1. Introduction

Surface modes (SMs) have been widely studied for boundaries between isotropic media [1–3], and they have been exhaustively analyzed for isotropic multilayered media [4–6]. It is well known that excitation of SMs by an incident wave requires to corrugate the surface or to use Attenuated Total Reflection (ATR) devices [3]. If we consider a metallic film surrounded by two dielectric media, SMs can be excited only at the lower interface. To excite more than one mode at isotropic structures, either a periodic corrugation or more than two boundaries are needed.

The propagation of surface modes has also been studied in anisotropic structures at a single interface [7–9], and excitation of SMs at a single corrugated uniaxial-metal interface has been analyzed in Refs. [10,11]. Recently, we have reported the case of a single flat interface between sodium nitrate (a negative uniaxial crystal) and a metal that, under adequate conditions and contrary to what happens for isotropic interfaces, supports a class of SMs that can be coupled directly to incident radiation [12,13].

Some works on multilayer structures that involve anisotropic media can be found in Ref. [14] (with a particular orientation of the principal dielectric tensor axes of the anisotropic layer) and in Ref. [15], where the first observation of enhanced optical second-harmonic generation due to excitation of long range surface plasmons at a multilayer structure is reported. The uniaxial-metal-dielectric configuration (which we study here) has also been considered in the literature, but in the case that the optic axis of the crystal belongs to the interface, thus excluding double resonances [16], or in order to study guided waves in anisotropic media Ref. [17,18]. Excitation of SMs in these structures has been studied in Ref. [19], but the analysis therein does not include double excitations.

Although many works deal with excitation of SMs at structures with more than one interface, we are not aware of studies addressing the problems of double excitations and of minimizing the distance between the propagation constants of the two SMs, for a structure of only two flat interfaces. Some studies on these phenomena can be found for gratings: Stewart and Gallaway, in an experimental study [20], observed the phenomenon of “reluctance” of two resonances to coincide; the same phenomenon was theoretically studied by Utagawa [21].

In this paper we present a theoretical study about SMs in structures formed by a metallic film, when the incidence medium is a uniaxial crystal and the substrate is an isotropic dielectric. We show that, with the adequate parameters, this configuration allows the excitation of two surface modes, both accessible to direct coupling; that is, without corrugating the surfaces or adding another layer, and that these two modes can be excited by an incident plane wave simultaneously (that is, at similar angles of incidence). We describe how to obtain the complex propagation of the surface modes, and how to optimize the excitation to obtain the zeroes of reflected energy. We then give several examples where the refractive index of the dielectric is chosen in a way that the two resonances can be excited at similar angles of incidence, by the incidence of a single plane wave from the uniaxial side. In this case we find almost simultaneous excitation, and the phenomenon of “reluctance” of the two complex propagation constants to coincide.

The organization of the paper is as follows. The theory is described in Section 2 where we briefly describe the steps to find the fields in the three media, the expressions for calculating the normal modes, the zeroes of reflected energy, and their associated fields. In Section 3 we describe in detail our numerical results, obtained by choosing the film parameters in a way that two resonances are located close to each other. Under these conditions, we will show the phenomenon of almost simultaneous “double resonance” takes place, together with the effect of “reluctance” between the propagation constants of the two modes. Concluding remarks are given in Section 4.

2. Theory

We consider a metallic film of width l, bounded by two semiinfinite media: an isotropic medium and a uniaxial medium. The coordinate system x - y - z is chosen in the following way: the incident waves are contained in the x - y plane, the x axis is parallel to the film, and the y axis points to the uniaxial medium. The uniaxial-metal and metal-dielectric surfaces are characterized, respectively, by y = 0 and y = -l. We deal with monochromatic fields, with harmonic time dependence exp(-iωt).

The isotropic region (y < -l) is characterized by a permittivity ∊ ₀ ∊_i and a permeability μ ₀ μ_i The metallic film (0 < y < l) is characterized by complex permittivity ∊ ₀ ∊_l and by permeability μ ₀ μ_l . In the above equations, ∊ ₀, μ ₀ are the permittivity and the permeability of vacuum.

The anisotropic side (y < 0) is a non lossy uniaxial medium characterized by a permeability μ ₀ μ_a , and a dielectric tensor $\widetilde{∊}$ = ∊ _⊥ Ĩ + (∊ _∥ - ∊ _⊥) ĉ ĉ. In the later expression, ∊ _⊥ and ∊ _∥ are the eigenvalues of $\widetilde{∊}$ Ĩ is the unit dyadic, and ĉ = (c_x,c_y,c_z ) = (sin θ_c cosφ_c , cosθ_c , sinθ_c sinφ_c ) is a unit eigenvector that corresponds to the non-repeated eigenvalue ∊_∥ ; it is called the optic axis.

