# Jones calculus

In optics, polarized light can be described using the Jones calculus, discovered by R. C. Jones in 1941. Polarized light is represented by a Jones vector, and linear optical elements are represented by Jones matrices. When light crosses an optical element the resulting polarization of the emerging light is found by taking the product of the Jones matrix of the optical element and the Jones vector of the incident light. Note that Jones calculus is only applicable to light that is already fully polarized. Light which is randomly polarized, partially polarized, or incoherent must be treated using Mueller calculus.

## Jones vectors

The Jones vector describes the polarization of light.

The x and y components of the complex amplitude of the electric field of light travel along z-direction, $E_x(t)$ and $E_y(t)$, are represented as

$\begin{pmatrix} E_x(t) \\ E_y(t)\end{pmatrix} = \begin{pmatrix} E_{0x} e^{i(kz- \omega t+\phi_x)} \\ E_{0y} e^{i(kz- \omega t+\phi_y)} \end{pmatrix} =\begin{pmatrix} E_{0x} e^{i\phi_x} \\ E_{0y} e^{i\phi_y} \end{pmatrix}e^{i(kz- \omega t)}$.

Here $\begin{pmatrix} E_{0x} e^{i\phi_x} & E_{0y} e^{i\phi_y} \end{pmatrix}^\top$ is the Jones vector ($i$ is the imaginary unit with $i^2=-1$). Thus, the Jones vector represents (relative) amplitude and (relative) phase of electric field in x and y directions.

The sum of the squares of the absolute values of the two components of Jones vectors is proportional to the intensity of light. It is common to normalize it to 1 at the starting point of calculation for simplification. It is also common to constrain the first component of the Jones vectors to be a real number. This discards the phase information needed for calculation of interference with other beams. Note that all Jones vectors and matrices on this page assumes that the phase of the light wave is $\phi = kz - \omega t$, which is used by Hecht. In this definition, increase in $\phi_x$ (or $\phi_y$) indicates retardation (delay) in phase, while decrease indicates advance in phase. For example, a Jones vectors component of $i$ ($=e^{i\pi/2}$) indicates retardation by $\pi/2$ (or 90 degree) compared to 1 ($=e^{0}$). Collett uses the opposite definition ($\phi = \omega t - kz$). The reader should be wary when consulting references on Jones calculus.

The following table gives the 6 common examples of normalized Jones vectors.

 Polarization Corresponding Jones vector Typical ket Notation Linear polarized in the x-direction Typically called 'Horizontal' $\begin{pmatrix} 1 \\ 0 \end{pmatrix}$ $|H\rangle$ Linear polarized in the y-direction Typically called 'Vertical' $\begin{pmatrix} 0 \\ 1 \end{pmatrix}$ $|V\rangle$ Linear polarized at 45° from the x-axis Typically called 'Diagonal' L+45 $\frac{1}{\sqrt2} \begin{pmatrix} 1 \\ 1 \end{pmatrix}$ $|D\rangle = \frac{1}{\sqrt2} ( |H\rangle + |V\rangle )$ Linear polarized at −45° from the x-axis Typically called 'Anti-Diagonal' L-45 $\frac{1}{\sqrt2} \begin{pmatrix} 1 \\ -1 \end{pmatrix}$ $|A\rangle = \frac{1}{\sqrt2} ( |H\rangle - |V\rangle )$ Right Hand Circular Polarized Typically called RCP or RHCP $\frac{1}{\sqrt2} \begin{pmatrix} 1 \\ -i \end{pmatrix}$ $| R\rangle = \frac{1}{\sqrt2} ( |H\rangle - i |V\rangle )$ Left Hand Circular Polarized Typically called LCP or LHCP $\frac{1}{\sqrt2} \begin{pmatrix} 1 \\ +i \end{pmatrix}$ $|L\rangle = \frac{1}{\sqrt2} ( |H\rangle + i |V\rangle )$

When applied to the Poincaré sphere (also known as the Bloch sphere), the basis kets ($|0\rangle$ and $|1\rangle$) must be assigned to opposing (antipodal) pairs of the kets listed above. For example, one might assign $|0\rangle$ = $|H\rangle$ and $|1\rangle$ = $|V\rangle$. These assignments are arbitrary. Opposing pairs are

• $|H\rangle$ and $|V\rangle$
• $|D\rangle$ and $|A\rangle$
• $|R\rangle$ and $|L\rangle$

The $|\psi\rangle$ ket is a general vector that points to any place on the surface. Any point not in the table above and not on the circle that passes through $|H\rangle, |D\rangle, |V\rangle, |A\rangle$ is collectively known as elliptical polarization.

