Free-electron laser

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https://en.wikipedia.org/wiki/Free-electron_laser 

 

The free-electron laser FELIX Radboud University, Netherlands.

A free-electron laser (FEL) is a (fourth generation) synchrotron light source producing extremely brilliant and short pulses of synchrotron radiation. An FEL functions and behaves in many ways like a laser, but instead of using stimulated emission from atomic or molecular excitations, it employs relativistic electrons as a gain medium. Synchrotron radiation is generated as a bunch of electrons passes through a magnetic structure (called undulator or wiggler). In an FEL, this radiation is further amplified as the synchrotron radiation re-interacts with the electron bunch such that the electrons start to emit coherently, thus allowing an exponential increase in overall radiation intensity.

As electron kinetic energy and undulator parameters can be adapted as desired, free-electron lasers are tunable and can be built for a wider frequency range than any type of laser, currently ranging in wavelength from microwaves, through terahertz radiation and infrared, to the visible spectrum, ultraviolet, and X-ray.

 

Schematic representation of an undulator, at the core of a free-electron laser.

The first free-electron laser was developed by John Madey in 1971 at Stanford University utilizing technology developed by Hans Motz and his coworkers, who built an undulator at Stanford in 1953, using the wiggler magnetic configuration. Madey used a 43 MeV electron beam and 5 m long wiggler to amplify a signal.

Beam creation

The undulator of FELIX.

To create an FEL, a beam of electrons is accelerated to almost the speed of light. The beam passes through a periodic arrangement of magnets with alternating poles across the beam path, which creates a side to side magnetic field. The direction of the beam is called the longitudinal direction, while the direction across the beam path is called transverse. This array of magnets is called an undulator or a wiggler, because the Lorentz force of the field forces the electrons in the beam to wiggle transversely, traveling along a sinusoidal path about the axis of the undulator.

The transverse acceleration of the electrons across this path results in the release of photons (synchrotron radiation), which are monochromatic but still incoherent, because the electromagnetic waves from randomly distributed electrons interfere constructively and destructively in time. The resulting radiation power scales linearly with the number of electrons. Mirrors at each end of the undulator create an optical cavity, causing the radiation to form standing waves, or alternately an external excitation laser is provided. The synchrotron radiation becomes sufficiently strong that the transverse electric field of the radiation beam interacts with the transverse electron current created by the sinusoidal wiggling motion, causing some electrons to gain and others to lose energy to the optical field via the ponderomotive force.

This energy modulation evolves into electron density (current) modulations with a period of one optical wavelength. The electrons are thus longitudinally clumped into microbunches, separated by one optical wavelength along the axis. Whereas an undulator alone would cause the electrons to radiate independently (incoherently), the radiation emitted by the bunched electrons is in phase, and the fields add together coherently.

The radiation intensity grows, causing additional microbunching of the electrons, which continue to radiate in phase with each other. This process continues until the electrons are completely microbunched and the radiation reaches a saturated power several orders of magnitude higher than that of the undulator radiation.

The wavelength of the radiation emitted can be readily tuned by adjusting the energy of the electron beam or the magnetic-field strength of the undulators.

FELs are relativistic machines. The wavelength of the emitted radiation, ${\displaystyle \lambda _{r}}$, is given by

${\displaystyle \lambda _{r}={\frac {\lambda _{u}}{2\gamma ^{2}}}\left(1+{\frac {K^{2}}{2}}\right)}$

or when the wiggler strength parameter K, discussed below, is small

${\displaystyle \lambda _{r}\propto {\frac {\lambda _{u}}{2\gamma ^{2}}}}$

where ${\displaystyle \lambda _{u}}$ is the undulator wavelength (the spatial period of the magnetic field), ${\displaystyle \gamma }$ is the relativistic Lorentz factor and the proportionality constant depends on the undulator geometry and is of the order of 1.

This formula can be understood as a combination of two relativistic effects. Imagine you are sitting on an electron passing through the undulator. Due to Lorentz contraction the undulator is shortened by a ${\displaystyle \gamma }$ factor and the electron experiences much shorter undulator wavelength ${\displaystyle \lambda _{u}/\gamma }$. However, the radiation emitted at this wavelength is observed in the laboratory frame of reference and the relativistic Doppler effect brings the second ${\displaystyle \gamma }$ factor to the above formula. In an X-ray FEL the typical undulator wavelength of 1 cm is transformed to X-ray wavelengths on the order of 1 nm by ${\displaystyle \gamma }$ ≈ 2000, i.e. the electrons have to travel with the speed of 0.9999998c.

