Frequency pulling

Frequency pulling is a phenomenon in laser physics . The effect comes about through the interaction of the light field with the laser medium and has the effect that the operating frequency of a laser is always between its cavity frequency and the transition frequency of the amplifying medium.

The frequency pulling equation

The equation that describes frequency pulling is

${\displaystyle \omega _{n}-\omega ={\frac {gc}{2\beta }}(\omega -\omega _{0})}$.

with the cavity frequency , the laser gain , the dipole frequency and the speed of light . ${\displaystyle \omega _{n}}$ ${\displaystyle g}$ ${\displaystyle \omega _{0}}$ ${\displaystyle c}$

From this equation it can be seen that the operating frequency of the laser must lie between the cavity frequency and the natural frequency of the oscillator. The laser frequency is drawn to the dipole frequency . In the classic Lorentz oscillator model, this dipole frequency is the natural frequency of the harmonic oscillator and corresponds to the transition frequency between the two laser levels in the semiclassical (quantum mechanical) model. ${\displaystyle \omega }$ ${\displaystyle \omega _{0}}$

Physical basics

Standing waves

In stationary laser operation, only certain longitudinal oscillation modes can develop within the resonator . Are formed standing waves of which one in complex notation for propagation in the direction of the axis and in the direction of polarization can be represented as follows -axis ${\displaystyle z}$ ${\displaystyle x}$

${\displaystyle {\vec {E}}(z,t)={\hat {x}}E_{n}\sin(k_{n}z)e^{-i\omega t}}$.

Here stands for the amplitude and for the wave number of the nth mode: ${\displaystyle E_{n}}$ ${\displaystyle k_{n}}$

${\displaystyle k_{n}=n\pi /L,\qquad n=1,2,3,\dots }$

Here stands for the length of the laser resonator. This expression for the wave number ensures that the electric field at the locations of the resonator mirrors is zero, which is a prerequisite for standing waves. The cavity frequency is now defined as ${\displaystyle L}$ ${\displaystyle \omega _{n}}$

${\displaystyle \omega _{n}=ck_{n}}$,

Derivation

There are two different approaches to expressing the effect of frequency pulling. On the one hand via the classic Lorentz oscillator model , which idealizes the binding of an electron to its atomic nucleus as a damped harmonic oscillator (two masses on a spring). The second approach is based on the semiclassical laser theory, which uses the Schrödinger equation to describe the electron behavior, but completely ignores the quantum character of light. Both approaches lead to the same result.

Classic approach

In the classical approach, the atomic nucleus-electron system is assumed to be a damped harmonic oscillator, which is driven by the electromagnetic wave described above. The differential equation of a damped, force- driven oscillator is ${\displaystyle {\vec {F}}}$

${\displaystyle m{\frac {d^{2}{\vec {x}}}{dt^{2}}}+2m\beta {\frac {d{\vec {x}}}{dt}}+m\omega _{0}^{2}{\vec {x}}={\vec {F}}}$

with the damping factor and the natural frequency . stands for the displacement of the electron from its equilibrium state. The natural frequency , which cannot be calculated from the classical model and is therefore an empirical value from a classical point of view, corresponds to the transition frequency of the electron between two neighboring energy levels and can be calculated quantum mechanically . The force that an electromagnetic field exerts on a particle's charge ${\displaystyle \beta }$ ${\displaystyle \omega _{0}}$ ${\displaystyle {\vec {x}}}$ ${\displaystyle \omega _{0}}$ ${\displaystyle {\vec {E}}}$ ${\displaystyle e}$

${\displaystyle {\vec {F}}=q{\vec {E}}}$

With the above definition of the electromagnetic field for a standing wave it follows

${\displaystyle {\frac {d^{2}{\vec {x}}}{dt^{2}}}+2\beta {\frac {d{\vec {x}}}{dt}}+\omega _{0}^{2}{\vec {x}}={\frac {q}{m}}{\hat {x}}E_{n}\sin(k_{n}z)e^{-i\omega t}}$.

This differential equation has the solution

${\displaystyle {\vec {x}}={\vec {a}}\sin k_{n}ze^{-iwt}}$.

The electromagnetic wave equation results in the case of an electromagnetic wave with direction of propagation in the laser medium with polarization and ohmic current ${\displaystyle z}$ ${\displaystyle {\vec {P}}}$ ${\displaystyle {\vec {j}}=\sigma {\vec {E}}}$

${\displaystyle \left({\frac {\partial ^{2}}{{\partial z}^{2}}}-{\frac {\sigma }{\epsilon _{0}c^{2}}}-{\frac {1}{c^{2}}}{\frac {\partial ^{2}}{{\partial t}^{2}}}\right){\vec {E}}({\vec {r}},t)={\frac {1}{\epsilon _{0}c^{2}}}{\frac {\partial ^{2}}{{\partial t}^{2}}}{\vec {P}}}$.

After solving this equation and assuming that the frequency of the incident light is close to the natural frequency of the Lorentz oscillator, one comes to the equation that describes frequency pulling: ${\displaystyle \omega }$ ${\displaystyle \omega _{0}}$

${\displaystyle \omega _{n}-\omega ={\frac {gc}{2\beta }}(\omega -\omega _{0})}$.

with the cavity frequency , the laser gain , the speed of light and the dipole frequency . ${\displaystyle \omega _{n}}$ ${\displaystyle g}$ ${\displaystyle c}$ ${\displaystyle \omega _{0}}$

Individual evidence

1. ^ ^{A b} P. W. Milonni, JH Eberly: Lasers (= Wiley Series in Pure and Applied Optics . Volume 7) John Wiley & Sons, 1988, ISBN 0-471-62731-3 , p. 82.
2. ↑ PW Milonni, JH Eberly: Lasers (= Wiley Series in Pure and Applied Optics . Volume 7) John Wiley & Sons, 1988, ISBN 0-471-62731-3 , p. 80.