Imaging issues for interferometric measurement of aspheric surfaces using CGH null correctors

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1 Invited Paper Iaging iue for interferoetric eaureent of apheric urface uing CGH null corrector Ping Zhou*, Ji Burge, Chunyu Zhao College of Optical Science, Univ. of Arizona, 63 E. Univ. Blvd, Tucon, AZ, USA 857 ABSTRACT Apheric urface are eaured uing tandard interferoeter coupled with coputer generated hologra (CGH) that copenate the apheric wavefront. Such yte can eaure coplex apheric hape with accuracy of a few nanoeter. However, the iaging propertie of the interferoeter-cgh cobination can provide liitation for data apping, reolution, and accuracy. Thee effect are explored, with an ephai on the diffraction effect that are unique to interferoetry. Keyword: CGH, iaging aberration, tranfer function, Talbot effect, phae oothing, edge diffraction. Introduction Apheric optical urface are often teted uing interferoeter with the help of coputer-generated hologra (CGH), becaue CGH can generate wavefront of virtually any deired hape. The CGH i deigned to tranfor a pherical wavefront fro an interferoeter to an apheric one that atche the hape of the urface under tet []. The CGH alo perfor another role. A an optical eleent, it relay the iage of the urface under tet to the interferoeter. The CGH i optiized for wavefront perforance, and not for good iaging. The firt order effect of iaging through CGH i the apping ditortion. Mapping ditortion can caue the lower order alignent error appear a high order wavefront error. Interferoetric eaureent error introduced by iaging ditortion have been dicued in detail by Selberg, Murphy and Zhao [-4]. The econd order iaging aberration due to the CGH can be decribed fully uing field curvature in the iage that arie fro the phae variation in the CGH. In general, the iage will uffer defocu and atigati which vary acro the focal plane. Thi aberration couple with diffraction to caue two principal proble with the eaureent: Real error in the urface will be blurred, which applie a low pa filter to the urface data. We treat thi effect, which we call phae oothing, a a function of the patial frequency uing a all-phae approxiation to the well-known Talbot iaging relation. The cobination of iaging aberration and the aplitude dicontinuity at edge how up in the eaureent a phae error. We treat thi effect uing Frenel integral. The field curve provide equivalent defocu projected into the two principal direction (S and T for an axiyetric yte). The two diffraction effect then have dependency on the orientation of the ripple or the edge with repect to the S and T direction. Thee are treated explicitly here uing equivalent propagation ditance and coordinate tranforation. A odel for diffraction effect for refractive null corrector wa developed by Zhao and Burge [5]. We further expand thi to include the pecial iue with diffractive null corrector. We provide background for the diffraction effect for coherent yte uch a interferoeter in ection. Section 3 preent a uary of the geoetry and iaging aberration pecific to CGH null teting of conic irror. The detailed derivation and analyi for thi ection are publihed by the author elewhere [6]. Section 4 preent the general ethod of coupling the diffraction effect with the iaging aberration that arie fro the ue of CGH. The pecific exaple of null teting a 4.- diaeter paraboloidal irror i provided in Section 5. Interferoetry XV: Technique and Analyi, edited by Catherine E. Tower, Joanna Schit, Katherine Creath, Proc. of SPIE Vol. 779, 779L SPIE CCC code: X//$8 doi:.7/.8647 Proc. of SPIE Vol L- Downloaded fro SPIE Digital Library on 9 Sep to Ter of Ue:

