Electromagnetic radiation special case
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. The term linear polarization (French: polarisation rectiligne ) was coined by
Augustin-Jean Fresnel in 1822.
[1] 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.
[2] 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
The
classical
sinusoidal plane wave solution of the
electromagnetic wave equation for the
electric and
magnetic fields is (cgs units)
E
(
r
,
t
)
=
|
E
|
R
e
{
|
ψ
⟩
exp
i
(
k
z
−
ω
t
)
}
{\displaystyle \mathbf {E} (\mathbf {r} ,t)=|\mathbf {E} |\mathrm {Re} \left\{|\psi \rangle \exp \left[i\left(kz-\omega t\right)\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
=
c
k
{\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
⟩
=
d
e
f
(
ψ
x
ψ
y
)
=
(
cos
θ
exp
(
i
α
x
)
sin
θ
exp
(
i
α
y
)
)
{\displaystyle |\psi \rangle \ {\stackrel {\mathrm {def} }{=}}\ {\begin{pmatrix}\psi _{x}\\\psi _{y}\end{pmatrix}}={\begin{pmatrix}\cos \theta \exp \left(i\alpha _{x}\right)\\\sin \theta \exp \left(i\alpha _{y}\right)\end{pmatrix}}}
is the
Jones vector in the x-y plane.
The wave is linearly polarized when the phase angles
α
x
,
α
y
{\displaystyle \alpha _{x}^{},\alpha _{y}}
are equal,
α
x
=
α
y
=
d
e
f
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
θ
sin
θ
)
exp
(
i
α
)
{\displaystyle |\psi \rangle ={\begin{pmatrix}\cos \theta \\\sin \theta \end{pmatrix}}\exp \left(i\alpha \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
⟩
=
d
e
f
(
1
0
)
{\displaystyle |x\rangle \ {\stackrel {\mathrm {def} }{=}}\ {\begin{pmatrix}1\\0\end{pmatrix}}}
and
|
y
⟩
=
d
e
f
(
0
1
)
{\displaystyle |y\rangle \ {\stackrel {\mathrm {def} }{=}}\ {\begin{pmatrix}0\\1\end{pmatrix}}}
then the polarization state can be written in the "x-y basis" as
|
ψ
⟩
=
cos
θ
exp
(
i
α
)
|
x
⟩
+
sin
θ
exp
(
i
α
)
|
y
⟩
=
ψ
x
|
x
⟩
+
ψ
y
|
y
⟩
{\displaystyle |\psi \rangle =\cos \theta \exp \left(i\alpha \right)|x\rangle +\sin \theta \exp \left(i\alpha \right)|y\rangle =\psi _{x}|x\rangle +\psi _{y}|y\rangle }
.
See also
References
Jackson, John D. (1998). Classical Electrodynamics (3rd ed.) . Wiley.
ISBN
0-471-30932-X .
^ A. Fresnel, "Mémoire sur la double réfraction que les rayons lumineux éprouvent en traversant les aiguilles de cristal de roche suivant les directions parallèles à l'axe", read 9 December 1822; printed in H. de Senarmont, E. Verdet, and L. Fresnel (eds.), Oeuvres complètes d'Augustin Fresnel , vol. 1 (1866), pp. 731–51; translated as "Memoir on the double refraction that light rays undergo in traversing the needles of quartz in the directions parallel to the axis",
Zenodo :
4745976 , 2021 (open access); §9.
^ Shapira, Joseph; Shmuel Y. Miller (2007).
CDMA radio with repeaters . Springer. p. 73.
ISBN
978-0-387-26329-8 .
External links
This article incorporates
public domain material from
Federal Standard 1037C .
General Services Administration . Archived from
the original on January 22, 2022.