Other types of polarization[edit] In addition to transverse waves, there are many wave motions where the oscillation is not limited to directions perpendicular to the direction of propagation. These cases are beyond the scope of the current article which concentrates on transverse waves (such as most electromagnetic waves in bulk media), however one should be aware of cases where the polarization of a coherent wave cannot be described simply using a Jones vector, as we have just done. Just considering electromagnetic waves, we note that the preceding discussion strictly applies to plane waves in a homogeneous isotropic non-attenuating medium, whereas in an anisotropic medium (such as birefingent crystals as discussed below) the electric and/or magnetic field may have longitudinal as well as transverse components. In those cases the electric displacement D and magnetic flux density B still obey the above geometry but due to anisotropy in the electric susceptibility (or in the magnetic permeability), now given by a tensor, the direction of E (or H) may differ from that of D (or B). Even in isotropic media, so-called inhomogeneous waves can be launched into a medium whose refractive index has a significant imaginary part (or extinction coefficient) such as metals; these fields are also not strictly transverse. Surface waves or waves propagating in a waveguide (such as an optical fiber) are generally not transverse waves, but might be described as an electric or magnetic transverse mode, or a hybrid mode. Even in free space, longitudinal field components can be generated in focal regions, where the plane wave approximation breaks down. An extreme example is radially or tangentially polarized light, at the focus of which the electric or magnetic field respectively is entirely longitudinal (along the direction of propagation).[1] For longitudinal waves such as sound waves in fluids, the direction of oscillation is by definition along the direction of travel, so the issue of polarization is not normally even mentioned. On the other hand, sound waves in a bulk solid can be transverse as well as longitudinal, for a total of three polarization components. In this case, the transverse polarization is associated with the direction of the shear stress and displacement in directions perpendicular to the propagation direction, while the longitudinal polarization describes compression of the solid and vibration along the direction of propagation. The differential propagation of transverse and longitudinal polarizations is important in seismology. Polarization state[edit] Over each cycle of a sinusoidal wave, the electric field vector (as well as the magnetic field) traces out an ellipse; note that a line and circle are special cases of ellipses. The shape and orientation of this ellipse (or line) defines the polarization state. The following figures show some examples of the evolution of the electric field vector (black), with time (the vertical axes), at a particular point in space, along with its x (red) and y (blue) components; at the base is the path traced by the vector in the transverse plane. Linear polarization diagram Linear Circular polarization diagram Circular Elliptical polarization diagram Elliptical In the leftmost figure above, the electric fields x and y components (according to the axes we have established) are exactly in phase. The net result is polarization along a particular direction (depending on the relative amplitudes of the two components) in the x-y plane over each cycle. Since the vector traces out a single line in the plane, this special case is called linear polarization. Most polarizing filters produce linear polarization from unpolarized light. In the middle figure, the x and y components still have the same amplitude but now are exactly ninety degrees out of phase. In this special case the electric vector traces out a circle in the plane, and is thus referred to as circular polarization. Depending on whether the phase difference is + or -90 degrees, it may be qualified as right-hand circular polarization or left-hand circular polarization, depending on ones convention. The more general case with the two components out of phase by a different amount, or 90 degrees out of phase but with different amplitudes [2] is called elliptical polarization because the electric vector traces out an ellipse (the polarization ellipse). This is shown in the above figure on the right. Again, the same ellipse shape can be produced either by a clockwise or counterclockwise rotation of the field, corresponding to distinct polarization states. Animation of a circularly polarized wave as a sum of two components Of course the orientation of the x and y axes in such a picture is arbitrary, and any state of polarization can be represented regardless. One would typically chose axes to suit a particular problem such as x being in the plane of incidence. Moreover, one can use as basis functions any pair of orthogonal polarization states. Beyond the linear polarizations we have used here, the most useful choice is right and left circularly polarized states. The Cartesian polarization decomposition is natural when dealing with reflection from surfaces, birefringent materials, or synchrotron radiation. The circularly polarized modes are a more useful basis for the study of light propagation in stereoisomers. Parameterization[edit] This section needs attention from an expert in Physics. Please add a reason or a talk parameter to this template to explain the issue with the section. WikiProject Physics (or its Portal) may be able to help recruit an expert. (February 2009) Polarisation ellipse2.svg For ease of visualization, polarization states are often specified in terms of the polarization ellipse, specifically its orientation and elongation. A common parameterization uses the orientation angle, ψ, the angle between the major semi-axis of the ellipse and the x-axis[3] (also known as tilt angle or azimuth angle[citation needed]) and the ellipticity, ε, the major-to-minor-axis ratio[4][5][6][7] (also known as the axial ratio). An ellipticity of zero or infinity corresponds to linear polarization and an ellipticity of 1 corresponds to circular polarization. The ellipticity angle, χ = arccot ε= arctan 1/ε, is also commonly used.[3] An example is shown in the diagram to the right. An alternative to the ellipticity or ellipticity angle is the eccentricity; however, unlike the azimuth angle and ellipticity angle, the latter has no obvious geometrical interpretation in terms of the Poincaré sphere (see below). Full information on a completely polarized state is also provided by the amplitude and phase of oscillations in two components of the electric field vector in the plane of polarization. This representation was used above to show how different states of polarization are possible. The amplitude and phase information can be conveniently represented as a two-dimensional complex vector (the Jones vector): \mathbf{e} = \begin{bmatrix} a_1 e^{i \theta_1} \\ a_2 e^{i \theta_2} \end{bmatrix} . Here a_1 and a_2 denote the amplitude of the wave in the two components of the electric field vector, while \theta_1 and \theta_2 represent the phases. The product of a Jones vector with a complex number of unit modulus gives a different Jones vector representing the same ellipse, and thus the same state of polarization. The physical electric field, as the real part of the Jones vector, would be altered but the polarization state itself is independent of absolute phase. The basis vectors used to represent the Jones vector need not represent linear polarization states (i.e. be real). In general any two orthogonal states can be used, where an orthogonal vector pair is formally defined as one having a zero inner product. A common choice is left and right circular polarizations, for example to model the different propagation of waves in two such components in circularly birefringent media (see below) or signal paths of coherent detectors sensitive to circular polarization. Regardless of whether polarization ellipses are represented using geometric parameters or Jones v
Posted on: Mon, 27 Jan 2014 14:15:56 +0000
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