The multi-pass solution for surface measurements with the help of a Shack-Hartmann sensor (SHS) on the basis of a Fizeau cavity enables fast access to surface deviation data due to the high speed of the SHS and easy referencing of the measured data through difference measurements. The multi-pass solution described in a previous publication [J. Schwider, Opt. Express 16, 362 (2008)], provides highly sensitive measurements of small displacements caused by thermal non-equilibrium states of the test set up. Here, we want to demonstrate how a pulsed thermal load changes the surface geometry. In addition the temporal response for different plate materials is monitored through a fast wave front measurement with very high sensitivity. The thermal load close to a delta-function in time will be applied from the back-side of a plane plate by heating a small Peltier element with a heat impulse of known order of magnitude. The development of the surface deviation on the time axis can be monitored by storing a set of successive deviation pictures.
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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. In contrast to this, the Shack-Hartmann wave front sensor (SHS) measures wave front deformations without using a reference wave [4–7]. Therefore, the wave front deformations of all optical elements will accumulate screening the wave front deformations due to a single surface. Only by subtracting SHS-measurement results of two surfaces the wave aberrations of the auxiliary optics of an optical test setup can be eliminated. Only after such a calibration step useful surface deviation data can be obtained. This is comparable to the situation of a Fizeau interferometer with the difference that in the ideal case the aberrations of the auxiliary optics are eliminated immediately and not in time succession as with the SHS. Such difference measurements will eliminate the deviations of the optical setup and they can therefore be used for the measurement of environmental disturbances in optical setups such as temperature variations or mechanical strain caused by gravitational forces or clamping.
2. Method for the measurement of deformations caused by thermal disturbances
Recently , we have shown that the combination of a SHS with a Fizeau cavity enables measuring results matching those of a Fizeau interferometer. The Fizeau philosophy can also be applied to a Fizeau cavity having reflective coated surfaces. By doing multi-pass measurements the impact of the aberrations of the auxiliary optics can be reduced and the sensitivity for surface deviations can be enhanced by the multi-pass transit of the wave fronts in the cavity. This feature of the multi-pass SHS enables the sensitive detection of very small changes of the geometry of the boundary surfaces of a cavity.
A null test can be performed with the wave front having passed p-times the Fizeau cavity in time-lapse mode. At the time t = 0 the deviation picture will indicate wave front deviations of null beside a small stochastic deviation caused by intensity noise of the source and detector and in addition by drifts of the optical components and air turbulence. With increasing time lapse there will only be a small increase of the deviation values due to instabilities and air turbulence. This self-reference regime can be exploited for studies of stress in glass plates due to temperature or mechanical strain.
If one of the cavity plates is heated or cooled from the backside with the help of a Peltier element there will be deformations of the surface profile which will show up in the measured optical path difference of a wave passing the cavity. Due to the enhancement of the multi-pass regime, all other changes of the aberrations can be neglected in comparison with the cavity contribution.
The thermal load introduced by heating one of the plates from the backside can be considered as a thermal point-disturbance. Since the glass has a non-null expansion constant the plate will be deformed. The heat propagation with time in the plate is described by the thermal conduction equation. The solutions of this differential equation strongly depend on the boundary conditions that are rather ill-defined under usual experimental conditions. In theory, solutions are derived for well-defined boundary conditions. Here, theory will only provide a guide-line for experimental tests. This is especially true because wave front tests measure the mechanical deformations caused by thermal loads on the cavity plates and not the temperature field itself. The measuring setup is given in Fig. 1 .
