MODELING OF RESONANCE EFFECTS IN ONE-DIMENSIONAL PERIODIC DIFFRACTION GRATINGS CONTAINING GRAPHENE STRIPS Part 1. MATHEMATICAL JUSTIFICATION OF THE SPECTRAL METHOD
Subject and Purpose. This paper presents a theoretical study of the interaction between monochromatic electromagnetic radiation and a one-dimensional periodic strip grating. The grating consists of periodically alternating perfectly conducting and graphene strips located at the boundary of a planar...
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Radio physics and radio astronomy |
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graphene one-dimensional periodic diffraction strip grating compact operator resonance Hilbert space surface conductivity |
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graphene one-dimensional periodic diffraction strip grating compact operator resonance Hilbert space surface conductivity Poyedinchuk, A. Ye. Melezhik, P. N. Brovenko, A. V. Khutoryan, E. M. Senkevych, O. B. Yashina, N. P. MODELING OF RESONANCE EFFECTS IN ONE-DIMENSIONAL PERIODIC DIFFRACTION GRATINGS CONTAINING GRAPHENE STRIPS Part 1. MATHEMATICAL JUSTIFICATION OF THE SPECTRAL METHOD |
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graphene one-dimensional periodic diffraction strip grating compact operator resonance Hilbert space surface conductivity |
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Article |
| author |
Poyedinchuk, A. Ye. Melezhik, P. N. Brovenko, A. V. Khutoryan, E. M. Senkevych, O. B. Yashina, N. P. |
| author_facet |
Poyedinchuk, A. Ye. Melezhik, P. N. Brovenko, A. V. Khutoryan, E. M. Senkevych, O. B. Yashina, N. P. |
| author_sort |
Poyedinchuk, A. Ye. |
| title |
MODELING OF RESONANCE EFFECTS IN ONE-DIMENSIONAL PERIODIC DIFFRACTION GRATINGS CONTAINING GRAPHENE STRIPS Part 1. MATHEMATICAL JUSTIFICATION OF THE SPECTRAL METHOD |
| title_short |
MODELING OF RESONANCE EFFECTS IN ONE-DIMENSIONAL PERIODIC DIFFRACTION GRATINGS CONTAINING GRAPHENE STRIPS Part 1. MATHEMATICAL JUSTIFICATION OF THE SPECTRAL METHOD |
| title_full |
MODELING OF RESONANCE EFFECTS IN ONE-DIMENSIONAL PERIODIC DIFFRACTION GRATINGS CONTAINING GRAPHENE STRIPS Part 1. MATHEMATICAL JUSTIFICATION OF THE SPECTRAL METHOD |
| title_fullStr |
MODELING OF RESONANCE EFFECTS IN ONE-DIMENSIONAL PERIODIC DIFFRACTION GRATINGS CONTAINING GRAPHENE STRIPS Part 1. MATHEMATICAL JUSTIFICATION OF THE SPECTRAL METHOD |
| title_full_unstemmed |
MODELING OF RESONANCE EFFECTS IN ONE-DIMENSIONAL PERIODIC DIFFRACTION GRATINGS CONTAINING GRAPHENE STRIPS Part 1. MATHEMATICAL JUSTIFICATION OF THE SPECTRAL METHOD |
| title_sort |
modeling of resonance effects in one-dimensional periodic diffraction gratings containing graphene strips part 1. mathematical justification of the spectral method |
| title_alt |
МОДЕЛЮВАННЯ РЕЗОНАНСНИХ ЕФЕКТІВ В ОДНОВИМІРНИХ ПЕРІОДИЧНИХ ДИФРАКЦІЙНИХ ҐРАТКАХ, ЩО МІСТЯТЬ СТРІЧКИ ГРАФЕНУ Частина 1. ОБГРУНТУВАННЯ СПЕКТРАЛЬНОГО МЕТОДУ |
| description |
Subject and Purpose. This paper presents a theoretical study of the interaction between monochromatic electromagnetic radiation and a one-dimensional periodic strip grating. The grating consists of periodically alternating perfectly conducting and graphene strips located at the boundary of a planar dielectric layer. The aim is to provide a mathematical justification for the spectral method analysis of resonance effects arising during the interaction of electromagnetic radiation with the strip grating.Methods and Methodology. The mathematical justification of the spectral method is based on the theory of non-self-adjoint compact operators in Hilbert spaces and the theory of compact analytic operator functions. In particular, we apply Keldysh’s theorems on the completeness of eigenvectors and associated vectors of non-self-adjoint compact operators, as well as the operator generalization of Rouché’s theorem for analytic operator functions.Results. The spectral approach to solving the diffraction problem of a one-dimensional periodic strip grating, which includes graphene strips, has received a rigorous mathematical treatment. It has