What is Diffraction?
Diffraction is generally defined as the bending of light around corners such that it spreads out and illuminates areas where a shadow is expected. In general it is hard to separate diffraction from interference since both occur simultaneously.
In this experiment, we can observe the bending phenomenon of light or diffraction that causes light from a coherent source interfere with itself and produce a distinctive pattern on the screen called the diffraction pattern. Diffraction is evident when the sources are small enough that they are relatively the size of the wavelength of light. You can see this effect in the diagram below. For large slits the spreading out is small and generally unnoticeable.
For a we shall assume the slit width a << D. x`D is the separation between slit and source.
We shall identify the angular position of any point on screen by ϑ measured from the slit centre which divides the slit by a2 lengths. To describe the pattern, we shall first see the condition for dark fringes. Also let us divide the slit into zones of equal widths a2. Let us consider a pair of rays that emanate from distances a/s from each other as shown.
The path difference exhibited by the top two rays shown is
ΔL = a2sin θ
Remember that this is a calculation valid only if D is very large. For more details about the approximation check out our article in the Young’s Double Slit experiment.
We can consider any number of ray pairings that start from a distance a2 from one another such as the bottom two rays in the diagram. Any arbitrary pair of rays a distance a2 can be considered. We shall see the importance of this trick in a moment.
For a dark fringe, the path difference must cause destructive interference; the path difference must be out of phase by λ2. (λ is the wavelength)
For the first fringe,
ΔL = λ2 = a2sin θ
λ = a sin θ
For a ray emanating from any point in the slit there exists another ray at a distance a2 that can cause destructive interference.
Thus, at θ = sin−1λa, there is destructive interference as any ray emanating from a point has a counterpart that causes destructive interference. Hence, a dark fringe is obtained.
For the next fringe, we can divide the slit into 4 equal parts of a/4 and apply the same logic. Thus, for the second minima
λ2 = a4 sin θ
2λ = a sin θ
Similarly for the nth fringe, we can divide the slit into 2n parts and use this condition as
nλ = a sin θ
The central maximum
The maxima lies between the minima and the width of the central maximum is simply the distance between the 1st order minima from the centre of the screen on both sides of the centre.
The position of the minima given by y (measured from the centre of screen) is:
For small ϑ, sin θ≈θ ,
λ = asin θ≈aθ
θ = yD = λa
y = λDa
The width of the central maximum is simply twice this value
Width of central maximum = 2λDa
Angular width of central maximum = 2θ = 2λa
The diffraction pattern and intensity graph is shown below.
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