Light beams, as electromagnetic waves, can carry both energy and momentum. Meanwhile, we know that momentum can be classified into linear and angular momentum, and there are two particular kinds of angular momenta: spin angular momentum (SAM) and orbital angular momentum (OAM). More than a century ago, Poynting proposed that the SAM is related to the photon spin, namely, the circularly polarized light carries an SAM of per photon[
Figure 1.(a) Linearly polarized light carries no SAM (left) and circularly polarized light carries an SAM of ±ℏ per photon (right). (b) Wavefront, intensity, and corresponding phase distribution of vortex beams with topological charge of l = 0, 1, 2.
Thus far, the most common mode for vortex beams is the Laguerre–Gaussian (LG) mode, because that mode is a solution of the Helmholtz equation in the cylindrical coordinate system under paraxial approximation[
In this review, a brief introduction about vortex beams is presented in Section 1, and approaches for generation and detection of OAM will be presented hereinafter. In Section 2, generation methods including classical optical elements, digital devices, photon sieve, metasurfaces, and other methods are summarized. In Section 3, detection methods by the use of interference, diffraction, geometric coordinate transformation, deep learning, and surface plasmon polariton are reviewed. Finally, we conclude and make a prospect in Section 4.
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2. Generation of Vortex Beams
Hitherto, to enable emerging applications, varieties of techniques have been proposed for the generation of vortex beams over the past few decades[
2.1. Using classical optical elements
2.1.1. Mode conversion
In the early years, laser beams carrying OAM were quickly developed both theoretically and experimentally. Particularly, Beijersbergen et al. first generated such beams experimentally by the use of two cylindrical lenses called a mode converter[
In order to obtain beams carrying OAM, it is necessary to introduce a phase difference of in two vertical directions for the conversion between the HG and LG modes. As shown in Fig. 2(a), the converter consists of two identical cylindrical lenses. Specifically, cylindrical lenses placed at the distance of and are known as and converters, respectively. The LG mode with azimuthal index and radial index can be generated by passing an HG mode with indices and through a mode converter. Despite that, optical vortices can be produced as such, the HG mode of high order is still required, and a complex optical system is inevitable. Moreover, in the case of an illuminating LG beam, the converter can reverse the sign of topological charge.
Figure 2.Schematic diagrams of (a)
2.1.2. Spiral phase plate
It is well known that phase modulation can be achieved through spatial variations of thickness or refractive index. In this section, we will introduce the SPP, as it is one of the most commonly used refractive optical elements, which is usually considered to be the basis of various other alternative phase-modulation methods. As shown in Fig. 2(b), SPP is a transparent plate with a spiral thickness profile, where the optical height is proportional to the azimuthal angle[
One of the prominent attributes of generating a helical beam by SPP is that the conversion efficiency is extremely high and can almost reach 100%. The SPP alleviates restriction on the incident light, for instance, a glass SPP can take high-power laser illumination without the requirement of an HG laser source or changing the direction of propagation. Nevertheless, since a supremely high precision is required for the manufacturing technique, the imperfection is also obvious due to the fact that the fabrication of the SPP with perfectly smooth variation in height is unattainable. Therefore, in most cases, approximate versions are manufactured in the implementation, where the smooth surface is replaced by many discrete steps[
Based on the aforementioned drawbacks, many approaches have been proposed over the past few decades. For instance, an adjustable SPP can achieve various values of topological charges and be used in a wide range of wavelength at the same time[
2.1.3. Computer-generated holograms
Holography is an approach to record and reconstruct the intensity as well as the phase of the optical field. Holographic optical elements such as fork grating and spiral zone plate are methods commonly used to generate optical vortices. It was Soskin et al. who first, to the best of our knowledge, proposed the concept of the fork grating[
Apart from the fork grating, the spiral zone plate is another computer-generated hologram that is mostly used to generate vortex beams[
Particularly, the spiral zone plate can be degraded to a regular Fresnel zone plate with . In practice, these computer-generated holograms are normally binarized for the fabrication of masks that can be given as follows:
The difference between the fork grating and the spiral zone plate that comes in the beam to generate the hologram is either a plane wave or a spherical wave. As mentioned above, refractive SPP increases the topological charges with an increasing thickness, whereas in the computer-generated holograms, only more lines in the pronged dislocation or spirals are required to be increased. Therefore, especially in the case of higher values of topological charges, the fabrication of computer-generated holograms can be much easier than that of SPP.
