Cardinal points, Focal points, Nodal points, Principal points, thin and thick lenses, neutralization, lens power formula, front and back vertex power

1. Raju KaitiRaju Kaiti
Consultant OptometristConsultant Optometrist
Dhulikhel Hospital, KUDhulikhel Hospital, KU

2. Thick-Lens Optics
•Baker
Optics, Refraction and Contact Lens
•AAO section 3
Optics
•A. H. Tunacliff
Optics
•Fincham Freeman
Principles of optics
•Hardy and Perrind
Optics and Refraction
• A.K Khurana
Ophthalmic optics and refraction (vol. 5)
•Duke Elder

3. INTRODUCTION
Are a set of six points situated on the
optical axis which are defined in such a way
so as to facilitate the image formation by the
lens system.

4. CARDINAL POINTS
Every Optical systems has 6 CARDINAL POINTS
1. Focal Points- Primary & Secondary
2. Principal Points- Primary & Secondary
3. Nodal Points- Primary & Secondary
*These points are defined following the tradition that light is
always supposed to travel from left to right in optical system.

5. SIGNIFICANCE
Knowing the location of cardinal points;
we can find out various information about the
image of an object produced by an optical
system.
•Information about;
 Size
 Location of image

6. FOCAL POINTS
can be defined as “ the points on the optical
axis to which the light rays that arrive
parallel to the optical axis, are brought to a
common focus after refraction.

7. PRIMARY FOCAL POINTS
Rays diverging form this point for a system
with positive power or converging to this
point for a system with negative power
before refraction, will emerge parallel to the
optical axis after refraction.

8. PRIMARY FOCAL POINTS

9. Focal Points
 Secondary focal point (F’):
Rays parallel to the optic axis before
refraction will converge to {or appear to
diverge from) this point after refraction.
Fig: showing focal points in a converging and diverging system

10. PRINCIPAL POINTS
 Are the points at the intersection of two imaginary rays,
created by extending the incident and the emergent rays.
H H1

11. PRINCIPAL POINTS AND PLANES
 A perpendicular drawn to the optical
axis from any other principal points
cross the optical axis at the main
principal point.
 The principal planes are the planes
normal to the optical axis, intersecting
the principal points on the optical axis.
 For all incident rays having equal
angles of incidence have their
principal points on the same plane.

12. PRINCIPAL PLANES
 Two Principle Planes (Primary & Secondary)
 Unit planes = magnification and size of 1st
& 2nd
is unity i.e; (1st
= obj. = 2nd
image)
All distances relating to optical systems are
measured from principal planes.

13. Relative position of the
principal points

14. • Thick lens in meniscus form the
principal planes are in front of
the lens with their spacing equal
to the lens thickness

15. • Thick lens in afocal form, for
which the equivalent power is
zero
• Occasionally met in contact lens
• The principal points and the
focal points are at infinity

16.  Convexo-plane
thick lens
 The principal
point coincides
with the vertex
at the front
surface while P’
lies at the
distance t’ from
the back surface

17. Fig: N and N’, Primary and Secondary nodal points
NODAL POINTS

18. NODAL POINTS
Are two axial points, such that a ray, directed at
the first will seem to emerge from the second
nodal point parallel to its original direction.
A ray directed to a nodal point at a certain angle,
will exit the lens at the same angle.
Graphically,extending the incident and emergent
ray towards the inside of the lane will intersect
the optical axis at the primary and second nodal
point.

19. NODAL POINTS
 When an optical axis is bounded on both sides by
the same medium (same n on both sides) then the
nodal points coincide with the principal points.
 For differing indices for converging system the
nodal points move towards the higher index side
and converse occurs for diverging system.
 When an optical system has equal curvature and
R.I on both sides, it will have a single principal
point and a single nodal point for one set of
incident ray and they will coincide.

20. A
F
F’
O
A’
O’
P P’

21. Ray tracing
 Many simple optical systems can be analyzed by just
tracing selected rays through the whole system of
optical components.
 All rays that pass through a focal point on one side of a lens will
emerge from the other side as parallel rays.
 Any ray passing through the center of a lens will continue
through the lens, and emerge out of the other side in a straight
line.
 All rays of a parallel bundle on one side of a lens go through a
single point in the focal plane on the other side, and visa versa.
This point is easily found by following where the ray that passes
through the center of the lens hits the focal plane.

