Undulator radiation characteristics

1
Undulator spectrum and brilliance dependence on
beta functions and straight section length
Josep Nicolas LLS - ALBA
Barcelona, 28 April 2004
2
Outline Spectrum of periodic insertion device
The periodic interference condition The
spectrum of a single period Beam effects on
radiation beam divergence Energy
dispersion Brilliance The problem of
brilliance Gauss-Schell beam approximation
optimization Optical limitations to phase
space Mirror, Monochromator slits
acceptance Efficiency through a pinhole
Final remarks
3
Periodic Insertion Device (ID)
Periodic Magnetic Field
Electron path
Ee 3 GeV
K 1.8
?U25 mm
LID 2 m ? N 80

one SINGLE period depends on the exact magnetic
field

N-pin comb function does NOT depend on the
magnetic field
Hint The idea of AB is to replace each point of
function A by the whole function B
4
Radiation a N-pin comb undulator
5
Radiation a N-pin comb undulator
6
Radiation a N-pin comb undulator
7
Radiation a N-pin comb undulator
8
Radiation a N-pin comb undulator
9
Radiation a N-pin comb undulator
10
Radiation a N-pin comb undulator
11
Radiation a N-pin comb undulator
?
Field at a given observation point
time domain
t
FT
frequency domain
?


h?

• In addition
• The wavelength depends on the viewing angle
• Each wavefront is centered at a different period

12
N-pin comb undulator
ONE period
Total field
?
?
?


t
t
t
FT
FT
FT
?
?
?




h?
h?
h?
Direct domain
13
First harmonic photon energy
ApproximationsSmall angleSmall K
IF2
Interference factor
N2
?
-?
h?
IV25 h?1 1.3 KeV N2 6400
14
Spectral Bandwidth
I.F.2
N2
N2
?
1
2
3
-?
h?
IV25 ??Z33 eV
??Z full width of the peak measured between the
two nearest zeros
15
Angular Distribution
n3
I.F.2
N2
?
-?
h?
IV25 sph 22/n µrad
16
Form Factor The radiation emitted in a single
period
ß is constant
electron path is a perfect sinusoidal function of
z only 1st harmonic
Time dependence of position is NOT a pure
sinusodal higher harmonics are
present Moreover Radiation depends on ß(t) and
dß(t)/dt derivation (high pass) enhances
higher harmonics
17
Relativistic horizon instantaneous emission cone
2/?
Vertical
18
Additional Nodes interference inside the period
Even harmonics
?

0
Odd harmonics
?
? 0

?
?



