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Constructive Interference

Occurs when waves are in phase, resulting in an increased amplitude and brighter light, a key concept in wave interference phenomena.
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The statement of the theorem

Let E1(r,t)\mathbf{E}_1(\mathbf{r}, t) and E2(r,t)\mathbf{E}_2(\mathbf{r}, t) be two monochromatic electric fields propagating in vacuum, defined by the solutions to the wave equation ∇2E−1c2∂2E∂t2=0\nabla^2 \mathbf{E} - \frac{1}{c^2}\frac{\partial^2 \mathbf{E}}{\partial t^2} = 0. Assume the fields can be represented as: \begin{align*} \mathbf{E}_1(\mathbf{r}, t) &= \mathbf{A}_1 e^{i (\mathbf{k}_1 \cdot \mathbf{r} - \omega t)} \mathbf{\hat{e}}_1 \\ \mathbf{E}_2(\mathbf{r}, t) &= \mathbf{A}_2 e^{i (\mathbf{k}_2 \cdot \mathbf{r} - \omega t)} \mathbf{\hat{e}}_2 \end{align*} where Aj\mathbf{A}_j are the amplitudes, kj\mathbf{k}_j are the wave vectors, and ω\omega is the angular frequency. The resultant electric field is Eres(r,t)=E1(r,t)+E2(r,t)\mathbf{E}_{\text{res}}(\mathbf{r}, t) = \mathbf{E}_1(\mathbf{r}, t) + \mathbf{E}_2(\mathbf{r}, t). Constructive interference occurs at a point r\mathbf{r} and time tt if the phase difference Δϕ=k1⋅r−k2⋅r+(k2−k1)⋅r\Delta \phi = \mathbf{k}_1 \cdot \mathbf{r} - \mathbf{k}_2 \cdot \mathbf{r} + (\mathbf{k}_2 - \mathbf{k}_1) \cdot \mathbf{r} satisfies the condition Δϕ=2πn\Delta \phi = 2\pi n, where n∈Zn \in \mathbb{Z}, and the polarization vectors e^1\mathbf{\hat{e}}_1 and e^2\mathbf{\hat{e}}_2 are aligned, such that the resultant amplitude is maximized: \begin{equation*} \left| \mathbf{E}_{\text{res}}(\mathbf{r}, t) \right| = \left| \mathbf{A}_1 + \mathbf{A}_2 \right| \end{equation*}. This condition is equivalent to the path difference ΔL=∣r1−r2∣⋅k^\Delta L = |\mathbf{r}_1 - \mathbf{r}_2| \cdot \mathbf{\hat{k}} satisfying ΔL=nλ\Delta L = n\lambda, where λ=2π/∣k∣\lambda = 2\pi/|\mathbf{k}| is the wavelength.
Source: Wikipedia