Theoretical analysis of Bloch mode propagation in an integrated chain of gold nanowires Download: 845次
1. INTRODUCTION
Localized surface plasmons (LSPs) are surface electromagnetic modes associated with the collective oscillations of the conducting electrons at the boundaries of metallic nanoparticles (MNPs) [1]. They have been extensively studied in recent years due to their potential applications in optical sensing nanodevices [2]. This is because the resonance wavelength of the LSP is highly dependent on the geometry of the particle and on the refractive index of the surrounding medium. Furthermore, LSP modes provide a highly confined electromagnetic field that can be used to probe a very small volume of matter [3].
Light propagation through periodic arrays of MNP or metallic nanowires (MNWs) has already been demonstrated in several previous works [4
The coupling effect can be well understood with an eigenmode scrutiny of the plasmonic modes in the MNP chain. However, the large majority of the analysis has been done above the light line in the dispersion relations [9–
In order to study the MNP mode excitation below the light line limit, MNP chains integrated on top of a dielectric waveguide have been proposed [1416" target="_self" style="display: inline;">–
Here we study the modes supported by a periodic array of MNWs below the light line in the dispersion relation and their excitation with the fundamental transverse magnetic (TM) mode of a dielectric waveguide placed in close proximity of the MNW array. To do so, we first analyze the isolated MNW chain embedded in a homogeneous dielectric medium, then on top of a dielectric substrate, and finally approaching a dielectric waveguide. The analysis was done by using the Fourier modal method (FMM) for nonperiodic structures.
The paper is organized as follows: after a general presentation of the theoretical model, eigenmode dispersion relations are computed and analyzed for a MNW chain embedded in three different background media: (1) in a homogeneous glass medium, (2) on top of a silicon nitride substrate, and (3) on top of a silicon nitride waveguide. Finally, we deeply study a practical implementation of the latter structure with calculations on beam propagation and spectral responses.
The numerical results show that there is a mode coupling effect between the dielectric and the plasmonic waveguide, an effect that could be applied in the design of an integrated plasmonic sensing device.
2. NUMERICAL METHOD
The numerical method developed in this work is grounded on the so-called FMM, also known as rigorous coupled wave analysis (RCWA), which relies on a rigorous electromagnetic model based on the description of the Maxwell equations in the frequency domain. It is based on the Fourier series expansions of the dielectric function and the electromagnetic field [19]. With this formulation is possible to find the eigenmodes supported by a chain of MNPs of any shape immersed in a multilayered medium (its dispersion relations) and also to simulate the beam propagation through the structure. By making use of the effective index method, it is also possible to extend the numerical model into a three-dimensional simulation.
The general solution of this method involves two main steps: Calculation of the eigenvalues and the eigenvectors of a matrix with constant elements that characterizes the diffracted wave propagation and coupling (the eigenvectors represent the characteristic modes of the periodic array) in a profile of the propagation axis, as well as the effective indices corresponding to these modes; The resolution of a linear system deduced by the boundary conditions for normal and tangential components of the electric and magnetic fields to reconstruct the total field.
The characteristic matrix of the multilayered structure is the product of the characteristic submatrices in each layer. A detailed description of this method can be found in the work published by Chateau and Hugonin [19] and also in other references [20–
2.4 A. Dispersion Relations
Since the multilayered structure to be analyzed includes a periodic array of metallic nanowires, it supports Bloch electromagnetic eigenmodes. These modes can be calculated by solving the general solution of the Maxwell equations in the frequency domain as a sum of propagative and counterpropagative plane waves. Taking into account the invariance of the structure along the
Fig. 1. (a) Schematic representation of a unitary cell used for the calculation of the dispersion relations as an eigenvalue problem. The periodicity is along the axis. (b) Scheme for the calculation of the beam propagation. A unitary cell contains a finite number of nanowires along the propagation direction ( axis), and the periodicity is now along the axis including PMLs.
It must be noted that as we are working in the frequency domain for a single section of the structure, the Bloch modes are the solution of an infinite number of nanowires in the reciprocal space, bounded in the transversal direction.
2.5 B. Beam and Mode Propagation
For the propagation of the field under a particular guided mode excitation, we make use of a similar procedure than that used before, but in this case we compute the modes perpendicularly to the propagation axis. In other words, the periodicity of the structure and the Fourier transform are taken along the
The structure is subdivided into several sections invariant along the propagation direction. Then we include perfectly matched layers (PMLs) at the top and bottom of these sections in order to absorb the light scattered by the section and to avoid any reflections at the edges of the finite computational window of size
To normalize the absorption in Eq. (
3. NUMERICAL EXAMPLES
In order to validate the numerical method and to show its implementation in a stratified system, we present the results of the modal analysis for three cases. The first [Fig.
Fig. 2. Schemes of (a) the periodical array of gold nanowires immersed in a homogeneous dielectric medium with refractive index . The height of the nanowires is , the width is , and the period is . (b) The same MNW chain on a substrate of refractive index , and (c) an integrated structure of MNW on a dielectric waveguide with core index .
3.3 A. Isolated MNW Chain
The calculated dispersion relation of the MNW array immersed in a homogeneous medium with refractive index
Fig. 3. (a) Dispersion relations for the quadrupolar (upper branch) and dipolar transversal (lower branch) Bloch modes. The propagation distance of the (b) quadrupolar branch is shorter than that of the (c) dipolar transversal mode.
