1.
Optical vortex
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An optical vortex is a zero of an optical field, a point of zero intensity. Research into the properties of vortices has thrived since a paper by John Nye and Michael Berry, in 1974. The research that became the core of what is now known as singular optics. In an optical vortex, light is twisted like a corkscrew around its axis of travel, because of the twisting, the light waves at the axis itself cancel each other out. When projected onto a surface, an optical vortex looks like a ring of light. This corkscrew of light, with darkness at the center, is called an optical vortex, the vortex is given a number, called the topological charge, according to how many twists the light does in one wavelength. The number is always an integer, and can be positive or negative, the higher the number of the twist, the faster the light is spinning around the axis. This spinning carries orbital angular momentum with the train. This orbital angular momentum of light can be observed in the motion of trapped particles. Interfering an optical vortex with a wave of light reveals the spiral phase as concentric spirals. The number of arms in the spiral equals the topological charge, Optical vortices are studied by creating them in the lab in various ways. An optical singularity is a zero of an optical field, the phase in the field circulates around these points of zero intensity. Vortices are points in 2D fields and lines in 3D fields, integrating the phase of the field around a path enclosing a vortex yields an integer multiple of 2π. This integer is known as the charge, or strength. A hypergeometric-Gaussian mode has a vortex in its center. The beam, which has the form ψ ∝ e i m ϕ e − r 2, is a solution to the wave equation consisting of the Bessel function. Photons in a hypergeometric-Gaussian beam have an angular momentum of mħ. The integer m also gives the strength of the vortex at the beams centre, spin angular momentum of circularly polarized light can be converted into orbital angular momentum

2.
Laser
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A laser is a device that emits light through a process of optical amplification based on the stimulated emission of electromagnetic radiation. The term laser originated as an acronym for light amplification by stimulated emission of radiation, the first laser was built in 1960 by Theodore H. Maiman at Hughes Research Laboratories, based on theoretical work by Charles Hard Townes and Arthur Leonard Schawlow. A laser differs from other sources of light in that it emits light coherently, spatial coherence allows a laser to be focused to a tight spot, enabling applications such as laser cutting and lithography. Spatial coherence also allows a laser beam to stay narrow over great distances, Lasers can also have high temporal coherence, which allows them to emit light with a very narrow spectrum, i. e. they can emit a single color of light. Temporal coherence can be used to produce pulses of light as short as a femtosecond, Lasers are distinguished from other light sources by their coherence. Spatial coherence is typically expressed through the output being a narrow beam, Laser beams can be focused to very tiny spots, achieving a very high irradiance, or they can have very low divergence in order to concentrate their power at a great distance. Temporal coherence implies a polarized wave at a single frequency whose phase is correlated over a great distance along the beam. A beam produced by a thermal or other incoherent light source has an amplitude and phase that vary randomly with respect to time and position. Lasers are characterized according to their wavelength in a vacuum, most single wavelength lasers actually produce radiation in several modes having slightly differing frequencies, often not in a single polarization. Although temporal coherence implies monochromaticity, there are lasers that emit a broad spectrum of light or emit different wavelengths of light simultaneously, there are some lasers that are not single spatial mode and consequently have light beams that diverge more than is required by the diffraction limit. However, all devices are classified as lasers based on their method of producing light. Lasers are employed in applications where light of the spatial or temporal coherence could not be produced using simpler technologies. The word laser started as an acronym for light amplification by stimulated emission of radiation, in the early technical literature, especially at Bell Telephone Laboratories, the laser was called an optical maser, this term is now obsolete. A laser that produces light by itself is technically an optical rather than an optical amplifier as suggested by the acronym. It has been noted that the acronym LOSER, for light oscillation by stimulated emission of radiation. With the widespread use of the acronym as a common noun, optical amplifiers have come to be referred to as laser amplifiers. The back-formed verb to lase is frequently used in the field, meaning to produce light, especially in reference to the gain medium of a laser. Further use of the laser and maser in an extended sense, not referring to laser technology or devices, can be seen in usages such as astrophysical maser

