# Nonlinear Optics

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## Nonlinear optics

A field of study concerned with the interaction of electromagnetic radiation and matter in which the matter responds in a nonlinear manner to the incident radiation fields. The nonlinear response can result in intensity-dependent variation of the propagation characteristics of the radiation fields or in the creation of radiation fields that propagate at new frequencies or in new directions. Nonlinear effects can take place in solids, liquids, gases, and plasmas, and may involve one or more electromagnetic fields as well as internal excitations of the medium. Most of the work done in the field has made use of the high powers available from lasers. The wavelength range of interest generally extends from the far-infrared to the vacuum ultraviolet, but some nonlinear interactions have been observed at wavelengths extending from the microwave to the x-ray ranges. *See* Laser

#### Nonlinear materials

Nonlinear effects of various types are observed at sufficiently high light intensities in all materials. It is convenient to characterize the response of the medium mathematically by expanding it in a power series in the electric and magnetic fields of the incident optical waves. The linear terms in such an expansion give rise to the linear index of refraction, linear absorption, and the magnetic permeability of the medium, while the higher-order terms give rise to nonlinear effects. *See* Absorption of electromagnetic radiation, Refraction of waves

In general, nonlinear effects associated with the electric field of the incident radiation dominate over magnetic interactions. The even-order dipole susceptibilities are zero except in media which lack a center of symmetry, such as certain classes of crystals, certain symmetric media to which external forces have been applied, or at boundaries between certain dissimilar materials. Odd-order terms can be nonzero in all materials regardless of symmetry. Generally the magnitudes of the nonlinear susceptibilities decrease rapidly as the order of the interaction increases. Second- and third-order effects have been the most extensively studied of the nonlinear interactions, although effects up to order 30 have been observed in a single process. In some situations, multiple low-order interactions occur, resulting in a very high effective order for the overall nonlinear process. For example, ionization through absorption of effectively 100 photons has been observed. In other situations, such as dielectric breakdown or saturation of absorption, effects of different order cannot be separated, and all orders must be included in the response. *See* Electric susceptibility, Polarization of dielectrics

#### Stimulated scattering

Light can scatter inelastically from fundamental excitations in the medium, resulting in the production of radiation at a frequency that is shifted from that of the incident light by the frequency of the excitation involved. The difference in photon energy between the incident and scattered light is accounted for by excitation or deexcitation of the medium. Some examples are Brillouin scattering from acoustic vibrations; various forms of Raman scattering involving molecular rotations or vibrations, electronic states in atoms or molecules, lattice vibrations or spin waves in solids, spin flips in semiconductors, and electron plasma waves in plasmas; Rayleigh scattering involving density or entropy fluctuations; and scattering from concentration fluctuations in gases. *See* Scattering of electromagnetic radiation

At the power levels available from pulsed lasers, the scattered light experiences exponential gain, and the process is then termed stimulated, in analogy to the process of stimulated emission in lasers. In stimulated scattering, the incident light can be almost completely converted to the scattered radiation. Stimulated scattering has been observed for all of the internal excitations listed above. The most widely used of these processes are stimulated Raman scattering and stimulated Brillouin scattering.

#### Self-action and related effects

Nonlinear polarization components at the same frequencies as those in the incident waves can result in effects that change the index of refraction or the absorption coefficient, quantities that are constants in linear optical theory. For example, propagation through optical fibers can involve several nonlinear optical interactions. Self-phase modulation resulting from the nonlinear index can be used to spread the spectrum, and subsequent compression with diffraction gratings and prisms can be used to reduce the pulse duration. The shortest optical pulses, with durations of the order of 6 femtoseconds, have been produced in this manner. Linear dispersion in fibers causes pulses to spread in duration and is one of the major limitations on data transmission through fibers. Dispersive pulse spreading can be minimized with solitons, which are specially shaped pulses that propagate long distances without spreading. They are formed by a combined interaction of spectral broadening due to the nonlinear refractive index and anomalous dispersion found in certain parts of the spectrum. *See* Soliton

#### Coherent effects

Another class of effects involves a coherent interaction between the optical field and an atom in which the phase of the atomic wave functions is preserved during the interaction. These interactions involve the transfer of a significant fraction of the atomic population to an excited state. As a result, they cannot be described with the simple perturbation expansion used for the other nonlinear optical effects. Rather they require that the response be described by using all powers of the incident fields. These effects are generally observed only for short light pulses, of the order of several nanoseconds or less. In one interaction, termed self-induced transparency, a pulse of light of the proper shape, magnitude, and duration can propagate unattenuated in a medium which is otherwise absorbing.

