Extraction of the Electromagnetic Parameters of a Metamaterial Using the Nicolson–Ross–Weir Method: An Analysis Based on Global Analytic Functions and Riemann Surfaces

Abstract

The characterization of electromagnetic metamaterials (MMs) plays a fundamental role in their engineering processes. To this end, the Nicolson–Ross–Weir (NRW) method is intensively used to recover the effective parameters of MMs, even though this is affected by the branch ambiguity problem. In this paper, we face this issue in the context of global analytic functions and Riemann surfaces. This point of view allows us to rigorously demonstrate the mathematical foundations of an algorithmic approach for avoiding the branch ambiguity problem, in which the phase unwrapping method is merged with K-K relations for recovering the effective parameters of an MM. In addition, exploiting the intimate relationship between the K-K relations and the Hilbert transform, a simple variant of the above algorithm is presented.

1. Introduction

Tailoring a material by adjusting its internal microstructure is the key to controlling the interaction between electromagnetic waves and the material itself, i.e., the key to ruling the properties of electromagnetic waves [1]. Electromagnetic metamaterials (MMs) are synthetic composites precisely based on this concept [1,2,3]. The MM’s ability to show exotic properties is due to its intimate structure and based on subwavelength-engineered units called meta-atoms [1,2,3]. However, for analysis and design purposes, an MM is modeled as a continuous medium [1,3]. This viewpoint, called homogenization procedure [1,3], allows replacing the detailed description of an electromagnetic MM with a small set of averaged parameters, named effective parameters [1,2,3,4]. Among the different approaches used for achieving this aim [5,6], the Nicolson–Ross–Weir (NRW) method is the most generally employed because of its theoretical simplicity and ease in algorithmic implementations [7,8,9,10,11,12,13,14,15]. However, despite the above considerations, it is well known that NRW is affected by some shortcomings that limit the validity of its findings, thus demanding special attention when the effective parameters are retrieved [16,17]. In particular, the branch ambiguity issue, i.e., the lack of uniqueness in the evaluation of the effective refractive index, $n_{eff}\left(\omega \right)$ [18,19], turns out to be the most critical and problematic of all, even though all the conditions for the employment of the NRW method are fully satisfied [18,19]. In the literature, a number of techniques based on Kramers–Kronig (K-K) relations [20,21,22,23,24] and on the phase unwrapping approach [25,26,27,28] have been developed to counter this problem. Concerning the rationale behind the use of K-K relations, it is a direct consequence of the causality principle, which makes it possible to determine the real part of the effective complex refraction index without any ambiguity, $\mathsf{Re}\left[N_{eff}\left(\omega \right)\right]=n_{eff}\left(\omega \right)$, from the knowledge of its imaginary part, $\mathsf{Im}\left[N_{eff}\left(\omega \right)\right]={\kappa }_{eff}\left(\omega \right)$ [29]. Regarding the phase unwrapping approach, its formal equivalence with the operation of analytic continuation of holomorphic functions has been demonstrated very recently in [30]. As clearly shown in [30], the branch ambiguity issue is rooted in the use of the multivalued complex logarithm $LOG(·)$ for solving the NRW relationship. To this aim, an analytic continuation is crucial for performing the proper inversion operation to solve it, thus avoiding any ambiguity in evaluating the effective refractive index $n_{eff}\left(\omega \right)$. Starting with the above-mentioned considerations, we extend the analysis conducted in [30], placing the NRW relation in the context of the global analytic functions and the Riemann surfaces. The main results of our study can be summarized as follows: (i) we provide a rigorous mathematical framework within handling the NRW relationship; (ii) we exploit this framework to derive a couple of phase unwrapping-based analytic continuation algorithms, with the first being based on the K-K relations and the second on the Hilbert transform, and they are both capable of solving the NRW equation while avoiding any ambiguity. The idea behind these algorithmic approaches, in the intuitive form, is that the Riemann surface of the global analytic logarithm contains all exact solutions of the NRW equation. More precisely, these solutions are regions of its Riemann surface. Hence, the K-K relations or the Hilbert transform can be used to localize these regions so that the principal complex logarithmic function, appropriately reshaped, can extract the solution of the NRW relationship based on their indications. Furthermore, the theory we propose in this study allows us to rigorously demonstrate some algorithmic procedures empirically proposed in the literature [20,21], broadening their interpretation. The paper is organized as follows: in Section 2, we review the cause of branch ambiguity in the NRW method (Section 2.1) and present a brief discussion on the K-K approach (Section 2.1.1) and the phase unwrapping method (Section 2.1.2) currently used to solve this issue. Afterward, (Section 2.2) we introduce the concept of a global analytic function and Riemann surfaces. First, we discuss the relationship between analytic continuation and global analytic functions (Section 2.2.1) and then the connection between global analytic functions and Riemann surfaces (Section 2.2.2). Finally, we use these theoretical tools to derive a couple of phase unwrapping methods for overcoming the NRW branch ambiguity issue, with the first being based on K-K relations and the second on the Hilbert transform (Section 2.3). In Section 3, their numerical performances are investigated by retrieving the effective refraction index of a couple of hypotetical MM samples and comparing their findings with those provided by the K-K relations and the phase unwrapping approach [30]. In Section 4, conclusions are drawn.

