Extracting Group Velocity Dispersion values using quantum-mimic Optical Coherence Tomography and Machine Learning | Scientific Reports

Signal to noise ratio

The neural networks were trained using datasets comprising of signals with different levels of noise: 45, 35, 30 and 25 dB and tested on experimental data in order to assess which noise level is most optimal. To give a sense of what each noise level looks like in a training signal, example spectra were generated for a single-interface object and with the maximum modulation depth. They are presented in Fig. 3e, g, i and k, with Fig. 3c showing a spectrum with no noise.

Figure 3

(a) One of the experimental spectra acquired for a 460-μm thick sapphire. (b) A B-scan showing a tilted sapphire glass. The vertical line represents autocorrelation peaks. (f, j, d, h, l) Dispersion maps of the sapphire glass obtained with a network trained with signals containing: no noise, 45, 35, 30 and 25 dB noise. All the dispersion maps are affected by the presence of the autocorrelation peaks: the network treats them as structural peaks and assigns GVD levels to them. I–IV mark four areas that can be discerned as a result. (c, e, g, i, k) Example training spectra showing what the simulated noise levels look like. (m) The 30th dispersion profile from all five predicted dispersion maps.

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The experimental dataset, obtained using the laboratory OCT system, contains 275 FFT stacks calculated from 275 experimental spectra that were obtained while transversely scanning a 460-μm thick sapphire glass. The spectra – one of which is shown in Fig. 3a – are linearised to remove the nonlinearity introduced by the spectrometer, filtered with a Gaussian window, and then Fourier transformed. The resulting A-scans are stacked one on top of another to form an XZ image called a B-scan presented in Fig. 3b. Because the sapphire was placed at an angle to the light propagation direction, the front and back surfaces of the sapphire are represented by two tilted lines. The vertical line located near the second sapphire surface corresponds to autocorrelation peaks. The autocorrelation peaks are inherent to OCT and originate from the interference of light reflected from the internal structures of the object. Because sapphire is a two-interface object, every A-scan contains one autocorrelation peak placed at a distance equal to the optical thickness of the sapphire.

The sapphire dataset is processed with a network trained with noiseless data and networks trained with noisy data. Each FFT stack from the dataset is transformed by a network into a dispersion profile. All the dispersion profiles are stacked one on top of another to form a dispersion map. Such predicted dispersion maps are shown in Fig. 3d corresponding to “noiseless” training and Fig. 3f, h, j and l corresponding to training with the noise levels of 45, 35, 30 and 25 dB, respectively.

Since the autocorrelation peak is treated by the neural network as a structural peak (as discussed in23, the dispersion profiles contain four distinct areas (marked with yellow I–IV in Fig. 3f), instead of expected three:

1. 1.

   Area I starts at 0 optical distance and ends at the location of the first interface. The GVD in this area should be equal to the GVD of the dispersion imbalance in the interferometer, \( \beta _{NL,\text {front}}^{(even)} \).

2. 2.

   Area II starts at the location of the first interface and ends at the location of the autocorrelation peak. The GVD in this area, \( \beta _{NL, 2}^{(even)}\), depends on the GVD of the first area and the object GVD, \( \beta _{NL,obj}^{(even)} \)23

   $$\begin{aligned} \beta _{NL, 2}^{(even)} = \frac{ L_{\textrm{front}} \beta _{NL, \textrm{front}}^{(even)} – L_{obj} \beta _{NL, obj}^{(even)} }{L_{\textrm{front}} – L_{obj}}, \end{aligned}$$

   (1)

   where \( L_{\textrm{front}} \) is the distance between 0 optical distance and the first object interface, and \( L_{obj} \) is the object thickness.

3. 3.

   Area III, situated between the autocorrelation peak and the second interface, should have the same GVD as area I.

4. 4.

   In the fourth area, area IV, located behind the second interface, the GVD is equal to 0 fs\(^2\)/mm.

All of the predicted dispersion maps contain all four areas, with areas I and III having a similar GVD value. It can be noticed that the more “noisy” the training, the better the predictions: the GVD values within all four areas become more uniform and the areas themselves smoother, with the smoothest map being the output of the 30dB network. To see that improvement on the level of a single dispersion profile, the 30th dispersion profile of the five predicted dispersion maps is plotted and shown in Fig. 3m. Also, in the predicted dispersion maps for 35dB, 30dB and 25dB networks, three areas are detected, instead of four, on the bottom of the maps: one between 0 optical distance and the first interface, one between the interfaces, and the third one behind the second interface. This is due to the fact that at these noise levels, the network treats the autocorrelation peak as noise and assigns GVD levels to just two peaks (the interface peaks), instead of three (the interface peaks and the autocorrelation peak).

The network trained on signals with signal-to-noise ratio (SNR) equal to 30dB was chosen to be used further, as it visibly shows the best results: the dispersion map is the “smoothest”. More detailed and parameter-focused analysis and comparison of the performance of the model trained with datasets corresponding to different SNR levels are presented in Section 3 in Supplement 1.

