Neural Network-Assisted Interferogram Analysis Using Cylindrical and Flat Reference Beams

Abstract

In this paper, we present the results of a comparative analysis of the sensitivity of interferograms to wavefront aberrations recorded with flat and cylindrical reference beams. Our results show that compared to classical linear interferograms based on flat wavefronts, cylindrical interferograms have at least 10% higher sensitivity for radially asymmetric types of aberrations and a 30% decrease in average absolute error for aberration recognition using a convolutional neural network. The use of cylindrical reference beams leads to an increase in the sensitivity of interferograms for detecting medium and strong aberrations.

1. Introduction

The task of measuring wavefront distortions is often encountered in optics, for example, in the design of ground-based astronomical telescopes, augmented reality systems, optical communication systems, industrial laser technology, medicine, etc. [1,2,3,4,5,6,7,8,9,10]. Almost always, the measurement of wavefront distortions is performed in order to compensate them, in particular, by means of adaptive or active optics [11,12,13,14]. The wavefront sensor is one of the main elements of the adaptive laser radiation correction system. Its task is to measure the aberrations of the wavefront and transmit the results of these measurements to the processing device. The main causes of wavefront aberrations are atmospheric turbulence [15], non-ideal shapes of the optical elements of the system, errors in system adjustment, etc. Today, there is a wide variety of wavefront sensors, including Shack-Hartman [16,17], pyramidal [18,19], fiber [20,21,22], and other sensors. One of the types of correlation sensors [23] are multiorder diffractive optical elements matched with the basis of Zernike polynomials [24,25,26].

Interferometry is one of the most accurate techniques for measuring wavefront aberrations [27]. However, it has its well-known limitations, such as the complexity of decoding interferograms and the sensitivity of the measuring equipment to vibrations. Traditionally, a flat wavefront has been used as the reference beam in interferometric sensors, as it allows for a simple algorithm based on the Fourier transform to be used for phase reconstruction [28,29]. Another area closely related to interferometry, where fringe demodulation techniques are also widely used, is fringe projection. Without using the interference principle for data recording, but at the same time actively using often identical data processing algorithms, these methods provide reliable information about the shape of objects. One of the current trends in further research aimed at improving these techniques is the optimization of the reference wave structure. Thus, it has recently been theoretically shown [30] that the optimal reference wave structure can be achieved by using circular-arc-shaped fringes centered at the epipole, providing the best phase sensitivities over the entire fringe pattern. In addition, it has been shown that the use of circular fringes in fringe projection profilometry is favorable since their central location serves as a good benchmark for shape reconstruction [31,32,33,34]. In addition, hexagonal stripes [35], gray-level coded patterns [36,37], and others are used. Returning to interferometry, it should be noted that in solving a number of problems, not only real but also virtual reference waves can be used [38,39].

Recent advancements in data mining and neural networks have enabled the extraction and recognition of the wavefront from both the intensity distribution [40,41,42,43,44,45,46] and real [47] and emulated [48] interference patterns. The current manuscript is a continuation of previous work [48], where neural networks were used to recognize the wavefront from conical interferograms. In that study, we utilized a system based on a digital micromirror device (DMD) and a technique for independent amplitude and phase modulation [49,50] to simulate the desired interference pattern with aberrations. In this work, two reference beams with a flat and cylindrical wavefront are considered to assess the sensitivity of the proposed interferograms. The use of reference beams with cylindrical wavefronts in the formation of interferograms is studied in order to improve the recognition of aberrations using a convolutional neural network.

Notably, the better the interferogram reflects the features of various aberrations, the more efficiently the recognition algorithm will work. In addition, it is necessary that the patterns corresponding to different types and levels of aberrations differ markedly. This possibility can be implemented in interference methods by choosing the type of reference beam.

2. Materials and Methods

2.1. Theoretical Foundations

Consider the Zernike circle polynomials, which are the complete set of orthogonal functions on a circle of unit radius [51]:

$$Z_{nm}\left(r,\mathsf{\phi }\right)=\sqrt{\frac{n+1}{\mathsf{\pi }{r}_0^2}}{R}_n^m\left(r\right)\left\{\begin{array}{c}\cos \left(m\mathsf{\phi }\right)\\ \sin \left(m\mathsf{\phi }\right)\end{array}\right\},$$

