Application of Machine Learning Methods for Identifying Wave Aberrations from Combined Intensity Patterns Generated Using a Multi-Order Diffractive Spatial Filter

Round 1

Reviewer 1 Report

Comments and Suggestions for Authors

In this paper, the authors have studied the application of ML techniques for identification of the wave aberration. This study has a certain reference value. In my opinion, the paper is well written. However, the following comments must be considered before it can be accepted.

1. In Section 1, what is the novelty of this work? The authors should explain it.
2. The lines in figure are hard to recognize when it is print. The figures can be changed.
3. How long does ML take?
4. Can the results be applied in industry? Please explain in abstract and conclusion.

Author Response

In this paper, the authors have studied the application of ML techniques for identification of the wave aberration. This study has a certain reference value. In my opinion, the paper is well written. However, the following comments must be considered before it can be accepted.

We are thankful for useful comments and suggestions, which allow us to improve the quality of the manuscript making it more clear for readers.

All changes in the manuscript are highlighted by yellow color.

Point 1.

In Section 1, what is the novelty of this work? The authors should explain it.

Response 1.

Thank you for your suggestion.

We have added the following additional explanation at the end of Section 1:

“A feature of combined intensity patterns is the conjunction of distributions formed on the basis of both standard and phase Zernike functions. Different types of distributions correspond to different diffraction orders formed at given locations of the focal plane. This approach allows obtaining significantly more information about the studied wave front, although it complicates the focal pattern. Machine learning methods are used to analyze the obtained complex combined distributions.”

Point 2.

The lines in figure are hard to recognize when it is print. The figures can be changed.

Response 2.

Thank you for your recommendation.

We have replaced the graph lines in Fig. 6. The blue solid line represents the “Training loss”, and the yellow dashed line represents the “Validation loss”.

Point 3.

How long does ML take?

Response 3.

We measured the training time and added the corresponding text to the article.

"For 80 training epochs, after no more than 27 minutes of training time on a single RTX 4070 Super GPU, the absolute recognition error (MAE) was achieved, which does not exceed 0.0028.”

Point 4.

Can the results be applied in industry? Please explain in abstract and conclusion.

Response 4.

Thank you for your valuable comment.

We have added the following text in Abstract:

“New advanced aberrometers and multichannel diffractive optics technologies can be used in industry for quality control of optical elements, assessing optical system alignment errors, and early-stage detection of eye diseases.”

We have added the following text in Conclusion:

“The obtained results may be useful for measuring and correcting wavefront aberrations in astronomical observations, microscopy, optical communication and coding, ophthalmology and other imaging and focusing systems.”

Author Response File: Author Response.pdf

Reviewer 2 Report

Comments and Suggestions for Authors

This paper presents a method for identifying wave aberrations using a multi-order diffractive spatial filter and machine learning. A dataset of combined intensity patterns was created, and a CNN based on the Xception architecture was trained. The model achieved a low mean absolute error and was validated through experiments, showing the effectiveness of the approach in optical systems.

Questions 1. How would the performance of the CNN model change if the size of the dataset was significantly increased or decreased? 2. Are there other machine - learning architectures that could potentially outperform the Xception - based CNN for this wave aberration identification task? 3. In real - world applications, how can the influence of environmental factors on the performance of the multi - order diffraction spatial filter be minimized? 4. Can the proposed method be extended to identify higher - order Zernike aberrations beyond the 4th order? 5. What is the potential impact of manufacturing imperfections in the diffractive optical elements on the accuracy of wave aberration identification? 

Author Response

This paper presents a method for identifying wave aberrations using a multi-order diffractive spatial filter and machine learning. A dataset of combined intensity patterns was created, and a CNN based on the Xception architecture was trained. The model achieved a low mean absolute error and was validated through experiments, showing the effectiveness of the approach in optical systems.

We are thankful for useful comments and suggestions, which allow us to improve the quality of the manuscript making it more clear for readers.

All changes in the manuscript are highlighted by yellow color.

 

Questions

Point 1.

How would the performance of the CNN model change if the size of the dataset was significantly increased or decreased?

Response 1.

We measured the training time and added the corresponding text to the article.

"For 80 training epochs, after no more than 27 minutes of training time on a single RTX 4070 Super GPU, the absolute recognition error (MAE) was achieved, which does not exceed 0.0028.”

By increasing the dataset due to new types of aberrations, the CNN will learn to recognize more types of distortions, but the training will take longer. If we increase the data density for each type of aberration (reducing the discretization step by the weight of the analyzed aberration), the CNN model will more accurately determine the degree of distortion, although it will require additional computing resources. The opposite situation occurs when reducing data: recognition accuracy decreases, training time decreases, and the risk of overtraining increases.

