Self-focusing is a non-linear optical process induced by the change in refractive index of materials exposed to intense electromagnetic radiation.A medium whose refractive index increases with the electric field intensity acts as a focusing lens for an electromagnetic wave characterised by an initial transverse intensity gradient, as in a laser beam. The peak intensity of the self-focused region keeps increasing as the wave travels through the medium, until defocusing effects or medium damage interrupt this process. Self-focusing of light was discovered by Gurgen Askaryan.
Nonlinear optics (NLO) is the branch of optics that describes the behaviour of light in nonlinear media, that is, media in which the polarization density P responds non-linearly to the electric field E of the light. The non-linearity is typically observed only at very high light intensities (values of atomic electric fields, typically 108 V/m) such as those provided by lasers. Above the Schwinger limit, the vacuum itself is expected to become nonlinear. In nonlinear optics, the superposition principle no longer holds.
In optics, the refractive index or index of refraction of a material is a dimensionless number that describes how fast light propagates through the material. It is defined as
In physics, electromagnetic radiation refers to the waves of the electromagnetic field, propagating (radiating) through space, carrying electromagnetic radiant energy. It includes radio waves, microwaves, infrared, (visible) light, ultraviolet, X-rays, and gamma rays.
Self-focusing is often observed when radiation generated by femtosecond lasers propagates through many solids, liquids and gases. Depending on the type of material and on the intensity of the radiation, several mechanisms produce variations in the refractive index which result in self-focusing: the main cases are Kerr-induced self-focusing and plasma self-focusing.
Kerr-induced self-focusing was first predicted in the 1960s as described by the formula , where n0 and n2 are the linear and non-linear components of the refractive index, and I is the intensity of the radiation. Since n2 is positive in most materials, the refractive index becomes larger in the areas where the intensity is higher, usually at the centre of a beam, creating a focusing density profile which potentially leads to the collapse of a beam on itself. Self-focusing beams have been found to naturally evolve into a Townes profile regardless of their initial shape.and experimentally verified by studying the interaction of ruby lasers with glasses and liquids. Its origin lies in the optical Kerr effect, a non-linear process which arises in media exposed to intense electromagnetic radiation, and which produces a variation of the refractive index
A ruby laser is a solid-state laser that uses a synthetic ruby crystal as its gain medium. The first working laser was a ruby laser made by Theodore H. "Ted" Maiman at Hughes Research Laboratories on May 16, 1960.
In physics, intensity is the power transferred per unit area, where the area is measured on the plane perpendicular to the direction of propagation of the energy. In the SI system, it has units watts per square metre (W/m2). It is used most frequently with waves, in which case the average power transfer over one period of the wave is used. Intensity can be applied to other circumstances where energy is transferred. For example, one could calculate the intensity of the kinetic energy carried by drops of water from a garden sprinkler.
Self-focusing occurs if the radiation power is greater than the critical power
In physics, power is the rate of doing work or transferring heat, the amount of energy transferred or converted per unit time. Having no direction, it is a scalar quantity. In the International System of Units, the unit of power is the joule per second (J/s), known as the watt in honour of James Watt, the eighteenth-century developer of the condenser steam engine. Another common and traditional measure is horsepower. Being the rate of work, the equation for power can be written:
where λ is the radiation wavelength in vacuum and α is a constant which depends on the initial spatial distribution of the beam. Although there is no general analytical expression for α, its value has been derived numerically for many beam profiles.The lower limit is α ≈ 1.86225, which corresponds to Townes beams, whereas for a Gaussian beam α ≈ 1.8962.
In physics, the wavelength is the spatial period of a periodic wave—the distance over which the wave's shape repeats. It is thus the inverse of the spatial frequency. Wavelength is usually determined by considering the distance between consecutive corresponding points of the same phase, such as crests, troughs, or zero crossings and is a characteristic of both traveling waves and standing waves, as well as other spatial wave patterns. Wavelength is commonly designated by the Greek letter lambda (λ). The term wavelength is also sometimes applied to modulated waves, and to the sinusoidal envelopes of modulated waves or waves formed by interference of several sinusoids.
In optics, a Gaussian beam is a beam of monochromatic electromagnetic radiation whose transverse magnetic and electric field amplitude profiles are given by the Gaussian function; this also implies a Gaussian intensity (irradiance) profile. This fundamental (or TEM00) transverse gaussian mode describes the intended output of most (but not all) lasers, as such a beam can be focused into the most concentrated spot. When such a beam is refocused by a lens, the transverse phase dependence is altered; this results in a different Gaussian beam. The electric and magnetic field amplitude profiles along any such circular Gaussian beam (for a given wavelength and polarization) are determined by a single parameter: the so-called waist w0. At any position z relative to the waist (focus) along a beam having a specified w0, the field amplitudes and phases are thereby determined as detailed below.
