To train the CNN model, a large amount of training data is normally needed. For example, Dong et al. uses 91 images as the training set in SRCNN [13]. However, the scanning imaging using THz-TDS is very time-consuming. In order to provide a large amount of training data for the CNN model, we use the key parameters of the used THz-TDS system to generate a large number of simulated THz-TDS images. Here, the used key parameters included the diameter of THz spot and the noise level of the used THz-TDS system. Figure 1 shows the flow of data processing in this work.

In this work, our aim is to obtain high resolution plane stress fields. The stress state at a point contains three stress components, σ_x, σ_y and τ_xy. These stress components are not independent, and should satisfy the following balance equation,

∂ σ x ∂ x + ∂ τ y x ∂ y = 0 ∂ τ x y ∂ x + ∂ σ y ∂ y = 0 (1)

We supply 30 different stress fields as the training sets, and another 5 stress fields are used as the test set. These training and test data contain different forms of stress distribution function.

Figure 2 presents optical setup of the used THz-TDS system. The THz-TDS system used a commercial THz time domain spectrometer (Terahertz photonics Co. Ltd. TPF15K). The THz radiation is generated and detected by two rotatable photoconductive antennas. The emitting antenna generates the polarized THz radiation, and the polarization direction can be adjusted by rotating the antenna. In the implementation process, we carried out data simulation and experimental measurement in the dark field. The dark field here refers to the configuration, in

which the polarization direction of the emitting antenna is perpendicular to that of the receiving antenna. The angle between the polarization direction of emitting antenna and the horizontal direction is φ, and the angle between the polarization direction of receiving antenna and the horizontal direction is φ + π/2. That is, the THz signal strength passing through the unstressed specimen under ideal conditions is zero (complete extinction). To obtain the high extinction rate, two polarizers are used in this system as shown in Figure 2. Their polarization directions are set to be consistent with those of the emitting and receiving antenna, respectively. In Figure 2, the lens is used to focus THz radiation on the specimen. After focusing, the diameter of the spot is about 5.5 mm, which is measured by the traditional blade method. The highly reliable frequency range of the used system was during 0.2 - 2.5 THz. For subsequent convenience, the horizontal direction will be referred as x-axis, and the vertical direction as y-axis.

As the THz radiation passes through the specimen under test, only part of it can get through the air-sample and sample-air interface. The classic Fresnel formula is used to calculate the transmission coefficient,

t a b = 2 N a N a + N b , t b a = 2 N b N b + N a , (2)

where N_a = 1 is the refractive index of air, N_b is the refractive index of the specimen, t_ab and t_ba represent the transmission coefficients at air-sample and sample-air interface, respectively.

According to the material mechanics, the principal stress σ₁, σ₂ and principal stress direction θ of a plane stress state can be calculated from σ_x, σ_y and τ_xy by

σ 1 σ 2 } = σ x + σ y 2 ± ( σ x − σ y 2 ) 2 + τ x y θ = 0.5 arctan ( − 2 τ x y σ x − σ y ) . (3)

According to the stress optic law, when a plane polarized beam is incident vertically on a specimen under plane stress state, it is divided into two polarized components along the two principal stress directions. The propagation velocities of the two plane-polarized beams are different in the specimen. This causes the material to behave as optical anisotropy, resulting in birefringence. The relation between principal stress and refractive index is expressed as

N 1 − N 0 = A σ 1 + B σ 2 N 2 − N 0 = A σ 2 + B σ 1 , (4)

where N₀ represents the initial refractive index in the stress-free state, N₁ and N₂ are the refractive index along the directions of σ₁ and σ₂, and the stress optic coefficients A = 2.03 × 10⁻⁹, B = 2.63 × 10⁻⁹ [11] for the material to be used in the subsequent experiments, Poly Tetra Fluoro Ethylene (PTFE). It should be noted that Equation (4) is valid only when the stress is less than the elastic limit of material.

When the two beams of plane polarized light pass through the specimen and reach the second polarizer, the two beams are combined on its polarization direction. Finally, the THz signal is detected through the receiving antenna. If the amplitude and phase of the received THz signal through stress-free specimen are unit and zero, respectively, following a deduction procedure similar to that in Ref. [12], the signal received through specimen under plane stress state E can be expressed as

E = sin 2 ( θ − φ ) ⋅ sin δ 1 − δ 2 2 ⋅ e i ( π 2 + δ 1 + δ 2 ) (5)

where

δ 1 = 2 π f d c ( A σ 1 + B σ 2 ) δ 2 = 2 π f d c ( A σ 2 + B σ 1 ) (6)

In Equation (6), δ₁ and δ₂ are the phase change of the THz radiations polarized along the direction of σ₁ and σ₂, respectively. Based on the above model in Equation (5), we can obtain a simulated ideal THz signal modulated by a plane stress state.