Other coherent effects involve changes of the propagation speed of a light pulse or production of a coherent pulse of light, termed a photon echo, at a characteristic time after two pulses of light spaced apart by a time interval have entered the medium. Still other coherent interactions involve oscillations of the atomic polarization, giving rise to effects known as optical nutation and free induction decay. Two-photon coherent effects are also possible.

#### Nonlinear spectroscopy

The variation of the nonlinear susceptibility near the resonances that correspond to sum- and difference-frequency combinations of the input frequencies forms the basis for various types of nonlinear spectroscopy which allow study of energy levels that are not normally accessible with linear optical spectroscopy.

Nonlinear spectroscopy can be performed with many of the interactions discussed earlier. Multiphoton absorption spectroscopy can be performed by using two strong laser beams, or a strong laser beam and a weak broadband light source. If two counterpropagating laser beams are used, spectroscopic studies can be made of energy levels in gases with spectral resolutions much smaller than the Doppler limit. Nonlinear optical spectroscopy has been used to identify many new energy levels with principal quantum numbers as high as 150 in several elements. *See* Resonance ionization spectroscopy, Rydberg atom

Many types of four-wave mixing interactions can also be used in nonlinear spectroscopy. The most widespread of these processes, termed coherent anti-Stokes Raman spectroscopy (CARS), offers the advantage of greatly increased signal levels over linear Raman spectroscopy for the study of certain classes of materials.

#### Phase conjugation

Optical phase conjugation is an interaction that generates a wave that propagates in the direction opposite to a reference, or signal, wave, and has the same spatial variations in intensity and phase as the original signal wave, but with the sense of the phase variations reversed. Several nonlinear interactions are used to produce phase conjugation.

Optical phase conjugation allows correction of optical distortions that occur because of propagation through a distorting medium. This process can be used for improvement of laser-beam quality, optical beam combining, correction of distortion because of mode dispersion in fibers, and stabilized aiming. It can also be used for neural networks that exhibit learning properties. *See* Optical phase conjugation

#### Photorefractive effect

The photorefractive effect occurs in many electrooptic materials. A change in the index of refraction in a photorefractive medium arises from the redistribution of charge that is induced by the presence of light. Charge carriers that are trapped in impurity sites in a photorefractive medium are excited into the material's conduction band when exposed to light. The charges migrate in the conduction band until they become retrapped at other sites. The charge redistribution produces an electric field that in turn produces a spatially varying index change through the electrooptic effect in the material. Unlike most other nonlinear effects, the index change of the photorefractive effect is retained for a time in the absence of the light and thus may be used as an optical storage mechanism. Storage times range from milliseconds to months or years, depending upon the material and the methods employed. *See* Traps in solids

Photorefractive materials are often used for holographic storage. In this case, the index change mimics the intensity interference pattern of two beams of light. Over 500 holograms have been stored in the volume of a single crystal of iron-doped lithium niobate. *See* Holography

Photorefractive materials are typically sensitive to very low light levels. The photorefractive effect is, however, extremely slow by the standards of optical nonlinearity. Because of their sensitivity, photorefractive materials are increasingly used for image and optical-signal processing applications. *See* Nonlinear optical devices

## Nonlinear Optics

the branch of physical optics that deals with the propagation of powerful light beams in solids, liquids, and gases and with their interaction with matter. The advent of lasers provided optics with sources of coherent radiation with a power of up to 10^{9}-10^{11} watts (W). In such a light field completely new optical effects take place, and the nature of known phenomena is substantially altered. The common feature of the new phenomena is that their nature depends on the light intensity. A strong light field changes the optical properties of the medium (the index of refraction *n* and the absorption coefficient) and, thus, the nature of the phenomenon. This explains the origin of the term “nonlinear optics”: if the optical properties of the medium become functions of the electric field intensity *E* in a light wave, then the polarization of the medium is a nonlinear function of *E*. Nonlinear optics has much in common with the nonlinear theory of oscillations, nonlinear acoustics, and other fields. The optics of weak light beams where the field is insufficient to produce an appreciable change in the properties of the medium is, of course, called linear optics.

