OP.OPTICS.ppt

Published in Australia by
The International Centre for Eyecare Education
First Edition 2002
Presbyopia Education Program 2002
All rights reserved. No part of this publication may be reproduced, stored in a retrieval system, or
transmitted in any form or by any means, without the prior permission in writing of:
ICEE Secretariat
PO Box 328
Randwick, Sydney
NSW 2031 Australia
Tel: (612) 9385 - 7435
Fax: (612) 9385 - 7436
Email: icee@ cclru.unsw.edu.au

1

2
Introduction 3
Acknowledgements 4
Contributors 5
Introduction to Ophthalmic Optics 6
Electromagnetic Spectrum 13
Refraction of light 20
Geometrical Optics 40
Image Formation 46
Thick Lenses 66
Spherical Lenses 84
Cylinder Lenses 93
Crossed-Cylinders 106
Toroidal Surface Imaging 112
High Refractive Errors 134
References 167
T A B L E O F C O N T E N T S

3
The Educators Presbyopia Education Program (EPEP) is intended to assist ophthalmic educators
with information and materials that may be useful in the teaching of refraction, dispensing, and
prescribing for the presbyope in ophthalmic schools, colleges, and other institutions.
The course of study includes:
Module 1 Ocular Anatomy and Refractive Errors
Module 2 The General Eye Examination
Module 3 The Refractive Examination
Module 4 Accommodation and Presbyopia
Module 5 Binocular Vision
Module 6 Ophthalmic Optics
Module 7 Spectacle Lenses
Module 8 Progressive Addition Lenses
The Modules contain most of the basic knowledge students will need as eyecare practitioners.
The teaching materials are intended to be in an easy-to-use format containing slides and
suggested accompanying text. The diagrams and photos provide visualisation for the concepts
presented.
Craig Butler
Director, Professional Education
I N T R O D U C T I O N

4
The International Centre for Eyecare Education (ICEE) was established to develop the global resources needed
to provide people everywhere with good vision and effective eyecare. Our role is to develop educational programs
where they do not exist, to provide resources where needed to help teachers teach, and to help people in need.
In the Presbyopia Education Program (PEP), we deal with a condition that affects 22% of the world’s population.
Presbyopia eventually affects everyone, yet the public understands little about it and is often resentful of this sign
of middle age. Many advances have been made in optical design of spectacles for presbyopia and it is important
that eyecare practitioners have the best correction options available. In many parts of the world the educational
programs and product availability are very rapidly advancing and ICEE, in this Essilor sponsored program, will
bring information and materials to help keep educators and practitioners up to date.
The goals of PEP are to help advance the refraction, dispensing, prescribing and patient management skills of
practitioners, particularly in the prescribing of progressive addition spectacles. The PEP for Educators will supply
infrastructure and materials for the teaching of refraction, dispensing and prescribing in relevant ophthalmic
schools, polytechnical colleges and other training institutions.
ICEE could not be successful in this endeavor without the assistance and generosity of a large number of
talented and dedicated people. To all those contributors of lectures, computer programs, videos and slides, we
say thank you. Your generosity of spirit will benefit many educators and students, and millions of patients
throughout the world.
The Program was also very fortunate to obtain the services of many committed people in writing, designing,
editing, producing and distributing the modules. In particular, the staffs of the Cooperative Research Centre for
Eye Research and Technology; the University of Waterloo, Canada; the Cornea and Contact Lens Research Unit
at the University of New South Wales; and the International Association of Contact Lens Educators, have
contributed substantially to this project through the donation of time, resources and editorial support. The
individuals involved are acknowledged in each module.
Essilor, one of the world’s leading ophthalmic companies, has generously funded this project. Their vision and
commitment to education and the community will help advance eyecare throughout the world. Their technical and
professional services staff have been extremely helpful.
All the contributors deserve recognition for their willingness to donate their knowledge, talent and time to the
advancement of international eyecare through this Program.
Brien A Holden
On behalf of the Management Committee of ICEE
A C K N O W L E D G E M E N T S

5
C O N T R I B U T O R S
PRODUCTION
CONTRIBUTORS
N N Coordinator
Barry Brown (2)
Maria Wong, BA (1)
Brad Ferguson, BVisCom (1)
Greta Spies (1)
EDITORIAL
Major Contributor
Graphics
N
1 Cooperative Research Centre for Eye Research and Technology (CRCERT) Sydney,
Australia
2 International Association of Contact Lens Educators (IACLE ) Sydney, Australia
3 International Centre for Eyecare Education (ICEE) Sydney, Australia
4 School of Optometry University of Waterloo Waterloo, Canada
5 College of Optometry University of Houston Houston, USA
6 Documentation Consultant Sydney, Australia
Noel Brennan
Chuan Ooi BOptom., MSc (Optom)
Stephen Kwok PhD
N
Kylie Knox, BA(Comm) (1)
Peter Fagan, BA(Hons), Dip Lib (6)
Editor
Coordinating Editor
Proof Readers
N
N
Assistant Editor
N
David Wilson, BEc, BA (Hons),
Dip Ed, FNAO
Caroline Llewellyn
(3)

6
Principles of Ophthalmic Optics

8
Geometrical Optics
• A representation of light path through optical system or
optical media
• Principle of rectilinear propagation
• Relies on simple geometric constructions

9
Nomenclature
• Optical Medium
– Space through which light travels
– Assume homogeneity in media
• Light Ray
– Represents direction of wavefront propagation
– Emanate from a point source passing through very small
aperture
– Perpendicular to the wavefront

10
Nomenclature
• Pencil
– A bundle of rays emanating from point source passing
through a limiting aperture
– Chief ray
• Beam
– A collection of pencils
– Arises from an extended source

12
Light as Electromagnetic Energy

13
Electromagnetic Spectrum
Violet
Blue
Green
Yellow
Orange
Red
400 nm 760 nm
Infrared
Radio
waves
Ultra-
violet
Gamma
rays X-Rays
Wavelength (nm)
10-2
Micro-
waves
10-1 1 101
104
106
108
1010
1012

