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1. REFRACTION THROUGH LENSES
A lens is a portion of a transparent refracting medium
bounded by two spherical surfaces or by one spherical
surface and a plane surface. Lenses are usually made of glass.
The line joining the centres of curvature of the two spherical
surfaces is know n as the principal axis.
If one of the surfaces is plane, the axis is a straight line
normal to the surface drawn through the centre of curvature
of the other surface. A plane through the axis is called the
principal section of the lens. Optical centre of a lens is a point
on the principal axis through which all the rays will pass,
when the incident and emergent paths are parallel to each
other. When the lens is thin the ray passing through the
optical centre is considered to go straight without deviation.
Fig. (3.1)
Chapter 3: REFRACTION THROUGH THIN LENSES
The following types of lenses are in common use (Fig.
3.1). The first three are convergent lenses and the last three
are divergent lenses;
1. Double convex or bi-convex lens
2. Plano-convex lens
3. Concavo-convex lens
4. Double concave or bi-coneave lens
5. Plano-concave lens
6. Convexo-concave lens.
2. REFRACTION THROUGH A LENS
Fig. 3.2
Consider a thin lens enclosing a medium of refractive
index. 2 and separating it from a medium of refractive index
Chapter 3: REFRACTION THROUGH THIN LENSES
1, on its two sides. Let R1 and R2 be the radii of curvature of
the two co-axial spherical surfaces and O is a point object
situated on the principal axis.
An image I' is formed by refraction at the first surface
and let its distance from the pole of the first surface be equal
to v'.
Then,
2
v

1
u

2  1
(i)
R1
The rays are refracted from the second surface of the lens.
The virtual image P may be regarded as the object for the
second surface and the final image is formed at I which lies in
the medium of refractive index 1. If the distance of the final
image from the, pole of the second surface is equal to v,
then,
1
v

2
v

1  2
R2
(ii)
In this Case, the rays are passing from the medium of
refractive, index 2 (i.e., lens) to the medium of refractive
index 1.
Adding (i) and (ii)
Chapter 3: REFRACTION THROUGH THIN LENSES
1
v

1
 1
1 
 (  2   1 )


u
 R1 R 2 
Dividing by l,
 1

1 1
1 
  ( 2  1 )


v u
1
 R1 R 2 
If the lens is placed in air 1= 1 and
(iii)
2
  , where  is the
1
refractive index of the material of the lens.
Then,
 1
1 1
1 
  (   1 )


v u
 R1 R 2 
(iv)
Note:
(1) It is to be remembered that these equations will hold true
only for paraxial rays and for a thin lens where the
thickness of the lens can be taken negligibly small as
compared to u, v, R1, and R2,
(2) While solving numerical problems, proper signs for u, v
R1, and R2, are to be used.
3. PRINCIPAL FOCI
In the formula,
Chapter 3: REFRACTION THROUGH THIN LENSES
 1
1 1
1 
  (   1 )


v u
 R1 R 2 
If u = 

1 1
 0
u 
 1
1
1 
 (   1 )


v
 R1 R 2 
(i)
This value of v is known as the second principal focal
length and the position of the image corresponding to the
axial point object lying at infinity is termed as second
Principal focus of the lens (Fig. 3.3).
Fig. 3-3
In a concave lens, f2 is -ve and in a convex lens it is +vs,
according to the sign convention.
In the case of a concave lens F2 in virtual and in the case
of a convex lens F2 is real.
Chapter 3: REFRACTION THROUGH THIN LENSES
 1
1
1 
 (   1 )


f2
 R1 R 2 

(ii)
Considering the case when the image is formed at
infinity,
v= 
1 1
 0
v 
 From the equation,

