1

• Principles of Geometric Optics– A fancy name for how rays are

used to understand shadows, mirrors, lenses and rainbows

• Shadows– Shadows and eclipses are

produced when light rays are blocked

– A pinhole camera also works by blocking rays

• Mirrors– Rays bounce off mirrors to

produce images (specular reflection)

• Why can't you see your reflection in a white piece of paper but you can in a mirror?

• Lenses, mirages and fiber optics– Lenses in cameras and contact

lenses create images by bending light rays (refraction)

– Mirages are also due to the bending of light rays.

– Another example of bending light is provided by fiber optics used in communications

• Rainbows– Rainbows are produced by the

spreading of the different wavelengths in white light (also involves bouncing and bending) as white light enters and exits water droplets

• Prisms create rainbows by spreading wavelengths

Physics 1230: Light and ColorChapter #2: Geometric Optics

Which level of physics is needed to explain what properties of light?

• Image formation - – ray theory

• Wavelength color, polarization and diffraction - – wave theory (electricity and

magneticsm)

• Interaction of light with atoms - – quantum theory of photons

• Constant speed of light no matter how fast the source or observer is moving - – special theory of relativity

A point light source emits rays in all directions radially outwards

The rays from twopoint light sources look like this

The rays only tell us which directionthe light goes in. We know that the light gets dimmer as you move further away from the light source. (Think of the sun. It would be blinding if we were closer to the sun)

Rays that ARE blocked by the book

Shadows appear when rays are blocked

Point light source

Book

Wall

Sha

dow

Rays that are NOT blocked by the book

2 point light sources Book

Wall

AB

Extra credit: what happens to the shadow if I move the screen back

from the book?

umbraumbra

brightbright

penumbrapenumbra

penumbrapenumbra

brightbrightunblocked

unblocked

blocked

The two parts of the penumbra each get light from only one of the two bulbs. The umbraumbra gets no light from either of the two bulbs. The bright region gets light from both of the bulbs.

Concept question

Shadows tell us:

• A) What direction the light is shining from

• B) That something is blocking the light

• C) That light travels in straight lines

• D) A, B, & C• E) A & C

• Extra Credit Opportunity: how do we prove this?

We can extend the definition of the umbraumbra and penumbrapenumbra to exist in space even without a wall or screen!

BookAB

umbraumbra

brightbright

penumbrapenumbra

penumbrapenumbra

Wall

brightbrightThe light from B doesn't reach this penumbra

The light from A doesn't reach this penumbra

We can think about large light sources as being composed of many small light sources

An "extended object" consists of many points. Each point on the object emits or reflects rays in all

directions (unless the object is a mirror)

Incident rays from a frosted light bulb

MANY reflected rays comefrom each point on Alex.This is diffuse reflection

The more rays that reach apoint the brighter the point

• This is why regions outside the penumbra and umbra are brighter– These regions get light rays

from both point light sources

• The more lights you turn on the brighter the reflected light from objects in the room– See rays at right

Lightsource 2

Lightsource 1

Reflected raysfrom light 1

Reflected raysfrom light 2

Your eye seesa brighter nosethan with either

light source alone

An extended light source such as the sun (or a large light bulb) also produces an umbra and penumbra in

empty space behind the Earth (or another object)

Sun

• All rays coming from point A on the sun between the two dashed rays are blocked by Earth

• All rays coming from point B on the sun between the two dotted rays are blocked by Earth

• The umbra gets no light from any portion of the sun

• The umbra gets smaller not larger further behind Earth since the Sun is larger than Earth

• The penumbra gets light from part of the sun

– If you look back from the penumbra you can see part of the sun

• When the moon passes completely into the umbra there is a total eclipse of the moon.

– When the moon passes into the penumbra there is a partial eclipse of the moon

A

B

UmbraPenumbra

Penumbra

Rays from this part of the sunDO reach the upper penumbra

Rays from this part of the sunDON'T reach the upper penumbrabecause they are blocked by Earth

Solar eclipse Geometry:

NOTE: The umbra is usually about 200km wide

http://www.mreclipse.com/Special/SEprimer.html

moon

Total solar eclipse

During a solar eclipse, the shadow of the Moon passes over the surface of the Earth. From the

Earth, we can see the moon blocking the light of the Sun. Looking at the demonstration, you may think that solar eclipses happen very often. The

Sun, Earth, and Moon must be lined up just right, in order for a solar eclipse to take place.

