2015-01-27 Manuel Tessmer [email protected] Minyi Zhong [email protected] Herbert Gross [email protected] Friedrich Schiller University Jena Institute of Applied Physics Albert-Einstein-Str. 15 07745 Jena Exercises Advanced Optical Design– Part 7 Solutions 7.1 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 the 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. b1) Remember that this system works as an achromate: the positive elements should be made of a crown type glass and the negative element must be made of a flint type glass. Next, make clear mathematically why a preferred choice of crown and flint glasses with Abbe numbers 21 d and refractive indices n22 n dn requires maximal difference dn and d , having the same sign and not the opposite! Therefore assume an achromate with total power F, buried surface with zero curvature, having outer curvatures C1 and C3 with n and as above. How do C1, C3 and dv turn out as functions of Ftot, and 1 ? Which choice relaxes the curvatures most? b2) For the flint glass: follow the flint glass section and look for the glass with the best performance. c) Use the first and the last lens surface to vignette the maximal field angle by 50 percent. Verify the setting by taking a look at the vignetting plot. Take care that the chief ray still approximately 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. 7.2 Schiefspiegler Mirror System Load the 2-mirror system Ex5-17 v1, which contains two curved mirrors M1, M2 to build a z-shaped setup with intermediate focus. Due to the simple spherical mirrors, which are used under a finite angle, the system performance is bad and suffers from astigmatism and coma. a) Correct the intermediate focus by using a toroidal mirror to replace the first mirror M1. Delete the second mirror M2. Hint: the toroidal mirror is modelled as biconic in Zemax without using the conic constants, only the X radius is variable. For this correction, the image plane should be defined in the plane of the intermediate focus (focus of the toroidal mirror), and the afocal-option must be removed. What are the remaining aberrations? (b) Add a second toroidal mirror at the same location of M2. Why are the radii of the two toroidal mirrors not perfectly the same after optimization? (c) To further improve the system, define the second mirror now as a freeform surface and optimize the quality of the system. Hint: define the second mirror as extended polynomial. See what surface sag it has after optimization, and what aberrations it corrects for the first mirror. Solution: a) Due to the astigmatic effect of the inclined spherical mirror, the two principal radii of curvature of the mirror are splitted no and the astigmatism is corrected (Zernike c5). The remaining limiting aberration is coma (Zernike c8). b) With two toroidal mirrors, the system looks like this: Due to the residual coma of the first mirror, the second mirrors shows some induced aberrations and therefore the optimization tries to compensate this and gives slightly different radii of curvature as result. The residual aberrations are again coma. c) The mirror M2 is defined as Extended Polynomial. In the Extra Data Editor, the normalization radius is fixed to be 10 mm. In the primary data, the radius of curvature is set to the fixed value of r = 100 mm. To correct for the astigmatism, only the terms with X0Y2 and X2Y0 are optimized in the first step. The result for the spot is now the same as in b). In a second step, more degrees of freedom are allowed and the second mirror is developed to compensate all residual aberrations be a free form shape. Due to the small numerical aperture, there are nearly no higher orders in the aberrations, therefore the system is quite good, if the 3rd order in the polynomial of the surface are included. If the radius effect is subtracted (by setting the radius on M2 to zero), the surface shape looks like the following figure. The deviation from a sphere therefore is primary toric with a small amount on coma. If the coefficients of the free form surface are inspected it is seen, that only a third order coma correction in y- direction is necessary to get this famous result. 7.3 Skew spherical aberration Usually spherical aberration is defined for an axis point and therefore is only considered on axis. If in a system an off-axis field point is considered, typically we have coma and the spot shows a certain asymmetry. The stop position influences the coma and there is a so called natural stop position, for which the asymmetry is disappearing. This setup than contains a residual off-axis spherical aberration, which is called skew spherical aberration and is an error of higher order. a) Establish a single lens of BK7 with plane front surface and a focal length of 100 mm at a single wavelength λ=780 nm with thickness of 20 mm and an object distance of 200 mm. Determine the image plane to be the paraxial Gaussian image plane on axis. Now set a finite field height of 10 mm. The stop of the system should be movable located in front of the system. The numerical aperture should be NA = 0.05. b) Establish a slider and move the stop position in the range 160...200 mm distance from the object. What is the value of the stop position, for which the coma vanishes? What is then the shape of the spot diagram? How does the ray caustic look in this case? What happens, if the numerical aperture is increased to NA = 0.15? Solution: a) Data: b) The coma Zernike coefficient of third order is zero for a distance of 178.08 mm. The shape of the spot is then nearly elliptical dur to the residual astigmatism. The ray caustic in this case is completly swymmetrical around the chief ray, although a spherical aberration can be seen.