19
Node distribution for ideal undulator radiation
n 1
n 2
Monochromatic radiation cone
y
y
x
x
n 3
n 4
y
y
x
x
500 µrad 500 µrad
20
Power spectrum of ideal undulator (3D function)
?2(?,x,0)
x
2K/g
y
Photon energy
x
?2(?,0,y)
y
Photon energy
?2
Photon energy
21
Analytical solutions
The form factor (it is evaluated only at ? n?1)
Ax, and Ay are factors for Electric Field
components
S1 and S2 are combinations of Bessel functions
that come from exp(isin()) terms in retarded
time Fourier transform of the electric field
22
Analytical solutions on Axis
Values on AXIS
23
sph
sX
24
sph
sX
sX
25
SX
SX
sph
sX
sX
Convolution in phase space each point
representing an electron is replaced by the
distribution of the radiation it emits
26
Effect of beam divergence on angular power
spectrum (angle-wise convolution) The integral
along constant ? line is preserved
?B2(?,x,0)
?B2(?,0,0)
?
x
2?1
?1
3?1
4?1
Angular distribution of electrons
x
?B2(?,x,0)
sX 5 µrad
?
27
Effect of beam divergence on angular power
spectrum (angle-wise convolution) The integral
along constant ? line is preserved
?B2(?,x,0)
?B2(?,0,0)
?
x
2?1
?1
3?1
4?1
Angular distribution of electrons
x
?B2(?,x,0)
sX 10 µrad
?
28
Effect of beam divergence on angular power
spectrum (angle-wise convolution) The integral
along constant ? line is preserved
?B2(?,x,0)
?B2(?,0,0)
?
x
2?1
?1
3?1
4?1
Angular distribution of electrons
x
?B2(?,x,0)
sX 20 µrad
?
29
Effect of beam divergence on angular power
spectrum (angle-wise convolution) The integral
along constant ? line is preserved
?B2(?,x,0)
?B2(?,0,0)
?
x
2?1
?1
3?1
4?1
Angular distribution of electrons
x
?B2(?,x,0)
sX 50 µrad
?
30
Effect of beam energy dispersion angular power
spectrum
Variations on the energy of electrons involve a
SCALING of the spectrum P(?) is approximated to
a Normal distribution but P(?) is not Scaling
in ?2(?) can be performed by shifting in ?2(log?)
?needed to apply convolution
?B2(?,0,0) for N inf.
?B2(?,0,0) for N inf.
?1(E)
log?1(E)
?1(E dE)
log?1(E dE)
31
Effect of beam energy dispersion angular power
spectrum (log?-wise convolution) The shape of
peaks changes from sinc2 to gaussian Higher
harmonics are more affected by energy dispersion
?B2(?,0,0)
x
?B2(?,x,0)
?
?1
3?1
?E/E 5x10-3
32
Effect of beam energy dispersion angular power
spectrum (log?-wise convolution) The shape of
peaks changes from sinc2 to gaussian Higher
harmonics are more affected by energy dispersion
?B2(?,0,0)
x
?B2(?,x,0)
?
?1
3?1
?E/E 10x10-3
33
Effect of beam energy dispersion angular power
spectrum (log?-wise convolution) The shape of
peaks changes from sinc2 to gaussian Higher
harmonics are more affected by energy dispersion
?B2(?,0,0)
x
?B2(?,x,0)
?
?1
3?1
?E/E 20x10-3
34
Effect of beam energy dispersion angular power
spectrum (log?-wise convolution) The shape of
peaks changes from sinc2 to gaussian Higher
harmonics are more affected by energy dispersion
?B2(?,0,0)
x
?B2(?,x,0)
?
?1
3?1
?E/E 40x10-3
The convenient framework for these problems is
phase space, and therefore Brilliance.
35
Bandwidth limits
Beam Divergence
Diffraction
Energy Spread
Bandwidth Limiting factor
Energy Spread
Beam Divergence
Energy Resolution
L
Diffraction
beta
b
Length
36
IV25 at ALBA
K1 eX 510-9 mrad n 9
Beam Divergence
Energy Spread
1.4 m
Diffraction
LID
ßX
29m
37
Brilliance photons with energy h? (0.01)
emitted per second from a point (x,y) and
direction (x,y)
Number of photons
Brilliance
second bandwidth unit area unit solid angle
photon distribution in transverse phase space
Spatial density Angular density
Phase space density
?
38
Distribution of electrons in phase space
Gaussian distribution ? Level curves are
ellipses Marginal integrals are the spatial
and angular distributions (also Gaussian)
Volume integral is the number of electrons in the
storage ring
Peak value does NOT depend on ßX
x
x
x
x
39
Distribution of Photons in phase space ? W(x,x)
NOT UNIQUE (redundant information) Marginal
integrals Spatial distribution Angular
distribution Volume integral The number of
photons emitted with that energy Heisenberg
uncertainty (or diffraction limit)
Flux density
x
x
Angular density
x
x
40
Distribution of Photons in phase space (Cont.)
Wigner historical and often used in wave optics
Kirkwood Signal theory framework ?approximates
ray tracing
x
x
Wigner distribution for a tuned photon energy
Many others (Gabor Windowed FT, Wavelets)
Phase space distribution can be obtained from
the electric field at a transverse plane, OR by
its Fourier transform (just one, thats
coherence) For numerical treatment, a discrete
transformation is desirable. The two
polarization components must be trated
separately. Phase terms cannot be neglected.
All this is needed only for diffraction limited
sources
41
Gaussian approximation (Gauss-Schell beams)
Approximating the angular distribution and
spatial distribution by normal distributions
Peak value DOES depend on L because nph N2
x, y
x
x
x, y
42
Gaussian approximation
Convolution
2SX
Analytical solution for convolution
x
2SX
x
Brilliance time evolution
43
Gaussian approximation
Large emittance or high photon energy
Small emittance or low photon energy
Long ID
Long ID
Short ID
Short ID
x
Small beta
x
x
x
x
big beta
x
x
x
What is lost for beam divergence can be
compensated by beam density
44
Peak brilliance based on gaussian approximation
(fixed eX and LID)
Horizontal
4pßX /LID
Brilliance on axis (A.U.)
Diffraction Limit
1
0
0
? / 4peX
1
Our 1st harmonic eX 1nmrad
45
Peak brilliance based on gaussian approximation
(fixed eX and LID)
Vertical
4pßY /LID
Brilliance on axis (A.U.)
Diffraction Limit
1
0
0
? / 4peY
9th
5th ? 0.01nmrad
7th
5th
3rd
9th
7th
1st ? 0.05 nmrad
46
Phase space for IV25 at ALBA
n 1 ßY,ßX 1m
47
Phase space for IV25 at ALBA
n 9 ßY,ßX 1m
48
Phase space for IV25 at ALBA
n 9 ßY LID/4p 0.16 m
49
Optical system
Optical system limitations to phase space
Light gathering for an undulator is not just
image formation because The undulator light
source is not punctual but linear The
magnification for different cross sections of a
linear light source depends on its relative
position Only one section of the undulator is
focused in a given image plane (located at sample
or detector position) The printfoot of light
coming from off-plane sections spreads because of
the defocusing
50
Phase space transformations by optical systems
Free space propagation
Focusing system
-x/f
x
x
dx
x
x
Slit
Monochromator
x
y
x
y
51
2. Propagation until focusing system
Diverging beam
1. Source
x
x
x
x
4. Propagation until image plane
Converging beam
3. Focusing system
x
x
x
x
52
The image plane of an extended source is not the
plane where the flux density is maximum
x
x
x
x
53
Optical system limitations to phase space
Source
Beam propagated to mirror plane
x
x
x
x
Other optical elements anti-propagated
Mirror anti-propagated to source
x
x
x
x
54
Light efficiency through a pinhole
55
The spectrum on sample is apodized
?
?
t
t
FT
FT
?
?




h?
h?
Without optical system
With optical system
56
Final remarks Spectrum of periodic insertion
device is analyzed in two steps The
periodic interference condition (generic)
The spectrum of a single period. Beam effects
on radiation have been analyzed L vs ßX
space is divided in three regions according to
bandwidth limitation factor (diffraction,
energy dispersion and beam divergence)
Brilliance optimization Gauss-Schell beam
Numerical calculation of angular distribution
Optical limitations to phase space Mirror,
Monochromator slits limit the used phase
space Efficiency through a pinhole is
establishes is different for different
undulator sections