1. Introduction

Metals are essential building blocks of photonic architectures. Their high level of free carriers screens out incoming electromagnetic fields, which, in terms of optical properties, translates into a negative permittivity. As a consequence, they are naturally used to reflect or confine electromagnetic fields. As we come to plasmonics, metallic materials are all the more indispensable since plasmonic modes exist at a metal-dielectric interface. At the end, they are fundamental to the flourishing field of nanophotonics and metamaterials, from early works on microwave artificial materials [1,2] to groundbreaking demonstration of negative-index materials [3,4], innovative transformation optics devices [5–7] or even nanoscale thermal management [8].

But metals are intrinsically lossy. At microwaves frequencies, electromagnetic fields do not penetrate much metallic components, and the aforementioned exciting concepts are achievable [2,3,5]. However, as higher frequencies are reached, dissipative losses increase and significant limitations are met [9,10]. As a concrete example, metal-based filtering elements cannot reach an ideal transmission of 100% [11,12]. Or, in spite of all the opportunities they open up, passive plasmonic waveguides can only support propagation over few tens of micrometers [13,14]. Even when absorption is desired, these high loss rates can still be problematic, as for perfect metamaterial absorbers, for which high quality resonances are sought after to provide for narrow-band thermal emitters or highly sensitive sensors [15–17].

This situation triggered the development of various strategies to reduce these losses [9,18,19]. An ideal metal-like material that exhibits absolutely no loss within a given frequency band can theoretically exist but remains to be found today [20,21]. A natural lead consists in looking for an alternative material with losses as low as possible. Metallic alloys can exhibit improved properties but only for very specific spectral bands [22–24]. Highly-doped semiconductors present relatively low carrier concentration, which results in lower loss rates but also in lower plasma wavelength [25,26]. If they present several advantages, such as easy on-chip integration [27], their plasmonic performances remain thereby lower than noble metals [28]. Another promising strategy consists in compensating losses with a gain medium [29,30], but these loss rates are so high that this approach is particularly difficult to implement. Finally, obtaining low-loss negative permittivity without free carriers is possible, and auspicious results have been obtained using phonon-polaritons in polar-dielectric crystals [31] at the expense of a limited range of possible wavelengths as well as lower fields enhancements than with metallic components [32]. Each of the above-mentioned strategies can prevail depending on the targeted application and the influence of parameters such as thermal resistivity, integration, chemical stability, fabrication or cost. Nonetheless, noble metals often persist as the most adapted choice [28,33].

In this paper, we propose to improve noble metals performances by integrating them within an architecture of optical resonant nanostructure whose loss rate can be tuned over several orders of magnitude, from millimeter waves to optical frequencies. As this strategy is particularly adapted to guided modes, we consider waveguide architectures to highlight the potential of this approach on a simple model before applying it to a photonic resonator. For both cases, the quality factors of the structures get boosted by a factor of at least $10^4$. This can be described as a dramatic drop in metallic loss rate by introducing an equivalent model of a very low-loss metal.

2. Low loss metallic waveguide

2.1 Noble metals and metallic waveguides

Noble metals electromagnetic response is generally well described by a Drude model, notably in the infrared range. It relies on two parameters: $\omega _p$, the plasma pulsation of the electronic gas, and $\gamma$, a normalized parameter inversely proportional to the mean collision time of an electron. The metallic relative permittivity is then given by $\varepsilon _r(\omega ) = 1 - \omega _p^2 / (\omega ^2+j\gamma \omega _p\omega )$. When reaching visible wavelengths, additional inter-band transitions appears making this model no longer adapted. In this paper, all metallic responses rely on Drude models apart from a few data points that we point out.

Let us now consider a planar metallic waveguide, its $\mathrm {TM}_0$ mode propagating along $z$ and the corresponding magnetic field $\boldsymbol {H} = H_y(x) e^{j\omega t-j\beta z} \boldsymbol {y}$. For perfect metallic walls, which corresponds to $\gamma = 0$ and infinite plasma frequency, $H_y$ is constant in the dielectric core and null in the metal. However, real Drude metals present a strictly positive $\gamma$ as well as a finite $\omega _p$. Therefore, electromagnetic fields interacts with the metallic electronic gas, inducing dissipative losses [9]. Hence, the shape of $H_y$ is modified as well as the propagation constant of the mode, which becomes complex and is written $\beta = \beta ' + j\beta ''$. A quality factor, defined as $Q = \beta ' / 2\beta ''$, is attributed to the mode. This definition of $Q$ is commonly used to quantify the decaying nature of propagating modes in waveguides or transmission lines [34].

