光学检测CH01课件

1.0 Basic Wavefront Aberration Theory For Optical Metrology Changchun Institute of Optics and Fine Mechanics and PhysicsDr.Zhang Xuejun 11.0 Basic Wavefront AberrationThe Principal purpose of optical metrology is to determine the aberrations present in an optical component or an optical system.To study optical metrology the forms of aberrations that might be present need to be understood.2The Principal purpose of opticFor most optical testing instruments,the test result is the difference between a reference(unaberrated)wavefront and a test(aberrated)wavefront.We usually call this difference the Optical Path Difference(OPD).OPDTest wavefrontReference wavefrontRayNote that the OPD is the difference between the reference wavefront and the test wavefront measured along the ray.3For most optical testing instr1.1 Sign ConventionlThe OPD is positive if the aberrated wavefront leads the ideal wavefront.In other word,a positive aberration will focus in front of the paraxial(Gaussian)image plane.Right Handed Coordinates:Z axis is the light propagation directionX axis is the meridional or tangential directionY axis is the sagittal direction41.1 Sign ConventionThe OPD is lThe distance is positive if measured from left to right.lThe angle is positive if it is in counterclockwise direction relative to Z axis.(+)(-)(+angle)(-angle)lSince most optical systems are rotationally symmetric,using polar coordinate is more convenient.XY x=cos y=sin 5The distance is positive if me1.2 Aberration Free SystemlIf the optical system is unaberrated or diffraction-limited,for a point object at infinity the image will not be a“point”,but an Airy Disk.lThe distribution of the irradiance on the image plane of Airy Disk is called Point Spread Function or PSF.lSince PSF is very sensitive to aberrations it is often used as an indicator of the optical performance.61.2 Aberration Free SystemIf tFirst maximumSecond maximumlDiameter to the first zero ring is called the diameter of Airy Disk:working wavelengthF#:f number of the system7First maximumSecond maximumDiaFinite conjugateNA:numerical ApertureNA=nsinuunF#W:Working F numberRule of thumb:for visible light,0.5 m,DAiry F#in microns8Finite conjugateunRule of thumx,y:coordinates measured in the exit pupilx0,y0:coordinates measured in the focal planeI0:intensity of incident wavefront(constant):wavelength of incident wavefrontf:focal length of the optical systemA:amplitude in the exit pupil(x,y):the phase transmission function in the exit pupilOPDPupil function9x,y:coordinates measured in lFor aberration free system,the PSF will be the square of the absolute of the Fourier transform of a circular aperture and it is given in the form of 1st order Bessel function.10For aberration free system,thThe fraction of the total energy contained in a circle of radius r about the diffraction pattern center is given by：11The fraction of the total enerrAngular Resolution-Rayleigh Criterion12rAngular Resolution-Rayleigh CGenerally a mirror system will have a central obscuration.If e is the ratio of the diameter of the central obscuration to the mirror diameter d,and if the entire circular mirror of diameter d is uniformly illuminated,the power per unit solid angle is given by13Generally a mirror system will1414 ,is in lp/mmThe Cut-Off frequency of an optical system is:15 ,is in lFeatures：Mirrors aligned on axisAdvantages：Simple and achromaticDisadvantages：Central obscuration and lower MTFSmaller FOV with long focal length Obscured System Unobscured SystemFeatures：Mirrors aligned off axisAdvantages：No obscuration and higher MTF；Larger FOV with long focal lengthAchromaticDisadvantages：Difficult to manufacture and assembly16Features：Obscured System 1.3 Spherical Wavefront,Defocus and Lateral ShiftA perfect lens will produce in its exit pupil a spherical wavefront converging to a point a distance R from the exit pupil.The spherical wavefront equation is:Sag equation 171.3 Spherical Wavefront,DefocDefocusOriginal wavefront:New wavefront:Defocus termIncreasing the OPD moves the focus toward the exit pupil in the negative Z direction.In other word,if the image plane is shifted along the optical axis toward the lens an amount z(z is negative),a change in the wavefront relative to the original spherical wavefront is:18DefocusOriginal wavefront:New unDepth of FocusRule of thumb:for visible light,0.5 m,Z (F#)2in micronsBy use of Rayleigh Criterion:The smaller the F#,or the larger the relative aperture,the smaller the Depth of Focus,so the harder the alignment.19unDepth of FocusRule of thumb:2020Lateral(Transverse)ShiftInstead of shifting the center of curvature along Z axis,we move it along X axis,then:For the same reason,if move along Y axis,then:21Lateral(Transverse)ShiftInstA general spherical wavefront:This equation represents a spherical wavefront whose center of