1. Introduction and background

Tunneling of electromagnetic radiation through metal films has captured the attention of researchers for a variety of reasons. Since the 1970s [1], for example, dielectric-metal-dielectric (DMD) coatings have been studied in the context of solar control and thermal emittance windows [2,3], and as transparent electrodes for displays [4]. In the late 1990s, Bloemer et al. [5] showed that periodic MD multilayers can exhibit wide bands of high transparency, and suggested their application as transparent conductors and radiation shields. Others have explored the analogy between optical tunneling and the quantum mechanical tunneling of electrons over potential barriers [6,7]. More recently, metal-dielectric stacks have been studied in the context of ‘superlenses’ [8,9], since they support plasmon-mediated tunneling of TM-polarized evanescent waves. Energy transport through such structures remains an active topic for study [10–12].

Symmetric DMD tri-layers have played a central role in the aforementioned research [1–4,6,7], including antireflection-coated MD stacks [5,11], which are essentially periodic structures with a DMD unit cell. Much of the knowledge about these structures derives from numerical studies which have shown, for example, that the use of high-index dielectric layers tends to enhance transparency [2,5,11]. Theoretical treatments, on the other hand, have typically relied on simplifications such as the use of a lossless metal assumption [7,13]. In our view, it has not been widely recognized that these various bodies of work can be unified within the framework of potential transmittance (PT) theory [14,15]. For example, PT theory explains [16] the observation that transmittance can be increased by sub-dividing a given thickness of metal into some number of thinner, appropriately spaced metal layers [5,8].

The goal of our work is to assess the limits of tunneling-based transparency, given a periodic DMD multilayer containing real, absorbing metal layers. In a previous, transfer-matrix-based numerical study [17], we used PT theory as a general framework for assessing maximum transmittance. Here, we provide an extended theoretical treatment, and derive a general equation that encapsulates the conditions for admittance-matched tunneling. This approach offers new insight into previously disjointed research on tunneling through MD stacks. Moreover, it directly addresses the important problem of maximizing transmittance (T) and minimizing reflectance (R) in such structures.

2. Admittance matching for minimum effective absorbance of a metal film

As first explained by Berning and Turner [14], the transparency of a thin absorbing film is highly dependent on its boundary conditions. In fact, there is a maximum potential transmittance (PT_MAX) that, for a sufficiently thin film, is much greater than the maximum transmittance suggested by the bulk optical absorption coefficient. To achieve PT_MAX at a given wavelength (which Berning and Turner termed as ‘inducing transmission’ [14]), one must design the surrounding media so that the exit optical admittance (viewed from the perspective of the absorbing film) attains a particular value determined by the optical constants and thickness of the film [15]. T = PT_MAX occurs when R = 0 for both left and right incidence [17], which illustrates that inducing transmission is essentially equivalent to admittance matching an absorbing film to the ambient media.

Consider a single absorbing film (with index N_m = n_m-iκ_m) embedded within an arbitrary assembly of otherwise lossless layers, as shown in Fig. 1(a) . The potential transmittance (PT = T/(1-R)) of the film depends on the properties (i.e. thickness and index) of the film itself and on the optical admittance presented by the exit assembly (Y_out = H_out/E_out), which determines the ratio of the magnetic to electric field at the output interface of the absorbing layer. For a given incident angle and state of polarization, PT_MAX is dependent on the properties of the absorbing film only, and can be calculated using the expressions provided previously [17].

 

Fig. 1 (a) Schematic showing a multilayer containing one absorbing layer. The optical admittance viewed from the perspective of the absorbing layer, looking towards the incidence and exit directions, is labeled as Y_in and Y_out, respectively. (b) The optimal (i.e. minimum) optical absorption coefficient, as defined in the text, is plotted versus Ag film thickness at a wavelength of 550 nm and for normal incidence. The bulk absorption coefficient for Ag at the same wavelength is represented by the red dotted line.

