1.1.

Characteristics of Orbital Angular Momentum Modes

Optical beams carrying OAM typically exhibit helical wavefronts. The pitch and handedness of the helix determine the topological charge and type (positive/negative) of the OAM beam.⁷ Various solutions of the Helmholtz wave equation can result in different kinds of beams that carry OAM. The differences arise based on the geometry or conditions that are used to solve the wave equation. Examples of some of the solutions with an OAM component are Laguerre–Gaussian (LG) modes,² Bessel modes,⁸ Mathieu modes,⁹ Ince–Gaussian modes,¹⁰ and hypergeometric-Gaussian modes.^11,12

The solutions of the paraxial wave equation in cylindrical coordinates $(\rho ,\varphi ,z)$ have a transverse scalar electric field given as

Fig. 1

Intensity (grey) and phase structures (coloured) of LG modes with azimuthal numbers $l=0$, $\pm 1$, $\pm 2$ and $p=0$.

Fig. 2

Intensity (grey) and phase structures (coloured) of LG modes with azimuthal numbers $l=0$, $\pm 1$, $\pm 2$ and $p=1$.

1.2.

Applications of Orbital Angular Momentum Modes

In Sec. 1, we briefly mentioned some of the applications of OAM modes. We discuss them in more detail in this section. In optical communications, the need for supporting higher data rates in a single optical fiber has led researchers to explore the path of mode division multiplexing (MDM). In conventional MDM, data are encoded in spatially orthogonal light beams based on the linearly polarized (LP) modes. The primary limitation with LP mode-based MDM communications is intermodal coupling, which essentially increases cross talk and degrades the signal-to-noise ratio (SNR). To address this issue, several research groups are exploring the possibility of using OAM modes as they promise lower intermodal coupling.^13,14 It is to be noted that OAM modes are actually unapproximated solutions of the optical fiber and are hence called “true modes” of the fiber.^15,16

Stimulated emission depletion microscopy is one of the most successful far-field imaging techniques, allowing access to resolution beyond the diffraction limit.¹⁷ It uses a Gaussian-shaped beam to excite the fluorophore and a redshifted, spatially overlapped donut-shaped beam to deplete the fluorescence everywhere except at the dark center of the depletion beam. OAM modes with a donut intensity structure are the “perfect” candidates for such applications.^18,19

Donut-shaped modes are also used in material processing, such as glass cutting, since the induced stress in the center of the irradiated area is minimal.²⁰ OAM modes have many diverse applications such as in optical trapping experiments to study biological interactions,²¹ as a rotational sensor for the remote detection of spinning objects based on rotational Doppler shift of the modes;²² in astronomy, where some of their properties are used in coronagraphs²³ and in quantum information processing to enlarge the information content that a qubit can transmit.²⁴

1.3.

Need for Detection and Sorting of Orbital Angular Momentum Modes

The need for higher data rate in optical communication has been met by various multiplexing schemes such as polarization division multiplexing, wavelength division multiplexing, and, as mentioned already, MDM. The different spatial modes in MDM are utilized as information-carrying channels and are copropagated through the same communication medium (free space or fiber). OAM modes of different topological charges are orthogonal to each other and hence form a suitable basis set of nonoverlapping channels for MDM.^25,26 Even in cases where only a single mode carrying information was used, the very process of propagation, whether through free space or fiber could cause the mode to deteriorate and leak into other modes. In such cases, the signal received at the detector can be considered to comprise a set of superimposed OAM modes, and it is important to separate and quantify the modal weight of the various modes present in the resultant beam.²⁷ In such cases, it becomes necessary to multiplex and demultiplex a number of modes and hence this has become an important research area in recent times. In addition, quantifying the modal weights, commonly called modal decomposition, has potential applications in wavefront reconstruction,²⁸ beam quality measurement ($M^2$),²⁹ and in the fiber-to-fiber coupling process.³⁰

2.1.

Phase Modulation Approach

In this section, various phase modulation-based schemes to generate beams with OAM are discussed. As mentioned in the previous section, a spatial variation of refractive index and/or thickness alters phase. Therefore, by this definition even refractive elements can be considered to be SLMs. While there are methods that use refractive elements, such as the work by Beijersbergen et al.⁶ that generated high-order LG modes directly from a laser using cylindrical lenses as mode convertors, this review focuses mainly on spatial light modulation based on interferometric, diffractive, or metaoptics to generate OAM beams. The one refractive element that will be discussed in detail is the SPP, as it is often the starting point for designs carried out using these alternative phase methods.

2.1.1.

Generation using refractive spiral phase plate

An SPP is a refractive element with a spiral thickness profile, as shown in Fig. 6. This element creates a complex wave with a spiral phase retardation.^44–46 One of the main advantages of using an SPP for generating a spiral wave is that the conversion efficiency can be close to 100%. However, the challenge lies in manufacturing a refractive element with a smooth variation in height. Therefore, in most implementations of the SPP, only an approximate version is fabricated.^47–52

Fig. 6

SPP for generation of LG beam with $\ell =1$.

Advances in focused ion beam milling technology enabled the fabrication of a refractive SPP directly on the tip of an optical fiber.⁵³ An electron beam microscope image of the fabricated element on a single-mode fiber (SMF) is shown in Fig. 7. Recent measurements on OAM modal purity using the modal decomposition method revealed that it is sufficient to have only three phase levels (for $\ell =1$) instead of a continuous phase variation for the generation of a fundamental OAM mode ($\ell =1$, $p=0$) with a high modal purity.⁵⁴ However, further studies are necessary to understand the effect of such an approximation on other applications such as optical trapping and communication. In addition, when the topological charge increases, the number of phase levels required to achieve a good accuracy also increases, which brings back the fabrication challenge. For these reasons, the SPP is usually modified using some technique into a simpler version and fabricated, instead of the continuous spiral pattern. These methods will be discussed in subsequent sections.

Fig. 7

Electron microscope image of SPP fabricated on the tip of an optical fiber (reprinted from Ref. 53).

2.1.2.

Generation using blazed and diffractive optical elements

Methods used to design complex light-generating elements always result in a continuous phase pattern, which we have seen is hard to fabricate. Often the analog phase is converted to a blazed structure using a modulo $2\pi$ operation. In other words, sections that change the phase of light by $2\pi$ are removed. Blazed elements still have smooth curved sides, as shown on the left of Fig. 8, that are almost impossible to make with conventional lithography techniques.

Fig. 8

Conversion from a Fresnel lens to a diffractive lens by quantizing the phase levels to binary levels (0 and $\pi$).

One way of making approximate blazed elements is by quantizing the continuous phase into $n$ discrete levels, resulting in what are called DOEs. These quantized phase profiles needs to be converted into GDSII format⁵⁵ in order to be fabricated with lithography tool. Theoretically, an eight-level quantization can give rise to an efficiency of 94.9% and a 16-level approximation to 98.7%.⁵⁶ However, it is still highly challenging to fabricate DOEs with more than two levels, as the process will require multiple lithography steps. The conversion from a blazed element to a binary (two-level) DOE is depicted in Fig. 8.

An SPP can also be created as a diffractive element. However, an SPP of charge 1 would have only two levels and would not be an accurate representation of the actual element. In this case, the azimuthal phase is split into sectors, with each sector being of constant phase, as shown in Fig. 9, resulting in a multilevel phase element.

Fig. 9

(a) Top view of an eight-level SPP and (b) a 3-D view of a 16-level SPP.

Binary DOEs are the easiest to fabricate and can also be mass-produced, which makes them affordable. It should be noted that the binarizing procedure decomposes the incoming field into its Fourier components in the Fourier plane resulting in the generation of several orders.⁴³ In the case of the SPP, the higher orders have higher topological charge values. This means that the efficiency (the fraction of power that goes into forming the desired beam) is poor. The first order will have at most 40.5% efficiency, for a binary element.

2.1.3.

Generation using holographic optical elements

As mentioned in the previous section, fabrication of a refractive element can be challenging. Holographic techniques can be used to overcome these challenges. Holography is the science of recording and reconstructing three-dimensional (3-D) information of an object.^57–59 When used to generate complex light, interference is carried out between the desired (hard-to-fabricate) phase distribution and a simple reference beam. The resulting pattern is usually simpler to fabricate. The hologram-recording process can be mathematically expressed as follows: two mutually coherent electromagnetic waves with complex amplitudes ${\mathrm{\Psi }}_1(x,y)=A_1(x,y)\exp [j{\mathrm{\Phi }}_1(x,y)]$ and ${\mathrm{\Psi }}_2(x,y)=A_2(x,y)\exp [j{\mathrm{\Phi }}_2(x,y)]$ are interfered to create a hologram, in which the relative phase difference is converted into an intensity distribution given as $I=C_1(x,y)+C_2(x,y)\exp \{j[{\mathrm{\Phi }}_1(x,y)-{\mathrm{\Phi }}_2(x,y)]\}+C_2(x,y)\exp \{-j[{\mathrm{\Phi }}_1(x,y)-{\mathrm{\Phi }}_2(x,y)]\}$, where $C_1(x,y)={|A_1(x,y)|}^2+{|A_2(x,y)|}^2$ and $C_2(x,y)=A_1(x,y)A_2(x,y)$. The terms $C_1(x,y)$ and $C_2(x,y)$ contain the information of the intensity of the two interfering waves, whereas the relative phase information $[{\mathrm{\Phi }}_1(x,y)-{\mathrm{\Phi }}_2(x,y)]$ is encoded in the second and third terms.

