**Nonlinear optics** (NLO) is the branch of

opticsOptics is the branch of physics which involves the behavior and properties of light, including its interactions with matter and the construction of instruments that use or detect it. Optics usually describes the behavior of visible, ultraviolet, and infrared light...

that describes the behavior of

lightLight or visible light is electromagnetic radiation that is visible to the human eye, and is responsible for the sense of sight. Visible light has wavelength in a range from about 380 nanometres to about 740 nm, with a frequency range of about 405 THz to 790 THz...

in

*nonlinear media*, that is, media in which the dielectric polarization

**P** responds nonlinearly to the

electric fieldIn physics, an electric field surrounds electrically charged particles and time-varying magnetic fields. The electric field depicts the force exerted on other electrically charged objects by the electrically charged particle the field is surrounding...

**E** of the light. This nonlinearity is typically only observed at very high light intensities (values of the electric field comparable to interatomic electric fields, typically 10

^{8} V/m) such as those provided by pulsed

laserA laser is a device that emits light through a process of optical amplification based on the stimulated emission of photons. The term "laser" originated as an acronym for Light Amplification by Stimulated Emission of Radiation...

s. In nonlinear optics, the

superposition principleIn physics and systems theory, the superposition principle , also known as superposition property, states that, for all linear systems, the net response at a given place and time caused by two or more stimuli is the sum of the responses which would have been caused by each stimulus individually...

no longer holds.
Nonlinear optics remained unexplored until the discovery of

Second harmonic generationAn optical frequency multiplier is a nonlinear optical device, in which photons interacting with a nonlinear material are effectively "combined" to form new photons with greater energy, and thus higher frequency...

shortly after demonstration of the first laser. (

Peter FrankenPeter A. Franken was an American physicist who contributed to the field of nonlinear optics. He was president of the Optical Society of America in 1977....

et al at

University of MichiganThe University of Michigan is a public research university located in Ann Arbor, Michigan in the United States. It is the state's oldest university and the flagship campus of the University of Michigan...

in 1961)

### Frequency mixing processes

NEWLINE

NEWLINE- Second harmonic generation
An optical frequency multiplier is a nonlinear optical device, in which photons interacting with a nonlinear material are effectively "combined" to form new photons with greater energy, and thus higher frequency...

(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. NEWLINE- 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.
NEWLINE- High harmonic generation
- Perturbative Harmonic Generation :Perturbative Harmonic Generation is a process whereby laser light of frequency ω and photon energy ħω can be used to generate new frequencies of light. The newly generated frequencies are integer multiples nħω of the original light's frequency...

(HHG), generation of light with frequencies much greater than the original (typically 100 to 1000 times greater) NEWLINE- Sum frequency generation
Sum-frequency generation is a non-linear optical process. This phenomenon is based on the annihilation of two input photons at angular frequencies \omega_1 and \omega_2 while, simultaneously, one photon at frequency \omega_3 is generated...

(SFG), generation of light with a frequency that is the sum of two other frequencies (SHG is a special case of this) NEWLINE- Difference frequency generation (DFG), generation of light with a frequency that is the difference between two other frequencies
NEWLINE- 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) NEWLINE- Optical parametric oscillation (OPO), generation of a signal and idler wave using a parametric amplifier in a resonator (with no signal input)
NEWLINE- Optical parametric generation (OPG), like parametric oscillation but without a resonator, using a very high gain instead
NEWLINE- Spontaneous parametric down conversion
Spontaneous parametric down-conversion is an important process in quantum optics, used especially as a source of entangled photon pairs, and of single photons.-Basic process:...

(SPDC), the amplification of the vacuum fluctuations in the low gain regime NEWLINE- Optical rectification
Electro-optic rectification is a non-linear optical process which consists in the generation of a quasi-DC polarization in a non-linear medium at the passage of an intense optical beam...

(OR), generation of quasi-static electric fields. NEWLINE- Nonlinear light-matter interaction with free electrons and plasmas

NEWLINE

### Other nonlinear processes

NEWLINE

NEWLINE- Optical Kerr effect
The Kerr effect, also called the quadratic electro-optic 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...

