Fresnel Equations

I. Theory introduction 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: n1 and glass: n2). 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 nr sinr nt sint 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: E Er Et B cos Br cos Bt cost The boundary conditions for TM waves: B Br Bt E cos Er cos Et cost Taking account of the relation between electric field and magnetic field: c E B B n Then the above boundary conditions for both wave modes can be presented as following: E Er Et TE : n1E cos n1Er cos n2 Et cost n1E n1Er n2 Et TM : E cos Er cos Et cost If we further employ Snell’s law to eliminate the angle of refraction θt while introducing the relative refractive index n=n2/n1 as shown below: 2 2 2 ncost n 1 sin t n sin In this way we can finally get the reflection coefficient r=Er/E and transmission coefficient t=Et/E for two modes as following: E cos n2 sin 2 r r E 2 2 TE : cos n sin E 2cos t t E cos n2 sin 2 E n2 cos n2 sin 2 r r E 2 2 2 TM : n cos n sin E 2ncos t t E n2 cos n2 sin 2 The above equations (9) and (10) are known as Fresnel equations. For non-planar interface, the scattering losses should be also taken into consideration when calculating both reflection and transmission coefficients. B. Total internal reflection and evanescent wave 1). Total internal reflection (TIR) After having got the coefficients for both reflection and transmission, we now turn to discuss the energy issue at the planar interface between two materials, that is, the power of incident beam Pi will be separated into reflected part Pr and transmitted part Pt, and the proportion for each part compared with the total energy of the incident wave is called reflectance (represented by R) and transmittance (represented by T) respectively. Here we give their mathematical expression without proof: 2 P E R r r 2 r Pi E P cos T t r 2 n t t 2 Pi cosi As we plot the incident angle θ versus R, and take the boundary between air (n=1) and glass(n=1.5) for instance, the reflectance R for both external and internal can be shown as below: [1] Figure 2. The reflectance of TM and TE modes for both external and internal reflection, nair=1, nglass=1.5. From Figure 2 we can see the external and internal Brewster angle, or so called polarizing angle, which is -1 expressed by θp=tan (nt/ni), in which nt and ni represent the index of refraction for the material of incident space and transmitted space separately. Additionally, we could also find that under the condition of internal reflection, the reflection coefficient rTM and rTE reaching to unity value not occurs at normal incidence. This -1 phenomenon is known as total internal reflection, in which the corresponding incident angle is θc=sin (nt/ni), the subscript ‘c’ indicates the specific name of this angle: critical angle. 2). Phase change of TIR mode Let us continue the topic of internal reflection. However, when the incident angle θ>θc, it means sinθ>n, with n=nt/ni, then the expression of reflection coefficient should be rewritten as followings: 2 2 Er cos i sin n rTE E cos i sin 2 n2 2 2 2 Er n cos i sin n rTM E n2 cos i sin 2 n2 For (13) and (14) either equation, r could be taken the form of r=±(a-ib)/(a+ib), then the phase of can be expressed by polar form: ei r ei2 ei Where the expression for r (polar form) and β in each mode are presented as followings: r ei2 2 2 TE : 1 sin n tan cos r ei2 ei2 2 2 TM : 1 sin n tan n2 cos Now we take the phase change of the E-field of the reflected wave into consideration with respect to the original phase of the incident wave, and we represent it as φ, then using the reflection coefficient and the wave equation of incident wave, the E-field of the reflected wave now is: i ik.rt ik.rt Er rE r e E0e r E0e Hence if we combine equations (16) - (18), we could get the phase change for each wave mode at interval of total internal reflection (θc<θ<π/2): sin 2 n2 2 2 tan 1 TE cos c : 2 2 2 1 sin n 2 2 tan TM n2 cos 3). Evanescent wave However, when under TIR mode, the electromagnetic field should be still continuous at the boundary of two mediums, akin to the transmitted wave, we call it as the “evanescent wave”. We will investigate the properties of evanescent wave by first looking at its wave equation as shown below. ik t .rt Et E0te We assume that the evanescent wave is propagating at x-z plane as shown in Figure 1, in which the E-field could be presented by x-y coordinate, then we can get: k t .r kt sint , cost , 0.x, y, 0 kt xsint y cost For the case of TIR mode, the angle of refraction θt can be presented by: sin 2 sin 2 cos 1 sin 2 1 i 1 t t n2 n2 So now the exponential factor is shown as following: sin sin 2 k .r k x ik y 1 t t n t n2 For convenience, we could also define a real and positive value: sin 2 k 1 t n2 Then the transmitted wave can be rewritten as: sin ixk it t n y Et E0te e e From equation (25) it is obvious that the evanescent wave has harmonic functions with invariable amplitude along x direction, but also decreasing exponentially along y direction. There exists a range that the energy of the evanescent wave will return to the first medium after having propagated at the second medium, this range is called penetration depth, which can be represented by equation (26), in which the wave amplitude is decreased to the 1/e of the original value. 1 y sin 2 2 1 n2 An exception occurs that the energy of evanescent wave can continue forward propagation, which is realized by placing an extra medium in contact with the medium of the evanescent wave, then the total internal reflection can be frustrated. The most common way to have frustrated TIR is to put two right-angle prism together with the diagonal surface facing to each other, as shown in Figure 3, in this way, the evidence of the evanescent fields, in which the field that leaks through the TIR surface, is provided. Figure 3. Comparison of TIR and frustrated TIR. C. Optical properties of materials 1). Dielectric a). Polarization of dielectric materials When we apply electromagnetic field (EM field) to dielectric material, there will be tiny displacement for electron regarding the position of nuclei, which will further produce an induced dipole. Then we introduce the dipole moment p which is the product of the displaced charge and the separated distance of negative and positive charge as below: p qr As we can see, the vector of the dipole moment is pointing from electrons to nuclei. Then we introduce another important notion, the polarization P, which represents the total dipole moment per unit volume, its expression is given by equation (28) as shown below, N is the number of dipole pairs per unit volume, e is the charge amount of a single electron. P Ner As the magnitude of the nuclei mass is far higher than that of electron mass, we can consider that the electrons are bind by nuclei via elastic force given by Hooke’s law. Furthermore, in an alternating EM field, electrons also oscillate, and the oscillation is actually a damping process because the kinetic energy of electron will decrease when colliding with other electrons. Therefore, we can use Newton’s classical mechanics to interpret the motion of oscillating electron by adding the above conditions into consideration, hence we can get the following equation to describe oscillation model of electron: dr d 2r Kr m eE m dt dt 2 In equation (29), K is the spring constant of the elastic model; m is the mass of electron; γ is a frictional constant, of which reciprocal is the relaxation time (time period between two collisions) of the free electron.

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