We apply a complete uncertainty analysis, not studied in the literature, to investigate the dependences of retrieved electromagnetic properties of two MM slabs (the first one with only split-ring resonators (SRRs) and the second with SRRs and a continuous wire) with single-band and dual-band resonating properties on the measured/simulated scattering parameters, the slab length, and the operating frequency. Such an analysis is necessary for the selection of a suitable retrieval method together with the correct examination of exotic properties of MM slabs especially in their resonance regions. For this analysis, a differential uncertainty model is developed to monitor minute changes in the dependent variables (electromagnetic properties of MM slabs) in functions of independent variables (scattering (S-) parameters, the slab length, and the operating frequency). Two complementary approaches (the analytical approach and the dispersion model approach) each with different strengths are utilized to retrieve the electromagnetic properties of various MM slabs, which are needed for the application of the uncertainty analysis. We note the following important results from our investigation. First, uncertainties in the retrieved electromagnetic properties of the analyzed MM slabs drastically increase when values of electromagnetic properties shrink to zero or near resonance regions where S-parameters exhibit rapid changes. Second, any low-loss or medium-loss inside the MM slabs due to an imperfect dielectric substrate or a finite conductivity of metals can decrease these uncertainties near resonance regions because these losses hinder abrupt changes in S-parameters. Finally, we note that precise information of especially the slab length and the operating frequency is a prerequisite for accurate analysis of exotic electromagnetic properties of MM slabs (especially multiband MM slabs) near resonance regions.
© 2012 OSA
Metamaterials (MMs) have attracted a huge research interest to scientific community in recent years since they have unusual electromagnetic properties not available in conventional materials [1–3], such as negative refraction , perfect lenses [2,5], potential applications in invisibility cloaks [6,7], sensors , and resonators . MMs are artificial media composed of periodically [2–7] or amorphously [10,11] oriented inhomogeneities (meta-atoms) embedded in a host conventional material. Periodically oriented MMs are similar to artificial dielectrics obtained by arranging a large number of identical conducting obstacles, simulating the behavior of a molecule (or a group of molecules) in an ordinary dielectric, in a regular three-dimensional (3-D) pattern . MMs are composed of unit cells with either metallic structures with various geometries on a host dielectric medium (metallic cut wires, split-ring-resonators (SRRs), etc [13–16].) or non-metallic ones [17,18]. Based on the optical property of the MM unit cells, simultaneous negative effective permittivity and negative effective permeability (or any of them) at a certain frequency band can be achievable. For instance, while metallic wire cut periodic array (continuous wires) behaves like plasma and possesses a negative effective permittivity over a broadband , SRR periodic array produces a negative effective permeability within a limited frequency range .
A brute force approach for analyzing the interaction of electromagnetic fields with MMs is to rigorously solve the macroscopic Maxwell’s equations considering inhomogeneous permittivity and permeability  by using standard numerical techniques such as the finite-difference time-domain method . In this circumstance, at a given frequency, any material (continuous or non-continuous) supporting only one propagating mode will generally exhibit a well-defined refractive index () [21,22], but not a normalized wave impedance () which highly depends upon the overall size of the MM . In some circumstances, however, the complex response of MMs to electromagnetic waves can be analyzed, as an equal problem, by the effective (continuous) medium approach considering a MM slab as a continuous material if the operating wavelength inside the MM slab is much larger than the spacing between the composite components (scatterers) and the size of respective components of the MM slab [23–26]. In latter circumstances in which effective medium approach could be applied, not only but also will have well-defined values not depending on the overall size of the MM .
Care must be given in using the continuous medium approximation. First, the electromagnetic field near the MM slab must satisfy the Floquet’s theorem and thus can be expanded in a set of transverse-magnetic and transverse-electric modes (Floquet or Bloch modes) . If the periodicity is small enough, higher-order Floquet modes are evanescent in the propagation direction, and reflected and transmitted fields at a certain distance from the structure are essentially a combination of zeroth order (fundamental) modes. Therefore, plane waves incident to a MM slab can be analyzed as if it is a homogeneous slab. Second, retrieved constitutive parameters of the analyzed MM slabs depend on the definite direction of the wave incident to the MM slab, and thus may be termed as wave parameters [19,24,28–33]. This dependence of optical parameters on the wave vector arises due to anisotropy or spatial dispersion .
