Abstract. Interferometric surface topography based on two-dimensional Fourier transform (2D FFT) is compared to phase stepping technique. New self-calibrating phase stepping method is developed and reported. Optimal filtering algorithm for 2D FFT is described. Both methods have been applied for determining the topography of a steel auxiliary plate (platen) using Michelson interferometer. The standard deviation of the difference between these two methods is found to be about 1 nm for short range spatial resolution (1 pixel) and about 0.6 nm for mid range (5x5 pixel window). The sub-nanometer resolution in surface topography is shown to be feasible in quasi-real-time mode with moderate performance hardware.
Keywords: optical interferometry, nano-metrology, gauge block, 2D FFT, 3D surface, phase stepping, fringe analysis, topography, interferogram processing, interferometer, interference pattern, dimensional metrology
1. INTRODUCTION
The length unit of SI (meter) is one of the most important units for science and technology. Practical realization of this unit is based on interferometric measurement of a gauge block relative to wave-length laser standards. Using modern frequency stabilized lasers the wave-length standard can be realized with accuracy of about 10^{-11}. Practical measurement of a gauge block is limited by accuracy of interferometric comparators. Typical uncertainty of those is usually about 10^{-7}, or 10^{-8} at the best. There is a big gap, therefore, between wave-length primary laser standards and practical artifact, such as gauge block (GB) measurement. To fill this gap some better optical interferometry is required. Interferometric measurement of gauge block (GB) is restricted by several factors. Fringe fraction determination by FFT method is one of the promising algorithms. Recently we have shown that measurement of the gauge block can be significantly improved if one can perform the following:
This kind of measurement is impossible without
detailed knowledge of the gauge block and auxiliary plate (platen) shape.
Therefore, interferometric topography in the optical gauge block metrology
becomes crucial for further improvement of the meter realization. There
are two alternative approaches in interferometric topography: (a) phase
stepping technique, i.e. dynamic interferogram processing, (b) fringe pattern
analysis, i.e. static single interferogram processing based on some phase
reconstruction method. Both methods have their advantages and disadvantages.
The first one (phase stepping) has better spatial resolution and is less
sensitive to intensity inhomogeneity of the interferometer field, but it
requires very accurate calibration of the phase stepping unit, which is
a quite difficult task. The second one (fringe pattern analysis) is faster,
self calibrating, but suffers from intensity inhomogeneity of the optical
field and has practical limitations on spatial resolution.
Several different methods has been developed for the fringe pattern processing. Among those we note linear fast Fourier transform (1D FFT), image FFT (2D FFT), direct data iterative fitting. Here we report our latest developments in the field of fringe interferogram processing. We compare results obtained with a phase shifting technique and various types of fringe pattern processing. We report a new self-calibrated phase stepping method for wave-front topography. We also report details of our optimum 2D FFT fringe-pattern processor.
2. EXPERIMENTAL
In our experiments we have used the Michelson type optical interferometer. As the detector we use the 480x640 pixel CCD camera with 8 bit digitizer. The output of the camera is stored and processed by the PC of a moderate processing performance (Pentium III, 500 MHz). The important part of the processing is original software specially developed for various scenarios of the fringe processing. The software is able to process single interferogram and perform phase stepping processing. The processed interferogram results in a 3D topographic map of the measured surface. The result is analyzed using our 3D surface view visualization software. The software is written in Microsoft Visual Basic (VB6) using object orientated approach and ActiveX technology, which among other advantages allows Client-Server communication and data exchange via network or internet.
2.1. Single Interferogram Processing
Static interference pattern (interferogram) can be processed by FFT or by direct data fitting. Historically, we have first developed the original direct data fitting algorithm. In this algorithm, the phase at the point of interest is found from direct least-square fit of the fringe signal, i.e. intensity of interferogram vs. pixel number in fringe direction. The fit is made by some modeling function. We successfully used a function of 5 parameters that takes in account the second order curvature of the gauge block (or platen). Further improvement in the method can be made if we also take in account some background fringe intensity variations. We have shown that the uncertainty of the procedure can be as low as 0.1 nm (or 0.0003 in fringe fraction) [4]. The advantage of this method is that it is not sensitive to local phase and intensity variations, and it usually results in repeatable within 0.1 nm read out. The disadvantage of the method is that it can produce some systematic offset if the gauge block is significantly different from an assumed model shape.
