Basic Telescope Optics
Andreas Quirrenbach
Landessternwarte
Universität Heidelberg
Andreas Quirrenbach Basic Telescope Optics 2
Optics and Telescopes
M. Born, E. Wolf, Principles of Optics
P. Léna, F. Lebrun, F. Mignard, Observational
Astrophysics
D.J. Schroeder, Astronomical Optics
R.R. Shannon, The Art and Science of Optical
Design
M.J. Kidger, Fundamental Optical Design
R.N. Wilson, Reflecting Telescope Optics I / II
Andreas Quirrenbach Basic Telescope Optics 3
Refraction at a Spherical Interface
Sign convention: all angles and distances in this diagram are positive
Andreas Quirrenbach Basic Telescope Optics 4
Basics of Paraxial Optics
Paraxial approximation: y and all angles are small
Law of refraction: nsin i = n'sin i', in paraxial approximation ni = n'i'
Points at distances s and s' from vertex are called conjugate points (image is conjugate to object)
If s or s' = , the conjugate distance is called focal length
Andreas Quirrenbach Basic Telescope Optics 5
Conjugate Points in the Paraxial
Region
B and B', Q and Q' are pairs of conjugate points
Transverse magnification: m = h'/h
Andreas Quirrenbach Basic Telescope Optics 6
Angular Magnification
Angular magnification: M = tan u' / tan u = s / s'
Andreas Quirrenbach Basic Telescope Optics 7
Power, Magnification, Lagrange
Invariant
Definition of power:
Transverse magnification:
Angular magnification:
Lagrange invariant:
In paraxial approximation:
fn
fn
Rnn
sn
snP
'''
''
snns
RsRs
hhm
''''
''''tan'tan
hnnh
mnn
ss
uuM
'tan''tan uhnunhH
''' uhnnhuH
Andreas Quirrenbach Basic Telescope Optics 8
Reflection at a Spherical Surface
Setting n' = 1 for reflection gives unified formulae for lenses and
mirrors
Andreas Quirrenbach Basic Telescope Optics 9
Basic Relations for Simple Optical
Systems
Power of two-surface system (thick lens, two-
mirror telescope):
Thin lens (d = 0):
Image scale:
Image size:
Focal ratio: F = f / D
Systems with small focal ratio (e.g., f / 1.5) are
called “fast” those with large focal ratio “slow”
2121 PPPPPnd
21
1121 )1(
RRnPPP
mm206265mm"f
S
"m86.4μm fx
Andreas Quirrenbach Basic Telescope Optics 10
Two-Mirror Reflecting Telescopes
(a) Cassegrain
(b) Gregorian
Andreas Quirrenbach Basic Telescope Optics 11
Normalized Parameters for Two-
Mirror Telescopes
Andreas Quirrenbach Basic Telescope Optics 12
Important Relations for Two-
Mirror Telescopes
Apply standard formulae to the secondary:
Solve for m, ρ, and k in turn:
Other relations:
221122
11112221ms
k
skRkRRs
m
mk
m
mk
km
1,
1,
1111 ,1 FFffmmPkPP
1,11 Fmk
Andreas Quirrenbach Basic Telescope Optics 13
Fermat's Principle
The optical path length of an actual ray between
any two points P0 and P1 is shorter than the
optical path length of any curve which joins
these points and lies in a neighborhood of it
Formulation as variation principle:
In (y,z) plane:
“Lagrange equation” for Fermat’s Principle:
0dsn
1
0
1
0
0,,1),( 2P
P
P
PdzzyyFdzyzyn
0
y
F
dz
d
y
F
Andreas Quirrenbach Astronomische Waarneemtechnieken 1 14
Rays between Conjugates at Finite
Distances via Convex Reflector
Andreas Quirrenbach Basic Telescope Optics 15
Derivation of Shape for Convex
Reflector (Finite Object Distance)
Fermat’s Principle:
From previous figure:
Some algebra:
Using , and defining :
(hyperbola, since ss' < 0)
sll 2
zsyl
ssdlsyd
,
,,
222
222
044 2
22
ss
ssss
ss zzy
2R
ssss
2421
ss
sse
012 222 zeRzy
Andreas Quirrenbach Basic Telescope Optics 16
Conic Sections General description:
Define conic constant:
• Oblate ellipsoid: K > 0
• Sphere: K = 0
• Prolate ellipsoid: 1 < K < 0
• Paraboloid: K = 1
• Hyperboloid: K < 1
012 222 zeRzy2eK
Andreas Quirrenbach Basic Telescope Optics 17
Definition of Sagittal (Dashed)
and Tangential (Continuous) Rays
tangential:
in plane
x = 0
sagittal:
out of plane
y = 0 in pupil plane
Andreas Quirrenbach Basic Telescope Optics 18
Ray Diagram for a Lens Showing
Spherical Aberration
Andreas Quirrenbach Basic Telescope Optics 19
Spot Diagrams through Focus for
Lens with Spherical Aberration
circle of least
confusion
Andreas Quirrenbach Basic Telescope Optics 20
Behavior of Rays in the Presence
of Astigmatism
Andreas Quirrenbach Basic Telescope Optics 21
Spot Diagrams through Focus for
Lens with Astigmatism
Andreas Quirrenbach Astronomische Waarneemtechnieken 1 22
Behavior of Rays in the Presence
of Coma
Andreas Quirrenbach Astronomische Waarneemtechnieken 1 23
Spot Diagrams through Focus for
Lens with Coma
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Ray from Distant Object Reflected
by Concave Mirror
Andreas Quirrenbach Basic Telescope Optics 25
Focal Length for Rays at Distance
r from Axis
From the geometry on the previous viewgraph:
For conic sections:
Inserting the second formula into the first:
tan2
tan1
2tan0
2
rrz
zKRr
drdzzKRzr
1
22 tan012
zKRrzKRzzf
122
1
20
2
Andreas Quirrenbach Basic Telescope Optics 26
Power Series
Power series for z and f from binomial series:
5
6
3
42
2
2
16
2
82
2/1
1
22
11
111
012
R
r
R
rR
r
R
rK
R
KK
Kz
zKRzr
3
42
16
31
4
1
2 R
rKK
R
rKRf
Andreas Quirrenbach Basic Telescope Optics 27
Transverse Spherical Aberration
at the Paraxial Focus
Andreas Quirrenbach Basic Telescope Optics 28
Transverse and Angular Spherical
Aberration
From the figure on the previous viewgraph:
Power series expansion:
Corresponding angular aberration:
zfr
fTSA
53
3131 4
5
2
3
82
TSATSA
KKKTSAR
r
R
r
323
3
133 FKTSAASAR
rR
Andreas Quirrenbach Basic Telescope Optics 29
Higher-Order Aberrations
From the formula on the previous page:
For a sphere with F = 1.19, TSA5 is 10% of
TSA3
Higher-order aberrations are even less
important for slower systems
In most cases considering third-order
aberrations is sufficient
22
2
64
33
4
33
35
F
K
R
rK
TSATSA
Andreas Quirrenbach Basic Telescope Optics 30
Path of Arbitrary Ray through
Refracting Surface
Q and Q' lie in the yz plane; B is on the surface
the chief ray passes through the origin
Andreas Quirrenbach Basic Telescope Optics 31
Optical Pathlength through
Refracting Surface
coscos
cos'cos
sinsin
2cos1
2cos1118
coscossincoscossin2
cos1sincos1sin2
coscos2
coscoscoscos2
2
4
223
2
2
222
nn
nn
nnysnnsOPL
nnb
Rssn
Rssn
RK
sn
sn
R
r
Rssn
Rssny
Rssn
Rssnyx
Rnn
sn
snx
Rnn
sn
sny
astigmatism
spherical
aberration
coma
= OPL (chief ray)
= 0 (Snell’s Law)
= 0 for tangential astigmatic image
= 0 for sagittal astigmatic image
Andreas Quirrenbach Basic Telescope Optics 32
Structure of Optical Path
Difference
Define Φ as optical path difference to chief ray:
From Φ one can compute the aberrations
• |TAS| = half-length of astigmatic line image =
diameter of astigmatic blur circle
• 3|TSC| = length of comatic flare = 1.5 width of
comatic flare
• |TSA| = radius of blur at paraxial focus = 2
diameter of circle of least confusion
• TDI = distortion
4
3
2
2
3
2
2
1
2
10 rAyxAyAxAyAyA
Andreas Quirrenbach Basic Telescope Optics 33
Third-Order Transverse
Aberrations for a Mirror Surface
Andreas Quirrenbach Basic Telescope Optics 34
Aberrations of a Paraboloid
Mirror in Collimated Light (m=0)
ASA = 0
ASC = θ / (16 F 2)
AAS = θ2 / (2F)
As we know, a paraboloid mirror images an on-
axis object perfectly (no spherical aberration)
The useable field size is given by coma and
astigmatism
The field size is larger for slower mirrors
Andreas Quirrenbach Basic Telescope Optics 35
Angular Aberrations of Paraboloid
Mirror
Andreas Quirrenbach Basic Telescope Optics 36
Two-Mirror Telescopes In the design of a two-mirror telescope, one can
choose the conic constants K1, K2 of the
primary and secondary such that there is no
spherical aberration
One solution is choosing K1 and K2 such that
each mirror produces a perfect on-axis image • K1 = 1 (paraboloidal primary)
• Hyperboloidal secondary
This is called a Classical Cassegrain Telescope
Andreas Quirrenbach Basic Telescope Optics 37
Aberration Coefficients for Two-
Mirror Telescopes
Andreas Quirrenbach Basic Telescope Optics 38
Angular Aberrations for Two-
Mirror Telescopes
(previous viewgraph)
Andreas Quirrenbach Basic Telescope Optics 39
Angular Aberrations of Classical
Cassegrain Telescopes Secondary is hyperboloid:
Coma is the same as for a single paraboloid
Astigmatism is about m times worse, but
usually still smaller than coma
κ = field curvature
2
11
2
mmK
Andreas Quirrenbach Basic Telescope Optics 40
Ritchey-Chrétien Telescopes
The choice of the conic constants to eliminate
both spherical aberration and coma gives a
large useable field
Many modern telescopes (e.g., Keck, VLT,
HST) have a Ritchey-Chrétien design
Both primary and secondary are hyperboloids
321
122
11
2
12
1 ,1
mm
mm
mm
mmKK
Andreas Quirrenbach Basic Telescope Optics 41
Ray Tracing Software
Optical systems are usually designed with the
help of ray tracing software
These packages allow the user to define an
optical system, trace rays through the optical
system, and provide output for a detailed
analysis
The most commonly used ray tracing packages
are Code V and Zemax
Andreas Quirrenbach Basic Telescope Optics 42
OSLO EDU
Sinclair Optics, the developers of the OSLO ray
tracing package, allow downloading of an
education version from their web page
This version is fully functional for systems with
up to ten surfaces
• This is not enough for a spectrograph or moderately
complicated lens design, but sufficient to analyze
most astronomical telescopes
All you need to know can be found at
http://www.sinopt.com