1. Introduction

The Fizeau interferometer outperforms most of the other interferometer arrangements concerning freedom from systematic errors because the quality of the illumination and detection optics influences the measuring result in a reduced manner. The reason behind this feature is the fact that the wedge fringes are produced between two boundary surfaces only, i. e., the imaging optics is positioned outside the interferometer cavity. In the limiting case of interferences at thin layers, i.e., in the case that the cavity length approaches zero, the error contribution of the imaging and illumination tract vanishes. This has been extensively exploited in the so called proof glass test and explains the tremendous success of this simple workshop method [6].

In the modern contact-less Fizeau interferometers this is only partly true since the resonator has in many cases a length of several cm’s. Therefore, the optical paths outside the resonator play an increasing role with increasing resonator length. This explains the requirement for well corrected illumination and detection optics to achieve satisfying accuracy of interferometric tests using the Fizeau interferometer. It is well known [1] that the illumination optics shall be corrected concerning spherical aberration and should in addition fulfil the sine-condition. Provided these conditions are fulfilled the light source can be slightly extended and fairly monochromatic to achieve fringes with sufficient contrast. In the past spectrum lamps with a selective line filter had been used. This results in smooth fringes enabling high measuring accuracy. The advent of the laser changed this because now arbitrary plane mirror arrangements could be used to obtain fringes. But the high degree of spatial coherence produces strong variations of the intensity due to dust diffraction effects [2, 3] which is also an indication for local phase variations originating outside the actual interferometer resonator. Therefore, the extended spatially incoherent spectrum lamp should be replaced by a slightly extended laser spot on a rotating scatterer since the detector having a time constant of e.g. 20-50ms averages out the speckle produced by the moving scatterer. But, the mechanical stability problem remains because this is one of the main disadvantages of most interferometers.

Wave front tests are considerably more flexible concerning the requirements on the light source and the rigidity of the test equipment. The wave front test after Hartmann [4] can be interpreted as physical ray-tracing where the local slope of a wave front is determined by measuring the lateral shift of spots in a plane some distance from a stop plane containing tiny holes. More recently [5, 6], the stop array has been replaced by a micro-lens array which splits the wave front into an array of sub-apertures. The disadvantage is that now the spots contain information about the wave front slopes integrated over the sub-apertures, but the advantage is that all light is used. This type of wave front sensor is known under the name of Shack-Hartmann Sensor (SHS). The lens array produces a spot array in the focal plane behind the lens array which is deformed from a regular pattern by the local slopes of the wave front under test in the sub-apertures of the micro-lenses. With the help of a detector array interfaced to a PC it is possible to obtain wave front information from the measured gradient values [7] in a very fast manner.

But, the SHS delivers the integral wave front of all the optical components between the light source and the SHS since in an un-calibrated version it works without a reference component or surface. Since the slope of the wave front is insensitive to piston movements and moreover if the evaluation speed is high enough the SHS-software is also removing tilt components of the wave front under test, the SHS is rather stable which explains its popularity in industrial environments.

But the big disadvantage is the big aberration bias produced through the numerous optical components to form the wave field according to the requirements of the actual SHS. Therefore, calibrations are necessary if high quality tests should be performed. In this way the reference surface or normal comes into play either before the measurement or directly after, i.e., the measurement of the reference surface has to be stored in the PC in order to enable the subtraction of the aberrations of the auxiliary optics. In case of the Fizeau interferometer the measured aberrations produced by the surface under test are only biased by the relatively small aberrations introduced by the reference surface which means that the total aberrations are small. This is the big difference between interferometry and SHS: while interferometry delivers intrinsically referenced data, the SHS delivers measurements only referenced to data that have been collected before or after the measurement. The time lapse between the measuring and the reference run might be the imminent accuracy problem if the room temperature can not be stabilized sufficiently and air turbulence contributes to wave aberration variations. For highest quality tests absolute measurements of the reference surface are mandatory for SHS-tests as well as for interferometric ones.

Here, it shall be shown how the Fizeau philosophy can contribute to the improvement of the SH-test at the example of a planeness test. Before we go into detail a remark concerning the achievable sensitivity of the SHS in relation to interferometric tests might be appropriate, because the spectrum of opinions is rather broad. From own comparative absolute sphericity tests with a sensor head similar to the head used in the present measurements it can be concluded that the SHS has an inferior sensitivity compared with two-beam interferometry [8]. Later on, it will be shown how the sensitivity of the test set-up can be improved without giving up the advantages of the SHS.

