TERM PAPER
PHY 153
LENS FORMULA AND ITS APPLICATIONS
ACKNOWLEDGEMENT
Firstly,I , Hita, student of B-TECH(HONRS.)-M-TECH(INT),thanks GOD
without who’s grace, this project work can never be completed.I am also
grateful to our proffessor Mr. Vikas Nanda for his numerous
suggestions,constructive criticisms&encouragement.My grateful thanks to
my brother Daksh and library faculties for their help in writing some
portions of the project work & helping me in the drawing of some
diagrams.I would also like to thank my colleagues Harpreet, Nidhi, Shivi,
Tanya, etc. for numerous stimulating discussions.Finally, I owe a lot to my
family - for their support , love and blessings.
ABSTRACT
Lens Formula(
) has various practical and beneficial uses &
applications. We can derive Lens Formula from the Lens Maker’s Formula
(1/f = (n-1) (1/R1-1/R2). From this formula, we can make the lenses of desired
focal length and use them appropriately .Lens is most important optical
component used in the making of cameras, microscopes, telescope,
projectors, etc. by making the various adjustments of object distance and image
distance from the lens used.
Sr. No.
1LENS?
1(B) 1(C) 1(D) 1(E) 1(F) -
TABLE OF CONTENTS
INTRODUCTION TO THE PROBLEM
5
CONSTRUCTION OF SIMPLE LENSES
TYPES OF SIMPLE LENSES
MAGNIFICATION OF THE LENSES
LENSMAKER’S FORMULA
THIN LENS EQUATION
PAGE No.
5
6
8
10
11
12
1(A) -
WHAT IS
1(G) 1(H) 1(I) 2(A) 2(B) 2(C) 2(D) 3-
IMAGING PROPERTIES
LENS FORMULA
DIAGRAMATIC APPROACH
THICK LENS FORMULA
THIN LENS FORMULA
NEWTON’S FORMULA
USES OF LENS FORMULA
REFERENCES
12
13
14
17
19
20
20
24
1- INTRODUCTION TO THE PROBLEM
To unveil the various facts, we, the curious science people, need different
instruments to think beyond the scope of book and to know more about our
nature. Likewise, to see distant objects like heavenly bodies (e.g. stars, planets),
study micro – organisms and to see the distant objects present on our planet, we
need to make various optical instruments like microscopes, astronomical
telescopes, terrestrial telescopes, etc. In telescopes, objective lens should be of
large focal length and aperture and eye piece of small focal length and aperture
while in case of compound microscopes, we need objective lens of small focal
length and small aperture and eye piece of large focal length. So, for the making
of lenses of different focal length, different refractive indexes, we need Lens
Formula.
Before we discuss Lens Formula, I would like to through some light on the lenses and its characteristics.
1(A) - WHAT IS LENS?
A lens is an optical device with perfect or approximate axial symmetry which
transmits and refracts light, converging or diverging the beam. A simple lens is a
lens consisting of a single optical element. A compound lens is an array of simple
lenses (elements) with a common axis; the use of multiple elements allows more
optical aberrations to be corrected than is possible with a single element.
Manufactured lenses are typically made of glass or transparent plastic. Elements
which refract electromagnetic radiation outside the visual spectrum are also called
lenses: for instance, a microwave lens can be made from paraffin wax.
Lenses can be used to focus light.
1(B) - Construction of simple lenses
Most lenses are spherical lenses: their two surfaces are parts of the surfaces of
spheres, with the lens axis ideally perpendicular to both surfaces. Each surface can
be convex (bulging outwards from the lens), concave (depressed into the lens), or
planar (flat). The line joining the centres of the spheres making up the lens
surfaces is called the axis of the lens. Typically the lens axis passes through the
physical centre of the lens, because of the way they are manufactured. Lenses may
be cut or ground after manufacturing to give them a different shape or size. The ens axis may then not pass through the
physical centre of the lens.
Toric or sphero-cylindrical lenses have surfaces with two different radii of
curvature in two orthogonal planes. They have a different focal power in different
meridians. This is a form of deliberate astigmatism.
More complex are aspheric lenses. These are lenses where one or both surfaces
have a shape that is neither spherical nor cylindrical. Such lenses can produce
images with much less aberration than standard simple lenses.
1(C)-TYPES OF SIMPLE LENSES
Lenses are classified by the curvature of the two optical surfaces. A lens is
biconvex (or double convex, or just convex) if both surfaces are convex, A lens
with two concave surfaces is biconcave (or just concave). If one of the surfaces is
flat, the lens is plano-convex or plano-concave depending on the curvature of the
other surface. A lens with one convex and one concave side is convex-concave or
meniscus. It is this type of lens that is most commonly used in corrective lenses.
If the lens is biconvex or plano-convex, a collimated or parallel beam of light
travelling parallel to the lens axis and passing through the lens will be converged
(or focused) to a spot on the axis, at a certain distance behind the lens (known as
the focal length). In this case, the lens is called a positive or converging lens.
