similarly to a mass attached to a spring) when the stretching or compression of the oscillator is large enough. Maxwell's equations are linear in vacuum Nonlinear optics (NLO) is a branch of optics that studies the case when optical properties of matter depend on the intensity of the input light. Nonlinear phenomena become relevant only when the input light is very intense. Typically, in order to observe nonlinear phenomena, an intensity of the electromagnetic field of light larger than 108 V/m (and thus comparable to the atomic electric field of Nonlinear optics (NLO) is a branch of optics that studies the case when optical properties of matter depend on the intensity of the input light. Nonlinear phenomena become relevant only when the input light is very intense. Typically, in order to observe nonlinear phenomena, an intensity of the electromagnetic field of light larger than 108 V/m (and thus comparable to the atomic electric field of ~1011 V/m) is required. In this case, the polarization density P responds non-linearly to the electric field E of light. In order to obtain an electromagnetic field that is sufficiently intense, laser sources must be used. In nonlinear optics, the superposition principle no longer holds, and the polarization of the material is no longer linear in the electric field intensity. Instead, in the perturbative limit, it can be expressed by a polynomial sum of order n. Many different physical mechanisms can cause nonlinearities in the optical behaviour of a material, i.e. the motion of bound electrons, field-induced vibrational or orientational motions, optically-induced acoustic waves and thermal effects. The motion of bound electrons, in particular, has a very short response timescale, so it is of particular relevance in the context of ultrafast nonlinear optics. The simplest way to picture this behaviour in a semiclassical way is to use a phenomenological model: an anharmonic oscillator can model the forced oscillations of a bound e Nonlinear optics (NLO) is a branch of optics that studies the case when optical properties of matter depend on the intensity of the input light. Nonlinear phenomena become relevant only when the input light is very intense. Typically, in order to observe nonlinear phenomena, an intensity of the electromagnetic field of light larger than 108 V/m (and thus comparable to the atomic electric field of ~1011 V/m) is required. In this case, the polarization density P responds non-linearly to the electric field E of light. In order to obtain an electromagnetic field that is sufficiently intense, laser sources must be used. In nonlinear optics, the superposition principle no longer holds, and the polarization of the material is no longer linear in the electric field intensity. Instead, in the perturbative limit, it can be expressed by a polynomial sum of order n. Many different physical mechanisms can cause nonlinearities in the optical behaviour of a material, i.e. the motion of bound electrons, field-induced vibrational or orientational motions, optically-induced acoustic waves and thermal effects. The motion of bound electrons, in particular, has a very short response timescale, so it is of particular relevance in the context of ultrafast nonlinear optics. The simplest way to picture this behaviour in a semiclassical way is to use a phenomenological model: an anharmonic oscillator can model the forced oscillations of a bound electron inside the medium. In this picture, the binding interaction between the ion core and the electron is the Coulomb force and nonlinearities appear as changes in the elastic constant of the system (which behaves similarly to a mass attached to a spring) when the stretching or compression of the oscillator is large enough. Maxwell's equations are linear in vacuum, so, nonlinear processes only occur in media. However, the theory of quantum electrodynamics (QED) predicts that, above the Schwinger limit, vacuum itself can behave in a nonlinear way. The description of nonlinear optics usually presented in textbooks is the perturbative regime, which is valid when the input intensity remains below 1014 W/cm2, which implies that the electric field is well below the intensity of interatomic fields. This approach allows to use a Taylor series to write down the polarization density as a polynomial sum. It is also possible to study the laser-matter interaction at a much higher intensity of light: this field is referred to as nonperturbational nonlinear optics or extreme nonlinear optics and investigates the generation of extremely high-order harmonics, attosecond pulse generation and relativistic nonlinear effects. where PNL is the nonlinear part of the polarization density, and n is the refractive index, which comes from the linear term in P. Note that one can normally use the vector identity is true in general, even for an isotropic medium. However, even when this term is not identically 0, it is often negligibly small and thus in practice is usually ignored, giving us the standard nonlinear wave equation: The above holds for χ ( 2 ) {\displaystyle \chi ^{(2)}} processes. It can be extended for processes where χ ( 3 ) {\displaystyle \chi ^{(3)}} is nonzero, something that is generally true in any medium without any symmetry restrictions; in particular resonantly enhanced sum or difference frequency mixing in gasses is frequently used for extreme or "vacuum" ultra-violet light generation. In common scenarios, such as mixing in dilute gases, the non-linearity is weak and so the light beams are focused which, unlike the plane wave approximation used above, introduces a pi phase shift on each light beam, complicating the phase-matching requirements. Conveniently, difference frequency mixing with χ ( 3 ) {\displaystyle \chi ^{(3)}} cancels this focal phase shift and often has a nearly self-canceling overall phase-matching condition,

