Polarization state | Ivan Lopushenko

This is a small app to compute and visualize parameters of either fully or partially polarized light from the supplied Stokes vector $S=[S_0, S_1, S_2, S_3]^T=[I, Q, U, V]^T$. Source code for the underlying MATLAB® class is available on GitHub and MATLAB Exchange.

In the following, we fully comply with M. Born and E. Wolf (1980), sections §1.4 and §10.8. For more details, please refer to these chapters or see theoretical reference at the bottom of the page.

Some short remarks before you begin working with the app:

Input are Stokes vector values (can be power in e.g. [W/m²], or in normalized dimensionless units). The main rule is that relations $S_0^2\geq S_1^2+S_2^2+S_3^2$ and $S_0\geq0$ must always hold;
Several input vectors can be provided to compare them;
Primary outputs are degrees of polarization and parameters of the polarization ellipse;
Additional outputs are 6 intensity values $I_H, I_V, I_D, I_A, I_R, I_L$ (accessible via tab switch below) and normalized Stokes vector values (accessible via Poincaré sphere);
Computed ellipse and Poincaré sphere depiction of the $S$ vector are presented on 2D and 3D plots below.

Attention: tabs are simultaneously updated with the latest submitted data.
Main results are returned in the table below.

ID	$$DoP$$	$$DoLP$$	$$DoCP$$	$$\mathrm{Azimuth}\:\psi$$ ⓘ	$$\mathrm{Ellipticity}\:\chi$$ ⓘ	$$\delta$$ ⓘ	$$\mathrm{Helicity}$$

Figure 1. Polarization ellipse for each input beam.

Figure 2. Depiction of the normalized Stokes vector on the Poincaré sphere.

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In case of the partially polarized light, this app plots polarization ellipse for the fully polarized component of light, which intensity is equal to $\sqrt{S_1^2+S_2^2+S_3^2}$. Here, your data is not normalized in any other way, and ellipses for all provided Stokes vectors are displayed with proper scale, which allows to compare them between each other both in terms of shape and in terms of amplitudes of the underlying light waves.

Theoretical reference

Measurement of Stokes vector

Using polarizer-compensator configuration (experimental setup with e.g. linear polarizer and quarter-wave plate) it is possible to measure several intensity values of the light beam: intensity of the horizontally polarized component $I_H$, vertically polarized component $I_V$, diagonally polarized component $I_D$, antidiagonal component $I_A$, right $I_R$ and left $I_L$ circularly polarized components. This is in agreement with Born and Wolf, §10.8, Eqns. (10) and (64). If light is unpolarized, then all components are equal to each other.

Most notably, in the experimental configuration with polarizer and compensating waveplate one would usually use angles $\theta$ and $\varepsilon$ of their rotation in order to describe the measured intensities: $I=I(\theta,\varepsilon)$. Then,

\[I_H=I(0,0), \:\: I_V=I\left(\dfrac{\pi}{2},0\right),\] \[I_D=I\left(\dfrac{\pi}{4},0\right), \:\: I_A=\left(\dfrac{3\pi}{4},0\right),\] \[I_R=I\left(\dfrac{\pi}{4},\dfrac{\pi}{2}\right), \:\: I_L=I\left(\dfrac{3\pi}{4},\dfrac{\pi}{2}\right).\]

Finally, Stokes vector can be computed as

\[S_0=I_H+I_V=I_D+I_A=I_R+I_L,\] \[S_1=I_H-I_V,\] \[S_2=I_D-I_A,\] \[S_3=I_R-I_L.\]

In the literature, Stokes vector can be defined somehow differently. A good overview is done by M. Mishchenko (2014) in Section 7.1. For further reading on polarized light, I would also recommend D. Goldstein (2011).

Degrees of polarization

Partially polarized light is characterized by its degree of polarization ($DoP$), Born and Wolf, §10.8, Eqn. (68):

\[DoP=\sqrt{S_1^2+S_2^2+S_3^2}/S_0.\]

In turn, it can be further decomposed into degrees of linear ($DoLP$) and circular ($DoCP$) polarization:

\[DoLP=\sqrt{S_1^2+S_2^2}/S_0,\] \[DoCP=\sqrt{S_3^2}/S_0=\vert S_3 \vert/S_0,\] \[DoP^2=DoLP^2 + DoCP^2.\]

Degrees of polarization are quite frequently employed in polarimetric applications.

