Tutorial Zemax 8: Correction II

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$1 High-NA Collimator An achromatic system is typically diffraction limited for a numerical aperture not larger then NA = 0.1. For a point on axis, by adding an aplanatic-concentric meniscus-shaped lens, the numerical aperture can be increased by a factor correponding to the refractive index of the lens.$

1 Tutorial Zemax 8: Correction II Correction II High-NA Collimator Zoom-System New Achromate and wide field system Correction II 8.1 High-NA Collimator An achromatic system is typically diffraction limited for a numerical aperture not larger then NA = 0.1. For a point on axis, by adding an aplanatic-concentric meniscus-shaped lens, the numerical aperture can be increased by a factor correponding to the refractive index of the lens. This principle is shown in the exercise 6.1 a) Load an achromate with focal length f = 100 mm from a vendors catalogue with a diameter of at least 30 mm. Take the wavelength to = nm. Reduce the entrance pupil diameter to achieve a diffraction limited system according to the Marechal criterion of W rms < /14. Add one aplanatic-concentric meniscus lens of a high-index medium of appropriate distance and thickness. b) Now add more meniscus lenses, until the image space NA is larger than 0.6. The thickness of these lenses should again be chosen appropriate to get manufacturable components (ratio of diameter to thickness not larger then 10). Show, that the spherical aberration contribution of the meniscus is zero for all orders of spherical aberration. What is the resulting free working distance of the final system? c) Now try to further improve the system by numerical optimization. Is it possible to get better performance? Think on the definition of proper constraints to get a reasonable and comparable system. Solution: a) In the CVI Melles&Griot catalog, the lens LAO is loaded as one possible f=100 achromate. The size is reduced with the help of the menue Tools/Misc/Scale lens with a factor of 0.1 and the wavelength is set to nm. Furthermore for a more comfortable layout, an additional surface is added in front of the lens to see the incoming rays. It is seen in the menue RMS vs defocus, that the numerical aperture of 0.15 in the image space (for this achromate, may be different for other choices).

$2 By try and error, the entrance pupil size is reduced to 23.66 mm with an image sided numerical aperture of 0.117 to get exactly a diffraction limited system.$

2 2 By try and error, the entrance pupil size is reduced to mm with an image sided numerical aperture of to get exactly a diffraction limited system. The choice of this menue options makes sense, because be changing the aperture, a small change of the best final plane occurs. If a refocussing is made with the Quick focus option, the criterium must be the same in the defocus diagram. There are small differences between a spot and a wavefront optimization. As a material for the aplanatic concentric meniscus lens, the glass LaSF35 is used, which has a refractive index of for the actual wavelength. The lens is placed in a distance of 1 mm and the thickness should be 3 mm. To get the desired shape of the bended meniscus, the front surface is forced to be aplanatic by the corresponding solve at the surface. For the rear surface, the solve option Marginal Ray Normal forces the marginal ray to pass this surface concentric. The resulting system now has a numerical aperture of NA=0.232.

$3 b) If now additional components are added, a factor of 0.6/0.232 = 2.59 is needed. Therefore a refractive index of at least 1.61 is necessary and must than be applied twice.$ If a ray trace of the marginal ray is calculated we see, that the paraxial and the real ray angle is quite different and that the real ray fulfilles the requirement of NA > 0.6 with a value of NA = 0.

If a ray trace of the marginal ray is calculated we see, that the paraxial and the real ray angle is quite different and that the real ray fulfilles the requirement of NA > 0.6 with a value of NA = 0.

3 3 b) If now additional components are added, a factor of 0.6/0.232 = 2.59 is needed. Therefore a refractive index of at least 1.61 is necessary and must than be applied twice. For example the glass F6 with an index of 1.63 is feasible. If the numerical aperture is evaluated, we see, that the paraxial numerical aperture angle is only NA = If a ray trace of the marginal ray is calculated we see, that the paraxial and the real ray angle is quite different and that the real ray fulfilles the requirement of NA > 0.6 with a value of NA = The reason for this strange deviation is the fact, that the aplanatic surfaces cause no spherical aberration. This means, that the intersection length in the image plane is the same as in the paraxial case. To get this goal, the real ray refraction must have a change in the angle of the ray, which has this large value here.