For air, n0 ≈ 1, n2 ≈ 4×10−23 m2/W for λ = 800 nm, and the critical power is Pcr ≈ 2.4 GW, corresponding to an energy of about 0.3 mJ for a pulse duration of 100 fs. For silica, n0 ≈ 1.453, n2 ≈ 2.4×10−20 m2/W, and the critical power is Pcr ≈ 2.8 MW.
Kerr induced self-focusing is crucial for many applications in laser physics, both as a key ingredient and as a limiting factor. For example, the technique of chirped pulse amplification was developed to overcome the nonlinearities and damage of optical components that self-focusing would produce in the amplification of femtosecond laser pulses. On the other hand, self-focusing is a major mechanism behind Kerr-lens modelocking, laser filamentation in transparent media,self-compression of ultrashort laser pulses, parametric generation, and many areas of laser-matter interaction in general.
Chirped pulse amplification (CPA) is a technique for amplifying an ultrashort laser pulse up to the petawatt level with the laser pulse being stretched out temporally and spectrally prior to amplification.
Kerr-lens modelocking (KLM) is a method of modelocking lasers via a nonlinear optical process known as the optical Kerr effect. This method allows the generation of pulses of light with a duration as short as a few femtoseconds.
In optics, an ultrashort pulse of light is an electromagnetic pulse whose time duration is of the order of a picosecond or less. Such pulses have a broadband optical spectrum, and can be created by mode-locked oscillators. They are commonly referred to as ultrafast events. Amplification of ultrashort pulses almost always requires the technique of chirped pulse amplification, in order to avoid damage to the gain medium of the amplifier.
Kelley is detuned downward or upward the center of gain line . Laser pulse propagation with slowly varying envelope is governed in gain medium by Nonlinear Schrodinger-Frantz-Nodvik equation.predicted that homogeneously broadened two-level atoms may focus or defocus light when carrier frequency
When is detuned downward or upward the the refractive index is changed. Noteworthy the red detuning leads to increase of index during saturation of resonant transition, i.e. to self-focusing, while for blue detuning the radiation is defocused during saturation :
where is stimulated emission cross section, is population inversion density before pulse arrival, and are longitudinal and transverse lifetimes of two-level medium, is propagation axis.
The laser beam with a smooth spatial profile is affected by modulational instability. The small perturbations caused by roughnesses and medium defects are amplified in propagation. This effect is referred to as Bespalov-Talanov instability . In a framework of nonlinear Shrodinger equation : .
The rate of the perturbation growth or instability increment is linked with filament size via simple equation: . Generalization of this link between Bespalov-Talanov increments and filament size in gain medium as a function of linear gain and detuning had been realized in .
Advances in laser technology have recently enabled the observation of self-focusing in the interaction of intense laser pulses with plasmas.Self-focusing in plasma can occur through thermal, relativistic and ponderomotive effects. Thermal self-focusing is due to collisional heating of a plasma exposed to electromagnetic radiation: the rise in temperature induces a hydrodynamic expansion which leads to an increase of the index of refraction and further heating.
Relativistic self-focusing is caused by the mass increase of electrons travelling at speed approaching the speed of light, which modifies the plasma refractive index nrel according to the equation
where ω is the radiation angular frequency and ωp the relativistically corrected plasma frequency .
Ponderomotive self-focusing is caused by the ponderomotive force, which pushes electrons away from the region where the laser beam is more intense, therefore increasing the refractive index and inducing a focusing effect.
The evaluation of the contribution and interplay of these processes is a complex task,but a reference threshold for plasma self-focusing is the relativistic critical power
where me is the electron mass, c the speed of light, ω the radiation angular frequency, e the electron charge and ωp the plasma frequency. For an electron density of 1019 cm−3 and radiation at the wavelength of 800 nm, the critical power is about 3 TW. Such values are realisable with modern lasers, which can exceed PW powers. For example, a laser delivering 50 fs pulses with an energy of 1 J has a peak power of 20 TW.
Self-focusing in a plasma can balance the natural diffraction and channel a laser beam. Such effect is beneficial for many applications, since it helps increasing the length of the interaction between laser and medium. This is crucial, for example, in laser-driven particle acceleration,laser-fusion schemes and high harmonic generation.
