Chapter 23: Geometric Optics

Review Basic Geometry!

Ray Approximation

• The rays are straight lines perpendicular to the wave fronts

• With the ray approximation, we assume that a wave moving through a medium travels in a straight line in the direction of its rays

Diffraction depends on SLIT WIDTH: the smaller the width,

relative to wavelength, the more bending and diffraction.

We will assume that λ<<d , where d is the diameter of the opening.

This approximation is good for the study of mirrors, lenses, prisms, etc.

Light Rays: Ignore Diffraction and

Interference of waves!

Reflection & Refraction

i r 1 1 2 2sin sinn n

Following the Reflected and

Refracted Rays •Ray is the incident ray.

•Ray is the reflected ray.

•Ray is refracted into the lucite.

•Ray is internally reflected in the lucite.

•Ray is refracted as it enters the air from the lucite. Section 35.5

Law of Reflection

• The normal is a line perpendicular to the surface

– It is at the point where the incident ray strikes the surface

• The incident ray makes an angle of θ1 with the normal

• The reflected ray makes an angle of θ1’ with the normal

Specular

Reflection

• Specular reflection is

reflection from a

smooth surface

• The reflected rays are

parallel to each other

• All reflection in this

text is assumed to be

specular

Diffuse

Reflection

• Diffuse reflection is

reflection from a rough

surface

• The reflected rays travel in

a variety of directions

• A surface behaves as a

smooth surface as long as

the surface variations are

much smaller than the

wavelength of the light

Law of Reflection

i r

Why are most materials Opaque?

(Opaque – Can’t see through)

They absorb light without re-emitting it. Vibrations given

by the light to their atoms and molecules are turned into

random kinetic energy – they become slightly warmer.

Opacity: Mirrors

Free electrons in opaque reflective surfaces

can vibrate, absorb & re-emit at any frequency.

Mirror Mirror

Transparency

Selective Absorption

Glass resonates strongly with UV and

infrared, absorbing those frequencies

while transmitting visible frequencies.

Refraction:

Bending Light into Focus

Refraction: Bending of Light

Transmitted through Materials

Light Bends because it Slows Down.

Atoms are Optical Tuning Forks

Light slows down as it travels through

glass because it takes time to be

absorbed and re-emitted.

Light Slows Down in Materials Light Bends Toward the Normal when going from a

medium of lower refractive index to one that has a

higher refractive index and visa versa.

lower n

higher n

Index of Refraction

cn

v

Vacuum: 1

Water: 1.33

Glass: 1.46

Diamond: 2.4

1n

The Index of Refraction • Refraction: Light Bends in

Transmission

• The speed of light in any

material is less than its speed

in vacuum

• The index of refraction, n,

of a medium can be defined

as

speed of light in a vacuumn

speed of light in a medium n

c λ

v λ

in vacuum

in a mediumn

λ λn

λ λ

• For a vacuum, n = 1

– We assume n = 1 for air

also

• For other media, n > 1

• n is a dimensionless number

greater than unity, not

necessarily an integer

Some Indices of Refraction

Frequency Doesn’t Change!

• As light travels from one medium to another, its frequency does not change

– Both the wave speed and the wavelength do change

– The wavefronts do not pile up, nor are created or destroyed at the boundary, so ƒ must stay the same

Snell’s Law of Refraction

1 1 2 2sin sinn n

Angles are always measured from the normal.

Snell’s Law – Example

Light is refracted into a crown glass slab. n1 = 1.00 and n2 = 1.52

If θ1 = 30.0o, θ2 = ?

θ2 = sin-1(n1 / n2) sin θ1 = 19.2o

The ray bends toward the normal, as expected because n2 > n1

Snell’s Law of Refraction

1 1 2 2sin sinn n

In general:

1 1 2 2n n

12 1

2

n

n

2 1 1 2If , then

measured from the normal!

n n

Prelab

Emerging Beam is Parallel to Incident Beam but

offset distance d, called the Lateral Shift and is the

subject of this week’s lab!

Fig. 35-15, p. 989

Following the Reflected and

Refracted Rays •Ray is the incident ray.

•Ray is the reflected ray.

•Ray is refracted into the lucite.

