1
/
3
2012

05

24
Prof.
Herbert Gross
Friedrich Schiller
Universit
y
Jena
Institut
e
of Applied Physics
Albert

Einstein

Str 15
07745 Jena
Exercises
Lecture Design and Correc
tion of Optical Systems
–
Part 3
Exercise
13
: Law of Refraction by F
ermat Principle
Derive the law of refraction by the Fermat principle of the smallest optical path length between two
points above and below a plane interface.
Exercise 14
: Group velocity from Sellmeier Dispersion
If short pulses are propagated thro
ugh transparent glass materials, the dispersion of the glass
changes the pulse shape. The group velocity of a pulse is defined by
d
dk
v
gr
2
1
1
The spreading of the pulse is controlled by the dispersion coefficient
gr
v
dv
d
d
k
d
D
1
2
1
2
2
Derive
the expression of the group velocity and the dispersion coefficient as a function of the glass
dispersion n(
).
The refractive indices of the glass BK7 are given by the following table:
in [
m]
n
0.3650100 1.5362680
0.4046600 1.5302390
0.4358300 1.5266850
0.4800000 1.5228290
0.4861300 1.5223760
0.5460700 1.5187220
0.5875600 1.5168000
0.5892900 1.5167280
0.6328000
1.5150890
0.6438500 1.5147190
0.6562700 1.5143220
0.7065200 1.5128920
0.8521100 1.5098030
1.0139800 1.5073080
1.0600000 1.5066880
1.5296000 1.5009070
2
1.9701000 1.4949480
2.3254
000 1.4892120
Draw a diagrams of n(
), dn(
)/d
and d
2
n(
)/d
2
by numerical differentiation
for BK7
.
Calculate the
derivative values for the wavelength
e
= 0.54607
m by numerical approaches and alternatively with
the help of a Sellmeier represen
tation. Compare the values and discuss the accuracy of both
methods.
Calculate of the group velocity v
gr
and the dispersion coefficient D
at
e
= 0.54607
m for the most
reliable method.
Exercise 15
: Depth Magnification
The equation for the axial ma
gnification reads
2
'
m
n
n
m
z
Derive this equation by differentiation of the lens makers formula. Show that the so called Herrschel
parameter
2
u
n
z
H
is invariant under paraxial imaging conditions.
Exercise 16
: Anamorphic
Collimation
A semiconductor laser diode emits a beam with two different numerical apertures in the x

z and y

z
section respectively.
The numerical apertures are NA
x
= 0.5
(fast axis)
and NAy = 0.
15 (slow axis).
Furthermore
the beam shows an intrinsic ast
igmatism
, the source point of the
fast axis lies on the chip
surface, the source point of the slow axis lies in the bulk.
Calculate the focallength of a lens, which collimates the x

z

section onto a diameter of 5 mm. It is
assumed, that the system fulfille
s the sine condition. Draw the principle ray path in both sections for
this case. Discuss some possible options for getting a collimation with small spherical aberration.
What kind of component can be used to collimate the beam after the first lens in the
y

z section too ?
Draw a sketch of the necessary layout. How can the system be extended for getting in addition a
circular cross section ?
Exercise 17
: Design of a Telephoto System
If a system combines a positive and a negative lens in a distance t,
it is possible to move the principal
plane in front of the system. The focal length f of the system is then larger than the overall size
L
. The
so called telephoto factor is the ratio between overall length of the system (first lens until image plane)
to t
he focal length
f
L
k
.
Calculate the individual focal lengths of both lenses as a function of given f, k and distance t between
the two lenses.
3
Discuss the theoretical and a practical ranges of validity of these equations. What is
a usefu
l range for
k in practice
?
Exercise
1
8
: Lens Endoscope
The Helmholtz

Lagrange invariant is defined by
)
'
sin(
'
'
u
y
n
H
for systems with finite image
location.
Derive the invariant for afocal systems as a function of the pupil diameter
D
p
and
the field
angle w.
The invariant is important for the layout of endoscopic systems. An endoscopic system may have a
field angle of 40° in water with refractive index n = 1.34 and a pupil diameter of D
p
= 0.5 mm. Behind
the front objective, relay systems ar
e located in an endoscope with length L = 200 mm and field
lenses of 4 mm diameter. Calculate the aperture angle in the intermediate images of the relay
systems. If one relay systems are ideally symmetric and consists of thin objective lenses and two thin
plane

convex field lenses, how many relay modules are necessary to get the total length L ?
Calculate the focal length of the field lens components.
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