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# Note for OPTI 517 with Professor Sasian at UA 2

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Date Created: 02/06/15

OPTISl7LensDengn Course Summary Proteep Mallik 121503 Note Christoph Baranec and I worked on the first section theory together Introduction OPTI 517 Lens Design was a course designed to help us learn how to use software to design lenses and lens systems In order to be able to do this we learnt basic concepts of imaging aberration aberration control and balance system performance ray tracing and tolerancing The software used to aid us in designing lenses included ZEMAX OSLO and Code V I used OSLO for HWl and Code V only partially for HW4 All other homework assignments were done using ZEMAX In the following sections a summary ofthe course is presented Imaging Concepts When mapping from one space to another space on needs to map points to points lines to lines and planes to planes To do this mathematically one uses the collinear transform equations Certain systems may display various types of symmetry Depending on the types of symmetry the collinear transform equations may be simplified When there is enough symmetry the system can be considered axially symmetric An example perfect axial symmetric system would be an infinite pinhole camera The General Collinear Transform Equations are a1X b1Y ch d1 a0X b0Y coZ do V a2Xb2YczZd2 a0X b0YcoZd0 a3X b3Y 032 d3 a0Xb0YcoZd0 The Collinear Transform Equations For an Axially Symmetric System are alX 002 do alY coZal0 CSZd3 coZal0 When one considers the origins to be used in the axially symmetric equations above one can make de nitions and further reduce those equations De ning the focal planes as the planes conjugate to in nity the transverse magni cation m as Y Y and the principle planes as the planes of unit magni cation one can consider the origins being at either the principle planes or the focal planes When one makes the principle planes the origins the collinear equations for axially symmetric systems reduce to the Gaussian equations When the focal planes are the origins the equations reduce to the Newtonian equations These are shown below The Gaussian Equations are l l Zf Zzli f m al m f ml f L 1 Z Z and the Newtonian Equations are E i f m Z 7 m f ZZ quot f f are the back and front focal lengths z z are the object and image distances m is the magni cation AK h F Z I O O Z F h V Gaussian reduction of system First Order Theory First order theory comes about by examining paraxial image formation image formation that happens near the axis where the relations Sinu Tanu u is observed In deriving the formulas that govern paraxial imaging on finds the refraction equation u nu 7 y I And the transfer equation y y n t One also finds the imaging equation n z nz 4 Using the relations f n I and f n 4 One then sees that paraxial image formation is mathematically congruent to collinear transformation Stemming from rst order imaging theory and the equations listed above are the ideas of pupils for the aperture stop The aperture stop is the limiting aperture of an imaging system The images of the stop in object space and image space are the entrance and exit pupil s respectively If there are any apertures other than the stop that clip incoming beams of light then the imaging system is displaying vignetting The location and size of the pupils is important for practical design considerations radiometry as well as aberration theory and control An imaging system s rst order properties can described by several signi cant on aXis points and distances Points Objectimage points conjugates to each other F rontrear focal points conjugate to in nity Nodal point Points of unit angular magni cation located at principle points if system in air Principal points Points of unit magni cation conjugate to each other Anti principal points Points of negative unit magni cation conjugate to each other Entranceexit pupil Images of aperture stop in objectimage space System vertices Edge of elements on optical aXis Distances F rontrear focal lengths distance from frontrear principle planes to