Self phase modulation

above: temporal course of the envelope of a pulse, below: frequency shift after propagation due to self-phase modulation. The front part of the pulse receives lower frequencies, the rear higher. The frequency shift is approximately linear in the center.

Self-phase modulation (SPM) (engl .: self phase modulation) is a nonlinear optical effect , which upon interaction of electromagnetic radiation with matter occurs. The radiation is spectrally symmetrically expanded by new frequency components.

Explanation

The cause of the self-phase modulation is the temporal Kerr effect . This states that the refractive index in media for high intensities is intensity-dependent: ${\displaystyle n}$

${\displaystyle n(I)=n_{0}+n_{2}\cdot I(t)}$

In optical media the non-linear refractive index coefficient is very low, so that the self-phase modulation only becomes relevant from light intensities of approx . ${\displaystyle n_{2}}$ ${\displaystyle I>10^{14}{\frac {\mathrm {W} }{\mathrm {m} ^{2}}}}$

The non-linear refractive index change mainly causes an intensity-dependent phase velocity . After traveling a distance in such a non-linear medium, this results in a non-linear phase shift of: ${\displaystyle l}$

${\displaystyle \Phi _{\mathrm {nl} }(t)=-\delta n\cdot l\cdot {\frac {\omega _{0}}{c}}}$

where represents the speed of light in vacuum and the carrier angular frequency. ${\displaystyle c}$ ${\displaystyle \omega _{0}}$

${\displaystyle \delta n=n_{2}\cdot I(t)}$

is the intensity-dependent change in the refractive index. The instantaneous frequency then becomes

${\displaystyle \omega (t)=\omega _{0}+\delta \omega (t)}$

with the time-dependent shift of the instantaneous frequency

${\displaystyle \delta \omega (t)={\frac {d\Phi _{\mathrm {nl} }(t)}{dt}}}$.

example

If one uses the frequently used model of a hyperbolic secant impulse as an example

${\displaystyle I(t)=I_{0}\cdot \operatorname {sech} ^{2}\!\left({\frac {t}{\tau _{0}}}\right)}$

becomes the nonlinear phase of the pulse

${\displaystyle \Phi _{\mathrm {nl} }(t)=-n_{2}\cdot l\cdot {\frac {\omega _{0}}{c}}\cdot I_{0}\cdot \operatorname {sech} ^{2}\!\left({\frac {t}{\tau _{0}}}\right)}$

thus the instantaneous frequency is shifted by

${\displaystyle \delta \omega (t)=-2\cdot n_{2}\cdot l\cdot {\frac {\omega _{0}}{c\cdot \tau _{0}}}\cdot I_{0}\cdot \operatorname {sech} ^{2}\!\left({\frac {t}{\tau _{0}}}\right)\cdot \operatorname {tanh} \!\left({\frac {t}{\tau _{0}}}\right)}$.

If you look at the last term, you can immediately see that new frequencies are generated symmetrically to the carrier frequency. It can also be seen that in the case of a positive (which is the case with common optical media, for example), new long-wave frequencies are generated in the front of the pulse and short-wave frequencies in its flank. ${\displaystyle n_{2}}$

In the time domain, however, the self-phase modulation does not change the amount of the pulse envelope , and especially the duration of the pulse. ${\displaystyle I(t)}$

Self-division

For pulses whose duration is of the same order of magnitude as the period of the carrier frequency, the above intensity-dependent refractive index also leads to an intensity-dependent group speed . This effect is called self-steepening (ger .: self steepening). The newly generated spectral components are then no longer arranged symmetrically around the original carrier frequency. In addition, the amount of the temporal impulse envelope no longer remains unchanged, but the impulse flattens out on its front and steepens on its back. This effect is named after this phenomenon.

Applications

• Optical solitons : If a light pulse propagates through a material (e.g. a glass fiber) it will i. A. broaden in time. The reason is the group velocity dispersion, in which different frequency components of the pulse have different velocities. If this dispersion is completely compensated for by the self-phase modulation, a temporal soliton is created.
• Pulse compression: The self-phase modulation gives a laser pulse additional frequency components and thus gains in bandwidth. In order to use the newly generated frequency components to shorten the pulse duration, the pulse must be freed of its chirp by means of dispersion compensation .

Further information

• Robert W. Boyd: Nonlinear Optics . 3. Edition. Academic Press, New York 2008, ISBN 978-0-12-369470-6 .