Madiha Amrani

Soufyane Khattou

Housni Al-Wahsh

Yamina Rezzouk

El Houssaine

El Boudouti
email: elboudouti@yahoo.fr

Cécile Ghouila-Houri

Abdelkrim Talbi

Abdellatif Akjouj

Leonard Dobrzynski

Bahram Djafari-Rouhani

Cecile Ghouila-Houri

Bound states in the continuum and Fano resonances in photonic and plasmonic loop structures

de niveau recherche, publiés ou non, émanant des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

Introduction

Bound states in the continuum (BICs) have brought significant attention over the last decades due to their important design principle to create systems with infinite quality factors in order to enhance light-matter interaction [START_REF] Hsu | Bound states in the continuum[END_REF]. BICs, also known as trapped modes, manifest as resonances with zero linewidths in lossless systems. In other words, they reside inside the continuous spectrum of extended states but remain perfectly confined in some parts of the system. The concept of BICs was first predicted in quantum mechanics by Von [START_REF] Von Neumann | Uber merkwiirdige diskrete Eigenwerte[END_REF] in 1929. Since then, BICs were found in various fields of physics such as photonics [START_REF] Bogdanov | Bound states in the continuum and Fano resonances in the strong mode coupling regime[END_REF][START_REF] He | Analogue of electromagnetically induced transparency in an all-dielectric double-layer metasurface based on bound states in the continuum[END_REF], acoustics [START_REF] Huang | Sound trapping in an open resonator[END_REF][START_REF] Amrani | Experimental evidence of the existence of bound states in the Continuum and Fano resonances in solid-liquid layered media[END_REF][START_REF] Quotane | Trapped-mode-induced Fano resonance and acoustical transparency in a one-dimensional solid-fluid phononic crystal[END_REF], mesoscopics [START_REF] Mrabti | Aharonov-Bohm-effect induced transparency and reflection in mesoscopic rings side coupled to a quantum wire[END_REF][START_REF] Bulgakov | Bound states in the continuum in open Aharonov-Bohm rings[END_REF]) and plasmonics [START_REF] Sun | Tunable plasmonic bound states in the continuum in the visible range[END_REF][START_REF] Qi | Plasmonic nanocavity for obtaining bound state in the continuum in silicon waveguides[END_REF][START_REF] Xie | Bound states in the continuum in a T-shape nanohole array perforated in a photonic crystal slab[END_REF]. Thanks to their high quality factors, they are very promising for many applications such as lasers [START_REF] Ha | Directional lasing in resonant semiconductor nanoantenna arrays[END_REF], filters [START_REF] Doskolovich | Integrated flat-top reflection filters operating near bound states in the continuum[END_REF]Cui et al. 2016) and sensors [START_REF] Wu | Giant enhancement of the Goos-Hänchen shift assisted by quasibound states in the continuum[END_REF][START_REF] Conteduca | Dielectric nanohole array metasurface for high-resolution nearfeld sensing and imaging[END_REF]. BICs can be categorized into several mechanisms based on their physical origin [START_REF] Hsu | Bound states in the continuum[END_REF]. Among them, one can cite, symmetry-protected BICs, Fabry-Perot (FP) BICs and Freidrich-Wintgen BICs which have been subsequently investigated theoretically and experimentally in different physical systems [START_REF] Amrani | Experimental evidence of the existence of bound states in the Continuum and Fano resonances in solid-liquid layered media[END_REF][START_REF] Zhao | Terahertz investigation of bound states in the continuum of metallic metasurfaces[END_REF][START_REF] Algorri | Strongly resonant silicon slot metasurfaces with symmetry-protected bound states in the continuum[END_REF][START_REF] Azzam | Formation of bound states in the continuum in hybrid plasmonic-photonic systems[END_REF][START_REF] Pankin | One-dimensional photonic bound states in the continuum[END_REF]). These modes are not observable from the continuous spectrum due to their non-radiative property with vanishing spectral linewidth. However, they can exist only under an appropriate choice of the geometrical or material parameters of the system and/or under specific external perturbations. Thus, by slightly detuning the system from the BIC conditions (for example, changing the geometrical parameters), the latter transforms to a quasi-BIC with a finite width. Generally, the quasi-BIC manifests itself as a Fano resonance when it falls in the vicinity of a zero of transmission. Let us recall that the Fano resonance is characterized by a resonance followed by an antiresonance over a narrow frequency range giving rise to an asymmetrical line profile [START_REF] Fano | Effects of configuration interaction on intensities and phase shifts[END_REF]. This resonance can be explained as the product of two processes of constructive and destructive interferences between a discrete state and a continuum background [START_REF] Fano | Effects of configuration interaction on intensities and phase shifts[END_REF][START_REF] Miroshnichenko | Fano resonances in nanoscale structures[END_REF].

Loop structures made of two arms are an excellent platform to explore BICs as well as Fano effects. For instance, Fano resonances have been studied in a mesoscopic loop with and without a quantum dot in one arm. The loop is placed under the action of a magnetic flux called Aharonov-Bohm (A-B) effect [START_REF] Wu | Persistent current and transmission probability in the Aharonov Bohm ring with an embedded quantum dot[END_REF][START_REF] Kobayashi | Fano resonance in a quantum wire with a sidecoupled quantum dot[END_REF][START_REF] Xiong | Fano resonance and persistent current of a quantum ring[END_REF]. Also, giant persistent currents and Fano resonances have been studied in single and double asymmetric rings with and without scatterers on both arms or in presence of stubs around the loop [START_REF] Hsueh | Resonant transport and giant persistent currents in double-asymmetric rings[END_REF][START_REF] Mrabti | Transmission gaps, trapped modes and Fano resonances in Aharonov Bohm connected mesoscopic loops[END_REF][START_REF] Voo | Fano resonance in transport through a mesoscopic two-lead ring[END_REF][START_REF] Al-Wahsh | Transmission gaps and sharp resonant states in the electronic transport through a simple mesoscopic device[END_REF]). Fano resonances have been obtained by tailoring the lengths of the wires constituting the system or the magnetic flux through the loop [START_REF] Bulgakov | Bound states in the continuum in open Aharonov-Bohm rings[END_REF][START_REF] Voo | Fano resonance in transport through a mesoscopic two-lead ring[END_REF][START_REF] Al-Wahsh | Transmission gaps and sharp resonant states in the electronic transport through a simple mesoscopic device[END_REF]. In plasmonic systems, different ring resonators based on metal-insulator-metal (MIM) structures and side branches attached to waveguides, have been proposed to study Fano resonances and for sensing applications [START_REF] Chou Chao | Multiple Fano resonance modes in an ultra-compact plasmonic waveguide-cavity system for sensing applications[END_REF][START_REF] Rakhshani | Fano resonances based on plasmonic square resonator with high figure of merits and its application in glucose concentrations sensing[END_REF][START_REF] Chen | Fano resonance in a MIM waveguide with double symmetric rectangular stubs and its sensing characteristics[END_REF][START_REF] Hajshahvaladi | Design of a highly sensitive tunable plasmonic refractive index sensor based on a ring-shaped nano-resonator[END_REF][START_REF] Liu | Tunable triple Fano resonance in MIM waveguide system with split ring resonator[END_REF]. In photonic systems, little works have been devoted to study BICs and Fano resonances in loop structures placed between two waveguides. For example, El [START_REF] El Boudouti | Experimental and theoretical evidence for the existence of photonic bandgaps and selective transmissions in serial loop structures[END_REF] have studied the possibility to realize superluminal phenomenon in an asymmetric loop structures as a consequence of the destructive interference between the waves in the two arms of the loop like in Mach-Zehnder interferometers [START_REF] Sánchez-Meroño | Fast light in unbalanced low-loss Mach-Zehnder interferometers[END_REF][START_REF] Zou | Temporal manipulation of light propagation via cross-intensity modulation in unbalanced fiber Mach-Zehnder interferometers[END_REF]. The same structure has been studied in acoustic tubes [START_REF] Robertson | Sound beyond the speed of light: measurement of negative group velocity in an acoustic loop filter[END_REF]. However, to our knowledge, an analytical and experimental study of BICs and Fano resonances in a single asymmetric photonic loop structure, has not been treated before.

