Prof. Vilas Shamrao Patil

1. Department of Physics
B.SC.II
Sem-IV.Paper-VIII
DSC-D2- Waves and Optics-II
Topic-Cardinal Points

2. 1.1 : Introduction
Cardinal Points
There are six very important points with respect to a
coaxial lens system (or thick lens) called Cardinal
Points viz.,
two principal points,
two focal points and
two nodal points.
The planes passing through these cardinal points and
perpendicular to the principal axis of the lens are
called cardinal planes.
The formation of image can be studied by considering
the refraction of the rays, starting from the object, at
various refracting surfaces of the lens system.
But generally the microscopes, telescopes and other
optical instruments employ combination of lenses to
avoid or minimize lens aberrations.
• Therefore, the study of formation of image by
considering the refraction at several surfaces
of lenses in the system becomes tedious job.
Therefore, Gauss, in 1841, suggested that a
coaxial lens system can be considered as a
single unit and using the knowledge of
cardinal points of the lens system, the image
formation can be studied in a very simple and
convenient way without considering the
refraction at individual refracting surfaces of
the lens system.
• All the distances like object distance, image
distance and focal lengths etc. are measured
from the hypothetical planes called principal
planes. Thus, concept of cardinal points is a
very convenient tool in the study of even
complicated optical systems.

3. 1.2 : Cardinal Points (Definitions) of an Optical System
(a) Principal points and Principal planes :
An incident ray OA parallel to the principal axis,
emerges from the lens system as BF2, where F2 is
second focal point in the image space. When these
rays (1 and 3) are extended to meet each other at H2,
then refraction of the ray at various surfaces in the
lens system is equivalent to a single refraction at H2.
Similarly refraction of ray F1C through lens system as
DE, may be
(
• considered as single refraction at N1. The planes H
P1 and H2P2 drawn, perpendicular to the principal
axis are called Principal Planes.
• "Thus principal planes (H1P1 and H2P2) are the
locus of points where single refraction due to lens
system is supposed to occur."
• "The points of intersection (P1 and P2) of the
principal planes with the principal axis are called
principal points."
• Here it should be noted that the incident rays (1
and 2) directed towards H1, appear to come from
H2 after emergence as ray (3 and 4), respectively.
So H2 is image of H1 or H1 and H2 are conjugate
points such that, HIP1 = H2P2.
• In other words, an object H1 in first principal plane
is imaged as H2 in the second principal plane with
unit lateral magnification (m = 1) and hence
principal planes are also called unit planes.
1.2 : Cardinal Points (Definitions) of an Optical System

4. • Focal points and Focal planes :
• First focal point (F 1) on the principal axis is a point in
object space. Such that the incident rays passing through
F1 emerge as parallel beam after passing through the
lens system.
• Second focal point (F2) on the principal axis is a point in
the image space where an incident parallel beam
converges at F2 after passing through the lens system.
• The planes passing through F1 and F2 and perpendicular
to the principal axis (viz. F1o and F2E) are called focal
planes.
• The distance of first focal point from first principal point
is called first focal length (f1) in the object space is F1P1 =
f1.
• Similarly the distance of second focal point from second
principal point is called second focal length ( f2) in the
image space i.e. F2P2=f2
• (c) Nodal points and Nodal planes :
• An incident ray G1N1 directed towards first nodal point
N1 on the axis, emerges as a parallel ray G2N2 from the
lens system, where N2 is second nodal point.
•
• Thus, for such conjugate pairs of rays, the lens
system behaves like a parallel slab i.e. it does not
produce any refraction. If Ѳ1 and Ѳ 2 are the
angles made by the incident and emergent rays
with the axis respectively, then the angular
magnification is given by
• 𝑚1=
tan Ѳ1
tan Ѳ2
• Since incident and emergent rays are parallel to each
other, tan Ѳ1 =tan Ѳ2 =1.
• Thus, nodal points N1 and N2 are conjugate pair of
points on the axis, having unit angular magnification.
1.2 : Cardinal Points (Definitions) of an Optical System