14
Energy of Light and Wavelength
In an isotropic medium, the velocity of propagation is constant. The wave
frequency (upper curve, right axis) decreases with wavelength. As a result,
the photon energy (lower curve, left axis) falls as the wavelength increases.
Similarly, photon energy rapidly increases with smaller wavelengths.
0 2000 4000 6000 8000 10000
Wavelength (nm)
Energy
(10
-18
J
oule)
Frequency
(Hz)
101
5
101
3
2
1 1011
Visible light
Ultraviolet
X-rays
Infrared

15
Intensity of Light and Amplitude
Both waves have the same frequency but the upper wave has double the
amplitude: hence, its intensity is four times that of the lower wave.
2a
a
I = 4a2
I = a2

16
Interaction of Light with an Object
 Transmission
 Absorption
 Reflection
 Refraction
 Diffraction

19
Specular Reflection of Light
Specular vs. diffuse reflection. In specular reflection, if an observer moves to
position A, the area of reflected light may be missed. In contrast when light is
diffusely reflected from a rough surface, some of the reflected light can still
be seen due to the scattering in different directions.
Specular
Diffuse
A
A

21
Light Velocity and the Medium
Refractive Index
Medium n Comment
Vacuum 1.0 Highest value
Air 1.0003 Usually 1.0 is used
Water 1.333 For 589.3 nm light
1.3427 For 404.7 nm light
Sucrose 1.420 50% aqueous solution
Quartz 1.544 Quartz is not isotropic
1.553
Diamond 2.417
Increasing
optical
density

22
Refractive Index
Refractive Index, n
= Velocity of light in vacuum = fl
Velocity of light in medium flm
= l
lm since frequency of light wave is constant
• Velocity of light in vacuum = 1, by definition
• Fraunhofer lines
• nF (Blue spectrum) 486.13 nm
• ne (Mercury green ) 546.07 nm
• nd (Helium yellow ) 587.56 nm
• nc (Red spectrum) 656.27nm

23
Propagation of Light Into a New Medium
MEDIUM 2
MEDIUM 1
Reflection
Absorption
Refraction
Transmission
Interfac
e
Incident
parallel
light
When light encounters a new medium, the
boundary is called the interface. By
convention, the incident medium (e.g., air)
is called medium 1, and the second
medium (e.g., glass) is called medium 2.

24
The Law of Reflection
Normal (perpendicular to
surface)
Medium 2
q1 q2
Incident
ray
Reflected
ray
Medium 1
q1 = q2

25
Refraction of Light
Normal (perpendicular to
surface)
i
i
Incident
ray
Refracted
ray
Medium 1
Medium 2
90°
Air
Glass

26
Refractive Index Change Causes Bending
i
i
Medium
1
Refracted
ray
Medium
2
90°
Faster
Incident
light
Slower Same
n
n 
new velocity
n < n  i > i slower
n = n  i = i same
n > n  i < i faster

27
Snell’s Law of Refraction
i
i'
Incident
ray
Refracted
ray
Medium
1
Medium
2
90°
Air
Glass

28
Refraction of Light
i
i'
Incident
ray
Refracted
ray
Medium 1
Medium 2
90°
Glass
Air

29
Total Internal Reflection
Ray incident at
critical angle ic
90°
Glass
Air
Medium
1
Medium
2
Refracted light
rays
All light rays
in this range
will refract
normally
All light rays in
this range will be
totally internal
reflected
Critical ray refracted at
90°

30
Refractive Index and Frequency
i
Medium
1
Refracted
rays
Medium
2
90°
n
n 
The higher
frequency light
is refracted
more
Incident
light

31
Constancy of Light Frequency
l1
l2
Two waves of different
velocities but same
frequency travel different
distances in the same time
interval. The total number
of cycles is the same but
the slower wave (lower)
travels less distance and
has a smaller wavelength.
f1 = f2
Medium 1
Medium 2

32
Refractive Index and Wavelength
The refractive index of a transparent material depends on the particular wavelength of
light. The values shown are for crown glass in air, and indicate that high frequency
light (small l) such as blue (left) has a higher refractive index than low frequency light
(large l) such as red (right). The corresponding velocity (right axis) indicates that blue
light is slowed down more than longer wavelength light.
1.5
1.51
1.52
Blue Yellow Red
Refractive
Index
Velocity
(10
m/s)
l = 486.1 nm 589.3 nm 656.3 nm
1.98
1.97
1.96
8

33
Dispersion of Light
i
Medium
1
90°
Medium
2
Incident rays
of different
wavelengths
Refracted rays
Longer wavelength
light is deviated
less from original
angle so has a
larger angle of
refraction i 
Shorter wavelength is
bent more towards the
normal so has a
smaller angle of
refraction i 
i 
Glass
Air

34
Dispersion of Light
In the dispersion of light, white light is refracted by the prism back into the air.
Because the prism has non-parallel sides, the emerging light is separated into
different directions according to incident frequency. The high frequency blue light is
bent more than the lower frequency red light. This causes all the component colours
contained in the incident white light to form a monochromatic spectrum.
Medium
2
i
Incident
white light
Dispersed
rays
90°
Medium
1
Medium
1
Glass
prism
Air
Air
The prism disperses the
incident white light into
individual colours
according to frequency;
the blue light is deviated
the most, red light the
least

35
Scattering of Light
Scattering of light. Incident light strikes molecules or particles in the medium
(the example here is sunlight travelling through the atmosphere). The energy of
the light is re-transmitted with no loss (upper right) or with some loss (left).
Sometimes the energy can be absorbed (lower right). The frequency of the
incident light (and hence its energy) is an important factor.

