measurement of small beam size by sr interferometer by prof. dr.toshiyuki mitsuhashi kek
TRANSCRIPT
Measurement of small beam size by SR interferometer
By Prof. Dr.Toshiyuki MITSUHASHI
KEK
Let us consider one single mode of photon (in the wavepocket) will be emitted from single electron as a pencil of light?
Simple physics of SR
Physical story of Schwinger’s theory for SR
Vector potential
)kr'tt(iexp'dtdr
2r,A
r,A x
2222
't1
2
1
'dt
dt'tK
Introducing temporal squeezing factor
6
't't
1
2
1dt'tK'tt
32
22
Then, t’ in the exponential is
To consider relativistic effect, spatial part in phaser represented by plane wave 1, then
'ttiexp'dt
2A
A x
ikrexp'ttiexp'dtdr2
)kr'tt(iexp'dtdr2r,A
r,A
x
x
Since Schwinger’s theory is focused into integration in temporal domain to get spectrum,
Spatial domain integral is also another possible way to discuss the spatial distribution of radiation.
ikrexpdr2
)ikrexp('ttiexp'dtdr2rA
rA x
According to single-valued function property of potential A, spatial integral result must give same result from temporal integral at same spatial point (as described in textbook for electromagnetic theory by Jackson).
Understanding for single mode of photon in bending radiation (according to K.J.Kim’s paradigm) by limiting the range of the integration.
x≈
y≈
≈
≈
n
eh
at’+bt’3
negligible
ve
ve
Understanding for single mode of photon in bending radiation (according to K.J.Kim’s paradigm) by limiting the range of the integration.
x≈
y≈
≈
≈
n
eh
The window emits a plane wave, and it propagates with the diffraction
at’+bt’3
negligible
ve
ve
Another approach
Schwinger’s approach:
introducing relativistic effect by temporal
squeezing factor. Relativistic effect
integrate the exponential Radiation
integration of the expnential Radiation
Lorenz traslation Relativistic effect
1/
Power distribution of
dipole radiation
d))cos()c/v(1(
))cos()c/v((
c16
vQ)(P
5
2
32
0)cos(c
v0
Power distribution as a function of is
given by;
P() be 0 at 0 ;
In the case of undulator
2=2+(rt’+)2
Understand of instantaneous opening through a simple diffraction from
squair mask in experiment using ATF
Beam sizes in ATF
Vertical beam size 5m
Horizontal beam size 32m
Both size is smaller than diffraction limited size (coherent volume size) at 500nm. Diffraction pattern is determined only by generalized pupil function of incident beam.
Question in horizontal instantaneous distribution.
Two dips at ±1/mast have no wavelength dependence.
We can observe effect of three peeks in the horizontal distribution?
1/
Power distribution of
dipole radiation
If the instantaneous horizontal intensity distribution has sharp cone as in figure in below, horizontal diffraction pattern will be determined by this distribution.
In the ATF 1/ opening angle is 0.4mrad and corresponding slit width is about 2.7mm. To measure the diffraction pattern by changing the slit width in both of the horizontal and the vertical, we may have a different diffraction pattern for the both direction.
1/mm
Vertical intensity distribution Horizontal intensity distribution?
Vertical diffraction pattern Horizontal diffraction pattern
?
2mmx2mm 3x3 4x4
5x5 6x6 7x7
ATF SR Profile 1.28GeV, 1.5x10e10 single bunch
8x8 9x9 10x10
11x11 12x12 14x14
15x15 16x16 17x17
18x18 19x19 20x20
21x21 22x22 23x23
24x24 25x25 26x26
27x27 28x28 29x29
30x30 30x30(Tshutter=30ms)
Experimental results shows diffraction pattern in the vertical and horizontal are quite same until the slit width of 12mm x 12mm. Beyond 12mm, in the horizontal direction, the curvature effect of field depth will be superimposed, and diffraction pattern will be smeared by this effect.
We can conclude not special difference between diffraction pattern in the vertical and the horizontal directions.
This means instantaneous intensity distribution seems same in the both directions.
How we can understand conclusion from diffraction experiment?
n(T=t-t)
n(T=t)
Observation of two wavepockets from single
electron those are radiated at T=t-t and T=t .
n(T=t-t)
n(T=t)
Observation of two wavepockets from single
electron those are radiated at T=t-t and T=t .
Cross section of such phenomena is
proportional to
n(T=t-t)
n(T=t)
Two independent electrons irradiate two independent
wavepockets .
Cross section of such phenomena will proportional to 2
electron1
electron 2
Consider again time advance and observation
t’
t’+
This term in the parenthesis seems strange, because this
term means observation direction must be depend to
time advance
By which reason we could not set detectors for simultaneous observation??
