Periodic Electromagnetic Lattices

Coupled plasmon modes have been observed experimentally in regularly spaced linear chains of gold nanoparticles [3.79] andin 2-D hexagonal arrays of silver nanoparticles [3.80]. The controlled propagation of plasmon excitations in tailored metal nanoparticle struc- 

Opals and Photonic-Band-Gap Materials 3-D ordered arrays of nanostractures with periods of the order of a fraction of the optical wavelength may show intense Bragg diffraction for specific wavelengths and diffraction angles. This phenomenon is the origin of the iridescent colors (opalescence) of opals. Opals consist of an fcc-like array of silica nanoparticles with sizes in the range 150-900 nm [3.82], with a size dispersion below 5% [3.83]. 

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The free electron resides in a quantized energy well, defined by k (in wave-numbers). This result Ccm be derived from the Schroedinger wave-equation. However, in the presence of a periodic array of electromagnetic potentials arising from the atoms confined in a crystalline lattice, the energies of the electrons from all of the atoms are severely limited in orbit and are restricted to specific allowed energy bands. This potential originates from attraction and repulsion of the electron clouds from the periodic array of atoms in the structure. Solutions to this problem were... 

Bravais lattice, 1. is an integer and a, is a basis vector of the periodic lattice. We assume that an electromagnetic field is produced by a current source J and we take... 

The fascinating phenomenon of surface plasmon resonance (SPR) occurs when an electromagnetic wave interacts with the conduction electrons of a metal (6). The periodic electric field of the electromagnetic wave causes a collective oscillation of the conductance electrons at a resonant frequency relative to the lattice of positive ions. Light is absorbed or scattered at this resonant frequency. The process of... 

Here p is a wavevector of an electromagnetic wave, which can propagates in our periodic structure (an eigenmode). Equations (12.13) state that, due to periodicity of the medium, the difference in wavevectors of possible modes should be equal to the lattice vector of the structure (p + qo) — (p — 0) = qo (remember, that, in a cholesteric, the period of the E-modulation is Po/2). Substituting F+ and F into Eq. (12.12) we find... 

Not all vibrations that occur in a solid lead to absorption of electromagnetic radiation. There are certain conditions, called selection rules, which must be fulfilled to get an absorption of light energy by molecular vibrations. In addition to the ordinary condition for IR activity (existence of a transition moment, see Krimm and Hummel ), we have lattice vibrations for which no periodicity in space is allowed, which means only frequencies with phase difference Acj) = 0 or infinite wavelength are observable. The quantity that gives the density of vibrational states falling into a certain interval, Ao), called the density of states Z(o)), plays an important role, since it is proportional to (d0)/dk) . Its maxima occur where the 03-k curve has a horizontal tangent. The rule is... 

Phonons are quantized vibrational waves, just as photons are quantized electromagnetic waves. In each case the energy of the quasi-particle is given by the famous Planck formula, E — hv, where v is the firequency of the light, in the case of the photon, or the frequency of the vibration, in the case of the phonon. Vibrational waves in a periodic one-dimensional lattice such as an ordered linear or helical polymer are periodic both in time and in space. Thus they possess both a frequency and a wave length, A. 

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