10.1 Wave nature of light

Light's wave nature explains phenomena like interference and diffraction that geometric ray optics can't account for. By treating light as electromagnetic waves with properties like wavelength and frequency, you can predict how light bends around obstacles, creates colorful patterns in thin films, and produces interference fringes. This topic lays the groundwork for wave optics and connects forward to ideas like wave-particle duality.

Wave-particle duality

Light doesn't fit neatly into one category. Sometimes it behaves like a wave (interference, diffraction), and sometimes it behaves like a stream of particles (photoelectric effect). Wave-particle duality captures this: light is both, and which behavior you observe depends on the experiment you're running. This idea bridges classical optics with quantum mechanics.

Light as electromagnetic waves

Light consists of oscillating electric and magnetic fields that propagate through space, perpendicular to each other and to the direction of travel. These waves are characterized by three properties:

• Wavelength ($\lambda$): the distance between successive crests
• Frequency ($f$): how many oscillations occur per second
• Amplitude: the maximum strength of the electric (or magnetic) field

Maxwell's equations describe this behavior mathematically. One key result from those equations is the speed of light in a vacuum:

$c = \frac{1}{\sqrt{\epsilon_0 \mu_0}}$

where $\epsilon_0$ is the permittivity of free space and $\mu_0$ is the permeability of free space. This gives $c \approx 3.0 \times 10^8 \text{ m/s}$. The wave model explains reflection, refraction, and diffraction.

Photon model of light

The wave model can't explain everything. In the photon model, light travels as discrete packets of energy called photons. Each photon carries energy:

$E = hf$

where $h$ is Planck's constant ($6.63 \times 10^{-34} \text{ J·s}$) and $f$ is the frequency. A photon also carries momentum:

$p = \frac{h}{\lambda}$

This model explains the photoelectric effect (where light ejects electrons from a metal surface) and Compton scattering (where photons bounce off electrons and lose energy). The quantization of light energy also explains why atoms have discrete energy levels.

De Broglie wavelength

De Broglie extended wave-particle duality to matter. Any particle with momentum $p$ has an associated wavelength:

$\lambda = \frac{h}{p}$

For a baseball, this wavelength is absurdly tiny and undetectable. But for an electron, it's large enough to produce observable diffraction patterns, which has been confirmed experimentally. This idea led directly to wave mechanics and the Schrödinger equation.

Interference of light

Interference occurs when two or more light waves overlap and combine. Where crests meet crests, you get constructive interference (brighter). Where crests meet troughs, you get destructive interference (darker). This superposition effect is one of the strongest pieces of evidence for light's wave nature.

Young's double-slit experiment

This is the classic demonstration of light interference. When coherent light passes through two narrow slits, it produces alternating bright and dark bands (fringes) on a distant screen.

• Bright fringes appear where the path difference between the two waves is a whole number of wavelengths (constructive interference).
• Dark fringes appear where the path difference is a half-wavelength off (destructive interference).

The position of the bright fringes on the screen is given by:

$y = \frac{m\lambda L}{d}$

where $m$ is the fringe order (0, ±1, ±2, ...), $\lambda$ is the wavelength, $L$ is the distance from the slits to the screen, and $d$ is the slit separation. Remarkably, this experiment can be performed with single photons fired one at a time, and the interference pattern still builds up over many detections.

Thin film interference

When light hits a thin transparent film (like a soap bubble or oil slick), some reflects off the top surface and some off the bottom. These two reflected waves can interfere with each other.

The condition for constructive interference in a thin film is:

$2nt = m\lambda$

where $n$ is the refractive index of the film, $t$ is the film thickness, and $m$ is an integer. Keep in mind that a phase shift of half a wavelength occurs when light reflects off a medium with a higher refractive index, which affects whether you use $m$ or $m + \frac{1}{2}$ in the equation.

Thin film interference is used in anti-reflective coatings on lenses and in optical filters. You can also use interference patterns to measure film thickness very precisely.

Michelson interferometer

The Michelson interferometer splits a beam of light into two paths using a beam splitter, sends them to mirrors, and recombines them. Any difference in the optical path length between the two arms creates an interference pattern.

• It can detect path length changes on the order of a single wavelength of light.
• It was historically used in the Michelson-Morley experiment, which disproved the existence of the luminiferous ether.
• Modern versions are the basis of LIGO, the detector that first observed gravitational waves.

Diffraction of light

Diffraction is the bending and spreading of light waves as they pass through openings or around obstacles. It becomes noticeable when the size of the opening or obstacle is comparable to the wavelength of light. Diffraction sets fundamental limits on the resolution of optical instruments like microscopes and telescopes.

