Are diffraction patterns additive?

[izzor]

Lets say I have three surfaces, one with pattern A, one with pattern B and the third with A and B overlapped. Will the diffraction pattern be a simple addition of the diffraction patterns from A and B?

 

[jtbell]

No. See here for example:

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

 

[blue_leaf77]

Suppose the transmissivities of surface $A$ and $B$ are $t_A(x,y)$ and $t_B(x,y)$, respectively. For far-field diffraction, the diffracted field is proportional to the Fourier transform of the objects transmissivity. One can then write for the relations between the diffracted fields with the corresponding transmissivity
$$
\begin{aligned}
u_A(u,v) \propto \textrm{FT}[t_A(x,y)] \\
u_B(u,v) \propto \textrm{FT}[t_B(x,y)]
\end{aligned}
$$
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
$$
u_{tot}(x,y) = u_A \ast u_B
$$.