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Wavelength of Undulator Radiation

The electron moves with a speed `v < c`

on a wavelike orbit through the undulator. The velocity component `v`

along the _{z}`z`

axis is smaller than `v`

:

If one computes the relativistic Doppler shift with this reduced velocity one finds the important formula for the wavelength of undulator radiation in the forward direction:

The quantity *K* is called the undulator parameter

It depends on the peak magnetic field B_{0} in the undulator and on the undulator period λ_{u}.

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Low-Gain and High-Gain FEL

In the low-gain FEL the light wave co-propagating with the relativistic electron beam is described by a plane electro-magnetic wave

The time derivative of the electron energy is

One can show that the condition for sustained transfer from electron to light wave leads to exactly the same wavelength as in undulator radiation:

To achieve light amplification in a low-gain FEL, the electron energy must exceed the resonance energy

where the resonant Lorentz factor is defined by

This means that electrons with this Lorentz factor would emit undulator radiation of exactly the incident wavelength λ.

The power gain length depends on peak current, electron beam emittance ε and beta function β in the form

If the FEL is seeded by an incident light wave of field amplitude E_{0} and power P_{0} the FEL power grows as

This is theoretically well understood and due to the fact that the field of the light wave obeys a third-order differential equation which has three independent eigenfunctions: the first one is exponentially growing, the second one is exponentially decaying, and the third one is oscillating along the undulator axis. At the entrance to the undulator the field amplitude of the light wave can be expressed as a linear superposition of the three eigenfunctions with equal coefficients. After two gain lengths the first eigensolution starts to dominate and determines the exponential rise.

###
BCS Surface Resistance

According to the Bardeen-Cooper-Schrieffer (BCS) theory of superconductivity the microwave surface resistance depends exponentially on temperature

Here Δ = 1.76 k_{B}T_{C} is the energy gap, k_{B} the Boltzmann constant, T_{C} the critical temperature, f_{0} the microwave frequency, and A a coefficient that depends on the London penetration depth and other material properties. The exponential temperature dependence is the reason why the high-field cavities are cooled with superfluid helium at 2K (R_{BCS} ≈ 10nΩ = 10^{-8} Ω) instead of using pressurized normal liquid helium at 4.4 K (R_{BCS} ≈ 1000 nΩ). The BCS surface resistance scales quadratically with the radio frequency f_{0} , hence it is advantageous to build superconducting cavities with low resonance frequency. The value of 1.3 GHz is a good compromise between low surface resistance and manageable size of the cavities.