Early holography involved the use of photopolymers, dichromated gelatin, and photorefractive materials, as these materials were required to store the intensity variation.⁵⁷ The intensity was created using an experimental setup. However, after the development of computers, holography evolved into computer-generated and digital holography.^57–62 In computer-generated holography (CGH), the entire hologram-recording process is carried out computationally and is, therefore, aberration-free. In addition, it is possible to create any exotic phase distribution with only imagination being the limit. By creating complex amplitudes with uniform intensity, the above intensity distribution can be reduced to $I=C_1+C_2\exp \{j[{\mathrm{\Phi }}_1(x,y)-{\mathrm{\Phi }}_2(x,y)]\}+C_2\exp \{-j[{\mathrm{\Phi }}_1(x,y)-{\mathrm{\Phi }}_2(x,y)]\}$. Consider the electromagnetic wave with a phase distribution ${\mathrm{\Phi }}_1(x,y)$ as the reference wave. When its conjugate illuminates $I$, the following beams are generated: $\exp [-j{\mathrm{\Phi }}_1(x,y)]\times (C_1+C_2\exp \{j[{\mathrm{\Phi }}_1(x,y)-{\mathrm{\Phi }}_2(x,y)]\}+C_2\exp \{-j[{\mathrm{\Phi }}_1(x,y)-{\mathrm{\Phi }}_2(x,y)]\})$. This reduces to ${\mathrm{\Psi }}_r=C_3\exp [j{\mathrm{\Phi }}_1(x,y)]+C_4\exp [-j{\mathrm{\Phi }}_2(x,y)]+C_4\exp \{-j[2{\mathrm{\Phi }}_1(x,y)-{\mathrm{\Phi }}_2(x,y)]\}$ upon simplification. If ${\mathrm{\Phi }}_1(x,y)$ is a carrier wave given as $\exp [-j2\pi (\lambda {)}^{-1}(s_{x}x+s_{y}y)]$, where $s_x$ and $s_y$ are the sines of the angle along the $x$ and $y$ directions, then the first term is the unmodulated light from the hologram, the second term is the object wave, which has the necessary phase distribution, and the last term is the conjugate of the object wave with twice the linear phase of the reference wave. The hologram-recording process (carried out with a computer) and the optical reconstruction of the resulting phase distribution are shown in Fig. 10. The advantage of the HOE is that it is possible to generate the 3-D phase as close as possible to the design with both amplitude (efficiency $\simeq 10\%$) and phase versions of the element with a penalty on only the efficiency.⁴³ In this section, we describe some of the most used flat HOEs to generate OAM modes. Apart from these, volume holograms can also be used to generate OAM modes.^63–66

Fig. 10

Hologram recording within computer and reconstruction in free space.

Spiral Fresnel zone plate

A spiral Fresnel zone plate is an HOE synthesized by the interference between a converging or diverging spherical wavefront with a spiral wavefront. The element can either be fabricated as an amplitude or a phase element, depending upon the efficiency requirement. It is necessary to convert the cosine modulation to a binary modulation in order to simplify fabrication.⁴³ The hologram formation can be mathematically expressed as follows: an electromagnetic wave with a complex amplitude ${\mathrm{\Psi }}_1(x,y)=\exp [j(\pi /\lambda z)(x^2+y^2)]$ is interfered with a mutually coherent spiral wavefront with a complex amplitude ${\mathrm{\Psi }}_2(x,y)=\exp [j\ell \theta ]$, where $\ell$ is the topological charge that indicates the number of azimuthal $2\pi$ phase cycles. The phase image of the spherical wave and the spiral wave with topological charges ($\ell =1$, $\ell =3$, and $\ell =5$) and the resulting interference patterns are shown in Fig. 11.

Fig. 11

Generation of holograms from interference between a spherical wave and spiral wave followed by binarization.

It can be understood from the above discussion that if the interference pattern $I=C_1+C_2\exp [j\{(\pi /\lambda f)(x^2+y^2)-\ell \theta \}]+C_2\exp [-j\{(\pi /\lambda f)(x^2+y^2)+\ell \theta \}]$ is illuminated by a conjugate of the spherical wave, the spiral wave can be generated. On the other hand, if the interference pattern is illuminated by a plane wave with constant phase, then the spiral phase will be generated at the focal plane of the spherical wave. Let us extract one of the interference components, namely $\exp [-j\{(\pi /\lambda z)(x^2+y^2)-\ell \theta \}]$ (assuming $C_2=1$), which upon multiplication with the quadratic phase function $\exp [j(\pi /\lambda z)(x^2+y^2)]$ will result in $\mathfrak{F}[\exp (-j\ell \theta )]$ in the Fourier plane, where $\mathfrak{F}$ is the Fourier transform operator. Therefore, depending upon the incident wavefront, interesting behavior can be obtained in the output. Again, upon binarizing the hologram, the diffracted light consists of multiple complex fields corresponding to the Fourier components. Interestingly, the Fourier components of the spiral Fresnel zone plate occur along the optical axis, as shown in Fig. 12.^67,68 However, the efficiency of the various Fourier components are different depending upon the phase value of the binary levels.

Fig. 12

Generation of spiral phases at different axial distances of a binary spiral Fresnel zone plate.

Fork grating

A fork grating is an HOE generated by interference between a spiral wave and a tilted plane wave. The hologram formation can be mathematically described as follows: an electromagnetic plane wave with a complex amplitude ${\mathrm{\Psi }}_1(x,y)=\exp [-j2\pi {(\lambda )}^{-1}(s_{x}x+s_{y}y)]$ is interfered with a mutually coherent spiral wavefront with a complex amplitude ${\mathrm{\Psi }}_2(x,y)=\exp (j\ell \theta )$, and the binary holograms for $\ell =1$ to 5 are shown in Figs. 13(a)–13(e), respectively. The holograms can be described as $I=C_1+C_2\exp [j\{2\pi {(\lambda )}^{-1}(s_{x}x+s_{y}y)-\ell \theta \}]+C_2\exp [-j\{2\pi {(\lambda )}^{-1}(s_{x}x+s_{y}y)-\ell \theta \}]$. Upon illumination by a conjugate plane wavefront, a spiral phase wave is generated, and if illuminated by a plane wave, then the spiral phase wavefront is diffracted away at an angle corresponding to the tilt of the original reference plane wave. When the fork grating is binarized and illuminated by a plane wave, the Fourier components of the grating are distributed over different lateral locations in the Fourier plane. The Fourier components with higher diffraction angles are associated with higher topological charges.^12,69–75 The diffraction of a plane wave by a binary fork grating is shown in Fig. 14. Therefore, it is possible to generate spiral wavefronts with different topological charges from the same element at the angles of the different diffraction orders. The orders will not have the same efficiency, however. The spiral Fresnel zone plate and fork grating are the most widely used HOEs for the generation of spiral wavefronts. Furthermore, both elements can be easily fabricated using lithography techniques or printed on transparent sheets using inkjet printers.^76,77 An increase in the topological charge results in an increase in the number of lines in the fork in case of an HOE, whereas in the diffractive version of the SPP, the thickness increases. For this reason, the fabrication of an HOE is often easier for any value of the topological charge. In the case of OAM generation from fiber lasers, it is possible to fabricate the forked grating directly on the tip of fiber. This is very compact and avoids free-space coupling losses. An electron microscope image of a forked grating fabricated directly on the tip of an optical fiber is shown in Fig. 15.⁵³

Fig. 13

Images of the binary fork holograms with topological charges: (a) $\ell =1$, (b) $\ell =2$, (c) $\ell =3$, (d) $\ell =4$, and (e) $\ell =5$.

Fig. 14

Generation of spiral phase waves with increasing topological charges with increasing order of diffraction.

Fig. 15

Electron microscope image of a fork grating fabricated using focused beam milling (reprinted from Ref. 53).

Trochoson

The name Trochoson is adapted from the Greek word “Trochos” meaning “ring” and is given to a spiral axicon, as first studied by Khonina et al.⁷⁸ The spiral axicon is a hologram obtained by interference between a spiral wave and a conical wave, as shown in the Fig. 16. The conical wave contains a distribution given as ${\mathrm{\Psi }}_3(x,y)=\exp [-j(2\pi /\lambda )\alpha {(x^2+y^2)}^{1/2}]$, where $\alpha$ is the base angle of the cone. Later, the synthesis of a spiral axicon for the generation of higher-order Bessel beams was reported.^79,80 As described earlier, it is possible to generate either a spiral wave or a spiral conical wave using different reconstruction beams.

