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Optics. Reflection Diffuse reflection Refraction Index of refraction Speed of light Snell’s law Geometry problems Critical angle Total internal reflection Brewster angle Fiber optics Mirages Dispersion. Prisms Rainbows Plane mirrors Spherical aberration

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Optics

Optics

  • Reflection

  • Diffuse reflection

  • Refraction

  • Index of refraction

  • Speed of light

  • Snell’s law

  • Geometry problems

  • Critical angle

  • Total internal reflection

  • Brewster angle

  • Fiber optics

  • Mirages

  • Dispersion

  • Prisms

  • Rainbows

  • Plane mirrors

  • Spherical aberration

  • Concave and convex mirrors

  • Focal length & radius of curvature

  • Mirror / lens equation

  • Convex and concave lenses

  • Human eye

  • Chromatic aberration

  • Telescopes

  • Huygens’ principle

  • Diffraction


Reflection

Reflection

Most things we see are thanks to reflections, since most objects don’t produce their own visible light. Much of the light incident on an object is absorbed but some is reflected. the wavelengths of the reflected light determine the colors we see. When white light hits an apple, for instance, primarily red wavelengths are reflected, while much of the others are absorbed.

A ray of light heading towards an object is called an incident ray. If it reflects off the object, it is called a reflected ray. A perpendicular line drawn at any point on a surface is called a normal (just like with normal force). The angle between the incident ray and normal is called the angle of incidence, i, and the angle between the reflected ray and the normal ray is called the angle of reflection, r. The law of reflection states that the angle of incidence is always equal to the angle of reflection.


Law of reflection

Law of Reflection

Normal line (perpendicular to surface)

r

i

reflected rays

incident rays

i = r


Diffuse reflection

Diffuse Reflection

Diffuse reflection is when light bounces off a non-smooth surface. Each ray of light still obeys the law of reflection, but because the surface is not smooth, the normal can point in a different for every ray. If many light rays strike a non-smooth surface, they could be reflected in many different directions. This explains how we can see objects even when it seems the light shining upon it should not reflect in the direction of our eyes. It also helps to explain glare on wet roads: Water fills in and smoothes out the rough road surface so that the road becomes more like a mirror.


Speed of light refraction

Speed of Light & Refraction

As you have already learned, light is extremely fast, about 3108 m/s in a vacuum. Light, however, is slowed down by the presence of matter. The extent to which this occurs depends on what the light is traveling through. Light travels at about 3/4 of its vacuum speed (0.75 c) in water and about 2/3 its vacuum speed (0.67 c) in glass. The reason for this slowing is because when light strikes an atom it must interact with its electron cloud. If light travels from one medium to another, and if the speeds in these media differ, then light is subject to refraction (a changing of direction at the interface).

Refraction of light waves

Refraction of light rays


Reflection refraction

At an interface between two media, both reflection and refraction can occur. The angles of incidence, reflection, and refraction are all measured with respect to the normal. The angles of incidence and reflection are always the same. If light speeds up upon entering a new medium, the angle of refraction, r, will be greater than the angle of incidence, as depicted on the left. If the light slows down in the new medium, r will be less than the angle of incidence, as shown on the right.

Reflection & Refraction

Reflected Ray

Reflected Ray

Incident Ray

Incident Ray

r

Refracted Ray

normal

normal

Refracted Ray

r


Axle analogy

Axle Analogy

Imagine you’re on a skateboard heading from the sidewalk toward some grass at an angle. Your front axle is depicted before and after entering the grass. Your right contacts the grass first and slows, but your left wheel is still moving quickly on the sidewalk. This causes a turn toward the normal. If you skated from grass to sidewalk, the same path would be followed. In this case your right wheel would reach the sidewalk first and speed up, but your left wheel would still be moving more slowly. The result this time would be turning away from the normal. Skating from sidewalk to grass is like light traveling from air to a more

overhead view

“optically dense” medium like glass or water. The slower light travels in the new medium, the more it bends toward the normal. Light traveling from water to air speeds up and bends away from the normal. As with a skateboard, light traveling along the normal will change speed but not direction.

sidewalk

grass

r


Index of refraction n

c v

n =

Index of Refraction, n

The index of refraction of a substance is the ratio of the speed in light in a vacuum to the speed of light in that substance:

Medium

Vacuum

Air (STP)

Water (20º C)

Ethanol

Glass

Diamond

n

1

1.00029

1.33

1.36

~1.5

2.42

n = Index of Refraction

c = Speed of light in vacuum

v = Speed of light in medium

Note that a large index of refraction corresponds to a relatively slow light speed in that medium.


Snell s law

Snell’s Law

i

ni

nr

r

Snell’s law states that a ray of light bends in such a way that the ratio of the sine of the angle of incidence to the sine of the angle of refraction is constant. Mathematically,

nisini = nr sinr

Here ni is the index of refraction in the original medium and nr is the index in the medium the light enters. iandrare the angles of incidence and refraction, respectively.