Wiggler strength parameter K

K, a dimensionless parameter, defines the wiggler strength as the relationship between the length of a period and the radius of bend,

${\displaystyle K={\frac {\gamma \lambda _{u}}{2\pi \rho }}={\frac {eB_{0}\lambda _{u}}{2\pi m_{e}c}}}$

where ${\displaystyle \rho }$ is the bending radius, ${\displaystyle B_{0}}$ is the applied magnetic field, ${\displaystyle m_{e}}$ is the electron mass, and ${\displaystyle e}$ is the elementary charge.

Expressed in practical units, the dimensionless undulator parameter is ${\displaystyle K=0.934\cdot B_{0}\,{\text{[T]}}\cdot \lambda _{u}\,{\text{[cm]}}}$.

Quantum effects

In most cases, the theory of classical electromagnetism adequately accounts for the behavior of free electron lasers. For sufficiently short wavelengths, quantum effects of electron recoil and shot noise may have to be considered.

FEL construction

Free-electron lasers require the use of an electron accelerator with its associated shielding, as accelerated electrons can be a radiation hazard if not properly contained. These accelerators are typically powered by klystrons, which require a high-voltage supply. The electron beam must be maintained in a vacuum, which requires the use of numerous vacuum pumps along the beam path. While this equipment is bulky and expensive, free-electron lasers can achieve very high peak powers, and the tunability of FELs makes them highly desirable in many disciplines, including chemistry, structure determination of molecules in biology, medical diagnosis, and nondestructive testing.

Infrared and terahertz FELs

The Fritz Haber Institute in Berlin completed a mid-infrared and terahertz FEL in 2013.

X-ray FELs

The lack of a material to make mirrors that can reflect extreme ultraviolet and x-rays means that FELs at these frequencies cannot use a resonant cavity like other lasers, which reflects the radiation so it makes multiple passes through the undulator. Consequently, in an X-ray FEL (XFEL) the output beam is produced by a single pass of radiation through the undulator. This requires that there be enough amplification over a single pass to produce an adequately bright beam.

Because of the lack of mirrors, XFELs use long undulators. The underlying principle of the intense pulses from the X-ray laser lies in the principle of self-amplified spontaneous emission (SASE), which leads to the microbunching. Initially all electrons are distributed evenly and emit only incoherent spontaneous radiation. Through the interaction of this radiation and the electrons' oscillations, they drift into microbunches separated by a distance equal to one radiation wavelength. Through this interaction, all electrons begin emitting coherent radiation in phase. All emitted radiation can reinforce itself perfectly whereby wave crests and wave troughs are always superimposed on one another in the best possible way. This results in an exponential increase of emitted radiation power, leading to high beam intensities and laser-like properties. Examples of facilities operating on the SASE FEL principle include the Free electron LASer in Hamburg (FLASH), the Linac Coherent Light Source (LCLS) at the SLAC National Accelerator Laboratory, the European x-ray free electron laser (EuXFEL) in Hamburg, the SPring-8 Compact SASE Source (SCSS) in Japan, the SwissFEL at the Paul Scherrer Institute (Switzerland), the SACLA at the RIKEN Harima Institute in Japan, and the PAL-XFEL (Pohang Accelerator Laboratory X-ray Free-Electron Laser) in Korea.

Self-seeding

One problem with SASE FELs is the lack of temporal coherence due to a noisy startup process. To avoid this, one can "seed" an FEL with a laser tuned to the resonance of the FEL. Such a temporally coherent seed can be produced by more conventional means, such as by high harmonic generation (HHG) using an optical laser pulse. This results in coherent amplification of the input signal; in effect, the output laser quality is characterized by the seed. While HHG seeds are available at wavelengths down to the extreme ultraviolet, seeding is not feasible at x-ray wavelengths due to the lack of conventional x-ray lasers.

In late 2010, in Italy, the seeded-FEL source FERMI@Elettra started commissioning, at the Trieste Synchrotron Laboratory. FERMI@Elettra is a single-pass FEL user-facility covering the wavelength range from 100 nm (12 eV) to 10 nm (124 eV), located next to the third-generation synchrotron radiation facility ELETTRA in Trieste, Italy.

In 2012, scientists working on the LCLS overcame the seeding limitation for x-ray wavelengths by self-seeding the laser with its own beam after being filtered through a diamond monochromator. The resulting intensity and monochromaticity of the beam were unprecedented and allowed new experiments to be conducted involving manipulating atoms and imaging molecules. Other labs around the world are incorporating the technique into their equipment.

