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Nonlinear optics (NLO) is the branch of
optics Optics is the branch of physics that studies the behaviour and properties of light, including its interactions with matter and the construction of instruments that use or detect it. Optics usually describes the behaviour of visible, ultrav ...
that describes the behaviour of
light Light or visible light is electromagnetic radiation that can be perceived by the human eye. Visible light is usually defined as having wavelengths in the range of 400–700 nanometres (nm), corresponding to frequencies of 750–420 t ...
in ''
nonlinear In mathematics and science, a nonlinear system is a system in which the change of the output is not proportional to the change of the input. Nonlinear problems are of interest to engineers, biologists, physicists, mathematicians, and many oth ...
media'', that is, media in which the polarization density P responds non-linearly to the
electric field An electric field (sometimes E-field) is the physical field that surrounds electrically charged particles and exerts force on all other charged particles in the field, either attracting or repelling them. It also refers to the physical field ...
E of the light. The non-linearity is typically observed only at very high light intensities (when the electric field of the light is >108 V/m and thus comparable to the atomic electric field of ~1011 V/m) such as those provided by
laser A laser is a device that emits light through a process of optical amplification based on the stimulated emission of electromagnetic radiation. The word "laser" is an acronym for "light amplification by stimulated emission of radiation". The ...
s. Above the
Schwinger limit In quantum electrodynamics (QED), the Schwinger limit is a scale above which the electromagnetic field is expected to become nonlinear. The limit was first derived in one of QED's earliest theoretical successes by Fritz Sauter in 1931 and discu ...
, the vacuum itself is expected to become nonlinear. In nonlinear optics, the
superposition principle The superposition principle, also known as superposition property, states that, for all linear systems, the net response caused by two or more stimuli is the sum of the responses that would have been caused by each stimulus individually. So tha ...
no longer holds.


History

The first nonlinear optical effect to be predicted was two-photon absorption, by Maria Goeppert Mayer for her PhD in 1931, but it remained an unexplored theoretical curiosity until 1961 and the almost simultaneous observation of two-photon absorption at
Bell Labs Nokia Bell Labs, originally named Bell Telephone Laboratories (1925–1984), then AT&T Bell Laboratories (1984–1996) and Bell Labs Innovations (1996–2007), is an American industrial research and scientific development company owned by mul ...
and the discovery of
second-harmonic generation Second-harmonic generation (SHG, also called frequency doubling) is a nonlinear optical process in which two photons with the same frequency interact with a nonlinear material, are "combined", and generate a new photon with twice the energy o ...
by
Peter Franken Peter A. Franken (November 10, 1928 – March 11, 1999) was an American physicist who contributed to the field of nonlinear optics. He was president of the Optical Society of America in 1977. In 1961, Professor Peter Franken and his coworkers in t ...
''et al.'' at
University of Michigan , mottoeng = "Arts, Knowledge, Truth" , former_names = Catholepistemiad, or University of Michigania (1817–1821) , budget = $10.3 billion (2021) , endowment = $17 billion (2021)As o ...
, both shortly after the construction of the first laser by Theodore Maiman. However, some nonlinear effects were discovered before the development of the laser. The theoretical basis for many nonlinear processes were first described in Bloembergen's monograph "Nonlinear Optics".


Nonlinear optical processes

Nonlinear optics explains nonlinear response of properties such as
frequency Frequency is the number of occurrences of a repeating event per unit of time. It is also occasionally referred to as ''temporal frequency'' for clarity, and is distinct from ''angular frequency''. Frequency is measured in hertz (Hz) which is eq ...
, polarization, phase or path of incident light. These nonlinear interactions give rise to a host of optical phenomena:


Frequency-mixing processes

*
Second-harmonic generation Second-harmonic generation (SHG, also called frequency doubling) is a nonlinear optical process in which two photons with the same frequency interact with a nonlinear material, are "combined", and generate a new photon with twice the energy o ...
(SHG), or ''frequency doubling'', generation of light with a doubled frequency (half the wavelength), two photons are destroyed, creating a single photon at two times the frequency. * Third-harmonic generation (THG), generation of light with a tripled frequency (one-third the wavelength), three photons are destroyed, creating a single photon at three times the frequency. * High-harmonic generation (HHG), generation of light with frequencies much greater than the original (typically 100 to 1000 times greater). * Sum-frequency generation (SFG), generation of light with a frequency that is the sum of two other frequencies (SHG is a special case of this). * Difference-frequency generation (DFG), generation of light with a frequency that is the difference between two other frequencies. * Optical parametric amplification (OPA), amplification of a signal input in the presence of a higher-frequency pump wave, at the same time generating an ''idler'' wave (can be considered as DFG). * Optical parametric oscillation (OPO), generation of a signal and idler wave using a parametric amplifier in a resonator (with no signal input). * Optical parametric generation (OPG), like parametric oscillation but without a resonator, using a very high gain instead. * Half-harmonic generation, the special case of OPO or OPG when the signal and idler degenerate in one single frequency, * Spontaneous parametric down-conversion (SPDC), the amplification of the vacuum fluctuations in the low-gain regime. * Optical rectification (OR), generation of quasi-static electric fields. * Nonlinear light-matter interaction with free electrons and plasmas.


Other nonlinear processes

* Optical
Kerr effect 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 chan ...
, intensity-dependent refractive index (a \chi^ effect). ** Self-focusing, an effect due to the optical
Kerr effect 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 chan ...
(and possibly higher-order nonlinearities) caused by the spatial variation in the intensity creating a spatial variation in the refractive index. ** Kerr-lens modelocking (KLM), the use of self-focusing as a mechanism to mode-lock laser. ** Self-phase modulation (SPM), an effect due to the optical
Kerr effect 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 chan ...
(and possibly higher-order nonlinearities) caused by the temporal variation in the intensity creating a temporal variation in the refractive index. ** Optical solitons, an equilibrium solution for either an optical pulse (temporal soliton) or
spatial mode A transverse mode of electromagnetic radiation is a particular electromagnetic field pattern of the radiation in the plane perpendicular (i.e., transverse) to the radiation's propagation direction. Transverse modes occur in radio waves and microwav ...
(spatial soliton) that does not change during propagation due to a balance between
dispersion Dispersion may refer to: Economics and finance *Dispersion (finance), a measure for the statistical distribution of portfolio returns *Price dispersion, a variation in prices across sellers of the same item *Wage dispersion, the amount of variatio ...
and the
Kerr effect 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 chan ...
(e.g. self-phase modulation for temporal and self-focusing for spatial solitons). ** Self-diffraction, splitting of beams in a multi-wave mixing process with potential energy transfer. *
Cross-phase modulation Cross-phase modulation (XPM) is a nonlinear optical effect where one wavelength of light can affect the phase of another wavelength of light through the optical Kerr effect. When the optical power from a wavelength impacts the refractive index, the ...
(XPM), where one wavelength of light can affect the phase of another wavelength of light through the optical Kerr effect. * Four-wave mixing (FWM), can also arise from other nonlinearities. *
Cross-polarized wave generation Cross-polarized wave (XPW) generation is a nonlinear optical process that can be classified in the group of frequency degenerate (four-wave mixing) processes. It can take place only in media with anisotropy of third-order nonlinearity. As a resul ...
(XPW), a \chi^ effect in which a wave with polarization vector perpendicular to the input one is generated. * Modulational instability. * Raman amplification *
Optical phase conjugation 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 typica ...
. * Stimulated Brillouin scattering, interaction of photons with acoustic phonons * Multi-photon absorption, simultaneous absorption of two or more photons, transferring the
energy In physics, energy (from Ancient Greek: ἐνέργεια, ''enérgeia'', “activity”) is the quantitative property that is transferred to a body or to a physical system, recognizable in the performance of work and in the form of ...
to a single electron. * Multiple photoionisation, near-simultaneous removal of many bound electrons by one photon. * Chaos in optical systems.


Related processes

In these processes, the medium has a linear response to the light, but the properties of the medium are affected by other causes: * Pockels effect, the refractive index is affected by a static electric field; used in electro-optic modulators. * Acousto-optics, the refractive index is affected by acoustic waves (ultrasound); used in
acousto-optic modulator An acousto-optic modulator (AOM), also called a Bragg cell or an acousto-optic deflector (AOD), uses the acousto-optic effect to diffract and shift the frequency of light using sound waves (usually at radio-frequency). They are used in lasers ...
s. *
Raman scattering Raman scattering or the Raman effect () is the inelastic scattering of photons by matter, meaning that there is both an exchange of energy and a change in the light's direction. Typically this effect involves vibrational energy being gained by ...
, interaction of photons with optical
phonon In physics, a phonon is a collective excitation in a periodic, elastic arrangement of atoms or molecules in condensed matter, specifically in solids and some liquids. A type of quasiparticle, a phonon is an excited state in the quantum mechani ...
s.


