Transmission of surface plasmon polaritons through a nanowire array: mechano-optical modulation and motion sensing

1. Introduction

Recently, plasmonics has made a great step toward the design of novel photonic devices [1,2]. The energy of the SPP, which is a light wave coupled to free electron oscillations in a metal, is strongly localized near the interface between a metal film and an insulator. The amplitude of the electromagnetic field decays exponentially with the distance from the interface that allows to create tiny devices for optical data processing and use SPPs in a number of sensing applications [3,4]. Unfortunately, the propagation length of SPPs usually does not exceed several tens of micrometers. None the less, it was shown for thin metal film waveguides that the propagation length tends to infinity as the film thickness decreases [5] and we are able to achieve actually long-range SPPs (LRSPPs) [6,7]. The next step was to control the intensity and phase of the SPP. A number of methods to do this have been proposed, including all-optical modulation [8–11]. All those techniques are pure optical, i.e. an applied voltage, an incident optical radiation, etc. change optical properties, e.g. the refractive index or polarization tensor, of the materials used. Here, a different technique is proposed.

We demonstrate how to control the intensity of the SPP by means of the array of nanowires placed above the metal film. Typical dimensions of nanowires are 5-100 nm in diameter (width) and 1-50 μm in length. Such a small size, combined with unique electrical, mechanical and optical properties, has attracted interest in the scientific community for their potential in different applications from microelectronic to nanooptics. We focus here on the mechanical properties of nanowires [12]. There are two main reasons for this. Firstly, the resonance frequency of mechanical oscillations is usually in the kilohertz or megahertz range and the amplitude of oscillations may exceed ten micrometers [12–14]. Secondly, nanowire cantilevers are very sensitive and can be used even for single-atom mass sensing and tiny force measurements [15–17]. The above features of nanowires allow to design a mechano-optical modulator (Fig. 1 ). If the radiation loss power is sufficiently small (less than 10%), the coupled-mode theory can be used to characterize the modulator. Such an approach gives a possibility to avoid time-consuming numerical simulations, obtain an analytical solution and analyze it.

 

Fig. 1 Schematic operation of the mechano-optic modulator, β is the SPP wavevector, h ₁>h ₂. (a). If the distance h between the nanowire array and the metal film is very large, there is no interaction between the SPP and the nanowires and we do not have any effect. (b). When the distance decreases, the effect of the nanowire array may be considered as a perturbation Δε of the dielectric constant of the waveguide and the coupling between guided and radiation modes occurs. Changing the distance h, one can control the intensity of the transmitted SPP.

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2. SPP dispersion (general relations)

In our paper, we discuss only the air-metal-air waveguide structure. This is due to two reasons: 1) we deal with the mechanics of nanowires and therefore the space for the wires to move is required and 2) we want to use LRSPPs, hence the waveguide structure should be symmetric because, for most asymmetric structures, it is impossible to excite LRSPPs at a wavelength of 800 nm or 1550 nm. In addition, we assume that the metal is lossless. It is a good approach, since the propagation length of LRSPPs is much greater than the longitudinal size of the nanowire array. The geometry of a thin metal film waveguide is presented in Fig. 2(a) . The structure under consideration supports two types of modes: guided modes and radiation modes. Radiation modes are characterized by the continuous spectrum, while guided modes by the discrete one [Fig. 2(b)].

 

Fig. 2 (a) Sketch of a thin metal film waveguide. The guide axis is chosen to coincide with the x-axis, the core is the metal with the dielectric function ε ₁. Also shown is a spatial distribution of the longitudinal electric field magnitude of guided and radiation modes (notations “S” and “AS” correspond to the symmetric and antisymmetric modes, respectively). (b). Dispersion curves of the SPP. The blue line corresponds to the symmetric guided mode, the red line to the antisymmetric guided mode. Radiation modes are placed in the region “RR”. The metal is assumed to be described by the general Drude model${\epsilon }_1={\epsilon }_{\text{r}}-{\omega }_{\text{p}}^2/{\omega }^2,$where ${\epsilon }_{\text{r}}$ is the high frequency dielectric constant and ${\omega }_{\text{p}}$ is the plasma frequency.