2. Theory

2.1. The Cause of Branch Ambiguity

The Nicolson–Ross–Weir (NRW) method is a well-established procedure first introduced to characterize electromagnetic materials [7] and later employed in the MM field [31]. Using the NRW method, the effective complex refractive index $N_{eff}\left(\omega \right)$ of an MM slab with effective thickness $d_{eff}$ is recovered via the inversion of the following relation (the $e^{-i\omega t}$ time convention is assumed and suppressed):

$$\begin{array}{c}\hfill e^{i{N}_{eff}\left(\omega \right)k_0{d}_{eff}}=\frac{S_{21}\left(\omega \right)}{1-S_{11}\left(\omega \right)R_{01}\left(\omega \right)}\end{array}$$

(1)

in which $k_0$, $S_{11}\left(\omega \right)$, $S_{21}\left(\omega \right)$, and $R_{01}\left(\omega \right)$ are, respectively, the free-space propagation constant, scattering parameters, and the reflection coefficient belonging to the effective medium slab related to the MM at hand [30]. Routinely, Equation (1) is solved by using the complex logarithm $LOG(·)=log(·)+2p\pi i$, where $log(·)$ is the principal logarithm and $p\in \mathbb{Z}$ is the branch index [32], from which we obtain, for $\mathsf{Re}\left[N_{eff}\left(\omega \right)\right]={\kappa }_{eff}\left(\omega \right)$ and $\mathsf{Im}\left[N_{eff}\left(\omega \right)\right]=n_{eff}\left(\omega \right)$, the following expressions.

$$\begin{array}{cc}\hfill n_{eff}\left(\omega \right)& =\frac{1}{k_0{d}_{eff}}\mathsf{Im}\left[log\left(\frac{S_{21}\left(\omega \right)}{1-S_{11}\left(\omega \right)R_{01}\left(\omega \right)}\right)\right]+2p\pi \hfill \end{array}$$

(2)

$$\begin{array}{cc}\hfill {\kappa }_{eff}\left(\omega \right)& =-\frac{1}{k_0{d}_{eff}}\mathsf{Re}\left[log\left(\frac{S_{21}\left(\omega \right)}{1-S_{11}\left(\omega \right)R_{01}\left(\omega \right)}\right)\right]\hfill \end{array}$$

(3)

Because of the presence of term p, which ranges in the set of integer numbers, refractive index $n_{eff}\left(\omega \right)$ turns out to be not uniquely determined, giving rise to the so-called branch ambiguity issue [18,19].

2.1.1. Surmounting the Branch Ambiguity Issue: Kramers–Kronig Relations

Since the effective dielectric permittivity ${ϵ}_{eff}\left(\omega \right)$ and the effective magnetic permeability ${\mu }_{eff}\left(\omega \right)$ of an MM must be causal quantities [29,33], we have it that their real and imaginary parts must obey Kramers–Kronig (K-K) relations [29,33]. This result suggests a way to overcome the lack of uniqueness afflicting Equation (2). The rationale can be outlined as follows: because of causality, we have it that the effective complex refraction index, $N_{eff}\left(\omega \right)=\sqrt{{ϵ}_{eff}\left(\omega \right){\mu }_{eff}\left(\omega \right)}$, must be analytic on the complex upper plane ${\mathbb{C}}^{\uparrow }$, with no zeros on this region [29,33]. This fact allows relating the real and the imaginary part of $N_{eff}\left(\omega \right)$ via K-K relations. More precisely, since the term ${\kappa }_{eff}\left(\omega \right)$ is uniquely determined from relation (3), it allows $n_{eff}\left(\omega \right)$ to be expressed as follows:

$$\begin{array}{c}\hfill n_{eff}\left(\omega \right)=1+\frac{2}{\pi }\mathcal{P}{\int }_0^{+\infty }\frac{{\omega }^{\prime }{\kappa }_{eff}\left({\omega }^{\prime }\right)}{{\omega }^{\prime 2}-{\omega }^2}d{\omega }^{\prime }\end{array}$$

(4)

from which $n_{eff}\left(\omega \right)$ can be computed without any ambiguity [30,33]. Exploiting the intimate relationship between K-K relations and the Hilbert transform:$\mathcal{H}[·]$

$$\mathcal{H}[·]=\frac{1}{\pi }\mathcal{P}{\int }_{-\infty }^{+\infty }\frac{[·]}{\omega -{\omega }^{\prime }}d{\omega }^{\prime }$$

(5)

it is possible to rewrite relation (4) as follows [29,34].

$$\mathsf{Re}\left[N_{eff}\left(\omega \right)\right]=1-\mathcal{H}\left[\mathsf{Im}\left[N_{eff}\left(\omega \right)\right]\right]$$

(6)

2.1.2. Surmounting the Branch Ambiguity Issue: Phase Unwrapping as Analytic Continuation

As discussed in [30], the branch ambiguity problem afflicting $n_{eff}\left(\omega \right)$ is caused by the incorrect handling of Equation (1). Here, we summarize the main theoretical considerations and results from that study. If we set the following:

$$\begin{array}{c}\hfill w=i{N}_{eff}\left(\omega \right)k_0{d}_{eff}\end{array}$$

(7)

$$\begin{array}{c}\hfill z=\frac{S_{21}\left(\omega \right)}{1-S_{11}\left(\omega \right)R_{01}\left(\omega \right)}\end{array}$$

(8)

relation (1) then becomes

$$\begin{array}{c}\hfill e^w=z.\end{array}$$

(9)

Now, if the complex exponential $e^{(·)}$ turns out to be a univalent function [32], it will be invertible, and its inverse will be unique [32]. Accordingly, relation (9) can be solved without ambiguity. However, it has been highlighted in [30] that $e^{(·)}$ holds this property only when the path (8) lies strictly inside a semi-open stright-line strip of $\mathbb{C}$, with width $2\pi$, parallel to ${\mathbb{R}}^{+}$. However, since the place where this image lies is tied to $N_{eff}\left(\omega \right)$, which is the unknown we are searching for, we must handle $e^{(·)}$ as if it was a non-univalent function [30]. Therefore, relation (9) can be only right-inverted [30,35]; i.e., relation (9) can be solved by evaluating an analytic function $h(·)$ such as the following.

$$e^{\left(h\right(z\left)\right)}=z\forall z\in \mathbb{D}$$

(10)

