Homework #9 Two slit and Diffraction grating
Equations:
The position "y" of order "m" bright fringe for a two slit pattern:   
In this equation, the fringes are viewed on a screen located a distance x from the two slits. The slits are separated by a distance d. In this equation, the sine is approximated by the tangent (y/x), which is a good approximation for very small angles. Locating the position "y" of order "m" bright fringe for a diffraction grating pattern requires two equations:
    and     
     In the first equation, m is the "order" of the bright fringe, d is the spacing between grating lines, and q is the diffraction 
     angle for the order being considered. To find the position "y" on a screen located a distance "x" from the grating use the second
     equation which relates the lengths of sides in a right triangle (see the diagram below). The small angle approximation can't be used
     because the diffraction grating produces large angles.

    The diagram below shows blue light diffracted by a grating. Three orders, the central one (m=0) and two located on one side 
     of the optical axis are shown. The angles for two of the diffraction orders are also shown (q1 for m=1 and q2 for m=2). 

HOMEWORK: 1. Briefly define the following terms.
a. interference fringe (bright or dark)
b. fringe order
2.A Young's double slit experiment is done with three different light sources: 400 nm, 500 nm and 600 nm. The slits are
50 microns apart and they are placed 2 meters from the screen. 
a. Find the locations (y position) of the first (m=1) bright fringe for each wavelength.
b. Make a general statement about the spacing of fringes as wavelength increases.
c. What colors do the three wavelengths correspond to? 3. Now using the 500 nm wavelength source, the experiment is repeated. This time, three sets of double slits with different separations are used. The slits are separated by 50, 100 and 150 microns. a. Find the locations (y positions) of the first (m=1_ bright fringe for each of the three sets of double slits.
b. Make a general statement about the spading of fringes as the separation of the two slits increases. 4. Suppose we have two diffraction gratings and a source of light of wavelength 400 nm. a. Find the diffraction angle for each of the following cases. You don't have to find the screen position, just the angle as it's shown in the figure above. Note what happens to the higher orders (larger values of m) when the lines are very close together!

line spacing (d)

order

diffraction angle

2 microns

1

2 microns

2

2 microns

3

1 micron

1

1 micron

2

1 micron

3

b. As this example shows, there is a maximum number of bright fringes formed for a given grating spacing and a given 
wavelength. Mathematically, we can explain this by noting that the sine of an angle is never more than 1. This means 
(look at the grating equation) that ml/d is never greater than 1, or that the largest m can be is d/l.  That is, 
the maximum number of orders you can see with a given grating with a given wavelength is d/l.
      a. How many orders can you see with 400 nm light and the grating with 2 micron spacing between the lines?
      b. How many orders could you see with the same grating with 500 nm light? with 600 nm light?

Note that there are ways to "shape" the diffraction grating so that you can control the number of fringe orders seen.