1、11.0 Basic Wavefront Aberration Theory For Optical Metrology1.0 Basic Wavefront Aberration Theory For Optical Metrology Changchun Institute of Optics and Fine Changchun Institute of Optics and Fine Mechanics and PhysicsMechanics and PhysicsDr. Zhang Xuejun 2The Principal purpose of optical metrology
2、 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.3For most optical testing instruments, the test result is the difference between a reference (unaberrated) wavefront a
3、nd 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.41.1 Sign ConventionlThe OPD is positive if the a
4、berrated 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 direction5lThe distan
5、ce 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 61.2 Aberration Free SystemlIf the o
6、ptical 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 use
7、d as an indicator of the optical performance.7First maximumSecond maximumlDiameter to the first zero ring is called the diameter of Airy Disk:working wavelengthF#: f number of the system#.F442DAiry8Finite conjugateNA:numerical ApertureNA=nsinu2NA1FDfFEP#unF#W: Working F number#)(Fm1FLLmWRule of thum
8、b: for visible light, 0.5 m, DAiry F# in microns92yxi22000eAFTfIyxPSF）),()(),(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: amplitude
9、in the exit pupil (x, y): the phase transmission function in the exit pupil),(),(),(yxieAyxw2yxOPDPupil function10l 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.11The fr
10、action of the total energy contained in a circle of radius r about the diffraction pattern center is given by：12rAngular Resolution-Rayleigh Criterion13Generally 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, an
11、d if the entire circular mirror of diameter d is uniformly illuminated, the power per unit solid angle is given by1415 , is in lp/mm#F1 The Cut-Off frequency of an optical system is:16Features：Mirrors aligned on axisAdvantages：Simple and achromaticDisadvantages：Central obscuration and lower MTFSmall
12、er 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 assembly171.3 Spherical Wavefront, Defocus and Lateral ShiftA perfect lens wi
13、ll produce in its exit pupil a spherical wavefront converging to a point a distance R from the exit pupil. The spherical wavefront equation is:2RyxyxW22),(2RrSag2Sag equation 18Defocus)(),(Z22OR2yxyxW2RyxyxW22N),(Original wavefront:New wavefront:222ZON2RyxyxWyxWyxW),(),(),(Defocus term222Z2RyxyxW),(
14、Increasing 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:19unDepth of Focus2#Zdefoc
15、us8(F4WyxW),(2#Z8(F1yxW),(Rule of thumb: for visible light, 0.5 m, Z (F#)2in micronsBy use of Rayleigh Criterion:2#Z(F2)222Z2RyxyxW),(2#22EP22EP2228(F18fD8RD2Ryx)The smaller the F#, or the larger the relative aperture, the smaller the Depth of Focus, so the harder the alignment.2021Lateral (Transver
16、se) Shift2222XRRzyx)()(Rx2RyxyxWX22),(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:Ry2RyxyxWY22),(XtiltRxX:YtiltRyY:22RyRx2Ryx2RyxyxWyX222Z22),(A general spherical wavefront:This equation represents a spherica
17、l wavefront whose center of curvature is located at the point ( X, Y, Z).RCRB2RACyBxyxAyxWyX2Z22,)(),(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.231.4 Transverse and Longitudinal Aberrati
18、onIn 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).1yxxdxLARrnyxWorT
19、AdxRrnyxWxyxWxnrRLAxyxWnrRTA22r022r022),(,),(),(),(24Wavefront aberration is the departure of actual wavefront from reference wavefront along the RAY.251.5 Seidel AberrationsIn a real optical system, the form of the wavefront aberrations can be extremly complex due to the random errors in design, fa
20、brication 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 Pisto
21、n. 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.xhbyxhbxhbyxhxbyxbhahxayxayxhW3522242232222221232221 )()()()(),(h: field coordinatesx, y: coordinates at exit pupil
22、2627If look the optical system from the rear end, we see exit pupil plane and image plane. ,)(HHHW ,HWW28Wavefront Aberration Expansion29)()()()()()()()()()(),()(,Distortion curvature Field mAstigmatis Coma SA3 Pistonterms order HighercosHW HW cosHW cosHW W HWTilt Defocus PistoncosHWWHWPistonWcosHWH
23、WW331122220222222313140404400111202022000003nmjmlkklm Classical Seidel Aberrations30W000W020W040W060W111W131W151W222W242What do aberrations look like?31W000W020W040W060W111W131W151W222W242W33332Field Curvature HW 22220 Where do aberrations come from?33Distortion cosHW 3311 34Astigmatism cosHW 222222
