Self-calibration of wavefront testing interferometers by use of diffractive elements Stephan Reichelta , Hans Tizianib , and Hans Zappea a
Laboratory for Micro-optics, Department of Microsystems Engineering – IMTEK, University of Freiburg, Georges-K¨ohler-Allee 102, 79110 Freiburg, Germany b
Institut f¨ ur Technische Optik, University of Stuttgart, Pfaffenwaldring 9, 70569 Stuttgart, Germany ABSTRACT Self-calibration techniques in interferometry imply the absolute determination of the interferometer’s systematic error without the need for a high-quality, precisely-known calibration standard. By taking separate measurements at different positions and under different orientations, the interferometer’s error can be determined in an absolute manner even with non-perfect optical elements. This paper discusses the versatile possibilities of using diffractive optical elements as calibration tools in wavefront testing interferometry. Keywords: Optical testing, Interferometry, Diffractive optical elements, Self-calibration, Fresnel zone plates, Multiplex holograms
1. INTRODUCTION 1.1. What is self-calibration? Interferometry is established as one of the standard means of testing smooth optics. Typical instruments are interferometers of the Fizeau, Twyman-Green or Mach-Zehnder type, all non-common path systems. The limitation of those systems is that, since the test wave and the reference wave pass through the system on different optical paths, even marginal inaccuracies in the interferometer’s optics influence the interfering waves in different ways. This, in turn, results in a systematic error which is part of the measurement. In high precision interferometry, therefore, calibration is a key issue in eliminating this systematic bias. In modern systems, the inaccuracies can often be of the same magnitude or even larger than the expected error of the optics to be measured. Thus in interferometry, following the toolmaker’s golden rule that the measurement capability has to be about ten times better than the required accuracy is usually impossible. Instead, sophisticated techniques, both in theory and experiment, are required to eliminate the systematic bias from the actual measurement. Of course, calibration can be done by comparing the unknown wavefront with an absolutely known, i.e. previously well-specified master piece. But to establish and verify a standard master piece is a science of its own. Interferometric self-calibration techniques offer a way out of this dilemma. Even with non-perfect optical elements, the interferometer’s systematic error can be determined in what we call an ”absolute manner”.1 This can be done by taking separate measurements at different positions and under different orientations followed by an appropriate evaluation of the separate measurements. The basic idea behind each method is to cancel out, more or less effectively, all those error contributions which originate from the measurement system. To summarize, the term self-calibration comprises all those techniques which manage this procedure without a previously well-specified standard surface.
1.2. Overview of state-of-the-art self-calibration methods Before we progress to diffractive optical elements and their prospects for calibration, some classical methods based on on reflective or refractive methods will be briefly summarized. E-mail:
[email protected], Phone: +49 (0)761 203 7519, Web: http://www.imtek.de/micro-optics
Interferometry XIII: Techniques and Analysis, edited by Katherine Creath, Joanna Schmit, Proc. of SPIE Vol. 6292, 629205, (2006) · 0277-786X/06/$15 · doi: 10.1117/12.680660
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N -position test In this method, the test piece is measured N times under different azimuthal orientations which are angularly incremented by 2π/N . An averaging of the N measurements cancels out all terms except those which are rotationally invariant and those of azimuthal order kN θ (k is an integer).2, 3 Therefore, the difference of a single measurement and the average of all measurements give the non-symmetric error components of the test piece. Recently, a further improvement of this technique has been developed.4 In practice, a number of N = 12 single measurements is a good compromise between the required measurement time and achievable accuracy. In particular, if there is no real focal point inside the test arm of the interferometer, as for diverging or aspherical waves, this method is qualified. Three-position method and its advanced improvements The three-position test proposed by Jensen5 is an established technique for wavefront testing interferometers with an outgoing spherical wave. Combining three measurements, two at confocal and one at cat’s eye positions, we obtain both the wavefront error of the spherical surface and that of the interferometer in an absolute manner. An uncertainty in the three position test occurs if the cat’s-eye data are rotated about a point which is not at the optical axis. As pointed out by Selberg,6 the accuracy can be further improved by taking a second series of confocal measurements, but oriented at 90◦ and 270◦ . By sharing the same cat’s-eye measurement for both confocal series and averaging over the two results (after reorientation of the second set), a potential astigmatism is canceled out. As a logical consequence, additional improvement will be achieved if the N -position test is included. Combining the six sets of opposite measurements with one cat’s-eye measurement yields the desired absolute results for the interferometer’s bias and the surface figure error with a reduced uncertainty. Random ball test In principle, ball averaging or random ball testing is an extension of the N -position averaging technique. The simple but ingenious idea is to replace the calibration artifact by an object with inherent symmetry. For calibration, a ball is randomly positioned at confocal position, with its center at the focus of the test objective. The method is based on the fact that the errors of a ball can be described by spherical harmonics and therefore, the wavefront aberrations introduced by figure errors of the ball will average to zero.7 A minor disadvantage of this technique is the large number of required measurements (50...100) which makes a device for automated random ball positioning indispensable and may result in a considerable summation of bad pixels.
