Lecture Notes 7- 203 Optics
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Transcript of Lecture Notes 7- 203 Optics
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Geometrical Optics /Mirror and Lenses
Outline
Reflection
Plane Mirrors
Concave/Convex Mirrors
Refraction
Lenses
Dispersion
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Geometrical Optics
In describing the propagationof light as a wave we need tounderstand:
wavefronts: a surface passingthrough points of a wave thathave the same phase andamplitude.
rays: a ray describes thedirection of wave propagation.A ray is a vector perpendicularto the wavefront.
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Reflection and RefractionWhen a light ray travels from one medium to another, part of theincident light is reflected and part of the light is transmitted at theboundary between the two media.
The transmitted part is said to be refracted in the second medium.http://www.geocities.com/CapeCanaveral/Hall/6645/propagation/propagation.html
*In 1678 the great Dutch physicist Christian Huygens (1629-1695) wrote a treatise calledTraite de la Lumiere on the wave theory of light, and in this work he stated that the wavefront
of a propagating wave of light at any instant conforms to the envelope of spherical waveletsemanating from every point on the wavefront at the prior instant. From this simple principleHuygens was able to derive the laws of reflection and refraction
incident ray reflected ray
refracted ray
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Types of ReflectionWhen light reflects from a
smooth surface, it undergoes
specular reflection (parallelrays will all be reflected in the
same direction).
When light reflects from a
rough surface, it undergoes
diffuse reflection (parallel rays
will be reflected in a variety ofdirections).
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The Law of ReflectionFor specular reflection the incident angle iequals the reflected angle r:
i = r(Known since 1000 BC)
The angles are
measured relative
to the normal,shown here as a
dotted line.
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Forming Images with a Plane MirrorA mirroris an object that reflects light. A plane mirroris simply a flatmirror. Plane mirrors are ground to be flat the flatter the moreexpensive. (Typically good ones have - where we use visible radiation- no hills or valleys larger than 500nm).
Consider an object placed at point P in front of a plane mirror. Animage will be formed at point P behind the mirror.
do diFor a plane mirror:
do = di and ho = hi
ho hi
do = distance from object to
mirror
di = distance from image to
mirror
ho = height of object
hi = height of image
vertex Q = do + di
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Images
An image is formed at the point where the rays of lightleaving the object either actually intersect or where theyappear to originate.
If the light rays actually do intersect, then the image is a realimage. If the light only appears to be coming from a point,but is not physically there, then the image is a virtual image.
We define the magnification, m, of an image to be:
o
i
o
i
d
d
h
hm ===
heightobject
heightimage
If m is negative, the image is inverted
(upside down).
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The image is called virtual because it doesnot really exist behind the mirror
Real image
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Plane MirrorsA plane mirror image has the following properties:
*The mirror in your bathroom is a piece of plate glass with a coating on thebackside so they are second surface mirrors.
The image distance equals the object distance.
The image is unmagnified.
The image is virtual. The image is not inverted.
Left and right are reversed
**The intensity of the reflected beam depends upon the angle ofincidence and the indices of refraction and they type of coating.
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To save expenses, you would like to buy the
shortest mirror that will allow you to see your entire
body. Should the mirror be (a) half your height (b)
two-thirds your height, or (c) equal to your height?
Does the answer depend on how far away from
the mirror you stand?
eye
you
mirror
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Spherical Mirrors
A spherical mirroris a mirrorwhose surface shape isspherical with radius of curvatureR. There are two types ofspherical mirrors: concave andconvex. **The principal axis (optical axis,vertex) is the straight line between C and themidpoint of the mirror
We will always orient the mirrorsso that the reflecting surface ison the left. The object will be on
the left.
concave
convex
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Focal PointWhen parallel rays are
incident upon a
spherical mirror, the
reflected rays intersect
at the focal point F.
For a concave mirror,
the focal point is in front
of the mirror.For a convex mirror, the
focal point is behind the
mirror.
The incident rays
diverge from the convex
mirror, but they trace
back to the focal point F.
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Focal LengthThe focal length fis the distance from the surface of
the mirror to the focal point. It can be shown that
the focal length is half the radius of curvature of the
mirror.
Sign Convention: the focal length is negative if thefocal point is behind the mirror.
For a concave mirror, f = R
For a convex mirror, f = R
(R is always positive)
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Ray TracingWe will use three
principal rays todetermine where an
image will be
located.
The parallel ray (P ray) reflects
through the focal point. The focal
ray (F ray) reflects parallel to theaxis, and the center-of-curvature
ray (C ray) reflects back along its
incoming path.
