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Karmaşık objeler içeren çok katmanlı periyodik yapılardan saçılan elektromanyetik dalgaların analizi için katmanlar arası akımları kullanan verimli bir integral denklem yöntemi

Year 2023, Volume: 38 Issue: 4, 2241 - 2250, 12.04.2023

Nilüfer Aslıhan Özdemir

https://doi.org/10.17341/gazimmfd.998432

Abstract

Bu makalede, katmanlı dielektrik ortamlara gömülü, üç boyutlu (3B), elektromanyetik (EM) dalgaların geçebildiǧi objelerden oluşan çok katmanlı ikili periyodik dizilerden saçılan EM dalgaları analiz etmek için yüzey integral denklem tabanlı, verimli bir yöntem önerilmektedir. Önerilen yöntemde, katmanlar arasında bulunan arayüzlerin üzerindeki eşdeğer elektrik ve manyetik akımları tanımlamak için yüzeysel eşdeğerlik ilkesi kullanılmakta, bu sayede katmanlı dielektrik ortamlar için Green fonksiyonunu hesaplama gerekliliği ortadan kaldırılmaktadır. Arayüzler, fiziksel veya kurgusal olabilir. Arayüzlerin fiziksel olduğu durumda, belli bir katmandaki arka plan ortamı, sonsuz bir ikili periyodik dizi için Green fonksiyonu hesaplamasının yeterli olduğu homojen, iki boyutlu (2B) sınırsız bir ortam olarak kabul edilebilir. Elde edilen momentler yöntemi (MoM) etkileşim matrisi blok-tridiagonal bir yapıya sahiptir; bu ise, hem matris dolumu, hem de matris denklemi çözümü için katman sayısı bakımından doğrusal olan bir hesaplama karmaşıklığına yol açar. Önerilen yöntem yoluyla elde edilen sayısal sonuçlar, FDTD (Zaman Alanında Sonlu Farklar) tabanlı ticari yazılım kullanılarak elde edilen sonuçlarla karşılaştırılarak doğrulanmıştır.

Keywords

Elektromanyetik saçılım Katmanlı dielektrik ortam Yüzeysel eşdeğerlik ilkesi Yüzey integral denklemi Momentler yöntemi Blok-tridiagonal matris İki boyutlu periyodik obje dizisi İki boyutlu periyodik Green fonksiyonu