Parametric processes

Nonlinear effects fall into two qualitatively different categories, parametric and non-parametric effects. A parametric non-linearity is an interaction in which the
quantum state In quantum physics, a quantum state is a mathematical entity that provides a probability distribution for the outcomes of each possible measurement on a system. Knowledge of the quantum state together with the rules for the system's evolution i ...
of the nonlinear material is not changed by the interaction with the optical field. As a consequence of this, the process is "instantaneous". Energy and momentum are conserved in the optical field, making phase matching important and polarization-dependent. See Section ''Parametric versus Nonparametric Processes'', ''Nonlinear Optics'' by Robert W. Boyd (3rd ed.), pp. 13–15.


Theory

Parametric and "instantaneous" (i.e. material must be lossless and dispersionless through the Kramers–Kronig relations) nonlinear optical phenomena, in which the optical fields are not too large, can be described by a
Taylor series In mathematics, the Taylor series or Taylor expansion of a function is an infinite sum of terms that are expressed in terms of the function's derivatives at a single point. For most common functions, the function and the sum of its Taylor ser ...
expansion of the
dielectric In electromagnetism, a dielectric (or dielectric medium) is an electrical insulator that can be polarised by an applied electric field. When a dielectric material is placed in an electric field, electric charges do not flow through the ma ...
polarization density (
electric dipole moment The electric dipole moment is a measure of the separation of positive and negative electrical charges within a system, that is, a measure of the system's overall polarity. The SI unit for electric dipole moment is the coulomb- meter (C⋅m). ...
per unit volume) P(''t'') at time ''t'' in terms of the
electric field An electric field (sometimes E-field) is the physical field that surrounds electrically charged particles and exerts force on all other charged particles in the field, either attracting or repelling them. It also refers to the physical field ...
E(''t''): :\mathbf(t) = \varepsilon_0 \left( \chi^ \mathbf(t) + \chi^ \mathbf^2(t) + \chi^ \mathbf^3(t) + \ldots \right), where the coefficients χ(''n'') are the ''n''-th-order susceptibilities of the medium, and the presence of such a term is generally referred to as an ''n''-th-order nonlinearity. Note that the polarization density P(''t'') and electrical field E(''t'') are considered as scalar for simplicity. In general, χ(''n'') is an (''n'' + 1)-th-rank
tensor In mathematics, a tensor is an algebraic object that describes a multilinear relationship between sets of algebraic objects related to a vector space. Tensors may map between different objects such as vectors, scalars, and even other tensor ...
representing both the polarization-dependent nature of the parametric interaction and the symmetries (or lack) of the nonlinear material.


Wave equation in a nonlinear material

Central to the study of electromagnetic waves is the
wave equation The (two-way) wave equation is a second-order linear partial differential equation for the description of waves or standing wave fields — as they occur in classical physics — such as mechanical waves (e.g. water waves, sound waves and ...
. Starting with
Maxwell's equations Maxwell's equations, or Maxwell–Heaviside equations, are a set of coupled partial differential equations that, together with the Lorentz force law, form the foundation of classical electromagnetism, classical optics, and electric circuits ...
in an isotropic space, containing no free charge, it can be shown that : \nabla \times \nabla \times \mathbf + \frac\frac\mathbf = -\frac\frac\mathbf^\text, where PNL is the nonlinear part of the polarization density, and ''n'' is the
refractive index In optics, the refractive index (or refraction index) of an optical medium is a dimensionless number that gives the indication of the light bending ability of that medium. The refractive index determines how much the path of light is bent, ...
, which comes from the linear term in P. Note that one can normally use the vector identity :\nabla \times \left( \nabla \times \mathbf \right) = \nabla \left( \nabla \cdot \mathbf \right) - \nabla^2 \mathbf and
Gauss's law In physics and electromagnetism, Gauss's law, also known as Gauss's flux theorem, (or sometimes simply called Gauss's theorem) is a law relating the distribution of electric charge to the resulting electric field. In its integral form, it st ...
(assuming no free charges, \rho_\text = 0), :\nabla\cdot\mathbf = 0, to obtain the more familiar
wave equation The (two-way) wave equation is a second-order linear partial differential equation for the description of waves or standing wave fields — as they occur in classical physics — such as mechanical waves (e.g. water waves, sound waves and ...
: \nabla^2 \mathbf - \frac\frac\mathbf = 0. For a nonlinear medium,
Gauss's law In physics and electromagnetism, Gauss's law, also known as Gauss's flux theorem, (or sometimes simply called Gauss's theorem) is a law relating the distribution of electric charge to the resulting electric field. In its integral form, it st ...
does not imply that the identity :\nabla\cdot\mathbf = 0 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: : \nabla^2 \mathbf - \frac\frac\mathbf = \frac\frac\mathbf^\text.


