---
title: "Long-range surface plasmons on duplex metal nanolayers"
authors: ["Valery N Konopsky"]
affiliation: "Institute of Spectroscopy, Fizicheskaya, 5, Troitsk, Moscow, 108840, RUSSIA."
journal: "Photonics and Nanostructures"
year: 2020
volume: ""
issue: ""
article_number: ""
pages: ""
doi: "10.1016/j.photonics.2020.100788"
type: journal-article
site_group: ""
url_abstract: ""
url_pdf: "https://valery.konopsky.com/kvnlocal/Long-range%20surface%20plasmons%20on%20duplex%20metal%20nanolayers_Konopsky_2020.pdf"
language: en
source_tex: "Z:\\ValeryData\\Valery_New\\my_articles\\PhotonicsAndNanostructures2020\\PhotonicsAndNanostructures\\manuscript\\Konopsky2PhN.tex"
source_pdf: "Z:\\ValeryData\\Valery_New\\my_articles\\PhotonicsAndNanostructures2020\\PhotonicsAndNanostructures\\published\\Long-range surface plasmons on duplex metal nanolayers_Konopsky_2020.pdf"
---
## Abstract

Planar structures with two metal nanolayers supporting long-range surface plasmons (LRSPs) are described. We introduce a design where a one-dimensional photonic crystal is used as a support for a planar dielectric film sandwiched between two metal nanolayers. This asymmetric design is compared with symmetrical where two metal nanolayers deposited on opposite sides of the planar dielectric film have the same semi-infinite dielectric environment from both sides. Such multilayer structures are useful when electrical power is supplied to the planar film through LRSP-supporting metal nanolayers on opposite sides of the film. This planar film may be, for example, an electro-optical crystal or an (organic) electroluminescent film. A method for designing such LRSP-supporting duplex metal structures based on an impedance approach is presented. Calculations of the optimal thickness of the metal-bounded dielectric film are given for both symmetrical and asymmetrical cases.

Long-range surface plasmons ,Photonic crystal waveguides ,Multilayer

# Introduction

Surface plasmons (SPs) are excitations of electromagnetic (EM) modes that exist near and propagate along a metal-dielectric interface [1]. They are bound to the interface by total internal reflection (TIR) on the dielectric side and reflection of the metal on the opposite side. A strong intrinsic damping of the EM field in a metal at optical frequencies is a reason for an unavoidable attenuation of the SPs during their propagation along the metal surface.

Despite the increase in the number of applications of SPs (for a recent review see [2]), damping in a metal has limited the success of most of these applications. To reduce this damping and increase the propagation length of the SPs, a structure allowing for the propagation of the long-range surface plasmon (LRSP) was invented in the early 1980s [3, 4]. In this approach, a metal nanolayer is imbedded between two dielectrics with the same (or very close) refractive indices (RIs), as shown in Fig. 1 (a). In practical applications, either the metal nanolayer must be imbedded in an integrated optical system whose sides have the same dielectric [5], or, if water is required on one side (for biosensor applications), a dielectric with an RI that is close to water’s RI ($n_e\simeq 1.33$) must be used as a support [6]. While in a gas environment (i.e., for $n_e\simeq 1$), a freely suspended nanofilm is necessary to obtain the same RI on each side [7].

To simplify the implementation of LRSP in practical applications, a one-dimensional (1D) photonic crystal (PC) was proposed as a substrate for a metal nanolayer [8]. This approach can be implemented for the environment with any RI, even for gas sensors [9, 10].

Here, we present the calculations for the design of multilayer systems, in which *two* metal nanolayers – one on each side of a thin dielectric film – support the long-term propagation of SPs. We will consider and compare two schemes: symmetrical, in which the dielectric film, bounded by two metal nanolayers, has the same external environment on each side; and asymmetrical, in which this film is deposited on a semi-infinite 1D PC.

The asymmetrical scheme (with 1D PC from one side) has introduced for the first time here, and we will show that it has characteristics similar to those of a symmetric one, but is more practical when the external environment is air. Both systems can be used in many applications in which the dielectric film is an active element driven by electrical power and the metal nanolayers are used as electrodes. Applications for which a small current is required, e.g., if the active element is a thin electro-optical crystal or an (organic) electroluminescent film, the electrical resistance of the metal nanolayers will not be a problem.

The real parts of solutions of the dispersion relation for the symmetrical scheme (i.e., without 1D PC) were theoretically derived in several articles, starting with the work of Economou, published in 1969 [11]. The field profiles and dispersion curves of coupled modes were analysed theoretically in [12, 13, 14]. It was shown that the interaction of modes in such a structure leads to the intersection of the light lines by the modes and to the mutual anti-crossings of the dispersion curves of the modes [13, 15].

An experimental test of the optical excitations of a layered structure with two metal layers was carried out by Kovacs and Scott in 1977 [16] and (in a completely symmetric configuration) by Welford and Sambles in 1987 [12]. Also, attempts were made to use the symmetric configuration for measurement of air gap width between metal layers [17]. Prism-coupled light emission from a tunnel junction between two metal layers was experimentally detected in [18].

For the practical implementation of such systems, it is important to check the imaginary parts of the dispersion curves, which determined the propagation lengths of the modes. The propagation lengths for the symmetrical system was derived by Stegeman and Burke in 1983 for a particular case where the refractive index (RI) of the film between metal nanolayers is the same as the RI of the external environment [19]. Systems with the film’s RI that is smaller and larger than the RI of the external environment were discussed in [20]. The absorption coefficients for one type of mode (non-long-range) in symmetrical plasmonic waveguide modulators were calculated numerically in [21].

