2016-06-09 Prof. Herbert Gross Mateusz Oleszko, Norman G. Worku Friedrich Schiller University Jena Institute of Applied Physics Albert-Einstein-Str 15 07745 Jena

Exercise: Lens Design I – Part 5

Exercise 5-1: 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.

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 amount for the geometrical and diffraction encircled energy inside the Airy diameter is obtained here?

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) System data and layout:

If the cardinal points of the lens are calculated, the unknown first distance is obtained paraxially as t1 = 25 – 3.280 = 21.72 mm If the lens is reverted in its orientation and the distance is optimized over the complete pupil, the optimal distance seems to be 21.88 mm.

b) If the numerical aperture is changed, the following steps are performed:

1. Reduce NA 2. Determine the Airy diameter out of the spot diagram window 3. Set the apertur in the image plane exactly to the Airy value 4. Calculate the estimated Strel 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 footprint diagram (with option: delete vignetted) 7. Calculate the diffraction encircled energy by the text output of the EE window.

Then the following table is obtained:

NA Strehl exact

Strehl estimated

relative error Airy radius geometrical EE inside Airy

diffraction EE inside Airy

0.1 0.019 0 0 0.003299 0.0389 0.0394

0.08 0.058 0 0 0.004123 0.0889 0.0889

0.07 0.053 0 0 0.004712 0.1449 0.1443

0.06 0.172 0.2158 0.255 0.005497 0.2514 0.3403

0.055 0.342 0.3662 0.0661 0.005996 0.3414 0.4611

0.051 0.486 0.4958 0.0198 0.006466 0.4467 0.5534

0.05 0.520 0.5277 0.0146 0.006600 0.4780 0.5650

0.045 0.676 0.6753 0.00104 0.007328 0.6887 0.6607

0.04 0.798 0.7940 0.00501 0.008244 1.0 0.7281

The relative error of the estimated Strehl ratio is smaller than 2% for NA < 0.051. Here the geometrical encircled energy is 45%, the diffraction calculated encircled energy 55%.

c) The universal plot is obtained with the following setting:

From the corresponding text window, we also get a limiting value of approximately NA = 0.04 for the diffraction limit.

Exercise 5-2: PSF scaling To check the Airy diameter formula, we establish a simple system. According to the formula

Dairy = 1.22 / NA we select a system with a perfect lens with the data:

- wavelength = 1 m - numerical aperture in the image space of NA = 0.61 = 1.22 / 2

With these numerical values, the Airy diameter must be exactly 1 m. A collimated input beam with 10 mm diameter therefore need a special focal length of the ideal lens to produce this angle. This is calculated by Zemax with optimization and a corresponding merit function with REAB = -0.61. We get a focal length and a final image distance of 6.495 mm.

Solution:

The corresponding cross section of the point spread function has its first zero exactly at 1m.

Now we select the focal length to generate a numerical aperture of NA = 0.5. According to the Rayleigh

length formula Re = / NA2 First zeros of axial distribution lie at z0= +/-2RE

Therefore we get a focusing distance of 2Re = 8 m to locate a zero of the PSF on axis. The focal length to get this aperture is f = 8.660 mm. Now the slider is used to find the zero point on axis. It is seen, that Zemax is not able to calculate the point spread function exactly: there is no zero point found.

Exercise 5-3: Kepler telescope

Establish a Kepler-type refractive telescope with a telescopic magnification of = 20. The positive front group should be built by 2 optimized bended lenses of SF6. The incoming collimated ray bundle is 60 mm, the considered wavelength is 550 nm. The negative group is a single plano-concave lens made of SF6.

a) What is the performance of the system on axis? Where is the limiting surface if the quality is considered?

b) Now introduce a finite field angle of 0.3° and inspect the spot performance on axis. What is the dominating aberration?

c) Now introduce a field lens in the intermediate image. Optimize the field lens by considering also

the off-axis field point. What is the result? Solution: a) Γ = 20 => f1 = 200mm, f2 = 10mm => marginal ray height at the second lens rear surface is

-1.5 We start with designing second plano-convex lens with the focal length 10mm

We note that the radius of second lens is 8.12 mm. Second is to design the front lens group appropiate starting conditions. We change focal length in the merit function to 200 mm.

After optimization

We combine both lens groups.

And reoptimize with the merit function. Changing the default merit function to optimize for angular radius. We should also change the general settings to afocal image space.

Our resulting system look like

b) With diffraction limited performance on axis. After establishing field point we see that the performence goes worse. The dominating problem is astigmatism.

c) To fix this problem we can add a field lens. Field lens is placed in the intermediate image

position where the ray boundles from each field point are well separated. Hence we could influence the aberrations of the field without changing on axis perfomance.

In the merit function we relax our constrains for focal length and marginal ray height, refreshing default merit function to consider the field point

After optimization we get following system

With nearly diffraction limited performance for both on-axis and field