 1
1 1
1 
  (   1 )


v u
 R1 R 2 
 1
1
1 
 (   1 )


u
 R1 R 2 
... (iii)
(iv)
This value of u is known as the first principal focal
length of the lens and is denoted by f1 (Fig 3.4).
Fig. 3.4
Chapter 3: REFRACTION THROUGH THIN LENSES
 1
1
1 
  (   1 )


f1
 R1 R 2 
(v)
Then :
1 1
1
 
v u
f
In a convex lens fl is -ve and in a concave lens fl is +vs,
according to the sign convention. Moreover, when the lens is
placed in air, the two focal lengths are equal in magnitude but
opposite in sign.
If the focal length of the lens is f then from equations
(iii), ,(ii,) and (v).
This is the lens equation and is applicable to a concave
or convex lens.
4. LEAST POSSIBLE DISTANCE BETWEEN AN OBJECT
AND ITS REAL IMAGE IN A CONVEX LENS
Fig. 3.5
Chapter 3: REFRACTION THROUGH THIN LENSES
Suppose I is the real image of a point object O formed
by convex lens (Fig. 3.5). Here, u is -ve and v is +ve. Let the
distance of the image from the lens he x and the distance
between, O and I be d.
u is -ve, v and f are +ve

v = +x and u = -(d-x)
From the formula
1 1
1
 
v u
f

1
1
1


x (d  x )
f

1
1
1


x (d  x ) f
d
1

x( d  x ) f

x 2  dx  df  0
For a real image the roots of the quadratic equation
should be real. Applying in this case the condition b 2-4ac>0
for a general, quadratic equation ax2 + bx + c = 0.
d2-4df > 0 ,
d2 > 4 df,
d > 4f
Chapter 3: REFRACTION THROUGH THIN LENSES
Thus, the distance between O and I should be more than
4f. In a special case when u = 2f, v is also equal to 2f and the
minimum distance between O and I is 4f.
5. DEVIATION PRODUCED BY A THIN LENS
A lens may be, considered to be, made up of large
number of prisms placed one above the other. As the function
of the lens is to, deviate the incident rays of light, it is
necessary to find the deviation produced by a particular
portion of the lens.
Fig, 3.6
Let a ray of monochromatic light parallel to the
principal axis be incident on a thin lens at a height h above
the axis and let f be the focal length of the lens. As the ray is
parallel to the principal axis, after refraction it will pass
through the second focus (Fig. 3.6).
The deviation suffered by the ray is given by
Chapter 3: REFRACTION THROUGH THIN LENSES
tan  = h/f
In the paraxial region  being small, tan  = 

h
f
Fig. 3-7
Next, consider a luminous point object O and its
corresponding image I (Fig. 3.7). Then, deviation suffered by
the ray OA incident at A. is given by
  OAL  AIL

h
h
1
1 1

  h     h( )
u v
f
v u

h
f
This shows that the deviation produced by a lens is
independent of the position of the object.
Chapter 3: REFRACTION THROUGH THIN LENSES
6. EQUIVALENT FOCAL LENGTH OF TWO THIN
LENSES SEPARATED BY A FINITE DISTANCE
Let fl and f2. be the focal lengths of two thin lenses L1
and L2 placed on-axially and separated by a distance d in air.
Let a ray IA of monochromatic light parallel to the common
aixs be incident on the first lens L1 at a height h1, above the
axis.
Fig. 3.8
This ray after refraction through the first lens is directed
towards F1, which is the second principal focus of L1 (Fig.
3.8). Then the deviation 1, produced by the first lens is given
by,  1 
h1
f1
The emergent ray from the first lens is refracted by the
second lens L2 at a height h2 and finally meets the axis at F.
Chapter 3: REFRACTION THROUGH THIN LENSES
Since the incident ray IA is parallel to the principal axis and
after refraction through the combination meets the axis at F, F
must be the second principal focus of the combination. The
deviation 2, produced by the second lens is given by  2 
h2
f2
The incident and the final emergent rays when produced
intersect at E. It is clear that a single thin lens placed at P2,
will produce the same deviation as the two constituent lenses
together. The lens of focal length P2F placed at P2 is termed
as the equivalent lens which can replace the two lens L1 and
L2. The deviation produced by the equivalent lens is  
h1
.
f
Where, f is the focal length of the equivalent lens.
  1   2
h1
h
h2
 1 
f
f1
f2
(i)
s AL1F1 and BL2F are similar
AL1
BL2