, intensity dependent refractive index (a $\backslash chi^\{(3)\}$ effect)NEWLINENEWLINE- Self-focusing
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...

, an effect due to the Optical Kerr effectThe Kerr effect, also called the quadratic electro-optic 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...

(and possibly higher order nonlinearities) caused by the spatial variation in the intensity creating a spatial variation in the refractive index NEWLINE- Kerr-lens modelocking
Kerr-lens modelocking 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....

(KLM), the use of Self-focusingSelf-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...

as a mechanism to mode lock laser. NEWLINE- Self-phase modulation
Self-phase modulation 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...

(SPM), an effect due to the Optical Kerr effectThe Kerr effect, also called the quadratic electro-optic 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...

(and possibly higher order nonlinearities) caused by the temporal variation in the intensity creating a temporal variation in the refractive index NEWLINE- Optical solitons
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:...

, An equilibrium solution for either an optical pulse (temporal soliton) or Spatial mode (spatial soliton) that does not change during propagation due to a balance between diffractionDiffraction refers to various phenomena which occur when a wave encounters an obstacle. Italian scientist Francesco Maria Grimaldi coined the word "diffraction" and was the first to record accurate observations of the phenomenon in 1665...

and the Kerr effectThe Kerr effect, also called the quadratic electro-optic 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...

(e.g. Self-phase modulationSelf-phase modulation 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...

for temporal and Self-focusingSelf-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...

for spatial solitons).

NEWLINE NEWLINE- Cross-phase modulation
Cross-phase modulation is a nonlinear optical effect where one wavelength of light can affect the phase of another wavelength of light through the optical Kerr effect.- Applications of XPM :...

(XPM) NEWLINE- Four-wave mixing
Four-wave mixing is an intermodulation phenomenon in optical systems, whereby interactions between 3 wavelengths produce a 4th wavelength in the signal. It is similar to the third-order intercept point in electrical systems...

(FWM), can also arise from other nonlinearities NEWLINE- Cross-polarized wave generation
Cross polarized wave 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...

(XPW), a $\backslash chi^\{(3)\}$ effect in which a wave with polarization vector perpendicular to the input one is generated NEWLINE- Modulational instability
In the field of nonlinear optics, modulational instability is a phenomenon whereby deviations from an optical waveform are reinforced by nonlinearity, leading to the generation of spectral-sidebands and the eventual breakup of the waveform into a train of pulses.-Initial instability and...

NEWLINE- Raman amplification
Raman amplification is based on the Stimulated Raman Scattering phenomenon, when a lower frequency 'signal' photon induces the inelastic scattering of a higher-frequency 'pump' photon in an optical medium in the nonlinear regime. As a result of this, another 'signal' photon is produced, with the...

NEWLINE- Optical phase conjugation
NEWLINE- Stimulated Brillouin scattering, interaction of photons with acoustic phonons
NEWLINE- Multi-photon absorption
Two-photon absorption is the simultaneous absorption of two photons of identical or different frequencies in order to excite a molecule from one state to a higher energy electronic state. The energy difference between the involved lower and upper states of the molecule is equal to the sum of the...

, simultaneous absorption of two or more photons, transferring the energyIn physics, energy is an indirectly observed quantity. It is often understood as the ability a physical system has to do work on other physical systems...

to a single electron NEWLINE- Multiple photoionisation, near-simultaneous removal of many bound electrons by one photon
NEWLINE- Chaos in Optical Systems
Optical Chaos is observed in many non-linear optical systems. One of the most common examples is a ring resonator.One of the most seminal works is published by Ikeda where chaotic behavior in a ring resonator was proposed and experiementally confirmed.Optical Chaos was an exciting field of...

NEWLINE

### 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:NEWLINE

NEWLINE- Pockels effect
The Pockels effect , or Pockels electro-optic effect, produces birefringence in an optical medium induced by a constant or varying electric field. It is distinguished from the Kerr effect by the fact that the birefringence is proportional to the electric field, whereas in the Kerr effect it is...

