Phy5645/Double pinhole experiment

(a) As stated in the problem, we assume that the denominators are approximately the same between the two waves. This is justified because the corrections are only on the order of $d/L\!$, and we are interested in the case where $d<<L\!$. We require that the numerators have the same phase, namely $kr_{+}-kr_{-}=2\pi n\!$. We expand the LHS with respect to $d\!$,

${\begin{aligned}k(r_{+}-r_{-})&\approx k\left\{\left({\sqrt {L^{2}+x^{2}+y^{2}}}+{\frac {yd}{2{\sqrt {L^{2}+x^{2}+y^{2}}}}}\right)\right.\\&-\left.\left({\sqrt {L^{2}+x^{2}+y^{2}}}-{\frac {yd}{2{\sqrt {L^{2}+x^{2}+y^{2}}}}}\right)\right\}\\&=k{\frac {dy}{\sqrt {L^{2}+x^{2}+y^{2}}}}\end{aligned}}$

Therefore,$k{\frac {dy}{\sqrt {L^{2}+x^{2}+y^{2}}}}=2\pi n\!$

and hence

${\frac {y}{\sqrt {L^{2}+x^{2}+y^{2}}}}=n{\frac {\lambda }{d}}\!$

(b) Let us work in units in which $k=1$; that is, we measure all lengths in units of $1/k\!$. We then set $d=20\!$ and $L=1000\!$. Here is the interference pattern. We first plot the pattern along the $y\!$ axis $(x=0):\!$

(c) Using the same parameters as before, we now plot the pattern on the $xy\!$ plane:

(d) We use the same parameters as in (b) and (c). First, we plot the pattern along the $y\!$ axis $(x=0):\!$

We now plot the pattern on the plane:

The main difference is the absence of the interference pattern.

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