Why we must do observations as a function of the time???
t’
detector2
detector1
detector2detector1
Source point t-t
detector1
The term t’+ must be replaced by t’2+
2222
't1
2
1
'dt
dt'tK
Same as vertical direction, temporal squeezing factor in horizontal must be given by
As a result, we reach same result for instantaneous angular distribution in horizontal direction as vertical one.
where,
But there exists no mode in horizontal direction, because electric vector points same direction in right and left.
6
't't
1
2
1dt'tK'tt
32
22
'ttiexp'dt2
A x
Understanding for single mode of photon in bending radiation (according to K.J.Kim’s paradigm) by limiting the range of the integration.
x≈
y≈
≈
≈
n
eh
The window emits a plane wave, and it propagates with the diffraction
at’+bt’3
negligible
ve
ve
29401100
ring floor
under ground
SR beam
electron beam orbit
Optical beamline for SR monitor
extraction mirror (Be)
mirror
mirror
lens for imaging
2900
source point
image
X-rays～ few 100 W
500
very hot radioactive environment
Beryllium extraction mirror
Photon Factory E=2.5GeV, 8.66m
Water cooling tube
Beryllium mirror
2mm
Photon energy (keV)
Photon energy (keV)
Beryllium extraction mirror for the B-factoryE=3.5GeV, 65m
Surface deformation for Be mirror of type used at KEKB
200W (ten times intencer) beam will come in Supper KEKB.
X-ray
Development of Diamond mirror
ANSYS simulation of temperature distribution of diamond mirror in copper holder, heated in a 2-mm horizontal ribbon on the mirror’s face.
Shape deformation of simulated mirror and holder under heating. (Colors represent deformation in z direction, perpendicular to mirror surface.)
Surface deformation of 1-mm thick single crystal diamond mirror due to 400 W applied over 20 mm width of mirror.
( a )
( b )
( c )
( d )
General design of the glass window. In this figure, (a): metal O-ring, (b): vacuum-side conflat flange, (c): optical glass flat, (d): air-side flange.
The ＜ glass window for the extraction of visible SR
Metal O-ring Metal O-ring
Delta seal
Glass window
Mirror with its holder used for the optical path
Installation of optical path ducts and boxes at the KEK B-factory
Uncertainty principal in imaging.
·x≥1
So, large opening of light will necessary to obtain a good spatial resolution.
General introduction of imaging
Aberration-free lens
Apochromat f=500 to 1000mm
Entrance aperture
Glan-tayler prism Band-pass filter
nm, nm
Magnification lens
Imaging system
Typical beam image observed by 500nm at the Photon factory (1992)
Decomvalution with MEM method by using the Wiener inverse filter
),(),(),(),( vuNvuFvuHvuG
),(
),(),(
),(),(
vu
vuvuH
vuHvuH
f
n2w
Fourier transform of blurred image G(u,v) in spatial frequency domain (u,v) is given by,
where H(u,v) is thought as a inverse filter (Fourier transform of PSF), F(u,v) is a Fourier transform of geometric image, and N(u,v) is a Fourier transform of noise in the image). The Wiener inverse filter Hw is given by,
where asterisk indicates the complex conjugate of H, n is a power spectra of the noise, and f is a power spectra of the signal.
Original image
Image after decomvolution
SR interferometer
To measure a size of object by means of spatial coherence of light (interferometry) was first proposed by H. Fizeau in 1868!
This method was realized by A.A. Michelson as the measurement of apparent diameter of star with his stellar interferometer in 1921.
This principle was now known as “ Van Cittert-Zernike theorem” because of their works;
1934 Van Cittert 1938 Zernike.
Spatial coherence and profile of the object Van Cittert-Zernike theorem
According to van Cittert-Zernike theorem, with the condition of light is 1st order temporal incoherent (no phase correlation), the complex degree of spatial coherence (xyis given by the Fourier
Transform of the spatial profile f(x,y) of the object (beam) at longer wavelengths such as visible light.
dxdyyx2iexp)y,x(f, yxyx
where xy are spatial frequencies given by;
0
yy
0
xx R
D ,
R
D
Typical arrangement for refractive interferometer
object
8m
80mm (
max)
double slit
Gran-Tayler prizm
Achromatic lens
Band-pass filter
Interferogram
DC
DStan ,
II
II
II
II2
d f
yDkcos1
f
)D(yacsin)II()D,y(I
1-
minmax
minmax
21
21
2
21
Typical interferogram in vertical direction at the Photon Factory (1994).
D=10mm
Result of spatial coherence measurement
(1994)
00
1
2
3
4
5
5 10 15 20
distance of double slit (mm)
Phase of the complex degree of spatial coherence vertical axis is phase in radian
-1.5 -1 -0.5 0 0.5 1 1.5 2 2.5-0.5
0
0.5
1
1.5
2
Vertical beam profile obtained by a Fourier transform of the complex degree of coherence.
