Vectorial differentiation of the profile of a light beam with respect to two spatial coordinates upon reflection from an interface at Brewster’s angle

Kashapov A.I., Porfiriev, A.P., Bezus, E.A., Bykov, D.A., Doskolovich, L.L.

For Russian citation (Opticheskii Zhurnal):

Кашапов А.И., Порфирьев А.П., Безус E.А., Быков Д.А., Досколович Л.Л. Векторное дифференцирование профиля светового пучка по двум пространственным координатам при отражении от границы раздела под углом Брюстера // Оптический журнал. 2026. Т. 93. № 2. С. 3–12. http://doi.org/10.17586/1023-5086-2026-93-02-03-12

Kashapov A.I., Porfirev A.P., Bezus E.A., Bykov D.A., Doskolovich L.L. Vectorial differentiation of the profile of a light beam with respect to two spatial coordinates upon reflection from an interface at Brewster’s angle [in Russian] // Opticheskii Zhurnal. 2026. V. 93. № 2. P. 3–12. http://doi.org/10.17586/1023-5086-2026-93-02-03-12

For citation (Journal of Optical Technology):

Abstract:

Subject of study. The operation of vectorial differentiation of an optical beam consisting in computing derivatives with respect to two spatial variables of the nonzero transverse electric field component of a linearly polarized incident beam in two transverse field components of the beam reflected from an interface at Brewster’s angle. Purpose. The development of a theoretical description and experimental confirmation of the optical implementation of vectorial differentiation for a linearly polarized optical beam incident on an interface at Brewster’s angle. Method. For the theoretical description of the vectorial differentiation operation, the transformation of a linearly polarized incident beam occurring upon reflection from an interface at Brewster’s angle was described using a vectorial transfer function. To validate the theoretical results, electromagnetic numerical simulations and optical experiment were carried out, demonstrating the application of vectorial differentiation for edge detection of phase objects. Main results. The feasibility of vectorial differentiation for a beam incident at Brewster’s angle was theoretically and experimentally confirmed. It was shown that by introducing a quarter-wave plate and a polarizer into the path of the reflected beam, isotropic vectorial differentiation can be achieved, for which the intensity of the reflected beam is proportional to the squared magnitude of the gradient of the electric field of a linearly polarized incident beam. Practical significance. The obtained results may be used for the development of novel optical systems for image processing.

Keywords:

optical differentiation, Brewster’s angle, transfer function, optical edge detection, optical experiment

Acknowledgements:

this work was supported by the Russian Science Foundation (Project № 24-12-00028), in the part related to the theoretical description of the vectorial differentiation and the optical experiment, and was carried out within the framework of the state assignment of the National Research Center “Kurchatov Institute”, in the part related to the implementation of the software for simulating the diffraction of an optical beam on an interface.

OCIS codes: 050.1940, 070.4560, 100.1160

References:

1. Silva A., Monticone F., Castaldi G., Galdi V., Alù A., Engheta N. Performing mathematical operations with metamaterials // Science. 2014. V. 343. P. 160–163. https://doi.org/10.1126/science.1242818

2. Mohammadi Estakhri N., Edwards B., Engheta N. Inverse-designed metastructures that solve equations // Science. 2019. V. 363. P. 1333–1338. https://doi.org/ 10.1126/science.aaw2498

3. Zhou Y., Zheng H., Kravchenko I.I., Valentine J. Flat optics for image differentiation // Nat. Photon. 2020. V. 14. P. 316–323. https://doi.org/10.1038/s41566-020-0591-3

4. Zhu T., Zhou Y., Lou Y., Ye H., Qiu M., Ruan Z., Fan S. Plasmonic computing of spatial differentiation // Nat. Commun. 2017. V. 8. P. 15391. https://doi.org/10.1038/ncomms1539

5. Doskolovich L.L., Kashapov A.I., Bezus E.A., Bykov D.A. Spatiotemporal optical differentiation and vortex generation with metal-dielectric-metal multilayers // Phys. Rev. A. 2022. V. 106. P. 033523. https://doi.org/10.1103/PhysRevA.106.033523

6. Doskolovich L.L., Kashapov A.I., Bezus E.A., Golovastikov N.V., Bykov D.A. Optical computation of the Laplace operator at oblique incidence using a multilayer metal-dielectric structure // Opt. Express. 2023. V. 31. P. 17050–17064. https://doi.org/10.1364/OE.489750

7. Wesemann L., Panchenko E., Singh K., Della Gaspera E., Gómez D.E., Davis T.J., Roberts A. Selective near-perfect absorbing mirror as a spatial frequency filter for optical image processing // APL Photon. 2019. V. 4. P. 100801. https://doi.org/10.1063/1.5113650

8. Youssefi A., Zangeneh-Nejad F., Abdollahramezani S., Khavasi A. Analog computing by Brewster effect // Opt. Lett. 2016. V. 41. P. 3467–3470. https://doi.org/ 10.1364/OL.41.003467

9. Нестеренко Д.В., Колесникова М.Д., Любарская А.В. Оптическое дифференцирование на основе эффекта Брюстера // Компьютерная оптика. 2018. Т. 42. № 5. С. 758–763. https://doi.org/10.18287/2412-6179-2018-42-5-758-763