2. DIFFRACTIVE OPTICAL ELEMENTS 2.1. General properties For wavefront shaping holograms, only the phase must be encoded into the hologram, since the amplitude is constant over the hologram. We restrict our discussion to phase-only holograms, since this is the most common type, due to its simple fabrication. The phase function φ of a diffractive element refers to the phase difference between the desired output wave and the incident wave, i.e. that difference before and behind the hologram φ(x, y) = φout (x, y) − φin (x, y). To describe the hologram and its behavior on the reconstructed wavefront, scalar grating theory is used throughout this paper. Within this scalar approach (where it is assumed that the period structure is large compared to the wavelength), the diffractive element is treated as a locally periodic linear grating, specified by either a local spatial frequency vector field ν (x, y) =
∇φ(x, y) ∇G(x, y) = 2πm0 λ0 m0
(1)
or a local grating vector field k(x, y) = ∇φ(x, y) = 2π ∇G(x, y), λ0
(2)
where λ0 , m0 denote the design wavelength and diffraction order (normally m0 = 1) and G is the recording eikonal of the diffractive element. As is known, the recording eikonal is related to the phase function by φ(x, y) = k0 G(x, y), with the free-space wave number k0 = 2π/λ0 .
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According to their symmetry, holograms can be classified into two types: inline and off-axis holograms. Table 1 shows some common designs used in interferometric applications. When using inline holograms, real and virtual images are reconstructed centered on the optical axis. Disturbances caused by twin images and the undiffracted zeroth order can occur. By using off-axis holograms, each diffraction order has its own, angularly separated optical axis, whereby ghost images can completely filtered out spatially. Table 1. Rough classification of most commonly used holograms in interferometry with regard to their symmetry.
Counterpart Wavefront
inline CGH
off-axis CGH
Aspheric null |ν | = f (r)
Aspheric null |ν | = f (x, y)
Axicon |ν | = f (r) = const.
Fresnel zone plate |ν | = f (r) ≈ 2a2 r
Linear grating |ν | = f (x) = const.
Cylinder null |ν | = f (x) ≈ 2a2 x
Cone Bessel beam
ideal lens | mirror spherical
prism | planar mirror tilted plane
cylinder lens | mirror cylindrical
In laser interferometry, wavefront shaping diffractive optical elements offer the following key advantages: Design freedom: Fabrication constraints given by classical refractive/reflective optics are irrelevant, since the phase function is encoded in a planar micro-structured surface and not realized by any large geometric shape. Nearly arbitrary wavefronts can be generated by encoding appropriate phase functions. Diffractive elements can be easily tailored to the specific requirements of the interferometric application. Conjugation feature: Diffractive elements with symmetrically shaped structures (rectangular, sinusoidal) always possess positive as well as negative diffraction orders. Even though this results in an efficiency loss, the phase conjugation feature can be advantageously used for beam splitting, error calibration, or compensation. Multi-functionality: Multiplexing allows the simultaneous realization of various optical functions with a single element. Methods are wavelength multiplexing, spatial multiplexing, angular multiplexing, and polarization multiplexing. Similarly, this property is very useful for beam splitting and error calibration. Dispersion insensitivity: The dispersion inherent to any diffractive optical element does not pose a difficulty, so long as laser interferometry with quasi-monochromatic light is used.
