Modelling surface plasmon resonances of arrays of nano particles on a dielectric substrate using discrete dipole approximation with surface interaction (DDA-SI)

Abstract

Analysis of surface plasmon resonance of arbitrary shape nano-particles on a substrate is important for many engineering applications. The strong optical absorption of noble metal nano-particles is due to the localized surface plasmon, which enables the development of novel applications such as surface enhanced Raman spectroscopy based biological sensing, optical transparency based sensors, and uni-directional nanoantennas. Size, shape, and distance between nano-particles on a surface are the key factors in the design of these structure. Usually, these systems utilize noble material such as gold which is deposited on a dielectric surface. Tuning the structures of the noble metals increases the local absorption e ciency of light within speci c frequency ranges, depending rely on the nal design of the system. Understanding the physics of plasmon phenomena and its relation with these parameters should be studied in depth. This thesis provides a theoretical study and a numerical validation of the coupled phenomena that occurs between gold nano-particles on a dielectric substrate where the nano-particles have di erent shapes and separation distances. Numerical study of nano-particles on a substrate is carried out using Discrete Dipole Approximation with Surface Interaction (DDA-SI) approach. The DDA-SI is based on the discretization of the nano-particles to dipoles for solving volume integral of Maxwell s equations and Green s dyadic tensor of electric eld within the dipoles to calculate the optical properties of arbitrarily shaped, non-homogeneous, anisotropic objects. There are some renowned open sources packages for DDAs. These packages are mostly used for calculating the interaction of particles in a free space with direct wave propagating. However, in order to calculate the light scattering from objects placed on a substrate, the substrate itself also is required to be approximated by the dipoles which increase the computational time remarkably. In this study, we used the Discrete Dipole Approximation with Surface Interaction (DDA-SI) which is an open source MATLAB based software package for calculation of optical properties of nano-particles (extinction, absorption and scattering) on a substrate. It can be used to investigate both the near- and far- eld e ects and accounts for the coupling between di erent particles on the surface. This study focuses on speci c geometries such as cube, spheroid and triangular geometries, each of varying sizes and separation distances from each other. Apart from studying the plasmon resonance of individual nano-particle on a dielectric substrate, a system with more nano-particles is scrutinized. It is found out that as the distance between particles decreases, the plasmon resonance frequency is pushed into the infrared region due to the inter-particle coupling, and the redshift becomes dominant. Furthermore, we found out that the coupling e ect becomes negligible if the ratio of their distance between nano-particles to the radius, c = d=a, is greater than three. During this study, the main DDA-SI toolbox is further developed, vecrorized and optimized numerically which named DDA-SI-3 in order to calculate the optical properties of noble metals. The imaginary component of refractive index for noble metals for larger wavelengths is large and make the calculation challenging. Therefore, we applied the new numerical method to calculate the linear system of DDA-SI in order to achieve more precise, faster and stable calculations. The structure of interaction matrix is studied and for this, speci c preconditioning matrices are extracted. It is also found out that the use of the least square method, with the proper preconditioning matrices for iteratively solving the linear system, yields results to achieve more accurate and relatively faster calculation.

Alt malzeme 7F uzeri iste ge ba gl yerle stirilmi s nano par cac klar n y 7F uzey plazmonik rezonans analizleri co gu m 7F uhendislik uygulamalar i cin 7F onem arz etmektedir. Soy metal nano par cac klar n g 7F osterdikleri g 7F u cl 7F u so gurma olay y 7F uzeyde g 7F u clendirilmi s, Raman spektroskopisi esasl , biyolojik sens 7F orler, optik saydaml k, esasl sens 7F orler ve tek y 7F onl 7F u nano antenler gibi bir cok 7F ozg 7F un uygulamalar n geli stirilmesini sa glayan b 7F olgesel hale getirilmi s y 7F uzey plazmonu sonucunda ger cekle sir. Y 7F uzey 7Fuzerine yerle stirilmi s nano par cac klar n tasar m ndaki kilit unsurlar boyut, sekil ve par cac klar aras ndaki uzakl k parametreleridir. Bu sistemlerde genel olarak dielektrik y 7F uzeye b rak lm s alt n gibi soy metallerden yararlan l r. Soy metal yap lar n n ayarlanmas belirli frekans aral klar nda s g n b 7F olgesel so gurulma verimlili gini artt r r. Bu cal smada, her biri farkl boyut ve ayr kl k uzakl g ndaki k 7F ubik, k 7F uremsi ve 7F u cgen bi cimli gibi belirli geometriler 7F uzerine odaklan lm st r. Dielektrik y 7F uzey 7F uzerindeki tekli nano par cac g n plazmonik rezonans cal smas n n yan s ra birden fazla nano par cac kl sistemler de irdelenmi stir. Nano par cac klar n ayr kl k uzakl klar azald k ca par cac klar aras ndaki kuplaj ve k rm z ya kayman n domine etti gi ve b 7Foylece plazmon rezonans frekans doru gunun k z l 7F otesi b 7F olgeye itildi gi bulunmu stur. Buna ek olarak, e ger nano par cac klar aras ndaki uzakl g n yar caplar na oran , c = d=a, 3ten b 7Fuy 7F uk olursa kuplaj etkisinin ihmal edilebilir oldu gunu ke sfettik. Bu cal sma boyunca, soy metallerin optik 7F ozelliklerinin hesaplanmas i cin as l kullan lan DDA-SI paketi daha fazla geli stirilerek vekt 7F or haline getirilmi s ve numerik olarak optimize edilerek DDA-SI-3 ad n alm st r. Soy metallerin k r lma indisinin sanal k sm n n uzun dalga boylar nda b 7F uy 7F uk olmas hesaplamalar zorla st r r. Bu sebeple, DDA-SI daki lineer sistemlerin hesaplamalar nda daha hassas, h zl ve stabil sonu clara ula san yeni bir numerik metot uygulad k. Bu metot ile etkile sim matrisi 7F uzerine cal s ld ve 7F ozel 7F on d 7F uzenleme matrisleri ortaya c kar ld . Ayr ca, lineer sistemin tekrarl (iterasyon) c 7F oz 7F um 7F unde en k 7F u c 7F uk kareler metodunun uygun 7F on d 7F uzenleme matrisleri ile kullan lmas sonucunda daha kesin ve g 7F orece daha h zl hesaplamaya ula st g ke sfedildi.