Friedrich Schiller University Jena

2013-01-05

Prof. Herbert Gross

Instituteof Applied Physics

Albert-Einstein-Str 15

07745 Jena

Exercise Solutions

Lecture Imaging and aberration Theory – Part 6

Exercise 6-1: Tele-photo System

Consider a tele-photo system (focal length: 100 mm). Both lenses be thin and have a distance of 30mm and are composed of glasses with refractive indices n=1.5 for the positive lens and n=1.8 for the negative lens.

Compute the individual focal lengths of the system under the precondition that the image field is flattened. Which is the second of the solutions to the ansatz?

Solution:

Condition for focal length:

Petzval condition:

Eliminate f'2 and solve for f'1:

Inserting that, one obtains f'2 = -42.35 mm.

Die zweite Lösung der quadratischen Gleichung liefert mit den Brennweiten f'1 = -70.83 mm und f'2 = +59.01 mm ein Objektiv vom Retrofokustyp.

Exercise 6-2: Spherical Aberration of a Plane Parallel Plate

Derive the axial spherical aberration, which corresponds to the difference between the real and the paraxial image location of a plane parallel plate with index n and thickness d. Calculate the lowest order of the aberration as a function of a small value of sin u. If the diameter in the Gaussian image plane should not be larger than 10 μm, calculate the greatest possible thickness of a plate in this approximation with refractive index of n = 1.48 for a numerical aperture of sinu = 0.8.

Solution:

The incidence angle is

If the ray height at the front side of the plate is given by y1, and the ray angle i' inside the plate is

the height at the rear surface is

and the intersection length of the image plane behind the plate is given by

The paraxial approximation is given by

Therefore the difference is the spherical aberration due to the plate

By Taylor expansion we get the expression

The diameter of the lateral blurr spot is given by

Therefore we have for the allowed thickness of the plate

Exercise 6-3: Achromate II

Consider an achromate made by a crown glass with index n1 = 1.573 and an Abbe number 1 = 57.4 and a flint glass with the corresponding data n2 = 1.689 and 2 = 31.2. The system should have the focal length f = 100 mm and the crown lens should be shaped symmetrical biconvex corresponding to the following figure.

a) Calculate the focal lengths of both thin lenses and the necessary radii of curvature.

b) Calculate the radii of curvature.

c) Why is it not possible to correct this system for spherical aberration at the boundary?

d) Consider a cemented 2-lens-component without achromatization which is used to focus collimated light from an object point in infinity on axis. If the component is turned around, which of the primary aberrations inclusive the two primary chromatic aberrations are changed, which are invariant and which have the constant value of zero?

Solution:

a) Focal length condition

Achromatic condition

gives with the material data for crownn1 = 1.5731 = 57.4

and flint glassn2 = 16892 = 31.2

the focal powers and focal lengthsf'1 = 45.64 mm

f'2 = -83.97 mm

b) crown lens symmetrical

r2 = -52.30 mm

flint lens :

resolved for r3 with r2=r1

c) Since the crown lens should be symmetric, the degree of freedom of bending the component is not available. This prevents the correction of spherical aberration.

aberration type / changed / invariant / constant = 0 / explanation
spherical / x
coma / x
astigmatism / x
field curvature / x
distortion / x
axial chromatic / x
lateral chromatic / x

Exercise 6-4: Stationary Phase Point

Consider a converging spherical wave with radius of curvature R = 100 mm. In the pupil plane, the diameter of the lens is DExP = 20 mm. The wave is observed from a point A on the axis with a distance z = 90 mm from the pupil.

Calculate the wave aberration as it is seen from the observation point as a function of the radial pupil coordinate r in the pupil. Calculate the corresponding function in paraxial approximation. Establish in addition an approximation for paraxial conditions and small values of the defocus z. Evaluate the wave aberration at the rim of the pupil for all three approaches.

Now consider the case, where an additional contribution of fourth order is added to the wave aberration with a coefficient a4. Calculate the values of a4 for all approaches in such a manner that in the observation point A the point at the rim of the pupil radius fulfils the condition of a stationary phase. What is the physical effect of this perturbation for the brightness of the focussed wave on axis?

Solution:

Wave aberration general along z-direction:

Paraxial approximation gives

If the value of the defocus has a small amount, we get

At the rim of the pupil with r = 10 mm we get:

W1 = - 0.0560246 mm

W2 = - 0.0555555 mm

W3 = - 0.05 mm

If now a fourth order contribution is added, the wave aberration reads in the various approximations

To get a point of stationary phase requires the vanishing of the derivative at the corresponding pupil height of r=10 mm. This delivers

A point of stationary phase would improve the brightness on axis considerably.

Exercise 6-4: Relation between Defocus and Zernike coefficient c4.

Derive the relationship for a slight defocusz of an image and the Zernike coefficient c4 = c2,0 for small numerical apertures NA. Use the relation

with rp to be the radial pupil coordinate scaled in wave lengths.

Solution

The setup looks as follows:

We have for a small defocussing two radii of the reference sphere R and R' with

The sag difference at the edge of the pupil reads

The defocus wave aberrations form this geometry

The numerical aperture is used to eliminate DExp and R by

and we have

On the other hand, the Zernike representation gives with a normalized radial pupil coordinate rp and c4 scaled in wavelengths.

If the image domain is located in the medium with index n, the wavelength is /n. Therefore in absolute units we get

The largest difference is obtained between the edge and the center and is

By equating both expressions we get