Nonlinearities as a wave-mixing process

The nonlinear wave equation is an inhomogeneous differential equation. The general solution comes from the study of
ordinary differential equations In mathematics, an ordinary differential equation (ODE) is a differential equation whose unknown(s) consists of one (or more) function(s) of one variable and involves the derivatives of those functions. The term ''ordinary'' is used in contrast ...
and can be obtained by the use of a Green's function. Physically one gets the normal
electromagnetic wave In physics, electromagnetic radiation (EMR) consists of waves of the electromagnetic (EM) field, which propagate through space and carry momentum and electromagnetic radiant energy. It includes radio waves, microwaves, infrared, (visible) ...
solutions to the homogeneous part of the wave equation: :\nabla^2 \mathbf - \frac\frac\mathbf = 0, and the inhomogeneous term :\frac\frac\mathbf^\text acts as a driver/source of the electromagnetic waves. One of the consequences of this is a nonlinear interaction that results in energy being mixed or coupled between different frequencies, which is often called a "wave mixing". In general, an ''n''-th order nonlinearity will lead to (''n'' + 1)-wave mixing. As an example, if we consider only a second-order nonlinearity (three-wave mixing), then the polarization P takes the form :\mathbf^\text = \varepsilon_0 \chi^ \mathbf^2(t). If we assume that ''E''(''t'') is made up of two components at frequencies ''ω''1 and ''ω''2, we can write ''E''(''t'') as :\mathbf(t) = E_1\cos(\omega_1t) + E_2\cos(\omega_2t), and using
Euler's formula Euler's formula, named after Leonhard Euler, is a mathematical formula in complex analysis that establishes the fundamental relationship between the trigonometric functions and the complex exponential function. Euler's formula states that ...
to convert to exponentials, :\mathbf(t) = \fracE_1 e^ + \fracE_2 e^ + \text, where "c.c." stands for
complex conjugate In mathematics, the complex conjugate of a complex number is the number with an equal real part and an imaginary part equal in magnitude but opposite in sign. That is, (if a and b are real, then) the complex conjugate of a + bi is equal to a - ...
. Plugging this into the expression for P gives :\begin \mathbf^\text &= \varepsilon_0 \chi^ \mathbf^2(t) \\ pt &= \frac \chi^ \left E_1\^2 + \left, E_2\^2\right)e^ + \text\right \end which has frequency components at 2''ω''1, 2''ω''2, ''ω''1 + ''ω''2, ''ω''1 − ''ω''2, and 0. These three-wave mixing processes correspond to the nonlinear effects known as
second-harmonic generation Second-harmonic generation (SHG, also called frequency doubling) is a nonlinear optical process in which two photons with the same frequency interact with a nonlinear material, are "combined", and generate a new photon with twice the energy o ...
, sum-frequency generation, difference-frequency generation and optical rectification respectively. Note: Parametric generation and amplification is a variation of difference-frequency generation, where the lower frequency of one of the two generating fields is much weaker (parametric amplification) or completely absent (parametric generation). In the latter case, the fundamental
quantum-mechanical Quantum mechanics is a fundamental theory in physics that provides a description of the physical properties of nature at the scale of atoms and subatomic particles. It is the foundation of all quantum physics including quantum chemistry, qu ...
uncertainty in the electric field initiates the process.