This happens only two to five times a year. Since the Moon's shadow is so small, compared

to the size of the Earth, a solar eclipse can be seen from only small portions of the Earth.

http://micro.magnet.fsu.edu/primer/java/scienceopticsu/solar/index.html

Map of the total solar eclipse, Aug 1, 2008

http://en.wikipedia.org/wiki/Image:Solar_eclipse_animate_(2008-Aug-01).gif

http://antwrp.gsfc.nasa.gov/apod/image/0311/112003lunareclipse_koehn.gif

Schedule:

Lunar eclipse (partial and total)

During a lunar eclipse, the moon passes through the shadow of the Earth. As we look at the moon from the Earth, it looks to us as if the shadow of the Earth is slowly covering the moon. You may think that lunar eclipses happen very often. However, the Sun, Earth, and Moon must be lined up just right, in order for a lunar eclipse to take place. This happens very rarely. In most years there are only two lunar eclipses that can be seen only from certain places on Earth. In a partial lunar eclipse, the moon passes through the penumbra or part of the umbra. In a total lunar eclipse, the moon is completely

within the umbra.

Extra Credit Opportunity: Based on what we know about eclipses, how do we prove this?

A pinhole camera works by blocking rays (demo)

Pinhole Camera

Light bulb

Image oflight bulb

blocked rays

• What is an image?• A real image is formed on a screen when one or

more rays from each point on the object reach the corresponding points on the screen and no other rays from other points on the object reach those points

• Notice that this image is upside down and left-right reversed.

Using shadows…

If a lens is used instead of a pinhole the image is brighter because many of the previously blocked rays are bent so that they arrive at the correct place on the screen image

Pinhole Camera

Light bulb

Image of light bulb

blocked rays

Not just ONE ray from the filament but MANY now arrive at the corresponding image point so the image is BRIGHTER

previously blocked rays

Camera with lens

#1: Why do we need lenses in the modern cameras?

#2: How do we make a photo of an object that does not emit visible light (i.e., ourselves)?

The object photographed with a pinhole camera does not have to be self-luminous!

Pinhole Camera

Alex

Image ofAlex

blocked rays

• Once again this image is upside down and left-right reversed. Early photographs (daguerreotypes) were always left-right reversed;

• Note the correspondence between the distances object-camera-screen and image vs. object sizes

One of many rays of light shining on Alex

Reflected raysoff the real Alex go through the holeand make the image

Finding an image by using rays is called ray tracing.Trace rays from the object through the pinhole in the

camera to find the image rather than trusting your intuition!

Is the image of Alex smaller or larger than the real Alex?a)Smallerb)Largerc)Same size

Is the image of Alex smaller or larger than the real Alex?a)Smallerb)Largerc)Same size

Extra Credit Project (20 points):

Construct a camera on your own (see textbook for details, pages 35 & 36)

Plasma frequency of silverPlasma frequency of silver

Materials like metals with many mobile electrons can cancel out the light wave field in the forward direction so there is no

transmission but only reflection at certain wavelengths.

• Metals reflect all waves below a certain frequency

– the plasma frequency - which varies from metal to metal

• Silver is particularly interesting because it reflects light waves at all visible frequencies

– Its plasma frequency is at the top of the violet so it reflects all of the wavelengths below and appears whitish

• Gold and copper have a yellowish-brownish color because they reflect greens, yellows and reds but not blues or violets

– Red and green make yellow

Plasma frequency of goldPlasma frequency of gold

Plasma frequency of copperPlasma frequency of copper

What is a mirror?

• Since silver is such a good reflector a coating of silver on glass makes a good (common) mirror.

• If the silver coating is thin enough the mirror can be made to transmit 50% of the light and to reflect the other 50%– This is called a half-silvered

mirror

– A half-silvered mirror used with proper lighting can show objects on one side or the other of the mirror

Law of specular reflection of a ray from a mirror

Mirror

This angle = this angle

The normal to the mirror is an imaginaryline drawn perpendicular to it from where the incident ray hits the mirror

Normal

• One of many rays from a light bulb hits Alex's chin.

• The ray from the light bulb is diffusely reflected off his chin. We show one of the many rays coming off his chin hitting a mirror.

– This is called an incident ray

• The incident ray undergoes specular reflection off the mirror

– Note the reflected ray

• Draw the normal to the mirror– The angle of incidence = the angle

of reflection

How is an image produced in a mirror?Part 1: Ray-tracing

• To find out how Bob "sees" Alex by looking in the mirror we trace rays which obey the law of reflection

– Consider an incident ray from Alex's chin which reflects according to the law of reflection at a specific point on the mirror and goes into Bob's eye.

– Note - it is not easy to construct this ray! You cannot arbitrarly choose a point on the mirror and expect that the law of reflection will be satisfied

– Bob will see only this reflected ray from Alex's chin.

