Visual Optics 2005/2006

1
Visual Optics 2005/2006

• Chapter 5
• Astigmatism The Cornea Astigmatic Eyes, Lenses
  Image Formation

2
Astigmatic (toric) lens - Meridional Powers
Fig 5.1, Page 5.1

• Meridional powers vary systematically from
  minimum to maximum.
• For any given meridian, lens power is constant
  across the lens.

3
Attributes of Regular Astigmatism
Page 5.1

• Principal meridians (maximum minimum power) are
  ____________
• Majority of ocular astigmatism is due to
  ____________________ (?n)
• Most toric corneal surfaces are a good
  approximation to regular astigmatic in the
  paraxial zone
• Therefore, most astigmatic eyes have regular
  astigmatism and can be adequately corrected with
  regular astigmatic lenses

4
Page 5.2
Lens Cross Regular Astigmatic Lens

• For most calculations with astigmatic lenses and
  eyes (e.g. effective power, ametropic correction
  based on A Femm ? Feye), lens cross notation
  should be used, NOT spherocylindrical notation.
• Rationale for using lens cross notation for
  calculations ___________________________________
  _____________________________

Fig. 5.2 - Lens cross representation of an
astigmatic lens
5
Page 5.2
Lens Cross Regular Astigmatic Lens

• An exception is determination of cylinder powers
  in non-principal meridians.
• Here, we can use a simple equation based on
  cylinder axis

Fig. 5.2 - Lens cross representation of an
astigmatic lens
6
Astigmatic Lens Powers in Non-Principal Meridians
Page 5.3
? measured from cylinder axis
Fig. 5.3 - Curvature of a plano-cylindrical lens
7
Variant on Example 5.1, using Cylinder Power (F)
only, not R
A plano-cylindrical lens has a power of ?4.50 D
axis 30 Determine cylinder power in the 55?
meridian The 55? meridian is 25? away from 30, so
? 25?
Page 5.4
Being close to the cylinder axis, the 55?
meridian has much lower power than the ?4.50 D
power meridian (120?) Power at 90 (60? from axis)
Power at 110 (80? from axis) Power at 40 (10?
from axis) Power at 70 (40? from axis)
?3.4 D
?0.8 D
?4.4 D
?0.14 D
8
Variant on Example 5.1, using a Spherocylinder
Page 5.4
A spherocylindrical lens has a power of 2.50
?3.75 D axis 90 Determine total lens power in the
30? meridian The 30? meridian is 60? away from
90, so ? 60? Spherical power is constant for
all meridians, so this will simply add on top of
the calculated cylinder power at 30
9
Revision - Standard Axis Notation
Page 5.5
Fig. 5.4 - Standard Axis Notation
Standard Axis notation seen from Practitioner
viewpoint
10
Standard Axis Notation on Trial Frame
Page 5.6
Standard axis notation is the same O.U.
Fig. 5.5 - Standard axis notation used on a
spectacle trial frame.
11
Standard Axis Notation Example
Page 5.6
Fig. 5.7 - A ?2.00 D cylinder with axis oriented
at 90 in front of the right eye.
12
Standard Axis Notation Example
Page 5.7
Oblique astigmatism is typically symmetrical
either side of 90 Parallel axes in an oblique
astigmat are very unusual
Fig. 5.8 - Standard axis notation for oblique
cylinders.
13
Sphere/Cyl Combo. Lens Cross Notation
Page 5.7
Fig. 5.9 - Compound lens and lens cross
representation
14
Sphere/Cyl Combo. Lens Cross Notation
Page 5.8
Fig. 5.10 - Meridional powers of a positive
sphere - negative cylinder combination
15
Origin of Spherocylindrical Notation
Page 5.8
Fig. 5.11 - The original spherocylinder form
16
Section 2  -  The Cornea
17
The Cornea
Page 5.14

