2012-05-08
Prof. Herbert Gross
Friedrich Schiller University Jena
Institute of Applied Physics
Albert-Einstein-Str 15
07745 Jena
Exercises
Lecture Design and Correction of Optical Systems – Part 2
Exercise 7: Focal Length of a Thick Lens
Use the formulas of the paraxial raytrace procedure
y j  y j 1  d j 1U j 1
i j  c j y j  U j 1 ,
i' j 
nj
n' j
ij
U ' j  U j  i j  i' j
to derive the equation for the focal length of a thick lens.
Exercise 8: Axial Image Shift in a Medium
Calculate the axial shift of the image position, if a raybundel with aperture sin(u) in a medium with
refractive index n is focussed into a medium with index n' exact and in paraxial approximation.
Exercise 9: Lagrange Invariant
An object with 2.5 mm diameter should be illuminated with a numerical aperture of NA = 0.3. If the
aplanatic corrected illumination system can accept a numerical aperture of NA = 0.9 of the light
source, what is the minimum size of the radiating area of the lamp ?
Exercise 10: Defocussed Telescope
An inverted afocal Galilean telescope is given which reduces the diameter of an incoming beam by a
factor of 5. Both used lenses have one plane surface, the positive lens with f 1 = 100 mm is made of a
material with refractive index n = 1.5, the negative lens with f 2 = -20 mm has n = 2.0.
Sketch the system with an orientation of the lenses, which is beneficial for the correction. Determine
the bending and the magnification parameter of both lenses. Calculate the paraxial ABCD-matrix of
the system for an arbitrary distance z between the lenses.
1/2
2
If the distance z between both lenses is misadjusted by Δz, the system is no longer afocal. Calculate
the change of the magnification and the residual outgoing ray angle for the case of a misadjustment of
Δ = 0.1 mm and an incoming beam diameter of 20 mm.
Exercise 11: Ball Lens
Derive the focal length of a ball lens. What is especially the formula of the focal length for the
refractive index n = 1.5 ?
If the ball lens is used symmetrical for an object in a finite distance, what is the overall length of the
imaging system ?
Derive the condition, that must be fulfilled, that an incoming plane wave is focussed onto the back
vertex point of a ball lens. What is the focal length of the ball lens for this special setup ? Where is the
principal plane in this case ?
Exercise 12: Sellmeier Dispersion Formula
The measured refractive indices of the glass BK7 are given by the following data:
0.3650100
0.4046600
0.4358300
0.4800000
0.4861300
0.5460700
0.5875600
0.5892900
0.6328000
0.6438500
0.6562700
0.7065200
0.8521100
1.0139800
1.0600000
1.5296000
1.9701000
2.3254000
1.5362680
1.5302390
1.5266850
1.5228290
1.5223760
1.5187220
1.5168000
1.5167280
1.5150890
1.5147190
1.5143220
1.5128920
1.5098030
1.5073080
1.5066880
1.5009070
1.4949480
1.4892120
where the first column represents the wavelength in μm, the second column gives the corresponding
indices.
Compute a numerical fit of these data by a 3-term Sellmeier dispersion formula. Check the model
representation for the given wavelengths 0.40466, 0.54607, 0.6328, 0.70652, 1.01398 μm and the
intermediate values λ=0.4, 0.5, 0.6, 0.7, 1.0 μm.
Discuss your results. Estimate the overall accuracy. Do you think, three Sellmeier terms fit the data
well ? What can be done to improve the results ?