Some features concerning calculation of the refractive index profile in the Wood′s lens

S.P. Skobelev; M.M. Kushneryov

doi:10.18127/j00338486-202610-18

Some features concerning calculation of the refractive index profile in the Wood′s lens

research article UDC 519.6; 535.4; 621.396.67 Pages: 223-232 doi.org/10.18127/j00338486-202610-18

¹ PJSC “Radiofizika” (Moscow, Russia); Moscow Institute of Physics and Technology (National Research University) (Moscow, Russia)
² Moscow Institute of Physics and Technology (National Research University) (Moscow, Russia)

Keywords: Inhomogeneous media geometric optics Wood’s lens refractive index profile integral equations numerical methods

Abstract

Some features of one known approach to calculation of the refractive index profile in the Wood′s lens are considered. The approach is based on the assumption that the square of the refractive index on the lens part located between the entry and exit points of a ray is determined by a linear function. The indicated feature allows one to calculate the index profile both as a function of coordinate of the entry point and as a function of coordinate of the exit point. It is shown that the indicated approach is not self-consistent because the indicated profiles do not coincide with each other in the common region of the coordinates.

A new algorithm designed for calculation of the piecewise-linear refractive index profile is proposed. The algorithm is based on subsequent calculation of the refractive index values in the exit points and coordinates of the latters with use of already available values for the index in the entry points and coordinates of the latters on the basis of solution of the rigorous integral equation formulated from the condition of equality of the eikonals for the rays passing through the lens.

A comparison has shown that the refractive index profile in the exit points calculated according to the reference approach reveals good coincidence with the results obtained with use of the new self-consistent algorithm proposed in the paper.

Received: 16.09.2026
Approved after review: 21.09.2026
Accepted: 30.09.2026

How to cite

Скобелев С.П., Кушнерёв М.М. Некоторые особенности, касающиеся расчета профиля показателя преломления в линзе Вуда* // Радиотехника. 2026. Т. 90. № 10. С. 223−232. DOI: https://doi.org/10.18127/j00338486-202610-18