Research

Biomedical

Basic research

Researchers have explored free-electron lasers as an alternative to synchrotron light sources that have been the workhorses of protein crystallography and cell biology.

Exceptionally bright and fast X-rays can image proteins using x-ray crystallography. This technique allows first-time imaging of proteins that do not stack in a way that allows imaging by conventional techniques, 25% of the total number of proteins. Resolutions of 0.8 nm have been achieved with pulse durations of 30 femtoseconds. To get a clear view, a resolution of 0.1–0.3 nm is required. The short pulse durations allow images of X-ray diffraction patterns to be recorded before the molecules are destroyed.  The bright, fast X-rays were produced at the Linac Coherent Light Source at SLAC. As of 2014 LCLS was the world's most powerful X-ray FEL.

Due to the increased repetition rates of the next-generation X-ray FEL sources, such as the European XFEL, the expected number of diffraction patterns is also expected to increase by a substantial amount. ^[23] The increase in the number of diffraction patterns will place a large strain on existing analysis methods. To combat this, several methods have been research in order to be able to sort the huge amount of data typical X-ray FEL experiments will generate.  While the various methods have been shown to be effective, it is clear that to pave the way towards single-particle X-ray FEL imaging at full repetition rates, several challenges have to be overcome before the next resolution revolution can be achieved. 

New biomarkers for metabolic diseases: taking advantage of the selectivity and sensitivity when combining infrared ion spectroscopy and mass spectrometry scientists can provide a structural fingerprint of small molecules in biological samples, like blood or urine. This new and unique methodology is generating exciting new possibilities to better understand metabolic diseases and develop novel diagnostic and therapeutic strategies.

Surgery

Research by Glenn Edwards and colleagues at Vanderbilt University's FEL Center in 1994 found that soft tissues including skin, cornea, and brain tissue could be cut, or ablated, using infrared FEL wavelengths around 6.45 micrometres with minimal collateral damage to adjacent tissue. This led to surgeries on humans, the first ever using a free-electron laser. Starting in 1999, Copeland and Konrad performed three surgeries in which they resected meningioma brain tumors. Beginning in 2000, Joos and Mawn performed five surgeries that cut a window in the sheath of the optic nerve, to test the efficacy for optic nerve sheath fenestration. These eight surgeries produced results consistent with the standard of care and with the added benefit of minimal collateral damage. A review of FELs for medical uses is given in the 1st edition of Tunable Laser Applications.

Fat removal

Several small, clinical lasers tunable in the 6 to 7 micrometre range with pulse structure and energy to give minimal collateral damage in soft tissue have been created. At Vanderbilt, there exists a Raman shifted system pumped by an Alexandrite laser.

Rox Anderson proposed the medical application of the free-electron laser in melting fats without harming the overlying skin. At infrared wavelengths, water in tissue was heated by the laser, but at wavelengths corresponding to 915, 1210 and 1720 nm, subsurface lipids were differentially heated more strongly than water. The possible applications of this selective photothermolysis (heating tissues using light) include the selective destruction of sebum lipids to treat acne, as well as targeting other lipids associated with cellulite and body fat as well as fatty plaques that form in arteries which can help treat atherosclerosis and heart disease.

Military

FEL technology is being evaluated by the US Navy as a candidate for an antiaircraft and anti-missile directed-energy weapon. The Thomas Jefferson National Accelerator Facility's FEL has demonstrated over 14 kW power output. Compact multi-megawatt class FEL weapons are undergoing research.

On June 9, 2009 the Office of Naval Research announced it had awarded Raytheon a contract to develop a 100 kW experimental FEL. On March 18, 2010 Boeing Directed Energy Systems announced the completion of an initial design for U.S. Naval use. A prototype FEL system was demonstrated, with a full-power prototype scheduled by 2018.

 

Degrees of freedom (physics and chemistry)

From Wikipedia, the free encyclopedia

https://en.wikipedia.org/wiki/Degrees_of_freedom_(physics_and_chemistry)

In physics and chemistry, a degree of freedom is an independent physical parameter in the formal description of the state of a physical system. The set of all states of a system is known as the system's phase space, and the degrees of freedom of the system are the dimensions of the phase space.

The location of a particle in three-dimensional space requires three position coordinates. Similarly, the direction and speed at which a particle moves can be described in terms of three velocity components, each in reference to the three dimensions of space. If the time evolution of the system is deterministic, where the state at one instant uniquely determines its past and future position and velocity as a function of time, such a system has six degrees of freedom. If the motion of the particle is constrained to a lower number of dimensions – for example, the particle must move along a wire or on a fixed surface – then the system has fewer than six degrees of freedom. On the other hand, a system with an extended object that can rotate or vibrate can have more than six degrees of freedom.

