Matrix formalism is a very systematic method to find the reflectance or transmittance in a stratified medium consisting of a pile of thin homogeneous films. Fitting experimental values of reflectance curve to expressions obtained from the matrix formalism method is an efficient method to estimate the refractive index (n) of a dielectric and/or the real and imaginary parts of a metal permittivity (e). In the next section, the method of matrix formalism is briefly reviewed with some examples to show how it can be applied in curve fitting to determine refractive indices or the metal permittivity. Applications to more complex structures such as planar waveguides and periodic grating are presented in sections 2 and 3, respectively.
Matrix formalism for the transverse electric and magnetic waves in stratified thin films
Maxwell equations will be applied at each interface between two homogeneous media to find the characteristic matrix defining a thin film. Let us consider figure 1 for a transverse electric (TE) wave with the E-field vector perpendicular to the plane of incidence for one thin homogeneous film.
Fig. 1. Electric field (E) and magnetic field (H) in each medium of refractive index n1, n2 and n3.
In figure 1, the H-field is related to the E-field using:
where eo and |o are referred to as the electric permittivity and the magnetic permeability, respectively. Letters i , r and t stand for incident, reflected and transmitted rays, respectively and the homogeneous medium is identified using numbers 1, 2 or 3. As both the E and H fields are continuous at boundary 1, one may write E1 and H1 as:
The system of equations (2) and (3) can be written under the matrix form as:
At interface 2, we merely write
By making use of the E-field amplitude phase shift, it can be shown that Ei2 and E’r2 can be expressed as:
where d2 is the thickness of the homogenous thin film and 0i2 is the angle defined as shown in figure 1. k2 is the wave-vector in the thin homogeneous film (medium 2) , which is given as
where X is the wavelength of the monochromatic incident light when propagating in a vacuum.
Equations (7) and (8) are used to express the tangential component of the H-field vector at interface 2 as:
Using Equations (7) and (8), Equations (6) and (11) are expressed under the matrix form and by matrix inversion one can show that:
Lastly, substituting Equation (12) into Equation (5) the E and H field components at interface 1 are related to those at interface 2 by:
The 2×2 matrix in equation (13) is the characteristic matrix (M2) of the homogenous thin film. Note that M2 is unimodular as its determinant is equal to 1. Assuming another film lying just underneath the thin film shown in figure 1, from Equation (13) we imply that field components E and H at interface 2 will be related to those at interface 3 by the matrix equation:
Substituting Equation (14) into Equation (13) one finds:
By applying this method repeatedly for a stratified system of N thin homogeneous thin films we can write:
for interfaces l = 2,3, … , N. (Born &
for interfaces l = 2,3, … , N. (Born & Wolf, 1980) show that the reflection and transmission coefficient amplitudes for a system of N-1 layers ( l = 2 to N) lying on a substrate of refractive index ns can be expressed from the matrix entries of the system matrix M as:
where r± is referred to as the reflection coefficient for the TE wave. Admittances Y1 and Ys for the incident medium and the substrate hosting the system of N-1 homogeneous thin films are given by:
For the case where the H-field is perpendicular to the plane of incidence (TM wave), the impedances YyYi and Ys must be replaced by Z1 , Zl and Zs , which are given by
Examples with dielectrics and metal thin films with some experimental results
Expressions derived in the previous section can be applied to find the reflectance curve of thin dielectric or metal films. They can be applied to fit experimental reflectivity data points to determine refractive indices of a dielectric film or metal film relative permittivity and even their thickness. Before we illustrate how it is used, let us apply Equation (17) for the simple case of Fresnel reflection coefficient amplitude for an interface between two semi-infinite media.
Interface between two semi-infinite media (Fresnel reflection coefficient)
This situation can be mimicked by setting d2 = 0 into Equation (13). In other words, interfaces 1 and 2 in Figure 1 collapse into one single interface separating two semi-infinite media of refractive index n1 and n2.
