Basic Wavefront Aberration Theory for Optical Metrology

1 APPLIED OPTICS AND Optical ENGINEERING, VOL. XlCHAPTER 1 Basic Wavefront Aberration Theoryfor Optical MetrologyJAMES C. WYANTO ptical Sciences Center, University of ArizonaandWYKO Corporation, Tucson, ArizonaKATHERINE CREATHO ptical Sciences CenterUniversity of Arizona, Tucson, ConventionsAberration-Free ImageSpherical Wavefront , Defocus, and Lateral ShiftAngular, Transverse, and Longitudinal AberrationSeidel AberrationsA. Spherical AberrationB. Coma C. Astigmatism D. Field Curvature E. Distortion Zernike PolynomialsRelationship between Zernike Polynomials and Third-OrderAberrationsPeak-to-Valley and RMS Wavefront AberrationStrehl RatioChromatic AberrationsAberrations Introduced by Plane Parallel PlatesAberrations of Simple Thin 48A.

2 Basic Properties48B. Spherical Aberration50C. Coma 51D. Astigmatism 52 XIV. General Aspheres52 References 531 Copyright 1992 by Academic Press, rights of reproduction in any form 0-12-408611-X28 JAMES C.

3 WYANT AND KATHERINE CREATHVI. ZERNIKE POLYNOMIALSO ften, to aid in the interpretation of Optical test results it is convenient toexpress Wavefront data in polynomial form. Zernike polynomials are oftenSagittal Focal SurfacePetzval SurfaceTangential Focal SurfaceFIG. 33. Focal surfaces in presence of field curvature and Basic Wavefront Aberration Theory 29 Barrel DistortionFIG. 34. Distortion,Pincushion Distortionused for this purpose since they are made up of terms that are of the sameform as the types of aberrations often observed in Optical tests (Zernike,1934).

4 This is not to say that Zernike polynomials are the best polynomialsfor fitting test data. Sometimes Zernike polynomials give a terrible represen-tation of the Wavefront data. For example, Zernikes have little value when airturbulence is present. Likewise, fabrication errors present in the single-pointdiamond turning process cannot be represented using a reasonable numberof terms in the Zernike polynomial. In the testing of conical Optical elements,additional terms must be added to Zernike polynomials to accuratelyrepresent alignment errors.

5 Thus, the reader should be warned that the blinduse of Zernike polynomials to represent test results can lead to polynomials have several interesting properties. First, they areone of an infinite number of complete sets of polynomials in two realvariables, and that are orthogonal in a continuous fashion over theinterior of a unit circle. It is important to note that the Zernikes areorthogonal only in a continuous fashion over the interior of a unit circle, andin general they will not be orthogonal over a discrete set of data points withina unit polynomials have three properties that distinguish them fromother sets of orthogonal polynomials.

6 First, they have simple rotationalsymmetry properties that lead to a polynomial product of the form(49)where G ( ) is a continuous function that repeats itself every 2 radians andsatisfies the requirement that rotating the coordinate system by an angle does not change the form of the polynomial. That is,(50)30 JAMES C. WYANT AND KATHERINE CREATHThe set of trigonometric functions(51)where m is any positive integer or zero, meets these second property of Zernike polynomials is that the radial functionmust be a polynomial in of degree n and contain no power of less than third property is that R(p) must be even if m is even, and odd if m is radial polynomials can be derived as a special case of Jacobipolynomials, and tabulated as Their orthogonality and normalizationproperties are given by(52)and(53)

7 It is convenient to factor the radial polynomial into(54)where is a polynomial of order 2(n - m). can be written generallyas(55)In practice, the radial polynomials are combined with sines and cosinesrather than with a complex exponential. The final Zernike polynomial seriesfor the Wavefront OPD W can be written as (56)where is the mean Wavefront OPD, and An, Bnm, and Cnm are individualpolynomial coefficients. For a symmetrical Optical system, the wave aberra-tions are symmetrical about the tangential plane and only even functions of are allowed.

8 In general, however, the Wavefront is not symmetric, and bothsets of trigonometric terms are III gives a list of 36 Zernike polynomials, plus the constant term.(Note that the ordering of the polynomials in the list is not universallyaccepted, and different organizations may use a different ordering.) Figures 35through 39 show contour maps of the 36 terms. Term #0 is a constant orpiston term, while terms # 1 and # 2 are tilt terms. Term # 3 represents , terms # 1 through # 3 represent the Gaussian or paraxial properties of1.

9 Basic Wavefront Aberration THEORY31n=1 FIRST- order PROPERTIESn=2 THIRD- order ABERRATIONSTILTFOCUSFIG. 35. Two- and three-dimensional plots of Zernike poly-nomials # 1 to # AND DEFOCUSCOMA AND TILTTHIRD- order SPHERICAL AND DEFOCUSFIG. 36. Two- and three-dimensional plots of Zernike polynomials #4 to # 37. Two- and three-dimensional plots of Zernike poly-nomials # 9 to # 38. Two- and three-dimensional plots of Zernike poly-nomials # 16 to # JAMES C. WYANT AND KATHERINE CREATHn=5,6 NINTH- & ELEVENTH- order ABERRATIONSFIG.

10 39. Two- and three-dimensional plots of Zernike polynomials # 25 to # Wavefront . Terms # 4 and # 5 are astigmatism plus defocus. Terms # 6and #7 represent coma and tilt, while term #8 represents third-orderspherical and focus. Likewise, terms # 9 through # 15 represent fifth-orderaberration, # 16 through # 24 represent seventh- order aberrations, and # 25through # 35 represent ninth- order aberrations. Each term contains theappropriate amount of each lower order term to make it orthogonal to eachlower order term.