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