– Other refelected rays from Alex's chin will miss his eye (see right)

– A ray from Alex's hair will reflect at one point on the mirror into Alex's eye (and satisfies the law of reflection)

Mirror

AlexBob looks atAlex's image

• To find the image of Alex we must learn how Bob’s eye (and our eyes) interpret rays

• Bob cannot directly know whether the rays entering his eyes have been reflected or not!

• We interpret all rays coming into our eye as traveling from a fictitious image in a straight line to our eye even if they are reflected rays!

• To find the virtual (fictitious) image of Alex’s chin we extend each reflected ray backwards in a straight line to where there are no real rays

– Extend the ray reflected into Bob's eye from Alex's chin backward behind the mirror.

– Extend the ray reflected towards Bob's chest (why?) from Alex's chin backward (dashed line) behind the mirror.

– The image of Alex's chin will be behind the mirror at the intersection of the two backward-extended reflected rays.

– Note all reflected rays from his chin intersect at the same image pt. when extended backwards

How is an image produced in a mirror?Part 2: The psychology of ray interpretation

Mirror

Alex Bob looks atAlex's image

• To find the location of his hair in the virtual image we extend any reflected ray from his hair backwards

How is an image produced in a mirror?Part 3: The meaning of a virtual image

• If we trace rays for every ray from every part of Alex which reflects in the mirror

– we get a virtual image of the real Alex behind the mirror. It is virtual because there is no light energy there, no real rays reach it, and it cannot be seen by putting a screen at its position!!

Virtual image of Alexis behind mirror

Mirror

Alex

• When all of the reflected rays from Alex's chin are traced backwards they all appear to come from the virtual image of Alex’s chin

– Hence Alex's image is always in the same place regardless of where Bob looks

• The image chin is behind the mirror by a distance = to the distance the real chin is in front of the mirror

– This is true for all parts of Alex's image– Alex's virtual image is the same size as

the real Alex – Alex's image is further away from Bob

than the real Alex

Bob looks atAlex's image

Bob sees Alex's imagein the same place when

he moves his head

For simple (flat) mirrors the image location is therefore predictable without knowing where the

observer's eye is and without ray-tracing

Mirror

Mirror Mirror

Mirror

Homework HW2 due today after the class

• HW #2 due today;• New homework HW3

assigned today: see the course web page;

• HW1 – graded (10 points - 100% maximum);

Mirror

Alex• Question: Where are

the images of Alex in the 2 mirrors?

a) At A onlyb) At B onlyc) At A and B onlyd) At C onlye) At A, B and C

Multiple mirrors - a virtual image can act as a real object and have its own virtual image

Mirror

A C

B

The virtual image at A acts as an object to produce the virtual image of C. It acts as an intermediate image. More precisely it is the red rays which reflect as green rays.

A few words about virtual images

• Here is the real Alex• Here are some (diffusely reflected)

diverging rays coming off his nose– They can be seen by eyes at various

locations

• We only know his nose is there because our eyes receive the rays

• Therefore, we would see an image (virtual) of Alex if those rays reached our eyes even when he wasn't there.

• Mirrors can provide those rays!• The (imaginary) extension of

(reflected) rays behind the mirror look just like the real rays from the real Alex

Mirror (incidentrays not shown)

Speed of Light in free space & in materials

Note: the fact that light has a unique speed was not obvious at the beginning . At that time, the real questions were:

          1. Is the speed of light infinite or finite?

          2. If the speed is finite, is it variable or a constant?

Everyday experience suggests that speed is very great and might be infinite.

How one can measure speed of light?

• General method: speed= distance/time

• The difficulty: when speed is very great, distance must be large and/or clock must have lots of resolution;

• Echo Method to measure the speed of a sound wave?

• Michelson: clock is rotating mirror wheel; result= 299,774 km/s

Echo method : measuring the speed of light

Reflection of waves occurs where the medium of propagation changes abruptly

• Part of the wave can be transmitted into the second medium while part is reflected back– You can hear someone from

outside the pool when you are underwater because sound waves are transmitted from the air through the water (withdifferent speed in each).

• When light waves are incident on a glass slab they are mostly transmitted but partly reflected (about 4%)!

Glass slab

Is the speed of light in the glass slab the same as in the free space???

No.

How can reflection require that the speed of the wave changes? We thought the speed of light was always

c = 3 x 108 m/s!

• The speed of an electromagnetic (EM) wave is constant (for every wavelength) in empty space!