frontrear focal points F rontrear focal distances distance from rst last surfaces to frontrear focal points Objectimage alistances distance from frontrear principle planes to objectimage Principal point separation distance from front principle to rear principle plane Seg uential Ray Tracing Local coordinates are used most often in ray tracing Surfaces are expressed in local coordinates and rays are traced between coordinate systems These surfaces are placed in sequence and the ray proceeds from the object to the image surface The ray is a unit vector s Li Mj Nk The coordinates of a point along the ray are found using the origin of the ray and its path length That is R R0 As where the coordinates of the end of the vector R are determined The most common surfaces are spherical lenses The following gure illustrates how rays are traced to and from such surfaces XI Yt 0 The ray is traced to an intersection with the surface The direction cosines of such a ray are known so only the length to the surface needs to be speci ed The transfer equation can be used to do this Once the transfer has been accomplished then the power equation is used to nd out how the ray is bent After that the transfer equation is used again to propagate the ray to the next surface This process can be done in an iterative way to check if the ray is hitting the surfaces at the appropriate locations During each of these transformations the ray tracing algorithm keeps track of the ray coordinates Introduction To Aberrations In the previous discussion it was seen that in collinear transformation and thusly first order imaging points where mapped to points lines to lines and planes to planes This is an ideal mathematical construct in which perfectly spherical wavefronts converge or diverge from object or image points This however is not the case with any real physical imaging system Geometrical deviation outside of diffraction from this form of mapping ie perfect spherical waves is considered an aberration The form of specific aberrations in an axial symmetric system can be described by the deviation from a perfectly spherical wavefront a reference sphere Aberration Function The mathematical form of the specific aberrations that would be associated with an axial symmetric optical system can me inferred from the symmetry displayed by on and off axis objects and images The form can also be found from rigorous mathematical considerations The wave aberration function is a scalar function that describes the optical path difference OPD between the reference sphere and the actual wavefront This function is found to be dependant on the aperture vector and the field vector p and H Aperture and field vectors I Aperture vector Field Vector Optical axis Exit pupil Image plane Figure borrowed from class notes prepared by Jose Sasian httpwwwopticsarizonaedusasianopti5171 Since the wavefront aberration function is scalar it can be considered a dot product of these vectors The result takes the form of WH p 6 VVzoon Wino2 Win Hp cos 6 W040p4 Wme3 cos 6 WmHzp2 cos2 6 WmHzp2 VV311H3p cos 6 VVAOOH4 Equation borrowed from class notes prepared by Jose Sasian httpwwwopticsarizonaedusasianopti5171 Note that the last two terms W020 Defocus W111 Tilt in the top row of the equation above describes first order ray results second order wave results These would go to zero at the image calculated from the Gaussian equations The first term W200 Field Dependant Phase in the first row represents a phase term that does not affect imaging Fourth Order Aberrations The second rows of terms in the above equation are defined as the fourth order aberrations 4Lh order combination of p and H They are W040 Spherical Aberration W131 Coma W222 Astigmatism W220 Field Curvature W311 Distortion W400 Field Dependant Phase All of the fourth order coefficients WW can be derived from geometrical considerations The results are given as nu39n39yc nFn399c Chromatic Aberrations The above equations describe the results for monochromatic light However monochromatic light is not used in most imaging systems In most imaging systems the light collected and imaged has some bandwidth