Summary of Geometric Optics Rules

1. Object distances, p are always positive (except in the case of more than one lens or mirror when the first image is on the far side of the second lens or other cases where you have a virtual object like object behind mirror).

2. Image distances, q, are positive for real images and negative for virtual images. 3. Real images form on the same side of the object for mirrors and on the opposite side for refracting surfaces (lenses).

Virtual images form on the opposite side of the object for mirrors and on the same side for refracting surfaces. 4. When an object faces a convex mirror or concave refracting surface the radius of curvature, R, is negative. When an

object faces a concave mirror or convex refracting surface the radius of curvature is positive.

Object object location

image location

image type

image orientation

sign of f sign of R (R1 for lens)

sign of q sign of m

Plane mirror anywhere opposite object

virtual same as object

f=∞ ∞ negative =+1

Concave mirror

inside f opposite virtual same positive positive negative positive

concave mirror

outside f same real inverted positive positive positive negative

convex mirror

anywhere opposite virtual same negative negative negative positive

converging lens (convex)

inside f same virtual same positive positive negative positive

converging lens

outside f opposite real inverted positive positive positive negative

diverging lens

anywhere same virtual same negative negative negative positive

Flat mirror images

• Your eyes tell you where/how big an object is• Mirrors and lenses can fool your eyes

• Place a point light source P in front of a mirror• If you look in the mirror, you will see the object as if it were at the

point P’, behind the mirror• As far as you can tell, there is a “mirror image” behind the mirror• For an extended object, you get an extended image• The distances of the object

from the mirror and the imagefrom the mirror are equal

• Flat mirrors are the onlyperfect image system(no distortion)

P’

Object Image

p q

P

Mirror

p q

Ch 36

Image Characteristics and Definitions

Object Imagep q

Mirror• The front of a mirror or lens is the side the light goes in• The object distance p is how far the object is in front of the mirror• The image distance q is how far the image is in front* of the mirror

• Real image if q > 0, virtual image if q < 0• The magnification M is how large the image is compared to the object

• Upright if positive, inverted if negative

h h’hM

h

If you place an object in front of a flat mirror, its image will beA) Real and upright B) Virtual and uprightC) Real and inverted D) Virtual and inverted

*back for lenses

CT - 1 In the morning you look at yourself in the mirror and you cannot see your feet. In order to see them in the mirror you should A. Move closer to the mirror and look down. B. Move backward. C. Give it up - you will never be able to see your feet in the mirror.

Spherical Mirrors• Typical mirrors for imaging are spherical mirrors – sections of a sphere

• It will have a radius R and a center point C• We will assume that all angles involved are small • Optic axis: an imaginary line passing through the center of the mirror

• Vertex: The point where the Optic axis meets the mirrorThe paths of some rays of light are easy to figure out• A light ray through the center will come back exactly on itself• A ray at the vertex comes back at the same angle it left• Let’s do a light ray coming in parallel to the optic axis:

• The focal point F is the place this goes through• The focal length f = FV is the distance to the mirror

• A ray through the focal pointcomes back parallel

C F

sin tan

V

R

X

FC FX 1 12 2CX R

f FV CV FC 12f R

f

Spherical Mirrors: Ray Tracing1. Any ray coming in parallel goes through the focus2. Any ray through the focus comes out parallel3. Any ray through the center comes straight back

C

12f R

• Let’s use these rules to find the image:

F

CPF

Do it again, but harder• A ray through the center won’t

hit the mirror• So pretend it comes from the center• Similarly for ray through focus• Trace back to see where they came from

Spherical Mirrors: Finding the Image

CV

• The ray through the center comes straight back• The ray at the vertex reflects at same angle it hits• Define some distances:

h

P

X

h’Q

Y

VP p

VQ q

PX h

QY h

CV R

12f R1 1 1

p q f

Magnification• Since image upside

down, treat h’ as negativeh q

h p

q

Mp

Convex Mirrors: Do they work too?

C

12f R

• Up until now, we’ve assumed the mirror is concave – hollow on the side the light goes in• Like a cave

• A convex mirror sticks out on the side the light goes in• The formulas still work, but just treat R as negative• The focus this time will be on the other side of the mirror• Ray tracing still works

FSummary:• A concave mirror has R > 0;

convex has R < 0, flat has R = • Focal length is f = ½R

• Focal point is distance f in front of mirror • p, q are distance in front of mirror of image, object

• Negative if behind

1 1 1

p q f

qM

p

Mirrors: Formulas and Conventions:• A concave mirror has R > 0; convex has R < 0, flat has R = • Focal length is f = ½R

• Focal point is distance f in front of mirror • p, q are distance in front of mirror of object/image