The THz signal simulated in 2.1 is totally ideal. In the physical experiment, the captured THz images by step scanning method are degraded due to Gaussian blur, rough scanning step and random noise. A model here is proposed to reflect these degradation effects in the physical experiments.

Firstly, since the THz beams generally obey the Gaussian distribution and spot diameter has been measured, a Gaussian kernel function G in Equation (7) is adopted to simulate the effect of Gaussian blur.

G ( x , y ) = 1 2 π m 2 exp [ − x 2 + y 2 2 m 2 ] , (7)

where m is the standard deviation of the Gaussian kernel. After a scanning mapping, we get a three-dimensional THz image of a × b × 1000. Here, a and b represent the width and height of the images, and 1000 is the signal length of the time-domain waveform at each point. In other words, we get 1000 two-dimensional images. Each image is called a layer. The spatial sampling spacing of the simulation field is 0.05 mm, which is the resolution we want to achieve. Then, the 0.05 mm-resolution images in each layer is blurred by the kernel in Equation (7). When using Equation (7), the value of m is key. The measured radius of THz spot is about 5.5 mm. We estimate 95% of the energy of the spot is concentrated in the area with a diameter of 5.5 mm. In order to consider possible randomness of spot diameter in practice, the standard deviation m takes random values in the range of 1 mm - 1.5 mm in the simulation processing.

Secondly, the step of the scanning imaging is one of the key factors affecting the resolution. In physical experiment, it will take about 40 hours to scan a 50 mm × 50 mm area with a step size of 0.5 mm by our used system, so it is not practical to carry out many experiments with a smaller step. So after Gaussian blurring, each layer with a resolution of 0.05 mmis down sampled to one tenth of the original size to simulate the effect of the limited size of scanning step. This method has also been used in the classic literature of SRCNN to simulate the limited spatial resolution in the original images [13].

Finally, the electromagnetic random noise is inevitable in the used THz-TDS system. Thus, we capture several THz signals through air, and then estimate the level of random noise. At last, each layer is added by a Gaussian random noise with a mean of 0, and standard variance of α. The standard variance is determined by the physically captured signal of dark field configuration, in which the ideal signal should be zero. Ten signals are collected to evaluate the noise level, and finally α take one percent of the maximum signal intensity.

In order to include the above three degradation factors, a degradation model in Equation (8) is adopted.

L = ( H ⊗ G ) D s + n , (8)

where H is the ideal THz signal, L is the degraded THz signal, G represents the Gaussian blurred kernel, ⊗ represents the convolution operation, D_s is a sub-sampler, and n is the additive noise with a noise level of α. Considering that THz waves may produce different levels of noise in practice, multiple levels of noise should be taken into the training set. The degraded THz signal from Equation (8) is considered to be a simulation of a physically experimental signal.

By combing the Fresnel model in Equation (2), the stress optic effect in Equation (4), and the degradation model in Equation (8), we can get a large number of simulated experimental THz signals through a plane-stressed specimen for training the neural network.

In 2.1, an original stress field is supposed, and it is used to produce an ideal signal. In 2.2, we simulate the degenerated terahertz images. In this part, we need to calculate the stress field from the degenerated images. In this way, the original and calculated stress fields, which respectively represent the high- and low-resolution images, compose a pair of learning data.

The stress field can be calculated from the captured THz signals at the two polarization configurations of φ = 0 and φ = π/4. According to Equation (5), the principal stress difference σ₁ − σ₂, the principal stress sum σ₁ + σ₂ and the principal stress direction θ can be calculated as

σ 1 − σ 2 = − c π f d ( A − B ) arcsin ‖ E ‖ φ = π / 4 cos 2 θ σ 1 + σ 2 = c π f d ( A + B ) ( δ φ = 0 − δ φ = π / 4 2 + π 2 ) θ = − 1 2 arctan ‖ E ‖ φ = 0 ‖ E ‖ φ = π / 4 (9)

In Equation (9), ||E||_φ=0 and ||E||_φ=π/4 are the amplitude of the received radiation when the polarization angle φ = 0 and φ = π/4. As indicated in Equation (9), we can obtain a set of σ₁ − σ₂, σ₁ + σ₂, and θ from each frequency. In the work, the average value at different frequencies is used as their measurement results.

