Transcript Optics I - Department of Applied Physics

Hong Kong Polytechnic University
Cylindrical Lenses and Astigmatism
Cylindrical Lens: Cylindrical lens is a section of a cylindrical rod.
One surface is cylindrical while the opposite is plane.
Optics 1----by Dr.H.Huang, Department of Applied Physics
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AB 
s  s
CL
s
Cylindrical Lenses and Astigmatism
S  S  P
Power of thin lens:
 1
1 
P  n2  n1   
 R1 R2 
P
n2  n1
R
s
Optics 1----by Dr.H.Huang, Department of Applied Physics
s
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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 S=1/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=S+P=1.0 D, so s=1/S=1.0 m to the right of the lens.
The line image length AB 
s  s
 0.25  1.0
CL 
 0.05  0.25 m
s
 0.25
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 S=1/s=1/(0.25) = 4.0 D,
Image vergence S=S+P=+6.0 D, so s=1/S= 0.167 m to the right of the lens.
Line image length AB 
s  s
 25  16.7
CL 
1  1.67 cm
s
 25
(b) Screen at x=s/2=8.35 cm.
Height: h 
 25  8.35
1  1.33 cm
 25
width:
w
waperture
2
 0.5 cm
Optics 1----by Dr.H.Huang, Department of Applied Physics
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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
@ can also be used
to replace X
Optics 1----by Dr.H.Huang, Department of Applied Physics
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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.
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Hong Kong Polytechnic University
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: S=1/s=1/(0.5)=2.0 D
vertical line image vergence S1=S+P1=2+7=+5.0 D, so s1=1/S1=0.20 m
horizontal line image vergence S2=S+P2=2+6=+4.0 D, so s2=1/S2=0.25 m
s2  s1
25  20
CL 
 4.0  1.0 mm
s1
20
s2  s1
25  20
V1V2 
CL 
 4  0.8 mm
s2
25
s  s1
s  s
d
CL  2
CL
s1
s2
H1 H 2 
s 
s
s
s1 
s2 
2s1s2
2  20 25

 22.2 cm
s2  s1
25  20
d  0.44 mm
Optics 1----by Dr.H.Huang, Department of Applied Physics
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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.
Optics 1----by Dr.H.Huang, Department of Applied Physics
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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 S=1/s=1/(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 spherocylindrical prescription.
Consider a “reduced eye”, that is, one whose optical behavior is wellapproximated 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:
CP90=5559=4.0 D
CP180=5557=2.0 D
Example:
Suppose the spectacle powers to be SP, since CP 
SP
1  xSP
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.
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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
Optics 1----by Dr.H.Huang, Department of Applied Physics
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Hong Kong Polytechnic University
Cylindrical Lenses and Astigmatism
Homework:
1. 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.
2. 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.
3. 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.
4. 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?
5. 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.
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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 aberrations—axial object points only
S; LCA
2(b). Oblique aberrations—off-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
LCAlens1  LCAlens2
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Aberration
Monochromatic Aberrations:
i3 i5 i7
Maclaurin’s expansion: sin i  i     ...
3! 5! 7!
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.
S  ay2  by4  cy6  ...
T  ky3
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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.
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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:
CT  3CS
Abbe’s sine condition is used for the elimination of coma
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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.
Optics 1----by Dr.H.Huang, Department of Applied Physics
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Aberration
Petzval, Tangential and Sagittal
Surfaces
PT  3PS
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,
h2
p
2
y
1

x 1 nx f x
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Aberration
Distortion: due to variation in lateral magnification
A rear positioned stop introduces pincushion
distortion.
A front positioned stop introduces barrel distortion.
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