Geometrical Optics

Chapter 24

OpticsThe study of light is called opticsSome highlights in the history of optics

Study of optics dates to at least third century BCEyeglasses invented around 1300Microscopes and telescopes invented around 1600

Applications depend on the ability of lenses and mirrors to focus light

Light is an electromagnetic wave and its wave nature needs to be accounted for

Geometrical OpticsApplies to the regime where light travels in straight-

line pathsEffects involving wave interference are not importantDescribes cases in which the wavelength of the light

is much smaller than the size of the objects in the light’s pathWavelength of visible light is less than 1µm

Describes many everyday applicationsIncluding the behavior of mirrors and lenses

RaysRays indicate the path and

direction of propagation of the light wave

In A, the waves pass through a large opening and, to a very good approximation, follow straight lines that pass through the opening

In B, the opening is about the same size or smaller than the wavelength of the light and needs wave optics to explain

Section 24.1

Wave Fronts

Wave front surfaces are determined by the crests and troughs of the wave

They are always perpendicular to the associated raysThe shape of a wave front depends on how the wave is

generated and the distance from the source

Section 24.1

Two Properties of LightThe motion of light along a light ray is reversible

If light can travel in one direction along a ray that connects point A to point B, light can also propagate in the reverse direction, from B to A

The perpendicular distance between two wave fronts is proportional to the speed of lightBecause of the way wave fronts are related to crest

and troughs of a wave

Section 24.1

Ray TracingLight from an object is used by your eye to form an

image of the objectWhen your eye combines the rays to form an image,

your brain extrapolates the rays back to their originThe method of following the individual rays as they

travel from an object to some other point is called ray tracing

Ray tracing involves the use of geometry

Section 24.1

Ray Tracing, cont.The figure shows a few

rays from the objectThere are an infinite

number of actual raysThe light waves

associated with all the rays contribute to the image formed by your eye

In most ray diagrams, we draw just a few rays from the top and bottom of the image

Section 24.1

Image FormationTwo problems must be considered to understand

how images are formedWhat happens to light rays when they reflect from a

surface such as a mirror or a piece of glassWhat happens to light rays when they pass across a

surface from one material to another such as when they pass from air into a piece of glass

You must also distinguish between a flat surface and a curved surface

Section 24.1

Reflection from a Plane MirrorLight rays travel in straight lines until they strike

somethingThe rays may be reflectedRays may be reflected from a plane mirror

A flat surface that reflects all or nearly all the light that strikes it

If the light is a plane wave, all the rays are parallel and strike a surface at many different points

Section 24.2

Reflection from a Plane Mirror, cont.

Characterize the reflection by a single rayThe normal (vertical dashed line in fig. B) is

perpendicular to the mirrorThe direction of the incoming and outgoing rays are

measured relative to the normal

Section 24.2

Law Of Reflection - DefinitionsThe incoming ray is called the incident ray

The angle it makes with the normal is called the angle of incidence, θi

The outgoing ray is called the reflected rayThe angle it makes with the normal is called the angle

of reflection, θr

The Law of Reflection says θi = θr

Reflection from a perfectly flat mirror is called specular reflection

Section 24.2

Diffuse ReflectionIf the reflecting surface is

rough, the reflections from each individual piece of the surface must be analyzed

An incident plane wave will give rise to many reflected rays propagating outward in many different directions

This is called diffuse reflection

Section 24.2

Image Formation – Plane MirrorAn image formed by a

plane mirror is shownTwo representative rays

are shown coming from the objectThere is an infinite

number of rays emanating from each point on the object

The rays that reflected from the mirror and reached your eyes form the image

Section 24.2

Image Formation, cont.To your eye, the location of the image is the point

from which these rays appear to emanateThis point can be found by ray tracingEach ray obeys the law of reflectionApplying geometry will allow the location of the

image to be found

Section 24.2

Image Formation, finalCharacteristics of the image

The distance from the object to the mirror is the same as the distance from the image to the mirror

The size (height) of the image, hi, is the same as the size (height) of the object, ho

The image is virtual The image point is located behind the mirror The light does not actually pass through the image