Chromatic aberrations describe the variations of the lens properties with color The chromatic aberrations considered where 5W111 axial color uniform change of image location across eld with wavelength and 5W020 lateral color change of magni cation with wavelength Higher order chromatic aberrations like spherochromatism where mentioned but not studied Glasses differ by their dispersion and index Dispersion is described by Abbe number v nd 7 lnf 7 nc The chromatic aberratio s are iven as le320 dIWl l L Sixth Order Aberrations As previously mentioned 43911 order aberrations are named by the order of their p and H combination They are also representative of 4Lh order geometrical considerations Both of these considerations can be taken to higher orders Because of symmetry these orders will all be even so the next order would be 63911 order aberrations The 63911 order aberrations are closely related to the 43911 order ones and in many cases display nearly the same functional form but of higher degree For example the departure from the reference sphere of 4th order spherical goes as p4 where 6Lh order spherical goes as p6 New forms are also added like sagittal amp tangential spherical aberration and elliptical coma Deriving the 6Lh order aberrations geometrically it is found that there are intrinsic and extrinsic contributions Intrinsic refers to a purely 6th order effect which is present regardless of other lower order aberrations Extrinsic refers to contributions to the 63911 order terms from 4Lh order terms For example the value of 43911 order spherical affects the value of 6Lh order spherical Thin Lens Contributions From Shack s notes it can be seen that the aberrations of a thin lens are as follows The above are in the form of the Seidel sums which are related to the wavefront aberration coefficients W Stop Shift Eg nations Also from Shack s notes the effect ofa stop shift can be seen as StcpShi Et d5 0 I d d9 SI SI Y dsuI2d 95 I d51v0 d 7d 9 Sm3 Y dCL0 Asphe c Contributions Lastly from Shack s notes the contribution of an aspheric surface on 4Lh order aberrations can be given as Asghe cContnhItlms dmo a 1 dWBl E 1 dm E dVbo0 1 dW l E le3200 dIWI1L Where a is given by a sc3y4An for a conic and 782 K the conic constant Aberration Reduction S gherical Aberration Spherical aberration can be described by either its Seidel sum or the equation provided earlier for its wavefront aberration coefficient They are respectively SI y4 I3 61 w040 18 A2 Auny where 61 is a structural aberration coef cient de ned through Shack s notes From the above it is noted that spherical aberration depends on the quartically on the extent of the aperture and cubically on the power of the element The Siedel sums are over all surfaces of a system so one way to reduce spherical aberration is to split a lens By Gaussian reduction thin lenses with no separation add in power Therefore SI would be reduced to a quarter of its original value when summed It is also found that spherical aberration is a function of the bending of a lens Bending of a lens does not cause spherical aberration to go to zero but it does reduce it The combination of bending and splitting can be used effectively 10 It was also seen in the homeworks that index plays a role in reducing spherical aberration It was also seen that at given conjugates aspheric surfaces could completely correct spherical aberration Aglanatic Points The terms for spherical aberration coma and astigmatism given by the wavefront aberration coefficients all depend linearly on Aun These three aberrations will all go to zero if there is either no refraction at a surface or the term Aun 0 Using this condition and the Gaussian imaging equation it is fount that object points defined by the position s n nROC will be imaged with no spherical aberration astigmatism or coma These points are called the aplanatic points Chromatic Aberration Chromatic aberration can be corrected through the use of an achromatic doublet a lens made of two different elements of two different glasses The doublet corrects for color through means of the Abbe number and power of the elements in