22. Ray Tracing (considering cardinal points)
 Ray OB, parallel to the system axis will appear to refract at
the second principal plane, it will then pass through the
second focal point F2.
 The ray OF1C passing through the first focal point F1 will
emerge from the system parallel to the axis, refracted at the
first principal plane.

23.  A third ray may be constructed from O to the first nodal point . This ray
appear to emerge from the second nodal point and would be parallel
to the entering ray. ( as n1=n2, nodal points coincide with principal
points)
 The intersection of three rays at point O' locates the image of point O.
 A similar construction for other points on the object would locate
additional image points which would lie along the O'A' arrow

24. CHANGES IN CARDINAL POINTS
 IN APHAKIA
 Eye becomes highly hyperopic
 Power of eye reduces from +60D to +44D
 Anterior focal point becomes23.2mm in front of
cornea
 Posterior focal point is about 31mm behind the
cornea i.e. 7mm behind the eyeball
 Nodal points are very near to each other and are
located about 7.75mm behind the anterior surface
of cornea
 The two principal points are almost at anterior
surface of cornea

25. CARDINAL POINTS IN….
 Nodal points in myopic eye is further away
from the retina.
 So, the image formed will be appreciably
larger than it would be in the emmetropic
eye and in spectacle corrected eye.
 The concept of schematic eye is strongly
dependent on cardinal points.
 In PSCC-hyperopic shift
 In nuclear sclerosis-myopic shift

26. LENS??
 “ an optical system of two refracting
interfaces where one or both of this is
curved ”
 Lens : Lentille (French)
Form of the lentil seed

27.  A lens is an optical medium bounded by two
surfaces, surrounded by two media
 Used to correct ametropia
 Must be designed to reduce secondary optical
problems e.g.; aberration

28. Classification;
 According to nature of its surfaces:
Most common – spherical (their surfaces are
portions of surfaces of sphere)
Other important forms; plane, cylindrical,
toroidal and aspherical surfaces
 According to its effect on light rays;
Converging / diverging
 Depending on its thickness;
Thick / thin

29. THIN
LENS

30. Various lens forms

31. Forms of lenses
 Thin lens is one where the aperture is
small compared to the radii of curvature
of its surfaces and where the thickness
can be ignored.
 Two classes according to curvature of
their surfaces;
Convex / positive
Concave / negative

32. CONVERGING SYSTEM(F>0)
 Positive lenses
 Against motion of images with movement
 Magnify images
 Thicker in centre and thinner at edge
 Adds positive vergence to eye
 Use to correct hyperopia

33. DIVERGING SYSTEM(F<O)
 Negative lenses
 With motion of images with movement
 Minify images
 Thinner at centre and thicker at edge
 Adds negative vergence to eye
 Use to correct myopia

34. Approximate power of thick lens
F=F1+F2
-can be measured with lens clock
-Only approximation because thickness
doesn’t enter the calculation

35. Measurement of Curvature
 Lens Clock (lens measure, lens gauge)
measures sagittal depth of lens surface

36. Lens surface curvatureLens surface curvature
SagSag
ChordChord

37. FORM OF LENSES

39. Lens specifications:
 Radius of curvature of first surface: r1
 Radius of curvature of second surface: r2
 Refractive index of lens material
 Centre thickness and edge thickness
 Aperture (diameter)
 Position of optical centre and optical axis

40.  Incident light: left to right;
 First surface/ second
surface
 Centre of curvature C1 /
C2
 Optical axis
 Principal axis
 Front and back
vertex(A1 and A2)
 Centre thickness t

41. Optical centre - O
 Defined as “ point on the lens axis,
where the nodal rays, as it passes
between the lenses, crosses or appears
to cross the principal axis of the system.
 For thin lens A1, A2 and O may be
considered as a single point.

42. Principal foci and focal length of a lens

43. Image formation:
 Lens - affects vergence or curvature of
wave fronts of the light incident upon it
• Image formation :- position and nature of
image, and its size relative to object size
depends on lens and the object position
 Two methods:
By scale drawing - graphical construction
By calculation - analytical method using
formulae.