Self-focusing can be induced by a permanent refractive index change resulting from a multi-pulse exposure. This effect has been observed in glasses which increase the refractive index during an exposure to ultraviolet laser radiation.Accumulated self-focusing develops as a wave guiding, rather than a lensing effect. The scale of actively forming beam filaments is a function of the exposure dose. Evolution of each beam filament towards a singularity is limited by the maximum induced refractive index change or by laser damage resistance of the glass.
Self-focusing can also been observed in a number of soft matter systems, such as solutions of polymers and particles as well as photo-polymers.Self-focusing was observed in photo-polymer systems with microscale laser beams of either UV or visible light. The self-trapping of incoherent light was also later observed. Self-focusing can also be observed in wide-area beams, wherein the beam undergoes filamentation, or Modulation Instability, spontaneous dividing into a multitude of microscale self-focused beams, or filaments. The balance of self-focusing and natural beam divergence results in the beams propagating divergence-free. Self-focusing in photopolymerizable media is possible, owing to a photoreaction dependent refractive index, and the fact that refractive index in polymers is proportional to molecular weight and crosslinking degree which increases over the duration of photo-polymerization.
The group velocity of a wave is the velocity with which the overall shape of the wave's amplitudes—known as the modulation or envelope of the wave—propagates through space.
Bremsstrahlung, from bremsen "to brake" and Strahlung "radiation"; i.e., "braking radiation" or "deceleration radiation", is electromagnetic radiation produced by the deceleration of a charged particle when deflected by another charged particle, typically an electron by an atomic nucleus. The moving particle loses kinetic energy, which is converted into radiation, thus satisfying the law of conservation of energy. The term is also used to refer to the process of producing the radiation. Bremsstrahlung has a continuous spectrum, which becomes more intense and whose peak intensity shifts toward higher frequencies as the change of the energy of the decelerated particles increases.
Ionization or ionisation, is the process by which an atom or a molecule acquires a negative or positive charge by gaining or losing electrons, often in conjunction with other chemical changes. The resulting electrically charged atom or molecule is called an ion. Ionization can result from the loss of an electron after collisions with subatomic particles, collisions with other atoms, molecules and ions, or through the interaction with electromagnetic radiation. Heterolytic bond cleavage and heterolytic substitution reactions can result in the formation of ion pairs. Ionization can occur through radioactive decay by the internal conversion process, in which an excited nucleus transfers its energy to one of the inner-shell electrons causing it to be ejected.
Zero-point energy (ZPE) is the difference between the lowest possible energy that a quantum mechanical system may have, and the classical minimum energy of the system. Unlike in classical mechanics, quantum systems constantly fluctuate in their lowest energy state due to the Heisenberg uncertainty principle. As well as atoms and molecules, the empty space of the vacuum has these properties. According to quantum field theory, the universe can be thought of not as isolated particles but continuous fluctuating fields: matter fields, whose quanta are fermions, and force fields, whose quanta are bosons. All these fields have zero-point energy. These fluctuating zero-point fields lead to a kind of reintroduction of an aether in physics, since some systems can detect the existence of this energy. However this aether cannot be thought of as a physical medium if it is to be Lorentz invariant such that there is no contradiction with Einstein's theory of special relativity.
Synchrotron radiation is the electromagnetic radiation emitted when charged particles are accelerated radially, i.e., when they are subject to an acceleration perpendicular to their velocity. It is produced, for example, in synchrotrons using bending magnets, undulators and/or wigglers. If the particle is non-relativistic, then the emission is called cyclotron emission. If, on the other hand, the particles are relativistic, sometimes referred to as ultrarelativistic, the emission is called synchrotron emission. Synchrotron radiation may be achieved artificially in synchrotrons or storage rings, or naturally by fast electrons moving through magnetic fields. The radiation produced in this way has a characteristic polarization and the frequencies generated can range over the entire electromagnetic spectrum which is also called continuum radiation.
Optical tweezers are scientific instruments that use a highly focused laser beam to provide an attractive or repulsive force, depending on the relative refractive index between particle and surrounding medium, to physically hold and move microscopic objects similar to tweezers. They are able to trap and manipulate small particles, typically order of micron in size, including dielectric and absorbing particles. Optical tweezers have been particularly successful in studying a variety of biological systems in recent years.