In this paper, we consider a simple photonic loop which is a good candidate to obtain BICs and Fano resonances. The loop is composed of two connected wires of lengths d 1 and d 2 . The length of the loop L = d 1 + d 2 is considered fix, whereas the detuning between the two wires = d 2 -d 1 is taken variable (Fig. 1). The asymmetric loop is placed between two semi-infinite leads (Fig. 1). Using the Green's function method [START_REF] Dobrzynski | Photonics[END_REF], we give a clear analytical demonstration of BICs in a loop under specific values of the detuning with an experimental confirmation in the radio-frequency performed with coaxial cables and a numerical validation in the infrared region based on finite element method (FEM) using Comsol Multiphysics. We demonstrate that the BICs only appear when the two arms of the loop are taken commensurate (i.e.,

d 2 d 1 = n m
where m and n are non-zero integers) in such a way that the electric field vanishes at the connecting points with the semi-infinite wires, leading to uncoupled stationary modes in the loop within the continuum states of the waveguide. These modes represent FP-type BICs [START_REF] Hsu | Bound states in the continuum[END_REF][START_REF] Sadreev | Interference traps waves in an open system: bound states in the continuum[END_REF]. By slightly shifting from BIC conditions, the FP-BICs collapse to specific Fano resonances with finite lifetime known as quasi-BICs with an anti-resonance shape. In addition, by taking into account the loss in the loop structure, we show that there exists a threshold value of where the absorption tends to a maximum value, this happens when the transmission and reflection coefficients become equal (around 22 % ). Also, we provide a comparison between the density of states (DOS) of the system which is a fundamental physical quantity and the derivative of the phase of the determinant of the scattering matrix, the so-called Friedel phase [START_REF] Friedel | The distribution of electrons around impurities in monovalent metals[END_REF] which is measurable in the transmission experiment. In particular, we show the similarities and differences between these two quantities as function of . Finally, considering plasmonic MIM waveguides and loop of nanometric size, we confirm our analytical results by FEM simulations of the plasmonic structure in the infrared domain and highlight the efficiency of the proposed design for gas sensing applications.

The rest of the paper is organized as follows: In Sect. 2, we give a brief review of the method of calculation based on the Green's function approach. We provide an analytical expression of the dispersion relation of BICs and a relationship between the DOS and the derivative of the Friedel phase. Section 3 is devoted to the analytical and experimental results in the radio-frequency, whereas section 4 provides numerical results of the plasmonic MIM structure in the infrared region. The conclusions are presented in Sect. 5.

Theoretical model: Green's function elements

We consider an infinite homogeneous isotropic dielectric wire characterized by its impedance Z and the relative permittivity . An electromagnetic wave field is propagating through the cable in the dielectric-field space between the center conductor and shield. In the limit where the widths of the wires are much smaller than their lengths, the telegrapher's equation describing microwave networks, the so-called TEM waves, is given by [START_REF] Dobrzynski | Photonics[END_REF] where c is the speed of light in vacuum, and is the angular frequency of the wave. Then, the corresponding Green's function between two points x and x ′ of this wire is defined by

The solution of Eq. ( 2) is given by [START_REF] Dobrzynski | Photonics[END_REF] where F = Z , k = √ ∕c , and j = √ -1 . Let us recall that √ = n defines the index of refraction of the coaxial. Its inverse called the velocity factor is given by

V F = 1 c √ LC
where L and C are the inductance and the capacitance per unit length of the cable, respectively. L and C are frequency dependent. As mentioned above, the validity of our results is subject to the requirement that the cross section of the waveguide is negligible compared to their length and to the propagation wavelength, then the assumption of mono-mode waveguide is satisfied. Before giving the theoretical results of the transmission and reflection coefficients as well as the dispersion relation and DOS, we will recall briefly the expression of the Green's function of the different wires from which the loop system is composed, namely:

(i) The inverse of the Green's function g -1 i (M i M i ) of a finite segment of length d i in the space of its interface M i ={0, d i } can be written as a ( 2 × 2 ) matrix (Dobrzynski et al. 2020) where

C i = cos(kd i ) , S i = sin(kd i ) , F = Z , and k = √ c ( i = 1, 2 ).
It should be noted that these and the following expressions remain valid even for a dispersive material (such as in plasmonics) where the permittivity is a function of . (ii) The inverse of the Green's function g -1 0 (MM) in the space of interfaces M ={1, 2} (Fig. 1) of an asymmetric loop is given by the superposition of two ( 2 × 2 ) matrices given in Eq. (4) [START_REF] Dobrzynski | Photonics[END_REF] for each wire i (1)

2 x 2 + 2 c 2 E(x)=0 (2) 2 x 2 + 2 c 2 G(x, x � )= (x -x � ) (3) G(x, x � )=-j e -jk|x-x � | 2F , (4) g -1 i (M i M i )= - FC i S i F S i F S i - FC i S i
, where a =-F(

C 1 S 1 + C 2 S 2 ) and b = F( 1 S 1 + 1 S 2
).

(iii) The inverse of the Green's function of a semi-infinite wire is given by [START_REF] Dobrzynski | Photonics[END_REF] where j = √ -1.

