2017-01-17

Prof. Herbert Gross

Sebastian Merx, Norman G. Worku

Friedrich Schiller University Jena

Institute of Applied Physics

Albert-Einstein-Str 15

07745 Jena

Lens Design II – Seminar 6 (Solutions)

Exercise 6-1: F-Theta Scan Lens

Scan lenses are used for moving a field point over the image area by rotating a mirror. If the

image height is direct linearly depending on the mirror angle, we have the so called f-theta

lenses. This is a special distortion approach to switch from a sin- to a tan-dependence of the

field height.

a) Establish a BK78-Lens with focal length f = 100 for the wavelength = 488 nm. To obtain

a model of the scanning unit, establish a plane mirror, which deviates the ray by 90° and

then in a distance of 40 mm a second mirror, which deviates the ray by -90°. If now the

lens has approximately a distance of 60 mm from the second mirror, the rotating first

mirror lies in the front focal plane of the lens. Therefore the chief ray will be telecentric in

the image space. Generate a multiconfiguration with the angles 45°, 46°, 47°, 48° and

49° for the first mirror and optimize the radii and the image distance.

b) Now add in the merit function the requirements, that the image heights increase with 3

mm per step and that the chief rays are telecentric. Remove the focal length target and

optimize also the distance between the lens and the mirrors.

c) Now add additional lenses for the system until you obtain a diffraction limited design for

all scan positions. What is the remaining telecentricity error for the final solution?

Solution: a) The setup looks like this:

b) c) In the case of two lenses, the setup is for example as follows. The largest scan positioin is not still good in quality.

If we have 3 lenses, the performance is nearly sufficient:

The residual error of telecentricity is 0.27 mrad:

Exercise 6-2 Correction of a Ploessl eyepice

An eyepiece is a system to view a finite intermediate image with the relaxed eye. Since the

human eye pupil is inside the body and a certain space is necessary to allow for the insertion

of eye glasses, the pupil of the eyepiece must be located remote. This enlarges the diameter

of the system and is a problem for the correction. The approach of Ploessl is in particular an

eyepice with two achromatic components with the crown glasses oriented towards the inner

air distance.

a) Setup a Ploessl eyepiece with focal length f = 40 mm for the wavelengths dFC out of two

achromates with glasses SF12 and BK7 with thicknesses 3 mm and 10 mm and equal

focal power respectively. The diameter of the intermediate image is 20 mm, the chief ray

path is telecentric here. The eye pupil should have a diameter of 2 mm. The two

achromates are positioned in a small distance of 1 mm.

b) Optimize the air distances in the system first for the basic data of pupil and focal lengths

using only air distance changes. Is the pupil aberration negligible?

c) Now optimize all radii for axis, zone and outer field point for an optimal performance. Is

the system diffraction limited? What are the dominating residual errors?

Solution: a) First a classical achromate with focal lengths f = 80 mm is established as a starting system.

In a second step, two achromates of these data are combined. The start value for the numerical aperture is here 2 mm / 2 / 40 mm = 0.025. The object space is declared to be telecentric, the image space to be afocal. The pupil is located at the final plane.

b) Now the air distances are optimized to match the pupil and the focal length as well as the eye pupil diameter.

The residual aberrations can be clearly seen for the outer field points. Therefore the pupil aberrations are considerable.

c) In a second step, all radii are optimized. The intersection points of the chief ray and the coma rays in the eye pupil are required to be 0 / +1 / -1. The air thickness is forced to be larger than 0.1 mm and the quality is required to be best for the angle deviation.

The system is not diffraction limited, but very nearby.

If the aberrations are investigated it is seen, that distortion (3.5%) and astigmatism (0.64 ) are th dominating residual aberrations.

Exercise 6-3: Schwarzschild Mirror Objective Lens

Mirror objective lenses have the advantage of a perfect color correction. If the systems are

circular symmetric, often the problem is the central obscuration, which lowers the contrast of

diffraction limited imaging. One attractive approach is invented by Schwarzschild, he

recommended a system of two mirrors, that are concentric. This setup has under certain

conditions a superior performance.

a) Establish a 2-mirror system for a collimated entrance pupil diameter of 4 mm, a

wavelength of 0.55 m. The first mirror has a radius of curvature of 4 mm, the second

concentric mirror has 12 mm. First only an axis point is considered, the pupil is located at

the first mirror. Determine the optimal image distance and evaluate the spot

performance. What is the numerical aperture of the system in the image space?

b) Now we introduce a finite field with angles of 1.4° and 1°. It is seen from the layout, that

there are conflicts by obscuration with both mirrors. Insert central obscurations at both

mirrors to avoid a truncation of the inner and outer aperture rays. What is the inner value

of the effective numerical aperture?

c) In the next step we want to optimize the performance. For this purpose introduce an

additional surface between the mirrors and guarantee by appropriate pick ups, that the

mirrors remain concentric. Inspect the Seidel aberrations and change the distance

between the mirrors. What is the observation? Is the system diffraction limited?

d) The system can be improved by making the main mirror M2 aspherical. To demonstrate

the difference we first calculate the cross section of the point spread function. What is the

Strehl ratio on axis?

No allow the M2 to have a conic constant and optimize kappa for the system together

with an optimized image position. Calculate the point spread function again. Is the

improvement helpful? Determine the increase of the side lobes and compare this effect

with the classical Airy pattern. What is the reason for this change?

Solution: a)The system looks like the following figure. The numerical aperture is 0.65, the performance is not diffraction limited.

b) First we define an inner central obscuration at the smaller mirror M1. A value of 0.8 mm guarantees, that the inner rays in the image space can pass. Secondly the main mirror M2 is limited by an inner radius of 2.3 mm to avoid trancation of the outer rays.

If we perform a simple raytrace with a marginal ray of relative height of py = 0.4, the ray height at the M1 is 0.8 corresponding to the inner rim. The inner numerical aperture is obtained to be 0.266.

c) The additional surface is located at a distance of -12 mm from the first mirror, the radius of the M2 is obtained as a pick up from this value. The additional thickness after the aretfical surface is obtained as a pickup from the M1. If now the distance after the M1 is changed, the systems remains concentric.

The Seidel aberration chart shows, that the sphrical aberration is not fully corrected in 3

rd order.

We introduce a merit function with the spherical aberration and coma and calculate a universal plot to detect the spherical aberration, if the concentricity is preserved. It is seen, that for a distance of 10.472 mm, the spherical aberration is corrected. The same result can be obtained by optimization. Correspondingly it is observed, that the coma is corrected as well. This result in Seidel approximation can be obtained analytically.

By inspecting the Spot diagram is it seen, that the system is not diffraction limited, the 3

rd order is not

sufficient to describe the performance completly. d) The point spread function look like this, the Strehl is approximately 17.2 %. If the conic constant and the image location is optimized, we get a Strehl of 96.7 %.

In the Airy pattern, the first sidelobe has a height of 1.7 %. Here we have due to the central obscuration a value of 7.12 %.