Recent progress in SR
interferometer-for small beam size measurement-

T. Mitsuhashi, KEKAgenda1. Brief introduction of beam size
measurement through SR interferometry.
2. Theoretical resolution of interferometry
3. Reflective interferometer formeasurement of beam size down
to 5μm range.
4. Imbalanced input method for measurement ofvery small beam size less than 5μm1. A brief introduction to beam
size measurement
through
SR interferometry

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 starwith his stellar interferometer in 1921.
This principle was now known as “ Van CittertZernike theorem” because of their works;
1934 Van Cittert
1938 Zernike.

Michelson’s stellar interferometer
Wilson mountain observatory

Spatial coherence and profile of the object
Van Cittert-Zernike theorem
According to van Cittert-Zernike theorem, with the condition oflight is 1st order temporal incoherent (no phase correlation), thecomplex degree of spatial coherence γ(υx,υy) is given by the Fourier
Transform of the spatial profile f(x,y) of the object (beam) at longer
wavelengths such as visible light.

γ (υ x , υ y ) = ∫ ∫ f ( x , y ) exp {− i ⋅ 2 ⋅ π (υ x ⋅ x + υ y ⋅ y )}dxdy
where υx,υy are spatial frequencies given by;

DyDx
υx =, υy =
λ ⋅R0
λ ⋅R0

Typical arrangement for refractive interferometer
double slit

Gran-Tayler prizm

80mm m( ax)Interferogram

object8m
Achromatic lensBand-pass filter2⎧⎛⎛ π ⋅ a ⋅ y ⋅ χ(D) ⎞⎫ ⎧⎛y⎞ ⎞⎫I( y, D ) = ∫ ( I1 + I 2 ) ⋅ ⎨sin c⎜⎟ ⎬ ⋅ ⎨1 + γ ⋅ cos⎜⎜ k ⋅ D ⋅ ⎜ + ψ ⎟ ⎟⎟ ⎬ dλλ ⋅f⎝⎠⎭ ⎩⎝f⎠ ⎠⎭⎩⎝⎛ 2 ⋅ I1 ⋅ I 2 ⎞ ⎛ I max − I min ⎞-1 S(D )
⎟⋅⎜⎟
γ=⎜
,ψ
=tan
⎜ I1 + I 2 ⎟ ⎜⎝ I max + I min ⎟⎠C(D )⎝⎠Typical interferogram in vertical direction at the Photon Factory (1994).
D=10mmResult of spatial coherence measurement(1994)

Phase of the complex degree of spatial coherence
vertical axis is phase in radianReconstruction of beam profile by Fouriertransform

Beam size (mm)Vertical beam profile obtained by a Fourier transform of
the complex degree of coherence.

Comparison between image

Beam profile taken with
an imaging system

Vertical beam profile obtained by Fourier Cosine transform155μm±0.6μm
0.8Vertical andhorizontal beam sizeat the Photon
Factory500nm|γ|0.6633nm

0.40.2004812162πD /λR020242832(mm - 1)(a) vertical
1261.2μm±2.6μm
0.8|γ|

designed beam size 263μm

0.6500nm
633nm

0.40.2002462πD /λR0-1(mm )(b) horizontal

81012

We can also evaluate the RMS. beam size
from one data of visibility, which is measured
at a fixed separation of double slit. TheRMS beam size σbeam is given by ,σ beam =λ⋅R01 ⎛1⎞⋅⋅ ln ⎜ ⎟2 ⎝γ⎠π⋅D

where γ denotes the visibility, whichmeasured at a double slit separation of D.

isTo consider that in the case to make an image,
the resolution is limited by diffraction which is aFourier transform using a given region of spatial
frequency space ( measurement in the real space).
In the case of interferometry, we can measure asmall beam size with limited region of spatial
frequency space by means of these two methods(measurement in the inverse space).

