. INTRODUCTIONiffractive optical elements (DOEs), or computer-enerated holograms (CGHs), are very powerful instru-ents for interferometric measurements of aspherical
urfaces because they can transform an incoming wave-ront into nearly any arbitrary numerically or mathemati-ally defined shape. Such CGHs, which consist of patternsf lines or rings, are now easily manufactured by usingquipment from the microelectronics industry or preci-ion task-oriented circular laser writing systems opti-ized for CGH fabrication.1 The precision of the CGH
abrication affects the accuracy and validation of the mea-ured results. However, errors and uncertainties duringhe CGH fabrication process result in errors in the dif-racted wavefront created by the hologram. Consequentlyhen the final hologram is used for optical testing, therecision of the measurement will be affected. The accu-acy of the fabrication of the hologram structure withresent-day equipment reaches several nanometers,hich allows formation of steep enough wavefronts with aeak-to-valley (PV) error of �1/20 waves and even less.2
he certification of the writing process of the CGH fabri-ation, which can reveal random errors of the writing pro-ess, has been demonstrated.3 The application of these in-ovations has allowed creation of precision CGHs for
nterferometric measurements of large aspherical optics,.g., 6.5-m f /1.25 and 8.4-m f /1.14 paraboloidal primaryirrors.4 In the past few years new types of diffractive el-
ments, such as combined (or split, cellular, multiplexed,ual-wave, for example) CGHs,5 which represent the al-ernate encoding of two or more wavefronts on the CGHperture split into regular strips or rings, have beenresented. CGHs of this kind can transform one inputavefront into several independent output wavefronts.
hese new kinds of elements have been widely investi-ated for testing optical surfaces and certification ofavefronts generated by the CGHs.5–7 However, these dif-
ractive elements also have properties of splitting wave-ronts and can be used as the basic component of an in-erferometer, splitting the measuring and referenceeams. This paper presents a new (as far as we know) de-ign, in which the principle of a Fizeau interferometer isow perfectly fulfilled, namely, by generating referencend measuring wavefronts on the last optical surface,hich carries a DOE. Several advantages of this designre discussed. We present a simple and general method ofspheric figure metrology using a combined CGH placedn the output beam of a conventional Fizeau interferom-ter. We examine this method experimentally by testing aeference f /0.68 spherical mirror. The experimental re-ults that we have obtained by the proposed method agreeell with those obtained by using a Fizeau interferometerith a standard transmission sphere.
. OPTICAL TESTING WITH AOMPUTER-GENERATED HOLOGRAM
nterferometers operate by generating two laser beams,ne as a reference and one as a test beam. The test beamnteracts with the optics under test, and the referenceeam is reflected by a reference surface. Test and refer-nce beams overlap each other inside the interferometern the plane of the CCD camera. Usually the referenceurfaces are flat or spherical; hence only plane andpherical surfaces can be measured.
For testing aspherical surfaces, a CGH is added to theest arm, acting as a null lens. The CGH null operates inouble path, first producing an aspheric test wavefront
06 Optical Society of America
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nd then recollimating the wavefront reflected from theested surface. This technique is used for measuring as-heres with existing Fizeau interferometers.8 A commononfiguration for using a CGH for optical testing withizeau interferometer is shown in Fig. 1. In this configu-ation, either on-axis or off-axis CGH nulls can be used. Ifhe interferometer operates with a plane output wave-ront, the CGH null is tilted by a small angle (less than°) with respect to the optical axis, so that the various dif-raction orders will be separated. The tilt of the CGH nulls necessary for elimination of the direct reflection (zerorder) from the CGH surface as well.
In this configuration, the interferometer measures theombination of the CGH null and the asphere, so the ac-uracy of the test depends on the quality of the CGH null.n the presented case, the major source of error is the sur-ace of the reference flat of the Fizeau interferometer andhe transmitted wavefront distortion (TWD) of the CGHubstrate.8,9 In practice, the TWD of the CGH substrateimits the precision of this type of interferometric mea-urement. Typical CGH substrate errors show low spatialrequencies, and these errors show the low-spatial-requency wavefront aberrations in the diffractedavefront.9 One method of eliminating the substrate er-
or is to measure its flatness first. The CGH substrate er-ors can be measured by using the zero-order diffractioneam. However, the zero-order diffracted wavefront is ex-
Fig. 1. Layout of a null CGH
Fig. 2. Configuration of a combined off-axis CGH optic
remely sensitive to duty-cycle variations of the phaseGH pattern.9 This phase error overwhelms the effects of
he CGH substrate error in the zero diffraction order andrevents substrate figure measurement. Thus, eliminat-ng the TWD of the CGH substrate leads directly to sig-ificantly improved measurement accuracy.