Unlike the inverse function, right-inverse function $\mathit{h}(·)$ is not unique [35]. Therefore, when $e^{(·)}$ is handled in this way, it is fundamental to use a suitable constraint condition to obtain a unique solution from inverting Equation (9) (i.e., from inverting (1)) [30]. In [30], it has been demonstrated that $h(·)$ can be computed via an analytic continuation of the principal complex logarithm $log(·)$ on the path traced in $\dot{\mathbb{C}}$ by term (8), which guarantees the fulfillment of condition $\mathsf{Im}\left(\mathit{h}\right(0\left)\right)=0$ that ensures the causality of the refractive index [30]. As a further result, in [30], it has been shown that the analytic continuation operation can be carried out by a phase unwrapping method using the algorithm developed by Oppenheim and Schafer for the homomorphic filtering, PUNWOS, [36]. It restores the continuity of the imaginary part of $h(·)$ along path (8) [30], thus unambiguously providing $n_{eff}\left(\omega \right)$ as follows.

$$\begin{array}{c}\hfill n_{eff}\left(\omega \right)=\frac{1}{k_0{d}_{eff}}\mathsf{Im}\left[h\left(z\right)\right]\end{array}$$

(11)

2.2. Analytic Continuation, Global Analytic Functions, and Riemann Surfaces

In this section, we will introduce the theoretical concepts of the global analytic function and Riemann surface. Using these mathematical tools, the K-K and phase unwrapping approaches can be merged together, thus providing an analytic continuation algorithmic methodology to solve Equation (1). Roughly speaking, the rationale is the following: In [30], it was shown that the solution of Equation (9) is given by the analytic continuation of the principal complex logarithm over the domain defined by the path described by Equation (8). Hence, different solutions of Equation (9) correspond to different analytic function elements. At this stage, the idea to exploit the global analytic functions and Riemann surface has its origin. A global analytic function is a pool containing all analytic continuations of its essential components—the analytic function elements, i.e., precisely the objects that are solutions of Equation (9). Accordingly, in Section 2.2.1, we will show that the set of all solutions of Equation (9) realizes the global analytic logarithm. In Section 2.2.2, we will discuss the concepts of Riemann surfaces and the lift operation (or lifting operation). We will show how the Riemann surface for the global analytic logarithm is realized by the Cartesian product between solutions of Equation (9) and points of the complex plane and how the lifting operation allows placing, in correspondence, the values assumed by the global analytic logarithm on regions of the Riemann surface with the values taken by the solution of Equation (9) on its domain of definition, i.e, the path (8). From this fundamental relation, we derive an algorithmic approach to solve Equation (9) in which the phase unwrapping method (i.e., the analytic continuation) is merged with the K-K relations and a variant based on the Hilbert transform. All topics will be presented in-depth, although topological details will be omitted to simplify the discussion.

2.2.1. Analytic Continuation and Global Analytic Functions

Roughly speaking, the analytic continuation is the operation with which it is possible to enlarge the domain of an analytic function $f(·)$, $\mathsf{\Omega }$. To formalize this idea, we provide the following definition [32].

Definition 1. 

An ordered pair $(\mathsf{\Omega },f(·\left)\right)$ where Ω is a domain ($\mathsf{\Omega }\subset \mathbb{C}$) and $f(·)$ is an analytic function is called an analytic function element. A pair $({\mathsf{\Omega }}_N,f_N(·))$ is the direct analytic continuation of a pair $({\mathsf{\Omega }}_1,f_1(·))$ if the following is the case.

$$\begin{array}{c}\hfill f_1\left(z\right)=f_N\left(z\right)\forall z\in {\mathsf{\Omega }}_1\cap {\mathsf{\Omega }}_N({\mathsf{\Omega }}_1\cap {\mathsf{\Omega }}_N\ne ⌀)\end{array}$$

(12)

Alternatively, they are indirect analytic continuations of each other if the following set of analytic function elements exist between elements $({\mathsf{\Omega }}_1,f_1(·))$ and $({\mathsf{\Omega }}_N,f_N(·))$:

$$\begin{array}{c}\hfill \left\{({\mathsf{\Omega }}_2,f_2(·)),\cdots ,({\mathsf{\Omega }}_{N-1},f_{N-1}(·))\right\}\end{array}$$

(13)

where $({\mathsf{\Omega }}_k,f_k(·))$ is the direct analytic continuation of $({\mathsf{\Omega }}_{k-1},f_{k-1}(·))$ for $k=2,\cdots ,N$ [32].

Based on the above statements, a global analytic function $\mathcal{G}$ is a non-empty set of pairs $(\mathsf{\Omega },f(·\left)\right)$ that are the direct or indirect analytic continuation of each other [32,35].The union $\mathbb{M}={\cup }_i{\mathsf{\Omega }}_i$ of all domains ${\mathsf{\Omega }}_i$ of all analytic function elements belonging to $\mathcal{G}$ is the domain of existence of $\mathcal{G}$ [32]. Each analytic function elements $(\mathsf{\Omega },f(·\left)\right)\in \mathcal{G}$ is called a branch of $\mathcal{G}$ [35]. The value $w_0$ provided by a pair $(\mathsf{\Omega },f(·\left)\right)\in \mathcal{G}$ at $z_0\in \mathsf{\Omega }$, i.e., $w_0=f\left(z_0\right)$, is called the value of $\mathcal{G}$ at $z_0$ [32]. A global analytic function $\mathcal{G}$ can have several branches over the same domain. This behavior follows directly from the indirect analytic continuation’s working, which can provide different results using different sequences of analytic function elements [32,35]. In that case, $\mathcal{G}$ is a multivalued function from $\mathbb{M}$ to $\mathbb{C}$ [32,35].

Remark 1. 