References

Vud R. Fizicheskaya optika. L.-M.: ONTI. 1936 (in Russian).
Majer V.V. Svet v opticheski neodnorodnoj srede: uchebnye issledovaniya. M.: FIZMATLIT. 2007 (in Russian).
Fel'd Ya.N., Benenson L.S. Antenno-fidernye ustrojstva. M.: Izdanie VVIA imeni professora N.E. Zhukovskogo. 1959 (in Russian).
Zelkin E.G., Petrova R.A. Linzovye antenny. M.: Sovetskoe radio. 1974 (in Russian).
Imbert M., Papió A., De Flaviis F., Jofre L., Romeu J. Design and performance evaluation of a dielectric flat lens antenna for millimeter-wave applications. IEEE Antennas and Wireless Propagation Letters. 2015. V. 14. P. 342−345. DOI: 10.1109/LAWP.2014.2363596.
Melendro-Jimenez J., Sanchez-Olivares P., Tamayo-Dominguez A., Sun X., Fernandez-Gonzalez J.M. 3D printed directive beam-steering antenna based on gradient index flat lens with an integrated polarizer for dual circular polarization at W-band. IEEE Transactions on Antennas and Propagation, 2023. V. 71. № 1. P. 1059−1064. DOI: 10.1109/TAP.2022. 3217177.
Song L.-Z., Ansari M., Qin P.-Y., Maci S., Du J., Guo Y.J. Two-dimensional wide-angle multibeam flat GRIN lens with a high aperture efficiency. IEEE Transactions on Antennas and Propagation. 2023. V. 71. № 10. P. 8018−8029. DOI: 10.1109/TAP.2023.3298143.
Ansari M., Song L., Qin P.-Y., Smith S. L., Guo Y.J. Advances in multibeam flat GRIN lens antennas: A promising 3D-printed design. IEEE Antennas and Propagation Magazine. 2026. V. 68. № 1. P. 47−58. DOI: 10.1109/MAP.2025.3588059.
Balabukha N.P., Menshikh N.L., Solosin V.S. Development of a plane lens for a tapered anechoic chamber. 2022 IEEE 8th All-Russian Microwave Conference (RMC). Moscow. Russian Federation, 2022. P. 187−190. DOI: 10.1109/RMC55984.2022.10079487.
Korolev D.P., Men'shih N.L., Fedorov S.A. Ploskaya linza kak kollimator dlya rupornoj bezekhovoj kamery. Sovremennaya elektrodinamika. 2025. T. 4. № 4 (18). S. 13−23. DOI: 10.24412/2949-0553-2025-418-13-23 (in Russian).
Chen X., Ma H.F., Yang X., Cheng Q., Jiang W.X., Cui T.J. X-band high directivity lens antenna realized by gradient index metamaterials. 2009 Microwave Conference. Singapore. 7−10 December 2009. P. 793−797. http://dx.doi.org/10.1109/APMC.2009.5384269.
Zhang Y., Mittra R., Hong W. On the synthesis of a flat lens using a wideband low-reflection gradient-index metamaterial. Journal of Electromagnetic Waves and Applications. 2011. V. 25. № 16. P. 2178−2187. https://doi.org/10.1163/156939311798147015.
Zhang S., Arya R.K., Pandey S., Vardaxoglou Y., Whittow W., Mittra R. 3D-printed planar graded index lenses. IET Microw., Antennas Propag. 2016. V. 10. № 13. P. 1411–1419. DOI: 10.1049/iet-map.2016.0013.
Maggiorelli F., Paraskevopoulos A., Vardaxoglou J.C., Albani M., Maci S. Profile inversion and closed form formulation of compact GRIN lenses. IEEE Open Journal of Antennas and Propagation. 2021. V. 2. P. 315−325. DOI: 10.1109/OJAP.2021.3059468.
Zhang S., Arya R.K., Whittow W.G., Cadman D., Mittra R., Vardaxoglou J.C. Ultra-wideband flat metamaterial GRIN lenses assisted with additive manufacturing technique. IEEE Transactions on Antennas and Propagation. 2021. V. 69. № 7. P. 3788−3799. DOI: 10.1109/TAP.2020.3044586.
Paraskevopoulos A., Maggiorelli F., Albani M., Maci S. Radial GRIN lenses based on the solution of a regularized ray congruence equation. IEEE Transactions on Antennas and Propagation. 2022. V. 70. № 2. P. 888−899. DOI: 10.1109/TAP.2021.3111315.
Whittaker T., Zhang S., Powell A., Stevens C.J., Vardaxoglou J.Y.C., Whittow W. 3D printing materials and techniques for antennas and metamaterials: A survey of the latest advances. IEEE Antennas and Propagation Magazine. 2023. V. 65. № 3. P. 10−20. DOI: 10.1109/MAP.2022.3229298.
Mikaelyan A.L. Ispol'zovanie sloistoj sredy dlya fokusirovki voln. Doklady AN SSSR. 1951. T. 81. № 4. S. 569−571 (in Russian).
Cornbleet S. Flat disc, radially nonhomogeneous, lenses. IEE Proc. 1980. V. 127. Pt. H. № 6. P. 330−334.
Kelleher K.S., Goatley C. Dielectric lens for microwaves. Electronics. 1955. № 8. P. 142−145.
Kushneryov M.M., Skobelev S.P. Dvumernaya zadacha elektromagnitnogo rasseyaniya na linze Mikaelyana, dva metoda resheniya i osobennosti fokusirovki. Fizicheskie osnovy priborostroeniya. 2020. T. 9. № 4(38). S. 38−47 (in Russian).
Veneckij A.S., Kaloshin V.A. Sintez gradientnoj linzovoj antenny s osevoj simmetriej. Radiotekhnika i elektronika. 1991. T. 36. № 12. S. 2301−2307 (in Russian).
Abella C., Vazquez J., Marin M., Peces J., Graham R., Crone G. Analysis and synthesis of planar non-homogeneous lenses. 1993 23rd European Microwave Conference. Madrid. Spain. 1993. Р. 893−895. DOI: 10.1109/EUMA.1993.336741.

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[1] Vud R. Fizicheskaya optika. L.-M.: ONTI. 1936 (in Russian).

[2] Majer V.V. Svet v opticheski neodnorodnoj srede: uchebnye issledovaniya. M.: FIZMATLIT. 2007 (in Russian).

[3] Fel'd Ya.N., Benenson L.S. Antenno-fidernye ustrojstva. M.: Izdanie VVIA imeni professora N.E. Zhukovskogo. 1959 (in Russian).

[4] Zelkin E.G., Petrova R.A. Linzovye antenny. M.: Sovetskoe radio. 1974 (in Russian).

[5] Imbert M., Papió A., De Flaviis F., Jofre L., Romeu J. Design and performance evaluation of a dielectric flat lens antenna for millimeter-wave applications. IEEE Antennas and Wireless Propagation Letters. 2015. V. 14. P. 342−345. DOI: 10.1109/LAWP.2014.2363596.

[6] Melendro-Jimenez J., Sanchez-Olivares P., Tamayo-Dominguez A., Sun X., Fernandez-Gonzalez J.M. 3D printed directive beam-steering antenna based on gradient index flat lens with an integrated polarizer for dual circular polarization at W-band. IEEE Transactions on Antennas and Propagation, 2023. V. 71. № 1. P. 1059−1064. DOI: 10.1109/TAP.2022. 3217177.