Fig. 16

Formation of spiral axicon hologram from interference between a spiral wave and a conical wave.

Based on the above three examples, there were numerous reports on the generation of spiral beams carrying OAM using different types of carrier waves.^81–83 The choice of the carrier waves decides how the spiral beams are delivered in 3-D space. This same principle has been used to generate a variety of exotic complex light with OAM, such as Mathieu beams,⁸⁴ achromatic vortices,⁸⁵ modulated optical vortices,⁸⁶ optical-vortex filaments,⁶³ spiraling higher-order Bessel beams,⁸⁷ beams with fractional topological charges,^88,89 and perfect vortex beam.^90,91 The disadvantages of employing an HOE for the generation of special beams carrying OAM, such as lower efficiency, are discussed in Ref. 7 and the references therein.

Implementation using active devices

It should be noted that all the phase techniques and elements (such as the holographic technique and the forked grating) described in the previous section can be implemented using an active SLM such as LCoS.^74,92,93 The required holograms are calculated using the mathematical expressions described in earlier sections for generation of OAM modes. Let us consider the generation of OAM modes of azimuthal order $(l)$ equal 1 to 6. The resulting phase patterns will be similar to those shown in Fig. 13. By illuminating the hologram with a reference Gaussian beam, the object is reconstructed and the OAM beams are generated at the far field, as shown in Fig. 17.

Fig. 17

OAM modes of azimuthal order $l$ from 1 to 6 generated using the sinusoidal gratings programmed in the SLM.

The azimuthal phase structure encoded with a sinusoidal grating provides the necessary phase structure to the diffracted orders. However, in this method, there is no control over the amplitude structure of the beam. For example, the OAM phase encoded in the sinusoidal gratings shown in Figs. 13(a)–13(e) generates additional intensity rings, which are not desirable. Scalar OAM modes with desired beam waist $w$, azimuthal order $l$, and radial order $p$, have both amplitude and phase structures. Such scalar modes can be generated using LCoS-phase-only SLMs by employing the algorithms proposed previously.^94–97 Among these, the Arrizón’s algorithm is widely used because it has been demonstrated to provide relatively high SNR⁹⁸ and requires a phase range of only $<1.17\pi$.

In Arrizón’s algorithm, any complex field $s(x,y)$ as expressed in Eq. (5) with amplitude function $a(x,y)$ ranging from 0 to 1 and phase function $\theta (x,y)$ can be generated using a CGH with the phase modulation function expressed in Eq. (6).

Eq. (5)

$$s(x,y)=a(x,y)\exp [i\theta (x,y)],$$

Eq. (6)

$$\mathrm{\Psi }(x,y)=f(a)\sin (\theta ),$$

where $f(a)$ has to be accurately determined to generate the complex field, as explained below.

When a light beam with electric field $U$ is incident on the LCoS device encoded with a phase modulation function, the beam is spatially modulated and the reflected beam from the device can be expressed in a Taylor–Fourier series expansion using the Jacobi–Anger identity⁹⁹ as

Eq. (7)

$$U\exp [\text{if}(a)\sin (\theta )]=U\sum _{m=-\infty }^{\infty }{J}_m[\mathit{f}(\mathit{a})]\exp (\mathit{i}m\theta ),$$

where $J_m$ is the Bessel function of first kind with order $m$. In the Fourier series expansion, the phase of the first component ($m=1$) is identical to the phase structure of the complex field. Hence, by equating the magnitude of the first component to the magnitude function of the complex field, as shown in Eq. (8), $\mathit{f}(\mathit{a})$ can be determined:

Eq. (8)

$$\mathit{f}(\mathit{a})=J_1^{-1}[Aa],$$

where $A$ is a constant. The maximum value of $A$ for which Eq. (8) will be satisfied is 0.5819. This corresponds to the first maximum of the first-order Bessel function $J_1$. Since the corresponding argument of the first-order Bessel function is 1.84, the required phase range of the SLM in this case is $-1.17\pi /2$ to $+1.17\pi /2$. Hence, by substituting Eq. (8) in Eq. (7), we generate a complex function $s(x,y)$ in the first harmonic of the above Fourier series. To achieve the spatial isolation of $s(x,y)$ from the other harmonics, a phase carrier $2\pi (u_{0}x+v_{0}y)$ is added to the phase of the encoded field $\theta$ [shown in Eq. (9)], where $u_0$ and $v_0$ are the spatial frequencies.⁹⁸

Eq. (9)

$$U\exp \{\text{if}(a)\sin [\theta +2\pi (u_{0}x+v_{0}y)]\}=U\sum _{m=-\infty }^{\infty }{J}_m[f(a)]\exp [im\theta +2\pi (u_{0}x+v_{0}y)].$$

The resultant first-order diffraction from the SLM can be written, as shown in Eq. (10).

Eq. (10)

$$Ua(x,y)\exp [i\theta (x,y)+2\pi (u_{0}x+v_{0}y)].$$

By following this algorithm, CGHs for OAM modes of azimuthal order $l=0$ to 5 and radial order $p=0$, 1 were generated. The corresponding intensity patterns were recorded with a camera and are shown in Figs. 18 and 19, respectively.

Fig. 18

Intensity profiles of zero-radial order OAM modes.

Fig. 19

Intensity profiles of first-radial order OAM modes.

Composite OAM modes, which are the co-axial superpositions of OAM modes, can also be generated using active LCoS SLMs.¹⁰⁰

2.1.4.

Generation using all dielectric metasurfaces

Optical metasurfaces^101,102 are artificial 2-D arrays of subwavelength meta-atoms that are designed to modify the amplitude, phase, and state-of-polarization of light. These devices are conceptually similar to the reflect-and-transmit antenna arrays,¹⁰³ which have been studied for decades in the microwave community. A typical meta-atom can have any shape that can support some form of resonance. When an electromagnetic wave passes through a patterned metastructure, the transmitted field amplitude and phase are modified according to the geometric parameters, such as height, length, and width, of the resonator. The transmission phase can be varied from 0 to $2\pi$ by changing one or more of these parameters. This results in the ability to fully manipulate optical wavefronts. The idea of using different structured surfaces to achieve spatially varying phase profiles has been used as early as 1993 in the microwave domain.¹⁰⁴ Early metasurfaces were, in fact, descendants of their microwave counterparts and were implemented using metallic meta-atoms. The major drawback of such metasurfaces is that they are made up of metallic structures, which are significantly lossy in the visible spectrum due to the strong absorption in metals.^105,106 However, these type of metasurfaces can be used in the infrared (IR) and higher wavelength spectrum as metals do not suffer from such losses at those wavelengths. Another popular method used in the design of metasurfaces is based on the Pancharatnam–Berry (geometric) phase. However, this method works only when the incident light is circularly polarized.¹⁰¹ On the other hand, dielectrics exhibit very low losses at optical frequencies and support a magnetic resonance mode close to the electric resonance mode in the visible and IR regions of the spectrum. These facts have steered the direction of metasurfaces from metallic structures toward all-dielectric ones, whenever applicable. A more comprehensive study on material platforms for optical metasurfaces can be found in the paper by Choudhury et al.¹⁰⁷

A typical meta-atom in a dielectric metasurface¹⁰⁸ is made up of a material with a high dielectric constant such as silicon or ${\mathrm{TiO}}_2$. When excited with light, the meta-atom can support electric and magnetic dipole resonances. By varying one or more of the geometrical parameters of the resonator, one can spectrally overlap the electric and magnetic resonances at the wavelength of interest and obtain a 0 to $2\pi$ phase coverage.¹⁰⁹ Some widely used geometries for meta-atoms are V-shaped antennas,¹¹⁰ cylindrical disks,¹⁰⁸ nanofin structures,¹¹¹ and cross shapes.¹¹² Other shapes are also possible. Rational design thinking can be applied to come up with advanced geometries for meta-atoms. Different conventional optical elements such as lenses, gratings, beam splitters, and waveplates have already been successfully realized using metasurfaces. In some cases, a single metasurface has provided the functionality that may only be achieved by a combination of several conventional refractive optical elements.¹¹³

Another advantage of metasurfaces is that many of the techniques used to design the phase are similar to that of DOEs. In essence, they both impart spatially varying phase to the incident light. Identifying these key similarities is very useful for a metasurface designer. For example, the procedure involved in designing a Fresnel zone lens (FZL) phase profile is identical for both metasurfaces and DOEs. The only difference comes in the physical realization of the devices. DOEs use spatially varying heights of a dielectric (such as glass or lithography resist) to emulate the phase profile of the FZL, whereas the metasurface uses 2-D arrays of meta-atoms with spatially varying dimensions to accomplish the same. Usually DOEs have only two-phase levels making them less efficient. Their efficiency can be significantly improved by increasing the number of phase levels, which increases the complexity of the fabrication process. According to scalar diffraction theory, a phase profile with 16 levels can achieve almost 99% efficiency.⁵⁶ Either multistep lithography or grayscale electron beam lithography would have to be used and both of these processes can be difficult and hard to reproduce without error. However, a metasurface with multiple phase levels is relatively easy to fabricate. The reason being that metasurfaces produce spatially varying phase by changing the lateral dimensions of the meta-atoms. This means no multistep or grayscale lithography is required. Such metasurfaces can, therefore, create elements with much higher efficiencies than binary DOEs.