The same analysis can be applied to multiple mirrors

Section 24.2

RefractionWhen a light ray strikes

a transparent material, some of the light is reflected and some is refractedThe reflected ray obeys

the Law of ReflectionThe refracted ray

passes into the materialThe incident angle is

now denoted as θ1

Section 24.3

Angle of RefractionThe direction of the refracted ray is measured by

using θ2 (refer to fig. 24.10)

The value of this angle depends on the incident angle and the speed of light in the material

The speed of light in a vacuum is 3 x 108 m/sWhen the light travels through a material substance,

its interactions with the atoms of the material slows down the wave

Section 24.3

Snell’s Law

The change in the speed of light from the vacuum to the material changes the direction of the wave

From the geometry of the waves in the material,

1 2sin sincv

Section 24.3

v = λ f

f constantv , λ change

Snell’s Law, cont.

Snell’s Law, cont.The ratio c/v is called the index of refraction and is

denoted by nn = c / vn is unitless

Then, sin θ1 = n sin θ2 This assumes the wave is incident in a vacuum

A more general statement can be applied to any two materials with indices of refraction n1 and n2

n1 sin θ1 = n2 sin θ2

This relationship is called Snell’s Law

Section 24.3

Speeds and n’s for Various Materials

Section 24.3

Applying Snell’s LawRefraction is also reversibleSnell’s Law applies whether light begins in the

material with the larger or smaller index of refractionPossible angles of refraction are always between 0°

and 90°The side with the larger index of refraction has the

smaller angle

Direction of Refracted RayLight is refracted toward

the normal when moving into the substance with the larger index of refraction

Light is refracted away from the normal when moving into the substance with the smaller index of refraction

Section 24.3

Total Internal ReflectionWhen light is incident

from the side with a higher index of refraction, it is bent away from the normal

As the incident angle gets larger, the refracted angle also increases

Eventually, θ2 will reach 90°

Section 24.3

Total Internal Reflection, cont.The angle of incidence for which the angle of

refraction is 90° is called the critical angleIf the angle of incidence is increased beyond the

critical angle, Snell’s Law has no solution for θ2 Physically, there is no refracted ray

This behavior is called total internal reflectionThis is only possible when the light is incident from the

side with the larger index of refraction

Section 24.3

Critical AngleFrom Snell’s Law, with θ2 = 90°, θ1 = θcrit

When the angle of incidence is equal to or greater than the critical angle, light is reflected completely at the interface

1 2

1

sincrit

nn

Section 24.3

Fiber OpticsTotal internal reflection is

used in fiber opticsOptical fibers are

composed of specially made glass and used to carry telecommunication signalsThese signals are sent as

light wavesThey are directed along

the fiber using internal reflection

Section 24.3

DispersionWhen light travels in a

material, the speed depends on the color of the light

This dependence of wave speed on color is called dispersion

Since the index of refraction is slightly different for each color, the angle of refraction will be different for each color

Section 24.3

Dispersion and Prisms

Dispersion is used by a prism to separate a beam of light into its component colors

There are two refractions with the prismThe red and blue show the extremes of the incident

beamsSection 24.3

Curved MirrorsA curved mirror can produce an image of an object

that is magnifiedThe image can be larger or smaller than the object

Magnified images are used in many applicationsTelescopesCar’s review mirrorMany others

Section 24.4

Ray Tracing – Curved MirrorA spherical mirror in one

in which the surface of the mirror forms a section of a spherical shell

The radius, R, of the sphere is the radius of curvature of the mirror

The mirror’s principal axis is the line that extends from the center of curvature, C, to the center of the mirror

Section 24.4

Concave Spherical MirrorProperties of concave

spherical mirrorsIncoming rays that are

close to and parallel to the principal axis reflect through a single point F F is the focal point It is located a distance ƒ,

the focal length, from the mirror

Rays that originate at the focal point reflect from the mirror parallel to the principal axis From reversibility of light

Section 24.4

Image From a Concave Mirror -- Examples

Section 24.4

Image from Concave Mirror – Ray Diagram

Trace rays emanating from the top of the objectThe rays all intersect at a single point

This is the top of the imageA similar result would be found from rays from other

parts of the object

Section 24.4

Drawing A Ray Diagram

Three rays are particularly easy to drawThere are an infinite number of actual rays

The focal rayFrom the tip of the object through the focal pointReflects parallel to the principal axis