Curvature point
Curvature point
Optical axis
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Ray Tracing Examples
concave convex
Virtual imageReal imageapplet mirror/lens
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Theorem of intersecting lines
iidRRd
hh
=
00
iidd
hh 00 =
iddf
111
0
+=
Mirror Equationi i
o o
R d d
d R d
=
with
f= R
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The Mirror Equation
The ray tracing techniqueshows qualitatively where theimage will be located. The
distance from the mirror to theimage, di, can be found fromthe mirror equation:
fdd io
111 =+
do = distance from object to
mirror
di = distance from image to
mirror
f= focal length
Sign Conventions:
do is positive if the object is in front of
the mirror (real object)
do is negative if the object is in back of
the mirror (virtual object)
di is positive if the image is in front of
the mirror (real image)
di is negative if the image is behind
the mirror (virtual image)
fis positive for concave mirrors
fis negative for convex mirrors
m is positive for upright images
m is negative for inverted images
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The Refraction of LightThe speed of light is different in different materials. Wedefine the index of refraction, n, of a material to be the ratio
of the speed of light in vacuum to the speed of light in thematerial:
n = c/v
When light travels from one medium to another, its velocityand wavelength change, but its frequency remainsconstant.
http://www.geocities.com/CapeCanaveral/Hall/6645/propagation/propagation.html
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Snells LawIn general, when light enters a new material its direction willchange. The angle of refraction 2 is related to the angle ofincidence 1 by Snells Law:
where v is the velocity of light in the medium.
Snells Law can also be written as:
n1sin1 = n2sin2
constantv
sinsin
22
11 ==
v
1
2
Air
GlassThe angles 1 and 2 aremeasured relative to the line
normal to the surface
between the two materials.
Normal line
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n = 1.2
n = 1.6
Example: Which way will the rays bend?
Which of these rays can be the refracted ray?
n = 1.4 n = 2
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Total Internal ReflectionWhen light travels from a medium with n1 > n2,there is an angle, called the critical angle c, atwhich all the light is reflected and none istransmitted. This process is known as total
internal reflection. The critical angle occurswhen 2= 90 degrees:
The incident ray is both reflected and
refracted. Total Internal Reflection
1
2sinn
nc =
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A pencil in a glass of water looks
bent due to the light refraction
A mirage is created due to
the bending of light. The
index of refraction of thehot air near the ground is
lower than the n of the
colder air on the top.
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Object in the sky appear to be shifted towards the zenith by a small amount.
This is due to the refractive effect of the atmosphere. This has been known
since the time of Ptlomey in Egypt in 150 BC.
ASTRONOMICAL REFRACTION: The displacements of astronomical objects by
atmospheric refraction.
These effects are many orders of magnitude larger than the accuracy of the best
astronomical position measurements, and so large that the mountings of mostastronomical telescopes are adjusted to minimize the effects of refraction.
http://www.sundog.clara.co.uk/rainbows/primrays.htm
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Light always bends toward the base of a triangular prism (if n of
the prism is higher than the ambient n).
Different colors bend differently. It means that n is different for
different colors. The separation of colors is called light
dispersion. http://www.wolles-website.de/teste_taeuschungen/taeuschungen_uebersicht.htm
n=1
n>1
Refraction in a Triangular PrismRefraction in a Triangular Prism
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Lenses
A lens is an object that
uses refraction to bend
light and form images
Light is reflected froma mirror. Light is
refracted through a
lens.
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Focal PointThe focal point of a lens is the place whereparallel rays incident upon the lens converge.
converging lens diverging lens
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Ray Tracing for Lenses
The P ray propagates parallel to the principal axis until it encounters thelens, where it is refracted to pass through the focal point on the far side of
the lens. The F ray passes through the focal point on the near side of the
lens, then leaves the lens parallel to the principal axis. The M ray passes
through the middle of the lens with no deflection.
Just as for mirrors we use
three rays to find the image
from a lens. The lens is
assumed to be thin.
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Ray Tracing Examples
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The Thin Lens EquationThe ray tracing technique shows qualitativelywhere the image from a lens will be located.The distance from the lens to the image, di,can be found from the thin-lens equation:
fdd io
111=+
Sign Conventions:
do is positive for real objects (from which light diverges)
do is negative for virtual objects (toward which light converges)
di is positive for real images (on the opposite side of the lens from theobject)
di is negative for virtual images (same side as object)
fis positive for converging (convex) lenses
fis negative for diverging (concave) lensesm is positive for upright images
m is negative for inverted images
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Lens makers formulaThe equation in the box is the thin lens equation. The
focal length is given by the lens makers formula:
This expression is good for a lens in air. The R-valuesare the radii of curvature of the first and second
surfaces of the lens. n is the refraction index. So fis
determined by construction: n and curvature Rs
ARE fixed by construction.
1 2
1 1 1( 1)nf R R
=
http://www.phy.ntnu.edu.tw/java/Lens/lens_e.html
**Not all lenses are thin lenses - Thick lens equation:
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DispersionIn a material, the velocity of light (and therefore the index ofrefraction) can depend on the wavelength. This is knownas dispersion. Blue light travels slower in glass and water than does
red light. (The shorter wavelengths are refracted by the greatestamount)
As a result of dispersion,different colors entering a
material will be refracted atdifferent angles.
Dispersive materials can beused to separate a light
beam into its spectrum (thecolors that make up the lightbeam). Example: prism