Design and fabrication of a metasurface with a specific phase profile

Designing a metasurface device can be divided into four main parts: (1) modeling meta-atom dimensions to achieve 0 to $2\pi$ phase coverage for the wavelength of interest, (2) generating the phase profile of the desired optical element, (3) converting the phase profile into metasurface GDSII layout, and (4) fabrication. Modeling the meta-atom starts with the choice of the meta-atom, e.g., cylinder and cross. After choosing an appropriate meta-atom, other geometric parameters of the meta-atom such as height, length, and lattice periodicity need to be optimized. This task is generally carried out using a commercial finite difference time domain (FDTD) solver such as Lumerical FDTD Solutions or CST FDTD Solver. Usually these geometric parameters are much smaller, compared to the operating wavelength. An initial rough simulation can be performed with approximate values for the geometric parameters of the meta-atom to obtain its transmission amplitude and phase. The next step in the design process is to vary one of the geometrical parameters over a range of values while keeping the remaining ones constant. The goal of this step is to obtain 0 to $2\pi$ phase coverage with uniformly high transmission amplitude across this entire range. This step can be thought to provide us with a lookup table containing the meta-atom dimensions for a desired transmission phase. It is the most challenging part of the simulation and this optimization usually requires several iterations to arrive at the final geometric parameters. It can typically take a few days to complete. Machine learning solutions are already making their way into solving this problem,^114,115 where models can predict the right geometrical parameters, given a desired operational wavelength. The second part in the design flow is the phase profile design, which has been discussed earlier.

The next step is to convert the phase profile from an image format into a metasurface layout in GDSII format. The phase profiles are usually generated as PNG, JPG or BMP files, where each pixel represents the transmission phase at that location. Each pixel can be considered to have a physical dimension when fabricated. On the other hand, in the metasurface layout, each pixel will be a meta-atom or a group of meta-atoms. Therefore, the phase images are converted into GDSII layout by a one-to-one mapping of the pixels in the phase profile to a GDSII layout, using the lookup table obtained from the FDTD simulations. It should be noted that each pixel in the layout is represented by a meta-atom that gives the desired transmission phase at that pixel location. An example layout for a spiral phase profile is shown in Fig. 20. Finally, this layout file is used to fabricate the metadevice using standard nanofabrication tools.

Fig. 20

GDSII layout of an SPP for generation vortex beam ($\ell =5$) based on cylindrical-type meta-atom. The increase in the size of the cross gives a transmission phase of 0 to $2\pi$.

Orbital angular momentum mode generation using metasurfaces

OAM generation in the IR region was first demonstrated by Genevet et al.¹¹⁶ in 2012, using a V-shaped antenna-based metasurface SPP. In the next year, the same design concept was applied to the THz region to fabricate OAM generators.¹¹⁷ Another modified version of the antenna was proposed to generate LG beams in THz domain.¹¹⁸ Geometric phase metasurfaces that are composed of periodic arrays of birefringent subwavelength meta-atoms with spatially varying orientation have been used to generate both scalar¹¹⁹ and vector¹²⁰ vortex beams. This type of metasurface works on the Pancharatnam–Berry phase phenomenon. Recently dielectric metasurfaces composed of Silicon nanofin structures have been used for polarization-sensitive generation and modulation of OAM beams.¹²¹ A new concept of metasurface OAM holography has also been demonstrated.¹²² This configuration is capable of reconstructing a range of distinct OAM-dependent holographic images from a single metasurface hologram. The metasurface was realized using gallium nitride (GaN) cylindrical pillars.

We have recently demonstrated a vortex beam generator^112,123 operating at 8.8 μm in the mid-IR spectral range using an all-dielectric metasurface composed of cross-shaped resonators. The meta-atoms can support both Mie-type electric and magnetic dipole resonances to realize spatial phase modulation and to completely suppress reflection losses. The schematic of the OAM generator is shown in Fig. 21(a). The length of the cross resonator is varied from 1 to $7.5\mu \mathrm{m}$ to achieve complete 0 to $2\pi$ transmission phase, as shown in Fig. 21(b). This figure is nothing but the phase lookup table described earlier and can be used to construct the desired SPP element or any phase element. The scanning electron microscope (SEM) image of fabricated SPP and the experimentally captured vortex beam are shown in Figs. 22(a) and 22(b), respectively.

Fig. 21

(a) Schematic representation of the vortex generator layout, silicon cross meta-atom in the inset. (b) Lookup table for IR cross resonators: the length $L$ of the cross arm is varied from 1 to $6.5\mu \mathrm{m}$ to achieve 0 to $2\pi$ transmission for IR design (reprinted from Ref. 112).

Fig. 22

(a) SEM image of meta-SPP for $l=+1$ topological charge; the inset shows the single building block. (b) Vortex beam image captured on the $64\times 64\text{pixel}$ FPA detector (reprinted from Ref. 112).

2.1.5.

Miscellaneous generation techniques

$Q$-plate

A $Q$-plate is a liquid crystal (LC) device having an azimuthal pattern of the LC molecular director around a central point, as shown in Fig. 23,¹²⁵ and works based on the spin–orbit interaction (SOI) of light in anisotropic-inhomogeneous medium. The $Q$-plate pattern is characterized mainly by the topological charge $q$ of the central singularity, which can be an integer or half-integer number. The $Q$-plates of topological charge $q$ allow the generation of light beams carrying an OAM of $\pm 2q\hslash$ per photon, with the OAM sign controlled by the input polarization state. A $Q$-plate with $q=\pm 1/2$ transforming right circularly polarized beam into left circularly polarized beam with an azimuthal phase $\exp (i\varphi )$ is shown in Fig. 23. Applying a suitable voltage across the device can reorient the LC molecules and hence modify the conversion condition leading to the generation of Hermite–Gaussian (HG) and LG beams, covering the whole “higher-order Poincaré sphere.”

Fig. 23

(a) Structure of the segmented $q=1/2$ plate, with indication of the LC director orientation. (b) Q-plate transforms right circularly polarized beam into left circularly polarized beam with an azimuthal phase $\exp (i\varphi )$ (reprinted from Ref. 124).

S-waveplate

This waveplate is a super-structured half-waveplate polarization converter created by femtosecond laser nanostructuring of glass.¹²⁶ It works on the principle of the SOI of light in a medium with form-birefringence. A beam of light without OAM but with right-circular polarization, when transmitted through the S-waveplate, changes to a left circular polarized beam with OAM, as shown in Fig. 24. While this generation method results in optical beams with fixed OAM, its main advantage is that it can be used for high-power and high-energy laser applications, in which other devices would be damaged.

Fig. 24

(a) Light with right circularly polarization and no OAM transmitted through the S-waveplate changes to left circularly polarized OAM beam. (b) Measured intensity profiles of left circularly polarized OAM beam (reprinted from Ref. 21 with the permission of AIP Publishing).

2.2.

Amplitude Modulation Approach

In the previous section, we discussed the generation of OAM modes using a phase modulation approach. Similarly, we can potentially generate OAM modes using an amplitude modulation approach as well. For example, OAM mode generation is possible using a DMD. A DMD consists of a 2-D array of pixels, which can only be turned ON or OFF. It is capable of projecting both binary and grayscale images at high frame rates. Phase patterns cannot be generated directly using a DMD. Therefore, the most common methods of generating such patterns use phase-only SLMs or fabricated DOEs. Special techniques have been developed by which a DMD can be used to create phase patterns.^127,128

In this section, we look at methods that generate complex amplitude patterns using binary amplitude modulation through the mirrors of a DMD. Several techniques based on holographic approaches have been explored in Sec. 2.1.3 of this paper. These principles are applied in the techniques used when DMDs are employed for generation of phase patterns. The slight modifications that need to be made as the DMD is a binary amplitude device are discussed in this section. For example, Brown and Lohmann^60,129 demonstrated binary holography and Lohmann and Paris¹³⁰ demonstrated binary computer-generated Fraunhofer holograms. While the paper was published in a pre-DMD era, the technique is of particular interest as they used binary amplitude-only holograms and produced desired phase patterns. This was achieved using a technique known as the “detour phase” technique. A seminal review on CGHs done by Tricoles⁶² discussed not only these holographic techniques but also their importance in applications such as medical diagnostics, imaging, data processing, and high energy physics. It should be noted that the desired patterns are generated in the far field of the holograms, using these techniques.