In this paper, we will present the real and imaginary parts of the dispersion curves for the symmetrical scheme and for the asymmetrical scheme (with 1D PC on one side), and show that only modes close to the light line can be long-range modes and may be promising for practical application. The equations for the thickness of the dielectric film, bounded by two metal nanolayers, that provides excitation of the long-range mode are presented for both types of schemes.

# Methods

The theoretical description of planar multilayer systems will be based on an impedance approach. The ‘normal impedance’ $Z$ is the ratio of the tangential components of the electric field to the magnetic field, which has the following forms (in a $j$th layer) for *s*- and *p*-polarizations [22, 23, 24]:
``` math
\begin{eqnarray}
Z_{s(j)}=
&\displaystyle\frac{1}{n_j \cos(\theta_j)}
&=\displaystyle\frac{1}{n_j\sqrt{1-(\rho/n_j)^2}}   
\[8pt]
Z_{p(j)}=
&\displaystyle\frac{\cos(\theta_j)}{n_j}   
&=\displaystyle\frac{\sqrt{1-(\rho/n_j)^2}}{n_j} \; .

\end{eqnarray}
```
Here and hereafter the numerical aperture $\rho=n\sin(\theta)$ will be used as an angle variable in a planar multilayer system. It is a unified angle variable for all layers, since according to Snell’s law $\rho=n_j\sin(\theta_j)$, for any $j$.

One of the advantages of using the impedance approach is the ability to represent the impedances of *several* layers as a *single* ‘apparent’ input impedance through a recursive relation (see e.g., (8) in [24]). For example, if a layer with an impedance of $Z_{(1)}$ and a thickness of $d_1$ is deposited on a semi-infinite medium $Z_{(e)}$, then together they can be considered to be a single medium with a total impedance $Z_{(e)\&(1)}$:

``` math
\begin{equation}
Z_{(e)\&(1)}=Z_{(1)}{\frac {{ Z_{(e)}}-iZ_{(1)}\tan(\alpha_{1})}{{Z_{(1)}-i{ Z}_{(e)}}\tan(\alpha_{1})}}\,,

\end{equation}
```
with $\alpha_1\!=\!k_{z(1)} d_1$ and $k_{z(1)}\!=\pm(2\pi/\lambda)n_1\sqrt{1-(\rho/n_1)^2}$. This procedure can be recursively continued: if the second layer is deposited on the first, (3) is applied again, where $Z_{(e)\&(1)}$ replaces $Z_{(e)}$ and so on.

![Two symmetrical schemes in which either (a) a single metal nanolayer (dark gray) is placed in an external environment with the same RI on both sides, or (b) two metal nanolayers that are separated by an active dielectric film (hatched) are placed in the same environment.](media/PhotonicsAndNanostructures2020/Fig1.eps)

*Two symmetrical schemes in which either (a) a single metal nanolayer (dark gray) is placed in an external environment with the same RI on both sides, or (b) two metal nanolayers that are separated by an active dielectric film (hatched) are placed in the same environment.*

##  Prerequisite for the existence of LRSP

Earlier it was shown that the tangential component of the electric field in the thin metal layer is equal to zero in the center of the layer, when the $p$-polarized electromagnetic wave, incident on this layer, has numerical aperture:
``` math
\begin{equation}
\rho_{1/2}= 
n_e
+
\frac{n_e^{3}}{2}\,\left[{\pi}{\frac {{d_\mathrm{m}}}{{\lambda}}}\right]^{2}\; ,

\end{equation}
```
where $d_\mathrm{m}$ is the thickness of the metal nanolayer, and $n_e$ is the refractive index of the external medium.

This equation appears in [8] (see (1) there), while the complete derivation of this equation appears in [24] (see (31) there). This condition must be satisfied for LRSP propagation to occur in a nanolayer whose RI has a large imaginary part ($\mathrm{Im}(n_\mathrm{m})\!\gg\!1$). The problem is to find waveguide structures containing a metal nanolayer that supports the propagation of a surface wave with a wave vector $k_y =  2\pi\rho_{1/2}/\lambda$. For the classic LRSP structure – a thin metal layer with the same medium on both sides (shown in Fig. 1 (a)) – this condition is satisfied automatically [24, 25].

For structures with duplex metal nanolayers, this prerequisite will be the same: we must find waveguide structures containing two metal nanolayers and supporting the propagation of a waveguide/surface wave with the same effective refractive index $\rho_ {1/2}$. The main purpose of this article is to derive equations for the thickness of a dielectric film, which provides the excitation of one of the coupled modes near $\rho_ {1/2}$.

![Two asymmetric schemes in which either (a) one metal nanofilm (dark gray) is deposited on a semi-infinite 1D PC, or (b) an active dielectric film (hatched), bounded by two metal nanofilms, is deposited to the same 1D PC.](media/PhotonicsAndNanostructures2020/Fig2.eps)

*Two asymmetric schemes in which either (a) one metal nanofilm (dark gray) is deposited on a semi-infinite 1D PC, or (b) an active dielectric film (hatched), bounded by two metal nanofilms, is deposited to the same 1D PC.*

## General dispersion relation and its solutions for film thicknesses

We start our consideration with a structure in which a waveguide film with an impedance of $Z_{(f)}$ is between media with impedances $Z_{(-)}$ and $Z_{(+)}$ (see the light gray marks in Fig. 1 (b)).