** History.** In “prelaser” optics it was considered firmly established that the basic characteristics of a light wave that determine the nature of its interaction with matter are the frequency or the wavelength λ, which is directly related to frequency, and the polarization of the wave. For the overwhelming majority of optical effects, the value of the electric intensity

*E*of the light field (or the radiant flux density I =

*cnE*π, where

^{2}/8*c*is the speed of light and

*n*is the index of refraction) had virtually no effect on the nature of the phenomenon. The index of refraction

*n*, the absorption coefficient, and the effective scattering cross section of the light appeared in handbooks without any indication of the light intensity for which they were measured, simply because no correlation with intensity had been observed. Only a few works may be noted in which attempts were made to study the effect of light intensity on optical phenomena. In 1923, S. I. Vavilov and V. L. Levshin detected a reduction in the absorption of light by uranium glass with increasing light intensity, and they reasoned that in a strong electromagnetic field a larger portion of the atoms (or molecules) are in an excited state and cannot absorb light. On the assumption that this is only one of a set of possible nonlinear effects in optics, Vavilov introduced the term “nonlinear optics.” The possibility of observing a number of nonlinear optical effects by means of photomultiplier tubes was examined theoretically in the 1950’s by G. S. Gorelik (USSR); one effect, the observation of the separation of an optical doublet by isolating the difference frequency lying in the microwave region (heterodyning of light), was carried out in 1955 by A. Forrester, R. Gudmundsen, and P. Johnson (USA).

The creation of lasers opened up extensive possibilities for the study of nonlinear optical phenomena. In 1961, P. Franken and his co-workers (USA) discovered the frequency-doubling effect for light in crystals (the generation of the second harmonic of the light). Frequency tripling (third harmonic generation) was observed in 1962. Fundamental results in the theory of nonlinear optical phenomena were obtained in the USSR and the USA from 1961 to 1963, laying the theoretical foundation for nonlinear optics. In 1962–63 the phenomenon of stimulated Raman scattering of light was discovered and explained. This gave the impetus for the study of other types of stimulated scattering, such as stimulated Brillouin and stimulated Rayleigh scattering.

The phenomenon of self-focusing of light beams was discovered in 1965. It was found that, in many cases, a powerful light beam propagating in a medium not only does not undergo the usual, so-called diffraction divergence but actually converges spontaneously. The phenomenon of self-focusing of electromagnetic waves was predicted in a general form by G. A. Askar’ian (USSR) in 1962. Optical experimentation was stimulated by the theoretical work of C. Townes and his co-workers (USA, 1964). The work of A. M. Prokhorov and his co-workers made a large contribution to understanding the nature of the phenomenon.

Parametric light generators, in which nonlinear optical effects are used to generate coherent optical radiation that can be varied smoothly in frequency over a broad range of wavelengths, were developed in 1965. Research on nonlinear phenomena associated with the propagation in a medium of ultrashort light pulses (with a duration down to 10^{-12} sec) was begun in 1967. Since 1969 techniques of nonlinear and active spectroscopy have also been developing; they use nonlinear optical phenomena to improve the resolution and enhance the sensitivity of spectroscopic methods of studying substances.