, the refractive index is affected by a static electric field; used in electro-optic modulatorElectro-optic modulator is an optical device in which a signal-controlled element displaying electro-optic effect is used to modulate a beam of light. The modulation may be imposed on the phase, frequency, amplitude, or polarization of the modulated beam...

s; NEWLINE- Acousto-optics
Acousto-optics is a branch of physics that studies the interactions between sound waves and light waves, especially the diffraction of laser light by ultrasound or sound in general.-Introduction:...

, the refractive index is affected by acoustic waves (ultrasound); used in acousto-optic modulatorAn acousto-optic modulator , also called a Bragg cell, uses the acousto-optic effect to diffract and shift the frequency of light using sound waves . They are used in lasers for Q-switching, telecommunications for signal modulation, and in spectroscopy for frequency control. A piezoelectric...

s. NEWLINE- Raman scattering
Raman scattering or the Raman effect is the inelastic scattering of a photon. It was discovered by Sir Chandrasekhara Venkata Raman and Kariamanickam Srinivasa Krishnan in liquids, and by Grigory Landsberg and Leonid Mandelstam in crystals....

, interaction of photons with optical phononIn physics, a phonon is a collective excitation in a periodic, elastic arrangement of atoms or molecules in condensed matter, such as solids and some liquids...

s;

NEWLINE

## ParametricA parametric process is an optical process in which light interacts with matter in such a way as to leave the quantum state of the material unchanged. As a direct consequence of this there can be no transfer of energy, momentum, or angular momentum between the optical field and the physical system...

processes

Nonlinear effects fall into two qualitatively different categories,

parametricA parametric process is an optical process in which light interacts with matter in such a way as to leave the quantum state of the material unchanged. As a direct consequence of this there can be no transfer of energy, momentum, or angular momentum between the optical field and the physical system...

and non-parametric effects. A parametric non-linearity
is an interaction in which the

quantum state 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 conserving in the optical field, making phase matching important; and polarization dependent.

### Theory

Parametric and lossy 'instantaneous' (i.e. electronic) nonlinear optical phenomena, in which the optical fields are not

too largePerturbation theory comprises mathematical methods that are used to find an approximate solution to a problem which cannot be solved exactly, by starting from the exact solution of a related problem...

, can be described by a

Taylor seriesIn mathematics, a Taylor series is a representation of a function as an infinite sum of terms that are calculated from the values of the function's derivatives at a single point....

expansion of the dielectric

Polarization densityIn classical electromagnetism, polarization density is the vector field that expresses the density of permanent or induced electric dipole moments in a dielectric material. When a dielectric is placed in an external electric field, its molecules gain electric dipole moment and the dielectric is...

(dipole moment per unit volume)

*P(t)* at time

*t* in terms of the electrical field:

$P(t)\; \backslash propto\; \backslash chi^\{(1)\}\; E(t)\; +\; \backslash chi^\{(2)\}\; E^2(t)\; +\; \backslash chi^\{(3)\}\; E^3(t)\; +\; \backslash cdots.$
Here, the coefficients χ

^{(n)} are the

*n*-th order

susceptibilitiesIn electromagnetism, the electric susceptibility \chi_e is a dimensionless proportionality constant that indicates the degree of polarization of a dielectric material in response to an applied electric field...

of the medium and the presence of such a term is generally referred to as an

*n*-th order nonlinearity. In general χ

^{n} is an

*n+1* order

tensorTensors are geometric objects that describe linear relations between vectors, scalars, and other tensors. Elementary examples include the dot product, the cross product, and linear maps. Vectors and scalars themselves are also tensors. A tensor can be represented as a multi-dimensional array of...

representing both the polarization dependent nature of the parametric interaction as well as the symmetries (or lack thereof) of the nonlinear material.

#### Wave-equation in a nonlinear material

Central to the study of electromagnetic waves is the

wave equationThe electromagnetic wave equation is a second-order partial differential equation that describes the propagation of electromagnetic waves through a medium or in a vacuum...

