Preface
To understand the mathematical notations in this document, you may want to have a look at the summary of mathematical notations and conventions before reading on.
Introduction
The EM Spin Visualizer is an interactive application to visualize 2D oscillations and their superpositions. It provides two “basis oscillations”, which can be scaled, phase-shifted, and then superimposed to get a resulting oscillation.
This mechanism can be seen as a model for the polarisation of electromagnetic radiation. But it is also closely related to the quantum-mechanical description of photon polarisation states. Thus, it builds a nice bridge between an abstract quantum-mechanical model and an easy-to-understand classical system. In particular, the application can be used to predict the physically observable polarisation properties of photons from the given amplitudes and phase-shifts of their basis states.
Feature Overview
The user interface is organized as a grid with three rows of visualizations:
The top row visualizes the two basis oscillations. By default, a horizontal and a vertical oscillation is used. But you can activate other basis oscillations by clicking on the basis oscillation visualization.
The middle row visualizes, how the basis oscillations are scaled and phase-shifted. To visualize these properties, circles with phase-angle indicators are used. The radius of the circle visualizes the scale factor, and the angle of the phase-angle indicator visualizes the phase-shift. The scale factors and phase-shifts can be modified by the control buttons on top. In addition, an auto-rotation of the phase-shifts can be activated with the
Play/Pausecheckbox.The bottom row shows the resulting oscillations: The individual scaled and phase-shifted oscillations are displayed at the left and right. The superposition of both oscillations is displayed in the center.
Scale and Phase Controls
For each basis oscillation, there are the following control buttons:
<<(double left arrow): Rotates the phase angle counterclockwise. This shifts the oscillation backward in time.>>(double right arrow): Rotates the phase angle clockwise. This shifts the oscillation forward in time.^(up arrow): Increases the scale factor of the oscillation, while simultaneously decreasing the scale factor of the other one, maintaining the normalization of scale factors.
Defining Oscillations as a Vector Space
It turns out, that the set of oscillations can mathematically be described as a vector space. A vector space is a set of elements, which can be added together and multiplied (“scaled”) by numbers, in our case complex numbers.
To define oscillations as a vector space, we have to define:
- the mathematical representation of oscillations
- how to add two oscillations
- how to multiply an oscillation with a scalar (a complex number)
These definitions have to fulfill the vector space axioms.
To highlight the relation to quantum mechanical systems, we will use the ket notation for the oscillation vectors. For example, a general oscillation will be denoted \({\ \color{blue} \ket{\psi}}\), and our horizontal and vertical basis oscillations will be denoted \({\ \color{blue} \ket{H}}\) and \({\ \color{blue} \ket{V}}\). As it turns out, these vectors can be handled exactly like quantum-mechanical state vectors, and they give exactly the right predictions for photon polarisation states.
Now let’s define the mathematical representation of oscillations: We define oscillations as cyclic functions, which map a phase-angle \(\tau\) to a point in 2D space. The set of all oscillations is given by the set of functions which match the following structure, where \(r_x, r_y \in \mathbb{R}_{\geq 0}\) and \(\alpha_x, \alpha_y \in \mathbb{R}\) are arbitrary constants and \(\vec{x}\), \(\vec{y}\) are the basis vectors of the 2D plane.
\[ \tau \mapsto r_x \, cos(\tau + \alpha_x) \, \vec{x} + r_y \, cos(\tau + \alpha_y) \, \vec{y} \]
When we feed in different phase-angles into such functions, we will get the oscillation path.
The horizontal and vertical basis oscillations used by the EM Spin Visualizer application can now be defined as follows:
\[ {\ \color{blue} \ket{H}} := \left( \tau \mapsto cos(\tau) \cdot \vec{x} \right) \]
\[ {\ \color{blue} \ket{V}} := \left( \tau \mapsto cos(\tau) \cdot \vec{y} \right) \]
Finally, let’s define the vector addition and scalar multiplication rules:
\[ {\ \color{blue} \ket{\psi_1}} + {\ \color{blue} \ket{\psi_2}} := ( \tau \mapsto {\ \color{blue} \ket{\psi_1}}(\tau) + {\ \color{blue} \ket{\psi_2}}(\tau) ) \]
\[ (r e^{i \alpha}) {\ \color{blue} \ket{\psi}} := ( \tau \mapsto r {\ \color{blue} \ket{\psi}}(\tau + \alpha) ) \]
These definitions exactly matche the behavior of the EM Spin Visualizer application:
- When two oscillations are superimposed, this is done by pointwise adding the 2D displacements.
- When an oscillation is scaled by a complex number with absolute value \(r\) and phase angle \(\alpha\), the absolute value \(r\) scales the 2D displacement, while \(\alpha\) causes a phase shift of the oscillation.
One can check, that these definitions fulfill the axioms of a vector space. This is left as an exercise for the reader :-).
App-Functionality in Terms of Oscillation Vectors
Now let’s interpret the EM Spin Visualizer application from the viewpoint of oscillation vectors.
The top row visualizes the basis oscillations by plotting the curve of all function values on the 2D plane. For small phase angles, the points are plotted in a faint color, and for large phase angles, a stronger color is used.
The pair of basis oscillations can be changed by clicking on a basis oscillation visualization. You can choose between the following pairs:
- \({\ \color{blue} \ket{H}}, {\ \color{blue} \ket{V}}\) - horizontal and vertical
- \({\ \color{blue} \ket{U}}, {\ \color{blue} \ket{D}}\) - diagonal up and diagonal down
- \({\ \color{blue} \ket{L}}, {\ \color{blue} \ket{R}}\) - left-circular and right-circular
These oscillations can be described as follows:
| Symbol | Definition |
|---|---|
| \({\ \color{blue} \ket{H}}\) | \(\tau \mapsto cos(\tau) \, \vec{x}\) |
| \({\ \color{blue} \ket{V}}\) | \(\tau \mapsto cos(\tau) \, \vec{y}\) |
| \({\ \color{blue} \ket{U}}\) | \(\tau \mapsto \sqrt{50\%} \, cos(\tau) \, \vec{x} + \sqrt{50\%} \, cos(\tau) \, \vec{y}\) |
| \({\ \color{blue} \ket{D}}\) | \(\tau \mapsto \sqrt{50\%} \, cos(\tau) \, \vec{x} + \sqrt{50\%} \, cos(\tau + 180°) \, \vec{y}\) |
| \({\ \color{blue} \ket{R}}\) | \(\tau \mapsto \sqrt{50\%} \, cos(\tau) \, \vec{x} + \sqrt{50\%} \, cos(\tau - 90°) \, \vec{y}\) |
| \({\ \color{blue} \ket{L}}\) | \(\tau \mapsto \sqrt{50\%} \, cos(\tau) \, \vec{x} + \sqrt{50\%} \, cos(\tau + 90°) \, \vec{y}\) |
The circles in the middle row can be interpreted as complex numbers \(a_1\) and \(a_2\). They are applied to the basis oscillations, forming new oscillation vectors \({\ \color{blue} \ket{\psi_1}}\) and \({\ \color{blue} \ket{\psi_2}}\). When we assume, that \({\ \color{blue} \ket{H}}\) and \({\ \color{blue} \ket{V}}\) are used as basis oscillations, the resulting oscillations are as follows:
\[ {\ \color{blue} \ket{\psi_1}} = a_1 {\ \color{blue} \ket{H}} \]
\[ {\ \color{blue} \ket{\psi_2}} = a_2 {\ \color{blue} \ket{V}} \]
The values of the numbers \(a_1\) and \(a_2\) come from two different sources. The first source is the user setting, and the second source is the auto-rotation. In combination, the visualized values \(a_1\) and \(a_2\) are determined as follows, assuming \(t\) is the running or frozen time, \(b_1\) and \(b_2\) are the user-defined contributions, and \(\omega\) is the speed of the auto-rotation:
\[ a_1 = e^{-i \omega t} \, b_1 \]
\[ a_2 = e^{-i \omega t} \, b_2 \]
The bottom row visualizes the oscillations \({\ \color{blue} \ket{\psi_1}}\), \({\ \color{blue} \ket{\psi_2}}\) and \({\ \color{blue} \ket{\psi_1}} + {\ \color{blue} \ket{\psi_2}}\) by plotting their function values at phase angle \(\tau = 0\). When the auto-rotation is activated, the complete oscillation path becomes visible.
Of course, the final oscillation can be written as follows, which shows that it has the same mathematical description as a superposition of quantum states.
\[ {\ \color{blue} \ket{\psi}} = a_1 {\ \color{blue} \ket{H}} + a_2 {\ \color{blue} \ket{V}} \]