2014-05-22
Prof. Herbert Gross
Dr. Manuel Tessmer
Friedrich Schiller University Jena
Institute of Applied Physics
Albert-Einstein-Str. 15
07745 Jena
Exercise:
Design and Correction of Optical Systems – Part 3
Exercise 3-1: the stationary phase approximation
The stationary phase approximation is an important tool for the computation of Fourier-domain signals in data analysis, and it is also a very important tool for understanding physical image formation and the influence of aberration.
Compute the time-Fourier domain of the following functions in the stationary phase approximation:
(a) (1.1)
(b) (1.2)
In any case, separate the Fourier integral into terms having a fast and those having a slowly varying phase. Next, approximate the exponent in the fast integral as a second-order polynomial in time and neglect all the higher-order terms. Approximate all slowly varying functions by their values where the fast exponent has zero slope!
Exercise 3-2: Stationary phase point for aberration
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?
Exercise 3-3: Marechal approximation to the Strehl ratio
The task of this exercise is to derive the Marechal approximation to the Strehl ratio.
The definition of the Strehl ratio, for uniformly illuminated pupil, is given to be
,
where r is integrated from 0 to 1 and phi from 0 to 2pi. Assume the W is a small number all over the whole pupil. Expand the above integral to 2nd order in W to obtain the Marechal approximation.
Hints: The integral of some quantity over the coordinates x and y gives the expectation value
Useful to do: start expanding the exponential itself, not the full expression.
Exercise 3-4: Microscope and the Lagrange invariant
A microscopic objective lens has a magnification of m = 100, the numerical aperture NA = 0.8 in water with refractive index n = 1.33 and a diameter of the intermediate image of Dim = 30 mm. The illumination is monochromatic at the wavelength = 546 nm.
a) Calculate the Airy diameter in the object space.
b) Calculate the size of the object which can be observed with this system.
c) What is the aperture angle in the intermediate image plane?
d) Is it possible for a diffraction limited correction of the system to image a liquid transparent sample with thickness z = 2 m sharply in every depth? (Use the depth of focus to solve this question.)