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Fundamentals of Planar Metamaterials and Subwavelength Resonators

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Symmetry Properties in Transmission Lines Loaded with Electrically Small Resonators

Part of the book series: Springer Theses ((Springer Theses))

Abstract

Electromagnetic metamaterials have been a subject of huge research worldwide in the electromagnetic and microwave communities since the first so-called left-handed structure was experimentally conceived in 2000. These efforts have been primarily motivated by the exotic (unusual) electromagnetic properties that these artificial media may feature, such as backward wave propagation. Meanwhile, electrically small resonators (ESRs) based on split rings, which are typical key building blocks to the synthesis of metamaterials, have been investigated broadly as well. The research activity on these resonant particles has been focused not only on the implementation of metamaterials, but also on the design of RF/microwave circuits and components. Thus, novel ideas have emerged in the microwave community to envisage devices inspired by metamaterial concepts with high performance, small size and/or new functionalities. In this regard, in the present thesis these metamaterial-based particles are not utilized to construct metamaterial structures, but to design innovative RF/microwave devices. Nonetheless, a brief introduction to metamaterials is provided for a historical standpoint and to make the thesis more complete, although reporting the current state-of-the-art is beyond the scope of this thesis.

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Notes

  1. 1.

    Since the international scientific community has not achieved a consensus in the definition of electromagnetic metamaterials [6], other artificial inhomogeneous structures with controllable electromagnetic properties (e.g. electromagnetic bandgaps—EBGs—which are based on the Bragg regime) are also sometimes regarded as metamaterials [2].

  2. 2.

    The subject has been known for a long time as artificial dielectrics, composite materials, or microstructured materials. The aim has always been to reproduce physical responses of known materials or to obtain some desirable responses not readily available in nature.

  3. 3.

    Periodicity is not fundamental, but eases the design.

  4. 4.

    If losses are absent, as is assumed throughout this chapter, \(\epsilon \) and \(\mu \) are real numbers.

  5. 5.

    The term NRI describes the particularity that the index of refraction in these media is negative, while the terminologies LH and BW will be addressed shortly.

  6. 6.

    Recall that losses are precluded in the whole chapter.

  7. 7.

    In the literature the propagation constant is represented by k or \(\beta \) [8]. Hereafter the symbol \(\beta \) will be used since it is the usual convention in microwave engineering.

  8. 8.

    Anti-parallel vectors are collinear vectors (lying in the same line) with opposite directions.

  9. 9.

    Backward-wave propagation has been known for decades. For instance, periodic structures support an infinite number of positive (forward) and negative (backward) space harmonics in addition to the fundamental space harmonic [7, 9]. The novelty of LH materials is that they are effectively homogeneous structures operating in a backward fundamental space harmonic [1, 2]. Forward/backward waves should not be confused with forward-/backward-traveling waves which are simply forward waves traveling in the positive/negative direction of propagation [10]. The latter are also known as positive- and negative-going waves [1].

  10. 10.

    This cutoff frequency is usually called plasma frequency in analogy to the permittivity function of a plasma.

  11. 11.

    In this thesis transmission-line metamaterials and metamaterial transmission lines are synonyms because this book deals with one-dimensional (1D) structures. When the same concept is extended to 2D and 3D dimensions, the structures are referred to as transmission-line metamaterials (i.e. metamaterial transmission lines apply only to 1D structures) [6].

  12. 12.

    Note that host transmission lines are commonly implemented on an ordinary dielectric substrate whose permittivity is different from the effective permittivity of the metamaterial.

  13. 13.

    Duality is defined here in terms of complementary response (e.g. the dual of a lowpass filter is a highpass filter) so that the series/shunt impedance of a network is proportional to the series/shunt admittance of the other network.

  14. 14.

    This ensures properly defined units in \(Z_s'\) (\(\Omega \)/m) and \(Y_{ p}'\) (S/m).

  15. 15.

    The calculation procedure of the dispersion relation of a periodic structure composed of a cascade of two-port unit-cell circuits is well-known (Appendix A).

  16. 16.

    The phase velocity is the slope of the line segment from the origin of the dispersion curve \(\omega (\beta )\) to a point in the curve, whereas the group velocity is the slope of the tangent to the dispersion curve at a point.

  17. 17.

    Alternatively, when \(|\beta |\) increases with frequency, \(v_{ p}\) and \(v_g\) are parallel and the propagation is forward. Otherwise the propagation is backward.

  18. 18.

    Wave propagation is allowed at \(\omega _{0}\) with infinite \(v_{ p}\) but (non-zero) finite \(v_g\).

  19. 19.

    The term balanced does not mean differential in this context.

  20. 20.

    The guided wavelength is obviously different from that guided in the host line (\(\beta \ne \beta _R\)).

  21. 21.