• The speed of light is slower than c in glass, water and other transparent media– (Einstein showed that light can

never travel faster than c)

• The speed of light in a medium is v = c/n, where n is a number larger than one called the index of refraction• n = 1.5 for glass• n = 1.3 for water• n = 1.5 for vegetable oil

• Light is reflected and transmitted at a boundary because– When a light wave travels in a

medium the electric field of the light jiggles the electrons in the medium.

– This produces new electric fields which can cancel or add to the original light wave both in the forward and backward directions

• These are the transmitted and reflected light waves

Material Refractive Index

Air 1.0008

Water 1.330

Glass 1.5

Diamond 2.417

Ruby 1.760

Oil 1.5

Refractive indices of different materials

Can we see a glass rod immersed into the oil with the same refractive index?

• A. Yes

• B. No

Refraction is the bending of a ray after it enters a medium where its speed is different

• A ray going from a fast medium to a slow medium bends towards the normal to the surface of the medium

• A ray going from a slow medium to a fast medium bends away from the normal to the surface of the medium

• The speed of light in a medium is v = c/n, where n is a number larger than one called the index of refraction andc = 3 x 108 m/s• n = 1.3 for glass• n = 1.5 for water

• Hence, a ray going into a medium with a higher index of refraction bends towards the normal and a ray going into a medium with a lower index of refraction bends away from the normal

Air (fast medium)

Glass orwater(slow)

Normal

Glass orwater(slow)

Normal Air (fast medium)

nair < nwater

1.0008 < 1.33

How about light going into a medium with exactly the same index of refraction?

Ray-bending together with our psychological straight-ray interpretation determine the location of images underwater

• The precise amount of bending is determined by the law of refraction (sometimes called Snell's law):

• ni sini = nt sint

• Here,i = angle between incident ray and normal,

• and t = angle between transmitted ray and normal

• ni and nt are the indices of refraction in the medium containing the incident ray and in the medium containing the transmitted ray

• Fig 2.49 Fisherman and fish

incident ray

transmitted ray

normal

image of fish for someone out of water

fish

• In order to observe the fish from outside the water a transmitted ray must enter your eye.

• You will think it comes from a point obtained by tracing it backwards,

• Extend any 2 of the many many transmitted rays from the fish backwards to find the image of the fish (where they intersect).

• The location of that image will be the same for any observer outside of the water.

What we see and how different it can be from what it seems to be

• The woman will see the underwater part of body being

a) Smaller than it really is;b) Much larger than it really is;c) Of natural size;

• The boy will see the underwater part of body being

a) Smaller than it really is;b) Much larger than it really is;c) Of natural size;

Total internal reflection is an extreme case of a ray bending away from the normal as it goes from a higher to a lower index of

refraction medium (from a slower to a faster medium)

Glass orwater(slow)

Normal

Air (fast medium)

Just below the critical angle for total internal reflection there is a reflected and a transmitted (refracted) ray

Glass orwater(slow)

Normal

Just above the critical angle for total internal reflection there is a reflected ray but no transmitted (refracted) ray

CriticalCriticalangleangle

For the glass-air interface

Total internal reflection

• Show that the internal reflection is a consequence of the Snell’s law

• The precise amount of bending is determined by the law of refraction (sometimes called Snell's law):

• ni sini = nt sint

• Here,i = angle between incident ray and normal,

• and t = angle between transmitted ray and normal

• ni and nt are the indices of refraction in the medium containing the incident ray and in the medium containing the transmitted ray

What we see and how different it can be from what it seems to be

• If the critical angle condition is satisfied, will the boy see the part of body above water:

a) yes;b) No.

• Extra Credit: Refractive index of water is 1.33;

What is the critical angle for the case of air-water interface?

0ptical fibers: how they work

• Application of total internal reflection phenomenon;

• Telecommunications;

• Can see inside a body of a living person – applications in medicine;

Total internal reflection

Mirages can occur if the index of refraction increases as the ray goes deeper into the slower medium?

• Refraction of light

• Light propagates along a curved line because of the multiple refraction phenomena

• However, in our mind, analyzing what we see, we do not take this into account and assume that light entering our eyes traveled along straight lines

Incident ray

High index

Higher index

Higher index

Morris, in Heavens Command, describes a view from Grosse Isle in the Gulf of St. Lawrence : "in the early morning sun the islands are inverted in mirage, and seem to hang there, suspended between sky and water." In the early morning, the air gets warm faster than the water, so there is warmer air above the cooler air that is lying just above the water. How do the light rays bend, resulting in the mirage Morris describes.

• Since the index of refraction decreases at higher altitudes the red ray coming from the island continuously bends away from the (vertical) normal.

