Hong Kong Polytechnic University Optics 1----by Dr.H.Huang, Department of Applied Physics1...

Optics 1----by Dr.H.Huang, Department of Applied Physics 1

Hong Kong Polytechnic University

Cylindrical Lens: Cylindrical lens is a section of a cylindrical rod. One surface is cylindrical while the opposite is plane.

Cylindrical Lenses and Astigmatism


Hong Kong Polytechnic University Cylindrical Lenses and Astigmatism

CLs

ssAB

s s

PSS

2112

11

RRnnP

Power of thin lens:

R

nnP 12



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=(n2n1)/R= (1.51)/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

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

(b) Screen at x=s/2=8.35 cm.

Height: width:

m25.005.025.0

0.125.0

CLs

ssAB

cm67.1125

7.1625

CLs

ssAB

cm33.1125

35.825

h cm5.02

aperture w

w



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



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

circle of least confusion: the nearest approximation to a focused image conjugate to a point object.

interval of Sturm

conoid of Sturm



Example: Let a composite cylindrical lens have perpendicular powers of +6.0180 and +7.090. 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 mm0.10.4

20

2025

1

1221

CLs

ssHH

mm8.0425

2025

2

1221

CLs

ssVV

s s1s2

cm2.222025

252022

12

21

2

2

1

1

ss

sss

CLs

ssCL

s

ssd

mm44.0d

s



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.



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 4.045/2.0135. 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.



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: CP90=5559=4.0 D CP180=5557=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 2.05/2.15180 and 4.20/+2.1590.

xSP

SPCP

1



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



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 +5.0090 and +10.00180, 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 +3.00/5.00120. Determine the power cross and Sturm interval of the lens for a point object 40 cm from the lens.



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

Aberration



Chromatic Aberrations: LCALCA is the axial (longitudinal) distance between two focal points for a given wavelength range.

Chromatic Aberrations: TCALinear 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.

Aberration

chromatic aberration of the eye



21 lenslens LCALCA

Aberration

Achromatic Doublet



Monochromatic Aberrations:

Maclaurin’s 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.

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sin753

iii

ii

Aberration

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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.

Aberration



Tangential coma and Sagittal coma satisfies:

Abbe’s sine condition is used for the elimination of coma

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

ST CC 3

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.

Aberration




The relative position of the image points formed from axis and off-axis object points.

The distance TTTS 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.

Aberration




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.

Aberration



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,

y

x xx fn

hp

1

2 1

2

Aberration

Petzval, Tangential and Sagittal Surfaces

PS3PT



Distortion: due to variation in lateral magnification

A rear positioned stop introduces pincushion distortion.

A front positioned stop introduces barrel distortion.

Aberration

Hong Kong Polytechnic University Optics 1----by Dr.H.Huang, Department of Applied Physics1...

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Transcript of Hong Kong Polytechnic University Optics 1----by Dr.H.Huang, Department of Applied Physics1...