. DESIGN CONCEPTur basic idea to increase the accuracy of the interfero-etric measurement consists of eliminating the influence
f the CGH substrate figure error. For this purpose, weropose to use an off-axis combined CGH, where the sub-trate plane carrying the diffractive structure is facingutside with respect to the interferometer. The combinedGH consists of two independent CGHs: One componentf the hologram is generating a plane wave (referenceeam), while the other component (null CGH) is generat-ng a test wavefront (test beam). This CGH can be com-osed of for example, a linear grating and a null CGH.he geometry for defining the CGH functions of both com-onents is shown in Fig. 2. The phase function of the nullGH component is derived by use of a geometrical modelf rays normal to the surface under test (aspheric sur-ace), as shown in Fig. 2(a). The null CGH function is theptical path difference (OPD) between O C S and OCS.
an asphere. BS, beam splitter.
: (a) null lens component, (b) linear grating component.
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174 J. Opt. Soc. Am. A/Vol. 23, No. 1 /January 2006 J.-M. Asfour and A. G. Poleshchuk
hoosing the reference point as the center yields the fol-owing function across the CGH10:
�1�x,y� = O��x,y�C�S� − O�x0,y0�CS. �1�
he reference beam function is the OPD between O�C�nd OC, as shown in Fig. 2(b). The grating spacing S isetermined in a way such that the angle of diffraction � isqual to the double of the angle of inclination of the CGHlane (Littrow angle):
� = �/S = 2�, �2�
here � is the wavelength.In this case, the diffracted light beam is exactly re-
urned back. The OPD for the reference beam is given by
�2�x,y� = O��x,y�C� − O�x0,y0�C = x tan���. �3�
rom Eqs. (1) and (3) we can see that the final OPD forhe null CGH, taking into account the reference beamPD, is given by
��x,y� = C��x,y�S� − C�x0,y0�S. �4�
igure 3 shows the layout of the null test on the basis ofhe combined CGH. In this optical scheme, both test andeference wavefronts are formed by means of diffraction
ig. 3. Layout of a null CGH test of an aspherical surface, usingeference flat.
Fig. 4. Formatio
nd by one single CGH. This makes it possible to elimi-ate the substrate heterogeneity and increases the testccuracy. Only one side of the substrate (the CGH plane)as to be fabricated with high quality. This combinedGH can be designed as an amplitude-only CGH or as aixed amplitude–phase CGH.The formation of a combined amplitude–phase CGH is
hown in Fig. 4. The first component [Fig. 4(a)] is built asphase CGH in order to achieve maximum diffraction ef-ciency. The second component [Fig. 4(b)] is a reflectivemplitude grating. The combination of these two ele-ents leads to an amplitude–phase CGH [Fig. 4(c)].The proposed test method is a hybrid of the two opticaleasurement methods: Fizeau test plate interferometry
nd the use of CGHs. The accuracy of the proposed test isimited by the quality of the single flat optical surface, byhe accuracy of the CGH structure location, and by thetching process. Structuring and etching can be producedith sufficiently high accuracy. This method is close to theGH test plate technique developed by Burge andnderson.11
It is interesting to compare the classical Fizeau inter-erometry with the developed test. The Fizeau principle iserfectly fulfilled if the combined CGH is an amplitude-nly element. For the case of a combined amplitude–hase CGH there might be a nonuniform etching depth,eading to a degradation of the test wavefront but without
ingle CGH working in a collimated interferometer beam without
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J.-M. Asfour and A. G. Poleshchuk Vol. 23, No. 1 /January 2006 /J. Opt. Soc. Am. A 175
nfluencing the reference wavefront. There is a similarituation with use of standard transmission spheres (TSs)n Fizeau interferometers. Here the reference wavefronts formed by reflection on the last surface acting as themaster surface,” commonly referred to as an aplanaticurface. The quality of the reference wavefront dependsn the figure quality of this master surface, although theuality of the transmitted test wavefront depends on theS lens design and the quality of all the lenses.The test beam with intensity I2 passes the null CGH
wice at the first order of diffraction. The reference beamith intensity I1 is formed by diffraction (in reflection) on
he linear grating with the spacing S calculated from Eq.2). The CGH is designed to diffract this reference beamo match an ideal test wavefront. The combined CGHhould be designed in a way such that its diffraction or-ers are spatially separated in the plane of the pinholespatial filter). If � is the full angular size of the pinhole,he spacing S is derived by S�� /�.
