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Coherence
By controlling the e-beam, i.e., its energy, time structure and emittance, and forcing it to
move in specific patterns by magnetic field structures, we have seen that we can vary the
properties of the synchrotron radiation in a wide range, including:





Energy spectrum
Flux
Brilliance (spectral brightness)
Polarization
Time Structure
Another important property of electromagnetic radiation is coherence, or phase order. Note
that coherence is not in general an inherent property of a source: A coherent part can always
be filtered out (see Fig. …. in Attwood). Different sources leave, however, very different
coherent brilliance (or flux) once this filtering is done, making the filtering more or less
worthwhile. We shall see that the filtering of undulator radiation can be very fruitful.
Phase order is measured in the beam of electromagnetic radiation both along the beam
(longitudinal coherence or time coherence), and perpendicular to the beam (transverse
coherence, or spatial coherence). We will describe them separately.
Longitudinal Coherence or Time Coherence or Coherence Length
Imagine a plane electromagnetic wave which consists of several waves with a distribution of
wavelengths, centered around 0 , having a spread of  . Imagine also that the beam is
frozen in time. We realize that, if the waves are totally in-phase at one point, they will
continuously go out of phase as we go along the beam because their wavelengths are
different. This is the only mechanism that can destroy a perfect longitudinal coherence. Thus:
Longitudinal coherence is directly related to spectral purity. We will define coherence length
below. First we ask the question: How many wavelengths does it take to bring the 0 wave

from being totally in phase with the 0 
wave to become totally out of phase? The
2
answer is M wavelengths:
1

M0  ( M  )(0 
)
2
2
This expression can be rearranged:
1
M 0
1
M
M0  M0  0 

 M0  0 
2
2
4
2
2
where we assume that M>>1. We solve for M:
M 
0

2
The coherence length (time coherence, or longitudinal coherence) is therefore defined:
lc  M0 
20

We realize immediately that bending magnet radiation, having a broad wavelength spectrum
is not a good candidate for filtering out coherent intensity. Undulator radiation, on the other
hand, emits in a narrow bandwidth has a good potential (It is the brilliance that matters). This
filtering is done in most beam lines, using monochromators.
Note that a factor of 2 difference appears in many books (e. g. Attwood), depending on how
the coherence is defined. The definition used here is the one in the Physics Handbook.
Transverse Coherence or Spatial Coherence or Coherence Width
The phase order in the direction perpendicular to the beam is also important in many cases.
The assumption in the previous section that we have a plane wave implicitly assumes perfect
transverse coherence: the phase is the same everywhere on the wavefront.
Perfect spatial coherence can be achieved if we have a point source. Then the phase at all
points on the spherical wavefront are correlated. We use diffraction to see how small a ‘point
source’ has to be. We know from the optics course, that we get a central diffraction maximum
after a circular aperture because beams from all over the opening are in phase. The criterion
for the first minimum is
 m  1.22

D
Thus, the smaller the diameter of the source, the larger the angle at which the phases are
correlated. Attwood does this using Heisenberg’s uncertainty principle, realizing that the
momentum of the photon becomes undefined as the position in the aperture becomes more
and more well-defined. This is one way to look at the diffraction phenomenon. Using rms
values he gets a similar expression:


2D
The angle is thus much smaller, and naturally the angle should be much smaller than the one
corresponding to the first diffraction minimum. The transverse coherence width at a distance z
from the aperture is then
lt 
z
2D
We see that if we had a perfect e-beam (all emission coming from the same position) we
would have perfect spatial coherence. If the emittance of the e-beam can be made so small
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that diffraction sets the limit for the angular distribution it is enough. The diffraction limit can
be reached at third generation sources, and it is reached in free electron lasers.
Otherwise one has to filter the beam using pinholes, so that the emittance is reduced to

in
2
both dimensions. The transmitted spatially coherent power


Pcoh  ( 2 )( 2 ) Pcen
d x x d y y
This is normally a substantial loss. If we want to have a small spectral bandwidth
increasing the longitudinal coherence from lc , cen


, thus
 2 
2



N

to
a
suitable
l

c
  



we get a corresponding reduction of the power by
lc, cen

N
lc

In general the available coherent power is
Pcoh,  / 



  ( 2 )( 2 ) N
P
d x x d y y
 cen
where the geometrical factor again stands for spatial filtering, N

stands for the spectral

filtering and  is a general beamline efficiency factor. It is this factor we are concerned with
when we discuss monochromators and X-ray optics.