References

  • Chew, W. C., Waves and Fields in Inhomogeneous Media, Wiley-IEEE Press, New York, A.B.D., 1999.
  • Michalski A. and Zheng D., Electromagnetic scattering and radiation by surfaces of arbitrary shape in layered media. Part I: Theory, IEEE Trans Antennas Propag., 38 (3), 335-344, 1990.
  • Kristensen P., Mork J., Lodahl P., and Hughes S., Decay dynamics of radiatively coupled quantum dots in photonoc crystal slabs, Phys. Rev. B, 83 (7), 2011, Art. no. 075305.
  • Okhmatovski V., Yuan M, Jeffrey I., Phelps R., A three-dimensional precorrected FFT algorithm for fast method of moments solutions of the mixed-potential integral equation in layered media, IEEE Trans. Microw. Theory Technol., 57 (12), 3505-3517, 2009.
  • Baba Y. and Rakov V., Electromagnetic models of the lightning return stroke, J. Geophys. Res., 112, 1-7, 2007.
  • Firoozy N. et al., A controlled experiment on oil release beneath thin sea ice and its electromagnetic detection, IEEE Trans. Geosci. Remote Sens., 56(8), 4406-4419, 2018.
  • Mautz J. R. and Harrington R. F., Electromagnetic scattering from a homogeneous material body of revolution, Arch. Elektr. Uebertrag., 33, 71-80, 1979.
  • Rao S. M. and Wilton D. R., E-Field, H-Field, and combined field solution for arbitrarily shaped three dimensional dielectric bodies, Electromagnetics, 10, 407-421, 1990.
  • Harrington R. F., Field Computation by Moment Method, Wiley-IEEE Press, Piscataway, NJ, USA, 1993.
  • Dardenne X. and Craeye C., Method of moments simulation of infinitely periodic structures combining metal with connected dielectric object,. IEEE Trans. Antennas Propag., 56 (8), 2372-2380, 2008.
  • Guerin N., Craeye C., and Dardenne X., Accelerated computation of the free space Green's function gradient of infinite phased arrays of dipole, IEEE Trans. Antennas Propag., 57 (10), 3430-3434, 2009.
  • Ozdemir N. A. and Craeye C., An efficient integral equation-based analysis of finite periodic structures in the optical frequency range, J. Opt. Soc. Am. A, 30 (12), 2510-2518, 2013.
  • Eibert T., Volakis J., Wilton D., and Jackson D., Hybrid FE/BI modeling of 3D doubly periodic structures utilizing triangular prismatic elements and an MPIE formulation accelerated by the Ewald transformation, IEEE Trans. Antennas Propag., 47 (5), 843-850, 1999.
  • Gallinet B., Kern A. M., and Martin O. J. F., Accurate and versatile modeling of electromagnetic scattering on periodic nanostructures with a surface integral approach, J. Opt. Soc. Am. A, 27 (10), 2261-2271, 2010.
  • Araujo M. G., Tabaoda J. M., Solis D. M., Rivero J., Landesa L., and Obelleiro F., Comparison of surface integral equation formulations for electromagnetic analysis of plasmonic nanoscatterers, Opt. Express, 20 (8), 9161-9171, 2011.
  • Marly N., Baekelandt B., De Zutter D., and Pues H. F., Integral equation modeling of the scattering and absorption of multilayered doubly-periodic lossy structures, IEEE Trans. Antennas Propag., 43 (11), 1281-1287, 1995.
  • Trintinalia L. C. and Ling H., Integral equation modeling of multilayered doubly periodic lossy structures using periodic boundary condition and a connection scheme, IEEE Trans. Antennas Propag, 52 (9), 2253-2261, 2004.
  • Aksun M. I. and Mittra R., Derivation of closed-form Green's functions for a general microstrip geometry, IEEE Trans. Microw. Theory Tech., 40 (11), 2055-2062, 1992.
  • Michalski K. A., Extrapolation methods for Sommerfeld integral tails, IEEE Trans. Antennas Propag., 46 (10), 1405-1418, 1998.
  • Boix R. R., Mesa F., and Medina F., Application of total least squares to the derivation of closed-form Green's functions for planar layered media, IEEE Trans. Microw. Theory Tech., 55 (2), 268-280, 2007.
  • Mosig J. R., The weighted averages algorithm revisited, IEEE Trans. Antennas Propag., 60 (4), 2011-2018, 2012.
  • Tajdini M. M. and Mosallaei H., Characterization of large array of plasmonic nanoparticles on layered substrate: dipole mode analysis integrated with complex image method, Opt. Express, 19 (2), 173-193, 2011.
  • Chen Y. P., Sha W. E. I., Choy W. C. H., Jiang L., and Chew W. C., Study on spontaneous emission in complex multilayered plasmonic system via surface integral equation approach with layered medium Green's function, Opt. Express, 20 (18), 20210-21, 2012.
  • Vujevic S., Krolo I., and Lovric D., Closed-form spectral-domain Green’s functions for infinitesimal current source in multilayer soil, IEEE Trans. Electromagn. Compat., 62 (6), 2814-2822, 2020.
