Phase matching

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The phase matching (engl. Phase matching ) is in the physics an important criterion for the phases of the light waves in many effects of non-linear optics , z. B. the frequency doubling and the sum frequency generation . In these processes, one or more light waves ( mostly laser due to the high light intensity required ) are sent through a propagation medium with non-linear susceptibility in order to generate one or more new light beams with a different frequency . In order for this to work, the waves involved must be "in phase", i.e. have a fixed relationship between the wave crests and troughs so that there is no destructive interference , which would lead to the cancellation of the waves and thus to the suppression of the effects.

For the wave vectors of the light waves, the relation

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

apply, where is the wave vector of the generated light and that of the original light. In principle, this equation does not have to contain an exact equality sign, since an approximate fulfillment ( ) also allows the generation of the non-linear effects, in practice, however, an attempt is always made to achieve equality, since then the intensity of the light generated is maximal. However, due to dispersion , all of these waves have a different index of refraction. This leads to the fact that the light waves in the medium inevitably run “out of phase”, that is to say that on average the waves are destructively superimposed. The condition is therefore also required for phase adjustment ${\displaystyle {\vec {k}}_{3}}$ ${\displaystyle {\vec {k}}_{1,2}}$ ${\displaystyle {\vec {k}}_{3}\approx {\vec {k}}_{1}+{\vec {k}}_{2}}$

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

necessary, whereby the refractive index is a function of the angular frequencies of the incoming light waves. In practice, this can only be achieved with birefringent materials. Beta-barium borate or potassium dihydrogen phosphate are mostly used, as these have both a suitable susceptibility and are birefringent. Need to phase adjustment to achieve the crystals according to its optical axis to be aligned so that the rays of light in the so-called phase matching angle (engl. Phase matching angle ) to one another. This angle must be determined geometrically for each crystal and each non-linear effect. ${\displaystyle n}$ ${\displaystyle \omega _{1,2}}$

Both of these conditions, the relationship of the wave vectors and the refractive indices are referred to as phase matching conditions (engl. Phase matching condition ), respectively.

A special case is the quasi-phase matching (engl. Quasi phase matching ) in the periodically poled crystals (z. B. lithium niobate ). Since the relation is not satisfied here, running in contrast to the phase matching by birefringence, the light waves over a certain distance ( "coherence length “) Out of phase. However, the periodic change in the crystal orientation prevents destructive interference from occurring, as the phase of the newly generated light waves reverses with the crystal orientation. This means that constructive summing is also possible over long propagation routes. The effect of the periodic polarity leads to a modification of the phase matching condition taking into account the polarity periodicity Λ ${\displaystyle {\vec {k}}_{3}={\vec {k}}_{1}+{\vec {k}}_{2}}$

${\displaystyle {\vec {k}}_{3}={\vec {k}}_{1}+{\vec {k}}_{2}-{\frac {\pi }{\Lambda }}m}$,

where the order indicates. For higher-order phase matching , destructive interference and thus lower conversion efficiency occur in the meantime, which is why this case is rarely used. In general, quasi-phase matching enables the use of crystals with high non-linear coefficients without restriction to birefringent properties. ${\displaystyle m=1,3,5,\dots }$ ${\displaystyle (m>1)}$

Further information

• Phase matching in the Encyclopedia of Laser Physics and Technology (Engl.)
• Robert W. Boyd: Nonlinear Optics . 3. Edition. Academic Press, New York 2008, ISBN 978-0-12-369470-6 .