Light Polarization & Manipulation

1. The Vectorial Nature of Light

Recall from Maxwell's equations that for a plane wave propagating in the $z$-direction, the electric field $\vec{E}$ must be transverse to the direction of propagation. This means it can oscillate in any direction within the $x$-$y$ plane. Polarization describes the specific path that the tip of the electric field vector traces in this plane over time.

The general form for the electric field of such a wave is:
$\vec{E}(z,t)=\text{Re}\left[\left(E_{0x}\hat{x}+E_{0y}\hat{y}\right)e^{i(kz-\omega t)}\right]$
Here, $E_{0x}$ and $E_{0y}$ are the complex amplitudes of the wave's components along the $x$ and $y$ axes. The polarization state is entirely determined by the relative magnitude and phase of these two components.

For a rigorous and visual introduction to the different types of polarization and how they arise from these components, please watch the first part of the following video.

The Jones Vector

As introduced in the video, we can conveniently represent the polarization state of a fully polarized wave using a two-component column vector called the Jones vector. We factor out the common propagation term $e^{i(kz-\omega t)}$ and focus on the complex amplitudes:
$\mathbf{J}=\left(\begin{array}{c}{E}_{0x}\\ E_{0y}\end{array}\right)$

The key polarization states are:

• Linear Polarization: $E_{0x}$ and $E_{0y}$ are in phase (or $\pi$ out of phase). The E-field oscillates along a fixed line.

  • Horizontal: $\left(\begin{array}{c}1\\ 0\end{array}\right)$
  • Vertical: $\left(\begin{array}{c}0\\ 1\end{array}\right)$
  • At 45°: $\frac{1}{\sqrt{2}}\left(\begin{array}{c}1\\ 1\end{array}\right)$

• Circular Polarization: $|E_{0x}|=|E_{0y}|$ and they are $\pm \pi /2$ out of phase. The E-field vector rotates in a circle of constant magnitude.

  • Left-Circular (LCP): $\frac{1}{\sqrt{2}}\left(\begin{array}{c}1\\ i\end{array}\right)$ (E-field rotates counter-clockwise when viewed facing the oncoming wave)
  • Right-Circular (RCP): $\frac{1}{\sqrt{2}}\left(\begin{array}{c}1\\ -i\end{array}\right)$ (E-field rotates clockwise)

• Elliptical Polarization: The most general case. The magnitudes and/or phase difference are arbitrary. The E-field vector traces an ellipse.

For a more detailed reference on these states, you can consult the following text.

2. Polarizers and Malus's Law

A polarizer is an optical element that transmits light of a specific polarization while blocking others.

How Polarizers Work

There are several physical mechanisms for polarization.

1. Dichroism (Selective Absorption): This is the principle behind Polaroid sheets. They contain long-chain polymer molecules aligned in one direction. The electric field component parallel to these chains drives electrons, dissipating its energy as heat (absorption). The component perpendicular to the chains is transmitted. The transmission direction is called the pass axis.

2. Reflection (Brewster's Angle): When unpolarized light reflects off a surface, the reflected light is generally partially or, at a specific angle, fully polarized.

To understand these mechanisms, watch the following video segments.

Key points from the video:

• Unpolarized light has an electric field that varies randomly in orientation. It can be modeled as two independent, orthogonal polarization components of equal average intensity. When it passes through an ideal polarizer, half the intensity is transmitted ($I=I_0/2$).
• Brewster's Law states that at a specific angle of incidence, ${\theta }_B$, the reflected light is completely linearly polarized. This angle is given by $\tan {\theta }_B=n_2/n_1$, where $n_1$ and $n_2$ are the refractive indices of the two media. At this angle, the reflected and refracted rays are perpendicular.

Malus's Law

What happens when light that is already linearly polarized passes through a second polarizer (often called an "analyzer")?

If the incident light has intensity $I_{in}$ and is linearly polarized at an angle $\theta$ relative to the polarizer's pass axis, the transmitted intensity $I_{out}$ is given by Malus's Law:

$I_{out}=I_{in}{\cos }^2\theta$

This is because only the component of the incident electric field parallel to the pass axis, $E_{in}\cos \theta$, is transmitted. Since intensity is proportional to the square of the electric field amplitude, the intensity is reduced by a factor of ${\cos }^2\theta$. The IIT-PAL video (LINK) provides a clear derivation and application of this from 44:28 to 53:57.

3. Jones Calculus: Matrices for Optical Elements

To analyze systems with multiple components, we extend the Jones vector formalism to Jones matrices. An optical element is represented by a 2x2 matrix that transforms the Jones vector of the incident light to that of the transmitted light.
${\mathbf{J}}_{out}=\mathbf{M}{\mathbf{J}}_{in}$
For a system of multiple elements, the total matrix is the product of individual matrices, applied in reverse order of how the light encounters them: ${\mathbf{M}}_{sys}={\mathbf{M}}_n\dots {\mathbf{M}}_2{\mathbf{M}}_1$.

