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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:
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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 Colorplate IX 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.
Fig. from zeiss.magnet.fsu.edu
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 λ0 , having a spread of Δλ . 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 λ0 wave
Δλ
from being totally in phase with the λ0 +
wave to become totally out of phase? The
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answer is M wavelengths:
2
1
Δλ
M λ0 = (M − )(λ0 +
)
2
2
This expression can be rearranged:
1
MΔλ Δλ0
1
MΔλ
Mλ0 = Mλ0 − λ0 +
−
≈ Mλ0 − λ0 +
2
2
4
2
2
where we assume that M>>1. We solve for M:
M=
λ0
Δλ
The coherence length (time coherence, or longitudinal coherence) is therefore defined:
lc = Mλ0 =
λ20
Δλ
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
θ m ≈ 1.22
λ
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 (page 304), 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 in one dimension:
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θ≈
λ
2π 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 ≈
λz
2π 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
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
both dimensions. The transmitted spatially coherent power
λ
in
2π
λ
λ
Pcoh = ( 2π )( 2π ) Pcen
d xθ x d yθ y
This is normally a substantial loss. If we want to have a small spectral bandwidth
Δλ
λ
, thus
⎛ λ2 ⎞
λ2
⎜
⎟
l
=
=
N
λ
increasing the longitudinal coherence from c , cen
to a suitable lc =
⎜ Δλ ⎟
Δλ
⎝
⎠
we get a corresponding reduction of the power by
lc, cen
Δλ
=N
lc
λ
In general the available coherent power is
Pcoh , λ / Δλ
λ
λ
Δλ
= η ( 2π )( 2π ) N
P
d xθ x d yθ y
λ cen
where the geometrical factor again stands for spatial filtering, N
Δλ
λ
stands for the spectral
filtering and η is a general beamline efficiency factor. It is this factor we are concerned with
when we discuss monochromators and X-ray optics.
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Photon orbital angular momentum ---- a little used property of light
The phase is displaced systematically over the wavefront. In the X-ray range it is investigated
by I. McNulty et al (APS) and in the radiowave region by Bo Thidé et al. (Uppsala),
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