The set $\mathcal{L}$ of all analytic function elements $(\mathsf{\Omega },h(·\left)\right)$ for which the following equation

$$e^{h\left(z\right)}=z\forall z\in \mathsf{\Omega }$$

(14)

holds is a global analytic function, $\mathcal{L}$, and is named the global analytic logarithm [32]. Its domain of existence is $\mathbb{M}=\dot{\mathbb{C}}$. Any branch of $\mathcal{L}$ has the form $log(·)=ln|·|+i\left[ar{g}_{\pi }(·)\right]$, where $ln|·|$ and $ar{g}_{\pi }(·)$ are the natural logarithm of the absolute value function and the principal argument function, respectively. Two distinct branches of $\mathcal{L}$ differ by $2p\pi i$ where $p\in \mathbb{Z}$ [32,35].

2.2.2. Global Analytic Functions and Riemann Surfaces

To handle a multivalued global analytic function $\mathcal{G}$ as a function in the ordinary sense, it becomes mandatory to redefine a suitable domain $\mathbb{S}\left(\mathcal{G}\right)$ for $\mathcal{G}$ and to operate in a way that it becomes an ordinary single-valued function from the new domain $\mathbb{S}\left(\mathcal{G}\right)$ to $\mathbb{C}$. To this aim, instead of dealing with analytic function elements $(\mathsf{\Omega },f(·\left)\right)$, where $\mathsf{\Omega }$ is an arbitrary domain, we have to consider pairs of the form $(\mathsf{\Delta },f(·\left)\right)$ where $\mathsf{\Delta }\subseteq \mathsf{\Omega }$ is a disk [35]. Between these elements, it is possible to introduce an equivalence relation [35,37].

Definition 2. 

Two pairs $({\mathsf{\Delta }}^{\prime },f^{\prime }(·))$ and $({\mathsf{\Delta }}^{″},f^{″}(·))$, with disks ${\mathsf{\Delta }}^{\prime }$ and ${\mathsf{\Delta }}^{″}$ centered in $z^{\prime }$ and $z^{″}$, respectively, are equivalent if the following is the case:

$$\left(z^{\prime }=z^{″}\right)\wedge \left(f^{\prime }(·)=f^{″}(·)\right)$$

(15)

in some neighborhood of the point $z^{\prime }.$

From it, we derive the possibility to partition $\mathcal{G}$ into particular subsets, called germs [37].

Definition 3. 

The set of all the equivalent pairs $(\mathsf{\Delta },f(·\left)\right)\in \mathcal{G}$ is called germ, centered at z, and denoted by the symbol ${\left[f\right]}_z$. The value of a germ ${\left[f\right]}_z$ at its center z, ${\overline{\left[f\right]}}_z$, is the value $w=f\left(z\right)$ given by any of its elements at z.

The germs can be collected together as sets [37].

Definition 4. 

The set of all germs in which all elements $(\mathsf{\Delta },f(·\left)\right)$ belonging to a global analytic function $\mathcal{G}$ can be collected is denoted by ${\mathcal{G}}^{*}$. The set of all germs in ${\mathcal{G}}^{*}$ with the same center z is denoted by ${\mathcal{G}}_z^{*}$.

Using the concept of a set of germs, we can introduce a theorem that formalizes what was first said about the number of values that a global analytic function $\mathcal{G}$ could assume in correspondence to a given point z of its domain of existence $\mathbb{M}$ (for its demonstration, see [32]).

Theorem 1 

(Poincaré–Volterra). Let $\mathcal{G}$ be a global analytic function. For all $z\in \mathbb{C}$ and ${\mathcal{G}}_z^{*}\in {\mathcal{G}}^{*}$, the cardinality of ${\mathcal{G}}_z^{*}$ is at most countable.

Based on the Poincaré-Volterra theorem, we are in the condition to introduce the concept of Riemann surface $\mathbb{S}\left(\mathcal{G}\right)$ for a global analytic function $\mathcal{G}$. It consists of a number of abstract sheets lying above $\mathbb{C}$. These sheets (or layers) can be built by associating an array to each $z\in \mathbb{C}$, for which its elements are germs belonging to set ${\mathcal{G}}_z^{*}$ with the center at z. The array can be empty (no germs at all) or, at most, have an infinite (countable) dimension [35,37]. In a more formal language, we have the following.

Definition 5. 

The set

$$\mathbb{S}\left(\mathcal{G}\right)=\left((z,{\left[f\right]}_z):z\in \mathbb{C},{\left[f\right]}_z\in {\mathcal{G}}_z^{*}\right)$$

(16)

is the Riemann surface related to $\mathcal{G}$.

Remark 2. 

In the case of the global analytic logarithm $\mathcal{L}$, its Riemann surface $\mathbb{S}\left(\mathcal{L}\right)$ is realized by all pairs $(z,{\left[f\right]}_z)$ where $z\in \dot{\mathbb{C}}$ and ${\left[f\right]}_z=log(·)+2p\pi i$ where $p\in \mathbb{Z}$. The p-th sheet of $\mathbb{S}\left(\mathcal{L}\right)$, ${\mathbb{S}}_p\left(\mathcal{L}\right)$, consists of all couples of $(z,log(·)+2p\pi i)$ with p fixed.

In other words, $\mathbb{S}\left(\mathcal{G}\right)$ is a subset of the Cartesian product between $\mathbb{C}$ and $\mathcal{G}$. Using the Riemann surface $\mathbb{S}\left(\mathcal{G}\right)$ in lieu of $\mathbb{M}$, it is possible to redefine $\mathcal{G}$ in a way that it realizes a function from its Riemann surface $\mathbb{S}\left(\mathcal{G}\right)$ to $\mathbb{C}$ [37].

Definition 6. 

The function $\widehat{\mathcal{G}}[·]$ is as follows:

$$\widehat{\mathcal{G}}\left[(z,{\left[f\right]}_z)\right]=w(z,{\left[f\right]}_z)\in \mathbb{S}\left(\mathcal{G}\right),w\in \mathbb{C},$$

(17)

where $w={\overline{\left[f\right]}}_z$ is called the lift of $\mathcal{G}$ on its Riemann surface $\mathbb{S}\left(\mathcal{G}\right)$.