Transmission and reflection coefficients and conditions for BICs

The photonic structure presented in Fig. 1a is composed by an asymmetric loop made of two wires of lengths d 1 and d 2 . The system is inserted between two semi-infinite wires. The inverse of Green's function of the whole photonic system is given by a linear superposition of the Green's functions of its constituents given above (Eqs. ( 5) and ( 6)) in the interface space M ={1, 2} leading to a ( 2 × 2 ) matrix, namely (Dobrzynski et al. 2020) Let us consider an incident wave U(x)=e -jkx launched in the left semi-infinite wire (Fig. 1). From Eq. ( 7), one can obtain the transmission coefficient in the right semi-infinite wire, namely, t =-2jFg(1, 2) , or equivalently

In the same way, the reflection coefficient in the left semi-infinite wire is given by r =-1 + 2jFg(1, 1) , or equivalently

where

L = d 2 + d 1 and = d 2 -d 1 .
From the expressions of t (Eq. ( 8)) and r (Eq. ( 9)), one can deduce the transmission and reflection rates as T = |t| 2 and R = |r| 2 . Therefore, in the absence of loss (i.e., k is real), one can check easily the conservation law R + T = 1 . However, in presence of loss, (i.e., k is complex) the absorption coefficient can be obtained from the equation

A = 1 -R -T.
It is well known that the eigenmodes of the system are given by the poles of the transmission coefficient t (Eq. ( 8)) or equivalently by the poles of the Green's function (Eq. ( 7)), namely This latter equation is a complex quantity. Its real part gives the position of the resonances in transmission and DOS, whereas its imaginary part is related to the width of the resonance. In general, it is not easy to simultaneously cancel the real and imaginary parts of Eq.( 10) at the same frequency. This will correspond to a bound state falling in the (5)

g -1 0 (MM)= ab ba , (6) g -1 s (i, i)=-jF, (7) g -1 (MM)= a -jF b ba -jF . (8) t = 4 sin(kL∕2) cos(k ∕2) 5 sin 2 (kL∕2)-sin 2 (k ∕2)+j4 sin(kL∕2) cos(kL∕2) . (9) r =- 5 sin 2 (kL∕2)+sin 2 (k ) 5 sin 2 (kL∕2)-sin 2 (k ∕2)+j4 sin(kL∕2) cos(kL∕2) , (10) 5 sin 2 (kL∕2)-sin 2 (k ∕2)+j4 sin(kL∕2) cos(kL∕2)=0.
continuum of states. In order to avoid the divergence of t, then its numerator should also vanish in such a way that t becomes finite. These two conditions can be fulfilled only if sin( kL 2 )=0 and sin( k 2 )=0 . Therefore, we get the following condition where m and n are two odd or two even integers (i.e., m + n = 2p ). This means that the lengths d 1 and d 2 should be commensurate with each other. To give a physical picture of the BICs situation, we have defined a unit length d 0 such that the lengths in the two arms of the loop are multiple of d 0 ( d 1 = md 0 and d 2 = nd 0 ). Now, the lowest (fundamental) BIC frequency f 0 will be defined such that the unit length d 0 accommodates half of the corresponding wavelength ( 0 = 2 ∕k =(2 √ c)∕f 0 ). Then, the number of wavelengths in the whole loop will be an integer multiple of 0 ( (m + n) 0 ∕2 = p 0 ), but this number can be divided among the two arms as integer or half-integer numbers of 0 . The higher (excited) BIC frequencies are the multiple of f 0 . Also, notice that the eigenfunction of the BIC will have zero amplitude at the junction points with the semi-infinite waveguides.

To express the conclusion differently, let us note that Eq. ( 10) can be written explicitly as where U n+m (S 0 ) and U n-m (S 0 ) represent Chebyshev Polynomials of the Second kind and S 0 = sin(kd 0 ∕2).

Therefore, the BICs are given by sin(kd 0 )=0 . These modes occur for Also, we can show easily that the transmission T reaches unity at the BIC condition. Let us mention that the single loop without the semi-infinite wires exhibits two degenerate modes given by sin(kL∕2)=0 (or sin(kd 0 ∕2)=0 ). Now, in presence of the semi-infinite wires, Eq. ( 12) shows that one of the two modes remains a BIC, whereas the second mode interacts strongly with the continuum states of the surrounding media (see below).

Density of states and Friedel phase

Another interesting quantity that can be deduced from the Green's function is the density of states (DOS). In order to provide an analytical comparison of the DOS with the phases involved in the scattering matrix, we consider the variation of DOS ( Δn( ) ) between the loop system depicted in Fig. 1 and a reference system formed out of the same volumes of the decoupled semi-infinite wires and the isolated loop. This quantity is given by For a lossless system, the expression of det[g(M, M)] can be derived from the matrix in Eq. ( 7) as follows

(11) d 1 d 2 = m n , (12)
sin(kd 0 ) 5 sin(kd 0 )U 2 n+m (S 0 )-sin(kd 0 )U 2 n-m (S 0 )+j8U n+m (S 0 ) cos[kd 0 (m + n)∕2] = 0, (13) = 2 m -n m + n L 2 and kL 2 = m + n 2 = p. (14) Δn( )= -1 d d Arg det g(M, M) g 0 (M, M)F 2 .
Also, the expression of det[g 0 (M, M)] is obtained from the matrix in Eq. ( 5), namely

Now, if we do not take into account the discrete eigenmodes of the isolated loop system (i.e, sin(kL∕2)=0 ), then Eq. ( 14) becomes simply

It is worth noticing that a slightly different expression of the variation of DOS (Δn � ( )) can be obtained by considering the difference between the structure in Fig. 1 anda reference system made of the infinite wire. This quantity can be derived by integrating the local DOS over the whole structure and subtracting the DOS of the infinite wire.

In addition to the information that can be deduced from the transmission and reflection coefficients, other supplementary informations can be deduced from the modulus and argument of the determinant of the scattering matrix defined as where t and r are the transmission and reflection coefficients given by Eqs. ( 8) and ( 9) respectively.

Indeed, we provide an analytical comparison between the scattering Friedel phase defined as [START_REF] Friedel | The distribution of electrons around impurities in monovalent metals[END_REF] f = Arg[det(S)] and Δn( ) . The determinant of the scattering matrix S (Eq. ( 18)) for the loop system is given by For the lossless system, one can show easily the well known relation satisfied by det(S) (i.e., |det(S)| = 1 ) and Hence, one can deduce from Eqs. ( 17) and (20) that However, for a lossy system, the analytical equation ( 21) is no longer valid. We will provide numerically and experimentally the effect of the presence of loss on the above results. In particular, we show that the absorption A can be also extracted from |det(S)| through the relation |det(S)| ≃ 1 -A , also A reaches a maximum value of 0.55 for a threshold value of th around the BIC where R = T ≃ 0.22 . In addition, we show that

15) det[g(M, M)] = - S 1 S 2 F 2 {5 sin 2 (kL∕2)-sin 2 (k ∕2)+j4 sin(kL∕2) cos(kL∕2)} . (16) det[g 0 (M, M)] = - S 1 S 2 4F 2 sin 2 (kL∕2) . (17) Δn( )=- 1 d d arctan 4 sin(kL∕2) cos(kL∕2) 5 sin 2 (kL∕2)-sin 2 (k ∕2) . (18) S = rt tr (19) det(S)=r 2 -t 2 = [3 sin 2 (kL∕2)+sin 2 (k ∕2)] 2 +[4 sin(kL∕2) cos(kL∕2)] 2 5 sin 2 (kL∕2)-sin 2 (k ∕2)+j4 sin(kL∕2) cos(kL∕2) 2 .
(20

) f = Arg[det(S)] = -2 arctan 4 sin(kL∕2) cos(kL∕2) 5 sin 2 (kL∕2)-sin 2 (k ∕2) . (21) d f d = 2 Δn( ).
It is worth noting that Eq. ( 21) has been obtained in mesoscopic systems in absence of loss [START_REF] Büttiker | Capacitance, admittance, and rectification properties of small conductors[END_REF][START_REF] Lee | Generic transmission zeros and in-phase resonances in time-reversal symmetric single channel transport[END_REF], however, there are little works on such relationship in presence of loss [START_REF] Khattou | Comparison of density of states and scattering parameters in coaxial photonic crystals: theory and experiment[END_REF].