contact The result is 1 system VlVI39VZ and 2 system 39VZVI39VZ There are multiple forms for the solution of an achromatic doublet Single glass lenses may be used in a system to produce single glass achromats These solutions are achieved through the use of their governing wavefront aberration coefficient equations provided above An example of this is the Huyghenian eyepiece Another way of achromatizing is the idea of a buried surface A buried surface has glass of the same refractive index on each side but with different dispersions A phantom stop may also be used to correct for color If a system displays axial color then lateral color can be modified The theory for this is found by exploiting the chromatic terms given by the stop shift equations provided The axial color can be corrected by merely moving the stop of a system While the stop is in this modified position axial color is corrected The stop is then moved back to its original position and the color correction is maintained Symmetry about the stop corrects the odd aberrations coma and distortion which also includes lateral color Coma As mentioned above coma is an odd aberration so it can be corrected by symmetry about the stop The equation for the wavefront aberration coefficient of coma can be rewritten from the form provided through the use of the Lagrange invariant In doing this it is found there is a dependence on ybary This then becomes a useful design variable 11 When an optical system displays the Abbe sine condition it is free of coma The Abbe sine condition can though of as the real marginal ray magni cation is equal to the paraxial magni cation or uu SinUSinU In the homework the affect of index on coma was examined Astigmatism Astigmatism becomes zero if the chief ray is not refracted It was also seen that astigmatism is zero at the aplanatic points Also if a surface is at an image there will be no astigmatism Field Curvature The equation for the wavefront aberration coefficient for field curvature has two terms One term is proportional to astigmatism and the other depends on the Petzval term The Petzval term is P C Aln The Petzval sum is 2P ln k p k ln 1 p l Z n nn nr which depends only on the index and radius For a system of thin lenses in air 1P k 2 MI where is the radius of the image This can be used to create a new achromat that takes advantage of new glass types By doing this on can make the radius of the image infinite ie a at field To have zero field curvature for a single element one could use a thick meniscus that has the same front and rear radii but where power is proportional to thickness To correct for field curvature one can use a field attener at the image Because of the field attener s location there will be no spherical astigmatism or coma Distortion Distortion can be controlled through the use of aspherics near the image plane or the bending of the field attener Distortion may also be corrected through use of symmetry about the stop 12 Lens Evolution This section is taken from my own homework There was series of lenses studied in this class This series highlights an evolution in lens design This evolution is one understanding of aberrations improved aberration control and better image quality This series will begin with the simplest form studied a single element and will lead up to the development of sophisticated multi element systems Landscape mar a nu DEE m a mm m man an am an an DEE Tim 2121 in ma 5 mm a um 21 MILLIMEYEES FEEEE m 52 m m LRVDW FIELD ELJEUFIHJEE DISYEIEYIEIN SUEFHEE W LENS HRS ND TITLE mm DEE 15 m TDYRL LENGTH qu macaw LENS HFIS ND YIYLE MEN DEC 15 M1 Maxmum FIELD IS 321 mm DEGREES a may MM LE was ND YIYLE 5 2m uNrTs 72E MILEngs The landscape lens is the first and simplest lens studied It however offers many insights into aberration control as well as trade offs The bending meniscus shape of the lens helps to control spherical aberration and coma The stop shift not only affects coma but also is used to control astigmatism Astigmatism is purposefully introduced to arti cially atten the tangential field Because of the single element made of a single piece of glass the