Polarization wave creation

The interaction between light and matter can be viewed as the creation of a coherent quantum superposition of initial and final electron states that has an associated polarization [3], as shown in Figure 1. The coherence between states with different wave vector requires an intermediate virtual state and the presence of a coherent phonon. A transition between the initial and final states may occur when the coherence of the system is broken either due to the finite width of an optical wave packet or by scattering from the environment. The transition results in the absorption of a photon and the creation of a hot electron-hole pair. Otherwise, the photon is re-radiated with a different phase and, perhaps, polarisation. 

As usual, the index k includes the wave vector k and the polarizations e [ and e2 perpendicular to k, and al (at) is the creation (annihilation) operator for photons of energy h(ok = he k 1. We must write in this formalism the operators, vector potential, and electric field which are involved in our calculations ... 

The simplest way to show the principal difference between the representations of plane and multipole photons is to compare the number of independent quantum operators (degrees of freedom), describing the monochromatic radiation field. In the case of plane waves of photons with given wavevector k (energy and linear momentum), there are only two independent creation or annihilation operators of photons with different polarization [2,14,15]. It is well known that QED (quantum electrodynamics) interprets the polarization as given spin state of photons [4]. The spin of photon is known to be 1, so that there are three possible spin states. In the case of plane waves, projection of spin on the... 

In turn, the monochromatic multipole photons are described by the scalar wavenumber k (energy), parity (type of radiation either electric or magnetic), angular momentum j 1,2,..., and projection m = —j,..., / [2,26,27]. This means that even in the simplest case of monochromatic dipole (j = 1) photons of either type, there are three independent creation or annihilation operators labeled by the index m = 0, 1. Thus, the representation of multipole photons has much physical properties in comparison with the plane waves of photons. For example, the third spin state is allowed in this case and therefore the quantum multipole radiation is specified by three different polarizations, two transversal and one longitudinal (with respect to the radial direction from the source) [27,28], In contrast to the plane waves of photons, the projection of spin is not a quantum number in the case of multipole photons. Therefore, the polarization is not a global characteristic of the multipole radiation but changes with distance from the source [22],... 

Considering Equation 6.38 again, we need to transform the Hamiltonian expression. Thus, if cos(k) and ss(k) are the frequency and the polarization vector for the classic modes with polarization s and wave vector k, respectively, we can define the phonon creation (aks+) and annihilation ( /,s ) operators as... 

Figure 4.20 shows that waves can be created in any direction v in a crystal. It is sufficient that the incident light be perpendicular to a crystal plate cut in a chosen direction. For a non-polarized incident ray, the crystal produces two light rays of mutually perpendicular polarizations D and D. If one of the two rays is suppressed, the crystal emits a ray that is perfectly polarized. This is the principle of the Nicol prism. In contrast, the creation of a wave in a selected direction s is not so simple. 

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