A new theory of cell phone lens designsDave ShaferSummaryThe highly deformed and very "wiggly"aspheric lensshapes in conventional cell phone camera lens designs areprobably due to having extremely high-order aberrationspartly compensate for some uncorrectable lower-orderaberrations,and that is because the designs are stuck in non-optimum low-orderaberration solution regions.I show an approach that is different and whichautomatically results in very smooth aspherics with minimal departures andallows a high performance design with only 4 lenses (three acrylic,one styrene).Cell phone camera lenses have many highly deformed aspheric surfaces andtypically every surface is aspheric.In a 5 element design that means 10 aspherics.These designs combine both a fast speed of f/2.0 or faster and a wide angle ofabout 75 degrees or more.Since they are nearly diffraction-limited over the fieldthat means very good aberration correction,with nearly complete control overall the 3rd and 5th order aberrations,as well as many of those higher-order thanthat.Since there are five 3rd order Seidel aberrations a design with 10 asphericshas very many more variables than are needed for 3rd-order correction.Theresult is that there are very many discrete 3rd-order solutions with aspherics thatwill look exactly the same,with respect to the first-order parameters.The first order lens configuration can be anything at all and with only 5aspherics,instead of 10,there will always be many 3d-order solutions that lookthe same to us in a lens drawing.I think it is deceptive to see some 4 or 5element designs that have the same first-order configuration and then assumethat they are basically the "same"design because they look the same to us.Theycould have very different aberration contributions going through the designs,because of the amounts and signs of the aspherics,even though they may havethe exact same radii,airspaces,etc.So the question for a designer is how to findthe "best"arrangement of aspherics in a design when there are so many morethan are needed for 3"d-order correction.The answer-of course-is that we need to look at the higher-orderaberrations,specifically the 5th-order,and try to find which of the very many 3rd_order solutions is the best one to build upon when we use the higher-orderaspheric terms to control the 5th and higher aberration orders.And here is whereit gets interesting.There is no guarantee that a particular 3d-order solution,usingthe 4th power aspheric terms,can be given complete control over all 9 of theindependent 5th-order aberrations simply by using all the 6th-order asphericterms.The "right"type of 3d order solution,not just any arbitrary one,is neededin order to correct for all the 3rd and 5th order aberrations,using the 4th and 6thpower aspheric terms.The proof of this is easy to find.I took a 5 lens configuration that I know hasvery good correction-shown below-for fast speeds and wide angles,with noaspherics,and added an aspheric to all the surfaces.It is possible to correct allthe 3d and 5th aberrations to zero using 10 aspherics but it is not a trivial exerciseto get there.TheNo nameUNITS:MFOCAL LENGTH =4 NA 0.25DES:Shafersolution space is very0.928non-linear.The newdesign looks basicallythe same as the designwithout aspherics.Yeta 6 element Double-Gauss design does notseem to be able to becorrected to zero forall the 3rd and 5th-order aberrationsdespite aspherics on every surface.So it seems that both the right first-orderconfiguration and the right 3d-order solution are needed to allow aspherics tocorrect all the 5th-order aberrations.That is,of course,not sufficient for thesefast speed wide-angle designs but it is necessary.Once I had a good 3d and 5th-order solution with 10 aspherics on 5 elements Itried seeing if a 4 lens solution is possible.It is and this is how it looks,shownwith f/2.0 and only a 30 degree field.This is an exact solution with all the 3rd andall 3rd and 5th order zeroUNITS:MMFOCAL LENGTH 4 NA 0.25DES:Shafer0.617Acrylic,styrene,acrylic,acrylicorder aberrations 0 plus a very few rays are also corrected to control some ofthe 7th-order aberrations.But there are still more variables available than areneeded.So there are multiple solutions(local minima)possible,although they