Post on 23-Aug-2019
transcript
US Particle Accelerator School
January 14-18, 2008
- Motivation
- Two-slit Interference – Young‟s experiment
- Diffraction from a single slit - review
- Extended Source – Partial Coherence
- The Mutual Coherence function
- Van-Cittert/Ziernike theorem
- Stellar Interferometers for SR applications
Michelson’s Interferometer – Theory and Practice
Motivation for Interferometry
Electron beam size can be very small
Need to measure beam size for optics verification,
machine monitoring and operation
Conventional imaging diffraction limited
sres~50 um visible
sres~10 um x-ray pinhole
What else can be used?
x-ray diffraction limit
visible diffraction limit
!11000/1011 9 m
Take advantage of light coherence properties
For a distributed source at finite distance
the light is only partially coherent
Interferometry enters world of wavefront physics and statistical optics
Motivation for Interferometry (cont’d)
)( txkie
(plane waves at infinity)
zkxkk zxˆˆ
dipole
oscillator
Interferometric Beam Size Measurement
Michelson (circa 1890)
a Orionis
0.047 sec
degree of coherence
Young’sdouble slit
(307cm) Mt. Wilson
Telescope Lens
Synchrotron Radiation (550nm)
degree of coherence
5cm double slit 15cm Lens
sourcepoint
screen
CCD array
visibility 0.8
Thomas Young
Albert Michelson
d
Field Pattern
Two-Slit Interference: Young’s Experiment
2sin
md
Two slits coherent
Two slits incoherent
Intensity Pattern: cos2(F)
dxeikx
FF ))()((
)cos(2)()( F FF
o
i
o
i
o EeEeEE
monochromatic light!
Intensity Pattern
Two-Slit Interference (cont’d)
Use a lens to concentrate image on screen
Note Contrast/Visibility
0.1
MinMax
MinMax
II
IIV
fully coherent!
Change phase/incidence angle, shift pattern phase
V=1.0 (no change)
Single-Slit Diffraction - Review
Plane wave incident on aperture - diffraction
Condition for first diffraction minima
2sin
2
a
a
field pattern at screen
Single-Slit: Electric Field Pattern
2sin
2
a
a
a
a
)sin(oEE
a sin
a
Field pattern
Field pattern is the Fourier Transform of the Aperture
o
o
o
ikxikxx
x
ikx
kx
kx
ikx
eedxe
ooo
o
)sin(
a
Intensity2
* )sin(
a
a oIEEI
Single-Slit: Intensity Pattern
(measure laser diffraction through pinhole in afternoon)
Intensity Pattern
Two-Slit Interference (cont’d)
Consider 2-slit interference with finite slit size:
V=1.0 (no change)
Important mathematical point: )2
(cos2)cos(1 2
Intensity pattern can be written:
Single-Slit Two-Slit
This is approximately the form we will work with (but visibility 1.0)
ad
)cos(1)sin(
2
d
a
aII o
)(cos)sin( 2
2
d
a
aII o
From Interference to Interferometry
Interferometry is used to measure coherence properties of light
Spatial Coherence
Temporal Coherence
coherence time is inversely proportional to line-width
coherence length is inversely proportional to size
thermal source emits wavepackets randomly in time
-emission is not coherent
laser source emits continuous wavetrain in time
-emission is coherent
a star emits plane-waves randomly in direction
-emission is not coherent at close distance
a point source emits sphereical-waves in all directions
-emission is coherent at long distances
Degree of Coherence
Light can be totally coherent, partially coherent or incoherent
Spatial Degree of CoherenceTemporal Degree of Coherence
Consider two colinear light waves E1(t) and E2(t)
P1 and P2 are two points in space and r=P1-P2
the correlation function is
The degree of coherence is found from a correlation function
if t>to then coherence is lost
and light waves do not interfere
)()()( *
2112 tt tEtE
if t<to (correlation time), then waves are coherent
and light interferes
Consider two waves and from two sources
the correlation function is now
if r>ro then coherence is lost
and light waves do not interfere
if r<ro (correlation length), then waves are coherent
and light interferes
)(1 kE
)(2 kE
12 is the degree of self-coherence
12 is the degree of mutual-coherence
)()()( 2
*
21112 PEPEr
Spatial Coherence - Two Sources
Visibility<1
sources are
partially coherent
slit separation
vis
ibili
ty
1.0
0
spatial coherence
0
preview: visibility is Fourier Transform
of source intensity
intensity
superposition
As slits separate
1) „cosine‟ frequency goes up
2) superposition decoheres
3) visibility decreases
Mutual Coherence of field at two points
)( 22 PE
)( 11 PE
Solve for intensity on the screen
ii
TT eEEeEEEEI 2121)( ii eEEeEEEEI 2121
2
2
2
1)(
Now take the time average
iiii eeeeEEEEI 0201
2
02
2
01)(
)cos()( 0201
2
02
2
01 EEEEI
2
02
2
01
210201 ),(2
EE
PPEE
II
IIVisibility
MinMax
MinMax
For E01=E02, ),( 21 PPII
IIVisibility
MinMax
MinMax
Mutual Coherence (cont’d)
),( 21 PPII
IIVisibility
MinMax
MinMax
For the case when I1=I2, we have
Where
has various descriptive labels
“mutual intensity”
“mutual coherence function”
“complex degree of coherence”
“correlation function”
“fringe parameter”
>< 2121 *),( EEePP i
...fringe parameter measurement is central to all problems involving coherence
Putting it all together...