. Starting with

Maxwell's equationsMaxwell's equations are a set of partial differential equations that, together with the Lorentz force law, form the foundation of classical electrodynamics, classical optics, and electric circuits. These fields in turn underlie modern electrical and communications technologies.Maxwell's equations...

in an isotropic space containing no free charge, it can be shown that:

$\backslash nabla\; \backslash times\; \backslash nabla\; \backslash times\; \backslash mathbf\{E\}\; +\; \backslash frac\{n^2\}\{c^2\}\backslash frac\{\backslash partial^2\}\{\backslash partial\; t^2\}\backslash mathbf\{E\}\; -\backslash frac\{1\}\{c^2\}\backslash frac\{\backslash partial^2\}\{\backslash partial\; t^2\}\backslash mathbf\{P\}^\{NL\},$
where

**P**^{NL} is the nonlinear part of the

Polarization densityIn classical electromagnetism, polarization density is the vector field that expresses the density of permanent or induced electric dipole moments in a dielectric material. When a dielectric is placed in an external electric field, its molecules gain electric dipole moment and the dielectric is...

and n is the

refractive indexIn optics the refractive index or index of refraction of a substance or medium is a measure of the speed of light in that medium. It is expressed as a ratio of the speed of light in vacuum relative to that in the considered medium....

which comes from the linear term in

**P**.
Note one can normally use the vector identity

$\backslash nabla\; \backslash times\; \backslash left(\; \backslash nabla\; \backslash times\; \backslash mathbf\{V\}\; \backslash right)\; =\; \backslash nabla\; \backslash left(\; \backslash nabla\; \backslash cdot\; \backslash mathbf\{V\}\; \backslash right)\; -\; \backslash nabla^2\; \backslash mathbf\{V\}$
and

Gauss's lawIn physics, Gauss's law, also known as Gauss's flux theorem, is a law relating the distribution of electric charge to the resulting electric field. Gauss's law states that:...

,

$\backslash nabla\backslash cdot\backslash mathbf\{D\}\; =\; 0,$
to obtain the more familiar

wave equationThe electromagnetic wave equation is a second-order partial differential equation that describes the propagation of electromagnetic waves through a medium or in a vacuum...

$\backslash nabla^2\; \backslash mathbf\{E\}\; -\; \backslash frac\{n^2\}\{c^2\}\backslash frac\{\backslash partial^2\}\{\backslash partial\; t^2\}\backslash mathbf\{E\}\; 0.$
For nonlinear medium

Gauss's lawIn physics, Gauss's law, also known as Gauss's flux theorem, is a law relating the distribution of electric charge to the resulting electric field. Gauss's law states that:...

does not imply that the identity

$\backslash nabla\backslash cdot\backslash mathbf\{E\}\; =\; 0$
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:

$\backslash nabla^2\; \backslash mathbf\{E\}\; -\; \backslash frac\{n^2\}\{c^2\}\backslash frac\{\backslash partial^2\}\{\backslash partial\; t^2\}\backslash mathbf\{E\}$#### 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 and can be solved by the use of a

Green's functionIn mathematics, a Green's function is a type of function used to solve inhomogeneous differential equations subject to specific initial conditions or boundary conditions...

. Physically one gets the normal electromagnetic wave solutions to the homogeneous part of the wave equation:

$\backslash nabla^2\; \backslash mathbf\{E\}\; -\; \backslash frac\{n^2\}\{c^2\}\backslash frac\{\backslash partial^2\}\{\backslash partial\; t^2\}\backslash mathbf\{E\}=\; 0.$
and the inhomogenous term

$\backslash frac\{1\}\{c^2\}\backslash frac\{\backslash partial^2\}\{\backslash partial\; t^2\}\backslash mathbf\{P\}^\{NL\},$
acts as a driver/source of the electromagnetic waves. One of the consequences of this is a nonlinear interaction will result in energy being mixed or coupled between different colors which is often called a 'wave mixing'.
In general an

