9.1 Nature of light

Wave-Particle Duality

Light behaves as both a wave and a particle, depending on the situation. This dual nature is one of the most important ideas in modern physics, and it underpins everything from how lenses work to how solar cells generate electricity.

Particle Theory of Light

The particle model treats light as a stream of discrete packets called photons. Each photon carries a specific amount of energy determined by its frequency:

$E = hf$

where $h$ is Planck's constant ($6.626 \times 10^{-34} \text{ J·s}$) and $f$ is the frequency of the light. Higher-frequency light (like ultraviolet) carries more energy per photon than lower-frequency light (like red).

The particle model is essential for explaining phenomena where light interacts with matter one photon at a time, such as the photoelectric effect and Compton scattering.

Wave Theory of Light

The wave model describes light as oscillating electric and magnetic fields that propagate through space. Wavelength and frequency are related by:

$c = \lambda f$

where $c$ is the speed of light in vacuum. This model accounts for interference, diffraction, and polarization, which are all patterns that only make sense if light behaves as a wave.

Complementarity Principle

Niels Bohr proposed the complementarity principle to reconcile these two pictures. The idea: wave and particle descriptions are mutually exclusive but both necessary. You'll never observe wave behavior and particle behavior simultaneously in the same measurement. The type of experiment you set up determines which aspect of light you detect. This principle extends beyond light and applies to matter as well (electrons also exhibit wave-particle duality).

Electromagnetic Spectrum

All light, whether visible or not, is electromagnetic radiation. The electromagnetic spectrum organizes these waves by wavelength and frequency, from low-energy radio waves to high-energy gamma rays. Different regions of the spectrum interact with matter in distinct ways, which is why each region has different applications.

Visible Light Range

Visible light is the narrow band of the spectrum that human eyes can detect, spanning wavelengths from about 380 nm (violet) to 740 nm (red). The familiar color order is red, orange, yellow, green, blue, indigo, violet. Human eye sensitivity peaks in the green-yellow region around 555 nm, which is why green lasers appear brighter than red or blue ones at the same power.

Infrared and Ultraviolet

These sit on either side of visible light:

• Infrared (IR): wavelengths from about 700 nm to 1 mm. You experience IR as heat radiation. Applications include thermal imaging, remote controls, and fiber-optic communication.
• Ultraviolet (UV): wavelengths from about 10 nm to 380 nm. UV light carries enough energy to cause sunburns and is used in sterilization, fluorescence analysis, and semiconductor manufacturing (photolithography).

X-rays and Gamma Rays

These are the highest-energy forms of electromagnetic radiation:

• X-rays (0.01 nm to 10 nm) penetrate soft tissue but are absorbed by bone, making them ideal for medical imaging. They're also used in X-ray crystallography to determine molecular structures.
• Gamma rays (wavelengths shorter than 0.01 nm) are emitted during radioactive decay and nuclear reactions. Both X-rays and gamma rays can ionize atoms, which makes them useful in radiation therapy but also potentially harmful to living tissue.

Properties of Light

Speed of Light

The speed of light in vacuum, $c \approx 3 \times 10^8 \text{ m/s}$, is a fundamental constant derived from Maxwell's equations of electromagnetism. Nothing carrying information can travel faster than $c$.

When light enters a material (glass, water, etc.), it slows down. The refractive index $n$ of a material tells you by how much:

$v = \frac{c}{n}$

For example, glass with $n = 1.5$ slows light to $2 \times 10^8 \text{ m/s}$.

Reflection and Refraction

Reflection occurs when light bounces off a surface. For smooth (specular) reflection, the angle of incidence equals the angle of reflection, both measured from the normal to the surface.

Refraction occurs when light crosses the boundary between two media with different refractive indices. The change in speed causes the light to bend. Snell's law describes this quantitatively:

$n_1 \sin\theta_1 = n_2 \sin\theta_2$

When light travels from a higher-index medium to a lower-index medium (say, glass to air), there's a critical angle beyond which all light reflects back into the denser medium. This is total internal reflection, and it's the principle behind fiber optics.

Diffraction and Interference

Diffraction is the bending of light waves around obstacles or through narrow openings. A single slit produces a pattern with a broad central maximum flanked by weaker secondary maxima.

Interference occurs when two or more light waves overlap. Where crests align, you get constructive interference (bright fringes); where a crest meets a trough, you get destructive interference (dark fringes). The classic double-slit experiment demonstrates both diffraction and interference and provides strong evidence for the wave nature of light.

Quantum Nature of Light

At atomic and subatomic scales, light behaves in ways that classical wave theory can't explain. Quantum optics introduces the idea that light energy comes in discrete packets.