44. Procedure for positive lens

45. Images formed by converging lenses:
 Object at infinity:
 Real, Inverted, Smaller than object, At F

46. Object at 2F
 Real, Inverted, Same size, At 2F

47. Object between 2F and F
 Real, Inverted, Larger than object,
Beyond 2F

48. Object at F
 Image at infinity

49. Object between F and lens
 Virtual, Erect, Larger than object, Behind
the object on the same side of the lens

50. Procedure for negative lens

51. Image formation by diverging
lens

52. Image formation by a diverging lens
 Characteristics of the image regardless of
object position:
Virtual, Erect, Smaller than object, Between
object and lens

53. Calculations of images:
-the conjugate foci formula -
 Two surfaces - subscript no 1 and 2
 L1 , L2 –reduced vergence on 2 surfaces
 Thin lens – considered as one refracting
surface followed by a second refracting
surface with negligible separation between
them

55.  For refraction at first surface;
 n1’ – n1 = n 1’ – n1
l1’ l1 r1
Or reduced vergence notation:
L1’ - L1 = F1
 For refraction at second surface;
 n2’ – n2 = n 2’ – n2
l2’ l2 r2
Or reduced vergence notation:
L2’ - L2 = F2

56.  Let n1’ = n2
 As thickness is negligible; reduced
vergence emerging from surface (1)
inside lens is same as reduced
vergence of light arriving at surface (2)
 i.e.:- L1’ = L2
 Hence:
L1’ - L1 + L2’ – L2 = F1 + F2 or
L2’- L1 = F1 + F2

57. L1’ / L2’
by dropping reduced vergence and using
l2’ = l’ = l1 = l
We get:- L’ – L = F = F1 + F2
Difference – change in reduced vergence
of light – hence power of lens

58.  General formula of lens
 Relates power of lens with reduced
vergence of incident and emergent
rays respectively
 Also called conjugate foci formula,
Because of fact, object and its real image
are interchangeable.

59.  As lens power is F = F1 +F2
 F1 = n’ - n 1 = ng – 1
r1 r1
 F2 = n2’ - n 2 = 1 - ng
r2 r2
 Power of lens in air = F = (ng – 1) 1 - 1
r1 r2
 Often know as lens maker formula
 In terms of radii of curvature; 1/r1 = R1 , 1/r2 = R2
F = (n – 1)(R - R )

60. Image magnification
 l = n/L , l’ = n’/L’
m = l’ = n’/L’ = L
l n/L L’
m = h’ / h = L / L’
incident reduced
vergence
emergent reduced
vergence

61. Thick lens
 Thickness of the lens
When the thickness of the lens cant
be regarded as negligible, the thin
lens equation and some thin lens
formulae are not valid and the lens
is described to be THICK LENS
With central thickness >0.1mm
The vergence change on transfer
cant be ignored

62. ○ Example
IOL
Some forms of spectacle lenses
Contact lens

63. Thick lens formulae
 The
refractive
indices of
the object
and image
space may not
be same.

64. Power of first surface
 F1=n’1-n1/r1=ng-n1/r1

65. Power of back surface
 F2=n’2-n2/r2=n’2-ng/r2

66. Equivalent power
 Fe=F1+F2- t’F1F2

67. Back vertex power
 F’v=Fe/1- t’F1

68. Front vertex power
 Fv = Fe/1-t’F2

69. Power of front surface F1=n’1-n1/r1=ng-n1/r1
Power of back surface F2=n’2-n2/r2=n’2-ng/r2
Equivalent power Fe=F1+F2- t‘F1F2
Back vertex power F’v=Fe/1- t’ F1
Back vertex focal length f’v=n’2/F’v
Front vertex power Fv = Fe/1-t’F2
Front vertex focal
length
fv = -n1/Fv
Summary of thick lens FORMULA:

70. Power Measurement
 Hand Neutralization
measures front vertex power of a lens
○ most meniscus lenses prohibit placing a trial
lens in contact with the back surface
can be used when lensometer is not working
or unavailable

71. Power Measurement
 Lensometer (a.k.a. lensmeter,
vertometer, focimeter)
measures vertex power of lens
and prism power
combines Badal system (I.e.,
standard lens) and Keplerian
telescope

72. Thank You !Thank You !
Suggestions !!Suggestions !!