In classical mechanics, the state of a point particle at any given time is often described with position and velocity coordinates in the Lagrangian formalism, or with position and momentum coordinates in the Hamiltonian formalism.

In statistical mechanics, a degree of freedom is a single scalar number describing the microstate of a system. The specification of all microstates of a system is a point in the system's phase space.

In the 3D ideal chain model in chemistry, two angles are necessary to describe the orientation of each monomer.

It is often useful to specify quadratic degrees of freedom. These are degrees of freedom that contribute in a quadratic function to the energy of the system.

Gas molecules

Different ways of visualizing the 6 degrees of freedom of a diatomic molecule. (CM: center of mass of the system, T: translational motion, R: rotational motion, V: vibrational motion.)

In three-dimensional space, three degrees of freedom are associated with the movement of a particle. A diatomic gas molecule has 6 degrees of freedom. This set may be decomposed in terms of translations, rotations, and vibrations of the molecule. The center of mass motion of the entire molecule accounts for 3 degrees of freedom. In addition, the molecule has two rotational degrees of motion and one vibrational mode. The rotations occur around the two axes perpendicular to the line between the two atoms. The rotation around the atom–atom bond is not a physical rotation. This yields, for a diatomic molecule, a decomposition of:

${\displaystyle N=6=3+2+1.}$

For a general, non-linear molecule, all 3 rotational degrees of freedom are considered, resulting in the decomposition:

${\displaystyle 3N=3+3+(3N-6)}$

which means that an N-atom molecule has 3N − 6 vibrational degrees of freedom for N > 2. In special cases, such as adsorbed large molecules, the rotational degrees of freedom can be limited to only one.

As defined above one can also count degrees of freedom using the minimum number of coordinates required to specify a position. This is done as follows:

1. For a single particle we need 2 coordinates in a 2-D plane to specify its position and 3 coordinates in 3-D space. Thus its degree of freedom in a 3-D space is 3.
2. For a body consisting of 2 particles (ex. a diatomic molecule) in a 3-D space with constant distance between them (let's say d) we can show (below) its degrees of freedom to be 5.

Let's say one particle in this body has coordinate (x₁, y₁, z₁) and the other has coordinate (x₂, y₂, z₂) with  z₂ unknown. Application of the formula for distance between two coordinates

${\displaystyle d={\sqrt {(x_{2}-x_{1})^{2}+(y_{2}-y_{1})^{2}+(z_{2}-z_{1})^{2}}}}$

results in one equation with one unknown, in which we can solve for z₂. One of x₁, x₂, y₁, y₂, z₁, or z₂ can be unknown.

Contrary to the classical equipartition theorem, at room temperature, the vibrational motion of molecules typically makes negligible contributions to the heat capacity. This is because these degrees of freedom are frozen because the spacing between the energy eigenvalues exceeds the energy corresponding to ambient temperatures (k_BT). In the following table such degrees of freedom are disregarded because of their low effect on total energy. Then only the translational and rotational degrees of freedom contribute, in equal amounts, to the heat capacity ratio. This is why γ=5/3 for monatomic gases and γ=7/5 for diatomic gases at room temperature.

However, at very high temperatures, on the order of the vibrational temperature (Θ_vib), vibrational motion cannot be neglected.

Vibrational temperatures are between 10³ K and 10⁴ K.

	Monatomic	Linear molecules	Non-linear molecules
Translation (x, y, and z)	3	3	3
Rotation (x, y, and z)	0	2	3
Total (disregarding Vibration at room temperatures)	3	5	6
Vibration	0	3N − 5	3N − 6
Total (including Vibration)	3	3N	3N

Independent degrees of freedom

The set of degrees of freedom X₁, ... , X_N of a system is independent if the energy associated with the set can be written in the following form:

${\displaystyle E=\sum _{i=1}^{N}E_{i}(X_{i}),}$

where E_i is a function of the sole variable X_i.

example: if X₁ and X₂ are two degrees of freedom, and E is the associated energy:

• If ${\displaystyle E=X_{1}^{4}+X_{2}^{4}}$, then the two degrees of freedom are independent.
• If ${\displaystyle E=X_{1}^{4}+X_{1}X_{2}+X_{2}^{4}}$, then the two degrees of freedom are not independent. The term involving the product of X₁ and X₂ is a coupling term that describes an interaction between the two degrees of freedom.

