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2019-01-30 Prof. Herbert Gross Uwe Lippmann, Yi Zhong Friedrich Schiller University Jena Institute of Applied Physics Albert-Einstein-Str 15 07745 Jena

Exercise Solutions 12:

Optical Design with Zemax for PhD - Advanced

Exercise 12-1: Tolerance sensitivity

The sensitivity of system parameters for small perturbations is one important information, which is needed for tolerancing a system. In this simple example, the basic idea and especially the difference between refractive and reflective elements is demonstrated. Furthermore the importance of adjusting or compensating of the image plane is addressed. a) Establish a 4f-system by using a focal length of f1/2 = 100 mm for every group. The first should be a plano-convex lens made of K5, the second should be a mirror. Add a plane folding

mirror after 50 mm of the lens. We are considering a wavelength of = 1064 nm and a numerical aperture of NA = 0.07. Calculate the spot diameter for this nominal system. b) Now change the air distances No 0, 2, 5 and 6 by 1 mm and list the corresponding changed spot diameters first in the same image plane and second after refocusing the system. Collect all the values in a table for better comparison. c) Perturb the system by changing the two curved surfaces by 1% and proceed with the spot changes as in b) d) Now perturb the system by tilting every of the surfaces separately by 30' and complete the table of b) with the data. Again use the original image plane and refocus the system. Explain and discuss the results. What are the shapes of the spot for the various perturbations? What kind of aberrations are therefore generated? Solution: a) The system layout is seen here:

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b) c) d) All the changed spot values are collected in the following table. a tilt of 30' corresponds to 0.5°. These values are set in the surface properties, after the surface the system should be re-aligned.

perturbation spot diameter

[m] in fixed plane

spot diameter in

[m] refocussed

nominal design 7.42

b t1 reduced by 1 mm 51.0 7.09

t2 reduced by 1 mm 7.42 7.42

t5 reduced by 1 mm 7.42 7.42

t6 reduced by 1 mm 50.1 (7.42)

c r2 enlarged by 0.51 mm 51.6 7.11

r6 enlarged by 2 mm 49.7 7.48

d surface 1 tilted by 30' 7.94 7.93

surface 2 tilted by 30' 8.35 8.30

surface 4 tilted by 30' 7.42 7.42

surface 6 tilted by 30' 7.67 7.56

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In the case of distance and radius changes the systems remains corrected and we are mainly producing defocus and a very small amount of spherical aberration. It is seen, that these values can be refocused without problems. If the image plane is not adjusted, the blurring is quite large. In the case of decentering, we are producing coma and a very small amount of astigmatism. Furthermore, due to a tilt, the image position is laterally shifted. It is seen, that a refocusing is not really helpful.

Alternative way of quickly obtaining the spot radius table: Use the tolerancing feature of Zemax. Open the Tolerance data editor and specify the image distance as a compensator. Then enter the desired perturbations to the system as given below:

Then run the tolerancing and set the options as follows:

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The spot radii without refocusing can be found in the resulting table:

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For refocusing set the compensator to „Optimize All (DLS)“:

Again, the spot radii for a refocused image plane can be found in the resulting table:

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Exercise 12-2: Tolerancing a split doublet In this example, the tolerances of a split doublet with an enlarged sensitivity at the inner surfaces are calculated. a) We load the split doublet LAPQ-100.0-20.0 out of the catalog of CVI Melles Griot. The wavelength is 248 nm, the material is fused silica, the numerical aperture is NA = 0.1. The system is nearly diffraction limited with good correction. We remove the solve of the image plane location and add a field point with angle 2°. Look at the Seidel bar diagram to get a feeling of the sensitivities of the surfaces. b) Open the tolerance editor and select the default tolerances. For simplicity we skip the tilts, the X decenters, the element tolerances, the surface irregularity and the Abbe number tolerance. The radius tolerance is switched to 'fringes'. How many tolerances are considered now? c) Now we calculate the sensitivities of the parameters and the tolerances of the complete system. We activate the tolerancing with 'sensitivity', 'RMS spot criterion' and select to overlay the graphics. The program is now calculating the Monte Carlo simulation impact of all the tolerances on the performance. Discuss the results.

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Solution: a) The system looks as follows:

The Seidel bar diagram shows a large difference in sensitivity of the surfaces. The spherical aberration contribution of the 3rd surface is larger by a factor of 8.5 in comparison to th 1st surface.

