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In the following paragraphs, the theory underlying multiplication and division with transformation optics is presented and described. At first, the key element of these operations, i.e., the circular-sector transformation, is introduced and developed in the paraxial approximation. Therefore, the possibility of combining many circular-sector transformations to either multiply or divide the OAM content of an input beam is shown, and the phase patterns of the corresponding optical elements are calculated.
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The optical layout of the system consists of a cascade of two optical elements: the former performs a conformal optical transformation, whereas the latter corrects the phase distortion owing to the different paths traveled by the distinct points of the beam and restores the desired phase profile. Conversely, owing to the invariance of the light path for time reversal, the second optical element performs the inverse optical transformation. The key element of OAM multiplication and division is represented by an optical transformation performing a conformal mapping of the whole circle onto a circular sector (Fig. 1a, d)36. By indicating with (r, ϑ) the polar coordinates on the input plane and with (ρ, φ) the polar reference frame on the second plane, this transformation operates a rescaling of the azimuthal coordinate:
$$ \varphi = \frac{\vartheta }{n} $$ (1) Fig. 1 Circular-sector transformation of OAM beams.
Schematics of OAM beam transformation with circular transformation optics for twofold (n = 2) a and threefold (n = 3) d circular-sector transformation. The first phase pattern (b.1, e.1) performs an n-fold circular-sector transformation, mapping the input intensity distribution onto a 2π/n arc. The second phase pattern (b.2, e.2) performs the required phase correction, retaining the compressed azimuthal phase distribution. Design parameters: a = 300 μm, b = 250 μm, and f = 20 mm. Numerical simulations of the propagation of an input Laguerre-Gaussian beam carrying ℓ = 2 (c.1, f.1), after illuminating the first element (c.2, f.2), at different z positions (z = 0.2 f (c.3, f.3), z = 0.4 f (c.4, f.4), z = 0.6 f (c.5, f.5), z = 0.8 f (c.6, f.6), up to the second optical element (c.7, f.7), and the output phase-corrected beam (c.8, f.8). Colors and brightness refer to the phase and intensity, respectively.After imposing the condition of conformity, the new radial coordinate is given by:
$$ \rho = a\left( {\frac{r}{b}} \right)^{ - \frac{1}{n}} $$ (2) with a and b being scaling parameters. By applying the previous transformation, an azimuthal phase gradient is mapped conformally onto a circular sector with amplitude 2π/n. To calculate the phase pattern of an optical element performing this transformation in the paraxial regime, we apply the stationary phase approximation37 to the Fresnel integral. The field U(ρ, φ) after a propagation length f along the coordinate z for an input plane-wave illuminating a phase-only optical element with phase function ΩS, n, located at z = 0, is given by:
$$ U(\rho , \varphi ) = \frac{{e^{ik\frac{{\rho ^2}}{{2f}}}}}{{i\lambda f}}{\iint} {e^{i\Omega _{S, n}\left( {r, \vartheta } \right)}e^{ik\frac{{r^2}}{{2f}}}e^{ - ik\frac{{r\rho }}{f}\cos \left( {\vartheta - \varphi } \right)}rdrd\vartheta } $$ (3) According to the stationary phase approximation37, the integral solution reduces to find the saddle points of the phase function in Eq. (3):