** Interaction of a strong light field with a medium.** The elementary process that is basic in the interaction of light with a medium is the excitation of an atom or molecule by the light field and the re-emission of light by the excited particle. The mathematical description of such processes consists of equations that associate the polarization

*℘*of a unit volume of the medium with the field intensity

*E*(the material equations). Linear optics is based on linear material equations, which for harmonic waves result in the relation

(1) *ρ* = *kE*

where *κ* is the dielectric susceptibility, which depends only on the properties of the medium. Equation (1) is the basis for the most important principle of linear optics, the superposition principle. However, theory based on (1) is unable to explain any of the nonlinear effects listed above. According to (1) the re-emitted field has the same frequency as the incident field, so that equation (1) does not describe the generation of optical harmonics. It follows from (1) that the index of refraction of a medium is independent of the intensity. This indicates that the material equation (1) is approximate: in practical terms, it can be used only for weak light fields.

The essence of the approximations on which (1) is based can be understood by turning to the classical model of an oscillator, which is used extensively in optics to describe the interaction of light with matter. According to this model, the behavior of an atom or molecule in a light field is equivalent to the vibrations

of an oscillator. The nature of the response of such an elementary atomic oscillator to a light wave can be determined by comparing the field intensity of the light wave with the intensity of the intra-atomic field *E*_{a} ≃ *e/a ^{2}* ≃10

^{8}-10

^{9}volts per cm (V/cm), where

*e*is the charge of the electron and

*a*is the atomic radius, which determines the binding forces in the atomic oscillator. In beams from nonlaser sources,

*E*≃ 1–10 V/cm (that is,

*E<*<E

_{a}£

_{a}), and the atomic oscillator can be regarded as harmonic (the restoring force is linearly related to the displacement). Equation (1) is a direct consequence of this. In beams from high-powered lasers,

*E*~ 10

^{6}-10

^{7}V/cm, and the atomic oscillator becomes anharmonic, or nonlinear (the restoring force is a nonlinear function of the displacement). The anharmonicity of the atomic oscillator makes the relationship between the polarization

*Ψ*and the field

*E*nonlinear; for

*(E/E*1, the polarization can be represented by a series expansion in the parameter

_{a}) <*E/E*

_{a}:(2) *℘ = kE + xE ^{2} + ΘE3 +*

The coefficients x, *θ*, and so on are called the nonlinear susceptibilities (in order of magnitude, x ~ 1/*E*_{a} and θ ~ _{1/Ea}^{2}). The material equation (2) is the basis of nonlinear optics. If a monochromatic light wave *E = A* cos *(ωt —kx)* is incident on the surface of a medium (where *A* is the amplitude, ω is the frequency, *k* is the wave number, *x* is the coordinate of a point along the direction of wave propagation, and *t* is the time), then according to (2) the polarization of the medium contains—in addition to the linear term *P*^{I}kA cos (ωi^{-} —*kx*) (linear polarization)—a nonlinear term of the second order:

The last term in (3) describes polarization that varies with a frequency 2ω—that is, the generation of a second harmonic. The generation of a third harmonic, as well as the dependence of the index of refraction on intensity, is described by the term *ΘE ^{3}* in (2), and so on.

The nonlinear response of the atomic oscillator to a strong light field is the most universal cause of nonlinear optical effects. However, there are other causes—for example, a change in the index of refraction *n* can be produced by heating the medium with laser radiation. A temperature change Δ*T* = α*E*^{2} is the absorption coefficient for the light) leads to the relation *n* = *n*_{0} + (∂*η*/∂*T*)Δ*T*. In many cases the electrostriction effect (compression of the medium in a light field *E*) also proves important. In the strong light field *E* of a laser the electrostriction pressure, which is proporional to *E ^{2}*, changes the density of the medium, and this can lead to the generation of acoustic waves. The self-focusing of light is sometimes attributable to thermal effects and electrostriction.

** Optical harmonics.** As the intense monochromatic radiation from a neodymium-doped glass laser (λ1 = 1.06 micron [μ]) passes through an optically transparent crystal of barium niobate, it is converted into radiation with a wavelength exactly half as great—that is, the second harmonic (λ

_{2}

*=*0.53 μ). Under certain conditions more than 60 percent of the energy of the incident radiation is converted into the second harmonic. Frequency doubling of the radiation from other lasers in the visible and infrared regions is observed. Frequency tripling of light (the third harmonic) has been recorded for a number of crystals and fluids. More complicated effects arise if two or more intense waves of different frequencies ω1 and ω

_{2}are propagating in a medium. Then, waves of combination frequencies (ω

_{1}+ ω

_{2}ω

_{1}—ω

_{2}, and so on) are produced in addition to the harmonics of each wave (2ω1, 2ω

_{2}, and so on).