2.1 Electric and magnetic fields

We consider incidence from the uniaxial side with an ordinary (subscript o) or an extraordinary (subscript e) uniform and monochromatic plane wave, with amplitude $C_o^{+}$ or $C_e^{+}$, respectively. The x component of its wave vector will be called α ₀.

2.1 .1 Regions for the propagation constant α₀

A uniform plane wave incident or reflected onto the surface propagates if the modulus of its propagation constant α ₀ is lower than a critical value. This value is different for ordinary or extraordinary waves [22] (see the constitutive relations: eqs. 3 and 4). Therefore, it is found that the real values of α ₀ may belong to one of three “regions” [13]:

In the above equations we have defined

and k ₀ = ω (μ ₀ ∊ ₀)^1/2 = 2π/λ ₀ the wave number in free space.

In this work we also consider complex values for α ₀ -which will be necessary to describe the modes supported by the structure- with Re(α ₀) in Region II, since in this case it is possible to excite surface modes at the uniaxial-metal interface [12,13]. Therefore, by choosing the refractive index of the dielectric in an adequate form, it will be possible to excite surface modes in the two interfaces, as it is described in Section 3. These excitations will be produced by the incidence of a single plane wave: if α ₀ belongs to Region II, the incident uniform plane wave must be ordinary if the uniaxial medium is negative, or extraordinary if it is positive.

2.1 .2 Incident, reflected and transmitted fields

The incident (superscript +) and reflected (superscript -) electric fields in the region y < 0 can be expressed

where $C_{o\mathit{,}e}^{\pm }$ are the amplitudes of the incident or reflected fields and ${\overrightarrow{k}}_o^{\pm },{\overrightarrow{k}}_e^{\pm }$ are the incident and reflected wavevectors (ordinary and extraordinary). It must be noted that only one wave is incident onto the first interface, since α ₀ belongs to region II, so if the uniaxial medium is negative it is chosen $C_e^{+}$ = 0; otherwise $C_o^{+}$ = 0.

In equation (2), ${\widehat{e}}_{\mathit{\text{oe}}}^{\pm }$ represent the polarization of the incident (or reflected) ordinary and extraordinary fields [22], which directions are ${\overrightarrow{e}}_o^{\pm }={\overrightarrow{k}}_o^{\pm }\times \hat{c}\mathrm{and}{e}_e^{\pm }=k_0^2{\mu }_a{∊}_{\perp }\hat{c}-({\overrightarrow{k}}_e^{\pm }·\hat{c}){\overrightarrow{k}}_e^{\pm }$ respectively, taking ${\hat{e}}_j^{\pm }=\frac{{\overrightarrow{e}}_j^{\pm }}{{({\overrightarrow{e}}_j^{\pm }·{\overrightarrow{e}}_j^{\pm }*)}^{\frac{1}{2}}}$.

The incident and reflected wave vectors have the same x component α ₀, and their y components can be found through the following dispersion relations, together with the criteria given in Ref. [13] to determine the square roots

The fields in the isotropic dielectric are given by

where ${\overrightarrow{k}}_i={\alpha }_0\hat{x}-{\gamma }_i\hat{y},$, is the transmitted wave vector, with ${\gamma }_i=\sqrt{k_0^2{\mu }_i{∊}_i-{\alpha }_0^2}$. We choose $\mathit{Re}\left({\gamma }_i\right)\ge 0\mathrm{if}\mid \frac{\mathit{Re}\left({\alpha }_0\right)}{k_0}\mid \le \sqrt{{\mu }_i{∊}_i}\mathrm{or}\mathit{Im}\left({\gamma }_i\right)>0\mathrm{if}\mid \frac{\mathit{Re}\left({\alpha }_0\right)}{k_0}\mid >\sqrt{{\mu }_i{∊}_i}.$ The complex amplitudes A_s and A_p are the TE and TM components of the transmitted waves, respectively.

The fields in the metallic film can be expressed as the sum of two waves: one of them propagates towards y < 0 and the other one propagates towards y < 0:

where the wave vectors (subscript l) are ${\overrightarrow{k}}_l={\alpha }_0\hat{x}-{\gamma }_l\hat{y},{\overrightarrow{k}}_l^{\prime }={\alpha }_0\hat{x}+{\gamma }_l\hat{y},\mathrm{with}{\gamma }_l=\sqrt{k_0^2{\mu }_l{∊}_l-{\alpha }_0^2}.$ As a convention, it was chosen Im(γ_l ) > 0.

