spherical and astigmatic lenses

1. Spherical & Cylindrical Lenses
Raju Kaiti, M. Optometry
Consultant Optometrist
Nepal Eye Hospital (NEH)

2. Lens: Definition
• A refracting media enclosed by two refracting
surfaces
– At least one curved surface

3. Types of Lens
• Contour of Lens Surface
– Depends upon the surfaces of revolution, which are formed by
rotating a plane curve about an axis with in its plane
• Lens Types
• Spherical
– Generated by rotating a circle or an arc about one of its
diameter as an axis of rotation
• Astigmatic Lenses
– Generated by rotating a straight line about another straight
line that is parallel to axis of rotation.
– Types
» Cross Cylinders
» Spherocylindrical
» Toric, Toroidal,
• Aspheric

4. Spherical Lenses
 Rotation of an arc whose axis passes thru the center of
the arc
 Earliest ophthalmic lens:- Biconvex
 Biconcave & Biconvex-Popular
 Later manufacturing skills improved
Flat lenses were produced
Plus lens having flat back surface
Minus lens having flat front surface

5. Spherical lens -Spherical lens -
constant curvature at all meridiansconstant curvature at all meridians
Spherical lens -Spherical lens -
constant curvature at all meridiansconstant curvature at all meridians
CC

6. Spherical Lens Forms
– Plano-concave
– Plano-convex
– Bi-convex
– Bi-concave

7. Spherical lens forms
– Periscopic (rarely used today)
Difficult to manufacture (steep
curvature)
• One of the first bent lenses
(convex front surface, concave
back surface)
• Plus Rx:
– -1.25 D base curve on back
• Minus Rx:
– +1.25 D base curve on front

8. Spherical lens forms
– Meniscus (Nitsche and Gunter)
• More Bent with +/- 6.00D BC
• Plus Rx:
– Base Curve = -6.00 Ds on the back
• Minus Rx:
– Base Curve = + 6.00 Ds on the front
BC = -6.00
BC = +6.00

9. Transverse Movements with
Lenses
• Convex Lens:
– Images of the objects swim opposite to lens
movement
• Concave Lens
– Images of the objects swim towards the movement of
the lens
• Best perceived with cross-line at a distance

10. Spherical Lens Identification

11. Convex lenses
• Magnification
• Thick center
• Thin periphery
• “Against Motion”
Concave lenses
• Minification
• Thick periphery
• Thin center
• “With Motion”

12. Power measurement:
Neutralization
Foci meter

13. Cylindrical Lenses
• Lens obtained by cutting a section
from a cylinder of a glass.
• Two principal meridians
– Power
– Axis
• A cylinder is produced by rotating
a st. line around an axis parallel to
first.
• Cylinder is plane along its axes
with maximum curvature at rt.
angle to it

14. Types of cylindrical lenses
• Plano cylinders
• Cross cylinder
• Sphero – cylinders
• Toric lenses
Axis meridian = the meridian of least curvature
Power meridian = the meridian of maximum curvature

15. Types
Plano Cylinder Spherocylinder
Toric

16. How to check for a cylindrical lens?
• Rotate the lens before a cross chart
– Either it’s a ‘+’ or ‘-‘ cylindrical lens,
• the vertical & horizontal lines move in opposite
direction – “Scissor reflex”

17. Characteristic features of Cylindrical lenses
• Two cylinders placed together with their cylindrical
axes parallel to one another
– gives a single cylinder whose power is equal to
sum of cylindrical lenses
• Two cylinders of equal and opposite sign with their
axes parallel
– neutralize each other.
• Two identical cylinders placed together with their axes
at rt. Angle to one another
– are equivalent to a sphere where power is equal to
either of cylinder.

18. Characteristic features of Cylindrical lenses
• Any single cylinder can be replaced
– by a sphere of the same power as the cylinder
combine with cylinder of equal but opposite
power
– so that of the original cylinder with its axis
perpendicular to the axis of first.
• Two unequal cylinders placed together with their
axes at rt. Angle to one another
– can be replaced by a sphere & a cylinder.

19. Clinical Implication
• Front Toric surface Vs Back toric surface
– A toric back surface decreases the amount of
meridional magnification compared with that obtained
by a toric front surface.
• The standard procedure
– is to grind sphere on front surface and toric back
surface.

