Rayleigh length to electron bunch emittance

[f77hacker]

TL;DR 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

 

[AndyW]

Hello,

Thank you for your question. Matching the spot size of a pulsed laser to a pulsed electron beam can be done by considering the Raleigh length and the emittance of the electron beam.

The Raleigh length, as you mentioned, is given by Z_r = w_o^2 * pi / lambda, where w_o is the spot size radius and lambda is the laser frequency. This represents the distance at which the laser beam diverges by a factor of e^2 from its initial spot size. On the other hand, the emittance of an electron beam is a measure of the spread of the electron trajectories in phase space. It is given by the normalized emittance, which is the product of the beam's transverse size and its angular spread.

To match the spot size to the emittance, we can use the following equation:

w_o = s * emittance_normalized / (L * Z_r)

where s is a scaling factor that takes into account the geometry of the interaction between the laser and electron beams. This equation can be derived by equating the Raleigh length to the transverse size of the electron beam at the interaction point.

To ensure that the emittance is within the limits given by emittance <= L * lambda / (4 * pi), we can rearrange the equation to get:

emittance_normalized <= (4 * pi * w_o * L * Z_r) / lambda

This shows that in order to match the spot size to the emittance, the normalized emittance must be smaller than the right-hand side of the above equation.

I hope this helps to clarify the details of matching the Raleigh length to the emittance in the context of Inverse Compton Scattering. Let me know if you have any further questions.