The Drude model of electrical conduction was proposed in 1900 by Paul Drude to explain the transport properties of electrons in materials. The model, which is an application of kinetic theory, assumes that the microscopic behavior of electrons in a solid may be treated classically and looks much like a pinball machine, with a sea of constantly jittering electrons bouncing and re-bouncing off heavier, relatively immobile positive ions.
The Kerr effect, also called the quadratic electro-optic (QEO) effect, is a change in the refractive index of a material in response to an applied electric field. The Kerr effect is distinct from the Pockels effect in that the induced index change is directly proportional to the square of the electric field instead of varying linearly with it. All materials show a Kerr effect, but certain liquids display it more strongly than others. The Kerr effect was discovered in 1875 by John Kerr, a Scottish physicist.
In theoretical physics, the (one-dimensional) nonlinear Schrödinger equation (NLSE) is a nonlinear variation of the Schrödinger equation. It is a classical field equation whose principal applications are to the propagation of light in nonlinear optical fibers and planar waveguides and to Bose-Einstein condensates confined to highly anisotropic cigar-shaped traps, in the mean-field regime. Additionally, the equation appears in the studies of small-amplitude gravity waves on the surface of deep inviscid (zero-viscosity) water; the Langmuir waves in hot plasmas; the propagation of plane-diffracted wave beams in the focusing regions of the ionosphere; the propagation of Davydov's alpha-helix solitons, which are responsible for energy transport along molecular chains; and many others. More generally, the NLSE appears as one of universal equations that describe the evolution of slowly varying packets of quasi-monochromatic waves in weakly nonlinear media that have dispersion. Unlike the linear Schrödinger equation, the NLSE never describes the time evolution of a quantum state. The 1D NLSE is an example of an integrable model.
In physics, a ponderomotive force is a nonlinear force that a charged particle experiences in an inhomogeneous oscillating electromagnetic field.
Self-phase modulation (SPM) is a nonlinear optical effect of light-matter interaction. An ultrashort pulse of light, when travelling in a medium, will induce a varying refractive index of the medium due to the optical Kerr effect. This variation in refractive index will produce a phase shift in the pulse, leading to a change of the pulse's frequency spectrum.
Photoacoustic imaging is a biomedical imaging modality based on the photoacoustic effect. In photoacoustic imaging, non-ionizing laser pulses are delivered into biological tissues. Some of the delivered energy will be absorbed and converted into heat, leading to transient thermoelastic expansion and thus wideband ultrasonic emission. The generated ultrasonic waves are detected by ultrasonic transducers and then analyzed to produce images. It is known that optical absorption is closely associated with physiological properties, such as hemoglobin concentration and oxygen saturation. As a result, the magnitude of the ultrasonic emission, which is proportional to the local energy deposition, reveals physiologically specific optical absorption contrast. 2D or 3D images of the targeted areas can then be formed.
In nonlinear optics, filament propagation is propagation of a beam of light through a medium without diffraction. This is possible because the Kerr effect causes an index of refraction change in the medium, resulting in self-focusing of the beam.
A pinch is the compression of an electrically conducting filament by magnetic forces. The conductor is usually a plasma, but could also be a solid or liquid metal. Pinches were the first type of device used for controlled nuclear fusion.
The Jaynes–Cummings model is a theoretical model in quantum optics. It describes the system of a two-level atom interacting with a quantized mode of an optical cavity, with or without the presence of light. It was originally developed to study the interaction of atoms with the quantized electromagnetic field in order to investigate the phenomena of spontaneous emission and absorption of photons in a cavity.
The Frank–Tamm formula yields the amount of Cherenkov radiation emitted on a given frequency as a charged particle moves through a medium at superluminal velocity. It is named for Russian physicists Ilya Frank and Igor Tamm who developed the theory of the Cherenkov effect in 1937, for which they were awarded a Nobel Prize in Physics in 1958.
In optics, the term soliton is used to refer to any optical field that does not change during propagation because of a delicate balance between nonlinear and linear effects in the medium. There are two main kinds of solitons:
In relativistic laser-plasma physics the relativistic similarity parameterS is a dimensionless parameter defined as
The semiconductor luminescence equations (SLEs) describe luminescence of semiconductors resulting from spontaneous recombination of electronic excitations, producing a flux of spontaneously emitted light. This description established the first step toward semiconductor quantum optics because the SLEs simultaneously includes the quantized light–matter interaction and the Coulomb-interaction coupling among electronic excitations within a semiconductor. The SLEs are one of the most accurate methods to describe light emission in semiconductors and they are suited for a systematic modeling of semiconductor emission ranging from excitonic luminescence to lasers.