A general condition for the existence of a waveguide/surface wave between two media with impedances $Z_{\mathrm{left}}$ and $Z_{\mathrm{right}}$ is:
``` math
\begin{equation}
Z_{\mathrm{left}}+Z_{\mathrm{right}}=0\, .
\end{equation}
```
In our case, shown in Fig. 1 (b), this condition takes the form:
``` math
\begin{equation}
Z_{(-)\&(f)}+Z_{(+)}=0\, .

\end{equation}
```
This equation is the general dispersion relation of our planar systems. To obtain the dispersion relation for a specific case, one should insert the appropriate impedances in this equation, using (3) when it is necessary.

By solving (6) and (3), we obtain the thicknesses of the dielectric film:
``` math
\begin{equation}
 d_\mathrm{f} =  \frac{\pi  {M}} {k_{z(f)}}+  \frac{1}{k_{z(f)}}\arctan\left({\frac{-i\left({  Z_{(-)}}+{
  Z_{(+)}} \right){Z_{(f)}}}{Z_{(f)}^{2}+{  Z_{(-)}} {  Z_{(+)}}}}\right),

\end{equation}
```
where $M$ is a whole number. This is a general equation to find thicknesses $d_\mathrm{f}$ for which waveguide modes of different orders $M$ exist in the structure, if the waveguide film with an impedance of $Z_{(f)}$ is between media with impedances $Z_{(-)}$ and $Z_{(+)}$ (as designated by the light gray marks in Fig. 1 (b)).

## Duplex LRSPs in symmetrical scheme

At first we find a specific solution for the symmetrical scheme, in which the dielectric film, bounded by two metal nanolayers, has the same external environment from both sides as it is shown in Fig. 1 (b). In this particular (quasi)symmetric case, we should use
``` math
\begin{eqnarray}
 
Z_{(-)}&=&Z_{(e)\&(m_1)}
 \\ 
Z_{(+)}&=&Z_{(e)\&(m_2)}\,,  \nonumber
\end{eqnarray}
```
obtained from (3), in the general dispersion relation (6) or in its solution for $d_\mathrm{f}$ (7). To satisfy the LRSP prerequisite for this structure, we must choose solutions with $\rho=\rho_{1/2}$, given by (4).

## Single LRSP in asymmetrical scheme with 1D PC

To find a solution for the 1D PC structure with *one* metallic nanolayer $m_1$ as shown in Fig. 2 (a), we must find the impedance of the semi-infinite 1D PC. This impedance was derived in [24], and, to provide a single source version of the related equations, its formula is presented in Appendix A. If the parameters of the metal nanolayer (thickness $d_{m_1}$ and RI $n_{m_1}$) are given, we may change the thickness $d_3$ of the last layer in the PC to obtain a solution with the desirable $\rho_{1/2}$ that satisfies the LRSP prerequisite.

Thus, as shown by the light gray marks in Fig. 2 (a), we again have a waveguide film with an impedance of $Z_{(3)}$ that is between media with impedances $Z_{(-)}=Z_{(PC)}$ and $Z_{(+)}=Z_{(e)\&(m_1)}$ and solution (7) yields the following thicknesses:
``` math
\begin{eqnarray}
 
d_3&=&(\pi{M}+\alpha_3)/k_{z(3)}\\
 \alpha_3&=&
\arctan{\left({\frac{-i\left({  Z_{(PC)}}+{
  Z_{(e)\&(m_1)}} \right){Z_{(3)}}}{Z_{(3)}^{2}+{  Z_{(PC)}} {  Z_{(e)\&(m_1)}}}}\right)}\,.  \nonumber
\end{eqnarray}
```

## Duplex LRSPs in asymmetrical scheme with 1D PC

To design the asymmetrical scheme (shown in Fig. 2 (b)) with *two* metal nanolayers, first we design the scheme with the single metal nanolayer $m_1$, e.g., as demonstrated in the previous subsection, and then we add a dielectric film $f$ and a second metal nanolayer $m_2$. The problem is that, to satisfy the LRSP prerequisite (4), the external environment (from point of view of the first metal nanolayer $m_1$) must have an impedance of $Z_e$ (as a semi-infinite medium with an RI of $n_e$). Thus, we must satisfy
``` math
\begin{equation}
Z_{(e)\&(m_2)\&(f)}=Z_{(e)}\, ,

\end{equation}
```
at some thickness of the dielectric film $d_\mathrm{f}$. The solution is
``` math
\begin{eqnarray}
d_\mathrm{f} &=&(\pi{M_f} - \alpha_{f})/k_{z(f)} \\

\alpha_{f}&=& 
 \arctan{ 
 \left(\frac{Z_{(f)}\left({  Z^2_{(e)}}-{Z^2_{(m_2)}} \right)\tan{(\alpha_{m_2})}} 
 {F}
 \right) }\,,   \nonumber
\end{eqnarray}
```
where $M_f$ is a whole number, $F=$\
$i Z_{(e)}\left(Z_{(f)}^{2}-  Z^2_{(m_2)}\right)\tan{(\alpha_{m_2})} + Z_{(m_2)} \left(   Z^2_{(e)} - Z_{(f)}^{2}\right)$ and $\alpha_{m_2}=k_{m_2}d_{m_2}=(2\pi/\lambda)n_{m_2} d_{m_2}\sqrt{1-(\rho/n_{m_2})^2}$.

Alternatively, the required film thickness can be derived from (7) with the following substitutions:
``` math
\begin{eqnarray}

Z_{(-)}&=&Z_{(PC)\&(d_3)\&(m_1)}
 \\ 
Z_{(+)}&=&Z_{(e)\&(m_2)}\,,  \nonumber
\end{eqnarray}
```
which should be obtained by recursively using the formula (3). Both approaches provide the same values for $d_\mathrm{f}$ in this asymmetrical scheme.

