Skip to main content Accessibility help

We use cookies to distinguish you from other users and to provide you with a better experience on our websites. Close this message to accept cookies or find out how to manage your cookie settings.

Close cookie message
×
Hostname: page-component-cd9895bd7-8ctnn Total loading time: 0 Render date: 2024-12-22T14:55:59.510Z Has data issue: false hasContentIssue false
Series:   Elements in Emerging Theories and Technologies in Metamaterials

Effective Medium Theory of Metamaterials and Metasurfaces

Published online by Cambridge University Press:  21 December 2021

Wei Xiang Jiang ,
Zhong Lei Mei  and
Tie Jun Cui
Wei Xiang Jiang
Affiliation:
Southeast University
Zhong Lei Mei
Affiliation:
Lanzhou University
Tie Jun Cui
Affiliation:
Southeast University

Summary

Metamaterials, including their two-dimensional counterparts, are composed of subwavelength-scale artificial particles. These materials have novel electromagnetic properties, and can be artificially tailored for various applications. Based on metamaterials and metasurfaces, many abnormal physical phenomena have been realized, such as negative refraction, invisible cloaking, abnormal reflection and focusing, and many new functions and devices have been developed. The effective medium theory lays the foundation for design and application of metamaterials and metasurfaces, connecting metamaterials with real world applications. In this Element, the authors combine these essential ingredients, and aim to make this Element an access point to this field. To this end, they review classical theories for dielectric functions, effective medium theory, and effective parameter extraction of metamaterials, also introducing front edge technologies like metasurfaces with theories, methods, and potential applications. Energy densities are also included.
Get access

Keywords

metamaterialeffective parameterseffective medium theorymetasurfaceartificial medium
Type
Element
Information
Series: Elements in Emerging Theories and Technologies in Metamaterials
Online ISBN: 9781108872386
Publisher: Cambridge University Press
Print publication: 03 February 2022