curvature is located at the point(X,Y,Z).The OPD is:This three terms are additive for the misalignment,some or all of them should be removed from the test result for different test configurations.22A general spherical wavefront:1.4 Transverse and Longitudinal AberrationIn general,the wavefront in the exit pupil is not a perfect sphere but an aberrated sphere,so different parts of the wavefront come to the focus in different places.It is often desirable to know where these focus points are located,i.e.,find(x,y,z)as a function of(x,y).231.4 Transverse and LongitudinaWavefront aberration is the departure of actual wavefront from reference wavefront along the RAY.24Wavefront aberration is the de1.5 Seidel AberrationsIn a real optical system,the form of the wavefront aberrations can be extremly complex due to the random errors in design,fabrication and alignment.According to Welford,this wavefront aberration can be expressed as a power series of(h,x,y):a3 term gives rise to the phase shift over that is constant across the exit pupil.It doesnt change the shape of the wavefront and has no effect on the image,usually called Piston.b1 to b5 terms have fourth degree for h,x,y when expressed as wavefront aberration or third degree as transverse aberration,usually called fourth-order or third order aberrations.h:field coordinatesx,y:coordinates at exit pupil251.5 Seidel AberrationsIn a rea2626If look the optical system from the rear end,we see exit pupil plane and image plane.27If look the optical system froWavefront Aberration Expansion28Wavefront Aberration ExpansionClassical Seidel Aberrations29Classical Seidel Aberrations29W000W020W040W060W111W131W151W222W242What do aberrations look like?30W000W020W040W060W111W131W151W2W000W020W040W060W111W131W151W222W242W33331W000W020W040W060W111W131W151W2Field CurvatureWhere do aberrations come from?32Field CurvatureWhere do aberraDistortion33Distortion33AstigmatismW22234AstigmatismW222343535ComaW13136ComaW13136Warren Smith,Modern Optical Engineering,P65Spherical Aberration W=W040 437Warren Smith,Modern Optical E+W=W040 4 W=W020 2 W=-1W020 2+W040 4Spherical Aberration+Defocus38+W=W0404W=W0202W=-1W020Through-focus Diffraction Image(With Spherical Aberration)39Through-focus Diffraction ImaglWavefront measurement using an interferometer only provides data at a single field point(often on axis).This causes field curvature to look like focus and distortion to look like tilt.Therefore,a number of field points must be measured to determine the Seidel aberration.lWhen performing the test on axis,coma should not be present.If coma is present on axis,it might result from tilt or/and decentered optical components in the system due to misalignment.lA common error in manufacturing optical surfaces is for a surface to be slightly cylindrical instead of perfectly spherical.Astigmatism might be seen on axis due to manufacturing errors or improper supporting structure.Important to know40Wavefront measurement using anCaustic41Caustic41Specifies the size of aberrationBasic form of aberrationThe aberrations of a given optical system depend on the system parameters such as aperture diameter,focal length,and field angle,as well as some specific configurations of the system.1.6 Aberration Coefficients42Specifies the size of aberrati4343The Lagrange Invariant The Lagrange Invariant holds at any plane between object and image.=At object plane:=At image plane:=For object at infinity：44The Lagrange InvariantThe LaParaxial Ray TracingSnells Law45Paraxial Ray TracingSnells LaL=Seidel Coefficient Table46L=Seidel Coefficient Table46Seidel Coefficient Calculation for a Singlelet47Seidel Coefficient CalculationCalculation by Zemax48Calculation by Zemax48Calculation by Seidel Coefficient Formula49Calculation by Seidel Coeffici5050The Thin Lens FormThe aberrations of a given optical system depend on the system parameters such as aperture diameter,focal length,and field angle,as well as some specific configurations of the system.The system parameters can be factored out of the aberration coefficients,leaving remaining factors which depend onlyupon the configuration of the system.These remaining factors we will call the structural aberration coefficients.51The Thin Lens FormThe aberrati5252The Structure Aberration CoefficientRoland V.Shack53The Structure Aberration CoeffThe Thin Lens BendingIt is possible to have a set of lenses with the same power and the same thickness but with different shapes.X：Minimum spherical aberrationIf Y is constant,thenIf object at infinity,Y=1,n=1.5,then54The Thin Lens BendingIt is posMinimum comaIf object at infinity,Y=1,n=1.5,thenX=-2X=-1X=+1X=+2For object at infinity,stop at thin lens,when lens power is fixed:55Minimum comaIf object at infinZemax ResultCalculation Using Thin Lens Form56Zemax ResultCalculation Using For object at infinity：=For