Analytical and experimental results in the radio-frequency domain

In what follows, we will illustrate the theoretical results discussed in Sect. ..). These modes appear as the intersection of the curves given by sin(kL∕2)=0 (white dashed lines) and sin(k ∕2)=0 (black dotted lines) in Fig. 2a. For any given value of the dimensionless frequency Ω=p and for lying between two BICs, one can notice the shrinking of the dips in the transmission spectra (Fig. 2a) given rise to a full transmission at the BICs; similarly the width of the resonances in the DOS become narrow giving rise to delta peaks at the BICs. These results confirm the analytical results found in Sect. 2.2. The first BIC falls at Ω=1 and = 0 for the pair ( m = 1, n = 1 ). In the vicinity of this particular point, BIC transforms to a quasi-BIC with a finite lifetime. Generally, the quasi-BIC manifests itself as a Fano resonance in the transmission and a narrow resonance in the DOS. In what follows, we will focus on the BIC associated to the pair ( m = 1, n = 1 ) at Ω=1 and around = 0.

In order to give a better insight about the behavior of the BICs and the associated Fano resonances in the transmission amplitude, we plot in Fig. 3 the transmission and DOS versus the dimensionless frequency Ω for = 0 and = 10 cm. For = 0 (Fig. 3a), the trans- mission coefficient shows ordinary Fabry-Pérot oscillations. The arrows indicate the positions of the BICs around Ω=1, 2, 3 where T = 1 . These modes are hidden and do not give any signature in the transmission spectra because the corresponding resonances have zero width. When we slightly shift the lengths of the two wires (i.e., ≃ 0 ), the BICs induce Fano resonances as shown in Fig. 3c for = 10 cm. This result is also clearly visible in the DOS spectra where we can notice for = 0 (Fig. 3b) a superposition of F-P broaden resonances and BICs (with infinite lifetime) indicated by vertical lines (delta peaks). For = 10 cm (Fig. 3d), the BICs become narrow resonances or quasi-BICs (with a finite life- time), whereas the broad resonances remain of F-P type. The resonance in Fig. 3c is of Fano shape. Indeed, a Taylor expansion of the transmission amplitude (Eq. ( 8)) around the resonance at Ω=1 (i.e., Ω=1 + with 𝜖 << 1 ) enables us to write T following the Fano- shape, namely where Γ= 2( ∕L) 4 16+6 2 ( ∕L) 4 -10 2 ( ∕L) 2 is the full width at half maximum of the resonance. This is a specific Fano resonance where the Fano parameter q = 0 Miroshnichenko et al. ( 2010); it appears as an anti-resonance in the transmission spectrum (Fig. 3c). Also, by detuning this resonance is quantified as electromagnetic induced transparency (EIT) resonance (Limonov et al. 2017). Indeed, the system behaving as an opaque system for ≃ 0 around Ω=p (Fig. 3c) becomes transparent at = 0 (Fig. 3a).

In Fig. 3e, we can notice a very good agreement between the exact results of the transmission coefficient (blue line) and the approximate result (red circles) obtained from Eq.( 22) around Ω=1 . Also, a Taylor expansion of the Δn( ) (Eq.( 17)) around the reso- nance at Ω=1 can be written as where B = ( ∕L) 2 16+6 2 ( ∕L) 4 -10 2 ( ∕L) 2 . The results of the approximate expression (Eq. ( 23)) are shown in Fig. 3f by open circles. These results are in accordance with the exact ones (solid lines).

From Eq. ( 22) and Figs. 3a and c one can see that for = 0 , T = 1 while for ≠ 0 , T = 0 at Ω=1 (i.e., = 0 ). Similarly, Eq. ( 23) and Fig. 3b andd show that for = 0 , Δn( ) → ∞ and for ≠ 0 , Δn( ) becomes finite ( Δn( )= 4B

Γ 2 ) at Ω=1. Now, we discuss the behavior of the transmission spectra when the dissipation in the wires constituting the loop is taken into consideration. Figure 4 provides the transmission and reflection amplitudes as well as the absorption versus the dimensionless frequency Ω for different values of . The arrow in Fig. 4a indicates the position of the BIC. The blue lines correspond to the theoretical results with loss, whereas the red circles give the experimental results. For = 0 (Fig. 4a), the transmission coefficient presents a maximum value around Ω=1 , where the BIC is hidden and does not give any signature in the transmis- sion spectrum, the corresponding resonance having a zero width. In this case, the reflection tends to zero and the absorption shows a weak value around 0.1. By slightly shifting from BIC condition (Fig. 4b), the FP-BIC transforms into a quasi-BIC with a Fano shape. Furthermore, one can observe from Figs. 4b-d that the width of the resonance increases and its intensity decreases in the transmission spectra as increases. The reflection increases with and reaches a maximum value of 0.55, while the absorption presents a maximum value of 0.56 around th = 20 cm and decreases for 𝛿>20 cm. The experimental results (open circles) are in good agreement with the theoretical ones (solid lines) in the presence of loss. It is worth noting that the loss affects considerably the shape of Fano resonances around Ω=1.

Figure 5 gives the theoretical variation of the transmission (red line), reflection (blue line) and absorption (green line) versus at Ω=1 . One can see that the absorption A increases for small values of and goes through a maximum A max ≃ 0.56 around th ≃ 20 cm and then decreases for 𝛿>20 cm. This behavior happens when the transmission and reflection become equal (i.e., T = R ≃ 0.22 ). These results show that the peak in the absorption can be controlled by detuning the lengths of the two arms.

To characterize better the BICs and Fano resonances, we give in Fig. 6 a comparison between Δn( ) which is a fundamental physical quantity and the derivative of the Frie- del phase which can be measured from the experiment as a function of the dimensionless

(22) T = ( + qΓ∕2) 2 2 + (Γ∕2) 2 , (23) Δn( )= B 2 + (Γ∕2) 2 ,
frequency Ω for different values of . The variation of DOS (green lines) is obtained from Eq. ( 17), while the derivative of the Friedel phase d f d (pink dashed lines) is given by Eq. ( 20). In the absence of loss, One can notice a very good agreement between 2 Δn( ) and d f d in accordance with Eq. ( 21) and confirm the results discussed in Sect. 2.3. As mentioned before, BIC appears as delta peak in the DOS with zero width as shown in Fig. 6a. However, the DOS still shows a broad resonance as illustrated in the inset of Fig. 6a. Now, by slightly shifting from the BIC (i.e., ≠ 0 ), a classical Breit Weigner resonance appears as a well defined peak characterized by a finite width and a finite intensity. Also, the width of the resonance increases with , while its intensity decreases.

