Editing Polarization (waves) (section)

== Implications for reflection and propagation ==

=== Polarization in wave propagation ===

In a [[vacuum]], the components of the electric field propagate at the [[speed of light]], so that the phase of the wave varies in space and time while the polarization state does not. That is, the electric field vector {{math|'''e'''}} of a plane wave in the {{math|+''z''}} direction follows:
<math display="block">\mathbf{e}(z + \Delta z, t + \Delta t) = \mathbf{e}(z, t) e^{ik (c\Delta t - \Delta z)},</math>

where {{mvar|k}} is the [[wavenumber]]. As noted above, the instantaneous electric field is the real part of the product of the Jones vector times the phase factor {{nowrap|<math>e^{-i\omega t}</math>.}} When an electromagnetic wave interacts with matter, its propagation is altered according to the material's (complex) [[index of refraction]]. When the real or imaginary part of that refractive index is dependent on the polarization state of a wave, properties known as [[birefringence]] and polarization [[dichroism]] (or [[Diattenuator|diattenuation]]) respectively, then the polarization state of a wave will generally be altered.

In such media, an electromagnetic wave with any given state of polarization may be decomposed into two orthogonally polarized components that encounter different [[propagation constant]]s. The effect of propagation over a given path on those two components is most easily characterized in the form of a complex {{val|2|×|2}} [[linear transformation|transformation]] matrix {{math|'''J'''}} known as a [[Jones calculus|Jones matrix]]:
<math display="block">\mathbf{e'} = \mathbf{J}\mathbf{e}.</math>

The Jones matrix due to passage through a transparent material is dependent on the propagation distance as well as the birefringence. The birefringence (as well as the average refractive index) will generally be [[Dispersion (optics)|dispersive]], that is, it will vary as a function of optical frequency (wavelength). In the case of non-birefringent materials, however, the {{val|2|×|2}} Jones matrix is the identity matrix (multiplied by a scalar [[phase factor]] and attenuation factor), implying no change in polarization during propagation.

For propagation effects in two orthogonal modes, the Jones matrix can be written as
<math display="block">\mathbf{J} = \mathbf{T}\begin{bmatrix}
  g_1 & 0 \\
    0 & g_2
\end{bmatrix}\mathbf{T}^{-1},</math>

where {{math|''g''{{sub|1}}}} and {{math|''g''{{sub|2}}}} are complex numbers describing the [[phase delay]] and possibly the amplitude attenuation due to propagation in each of the two polarization [[eigenmode]]s. {{math|'''T'''}} is a [[unitary matrix]] representing a change of basis from these propagation modes to the linear system used for the Jones vectors; in the case of linear birefringence or diattenuation the modes are themselves linear polarization states so {{math|'''T'''}} and {{math|'''T'''{{sup|−1}}}} can be omitted if the coordinate axes have been chosen appropriately.

==== Birefringence ====
{{Main|Birefringence}}

In a [[birefringent]] substance, electromagnetic waves of different polarizations travel at different speeds ([[phase velocity|phase velocities]]).  As a result, when unpolarized waves travel through a plate of birefringent material, one polarization component has a shorter wavelength than the other, resulting in a [[phase difference]] between the components which increases the further the waves travel through the material.  The Jones matrix is a [[unitary matrix]]: {{math|1= {{abs|''g''{{sub|1}}}} = {{abs|''g''{{sub|2}}}} = 1}}. Media termed diattenuating (or ''[[dichroic]]'' in the sense of polarization), in which only the amplitudes of the two polarizations are affected differentially, may be described using a [[Hermitian matrix]] (generally multiplied by a common phase factor). In fact, since {{em|any}} matrix may be written as the product of unitary and positive Hermitian matrices, light propagation through any sequence of polarization-dependent optical components can be written as the product of these two basic types of transformations.

[[File:Birefringence Stress Plastic.JPG|thumb|Color pattern of a plastic box showing [[Birefringence#Stress induced birefringence|stress-induced birefringence]] when placed in between two crossed [[polarizer]]s.]]
In birefringent media there is no attenuation, but two modes accrue a differential phase delay. Well known manifestations of linear birefringence (that is, in which the basis polarizations are orthogonal linear polarizations) appear in optical [[wave plate]]s/retarders and many crystals. If linearly polarized light passes through a birefringent material, its state of polarization will generally change, {{em|unless}} its polarization direction is identical to one of those basis polarizations. Since the phase shift, and thus the change in polarization state, is usually wavelength-dependent, such objects viewed under white light in between two polarizers may give rise to colorful effects, as seen in the accompanying photograph.

Circular birefringence is also termed [[optical activity]], especially in [[chiral]] fluids, or [[Faraday rotation]], when due to the presence of a magnetic field along the direction of propagation. When linearly polarized light is passed through such an object, it will exit still linearly polarized, but with the axis of polarization rotated. A combination of linear and circular birefringence will have as basis polarizations two orthogonal elliptical polarizations; however, the term "elliptical birefringence" is rarely used.

[[File:Birefringence.svg|thumb|upright=1.1|left|Paths taken by vectors in the Poincaré sphere under birefringence. The propagation modes (rotation axes) are shown with red, blue, and yellow lines, the initial vectors by thick black lines, and the paths they take by colored ellipses (which represent circles in three dimensions).]]