Horizontal beam size measurement±3μmVertical beam size measurement±1μm2.Theoretical resolution ofinterferometry
Uncertainty principle
in phase of lightUncertainty principal in imaging.
Δθ

Δθ/λ·Δx≥1,So, large opening of light will
necessary to obtain a good spatialresolution.

Uncertainty principal in interferometry ?

Uncertainty principal in interferometry
Function of the 1st orderinterferometery
Mode 1ψ1Mode 2 ψ2

ψ=ψ1+ψ2

Measure the correlation oflight phase in two modes

Uncertainty principal in interferometry
Function of the 1st orderinterferometery
Mode 1ψ1Mode 2 ψ2

ψ=ψ1+ψ2

Measure the correlation oflight phase in two modes

Uncertainty inPhase ΔφThe interference fringe will besmeared by the uncertainty of phase.
2⎧⎛⎞⎫⎧⎛ π ⋅ a ⋅ y ⎞⎫ ⎪y⎜⎟⎪I( y, D) = ∫ (I + I ) ⋅ ⎨sin c⎜⎟⎜⎬ ⎨1 + cos⎜ k ⋅ D + φ ⎟⎟ ⎬dφ12⎪Δφ⎝ λ ⋅ f ⎠⎭ ⎪⎩f⎩⎝⎠⎭

According to quantum optics,
Uncertainty principle concerning tophase is given by

Δφ·ΔN≥1/2
where ΔN is uncertainty of photonnumber.

We cannotobserve
interference fringewith small
number ofphotons!

Actually, different from imaging, wecan use large number of photons(intensity), so uncertainty in phase isvery small (this is the reason light
seems wave)

A comparison between imaging,
we can use large number of photons(intensity), so uncertainty in phase isvery small (this is the reason lightseems wave)
As a result, theoretical resolution isvery high, and practically resolution
will be limited by measurement error
such as baseline noise in detector.

1

Small size of thebeam will give agood visibility

1

0.8

0.6

I( y)

Stronglyinfluenced bybaseline noise!

0.4

0.2

−6

1.898×10

0

3−32

10y

1

233

1Δγ
Δσ ∝
⎛1⎞γ ⋅ ln⎜⎜ ⎟⎟⎝γ⎠

Δσ in μm

Error transferfrom Δγ to Δσwith constant Δγ

So, important point in small beam
size measurement
isHow to escape from noise invisibility measurement

1. Use larger separation of double slit

2. Use shorter wavelength
Both of this will reduce visibility ofinterferogram1. Use larger separation of double slit
limited by opening angle of SR
2. Use shorter wavelength
mainly limited by chromaticaberrations in focusing optics.

Refractive index of BK7 andSF2 as a function of wavelength

Elimination ofchromatic
aberration at
400nm is very
difficult due tolarge partial
dispersion ratioof glass

Chromatic aberration(longitudinal focal sift intypical achromatic
designF=600mm

Interferogram withchromatic aberration
and without
chromatic aberration.
λ=400nm, Δλ=80nm
Lens:achromat
D=45mm f=600mm

Δλ=80nmResults by normal refractive interferometerusing λ=400nm

We cannot see any difference
In coupling correction!

If the chromatic aberration at
400nm is measure source of error
in 5μm range beam size
measurement,

Use reflective optics!
Reflective system has no chromaticaberration.

3. Reflective interferometer

Possible arrangement for reflective optics forinterferometer
1. On axis arrangement
Newtonian arrengement of optics
Gran-tayler prismBand pass filter
Interferogram

Double slit

Optical flat

Parabolic mirror

Cassegrainian arrengement of optics

Interferogram
Band pass filter

Double slit

Hyperbolic mirror

Parabolic mirror
Gran-tayler prism2. Off axis arrangement
Herschelian arrengement of optics
Gran-tayler prismBand pass filter
Interferogram

Double slit

Optical flat (off axis)

Parabolic mirror

Measured interferogram
At ATF, KEK
Result of beam size is
4.73μm±0.55μm

The x-y coupling is controlled by the strength of the skew Q
at ATF

Remember same results by normal refractive
interferometer using λ=400nm

The reflective interferometer is more usefulthan refractive interferometer especiallyfor shorter wavelength range.
Actually, it is chromatic aberration-free,and reflectors are cheaper than lenses inlarge aperture.

