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Orbital angular momentum mode sorting based on a hybrid radial-angular hybrid lens

Junhe Zhou, Haoqian Pu, and Qiqi Wang

Open Access Open Access

Abstract

Orbital angular momentum (OAM) modes have their phase distribution as exp (jlθ), which resembles the plane wave in the Cartesian coordinates. Like the traditional lens, which can focus the plane wave on the focal plane, the angular lens can focus the OAM beam in the angular domain, albeit with a relatively long tail due to the unsatisfied angular focal condition for the non-ring shape beams. In this paper, a hybrid lens in the angular domain and the radial domain is proposed. The radial lens with the specific radially distributed phase guarantees the angular focal condition is met for the beams with an arbitrary beam waist or radial field distribution, which significantly improves the performance for the OAM modes sorting by the angular lens. The discrimination of the different OAM modes can be achieved efficiently based on such a single optical component, i.e., the proposed hybrid radial-angular lens, with the OAM modes inter-mode crosstalk as 3.7% when the topological charge difference is 3.

© 2022 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement

1. Introduction

Orbital angular momentum multiplexing (OAMM) has become one of the most promising techniques to further increase the optical communication system capacity [1]. Apart from optical communication [1], it can also be used for quantum information [2] and optical signal processing [3–4].

While the OAM mode generation can be accomplished via spiral phase plates (SPPs) [5], coherent combining with circularly arranged cores [6], ring resonators [7], spin angular momentum to OAM conversion [8], OAM modes to fiber modes index matching etc. [9], OAM mode sorting remains highly attractive and challenging, because OAM mode sorting enables intensity detection, which is still the main stream detection technique unless the expensive wave-front sensors are used. Numerous methods have been proposed to address the topic of OAM mode sorting, such as the interference method [10–11], the SPP/hologram or specific aperture-based diffraction methods [2,12,13], and the conformal transform method [14–15] etc. These methods have their unique advantages, but also have drawbacks. For example, the widely used SPP/hologram approach has a very high conversion/detection efficiency for the OAM modes, but one OAM mode will require a unique SPP, which makes the high volume multiplexing an expensive and difficult task. The conformal transform approach requires two elements. The first is used to complete the log-polar transform and the second is used to achieve phase correction. Recently, a method with an angular lens was proposed to sort the OAM modes [16–17]. The ingenious approach uses a lens with the parabola index distribution in the angular domain, and the OAM modes can be focused after arriving at the “angular focal plane”. The main drawback for the angular lens is that it is more suitable to sort the OAM modes with a specific radius [17], the performance and efficiency degrade as the beam radius does not meet the designed target, particularly for the non-ring-shape beams.

In this contribution, we propose a novel approach for the OAM modes discrimination. The idea is to the use the angular lens in combination with a radial lens, which can focus the beams with the OAM topological charges in the angular domain while avoiding the shortcoming of the previously reported pure angular lens [16–18]. The radial lens focuses the beam to a specific radius on the output plane, which is inversely proportional to the radius on the input plane. Such an arrangement will enable the waves with different radii on the input plane and the output plane to satisfy the angular lens focusing condition. The sorting efficiency and performance are significantly improved by implementing the proposed hybrid lens. With the proposed hybrid lens, it is possible to realize single-optical-component based OAM mode sorting. It should be noted that the hybrid lens is different from the spin–orbit transformation meta lens in [19] and the multi-focal hybrid lens in [20], as it has only one focal length and does not involve spin–orbit conversion.

2. Theory

The scalar beam propagation in free space can be characterized by

$$E({\rho ,\varphi ,z} )= \frac{{\exp ({jkz} )}}{{j\lambda z}}{e^{j\frac{k}{{2z}}{\rho ^2}}}\int\limits_0^{ + \infty } {\int\limits_0^{2\pi } {E({{\rho_1},{\varphi_1},0} )} } T({{\rho_1},{\varphi_1}} ){e^{j\frac{k}{{2z}}{\rho _1}^2}}{e^{ - j\frac{{k{\rho _1}\rho }}{z}\cos ({\varphi - {\varphi_1}} )}}{\rho _1}d{\rho _1}d{\varphi _1},$$
where ρ1, φ1, ρ, φ, are the polar coordinates on the input and output planes, z is the propagation distance, k is the wave number, λ is the wavelength, and T is the phase introduced by the lens.