3.1 Jones Matrix for a Polarizer

The following reading provides a rigorous derivation of the Jones matrix for a linear polarizer whose transmission axis is at an arbitrary angle $\theta$ with respect to the x-axis.

The resulting Jones matrix for a linear polarizer with its transmission axis at an angle $\theta$ is:
${\mathbf{M}}_{pol}(\theta )=\left(\begin{array}{cc}{\cos }^2\theta & \sin \theta \cos \theta \\ \sin \theta \cos \theta & {\sin }^2\theta \end{array}\right)$

3.2 Wave Plates (Retarders)

While polarizers work by selective absorption, wave plates work by selective phase shifting. They are made of birefringent materials, which have different refractive indices ($n_{fast}$ and $n_{slow}$) for orthogonal polarization directions (the fast and slow axes).

A wave plate of thickness $d$ introduces a phase difference $\Delta \varphi$ between the two components:
$\Delta \varphi =(k_{slow}-k_{fast})d=\frac{2\pi }{{\lambda }_0}(n_{slow}-n_{fast})d$
This phase shift modifies the polarization state of the light passing through it.

The most common types are:

• Half-Wave Plate ($\Delta \varphi =\pi$): If linearly polarized light enters a HWP, it emerges linearly polarized but rotated by twice the angle between the input polarization and the fast axis.
• Quarter-Wave Plate ($\Delta \varphi =\pi /2$): A QWP can convert linear polarization to circular polarization (and vice-versa). This happens when the input linear polarization is at 45° to the wave plate's axes.

The Jones matrix formalism is perfectly suited to describe these elements.

The key Jones matrices, with the fast axis at angle $\theta$, are:

• Half-Wave Plate: ${\mathbf{M}}_{HWP}(\theta )=\left(\begin{array}{cc}\cos (2\theta )& \sin (2\theta )\\ \sin (2\theta )& -\cos (2\theta )\end{array}\right)$
• Quarter-Wave Plate: ${\mathbf{M}}_{QWP}(\theta )=\left(\begin{array}{cc}{\cos }^2\theta +i{\sin }^2\theta & (1-i)\sin \theta \cos \theta \\ (1-i)\sin \theta \cos \theta & {\sin }^2\theta +i{\cos }^2\theta \end{array}\right)$

4. Optical Activity

Some materials, even when structurally isotropic (like a sugar solution), have the ability to rotate the plane of linearly polarized light. This phenomenon is called optical activity.

The physical origin of optical activity is molecular chirality (handedness). The molecules in an optically active substance have a three-dimensional structure that is different from its mirror image, like a left and right hand.

Because of this handedness, the material exhibits a different refractive index for left-circularly polarized light ($n_L$) versus right-circularly polarized light ($n_R$).

The following reading from MIT OCW provides a beautifully elegant derivation showing how this property leads to rotation.

As the text shows, by decomposing linearly polarized light into LCP and RCP components, applying the different phase shifts, and recombining, the net effect is described by a simple rotation matrix:
${\mathbf{M}}_{rot}(\theta )=\left(\begin{array}{cc}\cos \theta & -\sin \theta \\ \sin \theta & \cos \theta \end{array}\right)$
where the rotation angle $\theta$ is proportional to the path length and the difference in refractive indices, $\theta \propto L(n_L-n_R)$.

Conclusion

This lesson has formalized our understanding of light as a transverse electromagnetic wave. We've seen that polarization is a fundamental property that we can describe and manipulate with mathematical precision.

Key Takeaways:

• The polarization of light describes the oscillation direction of its electric field vector. States include linear, circular, and elliptical.
• Jones Calculus provides a rigorous framework, using 2x1 vectors for polarization states and 2x2 matrices for optical elements.
• Polarizers select a specific linear polarization via mechanisms like absorption (dichroism) or reflection (Brewster's angle). The transmitted intensity is governed by Malus's Law.
• Wave plates manipulate polarization by introducing a phase shift between orthogonal components due to birefringence. Quarter-wave plates can create circular polarization, and half-wave plates can rotate linear polarization.
• Optical activity is the rotation of linear polarization in chiral media, arising from a difference in refractive indices for left- and right-circularly polarized light.

This completes our module on Electromagnetic Waves and Optics. We have covered the generation of EM waves from Maxwell's equations, their properties, and the key wave phenomena of interference, diffraction, and now, polarization.

Preview of the Next Module:
In our next lesson, we will shift our focus from the physics of waves and fields to the mechanics of deformable matter. We will begin the module on Continuum and Fluid Mechanics. Our first topic will be to establish the mathematical tools needed to describe how continuous bodies deform under external forces, introducing the fundamental concepts of the stress and strain tensors.