2012-12-13
Prof. Herbert Gross
Friedrich Schiller University Jena
Institute of Applied Physics
Albert-Einstein-Str 15
07745 Jena
Exercise Solutions
Lecture Imaging and aberration Theory – Part 5
Exercise 5-1: Triplet System
A classical triplet system consists of two positive lenses L1, L3 and one negative lens L2 in the
sequence (+) (-) (+) as shown in the figure.
f1
f3
f2
It is assumed, that the lenses are thin and can be considered as close together. The outer positive
lenses are identical and of reversed orientation. They are made of a glass with high index n1 = n3 =
1.8 and the negative lens is made of a low index glass with n2 = 1.5 . Calculate the individual focal
lengths of the lenses, if the total focal length of the system is f' = 100 mm and the field of view is
flattened according to the Petzval theorem.
What can be done to control and correct the spherical aberration of the system ? If the negative lens
is exactly in the middle position between the positive lenses, why can be seen coma and distortion,
although the setup is symmetrical ? Why is the spherical aberration contribution of the identical outer
surfaces not equal ?
Exercise 5-2: Field flattening lens
Calculate a thick meniscus shaped lens with refractive index n = 1.5 with vanishing Petzval curvature
contribution, a focal length of f = 100 mm and a front radius of r1 = +10 mm. The back radius and the
thickness is to be determined.
Exercise 5-3: Galilei Telescope
1/2
2
A Galilean telescope consists of an afocal combination of a positive and a negative lens. The system
transforms the diameter of a collimated incoming beam by a magnification factor . The magnification
can be expressed by the focal lengths of the lenses as

f '
f '
(see exercise 10). What is the Petzval condition for a flattened field of view for this system ? Derive
the formula for the magnification for a Galilei telescope with flat field. If the available glass materials
have refractive indices in the range n = 1.45...2.0, what is the achievable range of magnifications ?
Exercise 5-4: Achromate and field flattening
Design an achromate with the focal length f = 50 mm with the crown glass PK2 for the positive lens in
front and the flint glass SF6 for the negative lens. The cemented surface should be plane, the
wavelengths should be d (578nm), C (656nm) and F(486nm).
a) Calculate the focal length of both lenses and the radii of curvature of the outer surfaces.
b) Calculate the residual chromatical error (secondary spectrum) as a difference of the image location
between the green and the blue/red. Calculate the primary and secondary chromatical aberration of a
simple lens of K5 for comparison. What is the improvement of the chromatical difference in the
intersection length ?
c) Compare the curvature of the image shell of Petzval for the achromate and the single lens. Which
is the system with the better performance ? Why is it impossible to flatten the achromate in this setup
?
The data of the materials are:
Glas
SF6
K5
PK2
486 nm
1.82775
1.52860
1.52372
587 nm
1.80518
1.52249
1.51821
656 nm
1.79609
1.51982
1.51576

25.4
59.5
65.05
Exercise 5-5: Derivation of the Coddington Equations
Derive the two Coddington equation for the astigmatic image locations.
Hint: The sagittal case can be obtained by considering an auxiliary axis through the center of
curvature of the surface, for the tangential case a single ray trace must be performed for two ray with
infinitesimal distance in the height at the surface.