L1 F1 L2 F1
or
h2 
h1
h2

f1 f1  d
h1 ( f 1  d )
f1
Substituting this value of h2. in equation (i),
(ii)
Chapter 3: REFRACTION THROUGH THIN LENSES
h1 h1 h1 ( f 1  d )


f
f1
f2 f1
So ,
f d
1
1

 1
f
f1
f2 f1
1
1
1
d



f
f1 f2 f2 f1
f
f
f1 f2
f1  f2  d
 f1 f2

Where  = d- (f1+f2) and is known as the optical
separation or Optical interval between the two lenses. It is
numerically equal to the first distance between the second
principal focus of the first lens and the first principal focus of
the second lens.
7. POWER OF A LENS
The power of a lens is the measure of its ability to
produce convergence of a parallel beam of light. A convex
lens of large focal length produces a small converging effect
on the rays of light, and a convex lens of small focal length
produces a large converging effect. Due to this reason, the
Chapter 3: REFRACTION THROUGH THIN LENSES
power of a convex lens is taken as +ve and a convex lens of
large focal length have low power and a convex lens of small
focal length has high power. On the other hand a concave
lens produces divergence. Therefore, its power is taken as
negative.
The unit in which power of a lens is measured is called
diopter (D). A convex lens of focal length one metre has a
power = + 1 diopter and a convex lens of focal length 2
metres has a power = + ½ diopter.
Mathematically,
power 
1
focal length in metres
If two lenses of focal lengths fl and f2, are in contact
1
1
1


F f1 f2
P  P1  P2
Where P1 and P2 are the powers of the two lenses and P is the
equivalent power.
When two thin lenses of focal lengths f1 and f2 are
placed co-axially and separated by a distance d, the
equivalent focal length f is given by;
Chapter 3: REFRACTION THROUGH THIN LENSES
1
1
1
d



f
f1 f2 f2 f1
P  P1  P2  dP1 P2
Where P, is the equivalent power.
Example 3. 1. Calculate the focal length of a double
convex lens, for which the radius of curvature of each surface
is 50 cm and refractive index of glass is 1.50.
Solution
Here, R1= +50 cm, R2 = -50 cm, µ = 1.50
1
1
1
 (   1 )(

)
f
R1 R2
1
1
1
 ( 1.5  1 ) (

)
f
50 50
f = 50 cm.
Example 3.2: Find the focal length of a plano convex lens,
the radius of the curved surface being 40 cm and =1.50.
Solution
Here, R1= , R2 = -40 cm, µ = 1.50
Chapter 3: REFRACTION THROUGH THIN LENSES
1
1
1
 (   1 )(

