Abstract — Localized surface plasmon resonance (LSPR) has been used to develop optical biosensors. Tuning the resonance wavelength to detect target biomolecules with a particular dipolar resonance is essential when designing LSPR biosensors. In this paper, the interaction of nanoparticles (NPs) with glass substrate (SiO2) for LSPR wavelength is investigated using the concept of the image-charge theory. Using the FDTD method, it is shown how the NP and substrate size change the plasmon wavelength. Next, this phenomenon is interpreted using the analytical electrostatic eigenvalue method. I. INTRODUCTION Localized surface plasmon (LSPR) resonance is associated with the collective resonances at the interface of a NP with a background media. This resonance leads to strong electric fields on the interface between the NP and its substrate at some wavelength called resonance or plasmon wavelength. Determining this wavelength is essential in LSPR applications such as surface enhanced Raman scattering (SERS) biosensors. Biomolecules are usually Raman scatterers resonating at a particular wavelength. Thus, the procedure to design a LSPR device with a desired resonance frequency is in demand for a variety of research areas. The resonance wavelength is affected by the strength of the NP-substrate interaction; thus, the material, shape, and physical dimension of both NP and media or substrate have remarkable impact on the plasmon spectrum [1]. Experiments show that increasing the background media dielectric factor red-shifts the plasmon resonance causing stronger Fano resonance [2], and accordingly more enhancements. This rule has been demonstrated for a silver- truncated tetrahedral NP, based on both the DDA (Discrete Dipole Approximation) numerical calculation method and experimental results [3]. The permittivity of the surrounding medium or refractive index per molecular binding, which is the second power of the permittivity, can be raised in at least three ways. First, larger molecules (e.g. proteins) produce larger shifts roughly in proportion to the mass of the molecules. Second, chromophores that absorb visible light couple strongly with the LSPR of NPs to produce large shifts and can be used to detect small molecules binding to protein receptors. Third, pairs of NPs that are separated by less than about 2.5 particle radii show plasmonic coupling and marked spectral shifts [4-5]. In another research, the DDA calculations implemented by Schatz et al. to model a 10 nm D. Mortazavi is with the School of Engineering, Deakin University, Geelong, VIC 3216, Australia (phone: +61352272795; fax: +613 52272167; e-mail: dmortaza@deakin.edu.au). A. Z. Kouzani is with the School of Engineering, Deakin University, Geelong, VIC 3216, Australia (e-mail: kouzani@deakin.edu.au). A. Kaynak is with the School of Engineering, Deakin University, Geelong, VIC 3216, Australia (e-mail: akaynak@deakin.edu.au). silver sphere either sinking into a mica substrate or surrounded by free vacuum, demonstrates that the plasmon wavelength of the sphere gets red-shifted when the sphere goes from free vacuum to being partially embedded in the mica [3]. Reducing the Au content also decreases the enhancement significantly [6]. To keep the enhancement factor in the reasonable range, we need to have a higher plasmon wavelength [7]. Although analytical methods such as Mie theory [8] are restricted to particular NP shapes, the electrostatic eigenmode method [9] has been developed to formulate the concept of image-charge for different shapes of NPs [10]. In this paper, the effect of substrate dimensions on different NP shapes including nano-ellipsoids, nano-triangles, and nano-diamonds are simulated using the FDTD method, and the results are interpreted using the analytical electrostatic eigenmode method [9, 11-14]. II. THEORETICAL BACKGROUND Assume a quasistatic approach using a spherical NP of radius a, irradiated by z polarized light of wavelength λ, in the long wavelength limit ሺa/λ ൏ Ͳ.ͳሻ. Due to this small size condition, the quasistatic estimation is a good estimation to solve electromagnetic Maxwell’s equations. Therefore, Maxwell’s equations can be replaced by Laplace’s equation [15]. Dipole resonance modes can be investigated through the electrostatic eigenmode approximation [9, 12]. In this formulation, the surface plasmon resonances are calculated using the oscillating surface charge distribution ߪ ௞ ሺݎሻ at resonance mode . For an arbitrary ensemble of NPs, the charge distributions can be written as a superposition of the normal modes of each NP of the ensemble as follows: ߪ ௞ ሺ࢘ሻ ൌ ஓ ౡ ଶగ ׯ ߪ൫࢘ ࢗ ൯ ሺ࢘࢘ ࢗ ሻ ห࢘࢘ ࢗ ห య ࢔ ෝ  ௤ . (1) Here, γ ୩ is k-th natural resonant mode (or eigenmode) of the nano-particle, ߝ ௠ ሺ ௞ ሻ is the permittivity of the metal at the െ ݐ resonance frequency  ௞ , and ߝ ௕ is the permittivity of the background medium [9, 12]. The total surface charge ߪሺ࢘, ሻ at frequency  is calculated as the weighted summation of the surface charges at every mode : ߪሺ࢘, ሻ ൌ ∑  ௞ ሺሻ ߪ ௞ ሺ࢘ሻ ୩ , (2) where  ௞ ሺሻ, the excitation amplitude of the െ ݐ resonance mode is given by:  ௞ ሺሻ ൌ ଶஓ ౡ ఌ  ሺఌ ೘ ൫ఠ ೖ ൯ఌ  ሻ ఌ  ൫ஓ ౡ ାଵ൯ାఌ ೘ ൫ఠ ೖ ൯൫ஓ ౡ ଵ൯ ࢖ ௞ .ࡱ ଴ . (3) Investigating Nanoparticle-Substrate Interaction in LSPR Biosensing using the Image-Charge Theory Daryoush Mortazavi, Abbas Z. Kouzani, Akif Kaynak 34th Annual International Conference of the IEEE EMBS San Diego, California USA, 28 August - 1 September, 2012 2363 978-1-4577-1787-1/12/$26.00 ©2012 IEEE