landscape lens features chromatic aberrations The lens also shows a significant amount of distortion And even though astigmatism is used to atten the tangential eld there is still field curvature and of course astigmatism But this simple lens gives acceptable sized spots to be a useful imaging device The simple rear landscape lens is shown above The lateral color is the limiting aberration At f 8 the spot sizes are large up to 400 microns for the 300 field Chevalier as a an DEE m m um um man an new 1a am has E5 2m 2 HE E um Em m1 2 a MILLIMEYERS FEREE UEFAEE 1m mar 21a a IHH 35 a 4 m as 2 LRVDLJY FIELD EUEURYUEE DISYEIETIDN FDY umsz LENS m w DEC 15 22121 M N TDYRL LENGTH LENS Hm ND YIVLE mm DEE l5 2an WIYS REE mums FIELD 2 HRS ND YITLE LENS HRS ND YITLE a DEC 15 2am mgxgmgmszLu IsrzziimarDEFEEEg 7 1 an m a The Chevalier lens is an attempt to improve on the existing landscape lens by making the meniscus an achromatic doublet While this helps to improve some of the chromatic aberrations it does not significantly reduce the spot size of the landscape lens The l3 distortion eld curvature remain at previous levels Astigmatism can still be used to arti cially atten the eld However one cannot both atten the eld and reduce coma in such a lens This is because both the front surface and the dispersive interface have overcorrected astigmatism and bending the lens reduces one contribution while increasing the other One could use the technique of a buried surface to partially correct for this problem The lens shown above has F2 and BK7 as the constituent glasses Periscoge m 3 Ian mg m a m m man an 4m 2m a an 121 M1 rm SE a a mummies mace m 55 m m um 21 I Q m as A mm new uEuHTuRE uxsmarmN W 1quot 5pm umsaw LENS M N as 5 22m a mm LENS HRS ND TIYLE MON DEE 15 2mm maxgmgmgxgtu sizaimmurugggzzg 7 LENS an ND YIYLE MW as 5 2m mm was HIDanNs EIELL 1 z LENGTH mu lair2 MM HRS ND YITLE a The periscope lens is the symmetric doubling of the landscape lens about the stop As mentioned previously the symmetry of this system signi cantly reduces the odd aberrations of this system including lateral color Distortion is drastically reduced However eld curvature arti cially attened with astigmatism still exists Even with these improvements the spot size is still on the same scale as that of the landscape lens If the lenses are bent steeper the stop is closer to the lens and the system necomes more compact The image quality is a lot better than the landscape lens simply because the level of distortion is so small The glass used here is BK7 Rapid Rectilinear as a um mm m m am NM man an m an m as rm 2m 2 BE 121 um rm 5a m a mummies pzacz 111R 55 Nu m UEFAEE 1m as 2 a Q IMF as 7 Lavnur FIELD uauamaz uIsruETIuN SFDY umsaw LENS M N as IS 22121 a mm HRS Na YITLE m 5295 MM LENS HRS Na YITLE MUN as 15 2am NgxgmgszgLu Isrzainmarugggzzg 7 LENS an ND nu HUN DEE l5 2an mm REE HIEEHNS E199quot 4 2 LENGTH The rapid rectilinear is also symmetric about the stop The rear half of the rapid rectilinear has large positive coma is corrected for spherical aberration and has a at eld This requires the Abbe numbers to be similar but the rfractive indices to be different The distortion is again well corrected however there is still no signi cant change in spot size This is one of the most popular photographic lenses ever made The glasses used here are LFl and F1 14 Schroeder Lens new In am as as 2 a Q m w m m m as u new 3 a mic man an 5 2121 212121 San 4w nu mummies mace WM M ma mm mm mm new uEuHTuRE uxsmamm SPOT DIHEEW rm Has ND ms LENS HRS ND TIYLE mum DEE 15 2mm maxgmwjxgtu sizaimmurugggzzg 7 LENS HRS ND YITLE MUN as 5 2 mm LENGTH n7 yam mm The Schroeder lens builds on the lessons learned from the symmetrical lenses already seen It however doubles two new achromats which were speci cally designed to signi cantly reduce eld curvature The result is a lens that enjoys the advantages of doubling including very low distortion as well as a at eld and little astigmatism By inspecting the OPD fans of this lens it is noted that the most apparent aberration is spherical The spot size associated with this lens is half as small as in the previous