The phenomenon described above, which is called optical harmonic generation, has much in common with the well-known frequency multiplication in nonlinear elements in radio equipment. At the same time, there is an essential difference: in optics the effects result from the interaction of waves rather than of oscillations. In a strong light field, according to (2), every atomic oscillator re-emits not only at the frequency of the incident wave but also at its harmonics. However, since the light is propagating in a medium whose dimensions *L* greatly exceed the wavelength λ (for visible light λ ~ 10^{-4} cm), the total effect of the generation of harmonics at the output depends on the phase relations between the fundamental wave and the harmonics within the medium; peculiar interference effects occur that can either enhance or weaken the effect. It has been found that the interaction of two waves of different frequencies, such as ω and 2ω, is greatest—and therefore the transfer of energy from the fundamental wave to the harmonics is a maximum—if their phase velocities are equal (the condition of phase matching). The conditions for phase matching can also be arrived at from quantum notions and correspond to the law for the conservation of momentum during the coalescence or decay of photons. For three waves, the matching condition is as follows: k_{3}*= =* k _{1}+ k_{2}, where k_{1}, k_{2}, and k_{3} are the momenta of the photons (in units of Planck’s constant *h)*.

At first glance, the matching conditions for a fundamental wave and its harmonics in a real dispersive medium seem to be unrealizable. The phase velocities of waves with different frequencies are equal only in a medium without dispersion. However, it was found that the absence of dispersion can be simulated by using the interaction of waves with different polarizations in an anisotropic medium (Figure 1). This method sharply increases the efficiency of nonlinear wave interactions. Whereas in 1961 the efficiency of optical frequency doubling was of the order of 10^{-10} to 10^{-l5}, in 1963 it had reached values of 0.2–0.3, and by 1973 it was close to 0.8.

Optical frequency multipliers have made possible substantial broadening of the area of application of lasers. Optical harmonic generation is used extensively to convert longwave laser radiation into the shorter-wavelength radiation. Industry in many countries is producing optical frequency multipliers made from neodymium-doped glass or yttrium aluminum garnet (YAG) with neodymium doping (λ = 1.06 μ), which make possible the production of high-power coherent radiation at wavelengths λ = 0.53 μ (second harmonic), λ = 0.35 μ (third harmonic), and λ = 0.26 μ (fourth harmonic). Crystals that have high nonlinearity (large values of *ξ*) and can satisfy the conditions for phase matching were chosen for this purpose. Illustrations of the current possibilities in this field are generators for the fifth harmonic and the ninth harmonic of the radiation of a neodymium laser (λ9 =1189 angstroms). In 1972 frequency multiplication was achieved experimentally in the vacuum ultraviolet region; in this case certain gases and metal vapors were used as the nonlinear medium.

** Self-focusing of light; self-action.** If the power of a light beam is sufficiently high (but quite moderate in terms of current laser technology), exceeding a certian critical value

*P*

_{cr}, self-narrowing of an initially parallel beam—rather than the usual diffraction broadening—is observed in the medium. The magnitude of

*Per*varies with the medium; for a number of organic fluids

*P*

_{cr}~ 10–50 kW, but in some crystals and optical glasses

*P*

_{cr}does not exceed a few watts.

Sometimes—for example, when the radiation from powerful pulsed lasers is propagating in fluids—the self-narrowing is like a “collapse” of the beam and is accompanied by such a rapid increase in the light field that it can cause structural light breakdown, phase transitions, and other changes in the state of the substance. In other cases, such as the propagation of radiation from continuous-wave (CW) gas lasers in glasses, the increase in

the field is also pronounced, although it is not as rapid. In a sense, self-focusing is similar to the focusing of a beam by an ordinary lens. However, there are essential differences beyond the focal point—for example, a self-focused beam can form quasi-stationary filaments (“wave guide” propagation), or a succession of focal points.