)cos()( 0201
2
02
2
01 EEEEI
Single-Slit Two-Slit
visibility factor (mutual coherence)beam intensity
(equal both slits)
2
0
=1
=0.66
=0.33
1 + cos(x)
x
MinMax
MinMax
II
IIVisibility
F
y
R
dy
R
acIyI
2cos1
2sin)( 0
Alan will derive in-depth with chromatic effects
Van-Cittert/Zernike Theorem
“Visibility is the Fourier transform of source intensity”
dyeyI yi 2)()( d=slit width
=wavelength
L=source to slits
I(y)=intensity distribution
frequencyspatialL
d
For a Gaussian, thermal-light source distribution
whereL
y
d
ss
2
2
2
2)( d
d
eds
= spatial frequency characteristic
In two dimensions:
dxdyeyxIvyxi
yxyx )(2
),(),(
(one dimension)
Fourier Transform Pairs
time domain
intensity domain
frequency domain
spatial-frequency domain
Fourier Transform
Fourier Transform
Synchrotron Radiation (550nm)
degree of coherence
5cm double slit 15cm Lens
sourcepoint
CCD array
visibility 0.8
Interferometeric Beam Size Measurement
For a Gaussian source
2
2
2)( d
d
eds
= spatial frequency characteristic
1) Measure (d) [visibility as a function of slit separation]
2) Solve for characteristic width sd
3) Infer beam size from: dy L ss 2/
y
d
L
s
s
2
Typical Stellar Interferometer for SR Measurements
5cm double slit f=1m lens
source
CCD
20um
10m
1mM
lineout
visibility
beam size
polarizer
550 +/- 5nm filter
Typical System Parameters
2
2
2)( d
d
eds
Visibility
Source size: sy=20um
Source-slit: L=10m
wavelength: =550nm mmL
y
d 4410202
1010550
2 6
9
s
s
d~40 mm
a~2 mm
Result of beam size is 210mcompliments: T. Mitsuhashi
Vertical beam size at the SR center of Ritsumeikan
university AURORA.
compliments: T. Mitsuhashi
Photon Factory Laboratory
Beam Size Measurement (cont’d)
From a single measurement at slit separation do
d
yLs
s
2
solve for 2
20
2)( d
d
o eds
)
1ln(
s o
d
d
From
2
)1
ln(
s
o
yd
L on-line diagnostic for slit separation do
KEK-B on-line system
PEP-II on-line system
SPEAR-3 coupling measurements
USPAS Simulator
Practical Issues
- Thermal distortion of mirrors – wavefront distortion
- Precision control of slit width (I1 and I2)
- Depth of field effects
- CCD camera linearity
- Table vibrations
- Readout noise
- Beam stability
- Numerical fitting
- Interferometers useful below the diffraction limit
- Two-slit Interference – Young‟s experiment
- Diffraction from a single slit
- Extended Source – Partial Coherence
- Visibility and the Mutual Coherence function
- Van-Cittert/Ziernike theorem: Fourier XFRM
- Stellar Interferometers for SR applications
Michelson’s Interferometer – Summary