20. Cylindrical lens- constant curvatureCylindrical lens- constant curvature alongalong each meridian; varyingeach meridian; varying
curvaturescurvatures betweenbetween meridiansmeridians
Cylindrical lens- constant curvatureCylindrical lens- constant curvature alongalong each meridian; varyingeach meridian; varying
curvaturescurvatures betweenbetween meridiansmeridians
CC
Principal meridians -Principal meridians - axisaxis andand powerpower
Oblique meridian -Oblique meridian - “power”:“power”:
FFαα == FFCC sinsin22
αα
αα

21. Aspheric lens- varying curvatureAspheric lens- varying curvature alongalong each meridianeach meridian

22. Crossed-Cylinder Lenses
• Has plus cylinder ground on the front surface and minus
cylinder ground on the back surface, with the axis 90°
apart
• Available: +/- 0.25, 0.50, 0.75, 1.00

23. Spherocylindrical LensSpherocylindrical LensSpherocylindrical LensSpherocylindrical Lens

24. Astigmatic Lenses
• Example : +3.50 –150 x 180
– Astigmatic : No point focus
• Forms Line focuses
– Interval of Sturm
+3.50 D+3.50 D
+2.00 D+2.00 D
28.5 cm
28.5 cm
50 cm50 cm

25. Astigmatic Lenses
• Plano-cylinder
– one plano surface, one cylindrical surface
plpl
plpl
FrontFront
plpl
-3-3
BackBack
plpl
-3-3
Compound:Compound:
pl -3.00 x090pl -3.00 x090

26. Astigmatic Lenses
• Toric
– one spherical surface, one cylindrical surface
with no plano meridian
+5+5
+5+5
FrontFront
-1-1
-4-4
BackBack
+4+4
+1+1
Compound:Compound:
+4.00 -3.00 x090+4.00 -3.00 x090

27. Astigmatic Lenses
• Bi-toric
– two cylindrical surfaces
– used in CL’s, but not spectacles
+5+5
+4+4
FrontFront
-3-3
-2-2
BackBack
+2+2
+2+2
Compound:Compound:
+2.00 sph!+2.00 sph!

28. Astigmatic Lenses
• Obliquely crossed cylinders
– Cylinders that are NOT 0 or 90 deg apart
require special solutions

29. Meridian - line along lens surfaceMeridian - line along lens surface
90 90
180 180
CCW
0
Specification of Cylinder AxisSpecification of Cylinder Axis

30. Transposition of
Spherocylindrical lenses
• Sph-cyl form from
Cross cyl.
– Write either cross cyl.
As the sphere
– To find the cylinder
• Substract the cross cyl.
chosen as the sphere
form the other cyl.
– Axis of sph. Cyl.Form
• same as axis of cross
cyl. Not chosen as
sphere.

31. Transposition of
Spherocylindrical lenses
• One sphero-cyl. Form
new Sph. Cylinder
– New Sphere = algebric
sum of old sphere and cyl.
Form
– New cyl. = old cyl with sign
change
– New axis at rt. Angle to the
old axis

32. Spherical Lens Identification & Marking
1. Straight edge test:
– Place the lens on straight surface.
– If lens is plano-
• Equal amount of light escapes beneath the edges
– If lens is cylindrical –
• Unequal light escape from the lens edge.
– At the Edge : ? Plus & Minus Lenses
•Cylindrical Lenses??

33. Neutralization:
• Method-1:
– Neutralize the two principle meridians separately by
placing lens of opposite power on the front surface
from trial lenses
– First neutralize axis meridian and then power
meridian.
– Subtract two meridians power = cylinder power.

34. Neutralization:
• Method-2
– Neutralize the lens along the axis meridian
(minimum minus or maximum plus)
• record the sphere power as before.
– Keep the neutralizing sphere in place &
• add known cylinder with their axis parallel to
the axis of the unknown lens until the lens is
neutralized along its power

35. Toric Lenses

•Introduction
•Types
•Transposition
•Identification
•Lens Form
•Lens Measure

36. Toric Surfaces
• Torus: Latin word
– Ring shaped moulding at the base of ancient stone
column.
• Toric lens –
– Meniscus form lens having –
• - 6.00 Ds back surface power for plus lenses or a
front surface power of +6.00 Ds for minus lenses.