# Results

SPs are *p-*polarized waves, and we should use (2) to determine the impedances of the layers in all previous equations. To be sure that these equations provide correct parameters for the design of duplex LRSPs, we obtain a graphical representation of the EM field profiles of duplex LRSPs in symmetrical and asymmetrical schemes.

## Propagation length and EM field profile of duplex LRSPs in the symmetrical scheme

The following symmetrical scheme is considered: two gold nanolayers are deposited on a dielectric layer (in this example, let the dielectric layer be a super-yellow light-emitting polymer) with an RI $n_\mathrm{f}=1.86$ at $\lambda=575$ nm. The thickness of both gold nanolayers is $d_{m_1}=d_{m_2}=12$ nm and their RI at this wavelength is $n_{m_1}=n_{m_2}=0.32+2.8i$. Air is the external environment on both sides ($n_e$=1.0003). From (4) we obtain $\rho_{1/2}=1.0024$ for these data, and from (7) we obtain the thickness of the dielectric film that supports waveguide propagation in this system at $\rho_{1/2}$: $\mathrm{Re}[d_\mathrm{f}]=163.43$ nm.

![The real and imaginary parts of the effective RI ρ(df) = Re[ρ] + iIm[ρ] (solid lines) obtained as solutions of the dispersion relation (6) for the symmetrical scheme. Corresponding propagation lengths are shown on the right side of the Im[ρ] graph. The real part of the solution df(Re[ρ]) of the equation (7) is represented by hollow circles. The vertical dashed line represents df, which corresponds to ρ1/2 for this system.](media/PhotonicsAndNanostructures2020/Fig3.eps)

*The real and imaginary parts of the effective RI ρ(df) = Re[ρ] + iIm[ρ] (solid lines) obtained as solutions of the dispersion relation (6) for the symmetrical scheme. Corresponding propagation lengths are shown on the right side of the Im[ρ] graph. The real part of the solution df(Re[ρ]) of the equation (7) is represented by hollow circles. The vertical dashed line represents df, which corresponds to ρ1/2 for this system.*

If we use pure real $\rho$ as the input argument in (7), but the system contains layers with complex RIs, then we get the complex $d_\mathrm{f}$ as the answer. We show that in the range where LRSPs exist, i.e., near $\rho=\rho_{1/2}$, the real part of this answer – $\mathrm{Re}\left[d_\mathrm{f}\left(\mathrm{Re}[\rho]\right)\right]$, provides a good approximation for practical implementation (7) or (12). To prove this, we should obtain an exact solution of the dispersion relation (6) for our system. To do this, we must use the complex $\rho$ and real $d_\mathrm{f}$ as input arguments for (6) and find such real and imaginary parts of $\rho$ that satisfy both the real and imaginary parts of the dispersion relation. This problem can be solved numerically, and the results for the symmetrical scheme, with given above parameters, are presented in Fig. 3. The solid line represents the exact solution for the complex $\rho$ as a function of the film thickness $d_\mathrm{f}$, while the hollow circles represent an approximated solution for $d_\mathrm{f}$ given by the real part of (7).

The imaginary part of the effective RI determines the surface wave propagation length:
``` math
\begin{equation}
L=\frac{1}{2\mathrm{Im}[k_y]}=\frac{\lambda}{4\pi\mathrm{Im}[\rho]}\, .

\end{equation}
```

From Fig. 3 it is seen that the curves of the approximated and exact solutions are very close near $\rho=\rho_{1/2}$ and, therefore, one can use the thickness $d_\mathrm{f}=163.43$ nm for the active film in this symmetrical system.

![The spatial distribution of the EM field components in a dielectric film bounded by two gold nanolayer when the duplex LRSPs are excited in the symmetrical scheme.](media/PhotonicsAndNanostructures2020/Fig4.eps)

*The spatial distribution of the EM field components in a dielectric film bounded by two gold nanolayer when the duplex LRSPs are excited in the symmetrical scheme.*

Now we have all the parameters of the system and we can calculate the spatial profile of the EM field components in this symmetrical scheme. The results are presented in Fig. 4. It can be seen that the tangential component of the optical field is indeed zero at the centers of the metal nanolayers, and the total optical field in the metal reaches a minimum. It should be noted that a similar picture can be obtained for a quasi-symmetrical scheme, in which the thicknesses of the nanolayers are slightly different and/or the metals are different (for example, gold and aluminum).

## Propagation length and EM field profile of duplex LRSPs in the asymmetrical scheme with 1D PC

As stated in subsection (2.5), we first develop a PC scheme with a single metal nanolayer $m_1$, that supports LRSP propagation and then add the dielectric film and the second nanolayer $m_2$. Let the first nanolayer again be gold with a thickness $d_{m_1}=12$ nm, and, therefore, the effective RI of the LRSP will be $\rho_{1/2}=1.0024$ as well. Let the 1D PC consist of alternating layers of $Ta_2O_5$ (with RI $n_1=2.08$) and $SiO_2$ (with RI $n_2=1.47$). We choose the thicknesses of the alternating layers according to [24] (see (20) there), namely, $d_1=87$ nm and $d_2=118.67$ nm (or, as a second-rate alternative, one can use the ‘quarter-wavelength’ layers: $\textstyle d_j=\lambda/[4(n_j^2-\rho_{1/2}^2)^{1/2}]$). To obtain the LRSP at $\rho_{1/2}$ with these thicknesses and this metal nanolayer, we replace the thickness of the last $Ta_2O_5$ layer from $d_1$ to $d_3=74.36$ nm, calculated from (9). We thus obtain an asymmetric scheme with one metal nanolayer $m_1$.

