Premed · Premed · Physics 2
Lecture 19: Wave Optics: Diffraction and Polarization
Physics II — Electromagnetism, Optics & Modern Physics
Learning Objectives
By the end of this lecture, students will be able to:
- Explain single-slit diffraction and calculate the positions of diffraction minima
- Describe the diffraction pattern from a diffraction grating and calculate principal maxima
- Apply the Rayleigh criterion to determine the resolving power of optical systems
- Explain the polarization of light and distinguish between polarization mechanisms
- Apply Malus's law to calculate transmitted intensity through polarizers
Lecture Content
I. Single-Slit Diffraction
When light passes through a narrow slit whose width a is comparable to the wavelength lambda, it spreads out rather than traveling in a straight line. This phenomenon is called diffraction. The resulting pattern on a distant screen has a broad central maximum flanked by much weaker secondary maxima.
Minima (dark fringes) occur at angles satisfying a sin(theta) = m lambda, where m = +/-1, +/-2, +/-3, ... (note that m = 0 is excluded, as it corresponds to the central maximum). At these angles, wavelets from different parts of the slit destructively interfere with one another.
The central maximum has an angular width of 2 lambda / a and contains about 84% of the total intensity. A wider slit produces a narrower central maximum, while a narrower slit produces a broader diffraction pattern. The secondary maxima are much weaker than the central peak: the first secondary maximum is only about 4.5% as intense. The intensity pattern follows the sinc-squared function: I(theta) = I_0 [sin(beta/2) / (beta/2)]^2, where beta = 2 pi a sin(theta) / lambda.
<image>A diagram of single-slit diffraction. A plane wave of wavelength lambda hits a slit of width a. Beyond the slit, the light spreads out. On a distant screen, the diffraction pattern is shown: a bright, wide central maximum flanked by progressively weaker secondary maxima separated by dark minima. The positions of the first and second minima are labeled at angles theta_1 = lambda/a and theta_2 = 2lambda/a. An intensity plot I(theta) versus theta is shown alongside, displaying the characteristic sinc-squared pattern with the dominant central peak and small side lobes.</image>
II. Diffraction Gratings
A diffraction grating consists of many equally spaced slits (or lines), with N slits separated by spacing d. Principal maxima (bright fringes) occur at angles satisfying d sin(theta) = m lambda, where m = 0, +/-1, +/-2, ..., the same condition as for double-slit interference.
The key differences from a double-slit setup are dramatic. The principal maxima are much sharper and brighter, with intensity proportional to N^2. Between adjacent principal maxima, there are (N - 1) minima and (N - 2) weak secondary maxima. The resolving power of a grating is R = mN, which determines its ability to distinguish closely spaced wavelengths. The minimum resolvable wavelength difference is Delta lambda = lambda / (mN).
Gratings are indispensable in spectroscopy, where they separate light into its component wavelengths far more effectively than a prism. They are used in spectrometers to analyze emission and absorption spectra. Higher-order maxima provide greater wavelength dispersion but may overlap with adjacent orders. Gratings come in two main types: transmission gratings (light passes through) and reflection gratings (light reflects off), with the latter used in most modern spectrometers. Typical line densities range from 300 to 1800 lines per millimeter.
III. Diffraction and Resolution
Diffraction places a fundamental limit on the resolving power of all optical instruments. A circular aperture of diameter D produces an Airy disk pattern: a central bright disk surrounded by concentric dark and bright rings. The angular radius of the first dark ring is theta = 1.22 lambda / D.
The Rayleigh criterion provides a practical definition of resolution: two point sources are just resolved when the central maximum of one falls on the first minimum of the other. The minimum resolvable angle is theta_min = 1.22 lambda / D. A smaller theta_min means better resolution, achieved by increasing the aperture D or using shorter wavelengths.
