Lens Design I Seminar 5

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1 Y. Sekman, X. Lu, H. Gross Friedrich Schiller University Jena Institute of Applied Physics Albert-Einstein-Str Jena Lens Design I Seminar 5 Exercise 5-1: PSF scaling (Homework) To check the Airy diameter formula, we establish a simple system. According to the formula Dairy = 1.22 λ / NA, we set up a system with a perfect lens with the data: - Wavelength λ = 1 μm - The incoming beam should be a collimated beam with a diameter of 10mm a) Calculate the focal length of the perfect lens to obtain an image space NA of 0.61 by Zemax optimization. b) Show the PSF of the system, what is the diameter of the Airy disk? c) Change the focal length of the perfect lens to obtain an image space NA of 0.5, calculate the Rayleigh length in this case. d) Use the slider to find a zero point of the PSF on axis, can the Rayleigh length be used to locate the zero point and why? Solution: a) Initial setup: MFE: Optimized: Focal length=6.495mm

2 b) First zero is at 1um so diameter of the Airy disk=2 um c) Re = / NA 2 we get a focussing distance of 2Re = 8 m to locate a zero of the PSF on axis. The focal length to get this aperture is f = mm. d) Now the slider is used to find the zeropoint on axis. It is seen, that Zemax is not able to calculate the point spread function exactly: there is no zero point found. Exercise 5-2: Strehl ratio and PSF vs spot size A single lens made of K5 with focal length f = 25 mm and thickness d = 5 mm is illuminated by a diverging beam with numerical aperture NA = 0.1. After the lens the light should be collimated. If the collimated beam is refocused without further aberrations, the point spread function is not diffraction limited. a) Calculate the accurate Strehl ratio, the estimated Strehl ratio and the geometrical and diffraction encircled energy inside the ideal Airy diameter. Show that the spherical aberration of this setup is exactly zero for all orders. b) If now the numerical aperture is reduced, the Marechal estimation becomes better. Calculate the largest NA, for which the relative error is smaller than 2%. What are the geometrical and diffraction encircled energy inside the Airy disk obtained here?

c) Show the Strehl ratio as a function of the numerical aperture as a

$What is the maximum value for getting a diffraction limited correction$ Solution: a) If the cardinal points of the lens are calculated, the

72 mm If the lens is reverted in its orientation and the distance is

3 c) Show the Strehl ratio as a function of the numerical aperture as a universal plot. What is the maximum value for getting a diffraction limited correction with DS > 0.8? Solution: a) If the cardinal points of the lens are calculated, the unknown first distance is obtained paraxially as t1 = = mm If the lens is reverted in its orientation and the distance is optimized over the complete pupil, the optimal distance seems to be mm. Setup: Add a paraxial lens: The following steps are performed: 1. Reduce NA 2. Determine the Airy diameter out of the spot diagram window

4 3. Set the aperture in the image plane exactly to the Airy value 4. Calculate the estimated Strehl ratio from the Zernike window 5. Calculate the accurate Strehl ratio from the Huygens PSF window with appropriate sampling 6. Calculate the geometrical encircled energy by the text output of the Geo. EE window. 7. Calculate the diffraction encircled energy by the text output of the Diff. EE window. NA Strehl exact Strehl estimated Rel. error Airy radius Geometrical EE inside airy Diffraction EE inside Airy % 6.6% % 7.7% % % 35% % % 59% % % 60% % % 63% d)

$In this exercise, consider a refractive Kepler telescope with a telescopic magnification of Γ = 20 for an incoming collimated ray$ bundle with 60mm diameter and a wavelength of 550nm.

bundle with 60mm diameter and a wavelength of 550nm.

The second group is a single plano-convex lens also made of SF6 with focal length f2 = +10mm.

5 For Ds>0.8, max NA=0.04 Exercise 5-3: Kepler Telescope A Kepler telescope is an afocal system made of two positive lens groups that has an internal focal point. In this exercise, consider a refractive Kepler telescope with a telescopic magnification of Γ = 20 for an incoming collimated ray bundle with 60mm diameter and a wavelength of 550nm. The positive front group should consist of two optimally bend lenses made of SF6. The second group is a single plano-convex lens also made of SF6 with focal length f2 = +10mm. a) Set up the system and evaluate the performance for an on-axis field point. Which surface contributes most to the Seidel aberrations of the system? b) Now consider a finite field angle of 0.3. How does the spot size change compared to the on-axis case? What is the dominating aberration? c) To improve the performance, introduce a field lens in the intermediate image plane. Optimize the field lens by considering also the off-axis field point. What is the final performance? Solution: Angle magnification: a) Paraxial setup:

6 Put the 2 nd lens (plano-convex, SF6, f=10) Put the 1 st lens (plano-convex, SF6, f=200) Split the 1 st lens and optimize:

7 b) Field angle of 0.3deg inserted Performance has dropped by a factor of ~20. Dominating field aberration is Astigmatism (Z5) and generated at the last surface

8 c) field lens is inserted at intermediate plane

Exercise 5-4: Galilei Telescope A Galilean telescope of magnification = 5 should be corrected in the following steps: a) Establish

b) Now change the requirements to an achromate of the negative focal length f = -25 mm.

focal length f = 125 mm and adjust the Galilean system to an afocal system. The field angle is 2.

9 Exercise 5-4: Galilei Telescope A Galilean telescope of magnification = 5 should be corrected in the following steps: a) Establish a classical achromate of the focal length f = 20 mm for an incoming ray bundle of 6 mm diameter made of BK7 and SF12. b) Now change the requirements to an achromate of the negative focal length f = -25 mm. Compare the solution with the results in a) c) Now combine the negative cemented lens with a commercial catalog achromate with focal length f = 125 mm and adjust the Galilean system to an afocal system. The field angle is 2. What is the dominating residual aberration of the system? d) Now as a final step optimize the system numerically by forcing the magnification and the focal length of the two groups to be constant. Can the system by improved? a)

$the system is diffraction limited as well 2. the spherical aberration has switched in sign 3.$ c) The positive achromate with focal length f = 125 mm should have a diameter of 5 x 5 mm = 25 mm.

c) The positive achromate with focal length f = 125 mm should have a diameter of 5 x 5 mm = 25 mm.

10 b) Invert all radii to negative. Then optimize for f=-25 In comparison to the positive focal length, we see here: 1. the system is diffraction limited as well 2. the spherical aberration has switched in sign 3. now the crown lens (BK7) is the negative lens and the flint lens the positive. c) The positive achromate with focal length f = 125 mm should have a diameter of 5 x 5 mm = 25 mm. We select the component LAO from CVI Melles Griot. The achromate is reversed and the intermediate distance of approximately f 2+f 1 = 100 mm is adjusted. The size of the achromate is enlarged to 34 mm diameter. The image space is defined to be afocal.

12 d)

Tutorial Zemax 3 Aberrations

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