• Most important ocular refracting component
• Power 43 D ? carries 2/3 of total ocular power
• Reason for large power contribution is the index
  difference across the anterior corneal surface
  (1.376 ? 1.000) OBJ
• Anterior corneal power (48.83 D) higher than
  total
• Posterior cornea must ? have negative power
• The majority of ocular astigmatism is due to the
  anterior cornea
• This is due to the large ?n, NOT due to higher
  anterior corneal toricity (radius difference
  between PMs)

18
Schematic eye to compare anterior and posterior
corneal contributions to total corneal power?
Page 5.14
19
Exact Eye Thick Lens
1.00
1.336
20
Corneal Surface Powers
Page 5.14
r1 7.7 mm
r2 6.8 mm
Fig 5.12 Page 5.15
21
The Cornea as a Thick Lens
Page 5.15
F2 ?5.88 D
F1 48.83 D
22
Corneal Back Vertex Power
Page 5.15
F2 ?5.88 D
F1 48.83 D
23
The Cornea
Page 5.15

• A good way to demonstrate the importance of a
  high ?n across the anterior corneal surface is to
  change the object medium
• Swimming with your eyes submerged in water, the
  new object medium has an index 1.333.
  Prediction??? New corneal power?

• With posterior corneal surface power unchanged in
  water (?5.88 D), the new equivalent power of the
  cornea is zero
• The eye becomes effectively 43 D hyperopic
  underwater
• Wearing a swimming mask restores the normal
  air/cornea interface

24
Corneal Surface Curvatures Powers
25
Example 5.6
Page 5.16
A patients right cornea differs from the
Gullstrand Exact Eye cornea in anterior corneal
radius only, which is 0.2 mm steeper. The left
cornea differs in posterior radius only, which is
0.2 mm flatter. Find differences in total
corneal power from the Exact Eye cornea
r2 7.0 mm
r2 6.8 mm
r1 7.7 mm
r1 7.5 mm
26
Page 5.16
(Exact F1 48.83 D)
(Exact Fe 43.05 D)
r2 6.8 mm
r1 7.5 mm
The right cornea is 1.31 D (3.04) stronger than
the Exact Eye cornea.
F2 ?5.88 D
F1 50.13 D
27
Page 5.16
(Exact F2 ?5.88 D)
(Exact Fe 43.05 D)
r2 7.0 mm
r1 7.7 mm
The left cornea is 0.17 D (0.39) stronger than
the Exact Eye cornea
F2 ?5.71 D
F1 48.83 D
28
Example 5.6
Page 5.16
A patients right cornea differs from the
Gullstrand Exact Eye cornea in anterior corneal
radius only, which is 0.2 mm steeper. The left
cornea differs in posterior radius only, which is
0.2 mm flatter. Find differences in total
corneal power from the Exact Eye cornea
r2 7.0 mm
r2 6.8 mm
r1 7.7 mm
r1 7.5 mm
F2 ?5.88 D
F2 ?5.71 D
F1 50.13 D
F1 48.83 D
OD Cornea
OS Cornea
Fe 44.36 D ? 1.31 D ?
Fe 43.22 D ? 0.17 D ?
29
Example 5.6
Page 5.16
A patients right cornea differs from the
Gullstrand Exact Eye cornea in anterior corneal
radius only, which is 0.2 mm steeper. The left
cornea differs in posterior radius only, which is
0.2 mm flatter. Find differences in total
corneal power from the Exact Eye cornea
OD Cornea
OS Cornea
Fe 44.36 D ? 1.31 D ?
Fe 43.22 D ? 0.17 D ?
For a given change in radius of curvature, the
anterior cornea produces an approximately 8 times
greater power change than the posterior cornea
30
Example 5.7
Page 5.17
Using the same numbers as the previous example,
but this time to induce 0.2 mm toricity in the
corneal surfaces
O.D.
O.S.
0.17 D astigmatism
1.36 D astigmatism
31
Corneal Astigmatism
Page 5.17
O.D. corneal radii
O.S. corneal radii
8.1 mm
6.8 mm
7.7 mm
6.4 mm
6.8 mm
7.7 mm
7.7 mm
6.8 mm
AnteriorCornea
PosteriorCornea
AnteriorCornea
PosteriorCornea
How much corneal astigmatism OD vs OS? (A) 2.41
D O.D., 2.41 D O.S. (B) 1.31 D O.D., 0.17 D
O.S. (C) 2.41 D O.D., 0.36 D O.S. (D) 0.36 D
O.D., 0.36 D O.S. (E) 2.41 D O.D., 1.31 D
O.S. (F) 1.31 D O.D., 2.41 D O.S.
32
Classification of Astigmatism by Ocular PMs
Page 5.18
33
Classification of Astigmatism by Ocular PMs
Page 5.18
Fig. 5.13 - Anterior corneal (a) radius, (b)
power vs. age
34
Classification of Astigmatism by Ocular PMs
Page 5.18