24、 W2223536ComaW131 cosHW 3131 37Warren Smith, Modern Optical Engineering, P65Spherical AberrationxWxnrRLAxWnrRTA22 W=W040 438+ W=W040 4 W=W020 2 W= -1W020 2 +W040 4Spherical Aberration + Defocus39Through-focus Diffraction Image(With Spherical Aberration)40lWavefront measurement using an interferomete
25、r 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 com
26、a 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
27、errors or improper supporting structure.Important to know41Caustic42 cosHWHWW3nmjmlkklm)(,),( Specifies 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 sp
28、ecific configurations of the system.1.6 Aberration Coefficients4344The Lagrange Invariant ynu-yunThe Lagrange Invariant holds at any plane between object and image.=-nuhynu-0unAt object plane:hu-n yu-n0un=At image plane:hunnuh=yun00-nyunFor object at infinity：45Paraxial Ray TracinguiuiyCRy inniSnell
29、s LawCnn)( ynuunyCunni)(R1C46L=R1CSeidel Coefficient Table47Seidel Coefficient Calculation for a Singlelet48Calculation by Zemax49Calculation by Seidel Coefficient Formula5051The Thin Lens FormThe aberrations of a given optical system depend on the system parameters such as aperture diameter, focal
30、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 coe
31、fficients.5253The Structure Aberration CoefficientRoland V. Shack54The Thin Lens BendingIt is possible to have a set of lenses with the same power and the same thickness but with different shapes.X：Y2abX0bY2aXdXddcYbXYaXI22I Minimum spherical aberrationIf Y is constant, thenY2n1n2Y2n21nn1nn1n4X22)()
32、()()()(714025112(1.5Y2n1n2X22.)(If object at infinity, Y=1, n=1.5, then55YefX0fYeXII Minimum coma801)(1.51)-1)(1.55121n1n12n1nn1nn12nX.()()()()()(If 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:56Zemax ResultCalculation Using
33、 Thin Lens Form57III#2222II2#131I3#0404FyuW16(FyuW256(FyW )For object at infinity：yun00-nyun=For thin lens is in air, n=1，rearrange the thin lens formula:2#2222#1313#040u ,FWuFWFW,)()(581.7 Zernike PolynomialsOften in optical testing, to better interpret the test results it is convenient to express
34、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 wavefro
35、nt aberrations. .)(),(2243210yxaxyayaxaayxW59Zernike 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: lnZ mn 0 whenmn whennddZZnmnmlmln111020
36、 60 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 th
37、e polynomials: lnZ illnlne)(RZ ilil)(ileee 61snmssmnn)!smn()!sm(! s)!sn()()(R2021 lnR can be expressed as:,where m n, l=n-2m. So Zernike term Unm can be expressed as: )mn(cossin)(RUmnnnm22 Where: sin function is used for n-2m0 cos function is used for n-2m 062 )mn(cossin)(RAUA),(Wmnnknnmnmknnmnmnm22
38、0000 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.634 th Zernike polynomials64Re-ordered Zernike polynomials (first 36 terms)6512354678Plots of Zernike polynomials #1# 8669101112131
39、415Plots of Zernike polynomials #9# 1567Plots of Zernike polynomials #16# 241617181920212223246833Plots of Zernike polynomials #25# 362526282729303231353469Zernike polynomials are easily related to classical aberrations. W( , ) is usually found the best least squares fit to the data points. Since Ze
40、rnike 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 square
41、s fit. )2(cossin)(2mnRAmnnnmAdvantages of using Zernike polynomials70Cautions 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 polynomia
42、ls 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.71Relationship Between Zernike polynomials and Seidel AberrationsThe first 9 Zernike polynomials are expressed as:The
43、same aberration can be expressed in Seidel form:72Using the identity:73741.8 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 wavefr
44、ont 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 R
45、MS wavefront error.RMS: “Root Mean Squares”, 2=RMS2PV=Wmax-Wmin75If 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
46、 coefficients. 2nmnm2nmk0nn0mnm2RMS n 2mfor 2n 2mfor 1term each of tscoefficienthe is AspolynomialZernike ofdegree the is kA)1n(2 76Strehl 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 th
47、e 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:77Marechal 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.