If the phase of the lens forms an angular lens, i.e.,

$$T({{\rho_1},{\varphi_1}} )= \exp \left( { - j\frac{N}{{4\pi }}{\varphi_1}^2} \right)\exp ({ - j\beta {\rho_1}} ),$$
where N is a scaling factor and β is a radial correction factor for the angular lens [16], one may expect the azimuthal convergence for the OAM modes. It is shown in the appendix (the correction factor is ignored in the analysis) that a ring shape beam with the radius of r1 on the input plane will concentrate in a circular sector with the radius of r2 on the output plane at the focal length f,
$$f = \frac{{2\pi k{r_1}{r_2}}}{N}.$$
The relative angle of the sector depends on the topological charge, i.e., the output wave concentrates on the sector with the radius r2 and rotates with the change of the topological charge.

When a ring shape beam with the radius of a is considered at the input and the output ring has the same radius a, $f = \frac{{2\pi k{a^2}}}{N}$. This is in accordance with the Eq. (6) in [17]. The rigorous and detailed derivation of Eq. (3) is shown in the appendix.

Equation (3) only works for the ring shape beams both on the input and output planes. In real applications, the input and the output beams usually do not have a single-ring shape and the different rings on the input and output planes have different angular focal lengths, and therefore, the performance is significantly degraded. Hence, the focused OAM modes have long tails [16–17].

We propose to implement the radial direction ring-lens [21] to concentrate the output beam on a ring shape area in combination with the angular lens. If the phase of lens forms a radial lens, i.e.,

$$T({{\rho_1},{\varphi_1}} )= \exp \left( { - j\frac{k}{{2f}}{{({{\rho_1} - a} )}^2}} \right).$$
At the focal length f, the beam $E({\rho ,\varphi ,f} )$ will concentrate as a ring with the radius of a. The combination of the radial lens and the angular lens will result in better OAM angular focusing in comparison to the pure angular lens, because the output beam has a definite ring radius.

The simulated results for the angular lens and the simple hybrid radial-angular lens are shown in Fig. 1. It can be seen from the figures that when the input beam is with the Gaussian shape, the output beam has a long tail if the pure angular lens is used. With the simple hybrid radial-angular lens, the tail will be reduced. However, the beam still has some side lobes on the output plane. This is because the input beam is not a ring shape beam. The waves with different radii at the input tend to have different focal lengths as suggested by Eq. (3).

 figure: Fig. 1.

Fig. 1. Simulation of the mode sorter with the simple hybrid lens with the angular lens and the radial direction ring lens. (a) The input beam is the Gaussian shape beam (b) the focus beam with a radial lens (c) the focused beam with a pure angular lens (d) the focused beam with the simple hybrid radial-angular lens. The parameters are as follows. The wavelength of the beam is 671 nm, the beam waist w0 is 1.9mm, the parameter N is 50, a is 1.9 mm, and β is 2/mm.

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In order to further improve the performance, i.e., to make Eq. (3) work for the waves with arbitrary radii on the input plane, we propose the focal radius of the ring a to be dependent on the radius on the input plane.

$$a({{\rho_1}} )= \frac{{r_0^2}}{{{\rho _1}}}.$$
In this way, the wave on the input beam with the radius of ${\rho _1}$ will concentrate on the ring with the radius of $\frac{{r_0^2}}{{{\rho _1}}}$ on the output plane. Therefore, one may expect the wave at different radii to have the same focal length $f = \frac{{2\pi ka({{\rho_1}} ){\rho _1}}}{N} = \frac{{2\pi k{r_0}^2}}{N}$.

The above-mentioned improved hybrid radial-angular lens has its working principle as shown in Fig. 2. By adopting the input radius dependent output focal radius, it is possible to guarantee the product of the wave radii on the input and output planes to be a constant, i.e., r02, so that Eq. (3) will work for a non-ring shape OAM beam.

 figure: Fig. 2.

Fig. 2. illustration of the focal condition for the improved hybrid radial-angular lens

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In summary, the phase shift for the improved hybrid lens is

$$\begin{array}{l} T({{\rho_1},{\varphi_1}} )= \exp \left( { - j\frac{k}{{2f}}{{\left( {{\rho_1} - \frac{{r_0^2}}{{{\rho_1}}}} \right)}^2}} \right)\exp \left( { - j\frac{N}{{4\pi }}{\varphi_1}^2} \right)\exp ({ - j\beta {\rho_1}} ),\\ f = \frac{{2\pi k{r_0}^2}}{N}. \end{array}$$
With the single optical component possessing the phase mask as Eq. (6), it is possible to realize OAM mode sorting efficiently.

3. Results and discussions

The OAM mode sorter with the proposed improved hybrid radial-angular lens is tested experimentally. The experimental setup is shown in Fig. 3.

 figure: Fig. 3.