been established that the diffraction field can be repre- sented as an expansion in eigenfunctions of the spectral problem, where the spectral parameter (eigenvalue) enters linearly into the boundary condition of conjugation on the graphene strips. The existence of the spectral problem solution has been proved in the case of small widths of the perfectly conducting strips. The completeness of the system of eigenfunctions (eigenvectors) in the corresponding Hilbert space has been demonstrated. As a consequence, in an unbounded region, there is a possibility to expand the diffraction field in resonance terms. An equation for resonance frequencies has been derived, indicating that the imaginary part of the spectral parameter equals the imaginary part of the surface conductivity of the graphene strips in the grating.Conclusions. The developed spectral method enables effective analysis of resonance effects that occur when electromagnetic radiation interacts with a one-dimensional periodic diffraction grating that includes graphene strips. This method can be used in the mathematical modeling of various devices and systems that utilize such gratings.Keywords: graphene; one-dimensional periodic diffraction strip grating; compact operator; resonance; Hilbert space; surface conductivityManuscript submitted 07.07.2025Radio phys. radio astron. 2025, 30(3): 163-173REFERENCES 1. Low, T., and Avouris, P., 2014. Graphene plasmonics for terahertz to mid-infrared applications. ACS Nano, 8(2), pp. 1086— 1101. DOI: https://doi.org/10.1021/nn406627u 2. Chandezon, J., Granet, G., Melezhik, P.N., Poyedinchuk, A.Ye., Sirenko, Yu.K. (ed.), Sjoberg, D., Strom, S. (ed.), Tuchkin, Yu.A., and Yashina, N.P., 2010. Modern theory of gratings. Resonant scattering: analysis techniques and phenomena. New York, Springer Science + Business Media, LCC. 3. Shestopalov, V.P., and Sirenko, Yu.K., 1989. Dynamical theory of gratings. Kiev: Naukova Dumka Publ. 4. Rotenberg, M., 1962. Application of Sturmian Functions to the Schrödinger Three-Body Problem: Elastic e+-H Scattering. Ann. Phys., 19(2), pp. 262—278. DOI: https://doi.org/10.1016/0003-4916(62)90219-1 5. Shestopalov, V.P., 1987. Spectral Theory and Excitation of Open Structures. Kiev: Naukova Dumka Publ. 6. Hanson, G. W., 2008. Dyadic Green’s functions and guided surface waves for a surface conductivity model of graphene. J. Appl. Phys., 103(6), 064302(8 pp.). DOI: https://doi.org/10.1063/1.2891452 7. Wood, R.W., 1935. Anomalous Diffraction Gratings. Phys. Rev., 48(12), pp. 928—936. DOI: https://doi.org/10.1103/PhysRev.48.928 8. Banach, S., 1948. Course of Functional Analysis. Kyiv: Radyanska Shkola Publ. (in Ukrainian). 9. Rouché, E., 1861. Mémoire on the Lagrange Series. Journal de l’École Polytechnique, 22, pp. 193—224. 10. Krein, M.G., 1947. On linear completely continuous operators in functional spaces with two norms. Zb. prac’ In-tu Mat. Akad. Nauk Ukr. RSR, 9, pp. 104—129 (in Ukrainian). |
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Видавничий дім «Академперіодика» |
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2025 |
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http://rpra-journal.org.ua/index.php/ra/article/view/1473 |
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oai:ri.kharkov.ua:article-14732025-09-16T09:06:09Z MODELING OF RESONANCE EFFECTS IN ONE-DIMENSIONAL PERIODIC DIFFRACTION GRATINGS CONTAINING GRAPHENE STRIPS Part 1. MATHEMATICAL JUSTIFICATION OF THE SPECTRAL METHOD МОДЕЛЮВАННЯ РЕЗОНАНСНИХ ЕФЕКТІВ В ОДНОВИМІРНИХ ПЕРІОДИЧНИХ ДИФРАКЦІЙНИХ ҐРАТКАХ, ЩО МІСТЯТЬ СТРІЧКИ ГРАФЕНУ Частина 1. ОБГРУНТУВАННЯ СПЕКТРАЛЬНОГО МЕТОДУ Poyedinchuk, A. Ye. Melezhik, P. N. Brovenko, A. V. Khutoryan, E. M. Senkevych, O. B. Yashina, N. P. graphene; one-dimensional periodic diffraction strip grating; compact operator; resonance; Hilbert space; surface conductivity Subject and Purpose. This paper presents a theoretical study of the interaction between monochromatic electromagnetic radiation and a one-dimensional periodic strip grating. The grating consists of periodically alternating perfectly conducting