From the pinhole camera … to the telescope and … from the telescope … to the optical interferometer
Pinhole
Camera obscura (cf. shoe box)
1 cm Angle = 1/114 radian
Projected image of the Sun Angle = 1/114 radian
114 cm
5 cm
5,7m = 5 x 114 cm
H
H S
D
The blur B of the hearts is proportional to H, and inversely proportional to S, which is itself propotional to D è B ÷ H / D
The angular resolution Φ = H / D
H
D
The surface brightness SB of the hearts is proportional to H2, and inversely proportional to S2, and thus to D2 è SB ÷ (H / D)2
H S
Quiz (?)
??? ???
I(ζ,η) = Heart(ζ ',η ') Solar disk(ζ −ζ ',η −η ') dζ 'dη '∫∫I(ζ,η) = Solar disk(ζ ',η ')Heart(ζ −ζ ',η −η ') dζ 'dη '∫∫
9 Fourier Optical Elements 9.3 Telescope applications
■ 9.3.1 Optical telescopes ■ 9.3.2 Coupled telescopes ■ 9.3.3 X-ray telescopes ■ 9.3.4 Radio-telescopes and radio-interferometers
9 Fourier Optical Elements 9.3 Telescope applications 9.3.1 Optical telescopes
(a)
(b) (c)
1 simple lens
1 double lens
9 Fourier Optical Elements 9.3 Telescope applications 9.3.1 Optical telescopes
Fobj Foc
αobj αoc
φ = E S = π D2 E / 4, f = F / D, E’ ~ f-2. G = αoc / αobj = Fobj / Foc.
9 Fourier Optical Elements 9.3 Telescope applications 9.3.1 Optical telescopes
Newton focus (a) and Primary focus (b) of optical telescopes
(a) (b)
9 Fourier Optical Elements 9.3 Telescope applications 9.3.1 Optical telescopes
(a) (b) Observers in the Prime focus cage
(c)
9 Fourier Optical Elements 9.3 Telescope applications 9.3.1 Optical telescopes
(a) Cassegrain, Ritchey-Chrétien or Strand foci of an optical telescope and (b) Coudé focus
(a) (b)
9 Fourier Optical Elements 9.3 Telescope applications 9.3.1 Optical telescopes
(a)
(b)
(a) Optical components of a Schmidt telescope and (b) big Schmidt telescope of Mont Palomar
9 Fourier Optical Elements 9.3 Telescope applications 9.3.1 Optical telescopes
ESO VLT (Paranal, Chile) See the URL: http://www.eso.org
9 Fourier Optical Elements 9.3 Telescope applications 9.3.1 Optical telescopes http://www.stsci.edu/
(a) (b)
VLTI
9 Fourier Optical Elements 9.3 Telescope applications
■ 9.3.1 Optical telescopes ■ 9.3.2 Coupled telescopes ■ 9.3.3 X-ray telescopes ■ 9.3.4 Radio-telescopes and radio-interferometers
9 Fourier Optical Elements 9.3 Telescope applications 9.3.3 X-ray telescopes
(a) X-ray telescope with double grazing reflection (P: paraboloid; H: hyperboloid); (b) simple paraboloid; (c) X-ray mirror of the Einstein satellite and (d) Cellular collimator.
(a)
(b)
(c)
(d)
9 Fourier Optical Elements 9.3 Telescope applications 9.3.3 X-ray telescopes
(a) (b) See the URLs: http://sci.esa.int/xmm/ and http://xmm.vilspa.esa.es/
9 Fourier Optical Elements 9.3 Telescope applications 9.3.3 X-ray telescopes
Entrance pupil P(x,y) and response function |h(ρ‘)|2 for an X-ray telescope (grazing light ray incidence).
9 Fourier Optical Elements 9.3 Telescope applications 9.3.4 Radio-telescopes and radio-interferometers
(a) Effelsberg radio-telescope (F: primary focus; F’: Gregory focus). (b) and (c) Photographs
(a) (b) (c)
9 Fourier Optical Elements
9.3 Telescope applications
9.3.4 Radio-telescopes and radio-interferometers
The Green Bank 300ft (West Virginia) … before and after 15/11/1988!
9 Fourier Optical Elements 9.3 Telescope applications 9.3.4 Radio-telescopes and radio-interferometers
(a), (b) and (c) : Photographs of the new Green Bank radio-telescope
(a) (b) (c)
9 Fourier Optical Elements 9.3 Telescope applications 9.3.4 Radio-telescopes and radio-interferometers
(a) Nançay radio-telescope
(a) (b)
9 Fourier Optical Elements 9.3 Telescope applications 9.3.4 Radio-telescopes and radio-interferometers
Arecibo radio-telescope
9 Fourier Optical Elements 9.3 Telescope applications 9.3.4 Radio-telescopes and radio-interferometers
The 500-metre aperture spherical telescope (FAST) in China's Pingtang county.