2.2 Three-dielectric metallic waveguides

Our approach consists in replacing the dielectric core of the waveguide by several layers of different optical indices chosen to concentrate the field within the middle of the guide. Its penetration in the metallic parts is thus limited, which results in a lower mode loss rate. For demonstration, a symmetric three-dielectric architecture is studied hereafter. It consists of a high index layer ($n_2 = 4.0$) placed between two low index layers ($n_1 = 1.4$) and surrounded by gold metallic walls. Theses two are ideal dielectric layers whose indexes $n_2$ and $n_1$ match mid infrared refractive indices of germanium [36] and silica [37], usual and mastered materials in thin film technology which present both a high index contrast and low losses. At the moment, loss and dispersion free dielectrics are considered to focus on the impact of the design on metallic losses. Figure 1(a) displays a schematic view of the structure and introduces the fill factor $f$, which corresponds to the proportion of high index dielectric in the core. Figure 1(b) depicts the form of the magnetic field $H_y$ for various fill factors. All electromagnetic responses presented in this Letter were obtained using a B-spline modal method that rigorously solves the Maxwell equations [38]. The one-dielectric cases ($f = 0$ and $f = 1$) exhibit slightly different forms of the field within the core but a very similar penetration in the metallic walls, whereas choosing an intermediate value for $f$ indeed permits a confinement of the field and a limited penetration. Figure 1(c) shows the evolution of the Q-factor as a function of fill factor. The resulting curve suggests that multi-dielectric guides allow to finely tune the quality factor and reach values more than three orders of magnitude higher than those of one-dielectric guides. For the considered guide width and wavelength ($h = {2}\;\mu \textrm {m}$, $\lambda = {4}\;\mu \textrm {m}$), the best results are obtained for $f=0.5$.

 

Fig. 1. Proposed three-dielectric waveguide architecture. $n_1$ and $n_2$ are set to 1.4 and 4.0. Wavelength in vacuum $\lambda$ and guide width $h$ are respectively set to ${4.0}\;\mu \textrm {m}$ and ${2.0}\;\mu \textrm {m}$. Permittivity of gold is obtained with a Drude model of parameter $\gamma = 0.048$ and $\omega _p = 1.2{\times }10^{16}\;\textrm {s}^{-1}$ [35]. Gold walls are considered semi-infinite. (a) Schematic view of the proposed structure. (b) Form of the normalized magnetic field $H_y$ of the $\mathrm {TM_0}$ mode for various values of fill factor $f$. (c) Dependency on $f$ of the mode quality factor $Q$.

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2.3 Quality factor of the TM0 mode

As illustrated in Fig. 2, $Q$ is also highly dependent on $h$. On the left part of Fig. 2(a), the red dashed line sketches the fill factor maximizing $Q$ at each guide width $h$. The right part of the figure depicts how $Q$ evolves with $h$ when traveling along this dashed line as well as for fixed fill factors. Considering the one-dielectric cases ($f = 0$, low index and $f = 1$, high index), we observe the quite intuitive increase of the quality factor of a less confined mode. While the mode becomes more extended, the penetration of the field in metal remains of the order of the skin depth. The proportion of the field in the metal thus decreases, resulting in a reduced modal loss rate. However the curves show that our architecture particularly enhances this effect. What was before close to a linear dependency to guide width $h$ becomes closer to an exponential as depicts the maximal value of $Q$. At the largest considered guide width, the three-dielectric architecture theoretically enables to achieve Q-factor as high as $10^{11}$ which is more than eight orders of magnitude higher than with a usual one-dielectric guide ($f = 0$, $f = 1$). Besides, it appears on Fig. 2(a) that keeping the fill factor to $f = 0.5$ provides for a quality factor rather close to its maximal value at a given width. For the sake of simplicity, the forthcoming analysis has been carried out at this fixed value.

 

Fig. 2. (a) Quality factor of the $\mathrm {TM_0}$ mode as a function of relative width $h/\lambda$ and fill factor $f$ for a ${4}\;\mu \textrm {m}$ vacuum wavelength. Other parameters are kept the same. The red dashed line corresponds to the fill factors maximizing the quality factor at each width. On the right, width dependency of this maximal quality factor along dependencies at fixed fill factors. (b) Evolution of the quality factor as a function of wavelength and relative width at fixed fill factor $f = 0.5$.

