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Elementary Optics
Astronomy 6525
Lecture 01
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Outline
The Perfect Telescope Diffraction-limited performance
Plate scale
The HST blunder Launched 1990, Fixed 1993
Simple optics
Telescope design
Types of telescopes
Ray Tracing
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What is a telescope?
Forms images of a distance object:
Key parameters: Collecting area
Effective focal length (f/# at focal plane)
Related parameters Plate scale (e.g. arcseconds/mm)
Image quality Geometric aberrations
Diffraction
Sensitivity: signal-to-noise ratio on a source Highly dependent upon instrument
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The Perfect Telescope Collects photons with 100%
transmission No obscurations Zero thermal emission and zero scattered
light
No geometrical aberrations But diffraction always present =>
no “point” sources Called diffraction-limited performance
DFWHM λ03.1≅
DDλθ 2.1≅
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Obscured Telescope For an obscured telescope the PSF, normalized to a peak of
unity is given by:
, = 2 − 2 11 −Here is the first order Bessel function of the first kind, = and =where and are the telescope and obscuration diameters respectively. Nominally is entered in units of / .
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0 0.5 1 1.5
0.2
0.4
0.6
0.8
x
Am
plit
ude
Diffraction Point Spread FunctionObscuration
10 %
4 %
0 1 2 3 4 5 6
0.06
0.04
0.02
0.0
Am
plit
ude
x (λ/D)
Obscuration is by area
None
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Diffraction PSF
No obscuration
10 %4 %
0 1 2 3 4 5 6
100
10-1
10-2
10-3
10-4
10-5
Am
plit
ude
x (λ/D)
Obscuration is by area
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Encircled Energy
No obscuration
10 %
4 %
0 1 2 3 4 5 6
1.0
0.8
0.6
0.4
0.2
0.0
Enc
ircl
ed E
nerg
y
x (λ/D)Obscuration is by area
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yGaussian vs. Airy Function
0 2 4 6 8 10
100
10-1
10-2
10-3
10-4
10-5
Am
plit
ude
x (λ/D)
10-6
10-7
Dsec/Dpri = 0.2 (4% obscuration)
Dsec/Dpri = 0
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Plate Scale
For Palomar: f/16 with 5 m primary
θ θx
f
diameterprimarylengthfocaleffectivef =#
f = focal lengthf # = f/D
x = θ f = θ D f #
mm388.0mm10516rad/"206265
"1 3 =×⋅=x
1” ↔ 0.388 mm Plate scale = 2.57”/mm in telescope focal plane
[Often reimaged to match detector pixel size.]
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When telescopes go bad HST: $2.5 billion and the optics were wrong! Very bad PR for NASA and Astronomy
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HST Primary Figuring Error
Actual Hyperboloidal Mirror
194 μm
2 μm
DesignedHyperboloid
1/2 μmSphere
Paraboloid
Focus is different for different height light rays
Focus is the same for different height light rays
Spheriod Paraboloid
Paraboloid
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HST Encircled Energy
Pre-fix HST performance.
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HST Spot Diagrams
0.2"
2"
Diffraction spot at 0.5 μm
As designed Actual (pre-fix)
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HST PSF Plots
Profiles of HST f/30 planetary camera normalized to thesame peak brightness for λ = 0.57 μm. The FWHM of thecore is 0.1” in both cases, but only 15% is contained in thespherically aberrated image core.
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Optics
Motivation
Thin lens
Telescopes Mixing conic sections
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Motivation
Why should we know about optics?
User viewpoint (observer) You will get better results if you know how your
experiment works and what its limitations are.
Builder’s viewpoint (experimenter) If you have someone design a system and build it for
you, there is little incentive for them to keep it simple (and cheap).
Pragmatists viewpoint (wage earner) You can make more money!
