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Diagram of the electric field of a light wave (blue), linear-polarized along a plane (purple line), and consisting of two orthogonal, in-phase components (red and green waves)

In electrodynamics, linear polarization or plane polarization of electromagnetic radiation is a confinement of the electric field vector or magnetic field vector to a given plane along the direction of propagation. See polarization and plane of polarization for more information.

The orientation of a linearly polarized electromagnetic wave is defined by the direction of the electric field vector.^[1] For example, if the electric field vector is vertical (alternately up and down as the wave travels) the radiation is said to be vertically polarized.

Mathematical description of linear polarization

The classical sinusoidal plane wave solution of the electromagnetic wave equation for the electric and magnetic fields is (cgs units)

E(r,t)=∣E∣Re{|ψ⟩exp⁡[i(kz−ωt)]}{displaystyle mathbf {E} (mathbf {r} ,t)=mid mathbf {E} mid mathrm {Re} left{|psi rangle exp left[ileft(kz-omega tright)right]right}}

B(r,t)=z^×E(r,t)/c{displaystyle mathbf {B} (mathbf {r} ,t)={hat {mathbf {z} }}times mathbf {E} (mathbf {r} ,t)/c}

for the magnetic field, where k is the wavenumber,

ω=ck{displaystyle omega _{}^{}=ck}

is the angular frequency of the wave, and c{displaystyle c} is the speed of light.

Here ∣E∣{displaystyle mid mathbf {E} mid } is the amplitude of the field and

|ψ⟩ =def (ψxψy)=(cos⁡θexp⁡(iαx)sin⁡θexp⁡(iαy)){displaystyle |psi rangle {stackrel {mathrm {def} }{=}} {begin{pmatrix}psi _{x}\psi _{y}end{pmatrix}}={begin{pmatrix}cos theta exp left(ialpha _{x}right)\sin theta exp left(ialpha _{y}right)end{pmatrix}}}

is the Jones vector in the x-y plane.

The wave is linearly polarized when the phase angles αx,αy{displaystyle alpha _{x}^{},alpha _{y}} are equal,

αx=αy =def α{displaystyle alpha _{x}=alpha _{y} {stackrel {mathrm {def} }{=}} alpha }.

This represents a wave polarized at an angle θ{displaystyle theta } with respect to the x axis. In that case, the Jones vector can be written

|ψ⟩=(cos⁡θsin⁡θ)exp⁡(iα){displaystyle |psi rangle ={begin{pmatrix}cos theta \sin theta end{pmatrix}}exp left(ialpha right)}.

The state vectors for linear polarization in x or y are special cases of this state vector.

If unit vectors are defined such that

|x⟩ =def (10){displaystyle |xrangle {stackrel {mathrm {def} }{=}} {begin{pmatrix}1\0end{pmatrix}}}

and

|y⟩ =def (01){displaystyle |yrangle {stackrel {mathrm {def} }{=}} {begin{pmatrix}0\1end{pmatrix}}}

then the polarization state can be written in the "x-y basis" as

|ψ⟩=cos⁡θexp⁡(iα)|x⟩+sin⁡θexp⁡(iα)|y⟩=ψx|x⟩+ψy|y⟩{displaystyle |psi rangle =cos theta exp left(ialpha right)|xrangle +sin theta exp left(ialpha right)|yrangle =psi _{x}|xrangle +psi _{y}|yrangle }.

See also

• Sinusoidal plane-wave solutions of the electromagnetic wave equation


• 
  Polarization
  
  
  
  • Circular polarization
  
  
  • Elliptical polarization
  
  
  • Plane of polarization
  
  
  
  


• Photon polarization

References

• 
  Jackson, John D. (1998). Classical Electrodynamics (3rd ed.). Wiley. ISBN 0-471-30932-X..mw-parser-output cite.citation{font-style:inherit}.mw-parser-output q{quotes:"""""""'""'"}.mw-parser-output code.cs1-code{color:inherit;background:inherit;border:inherit;padding:inherit}.mw-parser-output .cs1-lock-free a{background:url("//upload.wikimedia.org/wikipedia/commons/thumb/6/65/Lock-green.svg/9px-Lock-green.svg.png")no-repeat;background-position:right .1em center}.mw-parser-output .cs1-lock-limited a,.mw-parser-output .cs1-lock-registration a{background:url("//upload.wikimedia.org/wikipedia/commons/thumb/d/d6/Lock-gray-alt-2.svg/9px-Lock-gray-alt-2.svg.png")no-repeat;background-position:right .1em center}.mw-parser-output .cs1-lock-subscription a{background:url("//upload.wikimedia.org/wikipedia/commons/thumb/a/aa/Lock-red-alt-2.svg/9px-Lock-red-alt-2.svg.png")no-repeat;background-position:right .1em center}.mw-parser-output .cs1-subscription,.mw-parser-output .cs1-registration{color:#555}.mw-parser-output .cs1-subscription span,.mw-parser-output .cs1-registration span{border-bottom:1px dotted;cursor:help}.mw-parser-output .cs1-hidden-error{display:none;font-size:100%}.mw-parser-output .cs1-visible-error{font-size:100%}.mw-parser-output .cs1-subscription,.mw-parser-output .cs1-registration,.mw-parser-output .cs1-format{font-size:95%}.mw-parser-output .cs1-kern-left,.mw-parser-output .cs1-kern-wl-left{padding-left:0.2em}.mw-parser-output .cs1-kern-right,.mw-parser-output .cs1-kern-wl-right{padding-right:0.2em}
  

External links

• Animation of Linear Polarization (on YouTube)


• Comparison of Linear Polarization with Circular and Elliptical Polarizations (YouTube Animation)

 This article incorporates public domain material from the General Services Administration document "Federal Standard 1037C".