Resolution mismatch

It was tested how the neural network predictions are affected if the axial resolution of the input FFT stacks used for training, 4.08 μm, does not match the axial resolution of the OCT system.

FFT stacks for a three-interface object with GVD of −1500 fs\(^2\)/mm and 1000 fs\(^2\)/mm and corresponding to different axial resolutions, 2.72 to 8.16 μm, were generated and processed with the neural network. As shown in Fig. 4a, the predicted GVD level changes proportionally to the axial resolution: the smaller the resolution value, the smaller the predicted GVD and the bigger the resolution value, the bigger the predicted GVD.

Figure 4

Predicted dispersion profiles for a three-interface object with GVD of -1,500 fs\(^2\)/mm and 1,000 fs\(^2\)/mm corresponding to (a) different OCT system resolution values show the network’s potential for universal use if the resolution mismatch is accounted for. Predicted dispersion profiles in blue, with corresponding A-scans in orange, for (b) one 120-μm thick quartz glass (c) two 120-μm thick quartz glasses show similar GVD levels inside the objects and in front of the objects.

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The obtained predictions show that the network could be used with data acquired with an OCT system whose axial resolution does not match the ones used for training. In the presence of mismatch, the results are qualitative, and the influence of a mismatched resolution could potentially be corrected by applying a proportionality constant to get a more quantitative representation.

Two kinds of objects were imaged with the commercial system for which the mismatch between the experimental resolution (5.5 μm) and the simulated resolution (4.08 μm) is present: one 120-μm thick quartz glass and two 120-μm thick quartz glasses of the same kind (A-scans presented in Fig. 4b and c in orange). The predicted dispersion profiles (Fig. 4b and c in blue) show similar GVD levels for the inside of the glasses, around 25 fs\(^2\)/mm, and between 0 optical distance and the location of the first object interface, 680 fs\(^2\)/mm. The GVD levels are not constant within the corresponding areas and show rapid changes at boundaries – errors being the current limitation of our approach. However, the GVD levels heights look fairly consistent in the two dispersion profiles: the GVD level for the inside of the quartz glasses and around 0 optical distance are similar in both dispersion profiles.

GVD levels variability within the object

The neural network was tested in terms of prediction limitations with computer-generated signals representing three-layered objects for which all layers are of equal thickness and where each layer has a different GVD value (Fig. 5). The graphs in Fig. 5 labeled with 1 are the ground truth dispersion maps with 138 dispersion profiles each. The consecutive dispersion profiles in a dispersion map represent an increasing thickness of the object layers: 4 μm thickness of all three layers for the 0th dispersion profile and 306 μm thickness of all three layers for the 137th dispersion profile. The graphs in Fig. 5 labeled with 2 are predictions.

The first row in Fig. 5 shows the most realistic situations in the terms of the GVD values and their layer-to-layer variability in the object. The GVD values differ by 10 and 5 fs\(^2\)/mm between the layers of the object, resulting in: 50, 40, 45 fs\(^2\)/mm in Fig. 5a, 110, 100 and 105 fs\(^2\)/mm in Fig. 5b, and 1010, 1000 and 1005 fs\(^2\)/mm in Fig. 5c. The variability of GVD levels is higher in each consecutive row: in Fig. 5d, e the difference is 100 and 50 fs\(^2\)/mm, in Fig. 5g–i 200 and 100 fs\(^2\)/mm, in Fig. 5j–l 400 and 200 fs\(^2\)/mm and in Fig. 5m–o 1000 and 500 fs\(^2\)/mm with the minimum levels of GVD in each row being 100, 500 and 1000 fs\(^2\)/mm (corresponding to the middle layer).

Figure 5

(a1–o1) Ground truth dispersion maps, (a2–o2) Dispersion maps predicted using the 30 dB neural network. The titles of each pair of images state the GVD values of the layers of the object: the values increase from left to right in each row and the difference of GVD values within the object increases from top to bottom. The greater the variability of GVD values within the object, the better the predictions.

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The predictions of dispersion maps of objects with the least GVD variability fail to reproduce the correct values, but still show sharp changes at the locations of the interface, and therefore enable one to distinguish the layers. The larger the GVD variability is within the object, the better the predictions are, even for thicknesses as small as couple of micrometers. This behaviour is not surprising. Smaller layers and/or smaller GVD do not induce a change of the artefacts in the FFT stack big enough for the neural network to pick up, on the other hand, thicker layers and/or bigger GVD introduces a greater change in the appearance of the FFT stack making it much easier for the neural network to interpret.

A more quantitative and parameter-oriented analysis is found in Section 4 in Supplement 1.

