Propagation of Electromagnetic Waves in Thin Dielectric and Metallic Films Part 2

Wave propagation in dielectric films

Let us consider a dielectric film of thickness d6 overlaying the metal film in Figure 2. We will be assuming that the top surface of the overlaying dielectric is bounded by the semi-infinite air medium. The characteristic matrix M for the sub-system of four layers can be expressed as:

where M3, M4, M5 and M6 are the characteristic matrices for the index matching fluid layer, the glass slide, the metal thin film and the thin dielectric film, respectively. Each matrix in Eq.(33) is given by Eq.(26) for i = 3,4,5 and 6 and the reflectance for the p-polarized wave is given by Eq.(30). For this four layer system, q6′ in Eq.(30) should be replaced by q7′ (air) and m11, m12, m21 and m22 are the entries of the system matrix given by Eq.(33). The expression for q6 is given by Eq. (28) and is used in the computation of M6 for the dielectric film characteristic matrix.

Computation of reflectance with a thin dielectric film and experimental results

Eq. (30) can be used with the minor modifications discussed in section 2.1 to find the reflectance of the system in Fig. 2 with an extra dielectric film processed on the metal film. The dielectric film can support waveguide modes if the laser beam is directed at very precise incident angle 02. Let us consider a transparent polymer film with a real permittivity 86 = 2.30 processed on a silver film. The computation is done for a silver film that is 50 nm thick. Silver permittivity is assumed to be 85 = -18.0 +0.6i and the prism refractive index to be 2.15 (ZrO2) for He-Ne laser at X = 632.8 nm. We also assume that the metal film is directly coated on the prism long face and as a result we set d3 =d4 = 0. In other words M3 and M4 are expressed by identity matrices. Figure 7 is showing the reflectance curve for a dielectric film of different thickness that is overlaying the silver film coated on the high refractive index ZrO2 prism.

Fig. 7. a) Reflectance curve for a lossless dielectric film of 1.7 m overlaying a thin silver film b) Reflectance curve for a lossless dielectric film of 2.5 m overlaying a thin silver film

In figure 7a, a series of very sharp reflectivity drops occur in the reflectance curve for 02 within the range 35°-45°. These sharp reflectivity drops with small full width at half maximum (FWHM) are waveguide modes supported by the dielectric film. The last reflectivity dip with a larger FWHM near 02 ~ 50° is due to surface plasmon resonance (SPR) and is mostly depending upon the metal film properties and its thickness as discussed in section 1.2.2. A thicker dielectric film (c.f. fig. 7b) can support more waveguide modes and as a result the number of sharp reflectivity dip for 02 within the range 35° to 45° is expected to be greater. Note that the FWHM of the SPR dip remains at the same position as the metal film thickness was not changed. These waveguide modes do not propagate a very large distance as light is slightly attenuated when reflecting at the metal-dielectric film interface. Therefore, at precise angle 02 the incident light is probing the dielectric film locally before being reflected by the thin metal film. Nevertheless, the laser beam is simultaneously probing the metal and the dielectric films because it creates SPR on the thin metal film and waveguide modes are being supported by the dielectric film. In practice, dielectric films are not lossless (Podgorsek & Franke, 2002) and their permittivity should be expressed using a small imaginary part. Let us assume that each dielectric films in Fig. 7 have a permittivity of 86 = 2.30 +0.005j.

Fig. 8. a) Reflectance curve for a dielectric film (e6 = 2.30 +0.005j, d6 =1.7 ^m) overlaying the metal film. b) Reflectance curve for a dielectric film (e6 = 2.30 +0.005j, d6 =2.5 m) overlaying the metal film.

Note from figures 7 and 8 that the waveguide mode dips are greatly attenuated when a small imaginary part is assumed in the dielectric film permittivity. The dips at larger angles (near 45°) are getting smaller as the propagation distance into in the dielectric film is larger as 02 increases. Note that the SPR dip is not much affected by the imaginary part of 86. Essentially, the whole 4-layer system of prism material-silver film- dielectric film-air can be mounted on a rotary stage and the angle 02 can be varied using a set-up similar to that shown in figure 3. As it is difficult to obtain a large dynamic range in the measurements of reflectivity data points, scans must be done successively to cover a long range of incident angle. Figure 9 shows reflectance curves for a transparent layer of polyimide processed directly on silver films. Ranges of incident angle 02 where no noticeable change in reflectivity were observed are not shown. Only the dips in the reflectivity data points are fitted by Eq. (30).