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2.2. Brief discussion of CGH error types Several composite errors may be present in computer-generated holograms. Basically, they can be divided into three main categories: pattern errors of the hologram, figure errors of the hologram substrate, and profile depth errors of the diffractive structure. Their effect on the reconstructed wavefront has been studied in detail previously.8, 9 In this section, we briefly review only those errors which are important in the context of selfcalibration. 2.2.1. Hologram pattern distortion First we consider the hologram pattern distortion error inherent in the hologram. Due to any fabrication inaccuracies, i.e. position errors during exposure, an ideal hologram point P (x, y) may slightly be shifted with respect to the real point PD (x, y). Considering the overall hologram pattern, this distortion is described by the vector field y) = rD (x, y) − r(x, y), (3) ζ(x, where the vectors r, rD are the radius vectors of the points P, PD , respectively. The wavefront error WPD caused by a distortion of the hologram pattern is given by10 y) · ν (x, y) WPD (x, y) = −mR λ0 ζ(x, (4) which may alternatively expressed in terms of the phase or eikonal gradient as WPD (x, y) = −
mR λ0 mR ζ(x, y) · ∇φ(x, y) = − ζ(x, y) · ∇G(x, y) 2πm0 m0
(5)
From these equations it becomes clear that the local wavefront error WPD is directly proportional to the scalar product of the local distortion vector and the local spatial frequency vector. 2.2.2. Figure errors of hologram substrate Substrate figure errors typically have a low spatial frequency. Their effects on the resulting wavefront depend on the application of the hologram: when using the hologram in reflection, a surface figure error ∆s leads to a reflected wavefront error of WF = 2∆s, whereas in double pass transmission (substrate with refractive index n) an error of WF = 2(∆s1 − ∆s2 )(n − 1) is introduced. For most cases, the influence of the surface figure error is more critical when using the hologram in reflection than in transmission, even if the wavefront passes the hologram twice.
3. DIFFRACTIVE SELF-CALIBRATION METHODS The main idea behind diffractive calibration methods is to utilize not only the wave for which the hologram is designed for but also its conjugate wave. Already in the early years of computer-generated holography, this property was used for measuring the distortion of writing systems, primarily for linear gratings.11, 12 We start our discussion with calibration methods for linear gratings and Fresnel zone plates. These represent the simplest examples of arbitrary inline or off-axis holograms, which can be easily tested interferometrically. Moreover, these ordinary grating types are the basis of more advanced calibration techniques which are presented below.
3.1. Calibration by ordinary CGHs 3.1.1. Linear grating The easiest way to realize the diffractive calibration concept is by use of a linear grating which is tested in Littrow (autocollimation) configuration in which the incident plane wave is diffracted back along the same optical path.13 To determine the pattern distortion error of a linear grating, two of such measurements are required (cf. Fig. 1).
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!UTOCOLLIMATIONTESTOFALINEARGRATING AM
AM
PLANEWAVE FROM INTERFEROMETER M2
M2
Figure 1. Littrow mounting of a linear grating to perform an autocollimation test.
With WINT = WR + WO as the aberration of the of the interferometer, WF as the wavefront aberrations caused by surface figure error of the substrate and WPD as the wavefront aberration caused by pattern distortion, we can write for both Littrow arrangements W1
= WINT + WF + WPD+1 , = WINT + WF + WPD−1 .
W2
(6) (7)
Note that a sign inversion of the wavefront pattern distortion error WPD occurs if the wavefront is reconstructed in the complementary diffraction order. It can easily be seen from Eq. (4) that WPD−1 = −WPD+1 if mR changes in sign. Thus we are able to separate the wavefront pattern distortion error simply by subtracting the measured wavefronts pixel by pixel WPD−1 = 1/2 (W1 − W2 ) .
(8)
In our experimental studies, we tested a linear grating (chrome-on-glass type, 90 mm × 37 mm) with a period of p = 4.54 µm and a duty cycle of 0.75, which yields tilt angles of α±1 = ±4◦ and α±2 = ±8.02◦ . The results for the wavefront pattern distortion error of the linear grating extracted from both first and second order measurements are shown in Fig. 2.
100
200
50
100
0
0
:
:
z[nm]
z[nm]
-50
-100
-100
-1
-1
PV = 234.9 nm RMS = 33.1 nm
-0.5
0.5
0
1
-200
-1
-1
(a)
PV = 486.9 nm RMS = 62.2 nm
-0.5
0
0.5
1
(b)
Figure 2. Wavefront aberrations caused by a distorted pattern of a linear grating, (a) extracted from complementary first order measurements and (b) from second order measurements. The aberration scales linearly with the diffraction order. Depicted maps are wavefront aberrations.
The described test, however, only represents a partial calibration technique. All figure errors of the hologram substrate remain unknown which, in turn, leads to an uncertainty in the quality of the interferometer’s outgoing plane wave. Additional efforts are necessary to remedy this situation, for example by combination with the N -position test.
