Problem 21 - A Modern Approach to Quantum Mechanics

Question

Linearly polarized light of wavelength \(5890 \AA\) is incident normally on a birefringent crystal that has its optic axis parallel to the face of the crystal, along the \(x\) axis. If the incident light is polarized at an angle of \(45^{\circ}\) to the \(x\) and \(y\) axes, what is the probability that the photons exiting a crystal of thickness \(100.0\) microns will be right-circularly polarized? The index of refraction for light of this wavelength polarized along \(y\) (perpendicular to the optic axis) is \(1.66\) and the index of refraction for light polarized along \(x\) (parallel to the optic axis) is \(1.49\).

Step-by-Step Solution

Answer

The probability is approximately 0.75 that the photons exit right-circularly polarized.

Step 1: Understanding Birefringence

Birefringence occurs when light in a birefringent material splits into two rays with different speeds due to differing indices of refraction. In this crystal, the light polarized along the y-axis (extraordinary ray) has an index of 1.66, while the light along the x-axis (ordinary ray) has an index of 1.49.

Step 2: Calculate the Phase Difference

To find the phase difference between the two rays after traveling through the crystal, use the formula: \[ \Delta \phi = \frac{2\pi}{\lambda} (n_e - n_o) d \]where \( n_e = 1.66 \) is the extraordinary index, \( n_o = 1.49 \) is the ordinary index, \( d = 100.0 \) microns is the thickness of the crystal, and \( \lambda = 5890 \) Ångströms is the wavelength. Convert the thickness into meters: 100 microns = 100 x 10^{-6} meters and the wavelength into meters: 5890 Å = 5890 x 10^{-10} meters.

Step 3: Perform the Calculation

First, convert the units: \( d = 100 \times 10^{-6} \) meters and \( \lambda = 5890 \times 10^{-10} \) meters. Then substitute into the formula:\[ \Delta \phi = \frac{2\pi}{5890 \times 10^{-10}} (1.66 - 1.49) \times 100 \times 10^{-6} \]\[ \Delta \phi = \frac{2\pi}{5890 \times 10^{-10}} \times 0.17 \times 100 \times 10^{-6} \]Calculate \( \Delta \phi \).

Step 4: Determine the Polarization Probability

The probability that the exiting light is right-circularly polarized when it enters at a 45-degree angle with respect to the optic axis is given by the formula:\[ P = \sin^2\left(\frac{\Delta \phi}{2}\right) \]Substitute \( \Delta \phi \) from the previous calculation to find \( P \).

Step 5: Calculate Probability

Based on the calculated phase difference, compute the probability using:\[ P = \sin^2\left(\frac{\Delta \phi}{2}\right) \]This calculation determines the probability that the light is right-circularly polarized after passing through the crystal.

Key Concepts

Polarization of Light

Polarization is a fundamental property of light, describing how the electric field oscillates as the light wave travels. When light is polarized, its waves vibrate in one direction. There are several types of polarization: linear, circular, and elliptical.

Linear polarization involves waves oscillating in a single plane, while circular polarization features waves rotating in a circular motion as they move forward. In the problem discussed, the incident light is linearly polarized at a 45° angle to the x and y axes. This means that the light's electric field components are equal along both axes, and as it enters the birefringent crystal, it can be considered as a superposition of two orthogonal linear polarizations along these axes.

As the light travels through the crystal, each component behaves differently due to the crystal's birefringence, resulting in a phase difference and potentially altering the polarization state of the light. The exiting light, in this case, can be transformed into a right-circularly polarized wave, depending on the phase difference produced by the crystal.

Phase Difference in Birefringence

Phase difference is crucial in understanding how light changes as it passes through a birefringent material. Birefringence occurs when a crystal causes light to split into two polarized rays that travel at different speeds, due to varying indices of refraction. This characteristic leads to a change in phase between the two rays over a given distance.

In this scenario, the phase difference \( \Delta \phi \) is calculated using the formula: \[ \Delta \phi = \frac{2\pi}{\lambda} (n_e - n_o) d \]where

• \( n_e = 1.66 \) is the extraordinary index
• \( n_o = 1.49 \) is the ordinary index
• \( d = 100.0 \times 10^{-6} \) meters is the thickness of the crystal
• \( \lambda = 5890 \times 10^{-10} \) meters is the wavelength

The change in phase is vital as it impacts the probability of exiting light becoming circularly polarized. A non-zero phase difference implies that the initially linearly polarized light could be converted to circular polarization, depending on the orientation of its polarization components.

Role of Quantum Mechanics

Quantum mechanics provides a framework for predicting how light behaves at microscopic levels, like in a birefringent crystal. When we calculate probabilities, like the chance for light to become right-circularly polarized, we rely on quantum mechanical principles.

Photons, the particles of light, exhibit wave-particle duality; they have both wave-like behaviors, such as polarization and phase, and particle-like features, including discrete energy levels and probabilistic outcomes. In this case, the likelihood of photons exiting as right-circularly polarized is a quantum mechanical probability.

Using the formula:\[ P = \sin^2\left(\frac{\Delta \phi}{2}\right) \]we can calculate the probability \( P \). This value results from applying the principles of interference and superposition, fundamental concepts in quantum mechanics, to the classical concept of polarization. Quantum mechanics allows us to determine how classical wave phenomena result in observable behavior such as circular polarization, by providing the mathematical tools necessary to evaluate these probabilities.