The phenomenon of self-focusing occurs because the index of refraction of the medium changes in a strong light field. If the sign of the change in the index of refraction is such that it increases in the region occupied by the beam, the region becomes optically denser, and the peripheral rays are bent toward the center of the beam. In Figure 2, phase fronts and ray paths are depicted for a beam with limited cross section that is propagating in a medium with an index of refraction *n* = no + n_{2}*E ^{2}*, where

*n*

_{0}is the constant component that is independent of

*E*, and

*n*

_{2}> 0. Since the phase velocity of light is

*v = c/n*=

*c*/(

*n*

_{o}+

*n*

_{2}

*E*, the phase fronts are curved (the field

^{2})*E*is stronger on the axis than at the periphery) and the rays are bent toward the axis of the beam. Such nonlinear refraction can be so large (its “strength” increases with the concentration of the field) that for practical purposes it completely suppresses diffraction effects.

The inverse effect, or self-defocusing, occurs if the medium in the region occupied by the light beam becomes optically less dense (*n*_{2} < 0) because of the variation of the index of refraction with intensity. In this case a powerful laser beam diverges much more rapidly than a low-intensity beam. Nonlinear wave phenomena of the self-focusing and self-defocusing types in which the mean frequency and the mean wave number *k = ωη/c = 2π/λ* are almost unchanged are called self-action effects of the waves. In addition to self-action of waves that are modulated in space, the self-action of waves modulated in time is also studied in nonlinear optics.

The propagation of a light pulse in a medium having an index of refraction of the form *n* = no + n_{2}*E ^{2}* is accompanied by a distortion of its shape and by phase modulation. As a result the spectrum of the laser pulse is drastically broadened. The width of the spectrum of the radiation upon leaving the medium is hundreds or thousands of times greater than the width upon entering.

Self-action effects determine the basic characteristics of the behavior of powerful beams of light in most media, including the active media of the lasers themselves. In particular, the avalanche-like increase in the intensity of the light field with self-focusing in many cases produces optical breakdown of the medium.

An interesting question in the phenomenon of self-focusing is the behavior of the light beam beyond the focal point. A. M. Prokhorov and his co-workers drew attention to the important role of the motion of the focal points in self-focusing. In an actual laser pulse the power varies with time, and the focal length of the nonlinear lens varies correspondingly. As a result, the focus moves. The velocity of its motion can reach 10^{9} cm/sec. By taking into account the rapid motion of the foci along with the aberrations of the nonlinear lens, it is possible in many cases to construct a complete theory of the phenomenon of self-focusing.

** Self-induced transparency and nonlinear absorption.** Media that are opaque to weak radiation can become transparent to high-intensity radiation (clearing); on the other hand, transparent materials can “darken” under the action of intense radiation (nonlinear absorption). These are the most important features of the absorption of high-intensity light. They are explained by the dependence of the absorption coefficient on the intensity of the light.

If the intensity of radiation that is resonant with respect to an absorbing medium is high, a substantial portion of the particles in the medium pass from the ground state to an excited state, and the populations of its upper and lower levels are equalized (the saturation effect). To a saturation effect under conditions of equilibrium, some energy must be expended; therefore, the induced transparency of the medium is associated with certain energy losses in the light beam.

A different type of clearing effect, resonance self-induced transparency of the medium, is observed in a field of short light pulses whose duration is less than the characteristic relaxation time of the medium. In this case a short, intense light pulse passes through the medium, in general without experiencing any absorption, whereas weak quasi-continuous radiation at the same frequency is almost completely absorbed by the same medium. As a result of the interaction between such a very short light pulse and the medium, there is a sharp decrease in the group velocity of propagation of the light pulse and a change in its shape.

Nonlinear absorption effects are associated with the fact that during the interaction of intense radiation of frequency ω°with particles there is an appreciable probability for processes involving the simultaneous absorption of *m* quanta of frequency ω1, where *m* = ω°/ω1.