$$ \Phi \left( {r, \vartheta } \right) = \Omega _{S, n}\left( {r, \vartheta } \right) + k\frac{{r^2}}{{2f}} - k\frac{{r\rho }}{f}\cos \left( {\vartheta - \varphi } \right) $$ (4) The stationary condition $\nabla \Phi = 0$ leads to a system of partial derivatives of ΩS, n that is unknown. Substituting the transformation relations, i.e., Eqs. (1) and (2), into the partial derivatives of Eq. (4) and imposing the stationary condition, we obtain:
$$ \left\{ {\begin{array}{*{20}{c}} {\frac{{\partial \Omega _{S, n}}}{{\partial r}} + k\frac{r}{f} - k\frac{a}{f}\left( {\frac{r}{b}} \right)^{ - \frac{1}{n}}\cos \left( {\vartheta - \frac{\vartheta }{n}} \right) = 0} \\ {\frac{{\partial \Omega _{S, n}}}{{\partial \vartheta }} + k\frac{{ar}}{f}\left( {\frac{r}{b}} \right)^{ - \frac{1}{n}}\sin \left( {\vartheta - \frac{\vartheta }{n}} \right) = 0} \end{array}} \right. $$ (5) After integrating, we obtain the analytical expression for the phase function in the paraxial regime:
$$ \Omega _{S, n}(r, \vartheta ) = \frac{{2\pi ab}}{{\lambda f}}\left( {\frac{r}{b}} \right)^{1 - \frac{1}{n}} \cdot \frac{{\cos \left[ {\vartheta \left( {1 - \frac{1}{n}} \right)} \right]}}{{1 - \frac{1}{n}}} - k\frac{{r^2}}{{2f}} $$ (6) The corresponding phase corrector can be calculated by applying the same method to the inverse transformation. We obtain:
$$\Omega _{PC, n}(\rho , \varphi ) = \frac{{2\pi ab}}{{\lambda f}}\left( {\frac{\rho }{a}} \right)^{1 - n} \cdot \frac{{\cos \left[ {\varphi \left( {1 - n} \right)} \right]}}{{1 - n}} - k\frac{{\rho ^2}}{{2f}} $$ (7) which is basically the expression in Eq. (6) after the substitutions b↔a, n→1/n, and (r, ϑ)→(ρ, φ). Further details regarding the calculations of Eq. (6) and Eq. (7) are provided in Supplementary Material S1.
In Fig. 1a, d, schematics of the OAM beam evolution in the case of circular-sector transformations by a factor of n = 2 and n = 3 are depicted, respectively. The phase gradient of the input OAM beam is mapped over half and one-third of the entire circle, respectively. Numerical simulations are also reported (Fig. 1c, f), showing the evolution of the beam after illuminating the first optical element and during propagation in the range of the focal length, where the second element is placed for phase correction.
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OAM multiplication by a factor n basically consists of splitting the input beam into n copies and mapping each one over a set of n complementary circular sectors. Therefore, the phase pattern ΩM, n of the multiplier is described by the superposition of n circular-sector transformations {ΩS, n(j)}, j = 1, …, n, mapping the input beam over the corresponding circular sectors with amplitude 2π/n and centered on {(j - 1)2π/n}:
$$ \Omega _{M, n}(r, \vartheta ) = \arg \left\{ {\mathop {\sum}\limits_{j = 1}^n {e^{i\Omega _{S, n}^{(j)}}} } \right\} $$ (8) where:
$$ \Omega _{S, n}^{\left( j \right)}(r, \vartheta ) = \frac{{2\pi ab}}{{\lambda f}}\left( {\frac{r}{b}} \right)^{1 - \frac{1}{n}} \cdot \frac{{\cos \left[ {\vartheta \left( {1 - \frac{1}{n}} \right) + \left( {j - 1} \right)\frac{{2\pi }}{n}} \right]}}{{1 - \frac{1}{n}}} - k\frac{{r^2}}{{2f}} $$ (9) As demonstrated in Supplementary Material S2, the phase corrector of an n-fold circular transformation does not change if a shift of j2π/n, where j is an integer value, is performed on the azimuthal coordinate after rescaling. Therefore, all the circular-sector transformations {ΩS, n(j)} described by Eq. (9) share the same pattern for the phase corrector, which is given again by Eq. (7). In fact, the phase-corrector phase patterns in Fig. 2b.2 and Fig. 2e.2 of the twofold and threefold multipliers, respectively, are the same as those shown in Fig. 1b.2 and Fig. 1e.2 for the constituent twofold and threefold circular-sector transformations, respectively.
Fig. 2 Multiplication of OAM beams.