Fig. 3. Experimental setup. The 671nm laser has its beam expanded by an expander. A helical wavefront is added to the first SLM, which converts the beam into an OAM mode. The OAM mode is sorted by the second SLM which has the phase of the improved radial-angular lens on it. SLM: spatial light modulator. BS: beam splitter. CCD camera: charge coupled device camera

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A continuous wave laser at the wavelength of 671 nm is used as the light source. The laser beam is magnified by a laser-beam-expander and its intensity is adjusted by an attenuator. The laser beam is used to illuminate a SLM with the pixel size of 9.4 µm × 9.4 µm, and the SLM has the grid as 800 × 600. The beam acquires the helical wavefront from the first spatial light modulator (SLM) and becomes an OAM mode. The second SLM has its phase as Eq. (6), which is shown in Fig. 4, and becomes the proposed improved hybrid radial-angular lens. The parameters are as follows: N is 50, r0 is 0.6 mm, and β is 2/mm.

 figure: Fig. 4.

Fig. 4. Illustration of the phase distribution of the improved radial-angular hybrid lens. The parameters are as follows. The wavelength of the beam is 671 nm, the parameter N is 50, r0 is 0.6 mm, and β is 2/mm.

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With such a phase mask, it is possible to sort the OAM modes, whose output images are captured by a charge coupled device (CCD) camera as is shown in Fig. 5. The CCD camera has the grid as 2160 × 4096 and its grid size as 3.45 µm × 3.45 µm. As expected, the output wave concentrates on the sector with the center angle to be 0 if the input beam has the topological charge of 0. When the topological charge increases, the output beam rotates anti-clockwise. When the topological charge decreases, the output beam rotates clockwise. In comparison to the mode sorting performance by the pure angular lens as shown in Fig. 1 and Ref. [16–17], the sorted beam has a pencil shape and does not have a long tail in the azimuth direction [16–17]. Since the OAM modes with different topological charges occupy different sectors, the mode sorting process can be accomplished by conducting the measurement within the specific sectors, which is similar to [14] and is different from the cases with relatively complex algorithms in [22–23]. Hence, the proposed improved hybrid radial-angular lens does function as an efficient OAM mode sorter. If the beam is formed by the combination of two OAM modes with two different topological charges, the proposed OAM mode sorter can separate them in the azimuth domain. In the experiment, the combination of the OAM modes with the topological charges of -2 and +2 is generated by the first SLM, which has its pixels to be randomly filled with the two phase-patterns for equal probability [24]. As is shown on the bottom of Fig. 5, the beam formed by the combination of the +/-2 OAM modes is efficiently converted into two pencil-shape beams with two different rotation angles. In case the factional order OAM mode is used as the input, there will be multiple pencil-shape beams on the output plane, because the fractional order OAM modes can be decomposed as a series of integer order OAM modes, as is known and indicated in [25–26].

 figure: Fig. 5.

Fig. 5. Mode sorting with the proposed hybrid lens for the different OAM modes. The images are captured by the CCD camera.

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To quantitively investigate the performance of the proposed OAM sorter, the crosstalk matrix is shown in Fig. 6. The crosstalk matrix is evaluated as follows: the detector plane is divided into different sectors, with each corresponding to one OAM mode output. The ratio of the sorted mode power within the sector is defined as the correct detection probability of the mode. Due to the mode overlap at the output, the average correct detection probability for each mode is 49%. The average crosstalk between the neighboring OAM modes is 21%. The correct detection probability can be significantly increased to 92.5% and the neighboring mode crosstalk level can be significantly reduced to 3.7% if the topological charge separation of the OAM modes is 3 as is shown in Fig. 7. Such performance is far beyond the mode sorting capability for the pure angular-lens-based OAM mode sorter [16], which has the average crosstalk level of 16.5% with the topological charge separation of the OAM modes as 4.

 figure: Fig. 6.

Fig. 6. The table of crosstalk for different OAM modes

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 figure: Fig. 7.

Fig. 7. The table of crosstalk for different OAM modes with the topological charge separation of 3.

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To accurately sort the OAM modes without topological charge separation, i.e., the OAM mode …-2, -1, 0, 1, 2, … one may implement the OAM mode multiplier [4], which can convert the topological charge l to Ml, with M being an integer number, e.g, M = 3 and the OAM modes will be converted to …-6, -3, 0, 3, 6, … In this way, very high correct detection probability (92.5%) and low crosstalk (3.7%) can be achieved.

4. Conclusion

In summary, we have proposed a new hybrid lens which can focus the OAM modes in the angular direction. The hybrid lens consists of an angular lens with the quadratic phase in the azimuth domain and the radial lens whose focal radius is inversely proportional to the input wave radius. The hybrid lens can provide efficient OAM mode sorting with the pencil-shape output beam, whose azimuth angle rotates anti-clockwise as the topological charge increases. When the mode topological charge separation is 3, the crosstalk between the neighboring modes is only 4%. In comparison with the existing OAM mode sorters, only one optical component, i.e., the proposed hybrid radial-angular lens, is required during the sorting process, which significantly simplifies the OAM mode sorting system architecture and reduces the cost. An OAM multiplier can be inserted before the hybrid lens to further improve the performance. The hybrid lens can be realized by the SLM as is shown in this work, and it can also be realized through other means, such as the approaches by the holograms or the meta-lenses.

