NonInterferometric Testing.nb Optics 513 - James C. Wyant 1 Non-Interferometric Testing Introduction In tese notes four non-interferometric tests are described: (1) te Sack-Hartmann test, (2) te Foucault test, (3) te wire test, and (4) te Ronci test. Unlike most interferometric tests, tese four tests measure te slope of te wavefront error, rater tan te wavefront error itself. Te Sack-Hartmann test is especially useful for testing large telescope mirrors since if it is performed properly, atmosperic turbulence introduces little error into te test results. Te Foucault test is a powerful metod for qualitatively detecting small errors in optics, and it is simple and inexpensive to perform. Te wire test is a modification of te Foucault test tat makes it easier to obtain quantitative results. Te Ronci test can be tougt of as a multiple wire test. Wile te Ronci test will give many beautiful test results, te accuracy of te results is severely limited by diffraction. 8.2.11 Sack-Hartmann Test Te Sack-Hartmann test is essentially a geometrical ray trace tat measures angular, transverse, or longitudinal aberrations from wic numerical integration can be used to calculate te wavefront aberration. Figure 1 illustrates te basic concept for performing a classical Hartmann test. A Hartmann screen, wic consists of a plate containing an array of oles, is placed in a converging beam of ligt produced by te optics under test. One or more potograpic plates or solid-state detector arrays are placed in te converging ligt beam after te Hartmann screen. Te positions of te images of te oles in te screen as recorded on te potograpic plates or detector arrays give te transverse and longitudinal ray aberrations directly. It sould be noted tat if a single potograpic plate or detector array is used, bot te ole positions in te screen and te distance between te screen and plate must be known, wile if two potograpic plates or detector arrays are used, only te distance between te plates or detector arrays need be known. One advantage of te Hartmann test for te testing of telescope mirrors is tat effects of air turbulence will average out. Air turbulence will cause te spots to wander, but as long as te integration time is long compared to te period of te turbulence te major effect will be for te spots to become larger, and as long as te centroid of te spots can be accurately measured te turbulence will not introduce error in te measurement. Te oles in te Hartmann screen sould be made large enoug so diffraction does not limit te measurement accuracy, but not so large tat surface errors are averaged out.
NonInterferometric Testing.nb Optics 513 - James C. Wyant 2 Hartmann Screen Potograpic Plates #1 #2 Geometric Ray Trace Fig. 1. Classical Hartmann test. Single potograpic plate: must know (a) ole positions in screen, (b) distance between screen and plate. Multiple potograpic plates: must know te distance between plates. Figure 2 sows te results for testing a parabolic mirror at te center of curvature. Note tat te detectors must be kept away from te caustic or muc confusion can result. Once te transverse or longitudinal aberration is determined, te wavefront aberration can be determined. Outside Position Inside Position Fig. 2. Hartmann test of parabolic mirror near center of curvature.
NonInterferometric Testing.nb Optics 513 - James C. Wyant 3 Sack modified te Hartmann test by replacing te screen containing oles wit a lenslet array. In typical use te beam from te telescope is collimated and reduced to a size of a few centimeters and impinges on a lenslet array tat focuses te ligt onto a detector array as sown in Figure 3. Te positions of te various focused points give te local slope of te wavefront. Figure 4 sows potos of a Sack-Hartmann lenslet array. Fig. 3. Sack-Hartmann lenslet array measuring slope errors in an aberrated wavefront. Fig. 4. Potos of a Sack-Hartmann lenslet array. Te Sack-Hartmann wavefront sensor is widely used in adaptive optics correction of atmosperic turbulence. Figure 5 sows a movie of te focused spots from te Sack-Hartmann test dancing around due to atmosperic turbulence. S.mpeg Fig. 5. Movie made using Sack-Hartmann test to measure atmosperic turbulence.
