Electromagnetic Waves: Wave-Particle Duality, Physical Optics, and Interference Patterns

An Abridged "History" of Light

• The Dual Nature of Light: The debate over whether light is a particle or a wave has lasted over $300$ years. Depending on the experiment, light exhibits characteristics of both waves and particles.

• Isaac Newton (1704): Published "Opticks," describing light as a group of tiny particles called "corpuscles." This model explained reflection, refraction, and dispersion.

• Wave Theory Proponents: Certain properties like diffraction (light bending around objects) could only be explained by treating light as a wave. This theory is attributed to Christiaan Huygens, with contributions from Robert Hooke and Leonhard Euler.

• Thomas Young (1803): Conducted the Double Slit Experiment, which provided evidence that light acts as a wave.

• James Clerk Maxwell (1861): Published four equations of electromagnetism where light was treated specifically as a wave.

• Max Planck (1900): Explained Black Body Radiation by proposing that light is emitted only in quantized bits of energy, behaving like a particle. This marked the birth of quantum physics.

• Albert Einstein (1905): Published a paper on the photoelectric effect, confirming that light comes in discrete packets of energy. He later earned a Nobel Prize for this work.

• Gilbert Lewis (1926): Coined the term "photons" for these packets of light energy.

• The Fifth Solvay Conference (1927): A meeting in Leopold Park at the Institut International de Physique Solvay where the world's most prominent physicists discussed new quantum theory.     * Participants: A. Piccard, E. Henriot, P. Ehrenfest, E. Herzen, Th. De Donder, E. Schrödinger, J.E. Verschaffelt, W. Pauli, W. Heisenberg, R.H. Fowler, L. Brillouin, P. Debye, M. Knudsen, W.L. Bragg, H.A. Kramers, P.A.M. Dirac, A.H. Compton, L. de Broglie, M. Born, N. Bohr, I. Langmuir, M. Planck, M. Curie, H.A. Lorentz, Albert Einstein, P. Langevin, Ch. E. Guye, C.T.R. Wilson, and O.W. Richardson.

• Quantum Electrodynamics (QED): This theory fully integrated quantum physics with electricity and magnetism. Sin-Itiro Tomonaga, Julian Schwinger, and Richard Feynman received the Nobel Prize in $1965$ for describing the interactions between light and matter.

• Richard Feynman's Quote: "I want to emphasize that light comes in this form - particles… I'm telling you the way it does behave - like particles."

Properties of Electromagnetic Waves

• Foundational Laws:     * Ampere's Law: States that a current (caused by an electric field) generates a magnetic field.     * Faraday's Law: States that a changing magnetic field generates an electric field.

• Electromagnetic (EM) Waves: A changing electric field creates a changing magnetic field, which in turn creates a changing electric field, and so on. These time-varying fields travel as EM waves. They are pure energy and possess no mass.

• Structure: The electric and magnetic segments of an EM wave are perpendicular to each other and perpendicular to the direction of propagation.

• Creation via Accelerating Charges:     * In a broadcast radio or TV antenna on the vertical ($z$) axis, electrons are accelerated up and down by a changing voltage.     * This creates a changing electric field in the $z$ direction and a changing magnetic field in the $x-y$ plane.     * These changing fields propagate outward (e.g., along the $y$ axis) until absorbed by a material.

• The Speed of Light ($c$):     * Measured to be $2.9979 \times 10^8\,m/s$.     * Used in physics problems as $3.00 \times 10^8\,m/s$ in a vacuum.     * Equivalent to $671$ million miles per hour or $186,000$ miles per second.

• Wave Frequency and Wavelength:     * Frequency ($f$): Measures the number of oscillations the EM field makes per unit time.     * Period ($T$): The inverse of frequency ($T = \frac{1}{f}$).     * Wavelength ($\lambda$): Depends on wave velocity and frequency. In a vacuum: $c = \lambda f$.     * High Frequency: Results in short wavelength.     * Low Frequency: Results in long wavelength.

The Electromagnetic Spectrum

• Spectrum Overview: Visible light is only a small segment of the entire EM spectrum. As frequency increases, so does the energy of the wave.