The imposition of the boundary conditions (the continuity of the tangential components of the electric and magnetic fields), leads to an 8×8 linear system, from which we can obtain the following relation between the reflected and incident amplitudes:

The coefficients of the reflection matrix R̃ and the amplitudes of the fields in each medium can be easily found by solving the mentioned system of equations.

2.2 Propagation and excitation of surface modes

In this section we discuss the existence of surface modes for this structure, which are the solutions to the homogeneous problem. To do so, the propagation constant α ₀ should be considered as an unknown variable and so, it could take complex values. In addition, we will restrict ourselves to values of α ₀ which verify Re(α ₀) ∊ RegionII.

2.2.1 Normal modes

Surface modes are the solutions to the homogeneous problem, so the condition to find non trivial solutions to the problem in the absence of incident field leads to the condition

This condition may be expressed in the following way, by using the expression of the reflection coefficients found through boundary conditions

where η(α ₀), μ(α ₀) and ν(α ₀) depend on the constitutive parameters and on α ₀, but not on the film width l. Equation (9) can be numerically solved; the values of α ₀ that satisfy it are, in general, complex with a small imaginary part, and they are called the “poles” of the reflection matrix. The polarization of the surface modes can then be found by replacing these values in the boundary conditions.

For high values of the film width, equation (9) reduces to ν(α ₀) = 0. It can be easily seen that the solutions of this equation are the poles of two separate interfaces: the crystal-metal one and the metal-dielectric one. If the film width decreases, it should be expected to find solutions “associated” to each interface, although these solutions are no more independent.

It is important to note that the possibility of finding poles with Re(α ₀) in Region II depends critically on the parameters of the structure. In particular, only for certain orientations of the optic axis it is possible to find a pole in Region II “associated” to the crystal-metal interface; this fact has been studied in previous works for a single crystal-metal planar surface [12,13]. Besides, a pole “associated” to the metal-dielectric interface will be located in Region II only for a certain range of refractive indexes of the dielectric.

In this work we will study the possibility of exciting these two surface modes in Region II, as it is described in our numerical results.

2.2 .2 Excitation of surface modes

In order to excite these surface modes, the incident fields must be chosen in an adequate form. Let us consider that a wave with propagation constant α ₀ located in Region II is incident, with ordinary polarization (for a negative uniaxial medium) or extraordinary one (for a positive uniaxial medium). The propagation constant α ₀ must be chosen so that α ₀ ≈ Re(${\alpha }_0^p$) in order to have the possibility of exciting the surface mode. Besides, the incident wave must provide the polarization required by the surface mode.

It is well known that excitation of surface modes is accompanied by a decrease of the power reflected by the metal. In the present situation, the reflected power is associated only to the ordinary wave (negative uniaxial media) or to the extraordinary wave (positive uniaxial media), because the other wave is evanescent in Region II. Therefore, the condition that leads to zero reflected energy can be written

These conditions are the same that were obtained for the case of a single crystal-metal interface [13].

It can be shown that equations (10) and (11) lead to equations similar to (9). We call ${\alpha }_0^z$ the propagation constants that verify (10) or (11), which are, in general, complex. As it happens for the poles, it can be seen that for high values of the film width, equations (10) and (11) give two solutions, each of them associated to one of the two single interfaces. If the film width decreases, two solutions can also be found. Besides, our results show that each value of ${\alpha }_0^z$ (complex, in general) is located near a pole ${\alpha }_0^p$ in the complex plane α ₀. The imaginary part of ${\alpha }_0^z$ varies with the constitutive parameters, and total absorption of power can occur when ${\alpha }_0^z$ ∊ R (if the refractive index of the dielectric is lower than ${\alpha }_0^z$/k ₀). It is important to note that the existence of these modes and the possibility of exciting them depend on the constitutive parameters. In the following section we show that two resonances can be excited simultaneously.

3. Results

3.1 Double-resonance effect

Let us consider a crystal of sodium nitrate (∊ _⊥ = 2.51, ∊ _∥ = 1.78, μ_a = 1) and a metal film with constitutive parameters ∊_l = -21.6 + 1.4 i and μ_l = 1. The width of the film to wavelength ratio was varied between l/λ ₀ = 0.03 and l/λ ₀ = 0.3.

In this section we will present a phenomenon that may occur if the refractive index of the third media (dielectric) is chosen in an adequate form. Under certain conditions, the two resonances associated to each interface can be excited, and we will show here that, by tuning the film parameters, the angles of incidence for which these different excitations occur can be near enough as to have almost simultaneous excitation.