36
Absorption and Object Colour
Selective absorption of light. Left: Incident white light strikes an object which absorbs
all wavelengths except red which is reflected. Hence, the eye will perceive the object
as being coloured red when seen under white light. Right: When the same object is
seen under monochromatic light (green), all of the light is absorbed. Hence, the eye
will perceive the object as being black or grey in colour when seen under green light.
Red object Red object
B G R
White light
G
Monochromatic light

37
Polarization of Light
Specular reflection of light occurs when parallel incident light is reflected
as parallel light rays by the smooth surface. The “free” electrons in the
object do not pass on the incident energy onto atoms, but transmit the
light back out of the material at the same frequency. The object appears
shiny if visible frequencies are involved. In some materials such as glass
some light is still able to pass through (see light ray on the right).
Unpolarize
d light
Polarized
reflected light
Polarized
refracted
light

38
Polarization of Light
Polarization of light can occur when unpolarized light passes through a material that absorbs all
light vibrations except for one direction. The emerging light is linearly or plane polarized, that is,
it only vibrates in a direction parallel to the polarizing direction (here shown as a vertical
polarizer). If a second polarizer, sometimes called an analyzer, is placed in the light path but
with the polarizing direction perpendicular to the first, then no light will penetrate the analyzer.
In this situation, we say that the light has been extinguished.
Non-polarized
light
Polarized
light
Vertical
polarizer
Horizontal
polarizer

39
Diffraction of Light
As light passes through a pinhole, the edges bend the incident light. The bending leads to
interference between the light waves once they pass through the pinhole. This leads the
formation of a diffraction pattern: a bright central spot surrounded by dark and light rings.
George Airy (1801-1892) derived the mathematics of the spot formation, hence the circular
pattern is known as Airy’s disk. For simplicity, the light is assumed to be monochromatic.
Diffraction
pattern formed
on screen
Pinhole
Light
beam

40
Geometrical Optics
When light incident is on a circular opening whose dimensions
are large compared with the incident wavelength, the pattern
formed by the light rays can be predicted by simple geometry.
Opening

41
Sign Convention
Optic Axis
Direction of
travel of light
Object Image
f ’
f
+ve
-ve
-ve
+ve
-ve
+ve
Primary
Focal
Plane
Secondary
Focal
Plane

42
Vergence
Divergence
(negative vergence)

43
Light Propagation
The wavefronts of light from the distant object arriving at the spectacle
lens and eye are parallel; their curvature is essentially zero due to the
great distance to the object. Thus the light rays (arrows), which are
perpendicular to the plane wavefronts, are also parallel to each other.

44
Vergence
Vergence of light. Left: light emanating from a light source spreads out or is
divergent. Since vergence is measured from the wavefront to the source (at
left) it is negative. Right: light coming together to a focus is convergent. The
vergence is positive, since the focal point is to the right of the wavefronts.

45
Vergence & Refractive Index
Reduced Vergence, Ln
=. 1 .
Reduced distance
= . 1 . or . 1 .
l /n l ´/n´
= n or n´
l l ´
l = object distance, n = refractive index in object space
l ´ = image distancen´ = refractive index in image space.

46
Image Formation
Image formation by a convex lens: light rays from the object at left are
refracted by the lens and meet to form the image on the right. In this
illustrative example, the image is upside down and smaller. In general, the
exact size and location of the image depend on several factors such as
distance of the object to the lens, the media involved (hence refractive
indices), the curvature and thickness of the lens surface. Geometrical optics
is able to provide useful approximations of the image’s properties.
Convex
lens

47
Vergence and Image Formation

48
Light Rays and Image Formation
Increasing prismatic effect away from optic axis

49
Real and Virtual Images
An object is located to the left of a convex lens; the object can be self-luminous
(e.g., a lamp) or can be reflecting visible light. The object can be considered to be
comprised of many individual points. Each point acts as a source of many rays of
light, each ray travelling in a different direction as shown. Light rays from the
object are refracted by the lens to form an image. The further an object is from
the lens, the less important the object’s dimensions are, and distant objects can
be approximated as points. The location, size and nature (real or virtual) of the
image depends on the distance between the object and the lens.
Convex
lens

50
Image Formation
Sometimes the image and object space are the same. In this
example, the underwater object has an image that is also
underwater. The image of the object is displaced by refraction,
and the observer sees the object as being in another location.
Air
Water
n1
n2
The light rays
from the object
are refracted by
the water/air
interface
The observer thinks
that the fish is in the
position indicated by
the line of sight
Apparent
Position

51
Fermat’s Principle
Light from an object at point O is incident on a convex lens; after refraction
the light rays meet at a common point I. The light rays all obey Fermat’s
principle of least time by travelling in straight lines. All light rays have the
same optical path length and arrive at I at the same time. Even though the
ray on the optical axis (line joining O to I) seems to have the shortest path, it
has to travel through the thickest part of the lens; similarly, the furthest ray
has longer to travel through air but the shortest to travel through the lens.
O I

52
Geometrical Optics
Many light rays from an object (left) incident on a lens are collected and refracted to
form an image (right). Geometrical optics re-constructs this process as a group of
planar (two-dimensional) events, each calculated from simple geometry.

53
Geometrical Optics
When light incident is on a circular opening whose dimensions
are large compared with the incident wavelength, the pattern
formed by the light rays can be predicted by simple geometry.

54
Images and Ray Tracing
Ray tracing can be
used to estimate the
size and location of
the image (right). The
process involves
calculating how
various points on the
object (left) are
refracted, using
geometry and Snell’s
law, and applied in a
selected plane such
as the one shown.

