Lecture Notes 7- 203 Optics

7/27/2019 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

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Lecture Notes 7- 203 Optics