lenses The lenses used here are SK4 and LLF8 mm nE l5 2am LAan was Mmams BELL 4 z W 3 m m am an as 2 a a E ma a nu m m as 5 7 7 m 3 a m rs an m an 5 mm rm M a a mamas mug WE M W mm mm mm mamas mmmw SW BMW M HAS Nu mg LEN HRS Na YITLE LEN HRS Na YITLE m DE 1 2212 m DE 1 22m pgngE mm W 79mm 3 l mm mm a 7518 W nggmgmggu Isrzaiumarugggzzg 7 The Protar again uses the idea of symmetry but instead of two new achromats as used in the Schroeder the thick low powered front doublet with a strong cemented interface is mainly used to correct spherical The low power of the element does not add much eld curvature to the system The eld curvature is again corrected by the rear new achromat Paul Rudolph designed this lens in 1890 The spot sizes are slightly better than in the Schroeder lens 4 different lenses are used and they are BK7 SF4 LLF8 and SK4 from left to right 15 Tessar new In am as man m w m m new 3 a mic 7 an m an 2 m 7 2121 a a Mmtmzrzas pzacz m 55 was m mama 1m as 2 a m as 7 SPOT nxasaam annur FIELD uauamaz uIsruETIuN LENS m N as 5 2 a mm m Has ND ms HRS ND YITLE EgngE l5 2am LAan TEE 3911st in mm mm LENS HRS ND YITLE MUN as 15 2am mgxgmgmjzgtu Isrzaimwarugquzg 7 LENGTH 3 The Tessar improves on the Protar by making the front element an air spaced doublet as opposed to a cemented one As learned in the homework an air spaced doublet allows less zonal aberration The Tessar may also be thought of as triplet with a new achromat as its rear element The Tessar lens can correct all of the third order aberrations because it has seven degrees of freedom These are the 2 powers 2 bendings and 1 air space of the front element as well as the 2 outer curvatures of the second element and lastly the choice of glasses of the elements Image quality is vastly improved but the distortion is larger than in some of the earlier modified landscape lenses The same 4 lenses used for the Protar are used here Cooke Triplet Zn In V m w 959 m 71m 2 an 7 an m a mmtmzrzas PEEEE macs 1m m 1 mm m LENS MUN TDYFIL as 5 221m L vuur FIELD uEvaEE uxsmmiw HRS ND YITLE 55 57525 MM LENS HRS ND YITLE mm as 5 nggMgMngu Isrziaeaajgggn igrzygg LENS was ND ms Egg 2E l5 2m UNIYS b wz maim s LENGTH The Cooke triplet was designed by Dennis Taylor in 1893 It is difficult to design because changing any single surface affects all aberrations The Cooke triplet can also correct for all third order aberrations It may even be thought of as a corrected landscape lens Primary chromatic aberration lateral color Petzval sum and distortion are controlled by the three powers of the lenses and their respective separations Once these are solved for the bendings of the three lenses are variables used to correct the remaining three aberrations spherical coma and astigmatism Significant reduction of the spot size is now obtained The stop is at the second surface of the negative lens The image quality is very good with spot sizes a few times the diffraction limit The image quality can also be gauged by the 3050 rule in the MTF plots The levels of distortion and eld curvature are also very small Q m 2 2 16 Double Gauss new In am as as 15 1 0 4 m a m m m 11 5 new 2 a mic a a VERDE mama 1m ma nuzsm mm mm mamas uIsruETIuN SW BMW M HAS Nu mg LEN HRS Na YITLE LEN HRS Na YITLE m DE 1 2m m DE is me pgngE mm W 79mm 3 mm mm 5w gwmz MM nggmw gru Isrziiszzrngggzzg 7 mam Va 521 m an a 52 72 2m Mxrrmzrzas The double Gauss was designed originally as a telescope objective but it has never been used as one successfully The double Gauss follows the design of the Dialyte The Dialyte has 5 degrees of freedom 2 powers 2 bendings and the air space These are used to correct four aberrations Petzval sum spherical aberration axial color and astigmatism The symmetry corrects the odd aberrations coma distortion and lateral color The double Gauss uses different glasses from the Dialyte to get better Petzval curvature The double Gauss also uses a buried surface to help correct color The double Gauss is the best performing lens with an even significantly better spot size than seen with the Cooke triplet This lens has little field curvature and almost insignificant distortion