Schematics of OAM beam optical multipliers for twofold (n = 2) a and threefold (n = 3) d multiplication with transformation optics. The first phase pattern (b.1, e.1) performs an n-fold multiplication, splitting, and mapping the input azimuthal phase gradient over n complementary arcs. The second phase pattern (b.2, e.2) performs phase correction, retaining the azimuthal phase distribution. Design parameters: a = 300 μm, b = 250 μm, and f = 20 mm. Numerical simulations of the propagation of an input Laguerre-Gaussian beam carrying ℓ = 2 (c.1, f.1), after illuminating the first element (c.2, f.2), at different z positions (z = 0.2f (c.3, f.3), z = 0.4f (c.4, f.4), z = 0.6f (c.5, f.5), z = 0.8f (c.6, f.6), up to the second optical element (c.7, f.7), and the output phase-corrected beam (c.8, f.8). Colors and brightness refer to the phase and intensity, respectively.In Fig. 2a, d, schematics of the beam evolution in the case of multiplication by a factor of n = 2 and n = 3 are depicted, respectively. In Fig. 2c, f, the corresponding numerical simulations are reported, providing the evolution of the beam in the range of the focal length, where the second element is placed for the phase correction. As expected, the input beam is split into n equal contributions that are mapped onto complementary arcs, thus performing an n-fold multiplication of the input OAM. A conversion efficiency close to the unit can be achieved, which gradually decreases with increasing input OAM value (Supplementary Material S4) owing to a slight distortion of the output intensity ring distribution. In Fig. 2f.8, a slight deformation of the output beam can be observed, the shape of which is not perfectly axially symmetric. As discussed and demonstrated in detail in Supplementary Material S3, this distortion is caused by the twisted wavefront of the input OAM beam. In fact, the phase patterns in Eqs. (6) and (7) are calculated in the stationary phase approximation, assuming a uniform phase front in the input. On the other hand, OAM beams are endowed with a peculiar azimuthal phase, which affects the final intensity distribution of the transformed beam. However, as shown in the Supplementary Material, this effect can be mitigated and the conversion efficiency can be optimized by properly designing the transformation optics in terms of the focal length f and design parameters a and b.
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It could be interesting to divide the input beam into a bunch of n beams with OAMs equal to 1/n the OAM of the input beam, as depicted in Fig. 3a, d, for the case of twofold (n = 2) and threefold (n = 3) division, respectively. The OAM division operation can be understood considering OAM multiplication, e.g., Fig. 2d, in reverse. In Fig. 2d, the output threefold OAM beam is the combination of three complementary circular sectors of size 2π/3 that arise from the same number of combined threefold circular-sector transformations of the input beam (threefold multiplier). Thus, the input OAM beam carrying ℓ = 2 is split into three copies that are un-wrapped and projected onto three complementary arcs to achieve the final ℓ = 6 OAM beam. Considering the optical path in reverse, the three complementary circular sectors with size 2π/3 forming the OAM beam with ℓ = 6 are wrapped by a circular-sector transformation with n = 1/3 onto three OAM beams carrying the OAM value ℓ/3 = 2. By applying different tilt angles to the three arcs, it is possible to obtain three non-overlapping OAM beams as the output, as shown in Fig. 3d. Generalizing this approach to a division into n OAM beams, this is possible by defining the phase pattern of the OAM divider in the following way:
$$ \Omega _{D, n}(r, \vartheta ) = \arg \left\{ {\mathop {\sum}\limits_{j = 1}^n {e^{i\Omega _{S, 1/n, j}\left( {r, \vartheta } \right)}e^{ir \cdot \beta ^{(j)}\cos \left( {\vartheta - \vartheta ^{(j)}} \right)}} \Phi _n^{(j)}\left( \vartheta \right)} \right\} $$ (10) Fig. 3 Division of OAM beams.
