incident in a square aperture as illustrated in Fig. 3(b). Note that we have carefully aligned the beam at the center of the aperture to avoid asymmetries in the patterns. We have used a hologram type 1 for coding phase and amplitude as first proposed by Kirk et al [15] and first used for LG beams by Leach et al [16]. The Fourier transform was implemented by a 50-cm lens ($f_3$) and the collected diffracted light was imaged by a 20-mm lens ($f_4$) in a Charge Coupled Device (CCD) camera.

4. Experimental results and discussion

Panels presented in Fig. 4 show experimental results for the diffraction pattern of a LG beam by a square aperture for m from 0 to 12, −1, −2, −7, and −8. We observe a well-formed square optical lattice only for even values of m, confirming the numerical results. The sign of m does not change globally the shape of the pattern because the phase is invariant by $\pi$ rotation for a square aperture. Naturally, for m = 0 we have the usual square aperture diffraction pattern.

 

Fig. 4 Diffraction patterns corresponding to experimental results for integer topological charges.

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The experimental results shown in Fig. 4 can be obtained from summing the contributions of single slits and double slits, a result that in quantum optics is known as Born’s rule [17]. For simplicity we can associate a slit to each edge of the square. Note that both a square aperture and a square slit have very similar diffraction patterns.

Figure 5 shows the diffraction patterns for all slit combinations with m from 2 to 6. The overall pattern $P_{ABCD}$ is obtained through the following pattern summation [17]:

 

Fig. 5 Diffraction patterns produced by combinations of slits. Each slit combination is shown at the top.

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The patterns$P_i$, associated with the single slits, show that when m increases the number of fringes increases and all patterns are shifted as expected [18]. By contrast, the patterns $P_{ij}$, from the pairs of slits exhibit very different behaviors depending if they orthogonal or parallel. For the orthogonal slit configuration, it can be seen that the patterns comprise intensity peaks and the number of these peaks increases with the value of m but without any TC parity dependence. However, for the parallel slit configuration, it can be seen that there are two types of patterns, one for even values of m and the other for odd values of m. Such dependence on the TC parity can be associated to the fact that there is an odd or even multiple of $\pi$phase difference between the opposite slits for odd or even TC, respectively. This observation indicates that the patterns $P_{AC}$ and$P_{BD}$ have an important role in the optical intensity lattice formation processes that are dependent on the parity of m.

In Fig. 6 , we superimpose the intensity patterns, $P_{AC}$ and $P_{BD}$, for m = 5 and m = 6. In the first case, we observe that the intensity peaks in$P_{AC}$ and $P_{BD}$do not match. However, for m = 6 the peaks in the center of each pattern coincide. For even values of m this effect produces a shaped optical intensity lattice when all contributions of all patterns are taken into account. For odd values of m this effect smears out the optical lattice.

 

Fig. 6 Superimposed patterns, $P_{AC}$ (intensity) and $P_{BD}$ (intensity contours).

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Note that for the cases in which the intensity patterns $P_{AC}$ and $P_{BD}$ match in the center, the phase corresponding to the square optical intensity lattice will be similar to the one shown in Fig. 2(b).

Finally, we analyzed the evolution of the pattern for the azimuthal profile $r^m{e}^{im\varphi }$for fractional TC, varying m in increments of 0.1. Note that the use of an azimuthal profile instead of a true LG beam is a good approximation because the potential functions and the Laguerre have approximately the same behavior at the edges of the square. In the panels of Fig. 7 , we see the results of such simulation. It is clear that there is no well-formed lattice pattern between two patterns from two consecutive even values of m for any fractional TC.

 

Fig. 7 Diffraction patterns corresponding to the experimental results for the fractional topological charges

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5. Conclusions

We studied the Fraunhofer diffraction of LG beams by a square aperture. We observed that for even topological charge values, a truncated square optical intensity lattice, which is composed of a set of rotationally-symmetric intensity peaks, appears. By contrast, the resulting pattern is washed-out for odd topological charge values. To understand this diffraction pattern formation, we focused on the edge of the square rather than on the square aperture. Because the interference comes from the pairs, we decomposed the diffraction patterns of the square slit in patterns that originated from each slit and pair of slits. We found that the patterns that result from parallel slit configurations are responsible for two different types of patterns: one for odd and the other for even values of topological charge. In contrast to the case of odd values of the topological charge, for even topological charge values there is a good match between the intensity maxima in the center of the pattern. This is reflected in the formation of a square-truncated optical intensity lattice. Finally, we showed that there is a continuous transition between the two lattices that correspond to two consecutive even values of m, and there is no intermediate lattice between them for fractional TC.