Now we must find an acceptable thickness for the dielectric film that will support the LRSP propagation in two metal nanolayers. Let the second metal nanolayer $m_2$ be aluminum with a thickness of $d_{m_2}=15$ nm, and suppose the RI of the dielectric film is $n_\mathrm{f}=1.86$ (i.e., it is the super-yellow light-emitting polymer again). In this case, from (12) or from (7) and (12) one can obtain $Re[d_\mathrm{f}]=164.66$ nm, which is the thickness of the dielectric film supporting the propagation of the LRSP in the duplex metal nanolayers.

![The real and imaginary parts of the effective RI ρ(df) = Re[ρ] + iIm[ρ] (solid lines) obtained as solutions of the dispersion relation (6) for the asymmetrical scheme. Corresponding propagation lengths are shown on the right side of the Im[ρ] graph. The real part of the solution d(Re[ρ]) of the equation (7) is represented by hollow circles. The vertical dashed line represents df, which corresponds to ρ1/2 for this system.](media/PhotonicsAndNanostructures2020/Fig5.eps)

*The real and imaginary parts of the effective RI ρ(df) = Re[ρ] + iIm[ρ] (solid lines) obtained as solutions of the dispersion relation (6) for the asymmetrical scheme. Corresponding propagation lengths are shown on the right side of the Im[ρ] graph. The real part of the solution d(Re[ρ]) of the equation (7) is represented by hollow circles. The vertical dashed line represents df, which corresponds to ρ1/2 for this system.*

Again, to test this approximation and, in addition, to find the propagation length of this mode, we present the exact solution of the dispersion relation (6) with impedances (12) for the asymmetric system. The results are shown in Fig. 5. One can see that, as before, our approximation is very close to the exact solution near $\rho=\rho_{1/2}$.

This figure also shows that the propagation length has a noticeable maximum near the film thickness, which corresponds to $\rho_{1/2}$. This is due to the presence of a strongly damping metal (aluminium) in one of the duplex metal nanolayers. The profile of the electromagnetic field with a zero tangential component in the center of the aluminium nanolayer becomes noticeably more advantageous in this case.

The resulting spatial distribution of the components of the optical field in the final layers of the one-dimensional PC is shown in Fig. 6. Again, we see that the tangential component of the optical field is zero at the centers of the metal nanolayers $m_1$ and $m_2$ for the calculated film thickness $d_\mathrm{f}=164.66$ nm.

![The spatial distribution of the EM field components in the final layers of the PC (bordering the air), when the duplex LRSPs are excited in the asymmetrical scheme.](media/PhotonicsAndNanostructures2020/Fig6.eps)

*The spatial distribution of the EM field components in the final layers of the PC (bordering the air), when the duplex LRSPs are excited in the asymmetrical scheme.*

## Location of LRSP curves on dispersion plots

To be sure that long-range propagation in the structures with two metal nanolayers is possible only with an effective RI $\sim\!\rho_{1/2}$ (i.e., close to the light line), while other modes are experimentally impractical, it would be useful to calculate and present the general picture of the real and *imaginary* parts of the dispersion curves for structures under study. In order to obtain dispersion curves with practicable imaginary parts, we will use the experimental dispersion data [26] for complex RIs of metals (rather than the idealized Drude model) in the dispersion relation (6).

![(a,b) Dispersion curves for the symmetrical structure with one metal nanolayer, when the external environment is (a) air, and (b) a dielectric with RI nf (the dispersion of SPs on the surface of gold is given for reference). (c) Dispersion of the structure with duplex metal nanolayers, separated by the dielectric with RI nf, in air. The error bars of the dispersion curves are equal to the imaginary part of the effective RI.](media/PhotonicsAndNanostructures2020/Fig7.eps)

*(a,b) Dispersion curves for the symmetrical structure with one metal nanolayer, when the external environment is (a) air, and (b) a dielectric with RI nf (the dispersion of SPs on the surface of gold is given for reference). (c) Dispersion of the structure with duplex metal nanolayers, separated by the dielectric with RI nf, in air. The error bars of the dispersion curves are equal to the imaginary part of the effective RI.*

We start with the symmetrical scheme and, to get an insight into the genesis of the modes in the duplex metal structure, we shift the two metal nanolayers to each other and get one 24-nm nanolayer of gold in the air (see the color inset in Fig. 7(a)) or in a dielectric with RI $n_\mathrm{f}\sim1.86$ (see the color inset in Fig. 7(b)). It is well known that a metal nanolayer in symmetrical environment has two SP modes on both interfaces. When the thickness of the metal layer is small, these modes become coupled and shifted from the dispersion curve of SP on the only metal-dielectric interface.

Fig. 7(a) shows the dispersion curves of optical surface modes in 24-nm gold nanolayer in air. The dispersion curve of SPs on a single gold-air interface is also given as a reference. Hereafter, the error bars of the dispersion curves represent the imaginary part of the effective RI and are inversely proportional to the propagation length of modes (see (13)). It can be seen that the dispersion curves are shifted from the original SP curve (black) and form the long-range surface plasmon (LRSP) curve (red), and short-range surface plasmon (SRSP) curve (blue). One can see that only LRSPs are of practical interest, while the excitation of SRSPs (also as the excitation of ordinary SPs near SP resonance wavelength $\lambda_p$) is impractical due to their strong damping. Figure 7(b) is the same picture when dispersion-less air is replaced by a dielectric with RI $n_\mathrm{f}\sim1.86$.