and graphene strips located at the boundary of a planar dielectric layer. The aim is to provide a mathematical justification for the spectral method analysis of resonance effects arising during the interaction of electromagnetic radiation with the strip grating.Methods and Methodology. The mathematical justification of the spectral method is based on the theory of non-self-adjoint compact operators in Hilbert spaces and the theory of compact analytic operator functions. In particular, we apply Keldysh’s theorems on the completeness of eigenvectors and associated vectors of non-self-adjoint compact operators, as well as the operator generalization of Rouché’s theorem for analytic operator functions.Results. The spectral approach to solving the diffraction problem of a one-dimensional periodic strip grating, which includes graphene strips, has received a rigorous mathematical treatment. It has been established that the diffraction field can be repre- sented as an expansion in eigenfunctions of the spectral problem, where the spectral parameter (eigenvalue) enters linearly into the boundary condition of conjugation on the graphene strips. The existence of the spectral problem solution has been proved in the case of small widths of the perfectly conducting strips. The completeness of the system of eigenfunctions (eigenvectors) in the corresponding Hilbert space has been demonstrated. As a consequence, in an unbounded region, there is a possibility to expand the diffraction field in resonance terms. An equation for resonance frequencies has been derived, indicating that the imaginary part of the spectral parameter equals the imaginary part of the surface conductivity of the graphene strips in the grating.Conclusions. The developed spectral method enables effective analysis of resonance effects that occur when electromagnetic radiation interacts with a one-dimensional periodic diffraction grating that includes graphene strips. This method can be used in the mathematical modeling of various devices and systems that utilize such gratings.Keywords: graphene; one-dimensional periodic diffraction strip grating; compact operator; resonance; Hilbert space; surface conductivityManuscript submitted 07.07.2025Radio phys. radio astron. 2025, 30(3): 163-173REFERENCES 1. Low, T., and Avouris, P., 2014. Graphene plasmonics for terahertz to mid-infrared applications. ACS Nano, 8(2), pp. 1086— 1101. DOI: https://doi.org/10.1021/nn406627u 2. Chandezon, J., Granet, G., Melezhik, P.N., Poyedinchuk, A.Ye., Sirenko, Yu.K. (ed.), Sjoberg, D., Strom, S. (ed.), Tuchkin, Yu.A., and Yashina, N.P., 2010. Modern theory of gratings. Resonant scattering: analysis techniques and phenomena. New York, Springer Science + Business Media, LCC. 3. Shestopalov, V.P., and Sirenko, Yu.K., 1989. Dynamical theory of gratings. Kiev: Naukova Dumka Publ. 4. Rotenberg, M., 1962. Application of Sturmian Functions to the Schrödinger Three-Body Problem: Elastic e+-H Scattering. Ann. Phys., 19(2), pp. 262—278. DOI: https://doi.org/10.1016/0003-4916(62)90219-1 5. Shestopalov, V.P., 1987. Spectral Theory and Excitation of Open Structures. Kiev: Naukova Dumka Publ. 6. Hanson, G. W., 2008. Dyadic Green’s functions and guided surface waves for a surface conductivity model of graphene. J. Appl. Phys., 103(6), 064302(8 pp.). DOI: https://doi.org/10.1063/1.2891452 7. Wood, R.W., 1935. Anomalous Diffraction Gratings. Phys. Rev., 48(12), pp. 928—936. DOI: https://doi.org/10.1103/PhysRev.48.928 8. Banach, S., 1948. Course of Functional Analysis. Kyiv: Radyanska Shkola Publ. (in Ukrainian). 9. Rouché, E., 1861. Mémoire on the Lagrange Series. Journal de l’École Polytechnique, 22, pp. 193—224. 10. Krein, M.G., 1947. On linear completely continuous operators in functional spaces with two norms. Zb. prac’ In-tu Mat. Akad. Nauk Ukr. RSR, 9, pp. 104—129 (in Ukrainian). Предмет і мета роботи. Теоретично досліджується проблема взаємодії монохроматичного електромагнітного ви- промінювання з одновимірно періодичними стрічковими ґратками. Ґратки утворено ідеально провідними та графе- новими стрічками, що періодично повторюються, які розташовано на межі плоского діелектричного шару. Метою роботи є обґрунтування спектрального методу для дослідження резонансних ефектів, що виникають при взаємодії електромагнітного випромінювання зі стрічковими ґратками.