NonInterferometric Testing.nb Optics 513 - James C. Wyant 4 8.2.12 Foucault Test Te Foucault knife-edge test is one of te oldest and most common tests for determining longitudinal and transverse aberrations from wic te wavefront aberration can be determined. In te Foucault test, a knife edge placed in te image plane is passed troug te image of a point or slit source. Te observer's eye is placed immediately beind te edge and allowed to focus upon te exit pupil of te system under test, as sown in Fig. 6 for testing a concave mirror. As te knife edge passes troug te image, a sadow is seen to pass across te pupil. Te more compact te ligt distribution in te image, te more rapidly te sadow passes across te pupil. A perfect lens will ave one image point suc tat te entire pupil of te lens is seen to darken almost instantaneously wen te knife edge is passed troug te image. If te knife edge is moved longitudinally toward te lens from tis image point and again passed across te image, te sadow will be seen to travel across te aperture in te same direction in wic te knife edge is passed across te image, as illustrated in Fig. 7. Wen te knife edge is located beind te point image, an opposite motion of te sadow occurs. Te ultimate sensitivity of te test is attained by observing te motion of te sadow witin certain zones of te aperture as te knife edge is cut across te image. In practice, te knife edge is most often used to measure te zonal focus of different parts of an optical surface. Tis information is of greatest interest to an optician wo generally wants to know were te ig and low parts of a surface are. Surface to be Tested Slit Source Fig. 6. Foucault knife-edge test. Eye Knife Edge
NonInterferometric Testing.nb Optics 513 - James C. Wyant 5 Focus Observed Pattern Knife Edge Knife Edge Knife Edge Fig.7. Ray picture of te Foucault knife-edge test. It is not possible to describe in words te appearance of a Foucault sadow pattern for testing a general optical system. Te Foucault sadows ave to be observed to appreciate te sensitivity of te test for observing small slope errors; owever, it is possible to describe te sadows tat are obtained for te Seidel aberrations. It must be noted tat te following is simply a geometrical analysis, and due to diffraction, te sadows are not as distinct as described below.
NonInterferometric Testing.nb Optics 513 - James C. Wyant 6 First, let us look at te sadows produced for tird-order sperical aberration. For sperical aberration and defocus W = W 040 Hx 2 + y 2 L 2 + ε z 2 2 R 2 Hx2 + y 2 L. Te boundary of te geometrical sadow is obtained by setting te distance of te knife edge from te optical axis, d, equal to te transverse aberration for sperical aberration. Tus, for normalized pupil coordinates, te equation of te boundary of te sadow is d = R W y = 4 RW 040 y Hx 2 + y 2 L ε z y ; R Let us suppose, for a moment, tat te knife edge is located on te optical axis, so d = 0. Te solution to Eq. (1) ten becomes (1) y = 0 (2) and x 2 + y 2 = ε z 2 4 R 2 W 040 Te first solution is te equation of a straigt line troug te origin, and te second solution is te equation of a circle of radius ρ= i k j ε z 2 y 4 R 2 W 040 { 1ê2 z Te complete solution is illustrated in Fig. 8 for different values of e z. A procedure for finding W 040 is to first find te paraxial focus by noting te position were te central spot in te pattern just vanises. By measuring r as a function of e z, we obtain a measurement of W 040 using Eq. (4). (3) (4) Knife edge near paraxial focus Knife edge partway between paraxial and marginal focus Knife edge near marginal focus Fig. 8. Knife-edge test patterns due to tird-order sperical aberration, wen te knife edge is on te optical axis.
