Earlier work has shown that a narrow-frequency-band, wide-angle emission is produced by an array of metal patches supported on a thin dielectric layer covering a ground plane. The modes responsible for this emission are local plasmons trapped under the metal patches. As the dielectric layer thickness, h d , is increased, the resonant emission fades in strength because the plasmon modes can no longer be trapped under a single patch. Further increases in h d , making it comparable to the light wavelength in the dielectric layer, lead to a collection of new emission peaks. These are narrower than the one peak found for small h d but they are not well separated. We have found that some of these peaks can be suppressed over a narrow range of h d . This leaves one with well-separated, narrow-band emission peaks. We have identified the physical mechanism for this selective suppression of emission peaks. Among the central properties of plasmonic nanostructures are the dispersion relations and excitation probabilities of various plasmon resonances. In this paper, we examine these properties within a special class of nanostructures, probed in a particular way. The structures have three layers as illustrated in Fig. 1. On the bottom is an opaque ground plane of thickness h f . Since no light can pass through this layer, what supports it is optically irrelevant. On top of the ground plane, there is a uniform dielectric layer of thickness h d . Finally, the dielectric layer is covered with a two-dimensional (2D) (or one-dimensional) grating made from metal patches (or stripes) of height h p and various shapes. For instance, the cross section of a patch could be round or rectangular or more complex.As a probe of these systems, we consider linearly polarized light of wavelength λ incident from the air above the layers. Referring to Fig. 1, the incident light propagates along +x and is polarized alongŷ. We examine the reflection coefficient, R, of exiting light and look for resonant dips in R as a function of λ. We limit our study to long enough wavelengths (or short enough grating periods) so that the only possible reflected beam is along −x; i.e., so that no diffracted beam can propagate away from the sample. Then, by Kirchoff's law, 1 one has for emission (E), absorption (A), and reflection (R) probabilities,Hence, a resonant dip in reflection implies a resonant peak in absorption or emission. For various practical applications, one would like the resonances to be strong, narrow, and well separated. Systems such as those illustrated in Fig. 1 have been often studied (see Refs. 2-31). This list only includes papers where the examined resonances are at wavelengths less than about 10 μm. A simple result is found when h d is larger than the light's penetration depth in the metal but still much smaller than its wavelength in the dielectric. A sequence of isolated, narrow, resonant dips in R is then possible. These resonances are due to the excitation of transverse electromagnetic (TEM) waves trapped underneath individual metal st...