2.2. Using digital devices
As introduced above, optical vortices can be generated by holographic approaches such as a fork-shaped grating and a spiral zone plate. However, an optical element can only generate an optical vortex beam with specific topological charge, which is inconvenient in experiment. What makes such methods particularly attractive is the digital devices such as a commercial SLM that typically include liquid crystal displays, positive transparency film, and a DMD. The SLM consists of substantial liquid crystal molecules independently modulated by corresponding applied voltage. SLM is an electro-optical device that enables us to manipulate the amplitude and phase of the light field, which can be classified into two types as reflective SLM based on liquid crystal on silicon and transmissive SLM based on transparent liquid crystal displays[
Figure 3.Generation of optical vortex beam with digital devices: (a) SLM, (b) DMD. Schematic diagram of experimental apparatus of (a1) SLM and (b1) DMD. (a2) Phase holograms and corresponding far-field diffraction patterns. (a3) Generated vortex beams and OAM spectra[
DMD, as another candidate to generate vortex beams, is able to control incident light with high frame rate and a great number of spatial degrees of freedom. DMD is composed of tremendous high-speed digital switches such as aluminum reflective mirrors that can be modulated independently pixel by pixel. The resolution is determined by the number of pixels, and a small number of pixels may cause the quantization errors in encoding intensity and phase, and thus a larger number of micro-mirrors is preferable for high-quality beam generation. Binary data are sent to the static random access memory of DMDs and produce an electrostatic charge distribution, causing the deflection angle of each individual micro-mirror to be either or degrees, corresponding to “off” or “on” state. Beam shaping can be realized by switching the micro-mirrors on and off rapidly. Compared with liquid crystal displays absorbing up to 90% of the incident light with low transmittance and contrast, the DMD is a reflective SLM that can improve the image quality with higher throughput and superior diffraction efficiency[
Similarly, optical vortices can be dynamically generated by DMD with spatially controllable phase and intensity. One of the most commonly used techniques for phase modulation by use of DMD is Lee holography[
Compared with conventional techniques, generating OAM modes by DMD makes it possible to switch modes at a very high speed with less optical elements. The superpixel method can independently control the amplitude and phase of light with high resolution by combining a spatial filter and a DMD[
2.3. Using photon sieves
Apart from the aforementioned methods, a binary amplitude mask is usually used to shape both the phase and amplitude of lights; one example is the photon sieve composed of a large number of pinholes that can tightly focus the incident light. In 2001, Kipp et al. proposed a photon sieve with transmissive pinholes with designed diameters and distribution in order to overcome the limitation in spatial resolution, reduce unwanted diffraction orders, and further eliminate the scattered intensity by the conventional zone plate[
In 2015, Qiu et al. proposed an ultrahigh-capacity photon sieve containing subwavelength holes arranged in two structural orders of aperiodicity and randomness[
Figure 4.(a1) Schematic view of the spiral photon sieve. Vortex beams generated by (a2) spiral zone plate and (a3) spiral photon sieve[
Similarly, vortex beams can be produced by the use of electron sieves. In 2017, Yang et al. proposed the electron sieves with a rotationally symmetric structure to flexibly and systematically generate the electron vortex beams[
2.4. Using metasurfaces
In traditional methods, changing phase by accumulating the propagation distance makes it very bulky to generate optical vortices and restrains further development in integrated optics, and thus elements with smaller size are imperative to generate optical vortex beams for the OAM-based applications at miniature scales. Metasurface, an equivalent two-dimensional device, is constructed by artificial arrays of meta-atoms etched on the ultrathin plate that can control the phase, amplitude, and polarization state arbitrarily. The incident light can be imparted with a space-variant abrupt phase change through an ultrathin metasurface instead of acquiring the desired phase change via the propagation effect. The metasurface revolutionized the concept of light manipulation and paved the way to the construction of ultrathin optical devices.