We have used a 2D FFT based phase extracting procedure similar, in principle, to the ones developed by several groups and reported before. The basic steps of the processing are as follow. First, 2D FFT is applied to an interferogram image (strictly saying, to the range of interest in the interferogram) to produce 2D FFT transform. Following the literature we will call this transform the spectrum and its indexes the frequencies. Next, the 2D spectrum is filtered to remove image noise and other defects of interferogram. Next, the phase is calculated from reverse 2D FFT result. Finally, phase is unwrapped to remove ±pdiscontinues. In contrast with reported before algorithms, we tried to optimize the phase extraction by constructing an optimum filter for 2D FFT spectrum. Theory of the filtering of the 1D FFT spectrum has been developed for AC electrical signals. It can be shown that an optimal (least-square sense) filter can be constructed as
F(f)=S^{2}(f)/(S^{2}(f)+N^{2}(f)) (1)
where F(f) is the filter function, S(f) and N(f) are the spectra of the informative signal and noise, respectively. To construct the optimal filter we have to estimate noise and signal by some independent means and construct corresponding spectra. For the signal spectrum S(f) we use an estimation given by the Gauss shaped line with carrier frequency f_{0 }at the center. The width of the signal line depends on the bandwidth of the informative signal. This bandwidth is determined by the deviation of the measured object from an ideally flat surface. For practical reasons it is useful to consider some spatial ranges of interest in the topographic surface structure. We will consider two cases: (a) long and mid order structure (from 20 pixels up), and (b) short order (from 1 to several pixels). An optimum filter to extract long and mid order structure we will call the narrow-band filter. An optimum filter for extraction of the short order structure we will call the wide-band filter. For clarity first we describe construction of the filter for fringe direction (X-direction) of the 2D FFT spectrum. A particular line that crossing maximum of the 2D spectrum is shown in Fig.1. We start constructing filter with noise spectrum estimation. The noise estimation is made by a exponential trend to the data outside of the signal at f_{0} . Usually it results in some continuous flat noise spectrum decreasing toward the high-frequency end. Note that the Fig.1 is log scaled. For the short-band filter construction we consider that all the informative signal is carried by components close to the carrier frequency f_{0}. In this case the model signal spectrum is given by Gauss shape of about f_{0}/2 width at the noise level (Fig.1). Wider model signal results in wider filter which does not provide sufficient filtering at lowest frequency components, producing some heterodyning effect. The model narrow-band signal spectrum is labeled as "signal" in Fig.1. The optimal narrow-band filter found from the Eq.1 is shown in Fig.1 as a thick line.
Fig.1 Optimal filter for 2D FFT processor. Slice of the 2D spectrum in the Fringe direction.
To recover short range structure we should also process the high frequency edge. In this case we construct wide-band filter considering the signal band from f_{0} to the high-frequency end of the spectrum. The wide-band filter has the same shape at frequencies below f_{0} and is equal 1 for frequencies above f_{0}.
The optimum filter construction in other coordinate (Y-cut) is made in a similar manner as described above. Noise is approximated by exponential trend, signal is modeled by Gauss shape of same bandwidth. Some improvement in optimal filter construction can be achieved if we take in account the asymmetrical character of signal in X and Y directions of the interferogram. That is a general case as interferometric fringes are always predominant in one of the directions. Because of this the optimal filter is also asymmetrical.
The basics of the phase mapping using phase stepping technique is simple. First, several interferograms are produced, each corresponding to some fixed position of the phase stepper. After images are acquired, the processing is made to find a phase at each pixel. To deduce the phase from the output of each pixel, one has to know the exact phase at each phase step. Usually this is done by independent calibration of the phase stepper. Typical case of phase stepping device is a PZT mounted reference mirror of the interferometer. In this case phase stepping is made by applying some step voltage to PZT. The PZT response has to be independently calibrated in terms of sensitivity (nm/V). The problem of this approach is that PZT sensitivity is nonlinear function with some hysteresis. Also the PZT sensitivity is temperature dependent and not stable it time. The other problem is that purely linear stepping (without yaw and pitch) is quite difficult to achieve. We have tried an alternative approach in phase stepping, which provides some self-calibrating means. In this method, we acquire several arbitrary stepped interferograms into image array, and find phase at each of them by direct fitting of the fringe. Each image of the array contains information phase at any pixel point. This information can be extracted using spatial fringe. We select some reference point (pixel) on first image in the array, and make direct fit for this point using method described above (section 2.1). Note that the fit is done using spatial fringe for this particular interferogram. Repeating the procedure for all images in the array we find phase vs. array number at this particular pixel point. Thus, we have calibrated the array phase steps for this pixel.
