Editing Nonlinear optics (section)

==Parametric processes==
Nonlinear effects fall into two qualitatively different categories, [[Parametric process (optics)|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.<ref>{{Cite book
|last = Paschotta
|first = Rüdiger
|title = Encyclopedia of Laser Physics and Technology
|chapter = Parametric Nonlinearities
|chapter-url = http://www.rp-photonics.com/parametric_nonlinearities.html
|title-link = Encyclopedia of Laser Physics and Technology
|publisher = Wiley
|date = 2008
|isbn = 978-3-527-40828-3
|access-date = 2011-08-16
|archive-date = 2011-08-22
|archive-url = https://web.archive.org/web/20110822074013/http://www.rp-photonics.com/parametric_nonlinearities.html
|url-status = live
}}</ref>
<ref name="Boyd NLO">{{harvnb|Boyd|2008|pp=13–15 [{{GBurl|uoRUi1Yb7ooC|p=13}} 1.2.10 Parametric versus Nonparametric Processes]}}</ref>

===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 [[Perturbation theory|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''):
:<math>\mathbf{P}(t) = \varepsilon_0 \left( \chi^{(1)} \mathbf{E}(t) + \chi^{(2)} \mathbf{E}^2(t) + \chi^{(3)} \mathbf{E}^3(t) + \ldots \right),</math>
where the coefficients χ<sup>(''n'')</sup> are the ''n''-th-order [[Electric susceptibility|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<sup>(''n'')</sup> is an (''n''&nbsp;+&nbsp;1)-th-rank [[tensor]] representing both the [[Polarization (waves)|polarization]]-dependent nature of the parametric interaction and the [[Crystal symmetry|symmetries]] (or lack) of the nonlinear material.

====Wave equation in a nonlinear material====
Central to the study of electromagnetic waves is the [[Electromagnetic wave equation|wave equation]].  Starting with [[Maxwell's equations]] in an isotropic space, containing no free charge, it can be shown that
:<math>
\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},
</math>
where '''P'''<sup>NL</sup> 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
:<math>\nabla \times \left( \nabla \times \mathbf{V} \right) = \nabla \left( \nabla \cdot \mathbf{V} \right) - \nabla^2 \mathbf{V}</math>
and [[Gauss's law]] (assuming no free charges, <math>\rho_\text{free} = 0</math>),
:<math>\nabla\cdot\mathbf{D} = 0,</math>
to obtain the more familiar [[Electromagnetic wave equation|wave equation]]
:<math>
\nabla^2 \mathbf{E} - \frac{n^2}{c^2}\frac{\partial^2}{\partial t^2}\mathbf{E} = \mathbf{0}.
</math>
For a nonlinear medium, [[Gauss's law]] does not imply that the identity
:<math>\nabla\cdot\mathbf{E} = 0</math>
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:
:<math>
\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}.
</math>

====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:
:<math>\nabla^2 \mathbf{E} - \frac{n^2}{c^2}\frac{\partial^2}{\partial t^2}\mathbf{E} = \mathbf{0},</math>
and the inhomogeneous term
:<math>\frac{1}{\varepsilon_0 c^2}\frac{\partial^2}{\partial t^2}\mathbf{P}^\text{NL}</math>
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''&nbsp;+&nbsp;1)-wave mixing. As an example, if we consider only a second-order nonlinearity (three-wave mixing), then the polarization '''P''' takes the form
:<math>\mathbf{P}^\text{NL} = \varepsilon_0 \chi^{(2)} \mathbf{E}^2(t).</math>
If we assume that ''E''(''t'') is made up of two components at frequencies ''ω''<sub>1</sub> and  ''ω''<sub>2</sub>, we can write ''E''(''t'') as
:<math>\mathbf{E}(t) = E_1\cos(\omega_1t) + E_2\cos(\omega_2t),</math>
and using [[Euler's formula]] to convert to exponentials,
:<math>\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.},</math>
where "c.c." stands for [[complex conjugate]].  Plugging this into the expression for '''P''' gives
:<math>\begin{align}
     \mathbf{P}^\text{NL}
  &= \varepsilon_0 \chi^{(2)} \mathbf{E}^2(t) \\[3pt]
  &= \frac{\varepsilon_0}{4} \chi^{(2)} \left[{E_1}^2 e^{-i2\omega_1 t} + {E_2}^2 e^{-i2\omega_2 t} + 2E_1 E_2 e^{-i(\omega_1 + \omega_2)t} + 2E_1 {E_2}^* e^{-i(\omega_1 - \omega_2)t} + \left(\left|E_1\right|^2 + \left|E_2\right|^2\right)e^{0} + \text{c.c.}\right],
\end{align}</math>
which has frequency components at 2''ω''<sub>1</sub>, 2''ω''<sub>2</sub>, ''ω''<sub>1</sub>&nbsp;+&nbsp;''ω''<sub>2</sub>, ''ω''<sub>1</sub>&nbsp;−&nbsp;''ω''<sub>2</sub>, 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.