Appendix

In this appendix, the detailed derivation for the angular lens equation with the specific input ring and output ring radii (i.e., Eq. (3) in the main text) is provided. In addition, the radial phase distribution for the improved angular-radial lens is derived (i.e., Eq. (6) in the main text).

The scalar beam propagation in free space can be characterized by Eq. (1) (rewritten as Eq. (7))

$$E({\rho ,\varphi ,z} )= \frac{{\exp ({jkz} )}}{{j\lambda z}}{e^{j\frac{k}{{2z}}{\rho ^2}}}\int\limits_0^{ + \infty } {\int\limits_0^{2\pi } {E({{\rho_1},{\varphi_1},0} )} } T({{\rho_1},{\varphi_1}} ){e^{j\frac{k}{{2z}}{\rho _1}^2}}{e^{ - j\frac{{k{\rho _1}\rho }}{z}\cos ({\varphi - {\varphi_1}} )}}{\rho _1}d{\rho _1}d{\varphi _1},$$
where ρ1, φ1, ρ, φ, are the polar coordinates on the input and output planes, z is the propagation distance, k is the wave number, λ is the wavelength, and T is the phase introduced by the lens and is assumed as the radially quadratic phase
$$T({{\varphi_1}} )= \exp \left( { - \frac{N}{{4\pi }}{\varphi_1}^2} \right),$$
where N is a scaling factor. Without loss of generality, we assume N to be an even integer. Since,
$$exp ({ - jz\cos \varphi } )= \sum\limits_{n ={-} \infty }^{ + \infty } {{{({ - j} )}^n}{e^{jn\varphi }}{J_n}(z )} ,$$
and the approximated formula for Bessel function when z<<1
$$\begin{array}{l} {J_n}(z )\approx \sqrt {\frac{2}{{\pi z}}} \left( {\cos \left( {z - \frac{{2n + 1}}{4}\pi } \right) - \sin \left( {z - \frac{{2n + 1}}{4}\pi } \right)\frac{{({4{n^2} - 1} )}}{4}\frac{1}{{2z}}} \right)\\ = \sqrt {\frac{2}{{\pi z}}} Re \left( {{e^{j\left( {z - \frac{{2n + 1}}{4}\pi } \right)}}\left( {1 + j\frac{{({4{n^2} - 1} )}}{4}\frac{1}{{2z}}} \right)} \right)\\ \approx \frac{1}{2}\sqrt {\frac{2}{{\pi z}}} \left( {{e^{j\left( {z - \frac{{2n + 1}}{4}\pi } \right)}}{e^{j\frac{{({4{n^2} - 1} )}}{4}\frac{1}{{2z}}}} + {e^{ - j\left( {z - \frac{{2n + 1}}{4}\pi } \right)}}{e^{ - j\frac{{({4{n^2} - 1} )}}{4}\frac{1}{{2z}}}}} \right), \end{array}$$
we have
$$\begin{array}{l} E({\rho ,\varphi ,z} )\approx \frac{{\exp ({jkz} )}}{{j\lambda z}}\int\limits_0^{ + \infty } {\int\limits_0^{2\pi } {E({{\rho_1},{\varphi_1},0} )} } \exp \left( { - \frac{N}{{4\pi }}{\varphi_1}^2} \right)\sqrt {\frac{1}{{2\pi \frac{{k{\rho _1}\rho }}{z}}}} \\ \times \sum\limits_{n ={-} \infty }^{ + \infty } {{e^{jn({\varphi - {\varphi_1}} )}}\left( {{{({ - 1} )}^n}{e^{ - j\frac{\pi }{4}}}{e^{j\frac{{k{{({{\rho_1} + \rho } )}^2}}}{{2z}}}}{e^{j\frac{{({4{n^2} - 1} )}}{4}\frac{z}{{2k{\rho_1}\rho }}}} + {e^{j\frac{\pi }{4}}}{e^{j\frac{{k{{({{\rho_1} - \rho } )}^2}}}{z}}}{e^{ - j\frac{{({4{n^2} - 1} )}}{4}\frac{z}{{2k{\rho_1}\rho }}}}} \right)} {\rho _1}d{\rho _1}d{\varphi _1}\\ \approx \frac{{\exp ({jkz} )}}{{j\lambda z}}\int\limits_0^{ + \infty } {\int\limits_0^{2\pi } {E({{\rho_1},{\varphi_1},0} )} } \exp \left( { - \frac{N}{{4\pi }}{\varphi_1}^2} \right)\sqrt {\frac{1}{{2\pi \frac{{k{\rho _1}\rho }}{z}}}} \sum\limits_{n ={-} \infty }^{ + \infty } {{e^{jn({\varphi - {\varphi_1}} )}}{e^{j\frac{\pi }{4}}}{e^{j\frac{{k{{({{\rho_1} - \rho } )}^2}}}{z}}}{e^{ - j\frac{{({4{n^2} - 1} )}}{4}\frac{z}{{2k{\rho _1}\rho }}}}} {\rho _1}d{\rho _1}d{\varphi _1}. \end{array}$$
The approximation is based on the fact that ${e^{j\frac{{k{{({{\rho_1} + \rho } )}^2}}}{{2z}}}}$ oscillates much faster than ${e^{j\frac{{k{{({{\rho_1} - \rho } )}^2}}}{z}}}$ when ${\rho _1}$ and $\rho$ are comparable. Therefore, the former can be neglected in the integral.

