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Another advantage of both simulation with Matlab and the use of the PZT in the experimental setup is that it is possible to change the OPD by increments smaller than the resolution of the DRI autoconvolution method. While confirming previously published works in sections 3.1 to 3.7 using a bulk optics interferometer it was not immediately apparent that with small increments of OPD the autoconvolution peak does not move linearly across the detector pixels as expected. This means that while the linearity across the entire range is as described in section 3.6.3, the resolution is lower than anticipated. This behaviour was first observed while using the PZT within the experimental interferometer, with figure 3.27 illustrating the oscillatory nature of data, where the blue and red traces show the interferogram centre position by max() function and peak fitting respectively.

The rather significant impact of the observed non-monotonicity is that until it is resolved, resolution of DRI autoconvolution is lower than the previously stated 279 nm. This effect is not observed or explained in existing spectral interferogram literature.

Figure 3.27: Graph showing calculated interferogram centre position change with measurement path length change. Pixel with highest autoconvolution result (blue) and sub-pixel result using peak fitting (red).

This is an excellent example of where having simulation and real measurement data whose results match well is highly advantageous to solving of physical problems. Upon discovery of this effect, it was not known if this was an error with the PZT, errors within the physical setup or a problem inherent with the chosen signal processing methods.

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Important to correction of this oscillation is diagnosis of the cause. Through simulation with Matlab it can be determined if this non-monotonicity is a problem inherent with the signal processing methods used or if it is solely present in the real experimental data.

3.8.1 Non-monotonicity diagnosis through simulation

Using the discrete implementation equations described in section 3.5 allows generation of interferograms that closely match interferograms retrieved from the detector of the experimental bulk optics interferometer. Optical path difference is varied by changing the d term in equation 3.18. The simulated OPD is changed in 20 nm steps and the interferogram symmetry position calculated by max() function as well as peak fitting with the results displayed in 3.28 in blue and red respectively. The oscillations shown in the simulated data match the oscillations observed in the real data from figure 3.27. This demonstrates that the oscillations are inherent to the signal processing methods and are not an environmental or physical effect.

Figure 3.28: Autoconvolution result calculated with max function (blue) and peak fitting (red) for a range of simulated interferograms whose OPD changes in 20 nm increments over 3.5 µm.

3.8.2 Correction of simulated interferograms

With knowledge that the oscillations are inherent to the method, development of appropriate signal processing methods to limit these oscillations and return the absolute position encoding result to monotonicity is possible. Use of simulated interferograms for development of appropriate signal processing methods was chosen due to the lack of environmental effects (noise, DC offset, gain envelopes) on the interferograms as well as the quick turnaround to evaluate new methods.

The cause of the oscillations is hypothesised to be the changing intensities of the interferogram at the edge of the detector. As the instantaneous phase of the spectral interferogram changes while point of symmetry traverses the detector, the intensity at each edge of the detector varies at a different rate due

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to the increasing frequency of the interference as distance from the stationary phase point increases. This is supported by the fact that while the DRI OPD is zero the interferogram is centred on the detector for which the intensity at the edges of the detector changes at the same rate. For a case such as this the non-monotonicity is not apparent for approximately 2 µm either side of 0 µm OPD.

As a first attempt to eliminate these oscillations for simulated interferograms, a Hanning window is applied to make the intensity near the edges consistent. A Hanning window is chosen since the intensity tends to zero at the edges, unlike a Hamming window which never reaches zero. Equation 3.23 details discrete generation of a Hanning window of length N.

(

)

H =

w  

wherew_H 0.5 1 cos 2

ψ N

πψ

  

=  −  

 

  and index, ^ψ∈

[

]

3.23

Figure 3.29: Comparison of a Hanning window (blue) with a Hamming window (red)

The Hanning window is next split so that the edges of the detector are attenuated to zero while the centre remains unmodified which allows correction across the entire range of the interferometer. The split Hanning window, w , of length m is described as follows, with equation 3.24 re-indexing the _S Hanning window from equation 3.23 with a graph of the resulting window shown in figure 3.3:

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Figure 3.30: Split Hanning window described by equation 3.24.

This window,w , and the simulated interferogram, H , are then multiplied element by element as _S described in equation 3.25. This results in the vector J containing the windowed interferogram as shown in red in figure 3.31:

Figure 3.31: Original (blue) and Hanning windowed (red) simulated interferogram.

When this windowing operation is performed on multiple interferograms of increasing OPD, the effectiveness of windowing for centre position oscillation can be demonstrated. The blue trace in figure 3.32 is calculated from interferograms multiplied by the stretched Hanning window before

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autoconvolution is calculated. It shows much improved monotonicity when compared to the entirely non-monotonic red trace for autoconvolution of non-windowed interferograms. It can be demonstrated that this improvement in autoconvolution result is present across the entire 300 µm range of the simulated interferometer.

Figure 3.32: Autoconvolution fitted peak position for unmodified (red) and Hanning windowed (blue) series of interferograms.

Difficulties with this method are made apparent when a comparison is made between the simulated interferogram and a measured interferogram. Interferogram features such as DC offset, gain envelopes, noise, skew and distortion all exist in measured but not simulated interferograms. These can be seen on the left in figure 3.33.0.

Application of the stretched Hanning window to a set of measured interferograms does not result in correction of the non-monotonicity due to the differences between the measured and simulated interferograms. For this reason it is necessary to apply an additional signal processing step to measured interferograms before correction with windowing. This step to correct spectral interferogram shape is well described in the literature [97-101] and named regularisation and normalisation interchangeably.

Figure 3.33: A real interferogram (left) and simulated interferogram (right) to aid visualisation of their differences.

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3.8.3 Effect of regularisation on calculated centre position of real interferograms

A regularisation algorithm tailored to use with spectral interferograms is implemented in appendix 11.2.

Use of this algorithm before application of the Hanning window has been demonstrated to substantially reduce the non-monotonicity of the measured autoconvolution result. The data presented in figure 3.34 demonstrates this effectiveness, showing the autoconvolution result of autoconvolution of unmodified interferograms (blue) and the autoconvolution of interferograms corrected with regularisation and a stretched Hanning window. To obtain this data, interferograms were captured as 20 nm OPD increments were applied over a 20 μm range. This value was chosen for increments as being far below the current resolution of DRI.

Figure 3.34: Comparison of autoconvolution of unmodified (blue) and windowed and regularised (green) interferograms.

Elimination of non-monotonicity is obvious, improving the effective resolution of autoconvolution.

The windowed and regularised green trace shows far reduced oscillations and while there does appear to be small non-linearities it can now be said to be monotonic. This has been confirmed over the range of the interferometer.