Exercise: Lens Design I Part 4. Exercise 4-1: Astigmatism of oblique curved Mirrors

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1 Prof. Herbert Gross Mateusz Oleszko, Norman G. Worku Friedrich Schiller University Jena Institute of Applied Physics Albert-Einstein-Str Jena Exercise: Lens Design I Part 4 Exercise 4-1: Astigmatism of oblique curved Mirrors A focusing mirror suffers from astigmatism, if it is used under oblique illumination. Therefore an afocal combination of two curved folding mirrors with an intermediate focal point in a Z-constellation generates a strong astigmatism. a) Setup a system of the described kind with a distance between the mirrors of 100 mm for an incoming collimated ray bundle of 6 mm. The wavelength should be 1064 nm. The system should be formulated with a collimated outgoing beam with the help of pickups b) Find the minimum bending angle of the mirrors to avoid an obscuration at the second mirror by using the universal plot option c) The astigmatism of an obliquely used curved mirror splits symmetrically. Therefore in the current setup the location of the circle of least confusion remains in the midpoint between the two mirrors. Show this property by using the 2D universal plot with the distance of the midpoint and the bending angle as parameters. Is the absolute value of the spot size in this location changing with the bending angle? d) Determine the growing astigmatism as a function of the bending angle by a universal plot. Find the largest angle for reaching the diffraction limit e) If the second mirror is rotated around its y-axis instead of the x-axis, it deviates the beam n the x-direction. Can this kind of clocking be used to reduce the aberrations? Compare the spot diameter in both cases. What kinds of aberrations are seen? Solution: a) The system is established as follows (with some additional reference planes

2 b) The 1D universal plot is generated with the following data. The tilt angle at mirror 1 is used as a parameter, the axis ray is selected and the ray height at the second mirror is determined in global coordinates (RAGY insetad of the local REAY). From the text output, we get the limiting angle 1.72 for a ray height of 6 mm according to the beam diameter. c) The plot is confuguted as follows. It is seen from the plot, that the minimum value of the spot rms diameter remains constant. The absolute value increases with increasing bending angle.

3 d) The universal plot is established with the Zernike coefficient Z6. The ASTI operand don't wotk on axis, the Seidel aberration needs a finite field. To see the limiting value of the bending to get a diffraction limited imaging, the 1D universal plot is established with the Strehl reatio in the following format. The critical bending angle is 2.6.

4 e) If the tilt angle of the second mirror around the z-axis is set to 5, we get the layout

5 If both spot diagrams are plotted it is seen, that the remaining aberrations are astigmatism and coma. The rms value of the residual aberrations can be reduced by a factor of 2. The aberrations is a mixture of astigmatism and coma. Exercise 4-2: Focus in medium If a wave propagates in a medium, the effective wavelength is reduced by a factor 1/n. Some normalization aspect of the Zernikes therefore are critical. This example should show some effects in Zemax. a) Establish the Achromate LAO from the catalog of CVI Melles Griot. The incoming collimated beam diameter should be D = 20 mm and the wavelength is 550 nm. Add an aplanatic-concentric meniscus lens of an artificial medium with n = 2 and a thickness of 3 mm. Determine the optimal final image distance be quick focus. Calculate the Zernike coefficients and the spot diameter in the focus. b) Now insert in the image space a lens with medium n = 2, which is concentric to the focusing ray bundle in the paraxial region with a radius of R = 10 mm. The final distance should be optimized again by quick focus. Compare the wave Zernike coefficients and the spot size in this case. c) To verify the result, setup a similar example, where the focusing system is ideally perfect. What is now the change in comparison to b)? Solution: a) Data:

6 Performance: b) The data are now asfollows. Due to the residual aberrations of the achromate, the final distance and the radius of the concentric surface are not identical.

7 The spot size is decreased by a factor of two, the Zernike coefficients are constant. The explanation of this effect is as follows: In the medium, the phase aberration is preserved. But the effective wavelength is reduced to /n. Due to the law of refraction,every angle is reduced by a factor of n. Therefore the geometrical aberrations, which corresponds to small angle deviations, are reduced by 2. The Zernike coefficients are scaled on the original wavelength in air (this is a critical concept in Zemax). Therefore the phase difference is the same, but the effect on the image formation is reduced by 2 in the medium. c) The system looks as follows

8 The aberrations are now completely perserved by inserting the additional medium, because now the aberrations are really zero. Exercise 4-3: Simple Galilean telescope A Galilean telescope is an afocal system made of two lens groups without an internal focal point. If the diameter of a beam is enlarged, the first group is of negative power, the second one positive, both groups have a common virtual focal point. a) Establish a Galilean telescope of factor 4 for enlarging the incoming collimated ray bundle with diameter D = 6 mm at a wavelength of = nm. Both lenses are plane on one side, are made of SF59, the first group is a single lens with a focal length of f 1 = -25 mm for a thickness of 3 mm, the second positive lens has a thickness of 5 mm. Orient both lenses properly, adjust the system to by best for a collimated output beam. b) Is the system diffraction limited? Calculate the Zernike coefficient for spherical aberration. Inspect the Seidel aberrations and determine the surface with the largest contribution.

9 c) To improve the system, split the second positive group into two identical lenses of the double focal length. Is the system now better and diffraction limited? Now finally select the front surface of the first positive lens to be R5 = -200, which is oriented towards the focal point with smaller incidence angles. Readjust the system and look for the surface contributions and the overall performance. d) Set the stop of the system at the negative lens and introduce a field angle of 1. Check the realized change of the collimated beam diameter and calculate the telescopic magnification of the field angles. Solution: a) The system looks as follows. The radius of the first is found by the power-solve, after this the lens is reversed. The adjustment is obtained by adjustment of the angle radius. b) The system is not diffraction limited, the Zernike coefficient is c9 = 0.52, the system is defined to have an afocal image space to get an plane reference wave. The largest spherical contribution comes from the first plane surface of the positive lens.

10 c) The system is slightly improved, but not diffraction limited. The dominating spherical contribution of the plane surface is unchanged.

11 Now by selecting R5 = -200 the system is of quite good performance and diffraction limited.

12

13 d) The diameter of the marginal rays are yin = 3, yout = , r = yout/yin = The telescopic magnification is uin = , uout = , = = 1 / 4.012

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