In order to probe the electromagnetic properties of metamaterials, there are currently four methods. The first method (qualitative effective medium theory) determines the electromagnetic properties of MM slabs by using approximate analytical models . Although yielding accurate results for simple MM structures, this method is not much practicable for complex MM structures since it requires the preparation of four different MM slabs , and thus its accuracy may decrease with any inaccuracy in the fabricated MM slabs. The second method, which is suitable for numerical simulations but not in experiments, is to numerically calculate the ratios of the electric and magnetic fields over surfaces and edges inside MMs . The third method is to propose new homogenization methods by collecting the advantages of currently available homogenization schemes . The final and most prevalently used method is to retrieve the electromagnetic properties of MM slabs from measured/simulated scattering (S-) parameters [21,24,25,29,32,33,36,37]. It is suitable for both numerical simulations and experimental measurements. In applying this methodology, first S-parameters, containing amplitude and phase information, of the analyzed MM slab are obtained. Then, these parameters are substituted into theoretically derived expressions to determine the electromagnetic properties of that MM slab. Although new and high-capable state of the art network analyzers are produced, and these analyzers perform a wide band frequency measurements in a few seconds, methods using measured/simulated S-parameters suffer from a few drawbacks for accurate retrieval of electromagnetic properties of MM slabs. The first disadvantage is the presence of Fabry-Perot resonances, which is expected to be a constraint within a narrow frequency band. There are some methods available in the literature to overcome this drawback [38–41]. Another problem is the non-unique retrieval of electromagnetic properties of MM slabs as a result of the presence of complex exponentials in S-parameters. The methods in [42–51] can be employed for different purposes to remove this disadvantage. The final drawback of electromagnetic retrieval of MM slabs by methods based on S-parameters is the unfeasible measurement of the phase of S-parameters at very high frequencies [29,52–57]. As a solution to this problem, amplitude-only Fresnel measurements at multiple frequencies could be utilized [29,52–56]. Because, in the region of negative refraction, real parts of both relative complex permittivity () and relative complex permeability () depend mainly on the phase measurements, as another solution, complex S-parameter measurements based on an interferometric approach are more appealing for sensitive measurements .
After performing a survey over literature, we have noted that most of the S-parameter extraction methods [21,24–26,29,32,37,57] for determination of electromagnetic properties of MM slabs do not consider a full and descriptive analysis about the effect of any uncertainty in measured S-parameters, and the thickness of the analyzed MM slabs on the retrieved properties. The sensitivity analysis of the retrieval method  does not go beyond interrogating the effect of measured transmission S-parameter on the retrieved wave impedance of MM slabs. Besides, the uncertainty analysis in  is limited in the sense that it only analyzes the effect of measured S-parameters on the extracted (not the effect of MM slab thickness and operating frequency). A complete uncertainty analysis is very useful in selecting the most appropriate S-parameter method for accurate retrieval of electromagnetic properties of MM slabs with a given configuration. In addition to the uncertainties of S-parameters and the thickness, investigating the effect of any uncertainty in the operating wavelength seems to be useful for multiband MM structures [16,58,59]. In this research paper, we present the results of a complete uncertainty analysis for examining the dependence of retrieved electromagnetic properties of MM slabs on the uncertainties in S-parameters, the thickness and also the operating wavelength.
2. Scattering parameters of MM slabs
A typical problem of determining effective electromagnetic properties of MM slabs with different properties using forward and backward S-parameter measurements/simulations is depicted in Figs. 1 and 2 . A plane wave polarized in the direction propagates along the direction and is incident upon each cubic MM cell in Figs. 1 and 2. While the MM cells in Fig. 1 are single-band (SB), those in Fig. 2 are dual-band (DB) [16,58,59]. The MM cell (SB SRR MM slab) in Fig. 1(a) is composed of only edge-coupled SRRs asymmetric with respect to axis (symmetric to the vertical axis ) whereas the MM cell (DB SRR MM slab) in Fig. 2(a) consists two concentric open ring resonators symmetric with respect to axis. Besides, the MM structure (SB Composite MM slab) in Fig. 1(b) has asymmetric edge-coupled SRRs as well as a continuous wire in the direction of E-field, and the MM structure (DB Composite MM slab) in Fig. 2(b) has two concentric open ring resonators together with a continuous wire in the direction of E-field. Of these structures, those in Figs. 1(b) and 2(b) show negative refractive index within a certain frequency range or certain frequency ranges .
