A review of multiple optical vortices generation: methods and applications

Abstract

Optical vortices carrying orbital angular momentum (OAM) have attracted increasing interest in recent years. Optical vortices have seen a variety of emerging applications in optical manipulation, optical trapping, optical tweezers, optical vortex knots, imaging, microscopy, sensing, metrology, quantum information processing, and optical communications. In various optical vortices enabled applications, the generation of multiple optical vortices is of great importance. In this review article, we focus on the methods of multiple optical vortices generation and its applications. We review the methods for generating multiple optical vortices in three cases, i.e., 1-to-N collinear OAM modes, 1-to-N OAM mode array and N-to-N collinear OAM modes. Diverse applications of multiple OAM modes in optical communications and non-communication areas are presented. Future trends, perspectives and opportunities are also discussed.

Keywords

optical communications / optical vortices / orbital angular momentum (OAM) / mode-division multiplexing (MDM) / mode multicasting

Introduction

Optical vortices, also known as orbital angular momentum (OAM) beams, have been studied for decades. It was shown by Allen in 1992 that optical vortices comprising an azimuthal phase term exp(ilj), possess an OAM of lη per photon, where l is referred to topological charge and j is azimuthal angle [¹–³]. In recent years, optical vortices have seen wide applications in different areas, such as optical manipulation, optical trapping, optical tweezers, optical vortex knots, imaging, microscopy, sensing, metrology and quantum information processing [⁴–⁹]. Moreover, due to the intrinsic spatial orthogonality of OAM modes with different topological charge numbers, one promising prospect offered by optical vortices is in optical communications [¹⁰]. Accordingly, a series of research works on OAM modes multiplexing in optical communications have been reported recently in both free space and optical fibers [¹¹–²⁵].

In various optical vortices enabled applications, the generation of multiple optical vortices is always highly desired, such as optical communications, optical sensing and optical measurement [²⁶–³³]. By employing multiple optical vortices in optical communications, one can greatly increase the transmission capacity and spectral efficiency. However, in the conventional experiments, one element (e.g., spiral phase plate or spatial light modulator) can only provide one OAM mode [¹¹], which is not scalable. The cost and complexity of the system will rapidly grow with the number of the optical vortices and resultant required multiple optical elements. Thus, it is highly desirable to develop methods for generating a large number of optical vortices with less optical elements. Moreover, simultaneous generation of multiple OAM modes using a single element from a single input Gaussian beam is also an important basic function in an OAM multicasting system.

In this review article, we focus on the methods of multiple optical vortices generation and its applications. The remainder of this article is organized as follows. Section 2 presents the methods of multiple optical vortices generation. Section 3 focus on the application of multiple optical vortices. Finally, we give some brief discussions and perspectives of multiple optical vortices.

Methods of multiple optical vortices generation

Generally, there are three cases for the generation of multiple optical vortices in different applications, which are summarized in Fig. 1. The first one is 1-to-N collinear OAM modes generation. By specially designing an optical element, one can get multiple collinear OAM modes from a single input Gaussian mode, as shown in Fig. 1(a), which is usually employed in OAM mode multicasting scheme. The second one is 1-to-N OAM modes array. By inputting a Gaussian mode, one can get multiple OAM modes with preset different locations, which is illustrated in Fig. 1(b). The last one is N-to-N collinear OAM modes, which is used in OAM mode multiplexing scheme. Multiple Gaussian modes from different locations are incident on an optical element for generating multiple OAM modes, which is shown in Fig. 1(c). In this section, we will introduce multiple optical vortices generation methods for these three cases in detail.