Color is also well corrected Another metric used for image evaluation the MTF is also near diffraction limited even for significant fields for this lens The above design was allowed to be of the asymmetrical type similar to the one commercially used by Kodak in the past The glasses used are LAK33 and SF6 in this case Image Evaluation As noted in the previous section there were a few ways to see if a lens performance is good Some of the practical metrics used are the geometric spot size when large compared to the diffraction limited PSF field curvature plots distortion plots and raywave aberrations which indicate the amount of certain aberrations in a system These metrics work well when a system is not near diffraction limited when geometrical considerations dominate As the system s performance improves and nears or surpasses diffraction limited one is more aware of physical metrics to evaluate a system s performance This was alluded to when the MTF became a good metric to compare the performance of the Double Gauss Further metrics are now the PSF when it is larger or on the order of the geometrical spot size the Rayleigh resolving power criteria separation by an Airy disk Radii the Rayleigh M4 rule there is less M4 than worth of aberration in the system the Strehl Ratio the ratio of the real peak of the systems airy disk to the theoretical one and the previously mentioned MTF the contrast ratio as a function of spatial frequency passed by the lens 17 The above are just a sampling of some useful metric There are of course many other evaluation methods available that may or may not fall in the categories given above A useful criteria used by this author was that of encircled energy to determine the signal received by another optical system Below is given a table evaluating the image performance of all the classical lenses designed and discussed in class Aberration max Spot sizes Hm Field Curvature Distortion Landscape Lens 17 waves 153 243 459 Flat tangential 5 Chevalier Lens 18 waves 124 168 381 Flat tangential 6 New Achromat 5 waves 91 147 193 Flat sagittal 6 Periscopic Lens 18 waves 156 231 457 Flat tangential 05 Rapid Rectilinear 18 waves 167 265 467 S T not at 05 Schroeder Lens 7 waves 189 205 166 Sagittal at 05 Protar Lens 3 waves 114 149 162 Sagittal at 5 Dagor Lens 13 waves 111 232 623 Both at 6 Cooke Triplet 2 waves 11 44 50 Sagittal at 13 Double Gauss 2 waves 12 19 26 Both at 1 Tessar Lens 3 waves 61 46 91 Tangential at 2 Tolerancing In a conversation with a colleague today this author was told that if you do not like tolerancing do not go into lens design While like is a very opinionated word understanding and ability are certainly necessary to have any real success as a lens designer The majority if the course was spent tweaking numbers in a computer in order to get another desirable smaller or greater number as a result While doing this much was learned and gained in the Art and Science of Lens Design however grounding in practicality may or may not be lost This author perceives Tolerancing as can it be built and sometimes more importantly can it be built at a reasonable cost in a reasonable amount of time Tolerancing in the lens design code consisted of constant updating constant push and pull The computer can tell you what variables cause the most violence to your bottom line The variables are the physical aspects of the lens system and its assembly things like radii of curvature center thicknesses surface errors decenters tilts etc The computer can also take a bottom line and tell you what the tolerances on each variable may be The program can also simulate in a Monte Carlo fashion what percentage of lenses produced will fit the design made A typical tolerancing spreadsheet is shown below This is for the double Gauss system The tolerances are the defaults WQONMAUJNHJt Type Intl COMP 9 TWAVO TRAD 2 TRAD 3 TRAD 4 TRAD 5 TRAD 6 TRAD 7 TRAD 8 TRAD 9 TTHI 1 TTHI TTHI TTHI TTHI TTHI TTHI TTHI 8 TEDX 2 TEDY 2 TETX 2 TETY 2 TEDX 4 TEDY 4 TETX 4 TETY 4 TEDX 6 TEDY 6 TETX 6 TETY 6 TEDX 8 TEDY 8 TETX 8 TETY 8 TSDX 2 TSDY 2 TIRX 2 TIRY 2 TSDX 3 TSDY 3 TIRX 3 TIRY 3 TSDX 