It can be discussed how accurate is this calibration. The random uncertainty of this method is limited by linearity and noise of CCD that might be much better than that of PZT. Some indication on the stability of this procedure can be inferred from Fig.2. In this figure fringe fraction of the middle point of the interferogram is plotted vs. array number. Each point is obtained relative to 2 reference side points on interferograms as
F(i)=(F_{1}(i)+F_{2}(i))/2-F_{C}(i) (2)
where F is the fringe fraction,
F_{1},
F_{2
},
and
F_{C}
are the phases of fit at 2 reference points and the center point, i
is the array index. In ideal case, the Eq.2 should produce the same result
independently on interferogram phase. In real case, any inhomogeneity of
the interferometer field and noise of camera will result in some error.
We can evaluate this error by finding the standard deviation to the fringe
fraction F , which for data
presented on Fig.2, is about 0.00052. From this and other results we believe,
that arbitrary stepping can be calibrated with the random uncertainty similar
to that achieved in our previous direct data fitting experiments, i.e.
about 0.2 nm.
Fig.2. Fringe fraction reading stability of direct fitting method. Each point corresponds to phase stepped interferogram of array.
Using approach described above the phase stepped array can be calibrated at any pixel point. To find correct displacement of the object, it is necessary to calibrate steps in at least 3 points. We select 4 pixels on interferogram and calibrate steps on each of them (Fig.3).
Fig.3. Phase stepping technique. Calibration of the movement along interferometer axis, yaw and pitch.
The difference in displacement between pixel points 1 and 2 in Fig.3 gives pitch. Similarly, the displacement difference between points 3 and 4 gives yaw (Fig.3). After the displacement of the measuring plane is found we can reconstruct the phase at any arbitrary pixel of the image array. We do it in a standard way. The experimental data is fitted by cosine function to find the phase. The typical example of calibrated arbitrary stepping for one pixel point is presented in Fig.4. The standard deviation of the measured data from the fit was typically about 1%.
Fig.4 Arbitrary phase stepping for one pixel. Squares are the measured
fringe intensity, line is the cosine fit. Each experimental point corresponds
to one image of the interferometric array. The fringe fraction (X-axis)
is calibrated using our procedure (see section
2.2). Uncertainty of the measured point is about the size of the squares
in both X and Y directions.
2.3 Visualization
Visualization of the topographic surface map reconstructed from interferometric images is important part of the processing. The results of the processing should be easily evaluated and compared using some graphical 3D drawing software.
Suitable 3D visualization software requirements are
We have developed 3D Surface Viewer drawing
program based on OpenGL render. The whole processing including FFT, optimum
filtering, inverse FFT, and 3D surface drawing takes about 5 second for
256x256 images. For typical image acquisition time (5-20 s) the processing
goes real time. After processing is completed 3D surface animation, rotation,
zoom can be performed with the speed up to 10 fps (frames per second) in
true color screen mode.
3. RESULTS
To compare 2D FFT single interferogram processing
with phase stepping technique we have measured the same platen by both
methods. We deliberately selected a platen with mid order defects (some
scratches and oxidation spots) to see if the 2D FFT method will be able
to resolve them. In phase stepping case we used an array of 12 interferograms.
The result is shown in Fig.5 (left). In case of 2D FFT we have used 8 independently
measured interferograms (Fig.5, right). Both methods gave quite similar
shape. We note here that the topographic surfaces measured by both methods
includes plate shape as well as interferometer wave front error, which
are not possible to distinguish without additional experiments.
Fig.5. 3D surface plot of the platen measured with phase stepping technique (left) and optimum filter 2D FFT method. Local scratch on the left side is detected by both methods, as well as oxidation spots. (For scale see the slice plot in Fig.7)
The left side of the 3D surface plot (Fig.5) corresponds to 2D FFT wide-band
filter. Both methods have resulted in similar shape, including mid order
scratch on the left side of the plate. To demonstrate the difference between
two methods we include a differential plot (Fig.6) given by D(x,y)=S_{FFT}(x,y)-S_{STEP}(x,y),
where
S_{FFT}(x,y) and S_{STEP}(x,y) are
the topographic maps measured by 2D FFT and calibrated phase-stepping respectively.
Fig.6 shows both short (left plot) and mid (right plot) order difference
between the methods. Standard deviation of the topographic heights for
left plot is about 1.5 nm over the whole surface, whereas for left plot
it is about 0.6 nm.
Fig.6. 3D surface plot of the difference between data obtained by phase step technique and wide-band filter 2D FFT. Left plot is the short order difference, right plot is the same data smoothed by 5x5 pixels window. Mid order difference (right plot) has standard deviation of about 0.6 nm.
To see clearly the scale of the difference we also
present one line cut (slice plot) of the 3D surface plots of Figs.5
and 6. (Fig.7). We have also compared image processing with results of
direct data fit. Fig.7a shows result of the direct fit, narrow-band filtered
2D FFT and phase-stepping data. It is seen that narrow-band filtering has
resulted in the same mid order surface as phase-stepping method. Long order
given by direct fit is good enough only for some areas of the measured
platen.
