2016-5-1 Mateusz Oleszko, Chang Liu Friedrich Schiller University Jena Institute of Applied Physics Albert-Einstein-Str 15 07745 Jena

Solution: Lens Design II – Part 5

Exercise 5-1: Cooke triplet (Courtesy Kristina Uhlendorf) In this exercise a Cooke Triplet with a focal length of 52 mm will be optimized as well as the importance of vignetting for photographic lenses will be explained and analyzed.

a) Load file ‘Ex15.1 Cooke Triplet-1.zmx’. Optimize the system using the already defined merit function without changing glasses.

b) Investigate the impact of the glass choice of the negative lens on the system performance. c) Use the first and the last lens surface to vignette the maximal field angle by 50 percent. Take

care that the chief ray still passes the center of the pupil stop. What happens? Solution:

a) The Cooke triplet investigated here has a FoV 45.2 deg and an F-number of 3.5 with the object at

infinity. Five wavelength equally weigthed are defined: 450 nm, 500 nm, 550 nm, 600 nm and 650 nm. The reference wavelength is 550 nm.

The start system looks as follow:

There are exactly eight effective independent variables or degree of freedoms available for the control of optical properties. These major variables are six lens surface curvatures and the two airspaces between the lenses. The six curvatures can als be viewed as three lens powers and three lens bendings. Recall that there are seven primary aberrations (five monochromatic Seidel-aberrations, first-order longitudinal and lateral color). Thus, the Cooke triplet has just enough effective independent variables to correct all mentioned aberrations plus the focal length. But there are no variables available for controlling the higher-order aberrations. Therefore to balance the higher-order aberrations the low-order aberrations will be not corrrected to zero.

Glass selection A Cooke triplet is an achromat. Thus, each of the two positive elements must be made of a crown type glass and the negative element must be made of a flint type glass. For practical reasons and with no loss of performance, both positive crown elements are usually made of the same glass type. The airspaces in a Cooke Triplet are a strong function of the dispersion difference between the crown and the flint glass. A dispersion difference that is too small causes the lens elements to be jammed up. On the other side, a dispersion difference that is too large causes the system to be excessively stretched out. The difference in the refractive index also enters into the optical solution. To help reduce the Petzval sum to flatten the field the positive lenses should be made of a higher-index crown glass and the negative lens should be made of a somewhat lower flint. An excellent high-index crown glass of reasonable cost is N-LAF21. A first guess for a matching flint glass is N-SF15 (See glass map).

In the merit function besides the default merit function for spot size optimization the focal length, axial colour and overall length is defined. After running the optimization the following performance is achieved.

b) The first choice of the negative lens flint type glass was only a guess, to find the best glass choice we will now change the glass type along the glass line in the flint range by hand. (n-sf15 n-sf1 n-sf10 n-sf4 n-sf14). The best performance can be achieved with N-SF4.

c) Like the vast majority of camera lenses the Cooke Triplet is to have mechanical vignetting. Mechanical vignetting is useful for two reasons. First, the smaller lens elements reduce size, weight and cost. Second and more important, vignetting allows better performance. Here a vignetting of about 50 % is achieved setting the aperture of the front surface to 17 mm and of the last surface to 14 mm. With this choice the chief ray of the maximal FoV nearly passes the stop in the center. This is important to avoid a complete vignetted field when stopping down. After the final re-optimization the following system performance is achieved.

Exercise 5-2: New Achromate and wide field system A new achromate is a cemented component, that is optimized for large field applications. In this case, the Petzval condition is the basis for the refractive power distribution and the achromatization is no longer a correction goal. In this case, a large difference in the refractive indices is more important than a large spreading of the Abbe number.

a) Establish a classical achromate made of BK7 and SF6 with focal length f = 100 mm and an object at infinity from the scratch. Insert a stop 20 mm before the lens and select the wavelength 546.07 nm and an entrance pupil diameter of 4 mm. The system should be used on axis and for a maximum field angle of 20°. Optimize the radii for the default merit function and evaluate the system performance.

b) Extend the system to a finite imaging system with magnification m = -1 by doubling the system. The first group of lenses should by reversed and the stop is in the middle of the system to get a complete symmetric layout. What is the dominating aberration? Insert a second field point in the zone of the field at 14°.

c) Now the second part of the symmetric system is scaled down by a factor of 2. This delivers a system with a magnification of m= -0.5. Therefore we no longer have a symmetric layout. What happens with the system quality?

d) To improve the system add two meniscus lenses made of SK12 near to the cemented components towards the stop with the exact 1:2 scaling. Optimize the radii to improve the system while preserving the mirror-symmetry. Can the astigmatism be removed?

e) Now optimize the radii without any anti-symmetric constraints. Show, that the performance can be improved significantly. Finally enlarge the numerical aperture of the system to a value, that is just below the diffraction limit. To have some more degrees of freedom, the distances around the meniscus lenses are also set as variable. To guarantee a useful solution, the magnification (PMAG -0.5) and the overall length of the system (TOTR 190.5) must be forced

in the merit function to be constant. Show, that the field curvature is corrected by the meniscus shaped lenses to a very good flattened field.