Considering the contribution of the input wave with the radius of r1 and the wave at the output plane with the radius of r2, we have

$$E({{r_2},\varphi ,z} )\propto \int\limits_0^{2\pi } {\phi ({{\varphi_1}} )} T({{\varphi_1}} )\sum\limits_{n ={-} \infty }^{ + \infty } {({{e^{jn({\varphi - {\varphi_1}} )}}} ){e^{ - j\frac{{({4{n^2} - 1} )}}{4}\frac{z}{{2k{r_1}{r_2}}}}}} d{\varphi _1},$$
where $\phi ({{\varphi_1}} )$ is the azimuth distribution function for $E({{\rho_1},{\varphi_1},0} )$. Let $\frac{z}{{2k{r_1}{r_2}}} = \frac{\pi }{N}$, and n = gN + m (0=<m < N), we have
$$\begin{array}{l} \sum\limits_{n ={-} \infty }^{ + \infty } {({{e^{jn({\varphi - {\varphi_1}} )}}} )} {e^{ - j\frac{{\pi {n^2}}}{N}}}\\ = \sum\limits_{g ={-} \infty }^{ + \infty } {\sum\limits_{m = 0}^{N - 1} {({{e^{j({m + gN} )({\varphi - {\varphi_1}} )}}} )} {e^{ - j\frac{{\pi {{({m + gN} )}^2}}}{N}}}} \\ = \sum\limits_{m = 0}^{N - 1} {\left( {{e^{jm({\varphi - {\varphi_1}} )}}{e^{ - j\frac{{\pi {m^2}}}{N}}}} \right)} \sum\limits_{g ={-} \infty }^{ + \infty } {{e^{jgN({\varphi - {\varphi_1}} )}}} \\ = \sum\limits_{m = 0}^{N - 1} {\left( {{e^{jm({\varphi - {\varphi_1}} )}}{e^{ - j\frac{{\pi {m^2}}}{N}}}} \right)} \frac{{2\pi }}{N}\sum\limits_{g ={-} \infty }^{ + \infty } {\delta \left( {\varphi - {\varphi_1} - g\frac{{2\pi }}{N}} \right)} \\ = \frac{{2\pi }}{N}\sum\limits_{m = 0}^{N - 1} {\left( {{e^{jmg\frac{{2\pi }}{N}}}{e^{ - j\frac{{\pi {m^2}}}{N}}}} \right)} \sum\limits_{g ={-} \infty }^{ + \infty } {\delta \left( {\varphi - {\varphi_1} - g\frac{{2\pi }}{N}} \right)} . \end{array}$$
Based on Eq. (13), we have the integral in the azimuth domain as
$$\begin{array}{l} E({{r_2},\varphi ,z} )\propto I = \int\limits_0^{2\pi } {\phi ({{\varphi_1}} )} T({{\varphi_1}} )\sum\limits_{n ={-} \infty }^{ + \infty } {({{e^{jn({\varphi - {\varphi_1}} )}}} )} {e^{ - j\frac{{\pi {n^2}}}{N}}}d{\varphi _1}\\ = \frac{{2\pi }}{N}\int\limits_0^{2\pi } {\phi ({{\varphi_1}} )} \exp \left( { - \frac{N}{{4\pi }}{\varphi_1}^2} \right)\sum\limits_{m = 0}^{N - 1} {\left( {{e^{jm({\varphi - {\varphi_1}} )}}{e^{ - j\frac{{\pi {m^2}}}{N}}}} \right)} \sum\limits_{g ={-} \infty }^{ + \infty } {\delta \left( {\varphi - {\varphi_1} - g\frac{{2\pi }}{N}} \right)} d{\varphi _1}\\ = \frac{{2\pi }}{N}\sum\limits_{g ={-} \infty }^{ + \infty } {\phi \left( {\varphi - g\frac{{2\pi }}{N}} \right)\exp \left( { - \frac{N}{{4\pi }}{{\left( {\varphi - g\frac{{2\pi }}{N}} \right)}^2}} \right)\sum\limits_{m = 0}^{N - 1} {\left( {{e^{jmg\frac{{2\pi }}{N}}}{e^{ - j\frac{{\pi {m^2}}}{N}}}} \right)} } . \end{array}$$