The duty cycle of the linear grating is defined on the ba-is of the parity condition for the test and referenceeams’ intensities. The intensity of the diffracted refer-nce beam is defined by the diffraction amplitude of theinear grating,
I1 = I0�gratD2 sinc�mD�2, �5�
here �grat is the reflectivity of the linear grating mate-ial, m is the diffraction order, and D is the grating dutyycle defined by D=b /S. The intensity of the test beamafter double pass through the CGH) is defined by theero-order diffraction efficiency,12
I2 = I0�12�testD
4, �6�
here �1 is the diffractive efficiency of CGH 1 (null CGH)n +1st order and �test is the reflectance of the surface un-er test. The relative intensities of the test and referenceeam as a function of the duty cycle of the linear gratingre shown in Fig. 5, based on the parameters �1=0.4, �20.1, R=0.8 and �test=0.4 (curve 1), �test=0.1 (curve 2),nd �test=0.05 (curve 3). For these given values, the opti-al duty cycle is D=0.16. In this case, the light transmis-
ion of the test and reference arms of the interferometer
ig. 5. Relative output intensities of the test beam (curves 1, 2,) and reference beam (curve 4) as a function of the duty cycle ofhe linear grating. The reflectance of the tested surface is (1)test=40%, (2) �test=10%, (3) �test=5%.
s approximately I2 /I0=I1 /I0=1%. One can see that byhanging the duty cycle of the grating, it is possible toeach an optimal contrast of the final interferogram for aariety of materials with different reflectance (e.g., glass,ilicon, germanium, metallic mirrors, optical ceramics).
. FABRICATION PROCESS OF THEOMPUTER-GENERATED HOLOGRAM ANDXPERIMENTAL RESULTShe aim of the experiments was the verification of theroposed CGH design. The test was performed by using ahase-shifting Fizeau-type interferometer manufacturedy Zygo (GPI). The CGH has been designed and fabri-ated with the following parameters: diameter D50 mm, focal length f=150 mm, and tilt angle �=0.25°.s a surface under test we used a standard referencephere f /0.68 (from the GPI toolkit).
. Fabrication of the Computer-Generated Hologramhe CGH was written by using the circular laser writingystem (CLWS), built by the Institute of Automation andlectrometry in Novosibirsk, Russia.1 The CLWS is ca-able of writing up to 300-mm diameter CGHs with anbsolute accuracy of 100 nm across the full diameter. Theachine rotates the substrate at 600–800 rpm and uses
n interferometrically controlled linear air bearing writ-ng head, with a positioning precision of several nanom-ters. This machine also writes arbitrary patterns that doot have a circular symmetry, by using a coordinateransformation software and a high-quality angular en-oder with rapid-writing beam switching. We could cali-rate the writing system in a way such that nonrotation-lly symmetric patterns, especially the mentioned linearrating that generates the reference wavefront, are re-orded with the same precision as the circular patterns ofhe object wave hologram.
In fact, due to the 0.25° tilted off-axis design, the de-cribed combined CGH has slightly elliptical zones with ainimum spacing of Tmin=3.9 m. The 0.25° tilt angle
eads to a corresponding spacing of the linear gratingqual to S=72 m. This linear grating and the ellipticalones of the CGH are written simultaneously. The accu-acy of the resulting wavefronts formed by the combinedGH is defined by the writing pitch, which has been set to.2 m in our case. The value of the writing pitch is aompromise between the accuracy of the CGH structurend the velocity of the writing. The accuracy of the writ-ng in the angular direction is defined by the angular en-oder of the CLWS, which has approximately 2106 po-itions per revolution, corresponding to a spacing of0.1 m at 25-mm radius.The wavefront error produced by the CGH pattern at
he mth order of diffraction can be calculated by9
�W�x,y� = − m���x,y�
S�x,y�, �7�
here ��x ,y� is the positioning error perpendicular to theones and S�x ,y� is the spacing of the zones. Thus, with
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176 J. Opt. Soc. Am. A/Vol. 23, No. 1 /January 2006 J.-M. Asfour and A. G. Poleshchuk
=1, the maximum error of the test wavefront of the fab-icated CGH is less than 0.05� (PV), and the error of theeference wavefront is less than 0.003� (PV).
