32
P. R. Willmott
incoherent bunch
coherent bunch
n e
n e
Fig. 1.19 Incoherent and coherent radiation from an ensemble (bunch) of electrons. If the relative
longitudinal positions of the electrons bear no correlation to the radiation they emit, the relative
phases of that radiation are likewise random. From normal statistics, it can be shown that their
summed amplitude is equal to the amplitude of the radiation emitted from one electron multiplied
by the square root of the number of electrons (n e ) in the bunch. The intensity (proportional to the
square of the amplitude) is therefore proportional to n e . In contrast, the radiation induced by SASE
is fully coherent, as the emission from all electrons is in phase. The amplitudes of the radiation from
each of the n e electrons thus add linearly, resulting in an increase in intensity by n 2
e . Reproduced
from [3] with permission (Copyright 2019, John Wiley and Sons)
SASE is a runaway effect. The more the electrons bunch together into
microbunches, the stronger is the radiation they produce, leading to stronger associated forces on those electrons and still tighter microbunching.
An important property of the microbunches produced by SASE is the following.
In the case of normal SR, the emission of radiation from n e electrons within a
bunch is stochastic and uncorrelated, and hence the vector addition of the individual
amplitudes will follow normal statistics and, on average, be equal to A tot =
√
n e A,
where A is the amplitude of the contribution to the radiation of any one electron.
But the intensity is proportional to the square of the amplitude and is therefore
proportional to n e .
In the case of XFEL SASE radiation, the initial bunch is modified so that it
becomes a set of microbunches separated from their neighbours by the wavelength
of light produced by the undulator interference. It follows that the individual contributions to the radiation amplitude will all be more or less in phase and the electrons
will all emit coherently. Thus, the amplitudes add linearly as they all have the same
P. R. Willmott
incoherent bunch
coherent bunch
n e
n e
Fig. 1.19 Incoherent and coherent radiation from an ensemble (bunch) of electrons. If the relative
longitudinal positions of the electrons bear no correlation to the radiation they emit, the relative
phases of that radiation are likewise random. From normal statistics, it can be shown that their
summed amplitude is equal to the amplitude of the radiation emitted from one electron multiplied
by the square root of the number of electrons (n e ) in the bunch. The intensity (proportional to the
square of the amplitude) is therefore proportional to n e . In contrast, the radiation induced by SASE
is fully coherent, as the emission from all electrons is in phase. The amplitudes of the radiation from
each of the n e electrons thus add linearly, resulting in an increase in intensity by n 2
e . Reproduced
from [3] with permission (Copyright 2019, John Wiley and Sons)
SASE is a runaway effect. The more the electrons bunch together into
microbunches, the stronger is the radiation they produce, leading to stronger associated forces on those electrons and still tighter microbunching.
An important property of the microbunches produced by SASE is the following.
In the case of normal SR, the emission of radiation from n e electrons within a
bunch is stochastic and uncorrelated, and hence the vector addition of the individual
amplitudes will follow normal statistics and, on average, be equal to A tot =
√
n e A,
where A is the amplitude of the contribution to the radiation of any one electron.
But the intensity is proportional to the square of the amplitude and is therefore
proportional to n e .
In the case of XFEL SASE radiation, the initial bunch is modified so that it
becomes a set of microbunches separated from their neighbours by the wavelength
of light produced by the undulator interference. It follows that the individual contributions to the radiation amplitude will all be more or less in phase and the electrons
will all emit coherently. Thus, the amplitudes add linearly as they all have the same