Willebrord Snell


Snell s law derivation

1

A

n1

x

A

d

B

y

n2

B

2

Two parallel rays are shown. Points A and B are directly opposite one another. The top pair is at one point in time, and the bottom pair after time t. The dashed lines connecting the pairs are perpendicular to the rays. In time t, point A travels a distance x, while point B travels a distance y.

sin1= x/d, so x = d sin1

sin2 = y/d, so y = d sin2

Speed of A: v1 = x/ t

Speed of B: v2 = y/ t

Snell’s Law Derivation

Continued…


Snell s law derivation cont

1

A

n1

x

A

d

B

y

n2

B

2

Snell’s Law Derivation (cont.)

v1 x/t x sin1

=

=

=

So,

v2 y/t y sin2

v1/c sin11/n1sin1n2

=

=

=

v2/c sin21/n2sin2 n1

n1 sin1 = n2 sin2


Refraction problem 1

Refraction Problem #1

Goal: Find the angular displacement of the ray after having passed through the prism. Hints:

  • Find the first angle of refraction using Snell’s law.

  • Find angle ø. (Hint: Use Geometry skills.)

  • 3. Find the second angle of incidence.

  • Find the second angle of refraction, , using Snell’s Law

19.4712º

79.4712º

Air, n1 = 1

30°

10.5288º

Horiz. ray, parallel to base

ø

15.9º

Glass, n2 = 1.5


Refraction problem 2

Refraction Problem #2

Goal: Find the distance the light ray displaced due to the thick window and how much time it spends in the glass. Some hints are given.

20º

1

1. Find 1 (just for fun).

2. To show incoming & outgoing rays are parallel, find .

3. Find d.

4. Find the time the light spends in the glass.

Extra practice: Find  if bottom medium is replaced with air.

20º

H20

n1 = 1.3

20º

0.504 m

glass

10m

n2 = 1.5

5.2 ·10-8 s

d

H20

26.4º


Refraction problem 3

Refraction Problem #3

Goal: Find the exit angle relative to the horizontal.

19.8°

 =

36°

air

 = ?

glass

The triangle is isosceles.Incident ray is horizontal, parallel to the base.


Reflection problem

Reflection Problem

Goal: Find incident angle relative to horizontal so that reflected ray will be vertical.

 =

10º

50º

center of semicircular mirror with horizontal base


Brewster angle

n2

n1

b

b

Brewster Angle

The Brewster angle is the angle of incidence the produces reflected and refracted rays that are perpendicular.

From Snell, n1 sinb = n2 sin.

α = b since + =90º, andb + =90º.

β= since  + =90º, and + = 90º. Thus,

n1 sinb = n2 sin = n2 sin = n2 cosb

tanb = n2/n1

Sir David Brewster


Critical angle

Critical Angle

The incident angle that causes the refracted ray to skim right along the boundary of a substance is known as the critical angle, c. The critical angle is the angle of incidence that produces an angle of refraction of 90º. If the angle of incidence exceeds the critical angle, the ray is completely reflected and does not enter the new medium. A critical angle only exists when light is attempting to penetrate a medium of higher optical density than it is currently traveling in.

nr

ni

c

From Snell, n1 sinc = n2 sin90

Since sin90 = 1, we haven1 sinc = n2 and the critical angle is

nr

c = sin-1

ni


Critical angle sample problem

Critical Angle Sample Problem

Calculate the critical angle for the diamond-air boundary.

Refer to the Index of Refraction chart for the information.

c =sin-1(nr /ni)

= sin-1(1/2.42)

=24.4

Any light shone on this boundary beyond this angle will be reflected back into the diamond.

air

diamond

c


Total internal reflection

Total Internal Reflection

Total internal reflection occurs when light attempts to pass from a more optically dense medium to a less optically dense medium at an angle greater than the critical angle. When this occurs there is no refraction, only reflection.

n1

n2 >

n1

n2

 > c

Total internal reflection can be used for practical applications like fiber optics.


Fiber optics

Fiber Optics

Fiber optic lines are strands of glass or transparent fibers that allows the transmission of light and digital information over long distances. They are used for the telephone system, the cable TV system, the internet, medical imaging, and mechanical engineering inspection.

spool of optical fiber

Optical fibers have many advantages over copper wires. They are less expensive, thinner, lightweight, and more flexible. They aren’t flammable since they use light signals instead of electric signals. Light signals from one fiber do not interfere with signals in nearby fibers, which means clearer TV reception or phone conversations.

A fiber optic wire

Continued…


Fiber optics cont

Fiber Optics Cont.