•Ray is internally reflected in the lucite.

•Ray is refracted as it enters the air from the lucite. Section 35.5

Beam & Refraction Directions

• Possible directions of the

beam are indicated by rays

numbered 1 through 5

• The refracted rays are bent

away from the normal since

n1 > n2

Total Internal Reflection

2

1

sin C

n

n

2 90

1 1 2 2sin sinn n

The Critical Angle

Critical Angle

• There is a particular angle

of incidence that will

result in an angle of

refraction of 90°

– This angle of incidence is

called the critical angle, θC

2

1 2

1

sin (for )C

nθ n n

n

Fiber Optics • An application of internal reflection

• Plastic or glass rods are used to “pipe” light from one place to another

• Applications include:

– medical use of fiber optic cables for diagnosis and correction of medical problems

– Telecommunications

• A flexible light pipe is called an optical fiber

• A bundle of parallel fibers (shown) can be used to construct an optical transmission line

Critical Angle Sample Problem

A ray of light, emitted by a laser located beneath the surface of an

unknown liquid with air above it, undergoes total internal

refection as shown. What is the index of refraction for the liquid?

What is its likely identification?

If you pass white light through a prism,

it separates into its component colors.

R.O.Y. G. B.I.V long wavelengths short wavelengths

The index of refraction depends on

WAVELENGTH.

R.O.Y. G. B.I.V long wavelengths short wavelengths

The speed and wavelength change but

the FREQUENCY does NOT.

R.O.Y. G. B.I.V long wavelengths short wavelengths

Fr Frequency depends on the oscillating source!

Why does Violet Light bend more

than Red Light? Violet light slows down more because the atoms in the material

are tuned to higher frequencies. As the violet light travels

through glass it takes more time to be absorbed and re-emitted.

Variation of Index of Refraction with Wavelength

• This dependence of n on λ

is called dispersion

• The index of refraction for

a material generally

decreases with increasing

wavelength

• Violet light bends more

than red light when passing

into a refracting material

speed of light in a vacuum

nspeed of light in a medium n

c λ

v λ

Refraction in a Prism

•Since all the colors have different angles of deviation, white light will spread out into a spectrum.

– Violet deviates the most.

– Red deviates the least.

– The remaining colors are in between.

Section 35.7

Dispersion via Diffraction

: sin , 0,1,2,3constructive d m m

If you pass white light through a prism,

it separates into its component colors.

R.O.Y. G. B.I.V long wavelengths short wavelengths

The index of refraction depends on

WAVELENGTH.

R.O.Y. G. B.I.V long wavelengths short wavelengths

The speed and wavelength change but

the FREQUENCY does NOT.

R.O.Y. G. B.I.V long wavelengths short wavelengths

Fr Frequency depends on the oscillating source!

Why does Violet Light bend more

than Red Light? Violet light slows down more because the atoms in the material

are tuned to higher frequencies. As the violet light travels

through glass it takes more time to be absorbed and re-emitted.

Variation of Index of Refraction with Wavelength

• This dependence of n on λ

is called dispersion

• The index of refraction for

a material generally

decreases with increasing

wavelength

• Violet light bends more

than red light when passing

into a refracting material

speed of light in a vacuum

nspeed of light in a medium n

c λ

v λ

Angle of Deviation

• Since all the colors have different angles of deviation, white light will spread out into a spectrum – Violet deviates the most

– Red deviates the least

– The remaining colors are in between

Dispersion Sample Problem

The index of refraction for

violet light in silica flint glass

is 1.66, and that for red light is

1.62. What is the angular

dispersion of visible light

passing through a prism of

apex angle 60.0° if the angle of

incidence is 50.0°? red (660

nm) violet (410 nm)

1 1 2 2sin sinn n Use Snell’s Law twice and some

geometry! Angles are always

measured from the normal.

Thin Film

Interference

Interference in Thin Films When reflecting off a medium of greater refractive index, a

light wave undergoes a phase shift of ½ a wavelength.

Wave 1 undergoes a phase shift of 180 degrees.

From Low to High,

a phase change of

pi!

From High to Low,

a phase change?

NO!