1-to-N collinear OAM modes

By designing sliced phase patterns, Yan et al. demonstrated multiple collinear OAM modes generation with a spatial light modulator (SLM) [³⁴]. Moreover, they also experimentally demonstrated data-carrying multicasting five and seven OAM channels from a single-input OAM channel. The approach is illustrate in Fig. 2. It has been shown that an angular amplitude aperture of central angle q with M-fold rotational symmetry can distribute energy from the input OAM beam of charge l to multiple OAM beams having equally spaced OAM charge number of {…, −kM+ l, …, −M+ l, l, M+ l, …, kM+ l, … } (k is an integer). Note that in the first row of Fig. 2, the aperture’s transmission part has a constant phase value b₀, which results in a sinc²-like OAM charge spectrum, centered at input OAM charge l. In the second row of Fig. 2, the aperture’s transmission part is complementary to that of the first aperture, and it produces a sinc²-like OAM charge spectrum centered at l - 6. In the last row, the sliced phase pattern can be viewed as the superposition of the transmission parts of the above two amplitude-phase apertures. As a result, the output spectrum of the sliced phase pattern is a coherent addition of those two previous output spectra. The parameter b₀ is optimized such that most of the multicasting OAM channels have equalized power except for the two channels at the wings of the spectrum. By using this approach, 7 equally spaced OAM modes are successfully generated. The experimental results are shown in Fig. 3. By using a pattern of more slices, seven OAM channels (l= 6, 9, 12, 15, 18, 21, 24) are generated in the experiment. The phase pattern consists of three sliced regions, where there are two parameters, b₁ and b₂, to be optimized to equalize the power of the multicasting channels. The intensity patterns of before multicasting OAM mode and after multicasting OAM modes are shown in Fig. 3(a). Figure 3(b) illustrates the power distribution after multicasting.

By using the above method, one can only generate equally spaced OAM modes. However, in many applications, it is also important to generate randomly spaced OAM modes. For the simultaneous generation of n OAM modes \{{{\displaystyle l}}_{\mathrm{1}},{{\displaystyle l}}_{\mathrm{2}},\mathrm{...},{{\displaystyle l}}_n\}, the mathematical description of the desired transmission function can be expressed as

(1)f(\varphi )=\underset{m=\mathrm{1}}{\overset{n}{{\displaystyle \Sigma }}}{{\displaystyle A}}_{{{\displaystyle l}}_m}\hspace{0.17em}\exp (\mathrm{i}{{\displaystyle l}}_m\varphi ),

where the complex number \{{{\displaystyle A}}_{{{\displaystyle l}}_m}\} comprises the weight coefficients. Normally f(\phi ) is a complex form, including both amplitude and phase. To make it simplified, our work focuses on how to reduce the complex form to a phase-only function with tolerable loss in efficiency. The phase-only approximate function can be defined as g(\phi )=\exp [i\varphi (\phi )] with the phase function

(2)\phi (\varphi )=\mathrm{Re}\left\{-\mathrm{i}\ln \left[\underset{m=\mathrm{1}}{\overset{n}{{\displaystyle \Sigma }}}{{\displaystyle B}}_{{{\displaystyle l}}_m}\hspace{0.17em}\exp (i{{\displaystyle l}}_m\varphi )\right]\right\}.

In Eq. (2), Re{ } means “real part of,” and {{\displaystyle B}}_{{{\displaystyle l}}_m}is a decisive factor for \phi (\varphi ). Discarding the imaginary part of the right-hand side of Eq. (2) is equivalent to setting amplitude to unity, which ensures g(\varphi ) is a phase-only function. Expand g(\varphi )in Fourier series:

(3)g(\varphi )=\underset{m=-\infty }{\overset{\infty }{{\displaystyle \Sigma }}}{{\displaystyle C}}_m\exp (im\varphi ),

where the decomposition coefficient is {{\displaystyle C}}_m={\displaystyle \frac{\mathrm{1}}{\mathrm{2}\pi }}{{\displaystyle \int }}_{\mathrm{0}}^{\mathrm{2}\pi }g(\varphi )\times \exp (-im\varphi )\mathrm{d}\varphi. To achieve high efficiency, the phase-only function g(\varphi ) and the original function f(\varphi ) should be with little difference. To evaluate the difference, we introduce a parameter of relative root-mean-square error (R-RMSE):

(4)\mathrm{R}-\mathrm{R}\mathrm{M}\mathrm{S}\mathrm{E}=\sqrt{{\displaystyle \frac{{{\displaystyle \Sigma }}_{m-\mathrm{1}}^n{({\left|{{\displaystyle C}}_{{{\displaystyle l}}_m}\right|}^{\mathrm{2}}-{\left|{{\displaystyle A}}_{{{\displaystyle l}}_m}\right|}^{\mathrm{2}})}^{\mathrm{2}}}{n{{\displaystyle \Sigma }}_{m-\mathrm{1}}^n{\left|{{\displaystyle C}}_{{{\displaystyle l}}_m}\right|}^{\mathrm{2}}}}}.

