Extended depth of field through wave-front codingEdward R.Dowski,Jr.,and W.Thomas CatheyWe designed an optical-digital system that delivers near-diffraction-limited imaging performance with alarge depth of field.This system is the standard incoherent optical system modified by a phase maskwith digital processing of the resulting intermediate image.The phase mask alters or codes the receivedincoherent wave front in such a way that the point-spread function and the optical transfer function donot change appreciably as a function of misfocus.Focus-independent digital filtering of the intermediateimage is used to produce a combined optical-digital system that has a nearly diffraction limitedpoint-spread function.This high-resolution extended depth of field is obtained through the expense ofan increased dynamic range of the incoherent system.We use both the ambiguity function and thestationary-phase method to design these phase masks.Key words:Extended depth of field,extended depth of focus,wave-front coding.1.Introductionregions of zeros,digital processing can be used toExtending the depth of field of incoherent opticalrestore the sampled intermediate image.Further,systems has been an active research topic for manybecause the OTF is insensitive to misfocus.the sameyears.The majority of the literature on this topicdigital processing restores the image for all values ofhas concerned methods ofemploying an optical power-misfocus.This combined optical-digital system pro-absorbing apodizer,with possible+m phase varia-duces a PSF that is comparable to that of the diffrac-tions,on a standard incoherent optical system as ation-limited PSF but over a far larger region of focus.means to increase the depth of field.1-5 These meth-We term the general process of modifying the incoher-ods have all suffered from two significant deficiencies:ent optical system and the received incoherent wavea decrease of optical power at the image plane and afront,by means of a phase mask,wave-front coding.decrease of image resolution.A unique method ofBy modifying only the phase of the received waveachieving an extended depth of field without anfront,general wave-front coding techniques maximizecoming of this method is that the focus must be variedoptical power at the image plane.during exposure.When designing extended-depth-of-field systems,We describe a novel method for extending the depthwe make two main assumptions.The first is that theof field of incoherent optical systems that does notincoherent optical system is being modified by asuffer from the significant deficiencies of earlierrectangularly separable phase mask.This leads to amethods.Our method employs a phase mask torectangularly separable PSF and OTF.Second,wemodify the incoherent optical system in such a wayassume that any resulting image will be an intermedi-that the point-spread function (PSF)is insensitive toate image.This intermediate image will requiremisfocus,while the optical transfer function (OTF)digital processing.This second assumption followshas no regions of zero values within its passband.from our belief that the best performance is obtainedThe PSF of the modified optical system is not directlyby optimum preprocessing through optics,followed bycomparable to that produced from a diffraction-optimum digital postprocessing.7 These preprocess-limited PSF.However,because the OTF has noing and postprocessing stages are optimum in thesense that each is matched to the other in order tosolve an interesting problem.Our solution to extended-depth-of-field systems re-The authors are with the Imaging Systems Laboratory.Depart-lies on the theory of the ambiguity function8-10 andment of Electrical Engineering.University of Colorado,Boulder,Colorado 80309.the stationary-phase11-13 method.The ambiguityReceived 12 September 1994:revised manuscript received 5function can be used as a polar display of the OTF's ofDecember 1994.a rectangularly separable incoherent optical system0003-6935/95/111859-08S06.00/0.as a function of misfocus.10 Extended-depth-of-field1995 Optical Society ofAmerica.systems can be noticed almost by inspection of their10 April 1995 Vol.34,No.11 APPLIED OPTICS1859corresponding ambiguity functions.The stationary-plane.The quantity fis the focal length of the lens.phase method permits the design of phase masksThe wave number is given by k,and the traditionalwhose corresponding ambiguity functions have de-misfocus aberration constant is given by W2o.Thesired extended-depth-of-field qualities.traditional or Hopkins criterion for misfocust6 isIn Section 2 we outline this method of designingequivalent to 1.phase masks for extended-depth-of-field systems.The ambiguity function related to this generalThis design leads to the cubic-phase-modulation(cubic-mask can be used as a polar display of the OTF for allpm)mask,which is introduced in Section 3.Throughvalues of misfocus.10 The ambiguity function of thesimulated measurement of the width of the combinedmask Rx is given by 8.9optical-digital PSF,as well as through simulations ofimaging a spoke target,we show that this method canproduce an incoherent optical system with a largedepth of field with near-diffraction-limited imagingperformance.A complete mathematical derivationof the cubic-pm mask function,based on the station-ary-phase method,can be found in Appendix A.From Eqs.(2)and (4)the ambiguity function can beshown to be related to the OTF of the system gener-2.Design of Extended-Depth-of-Field SystemsThrough the use of the ambiguity function and theHu,)=A(u,/).