As mentioned previously, the relation between the DOS and the derivative of the Friedel phase d f d are exact for a lossless system. However, the presence of loss can affect significantly the above relations. Indeed, from Fig. 5, we can expect different behaviors of det(S) around = th where R = T . In order to give a better insight about the effect of dissipation on these two quantities, we have plotted in Fig. 7 a comparison between Δn( ) (green dashed lines) and -e. Indeed, for = th we have found that det(S) vanishes (i.e., t ≃±r ) at Ω=1 and its argument becomes ill-defined. However, for 𝛿<𝛿 th ( 𝛿>𝛿 th ), Img[det(S)] changes sign at Ω=1 , whereas Re[det(S)] remains positif (negative). Therefore, Arg[det(S)] decreases (increases) around Ω=1 for 𝛿<𝛿 th ( 𝛿>𝛿 th ) as it is shown in the supplementary material (Figs. S1a,b). However, the variation of the number of states given by Arg[g(MM)] increases monotonically as a function of Ω whatever the value of (See Fig. S1 c). Conse- quently, for 𝛿<𝛿 th , Δn( ) is different from It is worth noting that the other BICs in Fig. 2 present the same behavior as for the pair ( m = 1, n = 1 ). An example is given in the supplementary material SM2 (Fig. S2) for the pair ( m = 5, n = 3 ). Also, we have checked theoretically and experimentally that the absorption can be accurately derived from |det(S)| (i.e., A ≃ 1 -|det(S)| ). This result is given in the sup- plementary material (Fig. S1d) for = 20 cm. Finally, let us mention that the threshold value th decreases as far as the strength of the attenuation ( ′′ ) decreases. However, the results in Fig. 5 remain qualitatively the same and the relationship between DOS and d f d remains also valid for 𝛿<𝛿 th and 𝛿>𝛿 th .

Analytical and numerical results in the infrared domain

In this section, we give another analytical and numerical application of the results presented in the previous section to plasmonic nano-waveguides operating in the infrared domain. Also, we show the usefulness of the Fano resonances for sensing applications. Therefore, we consider a plasmonic structure composed of a rectangular metal-insulator-metal (MIM) ring resonator inserted between two semi-infinite MIM waveguides (Fig. 1c). The length of the rectangular ring is fixed to L = d 1 + d 2 = 1400nm, whereas the detuning between the two arms = d 2 -d 1 is chosen variable. The waveguides are filled with air, while the surrounding metal is made of silver with a dielectric constant m given by the Drude model [START_REF] Maier | Plasmonics: Fundamentals and Applications[END_REF][START_REF] Zhang | Analytical method for metal-insulator-metal surface plasmon polaritons waveguide networks[END_REF] where ∞ = 3.7 is the infinite frequency permittivity, = 2.634 × 10 13 rad/s is the electron collision frequency, p = 1.38 × 10 16 rad/s the bulk plasma frequency of silver, and is the angular frequency of the incident light. These parameters are obtained by fitting the experimental data [START_REF] Han | Plasmon-induced transparency with detuned ultracompact Fabry-Perot resonators in integrated plasmonic devices[END_REF] at the infrared domain. The width of the MIM waveguide has been chosen to be d=50 nm in order to ensure that only the fundamental transverse magnetic ( TM 0 ) mode is supported.

(24) m = ∞ - 2 p 2 + j ,
The plasmonic device presented in Fig. 1c is 2D system in the (x, y) plane and infinite in the z-direction. This design is the most used in MIM waveguides simulations as it is less time consuming. However, for practical processing, the MIM waveguides are 3D as they present a finite height "h" along the z-direction and they are deposited on SiO 2 (or Si) substrate and terminated by air. The finite depth of the system and the surrounding media introduce a significant effect on loss in the system. Nevertheless, it was shown that above a certain height ( h = 50 nm), 2D and 3D models provide almost similar results in particu- lar as concerns the positions of the resonant modes in the transmission spectra [START_REF] Niu | Nanofocusing of the free-space optical energy with plasmonic Tamm states[END_REF]; however, their shapes and widths are considerably affected. The analytical results of section 2.2 will be compared with numerical simulations based on a two-dimensional (2D) finite element method used to simulate the transmission spectra through the plasmonic device depicted in Fig. 1c using Comsol Multiphysics software.

First, it is necessary to introduce the surface plasmon polariton (SPP) propagation wavevector k sp and the impedance of the MIM waveguide Z. Here, we focus on the MIM waveguide which only supports the symmetric TM 0 mode and for sufficiently small dielectric thickness d (i.e., d ≪𝜆 sp ), the wavevector k sp can be obtained from a Taylor expansion of the dispersion relation of the MIM system, namely [START_REF] Maier | Plasmonics: Fundamentals and Applications[END_REF] where d and m are the dielectric constants of the dielectric and the metal respectively. k 0 = c is the wave vector in vacuum and c is the speed of light in vacuum. Another interesting physical quantity in the Green's function method is the impedance Z of the MIM waveguide, this quantity is given by [START_REF] Maier | Plasmonics: Fundamentals and Applications[END_REF], [START_REF] Zhang | Analytical method for metal-insulator-metal surface plasmon polaritons waveguide networks[END_REF] By using Eqs. ( 25) and ( 26), the calculation of the transmission coefficient and the dispersion relation as well as DOS are similar to those presented in Sect. 2.2, just we need to replace in Eq. ( 3), k by k sp (Eq. ( 25)) and F by Z where Z is given by Eq. ( 26). Let us mention that the definition of the parameter F in the plasmonic system is the inverse of that of the photonic system (Eq. ( 4)) as the Green's function of the latter is defined from the electric field, while that of the former is defined from the magnetic field.

In order to excite the symmetric guided plasmon (with respect to the symmetry plane in the middle of the MIM-guide) in which we are interested, a boundary mode analysis is set prior to the wave propagation study at the inlet/outlet of the structure. This step solves for the fields and associated propagation constant k sp ≡ k x which can propagate in the MIM structure. The frequency domain which we are interested in here corresponds to the Infrared (IR) domain ranging from 1200 to 2600 nm. In this frequency interval, the waveguide is strictly mono-mode and antisymmetric [START_REF] Bah | Response functions in layered dielectric media[END_REF]. It corresponds to a SPP-like mode in which the electric field is strongly confined along the dielectric-metal interface and decays exponentially away from it [START_REF] Noual | Nanoscale plasmon waveguide including cavity resonator[END_REF][START_REF] Noual | Plasmonic-induced transparency in a MIM waveguide with two side-coupled cavities[END_REF]. Concerning the employed mesh, we set the mesh maximal step size in the dielectric as Δ ≤ 0 ∕6 (i.e., six elements per wavelength) whilst in the metal at least 5 to 6 points per the silver metal skin depth ∼ 20 nm are to be set, so we chose Δ∼2 nm to guarantee a good numerical convergence. On the other hand, we take numerically benefit from the ports boundary conditions to compute the transmission/reflection spectra. This comes from the scattering matrix of the structure S (Eq. ( 18)) which is automatically set whenever ports are used in Comsol.

Similarly to the photonic case, we plot in Figs. 8a and b the analytical and numerical variation of the transmission intensity (in color scale) versus the wavelength and the detuning with L = d 1 + d 2 = 1400 nm, respectively. Also, in this optical range, we prefer to use the wavelength instead of frequency. Here, we focus on the second BIC corresponding to ( m = 2, n = 2 ) which falls in the near infrared domain. One can see that the BIC falls at = 1707 nm for = 0 . By detuning from the BIC position, the latter transforms to a Fano resonance with a finite width as shown in Fig. 8. These results are qualitatively similar to those found in the photonic loop system (Fig. 2a and the inset in Fig. 3) but in the infrared domain. Also, one can notice that the transmission intensity does not reach unity ( T max = 0.8 ) ( 25)

k sp = k 0 � d - 2 d √ d -m m k 0 d , (26) Z = k sp d d .
because of the intrinsic loss in the MIM waveguides. We can see a good agreement between the analytical (Fig. 8a) and numerical (Fig. 8b) results.

