the case of single-slit diffraction. A mathematical representation of Huygens' principle can be used to start an equation. Consider a monochromatic complex plane wave of wavelength λ incident on a slit of width a. If the slit lies in the x′-y′ plane, with its center at the origin, then it can be assumed that diffraction generates a complex wave ψ, traveling radially in the r direction away from the slit, and this is given by: Let be a point inside the slit over which it is being integrated. If is the location at which the intensity of the diffraction pattern is being computed, the slit extends from to , and from to . The distance r from the slot is: Assuming Fraunhofer diffraction will result in the conclusion . In other words, the distance to the target is much larger than the diffraction width on the target. By the binomial expansion rule, ignoring terms quadratic and higher, the quantity on the right can be estimated to be: It can be seen that 1/r in front of the equation is non-oscillatory, i.e. its contribution to the magnitude of the intensity is small compared to our exponential factors. Therefore, we will lose little accuracy by approximating it as 1/z. To make things cleaner, a placeholder C is used to denote constants in the equation. It is important to keep in mind that C can contain imaginary numbers, thus the wave function will be complex. However, at the end, the ψ will be bracketed, which will eliminate any imaginary components. Now, in Fraunhofer diffraction, is small, so (note that participates in this exponential and it is being integrated). In contrast the term can be eliminated from the equation, since when bracketed it gives 1. (For the same reason we have also eliminated the term ) Taking results in: It can be noted through Euler's formula and its derivatives that and . where the (unnormalized) sinc function is defined by . Now, substituting in , the intensity (squared amplitude) of the diffracted waves at an angle θ is given by: Multiple slits Let us again start with the mathematical representation of Huygens' principle. Consider slits in the prime plane of equal size and spacing spread along the axis. As above, the distance from slit 1 is: To generalize this to slits, we make the observation that while and remain constant, shifts by Thus and