Data Analysis for the Surface Topography

The internal structures of a sample are distinguishable with the FF-OCT tomographic imaging. The locations of internal interfaces can be clearly revealed by the envelope signal in the A-scan signals or B-scan images. The layers distribution underneath the sample surface can be evaluated.

The OCT depth-resolution was measured as the FWHM width of a main envelope signal (at sample surface) or the OCT axial PSF profile. This is based on the fact that the spatial resolution of an imaging system is characterised by the PSF, which describes the response of the imaging system to a point object. An envelope signals representing the axial PSF in the simple FF-OCT system is derived from the acquired interferogram with well-separated

Fig. 3.5 GUI for browsing the FF-OCT B-scan images.

over-sampling points. They have a perfect fidelity and their peak positions can reach a sub-micrometre precision. The axial resolution is one of the inherent characteristics of an imaging system. It is determined only by the width of the axial PSF, which stays at the micrometre scale in this simple FF-OCT system.

For the OCT tomographic imaging, the location of a sample interface is measured by the position of the highest sampling point of the axial PSF (envelope). Alternatively, the position of the strongest sampling point of the zero-order beat cycle of the interferogram can be taken as the location of the interface, as shown in Fig. 3.6. Since the sampling points in the originally acquired interference signal are distinct in terms of the axial position and the intensity, the position deviation of the strongest sampling point from the true peak point (best-focus or highest contrast position) should be within the sampling depth interval. In the case of more than 16 sampling points per beat cycle in the interferogram, the theoretical depth interval is less than 25 nm, which indicates the theoretical imaging precision in the axial direction. Moreover, interpolation between sample points can be used to further increase the imaging precision beyond the sampling interval [146]. In practice, the imaging precision is limited by the systematic noise and sampling spacing (or step size) errors of instruments.

Fig. 3.6 (a) Zero-order beat cycle in a typical measured interferogram, (b) the zoomed-in central interferogram with the highlighted strongest sampling point and true peak point.

b. Determination of true peak position

The data analysis with the envelope detection for tomographic imaging may neither have sufficient precision to detect nanometre scale structural elements, nor be able to differentiate interfaces of optically thin layers, as the resultant A-scan signal could have connected / combined envelopes, of which the maxima are not well separated. By searching for the maxima positions of the central beat cycles, both the determination of the areal surface topography and the investigation of optically thin layers with nanometre scale precision are allowed. The same FF-OCT experimental procedures were followed to acquire a 3-D data cube. The number of sampling points per beat cycle was sufficient to permit a good fidelity in the reconstruction of an interference signal. Low noise is required in the measured OCT signals, so that the assumption of local linearity of the sampling interferogram is valid.

By using this method, the surface height was obtained, based on the true peak separation of the measured and the reference signals. Usually the highest contrast interference signal extracted from the acquired 3-D data cube was chosen as the reference signal.

Interpolation is a method of constructing new data points based on a discrete set of known data points. With the spline interpolation and a valuea, a new interferogram can be reconstructed (interpolated) with a displacement ofafrom the old interferogram. This is equivalent to the shift of the interferogram from its initial axial positionz_sto a new position

z_s−a(Fig. 3.7 (a) see below). It is also possible to derive the displacement with known raw and shifted interference signals.

A minimum search algorithm was used with the spline interpolation to find displacement

a, so that the new interpolated interferogram and the reference signal can be approximate. The searched minimum displacementais obtained by using the Nelder-Mead simplex algorithm

Fig. 3.7 Schematic diagram of position search with interpolation. (a) A zero-order interferogram cycle in dark blue with the peak at z_s and its interpolated interferogram cycle in light blue with the peak at zs−a (their peak displacement isa); (b) a reference zero-order cycle in green with the peak atz_r,Nmeasured cycless₁,s₂,s₃... s_Nwith respective peak displacements ofa₁,a₂,a₃... a_N from the reference peak

[216, 217]. This algorithm is a direct search optimisation method which works by minimising an objective function: ∆2= n

∑

i=0 [I₀(z_s[i])−b·I_OCT(z_s[i]−a)]2,i=0,1,2, ...,n. (3.17) in which∆2is the error metric that is minimised in the minimum search algorithm,Ire f is the selected reference interference,I_OCT is the OCT measured signal at pixel(p_x,p_y),ndenotes the number of sampling points in the interferogram,zs[i]is the axial position at point number

i, and the coefficientsaandbrepresent respectively the axial shifting and the magnification of the OCT sample signal. Note that ais the searched displacement value as well as the desired surface height. Two commands “interp1” and “fminsearch” were used to implement the interpolation technique and the minimum search algorithm.

All interference signals of the measured 3-D data cube were processed with this method to find the displacementsa₁,a₂,a₃... a_N for cycless₁,s₂,s₃... s_N from the reference signal, as demonstrated in Fig. 3.7 (b). The resultant displacement values were taken as surface heights of measured interference signals relative to the reference peak position. They were stored as a 2-D topographic image, in which the surface height of a pixel was indicated by the scale of colour. The image can be used to interpret the surface topography by the colour variations. It is also possible to depict a 3-D representation of the surface on which superficial characteristics can be directly distinguished.

c. Phase-Unwrapping

The phases of the measured sampling points are inherently ambiguous outside of a 2πrange; the variations of surface heights acquired from the minimum search algorithm lie within a range of λ0

2 corresponding to a single beat-cycle [210]. To measure height variation larger than this limited range, the fractional phases of measured sampling points have to be unwrapped to retrieve height information.

The wrapped phase can be corrected (unwrapped) by adding or subtracting 2π to the phase of one of the data points so that the phase difference is less thanπ. The unwrapping of phases needs to be repeated for all data points until the phase difference between all adjacent data points is less thanπ [218]. It is more complicated to fix the wrapped phases for a 2-D topographic image.

A 2-D phase-unwrapper based on a region-merging principle [219, 220] was adopted in the 2-D phase-unwrapping process and corrected surface heights were obtained with wrapped phases eliminated. Fig. 3.8 demonstrates the phase-unwrapping of an example topographic image in Fig. 3.8 (a) with wrapped phases. Note that there still exists a residual wrapped phase shift in the topographic images in Fig. 3.8 (b) and (c), suggesting that a more robust unwrapping algorithm might be needed for more accurate measurements.

A flowchart presented in Fig. 3.9 summarises the data analysis with both tomographic imaging and surface topography of a 3-D data cube acquired with the simple FF-OCT system. In tomographic imaging, the FF-OCT data analysis is to illustrate sample internal structures. B-scan images and A-scan signals are obtained by the determination of peak positions of envelopes of FF-OCT signals. In surface topography, the FF-OCT data analysis is to describe surface characteristics. Surface maps are obtained by the determination of peak positions of zero-order interferogram cycles.

(a) (b) (c) Height [nm] Height [nm] Height [nm] 50 -50 0 -50 0 -100 -50 0 -100

Raw Phase-unwrapped Tilt-corrected

Fig. 3.8 (a) An example topographic image with wrapped phases, (b) the phase-unwrapped topographic image, (c) the tilt-corrected topographic image.

Tomography Imaging Surface Topography Signal Smoothing Envelope Detection Interpolation Minimum Search Phase Unwrap 2-D and 3-D Surface Maps B-scan Images A-scan Signals FF-OCT 3-D Data Cube Determination of Peak Position of the Envelope of Interferogram Determination of Peak Position of

the Zero Order Interferogram Cycle

Fig. 3.9 Flowchart of FF-OCT data analysis with tomographic imaging and surface topography.

In document Development of a simple full field optical coherence tomography system and its applications (Page 83-88)