Invited PaperWavefront coded zoom lens with a single moving elementMads Demenikov,Ewan Findlay,Andrew R.Harvey*aHeriot Watt University,Imaging Concepts Group,School of Engineering and Physical Sciences,Edinburgh,EH14 4AS,Scotland,UK;STMicroelectronics,33 Pinkhill,Edinburgh,EH12 7BF,Scotland,UKABSTRACTTraditional zoom lens designs employ multiple moving lens elements to provide simultaneous control of focal length andfocal plane.We present an example of a simplified and compact zoom lens design by employing a single movingelement to control only the focal length.In this zoom lens,wavefront coding is used to control the defocus.We describethe principles of operation with special attention to image processing.We simulate imaging and image restorationcapabilities and present that the zoom lens provides high imaging quality.Keywords:Zoom lenses,wavefront coding,extended depth of field,optical design,design miniaturization,designsimplification,image processing.1.INTRODUCTIONThere are two major problems with traditional zoom lenses;the variator,which is responsible for "zooming",introduces(A)defocus,and(B)variations in optical aberrations [1].To solve problem A,most modern zoom lenses are designed with at least one extra moving element (a compensator)which is responsible for compensating the defocus introduced by the variator [2].Such designs,called mechanicallycompensated,are not suitable for ultra compact designs;the compensator requires complex and precise control.To solveproblem B.most modern zoom lenses are designed with many lens elements (some of them aspherical)which areresponsible for compensating the variations in optical aberrations.Such designs are neither suitable for ultra compactdesigns;the elements require extra space.Prischepa and Dowski [3]have reported a compact mechanically compensated zoom lens that employed wavefrontcoding (WFC)to enable a simplified mechanically compensated zoom lens with two moving single aspherical lenses,hereby solving problem B such that it is possible to obtain a compact design.Similar simplification where aberrationcorrecting lens elements have been replaced with WFC has been demonstrated with infrared imaging systems [4-5].Higher order aberrations such as coma and astigmatism have been demonstrated to be mitigated with circular symmetricphase filters in [6].In this paper we desire to solve problem A and B such that it is possible to design ultra compact zoom lenses.Wedescribe how to solve problem A by employing WFC [7]or other techniques [8-10]to increase depth of focus(DOF)inorder to compensate the defocus introduced by the variator.In this way,we describe how the role of the compensator isreplaced by WFC,such that zoom lenses with a single moving element are obtained.Furthermore,we describe howincreased DOF can be used to miniaturize zoom lenses with a single moving element.To our knowledge,WFC or othertechniques to increase DOF have not previously been employed in zoom lenses to replace a compensator and decreasethe length.We describe how to solve problem B by using a symmetric design and inserting additional thin lens elementswith aspherical surfaces in order to compensate for variations in optical aberrations introduced by the variator.*A.R.Harvey@hw.ac.uk;phone +44(0)131 451 3356;fax+44(0)131 451 4155;www.ece.eps.hw.ac.uk/~arharveyOptical Design and Engineering Il,edited by Laurent Mazuray,Rolf Wartmann,Andrew Wood,CCC code:0277-786XW08/s18·doi:10.1117/12.797756Proc.of SPIE Vol.7100 71000D-1In section 2,we conduct a new analysis of zoom lenses with a single moving element,from which we clearly see howthe length is enlarged by the zoom-factor,but most importantly how the length can be reduced by decreasing the sensorsize and increasing the DOF.In section 3,we demonstrate the effects on the length of zoom lenses with a single movingelement by increasing the zoom factor and DOF.In section 4,we demonstrate an example of a ray-traced design of aminiaturized 2.3X optical zoom lens with a single moving element using WFC.2.THEORY OF ZOOM LENSES WITH A SINGLE MOVING ELEMENTVariation of lens power can be achieved by changing the separation between two lens elements according to the well-known equation=+-d,where and are the powers of the two lens elements and d is the lensdisplacement [2].The back focal length of the combined lens system is[2]:BFLAB(d)=(1-d)/B(1)An asymmetric two-lens system as above suffers from a rapid variation in optical aberrations during zooming.Twozoom-lens configurations (with small variations in various optical aberrations with variation of lens power due tosymmetry)with two stationary lenses(A and C)and a moving lens B have been described in [1]and are shown in figure1.Long EFL0Short EFLFig.1.Two zoom lens configurations with one moving element and a total of 3 elements showing the defocus related to themovement of lens element B.The defocus is exaggerated to show the principle and the direction.Typically,the defocusfor the two zoom lens configurations is not identical in magnitude as also indicated by the dashed curveThe zoom factor of a zoom lens is the ratio between the maximum and minimum effective focal lengths,Z=/min.Denoting the ratio between the effective focal length displacement as M when lens A and B are incontact with each other and when displaced by d=S,the powers of lens A and B are [1,2]:=(R-1)/(RS)B=(1-R2)/(RS)(2)Proc.of SPIE Vol.7100 71000D-2where R=M.Lens A and B can be:positive and negative(+-),or negative and positive(-+);the +-configuration hasM=Z and the -+configuration has M=1/Z.Defocus formed by lens A and B,as well as formed by lens A,B and C,depend on the lens displacement d.The defocus formed by lens A,B and C can be found by finding the power oflens C.The focal length of a combined system of two lenses is [2]f=ff+f-d),from which the focallength of a combined system of 3 lenses can be found using a re-occurrence procedure.Denoting the focal length of acombined 3-lens system at the widest field of viewthe power is found with(2)toyield the simple expression:(3)From (3)it is seen that if the effective focal length at wide field-of-view is infinity,then an afocal zoom lens is obtained.Two zoom lens configurations will in the following be denoted +-and-+-,which refer to the signs of the powers of thelenses for an afocal system,although it should be noted that the-+-configuration usually has a positive lens power in therear for a focusing system.For three lenses,the back focal length as a function of lens separationS and lensdisplacementd is found with geometrical optics.Thetransfer-matrix of thelens is1001The focused light rays are given bywherey andy are the light ray height,whilst and are light ray angle.Focusing at infinity (=0,Vy)by solvingfor y'=0 yields the non-trivial solution fory0:hu.comBFLABC(d,S)=-(4)which reduces to (1)when c=0ands=d.By insertion of (2)and (3)into (4),the back focal length atd=0 andd=S is:By subtracting (5)from(4)the defocus is obtained as a function of d with zeroes at d=0 and d=S.The maximumdefocus when focusing at infinity for the two lens configurations is:(6)This equation shows that the defocus is approximately inverse proportional to the lens separation S and hence whyzoom lenses with a single moving element are difficult to make ultra compact;for a given zoom factor and a desiredminimum focal length the defocus can simply become too large to be tolerated in a traditional imaging system.The lensseparation.S for the two zoom lens configurations is found from (6).The total length of the two zoom lensconfigurations is L=S+BFL which can be written as:Proc.of SPIE Vol.7100 71000D-3