The smaller of the R-RMSE, the better performance of the phase-only element we can achieve.

Since the weight coefficients \{{{\displaystyle A}}_{{{\displaystyle l}}_m}\} is settled at first, the parameter R-RMSE is determined by \{{{\displaystyle C}}_{{{\displaystyle l}}_m}\} or {{\displaystyle B}}_{{{\displaystyle l}}_m}. Then it becomes a simple minimization problem. We need to find the suitable \{{{\displaystyle B}}_{{{\displaystyle l}}_m}\} for minimizing R-RMSE. To solve the problem, Lin et al. proposed an iterative algorithm, which is a spontaneous optimization algorithm [³⁵]. It is a highly effective method for generating multiple OAM modes with a single phase-only element. However, when we use this method to generate more than 10 OAM modes, the performance of the algorithm gets worse, mainly because the initial set of parameters \{{{\displaystyle B}}_{{{\displaystyle l}}_m}^{\mathrm{0}}\} are set equal to \{{{\displaystyle A}}_{{{\displaystyle l}}_m}\}. The unsuitable choice of \{{{\displaystyle B}}_{{{\displaystyle l}}_m}^{\mathrm{0}}\} will lead to immature convergence of the iterative algorithm.

Based on the iterative algorithm, our group proposes a pattern search assisted iterative (PSI) algorithm to simultaneously generate multiple OAM modes using a single phase-only element [³⁶]. The PSI algorithm shows a favorable operation performance for generating 100 randomly spaced OAM modes with high diffraction efficiency (>93%), low R-RMSE and low standard deviation. Figure 4 shows the simulation results of the generated randomly spaced 100 OAM modes by PSI algorithm.

Moreover, we study the convergence of the PSI algorithm. The iterative process in the PSI algorithm is convergent. Shown in Fig. 5 are the convergence curves of R-RMSE. Figure 5(a) shows the R-RMSE convergence curve for generating 20 randomly spaced equal-power OAM modes. One can easily see from the curve that the R-RMSE is convergent after 65 iterations. The R-RMSE convergence curve for generating 50 randomly spaced equal-power OAM modes is depicted in Fig. 5(b). One can also clearly see that the R-RMSE is convergent after 91 iterations.

To evaluate the algorithm under a more realistic scenario, we study the performance of the phase patterns loaded onto the practical SLM. Here, we take commercially available Holoeye PLUTO phase only SLM as an example. The resolution of the SLM is 1920´1080 pixels with 256 gray levels covering 0−2p phase modulation. By using the PSI algorithm, we get realistic SLM phase patterns for generating 20 randomly spaced equal-power OAM modes. The phase pattern loaded onto SLM has 1080´1080 pixels with 256 gray levels, which is shown in Fig. 6(a). We then compare the performance of the realistic SLM phase pattern for generating OAM modes. The simulation results of the OAM spectra are depicted in Fig. 6(b). The OAM spectra of the original phase patterns (i.e., no consideration of realistic SLM) using PSI algorithm is also shown for reference. Figure 6(b) shows the OAM spectra of the original phase pattern (blue one) and realistic SLM phase pattern (red one) using the PSI algorithm. By comparing the simulation results shown in Fig. 5, one can see that the performance of realistic SLM phase pattern is slightly degraded compared with the original phase pattern.

In addition, we can also manipulate the relative power distribution of the generated OAM modes simply by setting the initial weight coefficients in the PSI algorithm. The simulation results of power manipulation of OAM modes are shown in Fig. 7.

