Transcript Optics in Astronomy

Optics in Astronomy
- Interferometry -
Oskar von der Lühe
Kiepenheuer-Institut für Sonnenphysik
Freiburg, Germany
Contents
•
•
•
Fundamentals of interferometry
– Why interferometry?
Concepts
of
interferometry
– Diffraction-limited imaging
– A Young‘s interferometry
interference experiment
Practical
– Propagation of light and coherence
– Theorem of van Cittert - Zernike
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What is interferometry?
• superposition of electromagnetic waves
– at (radio and) optical wavelengths
– infrared:  = 20 µm ... 1µm
– visible:  = 1 µm ... 0.38 µm
• which emerge from a single source
• and transverse different paths
• to measure their spatio-temporal coherence properties
• Topic
Why interferometry?
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 coherent
combination of several separated telescopes
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Diffraction limited imaging
D
focal length F
telescope aperture
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X  2.44
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F
D
4
A Young‘s interference experiment
X  2.44
x 
F
D
diameter D
Screen
F
B
baseline B
Source
Mask
wavelength 
focal length F
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Two-way
interferometer I
Change of
baseline
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Two-way
interferometer II
Change of
element diameter
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Two-way
interferometer IIa
Change of
element diameter
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Two-way
interferometer III
Change of
wavelength
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Dependence on source position
Screen
Source
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Mask
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Two-way
interferometer IV
Change of source
position
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Dependence on internal delay
Screen
delay 
Source
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Mask
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Two-way
interferometer V
Change of
internal delay
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Monochromatic e. m. waves
Time-dependent,
monochromatic e.m. field:
Time-independent e.m. field
fulfils Helmholz equation:
V (r , t )  U (r ) exp   j t 

position in space r
time t
circular frequency   2
( 2  k 2 ) U (r )  0 ;
k

c

2

U ,V  C
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Propagation of field


r1

r0
r
Kirchhoff-Fresnel integral, based on Huyghens‘ principle
U (ro ) 
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1
1

U
(
r
)
exp
2j

 1

j 
r

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r
 ( ) ds


15
Non-monochromatic waves
Spectral decomposition:
1
V (r , t )  U (r ) exp   j t  d
Propagation of a
general e.m. wave:
V (r0 , t ) 
Condition of quasimonochromatism:

Propagation of a quasimonochromatic wave:
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
V (ro , t ) 
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 
r   ( )
V
r
,
t

  t  1 c  2cr ds





r

V  r1, t    ( ) ds
j r 
c
1
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Intensity at point of superposition
I (r , t ) 
V1 (r , t )  V2 (r , t )
2


 I1 (r , t )  I 2 (r , t )  2 Re V1 (r , t )V2* (r , t )

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.
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Concepts of coherence I
(terminology from J. W. Goodman, Statistical Optics)
(r1, r2 , t1, t2 ) : V (r1, t1 )  V * (r2 , t2 )
mutual intensity:
temporally stationary conditions,
with  = t2 - t1
complex degree of coherence:
(r1, r2 , t1, t 2 )  (r1, r2 , t1, t1   )
: 12 ( )
 12 ( ) :

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12 ( )
11(0)  22 (0) 1/ 2


V r1, t1   V * r2 , t1   


2
2
V r1, t1  V r2 , t1   
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Concepts of coherence II
self coherence (temporal coherence):
11  : (r1, r1, )
 V (r1, t1 )  V * (r1, t1   )
complex degree of self coherence:
 11 ( )
mutual intensity (spatial coherence):
J12  12 0
 V (r1, t1 )  V * (r2 , t1 )
complex coherence factor:
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12   12 (0)
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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.
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Temporal coherence
S1
2R
2Z
1
2
S2
1

 c :   11 ( ) 2 d
coherence time:

refined monochromatic condition:
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c 
2

  lc : c c
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Two-way
interferometer VI
Extended spectral
distribution of a
point source
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Two-way
interferometer VII
Extended spectral
distribution source
and delay errors
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Transport of coherence
1
r1
S1  x1  r1
r2
1
J ( x1, x2 ) 
2
x2
2
 (1 )  (2 )
 2

J
(
r
,
r
)
exp

j
(
S

S
)
  S S
1 2
2
1  d 1d 2



1
2
 
1
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S2  x2  r2
x1
1
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Extended sources
Screen
Source
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Mask
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Extended, incoherent sources
Most astronomical sources of light have thermal origin. The processes
emitting radiation are uncorrelated (incoherent) at the atomic level.
Mutual intensity at the surface of
an incoherent source:
2
J (r1, r2 ) 
I (r1 )  (r1  r2 )

Transport of mutual intensity to the interferometer:
J ( x1, x2 ) 


1  (1 )  (2 )
 2

I (r1 ) exp  j
( S 2  S1 ) d 1

S1S 2



1
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Theorem of van Cittert - Zernike
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  are set to

unity. Linear distances
in the source surface are replaced by
x

apparent angles  . The transport equation can then be simplified:
J ( x1, x2 ) 

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 2



I
(

)
exp
j

x




y

d



 source
 

1

 x1  x2 
I
(

)
exp

2

j


 d


 source
  

1
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Intensity distribution in the Young‘s intererometer

I (r )  I1 (r )  I 2 (r )  2 Re V1 (r , t )V2* (r , t )

 
 Br 
 I1 (r )  I 2 (r )  2 ReJ  x1 , x2 cos 2

F 

 

 B  r 
 2 A(r ) 1  12 cos 2
 
F  


with
12



 
B

B  
      I  exp   2j   d  I  d
 


The complex coherence factor µ is often called the complex visibility.
It fulfils the condition   1
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Two-way
interferometer VII
Spatially extended
source - double star
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Two-way
interferometer VII
Spatially extended
source - limb
darkened stellar disk
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Extended sources - not unique?
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van Cittert - Zernike theorem
Source
intensity
Response to a point
source in direction
of 
 

B  x    



I  x   Re I   exp 2j    d 
z
 




Bx
B   

 Reexp 2j
I   exp  2j  d 

z



Observed
Intensity
Instrumental
cosine term
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2D Fourier transform of source
intensity at angular frequency
B/ (visibility function)
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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
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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.
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