By assuming that each cell in Figs. 1 and 2 is reciprocal (because it is composed of passive components ) and that the reference harmonic time dependence is , implying that imaginary parts of and are both greater than zero for a passive medium in the investigated frequency region, forward and backward reflection and transmission S-parameters for each homogeneous MM slab (actually, MMs have an inhomogeneous structure) in Figs. 1 and 2 can be written [21,26,36,37,42] after the application of boundary conditions of continuous transverse electric and magnetic field components at and
3. Simulation results
We use the unit cell dimensions in  as for the dimensions of SB unit MM cells in Fig. 1 to simulate S-parameters whereas we analyze a different structure for the DB unit MM cells in Fig. 2. Each unit cell in Figs. 1 and 2 is cubic with a cell dimension of mm. A 0.25 mm thick substrate has the following properties and loss tangent of 0.0002 for the cells in Fig. 1, and and loss tangent of 0.0023 for the cells in Fig. 2. Perfect electric conductor (PEC) SRRs and wire, each having a thickness of 17, are assumed to be positioned on opposite sides of the substrate of each cell in Figs. 1 and 2. The wire height is 2.5 mm for each cell in Figs. 1 and 2, and the wire width of the cells in Fig. 1 is 0.14 mm and is 0.90 mm for the cells in Fig. 2. The outer ring length of the SRR in each cell in Figs. 1 and 2 is 2.2 mm, and while both rings in Fig. 1 have a linewidth of 0.2 mm, those in Fig. 2 have a linewidth of 0.1 mm. The gap open in the direction in each ring in Figs. 1 and 2 is 0.3 mm, and the gap between the inner and outer rings is 0.15 mm for cells in Fig. 1 and 0.10 mm for the cells in Fig. 2. These unit cells are denoted as low-loss MM unit cells in the explanation of the results in our manuscript. When the loss tangent of substrate of MM slabs is increased to 0.02, and PEC of SRRs and wire is changed to copper with an electrical conductivity of (S/m), they are designated as lossy unit cells for referencing the results.
We utilize the CST Microwave Studio simulation program based on finite integration technique  to simulate S-parameters for each unit cell in Figs. 1 and 2. The structure is excited by an electromagnetic plane wave with an electric field vector oriented along -axis, magnetic field vector oriented along -axis, and propagation vector oriented along -axis. Whereas waveguide ports are assumed perpendicular to the -direction, periodic boundary conditions are used along - and -directions . For brevity, simulated S-parameters of only lossy cells in Figs. 1 and 2 are shown in Figs. 3 and 4 .
4. Retrieved electromagnetic properties
In this section, we present the retrieved electromagnetic properties of homogenized MM slabs with details of their geometries discussed in Section 3 using their simulated S-parameters (only lossy ones are shown in Figs. 3 and 4). We apply two different approaches in the retrieval process. The first approach is based upon extraction of electromagnetic properties from explicit analytical expressions , while the second approach uses dispersion models (Lorentz and Drude types [3,14,63–66]) to predict accurate electromagnetic properties . The advantages and drawbacks of each approach will be discussed where appropriate.
4.1. The analytical approach
The analytical approach  extracts the electromagnetic properties of homogenized MMs in Figs. 1 and 2 using explicit expressions. To apply this approach, from Eqs. (1) and (2), and are determined by simulated/measured S-parameters as
It is known that for a homogenized MM slab, its and can be uniquely retrieved from S-parameter measurements [21,26,37] without considering the value. For a passive (source-free) medium, the sign before the square root in the expression of in Eq. (3) can be ascertained by . Besides, the correct choice of for unique retrieval of can be realized by different techniques [42–51]. Finally, and are computed from
Utilizing the expressions in Eqs. (3)-(5) and simulated S-parameters, for each low-loss and lossy MM slab in Figs. 1 and 2, we first retrieve and and then determine and . For the purpose of conciseness, we show these parameters of only the lossy SB&DB MM slabs in Figs. 5 , 6 , 7 , and 8 using their simulated S-parameters in Figs. 3 and 4. The results and discussion of the uncertainty analysis for low-loss MM slabs are presented in next section. Note that because our formulation in Section 2 assumes a time dependence of while the CST software package considers the for its computations, we negated the phase value of simulated S-parameters for the retrieval of electromagnetic properties in the remainder part of our study.