NonInterferometric Testing.nb Optics 513 - James C. Wyant 7 If te knife edge is not located on te optical axis, te simplified solutions given in Eq. (3) are not valid, and te cubic Eq. (1) must be considered. Drawings sowing typical solutions are sown in Fig. 9. A B A Knife-edge aead of marginal focus B Knife-edge beind of paraxial focus Fig. 9. Knife-edge test pattern for tird-order sperical wen knife-edge is not on te optical axis. Next, we will consider te knife edge test for coma. For coma and defocus W = W 131 y o y Hx 2 + y 2 L + ε z 2 2 R 2 Hx2 + y 2 L Because of te asymmetry of coma, te pattern depends upon weter te knife edge is parallel to te x or te y axis. If te knife edge is parallel to te x axis, te equation of te boundary of te sadow is d = R W y = R W 131 y o Hx 2 + 3 y 2 L ε z y R wic is te equation of an ellipse, wile if te knife edge is parallel to te y axis, te equation of te boundary of te sadow is (5)
NonInterferometric Testing.nb Optics 513 - James C. Wyant 8 d = R W x = 2 R W 131 y o xy ε z x R (6) wic is te equation of a yperbola. Drawings sowing typical test patterns for coma are sown in Fig. 10. Knife edge parallel x-axis Inside paraxial focus At paraxial focus Knife edge parallel y-axis Inside paraxial focus At paraxial focus Fig. 10. Knife-edge test pattern due to coma. Te knife-edge test can be used to measure astigmatism in a lens. For astigmatism W = W 222 y o 2 y 2 + ε z 2 2 R 2 Hx2 + y 2 L It is important tat te knife edge is not parallel to eiter te tangential or te sagittal plane wile making te measurement. If te knife edge is parallel to eiter te y or te x axis, te equations for te sadow boundaries would be
NonInterferometric Testing.nb Optics 513 - James C. Wyant 9 d = R or d = R W x W y = ε z x R = J 2 R W222 y 2 o + εz R N y wic are just te equations of straigt lines, so te astigmatic wavefront would be indistinguisable from a sperical wavefront. However, if we put te knife edge in te beam at an angle to te x axis, ten d =- e x Sin[a] + e y Cos[a]. Terefore d = ε z R x Sin@αD J 2 R W222 y 2 o + εz R N y Cos@αD Hence, te angle of te sadow is not generally te same as te knife edge, and te angle canges as te knife edge is moved along te axis. To measure te longitudinal aberration, it is convenient to place over te surface being tested a mask aving two symmetrically located apertures to define te zone being measured. Te knife edge is ten sifted longitudinally until it cuts off te ligt troug bot apertures simultaneously. Te knife edge is ten at te focus of te zone defined by te mask. Just as for te Hartmann test, it is possible to obtain te wavefront from eiter te transverse or te longitudinal aberrations. If te slope, b, of te wavefront is calculated from te transverse or longitudinal aberration, numerical integration can be used to calculate te wavefront profile across te wavefront. If b i and x i are te slope and real coordinate (not normalized) of te it point across te wavefront, te wavefront aberration is given by (7) (8) n W = 1 2 Hβ i 1 +β i L Hx i x i 1 L i=1 (9) It sould be noted tat if bot te ligt source and te knife edge are moved togeter, as is often te case for testing mirrors, te effective motion of te knife edge is twice te actual motion. Te advantages of te knife edge test are tat it is simple and convenient, and no accessory optics are required. Also, te wole surface can be viewed at once, and te test is sensitive. Te disadvantages of te test are tat it is sensitive to slopes rater tan eigts, and it measures slopes in a single direction wit a single orientation of te knife edge. Because of diffraction, te sadow position can be difficult to define. One way of improving quantitative measurements performed using te knife edge is to use a pase knife edge instead of a density knife edge. In a pase knife edge bot parts of te knife are transparent, but one side introduces a pase difference, preferably 180. In tis case te diffraction pattern produced by te knife edge is symmetric and if te pase difference is exactly 180 te center of te diffraction pattern is dark. Te most important feature is tat since te diffraction is symmetrical, te center of te pattern can be identified.
NonInterferometric Testing.nb Optics 513 - James C. Wyant 10 8.2.13 Wire Test Te wire test is te same as te Foucault test except te knife edge is replaced wit a wire (or slit). Often te wire is a strand of air. Te wire test is inferior to te Foucault test as far as obtaining qualitative data, but it is superior as far as obtaining quantitative data since te wire gives a sarper boundary tan te knife edge. Figure 11 sows some computer-generated images tat would be obtained using te wire test. Te images of te wire are described by te same equations given above for te sadow boundaries in te Foucault test. Figure 12 sows experimental results for using te wire test to evaluate a parabolic mirror at te center of curvature. Fig. 11a. Wire test pattern sowing effect of focus for tird-order sperical aberration (from R. V. Sack).
NonInterferometric Testing.nb Optics 513 - James C. Wyant 11 Fig. 5-11b. Wire test pattern sowing effect of lateral displacement of wire for tird-order sperical aberration (from R. V. Sack).