Single-slit diffraction

When light passes through a single narrow slit of width $a$, it spreads out and produces a pattern with a broad central maximum and weaker side fringes. The dark fringes (minima) occur at angles given by:

$\sin \theta = \frac{m\lambda}{a}$

where $m = \pm 1, \pm 2, ...$ (not zero; $m = 0$ is the central maximum). The intensity distribution across the pattern is:

$I(\theta) = I_0 \left(\frac{\sin \alpha}{\alpha}\right)^2$

where $\alpha = \frac{\pi a \sin \theta}{\lambda}$. This pattern is a direct consequence of the Huygens-Fresnel principle, which treats every point in the slit as a source of secondary wavelets that interfere with each other.

Diffraction gratings

A diffraction grating has many closely spaced parallel slits (often thousands per centimeter). Because many beams interfere simultaneously, the bright fringes become very sharp and well-defined, making gratings excellent tools for separating light into its component wavelengths.

The grating equation is:

$d \sin \theta = m\lambda$

where $d$ is the spacing between adjacent slits. The more slits a grating has, the sharper its spectral lines and the better it can resolve closely spaced wavelengths. Gratings are widely used in spectroscopy.

X-ray diffraction

X-rays have wavelengths comparable to the spacing between atoms in crystals (on the order of angstroms). When X-rays hit a crystal, the regularly spaced atomic planes act like a diffraction grating. Constructive interference occurs when Bragg's law is satisfied:

$2d \sin \theta = n\lambda$

where $d$ is the spacing between crystal planes. X-ray diffraction reveals crystal structure, bond lengths, and molecular geometry. It was famously used to determine the structure of DNA.

Polarization of light

Polarization describes the direction in which the electric field oscillates in a light wave. Unpolarized light (like sunlight) has electric field oscillations in all directions perpendicular to the direction of travel. Polarization is a property unique to transverse waves, so its existence confirms that light is a transverse wave.

Linear vs circular polarization

• Linear polarization: the electric field oscillates in a single plane. Think of a wave on a string moving only up and down.
• Circular polarization: the electric field vector rotates in a circle as the wave propagates. The tip of the electric field traces out a helix.
• Elliptical polarization: a more general case where the electric field traces an ellipse. Both linear and circular are special cases of elliptical.

Natural light is typically unpolarized, containing all possible orientations of the electric field randomly mixed together.

Polarizers and filters

A polarizer is an optical element that transmits only light with a specific polarization direction. When polarized light passes through a second polarizer (called an analyzer), the transmitted intensity follows Malus's law:

$I = I_0 \cos^2 \theta$

where $\theta$ is the angle between the light's polarization direction and the polarizer's transmission axis. At $\theta = 90°$, no light gets through.

Practical applications include:

• Polarizing sunglasses, which block horizontally polarized glare from reflective surfaces
• LCD screens, which use polarizers and liquid crystals to control which light passes through each pixel

Brewster's angle

When light hits a surface at a specific angle called Brewster's angle, the reflected light is completely polarized perpendicular to the plane of incidence. This angle is given by:

$\tan \theta_B = \frac{n_2}{n_1}$

where $n_1$ and $n_2$ are the refractive indices of the two media. For glass ($n \approx 1.5$) in air, Brewster's angle is about 56°. The transmitted light at this angle is partially polarized. Brewster's angle is used in laser cavities to minimize reflection losses at optical surfaces, and it explains why glare off water or roads is mostly horizontally polarized.

Dispersion of light

Dispersion occurs because the refractive index of a material depends on wavelength. Different colors of light travel at different speeds in a medium, which causes them to refract by different amounts. This is why a prism separates white light into a spectrum.

Prism dispersion

When white light enters a prism, shorter wavelengths (violet) bend more than longer wavelengths (red) because the refractive index is higher for shorter wavelengths. The result is a spread-out spectrum of colors.

• The amount of dispersion depends on the prism material. The Abbe number quantifies how strongly a material disperses light (lower Abbe number means more dispersion).
• Newton's experiments with prisms showed that white light is a mixture of all visible colors, and that individual colors cannot be further separated.

Rainbows and spectral colors

Rainbows form when sunlight enters water droplets, undergoes dispersion and internal reflection, and exits at specific angles.

• The primary rainbow appears at about 42° from the antisolar point (the point directly opposite the sun from your perspective), with red on the outside and violet on the inside.
• The secondary rainbow appears at about 51°, with the color order reversed, because light reflects twice inside the droplet.
• Alexander's dark band is the darker region between the two rainbows where no light is directed toward the observer.
• Faint supernumerary bows just inside the primary rainbow are caused by interference effects.