3.
Helix
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A helix is a type of smooth space curve, i. e. a curve in three-dimensional space. It has the property that the tangent line at any point makes a constant angle with a line called the axis. Examples of helices are coil springs and the handrails of spiral staircases, a filled-in helix – for example, a spiral ramp – is called a helicoid. Helices are important in biology, as the DNA molecule is formed as two intertwined helices, and many proteins have helical substructures, known as alpha helices, the word helix comes from the Greek word ἕλιξ, twisted, curved. Helices can be either right-handed or left-handed, handedness is a property of the helix, not of the perspective, a right-handed helix cannot be turned to look like a left-handed one unless it is viewed in a mirror, and vice versa. Most hardware screw threads are right-handed helices, the alpha helix in biology as well as the A and B forms of DNA are also right-handed helices. The Z form of DNA is left-handed, the pitch of a helix is the height of one complete helix turn, measured parallel to the axis of the helix. A double helix consists of two helices with the axis, differing by a translation along the axis. A conic helix may be defined as a spiral on a conic surface, an example is the Corkscrew roller coaster at Cedar Point amusement park. A circular helix, has constant band curvature and constant torsion, a curve is called a general helix or cylindrical helix if its tangent makes a constant angle with a fixed line in space. A curve is a general helix if and only if the ratio of curvature to torsion is constant, a curve is called a slant helix if its principal normal makes a constant angle with a fixed line in space. It can be constructed by applying a transformation to the frame of a general helix. Some curves found in nature consist of multiple helices of different handedness joined together by transitions known as tendril perversions, in mathematics, a helix is a curve in 3-dimensional space. The following parametrisation in Cartesian coordinates defines a particular helix, Probably the simplest equations for one is x = cos , y = sin , z = t. As the parameter t increases, the point traces a right-handed helix of pitch 2θ and radius 1 about the z-axis, in cylindrical coordinates, the same helix is parametrised by, r =1, θ = t, h = t. A circular helix of radius a and slope b/a is described by the following parametrisation, another way of mathematically constructing a helix is to plot the complex-valued function exi as a function of the real number x. The value of x and the real and imaginary parts of the function value give this plot three real dimensions, except for rotations, translations, and changes of scale, all right-handed helices are equivalent to the helix defined above. The equivalent left-handed helix can be constructed in a number of ways, in music, pitch space is often modeled with helices or double helices, most often extending out of a circle such as the circle of fifths, so as to represent octave equivalency

4.
Paraxial approximation
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In geometric optics, the paraxial approximation is a small-angle approximation used in Gaussian optics and ray tracing of light through an optical system. A paraxial ray is a ray which makes an angle to the optical axis of the system. Generally, this allows three important approximations for calculation of the path, namely, sin θ ≈ θ, tan θ ≈ θ. The paraxial approximation is used in Gaussian optics and first-order ray tracing, ray transfer matrix analysis is one method that uses the approximation. In some cases, the approximation is also called paraxial. The approximations above for sine and tangent do not change for the second-order paraxial approximation, the second-order approximation is accurate within 0. 5% for angles under about 10°, but its inaccuracy grows significantly for larger angles. For larger angles it is necessary to distinguish between meridional rays, which lie in a plane containing the optical axis, and sagittal rays. Paraxial Approximation and the Mirror by David Schurig, The Wolfram Demonstrations Project

5.
Angular momentum
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In physics, angular momentum is the rotational analog of linear momentum. It is an important quantity in physics because it is a conserved quantity – the angular momentum of a system remains constant unless acted on by an external torque. The definition of momentum for a point particle is a pseudovector r×p. This definition can be applied to each point in continua like solids or fluids, unlike momentum, angular momentum does depend on where the origin is chosen, since the particles position is measured from it. The angular momentum of an object can also be connected to the angular velocity ω of the object via the moment of inertia I. However, while ω always points in the direction of the rotation axis, Angular momentum is additive, the total angular momentum of a system is the vector sum of the angular momenta. For continua or fields one uses integration, torque can be defined as the rate of change of angular momentum, analogous to force. Applications include the gyrocompass, control moment gyroscope, inertial systems, reaction wheels, flying discs or Frisbees. In general, conservation does limit the motion of a system. In quantum mechanics, angular momentum is an operator with quantized eigenvalues, Angular momentum is subject to the Heisenberg uncertainty principle, meaning only one component can be measured with definite precision, the other two cannot. Also, the spin of elementary particles does not correspond to literal spinning motion, Angular momentum is a vector quantity that represents the product of a bodys rotational inertia and rotational velocity about a particular axis. Angular momentum can be considered an analog of linear momentum. Thus, where momentum is proportional to mass m and linear speed v, p = m v, angular momentum is proportional to moment of inertia I. Unlike mass, which only on amount of matter, moment of inertia is also dependent on the position of the axis of rotation. Unlike linear speed, which occurs in a line, angular speed occurs about a center of rotation. Therefore, strictly speaking, L should be referred to as the angular momentum relative to that center and this simple analysis can also apply to non-circular motion if only the component of the motion which is perpendicular to the radius vector is considered. In that case, L = r m v ⊥, where v ⊥ = v sin θ is the component of the motion. It is this definition, × to which the moment of momentum refers