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Cui, TJ, Tang, WX, Yang, XM, Mei, ZL, and Jiang, WX (2016), Metamaterials: Beyond Crystals, Noncrystals, and Quasicrystals. CRC Press, Boca Raton, FL.Google Scholar
Shelby, RA, Smith, DR, and Schultz, S (2001), Experimental verification of a negative index of refraction, Science, 292(5514), 77.CrossRefGoogle ScholarPubMed
Yen, TJ, Padilla, WJ, Fang, N et al. (2004), Terahertz magnetic response from artificial materials, Science, 303(5663), 1494.CrossRefGoogle ScholarPubMed
Jiang, WX, Qiu, C-W, Han, TC et al. (2013), Broadband all-dielectric magnifying lens for far-field high-resolution imaging, Adv. Mater., 25(48), 6963–6968.Google Scholar
Pendry, JB, Schurig, D, and Smith, DR (2006), Controlling electromagnetic fields, Science, 312(5781), 1780.Google Scholar
Schruig, D, Mock, JJ, Justice, BJ et al. (2006), Metamaterial electromagnetic cloak at microwave frequencies, A Science, 314(5801), 977.CrossRefGoogle Scholar
Jiang, WX, Qiu, C-W, Han, T, Zhang, S, and Cui, TJ, (2013) Creation of ghost illusions using wave dynamics in metamaterials, Adv. Funct., 23(32), 4028–4034.CrossRefGoogle Scholar
Smith, DR and Pendry, JB (2006), Homogenization of metamaterials by field averaging, J. Opt. Soc. Am. B, 23(3), 391.CrossRefGoogle Scholar
Smith, DR, Vier, DC, Koschny, T, and Soukoulis, CM (2005), Electromagnetic parameter retrieval from inhomogeneous metamaterials, Phys. Rev. E, 71(3), 036617.Google Scholar
Lorentz, HA (1952), The Theory of Electrons and Its Applications to the Phenomena of Light and Radiative Heat, 2nd ed., Dover, Mineola, NY.Google Scholar
Bohren, CF and Huffman, DR (1998), Absorption and Scattering of Light by Small Particles, Wiley, New York.Google Scholar
Rakic, AD, Djurisic, AB, Elazar, JM, and Majewski, ML (1998), Optical properties of metallic films for vertical-cavity optoelectronic devices, Appl. Opt., 37(22), 5271.Google Scholar
Lorentz, HA (1880), Ueber die Beziehung zwischen der Fortpflanzungsgeschwindigkeit des Lichtes und der Korperdichte, Ann. Phys., 9, 641.Google Scholar
Lorenz, L (1880), Ueber die Refractionsconstante, Ann. Phys., 11, 70.Google Scholar
Jackson, JD (1998), Classical Electrodynamics, 3rd ed., Wiley, New York.Google Scholar
Born, M and Wolf, E (1999), Principles of Optics: Electromagnetic Theory of Propagation, Interference and Diffraction of Light, 7th ed., Cambridge University Press, Cambridge UK.Google Scholar
Kittel, C (2005), Introduction to Solid State Physics, 8th ed., Wiley, New York.Google Scholar
Maxwell Garnett, JC (1904), Colours in metal glasses and in metallic films, Philos.Trans. R. Soc., London A, 203, 385.Google Scholar
Smith, GB (1977), Dielectric constants for mixed media, J. Phys. D, 10(4), L39.Google Scholar
Jayannavar, AM and Kumar, N (1991), Generalization of Bruggeman’s unsymmetrical effective-medium theory to a three component composite, Phys. Rev. B, 44(21), 12014.CrossRefGoogle ScholarPubMed
Merrill, WM, Diaz, RE, LoRe, MM, Squires, MC, and Alexopoulos, NG (1999), Effective medium theories for artificial materials composed of multiple sizes of spherical inclusions in a host continuum, IEEE Trans. Antennas Propag., 47(1), 142.Google Scholar
Chettiar, UK and Engheta, N (2012), Internal homogenization: Effective permittivity of a coated sphere, Opt. Express, 20(21), 22976.CrossRefGoogle ScholarPubMed
Giovampaola, CD and Engheta, N (2014), Digital metamaterials, Nat. Mater., 13(12), 1115.Google Scholar
Ma, HF and Cui, TJ (2010), Three-dimensional broadband ground-plane cloak made of metamaterials, Nat. Commun. 1, 21.Google Scholar
Jiang, WX and Cui, TJ (2011), Radar illusion via metamaterials, Phys. Rev. E, 83(2).Google Scholar
Jiang, WX, Cui, TJ, Yang, XM, Ma, HF, and Cheng, Q (2011), Shrinking an arbitrary object as one desires using metamaterials, Appl. Phys. Lett., 98(20).CrossRefGoogle Scholar
Smith, DR, Schultz, S, Markos, P, and Soukoulis, CM (2002), Determination of effective permittivity and permeability of metamaterials from reflection and transmission coefficients, Phys. Rev. B, 65(19), 195104.Google Scholar
Pendry, JB, Holden, AJ, Stewart, WJ, and Youngs, I (1996), Extremely low frequency plasmons in metallic mesostructures, Phys. Rev. Lett., 76(15), 4773.CrossRefGoogle ScholarPubMed