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2.3.1 Limitations

In practice, dielectric losses should be accounted for such high values of quality factors, as they may no longer be negligible. As an example, when using experimental data of SiO$_{2}$ and Ge indices for $n_1$ and $n_2$ - which amounts to consider losses in dielectric 1, $Q$ increases at a slower pace and reaches $10^7$ instead of $10^{11}$, which remains yet four order of magnitude higher than with a conventional design. The actual limit on $Q$ is actually set by losses in the high index region, which is consistent as the field gets confined in the latter. If losses are indeed added to dielectric 2, the quality factor observes the same increase in $h$ at low width and then stagnates at $Q_{max} = n_2 / (2k_2)$, which would be the quality factor associated to a plane wave propagating in an infinite substrate of dielectric 2. The reader will find a detailed discussion on the impact of dielectric losses, dispersion as well as increased metallic losses - because of adhesive layers for example - in Supplement 1.

At low widths, the curves show a lower limit, close to $h = 0.1\lambda$, below which the multidielectric slicing does not affect the loss rate of the mode.

2.3.2 Wavelength dependency

The reader might have noted on Fig. 2 that guide widths are given relatively to the vacuum wavelength. This choice comes from the rather low wavelength dependency of this design. Albeit all previous results were presented at ${4}\;\mu \textrm {m}$, Fig. 2(b) demonstrates that very similar performances can be obtained over a large spectral band: from visible to at least millimeter waves. The two dielectric indices $n_1$ and $n_2$ were kept the same throughout this spectral range. In practice, these would depend on the actual choice of materials, but we assume that it is possible to find a couple of dielectrics that present similar properties in each given spectral band; that is to say a large index contrast as well as losses as low as possible. If SiO$_{2}$ and Ge are adapted to near and mid infrared, it would be for example possible to use ZnS and Ge for far-infrared and SiO$_{2}$ and Sapphire for THz. Concerning metallic response, at high frequencies - below $\lambda = {700}\;\textrm {nm}$ - the apparition of interband metallic transitions invalidates the aforementioned Drude model. Therefore, a more exhaustive Brendel-Bormann model is used for gold within this specific spectral band [39]. In this range, performance slightly deteriorates because of this high rise in metallic loss rate. Moreover, the increase in the ratio $\delta / \lambda$ at short wavelength, where $\delta$ denotes the metallic skin depth, also hinders the loss rate of the guided mode at a given ratio $h / \lambda$. Nonetheless, Fig. 2(b) illustrates that despite these constraints, the proposed design still offers in this region the possibility of low-loss propagation.

2.4 Low loss equivalent metallic model

In order to offer a different physical insight into this three-dielectric architecture, the very high quality factors of the three-dielectric waveguide are interpreted as a dramatic drop in the metallic losses and described by an equivalent model. Indeed, for each width $h$, keeping all other parameters unchanged, it is possible to obtain the same propagation constant $\beta$ with a one-dielectric architecture consisting of a core of equivalent optical index $n_{eq}$ and a equivalent metal of Drude dissipative parameter $\gamma _{eq}$. $n_{eq}$ and $\gamma _{eq}$ have been found after successive iterations with a lower limit at 0.01% for the residual relative differences of $\beta '$ and $Q = \beta '/(2\beta '')$. Figures 3(a) and (b) depict the width dependency of respectively $n_{eq}$ and $\gamma _{eq}$. For most of the width range, $n_{eq}$ is found to be very close to the effective index of the three-dielectric waveguide mode defined as $n_{eff} = \beta '/k_0$, where $k_0$ designates the wavevector in vacuum. By definition, $n_{eff}$ also corresponds to the effective index of the one-dielectric equivalent model. The proximity between the latter and $n_{eq}$ is consistent with the equality between the effective index of the $\mathrm {TM_0}$ mode of a perfect metallic waveguide and its dielectric core index. The remaining small difference when considering real metals is precisely due to the penetration of fields in the metal. At low width, this penetration is no longer negligible and the strong increase in $n_ {eff}$ is a consequence of the plasmonic nature of the highly confined $\mathrm {TM_0}$ mode. In this region $n_{eq}$ converges towards 1.87, which corresponds to the harmonic mean of the permittivities of the three-dielectric core. Indeed, at low width $h$, the homogeneization theory can be applied. On the contrary, at large widths, $n_{eq}$ converges towards $n_2 = 4$, which is coherent with a field that is increasingly concentrated in the high index section.

As for $\gamma _{eq}$, as shown on Fig. 3(b), it can be tuned over seven orders of magnitude and becomes as low as $10^{-8}$, in which case the resulting equivalent metal is nearly lossless. However, since we choose not to modify the plasma frequency $\omega _p$ of its corresponding Drude model, one have to keep in mind that only one of the two natural conditions for perfect metallic behavior in Drude formalism ($\gamma \rightarrow 0$ and $\omega _p \rightarrow \infty$) is fulfilled. We therefore denotes it as almost perfect. Equation (1) provides for an estimation of $\gamma _{eq}$ that offers a good physical insight.

