An optical vortex, or a phase singularity, is a unique point within an optical field where light intensity reaches zero due to the twisting of light waves, resembling a corkscrew pattern. It is one type of optical singularity or zero of an optical field, that has a spiral phase wave front around a singularity point where the phase is undefined. This twisting of light waves is a consequence of variations in the phase of the light wave as it propagates through space. Due to the twist, light waves along the central axis cancel each other out. This twisting imparts a distinctive spatial phase distribution, resulting in a central dark hole when projected onto a plane, as seen in Figure 1.

Figure 1

Optical vortices are assigned a topological charge, an integer that quantifies the number of twists occurring within a single wavelength. The charge can be either positive or negative, depending on the twist direction, with higher charges indicating faster light rotation around the vortex's axis.

This rotation carries orbital angular momentum (OAM) with the light wave, which is associated with the polarization of light. Spin angular momentum often generates circularly polarized light, whereas orbital angular momentum (OAM) manifests in the rotational motion of trapped particles. Interference between an optical vortex and a plane wave reveals helical phase patterns with a number of spiral arms equal to the topological charge.

Optical vortices are studied by creating them in the laboratory. It can be generated directly within a laser, or the transformation of a laser beam into an optical vortex. This transformation is achieved through various methods, such as:

1. Spiral-phase delay structures
2. Computer-generated holograms
3. Birefringent vortices in materials

Generation of Optical Vortex using Spiral-phase delay structures

Using spiral phase plates is the most obvious approach for the generation of optical vortices. The spiral phase plate (SPP) is an optical element with increasing optical thickness in the azimuthal direction. This direction refers to the rotation around a circular or spherical object, measured in the horizontal plane, typically centered around the plate.

Figure 2

Spiral phase plates can transform the input Gaussian beam into an output vortex beam. It is important to ensure that the variation in optical thickness, ranging from its maximum to minimum, matches the specific wavelength (λ) designated for the phase plate.

When a beam with a uniform phase front passes through the plate, its azimuthal angle spans from 0 to 2π, resulting in the generation of an optical vortex beam with a unit topological charge. Changing the sign of the topological charge is achieved by reversing (flipping) the phase plate.

Spiral phase plates operate by applying a direct phase shift to the incoming light. These plates are constructed from a piece of transparent material with gradually increasing, spiraling thickness. In optically dense mediums, light slows down, requiring more time to traverse a specific distance compared to its speed in air. When discussing phase shifts, it is more convenient to consider the shift in terms of distance rather than time. This interpretation gives rise to the notion of optical path length, the apparent distance that light appears to travel within any given medium.

Where, n, the index of refraction, varies with distance s. 

A thicker plate corresponds to a longer optical path length and a more significant phase shift. The spiral thickness of the phase plate generates the spiraling phase pattern characteristic of an optical vortex. For the phase plate to function properly, it must be precisely shaped and smoothed, accurate to a fraction of a wavelength.

Generation of Optical Vortex using Computer-generated Holograms

Utilizing computer-generated holograms (CGHs) is another effective technique for creating beams with a specific singular phase distribution. Computer-generated holograms (CGH’s) are diffractive-optical elements that enable the development of wave-optical display systems fully controlled by computers. They are specialized types of holograms that are created and calculated using computer algorithms. Unlike traditional holograms that are recorded on photosensitive materials using laser beams, CGHs are purely digital and exist in the form of numerical data. These numerical data represent complex patterns of light waves that can be used to reconstruct three-dimensional scenes or create specific optical effects.

CGHs are diffractive-optical elements because they rely on the principle of diffraction, where light waves are bent or spread out as they pass through or around obstacles. In the case of CGHs, the digital patterns encode the information required for diffraction, allowing the reconstruction of complex wavefronts.

CGHs enable precise control over the phase and amplitude of light waves. By manipulating these properties, CGHs can create specific wavefronts that result in unique optical effects, such as creating 3D holographic images, shaping laser beams, or generating intricate light patterns. They find applications in holographic displays, optical testing, laser beam shaping, and virtual reality.

Optical vortex can be written as:

where l is the topological charge

θ is an angle in the plane transverse to the direction of propagation 

Consider a plane wave u,

If the recording system is located at z=0, then the intensity distribution can be found by squaring the sum of the two amplitude functions:

A fourier transform gives the transmittance function that used to create the diffraction gratings.

where α is the amplitude of the phase modulation, T0 is the constant absorption coefficient of the hologram, and Λ is period of the grating (fringe spacing).