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While not typically stated in terms of absorption, one implication of PT theory is that the absorbance of a thin film can be significantly lower than the value predicted by the bulk optical constants. This is especially true for very thin films, but the possibility of reduced absorption (with appropriate admittance matching) extends to surprisingly large film thickness. To illustrate, we compared the minimum effective absorption coefficient (α_min) for an Ag thin film to the bulk optical absorption coefficient (α_m = 4πκ_m/λ) for Ag. Optical constants of Ag were modeled using the Lorentz-Drude expressions provided by Rakic et al. [18]. Noting that PT is the ratio of power flux at the entrance and exit interface of the absorbing film, it follows that PT_MAX = exp(-α_mind_m), or rearranging:

As an example, Fig. 1(b) shows a plot of α_min versus Ag film thickness at a wavelength of 550 nm and for normal incidence. Remarkably, for a 10-nm-thick Ag film the minimum effective absorption coefficient is 2 orders of magnitude lower than the bulk absorption coefficient. Note that a multilayer containing an arbitrary number of 10-nm-thick Ag films can have absorbance embodied by this same α_min, provided that the Ag films are separated by appropriate dielectric layers to produce an optimal admittance match. Band-limited admittance matching is the reason for the surprisingly high transparency of MD stacks containing many skin depths of metal [5].

Implicit above, and to PT theory in general, is the assumption that the bulk refractive index N_m = n_m-iκ_m remains valid for describing the optical properties of the thin film. As is well known, very thin metal films can exhibit optical properties that deviate from bulk values, such as a higher effective extinction coefficient arising from electron scattering at grain boundaries. Furthermore, quantum confinement effects cannot be neglected for length scales less than ~10 nm [19]. Nevertheless, carefully deposited Ag films have been shown [19,20] to exhibit bulk properties for thickness as low as ~10-12 nm. Based on this, we restrict our theoretical analysis to films > 10 nm thick, and assume that bulk optical constants can be applied in this range. In practice, deviation from bulk properties is possible depending on the film deposition technique, and should be considered.

As mentioned, to achieve PT = PT_MAX requires that the exit admittance Y_out is set to a specific, optimal value Y_op = X_op + iZ_op. In the following, all admittances are expressed in free-space units (i.e. normalized to the admittance of free space). Closed-form expressions for X_op and Z_op at normal-incidence are provided by Macleod [15]. Those expressions can be generalized for non-normal incidence as follows:

Figure 2(a) shows a representative plot of X_op and Z_op versus Ag film thickness, at a wavelength of 550 nm and for normal incidence. For reasons that will be explained in the following section, the ratio Z_op/X_op is also plotted. As expected [15], Y_op tends toward N_m* = n_m + iκ_m in the limit of a thick metal film (for example, the Rakic model predicts N_m = 0.1342-i3.1688 for Ag at 550 nm wavelength, as indicated by the dashed lines in the figure). Figure 2(b) plots the same quantities versus the normalized transverse wave vector, for a 30 nm thick Ag film at the same wavelength and for both TE and TM polarization. For a fixed metal film thickness, the optimal admittance (and the ratio Z_op/X_op) shows relatively little variation versus the transverse wave vector (i.e. angle).

 

Fig. 2 (a) Components of the exit admittance (X_op and Z_op) that produce PT = PT_MAX for normal incidence are plotted versus Ag film thickness, at a wavelength of 550 nm. The green curve shows the ratio Z_op/X_op. The dashed curves show the real (blue) and imaginary (red) parts of the Ag refractive index at the same wavelength. (b) The same quantities are plotted versus normalized transverse wave vector, for a 30-nm thick Ag film at 550-nm wavelength. The dashed and solid curves correspond to TM and TE polarization, respectively.

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3. Admittance matching conditions for a periodic DMD multilayer

Consider next a symmetric DMD unit cell embedded between identical incident and exit media, as shown schematically in Fig. 3(a) . As explained previously [17], conditions (i.e. particular combinations of λ, n₁, d₁, N_m, d_m, n₂, and θ₂) that maximize transmittance (i.e. that produce T = PT_MAX) for the unit cell also correspond to conditions for maximum transmittance through the periodic multilayer based on that unit cell.

 

Fig. 3 (a) Geometry for plane wave incidence on a symmetric DMD unit cell embedded between identical incidence and exit media. The admittance viewed from the perspective of the absorbing metal film (Y_out) is that for a thin film (n₁) on a substrate (n₂). (b) Schematic illustration of a periodic MD stack composed of DMD unit cells. Conditions (i.e. n₁, d₁, N_m, d_m, n₂, θ₂) that admittance match the unit cell also result in an admittance matched multilayer.