The neural network form adopted in this paper is similar to SRCNN in Ref. [13]. Instead of training the gray value of the image or the Y channel in the YCbCr channels, this paper realizes the end-to-end training from low-resolution stress field to high resolution stress field. As indicated in Figure 3, the mapping of low-resolution stress field σ_L to high resolution stress field σ_H is realized by three layers convolution operation. These three convolution operations are Patch Extraction and Representation, Non-linear Mapping and Reconstruction, as proposed in Ref. [13]. We will detail these three layers in turn.

Feature Extraction: This operation extracts (overlapping) patches from the stress field σ_L of three channels with low resolution, and represents each patch as a n₁-dimensional vector.

F 1 ( σ L ) = max ( 0 , W 1 ⊗ σ L + B 1 ) , (10)

where W₁ and B₁ represent the filters and biases, respectively. Here W₁ is of a size 3 × f₁ × f₁ × n₁, and B₁ is an n₁-dimensional vector.

Non-linear mapping: this operation nonlinearly maps each n₁-dimensional vector onto another n₂-dimensional vector by the following Equation (7). Each mapped vector is conceptually the representation of a high resolution patch.

F 2 ( σ L ) = max ( 0 , W 2 ⊗ F 1 ( σ ) L + B 2 ) , (11)

where W₂ is of a size n₁ × 1 × 1 × n₂, and B₂ is n₂-dimensional.

Reconstruction: This operation aggregates the above n₂-dimensional patchwork representations to produce the final high resolution stress field using

F ( σ L ) = W 3 ⊗ F 2 ( σ L ) + B 3 , (12)

where W₃ is of a size n₂ × f₃ × f₃ × 3, and B₃ is a three-dimensional vector.

Learning the end-to-end mapping function F requires the estimation of parameters Θ = {W₁, W₂, W₃, B₁, B₂, B₃}. The mean square error function shown in Equation (13) is used as the loss function to minimize the loss between the trained stress field F(σ_L) and the real stress field σ_H.

L ( Θ ) = 1 N ∑ i = 1 N ‖ F ( σ L i , Θ ) − σ H i ‖ 2 , (13)

where N is the total number of sub-field in the training process.

In the training phase, we set f₁ = 9, f₃ = 5, n₁ = 64 and n₂ = 32 in our implementation. The filter weights of each layer are initialized by drawing randomly from a Gaussian distribution with zero mean and standard deviation 0.001. The learning rate is 10⁻⁴ for the first two layers, and 10⁻⁵ for the last layer as suggested by Ref. [13].

In order to facilitate the display of the training set and test set, we transfer the normalized principal stress difference, principal stress sum and principal stress direction into RGB three color image channels respectively. Figure 4 presented a visual display of the training set and test set.

We train the models in a Intel(R) Core(TM) i5-9400F CPU and NVIDIA GeForce RTX 2060 GPU with Caffe package [22].

In this numerical experiment, the stress field conforming to a function presented in Equation (14) is selected to demonstrate the super resolution ability of the trained neural network. The corresponding principal stress difference σ₁ − σ₂, principal stress sum σ₁ + σ₂ and principal stress direction θ can be calculated by using Equation (3).

σ x = 2 sin ( x 25 ) + 0.1 σ y = − 1 625 y 2 sin ( x 25 ) − 0.2 τ x y = − 2 25 y cos ( x 25 ) + 0.1 , (14)

Figures 5-7 shows the super resolution effect of principal stress difference, principal stress sum, and principal stress direction, respectively. In these three figures, (a) sub-figures show the ideal high resolution fields obtained by Equation (14), (b) ones show the degraded low resolution fields simulated by the method in Section 2.1, and (c) ones show the super resolution fields obtained from the trained neural network. There is a sharp jump near y = −5.5 mm in Figure 7(a), which is an ideal position for us to observe the super resolution effect. In Figure 7(b), the jump become blurred and smooth due to Gaussian blur of large focus spot and random noise. In Figure 7(c), the jump is partially restored. Obviously, this jump in Figure 7(c) is sharper than that in Figure 7(b). This demonstrates the ability of the trained neural network to a certain extent. However, after all, the super-resolution of the network is limited, and it is impossible to recover all the details perfectly. For example, at the location of x = 17 mm, y = −5.5 mm in Figure 7(a), the field has sharp jump in both x and y directions. After super reconstruction, the horizontal change in the vicinity x = 17 mm along y = −5.5 mm line in Figure 7(c) is still more blurred than that in Figure 7(a).

