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Huygens' Principle

Light propagates as if emitted from every point along a wavefront, explaining phenomena like reflection and refraction. It’s a fundamental concept in physical optics.
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The statement of the theorem

Let E(x,t)\mathbf{E}(\mathbf{x}, t) be the electric field satisfying the homogeneous wave equation (∂2∂t2−c2∇2)E=0\left(\frac{\partial^2}{\partial t^2} - c^2 \nabla^2\right) \mathbf{E} = 0 in R3\mathbb{R}^3. Define the initial wavefront S(t0)S(t_0) as the surface where E(x,t0)=E0(x)\mathbf{E}(\mathbf{x}, t_0) = \mathbf{E}_0(\mathbf{x}). Huygens' Principle asserts that the field E(x,t)\mathbf{E}(\mathbf{x}, t) at a point x\mathbf{x} and time tt is given by the superposition integral over the initial surface S(t0)S(t_0): E(x,t)=14πρ∮S(t0)E0(x′)1ρ⋅n dS′\mathbf{E}(\mathbf{x}, t) = \frac{1}{4\pi\rho} \oint_{S(t_0)} \mathbf{E}_0(\mathbf{x}') \frac{1}{\rho} \cdot \mathbf{n} \, dS' where ρ=∣x−x′∣\rho = |\mathbf{x} - \mathbf{x}'| is the distance, x′\mathbf{x}' is a point on S(t0)S(t_0), and n\mathbf{n} is the unit normal vector to S(t0)S(t_0) pointing towards x\mathbf{x}. This formulation captures the constructive interference of secondary wavelets.