*n*-th order will lead to

*n+1*-th wave mixing. As an example, if we consider only a second order nonlinearity (three-wave mixing), then the polarization,

*P*, takes the form

$P^\{NL\}=\; \backslash chi^\{(2)\}\; E^2(t).$
If we assume that

*E(t)* is made up of two colors at frequencies

*ω*_{1} and

*ω*_{2}, we can write

*E(t)* as

$E(t)\; =\; E\_1e^\{-i\backslash omega\_1t\}+E\_2e^\{-i\backslash omega\_2t\}\; +\; c.c.$
where

*c.c.* stands for

complex conjugateIn mathematics, complex conjugates are a pair of complex numbers, both having the same real part, but with imaginary parts of equal magnitude and opposite signs...

. Plugging this into the expression for

*P* gives

$\backslash begin\{align\}\; P^\{NL\}=\; \backslash chi^\{(2)\}\; E^2(t)\; \&=\; \backslash chi^\{(2)\}\; [\; |E\_1|^2e^\{-i2\backslash omega\_1t\}+|E\_2|^2e^\{-i2\backslash omega\_2t\}\backslash \backslash \; \&\backslash qquad+2E\_1E\_2e^\{-i(\backslash omega\_1+\backslash omega\_2)t\}\backslash \backslash \; \&\backslash qquad+2E\_1E\_2^*e^\{-i(\backslash omega\_1-\backslash omega\_2)t\}\backslash \backslash \; \&\backslash qquad+2\backslash left(|E\_1|+|E\_2|\backslash right)e^\{0\}],\; \backslash end\{align\}$
which has frequency components at

*2ω*_{1},

*2ω*_{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 generationAn optical frequency multiplier is a nonlinear optical device, in which photons interacting with a nonlinear material are effectively "combined" to form new photons with greater energy, and thus higher frequency...

,

Sum frequency generationSum-frequency generation is a non-linear optical process. This phenomenon is based on the annihilation of two input photons at angular frequencies \omega_1 and \omega_2 while, simultaneously, one photon at frequency \omega_3 is generated...

, Difference frequency generation and

Optical rectificationElectro-optic rectification is a non-linear optical process which consists in the generation of a quasi-DC polarization in a non-linear medium at the passage of an intense optical beam...

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-mechanicalQuantum mechanics, also known as quantum physics or quantum theory, is a branch of physics providing a mathematical description of much of the dual particle-like and wave-like behavior and interactions of energy and matter. It departs from classical mechanics primarily at the atomic and subatomic...

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

$E\_j(\backslash mathbf\{x\},t)\; =\; e^\{i(\backslash omega\_j\; t\; -\; \backslash mathbf\{k\}\_j\; \backslash cdot\; \backslash mathbf\{x\})\},$
at position

$\backslash mathbf\{x\}$, with the wave vector

$\backslash mathbf\{k\}\_j\; =\; n(\backslash omega\_j)\backslash omega\_j/c$, where

$c$ is the velocity of light and

$n(\backslash omega\_j)$ the index of refraction of the medium at angular frequency

$\backslash omega\_j$. Thus, the second-order polarization angular frequency

$\backslash omega\_3$ is

$P^\{(2)\}\; (\backslash mathbf\{x\},\; t)\; \backslash propto\; E\_1^\{n\_1\}\; E\_2^\{n\_2\}\; e^\{i\; (\backslash omega\_3\; t\; -\; (m\_1\; \backslash mathbf\{k\}\_1\; +\; m\_2\; \backslash mathbf\{k\}\_2)\backslash cdot\backslash mathbf\{x\})\}.$
At each position

$\backslash mathbf\{x\}$, the oscillating second-order polarization radiates at angular frequency

$\backslash omega\_3$ and a corresponding wave vector

$\backslash mathbf\{k\}\_3\; =\; n(\backslash omega\_3)\backslash omega\_3/c$. Constructive interference, and therefore a high intensity