Another function of paramount importance is the following.

Definition 7. 

The function $\widehat{\mathcal{P}}[·]$

$$\widehat{\mathcal{P}}\left[(z,{\left[f\right]}_z)\right]=z(z,{\left[f\right]}_z)\in \mathbb{S}\left(\mathcal{G}\right),z\in \mathbb{C}$$

(18)

is called the projection of $\mathbb{S}\left(\mathcal{G}\right)$ to $\mathbb{C}$.

To clarify the idea behind the lift operation, let us consider set $\widehat{\mathsf{\Omega }}$ of all pairs $(z,{\left[f\right]}_z)$ such that $z\in \overline{\mathsf{\Omega }}$, and ${\left[f\right]}_z$ is germ-derived by $(\overline{\mathsf{\Omega }},\overline{f}(·))$. This set, $\widehat{\mathsf{\Omega }}$, is a copy of $\mathsf{\Omega }$ on $\mathbb{S}\left(\mathcal{G}\right)$ (we recall that $\widehat{\mathcal{P}}\left[(z,{\left[f\right]}_z)\right]=z\in \overline{\mathsf{\Omega }}$). Let $(z,{\left[f\right]}_z)$ be an element of $\widehat{\mathsf{\Omega }}$; then, according to definition (6), we have the following.

$$\widehat{\mathcal{G}}\left[(z,{\left[f\right]}_z)\right]={\overline{\left[f\right]}}_z=\overline{f}\left(z\right)$$

(19)

In otherwords, $\widehat{\mathcal{G}}[·]$ assigns to each point in $(z,{\left[f\right]}_z)\in \widehat{\mathsf{\Omega }}\subset \mathbb{S}\left(\mathcal{G}\right)$ precisely the same value that $\overline{f}(·)$ assigns to each point in $z\in \overline{\mathsf{\Omega }}\subset \mathbb{C}$. In sum, we have that the lift of $\mathcal{G}$ on its Riemann surface, $\widehat{\mathcal{G}}[·]$, and it is restricted to $\widehat{\mathsf{\Omega }}$ and behaves exactly as $\overline{f}(·)$ on $\overline{\mathsf{\Omega }}$ [37].

Remark 3. 

The lift of $\mathcal{L}$ on $\mathbb{S}\left(\mathcal{L}\right)$ is given by the following.

$$\widehat{\mathcal{L}}\left[(z,{\left[f\right]}_z]\right)]={\overline{\left[f\right]}}_z=log\left(z\right)+2p\pi i$$

(20)

If $(\mathsf{\Omega },f(·\left)\right)\in \mathcal{L}$, the set $\widehat{\mathsf{\Omega }}$ of all pairs

$$(z,log(·)+2p\pi i),z\in \mathsf{\Omega }$$

(21)

is a copy of Ω on $\mathbb{S}\left(\mathcal{L}\right)$. The integers p fulfilling Equation (21) define sheets ${\mathbb{S}}_p\left(\mathcal{L}\right)$ where $\widehat{\mathsf{\Omega }}$ lies.

2.3. Kramers–Kronig Relations and Phase Unwrapping Approach: A Unique Point of View within the Framework of the Global Analytic Functions on Riemann Surfaces

As stated in Section 2.1.1 and Section 2.1.2, Equation (1) is solved by evaluating $h(·)$ via the K-K relations or by the phase unwrapping approach, and both methods are equivalent to analytic continuation. It is quite straightforward to encompass these methods in the theoretical framework given in Section 2.2.2. Specifically, from Remark 1, the set of all the solutions of Equation (9) is a global analytic function, i.e, the global analytic logarithm $\mathcal{L}[·]$; from Remark 2, we have the explict structure of its Riemann surface $\mathbb{S}\left(\mathcal{L}\right)$; finally, from Remark 3, the values assumed by the lift of the global analytic logarithm $\widehat{\mathcal{L}}\left[(·)\right]$ on sub-domain $\widehat{\mathbb{D}}$ established by pairs $(z,{\left[f\right]}_z])$ are equal to the values taken by $h(·)$ on $\mathbb{D}$; i.e., between the lift of global analytic logarithm, $\widehat{\mathcal{L}}[·]$, and the right inverse $h(·)$, relation (20) must hold.

$$\widehat{\mathcal{L}}\left[(z,{\left[f\right]}_z]\right)]={\overline{\left[f\right]}}_z=h\left(z\right)\forall z\in \mathbb{D}$$

(22)

That is, we have the following.

$${\overline{\left[f\right]}}_z=log\left(z\right)+2p\pi i=h\left(z\right)\forall z\in \mathbb{D}$$

(23)

Only in the context of the theory described above can the equalities (22) and (23) be derived and have meaning. Since $log(·)=ln|·|+i\left[ar{g}_{\pi }(·)\right]$, we can write the following.

$$\mathsf{Im}\left[h\left(z\right)\right]=[ar{g}_{\pi }\left(z\right)+2p\pi ]=\forall z\in \mathbb{D}$$

(24)

Considering Equation (8), the above relation can be written as follows.

$$\mathsf{Im}\left[h\left(\frac{S_{21}\left(\omega \right)}{1-S_{11}\left(\omega \right)R_{01}\left(\omega \right)}\right)\right]=\left[ar{g}_{\pi }\left(\frac{S_{21}\left(\omega \right)}{1-S_{11}\left(\omega \right)R_{01}\left(\omega \right)}\right)+2p\pi \right]$$

(25)

With respect to integers p that specify the sheets of $\mathbb{S}\left(\mathcal{L}\right)$ where $(\mathbb{D},h(·\left)\right)$ is embedded, they are unknowns that have to be determined. By means of Equation (11), we can write Equation (25) as follows.