55
Ray Tracing
Light from a distant object located to the left of a convex lens is incident
in the form of parallel rays. This means that the wavefronts are parallel to
each other and have zero curvature due to the distance from the object
to the lens. Paraxial rays travel close to the principal axis; light rays
incident near the edge of the lens are called marginal rays. Ray tracing of
individual light rays can be used to determine the location and height of
the image formed after the light travels through the lens.
Marginal
Ray
Paraxial
ray
Optical Axis

56
Convex (Positive) Lens
Left: overlapping circles define the
surface curvatures of a bi-convex
lens. Below: converging refracted
light (from parallel incident light)
defines the focal point F.
Focal Length
f’

57
Concave (Negative) Lens
Left: the surface curvatures of a bi-
concave lens are defined by non-
overlapping circles. Below: the focal
point is defined by the apparent
source of divergent light, refracted
from parallel incident light.
Focal Length
Light diverges
from apparent
source at left
(focal point)

59
Principal Rays - Convex Lens
The three principal rays originate from the same point on the object: any point can
be chosen. The distance between the object and lens is greater than the focal
length. Ray 1 is incident parallel to the optical axis and is refracted through point f ;
ray 2 passes undeviated through the lens centre; and ray 3 passes through point f
and is refracted parallel to the optical axis.
f’
1
2
3
f

60
Ray Tracing - Convex Lens
Left: light from an object located to the left of a convex lens (further than the focal
length) forms a real, inverted image. The image size is reduced, but as the object gets
closer to the lens, the image becomes larger. At a distance equal to 2f, the image is the
same size; the image size continues to increase up to a distance of f. Right: (a) parallel
light from a distant object forms a spot image at f . (b) A point source at F forms
parallel light after refraction.

61
Ray Tracing - Convex Lens
Light from a distant object located to the left of a convex lens is incident in
the form of parallel rays. This means that the wavefronts are parallel to each
other and have zero curvature due to the distance from the object to the
lens. Paraxial rays travel close to the principal axis; light rays incident near
the edge of the lens are called marginal rays. Ray tracing of individual light
rays can be used to determine the location and height of the image formed
after the light travels through the lens.

62
For a concave lens, two principal rays can locate the image. The distance between the
object and lens is greater than the second focal length; note that f  is on the same side
of the lens as the incident light. Ray 1 is incident parallel to the optical axis and is
refracted seeming to diverge from point f ; ray 2 passes undeviated through the lens
centre. An observer to the right sees the virtual image which is upright and smaller.
Principal Rays - Concave Lens
f
f
1
2
Virtual
Image
Ray Tracing - Concave Lens

63
Ray Tracing - Concave Lens
Light from a distant object located to the left of a concave lens is
incident in the form of parallel rays. This means that the wavefronts are
parallel to each other and have zero curvature due to the distance from
the object to the lens. Paraxial rays travel close to the principal axis;
light rays incident near the edge of the lens are called marginal rays. Ray
tracing of individual light rays can be used to determine the location and
height of the image formed after the light travels through the lens.

64
Lens Notation and Rules for Refraction
• Power
– F1, F2: Refracting power of front & back surfaces
– Fv, Fv’: Front & back vertex power

66
Back Vertex Power - Overview
f ’
O’
n’
A1
n
A2
t
o’ fv
’
l

67
BVP - Front Surface
f ’
O’
n’
A1
n
A2
t
l1’ fv
’
l
n

68
BVP - Back Surface
f ’
O’
n’
A1
n
t
o’ fv
’
l2
A2
























-
+
=
1
'
1
2
'
.
1 F
n
t
F
F
F v

69
Front Vertex Power
Front Vertex Power, Fv
Fv = F1 + F2 .
1 - (t/n’)F2












70
FA , Fv’ and Fv
Lens Specifications
F1= +8.00D; F2= -6.00D; t=2.00mm; n=1.523
Approximate Power
FA= (+8.00) + (-6.00) = +2.00D
Back Vertex Power
Fv’= (-6.00) + . (+8.00) = +2.08D
1- (0.002/1.523).(+8.00)
Front Vertex Power
Fv = (+8.00) + . (-6.00) = +1.94D
1 - (0.002/1.523).(-6.00)

71
FA , Fv
’ and Fv
Lens Specifications
F1= +14.00D; F2= -4.00D; t=8.00mm; n=1.523
Approximate Power
FA= (+14.00) + (-4.00) = +10.00D
Back Vertex Power
Fv’= (-4.00) + . (+14.00) = +11.11D
1- (0.008/1.523).(+14.00)
Front Vertex Power
Fv = (+14.00) + . (-4.00) = +10.19D
1 - (0.008/1.523).(-4.00)

72
Usefulness of BVP
• Not influence by the lens form
• Secondary focal point (fv
’) coincides with far point
• Simple manipulation of the value to obtain the BVP at a
different vertex distance

73
Effective Power
A2
d
5mm
fv
’ [Lens A]=100mm
Lens B,
Fv
’=?
Vd = 10 mm fv
’ [Lens B]=95mm
Lens A,
Fv
’=+10.00D
vd = 15 mm
A2

74
• Do we always need a spherical correction?
– Non-spherical prescriptions
– Aberration corrected lenses
– Specialized optical requirements
• If we don’t want a spherical prescription…
– Cylindrical lenses
– Toroidal (Toric) lenses
– Aspheric lenses
Spherical vs. Toric

76
Lens Power
Thin lens power, (or nominal power, FA)
FA = F1 + F2
Thick lens power, (or nominal power, FA)
FA = F1 + F2 - t/n(F1F2)
where t = thickness of the lens,
n = the refractive index of the lens material

77
If y is small (paraxial),
µ1 ˜ y/ l1 ,
µ2 ˜ y/ l2, and
ø ˜ y/r
i = ø-µ1
i = ø-µ2
n i = n i (Snell's Law)
n' (ø-µ2) = n (ø-µ1)
n (y/r - y/l2) = n (y/r - y/l1)
Cancelling, and rearranging gives
n /l2- n /l1= (n -n )/r
which we call F, the power of the surface
'
'
'
' '
'
Paraxial Optics
n n
µ µ
i
i
y
r
O O
c
1
1
'
2
'
2
ø
1 2
l l

78
Paraxial Optics
n n
µ µ
i
i
y
r
O O
c
1
1
'
2
'
2
ø
1 2
l l
If y is small (paraxial),
µ1 ˜ y/ l1 ,
µ2 ˜ y/ l2, and
ø ˜ y/r
i = ø-µ1
i = ø-µ2
n i = n i (Snell's Law)
n' (ø-µ2) = n (ø-µ1)
n (y/r - y/l2) = n (y/r - y/l1)
Cancelling, and rearranging gives
n /l2- n /l1= (n -n )/r
which we call F, the power of the surface
'
'
'
' '
'