Let $\varphi = \frac{{2\pi l}}{N},l = 0\ldots N - 1$, which means samples are taken evenly on the azimuth direction on the output plane, and let g = g’N + m’ (0=<m’<N), and consider the fact that $\varphi = \varphi \pm 2\pi g^{\prime}$, we have
$$\begin{array}{l} I\left( {\varphi = \frac{{2\pi l}}{N}} \right) = \frac{{2\pi }}{N}\sum\limits_{g^{\prime} ={-} \infty }^{ + \infty } {\sum\limits_{m^{\prime} = 0}^{N - 1} {\phi \left( {\frac{{2\pi }}{N}({l - m^{\prime}} )} \right)\exp \left( { - \frac{N}{{4\pi }}{{\left( {\frac{{2\pi }}{N}({l - m^{\prime}} )} \right)}^2}} \right)\sum\limits_{m = 0}^{N - 1} {\left( {{e^{jmm^{\prime}\frac{{2\pi }}{N}}}{e^{ - j\frac{{\pi {m^2}}}{N}}}} \right)} } } \\ \propto \sum\limits_{m^{\prime} = 0}^{N - 1} {\phi \left( {({l - m^{\prime}} )\frac{{2\pi }}{N}} \right)\exp \left( { - {{({l - m^{\prime}} )}^2}\frac{\pi }{N}} \right)\sum\limits_{m = 0}^{N - 1} {\left( {{e^{jmm^{\prime}\frac{{2\pi }}{N}}}{e^{ - j\frac{{\pi {m^2}}}{N}}}} \right)} } . \end{array}$$
Let $k = l - m^{\prime}$, we have
$$\begin{array}{l} I\left( {\varphi = \frac{{2\pi l}}{N}} \right) \propto \sum\limits_{k = 0}^{N - 1} {\phi \left( {\frac{{2\pi k}}{N}} \right)\exp \left( { - \frac{{{k^2}\pi }}{N}} \right)\sum\limits_{m = 0}^{N - 1} {\left( {{e^{jm({l - k} )\frac{{2\pi }}{N}}}{e^{ - j\frac{{\pi {m^2}}}{N}}}} \right)} } \\ = \frac{{2\pi }}{N}{e^{ - j\frac{{\pi {l^2}}}{N}}}\sum\limits_{k = 0}^{N - 1} {\phi \left( {\frac{{2\pi k}}{N}} \right)} {e^{ - j\frac{{2\pi kl}}{N}}}\sum\limits_{m = 0}^{N - 1} {{e^{ - j\frac{{\pi {{({k + m - l} )}^2}}}{N}}}} \\ = \frac{{2\pi }}{{\sqrt N }}{e^{ - j\frac{\pi }{4}}}{e^{ - j\frac{{\pi {l^2}}}{N}}}\sum\limits_{k = 0}^{N - 1} {\phi \left( {\frac{{2\pi k}}{N}} \right)} {e^{ - \frac{{2\pi kl}}{N}}}. \end{array}$$
Equation (16) suggests that the two terms constitute the discrete Fourier transform (DFT) pairs, i.e.,
$$I\left( {\varphi = \frac{{2\pi l}}{N}} \right) = \frac{{2\pi }}{{\sqrt N }}{e^{ - j\frac{\pi }{4}}}{e^{ - j\frac{{\pi {l^2}}}{N}}}DFT\left( {\phi \left( {\frac{{2\pi k}}{N}} \right)} \right).$$
So, with the change of the topological charge, one may expect the field concentration on the different angles due to the property of N point DFT. In order to guarantee the interference for all r1 and r2 at the distance z, one must ensure
$${r_1}{r_2} = \frac{{Nz}}{{2k\pi }},$$
which is Eq. (3) in the main text. To make sure that the wave with the radius of r1 on the input plane goes to the radius of r2 on the output plane, the radial lens is introduced at r1 with the target output radius as
$${r_2}({{r_1}} )= \frac{{r_0^2}}{{{r_1}}},$$
with
$$r_0^2 = \frac{{Nz}}{{2k\pi }},$$
which is Eq. (6) in the main text.