For i from 1 to N, the value of the ith degree of freedom X_i is distributed according to the Boltzmann distribution. Its probability density function is the following:

${\displaystyle p_{i}(X_{i})={\frac {e^{-{\frac {E_{i}}{k_{B}T}}}}{\int dX_{i}\,e^{-{\frac {E_{i}}{k_{B}T}}}}}}$,

In this section, and throughout the article the brackets ${\displaystyle \langle \rangle }$ denote the mean of the quantity they enclose.

The internal energy of the system is the sum of the average energies associated with each of the degrees of freedom:

${\displaystyle \langle E\rangle =\sum _{i=1}^{N}\langle E_{i}\rangle .}$

Quadratic degrees of freedom

A degree of freedom X_i is quadratic if the energy terms associated with this degree of freedom can be written as

${\displaystyle E=\alpha _{i}\,\,X_{i}^{2}+\beta _{i}\,\,X_{i}Y}$,

where Y is a linear combination of other quadratic degrees of freedom.

example: if X₁ and X₂ are two degrees of freedom, and E is the associated energy:

• If ${\displaystyle E=X_{1}^{4}+X_{1}^{3}X_{2}+X_{2}^{4}}$, then the two degrees of freedom are not independent and non-quadratic.
• If ${\displaystyle E=X_{1}^{4}+X_{2}^{4}}$, then the two degrees of freedom are independent and non-quadratic.
• If ${\displaystyle E=X_{1}^{2}+X_{1}X_{2}+2X_{2}^{2}}$, then the two degrees of freedom are not independent but are quadratic.
• If ${\displaystyle E=X_{1}^{2}+2X_{2}^{2}}$, then the two degrees of freedom are independent and quadratic.

For example, in Newtonian mechanics, the dynamics of a system of quadratic degrees of freedom are controlled by a set of homogeneous linear differential equations with constant coefficients.

Quadratic and independent degree of freedom

X₁, ... , X_N are quadratic and independent degrees of freedom if the energy associated with a microstate of the system they represent can be written as:

${\displaystyle E=\sum _{i=1}^{N}\alpha _{i}X_{i}^{2}}$

Equipartition theorem

In the classical limit of statistical mechanics, at thermodynamic equilibrium, the internal energy of a system of N quadratic and independent degrees of freedom is:

${\displaystyle U=\langle E\rangle =N\,{\frac {k_{B}T}{2}}}$

Here, the mean energy associated with a degree of freedom is:

${\displaystyle \langle E_{i}\rangle =\int dX_{i}\,\,\alpha _{i}X_{i}^{2}\,\,p_{i}(X_{i})={\frac {\int dX_{i}\,\,\alpha _{i}X_{i}^{2}\,\,e^{-{\frac {\alpha _{i}X_{i}^{2}}{k_{B}T}}}}{\int dX_{i}\,\,e^{-{\frac {\alpha _{i}X_{i}^{2}}{k_{B}T}}}}}}$

${\displaystyle \langle E_{i}\rangle ={\frac {k_{B}T}{2}}{\frac {\int dx\,\,x^{2}\,\,e^{-{\frac {x^{2}}{2}}}}{\int dx\,\,e^{-{\frac {x^{2}}{2}}}}}={\frac {k_{B}T}{2}}}$

Since the degrees of freedom are independent, the internal energy of the system is equal to the sum of the mean energy associated with each degree of freedom, which demonstrates the result.

Generalizations

The description of a system's state as a point in its phase space, although mathematically convenient, is thought to be fundamentally inaccurate. In quantum mechanics, the motion degrees of freedom are superseded with the concept of wave function, and operators which correspond to other degrees of freedom have discrete spectra. For example, intrinsic angular momentum operator (which corresponds to the rotational freedom) for an electron or photon has only two eigenvalues. This discreteness becomes apparent when action has an order of magnitude of the Planck constant, and individual degrees of freedom can be distinguished.

 

Nonlinear optics

From Wikipedia, the free encyclopedia

 

Structure of KTP crystal, viewed down b axis, used in second harmonic generation.

Nonlinear optics (NLO) is the branch of optics that describes the behaviour of light in nonlinear media, that is, media in which the polarization density P responds non-linearly to the electric field E of the light. The non-linearity is typically observed only at very high light intensities (values of atomic electric fields, typically 10⁸ V/m) such as those provided by lasers. Above the Schwinger limit, the vacuum itself is expected to become nonlinear. In nonlinear optics, the superposition principle no longer holds.