To give a better insight about the behavior of BIC in the transmission spectra, we present in Fig. 9 the transmission versus the wavelength for = 0 (Fig. 9a) and = 50 nm (Fig. 9b). For = 0 , one can notice the existence of a BIC with zero width around r = 1707 nm indi- cated by the black arrow. Now, in order to show the existence of a Fano resonance, we detune slightly from = 0 , this is shown in Fig. 9b for = 50 nm. The analytical (open triangles) and numerical (solid lines) results are compared together in Figs. 9a andb revealing a good agreement between them which validate the accuracy of the analytical results. Also, we have checked that the absorption coefficient can reach a maximum value around A = 0.53 for th = 38.5 nm similarly as in Fig. 5 (see Fig. S3 in the supplementary material SM3).

Plasmonic MIM waveguides coupled side with a ring resonator have been considered as one of the most promising ways for developing nanoscale integrated optical circuits and they are appropriate candidates for refractive index sensing. Indeed, Fano resonance is extremely sensitive to changes in the refractive index in the waveguide [START_REF] Hayashi | Waveguide-coupled surface plasmon resonance sensor structures: Fano lineshape engineering for ultrahigh-resolution sensing[END_REF], which provides a very promising pathway to achieve a high sensitive sensor efficiency. The sensitivity and the figure of merit (FOM) are the most important factors that characterize the sensor. The sensitivity is determined by the shift of the resonance wavelength Δ due to the changes in the refractive index Δn , and is defined as whereas, FOM is given by where FWHM is the full width at half maximum of the resonance dip.

The transmission amplitudes are calculated and simulated using different filling media in the waveguide in order to investigate the effect of the refractive index n on Fano resonance for = 50 nm. The analytical (open circles) and numerical results are plotted in Fig. 9c for different refractive index from n = 1 to n = 1.1 with a step of 0.02, leading to a shift of the spectra when n increases. Figure 9d and FOM = 94 around = 2393 nm. However, in this case the numerical results do not match sufficiently with the analytical ones. These results are given in the supplementary material SM3 (Figs. S4a andb). Table 1 gives a comparison of our results on the sensitivity and FOM with those of other plasmonic refractive index sensors based on a MIM waveguide with a side-coupled ring resonator published in the literature. In comparison to other similar sensors studied recently, the proposed design shows very good performances. It is worth noting that the plasmonic sensor can also sense the refractive index change of water and bio-cells and therefore can be used as a biosensor [START_REF] Zhang | Plasmonic refractive index sensor with high figure of merit based on concentric-rings resonator[END_REF]. The compact design of the sensor makes it suitable to be integrated into the on-chip optical sensing platforms.

Conclusion

In summary, we have given an analytical evidence about the existence of BICs and Fano resonances in a single loop made of two arms of lengths d 1 and d 2 with both an experi- mental validation in the radio-frequency domain using coaxial cables and a numerical validation in the infrared domain using plasmonic MIM waveguides. The whole system is inserted between two semi-infinite wires. The analytical results obtained with the help of the Green's function method are confirmed by the experimental measurements using the coaxial cables in the radio-frequency regime. We have demonstrated analytically that the loop system may exhibit BICs for particular values of such that d 1 and d 2 are com- mensurate. These modes appear as resonances with zero width in the transmission spectra, as well as in the DOS for lossless system. By slightly shifting from that corresponding to BICs, one can obtain Fano resonances with finite width known as quasi-BICs with an anti-resonance shape. Additionally, for a lossy system and depending on the detuning between the two arms, the absorption may goes to a maximum value of around 0.56 for a threshold th = 22 cm. This behavior occurs when the transmission and reflection become equal ( R = T ≃ 0.22 ). Also, we have given a comparative study of DOS and Friedel phase (or equivalently the argument of the determinant of the scattering matrix). In particular, we have shown that for a lossless system, the DOS is proportional to the derivative of the Friedel phase, while for a lossy system, one can distinguish two cases: i) for 𝛿<𝛿 th , the Friedel phase decreases around certain frequencies giving rise to a negative enlarged delta peak in the derivative of the Friedel phase and the DOS becomes different from the latter. ii) For 𝛿>𝛿 th , the DOS is equivalent to the derivative of the Friedel phase. Also, we have demonstrated that the absorption coefficient can be extracted from the modulus of det(S) which is a measurable quantity. Finally, we have given a transposition of these results to plasmonic MIM waveguides operating in the infrared domain by taking properly the lengths of the two arms of the loop at the nanoscale. The last structure is proposed to realize an efficient refractive index sensor. Here also the analytical results (obtained by the Green's function approach) are in good agreement with the numerical results obtained by the finite element method. The results presented in this work can be transposed to acoustic waveguides in the audible frequency domain [START_REF] Robertson | Sound beyond the speed of light: measurement of negative group velocity in an acoustic loop filter[END_REF]).

Fig. 1 a

 1 Fig. 1 a Schematic representation of the asymmetric loop with different lengths d 1 and d 2 inserted between two semi-infinite leads. b Experimental setup for the study of the loop system performed with standard coaxial cables and a network vectorial analyzer. The lengths are indicated in accordance with (a). c Schematic representation of the plasmonic nanowaveguide loop used in Comsol simulation

  d f d can exhibit a different behavior in comparison with DOS depending on whether 𝛿<𝛿 th or 𝛿>𝛿 th (see below).

(

Fig. 2

 2 Fig. 2 Variation of transmission (a) and DOS (b) (with color scale) versus the dimensionless frequency Ω and . The white circles indicate the positions of the BICs with different pairs (m, n). The black heavy curves and the white dashed lines in (a) represent the modes given by sin(k ∕2)=0 and sin(kL∕2)=0 , respectively

Fig. 3

 3 Fig. 3 Transmission and DOS versus the dimensionless frequency Ω=kL∕2 for a loop with = 0 (a), (b) and = 10 cm (c), (d). (e) and (f) Comparison of the approximate results obtained by the Taylor expansion of the transmission and DOS (open circles) around the resonance with the exact calculations (solid lines).The inset shows a zoom of the transmission with color scale around the mode (1, 1) at Ω=1

  lines) for different values of around th . For = 0 , Fig.7ashows that the BIC

Fig. 4

 4 Fig. 4 Transmission, reflection and absorption versus the dimensionless frequency Ω for different values of . The blue lines are the theoretical results, whereas the open circles represent experimental data

  the phase drop of Arg[det(S)] giving rise to a negative enlarged delta peak as shown in Figs. 7b and c for = 10 cm and = 20 cm respectively. Whereas, for 𝛿>𝛿 th , Δn( ) becomes almost similar to d f d as shown in Figs. 7d and e for = 30 cm and = 40 cm respectively.