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Thin Lens Equations Thin Lens Formula:
fpq111 =−
p q
f
f = focal lengthp = object distance
(neg. when to left of lens)q = image distance
(pos. when to right of lens)
Newton’s Formula:2fyx =⋅
x y
ff
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Thin Lens Equations (cont’d) Lensmaker’s Formula:
( )
−−=
21
111
1
rrn
f
n = refractive indexr1, r2 = radii of curvature
r1 r2
( r > 0) r < 0
For a convex lens: r1 > 0, r2 < 0 => f > 0
concave lens: r1 < 0, r2 > 0 => f < 0
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Telescopes
Refractors Chromatic aberration
Must be internally flaw free
Must support from the side
Reflectors (astronomers choice) Typically have central obscuration
Have spiders to support secondary (diffraction spikes)
Object Image
Object not at infinity!
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The Parabolic Mirror Consider light from a very distant spot on the optical axis
A
cd
A parallel wavefront passes A in phase. We want it to arrive at the focus still in phase. Therefore, all paths from A to the focus must be the same length.
A parabola is the locus of points equidistant from a point and a line. Therefore, c = d and the distance from A to the focus is a constant.
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Parabolic Mirror (cont’d)
A parabola will form a perfect (geometric) image at the focus.
NOTE: This is only for rays parallel to the axis. Off-axis rays will not be as good.
Rays from off-axis source
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Other Conics
Ellipse (e < 1):
Sphere (e = 0):
Hyperbola (e > 1):
tangentline bisectsangle
F′ F
reflected ray
P
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Gregorian Telescope Conic sections produce perfect (geometric) images and can
be strung together to form complex systems.
Parabolic primary produces a perfect image at #1.
Ellipsoidal secondary transfers a perfect image to #2.
An erect image is produced.
Focus #1Focus #2
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Cassegrain Telescope
Focus #1Focus #2
Parabolic primary produces a perfect image at #1.
Hyperbolic secondary relays the virtual image at #1 to a real image at #2.
Greater compactness than Gregorian telescope.
But - hyperbolic secondary is hard to make and off-axis performance is not terribly good.
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Designing a Cassegrain Telescope Start by picking the aperture and final focal length.
This gives f-ratio (f#) and plate scale
The final focal length is fpm where m = magnification produced by the secondary. fp is the primary focal length. m = 1 for flat.
p q
pqm =
1
1
−+=
mmes
111
−
−=
qpfs ss fr 2=
Relational equations:
p, q > 0 (convex)
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Designing a Cassegrain (cont’d)
Effective focal length = focal length of telescope
feff = fpm
s b
b = backfocal distance(> 0 as shown)
bsq +=
sfp p −= pqm =
1+−
=m
bmfs p&
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Cassegrain Examples
f/2.2 primary, f/13.4 telescope magnification: 6 plate scale: 15.4/D(m) ′′/mm
f/1.3 primary, f/13.4 telescope magnification: 10.3 plate scale: 15.4/D(m) ′′/mm
f/2.2 primary, f/4.6 telescope magnification: 10.3 plate scale: 44.8/D(m) ′′/mm
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Other Telescope designs
Dall-Kirkham: Make secondary mirror a sphere
Adjust “figure” of primary to compensate (remove spherical aberration)
Bad off-axis performance
Ritchey-Chrétien Telescope Design used for all large telescopes
Reduce off-axis aberrations by Slightly flattening primary (hyperbolic)
slightly flattening rim of secondary (hyperbolic)
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Schematics of TelescopesNewtonian
Keplerian
Herschelian
Cassegrain,Ritchey-Chrétien,Dall-Kirkham
Mersenne
Gregorian
Schmidt Bouwers-Maksutov
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Telescope TypesType Primary Optics Secondary Optic
Keplerian Sphere or parabola None
Herschelian Off-axis parabola None
Newtonian Parabola Diagonal Flat
Gregorian Parabola Ellipse
Mersenne Parabola Parabola
Cassegrain Parabola Hyperbola
Ritchey-Chrétien Modified parabola Modified hyperbola
Dall-Kirkham Ellipse Sphere
Schmidt Aspheric refractor Sphere
Bouwers-Maksutov Refractive meniscus Sphere