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Double Slit Diffraction

Module by: Paul Padley

Summary: we derive the expression describing double slit diffraction

Two Slit Diffraction

Now we consider the case of two slit diffraction.

Figure 1
Figure 1 (double-slit.png)
Notice that the x axis has been drawn through the lower slit. Then the field at the distant point is just the sum of the field from the two slits. Thus we can use our solution to single slit diffraction for each slit and add them together E = ε L a R 1 s i n c β e i ( k R 1 − ω t ) + ε L a R 2 s i n c β e i ( k R 2 − ω t ) E = ε L a R 1 s i n c β e i ( k R 1 − ω t ) + ε L a R 2 s i n c β e i ( k R 2 − ω t )

Now we we will define R = R 1 R = R 1 and use R 2 = R − d sin θ R 2 = R − d sin θ E = ε L a R s i n c β e i ( k R − ω t ) + ε L a R − d sin θ s i n c β e i ( k R − k d sin θ − ω t ) E = ε L a R s i n c β e i ( k R − ω t ) + ε L a R − d sin θ s i n c β e i ( k R − k d sin θ − ω t ) Now we can ignore the d sin θ d sin θ in the denominator, as it will not have a significant effect on that. However in the exponent, we can not ignore it, since it could significantly affect the phase of the harmonic function. Lets define α = ( k d sin θ ) / 2 α = ( k d sin θ ) / 2 so now we can write: E = ε L a R s i n c β e i ( k R − ω t ) + ε L a R s i n c β e i ( k R − 2 α − ω t ) E = ε L a R s i n c β e i ( k R − ω t ) + ε L a R s i n c β e i ( k R − 2 α − ω t ) and start rearranging: E = ε L a R s i n c β [ 2 cos α ] e i ( k R − α − ω t ) E = ε L a R s i n c β [ 2 cos α ] e i ( k R − α − ω t ) This is very similar to the case of single slit diffraction except that you now get a factor 2 cos α 2 cos α included and a phase shift in the harmonic function.

So we can see immediately the intensity is I = 4 I 0 cos 2 α s i n c 2 β I = 4 I 0 cos 2 α s i n c 2 β recall α = ( k d sin θ ) / 2 α = ( k d sin θ ) / 2 and β = ( k a sin θ ) / 2 β = ( k a sin θ ) / 2 If d d goes to 0 then expression just becomes the expression for single slit diffraction. If a goes to 0 then the expression just becomes that for Youngs double slit. The double slit diffraction is just the product of these two results. (Hey cool!)

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