Signal processing improvements

This section discusses the possible alternative methods of signal processing to achieve improved range, resolution, measurement rate and simulation accuracy. The signal processing methods shown here have had initial investigations performed with simulated interferograms but the effects of real interferograms on these approaches have not yet been explored fully.

8.3.1 Virtual reference interferometry

The range of DRI spectral interferograms for absolute position measurement is limited by the ability of autoconvolution to resolve the position of the interferogram point of symmetry as previously illustrated in figure 3.12. As the d value changes and the interferogram point of symmetry approaches the edge of the detector, the autoconvolution result distorts until a clear peak is no longer present.

In the case of virtual reference interferometry (VRI), the author describes spectral interferograms having a visible point of symmetry as balanced spectral interferograms (BSI) and interferograms having an increasing frequency with wavenumber but no point of symmetry are described as unbalanced spectral interferograms (USI) [79, 80]. In the case of VRI, range is extended by point-wise multiplication of a unbalanced spectral interferogram (USI) with a simulated USI of opposing imbalance has been shown to result in amplitude modulation of the interferograms from which a balanced spectral interferogram (BSI) may be obtained after low pass filtering of the amplitude modulated signal.

Figure 8.2: Simulated spectral interferogram for an OPD offset of +500 μm

Initial Matlab simulations using VRI for DRI suggest that this method is applicable if a simulated USI of known d is used. The resulting BSI offset allows calculation of the total distance from the balance point.

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Take the example interferogram shown in figure 8.2 for which convolution with itself does not yield an autoconvolution vector with a resolvable peak. However, point-wise multiplication of the interferogram for d=500 μm by an interferogram with a value of d= −500 μm results in a valid autoconvolution result.

This is shown in blue in figure 8.3. Also seen in figure 8.3 are autoconvolution results for pointwise multiplication of d=520 and d= −500 (green) as well as d=540 and d= −500 (red). This shows that it may be possible to extend DRI range further than previously expected by multiplication of unbalanced spectral interferograms by a spectral interferogram with an opposite sign to the OPD, a method previously described as virtual reference interferometry. Further work to verify this method with measured interferograms remains to be performed.

Figure 8.3: Simulated interferograms having OPDs of 500 (blue), 520 (green) and 540 (red) μm are point-wise multiplied by an interferogram with an OPD of -500 μm. This allows determination of absolute position for an interferogram whose autoconvolution normally does not result in a useful peak.

8.3.2 Fast Fourier transform methods of phase extraction

FFT based methods of phase extraction have been demonstrated previously for interferometry in the case of FFT profilometry as well as determination of phase in spectral interferometry [112].

This section details observations from initial investigations into FFT based phase extraction for DRI, with a proposed method outlined, initial results displayed and possible advantages and disadvantages discussed.

The following flow diagram shows the proposed signal processing flow for FFT calculation of DRI phase:

Figure 8.4: Flow diagram showing initial signal processing steps to retrieve phase information from DRI interferograms using FFT.

Starting with an interferogram, H, of length m, whose pixels are indexed by the variable i, the FFT of the interferogram is calculated.

Band-pass filtering is achieved by setting lower and upper ranges of the FFT values to zero. The upper and lower frequency cut-offs are f and _U f respectively and can be values between 1 and _L m depending on

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the desired bandwidth to pass. For the purpose of this initial simulated attempt, f and _U f are set to 10 _L and 6000 where m is 8192. The IFFT of this data is a vector, H′, containing real and imaginary components.

The complex logarithm of H′ is taken before finally the imaginary component of the signal is separated from the real component. This results in the instantaneous phase for each pixel of the original interferogram, varying from –π to π. This result is shown in Figure 8.5 where the frequency of the phase wraparounds can be seen to reduce towards the middle after which they increase again.

Figure 8.5: Instantaneous phase across the 8192 pixels of a simulated interferogram calculated by FFT methods.

Figure 8.6 demonstrates how the phase of a single pixel of the interferogram changes, with this graph showing the phase at pixel 6500 (blue), 6600 (red) and 6700 (green) as the simulated interferogram OPD is incremented by 1 nm steps over a 1000 nm range. It is expected that if necessary, phase shifting algorithms can be implemented with this data to account for the changing frequency of the wraparounds with wavelength and measurement mirror translation.

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Figure 8.6: Instantaneous phase for a single pixel over 1000 nm for three different pixels (6500 (blue), 6600 (green) and 6700 (red)) separated by 100 pixels.

Also potentially useful is the unwrapped phase of the interferogram, as shown in Figure 8.7. It is likely that this will suffer from edge effects as mentioned by Hlubina et al. in their 2001 paper on FFT analysis of spectral interferograms [112]. Also warranting further investigation is the unwrapped instantaneous interferogram phase (Figure 8.7) and use of FFT for real phase, an initial example of which is shown in 8.8.

Of particular interest is the much reduced computational overhead of FFT-based phase extraction when compared to template matching, with an i7 CPU able to calculate phase by FFT at 220 Hz compared to 15-25 Hz for template matching.

Figure 8.7: Unwrapped instantaneous phase

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Figure 8.8: Phase of real interferograms as 40 nm OPD changes were introduced using a PZT upon which the measurement mirror was mounted.

An inherent problem with this method which must yet be explored is the changing frequency of the instantaneous phase wraparounds across the width of the detector. For this reason it may be necessary to calculate instantaneous phase as described above and then apply a phase shifting algorithm such as the 7-point phase shifting algorithm described for spectral interferograms by Cohen-Sabban [50].

8.3.3 Noise immunity of DRI signal processing methods

The range and resolution of autoconvolution and template matching for ideal simulated interferograms are considered in sections 3.5 and 4.4 respectively. This is followed in sections 3.6 and 4.5 with use of these signal processing methods for non-ideal, measured interferograms containing features such as noise, imperfect visibility and low frequency envelopes. Future application of the DRI measurement method to measurands having higher surface slopes and optically rougher surfaces will lead to cases where the range and resolution begin to deteriorate due to poor interferogram visibility caused by reduction in measurement arm intensity. It is important to understand the extent of the immunity DRI has to these changes. Addition of Gaussian white noise and changes to interferogram visibility are difficult to control in a repeatable manner experimentally and so simulation of these factors is an attractive method to study their effect on DRI range and resolution more completely.

Figure 8.9 shows such a simulated interferogram with the addition of white noise, DC offset, reduced visibility and low frequency envelopes. The impact of the signal-to-noise ratio resulting from the noise and visibility are of particular interest due to their strong link to lower visibility samples and fibre polarisation and the effect this will have on their measurement.

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Figure 8.9: Simulating interferogram with addition of white noise, DC offset, reduced visibility and low frequency envelopes.