Huang, F, Jiang, T, Ran, LX, and Chen, HS (2004), Experimental confirmation of negative refractive index of a metamaterial composed of Ω-like metallic patterns, Appl. Phys. Lett., 84(9), 1537–1539.Google Scholar
Ran, L, Huangfu, J, Chen, HS et al. (2005), Experimental study on several left-handed metamaterials, PIER, 51, 249.CrossRefGoogle Scholar
Chen, HS, Ran, LX, Jiang, T et al. (2004), Left-handed materials composed of only S-shaped resonators, Phys. Rev. E, 70(5), 057605.Google Scholar
Chen, HS, Ran, LX, Jiang, T et al. (2005), Magnetic properties of S-shaped split-ring resonators, PIER, 51, 231.CrossRefGoogle Scholar
Chen, HS, Ran, LX, Jiang, T et al. (2005), Negative refraction of a combined double S-shaped metamaterial, Appl. Phys. Lett., 86(15), 151909.Google Scholar
Chen, X, Grzegorczyk, TM, Wu, B-I, Jr, Pacheco J, and Kong, JA (2004), Robust method to retrieve the constitutive effective parameters of metamaterials, Phys. Rev. E, 70(1), 016608.CrossRefGoogle ScholarPubMed
Li, Z, Aydin, K, and Ozbay, E (2009), Determination of the effective constitutive parameters of bianisotropic metamaterials from reflection and transmission coefficients, Phys. Rev. E, 79(2), 026610.Google Scholar
Hou, LL, Chin, JY, Yang, XM et al. (2008), Advanced parameter retrievals for metamaterial slabs using an inhomogeneous model, J. Appl. Phys., 103(6), 064904.CrossRefGoogle Scholar
Kong, J. A. (2002), Electromagnetic wave interaction with stratified negative isotropic media, Prog. Electromagn. Res. PIER, 35, 1.Google Scholar
Smith, DR, Schultz, S, Markos, P, and Soukoulis, CM (2002), Determination of effective permittivity and permeability of metamaterials from reflection and transmission coefficients, Phys. Rev. B, 65(19), 195104.Google Scholar
Yee, KS (1966), Numerical solution of initial boundary value problems involving Maxwell’s equations in isotropic media, IEEE Trans. Antennas Propag. 14(3), 302.Google Scholar
Smith, DR and Pendry, JB (2006), Homogenization of metamaterials by field averaging, J. Opt. Soc. Am. B, 23(3), 391.Google Scholar
Liu, R, Cui, TJ, Huang, D, Zhao, B, and Smith, DR (2007), Description and explanation of electromagnetic behaviors in artificial metamaterials based on effective medium theory, Phys. Rev. E, 76(2), 026606.CrossRefGoogle ScholarPubMed
Kittel, C (1996), Introduction to Solid State Physics, 7th ed., Wiley, New York.Google Scholar
Xiong, XYZ, Jiang, LJ, Markel, VA, and Tsukerman, I (2013), Surface waves in three-dimensional electromagnetic composites and their effect on homogenization, Opt. Express, 21(9), 10412.Google Scholar
Tsukerman, I (2011), Effective parameters of metamaterials: a rigorous homogenization theory via Whitney interpolation, J. Opt. Soc. Am. B, 28(3), 577.Google Scholar
Pors, A, Tsukerman, I, and Bozhevolnyi, SI (2011), Effective constitutive parameters of plasmonic metamaterials: homogenization by dual field interpolation, Phy. Rev. E, 84(1), 016609.Google Scholar
Gozhenko, VV, Amert, AK, and Whites, K (2013), Homogenization of periodic metamaterials by field averaging over unit cell boundaries: use and limitations, New J. Phys., 15, 043030.CrossRefGoogle Scholar
Amert, AK, Gozhenko, VV, and Whites, KW (2012), Calculation of effective material parameters by field averaging over lattices with non-negligible unit cell size, Appl. Phys. A, 109(4), 1007.Google Scholar
Smith, DR, Schultz, S, Markos, P, and Soukoulis, CM (2002), Determination of effective permittivity and permeability of metamaterials from reflection and transmission coefficients, Phys. Rev. B, 65(19), 195104.Google Scholar
Cui, TJ, Smith, DR, and Liu, RP (2010), Metamaterials: Theory, Design, and Applications, Chapters 3 and 4, Springer US, New York.Google Scholar
Landau, LD and Lifshitz, EM (1984), Electrodynamics of Continuous Media, 2nd ed. Pergamon Press, New York.Google Scholar
Cui, TJ and Kong, JA (2004), Time-domain electromagnetic energy in a frequency-dispersive left-handed medium, Phys. Rev. B, 70(20), 205106.CrossRefGoogle Scholar
Ruppin, Q (2002), Electromagnetic energy density in a dispersive and absorptive material, Phys. Lett. A, 299(2–3), 309–312.Google Scholar
Boardman, AD and Marinov, K (2006), Electromagnetic energy in a dispersive metamaterial, Phys. Rev. B, 73(16), 165110.Google Scholar
