Diffraction from slits
Diffraction processes affecting waves are amenable to quantitative description and analysis.
Such treatments are applied to a wave passing through one or more slits whose width is specified as a proportion of the wavelength.
Because diffraction is the result of addition of all waves (of given wavelength) along all unobstructed paths, the usual procedure is to consider the contribution of an infinitesimally small neighborhood around a certain path (this contribution is usually called a wavelet) and then integrate over all paths (= add all wavelets) from the source to the detector (or given point on a screen).
Thus in order to determine the pattern produced by diffraction, the phase and the amplitude of each of the wavelets is calculated.
That is, at each point in space we must determine the distance to each of the simple sources on the incoming wavefront.
If the distance to each of the simple sources differs by an integer number of wavelengths, all the wavelets will be in phase, resulting in constructive interference.
If the distance to each source is an integer plus one half of a wavelength, there will be complete destructive interference.
Usually, it is sufficient to determine these minima and maxima to explain the observed diffraction effects.
The simplest descriptions of diffraction are those in which the situation can be reduced to a two-dimensional problem.
For light, we can often neglect one dimension if the diffracting object extends in that direction over a distance far greater than the wavelength.
In the case of light shining through small circular holes we will have to take into account the full three-dimensional nature of the problem.
Several qualitative observations can be made of diffraction in general: The problem of calculating what a diffracted wave looks like, is the problem of determining the phase of each of the simple sources on the incoming wave front.
To make this statement more quantitative, consider a diffracting object at the origin that has a size
For definiteness let us say we are diffracting light and we are interested in what the intensity looks like on a screen a distance
At some point on the screen the path length to one side of the object is given by the Pythagorean theorem If we now consider the situation where
To further simplify things: If the diffracting object is much smaller than the distance
, the last term will contribute much less than a wavelength to the path length, and will then not change the phase appreciably.
Multiple-slit arrangements can be mathematically considered as multiple simple wave sources, if the slits are narrow enough.
The simplest case is that of two narrow slits, spaced a distance
Maxima in the intensity occur if this path length difference is an integer number of wavelengths.
where The corresponding minima are at path differences of an integer number plus one half of the wavelength:
For an array of slits, positions of the minima and maxima are not changed, the fringes visible on a screen however do become sharper, as can be seen in the image.
A mathematical representation of Huygens' principle can be used to start an equation.
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:
If (x, 0, z) is the location at which the intensity of the diffraction pattern is being computed, the slit extends from
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.
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.
