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Introduction

Jones Calculus is an extremely compact mathematical description of polarised light. Unlike Mueller calculus (Stokes Vectors) it can not be used to describe unpolarised light, but can describe the phase.

Jones Vector

The Jones vector is a two element column vector describe the amplitude and phase of electric field along the $X$ and $Y$ direction for a beam of light travelling along $Z$.

$$\mathbf{J}=\left[ \begin{array}{c}
E_{x} \\ E_{y} \end{array} \right] = e^{i2\pi \omega t} \left[ \begin{array}{c} A_{x} e^{i \varepsilon_{x}} \\ A_{y} e^{i \varepsilon_{y}} \end{array} \right]
$$

where $t$ is time, $\omega$ angular frequency, $\varepsilon$ is phase, $A$ is amplitude, and $E$ is the electric field. For many calculations, the time variation can be dropped as well as any overall constant phase term.

The intensity of a beam is given by

$$I=A_{x}^{2}+A_{y}^{2}$$

(This assumes the detector is polarisation insensitive, and this is usually true.)

The Jone’s vector is then often normalised, so the intensity is unity and the vector is written in its simplest form. Examples of this are
$$\left[ \begin{array}{c} 1 \\ 1 \end{array} \right] \textrm{becomes normalised to } \frac{1}{\sqrt{2}}\left[ \begin{array}{c} 1 \\ 1 \end{array} \right]$$
Some people also shorten $e^{i \pi / 2}$ to $i$ so

$$\left[ \begin{array}{c} e^{i \frac{\pi}{2}} \\ 1 \end{array} \right] \textrm{becomes normalised to } \frac{1}{ \sqrt{2}}\left[ \begin{array}{c} i \\ 1 \end{array} \right]$$

although personally I prefer the normalised form with the exponential left in.

The table below contains some examples of common Jones vectors

Name	Normalised	Full
$ \leftarrow$ Horz. linear polarised	$\tabvect{1}{0}$	$ \tabvect{A_{x} e^{i \varepsilon_{x}}}{0}$
$ \uparrow$ Vert. linear polarised	$\tabvect{0}{1}$	$\tabvect{0}{A_{y} e^{i \varepsilon_{y}}}$
$ \nearrow 45^{\circ}$	$\frac{1}{\sqrt{2}} \tabvect{1}{1}$	$\tabvect{A_{x} e^{i \varepsilon_{x}}}{A_{x} e^{i \varepsilon_{x}}}$
$\nwarrow -45^{\circ}$	$\frac{1}{\sqrt{2}} \tabvect{1}{-1}$	$\tabvect{A_{x} e^{i \varepsilon_{x}}}{-A_{x} e^{i \varepsilon_{x}}}$
General linear	$\tabvect{\cos R}{\pm \sin R }$	$\tabvect{A_{x} e^{i \varepsilon_{x}}}{A_{y} e^{i \varepsilon_{y}}}$
$$\circlearrowright$$ Circular	$\frac{1}{\sqrt{2}} \tabvect{-i}{1}$	$\tabvect{A_{x} e^{i \varepsilon_{x}}}{A_{x} e^{i \left( \varepsilon_{x}+\pi / 2 \right)}}$
$$ \circlearrowleft $$ Circular	$\frac{1}{\sqrt{2}} \tabvect{i}{1}$	$\tabvect{A_{x} e^{i \varepsilon_{x}}}{A_{x} e^{i \left( \varepsilon_{x}- \pi /2 \right)}}$
General	$\frac{1}{\sqrt{2}} \tabvect{\cos \left( R \right) e^{-i \gamma / 2}}{\sin \left( R \right) e^{i \gamma /2}}$	$\tabvect{A_{x} e^{i \varepsilon_{x}}}{A_{y} e^{i \varepsilon_{y}}}$

$R$ is defined as

$$R=\vert \arctan \left( \frac{A_{y}}{A_{x}} \right) \vert$$

and $ \gamma$ as

$$\gamma = \varepsilon_{y} -\varepsilon_{x}$$

Jones Matrix

When light then passes through a polarisation sensitive device, the polarisation state will change. The new state, $ \mathbf{\acute{J}} $, is calculated by multiplying the old state, $ \mathbf{J}$, by a 2×2 matrix, $ \mathbf{M}$, the Jones Matrix.