Two primary approaches to design the metasurfaces are based on Pancharatnam–Berry (PB)/geometry phase, dynamic phase, and detour phase[
Figure 5.Generation of optical vortices through metasurfaces based on (a) dynamic phase[
Other than the geometry phase, another dominant approach to design a metasurface is modifying the transmission phase by changing the meta-atoms, known as dynamic phase. The geometries of meta-atoms can be any shape ranging from cross slits[
Moreover, vortex beams can be generated by metasurfaces based on the combination of dynamic and geometry phase. Figure 5(c) shows a metasurface as a J plate composed of rectangular-shaped nanoantennas with different sizes, which coverts right and left circular polarizations into states with independent values of OAM[
Except for generation of a single vortex beam, as illustrated in Fig. 5(d), metasurfaces composed of three parts can generate multiple optical vortex beams at different focal planes, combining the functionalities of a lens and an SPP[
2.5. Using other techniques
Most traditional devices can be damaged with high-power laser illumination; hence approaches such as the foil-made light fan and -plate were introduced to produce optical vortex beams[
Similar to the half- and quarter-wave plate, the Jones matrix of a -plate can be written as[
Noting that a vortex light field with an opposite circular polarization plus an azimuthal phase is transformed, the topological charge of which is . Analogously, in the case of a left-handed circularly polarized incident light, the sign of the topological charge can be reversed, i.e., the output helical wavefront is determined by the incident polarization state.
Despite the aforementioned optical elements known as passive methods, direct generation of vortex beams can also be achieved from the solid-state laser cavity by active methods, i.e., optical vortex lasers. Generally, techniques used to directly produce vortex beams from the laser resonator can be categorized as annular beam pumping, central damaged cavity mirror, thermally induced mode aperture in side pumped vortex laser, and computer-controlled vortex laser. The pump beam can be shaped into an annular beam through the capillary fiber in order to produce vortex output, where the sign of topological charge can be controlled by the intracavity etalon angle. Spatial intensity matching between gain volume in the laser resonator and modes can be achieved by annular beam pumping, realizing the generation of modes with high value of up to 200[
Moreover, despite optical vortex beams or electron vortex beams, plasmonic waves can possess OAM as well, known as a plasmonic vortex. Surface plasmon polaritons are collective oscillation of incident electromagnetic waves coupled with electrons on the metal surface confined to the interface that exhibit evanescence in the direction perpendicular to the interface. The energy of the plasmonic field is confined in a subwavelength scale that can be used to surmount the conventional diffraction limit and provide colossal local field intensity. To generate plasmonic vortices with specific topological charge, elements such as plasmonic metasurfaces and plasmonic vortex lenses have been researched[
3. Detection of OAM
Vortex beams have been widely used in the fields of optical communication[
3.1. Using interference method
There are many methods to measure the OAM of vortex beams by the principle of interference, among which the most basic one is interfering with a tilted plane wave so that a fork-shaped interference pattern can be exhibited[
The OAM of the vortex beam can be detected by the use of double-slit interference[
Figure 6.(a) Left to right: phase distribution of a vortex beam with l = 1, the spiral phase passing through the double slits, and interference intensity distribution[
In addition, other methods have been proposed to detect the OAM of vortex beams such as dynamic angular double slits[
The Mach–Zehnder interferometer with Dove prisms can also be used to measure OAM states of vortex beams[
3.2. Using diffraction method
Typically, detection of OAM based on diffraction can be divided into the aperture diffraction method[
Figure 7.(a) Diffraction patterns of triangular aperture[
A cylindrical lens is a kind of aspheric lens that can effectively reduce spherical and chromatic aberration with the function of one-dimensional amplification. As mentioned in the second section, a mode converter consisting of a cylindrical lens can convert an HG beam into an LG beam and vice versa. Based on the theory, detection of OAM can be realized by the use of a cylindrical lens[
One can note that vortex beams can be generated using computer-generated holograms. A vortex beam with topological charge can be generated at the first diffraction order with a Gaussian beam passing through an -dislocations fork-shaped grating. Similarly, a vortex beam with can be transformed into a fundamental mode Gaussian beam by eliminating its helical phase through an -dislocations fork-shaped grating, and thus the detection of OAM can be realized. However, this method is only appropriate for vortex beams with specific OAM, and multiple holograms are needed for the detection of different OAM modes, resulting in the low efficiency. Therefore, more complex holograms were proposed by superimposing multiple holograms based on the Daman grating, making it possible to measure multiple topological charges through a single grating[