Fig.7. 2D slice of the surfaces obtained by phase stepping technique, 2D FFT processing and direct fit. Plot corresponds to one line of the interferometric image. 2D FFT processing is made with optimum filter narrow-band (a), and wide-band (b). The difference between phase stepping and 2D FFT (b) has standard deviation s of about 1 nm.
In Fig.7 we show the slice of the surface data obtained by wide-band filtered 2D FFT and phase-stepping method. Using the wide-band filter it was possible to recover some short order structure. The difference between two methods is shown on the lower part of Fig.7b. We believe that this difference resulted from (i) non-calibrated residual of pixel-to-pixel noise, (ii) noise of electrical circuit of CCD video channel, (iii) digitizer noise. Those sources together in our case produce about 1% of the random noise. From this value we can evaluate the noise limitation of the our hardware. To do this we have to estimate a transfer coefficient between amplitude input noise and output measured phase. We can do it by numerical model of the interferogram with random noise and applying our standard 2D FFT procedure to modeled interferometric image. For the wide-band filter, the noise of 1% results in about 0.1% random phase noise (0.3 nm in terms of length). This value is close to the s value obtained for the data in Fig.7b. Therefore, we consider this difference (Fig.7b) as the random residual noise with the standard deviation of about 1 nm. This value is the random uncertainty in short order structure reconstruction. For mid order structure the random uncertainty is about 0.6 nm. The value is obtained as standard deviation of the topographic difference smoothed by 5x5 pixels window. The corresponding 3D surface plot is presented in Fig.6. The results indicate that the mid order topography of the platen can be measured with sub-nanometer random uncertainty.
4. CONCLUSIONS
Comparison of the 2D FFT processing with data of
phase stepping measuring technique shows good sub-nanometer agreement for
extracting long and mid order surface heights. If the least-square optimum
filter of wide-band is used in 2D FFT method, the short order (1 pixel)
heights of the measured object can be found with random uncertainty of
about 1 nm. The standard deviation of the 2D FFT data relative to the phase-stepped
reference data is found to be about 0.5 nm for mid order (5x5 pixel smoothed)
surface heights. Even moderate performance hardware (CCD and PC) when combined
with optimum 2D FFT processing allows real time 3D surface reconstruction
with random uncertainty lying in sub-nanometer range. Possible applications
of the methods discussed: (i) accurate to sub-nanometer level gauge block
measurements, (ii) step gauges measurement, (iii) interferometric characterization
of optical parts, (iv) shape measuring with fringe projecting technique.
We tried to use general terms in accordance with Academic Dictionaries and WIKI. Nevertheless some terms are also in use by other quantities, such as surface map is in use for weather map, etc. To avoid misunderstanding we include meanings of some terms used above.
1D, 2D, 3D | one-dimensional, two-dimensional, three-dimensional |
3D surface | graphical representation of the surface in space |
FFT | Fast Fourier Transform |
PZT | piezo-ceramic transducer |
gauge block | length standard, definition is given by ISO 3650, see appendix |
platen | auxiliary plate required for gauge block measurement as defined in ISO 3650, see appendix |
profile | 2D graphical representation of the surface heights along some line in surface plane (also called slice of surface) |
surface topography | configuration of a measured surface, microscopic surface features |
contour | line of equal height on 3D surface |
orthogonal slicer |
2D viewing facility of software, graphical representation of profile (slice of the surface) |
interferometric
image |
interferogram, interference pattern, 2D image taken from optical interferometer, see shots |
fringe | interference pattern of alternating light and dark bands that arises when homogeneous light beams overlap |
phase | phase of interference pattern that contain object length information |
phase map | set of data calculated from the interference pattern, which defines phase vs. point location |
phase unwrapping | removing discontinuities from the phase map |
topographic map | digital elevation map, map of
the surface heights, calculated from unwrapped phase map,
microscopic surface features |
phase stepping | phase measuring technique that dinamicaly determines phase by introducing some phase shift into intrferometer (also known as phase shifting) |
OpenGL | low level graphical library from Silicon Graphics, Inc. (included into Win98 and later) |
VB6, VB.NET | Visual Basic, programming language from Microsoft Corporation |
ActiveX | Microsoft technology that among other things provides Server-Client communication between objects/programs, see also COM |
Gauge Block length is defined by International Standard ISO 3650:1998(E) Geometrical Product Specifications Length standards Gauge blocks
Length of a gauge block is the perpendicular distance between
any particular point of the measuring face and the planar surface of an auxiliary
plate of the same material and surface texture upon which the other measuring
face has been wrung (ISO 3650:1998(E) definition)
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