$\backslash omega\_3$ field, will occur only if

$\backslash mathbf\{k\}\_3\; =\; m\_1\; \backslash mathbf\{k\}\_1\; +\; m\_2\; \backslash mathbf\{k\}\_2.$
The above equation is known as the

*phase matching condition*. Typically, three-wave mixing is done in a birefringent crystalline material (I.e., the

refractive indexIn optics the refractive index or index of refraction of a substance or medium is a measure of the speed of light in that medium. It is expressed as a ratio of the speed of light in vacuum relative to that in the considered medium....

depends on the polarization and direction of the light that passes through.), where 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 opticsCrystal optics is the branch of optics that describes the behaviour of light in anisotropic media, that is, media in which light behaves differently depending on which direction the light is propagating. The index of refraction depends on both composition and crystal structure and can be...

). 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.NEWLINENEWLINENEWLINENEWLINENEWLINENEWLINENEWLINENEWLINENEWLINENEWLINENEWLINENEWLINENEWLINENEWLINENEWLINENEWLINENEWLINENEWLINENEWLINENEWLINENEWLINENEWLINENEWLINENEWLINENEWLINENEWLINENEWLINENEWLINENEWLINENEWLINENEWLINENEWLINENEWLINENEWLINENEWLINENEWLINENEWLINENEWLINENEWLINENEWLINENEWLINENEWLINENEWLINENEWLINENEWLINENEWLINENEWLINENEWLINENEWLINENEWLINENEWLINENEWLINE

Phase-matching types($\backslash lambda\_p\; \backslash leq\; \backslash lambda\_s\; \backslash leq\; \backslash lambda\_i$) Polarizations | Scheme |
---|

Pump | Signal | Idler | |
---|

e | o | o | Type I |

e | o | e | Type II (or IIA) |

e | e | o | Type III (or IIB) |

e | e | e | Type IV |

o | o | o | Type V |

o | o | e | Type VI (or IIB or IIIA) |

o | e | o | Type VII (or IIA or IIIB) |

o | e | e | Type VIII (or I) |

NEWLINENEWLINE
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 phasematching 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 that is not parallel with the propagation vector. This would lead to beam walkoff which limits the nonlinear optical conversion efficiency. Two other methods of phase matching avoids beam walkoff by forcing all frequencies to propagate at a 90 degree angle with respect to the optical axis of the crystal. These methods are called temperature tuning and

quasi-phase-matchingQuasi-phase-matching is a technique in nonlinear optics which allows a positive net flow of energy from the pump frequency to the signal and idler frequencies by creating a periodic structure in the nonlinear medium. Momentum is conserved, as is necessary for phase-matching, through an additional...

.
Temperature tuning is where 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 is controlled at a certain temperature 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 poledPeriodic poling is a formation of layers with alternate orientation in a birefringent material. The domains are regularly spaced, with period in a multiple of the desired wavelength of operation. The structure is desired to achieve quasi-phase-matching in the material.Periodically poled crystals...

. 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 modulationSelf-phase modulation 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...

(emulated by second order processes) of the signal and an

optical parametric amplifierAn optical parametric amplifier, abbreviated OPA, is a laser light source that emits light of variable wavelengths by an optical parametric amplification process.-Optical parametric generation :...

can be integrated monolithically.

## Higher-order frequency mixing

The above holds for

$\backslash chi^\{(2)\}$ processes. It can be extended for processes where

$\backslash chi^\{(3)\}$ is nonzero, something that is generally true in any medium without any symmetry restrictions. Third-harmonic generation is a

$\backslash chi^\{(3)\}$ process, although in laser applications, it is usually implemented as a two-stage process: first the fundamental laser frequency is doubled and then the doubled and the fundamental frequencies are added in a sum-frequency process. The Kerr effect can be described as a

$\backslash chi^\{(3)\}$ as well.
At high intensities the

Taylor seriesIn mathematics, a Taylor series is a representation of a function as an infinite sum of terms that are calculated from the values of the function's derivatives at a single point....