Fiber optics are often long strands of very pure glass. They are very thin, about the size of a human hair. Hundreds to thousands of them are arranged in bundles (optical cables) that can transmit light great distances. There are three main parts to an optical fiber:

  • Core- the thin glass center where light travels.

  • Cladding- optical material (with a lower index of refraction than the core) that surrounds the core that reflects light back into the core.

  • Buffer Coating- plastic coating on the outside of an optical fiber to protect it from damage.

Continued…


Fiber optics cont1

Light travels through the core of a fiber optic by continually reflecting off of the cladding. Due to total internal reflection, the cladding does not absorb any of the light, allowing the light to travel over great distances. Some of the light signal will degrade over time due to impurities in the glass.

Fiber Optics (cont.)

There are two types of optical fibers:

  • Single-mode fibers-transmit one signal per fiber (used in cable TV and telephones).

  • Multi-mode fibers- transmit multiple signals per fiber (used in computer networks).


Mirage pictures

Mirage Pictures

Mirages


Mirages

Mirages

Mirages are caused by the refracting properties of a non-uniform atmosphere.

Several examples of mirages include seeing “puddles” ahead on a hot highway or in a desert and the lingering daylight after the sun is below the horizon.

More Mirages

Continued…


Inferior mirages

Inferior Mirages

A person sees a puddle ahead on the hot highway because the road heats the air above it, while the air farther above the road stays cool. Instead of just two layers, hot and cool, there are really

many layers, each slightly hotter than the layer above it. The cooler air has a slightly higher index of refraction than the warm air beneath it. Rays of light coming toward the road gradually refract further from the normal, more parallel to the road. (Imagine the wheels and axle: on a light ray coming from the sky, the left wheel is always in slightly warmer air than the right wheel, so the left wheel continually moves faster, bending the axle more and more toward the observer.) When a ray is bent enough, it surpasses the critical angle and reflects. The ray continues to refract as it heads toward the observer. The “puddle” is really just an inverted image of the sky above. This is an example of an inferior mirage, since the cool are is above the hot air.


Superior mirages

Superior Mirages

Superior mirages occur when a layer of cool air is beneath a layer of warm air. Light rays are bent downward, which can make an object seem to be higher in the air and inverted. (Imagine the wheels and axle on a ray coming from the boat: the right wheel is continually in slightly warmer air than the left wheel. Thus, the right wheel moves slightly faster and bends the axle toward the observer.) When the critical angle is exceeded the ray reflects. These

mirages usually occur over ice, snow, or cold water. Sometimes superior images are produced without reflection. Eric the Red, for example, was able to see Greenland while it was below the horizon due to the light gradually refracting and following the curvature of the Earth.


Sunlight after sunset

Lingering daylight after the sun is below the horizon is another effect of refraction. Light travels at a slightly slower speed in Earth’s atmosphere than in space. As a result, sunlight is refracted by the atmosphere. In the morning, this refraction causes sunlight to reach us before the sun is actually above the horizon. In the evening, the

Sunlight after Sunset

Apparent position of sun

Observer

Actual position of sun

Earth

Atmosphere

sunlight is bent above the horizon after the sun has actually set. So daylight is extended in the morning and evening because of the refraction of light. Note: the picture greatly exaggerates this effect as well as the thickness of the atmosphere.

Different “shapes” of Sun


Dispersion of light

Dispersion of Light

Dispersion is the separation of light into a spectrum by refraction. The index of refraction is actually a function of wavelength. For longer wavelengths the index is slightly small. Thus, red light refracts less than violet. (The pic is exaggerated.) This effect causes white light to split into it spectrum of colors. Red light travels the fastest in glass, has a smaller index of refraction, and bends the least. Violet is slowed down the most, has the largest index, and bends the most. In other words: the higher the frequency, the greater the bending.

Animation


Atmospheric optics

Atmospheric Optics

There are many natural occurrences of light optics in our atmosphere.

One of the most common of these is the rainbow, which is caused by water droplets dispersing sunlight. Others include arcs, halos, cloud iridescence, and many more.

Photo gallery of atmospheric optics.


Rainbows

A rainbow is a spectrum formed when sunlight is dispersed by water droplets in the atmosphere. Sunlight incident on a water droplet is refracted. Because of dispersion, each color is refracted at a slightly different angle. At the back surface of the droplet, the light undergoes total internal reflection. On the

Rainbows

way out of the droplet, the light is once more refracted and dispersed. Although each droplet produces a complete spectrum, an observer will only see a certain wavelength of light from each droplet. (The wavelength depends on the relative positions of the sun, droplet, and observer.) Because there are millions of droplets in the sky, a complete spectrum is seen. The droplets reflecting red light make an angle of 42o with respect to the direction of the sun’s rays; the droplets reflecting violet light make an angle of 40o.