3. Challenges occurring with reflected light SHS measurements
The measurement of wave aberrations in reflected light are very often disturbed by spurious reflections form surfaces being in the illuminated light path. Although the contribution of spurious reflections to the measuring field is much reduced through a spatial filter in front of the detector disturbing contributions from spherical surfaces survive the filtering process since there is in most cases a part of the spherical surface in-line with the wave to be measured. In case of reflective coated Fizeau plates the number of such contributions is even higher. An overview over such contributions in our measuring set up depicts Fig. 2 . In the actual measuring set up a spatial filter in the back focal plane of the collimator is used (see Fig. 1). This filter will remove reflections from the back side of the Fizeau plates but can only reduce the reflections from curved surfaces in their spatial extent. In Fig. 3 this behavior is demonstrated. Without spatial filter in the back focal plane of the collimator all disturbances will reach the detector array. The removal of the spatial filter allows also for the adjustment of the best focus position of the camera lens because the waves having undergone a different number of passes through the Fizeau resonator have an increasing angle offset. The coincidence of the plate images on the detector array show up as a near coincidence of the sub-aperture foci of the individual micro-lenses of the SHS. In upper part of Fig. 3a – Fig. 3c the spatial filter has been removed from the light path and the Fizeau cavity was blocked by inserting a screen either in front (Fig. 3b) or between (Fig. 3c) the Fizeau plates showing in this way the impact of the spurious reflections 1 through 4. The insertion of the spatial filter reduces mainly the spatial region being disturbed on the detector array but not so much the intensity contribution to the spot field concerning single spots. Since the multiple passes through the Fizeau resonator result in strongly decreasing intensities, the ratio of the signal to the parasitic contributions will greatly deteriorate with an increasing number of passes through the resonator which means a loss in the dynamic measuring range.
In the lower part of the Fig. 3 (Figs. 3d and 3e) the spatial filter is inserted and in this way the number of overexposed spots for the SHS is greatly reduced. Nevertheless, the distortions could hamper a special software function, i. e., a tilt evaluation which is installed in the SHS by using one blind sub-aperture which enables the measurement of the absolute tilt. The latter feature helps with the adjustment of the p-th reflex in the center of the spatial filter through the indication of a minimal aberration. The impact of misalignments on the measuring result can be reduced in this way and be kept below the repeatability value of the measurement.
The software of OPTOCRAFT GmbH for the SHS allows for the subtraction of “dark-field” intensity data which comprise the spurious intensities described in Figs. 2 and 3 which can be stored when the Fizeau resonator is obstructed by an opaque screen. This improves the performance of the sensor very much.
For long term measurements it is necessary to control the repeatability and stability of the measured wave front deviations. This can be done by referencing currently measured data to the data at the beginning of the measuring run. Since the multi-pass method  uses an on-axis geometry for the measured wave through a suitable rotation of the Fizeau resonator it is plausible to assume a minimum of aberrations caused by the auxiliary optics (see Fig. 1).
In order to secure the repeatability of the measuring set up we have made two measuring runs: (1) a single reflection from the entrance mirror of the Fizeau (see Figs. 4 ) and (2) a run with p = 30 (p being the number of passes in the Fizeau resonator (s. Figs. 1 and 2)) for a resonator built of a Zerodur mirror in combination with a fused silica mirror (see Fig. 5 ). In both cases the peak to valley (PV) and root mean square (RMS) data versus time are shown.
In both measurement series an increase of the PV and RMS figures with elapsed time can be observed. Obviously, there is a slow drift of the mirror position relative to the incoming wave field combined with some irregular deviations which could be attributed to air turbulence in the collimator. In Fig. 4 there is no resonator structure present which points in the direction of air turbulence and slow variation of the aberration data of the auxiliary optics. With increasing time lapse there is also a greater variability of the difference data from the start values. One reason could be a slow change of inhomogeneity cells in the air volume of the auxiliary optics.
In the repeatability measurement using the multi-pass resonator geometry (Fig. 5) the deviations are clearly bigger and also the drift is stronger compared to the case of Fig. 4 (which is by the way the normal case for SHS calibrations). Since the resonator length has been chosen to be about 1mm, air turbulence within the resonator is probably not the main reason for the drift and the increased variability of the measured aberrations. The only remaining reason is strong tilt variability due to the multi-pass arrangement. Therefore, drifts and accidental tilts of the resonator mirrors will lead to enhanced wave front tilts which increase the influence of aberrations due to air turbulence and aberrations of the auxiliary optics. So far, the resonator has not been stabilized by using a spacer etalon which would keep the relative positions of the resonator mirrors fixed. The disadvantage of such a spacer would be a severe loss of flexibility in our experimental set up. Furthermore, there would be the danger of damaging the mirror coatings which is the reason we refrained from such measures.
