Michelson's Interferometer – Theory and Practice - US Particle ...

Michelson’s Interferometer – Theory and Practice 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

Motivation for Interferometry  Electron beam size can be very small

 

11 10 9 / 1000  1m!

visible diffraction limit

x-ray diffraction limit

 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?

Motivation for Interferometry (cont’d)

Take advantage of light coherence properties

dipole oscillator

e

 i ( k  x t )

(plane waves at infinity)

 For a distributed source at finite distance the light is only partially coherent

 k  k x xˆ  k z zˆ

 Interferometry enters world of wavefront physics and statistical optics

Interferometric Beam Size Measurement Michelson (circa 1890)

Thomas Young

degree of coherence a Orionis 0.047 sec Young’sdouble slit (307cm)

Mt. Wilson Telescope Lens

Synchrotron Radiation (550nm)

screen Albert Michelson

visibility 0.8

degree of coherence sourcepoint

5cm double slit

15cm Lens

CCD array

Two-Slit Interference: Young’s Experiment

Field Pattern E  Eo e i (F )  Eo e i ( F )  2Eo cos(F)

d



d sin   m

 ( (F)   (F))e



ikx

dx



2

monochromatic light!

Intensity Pattern: cos2(F) Two slits coherent Two slits incoherent

Two-Slit Interference (cont’d) Use a lens to concentrate image on screen Intensity Pattern

Note Contrast/Visibility V 

I Max  I Min  1.0 I Max  I Min

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 field pattern at screen



a a  sin   2 2

Single-Slit: Electric Field Pattern Field pattern a  sin   2 2



a

E  Eo

a

e ikxo  e ikxo sin(kxo )  x e dx  ikxo  kxo o ikx

a

a sin  

Field pattern is the Fourier Transform of the Aperture xo

sin(a )

Single-Slit: Intensity Pattern

Intensity

 sin(a )  I  E * E  I o    a  a



(measure laser diffraction through pinhole in afternoon)

2

Two-Slit Interference (cont’d) Consider 2-slit interference with finite slit size: Intensity Pattern a

 sin(a )  2 I  I o    cos (d )  a  2

d

V=1.0

(no change)



Important mathematical point:

1  cos( )  2 cos 2 ( ) 2

Intensity pattern can be written:

 sin(a )  I  I o    1  cos(d )   a  2

Single-Slit

Two-Slit

This is approximately the form we will work with (but visibility 1.0)

From Interference to Interferometry Interferometry is used to measure coherence properties of light Spatial Coherence coherence length is inversely proportional to size 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

Temporal Coherence coherence time is inversely proportional to line-width thermal source emits wavepackets randomly in time -emission is not coherent laser source emits continuous wavetrain in time -emission is coherent

Degree of Coherence Light can be totally coherent, partially coherent or incoherent

The degree of coherence is found from a correlation function

Temporal Degree of Coherence

Spatial Degree of Coherence

Consider two colinear light waves E1(t) and E2(t)

  Consider two waves E1 (k ) and E 2 (k ) from two sources

the correlation function is 12 (t )  E1 (t ) E2* (t  t )

* the correlation function is now 12 (r )  E1 ( P1 ) E2 ( P2 )

12 is the degree of self-coherence

P1 and P2 are two points in space and r=P1-P2

12 is the degree of mutual-coherence

if tro then coherence is lost and light waves do not interfere

Spatial Coherence - Two Sources

intensity superposition

sources are partially coherent

preview: visibility is Fourier Transform of source intensity 1.0 visibility

As slits separate 1) „cosine‟ frequency goes up 2) superposition decoheres 3) visibility decreases

Visibility has various descriptive labels “mutual intensity” “mutual coherence function” “complex degree of coherence” “correlation function” “fringe parameter”

...fringe parameter measurement is central to all problems involving coherence

Putting it all together...

2 2 I ( )  E01  E02  E01E02  cos(   )

  2a    2d  I ( y )  I 0 sin c y   1   cos y  F   R     R   Single-Slit beam intensity (equal both slits)

Two-Slit visibility factor (mutual coherence) 1 + cos(x) 2

Visibility 

I Max  I Min  I Max  I Min

=0.33

0 Alan will derive in-depth with chromatic effects

=1 =0.66

x

Van-Cittert/Zernike Theorem

“Visibility is the Fourier transform of source intensity”

 ( )   I ( y)e i 2 y dy

I(y)=intensity distribution

d   spatial  frequency L In two dimensions:

 ( x , v y )   I ( x, y)e

i 2 ( x x  y y )

d=slit width =wavelength L=source to slits

dxdy

For a Gaussian, thermal-light source distribution d 2

 (d )  e where

sd 

2s d2

2s y

L

(one dimension) = spatial frequency characteristic

Fourier Transform Pairs

Fourier Transform

time domain

frequency domain

Fourier Transform

intensity domain

spatial-frequency domain

Interferometeric Beam Size Measurement Synchrotron Radiation (550nm)

visibility 0.8

degree of coherence sourcepoint

5cm double slit

15cm Lens

CCD array

For a Gaussian source d 2

 (d )  e

2s d2

sd 

L = spatial frequency characteristic 2s y

1) Measure (d) [visibility as a function of slit separation] 2) Solve for characteristic width sd 3) Infer beam size from: s y  L / 2s d

Typical Stellar Interferometer for SR Measurements

polarizer 550 +/- 5nm filter CCD

lineout

10m

1m M

source 20um

visibility 5cm double slit

f=1m lens beam size

Typical System Parameters

d 2

Source size: sy=20um Source-slit:

L=10m

wavelength: =550nm

Visibility

 (d )  e

2s d2

L 550  10 9  10 sd    44mm 2s y 2  20  10 6

a~2 mm d~40 mm

Result of beam size is 210m compliments: 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 02

 (d o )  e From

2s d2

solve for s d 



2 sy   Ls d

sy 

L d o

do 1 ln( )

1 ln( )



2

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

Michelson’s Interferometer – Summary

- 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