This phenomenon can only be found if the two modes are located in Region II (because the modes at the first interface are always located in that region). Therefore the refractive index of the dielectric should verify $n_i=\sqrt{{\mu }_i{∊}_i}<\sqrt{{\mu }_a{∊}_{\perp }}$. Besides, n_i should be chosen to give similar values of α ₀ where the resonances are excited. In this example we have selected several values of ∊_i , so that the real part of the pole corresponding to the single metal-dielectric interface is located near the real part of the pole of a single crystal-metal interface.

The orientation of the optic axis is given by φ_c = 20° and θ_c = 25°, and 1.94 < ∊_i < 1.96.

 

Fig. 1. Poles of the reflection matrix and zeroes of R_oo in the complex plane α ₀. The constitutive parameters of the structure are: ∊ _⊥ = 2.51, ∊ _∥ = 1.78, μ_a = 1, φ = 20°, θ_c = 25°, ∊_l = -21.6+1.4i, μ_l = 1, μ_i = 1 and 0.03 < l/λ ₀ < 0.3. Several values of ∊_i have been chosen: a) 1.94, b) 1.952, c) 1.953, d) 1.954 and e) 1.96.

Download Full Size | PDF

In this case, the pole associated to a single crystal-metal interface is ${\alpha }_{02}^p$/k ₀ = 1.463 + 0.0129i, while the pole corresponding to a dielectric-metal interface varies with ∊_i ; for example, it takes the value ${\alpha }_{01}^p$ ≈ 1.464 + 0.0047 i for ∊_i = 1.953. These values are the limit values for the poles of the film when the width tends to infinity.

The poles associated to a film characterized by the mentioned parameters in the complex plane α ₀/k ₀, can be appreciated in Figure 1a, where we have chosen ∊_i = 1.94. The film width is varied between l/λ ₀ = 0.03 and l/λ ₀ = 0.3. The figure shows two branches of poles; there are two poles for each value of l/λ ₀ and, for high values of l/λ ₀, the poles tend to the limit values given above. The poles in the left branch tend to the metal-dielectric pole, and the ones in the right branch tend to the crystal-metal value. On the other hand, a very interesting phenomenon can be observed in the same figures. The two branches of poles seem to “repel” each other; and therefore the branches do not cross.

Figure 1a also shows the complex zeroes of R_oo associated to each value of l/λ ₀. Again, two branches of zeroes appear, with real parts similar to the real part of the poles. When the film width increases, the zeroes of the left branch tend to the metal-dielectric interface zero: ${\alpha }_{01}^z$ = ${\alpha }_{01}^p$, while the zeroes of the right one tend to ${\alpha }_{02}^z$ ≈ 1.463 - 0.00104i (which coincides with the zero associated to a single crystal-metal interface). Each branch crosses the real axis α ₀/k ₀ once. Therefore, there are two values of α ₀/k ₀ for which there is zero reflected energy, for different values of l/λ ₀.

Let us analyze what happens if the refractive index of the isotropic dielectric medium is slightly varied. Figures 1b to 1e show the poles and zeroes when ∊_i = 1.952, 1.953, 1.954 and 1.96, respectively. These figures, together with Figure 1a, evidence an interesting evolution of poles and zeroes trajectories. First the two branches of poles get closer while ∊_i increases (Figure 1b and 1c), and for higher values of ∊_i the left side of both branches “join” together, and the right side of them do so (Figure 1d), so two “new branches” are formed. For this later value of ∊_i , the left branch tends to the crystal-metal pole, and the other one tends to the metal-dielectric pole. If ∊_i increases, these branches separate from each other (Figure 1e). Analogously, the zeroes present a similar behavior when ∊_i is varied: Figure 1a shows two branches, which “join” and form two new branches (Figures 1b to 1e).

The repulsion between the two modes and the phenomenon described above is a more general property of resonances. We have found the two complex resonant frequencies associated to a system of two serial RLC circuits coupled by a mutual inductance M, and we have obtained that, varying the parameters of the circuits in an adequate form, the resonant frequencies move in the complex plane “repelling” each other in the same way as it is shown in the present case.

3.2 Total absorption of power

Figures 1a to 1e show the existence of particular values for α ₀/k ₀ that could give zero reflected power. Given a set of constitutive parameters, these real values exist only for critical values of the film width. Therefore, if the film width takes one of these critical values, and an ordinary plane wave with propagation constant α ₀ is incident onto the structure, all the incident power will be absorbed by the film (the transmitted wave is evanescent).

