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André Gide
http://www.aeos.ulg.ac.be/teaching.php
Astrophysics and Space Techniques (2) ASTR0004-2 Surdej Jean
Slides of the 4th lecture “Astrophysics and Space Techniques” (2015-16)
And 1st lecture “Observing the sky”. Un ancien vidéo de ce cours en
français est accessible via le lien http://orbi.ulg.ac.be/handle/2268/74101
fichier pdf taille (octets) : 1698109
Introduction to optical/IR
interferometry
Jean Surdej
([email protected])
http://hdl.handle.net/2268/155589
Telescope
Airy disk
detector
2.44 λ/D1!
diameter (D1)
• !The!image!of!a!star!is!like!a!dot!!
Telescope
Airy disk
detector
2.44 λ/D2!
diameter (D2)
• !The!image!of!a!star!is!s2ll!like!a!dot!!
•  Need for very large telescopes !!!
• !!H.!Fizeau!and!E.!Stephan!(1868@1870):!
“In!terms!of!angular!resolu2on,!two!small!apertures!distant!of!
B!are!equivalent!to!a!single!large!aperture!of!diameter!B”!
Telescope 1
Telescope 2
Recombiner
Baseline (B)
Telescope 1
Telescope 2
Recombiner
Interference fringes
Inter@fringe= λ/B!
Baseline (B)
B!>!D!
=
⊗
I (ζ ,η ) = ∫∫ PSF (ζ − ζ ' ,η −η ' ) O(ζ ' ,η ' ) dζ ' dη ' = PSF (ζ ,η ) ⊗ O(ζ ,η )
TF(I(ζ , η ))(u, v) = TF(PSF(ζ , η ))(u, v) ⋅ TF(O(ζ , η ))(u, v)
u = Bu / λ, v = Bv / λ
O(ζ , η ) = TF(−1)TF(O(ζ , η )) = TF(−1)(TF(I(ζ , η )) / TF(PSF(ζ , η )))
=
⊗
*
TF ( PSF (ζ ,η ))(u, v) = ∫∫ A ( x, y)
A ( x + u, y + v) dx dy
An introduction to optical/IR interferometry
■ 
■ 
■ 
■ 
■ 
■ 
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1 Introduction
2 Reminders
3 Brief history of stellar diameter measurements
4 Interferometry with two independent telescopes
5 Light coherence (Zernicke-van Cittert theorem)
6 Examples of optical interferometers
7 Results
8 Three important theorems (Fundamental theorem,
Convolution theorem and Wiener-Khintchin
theorem)!
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An introduction to optical/IR interferometry
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1 Introduction
Δ
z
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An introduction to optical/IR interferometry
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1 Introduction
ρ=R/z
(1.1)
Δ = 2ρ
z
F = f / ρ2
Teff = (F/σ)1/4 = (f / σ ρ2)1/4
2R
(1.2)
(1.3)
14
An introduction to optical/IR interferometry
■ 
2 Reminders
■ 
2.1. Representation of an electromagnetic wave
T = 1/ν
E
t, z
E = a cos[2π (ν t - z / λ)] (2.1.1)
where λ = c T = c / ν. (2.1.2)
λ
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An introduction to optical/IR interferometry
■ 
2.1. Representation of an electromagnetic wave
E = Re{ a exp[i2π(νt - z / λ) ]}
E = Re{ a exp[-i φ] exp[i2πνt]}
where
φ = 2π z / λ.
E = a exp[-i φ] exp[i2πνt]
(2.1.3)
(2.1.4)
(2.1.5)
(2.1.6)
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An introduction to optical/IR interferometry
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2.1. Representation of an electromagnetic wave
with
E = A exp[i2πνt]
(2.1.7)
A = a exp[-i φ]
(2.1.8)
ν ~ 6 1014 Hz for λ = 5000 Å
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An introduction to optical/IR interferometry
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2.1. Representation of an electromagnetic wave
1
2
〈 E 〉 = lim
T →∞ 2T
2
2
〈E 〉 = a
+T
2
∫ E dt
(2.1.9)
−T
/2
I = A A* = (A(2 = a 2 .
(2.1.10)
(2.1.11)
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An introduction to optical/IR interferometry
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2.2. The Huygens-Fresnel principle
σ = 2.44 λ / d
(2.2.1)
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An introduction to optical/IR interferometry
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2.3. Atmospheric turbulences
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An introduction to optical/IR interferometry
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2.3. Atmospheric turbulences
Fried parameter: r0
Coherence time (Greenwood time) : τ0 = 0,314 r0/v
Seeing : β = 0,98 λ / r0
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An introduction to optical/IR interferometry
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2.3. Atmospheric turbulences
Speckles
β+
r0
Short exposure
vs
Long exposure
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An introduction to optical/IR interferometry
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2.2. The Huygens-Fresnel principle
1st experiment!