** Nonlinear optics and spectroscopy; parametric light generator.** The development of nonlinear optics has made possible improvement of the methods of optical spectroscopy and development of the fundamentally new methods of nonlinear and active spectroscopy. An important problem of absorption spectroscopy is the production of a suitable frequency-tunable light source. Nonlinear optics provides a radical solution to the problem: the possibilities include not only the addition of photons in a nonlinear medium but also the inverse process—the coherent decay of a photon of frequency Ω into two photons of frequencies ω

_{1}and ω

_{2}that satisfy the condition ω1 + <t>2- The process proceeds efficiently if the condition for wave matching is satisfied simultaneously: k

_{t}= k

_{1}+ k

_{2}.

The operation of the parametric light generator is based on this principle. For a fixed frequency Ω (the pumping frequency), the frequencies ω_{1} and ω_{2} can be varied within wide limits (only their sum must be kept constant) by changing the parameters of the medium that affect the fulfillment of the phase matching conditions. With such generators it is now possible to cover the longwave part of the visible region and the near infrared region. Parametric light generators for the far infrared region also have been produced. The parametric light generator is a convenient light source for absorption spectrometers; with its advent, optics gained a tunable, stable, easily controlled source of coherent radiation (frequency or amplitude modulation of the radiation can be produced by applying an electric field to a nonlinear crystal).

The methods of nonlinear optics open up new possibilities for the development of correlation spectrographs and spectrographs with spatial resolution of the spectrum. Figure 3 shows a diagram of a nonlinear spectrograph with spatial resolution of the spectrum, which takes advantage of the fact that the dispersion

of the directions of phase matching in nonlinear crystals (Figure 1) can be greater than the ordinary dispersion of the substance. Spectral analysis in this case is accompanied by an increase in the frequency of the light, which is particularly favorable for spectral studies in the infrared region, and by amplification of the signal under study.

** Conversion of signals and images.** The frequency-addition effect, which is the basis of the operation of the spectrograph described above, has other uses. One of them is the recording of weak signals in the infrared region. If a frequency ω

_{x}lies in the infrared and ω

_{p}is in the visible, then the sum frequency Ω will also fall in the visible, and the conversion factor can be much greater than 1. In the visible range, however, the signal is recorded by means of a high-sensitivity photomultiplier (PM). A system consisting of a nonlinear crystal (in which frequency addition takes place) and a PM is a sensitive receiver of infrared radiation; such detectors are used in infrared astronomy. With this arrangement, it is possible not only to record a signal but also to convert an image from the infrared region to the visible region.

** Conclusion.** The methods of nonlinear optics pervade all the traditional branches of optics and are the basis for a number of new trends (nonlinear rotation of the plane of polarization, nonlinear scattering, nonlinear diffraction, nonlinear magneto-optics, and so on). As the intensity of the light field is increased, still newer nonlinear processes are being discovered. Unfortunately, the limiting light field that can be used experimentally is determined by the breakdown of the medium or by the change in its optical properties under the action of the light, rather than by the possibilities of laser technology.

In the first stage of development of nonlinear optics, the wavelengths used ranged from 1.06 to 0.3 μ. The transition to CO_{2} lasers (λ = 10.6 μ) led to the discovery of nonlinearity associated with the behavior of charge carriers in semiconductors (in the visible range it is virtually undetectable) and to the discovery of new nonlinear materials. With powerful sources of ultraviolet radiation, it is possible to study nonlinear absorption in crystals and fluids with a wide forbidden band, frequency multiplication in the vacuum ultraviolet region, and the development of ultraviolet lasers with optical pumping. In 1971, coherent nonlinear effects were observed for the first time in the X-ray region.

Advances in nonlinear optics have stimulated similar research in plasma physics, acoustics, and radio physics and have also created interest in the general theory of nonlinear waves. New avenues of research have appeared in solid-state physics that are associated with the study of nonlinear materials and the optical strength of solids and fluids. It is possible that some features of the characteristics of quasars are due to nonlinear optical phenomena in interstellar plasma. The attainment of laser radiation intensities such that nonlinear optical phenomena can be observed in a vacuum has not been ruled out.

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