Section 24.4

Drawing A Ray Diagram, cont.The parallel ray

From the tip of the object parallel to the principal axisReflects through the focal point

The central rayFrom the tip of the object through the center of

curvature of the mirrorReflects back on itself

The three rays intersect at the tip of the image

Section 24.4

Properties of an ImageMagnification is the ratio of the height of the image,

hi, to the height of the object, ho

By convention, the image height of an inverted image is negativeTherefore, the magnification is also negative

Images smaller than the object are said to be reduced

Section 24.4

i

o

hm

h

Real vs. Virtual ImagesIf the rays that form the image all pass through a

point on the image, the image is called a real imageReal images and virtual images differ

Light rays only appear to emanate from a virtual image, they do not actually pass through the image For a real image, the light rays do actually pass through the

image

An object and its real image are both on the same side the mirror A virtual image is located behind the mirror while the object

is in frontSection 24.4

Concave Mirror and Virtual Images

Use ray tracing to find the image when the object is close to the mirrorCloser than the focal point

Use the same three raysThe rays do not intersect at any point in the front of the

mirror

Section 24.4

Virtual ImagesExtrapolate the rays back behind the mirrorThey intersect at a single image pointThe rays appear to emanate from the image point

behind the mirrorThe image is virtual because light does not actually

pass through any point on the imageThe object and its image are on different sides of the

mirrorThe image is upright and enlarged

Section 24.4

Rules for Ray Tracing – Mirrors Construct a figure showing the mirror and its

principal axisThe figure should also show the focal point and the

center of curvatureDraw the object at the appropriate point

One end of the object will often lie on the principal axisDraw three rays that emanate from the tip of the

objectThe focal ray passes through the focal point and

reflects parallel to the principal axis

Section 24.4

Rules, cont.Three rays, cont.

The parallel ray is parallel to the principal axis and reflects through the focal point

The central ray passes through the center of curvature of the mirror and reflects back through the tip of the object

The point where the three rays intersect is the image pointThis point may be in front of the mirror giving a real imageThis point may be in back of the mirror giving a virtual

image Found by extrapolation of the rays behind the mirror

Section 24.4

Rules, finalThis ray-tracing procedure can be repeated for any

desired point on the objectThis allows you to find other points on the imageIt is usually sufficient to consider just the tip of the

imageOther points may be used if needed

Section 24.4

Ray Tracing – Convex Spherical MirrorsA mirror that curves away

from the object is called a convex mirror

The center of curvature and the focal point lie behind the mirror

After striking the convex surface, the reflected rays diverge from the mirror axis

The parallel rays converge on an image point behind the mirrorThis is the focal point, F

Section 24.4

Ray Tracing – Convex Mirrors, cont.The same three rays are

used as were used for concave mirrors

The focal ray is directed toward the focal point but is reflected at the mirror’s surface, so doesn’t go through F

The three rays extrapolate to a point behind the mirrorProduces virtual image

Section 24.4

Mirror Equation

Geometry can be used to find the characteristics of the image quantitatively

The distance from the object to the mirror is so The distance from the image to the mirror is si The given rays produce similar triangles

Section 24.4

Mirror Equation and Focal LengthFrom the similar triangles,

For an object at (approximately) infinity, 1/so = 0But an “infinite” object will produce parallel raysParallel rays all intersect at the focal pointTherefore, the focal length can be found from the

radius of curvature of the mirror

1 1 2

o is s R

ƒ2R

Section 24.4

Mirror Equation and MagnificationThe mirror equation can be written in terms of the

focal length

The magnification can also be found from the similar triangles shown in fig. 24.30

1 1 1ƒo is s

Section 24.4

i i

o o

h sm

h s

Sign ConventionsAll diagrams with mirrors should be drawn with the

light ray incident on the mirror from the leftThe object distance is positive when the object is to

the left of the mirror and negative if the object is to the right (behind) of the mirror

The image distance is positive when the image is to the left of the mirror and negative if the image is to the right (behind) of the mirrorThe image distance is positive for real images and

negative for virtual images

Section 24.4

Sign Conventions, cont.The focal length is positive for a concave mirror and

negative for a convex mirrorFor a concave mirror, ƒ = R / 2For a convex mirror, ƒ = - R / 2

