Polarization

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

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

The representation of the polarization in $\left(\begin{array}{c}{E}_h\\ E_v\end{array}\right)$ basis is obtained by applying Jones conversion: $\overrightarrow{J}=\left(\begin{array}{c}\cos \frac{\theta }{2}\\ e^{i\varphi }\sin \frac{\theta }{2}\end{array}\right)$. The same can also be noted: $\cos \frac{\theta }{2}|H⟩+e^{i\varphi }\sin \frac{\theta }{2}|V⟩$.

For instance, the following defines a polarization with $\theta =\frac{\pi }{2},\varphi =\frac{\pi }{4}$ corresponding to Jones vector: $\left(\begin{array}{c}\cos \frac{\pi }{4}\\ e^{i\frac{\pi }{4}}\sin \frac{\pi }{4}\end{array}\right)$

>>> 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	$(\varphi ,\theta )$	Jones vector
H	$(0,0)$	$\left(\begin{array}{c}1\\ 0\end{array}\right)$
V	$(\pi ,0)$	$\left(\begin{array}{c}0\\ 1\end{array}\right)$
D	$(\frac{\pi }{2},0)$	$\frac{1}{\sqrt{2}}\left(\begin{array}{c}1\\ 1\end{array}\right)$
A	$(\frac{\pi }{2},\pi )$	$\frac{1}{\sqrt{2}}\left(\begin{array}{c}1\\ -1\end{array}\right)$
L	$(\frac{\pi }{2},\frac{\pi }{2})$	$\frac{1}{\sqrt{2}}\left(\begin{array}{c}1\\ i\end{array}\right)$
R	$(\frac{\pi }{2},\frac{3\pi }{2})$	$\frac{1}{\sqrt{2}}\left(\begin{array}{c}1\\ -i\end{array}\right)$

>>> 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.BasicState("|{P:H},{P:V}>")
>>> st2 = pcvl.BasicState("|{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.BasicState("|{P:H},0,{P:V}>").has_polarization
True
>>> pcvl.BasicState("|{P:V},0,1>").has_polarization
True
>>> pcvl.BasicState("|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.