Chromatic aberration

Chromatic aberration is a lens defect caused by dispersion. Because a lens bends different wavelengths by different amounts, it focuses them at slightly different points.

• Axial chromatic aberration: different colors focus at different distances along the optical axis.
• Lateral chromatic aberration: different colors form images at slightly different positions, causing color fringing at the edges of an image.

This is corrected using achromatic doublets (two lenses of different glass types cemented together) or diffractive optical elements.

Coherence of light

Coherence measures how well-correlated light waves are with each other. High coherence means the waves maintain a stable phase relationship, which is necessary for producing clear interference patterns. Lasers have high coherence; incandescent bulbs have very low coherence.

Temporal vs spatial coherence

• Temporal coherence describes how well a wave maintains its phase over time (or equivalently, along its direction of travel). It's related to how monochromatic the source is. A source with a narrow spectral bandwidth has high temporal coherence.
• Spatial coherence describes how well-correlated the wave is at different points across a wavefront. It's related to the size of the source. A small, distant source (like a star) has high spatial coherence.

Perfect coherence is an idealization. Lasers come closest in practice.

Coherence length

The coherence length is the maximum path difference over which two waves from the same source can still produce visible interference. It's given by:

$L_c = \frac{c}{\Delta f}$

where $\Delta f$ is the spectral bandwidth of the source. A highly monochromatic laser might have a coherence length of many meters, while a white LED might have a coherence length of only a few micrometers. This parameter determines, for example, the maximum sample thickness you can probe with an interferometer.

Applications in interferometry

Coherence is central to many precision measurement techniques:

• Michelson interferometers measure tiny displacements using coherent light.
• Optical coherence tomography (OCT) uses low-coherence light to produce high-resolution cross-sectional images of biological tissue (widely used in ophthalmology).
• Stellar interferometry combines light from multiple telescopes to achieve angular resolution far beyond what a single telescope can provide.
• Holography requires highly coherent light to record both the amplitude and phase of a wavefront.
• Fiber optic gyroscopes use the Sagnac effect with coherent light to sense rotation.

Quantum nature of light

At the microscopic level, light reveals particle-like properties that classical wave theory cannot explain. The quantum description of light treats it as quantized electromagnetic radiation, where energy and momentum come in discrete amounts. This framework is essential for understanding how light interacts with matter.

Photoelectric effect

When light shines on a metal surface, electrons can be ejected. The key observations are:

1. Below a certain threshold frequency, no electrons are emitted regardless of light intensity.
2. Above the threshold, the maximum kinetic energy of emitted electrons depends on frequency, not intensity.
3. Increasing intensity increases the number of electrons emitted, not their energy.

Einstein explained this by proposing that light consists of photons, each with energy $E = hf$. The maximum kinetic energy of an emitted electron is:

$K_{max} = hf - \phi$

where $\phi$ is the work function (the minimum energy needed to free an electron from the material). This result was a major piece of evidence for the quantization of light.

Compton scattering

When a high-energy photon (typically an X-ray) collides with an electron, the photon loses some energy and its wavelength increases. The change in wavelength is:

$\Delta \lambda = \frac{h}{m_e c}(1 - \cos \theta)$

where $m_e$ is the electron mass and $\theta$ is the scattering angle. This result can only be explained by treating the photon as a particle with definite momentum, providing strong evidence for the particle nature of light.

Wave function and probability

In quantum mechanics, the state of a photon (or any quantum particle) is described by a wave function. The square of the wave function's magnitude gives the probability density of finding the photon at a particular location.

This means you can't predict exactly where a single photon will land, but you can predict the statistical pattern that many photons will form. This probabilistic interpretation explains phenomena like quantum tunneling and is the foundation for understanding atomic transitions and laser operation.

Light-matter interactions

When light encounters matter, several things can happen depending on the photon energy and the material's properties. These interactions are the basis of spectroscopy, optoelectronics, and photochemistry.

Absorption and emission

• Absorption occurs when a photon's energy matches the energy difference between two states in an atom or molecule. The photon is absorbed and the atom jumps to a higher energy state.
• Spontaneous emission happens when an excited atom randomly drops to a lower state, releasing a photon.
• Stimulated emission occurs when an incoming photon triggers an excited atom to emit a second photon with the same frequency, phase, and direction.

The absorption of light through a material follows the Beer-Lambert law:

$I = I_0 e^{-\alpha x}$

where $\alpha$ is the absorption coefficient and $x$ is the distance traveled through the material. This law explains why materials have characteristic colors and is the basis of absorption spectroscopy.