![Dispersion curves of the structure with duplex metal nanolayers near the light line (nair) for (a) the symmetrical external environment, and for (b) the asymmetrical environment with a semi-infinite 1D PC on one side. The error bars of the dispersion curves are equal to the imaginary part of the effective RI.](media/PhotonicsAndNanostructures2020/Fig8.eps)

*Dispersion curves of the structure with duplex metal nanolayers near the light line (nair) for (a) the symmetrical external environment, and for (b) the asymmetrical environment with a semi-infinite 1D PC on one side. The error bars of the dispersion curves are equal to the imaginary part of the effective RI.*

When the gold nanolayer is split into two parts, and the width of the dielectric film between them becomes thick enough (in our case $d_\mathrm{f}=163.43$ nm), a guided mode appears, which expands from one light line ($n_\mathrm{f}$) to another ($n_\mathrm{air}$) – see a blue line in Fig. 7(c). Figure 7(c) represents the dispersion curves for the symmetrical structure under study: one can see that the guided mode (denoted as GM-2) forms an anti-crossing with the LRSP-1 mode. The locations of unperturbed curves are schematically shown by dashed lines, and a crossing point is shown by a dashed circle. Due to the repulsion of these dispersion curves, the mode LRSP-1 shifts to the light line and becomes the long-range propagating mode near $\lambda=575\,$nm.

An enlarged view of this area is shown in Fig. 8(a), while Fig. 8(b) shows the dispersion curves of asymmetrical structure in this area with $d_\mathrm{f}=164.66$ nm and with other parameters specified in subsection (3.2). It can be seen that the LRSP-1 mode also becomes a long-range propagating mode near $\lambda=575\,$nm due to the anti-crossing with the guided mode GM-2. Thus, from these dispersion plots it is seen that for both types of structures, only modes close to the light line can be long-range propagating modes, while other modes are experimentally impracticable due to their strong damping.

# Discussion

The obtained results allow the design of symmetrical and asymmetrical structures supporting LRSPs with an effective RI near the light line for any given wavelength. For the given above parameters, at $\lambda=575$ nm, the propagation length of duplex LRSPs is about 170 $\mu$m for the symmetrical scheme (see Fig. 3) or about 80 $\mu$m for the asymmetrical scheme with the 1D PC on one side (see Fig. 5) . These are the theoretical propagation lengths that consider only the internal damping of the LRSPs in the metals. If there is a prism near the surface for decoupling of the light from the structures (Kretschman or Otto configuration), the radiation loss would further decrease the propagation length.

Such propagation lengths would allow one to use of the metal nanolayers as electrodes for applying voltage to the film (if the film is an electro-optical crystal) or to inject current into the film (if the film is an organic electroluminescent film). There are several ways to incorporate a thin-film electro-optical crystal ($LiNbO_3$) as an upper layer into the 1D PC structure [27], but as far as we know, no experiments applying a voltage to such a crystal throughout its thickness have been carried out so far. One can also use an electro-optical polymer film (instead of a crystal) as an active element. An electro-optical polymer light modulator with a single metal nanolayer supporting LRSP was studied numerically in [28]. The approaches presented here make it possible to create a large electric field inside this thin-film crystal (or film) by applying moderate voltage on the metal electrodes.

The second application possibility is a current-driven light source with current injection into the electroluminescent film through the metal nanolayers. Until now there has been one proposal to use LRSPs in a current-driven semiconductor light source in which the first electrode is a metal nanolayer, while the second is a doped semiconductor layer (with an active layer containing quantum wells between them) [29]. The presented schemes with the two metal electrodes may be used also for a non-crystalline active region, e.g., an organic electroluminescent film.

# Conclusion

In this paper, we have presented formulas allowing one to design symmetrical and asymmetrical structures consisting of double metal nanolayers separated by an active dielectric film that support the propagation of long-range surface plasmons. The required film thickness is given by (7) for the symmetrical scheme and by (12) for the asymmetrical scheme with a 1D PC on one side. Possible practical applications of such schemes would be in waveguide-based electro-optical modulators or current-controlled light sources.

# 

The impedance of a semi-infinite 1D PC is [24]:
``` math
Z_{(PC)}=-\frac{i}{2} {\frac{\left( (  Z_{(2)}^{2}-  Z_{(1)}^{2} )\tan
(\alpha_{2})\tan(\alpha_{1})\pm\sqrt {{s}}\right)}
{{  Z_{(2)}} \tan(\alpha_{1})+Z_{(1)}\tan(\alpha_{2})}}  \,,
```
where
``` math
\begin{eqnarray}
s=-4 {  Z_{(1)}} {  Z_{(2)}} \left ({  Z_{(2)}} \tan({  \alpha_1})+{  Z_{(1)}} 
\tan({  \alpha_2})\right ) 
\nonumber \\ \qquad 
\left ({  Z_{(1)}} \tan({  \alpha_1})+{  Z_{(2)}}\tan({  \alpha_2})\right )+
 \nonumber \\ \qquad 
 \left [\left (Z_{(2)}^2-Z_{(1)}^{2}
\right )\tan({  \alpha_1})\tan({  
\alpha_2})\right ]^2
\nonumber
\end{eqnarray}
```
and $\alpha_j=k_{z(j)}\,d_j=\pm(2\pi/\lambda)n_j d_j \sqrt{1-(\rho/n_j)^2}$.

# Acknowledgment

This work has been performed as a part of the State assignment for the Institute of Spectroscopy of the Russian Academy of Sciences.

# References

10 url urlprefix href

H. Raether, Surface Plasmons, Springer, Berlin, 1988.

M. I. Stockman, K. Kneipp, S. I. Bozhevolnyi, S. Saha, A. Dutta, J. Ndukaife, N. Kinsey, H. Reddy, U. Guler, V. M. Shalaev, et al., Roadmap on plasmonics, Journal of Optics 20 (4) (2018) 043001.