Interference in Thin Films • The wavelength of ray 1 in the

film is /n

• For constructive interference

2t = (m + ½) /n (m = 0, 1, 2 …)

This takes into account both the difference in optical path length for the two rays and the 180° phase change

• For destructive interference

2t = m/n (m = 0, 1, 2 …)

A thin film of gasoline floats on a puddle

of water. Sunlight falls almost

perpendicularly on the film and reflects

into your eyes a yellow hue. Interference

in the the thin gasoline film has eliminated

blue (469nm in vacuum) from the

reflected light. The refractive indices of

the blue light in gasoline and water are

1.40 and 1.33 respectively.

Determine the minimum nonzero

thickness of the film.

What color do you see?

Problem: Thin Films

Thin Film Interference

The light reflected from a soap bubble

(n = 1.40) appears red ( = 640 nm). What is the minimum thickness (in nm)?

a. 124

b.104

c. 114

d.134

e. 234

How are Rainbows Formed?

Dispersion: Raindrops Act like Prisms

• A ray of light strikes a drop

of water in the atmosphere

• It undergoes both reflection

and refraction

– First refraction at the

front of the drop

• Violet light will

deviate the most

• Red light will deviate

the least

The Rainbow

• At the back surface the light is reflected

• It is refracted again as it returns to the front surface and moves into the air

• The rays leave the drop at various angles

– The angle between the white light and the most intense violet ray is 40°

– The angle between the white light and the most intense red ray is 42°

Observing the Rainbow

• If a raindrop high in the sky is observed, the red ray is seen

• A drop lower in the sky would direct violet light to the observer

• The other colors of the spectra lie in between the red and the violet

The droplets form a circular arc, with each droplet within the arc

dispersing light and reflecting it back towards the observer with

the greatest concentration of outgoing rays found at these 40-42

degree angles of deviation. Every droplet within the arc is

refracting and dispersing the entire visible light spectrum

(ROYGBIV).

Rainbow facts

• an observer is in a position to see only a single color from any one droplet of water.

• your rainbow is slightly different from the rainbow seen by others

• your rainbow moves with you

• disk within the bow is brighter because of overlapping of multiple refractions (which don’t occur outside the disk)

A line drawn from your eye to the top of the rainbow forms a 42-

degree angle with the imaginary line from the sun through your

eye. (If there is a secondary rainbow, it forms an angle of 51-

degrees). Because these angles determine the position of the

rainbow in the sky, it will sink as the sun rises and rise as the sun

sinks. At some points, the entire rainbow, not just the bottom half,

will be below the horizon where you can't see it. That's why you'll

never see a summer rainbow at midday.

Double Rainbow

• The secondary rainbow is fainter than the primary

• The secondary rainbow arises from light that makes two reflections from the interior surface before exiting the raindrop

• Higher-order rainbows are possible, but their intensity is low

• Halos are caused by the light of the sun or moon passing through a very thin layer of

cirruform (ice-crystal) clouds in the upper atmosphere. The ice crystals refract the light

of the moon, similar to the way water droplets in the lower atmosphere can refract

sunlight to produce a rainbow. Just like a rainbow, strong halos can have bands of color

in them, due to slightly different refractive properties of the ice crystals for different

colors. Essentially, halos ARE rainbows caused by primary refraction in ice crystals.

• Some interesting facts about halos: Halos always occur exactly 22 degrees away from

the sun or moon. Occasionally, intense halos can be double halos, just as intense

rainbows can be doubled. Intense halos can also produce "moondogs" or "sundogs,"

very bright regions on the halo evenly spaced at 90 degree intervals around the halo.

Physics Fun on an Airplane

Always sit on the side opposite the

sun when traveling north-south!!

Why is the Sky Blue?

Galileo

In the early 17th century, many scientists believed that there was no

such thing as the "speed of light"; they thought light could travel any

distance in no time at all. Galileo disagreed, and he came up with an

experiment to measure light's velocity: he and his assistant each took

a shuttered lantern, and they stood on hilltops one mile apart. Galileo

flashed his lantern, and the assistant was supposed to open the shutter

to his own lantern as soon as he saw Galileo's light. Galileo would

then time how long it took before he saw the light from the other

hilltop. The problem was that the speed of light is simply too fast to

be measured this way; light takes such a short time (about 0.000005

seconds, in fact) to travel one mile that there's no way the interval

could have been measured using the tools Galileo had.