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An introduction to optical/IR interferometry
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3 Brief history of stellar diameter measurements
a) Galileo (1632)
2
z
><
Δ = 2ρ = D / z
D
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An introduction to optical/IR interferometry
■ 
3 Brief history of stellar diameter measurements
b) Newton:
V! - V = -5 log (z / z!) ,
Δ = 2 R! / z
Δ ~ 2 10-3
,
(8 10-3
).
(3.1)
(3.2)
(3.3)
c) Fizeau-type interferometry
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An introduction to optical/IR interferometry
■ 
4 Interferometry with two independent telescopes
a) Young s double hole experiment (24-11-1803)
Σ x
|PP1| - |PP2| = nλ
(4.1)
P1 (B/2,0,0)
B
x
P(x,y,z)
.
φ=x/z=λ/B
z
(4.6)
P2 (-B/2,0,0)
I
28
An introduction to optical/IR interferometry
■ 
4 Interferometry with two independent telescopes
b) Fizeau … the father
of stellar interferometry
(1868)
If Δ ≥ φ/2 = λ / (2B), (4.7)
1
1
2
Δ
fringe disappearance!
Fringe visibility:
$ Imax − Imin '
υ =&
)
I
+
I
% max min (
29
€
An introduction to optical/IR interferometry
■ 
4 Interferometry with two independent telescopes
b) Fizeau … the father of stellar interferometry (1868)
2nd experiment!
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An introduction to optical/IR interferometry
■ 
4 Interferometry with two independent telescopes
b) Fizeau … the father of stellar interferometry (1868)
Stéphan, 1873
Δ << 0,16
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An introduction to optical/IR interferometry
■ 
Marseille 80 cm telescope
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An introduction to optical/IR interferometry
■ 
4 Interferometry with two independent telescopes
b) Fizeau … the father of stellar interferometry (1868)
•  Michelson, 1890 (satellites
of Jupiter)
•  Michelson and Pease (1920)
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An introduction to optical/IR interferometry
■ 
4 Interferometry with two independent telescopes
b) Fizeau … the father of stellar interferometry (1868)
•  Anderson
•  Brown and
Twiss (1956)
•  Radio Interferometry
(1950)
35
An introduction to optical/IR interferometry
■ 
■ 
5 Light coherence
5.1 Quasi monochromatic light (waves and wave
groups)
λ ± Δλ
ν ± Δν
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An introduction to optical/IR interferometry
■ 
■ 
5 Light coherence
5.1 Quasi monochromatic light (waves and wave
groups)
I = 〈V (t )V (t )〉
∗
(5.1.1)
ν + Δν
V ( z, t ) =
a(ν ' ) exp(i 2Π(ν ' t − z / λ ' ))dν '
∫
ν ν
(5.1.2)
−Δ
exp(-i2Π(νt-z/λ)) exp(i2Π(νt-z/λ))
V (z,t) = A(z,t)exp(i2Π(νt − z / λ))
(5.1.3)
ν + Δν
A( z, t ) =
a(ν ' ) exp(i 2Π((ν '−ν )t − z (1 / λ '−1 / λ )))dν '
∫
ν ν
−Δ
(5.1.4)
37
An introduction to optical/IR interferometry
■ 
■ 
5 Light coherence
5.1 Quasi monochromatic light (waves and wave
groups)
1/ν
1/Δν
t
z
λ
λ2 / Δλ
38
An introduction to optical/IR interferometry
■ 
■ 
5 Light coherence
5.1 Quasi monochromatic light (waves and wave
groups)
λ0 = 2.2µm
λ [2.07 ; 2.33]µm
Δλ = 0.13µm
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An introduction to optical/IR interferometry
■ 
■ 
5 Light coherence
5.2 Fringe visibility
I q = 〈V q (t )V q (t )〉 (5.2.1)
∗
P1
S
*
P2
V
q
V
q
(t ) = V 1 (t − t1q) + V 2 (t − t 2q)
V1(t)
(5.2.2)
Iq ?
+
+
+q
V2(t)
(t ) = V 1 (t ) + V 2 (t − τ )(5.2.3)
τ = t 2q − t1q (5.2.4)
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An introduction to optical/IR interferometry
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■ 
I
q
5 Light coherence
5.2 Fringe visibility
= I + I + 2 I Re{γ 12 (τ )}
γ
(τ ) = 〈V 1 (t )V 2 (t − τ )〉 / I
∗
12
(5.2.5)
γ
(5.2.6)
(τ ) = 〈 A1 ( z , t ) A2 ( z , t − τ )〉 exp(−i 2Πντ ) / I
∗
12
(5.2.7)
If
τ 〈〈1 / Δν
γ
12
(τ ) = γ
12 (τ =0) exp(i β 12 −i2Πντ )
(5.2.8)
41
An introduction to optical/IR interferometry
■ 
■ 
5 Light coherence
5.2 Fringe visibility
I
€
q
(β
= I + I + 2 I γ12 (0) cos
( I max − I min
υ = &&
' I max + I min
%
## = γ (0)
12
$
12
− 2Πντ
)
(5.2.9)
(5.2.10)
42