Reconstruction of beam profile by Fourier transform (1996)
Beam size (mm)
Beam profile taken with an imaging system
Comparison between image
Vertical beam profile obtained by Fourier Cosine transform
55m± 0.6m
500nm
633nm
(a) vertical
261.2m± 2.6m
designed beam size 263m
500nm
633nm
(b) horizontal
2D /R0 (mm - 1)
0
0.2
0.4
0.6
0.8
1
0 4 8 12 16 20 24 28 32
2D /R0 (mm - 1)
0
0.2
0.4
0.6
0.8
1
0 2 4 6 8 10 12
Vertical and horizontal beam size at the Photon Factory
We can also evaluate the RMS. beam size from one data of visibility, which is measured at a fixed separation of double slit. The
RMS beam size beam is given by ,
where denotes the visibility, which is
measured at a double slit separation of D. To consider that in the case to make an image, the resolution is limited by diffraction which is a Fourier transform using a given region of spatial frequency space ( measurement in the real space). In the case of interferometry, we can measure a small beam size with limited region of spatial frequency space by means of these two methods (measurement in the inverse space).
1
ln21
D
R0beam
±3m
Horizontal beam size measurement
±1m
Vertical beam size measurement
≈≈
Incoherent field depth in horizontal beam size measurement
Longitudinal depth effect in horizontal beam size measurementelectrons in the longitudinal
depth emits the photons at different times and different positions independently
CCD observes a temporal average of interferogram
Observation axis
An example of simulation of horizontal spatial coherence in KEK B factory. A solid line denotes a spatial coherence with the longitudinal depth effect, and a dotted line denotes that of without longitudinal depth effect. A beam size is 548m and bending radius is 580m.
Without depth effect
With depth effect
Longitudinal field depth effect in horizontal beam size measurement at ATF
Without field depth
With field depth
Theoretical resolution of interferometry
Uncertainty principlein phase of light
Mode 1
Mode 2
Measure the correlation of light phase in two modes
Function of the 1st order interferometer
2 fog modes of single photon
Uncertainty principal in imaging.
·x≥1
So, large opening of light will necessary to obtain a good spatial resolution.
What is Uncertainty principal in interferometry ?
Mode 1
Mode 2
Measure the correlation of light phase in two modes
Function of the 1st order interferometery
Uncertainty principal in interferometry
Mode 1
Mode 2
Measure the correlation of light phase in two modes
Function of the 1st order interferometery
Uncertainty principal in interferometry
Uncertainty in Phase
The interference fringe will be smeared by the uncertainty of phase.
d)()f
ykDcos(1
f
aycsin)II()D,y(I
2
21
According to quantum optics,
In the large number limit, uncertainty principle concerning to phase is given by
·N≥1/2
where N is uncertainty of photon number.
Using the wavy aspect of photon in small number of photons, Forcibly ;
From uncertainty principal
·≥1/2,
then,
≥1/(2·
Even in the case of coherent mode, interference fringe will be smeared by the uncertainty of phase.
d)()f
ykDcos(1
f
aycsin)II()D,y(I
2
21
3 2 1 0 1 2 30
0.2
0.4
0.6
0.8
11
1.898 106
I y( )
33 y
Interference fringe with no phase fluctuation
3 2 1 0 1 2 30
0.2
0.4
0.6
0.8
11
1.947 106
II y( )
33 y
Interference fringe with uncertainty of phase /2
We can feel the visibility of interference fringe will reduced by uncertainty of phase under the small number of photons. But actually, under the small number of photons, photons are more particle like, and difficult to see wave-phenomena.
Actually, we can have sufficient photons for an interferogram, and theoretical limit due to theoretical limit due to the phase uncertainty is negligible smallthe phase uncertainty is negligible small.
In actual optical component, wavefont error is better than /10, this error corresponds to /50 p-v (0.126rad) over 2mm x 2mm area this systematic error in the phase can introduce a reducing of the visibility by 0.9994. This visibility corresponds to the object size of 0.26m
In actual case, we cannot observe interference fringe with small number of photons!
Points for small beam size at low ring current
1.Use larger separation of double slit
2. Use shorter wavelength
Points for small beam size at low ring current
1.Use larger separation of double slit
limited by opening angle of SR
about 10mrad at visible region.
2.Use shorter wavelength
limited by aberrations in focusing optics.
In the small ring current range, we use a wider band width (80nm) of band-pass filter to obtain sufficient intensity for the interferogram.
The use of wider band width at shorter wave length such as 400nm, the most significant error arises from the chromatic aberration in the refractive optics.
