The VLT Interferometer - ESO

2. Signal-To-Noise Considerations
I will use the following notations:
Aperture Diameter
Equivalent Equal
Apertures Diameter:
Aperture Area:
Equivalent Equal
Apertures Area:
Diameter Airy Disk (radian)
( a is a constant):
Infrared emissivity:
Equivalent Equal
LARGE TELESCOPE
D
6
S
(1'
aiD
E
SMALL TELESCOPE
d
Apertures Emissivity: e e
Transmission: T == I-E t == I-e
Effective Focal Length: F f
f-ratio: FR = D /F fr=d/f
Now I will consider two ways of combining the light of the two telescopes. In Case I the Airy
disks are made equal in size in the combined focal plane. This implies equal {-ratios and
hence unequal effective focal lengths. In Case II the image scales are made equal implying
equal effective focal lengths and hence unequal Airy disk sizes. In Case I all of the light from
a point source is allowed to interfere in both telescopes, in Case II only a fraction (s/S) of
the light of the smallest telescope interferes with all of the light of larger telescope. Below
I compare the signals and noise in the two cases for E=e (for simplicity I have assumed a
constant intensity within the Airy disk and a zero intensity outside):
CASE I CASE II
Characteristics: Equal Size Airy Disks Equal hnage Scales
Equal f-ratios Equal Focal lengths
FR = fr = D IF = d/f F=f
Cannot Preserve Pupil Can Preserve Pupil
Configuration Configuration
Very small FOV Can have large FOV
TELESCOPE: LARGE SMALL LARGE SMALL
Linear diameter
Airy disk: a/FR a/FR a/FR a/fr
Point source
Brightness in
0:8 2 focal plane:
(a is a constant)
o:S 0: s 0: S /8
Fringe Visibility
inside a/FR: 2vsSI(8 + 8) 288/(82 + 8 2 Fringe signal:
Thermal Background
2aVSS
)
20:8
Brightness in
Focal Plane:
(13 is a constant)
f3F R2 + f3F R2 f3F R2 + f3fr2
s
aid
e
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