Phase matching

The above ignores the position dependence of the electrical fields. In a typical situation, the electrical fields are traveling waves described by :E_j(\mathbf,t) =E_ e^ + \text at position \mathbf, with the wave vector \, \mathbf_j\, = \mathbf(\omega_j)\omega_j/c, where c is the velocity of light in vacuum, and \mathbf(\omega_j) is the index of refraction of the medium at angular frequency \omega_j. Thus, the second-order polarization at angular frequency \omega_3=\omega_1+\omega_2 is :P^(\mathbf, t) \propto E_1^ E_2^ e^ + \text At each position \mathbf within the nonlinear medium, the oscillating second-order polarization radiates at angular frequency \omega_3 and a corresponding wave vector \, \mathbf_3\, = \mathbf(\omega_3)\omega_3/c. Constructive interference, and therefore a high-intensity \omega_3 field, will occur only if :\vec_3 = \vec_1 + \vec_2. The above equation is known as the ''phase-matching condition''. Typically, three-wave mixing is done in a birefringent crystalline material, where the
refractive index In optics, the refractive index (or refraction index) of an optical medium is a dimensionless number that gives the indication of the light bending ability of that medium. The refractive index determines how much the path of light is bent, ...
depends on the polarization and direction of the light that passes through. The polarizations of the fields and the orientation of the crystal are chosen such that the phase-matching condition is fulfilled. This phase-matching technique is called angle tuning. Typically a crystal has three axes, one or two of which have a different refractive index than the other one(s). Uniaxial crystals, for example, have a single preferred axis, called the extraordinary (e) axis, while the other two are ordinary axes (o) (see crystal optics). There are several schemes of choosing the polarizations for this crystal type. If the signal and idler have the same polarization, it is called "type-I phase matching", and if their polarizations are perpendicular, it is called "type-II phase matching". However, other conventions exist that specify further which frequency has what polarization relative to the crystal axis. These types are listed below, with the convention that the signal wavelength is shorter than the idler wavelength. Most common nonlinear crystals are negative uniaxial, which means that the ''e'' axis has a smaller refractive index than the ''o'' axes. In those crystals, type-I and -II phase matching are usually the most suitable schemes. In positive uniaxial crystals, types VII and VIII are more suitable. Types II and III are essentially equivalent, except that the names of signal and idler are swapped when the signal has a longer wavelength than the idler. For this reason, they are sometimes called IIA and IIB. The type numbers V–VIII are less common than I and II and variants. One undesirable effect of angle tuning is that the optical frequencies involved do not propagate collinearly with each other. This is due to the fact that the extraordinary wave propagating through a birefringent crystal possesses a
Poynting vector In physics, the Poynting vector (or Umov–Poynting vector) represents the directional energy flux (the energy transfer per unit area per unit time) or ''power flow'' of an electromagnetic field. The SI unit of the Poynting vector is the watt p ...
that is not parallel to the propagation vector. This would lead to beam walk-off, which limits the nonlinear optical conversion efficiency. Two other methods of phase matching avoid beam walk-off by forcing all frequencies to propagate at a 90° with respect to the optical axis of the crystal. These methods are called temperature tuning and quasi-phase-matching. Temperature tuning is used when the pump (laser) frequency polarization is orthogonal to the signal and idler frequency polarization. The birefringence in some crystals, in particular lithium niobate is highly temperature-dependent. The crystal temperature is controlled to achieve phase-matching conditions. The other method is quasi-phase-matching. In this method the frequencies involved are not constantly locked in phase with each other, instead the crystal axis is flipped at a regular interval Λ, typically 15 micrometres in length. Hence, these crystals are called periodically poled. This results in the polarization response of the crystal to be shifted back in phase with the pump beam by reversing the nonlinear susceptibility. This allows net positive energy flow from the pump into the signal and idler frequencies. In this case, the crystal itself provides the additional wavevector ''k'' = 2π/Λ (and hence momentum) to satisfy the phase-matching condition. Quasi-phase-matching can be expanded to chirped gratings to get more bandwidth and to shape an SHG pulse like it is done in a dazzler. SHG of a pump and self-phase modulation (emulated by second-order processes) of the signal and an
optical parametric amplifier An optical parametric amplifier, abbreviated OPA, is a laser light source that emits light of variable wavelengths by an optical parametric amplification process. It is essentially the same as an optical parametric oscillator, but without the optic ...
can be integrated monolithically.