Characteristic matrix in Equation (13) can be used to find the matrix system for two semi-infinite media. Setting for d2 = 0, the matrix system for the two semi-infinite media becomes the identity matrix as h2 equals 0. This means that m11 = m22 = 1 and m12 = m21 = 0. Substituting the matrix entries into Equation (17) one obtains:
In the previous equation we use ns = n2 and 0N = 02 for this single interface system. For the TM wave, it can be shown that:
We then retrieve the results for the Fresnel reflection coefficients. Results for the transmission coefficient amplitude (t) can be obtained in the same manner.
Reflectance curve for a thin metallic film of silver or gold (surface plasmons)
A matrix approach is used to compute the reflectance of a thin film coupled to the hypotenuse of a right angle prism. The system shown in Figure 2 can be modeled by using three characteristic matrices for the matching fluid, the glass slide, the metal film and then accounting for the various Fresnel reflection losses at both the entrance and output face of the prism.
Fig. 2. Path of a laser beam propagating through all interfaces bounded by two given media. For a one way trip the media are (1) air, (2) glass, (3) matching fluid (greatly exaggerated), (4) glass (slide), (5) metal film (Au of Ag) and (6) air.
(Levesque, 2011) expressed the characteristic matrix M of the sub-system of three layers in Figure 2 as
where M3, M4 and M5 are the characteristic matrices for the index matching fluid layer, the glass slide and the metal thin film, respectively. Each of these matrices is given by
where i =3, 4 or 5. fli and qi for p-polarized light are expressed as
respectively, where Ei ( = ni2) is the relative permittivity of the material. Using Snell’s law, note that
We are assuming all media to be non-magnetic and d3, d4 and d5 are the thicknesses of the matching fluid, glass slide and the metal film, respectively. 83, 84 and 85 (= 8’5 +i85”) are the relative permittivity for the matching fluid, the glass slide and the metal film, respectively. 85′ and 85′ are respectively, the real and imaginary parts of the metal film relative permittivity. By taking into account the Fresnel reflection losses F1 at the input and output faces of the glass prism, the reflectance for the p-polarized light Roet is given by:
where mij are the entries of matrix M and F^l is given by
In previous equation n2 is the refractive index of the prism. Investigations on optical reflectivity were done on glass slides which were sputtered with gold or silver. These glass slides were pressed against a right angle prism long face and a physical contact was then established with a refractive index matching fluid. The prism is positioned on a rotary stage and a detector is measuring the signal of the reflected beam after minute prism rotations of roughly 0.03°. The p-polarized light at X = 632.8 nm is incident from one side of a glass prism and reflects upon thin metal films as shown in figure 3. As exp (-jrot) was assumed in previous sections, all complex permittivity 8 must be expressed as 8 = 8′ + j8”.
Fig. 3. Experimental set-up to obtain reflectivity data points.
If no film is coating the glass slide, a very sharp increase in reflectivity is expected when 04 approaches the critical angle. This sudden increase would occur at 0c = sin-1(1/n2) ~ 41.3°. The main feature of the sharp increase in the reflectivity curve is still obvious in the case of a metalized film. This is so as the penetration of the evanescent field is large enough to feel the presence of air bounding the thin metal film. As silver or gold relative permittivity (optical constant) is complex, cos05 becomes complex in general and as a result 05 is not represented in Fig. 2. This means physically that the field penetrates into the metal film and decays exponentially through the film thickness. At an optimum thickness, the evanescent field excites charge oscillations collectively at the metal-film-air surface (c.f.fig.2), which is often used to probe the metal surface. This phenomenon known as Surface Plasmon Resonance (Raether, 1988; Robertson & Fullerton, 1989; Welford, 1991) is occurring at an angle of 02 that is a few degrees greater than 0c. For a He-Ne laser beam at X = 632.8 nm, that is incident from the prism’s side (c.f.fig.2) and then reflecting on silver or gold metal films, surface plasmons (SP) are excited at 02 near 43° and 44°, respectively. At these angles, the incident light wave vector matches that of the SP wave vector. At this matching condition, the incident energy delivered by the laser beam excites SP and as a result of energy conservation the reflected beam reaches a very low value. At an optimal thickness, the reflectance curve displays a very sharp reflectivity dip. Figure 4 shows the sudden increase at the critical angle followed by a sharp dip in the reflectance curve in the case of a gold film of various thicknesses, which is overlaying the glass slide.