2 . Diffraction effect for coherent iaging yte.. Interferoeter iaging yte The iaging perforance through the CGH i difficult to evaluate becaue it i part of a coherent yte. For tandard incoherent iaging, light i aued to coe fro each point in the object and propagate in all direction. Soe of the light ake it into the optical yte, and i ued for iaging. The aperture top in the yte define which ray fro the object will be ued to create the iage. If you change the top, you change the iaging perforance [5]. The iaging for ot interferoeter i fundaentally different. The light tart at a point ource, and follow only refraction and reflection to the urface under tet, then to the final iage. The aperture top in the yte doe not noinally define the light that ake it to the iage. For thi fully coherent yte, it i convenient to think of one ray per point on the irror. The ray will propagate to the iage plane and define the iage. Only for the cae of large lope error or high frequency ripple doe the top have any effect. Thi fully coherent odel leave no roo for iaging aberration to degrade the iage. Each ray will interect the iage plane at a harp geoetric point. We treat the iaging yte uing an infiniteial bundle of ray, becaue when a ray hit a point with tructure there i a bundle of ray coing out due to cattering and diffraction. Thee ray originate fro each point on the urface under tet. The width of the bundle can be defined in the coputer odel uing the ize of the top at the focu in the interferoeter. The top, acting a a low-pa filter, liit the aount of the light that can ake it to the detector. Interferoeter iaging aberration are well known to degrade perforance in other way. Such aberration caue retrace error where large lope variation in the wavefront under tet couple with aberration to create phae error that would reult in an incorrect eaureent [3]. Thi effect can be fully copenated if the aberration are well known. For incoherent iaging yte, diffraction effect are wahed out. For coherent iaging yte, diffraction effect influence interferoetric eaureent by oothing actual phae irregularitie and introducing phae error due to edge diffraction... Phae oothing analyi uing Talbot odel Diffraction effect caue a change in high patial frequency coponent, which we call phae oothing. Phae oothing can be tudied uing the Talbot iaging theory [7, 8]. Talbot iaging i a diffraction phenoenon that occur for any wavefront with a periodic coplex aplitude. If a wavefront with coplex aplitude of a period of d i illuinated by colliated light, then that ae coplex aplitude i fored by free pace diffraction at integer ultiple of the Talbot ditance zt = d λ. A a inuoidal phae pattern propagate, it will cycle through a revere contrat aplitude pattern, a conjugate phae pattern, a pure aplitude pattern, then back to the original phae pattern. If a wavefront with all phae ripple of W ( W << ) wave propagate a ditance of L in colliated pace, then the agnitude of the ripple will be attenuated to W. A tranfer function can be defined a W ' L πλl TF = = co π = co W z T d, () which can be ued to predict the attenuation in high frequency (all cale) ripple due to propagation. The Talbot ditance z T i defined for a colliated bea. For a pherical bea, it i convenient to convert it into an equivalent colliated bea and then ue the Talbot effect to predict how wavefront change due to propagation. The diffraction pattern for a pherical bea i the ae a that oberved for a colliated bea, except that the diffraction pattern occur at the effective propagation ditance L e, and it i caled in the tranvere dienion. Proc. of SPIE Vol L- Downloaded fro SPIE Digital Library on 9 Sep to Ter of Ue:

3 Figure. Propagation in a converging pace i converted to equivalent propagation in a colliated pace. A hown in Figure, a converging wavefront tarting with radiu of curvature R propagate to the poition where it ha a radiu of curvature R, the effective propagation ditance L e can be defined a R( R R) Le =. () R The effective propagation ditance L e can replace the propagation ditance L in Eq. () to calculate the tranfer function for the wavefront that propagate in a converging or diverging bea..3. Edge effect uing Frenel integral Edge diffraction fro the tet urface occur when it edge i not in focu. The diffraction at the edge of the aperture can be odeled a the Frenel knife-edge diffraction. The real and iaginary part of the electric field ditribution can be found by evaluating the Frenel integral [9]. Figure how the aplitude and phae fluctuation due to the edge diffraction. Both aplitude and phae have a rapid ocillation a the ditance fro the edge becoe large. The beauty of thee picture i that it work for any geoetry and only cale with λ L, where L i the defocu error in a colliated wavefront pace or the equivalent propagation ditance in a non-colliated wavefront pace. a) b) Figure. The aplitude and phae variation calculated fro the Frenel integral for the cae of diffraction of colliated light fro a knife edge. The everity of effect of edge diffraction can be decribed by the diffracted edge ratio: 5 λl ε =, (3) r iage Proc. of SPIE Vol L-3 Downloaded fro SPIE Digital Library on 9 Sep to Ter of Ue:

4 where r iage i the iage ize in radiu at the interediate iage. Here we arbitrary chooe 5 λl to calculate the diffracted edge ratio, which correpond to about ix ripple fro the edge. For exaple, if the interediate iage i 5 in radiu and it i 79 out of focu, then the diffracted edge ratio i.5 for the wavelength of 633 n. Alot all of the edge effect occur in the outer 5% of the radiu. The phae error due to the edge diffraction i 3.7 n a hown in Figure 3. RMS = 3.66 n Figure 3. Phae error caued by edge diffraction. The diffracted edge ratio i 5% and the RMS phae error i 3.7 n. 3. Interferoetric eaureent of apheric irror uing CGH null corrector 3.. Tet geoetry Here, we apply the reult fro a generalized odel and analyi for CGH teting of conic irror []. Figure 4 how the geoetry of CGH null tet for a conic urface. The hologra i placed ditance h away fro the paraxial center of curvature to avoid the cautic area. The CGH ue the power carrier to eparate diffraction order, and the light focue at the point E. The phae function of the CGH i derived uing thi geoetrical odel. Thi odel i contructed by coputing the abolute path length along the direction of propagation. A ray that i perpendicular to the tet urface i traced to the hologra and then to the focu point E. Figure 4. Tet geoetry uing a hologra to tet a conic urface. The hologra ue power carrier to eparate the unwanted diffraction order. Proc. of SPIE Vol L-4 Downloaded fro SPIE Digital Library on 9 Sep to Ter of Ue:

5 The conic optical urface i given by r z(r)= R+ R K + r ( ), (4) where r i the radial poition on the tet urface, R i the radiu of curvature and K i conic contant. The phae function of the CGH i decribed by geoetry, and approxiated to the fourth order: p Φ ( r ) = ( R h) + p AB- co ( φ) ( h + hp) r ( h h p h Kp + hp + hkp + hkp R Kp R) = + pr 8p R 3 4 r 4. (5) 3.. Iaging ditortion The firt order effect of iaging through CGH i the apping ditortion, which ean the apping fro the tet optic to the interferogra i not linear. Null corrector are optiized to generate an apheric wavefront but not necearily to provide unifor apping fro the tet optic to the iage plane. The iaging ditortion fro a null corrector will caue three coplication: one i that the urface defect appear hifted; the econd i that lower order alignent error appear a higher order wavefront error; the third proble i that diffraction effect, which depend on patial frequency, will vary a the agnification change with poition and orientation. For an axiyetric yte, the apping ditortion between the radial poition on the irror r and the radial poition on the CGH r CGH i approxiately 3 rcgh (r )= ar + ar, (6) where a and a are contant, and are related to the agnification and ditortion. Mapping ditortion alo caue the nonlinear agnification in the radial and circuferential direction. We define the agnification in polar coordinate, with φ a the polar angle and α a the aziuthal angle. The radial agnification to polar coordinate dφ/dr be decribed a dφ dφ drcgh = = c + cr, (7) dr drcgh dr and the circuferential or aziuthal agnification to polar coordinate i rcghdαcgh rcgh = = c + cr, (8) rdα r where c, c, c are contant and they are related to the tet geoetry and the paraeter of the conic urface Field curve We deterine the field curvature fro the two econd derivative for the axiyetric hologra phae function Φ. One i in the radial direction C t and the other C i in the circuferential direction, perpendicular to the radial direction. The two curvature in the hologra coordinate are C C t ( r ) ( r ) 3 Φ Φ dr Φ d r CGH dr CGH drcgh dr drcgh d d d = = dr dr dr. (9) dφ dφ dr = = r dr r dr dr CGH CGH CGH CGH The two principal curvature C t and C bring the light to the tangential and agittal focal plane for the axiyetric yte. Thi can be confuing becaue the tangential focu by thi definition coe fro curvature in the radial direction and the agittal focu coe fro the curvature in the circuferential direction. We aintain the T and Proc. of SPIE Vol L-5 Downloaded fro SPIE Digital Library on 9 Sep to Ter of Ue:

6 S definition for the field curve, but refer to C t a the radial curvature and C a the circuferential curvature. Thi i illutrated in Figure 5. Figure 5: A ketch of the tangential and agittal curvature for an axiyetric hologra. When a all bundle of ray originating fro a point on the tet urface are incident on the hologra, the ray ee different curvature. The fan in the radial and circuferential direction will focu onto eparate curved urface becaue of the field curvature and atigati iaging aberration fro the CGH. Thee field curvature can be obtained in coercial raytracing oftware or calculated by uing generalized Coddington equation []. The generalized Coddington equation for diffractive optic i co φ co θ = Ct t' t, () = C ' where t and are the phyical ditance fro the tet urface to the hologra, and t and are the ditance fro the hologra to the tangential iage point and agittal iage point. To evaluate the coupling of diffraction with the field curvature, we create an iage at an equivalent colliated pace by inerting an ideal len with a focal len of f at one focal length away fro the point focu. Thi contruction i ueful becaue it iplifie the analyi, but it i not neceary. The focal length itelf i arbitrary, and will be noralized out of the reult. An ideal len preerve the field curve and create a colliated wavefront, and telecentric iage. We how elewhere that the equivalent propagation ditance in colliated pace aintain the diffraction perforance for general converging or diverging light [8]. When the ditance between the hologra and the paraxial focu h and the ditance between the hologra and the point focu p are all copared to the radiu curvature of the tet piece R, the field curve are otly quadratic in field. We can ue the geoetric iaging relationhip to calculate the field curve. Keeping the doinant quadratic ter, the two field curve are 3 f K ti ' r pr f K i ' r pr. () For axiyetric urface, the inide and outide edge are alway circuferential, which ean the diffraction effect fro the edge i alway radial and caued by the T-coponent of the field curve. We can ue the field curve to analyze the phae ripple and edge effect in the principal radial and circuferential direction. For non-axiyetric urface, the boundary could be in general direction, o the diffraction calculation ut ue the cobination of the two field curve. Proc. of SPIE Vol L-6 Downloaded fro SPIE Digital Library on 9 Sep to Ter of Ue:

7 4. Coupling of diffraction effect with field curve Fundaentally, the field curve how defocu in the iage pace a function of field poition and orientation. The diffraction effect can be treated at each field point and orientation a propagation of light fro it focued poition given by the field curve to the focal plane. Thi i apparent for the principal S and T orientation. The cae with general orientation i provided elewhere [5,6]. We iplify the proble uing the concept of ideal lene to tranfer fro one plane to another. If the wavefront ripple are iaged correctly fro one pace to another, there i no attenuation fro the Talbot effect. The agnitude of the aplitude and phae ripple will be preerved. For the plane that i out of focu, the Talbot effect decribed above can be ued to deterine the change in phae due to the propagation. In any re-iaged optical pace, the propagation ditance equal to the out of focu ditance. The patial period of the ripple ut correctly include the agnification of the iage. The ae principle applie for the field curve, but thi include variation acro the iage of the tet urface and it include a defocu in the circuferential and radial direction. The defocu acro the field i given by the field curve, and the two curve correpond to the iage with ripple that are radial or circuferential. Interferoeter can only focu one plane at a tie o that the other field will be out of focu. To evaluate the phae oothing effect at an arbitrary iage plane, we ue the fact that the longitudinal agnification in ditance and tranvere agnification of the patial period cancel with each other. If we relay the iage with a local agnification, the patial period of the wavefront ripple d will be agnified by. Auing the field curvature are not corrected by the interediate optic, the aount of defocu for any field point i given by the longitudinal agnification, which i equal to. The effect of propagation, a een in the tranfer function, i invariant to agnification: W ' πλ z πλ z πλ z TF = = co co co = =. () W d ( d) d Note that both the agnification and z in the above equation vary acro the field. They are a function of poition and orientation. The radial and circuferential agnification are given in Eq. (7-8). Thi ean that we can evaluate the wavefront oothing at any interediate iaging pace and the reult will be the ae. If d and z are in the pace of the tet urface, then we can noralize the period d with the tet urface ize a and the tranfer function becoe πλ z( r ) πλ z( r) f noralized TF( r ) = co co = (3) r ( ) d 4 ar ( ) The noralized frequency ha a unit of cycle/diaeter. The tranfer function depend on the field curve and the local agnification at each field point. Thi tranfer function i given for the principal orientation, radial and circuferential for the axiyetric CGH. The developent of the tranfer function for inuoidal ripple in an arbitrary direction i provided elewhere [6]. The non-linear apping fro the CGH create ditorted iaging where feature near the edge of the irror are proportionally larger. The difficult for optical teting coe about near the center where the actual feature are greatly de-agnified. A contant frequency at the irror will appear to have higher frequency variation near the center. 5. Exaple In thi paper, we ue a 4. parabola with a radiu of curvature of 6 a an exaple to look at it iaging perforance. The CGH i placed 5 below the paraxial focu ( h = 5 ), and the point focu after the CGH i 48 above the hologra ( p = 48 ). The CGH i 95 diaeter. It phae function i hown in Figure 6. 6 We expand the phae function into the Taylor erie in r to the order r, which atche the exact phae function well. Uing the truncated Taylor erie can iplify the field curvature calculation. Proc. of SPIE Vol L-7 Downloaded fro SPIE Digital Library on 9 Sep to Ter of Ue:

8 .4 CGH phae function. exact ix order approxiation phae () radial poition on irror () 6 Figure 6. The CGH phae: exact for and r approxiation. Thi plot i for a CGH ued to tet a 4. parabola with a radiu of curvature of 6. The CGH i place 5 below the paraxial focu, and the point focu after the CGH i 48 above hologra. The iage ditortion fro the tet urface to the CGH i defined by the apping hown in Figure radial poition on CGH () radial poition on the tet urface [] Figure 7. CGH apping function howing the relationhip between the tet urface poition and poition on hologra. The radial curvature C t and the circuferential C curvature, defined a directional derivative of the phae function are given in Figure 8. Proc. of SPIE Vol L-8 Downloaded fro SPIE Digital Library on 9 Sep to Ter of Ue:

9 .5 x -3 C t C.5 [/] radial poition on the hologra [] Figure 8. The radial C t and the circuferential C curvature for the CGH ued to tet the 4.- paraboloidal irror for the cae where the CGH poition i 5 below the paraxial focu, and the point focu after the CGH i 48 above the hologra. To treat the hologra a coherent iaging yte, we odel the parabola a the object, and it field i fro - to. The aperture top i located at the point iage of the interferoeter. Figure 9 how a ray trace odel by Zeax. The oftware ut have ray aiing turned on o the ray bundle will be correctly defined by thi top, and the top ize i diaeter, which pae lope up to 3.5 µrad and patial frequencie up to 7 cycle/diaeter. An arbitrary paraxial len with a focal length of i placed away fro the point focu to iage the tet urface through the hologra o that the wavefront i in a colliated pace. The paraxial len can be aued to have an ideal iaging and will not introduce field curve. The paraxial len wa verified to faithfully reproduce the field curve. In an actual interferoetric eaureent, the optic inide the interferoeter will act a thi paraxial len and re-iage the field curve to the detector plane. The optic inide the interferoeter ay not be perfect and can caue additional aberration, but in thi paper we conider only field curve introduced by the hologra. Object point on irror urface Aperture top located at the interferoeter focu Ideal len CGH Figure 9. Layout of a null tet for a parabola with a paraxial len to create an equivalent colliated wavefront pace. If we chooe the paraxial len with a focal length of, the field curve are hown in Figure. The difference between the T and S field curve at the full field i 46 a calculated in Zeax. Proc. of SPIE Vol L-9 Downloaded fro SPIE Digital Library on 9 Sep to Ter of Ue:

10 Figure. The field curve with the ue of a paraxial len with the focal length of. The radial and circuferential agnification due to the CGH apping ditortion are hown in Figure t Radial poition on the irror Figure. The radial and circuferential agnification for the CGH ued to tet 4.- parabola with the radiu of curvature of 6. The CGH i placed 5 below the paraxial focu of the parabola. The effect of diffraction couple both the equivalent propagation given by the field curve and the variation in patial frequency, given by the agnification. The perforance i alo clearly a function of the choice of focu, which i achieved in hardware by hifting the detector array. We illutrate thee iue with the exaple of the 4. parabola. We evaluate the diffraction effect by etting the focu at three different place: radial focu at the inide edge of the irror, 6 fro the center, focu at the outer edge, and halfway between. Thi i illutrated in Figure. When the interferoeter focue at the inner edge, the T curve at 6 field i in focu and the other are not. Proc. of SPIE Vol L- Downloaded fro SPIE Digital Library on 9 Sep to Ter of Ue:

11 5 irror outer edge Field [] 5 t irror inner edge 5 focu outer edge focu inner edge [] Figure. The field curve at the interediate iage plane. We could chooe to et the interferoeter at the inner edge of the irror, defined by the central hole, at the outer edge, or any place between. The tranfer function for phae ripple at the outer edge in radial and circuferential direction are hown in Figure 3(a). There i not uch oothing effect at the outer edge in both diffraction becaue even though the outer edge i out of focu for both S and T, the patial ripple are greatly agnified, and the oothing goe a the fourth power of patial frequency. When the interferoeter focue at the outer edge of the tet urface (at the full field of tangential field curve), the TF function at the inner edge are given in Figure (c). Thi i coniderably wore. We alo how a cae halfway between the two. focu at inner edge T focu at field 35 T focu at outer edge T Tranfer function T outeredge Tranfer function T outeredge Tranfer function T outeredge -.5 S outeredge -.5 S outeredge -.5 S outeredge T inneredge T inneredge T inneredge S inneredge Frequency [cycle/diaeter] S inneredge Frequency [cycle/diaeter] S inneredge Frequency [cycle/diaeter] (a) (b) (c) Figure 3. Tranfer function: a) the interferoeter focue at the inner edge of the irror; b) the interferoeter focue at the field 35; c) the interferoeter focue at the outer edge of the irror. For the frequency of cycle/diaeter, the tranfer function in the t and direction at different field poition are given in Figure 4. The analyi of the edge diffraction can be evaluated uing the T field curve. For the 4. parabola, the diffracted edge ratio, defined a Eq.(3) i hown in Figure 5. The difference of the T field curve at the outer and inner edge i 46. The iage ize i 9.8 radiu, o the diffracted edge ratio at the inner edge i 3.6% when the interferoeter focue the outer edge. Thi ratio i the ae at the outer edge when the interferoeter focue the inner edge. A the interferoeter focue at different field poition, the diffracted edge ratio for both the inner and outer edge are hown in Figure 5. Proc. of SPIE Vol L- Downloaded fro SPIE Digital Library on 9 Sep to Ter of Ue:

12 focu at inner edge T focu at field 35 T focu at outer edge T Tranfer function t Tranfer function t Tranfer function t field poition field poition field poition Figure 4. For cycle/diaeter, the tranfer function in the t and direction at different field poition inner edge outer edge.3 Diffracted edge ratio field pition that i in focu () Figure 5. The diffracted edge ratio a a function of the focu. 6. Concluion We how pecifically how the iaging propertie of CGH null corrector couple with diffraction to caue eaureent error. Uing the tet of a 4.- diaeter irror a an exaple, we how how thee effect are itigated by balancing the field curve due to the phae curvature with the agnification variation. If the yte focu i et to the inide edge of the irror under tet, then the fact that the outer edge i defocued for both principal direction i balanced with the increaed agnification that ake the feature on the irror appear proportionally larger. REFERENCE [] D. Malacara, Optical Shop Teting, nd ed. Wiley (99). [] L. A. Selberg, Interferoeter accuracy and preciion, Proc. SPIE 4, 4-3 (99) [3] P. E. Murphy, T. G. Brown, and D. T. Moore, Meaureent and calibration of interferoetric iaging aberration, Applied Optic Vol.39, No.34, () [4] C. Zhao and J. H. Burge, Orthonoral vector polynoial in a unit circle application: fitting apping ditortion in a null tet, Proc. SPIE 746, (9). [5] C. Zhao, and J. H. Burge, Iaging aberration fro null corrector, Proc. SPIE 673,673L (7) [6] P. Zhou, J. H. Burge and C, Zhao, Diffraction effect for interferoetric eaureent due to field aberration, to be ubitted to Optic Expre (). [7] P. Zhou and J. H. Burge, Liit for interferoeter calibration uing the rando ball tet, Proc. of SPIE 746, 746U (9). [8] P. Zhou and J. H. Burge, Analyi of wavefront propagation uing the Talbot effect, ubitted to Applied Optic (). [9] J. Goodan, Introduction to Fourier Optic, Robert and Copany Publiher, 88-9 (5). [] J. H. Burge, Advanced technique for eauring priary irror for atronoical telecope, PhD diertation, Univerity of Arizona, Tucon, AZ (993) [] C. Zhao and J. H. Burge, Generalized Coddington equation for refractive/diffraction hybrid urface, SPIE IODC (). Proc. of SPIE Vol L- Downloaded fro SPIE Digital Library on 9 Sep to Ter of Ue: