1. Introduction

Quantum optical coherence tomography (Q-OCT) [1,2] is a fourth-order interferometric sectioning scheme that employs frequency-entangled photon pairs [3] as the light source and Hong-Ou-Mandel interferometer as the analyzer [4,5]. It is the non-classical counterpart of optical coherence tomography (OCT) [6], and has two important features: one is the entirely quantum property of nonlocal dispersion cancellation, the other is the improvement in the axial resolution by a factor of two [7–10]. Because this technology is non-destructive, and can visualize structures at a micro scale, it has a broad application prospects in the areas such as medical diagnosis, surgical guidance, or characterization of polymer microstructures, etc [11–14].

However, such a huge increase in resolution is not easy to implement. There are several obstacles that have been discussed, such as the long scanning time caused by coincidence detection, image-scrambling artefacts –– disturbing dips or peaks induced by photon wave packets interference, etc. By far, several strategies have been proposed for overcoming the above drawbacks [15], such as broadband-pumped Q-OCT [2,16], spectrally-resolved Q-OCT [17], Fourier domain Q-OCT (Fd-Q-OCT) [18], etc. Among the methods, Fd-Q-OCT proposed by Ref [18]. has shown great advance. As the true spectral equivalent of the existing “time-domain” Q-OCT, it employs the full joint spectrum intensity as the measure and provides an artefact-free A-scan through algorithms. It has great prospects in terms of scanning time and mechanical stability since it does not require moving any equipment in the set-up.

However, the above method [18] only solves the problem under the condition that the system is operated in its perfect state and will not be affected by various types of noise. Obviously, the statistical nature of the quantum state of light leading to the truth that the measure of the correlation function would be affected by intrinsic fluctuations. According to the quantum theory, such affection would still exist even if the impletation of the quantum scheme is close to its perfect status [19,20]. Hence, the benefits provided by quantum correlation could be largely limited, and the issue has not been closely investigated in the Fd-Q-OCT scheme by far. In our work, we present an evaluation of such affection and provide a solution at the same time. Specifically, we focus on the joint spectrum corrupted by noise and numerically show that it would give misleading information about the object. We consider the quantum fluctuation induced by the fluctuation of the quantum state.

The solution we present is enabled by deep learning method. When talking about the various application of the high-quality imaging technique, the role of denoising algorithms is non-negligible. They have proven to be an effective and low-consuming support for different imaging tasks [21–24]. With the development of machine learning in recent years, the capability of denoising algorithms is greatly enhanced by being equipped with the algorithmic structures such as well-trained deep neural networks [25–30]. As discussed in the above, the affection of noise on Fd-Q-OCT has not been investigated. Nor does the problem whether the denoising tool such as deep neural network would reveal the information from the noisy quantum spectrum. Obviously, if the answer is yes, one could employ such a denoiser in Fd-Q-OCT and make a big step toward the real application of it. Here, we numerically confirm the answer.

The rest of the paper is arranged as follows. In Section 2, we theoretically analyze the affection of quantum fluctuation, and demonstrate the calculation result through simulations. In section 3, we propose a DNN model and verify its denoising ability against quantum fluctuation. For this type of noise, the model performs well. Finally, the discussion and conclusion are provided in Section 4.

2. Effect of noise on Fourier domain quantum optical coherence tomography

We consider a theoretical scheme of Fd-Q-OCT with quantum fluctuation. The schematic diagram is shown in Fig. 1, which is based on the theory of Ref [18]. The pump light irradiates the nonlinear crystal (NLC) and generates entangled photon pairs through the process of Spontaneous Parametric Down-Conversion (SPDC) [3]. One photon is called signal photon which illuminates the object in one arm of the interferometer (object arm). The other is called idler photon which is directly reflected by the reference mirror in the other arm of the interferometer (reference arm). Both photons interfere at the beam splitter, and the spectral relationship of them is measured using two spectrometers connected by a coincidence counter. The output of the measurement is termed by joint spectrum intensity (JSI). All the information about the internal structures of the object is encoded in JSI and can be retrieved through Fourier transformation. In the ideal case, JSI can be denoted as C1(ν₁, ν₂). If the quantum fluctuation of this system is considered, the simulated result of coincidence rate will be affected, which can be denoted as C2(ν₁, ν₂). Here ν₁ and ν₂ are used throughout the calculations to distinguish the photons in a pair and defined as the angular frequencies detuned from half of the central angular frequency of pumping laser. They can be expressed as ν₁=ω₁-ω_p/2 and ν₂=ω₂-ω_p/2, where ω₁ and ω₂ are the angular frequencies of signal photon and idler photon, respectively, and ω_p is the central angular frequency of pumping laser which makes the central angular frequency of entangled photon pairs is ω_p/2.