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For the single unit cell, the exit admittance from the perspective of the metal layer is that presented by a single thin film (with real index n₁) on an infinitely thick substrate (with real index n₂) [15]:

Note that X_op and Z_op are real numbers by definition, and that η₂ is a real number for all cases considered below (i.e. lossless ambient media with real incident angle). Thus, in all cases for which η₁ is purely real (i.e. for non-evanescent wave solutions in dielectric layers n₁), solutions to Eq. (6) are restricted to the following range:

3.1. Normal incidence in air

We first consider the simplest but practically important case of normal incidence from an ambient air medium (i.e. n₂ = 1). For normal incidence, the admittance of a medium in free-space units is simply equal to its refractive index. From the preceding discussion, solutions to Eq. (6) are possible provided (n₁ - 1/n₁) > 2Z_op/X_op. Given typical values of X_op and Z_op for a thin Ag film in the visible range (see Fig. 2), this implies that high values of dielectric index (n₁ > 4) are necessary to achieve a perfect admittance match. Moreover, with fixed n₂ and for a given metal layer at a given wavelength, only a specific value of n₁ results in a solution to Eq. (6). This observation was made previously [17], but in that case solely on the basis of a transfer-matrix numerical study. Solutions to Eqs. (6) and (7) were obtained as a function of Ag film thickness using a commercial software tool (Matlab), and representative data is shown in Figs. 4(a) and 4(b).

 

Fig. 4 (a) The dielectric index required for admittance-matched tunneling of normally incident waves through a periodic DMD stack (with Ag metal layers and air ambient) is plotted for three different wavelengths. The symbol indicates the data point used in subsequent examples. (b) The minimum dielectric layers thickness that results in admittance-matched tunneling when combined with the dielectric indices in part (a). (c) Predicted transmittance for a 1-period (blue solid line) and 20-period (blue dashed line) DMD multilayer with d_m = 25 nm, n₁ = 4.732 and d₁ = 17.5 nm (as indicated by the symbols in parts (a) and (b)). The red dashed lines indicate PT_MAX for each case. (d) Predicted reflectance for the structures in part (c). Note that the reflectance diverges at 550 nm, indicating a perfect admittance match for any number of periods.

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The results in Fig. 4(a) provide a theoretical underpinning for, and are in exact agreement with, the transfer-matrix results reported previously (see Fig. 5 in [17]), other than some numerical noise in the earlier results. Clearly, an impractically large dielectric index is required to facilitate a perfect admittance match for the Ag film at normal incidence. Nevertheless, it is illustrative to consider the implications of the solutions to Eqs. (6) and (7). As an example, Figs. 4(c) and 4(d) show the predicted T, PT_MAX, and R for a specific admittance-matched case (λ = 550 nm, and assuming a unit cell with d_m = 25 nm, n₁ = 4.732 and d₁ = 17.5 nm) indicated by the symbols in Figs. 4(a) and 4(b). T and R were calculated using a transfer matrix technique, and PT_MAX was calculated using previously described expressions [17]. As expected, a perfect admittance match (i.e. T = PT_MAX and R = 0) is verified at λ = 550 nm. This perfect admittance match holds, in principle, for a DMD multilayer comprising an arbitrary number of such unit cells. To illustrate this, data for both 1- and 20-period cases are shown.

 

Fig. 5 (a) Incident angle that results in admittance-matched tunneling of TE waves is plotted versus Ag film thickness, for n₁ = 2.3 and n₂ = 1, and at two different wavelengths. The dotted curve plots the tunneling angle for λ = 550 nm, n₁ = 2.3, and n₂ = 1.5. The symbol indicates the data point used in subsequent examples. (b) The minimum dielectric layers thickness that results in admittance-matched tunneling when combined with the indices and tunneling angles from part (a). (c) Predicted transmittance versus incident angle for a 1-period (blue solid line) and 10-period (blue dashed line) DMD multilayer with n₁ = 2.3, n₂ = 1, λ = 550 nm, d_m = 25 nm, and d₁ = 53.7 nm (as indicated by the symbols in parts (a) and (b)). The red dashed lines indicate PT_MAX for each case. (d) Predicted reflectance for the structures in part (c). Note that the reflectance diverges at 75.01 degrees, indicating a perfect admittance match.