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b)

We have now totally 13 tolerances: 4 radii 4 decenterings 3 thicknesses 2 refractive indices

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c)

The text output now contains all the relevant information on the tolerances: 1. block: sensitivities: The spot diameters are averaged over axis and field. We also see, that the image location as a compensator must be considered and what amount of movement is necessary. The enlargement of the spot diameter is expected to be 18 % in this case. The 'worst offenders' are listed in decreasing order of impact. This shows, where the main tightening of tolerances can help most. In this case as expected from the Seidel diagram, the decentering tolerance of the 3rd surface is the most critical tolerance. On the other side it is seen, that the radius tolerance can be made more relaxed for the 1st, 2nd and 4th surface.

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2. block: Monte Carlo results Here the statistical model is used to calculate the results on a more rigorous basis. Especially all possible compensations and interactions are considered. We see the individual results of all trials and the statistics. In this case a yield of 90% can be extected, for which the spot

diameter is better than 10 m.

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3. overlay graphics

The nominal spot diameters are 1.7 and 13.6 m on axis and in the field. The overlay graphics overlays the spots for axis, the zone and field with both signs in one diagram and also overlays the numbers.

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The overlay feature can be used with other analysis features too, more useful than overlaying spots might be the overlay plotting of the MTF for every Monte Carlo Trial to get a feel for the as-built performance of the system.

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Exercise 12-3: Astigmatic clocking

If lenses have cylindrical or toroidal surface shapes, they produce astigmatism on the axis. Since the sign of the astigmatism can be used to compensate this effect, the lenses are rotated in practice around the optical axis. This procedure is called clocking and can correct the astigmatism, if the sizes and the angles of the residual errors are appropriate. a) Establish a system that is decomposed by three plates with thickness 1 mm and a final ideal lens of 100 mm focal length. The plates have residual radii of curvature due to manufacturing errors of: 1. R = 6000 / -5000 2. R = 8000 / -6000 3. R = 7000 / -9000 The second and the third lens are rotated around the z-axis. The distances between the lenses

is 1 mm, the wavelength is = 0.55 µm, the entrance pupil diameter is 10 mm and we are only looking on axis. b) Establish a merit function with the total amount of astigmatism and defocus by calculating the corresponding Zernike coefficients. Generate a 2-dimensional universal plot with the total astigmatism as a function of the two rotational angles. c) Now optimize the two azimuthal rotational angles to compensate the astigmatism. Is the correction perfect? Is the defocus corrected simultaneously? Solution: a) The system look as follows

b) Merit function: For the astigmatism, the Zernike 5 and 6 are calculated and the sqrt of the sum of squares is determined.

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The universal plot shows, that there are two values of the angles, which allow for a good correction. There are two solution spaces, which are anti-symmetrical.

c) The best values are obtained for the angles 111.77° and -63.85°. The residual value of the astigmatism is nearly perfect. The defocus is still finite and unchanged.

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Exercise 12-4: Alignment simulation

a) Establish for the wavelength 632.8 nm a system of beam expansion with a final focusing lens. The incoming beam has a diameter of 2 mm. The beam expander is composed of two single lenses with focal lengths -25 mm and 125 mm. The final focusing single lens has a focal length of 150 mm. Calculate the spot size in the optimal image plane on axis as the reference number for the alignment simulation. b) Now simulate the worst case alignment if it is assumed, that the groups are centered in the sequence 1-2-3 and the residual accuracy of the spot position is 0.5 mm. Determine the residual lateral decenter values of the 3 groups in this case. Calculate the broadening of the spot for this case. What are the wave aberrations on axis now? Solution: a) We select the following lenses from CVI Melles Griot. They are oriented in the optimal direction to achieve best performance.

The spot size is extremly good with 2.5 µm radius.

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b) Starting system: all three lenses are defined to by decentered as groups, therefore the global reference is preserved. In the merit function, the REAY operator describes the chief ray deviation from the axis as criterion.

Now the three lenses are moved in y-direction to get a shift in the image plane of:

1. Lens 1: y' = +0.5 mm

2. Lens 2: y' = -0.5 mm

3. Lens 3: y' = +0.5 mm The corresponding decenter values are -0.414 / -0.833 / +1.001 mm.

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The spot is now increased to 6.7 µm. The dominating wave aberrations are tilt, astigmatism and coma.