Fig. 4. Reflectance curves for gold films of various thicknesses d5 obtained from Eq.(30). We used ds = 10000 nm and d = 1000000 nm (1mm), n2 =1.515, n3 =1.51, n4 =1.515 and 85 = -11.3+3j.
Reflectance curves for gold films sputtered on glass slides show a sudden rise at the critical angle 0c followed by a sharp drop reaching a minimum near 44°. For all film thicknesses, a sudden rise occurs at the critical angle. Note that the reflectance curve for a bare glass slide (d5 = 0 nm) is also shown in figure 4. At smaller thicknesses, the electromagnetic field is less confined within the metallic film and does penetrate much more into the air. The penetration depth of the electromagnetic field just before reaching the critical angle (0 < 41.3°) is indicated by a lower reflectance as d5 gets closer to zero, as shown in Fig.4. The reflectivity drop beyond 0c is known as Surface Plasmon Resonance (SPR). SPR is discussed extensively in the literature and is also used in many applications. Good fitting of both regions displaying large optical intensity change is also useful in chemical sensing devices. As a result fitting of both regions is attempted using the exact function curve without any approximations given by Eq. (30). Eq. (30) is only valid for incident plane wave. Therefore, the reflectivity data points were obtained for a very well-collimated incident laser beam. A beam that is slightly converging would cause more discrepancy between the curve produced from Eq. (30) and the reflectivity data points. Although the Fresnel loss at the transparent matching fluid-glass slide interface is very small, it was taken into account in Eq. (30), using d3=10 000nm (10 ^m) in matrix M3. The theoretical reflectance curve is not affected much by the matching fluid thickness d3. It was found that d3 exceeding 50 ^m produces larger oscillations in the reflectance curve predicted by Eq.(30). As the oscillations are not noticeable amongst the experimental data points, the value of d3 = 10 ^m was deemed to be reasonable. A function curve from Eq. (30) is generated by changing three output parameters 85′, 85′ and d5. The sum of the squared differences (SSQ) between RDet and the experimental data points Ri is calculated. The best fit is determined when the SSQ is reaching a minimum. The SSQ is defined as:
where i is a subscript for each of the N data points from the data acquisition. Each sample was placed on a rotary stage as shown in Figure 3 and a moving Si-pin diode is rotating to track down the reflected beam to measure a DC signal as a function of 02. The reflectivity data points and typical fits are shown in Figure 5.
In the fit in Fig. 5a, we used n2 =ru = 1.515, n = 1.47(glycerol) for red light, d3 = 10 000 nm, 85 =-11.55+3.132j and d5 =43.34 nm.
In the fit in Fig. 5b, we used n2 =n4 = 1.515, n = 1.47(glycerol) for red light, d3 = 10 000 nm, 85 = -10.38+2.22j and d5 = 53.8 nm. The three output parameters ( 85′, 85" and d5) minimizing the SSQ determine the best fit. Plotting the SSQ in 3D as a function of 85′ and d5 at 85" = 3.132 shows there is indeed a minimum in the SSQ for the fit shown in Fig.5a. Figure 6 shows a 3D plot of the SSQ near the output parameters that produced (Levesque, 2011) the best fit in Fig. 5a. 3D plots at values slightly different from 85" = 3.132 yield larger values for the minimum.
Wave propagation in a dielectric waveguide
In this section, we apply the matrix formalism to a dielectric waveguide. We will describe how the reflectance curve changes for a system such as the one depicted in Fig. 2 if a dielectric film is overlaying the metal film. It will be shown that waveguide modes can be excited in a dielectric thin film overlaying a metal such as silver or gold and that waveguide modes supported by the dielectric film depend upon its thickness.
Fig. 5. Reflectivity data points (+) and a fit (solid line) produced from Eq.(30) for two different gold films.
Fig. 6. 3D plot of SSQ as a function of two output parameters at a given value of 85" (= 3.132). We assumed the glycerol layer (d3) to be 10 ^m and the thickness of the glass slide is 1 mm (d4). The SSQ reaches a minimum of 0.01298 for e5′ =-11.55, 85" = 3.132 and d5 =43.34 nm.