This phenomenon can be appreciated in Figure 2, where the reflected power is plotted as a function of the propagation constant α ₀, for ∊_i = 1.94 and several values of the film width. Each curve presents two dips, at different angles of incidence. Note that the right dip on each of the curves is associated to the modes in the right branch of Figure 1a. From this figure we conclude that these dips are associated to the excitation of two surface modes: the value of α ₀ where each dip occurs is located near the real part of the corresponding pole and zero. The total absorption of the incident power is obtained for l/λ ₀ = 0.09, at α ₀ ≈ 1.445. This curve presents another important dip at α ₀ ≈ 1.482, where less than 20% of the power is reflected. By tuning the film width, the two dips get closer. For example, if l/λ ₀ = 0.13, the two dips occur at angles of incidence that differ in one degree. Less than 5% of the energy is reflected for this film width at each dip, and less than 10% is reflected in all the range between the two dips; this situation corresponds to an almost simultaneous excitation. When l/λ ₀ takes higher values, the left dip tends to disappear, and the right one reaches a total absorption for l/λ ₀ = 0.15, at α ₀ ≈ 1.464. Finally, for higher values of the film width, one dip is found.

 

Fig. 2. Reflected power as a function of α ₀/k ₀ in the region of excitation of surface modes, for several values of the ratio l/λ ₀, for the incidence of an ordinary wave. The other parameters are the same as in the previous figures.

Download Full Size | PDF

3.3 Behavior of the fields

The excitation of surface modes is associated to the enhancement of the field at the interface [1,3]. In what follows we define the field enhancement factor as

where |E⃗(y)| is the modulus of the total electric field evaluated at y, inside either of the three media.

Figure 3 shows that the field is enhanced at the crystal-metal surface. In this figure we have plotted the field enhancement factor at y = 0, on the crystal side, as a function of α ₀/k ₀ and of the film width. Two peaks are observed, with their maxima located around the same values of α ₀/k ₀ for which the reflected power has minima (Figure 2). The peaks get closer while the film width increases, and for higher values of l/λ ₀ the two peaks merge; in this situation the field enhancement reaches a value of ≈ 45.

In a similar way, we may analyze the field enhancement at y = l, on the isotropic dielectric interface. Figure 4 shows the field enhancement factor evaluated at that interface, as a function of α ₀/k ₀ and l/λ ₀. We observe that the field enhancement factor has a peak located around the value of α ₀ associated with the left branch of the modes. The other peak exists only for very low values of l/λ ₀. This field enhancement decreases when l/λ ₀ increases.

It is well known that to excite a surface plasmon in (isotropic) ATR devices or gratings, the fields must be TM-polarized. In the present situation, we have found the fields at the two interfaces to be TM-polarized also (not plotted here). This fact is clearly due to the existence of the mode at the metal-dielectric interface.

 

Fig. 3. Field enhancement factor on the crystal (y = 0) as a function of α ₀/k ₀ in Region II and of l/λ ₀, for the incidence of an ordinary wave. The other parameters are the same as in the previous figures.

Download Full Size | PDF

 

Fig. 4. Field enhancement factor evaluated at y = -l, as a function of α ₀/k ₀ and of l/λ ₀, when incidence of an ordinary wave. The other parameters are the same as in the previous figures.

Download Full Size | PDF

However, we have found that the incident field is mainly polarized in the TE mode in all the region of excitation of surface modes, so a high polarization conversion between the incident and reflected fields near the crystal-metal interface takes place. We have also observed that the excitation of surface modes is associated with the enhancement of the evanescent extraordinary fields at the first interface; a result that has also been obtained for the case of a single crystal-metal interface [13].

4. Summary

The “double” propagation and excitation of surface modes has been studied in a structure formed by a metallic film that separates a uniaxial medium from an isotropic dielectric, when incidence is from the uniaxial side. For particular values of the refractive index of the dielectric and particular orientations of the optic axis, this structure supports two normal modes, with their propagation constants located in the radiative region. By tuning the constitutive parameters, we have found that the propagation constants of the normal modes can be located very near each other although they never coincide; in this situation we have shown the phenomenon of “reluctance” of the two normal modes to coincide. These modes can be excited together directly from the uniaxial side -without adding another interface-; these excitations are accompanied by an important absorption of power by the film and by the field enhancement at the surfaces. We have reported, through several examples, that the angles of incidence where the resonances take place can be near enough so as to have almost simultaneous excitation. This phenomenon could be of interest in many applications; for example, when high field enhancements are required, in experiences involving finite beams.