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If $\beta >k,$ one should solve the dispersion equation [18,19]

Here, ${\kappa }_1=\sqrt{{\beta }^2-k^2{\epsilon }_1},$ ${\kappa }_2=\sqrt{{\beta }^2-k^2}$ (β is the SPP wavenumber) and the notation $k=\omega /c$ is used for the sake of brevity. The plus sign in Eq. (1) corresponds to the antisymmetric guided mode and the minus sign to the symmetric guided mode. By the symmetry we mean that the amplitude of the longitudinal electric field $(E_x)$ does not exhibit a zero inside the film – conversely, the antisymmetric guided mode has a zero [Fig. 2(a)], i.e. the symmetry of charges is supposed. According to the above definition, LRSPPs have an antisymmetric distribution of the longitudinal electric field, whereas the complex amplitude of the magnetic field is written as

The complex amplitude of the magnetic field of the symmetric guided mode is expressed as

If $\beta <k,$ ${\kappa }_2$ becomes imaginary $({\kappa }_2=i{\chi }_2)$ and the field does not decay exponentially with the distance from the film. Solving Maxwell equations [20,21], we get for the antisymmetric radiation mode:

3. Coupled-mode equations

Coupled-mode theory is well suited for the description of the interaction of the SPP with the nanowire array. This theory is well described in Ref [20–23], therefore we concentrate only on most important points. At first, we have to normalize modes. Since the radiation losses interest us, the amplitude coefficients should be related to the power carried by the mode. For each pair ν and μ of guided modes, we require [20] $c/(2\pi )\times {\displaystyle {\int }_{-\infty }^{+\infty }-E_z^{\nu }(z)H_y^{\mu *}(z)\text{\hspace{0.17em} d}z={\delta }_{\nu ,\mu }},$ and for each pair of radiation modes: $c/(2\pi )\times {\displaystyle {\int }_{-\infty }^{+\infty }-E_z^{\nu }(\beta ,z)H_y^{\mu *}({\beta }^{\prime },z)\text{\hspace{0.17em} d}z={\delta }_{\nu ,\mu }\delta (\beta -{\beta }^{\prime })},$ where ${\delta }_{\nu ,\mu }$ is the Kronecker symbol and $\delta \left(\beta \right)$ is the Dirac delta-function.

Now we can obtain normalized field amplitudes using the exact equations for the power flow [24]. For the antisymmetric guided mode, we have:

Here, the subscript indicates the guided (“g”) or radiation (“r”) mode. Assuming β to be positive, introduce now amplitude coefficients A (for the forward-running mode $(\beta >0)$) and B (for the backward-running mode $(\beta <0)$), so that

In general case, the system of coupled mode equations is written as [20,21]

We have an infinite number of differential equations, since we have an infinite number of radiation modes. To solve the problem, we have to simplify Eq. (12). Assume the relative change of amplitude $A_{\text{g}}^{\text{AS}}$ of the transmitted SPP to be much less than unity $\left(\left[A_{\text{g}}^{\text{AS}}(0)-A_{\text{g}}^{\text{AS}}(Nl)\right]/A_{\text{g}}^{\text{AS}}(0)\ll 1\right)$, i.e. assume a weak coupling regime $\left(\left|A_{\text{g}}^{\text{AS}}\right|\gg \left|A_{\nu }\right|,\left|B_{\nu }\right|\right)$, and only one amplitude coefficient remains in the right-hand part of Eq. (12) [21] (Ch. 3, Sec. 3.4):

Integration of Eq. (13) gives the exact expressions for the amplitude coefficients A and B:

Also, there is a coupling between the guided symmetric and guided antisymmetric modes and a coupling between the forward-running and backward-running guided antisymmetric modes and about $\left({\left|A_{\text{g}}^{\text{S}}(Nl)\right|}^2+{\left|B_{\text{g}}^{\text{S}}(0)\right|}^2+{\left|B_{\text{g}}^{\text{AS}}(0)\right|}^2\right)$ of the initial energy is lost. Usually, in the case of the nanowire array, these kinds of losses are neglected in comparison with the radiation losses.

4. Transmission of SPPs through a nanowire array

4.1 Nanobelt array

The coupled-mode theory gives a possibility to consider nanowires with arbitrary parameters. Nevertheless, we start with the analysis of nanowires with a rectangular cross-section (nanobelts). The reason for this is that an exact analytical solution can be found and major properties may be analyzed. So the geometry under consideration consists of N identical nanobelts placed above the Au film of thickness d = 2a = 25 nm, they are separated by a distance l from each other, the width of each belt is u and the height is b (Fig. 3(a) ). The permittivity of nanobelts ${\epsilon }_{\text{d}}$ equals 3.84 that corresponds to ZnO at a light wavelength of 800 nm.

 

Fig. 3 (a). Sketch of the structure under investigation (the nanobelt array near the Au film). (b). Spatial spectrum of the function f(x), $F(\beta )=\left|\widehat{f}(\beta )\right|$, ${\beta }_{\text{g}}$ is the wavenumber of the guided antisymmetric mode (superscript “AS” is omitted for the sake of brevity). (c). Dependence of the coupling coefficients on β. (d). Radiation loss power spectrum, β is assumed to be positive, so all curves are located in the right-hand part of the plot. For all figures, the following parameters are used: h = 500 nm, u = b = 80 nm, l = 500 nm, N = 10, 2πc/ω = 800 nm.