<!-- The following note is taken (with permission) from Han-Kwang Nienhuys's PhD thesis "Femtosecond mid-infrared spectroscopy of water" (2002). -->
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 mechanics|quantum-mechanical]] uncertainty in the electric field initiates the process.

===Phase matching===
[[File:Sellmeier-equation.png|thumb|right|Most transparent materials, like the [[BK7 glass]] shown here, have [[normal dispersion]]: the [[index of refraction]] decreases [[monotonic]]ally 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 <math>\omega'=2\omega</math> and  <math>\mathbf{k}' = 2\mathbf{k}</math> in these materials. [[Birefringent]] materials avoid this problem by having two indices of refraction at once.<ref>{{harvnb|Boyd|2008|loc=[{{GBurl|uoRUi1Yb7ooC|p=79}} 2.3. Phase Matching]}}</ref>]]
The above ignores the position dependence of the electrical fields. In a typical situation, the electrical fields are traveling waves described by

:<math>E_j(\mathbf{x},t) =E_{j,0} e^{i(\mathbf{k}_j \cdot \mathbf{x} - \omega_j t)} + \text{c.c.}</math>

at position <math>\mathbf{x}</math>, with the [[wave vector]] <math>\|\mathbf{k}_j\| = \mathbf{n}(\omega_j)\omega_j/c</math>, where <math>c</math> is the velocity of light in vacuum, and <math>\mathbf{n}(\omega_j)</math> is the index of refraction of the medium at angular frequency <math>\omega_j</math>. Thus, the second-order polarization at angular frequency <math>\omega_3=\omega_1+\omega_2</math> is

:<math>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.}</math>

At each position <math>\mathbf{x}</math> within the nonlinear medium, the oscillating second-order polarization radiates at angular frequency <math>\omega_3</math> and a corresponding wave vector <math>\|\mathbf{k}_3\| = \mathbf{n}(\omega_3)\omega_3/c</math>. Constructive interference, and therefore a high-intensity <math>\omega_3</math> field, will occur only if

:<math>\vec{\mathbf{k}}_3 = \vec{\mathbf{k}}_1 + \vec{\mathbf{k}}_2.</math>

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.
{| class="wikitable" style="text-align:center"
|+Phase-matching types
|+(<math>\lambda_p \leq \lambda_s \leq \lambda_i</math>)
|-
! colspan="3" | Polarizations
! rowspan="2" | 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,<ref>{{cite journal|last=Abolghasem|first=Payam|author2=Junbo Han |author3=Bhavin J. Bijlani |author4=Amr S. Helmy |title=Type-0 second order nonlinear interaction in monolithic waveguides of isotropic semiconductors|journal=Optics Express|year=2010|volume=18|issue=12|pages=12681–12689|doi=10.1364/OE.18.012681|pmid=20588396|bibcode=2010OExpr..1812681A|doi-access=free}}</ref> 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 [[Periodic poling|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''&nbsp;=&nbsp;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 [[Acousto-optic programmable dispersive filter|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.