Schematics of OAM-beam optical dividers for twofold (n = 2) a and threefold (n = 3) d division with transformation optics. The first phase pattern (b.1, e.1) performs an n-fold division, splitting the input azimuthal gradient into n complementary arcs that are wrapped and mapped over distinct whole circles at desired positions. The second phase pattern (b.2, e.2) performs phase correction, retaining the azimuthal phase distribution for each output beam. Design parameters: a = 300 μm, b = 300 μm, f = 20 mm, and spatial frequency β(j) = 0.3 μm-1, j = 1, .., 3. Numerical simulations of the propagation of an input Laguerre-Gaussian beam carrying ℓ = 4 (c.1) or ℓ = 6 (f.1), after illuminating the first element (c.2, f.2), at different z positions (z = 0.2 f (c.3, f.3), z = 0.4f (c.4, f.4), z = 0.6 f (c.5, f.5), z = 0.8 f (c.6, f.6), up to the second optical element (c.7, f.7), and the output phase-corrected beams (c.8, f.8). Colors and brightness refer to the phase and intensity, respectively.where ΩS, 1/n, j is the phase pattern of a 1/n-fold circular-sector transformation centered on (j - 1)2π/n, j = 1, …, n:
$$ \Omega _{S, 1/n, j}(r, \vartheta ) = \frac{{2\pi ab}}{{\lambda f}}\left( {\frac{r}{b}} \right)^{1 - n} \cdot \frac{{\cos \left[ {\left( {\vartheta - \left( {j - 1} \right)\frac{{2\pi }}{n}} \right)\left( {1 - n} \right)} \right]}}{{1 - n}} - k\frac{{r^2}}{{2f}} $$ (11) and Фn(j) is a phase mask selecting the desired circular sector with its center at (j - 1)2π/n and a size of 2π/n:
$$ \Phi _n^{(j)}\left( \vartheta \right) = \Theta \left( {\vartheta - \frac{{2\pi }}{n}(j - 1) + \frac{\pi }{n}} \right)\Theta \left( {\frac{{2\pi }}{n}(j - 1) + \frac{\pi }{n} - \vartheta } \right) $$ (12) where Θ(·) is the Heaviside function (Θ(x) = 1 when x ≥ 0 and Θ(x) = 0 otherwise). According to Eq. (10), the total phase pattern of the OAM divider is basically the composition of n complementary and non-overlapping phase patterns, each defined over a circular sector spanning an angle equal to 2π/n and centered on (j - 1)2π/n, j = 1, …, n. Specifically, each zone is illuminated by 1/n of the input beam, and it is designed to wrap the impinging arc around the entire circle (1/n-fold circular-sector transformation). In addition, spatial frequency carriers {(β(j), ϑ(j))} are introduced in Eq. (10), to spatially separate the n output beams and locate them at defined positions (ρ(j), ϕ(j)) in the focal plane according to:
$$ \left\{ {\begin{array}{*{20}{c}} {\rho ^{(j)} = \frac{f}{k}\beta ^{(j)}} \\ {\phi ^{(j)} = \vartheta ^{(j)} = (j - 1)\frac{{2\pi }}{n}} \end{array}} \right. $$ (13) where j = 1, …, n and f is the focal length of the optical transformation. Then, the phase corrector for the divider in Eq. (10) is given by the composition of n complementary and non-overlapping phase correctors, each defined over a circular sector spanning an angle equal to 2π/n and centered at the positions given by (ρ(j), ϕ(j)):
$$ \begin{array}{l}\Omega _{D - PC, n}(\rho , \varphi ) = \\ \arg \left\{ {\mathop {\sum}\limits_{j = 1}^n {e^{i\Omega _{PC, 1/n, j}\left( {\rho \prime _j, \varphi \prime _j} \right)}} \Phi _n^{(j)}\left( \varphi \right)} \right\}\end{array} $$ (14) where (ρ'j, φ'j) are polar coordinates centered on (ρ(j), ϕ(j)) given by Eq. (13), and:
$$ \begin{array}{l}\Omega _{PC, 1/n, j}\left( {\rho \prime _j, \varphi \prime _j} \right)\\ = \frac{{2\pi ab}}{{\lambda f}}\left( {\frac{{\rho \prime _j}}{a}} \right)^{1 - \frac{1}{n}} \cdot \frac{{\cos \left[ {\left( {\varphi \prime _j - \left( {j - 1} \right)\frac{{2\pi }}{n}} \right)\left( {1 - \frac{1}{n}} \right)} \right]}}{{1 - \frac{1}{n}}} - k\frac{{\rho ^2}}{{2f}}\end{array} $$ (15) In Fig. 3c, f, numerical calculations are reported in the case of n = 2 and n = 3, showing the evolution of the beam in the range of the focal length. As expected, the input beam is split into n equal contributions that are mapped onto distinct circles, thus performing an n-fold division of the input OAM into a set of n OAM beams. In principle, the OAM divider can be generalized and properly designed to decompose the input OAM into many beams carrying different values of the OAM and whose total sum equals the OAM value of the input beam.