In a study published by Piestun et al.,¹³¹ on-axis CGHs were used to modulate phase in the Fresnel plane, using a Fresnel transform and progressive binarization. In 2010, Ren et al.¹³² succeeded in generating LG beams using a DMD. In this case, the DMD was loaded with computer-generated fork-like patterns of different topological charges. The patterns were first corrected for intensity because the gamma curve of the DMD is not linear. Goorden et al.¹³³ reported the superpixel method in 2014. We look at three of the most commonly used techniques for phase generation using a DMD. An LG beam of charge 1 is used as a test intensity and phase pattern to compare the various methods. It should be noted that the DMD can only generate the desired phase pattern if it is placed in the relevant experimental setup.

2.2.1.

Superpixel method

In this method, square matrices of pixels were combined into superpixels. By controlling the index and number of the pixels in a superpixel that are turned ON, the required amplitude and phase values can be achieved. The experiment is set up such that each individual pixel in a superpixel possesses a phase prefactor. The field at all points can be represented in phasor form using this approach. The method requires a typical $4f$ system, as seen in Fig. 25(a). The DMD (with required pattern) lies in the input plane. A spatial filter in the back focal plane filters out a particular order, which will form the image in the output plane.¹³³ The pattern on the DMD consists of several superpixels, with each superpixel representing the intensity and phase of one point at the output plane. In an $n\times n$ superpixel, each individual pixel contains a phase prefactor, as a result of the optics following it [see Fig. 25(b)].

Fig. 25

Superpixel method: (a) setup for superpixel method and (b) superpixel with phase prefactors.

The field at any point on the output plane can be represented as a vector sum considering each ON pixel as a vector of intensity $1/16$ and phase as the complex angle. Turning certain pixels ON will give a resultant field $E_{\text{superpixel}}$, whose intensity and phase will depend, respectively, on how many and which pixels are ON.

The aperture in the back focal plane in the $4f$ system plays a major role in this method. Its position is calculated such that the pixels within each superpixel contain specific phase prefactors. For an $n\times n$ superpixel, the location of the aperture is given as $(x,y)=(\pm a,\pm na)$, where $a=\lambda f/n^{2}d$. Here $\lambda$ is the wavelength of light used, $f$ is the focal length of the first lens in the $4f$ system, and $d$ is the pixel dimension [micromirror pitch of the DMD (Texas Instruments 6500 evaluation module), $\sim 7.6\mu \mathrm{m}$].

A way by which to use this method is described. Initially, the value of $n$ is chosen, say $n=4$. Then, we generate a lookup table that contains all 6561 possible vectors and the pixel combinations for them. The correct combination is pulled out of the table for each desired value of the field at each point in the target plane. We use these calculations to generate a $400\times 400$ pixel ${\mathrm{LG}}_{10}$ beam. One limitation is the trade-off between the size of the desired field and resolution.

Figure 26(a) shows the DMD pattern generated using the superpixel method for ${\mathrm{LG}}_{10}$ beam. We conducted an experiment to verify and obtained the output shown in Fig. 26(b). The first image is that of an $400\times 400\text{pixel}$ LG beam interfered with a reference beam. The second image is a zoomed-in version of the same interference pattern near the phase singularity. We can clearly see the fork pattern, confirming the spiral phase.

Fig. 26

Superpixel pattern and output: (a) pattern generated for LG1 beam and (b) interference of LG1 beam generated using the superpixel method and a Gaussian beam.

2.2.2.

Lee method

This is essentially a holographic technique, where the hologram is generated using the desired pattern, and it is binarized on the basis of the intensity and phase at each point.^134,135 The setup is similar to that of the superpixel method, i.e., a $4f$ setup with an aperture in the Fourier plane to select a particular order of interest. The equation used to generate the binary hologram is given below. The binary hologram thus generated is shown in Fig. 27(a).

Eq. (11)

$$h(x,y)=\frac{{\mathrm{mod}}_{2\pi }[\varphi (x,y)+2\pi (kx+ky)]-\pi }{2\pi }<=\frac{{\sin }^{-1}\{abs[E(x,y)]\}}{\frac{\pi }{2}}.$$

In this equation, $h(x,y)$ is the binary value of each point on the DMD pattern; $E(x,y)$ and $\varphi (x,y)$ are the normalized amplitude and phase at each point in the desired pattern, respectively; $k$ is the carrier frequency that determines both the fringe width of the pattern and the separation of the different orders in the Fourier plane. The mod or modulo operation of the phase term with respect to $2\pi$ gives the remainder of the phase term when divided by $2\pi$, or, in other words, we wrap the phase. Hence, we subtract $\pi$ from it to make the average value 0.

Fig. 27

Lee pattern and output for a $400\times 400\text{pixel}$ LG1 beam: (a) pattern generated using Lee holography and (b) experimental output of Lee holography.

All these techniques are very susceptible to alignment of both the pinhole and the DMD. This is clear as seen by the experimentally generated ${\mathrm{LG}}_{10}$ beam using this method shown in Fig. 27(b). The nonuniformity of intensity along the azimuthal direction is an artifact of the misalignment stemming from the aperture positioning and the way the DMD is held.

2.2.3.

Binary Fraunhofer holography

This method is best explained by referring to a simple explanation of binary diffraction gratings. Two rays leaving two adjacent grating slits and going into the first diffraction order at the far field have a path difference of one wavelength.⁶⁰ Their detour phase is $2\pi$. However, if a slit is slightly out of position, the emerging wavefront will diffract differently, thereby creating a different intensity and phase pattern in the far field. It is this principle of detour phase that is used to generate any phase pattern.¹³⁰

In Fig. 28(a), $u(x,y)$ is the desired ${\mathrm{LG}}_{10}$ pattern, which will be obtained by the Fourier transform of $U({\nu }_x,{\nu }_y)$. This expression is encoded as a hologram by interfering the pattern with a reference wave. The resultant pattern is binarized $H({\nu }_x,{\nu }_y)$ and this is what is loaded on the DMD or holographic plane. To simulate a 2-D grating, whose slits are adjustable, the DMD is divided into groups of pixels that we call cells. While this may seem similar to the superpixel or Goorden technique, it should be noted that those techniques are $4f$ methods, whereas the binary Fraunhofer technique is a $2f$ one. A cell in the holographic plane represents each element in $U({\nu }_x,{\nu }_y)$. And in turn, each element in $U({\nu }_x,{\nu }_y)$ will be represented by say $n\times n$ pixels. Figure 28(b) is a representation of a cell. As with the Goorden technique, there is a clear trade-off between the number of pixels within a cell and the overall resolution with which a pattern can be generated.

Fig. 28

Binary Fraunhofer holography method: (a) schematic representation of binary Fraunhofer holography method and (b) cell.

Figure 29(a) shows the pattern generated for an ${\mathrm{LG}}_{10}$ beam using the binary Fraunhofer holography method, and Fig. 29(b) shows the output in the Fourier plane. The zeroth order is bright and contains no phase information, just like a regular hologram. A zoomed version of the interference pattern obtained with one of the orders is shown as an inset. The fork pattern shows that the beam generated contains a spiral phase.

Fig. 29

Experimental output of binary Fraunhofer holography: (a) pattern for LG1 beam and (b) interference of ${\mathrm{LG}}_{10}$ beam and a reference beam.

Three techniques for using an amplitude element such as a DMD to generate complex amplitude patterns have been discussed. All these methods typically have a poor efficiency compared to direct phase modulation achieved using an active LCoS device. However, the DMD offers much faster switching, and the device itself is much cheaper than the LCoS device. Of the three techniques described above, the Lee method is most commonly used. However, the binary Fraunhofer method is simpler to set up due to the fact that it is a $2f$ method.

2.3.

Hybrid Approach

So far, a variety of active and passive approaches for generating OAM modes have been discussed, including diffractive elements, holographic elements, and DMDs. The serious drawback of all the methods described so far is that the OAM mode is generated in free space and is not amenable for coupling into an optical fiber. In this section, we discuss an all-fiber technique for the excitation of an OAM mode using a fused fiber coupler. Such an approach would be quite valuable for applications such as optical fiber communication.