37. Toric Surfaces
• Toric lens: Two principle
curvature on one surface
1. Base Curve
lowest numerical curve
1. Cross curve
– highest numerical curve
• Other surface = spherical
curve (convex/concave)
Base curve @ H / Cross curve @VBase curve @ H / Cross curve @V
Sphere curveSphere curve

38. Types of Toric surfaces
1. Tyre formation:-
– surface formed by rotating a
generating curve (AVR)
• such that the axis of rotation AaA’
&the vertex (v) lies in opposite side
of the center of curvature ‘C’
– ‘C’does not lie on the axis of
rotation.
– This type of surface is mostly
used in ophthalmic lenses.

39. Types of toric surfaces
2. Barrel formation toroidal
surface:
– The axis of rotation
AaA’, lies between
• the vertex of the
generating arc(AVR) &
• its center (c)

40. Types of toric surfaces
3. Capstan surface formation:
– The axis of rotation AaA’
the center of curvature (C)
of the generating curve lie
on opposite side of vertex
(v).
– This type of surface is
used to form solid bifocals.

41. Transposition
• A toric lens can be written as:
Base curve / cross curve
Sphere curve
• For a given prescription of sphero-cylinder where,
– Base curve=Given
– Cross curve= Base curve + Cylindrical component
– Sphere curve= Spherical Component – Base curve

42. Transposition
• Transposition to toric form when sphere curve given,
– Base curve = Spherical Component – Spherical
Curve
– Cross curve = Base curve + Cylindrical component
• Transposition from toric form to sphero-cylinder form
– Sphere = Base curve + Sphere curve
– Cylinder curve =Cross curve – Base curve
Axis same as axis of cross curve.

43. Lens Form
• Cross cyl form  +8.50 DCXV / +9.25DCXH
• Sph cyl form  +8.50 DS / +0.75 DCXH
+9.25 DS / -0.75 DCXV
• Toric form  With base curves 6.00D
+6.00 DCXV/+6.75 DCXH (1)
+2.50DS
+15.25 DS________
+6.00 DCXV/+6.75 DCXH
 With base curves 6.00D
+14.50 DCXV/+15.25 DCXH
+2.50DS
+6.00 DS________ (1)
+2.50 DCXV/+3.25 DCXH
(1) represents Bi- convex toric

44. Lens Measure
• Instrument
– lens measure, lens clock or lens gauge with three pins.
– Chord length constant for the instrument (h)
– Central pin position- sagitta (S)
• For reading of D, relationship between refracting power, n,
s, h.
– F= 2(n-1)s/h2
• Curve gauge, used to check the curvature of a surfacing
lap.
• Refracting power of each lens surface found & added that
gives approximate power of the lens.

45. Specification of Cylinder Axis
• With the Rule –
– the minus axis is within 30° of the 180° meridian
• Against the Rule –
– the minus axis is within 30° of the 90° meridian
• Oblique cylinder –
– the minus axis is between 30° and 60° or 120° and
150°

46. Prescription Writing
• Spherical power first
• Then, cylinder power
• Then, the cylinder axis
– Eg: + 3.50 Ds / -1.50 Dc @ 1800
• Can be written in
– ? plus-cylinder or minus-cylinder
• Optical cross can help with visualization
Example +2.00Ds /+1.50 Dc x 090

47. Astigmatic Lenses
• Optical cross diagrams
FrontFront BackBack
Total PowerTotal Power
((ApproximateApproximate))
Example +2.00 +1.50 x 090

48. Three-Step Rule for Transposition
1. Add the sphere and cylinder power algebraically
2. Change the sign of the cylinder
3. Rotate the cylinder axis 090°
–Eg:
–Minus Cylinder : + 3.50 Ds / -1.50 Dc @ 1800
–Plus Cylinder : +2.00 Ds/ +1.50 Dc x 0900

49. Pearls of Writing Standard Prescription
1. Right eye (OD) always first
2. Dioptric values always carried to 2nd
decimal point
3. Axis specified in three digits (x 016)
4. Fill in SPH or DS if no cylinder
• ? ? Do not use the degree ( ° ) symbol