4. Differences between Fizeau-SH test and Fizeau interferometry
The differences between interferometric and SH-tests shall be discussed for the example of a Fizeau-test of plane surfaces which is one of the simplest cases and can stand for the typical optical test needs.
On the one hand, the working principle of the SHS allows for reference-less measurements of wave fronts or more precisely of the gradient field of wave fronts. This seems to be a big advantage which is today heavily exploited in optical measurements of aberrations. Since there is no reference wave front involved the set up is very stable especially against piston movements of reflecting optical parts. Also, the change of direction of a wave front will in many cases not be of utmost interest and is commonly eliminated from the final wave front result. The disadvantage of the SH test is the fact that all optical elements between source and detector will contribute to the wave front shape. Commonly, the test sample will only contribute on the same level of magnitude to the wave front deviations as all other elements. Since the beam cross section will undergo several transformations on the way from the source to the detector in order to match the requirements of the test sample there is at least a type of a collimator and a telescopic imaging system necessary. In addition one cannot neglect the air volume of the test set up which will also contribute to the final measuring result in form of stochastic wave aberrations. Therefore, calibrations are necessary which have to be done before or after the measurement of the test sample. The main point is the time lapse between the measurements and in general the necessary changes in the state of adjustment between test and calibration run. The calibration of a SH test on a level of better than λ/50 will be a very big challenge.
On the other hand, interferometric tests rely on the measurement of wave front differences, i.e., the interference pattern depends on the phase difference of an object wave front from a reference wave front. In case of the Fizeau test this means that one measures the sum of the deviations of two surfaces from ideal mathematical planes. If the reference surface is ideally plane (consider for example a liquid mercury mirror at rest ) the measuring result delivers immediately the deviations of the test sample. This happens in a single run and no time delay between two runs has to be considered. This is a very substantial advantage of interferometry since it is self-referencing. Especially, the optics outside the Fizeau resonator established by the two surfaces will only contribute to the measuring error for perpendicular incidence via the angle aberrations u in the following way:
For very small resonator lengths z this error can be neglected. If in addition a parallel adjustment of the Fizeau mirrors is maintained - fluffed out fringe field - then the ray path of the two interfering waves is nearly identical up to the detection plane which means that all aberrations cancel out. The latter is also true for the contribution of air turbulence to the aberrations. The price, which has to be paid, is the requirement for a stable mounting since otherwise the interferogram would be wiped out by vibrations. The sensitivity of interferometers to mechanical vibrations can also be considered as a convenient indicator for the stability of the measuring set up. In contrast to this the SHS does not deliver such a clear indication because, e.g., piston movements of mirrors cannot be detected.
The coherence requirements for the used light source are very relaxed for the SHS. The spectral bandwidth can be several nanometers without loss in the definition of the spots because chromatic aberrations of the lens array will lead to a symmetric halo. The lateral extension of an incoherent light source must fulfill the requirements of the vanCittert-Zernike theorem in the sense that each subaperture of the lens array should be coherently illuminated. The resulting spot will be a convolution of the source distribution with the Airy disk of a single micro-lens. Because of this, the shape of the transverse source field distribution should have inversion symmetry to secure high accuracy for the calculation of the center of gravity for the spots of the SHS.
In the case of the Fizeau interferometer the coherence length has to be greater than the cavity length which is guaranteed if a suitable laser is used. The spatial coherence can be reduced via a rotating scatterer in the light path. This helps to get rid off the dust diffraction patterns enabling smooth cos-type interference fringes.
