A6525: Lecture - 01
Elementary Optics
Astronomy 6525
Lecture 01
Outline
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The Perfect Telescope
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Plate scale
The HST blunder
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Diffraction-limited performance
Launched 1990, Fixed 1993
Simple optics
Telescope design
Types of telescopes
Ray Tracing
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A6525: Lecture - 01
What is a telescope?
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Forms images of a distance object:
Key parameters:
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Collecting area
Effective focal length (f/# at focal plane)
Related parameters
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Plate scale (e.g. arcseconds/mm)
Image quality
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Geometric aberrations
Diffraction
Sensitivity: signal-to-noise ratio on a source
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Highly dependent upon instrument
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The Perfect Telescope
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Collects photons with 100%
transmission
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No obscurations
Zero thermal emission and zero scattered
light
No geometrical aberrations
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But diffraction always present =>
no “point” sources
Called diffraction-limited performance
FWHM ≅
1.03λ
D
θD ≅
Elementary Optics
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1.2λ
D
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Obscured Telescope
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For an obscured telescope the PSF, normalized to a peak of
unity is given by:
,
Here
=
2
−
1
1−
2
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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A6526 - Lecture 1
Diffraction Point Spread Function
Obscuration
None
0.06
0.8
Amplitude
4%
Amplitude
0.04
0.6
10 %
0.4
0.2
0.02
0
0.5
1
1.5
x
0.0
0
1
2
3
x (λ/D)
4
5
6
Obscuration is by area
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Diffraction PSF
100
No obscuration
Amplitude
10-1
4%
10-2
10 %
10-3
10-4
10-5
0
1
2
3
x (λ/D)
4
5
6
Obscuration is by area
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Encircled Energy
1.0
Encircled Energy
0.8
No obscuration
4%
0.6
10 %
0.4
0.2
0.0
0
Elementary Optics
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2
3
x (λ/D)
8
4
5
6
Obscuration is by area
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Gaussian vs. Airy Function
y
100
10-1
Dsec/Dpri = 0
Dsec/Dpri = 0.2
(4% obscuration)
Amplitude
10-2
10-3
10-4
10-5
10-6
10-7
0
2
4
6
8
10
x (λ/D)
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Plate Scale
f
θ
f #=

x
θ
effective focal length
primary diameter
f = focal length
f # = f/D
x = θ f = θ D f#
For Palomar: f/16 with 5 m primary
1"
x=
16 ⋅ 5 × 103 mm = 0.388 mm
206265" / rad
 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
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HST: $2.5 billion and the optics were wrong!
Very bad PR for NASA and Astronomy
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HST Primary Figuring Error
Sphere
194 μm
Paraboloid
2 μm
1/2 μm
Designed
Hyperboloid
Actual Hyperboloidal Mirror
Spheriod
Focus is different for
different height light rays
Elementary Optics
Paraboloid
Paraboloid
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Focus is the same for
different height light rays
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HST Encircled Energy
Pre-fix HST performance.
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HST Spot Diagrams
As designed
Actual (pre-fix)
0.2"
2"
Diffraction spot at 0.5 μm
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HST PSF Plots
Profiles of HST f/30 planetary camera normalized to the
same peak brightness for λ = 0.57 μm. The FWHM of the
core is 0.1” in both cases, but only 15% is contained in the
spherically aberrated image core.
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Optics
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Motivation
Thin lens
Telescopes
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Mixing conic sections
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Motivation
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Why should we know about optics?
User viewpoint (observer)
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Builder’s viewpoint (experimenter)
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You will get better results if you know how your
experiment works and what its limitations are.
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)
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You can make more money!
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Thin Lens Equations
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Thin Lens Formula:
1 1 1
− =
q p f
q
p
f = focal length
p = object distance
(neg. when to left of lens)
q = image distance
(pos. when to right of lens)
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f
Newton’s Formula:
f
x⋅ y = f 2
x
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f
y
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Thin Lens Equations (cont’d)
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Lensmaker’s Formula:
1 1 
1
= (n − 1)  − 
f
 r1 r2 
n
= refractive index
r1, r2 = radii of curvature
For a
convex lens:
r2
r1
( r>0
) r<0
r1 > 0, r2 < 0 => f > 0
concave lens: r1 < 0, r2 > 0 => f < 0
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Telescopes
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Refractors
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Chromatic aberration
Must be internally flaw free
Must support from the side
Reflectors (astronomers choice)
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Typically have central obscuration
Have spiders to support secondary (diffraction spikes)
Object
Image
Object not at infinity!
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The Parabolic Mirror
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Consider light from a very distant spot on the optical axis
d
c
A
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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)
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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
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Ellipse (e < 1):
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Hyperbola (e > 1):
reflected
ray
P
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Sphere (e = 0):
F
F′
tangent
line bisects
angle
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Gregorian Telescope
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Conic sections produce perfect (geometric) images and can
be strung together to form complex systems.
Focus #2
Focus #1
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Parabolic primary produces a perfect image at #1.
Ellipsoidal secondary transfers a perfect image to #2.
An erect image is produced.
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Cassegrain Telescope
Focus #2
Focus #1
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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
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Start by picking the aperture and final focal length.
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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
Relational equations:
q
m +1
m=
es =
p
m −1
1 1
fs =  − 
 p q
p, q > 0
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−1
rs = 2 f s
(convex)
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Designing a Cassegrain (cont’d)
b = back
focal distance
(> 0 as shown)
s
q = s+b
p = fp − s
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&
m=
b
q
p
s=
mf p − b
m +1
Effective focal length = focal length of telescope
feff = fpm
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Cassegrain Examples
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f/2.2 primary, f/13.4 telescope
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f/1.3 primary, f/13.4 telescope
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magnification: 10.3
plate scale:
15.4/D(m) ′′/mm
f/2.2 primary, f/4.6 telescope
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Elementary Optics
magnification: 6
plate scale:
15.4/D(m) ′′/mm
magnification: 10.3
plate scale:
44.8/D(m) ′′/mm
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Other Telescope designs
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Dall-Kirkham:
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Make secondary mirror a sphere
Adjust “figure” of primary to compensate (remove
spherical aberration)
Bad off-axis performance
Ritchey-Chrétien Telescope
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Design used for all large telescopes
Reduce off-axis aberrations by
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Slightly flattening primary (hyperbolic)
slightly flattening rim of secondary (hyperbolic)
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Schematics of Telescopes
Herschelian
Newtonian
Keplerian
Gregorian
Mersenne
Cassegrain,
Ritchey-Chrétien,
Dall-Kirkham
Schmidt
Bouwers-Maksutov
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Telescope Types
Type
Keplerian
Herschelian
Newtonian
Gregorian
Mersenne
Cassegrain
Ritchey-Chrétien
Dall-Kirkham
Schmidt
Bouwers-Maksutov
Elementary Optics
Primary Optics
Sphere or parabola
Off-axis parabola
Parabola
Parabola
Parabola
Parabola
Modified parabola
Ellipse
Aspheric refractor
Refractive meniscus
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Secondary Optic
None
None
Diagonal Flat
Ellipse
Parabola
Hyperbola
Modified hyperbola
Sphere
Sphere
Sphere
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