Методи та методологія. Для обґрунтування спектрального методу використано результати теорії несамоспряже- них компактних операторів у гільбертових просторах і теорії компактних аналітичних оператор-функцій. Зокрема, теореми Келдиша про повноту власних і приєднаних векторів несамоспряжених компактних операторів і операторне узагальнення теореми Руше для аналітичних оператор-функцій.Результати. Наведено строге математичне трактування спектрального підходу до розв’язання задач дифракції на одновимірно періодичних стрічкових ґратках, що містять стрічки графену. Встановлено, що дифракційне поле можна задати у вигляді ряду за власними функціями спектральної задачі, в якій спектральний параметр (власне значення) лінійно входить у граничну умову спряження на графенових стрічках. Доведено існування розв’язань спектральної задачі для випадку малих ширин ідеально провідних стрічок. Доведено повноту системи власних функцій (векторів) у відповідному гільбертовому просторі. І, як наслідок, обґрунтовано представлення дифракційного поля в нескінчен- ній області у вигляді ряду резонансних членів. Отримано рівняння для резонансних частот — рівність уявних частин спектрального параметра та поверхневої провідності графенових стрічок ґратки.Висновки. Розроблений спектральний метод дозволяє ефективно досліджувати резонансні ефекти, що супрово- джують взаємодію електромагнітного випромінювання з одновимірно періодичними дифракційними ґратками, які містять стрічки графену. Його можна застосовувати для математичного моделювання різних приладів і пристроїв, що використовують стрічкові ґратки, які містять стрічки графену.Ключові слова: графен, одновимірно періодичні дифракційні стрічкові ґратки, компактний оператор, резонанс, гіль- бертовий простір, поверхнева провідністьСтаття надійшла до редакції 07.07.2025Radio phys. radio astron. 2025, 30(3): 163-173БІБЛІОГРАФІЧНИЙ СПИСОК 1. Low, T., and Avouris, P., 2014. Graphene plasmonics for terahertz to mid-infrared applications. ACS Nano, 8(2), pp. 1086— 1101. DOI: 10.1021/nn406627u 2. Chandezon, J., Granet, G., Melezhik, P.N., Poyedinchuk, A.Ye., Sirenko, Yu.K. (ed.), Sjoberg, D., Strom, S. (ed.), Tuchkin, Yu.A., and Yashina, N.P., 2010. Modern theory of gratings. Resonant scattering: analysis techniques and phenomena. New York, Springer Science + Business Media, LCC. 3. Shestopalov, V.P., and Sirenko, Yu.K., 1989. Dynamical theory of gratings. Kiev: Naukova Dumka Publ. 4. Rotenberg, M., 1962. Application of Sturmian Functions to the Schrödinger Three-Body Problem: Elastic e-H Scattering.Ann. Phys., 19(2), pp. 262—278. DOI: 10.1016/0003-4916(62)90219-1 5. Shestopalov, V.P., 1987. Spectral Theory and Excitation of Open Structures. Kiev: Naukova Dumka Publ. 6. Hanson, G. W., 2008. Dyadic Green’s functions and guided surface waves for a surface conductivity model of graphene. J. Appl. Phys., 103(6), 064302(8 pp.). DOI: 10.1063/1.2891452 7. Wood, R.W., 1935. Anomalous Diffraction Gratings. Phys. Rev., 48(12), pp. 928—936. DOI: 10.1103/PhysRev.48.928 8. Banach, S., 1948. Course of Functional Analysis. Kyiv: Radyanska Shkola Publ. (in Ukrainian). 9. Rouché, E., 1861. Mémoire on the Lagrange Series. Journal de l’École Polytechnique, 22, pp. 193—224. 10. Krein, M.G., 1947. On linear completely continuous operators in functional spaces with two norms. Zb. prac’ In-tu Mat. Akad. Nauk Ukr. RSR, 9, pp. 104—129 (in Ukrainian). Видавничий дім «Академперіодика» 2025-09-11 Article Article application/pdf http://rpra-journal.org.ua/index.php/ra/article/view/1473 10.15407/rpra30.03.163 РАДИОФИЗИКА И РАДИОАСТРОНОМИЯ; Vol 30, No 3 (2025); 163 RADIO PHYSICS AND RADIO ASTRONOMY; Vol 30, No 3 (2025); 163 РАДІОФІЗИКА І РАДІОАСТРОНОМІЯ; Vol 30, No 3 (2025); 163 2415-7007 1027-9636 10.15407/rpra30.03 en http://rpra-journal.org.ua/index.php/ra/article/view/1473/pdf Copyright (c) 2025 RADIO PHYSICS AND RADIO ASTRONOMY |