• Order of Wavelength (Longest to Shortest): Radio Waves, Microwaves, Infrared, Visible Light, Ultraviolet, X-rays, Gamma rays.

• Scales of Wavelength:     * Radio Waves: $10^0\text{ to }10^3\,m$ (Buildings, Humans).     * Microwaves: $10^{-2}\,m$ (Butterflies).     * Infrared: $10^{-5}\,m$ (Needle Point).     * Visible Light: $0.5 \times 10^{-6}\,m$ (Protozoans).     * Ultraviolet: $10^{-8}\,m$ (Molecules).     * X-rays: $10^{-10}\,m$ (Atoms).     * Gamma Rays: $10^{-12}\,m$ (Atomic Nuclei).

• White Light: Composed of many frequencies (colors). Dark red light has a frequency of approximately $4.29 \times 10^{14}\,Hz$.

• Specific Properties Calculations:     * Example 1 (Wavelength): For $f = 4.29 \times 10^{14}\,Hz$ and $c = 3.00 \times 10^8\,m/s$, $\lambda = \frac{3.00 \times 10^8}{4.29 \times 10^{14}} \approx 700\,nm$.     * Example 2 (Frequency): For $\lambda = 630\,nm$, $f = \frac{3.00 \times 10^8}{630 \times 10^{-9}} \approx 4.76 \times 10^{14}\,Hz$.

Reflection, Refraction, and Dispersion

• Reflection: Light reflects off surfaces at an angle equal to the incident angle. This mirrors the behavior of a particle bouncing off a wall.

• Refraction: The bending of light as it enters a different medium at an angle. This occurs because the part of the wave entering first changes speed before the rest of the ray.     * Index of Refraction ($n$): The ratio between the speed of light in a vacuum ($c$) and the speed in a medium ($v$). $n = \frac{c}{v}$.     * Examples of $n$: Water ($1.33$), Diamond ($2.42$).     * If a ray enters parallel to the normal (straight in), no refraction occurs.

• Dispersion: The separation of white light into constituent colors (seen in prisms and rainbows).     * The index of refraction varies with the color (wavelength) of the light.     * Red light travels faster in transparent materials and refracts the least.     * Violet light travels slower and refracts the most.     * Relationship: As wavelength increases, refractive index decreases ($\lambda \uparrow, n \downarrow$).

Polarization

• Definition: The electric field vectors of an EM wave lie in a plane perpendicular to the wave's motion, called the plane of polarization.

• Unpolarized Light: Emitted by the sun; contains electric field vectors in random orientations.

• Polarizing Filters: Long organic polymers allow electrons to move along their length. If the electric field is parallel to the polymer, the wave is absorbed. If perpendicular, it passes through.

• Sunglasses: Use polarizing filters to block horizontally polarized light (glare) that reflects off horizontal surfaces like water.

• Intensity Reduction:     * The first polarizer reduces unpolarized light intensity by half ($\frac{1}{2} I_{unpolarized}$).     * Malus's Law: For a second polarizer at an angle $\theta$, $I = I_0 \cos^2(\theta)$.     * If two polarizers are at $90^{\circ}$, no light passes through.

• Nature of Waves: Only transverse waves can be polarized; longitudinal waves (sound) cannot.

Diffraction and Interference

• Diffraction: The bending of waves around obstacles or through small openings.     * Huygens' Principle: Every wave front consists of an infinite line of point sources.     * Passing through a narrow slit makes one point source act as a radial emitter.     * A wider slit allows multiple sources through, causing interference.

• Superposition Principle: When two waves meet, the resulting displacement is the sum of individual displacements.     * Constructive Interference: Waves are "in phase" (crests align with crests), resulting in a larger amplitude.     * Destructive Interference: Waves are "out of phase" (crests align with troughs), resulting in cancellation.     * Partial Interference: Waves are partially out of phase, leading to partial cancellation.

Young's Double Slit Experiment

• Experiment (1801): Thomas Young proved light is a wave by creating an interference pattern with two narrow slits.

• Pattern: Alternating bright lines (maxima) and dark lines (fringes) that decrease in intensity from the center.