(1)

where $R_n^m\left(r\right)$ is the Zernike radial polynomials, m and n are integers,

$$R_n^m\left(r\right)={\displaystyle \sum _{s=0}^{\left(n-m\right)/2}\frac{{\left(-1\right)}^s\left(n-s\right)!}{s!\left((n+m)/2-s\right)!\left((n+m)/2-s\right)!}}{\left(\frac{r}{r_0}\right)}^{n-2s},$$

(2)

$A_n$ is the normalizing factor:

$$A_n=\sqrt{\frac{n+1}{\mathsf{\pi }}}.$$

(3)

Wavefront aberrations encountered in optical systems are usually described in terms of Zernike functions as follows:

$$E_W\left(r,\mathsf{\phi }\right)=\exp \left[iW\left(r,\mathsf{\phi }\right)\right],$$

(4)

$$W\left(r,\mathsf{\phi }\right)=2\mathsf{\pi }{\displaystyle \sum _{n=0}^{n_{\max }}{\displaystyle \sum _{m=0}^n{c}_{nm}}{Z}_{nm}\left(r,\mathsf{\phi }\right)},$$

(5)

where c_nm is the weight of the wavefront aberration,

The interference of two fields is registered by the intensity of the following distribution:

$$I\left(x,y\right)={\left|E_W\left(x,y\right)+E_B\left(x,y\right)\right|}^2$$

(6)

where $E_W\left(x,y\right)$ is analyzed field, e.g., aberrated wavefront, $E_B\left(x,y\right)$ is the reference beam.

As a reference beam, as a rule, some standard wavefronts are used (for example, flat, spherical, or vortex):

$$E_B\left(x,y\right)=\exp \left[i{B}_p\left(x,y\right)\right]$$

(7)

In this paper, we consider two types of reference beams: a flat tilted beam (Figure 1) $B_{p=1}\left(x,y\right)=\mathsf{\alpha }x+\mathsf{\beta }y$ (p is the reference beam type; α, β are parameters that determine the beam tilt angle), and a cylindrical beam (Figure 2) $B_{p=2}\left(x,y\right)=\mathsf{\alpha }{x}^3$ (α is the parameter that determines the angle of inclination of the cylindrical front to the optical axis). The choice of the cubic beam type is related to its wide use in recent years for wavefront coding in order to increase the depth of focus [51,52,53,54].

Thus, in the case under consideration, the intensity of interferograms (6) is proportional to the following expression:

$$I_p\left(x,y\right)\cong 1+\cos \left[W\left(x,y\right)-B_p(x,y)\right]$$

(8)

Furthermore, the frequency of the interferogram fringes will mainly depend on the parameters of the reference beam, and the complexity of the pattern being formed is determined both by the type of reference beam and by the aberration of the wavefront.

In order to quantitatively estimate the sensitivity of recognition of the types of interferograms under consideration, we introduce the parameter S, which corresponds to the value of the root-mean-square error (RMSE) of the p-type interferogram formed by an aberrated wavefront $I_p(x,y),$ from the interferogram corresponding to the absence of aberrations $I_{0p}(x,y)$:

$$S_p=\sqrt{{\displaystyle \iint {\left[I_p(x,y)-I_0{}_p(x,y)\right]}^{2}dxdy}/{\displaystyle \iint I_{0p}^2(x,y)dxdy}}$$

(9)

Increasing the deviation for one type of interferogram compared to another should lead to an increase in recognition efficiency. This assumption is confirmed by the more stable recognition of interferograms with a conical reference wave compared to a flat reference wave [48]. For clarity, the value Δ is introduced as the sensitivity of interferogram recognition by machine learning methods:

$$\Delta =(S_2-S_1)/c_{nm}$$

(10)

corresponding to the difference in RMSE values of cylindrical (p = 2) and linear (p = 1) interferograms and average sensitivity:

$$\overline{\Delta }=({\overline{S}}_2-{\overline{S}}_1)/c_{nm}$$

(11)

where ${\overline{S}}_p={\displaystyle \sum _{q=1}^Q{S}_{pq}}$, p is the interferogram type, q is the aberration type, and Q is the number of considered aberrations (Q = 8) for different types and levels of aberration. If Δ ≥ 0, then the sensitivity of the cylindrical interferogram is higher than the linear one; otherwise, the sensitivity is lower.

For clarity, the information content parameter is introduced as

$$E={\overline{S}}_2/{\overline{S}}_1-1$$

(12)

for cylindrical and linear interferograms for different types and levels of aberrations with a variation of the parameter α, which shows how many times one type of interferogram is more effective for recognizing aberrations from interferograms using convolutional neural networks.

