Design of metallic nanocylinder array waveguide for controlling resonant wavelength shift

A metallic nanocylinder array was previously proposed for the propagation of localized light generated in nanosized objects. However, the plasmon resonant wavelength for each cylinder shifts significantly with the propagation of light. In this study, a metallic nanoelliptic cylinder waveguide is investigated to control the plasmon resonant wavelength shift. The finite-difference frequency-domain (FDFD) electromagnetic solver is used to reduce the computational time. The relationship between the cross-section of the cylinder and the plasmon resonance for the metal is described with the hydrodynamic Drude model. The effectiveness of the proposed waveguide in controlling the resonant wavelength shifts is confirmed by examining the plasmon propagation for 30 nanocylinders.

✉ Email: ohnuki.shinichiro@nihon-u.ac.jp A metallic nanocylinder array was previously proposed for the propagation of localized light generated in nanosized objects. However, the plasmon resonant wavelength for each cylinder shifts significantly with the propagation of light. In this study, a metallic nanoelliptic cylinder waveguide is investigated to control the plasmon resonant wavelength shift. The finite-difference frequency-domain (FDFD) electromagnetic solver is used to reduce the computational time. The relationship between the cross-section of the cylinder and the plasmon resonance for the metal is described with the hydrodynamic Drude model. The effectiveness of the proposed waveguide in controlling the resonant wavelength shifts is confirmed by examining the plasmon propagation for 30 nanocylinders.
Introduction: In recent years, surface plasmons generated on nanosized metals have been applied to waveguides and antennas because they allow the localization of light in the vicinity of metals, beyond the diffraction limit [1][2][3][4][5][6].
A metallic cylinder array was investigated as a plasmon device for the propagation of localized light generated in nanosized objects [7]. However, the plasmon resonant wavelength for each cylinder shifts significantly with the propagation of light. If the localized plasmon can only be excited at specific wavelengths, then the propagation distance can be reduced. Therefore, it is desirable to design a structure that can suppress and control the resonant wavelength shift. In this study, a metallic elliptic nanocylinder array waveguide, featuring a controllable resonant wavelength shift with light propagation, is investigated. First, the propagation characteristics of a circular cylinder array are elucidated from the wavelength response. To clarify the relationship between the crosssection of the cylinder and the plasmon resonant wavelength, the electric scattering field for an elliptic cylinder is analyzed. Using the characteristics of the elliptic cylinder, an elliptic cylinder array waveguide is examined for controlling the resonant wavelength shift. Furthermore, the plasmon propagation characteristics of a long cylinder array, comprising 30 nanocylinders, are discussed.
Finite-difference frequency-domain method: Herein, the steady-state plasmon propagation characteristics are evaluated, using the finitedifference frequency-domain (FDFD) electromagnetic solver [8][9][10][11] to reduce the computational time. For a detailed understanding of the plasmon resonance characteristics, the medium of the metallic nanocylinder is modelled based on the hydrodynamic Drude model [12][13][14][15][16][17]; this model can be used to predict the response of plasmonic devices reliably because it considers the nonlocal effects, which are important in nanostructures.
In the FDFD, the time dependence is assumed to be of the form-e jωt , where j is the imaginary unit, ω is the angular frequency, and t is time. Maxwell's equations in the frequency domain can be rewritten as follows [18]: Here, E is the electric field intensity, H is the magnetic field intensity, P is the polarization, J is the impressed current density, and ε 0 and μ 0 are the permittivity and permeability in the vacuum, respectively. To incorporate the polarization P, the hydrodynamic Drude model of a metallic where β is expressed in terms of the Fermi velocity, γ is a parameter that represents decay due to electron collision, and ω p is the plasma frequency. The nonlocal effect of electrons is expressed by the first term in (3). This effect is important for the reliable verification of plasmon resonance because it can represent the diffusion of electrons over the nanostructured surface. The central difference is applied to the spatial differentiation term in the wave equation obtained by (1) and (2) and the equation of motion of an electron (3). The entire computational space is discretized using the Yee cell; the linear equation form,Āx = b, can be obtained. Here,Ā is the operator matrix, x is the unknown vector that describes the unknown electric field and polarization, and b is the electric current source. The spatial distribution of the electric field and polarization can be obtained by solving the linear equation, using the FDFD method in Matlab.
