Polarisation of Light and the Analogy to the Quantum Eraser

Experimental Goal

To generate circularly and elliptically polarized light from linearly polarized light using a quarter-wave ( $\lambda/4$ ) plate. Initially, linearly polarized light can be represented by a single oscillating electric field, for example, $E(z,t)=E_0\cos(\omega t-kz)\hat{x}$ A quarter-wave plate ( $\lambda/4$ plate) is designed to introduce a phase difference of $\frac{\Delta}{\varphi}=\frac{\pi}{2}$ between two orthogonal components of the electric field. If an incident linearly polarized wave has components $Ex=E{_{}0x}\cos(\omega t-kz)$ and $E_y=E_{0y}\cos(\omega t-kz)$ (e.g., if initially polarized at an angle to the crystal axes), after passing through the plate, the components become: $E_{x^{\prime}}=E{_0x}\cos(\omega t-kz)Ey^{\prime}=E_{0y}\cos(\omega t-kz+\frac{\pi}{2})$ If the incident light is linearly polarized at $45^\circ$ to the fast/slow axes of the plate, then $E_{0x} = E_{0y}$ and the output will be circularly polarized. If E_{0x} \neq E{0y} $or if a different phase difference $ \Delta $is introduced, (e.g., by a wave plate of different thickness or if the incident polarization is not at$ 45^\circ $to the optical axeis, the light will be ellipically polarized. The thickness$ d $required for a quarter-wave plate is given by the relation d required for a quarter-wave plate is given by the relation $ (n_{e}-n_{o})d=\frac{\lambda_0}{4} $, where$ \lambda_0 $is the vacuum wave length and$ n_e $and$ n_{o} $are the extraordinary and ordinary refractive indices of the plate material, respsctively.e vacuum wavelength and$
To detect these polarization states using an analyzer. An analyzer, typically a polaroid filter, is employed to selectively transmit or absorb light components based on their polarization, thereby allowing for the identification and quantitative analysis of the final polarization state.
To use linearly polarized light to illustrate the quantum eraser effect. This involves setting up an experiment where 'which-path' information about photons can be gained or lost, demonstrating how the presence or absence of this information impacts interference patterns, thereby highlighting the wave-particle duality and the role of measurement in quantum mechanics.

Theoretical Background: Birefringence

Definition: Birefringence, also known as double refraction (Doppelbrechung), is an optical property observed in certain anisotropic crystals. When unpolarized light illuminates such a crystal, it is split into two distinct partial beams: the ordinary (o-ray) and the extraordinary (e-ray). These two rays not only propagate in different directions but are also linearly polarized perpendicular to each other and travel through the crystal at different phase velocities. This difference in velocity is a direct consequence of their experiencing different refractive indices among the ordinary refractive index ( $n_o$ )and the extraordinary refractive index ( $n_e$ ) within the crystal. Specifically, the phase velocity $v$ of light in a material is inversely proportional to its refractive index $n$ ( $v = c/n$ ), meaning the ordinary ray ( $vo = c/no$ ) and extraordinary ray ( $ve = c/ne$ ) propagate at different speeds.
Illustration: Figure 1 typically depicts this phenomenon, showing an incident unpolarized light ray splitting upon entering a birefringent crystal, with the o-ray and e-ray following distinct paths and exhibiting orthogonal linear polarizations.
Cause: Birefringence originates from the optically anisotropic structure of the crystal lattice, which is determined by its specific crystal growth and internal molecular arrangement. In anisotropic materials, the distribution of electron clouds around atoms is not uniform in all directions. As a result, the interaction between the oscillating electric field vector of incident light and these electron clouds varies depending on the orientation of the electric field with respect to the crystal axes. This directional dependence leads to varying electrical susceptibility and permittivity in different directions, causing the refractive index of the crystal to be dependent on both the direction of light propagation and its polarization state.
Optically Isotropic vs. Anisotropic Materials:
- Isotropic materials: These are substances (e.g., gases, liquids, glasses, cubic crystals) where the refractive index is the same in all directions. Light propagating through such materials does not exhibit birefringence because the interaction with the material is uniform regardless of the light's polarization or propagation direction.
- Anisotropic materials: Birefringent crystals are intrinsically optically anisotropic. Their refractive index is not constant, but rather varies with the direction of light propagation and the orientation of the light's electric field vector.
Optical Axis: All birefringent crystals possess one or two specific directions known as optical axes. When light propagates precisely along an optical axis, it travels without experiencing birefringence; both the ordinary and extraordinary rays move at the same velocity and thus encounter the same refractive index ( $n_o = n_e$ ). Consequently, no splitting or differential phase shift between the rays occurs. Crystals with a single optical axis are classified as uniaxial, while those with two optical axes are termed biaxial.