2.2. Optical Setup

Our work employed the experimental setup (Figure 3) based on a Mach-Zehnder interferometer with the digital micromirror device (DMD) and the Fourier-plane filtered 4f-system [49,50] in one of its optical paths. The reference beam in the second optical path of the setup was utilized to monitor the phase distribution of the optical field formed by the DMD and 4f-system. Each of the optical paths of the interferometer was blocked by an opaque screen (shutter) according to the conditions of the experiment.

The DMD device allows only binary modulation. In our experimental setup, the Texas Instruments Light Crafter DLP6500FYE binary modulator (1920 × 1080 pixels with a size of 7.56 μm) operated by the DLPC900 controller was used. Each micromirror is located on a corresponding cell of the CMOS matrix and can be in one of two positions, +β or −β, which correspond to logical 1 and 0, respectively. Despite the binary limitation, it is possible to achieve independent amplitude-phase modulation by conjugating the DMD with an input plane of the 4f-system with spatial Fourier filtering. We used the method of obtaining independent amplitude and phase modulation from binary Lee holograms [55] to generate the patterns. This independent wavefront modulation of the amplitude and phase greatly expands the possibilities of the experiment. The standard two-arm interferometer scheme is a straightforward option, where the object arm contains a DMD for shaping the wavefront with predefined aberrations, and the reference beam containing a flat wavefront is directed to the matrix photodetector by the beam splitter. Another option is to use a single-arm scheme for forming interferograms with predefined aberrations and the reference beam, which simplifies the system and increases experiment accuracy and vibration resistance.

In the single-arm scheme, two beams with different amplitude and phase configurations can be implemented simultaneously using the same pattern on the DMD, in which the interference field of these beams is encoded. If one of these beams has a different wavefront tilt from the other, they can be spatially separated in the Fourier plane of the 4f-system. Figure 4 shows the Fourier spectra for the instances where the field is generated with the quadrafoil aberration (c₄₄ = 1) (a) and its interference field with the plane tilted wave α = 10π (b) and α = 50π (c). The separation of diffraction suborders (Figure 4b,c) proves the formation of the interference pattern in a single-arm scheme is physically identical to the classical two-arm interference. The presence of wavefronts with quadrafoil aberration and linear tilt in the diffraction suborders was confirmed in the two-arm scheme with sequential blocking of each of them.

In our experimental setup, interferograms for the training of the neural network were formed only in the objective beam (the single-arm scheme), and the reference beam was used to validate the correct phase distribution, as shown in Section 3.2. The amplitude-phase distributions with a given type and level of aberrations were encoded into binary patterns as the sum of the complex fields of the wavefront aberration and the reference beam: E_w(x, y) = E_B(x, y).

In order to experimentally confirm the coincidence between the interferograms synthesized in the single-arm scheme and the classical two-arm scheme with further quality estimation, the following experiment was performed. The interference fields of the plane tilted wave and complex wave containing the quadrafoil aberration synthesized in the single-arm scheme as well as the interference fields of the synthesized complex wave containing the quadrafoil aberration and the reference plane wave in the two-arm scheme were recorded. The local least squares method was applied to demodulate the interference fringes [56,57]. Figure 5 demonstrates the target amplitude and phase distributions encoded in the DMD pattern, the interferograms for single-arm and two-arm schemes, and the corresponding reconstructed phases.

The reconstructed phase distribution quality for the low carrier frequency interferograms (α = 10π and α = 14π) is lower than for the high-frequency case (α = 50π). This is due to the limitation of the reconstruction method, the main condition for it is that the object wave has smaller local variations than the reference wave. Therefore, in some cases, with a small angle of inclination of the reference wavefront to the optical axis of the interferogram, there may be a deterioration in the quality of the object wavefront reconstruction. Nevertheless, the low carrier frequency does not have a negative impact on the ability of the neural network to recognize aberrations.