• Mathematical Conditions:     * Constructive (Bright spots): The path length difference ($d \sin(\theta)$) must be an integer multiple of the wavelength ($m \lambda$). Formula: $d \sin(\theta) = m \lambda$, where $m = 0, 1, 2, \dots$.     * Destructive (Dark spots): Paths differ by a half-wavelength. Formula: $d \sin(\theta) = (m + \frac{1}{2}) \lambda$.

• Variables:     * $d$: Distance between slits.     * $L$: Distance to viewing screen.     * $x$: Distance from central midpoint to the fringe on the screen.

• Small Angle Approximation (\theta < 15^{\circ}):     * $\tan(\theta) \approx \sin(\theta) \approx \theta$.     * Location of maxima: $x \approx \frac{m \lambda L}{d}$.

• Diffraction Grating: Consists of thousands of equally spaced slits. Produces much sharper and narrower maxima. Distance between lines ($d$) is found by taking the inverse of lines per mm (e.g., $600\text{ lines/mm} \rightarrow d = 1.67 \times 10^{-6}\,m$).

Single Slit Diffraction

• Phenomenon: A single slit of width $d$ produces a wide central maximum followed by secondary maxima and dark minima.

• Mechanism: Based on Huygens' Principle, point sources within the single slit interfere with one another.

• Dark Fringes (Minima) Formula: $d \sin(\theta) = m \lambda$. Note that in this context, $d$ is the width of the single slit.

• Central Maximum Width: $2 \times \frac{\lambda L}{d}$. As the slit ($d$) gets narrower, the central maximum spreads out (diffracts more).

• Resolution: Large apertures (like eagle eyes or telescope lenses) increase resolution by reducing the diffraction spread of light.

• AP Physics 2 Notation Mapping:     * Path length difference ($\Delta L$): $d \sin(\theta)$.     * Width of slit: $w$.     * Location: $x$.

Thin Film Interference

• Principles: Combined effects of refraction, reflection, and interference in layers of material ($10^{-9}\,m\text{ to }10^{-6}\,m$ thick).

• Phase Shifts on Reflection:     * Reflection off a more dense medium (higher $n$): $180^{\circ}$ phase shift ($\frac{\lambda}{2}$).     * Reflection off a less dense medium (lower $n$): No phase shift ($0$).     * Transmitted waves: Never experience a phase shift upon entering a medium.

• Wavelength in Film: $\lambda_{film} = \frac{\lambda_{air}}{n_{film}}$.

• Case 1: Film Coating (e.g., Anti-glare lens): Indices increase (n_{air} < n_{film} < n_{glass}). Both reflections shift by $\frac{\lambda}{2}$, so they start back "in phase."     * Constructive: $2t = m \lambda_{film}$.     * Destructive: $2t = (m + \frac{1}{2}) \lambda_{film}$.

• Case 2: Bubble (e.g., Soap bubble): Indices follow a Low $\rightarrow$ High $\rightarrow$ Low pattern (n_{air} < n_{soap} > n_{air}). Only the top reflection shifts by $\frac{\lambda}{2}$, meaning reflections are naturally "out of phase."     * Constructive: $2t = (m + \frac{1}{2}) \lambda_{film}$.     * Destructive: $2t = m \lambda_{film}$.

• Calculations:     * Minimum thickness for no reflection in soap ($n=1.4, \lambda=560\,nm$): Use destructive bubble formula ($2t = 1 \times \lambda_{film}$), resulting in $t = 200\,nm$.     * Anti-glare minimum thickness ($n_{film}=1.3, n_{glass}=1.52, \lambda=650\,nm$): Use destructive film formula ($2t = \frac{1}{2} \lambda_{film}$), resulting in $t = 125\,nm$.

Questions & Discussion

• Q: What is produced by a changing magnetic field?     * A: An electric field (Faraday's Law).

• Q: What is light with a wavelength slightly shorter than $400\,nm$ called?     * A: Ultraviolet light.

• Q: Which component of an EM wave interacts most strongly with matter?     * A: The Electric Field.

• Q: What happens to a diffraction pattern if the wavelength is decreased?     * A: Interference fringes move closer to the central maximum.

• Q: What is responsible for colors on an oil slick?     * A: Reflection, refraction, and interference.