Reflection & Refraction of Light
Advanced University‑Level Notes (MIT‑style)
Goal – Build a rigorous, first‑principles understanding of how light behaves at material interfaces, from Maxwell’s equations to practical design of optical coatings.
1. Historical & Conceptual Context
| Milestone | Key Insight | Relevance |
|---|---|---|
| 1600 – Huygens’ wavefront theory | Light as waves | Foundation for interference & diffraction |
| 1687 – Snell’s law | Empirical law of refraction | Shows dependence on optical path length |
| 1801 – Fresnel’s equations | Polarization–dependent reflection | Connects to Maxwell’s electromagnetic theory |
| 1844 – Maxwell’s equations | Unified EM field theory | Derives wave equation & boundary conditions |
| 1950s–1960s | Optical coatings & waveguides | Practical engineering of reflection & transmission |
2. Maxwell’s Equations in Linear Media
[ \begin{aligned} \nabla!\cdot!\mathbf{D} &= \rho_{!f}, & \nabla!\cdot!\mathbf{B} &= 0,\[2pt] \nabla\times\mathbf{E} &= -\dfrac{\partial\mathbf{B}}{\partial t}, & \nabla\times\mathbf{H} &= \mathbf{J}_{!f} + \dfrac{\partial\mathbf{D}}{\partial t}. \end{aligned} ]
For a source‑free region ((\rho_f=\mathbf{J}_f=0)) and linear, isotropic, homogeneous media:
[ \begin{aligned} \mathbf{D} &= \varepsilon \mathbf{E},\ \mathbf{B} &= \mu \mathbf{H}, \end{aligned} \qquad \varepsilon = \varepsilon_0 \varepsilon_r,;;\mu=\mu_0 \mu_r. ]
2.1 The Wave Equation
Taking the curl of Faraday’s law and substituting Ampère’s law:
[ \nabla \times (\nabla \times \mathbf{E}) = -\mu \varepsilon,\frac{\partial^2 \mathbf{E}}{\partial t^2}. ]
Using (\nabla \times (\nabla \times \mathbf{E}) = \nabla(\nabla!\cdot!\mathbf{E}) - \nabla^2\mathbf{E}) and noting (\nabla!\cdot!\mathbf{E}=0) in a source‑free region, we obtain
[ \nabla^2 \mathbf{E} - \mu\varepsilon,\frac{\partial^2 \mathbf{E}}{\partial t^2}=0. ]
A plane‑wave solution:
[ \mathbf{E}(\mathbf{r},t)=\mathbf{E}_0,e^{i(\mathbf{k}!\cdot!\mathbf{r}-\omega t)}. ]
Inserting gives the dispersion relation
[ |\mathbf{k}|^2 = k^2 = \mu\varepsilon,\omega^2 \equiv \frac{\omega^2}{v^2}, \qquad v=\frac{1}{\sqrt{\mu\varepsilon}}. ]
Defining the refractive index
[ n \equiv \frac{c}{v} = n_r + i n_i, ]
with (c=1/\sqrt{\mu_0\varepsilon_0}).
For lossless media (n_i=0) and (n=n_r=\sqrt{\varepsilon_r\mu_r}).
3. Boundary Conditions at an Interface
Consider two media (1) and (2) separated by a planar boundary (z=0). Let the interface plane be the (xy)-plane. For time‑harmonic fields ((e^{-i\omega t}) convention), Maxwell’s equations give the following continuity conditions:
| Quantity | Continuity |
|---|---|
| Tangential (\mathbf{E}) | (\mathbf{E}_1^{\parallel} = \mathbf{E}_2^{\parallel}) |
| Tangential (\mathbf{H}) | (\mathbf{H}_1^{\parallel} = \mathbf{H}_2^{\parallel}) |
| Normal (\mathbf{D}) | (\mathbf{D}_1^{\perp} = \mathbf{D}_2^{\perp}) |
| Normal (\mathbf{B}) | (\mathbf{B}_1^{\perp} = \mathbf{B}_2^{\perp}) |
These four conditions are the cornerstone for deriving Fresnel’s equations. They encapsulate the physical fact that there are no surface charges or currents at a clean, non‑conducting interface.
4. Plane Wave Incidence on a Planar Interface
Let an incident plane wave in medium 1 strike the interface at angle (\theta_i). The wavevector (\mathbf{k}_i) lies in the (xz)-plane:
[ \mathbf{k}_i = k_1(\sin\theta_i,\hat{\mathbf{x