For the calibration of interferometers and of the SHS it is necessary to have a material normal surface of high quality which has been measured in an absolute manner in order to obtain the deviations from a mathematical plane or sphere [1,9–13]. Such surfaces with known absolute deviations shall be denoted as normal surfaces. Here, we will concentrate on planes and spheres which are the most abundant surface types in the optical practice.

2. Fizeau-type Shack-Hartmann-Test

A SHS-test shall be denoted as a Fizeau-type test if the two plane surfaces are simultaneously in the light path as depicted in Fig. 1.

 

Fig. 1. SHS-setup for performing a Fizeau-type flatness test (The reflected light beams are deflected by 90° by means of the mirror surface of the prism positioned behind the slit aperture which is possible when the plate combination is slightly tilted out of the drawing plane)

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The setup of Fig. 1 is in fact a Fizeau interferometer with the exception that the angle between the test surfaces is large enough to allow for a clear separation of the waves reflected from the two plane surfaces under test by means of a slit aperture in the focal plane of the collimator. Through a slight tilt out of the plane of the drawing the back reflected light can be reflected via a plane mirror directly onto the SHS after a further collimation. The SHS is positioned at the conjugate of the Fizeau plate pair. The test procedure works as follows: (1) the rotation stage on which the two plane surface mounts are fixed is turned in such a way that the wave from the first plane surface is allowed to pass the slit aperture and is detected by the SHS and stored in the PC-memory and (2) after this the rotation stage is turned to allow the measurement of the plane surface of the second plate. The measuring result is then the difference of the two wave fronts following from the two SHS measurements.

 

Fig. 2. Schematic ray path of the waves reflected from the two surfaces A, B with the absolute deviations from planeness a, b. It is assumed that the back side of the plates forms a small angle with the surfaces under test enabling suppression of the backside reflections by spatial filtering operations.

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The measured wave aberration is then:

In this notation W_ob are the aberrations of the illumination and observation tract outside the Fizeau resonator, z₀ is the distance of the mathematical reference planes, z is the actual air gap between the two boundary surfaces under test, n is the refractive index of the first glass plate, and a, b are the absolute deviations of the surfaces A,B from their respective mathematical reference planes. For simplicities sake the refractive index of air has been put to one.

As in the Fizeau-test, the measured quantity is the sum of the surface deviations beside an irrelevant linear function z₀ of the lateral coordinates of the plates. Due to the special geometry, the sum of the deviations is measured where the sign of the deviations is taken positive in the direction of the outer normal. The mathematical identity to the Fizeau results enables, e.g., the application of the 3 plate absolute test for planeness since the solvability of the system of 3 equations for the unknowns a, b, c of three plates A, B, C presumes that deviation sums are measured (see ref. [1]).

It is now necessary to discuss the consequences of the proposed procedure. In order to get access to the two waves separately it is necessary to turn the plate combination by a small angle. Therefore, it is necessary to multiply the deviations with a cosine-factor very close to one. Now, the question is how big this correction is.

A clue is delivered by the resolution of the CCD-detector array of the SHS because the cut-off frequency of this detector limits the shortest spatial period of the wave aberrations. The cut-off frequency determines the width of the spatial filter in the focal plane of the collimator of 500 mm focal length. For an estimate let us assume that the surface deviations are decomposed into a Fourier series of surface waves. With a SHS having about 50 sub-apertures per a diameter of 50mm in plate space this means a highest detectable spatial frequency of about 0.5/mm. With a collimator having 500mm focal length this amounts to a stop aperture smaller than 1 mm or an angle of <10^-3 rad for λ=543nm. For the convenience of the experimental work we used a stop diameter of 1.5mm. This means that a cosine factor differs only by approximately 5·10^-7 from 1 which accounts for the projection of the surface deviation onto the light propagation direction. Such small corrections can be neglected since a good plane surface has deviations of the order of λ/20 or smaller.

Because of the inferior accuracy of the SHS compared with the normal Fizeau interferometer equipped with phase shifting technique the advantage to make the measurement with the help of the SHS would not be very convincing since the elimination of the aberrations of the auxiliary optics might spoil the accuracy because the repeatability of the SHS-test is limited. It has been pointed out for the case of a Twyman-Green surface measuring interferometer [14] that a test with an RMS-accuracy better than λ/100 requires an optical setup with a maximum of aberrations well below 1λ. The aberration bias in case of the Twyman-Green and the SHS are comparable concerning the number of components and their respective aberration load.