$$\mathbf{\acute{J}}=\tabmatrix{m_{11}}{m_{12}}{m_{21}}{m_{22}} \mathbf{J}$$

The Jones matrix for a co-ordinate rotation by angle $ \theta$ about the $Z$ axis is given by

$$\mathbf{M_{rotate}}\left(\theta \right) = \tabmatrix{\cos \theta}{\sin \theta}{-\sin \theta}{\cos \theta}$$

If we consider a wave-plate, the refractive index along the $ Y$ axis, $ n_{Y}$, is different from that of the $X$ axis, $n_{X}$, i.e., the ordinary, $n_{o}$, and extra-ordinary, $n_{e}$, refractive indices. If the wave-plate is of thickness $d$ and the wavelength is $\lambda$, the phase delay between the two axes is the $\phi = \frac{2 \pi}{\lambda} d \Delta n$. Where $\Delta n = n_{e} – n_{o}$, i.e., the birefringence. A linear wave-plate can then be described as

$$\mathbf{M_{retard}}\left( \phi \right) =\tabmatrix{e^{i \phi/2}}{0}{0}{e^{-i \phi /2}}$$

$ \phi = \pi$ for a half-wave plate and $ \phi= \pi /2$ for a quarter wave-plate. To calculate the Jones matrix for an arbitrary optical axis the rotator matrix is used to transform the co-ordinate system to that of the wave-plate, and then another transforms it back again to the original system.

$$\mathbf{M_{retard}}\left( \phi , \theta \right) = {M_{rotate}}\left(-\theta \right)\mathbf{M_{retard}}\left( \phi \right){M_{rotate}}\left(\theta \right)$$

$$\mathbf{M_{retard}}\left( \phi , \theta \right)=\tabmatrix{\cos^{2} \left(\theta \right) e^{i \phi /2}+\sin^{2} \left(\theta \right) e^{-i \phi /2}}{\cos \left(\theta \right) \sin \left(\theta \right) 2i \sin \left(\phi/2 \right)}{\cos \left(\theta \right) \sin \left( \theta \right) 2i \sin \left(\phi/2 \right)}{\cos^{2} \left( \theta \right) e^{-i \phi /2}+sin^{2} \left( \theta \right) e^{i \phi /2}}$$

An example of common Jones matrices are given in the table below.

To calculate the effect of an optical component, simply multiply the Jones matrices together and the correct input vector. If we have multiple components, they can be modelled as a chain of matrices.

For example, if we take a horizontal linearly polarised input and pass this through a half wave-plate at $45^{\circ}$ and then a vertically aligned polariser we get

$$\tabmatrix{0}{0}{0}{1} \tabmatrix{1}{0}{0}{1} \tabvect{1}{0} = \tabvect{0}{1}$$

As we expect, the light is now vertical linearly polarised.

Ideal isotropic non-absorbing material	$\tabmatrix{1}{0}{0}{1}$
Absorbing isotropic material with transmittance $p$	$\tabmatrix{\sqrt{p}}{0}{0}{\sqrt{p}}$
Horizontal polariser	$\tabmatrix{1}{0}{0}{0}$
Vertical polariser	$\tabmatrix{0}{0}{0}{1}$
$latex 45^{\circ}$ polariser	$\tabmatrix{0.5}{0.5}{0.5}{0.5}$
$latex -45^{\circ}$ polariser	$\tabmatrix{0.5}{-0.5}{-0.5}{0.5}$
General polariser at angle $\theta$	$\tabmatrix{\cos^{2} \left(\theta \right)}{\cos \left(\theta \right) \sin \left(\theta \right)}{\cos \left(\theta \right) \sin \left(\theta \right)}{\sin^{2} \left(\theta \right)}$
Waveplate with horizontal fast axis, retardance $\phi$	$\tabmatrix{e^{i \phi /2}}{0}{0}{e^{-i \phi /2}}$
Waveplate with vertical fast axis, retardance $\phi$	$\tabmatrix{e^{-i \phi /2}}{0}{0}{e^{i \phi /2}}$
Waveplate with fast axis angle $\theta$, retardance $\phi$	$$\tabmatrix{\cos^{2} \left(\theta \right) e^{i \phi /2}+\sin^{2} \left(\theta \right) e^{-i \phi /2}}{\cos \left(\theta \right) \sin \left(\theta \right) 2i \sin \left(\phi/2 \right)}{\cos \left(\theta \right) \sin \left( \theta \right) 2i \sin \left(\phi/2 \right)}{\cos^{2} \left( \theta \right) e^{-i \phi /2}+sin^{2} \left( \theta \right) e^{i \phi /2}}$$