6.
Circular polarization
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In electrodynamics the strength and direction of an electric field is defined by its electric field vector. In the case of a polarized wave, as seen in the accompanying animation. At any instant of time, the field vector of the wave describes a helix along the direction of propagation. Circular polarization is a case of the more general condition of elliptical polarization. The other special case is the linear polarization. The phenomenon of polarization arises as a consequence of the fact that light behaves as a transverse wave. On the right is an illustration of the electric field vectors of a circularly polarized electromagnetic wave, the electric field vectors have a constant magnitude but their direction changes in a rotary manner. Given that this is a wave, each vector represents the magnitude. Specifically, given that this is a circularly polarized plane wave, refer to these two images in the plane wave article to better appreciate this. This light is considered to be right-hand, clockwise circularly polarized if viewed by the receiver, as a result, the magnetic field vectors would trace out a second helix if displayed. Circular polarization is often encountered in the field of optics and in this section, refer to the second illustration on the right. The vertical component and its plane are illustrated in blue while the horizontal component. Notice that the horizontal component leads the vertical component by one quarter of a wavelength. The result of this alignment is that there are select vectors, corresponding to the helix, to appreciate how this quadrature phase shift corresponds to an electric field that rotates while maintaining a constant magnitude, imagine a dot traveling clockwise in a circle. Consider how the vertical and horizontal displacements of the dot, relative to the center of the circle, now referring again to the illustration, imagine the center of the circle just described, traveling along the axis from the front to the back. The circling dot will trace out a helix with the displacement toward our viewing left, the next pair of illustrations is that of left-handed, counter-clockwise circularly polarized light when viewed by the receiver. Because it is left-handed, the horizontal component is now lagging the vertical component by one quarter of a wavelength rather than leading it. To convert a given handedness of polarized light to the other one can use a half-wave plate

7.
Spin angular momentum of light
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The spin angular momentum of light is the component of angular momentum of light that can be associated with the waves circular or elliptical polarization. An electromagnetic wave is said to have circular polarization when its electric and magnetic fields rotate continuously around the beam axis during the propagation, the circular polarization is left or right depending on the field rotation direction. This SAM is directed along the beam axis, the above figure shows the instantaneous structure of the electric field of left and right circularly polarized light in space. The green arrows indicate the propagation direction, the mathematical expressions reported under the figures give the three electric-field components of circularly polarized plane wave propagating in the z direction, in complex notation. Monochromatic-wave case, S = ϵ02 i ω ∫ d 3 r, in particular, this expression shows that the SAM is nonzero when the light polarization is elliptical or circular, while it vanishes if the light polarization is linear. In this case, for a photon the SAM can only have two values, S z = ± ℏ. The corresponding eigenfunctions describing photons with well defined values of SAM are described as circularly polarized waves, helmholtz equation Orbital angular momentum of light Photon polarization Spin polarization

8.
Orbital angular momentum of light
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The orbital angular momentum of light is the component of angular momentum of a light beam that is dependent on the field spatial distribution, and not on the polarization. It can be split into an internal and an external OAM. The internal OAM is an origin-independent angular momentum of a beam that can be associated with a helical or twisted wavefront. The external OAM is the origin-dependent angular momentum that can be obtained as cross product of the beam position. A beam of light carries a linear momentum P, and hence it can be attributed an external angular momentum L e = r × P. This external angular momentum depends on the choice of the origin of the coordinate system, if one chooses the origin at the beam axis and the beam is cylindrically symmetric, the external angular momentum will vanish. The external angular momentum is a form of OAM, because it is unrelated to polarization, a more interesting example of OAM is the internal OAM appearing when a paraxial light beam is in a so-called helical mode. Helical modes of the field are characterized by a wavefront that is shaped as a helix, with an optical vortex in the center. The helical modes are characterized by a number m, positive or negative. If m =0, the mode is not helical and the wavefronts are multiple disconnected surfaces, for example, a sequence of parallel planes. If m = ±1, the handedness determined by the sign of m, the wavefront is shaped as a single helical surface, with a step length equal to the wavelength λ. If | m | ⩾2, the wavefront is composed of | m | distinct but intertwined helices, with the length of each helix surface equal to | m | λ. The integer m is also the so-called topological charge of the optical vortex, Light beams that are in a helical mode carry nonzero OAM. In the figure to the right, the first column shows the beam wavefront shape, the second column is the optical phase distribution in a beam cross-section, shown in false colors. The third column is the intensity distribution in a beam cross-section. Photons corresponding to such a beam each have an OAM of m ℏ directed along the beam axis, as an example, any Laguerre-Gaussian mode with rotational mode number l≠0 has such a helical wavefront. The i -superscripted symbols denote the components of the corresponding vectors. For a monochromatic wave this expression can be transformed into the following one and this expression is generally nonvanishing when the wave is not cylindrically symmetric