Fig. 5

 5 Fig. 5 Theoretical variation of the transmission (red line), reflection (blue line) and absorption (green line) versus at the resonance Ω=1 in Fig. 4. Open circles correspond to experimental data. (Color figure online)

Fig. 6

 6 Fig. 6 Theoretical variation of the derivative of the Friedel phase d f d (pink dashed line), and the variation of DOS Δn( ) (green solid lines) for different values of in a lossless system. The position of the BIC at = 0 is indicated by the vertical dashed line (delta peak) in (a)

Fig. 7

 7 Fig. 7 Same as in Fig. 6 but for a lossy system. The theoretical results of d f d (pink solid lines) are validated by the experimental data (red open circles). The variation of DOS ( Δn( ) ) is plotted with green dashed lines

Fig. 8 aFig. 9

 89 Fig. 8 a Analytical and b numerical variation of the transmission (with color scale) versus the wavelength and the detuning . The white circle indicates the position of the BIC

Table 1

 1

	Comparison results rectangular ring with previous of the proposed plasmonic	Structure	Year	(nm/RIU) Sensitivity	FOM
	works	Ref Tang et al. (2017)	2017	1000	63
		Ref Zhang et al. (2018)	2018	1700	60.7
		Ref Butt et al. (2019)	2019	1367	25
		Ref Rahmatiyar et al. (2020)	2020	1295	159.6
		Ref Zhu and Wu (2021)	2021	825	21.54
		Ref Al Mahmud et al. (2021)	2021	2713	35.1
		This work	2021	2400	95

Funding This study was funded by the National Center of Scientific and Technical Research (CNRST), Morocco.

Availability of data and materials

The data that support the findings of this study are available from the corresponding author upon reasonable request.

Code availability

The codes that support the findings of this study are available from the corresponding author upon reasonable request.

Author Contributions M. Amrani, S. Khattou

Declarations

Conflict of interest The authors declare no competing interests.

Consent to participate All authors accepted.

Consent for publication All authors accepted.


L Achrait-Furlan

T Lasri

A Mamouni

MICROWAVE TECHNIQUE FOR DETECTION, LOCATION, AND SIZING OF BURIED OBJECTS

Keywords: microwa¨es, nonin¨asi¨e, reflection coefficient, heterogeneous medium, buried objects

The detection, location, and sizing of immersed objects in a host medium are tasks of interest in many fields. One of the most suitable tools for the nonin¨asi¨e measurements of buried objects is the use of microwa¨es. We propose in this paper a microwa¨e method based on the knowledge of the reflection coefficient.

I. INTRODUCTION wx

The detection andror identification of buried objects 1᎐3i s a very important problem for environmental, military, and humanitarian purposes. Several fields, such as geophysics, civil engineering, medicine, or the food industry, are concerned.

The techniques principally employed are based on the use Ž. of ground-penetrating radars GPRs which are now com-Ž monly used in many applications mines, cables, or pipe . detection . Nevertheless, for the detection of small objects at shallow depth, this method is not as efficient as expected. For these kinds of studies, we propose an alternative method based on the measurement of the reflection coefficient of the medium under test. In particular, modeling studies have been performed in order to predict the reflection coefficient of different structures. These theoretical results have been compared to experimental ones, obtained with an S-parameter Ž. measurement system SPMS-2450 developed in the laborawx tory 4 . Examples are used to illustrate the method to determine the location and dimensions of a buried object.

II. SIMULATION AND MEASUREMENT METHODS

In order to detect nonmetallic mines or to localize cables, pipes, or parasitic inclusions in a host medium, ground-Ž. penetrating radars GPRs are generally used.

We present here a complementary method based on the wx measurement of the reflection coefficient at 2.45 GHz 5᎐7. The test setup presented in Figure 1 consists of a PVC tank, 150 = 150 = 150 mm [START_REF] Cashman | Imaging of buried objects from multi-look, multifrequency radar data in the Fourier domain, including antenna effects[END_REF] , where the inclusion to be characterized is inserted. Three rectangular apertures have been made on three walls of the box, so that the microwave signal is Ž. applied to the test structure through a probe. The probes P i are rectangular waveguides filled with a low-loss dielectric material. The experiments have been performed using inclusions which are hollow plastic pieces that can be filled with various dielectric materials. The measurement of the reflec-<< j S 11 tion coefficient S s Se of the device under test is 11 11 made by using a system, operating at 2.45 GHz, developed in the laboratory, called the S-parameter measurement system Ž. SPMS-2450 . Two different methods are used to predict the reflection wx coefficient. The first one, called the modal method 8 and based on the continuity of the propagation modes between the waveguide and a dissipative medium, is well suited to study homogeneous or multilayer materials. In the case of more complicated structures, such as inclusions in a host medium, we make use of the three-dimensional electromagnetic simulation software, Ansoft's High Frequency Structure Ž. Simulator HFSS .

III. RESULTS

A. Estimation of the Inclusion Cross Section. In this work, the inclusions considered are cubic or cylindrical plastic-made hollow pieces filled with glycerol or water, whose permittivities at 2.45 GHz are, respectively, U s 5 y j0.8 and U s 77 g w Ž U y j12. The host medium is sand s 3.2 y j0.05 at s .

GHz .

The cubic pieces have either a 33 or 40 mm side, while the cylindrical pieces considered are such that: diameter s height s either 33 or 40 mm. Ox-axis Fig. 1 , and for each position, the reflection coefficient is recorded. The height of the cylinder is along the propagation axis Oz, and the circular cross section is parallel to the cross section of the waveguide. In these graphs, we have also reported the results obtained when the tank is only Ž. Ž filled with a homogeneous without inclusion medium sand . or glycerol .

We note that the presence of the inclusion is clearly observed. Moreover, the influence of the geometrical shapes and sizes is also evidenced. However, even if the differentiation between two geometrical shapes of the same size is difficult, it appears that both the magnitude and phase of S 11 depend on the geometrical shape of the piece under investigation.

Due to the size of the tested inclusions compared to the Ž 2 . aperture of the waveguide probe 58 = 28 mm , the magnitude of the reflection coefficient at x s 0 is very far from the Ž one obtained when only glycerol is present homogeneous . medium . In this case, it is very difficult to use the magnitude < Ž. < of the reflection coefficient Sxto distinguish the shape 11 and size of the inclusions.

The same experiments are made with the inclusion located at z s 5 mm from the interface waveguide᎐host dissi-Ž. pative medium Fig. 3 . The conclusions drawn from the responses recorded are very close to those stated for z s 0 mm. In particular, these results point out that the subsurface location and sizing are conceivable. Before reporting the results of the estimated cross section of the different tested inclusions, we present the method used to determine these sections.

One can note that the shape of the plots presented in Figures 2 and3 appears like a Gaussian form. Thus, by means of a currently used method, which consists of measuring the y3 dB width in order to characterize the width of the radiation pattern of an antenna, it is possible to obtain an estimation of the dimensions of the object under test.

Ž. It can also be seen that the width of the Sxcurves 11 depends on the dimensions of the inclusions: the width gets larger as the dimensions of the inclusion increase. Therefore, Ž the measurement of the width of the curves both in magni-. tude and phase can lead to an approximation of the inclu-Ž. sion cross section Table 1 .