)
f
R1 R2
1
1 1
 ( 1.5  1 ) ( 
)
f
 40
f = 80 cm.
10. SPHERICAL ABERRATION IN A LENS
The presence of spherical aberration in the image
formed by a single lens is illustrated in Fig. (3.11). O is a
point object on the axis of the lens and Ip and Im are the
Fig. 3-11
images formed by the paraxial and marginal rays respectively.
It is clear from the figure that the paraxial rays of light form
the image at a longer distance from the lens than the marginal
rays. The image is not sharp at any point on the axis.
However, if the screen is placed perpendicular to the axis at
AB, the image appears to be a circular patch of diameter AB.
At positions on the two sides of AB, the image patch has a
Chapter 3: REFRACTION THROUGH THIN LENSES
larger diameter. This patch of diameter AB is called the circle
of least confusion, which corresponds to the position of the
best image. The distance Im IP, measures the longitudinal
spherical aberration. The radius of the circle of least
confusion measures the lateral spherical aberration. When the
aperture of the lens is relatively large compared to the focal
length of the lens, the come of the rays of light refracted
through the different zones of the lens surface are not brought
to focus at the same point on the axis.
The marginal rays come to focus at a nearer point Im and
the axial rays come to focus at a farther point Ip. Thus, for an
object point O on the axis, the image extends over the length
Im Ip. This effect is called spherical aberration and arises due
to the fact that different annular zones have different focal
lengths. The spherical aberration produced by a concave lens
is illustrated in Fig. (3.12).
Chapter 3: REFRACTION THROUGH THIN LENSES
Fig. (3.12)
The spherical aberration produced by a lens, depends on
the distance of the object point and varies approximately as
the square of the distance of the object ray, above the axis of
the lens. The spherical aberration produced by a convex lens
is positive and that produced by a concave lens is negative.
127. COMA
The effect of rays from an object point not situated on
the axis of the lens results in an aberration called coma.
Fig. 3.13
Comatic aberration is similar to spherical aberration in that
both are due to the failure of the lens to bring all rays from a
point object to focus at the same point. Spherical aberration
refers to object points situated on the axis whereas comatic
aberration refers to object points situated off the axis. In the
Chapter 3: REFRACTION THROUGH THIN LENSES
case of spherical aberration, the image is a circle of varying
diameter along the axis and in the case of comatic aberration
the image is comet-shaped and hence the name coma. Fig.
3.13 illustrates the effect of coma. The resultant image of a
distant point off the axis is shown in the side figure. The rays
of light in the tangential plane are represented in the figure.
Fig. 3.14 illustrates the presence of coma in the image due to
a point object situated off the axis of the lens. Rays of light
getting refracted through the centre of the lens (ray 1) meets
the screen XY at the point P.
Fig. 3.13
Rays 2, 2 ; 3, 3 etc., getting refracted through the outer zones
of the lens come to focus at points Q, R, S, etc., nearer the
lens and on the screen overlapping circular patches of
gradually increasing diameter are formed. The resultant
Chapter 3: REFRACTION THROUGH THIN LENSES
image of the point is comic-shaped as indicated in the side
figure.
Let 1, 2, 3 etc., be the various zones of the lens [Fig.
3.14(a).]
Rays of light getting refracted through them different zones
give
Fig. 3.14
rise to circular patches of light 1', 2’, 3' etc. The screen is
placed perpendicular to the axis of the lens and at the position
where the central rays come to focus [Fig. 3.14 (b)]. Like
spherical aberration comatic aberration produced by a single
lens can also be corrected by properly choosing the radii of
curvature of the lens surfaces. Coma can be altogether
eliminated for a given pair of object and image points
whereas spherical aberration cannot be completely corrected.
Chapter 3: REFRACTION THROUGH THIN LENSES
Further, a lens corrected for coma will not be free from
spherical aberration and the one corrected for spherical
aberration will not be free from coma. Use of a stop or a
diaphragm at the proper position eliminates coma.
Coma is the result of varying magnification for rays
refracted through different zones of the lens. For example, in
Fig. 3.14, rays of light getting refracted through the outer
zones come to focus at points nearer the lens. Hence the
magnification of the image due to the outer zones is smaller
than the inner zones and in this case coma is said to be
negative. On the other hand if the magnification produced in
an image due to the outer zones is greater, coma is said to be
positive.
According to Abbe, a German optician, coma can be
eliminated if a lens satisfies the Abbe's since condition viz.
 1 y 1 sin  1   2 y 2 sin  2
where 1 , y1 and 1 refer to the refractive index, height of the
object above the axis and the slope angle of the incident ray
of light. Similarly 2, y2 and 2 refer to the corresponding
quantities in the image medium. The magnification of the
image is given by
Chapter 3: REFRACTION THROUGH THIN LENSES
y 2  1 sin 1