The Speed of Light? • 186,000 miles per second

• 300,000 kilometers per second

• 3 x 108 m/s

• first successfully determined by Danish astronomer Ole Roemer in 1675: 2.3 x 108 m/s

• First Terrestrial Measurement by Fizeau in 1849: 2.9979 x 108 m/s

• In 1926, Michelson used a rotating prism to measure the time it took light to make a round trip from Mount Wilson to Mount San Antonio in California, a distance of about 22 miles (36 km). The precise measurements yielded a speed of 186,285 miles per second (299,796 kilometres per second).

Huygens’s Principle

Construction for a Plane Wave

• Huygens assumed that light is a form of wave motion rather than a stream of particles

• All points on a given wave front are taken as point sources for the production of spherical secondary waves, called wavelets, which propagate outward through a medium with speeds characteristic of waves in that medium

• After some time has passed, the new position of the wave front is the surface tangent to the wavelets

Huygens’s Construction for a

Spherical Wave • The inner arc represents

part of the spherical wave

• The points are

representative points

where wavelets are

propagated

• The new wavefront is

tangent at each point to

the wavelet

Huygens’s Principle Prove the

Laws of Reflection & Refraction

Huygens’s Principle and the Law of

Reflection

• Triangle ABC is congruent

to triangle ADC

• cos g = BC / AC

• cos g’ = AD / AC

• Therefore, cos g = cos g’

and g g’

• This gives θ1 = θ1’

• This is the law of

reflection

Huygens’s Principle and the Law

of Refraction • Ray 1 strikes the

surface and at a time interval Δt later, ray 2 strikes the surface

• During this time interval, the wave at A sends out a wavelet, centered at A, toward D

Huygens’s Principle and the Law of

Refraction

• The wave at B sends out a

wavelet, centered at B, toward C

• The two wavelets travel in

different media, therefore their

radii are different

• From triangles ABC and ADC,

we find

1 21 2sin and sin

BC v t AD v tθ θ

AC AC AC AC

1 1 1 2

2 2 2 1

sin

sin

v c n n

v c n n1 1 2 2 sin sin n θ n θ

Why aren’t images of objects

produced on the wall without a

lens or hole?

Why aren’t images of objects

produced on the wall without a

lens or hole?

Law of Reflection

Construction of an Optical Fiber

• The transparent core is

surrounded by

cladding – The cladding has a lower n

than the core

– This allows the light in the

core to experience total

internal reflection

• The combination is

surrounded by the

jacket

The heating effect of a medium such as glass or the Earth’s

atmosphere that is transparent to short wavelengths but opaque

to longer wavelengths: Short get in, longer are trapped!

Lenses and Mirrors

Image Formation

Real and Virtual Images

Virtual Images can not be displayed onto screens.

Real images can be displayed on screens

Notation for Mirrors and Lenses The object distance is the distance

from the object to the mirror or lens: Denoted by p

The image distance is the distance from the image to the mirror or lens: Denoted by q

The lateral magnification of the mirror or lens is the ratio of the image height to the object height

Denoted by M

The focal point: f The radius of curvature: R

1 1 2 1

p q R f

FRONT is on the same side as the object and BACK is the other side!

The Principal axis goes through the focal point and the center of curvature of the lens or mirror!!

Image height

Object height

'hM

h

Flat Mirrors Make Virtual Images

Virtual Image: An image that cannot be projected onto a surface.

A virtual image only appears like light rays came from the

location of the image, they are not really there.

Flat mirrors make Virtual Images.

Images Formed by

Flat Mirrors Light rays leave the source and

are reflected from the mirror

Images are always located by

extending diverging rays back to

a point at which they intersect

One ray starts at point P, travels

to Q and reflects back on itself

Another ray follows the path PR

and reflects according to the law

of reflection

h’ = h for all images

Flat mirrors make virtual images

Reversals in a Flat Mirror

A flat mirror produces

an image that has an

apparent left-right

reversal

For example, if you

raise your right hand

the image you see

raises its left hand

Properties of the Image Formed

by a Flat Mirror – Summary

The image is as far behind the mirror as the object is in front |p| = |q|

The image is unmagnified The image height is the same as the object height

h’ = h and M = 1

The image is virtual

The image is upright It has the same orientation as the object

There is a front-back reversal in the image

Mirror Reflection

Convex & Concave

Convex Mirror Convave Mirror

“Object” on the left, image on the right.