To achive this result, the focussing must be made identical.

4 4 If a higher index glass is used, for example SF6, we get a higher numerical aperture also for the paraxial case. If the Zernike coefficients are calculated for the complete system and tfor the achromate alone (surface 4), we get the same spherical coefficients (c 9,c 16,c 25 ). To achive this result, the focussing must be made identical. Due to the large difference in th enumerical aperture, the defocus coefficient c 4 is not identical, if the Quick focus is used as a criterion. If the last distance is change by 0.00m mm, the c 4 coefficient is nearly equal and then also the the spherical aberrations agree. The free working distance now is on axis mm (last air thickness. If the cursor is set onto the intersection point of the last surface we see, that the z-difference from this point until the image plane is only 8.7 mm. This is the real free working distance.

63 must be set invariant. The optimization has made only very small changes to the radii.

5 5 c) If now the radii of the new three lenses are defined as variables and a default merit function is used, the Strehl definition can be improved from S = to S = 1. To have a fair comparison, the focal length of f = must be set invariant. The optimization has made only very small changes to the radii. By inspection of the Seidel diagram it is seen, that now the spherical contributions of the surfaces have changed to correct the residual errors of the achormate, which is not changed here.

a) Determine the second and the third focal length and the corresponding distances of the lenses.

6 6 8.2 Zoom-System A zoom system with afocal setup should be implemented with 3 lenses, a plus-minus-plus setup a zoom range of 9. The input diameter of the collimated beam is 12 mm, the first focal length should be f 1 = 100 mm. The zoom factor should be located symmetrically around the magnification = 1. a) Determine the second and the third focal length and the corresponding distances of the lenses. Formulate a merit function, that forces the corresponding conditions and values of the different zoom positions. Take in mind, that the overall length of the system cannot be constant and that the system should be afocal for every zoom position. Establish the system by ideal lenses as a multi-configuration system with 3 zoom positions. b) The paraxial layout of the system can also be calculated by the set of formulas of the lecture. Prove the numerical results by the analytical solution. c) Change the different lenses to real single lenses of standard glasses and real air distances and perform a multi-configuration optimization and extend the multiconfiguration to 5 zoom positions. The default wavelength = 0.55 m is chosen. What can be said about the performance of the system? d) Now to improve the result, the moving components should be build as achromatic doublets. Try to find appropriate focal lengths in the vendor catalogs. Re-optimize the system. Can an overall diffraction limited design be obtained? What is the limiting reason? Solution: a) The afocal system is symmetrical for the = 1 constellation. Therefore also the third lens should have the focal length f 3 = 100 mm. In the extreme moving positions of the system, one thickness shrinks to zero. The unknown parameters are the overall lengths of the system and the focal length f 2 of the moving negative lens. The multi configuration editor looks like the following picture: and in the merit function the afocality and the different magnifications are required by defining the corresponding operands for the marginal ray direction (PARB) and height (PARY) for every configuration.

04mm, f2 mm F 25 which are identical to the numerical values. The following figure again shows the basic principle of the layout.

7 7 In principle, we have in the merit function 4 variables and 6 requirements. But due to the symmetry, the problem has a unique exact solution. b) The set analytical formulas give the results 2 max 1 2 L 2tm 2 100mm 100mm F1 max max 1 2 tmax tmin 100mm mm F 3 1 max F1 max mm mm, f2 mm F 25 which are identical to the numerical values. The following figure again shows the basic principle of the layout. In particular it should be taken in mind, that the symmetrical setup has a significant larger total length of the system. An invariant setup is physically impossible. = 1/3 f 1 f 3 f 2 f 1 f 3 = 1 f 2 f 1 f 3 = 3 f 2 L f 2 f 1

8 c) First, the two additional configurations are established. The corresponding marginal ray heights are forced to be 4 and 12 mm corresponding to magnifications of 2/3 and 2.