In the case of the SHS an intensity pattern is detected. Therefore all intensity variations across the wave front will reduce the dynamic range of the SHS. This is especially true if spurious intensity contributions from parasitic reflections blind a certain amount of the sub-apertures. In the case of interferometry spurious intensities in the field of view will only matter if their intensity modulates during the phase shifting run. This means that they have to be coherent with the reference wave in order to modulate. By a proper choice of the degree of coherence it is in most cases possible to tolerate such spurious intensities. Furthermore, an intensity variation across the field of view will not severely influence the modulation degree of the interference pattern since this quantity depends only on the square root of the intensity ratio of the two interfering wave fields. This is the background for the stability of phase shifting evaluations of interferograms. The disadvantage of vibration sensitiveness of interferometry is to some extent outweighed by the information on the mechanical stability of the measuring setup which is not directly accessible with the SHS. Direction instability will be one of the causes for the limited long term repeatability of the SHS because the referencing is done in time succession by subtracting wave aberrations of the order of one wavelength (see Figs. 4 and 5). The time delay between referencing and measuring run will result in an error floor due to turbulence in the air volume of the auxiliary optical system and also due to drifts in the mean light direction. The error floor level is increasing with time. A stable level has not been reached during the measuring series shown in Figs. 4 and 5 but has reached approximately the PV-λ/10 level after more than 1 hour. Similar results have been reported by others  for the measurement of aberrations or mirror deviations in reflected light.
It is obvious that also interferometric measurements of surfaces or components make calibrations necessary since the intrinsic reference piece or normal will in most cases have deviations of the same order of magnitude as the test sample. An exceptional normal surface for plane surfaces could be a liquid mirror . But the calibration philosophy of interferometry is substantially different since the impact of the collimation and imaging optics on the basic error level can be avoided which is not the case for the SHS. Calibration in surface measuring interferometry means absolute testing, i.e., removal of all contributions of the measuring set up through the combination of several relative measurements . As we have reported , it is also possible to obtain high sensitivity with multi-pass SH tests but the stability compared to interferometric tests will be inferior. The most favorable feature of the multi-pass test is the possibility of reflected light measurements compared to the proposed frequency comb solution .
5. Thermal stress detection by using a multi-pass Fizeau resonator
The high sensitivity of the multi-pass SH-test using a Fizeau cavity with high gain can be put to a test by measuring small surface changes occurring under thermal loads. Local temperature changes will lead to deformations of the glass body. Since the mechanical stress exerted by temperature gradients is a very involved physical problem we have in our experiments applied a heat source in form of a Peltier element at the backside of the last Fizeau mirror (see Fig. 1). The Peltier element has an area of 4x4mm2 and it will provide a good approximation to a thermal point load if heated for a short time period (e.g. 20 sec.). In a multi-pass adjustment with p = 25 or even p = 30 the time development of the surface deviations were measured over a period of approximately 200-300 sec.
As a start a mirror combination BK7 with fused silica has been chosen where the load was applied to the BK7-mirror from the backside. The theory of the heat conduction [9,10] can only give a guide-line for the time behavior of a Gaussian start distribution which is the most realistic spatial distribution for a point load. Since the material parameters are independent on the spatial coordinates the time development of such a starting distribution can be described by:
For a thermal δ-load at the time t = 0 one obtains:Fig. 6 where some deformation states have been recorded. The time response curve is shown in the diagram of Fig. 7 .
In our measurements the difference of the wave aberrations between an initial state before applying the thermal load and an aberrations after some time has elapsed counted from the start of the heating with the Peltier element. Here we have applied a point-like load at the center of one of the plates of a multi-pass Fizeau resonator for a rather short time period. First, a BK7 plate and in a second experiment quartz glass has been exposed to thermal stress.
At the beginning of the heating period the propagating heat causes a concave deformation of the plate in the center which has to be interpreted as a surface bending effect. This is plausible because the back surface of the glass body is exposed to the heat at first and is therefore expanding which leads to a concave bending of the front surface. After the Peltier element is switched off the heat mountain broadens and loses height. In addition, the heat propagates also into the glass body along the surface normal leading to a thermal expansion especially at the center. In a later phase the bending is decreasing more swiftly because it is a surface effect and a central buckling of the heated volume will dominate although on a much lower deviation level.