3. Enhancement of the sensitivity using multiple reflections

Therefore, the aim is to enhance the measuring sensitivity for the surface deviations without the disturbing influence through the contributions of the auxiliary optics. This goal can be reached if a Fizeau resonator with sufficient gain is used.

It is of course known that multiple beam interference offers such increased phase sensitivity. But phase shift evaluations in reflected light suffer from two problems: (1) the intensity distribution shows strong intensity modulation only at the resonance peaks making phase shift techniques unattractive and difficult [15] and (2) reflected light arrangements have a strong background bias especially if the reflecting coatings show strong absorption. The latter is due to the strong dominance of the first reflected wave. Under the best conditions, the sum over all other reflected waves delivers an amplitude equal to the first reflected amplitude [16].

So, multiple beam interferometers commonly prefer transmitted light arrangements [17, 18] but this is restricted to partially reflecting mirrors only. A solution for reflected light has been given by Langenbeck [19] where the multiply reflected light in a Fizeau resonator is either superimposed by the light of the reference arm in a Michelson interferometer or where one of the low order beams is used as reference. In this way also opaque strongly reflecting surfaces can be tested within a multiple beam Fizeau interferometer. But, the disadvantage of the latter method is the strong dependence of the measured wave aberrations on the aberrations of the collimator and the imaging optics since strongly differing ray paths are taken through the optics. In addition the interference pattern is increased in its carrier frequency by the same multiplicative factor as the deviations of the Fizeau resonator. It is then necessary to transform the interference pattern to a low frequency one for the actual measurement which can be done by Moire methods [19]. This frequency transformation does not free the final result from the aberrations of the interferometer optics although due to the enhanced sensitivity through the multiple reflections their influence is somewhat reduced.

Here, the advantage of the SHS can be brought into play because it allows the superposition of wave fronts a posteriori which can be done in an on-axis geometry by turning the resonator bringing the selected reflection on-axis. Also, big intensity values can automatically be scaled to the dynamic range of the photo-detector by adapting the integration time of the CCD-camera. This is done in an iterative manner in case of the SHS of OPTOCRAFT GmbH which has been used in our experiments.

Again a two step procedure is carried out: (1) one of the first reflected wave fronts is measured by adequate turning the rotation table and (2) the wave having undergone N reflections in our tilted Fizeau resonator is adjusted by turning the rotating table carrying the Fizeau resonator enabling the latter wave to pass the slit aperture on axis. This has been illustrated in Fig. 3.

 

Fig. 3. Enhancement of surface aberrations through multiple reflections in a Fizeau resonator. At the left: Reflected waves due to semitransparent highly reflecting reference and highly reflecting mirror under test. At the right: Perspective display of the waves coming to a focus in the focal plane of the collimator objective.

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In accordance with the scheme of Fig.1 the plane surface combination without coatings has now to be replaced by the two highly reflecting mirrors of the multiple beam Fizeau resonator.

Following our procedure of storing in a first step the aberrations of one of the first reflected waves (0-th order see Fig. 3) in the online PC and then adjusting in a second step the N-th order reflected wave on axis by turning the rotation stage for the SHS measurement the necessary wave aberrations are measured.

By modifying equ. (1) accordingly we obtain for the combination of the waves having undergone M and N (M<N) reflections:

Since the evaluation is carried out in our two step procedure strictly on axis the aberrations of the auxiliary collimation and imaging optics cancel. Even if there might be a small additional uncertainty due to small misalignments, time depending air turbulence, and due the noise of our SHS, the small aberrations of the auxiliary optics can be neglected in most cases since the deviation sum of the surfaces under test contributes typically by one order of magnitude more to the measuring result.

As it can be inferred from Fig. 3 on the left there is a smear of the surface information in a direction perpendicular to the edge of the air wedge formed by the two high reflecting mirrors which is denoted as “walk-off”. To keep this walk-off small it is necessary to reduce the resonator length to values below 1mm which is discussed below in more detail. Since the SHS measures only global aberrations this restriction can easily be tolerated.