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  and the discovery of second-harmonic generation by Peter Franken et al. at University of Michigan, 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, polarization, phase or path of incident light. These nonlinear interactions give rise to a host of optical phenomena:

Frequency-mixing processes

• Second-harmonic generation (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, intensity-dependent refractive index (a ${\displaystyle \chi ^{(3)}}$ effect).
  • Self-focusing, an effect due to the optical Kerr effect (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 (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 (spatial soliton) that does not change during propagation due to a balance between dispersion and the Kerr effect (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 (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 (XPW), a ${\displaystyle \chi ^{(3)}}$ effect in which a wave with polarization vector perpendicular to the input one is generated.
• Modulational instability.
• Raman amplification
• Optical phase conjugation.
• Stimulated Brillouin scattering, interaction of photons with acoustic phonons
• Multi-photon absorption, simultaneous absorption of two or more photons, transferring the energy 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 modulators.
• Raman scattering, interaction of photons with optical phonons.

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 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.

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 expansion of the dielectric polarization density (electric dipole moment per unit volume) P(t) at time t in terms of the electric field E(t):

${\displaystyle \mathbf {P} (t)=\varepsilon _{0}(\chi ^{(1)}\mathbf {E} (t)+\chi ^{(2)}\mathbf {E} ^{2}(t)+\chi ^{(3)}\mathbf {E} ^{3}(t)+\ldots ),}$

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 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. Starting with Maxwell's equations in an isotropic space, containing no free charge, it can be shown that

${\displaystyle \nabla \times \nabla \times \mathbf {E} +{\frac {n^{2}}{c^{2}}}{\frac {\partial ^{2}}{\partial t^{2}}}\mathbf {E} =-{\frac {1}{\varepsilon _{0}c^{2}}}{\frac {\partial ^{2}}{\partial t^{2}}}\mathbf {P} ^{\text{NL}},}$

where P^NL is the nonlinear part of the polarization density, and n is the refractive index, which comes from the linear term in P.

Note that one can normally use the vector identity

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

and Gauss's law (assuming no free charges, ${\displaystyle \rho _{\text{free}}=0}$),

${\displaystyle \nabla \cdot \mathbf {D} =0,}$

to obtain the more familiar wave equation

${\displaystyle \nabla ^{2}\mathbf {E} -{\frac {n^{2}}{c^{2}}}{\frac {\partial ^{2}}{\partial t^{2}}}\mathbf {E} =0.}$

For nonlinear medium, Gauss's law does not imply that the identity

${\displaystyle \nabla \cdot \mathbf {E} =0}$

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:

${\displaystyle \nabla ^{2}\mathbf {E} -{\frac {n^{2}}{c^{2}}}{\frac {\partial ^{2}}{\partial t^{2}}}\mathbf {E} ={\frac {1}{\varepsilon _{0}c^{2}}}{\frac {\partial ^{2}}{\partial t^{2}}}\mathbf {P} ^{\text{NL}}.}$

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 obtained by the use of a Green's function. Physically one gets the normal electromagnetic wave solutions to the homogeneous part of the wave equation:

${\displaystyle \nabla ^{2}\mathbf {E} -{\frac {n^{2}}{c^{2}}}{\frac {\partial ^{2}}{\partial t^{2}}}\mathbf {E} =0,}$

and the inhomogeneous term

${\displaystyle {\frac {1}{\varepsilon _{0}c^{2}}}{\frac {\partial ^{2}}{\partial t^{2}}}\mathbf {P} ^{\text{NL}}}$

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

${\displaystyle \mathbf {P} ^{\text{NL}}=\varepsilon _{0}\chi ^{(2)}\mathbf {E} ^{2}(t).}$

If we assume that E(t) is made up of two components at frequencies ω₁ and ω₂, we can write E(t) as

${\displaystyle \mathbf {E} (t)=E_{1}\cos(\omega _{1}t)+E_{2}\cos(\omega _{2}t),}$

and using Euler's formula to convert to exponentials,

${\displaystyle \mathbf {E} (t)={\frac {1}{2}}E_{1}e^{-i\omega _{1}t}+{\frac {1}{2}}E_{2}e^{-i\omega _{2}t}+{\text{c.c.}},}$

where "c.c." stands for complex conjugate. Plugging this into the expression for P gives