 

Fig. 1. The schematic diagram of Fd-Q-OCT with quantum fluctuation.

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The spectrum [31,32] of entangled photon pairs produced by SPDC can be written as

The spectrometer here is implemented by the combination of a diffraction grating and a linear detector array for certifying different frequency component. If the photon with angular frequency ν₁ is collected by spectrometer 1 and the photon with angular frequency ν₂ is collected by spectrometer 2, the correlated state of light source can be given by

Then we derive a formula to calculate the variance of the coincidence rate for the photon pairs with angular frequencies ν₁ and ν₂. For simplicity, the possible fluctuations from the detectors are not considered [20,33,34]. We define an operator

To study the quantum fluctuation in Fd-Q-OCT, we need to calculate the variance of the operator $\widehat S$. The quantum fluctuation of the coincidence rate Δc1(ν₁, ν₂) obtained by detector 1 and detector 2 is

Usually, the experimental noise is distributed in Gaussian. Therefore, we simulate the joint spectrum C2(ν₁, ν₂) as C2(ν₁, ν₂) = C1(ν₁, ν₂) + ℕ (0, ΔC1(ν₁, ν₂)), where ℕ (0, ΔC1(ν₁, ν₂)) represents a Gaussian random variable with mean zero and variance ΔC1(ν₁, ν₂). An artefact-free A-scan is obtained by calculating the minimum values for every column in an FFT stack, and the FFT stack is a two-dimensional matrix obtained by stacking the results of Fourier transform on the continuous main diagonal (ν₁=-ν₂=ν) and auxiliary diagonal (ν₁=-ν₂+Δν=ν) spectra of C1(ν₁, ν₂) [18]. Due to the disturbance of quantum fluctuation, the simulated joint spectrum C2(ν₁, ν₂) will provide misleading information about the internal structure of the object. The examples are given below.

Now, we present the numerical results for single-layer and double-layer objects to display the effect of quantum fluctuation. Basically, for an object consisting of N dispersive layers, the transfer function can be expressed as

In our simulations, the central wavelength of entangled photon pairs produced by SPDC is 812 nm. The spectral width of pump light σ_p is 25 THz and entanglement parameter σ_d is 55 THz in Eq. (1). This results in a 20.6 nm full width at half maximum (FWHM) in the anti-diagonal direction and 45.3 nm in the diagonal direction. The two materials simulated are fused-silica and ZnSe. At 812 nm, the group refractive index n of fused-silica is 1.5, and its GVD coefficient β₂₂ is 25 fs²/mm. Correlation coefficients of ZnSe are n = 2.717 and β₂₃= 500 fs²/mm [7]. The algorithm we use for peak detection can be described by a three-step procedure. Firstly, divide the spectral domain into a series of intervals whose corresponding geometry length is shorter than the estimated thickness of the layers. Then, the maximum of spectral value in each interval can be found by comparing the adjacent values. Secondly, sort the local maximums in order of their values. Then, the maximums with relatively big values are the structure peaks we search for. Thirdly, mark the preceding maximums and record their positions which can be used to find the distance between the interfaces. In the following results (Fig. 2 to Fig. 5), “${\times} $” indicates the peaks found under ideal condition, and “${\bullet} $” indicates the peaks found under the disturbance of quantum fluctuation. The interferogram is sampled into 1024 spectral channels linearly spaced in angular frequency ν and the total angular frequency range collected is 200×10¹² rad/s, and this in turn sets the sampling interval in the z-domain Δz≈2.36 µm.

 

Fig. 2. The effect of quantum fluctuation on the single-layer object. (a) The main diagonal spectrum (the spectrum when ν₁=-ν₂=ν, the followings are the same) of C1(ν₁, ν₂). (b) The main diagonal spectrum of C2(ν₁, ν₂). (c) The blue line represents the first row of F1 and the brown line represents the first row of F2. (d) The imaging results of a 50 µm thick fused-silica under ideal condition (the blue line) and under the disturbance of quantum fluctuation (the brown line). For the convenience of displaying the imaging result, the abscissa axis is only intercepted to 500 µm, and the followings are the same.

