Set Theory: Introduction

Abstract

A. Fresnel Equations Since we have decided to build up an plasmonics sensor in which the amplitude of the reflected beam (reflected intensity or reflectance) from gold-dielectric interface is detected, and also a more precise way to detect surface plasmon resonance is to detect phase changing. Where both the theory of reflected amplitude and phase changing are based on Fresnel equations. Hence we would like to introduce the theory for our master project starting from Fresnel equations. Let us first consider an incident beam at the boundary between two materials with different index of refraction (for example, air: n 1 and glass: n 2). We will discuss two different conditions for both TM (transverse magnetic) and TE (transverse electric) mode waves. Figure 1 [1] shows the picture of the incident, reflected and transmitted waves at an planar interface for TE (left) and TM (right) mode respectively. Figure 1. Left: incident wave of TE mode (electric field is perpendicular to the plane of incidence) at the interface; Right: incident wave of TM mode (magnetic field is perpendicular to the plane of incidence). We can see that "E" represents electric field, "B" represents magnetic field, "Xr" represents the reflected components, "Xt" represents the transmitted components. On the basis of law of reflection and law of refraction (Snell's law): r     t t r r n n   sin sin   We first introduce the definition of the required boundary condition: the components (both electric field and magnetic field) parallel to the interface should be continuous when crossing the boundary. The boundary conditions for TE waves:        t t r t r B B B E E E    cos cos cos 