Here due to the asymmetric use in the extreme zoom positions, the forced symmetry in the shape of the first and the third lens is given up.

8 8 c) First, the two additional configurations are established. The corresponding marginal ray heights are forced to be 4 and 12 mm corresponding to magnifications of 2/3 and 2. The multi configurations editor and the layouts now looks like the following figures. In the next step, all lenses are generalized to singlets and the system is optimized numerically. Here due to the asymmetric use in the extreme zoom positions, the forced symmetry in the shape of the first and the third lens is given up. The following points should be taken into account: - the flag afocal image space is set - the bending of the lenses is realized by setting the focal length to remain constant with the help of a solve for the rear radii. - as lens thicknesses, a first guess of 3, 2 and 7 mm are chosen - as material BK7 is used - to get realistic solutions, now in the first and the last configuration the zero-thicknesses are set to variable to avoid non-physical solutions - in addition to the merit function of the paraxial layout, a default merit function with angle radius as criterion and air thicknesses larger than 0.5 mm are forced. If the optimization is run it is seen, that due to the changed diameters of the ray bundel for the different zoom positions, a collision of the second and the third lens takes place in the last configuration. Therefore a minimum air thickness of 3 mm is used in the merit function. Now the optimization run gives the following solution.

$9 If now the spot diagram is observed it is seen, that the performance is significantly far off from beeing diffraction limited.$

9 9 If now the spot diagram is observed it is seen, that the performance is significantly far off from beeing diffraction limited. d) In a last step, the three lenses are substituted by achromates to generate a larger number of degree of freedom to correct the system. As an example, the lenses AAP of CVI Melles Griot and from the Edmund Optics are possible candidates. To guarantee that the focal lengths of the doublet components remain constant, three lines with the EFLY are defined to specify the partial focal lengths to the lens-groups. In a first step, only the air distances are allowed to be varied. It is seen, that the diameter of the third lens is too small and the thicknesses must be enlarged. The results looks fine only for the configurations 2-4.

10 In a third step, also the radii are allowed to be changed

$Then the results looks better, but is still not diffraction$ The reason for this bad result is, that the overall length of

10 10 In a third step, also the radii are allowed to be changed and the thicknesses are appropriate. Then the results looks better, but is still not diffraction limited. The reason for this bad result is, that the overall length of the system is quite short for the ambitious zoom factor and numerical aperture. In reality, in addition the broadband behavior for several wavelength and the correction for finite field points must be considered. This shows, that the correction of zoom system is quite complicated.

$In this case, the Petzval condition is the basis for the refractive power distribution and the achromatization is no longer a correction goal.$

11 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 nm and an entrance pupil diameter of 4 mm. The system should be used on axis andfor 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= 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 mensicus shaped lenses to a very good flattened field. Solution: a) The approach delivers the follwing system: The performance is nearly diffraction limited, a residual astigmatism is seen for the field point. b) the extension of the systems gives the following layout.

12 The astigmatism is enlarged in the field, this is seen at the line-shaped focus

c) The scaling can not be performed by a simple single command in Zemax.

the values, the radii are calculated by setting a pick up of the corresponding

12 12 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 simple single command in Zemax. Therefore for the values of the distances and thicknesses are set by hand to half the values, 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:

13 13 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:

14 14 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:

15 15 The rms-plot against the focus shows nearly coincident image locations over the field. This means, that the field curvature is well corrected.

1.1 Singlet. Solution. a) Starting setup: The two radii and the image distance is chosen as variable.

1.1 Singlet. Solution. a) Starting setup: The two radii and the image distance is chosen as variable. 1 1.1 Singlet Optimize a single lens with the data λ = 546.07 nm, object in the distance 100 mm from the lens on axis only, focal length f = 45 mm and numerical aperture NA = 0.07 in the object space.