Representation of Diffraction by Vector Summation
The method applied here to calculate the intensity distribution of the one-slit diffraction is based on Richard P. Feynman's concept of integrating over paths. Each path contributes a vector with an angle corresponding to the length of the path. The sum of all the vectors gives the amplitude and the square of the amplitude of the intensity at a given point on the screen. You can observe how the changes of the intensity develop from the addition of vectors.[more]
The accuracy of the calculated intensity depends on the number of paths. Here there are slight differences to the theoretical values of the Fraunhofer diffraction, but this is balanced here by geometrical clarity. The method offers a comprehensive approach and can be easily verified with comparatively little mathematical effort.[less]
The space in the slit is divided by equally spaced points. A path is assumed to consist of two rectilinear parts, from the source of light to one of the points in the slit and from this point to a screen. The length of a path is calculated geometrically. Feynman described a particle by its wave vector. Along the path the vector rotates and reaches the same phase angle after a difference of a wavelength . To calculate the resulting angle at the end of the path one therefore needs to know only the fractional part of the length of the path divided by . This is done for the paths and the vector sum of the wave vectors of each path gives the amplitude at the endpoint of all the paths. The intensity is the square of the amplitude divided by .
The formula for determining the position of minima or maxima can also be derived. The phase-difference between two adjacent paths is where is the width of the slit, is the distance from the slit to the screen, is the position of the screen relative to the optical axis, and we use the approximations and for small values of .
For the first minimum, the sum of all vectors must give , hence , which gives , so, for large enough , .
R. P. Feynman, QED: The Strange Theory of Light and Matter, Princeton: Princeton University Press, 1985.