4 TSDY 4 TIRX 4 lCNUIJkUJN Int2 AbbmmmmmmmmxoxooxoqxlxlxluuuuuummmmmxooqqmmmmcxmmmUAAAOO 18 Min 20000000E000 63280000E001 50000000E002 50000000E002 50000000E002 50000000E002 50000000E002 50000000E002 50000000E002 50000000E002 50000000E002 50000000E002 50000000E002 50000000E002 50000000E002 50000000E002 50000000E002 50000000E002 50000000E002 50000000E002 20000000E001 20000000E001 50000000E002 50000000E002 20000000E001 20000000E001 50000000E002 50000000E002 20000000E001 20000000E001 50000000E002 50000000E002 20000000E001 20000000E001 50000000E002 50000000E002 50000000E002 50000000E002 50000000E002 50000000E002 50000000E002 50000000E002 50000000E002 50000000E002 50000000E002 Max 20000000E000 00000000E000 50000000E002 50000000E002 50000000E002 50000000E002 50000000E002 50000000E002 50000000E002 50000000E002 50000000E002 50000000E002 50000000E002 50000000E002 50000000E002 50000000E002 50000000E002 50000000E002 50000000E002 50000000E002 20000000E001 20000000E001 50000000E002 50000000E002 20000000E001 20000000E001 50000000E002 50000000E002 20000000E001 20000000E001 50000000E002 50000000E002 20000000E001 20000000E001 50000000E002 50000000E002 50000000E002 50000000E002 50000000E002 50000000E002 50000000E002 50000000E002 50000000E002 50000000E002 50000000E002 19 46 TIRY 4 4 50000000E002 50000000E002 47 TSDX 5 5 50000000E002 50000000E002 48 TSDY 5 5 50000000E002 50000000E002 49 TIRX 5 5 50000000E002 50000000E002 50 TIRY 5 5 50000000E002 50000000E002 51 TSDX 6 6 50000000E002 50000000E002 52 TSDY 6 6 50000000E002 50000000E002 53 TIRX 6 6 50000000E002 50000000E002 54 TIRY 6 6 50000000E002 50000000E002 55 TSDX 7 7 50000000E002 50000000E002 56 TSDY 7 7 50000000E002 50000000E002 57 TIRX 7 7 50000000E002 50000000E002 58 TIRY 7 7 50000000E002 50000000E002 59 TSDX 8 8 50000000E002 50000000E002 60 TSDY 8 8 50000000E002 50000000E002 61 TIRX 8 8 50000000E002 50000000E002 62 TIRY 8 8 50000000E002 50000000E002 63 TSDX 9 9 50000000E002 50000000E002 64 TSDY 9 9 50000000E002 50000000E002 65 TIRX 9 9 50000000E002 50000000E002 66 TIRY 9 9 50000000E002 50000000E002 67 TIRR 2 0 20000000E001 20000000E001 68 TIRR 3 0 20000000E001 20000000E001 69 TIRR 4 0 20000000E001 20000000E001 70 TIRR 5 0 20000000E001 20000000E001 71 TIRR 6 0 20000000E001 20000000E001 72 TIRR 7 0 20000000E001 20000000E001 73 TIRR 8 0 20000000E001 20000000E001 74 TIRR 9 0 20000000E001 20000000E001 75 TIND 2 0 l0000000E004 10000000E004 76 TIND 4 0 l0000000E004 10000000E004 77 TIND 6 0 l0000000E004 10000000E004 78 TIND 8 0 l0000000E004 10000000E004 79 TABB 2 0 l0000000E002 10000000E002 80 TABB 4 0 l0000000E002 10000000E002 81 TABB 6 0 l0000000E002 10000000E002 82 TABB 8 0 l0000000E002 10000000E002 Then an inverse sensitivity algorithm was run to produce the following output for the Monte Carlo simulation This is what is used to see whether the system performance is within speci cations The merit function values are compared to the nominal value and a measure is made of how far the merit function value varies from the nominal value 20 Monte Carlo Analysis Number of trials 20 Statistics Normal Distribution Trial Merit Change 1 0014637 0002700 2 0041403 0029467 3 0012640 0000704 4 0037177 0025241 5 0017331 0005394 6 0014496 0002560 7 0017118 0005182 8 0017557 0005621 9 0019373 0007437 10 0015249 0003313 11 0025531 0013594 12 0018644 0006707 13 0025227 0013291 14 0013524 0001588 15 0018618 0006682 16 0026687 0014751 17 0013802 0001866 18 0016928 0004992 19 0013237 0001301 20 0017550 0005614 Nominal 0011936 Best 0012640 Worst 0041403 Mean 0019836 Std DeV 0007610 Compensator Statistics Change in back focus Minimum 0 174974 Maximum 0290420 Mean 0025 876 Standard DeViation 0 109144 For this run one can see that about 50 of the lenses are within 5 of the nominal value of the merit function This is a reasonable performance level 21 REFERENCES 1 httpwww0pticsarizonaedusasianoptiS17 2 Kingslake R Lens Design Fundamentals Academic Press San Diego CA 1978 3 Shannon R The Art and Science of Optical Design Cambridge University Press 1997 4 Shack R OPTI518 Intro to Lens Design Took class in Spring 2002

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