Combination of Lenses

Equivalent lens: When a single lens produces an image of a given object at the
same place and of same size as like formed by the number of lenses together, that
single lens is called equivalent lens to the number of lenses.

Equivalent focal length: The focal length of such a single lens is known as the
equivalent focal length.

Deviation produced by a thin lens:

Let a ray of monochromatic light , parallel to the principal axis, be incident on a

thin lens at 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 as shown in the figure. The deviation suffered by the ray is given by,

In the paraxial region, is small,

Hence,
So,

In the following discussion it shows that the deviation of any ray due to refraction
through a thin lens is independent of the angle of incidence and depends upon (1)
the height of the point above the principal axis where the deviation occurs and (2)
value of the focal length of the lens.
Equivalent focal length of two lenses in contact (by the method of
deviation):
L1 and L2 are two thin convex lenses of focal lengths f1 and f2. They are placed in
contact with each other on the same common axis as shown in the figure.
Let a parallel ray strike the first lens at a height h1 above the principal axis. The ray
is refracted and falls on the second lens. Since the lenses are thin and are in
contact with each other, the ray emerging from the first lens after refraction will
fall on the second lens at the same height h1 above the principal axis. Let the
deviations produced by the first and the second lens be . Since the
deviations produced are in the same direction, the total deviation is given by,
…………………(1)

Now,
= ………………..(2)
and, = …………………(3)
Putting value of eqn (2) and (3) in eqn (1),

 + (as given, = )
 ( ) ………………(4)

If the combination is replaced by their Equivalent lens of focal length f then the
deviation,

…………………………(5)

Comparing two values of , from equation (4) and (5), we obtain,

This is the expression of the equivalent focal length of two lenses are placed in
contact.

Equivalent focal length of two thin coaxial lenses separated by a finite

distance (by the method of deviation):
Two thin convergent lenses, L1 and L2 of focal lengths f1 and f2 are placed coaxially
in air separated by a finite distance d as shown in figure.
Let a ray of monochromatic light, parallel to the common axis be incident on the
first lens L1 at a point A, a height h1 above the axis. After refraction through the
lens, the ray is directed towards its second principal focus F1. Then the deviation
produced by the first lens is given by,

= …………….….(1)
The emergent ray from the first lens, before reaching the point F1, meets the
second lens at a height h2 above the axis and suffers further refraction through it,
finally meeting the axis at the point F. The deviation produced by the second
lens is given by,

= ………………(2)

Since the deviations, are in the same direction, the total deviation
suffered by the incident ray due to the refraction through the combination of
lenses is given by,
= + ……………(3)
The incident ray IA is parallel to the principal axis. Hence, at point F, where the
final emergent ray BF intersects the principal axis, is the (second) principal focus
of the optical system. When extended line of IA and BF intersects at point E2, let
E2H2 be a plane through E2 transverse to the principal axis.
Then the optical system may be replaced by a single lens of focal length f= H2F
and placed in the position E2H2. Under this condition, rays parallel to the principal
axis, after refraction through this hypothetical lens, would come to focus at F. This
lens will be termed as the equivalent lens for parallel rays incident on the first
lens. So, H2F is called the equivalent focal length of the combination and the plane
E2H2 is called the (second) principal plane of the optical system.
The deviation produced by the equivalent lens is given by,

= ………….…(4)
Now substituting the values of eqn (1), (2) and (4) in eqn (3), we get,

+ ………..…..(5)

From similar triangles & , we get,

Or,
Or,
……………..(6)
 Now substituting this value of in eqn (5),

Or,

This is the expression of the equivalent focal length of two lenses are separated
by a finite distance d.

Power of a Lens:
The power of a lens is defined as the ability of the lens to converge a beam of light
and is measured by the amount of convergence produced to a parallel beam of
light. A convex lens of small focal length produces large converging effect to a
beam of light while a convex lens of large focal length produces small converging
effect to the same beam. So, we can say that power of the lens is inversely
proportional to the focal length of the lens. That is, power of a lens, P

Thus, a convex lens of small focal length has high power and a convex lens of large
focal length has low power. So, power of the lens can be taken as the reciprocal of
the focal length.

The unit in which power is measured is called Diopter (D). Mathematically,

Power= Dioptre
= Dioptre
A convex lens of focal length 1meter has a power = +1 diopter and focal length of
2 meter has a power = + diopter. Since a convex lens produces convergence, its
power is taken as positive while the power of a concave lens is taken as negative
as it produces divergence.

Power of combination of lenses:

If two lenses of focal lengths are in contact,

• Equivalent focal length,
• Equivalent Power, P =
Where and are the powers of the individual lenses.
If two lenses of focal lengths are placed coaxially and separated by a
distance d, then the relations for equivalent focal length and equivalent power are
given by,
• Equivalent focal length,

Equivalent Power, P =

Cardinal points:
Basically for thin lenses, we measure the distances from the Centre of the lens or
optical Centre. But in the case of refraction through a thick lens or mix of two
lenses or a system of coaxial lenses, it doesn’t work. There are actually six points
taken together are known as the cardinal points of an optical system.
These are:
1. Two focal points / Two principal foci
2. Two principal points and
3. Two Nodal points
If the medium on both sides of the lenses are same, the positions of the nodal
points coincide with the positions of the principal points. In that case, we consider
four points instead of six points in case of refraction through an optical system.
Principal points:
In the case of thin lenses the distances are always measured from the Centre of
the lens. The distance from the Centre of the lens to either focus gives the focal
length. But if a thick lens is used the distance from the Centre to one focus is
different from the distance to the other focus. To overcome this difficulty, in 1841
Gauss proved that any number of coaxial refracting systems can be treated as one
unit and the simple formula for thin lenses can be applied to this.
In the case of the combination of lenses, the point H1 from the first surface is
called the first principal point and similarly the point H2 from the second surface
is called the second principal point. Planes drawn perpendicular to the axis and
passing through these points are called first and second principal planes
respectively. These principal points are two special points on the lens axis that
helps to measure object distance and image distance in complex lens system.

Nodal points:
If the medium on one side of the lens is different from that on the other, it
becomes necessary to consider two more points on the axis. Like principal points,
these are also a pair of conjugate points so that if a ray be allowed to pass
through or travel towards the first point N1, then its corresponding emergent ray
from the lens will be parallel to the incident ray and will appear to diverge from or
pass through the second point N2. N1 and N2 are respectively called the first and
second nodal points.

Figure: Nodal points