${\displaystyle {\begin{aligned}\mathbf {P} ^{\text{NL}}=\varepsilon _{0}\chi ^{(2)}\mathbf {E} ^{2}(t)&={\frac {\varepsilon _{0}}{4}}\chi ^{(2)}{\Big [}{E_{1}}^{2}e^{-i2\omega _{1}t}+{E_{2}}^{2}e^{-i2\omega _{2}t}\\&\qquad +2E_{1}E_{2}e^{-i(\omega _{1}+\omega _{2})t}\\&\qquad +2E_{1}{E_{2}}^{*}e^{-i(\omega _{1}-\omega _{2})t}\\&\qquad +\left(|E_{1}|^{2}+|E_{2}|^{2}\right)e^{0}+{\text{c.c.}}{\Big ]},\end{aligned}}}$

which has frequency components at 2ω₁, 2ω₂, ω₁ + ω₂, ω₁ − ω₂, and 0. These three-wave mixing processes correspond to the nonlinear effects known as second-harmonic generation, 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 uncertainty in the electric field initiates the process.

Phase matching

Most transparent materials, like the BK7 glass shown here, have normal dispersion: the index of refraction decreases monotonically as a function of wavelength (or increases as a function of frequency). This makes phase matching impossible in most frequency-mixing processes. For example, in SHG, there is no simultaneous solution to ${\displaystyle \omega '=2\omega }$ and ${\displaystyle \mathbf {k} '=2\mathbf {k} }$ in these materials. Birefringent materials avoid this problem by having two indices of refraction at once.

The above ignores the position dependence of the electrical fields. In a typical situation, the electrical fields are traveling waves described by

${\displaystyle E_{j}(\mathbf {x} ,t)=E_{j,0}e^{i(\mathbf {k} _{j}\cdot \mathbf {x} -\omega _{j}t)}+{\text{c.c.}}}$

at position ${\displaystyle \mathbf {x} }$, with the wave vector ${\displaystyle \|\mathbf {k} _{j}\|=\mathbf {n} (\omega _{j})\omega _{j}/c}$, where ${\displaystyle c}$ is the velocity of light in vacuum, and ${\displaystyle \mathbf {n} (\omega _{j})}$ is the index of refraction of the medium at angular frequency ${\displaystyle \omega _{j}}$. Thus, the second-order polarization at angular frequency ${\displaystyle \omega _{3}=\omega _{1}+\omega _{2}}$ is

${\displaystyle P^{(2)}(\mathbf {x} ,t)\propto E_{1}^{n_{1}}E_{2}^{n_{2}}e^{i[(\mathbf {k} _{1}+\mathbf {k} _{2})\cdot \mathbf {x} -\omega _{3}t]}+{\text{c.c.}}}$

At each position ${\displaystyle \mathbf {x} }$ within the nonlinear medium, the oscillating second-order polarization radiates at angular frequency ${\displaystyle \omega _{3}}$ and a corresponding wave vector ${\displaystyle \|\mathbf {k} _{3}\|=\mathbf {n} (\omega _{3})\omega _{3}/c}$. Constructive interference, and therefore a high-intensity ${\displaystyle \omega _{3}}$ field, will occur only if

${\displaystyle {\vec {\mathbf {k} }}_{3}={\vec {\mathbf {k} }}_{1}+{\vec {\mathbf {k} }}_{2}.}$

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 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.

Phase-matching types

(${\displaystyle \lambda _{p}\leq \lambda _{s}\leq \lambda _{i}}$)

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 (or type 0, or "zero")
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)

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 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 can be integrated monolithically.

Higher-order frequency mixing

The above holds for ${\displaystyle \chi ^{(2)}}$ processes. It can be extended for processes where ${\displaystyle \chi ^{(3)}}$ is nonzero, something that is generally true in any medium without any symmetry restrictions; in particular resonantly enhanced sum or difference frequency mixing in gasses is frequently used for extreme or "vacuum" Ultra Violet light generation. In common scenarios, such as mixing in dilute gases, the non-linearity is weak and so the light beams are focused which, unlike the plane wave approximation used above, introduces a pi phase shift on each light beam, complicating the phase matching requirements.

Conveniently, difference frequency mixing with ${\displaystyle \chi ^{(3)}}$ 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 ${\displaystyle \chi ^{(3)}}$ all four frequencies are mixing simultaneously, as opposed to sequential mixing via two ${\displaystyle \chi ^{(2)}}$ processes.