The air film aberrations between the two optical surfaces in the balance of the measured wave aberrations cannot so easily be separated from the bending and buckling effects of the glass body. The air film between the plates will warm up and the deformation values increase because the refractive index  is decreasing. This contribution will add a wave aberration to the buckling effect caused by the thermal expansion of the glass body in the central region. For an air gap of 1mm length an increase by 1°C might already produce a wave aberration of the order seen in Fig. 6 (c_1) and Fig. 6d_1).
Furthermore, two plane mirrors were measured both made of fused silica. The geometrical dimensions were similar to the dimensions of the BK7-glass plate. Since the thermal expansion constants are greatly different the peak deformation is much smaller and also the time constant of the temperature decay is different (see Fig. 8 ) since the heat conductivity of fused silica is greater than that of BK7.
The general time response is quite similar although there is more than 1 order of magnitude in the thermal expansion coefficient of BK7 in relation to fused silica. A set of typical deformation pictures is given in Fig. 9 . In this case the surface bending effect is dominating. A volume effect could not be detected because of the greater heat conductivity of quartz glass and the fact that the bending effect is much smaller so that the error floor screens smaller effects.
The limited stability of the self-referencing method especially under multi-pass conditions does not allow for the exploitation of the gain in the measuring sensitivity of the multi-pass technique in full extent. In the long term the error level is too great to allow for a similar thermal stress measurement of Zerodur. The reason behind it is the ratio of the expansion coefficients:
BK7:fused silica:Zerodur = 7,1-8,3 10−6/K: 0,54 10−6/K:5 10−8/K.
In order to measure the mechanical deformation of a Zerodur plate under such a thermal load a repeatability value below 0.5nm would be required for the peak to valley figures. With the equipment used at the moment this was not possible.
It has been shown that the multi-pass SH-technique works sufficiently to achieve temperature time profiles for normal glass and also for fused silica. In the short term of up to 100-200sec it is possible to follow the heating and cooling process for BK7 and fused silica.
However, long term measurements of wave fronts reflected from a single mirror showed variations and drift effects in the order of PV = λ/67, RMS = λ/357; at λ = 532nm. It turned out that those deviations of the wave front aberrations become considerably bigger for a wave front reflected p = 30-times, i.e.: PV = λ/14, RMS = λ/100; λ = 532nm. Instead of a gain in sensitivity by a factor of 30 only a factor of about 6 could be achieved.
The reason for this reduction in sensitivity is probably mechanical instability. This means that the mechanical stiffness against vibrations and drifts has to be p-times that of the single pass geometry if the full sensitivity gain shall be maintained. The stability problems in our experiment can be made visible with the help of a tilt indication which is implemented in our SHS. Because a reference measurement at the beginning of a time series is used, the tilt indication shows how far the wave front normal has drifted or jumped between successive measurements. As it can be inferred from Figs. 4 and 5 the PV- and RMS-values are increasing steadily from the start value. The adjustment necessities do so far not allow for a sufficiently rigid combination of two highly reflective plates to a wedge resonator. For the measurement of the thermal time behavior of low expansion glasses or ceramics it would be necessary to reduce environmental influences by three measures (1) using a stiff wedge etalon between the plates, (2) clamping the rotational degree of freedom of the Fizeau wedge after adjustment, and (3) putting the whole optical set up in a vacuum vessel to get rid of air turbulence. If all these elaborate measures were taken then the only candidate for tilt would be the holder of the mono-mode fiber end. This could be controlled beforehand by monitoring the ray stability with the help of the tilt indication feature of the SHS and long term repeatability measurements.
Help and assistance with the SHS-software by OPTOCRAFT GmbH is greatly appreciated. The authors would like to thank for a helpful criticism by one reviewer which helped us to locate a sign error in the aberration data which led to an erroneous interpretation of our measuring results of Figs. 6 and 9 in the first version of this publication.
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