One can visualize the case of linear birefringence (with two orthogonal linear propagation modes) with an incoming wave linearly polarized at a 45° angle to those modes. As a differential phase starts to accrue, the polarization becomes elliptical, eventually changing to purely circular polarization (90° phase difference), then to elliptical and eventually linear polarization (180° phase) perpendicular to the original polarization, then through circular again (270° phase), then elliptical with the original azimuth angle, and finally back to the original linearly polarized state (360° phase) where the cycle begins anew. In general the situation is more complicated and can be characterized as a [[coordinate rotation|rotation]] in the Poincaré sphere about the axis defined by the propagation modes. Examples for linear (blue), circular (red), and elliptical (yellow) [[birefringence]] are shown in the figure on the left. The total intensity and degree of polarization are unaffected. If the path length in the birefringent medium is sufficient, the two polarization components of a collimated beam (or [[Ray (optics)|ray]]) can exit the material with a positional offset, even though their final propagation directions will be the same (assuming the entrance face and exit face are parallel). This is commonly viewed using [[calcite]] [[crystal]]s, which present the viewer with two slightly offset images, in opposite polarizations, of an object behind the crystal. It was this effect that provided the first discovery of polarization, by [[Erasmus Bartholinus]] in 1669.

==== Dichroism ====
Media in which transmission of one polarization mode is preferentially reduced are called ''[[dichroism|dichroic]]'' or ''diattenuating''. Like birefringence, diattenuation can be with respect to linear polarization modes (in a crystal) or circular polarization modes (usually in a liquid).

Devices that block nearly all of the radiation in one mode are known as ''{{dfn|polarizing filters}}'' or simply "[[polarizer]]s". This corresponds to {{math|1= ''g''{{sub|2}} = 0}} in the above representation of the Jones matrix. The output of an ideal polarizer is a specific polarization state (usually linear polarization) with an amplitude equal to the input wave's original amplitude in that polarization mode. Power in the other polarization mode is eliminated. Thus if unpolarized light is passed through an ideal polarizer (where {{math|1= ''g''{{sub|1}} = 1}} and {{math|1= ''g''{{sub|2}} = 0}}) exactly half of its initial power is retained. Practical polarizers, especially inexpensive sheet polarizers, have additional loss so that {{math|''g''{{sub|1}} < 1}}. However, in many instances the more relevant figure of merit is the polarizer's [[degree of polarization]] or [[extinction ratio]], which involve a comparison of {{math|''g''{{sub|1}}}} to {{math|''g''{{sub|2}}}}. Since Jones vectors refer to waves' amplitudes (rather than [[Irradiance|intensity]]), when illuminated by unpolarized light the remaining power in the unwanted polarization will be {{math|(''g''{{sub|2}}/''g''{{sub|1}}){{sup|2}}}} of the power in the intended polarization.

=== Specular reflection ===
In addition to birefringence and dichroism in extended media, polarization effects describable using Jones matrices can also occur at (reflective) interface between two materials of different [[refractive index]]. These effects are treated by the [[Fresnel equations]]. Part of the wave is transmitted and part is reflected; for a given material those proportions (and also the phase of reflection) are dependent on the [[angle of incidence (optics)|angle of incidence]] and are different for the ''s''- and ''p''-polarizations. Therefore, the polarization state of reflected light (even if initially unpolarized) is generally changed.

[[File:Brewster-polarizer.svg|class=skin-invert-image|right|thumb|upright=1.25|A stack of plates at Brewster's angle to a beam reflects off a fraction of the ''s''-polarized light at each surface, leaving (after many such plates) a mainly ''p''-polarized beam.]]
Any light striking a surface at a special angle of incidence known as [[Brewster's angle]], where the reflection coefficient for ''p''-polarization is zero, will be reflected with only the ''s''-polarization remaining. This principle is employed in the so-called "pile of plates polarizer" (see figure) in which part of the ''s''-polarization is removed by reflection at each Brewster angle surface, leaving only the ''p''-polarization after transmission through many such surfaces. The generally smaller reflection coefficient of the ''p''-polarization is also the basis of [[polarized sunglasses]]; by blocking the ''s''- (horizontal) polarization, most of the glare due to reflection from a wet street, for instance, is removed.<ref name=Hecht2002 />{{rp|348–350}}

In the important special case of reflection at normal incidence (not involving anisotropic materials) there is no particular ''s''- or ''p''-polarization. Both the {{mvar|x}} and {{mvar|y}} polarization components are reflected identically, and therefore the polarization of the reflected wave is identical to that of the incident wave. However, in the case of circular (or elliptical) polarization, the handedness of the polarization state is thereby reversed, since by [[Circular polarization#Left/Right|convention]] this is specified relative to the direction of propagation. The circular rotation of the electric field around the {{mvar|''x-y''}} axes called "right-handed" for a wave in the {{math|+''z''}} direction is "left-handed" for a wave in the {{math|−''z''}} direction. But in the general case of reflection at a nonzero angle of incidence, no such generalization can be made. For instance, right-circularly polarized light reflected from a dielectric surface at a grazing angle, will still be right-handed (but elliptically) polarized. Linear polarized light reflected from a metal at non-normal incidence will generally become elliptically polarized. These cases are handled using Jones vectors acted upon by the different Fresnel coefficients for the ''s''- and ''p''-polarization components.