Rainbow images


Primary rainbow

Primary Rainbow


Secondary rainbow

The secondary rainbow is a rainbow of radius 51, occasionally visible outside the primary rainbow. It is produced when the light entering a cloud droplet is reflected twice internally and then exits the droplet. The color spectrum is reversed in respect to the primary rainbow, with red appearing on its inner edge.

Secondary

Secondary Rainbow

Primary

Alexander’s dark region


Supernumerary arcs

Supernumerary Arcs

Supernumerary arcs are faint arcs of color just inside the primary rainbow. They occur when the drops are of uniform size. If two light rays in a raindrop are scattered in the same direction but have take different paths within the drop, then they could interfere with each other constructively or destructively. The type of interference that occurs depends on the difference in distance traveled by the rays. If that difference is nearly zero or a multiple of the wavelength, it is constructive, and that color is reinforced. If the difference is close to half a wavelength, there is destructive interference.


Real vs virtual images

Real vs. Virtual Images

Real images are formed by mirrors or lenses when light rays actually converge and pass through the image. Real images will be located in front of the mirror forming them. A real image can be projected onto a piece of paper or a screen. If photographic film were placed here, a photo could be created.

Virtual images occur where light rays only appear to have originated. For example, sometimes rays appear to be coming from a point behind the mirror. Virtual images can’t be projected on paper, screens, or film since the light rays do not really converge there.

Examples are forthcoming.


Plane mirror

Object

Rays emanating from an object at point P strike the mirror and are reflected with equal angles of incidence and reflection. After reflection, the rays continue to spread. If we extend the rays backward behind the mirror, they will intersect at point P’, which is the image of point P. To an observer, the rays appear to come from point P’, but no source is there and no rays actually converging there . For that reason, this image at P’ is a virtual image.

Plane Mirror

P

P’

Virtual Image

do

di

O

I

The image, I, formed by a plane mirror of an object, O, appears to be a distance di , behind the mirror, equal to the object distance do.

Animation

Continued…


Plane mirror cont

Plane Mirror (cont.)

Two rays from object P strike the mirror at points B and M. Each ray is reflected such that i = r.

P’

Triangles BPM and BP’M are congruent by ASA (show this), which implies that do= di and h = h’. Thus,the image is the same distance behind the mirror as the object is in front of it, and the image is the same size as the object.

do

di

P

B

h’

M

h

Image

Object

object

image

Mirror

With plane mirrors, the image is reversed left to right (or the front and back of an image ). When you raise your left hand in front of a mirror, your image raises its right hand. Why aren’t top and bottom reversed?


Concave and convex mirrors

Concave and Convex Mirrors

Concave and convex mirrors are curved mirrors similar to portions of a sphere.

light rays

light rays

Concave mirrors reflect light from their inner surface, like the inside of a spoon.

Convex mirrors reflect light from their outer surface, like the outside of a spoon.


Concave mirrors

Concave Mirrors

  • Concave mirrors are approximately spherical and have a principal axis that goes through the center, C, of the imagined sphere and ends at the point at the center of the mirror, A. The principal axis is perpendicular to the surface of the mirror at A.

  • CA is the radius of the sphere,or the radius of curvature of the mirror, R.

  • Halfway between C and A is the focal point of the mirror, F. This is the point where rays parallel to the principal axis will converge when reflected off the mirror.

  • The length of FA is the focal length, f.

  • The focal length is half of the radius of the sphere (proven on next slide).


R 2 f

To prove that the radius of curvature of a concave mirror is twice its focal length, first construct a tangent line at the point of incidence. The normal is perpendicular to the tangent and goes through the center, C. Here, i = r = .By alt. int. angles the angle at C is also ,andα= 2β. s is the arc length from the principle axis to the pt. of incidence. Now imagine a sphere centered at F with radius f. If the incident ray is close to the principle axis, the arc length of the new sphere is about the same ass. Froms = r, we haves = rβand s  fα= 2fβ.Thus,rβ 2fβ, andr = 2f.

r = 2f

tangent line

s

f

C

F

r


Focusing light with concave mirrors

Focusing Light with Concave Mirrors

Light rays parallel to the principal axis will be reflected through the focus (disregarding spherical aberration, explained on next slide.)

In reverse, light rays passing through the focus will be reflected parallel to the principal axis, as in a flood light.

Concave mirrors can form both real and virtual images, depending on where the object is located, as will be shown in upcoming slides.


Spherical aberration

C

F

Spherical Aberration

F

C

Spherical Mirror

Parabolic Mirror

Only parallel rays close to the principal axis of a spherical mirror will converge at the focal point. Rays farther away will converge at a point closer to the mirror. The image formed by a large spherical mirror will be a disk, not a point. This is known as spherical aberration.