  • Yang H. and Yilmaz A. E., A generalized singularity subtraction method for evaluating layered medium Green’s functions, USNC/URSI Radio Science Meeting, Boston, MA, A.B.D., 81-82, 8-13 Temmuz, 2018.
  • Jia Y., Yu Z., Dai J., and Liu Q. H., Fast simulation of scattering problem for magnetodielectric materials with general anisotropy in layered media, IEEE Trans. Antennas Propag., 64 (11), 4785-4793, 2016.
  • Li. X and Okhmatovski V., Uniformly accurate electrostatic layered medium Green’s function approximation via scattered field formulation, IEEE 20th Conference on Electrical Performance of Electronic Packaging and Systems (EPEPS), San Jose, CA, USA, 5-7 Ekim, 2020.
  • Blezynski E. H., Blezynski M. K., Jaroszewicz T., Johnson W. A., Rivero J., Vipiana F., and Wilton D., Analytical treatment of the near-field term of the Green fuction of planarly stratified media, 2018 International Conference on Electromagnetics in Advanced Applications (ICEAA), Cartagena, Kolombiya, 683-686, 10-14 Eylül, 2018.
  • Blezynski E. H., Blezynski M. K., Jaroszewicz T., Johnson W. A., Rivero J., Vipiana F., and Wilton D., Nonsingular Laplacian representation of the asymptotic part of the layered medium Green function in the mixed potential formulation, 2019 International Conference on Electromagnetics in Advanced Applications (ICEAA), Granada, İspanya, 1080-1083, 9-13 Eylül, 2019.
  • Yang H. and Yilmaz A. E., A log-scale interpolation method for layered medium Green’s functions, IEEE International Symposium on Antennas and Propagation & USNC/URSI National Radio Science Meeting, Boston, MA, A.B.D., 2495-2496, 8-13 Temmuz, 2018.
  • Li D., Wilton D. R, Jackson D. R., Chen J., and Wang H., Efficient computation of Green’s functions for lossy uniaxial anisotropic layered media, Radio Sci. 54 (3), 196-214, 2019.
  • Hall R. C., Mittra R., and Mitzner K. M., Analysis of multilayered periodic structures using generalized scattering matrix theory, IEEE Trans. Antennas Propag., 36 (4), 511-517, 1998.
  • Khali A. I. and Steer M. B., A generalized scattering matrix method using the method of moments for electromagnetic analysis of multilayered structures in waveguide, IEEE Trans. Antennas Propag., 47 (11), 2151-2157, 1999.
  • ElMahgoub K., Yang F., and Elsherbeni A. Z., FDTD/GSM analysis of multilayered periodic structures with arbitrary skewed grid, IEEE Trans. Antennas Propag., 59 (12), 3264-3271, 2011.
  • Ozdemir N. A., Craeye C., Ehrhardt K., and Aradian A., Efficient integral equation approach for metamaterials made of core-shell nanoparticles at optical frequencies, 7th European Conference on Antennas and Propagation (EUCAP 2013), Gothenburg, İsveç, 1840-1842, 8-11 Nisan, 2013.
  • Harrington R. F., Time-Harmonic Electromagnetic Fields, Wiley-IEEE Press, Piscataway, NJ, USA, 2001.
  • Malassis L., Masse P., Treguer-Delapierre M., Mornet S., Weisbecker P., Kravets V., Grigorenko A., and Barois P., Bottom-up fabrication and optical characterization of dense films of meta-atoms made of core-shell plasmonic nanoparticles, Langmuir, 29 (5), 1551-1561, 2013.
  • LUMERICAL Solutions Inc., https://www.lumerical.com/tcad-products/fdtd/.
  • Craeye C., Radu X., Schuchinsky A., and Capolino F., Fundamentals of method of moments for artificial materials, Metamaterials Handbook, Editor: Capolino F., Taylor and Francis, New York, A. B. D., 2009.
  • Press W., Teukolsky S., Vetterling W., Flannery B., Numerical Recipes in C, Cambridge University Press, Cambridge, İngiltere, 2007.
  • SOPRA database, http://www.sopra-sa.com.
  • Bass M., De Cusatis C., Enoch J., Lakshminarayanan V., Li G., MacDonald C., Maha-jan V., Van Stryland E., Handbook of Optics, McGraw-Hill, New York, A. B. D., 2009.
  • Palik E. D., Handbook of Optical Constants of Solids, Academic Press, Cambridge, A. B. D., 1991.
  • Rao S., Wilton D., and Glisson A., Electromagnetic scattering by surfaces of arbitrary shape, IEEE Trans. Antennas Propag., 30 (3), 409-418, 1982.
  • Newman E., Generation of wide-band data from the method of moments by interpolating the impedance matrix [EM problems], IEEE Trans. Antennas Propag., 36 (12), 1820-1824, 1988.
  • Pozar D. M. and Schaubert D. H., Scan blindness in infinite phased arrays of printed dipoles, IEEE Trans. on Antennas and Propag., 32 (6), 602-610, 1984.