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Figure 2 presents the numerical data obtained when a single layer (N = 1) of 50 µm thick fused-silica is positioned 130 µm away from zero OPD point. F1 denotes the FFT stack of C1(ν₁, ν₂), i.e., the Fourier transform of 70 consecutive off-diagonal spectra of C1(ν₁, ν₂), where the height of artifact peaks goes to 0 for some certain spectral shifts. F2 is the FFT stack of C2(ν₁, ν₂), where the minimum height of artifact peaks may no longer be zero and the height of structural peaks is no longer a fixed value. Figure 2(a) shows the main diagonal spectrum of C1(ν₁, ν₂), which can be denoted as C1[0]. If quantum fluctuation is introduced, the noisy spectrum C2(ν₁, ν₂) is obtained. Figure 2(b) shows its main diagonal spectrum, which can be denoted as C2[0]. Figure 2(c) shows the Fourier transforms of C1[0] (the blue line) and C2[0] (the brown line). Obviously, quantum fluctuation has little effect on the periodicity of its joint spectrum. The positions of structural peaks and artifact peaks remain unchanged. The large peak at the zero geometrical distance corresponds to the Fourier transform of the instensity sum of the light fields reflected by the interfaces. In the following results, similar peaks also emerges at the zero geometrical distance. They do not shift with the position of samples, so their affection on the structure peaks can be avoided by properly choosing the sample position.

Then we calculate the minimum values for every column in an FFT stack and search structural peaks based on their heights. The result is shown in Fig. 2(d). By comparison, one could find that the noise only affects the height of the peaks, and does not change the positions of them. In the following, the effect of the reconstruction is evaluated by the improvements in the recognition accuracy of a series samples. The recognition accuracy is defined by the number of the samples whose structure peaks are correctly recognized over the total number of the samples. We consider the recognition accuracies in two precisions. For the first case, if the structure peaks lie in the positions no more than one sampled frequency channel away from the standard peak positions, they are thought to be correctly recognized. We call it the Δz precision. For the second case, only the structure peaks lie in the same positions with the standard peak positions are thought to be correctly recognized. This is a higher requirement than the first, and we call it the ideal precision.

Through 1000 repeated simulations, the recognition accuracy of structural peaks in Δz precision is close to 100 percent. Thus, the A-scan results based on the peak positions would not be significantly influenced. In conclusion, quantum fluctuation has a limited effect on the spectrum of single-layer objects, and does not affect the measurements of their thickness.

 

Fig. 3. The effect of quantum fluctuation on the double-layer object. (a) The main diagonal spectrum of C1(ν₁, ν₂). (b) The main diagonal spectrum of C2(ν₁, ν₂). (c) The blue line represents the first row of F1 and the brown line represents the first row of F2. (d), (e), (f) represent three different simulation imaging results (logarithmic form).

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Next, we simulate the A-scan of a double-layer object. It is positioned 130 µm away from 0 OPD and consisted of a 85 µm thick fused-silica on top of a 20 µm thick ZnSe. Thus, the structure peaks of the sample will not be covered by the large peak at zero geometrical distance position (also the zero OPD position). Figure 3(a) and 3(b) correspond to the main diagonal spectrum of C1(ν₁, ν₂) and C2(ν₁, ν₂), respectively. Figure 3(c) shows the first row of F1 (the blue line) and the first row of F2 (the brown line). Compared with single-layer object, the spectra of double-layer object are more complex. Relatively low peaks are submerged by quantum fluctuation and new peaks appear, which affects the final results. Figure 3(d), 3(e) and 3(f) shows the examples of imaging results of repeated numerical simulations for this double-layer object. For the convenience of showing the details of peaks, we add one to the value of vertical axis, and then take the logarithm to obtain the logarithmic form of intensity. There are nine peaks, three of which are structural peaks marked by “${\times} $”, and the rest are artifacts. The positions of peaks marked by “${\bullet} $” may be the same as those marked by “${\times} $”, such as the results shown in Fig. 3(d). This means that the judgment of the object interfaces and thickness could be accurate. However, through 1000 repeated simulations, the probability of this in Δz precision is about 27.7%. More frequently, the positions of marked structural peaks can also be different, such as the results shown in Fig. 3(e) and 3(f). Structural peaks can be submerged by quantum fluctuation and cannot be identified, and artifacts may also be identified as structural peaks.

In Fig. 3(e), the structural peak at optical distance of 130 µm is submerged, and the artifact at optical distance of 220.92 µm is identified as a structural peak. In Fig. 3(f), both structural peaks at optical distances of 130 µm and 257.5 µm are submerged, and the artifacts at optical distances of 27.12 µm and 284.67 µm are identified as structural peaks. Other artifact peaks may also be identified as structural peaks, which means that even if the same object is simulated, there are many possibilities for the imaging results of this object obtained.

For double-layer objects, the influence of quantum fluctuation on the spectra is not negligible. The accuracy of 27.7% is far from enough to meet the needs of practical applications. For specific tasks, such as the detection of double-layer dispersion objects whose interfaces contain important information, the above Fd-Q-OCT scheme would fail. It is very important to seek an effective denoising method to eliminate the damage of quantum fluctuation.