The reconstructed phase distributions in Figure 5 have less smooth phase variation in the center of the distributions compared to the target phase distribution. This effect is caused by the limitations on quantization levels induced by the Lee holography method and amplitude-phase modulation with DMD [50]. Root mean square error (RMSE) was used as a quality metric for reconstructed phase distributions. The table under Figure 5 shows RMSE values for the phase distributions obtained for single-arm and two-arm schemes with respect to the target phase distribution. The error for interferograms with α = 10π and α = 14π is higher than for interferograms with α = 50π due to the difficulty of demodulating such interference fringes. With the single-arm scheme, there was a decrease in RMSE of 0.12 rad in the case of a high-frequency interferogram and by 0.08 rad in the case of a low-frequency interferogram compared to the results obtained with the two-arm scheme. It can be explained by additional factors affecting the quality of reconstruction in the two-arm scheme, such as the quality of the reference wavefront, induced by imperfections of optical elements in the reference arm, and external vibrations, which cause changes in the angle between the reference and object waves. The quality of the phase distributions is higher due to the elimination of additional artifacts introduced by the second arm of the interferometer.

In summary, the quality of the reconstructed object wavefront from interferograms in our scheme is affected by: (1) the quality of the wavefront; (2) the modulation error of the target distribution; and (3) the reconstruction error of the object wavefront. In the case of a two-arm scheme, the quality of the reference wavefront and vibrations, which can cause dynamic changes in the angle between the reference and object waves during registration, is also influenced. In addition, the formation of interference fields with low carrier frequency and their further demodulation is simpler from the experimental point of view in the single-arm scheme. Therefore, we experimentally verified the capability of the single-arm scheme to generate interference fields used to train a neural network for aberration recognition. It was also shown that the use of a single-arm scheme for generating such interference fields allows us to provide a better quality of the object wavefront compared to the two-arm scheme.

2.3. Neural Network

For this work, a neural network architecture based on Xception was used [58,59,60]. A distinctive feature of this model is the use of depthwise separable convolution.

Since we are dealing with a regression problem, the use of such metrics as accuracy, confusion matrix, and other metrics applicable to classification problems is not possible. To control the accuracy in this case, the mean absolute error function was used as a loss function. The training of the neural network was implemented using a Nvidia Tesla K80 video card with 12 GB of built-in DDR5 video memory. Libraries for machine learning—the Keras package. The essence of the “depthwise separable convolution” method is to split the transformations over the input tensor on each layer into two stages:

-

at the first stage, the tensor is convoluted with a 1 × 1 kernel. This operation is called “pointwise convolution.”

-

at the second stage, standard convolution is performed. After that, the results of the convolution of the tensor obtained as a result of applying the “pointwise convolution” operation to the input tensor are summed up with all the filters of the layer and form the output tensor of the layer. This operation is called “depthwise convolution.”

This technique allows you to significantly reduce the number of weights in the neural network, which significantly affects the learning rate required for training resources and the weight of the pre-trained model. Thus, the Xception architecture (Figure 6) is the best choice, providing sufficient training accuracy while requiring significantly fewer resources.

Since the given problem is a regression problem and the Xception architecture was developed to solve the classification problem, the original architecture was modified. One fully connected layer was added, the dimension of which is equal to the number of Zernike functions (chosen to be 8).

To obtain continuous values of the predicted coefficients, a linear activation function was used. To improve the convergence of the optimization algorithm, the network architecture was supplemented with a dropout layer in front of the output semi-connected layer.

For each type of interferogram, a data set was prepared corresponding to the superposition of two aberrations. The dataset consists of 6000 images. 4000, 1000, and 1000 images were used for training, testing, and validation, respectively. The value of c_nm in the range from 0 to 1 and the type for the superposition of two aberrations were chosen randomly. Single-channel images of 299 × 299 pixels of two types were fed to the input of the neural network: linear interferograms as well as cylindrical interferograms.

3. Results

3.1. Numerical Simulation

We consider the formation of a linear and cylindrical (cubic) interferogram with the reference beam tilt angle α divisible by 2π, thus matching the frequency parameter of the interferogram with the value of the phase shift of the reference beam transmission function. Figure 1 and Figure 2 show that the parameter α becomes consistent with the phase shift at a multiplicity of 2π. Thus, the phase passes values from 0 to 2π α/π times, and, accordingly, the ideal interferogram (Z₀₀) in the resulting plane will be represented as α/π full interference fringes.

The interferograms for each of the aberrations up to the 4th order were calculated.

Table 1. Interferogram patterns (α = 2π, β = 0) for aberrations in the form of separate Zernike functions.