• With-the-Rule (wtr) Astigmatism
• Power greater in the vertical meridian (90 ? 15?)
• In early decades, average astigmatic cornea
  steeper at 90? wtr corneal astigmatism (highest
  magnitudes up to age 30)
• Because anterior cornea is predominant cause of
  ocular astigmatism ? most young astigmats have
  wtr total ocular astigmatism

35
Classification of Astigmatism by Ocular PMs
Page 5.18

• Against-the-Rule (atr) Astigmatism
• Power greater in the horizontal meridian (180 ?
  15?)
• Difference between average H and V corneal power
  decreases after 30
• This produces an increase in incidence of atr
  corneal astigmatism
• In 70s, mean horizontal power becomes steeper
  than vertical, making atr corneal astigmatism
  more common

36
Classification of Astigmatism by Ocular PMs
Page 5.18

• Overall trends With-the-Rule versus
  Against-the-Rule Corneal Astigmatism
• Over 90 of astigmatic infants have wtr
  astigmatism
• By age 50, this has decreased to 80
• Wtr rapidly decreases to be overtaken by atr from
  age 50 - 75

37
Total Ocular Astigmatism vs. Age
Wtr-atr crossover at age 45
Fig. 5.14
Page 5.20
38
Corneal vs. Total Ocular Astigmatism
Fig. 5.14
Page 5.20
Crossover occurs much earlier for total ocular
astigmatism than corneal Difference is due to
intraocular astigmatism (physiological
astigmatism) Intraocular astigmatism averages
0.50 to 0.75 D in non-oblique astigmats This
skews the incidence of total ocular astigmatism
toward atr, and produces the earlier crossover
between wtr/atr vs. corneal astigmatism
39
Corneal vs. Total Ocular Astigmatism
Fig. 5.14
Page 5.20
Oblique astigmatism much less common than wtr or
atr Lowest incidence in early life remains
constant after age 30 Oblique wtr means greatest
power closer to vertical than horizontal
40
The Tear Film as the Anterior Corneal "Surface"
Page 5.22
Fig. 5.15 - The tear film and cornea can be
treated as two thick lenses separated by air
41
The Tear Film as the Anterior Corneal "Surface"
Page 5.22
The tear film is very thin and conforms exactly
to anterior corneal shape It therefore has
negligible effect on anterior corneal power
42
Section 3 Astigmatic Eyes, Lenses, Image
Formation
43
Page 5.29
Astigmatic Image Formation
44
Astigmatic Image Formation
Page 5.29
Fig. 5.17 - Reduced eye (accommodation relaxed)
showing astigmatic image formation. For parallel
incident light (distant object), focal lines
coincide with the second principal foci of the
principal meridians. y pupil diameter.
45
Astigmatic Image Formation
Page 5.29
The astigmatic reduced eye is represented by a
toric reduced surface coincident with the
pupil Reduced surface power is always positive in
both PMs (/ toric) Ocular astigmatism usually
arises predominantly from the anterior cornea,
but the reduced eye representation shows only
total astigmatism The astigmatic image of a point
is the paraxial idealization two focal lines and
a circle of least confusion at the dioptric
center of the IO Sturm
46
Positions Diameters of Focal Lines COLC
47
Positions Diameters of Focal Lines COLC
Page 5.31
Fig 5.18 - (a) Path of parallel incident light
rays refracted through an astigmatic eye (or /
lens) with greatest power in the vertical
meridian.
Figure shows with-the-rule astigmatism (greater
power in the 90? PM) Anterior focal line is
horizontal (? to more powerful PM) The astigmatic
image of a point is the paraxial idealization
two focal lines and a circle of least confusion
at the dioptric center of the IO Sturm
48
Positions Diameters of Focal Lines COLC
Page 5.31
The width of the horizontal focal line is
produced by all the vertical sections across the
width of the pupil Likewise, the height of the
vertical focal line is produced by all the
horizontal sections from top to bottom of the
pupil
49
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50
Positions Diameters of Focal Lines COLC
Page 5.31
Fig 5.18 - (a) Path of parallel incident light
rays refracted through an astigmatic eye (or /
lens) with greatest power in the vertical
meridian. (b) Refracted ray paths for the two
principal meridians superimposed in a single
plane.
Use this construction to derive equations for
focal line length and COLC diameter. All based on
similar triangles between pupil diameter and
focal line/COLC length
51
Positions of Focal Lines COLC
Page 5.31
Distant object
52
Positions of Focal Lines COLC
Page 5.31
Distant object
53
Diameters of Focal Lines COLC
Page 5.31
?
54
Diameters of Focal Lines COLC
Page 5.31
?
55
Diameters of Focal Lines COLC
Page 5.31
?
Distant object
56
Diameters of Focal Lines COLC
Page 5.31
?
?
Distant object
57
Diameters of Focal Lines COLC
Page 5.31
?
58
Diameters of Focal Lines COLC
Page 5.31
?
Distant object
59

• Example 5.9
• An astigmatic eye, represented in reduced eye
  form, has meridional powers of 64 D along the
  75? principal meridian and 61 D along the 165?
  principal meridian.
• For a 4 mm pupil and a distant point source,
  characterize the image quantitatively.
• (b) What type of astigmatism does this eye have?

Page 5.35
(a) F1 75? PM F2 165? PM, L 0 so
use power terms F, not L?
Answer (b) first Higher power at 75 than 165 ?
with-the-rule astigmatism
60
Positions FLs COLC
Page 5.35
(a) F1 (75? PM) 64 D F2 (165? PM) 61 D