Parabolic mirrors don’t have spherical aberration. They are used to focus rays from stars in a telescope. They can also be used in flashlights and headlights since a light source placed at their focal point will reflect light in parallel beams. However, perfectly parabolic mirrors are hard to make and slight errors could lead to spherical aberration. Continued…


Spherical vs parabolic mirrors

Spherical vs. Parabolic Mirrors

Parallel rays converge at the focal point of a spherical mirror only if they are close to the principal axis. The image formed in a large spherical mirror is a disk, not a point (spherical aberration).

Parabolic mirrors have no spherical aberration. The mirror focuses all parallel rays at the focal point. That is why they are used in telescopes and light beams like flashlights and car headlights.

SPHERICAL

vs.

PARABOLIC


Concave mirrors object beyond c

image

Concave Mirrors: Object beyond C

The image formed when an object is placed beyond C is located between C and F. It is a real, inverted image that is smaller in size than the object.

object

C

F

Animation 1

Animation 2


Concave mirrors object between c and f

image

Concave Mirrors: Object between C and F

The image formed when an object is placed between C and F is located beyond C. It is a real, inverted image that is larger in size than the object.

object

C

F

Animation 1

Animation 2


Concave mirrors object in front of f

image

Concave Mirrors: Object in front of F

The image formed when an object is placed in front of F is located behind the mirror. It is a virtual, upright image that is larger in size than the object. It is virtual since it is formed only where light rays seem to be diverging from.

object

C

F

Animation


Concave mirrors object at c or f

The image will be formed at C also, but it will be inverted. It will be real and the same size as the object.

No image will be formed. All rays will reflect parallel to the principal axis and will never converge. The image is “at infinity.”

Concave Mirrors: Object at C or F

What happens when an object is placed at C?

Animation

What happens when an object is placed at F?


Convex mirrors

light rays

Convex Mirrors

  • A convex mirror has the same basic properties as a concave mirror but its focus and center are located behind the mirror.

  • This means a convex mirror has a negative focal length (used later in the mirror equation).

  • Light rays reflected from convex mirrors always diverge, so only virtual images will be formed.

  • Rays parallel to the principal axis will reflect as if coming from the focus behind the mirror.

  • Rays approaching the mirror on a path toward F will reflect parallel to the principal axis.


Convex mirror diagram

Convex Mirror Diagram

The image formed by a convex mirror no matter where the object is placed will be virtual, upright, and smaller than the object. As the object is moved closer to the mirror, the image will approach the size of the object.

object

image

F

C


Mirror lens equation derivation

s

 =

s

r

 

do

s

 

di

Mirror/Lens Equation Derivation

From PCO,  =  + , so 2 = 2 + 2. From PCO,  = 2 + , so - = -2 - .

Adding equations yields 2 -  = .

P

Froms = r, we haves = rβ, s  diα,and s  diα(for rays close to the principle axis). Thus:

object

s

T

O

C

image

di

do

(cont.)


Mirror lens equation derivation cont

Mirror/Lens Equation Derivation (cont.)

From the last slide,  = s/r,  s/d0,  s/di, and2β -  = . Substituting into the last equation yields:

P

2s

s

s

-

object

=

s

r

di

do

2

1

1

=

+

r

T

O

di

do

C

image

1

1

2

=

+

2f

di

do

di

1

1

1

=

+

f

di

do

do

The last equation applies to convex and concave mirrors, as well as to lenses, provided a sign convention is adhered to.


Mirror sign convention

Mirror Sign Convention

f = focal length

di= image distance

do = object distance

1

1

1

=

+

f

di

do

+ for real image

- for virtual image

di

+ for concave mirrors

- for convex mirrors

f


Magnification

hi

m =

ho

Magnification

By definition,

m = magnification

hi = image height (negative means inverted)

ho = object height

Magnification is simply the ratio of image height to object height. A positive magnification means an upright image.


Magnification identity

hi

-di

m =

=

do

ho

Magnification Identity:

To derive this let’s look at two rays. One hits the mirror on the axis. The incident and reflected rays each make angle relative to the axis. A second ray is drawn through the center and is reflected back on top of itself (since a radius is always perpendicular to an tangent line of a

circle). The intersection of the reflected rays

determines thelocation of

the tip of the image. Our result follows from similar triangles, with the negative sign a consequence of our sign convention. (In this picture hiis negative and diis positive.)

object

ho

C

image, height = hi

di

do


Mirror equation sample problem

C

F

Mirror Equation Sample Problem

Suppose AllStar, who is 3 and a half feet tall, stands 27 feet in front of a concave mirror with a radius of curvature of 20 feet. Where will his image be reflected and what will its size be?

di =

15.88 feet

hi =

-2.06 feet


Mirror equation sample problem 2

F

C

Mirror Equation Sample Problem 2

Casey decides to join in the fun and she finds a convex mirror to stand in front of. She sees her image reflected 7 feet behind the mirror which has a focal length of 11 feet. Her image is 1 foot tall. Where is she standing and how tall is she?

do =

19.25 feet

ho =

2.75 feet


Lenses

Lenses

Lenses are made of transparent materials, like glass or plastic, that typically have an index of refraction greater than that of air. Each of a lens’ two faces is part of a sphere and can be convex or concave (or one face may be flat). If a lens is thicker at the center than the edges, it is a convex, or converging, lens since parallel rays will be converged to meet at the focus. A lens which is thinner in the center than the edges is a concave, or diverging, lens since rays going through it will be spread out.