79
Positive Thin Lens Forms
F1=+2.00
F2=+2.00
FA = +4.00
F1=+4.00
F2= plano
FA = +4.00
F1=+6.00
F2= -2.00
FA = +4.00
F1= Power of the First Surface
F2= Power of the Second Surface
FA = Nominal Power of the Lens
where
Biconvex Planoconvex Meniscus

80
Negative Thin Lens Forms
F1= Power of the First Surface
F2= Power of the Second Surface
FA = Nominal Power of the Lens
where

81
Base Curves
Positive
Spheres on a
-4 Base
Negative
Spheres on a
+6 Base

82
Refractive Power
Refractive Power, F
= Reduced Image Vergence - Reduced Object Vergence
= L’- L
L’ = n’ and L = n
l’ l
For a single spherical surface, F
=n’- n = n’ - n
r l’ l

83
Different Powers
The power of an ophthalmic lens may be given by:
Approximate Power, FA
Back Vertex Power, Fv
’
Front Vertex Power, Fv
Equivalent Power, Fe

84
Spherical Lenses
– Ideally produce a point image from a point source object
– Have a radius of curvature, the reciprocal of which describes
the lens curvature
– The spherical surface changes the vergence of the incident
light at the interface between refractive indices
– Ideally have a single power for the entire lens, and this is
true in paraxial conditions and/or for thin lenses

85
Spherical Lenses
• Spherical Lens
-Radius of curvature equal in all meridians
-Power is equal in all meridians
-Forms a point focus for all incident rays

87
Refractive Power and Vergence
Refractive Power (units of dioptres)
F = (Reduced image vergence) - (Reduced
object
vergence)
= L’ - L
where L’ = n’ and L = n
l’ l
For a single spherical surface (or interface):
F = n’ - n = n’ - n
r l’ l

88
Focal Length
Distant parallel light (object at infinity to left) incident on the curved interface
focuses at the point F’, termed the second principal focus. The distance from the
surface to F’ is called the second focal length, f’. Similarly, light emanating from
point F, the first principal focus is refracted as parallel light (image at infinity to the
right). Distance from the surface to F is the first focal length, f.

89
Spherical Lens
A positive spherical lens is equivalent to two interfaces in series, each with a radius
of curvature. The light is incident from a point to the left and is refracted by the first
interface (A) from air into the lens medium (image space). The light rays now become
incident at the second interface (B) - so the lens medium now acts as the object
space. The light is then refracted into air.

90
Thin Lens Power
The thin lens power is a function of n, the refractive index of the lens material and
the radius of curvature of the front and back surface, r1 and r2 respectively. The
sign of r1 and r2 determines the sign of FA. For a biconvex lens (left), r1 > 0 and r2 <
0 hence FA is a positive power. For a biconcave lens (right), r1 < 0 and r2 > 0 hence
FA is a negative power.



-



-
=
2
1
1
1
1
r
r
n
FA )
(
r2 r1 r1 r2
Lens Maker’s
Formula

91
Thin Lens Power
The figure shows the imaging of a positive spherical lens, of radius of curvature r. The
principal meridians are seen at 90 and 180. The principal meridians of a lens are always
positioned at right angles to each other, whatever the orientation of the lens in front of the eye.

92
Thin vs Thick Lens
Left: The thin lens is assumed to have “zero” thickness; hence the
refraction of light rays is assumed to occur with respect to a single plane
at the midline position. Right: Refraction of light calculated for a thick lens
takes into account the deviation of light at the front and back lens
surfaces.

93
Cylindrical Lenses
• Non-spherical / Cylindrical Lens
-Radius of curvature NOT equal in all meridians
-Power is NOT equal in all meridians
Cylinder - power in one meridian is plano
Toric / Toroidal - both meridians have power,
but power is not the equal
-DOES NOT form a point focus for all incident rays

94
Cylinders vs Torics
Cylindrical Lens Toric Lens

96
Slicing up Spheres
A
A’
B B’
Slicing along AA’ Slicing along BB’
A A’ B B’
rs rs

97
Slicing up Cylinders
A
A’
B B’
Slicing along AA’ Slicing along BB’
A A’ B B’
rc Plano

98
Power & Axis
A
A’
B B’
• BB’ represents the axis meridian
– ie. The meridian of plano power
AA’ represents the power meridian
ie. The meridian of maximum power
The axis & power meridians are always perpendicular

99
OD OS
0
180
90
45
135
0
180
90
45
135
Axis Notation
• Confrontational view
• Horizontal = 180; Vertical = 90
• The axis is never >180
Nasal
Temporal
Temporal

100
Optometric Notation of Cylinders
• Which is the power meridian?
• Which is the axis meridian?
Plano
+2.00

101
Power Diagrams for Cylinders
• Power @ 90 = Plano; Power @ 180 = +2.00D
• Plano / +2.00 x 90
• Power @ 180 = Plano; Power @ 90 = +4.00D
• Plano / +4.00 x 180
Plano
+2.00
1
Plano
+4.00
2

102
• In the power meridian
– where l is the distance of the point object from the lens, l 'of the
line image. Fc is the power in the power meridian and fc is the
focal length.
– Use this to find the distance from the lens at which the image is
produced.
Cylindrical Calculations
c
c
f
l
l
F
L
L
/
1
/
1
/
1 '
'
+
=

+
=

103
Example
A point source is placed 1m from a cylindrical lens of +10D power, cylinder
axis horizontal and 40mm in diameter. What is the position, length and
direction of the line focus?
The set-up…..
The image is in the same plane as
the axis ie. horizontal
( )
m
l
F
l
l c
111
.
0
9
/
1
10
1
/
1
/
1
/
1
'
'
=
=

+
-
=
+
=

104
Height of the Image
l l’
y x
 l=1m l ’ =0.111m y=40mm
• x=image height
• By similar triangles:
l / y = ( l + l ’ )/ x
1 / 40 = 1.111 / x
x = 1.111 / 0.025 = 44.44mm
Normal
Normal

