Polarization state vector

Fig. 3—Measurement of surface by HDI surface reflectance analyzer. In electromagnetic radiation (light), the polarization direction is defined as the direction of the electric field vector. The incident polarization of the light can be controlled. The instrument uses a variety of detectors to analyze the reflected polarization state of the light. (U.S. Patent 6,134,011). (a) Plane of the disk The SRA uses a fixed 60 degree (from the surface normal) angle of incidence. The plane of incidence is the same as the paper plane (b) Pit on a surface detected by reflected light channels of HDI instrument (c) Scratches on disk surface measured by HDI surface reflectance analyzer (d) Particles on the surface of disk detected by reflected light (black spot) and by scattered light (white spot) [8].

Each polarization state, considered as a vector can be described in terms of its components, i.e. 

As with polarization, the orientation of spin is represented by state vectors. Just like ix and jy which serve as base vectors to decompose polarization vectors such as the diagonal vector considered above, a spin state like Sy can be represented by a vector s+, which is a linear combination of two base vectors s+ and s. This decomposition could be formulated, by analogy as... 

The photon state for a given momentum is characterized by the polarization unit vector e. On expanding e in terms of the unit vectors (53) and (54)... 

In the conventional formalism, as the p term does not appear, B L Jo. Now, in Roscoe s framework, as p 0, B is not perpendicular to the current flow, and therefore has a component in the direction of the current flow. It has been shown that the magnetization effects similar to the inverse Faraday effect (IFF) can be expected for appropriate polarization states of the transmitted radiation. Moreover, a massive vector boson can be constructed from the electromagnetic field so that it can be interpreted only as a nonzero mass photon. Here, the model suggested for photon can be interpreted as a bound system with discrete mass and frequency states. This may have important role in explaining redshift phenomena. 

The Stokes parameters for the polarization of an electron beam can be represented in a Cartesian basis which also provides a convenient pictorial view for the polarization state of an electron beam. Since the polarization of an ensemble of electrons requires the determination of spin projections along preselected directions, the classical vector model of a precessing spin will first be discussed. Here the spin is represented by a vector s of length 3/2 (in atomic units) which processes around a preselected direction, yielding as expectation values the projections (in atomic units, see Fig. 9.1)... 

In the discussion of light polarization so far the Cartesian basis and spherical basis have been considered. Because the linear polarization might be tilted with respect to the (ex, e -basis, a third basis system has to be introduced against which such a tilted polarization state can be measured via its non-vanishing components. This coordinate system is called (e e and its axes are rotated by +45° with respect to the previous ones. This leads to a third representation of the arbitrary vector b ... 

The consequence of dispersion in wavelength is that the polarization properties of the electric vector will fluctuate randomly in time. The parametric mapping of the electric vector shown in Figure 1.2 will produce blurred contours and the light will be partially polarized. If the light shows no preference towards a particular polarization state, it is referred to as unpolarized, or natural light. The Stokes vector for this case is... 

Unit vector defining the basis set for a polarization state, (1.54), (1.55) segmental absorption tensor, (5.4). 

Table 3.2-1 Stokes vectors describing the possible polarization states of light...

In order to proceed beyond a qualitative description of how a PTR operates, it is convenient to use a mathematical description of coherent polarization states, which are a good approximation to the output of solid-state near-millimeter sources. The Jones vector formalism is well known (Hecht and Zajac, 1979, pp. 268-270 LeSurf, 1990) and well suited to the present purpose. Any transverse polarization vector can be represented by an equation of the form E = -I- E y), where H and V are... 

In the last step the erroneous state vector is projected onto the subspace C to recover the initial information. Projection is a non-unitary process which cannot be achieved through a Hamiltonian process, but requires the introduction of irreversibility. To this end, we make use of a path which is symmetric with the pumping step, and consists in two stimulated and one spontaneous emissions. To be more explicit, we apply two left circularly polarized lasers slightly detuned from the transitions (60/ <—> 5d, j = 3/2) and (5d, j = 3/2 <—> 5p, j = 3/2). Due to these laser fields, the atom is likely to fall towards the ground state and emit two stimulated and one spontaneous photons. 

With the help of the selection rules, one can infer that, if a circularly right-polarized spontaneous photon is emitted, the only states to be coupled to the ground level are I71) and 72) to u ) and z/2), respectively (see Fig. 6). This means that the emission of a right polarized spontaneous photon brings the "correct" part of the state vector back into Hi = Span // ), On the contrary, the other cases - "left polarized", "linearly polarized spontaneous photon", or "no photon at all" - do not lead to the right projection process. 

In complete analogy to the atomic collective spin variables, the polarization state of a pulse of light can be described by a vector, the so-called Stokes vector. For light propagating along the 2-axis we define... 

In classical terms, radiation is represented by an electromagnetic wave. The polarization of plane-wave radiation is defined by the way the oscillating electric field evolves in space, in a plane perpendicular to the direction of propagation. The most general polarization state is called elliptical polarization [23], but for luminescence applications the subset of linear polarization states usually suffices. In these cases the electric field vector oscillates along a well defined direction in a plane perpendicular to the direction of propagation. This direction is the polarization direction, and radiation with this characteristic is said to be linearly polarized. 

As an example, consider the simple case of two single-mode fields of equal frequencies and polarizations. Assume that there are initially n photons in field 1 and m photons in the field 2, and that the state vectors of the fields are the Fock states /j) = n) and v /2) = m). The initial state of the two fields is the direct product of the single-field states, v /) = n) m). Inserting Eq. (27) into Eq. (28) and taking the expectation value with respect to the initial state of the fields, we find... 

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