Point 2.

Are there other machine - learning architectures that could potentially outperform the Xception - based CNN for this wave aberration identification task?

Response 2.

The use of residual connections (ResNet50 Architecture) can provide high accuracy in classifying aberration types and their magnitudes from intensity distributions with several diffraction orders obtained using multichannel DOEs. While Xception's depthwise separable convolutions enable highly expressive feature extraction at the cost of increased sensitivity to optimization dynamics, ResNet50’s use of residual connections and standard convolutional blocks provides more stable and robust gradient flow during early training phases.

We have added the following text in Discussion:

Further prospects are related to conducting theoretical research on the selection of the optimal architecture (ResNet, Xception, MobileNet, EfficientNet, etc.) of a neural network, using the resulting knowledge base with images of multi-channel pictures.

Point 3.

In real - world applications, how can the influence of environmental factors on the performance of the multi - order diffraction spatial filter be minimized?

Response 3.

We have added the following text in Discussion:

“In real applications, the main problem may be the misalignment of the diffraction filter and focusing system. In this case, two approaches to solving this problem are possible. The first is to create a monolithic filter-lens system or combine the filter and lens on one diffraction element. The second approach is based on the fact that any errors introduced by environmental factors can be taken into account in additional training of the used neural network.”

Point 4.

Can the proposed method be extended to identify higher - order Zernike aberrations beyond the 4th order?

Response 4.

We have added the following text in Section 3.1:

“It is possible to extend the proposed method to detect Zernike aberrations of higher order than 4th and with a weight greater than 0.5 wavelength by adding new diffraction orders in optical element. Dynamic restructuring of the optical element, including using SLM, allows changing both the set of analyzed types of aberrations and varying their magnitude.”

Point 5.

What is the potential impact of manufacturing imperfections in the diffractive optical elements on the accuracy of wave aberration identification? 

Response 5.

We have added the following text in Discussion:

“In case of manufacturing errors of the diffractive optical elements, somewhat distorted distributions will be formed in the focal plane of the lens. However, the factors of such distortion can be taken into account and leveled out by retraining of the neural network. For example, the works are known [Peng Y, Fu Q, Amata H, Su S, Heide F, Heidrich W. Computational imaging using lightweight diffractiverefractive optics. Opt Express 2015; 23(24): 31393-31407. DOI: 10.1364/OE.23.031393; Nikonorov AV, Petrov MV, Bibikov SA, Kutikova VV, Morozov AA, Kazanskiy NL. Image restoration in diffractive optical systems using deep learning and deconvolution. Computer Optics 2017; 41(6): 875-887. DOI: 10.18287/2412-6179-2017-41-6-875-887; Khonina S.N., Kazanskiy N.L., Oseledets I.V., Nikonorov A.V., Butt M.A., Synergy between Artificial Intelligence and Hyperspectral Imagining—A Review, Technologies, 12, 163-(32p) (2024), https://doi.org/10.3390/technologies12090163] in which errors in image formation were compensated by subsequent processing using machine learning methods.”

Author Response File: Author Response.pdf

Reviewer 3 Report

Comments and Suggestions for Authors

This paper presents an innovative method for wavefront aberration identification using multi-order diffractive spatial filters integrated with an Xception-based convolutional neural network. The work demonstrates novelty in theoretical modeling, experimental validation, and algorithm design. The results that the recognition error was shown to be independent of both the type and weight of the aberrations analyzed are impressing. This work can provide a valuable reference for the interdisciplinary field of optical detection and machine learning. 

Here are my specific recommendations for improving the manuscript:

1. The experimental schematic in Figure 1 appears overly complex and reliant on high-end equipment (DMD, SLM, etc.). Given the prohibitive costs of these components for many research groups, the authors should discuss potential cost-effective alternatives for implementing their method.
2. Figure 6 lacks essential axis labels and measurement units, significantly hindering interpretability.
3. Critical variables in Equations (1)-(3) remain undefined. r，φ，m，n
4. To strengthen the machine learning context, consider citing recent advances at the optics-AI intersection in quantum metrology:
   [1] Wang, Z.; Lu, J.; Liu, Z.; et al. Neural network assisted magnetic moment measurement using an atomic magnetometer. IEEE Trans. Instrum. Meas. ​​2025​​, 74, 1–10.
   [2] Ge, X.; Liu, G.; Fan, W.; et al. Decoupling measurement and closed-loop suppression of transverse magnetic field drift in a modulated double-cell atomic comagnetometer. Measurement ​​2025​​, 250, 117123.
   [3] Qin, J.N.; Xu, J.X.; Jiang, Z.Y.; et al. Enhanced all-optical vector atomic magnetometer enabled by artificial neural network. Appl. Phys. Lett. ​​2024​​, 125, 102405.
   [4] Huang, J.; Zhuang, M.; Zhou, J.; et al. Quantum metrology assisted by machine learning. Adv. Quantum Technol. ​​2024​​, 7, 2300281.

