Number of periods

An electron passing through an undulator with Nu
periods produces a wave train with Nu oscillations
and a time duration of T = Nuλ1/c.
Spectrum

Due to its finite length, this wave train is not
monochromatic but contains a frequency spectrum
which is obtained by Fourier transformation
Lineshape
Angular spread
Higher harmonic wavelength

Spectral density for on axis radiation
Energy


Example of a computed photon energy spectrum of undulator radiation
for an undulator with 10 periods.
The spectral energy of the mth harmonic that is emitted into the solid
angle ΔΩm. = 1000, the undulator has the period λu = 25 mm and the
parameter K = 1.5. Note that the energy ratios Um/U1 depend only on the
harmonic index m and the undulator parameter K, but not on  nor on λu
Comparison
Bibliography

G. Margaritondo, Y. Hwu and G. Tromba
“Synchrotron light: From basics to coherence and
coherence-related applications”
Again on Brilliance



A source can be made brighter by increasing the flux, by decreasing the
size or by enhancing the angular collimation
The average flux emitted by each single circulating electron is fixed by
the electron motion parameters.
One could, however, increase the number of circulating electrons, i.e., the
stored current in the ring. Unfortunately, the improvements in that sense
practically saturated at 1 ampere in the 1980's
Coherence



A point-like single-wavelength (monochromatic) source is a
coherent source.
But what happens if the source is no longer monochromatic, or
no longer point-like, or both?
The fringes will be blurred and, beyond a certain point, no longer
visible. This point marks the difference between coherent and
non-coherent sources
Two point source






The theory of diffraction tells us that
the angular distance between two
adjacent fringes is about /d radians.
If this value is substantially larger than
Sz/D, then the superposition of the two
patterns gives a somewhat blurred but
still clearly visible set of fringes.
The condition for source coherence,
known as “lateral coherence" or
“spatial coherence“, is Sz/D < /d, or Sz
(d/D) <
Note that (d/D)  z, where z is the illumination angle of each of the two
slits.
Thus, the condition for spatial coherence can be written: Sz z< 
This equation implies that while reducing the source size Sz we improve
not only the source brightness but also the spatial coherence.
Angular divergence




Suppose that each one of the two point source
has an angular divergence z.
Only a portion of this angular emission can be
used to produce a detectable fringe pattern.
This portion is /Sz. This implies that of the entire
emission over the angular range z only a fraction
(/Sz)/ z = /(Sz z ) can be used to produce
coherence-requiring phenomena
By increasing the source collimation, i.e., by
decreasing z, one increases this fraction
Coherent power




The coherent power of the source is the fraction of
the emitted power that can be used to produce
coherence-requiring phenomena
When the brightness is increased, then the
coherent power is also enhanced.
Also note that the coherent power increases with
the square of the wavelength.
Reaching high spatial coherence is thus more
difficult for X-rays than for visible light
Diffraction limit





How much spatial coherence can be obtained?
If Sz z (or Sy y ) equals the wavelength, the
source is fully coherent
This is the so-called diffraction limit.
Sources of the class of ELETTRA, BESSY-II are fully
coherent down to wavelengths of the order of 100
nm. The Swiss Light Source arrives down to 10 nm.
LCLS (FEL) arrives to 0.1 nm!!
Longitudinal or temporal coherence




This is the coherence condition related to the nonmonochromaticity of the source, due to its bandwidth
The first-order fringe for  occurs at the angle /d radians,
and that for + at (+ )/d radians.
These fringes are difficult to observe in the superposition
pattern if they are too much shifted from each other. On the
contrary, if /d (+)/d then they are blurred but visible.
This implies << 1
Coherence length






In some applications of longitudinal coherence, what
matters is the so called coherence length
This notion can be understood by noting that two waves of
wavelengths  and +  , which happen to be in phase at
a certain point in space, will become out of phase beyond
this point.
Specifically, they will be totally out of phase (i.e., the
maximum on one wave coincides with the minimum of the
other) after a distance Lc such that
Lc/ - Lc/(+  ) = 1/2,
which for a small gives Lc /2 = 1/2 , or
Usually, depending also from applications, the
coherence length of synchrotron radiation is not
enough and a monochromator is needed
Model for a chaotic source (no coherence!)




Consider a particular excited atom radiating light of
frequency 0
We can consider a wave train of electromagnetic
radiation steadily emanating from the atom until it
suffers a collision.
During a collision, the energy levels of the radiating
atom are shifted by the forces of interaction between
the two colliding atoms. Thus the radiated wave train is
interrupted for the duration of the collision.
When the wave of frequency 0 is resumed after the
collision, its characteristics are identical to those that it
had prior to the collision, except that the phase of the
wave is unrelated to the phase before the collision
Phase

the phase (t) remains constant during periods of
free flight but changes abruptly each time a
collision occurs
Intensity distribution
Relevance of coherence for
crystallography
Important of the phase information
Important of the phase information II
Important of the phase information III
Application of coherent radiation
A.
Cianchi
Introduction to SPARC_LAB
117
Coherent emission due to bunching!



Synchrotron radiation is emitted into a broad spectrum
with the lowest frequency equal to the revolution
frequency and the highest frequency not far above the
critical photon energy.
At low photon frequencies we may observe an
enhancement of the synchrotron radiation beyond
intensities predicted by the theory of synchrotron
radiation as discussed so far.
For photon wavelengths equal and longer than the
bunch length, we expect therefore all particles within a
bunch to radiate coherently and the intensity to be
proportional to the square of the number Ne of
particles rather than linearly proportional as is the case
for high frequencies. This quadratic effect can greatly
enhance the radiation since the bunch population can
be from 108–1011 electrons.
Coherent emission II
λ


1
λ
Generally such radiation is not emitted from a storage ring beam because
radiation with wavelengths longer than the vacuum chamber dimensions
are greatly damped and will not propagate along a metallic beam pipe.
Much shorter electron bunches of the order of 1-2 mm and the
associated coherent radiation can be produced in linear accelerators
where a significant fraction of synchrotron radiation is emitted
spontaneously as coherent radiation.