D. Sarid, Long-range surface-plasma waves on very thin metal films, Phys. Rev. Lett. 47 (26) (1981) 1927–1930.

A. E. Craig, G. A. Olson, D. Sarid, Experimental observation of the long-range surface-plasmon polariton, Opt. Lett. 8 (7) (1983) 380–382.

P. Berini, R. Charbonneau, S. Jetté-Charbonneau, N. Lahoud, G. Mattiussi, Long-range surface plasmon-polariton waveguides and devices in lithium niobate, Journal of Applied Physics 101 (11) (2007) 113114.

O. Krupin, H. Asiri, C. Wang, R. N. Tait, P. Berini, Biosensing using straight long-range surface plasmon waveguides, Optics Express 21 (1) (2013) 698–709.

P. Berini, R. Charbonneau, N. Lahoud, Long-range surface plasmons on ultrathin membranes, Nano Lett. 7 (5) (2007) 1376–1380.

V. N. Konopsky, E. V. Alieva, Long-range propagation of plasmon polaritons in a thin metal film on a one-dimensional photonic crystal surface, Phys. Rev. Lett. 97 (25) (2006) 253 904.

E. Alieva, V. Konopsky, D. Basmanov, S. Sekatskii, G. Dietler, Blue surface plasmon propagation along thin gold film–gas interface and its use for sensitive nitrogen dioxide detection, Optics Communications 309 (2013) 148–152.

V. N. Konopsky, E. V. Alieva, Long-range plasmons in lossy metal films on photonic crystal surfaces, Opt. Lett. 34 (4) (2009) 479–481.

E. Economou, Surface plasmons in thin films, Physical Review 182 (2) (1969) 539.

K. Welford, J. Sambles, Coupled surface plasmons in a symmetric system, Journal of Modern Optics 35 (9) (1988) 1467–1483.

M. Gilmore, B. Johnson, Forbidden guided-wave plasmon polaritons in coupled thin films, Journal of Applied Physics 93 (8) (2003) 4497–4504.

L. H. Smith, M. C. Taylor, I. R. Hooper, W. L. Barnes, Field profiles of coupled surface plasmon-polaritons, Journal of Modern Optics 55 (18) (2008) 2929–2943.

N. Chen, C. Lu, Y. Huang, C. Liao, W. Ke, B. Huang, Properties of coupled surface plasmon-polaritons in metal-dielectric-metal structures, Journal of Applied Physics 112 (3) (2012) 033111.

G. Kovacs, G. Scott, Optical excitation of surface plasma waves in layered media, Physical Review B 16 (4) (1977) 1297.

P.-T. Wu, M.-C. Wu, C.-M. Wu, Measurement of the air gap width between double-deck metal layers based on surface plasmon resonance, Journal of Applied Physics 107 (8) (2010) 083111.

S. Ushioda, J. Rutledge, R. Pierce, Prism-coupled light emission from tunnel junctions, Physical Review Letters 54 (3) (1985) 224.

G. Stegeman, J. Burke, Long-range surface plasmons in electrode structures, Applied Physics Letters 43 (3) (1983) 221–223.

J. Yoon, S. H. Song, S. Park, Flat-top surface plasmon-polariton modes guided by double-electrode structures, Optics Express 15 (25) (2007) 17151–17162.

V. E. Babicheva, R. Malureanu, A. V. Lavrinenko, Plasmonic finite-thickness metal–semiconductor–metal waveguide as ultra-compact modulator, Photonics and Nanostructures-Fundamentals and Applications 11 (4) (2013) 323–334.

L. Brekhovskikh, Waves in Layered Media, Academic, New-York, 1980.

E. Delano, R. Pegis, Methods of syntesis for dielectric multilayer filters, in: E. Wolf (Ed.), Progress in Optics, Vol. VII, North-Holland, Amsterdam, 1969, Ch. 2, pp. 67–137 (see pp.77, 130).

V. N. Konopsky, Plasmon-polariton waves in nanofilms on one-dimensional photonic crystal surfaces, New J. Phys. 12 (2010) 093 006.

F. Yang, J. R. Sambles, G. W. Bradberry, Long-range surface modes supported by thin films, Phys. Rev. B 44 (11) (1991) 5855–5872. doi:10.1103/PhysRevB.44.5855.

E. D. Palik, Handbook of Optical Constants of Solids, Academic, London, 1985.

T. Kovalevich, D. Belharet, L. Robert, M.-S. Kim, H. P. Herzig, T. Grosjean, M.-P. Bernal, Experimental evidence of Bloch surface waves on photonic crystals with thin-film $\mathrm{LiNbO_3}$ as a top layer, Photonics Research 5 (6) (2017) 649–653.

X. Shi, S. Zheng, H. Chi, X. Jin, X. Zhang, A wideband electro-optic modulator based on long range surface plasmon resonances, Journal of Optics 13 (12) (2011) 125001.

V. Konopsky, Long-range surface plasmon amplification with current injection on a one-dimensional photonic crystal surface, Optics Letters 40 (10) (2015) 2261–2264.

## References

1. H. Raether, Surface Plasmons, Springer, Berlin, 1988.

2. M. I. Stockman, K. Kneipp, S. I. Bozhevolnyi, S. Saha, A. Dutta, J. Ndukaife, N. Kinsey, H. Reddy, U. Guler, V. M. Shalaev, et al., Roadmap on plasmonics, Journal of Optics 20 (4) (2018) 043001.