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3.1.2. Fresnel zone plate To self-calibrate an interferometer with a spherical outgoing wave by means of a Fresnel zone plate, at least five measurements are required. For the sake of completeness, we just summarize the fundamentals of the method: it is a combination of the conjugate diffraction order measurement principle with the three-position Jensen method. A more detailed discussion is given in a reference paper.14 Fig. 3 illustrates the five measurements. With the measured wavefronts W1 , W2 taken at the intrafocal positions of the reflective zone plate, W3 , W4 at extrafocal positions and W5 at the cat’s-eye position, these may be written as W1 W2 W3 W4 W5
= = = = =
0 0 0 WFZM if + WO + WR , π 0 WFZM if + WO + WR0 , 0 0 0 WFZM ef + WO + WR , π 0 0 WFZM ef + WO + WR , 0 0 π WR + 1/2 WO + WO ,
(9) (10) (11) (12) (13)
&IVE POSITIONTESTWITHA&RESNELZONEMIRROR&:- #ONVERGER LENS
&:- ²
PLANEWAVE FROM INTERFEROMETER
where WRβ denotes the wavefront contribution of the reference arm, WOβ that of the test arm without the test β piece and WFZM that of the tested zone mirror alone. The wavefront error WFZM consists of those caused by pattern distortion WPD and of those caused by figure error WF . The angle β indicates the azimutal orientation of the FZM relative to the interferometer.
&@ EXTRAFOCAL M2
INTRAFOCAL M2 &:- ²
² &@ INTRAFOCAL M2
EXTRAFOCAL M2
Equations (9)-(13) can be solved for the wavefront #ATSEYE aberrations caused by pattern distortion WPD as well as POSITION for those caused by figure errors WF of the zone plate, &@ yielding both error parts in an absolute manner 0 WPD+1 = 1/4 W1 + W 2 − W 3 − W4 , (14) Figure 3. Five-position test of a Fresnel zone mirror (FZM) for self-calibration of an interferometer with spherWF0 = 1/4 W1 + W 2 + W 3 + W4 ical outgoing wave. −1/2 W5 + W 5 , (15) where the bar indicates a rotation of the wavefront by 180◦ . Additionally, it gives the systematic bias of the interferometer WINT = WR + WO 0 WINT = 1/4 W1 − W 2 − W 3 + W4 + 1/2 W5 + W 5 . (16) The diffractive self-calibration method has been compared with the standard three-position test. Both tests were performed successively in the same interferometer, without changing or readjusting any optics of the interferometer. The results of both tests indicate excellent agreement as shown in Fig. 4, with tilt and power removed. The difference between the two measurements has a rms deviation of 2.5 nm, which indicates for the so-called “absolute” results a measurement rms-uncertainty of ∼ λ/250.
3.2. Calibration by multiplex CGHs Thus far we have restricted our discussion to “ordinary” or “simple” holograms, where we use this term for holograms for which refractive or reflective counterparts exist, e.g. a glass wedge or ideal (aspherical) lenses or mirrors. However, holograms offer an additional way to take advantage of a further degree of freedom in design and application of diffractive optics. One special feature is multi-functionality, which is achievable by using multiplex encoding techniques. Multiplexing means in this context the simultaneous generation of several
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5 0
0
0
-50 :50
-50 :50
:,o -10
-5
z[nm] PV = 136.8 nm RMS = 23.2 nm
(a)
z[nm] PV62.1
z[nm]
______ 0.5 1
PV = 137.4 nm
__________ RMS23.Onm
(b)
RMS = 2.5
(c)
Figure 4. Systematic bias of a Fizeau interferometer: (a) Self-calibration by a spherical mirror, (b) self-calibration by a reflective zone plate and (c) pixel-by-pixel difference of (a) and (b). Note the different scaling in (c). Again, depicted maps are wavefront aberrations.