Fig. 9. Reflectivity data points of the 4-layer system and fits (solid lines) from Eq. (30) for a) ZrO2 prism-Ag-polyimide film b) glass prism-Ag-polyimide film

Using a method based on the optimization of the sum of square (SSQ) as presented in section 1.2.2, thicknesses and the complex permittivities of both films can be estimated. Values obtained from the minimization of the SSQ are given in table 1. Uncertainties are estimated from a method described by (Levesque et al., 1994).

Prism	ZrO2	Glass
	-17.41±0.10 0.2 ±0.1 615 ±15A	-18.1 ±0.1 1.36 ±0.08 146 ±5A
	2.495±0.001 0.011±0.003 1.488±0.004jum	2.230±0.002 0.0017±0.0002 1.723±0.003um

Table 1. Thicknesses and permittivities of the silver (Ag) and polyimide (Pi) films.

Diffraction efficiency (DE) in dielectric periodic grating structures

Abrupt changes in reflectivity or transmission were first observed in gratings as early as 1902 (Wood, 1902). These so-called anomalies in diffraction efficiency (DE) occurring over an angle range or a wavelength spectrum are very different from the normally smooth diffraction curves. These abrupt changes in DE led researchers to design and investigate resonant filters for applications in many devices including gratings.

Rigorous coupled wave analysis (RCWA) has been used extensively (Moharam et al., 1995; Lalanne & Morris, 1996; Lenaerts et al., 2005) to calculate diffraction efficiencies (DE) in waveguide structures. The application of RCWA to resonant-grating systems has been investigated mostly for both the TE and TM polarization. In this section, the basic binary dielectric rectangular-groove grating is treated with careful considerations on the computation of DE. The results obtained for binary dielectric rectangular-groove grating are also applied to metallic grating. Introduction to photonic bandgap systems are discussed and some examples are presented at the end of this section.

Theory of coupled wave analysis

As the numerical RCWA method is introduced extensively in the literature, only the basics equations will be presented in this section. Computation will be done for the TM wave on ridge binary grating bounded by two semi-infinite dielectric media of real permittivities e and e3. The type of structures presented in this section is depicted in figure 10.

Fig. 10. Basic structure of the binary rectangular-groove grating bounded by two semi-infinite dielectrics.

The relative permittivity e(x) of the modulated region shown in figure 10 is varying periodically along the x-direction and is defined as:

where es is the sth Fourier component of the relative permittivity in the grating region (0< z <h), which can be complex in the case of metallic gratings. The incident normalized magnetic field that is normal to the plane of incidence (cf. fg.10) is given by:

where 

is the incident angle with respect to the z-axis as shown in figure 10.

The normalized solutions in regions 1 (z < 0) and 3 (z > h) are expressed as:

where kxi is defined by the Floquet condition, i.e.,

 

In previous equations, A is the grating spacing,

is the refractive index of medium 1 and

is the refractive index of medium 3.

Ri and Ti are the normalized electric-field amplitudes of the ith diffracted wave in media 1 and 3, respectively. In the grating region (0 < z < h) the tangential magnetic (y-component) and electric (x-component) fields of the TM wave may be expressed as a Fourier expansion:

where Uyi (z) and Sxi (z) are the normalized amplitudes of the ith space-harmonic which satisfy Maxwell’s equations, i.e.,

where a temporal dependence of exp ( jmt ) is assumed (j2 = -1) and m is the angular optical frequency. so and | are respectively the permittivity and permeability of free space. As the exp (jmt) is used, all complex permittivity must be expressed under 8 = 8′ – j8”. Substituting the set of equations (40) into Maxwell’s equations and eliminating Ez, the coupled-wave equations can be expressed in the matrix form as:

where z’ equals koz.

Previous equations under the matrix form can be reduced to

where B = KxE-1Kx – I. E is the matrix formed by the permittivity elements, Kx is a diagonal matrix, with their diagonal entries being equal to kxm / ko and I is the identity matrix. The solutions of Eq. (43) and the set of Eq. (42) for the space harmonics of the tangential magnetic and electric fields in the grating region are expressed as:

where, w,i,,m and qm are the elements of the eigenvector matrix W and the positive square root of the eigenvalues of matrix G (=-EB), respectively. The quantities cm+ and cm – are unknown constants (vectors) to be determined from the boundary conditions. The amplitudes of the diffracted fields Ri and Ti are calculated by matching the tangential electric and magnetic field components at the two boundaries. Using Eqs. (35) , (36), (44) and the previously defined matrices, the boundary conditions at the input boundary (z = 0) are:

and

where X and Z1 are diagonal matrices with diagonal elements exp(-jkoqmh) and k1zi/(n12 ko), respectively. c+ and c- are vectors of the diffracted amplitude in the ith order. From (42) and (44), it can be shown that

where vm,i are the elements of the product matrix with Q being a diagonal matrix with diagonal entries ql.

