Optics I

1
Cylindrical Lenses and Astigmatism
Cylindrical Lens Cylindrical lens is a section
of a cylindrical rod. One surface is cylindrical
while the opposite is plane.
2
Cylindrical Lenses and Astigmatism
Power of thin lens
s
s?
3
Cylindrical Lenses and Astigmatism
Example A thin plano-cylindrical lens in air has
a radius of curvature of 10 cm, a refractive
index of 1.50 and an axial length of 5 cm. Light
from a point object is incident on the convex,
cylindrical surface from a distance of 25 cm to
the left of the lens. Find the position and
length of the line image formed by the lens.
Solution Object vergence S1/s 1/(?0.25)
?4.0 D Power of lens P(n2?n1)/R
(1.5?1)/0.1 0.5/0.1 5.0 D Image
vergence S?SP1.0 D, so s?1/S?1.0 m to the
right of the lens. The line image length
Example A thin, cylindrical lens of 10.0 D
power and a vertical cylinder axis is located 25
cm from a point source of light. A square
aperture 1 cm on a side is placed directly in
front of the lens. (a) Describe the image of the
point source formed by the lens. (b) Describe the
light pattern on a screen positioned halfway
between lens and line image. Solution (a)
Object vergence S1/s1/(?0.25) ?4.0 D,
Image vergence S?SP6.0 D, so s?1/S? 0.167 m
to the right of the lens. Line image
length (b) Screen at xs?/28.35 cm.
Height
width
4
Cylindrical Lenses and Astigmatism
Power Cross An astigmatic eye, while possesses
predominately spherical optics, might have a
cylindrical lens component whose axis can be in
any direction. Convention angles are measured
counterclockwise from the positive x-axis. The
coordinate system is the one seen by the
examiner, not by the patient.
use 180 instead of 0 to avoid confusion with zero
power
_at_ can also be used to replace X
5
Cylindrical Lenses and Astigmatism
Combining Cylindrical Powers Two cylindrical
lens in contact powers along parallel axes
simply add algebraically and powers along
perpendicular axes remain independent of one
another.
circular aperture
square aperture
interval of Sturm conoid of Sturm
circle of least confusion the nearest
approximation to a focused image conjugate to a
point object.
6
Cylindrical Lenses and Astigmatism
Example Let a composite cylindrical lens have
perpendicular powers of 6.0?180 and 7.0?90. A
point object is situated at 50 cm from the lens.
Line images then form as follows A vertical line
due to the 7.0 D power and a horizontal line
image due to the 6.0 D power. Assume a circular
aperture of 4.0 mm diameter (Under bright
lighting conditions this is roughly the diameter
of the pupil of the eye, which serves as the
limiting aperture for the eye.). Determine the
lengths of the two line images and describe the
circle of least confusion. Solution Object
vergence S1/s1/(?0.5)?2.0 D vertical
line image vergence S1?SP1?275.0 D, so
s1?1/S1?0.20 m horizontal line image
vergence S2?SP2?264.0 D, so s2?1/S2?0.25
m
s?
s
s1?
s2?
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Cylindrical Lenses and Astigmatism
Example What combination of a spherical and a
cylindrical lens produces a lens with vertical
power axis of 10.0 D and horizontal power axis
of 3.0 D? Solution The cylindrical lens
may have one axis at zero power while the
spherical lens must have identical powers along
both axes.
8
Cylindrical Lenses and Astigmatism
Example Consider a toric lens whose two
surfaces are spherical and toroidal. The
spherical side has a power of 6.0 D, while the
toroidal side is described by 4.0?45/?2.0?135.
Determine (a) the power cross of the lens, and
(b) the interval of Sturm for a point object at
50 cm from the lens. Solution Object
vergence S1/s1/(?0.5)?2.0 D
The 4.0 D power produces an image vergence of
2.0 D and the image distance is 50 cm. Image is
tilted at 135? from the horizontal. The 10 D
power produces an image vergence of 8.0 D and
the image distance is 12.5 cm. Image is tilted at
45? from the horizontal. The interval of Sturm is
37.5 cm.
A toroid with two circular cross sections of
radii r1 and r2.
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Cylindrical Lenses and Astigmatism
Ocular Astigmatism Astimagtism occurs in the eye
due to an additional cylindrical curvature in
addition to the spherical curvature. The
correction can be made in the form of a
sphero-cylindrical prescription. Example
Consider a reduced eye, that is, one whose
optical behavior is well-approximated by
refraction at a single surface at the cornea,
separating air from the vitreous fluid of
refractive index 4/3. The axial length of this
eye is 24.24 mm and its measured powers along the
vertical and horizontal axes are 59 D ad 57 D,
respectively. What spectacle correction is
required for good, distant vision if the
correcting lens is to be 12 mm in front of the
cornea? Solution To form an image of a
distant object on retina, S?n/s?55.0 D
The contact lens correction power at the cornea
should be CP9055?59?4.0 D
CP18055?57?2.0 D Suppose the
spectacle powers to be SP, since we
obtain SP90?4.20 D and SP180?2.05 D. There
are two combinations of a spherical and
cylindrical surface to result in this correction.
In standard notation, they are ?2.05/?2.15?180
and ?4.20/2.15?90.
10
Cylindrical Lenses and Astigmatism
Classification of astigmatism in an
unaccommodated eye Assume the astigmatic line
images are along horizontal and vertical axes and
the vertical line image results from the stronger
convergence. (a) CMA (compound myopic
astigmatism) both lines images fall short of
retina (b) SMA (simple myopic astigmatism)
horizontal line image on retina (c) MXA (mixed
astigmatism) retina between two line images (d)
SHA (simple hyperopic astigmatism) vertical line
image on retina (e) CHA (compound hyperopic
astigmatism) both line images beyond retina
a SHA eye looking at H
a SMA eye looking at H
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Cylindrical Lenses and Astigmatism