Author Response

This paper presents an innovative method for wavefront aberration identification using multi-order diffractive spatial filters integrated with an Xception-based convolutional neural network. The work demonstrates novelty in theoretical modeling, experimental validation, and algorithm design. The results that the recognition error was shown to be independent of both the type and weight of the aberrations analyzed are impressing. This work can provide a valuable reference for the interdisciplinary field of optical detection and machine learning. 

We are thankful for useful comments and suggestions, which allow us to improve the quality of the manuscript making it more clear for readers.

All changes in the manuscript are highlighted by yellow color.

Here are my specific recommendations for improving the manuscript:

Point 1.

The experimental schematic in Figure 1 appears overly complex and reliant on high-end equipment (DMD, SLM, etc.). Given the prohibitive costs of these components for many research groups, the authors should discuss potential cost-effective alternatives for implementing their method.

Response 1.

Thanks for the great comment. Indeed, we did not provide an explanation in the manuscript about possible practical implementation of the proposed technique. The setup in this paper is only necessary to validate the experimental data and obtain an experimental dataset. Ultimately, if you have a trained convolutional neural network, it would be possible to replace the SLM with a static DOE made of any suitable material and determine the wavefront aberrations by one intensity distribution. We have added the following paragraph to Section 2.4:

“The experimental setup used in this study was designed to validate the proposed aberration detection method using DOEs. The configuration allows for modifications - for instance, the DMD can be replaced with LCoS SLM. The experimental results show strong agreement with numerical simulations, confirming the method's validity and making the data suitable for training convolutional neural networks. Since the developed DOE is used as a static optical element, it can be transferred onto a photosensitive material using lithography or holography techniques. This enables the replacement of programmable SLMs with static DOEs for industrial applications. In such a configuration, the test sample illuminated by a plane wave serves the function of the DMD, while the SLM is replaced by a fabricated DOE and the intensity distribution is registered by the matrix photodetector in single-shot exposure. In this way the system can be realized for mass production and real aberration measurement applications.”

Point 2.

Figure 6 lacks essential axis labels and measurement units, significantly hindering interpretability.

Response 2.

Thank you for your recommendation.

We have replaced Fig. 6. The figure represents the "MAE value of the Loss function for each epoch during the training process". The blue solid line represents the “Training loss”, and the yellow dashed line represents the “Validation loss”. We have added the main axis labels (MAE Value and Epochs). The units of measurement are dimensionless.

Point 3.

Critical variables in Equations (1)-(3) remain undefined. r，φ，m，n

Response 3.

Thank you for your comment.

We have added the following text in Section 2:

“(1), where r and φ are polar coordinates, С_nm is the aberration weight coefficient, Z_nm(r,φ ) is the Zernike functions of order (n,m), n is radial index, m is azimuthal index [Lakshminarayanan, V., Fleck, A. Zernike polynomials: a guide. Journal of Modern Optics, 58(7), 545–561 (2011). https://doi.org/10.1080/09500340.2011.554896; Kuo Niu and Chao Tian, Zernike polynomials and their applications, 2022 J. Opt. 24 123001]…”

Point 4.

To strengthen the machine learning context, consider citing recent advances at the optics-AI intersection in quantum metrology:

Response 4.

Thank you for your recommendation!

We have added the following text in Discussion:

The advantages of using ML in optical systems are considerable, including process automation, high accuracy, and flexibility, including in tasks at the intersection of optics and artificial intelligence in quantum metrology [Wang, Z.; Lu, J.; Liu, Z.; et al. Neural network assisted magnetic moment measurement using an atomic magnetometer. IEEE Trans. Instrum. Meas. ​​2025​​, 74, 1–10; Ge, X.; Liu, G.; Fan, W.; et al. Decoupling measurement and closed-loop suppression of transverse magnetic field drift in a modulated double-cell atomic comagnetometer. Measurement ​​2025​​, 250, 117123; Qin, J.N.; Xu, J.X.; Jiang, Z.Y.; et al. Enhanced all-optical vector atomic magnetometer enabled by artificial neural network. Appl. Phys. Lett. ​​2024​​, 125, 102405; Huang, J.; Zhuang, M.; Zhou, J.; et al. Quantum metrology assisted by machine learning. Adv. Quantum Technol. ​​2024​​, 7, 2300281]

