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jtbell

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http://hyperphysics.phy-astr.gsu.edu/hbase/phyopt/mulslid.html

The three-slit pattern is not the sum of the single-slit and two-slit patterns. Nor is the five-slit pattern the sum of the two-slit and three-slit patterns. See here for the general formula for N "narrow" slits (Fraunhofer approximation):

http://hyperphysics.phy-astr.gsu.edu/hbase/phyopt/gratint.html

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blue_leaf77

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$$

\begin{aligned}

u_A(u,v) \propto \textrm{FT}[t_A(x,y)] \\

u_B(u,v) \propto \textrm{FT}[t_B(x,y)]

\end{aligned}

$$

If the two surfaces are overlapped, provided the thickness of each surfaces is much smaller than the wavelength, the total transmissivity can be modeled as the product between the individual ones. Thus ##t_{tot}(x,y) = t_A(x,y)t_B(x,y)##. Following convolution theorem, the diffracted field of the overlapping surfaces will be proportional to the convolution between ##u_A(u,v)## and ##u_B(u,v)##, thus

$$

u_{tot}(x,y) = u_A \ast u_B

$$.

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