How Does Incident Angle Affect Two-Slit Interference Patterns?

[luijsu]

Homework Statement

Consider a plane wave (of wavelength λ) incident on a wall at an angle Φ = 30. There are two slits in the wall separated by a distance d=10λ. Each slit has width a<<λ. Rays emerging from the slits propagate to a distant screen where an interference/diffraction pattern may be seen.

a. For rays emerging from the slits at the angle θ, calculate the total path length difference in terms of θ, Φ, a, and λ

b. For what angle θ will we find the "central maximum"?

c. For what angle θ will we find the first interference minimum? Note: There will be a "first minimum" on each side of the central maximum. Find one of these.

Homework Equations

I'm not really sure. Maybe the equation for the intensity of a two-slit interference-diffraction pattern.

The Attempt at a Solution

The problem is, I have no idea what to make of a. I'm guessing for b and c, that I'm supposed to find the angles at which intensity will be a maximum and minimum, but I don't even have the path difference. I cannot figure out how the fact that the wave is incident affects the pattern.

 

[Doc Al]

I cannot figure out how the fact that the wave is incident affects the pattern.

What if the wave was normal to the wall (Φ = 0)? What would be the phase difference at the slits? (That's the usual situation.)

Since here the wave is incident at an angle, the light entering one slit had to travel an extra distance just in getting to the slits. Figure out that extra path length. (The rest of the analysis is standard for slit patterns.)

 

[luijsu]

So, would the path difference just be 10λsinΦ + (10λ+a)sinθ? Is the trig right?

 

[Doc Al]

So, would the path difference just be 10λsinΦ + (10λ+a)sinθ? Is the trig right?

That looks OK, assuming you define your angles with respect to the normal. (A diagram would help avoid confusion, since angles can be left or right of the normal.) But why did you add an "a" to one, but not the other? (I would just ignore the slit width for the purposes of finding the two-slit interference pattern.)