The fused-coupler approach is based on selective coupling between different modes in two dissimilar fibers. Previously, such a technique has been used to achieve coupling between the fundamental $H{E}_{11}$ mode in an SMF and a selected higher-order LP mode group in a few-mode fiber (FMF).¹³⁶ However, the selective excitation of OAM modes within a mode group in typical step-index FMFs cannot be achieved using such couplers. This is due to the fact that the targeted modes in the FMF nearly degenerate with other OAM modes of the same $|l|$ ($l>1$) consisting of even/odd HE or EH eigenmodes, or, in the case of $l=1$, with the transverse electric ($T{E}_{01}$)/transverse magnetic modes ($T{M}_{01})$ modes. Since breaking such degeneracy in FMFs is essential for the excitation as well as stable propagation of the OAM modes,¹³⁷ a solid core fiber may be used. The solid core fiber breaks the degeneracy between the desired $H{E}_{21}$ modes and neighboring $T{E}_{01}$ or $T{M}_{01}$ modes, thereby allowing the preservation of the excited OAM mode without significant intermodal coupling.¹³⁷ Phase matching between the $H{E}_{11}$ mode in the SMF and the desired mode in the solid core fiber can be achieved by pretapering the SMF and then fusing both fibers.¹³⁶ As such, it provides a promising alternative pathway to directly couple the traditional Gaussian beam to the desired OAM modes with high purity and good stability in an entirely integrated scheme. In addition, this all-fiber device could potentially generate OAM modes with higher charge as the above phase-matching technique could be extended to other OAM modes supported by the solid core fiber¹³⁸

Based on simulations, the fused coupler described above is fabricated using the modified flame-brushing technique.¹³⁹ A schematic of the experimental setup used for characterization of the fabricated fused coupler is shown in Fig. 30. Light from a 1550-nm laser source (spectral width $\approx 0.08\mathrm{nm}$) is split into two arms using a commercial 3-dB coupler. Light from one arm of the 3-dB coupler is used as a reference for the interference setup to analyze the OAM beam at the output of the fused coupler. The output beam from the solid core fiber is collimated with a lens ($f=6\mathrm{mm}$) and imaged using a CCD camera (MicronViewer-7290A). The output beam from the reference SMF and solid core fiber (cleaved at $\approx 40\mathrm{cm}$ beyond the interaction region of coupler) are combined using a free-space beam splitter to form the interference pattern. The interference setup (shown in the dotted box) is used to determine the charge $\pm 1$ of the generated OAM mode by observing its characteristic fork and spiral patterns.

Fig. 30

Experimental setup for characterizing the generated OAM beam: solid lines are propagation path of light through the fiber; dashed lines are that of light through free space. PC, polarization controller; L1 and L2, collimating lens; FBS, free space beam splitter; QWP, quarter waveplate; and PBD, polarization beam displacer.

The above fused fiber coupler technique has been extended by different groups for exciting the OAM modes in fibers with different index profiles and also in photonic lanterns. Heng et al.¹⁴⁰ reported the generation of an OAM mode of charge 1 in a graded index FMF with a mode purity of 95% over 100 nm (1500 to 1600 nm) bandwidth. Wen et al.¹⁴¹ reported amplification of OAM modes generated using fused couplers, which have an 160-nm bandwidth (1480 to 1640 nm). This technique was further scaled for exciting the higher-order circularly polarized OAM modes $(l=\pm 2)$ using a ring core fiber,¹⁴² and a modal power extinction ratio of 12.9 dB was reported with respect to the neighboring modes. The fused coupler technique is employed in making mode-selective photonic lanterns (MSPLs) exciting five OAM modes in a ring core fiber with a bandwidth of 500 nm (1000 to 1500 nm).¹⁴³ More recently, Li et al.¹⁴⁴ have reported on the simulation of MSPL for three OAM modes with a modal cross talk of less than $-24\mathrm{dB}$. Such results help in highlighting the immense potential of the mode-selective fused fiber coupler approach for the generation of desired OAM modes.

2.4.

Generation of Optical Vector Beams

The fact that the state of polarization (SoP) across an optical beam need not be uniform at different spatial locations at the same time or at different times led to the realization of vector optical beam (VOB). VOBs are a class of solutions to the vector wave equation in the paraxial limit. These solutions along with axial symmetry around the beam axis are known as cylindrical vector beams (CVBs). Radial and azimuthal polarization can be considered as special cases of CVBs. The spatial inhomogeneity of the SoP of the optical beams can be cylindrically symmetric or asymmetric, leading to unconventional polarization states depending on the generation method. An authoritative review of this area of research presented previously ^145–148 provides details from fundamental aspects to emerging applications. The exploration of CVB capabilities is still largely limited to optical physicists and has recently been used as a tool to access electromagnetic properties of matter, including a novel way of studying and controlling edge currents in topological materials and for Floquet engineering of nonequilibrium states of matter.¹⁴⁹

In addition to the spatial variation in the SoP, it is possible to modify the phase front of the optical beam either independently or together. In the first situation, the wavefront can be made helical or in some alternate shape determined by a polynomial or special mathematical functional form. For example, this could be used to generate optical beams with vortices or beams that are Airy, HG or some other exotic form with unique phase structure. Such demonstrations have proved to be useful for several obvious and unforeseen applications in the classical and quantum domains. The following discussion provides a detailed, current status account of the generation, detection, and manipulation of optical vortex beams (OVB) with an OAM DoF.

Ensuring that the SoP and the phase DoFs of an optical beam are nonseparable results in beams that embed both characteristics. Such beams are known as “vector-vortex” beam (VVB),^150,151 polarization singular (PS) beams,^152,153 Poincaré beam (PB),¹⁵⁴ spin–orbit beams,¹⁵⁵ or classically entangled beams.¹⁵⁶ In all these paraxial optical beams, the SoP shows spatial variation due to the underlying variations in the phase structure. Under special circumstances, the spatial variation can also lead to the appearance of geometric phase due to variations in the SoP and can be understood as arising due to optical SOI.¹⁵⁷ SOI refers to the nonseparably coupled spin and OAM of a paraxial light beam. Mapping the phase-polarization structure on the Poincaré sphere, these beams are also known as PBs, a 2-D projection of the Poincaré sphere on any given plane. For example, a projection of the top half of the Poincaré sphere onto the (${\mathrm{S}}_1$–${\mathrm{S}}_2$) Stokes parameter plane leads to a beam, whose SoP in the beam cross section varies radially outward with different ellipticities from the axial (${\mathrm{S}}_3$) point. Such a construct leads to the appearance of “C-point” and “L-line” polarization singularities, corresponding, respectively, to a point where the orientation of the polarization ellipse is undetermined and a symmetric ring around which the spin direction is undetermined. The C-point corresponds to purely circular polarization state (${\mathrm{S}}_3=1$), also known as polarization vortex, and the L-line is where the SoP is linear (${\mathrm{S}}_3=0$). The appearance of C-point and L-line polarization singularities and spatially varying (vector) SoP gives the optical beams the appropriate names mentioned.

The lowest-order polarization singularities and the associated topological structure in the ellipse field typically classified as lemon, star, and monstar^158,159 are shown in Figs. 31(a)–31(f) along with the first higher-order spiral, node, and saddle structures. They are characterized by their index $I_c=\pm 1/2$, $\pm 1$, denoting the rotation of the disinclination around the C-point. Extending beyond the half-turn around the C-point, higher-order disclination indices $(I_c=\pm 2,3)$ have also been investigated and were recently experimentally demonstrated. Such optical beams are represented in higher-order spherical space, also known as the C-point sphere.^160,161

Fig. 31

Topological structures associated with singular points: (a) lemon, (b) monstar, and (c) star of index $\pm 1/2$ and (d) spiral, (e) node, and (f) saddle of index $\pm 1$.

The C-point singularity and the surrounding elliptical-linear SoP in the beam cross section are formed by superposing a helical-phase optical vortex (LG) beam and a plane wave Gaussian beam in two orthogonal circular SoPs.¹⁶² Alternatively, they may be formed via superposition of two LG beams in orthogonal circular SoP.¹⁶¹ Mathematically, they are represented as^161,162

2.4.1.

Generation methods

As described above, one of the characteristic features of vector wave fields is the appearance of C-point and L-line polarization singularities and disclinations^163–166 in the transverse cross section due to on-axis or off-axis superposition of two or more wave fields with different phase-polarization structures. In a field of elliptical polarization, C-point singularities are locations where the elliptical SoP degenerates to circular SoP, wherein the azimuth of the circular SoP is undefined. On the other hand, the L-line singularities correspond to locations where the polarization ellipse degenerates to linear SoP with undefined ellipse handedness. It is important to note that the C-point and L-line singularities are fundamental features of vector wave fields, where the SoP of the beam field is spatially inhomogeneous and elliptically polarized. These are an important source of information to explore a variety of fundamental aspects of light, light–matter interaction, and their applications.^166,167 These unique signature field patterns were first observed in the microwave region^168–170 and were subsequently measured and studied in several optical systems, including in speckle fields, in polychromatic fields, and in scattered radiation.¹⁶⁷ The speckle beam fields with a large number of randomly oriented polarization singularities are generated by propagation of polarized light through an inhomogeneous and weakly birefringent medium such as a multimode optical fiber. This results in an elliptically polarized field in the output rich in polarization singularities.¹⁷¹ Such elliptically polarized light fields with myriad of randomly located PS C-points and L-lines act as markers in a number of applications, including characterization of rough surfaces and biological tissues.¹⁶⁷