[7] Song L.-Z., Ansari M., Qin P.-Y., Maci S., Du J., Guo Y.J. Two-dimensional wide-angle multibeam flat GRIN lens with a high aperture efficiency. IEEE Transactions on Antennas and Propagation. 2023. V. 71. № 10. P. 8018−8029. DOI: 10.1109/TAP.2023.3298143.

[8] Ansari M., Song L., Qin P.-Y., Smith S. L., Guo Y.J. Advances in multibeam flat GRIN lens antennas: A promising 3D-printed design. IEEE Antennas and Propagation Magazine. 2026. V. 68. № 1. P. 47−58. DOI: 10.1109/MAP.2025.3588059.

[9] Balabukha N.P., Menshikh N.L., Solosin V.S. Development of a plane lens for a tapered anechoic chamber. 2022 IEEE 8th All-Russian Microwave Conference (RMC). Moscow. Russian Federation, 2022. P. 187−190. DOI: 10.1109/RMC55984.2022.10079487.

[10] Korolev D.P., Men'shih N.L., Fedorov S.A. Ploskaya linza kak kollimator dlya rupornoj bezekhovoj kamery. Sovremennaya elektrodinamika. 2025. T. 4. № 4 (18). S. 13−23. DOI: 10.24412/2949-0553-2025-418-13-23 (in Russian).

[11] Chen X., Ma H.F., Yang X., Cheng Q., Jiang W.X., Cui T.J. X-band high directivity lens antenna realized by gradient index metamaterials. 2009 Microwave Conference. Singapore. 7−10 December 2009. P. 793−797. http://dx.doi.org/10.1109/APMC.2009.5384269.

[12] Zhang Y., Mittra R., Hong W. On the synthesis of a flat lens using a wideband low-reflection gradient-index metamaterial. Journal of Electromagnetic Waves and Applications. 2011. V. 25. № 16. P. 2178−2187. https://doi.org/10.1163/156939311798147015.

[13] Zhang S., Arya R.K., Pandey S., Vardaxoglou Y., Whittow W., Mittra R. 3D-printed planar graded index lenses. IET Microw., Antennas Propag. 2016. V. 10. № 13. P. 1411–1419. DOI: 10.1049/iet-map.2016.0013.

[14] Maggiorelli F., Paraskevopoulos A., Vardaxoglou J.C., Albani M., Maci S. Profile inversion and closed form formulation of compact GRIN lenses. IEEE Open Journal of Antennas and Propagation. 2021. V. 2. P. 315−325. DOI: 10.1109/OJAP.2021.3059468.

[15] Zhang S., Arya R.K., Whittow W.G., Cadman D., Mittra R., Vardaxoglou J.C. Ultra-wideband flat metamaterial GRIN lenses assisted with additive manufacturing technique. IEEE Transactions on Antennas and Propagation. 2021. V. 69. № 7. P. 3788−3799. DOI: 10.1109/TAP.2020.3044586.

[16] Paraskevopoulos A., Maggiorelli F., Albani M., Maci S. Radial GRIN lenses based on the solution of a regularized ray congruence equation. IEEE Transactions on Antennas and Propagation. 2022. V. 70. № 2. P. 888−899. DOI: 10.1109/TAP.2021.3111315.

[17] Whittaker T., Zhang S., Powell A., Stevens C.J., Vardaxoglou J.Y.C., Whittow W. 3D printing materials and techniques for antennas and metamaterials: A survey of the latest advances. IEEE Antennas and Propagation Magazine. 2023. V. 65. № 3. P. 10−20. DOI: 10.1109/MAP.2022.3229298.

[18] Mikaelyan A.L. Ispol'zovanie sloistoj sredy dlya fokusirovki voln. Doklady AN SSSR. 1951. T. 81. № 4. S. 569−571 (in Russian).

[19] Cornbleet S. Flat disc, radially nonhomogeneous, lenses. IEE Proc. 1980. V. 127. Pt. H. № 6. P. 330−334.

[20] Kelleher K.S., Goatley C. Dielectric lens for microwaves. Electronics. 1955. № 8. P. 142−145.

[21] Kushneryov M.M., Skobelev S.P. Dvumernaya zadacha elektromagnitnogo rasseyaniya na linze Mikaelyana, dva metoda resheniya i osobennosti fokusirovki. Fizicheskie osnovy priborostroeniya. 2020. T. 9. № 4(38). S. 38−47 (in Russian).

[22] Veneckij A.S., Kaloshin V.A. Sintez gradientnoj linzovoj antenny s osevoj simmetriej. Radiotekhnika i elektronika. 1991. T. 36. № 12. S. 2301−2307 (in Russian).

[23] Abella C., Vazquez J., Marin M., Peces J., Graham R., Crone G. Analysis and synthesis of planar non-homogeneous lenses. 1993 23rd European Microwave Conference. Madrid. Spain. 1993. Р. 893−895. DOI: 10.1109/EUMA.1993.336741.

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