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Coupling coefficients $K_{\eta }^z(\beta ,x)$ and $K_{\eta }^x(\beta ,x)$ become periodic functions of x for $x\in \left[0,Nl\right]$ and are zero elsewhere, i.e. they can be represented as $K_{\eta }^z(\beta ,x)=K_{\eta }^z(\beta )f(x)$ and $K_{\eta }^x(\beta ,x)=K_{\eta }^x(\beta )f(x).$ The spatial spectrum of the function f(x) has the form (Fig. 3(b))

Here, $\mathrm{\Delta }\epsilon ={\epsilon }_{\text{d}}-1$, if $a+h<z<a+h+b,$and$\mathrm{\Delta }\epsilon =0$otherwise, the subscript η indicates the radiation symmetric (“rS”) or radiation antisymmetric (“rAS”) mode and $E_z^{\eta }(\beta ,z)$ $(E_x^{\eta }(\beta ,z))$ is the complex amplitude of the normalized electric field of that mode, $E_z^{\text{gAS}}(\beta ,z)$ $(E_x^{\text{gAS}}(\beta ,z))$ corresponds to the normalized electric field of the guided antisymmetric mode. Finally, the spatial spectrum of $K_{\eta }^z(\beta ,x)$ $(K_{\eta }^x(\beta ,x))$ is the product of $\widehat{f}(\beta )$ and $K_{\eta }^z(\beta )$ $(K_{\eta }^x(\beta ))$. Using Eq. (15), obtain the radiation loss power spatial spectrum [Fig. 3(d)]. The maximum of radiation is at $\beta ={\beta }_{\text{max}}\approx 4.8\times {10}^4{\text{cm}}^{-1}$that is very close to the position of the maximum of the function $F(\beta +{\beta }_{\text{g}}^{\text{AS}})=\left|\widehat{f}(\beta +{\beta }_{\text{g}}^{\text{AS}})\right|.$ At the first sight, the second maximum is expected at $\beta ={\beta }_{\text{g}}^{\text{AS}}$ but $K_{\eta }^z(\beta )$ $(K_{\eta }^x(\beta ))$ rapidly goes to zero as β goes to ${\beta }_{\text{g}}^{\text{AS}}$ [Fig. 3(c)] and so this maximum is far less than the first one. The total radiation loss power equals 6.8% and is mainly related to the backward-running radiation modes.

4.2 Circular nanowires

To consider nanowires with a circular cross-section [Fig. 4(a) ], one should again use Eq. (15). The radiation loss power spectrum is shown in Fig. 4(b). It is not surprising that the shape of the spectrum is the same as that in Fig. 3(d). This is because the diameter of nanowires (the width and height of belts) is much less than the SPP wavelength $2\pi /\beta ,$ the period of the structure l is compared with $2\pi /\beta$ and N is sufficiently large (N = 10), therefore the width of the maximum $\mathrm{\Delta }{\beta }_{\max }$ is mainly determined by the length of the nanowire array Nl. Using a simple formula $\mathrm{\Delta }{\beta }_{\max }\mathrm{\Delta }x\approx 2\pi ,$ we get $\mathrm{\Delta }{\beta }_{\max }\approx 1.3\times {10}^4{\text{cm}}^{-1}$ but $R(\beta )\propto {\left|A(\beta )\right|}^2$ and we should divide $\mathrm{\Delta }{\beta }_{\max }$ by 2 to estimate it. Finally, $\mathrm{\Delta }{\beta }_{\max }\approx 9\times {10}^3{\text{cm}}^{-1}$ for both circular nanowires [Fig. 4(b)] and nanobelts [Fig. 3(d)]. As for the position of the maximum ${\beta }_{\max },$ it is approximately equal to $({\beta }_{\text{g}}^{\text{AS}}-2\pi /l)\approx 4.8\times {10}^4{\text{cm}}^{-1}$ in both cases. The total radiation loss power (4.3%) is less than for the nanobelt array (6.8%), since the cross-sectional area $\pi r^2\approx 5.0\times {10}^{-11}{\text{cm}}^2$ is less than in the case of nanobelts $(ub\approx 6.4\times {10}^{-11}{\text{cm}}^2).$

 

Fig. 4 (a). Sketch of the nanowire array above the Au film. (b). Spatial spectrum of the radiation loss power for the array of nanowires with a circular cross-section, r = 40 nm, h = 500 nm, l = 500 nm, N = 10, 2πc/ω = 800 nm. The normalized total radiation loss power is equal to 4.3%.