Photons and Energy Quanta

A photon is the smallest unit of light energy. Its energy depends on frequency:

$E = hf$

and its momentum is:

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

The quantization of light energy was first proposed by Planck to explain blackbody radiation, the spectrum of light emitted by hot objects. Classical theory predicted that a hot object should radiate infinite energy at short wavelengths (the "ultraviolet catastrophe"). Planck's quantum hypothesis resolved this by requiring energy to come in discrete chunks.

Photoelectric Effect

When light shines on a metal surface, electrons can be ejected. This is the photoelectric effect, and Einstein explained it by treating light as photons. The kinetic energy of an ejected electron is:

$KE = hf - W$

where $W$ is the work function (the minimum energy needed to free an electron from the metal). A key prediction: if the photon frequency is too low, no electrons are emitted regardless of intensity. This was a major piece of evidence for the particle nature of light. Practical applications include photovoltaic cells and light sensors.

Compton Effect

In Compton scattering, a photon collides with a free electron and loses some energy, emerging with a longer wavelength. The wavelength shift depends on the scattering angle:

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

where $m_e$ is the electron mass. This result only makes sense if you treat the photon as a particle with definite momentum, providing further evidence for light's particle nature. The quantity $\frac{h}{m_e c} \approx 2.43 \times 10^{-12} \text{ m}$ is called the Compton wavelength of the electron.

Light Sources

Different mechanisms produce light with very different characteristics. Understanding these mechanisms helps you predict the spectrum, coherence, and intensity of the light produced.

Thermal Radiation

Any object with a temperature above absolute zero emits electromagnetic radiation. The hotter the object, the more radiation it emits and the shorter the peak wavelength. Two key laws describe this:

• Stefan-Boltzmann law: Total radiated power is proportional to $T^4$.
• Wien's displacement law: The peak wavelength shifts to shorter values as temperature increases.

Examples include incandescent bulbs (around 2700 K), the Sun's surface (about 5800 K), and any heated object you can see glowing.

Fluorescence vs. Phosphorescence

Both involve absorbing light and re-emitting it, but they differ in timing:

• Fluorescence: Re-emission happens almost immediately (nanoseconds to microseconds). Used in fluorescent lamps and biomedical imaging.
• Phosphorescence: Re-emission is delayed (milliseconds to hours) because the electron gets temporarily trapped in a different energy state. This is what makes glow-in-the-dark materials work.

In both cases, the emitted light has a longer wavelength (lower energy) than the absorbed light.

Lasers and Coherent Light

LASER stands for Light Amplification by Stimulated Emission of Radiation. A laser produces light that is:

• Coherent: all the waves are in phase
• Monochromatic: a single wavelength (or very narrow range)
• Directional: the beam spreads very little

Lasers work by creating a population inversion (more atoms in an excited state than the ground state) and then using stimulated emission to produce a cascade of identical photons. Types include gas lasers (like helium-neon), solid-state lasers, and semiconductor (diode) lasers. Applications span medicine, manufacturing, telecommunications, and scientific research.

Optical Phenomena

Polarization of Light

Ordinary (unpolarized) light has electric field oscillations in all directions perpendicular to the direction of travel. Polarization restricts these oscillations:

• Linear polarization: oscillations confined to a single plane
• Circular and elliptical polarization: the electric field vector rotates as the wave propagates

A polarizer transmits only the component of light aligned with its transmission axis. Malus's law gives the transmitted intensity when polarized light passes through a second polarizer:

$I = I_0 \cos^2\theta$

where $\theta$ is the angle between the polarization direction and the polarizer's axis.

Dispersion and Rainbows

Dispersion occurs because a material's refractive index varies slightly with wavelength. Shorter wavelengths (violet) bend more than longer wavelengths (red). When white light passes through a prism, dispersion separates it into its component colors.

Rainbows form by the same principle: sunlight enters water droplets, refracts, reflects off the back surface, and refracts again on exit. Each wavelength exits at a slightly different angle, producing the familiar color band. Dispersion also causes chromatic aberration in lenses, where different colors focus at slightly different points.

Scattering and Sky Color

Rayleigh scattering occurs when light interacts with particles much smaller than its wavelength (like nitrogen and oxygen molecules in the atmosphere). The intensity of scattered light is proportional to $1/\lambda^4$, meaning blue light (short wavelength) scatters much more strongly than red light. This is why the sky appears blue overhead.

At sunrise and sunset, sunlight travels through a much thicker layer of atmosphere. Most of the blue light has scattered away before reaching your eyes, leaving the reds and oranges that color the horizon.

Mie scattering applies to larger particles (comparable to the wavelength) and doesn't depend as strongly on wavelength, which is why clouds (made of larger water droplets) appear white.

Light Interactions with Matter

Absorption and Emission

When a photon's energy matches the energy gap between two states in an atom or molecule, the photon can be absorbed, promoting the system to a higher energy level. The system can then emit a photon when it drops back down.

