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Finding highest order number for a diffraction grating

  1. Dec 6, 2009 #1
    1. The problem statement, all variables and given/known data
    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?


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


    3. 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?
     
  2. jcsd
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