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1、1,1.0 Basic Wavefront Aberration Theory For Optical Metrology,Changchun Institute of Optics and Fine Mechanics and Physics,Dr.Zhang Xuejun,2,The Principal purpose of optical metrology is to determine the aberrations present in an optical component or an optical system.To study optical metrology the
2、forms of aberrations that might be present need to be understood.,3,For 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).,Note that the OPD is t
3、he difference between the reference wavefront and the test wavefront measured along the ray.,4,1.1 Sign Convention,The 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 Coord
4、inates:Z axis is the light propagation directionX axis is the meridional or tangential directionY axis is the sagittal direction,5,The distance is positive if measured from left to right.The angle is positive if it is in counterclockwise direction relative to Z axis.,Since most optical systems are r
5、otationally symmetric,using polar coordinate is more convenient.,x=cosy=sin,6,1.2 Aberration Free System,If 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.The distribution of the irradiance on the image plane of
6、 Airy Disk is called Point Spread Function or PSF.Since PSF is very sensitive to aberrations it is often used as an indicator of the optical performance.,7,Diameter to the first zero ring is called the diameter of Airy Disk,:working wavelengthF#:f number of the system,8,Finite conjugate,NA:numerical
7、 ApertureNA=nsinu,F#W:Working F number,Rule of thumb:for visible light,0.5m,DAiry F#in microns,9,x,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:a
8、mplitude in the exit pupil(x,y):the phase transmission function in the exit pupil,10,For 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.,11,The fraction of the total energy con
9、tained in a circle of radius r about the diffraction pattern center is given by：,12,r,Angular Resolution-Rayleigh Criterion,13,Generally 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 mi
10、rror of diameter d is uniformly illuminated,the power per unit solid angle is given by,14,15,is in lp/mm,The Cut-Off frequency of an optical system is:,16,Features：Mirrors aligned on axisAdvantages：Simple and achromaticDisadvantages：Central obscuration and lower MTFSmaller FOV with long focal length
11、,Obscured System,Unobscured System,Features：Mirrors aligned off axisAdvantages：No obscuration and higher MTF；Larger FOV with long focal lengthAchromaticDisadvantages：Difficult to manufacture and assembly,17,1.3 Spherical Wavefront,Defocus and Lateral Shift,A perfect lens will produce in its exit pup
12、il a spherical wavefront converging to a point a distance R from the exit pupil.The spherical wavefront equation is:,Sag equation,18,Defocus,Original wavefront:,New wavefront:,Defocus term,Increasing the OPD moves the focus toward the exit pupil in the negative Z direction.In other word,if the image
13、 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:,19,Depth of Focus,Rule of thumb:for visible light,0.5m,Z(F#)2in microns,By use of Rayleigh Criterion:,The smaller the F#,or the larger the re
14、lative aperture,the smaller the Depth of Focus,so the harder the alignment.,20,21,Lateral(Transverse)Shift,Instead 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:,22,A general spherical wavefront:,This equation represents
15、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.,23,1.4 Transverse and Longitudinal Aberration,In general,the wa
16、vefront 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).,24,Wavefront aberration is the departure of ac
17、tual wavefront from reference wavefront along the RAY.,25,1.5 Seidel Aberrations,In 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 po
18、wer 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
19、as transverse aberration,usually called fourth-order or third order aberrations.,h:field coordinatesx,y:coordinates at exit pupil,26,27,If look the optical system from the rear end,we see exit pupil plane and image plane.,28,Wavefront Aberration Expansion,29,Classical Seidel Aberrations,30,What do a
20、berrations look like?,31,32,Field Curvature,Where do aberrations come from?,33,Distortion,34,Astigmatism,W222,35,36,Coma,W131,37,Warren Smith,Modern Optical Engineering,P65,Spherical Aberration,W=W0404,38,+,W=W0404,W=W0202,W=-1W0202+W0404,Spherical Aberration+Defocus,39,Through-focus Diffraction Ima
21、ge(With Spherical Aberration),40,Wavefront 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 aberrat
22、ion.When 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.A common error in manufacturing optical surfaces is for a surface to be slightly cylindrical instead of perfectl
23、y spherical.Astigmatism might be seen on axis due to manufacturing errors or improper supporting structure.,Important to know,41,Caustic,42,Specifies the size of aberration,Basic form of aberration,The aberrations of a given optical system depend on the system parameters such as aperture diameter,fo
24、cal length,and field angle,as well as some specific configurations of the system.,1.6 Aberration Coefficients,43,44,The Lagrange Invariant,The Lagrange Invariant holds at any plane between object and image.,=,For object at infinity：,45,Paraxial Ray Tracing,Snells Law,46,L=,Seidel Coefficient Table,4
25、7,Seidel Coefficient Calculation for a Singlelet,48,Calculation by Zemax,49,Calculation by Seidel Coefficient Formula,50,51,The Thin Lens Form,The 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 con
26、figurations 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.,52,53,The Structure Aberration Coefficient,R
27、oland V.Shack,54,The Thin Lens Bending,It is possible to have a set of lenses with the same power and the same thickness but with different shapes.,X：,Minimum spherical aberration,If Y is constant,then,If object at infinity,Y=1,n=1.5,then,55,Minimum coma,If object at infinity,Y=1,n=1.5,then,For obje
28、ct at infinity,stop at thin lens,when lens power is fixed:,56,Zemax Result,Calculation Using Thin Lens Form,57,For object at infinity：,=,For thin lens is in air,n=1，rearrange the thin lens formula:,58,1.7 Zernike Polynomials,Often in optical testing,to better interpret the test results it is conveni
29、ent 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 represen
30、t the wavefront aberrations.,59,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:,60,can be expressed as the product of two functions.
31、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:,61,can be expressed as:,where mn,l=n-2m.So
32、 Zernike term Unm can be expressed as:,Where:sin function is used for n-2m0 cos function is used for n-2m0,62,So 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.,63,4 th Zernike polynomi
33、als,64,Re-ordered Zernike polynomials(first 36 terms),65,1,2,3,5,4,6,7,8,Plots of Zernike polynomials#1#8,66,9,10,11,12,13,14,15,Plots of Zernike polynomials#9#15,67,Plots of Zernike polynomials#16#24,16,17,18,19,20,21,22,23,24,68,33,Plots of Zernike polynomials#25#36,25,26,28,27,29,30,32,31,35,34,6
34、9,Zernike 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
35、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 polynomials,70,Cautions of using Zernike polynomials,Mid or high frequency errors might be“smoothed out”.For example the Diamo
36、nd 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.
37、,71,Relationship Between Zernike polynomials and Seidel Aberrations,The first 9 Zernike polynomials are expressed as:,The same aberration can be expressed in Seidel form:,72,Using the identity:,73,74,1.8 Peak to Valley and RMS Wavefront Aberration,Peak to Valley(PV)is simply the maximum departure of
38、 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
39、 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=RMS2,PV=Wmax-Wmin,75,If the wavefront errors are expressed in the form of Zernike polynomials,by using orthogonal property the 2 is
40、 simply:,The RMS or variance of the wavefront error is simply the linear combination of the squares of its Zernike polynomial coefficients.,76,Strehl Ratio,The ratio of the intensity at the Gaussian image point(the origin of the reference sphere is the point of maximum intensity in the observation p
41、lane)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 n
42、eglected,then the Strehl ratio will be:,77,Marechal 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.,