3. Deep-learning-based quantum optical coherence tomography with noise

Deep learning has been proven to perform well in image denoising. In order to introduce deep learning, we define a model to describe the denoising process. The objective is to obtain noise-free imaging information from the noisy imaging information. A typical model can be defined as

 We consider a series of double-layer dispersion objects with uniform thicknesses as the standard samples. Specifically, the thickness of fused-silica is evenly distributed in the range of 50 µm to 100 µm, and the thickness of ZnSe is 20 µm. Other parameters are the same as the previous simulations. The simulated raw data of the samples are obtained by the Fd-Q-OCT model with noise in the above, employed as the analogue output of an ideal Fd-Q-OCT system. The standard spectrum data are obtained by the theoretical Fd-Q-OCT model in Section 2. Since the imaging result of Fourier transform is symmetrical about the zero optical distance line, each structural peak in the positive distance range will have its mirror image in the negative distance range. So we only intercept the spectrum line in the positive part, which contains 512 sampled channels. Then, the dataset for denoising can be given by the labled raw spectrums, whose lables are the corresponding standard spectrum data. The destruction of the quality of Fd-Q-OCT is apparent.

The structure of the DNN modle we conside is shown in Fig. 4. It contains two hidden layers and one output layer. The training dataset and the validation dataset for the DNN is composed of 500 and 125 randomly-picked samples in the denosing dataset respectively. Because the spectrum lines are preprocessed as 1×512 vectors, all hidden layers and the output layer are set to contain 512 neurons. To train the network, we feed training data set into the neural network, and optimize the weighting factors and bias that connect every two neurons in two neighboring layers. To monitor the training process, we also input validation data into the DNN model after each epoch of training. The similarity between the output and the standard specturm is described by the loss function and it is here chosen as the mean square error (MSE) [35]. The activate function of these neurons is rectified linear units (ReLU) which supports a faster and effective training of deep neural architectures on large and complex datasets compared with sigmoid function [36]. In order to avoid overfitting [37], dropout is added between every two layers (dropout rate equals to 0.2).

 

Fig. 4. Framework of DNN. The number of neurons in each layer is marked in the figure.

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The optimization of this DNN is adaptive moment estimation (Adam) [38]. Our model is trained by minimizing the loss function. Determining the optimal hyperparameters of DNN means the end of training, and the model can be used to reconstruct the actual spectrum information from the noisy spectrum. The program was implemented using Python version 3.6 and implemented using Keras framework based on TensorFlow.

The denoising results of Fd-Q-OCT with quantum fluctuation are shown in Fig. 5(a)–5(c). The blue lines denote ground-truth spectrums. The brown lines correspond to noisy spectrums simulated by Fd-Q-OCT. They are corrupted by fluctuation of the correlation function of quantum states, leading to the misjudging of the depth information. As shown in Fig. 5(d), as the number of training epochs increases, the loss function values of the training dataset and the validation dataset both decrease until they become stable. The training terminates after five hundred epochs, and we can see that the value of MSE finally converges. The predicted specturm using the DNN model is shown as the green lines. The objects shown in Fig. 5(a) and Fig. 5(b) correspond to Fig. 3(e) and Fig. 3(f), respectively, whose thicknesses of fused-silica are both 85 µm. Although there is a big difference between the brown lines, after denoising by the DNN model, the green lines in the second row of Fig. 5(a) and 5(b) are both coincident with the ground-truth. The thickness of fused-silica of the object shown in Fig. 5(c) is 55 µm, and it can also be perfectly reconstructed. Based on the reconstructed spectrum, we can accurately recognize structural peaks, that is, the peaks marked with “${\bullet} $” in the green lines and the peaks marked with “${\times} $” in the blue lines correspond to the same optical distances. By reconstructing 2500 spectrums of the samples with uniformly distributed thicknesses of fused-silica between 50 µm and 100 µm, the recognition accuracy of structural peaks in Δz precision has increased from 26.68% to nearly 100%. It demonstrates the ability of the DNN model we proposed to effectively reduce the quantum fluctuation and reconstruct structural peaks of objects.

 

Fig. 5. (a-c) The denoising results of Fd-Q-OCT with quantum fluctuation. Blue lines: The ground-truth imaging data. Brown lines: simulated imaging data corrupted by quantum noise. Green lines: reconstructed imaging data using the DNN model shown in Fig. 4. (d) The performance of the DNN on suppressing the affection of quantum fluctuation with respect to epochs. Y-axis is the average value of MSE. In this part, we do five hundred epochs of training in total. The curve marked by train loss (val loss) is obtained by training dataset (validation dataset).