Moreover, one can use SLMs to fully control the amplitude and phase of the input light beam, which could also be employed for the generation of multiple optical vortices. In 2014, our group proposes a simple method to arbitrarily manipulate the amplitude and phase of the incoming light beam with two phase-only SLMs without using any phase iterative algorithm [³⁷]. The concept and principle are illustrated in Fig. 8(a). We set the polarization direction of the input light A₀ 45° with respect to the x direction. The working direction of polarization-dependent SLM1 is x direction with a phase distribution {\varphi }_1(x,y). After the SLM1, the light is half modulated, with the x direction distribution {\displaystyle \frac{\sqrt{\mathrm{2}}}{\mathrm{2}}}{{\displaystyle A}}_{\mathrm{0}}\exp (\mathrm{i}{{\displaystyle \varphi }}_{\mathrm{1}}(x,y)) and y direction distribution {\displaystyle \frac{\sqrt{\mathrm{2}}}{\mathrm{2}}}{{\displaystyle A}}_{\mathrm{0}}. And then, the light beam passes through a polarizer with the polarization direction 45° deviation from the x direction. Thus, the electrical field of light beam becomes

(5)E(x,y)={\displaystyle \frac{\sqrt{\mathrm{2}}}{\mathrm{2}}}{{\displaystyle A}}_{\mathrm{0}}(\exp (\mathrm{i}{{\displaystyle \varphi }}_{\mathrm{1}}(x,y)))\times {\displaystyle \frac{\sqrt{\mathrm{2}}}{\mathrm{2}}}+{\displaystyle \frac{\sqrt{\mathrm{2}}}{\mathrm{2}}}{{\displaystyle A}}_{\mathrm{0}}\times {{\displaystyle A}}_{\mathrm{0}}={\displaystyle \frac{\mathrm{1}}{\mathrm{2}}}{{\displaystyle A}}_{\mathrm{0}}(\exp (\mathrm{i}{{\displaystyle \varphi }}_{\mathrm{1}}(x,y))+\mathrm{1}).

After the polarizer, the light comes through the SLM2 with a phase distribution {{\displaystyle \varphi }}_{\mathrm{2}}(x,y). The working direction of polarization-dependent SLM2 keeps the same as the light polarization after the polarizer, which is actually enabled by adjusting the light polarization after the polarizer via a half-wave plate placed between the polarizer and SLM2. So the electrical field of light after the SLM2 becomes

(6)E(x,y)={\displaystyle \frac{\mathrm{1}}{\mathrm{2}}}{{\displaystyle A}}_{\mathrm{0}}(\exp (\mathrm{i}{{\displaystyle \varphi }}_{\mathrm{1}}(x,y))+\mathrm{1})\exp (\mathrm{i}{{\displaystyle \varphi }}_{\mathrm{1}}(x,y))={{\displaystyle A}}_{\mathrm{0}}\cos \left({\displaystyle \frac{{{\displaystyle \varphi }}_{\mathrm{1}}(x,y)}{\mathrm{2}}}\right)\exp \left(\mathrm{i}{\displaystyle \frac{{{\displaystyle \varphi }}_{\mathrm{1}}(x,y)+\mathrm{2}{{\displaystyle \varphi }}_{\mathrm{2}}(x,y)}{\mathrm{2}}}\right).

From Eq. (6), one can easily find that the amplitude distribution is determined by {{\displaystyle \varphi }}_{\mathrm{1}}(x,y), and the phase distribution is determined by {\displaystyle \frac{\mathrm{1}}{\mathrm{2}}}{{\displaystyle \varphi }}_{\mathrm{1}}(x,y)+{{\displaystyle \varphi }}_{\mathrm{2}}(x,y). Hence, by changing the phase distribution patterns written onto the SLM1 and SLM2, we can manipulate the amplitude and phase independently at the same time. By using this method, we show the successful generation of multiple collinear OAM modes. The theoretical and experimental results are shown in Fig. 8(b). In addition, we can arbitrarily generate different types of light beams as desired, such as Laguerre-Gaussian (LG) beams and Bessel beams. Figure 9 shows the intensity distributions of the generated LG beams and Bessel beams. The obtained results shown in Fig. 9 indicate the feasibility of arbitrary manipulation of amplitude and phase using the proposed approaches.

