2012-11-12
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 3
Exercise 3-1: Astigmatic Wave Aberration
Consider a wave aberration with an astigmatic and an defocus part of the form


W ( x, y )  c20  x 2  y 2  c22  y 2
Calculate the transverse aberration in x and y for a radius of the reference sphere R = 100 mm.
Derive the general equation of the transverse aberrations in polar coordinates r, for a ring sampling
in the pupil by eliminating the angle . This equation gives the shape of a spot in the image plane.
What kind of figure is obtained ? Determine the values of the defocus coefficient c20, for which the
focal lines are obtained. What is the shape of the spot for a defocus in the middle between these two
focal lines ?
Exercise 3-2: Spherical Mirror
Derive the expression for the focal length of a spherical mirror. Calculate the formula of the axial
spherical aberration for a collimated incoming beam.
Exercise 3-3: Mirror Aberrations
A spherical mirror with radius r and object intersection length s has for a ray in the height y p the wave
aberration.
W(yp )
y 4p  1 1 2
  
4r  r s 
for spherical aberration. Derive the general expression of the transverse aberration y’ in the image
plane. Calculate the Airy spot diameter, if the z-distance at the mirror surface can be neglected.
If the mirror radius is r = -500 mm, the wavelength  = 633 nm and the diameter of the incoming beam
2yp = 40 mm what is the range of intersections lengths s', were the system is diffraction limited with a
geometrical spot, that is smaller than the diffraction spot size ?
1/2
2
Exercise 3-4: Spherical Aberration of a single lens
A thin plane-convex lens with focal length f' = 100 mm is used to focus a collimated beam with
diameter D = 10 mm and wavelength  = 1.06 m.
Draw a sketch of the lens in the optimal setup and explain, why this orientation is advantageous.
Calculate the magnification parameter M and the bending parameter X of the lens.
The surface contribution of the primary spherical wave aberration of a single lens can be expressed
by the formula
2
 3



2 n2  1
n2  n  1 2 
1
n
n

2



As 

M 
M
X 

32n( n  1) f 3  n  1 n  1 
n2
n2





Calculate this coefficient for the refractive indices n = 1.4 and n = 2.0. Which choice of refractive index
is more advantageous ?
Calculate the transverse aberration y’ in the image plane in the one-dimensional case with the pupil
coordinate yp by using this formula in the form WSPH y p  As  y 4p for both indices. Is the system


 
diffraction limited ? Compare the geometrical spot size and the diffraction Airy diameter.
Exercise 3-5: 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.