• Think of your car with right wheels on a good road and left wheels on the muddy terrain

– At the top of its trajectory critical reflection occurs and the reflected ray then refracts towards the normal on its way down until it reaches the eye.

– The brain then interprets this curved ray as a straight ray tangent to the curved one at the point of entry into the eye.

– Hence, the island appears to be in the sky.

Hot air, low density, index of refraction closer to 1, FAST medium

Grosse Isle

Image of Grosse Isle

Sky

Water

Grosse Isle

Sky

Cool air, higher density, index of refraction > 1,SLOW medium

Cool air

Hot air

A closer look at the curved ray from Grosse Isle to the eye of the beholder

• As the curved ray moves up and bends towards the horizon it is bending away from the normal– Think of marching soldiers leaving

deepest mud

Slow

Fast

High index

Highest index

• At the top of its trajectory the ray undergoes total internal reflection

• Then the curved ray begins to move down and towards the normal. – We can see sun after it is below the

geometrical horizon:

More common mirages occur when the lower air is hotter than the cooler air

• When the road (or desert) is hot the air is thin close to the road (or desert) so the density is lower.

• The density rises slowly above the road to heights which are cooler.

• The index of refraction of air increases as the density increases.

• A mirage results because we interpret the last ray to enter our eye as having traveled a straight line rather than a curve

Higher tempmeans lower

densityLower index Faster light speed

Lower tempmeans higherdensity

Higher index Slower light speed

Ray bends away from normal

HOT ROAD

Dispersion is responsible for rainbows

• Dispersion causes the spreading of all of the colors in white light

• This can be done by a prism to create a spectrum or by raindrops to create a rainbow

• Dispersion occurs because the speed of short wavelength light (blues) is slightly slower than that of long wavelengths (reds) in glass or water

• Hence short wavelengths (blues) bend more towards the normal than long wavelengths (reds) when white light enters glass or water. Blues also bend more away from the normal than reds when leaving glass or water

What is the normal to a curved surface and how is it used to find rays?

• To find the normal to a curved surface at a point where a ray hits that surface (and will be reflected or refracted)– First draw a tangent line to the

curve (or tangent plane to the surface)

– The normal is perpendicular to that line or plane and going through the point

– Once you have drawn the normal you can draw the reflected or refracted ray

How does a single raindrop contribute to a rainbow?

• White light enters the waterdrop– Remember white light contains rays of all

wavelengths

• The blue ray is refracted closer to the normal than the red ray

– Remember light travels slower in water than in air

• This greater bending of the blue ray than the red ray is called dispersion

– We have not shown the green, yellow, and orange rays but they each bend by amounts more than the red and less than the blue

• All of the rays reflect off the inside of the raindrop and then undergo another dispersion as they exit the raindrop

– The laws of reflection and refraction are always obeyed

Raindrop

Dispersion occurs here during refraction

white lightcomes in

Reflections

Dispersion occurs here during refraction

A spectrum ofcolors comes out

How we see all the colors in the rainbow?

Primary & Secondary Rainbows

Raindrop

Dispersion occurs here during refraction

white lightcomes in

Reflections

Dispersion occurs here during refraction

A spectrum ofcolors comes out

Secondary

Primary

Note inverted color sequence;

Question about the Rainbow

• The colors in rainbow are caused by:

• (A) Dispersion of light;• (B) Absorption of light;• (C) Total internal reflection

and dispersion of light;• (D) Light sources of

different color;• (E) Scattering;

Scattering of light: how we see fog, clouds, milk

• Clouds –water droplets, crystals of ice, or both;

• Fog – water droplets;• There are tiny droplets in

milk too;

• Why we see clouds, fog, and milk being of more or less white color?

• Small droplets scatter light of all colors reflected in all directions;

Question

• A pen semi-immersed in water appears bent because of light

• A. reflection;• B. refraction;• C. dispersion;• D. A,B, &C;• E. total internal

reflection;

Demonstrations

• Rainbow;• Dispersion of light

after going through a prism;

• Retro-reflector;• Half-silvered mirror;

Extra credit project: magic box or periscope (p.44 & p.52 of the textbook)

• 20 points extra credit;• To be submitted before the

last exam;• Magic box: can show on

Pearl street

• Periscope – how could we have skyscrapers & perfect mountain views in Boulder at the same time;

Read chapter 2 of the textbook

• Wave propagation in different media and reflection/refraction at interfaces (1st HW problem);

• Mirages;• Sunsets;• Subsuns;• Sun pillars;• Reflections of diamonds;• Dry vs. wet road and diffuse vs.

specular reflection;• Sun dogs, halos, & more;

• HW #3 is due on Thursday;

• Chapter # 3: mirrors and lenses;