105
Combining Cylindrical Lenses
• Combination of two cylindrical lenses
– Creates power in meridians at right angles
– Images two line foci
– Separation of the line foci dependent on the power in each
meridian

106
Crossed-cylinders
+2.00  90 +2.00  180
3 dimensional representation
First scenario, where axis of one
cylindrical lens is perpendicular to the
other cylindrical lens
= +2.00DS

107
Power Diagram of Two Crossed Cylinders
Two Thin Cylindrical Lenses of
Equal Power and Perpendicular
Axes In Contact
+2.00DC  090
+2.00
PL
Sphere
+2.00
+2.00
+2.00DC  180
+2.00
PL
+2.00D  180 plus +2.00D  090
Equals +2.00DS

108
Two Crossed Cylinders
Two Thin Lenses In Contact
+9.00DC  090
+9.00
PL
Toroidal Surface
+3.00
+9.00
+3.00DC  180
+3.00
PL
Two Crossed Cylinders
+3.00D  180 / +9.00D  090
This is written in crossed-cylinder form

109
Spheres & Positive Cylinders
+6.00DC x 090
Toroidal Surface
+3.00
+9.00
+6.00
PL
+3.00DS
+3.00
+3.00
A Sphere & A Cylinder
+3.00 / +6.00  090
This is written in positive-cylinder form

110
Spheres & Negative Cylinders
+9.00DS -6.00DC x 180
PL
-6.00
Toroidal Surface
+3.00
+9.00
+9.00
+9.00
A Sphere & A Cylinder
+9.00 / -6.00  180
This is written in negative-cylinder form

111
Transposing
• Power of toric lens normally expressed:
– Spherical power + Cylindrical power
– ‘Sphero-cyl form’
• If Fmax = Meridian of Max power
Fmin = Meridian of Min power
• Then
C = Fmax - Fmin &
Ftotal = Fmax + Fmin = 2S + C
S = (Ftotal - C ) / 2

112
Toroidal Surface Imaging
• A toroidal surface produces two line foci
• Line foci are formed due to power in principal meridians
• Pencil formed is termed
– astigmatic pencil
• Separation of line foci is termed
– Interval of Sturm

113
Toroidal surface Imaging
Interval of Sturm
(axial astigmatism)
Lens 2 - Power Surface
Lens 1 - Power Surface

114
Example
Two perpendicularly crossed cylinders:
a. +10.00DC x 180
b. +5.00DC x 90
Find the position of the two line foci for a distant object
For a distant object L = 0
Considering Lens (a)
L = 0
L’ = Fa = +10
l ’ = 1/10 = 0.10m = 10cm
Considering Lens (b)
L = 0
L’ = Fb = +10
l ’ = 1/5 = 0.20m = 20cm
Since Lens (a) has
axis 180
a horizontal line focus
is produced at 10cm
Since Lens (b) has
axis 90
a vertical line focus
is produced at 20cm

115
Interval of Sturm
Line foci
Principal
axis
S
Interval
of Sturm
Circle of least confusion
A
B

116
Image Formation
Max power meridian @ 90
Min power meridian @ 180

117
Circle of Least Confusion
• Position at which the astigmatic pencil is circular
• Provides best ‘compromise’ vision with sph-cyl combination
• CLC lies, dioptrically, half way between the two line foci

118
Example
A sphero-cylindrical lens has powers of +8.00 / -2.50D  180.
If the back surface is Plano, what are the powers of the principal
meridians of the front surface? What would be the powers in crossed
cylinder form? And positive sph-cyl form?
Back surface = Plano power is all on the front surface
+8.00 / -2.50D x 180
+8.00
+8.00
PL
-2.50
Powers in principal meridians +5.50D @ 90 +8.00D @ 180
Crossed Cylinder Form +5.50D x 180 / +8.00D x 90
Positive Cylinder Form +5.50 / +2.50 x 90

119
Example
A sphero-cylindrical lens has powers of +8.00 / -2.50D  180.
Where are the line foci & circle of least confusion located for an object at
0.5m in front of the lens?
Powers in principal meridians
For an object at 0.5m
For the vertical line focus
l = -0.5 L = 1/ (-0.5) = -2
FH = +8.00D
From, L’ = L + F LH’ = (-2) + (+8) = +6
lH’ = 1/6 = 16.7cm
For the horizontal line focus FV = +5.50D
LV’ = +3.5 lV’ = 1/3.5 = 28.6cm
For the circle of least confusion Lclc’ = ( LH’ + LV’) /2 = 9.5/2 = 4.75
lclc’ = 1/(4.75) = 21.1cm
+5.50D @ 90 +8.00D @ 180

121
Toroidal Surface
• Powered surfaces in perpendicular meridians
• Power is not equal
• Second powered surface is generated by an arc whose centre of
rotation is in the same plane as the arc
• Principal meridians are the meridians of minimum & maximum
power
• Both meridians contain power (compare with cylindrical surface)

122
Generation of a Barrel Surface
rz
ry
NB. Centre of curvature of rz  ry
both meridians have power but power
of rz surface  power of ry surface

123
Generation of a Tyre Surface
rz
ry

124
Generation of a Capstan Surface
rz
ry

125
Crossed Cylinders
Second scenario, where axis of one
cylindrical lens is not perpendicular to the
other cylindrical lens
+3.00/+1.00  90 +3.00
+4.00
+0.50
-0.75
+0.50/-1.25  150

126
• AA’ is plane
• EE’ is circular, radius r
• HH’ is elliptical, radius rq
• Approximate sag formula:
– s = y2 / r
– s = (EP2) / r = (HP2) / rq
– rq = r (HP2) / (EP2)
• since (EP)/(HP) = sinq and 1/r = R
The Cylindrical Surface
A
q
A’
H
H’
E
E’
P
s
Rq = R.sin2 q or Fq = F.sin2 q