Funding

National Natural Science Foundation of China (61775168).

Disclosures

The authors declare no conflicts of interest.

Data availability

Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.

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Data availability

Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.

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Figures (7)
Fig. 1.
Fig. 1. Simulation of the mode sorter with the simple hybrid lens with the angular lens and the radial direction ring lens. (a) The input beam is the Gaussian shape beam (b) the focus beam with a radial lens (c) the focused beam with a pure angular lens (d) the focused beam with the simple hybrid radial-angular lens. The parameters are as follows. The wavelength of the beam is 671 nm, the beam waist w0 is 1.9mm, the parameter N is 50, a is 1.9 mm, and β is 2/mm.

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Fig. 2.
Fig. 2. illustration of the focal condition for the improved hybrid radial-angular lens

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Fig. 3.
Fig. 3. Experimental setup. The 671nm laser has its beam expanded by an expander. A helical wavefront is added to the first SLM, which converts the beam into an OAM mode. The OAM mode is sorted by the second SLM which has the phase of the improved radial-angular lens on it. SLM: spatial light modulator. BS: beam splitter. CCD camera: charge coupled device camera

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Fig. 4.
Fig. 4. Illustration of the phase distribution of the improved radial-angular hybrid lens. The parameters are as follows. The wavelength of the beam is 671 nm, the parameter N is 50, r0 is 0.6 mm, and β is 2/mm.

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Fig. 5.
Fig. 5. Mode sorting with the proposed hybrid lens for the different OAM modes. The images are captured by the CCD camera.

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Fig. 6.
Fig. 6. The table of crosstalk for different OAM modes

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Fig. 7.
Fig. 7. The table of crosstalk for different OAM modes with the topological charge separation of 3.

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Equations (20)