Although the maximum angle between the surface normal and the light direction of the probing beam is also increased by a factor of 2N the cosine factor can be put to 1. Approximately it can be said that the probe beam gathers the path difference with a varying sensitivity factor cos2Nα which differs from 1 only by 10^-3 if a maximum order of 30 is assumed which is still tolerable without further corrections because it can rightly be assumed that the deviations of the good plane surfaces are very much below one wavelength of the illuminating light. In contrast to an interferometric test the variation of the inclination angle does here not matter because no direct phase measurement is made. Therefore, the phase mismatch of the reflected waves relative to each other does not affect the evaluation since no referencing of the emerging waves to a fixed value is carried out.

4. Experimental verification

In our experiments highly reflecting dielectric or metallic coatings have been used where the surface hit first has to be dielectric coated to allow access to the waves having undergone N reflections in the Fizeau resonator. The second surface could be opaque but should be highly reflecting as it would be the case with a surface having an Al-coat or if the material shows already high reflectivity as in case of wafers made from semi-conducting materials.

At first, it has to be shown that the repeatability is high enough to enable a sufficient cancellation of the contributions of the auxiliary optics. In this connection it is of course necessary to keep the aberrations of the auxiliary optics, i.e., all optics outside the Fizeau resonator, small which is also true in case of interferometric high quality tests. Here, normal achromats have been used. The total aberrations of our test setup are shown in Fig. 4.

 

Fig. 4. Basic aberration of the optical system measured against the first Fizeau reflection. The contour line distance is 0.5 waves.

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In a repeatability test the results of two consecutive measurements with the SHS are subtracted from each other delivering the result of Fig. 5.

 

Fig. 5. Repeatability test for the SHS, Zernike-fit of degree 20, please note the change in the contour line distance by a factor of 50 compared to Fig. 4. The contour line distance is 0.01 waves.

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This repeatability test implies that an enhancement factor N of more than one order of magnitude would be desirable to guarantee accuracy levels obtained with phase shifting interferometry which is of the order λ/500 rms or better. The latter would be necessary if the SHS shall be applied to absolute tests where several wave fronts contribute to the final accuracy.

Fig. 6 shows two test results with enhancement factors of 10 and 15 which are in fact control patterns during the adjustment of the Fizeau resonator with corresponding positions of the turning stage carrying the Fizeau resonator.

 

Fig. 6. Tilt indication for the case of the enhancement factors 10 (on the left) and 15(on the right) In this picture the measured wave aberration is shown in a mod 2π representation. The absence of big tilts can be inferred from the near parallel section through the wave front deformation. In case of a remaining tilt a picture similar to a fringe interferogram would be seen.

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The screen (s. Fig. 6) with the referenced deviations delivers a very sensitive criterion for the exact adjustment of the reflexes within the light path through the optics. In this way it is guaranteed that the contribution of the auxiliary optics is identical.

Depending on the increase of the aberrations introduced by the auxiliary optics with small tilt differences of consecutive runs there remains an additional uncertainty in the final result. Here, it is very helpful if the tilt of the incoming waves in the consecutive runs is controlled and is made equal by careful adjustment. To show this influence more clearly it is possible to compare results with and without careful adjustment supporting this expectation (see Fig. 7).

In a series of measurements with increasing multiplication factors the enhancement effect can be shown (see Fig. 8). The peak/valley values as well as the RMS figures show the functional behaviour with sufficient reliability as can be inferred from the ratios of the p/v-values for the 10-times and 30-times enhancement pictures. This ratio has the value 2.96 instead of 3, which gives an indication of the error margin of 1.3 %. This error is of the same order as it has been shown in the repeatability test and is probably due to the measuring uncertainty of the SHS in general.

Figure 9 shows the evaluation of different plate combinations, i.e., the combinations AC, AD, and BD. The plates C,D were Al-coated test plates which could not be measured in transmitted light. The stronger damping of the plate combinations allowed only the use of the 20-th order which results in a contour line distance of λ/80.

 

Fig. 7. Difference between two runs (cont. line dist. λ/20) and a hard screen copy of the best adjustment (on the right). The first modulo (2π)-picture from the left shows the wave front without tilt removal. In a second run about 10 carrier fringes have been added to show the influence.

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Fig. 8. Deviation sums of the two surfaces AB, i.e., (a+b): AB(N=10) left; AB(N=20) middle, AB(N=30) right (which corresponds to enhancement factors of 10/20/30, or in wavelengths λ/20, λ/40, λ/60 or contour line distances: λ/40; λ/80, λ/120).

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Fig. 9. Deviation sums of two surfaces AC, i.e., (a+c); surfaces AD, i.e. (a+d) and Surfaces BD, i.e., (b+d): AC (N=20) left, AD (N=20) middle, BD (N=20) right, from contour line to contour line the surface deviations are varying by λ/80.