The Kerr effect can be described as a ${\displaystyle \chi ^{(3)}}$ 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, 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 a few KeV. 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), KDP (potassium dihydrogen phosphate), 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

Vortex photon (blue) with linear momentum ${\displaystyle \mathbf {P} =\hbar \mathbf {k} }$ and angular momentum ${\displaystyle L=\pm \hbar \ell }$ are reflected from perfect phase-conjugating mirror. Normal to mirror is ${\displaystyle {\vec {n}}}$ , propagation axis is ${\displaystyle {\vec {z}}}$. Reflected photon (magenta) has opposite linear momentum ${\displaystyle \mathbf {P} =-\hbar \mathbf {k} }$ and angular momentum ${\displaystyle L=\mp \hbar \ell }$. Because of conservation laws PC mirror experiences recoil: the vortex phonon (orange) with doubled linear momentum ${\displaystyle \mathbf {P} =2\hbar \mathbf {k} }$ and angular momentum ${\displaystyle L=\pm 2\hbar \ell }$ is excited within mirror.

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 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 is implemented via stimulated Brillouin scattering, four-wave mixing, three-wave mixing, static linear holograms and some other tools.

Comparison of a phase-conjugate mirror with a conventional mirror. With the phase-conjugate mirror the image is not deformed when passing through an aberrating element twice.

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:

${\displaystyle \Xi _{j}(\mathbf {x} ,t)={\frac {1}{2}}E_{j}(\mathbf {x} )e^{i(\omega _{j}t-\mathbf {k} \cdot \mathbf {x} )}+{\text{c.c.}},}$

where E_j 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:

${\displaystyle P_{\text{NL}}=\varepsilon _{0}\chi ^{(3)}(\Xi _{1}+\Xi _{2}+\Xi _{3})^{3},}$

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:

${\displaystyle P_{\omega }={\frac {1}{2}}\chi ^{(3)}\varepsilon _{0}E_{1}E_{2}E_{3}^{*}e^{i(\omega t-\mathbf {k} \cdot \mathbf {x} )}+{\text{c.c.}}}$

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

${\displaystyle E_{4}={\frac {i\omega l}{2nc}}\chi ^{(3)}E_{1}E_{2}E_{3}^{*},}$

where c is the speed of light. If the pump beams E₁ and E₂ are plane (counterpropagating) waves, then

${\displaystyle E_{4}(\mathbf {x} )\propto E_{3}^{*}(\mathbf {x} ),}$

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 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:

${\displaystyle \mathbf {S} _{\text{out}}(\mathbf {r} ,t)=-\mathbf {S} _{\text{in}}(\mathbf {r} ,t),}$

("in" means incident field, "out" means reflected field) where

${\displaystyle \mathbf {S} (\mathbf {r} ,t)=\epsilon _{0}c^{2}\mathbf {E} (\mathbf {r} ,t)\times \mathbf {B} (\mathbf {r} ,t),}$

which is a linear momentum density of electromagnetic field. In the same way a phase-conjugated wave has an opposite angular momentum density vector ${\displaystyle \mathbf {L} (\mathbf {r} ,t)=\mathbf {r} \times \mathbf {S} (\mathbf {r} ,t)}$ with respect to incident field:

${\displaystyle \mathbf {L} _{\text{out}}(\mathbf {r} ,t)=-\mathbf {L} _{\text{in}}(\mathbf {r} ,t).}$

The above identities are valid locally, i.e. in each space point ${\displaystyle \mathbf {r} }$ in a given moment ${\displaystyle t}$ for an ideal phase-conjugating mirror.

Quantum picture

In quantum electrodynamics the photon with energy ${\displaystyle \hbar \omega }$ also possesses linear momentum ${\displaystyle \mathbf {P} =\hbar \mathbf {k} }$ and angular momentum, whose projection on propagation axis is ${\displaystyle L_{z}=\pm \hbar \ell }$, where ${\displaystyle \ell }$ is topological charge of photon, or winding number, ${\displaystyle \mathbf {z} }$ is propagation axis. The angular momentum projection on propagation axis has discrete values ${\displaystyle \pm \hbar \ell }$.

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):

${\displaystyle \mathbf {P} _{\text{out}}=-\hbar \mathbf {k} =-\mathbf {P} _{\text{in}}=\hbar \mathbf {k} ,}$

${\displaystyle {L_{z}}_{\text{out}}=-\hbar \ell =-{L_{z}}_{\text{in}}=\hbar \ell .}$

Nonlinear optical pattern formation

Optical fields transmitted through nonlinear Kerr media can also display pattern formation 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.

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 SHG materials

Dark-red gallium selenide in its bulk form

Ordered by pump wavelength:

• 800 nm: BBO
• 806 nm: lithium iodate (LiIO₃)
• 860 nm: potassium niobate (KNbO₃)
• 980 nm: KNbO₃
• 1064 nm: monopotassium phosphate (KH₂PO₄, KDP), lithium triborate (LBO) and β-barium borate (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)