Aberration Type ${\mathit{Z}}_{\mathit{n}\mathit{m}}\left(\mathit{x},\mathit{y}\right)$ Phase W(x,y) $\mathbf{(}{\mathit{c}}_{\mathit{n}\mathit{m}}\mathbf{=}\mathbf{0.5}\mathbf{)}$	Linear ${\mathit{I}}_{\mathit{p}\mathbf{=}\mathbf{1}}\left(\mathit{x}\mathbf{,}\mathit{y}\right)$ Interferogram $\mathbf{(}{\mathit{c}}_{\mathit{n}\mathit{m}}\mathbf{=}\mathbf{0.1}$, ${\mathit{c}}_{\mathit{n}\mathit{m}}\mathbf{=}\mathbf{0.5}\mathbf{)}$	Cylindrical ${\mathit{I}}_{\mathit{p}\mathbf{=}\mathbf{2}}\left(\mathit{x}\mathbf{,}\mathit{y}\right)$ Interferogram $\mathbf{(}{\mathit{c}}_{\mathit{n}\mathit{m}}\mathbf{=}\mathbf{0.1}$, ${\mathit{c}}_{\mathit{n}\mathit{m}}\mathbf{=}\mathbf{0.5}\mathbf{)}$
$Z_{11}\left(x,y\right)$- tilt
$Z_{22}\left(x,y\right)$- astigmatism
$Z_{20}\left(x,y\right)$- defocus
$Z_{33}\left(x,y\right)$- trefoil
$Z_{31}\left(x,y\right)$- coma
$Z_{44}\left(x,y\right)$- quadrafoil
$Z_{42}\left(x,y\right)$- secondary ast.
$Z_{40}\left(x,y\right)$- spherical

shows the patterns of the considered interferograms at α = 2π for aberrations in the form of separate Zernike functions.

As can be seen from the interferograms shown in Table 1, the cylindrical interferogram is less sensitive to weak aberrations, however, both types demonstrate significant differences in patterns (distortion of interference lines) at high levels of aberrations ($c_{nm}=0.5$), i.e., they can be successfully used to analyze aberrations in a situation where expansion in terms of Zernike functions leads to noticeable recognition errors [61].

Consider a cubic interferogram using the example of a coma-type aberration Z₃₁ with an aberration value $c_{nm}=0.5$ and different values of the tilt angle parameter of the reference cylindrical beam α (Figure 7).

It should be noted that the use of a reference cylindrical beam, in contrast to a flat beam in interferograms, leads to a complication of the intensity distribution pattern with an increase in the alpha α. This consideration is detailed in the Section 4.

It is assumed that interferograms using a structured reference beam corresponding to a cylindrical (cubic) carrier make it possible to form a more complex pattern of intensity distribution (especially in the peripheral part) in the resulting plane of the optical system under study and make it possible to more accurately detect and interpret aberrations in analyzed wavefronts, including by training neural networks.

For each type of interferogram, a data set was prepared corresponding to the superposition of two aberrations. The value of c_nm in the range from 0 to 1 and the type for the superposition of two aberrations were chosen randomly. Examples of generated linear and cylindrical interferograms for various types and levels of aberrations with a variable parameter α are presented in

Table 2. Examples of generated linear and cylindrical interferograms for various types and levels of aberrations with a variable parameter α.

Images of Linear Interferograms			Images of Cylindrical Interferograms
2π	4π	6π	2π	4π	6π
$0.3Z_{20}\left(x,y\right)+0.2Z_{44}\left(x,y\right)$
$0.4Z_{22}\left(x,y\right)+0.1Z_{33}\left(x,y\right)$
$0.3Z_{42}\left(x,y\right)+0.2Z_{20}\left(x,y\right)$

.

Table 2 shows that the tilt angle of the flat reference beam α significantly affects the structure of the linear interferogram pattern. However, the image of a cylindrical interferogram retains its structure, with noticeable changes only in the peripheral part. This property of the proposed cylindrical interferogram gives an advantage in deciphering and unambiguously interpreting aberrations in the analyzed wavefront. In turn, unambiguity can help reduce errors in recognizing aberrations of the analyzed wavefront from interferograms with machine learning methods and neural networks.

3.2. Experimental Formation of Interferograms

Table 3. Interferogram patterns (α = 2π, β = 0) for aberrations as separate Zernike functions.