• Homework
• Light rays emanating in air from a point object
  on axis strike a plano-cylindrical lens with its
  convex surface facing the object. Describe the
  line image by length and location if the lens has
  a radius of curvature of 5 cm, a refractive index
  of 1.60, and an axial length of 7 cm. The point
  object is 15 cm from the lens.
• A plano-cylindrical lens in air has a radius of
  curvature of 10 cm, a refractive index of 1.50,
  and an axial length of 5 cm. Light from a point
  object is incident on the concave, cylindrical
  surface from a distance of 25 cm to the left of
  the lens. Find the position and length of the
  line image formed by the lens.
• Determine the interval of Sturm for a composite
  cylindrical lens with perpendicularly oriented
  powers, given by 5.00?90 and 10.00?180, when
  the lens is illuminated by rays of light from a
  point source object at 30.00 cm from the lens. A
  circular aperture of 1.00 cm diameter is
  positioned at the lens. Also describe the circle
  of least confusion.
• What combination of a spherical and a cylindrical
  lens produces a composite lens with vertical
  power axis of 6.00 D and horizontal power axis
  of 2.00 D?
• One side of a lens is spherical with a power of
  8.00 D, and the other side is toric, described
  by 3.00?/?5.00?120. Determine the power cross
  and Sturm interval of the lens for a point object
  40 cm from the lens.

12
Aberration
Introduction Aberration departure of an image
from perfection Chromatic Aberration variation
of the refractive index of a material with
wavelength Monochromatic Aberrations deviation
even if monochromatic Classification of
Aberrations 1(a). Due to the material of the
lens (chromatic aberrations) LCA
longitudinal chromatic aberrations TCA
transverse chromatic aberrations 1(b). Due to
the form of the lens (monochromatic aberrations)
S spherical aberration C
coma A oblique astigmatism
P curvature of field (Petzval curvature)
D distortion 2(a). Axial aberrationsaxial
object points only S LCA 2(b).
Oblique aberrationsoff-axis objects points only
TCA C A P D
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Aberration
Chromatic Aberrations LCA LCA is the axial
(longitudinal) distance between two focal points
for a given wavelength range. Chromatic
Aberrations TCA Linear TCA the difference
between the size of the image in red (C) and blue
(F) light. Angular TCA the angle ? between the
emerged red and blue rays.
chromatic aberration of the eye
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Aberration
Achromatic Doublet
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Aberration
Monochromatic Aberrations Maclaurins
expansion The Gaussian theory is the first
order theory. For larger inclination angles,
third order theory has to be used. Departures
from the first order theory are known as
monochromatic aberrations. Spherical
Aberrations (a) Longitudinal and transverse
spherical aberrations. Fp and Fm are the focal
points of the paraxial and marginal rays
respectively. (b) Graphical representation of
the longitudinal spherical aberration.
16
Aberration
Minimize Spherical Aberrations The spherical
aberration is reduced when the convex surface
faces the distant object. In general, spherical
aberration will be minimized when the refraction
is equally shared between the surfaces.
17
Aberration
Coma Coma occurs with light from an off-axis
object point and produces a comet-like spread of
light in the image plane. Off-axis spherical
aberration
Tangential coma and Sagittal coma satisfies
Abbes sine condition is used for the
elimination of coma
18
Aberration
Oblique Astigmatism Oblique astigmatism arises
from the fact that the fans of rays in the two
meridian DE and AC have different angles of
incidence at the lens, giving rise to different
foci along the direction of propagation.
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Aberration
Oblique Astigmatism The relative position of the
image points formed from axis and off-axis object
points. The distance T?TT?S between the two
focal lines in the astigmatic pencil is referred
to as the astigmatic difference in the image.
The difference between the emergent vergences in
the tangential and sagittal meridians is called
the astigmatic error for the particular obliquity
of the incident light.
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Aberration
Oblique Astigmatism T and S shells. The discs
of least confusion lie on the shell marked D.
The magnitude of the astigmatic difference TS
depends on the form of the lens (known as
bending). For object point at fairly small
angular distance from the axis, astigmatism is
less significant than coma, while the reverse is
true at larger angles.
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Aberration
Petzval, Tangential and Sagittal Surfaces
Curvature of Field A planar object is imaged on
a curved surface. Petzval curvature may be
measured in terms of the distance p along the
chief ray between the Petzval surface and the
ideal image plane. The Petzval sum is,
22
Aberration
Distortion due to variation in lateral
magnification A rear positioned stop introduces
pincushion distortion. A front positioned stop
introduces barrel distortion.