If the lens is biconcave or plano-concave, a collimated beam of light passing
through the lens is diverged (spread); the lens is thus called a negative or diverging
lens. The beam after passing through the lens appears to be emanating from a
particular point on the axis in front of the lens; the distance from this point to the
lens is also known as the focal length, although it is negative with respect to the
focal length of a converging lens.
Convex-concave (meniscus) lenses can be either positive or negative, depending
on the relative curvatures of the two surfaces. A negative meniscus lens has a
steeper concave surface and will be thinner at the centre than at the periphery.
Conversely, a positive meniscus lens has a steeper convex surface and will be
thicker at the centre than at the periphery. An ideal thin lens with two surfaces of
equal curvature would have zero optical power, meaning that it would neither
converge nor diverge light. All real lenses have a nonzero thickness, however,
which affects the optical power. To obtain exactly zero optical power, a meniscus
lens must have slightly unequal curvatures to account for the effect of the lens'
thickness.
1(D) – MAGNIFICATION OF THE LENS
The magnification of the lens is given by:
M=-v/u=f/f-u ,
where M is the magnification factor; if |M|>1, the image is larger than the object. Notice the sign convention
here shows that, if M is negative, as it is for real images, the image is upside-down with respect to the object.
For virtual images, M is positive and the image is upright.
The formulas above may also be used for negative (diverging) concave lens by using a negative focal length (f), but for these
lenses only virtual images can be formed.
1(E) – LENSMAKER’S FORMULA
The focal length of a lens in air can be calculated from the lensmaker's equation:
1/f=(n-1)[1/R1-1/ R2+(n-1)d/nR1 R2],
where
f is the focal length of the lens,
n is the refractive index of the lens material,
R1 is the radius of curvature of the lens surface closest to the light source,
R2 is the radius of curvature of the lens surface farthest from the light source, and
d is the thickness of the lens (the distance along the lens axis between the two surface vertices).
Sign convention of lens radii R1 and R2
The signs of the lens' radii of curvature indicate whether the corresponding surfaces are convex or concave. The sign
convention used to represent this varies, but in this article if R1 is positive the first surface is convex, and if R1 is
negative the surface is concave. The signs are reversed for the back surface of the lens: if R2 is positive the
surface is concave, and if R2 is negative the surface is convex. If either radius is infinite, the corresponding
surface is flat.
1(F) – THIN LENS EQUATION
If d is small compared to R1 and R2, then the thin lens approximation can be made.
For a lens in air, f is then given by
1/f=(n-1)(1/ R1 -1/ R2 )
The focal length f is positive for converging lenses, and negative for diverging
lenses. The value 1/f is known as the optical power of the lens, measured in
dioptres, which are units equal to inverse meters (m ).
Lenses have the same focal length when light travels from the back to the front as
when light goes from the front to the back, although other properties of the lens,
such as the aberrations are not necessarily the same in both directions.
1(G) – IMAGING PROPERTIES
As mentioned above, a positive or converging lens in air will focus a collimated
beam travelling along the lens axis to a spot (known as the focal point) at a
distance f from the lens. Conversely, a point source of light placed at the focal
point will be converted into a collimated beam by the lens. These two cases are
examples of image formation in lenses. In the former case, an object at an infinite
distance (as represented by a collimated beam of waves) is focused to an image at
the focal point of the lens. In the latter, an object at the focal length distance from
the lens is imaged at infinity. The plane perpendicular to the lens axis situated at a
distance f from the lens is called the focal plane.
If the distances from the object to the lens and from the lens to the image are u and
v respectively, for a lens of negligible thickness, in air, the distances are related by
the thin lens formula.
1(H) – LENS FORMULA
The relationship between distance of the object (u), distance of the image (v) and focal
length (f) of the lens is called lens formula or lens equation.
-------- Lens formula
This lens formula is applicable to both convex and concave lenses.
What this means is that, if an object is placed at a distance u (u>f) along the axis
in front of a positive lens of focal length f, a screen placed at a distance v behind
the lens will have a sharp and real image of the object projected onto it.
If u < f, v becomes negative, the image is apparently positioned on the same side
of the lens as the object. Although this kind of image, known as a virtual image,
cannot be projected on a screen.
1(I) – DIAGRAMATIC APPROACH
PRODUCTION OF A REAL IMAGE BY A CONVEX LENS
Figure 1 I(a) shows how a positive lens makes an image. The image is produced by all of the light from each
point on the object falling on a corresponding point in the image. If the arrow on the left is an illuminated
object, an image of the arrow will appear at right if the light coming from the lens is allowed to fall on a piece of
paper or a ground glass screen. A positive lens inverts the image.
PRODUCTION OF VIRTUAL IMAGE BY A CONVEX LENS
. An image will be produced to the right of the lens only if . If , the lens is unable to converge the rays from the
image to a point, as is seen in figure 1I (b). However, in this case the backward extension of the rays
converge at a point called a virtual image, which in the case of a positive lens is always farther away from the
lens than the object. The thin lens formula still applies if the distance from the lens to the image is taken to be
negative. The image is called virtual because it does not appear on a ground glass screen placed at this point.