9 Fourier Optical Elements 9.3 Telescope applications 9.3.4 Radio-telescopes and radio-interferometers
Antenna diagram of a radio-telescope. (a) Excitation of the antenna for a point-like object inclined by an angle θ with respect to the central axis; (b) sensitivity of the mirror + antenna in polar coordinates (or graph of the response function in polar coordinates assuming that the antenna is point-like; (c) real diagram corresponding to λ / D ~ 1 / 10 and showing the back-side lobe B.
(a) (b) (c)
9 Fourier Optical Elements 9.3 Telescope applications 9.3.4 Radio-telescopes and radio-interferometers
Radio-interferometer: (a) Connection diagram; (b) Antenna diagram (graph in polar coordinates of the response function).
(a) (b)
~2
9 Fourier Optical Elements 9.3 Telescope applications 9.3.4 Radio-telescopes and radio-interferometers
( ) ( )2
2 2sin( ) 2 ( ) cos
pde p d O p pD
pdπ
ππ
⎛ ⎞⎡ ⎤= ∗⎜ ⎟ ⎣ ⎦
⎝ ⎠
( )2
2 sin 1 1( ) 2 ( ) ( ) cos(2 )
2 2R
pde p d O p dp O p pD
pdπ
ππ
⎛ ⎞ ⎡ ⎤= + ∗⎜ ⎟ ⎢ ⎥⎣ ⎦⎝ ⎠∫
( )1( ) Re ( ) exp(2 )
2e p A B O p i pDπ⎡ ⎤= + ∗⎢ ⎥⎣ ⎦
A= 2d 2sin π pd( )π pd
⎛
⎝
⎜⎜
⎞
⎠
⎟⎟
2
and B = 12
O( p)dpR∫
(9.3.10)
(9.3.11)
(9.3.12)
(9.3.13)
,
,
,
,
9 Fourier Optical Elements 9.3 Telescope applications 9.3.4 Radio-telescopes and radio-interferometers
⎥⎦
⎤⎢⎣
⎡ += )))((_()2cos(21
)( DrOTFpDBApe π
( )( )1( ) Re ( )exp 2 ( )
2 Re p A B O r i p r D drπ⎡ ⎤= + −⎢ ⎥⎣ ⎦∫
γ(D) = (emax - emin) / (emax + emin),
γ(D) = TF_(O(r))(D) / (2B) = TF_(O(r))(D) / ∫O(p) dp.
(9.3.14)
(9.3.15)
(9.3.16)
(9.3.17)
,
,
9 Fourier Optical Elements 9.3 Telescope applications 9.3.4 Radio-telescopes and radio-interferometers
(a) (b)
(c) (d) (e)
Radio-telescope interferometer (a) Connection diagram; (b) Entrance pupil; (c) Response function; (d) Antenna diagram; (e) Trace of the main lobe over the celestial sphere.
9 Fourier Optical Elements 9.3 Telescope applications 9.3.4 Radio-telescopes and radio-interferometers
∑∫∫−
=
+−−
+−−−−
−−=1
0
2)1(
2
2)1(
2
2/
2/0,)2exp()2exp(),(
N
n
naaNd
naaNd
d
drdxpxidyqyiqph ππ
∑∫−
=
+−−
+−−− −=
1
0
2)1(
2
2)1(
20,
)2exp()sin(
),(N
n
naaNd
naaNdrdxpxi
qdqd
dqph πππ
( )( ) ( )( )1
,00
exp ( 1) 2 exp ( 1) 2sin( )( , )
2
N
rn
i p d N a na i p d N a naqdp q dqd i ph
π πππ π
−
=
− + − − − + − −=
−∑
(9.3.18)
(9.3.19)
(9.3.20)
(9.3.21)
,
,
,
,
1
,00
( 1)2( , )
N
rn
ax N nayx y
d dP−
=
⎡ ⎤+ − −⎢ ⎥⎡ ⎤=Π Π ⎢ ⎥⎢ ⎥⎣ ⎦ ⎢ ⎥⎣ ⎦
∑
9 Fourier Optical Elements 9.3 Telescope applications 9.3.4 Radio-telescopes and radio-interferometers
( ) ( )1
2,0
0
sin( ) sin( )( , ) exp ( 1) exp 2
N
rn
qd pdh p q d i pa N i panqd pdπ π
π ππ π
−
=
⎛ ⎞⎛ ⎞= − −⎜ ⎟⎜ ⎟
⎝ ⎠⎝ ⎠∑
x = iπap and r = exp(-2x) ,
(9.3.22)
(9.3.23)
.
