1 X-Ray Sources at Large-Scale Facilities
13
win width, and λ//λ can easily exceed 10
4 —the relative bandwidth of a Si(111)
single crystal is determined by the extinction depth and is approximately 1.4 × 10
−4 .
The longitudinal coherence length can thus be several micrometres, even in the hard
X-ray regime.
The transverse coherence length l
(t)
c (also called the spatial coherence length)
results from the interference of waves having the exact same wavelength but with
slightly different directions of propagation. This arises because all sources have a
finite size D and a non-zero divergence θ (that is, a non-zero emittance), as shown
in Fig. 1.9b. In this case,
l
(t)
c = λ/2θ = λR/2D ,
(1.15)
where D is the linear size of the finite source and R is the distance from the source to
the observation point. If we assume the source has a Gaussian profile, determination
of D requires integration of interference contributions across the entire source’s
intensity distribution. It emerges that l
(t)
c is related to the standard deviation of the
beam size σ x,y by
l
(t)
c = λR/(2π
1/2
σ x,y ) ,
(1.16)
or, in practical units
l
(t)
c [µm] = 28.21
λ
◦
A R [m]
σ x,y [µm]
.
(1.17)
Note that the transverse coherence length can be made larger by judicious use of slits
limiting the apparent source size and divergence, but obviously at the cost of flux.
Beamlines such as coherent lensless imaging beamlines tend to be very long in order
to maximize R.
In the vertical direction, the source size at an undulator of, say, 2-m length, is of
the order of σ y = 2 µm. For 1-Å radiation, this yields a vertical spatial coherence
for an observer at 40 m of l
(t,y)
c
= 564 µm. The horizontal spatial coherence has
traditionally been two orders of magnitude smaller than this, due to the very much
larger extent of the electron beam in the orbital plane. The electron beam source
size in DLSR storage rings in the horizontal direction may, however, be as small as
σ x = 10 µm; the corresponding coherence length l
(t,x)
c
is of the order of 0.1 mm.
1.3 Sources of Synchrotron Radiation
In this section, the three sources of SR are semi-quantitatively described, while the
differences between the radiation produced by bending magnets and wigglers on the
one hand, and undulators on the other, are presented in a heuristic manner.
13
win width, and λ//λ can easily exceed 10
4 —the relative bandwidth of a Si(111)
single crystal is determined by the extinction depth and is approximately 1.4 × 10
−4 .
The longitudinal coherence length can thus be several micrometres, even in the hard
X-ray regime.
The transverse coherence length l
(t)
c (also called the spatial coherence length)
results from the interference of waves having the exact same wavelength but with
slightly different directions of propagation. This arises because all sources have a
finite size D and a non-zero divergence θ (that is, a non-zero emittance), as shown
in Fig. 1.9b. In this case,
l
(t)
c = λ/2θ = λR/2D ,
(1.15)
where D is the linear size of the finite source and R is the distance from the source to
the observation point. If we assume the source has a Gaussian profile, determination
of D requires integration of interference contributions across the entire source’s
intensity distribution. It emerges that l
(t)
c is related to the standard deviation of the
beam size σ x,y by
l
(t)
c = λR/(2π
1/2
σ x,y ) ,
(1.16)
or, in practical units
l
(t)
c [µm] = 28.21
λ
◦
A R [m]
σ x,y [µm]
.
(1.17)
Note that the transverse coherence length can be made larger by judicious use of slits
limiting the apparent source size and divergence, but obviously at the cost of flux.
Beamlines such as coherent lensless imaging beamlines tend to be very long in order
to maximize R.
In the vertical direction, the source size at an undulator of, say, 2-m length, is of
the order of σ y = 2 µm. For 1-Å radiation, this yields a vertical spatial coherence
for an observer at 40 m of l
(t,y)
c
= 564 µm. The horizontal spatial coherence has
traditionally been two orders of magnitude smaller than this, due to the very much
larger extent of the electron beam in the orbital plane. The electron beam source
size in DLSR storage rings in the horizontal direction may, however, be as small as
σ x = 10 µm; the corresponding coherence length l
(t,x)
c
is of the order of 0.1 mm.
1.3 Sources of Synchrotron Radiation
In this section, the three sources of SR are semi-quantitatively described, while the
differences between the radiation produced by bending magnets and wigglers on the
one hand, and undulators on the other, are presented in a heuristic manner.