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Diffraction

The bending of waves around obstacles or through apertures, illustrating wave nature and the wave-particle duality of light.
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The statement of the theorem

Let E(r,t)E(\textbf{r}, t) be the complex electric field amplitude at position r=(x,y,z)\textbf{r} = (x, y, z) and time tt. Assume the field propagates from an aperture plane z=0z=0 to a observation point r=(x,y,z)\textbf{r} = (x, y, z) where z>0z>0. Let E0(r⊥)E_0(\textbf{r}_\bot) be the field amplitude on the aperture plane r′=(x′,y′,0)\textbf{r}' = (x', y', 0), where r⊥=(x′,y′)\textbf{r}_\bot = (x', y'). The field E(r,t)E(\textbf{r}, t) at r\textbf{r} is given by the Kirchhoff-Huygens diffraction integral:\n\nE(x,y,z)=1iλ∫AE0(x′,y′)[eikRR] dx′dy′E(x, y, z) = \frac{1}{i\lambda} \int_{A} E_0(x', y') \left[ \frac{e^{ikR}}{R} \right] \, dx' dy' \n\nwhere k=ω/ck = \omega/c is the wave number, λ\lambda is the wavelength, AA is the aperture area, and R=(x−x′)2+(y−y′)2+z2R = \sqrt{(x-x')^2 + (y-y')^2 + z^2} is the distance from the source point (x′,y′,0)(x', y', 0) to the observation point (x,y,z)(x, y, z). For the Fraunhofer approximation (far-field), where zz is large and the phase term is approximated by the planar wave phase exp⁡(ik(x−x′)2+(y−y′)22z)\exp\left(i k \frac{(x-x')^2 + (y-y')^2}{2z} \right), the integral simplifies to:\n\nE(x,y,z)≈eikziλz∫AE0(x′,y′)eikz(xx′+yy′) dx′dy′E(x, y, z) \approx \frac{e^{ikz}}{i\lambda z} \int_{A} E_0(x', y') e^{i\frac{k}{z} (x x' + y y')} \, dx' dy'
Source: Wikipedia