$$k_0{d}_{eff}{n}_{eff}\left(\omega \right)=\left[ar{g}_{\pi }\left(\frac{S_{21}\left(\omega \right)}{1-S_{11}\left(\omega \right)R_{01}\left(\omega \right)}\right)+2p\pi \right]$$

(26)

Because term $n_{eff}\left(\omega \right)$ can be obtained from ${\kappa }_{eff}\left(\omega \right)$ via the K-K relation (4), p can be evaluated as follows.

$$p=\frac{1}{2\pi }\left[k_0{d}_{eff}\left(1+\frac{2}{\pi }\mathcal{P}{\int }_0^{+\infty }\frac{{\omega }^{\prime }{\kappa }_{eff}\left({\omega }^{\prime }\right)}{{\omega }^{\prime 2}-{\omega }^2}d{\omega }^{\prime }\right)-ar{g}_{\pi }\left(\frac{S_{21}\left(\omega \right)}{1-S_{11}\left(\omega \right)R_{01}\left(\omega \right)}\right)\right]$$

(27)

Equation (25) highlights an important concept: $\mathsf{Im}\left[h\right(·\left)\right]$ can be determined from the knowledge of the argument of the principal logarithm, $ar{g}_{\pi }(·)$, properly compensated by phase term $2p\pi$, where unknown integer p can be evaluated by using K-K relations by using Equation (27). Once we determined $\mathsf{Im}\left[h\right(·\left)\right]$, the effective refractive index $n_{eff}\left(\omega \right)$ can be evaluated via Equation (11). In sum, using the framework of global analytic functions, we rigorously demonstrated and derived a phase unwrapping approach based on Kramers–Kronig relations, which provides an analytic continuation solution $h(·)$ for Equation (1) based on $log(·)$ and restoring the continuity of its imaginary part, $\mathsf{Im}\left[h\right(·\left)\right]$ [30], given by $ar{g}_{\pi }(·)$. As an alternative, using the Hilbert transform of ${\kappa }_{eff}\left(\omega \right)$ from Equation (5), we have the following.

$$p=\frac{1}{2\pi }\left[k_0{d}_{eff}\left(1-\mathcal{H}\left({\kappa }_{eff}\left(\omega \right)\right)\right)-ar{g}_{\pi }\left(\frac{S_{21}\left(\omega \right)}{1-S_{11}\left(\omega \right)R_{01}\left(\omega \right)}\right)\right]$$

(28)

3. Results

As derived in the above section, relations (27) and (28) represent two phase unwrapping approaches based on the K-K relation (4) and the Hilbert transform (6), respectively. We named PUNW${}_{KK}$ and PUNW${}_{HT}$ the related numerical procedures. Regarding the PUNW${}_{KK}$ method, the numerical evaluation of the involved K-K integral has been performed using the Simpson quadrature method described in [38], while for the PUNW${}_{HT}$ procedure, the Hilbert Transform has been computed by using the FFT technique described in [34]. For both methods, the parameter p has been determined, forcing it to assume an integer value selected to ensure the continuity of $ar{g}_{\pi }(·)$ [36]. To assess their numerical performances, we compared these with those provided by the numerical evaluation of the K-K relation (4) using the same numerical integration algorithm adopted for PUNW${}_{KK}$ [38] and the phase unwrapping PUNWOS method [36] using the implementation described in [30]. To this end, we have considered recovering the effective refractive index $n_{eff}\left(\omega \right)$ of two theoretical double negative MM slabs, in which relative permeability ${\mu }_r\left(\omega \right)$ and permittivity ${ϵ}_r\left(\omega \right)$ are described by the following Lorentzian model [3].

$$\begin{array}{cc}\hfill {\mu }_r\left(\omega \right)& ={\mu }_{\infty }+\sum _{n=1}^{np}\frac{({\mu }_{s,n}-{\mu }_{\infty }){\omega }_{0m,n}^2}{{\omega }_{0m,n}^2-{\omega }^2+i{\gamma }_{m,n}\omega }\hfill \end{array}$$

(29)

$$\begin{array}{cc}\hfill {ϵ}_r\left(\omega \right)& ={ϵ}_{\infty }+\sum _{n=1}^{np}\frac{({ϵ}_{s,n}-{ϵ}_{\infty }){\omega }_{0e,n}^2}{{\omega }_{0e,n}^2-{\omega }^2+i{\gamma }_{e,n}\omega }\hfill \end{array}$$

(30)

The parameters of the two models are reported in Table 1. This modeling choice is justified because: (i) Lorentzian models describe the dispersive nature of split-ring resonators and strip wires [3], and (ii) the superposition of Lorentzian functions describes every causal function [39]. The MATLAB environment has been employed to code all methods. All numerical experiments were performed on a 2.0 GHz Intel Xeon workstation. The percentage error (PE) is defined as follows:

$$\frac{\parallel n_{ex}\left(\omega \right)-n_{eff}{\left(\omega \right)\parallel }_2}{\parallel n_{ex}{\left(\omega \right)\parallel }_2}\%$$