Higher-order frequency mixing

The above holds for \chi^ processes. It can be extended for processes where \chi^ 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 \chi^ cancels this focal phase shift and often has a nearly self-canceling overall phase-matching condition, which relatively simplifies broad wavelength tuning compared to sum frequency generation. In \chi^ all four frequencies are mixing simultaneously, as opposed to sequential mixing via two \chi^ processes. The Kerr effect can be described as a \chi^ as well. At high peak powers the Kerr effect can cause filamentation of light in air, in which the light travels without dispersion or divergence in a self-generated waveguide. At even high intensities the
Taylor series In mathematics, the Taylor series or Taylor expansion of a function is an infinite sum of terms that are expressed in terms of the function's derivatives at a single point. For most common functions, the function and the sum of its Taylor ser ...
, which led the domination of the lower orders, does not converge anymore and instead a time based model is used. When a noble gas atom is hit by an intense laser pulse, which has an electric field strength comparable to the Coulomb field of the atom, the outermost electron may be ionized from the atom. Once freed, the electron can be accelerated by the electric field of the light, first moving away from the ion, then back toward it as the field changes direction. The electron may then recombine with the ion, releasing its energy in the form of a photon. The light is emitted at every peak of the laser light field which is intense enough, producing a series of
attosecond An attosecond (symbol as) is a unit of time in the International System of Units (SI) equal to 1×10−18 of a second (one quintillionth of a second). For comparison, an attosecond is to a second what a second is to about 31.71 billion years.
light flashes. The photon energies generated by this process can extend past the 800th harmonic order up to a few K eV. This is called high-order harmonic generation. The laser must be linearly polarized, so that the electron returns to the vicinity of the parent ion. High-order harmonic generation has been observed in noble gas jets, cells, and gas-filled capillary waveguides.


Example uses


Frequency doubling

One of the most commonly used frequency-mixing processes is frequency doubling, or second-harmonic generation. With this technique, the 1064 nm output from Nd:YAG lasers or the 800 nm output from Ti:sapphire lasers can be converted to visible light, with wavelengths of 532 nm (green) or 400 nm (violet) respectively. Practically, frequency doubling is carried out by placing a nonlinear medium in a laser beam. While there are many types of nonlinear media, the most common media are crystals. Commonly used crystals are BBO (
β-barium borate Barium borate is an inorganic compound, a borate of barium with a chemical formula BaB2O4 or Ba(BO2)2. It is available as a hydrate or dehydrated form, as white powder or colorless crystals. The crystals exist in the high-temperature α phase and ...
), KDP (
potassium dihydrogen phosphate Monopotassium phosphate (MKP) (also, potassium dihydrogenphosphate, KDP, or monobasic potassium phosphate) is the inorganic compound with the formula KH2PO4. Together with dipotassium phosphate (K2HPO4.(H2O)x) it is often used as a fertilizer, f ...
), KTP ( potassium titanyl phosphate), and lithium niobate. These crystals have the necessary properties of being strongly birefringent (necessary to obtain phase matching, see below), having a specific crystal symmetry, being transparent for both the impinging laser light and the frequency-doubled wavelength, and having high damage thresholds, which makes them resistant against the high-intensity laser light.


Optical phase conjugation

It is possible, using nonlinear optical processes, to exactly reverse the propagation direction and phase variation of a beam of light. The reversed beam is called a ''conjugate'' beam, and thus the technique is known as optical phase conjugation (also called ''time reversal'', ''wavefront reversal'' and is significantly different from '' retroreflection''). A device producing the phase-conjugation effect is known as a phase-conjugate mirror (PCM).


Principles

One can interpret optical phase conjugation as being analogous to a real-time holographic process. In this case, the interacting beams simultaneously interact in a nonlinear optical material to form a dynamic hologram (two of the three input beams), or real-time diffraction pattern, in the material. The third incident beam diffracts at this dynamic hologram, and, in the process, reads out the ''phase-conjugate'' wave. In effect, all three incident beams interact (essentially) simultaneously to form several real-time holograms, resulting in a set of diffracted output waves that phase up as the "time-reversed" beam. In the language of nonlinear optics, the interacting beams result in a nonlinear polarization within the material, which coherently radiates to form the phase-conjugate wave. Reversal of wavefront means a perfect reversal of photons' linear momentum and angular momentum. The reversal of
angular momentum In physics, angular momentum (rarely, moment of momentum or rotational momentum) is the rotational analog of linear momentum. It is an important physical quantity because it is a conserved quantity—the total angular momentum of a closed syst ...
means reversal of both polarization state and orbital angular momentum. Reversal of orbital angular momentum of optical vortex is due to the perfect match of helical phase profiles of the incident and reflected beams.
Optical phase conjugation 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 typica ...
is implemented via stimulated Brillouin scattering, four-wave mixing, three-wave mixing, static linear holograms and some other tools. The most common way of producing optical phase conjugation is to use a four-wave mixing technique, though it is also possible to use processes such as stimulated Brillouin scattering.