The CGH had been fabricated on a 60-mm fused silicaubstrate with 1/20-wave PV surface quality. The secondurface had approximately 1–2 waves PV quality. Theubstrate wedge was approximately 4–5 min. A chromiumayer was deposited onto the high-quality surface and wastructured using a resistless technology by direct laserriting onto chromium films.2 Figure 6(a) shows a photo-raph of the chromium pattern of the CGH null fabricatedy direct writing with the CLWS. The linear grating haspproximately S=72 m spacing and b=18 m lineidth. The CGH null component as chrome on glassattern diffracts approximately 10% into the first ordern transmission, and the linear grating diffracts 1% intohe first order in reflection. This amplitude CGH can besed for testing surfaces with high reflectivity ��test40% �.For testing glass surfaces with a low reflectivity ��test5% � we need to achieve a high diffraction efficiency
or the transmitted object wavefront and a low efficiency
ig. 6. Microscope images with transmission illumination: (a) cmplitude–phase CGH.
Fig. 7. 3D image of the central
or the linear grating, thus obtaining similar amplitudesor the reference and the test wave. Therfore the CGH haseen processed in the following way: First, we transferredhe CGH structures into the substrate by ion etching. Inrder to keep the chromium pattern of the linear grating,e covered each single line of the linear grating with ahotoresist masking process. Then the uncoated chro-ium was removed, and, finally the photoresist, whichad protected the linear grating, was removed.One should add that for the case of a combined ampli-
ude to phase CGH used for testing low-reflective glassurfaces, the geometrical plane of origin of the objectavefront is no longer well defined. Thus the Fizeau prin-
iple might be affected in theory. But it is still valid that aighly accurate flatness is required only on the last sub-trate plane. With this precision flatness, the binaryhase step level inside the substrate is then generated byon etching.
Figure 6(b) shows a photograph of the resulting com-ined amplitude–phase CGH after ion etching and afteremoval of the chromium layer from the phase CGH area.igure 7 shows an image of the central part of the fabri-
part of the chromium mask fabricated with the CLWS, (b) final
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J.-M. Asfour and A. G. Poleshchuk Vol. 23, No. 1 /January 2006 /J. Opt. Soc. Am. A 177
ated combined CGH taken with a white-light interfero-etric microscope and processed in a three-dimensional
3D) representation. Hence, an amplitude-phase CGH haseen fabricated with approximately 40% diffraction effi-iency into the first order of diffraction and 1% efficiencyn reflection. This CGH can be used for testing surfacesith low reflectivity.
. Characterization of the Computer-Generatedologram Nullo prove the feasibility of the concept, the fabricated CGHull was mounted into a standard Zygo-type bayonetousing and was fixed at the place of the transmissionphere in the Zygo interferometer [Fig. 8(a)]. The opticalest layout was chosen according to Fig. 3. As a first step,he alignment procedure (tilt of CGH at 0.25.) was made.igure 8(b) shows the “view” mode of the alignment pat-ern. One can see several orders of diffraction caused byhe linear grating of the CGH.
ig. 8. (a) CGH (left), mounted directly into a Zygo Fizeau inteith CGH null.
Fig. 9. (a) Interferogram and (b) phase map of a reference sp
The first order of diffraction indicated in Fig. 8(b) haso be aligned into the center of the crosshair. The align-ent of this kind of CGH is comparable with the well-
nown procedure used in alignment of commonly used re-ractive reference lenses. As this hologram is working in aollimated beam, there is no need to align it transversally.rincipally one could think about a reference wavefront
hat is reflecting back an incoming spherical wave. But inhis case, the compatibility with standard interferometerss lost for two reasons: First, the incoming beam needs toe spherical instead of parallel; and second, the bayonetolder would require additional adjustments in the X anddirections.A typical interferogram produced by testing a f /0.68
pherical reference mirror is shown in Fig. 9(a). The in-ensity distribution of the reference beam shows veryood uniformity, and the contrast of the fringes is goodnough. A 3D plot of the phase map is shown in Fig. 9(b).he resulting precision of the measurement is P-V=0.06aves and rms=0.01 waves.
eter, and lens surface (right) under test. (b) Alignment pattern
l mirror with f /0.68, measured with the combined CGH null.