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A limit of performing optical operations with transformation optics is the need for at least two optical elements, i.e., transformer and phase corrector. As widely experienced in the case of OAM sorting with a log-pol transformation, the presence of two confocal elements to be perfectly aligned, coaxial and coplanar, could require arduous efforts for alignment and could be detrimental to miniaturization and integration. To simplify the alignment process and improve the compactness of the optical architecture, we designed the optical configuration to incorporate the two elements onto a single optical platform. In this configuration, the substrate is illuminated twice: after crossing the first zone, while performing multiplication/division, the beam is back-reflected with a mirror, placed at half the focal length, and made to impinge on the second region providing the phase correction. By adding a tilt term to the first phase pattern, the back-reflected beam does not overlap with the input one, and the two optical elements, i.e., transformer and phase corrector, can be fabricated side-by-side on the same substrate. This solution makes the alignment operation significantly easier, as the number of degrees of freedom is remarkably reduced and the two elements are by-design aligned and parallel to each other. We fabricated compact diffractive optics performing either twofold or threefold multiplication and division. The focal length was fixed to 20 mm; therefore, the distance between the optics and the mirror was reduced to 1 cm. Adding to the first pattern a spatial frequency β of ~0.5 μm-1, the centers of the two phase elements were found to be ~1 mm from each other. For the multiplier, this result is achieved by adding a phase term βx to the phase in Eq. (8) and centering the phase corrector at a distance βf/k from the multiplier center. As far as the divider is concerned, as shown in Eq. (10), spatial frequency carriers are already included to separate the different OAM beams in the output; therefore, it is sufficient to choose the moduli β(j) to avoid overlap of the phase-correcting zones with the divider pattern. The radius of the first zone was set at approximately 600 μm, whereas the radii of the phase correctors were chosen to avoid overlapping the first pattern (see Fig. 4 and Fig. 5). In more detail, for both multipliers, we set the parameters a and b to 300 μm and 250 μm, respectively, whereas for the dividers we chose the values a = 250 μm and b = 400 μm.
Fig. 4 Diffractive optics for twofold multiplication.
Compact twofold multiplier: inspection via optical microscope (a, c.2—top view) and SEM analysis (b, c.1—tilted views). (b.1) SEM inspection of the multiplier central zone and details (b.2, b.3). (c.2) Dark-field optical analysis of the phase-corrector central region, and further details obtained via SEM (c.1). -
The designed optical elements were fabricated as surface-relief phase-only diffractive optics using high-resolution electron-beam lithography (EBL) over a resist layer. By locally controlling the released electronic dose, a different dissolution rate is induced point-by-point on the exposed polymer with a resolution of few nanometers, giving rise to a spatially variant resist thickness after the development process38. A dose-depth correlation curve (contrast curve) is required to calibrate the correct electron dose to assign to obtain the desired thickness. For a phase pattern Ω(x, y), the depth d(x, y) of the exposed zone for normal incidence in air is given by:
$$ d\left( {x, y} \right) = \frac{\lambda }{{n_R\left( \lambda \right) - 1}} \cdot \frac{{2\pi - \Omega \left( {x, y} \right)}}{{2\pi }} $$ (16) where λ is the incident wavelength and nR(λ) is the corresponding refractive index of the resist. In this work, all diffractive optics were fabricated by patterning a layer of negative resist (AR-N 7720.30, Allresist) spin-coated onto a 1.1 mm-thick ITO-coated soda lime float glass substrate. At the experimental wavelength of the laser (λ = 632.8 nm), the refractive index of the resist polymer was assessed to be nR = 1.679. From Eq. (16), the maximal depth of the surface-relief pattern was found to be 928.5 nm, with a thickness resolution of Δd = 3.64 nm. The quality of the fabricated phase-only optical elements was assessed with scanning electron microscopy (SEM) and optical microscopy. In Fig. 4, SEM inspections are reported, referring to the compact two fold multiplier. In Fig. 5, the inspections refer to the compact threefold divider.
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The optical response of the fabricated optical elements has been analyzed for illumination under integer OAM beams generated with an LCoS spatial light modulator (SLM) using a phase and amplitude modulation technique39. Once properly filtered and resized, the OAM beam illuminated the first zone of the sample, performing either n-fold multiplication or division, as shown in the experimental layout of the optical bench in Fig. 6a. A mirror was placed on a kinematic mount, and its position could be finely controlled with a micrometric translator to set the distance from the multiplier/divider to equal to half the focal length of the first element, i.e., 1 cm. After back reflection, the transformed beam illuminated the optics again on the phase-correcting zones, as shown in the schemes in Fig. 6b, and the far-field was collected by a second camera placed at the back focal plane of a Fourier lens. To check the OAM content of the output beams, a Mach-Zehnder interferometric bench was added to the described optical setup, as depicted in Fig. 6a.