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Fig. 6. The dependence of the recognition accuracy of the DNN model on the size of training dataset under the two precision standards. The precision of brown line (precision 1) corresponds to Δz precision. The precision of green line (precision 2) is ideal precision. The small subgraph shows the details when the training dataset is less than 500.

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In order to study the dependence of the recognition accuracy of the DNN model on the size of training dataset, we change the number of samples contained in the training dataset, and the result is shown in Fig. 6. In Δz precision, when the training dataset contains 120 samples, the accuracy rate has exceeded 99%. This means that we can use few objects to achieve desired reconstruction effect. For some tasks that require higher accuracy, for example in ideal precision, the recognition accuracy of structure peaks is shown in the green line of Fig. 6. With this precision, the recognition accuracy of structure peaks before training is 16.68%. When the size of training dataset is increased to 500, the recognition accuracy rate reaches 71.04%. Then as the training objects increase, the recognition accuracy rate increases slowly, and finally stabilizes at about 80%.

Besides the above denoising scheme for the interferometry of Fd-Q-OCT, the DNN model can also be applied to suppress the noise of the landscape obtained by scanning the samples point-by-point. For example, we simulate the case with the aid of the famous MNIST handwriting digit database [39]. Suppose that a series of dispersion objects can be generated according to the MNIST database. Specifically, the grayscale information of pixels of one figure in the database is mapped to the actual thickness of the first layer of the corresponding position of an object, h_actual = grayscale value/5 + h₀ (h₀ is a constant and we set it to be 20 µm here), and the thickness of the second layer is 20 µm [40]. The images of objects can be obtained by scanning the objects point-by-point. Other parameters are set as follows. The central wavelength of entangled photon pairs is 1560 nm and σ_p= 32.9 THz, σ_d= 55 THz. The first layer is quartz. It’s position 120 µm away from 0 OPD with a group refractive index of n₁= 1.46 and a GVD coefficient, β₂₂=-28 fs²/mm [18]. To avoid the coincidence of the structural peaks with the artifacts, the parameters of the second layer as substrate are n₂= 1.98, β₂₃=-20 fs²/mm. We also need to change the number of neurons in each layer of the DNN model to 784 and add a reshaping layer at the output end shaping the 1×784 vector into 28×28 image to clearly show the reconstruction effect. The denoising result is shown in Fig. 7(a). The images in the first row denote ground-truth images without noise. The images in the second row correspond to noisy images simulated by Fd-Q-OCT. As shown in Fig. 7(b), The training terminates after one thousand epochs, and we can see that the value of MSE finally converges. The predicted results of the noisy images are shown in the last row of Fig. 7(a). They are similar to the images in the first row, that is, we can get accurate information about object thicknesses. It demonstrates the ability of the DNN model we proposed to effectively reconstruct information of two-dimensional objects.

 

Fig. 7. (a) The denoising results of Fd-Q-OCT with quantum fluctuation. The pixel values of each image represent the thicknesses of the first layer of this object whose colorbar is on the right (in µm). First row: the ground-truth images. Second row: simulated images corrupted by quantum noise. Last row: reconstructed images using the DNN model. (b) The performance of the DNN on suppressing the affection of quantum fluctuation with respect to epochs. Y-axis is the average value of MSE. In this part, we do one thousand epochs of training in total. The curve marked by train loss (val loss) is obtained by training dataset (validation dataset).

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4. Summary

In summary, we have theoretically analyzed the affection of quantum fluctuation on Fourier domain quantum optical coherence tomography, the deep-learning method has been proposed to suppress the effect. Our results have shown that the noise induced by fluctuation of quantum states does not affect the measurements of thickness for the single-layer objects. However, as the number of object layers increases, the spectrum becomes complicated, and the disturbance caused by the quantum fluctuation would lead to wrong judgments of the object information. Such an affection is underlied by the quantum nature of the Fd-Q-OCT scheme. Therefore, it would exist even if the whole physical setup is ideal and perfectly described by the model in second section. As a solution to the problem, we have proposed a DNN model for denoising, and proved that it can reconstruct A-scan images of the standarded objects destroyed by fluctuation of quantum states. Hence, we can conclude that the DNN model could partially capture the noise information of the theoretical Fd-Q-OCT setup. Then, for the given objects to be measured, the affection of the theorectical noise we consider on their ideal spectrums can be effectively supressed by the DNN model so that their real spectrums can be well predicted (supported by the performace of our well-trained DNN model when fed with the test dataset). We believe this study will promote the practical application of Fd-Q-OCT.