Beta Phase: Square45 is currently in beta testing. Expect some features or content to be incomplete or missing.
45

Wavefronts

An imaginary surface connecting points of equal phase (and therefore equal amplitude) of a wave. Crucial for understanding wave propagation in optics.
📜

The statement of the theorem

Let r∈R3\mathbf{r} \in \mathbb{R}^3 be the spatial position vector and Φ(r,t)\Phi(\mathbf{r}, t) be the phase function of an electromagnetic wave, satisfying the Helmholtz equation ∇2Φ+k2Φ=0\nabla^2 \Phi + k^2 \Phi = 0 (for monochromatic waves, k=ω/ck = \omega/c). A wavefront Σ\Sigma at time t0t_0 is defined as the level set of the phase function: Σ={r∈R3∣Φ(r,t0)=C}\Sigma = \{\mathbf{r} \in \mathbb{R}^3 \mid \Phi(\mathbf{r}, t_0) = C \}. The geometry of this surface is governed by the eikonal equation, which states that the magnitude of the wave vector k=∇Φ\mathbf{k} = \nabla \Phi must satisfy: \n\n∣∇Φ(r,t0)∣=k|\nabla \Phi(\mathbf{r}, t_0)| = k \n\nThis implies that the wavefront Σ\Sigma is an isophase surface, and its normal vector n\mathbf{n} is proportional to the gradient of the phase: n∝∇Φ\mathbf{n} \propto \nabla \Phi. Furthermore, the propagation of the wavefront is described by the characteristic manifold of the wave equation, where the phase Φ\Phi is a solution to the Hamilton-Jacobi equation derived from the eikonal approximation.