50. 50
Astigmatism From Lens Tilt
• Tilting a lens
– new sphere becomes stronger than old
sphere
• minus lens becomes more minus
• plus lens becomes more plus
– cylinder will be induced
• same sign as the sphere
• axis equal to meridian of rotation

51. 51
Pantoscopic TiltPantoscopic Tilt
180180

52. 52
FaceformFaceform
(a.k.a. wrap, goggle effect)(a.k.a. wrap, goggle effect)
9090 9090

53. 53
Astigmatism From Lens Tilt
• Tilting a spherical lens
– new sphere power given by
FIC = FNS tan2
α
FNS = FOS (1+ )
sin2
α
2n
induced cylinder power given by

54. 54
ProblemProblem
A +8.00 D lens has 20 degA +8.00 D lens has 20 deg pantoscopicpantoscopic tilt.tilt.
What is the new power?What is the new power?
F = F
n
= +
N S O S (
s in
)
( ) (
s in
[ . ]
)
1
2
8 0 0 1
2 0
2 1 5 2 3
2
2
+
+
α
.
= +8.31 D

55. 55
ProblemProblem (continued)(continued)
F = F
= + 8 . 3 1
I C N S t a n
( ) t a n
2
2
2 0
α
= +1.10 D
New lens power:
+8.31 +1.10 x180

56. 56
ProblemProblem
A -10.00 D lens has 10 degA -10.00 D lens has 10 deg faceformfaceform tilt.tilt.
What is the new power?What is the new power?
F = F
n
=
N S O S (
s in
)
( ) (
s in
[ . ]
)
1
2
1 0 0 0 1
1 0
2 1 5 2 3
2
2
+
− +
α
.
= -10.10 D

57. 57
ProblemProblem (continued)(continued)
F = F
= 1 0 . 1 0
I C N S t a n
( ) t a n
2
2
1 0
α
−
= -0.31 D
New lens power:
-10.10 -0.31 x090

58. Spherical Equivalent/ Significance
• “Average” power of an ophthalmic lens
– Determined by combining one-half the cylindrical
power with the spherical power
• E.g.: + 4.50 Ds/ -1.50 Dc x 040
• ?? Spherical Equivalent = ??
• Adjusting the cylinder and sphere to ease patient
adaptation to a new spectacle Rx
• Determining the total plus at near
– when a presbyope’s spectacle Rx is changing

59. Positive Vs Negative Toric
• Positive Toric:- Net positive power with constant back
curve and toroidal front curves
• Negative Toric: constant front surface and toroidal back
surface
• All corrected curves spherocyl lenses were designed in
positive toric form
• Modern lenses are redesigned for negative toric

60. Why Negative Toric?
• Easier to incorporate near addition power in the
front surface, cylinder being at the back surface
• Meridianal differences in Spectacle
magnification
• SM=
– Negative toric- F1 constant
• Thickness variations hidden inside frame
1
1-F1(t/n)
1
1-hFv
First negative
toric single
vision lens-
Tillyer
Mastrpiece
Shape Factor Power Factor

61. Aspheric Lenses
• Axially or Rotationally symmetrical
surface
• Rotation of a portion of a parabola
and ellipse or a hyperbola.
• non circular cross section
• Curvature goes on reducing from
center to edges
• Drop in power towards the
periphery

63. • The principal use of aspheric lens design
is the reduction or elimination of optical
aberrations produced by looking through
an ophthalmic lens obliquely.

64. Aspheric Lens-
corrected off axis
astigmatism
Spheric Lens
Off axis astigmatism
present

65. Purposes for using an
Aspheric Design
• To be able to optically correct lens
aberrations
• To allow the lens to be made flatter,
thereby reducing magnification &
making it more attractive.
• To produce a thinner & lighter lens
• To make a lens with progressive optics.

66. • The asphericity is normally provided on front
surface and back surface is reserved for cylinder
if present.
• The asphericity is selected to correct off axis
aberration.
• For plus lens, aspheric surface flattens towards
periphery in oblate ellipse form.
• For minus lens, aspheric front surface steepens
towards periphery in prolate ellipse

67. • By taking high index material
further wt and thickness can
be reduced.
• Use
– High Prescription lenses
≥±4.00 D.
– Funduscopic Lenses
– Low Vision Devices