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 Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle d = 20\!} and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle L = 1000\!} . Here is the interference pattern. We first plot the pattern along the $y\!$ axis Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (x = 0):\!}

(c) Using the same parameters as before, we now plot the pattern on the Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle xy\!} plane:

(d) We use the same parameters as in (b) and (c). First, we plot the pattern along the $y\!$ axis Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (x = 0):\!}

We now plot the pattern on the plane:

The main difference is the absence of the interference pattern.

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