1-to-N OAM mode array

OAM mode array has found its applications in many areas, such as optical manipulation, three-dimensional scanning interferometry, and quantum processing. In 2010, Moreno et al. employed Dammann vortex grating for OAM mode array generation [³⁸]. By combining Dammann grating with vortex phase distribution, one can get 1D Dammann vortex grating, as shown in Fig. 10(a). 2D Dammann gratings is formed by multiplying the 1D Dammann vortex grating by a 90° rotated version, which is shown in Fig. 10(c). By using Dammann vortex grating, 7´7 OAM mode array is successfully generated.

By employing the above method, the same group fabricates 2D Dammann grating for generating 25 OAM modes using UV lithography technique [³⁹]. The microscopic images of the fabricated grating are shown in Figs. 11(a) and 11(b). The intensity distributions of generated OAM modes by the 2D Dammann grating are shown in Figs. 11(c)−11(f).

Besides, our group presents a simple and compact on-chip OAM mode array emitter on silicon photonics platforms [⁴⁰]. The principle relies on three-plane-wave interference. We design, fabricate and demonstrate an on-chip OAM mode array emitter consisting of three parallel waveguides with etched tilt gratings, as shown in Fig. 12. The tilt gratings facilitate flexible light emission in a wide range of directions, enabling the generation of OAM mode array above the silicon chip. The OAM mode array is also known as optical vortex (OV) lattice.

The fabricated on-chip OV lattice emitter is experimentally investigated. The experimental configuration for observing the on-chip generation of the OV lattice is illustrated in Fig. 13(a). Figure 13(b) shows the measured near-field intensity distribution of y-polarization light with 10° magnification coming out from the OV lattice emitter at the wavelength of 1550 nm. One can clearly see the three bright light spots emitted from the three tilt gratings. Figure 13(c) plots the measured far-field intensity distribution of y-polarization light with 40° magnification generated from the on-chip OV lattice emitter, which indicates the phenomenon of three-plane-wave interference and a network of dark spots. When interfering the generated OV lattice in Fig. 13(d) with another plane wave, there emerges a network of fork-like fringe patterns. One can indicate from Figs. 13(c) and 13(d) that each fork-like fringe pattern corresponds to a dark spot, verifying the successful generation of the OV lattice using the designed and fabricated compact on-chip OV lattice emitter.

N-to-N collinear OAM modes

N Gaussian modes from different angles are incident on optical element for generating N collinear OAM modes, which is always demanded in OAM mode multiplexing scheme. Here we introduce two methods for N-to-N collinear OAM modes.

The first one is to use Dammann vortex grating, which is the same with the previous section. Figure 14 shows a schematic of the Dammann vortex grating enabled multiple collinear OAM modes generation and multiplexing [⁴¹]. Gaussian beams with plane wavefronts are incident on the Dammann vortex grating at the angles of its diffraction orders. Therefore, along the zeroth order of diffraction, all the incident Gaussian beams are transformed into different OAM modes in the same direction. Moreover, one can also use Dammann vortex grating for OAM modes demultiplexing which is also shown in Fig. 14.

By using Dammann vortex grating, Lei et al. demonstrate independent collinear OAM channel generation, transmission and simultaneous detection [⁴¹]. 80 Tbit/s transmission capacity is achieved with uniform power distributions along all channels, with 1600 individually modulated QPSK/16-QAM data channels multiplexed by 10 OAM states, 80 wavelengths and two polarizations. The experimental results are shown in Fig. 15.

The second method is well-known as the OAM mode sorter, which is used to sort different OAM modes by two static optical elements [⁴²–⁴⁴]. The OAM mode sorter performs a log-polar to Cartesian coordinate transformation, which can convert the helically phased OAM beam into a beam with a transverse phase gradient. Then, a subsequent lens focuses each input OAM state to a different lateral position, as shown in Fig. 16 [⁴⁵]. By using the OAM mode sorter, N multiplexed collinear OAM modes each with a different topological charge, can be transformed into N laterally separated and elongated spots. Importantly, when the OAM mode sorter is used in the reverse direction, N laterally separated and elongated spots can generate N collinear OAM modes, which can be used for OAM mode multiplexing. By employing OAM mode sorter, OAM modes multiplexing transmission is experimentally demonstrated. The experimental results of multiple OAM modes generation with OAM mode sorter are shown in Fig. 17.