thin lens is in air,n=1，rearrange the thin lens formula:57For object at infinity：=For t1.7 Zernike PolynomialsOften in optical testing,to better interpret the test results it is convenient to express wavefront data in polynomial form.Zernike polynomials are often used for this purpose since they contain terms having the same forms as the observed aberrations(Zernike,1934).Nearly all commercial digital interferometers and optical design softwares use Zernike polynomials to represent the wavefront aberrations.581.7 Zernike PolynomialsOften Zernike polynomials have some interesting properties,If is Zernike polynomial terms of nth degree and we discuss within a unit circle:These polynomials are orthogonal over the continuous interior of the unit circle:59Zernike polynomials have some can be expressed as the product of two functions.One depends only on the radial coordinate and the other depends only on the angular coordinate .n and l are either both even or both odd.It has rotational symmetry property.Rotating the coordinate system by an angle doesnt change the form of the polynomials:60 can be expressed as the pro can be expressed as:,where m n,l=n-2m.So Zernike term Unm can be expressed as:Where:sin function is used for n-2m0 cos function is used for n-2m 061 can be expressed as:,whereSo the wavefront aberration can be expressed as a linear combination of Zernike circular polynomials of kth degree:Where Anm is the coefficient of Zernike term Unm.62So the wavefront aberration ca4 th Zernike polynomials634 th Zernike polynomials63Re-ordered Zernike polynomials(first 36 terms)64Re-ordered Zernike polynomials12354678Plots of Zernike polynomials#1#86512354678Plots of Zernike polyn9101112131415Plots of Zernike polynomials#9#15669101112131415Plots of Zernike Plots of Zernike polynomials#16#2416171819202122232467Plots of Zernike polynomials#33Plots of Zernike polynomials#25#36252628272930323135346833Plots of Zernike polynomialsZernike polynomials are easily related to classical aberrations.W(,)is usually found the best least squares fit to the data points.Since Zernike polynomials are orthogonal over the unit circle,any of the terms:also represents individually a best least squares fit to the data.Anm is independent of each other,so to remove defocus or tilt we only need to set the appropriate coefficients to zero without needing to find a new least squares fit.Advantages of using Zernike polynomials69Zernike polynomials are easilyCautions of using Zernike polynomialsMid or high frequency errors might be“smoothed out”.For example the Diamond Turned surface profile can not be accurately expressed by using reasonable number of Zernike terms.Zernike polynomials are orthogonal only over the continuous interior of an unit circle,generally not orthogonal over the discrete set of data points within a unit circle or any other aperture shape.70Cautions of using Zernike polyRelationship Between Zernike polynomials and Seidel AberrationsThe first 9 Zernike polynomials are expressed as:The same aberration can be expressed in Seidel form:71Relationship Between Zernike pUsing the identity:72Using the identity:7273731.8 Peak to Valley and RMS Wavefront AberrationPeak to Valley(PV)is simply the maximum departure of the actual wavefront from the desired wavefront in both positive and negative directions.While using PV to specify the wavefront error is convenient and simple,but it can be misleading.It tells nothing about the whole area over which the error are occurring.An optical system having a large PV error may actually perform better than a system having a small PV.It is more meaningful to specify wavefront quality using the RMS wavefront error.RMS:“Root Mean Squares”,2=RMS2PV=Wmax-Wmin741.8 Peak to Valley and RMS WavIf the wavefront errors are expressed in the form of Zernike polynomials,by using orthogonal property the 2 is simply:The RMS or variance of the wavefront error is simply the linear combination of the squares of its Zernike polynomial coefficients.75If the wavefront errors are exStrehl RatioThe ratio of the intensity at the Gaussian image point(the origin of the reference sphere is the point of maximum intensity in the observation plane)in the presence of aberration,divided by the intensity that would be obtained if no aberration were present,is called the Strehl ratio,the Strehl definition,or the Strehl intensity.The Strehl ratio is given by:If the aberrations are so small that the third-order and higher-order terms can be neglected,then the Strehl ratio will be:76Strehl RatioThe ratio of the iMarechal Criterion Once Strehl Ratio at diffraction focus has been determined,we can use Marechal Criterion to evaluate the system.It says that a system is regarded as well corrected if the Strehl Ratio is 0.8,which corresponds to a RMS wavefront error/14.77Marechal Criterion Once Strehl