Diffraction gratings are basically multi-slit patterns with a uniform spacing be

Question

Diffraction gratings are basically multi-slit patterns with a uniform spacing between each slit. A similar analysis to the double slit yields the same expression for the diffraction maxima as found: dsin?m=m?, where d is the slit separation.

A standard Compact Disc has many closely-spaced grooves, and when help up in sunlight produces a multi-slit diffraction pattern on a screen 1.0 m away. If the 1st-order diffraction maxima for red (?r = 800 nm) and blue (?b = 400 nm) light are separated by 40 cm ont he screen, what us the groove separation on the CD (you should make the small-angle approximation)?

Any help would be great. As a final answer I got 1.0*10^-6 m. I am just not sure if I completed this problem correctly.

Explanation / Answer

By small-angle approximation,

d*?=m? => d*(delta?) = m*(delta?)

Since m=1, delta? = d/(1 m) and delta? = 400 nm,

we get d^2 = 32.45 micro-metres400*10^-9 => d= 6

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