Applications of multiple optical vortices

Multiple optical vortices are usually employed in optical communications. The generation of multiple OAM modes using a single element from a single input Gaussian beam is an important basic function in an OAM mode multicasting system. By using the proposed PSI algorithm, our group experimentally demonstrates 1-to-34 multicasting of an optical signal from single Gaussian mode to multiple OAM modes [⁴⁶]. All 34-fold multicasted OAM channels show relatively low crosstalk (<-10 dB) from their neighboring OAM modes and achieve BER less than 2e−3. The concept and OAM mode spectrum are shown in Fig. 18.

Moreover, by designing and optimizing the complex phase pattern through the adaptive correction of feedback system, the power of each multicasting OAM channel can be arbitrarily controlled. We experimentally demonstrate power-controllable multicasting from a single Gaussian mode to 6 OAM modes with different target power distributions [⁴⁷]. The concept and experimental results are shown in Fig. 19.

In addition, we also demonstrate OAM mode multicasting under atmospheric turbulence condition. By employing an adaptive optics closed loop consisting of a wavefront detector and a wavefront corrector, we can successfully compensate the phase distortions [⁴⁸]. The compensation can reduce power fluctuations of multicasted OAM channels and inter-channel power crosstalk, achieving a favorable communication performance. The concept and experimental results are shown in Fig. 20.

High-order Bessel beams are considered as a typical kind of light beam that can carry OAM. Remarkably, Bessel beam is widely known as a self-reconstructing light beam, which can reconstruct its electric field after passing through an obstruction. By using the proposed PSI algorithm designed phase pattern combining with axicon phase distribution, our group reports data multicasting from a single Gaussian mode to multiple Bessel modes using a single phase-only spatial light modulator [⁴⁹]. Under the obstructed path conditions, obstruction-free data-carrying N-fold Bessel modes multicasting is also demonstrated in the experiment. The experimental setup and results are shown in Fig. 21.

Besides optical communication applications, multiple optical vortices can also be used in non-communication areas. In 2014, Lavery et al. observed the rotational Doppler shift from a white-light source after backscattered by a spinning object [³¹]. They show that the magnitude of this shift is dependent upon the OAM of the light, and that superposition of different OAM states give rise to multiple sidebands on the shifted frequency. The observability of the frequency shift for white-light illumination highlights the potential of this rotational Doppler effect as the basis of a rotational sensor using back-scattered light. Figure 22 shows the experimental setup and measured results.

Trends, perspectives and challenges

In recent years, multiple optical vortices have shown great potential in optical communication systems. In this article, we review the methods of multiple optical vortices generation and its emerging applications. Different methods have been employed for multiple optical vortices generation, such as search assisted iterative (PSI) algorithm, Dammann vortex grating, and OAM mode sorter. The experimental results show favorable performance in different applications. Figure 23 shows the trends, perspectives and opportunities of multiple optical vortices generation. In this review, we only introduce the generation of multiple OAM modes. However, other mode bases, such as LG modes, Hermite-Gaussian (HG) modes, Bessel modes and vector modes may also be desired in practical applications [⁵⁰–⁶³]. Diverse methods for the generation of multiple other mode bases are of great interest. Moreover, in the existing experiments, SLM is usually employed to control the input spatial light for generating multiple OAM modes. With future improvement, photonic integrated devices on different platforms (e.g., silicon platform) may also be designed and fabricated, which will make the system much more compact, flexible and low cost. Additionally, metamaterials and metasurfaces could also be employed to manipulate the spatial structure of light beams, which will extend the applications of multiple optical vortices and multiple other spatial modes [⁶⁴–⁷¹].

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