Examples
Wave Optics
1- A double-slit experiment is set up using a helium-neon laser ( λ = 633
nm). Then a very thin piece of glass ( n= 1.50 ) is placed over one of the
slits. Afterward, the central point on the screen is occupied by what had
been the m = 10 dark fringe. How thick is the glass?
For constructive interference:
   2
x


   0  2 m  or
2

x


For destructive interference:
   2

2

x

x


   0  2( m  2 ) or
1
 0
2
m
1
2
0
2
m
2- Light of wavelength 600 nm passes through a double slit and is viewed on a
screen 2.0 m behind the slits. Each slit is 0.040 mm wide and they are separated by
0.200 mm. How many bright fringes are seen on the screen?
3- Light consisting of two nearly equal wavelengths λ +λ and λ, where λ << λ , is
incident on a diffraction grating. The slit separation of the grating is d.
A. Show that the angular separation of these two wavelengths in the mth order is
 

d
/ m  
2
2
b. sodium atoms emit light at 589.0 nm and 589.6 nm. What are the first-order
and second-order angular separations ( in degrees) of these two wavelengths
for a 600 line/mm grating?
4- The Figure shows two nearly overlapped intensity peaks of the sort you might
produce with a diffraction grating . As a practical matter, two peaks can just
barely be resolved if their spacing y equals the width w of each peak, where w is
measured at half of the peak’s height. Two peaks closer together than w will
merge into a single peak. We can use this idea to understand the resolution of
diffraction grating.
A. In small angle approximation, the
position of the m=1 peak of diffraction
grating falls at the same location as the
m =1 fringe of a double slit: y1 = λL/d.
Suppose two wavelengths differing by
λ pass through a grating at the same
time. Find an expression for y, the
separation of their first-order peaks.
B. We noted that the widths of the bright fringes are proportional to 1/N, where N
is the number of slits in the grating. Let’s hypothesize that the fringe width is
w = y1 / N. Show that this is true for the double-slit pattern. We’ll then assume it to
be true as N increases.
c. Use your results from parts a and b together with the idea that ymin = w to find an
expression for λmin, the minimum wavelength separation ( in first order ) for which
the diffraction fringes can barely be resolved.
d. Ordinary hydrogen atoms emit red light with a wavelength of 656.45 nm. In
deuterium, which is a “heavy” isotope of hydrogen, the wavelength is 656.27 nm.
What is the minimum number of slits in a diffraction grating that can barely resolve
these two wavelengths in the first-order diffraction pattern?
5. The Figure shows a plane wave approaching a
diffraction grating at angle .
a. Show that the angles m for constructive interference
are given by the grating equation
d sin  m  sin    m 
, m  0 ,  1,  2 ,......
Angles are considered positive if they are above the
horizontal line, negative if below it.
b. The two first-order maxima, m = +1 and m = -1, are no longer symmetrical about
the center. Find 1 and -1 for 500 nm light incident on a 600 line/mm grating at
 = 30.