• Negative if behind• For all mirrors (and lenses as well):• The radius R, focal length f, object distance p, and image

distance q can be infinity, where 1/ = 0, 1/0 =

1 1 1

p q f

12f R

Light from the Andromeda Galaxy bounces off of a concave mirror with radius R = 1.00 m. Where does the image form?A) At infinity B) At the mirrorC) 50 cm left of mirror D) 50 cm right of mirror

12 50 cmf R

2 Mlyp

• Concave, R > 01 1 1

q f p

10

50 cm

50 cmq

Ex- (Serway 36-25) A spherical mirror is to be used to form, on a screen located 5 m from the object, an image 5 times the size of the object. (a) Describe the type of mirror required (concave or convex). (b) What s the required radius of curvature of the mirror? (c) Where should the mirror be placed relative to the object?

Solve on board

Images of Images: Multiple Mirrors• You can use more than one mirror to make images of images

• Just use the formulas logicallyLight from a distant astronomical source reflects from an R1 = 100 cm concave mirror, then a R2 = 11 cm convex mirror that is 45 cm away. Where is the final image?

1 1 1

2 2 2

1 1 1

1 1 1

p q f

p q f

1 50 cmf 2 5.5 cmf

1

1 1 1

50 cmq

1 50 cmq

45 cm5 cm

2 5 cmp

2

1 1 1

5 cm 5.5 cmq

2 55 cmq

10 cm

Refraction and Images• Now let’s try a spherical surface between two regions with

different indices of refraction• Region of radius R, center C, convex in front:Two easy rays to compute:• Ray towards the center continues straight• Ray towards at the vertex follows Snell’s Law

n1

n2

Ch

P

X

p

q

1 1 2 2sin sinn n

1

2

R

1 2 2 1n n n n

p q R

• Magnification:

1

2

n qM

n p

Q

Y

h’

Comments on Refraction• R is positive if convex (unlike reflection)

• R > 0 (convex), R < 0 (concave), R = (flat)• n1 is index you start from, n2 is index you go to• Object distance p is positive if the object in front (like

reflection)• Image distance q is positive if image is in back (unlike

reflection)We get effects even for a flat boundary, R = • Distances are distorted:

n1

n2

h

P

X

p

Q

Y

q

2

R

1 2 2 1n n n n

p q R

1 2 0n n

p q

2

1

nq p

n

• No magnification: 1 2

2 1

n n pM

n p n

1

1

2

n qM

n p

Warmup 25

CG36.16 page 1125

Flat Refraction 2

1

nq p

n

A fish is swimming 24 cm underwater (n = 4/3). You are looking at the fish from the air (n = 1). You see the fishA) 24 cm above the water B) 24 cm below the waterC) 32 cm above the water D) 32 cm below the waterE) 18 cm above the water F) 18 cm below the water

24 cm

• R is infinity, so formula above is valid• Light comes from the fish, so the water-side is the front• Object is in front• Light starts in water• For refraction, q tells you

distance behind the boundary

24 cmp

1

2

4 3

1

n

n

1 24 cm

4 3q

18 cm

18 cm

CT – 2 A parallel beam of light is sent through an aquarium. If a convex glass lens is held in the water, it focuses the beam

A. closer to the lens than B. at the same position as C. farther from the lens than outside the water.

Double Refraction and Thin Lenses• Just like with mirrors, you can do double refraction

• Find image from first boundary• Use image from first as object for second

We will do only one case, a thin lens:• Final index will match the first, n1 = n3

• The two boundaries will be very close

n1 n2 n3

Where is the final image?• First image given by:

• This image is the object for the second boundary:• Final Image location:• Add these:

1 2 2 1

1 1

n n n n

p q R

2 1 1 2

2 2

n n n n

p q R

1 12 1

1 2

1 1n nn n

p q R R

2

1 1 2

1 1 1 11

n

p q n R R

p

n1 n2 n1

1 2q p

Thin Lenses (2)

• Define the focal length:• This is called lens maker’s equation

• Formula relating image/object distances• Same as for mirrors

Magnification: two steps• Total magnification is product• Same as for mirrors

2

1 1 2

1 1 1 11

n

p q n R R

2

1 1 2

1 1 11

n

f n R R

1 1 1

p q f

1 11

2

n qM

n p 2

21 2

n qM

n p

1 2M M M 1

2

qq

pp

1 2q p

qM

p

Using the Lens Maker’s Equation

• If you are working in air, n1 = 1, and we normally call n2 = n.

• By the book’s conventions, R1, R2 are positive if they are convex on the front

• You can do concave on the front as well, if you use negative R• Or flat if you set R =

2

1 1 2

1 1 11

n

f n R R

If the lenses at right are made ofglass and are usedin air, which one definitely has f < 0?

