1.1 Singlet. Solution. a) Starting setup: The two radii and the image distance is chosen as variable.

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Erick Morgan
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1 1.1 Singlet Optimize a single lens with the data λ = 546.07 nm, object in the distance 100 mm from the lens on axis only, focal length f = 45 mm and numerical aperture NA = 0.07 in the object space.

$b) Now start the optimization with a lens and an image distance, which is near to the solution. Is the optimized lens diffraction limited in its performance?$

1 1 1.1 Singlet Optimize a single lens with the data λ = nm, object in the distance 100 mm from the lens on axis only, focal length f = 45 mm and numerical aperture NA = 0.07 in the object space. The lens should be made of the Schott glass N-K5 and has a thickness of 5 mm. a) Try to start from a plane plate approach to find the best lens bending solution. b) Now start the optimization with a lens and an image distance, which is near to the solution. Is the optimized lens diffraction limited in its performance? c) One possibility to improve the result is to use an aspherical lens. The first approach is to use the rear surface with a conic constant to allow the program a conic section as solution. Is this sufficient to get a diffraction limited solution? d) Now enlarge the numerical aperture by a factor of two. Re-optimize the system. What about the diffraction limited performance? Use an aspherical coefficient of 4th order to improve the system. What is the result? e) Now introduce a finite object size of diameter 10 mm. What is the dominant aberration for the offaxis field points? Can the system by made diffraction limited by re-optimization, for example with more aspherical constants? What can be done to get a better performance? Solution a) Starting setup: The two radii and the image distance is chosen as variable. Merit function: the default spot diameter and the focal length (EFFL) is required. If the optimization is started, no useful result is obtained: b) If a starting system for example with the following data are used:

$diffraction limited: c) Now the conic constant of the back surface is chosen as$ variable and the optimization is run again.

2 we get the correct solution: The spot diagram shows, that the singlet lens is not diffraction limited: c) Now the conic constant of the back surface is chosen as variable and the optimization is run again. The solution is found after several iterations, the merit function is near to zero. Therefore the system must be diffraction limited

d) The numerical aperture is changed to NA =

$The system is still diffraction limited, but$ aperture.

3 d) The numerical aperture is changed to NA = 0.14 and the optimization is run again. The system is still diffraction limited, but a little bit worse in comparison to the lower aperture. The exact result depends on the number of rings in the pupil for the spot optimization.

4 If now an aspherical coefficient of the 4th order is included, the result is as good as in c). e) If a finite field size of diameter 10 mm is introduced (y = 5 mm), we get the following spot diagram. The shape of the field spot is dominated by a coma contribution.

5 Correspondingly the ray fan plot shows a strong quadratic contribution The Zernike coefficients show, that there is a considerable contribution of astigmatism (term no 5) too: Now we reoptimize the system and allows step by step more constants for the aspherical surface. The performance is improved a little bit, but a diffraction limited system is not obtained.

6 1.2 Influence of initial system Solution: Consider a system with two lenses, made of BK7 and K5 at nm. Both lenses have the thickness 5 mm, between both lenses the air distance is 1 mm. The incoming ligth bundle is collimated with diameter D = 20 mm, the focal length should be f = 30 mm. The optimization result now depends strongly on the initail values of the radii. Try and compare the following possibilities: a) start with plane surfaces only b) start with a final radius of R4 = -20 c) start with a first radius R1 = +20 The results as layouts and with the corresponding spot rms radius values are collected in the following table No Layout Start radii a Rms spot size b c

1.3 Influence of criteria selection Load the achromate LAO-100.1-50.0 from the catalogue of CVI Melles Griot. Select the single wavelength 632.