Name	Normalised	Full
\( \leftarrow\) Horz. linear polarised	\(\tabvect{1}{0}\)	\( \tabvect{A_{x} e^{i \varepsilon_{x}}}{0}\)
\( \uparrow\) Vert. linear polarised	\(\tabvect{0}{1}\)	\(\tabvect{0}{A_{y} e^{i \varepsilon_{y}}}\)
\( \nearrow 45^{\circ}\)	\(\frac{1}{\sqrt{2}} \tabvect{1}{1}\)	\(\tabvect{A_{x} e^{i \varepsilon_{x}}}{A_{x} e^{i \varepsilon_{x}}}\)
\(\nwarrow -45^{\circ}\)	\(\frac{1}{\sqrt{2}} \tabvect{1}{-1}\)	\(\tabvect{A_{x} e^{i \varepsilon_{x}}}{-A_{x} e^{i \varepsilon_{x}}}\)
General linear	\(\tabvect{\cos R}{\pm \sin R }\)	\(\tabvect{A_{x} e^{i \varepsilon_{x}}}{A_{y} e^{i \varepsilon_{y}}}\)
$$\circlearrowright$$ Circular	\(\frac{1}{\sqrt{2}} \tabvect{-i}{1}\)	\(\tabvect{A_{x} e^{i \varepsilon_{x}}}{A_{x} e^{i \left( \varepsilon_{x}+\pi / 2 \right)}}\)
$$ \circlearrowleft $$ Circular	\(\frac{1}{\sqrt{2}} \tabvect{i}{1}\)	\(\tabvect{A_{x} e^{i \varepsilon_{x}}}{A_{x} e^{i \left( \varepsilon_{x}- \pi /2 \right)}}\)
General	\(\frac{1}{\sqrt{2}} \tabvect{\cos \left( R \right) e^{-i \gamma / 2}}{\sin \left( R \right) e^{i \gamma /2}}\)	\(\tabvect{A_{x} e^{i \varepsilon_{x}}}{A_{y} e^{i \varepsilon_{y}}}\)

Ideal isotropic non-absorbing material	\(\tabmatrix{1}{0}{0}{1}\)
Absorbing isotropic material with transmittance \(p\)	\(\tabmatrix{\sqrt{p}}{0}{0}{\sqrt{p}}\)
Horizontal polariser	\(\tabmatrix{1}{0}{0}{0}\)
Vertical polariser	\(\tabmatrix{0}{0}{0}{1}\)
$latex 45^{\circ}$ polariser	\(\tabmatrix{0.5}{0.5}{0.5}{0.5}\)
$latex -45^{\circ}$ polariser	\(\tabmatrix{0.5}{-0.5}{-0.5}{0.5}\)
General polariser at angle \(\theta\)	\(\tabmatrix{\cos^{2} \left(\theta \right)}{\cos \left(\theta \right) \sin \left(\theta \right)}{\cos \left(\theta \right) \sin \left(\theta \right)}{\sin^{2} \left(\theta \right)}\)
Waveplate with horizontal fast axis, retardance \(\phi\)	\(\tabmatrix{e^{i \phi /2}}{0}{0}{e^{-i \phi /2}}\)
Waveplate with vertical fast axis, retardance \(\phi\)	\(\tabmatrix{e^{-i \phi /2}}{0}{0}{e^{i \phi /2}}\)
Waveplate with fast axis angle \(\theta\), retardance \(\phi\)	$$\tabmatrix{\cos^{2} \left(\theta \right) e^{i \phi /2}+\sin^{2} \left(\theta \right) e^{-i \phi /2}}{\cos \left(\theta \right) \sin \left(\theta \right) 2i \sin \left(\phi/2 \right)}{\cos \left(\theta \right) \sin \left( \theta \right) 2i \sin \left(\phi/2 \right)}{\cos^{2} \left( \theta \right) e^{-i \phi /2}+sin^{2} \left( \theta \right) e^{i \phi /2}}$$