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Polarization (waves)
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==== Transverse electromagnetic waves ==== [[File:Electromagnetic wave2.svg|class=skin-invert-image|thumb|center|upright=2.6|A "vertically polarized" electromagnetic wave of wavelength λ has its electric field vector {{math|'''E'''}} (red) oscillating in the vertical direction. The magnetic field {{math|'''B'''}} (or {{math|'''H'''}}) is always at right angles to it (blue), and both are perpendicular to the direction of propagation ({{math|'''z'''}}).]] [[Electromagnetic waves]] (such as light), traveling in free space or another [[Homogeneity (physics)|homogeneous]] [[isotropic]] [[Attenuation|non-attenuating]] medium, are properly described as [[transverse waves]], meaning that a plane wave's electric field vector {{math|'''E'''}} and magnetic field {{math|'''H'''}} are each in some direction perpendicular to (or "transverse" to) the direction of wave propagation; {{math|'''E'''}} and {{math|'''H'''}} are also perpendicular to each other. By convention, the "polarization" direction of an electromagnetic wave is given by its electric field vector. Considering a monochromatic [[plane wave]] of optical frequency {{mvar|f}} (light of vacuum wavelength {{mvar|λ}} has a frequency of {{math|1=''f'' = ''c/λ''}} where {{mvar|c}} is the speed of light), let us take the direction of propagation as the {{mvar|z}} axis. Being a transverse wave the {{math|'''E'''}} and {{math|'''H'''}} fields must then contain components only in the {{mvar|x}} and {{mvar|y}} directions whereas {{math|1=''E<sub>z</sub>'' = ''H<sub>z</sub>'' = 0}}. Using [[Complex number|complex]] (or [[phasor]]) notation, the instantaneous physical electric and magnetic fields are given by the [[real part]]s of the complex quantities occurring in the following equations. As a function of time {{mvar|t}} and spatial position {{mvar|z}} (since for a plane wave in the {{math|+''z''}} direction the fields have no dependence on {{mvar|x}} or {{mvar|y}}) these complex fields can be written as: <math display="block">\vec{E}(z, t) = \begin{bmatrix} e_x \\ e_y \\ 0 \end{bmatrix}\; e^{i2\pi \left(\frac{z}{\lambda} - \frac{t}{T}\right)} = \begin{bmatrix} e_x \\ e_y \\ 0 \end{bmatrix}\; e^{i(kz - \omega t)}</math> and <math display="block">\vec{H}(z, t) = \begin{bmatrix} h_x \\ h_y \\ 0 \end{bmatrix}\; e^{i2\pi \left(\frac{z}{\lambda} - \frac{t}{T}\right)} = \begin{bmatrix} h_x \\ h_y \\ 0 \end{bmatrix}\; e^{i(kz - \omega t)},</math> where {{math|1=λ = λ{{sub|0}}/''n''}} is the wavelength {{em|in the medium}} (whose [[refractive index]] is {{mvar|n}}) and {{math|1=''T'' = 1/''f''}} is the period of the wave. Here {{mvar|e<sub>x</sub>}}, {{mvar|e<sub>y</sub>}}, {{mvar|h<sub>x</sub>}}, and {{mvar|h<sub>y</sub>}} are complex numbers. In the second more compact form, as these equations are customarily expressed, these factors are described using the [[wavenumber]] {{math|1=''k'' = 2π''n''/''λ''{{sub|0}} }} and [[angular frequency]] (or "radian frequency") {{math|1=''ω'' = 2π''f''}}. In a more general formulation with propagation {{em|not}} restricted to the {{math|''+z''}} direction, then the spatial dependence {{math|''kz''}} is replaced by {{math|{{vec|''k''}} ∙ {{vec|''r''}}}} where {{mvar|{{vec|k}}}} is called the [[wave vector]], the magnitude of which is the wavenumber. Thus the leading vectors {{math|'''e'''}} and {{math|'''h'''}} each contain up to two nonzero (complex) components describing the amplitude and phase of the wave's {{mvar|x}} and {{mvar|y}} polarization components (again, there can be no {{mvar|z}} polarization component for a transverse wave in the {{math|+''z''}} direction). For a given medium with a [[wave impedance|characteristic impedance]] {{mvar|η}}, {{math|'''h'''}} is related to {{math|'''e'''}} by: <math display="block">\begin{align} h_y &= \frac{e_x}{\eta} \\ h_x &= -\frac{e_y}{\eta}. \end{align}</math> In a dielectric, {{mvar|η}} is real and has the value {{math|''η''<sub>0</sub>/''n''}}, where {{mvar|n}} is the refractive index and {{math|''η''<sub>0</sub>}} is the [[impedance of free space]]. The impedance will be complex in a conducting medium. Note that given that relationship, the [[dot product]] of {{math|'''E'''}} and {{math|'''H'''}} must be zero: <math display="block">\begin{align} \vec{E}\left(\vec{r}, t\right) \cdot \vec{H}\left(\vec{r}, t\right) &= e_x h_x + e_y h_y + e_z h_z \\ &= e_x \left(-\frac{e_y}{\eta}\right) + e_y \left(\frac{e_x}{\eta}\right) + 0 \cdot 0 \\ &= 0, \end{align}</math> indicating that these vectors are [[orthogonal]] (at right angles to each other), as expected. Knowing the propagation direction ({{math|+''z''}} in this case) and {{mvar|η}}, one can just as well specify the wave in terms of just {{mvar|e<sub>x</sub>}} and {{mvar|e<sub>y</sub>}} describing the electric field. The vector containing {{mvar|e<sub>x</sub>}} and {{mvar|e<sub>y</sub>}} (but without the {{mvar|z}} component which is necessarily zero for a transverse wave) is known as a [[Jones vector]]. In addition to specifying the polarization state of the wave, a general Jones vector also specifies the overall magnitude and phase of that wave. Specifically, the [[intensity (physics)|intensity]] of the light wave is proportional to the sum of the squared magnitudes of the two electric field components: <math display="block"> I = \left(\left|e_x\right|^2 + \left|e_y\right|^2\right) \, \frac{1}{2\eta} </math> However, the wave's ''state of polarization'' is only dependent on the (complex) ratio of {{mvar|e<sub>y</sub>}} to {{mvar|e<sub>x</sub>}}. So let us just consider waves whose {{math|1={{abs|''e<sub>x</sub>''}}<sup>2</sup> + {{abs|''e<sub>y</sub>''}}<sup>2</sup> = 1}}; this happens to correspond to an intensity of about {{val|.00133|ul=W|up=m2}} in free space (where {{math|1=''η'' = ''η''<sub>0</sub>}}). And because the absolute phase of a wave is unimportant in discussing its polarization state, let us stipulate that the phase of {{mvar|e<sub>x</sub>}} is zero; in other words {{mvar|e<sub>x</sub>}} is a real number while {{mvar|e<sub>y</sub>}} may be complex. Under these restrictions, {{mvar|e<sub>x</sub>}} and {{mvar|e<sub>y</sub>}} can be represented as follows: <math display="block">\begin{align} e_x &= \sqrt{\frac{1 + Q}{2}} \\ e_y &= \sqrt{\frac{1 - Q}{2}}\, e^{i\phi}, \end{align}</math> where the polarization state is now fully parameterized by the value of {{mvar|Q}} (such that {{math|−1 < ''Q'' < 1}}) and the relative phase {{mvar|ϕ}}.
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