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3.2. Admittance-matched tunneling of TE-polarized propagating waves

Given the difficulty of achieving a perfect match at normal incidence, it is interesting to consider tunneling at off-normal incidence. We first consider structures with n₁ > n₂, such as the air-ambient case in the previous section. From above, admittance matching requires a sufficiently high value of the ratio η₁/η₂. Interestingly, for a given n₂ and n₁ > n₂, this ratio increases with increasing incident angle (θ₂) for TE polarization, but decreases with increasing incident angle for TM polarization. This implies that when n₁ > n₂, solutions to Eq. (6) are possible for off-normal incidence of TE-polarized light only. Tunneling of this kind was described by Hooper et al. [7], but not in the context of the PT theory formalism employed here.

As an example, solutions to Eqs. (6) and (7) were obtained for fixed indices n₁ = 2.3 and n₂ = 1, with results at two different wavelengths plotted in Figs. 5(a) and 5(b). For a given wavelength and Ag film thickness, admittance-matched tunneling of TE-polarized light is predicted at a specific incident angle. To further illustrate, a specific data point from these solutions (λ = 550 nm, d_m = 25 nm, d₁ = 53.71 nm) was used to generate plots of T and R versus incident angle, shown in Figs. 5(c) and 5(d). As expected, admittance-matched tunneling occurs at the incident angle of 75.01 degrees indicated by the symbol in Fig. 5(a). As for the normal-incidence case, this admittance match extends to a multilayer with arbitrary number of unit cells. The 1- and 10-period cases are shown as examples.

To our knowledge, this efficient tunneling of off-normal TE-polarized waves through multi-period DMD stacks has not been studied previously. It might have implications for the realization of polarizing and angularly selective filters. However, it should be noted that the perfect admittance match is dependent on the symmetry of the structures considered, including the assumption of identical input and exit media. The air-ambient case considered would certainly present practical challenges. Solutions exist for higher values of n₂; example data is shown for n₂ = 1.5 in Fig. 5(a), and could represent a DMD stack symmetrically embedded between glass plates. Note, however, that the solutions lie at even higher incident angles in this case. For propagating waves in an external air medium to access these tunneling angles, coupling prisms [7] or other means of momentum matching would be required.

3.3. Admittance-matched tunneling of TM-polarized evanescent waves

Plasmon-mediated tunneling through symmetric, single-period DMD stacks was originally reported by Dragila et al. [21] and later by Hayashi et al. [6]. Analogous tunneling through multi-period DMD stacks was studied experimentally by Tomita et al. [22] and theoretically by Feng et al. [13], who proposed optical filters and modulators based on the concept. Relevant to the present discussion, the low transmittance reported in [22] can be attributed in part to non-optimal admittance matching in their structures.

Consider the structure in Fig. 3 for cases when n₂ > n₁, which might represent a periodic DMD multilayer coupled at its entrance and exit faces by high index prisms. This geometry allows plane waves in the ambient media to couple with evanescent waves in the dielectric layers, thereby enabling a straightforward analysis of the Poynting power flow associated with surface-plasmon-mediated tunneling [13,17]. Excitation of evanescent waves occurs for θ₂ > θ_C = sin⁻¹(n₁/n₂), where θ_C is the critical angle for total internal reflection. Assuming n₁ and n₂ are real, then η₁ and sinδ₁ are both purely imaginary (i.e. when θ₂ > θ_C), so that the admittance expressed by Eq. (5) has the same general form for evanescent waves as it had for the propagating wave cases discussed above. This means that, assuming lossless dielectrics, Eqs. (6) and (7) are valid for tunneling of both evanescent and propagating waves. When n₂ > n₁, however, η₁/η₂ increases for TM waves and decreases for TE waves (i.e. as the incident angle is increased). Thus, opposite to the situation described in the previous section, here solutions to Eq. (6) are expected for TM-polarized waves only.

The existence and the nature of the solutions to Eq. (6) were found to depend on the stack parameters (λ, d_m, n₁, n₂). However, typically two solutions were found in the evanescent range for a given Ag film thickness and wavelength, as shown by the representative data in Figs. 6(a) and 6(b). Note that the transverse wave vector is defined by k_t = (2π/λ)n₂sinθ₂, and that waves in the n₁ layers are evanescent when (k_t/k₀) > n₁. We assumed n₁ = 1.631 and a fictitious coupling medium with n₂ = 4, to enable comparison with similar structures studied by Feng et al. [13]. For some combinations of λ and d_m (such as for λ = 550 nm and d_m < ~42 nm in Fig. 6(a)), no solutions are found.