Figure 3

Interference patterns resulting from the interaction between plane waves and optical vortices exhibit a distinctive fork shape at their center. The vortex's charge can be identified by counting the number of forks or by subtracting one from the count of prongs (the distinct arms or branches in the fork-shaped pattern that is created). Since these patterns act as diffraction gratings, passing a plane wave through them generates multiple vortices with charges such as {. . . , −l, 0, +l,. . . }, where the negative charges exhibit a phase ramp in the opposite direction.

Generation of Optical Vortex using Birefringent Vortices

The generation of optical vortices using birefringent vortices involves exploiting the birefringent properties of certain materials to create structured light beams with a helical phase front. Birefringence is the property of a material to have different refractive indices for light polarized along different axes. Passing light through such materials enables the introduction of a phase singularity that creates an optical vortex.

Steps for generation of optical vortex

1. Select a suitable birefringent material, such as a uniaxial crystal or a liquid crystal, which exhibits different refractive indices for different polarizations.
2. Create a phase mask that introduces a helical phase structure. This can be done mathematically to produce the desired phase profile that corresponds to an optical vortex. The helical phase structure is what gives rise to the vortex pattern in the resulting light beam.
3. Direct the light through the birefringent material. As the light passes through the material, the different polarizations experience different refractive indices, introducing a relative phase shift between them.
4. Manipulate polarization of light using the birefringent properties. This polarization manipulation is crucial because the differential phase shift between polarizations creates the helical phase structure necessary for the optical vortex.
5. When the light waves with different polarizations exit the birefringent material, they interfere and superpose. The interference pattern results in a structured light beam with an optical vortex at its center.

Optical vortices, primarily characterized by their phase properties, cannot be detected from its intensity profile alone, especially when vortex beams of the same order exhibit similar intensity distributions. Therefore, a diverse set of interferometric techniques are employed to study and understand them.

One straightforward approach involves interfering an optical vortex beam with an inclined plane wave, resulting in the formation of a fork-like interferogram as seen in Figure 3. 

By carefully analyzing the number of forks within the pattern and their relative orientations, it becomes possible to precisely determine the vortex order, which is the number of times the phase of the wave wraps around the singularity point, and its corresponding sign.

Another method involves transforming a vortex beam into its characteristic lobe structure, the specific pattern of intensity distribution in the cross-section of the beam, by allowing it to pass through a tilted lens. This transformation occurs due to self-interference between different phase points within the vortex.

Figure 4

When a vortex beam with order 'l' passes through a tilted convex lens, it will create a pattern of intensity lobes around the depth of focus. The number of these lobes is given by 'n = l + 1'. Each lobe represents a discrete intensity peak in the transverse intensity profile of the beam as seen in figure 4.

The orientation of these lobes, whether they form a right-handed or left-handed diagonal pattern, provides information about the sign of the orbital angular momentum associated with the vortex beam. Right-handed and left-handed diagonal lobes correspond to positive and negative orbital angular momentum orders, respectively.

Applications of Optical Vortex

Optical vortices have a wide range of applications across various fields, including communications and imaging:

• Direct Observation of Extrasolar Planets: Optical vortex coronagraphs are used to observe extrasolar planets that are typically challenging to detect due to the brightness of their parent stars. These coronagraphs enable the direct observation of planets with low contrast ratios to their parent stars.
• Optical Tweezers for Particle Manipulation: Optical vortices are employed in optical tweezers to manipulate micrometer-sized particles like cells. They allow for the rotation of particles in orbits around the axis of the vortex beam. Optical vortex tweezers have also been used to create micro-motors.
• Communication Bandwidth Enhancement: Optical vortices can significantly improve communication bandwidth. Twisted radio beams, for instance, can increase radio spectral efficiency by utilizing different vortical states. This orbital angular momentum-based multiplexing has the potential to increase system capacity and spectral efficiency in millimeter-wave wireless communication.
• Quantum Computing: Optical vortices, with their theoretically infinite number of states in free space, hold promise for quantum computing applications. These states could enable faster data manipulation, and they are of interest to the cryptography community for higher bandwidth communication.
• Super-Resolution Microscopy: In optical microscopy, optical vortices are used to achieve spatial resolution beyond the normal diffraction limits. Techniques like Stimulated Emission Depletion (STED) Microscopy utilize the low-intensity singularity at the center of the vortex beam to deplete fluorophores around a specific area without affecting the target area.
• Study of Quantum Vortices: Optical vortices can be resonantly transferred into polariton fluids of light and matter to study the dynamics of quantum vortices under various interaction regimes, both linear and nonlinear.
• Non-local Correlations in Quantum Photon Pairs: Optical vortices can be identified in the non-local correlations of entangled photon pairs, contributing to the study of quantum phenomena. 

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