( )2,0
sin( ) sin( ) 1 exp( 2 )( , ) exp ( 1)
1 exp( 2 )rqd pd xN
h p q d x Nqd pd xπ ππ π
⎛ ⎞⎛ ⎞ − −= −⎜ ⎟⎜ ⎟ − −⎝ ⎠⎝ ⎠
(9.3.24) ,
2,0
sin( ) sin( ) exp( ) exp( )( , )
exp( ) exp( )rqd pd xN xN
h p q dqd pd x xπ ππ π
⎛ ⎞⎛ ⎞ − −= ⎜ ⎟⎜ ⎟ − −⎝ ⎠⎝ ⎠
2,0
sin( ) sin( ) sin( )( , )sin( )r
qd pd N pah p q dqd pd paπ π ππ π π
⎛ ⎞⎛ ⎞⎛ ⎞= ⎜ ⎟⎜ ⎟⎜ ⎟
⎝ ⎠⎝ ⎠⎝ ⎠
,
.
(9.3.25)
(9.3.26)
9 Fourier Optical Elements 9.3 Telescope applications 9.3.4 Radio-telescopes and radio-interferometers
.
(9.3.27)
⎟⎟⎠
⎞⎜⎜⎝
⎛⎟⎟⎠
⎞⎜⎜⎝
⎛⎟⎟⎠
⎞⎜⎜⎝
⎛⎟⎟⎠
⎞⎜⎜⎝
⎛=
)sin()sin()sin()sin(
)(),(2222
220,
2
papa
paNpaN
pdpd
qdqd
dNqphr ππ
ππ
ππ
ππ
Central peak + intermediate peaks
Response function of a single aperture
N apertures!
9 Fourier Optical Elements 9.3 Telescope applications 9.3.4 Radio-telescopes and radio-interferometers
Mills cross. (a) Disposition of the elements; (b) and (c) Traces over the celestial sphere of the antenna diagram when the two branches R1 and R2 are connected in phase (b) and in phase opposition (c); (d) trace over the celestial sphere of the difference diagram
(a) (b) (c) (d)
9 Fourier Optical Elements 9.3 Telescope applications 9.3.4 Radio-telescopes and radio-interferometers
PM,0(x,y) = Pr,0(x,y) + Pr,0(y,x), (9.3.28)
P’M,0(x,y) = Pr,0(x,y) - Pr,0(y,x). (9.3.29)
hM,0(p,q) = hr,0(p,q) + hr,0(q,p), (9.3.30)
h’M,0(p,q) = hr,0(p,q) - hr,0(q,p). (9.3.31)
9 Fourier Optical Elements 9.3 Telescope applications 9.3.4 Radio-telescopes and radio-interferometers |hc(p,q)|2 = |hM,0(p,q)|2 - |h’M,0(p,q)|2. (9.3.32)
|hc(p,q)|2 = 2[hr,0(p,q) h*
r,0(q,p) + h*r,0(p,q) hr,0(q,p)], (9.3.33)
|hc(p,q)|2 = 4 Re [hr,0(p,q) h*
r,0(q,p)]. (9.3.34)
. (9.3.35)
Interference terms
( ) ( ) ( )2 2
2 4 sin sin sin( ) sin( ), 4sin( ) sin( )c
pd qd N pa N qah p q dpd qd pa qaπ π π π
π π π π
⎛ ⎞ ⎛ ⎞= ⎜ ⎟ ⎜ ⎟
⎝ ⎠ ⎝ ⎠
9 Fourier Optical Elements
The LOFAR 'superterp'. View of the heart of the radio-interferometer located near Exloo, Holland (1,3 - 30m, 20 000 antenna, B ≈ 1000 km, 300 000 m2) .
9 Fourier Optical Elements
The 'SKA' (Square Kilometer Array) project: phased radio interferometer (Australia / South Africa, Base ≈ 3 000 km, 1 000 000 m2, 2024).
9 Fourier Optical Elements The Nançay
radio-helio- graphs (France)
See the URL: http://www.obs-nancay.fr/html_an/a_rh.htm
9 Fourier Optical Elements
Atacama Large Millimeter Array (Chile)
See the URL: http://www.eso.org/projects/alma/
View of North of Chile (space shuttle)
Geographic sites of the VLT and ALMA
9 Fourier Optical Elements
Atacama Large Millimeter Array (Chile)
See the URL: http://www.eso.org/projects/alma/
Aerial view of the ALMA site
Panoramic view of the ALMA site in Chajnantor
9 Fourier Optical Elements
Atacama Large Millimeter Array (Chile)
See the URL: http://www.eso.org/projects/alma/
Futuristic model of ALMA
9 Fourier Optical Elements
Atacama Large Millimeter Array (Chile)
See the URL: http://www.eso.org/projects/alma/
Futuristic models of ALMA
9 Fourier Optical Elements
Atacama Large Millimeter Array (Chile)
See the URL: http://www.eso.org/projects/alma/
9 Fourier Optical Elements
Atacama Large Millimeter Array (Chile)
See the URL: http://www.eso.org/projects/alma/