📘 How does light travel as a wave?

What you’ll learn

1. Electromagnetic Wave EquationDerive the electromagnetic wave equation and identify its speed and polarization properties.The wave equation follows from taking the curl of Faraday's and Ampere's laws. Its solutions are transverse waves with E and B perpendicular to the propagation direction. Phase velocity equals c when the medium is vacuum. Polarization direction is fixed by the E-field orientation. This foundation links optics to Maxwell's full theory.
2. Superposition and CoherenceApply the principle of superposition and quantify coherence length for interference visibility.Superposition produces constructive or destructive interference according to path difference. Coherence length equals λ²/Δλ for quasi-monochromatic sources. Visibility drops when path difference exceeds this length. Spatial coherence across the wavefront is set by source size. These limits determine whether fringes appear in any given setup.
3. Young's Double-Slit ExperimentCalculate fringe positions and intensities in the double-slit geometry.Path difference δ = d sinθ produces bright fringes when δ = mλ. Intensity follows I = 4I₀ cos²(δ/2). The pattern is modulated by single-slit diffraction envelope. Fringe spacing is inversely proportional to slit separation. The experiment quantifies wavelength from measurable geometry.
4. Thin-Film InterferencePredict reflected color from film thickness and refractive index.A π phase shift occurs only at the higher-index interface. Condition for destructive interference in reflection is 2nt = mλ when both reflections have the same shift. Soap films therefore appear dark at t → 0. Newton's rings and anti-reflection coatings follow the same rule. Thickness maps are obtained by counting fringes under monochromatic light.
5. Single-Slit DiffractionDerive the single-slit intensity pattern and locate its minima.Each infinitesimal strip across the slit contributes a phasor. The resultant amplitude is proportional to sinc(β) where β = (π a sinθ)/λ. Zeroes occur at a sinθ = mλ, m ≠ 0. The central maximum contains 90 % of the power. Narrower slits produce wider patterns, illustrating the uncertainty relation between position and momentum.
6. Diffraction GratingsCalculate diffraction angles and resolving power of a grating.The grating equation d(sinθ_i + sinθ_m) = mλ locates principal maxima. Resolving power R = mN where N is total lines illuminated. Blazed gratings concentrate energy into one order by shaping groove angle. Overlapping orders are separated with filters or cross-dispersion. Spectroscopy in astronomy relies on these devices.
7. Polarization StatesDescribe linear, circular, and elliptical polarization with Jones vectors.Any polarization state is a linear combination of two orthogonal basis states. Jones calculus multiplies 2×1 vectors by 2×2 matrices for each optical element. Quarter-wave plates convert linear into circular light. Malus's law governs intensity through successive polarizers. Stress-induced birefringence reveals internal strains in transparent materials.
8. Fresnel Reflection CoefficientsApply Fresnel equations to calculate amplitude and phase shifts at dielectric interfaces.r_s and r_p differ because boundary conditions on E_parallel and D_perp are distinct. Phase shift of π occurs for external reflection when the incident medium has lower index. Internal reflection beyond the critical angle yields evanescent waves. These coefficients govern anti-reflection coatings and polarizing beam splitters.
9. Thin Lenses and Image FormationUse the lens equation and magnification formula to locate images.1/f = 1/s_o + 1/s_i holds for paraxial rays. Positive f denotes converging lenses. Lateral magnification m = –s_i/s_o. Ray-tracing rules locate image points without calculation. Thick-lens corrections become necessary when object or image lies inside the lens thickness.
10. Aberrations and LimitsIdentify primary aberrations and estimate their blur sizes.Seidel coefficients quantify spherical, coma, astigmatism, field curvature, and distortion. Aspheric surfaces or achromatic doublets cancel spherical and chromatic errors. The diffraction limit remains after aberrations are removed. Wavefront error is measured in fractions of a wavelength.
11. Rayleigh CriterionApply the Rayleigh criterion to calculate resolution of circular apertures.The Airy radius is 1.22λ/D. Two sources are resolved when their separation equals this radius. Apodization trades peak intensity for narrower central lobe. Adaptive optics on ground-based telescopes approach the diffraction limit by correcting wavefront tilt in real time.
12. Fourier Optics and Spatial FilteringExplain how a lens performs a Fourier transform and how spatial filters modify images.The field in the focal plane is the Fourier transform of the input transparency. Multiplication by a filter function followed by a second transform yields convolution. High-pass filters accentuate edges; low-pass filters remove speckle. This framework underpins holography and modern computational imaging.

Questions this course answers

Grounded in trusted sources

• nist.gov
• nasa.gov
• aip.org
• OpenStax University Physics Volume 3 — Electromagnetic Waves
• NIST — SI units and optical metrology
• NASA — electromagnetic spectrum educational resources

Every Wunder lesson is built from real, reputable sources — never invented.

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