NonInterferometric Testing.nb Optics 513 - James C. Wyant 12 Wire test experimental results for parabolic mirror tested at center of curvature Close-up sowing diffraction pattern Fig. 12. Experimental results obtained using te wire test to evaluate a parabolic mirror at te center of curvature. 8.2.14 Ronci Test In te Ronci test, a low-frequency grating, called a Ronci ruling, is substituted for te knife edge used in te Foucault test, or te wire used in te wire test. Te test may be understood by considering te Ronci ruling as equivalent to multiple wires. A typical experimental arrangement for utilizing a Ronci ruling is sown in Fig. 13. Instead of a single wire, we now ave several wires to consider, and instead of a single sadow, we now ave several sadow boundaries. In our analysis we will initially neglect diffraction.
NonInterferometric Testing.nb Optics 513 - James C. Wyant 13 Convergent Wavefront Eye Ronci Ruling Fig. 13. Ronci test. Let te Ronci ruling be inserted in te beam a distance e z from te paraxial focus. Suppose te rulings make an angle a wit respect to te x axis. If we let d be te grating spacing and m be an integer (wic we sall call te order number), ten te equation for te sadow boundaries is md= ε x Sin@αD +ε y Cos@αD = R W x Sin@αD R W y Cos@αD First, consider a perfect lens. If a = 90, ten te equation for te sadow boundaries is (10) md= R W x = ε z x R (11) Hence, te pattern observed in te Ronci test of a perfect lens is a series of straigt lines corresponding to different values of m, as sown in Fig. 14. Te number of lines in te field depends upon te distance between te Ronci ruling and te focus.
NonInterferometric Testing.nb Optics 513 - James C. Wyant 14 Ruling near focus Ruling away from focus Fig. 14. Ronci test patterns of a perfect lens. Let us now consider te situation were te wavefront as tird-order sperical aberration. Te equation for te sadow boundaries is te same as Eq. (1) given for te Foucault test except now d is replaced wit md. Figure 15 gives typical Ronci test patterns for tird-order sperical aberration.
NonInterferometric Testing.nb Optics 513 - James C. Wyant 15 Fig. 5-15. Ronci test patterns for tird-order sperical aberration (from R.V.Sack). Likewise, te Ronci pattern for coma is obtained from Eqs. (5) and (6) by replacing d wit md. Te resulting elliptical and yperbolic patterns are sown in Fig. 5-16. Te beavior of te Ronci pattern for astigmatism is te same as for te Foucault test described above. Figure 5-17 sows te corresponding pattern of an astigmatic wavefront as te Ronci ruling is moved troug focus.
NonInterferometric Testing.nb Optics 513 - James C. Wyant 16 Fig. 5-16. Ronci test patterns for coma (from V. Ronci, "Forty Years of History of a Grating Interferometer," Applied Optics 3, 437-451, 1964). Fig. 5-17. Ronci test patterns for astigmatism (from V.Ronci,"Forty Years of History of a Grating Interferometer," Applied Optics 3, 437-451, 1964). Generally, te Ronci ruling is not used wit a point or slit source, but rater te ruling is illuminated by a diffuse source from beind as sown in Fig. 5-18. Te ruling ten acts as a multiple slit source, wit te ligt from eac opening in te grating producing a test pattern identical to tat of te oter openings. Surface to be Tested Diffuse Source Fig. 5-18. Te use of Ronci rulings wit a diffuse source. Eye Ronci Ruling Unfortunately, diffraction effects are readily observed wen using Ronci rulings to test optics. In te Ronci test, te ruling acts as a diffraction grating to create multiple images of te surface being tested. If te gratings are
NonInterferometric Testing.nb Optics 513 - James C. Wyant 17 coarse, say less tan l0 lines/mm, te multiple images due to diffraction will lie close togeter, and will cause only a sligt perturbation of te sadow pattern. If ig-frequency gratings, say greater tan l00 lines/mm are used, te various images of te surface being tested will be clearly distinguised. It can be sown tat te patterns obtained in a Ronci test are essentially te same wen te various images of te source are clearly resolved as tey are wen diffraction can be totally neglected. In te intermediate situation were te various images of te test surface all overlap, te resultant patterns are difficult to analyze. Te advantages of a Ronci test are tat te test is simple and will work wit a wite ligt source. Te disadvantage is tat it does not give te wavefront directly, and for a single Ronci ruling orientation, slope in only one direction is obtained. Also, te diffraction effects are very troublesome and limit te accuracy of te test.