Propagation of Electromagnetic Waves in Thin Dielectric and Metallic Films Part 3

Photonic bandgap in metallic periodic gratings

Resonant surface plasmon (SP) coupling involving metallic periodic gratings has been extensively studied over the past years and more recently in work looking at photonic devices (Park et al., 2003; Barnes et al. 2003; Ebbesen et al., 1998; Ye & Zhang, 2004) surface-enhanced Raman scattering (Sheng et al., 1982) and photonic bandgaps (Barnes et al., 1996). Corrugated surfaces are commonly produced by direct exposure of a photoresist film to a holographic interference pattern. There is some experimental evidence that owing to nonlinear response of the photoresist, this technique leads to the presence of higher harmonics in addition to the fundamental pattern that is inscribed (Gallatin, 1987; Pai & Awada, 1991). The higher harmonics can then influence the propagation of the SP on the metallic periodic grating and, in particular, can generate a bandgap in the plasmon dispersion curve.

Generating a photonic bandgap with two metallic periodic gratings

Let us consider two metallic sinusoidal gratings with vectors K1 (= 2ft/ A1) and K2 (=2ft/A2) inscribed at the same location on the film surface. One grating acts as a coupler that allows light to generate SPs while the second grating creates a bandgap in the dispersion curve for the SP propagation. Herein, we consider the case K2 = 2Kl. More complicated cases such as K2 < 2Kl have also been investigated and may be found in the literature (Levesque & Rochon , 2005). The SP dispersion curve for a uniform silver or gold film in the absence of a gap is shown in Figure 13a and is described by:

where kSP is the wave vector of the SP modes coupled at the surface and sm and sd are the permittivities of the metal and dielectric material (air). The dispersion line for light incident at an angle 0 and scattered by a vector Kl is given by:

Fig. 13. a) SP dispersion curves for one periodic grating of Bragg vector Kl b) Normalized reflectance (Rp/Rs) curve for a single metallic grating with A ~ 755 nm.

Note from Figure 13a that at normal incidence (0 = 0°) SPs will be excited at a single wavelength from a loss or gain of light momentum by the grating Bragg vector Kl = 2ft/ A. Scattering of incident light from the metallic grating at a given incident angle can fulfill the phase-matching condition (kSP = klight) for SP excitation. As 0 increases from zero SPs can be generated if light scatters by a Bragg vector + Kl, i.e., two valleys will form for 0 > 0° and 0 < 0° as shown in Figure 13b. Experimentally, a fairly sharp dip in the reflectivity curve (Rp) was observed for the p-polarized light but not for the s-polarized light (Rs). To emphasize the SP contribution, the Rp reflectance curves were normalized to Rs in the range 600-900 nm spectral range. Curves shown in Figure 13b were predicted by DER computations shown in Figure 12.

The SP dispersion curves for the doubly corrugated surfaces and light lines are shown in Figure 14a.

It can be seen from Figure 14a) that two SPs can be generated at normal incidence as a bandgap is being created by the grating of Bragg vector K2. As a result, SPs can be generated at m = roL and m = m2. This means the band will open as shown in Figure 14b, where two minima are shown in the experimental data points for all incident angles (Levesque & Rochon , 2005). Each of these minima corresponds to SP excitation at the air-metal grating surface when light is scattered by Bragg vector +Kl.

Fig. 14. a) SP dispersion curves for two superimposed periodic grating of Bragg vector K1 and K2 b) Normalized reflectance curve for a doubly corrugated surface with Ai ~ 755 nm and A2 ~ 375 nm.

Processing of the single and double metallic corrugated surfaces

Surface corrugations with selected pitches (Bragg vectors) can be produced on azopolymer films by direct exposure of an interference pattern from two coherent light beams at X = 532 nm, as shown in Figure 15a. The two desired spacing are obtained by adjusting the angle 9 (c. f fig.15a) between the writing beams, and their depth is determined by their exposure time. The films under investigation have two superimposed sinusoidal gratings with vectors K1 and K2. These azopolymer films were prepared on glass slides and then coated with a 50 nm thick gold film by sputter.

Fig. 15. a) Experimental set-up to produce corrugated metallic grating. b) Atomic force microscope image of a double metallic grating. Pitches here are 700 and 375 nm with their respective depths of 19+ 1 nm and 7.0+0.5 nm.

The surface profile s(x) shown in the atomic force microscope image in Figure 15b can be represented as

where x is the spatial coordinate, h1 and h2 are the amplitudes of the two harmonic components, and is their relative phase.

Conclusion

The matrix formalism was shown to be efficient to predict the reflectance curves of both uniform films and periodic corrugated surfaces. It was shown in this topic that the reflectance derived from the matrix formalism allows precise determinations of refractive indices and thickness when it is fitted to experimental data points. The principle to determine a good fit from the minimization of the sum of squares was presented in some details. The application of the sum of squares in more complex structures involving transparent overlaying films were also introduced along with waveguide modes. It was also shown that the matrix formalism can be used in numerical techniques and can be applied to periodic gratings to predict diffraction efficiencies. Systems of metallic periodic gratings were discussed and it was shown that photonic bandgap can be produced by superposition of two inscribed corrugated surface on an azopolymer film. The modulated films were made by holographic technique to write surface relief structures. One grating is written to have a spacing vector K2 to generate a bandgap in the SP dispersion curve. A second grating with grating spacing vector K1 is superimposed and allows the coupling of the incident light to generate the SP itself.