Luan, PG, Wang, YT, Zhang, S, and Zhang, X (2011), Electromagnetic energy density in a single-resonance chiral metamaterial, Opt. Lett., 36(5), 675.Google Scholar
Luan, PG (2009), Power loss and electromagnetic energy density in a dispersive metamaterial medium, Phys. Rev. E, 80(4), 046601.Google Scholar
Loudon, R (1970), The propagation of electromagnetic energy through an absorbing dielectric, J. Phys. A: Gen. Phys., 3(3), 233.Google Scholar
Fung, PCW and Young, K (1978), Electric energy density in a dissipative medium by circuit analog, Am. J. Phys., 46(1), 57.Google Scholar
Tretyakov, SA (2005), Electromagnetic field energy density in artificial microwave materials with strong dispersion and loss, Phys. Lett. A, 343(1–3), 231–237.Google Scholar
Civelek, C and Bechteler, TF (2008), Lagrangian formulation of electromagnetic fields in nondispersive medium by means of the extended Euler–Lagrange differential equation, Int. J. Eng. Sci., 46(12), 1218–1227.Google Scholar
Kong, JA (1986), Electromagnetic Wave Theory, Wiley, New York.Google Scholar
Kuester, EF, Mohamed, MA, Piket-May, M, and Holloway, CL (2003), Averaged transition conditions for electromagnetic fields at a metafilm, IEEE Trans. Antennas Propag., 51(10), 2641–2651.Google Scholar
Holloway, CL, Mohamed, MA, Kuester, EF, and Dienstfrey, A (2005), Reflection and transmission properties of a metafilm: with an application to a controllable surface composed of resonant particles, IEEE Trans. Antennas Propag., 47(4), 853–865.Google Scholar
Holloway, CL, Kuester, EF, Gordon, JA et al. (2012), An overview of the theory and applications of metasurfaces: the two-dimensional equivalents of metamaterials, IEEE Trans. Antennas Mag., 54(2), 10–35.Google Scholar
Patel, AM and Grbic, A (2011), A printed leaky-wave antenna based on a sinusoidally-modulated reactance surface, IEEE Trans. Antennas Propag., 59(6), 2087–2096.Google Scholar
Patel, AM and Grbic, A (2011), Modeling and analysis of printed-circuit tensor impedance surfaces, IEEE Trans. Antennas Propag., 59(1), 2087–2096.CrossRefGoogle Scholar
Patel, AM and Grbic, A (2013), Effective surface impedance of a Printed-Circuit Tensor Impedance Surface (PCTIS), IEEE Trans. Microwave Theory Tech., 61(4), 1403–1413.Google Scholar
Patel, AM and Grbic, A (2014), Transformation electromagnetics devices based on Printed-Circuit Tensor Impedance Surfaces, IEEE Trans. Microwave Theory Tech., 62(5), 1102–1111.Google Scholar
Falcone, F, Lopetegi, T, Laso, MAG et al. (2004), Babinet principle applied to the design of metasurfaces and metamaterials, Phys. Rev. Lett., 93(19), 197401.Google Scholar
Yu, N, Genevet, P, Cats, MA et al. (2011), Light propagation with phase discontinuities: generalized laws of reflection and refraction, Science, 334(6054), 333.CrossRefGoogle ScholarPubMed
Pfeiffer, C and Grbic, A (2013), Metamaterial Huygens’ surfaces: tailoring wave fronts with reflectionless sheets, Phys. Rev. Lett., 110(19), 197401.Google Scholar
Collin, RE (1960), Field Theory of Guided Waves, McGraw-Hill, New York.Google Scholar
Kang, M, Feng, TH, Wang, HT, and Li, J (2012), Wave front engineering from an array of thin aperture antennas, Opt. Express, 20(14), 15882.Google Scholar
Ni, X, Kildishev, AV, and Shalaev, VM (2013), Metasurface holograms for visible light, Nat. Commun., 4, 2807.Google Scholar
Kock, W (1968), Microwave holography, Microwaves, 7(12), 46–54.Google Scholar
Fong, BH, Colburn, JS, Ottusch, JJ, Visher, JL, and Sievenpiper, DF (2010), Scalar and tensor holographic artificial impedance surfaces, IEEE Trans. Antennas Propag., 58(10), 3212.Google Scholar
Sievenpiper, D, Schaffner, JH, Song, HJ, Loo, RY, and Tangonan, G (2003), Two-dimensional beam steering using an electrically tunable impedance surface, IEEE Trans. Antennas Propag., 51(10), 2713–2722.Google Scholar
Oliner, AA and Hessel, A (1959), Guided waves on sinusoidally-modulated reactance surfaces, IEEE Trans. Antennas Propag., 7(5), 201–208.CrossRefGoogle Scholar
Wan, X, Chen, TY, Zhang, Q et al. (2016), Manipulations of dual beams with dual polarizations by full-tensor metasurfaces, Adv. Opt. Mat., 4(10), 1567–1572. doi: https://doi.org/10.1002/adom.201600111.Google Scholar
Chen, K, Feng, YJ, Monticone, F et al. (2017), A reconfigurable active Huygens’ metalens, Adv. Mat., 29(7), 1606422.Google Scholar