(31)

where ${\parallel ·\parallel }_2$ is the 2-norm [40], $n_{ex}\left(\omega \right)=\mathsf{Re}\left[\sqrt{{ϵ}_r\left(\omega \right){\mu }_r\left(\omega \right)}\right]$ and $n_{eff}\left(\omega \right)$ are the exact and recovered refractive indexes, respectively, which have been used as criteria to evaluate the efficiency of the considered recovering procedures. Moreover, computational time t has been evaluated. However, this last datum has to be considered as only indicative of the order of magnitude rather than the exact time involved in each considered retrieval procedure since such knowledge would require the development of numerical codes optimized for a given CPU in order to achieve the best possible computational performances [41]. Figure 1 shows path $\gamma \left(\omega \right)$ traced by Equation (8) for the first slab, with $d_{eff}=180$ nm, which is characterized by a single pole Lorentzian model for pair $\left({ϵ}_r\left(\omega \right),{\mu }_r\left(\omega \right)\right)$. As pointed out in [30], this trend for $\gamma \left(\omega \right)$ confirms that this is an electrically thick slab because $\gamma \left(\omega \right)$ repeatedly intersects ray ${\mathbb{R}}_{\pi }$, i.e., the branch cut for the principal complex logarithm $log(·)$ [30,32]. Accordingly, the complex exponential function in equation $e^{(·)}$ in (1) lacks univalence for the considered case, thus indicating that $n_{eff}\left(\omega \right)$ must be evaluated via the right-inverse operation [30]. Table 2 shows the values of PE and t as a function of the number of sample points $n_s$ in frequency band $[0,1]$ PHz for all the considered recovering methods.

Table 1. Parameters for the Lorentzian model adopted in this study.

	First slab: $\mathit{n}\mathit{p}=1$, ${\mathit{d}}_{\mathit{e}\mathit{f}\mathit{f}}=180$ nm
${ϵ}_{\infty }$	1.8	${\mu }_{\infty }$	1.0
${ϵ}_{s,1}$	2.0	${\mu }_{s,1}$	1.3
${\omega }_{oe,1}$	$2\pi ·0.695$ PHz	${\omega }_{om,1}$	$2\pi ·0.7$ PHz
${\gamma }_{e,1}$	$0.08$ PHz	${\gamma }_{m,1}$	$0.05$ PHz
	Second slab: $np=2$, $d_{eff}=300$ nm
${ϵ}_{\infty }$	1.8	${\mu }_{\infty }$	1.0
${ϵ}_{s,1}$	2.5	${\mu }_{s,1}$	1.8
${ϵ}_{s,2}$	2.0	${\mu }_{s,1}$	1.3
${\omega }_{oe,1}$	$2\pi ·0.695$ PHz	${\omega }_{om,1}$	$2\pi ·0.7$ PHz
${\omega }_{oe,2}$	$2\pi ·0.895$ PHz	${\omega }_{om,2}$	$2\pi ·0.899$ PHz
${\gamma }_{e,1}$	$0.09$ PHz	${\gamma }_{m,1}$	$0.07$ PHz
${\gamma }_{e,2}$	$0.08$ PHz	${\gamma }_{m,2}$	$0.05$ PHz

Figure 1. Path $\gamma \left(\omega \right)$ traced by Equation (8) on the complex punctured plane $\dot{\mathbb{C}}$ for the slab with $d_{eff}=180$ nm. Green diamond marker: path starting point, $\omega =0$ Hz. Red diamond marker: path end point, $\omega =1$ PHz.

Table 2. Percentage Error (PE) and computational time t for the slab with $d_{eff}=180$ nm.

${\mathit{n}}_{\mathit{s}}$	PE
	PUNWOS	K-K	PUNW${}_{\mathbf{KK}}$	PUNW${}_{\mathbf{HT}}$
512	$303.88$	$13.97$	$1.51\times {10}^{-3}$	$1.51\times {10}^{-3}$
1024	$260.49$	$13.67$	$1.11\times {10}^{-3}$	$1.11\times {10}^{-3}$
2048	$130.14$	$13.61$	$2.63\times {10}^{-3}$	$2.63\times {10}^{-3}$
4096	$2.01\times {10}^{-3}$	$13.63$	$2.01\times {10}^{-3}$	$2.01\times {10}^{-3}$
${\mathbf{n}}_{\mathbf{s}}$	t (s)
	PUNWOS	K-K	PUNW${}_{\mathbf{KK}}$	PUNW${}_{\mathbf{HT}}$
512	$0.25$	$0.29$	$0.35$	$0.34$
1024	$0.28$	$0.42$	$0.34$	$0.33$
2048	$0.26$	$1.01$	$0.54$	$0.69$
4096	$0.33$	$2.92$	$1.16$	$0.35$

In correspondence to starting value $n_s=512$, we can note that the PE value turns out to be high for the K-K approach and very high for the PUNWOS method, whereas in the cases of the PUNW${}_{KK}$ and PUNW${}_{HT}$ methods, it is already factually negligible. In correspondence to $n_s=1024$, the value of this parameter related to the PUNW${}_{HT}$ and the PUNW${}_{KK}$ methods is still factually insignificant, whereas in the cases of the PUNWOS and K-K methods, it turns out to be comparable to the previous case. While increasing to $n_s=2048$, the above trend remains stationary for all methods. Finally, for $n_s=4096$, we have three unwrapping methods (PUNWOS, PUNW${}_{KK}$ and PUNW${}_{HT}$) that provide a factually insignificant PE, whereas we have no improvements for the PE in the case of the K-K method. As indicated by the PE parameter, values reported in Table 2 are confirmed by the results shown in Figure 2, which reports the plot of the retrieved refractive index $n_{eff}\left(\omega \right)$ compared with the exact refractive index $n_{ex}\left(\omega \right)$ for the four recovering methods for all the considered $n_s$ values. Figure 3 shows the sequence of p integers needed to give continuity to $ar{g}_{\pi }(·)$ provided by PUNWOS, PUNW${}_{HT}$, and PUNW${}_{KK}$ for $n_s=2048$. The high PE related to the PUNWOS method is clearly due to its incorrect calculation of the p sequence until an adequate number of sample points $n_s$ is used. Figure 4 shows path $\gamma \left(\omega \right)$ traced by Equation (8) for the slab with $d_{eff}=300$ nm. This last case we analyzed is characterized by a double pole Lorentzian model for its pair $\left({ϵ}_r\left(\omega \right),{\mu }_r\left(\omega \right)\right)$. Since for this slab, the resulting path $\gamma \left(\omega \right)$ intersects ${\mathbb{R}}_{\pi }$ many times more than in the previously considered case, the number of integers p that has to be computed to recover the continuity of $ar{g}_{\pi }(·)$ will be higher than previously performed, thus providing a more challenging test for the considered recovering approaches. The results reported in Table 3 validate the above hypothesis. For this test case, only the PUNW${}_{HT}$ method provides values for PE that are factually negligible for all considered values of $n_s$. Lower performances characterize all other remaining methods. More precisely, the PUNWOS method reaches a value of PE equal to those characterizing PUNW${}_{HT}$ only when the number of sample point in the $[0,1.5]$ PHz band increases to $n_s=$ 16,384. On the other hand, for K-K and PUNW${}_{KK}$ methods, the PE is always high, thus providing a recovered $n_{eff}\left(\omega \right)$ of lower quality, as demonstrated by the results reported in Figure 5, which shows the plot of $n_{eff}\left(\omega \right)$ compared with $n_{ex}\left(\omega \right)$, as a function of $n_s$ over the $[0,1.5]$ PHz frequency range. The results shown in Figure 6 confirm what has been stated about the number of p integers needed to ensure the continuity of the argument of $log(·)$, meaning that the number of Riemann sheets involved by $h(·)$ increase (as expected) with an increase in the slab’s thickness. Moreover, in this case, the high PE related to the PUNWOS method is clearly due to its incorrect calculation of the p sequence until an adequate number of sample points $n_s$ is used.