Computational result: Figure 1 shows the computational model, wherein the cylinder array is placed inside the vacuum. The computational domain is enclosed in a convolutional perfectly matched layer (CPML) and discretized using small square cells of size 0.1 nm. The cross-section of the cylinder is an ideal circle with radius r = 5 nm, and the distance d between the cylinders is 10 nm. The cylinder, made of gold, is modelled by the staircase approximation and its nonlocal property is represented by the hydrodynamic Drude model. The light source is assumed to be an electric dipole source with a radius same as that of the cylinder, to consider localized surface plasmon. The plasmon excitation mode is assumed to be the longitudinal mode (θ = 90°), wherein the dipole source is parallel to the direction of the cylinder array. The incident wavelength is varied from 400 to 600 nm. The observation point A is between the second and third cylinders, and the observation point B is between the sixth and seventh cylinders. Figure 2 shows the electrical fields normalized with the maximum values. The electric fields at points A and B exhibit peaks at 530 and 558 nm, respectively. Evidently, the resonant wavelength shifts to the longer wavelength side with plasmon propagation. Here, we investigate the origin of the resonance wavelength shift. The interaction between the charges on different cylinders can be explained by the spring model. The strength of these 'springs' is determined by the strength of the interaction, which is also affected by cylinders other than the nearest ones. Therefore, the resonance wavelength changes because the spring coefficient changes depending on the location.
To clarify the relationship between the cross-section of the cylinder and the resonant wavelength, the electric fields for an elliptic cylinder Here, the direction of the major axis changes with respect to the incident electric field. The incident electric field and major axis are in the same direction for Pattern 1 and orthogonal for Pattern 2. Figure 4 shows the wavelength response for the electric field at the observation points, which are parallel and perpendicular to the major axis in Pattern 1 and Pattern 2, respectively. As the length of the major axis increases, the resonant wavelength shifts towards longer wavelengths in Pattern 1 and towards shorter wavelengths in Pattern 2. By changing the cross-section of the cylinder, to an ellipse, and the direction of the major axis, with respect to the incident electric field, the resonant wavelength can be controlled for each short and long wavelength. Figure 5 shows the computational model comprising elliptic cylinders. The plasmon excitation mode is assumed to be the longitudinal mode (θ = 90°). The major axis is perpendicular to the direction of the cylinder array, and it is gradually increased by r, which is the difference between the radii of the cylinders in the array. The observation points are the same as those in the case of the circular cylinder array. Figure 6 shows the electric fields for r = 1 nm. For the longitudinal mode (with r = 1 nm), the resonant wavelengths at points A and B (538 and 540 nm, respectively) are approximately equal. Figure 7 shows the electric fields for the longitudinal modes for r = 2 nm. For this case, the resonant wavelengths at points A and B (534 and 528 nm, respectively) differ significantly from those for r = 1 nm. As the major axis of the elliptic cylinder increases, the resonant wavelength at observation Here, changing the shape corresponds to adjusting the spring coefficient, which is determined by the interaction between the cylinders. Another method for controlling the resonance wavelength is to change the distance between the cylinders. In addition, it has been reported that the resonance wavelength can be controlled by coating the metal with dielectric material [19]. Thus, by combining these methods, it is possible to relax the constraint ( r = 1 nm). Figure 8a and b show the electric field distributions for an ideal cylinder array with an incident wavelength of 530 nm and an elliptic cylinder with the incident wavelength of 534 nm, respectively. It can be seen that when the cross-section is an ideal circle, the plasmon resonant wavelength shifts significantly with propagation and cannot be controlled. The electric field around the eighth cylinder decreases. For an elliptical cross-section, the plasmon resonant wavelength shift is suppressed, and the electric field around all the cylinders is higher than that around the cylinders of an ideal cylinder array.
To validate the ability of the elliptic cylinder array to control the plasmon resonant wavelength shift for long propagation, the electric fields near 30 cylinders (Figure 9) are investigated. The observation planes are considered to be at the centre of the cylinders, and the excitation mode  Figure 10 shows the wavelength response of the electric field for the circular cylinders at the observation planes. The electric field intensity is computed by the average of the wavelength response at the observation plane. For an ideal circular crosssection, the resonant wavelength that facilitates the high electric field varies with the plasmon propagation. Here, the resonant wavelengths of the fifth and tenth cylinders are 550 and 565 nm, respectively. It is necessary to select the resonant wavelength with the highest propagation efficiency, according to the propagation distance, because the resonant wavelength is shifted. Figure 11 shows the wavelength response of the electric field for the elliptic cylinders at the observation planes, for r = 1 nm. The resonant wavelength shift is suppressed for the 30 cylinders. If the incident wavelength is fixed at approximately 540 nm, the propagation efficiency is observed to be always high after the fifth cylinder.
Conclusion: In this study, a metallic elliptic nanocylinder array waveguide is proposed for controlling the plasmon resonant wavelength shift. The relationship between the cross-section of the cylinder and the plasmon resonant wavelength is clarified and exploited in the proposed waveguide. The plasmon resonant wavelength shift can be controlled by increasing the major axis of the elliptic cylinder array. To determine the long-distance propagation characteristics, the plasmon resonance is investigated for 30 nanocylinders. The elliptic cylinder array was found to propagate light more efficiently when the wavelength of the excited light was predetermined. In the future, it will be necessary to examine the propagation of local plasmons, with controlled resonant wavelengths, by this waveguide. In addition, the wavelength shift for the first few cylinders will be evaluated by changing the shape, position, and spacing between the cylinders.