Optics in Astronomy - Interferometryastro.u-szeged.hu/oktatas/mtech/I-07/Interferometry_01.pdf · • Practical interferometry ... – Diffraction-limited imaging – A Young‘s

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Optics in Astronomy- Interferometry -

Oskar von der LüheKiepenheuer-Institut für Sonnenphysik

Freiburg, Germany

September 2002 Optics in Astronomy 2

Contents

• Fundamentals of interferometry• Concepts of interferometry• Practical interferometry

• Fundamentals of interferometry– Why interferometry?– Diffraction-limited imaging– A Young‘s interference experiment– Propagation of light and coherence– Theorem of van Cittert - Zernike


What is interferometry?• superposition of electromagnetic waves

– at (radio and) optical wavelengths– infrared: λ = 20 µm ... 1µm– visible: λ = 1 µm ... 0.38 µm

• which emerge from a single source• and transverse different paths• to measure their spatio-temporal coherence properties

• Why interferometry?– to increase the angular resolution in order to– compensate for atmospheric and instrumental aberrations

• speckle interferometry• diluted pupil masks

– overcome the diffraction limit of a single telescope by coherentcombination of several separated telescopes

• Topic of this lecture on interferometry?– to increase the angular resolution in order to– compensate for atmospheric and instrumental aberrations

• speckle interferometry• diluted pupil masks

– overcome the diffraction limit of a single telescope by coherentcombination of several separated telescopes


Diffraction limited imaging

D

DFX λ44.2=∆

telescope aperture

focal length F


A Young‘s interference experiment��

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SourceMask

Screen

diameter D

baseline B

focal length F

wavelength λλλλ

BFx λ=∆

DFX λ44.2=∆


Two-way interferometer I

Change of baseline


Two-way interferometer II

Change of element diameter


Two-way interferometer IIa

Change of element diameter


Two-way interferometer III

Change of wavelength


Dependence on source position��

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SourceMask

Screen


Two-way interferometer IV

Change of source position


Dependence on internal delay��

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SourceMask

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delay δδδδ


Two-way interferometer V

Change of internal delay


Monochromatic e. m. waves

[ ]tjrUtrV ωω −= exp)(),(Time-dependent, monochromatic e.m. field:

πνω 2frequency circular time

spacein position

=t

r

Time-independent e.m. field fulfils Helmholz equation: λ

πωω

2;0)()( 22 ===+∇c

krUk

∈VU , Ī


dsrjr

rUj

rU o ∫∫∑

= )(2exp1)(1)( 1 ϑχ

λπ

λ ωω

Propagation of field

Kirchhoff-Fresnel integral, based on Huyghens‘ principle

Σ

0r

1rr


Non-monochromatic waves

[ ] ωωπ ω

οdtjrUtrV −= ∫

∞exp)(1),(Spectral decomposition:

∫∫∑

−= ds

crcrtrV

ttrV

πϑχ

∂∂

2)(,),( 10

Propagation of a general e.m. wave:

ϖω<<∆Condition of quasi-monochromatism:

Propagation of a quasi-monochromatic wave:

∫∫∑

−= ds

crtrV

rjtrV o )(,1),( 1 ϑχ

λ


Intensity at point of superposition

{ }),(),(Re2),(),(

),(),(),(

*2121

221

trVtrVtrItrI

trVtrVtrI

++=

=+=

The intensity distribution at the observing screen of the Young‘s interferometer is given by the sum of the intensities originating from the individual apertures plus the expected value of the cross product (correlation) of the fields.


),(),(:),,,( 22*

112121 trVtrVttrr ⋅=Γ

Concepts of coherence I(terminology from J. W. Goodman, Statistical Optics)

mutual intensity:

temporally stationary conditions, with τ = t2 - t1 )(:

),,,(),,,(

12

11212121τ

τΓ=

+Γ=Γ ttrrttrr

[ ]( ) ( )

( ) ( )212

211

12*

11

2/12211

1212

,,

,,

)0()0()(:)(

τ

τ

ττγ

+

+⋅=

Γ⋅ΓΓ=

trVtrV

trVtrV

complex degree of coherence:


Concepts of coherence II

( )),(),(

),,(:

11*

11

1111

τ

ττ

+⋅=

Γ=Γ

trVtrV

rr

)(11 τγ

self coherence (temporal coherence):

complex degree of self coherence:

( )),(),(

0

12*

11

1212

trVtrV

J

⋅=

Γ=

)0(1212 γµ =

mutual intensity (spatial coherence):

complex coherence factor:


Concepts of coherence III

• Coherence is a property of the e.m. field vector!