In order to exhibit the super-resolution ability of the proposed algorithm in more detail, Figure 8 shows the stress components when x = −25.5 mm. In this figure, the solid line shows the ideal high resolution components, the spot line

shows the blurred low resolution components simulated by the method in Section 2.1, and the dotted line shows the super resolution components obtained from the trained neural network. Obviously, the random variation of the simulated field (i.e. the spot line) is depressed in the super resolution result (i.e. the dotted line). So the trained neural network has the ability of depressed random noise. Figure 8(c) more intuitively shows the ability of the trained neural network to restore the sharp jump.

In this section, a diametrical loaded disk is chosen to demonstrate the ability of the proposed super resolution algorithm in physical experiment. Figure 9 shows the setup of the diametrical loaded disk experiment. The loading device is installed on a two-dimensional scanning platform to facilitate us to obtain two-dimensional terahertz spectral fields. In the experiment, a scanning step of 0.5 mm is adopted.

According to elastic mechanics theory, the analytical stress distribution of the loaded disk should be

{ σ x = 2 p π t { ( r + y ) x 2 [ ( r + y ) 2 + x 2 ] 2 + ( r − y ) x 2 [ ( r − y ) 2 + x 2 ] 2 − 1 2 r } σ y = 2 p π t { ( r + y ) 3 [ ( r + y ) 2 + x 2 ] 2 + ( r − y ) 3 [ ( r − y ) 2 + x 2 ] 2 − 1 2 r } τ x y = 2 p π t { ( r + y ) 2 x [ ( r + y ) 2 + x 2 ] 2 − ( r − y ) 2 x [ ( r − y ) 2 + x 2 ] 2 } , (15)

where p is the applied force, r and t are the radius and the thickness of the disk, and x, y is the coordinate in space. The corresponding principal stress difference σ₁ − σ₂, principal stress sum σ₁ + σ₂ and principal stress direction θ can be calculated by using Equation (3). In this experiment, the disk material is PTFE. The experimental parameters are presented in Table 1.

Figure 10 presented the high resolution stress field distribution from Equation (15). Figure 11 shows the low-resolution stress fields from the actual THz signal by Equation (9). In Figure 10(a), there are two approximately circular bright spots near the loading points. In Figure 10(b), there are two dark spots. By comparison, in Figure 11(a) and Figure 11(b), it is found that the bright and dark spots in the low resolution field become blurred, and their boundaries become smooth. Furthermore, comparing Figure 10(c) and Figure 11(c), we can find that the mutation in the θ field near the loading points in the low-resolution fields is blurred. These should be caused by large terahertz Gaussian spot, inevitable random noise and limited scanning step.

In order to restore more details of the stress fields, the trained network is adopted to obtain its super resolution counterpart. Figure 12 shows the super resolution stress fields. In order to show the super resolution effect in more detail, Figure 13 shows the principal stress difference when y = 0 mm and the principal stress direction when x = 12.5 mm, respectively. After super resolution network processing, the boundary of bright and dark spots in the Figure 12(a) and Figure 12(b) become sharper than that in Figure 11(a) and Figure 11(b). The edge of angle mutation in the Figure 12(c) also becomes clear. Figure 13 demonstrates the noise depression capability of the used neural network. Furthermore, it is obvious the super resolution results are closer to the analytical solution in Figure 13.

In order to quantify the effect of the trained neural network, error analysis is performed on simulated data and experimental data. The mean square error of principal stress difference, principal stress sum, and principal stress direction are calculated respectively. The mean square error here refers to the average relative error of the LR result and the SR result with respect to the analytical solution at each point. They can be calculated by

error = ∑ i = 1 N ‖ σ i − σ H i ‖ 2 ∑ i = 1 N ‖ σ H i ‖ 2 , (14)

where σ_i is the stress components of LR and SR, and σ_Hi is those of HR. In addition, we also calculate a mean square error of the three channels. Table 2 lists the results. The datum in Table 2 indicates that the SR result obtain a closer approaching to the analytical solution. This shows the effectiveness of the proposed super-resolution algorithm to a certain extent.