3.3. Using geometric coordinate transformation
Separation of OAM modes by the use of geometric coordinate transformation is also an effective method for vortex beams detection, the basis of which is the conversion from OAM into linear momentum[
Figure 8.(a) Phase profiles of the transforming and the phase-correcting optical element (top). Schematic of the optical system (bottom)[
As introduced above, the log-polar transformation truncates the azimuthal light field of the vortex beam and transforms it into a plane wave with only a finite length, and thus focusing spots produced by vortex beams with adjacent topological charges overlap slightly due to the existence of the diffraction limit, causing mutual crosstalk. Based on this, Chen et al. proposed the concept of spiral transformation[
Log-polar transformation requires two phase elements for mode separation, and the alignment of these two elements is very strict. In order to improve the miniaturization of the optical system, more designs have been proposed, such as integration of expansion and correction and non-paraxial log-polar transformation[
3.4. Using deep learning
Recently, deep learning has been widely used in the fields of image processing with its powerful advantages of data analysis and information processing by the use of local connection, shared weight, pooling, and multi-layer structure[
In the free-space communication system, atmospheric turbulence can cause wavefront distortion of the transmitted beams, which is a serious challenge for OAM-based communication systems. Owing to the rapid development of machine learning, especially deep learning technology, vortex beams detection based on the convolutional neural network (CNN) has gradually emerged in recent years[
Figure 9.(a) Numerical model of the OAM-based communication system. CCD camera, charge-coupled device camera[
In addition, Liu et al. proposed a superhigh-resolution recognition scheme based on deep learning with fractional topological charges of the 0.01 level. Through changing the phase, the same OAM mode can get different intensity maps, which can generate a large number of intensity maps and send them to neural network training, as shown in Fig. 9(c)[
As a consequence, with the advantages of low cost, high speed, high precision, low device processing difficulty, and no redundant optical equipment, the OAM recognition scheme based on deep learning is still a research hotspot in the field of OAM optical communication. The technology has potential application value for realizing large-capacity optical communication networks based on vortex beams in the future.
3.5. Using surface plasmon polaritons
In conventional methods, regulating the light field by accumulating propagation distance makes it rather bulky, and it is difficult to break the diffraction limit. Owing to the excellent ability of manipulating the nanoscale field, surface plasmon polaritons have aroused interest to detect the OAM of vortex beams, such as on-chip plasmonic nanogratings[
Figure 10.Methods for detecting OAM modes based on surface plasmon polariton. (a) A semi-ring plasmonic nanoslit[
Later, Chen et al. proposed another kind of plasmonic nanograting with fringes to couple OAM modes into two bifurcated surface plasmon polaritons with different splitting angles. As shown in Fig. 10(b), the splitting angles of plasmonic waves are strongly dependent on the topological charges of incident vortex beams, and thus OAM modes can be detected by different angles without particular alignment[
Vortex beams carrying OAM have broadened the perspectives in the optical realm owing to unique properties. In this review, we have summarized approaches and recent advances in the field of vortex beams generation and OAM detection. Classical optical elements such as SPP can generate vortex beams with extremely high conversion efficiency, whereas the requirement for manufacturing techniques is rather high. With the development of liquid crystal technology, digital devices are commonly used to produce vortex beams for their versatility. Moreover, binary amplitude masks such as photon and electron sieves can also be adopted for the generation of vortex beams. Nanofabrication technology makes it possible to generate such beams via ultrathin optical devices such as metasurfaces. As for the detection methods, the interferometer and diffraction grating can only detect a single topological charge at a time. In recent years, methods using geometric coordinate transformation, deep learning, and on-chip plasmonic nanogratings can further realize the recognition of complex OAM modes. We review the methods for generating and detecting the OAM in a concise way to facilitate further research on the vortex beams.
Notwithstanding the aforementioned approaches, there still exist challenges for both generation and detection of OAM. For example, the switch speed of SLMs usually hinders real-time applications in experiments. Promising advances can be expected once the technology becomes more mature. As for the detection methods, surface plasmon polaritons, for instance, have certain limitations such as strict alignment of incident lights, despite that such a method can promote the development of integrated photonic systems. Hopefully, vortex beams in light, as well as other forms of waves, will continue to thrive and enable new applications in many other fields[
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