Nesterenko D.V., Kolesnikova M.D., Lyubarskaya A.V. Optical differentiation based on the Brewster effect [in Russian] // Computer Optics. 2018. V. 42. № 5. P. 758–763. https://doi.org/10.18287/2412-6179-2018-42-5-758-763

10. Zhu T., Lou Y., Zhou Y., Zhang J., Huang J., Li Y., Luo H., Wen S., Zhu S., Gong Q., Qiu M., Ruan Z. Generalized spatial differentiation from the spin Hall effect of light and its application in image processing of edge detection // Phys. Rev. Appl. 2019. V. 11. P. 034043. https://doi.org/10.1103/PhysRevApplied.11.034043

11. Zhu T., Huang J., Ruan Z. Optical phase mining by adjustable spatial differentiator // Adv. Photon. 2020. V. 2. P. 016001. https://doi.org/10.1117/1.AP.2.1.016001

12. Doskolovich L.L., Kashapov A.I., Bezus E.A., Bykov D.A. Vectorial spatial differentiation of optical beams with metal–dielectric multilayers enabled by spin Hall effect of light and resonant reflection zero // Opt. Laser Technol. 2025. V. 181. P. 111884. https://doi. org/10.1016/j.optlastec.2024.111884

13. Bykov D.A., Doskolovich L.L., Soifer V.A. Temporal differentiation of optical signals using resonant gratings // Opt. Lett. 2011. V. 36. P. 3509–3511. https://doi.org/10.1364/OL.36.003509

14. Dong Z., Si J., Yu X., Deng X. Optical spatial differentiator based on subwavelength high-contrast gratings // Appl. Phys. Lett. 2018. V. 112. P. 181102. https://doi.org/10.1063/1.5026309

15. Bykov D.A., Doskolovich L.L., Morozov A.A., Podlipnov V.V., Bezus E.A., Verma P., Soifer V.A. Firstorder optical spatial differentiator based on a guidedmode resonant grating // Opt. Express. 2018. V. 26. P. 10997–11006. https://doi.org/10.1364/OE.26.010997

16. Yang W., Yu X., Zhang J., Deng X. Plasmonic transmitted optical differentiator based on the subwavelength gold gratings // Opt. Lett. 2020. V. 45. P. 2295–2298. https://doi.org/10.1364/OL.390566

17. Huang J., Zhang J., Zhu T., Ruan Z. Spatiotemporal differentiators generating optical vortices with transverse orbital angular momentum and detecting sharp change of pulse envelope // Laser Photon. Rev. 2022. V. 16. P. 2100357. https://doi.org/10.1002/lpor.202100357

18. Doskolovich L.L., Bykov D.A., Bezus E.A., Soifer V.A. Spatial differentiation of optical beams using phase-shifted Bragg grating // Opt. Lett. 2014. V. 39. P. 1278–1281. https://doi.org/10.1364/OL.39 .001278

19. Liu Y., Huang M., Chen Q., Zhang D. Single planar photonic chip with tailored angular transmission for multiple-order analog spatial differentiator // Nat. Commun. 2022. V. 13. P. 7944. https://doi.org/10.1038/s41467-022-35588-5

20. Сафронов К.Р., Бессонов В.О., Федянин А.А. Оптимизация многослойных фотонных структур с помощью искусственных нейронных сетей для получения заданного оптического отклика // Письма в ЖЭТФ. 2021. Т. 114. № 6. С. 360–364. https://doi.org/10.31857/S123456782118004X

Safronov K.R., Bessonov V.O., Fedyanin A.A. Optimization of multilayer photonic structures using artificial neural networks to obtain a target optical response // JETP Lett. 2021. V. 114. P. 321–325. https://doi.org/10.1134/S0021364021180119

21. Мусорин А.И., Шорохов А.С., Чежегов А.А., Балуян Т.Г., Сафронов К.Р., Четвертухин А.В., Грунин А.А., Федянин А.А. Подходы фотоники для реализации нейроморфных вычислений // УФН. 2023. Т. 193. С. 1284–1297. https://doi.org/10.3367/UFNr. 2023.07.039505

Musorin A.I., Shorokhov A.S., Chezhegov A.A., Baluyan T.G., Safronov K.R., Chetvertukhin A.V., Grunin A.A., Fedyanin A.A. Photonics approaches to the implementation of neuromorphic computing // Phys. Usp. 2023. V. 66. P. 1211–1223. https://doi.org/10.3367/UFNe.2023.07.039505

22. Levkovskaya V.M., Kharitonov A.V., Kharintsev S.S. Time-varying materials for analog optical computing // J. Opt. Technol. 2024. V. 91. № 5. P. 293–299. https://doi.org/10.1364/JOT.91.000293

23. Verhoglyad A., Zavyalova M., Kachkin A., Kokarev S.A., Korolkov V.P. Circular laser writing system for generating phase and amplitude microstructures on spherical surfaces // Sens. Syst. 2015. V. 9–10. P. 45–52.