Polarization ellipse and Poincaré sphere

Fully polarized light can be characterized with polarization ellipse.

Each ellipse is a result of a superposition of two orthogonally linearly polarized waves with phase difference $\delta$ between them. Assuming that we deal with plane time-harmonic wave propagating along $z$ direction, the electromagnetic wave field is then described as $\mathbf{E}=(E_x,E_y,0)$, Born and Wolf, §1.4, Eqn. (12):

\[E_x = a_1 \cos{(\tau+\delta_1)}, \:\: E_y = a_2 \cos{(\tau+\delta_2)},\] \[\tau=\omega t - \mathbf{kr}, \:\: \delta = \delta_2-\delta_1.\]

Here, amplitudes $a_1$ and $a_2$ of both orthogonally polarized light waves are introduced. We would allow ourselves not to comment upon all other variables, directing the inquiring reader to the given reference for further reading.

Depending on the amplitudes and phase difference, polarization ellipse can take different forms. Its axes are generally not parallel to either $Ox$ or $Oy$ of the Cartesian coordinate system, and can be rotated by azimuth angle $\psi$ with respect to them (see Figure 3):

Figure 3. Polarization ellipse and its parameters. Arrow corresponds to the helicity direction. Picture source: wiki.

Figure 4. Definition of the Poincaré sphere. Picture source: wiki.

Larger $a$ and smaller $b$ ellipse semi-axes, as well as ellipticity $\chi$ and azimuth $\psi$ are related to Stokes vector in the following way, Born and Wolf §1.4, Eqns. (23),(32),(43),(45a)-(45c):

\[S_0=a_1^2+a_2^2=a^2+b^2,\] \[S_1=S_0\cos2\chi\cos2\psi,\] \[S_2=S_0\cos2\chi\sin2\psi,\] \[S_3=S_0\sin2\chi,\] \[\mathrm{tg}\chi=\mp b/a.\]

These relations, resembling connection between Cartesian and spherical coordinate systems, allow to depict Stokes vector as point on the sphere. This depiction of the polarization state is known as Poincaré sphere.

Obvious special cases are the so-called degenerate states of the polarization ellipse, which are available in the current tool via “Load demo data” button. In particular, in the 1st degenerate state (Linear Horizontal Polarization, LHP) ellipse is reduced to the line aligning with $Ox$ Cartesian axes, in the 2nd (Linear Vertical Polarization, LVP) - to the line aligning with $Oy$ Cartesian axes, in the 3rd (linear diagonal polarization, L+45) and 4th (linear antidiagonal polarization L-45) - to the lines rotated at $\psi_D=\pi/4$ and $\psi_A=3\pi/4$, correspondingly, and finally, in 5th (right circular polarization, RCP) and 6th (left circular polarization, LCP) ellipse is reduced to the circle which one can follow either clockwise or counter-clockwise.

Wolf’s coherency matrix

It is worth noting that Stokes parameters bijectively (one-to-one) correspond to the coherency matrix introduced by Emil Wolf and which can be referenced in Born and Wolf, §10.8, Eqn. (63). In turn, this matrix is directly related to the concept of density matrix in quantum description of light: Wolf’s matrix is in fact a density matrix of the single photon. This is an important connection between wave and quantum properties of light which manifests itself through polarization. In the source code of this app, elements of the Wolf’s coherency matrix are also evaluated, and there are plans on implementing web interface for the convenient coherency matrix analysis in this app - in the tab next to intensity values. For now, we conclude our short theoretical overview and provide all necessary references.

Source code

Polarization state calculator. Preview version 0.5, 31.03.2024.
This tool has been inspired by fundamental relation between polarization and quantum properties of light, and by processing of polarimetric measurements data at OPEM Unit within Academy of Finland and MetaHiLight projects.
Author warmly acknowledges support and feedback from colleagues and friends.
Built with MATLAB Coder and Emscripten. Runs locally in the web browser with WebAssembly.

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