Lateral Magnification

Magnification does not always mean bigger, the

size can either increase or decrease.

M>1: Increase M<1: Decrease

Positive: Upright Negative: Inverted

Image height

Object height

'hM

h

Focal Length Shown by

Parallel Rays

Focal Length& Radius of

Curvature

When the object is very far

away, then p → ∞ and the

incoming rays are essentially

parallel

In this special case, the image

point is called the focal point

The distance from the mirror to

the focal point is called the

focal length

The focal length is ½ the radius of

curvature

R = 2f

Ray Diagrams:Concave Mirrors

Ray 1 is drawn from the top of the object parallel to the principal axis and is reflected through the focal point, F

Ray 2 is drawn from the top of the object through the focal point and is reflected parallel to the principal axis

Ray 3 is drawn through the center of curvature, C, and is reflected back on itself

The intersection of any two of the rays at a point locates the image. The third ray serves as a check of the construction

Concave Mirror, p > R

The center of curvature is between the object and the concave mirror surface (f >0)

The image is real (q>0)

The image is inverted (M<0)

The image is smaller than the object (absM<1)

1 1 2 1

p q R f

Concave Mirror, p < f

The object is between the mirror surface and the focal point (p>0)

The image is virtual (q<0)

The image is upright (M>0)

The image is larger than the object (M>1)

Ray Diagrams:Convex Mirrors

Ray 1 is drawn from the top of the object parallel to the principal axis and is reflected away from the focal point, F

Ray 2 is drawn from the top of the object toward the focal point and is reflected parallel to the principal axis

Ray 3 is drawn through the center of curvature, C, on the back side of the mirror and is reflected back on itself

Convex Mirror

The object is in front of a convex mirror (p>0)

The focal point distance q is negative (q <0)

The image is always virtual and upright (M>0)

As the object distance decreases, the virtual image size increases

The image is smaller than the object (0<M<1)

Sign Conventions: Mirrors

1 1 2 1

p q R f

Optics Activity Fun with Mirrors!!!

Lenses

Image formation is a consequence of

light traveling in straight lines

The first camera—the pinhole camera—

illustrates this fact.

Lenses

A lens nicely bends the straight-line paths

of light.

Lenses

A converging lens can project an image.

Lenses Key features of lenses principal axis

line joining the centers of curvature of the two lens surfaces

focal point point at which all the light rays come together

focal length distance between the center of the lens and either

focal point

Lens Refraction

Converging & Diverging

Converging Lens Diverging Lens

Lenses Lenses

two common types converging (convex) lens

thicker at the center than edges

converges light

diverging (concave) lens thinner at the center than edges

diverges light

Focal Length:Converging Lens

Focal Length:Diverging Lens

Converging Thin Lens Shapes

These are examples

of converging lenses

They have positive

focal lengths

They are thickest in

the middle

Diverging Thin Lens Shapes

These are examples

of diverging lenses

They have negative

focal lengths

They are thickest at

the edges

Signs for Thin Lenses

1 1 2 1

p q R f

'h qM

h p

Compare Signs for Mirrors

and Thin Lenses

Thin Lenses

Ray Diagram for Converging

Lens, p > f

The image is real (q>0)

The image is inverted (M<0)

The image is on the back side of the lens (q>0)

Ray Diagram for Converging

Lens, p < f

The image is virtual (q < 0)

The image is upright (M>0)

The image is larger than the object (M>1)

The image is on the front side of the lens (q<0)

Ray Diagram for Diverging

Lens

For a diverging lens, the image is always virtual

and upright (M>0)

This is regardless of where the object is placed

The image is on the front side of the lens (q<0)

Optics Activity Fun with Lenses!!!!

Combination of Thin Lenses,

example