These results show that it is possible to evaluate the width of the inclusion. However, it can be noted that the accuracy of this determination depends on the shape of the tested Ž. object and on the parameter used magnitude or phase . When small objects are to be detected, we obtain a better result when phase values of S are used. The mean value for 11 the error on the cross-section estimation is about 15% when the phase is used instead of about 25% when the magnitude is considered.

As a conclusion, we have shown that it is possible to locate Ž. the inclusion along the Ox-axis and estimate its cross section in a host medium through the determination of the reflection coefficient.

B. Influence of the Inclusion Location in Depth.

In the previous section, it was demonstrated that the location of an inclusion along the scanning axis can be achieved. In this paragraph, we propose to describe the influence of the inclusion depth on the location possibilities.

When the cross section of the inclusion is very large compared to the waveguide aperture, one can approximate the structure to be tested to a three-layer one. The modal method can then be applied, and allows us to have a quick answer about the dependence of the reflection coefficient versus the layered structure.

Figure 4 gives the results obtained when a three-layer medium and a structure which is made of an inclusion buried in a homogeneous environment are considered. The materi-Ž. als selected for these tests are sand host medium and water Ž. inclusion . Ž.Ž ࡗ: modal method, B: HFSS three layers , q: HFSS host medium . q inclusion Three kinds of simulation data have been used to plot these graphs. The software HFSS provides the responses for the two cases proposed, while the modal method deals only with the three-layer medium. The fourth curve given in this figure corresponds to the reflection coefficient measurement, by means of the SPMS-2450, of the host medium᎐inclusion structure. Measurements of a three-layer medium have not been conducted in this study. Despite the aperture of the probe being larger than the cross section of the inclusion under test, we note that the heterogeneous structure can be approximated to a three-layer model. The HFSS simulation results for the host medium᎐inclusion structure and for the three-layer material are roughly the same. The shapes of the variations of the measured Ž. Ž SPMS-2450 and simulated modal method and HFSS . method reflection coefficient versus z agree for depths below 15 mm. Another reason that explains the discrepancies between the measured and computed data is related to the dimension of the tank. These have been initially chosen so that the edge effects can be neglected at 2.45 GHz when the homogeneous material that fills the box is water or glycerol. In the case of sand, the dimensions required to fulfill this condition are larger than the actual ones. We have not rebuilt a box especially for this material, knowing that, in most cases, we have to deal with moist sand for which smaller dimensions are needed. The aim of this experiment is just to have an idea of the behavior of the reflection coefficient when testing this kind of material. Many kinds of inclusions with various shapes and sizes have been tested by this method wx 6, 7 .

We present in Figure 5 the simulation results obtained when two different probes are considered. On this graph, two Ž. zones have been defined: a visibility zone VZ corresponding to the z-value from which the reflection coefficient tends Ž. toward the host medium's one, and an ambiguity zone AZ related to a zone where there is some doubt about the Ž. presence or absence of an inclusion.

One can observe that, when the magnitude of S is 11 considered, the inclusion is always seen, using either P or P to have a VZ around 16 mm. For P , a VZ up to 40 mm is 1 obtained, and the presence of an AZ is seen around 20 mm. An important issue that is also emphasized here is the ambiguity that can arise when a location, in depth, is aimed. A solution to overcome this problem is to use several probes. To illustrate the method to remove the uncertainty of the inclusion position, we refer to Figure 5. Due to the nonmonotonous shape of the magnitude of the reflection coefficient, for a magnitude value, we may have two possible Ž. depths Z and Z . To choose between these two candi-0 1

dates, we make use of the results given by another probe Ž. here, P .

2

The knowledge of the phase of the reflection coefficient sometimes may be sufficient to solve this problem without Ž having recourse to a measurement with another probe Fig.

. The same kind of test has been made in the case of an inclusion filled with glycerol. The measurement results given in Figure 6 lead to similar conclusions.

IV. CONCLUSION

This paper shows the ability to detect an object in a host medium and to obtain an estimation of its cross section, using a noninvasive microwave technique which consists of the reflection coefficient measurement.

The dielectric inclusions tested are filled with low-or high-loss dielectric materials, which allow us to obtain low-or Ž. high-permittivity contrasts between the host medium sand and the dielectric inclusion. In both cases, we obtain acceptable visibility zones, and the cross section of the object is estimated with an accuracy of about 20% in most cases.

The accuracy of the measurements can be increased by using probes with different sizes and dielectric characteristics. This method is a complementary one to the GPR method.

Figure 1 Figure 2 1 .

 121 Figure 2 Reflection coefficient simulated by means of HFSS versus Ž. the lateral position x of the inclusion filled with glycerol, immersed Ž. in sand, and located at the surface z s 0 . Probe used: P , F s 2.45 1 Ž. Ž. Ž . GHz. a Magnitude of S . b Phase of S . B: cube side 40 mm , 11 11 Ž. Ž . *: cube side 33 mm , q: cylinder diameter s height s 40 mm , ࡗ: Ž. cylinder diameter s height s 33 mm

Figure 3

 3 Figure 3 Reflection coefficient simulated by means of HFSS versus Ž. the lateral position x of the inclusion filled with glycerol, immersed Ž. in sand, and located at z s 5 mm. Probe used: P , F s 2.45 GHz. a 1 Ž. Ž . Magnitude of S . b Phase of S . B: cube side: 40 mm , *: cube 11 11Ž. Ž . side: 33 mm , q: cylinder diameter s height s 40 mm , ࡗ: cylin-Ž. der diameter s height s 33 mm

Figure 4

 4 Reflection coefficient versus the longitudinal position z Ž. of the cubic inclusion side: 40 mm filled with water, immersed in Ž. sand, and located at x s 0 mm. Probe used: P , F s 2.45 GHz. Magnitude of S . b Phase of S . *: measurements SPMS-2450 , 11 11

Figure 5

 5 Figure 5 Reflection coefficient simulated by means of HFSS versus Ž. Ž . the longitudinal position z of the cubic inclusion side: 40 mm filled with water, immersed in sand, and located at x s 0 mm, Ž. Ž. F s 2.45 GHz. a Magnitude of S . b Phase of S .* : P , ࡗ: P 11 11 1 2

5 .Figure 6

 56 Figure 6 Reflection coefficient simulated by means of HFSS versus Ž. Ž . the longitudinal position z of the cubic inclusion side: 40 mm filled with glycerol, immersed in sand, and located at x s 0 mm, Ž. Ž. F s 2.45 GHz. a Magnitude of S . b Phase of S .* : P , ࡗ: P 11 11 1 2

TABLE 1 Determination of the Inclusion Cross Section

 1

			Ž. Cube mm		Ž. Cylinder mm	Ž. Cube mm		Ž. Cylinder mm
		Inclusion	Ž. Side s 40 mm	Ž	. d s h s 40 mm	Ž. Side s 33 mm	Ž	. d s h s 33 mm
	Z s 0 mm	Dimension	62		54	43		35
		measured
		<< using S 11
		Dimension	36		32.5	38		27
		measured
		using S 11
	Z s 5 mm	Dimension	60		48	34		34
		measured
		<< using S 11
		Dimension	28		34	34		34
		measured
		using S 11