Figure 2. Plot of $n_{eff}\left(\omega \right)$, as a function of $n_s$, and $n_{ex}\left(\omega \right)$ for the MM with $d_{eff}=180$ nm over the frequency band $[0,1]$ PHz. Top left: PUNWOS method; top right: PUNW${}_{HT}$ method; bottom left: K-K relations; bottom right: PUNW${}_{KK}$ method. The insets magnify $n_{eff}\left(\omega \right)$ around its peaks.

Figure 3. Values of the integer p over the band $[0,1]$ PHz ($n_s=2048$). Left: PUNWOS; centre: PUNW${}_{HT}$; right: PUNW${}_{KK}$.

Figure 4. Path $\gamma \left(\omega \right)$ traced by (8) on the complex punctured plane $\dot{\mathbb{C}}$ for the slab with $d_{eff}=300$ nm. Green diamond marker: path starting point, $\omega =0$ Hz. Red diamond marker: path end point, $\omega =1.5$ PHz.

Table 3. Percentage error (PE) and computational time t for the slab with $d_{eff}=300$ nm.

${\mathit{n}}_{\mathit{s}}$	PE
	PUNWOS	K-K	PUNW${}_{\mathbf{KK}}$	PUNW${}_{\mathbf{HT}}$
1024	$206.03$	$6.86$	$3.90$	$5.76\times {10}^{-4}$
2048	$503.37$	$6.51$	$3.19$	$5.62\times {10}^{-4}$
4096	$464.54$	$6.48$	$2.84$	$5.81\times {10}^{-4}$
8192	$485.42$	$6.72$	$3.93$	$7.73\times {10}^{-4}$
16,384	$8.85\times {10}^{-4}$	$6.86$	$3.86$	$8.85\times {10}^{-4}$
${\mathbf{n}}_{\mathbf{s}}$	t (s)
	PUNWOS	K-K	PUNW${}_{\mathbf{KK}}$	PUNW${}_{\mathbf{HT}}$
1024	$0.30$	$0.41$	$0.37$	$0.36$
2048	$0.31$	$0.93$	$0.50$	$0.37$
4096	$0.38$	$3.18$	$1.30$	$0.27$
8192	$0.40$	$11.13$	$3.84$	$0.35$
16,384	$0.37$	$45.04$	$14.97$	$0.33$

Figure 5. Plot of $n_{eff}\left(\omega \right)$, as a function of $n_s$ and $n_{ex}\left(\omega \right)$ for the MM with $d_{eff}=300$ nm over frequency band $[0,1.5]$ PHz. Top left: PUNWOS method; top right: PUNW${}_{HT}$ method; bottom left: K-K relations; bottom right: PUNW${}_{KK}$ method. The insets magnify $n_{eff}\left(\omega \right)$ around its peaks.

Figure 6. Values of the integer p over the $[0,1.5]$ PHz ($n_s=8196$). Left: PUNWOS; centre: PUNW${}_{HT}$; right: PUNW${}_{KK}$.

4. Conclusions

In this work, we extend the analysis conducted in [30], placing the NRW relation in the context of the global analytic functions and the Riemann surfaces, thus providing a rigorous mathematical framework for handling the NRW relationship. In this context, the branch ambiguity problem finds its correct interpretation in terms of the number of Riemann surface sheets $\mathbb{S}\left(\mathcal{L}\right)$ on which the right-inverse $h(·)$, the solution of the NRW Equation (1), lies. Furthermore, we exploit this framework to derive a couple of phase unwrapping-based analytic continuation algorithms: the first, named PUNW${}_{KK}$, is based on K-K relations, and the second, named PUNW${}_{HT}$, is based on the Hilbert transform; both are capable of solving the NRW equation and avoid any ambiguity. To validate their numerical performances, some numerical experiments conducted on a couple of theoretical negative refractive index (NRI) slabs have been carried out. The close agreement obtained between the recovered and the exact refraction index provided by the PUNW${}_{HT}$ approach demonstrates its superiority over the other recovery methods considered in this study and confirms the validity of our theoretical analyses. Although the results provided by the PUNW${}_{KK}$ were less satisfactory than those given by PUNW${}_{HT}$, the use of K-K relations merged with the phase unwrapping approach enhances the results these relations provides. Finally, to conclude, we point out that the theory we have given in this work rigorously demonstrates some algorithmic procedures empirically proposed in the literature [20,21], broadening their meaning.