Finding highest order number for a diffraction grating

[Oijl]

Homework Statement

Light of wavelength 680 nm is incident normally on a diffraction grating. Two adjacent maxima occur at angles given by sin θ = 0.2 and sin θ = 0.3, respectively. The fourth-order maxima are missing.

Slit separation d = 6.8 µm
Slit width a = 1.7 µm

What are the largest, second largest, and third largest values of the order number m of the maxima produced by the grating?

Homework Equations

((y is wavelength))
[diffraction grating, maxima]: dsinθ=my
[single slit, minima]: asinθ=my

The Attempt at a Solution

I know, from the previous part of this problem, that d = 4a. So, the equation for maxima for the grating can be rewritten

4asinθ=my

I know that sinθ can at most be 1. Therefore, the equation for greatest m is 4a=my, and this solves where m = 10.

So I know the greatest value of m is 10. But how can I figure the penultimate and antepenultimate values of m? If I decrease m by one, so that m = 9, I have the equation

4asinθ=9y

for which a θ exists that will solve it. And for any m less than 10, there is a θ that will make the statement (4asinθ=my) true.

But the answer is not m = 10, 9, 8. How do I find the other order numbers?

 

[Steve4Physics]

This is a very old (12+ years at the time of answering) question. Since there is no possibility of guiding the OP through the solution, here’s a fairly complete explanation.

Using ‘nλ = dsinθ’ the diffraction grating orders are at:
sinθ₁ = 1*680e-9 / 6.8e-6 = 0.1
sinθ₂ = 2*680e-9 / 6.8e-6 = 0.2
.
.
sinθ₁₀ = 10*680e-9 / 6.8e-6 = 1

Some of the diffraction grating orders will be missing. These are the ones which coincide with the minima of the single-slit diffraction pattern. For slit width = a, the single-slit minima are given by mλ = asinφ and there will be two of them:
sinφ₁ = 1*680e-9 / 1.7e-6 = 0.4
sinφ₂ = 2*680e-9 / 1.7e-6 = 0.8

As a result, the 4th and 8th order diffraction grating maxima (at sinθ₄ = 0.4 and sinθ₈ = 0.8) will both be missing.

n=10 is a limiting case. The 10th order cannot actually be seen in practice. The highest order visible is the 9th . So the three highest orders of grating maximum are 9, 7 and 6.

(If you wanted to argue that the 10th order of the grating should count, the answer would be 10, 9, and 7.)