The Huygens-Fresnel Principle may be considered the fundamental postulate of wave optics. This is the principle by which any wavefront may be modeled as a [superposition](Superposition%20principle.md) of _wavelets_ as first presented by Huygens^[Huygens C.,translation by S. P. Thompson, _Treatise on Light_ ]. The mathematical formulation by which we understand this principle in a contemporary context is first given by Fresnel hence why we generally refer to the _Huygens-Fresnel Principle_. Given below is a schematic representation Huygen's principle. Here each point in space along a wavefront may be modeled as a source of a [spherical wave](Spherical%20wave.md) proportional to $e^{ikR}/R.$ This expression _approximates_ a possible solution to [Maxwell's equations](Maxwell's%20equations.md) for large values of $R.$  (image adapted from Peatross J., Ware M. _Physics of Light and Optics_) For a Homogeneous medium the choice of sources that compose the wavefront is arbitrary and does not require the use of infinitessimal wavelets. # Diffraction Integral The mathematical formulation of the Huygens principle is naturally motivated by the need to model [[diffraction]] since diffraction is the result of mutual interference between waves. Our goal is to find the form of the light field due to this interference. In particular, the resulting [light-field](Light-field.md) due to interfering wavelets as postulated by Fresnel is given by the following [diffraction integral](Diffraction%20integral.md). $E(x,y,z)=-\frac{i}{\lambda}\int\int dx'dy'E(x',y',0)\frac{e^{ikR}}{R}$ where $R=\sqrt{(x-x')^2+(y-y')^2+z^2}$ and $\lambda$ is the wavelength corresponding with wave number $k=\frac{2\pi}{\lambda}.$ The integral may be unbounded in free space or over a given aperture. In keeping with the original formulation by Fresnel, here we ignore the [obliquity factor](Diffraction%20integral.md#Obliquity%20factor) by setting it to $1.$ This integral is an exact solution of the [scalar Helmholtz equation](Helmholtz%20equation%20(optics).md#Scalar%20diffraction%20theory) however, it only approximates a solution to the [vector Helmholtz equation](Helmholtz%20equation%20(optics).md). #Electromagnetism/Optics/waveOptics