## Jones matrices

The Jones matrices are the operators that act on the Jones Vectors as listed above. These matrices are implemented by various optical elements such as lenses, beam splitters, mirrors, etc. The following table gives examples of Jones matrices for polarizers:

 Optical element Corresponding Jones matrix Linear polarizer with axis of transmission horizontal[1] $\begin{pmatrix} 1 & 0 \\ 0 & 0 \end{pmatrix}$ Linear polarizer with axis of transmission vertical[1] $\begin{pmatrix} 0 & 0 \\ 0 & 1 \end{pmatrix}$ Linear polarizer with axis of transmission at ±45° with the horizontal[1] $\frac{1}{2} \begin{pmatrix} 1 & \pm 1 \\ \pm 1 & 1 \end{pmatrix}$ Right circular polarizer[1] $\frac{1}{2} \begin{pmatrix} 1 & i \\ -i & 1 \end{pmatrix}$ Left circular polarizer[1] $\frac{1}{2} \begin{pmatrix} 1 & -i \\ i & 1 \end{pmatrix}$ Linear polarizer with axis of transmission at angle $\theta$ with the horizontal. (Shown construction from rotating up from the horizontal into the polarizing element, the polarizing element, and then rotating back down into the horizontal.) $\begin{pmatrix} \cos^2(\theta) & \cos(\theta)\sin(\theta) \\ \sin(\theta)\cos(\theta) & \sin^2(\theta) \end{pmatrix} =$ $\begin{pmatrix} \cos(\theta) & -\sin(\theta) \\ \sin(\theta) & \cos(\theta) \end{pmatrix} \begin{pmatrix} 1 & 0 \\ 0 & 0 \end{pmatrix} \begin{pmatrix} \cos(-\theta) & -\sin(-\theta) \\ \sin(-\theta) & \cos(-\theta) \end{pmatrix}$

## Phase retarders

Phase retarders introduce a phase shift between the vertical and horizontal component of the field and thus change the polarization of the beam. Phase retarders are usually made out of birefringent uniaxial crystals such as calcite, MgF2 or quartz. Uniaxial crystals have one crystal axis that is different from the other two crystal axes (i.e., ninj = nk). This unique axis is called the extraordinary axis and is also referred to as the optic axis. An optic axis can be the fast or the slow axis for the crystal depending on the crystal at hand. Light travels with a higher phase velocity through an axis that has the smallest refractive index and this axis is called the fast axis. Similarly, an axis which has the highest refractive index is called a slow axis since the phase velocity of light is the lowest along this axis. Negative uniaxial crystals (e.g., calcite CaCO3, sapphire Al2O3) have ne < no so for these crystals, the extraordinary axis (optic axis) is the fast axis whereas for positive uniaxial crystals (e.g., quartz SiO2, magnesium fluoride MgF2, rutile TiO2), ne > n o and thus the extraordinary axis (optic axis) is the slow axis.

Any phase retarder with fast axis vertical or horizontal has zero off-diagonal terms and thus can be conveniently expressed as

$\begin{pmatrix} e^{i\phi_x} & 0 \\ 0 & e^{i\phi_y} \end{pmatrix}$

where, $\phi_x$ and $\phi_y$ are the phases of the electric fields in $x$ and $y$ directions respectively. In the phase convention $\phi = kz - \omega t$, the relative phase between the two waves when represented as $\epsilon = \phi_y - \phi_x$ suggests that a positive $\epsilon$ (i.e., $\phi_y$ > $\phi_x$) means that $E_y$ doesn't attain the same value as $E_x$ until a later time i.e., $E_x$ leads $E_y$. Similarly, if $\epsilon < 0$ i.e., $\phi_x$ > $\phi_y$, $E_y$ leads $E_x$. For e.g., if the fast axis of a quarter wave plate is horizontal, this suggests that the phase velocity along the horizontal direction is faster than that in the vertical direction i.e., $E_x$ leads $E_y$. Thus, $\phi_x < \phi_y$ which for a quarter wave plate suggests that $\phi_y = \phi_x + \pi/2$.