Author Response File: Author Response.pdf

Reviewer 4 Report

Comments and Suggestions for Authors

The authors develop a machine learning approach fo calculation and identification of aberrations on a wavefront. The original method is based on the use of SLM allowing to recognize aberrations with the Zernike basis. It is realized with a multichannel DOE on DMD-SLM. The developments of the Zernike functions is well introduced as well as the 49 channels filters of the Fourier correlator. I outline the following points to consider in the final form of the paper: 

The DOE is the important holographic component for the experiment. Detail which type of binary hologram is used for amplitude and phase coding. Multilevel phase coding? - Maximum efficiency - limited spatial resolution. The optical setup generates aberrations due to Fourier lenses- SLMs. It is not clear if the DOE takes account of these aberrations. Which criteria in comparaison with the measured wave aberrations.

The paper consider the superposition of two wave aberrations. Does the operating principle apply to a higher number of superposed aberrations as observed in applications. The aberrations are in the range of +or - 0.5 lambda. Does the DOE approach also operates with much higher values of wave aberrations. Also precise the operating conditions for measuremment of time varying wave aberrations on the incident wavefront. 

Coherent speckle noise may contribute to limit the sensitivity of the measurements and identification of aberrations and SNR ?  

To conclude the manuscript brings significant new results and original methods based on DOE-SLMs for high sensitivity wavefront aberrations measurements. The results are of interest for applications but it is required to take account of comments and questions of the review in the final form  to be published in the journal.   

Author Response

The authors develop a machine learning approach fo calculation and identification of aberrations on a wavefront. The original method is based on the use of SLM allowing to recognize aberrations with the Zernike basis. It is realized with a multichannel DOE on DMD-SLM. The developments of the Zernike functions is well introduced as well as the 49 channels filters of the Fourier correlator.

We are thankful for useful comments and suggestions, which allow us to improve the quality of the manuscript making it more clear for readers.

All changes in the manuscript are highlighted by yellow color.

I outline the following points to consider in the final form of the paper: 

Point 1.

The DOE is the important holographic component for the experiment. Detail which type of binary hologram is used for amplitude and phase coding.

- Multilevel phase coding

- Maximum efficiency

- limited spatial resolution

Response 1.

Thank you for your valuable comment.

We have added the following text in Section 3.1:

“The partial coding method is multilevel phase coding, oriented towards application with spatial light modulators, is as follows: (12), (13), (14) where  τ(x,y) is the initial amplitude-phase transmission function; α is the parameter responsible for the threshold amplitude value at which a phase jump will be added to the point; μ is the magnitude of the phase jump;  S_ij is a pseudo-random variable, the sign of which determines the magnitude of the phase jump;  τ^~(x,y) is the calculated phase transmission function.”

 “Figure 4 shows the phase of the calculated phase DOE (12) with the parameter value α=1/π, which makes it possible to calculate a multichannel DOE with minimal error and increased diffraction efficiency [Khonina SN, Kotlyar VV, Soifer VA Techniques for encoding composite diffractive optical elements. Proc. SPIE 2003; 5036:493-498. DOI: 10.1117/12.498521]. The maximum diffraction efficiency is 60%, while the error in forming the intensity pattern in the focal plane of the DOE is 0.2.”

 Limited spatial resolution was discussed in section 3.1 previously:

“The physical size of the proposed DOE is 5×5 mm^2, provided that the size of one pixel is approximately 12×12 μm ^2. It is worth noting that currently the most accessible technologies for the manufacture of multi-level DOEs are limited in resolution to approximately 1 μm [Skidanov RV, Moiseev OY, Ganchevskaya SV. Additive Process for Fabrication of Phased Optical Diffraction Elements. J Opt Technol 2016; 83:23–25; Khonina SN, Kazanskiy NL, Butt MA. Grayscale Lithography and a Brief Introduction to Other Widely Used Lithographic Methods: A State-of-the-Art Review. Micromachines 2024; 15: 1321. DOI: 10.3390/ /mi15111321]. The resolution of the proposed 49-channel filter is more than 10 times greater than the critical resolution, ensuring a relatively simple process of applying a diffraction pattern and the possibility of practical manufacture of the DOE. At the same time, the proposed multichannel DOE can be easily implemented using both available transmissive and reflective SLMs with a resolution of more than 1000×700 pixels [Suchkov N, Fernández EJ, Martínez-Fuentes JL, Moreno I, and Artal P. Simultaneous aberration and aperture control using a single spatial light modulator. Opt Express 2019; 27: 12399-12413; Khonina SN, Karpeev SV, Butt MA Spatial-light-modulator-based multichannel data transmission by vortex beams of various orders. Sensors 2021; 21: 2988. DOI: 10.3390/s21092988].”