Year 2023, Volume: 38 Issue: 4, 2241 - 2250, 12.04.2023

Nilüfer Aslıhan Özdemir

https://doi.org/10.17341/gazimmfd.998432

Abstract

References

  • Chew, W. C., Waves and Fields in Inhomogeneous Media, Wiley-IEEE Press, New York, A.B.D., 1999.
  • Michalski A. and Zheng D., Electromagnetic scattering and radiation by surfaces of arbitrary shape in layered media. Part I: Theory, IEEE Trans Antennas Propag., 38 (3), 335-344, 1990.
  • Kristensen P., Mork J., Lodahl P., and Hughes S., Decay dynamics of radiatively coupled quantum dots in photonoc crystal slabs, Phys. Rev. B, 83 (7), 2011, Art. no. 075305.
  • Okhmatovski V., Yuan M, Jeffrey I., Phelps R., A three-dimensional precorrected FFT algorithm for fast method of moments solutions of the mixed-potential integral equation in layered media, IEEE Trans. Microw. Theory Technol., 57 (12), 3505-3517, 2009.
  • Baba Y. and Rakov V., Electromagnetic models of the lightning return stroke, J. Geophys. Res., 112, 1-7, 2007.
  • Firoozy N. et al., A controlled experiment on oil release beneath thin sea ice and its electromagnetic detection, IEEE Trans. Geosci. Remote Sens., 56(8), 4406-4419, 2018.
  • Mautz J. R. and Harrington R. F., Electromagnetic scattering from a homogeneous material body of revolution, Arch. Elektr. Uebertrag., 33, 71-80, 1979.
  • Rao S. M. and Wilton D. R., E-Field, H-Field, and combined field solution for arbitrarily shaped three dimensional dielectric bodies, Electromagnetics, 10, 407-421, 1990.
  • Harrington R. F., Field Computation by Moment Method, Wiley-IEEE Press, Piscataway, NJ, USA, 1993.
  • Dardenne X. and Craeye C., Method of moments simulation of infinitely periodic structures combining metal with connected dielectric object,. IEEE Trans. Antennas Propag., 56 (8), 2372-2380, 2008.
  • Guerin N., Craeye C., and Dardenne X., Accelerated computation of the free space Green's function gradient of infinite phased arrays of dipole, IEEE Trans. Antennas Propag., 57 (10), 3430-3434, 2009.
  • Ozdemir N. A. and Craeye C., An efficient integral equation-based analysis of finite periodic structures in the optical frequency range, J. Opt. Soc. Am. A, 30 (12), 2510-2518, 2013.
  • Eibert T., Volakis J., Wilton D., and Jackson D., Hybrid FE/BI modeling of 3D doubly periodic structures utilizing triangular prismatic elements and an MPIE formulation accelerated by the Ewald transformation, IEEE Trans. Antennas Propag., 47 (5), 843-850, 1999.
  • Gallinet B., Kern A. M., and Martin O. J. F., Accurate and versatile modeling of electromagnetic scattering on periodic nanostructures with a surface integral approach, J. Opt. Soc. Am. A, 27 (10), 2261-2271, 2010.
  • Araujo M. G., Tabaoda J. M., Solis D. M., Rivero J., Landesa L., and Obelleiro F., Comparison of surface integral equation formulations for electromagnetic analysis of plasmonic nanoscatterers, Opt. Express, 20 (8), 9161-9171, 2011.
  • Marly N., Baekelandt B., De Zutter D., and Pues H. F., Integral equation modeling of the scattering and absorption of multilayered doubly-periodic lossy structures, IEEE Trans. Antennas Propag., 43 (11), 1281-1287, 1995.
  • Trintinalia L. C. and Ling H., Integral equation modeling of multilayered doubly periodic lossy structures using periodic boundary condition and a connection scheme, IEEE Trans. Antennas Propag, 52 (9), 2253-2261, 2004.
  • Aksun M. I. and Mittra R., Derivation of closed-form Green's functions for a general microstrip geometry, IEEE Trans. Microw. Theory Tech., 40 (11), 2055-2062, 1992.
  • Michalski K. A., Extrapolation methods for Sommerfeld integral tails, IEEE Trans. Antennas Propag., 46 (10), 1405-1418, 1998.
  • Boix R. R., Mesa F., and Medina F., Application of total least squares to the derivation of closed-form Green's functions for planar layered media, IEEE Trans. Microw. Theory Tech., 55 (2), 268-280, 2007.
  • Mosig J. R., The weighted averages algorithm revisited, IEEE Trans. Antennas Propag., 60 (4), 2011-2018, 2012.
  • Tajdini M. M. and Mosallaei H., Characterization of large array of plasmonic nanoparticles on layered substrate: dipole mode analysis integrated with complex image method, Opt. Express, 19 (2), 173-193, 2011.
  • Chen Y. P., Sha W. E. I., Choy W. C. H., Jiang L., and Chew W. C., Study on spontaneous emission in complex multilayered plasmonic system via surface integral equation approach with layered medium Green's function, Opt. Express, 20 (18), 20210-21, 2012.