optical functions by a single element, which can not be achieved by conventional smooth optical surfaces. Here we utilize this capability for calibration means. Such a hologram was proposed for the first time by Fercher,10 who used a dual synthetic hologram for partial self-compensation of pattern distortion errors by passing both reference and object wave through the hologram at a non-zero diffraction order. Later, a similar approach but for partial self-calibration, was used by Schwider,15 who encoded both a spherical auxiliary wave for calibration and an aspherical test wave into one hologram. But these ingenious ideas for compensation and thus calibration of the hologram errors suffer from the fact that the uncertainty is increased with the difference in the k-vectors of both simultaneously encoded phase functions, cf. Eq. (5). The method presented here is based on a multiplex computer-generated hologram as well. The hologram basically reconstructs two kinds of wavefronts: (a) a non-regularly shaped, e.g. aspherical wave which cannot be tested absolutely by conventional methods; and (b) auxiliary waves which are suited for self-calibration because of their basic form. To give examples, planar, spherical, and cylindrical waves represent such regular wavefronts which propagate without changing their rudimentary shape. The idea is to measure the auxiliary waves and, with these aberrations, predict the errors of the non-regularly shaped test wave. Based on these considerations, we can define the following primary conditions which must be fulfilled to achieve a self-calibration by use of multiplex diffractive optical elements: (1) Appropriate encoding and simultaneous generation of the combined diffractive pattern which reconstructs the test wave as well as auxiliary waves, a concern for hologram design and fabrication. In design, attention has to be paid to assure well-suited amplitude-weighting for superposition encoding or a proper subaperture size, geometry, and arrangement for alternating encoding. Any fabrication process, on the other hand, results in certain inaccuracies. These deviations from the ideal shape must be locally the same for both diffractive structures. (2) Unique separation and complete determination of the wavefront errors caused by the diffractive element, a concern of tricky testing methods. By measuring the wavefronts reconstructed by the hologram in complementary diffraction orders, the sign of the wavefront pattern distortion aberration reverses, and therefore it can be effortlessly separated from other error contributions. Other aberrations, for example surface figure errors or phase depth variations, have to determined by other means. (3) Possibility of determining the pattern error component in direction of the test wave’s k-vector from the calibration measurements of the auxiliary waves. Due to encoding and testing, it must be guaranteed that the distortion vector ν can be measured in magnitude and direction. There are basically two kinds of encoding methods: alternating encoding of several phase functions into hologram subapertures (spatial multiplexing) and the complex superposition of wave fields (angular multiplexing). Both techniques have own pros and cons regarding sufficient cross-talk filtering and fabrication constraints.
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Sufficient suppression of cross-talk noise due to disturbing diffraction orders is the main design problem which must be solved. The most favorable encoding technique for the multiplex holograms depends on several aspects: similarity of the phase functions to be encoded, diffraction efficiency (ratio of desired to disturbed orders), number of diffractions by the hologram (single pass/reflection or double pass), size of the pinhole in the filter plane, fabrication constraints etc. These secondary requirements have to be met too. It has been shown that there is no general solution for the encoding problem. After an appropriate design and fabrication of the multiplex CGH, the calibration procedure consists of the following basic steps: Calibration #1: Self-calibration of the multiplex hologram by easy-to-measure wavefronts. Use of the conjugate waves concept to separate the errors due to hologram pattern distortion. Use of an additional method to determine the surface figure error of the hologram substrate. (The multiplex CGH operates in ‘auxiliary calibration mode’.) Error weighting: Transferring the errors to the non-regular shaped wavefront. Calculation of a calibration function for the complex wavefront by use of the design data (local spatial frequency differences). Calibration #2: Calibration of the interferometric setup with the multiplex hologram. With the known errors of the multiplex hologram (calibration function), the interferometer’s bias can be completely determined. (The multiplex CGH operates in ‘null calibration mode’.) Measurement: Measurement of the actual test piece with the calibrated interferometer setup. 3.2.1. Multiplex CGHs with preferred symmetry If the phase function of the test wave is symmetrical, we can take advantage of this by choosing an auxiliary phase function with same symmetry. In this case, the diffractive zones of both single patterns run parallel to each other, which means that previous condition (3) is fulfilled since knull kaux . Then only one auxiliary phase function is required. Useful combinations of test and auxiliary wave are for instance: aspherical/spherical waves, conical/spherical waves, cylindrical/tilted plane waves. The transfer operation of the procedure’s step 2 can then be written as (WPD + WF )null = γ WPD aux + β WF aux . (17) with the weighting operators γ β
(mR |ν |)null , (mR |ν |)aux 2 12 1 − 12 mR λ0 |ν | aux = , 2 1 − 12 mR λ0 |ν | null
=
(18)
(19)
where γ refers to the different spatial frequencies of the phase functions and β refers to slightly different incident angles onto the hologram substrate. (Note that all W and ν are two-dimensional functions of the cartesian coordinates (x, y); for the sake of better readability we’ve omitted these.) The multiplex calibration technique was first experimentally evaluated with a multiplex Fresnel zone mirror (‘Twin-FZM’).16 Since both spherical waves encoded in the Twin-FZM can be measured with the five-position test, the comparison of calculated and actual wavefront aberrations of the second spherical wave is a most stringent criterion of the method’s capability. Figure 5 shows the absolutely measured wavefront pattern distortion error WPD of both in the Twin-FZM encoded spherical wavefronts. We observed a spoke-shaped aberration that is typical for a circular laser writing system. As expected, the magnitude of the aberration differs slightly according to the difference in spatial frequencies. Applying the conversion from WPD1 to WPD2 , we obtain the predicted aberration of the 2nd spherical wavefront. This compares quite favorably with the result of the absolute measurement of the spherical wave #2. The wavefront difference between predicted and measured aberration WPD2 depicted in Fig. 5(c) has an rms-value of 3.1 nm (∼ λ/200) only, where except from phase shifting errors and diffraction artifacts no typical aberrations are present.