These considerations govern the performance of many instruments. Telescope resolution improves with larger mirrors or lenses. Microscope resolution improves with shorter wavelengths, which is why UV and electron microscopes can resolve smaller features. The human eye, with a pupil diameter of about 5 mm, has a resolution of approximately 1 arcminute (about 0.02 degrees) at lambda = 550 nm. Electron microscopes achieve atomic resolution because fast electrons have de Broglie wavelengths of roughly 0.01 nm.
<image>Panel A: The Airy disk pattern from a circular aperture — a bright central disk surrounded by concentric rings. The angular radius to the first dark ring is theta = 1.22 lambda/D. Panel B: The Rayleigh criterion illustrated with two point sources. Top: well-resolved (widely separated Airy disks). Middle: just resolved (the central maximum of one aligns with the first minimum of the other — the Rayleigh limit). Bottom: unresolved (the two Airy patterns overlap and cannot be distinguished as two separate sources). Intensity profiles are shown alongside each case.</image>
IV. Polarization of Light
Light is a transverse wave in which the electric field oscillates perpendicular to the direction of propagation. Unpolarized light has an E-field direction that is random and changes rapidly, with all transverse directions equally represented. Most natural light sources, such as the sun and incandescent bulbs, produce unpolarized light. Linearly polarized light has the E-field oscillating in a single plane. Circular and elliptical polarization describe situations in which the E-field vector rotates as the wave propagates.
V. Methods of Polarization
Polarization by selective absorption (dichroism) uses a polarizing filter (Polaroid) that transmits only the component of the electric field along its transmission axis. When unpolarized light passes through a polarizer, the intensity is reduced by half: I = I_0/2. Malus's law governs what happens when polarized light passes through a second polarizer (analyzer): I = I_0 cos^2(theta), where theta is the angle between the polarization direction and the analyzer's transmission axis. Two crossed polarizers (theta = 90 degrees) block all light (I = 0). Interestingly, inserting a third polarizer at 45 degrees between two crossed polarizers allows some light to pass through.
Polarization by reflection occurs at any interface, but at Brewster's angle, the reflected light is completely polarized. Brewster's law gives tan(theta_B) = n_2/n_1. At this angle, the reflected and refracted rays are perpendicular, and the reflected light is polarized parallel to the surface. Polaroid sunglasses exploit this by blocking horizontally polarized glare from reflective surfaces.
Polarization by scattering occurs when light is scattered at 90 degrees, producing completely polarized light. This is why the sky appears polarized, a fact exploited by some animals for navigation.
Polarization by birefringence (double refraction) occurs in crystals such as calcite and quartz that have two different indices of refraction for different polarization directions. An unpolarized beam entering such a crystal splits into two polarized beams: the ordinary and extraordinary rays.
<image>Three panels on polarization. Panel A: Unpolarized light (shown with arrows in all transverse directions) passes through a polarizer with a vertical transmission axis, producing vertically polarized light (only vertical arrows). The transmitted intensity is I_0/2. Panel B: The vertically polarized light then hits an analyzer with its transmission axis at angle theta to the vertical. The transmitted intensity is (I_0/2)cos^2(theta) (Malus's law). Panel C: Polarization by reflection at Brewster's angle — a ray hits a glass surface at theta_B. The reflected ray is horizontally polarized (E oscillates parallel to the surface), and the refracted ray is partially polarized. The reflected and refracted rays are perpendicular (90 degrees apart).</image>
VI. Applications of Polarization
Polarized sunglasses block horizontally polarized glare from reflective surfaces such as water and roads. LCD displays use liquid crystals sandwiched between crossed polarizers to control light transmission electronically. Stress analysis (photoelasticity) reveals stress distributions in transparent materials by placing them between crossed polarizers, producing colorful fringe patterns.
Optical activity is the property of certain substances, such as sugar solutions and quartz, to rotate the plane of polarization. Polarimetry measures this rotation to determine sugar concentration, which is important in the food industry and clinical laboratories. The specific rotation depends on concentration, path length, and wavelength.
3D movies use polarized light (either linear or circular) to send different images to each eye. Polarization microscopy is used in geology for mineral identification and in biology for examining birefringent structures such as collagen, starch granules, and crystals in urine samples.