    Semi-lumped components are planar elements whose physical dimensions are usually restricted to be smaller than a quarter of wavelength. The components may also be designated to as lumped elements when the dimensions are smaller than one-eighth wavelength. These terminologies are in contrast to distributed (transmission line) elements in which the physical size is comparable to or larger than the wavelength [10, 18]. The advantage of electrically small planar (i.e. semi-lumped or even lumped) elements is that their behavior may be approximate by ideal (sizeless) lumped parameters using circuit theory.

  22. 22.

    Subwavelength resonators are those whose physical size is a small portion of the wavelength, and accordingly they may be referred to as semi-lumped or lumped resonators when composed of semi-lumped or lumped inductors and capacitors, respectively [18].

  23. 23.

    Note that CRLH lines based on the CL-loaded approach are also resonant by nature despite the fact that self-resonant elements are not used.

  24. 24.

    Canonical refers to the simplest or standard form.

  25. 25.

    Simultaneously positive or negative values of \(\epsilon \) and \(\mu \) are necessary and sufficient for wave propagation, as was illustrated in Fig. 2.1 (operation in the long wavelength regime is implicitly fulfilled). By contrast, the same sign of \(X_s\) and \(B_{ p}\) (or different sign of \(X_s\) and \(X_{ p}=-1/B_{ p}\)) cannot guarantee wave propagation, unless the operation band is explicitly restricted to the long wavelength regime, as highlighted in Fig. 2.5. Note that possible impedance mismatch with feeding sources at the port terminals in practical situations resulting in reflections is not considered in the aforementioned assertions.

  26. 26.

    The electromagnetic field distribution of these lines is analyzed in Sect. 4.1.

  27. 27.

    The valid frequency range of the models is evaluated throughout the thesis by comparing circuit simulations with full-wave electromagnetic simulations and measurements.

  28. 28.

    Two networks are (formally and numerically) dual if \(Z_{ {ii}}=Y_{ {ii}}\) and \(Z_{ij}=-Y_{ij} (i\ne {j})\), where \(Z_{ij}\) are the impedance matrix parameters of one network and \(Y_{ij}\) are the admittance matrix parameters of its dual one [18].

  29. 29.

    Besides the fact that complementary structures are defined in single planes, coplanar strips (CPS) are the complementary counterpart of a CPW.

  30. 30.

    Immittances are either impedances or admittances [18].

  31. 31.

    The propagation constant and the characteristic impedance obtained from the theory of periodic structures is still valid for a single unit cell.

  32. 32.

    Besides the broad definition, a distinction between metamaterial-based and metamaterial-inspired has been proposed [32].

  33. 33.

    Split rings are also sometimes called open rings [33].

  34. 34.

    SRR is exclusively used here to designate the original split-ring resonator topology [13].

  35. 35.

    On the contrary, open split-ring resonators have well-defined port terminals, e.g. the open split-ring resonator (OSRR) [34] or the open complementary split-ring resonator (OCSRR) [35].

  36. 36.

    The radiation of split-ring resonators can be quantified by the induced dipole moments through an equivalent radiation resistance. Nevertheless, since the dimensions of the dipoles are small relative to the wavelength, in general radiation may be neglected [5, 37].

  37. 37.

    Polarization is related here to the generation of local (microscopic) dipole moments in the presence of external fields (like medium electric and magnetic polarizations accounted for by \(\epsilon \) and \(\mu \), respectively, at the macroscopic level), and not to wave polarization.

  38. 38.

    A bidimensional object possesses inversion symmetry when is invariant by rotating 180\(^\circ \) taking its center as the rotation axis.

  39. 39.

    The overall size of the SRR is typically about one-tenth of the wavelength [3].

  40. 40.

    Nevertheless, the particle may be magnetically excited if the applied magnetic fields in the individual loops are in opposite directions to each other. In fact, in this thesis the ELC resonator is magnetically coupled to transmission lines.

  41. 41.

    Despite the nomenclature, the SRR is the complementary of the CSRR as well.

  42. 42.

    A slot resonator is sometimes referred to as a defected ground structure (DGS) or patterned ground structure (PGS) when it is etched in the ground plane.

  43. 43.

    The first-neighbor approximation assumes that the field strength declines so rapidly away from a resonator that it is too small beyond the nearest neighbor. The accuracy of this first-order coupling is dependent on the resonant elements [3, 5] and the spacing between resonators.

  44. 44.

    Slow-wave effects are usually accounted for by the resulting phase velocity reduction factor (or slow-wave factor).

  45. 45.

    However, propagation through electrically-coupled metallic rods have been known for a long time [66].

  46. 46.

    In a strict sense, however, multilayer structures cannot be dual.

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    Jordi Naqui

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Naqui, J. (2016). Fundamentals of Planar Metamaterials and Subwavelength Resonators. In: Symmetry Properties in Transmission Lines Loaded with Electrically Small Resonators. Springer Theses. Springer, Cham. https://doi.org/10.1007/978-3-319-24566-9_2

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