Convex (Converging) Lens

Concave (Diverging) Lens


Lenses focal length

Lenses: Focal Length

  • Like mirrors, lenses have a principal axis perpendicular to their surface and passing through their midpoint.

  • Lenses also have a vertical axis, or principal plane, through their middle.

  • They have a focal point, F, and the focal length is the distance from the vertical axis to F.

  • There is no real center of curvature, so 2F is used to denote twice the focal length.


Ray diagrams for lenses

F

2F

2F

F

F

2F

2F

F

Ray Diagrams For Lenses

When light rays travel through a lens, they refract at both surfaces of the lens, upon entering and upon leaving the lens. At each interface the bends toward the normal. (Imagine the wheels and axle.) To simplify ray diagrams, we often pretend that all refraction occurs at the vertical axis. This simplification works well for thin lenses and provides the same results as refracting the light rays twice.

Reality

Approximation


Convex lenses

F

2F

2F

F

F

2F

2F

F

  • Rays traveling parallel to the principal axis of a convex lens will refract toward the focus.

  • Rays traveling directly through the center of a convex lens will leave the lens traveling in the exact same direction.

Convex Lenses

F

2F

2F

F

Rays traveling from the focus will refract parallel to the principal axis.


Convex lens object beyond 2f

image

Convex Lens: Object Beyond 2F

The image formed when an object is placed beyond 2F is located behind the lens between F and 2F. It is a real, inverted image which is smaller than the object itself.

object

F

2F

2F

F

Experiment with this diagram


Convex lens object between 2f and f

image

Convex Lens: Object Between 2F and F

The image formed when an object is placed between 2F and F is located beyond 2F behind the lens. It is a real, inverted image, larger than the object.

object

F

2F

2F

F


Convex lens object within f

image

Convex Lens: Object within F

The image formed when an object is placed in front of F is located somewhere beyond F on the same side of the lens as the object. It is a virtual, upright image which is larger than the object. This is how a magnifying glass works. When the object is brought close to the lens, it will be magnified greatly.

F

2F

2F

F

object

convex lens used as a magnifier


Concave lenses

F

2F

2F

F

F

2F

2F

F

F

2F

2F

F

Concave Lenses

  • Rays traveling parallel to the principal axis of a concave lens will refract as if coming from the focus.

  • Rays traveling directly through the center of a concave lens will leave the lens traveling in the exact same direction, just as with a convex lens.

Rays traveling toward the focus will refract parallel to the principal axis.


Concave lens diagram

Concave Lens Diagram

No matter where the object is placed, the image will be on the same side as the object. The image is virtual, upright, and smaller than the object with a concave lens.

object

F

2F

2F

F

image

Experiment with this diagram


Lens sign convention

Lens Sign Convention

f = focal length

di = image distance

do = object distance

1

1

1

+

=

f

di

do

+ for real image

- for virtual image

di

+ for convex lenses

- for concave lenses

f


Lens mirror sign convention

Lens/Mirror Sign Convention

The general rule for lenses and mirrors is this:

+ for real image

- for virtual image

di

and if the lens or mirror has the ability to converge light, fis positive. Otherwise, fmust be treated as negative for the mirror/lens equation to work correctly.


Lens sample problem

Tooter, who stands 4 feet tall (counting his snorkel), finds himself 24 feet in front of a convex lens and he sees his image reflected 35 feet behind the lens. What is the focal length of the lens and how tall is his image?

f =

hi =

Lens Sample Problem

F

2F

2F

F

14.24 feet

-5.83 feet


Lens and mirror applet

Lens and Mirror Applet

This application shows where images will be formed with concave and convex mirrors and lenses. You can change between lenses and mirrors at the top. Changing the focal length to negative will change between concave and convex lenses and mirrors. You can also move the object or the lens/mirror by clicking and dragging on them. If you click with the right mouse button, the object will move with the mirror/lens. The focal length can be changed by clicking and dragging at the top or bottom of the lens/mirror. Object distance, image distance, focal length, and magnification can also be changed by typing in values at the top.

Lens and Mirror Diagrams


Convex lens in water

Glass

H2O

Glass

Air

Convex Lens in Water

Because glass has a higher index of refraction that water the convex lens at the left will still converge light, but it will converge at a greater distance from the lens that it normally would in air. This is due to the fact that the difference in index of refraction between water and glass is small compared to that of air and glass. A large difference in index of refraction means a greater change in speed of light at the interface and, hence, a more dramatic change of direction.