If we can not use more shorter wave
length,How we can do for more smaller beam
size measurement?
110.8

0.6

I ( y)
0.4

0.2−6

1.898 ×1003−3210y

1233

Result of visibility for beam size 5.8μm (l=550nm) with several
separation of double slit.

Result of visibility for beam size 5.8μm (l=550nm) with several
separation of double slit.

We hardly recognize saturationin visibility from this figure, let
us convert visibility into beamsize!

Convert visibility into beam size. We can see clear saturation insmaller double slit range which has visibility near 1.

Saturation is significant invisibility better than 0.9

0.86
0.92

4. Imbalanced input method
Another method to escape from noiseformore small beamsize measurement

Let’s us consider equation for interferogram.
2⎛⎧⎛y⎞ ⎞⎫⎛ π ⋅ a ⋅ y ⋅ χ(D) ⎞⎫ ⎧I( y, D ) = ∫ ( I1 + I 2 ) ⋅ ⎨sin c⎜⎟ ⎬ ⋅ ⎨1 + γ ⋅ cos ⎜⎜ k ⋅ D ⋅ ⎜ + ψ ⎟ ⎟⎟ ⎬ dλλ⋅f⎝f⎠ ⎠⎭⎝⎠⎭ ⎩⎩⎝⎛ 2 ⋅ I1 ⋅ I 2 ⎞ ⎛ I max − I min ⎞-1 S(D )⎟⋅⎜⎟,ψ=tanγ=⎜⎜ I1 + I 2 ⎟ ⎜⎝ I max + I min ⎟⎠C(D )⎝⎠In this equation, the term “γ” has not only real partof complex degree of spatial coherence but also
intensity factor!

⎛ 2 ⋅ I1 ⋅ I 2 ⎞ ⎛ I max − I min ⎞⎟⋅⎜⎟⎟γ =⎜⎜⎜ I1 + I 2 ⎟ ⎝ I max + I min ⎠⎠⎝If I1=I2, γ is just equal to real part of complex
degree of spatial coherence , but if I1 ≠ I2, we musttake into account of intensity factor;

2 ⋅ I1 ⋅ I 2
I1 + I 2
This intensity factor is always smaller than 1for I1 ≠ I2.

⎛ 2 ⋅ I1 ⋅ I 2 ⎞ ⎛ I max − I min ⎞⎟⋅⎜
⎟γ =⎜⎜ I1 + I 2 ⎟ ⎜⎝ I max + I min ⎟⎠⎝⎠Since intensity factor is smaller than 1 for I1 ≠I2, the “γ” will observed smaller than real partof 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!

γ=0.8γ=0.9

We can use magnification range up to 2for I1 : I2=1 : 0.2 or 3 for 1 : 0.05.

In interferometry, we canmagnify beam size by very
simple way;
applying imbalance input for
double slit!

Setup for imbalanced input by half ND filter
Herschelian arrengement of optics
Gran-tayler prismBand pass filter
Interferogram

half NDfilter

Double slit

Optical flat (off axis)

Parabolic mirror

Appling unbalance method for D=30mm.
I1 : I2 =0.853:0.249

We hardly recognize effect of
unbalanced input for saturationin visibility from this figure, let
us convert visibility into beamsize!

Unbalanced

Further result ofunbalanced technique,please hear presentationof Dr. Mark Boland

Conclusion
Smallest result of beam size at ATF is 4.7μm withreflective SR interferometer using double slit
separation of 45-55mm, λ=400nm. This size isalmost small limit with equal input method.
When we will apply imbalanced method;
With magnification factor 2
With magnification factor 3

2.4μm
1.6μm

We are waiting beam size in this range!

Thank you very muchfor your attention.