Intensity from a multi slit grating

[UrbanXrisis]

The intensity from a multi slit grating with 2N+1 slites is given by:

$$I(Y)=\frac{M^2}{2R^2} \frac{sin^2 \left[(2N+1) \frac{x}{2}\right]}{sin^2 \left[\frac{x}{2}\right]}$$

where $$x=\frac{2 \pi Y d}{D \lambda}$$

I am to show how to find the limiting case of just two slites and show that in this limit the formula is exactly the same as:

$$I(Y)=\frac{2M^2}{D^2} cos^2 \left(\frac {\pi d sin\theta}{\lambda}\right)$$

so what I did:

$$I(Y)=\frac{M^2}{2R^2} \frac{sin^2 \left[(2N+1) \frac{x}{2}\right]}{sin^2 \left[\frac{x}{2}\right]}$$

$$=\frac{M^2}{2R^2} \frac{sin^2 \left[ \frac{2 \pi \lambda d}{D \lambda} \right]}{sin^2 \left[\frac{\pi \lambda d}{D \lambda} \right]}$$


I could turn this into cos/cos but I don't know how to get it into cos(sin) any ideas?

 

[member 553083]

The answer is actually:I(Y)=\frac{M^2}{2R^2} \frac{sin^2 \left[(2N+1) \frac{x}{2}\right]}{sin^2 \left[\frac{x}{2}\right]}=\frac{M^2}{2R^2} \frac{sin^2 \left[ \frac{2 \pi \lambda d}{D \lambda} \right]}{sin^2 \left[\frac{\pi \lambda d}{D \lambda} \right]}=\frac{M^2}{2R^2} \frac{cos^2 \left[ \frac{\pi \lambda d}{D \lambda} \right]}{sin^2 \left[\frac{\pi \lambda d}{D \lambda} \right]}=\frac{M^2}{2R^2} cos^2 \left(\frac {\pi d sin\theta}{\lambda}\right)=\frac{2M^2}{D^2} cos^2 \left(\frac {\pi d sin\theta}{\lambda}\right)