7 1.3 Influence of criteria selection Load the achromate LAO from the catalogue of CVI Melles Griot. Select the single wavelength nm and introduce a field angle of 3 and reduce the aperture to a diameter of 15 mm. a) Optimize the system by changing only the radii of curvature for different performence criteria. Select first a spot optimization and set the focal length to be f = 100. The image distance should be a variable to allow for perfect focussing. Take 9 rings of the pupil sampling b) Now perform the same task by using the rms-wavefront criterion according to the centroid of the spot. c) Now select the Strehl definition. A 'quick focus' or other pre-optimization should be called before the optimization is started. Calculate the PSF in the Huygens approximation. Explain the dominant aberrations with the help of the wavefront and the PSF. Evaluate the difference in the Strehl ratio between the chief-ray location, the peak and the centroid definition. d) Compare the various results concerning the proposed quality data and the computational time for the option 'automatic'. Solution: a) System result with the spot criterion: b) c) The wavefront and the PSF looks like the following figures. The dominant aberration is coma and astigmatism, this can also be seen from the Zernike expansion. The relative intensity changes considerably from the chief ray position (y=0, intensity I=0.245) to the centroid (y = µm, I = 0.155) to the peak (y = µm, I = 0.273) due to the asymmetry. It is seen, that the estimated Strehl calculation in the Zernike menue don't work in this case. The reason for this is the very large wave aberration at the y-edge, which leads to a large rms-value. The optimization is quite slow, the reasons are the more complicated calculation of the criterion and a slow convergence. If the starting system is too far from the final local minimum, the algorithm breaks down.

d) The following table summarizes the result of

problems concidering the time, the convergence

It is also seen No a Spot spot rms radius [µm]

8 d) The following table summarizes the result of the calculations in a) to c) It is seen, that the results are quite different. EWspecially the Strehl optimization makes large problems concidering the time, the convergence and the necessary starting configuration. It is also seen No a Spot spot rms radius [µm] Wrms [λ] Strehl [%] Optimization time [s] b

9 c ca. 300

$Achromat in convergent light The classical achromate solution assumes collimated incoming light. It can be made nearly diffraction limited, if the numerical aperture is not larger than NA = 0.1.$

10 Achromat in convergent light The classical achromate solution assumes collimated incoming light. It can be made nearly diffraction limited, if the numerical aperture is not larger than NA = 0.1. If now the initial ray bundle is convergent or divergent and the achromate should focus the beam still faster by an increment of 0.1, we get quite similar conditions. The only observation in the optimization of the system is a modified bending of the achromate towards the ray path. Establish an achromatic system for an incoming convergent ray bundle with NA = 0.2, that is focussed by NA = 0.3. Inspect the performance of the system. The EPD is 10 mm and the wavelengths are FDC. In this exercise an achromat is designed that achieves its optimal performance in convergent light. Therefore, an ideal lens is used that generates the convergent beam. Afterwards, an achromate is designed to further increase the numerical aperture, while keeping the achromatic correction. In particular, this means that the red and blue wavelength achieve a common focal point and additionally the spherical aberration for the marginal ray is corrected in the green. In comparison to previous exercises not a specific focal length of the achromat is given, than rather the growth in numerical aperture. In order to solve this task, either a precalculation can be performed to calculate the focal lengths of the ideal lens and the achromat or one can specify the NA constraints in Zemax directly. Here the second option will be used. Therefore the REAB operand in the merit function will be used (REAB = Real ray y-direction cosine of the ray after refraction from the surface defined by Surf at the wavelength defined by Wave.) The REAB command for the marginal ray equals the numerical aperture. The system before optimization looks as follows. In the merit function the first REAB commands is placed to force the NA after the ideal lens to be 0.2. Therefore additionally the focal length of the ideal lens is set to be variable. Moreover, instead of setting the focal length of the achromat a second REAB command is placed. It forces the numerical aperture after the second lens to be NA=0.3. Moreover the classical achromat constraints are set: AXCL to force the red and the blue wavelength to a common focal point and the PARY and REAY command to correct the marginal spherical aberration in the green. After optimization the following result is obtained

$According to the spot diagram a nearly diffraction limited performance is achieved. In comparison to classical achromats a changed bending of the surfaces can be observed.$

11 According to the spot diagram a nearly diffraction limited performance is achieved. In comparison to classical achromats a changed bending of the surfaces can be observed. Most apparently the intermediate and final surface are weaker bended. The longitudinal aberration plot proves the classical achromat performance. The secondary spectrum, the spherochromatism, and the zonal error are remaining.

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