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178 J. Opt. Soc. Am. A/Vol. 23, No. 1 /January 2006 J.-M. Asfour and A. G. Poleshchuk
. CONCLUSIONShybrid optical measurement method was developed in
hich the test plate of a Fizeau interferometer is com-ined with a computer-generated hologram (CGH). Thisethod meets the extreme challenges of high-precisioneasurements of aspherical optics. The technique can besed with standard commercial interferometers. TheGH test plate can be used to measure any spherical orspherical surfaces with convex or concave shape.
CKNOWLEDGMENTShe authors thank Anatoly Malishev and Vadimherkashin of the Institute of Automation and Electrom-try for their assistance in CGH fabrication and Arnoöhler from ZygoLOT Inc. for his assistance in testing theGH. The authors’ e-mail addresses [email protected] and [email protected].
EFERENCES1. A. G. Poleshchuk, E. G. Churin, V. P. Koronkevich, V. P.
Korolkov, A. A. Kharissov, V. V. Cherkashin, V. P. Kiryanov,A. V. Kiryanov, S. A. Kokarev, and A. G. Verhoglyad, “Polarcoordinate laser pattern generator for fabrication ofdiffractive optical elements with arbitrary structure,” Appl.Opt. 38, 1295–1301 (1999).
2. V. V. Cherkashin, E. G. Churin, V. P. Korolkov, V. P.Kooronkevich, A. A. Kharissov, A. G. Poleshchuk, and J. H.Burge, “Processing parameters optimization forthermochemical writing of DOEs on chromium films,” inProc. SPIE 3010, 168–179 (1997).
3. A. G. Poleshchuk, V. P. Korolkov, V. V. Cherkashin, and J.Burge, “Methods for certification of CGH fabrication,” inDiffractive Optics and Micro-optics, Vol. 75 of OSA Trendsin Optics and Photonics Series (Optical Society of America,2002), pp. 438–440.
4. J. H. Burge, L. R. Dettmann, and S. C. West, “Nullcorrectors for 6.5-m f /1.25 paraboloidal mirrors,” inFabrication and Testing of Aspheres, Vol. 24 of OSA Trendsin Optics and Photonics Series (Optical Society of America,1999), pp. 182–186.
5. A. G. Poleshchuk, J. H. Burge, and E. G. Churin, “Designand application of CGHs for simultaneous generation inseveral specified wavefronts,” in Diffraction OpticalElements (DOE)-1999, Vol. 22 of EOS Topical MeetingDigest Series (European Optical Society, 1999), pp.155–156.
6. M. Beyerlein, N. Lindlein, and J. Schwider, “Dual-wave-front computer-generated holograms for quasi-absolutetesting of aspherics,” Appl. Opt. 41, 2440–2447 (2002).
7. S. Reichelt and H. J. Tiziani, “Twin-CGHs for absolutecalibration in wavefront testing interferometry,” Opt.Commun. 220, 23–32 (2003).
8. S. M. Arnold, L. C. Maxey, J. E. Rogers, and R. C. Yoder,“Figure metrology of deep aspherics using a conventionalinterferometer with CGH null,” in Proc. SPIE 2536,106–116 (1996).
9. Yu-C. Chang and J. H. Burge, “Error analysis for CGHoptical testing,” in Proc. SPIE 3782, 358–366 (1999).
0. T. Kim, J. H. Burge, Y. Lee, and S. Kim, “Null test for ahighly paraboloidal mirror,” Appl. Opt. 43, 3614–3618(2004).
1. J. H. Burge and D. S. Anderson, “Full apertureinterferometric test of convex secondary mirrors usingholographic test plates,” in Proc. SPIE 2199, 181–192(1994).
2. A. G. Poleshchuk, “Diffractive light attenuators withvariable transmission,” J. Mod. Opt. 45, 1513–1522 (1998).
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