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B Integral

## Summary

In nonlinear optics, B-Integral is a measure of the nonlinear phase shift of light. It calculates the exponential growth of the least stable spatial frequency in a laser beam, and is the numerical equivalent of the nonlinear phase shift along the laser system's optical axis.

In a multipass laser system as a cumulative measure of the nonlinear interaction,[1] this integral is given by:

${\displaystyle B={\frac {2\pi }{\lambda }}\int \!n_{2}I(z)\,dz\,}$

where ${\displaystyle I(z)}$ is the optical intensity along the beam axis, ${\displaystyle z}$ the position in beam direction, and ${\displaystyle n_{2}}$ the nonlinear index quantifying the Kerr nonlinearity. As ${\displaystyle n_{2}I(z)}$ is the nonlinear change in the refractive index, one easily recognizes the B integral to be the total on-axis nonlinear phase shift accumulated in a passage through the device. The B integral is frequently used in the context of ultrafast amplifiers, e.g. for optical components such as the Pockels cell of a regenerative amplifier.