A well-defined optical beam field containing an elementary polarization cell embedded with one C-point bounded by an L-contour has also been demonstrated and applications using such beams are emerging. The polarization singularity features in a beam are associated with the optical AM of beam fields and appear due to the simultaneous presence of both the spin and OAM DoF in a beam. Optical beams with components of both SAM ($\pm \sigma =0$, 1) and OAM ($\pm l=0$, 1, 2) attributes present simultaneously, contribute to the total angular momentum (TAM) ($\pm \sigma l$) of the beam. Such beams have been generated using different methods and technology platforms.^150–157 Known initially as VVBs, the optical singularities are formed around a point where a scalar vortex is centered in at least one of the scalar components of the vector wave field. Propagating a polarized light beam through a two-mode optical fiber (TMF) instead of a multimode optical fiber generates a well-defined and controllable optical beam containing an elementary polarization cell embedded with one C-point bounded by L-contour.¹⁵² Such beams are formed due to the linear superposition of a collinearly propagating vortex mode and the fundamental mode in the optical fiber. Such beams have also been generated using geometric phase elements, such as subwavelength gratings,^172,173 wherein the Pancharatnam phase was shown to play a critical role in the AM of the beam.¹⁷⁴ The role played by the geometric phase in such beams is indicated by its evolution during propagation, which indicates that VVBs carry OAM. This led to the realization that VVBs can also be identified by the presence or absence of TAM carried by the beam, and geometric phase has been shown to play a critical role in this beam attribute. In addition to TMFs,^{150–153,162} VVBs have also been generated using a variety of methods, including interferometry,^{145–147,175} LC Q-plates,^125,176 optical crystals,^177–179 stressed optical element,^154,180 and, more recently, using structured metasurfaces^181–183 and in plasmonic-fields.¹⁸⁴

Generation of the family of VVBs using LCoS devices^74,185–187 and their recent off-shoot the DMDs^132,188 were notably missed in the above discussion as they belong to active and flexible generation methods. DMDs are finding more and more uses in research labs for the generation of a wide variety of complex beams. They are cheaper and faster than LCos devices. Though LCoS devices are considered expensive compared to other optical elements used in this area of research, the flexibility they bring to generating a large variety of phase-polarization structured optical beam far outweigh the cost. Moreover, they have almost become a staple in all optics-photonics research laboratories involved in the above research activity, which is rapidly pushing the cost down. Having been around for more than a decade now, a number of research articles have been reported on the use of LCoS SLMs to generate vortex beams, vector beams, VVBs, and a large number of phase-polarization-intensity-modulated complex light beams with custom purpose. Most of these are reviewed authoritatively in the references.^74,185–187 More interestingly, the spatially inhomogeneous polarization and helical-phase wavefront can coexist in the same light beam and thus form cylindrical VVBs or PBs of different orders.¹⁸⁹ Its perfect version¹⁹⁰ generated using SLMs provides additional DoFs with a large variety of patterns and offers an almost unlimited experimental flexibility for beam and thereby material manipulation. In addition to manipulation of the phase-polarization pattern of the light beam, the underlying geometric phase aspect of the beam has also been modified to generate PBs.¹⁹¹ Needless to emphasize, several alternate methods are being actively pursued as well.

Based on the formalism given in the introduction part of this section, we have generated the lowest order and the first few higher-order VVBs using a superposition of vortex (LG) beams and a Gaussian beam with suitable spin and orbital AM (Figs. 32 and 33). The Poynting vector density and the total AM¹⁹² of such beams are quite complex in the cross-section. In addition to integer charges, we were able to generate fractional charge VVBs, which could find potential application elsewhere. Decomposition of the beams into their fundamental constituents is typically carried out via interferometry, polarimetry,^{150–153,192} or vector decomposition using LCoS SLMs.^74,185–187

Fig. 32

PS beams generated using LCoS SLM and measured using polarimetry. Topological pattern of (a) star, (b) lemon, (c) fractional charge, and (d) node. The SoP of the beams measured (red and green colors) are plotted overlaying the beam intensity (yellow color); white color dots are the C-points and white color dotted rings are the L-line in the beam cross section. Inset shows the phase map of the beams highlighting helical phase structure, with yellow and blue colors corresponding to 0 and $\pi$ phases, respectively.

Fig. 33

(a)–(c) Higher-order PS beams generated using LCoS SLM; (a’)–(c’) corresponding phase maps.

A brief overview of the current status of vector-vortex optical beams generated with contributions from both spin and OAM DoFs has been presented in this section. The lower, higher, and fractional order VVB results presented here are generated using LCoS SLMs. In all cases, the appropriate circular polarization of the input beam and phase mask are projected on the LCoS SLM to generate the vector beam. The spatial polarization characteristics of the beams are obtained using imaging Stokes parameter measurements. Such phase-polarization structured optical beams are beginning to find applications in optical microscopy, communication, and material characterization.

3.1.

Geometric Transformation Method

One sorting technique that is used to find the weights of individual modes present in the incident beam is based on a Cartesian to log-polar transformation.^197,198 While this is an elegant method, it has some drawbacks, such as the limited range of azimuthal order modes that it can sort at a time and its inability to determine the weights of radial order modes. Owing to the use of two active LCoS devices, the overall efficiency was also quite poor. However, it is one of the most popular techniques used currently, with the lossy LCoS devices replaced either by diffractive or by refractive elements. The method is described in detail below.

This method is based on the fact that a set of superimposed plane waves propagating with different wave vectors can be focused into distinguishable spots at the focal plane of a convex lens, provided that each sees an additional phase shift that is a multiple of $2\pi$ across the lens. This is to ensure that the difference in spot position is comparable to the Rayleigh resolution limit.^199,200 The location of the spot depends on the topological charge of the OAM mode (Fig. 34).

Fig. 34

Configuration of an optical log-polar transformation-based mode sorter and its working principle. The leftmost image indicates the input, which could be the superposition of a number of OAM beams of different charges. This goes through the first element of the log-polar transformation, which unwraps the azimuthal phase profile of the OAM mode. The second phase element lies at the Fourier transform plane and provides the necessary phase correction that allows focusing of the beam into spots for each OAM present in the incident beam. This is indicated on the rightmost part of the figure. The color map of the two phase elements indicates the phase variation across the elements, with blue being 0 and red having the value $2\pi$.

The basis of this sorting technique is the conversion of OAM into linear momentum. The method requires two elements. The first unwraps the phase profile of an OAM mode azimuthally along the ring-shaped nonzero intensity region. Modes with higher azimuthal index number correspond to plane waves with larger wave vector.

The unwrapping operation is carried out by an optical log-polar transformation.²⁰¹ The amplitude of the unwrapped beam is obtained at the Fourier plane of the transformation element. The residual phase distortion introduced by the transformation element is compensated by a second element that provides the required phase correction at the Fourier plane to retrieve the phase ramp of the unwrapped beam. The setup is completely reversible, which means that a spatial array of Gaussian focal spots at the focal plane of the lens will be converted into superimposed plane waves with different tilt. These beams then pass through the phase corrector and the transformation element to produce multiplexed OAM beams. The phase profile of the transformation element ${\theta }_1(x;y)$ and phase correction element ${\theta }_2(u;v)$ are given as^202,203

The parameter $d$ defines the length of the unwrapped beam and is selected such that the unwrapped beam covers 80% of the width of the corrector. The parameter $a=d/2\pi$ ensures that the unwrapped beam is fully mapped on the width of $d$. The parameter $b$ allows independent control of the translation of the unwrapped beam in the $u$ direction. The $f$ is the focal length of the Fourier transforming lens used. Figure 35 illustrates the results of simulation carried out to demonstrate the demultiplexing process.

Fig. 35

The simulation results for various OAM mode inputs.

One limitation of this method is that there will be a slight overlap of OAM mode content in the output, especially while sorting consecutive OAM modes. This limits the use of this system for single-photon applications. However, the resolution can be improved by incorporating a fan-out element at the output of unwrapper and phase corrector element, followed by a fan-out phase correction element. Therefore, four optical elements are required to improve the resolution. Recent research work²⁰⁴ has demonstrated that the fan-out element can be integrated along with the phase profile of the transformation element that creates multiple coherent copies of the unwrapped beam at the Fourier plane of the unwrapper. The second element has to correct the relative phase between these copies at the Fourier plane of the fan-out element.^205–207 Recent research work has also demonstrated that performing a spiral transformation instead of a log-polar transformation can also improve the resolution of the mode-sorter output.²⁰⁸ However, all these methods improve resolution at the expense of making the setup more complex.

The log-polar technique has proved to be one of the most used methods of carrying out mode sorting. While refractive elements are the most efficient, most researchers use phase diffractive elements. These elements are either LC SLMs or elements fabricated using a variety of lithography and etching techniques. Although the log-polar optical transformation is efficient, the need for two elements to complete the demultiplexing still makes the entire operation occupy a large footprint. Extensions to this work, therefore, focused on miniaturing the setup, by integrating the two optical components into a single DOE.^204,209 This was achieved with a neat but simple trick. As the geometric transformation element acts on a ring of light, the central part of the element is unused. This region was filled with the second element structure. In other words, the first element occupied the outer ring of the new combined DOE, whereas the second one lay at the center. The light that had traversed the first element (outer ring) was reflected and made to travel through the second element (central region).

3.2.

Optical Correlation Technique

In contrast to the mode-sorting techniques outlined above, modal decomposition techniques create a representation of the complex beam as a linear summation of the eigenmodes. Some of the modal decomposition techniques include the spatial and spectral $(S^2)$ imaging technique,²¹⁰ numerical methods,^211,212 phase extraction methods,^213,214 and the optical correlation technique.^74,215,216 The relative merits of these techniques are mentioned below.