• It is important to consider the state of polarisation of the light as orthogonal states of polarisation do not interfere (laws of Fresnel-Arago).

• Optical designs of interferometers which change the state of polarisaiton differently in different arms produce instrumental losses of coherence and therefore instrumental errors which need calibration.


Temporal coherence

1

2R

2Z

S2

S1

1

2

∫∞

∞−= ττγτ dc

211 )(:

coherence time: ωπτ

∆≈ 2

c

refined monochromatic condition: cc cl τδ :=<<


Two-way interferometer VI

Extended spectral distribution of a

point source


Two-way interferometer VII

Extended spectral distribution source

and delay errors


Transport of coherence

∫∫ ∫∫∑ ∑

−−=

1 1

2112212

2

1

121 )(2exp),()()(),( σσ

λπ

λϑχ

λϑχ ddSSjrrJ

SSxxJ

1r

2r

1ϑ

2ϑ

1x

2x

111 rxS −=

222 rxS −=

1Σ 2Σ


Extended sources��

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Extended, incoherent sourcesMost astronomical sources of light have thermal origin. The processes emitting radiation are uncorrelated (incoherent) at the atomic level.

)()(),( 211

2

21 rrrIrrJ −= δπλMutual intensity at the surface of

an incoherent source:

∫∫∑

−−=

1

112121

2121 )(2exp)()()(1),( σ

λπϑχϑχ

πdSSjrI

SSxxJ

Transport of mutual intensity to the interferometer:


( )

∫∫

∫∫

−−=

∆+⋅∆=

source

21

source21

2exp)(1

2exp)(1),(

ϑλ

ϑπϑπ

ϑδαλπϑ

π

dxxjI

dyxjIxxJ

Celestial sources have large distances S compared to their dimensions. Differences in S can be expanded in first order to simplify the propagation equation. Inclination terms χ are set to unity. Linear distances in the source surface are replaced by apparent angles . The transport equation can then be simplified:

xϑ

Theorem of van Cittert - Zernike


Intensity distribution in the Young‘s intererometer{ }

( ){ }

( ) ( )∫∫∫∫

−=

=

⋅+=

⋅++=

++=

ϑϑϑϑλ

πϑλ

µµ

λπµ

λπ

dIdBjIB

FrBrA

FrBxxJrIrI

trVtrVrIrIrI

2exp

with

2cos1)(2

2cos,Re2)()(

),(),(Re2)()()(

12

12

2121

*2121

The complex coherence factor µ is often called the complex visibility.

It fulfils the condition 1≤µ



Spatially extended source - double star



Spatially extended source - limb

darkened stellar disk


Extended sources - not unique?


van Cittert - Zernike theorem

2D Fourier transform of source intensity at angular frequency B/λ (visibility function)

Instrumental cosine term

Observed Intensity

Response to a point source in direction of α

( ) ( )

( )

−=

−=

∫∫

∫∫

ααλ

παλ

π

ααλ

πα

dBjIzxBj

dzxBjIxI

2exp2expRe

2expRe

Source intensity


What does the vCZT mean?

An interferometer projects a fringe onto the source‘s intensity distribution

The magnitude of the fringe amplitude is given by the structural content of the source at scales of the fringe spacing

The phase of the fringe is given by the position of the fringe which maximises the small scale signal


What have we learned?• An astronomical interferometer measures spatio-temporal

coherene properties of the light emerging from a celestial source.

• The spatial coherence properties encodes the small scale structural content of the intensity distribution in celestial coordinates.

• The temporal coherence properties encodes the spectral content of the intensity distribution.

• The measured interferometer signal depends on structural and spectral content.

Documents

Optics in Astronomy - Interferometryastro.u-szeged.hu/oktatas/mtech/I-07/Interferometry_01.pdf · • Practical interferometry ... – Diffraction-limited imaging – A Young‘s