 

Fig. 3. One-dielectric equivalent model of the three-dielectric waveguide at $f = 0.5$, $n_1 = 1.4$ and $n_2 = 4.0$. Dependency on the guide width $h$ of: (a) the $\mathrm {TM_0}$ effective index $n_{eff}$ and parameters used in the equivalent model: the dielectric index $n_{eq}$ and ((b)) the Drude metallic dissipative parameter $\gamma _{eq}$.

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3. Low loss photonic resonator

As an application of this concept to a resonant nanostructure, we opted for a guided-mode resonator (GMR) [11,40], which, as shown on the inset of Fig. 4, is based on a metallic grating coupled to the aforementioned three-dielectric waveguide. Under illumination, this structure resonates when a grating-diffracted order coherently couples with a guided mode. Knowing the effective index of a mode $n_{eff}$, the largest corresponding resonant wavelength at normal incidence can be estimated as $\lambda _{r} = p \times n_{eff}$, where $p$ designates the period of the grating. The polarization of the incoming light is transverse magnetic (TM) to correctly excite the $\mathrm {TM_0}$ mode of the guide. Finally, the width of the gold ribbon $L$ is tuned to achieve perfect impedance matching between the resonator and free space. In an impedance-matched absorption resonance, all incoming energy is transferred to the guided mode. The quality factor of the resonant peak, defined as $Q_{GMR} = \lambda _r/\Delta \lambda$ with $\Delta \lambda$ the full width at half maximum, is therefore directly related to the quality factor of the propagating mode introduced earlier. Both terms are denoted as $Q$ in the rest of the article.

3.1 Control of the quality factor of the resonant absorption

Figure 4 presents how the absorption spectrum of this three-dielectric GMR is affected by a change in its width $h$, proving this design to be as versatile in terms of loss control as predicted by the waveguide analysis. The peak quality factor evolves again exponentially with structure width as depicted in the right part of the figure and the curve is very similar to Fig. 2(a). Ultimately, this means that quality factors exceeding one million are achievable in absorption with such structures. Moreover, changing other parameters such as fill factor $f$ or period $p$ offers supplemental degrees of freedom to adjust the resonant wavelength over a wide spectral band, by respectively changing either the effective index of the resonant mode or the horizontal wavevector of the diffracted orders. As it is the case for the waveguide architecture, $Q$ depends also a lot on fill factor $f$. Even if not identical - one of walls of the GMR waveguide is different - we except the dependency to be similar to Fig. 2(a). For example the fill factor maximizing $Q$ approaches $0.65$ for $h = {1}\;\mu \textrm {m}$, $0.55$ for $h = {1.5}\;\mu \textrm {m}$ and finally $0.5$ for $h = {1.9}\;\mu \textrm {m}$. $f = 0.5$ remains thus close to the optimum and we choose to keep this value in order to remain consistent with the rest of the manuscript.

 

Fig. 4. Absorption spectra under TM normal illumination of three-dielectric GMRs of various widths $h$. The corresponding architecture is depicted with relevant new parameters: the period $p$ and the width of the gold ribbon $L$. The fill factor $f$ and guide width $h$ remain as previously defined. As before, $f$ is kept fixed at 0.5. Gold upper ribbon is ${50}\;\textrm {nm}$ thick and considered semi-infinite at the bottom. Each colored curve is reported on the right graph which represents the Q-factor dependency on width $h$. Other parameters are given in $\mu \textrm {m}$ in the following order: ($\textbf {h}$, $p$, $L$): (0.70, 2.10, 0.557); (1.0, 1.75, 0.248); (1.3, 1.39, 0.403); (1.6, 1.23, 0.157); (1.9, 1.15, 0.118).

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As it was the case for waveguides, the quality factor of an usual one-dielectric GMR also increases with width $h$, but again this evolution remains limited compared to the three-dielectric architecture. As an example, at width $h = {1.5}\;\mu \textrm {m}$, the latter exhibits a 400 times higher Q. Supplement 1 provides for more information on this comparison as well as an additional analysis of the impact of dielectric losses on theses resonances and a absorption spectrum over a larger spectral band.