127
The Cylindrical Surface
+10.00 DC  90
Axis 90
q=90
Fq=+10.00
q=60
Fq=+7.50
q=45
Fq=+5.00
q=30
Fq=+2.50
q=0
Fq=0
q=60
Fq=+7.50
q=45
Fq=+5.00
q=30
Fq=+2.50
Fq = F.sin2 q

128
Mathematical Method
Axis F1
Axis F2
Axis Combined
Max F1
Max F2
Max Comb
a q
Axis combined = minimum power meridian = S + C
Max combined = Resultant cylinder power = C
Resultant cylinder axis = q
Axis F1
Axis F2
a

129
Obliquely Crossed Cylinders
• To find ‘S’ resolve F1 to meridian q and F2 to meridian q.
ie.
– Power of F1 at q= F1.sin2q
– Power of F2 at q= F2. sin2(a - q)
– S = F1.sin2q+ F2. sin2(a - q)
• To find ‘S+C’, resolve F1 and F2 to meridian (90 + q). ie.
– F1 at (90+ q)=F1. sin2(+q)= F1. cos2 q
– F2 at (90+ q)=F2. sin2(+(a-q)= F1. cos2 (a - q)
– S + C = F1. cos2 q + F1. cos2 (a - q)

130
• C = (S + C) - S ie.
– C = (F1. cos2 q + F1. cos2 (a - q))
- (F1.sin2q+ F2. sin2(a - q))
– C = F1. cos2q+ F2. cos2(a-q)
– which can be rewritten as
– C = F1. cos2q+ F2. cos2acos2q+ F2. sin2asin2q
• This expression can then be differentiated (C/q) and simplified to:
tan2q = F2.sin2a
F1 + F2.cos2a

131
• When q is known, S may be found by substitution:
S = F1.sin2q+ F2. sin2(a - q)
• When q and S are known, C may be found from:
– (S + C) + S = F1 + F2
– Rearranged: C = (F1 + F2) - 2S

132
• Summary of obliquely crossed cylinder procedure by
calculation
1) Transpose into positive sphero-cylinder form (if
necessary)
2) Only consider cylindrical component in calculation
3) Select cyl with axis nearest to 0 or 180 as F1
4) Calculate a=Axis F2 - Axis F1
5) Find q
6) Find S
7) Find C
8) Calculate resultant cylinder axis from (q + axis F1)
9) Correct sphere from step 1 (if necessary)

133
Example
A. +3.00/+1.00  90
B. +0.50/-1.25  150
1. Convert B to -0.75/+1.25  60
2. Spherical components add to +2.25.
3. Now use F1=+1.25 x 60 and F2=+1.00  90
4. a=30°
5. q=13.16°
6. S= 0.15
7. C= 1.95
8. Axis = 73
9. Final Rx= +2.25/+2.00  73 (rounded)

134
High Refractive Errors
• Aphakic patients
– Constitute greatest proportion of high positive powers
– Absence of crystalline lens due to
• surgery, trauma (subluxation), congenital
– Rx usually >+10.00D
– Less common in ‘90’s due to increased use of IOLs
– Cosmetics: Increased lens weight & thickness, increased
magnification, decreased field of view
• High Minus
– Congenital for high myopia
– Cosmetics: Increased edge thickness, increased weight, minification

135
Aphakes: Optical Consequences of Intraocular
Surgery
• Corneal distortion
– high cylinders &/or irregular astigmatism
– Size / shape of pupil
– Pupil mobility
– Loss of accommodation
– Reduction in refractive power of eye

136
Aspherics
• Blended Aspherics
– Series of spherical flattening curves
– Zones at curve change are blended
• Aspheric Curve
– Front surface curve (usually) derived from rotation of a conic
section
– Reduces oblique astigmatism
– Rapid increase in tangential radius cf. sagittal radius
– Plus powers - flatten towards periphery
– Minus powers - steepen towards periphery
– Overall, creates lens of a flatter form

137
Formulation of Aspheric Surface
• When the vertex is taken at the origin, the mathematical
representation of the conic surface is given by:
y2 = 2 r0 x - p x2
– at the origin, r0 is the vertex radius
– p is the value that describes the type of curve, shape and
degree of asphericity

138
Formulation of Aspheric Surface
x
y
r0
• When p=1 the formulation gives the sag formula:
– px2 - 2 r0x + y2= 0
– For which there are two solutions: x = r0   (r0
2 - y2)
• All curves have the same r0
y2 = 2 r0 x - p x2
• For values of p:
– p > 1 Oblate Ellipse
– p = 1 Sphere
– 0 < p < 1 Prolate Ellipse
– p = 0 Parabola
– p < 0 Hyperbola

139
Spectacle Correction of Aphakes
• Inherent problems
– Distance of correction from eye
– Static position of spectacle lens
• Optical problems
– Increased retinal image size; Reduced field of view; Ring scotoma
– Increased in required ocular rotations
– Increased demands on convergence
– Increased lens aberrations
– Apparent motion of objects in the field of view
– Appearance of the wearer

140
Increased Retinal Image Size
• Spectacle magnification can be expressed in terms of the ratio of the
‘corrected’ retinal image height to that of the emmetrope. ie.
Retinal image height in aphake = 23.23 = 58.64 = 1.36
Retinal image height in emmetrope 17.05 43.05
– ie. 36% increase in retinal image height if spectacles placed at
anterior focal point

141
Increased Ocular Rotations
• Ocular rotation
– Angle through which the eye moves to fixate from one point
to another
• Compare the magnitude with & without spectacles
• Due to prismatic effect of lenses
– Plus lens wearers require increased ocular rotations
– Minus lens wearers require decreased ocular rotations
• High Rx in aphakia demands a significant amount of increased
ocular rotation

142
Aspheric Surface & Aberrations
• Astigmatism of oblique incidence
– Controlled
– Rate of change of tangential & sagittal radii
• Distortion
– Pincushion effect
– Minimising OA, minimises distortion
– Relative improvement over spherical surfaces
• Chromatic Aberration
– Primarily TCA