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4.3 Modulation of the intensity of the transmitted SPP

Consider now the dependence of the total radiation loss power on the gap between the nanowire array and the metal film. Since there is no significant difference between nanobelts and circular nanowires in the radiation loss spatial spectrum, we again move to the nanobelt array for the sake of simplicity. Consider the expression for the coupling coefficient $K_{\eta }^z(\beta )$ ($K_{\eta }^x(\beta )$ is discussed in the same manner):

Here, η again should be replaced by “rAS” or “rS”, depending on the mode considered, and ${\kappa }_2^{\text{gAS}}=\sqrt{{({\beta }_{\text{g}}^{\text{AS}})}^2-k^2}$. We have an exponentially decaying term $\exp (-{\kappa }_2^{\text{gAS}}h),$ consequently, the radiation loss power decreases as h increases. However, there are oscillating terms in braces and therefore oscillations in $K_{\eta }^z(h,\beta )$ as a function of h are expected. The form of the dependence can be approximately expressed as $\exp (-{\kappa }_2^{\text{gAS}}h)\left[1+\xi \cos ({\chi }_2^{\eta }h+\theta )\right]$ $(\left|\xi \right|<1).$

Results of direct calculations show that it is possible to achieve a modulation depth of 5% with only 60 nm amplitude of mechanical oscillations for u = b = 80 nm (Fig. 5 ). Such a high sensitivity of the system can be used for the detection of the mechanical motion of nanowires. Moreover, if the intensity of the incident SPP is sufficiently low, optical forces, which act on the nanowires, are negligible and the SPP does not affect the nanowire motion, therefore precise measurements can be performed.

 

Fig. 5 Dependence of the normalized total radiation loss power on the gap between the nanobelt array and the metal film for different values of the nanobelt transverse size, l = 500 nm, N = 10, 2πc/ω = 800 nm.

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It should be noted that the modulation depth is bounded above (Fig. 5) that follows from Eq. (19) and can be easily explained. The total radiation loss power is proportional to the coupling coefficient $K_{\eta }^z(h,\beta )$ that contains expression $\mathrm{\Delta }\epsilon E_z^{\text{gAS}}(z)E_z^{\eta \text{*}}(\beta ,z)$ under the integral sign. Note that $b<<1/{\kappa }_2^{\text{gAS}},$ $b<<2\pi /{\chi }_2^{\eta }$ and $z>a,$ hence ${\displaystyle \int \mathrm{\Delta }\epsilon E_z^{\text{gAS}}(z)E_z^{\eta \text{*}}(\beta ,z)\text{d}z}\approx b({\epsilon }_{\text{d}}-1)E_z^{\text{gAS}}(h)E_z^{\eta \text{*}}(\beta ,h)$ where $E_z^{\text{gAS}}(z)$ is an exponential function and $E_z^{\eta }(\beta ,z)$ is a periodic function of z. The periodicity of the array gives a possibility to take into account only a narrow interval of β [Fig. 2(d)] where $\beta \approx const={\beta }_{\text{max}}$. Thus, the total radiation loss power is approximately proportional to $E_z^{\text{gAS}}(h)E_z^{\eta \text{*}}({\beta }_{\text{max}},h)\approx const\times \exp (-{\kappa }_2^{\text{gAS}}h)\left[1+\cos ({\chi }_2^{\eta }h+\theta )\right]$ that is in a good agreement with the exact dependence (Fig. 5).

To increase the maximum possible modulation depth, one should: (1) increase the number of nanowires [Fig. 6(a) ], (2) use high-permittivity nanowires, e.g. semiconductor nanowires [Fig. 6(b)], (3) increase the width, height or radius of nanowires [Fig. 6(a), 6(b)].

 

Fig. 6 (a). Dependence of the normalized total radiation loss power on the number of nanobelts, h = 500 nm, l = 500 nm, N = 10, 2πc/ω = 800 nm. (b). Dependence of the normalized total radiation loss power on the permittivity of the nanobelts, h = 500 nm, l = 500 nm, N = 5, 2πc/ω = 800 nm.

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The use of high-permittivity nanowires with a large cross-section area may degrade mechanical properties of the system, e.g. decrease the resonant frequency of mechanical oscillations. Thus, one should reach a compromise between the speed and the sensitivity. A comprehensive study of this problem is the goal of our future work.

5. Conclusion

In conclusion, we have brought together the mechanics of nanowires and the guiding properties of plasmonic waveguides, proposed a compact machano-optical modulator and characterized it analytically with the help of the coupled-mode theory. The modulator is based on a nanowire array placed above a thin metal film. The intensity of the SPP is modulated by changing the distance between the nanowire array and the film. The maximum possible modulation depth depends strongly on the number of nanowires and their parameters and thus it can be varied in a wide range. The longitudinal size of the device is of the order of a few micrometers and depends on the number of nanowires and the distance between them, while the transverse size is dictated by the length of nanowires. Moreover, the proposed technique can be used for the detection of the mechanical motion of nanowires and for the measurement of their oscillation amplitude.