# Homework Help: Optics - diffraction gratings and blaze angles

1. Mar 28, 2007

### Brewer

1. The problem statement, all variables and given/known data
Light is incident on the reflecting surface tilted at a blaze angle $$\gamma$$ with respect to the horizontal non-reflecting screen as shown in the figure (below). The angle of incidence with respect to the screen normal is $$\theta_i$$.

Consider two points along the reflecting surface with the distance z between them. Show that the path difference between the waves incident at these two points and then diffracted in the direction with the angle $$\theta$$ (with respect to the screen normal) can be calculated as:

$$\Delta = z[sin(\theta + \gamma) - sin(\theta_i - \gamma)]$$

2. Relevant equations
I'm not sure - I think maybe simple geometry and tig can be used.

3. The attempt at a solution
I've drawn a diagram (not here) with the setup of the question, and all rays and important lines on it. From this I have extracted a triangle with base z (which is also the hypotenuse), and the longer of the two sides is $$\Delta$$, as shown below.

Now from this I can see that $$\Delta$$ = zsin$$\phi$$, so this indicates to me that I have find $$\phi$$ in terms of the other angles (or sin$$\phi$$ in terms of the sine's of the other angles). However I am struggling to do this, I can't see a relationship, unless I think of vector addition, and say that $$z[sin(\theta + \gamma) - sin(\theta_i - \gamma)] = zsin\phi$$, which I think is correct. Then is sufficient to say that I found this out from the diagram and geometry? Or do you believe that the question is looking for more of an algebraic approach to the solution?