  • Vujevic S., Krolo I., and Lovric D., Closed-form spectral-domain Green’s functions for infinitesimal current source in multilayer soil, IEEE Trans. Electromagn. Compat., 62 (6), 2814-2822, 2020.
  • Yang H. and Yilmaz A. E., A generalized singularity subtraction method for evaluating layered medium Green’s functions, USNC/URSI Radio Science Meeting, Boston, MA, A.B.D., 81-82, 8-13 Temmuz, 2018.
  • Jia Y., Yu Z., Dai J., and Liu Q. H., Fast simulation of scattering problem for magnetodielectric materials with general anisotropy in layered media, IEEE Trans. Antennas Propag., 64 (11), 4785-4793, 2016.
  • Li. X and Okhmatovski V., Uniformly accurate electrostatic layered medium Green’s function approximation via scattered field formulation, IEEE 20th Conference on Electrical Performance of Electronic Packaging and Systems (EPEPS), San Jose, CA, USA, 5-7 Ekim, 2020.
  • Blezynski E. H., Blezynski M. K., Jaroszewicz T., Johnson W. A., Rivero J., Vipiana F., and Wilton D., Analytical treatment of the near-field term of the Green fuction of planarly stratified media, 2018 International Conference on Electromagnetics in Advanced Applications (ICEAA), Cartagena, Kolombiya, 683-686, 10-14 Eylül, 2018.
  • Blezynski E. H., Blezynski M. K., Jaroszewicz T., Johnson W. A., Rivero J., Vipiana F., and Wilton D., Nonsingular Laplacian representation of the asymptotic part of the layered medium Green function in the mixed potential formulation, 2019 International Conference on Electromagnetics in Advanced Applications (ICEAA), Granada, İspanya, 1080-1083, 9-13 Eylül, 2019.
  • Yang H. and Yilmaz A. E., A log-scale interpolation method for layered medium Green’s functions, IEEE International Symposium on Antennas and Propagation & USNC/URSI National Radio Science Meeting, Boston, MA, A.B.D., 2495-2496, 8-13 Temmuz, 2018.
  • Li D., Wilton D. R, Jackson D. R., Chen J., and Wang H., Efficient computation of Green’s functions for lossy uniaxial anisotropic layered media, Radio Sci. 54 (3), 196-214, 2019.
  • Hall R. C., Mittra R., and Mitzner K. M., Analysis of multilayered periodic structures using generalized scattering matrix theory, IEEE Trans. Antennas Propag., 36 (4), 511-517, 1998.
  • Khali A. I. and Steer M. B., A generalized scattering matrix method using the method of moments for electromagnetic analysis of multilayered structures in waveguide, IEEE Trans. Antennas Propag., 47 (11), 2151-2157, 1999.
  • ElMahgoub K., Yang F., and Elsherbeni A. Z., FDTD/GSM analysis of multilayered periodic structures with arbitrary skewed grid, IEEE Trans. Antennas Propag., 59 (12), 3264-3271, 2011.
  • Ozdemir N. A., Craeye C., Ehrhardt K., and Aradian A., Efficient integral equation approach for metamaterials made of core-shell nanoparticles at optical frequencies, 7th European Conference on Antennas and Propagation (EUCAP 2013), Gothenburg, İsveç, 1840-1842, 8-11 Nisan, 2013.
  • Harrington R. F., Time-Harmonic Electromagnetic Fields, Wiley-IEEE Press, Piscataway, NJ, USA, 2001.
  • Malassis L., Masse P., Treguer-Delapierre M., Mornet S., Weisbecker P., Kravets V., Grigorenko A., and Barois P., Bottom-up fabrication and optical characterization of dense films of meta-atoms made of core-shell plasmonic nanoparticles, Langmuir, 29 (5), 1551-1561, 2013.
  • LUMERICAL Solutions Inc., https://www.lumerical.com/tcad-products/fdtd/.
  • Craeye C., Radu X., Schuchinsky A., and Capolino F., Fundamentals of method of moments for artificial materials, Metamaterials Handbook, Editor: Capolino F., Taylor and Francis, New York, A. B. D., 2009.
  • Press W., Teukolsky S., Vetterling W., Flannery B., Numerical Recipes in C, Cambridge University Press, Cambridge, İngiltere, 2007.
  • SOPRA database, http://www.sopra-sa.com.
  • Bass M., De Cusatis C., Enoch J., Lakshminarayanan V., Li G., MacDonald C., Maha-jan V., Van Stryland E., Handbook of Optics, McGraw-Hill, New York, A. B. D., 2009.
  • Palik E. D., Handbook of Optical Constants of Solids, Academic Press, Cambridge, A. B. D., 1991.
  • Rao S., Wilton D., and Glisson A., Electromagnetic scattering by surfaces of arbitrary shape, IEEE Trans. Antennas Propag., 30 (3), 409-418, 1982.
  • Newman E., Generation of wide-band data from the method of moments by interpolating the impedance matrix [EM problems], IEEE Trans. Antennas Propag., 36 (12), 1820-1824, 1988.
  • Pozar D. M. and Schaubert D. H., Scan blindness in infinite phased arrays of printed dipoles, IEEE Trans. on Antennas and Propag., 32 (6), 602-610, 1984.
There are 46 citations in total.