3. D. Sarid, Long-range surface-plasma waves on very thin metal films, Phys. Rev. Lett. 47 (26) (1981) 1927--1930.

4. A. E. Craig, G. A. Olson, D. Sarid, Experimental observation of the long-range surface-plasmon polariton, Opt. Lett. 8 (7) (1983) 380--382.

5. P. Berini, R. Charbonneau, S. Jett'e-Charbonneau, N. Lahoud, G. Mattiussi, Long-range surface plasmon-polariton waveguides and devices in lithium niobate, Journal of Applied Physics 101 (11) (2007) 113114.

6. O. Krupin, H. Asiri, C. Wang, R. N. Tait, P. Berini, Biosensing using straight long-range surface plasmon waveguides, Optics Express 21 (1) (2013) 698--709.

7. P. Berini, R. Charbonneau, N. Lahoud, Long-range surface plasmons on ultrathin membranes, Nano Lett. 7 (5) (2007) 1376--1380.

8. V. N. Konopsky, E. V. Alieva, Long-range propagation of plasmon polaritons in a thin metal film on a one-dimensional photonic crystal surface, Phys. Rev. Lett. 97 (25) (2006) 253,904.

9. E. Alieva, V. Konopsky, D. Basmanov, S. Sekatskii, G. Dietler, Blue surface plasmon propagation along thin gold film--gas interface and its use for sensitive nitrogen dioxide detection, Optics Communications 309 (2013) 148--152.

10. V. N. Konopsky, E. V. Alieva, Long-range plasmons in lossy metal films on photonic crystal surfaces, Opt. Lett. 34 (4) (2009) 479--481.

11. E. Economou, Surface plasmons in thin films, Physical Review 182 (2) (1969) 539.

12. K. Welford, J. Sambles, Coupled surface plasmons in a symmetric system, Journal of Modern Optics 35 (9) (1988) 1467--1483.

13. M. Gilmore, B. Johnson, Forbidden guided-wave plasmon polaritons in coupled thin films, Journal of Applied Physics 93 (8) (2003) 4497--4504.

14. L. H. Smith, M. C. Taylor, I. R. Hooper, W. L. Barnes, Field profiles of coupled surface plasmon-polaritons, Journal of Modern Optics 55 (18) (2008) 2929--2943.

15. N. Chen, C. Lu, Y. Huang, C. Liao, W. Ke, B. Huang, Properties of coupled surface plasmon-polaritons in metal-dielectric-metal structures, Journal of Applied Physics 112 (3) (2012) 033111.

16. G. Kovacs, G. Scott, Optical excitation of surface plasma waves in layered media, Physical Review B 16 (4) (1977) 1297.

17. P.-T. Wu, M.-C. Wu, C.-M. Wu, Measurement of the air gap width between double-deck metal layers based on surface plasmon resonance, Journal of Applied Physics 107 (8) (2010) 083111.

18. S. Ushioda, J. Rutledge, R. Pierce, Prism-coupled light emission from tunnel junctions, Physical Review Letters 54 (3) (1985) 224.

19. G. Stegeman, J. Burke, Long-range surface plasmons in electrode structures, Applied Physics Letters 43 (3) (1983) 221--223.

20. J. Yoon, S. H. Song, S. Park, Flat-top surface plasmon-polariton modes guided by double-electrode structures, Optics Express 15 (25) (2007) 17151--17162.

21. V. E. Babicheva, R. Malureanu, A. V. Lavrinenko, Plasmonic finite-thickness metal--semiconductor--metal waveguide as ultra-compact modulator, Photonics and Nanostructures-Fundamentals and Applications 11 (4) (2013) 323--334.

22. L. Brekhovskikh, Waves in Layered Media, Academic, New-York, 1980.

23. E. Delano, R. Pegis, Methods of syntesis for dielectric multilayer filters, in: E. Wolf (Ed.), Progress in Optics, Vol. VII, North-Holland, Amsterdam, 1969, Ch. 2, pp. 67--137 (see pp.77,130).

24. V. N. Konopsky, Plasmon-polariton waves in nanofilms on one-dimensional photonic crystal surfaces, New J. Phys. 12 (2010) 093,006.

25. F. Yang, J. R. Sambles, G. W. Bradberry, Long-range surface modes supported by thin films, Phys. Rev. B 44 (11) (1991) 5855--5872. doi:10.1103/PhysRevB.44.5855.

26. E. D. Palik, Handbook of Optical Constants of Solids, Academic, London, 1985.

27. T. Kovalevich, D. Belharet, L. Robert, M.-S. Kim, H. P. Herzig, T. Grosjean, M.-P. Bernal, Experimental evidence of Bloch surface waves on photonic crystals with thin-film $LiNbO_3$ as a top layer, Photonics Research 5 (6) (2017) 649--653.

28. X. Shi, S. Zheng, H. Chi, X. Jin, X. Zhang, A wideband electro-optic modulator based on long range surface plasmon resonances, Journal of Optics 13 (12) (2011) 125001.

29. V. Konopsky, Long-range surface plasmon amplification with current injection on a one-dimensional photonic crystal surface, Optics Letters 40 (10) (2015) 2261--2264. thebibliography.

---
Generated: 2026-08-30 from Z:\ValeryData\Valery_New\my_articles\PhotonicsAndNanostructures2020\PhotonicsAndNanostructures\manuscript\Konopsky2PhN.tex, validated against Z:\ValeryData\Valery_New\my_articles\PhotonicsAndNanostructures2020\PhotonicsAndNanostructures\published\Long-range surface plasmons on duplex metal nanolayers_Konopsky_2020.pdf.