, 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 light flashes. The photon energies generated by this process can extend past the 800th harmonic order up to 1300

eVIn physics, the electron volt is a unit of energy equal to approximately joule . By definition, it is equal to the amount of kinetic energy gained by a single unbound electron when it accelerates through an electric potential difference of one volt...

. This is called

high-order harmonic generation- Perturbative Harmonic Generation :Perturbative Harmonic Generation is a process whereby laser light of frequency ω and photon energy ħω can be used to generate new frequencies of light. The newly generated frequencies are integer multiples nħω of the original light's frequency...

. 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.

### 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 lasersTi:sapphire lasers are tunable lasers which emit red and near-infrared light in the range from 650 to 1100 nanometers. These lasers are mainly used in scientific research because of their tunability and their ability to generate ultrashort pulses...

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), KDP (potassium dihydrogen phosphate), KTP (

potassium titanyl phosphatePotassium titanyl phosphate or KTP is a nonlinear optical material which is commonly used for frequency doubling diode pumped solid-state lasers such as Nd:YAG and other neodymium-doped lasers. The material has a relatively high optical damage threshold , a great optical nonlinearity and excellent...

), and

lithium niobateLithium niobate is a compound of niobium, lithium, and oxygen. Its single crystals are an important material for optical waveguides, mobile phones, optical modulators and various other linear and non-linear optical applications.-Properties:...

. These crystals have the necessary properties of being strongly

birefringentBirefringence, or double refraction, is the decomposition of a ray of light into two rays when it passes through certain anisotropic materials, such as crystals of calcite or boron nitride. The effect was first described by the Danish scientist Rasmus Bartholin in 1669, who saw it in calcite...

(necessary to obtain phase matching, see below), having a specific crystal symmetry and of course being transparent for both the impinging laser light and the frequency doubled wavelength, and have high damage thresholds which make them resistant against the high-intensity laser light. However, organic polymeric materials are set to take over from crystals as they are cheaper to make, have lower drive voltages and superior performance. {{Citation needed|date=May 2010}}

### 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

*retroreflection*A retroreflector is a device or surface that reflects light back to its source with a minimum scattering of light. An electromagnetic wave front is reflected back along a vector that is parallel to but opposite in direction from the wave's source. The device or surface's angle of incidence is...

).
One can interpret this nonlinear optical interaction 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 off 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.
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. A device producing the phase conjugation effect is known as a phase conjugate mirror (PCM).
For the four-wave mixing technique, we can describe four beams (

*j* = 1,2,3,4) with electric fields:

$\backslash Xi\_j(\backslash mathbf\{x\},t)\; =\; \backslash frac\{1\}\{2\}\; E\_j(\backslash mathbf\{x\})\; e^\{i\; (\backslash omega\_j\; t\; -\; \backslash mathbf\{k\}\backslash cdot\backslash mathbf\{x\})\}\; +\; \backslash mbox\{c.c.\}$
where

*E*_{j} are the electric field amplitudes. Ξ

_{1} and Ξ

_{2} are known as the two pump waves, with Ξ

_{3} being the signal wave, and Ξ

_{4} 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\_\{\backslash mbox\{NL\}\}\; =\; \backslash epsilon\_0\; \backslash chi^\{(3)\}\; (\backslash Xi\_1\; +\; \backslash Xi\_2\; +\; \backslash Xi\_3)^3\backslash $
resulting in generation of waves with frequencies given by ω = ±ω

_{1} ±ω

_{2} ±ω

_{3} in addition to third harmonic generation waves with ω = 3ω

_{1}, 3ω

_{2}, 3ω

_{3}.
As above, the phase-matching condition determines which of these waves is the dominant. By choosing conditions such that ω = ω

_{1} + ω

_{2} - ω

_{3} and

**k** =

**k**_{1} +

**k**_{2} -

**k**_{3}, this gives a polarization field:

$P\_\backslash omega\; =\; \backslash frac\{1\}\{2\}\; \backslash chi^\{(3)\}\; \backslash epsilon\_0\; E\_1\; E\_2\; E\_3^*\; e^\{i(\backslash omega\; t\; -\; \backslash mathbf\{k\}\; \backslash cdot\; \backslash mathbf\{x\}\; )\; \}\; +\; \backslash mbox\{c.c.\}.$
This is the generating field for the phase conjugate beam, Ξ

