TAKS II: Undulation Theory
Anjar Sari (1017016300991)
Physic Education V

UNDULATION THEORY


To estimate wavelength of undulator radiation apply special relativity twice
1. Moving system: undulator period appears shortened by length contraction emission of dipole radiation with wavelength

2. Lorentz transformation into lab. System

Comparison of Quantum Laser and Free-Electron Laser
→ quantized energy levels
→ energy pump to create population inversion
→ stimulated emission of radiation
→ optical resonator
→ electron energy not quantized
→ energy source: kinetic energy of beam
→ stimulated emission of radiation
→ optical resonator or SASE







Undulator Radiation
 Electron motion in undulator
 Lorentz transformation into moving coordinate system
 Power of undulator radiation
 Line shape of undulator radiation



Undulator field approx. Harmonic
Plane




Total relativistic energy of the electron

Firs oder solution:


Electron travels on a sine-like trajectory

Undulator parameter

Transverse velocity and maximum divergence angle of trajectory


Synchrotron radiation is emitted inside a cone with opening angle
Undulator: particle trajectory stays within this cone

Note: Radiation field amplitudes from various sections of andulator trajectory overlap
and interfere =>radiation is nearly monochromatic (fundamental frequency plus higher
harmonics).
Motion in second order
On sine-trajectory:


Insert for the first-order solution, then the average z velocity is:


Note: z velocity oscillates about the average
with
Electron trajectory in second order


Lorentz transformation into moving coordinate system
Coordinate system (x*, y*, z*) moving with average z velocity of electron:






Mainly transverse harmonic oscillation with frequency
Small superimposed longitudinal oscillation generates odd higher harmonics in moving system: electron emits dipole radiation



Lorentz transformation of photon energy
Use and


Wavelength of undulator radiation


Undulator radiation of an electron with v = 0.9 c
Radiation power in moving system computed with Larmor formula




Radiation power in laboratory system is the same:


Line shape of undulator radiation
Electron passing an undulator with Nu periods produces wave train with Nu oscillations.


Spectral intensity :