Nonlinear optics (NLO) is the branch of
opticsOptics is the branch of physics which studies 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 behaviour of
lightLight is electromagnetic radiation, particularly radiation of a wavelength that is visible to the human eye ....
in
nonlinear media, that is, media in which the dielectric polarization
P responds nonlinearly to the
electric fieldIn physics, the space surrounding an electric charge or in the presence of a time-varying magnetic field has a property called an electric field. This electric field exerts a force on other electrically charged objects...
E of the light. This nonlinearity is typically only observed at very high light intensities such as those provided by pulsed
laserA laser is a device that emits light through a process called stimulated emission. Laser light is usually spatially coherent, which means that the light either is emitted in a narrow, low-divergence beam, or can be converted into one with the help of optical components such as lenses...
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,So that if input A produces response X and input B produces response Y then input produces response .Mathematically, for all linear systems F = y, where x is some...
no longer holds.
Frequency mixing processes
- 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)
- Third harmonic generation (THG), generation of light with a tripled frequency (one-third the wavelength) (usually done in two steps: SHG followed by SFG of original and frequency-doubled waves)
- High harmonic generation
- Perturbative Harmonic Generation :Perturbative Harmonic Generation is a process where by laser lightof frequency ω and photon energy ħω can be usedto generate new frequencies of light...
(HHG), generation of light with frequencies much greater than the original (typically 100 to 1000 times greater)
- Sum frequency generation
Sum-frequency generation is an example of a second order non-linear optical process.This phenomenon is based on the annihilation of two input photons at frequencies and while, simultaneously, one photon at frequency is generated....
(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
- Spontaneous parametric down conversion
Spontaneous parametric down-conversion is an important process in quantum optics. A nonlinear crystal splits incoming photons into pairs of photons of lower energy whose combined energy and momentum are equal to the energy and momentum of the original photon...
(SPDC), the amplification of the vacuum fluctuations in the low gain regime
- Optical rectification
Electro-optic rectification is a non-linear optical process which consists in the generation of a DC polarization in a non-linear medium at the passage of an intense optical beam...
(OR), generation of quasi-static electric fields.
Other nonlinear processes
- 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
- 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...
- 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)
- 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), a effect
- Optical soliton
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:...
s
- 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)
- Four-wave mixing
Four-wave mixing is an intermodulation distortion in optical systems, similar to the third-order intercept point in electrical systems.Four-wave mixing can be compared to the intermodulation distortion in standard electrical systems...
(FWM), can also arise from other nonlinearities
- 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 effect in which a wave with polarization vector perpendicular to the input one is generated
- 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...
- 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...
- Optical phase conjugation
- Brillouin scattering
Brillouin scattering, named for Léon Brillouin, occurs when light in a medium interacts with time dependent density variations and changes its energy and path. The density variations may be due to acoustic modes, such as phonons, magnetic modes, such as magnons, or temperature gradients...
, interaction of photons with acoustic phonons
- Optical phase conjugation
- Two-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 photons, transferring the energyIn physics, energy is a scalar physical quantity that describes the amount of work that can be performed by a force, an attribute of objects and systems that is subject to a conservation law...
to a single electron
- Multiple photoionisation
Photoionization is the physical process in which an incident photon ejects one or more electrons from an atom, ion or molecule.The ejected electrons, known as photoelectrons, carry information about their pre-ionised states. For example, a single electron can have a kinetic energy equal to the...
, near-simultaneous removal of many bound electrons by one photon
- 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...
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 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 direction of the modulated beam...
s;
- 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.
- Raman scattering
Raman scattering or the Raman effect is the inelastic scattering of a photon. Discovered by Sir Chandrasekhara Venkata Raman in liquids and by Grigory Landsberg and Leonid Mandelstam in crystals...
, interaction of photons with optical phononIn physics, a phonon is a quantized mode of vibration occurring in a rigid crystal lattice, such as the atomic lattice of a solid. The study of phonons is an important part of solid state physics, because phonons play a major role in many of the physical properties of solids, including a material's...
s;
Frequency-mixing processes
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 special crystal in a laser beam under a well-chosen angle. 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 types of material, such as calcite crystals or boron nitride, depending on the polarization of the light...
(necessary to obtain phase matching, see below), having a specific crystal symmetry and of course being transparent for and 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.
