Question

Your artist friend is designing an exhibit inspired by circular-aperture diffraction. A pinhole in a red zone is going to be illuminated with a red laser beam of wavelength $670nm$, while a pinhole in a violet zone is going to be illuminated with a violet laser beam of wavelength $410nm$. She wants all the diffraction patterns seen on a distant screen to have the same size. For this to work, what must be the ratio of the red pinhole’s diameter to that of the violet pinhole?

Short answer

Ratio of red pinhole diameter to violet pinhole ,$\frac{D_{\text{red}}}{D_{\text{violet}}}=1.63$

Step-by-step solution

Laser Beam

A laser beam is a single-wavelength stream of concentrated, coherent light.

Find ratio

In order for the diffraction patterns from the two pinholes to have the same size, the angle of the first minimum in the two patterns must be the same, where this angle is given by the following relation

${\theta }_1=\frac{1.22\lambda }{D}$

When you multiply ${\theta }_1$by the number of pinholes, you get

$\frac{1.22{\lambda }_{\text{red}}}{D_{\text{red}}}=\frac{1.22{\lambda }_{\text{violet}}}{D_{\text{violet}}}$

rearranging the equations,

$$\begin{gathered} \frac{D_{\text{red}}}{D_{\text{violet}}}=\frac{{\lambda }_{\text{red}}}{{\lambda }_{\text{violet}}} \\ =\frac{670\mathrm{nm}}{410\mathrm{nm}} \\ =1.63 \end{gathered}$$