y 1  2 sin 2
Elimination
of
coma
is
possible
(i)
if
the
lateral
magnification is the same for all rays of light irrespective of
the slope angle 1 and 2 thus , can be eliminated if,
sin 1
is
sin 2
a constant because 1/2 is constant . A lens, that satisfies the
above condition is called an aplanatic lens
9. ASTIGMATISM
Astigmatism, similar to coma, is the aberration in the
image formed by a lens, of object points off the axis. The
difference between astigmatism and coma, however, is that in
coma the spreading of the image takes place in a plane
perpendicular to the lens axis and in astigmatism the
spreading takes place along the lens axis. Astigmatism
discussed in this article is different from the one treated in
defective vision.
Fig. (3.15) illustrates the defect of astigmatism in the
image of a point B situated off the axis. Two portions of the
cone of rays of light diverging from the point B are taken.
Chapter 3: REFRACTION THROUGH THIN LENSES
Fig. 3.15
The cone of the rays of light refracted through the
tangential (vertical) plane BMN comes to focus at a point P1
Nearer the lens and the cone of rays refracted through the
sagittal ( horizontal) plane BRS comes to focus at the P2 away
from the lens. All rays pass through a horizontal line passing
through P1 called the primary image and also through a
vertical line passing through P2 called the secondary image.
The refracted beam has an elliptical cross-section, which ends
to a horizontal line at P1 and a vertical line at P2. The
cross-section of the refracted beam is circular at some point
between the primary and the secondary images and this is
called the circle of least confusion. If a screen is held
perpendicular to the refracted beam between the points P1 and
P2, the shape of the image at different positions is as shown in
Fig. 3.16.
Chapter 3: REFRACTION THROUGH THIN LENSES
Fig. 3.16
The focus of the primary images of all points in the object
plane gives the surface of revolution about the lens axis and
is called the primary image surface. Similarly, the locus of the
secondary images gives the secondary image surface.
Fig. 3.17
The surface of best focus is given by the locus of the circles
of least confusion. The primary and the secondary image
surfaces and the surface of best focus are illustrated in Fig.
(3.17). P1 and P2 are the images of the object point B. TPN
and SPR are the first and the second image surfaces and KPL
Chapter 3: REFRACTION THROUGH THIN LENSES
is the surface of beat focus. The three surfaces touch at the
point P on the axis. Generally, the surface of best focus is not
plane but curved as shown. This defect is called the curvature
of the field. The shape of the image surfaces depends on the
shape of the lens and the position of the stops. If the primary
image surface is to the left of the secondary image surface,
astigmatism is said to be positive, otherwise negative. By
using a convex and a concave lens of suitable focal lengths
and separated by a distance, it is possible to minimise the
astigmatic difference and such a lens combination is called an
anastigmat.
10. CURVATURE OF THE FIELD
The image of an extended plane object due to a single
lens is not a flat one but will be a curved surface. The central
portion of the image nearer the axis is in focus but the outer
Fig. 3.18 (a)
Chapter 3: REFRACTION THROUGH THIN LENSES
regions of the image away from the axis are blurred. This
defect is called the curvature of the field. This defect. is due
to the fact that the paraxial focal length is greater than the
marginal focal length.
Fig. 3.18 (b)
This aberration is present even if the aperture of the lens
is reduced by a suitable stop, usually employed to reduce
spherical aberration, coma and astigmatism. Fig. 3.18
illustrates the presence of curvature' of the field in the image
formed by a convex lens. A real image formed by a convex
lens curves towards the lens [Fig, 3.18(a) and a virtual image
curves away from the lens [Fig. 3.18 (b)]. Fig. 3.19 represents
the curvature of the field present in the image formed by a
concave lens.
Chapter 3: REFRACTION THROUGH THIN LENSES
Fig. (3.19)
11. DISTORTION
The failure of a lens, to form a point image due to a
point object is due to the presence of spherical aberration,
coma and astigmatism. The variation in the magnification
produced by a lens for different axial distances results in the
aberration called distortion. This aberration is not due to the
lack of sharpness in the ‘image’. Distortion is of two types
viz. (a) pin-cushion distortion and (b) barrelshaped distortion.
In pin-cushion distortion, the magnification increases with
increasing axial distance and the image of an object [Fig 3.20
(a)] appears as shown in Fig. 3.20 (b). On the other hand, if
the magnification decreases with increasing axial distance, it
results in barrel-shaped distortion and the image appears as
shown in Fig. 3.20 (c).
Chapter 3: REFRACTION THROUGH THIN LENSES
Fig. 3.20
In the case of optical instruments intended mainly for
visual observation, a little amount of distortion may be
present
but it
must be completely eliminated in a
Fig. 3.21 (a)
photographic camera lens, where the magnification of the
various regions regions of the object must be the same. In the
Chapter 3: REFRACTION THROUGH THIN LENSES
absence of stops, which limit the cone of rays of light striking
the lens, a single lens is free from distortion. But, if stops are
used, the resulting image is distorted. If a stop is placed
before the lens the distortion is barrel-shaped [Fig. 3.21 (a)]
and if a stop is placed after the lens, the distortion is pincushion type [Fig. 3.21 (b)]. To eliminate distortion, a stop is
placed in between two symmetrical lenses, so that the pincushion distortion produced by the first lens is compensated
by the barrel-shaped distortion produced by the second lens
[Fig. 3.21 (c)]. Projection and camera-lenses are constructed
in this way.
Fig. 3.21 (c)