Fig. 6 Scheme of the experimental setup.
a Scheme of the experimental setup used for the optical characterization of the fabricated dividers and multipliers. The laser beam is linearly polarized (P) and expanded (f1 = 3.5 cm, f2 = 12.5 cm). A beam splitter (BS) is used to generate a reference Gaussian beam for the interferometric bench. The SLM first order is filtered (D) and resized (f3 = 25.0 cm, f4 = 12.5 cm) before illuminating the optical element as in b. A mirror (M) is used for back-reflection onto the phase-correcting pattern. A second beam splitter is used both to check the input beam and collect the output on two different cameras. The second camera is placed on the back focal plane of a fifth Fourier lens (f5 = 10.0 cm), and it is used to collect the output beam intensity and its interferogram (f6 = 20.0 cm). b Compact optical configurations for the twofold multiplier and the twofold divider. The two elements, i.e., tilting divider/multiplier and phase corrector, are placed on the same substrate, and a mirror (M) is used for back-reflection.The optical response of the fabricated optics was tested for the multiplication and division of optical beams with integer OAM values. The OAM of the output beam was analyzed from the interference pattern, with a reference arm obtained by splitting the Gaussian output of the laser. By counting the number of arms in the generated spiralgrams, it is possible to infer the OAM of the output beam and prove the expected multiplication or division of the initial OAM state. The optical characterization of the fabricated samples confirmed the expected capability to perform optical multiplication and division of the OAM of the input beam.
In Fig. 7, the optical characterization is reported for the twofold multiplier. The optical response was characterized for input OAM beams with ℓ in the range from -4 to +4 (Fig. 7a), 0 excluded. As expected, the interference pattern of the output beam shows twice the number of spiral arms of the input interference pattern, confirming the duplication of the input OAM value (Fig. 7c). The same analysis was performed for the threefold multiplier for input OAM values in the range from -3 to +3, 0 excluded (Fig. 8a). As shown in Fig. 8c, the number of output arms in the interferograms is three times the number of input arms, as expected.
Fig. 7 Optical characterization of the twofold multiplier for input OAM beams with ℓ values in the range from -4 to +4, 0 excluded.
a Interferogram of the input OAM beams. Intensity pattern b and interferogram c of the output beam. As expected, the number of spiral arms in the output interference pattern is equal to twice the number in the input interferogram.Fig. 8 Optical characterization of the threefold multiplier for input OAM beams with ℓ values in the range from -3 to + 3, 0 excluded.
a Interferogram of the input OAM beams. Intensity pattern b and interferogram c of the output beam. As expected, the number of spiral arms in the output interference pattern is equal to three times the number in the input interferogram.In Fig. 9, the experimental output of the twofold divider is reported for input OAM beams with even ℓ values in the range from -8 to +8, 0 excluded (Fig. 9a). As expected, the input beam is split into two output beams (Fig. 9b), with an OAM equal to half the input value. In Fig. 9c, the interferogram is reported for only one out of the two beams, and the interference pattern confirms the capability of the designed optics to divide the input OAM into two. The same analysis was performed for the threefold divider, as reported in Fig. 10. In this case, the output is composed of three OAM beams (Fig. 10b), each carrying one-third of the input OAM, as demonstrated by the experimental interferograms (Fig. 10c) for input ℓ in the range from -9 to +9, with a step of 3, 0 excluded (Fig. 10a).
Fig. 9 Optical characterization of the twofold divider for input OAM beams with even ℓ values in the range from -8 to +8, 0 excluded.
a Interferogram of the input OAM beams. Intensity pattern b and interferogram c of the output beam. As expected, the number of spiral arms in the output interference pattern is equal to half the number in the input interferogram.Fig. 10 Optical characterization of the threefold divider for input OAM beams with ℓ values in the range from -9 to +9, with a step of 3, 0 excluded.
a Interferogram of the input OAM beams. Intensity pattern b and interferogram c of the output beam. As expected, the number of spiral arms in the output interference pattern is equal to one-third the number in the input interferogram.