At z = h, the boundary conditions are:

and

where Z3 is the diagonal matrix with diagonal elements k3zi/ (n32 ko). Multiplying each member of Eq. (48) by -jZ3 and using Eq. (49) to eliminate T vectors c- and c+ are related by:

Multiplying each member of Eq. (45) by jZ1 and using Eq. (46) to eliminate Ri a numerical computation can be found for c+ by making use of Eq.(50), that is:

where

Note in Eq. (51) that 8i/0 is a column vector. In the case of a solution truncated to the first negative and positive orders,

assuming the incident wave to be a plane wave. In this particular case

where Z1(2,2) is the element on line 2 and column 2 of matrix Z1. Finally, the vector on the right-hand side of Eq.(54) is applied to the inverse matrix of C to find the column vector for the diffracted amplitude c+ from Eq. (51). Then c- is found from Eq. (50) and the normalized electric field amplitudes for Ri and Ti can be found from Eqs. (48) and (49). Substituting Eq. (34) and Eq.(44) into Maxwell’s equations and eliminating Ez , it can be shown that

Eq. (55) is one of the two coupled-wave equations involving the inverse permittivity for the case of TM polarization only. In the conventional formulation (Wang et al., 1990; Magnusson & Wang, 1992; Tibuleac & Magnusson, 1997) the term el1, is treated by taking the inverse of the matrix E defined by the permittivity components (Moharam & Gaylord, 1981), with the i, p elements being equal to )(i-p). In the reformulation of the eigenvalue problem (Lalanne & Morris, 1996), the term eel1, is considered in a different manner by forming a matrix A of the inverse-permittivity coefficient harmonics for the two regions inside the modulated region. Fourier expansion in Eq.(34) is modified to:

where (1/s)s is the sth Fourier component of the relative permittivity in the grating region. Since the coupled-wave equations do not involve the inverse of the permittivity in the coupled-wave equations for the TE wave, matrix A is not needed in numerical computations and the eigenvalue problem is greatly simplified in this case. As a result, solutions for the TE wave are more stable in metallic lamellar gratings.

Only the DE in reflection and transmission for zeroth order are computed in the examples that will be discussed throughout this section. The diffraction efficiencies in both reflection (DEr) and transmission (DET) are defined as:

and

Examples with binary dielectric periodic gratings

Let us consider a binary rectangular-groove grating with real permittivity sl and sh as shown in figure 10. In the case of notch filters the higher permittivity value sh (A/2< x < A) is greater than sl (0< x < A / 2). Figure 11 shows the numerical computation for DE from the RCWA formulation for the TM wave when only three orders (m = -1, 0, 1) are retained in the computation.

Fig. 11. DEr and DET for a binary dielectric periodic grating for sl =4.00, sh = 4.41, A =314 nm, nL =1.00 (air), n3 =1.52 (glass) and h = 134 nm.

From the principle of energy conservation, the sum of DER and DET must be equal to unity. This principle is useful to decide if the number of orders retained in the computation is sufficient. As no deviation from unity is seen in the sum of DER and DET in figure 11, three orders is deemed to be enough to describe the diffraction efficiencies within this narrow wavelength spectrum. At a wavelength of roughly 511.3 nm all the optical energy is reflected back in the opposite direction from that of the incident light. As a result, DET is reaching a zero value as destructive interferences occur within the grating at this precise wavelength value of 511.3 nm.

Examples with metallic periodic gratings

The theory presented in section 3.1 can be applied to metallic periodic gratings. For the TM wave many terms need to be retained in the calculation to reach convergence (Li & Haggans, 1993). For the sake of saving time, a fairly accurate computation is reached after retaining ten orders when using the reformulated eigenvalue problem (Lalanne & Morris, 1996). Figure 12 shows DER for a metallic periodic grating using a 3D plot. Metallic periodic grating are used to excite surface plasmons (SP) to improve Surface-enhanced-Raman-Scattering (SERS) sensor performances (Sheng et al., 1982). At a given wavelength X the reflectivity of the metallic grating should be symmetric with the incident angle 0. If a reflectivity drop occurs due to SP at 0, the metallic periodic grating should display a similar drop at -0. Note that two minima occur on either side of normal incidence (0 = 0°) and one single minimum is displayed at normal incidence for X ~ 630 nm. Basically each minimum in DEr forms two valleys which crisscross at normal incidence and X ~ 630 nm. This point will become important in the next section where photonic band gap is discussed.

Fig. 12. 3D plot of DER for a periodic metallic grating. In the calculation, we used n1 = 1, 83 =e2 = -17.75 -0.7j, A = 600nm, )l = -17.75 – 0.7j, )h =1, and h = 10.5 nm.