Aberration Type ${\mathit{Z}}_{\mathit{n}\mathit{m}}\left(\mathit{x},\mathit{y}\right)$ Phase W(x,y) $\mathbf{(}{\mathit{c}}_{\mathit{n}\mathit{m}}\mathbf{=}\mathbf{0.5}\mathbf{)}$	Linear ${\mathit{I}}_{\mathit{p}\mathbf{=}\mathbf{1}}\left(\mathit{x},\mathit{y}\right)$ Interferogram $\mathbf{(}{\mathit{c}}_{\mathit{n}\mathit{m}}\mathbf{=}\mathbf{0.1}$, ${\mathit{c}}_{\mathit{n}\mathit{m}}\mathbf{=}\mathbf{0.5}\mathbf{)}$	Cylindrical ${\mathit{I}}_{\mathit{p}\mathbf{=}\mathbf{2}}\left(\mathit{x}\mathbf{,}\mathit{y}\right)$ Interferogram $\mathbf{(}{\mathit{c}}_{\mathit{n}\mathit{m}}\mathbf{=}\mathbf{0.1}$, ${\mathit{c}}_{\mathit{n}\mathit{m}}\mathbf{=}\mathbf{0.5}\mathbf{)}$
$Z_{11}\left(x,y\right)$- tilt
$Z_{22}\left(x,y\right)$- astigmatism
$Z_{20}\left(x,y\right)$- defocus
$Z_{33}\left(x,y\right)$- trefoil
$Z_{31}\left(x,y\right)$- coma
$Z_{44}\left(x,y\right)$- quadrafoil
$Z_{42}\left(x,y\right)$- secondary ast.
$Z_{40}\left(x,y\right)$- spherical

and

Table 4. Examples of experimentally obtained linear and cylindrical interferograms for various types and levels of aberrations.

Images of Linear Interferograms			Images of Cylindrical Interferograms
2π	4π	6π	2π	4π	6π
$0.3Z_{20}\left(x,y\right)+0.2Z_{44}\left(x,y\right)$
$0.4Z_{22}\left(x,y\right)+0.1Z_{33}\left(x,y\right)$
$0.3Z_{42}\left(x,y\right)+0.2Z_{20}\left(x,y\right)$

show examples of experimentally obtained linear and cylindrical interferograms for various types and levels of aberrations. Since conical interferograms showed [48] a significant difference in sensitivity for high-frequency and low-frequency types, both types were used in the analysis of experimental patterns.

As can be seen from the interferograms given in Table 4, both types demonstrate significant differences in patterns (distortion of interference lines) at high levels of aberrations ($c_{nm}=0.5$) and are highly consistent with the results obtained in numerical simulation.

The interferograms were obtained in the single-arm scheme (see Figure 3, with the reference beam arm blocked by the shutter). The interference of the two fields was encoded in the pattern calculation, which was supplied to the DMD modulator. This approach simplifies the registration of complex distributions and avoids the influence of vibrations. In order to confirm the physical analogue of the amplitude-phase field distribution for the single-arm scheme, the second arm of the experimental setup was engaged, and the interference distributions of the interferogram and the tilted plane wave were recorded.

Table 5. The model phase distribution, the reconstructed phase, and corresponding distribution interferogram for angle α = 6π.

Model Phase Distribution	Reconstructed Phase Distribution	Interferogram
$0.3Z_{20}\left(x,y\right)+0.2Z_{44}\left(x,y\right)$
$0.4Z_{22}\left(x,y\right)+0.1Z_{33}\left(x,y\right)$
$0.3Z_{42}\left(x,y\right)+0.2Z_{20}\left(x,y\right)$

shows the model and reconstructed phase distributions, the cylindrical interferograms for the angle α = 6π with different types of aberration superposition.

Since a complex interference field of two beams containing amplitude and phase distributions is encoded on the DMD to form interferograms, a two-arm interferometer scheme was used to control the resulting phase distributions.

A set of digital interferograms was recorded for each DMD pattern containing aberration for further averaging. Processing of the digital hologram series was performed as follows: The phase was reconstructed from the set of holograms for one DMD pattern by the local least squares method. The phase distributions were multiplied by a binary mask to ensure that noise was excluded in further processing. The phase difference between the fields recorded with and without the introduced aberration was then calculated. Next, two stages of averaging the phase difference distributions were performed to eliminate the reconstruction error associated with external vibrations. The first stage of averaging consisted of calculating the mean value for each distribution and subtracting it from the corresponding phase. The second step was to calculate the arithmetic mean of the entire set of phase distributions for one pattern.

Table 5 shows the model and reconstructed phase distributions, the cylindrical interferograms for the angle α = 6π with different types of aberration superposition. The reconstructed phase distributions in Table 5 obtained from experimental interferograms with a two-arm scheme match with the simulated phase distributions with the error of the object wavefront reconstruction method and distortions induced by imperfections of the experimental scheme such as beam quality and interference fringes period instability. The results confirm that complex interference fields were obtained with the amplitude-phase modulation technique [50].