3.2 Metallic equivalent model for a photonic resonator

The equivalent model introduced for the waveguide architecture is easily transferable to this resonator. It is possible to get a very similar response under TM normal illumination with a one-dielectric structure of equivalent set of parameters ($\gamma _{eq}$, $n_{eq}$), whose gold ribbon width is tuned for impedance adaptation. An example is given on Fig. 5, where a very good matching can be observed for a structure of width $h = {1.5}\;\mu \textrm {m} = {0.375}\lambda _r$. The magnetic field maps are given for both structures and present a very high and similar level of enhancement $|\vec {H_y}|/|\vec {H_0}|$. As a comparison, the latter here reaches 600 against only 30 for a conventional one-dielectric structure with a real loss rate $\gamma _{Au}$ (see Supplement 1). The value of $n_{eq}$ fits almost perfectly the waveguide analysis performed on Fig. 3. As for $\gamma _{eq}$, the corresponding value differs a bit, probably because of the presence of radiative damping terms in the GMR, but still remains of the same order of magnitude. This confirms the relevance of the equivalent model of these structures: this design presents an extremely low and tunable metallic loss rate at the resonance.

 

Fig. 5. Metallic equivalent model adapted to GMR. The absorption – black plain curve – of a three-dielectric structure of parameters $h = {1.5}\;\mu \textrm {m}$, $p = {1.27}\;\mu \textrm {m}$, $L = {186}\;\textrm {nm}$ is given alongside the response – blue dashed curve – of a corresponding one-dielectric structure of equivalent parameters $\gamma _{eq} = 1.3{\times }10^{-4}$ and $n_{eq} = 3.14$ with an adapted ribbon width $L = {85.0}\;\textrm {nm}$. The period and width are unchanged. The maps of the magnetic field enhancement at the resonance are also depicted.

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4. Concentration of the absorption in a low absorbing layer

As an opening for this paper, we have chosen to illustrate the applicability of this general concept with an adaptation of the GMR architecture where all the absorption gets concentrated within a thin active layer. Figure 6 portrays the introduction of such a layer at the center of the dielectric core within the considered three-dielectric design (Fig. 6(a)) and a conventional one-dielectric GMR architecture for reference (Fig. 6(b)). For comparison, the spectra and absorption localization with and without losses in this thin layer are depicted. Losses in the active layer are represented by the imaginary part of the corresponding complex index $n_a = n_2 + \mathbf {j}\kappa$. Its real part corresponds to the index of the surrounding dielectric.

Adding this active layer to the conventional one-dielectric GMR does not affect much its response. Only $6\%$ gets absorbed within the layer. As for the three-dielectric architecture, this important new source of losses drastically reduces the quality factor of the peak ($6.7{\times }10^{4}$ against 640). However, nearly all the absorption now occurs in this active layer: $97\%$ for the proposed design. Besides, the location of this active layer is not crucial since positioning it on the border of the high index region still exhibits $95\%$ of absorption in the latter, far more than with a one-dielectric GMR.

 

Fig. 6. (a) A thin layer ($n_a$, $h_a$) is introduced at the center of the three-dielectric architecture from Fig. 5. The imaginary part of its refractive index $\kappa$ is either set to 0, in which case the structure is not altered, or set to 0.1, which corresponds to a low absorption rate (the sole active layer would only absorb $0.3\%$). (b) The same process is proposed as reference for a conventional one-dielectric GMR. Akin Fig. 3, its dielectric index is set to $n_{eq} = 3.14$, but $\gamma$ is kept at $\gamma _{Au}$. For both architectures, the gold ribbon width $L$, as well as slightly the period $p$, are adjusted between cases $\kappa =0$ and $\kappa =0.1$ to keep an absorption peak at ${4}\;\mu \textrm {m}$. The absorption spectra are depicted for the four designs alongside the distribution of absorption between the various layers of the structure.

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5. Conclusion

We have presented in this paper a versatile and original design to control optical dissipative losses in metal-dielectric waveguides and resonant photonic nanostructures over several orders of magnitude. An equivalent model where the metallic elements are replaced by metals of extremely low loss - thus denoted as almost perfect - has been proposed and presents a very similar optical response. Practically, the potential of this design has been demonstrated by opening perspectives on absorption control. The high narrowness of the corresponding resonances now constitutes a challenge for optical characterization as we are considering an experimental demonstration. Moreover, an advanced investigation of the angular dependency of this GMR based architecture and the adaptation of this strategy to resonators presenting various angular responses are under progress to offer an optical behavior adapted to the intended application. In this respect, various practical designs are envisaged. Taking a quantum well as active layer, the absorption control might for example be an interesting approach for improving infrared photodetector efficiencies. Taking advantage of the loss control, ultra-narrow filtering elements of diminished losses are also considered. Finally, the very high field enhancement factors allows to seriously consider applications to non linear optics where high field concentration is crucial, such as second harmonic generation.