5. Fig.
(a) Lateral magnification, m =
A1B1
𝐴𝐵
=
h2
ℎ1
------------15
(b) Axial magnification, mL =
dv
𝑑𝑢
≡
dx2
𝑑𝑥1
Now differentiating Newton's formula,
x1x2=f1f2,
(a) Lateral magnification,
m =
A1B1
𝐴𝐵
=
h2
ℎ1
----15
(b) Axial magnification,
mL =
dv
𝑑𝑢
≡
dx2
𝑑𝑥1
we get, x1dx2 + x2dx1 = 0
divide by dx1
x1dx2
dx1
+ x2=0,
x1dx2
dx1
= - x2
mL =
𝐝𝐱𝟐
𝒅𝒙𝟏
= −
𝐱𝟐
𝒙𝟏
---------16
But from eqn. (1), we have,
AB
𝐴1𝐵1
=
h1
ℎ2
=
x1
𝑓1
(becoz
AB
𝐴1𝐵1
=
x1
𝑓1
)
𝐱𝟏 = 𝒇𝟏.
𝐡𝟏
𝒉𝟐
=
𝐟𝟏
𝒎
(
𝐡𝟏
𝒉𝟐
=
𝟏
𝒎
)---17
Similarly, from eqn. (2), we
get,
AB
A1B1
=
f2
x2
x2 = 𝑓2.
A1B1
𝐴𝐵
= 𝑓2𝑚-----18--
from 15
Putting x1 and x2 from 17 and 18 in 16
𝐦𝑳 = −
𝒇𝟐𝒎
𝒇𝟏
𝒎
= −𝒎𝟐 𝒇𝟐
𝒇𝟏
But from 12 we get (
𝑓2
𝑓1
= −
𝜇2
𝜇1
)
put
We get
𝐦𝑳 = -𝒎𝟐
(−
𝝁𝟐
𝝁𝟏
) = 𝒎𝟐
(
𝝁𝟐
𝝁𝟏
)

6. (c) The angular magnification, 𝜶 =
𝜃2
𝜃1
but from (11)
𝜃1
𝜃2
=
𝜇2ℎ2
𝜇1ℎ1
putting we get
𝛼 =
𝜇1ℎ1
𝜇2ℎ2
(because 𝜇1ℎ1𝜃1 = 𝜇2ℎ2𝜃2)
=
𝜇1
𝜇2
.
h1
ℎ2
=
𝜇1
𝜇2
.
1
𝑚
from equ.15
𝜶𝒎 =
𝝁𝟏
𝝁𝟐
From equ.20 we get 𝐦𝑳 = -𝒎𝟐(−
𝝁𝟐
𝝁𝟏
) = 𝒎𝟐(
𝝁𝟐
𝝁𝟏
)
𝐦𝑳 = 𝒎𝟐(
𝝁𝟐
𝝁𝟏
) ,
𝐦𝑳
𝒎𝟐= (
𝝁𝟐
𝝁𝟏
) inverting----20
𝝁𝟏
𝝁𝟐
=
𝑚2
𝑚𝐿
but from above equ. 𝜶𝒎 =
𝝁𝟏
𝝁𝟐
i.e. . 𝜶𝒎 =
𝝁𝟏
𝝁𝟐
=
𝑚2
𝑚𝐿
𝛼. 𝑚𝐿 = 𝑚 ------------21
This is the relationship between linear, axial, and angular
magnifications.
If the lens system is in air, then 𝜇1= 𝜇2.
𝛼. 𝑚 = 1 .... From eqn. (21)
𝑚𝐿 = 𝑚2
.... From eqn. (20)
Thus, angular magnification is inversely proportional to the
linear magnification (𝜶 =
𝟏
𝒎
). Also lateral magnification is
equal to square of linear magnification.