Solution: a) The setup of a classical achromate is very similar to the exercise 5-2 and requires the 4 lines with EFFL, AXCL, RAYY, PARY as requirements. The approach delivers the following system, the thicknesses of the lenses are selcted arbitrary:

The performance is nearly diffraction limited, a residual astigmatism is seen for the field point.

b) Before the system if generalized to a symmetrical one, it should be noticed, that the object is then of finite distance with a size of y = 35.3 mm according to the image size in the new achromate and the numerical aperture defined by NA = D/2/F = 0.02. The extension of the systems gives the following layout.

The astigmatism is enlarged in the field, this is seen at the line-shaped focus at 20° field.

c)

The scaling can not be performed by a scaling factor 0.5 for a limited number of surfaces in Zemax. The radii are calculated by setting a pick up of the corresponding surface with a scaling factor of -2 (not -0.5!). It is seen, that the astigmatism for the outer field is still the limiting aberration. d) The system now looks as follows. For optimization, the focal length, which is now in the range of 66 mm should be fixed in the merit function.

It is seen, that the performance is much better, there is nearly no longer astigmatism inside. e) A complete arbitrary optimization of the radii gives the following result. The exact values depends on the initial values of the calculation. Usually, the first lens should be bended different and 'looks' towards the stop.

In the second step, the entrance pupil diameter is enlarged to a value of 7 mm. In the merit function the two requirements are added:

The rms-plot against the focus shows nearly coincident image locations over the field. This means, that the field curvature is well corrected.

Exercise 5-3: Anamorphotic Diode collimator A semiconductor diode with wavelength 650 nm and the divergence / aperture values 0.4 / 0.1 in the fast and slow axis respectively should be collimated in a circular beam with a diameter of approximately 8 mm. The collimated beam is now focused into a fiber with numerical aperture of NA = 0.1.

semiconductor

diode

NAy = 0.4

NAx = 0.1

= 650 nm fiber

NA = 0.1

L1

aspherical

collimator

fast axis

L2

cylindrical lens

L3

L4

focussing lens

circular beamD = 8 mm

cylindrical lens

Find a solution for this problem with only available catalog lenses. Is the setup diffraction limited? Explain the shape of the residual spot pattern. What are the reasons for the residual aberrations in the system? What can be done to further improve the result? Discuss possible steps to get a shorter system. What are the consequences of a compact layout?

Solution If the desired beam diameter after the collimation of the fast axis is 8 mm, the focal length of the first lens is

mmNADf y 10/2/

Since the numerical aperture of the fast axis is high, it is recommended to use an aspherical collimator lens, which is corrected for spherical aberration on axis. If such a lens is found in the lens catalogs, it must be considered: 1. the lens should be used without cover glas plate 2. if a working wavelength near to the 650 nm is found, it is an advantage Possible solution: Catalog Asphericon, lens with the No A12-10HPX

Necessary steps to process this lens: 1. load the lens 2. turn around 3. set NA to 0.4 and vignetting factors in field menu to VCX = 0.75. Alternatively, the front surface of the collimating lens can be established by an elliptical aperture. If the axes of the ellipse are set in a ratio of 1:4, the desired light cone is obtained in approximation. In this case exactly the tan(u) values are related and therefore the numerical apertures as sin(u) values are only roughly obtained.

4. change wavelength to 650 nm 5. optimize first distance to collimate this wavelength (default merit function, with criterion: direction cosines). Alternatively, the option QUICK ADJUST can be used with the first distance as variable and the angle spot as an afocal criterion.

A footprint diagram shows the elliptical beam cross section behind the lens.

In the next step, a Galilean telescope with factor = 4 must be found to enlarge the diameter of the x-section to the same value as in the y-section. First a negative cylindrical lens with a rather short focal length must be found. Possible solution: Lens with 1 inch negative focal length in the catalog of Melles Griot: RCC-25.4-12.7-12.7-C

The lens is inserted behind the collimating asphere and rotated around the x-axis by 90° to work in the x-section.

The distance to the collimator is not very relevant and is fixed to be 5 mm.

For a Galiean telescope with factor 4, the second lens must have a focal length of 4x25.1 mm = 100.4 mm. In the same lens catalog one can found the following lens: RCX-40.0-20.0-50.9-C

The lens is inserted, turned around to get a better performance and also tilted by 90° in the azimuth. A first guess gives a distance of 100-25=75 mm between the telescope lenses to get a collimated x-section. But from the spot diagram with direction cosine option it is seen, that the angle distribution is not equal in both sections. Due to the finite positions of the principal planes of the lenses, the distance must be optimized with an angle criterion default merit function. Again as an alternative, the QUICK ADJUST feature can be used to find the optimal lens distance in the telescope. Spot diagram before and after this focussing operation with the same scale:

The footprint diagram now shows a rather circular cross section. The residual error can be neglected and comes from the fact, that for this wavelengths, the catalog focal lengths are not exact.

The data are now the following:

To focus the beam into a fiber with numerical aperture 0.1, the focal length must be not smaller than f = 4.32 mm / 0.1 = 43.2 mm. A lens of approximately this size can be found in the catalog of Melles Griot as an achromate. This helps in getting a better correction: LAO-44.0-14.0

This lens is inserted to complete the system. Finally the last distance is optimized to get a minimal spot size. It is seen, that the spot is nearly diffraction limited.