The $S^2$ imaging technique involves a complex experimental setup (broadband source or tunable laser, optical spectrum analyzer, and a two-axis mechanical shifter) and requires a long measurement time.²¹⁰ The crucial element of numerical method-based modal decomposition is the optimization algorithm such as the Gerchberg–Saxton algorithm,²¹¹ stochastic parallel gradient descent algorithm,²¹² and hybrid genetic algorithm.²¹⁷ These optimization algorithms easily suffer from initial value sensitivity and local minima problems. The ring technique is one of the modal decomposition techniques, which does not use any optimization algorithm.²¹⁸ However, the drawback of this technique is it works only for the zero radial order OAM modes.

Phase extraction methods, such as phase-shifting digital holography,²¹³ require a reference beam to extract the phase structure of the complex field. Primary limitations of this technique are that it requires a stable interferometric setup and a coherent laser source at the receiver end. On the other hand, optical correlation-based modal decomposition has been demonstrated to address a wide range of the spectrum, although the accuracy may be limited by the specific algorithm that is used. For example, Xie et al.²¹⁹ experimentally demonstrated the 15-dB power extinction between the zeroth and first radial order LG modes using an optical correlation technique. The power extinction in their demonstration is limited by the fact that they have used only the phase structure of the LG modes in the decomposition process. Optical correlation-based modal decomposition considering both amplitude and phase structures gives accurate modal weights of the complex beam. Such a technique has been employed to find the modal weights of the optical fiber output in the LP basis.²¹⁵ This optical correlation technique can also be used to find the weights of an optical fiber output in OAM basis having both azimuthal and radial order modes. As discussed previously, OAM and SAM are coupled together for fiber modes. Hence, in order to implement optical correlation, OAM and SAM have to be decoupled.

In this section, we discuss optical correlation techniques for scalar OAM modes. It should be noted that any scalar light beam ($U$) can be represented by a superposition of $L{G}_{l,p}$ modes with corresponding complex weights $W_{l,p}$, as shown in Eq. (15). The complex weight of a mode is calculated by optically correlating the input beam with the complex conjugate of the mode, as shown in Eq. (16).

Equation (16) is essentially a dot-product operation between the input beam and the complex conjugate of the mode. This operation is performed experimentally by using an SLM and a convex lens.²²⁰

The input beam $L{G}_{l,p}$ is multiplied with its complex conjugate mode using an SLM by encoding necessary holograms. To perform the integral operation, the reflected beam is propagated through the convex lens. The electric field corresponding to the reflected beam at the Fourier plane of the convex lens is given in Eq. (17).^57,220

The center of the Fourier plane $U_f(0,0)$, which is commonly referred as the DC component, gives the weight of the $L{G}_{l,p}$ mode. Since the camera captures the intensity, the square root of the observed intensity is calculated to obtain the field weight. However, it should be noted that the phase information of the beam is lost in this process and, as such, the relative phase between different modes is not considered.

Based on the above optical correlation algorithm, modal decomposition of azimuthal, as well as radial order LG modes, is performed. The schematic of the experimental setup used for implementing the optical correlation technique is shown in Fig. 36.²²⁰ One half of the SLM is programmed to generate the required LG beam and the other half is programmed to perform the dot-product operation as part of the modal decomposition.

Fig. 36

Experimental setup for implementing the optical correlation technique. DFB, distributed feedback laser; $L_1$ ($\text{focal length}=10\mathrm{cm}$) and $L_2$ ($\text{focal length}=20\mathrm{cm}$), convex lens; PH, pinhole; P, polarizer; SLM, spatial light modulator; M, mirror; LLF, laser line filter at 1064 nm; CCD, charge-coupled device camera.

In the above experiments, high extinction (27 dB, limited by the camera bit resolution) between the matched and the orthogonal cases are achieved when two radial order modes ($p=0$, 1) with two azimuthal order modes ($l=0$, 1) are used. To explore the scalability to higher-order modes, the above optical correlation experiments are extended to higher azimuthal order modes ($l=-4$ to $+4$). The observations from such experiments are summarized using a parity plot, as shown in Fig. 37.²²⁰ For simplicity, a constant decomposition CGH radius $R$ of 1.0 mm is used for these experiments. The diagonal elements of the parity plot represent the power measured when the generated and decomposition CGH corresponding to LG modes are matched and the off-diagonal elements represent the power coupled to neighboring modes. The observations from Fig. 37 are the following: when LG modes with $p=0$ and $l=-4$ to $+4$ are generated and correlated with matched decomposition CGH patterns (corresponding to diagonal values in Q1 quadrant), the normalized power is uniform and maximum. However, a finite amount of power is observed when correlated with the orthogonal $p=1$ decomposition CGH patterns (corresponding to diagonal values in Q2). For $l=0$ mode, an extinction of $\approx 23\mathrm{dB}$ is observed. This degradation of extinction compared to an expected value of 27 dB is due to the nonoptimal decomposition CGH radius of 1 mm used in the above measurements. Furthermore, an increase in the power measured is observed as we increase the azimuthal number ($l$). For example, when $L{G}_{-40}$ is generated, we noticed only 15.5 dB extinction between $L{G}_{-40}$ and $L{G}_{-41}$ modes. This degradation in extinction for higher-order azimuthal modes is once again attributed to the wavefront aberration introduced by the SLM, as well as the associated optics, for a beam of higher-order transverse mode such as the LG modes studied here.^221,222

Fig. 37

Normalized power measured from optical correlation technique for different combinations of generated and decomposition $LG$ modes with radial mode order $p=0$, 1 and azimuthal mode order $l=-4$ to $+4$. Diagonal elements represent parity and the off-diagonal elements represent the magnitude of coupling to neighboring modes. Q1–Q4 corresponds to the different quadrants in the above plot.

Interestingly, when LG modes with $p=1$ are generated, the power coupling to the $p=0$ LG modes is much lower (extinction of 27 dB, corresponding to diagonal values in Q4). Specifically, when $L{G}_{-41}$ is generated, 27-dB extinction between $L{G}_{-41}$ and $L{G}_{-40}$ modes is observed (which is better than the converse case). This is not surprising since the optimal decomposition CGH radius is used for this case compared to the above case.

It should be noted that a similar optical correlation technique has been reported recently by other groups. A collinear phase-shifting holographic method²¹³ is reported with a modal extinction ratio of 24 dB between $L{G}_{10}$ and $L{G}_{11}$. Another optical correlation technique based on a phase-retrieval method²²³ is reported for decomposing $L{G}_{lp}(l=0,0\le p\le 2)$ modes with $<10\mathrm{dB}$ modal extinction ratio between the mode of interest and the neighboring modes. An optical correlation technique based on intensity flattening²²⁴ is reported for decomposing $L{G}_{lp}(l=0,0\le p\le 7)$ radial order modes. Unfortunately, modal extinction ratios of the decomposed modes are not presented in the article. A maximum visibility (ratio of the power in the desired mode to the power in all the modes) of 99.1% is reported. Finally, a radial mode sorter²²⁵ has been reported for decomposing $L{G}_{lp}(-2\le l\le \mathrm{2,0}\le p\le 1)$ modes with a maximum modal extinction ratio of 10 dB.

In comparison to the above methods, our method is tested for $L{G}_{l,p}(-4\le l\le 4,0\le p\le 1)$ modes having both azimuthal and radial orders.²²⁰ A modal extinction ratio of 27 dB is obtained (limited by our camera resolution), irrespective of the input mode, which is greater than any of the extinction ratios of the above-mentioned schemes. A visibility of 99.9% is obtained in our experiments,²²⁰ which is also greater than the visibility obtained in the optical correlation technique based on intensity flattening.²²⁴

So far, modal decomposition using an optical correlation algorithm is discussed for the case wherein only a single mode is generated. In several practical applications, including MDM systems, modal decomposition has to be performed for composite (two or more) modes. As such, the above investigation is extended to the case of a composite beam consisting of 10 different LG modes (azimuthal mode indices $l=-2$ to $+2$ and radial mode indices $p=0$, 1). A sample modal distribution is chosen such that the generated mode weights (denoted as markers in Fig. 38²²⁰) are halved for consecutive azimuthal mode index (both positive and negative values). As in the above case, the decomposed modal weights agree well with the generated modal weights across the entire mode spectrum. The relatively high error observed for the $L{G}_{-20}$ and $L{G}_{20}$ modes are due to the nonoptimal radius of the decomposition CGH, as explained previously. However, the intensity structure of the reconstructed beam [illustrated in Fig. 38(c)] based on the experimentally obtained modal weights closely resembles the generated composite beam of Fig. 38(a).

Fig. 38

(a) Intensity structure of the generated composite beam consisting of 10 $LG$ modes. (b) Weights of the generated (markers) and decomposition (bars) $LG$ mode spectrum for radial orders $p=0$, 1 plotted as a function of different azimuthal mode indices $l=-4$ to $+4$. (c) Intensity structure simulated using the experimentally obtained modal weights.