144
Motion of Objects
• “Swim”
– Patient experiences ‘against’ motion in periphery of vision
– Avoided by moving eyes
– Moving eyes introduces distortion as a factor
– Overcoming “Swim”:
• Small head & eye movements combined
• Appearance
– Magnified eyes & awkwardness of movements
• Convergence
– Base-out effect when converging for near (Rx centered for
distance)

145
Lenticular Lenses
• Reduced aperture lens
• Manufactured to reduce weight of high plus & high minus lenses
• Particularly useful for large frames
– full aperture spherical lenses may be impossible to
manufacture
• Not cosmetically pleasing

146
Positive Lenticular Lenses
• Alternative to producing an aspheric lens
• Essentially a ‘plus’ button with a plano or low power carrier lens
• Reduced aperture  Decreased field of view
• Must be fitted as close to the eye as possible
• “Bulls Eye” or “Fried Egg” effect

147
Cemented Convex Lenticulars
• Knife edge button cemented to afocal carrier
• Aperture size ~30mm
• Toric surfaces

148
Solid Convex Lenticulars
• Glass & plastic materials
• Manufacture similar to solid bifocals

149
Fused Convex Lenticulars
• Obtained by fusing high index glass into crown glass countersink

150
High Myopes
• Lens weight & Thickness
• Minification of the image
• Field of view
• Cosmetic appearance

151
Lenticular Lenses for High Myopia
• Myo-Disc
– Created by grinding small concave disc on back surface of a
plano lens
– Diameter of powered portion ~30mm
– Carrier may be plano or low plus
– “Bulls Eye” effect
– Useful for myopia >~12.00D

153
Vertex Distance
What power lens is required at
12mm to correct a myope who
requires -12D at 20mm?
By ray tracing
L1 = 0
L1’ = -12
L2 = -10.95
Myopes require lower powers closer to the eye
Hyperopes require higher powers closer to the eye

154
Accommodation
A myope wearing -6D at a vertex
distance of 15mm wishes to see a
target at 33cm from the eye. How
much accommodation is required?
Effective Power at the eye L2 is -
5.50
Tracing a ray from 33cm
L1 = 1/31.5 = -3.174
L1’ = -9.17
L2 = -8.06
Accommodation required is
-5.50 - (-8.06) = 2.56D,
Which is considerably less than
the 3.03D required to view at
33cm.

155
Accommodation
• Effect of spectacle lens positions on accommodative demand.
• Example:
• A hypermetrope wearing +8.00D at 10mm requires 2.36D of
accommodation to see at 50cm.
• If the frame is worn at 20mm, only +1.99D of accommodation is
required

156
Sphero-cylinders
• Effectivity and sphero-cylinders
• Example: A prescription of +11.00/+3.00 x 180 is worn at a
vertex distance 12mm. What will be the required prescription
be at a vertex distance of 15 mm?
• Answer: +10.65/+2.79 x 180
• Note the cylindrical component has changed.

157
Sphero-cylinders
• Near vision through high-powered sphero-cylinders
• Example: A prescription of +4.00/+4.00 x 180 is worn at a vertex
distance 20mm. How much accommodation is required to see
an object at 20cm?
• Answer: +5.86/+1.09 x 180
• Note: accommodative demand has a cylindrical component

158
Prism
• Differential vertical prismatic effects at near with aniseikonia
– Because the visual point on the lens often changes by 1.5cm
from distance or near viewing, a vertical prism imbalance will
occur with even mild vertical aniseikonia.
– For example, if the aniseikonia in the vertical meridian is 1D,
the resultant prismatic effect is 1.5 (Prentice’s Rule)

159
Aniseikonia
– Definition: Difference in size of ocular images between eyes
– Ocular image includes retinal image plus its representation in the
visual cortex
– Absolute size is impossible to measure, so relative sizes are
important
– Significant effects if size is different by >1%
– Difficulties in calculation and implementation

160
Leaf Room
– Good aid in understanding aniseikonia
– Use meridional magnifying lenses (size lenses)
– Overall change in image in one eye gives no effect on leaf
room shape
– Vertical magnification- gives ‘induced’ effect, as if horizontal
magnification in the other eye
– 0.25% change is detectable.

161
Formula for Spectacle Magnification
SM = x
1
1 - (t/n) F1
1
1 - d F'v
Influence of varying parameters on SM in approximate order of importance
i) increased power, Fv’ :- increased magnification (+) increased minification (-)
ii) increased vertex distance, d :- increased magnification (+) increased
minification (-) iii) increased surface power F1 :- increased magnification (+)
reduced minification (-)
iv) increased thickness, t :- increased magnification (+) reduced minification (-)
v) higher refractive index, n :- reduced magnification (+) increased minification (-
)

162
Horizontal Magnification
B
C
A
B
D
E
F
G
H
A
C
D
E
F
G
H
L R
Horizontal Magnification in RE
(Equals vertical magnification in LE)

163
Oblique Magnification
C
A
B
D
E
F
G
H
A
B
C
D E
F
G
H
L R

164
RSM and SM
• Relative Spectacle Magnification (RSM) is correct theoretical
method for estimating image differences between eyes.
• In reality, we only have Spectacle Magnification (SM) that we
can clinically calculate.

165
Clinical Assessment
• There are a number of techniques for measuring
aniseikonia that have been devised, but none are widely
used.
• Calculations of spectacle magnification give us clues as to
the important components in determining retinal image size

166
Conclusion
• This module has introduced the principles of ophthalmic
optics basic theories of spectacle lenses. Modules 7 and 8
will expand on these principles and theories and apply
them to clinical practice.

167
References
Freeman, M. Optics 10th Edition. Butterworth-Heinemann London
Jalie M (1999). Ophthalmic Lenses and Dispensing Butterworth- Heinemann, Oxford
Jalie M (1988). The Principles of Ophthalmic Lenses. 4th ed. Association of British Dispensing Opticians, London..
Wilson D. Practical Optical Dispensing. Strathfield: OTEN-DE; 1999.

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