2014-12-09 Manuel Tessmer M.Tessmer@uni-jena.dee Minyi Zhong minyi.zhong@uni-jena.de Herbert Gross herbert.gross@uni-jena.de Friedrich Schiller University Jena Institute of Applied Physics Albert-Einstein-Str. 15 07745 Jena Exercises Advanced Optical Design Part 5 Solutions 5.1 Pinhole Camera lens A camera lens with a pinhole like entrance pupil should be established. This is a suitable system for security applications, due to the pinhole character the camera is hardly seen from outside. The light enters the system through a small pinhole and the lens system is located behind, hidden for the observer. The data of the system are as follows: for the wavelengths dfc a lens should be developed with a focal length of f = 10 mm and a remote entrance pupil of diameter D = 1 mm. As maximum field angle 20 are required. a) Establish a system with a single lens and one cemented component to start the system. The merit function is to be formulated for axis, zone and outer field point. For the glass selection we choose BK7 and SK16 / SF5. Correct the system as good as possible. Is the performance diffraction limited? b) Now the requirements are more complicated: the field is enlarged to 28 and the condition of a telecentric image space is fixed. Due to the telecentricity the diameters are growing and a retro-focus type system is needed. Therefore we need for the same quality of correction one more lens. This additional single lens is added between the two lenses and is made of PK50 to get a good color correction. Optimize the system. (Hint: make the air thickness between three lenses and all the radii into variables, and use REAB to force image-side telecentricity for fields 0, 20, 28.) c) What are the largest residual aberrations? Determine the largest field position with a diffraction limited image. Show the residual variation of the telecentricity over the field (Hint: universal plot, field angel vs. RMS spot radius). What is the largest deviation value? Solution: a) The system looks as follows.
The quality is diffraction limited: b) Now we increase the field to 28 and 20 for the zone. Furthermore we add two lines in the merit function with REAB for the two field points to get a zero chief ray angle in the image. With the additional lens we get the result:
The performance is diffraction limited, only in the outer field part the quality is worse.
From the representation of the spot rms as a function of the field size it is seen, that the diffraction limit is exceeded at 22.4 (white light). If we establish only one field point with 28 and generate a universal plot of the chief ray angle vs the field size, we see the variation of the telecentricity. The largest value occurs at 12 with approx. 5.5 mrad.
5.2 Crossed Aspherical Cylindrical Lenses Aspherical cylindrical lenses are components, which are able to focus collimated light at a high numerical aperture. With two crossed lenses, it is in principle possible to focus light in both cross sections x and y. In the case of high numerical apertures, the decoupling of the x- and the y-section is no longer valid. Therefore, a system of crossed lenses is more complicated. This is presented in this task. a) Load the aspherical cylindrical lenses C 50-40 HPX and C 45-32 HPX from the lens catalog of Asphericon. The wavelength is = 780 nm, the collimated input beam has a diameter of 40 mm. Establish first the longer focal length focusing in y-section and then after a short distance the short focal length to focus in x-section. Determine the image distance to have a good focusing in the x-section only and the distance between the lenses to have a good y-focusing. Hint: use TRCX, TRCY in the merit function to optimize the spot size in x or y section. b) Calculate the spot diagram and the wavefront for the system. Explain the results. c) Calculate also the transverse aberrations. Why are differences seen between the x and the y- section? d) Now reduce the input beam diameter to 10 mm and calculate the point spread function. Why is the spot not circular symmetric? e) To correct the residual errors, introduce a thin plate after the system and define one of the surfaces as a sagittal Zernike surface. Solution: Correct the shape of this surface by taking 33 Zernike polynomial terms into account Hint: In order to optimize the Zernike surface, allow the indices of even symmetry to be variables: Zernike terms no. 4-5, 6-12, 13-17, 18-21, 22-28, 29-32, 33. Don t forget a circular symmetric spot is required! Is the system now diffraction limited? How large can the input beam diameter be made to remain diffraction limited result? a) The System looks like the following. It is necessary to rotate the second lens by 90 around the z-axis. Furthermore it is important, that a pure x-section focussing in the default merit function or the quick focus option does not perform the correct result: it is only the corresponding diameter of the spot evaluated, but for a complete x-y-filling of the pupil. Therefore it is necessary to define the operands TRCX for only Px sampling and TRCY for only Py sampling.
An alternative is the determination of the distances by forcing the marginal ray heights in the x- and y- section to be zero. This can be done paraxial or real. b) The spot diagram and the wavefront shows a rather good correction only along the axes, in the diagonal direction, the performance is poor.
The correction in the x-section is perfect, but in the y-section, we get a residual spherical aberration. The reason for this is the second lens, which acts as a plane plate for the strongly and nearly perfect focussed rays in the y-section of the first lens. This can also be seen from the Seidel contributions. In the wavefront
plot it is also seen, that in the x-section, we really have deep blue but in the y-section, the residual aberration produces a brighter color. c) If the aperture is reduced, we get the point spread function (512/128 sampling points): If the marginal rays in x- and y are inspected, we have due to the different focal lengths the numerical apertures NAx = 0.156, NAy = 0.125 Therefore there are two reasons for the difference in the two cross sections: 1. the different numerical aperture 2. the different spherical aberration d) If the corrector plate is inserted, we select 33 Zernike polynomials and allow the indices of even symmetry to be changed. In addition, the focus correction is allowed. Therefore the indices 4 5,6 12,13 17,18 21,22 28,29 32,33 are made variable. The merit function is now also changed and a circular symmetric spot is required.
The result looks as follows: The system now is quite good corrected. The ellipticity of the Airy pattern demonstrates the different apertures in the two sections.
By enlarging the diameter of the aperture it is seen, that for a value of approximately 20 mm the system exceeds the diffraction limit. The reason for this is, that th available 36 Zernikes only contains a certain limited slope in the edges of the pupil.