f?1
??C
f?2
F?1
F?2
61
Diameters FLs COLC
Page 5.35
(a) F1 (75? PM) 64 D F2 165? PM, 61
D
f?1
??C
f?2
F?1
F?2
62
Page 5.37
Effect of a Stenopaic Slit on the Astigmatic
Image
63
Effect of a Stenopaic Slit on the Astigmatic
Image
Page 5.37
Fig. 5.19
64
Effect of a Stenopaic Slit on the Astigmatic
Image
Page 5.38
Fig. 5.20
65
Fig. 5.21 Page 5.39
Effect of a Stenopaic Slit on the Astigmatic
Image
66
Image Formation for an Extended (Finite) Object
Page 5.40
Fig. 5.22 - Formation of astigmatic focal lines
by three points (A, B and C) on an extended
object
67
Image Formation for an Extended (Finite) Object
? moving closer to real world objects
Page 5.40
FV
FH
A
B
C
ASTIGMATIC REDUCED SURFACE
Deviation at reduced surface Snells Law n sin i
n? sin i ?
EXTENDED OBJECT
i ? 0.75 i
68
Image Formation for an Extended (Finite) Object
Page 5.40
C?V
FV
FH
B?V
A?V
ASTIGMATIC REDUCED SURFACE
EXTENDED OBJECT
If retina at F?V ? see three horizontal lines as
image
69
Image Formation for an Extended (Finite) Object
Page 5.40
C?H
C?V
FV
FH
B?H
B?V
A?V
A?H
ASTIGMATIC REDUCED SURFACE
EXTENDED OBJECT
If retina at F?H ? see three vertical lines as
image
70
Image Formation for an Extended (Finite) Object
Page 5.40
C?H
C?C
C?V
FV
FH
B?H
B?V
B?C
A?V
A?C
A?H
ASTIGMATIC REDUCED SURFACE
EXTENDED OBJECT
If retina at L?C ? see three COLCs as image
71
Example 5.10 - Image Formation Extended Object
Page 5.42
Astigmatic Eye with 2 D astigmatism
Eye viewing a finite (extended) distant object.
Describe the retinal image if reduced axial
length is (a) 21.51 mm (b) 22.22 mm (c) 21.86 mm
Focus of vertical meridian
Focus of horizontal meridian
Dioptric midpoint focal plane for COLCs
72
Example 5.10 - Image Formation Extended Object
Astigmatic Eye with 2 D astigmatism
EXTENDED OBJECT
73
Example 5.10 - Image Formation Extended Object
f ?90 21.51 mm f ?180 22.22 mm ? ?COLC
21.86 mm
Page 5.42
(a) Reduced axial length 21.51 mm Coincides with
focal plane for vertical meridian ? each image
point stretched out into a horizontal focal line
Image for Retina at F?90
74
Example 5.10 - Image Formation Extended Object
f ?90 21.51 mm f ?180 22.22 mm ? ?COLC
21.86 mm
(b) Reduced axial length 22.22 mm Coincides with
focal plane for horizontal meridian ? each image
point stretched out into a vertical focal line