Wang, SM, Wu, PC, Su, VC et al. (2017), Broadband achromatic optical metasurface devices, Nat. Commun., 8,187.Google Scholar
Tittl, A, Leitis, A, Liu, MK et al. (2018), Imaging-based molecular barcoding with pixelated dielectric metasurfaces, Science, 360(6393), 6393.Google Scholar
Yuan, YY, Zhang, K, Ding, XM et al. (2019), Complementary transmissive ultra-thin meta-deflectors for broadband polarization-independent refractions in the microwave region, Photonics Res., 7(1), 80–88.Google Scholar
Chen, MLLN, Jiang, LJ, and Sha, WEI (2017) Ultrathin complementary metasurface for orbital angular momentum generation at microwave frequencies, IEEE Trans. Antennas Propag., 66(1), 396–400.Google Scholar
Xu, P, Jiang, WX, Wang, SY, and Cui, TJ (2018), An ultrathin cross-polarization converter with near unity efficiency for transmitted waves, IEEE Trans. Antennas Propag., 66(8), 4370–4373.CrossRefGoogle Scholar
Chen, K, Ding, G, Hu, G, et al. (2020), Directional Janus metasurface, Adv. Mater., 32(2), 1906352.Google Scholar
Tao, Z, Jiang, WX, Ma, HF, and Cui, TJ (2019), High-gain and high-efficiency GRIN metamaterial lens antenna with uniform amplitude and phase distributions on aperture, IEEE Trans. Antennas Propag., 66(1), 16–22.Google Scholar
Zhang, N, Jiang, WX, Ma, HF, Tang, WX, and Cui, TJ (2019), Compact high-performance lens antenna based on impedance-matching gradient-index metamaterials, IEEE Trans. Antennas Propag., 67(2), 1323–1328.Google Scholar
Noginov, MA, Barnakov, YA, Zhu, G et al. (2009), Bulk photonic metamaterial with hyperbolic dispersion, Appl. Phys. Lett., 94(15), 151105.Google Scholar
Asai, H, Savel’ev, S, Kawabata, S, and Zagoskin, AM (2015), Effects of lasing in a one-dimensional quantum metamaterial, Phys. Rev. B, 91(13), 134513.Google Scholar
You, JW, Bongu, SR, Bao, Q, and Panoiu, NC (2019), Nonlinear optical properties and applications of 2D materials: theoretical and experimental aspects, Nanophotonics, 8(1), 63.Google Scholar
Cui, TJ, Qi, MQ, Wan, X, Zhao, J, and Cheng, Q (2014), Coding metamaterials, digital metamaterials and programmable metamaterials, Light-Science and Applications, 3, e218.CrossRefGoogle Scholar
Li, LL, Cui, TJ, Ji, W et al. (2017), Electromagnetic reprogrammable coding-metasurface holograms, Nat. Commun., 8, 197.Google Scholar
Zhang, XG, Tang, WX et al. (2018), Light-controllable digital coding metasurfaces, Adv. Sci., 5(11), 1801028.CrossRefGoogle ScholarPubMed
Zhang, XG, Jiang, WX, Jiang, HL et al. (2020), An optically driven digital metasurface for programming electromagnetic functions, Nat. Electron., 3(3), 165–171.Google Scholar
Cui, TJ, Liu, S, Bai, GD, and Ma, Q (2019), Direct transmission of digital message via programmable coding metasurface, Research, 2019, 2584509.Google Scholar
Cui, TJ (2018), Microwave metamaterials, Natl. Sci. Rev. 5(2), 134–136.Google Scholar
Cui, TJ, Liu, S, and Zhang, L (2017), Information metamaterials and metasurfaces, J. Mater. Chem. C, 5(15), 3644–3668.Google Scholar

Save element to Kindle

To save this element to your Kindle, first ensure [email protected] is added to your Approved Personal Document E-mail List under your Personal Document Settings on the Manage Your Content and Devices page of your Amazon account. Then enter the ‘name’ part of your Kindle email address below. Find out more about saving to your Kindle.

Note you can select to save to either the @free.kindle.com or @kindle.com variations. ‘@free.kindle.com’ emails are free but can only be saved to your device when it is connected to wi-fi. ‘@kindle.com’ emails can be delivered even when you are not connected to wi-fi, but note that service fees apply.

Find out more about the Kindle Personal Document Service.

Effective Medium Theory of Metamaterials and Metasurfaces
Available formats
×

Save element to Dropbox

To save content items to your account, please confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your account. Find out more about saving content to Dropbox.

Effective Medium Theory of Metamaterials and Metasurfaces
Available formats
×

Save element to Google Drive

To save content items to your account, please confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your account. Find out more about saving content to Google Drive.

Effective Medium Theory of Metamaterials and Metasurfaces
Available formats
×