The results and discussion of the uncertainty analysis for low-loss MM slabs are presented in next section. Note that because our formulation in Section 2 assumes a time dependence of while the CST software package considers the for its computations, we negated the phase value of simulated S-parameters for the retrieval of electromagnetic properties in the remainder part of our study.
It is noticed from Fig. 5 that the lossy SB&DB SRR MM slabs have only positive over the whole frequency band while the lossy SB&DB Composite MM slabs have a negative over some narrow frequency regions (approximately between 8 and 12 GHz for SB, 6.0-8.5 GHz and 9.5-10.5 GHz for DB). It is well-known that a MM structure composed of only SRRs cannot possess a negative [21,24]. In addition, it is seen from Figs. 5–8 that the dependences of retrieved , , , and of the lossy SB Composite MM slab are in good agreement with those of Fig. 3 in . However, we note from Fig. 7 that the retrieved of all MM slabs violates the passivity condition () over a few frequency regions such as 7.0-8.0 GHz and 10.0-11.0 GHz for DB SRR MM slab. This problem (violation of locality conditions) is attributed to spatial dispersion effects arising from the discreteness of the conducting elements that are repeated periodically in a non-homogeneous metamaterial bulk, which thus behaves as a non-local medium, and is seen in other articles [24–26,62]. These effects mostly manifest for large and values, when the amplitude and phase of the fields inside MM slabs vary quickly. This leads to a strong non-uniformity in the current distribution and local fields in the unit cell, thus making it difficult for the numerical code to account for such strong field variations properly. Finally, we note from Figs. 7(a) and 7(b) that the inverted red Gaussian-like curve for of the SB SRR MM slab at GHz is on the right side of a similar curve for of the SB Composite MM slab at GHz. This circumstance arises from the fact that MM slabs can be considered as resonating structures [67–69], and thus their resonance frequencies downshift with the inclusion of any lossy sample such as cut wires.
The preceding discussions for the lossy SB&DB MM slabs are also valid for low-loss SB&DB MM slabs. Therefore, the dependences of , , , and for low-loss MM slabs are not shown for conciseness. Although the analytical approach extracts electromagnetic properties of MMs faster, it does not clean out and thus not tolerate the inaccuracies () arising when the accuracy of our simulation program is not so high.
4.2. The dispersion model approach
This approach overcomes the problem of the analytical approach, discussed above, by fitting the simulated S-parameters to those obtained from the Drude and Lorentz type dispersion models, which describe local continuous media. In this way, any limited accuracy of simulation programs is handled by optimizing this fitting. Based on the Drude and Lorentz type dispersion models, expressions of and for any MM cell in Fig. 1 read as
We note from Figs. 5–8 that there are two frequency regions at which the MM cells in Fig. 2 resonates (DB). Therefore, the expressions of and in Eq. (6) cannot reflect such multiple resonance phenomenon. To meet this demand, similar to the Eqs. (20) and (21) in , we changed the form of the expressions in Eq. (6) toEqs. (6) and (7).
By accounting for the frequency dependences of the MM slabs in Figs. 1 and 2 (SB or DB), by substituting and in Eqs. (6) and (7), sequentially, into Eqs. (5), (2), and (1), and by minimizing the difference between the S-parameters obtained from simulation and from the homogenized-cell representation, it becomes possible to get error-removed electromagnetic properties as well as the values of the unknowns in Eqs. (6) and (7) . In , the authors utilized the differential evolution (DE) algorithm which is based on vector differences (lower value) for perturbing an older vector population to obtain a newer vector population [71,72]. Although it is faster and needs less number of iterations than Nonlinear-least squares based on Levenberg-Marquardt method and genetic algorithms, can be applicable to nonlinear, non-differentiable, and multidimensional continuous space functions, and does not make any assumption (if not, few assumptions) about the problem being optimized (metaheuristic algorithm), the DE algorithm requires correct selection of the differential weight (), the crossover probability (), and the population size (). With different initial combinations of , , and , the DE algorithm produces different results  for the same optimization problem. Because there is more than one unknown to be optimized in Eqs. (6) or (7), because the expressions of our problem in Eqs. (6) and (7) are nonlinear, and because it is possible to find some ranges for the lower and upper bounds of unknowns in Eqs. (6) and (7), in this paper, we will apply the “fmincon” function of MATLAB® , as one of the nonlinear minimization functions, for our multidimensional constrained nonlinear minimization problem. For the determination of the ranges of some unknowns, we could directly utilize the dependences of electromagnetic properties shown in Figs. 7 and 8 in addition to physical bounds of and . For example, for SB SRR and Composite MM slabs we can setFigs. 7 and 8 and from the dispersive model expressions in Eqs. (6) and (7), we determine the ranges of and asFigs. 7 and 8, respectively. Due to finite accuracy of our simulation program (CST), we arranged the lower and upper bounds of and using a 5 percent offset criterion.