arises, as to whether the assumption that the linear equations of free aether are a first approximation, obtained by the omission of non - linear terms , would The results above obtained have been derived from the correlation developed in § 106 , up to the first order of the small quantity v/c , between the equations for aethereal vectors here represented by (f', g', h') and (a', b', c') referred to the axes (x', y', z') at rest in the aether and a time t" , and those for related aethereal vectors represented by ( f, g, h) and ( a, b, c ) referred to axes (x', y', z') in uniform translatory motion and a time t'. But we can proceed further, and by aid of a more complete transformation institute a correspondence which will be correct to the second order. We derive the result, correct to the second order, that if the internal forces of a material system arise wholly from electrodynamic actions between the systems of electrons which constitute the atoms, then an effect of imparting to a steady material system a uniform velocity of translation is to produce a uniform contraction of the system in the direction of the motion, of amount ϵ − 1 2 {\displaystyle \epsilon ^{-{\frac {1}{2}}}} or 1 − 1 2 v 2 / c 2 {\displaystyle 1-{\frac {1}{2}}v^{2}/c^{2}} . {\displaystyle V-{\frac {v}{\mu ^{2}}}-\left({\frac {1}{\mu }}-{\frac {1}{\mu ^{3}}}\right){\frac {v^{2}}{c}}.} The second term in this expression is the Fresnel effect, and the remaining term is its second order correction on our hypothesis which includes Michelson's negative result. It can be said on the other side that this view of aethereal action does not directly cover gravitational phenomena, unless the rather artificial pulsatory theory of gravity is allowed. [ 6 ] But there is another aspect of the matter. The equations of the free aether, as revealed by MacCullagh's optical analysis, are linear equations: they in fact must be so if all kinds of radiations are to travel with the same speed in the celestial spaces. In Maxwell's hands, equivalent relations with the appropriate generalization were arrived at on the electric side, and formed a basis for the explanation of the whole plexus of electrodynamic and optical phenomena. The question arises whether there is anything to gainsay a view that this simple Why then should not relatively minute phenomena like gravitation be involved in similar non-linear terms, or terms involving differentials of higher orders, in the analytical specification of the free aether, which are as insignificant compared with the main fully ascertained linear terms as is the gravitation between two electric systems compared with their mutual electric forces? Against this there is a subjective reluctance to disturb the ideal simplicity of the aethereal scheme: but there is no help for that if its content is not sufficiently extensive for the facts. Of more weight is the circumstance that a train of radiation from a distant star would change its form as it advanced across  ​ space, that there would in fact be optical dispersion in the free aether if such second-order terms existed. The amount of such dispersion that would be at all allowable is known to be excessively minute, from the circumstance that celestial bodies on emerging from eclipse or occupation show no changes of colour: the smallness of the amount that would be required may be estimated by comparing the electric force between two ions with their gravitational attraction. Unless the effects of such terms of higher order, in the equations of aethereal activity, increased enormously in importance at molecular distances, relatively to the main linear terms, the proposition that the interactions of molecules are mainly of electric quality would remain valid: now such increase of importance does not seem likely as regards the mechanism of gravitation, for gravity and electric force both obey the same law of the inverse square of the distance, a law which in fact belongs, of mathematical necessity, to the steady permanent interactions between any kinds of molecular nuclei of elastic or motional disturbance in an extended medium, which are of the type of simple poles. A question of some interest arises, as to whether the assumption that the linear equations of free aether are a first approximation, obtained by the omission of non-linear terms, would imply a virtual recognition of structure in that medium. A presumption of this kind would be useless except for purposes of vivid illustration after the manner of mechanical models, so long as there is absolutely no means of experimenting on the properties of free aether: and this practically comes to the same thing as taking such structure to be non-existent. 121 . There is thus little to be urged in favour of leaving this loophole for the explanation of gravitation. The size of a molecule would also be rendered determinate if residual non-linear terms in the aethereal equations became sensible at intermolecular distances. Thus, these saving hypotheses being excluded, if the atoms of matter were constituted electrically, and the forces between them were wholly of electric origin, there would be nothing to determine the scale of an isolated system as regards time and space: and different systems need not be always of the same scale of magnitude as regards their atomic structure. 123 . A similar deficiency of definite scale would also be expected to exist in any hydrodynamical theory or illustration which would construct an atom out of vortex rings. In the above considerations there is strong evidence that gravitation is not to be expected to be appreciably involved within the scheme which suffices to cover the phenomena of electrodynamics and optics. The introduction of the time-relation inherent in pulsating nuclei seems still to be the only obvious way of representing it, in default of its arising from second-order terms in the dynamical relations of the aether.