The object and image heights are positive if the object/image is upright and negative if it is inverted

Section 24.4

Sign Convention, Summary

Section 24.4

LensesA lens uses refraction to form an imageTypical lenses are composed of glass or plasticThe refraction of the light rays as they pass from the

air into the lens and then back into the air causes the rays to be redirectedAlthough refraction occurs at both surfaces of the lens,

for simplicity the rays are drawn to the center of the lens

Section 24.5

Lenses, Focal PointParts B and C show the

simplification of the single deflection of the rays

Parallel rays close to the principal axis intersect at the focal pointThis is true for incident

rays from either side of the lens

The focal points are at equal distances on the two sides of the lens

Section 24.5

Spherical LensesThe simplest lenses

have spherical surfacesThe radii of curvature of

the lenses are called R1 and R2

The radii are not necessarily equal

Section 24.5

Types of Lenses

Converging lensesAll the incoming rays parallel to the principal axis intersect

at the focal point on the opposite sideDiverging lenses

All the incoming rays parallel to the principal axis intersect at the focal point on the same side as the incident rays

Section 24.5

Focal Point – Diverging LensThe parallel incident

rays from the left are refracted away from the axis

The rays on the right appear to emanate from a point F on the left side of the lens

This point F is one of the focal points of the lens

Section 24.5

Image from a Converging LensAn infinite number of

rays emanate from the object

For simplicity, choose three rays that are easy to draw

Start at the tip of the object

Section 24.5

Rays for a Converging LensThe parallel ray is initially

parallel to the principal axisRefracts and passes

through the focal point on the right (FR)

The focal ray passes through the focal point on the left (FL)Refracts and goes parallel

to the principal axis on the right

The center ray passes through the center of the lens, C

Section 24.5

Rays, cont.If the lens is very thin, the center ray is not deflected

by the lensThese three rays come together at the tip of the

image on the right of the lensIn this case, the image is invertedThe image is real

The rays pass through the image

Section 24.5

Rules for Ray Tracing – Lenses Construct a figure showing the lens and its principal

axisThe figure should also show the focal points on both

sides of the lensDraw the object at the appropriate point

One end of the object will often lie on the principal axisDraw three rays that emanate from the tip of the

objectThe parallel ray is initially parallel to the principal axis

and after refraction passes through one of the focal points

Section 24.5

Rules, cont.Three rays, cont.

The focal ray is directed at the other focal point and after refraction the ray is parallel to the principal axis

The central ray passes through the center of the lens and is not deflected

The point where the three rays or their extrapolation intersect is the image pointIf the rays actually pass through the lens, the image is

realIf the rays do not pass through the lens, the image is

virtual

Section 24.5

Rules, finalReal image

When a lens forms a real image, the object and image are on opposite sides of the lens

Virtual imageWhen a lens forms a virtual image, the object and

image are on the same side of the lensAll other rays that pass through the lens will also

pass through the image

Section 24.5

Ray Tracing – Diverging LensFollow the rules for ray

tracing for lensesSince the refracted rays

do not intersect on the right side of the lens, extrapolate the rays back to the left side of the lens

The extrapolations do intersect

The point of intersection is the image point at the tip of the image

Section 24.5

Sign Conventions – Lenses, Diagram

Sign Conventions for LensesAssume light travels through the lens from left to

rightThe object will always be located to the left of the lens

The object distance is positive when the object is to the left of the lensAccording to the first convention, the object distance

will always be positiveThe image distance is positive when the image is to

the right of the lens and negative if the image is to the left of the lens

Section 24.5

Sign Conventions, cont.The focal length is positive for a converging lens and

negative for a diverging lensThe object height is positive if the object extends

above the axis and is negative if the object extends below

The image height is positive if the image is extends above the axis and is negative if the image extends below

Section 24.5

Thin-Lens EquationGeometry can be used

to find a mathematical relation for locating the image produced by a converging lens

The shaded triangles are pairs of similar triangles

Section 24.5

Thin-Lens Equation and MagnificationThe thin-lens equation is found from an analysis of

the similar triangles

The magnification can also be found from the similar triangles shown

These results are identical to the results found for mirrors

1 1 1ƒo is s

Section 24.5

i i

o o

h sm

h s