Four-wave mixing technique

For the four-wave mixing technique, we can describe four beams (''j'' = 1, 2, 3, 4) with electric fields: :\Xi_j(\mathbf,t) = \frac E_j(\mathbf) e^ + \text, where ''Ej'' are the electric field amplitudes. Ξ1 and Ξ2 are known as the two pump waves, with Ξ3 being the signal wave, and Ξ4 being the generated conjugate wave. If the pump waves and the signal wave are superimposed in a medium with a non-zero χ(3), this produces a nonlinear polarization field: :P_\text = \varepsilon_0 \chi^ (\Xi_1 + \Xi_2 + \Xi_3)^3, resulting in generation of waves with frequencies given by ω = ±ω1 ± ω2 ± ω3 in addition to third-harmonic generation waves with ω = 3ω1, 3ω2, 3ω3. As above, the phase-matching condition determines which of these waves is the dominant. By choosing conditions such that ω = ω1 + ω2 − ω3 and k = k1 + k2 − k3, this gives a polarization field: :P_\omega = \frac \chi^ \varepsilon_0 E_1 E_2 E_3^* e^ + \text This is the generating field for the phase-conjugate beam, Ξ4. Its direction is given by k4 = k1 + k2 − k3, and so if the two pump beams are counterpropagating (k1 = −k2), then the conjugate and signal beams propagate in opposite directions (k4 = −k3). This results in the retroreflecting property of the effect. Further, it can be shown that for a medium with refractive index ''n'' and a beam interaction length ''l'', the electric field amplitude of the conjugate beam is approximated by :E_4 = \frac \chi^ E_1 E_2 E_3^*, where ''c'' is the speed of light. If the pump beams ''E''1 and ''E''2 are plane (counterpropagating) waves, then :E_4(\mathbf) \propto E_3^*(\mathbf), that is, the generated beam amplitude is the complex conjugate of the signal beam amplitude. Since the imaginary part of the amplitude contains the phase of the beam, this results in the reversal of phase property of the effect. Note that the constant of proportionality between the signal and conjugate beams can be greater than 1. This is effectively a mirror with a reflection coefficient greater than 100%, producing an amplified reflection. The power for this comes from the two pump beams, which are depleted by the process. The frequency of the conjugate wave can be different from that of the signal wave. If the pump waves are of frequency ω1 = ω2 = ω, and the signal wave is higher in frequency such that ω3 = ω + Δω, then the conjugate wave is of frequency ω4 = ω − Δω. This is known as ''frequency flipping''.


Angular and linear momenta in optical phase conjugation


Classical picture

In ''classical Maxwell electrodynamics'' a phase-conjugating mirror performs reversal of the
Poynting vector In physics, the Poynting vector (or Umov–Poynting vector) represents the directional energy flux (the energy transfer per unit area per unit time) or ''power flow'' of an electromagnetic field. The SI unit of the Poynting vector is the watt p ...
: :\mathbf_\text(\mathbf,t) = -\mathbf_\text(\mathbf,t), ("in" means incident field, "out" means reflected field) where :\mathbf(\mathbf,t) = \epsilon_0 c^2 \mathbf(\mathbf,t) \times \mathbf(\mathbf,t), which is a linear momentum density of electromagnetic field.A. Yu. Okulov, "Angular momentum of photons and phase conjugation", J. Phys. B: At. Mol. Opt. Phys. v. 41, 101001 (2008)
In the same way a phase-conjugated wave has an opposite angular momentum density vector \mathbf(\mathbf,t) = \mathbf \times \mathbf(\mathbf,t) with respect to incident field:A. Yu. Okulov, "Optical and Sound Helical structures in a Mandelstam–Brillouin mirror". JETP Lett., v. 88, n. 8, pp. 561–566 (2008)
.
:\mathbf_\text(\mathbf,t) = -\mathbf_\text(\mathbf,t). The above identities are valid ''locally'', i.e. in each space point \mathbf in a given moment t for an ''ideal phase-conjugating mirror''.