Details

Primary Language Turkish
Subjects Engineering
Journal Section Makaleler
Authors

Nilüfer Aslıhan Özdemir 0000-0003-0904-319X

Publication Date April 12, 2023
Submission Date November 3, 2021
Acceptance Date October 23, 2022
Published in Issue Year 2023 Volume: 38 Issue: 4

Cite

APA Özdemir, N. A. (2023). Karmaşık objeler içeren çok katmanlı periyodik yapılardan saçılan elektromanyetik dalgaların analizi için katmanlar arası akımları kullanan verimli bir integral denklem yöntemi. Gazi Üniversitesi Mühendislik Mimarlık Fakültesi Dergisi, 38(4), 2241-2250. https://doi.org/10.17341/gazimmfd.998432
AMA Özdemir NA. Karmaşık objeler içeren çok katmanlı periyodik yapılardan saçılan elektromanyetik dalgaların analizi için katmanlar arası akımları kullanan verimli bir integral denklem yöntemi. GUMMFD. April 2023;38(4):2241-2250. doi:10.17341/gazimmfd.998432
Chicago Özdemir, Nilüfer Aslıhan. “Karmaşık Objeler içeren çok Katmanlı Periyodik yapılardan saçılan Elektromanyetik dalgaların Analizi için Katmanlar Arası akımları Kullanan Verimli Bir Integral Denklem yöntemi”. Gazi Üniversitesi Mühendislik Mimarlık Fakültesi Dergisi 38, no. 4 (April 2023): 2241-50. https://doi.org/10.17341/gazimmfd.998432.
EndNote Özdemir NA (April 1, 2023) Karmaşık objeler içeren çok katmanlı periyodik yapılardan saçılan elektromanyetik dalgaların analizi için katmanlar arası akımları kullanan verimli bir integral denklem yöntemi. Gazi Üniversitesi Mühendislik Mimarlık Fakültesi Dergisi 38 4 2241–2250.
IEEE N. A. Özdemir, “Karmaşık objeler içeren çok katmanlı periyodik yapılardan saçılan elektromanyetik dalgaların analizi için katmanlar arası akımları kullanan verimli bir integral denklem yöntemi”, GUMMFD, vol. 38, no. 4, pp. 2241–2250, 2023, doi: 10.17341/gazimmfd.998432.
ISNAD Özdemir, Nilüfer Aslıhan. “Karmaşık Objeler içeren çok Katmanlı Periyodik yapılardan saçılan Elektromanyetik dalgaların Analizi için Katmanlar Arası akımları Kullanan Verimli Bir Integral Denklem yöntemi”. Gazi Üniversitesi Mühendislik Mimarlık Fakültesi Dergisi 38/4 (April 2023), 2241-2250. https://doi.org/10.17341/gazimmfd.998432.
JAMA Özdemir NA. Karmaşık objeler içeren çok katmanlı periyodik yapılardan saçılan elektromanyetik dalgaların analizi için katmanlar arası akımları kullanan verimli bir integral denklem yöntemi. GUMMFD. 2023;38:2241–2250.
MLA Özdemir, Nilüfer Aslıhan. “Karmaşık Objeler içeren çok Katmanlı Periyodik yapılardan saçılan Elektromanyetik dalgaların Analizi için Katmanlar Arası akımları Kullanan Verimli Bir Integral Denklem yöntemi”. Gazi Üniversitesi Mühendislik Mimarlık Fakültesi Dergisi, vol. 38, no. 4, 2023, pp. 2241-50, doi:10.17341/gazimmfd.998432.
Vancouver Özdemir NA. Karmaşık objeler içeren çok katmanlı periyodik yapılardan saçılan elektromanyetik dalgaların analizi için katmanlar arası akımları kullanan verimli bir integral denklem yöntemi. GUMMFD. 2023;38(4):2241-50.