How Does Changing Hole Shapes Affect Diffraction Patterns?

[Haftred]

Homework Statement

I am trying to solve the following homework problem:

1) A laser beam of light with wavelength λ=600nm shines perpendicular to a slide that has a
rectangular pattern of small holes that can be viewed as point sources. The spacing
between the holes is a=40μm and b=30μm along the x and y directions, respectively. The
light passing through the slide is projected on a screen 2 meters behind the slide.
Determine the positions of the spots on the screen using the small angle approximation.
Describe how the answer to problem 1 changes if the holes are squares of side 10μm.

Homework Equations

Sigma[e^i(Delta(k)*T]*S(g)

S(g) = integral[(e^(G dot r)*A(r))dr]

The Attempt at a Solution

For the first part, I solved for G, which is the basis for the reciprocal lattice. Since a1 = 40 um and a2 = 30 um, I used the formula which states that a(i) dot b(j) = 2Pi (kronecker delta i,j), so b1 = 2Pi / 40 and b2 = 2Pi / 30. I am confused how I use these reciprocal lattice vectors. Also, do I solve for S(g) in the first part as well as the second part?

Thanks

 

[Jhero]

for your question! It seems like you're on the right track with using the small angle approximation to solve this problem. To answer your first question, the reciprocal lattice vectors are used to determine the positions of the spots on the screen. The spots will correspond to the diffraction pattern produced by the rectangular pattern of holes on the slide. You can use the small angle approximation to calculate the positions of the spots on the screen.

To answer your second question, you will need to solve for S(g) in both parts of the problem. In the first part, you will use the reciprocal lattice vectors to calculate the diffraction pattern produced by the rectangular holes. In the second part, you will need to recalculate S(g) using the new parameters for the square holes. This will give you the diffraction pattern for the square holes, which will have different positions on the screen compared to the rectangular holes.

I hope this helps clarify things for you. Keep up the good work!