Fraunhofer diffraction from gratings
In this exercise we use a two-dimensional grating consisting of many straight and equidistant
lines in a plane (a slide).
We perform Fraunhofer diffraction which means a parallel incident beam entering the object, and
we observe the diffraction pattern far away from the object. Depending on the scattering angle we
now observe constructive or destructive interference.
While Bragg’s law is given by the expression 2d sin   n , we get for a two-dimensional grating
that d sin 1  sin  2   n .
Here  , 1 are  2 are the Bragg angle, the angle of incidence and the angle of exit,  is the
wavelength, d the distance between the lines in the grating (periodicity), and n is an integer.
Bragg’s law applies for a three dimensional periodic object where the scattering planes act as
partly reflecting planes, and thus the angle of reflection must be the same as the angle of
incidence.
In studying two-dimensional gratings we usually select 1  0 , and thus we observe the first order
interference maximum (n = 1) at   d sin  2 .
We are going to determine experimentally the wave-length of the red Balmer line of hydrogen by
measuring  2 using a grating with d=847 nm
By solving the Schrödinger equation for the hydrogen atom we find that the wave-length of this
red line is 656.3 nm.
Prism and refractive index
A beam of parallel light from a hydrogen lamp enters a triangular glass prism with all three
angles equal to 60  . We measure the deflection angle for a given color after having rotated the
prism so that this angle is the smallest possible. Then, according to Snell’s law the refractive
index n is given by:
n
60  
2
60
sin
2
sin
Here  is the deflection angle. Please, measure the deflection angle and determine n for the red,
blue-green and blue-violet lines at 656.3 nm, 486.1 nm and 434.1 nm.
Lenses and Microscopy
The lens equation is given by
1 1 1
 
f s s´
Here f is the focal length, s the distance from the object to the lens and s ' the distance from the
lens to the sharp (focused) image.
The magnification is given by M 
s'
s
It follows from these two equations that M 
s'  f
as stated in the text book where v is used
f
instead of s ' .
We will test experimentally the lens equation on an optical bench for a lens with positive f which
means a convex lens. We will use a lens with f  150 mm and we will search for the sharp
image for the following three values of s: 1.5 f, 2.0 f and 3.0 f.
Furthermore we will demonstrate that the image formed by the first lens acts as the object of the
next lens, or more generally, the image formed by the n-th lens acts as the object of lens n+1
when there are many lenses after each other.
The lensmaker’s formula; chromatic and spherical aberration
Assume a lensmaker intends to make a lens with focal length f, and glass with refractive index n
is available. To achieve his goal he machines the surfaces so that the radii of curvatures of them,
R1 and R2 , obey the following equation:
 1
1
1 
 .
 n  1 
f
 R1 R2 
The refractive index n varies with wav length (color) of the light. Thus, the focal length varies
with the wave length, thereby explaining the chromatic aberration. The explanation for the
spherical aberration is that the curvatures of surfaces are not ideal. Usually the surfaces are
spherical, resulting in that light near the periphery of the lens is refracted more than light near the
central zone of the lens.
JEVT 14.01.15