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E ( ρ , φ , z ) = exp ⁡ ( j k z ) j λ z e j k 2 z ρ 2 ∫ 0 + ∞ ∫ 0 2 π E ( ρ 1 , φ 1 , 0 ) T ( ρ 1 , φ 1 ) e j k 2 z ρ 1 2 e − j k ρ 1 ρ z cos ⁡ ( φ − φ 1 ) ρ 1 d ρ 1 d φ 1 ,
T ( ρ 1 , φ 1 ) = exp ⁡ ( − j N 4 π φ 1 2 ) exp ⁡ ( − j β ρ 1 ) ,
f = 2 π k r 1 r 2 N .
T ( ρ 1 , φ 1 ) = exp ⁡ ( − j k 2 f ( ρ 1 − a ) 2 ) .
a ( ρ 1 ) = r 0 2 ρ 1 .
T ( ρ 1 , φ 1 ) = exp ⁡ ( − j k 2 f ( ρ 1 − r 0 2 ρ 1 ) 2 ) exp ⁡ ( − j N 4 π φ 1 2 ) exp ⁡ ( − j β ρ 1 ) , f = 2 π k r 0 2 N .
E ( ρ , φ , z ) = exp ⁡ ( j k z ) j λ z e j k 2 z ρ 2 ∫ 0 + ∞ ∫ 0 2 π E ( ρ 1 , φ 1 , 0 ) T ( ρ 1 , φ 1 ) e j k 2 z ρ 1 2 e − j k ρ 1 ρ z cos ⁡ ( φ − φ 1 ) ρ 1 d ρ 1 d φ 1 ,
T ( φ 1 ) = exp ⁡ ( − N 4 π φ 1 2 ) ,
e x p ( − j z cos ⁡ φ ) = ∑ n = − ∞ + ∞ ( − j ) n e j n φ J n ( z ) ,
J n ( z ) ≈ 2 π z ( cos ⁡ ( z − 2 n + 1 4 π ) − sin ⁡ ( z − 2 n + 1 4 π ) ( 4 n 2 − 1 ) 4 1 2 z ) = 2 π z R e ( e j ( z − 2 n + 1 4 π ) ( 1 + j ( 4 n 2 − 1 ) 4 1 2 z ) ) ≈ 1 2 2 π z ( e j ( z − 2 n + 1 4 π ) e j ( 4 n 2 − 1 ) 4 1 2 z + e − j ( z − 2 n + 1 4 π ) e − j ( 4 n 2 − 1 ) 4 1 2 z ) ,
E ( ρ , φ , z ) ≈ exp ⁡ ( j k z ) j λ z ∫ 0 + ∞ ∫ 0 2 π E ( ρ 1 , φ 1 , 0 ) exp ⁡ ( − N 4 π φ 1 2 ) 1 2 π k ρ 1 ρ z × ∑ n = − ∞ + ∞ e j n ( φ − φ 1 ) ( ( − 1 ) n e − j π 4 e j k ( ρ 1 + ρ ) 2 2 z e j ( 4 n 2 − 1 ) 4 z 2 k ρ 1 ρ + e j π 4 e j k ( ρ 1 − ρ ) 2 z e − j ( 4 n 2 − 1 ) 4 z 2 k ρ 1 ρ ) ρ 1 d ρ 1 d φ 1 ≈ exp ⁡ ( j k z ) j λ z ∫ 0 + ∞ ∫ 0 2 π E ( ρ 1 , φ 1 , 0 ) exp ⁡ ( − N 4 π φ 1 2 ) 1 2 π k ρ 1 ρ z ∑ n = − ∞ + ∞ e j n ( φ − φ 1 ) e j π 4 e j k ( ρ 1 − ρ ) 2 z e − j ( 4 n 2 − 1 ) 4 z 2 k ρ 1 ρ ρ 1 d ρ 1 d φ 1 .
E ( r 2 , φ , z ) ∝ ∫ 0 2 π ϕ ( φ 1 ) T ( φ 1 ) ∑ n = − ∞ + ∞ ( e j n ( φ − φ 1 ) ) e − j ( 4 n 2 − 1 ) 4 z 2 k r 1 r 2 d φ 1 ,
∑ n = − ∞ + ∞ ( e j n ( φ − φ 1 ) ) e − j π n 2 N = ∑ g = − ∞ + ∞ ∑ m = 0 N − 1 ( e j ( m + g N ) ( φ − φ 1 ) ) e − j π ( m + g N ) 2 N = ∑ m = 0 N − 1 ( e j m ( φ − φ 1 ) e − j π m 2 N ) ∑ g = − ∞ + ∞ e j g N ( φ − φ 1 ) = ∑ m = 0 N − 1 ( e j m ( φ − φ 1 ) e − j π m 2 N ) 2 π N ∑ g = − ∞ + ∞ δ ( φ − φ 1 − g 2 π N ) = 2 π N ∑ m = 0 N − 1 ( e j m g 2 π N e − j π m 2 N ) ∑ g = − ∞ + ∞ δ ( φ − φ 1 − g 2 π N ) .
E ( r 2 , φ , z ) ∝ I = ∫ 0 2 π ϕ ( φ 1 ) T ( φ 1 ) ∑ n = − ∞ + ∞ ( e j n ( φ − φ 1 ) ) e − j π n 2 N d φ 1 = 2 π N ∫ 0 2 π ϕ ( φ 1 ) exp ⁡ ( − N 4 π φ 1 2 ) ∑ m = 0 N − 1 ( e j m ( φ − φ 1 ) e − j π m 2 N ) ∑ g = − ∞ + ∞ δ ( φ − φ 1 − g 2 π N ) d φ 1 = 2 π N ∑ g = − ∞ + ∞ ϕ ( φ − g 2 π N ) exp ⁡ ( − N 4 π ( φ − g 2 π N ) 2 ) ∑ m = 0 N − 1 ( e j m g 2 π N e − j π m 2 N ) .
I ( φ = 2 π l N ) = 2 π N ∑ g ′ = − ∞ + ∞ ∑ m ′ = 0 N − 1 ϕ ( 2 π N ( l − m ′ ) ) exp ⁡ ( − N 4 π ( 2 π N ( l − m ′ ) ) 2 ) ∑ m = 0 N − 1 ( e j m m ′ 2 π N e − j π m 2 N ) ∝ ∑ m ′ = 0 N − 1 ϕ ( ( l − m ′ ) 2 π N ) exp ⁡ ( − ( l − m ′ ) 2 π N ) ∑ m = 0 N − 1 ( e j m m ′ 2 π N e − j π m 2 N ) .
I ( φ = 2 π l N ) ∝ ∑ k = 0 N − 1 ϕ ( 2 π k N ) exp ⁡ ( − k 2 π N ) ∑ m = 0 N − 1 ( e j m ( l − k ) 2 π N e − j π m 2 N ) = 2 π N e − j π l 2 N ∑ k = 0 N − 1 ϕ ( 2 π k N ) e − j 2 π k l N ∑ m = 0 N − 1 e − j π ( k + m − l ) 2 N = 2 π N e − j π 4 e − j π l 2 N ∑ k = 0 N − 1 ϕ ( 2 π k N ) e − 2 π k l N .
I ( φ = 2 π l N ) = 2 π N e − j π 4 e − j π l 2 N D F T ( ϕ ( 2 π k N ) ) .
r 1 r 2 = N z 2 k π ,
r 2 ( r 1 ) = r 0 2 r 1 ,
r 0 2 = N z 2 k π ,
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