Point 2.

The optical setup generates aberrations due to Fourier lenses- SLMs. It is not clear if the DOE takes account of these aberrations. Which criteria in comparaison with the measured wave aberrations.

Response 2.

You are absolutely right, that the optical setup generates aberrations. We take these aberrations into account and encode them in the DMD. Thus, no additional criteria are required in comparison with the measured wave aberrations.

Setup aberrations was discussed in section 2.4 previously:

“The aberrations inherent to the system, including those from the DMD, SLMs, lenses, and reflectors, were reconstructed by the off-axis holography methods and encoded into the DMD patterns. This procedure allowed us to correct for these aberrations. The process was repeated six times to mitigate the influence of interference oscillations between the two arms of the setup.”

Point 3.

The paper consider the superposition of two wave aberrations. Does the operating principle apply to a higher number of superposed aberrations as observed in applications. The aberrations are in the range of +or - 0.5 lambda. Does the DOE approach also operates with much higher values of wave aberrations.

Response 3.

Thank you for this comment!

We have added the following text in Conclusions:

“This approach is applicable to a larger number of aberrations superpositions. This is confirmed by the successful detection up to 5 aberrations superpositions separately by a filter matched to Zernike functions (5) [Volotovskiy, S., Khorin, P., Dzyuba, A. et al. Adaptive Compensation of Wavefront Aberrations Using the Method of Moments. Opt. Mem. Neural Networks 33 (Suppl 2), S359–S375 (2024). https://doi.org/10.3103/S1060992X24700644] and a filter matched to wave aberrations (6) [Khorin, PA, Porfirev AP, Khonina, SN Adaptive detection of wave aberrations based on the multichannel filter / Pho-tonics, 9(3), 204 (2022). https://doi.org/10.3390/photonics9030204].”

 We have added the following text in Section 3.1:

“It is possible to extend the proposed method to detect Zernike aberrations of higher order than 4th and with a weight greater than 0.5 wavelength by adding new diffraction orders in optical element. Dynamic restructuring of the optical element, including using SLM, allows changing both the set of analyzed types of aberrations and varying their magnitude.”

Point 4.

Also precise the operating conditions for measuremment of time varying wave aberrations on the incident wavefront. 

Response 4.

Thank you for your recommendation!

We have added the following text in Discussion:

“Correct measurement of time-varying wave aberrations in the incident wave front is related to the technical characteristics of the recording and modulating devices. In the conditions of implementation of the developed spatial filters using a spatial light modulator (SLM), a refresh rate of 120 Hz is possible for most liquid crystal SLMs. The use of a digital micromirror device (DMD) provides a much higher refresh rate (up to 32 kHz), but the binary nature of the device implies additional coding of the output masks.”

We have added the following text in Section 2.4:

At the same time, stabilization against vibration interference is provided.

“In the experiment, we use the single-beam setup, which is resistant to mechanical vibrations. In order to have repeatable results, it is enough to fix all elements on the same platform. Unlike the two-beam schemes with beam separation by different optical elements, the proposed scheme does not require an additional vibration isolation system. Also in favor of stability is the advantage of a wide beam with a diameter of 5.12 mm. Aberration correction is mainly required for optical elements with larger optical aperture.”

Point 5.

Coherent speckle noise may contribute to limit the sensitivity of the measurements and identification of aberrations and SNR?

Response 5.

Thank you for your helpful question.

We have added the following text in Discussion:

Yes, of course, the presence of coherent speckle noise produces blurring of the focal spots. However, when comparing experimental data with numerical modeling, we did not find significant blurring and differences in the generated pattern distributions. Judging from our data, the approach proposed in the manuscript is robust to coherent speckle noise because they are cut off by spatial filtering as high-frequency components.

To conclude the manuscript brings significant new results and original methods based on DOE-SLMs for high sensitivity wavefront aberrations measurements. The results are of interest for applications but it is required to take account of comments and questions of the review in the final form to be published in the journal.   

Author Response File: Author Response.pdf

Round 2

Reviewer 2 Report

Comments and Suggestions for Authors

The paper can be accepted.