Self Amplified Spontaneous Emission
For wavelengths in the ultraviolet and X-ray regime the start-up of the FEL process by seed radiation is not readily done due to the lack of suitable lasers. Seeding by a high harmonic of an optical laser is a widely discussed idea. The process of Self-Am
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For wavelengths in the ultraviolet and X-ray regime the start-up of the FEL process by seed radiation is not readily done due to the lack of suitable lasers. Seeding by a high harmonic of an optical laser is a widely discussed idea. The process of Self-Amplified Spontaneous Emission SASE permits the startup of lasing at an arbitrary wavelength, without the need of external radiation. The SASE mechanism was proposed and theoretically explored in the early 1980s [1, 2, 3, 4] but it took 20 years before it could be verified experimentally at visible and ultraviolet wavelengths. The most intuitive explanation of SASE is that the electrons produce spontaneous undulator radiation in the first section of a long undulator magnet which serves then as seed radiation in the main part of the undulator. The FEL can also be started by a periodic charge density modulation in the electron beam, as discussed in Sect. 5.3. The bunches coming from the accelerator do not possess such a modulation at the light wavelength. But due to the fact that they are composed of a large number of randomly distributed electrons a white noise spectrum is generated which has a spectral component within the FEL bandwidth (see Appendix F). This component will be amplified according to Fig. 5.3. The above two interpretations of SASE are physically equivalent: seeding by spontaneous undulator radiation or FEL start-up by the proper Fourier component of the stochastic density modulation in the electron beam. Randomness is obviously essential in the second model of the SASE process but it is equally important in the first model. It must be noted that the emission of undulator radiation by a bunch much longer than the light wavelength would be impossible if the longitudinal particle distribution were perfectly uniform, in the extreme case, if the electron beam current would be a perfect direct current (dc current). A perfect dc current moving on a sinusoidal orbit through the undulator magnet cannot emit any radiation because there are no oscillating charges. Likewise, a perfect dc current circulating in an electron synchrotron or storage ring would be unable to emit ordinary synchrotron radiation.
P. Schm¨ user, et al.: Self-Amplified Spontaneous Emission, STMP 229, 103–120 (2008) c Springer-Verlag Berlin Heidelberg 2008 DOI 10.1007/978-3-540-79572-8 7
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7 Self-Amplified Spontaneous Emission
7.1 Computation of the SASE Process in the 1D Theory 7.1.1 Solution of the Third-Order Equation In this section the equivalent current density modulation ˜j1 arising from the random time distribution of the electrons in the bunch is used as an input for calculating the time evolution of the FEL power by means of the method discussed in Sect. 5.3. The initial conditions are according to (5.14) ⎞ ⎞ ⎛ ⎛ ˜x (0) E 0 ˜x (0) ⎠ = ⎝ −1 ⎠ μ0 c K ˜j1 (0) . ⎝E 4γr ˜ (0) i 2ku η E x ˜j1 (0)/(4γr ) by introducing It is convenient to factor out the driving term μ0 c K new coefficients dj ⎞ ⎛ ⎞ ⎛ ⎞−1 ⎛ d1 1 1 1 0 ⎝ d2 ⎠ = ⎝ α1 α2 α3 ⎠ · ⎝ −1 ⎠ . (7.1) d3 α12 α22 α32 i 2ku η Then
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