 

Fig. 6 (a) Normalized transverse wave vectors that result in admittance-matched tunneling of TM waves is plotted versus Ag film thickness, for n₁ = 1.631, n₂ = 4, and two different wavelengths (λ = 500 nm and λ = 550 nm). The symbols indicate data points used in subsequent examples. (b) The minimum dielectric layers thicknesses that result in admittance-matched tunneling when combined with the indices and tunneling angles from part (a). (c) Predicted transmittance for a 1-period DMD multilayer for λ = 500 nm, d_m = 50 nm, n₁ = 1.631, n₂ = 4, and various d₁ as indicated by the labels. Note that d₁ values corresponding to the data points indicated by the symbols in part (b) result in admittance-matched tunneling at the transverse wave vector values indicated by the corresponding symbols in part (a). The red dashed line indicates PT_MAX. (d) Predicted reflectance for the structures in part (c). Note that the reflectance diverges for the two cases that produce a perfect admittance match.

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The implications of the solutions to Eqs. (6) and (7) are analogous to those for the propagating wave cases above. As an example, T and R were plotted in Figs. 6(c) and 6(d) for a particular admittance-matched case (λ = 500 nm, d_m = 50 nm). Consistent with the solutions in Figs. 6(a) and 6(b), dielectric thicknesses d₁ = 12.77 nm and 46.07 nm result in a perfect admittance match (T = PT_MAX and R = 0) at normalized wave vector values of 2.18 and 3.61, respectively. Other values of d₁ result in a partial tunneling peak at a different angle, but without a perfect admittance match. The data for d₁ = 20 nm is shown as an example.

From an experimental perspective, coupling prisms with n₂ = 4 are not practical. Figure 7 shows results for a more practical combination of indices, n₁ = 1.38 and n₂ = 1.515, representing the MgF₂-based tunneling structures studied by Dragila et al. [21].

 

Fig. 7 As in Fig. 6, except with n₂ = 1.515, n₁ = 1.38, and for a 632.8 nm wavelength. (a) Incident angle in external medium (θ₂) that produces a perfect admittance match. (b) Dielectric layers thickness that produces admittance matched tunneling when combined with the incidence angles from part (a). (c) Transmittance versus incidence angle for a 1 period DMD structure, with λ = 632.8 nm, d_m = 25 nm, and d₁ set to the values indicated by the symbols in part (b), verifying a tunneling peak at the corresponding angles from part (a). (d) Reflectance versus incidence angle for the same conditions as in part (c).

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As mentioned in the introduction, much of the recent interest in plasmon-mediated tunneling through DMD structures is motivated by their potential to transmit the evanescent fields from an object [8]. In these superlens applications, there is a need to design MD structures offering both high transmittance and low reflectance, ideally for a wide range of transverse wave vectors and over the entire operational wavelength range of the lens. In particular, significant reflectance of energy back towards the object contributes to image distortion [10]. The model system studied above (a symmetric DMD stack, symmetrically embedded in lossless ambient media) is not expected to capture all pertinent details of the superlens. For example, transfer of energy associated with the evanescent fields of an object necessarily involves optical absorption by the detector (at the image plane) and finite reflection from the lens [12]. Nevertheless, the prism-coupled DMD stack can provide insight regarding the qualitative nature of tunneling through the multilayer superlens [22,23].

4. Summary and conclusions

Using potential transmittance theory, we derived a general equation describing the conditions for admittance-matched tunneling through periodic DMD multi-layers. We furthermore verified its applicability to tunneling problems involving both propagating and evanescent waves. For normal incidence, the equation predicts that a perfect match occurs only for specific and large values of the dielectric refractive index. For off-normal incidence and assuming fixed dielectric and ambient medium indices, the equation predicts that a perfect admittance match occurs at specific tunneling angles. For TE-polarized light, solutions are found in the propagating wave regime. For TM-polarized light solutions are found in the evanescent-wave regime. To our knowledge, these various classes of tunneling have not previously been unified within a single theoretical framework.

In all cases, matching is predicted to occur at a single wavelength and/or tunneling angle. It seems probable that using two or more dielectrics would enable structures that are matched at more than one wavelength or tunneling angle. The rich body of literature on multi-layer anti-reflection coatings and on induced transmission filters [14–16] might provide substantial guidance in this regard, but such studies are left for future work.