To identify the nature of each eigenmode branch, we calculated the energy density maps and their corresponding electric field lines distribution at the edge of the first Brillouin zone (at the Bragg condition) (Fig.
Fig. 4. Energy density maps and electric field distribution at the Bragg condition for (a) the quadrupolar Bloch mode at and (b) the dipolar transversal Bloch mode at . The corresponding squares show the phase distributions and orientation of the charges.
The propagation distance (
We must remark that when the height of the nanowires is decreased, for example at
3.4 B. MNW Chain on a Substrate
When the MNW array is placed on a dielectric substrate (
Fig. 5. (a) Dispersion relation of the MNW chain on a dielectric substrate ( , ). The top and bottom branches belong to the quadrupolar and transversal Bloch modes, respectively. The middle branch corresponds to the excitation of the SPP at the interface between the metallic nanowires and the substrate. Energy density maps and electric field distributions at the Bragg condition for (b) the quadrupolar mode at , (c) the SPP-like mode at , and (d) the dipolar transversal mode at . The charge distribution in (c) exhibits a dipolar longitudinal interaction between the MNW only at the metal–substrate interface.
The middle branch belongs to the excitation of a surface plasmon polariton (SPP)-like mode. This is because the applied field induces longitudinal dipoles at the metal–substrate interface, and since the separation of the MNWs is smaller than their width (
The previous results suggest that if a dielectric waveguide is approached to the structure, it is possible to have a directional coupling between the modes of the MNW chain and the modes of the dielectric waveguide, as will be discussed in the next numerical example.
3.5 C. MNW Chain in an Integrated Structure
From the previous results, we concluded that under certain conditions, the MNW chain behaves as an optical waveguide supporting Bloch modes. These modes can be coupled to the modes of a dielectric waveguide placed in close proximity to the MNW chain if their respective
In the dispersion curves of Fig.
Fig. 6. Dispersion curves of the integrated structure (red lines), the isolated dielectric waveguide (blue), and the isolated MNW chain (green lines). The quadrupolar Bloch mode (inset) is coupled to the dielectric waveguide at , generating antisymmetric and symmetric supermodes. The dipolar transversal mode does not cross the fundamental TM0 mode of the dielectric waveguide.
4. DIRECTIONAL COUPLING AND BEAM PROPAGATION
As was previously demonstrated, the MNW chain supports quadrupolar and dipolar transversal Bloch modes. The dispersion curves reveal that an efficient directional coupling with a dielectric waveguide is expected with the quadrupolar branch. For the dipolar transversal mode, the field is confined between the nanowires, so an energy beating with the evanescent field of the dielectric waveguide is also expected.
Since the numerical method allows us to determine the beam propagation through the integrated structure, a deeper study of the integrated structure can be realized. To this purpose, we calculate the normalized transmission and reflection at the input and output of the dielectric waveguide, as well as the absorption of the structure in a spectral range from 400 to 1500 nm. Near-field maps of the amplitude of the
The spectral curve of Fig.
Fig. 7. Transmission, reflection, and absorption spectra for the integrated structure. In the transmission curve, the quadrupolar mode is excited at , and the constructive interference of the dipolar transversal mode is positioned at . The minimum at is a cavity resonance effect. In the reflection curve, Bragg reflections are located at , , and at .
Efficient excitations of the quadrupolar and dipolar transversal MNW chain modes are observed respectively at
Fig. 8. Amplitude maps of the component of the electromagnetic field corresponding to (a) the excitation of the quadrupolar mode ( ) and (b) the interference of the dipolar transversal mode with the fundamental TM0 mode of the dielectric waveguide at .
Concerning the excitation of the dipolar transversal mode, the directional coupling is not the main process since no anticrossing phenomena occur in the dispersion curves. Nevertheless, as is depicted in Fig.
Also, as was previously predicted in the plots of Fig.
5. CONCLUSIONS
With the use of the FMM for stratified mediums, we calculated the dispersion relation of the Bloch modes in a chain of MNWs integrated on top of a dielectric waveguide. A quadrupolar and a dipolar transversal LSP Bloch mode were found along the MNW chain. When the nanowires are placed on a substrate, a third resonance branch arises below the substrate light line due to the excitation of an SPP at this interface which is also a propagation mode. Besides the dispersion relation, the numerical method allowed us to obtain near-field maps as well as to compute the transmission, reflection, and absorption spectra.
We demonstrated that the MNW chain behaves as a waveguide for LSPs, whose modes can be excited with the fundamental mode of a dielectric waveguide. A coupling of the energy inside the nanowire array is achievable with short coupling lengths even without perfect matching of the
Since the eigenmodes supported by the nanowires exhibit strong LSP resonances, the analyzed structure may serve as a nanoscale integrated device for sensing applications, useful for biological or chemical detection in Raman or localized surface plasmon resonance spectroscopy methods.
The proposed integrated structure opens a direction for new optical waveguide designs, potentially helpful on the efficient excitation of LSPs and strong field enhancement.
[1]
[20] P. Lalanne, E. Silberstein. Fourier-modal methods applied to waveguide computational problems. Opt. Express, 2000, 25: 1092-1094.
Article Outline
Ricardo Tellez-Limon, Mickael Fevrier, Aniello Apuzzo, Rafael Salas-Montiel, Sylvain Blaize. Theoretical analysis of Bloch mode propagation in an integrated chain of gold nanowires[J]. Photonics Research, 2014, 2(1): 01000024.