Fluorescence and phosphorescence

Both involve a material absorbing light and re-emitting it at a longer wavelength (lower energy). The difference is timing:

• Fluorescence is fast (nanoseconds). The excited electron drops back down almost immediately.
• Phosphorescence is slow (milliseconds to hours). The electron gets trapped in a metastable state before eventually emitting.

The energy difference between the absorbed and emitted photons is called the Stokes shift. These processes are used in fluorescent lighting, biological imaging (fluorescent markers), and glow-in-the-dark materials.

Lasers and stimulated emission

LASER stands for Light Amplification by Stimulated Emission of Radiation. A laser works through these key steps:

1. Pumping: Energy is supplied to the gain medium (by light, electricity, etc.) to excite atoms to higher energy states.
2. Population inversion: More atoms end up in the excited state than the ground state, which doesn't happen naturally.
3. Stimulated emission: Photons passing through the medium trigger excited atoms to emit identical photons, amplifying the light.
4. Optical feedback: Mirrors on either end of the gain medium form a resonant cavity, bouncing light back and forth to build up intensity. One mirror is partially transparent, letting the laser beam out.

The result is light that is highly coherent, monochromatic, and directional.

Measurement and detection

Detecting and measuring light accurately is essential across science and technology. Different detectors are suited to different applications depending on the required sensitivity, speed, and wavelength range.

Photomultiplier tubes

Photomultiplier tubes (PMTs) are extremely sensitive light detectors. A photon strikes a photocathode, ejecting an electron via the photoelectric effect. That electron is then accelerated through a series of electrodes (dynodes), knocking out more electrons at each stage. This cascade can produce a gain exceeding $10^8$, making PMTs capable of detecting single photons. They offer time resolution better than 1 nanosecond and are used in nuclear physics, astronomy, and medical imaging.

Charge-coupled devices

A CCD (charge-coupled device) is a semiconductor chip with an array of light-sensitive pixels. Each pixel accumulates charge proportional to the amount of light hitting it. After exposure, the charges are read out row by row.

CCDs are characterized by:

• Quantum efficiency: the fraction of incoming photons that generate a detectable signal
• Dynamic range: the ratio of the brightest to dimmest signal the sensor can capture
• Read noise: electronic noise introduced during readout

CCDs are used in digital cameras, astronomical telescopes, and scientific imaging instruments.

Spectroscopy techniques

Spectroscopy separates light into its component wavelengths to reveal information about a source or sample:

• Absorption spectroscopy: measures which wavelengths a sample absorbs, identifying its chemical composition.
• Emission spectroscopy: analyzes light emitted by excited atoms or molecules (used to identify elements in stars, for example).
• Raman spectroscopy: detects the small wavelength shifts caused by inelastic scattering of monochromatic light, revealing molecular vibration modes.
• Fourier transform spectroscopy: uses an interferometer to collect data across all wavelengths simultaneously, then applies a Fourier transform to extract the spectrum. This approach offers high resolution and sensitivity.

Applications of wave optics

The principles of interference, diffraction, and coherence enable a wide range of technologies that go well beyond basic optics experiments.

Fiber optics

Optical fibers transmit light over long distances by trapping it inside a thin glass or plastic core using total internal reflection. The core has a higher refractive index than the surrounding cladding, so light that enters at a shallow enough angle bounces along the fiber without escaping.

• Single-mode fibers have a very small core (about 9 μm) and carry one mode of light. They're used for long-distance telecommunications because they have low signal dispersion.
• Multi-mode fibers have a larger core (50-62.5 μm) and carry multiple modes. They're used for shorter distances, like within buildings.

Fiber optics also have applications in medical endoscopy and sensing.

Holography

A hologram records not just the intensity of light (like a photograph) but also the phase information. This is done by recording the interference pattern between a reference beam and light reflected from an object, both from the same coherent laser source.

When the hologram is illuminated with the reference beam, it reconstructs the original wavefront, producing a true three-dimensional image. Applications include security features on credit cards, data storage, and holographic microscopy.

Optical computing

Optical computing uses photons instead of electrons to process information. The potential advantages include higher speed (photons travel at the speed of light) and lower power consumption for certain operations.

• Optical logic gates rely on nonlinear optical effects to perform computations.
• Major challenges remain in miniaturization and integration with existing electronic systems.
• Quantum optical computing explores using individual photons as quantum bits (qubits) for quantum information processing.

This field is still largely in the research stage, but it represents a promising direction for future computing technology.