_{4}. Its direction is given by

**k**_{4} =

**k**_{1} +

**k**_{2} -

**k**_{3}, and so if the two pump beams are counterpropagating (

**k**_{1} = -

**k**_{2}), then the conjugate and signal beams propagate in opposite directions (

**k**_{4} = -

**k**_{3}). This results in the retroreflecting property of the effect.
Further, it can be shown for a medium with refractive index

*n* and a beam interaction length

*l*, the electric field amplitude of the conjugate beam is approximated by

$E\_4\; =\; \backslash frac\{i\; \backslash omega\; l\}\{2\; n\; c\}\; \backslash chi^\{(3)\}\; E\_1\; E\_2\; E\_3^*$
(where

*c* is the speed of light). If the pump beams

*E*_{1} and

*E*_{2} are plane (counterpropagating) waves, then:

$E\_4(\backslash mathbf\{x\})\; \backslash propto\; E\_3^*(\backslash mathbf\{x\});$
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 ω

_{1} = ω

_{2} = ω, and the signal wave higher in frequency such that ω

_{3} = ω + Δω, then the conjugate wave is of frequency ω

_{4} = ω - Δω. This is known as

*frequency flipping*.

## Common SHG materials

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NEWLINE- 800 nm light : BBO
NEWLINE- 806 nm light : lithium iodate (LiIO
_{3}) NEWLINE- 860 nm light : potassium niobate (KNbO
_{3}) NEWLINE- 980 nm light : KNbO
_{3} NEWLINE- 1064 nm light : monopotassium phosphate
Monopotassium phosphate -- 24 -- is a soluble salt which is used as a fertilizer, a food additive and a fungicide. It is a source of phosphorus and potassium. It is also a buffering agent...

(KH_{2}PO_{4}, KDP), lithium triborateLithium triborate LBO is a non-linear optics crystal. It has a wide transparency range, moderately high nonlinear coupling, high damage threshold and desirable chemical and mechanical properties. This crystal is often used for second harmonic generation of Nd:YAG lasers...

(LBO) and β-barium borate (BBO). NEWLINE- 1300 nm light : gallium selenide
Gallium selenide is a chemical compound. It has a hexagonal layer structure, similar to that of GaS. It is a photoconductor, a second harmonic generation crystal in nonlinear optics, and has been used as a far-infrared conversion material at 14-31 THz and above.-Uses:It is said to have potential...

(GaSe) NEWLINE- 1319 nm light : KNbO
_{3}, BBO, KDP, potassium titanyl phosphatePotassium titanyl phosphate or KTP is a nonlinear optical material which is commonly used for frequency doubling diode pumped solid-state lasers such as Nd:YAG and other neodymium-doped lasers. The material has a relatively high optical damage threshold , a great optical nonlinearity and excellent...

(KTP), lithium niobateLithium niobate is a compound of niobium, lithium, and oxygen. Its single crystals are an important material for optical waveguides, mobile phones, optical modulators and various other linear and non-linear optical applications.-Properties:...

(LiNbO_{3}), LiIO_{3}, and ammonium dihydrogen phosphateAmmonium dihydrogen phosphate , or monoammonium phosphate, NH4H2PO4, is formed when a solution of phosphoric acid is added to ammonia until the solution is distinctly acidic. It crystallizes in tetragonal prisms. Monoammonium phosphate is often used in the blending of dry agricultural fertilizers...

(ADP)

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## See also

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NEWLINE- Born-Infeld action
NEWLINE- Filament propagation
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....

NEWLINE- :Category:Nonlinear optical materials
NEWLINE- Parametric process (optics)
A parametric process is an optical process in which light interacts with matter in such a way as to leave the quantum state of the material unchanged. As a direct consequence of this there can be no transfer of energy, momentum, or angular momentum between the optical field and the physical system...

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## External links

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