Double Slit Diffraction with Angled Light

AI Thread Summary
The discussion revolves around solving for the position of interference maxima in a double slit diffraction setup with angled light. The user has established equations for sinθ and tanθ but is uncertain about fixing θ at 40 degrees while needing to solve for θo. There is confusion regarding the relevance of the given value of β and how to apply small-angle approximations in this context. Suggestions include using plane geometry to analyze path differences and creating a diagram to clarify the relationships involved. Ultimately, the user seeks clarity on deriving θo without knowing the wavelength or slit separation.
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Homework Statement
Suppose that a laser beam hits a double-slit apparatus at an angle of β = 40 degrees with respect to the normal. The screen is a distance L away, with slit width a and slit separation d.

Derive a symbolic expression for locating the double-slit interference maxima. At what angle θo is the center of the interference pattern located on the screen? Is the interference pattern symmetric about that angle?
Relevant Equations
dsinθ = mλ (maxima)
tanθ = ym / L
Given that [y][/m] is equal to the position of the interference maxima and is the variable I’m solving for. I got as far as setting up two expressions, one for sinθ and the other for tanθ.

sinθ = (mλ)/d
tanθ = ym/L

My first instinct was to set θ = 40 degrees. By relating sinθ and tanθ to each other through cosθ, I get:

ym = (Lmλ) / (dcos40)

However, I don’t think this makes sense if I have to solve for θo in the second part of the question, so θ must not be fixed. I’m confused about how I am supposed to use the given value of β, but I assume it must be relevant.

If I were able to use small-angle approximation, sinθ ≈ tanθ, but does this still apply here?
 

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There are many ways to do this. Personally I would look at the two paths and figure the difference in distance using plane geometry. ( Make a good drawing of the path near the plate. ) Then write down the condition for each max assuming β to be held fixed
 
This drawing is from my notes, but this is also what I was thinking for this problem. (Ignore the s1, that’s just a reference point.) I‘m still a bit confused by what you mean with “the condition for each max assuming β to be held fixed.” In this case, β is explicitly held fixed at 40 degrees, no? Is there an assumption I’m making without realizing it?
 

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Yes I meant that you are interested in solving for θm (in fact only for θ0 ) in terms of the other parameters.
 
I see what you mean and why that is the goal, but I‘m still short-circuiting on what to do in order to get there. For instance, I suppose I could say θm = sin-1[(mλ)/d] = tan-1(ym/L), but is that really a sufficient symbolic expression?

And assuming that it is, how does that help me find θo since I don’t know wavelength or slit separation?
 
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