Image for Retina at F?180
75
Example 5.10 - Image Formation Extended Object
f ?90 21.51 mm f ?180 22.22 mm ? ?COLC
21.86 mm
(c) Reduced axial length 21.86 mm Coincides with
COLC plane (dioptric midpoint of focal planes ?
each image point becomes a circle (COLC)
Image for Retina at L?COLC
76
Example 5.10 Image Formation - Finite Object
EXTENDED OBJECT
Retina at F?90
Retina at F?180
Retina at L?COLC
77
Example 5.10 Image Formation - Finite Object
EXTENDED OBJECT
Retina at F?90
Retina at F?180
Retina at L?COLC
78
Example 5.10 Image Formation - Finite Object
EXTENDED OBJECT
Retina at F?90
Retina at F?180
Retina at L?COLC
79
Clinical Classification of Regular Astigmatism
Fig. 5.24 Page 5.45
80
Example 5.11 Correcting Astigmatism
81
Example 5.11
Fig 5.24 Page 5.45
82
Fig 5.24 Page 5.45
Example 5.11 Correcting Astigmatism
?5
?3
180? meridian A    Femm  ?  Fe    60  ?  63
    ?3 D myopia (refractive).
90? meridian A    Femm  ?  Fe    60  ?  65 
   ?5 D myopia (refractive).
83
Fig 5.24 Page 5.45
Example 5.11
Add ?3 DS in front of eye. What does the patient
then see on the VA chart?
84
Fig 5.24 Page 5.45
Example 5.11
Add ?5 DS in front of eye. What does the patient
then see on the VA chart?
85
One of the two appearances below is the valid
starting point for an all focal-line based
astigmatic subjective refraction. Which one?
86
Clinical Aspects of Astigmatism
Page 5.47

• In real-world vision, looking at complex objects,
  the uncorrected astigmat will not always be
  focusing focal lines or COLCs on the retina
• All other parts of the interval of Sturm are
  elliptical in cross-section
• ? A feature of uncorrected astigmatic vision is
  elliptical elongation or distortion of images
• This effect increases with magnitude of
  (uncorrected) astigmatism and with pupil diameter

87
Accommodation in the Uncorrected Astigmat
Page 5.47

• To obtain optimum uncorrected vision, astigmats
  will (subconsciously) try to place the COLC on
  the retina
• In orientation-dominant objects, the astigmats
  visual system may try to place one or other focal
  line on the retina especially in city
  environments, where much of the landscape is
  horizontals and verticals
• How do our five clinical types of astigmat fare
  when uncorrected?