Following the procedure just discussed and applying the “fmincon” function of MATLAB, we extract the electromagnetic properties of the low-loss and lossy SB&DB homogenized SRR and Composite MM cells whose details are presented in Section 3. In the analysis, after some optimization, we decided to use “Interior-point” algorithm and set the number of maximum iterations to 1,000 and maximum function evaluations to 10,000. Table 1 presents the optimized ,,,,,, and values for the analyzed lossy SB&DB SRR and Composite MM slabs, and Figs. 9 , 10 , 11 , and 12 demonstrate the computed , , and over 6-14 GHz using the optimized values in Table 1.
It is seen from Figs. 5–12 that the electromagnetic properties retrieved by the dispersion model approach are smoother than those by the analytical approach. This smoothing property resembles to that used for suppressing the Fabry-Perot artifacts arising from artificial Lorentzian poles . Most importantly, we note from Figs. 7–12 that extracted and of both lossy SB&DB SRR and Composite MM slabs by the dispersion model approach are in good agreement with the constraint on passive media ( and ), while those by the analytical model do not.
5. Uncertainty analysis
In the retrieval of electromagnetic properties in Figs. 5–12, we have assumed that there is not any inaccuracy in the measured S-parameters. However, in practical applications, the magnitude and phase of each S-parameter have some inaccuracies associated with them [75–79], and in turn these uncertainties may hinder accurate retrieval of electromagnetic properties of MM slabs. In addition, in most applications in the electromagnetic characterization of MM slabs, their thickness is arranged so that it is quite less than the operating wavelength in order to simplify the complexity of analysis (continuous medium approach). In this circumstance, the uncertainty in slab length measurements considerably increases even such measurements have been performed by a high precision micrometer. Finally, to fully characterize the properties of MM slabs (especially for multiband MM slabs [16,58,59]) near some specific frequency bands such as a band of negative refractive index, it is beneficial to take into consideration the uncertainty in the operating frequency. Since the electromagnetic properties of MM slabs are all linked to S-parameters, the slab length, and the operating frequency (see Eqs. (1)-(5)), any uncertainty in these quantities may seriously affect the retrieved electromagnetic properties. Accordingly, an appropriate metric tool considering the effects of any uncertainties in these parameters should be devised. The sensitivity analyses given in [37,57] do not attempt to analyze all these effects. Only a complete uncertainty analysis, which is overlooked in most MM retrieval methods based on S-parameters in the literature, can fill in this gap. Therefore, in this section, we apply a comprehensive uncertainty analysis for investigation the effects of uncertainties in S-parameters, the slab length, and the operating frequency on the retrieved electromagnetic properties of MM slabs in Figs. 1 and 2.
The differential uncertainty model, which examines the effect of minute changes in the independent variables on dependent ones, will be applied for our analysis [75–77,79–81]. In this model, a total differential is taken for each dependent variable (for our case, and ) as functions of independent variables [for our case, the main factors contributing to the error budget are assumed to be , , , , , and ]. Since the value of a term of the total differential of a dependent variable can be negative while others being positive or vice versa, each term of the total differential is first squared and then summed, and finally, the summation is square-rooted. We also note that since our uncertainty analysis considers the total differential of a dependent variable, it is assumed that the dependent variables are analytic (or satisfy the Cauchy-Riemann equations [82,83]) over the region of interest with respect to the differentiation parameters. Following the procedure in , we findEq. (10) are given as
Since the explicit expressions in Eqs. (10)-(15) require the values of and for their evaluation, we utilize the retrieved electromagnetic properties by the analytical and dispersive model approaches (some are given in Figs. 5–12 in Section IV). Then, we draw the dependence of over frequency using those retrieved values.