• Spontaneous emission happens randomly and produces incoherent light.
• Stimulated emission is triggered by an incoming photon and produces a photon identical to the trigger (this is the basis of lasers).

The Beer-Lambert law describes how light intensity decreases as it passes through an absorbing solution, and it's widely used in chemistry and biology for concentration measurements.

Transmission vs. Opacity

Materials fall on a spectrum from transparent to opaque:

• Transparent materials (like clear glass) transmit light with minimal absorption or scattering.
• Translucent materials (like frosted glass) transmit light but scatter it, so you can't see clear images through them.
• Opaque materials block light entirely through absorption or reflection.

How much light a material transmits depends on its composition, thickness, and the wavelength of the light. A material can be transparent to visible light but opaque to UV, for instance.

Luminescence and Fluorescence

Luminescence is a broad term for light emission that isn't caused by high temperature (unlike thermal radiation). Types include:

• Chemiluminescence: light from chemical reactions (glow sticks)
• Bioluminescence: light from biological organisms (fireflies, deep-sea fish)
• Electroluminescence: light from electrical excitation (LEDs)

In fluorescence specifically, the emitted photon always has a longer wavelength than the absorbed photon. The difference between the absorption and emission wavelengths is called the Stokes shift. This property is exploited in fluorescent microscopy, where you can illuminate a sample with one color and detect the emitted fluorescence at a different color.

Measurement of Light

Intensity and Luminous Flux

Two related but distinct quantities:

• Luminous intensity measures the power of light emitted per unit solid angle, in candelas (cd).
• Luminous flux measures the total light output of a source, in lumens (lm).

The inverse square law states that intensity from a point source decreases as the square of the distance: double the distance, and intensity drops to one-quarter. This is critical for lighting design and understanding how brightness falls off with distance.

Spectroscopy Basics

Spectroscopy studies how matter absorbs, emits, or scatters light at different wavelengths. By analyzing the resulting spectrum, you can identify elements, determine molecular structures, and measure concentrations.

• Absorption spectroscopy: measures which wavelengths a sample absorbs
• Emission spectroscopy: measures which wavelengths a heated or excited sample emits
• Raman spectroscopy: measures wavelength shifts caused by molecular vibrations

Each element produces a unique set of spectral lines, acting like a fingerprint. Resolution (the ability to distinguish closely spaced lines) and sensitivity (the ability to detect weak signals) are the key performance parameters.

Photometry vs. Radiometry

These are two different frameworks for measuring light:

• Radiometry measures the absolute power of electromagnetic radiation at all wavelengths, using units like watts and joules.
• Photometry measures light as the human eye perceives it, weighting the measurement by the eye's sensitivity curve. Photometric units include lumens and lux.

A green light source and an infrared source could have the same radiometric power (same number of watts), but the green source would have a much higher photometric value because your eyes are sensitive to green but can't see infrared at all. Converting between the two systems requires knowing the spectral distribution of the light.

Applications of Light

Fiber Optics

Optical fibers transmit information as pulses of light through thin, flexible glass or plastic strands. The light stays inside the fiber through total internal reflection.

• Single-mode fibers have a very small core and carry one light path, allowing signals to travel long distances with minimal distortion. Used in long-haul telecommunications.
• Multi-mode fibers have a larger core and carry multiple light paths. They're cheaper but limited to shorter distances.

Advantages over copper wire include higher bandwidth, lower signal loss over distance, and immunity to electromagnetic interference.

Holography Principles

Holography records not just the intensity of light (like a photograph) but also the phase information, creating a true three-dimensional image. The process requires:

1. A coherent light source (laser) split into two beams
2. One beam (the reference beam) goes directly to the recording medium
3. The other beam (the object beam) reflects off the subject
4. The two beams create an interference pattern on the recording medium

To view the hologram, you illuminate it with a beam similar to the original reference beam, and the recorded interference pattern reconstructs the 3D image. Applications include security features on credit cards, data storage, and artistic displays.

Optical Imaging Techniques

Optical imaging uses light to create visual representations of objects, from the very small (microscopy) to the very distant (telescopy).

Traditional optical systems are limited in resolution by diffraction. The Abbe diffraction limit sets the smallest feature you can resolve at roughly half the wavelength of light used. For visible light, that's around 200 nm.

Advanced techniques push past this limit:

• Confocal microscopy uses a pinhole to reject out-of-focus light, improving contrast and enabling 3D imaging.
• Super-resolution microscopy techniques (like STED and PALM) achieve resolution below the diffraction limit, earning the 2014 Nobel Prize in Chemistry.

Modern optical imaging increasingly combines traditional optics with digital image processing and computational methods.