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5 0
-5
:,o-10 z[nm] PV = 54.2 nm
RMS=3.1
(a)
(b)
(c)
Figure 5. Wavefront aberrations caused by hologram pattern distortion WPD+1 of a complex superposition encoded TwinFZM. (a) Test result of the spherical wave #1 (FZM with confocal positions at zc1 = ±92 mm) and (b) test result of the spherical wave #2 (FZM with zc2 = ±120 mm). Tilt and power have been subtracted. (c) Residual uncertainty between calculated and measured wavefront pattern distortion errors WPD2 of the complex superposition encoded Twin-FZM.
This promising concept opens the door for an overall calibration of any CGH-null test setups. Recently, we have developed, to our knowledge, the first complete absolute test for rotationally symmetric aspheric surfaces.17 The determined interferometer’s bias includes all errors introduced by the null lens. The multiplex CGH reconstructs a spherical auxiliary wave as well as an aspherical wave. The calibration procedure begins with testing the multiplex hologram in its ‘auxiliary calibration mode’, that is here the spherical mode. By performing the five-position test for Fresnel zone mirrors described above the two error parts of the diffractive element can be separated from each other and from those of the interferometer. Thus, WPD and WF introduced by the Fresnel zone mirror are absolutely specified. Next, these error parts are transferred to those who are introduced by the aspheric mirror. For this weighting operation the non-proportional imaging of aspheric master must be taken into account. After that, the asphere under test is replaced by the multiplex CGH to calibrate the null test setup. The hologram then operates in the ‘null calibration mode’ – it forms a diffractive master of the asphere. Last, the actual asphere (or a series of aspheres of same type) is measured with the calibrated setup. 3.2.2. Arbitrary multiplex CGHs In the case of freeform aspheres, there is no preferred symmetry, neither in the test piece nor in the CGH-null lens. To apply the rules outlined in the beginning of this section, three phase functions (two mutually orthogonal oriented linear gratings and an arbitrary aspheric phase function) have to be simultaneously encoded in the y) are determined by the linear grating multiplex CGH. The x-components of pattern distortion vector field ζ(x, with kLG1 parallel to the x-axis, where the y-components are determined by the other grating with kLG2 parallel y) is identified in magnitude and direction. to the y-axis. Then ζ(x, In short, for a complete calibration of a interferometric null test setup for freeform aspheres, seven measurements are required: five calibration measurements to obtain the error of the diffractive master (crossed grating) and that of the plane output wave of the interferometer and one calibration measurement for the test setup with the calibrated multiplex CGH operating in the aspheric mode. Finally, the measurement of the actual freeform asphere is performed.
4. CONCLUSION Different techniques for self-calibration by use of diffractive elements have been presented. It has been pointed out that diffractive optical elements possess significant potential for interferometric optics testing, since they can be designed very flexibly and multi-functionally. Besides their established use as null optics for asphere metrology, computer-generated holograms can serve as versatile calibration tools in interferometry. Especially their potential for multiplexing and their unique property of reconstructing conjugate diffraction orders paves the way for further advances in high-precision interferometry.
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5. ACKNOWLEDGEMENTS The authors wish to gratefully acknowledge the German government (BMBF) and industrial partners for their financial support. A sincere thank to Christof Pruss and Rolf Freimann for many fruitful discussions.
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