Unlike the real image seen in figure 1I (a), the virtual image is not inverted.
PRODUCTION OF A VIRTUAL IMAGE BY A CONCAVE LENS
A negative lens is thinner in the center than at the edges and produces only virtual images. As seen in figure 1I(c
, the virtual image produced by a negative lens is closer to the lens than is the object. Again, the thin lens
formula is still valid, but both the distance from the image to the lens and the focal length must be taken as
negative. Only the distance to the object remains positive.
Thus, Lens Formula is valid for both convex and concave lenses for real and virtual objects and images.
For the case of lenses that are not thin, or for more complicated multi-lens optical systems, the same formulas
can be used, but u and v are interpreted differently. If the system is in air or vacuum, u and v are measured
from the front and rear principal planes of the system, respectively.
2(A) – THICK LENS
FORMULA
The above figure is
Figure 2.5, p. 13, from Schroeder (1987). Applying the equation of
paraxial refraction with
(air) to each surface
gives
(1)
(2)
,
Using
Adding (1)
(3)
and (3) gives
For
(4)
conjugate points,
As is derived
(5)
by Morgan,
The
(6)
arelens
at
ends of the
(7)
(8)
2(B) - THIN LENS FORMULA
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(3)
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Lens Maker’s Formula (1/f=(n-1)(1
Lens Formula connects focal length of the lens, f with refractive index of the medium n.
2(C) – NEWTON’S FORMULA
When x1 & x2 are the distances of object and image from the principal focus of
the lens, then lens formula becomes NEWTON’S FORMULA i.e.x1x2=f2
2(D) – USES OF LENS FORMULA
1. IN MAGNIFYING GLASS:A single convex lens mounted in a frame with
a handle or stand is a magnifying glass.
If u < f, v becomes negative, the image is apparently positioned on the
same side of the lens as the object. Although this kind of image, known as a
cannot be projected on a screen. A magnifying glass creates this kind of image.
2. IN CAMERA AND PHOTOGRAPHY:
If an object is placed at a distance u (u>f) along the axis in front of a
positive lens of focal length f, a screen placed at a distance v behind
the lens will have a sharp and real image of the object projected onto it.
If the lens-to-screen distance v is varied slightly, the image will become less
sharp). This is the principle behind photography. The image in this case is known
as a real image.We can adjust image distances by focussing in the objects by
virtual image,
making different arrangements.
3. Lenses are used as prosthetics for the correction of visual impairments such as myopia, hyperopia, presbyopia
, and astigmatism. (e.g. corrective lens, contact lens, eyeglasses.) Most lenses used for other purposes have strict ax
ial symmetry; eyeglass lenses are only approximately symmetric. They are usually shaped to fit in a roughly
oval, not circular, frame; the optical centers are placed over the eyeballs; their curvature may not be axially
symmetric to correct for astigmatism. Sunglasses lenses may be designed to attenuate light without refraction.
4. Another use is in imaging systems such as a monocular, binoculars, telescope, spotting scope, telescopic gun
sight, theodolite, microscope, camera (photographic lens) and projector. Some of these instruments produce a virtual
image when applied to the human eye; others produce a real image which can be captured on photographic film o
r an optical sensor.
5. Convex lenses produce an image of an object at infinity at their focus; if the sun is imaged, all the infrared energy
incident on the lens is concentrated on the small image. A large lens will concentrate enough energy to heat
an inflammable object on which the image falls to burning point. Such lenses, which do not need to be even
approximately optically accurate, have been used as burning-glasses for hundreds of years. A modern
application is the use of relatively large lenses to concentrate solar energy on relatively small photovoltaic cells,
harvesting more energy without the need to use larger, more expensive, cells.
Radio astronomy and radar systems often use dielectric lenses, commonly called a lens antenna to refract electromag
netic radiation into a collector antenna. The Square Kilometre Array radio telescope, scheduled to be operational by
2020, will employ such lenses to get a collection area nearly 30 times greater than any previous antenna.
3 – REFERENCES
General References
Hecht, Eugene (1987). Optics (2nd ed. ed.), Addison Wesley. ISBN 0-201-11609-X. Chapters 5 & 6.
Greivenkamp, John E. (2004). Field Guide to Geometrical Optics. SPIE Field Guides vol. FG01, SPIE. ISBN 08194-5294-7.
Book References
Morgan, J. Introduction to Geometrical and Physical Optics. New York: McGraw-Hill, p. 57, 1953.
Schroeder, D. J. Astronomical Optics, 2nd ed. San Diego: Academic Press, 1999.
Wave site References
Steinhaus, H. Mathematical Snapshots, 3rd ed. New York: Dover, pp. 93-94, 1999.
http://en.wikipedia.org/wiki/Lens_(optics)
http://hyperphysics.phy-astr.gsu.edu/Hbase/geoopt/lenmak.html
http://scienceworld.wolfram.com/physics/LensFormula.html
http://physics.nmt.edu/~raymond/classes/ph13xbook/node35.html