(5stationary-phase method,phase masks for an ex-tended-depth-of-field incoherent optical system areOr,the projection of the point (u,u/)of the ambigu-readily found.The ambiguity function is an analyti-ity function onto the horizontal u axis yields the OTFcal tool that permits us to observe and to design OTFsfor spatial frequency u and misfocus In this waythe two-dimensional ambiguity function can be usedfor all values of misfocus at the same time.Theto determine the one-dimensional OTF for all valuesstationary-phase method provides the analytical flex-of misfocus.ibility needed to consider only phase masks in thisdesign process.As an example of the utility of the ambiguityConsider a one-dimensional unit-power phase maskfunction approach to visualizing misfocus OTF's,con-or phase function,in normalized coordinates,such assider the standard rectangularly separable incoher-ent optical system.Such a system is formed with arectangular-pupil or mask function.Calculation ofexpl for≤1the magnitude of the ambiguity function of this1one-dimensional rectangular function leads to the0otherwiseimage shown in Fig.1.In this image,regions oflarge power are given by dark shades.The majoritywherej=-1 and e(x is some unspecified nonlinearof power in the ambiguity function of the rectangularfunction.Knowledge of this phase function deter-aperture is concentrated along the v=0 axis,whichmines the PSF and the OTF of the incoherent opticalcorresponds to the in-focus OTF.The radial line insystem for all values of misfocus.14.15 We assumethis figure has a slope of /2.that a two-dimensional rectangularly separable phasemask will be used in practice.The one-dimensionalOTF,as a function of misfocus,is given by1510with spatial frequency u and misfocus parameterThe symbol denotes the complex conjugate.The器misfocus parameter,is dependent on the physicallens size as well as the focus state:1where L is the one-dimensional length of the lens-15aperture and A is the wavelength of the light.The0u axisdistance d is measured between the object and thefirst principal plane of the lens,and d,is the distanceFig.1.Ambiguity function of a rectangular aperture.The radialline has a slope of /2,corresponding to an OTF with misfocus=between the second principal plane and the image1860 APPLIED OPTICS Vol.34,No.11 10 April 1995where the constant a controls the phase deviation.The OTF of the incoherent system related to thisfunction can be approximated as1/2u=0See Appendix A for a derivation of this result.Theapproximation of the OTF is independent of misfocus.This can be inferred from the ambiguity functionrelated to the cubic-pm mask,with a 90,shown inFig.3.The cubic-pm ambiguity function has uni-Normalized Spatial Frequencyform nonzero values distributed about the u axis.Radial lines through the origin of this ambiguityFig.2.Misfocus OTF of the standard optical system with misfo-function have nearly the same values as a function ofcus parameterψ=T2/2.angle for a broad range of angles.Hence the cu-bic-pm mask should form an extended-depth-of-fieldincoherent optical system.Figure 4 shows a compari-Figure 2 shows a misfocused OTF related to theson between the stationary-phase approximation andrectangular aperture,or standard optical systemthe actual calculated OTF obtained by Eq.(2).TheThe misfocus parameter for this OTF is=m2/2.smooth curve in this figure is the approximation of theFrom Eq.(5)the ambiguity function of Fig.1 along themagnitude of the OTF;the other is the calculatedradial line with a slope of /2 describes this OTF.magnitude of the OTF.For this figure the constant oBy inspection of these two figures we can confirm thisof Eq.(6)was also selected as 90,and the misfocusrelationship between the OTF and the ambiguityparameter,.业，was set to 15.The approximationfunctionholds for other values of misfocus as well as for theExtended-depth-of-field systems,or systems thatphase ofthe OTF.See Fig.5.This figure is a plot ofare insensitive to changes of focus,have ambiguitythe magnitude of three misfocused OTF's related tofunctions that are not a function ofthe second param-the cubic-pm mask,with o =90.The misfocuseter,here given as v.From Eq.(5),ambiguity func-tions that are independent ofthe second parameter,v.values of the three OTF's are equal to 0,15,and 30.lead to OTF's that are invariant to misfocus InThese OTF's are nearly constant with misfocus andpractice,extended-depth-of-field systems are thosehave no zeros.This is what makes it possible to usewith ambiguity functions approximately independentone focus-independent digital filter to restore theof vover a relatively wide angular region about the uintermediate image.Figure 6 shows the dramaticaxis.From the ambiguity function of Fig.1 we canvariation of the OTF's of the standard optical systemimmediately notice that a rectangular-pupil functionwith the same misfocus values.Also,the verticaldoes not describe an extended-depth-of-field system.scale in Fig.6 is different from that of Fig.5.By careful selection of the nonlinear function x ofthe general mask given in Eq.(1)a phase function that60produces an ambiguity function with the desiredextended-depth-of-field characteristics can be found.We term this mask the cubic-phase-modulation(cubic-pm)mask.Section 3 describes this mask.See Ap-pendix A for a derivation.203.Cubic-Phase-Modulation Phase Mask器Modification of a standard incoherent optical systemby a cubic-pm phase mask produces intermediateimages that are insensitive to misfocus.Conceptu-20ally simple filtering techniques applied to these inter-mediate images form a complete system that images40with high resolution and a large depth of field.Thecubic-pm mask,in normalized coordinates,is given by6020u axisFig.3.Magnitude of the ambiguity function of the cubic-pmfunction with a 90.Radial lines through this function areotherwiseinsensitive to angle for a broad range of angles.10 April 1995 Vol.34.No.11 APPLIED OPTICS 1861