In the opposite convention $\phi = \omega t - kz$, the relative phase when defined as $\epsilon = \phi_x - \phi_y$ suggests that a positive $\epsilon$ means that $E_y$ doesn't attain the same value as $E_x$ until a later time i.e., $E_x$ leads $E_y$.

Phase retarders Corresponding Jones matrix
Quarter-wave plate with fast axis vertical[2][note 1]

$e^{i \pi/4} \begin{pmatrix} 1 & 0 \\ 0 & -i \end{pmatrix}$

Quarter-wave plate with fast axis horizontal[2]

$e^{i \pi/4} \begin{pmatrix} 1 & 0 \\ 0 & i \end{pmatrix}$

Half-wave plate with fast axis at angle $\theta$ w.r.t the horizontal axis[3]

$\begin{pmatrix} \cos2\theta & \sin2\theta \\ \sin2\theta & -\cos2\theta \end{pmatrix}$

Any birefringent material (phase retarder)[4]

$\begin{pmatrix} e^{i\phi_x} \cos^2\theta+e^{i\phi_y} \sin^2\theta & (e^{i\phi_x}-e^{i\phi_y}) e^{-i\phi} \cos\theta \sin\theta \\ (e^{i\phi_x}-e^{i\phi_y}) e^{i\phi} \cos\theta \sin\theta & e^{i\phi_x} \sin^2\theta+e^{i\phi_y} \cos^2\theta \end{pmatrix}$

The special expressions for the phase retarders can be obtained by using the general expression for a birefringent material. In the above expression:

• Phase retardation induced between $E_x$ and $E_y$ by a birefringent material is given by $\phi_y - \phi_x$
• $\theta$ is the orientation of the fast axis with respect to the x-axis.
• $\phi$ is the circularity (For linear retarders, $\phi$ = 0 and for circular retarders, $\phi$ = ± $\pi$/2, $\theta$ = $\pi$/4. For elliptical retarders, it takes on values between - $\pi$/2 and $\pi$/2).

## Rotated elements

Assume an optical element has its optic axis perpendicular to the surface vector for the plane of incidence and is rotated about this surface vector by angle θ/2 (i.e., the principal plane, through which the optic axis passes, makes angle θ/2 with respect to the plane of polarization of the electric field of the incident TE wave). Recall that a half-wave plate rotates polarization as twice the angle between incident polarization and optic axis (principal plane). Therefore, the Jones matrix for the rotated polarization state, M(θ), is

$M(\theta )=R(\theta )\,M\,R(-\theta ),$
where $R(\theta ) = \begin{pmatrix} \cos \theta & -\sin \theta \\ \sin \theta & \cos \theta \end{pmatrix}.$

This agrees with the expression for a half-wave plate in the table above. These rotations are identical to beam unitary splitter transformation in optical physics given by

$R(\theta ) = \begin{pmatrix} r & t'\\ t & r' \end{pmatrix}$

where the primed and unprimed coefficients represent beams incident from opposite sides of the beam splitter. The reflected and transmitted components acquire a phase θr and θt, respectively. The requirements for a valid representation of the element are [5]

$\theta_\text{t} - \theta_\text{r} + \theta_\text{t'} - \theta_\text{r'} = \pm \pi$

and $r^*t' + t^*r' = 0.$

Both of these representations are unitary matrices fitting these requirements; and as such, are both valid.

## Notes

1. ^ The prefactor $e^{i\pi/4}$ appears only if one defines the phase delays in a symmetric fashion; that is, $\phi_x = -\phi_y = \pi/4$. This is done in Hecht[2] but not in Fowles.[1] In the latter reference the Jones matrices for a quarter-wave plate have no prefactor.

## References

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2. ^ a b c Hecht, E. (2001). Optics (4th ed.). p. 378. ISBN 0805385665.
3. ^ Gerald, A.; Burch, J.M. (1975). Introduction to Matrix Methods in Optics (1st ed.). John Wiley & Sons. ISBN 0471296856.
4. ^ Obtainment of the polarizing and retardation parameters of a non-depolarizing optical system from the polar decomposition of its Mueller matrix, Optik, Jose Jorge Gill and Eusebio Bernabeu,76, 67-71 (1987).
5. ^ Am. J. Phys. 57 (1), 66 (1988).
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