5.1. Uncertainties in S-parameters
Figures 13 , 14 , 15 , 16 , 17 , and 18 demonstrate, respectively, the dependences and over GHz for the lossy SB&DB SRR and Composite MM slabs, and the same dependences for the low-lossq and lossy SB Composite MM slabs. It is noted from Fig. 15(a) that for SB&DB SRR slabs, only the dependence is shown since is set to zero by the dispersion model approach. To better interpret these dependences, we also illustrate important largest uncertainty levels and their reasons for both the lossy SB&DB SRR and Composite MM slabs in Tables 2 and 3 . Here, the errors due to S-parameters are taken as and , , and .
- 1) The dependences of both real and imaginary parts of and drastically increase at some frequency ranges due to relatively small (approximately zero) values of and which directly enter into the expression of the denominator of Eq. (10). This increase completely depends on how small the values of and are and is denoted as the first reason of larger dependences in Tables 2 and 3. For instance, it is seen from Figs. 14(a) and 16(a) that the dependence of the lossy SB Composite MM slab determined from analytical and dispersive models is relatively lower over the whole frequency band except for GHz which corresponds to a zero value of for the Composite MM slab as seen from Figs. 7(b) and 11(b). On the other hand, for example, it is noticed from Figs. 13(b) and 15(b) and Table 2 that values of the dependence of the lossy DB SRR MM slab determined from analytical and dispersive models approximately at GHz and GHz are significantly lower than those nearly at GHz and GHz. This discrepancy comes from the fact that at some frequencies the values of and drastically decrease to zero, while at other frequencies their values are comparatively larger than, but still approximately zero. In general, we can state that the value of the analyzed dependence of a MM slab is relatively larger at frequencies where that dependence slowly varies with frequency (smaller frequency rate (derivative/slope) of that dependence).
- 2) The dependences and noticeably increase when there result large variations in the magnitudes and phases of reflection and transmission S-parameters as shown in Figs. 3 and 4. This result is denoted as the second reason for larger dependences in Tables 2 and 3. In particular, the uncertainties and (and also their imaginary parts) are perceptible at approximately GHz and GHz for the lossy DB Composite MM slab, as can be seen from the insets in Figs. 14(a) and 16(a) and also from Table 1. These frequency regions correspond to the resonance regions of the lossy DB Composite MM slab, and fast changes within those regions are the main characteristics of MM slabs just like any other resonating structures  such as series or parallel RLC resonant (lump parameter) circuit , transmission-lines (distributed parameters) with lengths of and , Fabry-Perot resonators , and cavity resonators .
- 3) The increased value of the analyzed dependences and/or at some frequencies due to the second reason is remarkably lower than that due to the first reason. For example, Figs. 17(a) and 17(b) demonstrate the dependence of lossy SB&DB SRR and Composite MM slabs extracted from analytical and dispersive model approaches. Comparison the dependences in Fig. 17 with those in Figs. 13(b)-16(b) indicates that the effect of approximately zero value of significantly alters the uncertainty . This circumstance is also valid for the uncertainty in which the dependence for different MM slabs is not shown for brevity.
- 4) While the dependences and of the lossy SB&DB SRR and Composite MM slabs obtained from the analytical approach drastically increase at some frequency regions, the same dependences obtained from the dispersive model do not have any large value at all at those frequencies. As an example of this, from Figs. 14(a) and 16(a) we note that the dependence of lossy DB Composite MM slab determined from the analytical approach extremely increases approximately at 8.2 GHz corresponding with as seen from Fig. 7(b), whereas that from the dispersion model does not. This circumstance arises from the fact that the CST simulation program utilized in our analysis has finite accuracy in simulations (as other simulation programs) and that while analytical model does not consider the constraint , the dispersive model does.
- 5) Results given between 1) and 4) for lossy MM slabs are also valid for low-loss MM slabs.
- 6) As seen from Figs. 18(a) and 18(b), an increase of loss factor, arising from an increase due to imperfect dielectric used as a substrate as well as finite conductivity of metals, of the SB&DB Composite MM slabs accompanies with a decrease of the overall uncertainty in the determination of and . This point is linked to the above second point since any loss present inside a resonating structure diminishes the number of radians through which the lossy structure oscillates as its energy decays to of its initial amplitude  and thus decreases the quality factor, limiting a rapid change in the measured S-parameters as well as and values.