GVD value predictions for sapphire and BK7

The neural network trained on signals with an SNR of 30 dB is used further to test if it is able to correctly determine the GVD values from the experimental data. Using the laboratory OCT system, we acquired 300 A-scans at one position for the 460-μm thick sapphire glass and a 1000-μm thick BK7 glass. Such A-scans, when stacked one on top of another, form an M-scan. The M-scan for the sapphire is shown in in Fig. 6a and the M-scan for the BK7 in Fig. 6e – in both cases the autocorrelation peaks were removed from every A-scan.

Figure 6

Sapphire and BK7: (a, e) an M-scan containing 300 A-scans with the autocorrelation peaks removed, (b, f) dispersion profiles corresponding to each A-scan in the M-scan, (c, g) dispersion profiles for which the standard deviation (STD) calculated from the GVD values in the green rectangle is the smallest, the inset is a closeup of the area in the plot marked with the green rectangle, (d, h) is the dispersion profile obtained by taking the mean of all the dispersion values, again the inset is a closeup of the area marked with the green rectangle. The twice bigger length of the x axis in the dispersion maps and dispersion profiles is a by-product of the FFT stack generation.

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The corresponding GVD levels predictions are shown in Fig. 6b and f. The GVD value for sapphire at 840 nm is 53.2 fs\(^2\)/mm and the GVD value for BK7 is 40.98 fs\(^2\)/mm. Consequently, the GVD level in the area between the object interfaces for both the sapphire and BK7 should be very close to 0, resulting in light red colour in the dispersion map. That area in the obtained dispersion maps is predominantly light blue indicating that the GVD values are predominantly negative, which may be due to the reversed order of interfaces in the A-scan. When the order of interfaces is reversed in the A-scan, the first peak represents the back-surface of the object and the second peak represents the front surface. In such a situation, it is the first peak whose width is changed due to layer GVD which means that the accumulation of dispersion happens in the opposite, negative direction. Also, the GVD values in that area become positive at some point confirming what was established in the previous section: for smaller GVD values the network loses accuracy.

In Fig. 6c and g, we see dispersion profiles for which the standard deviation calculated from the GVD values in the range marked with the green rectangle is the smallest. Consequently, these dispersion profiles show the most “constant” GVD levels in all the acquired data. One can see in the bottom inset which shows the area inside the black rectangle, that still, the most “constant” GVD level varies in its height. The mean GVD value in such a case is calculated to be −46 ± 26 fs\(^2\)/mm for sapphire and −29 ± 27 fs\(^2\)/mm for BK7. As expected, both values are negative and burdened with a big error.

Figure 6d and h show a mean dispersion profile calculated from all the dispersion profiles on the corresponding maps. The light orange areas represent the standard deviation. The range of GVD values in the bottom inset, which shows the area inside the second green rectangle, is again broad. The mean of all the GVD values from all the dispersion profiles is calculated to be −50.3 ± 33.6 fs\(^2\)/mm for sapphire and −20.0±39 fs\(^2\)/mm for BK7. Again, as expected, both values are negative and burdened with a larger error than in the previous case.

In both cases, the mean GVD value is very close to the literature value for sapphire. For BK7, the mean values are further from the literature value, but remain within the calculated error.

Dispersion maps of a grape and cucumber

The dispersion maps are produced with the same neural network (trained with signals with SNR of 30dB) for biological objects: a grape (Fig. 7a), imaged with the laboratory OCT system, and a cucumber placed on a 120 μm thick cover glass, imaged with the commercial OCT system.

Figure 7

(a) Dispersion map of the grape, (b) the B-scan where the dispersion imbalance in the interferometer is not compensated. (c) An image for which the unbalanced dispersion was compensated. The twice bigger length of the x axis in the dispersion maps is a by-product of the FFT stack generation.

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Figure 8

Dispersion maps of a cucumber obtained using the neural network trained with signals with (a) SNR = 30 dB, (b) and SNR=45dB show the boundaries of the imaged object, but the predicted GVD levels remain random. (c) The image of the cucumber. The twice bigger length of the x axis in the dispersion maps is a by-product of the FFT stack generation.

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The predicted dispersion map of the grape shows a positive GVD area in front of the object, but the GVD levels for the inside of the grape remain random. Fig. 7b shows a B-scan obtained by Fourier transforming spectra used to calculate FFT stacks for the neural network: these spectra were linearised to only remove the nonlinearity coming from the spectrometer. In Fig. 7c, a B-scan is shown for which the spectra were fully linearised: the nonlinearities introduced by both the spectrometer and the unblalanced dispersion were removed.

The image of the cucumber (Fig. 8c) has a visibly lower SNR than the image of the grape. Since the intensity of both images are normalised, the comparison of the maximum values on the colourbars gives an indication on the difference in the level of noise with respect to the signal. For low-SNR input data, the dispersion maps, either obtained with the network trained with low-SNR signals (Fig. 8a) or high-SNR signals (Fig. 8b), are only able to distinguish the boundaries of the cucumber and the front of the cover glass on which the cucumber slice was placed on. For some A-scans, brighter elements inside the cucumber lead to the appearance of more dispersion levels in the dispersion profiles in those areas. However, the obtained values remain random.