Theory
A number of nonlinear optical phenomena can be described as frequency-mixing processes. If the induced dipole moments of the material respond instantaneously to an applied electric field, the dielectric polarization (dipole moment per unit volume) at time in a medium can be written as a
power seriesIn mathematics, a power series is an infinite series of the formwhere a
n represents the coefficient of the nth term, c is a constant, and x varies around c...
in the electrical field:
Here, the coefficients are the -th order
susceptibilitiesThe electric susceptibility χe of a dielectric material is a measure of how easily it polarizes in response to an electric field. This, in turn, determines the electric permittivity of the material and thus influences many other phenomena in that medium, from the capacitance of...
of the medium. For any three-wave mixing process, the second-order term is crucial; it is only nonzero in media that have no inversion symmetry. If we write
where c.c. denotes the
complex conjugateAs found in mathematics, a complex conjugate is most simply defined as one of a pair of complex numbers, each having the same real parts but with imaginary parts that differ in sign; e.g. 3 + 4i and 3 - 4i are complex conjugates...
(
E1 and
E2 being the incident beams of interest), the second-order term in the above expansion will read
where the summation is over
The six combinations correspond, respectively, to the second harmonic of , the second harmonic of , the optically rectified signals of and , the difference frequency, and the sum frequency. A medium that is thus pumped by the fields and will radiate a field with an angular frequency .
Note: in this description, is a scalar. In reality, is a
tensorTensors are geometrical entities introduced into mathematics and physics to extend the notion of scalars, vectors, and matrices. Many physical quantities are naturally regarded, not as vectors themselves, but as correspondences between one set of vectors and another...
whose components depend on the combination of frequencies.
Parametric generation and amplification is a variation of difference frequency generation, where the lower-frequency 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 is a set of principles describing the physical reality at the atomic level of matter and the subatomic . These descriptions include the simultaneous wave-like and particle-like behavior of both matter and radiation...
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
at position , with the wave vector , where is the velocity of light and the index of refraction of the medium at angular frequency . Thus, the second-order polarization angular frequency is
At each position , the oscillating second-order polarization radiates at angular frequency and a corresponding wave vector . Constructive interference, and therefore a high intensity field, will occur only if
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 indexThe refractive index of a medium is a measure of how much the speed of light is reduced inside the medium. For example, typical soda-lime glass has a refractive index close to 1.5, which means that in glass, light travels at 1 / 1.5 = 2/3 the speed of light in a vacuum...
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.
Phase-matching types
| 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) |
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 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 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 are...
. 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 processes. It can be extended for processes where is nonzero, something that is generally true in any medium without any symmetry restrictions. Third-harmonic generation is a 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 as well.
At high intensities the
Taylor seriesIn mathematics, the Taylor series is a representation of a function as an infinite sum of terms calculated from the values of its derivatives at a single point. It may be regarded as the limit of the Taylor polynomials. Taylor series are named after the English mathematician Brook Taylor...
, 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. By definition, it is equal to the amount of kinetic energy gained by a single unbound electron when it accelerates through an electrostatic potential difference of one volt...
. This is called
high-order harmonic generation- Perturbative Harmonic Generation :Perturbative Harmonic Generation is a process where by laser lightof frequency ω and photon energy ħω can be usedto generate new frequencies of light...
. 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.
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
retroreflectionA 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:
where
Ej 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:
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 =
k1 +
k2 -
k3, this gives a polarization field:
This is the generating field for the phase conjugate beam, Ξ
4. 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 for a medium with refractive index
n and a beam interaction length
l, the electric field amplitude of the conjugate beam is approximated by
(where
c is the speed of light). If the pump beams
E1 and
E2 are plane (counterpropagating) waves, then:
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
- 806 nm light : lithium iodate (LiIO3)
- 860 nm light : potassium niobate (KNbO3)
- 980 nm light : KNbO3
- 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...
(KH2PO4, 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).
- 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)
- 1319 nm light : KNbO3, BBO, KDP, potassium titanyl phosphate
Potassium 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:...
(LiNbO3), LiIO3, 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 quadratic prisms. Monoammonium phosphate is often used in the blending of...
(ADP)
See also
- Born-Infeld action
- 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....
- :Category:Nonlinear optical materials
External links