Rayleigh length to electron bunch emittance

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Summary:

How do you match a laser waist to an electron bunch emittance in context of Inverse Compton Scattering?
Hello,

Context is Inverse Compton Scattering with an pulsed electron beam interacting with a pulsed laser.

A laser has a Raleigh length of Z_r = w_o^2 * pi / lambda
w_o is the the radius of the spot size, lambda is the laser frequency.

I want to match this spot size to a pulsed electron bunch traveling with Lorentz factor of L. Usually this equation is given by

waist(s) = s * emittance_normalized / ( w_o * L)

answers I've read indicate that

emittance <= L * lambda / (4 *pi)

Can someone show the details on how to match the Raleigh to the emittance?

Thanks
 

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