88
Page 5.47
Take the eyes that we used to define the five
clinical types of astigmatism
89
Fig 5.24 Page 5.45

• The uncorrected CHA can accommodate to bring any
  part of the Interval of Sturm to the retina, IF
• Amplitude of Accommodation is sufficient
• The young CHA should ? have reasonable vision
  when uncorrected
• The downside is fatigue, due to constant
  refocusing to obtain optimum clarity

Accommodation
90
Page 5.47
Take the eyes that we used to define the five
clinical types of astigmatism
91
Fig 5.24 Page 5.45

• The uncorrected SHA already has one focal line on
  the retina without accommodation
• Accommodation can bring any other part of the
  Interval of Sturm to the retina
• The demand on Amp Accom is less than for a CHA
  with the same magnitude of astigmatism
• Fatigue is still an issue because any part of the
  IOS can be moved to the retina

Accommodation
92
Page 5.47
Take the eyes that we used to define the five
clinical types of astigmatism
93
Fig 5.24 Page 5.45

• For the uncorrected MxA vision depends in part
  where the COLC is located (relative to the
  retina) in distance vision
• If it is in front, vision will be worse because
  only the posterior FL can be moved to the retina
• If it is on or behind the retina, the patient has
  a choice of COLC or posterior FL

94
Page 5.47
Take the eyes that we used to define the five
clinical types of astigmatism
95
Fig 5.24 Page 5.45

• With the posterior FL on the retina, the SMA
  obtains some clear vision at distance
• Accommodation is no help because it moves the
  entire IOS in front of the retina
• Poor uncorrected vision rather than accommodative
  fatigue is the main symptom for the uncorrected
  myopic astigmat

96
Page 5.47
Take the eyes that we used to define the five
clinical types of astigmatism
97
Fig 5.24 Page 5.45

• Of all uncorrected astigmats, the CMA will have
  the worst distance vision, but no problem with
  accommodative fatigue

98
Vision in Astigmatism (w BVS) vs. Spherical
Ametropia
Fig. 5.25 Page 5.48
Uncorrected spherical myope
Vision in the uncorrected 2D myope is identical
to that of the 4 D astigmat with COLC on the
retina COLC size is the basis for predicting
magnitude of astigmatism Move the COLC to the
retina with sphere. Worse vision correlates with
higher astigmatism
Uncorrected astigmat with COLC on retina
99
Clinical Aspects of Astigmatism
Page 5.49
Importance of Axis Direction

• For general distance vision in city/indoor
  environments the wtr and atr astigmat should be
  better off than the oblique astigmat because much
  of the environment is made up of horizontals and
  verticals
• Reading performance in the uncorrected astigmat
  will be better if the patient can move vertical
  focal lines to the retina

100
Fig. 5.26, page 5.50
(a) Emmetropia
(b) Uncorrected astigmatism horizontal FLs on
retina
(b) Uncorrected astigmatism vertical FLs on
retina
101
Who will have an easier time reading (near
vision) a 2 D wtr CMA or 2 D atr CHA, if for
distance vision both have the COLC 3 D from the
retina?
2 D wtr CMA
2 D atr CHA
102
Who will have an easier time reading distant
street signs a young 2 D wtr CHA or 2 D atr CHA,
if for distance vision both have the COLC 3 D
behind the retina?
2 D wtr CHA
2 D atr CHA
103

• end

104
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105
Using the Visual Axis, the estimate tends to be
high. The currently accepted value for ?
(pupillary and PLS) is 2.75? per ¼ mm
displacement