3.3. Neural Network Training

The neural network based on Xception [58,59,60] architecture was trained using a dataset corresponding to the superposition of two aberrations (6000 images: 4000, 1000, and 1000 images were used for training, testing, and validation, respectively), examples of which are presented in Table 2.

Upon completion of the learning process lasting 80 epochs for the mean absolute error (MAE) function, results were obtained on linear-type and cylindrical-type interferograms (Figure 7) with the parameter α = 2π. As a result of this learning process (Figure 8b), we get a model capable of making a prediction in the form of an array of eight elements, where each element corresponds to a level (or power) of an aberration type c_nm. Predictions are made using only an image of an interferogram as input (Figure 8a).

3.3.1. Modeling Dataset

In several numerical experiments, the recognition errors of the superposition of aberrations were obtained from interferograms with a tilt angle of the reference beam in the range from 2π (Figure 9) to 6π (Figure 10). The results are presented in

Table 6. Mean absolute recognition error (MAE) for model interferograms of various types.

Type Angle	Linear	Cylindrical
α = 2π	0.0063	0.0050
α = 4π	0.0065	0.0047
α = 6π	0.0068	0.0050

.

It has been found that a cylindrical interferogram makes it possible to reduce the aberration recognition error by 25% compared to a linear one, regardless of the beam tilt angle α. Moreover, it is confirmed that the recognition error is linearly related to the calculated parameter (informativeness) based on the RMSE of the reference interferogram compared to the aberrated one.

Thus, the use of cylindrical interferograms makes it possible to improve the results of the recognition of types and levels of aberrations—the average absolute error decreases by more than 30% (from 0.0068 for linear interferograms to 0.0047 for cylindrical interferograms).

3.3.2. Experimental Dataset

The neural network based on the Xception [58,59,60] architecture was trained using a dataset consisting of experimentally obtained interferograms corresponding to the superposition of two aberrations. A fragment of the dataset is presented in Table 4 and

Table 7. Examples of experimentally obtained linear and cylindrical interferograms for various types and levels of aberrations.

	Images of Cylindrical Interferograms ${\mathit{D}}_{\mathbf{20}}{\mathit{Z}}_{\mathbf{20}}\left(\mathit{x},\mathit{y}\right)+{\mathit{D}}_{\mathbf{44}}{\mathit{Z}}_{\mathbf{44}}\left(\mathit{x},\mathit{y}\right)$
$D_{20}$	0.30
$D_{44}$	0.05	0.15	0.25	0.35	0.45
Interf.

.

In several numerical experiments, the recognition errors of the superposition of aberrations were obtained from experimental cylindrical interferograms with a reference beam tilt angle in the range from 2π to 6π (Figure 11). The results are presented in

Table 8. Mean absolute recognition error for cylindrical interferograms from the model and experimental dataset.

Type Angle	Cylindrical Model Interferogram	Cylindrical Experimental Interferogram
α = 2π	0.0050	0.0049
α = 4π	0.0047	0.0046
α = 6π	0.0050	0.0052

.

4. Discussion

In this section, we discuss the relationship between an increase in the sensitivity of interferograms due to the structural complexity of the reference beam and a decrease in MAE recognition by the neural network.

In order to quantitatively estimate the sensitivity of recognition of the interferogram under consideration, we calculate the values of the parameters S₁ and S₂, which correspond to the value (9) of the RMSE of the p-type interferogram formed by an aberrated wavefront $I_p(x,y),$ from the interferogram corresponding to the absence of aberrations $I_{0p}(x,y)$.

Figure 12 shows the RMSE values (9) of standard (without aberrations) cylindrical interferograms compared to aberrated ones. On average, the RMSE has a uniform distribution among the first 8 types of aberrations, except for radially symmetric ones. For aberrations of the Z₂₀ и Z₄₀ types, corresponding to defocusing and spherical aberration, an almost twofold increase in the RMSE is observed over the entire range of c_nm from 0.1 to 1.

Figure 13 shows the average values of the RMSE of linear and cylindrical interferograms. It has been found that the RMSE increases with an increase in the aberration level c_nm and barely depends on the tilt angle of the reference beam α. However, for cylindrical interferograms, the best sensitivity is at α = 4π–6π, while for a flat reference beam at α = 2π. This narrows the applicability range of a flat reference beam. In addition, at levels of aberration c_nm ≥ 0.5, the sensitivity of cylindrical interferograms is higher than that of linear ones.

