# Nonlinear optics

In physics, non-linear optics ( NLO for short ) is a sub-area of ​​the optics of electromagnetic waves , in which the relationship between electric field and electric polarization in a medium is not linear, but rather of a higher degree .

## Basics

The starting point for modern optical descriptions are the Maxwell equations , which among other things form a mathematical formalism for describing electromagnetic waves in a vacuum and in matter. If an electromagnetic wave propagates in a medium, the electrons in it are stimulated to vibrate and in turn send out new waves. This is described by the electrical flux density :

${\ displaystyle {\ vec {D}} = \ varepsilon _ {0} {\ vec {E}} + {\ vec {P}}}$

Here is the electric field constant , the electric field of the wave and the electric polarization . For low intensities it is approximately the case that the polarization increases linearly with the electric field: ${\ displaystyle \ varepsilon _ {0}}$${\ displaystyle {\ vec {E}}}$${\ displaystyle {\ vec {P}}}$

${\ displaystyle {\ vec {P}} = \ varepsilon _ {0} \ chi {\ vec {E}}}$

where represents the electrical susceptibility . For very high intensities, however, this no longer applies and higher-order terms must be taken into account, since the intensity is proportional to the square of the electric field and the electric polarization cannot increase linearly at will: ${\ displaystyle \ chi}$

${\ displaystyle {\ vec {P}} = \ varepsilon _ {0} \ sum _ {n} \ chi ^ {(n)} {\ vec {E}} ^ {n} = \ varepsilon _ {0} \ left [\ chi ^ {(1)} {\ vec {E}} + \ chi ^ {(2)} {\ vec {E}} ^ {2} + \ chi ^ {(3)} {\ vec { E}} ^ {3} + \ dots \ right]}$

This is generally a higher order tensor . The wave equation resulting from the introduction of higher order terms is: ${\ displaystyle \ chi ^ {(n)}}$

${\ displaystyle \ left (\ Delta - {\ frac {n ^ {2}} {c ^ {2}}} {\ frac {\ partial ^ {2}} {\ partial t ^ {2}}} \ right ) {\ vec {E}} = \ mu _ {0} {\ frac {\ partial ^ {2}} {\ partial t ^ {2}}} {\ vec {P}} ^ {NL}}$

It is the Laplace operator , n the refractive index of the medium, c the speed of light and the sum of all non-linear terms of the polarization. ${\ displaystyle \ Delta}$${\ displaystyle {\ vec {P}} ^ {NL}}$

## Effects and Applications

Frequency doubling
Sum frequency generation
Difference frequency generation

Light as an electromagnetic wave is generally represented by a spatially and temporally oscillating function:

${\ displaystyle {\ vec {E}} ({\ vec {r}}, t) = {\ vec {E}} _ {0} \ cdot \ cos {({\ vec {k}} \ cdot {\ vec {r}} - \ omega t)} = {\ frac {1} {2}} {\ vec {E}} _ {0} \ cdot (e ^ {i ({\ vec {k}} \ cdot {\ vec {r}} - \ omega t)} + e ^ {- i ({\ vec {k}} \ cdot {\ vec {r}} - \ omega t)})}$

with the location , the time t , the wave vector , the angular frequency and the amplitude . Inserting this function or superpositions of different light waves with different frequencies in the nonlinear electrical polarization yields different terms in which new frequencies are contained. However, not all of the effects appearing in this calculation appear at the same time. Due to the frequency dependence of the refractive index , i.e. the dispersion , light with different frequencies has different phase velocities in a medium . This leads to destructive interference of the waves. For the desired effect to occur, the phase adjustment condition for the frequencies involved must be met: ${\ displaystyle {\ vec {r}}}$ ${\ displaystyle {\ vec {k}}}$ ${\ displaystyle \ omega}$${\ displaystyle {\ vec {E}} _ {0}}$

${\ displaystyle n (\ omega _ {1}) = n (\ omega _ {2})}$

This means that the refractive indices of the two light waves with the circular frequencies and must be the same. This condition can only be achieved by means of birefringent materials by choosing the optical polarizations of the light waves appropriately. ${\ displaystyle \ omega _ {1}}$${\ displaystyle \ omega _ {2}}$

Some nonlinear optical effects are:

Nonlinear optics or optically nonlinear materials are used in the construction of optical switches and components. So are z. B. in green laser pointers often diodes that emit infrared light, which is used to pump Nd: YVO 4 lasers (wavelength 1064 nm, infrared), which in turn is frequency-doubled with a nonlinear crystal (wavelength 532 nm, green). They can also be used as memory in (digital) optical data and image processing.

## Media with non-linear effects

Nonlinear optical effects only occur in media in which the terms with susceptibilities of the order greater than or equal to 2 do not vanish, i.e. are not equal to zero. Second-order effects are mostly crystals that also have a piezo effect . The most common crystals with second order nonlinearity are: