Lecture 2: Co-axial optical system and six cardinal points

P.A. Nagpure
Department of Physics
Shri Shivaji Science College, Amravati
Lenses
(Lecture – II)

 Single lens are rarely used for image formation because they
suffer from various defects.
 In optical instruments, such as microscopes, telescopes,
camera etc. the combination of lenses are used for forming
images of objects.
 An optical system consists of number of lenses placed apart,
and having a common principal axis.
 The image formed by such a coaxial optical system is good
and almost free of aberrations.
Co-axial Optical System
P.A. Nagpure

 A combination two or more lenses having a common
principal axis on which the centre of curvature of all the
spherical surfaces lie is called coaxial system of lenses.
 In a coaxial system thickness of the lens system cannot be
neglected. This is also true in the case of a thick lens.
Principal Axis
L1 L2 L4
L3
P.A. Nagpure

 The thin lens formula cannot be applied as such to a
coaxial system because in this formula the thickness of the
lens has been neglected.
 Further, since a thin lens consists of two refracting surfaces
a coaxial system will have a large number of refracting
surfaces.
 In order to determine the size and position of the image,
one has to consider the refraction at each surface
separately.
 It is very tedious to determine the size and position of the
image in a coaxial system similar to the case of a thick lens.
 In order to overcome this difficulty six cardinal points of an
optical system were suggested by Gauss in 1841.
P.A. Nagpure

Cardinal points (or Gauss points):
• Six cardinal points greatly simplify the study of the formation of
images and the tracing of conjugate rays.
• With the help of these points, the exact details of refraction
within the system need not to be considered and the coaxial
system of lenses may be treated as a single unit.
• The position and size of image of an object may then directly be
determined by the simple formulae for thin lenses, however
complicated the system may be.
P.A. Nagpure

Cardinal points (or Gauss points):
• The six cardinal points of an optical coaxial system are
 Two focal points
 Two principal points
 Two nodal points
P.A. Nagpure

Two Focal Points and Focal Planes
S1 S2
X Y
Figure indicates a coaxial system with the principal axis as XY.
Inside the space there may be a number of coaxial lenses.
The actual passage of the refracted rays within this coaxial
system is not known and has been shown by dotted line. P.A. Nagpure

First Focal Point and Focal Plane
Consider a point object situated at F1 on the principal axis such
that all the paraxial rays incident from it on the system are
rendered parallel to the principal axis after emergence. Such
an incident and an emergent ray have been shown by mark ‘1’
in Fig.
F
1
S1 S2
X Y
1
1
P.A. Nagpure

First Focal Point and Focal Plane
Then F1 is called the first focal point or the first principal focal
point of the optical system.
A plane passing through F1 and perpendicular to the principal
axis XY is called the first focal plane.
 All the rays from F1 will form a point image at infinity.
F
1
S1 S2
X Y
1
1
P.A. Nagpure

Second Focal Point and Focal Plane
Similarly parallel rays (shown by mark ‘2’) will converge at F2
after emergence from the coaxial system.
The point F2 where the emergent beam cuts the principal axis
is called second focal point.
The plane passing through F2 perpendicular to the principal
axis is called second focal plane.
F2
S2S1
YX
2
2
P.A. Nagpure

First Principal Point and Principal Plane
F
1
S1 S2Q1
X Y
1
1
Let a point F1 is the first focal point of the optical system. Hence
the paraxial rays incident from it on the system are rendered
parallel to the principal axis after emergence.
 Now if the incident and emergent rays are extended between
the surfaces S1 and S2, they meet at point Q1 .
P.A. Nagpure

First Principal Point and Principal Plane
F
1
S1 S2Q1
P
1
X Y
1
1
f
1
The plane passing through the point Q1 and perpendicular to
principal axis, intersect the principal axis at point P1.
The point P1 is called first principal point and the plane is called
first principal plane.
P.A. Nagpure

Let a point F2 is the second focal point of the optical system.
Hence the incident parallel rays will converge at point F2 after
emergence from the coaxial system.
Now if the incident and emergent rays are extended between the
surfaces S1 and S2, they meet at point Q2 .
F2
S2S1 Q 2
YX
2
2
Second Principal Point and Principal Plane
P.A. Nagpure

F2
S2S1 Q 2
P2
YX
2
2
f2
Second Principal Point and Principal Plane
The plane passing through the point Q2 and perpendicular to
principal axis, intersect the principal axis at point P2.
The point P2 is called first principal point and the plane is called
second principal plane.
P.A. Nagpure

F2
S2S1 Q 2
P2
Y
2
2
f2
F
1
X
1
1
P1
Q 1
f1
It is seen from the figure that two incident rays are directed
towards point Q1 in first principal plane, after refraction seems to
be come from point Q2 in second principal plane. Therefore, Q2 is
the image of Q1 . Thus Q1 and Q2 are conjugate points. It is also
seen that P1Q1 = P2Q2. Hence for the images at principal planes,
the lateral magnification is +1 .
P.A. Nagpure

A pair of conjugate points on the principal axis of the optical
system having unit positive angular magnification are called the
nodal points of the optical system and plane passing through
these nodal points and perpendicular to the principal axis are
called nodal planes. i.e.
Nodal Points and Nodal Planes
Where: α1 and α2 are the angles made by the incident and emergent ray
with the nodal planes respectively.
P.A. Nagpure

 It simply means that the a light ray directed towards one of these
nodal points (N1), after refraction through the optical system, will
appear to emerge from the second nodal point (N2) parallel to the
direction of incident ray .
 From the figure it can be shown that P1N1 = P2N2.
Nodal Points and Nodal Planes
F
2
S2
S1
P2
Y
1
1
F
1
X
P1 N1 N2
a1 a2
P.A. Nagpure
P.A. Nagpure

Construction of the image using cardinal points
P.A. Nagpure
 By knowing the cardinal points of an co-axial optical system,
the image corresponding to any object placed on the
principal axis of the system can be constructed.
 It need not necessary to know the position and curvatures
of the refracting surfaces or nature of intermediate media.

P.A. Nagpure
F2P2
Y
F1
X
P1 N1 N2
 Let F1 , F2 be the principal foci, P1 , P2 be the principal points and
N1 , N2 be the nodal points of the optical system. (Figure)
 AB is a linear object on the axis.
 In order to find the image of the point A we make following
constructions.

P.A. Nagpure
F2P2
Y
1
1
F1
X
P1 N1 N2
a1 a2
B
A
A1
B1
Q1 Q2
1. A ray AQ1 ray is drawn parallel to principal axis touching
principal plane at Q1 . The conjugate ray proceed from Q2 , a
point from second principal plane such that P1Q1 = P2Q2 and
will pass through the second focal point.
R1 R2

P.A. Nagpure
F2P2
Y
1
1
F1
X
P1 N1 N2
a1 a2
B
A
A1
B1
Q1 Q2
2. Second ray AF1R1 is drawn passing through the first focal
point and touching principal plane at R1 . The conjugate ray
proceed from R2 , a point from second principal plane such
that P1R1 = P2R2 and it will be parallel to the principal axis.
R1 R2

P.A. Nagpure
F2P2
Y
1
1
F1
X
P1 N1 N2
a1 a2
B
A
A1
B1
Q1 Q2
3. Third ray AN1 is drawn passing through the first nodal point
N1. The conjugate ray proceed from N2 , the second nodal
point and it will be parallel to the incident ray AN1.
The point of the intersection of any of the above two ray wil
give the image of A. Let it be A1. If perpendicular drawn on to
the axis, It gives the image A1B1. of object AB.
R1 R2

Lecture 2: Co-axial optical system and six cardinal points

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