For clarity, the value Δ is introduced—the sensitivity of interferogram recognition by machine learning methods (11): if Δ ≥ 0, then the sensitivity of the cylindrical interferogram is higher than the linear one, otherwise the sensitivity is lower.

Figure 14 shows diagrams of the average sensitivity for various types of aberrations (Q = 8) in cylindrical and linear interferograms with variations in the parameters α and c_nm. It can be seen from the graphs in Figure 14 that, for small aberrations, the cylindrical interferogram is less sensitive than the linear one. The cylindrical interferogram is more sensitive than the linear one for aberrations with levels of c_nm ≥ 0.3.

For clarity, we will calculate the information content (12) of cylindrical and linear interferograms for different types and levels of aberrations with a variation of the parameter α, which shows how many times one type of interferogram is more effective for recognizing aberrations from interferograms using convolutional neural networks.

With an average value of aberrations, $0.3<c_{nm}<0.7$ cylindrical interferograms are more sensitive than linear ones, regardless of the parameter α. At high levels of aberrations, $c_{nm}>0.7$ cylindrical interferograms are also consistently more sensitive than linear ones.

Figure 15 shows diagrams of information content (12) E of cylindrical and linear interferograms for different types and levels of aberrations with variations in the parameter α.

Thus, the maximum information content of cylindrical interferograms is achieved when detecting medium (≥0.5) and strong aberrations (>0.7). In this case, the cylindrical interferogram is 15% more informative than the linear one. The average information content in detecting medium and strong aberrations, regardless of the frequency of the interferogram, is 9%.

A 15% increase in information content parameter (Figure 16) for certain wave aberrations is shown (defocusing, astigmatism, spherical aberration, and 2nd order astigmatism) when used as a reference beam, a cubic cylindrical wavefront instead of a flat wavefront. Moreover, the use of cylindrical interferograms makes it possible to improve the results of neural network recognition (Table 6). The average absolute error decreases in the range of 20% to 30%.

It is worth noting that, among the limitations of the proposed approach, one can single out the influence of noise and other artifacts that affect the accuracy of the results of the recognition of aberrations. In addition, a reference plane beam can be easily created experimentally, and any structural reference beams (including those with a cylindrical wavefront) require additional optical elements, which is also a limitation of the interference method.

Thus, we can talk about some correlations between sensitivity and MAE recognition by the neural network. This makes it possible to predict which types of reference beams will give the best result before training the neural network.

5. Conclusions

A study was made on the sensitivity of interferograms formed using structured reference beams with a flat and cylindrical wavefront. The maximum sensitivity of cylindrical interferograms is achieved by detecting medium and strong aberrations. In this case, the cylindrical interferogram is more sensitive than the linear one by 0.03. The average sensitivity for detecting strong aberrations, regardless of the tilt angle α, is 0.025.

The increased sensitivity and information content of the proposed interferograms compared to classical (linear) ones based on a flat wavefront is confirmed by a decrease in the RMSE values and an expanded range of reference beam parameters, which allows more accurate tuning of the wavefront sensor to a specific range of the level of wave aberration.

The use of reference beams with cylindrical wavefronts is proposed to improve the recognition of aberrations from interferograms using convolutional neural networks. The sensitivity of interferograms (the change in the aberrated interferogram relative to the reference one) when using a cylindrical reference beam increases by at least 10% compared to a flat reference beam for radially asymmetric types of aberrations, and the mean absolute error of aberration recognition decreases from 0.0068 to 0.0047.

A comparative analysis between the use of a flat and cylindrical reference beam for wave aberration recognition from interferograms using neural networks showed that the average absolute error is reduced by more than 30%.

It should be noted that a cylindrical interferogram with a reference beam tilt angle α in the range from 2π to 6π turned out to be 15% more efficient than a high-frequency conical interferogram. This conclusion was made on the basis of data on the mean absolute recognition error [48] of conical interferograms.

Considering that optical interferometry is already widely used in many applications, such as precision measurements, astrophysics, seismology, quantum informatics, biomedical imaging, and digital contouring and deformation analysis in mechanics, an extension of the method with the help of various reference beams in combination with intelligent data analysis will preserve the advantages of the wavefront sensor and eliminate some of the disadvantages, depending on the specific application task.