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Optical lenses and lens system
By Matthias Heyner and Nicolas Schleif
Group 2 pair 4
<<Optical lenses are spherical surfaces which are light refractive media. If the refractive
indices of lenses are greater than the environment, converging lenses collect and diverging
lenses spread a light beam which runs originally parallel.>>
Focal length of a lens depends on the radius, r and the refractive index of the material.
1 1
1
 (n  1)     Where ri is positive if the surface relating to it is convex (seen from
f
 r1 r2 
outside) otherwise it is negative. (For a planar surface, ri   ) we see that, if the relative
refractive index changes, the focal length changes too.
Important rays of converging lenses:
 A ray parallel to the optical axis will pass through the focal point on the other side of
the lens after being refracted
 A ray passing through the focal point will run parallel to the optical axis after being
refracted by the lens
 A ray passing though the optical centers of the lens will not change its direction
1 1 1
  ;
f t K
The magnification of a lens is the ratio of the signs of the image K and the object T: N=K/T
The pathway of important rays through diverging lens when object is located in front of
the diverging lens
 A ray parallel to the optical axis will pass through the lens as if it came from the focal
point on the incidence side
 A ray moving towards the focal point on the refraction side will run parallel to the
optical axis after being refracted by the lens
 A ray passing through the optical center of the lens will not change its direction
1 1 1
L
  
f
f1 f 2 f1  f 2
If thin lenses are near each other then the last term is negligible. The common focal length F
is the measured and can be calculated:
1
1
1
f f

 ; fd  c
f
fd fc
fc  f
The power of the lens, called Diopter, is determined by:
1
Determination of focal length by the Bessel and Abbe method
Bessel method
Fix the object and the screen on the optical bench after finding the magnified and reduced
images of an object by moving converging lens.
Two positions of the lens are symmetrical to the planes of object and image
e a
e a
e2  a 2 2
t   ;K   ; f 
; a  e 2  4  et
2 2
2 2
4e
The Bessel method is used to determine the focal length of the thin lenses.
Abbe method
Fix the lens in the middle of optical bench and find images of an object of different distance
N N
f d 1 2
N1  N 2
Great advantage is that not the distance but the size of the objects and images are measured.
Experiments
1. Determine the focal length of converging lens with the Bessel method
2. Fill “air lens” with H 2O and determine the focal length of the modified lens by Abbe
method
3. To study the result of cataract formation put a few drops of milk into the water and
note the deterioration of the image
4. To study the effect of the crystalline body on the focal length of this thick lens with
the Abbe method
The human eye as an optical system
Receptors evolved for sensing the electromagnetic waves of the wavelength 400-700nm.
Receptors allow us to receive light signals. The relative quantity of light reflected from a
 n 1
medium of refractive index n in the event of incidence normal to the surface is Rref  

 n 1
The clearness of our vision for objects at different distances is made possible by means of
crystalline lens; the focal length can be changed by special muscles. This phenomenon is
called accommodation.
The furthest point we can see is called far-point, the nearest near-point. The smallest distance
at which an object can be seen is called the distance of clear light.
2
Resolution of the eye
Resolution of the eye is the smallest angle at which two different points can be seen. This
limiting angle is ~0.0017°.
c
The distance d between the macula lutea and the blind spot is d  a 
b
2
The limit of resolution is determined by the wavelength of light. In short sighted eye, the
image is formed behind the retina.
Depth of focus
The finite resolution of the eye is that even a well-functioning eye can not differentiate
between images formed on the retina from objects at different depths.
The distance Δ at which the images of the point are clear is called the depth of focus. Then the
diaphragm is used, the improvement in vision is correlated with the greater depth of focus
relating to rings of given radius n. The distance Δ can be increased by decreasing the focal
length l and by increasing the distance of the object t:
t f
  2r t 
R f
Modifying effect on vitreous body on focal length
The focal lens of a thin lens depends on the radius of the curvature of the boundary surfaces r
and on the refractive index of the material of the lens relative to those of the materials of the
environment. Sine and tangent of small angels α ~sin α~tan α.
 n 
n

For the deviation of the light beam, we have   1 - 0   1    1  
 n1 
 n1 
For light beams passing through the lens at small angles to the optical axis, the
h h
h h
relationships    1   2 ; 1   1  1   and   1   2   are fulfilled.
t r1
r1 r2
n n n  n0 n  n1

This leads to: 0  1 
t
k
r1
r2
If the object is at an infinite distance, the image is formed at the focus on the image side. The
n1
focal length on the image side is: FK 
n  n0 n  n1

r1
r2
If there is a medium of the dame refractive index n0 on both sides of the lens, then n  n0 ~ 1
1
n  nwater  1,33
n0  1
and f K (air ) 
;
if
and
then
n 1 n 1

r1
r2
1,33
f K ( water) 
n  1,33 n  1,33

r1
r2
3
Experiments:
1. To demonstrate the dependence of the focal depth on the distance of the object, find
the clear reduced and magnified images of an object in the Bessel arrangement. Fix the
lens in the positions relating to these images formations and, by moving the object
through systematic (e.g. ±0,5cm) distances, find (and note) the object distances at
which you obtain rather well recognizable images. What is the ratio of the two focal
depths relating to the two selected positions?
2. To learn the effect of a diaphragm, find the clear reduced image of an object by an
amount (about 5-8cm) such that a clear visible halo is produces around the clear
discernible image by decreasing the diameter of the diaphragm. Note the greatest
diameter of the diaphragm. Note the greatest diameter of the diaphragm at which you
obtain an acceptable clear (e.g. legible) image.
3. Find the magnified image of the object too, fix the lens, and repeat the steps described
in task 3. In which case do you have to make the greater correction by means of the
diaphragm and why?
4. Place a diaphragm in front of your eye and note how the clarity of the image of a
distant object (e.g. and ophthalmic table) is improved when you decrease the diameter
of the diaphragm.
5. By using Fig 14.8 determine the distance of the blind spot from the macula lutea in
your right eye
Fig.14.8
4