Quantum picture

In ''quantum electrodynamics'' the photon with energy \hbar \omega also possesses linear momentum \mathbf = \hbar \mathbf and angular momentum, whose projection on propagation axis is L_\mathbf = \pm \hbar \ell, where \ell is ''topological charge'' of photon, or winding number, \mathbf is propagation axis. The angular momentum projection on propagation axis has ''discrete values'' \pm \hbar \ell. In ''quantum electrodynamics'' the interpretation of phase conjugation is much simpler compared to ''classical electrodynamics''. The photon reflected from phase conjugating-mirror (out) has opposite directions of linear and angular momenta with respect to incident photon (in): :\begin \mathbf_\text &= -\hbar \mathbf = -\mathbf_\text = \hbar\mathbf, \\ _\text &= -\hbar \ell = -_\text = \hbar \ell. \end


Nonlinear optical pattern formation

Optical fields transmitted through nonlinear Kerr media can also display
pattern formation The science of pattern formation deals with the visible, ( statistically) orderly outcomes of self-organization and the common principles behind similar patterns in nature. In developmental biology, pattern formation refers to the generation of ...
owing to the nonlinear medium amplifying spatial and temporal noise. The effect is referred to as optical modulation instability. This has been observed both in photo-refractive, photonic lattices, as well as photo-reactive systems. In the latter case, optical nonlinearity is afforded by reaction-induced increases in refractive index. Examples of pattern formation are spatial solitons and vortex lattices in framework of
nonlinear Schrödinger equation 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 non ...
.


Molecular nonlinear optics

The early studies of nonlinear optics and materials focused on the inorganic solids. With the development of nonlinear optics, molecular optical properties were investigated, forming molecular nonlinear optics. The traditional approaches used in the past to enhance nonlinearities include extending chromophore π-systems, adjusting bond length alternation, inducing intramolecular charge transfer, extending conjugation in 2D, and engineering multipolar charge distributions. Recently, many novel directions were proposed for enhanced nonlinearity and light manipulation, including twisted chromophores, combining rich density of states with bond alternation, microscopic cascading of second-order nonlinearity, etc. Due to the distinguished advantages, molecular nonlinear optics have been widely used in the biophotonics field, including bioimaging, phototherapy, biosensing, etc.


Common second-harmonic-generating (SHG) materials

Ordered by pump wavelength: * 800 nm: BBO * 806 nm: lithium iodate (LiIO3) * 860 nm: potassium niobate (KNbO3) * 980 nm: KNbO3 * 1064 nm:
monopotassium phosphate Monopotassium phosphate (MKP) (also, potassium dihydrogenphosphate, KDP, or monobasic potassium phosphate) is the inorganic compound with the formula KH2PO4. Together with dipotassium phosphate (K2HPO4.(H2O)x) it is often used as a fertilizer, f ...
(KH2PO4, KDP), lithium triborate (LBO) and
β-barium borate Barium borate is an inorganic compound, a borate of barium with a chemical formula BaB2O4 or Ba(BO2)2. It is available as a hydrate or dehydrated form, as white powder or colorless crystals. The crystals exist in the high-temperature α phase and ...
(BBO) * 1300 nm: gallium selenide (GaSe) * 1319 nm: KNbO3, BBO, KDP, potassium titanyl phosphate (KTP), lithium niobate (LiNbO3), LiIO3, and ammonium dihydrogen phosphate (ADP) * 1550 nm: potassium titanyl phosphate (KTP), lithium niobate (LiNbO3)


See also

* Born–Infeld model * Filament propagation * :Nonlinear optical materials


Further reading


Encyclopedia of laser physics and technology
with content on nonlinear optics, by Rüdiger Paschotta




Robert Boyd plenary presentation: Quantum Nonlinear Optics: Nonlinear Optics Meets the Quantum World
SPIE Newsroom * Boyd, R. W. 020 Nonlinear optics, 4th ed. edn, Academic, London.


References

{{DEFAULTSORT:Nonlinear Optics Optics