Convex lens made of water

Convex Lens Made of Water

Glass

Since water has a higher index of refraction than air, a convex lens made of water will converge light just as a glass lens of the same shape. However, the glass lens will have a smaller focal length than the water lens (provided the lenses are of same shape) because glass has an index of refraction greater than that of water. Since there is a bigger difference in refractive index at the air-glass interface than at the air-water interface, the glass lens will bend light more than the water lens.

Air

n = 1.5

H2O

Air

n = 1.33


Air water lenses

Air & Water Lenses

On the left is depicted a concave lens filled with water, and light rays entering it from an air-filled environment. Water has a higher index than air, so the rays diverge just like they do with a glass lens.

Air

Concave lens made of H2O

To the right is an air-filled convex lens submerged in water. Instead of converging the light, the rays diverge because air has a lower index than water.

H2O

Convex lens made of Air

What would be the situation with a concave lens made of air submerged in water?


Chromatic aberration

As in a raindrop or a prism, different wave-lengths of light are refracted at different angles (higher frequency ↔ greater bending). The light passing through a lens is slightly dispersed, so objects viewed through lenses will be ringed with color. This is known as chromatic aberration and it will always be present when a single lens is used. Chromatic aberration can be greatly reduced when a convex lens is combined with a concave lens with a different index of refraction. The dispersion caused by the convex lens will be almost canceled by the dispersion caused by the concave lens. Lenses such as this are called achromatic lenses and are used in all precision optical instruments.

Chromatic Aberration

Chromatic Aberration

Achromatic Lens

Examples


Human eye

Human eye

The human eye is a fluid-filled object that focuses images of objects on the retina. The cornea, with an index of refraction of about 1.38, is where most of the refraction occurs. Some of this light will then passes through the pupil opening into the lens, with an index of refraction of about 1.44. The lens is flexi-

Human eye w/rays

ble and the ciliary muscles contract or relax to change its shape and focal length. When the muscles relax, the lens flattens and the focal length becomes longer so that distant objects can be focused on the retina. When the muscles contract, the lens is pushed into a more convex shape and the focal length is shortened so that close objects can be focused on the retina. The retina contains rods and cones to detect the intensity and frequency of the light and send impulses to the brain along the optic nerve.


Hyperopia

The first eye shown suffers from farsightedness, which is also known as hyperopia. This is due to a focal length that is too long, causing the image to be focused behind the retina, making it difficult for the person to see close up things.

The second eye is being helped with a convex lens. The convex lens helps the eye refract the light and decrease the image distance so it is once again focused on the retina.

Hyperopia usually occurs among adults due to weakened ciliary muscles or decreased lens flexibility.

Hyperopia

Formation of image behind the retina in a hyperopic eye.

Convex lens correction for hyperopic eye.

Farsighted means “can see far” and the rays focus too far from the lens.


Myopia

The first eye suffers from nearsightedness, or myopia. This is a result of a focal length that is too short, causing the images of distant objects to be focused in front of the retina.

The second eye’s vision is being corrected with a concave lens. The concave lens diverges the light rays, increasing the image distance so that it is focused on the retina.

Nearsightedness is common among young people, sometimes the result of a bulging cornea (which will refract light more than normal) or an elongated eyeball.

Myopia

Formation of image in front of the retina in a myopic eye.

Concave lens correction for myopic eye.

Nearsighted means “can see near” and the rays focus too near the lens.


Refracting telescopes

Refracting telescopes are comprised of two convex lenses. The objective lens collects light from a distant source, converging it to a focus and forming a real, inverted image inside the telescope. The objective lens needs to be fairly large in order to have enough light-gathering power so that the final image is bright enough to see. An eyepiece lens is situated beyond this focal point by a distance equal to its own focal length. Thus, each lens has a focal point at F. The rays exiting the eyepiece are nearly parallel, resulting in a magnified, inverted, virtual image. Besides magnification, a good telescope also needs resolving power, which is its ability to distinguish objects with very small angular separations.

Refracting Telescopes

F


Reflecting telescopes

Galileo was the first to use a refracting telescope for astronomy. It is difficult to make large refracting telescopes, though, because the objective lens becomes so heavy that it is distorted by its own weight. In 1668 Newton invented a reflecting telescope. Instead of an objective lens, it uses a concave objective mirror, which focuses incoming parallel rays. A small plane mirror is placed at this focal point to shoot the light up to an eyepiece lens (perpendicular to incoming rays) on the side of the telescope. The mirror serves to gather as much light as possible, while the eyepiece lens, as in the refracting scope, is responsible for the magnification.