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5. Discussion of the experimental results

Our first experiment clearly shows the validity of the enhancement concept based on multiple reflections within a Fizeau resonator and the detection of the wave aberrations by means of a SHS. But there is room for improvements concerning the achievable accuracy.

As has been pointed out [14] the amount of basic aberrations of the imaging and collimation optics has an impact on the achievable accuracy since adjustments and other aberrations become more critical compared to a well designed set up. In our case the basic aberrations are strong astigmatism of p/V about 2 waves which is probably produced through the deflecting mirror in the observation path (see Fig. 1). If this mirror has a spherical deformation it will produce astigmatism when hit under 45°. Although the amount of aberration in a repeatability test is two orders of magnitude smaller (see Fig. 5) it shows a similar functional character which supports our argument. A repeatability test with a slight additional tilt supports these findings. The measured aberration is in this case one order of magnitude bigger than in a repeatability test without change of the state of adjustment as should be expected and in addition once again the functional character is also having astigmatic symmetry.

Therefore, it seems necessary to reduce the basic aberrations of the optical imaging system if high quality surface tests are envisaged as in absolute tests for planeness. Since one candidate for causing astigmatic aberration is the skew reflection at a small mirror in the focal plane of the collimator it would be possible to avoid the use of a folding mirror if the illuminating light is delivered by a single mode fibre. In this case the observation could be made without such a mirror. The other elements in our set up are achromats which should be sufficiently corrected. The SHS allows anyways the control of the quality on the λ/10-level which should suffice in our case.

The SHS in our experiment uses micro-lenses with 150µm diameter and a focal length of 4.5 mm. It had been designed for the measurement of steep aspheric wave fronts. This results in a Fresnel number of about 10. For the measurement of small aberrations a longer focal length (say e.g. by a factor 2) would improve the sensitivity without causing problems with the definition of the focal spots or with ambiguities due to overriding the sub-aperture limits [7] Which might become a problem if the spots are too blurred.

The number of usable reflections between the resonator mirrors is limited by the following effects: (1) the intensity of the N-th reflection decreases with (R₁R₂)^N and (2) the lateral walk-off on the mirror surface may become too big for reliable measurements.

The span of intensities which can be worked with the SHS of OPTOCRAFT GmbH without an optical manipulation of the light source is about two to three orders of magnitude in integration time constant. With the plate-coatings at our disposal it means 20–30 reflections. Concerning the walk-off problem the following equation can be used from literature [16]: Δρ={2N ²-4N+3/2}zα Where z is the mean mirror distance, α is the tilt angle of the plates, and Δρ is the lateral walk-off. In our experiment the walk-off is typically below 1 mm but could be reduced to 0.5 mm by careful adjustment of the resonator length which is of the same order of magnitude as the diameter of one sub-aperture defined by the diameter of a micro-lens..

A further improvement would be possible if consecutive runs and not only intensity frames were averaged. The reason behind this argument is the measuring regime where two consecutive wave front measurements are subtracted from each other since in this case the influence of air turbulence will cause additional wave aberrations which should be averaged out if their contribution has a stochastic character. Averaging of intensity frames on the other hand will mainly reduce electronic noise but would lead to a broadening of the focal spots and therefore might not necessarily increase the measuring accuracy.

6. Conclusion

The proposed accuracy enhancement method in reflected light uses the advantages of the SHS, i.e., the separate measurement of wave fronts reflected from the surfaces forming a Fizeau resonator. This is made possible by introducing a small tilt between the plates. The single reflections are then brought on-axis by means of a turning stage the Fizeau resonator is mounted on. To avoid lateral shears the axis of rotation should go through the middle of the small resonator layer. There is no correction of the measured surface deviations due to this small rotation necessary as long as the deviation sum of the two surfaces under test keeps below one wave of the illuminating light. With dielectric coating due to their high reflectivity values enhancement factors of 30 and for combinations with one Al-coated surface factors of 20 are achievable without using external intensity manipulations which could introduce additional errors.

So far the concept has been applied to the planeness test but it could also be applied to the spherical Fizeau resonator provided: (1) the reference surface is separable from the illuminating beam shaper, (2) a complimentary coated surface to the reference surface having a radius of curvature differing only by a part of 1 mm is available. Also in this case the beam shaping system should be well corrected concerning spherical aberration and coma.