1.5
1.55
1.6
1.65
1.7
1.75
300 400 500 600 700 800
BK7SF2
wavelength (nm)
Elimination of chromatic
aberration at 400nm is very difficult due to large partial
dispersion ratio of glass
Chromatic aberration
(longitudinal focal sift in typical achromat design
F=600mm
Interferogram with chromatic aberration and without chromatic aberration.
=400nm, =80nm
Lens:achromat
D=45mm f=600mm
80nm
Results by normal refractive interferometer using =400nm
We cannot see any difference
In coupling correction!
Under the weak-intensity input,
chromatic aberration at 400nm is measure source of error in 5m range beam size measurement.
Use reflective optics!
Reflective system has no chromatic aberration.
Double slit
Newtonian arrengement of optics
Optical flat Parabolic mirror
Band pass filter
Gran-tayler prism
Interferogram
Possible arrangement for reflective optics for interferometer
1. On axis arrangement
Cassegrainian arrengement of optics
Hyperbolic mirror Parabolic mirror
Band pass filter
Gran-tayler prism
Interferogram
Double slit
Herschelian arrengement of optics
Optical flat (off axis) Parabolic mirror
Band pass filter
Gran-tayler prism
Interferogram
Double slit
2. Off axis arrangement
Measured interferogram
Result of beam size is 4.73m±0.55m
The x-y coupling is controlled by the strength of the skew Q at ATF
Remember same results by normal refractive interferometer using =400nm
Not only for beam size measurement in small beam current, but also in the most case. The reflective interferometer is more useful than refractive interferometer especially for shorter wavelength range.
Actually, it is chromatic aberration-free, and reflectors are cheaper than lenses in large aperture.
Imbalanced input method for measurement of very small
beam size less than 5m
(2010-2012)
Beam size (mm)
Spat
ial c
oher
ence
=400nm
D=45mm
5m
Spatial coherence (visibility) and beam size
Error transfer from to
with constant
i
n
mLet assume we measured with 1% error, and use a typical conditions for wavelength, distance and separation of double slit.
To allow error in beam size 1m, we can measure at =0.98
1
ln8
1
D
F
0
50
100
150
200
250
0 20 40 60 80 100
intensity of input light (arbitrary unit)
We often have a nonlinearity near by baseline
Convert visibility into beam size. We can see clear saturation in smaller double slit range which has visibility near 1.
Bea
m s
ize
(m
)
D (mm)
Bea
m s
ize
(m
)
D (mm)
Saturation is significant in visibility better than 0.9
0.92
DC
DStan ,
II
II
II
II2
d f
yDkcos1
f
)D(yacsin)II()D,y(I
1-
minmax
minmax
21
21
2
21
minmax
minmax
21
21
II
II
II
II2
Let’s us consider equation for interferogram.
In this equation, the term “” has not only real part of complex degree of spatial coherence but also
intensity factor!
If I1=I2, is just equal to real part of complex degree of spatial coherence , but if I1 ≠ I2, we must take into account of intensity imbalance factor;
21
21
II
II2
This intensity factor is always smaller than 1 for I1 ≠ I2.
0
0.2
0.4
0.6
0.8
1
00.20.40.60.81
unbalance factor
Since intensity factor is smaller than 1 for I1 ≠
I2, the “” will observed smaller than real part of complex degree of spatial coherence.
This means beam size will observed larger than primary size and we know ratio between observed size and primary size.
This is magnification!
minmax
minmax
21
21
II
II
II
II2
Tra
nsve
rse
mag
nifi
cati
on
Imbalance factor M
Tra
nsve
rse
mag
nifi
cati
on
Imbalance factor M
Herschelian arrengement of optics
Optical flat (off axis) Parabolic mirror
Band pass filter
Gran-tayler prism
Interferogram
Double slit
half ND filter
Setup for imbalanced input by half ND filter
Bea
m s
ize
(m
)
D (mm)
Bea
m s
ize
(m
)
D (mm)
Unbalanced
1 : 1
1 : 0.1
1 : 0.01
Visibility in imbalance input
Tra
nsve
rse
mag
nifi
catio
n
Imbalance factor M
Tra
nsve
rse
mag
nifi
catio
n
Imbalance factor M
Appropriate magnification is limited by wavefront error of optical components
By balance input interferometer,
We can measure the beam size down to 3-5m.
Introducing a magnification using imbalance input for interferometer,
We might be can extend this limit down to little bit less than 1m.
Do not forget, imbalance method will not increase information of spatial coherence!
It is only for convenience of measurement as well as the magnification in telescope.
Do not exceed appropriate rangeDo not exceed appropriate range!
Thank you very much for your attention.