Reflecting Telescopes


Huygens principle

Christiaan Huygens, a contemporary of Newton, was an advocate of the wave theory of light. (Newton favored the particle view.) Huygens’ principle states that a wave crest can be thought of as a series of equally-spaced point sources that produce wavelets that travel at the same speed as the original wave. These wavelets superimpose with one another. Constructive interference occurs along a line parallel to the original wave at a distance of one wavelength from it. This principle explains diffraction well: When light passes through a very small slit, it is as if only one of these point sources is allowed through. Since there are no other sources to interfere with it, circular wavefronts radiate outwards in all directions.

Huygens’ Principle

Christiaan Huygens

Applet showing reflection and refraction Huygens style


Diffraction single slit

screen

P

Diffraction: Single Slit

Light enters an opening of width aand is diffracted onto a distant screen. All points at the opening act as individual point sources of light. These point sources interfere with each other, both constructively and destructively, at different points on the screen, producing alternating bands of light and dark. To find the first dark spot, let’s consider two point sources: one at the left edge, and one in the middle of the slit. Light from the left point source must travel a greater distance to point P on the screen than light from the middle point source. If this extra distance is a half a wavelength, /2,destructive interference will occur at P and there will be a dark spot there.

Extra distance

a/2

applet

a

Continued…


Single slit cont

Single Slit (cont.)

Let’s zoom in on the small triangle in the last slide. Since a/2 is extremely small compared to the distanced to the screen, the two arrows pointing to P are essentially parallel. The extra distance is found by drawing segment AC perpendicular to BC. This means that angle A in the triangle is also . Since AB is the hypotenuse of a right triangle, the extra distance is given by (a/2)sin.Thus, using

(a/2)sin = /2, or equivalently,asin = , we can locate the first dark spot on the screen. Other dark spots can be located by dividing the slit further.

To point P

C

Extra distance

To point P

A

B

a/2


Diffraction double slit

screen

P

Light passes through two openings, each of which acts as a point source. Here a is the distance between the openings rather than the width of a particular opening. As before, if d1 - d2 = n (a multiple of the wavelength), light from the two sources will be in phase and there will a bright spot at P for that wavelength. By the Pythagorean theorem, the exact difference in distance is

Diffraction: Double Slit

d1

d2

L

d1 - d2 = [L2 + (x + a/2)2]½ - [L2 + (x - a/2)2]½

Approximation on next slide.

a

Link 1Link 2

x


Double slit cont

screen

P

Double Slit (cont.)

In practice, L is far greater than a, meaning that segments measuring d1and d2 are virtually parallel. Thus, both rays make an angle  relative to the vertical, and the bottom right angle of the triangle is also  (just like in the single slit case). This means the extra distance traveled is given byasin. Therefore, the required condition for a bright spot at P is that there exists a natural number, n, such that:

d1

d2

L

asin = n

If white light is shone at the slits, different colors will be in phase at different angles.

a

Electron diffraction


Diffraction gratings

A different grating has numerous tiny slits, equally spaced. It separates white light into its component colors just as a double slit would. When asin = n,light of wavelengthwill be reinforced at an angle of. Since different colors have different wavelengths, different colors will be reinforced at different angles, and a prism-like spectrum can be produced. Note, though, that prisms separate light via refraction rather than diffraction. The pic on the left shows red light shone through a grating. The CD acts as a diffraction grating since the tracks are very close together (about 625/mm).

Diffraction Gratings


Credits

Credits

Snork pics: http://www.geocities.com/EnchantedForest/Cottage/7352/indosnor.html

Snorks icons: http://www.iconarchive.com/icon/cartoon/snorks_by_pino/

Snork seahorse pic: http://members.aol.com/discopanth/private/snork.jpg

Mirror, Lens, and Eye pics:

http://www.physicsclassroom.com/Refracting Telescope pic: http://csep10.phys.utk.edu/astr162/lect/light/refracting.html Reflecting Telescope pic: http://csep10.phys.utk.edu/astr162/lect/light/reflecting.html Fiber Optics: http://www.howstuffworks.com/fiber-optic.htm

Willebrord Snell and Christiaan Huygens pics: http://micro.magnet.fsu.edu/optics/timeline/people/snell.html Chromatic Aberrations: http://www.dpreview.com/learn/Glossary/Optical/Chromatic_Aberrations_01.htm Mirage Diagrams: http://www.islandnet.com/~see/weather/elements/mirage1.htmSir David Brewster pic: http://www.brewstersociety.com/brewster_bio.html Mirage pics: http://www.polarimage.fi/http://www.greatestplaces.org/mirage/desert1.htmlhttp://www.ac-grenoble.fr/college.ugine/physique/les%20mirages.htmlDiffuse reflection: http://www.glenbrook.k12.il.us/gbssci/phys/Class/refln/u13l1d.htmlDiffraction: http://hyperphysics.phy-astr.gsu.edu/hbase/phyopt/grating.html


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