5.2. Uncertainties in slab thickness and operating frequency
For the dependences in Figs. 13–18, we considered that and . Now we focus on monitoring the dependences and over frequency. For example, Figs. 19 , 20 , 21 , 22 , 23 , and 24 demonstrate, respectively, the dependences , , , and over GHz for the lossy SB&DB Composite MM slabs, and the same dependences for the low-loss and lossy SB Composite MM slab, since these slabs demonstrate negative refractive indices over some certain frequency band(s).
- 1) The Fabry-Perot artifacts, arising from artificial Lorentzian poles , of and by the analytical approach are removed by the dispersive model approach. For example, the artifacts of and for the lossy SB Composite MM slab near 9.5 GHz in Figs. 19(a) and 20(a) are removed in Figs. 21(a) and 22(a).
- 3) The dependences ,, , and in Figs. 19–22 extracted from analytical and dispersive model approaches for the lossy SRR and Composite MM slabs exhibit similar behaviors and patterns contrary to the dependences and/or in Figs. 13–16. This difference mainly comes from the effect of inclusion of four uncertainty parameters , , , and into the uncertainty analysis of each and/or dependence while , , , and are only considered as themselves and individual in the uncertainty analysis.
- 4) Values of the dependences and in Figs. 20 and 22 are significantly lower (or negligible on practical grounds) than those and in Figs. 19 and 21. This effect is a result of dominant or insignificant relative changes in operating frequency and slab length over their nominal values in the expression of in Eq. (2). For example, while a 5 percent change in a slab length of mm is perceivable and effective in the dependences and , the same change in operating frequency of GHz is not much sensible. Nonetheless, for MM slabs resonating at lower frequencies and having larger MM slab lengths, dependences and become significant and should be taken into account in the error budget.
- 5) The results given between 1) and 4) for lossy Composite MM slabs are also valid for low-loss SRR and Composite MM slabs as well as lossy SRR MM slabs.
- 6) The impact of slab losses alters the dependences , and especially for and near resonance frequencies (Figs. 23 and 24). This means that accurate knowledge of MM slab length as well as the operating frequency is a prerequisite for accurate retrieval of exotic electromagnetic properties (such as negative) of MM slabs, especially for multiband MM slabs [16,58,59].
A complete uncertainty analysis has been performed to monitor the dependences on retrieved electromagnetic properties of MM slabs. Two different slabs composed of only SRR (SRR MM slab) and of SRR with a continuous wire (Composite MM slab) with SB&DB properties are utilized for this purpose. We applied two different techniques to extract the electromagnetic properties of these slabs with different loss factors. While the first technique (analytical approach) retrieves the electromagnetic properties quicker, sometimes retrieved complex permittivity by this technique in resonance regions does not comply with passivity requirement for passive materials when the accuracy of employed simulation program is not so high. In fact, the number of lines per wavelength in the spatial discretization of the cubic SRR should be high enough to ensure that the transit time of the signal over the computation cell is much shorter than the characteristic times of the physical processes which occur inside the SRR to avoid numerical artifacts and non-causal effects. Analogously, in the time discretization the time step should be much smaller than the transit time along the cells to guarantee numerical stability and control parasitic oscillations. Then to provide accurate responses the computer running time becomes prohibitively large. Besides, the second technique (dispersion model approach) does remove these accuracies for the sake of increased time and of complexity in using an optimization algorithm. In our uncertainty analysis, after extracting the electromagnetic properties of the low-loss and lossy SB&DB SRR and Composite MM slabs using two approaches, we applied a differential uncertainty model, as a versatile uncertainty analysis method used for the analysis of conventional materials, to examine the dependences of retrieved electromagnetic properties of MM slabs on the uncertainties in S-parameters, the slab length as well as the operating frequency. Our uncertainty analysis shows that when either the values of retrieved electromagnetic properties approach to zero or measured/simulated S-parameters have a sharp change over frequency such as negative refractive index region, the uncertainty in extracted electromagnetic properties of MM slabs significantly increases. Any low-loss or medium-low present in the MM slabs decreases the uncertainties in electromagnetic properties arising from abrupt changes in measured/simulated S-parameters. Finally, we note that exact information about especially the slab length together with the operating frequency is an essential parameter to investigate the correct and exotic electromagnetic features of SRR and Composite MM slabs.
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