# Polarization

Polarization encoding is stored in Annotated Basic State objects as a special P annotation.

Their value follows Jones calculus. Annotations values are represented by two angles $$(\theta, \phi)$$.

The representation of the polarization in $$\begin{pmatrix}E_h\\E_v\end{pmatrix}$$ basis is obtained by applying Jones conversion: $$\overrightarrow{J}=\begin{pmatrix}\cos \frac{\theta}{2}\\e^{i\phi}\sin \frac{\theta}{2}\end{pmatrix}$$. The same can also be noted: $$\cos \frac{\theta}{2}\ket{H}+e^{i\phi}\sin \frac{\theta}{2}\ket{V}$$.

For instance, the following defines a polarization with $$\theta=\frac{\pi}{2},\phi=\frac{\pi}{4}$$ corresponding to Jones vector: $$\begin{pmatrix}\cos \frac{\pi}{4}\\e^{i\frac{\pi}{4}}\sin \frac{\pi}{4}\end{pmatrix}$$

>>> p = pcvl.Polarization(sp.pi/2, sp.pi/4)
>>> p.project_ev_eh()
(sqrt(2)/2, sqrt(2)*exp(I*pi/4)/2)


It is also possible to use H, V, D, A, L and R as shortcuts to predefined values:

Code

$$(\phi,\theta)$$

Jones vector

H

$$(0,0)$$

$$\begin{pmatrix}1\\0\end{pmatrix}$$

V

$$(\pi,0)$$

$$\begin{pmatrix}0\\1\end{pmatrix}$$

D

$$(\frac{\pi}{2},0)$$

$$\frac{1}{\sqrt 2}\begin{pmatrix}1\\1\end{pmatrix}$$

A

$$(\frac{\pi}{2},\pi)$$

$$\frac{1}{\sqrt 2}\begin{pmatrix}1\\-1\end{pmatrix}$$

L

$$(\frac{\pi}{2},\frac{\pi}{2})$$

$$\frac{1}{\sqrt 2}\begin{pmatrix}1\\i\end{pmatrix}$$

R

$$(\frac{\pi}{2},\frac{3\pi}{2})$$

$$\frac{1}{\sqrt 2}\begin{pmatrix}1\\-i\end{pmatrix}$$

>>> p = pcvl.Polarization("D")
>>> p.theta_phi
(pi/2, 0)
>>> p.project_ev_eh())
(sqrt(2)/2, sqrt(2)/2)


Defining states with polarization is then simply to use the Annotation P:

>>> st2 = pcvl.AnnotatedBasicState("|{P:H},{P:V}>")
>>> st2 = pcvl.AnnotatedBasicState("|{P:(sp.pi/2,sp.pi/3)>")


If polarization is used for any photon in the state, the state is considered as using polarization:

>>> pcvl.AnnotatedBasicState("|{P:H},0,{P:V}>").has_polarization
True
>>> pcvl.AnnotatedBasicState("|{P:V},0,1>").has_polarization
True
>>> pcvl.AnnotatedBasicState("|1,0,1>").has_polarization
False


Note

To simplify the notation:

• linear polarization can be defined with a single parameter: {P:sp.pi/2} is equivalent to {P:(sp.pi/2,0}

• if the polarization annotation is omitted for some photons, these photons will be considered as having a horizontal polarization.

See Polarization Object code documentation.