A B C

D Light entering on the left:• We want R1 < 0: first

surface concave on left• We want R2 > 0: second

surface convex on left

• If f > 0, called a converging lens• Thicker in middle

• If f < 0, called a diverging lens• Thicker at edge

• If you turn a lens around, its focal length stays the same

Ray Tracing With Converging Lenses• Unlike mirrors, lenses have two foci, one on each side of the lens• Three rays are easy to trace:

1. Any ray coming in parallel goes through the far focus2. Any ray through the near focus comes out parallel3. Any ray through the vertex goes straight through

f f

F F

• Like with mirrors, you sometimes have to imagine a ray coming from a focus instead of going through it

• Like with mirrors, you sometimes have to trace outgoing rays backwards to find the image

Ray Tracing With Diverging Lenses• With a diverging lens, two foci as before, but they are on the wrong

side• Still can do three rays1. Any ray coming in parallel comes from the near focus2. Any ray going towards the far focus comes out parallel3. Any ray through the vertex goes straight through

f f

F F

• Trace purple ray back to see where it came from

Lenses and Mirrors Summarized

R > 0 p > 0 q > 0 f

mirrorsConcave

frontObject in front

Image in front

lensesConvex

frontObject in front

Image in back

• The front of a lens or mirror is the side the light goes in

2

1 1 2

1 1 11

n

f n R R

12f R

1 1 1

p q f

h qM

h p

Variable definitions:

• f is the focal length• p is the object distance from lens• q is the image distance from lens• h is the height of the object• h’ is the height of the image• M is the magnification

Other definitions:• q > 0 real image• q < 0 virtual

image• M > 0 upright• M < 0 inverted

Warmup 25

Ex- A transparent sphere of unknown composition is observed to form an image of the Sun on the surface opposite to the Sun. What is the refractive index of the sphere?

Ex - A transparent photographic slide is placed in front of a converging lens that has a focal length of 2.44 cm. The lens forms an image of the slide 12.9 cm from the slide. How far is the lens from the slide if the image is (a) real and (b) virtual.

Solve on Board

Imperfect Imaging• With the exception of flat mirrors, all imaging systems are imperfect• Spherical aberration is primarily concerned with the fact that the small

angle approximation is not always valid

F

• Chromatic Aberration refers to the fact that different colors refract differently

F

• Both effects can be lessened by using combinations of lenses• There are other, smaller effects as well

Angular Size & Angular Magnification• To see detail of an object clearly, we must:

• Be able to focus on it (25 cm to for healthy eyes, usually best)• Have it look big enough to see the detail we want

• How much detail we see depends on the angular size of the object

d

0h

0 h d

Two reasons you can’t see objects in detail:1. For some objects, you’d have to get closer than your near point

• Magnifying glass or microscope2. For others, they are so far away, you can’t get closer to them

• TelescopeGoal: Create an image of an object that has• Larger angular size• At near point or beyond (preferably )

Angular Magnification:how much bigger the

angular size of the image is

0m

F

The Simple Magnifier• The best you can do with the naked eye is:

• d is near point, say d = 25 cm• Let’s do the best we can with one converging lens• To see it clearly, must have |q| d

h

0 h d

h’

-q

h q

1 1 1

p q f

p

h h

q p

1 1

hf q

1 1h

f q

0

m

d d

f q

• Maximum magnification when |q| = d• Most comfortable when |q| = • To make small f, need a small R:

• And size of lens smaller than R• To avoid spherical aberration, much smaller• Hard to get m much bigger than about 5

max 1d

mf

d

mf

Fe

The MicroscopeA simple microscope has two lenses:• The objective lens has a short focal length and produces a large,

inverted, real image• The eyepiece then magnifies that image a bit more

• Since the objective lens can be small, the magnification can be large• Spherical and other aberrations can be huge

• Real systems have many more lenses to compensate for problems• Ultimate limitation has to do with physical, not geometric optics

• Can’t image things smaller than the wavelength of light used• Visible light 400-700 nm, can’t see smaller than about 1m

Fo

The TelescopeA simple telescope has two lenses sharing a common focus• The objective lens has a long focal length and produces an

inverted, real image at the focus (because p = )• The eyepiece has a short focal length, and puts the image back at

(because p = f)

Angular Magnification:• Incident angle:• Final angle:• The objective lens is made as large as possible

• To gather as much light as possible• In modern telescopes, a mirror replaces the objective lens• Ultimately, diffraction limits the magnification (more later)

• Another reason to make the objective mirror as big as possible

F

fofe

0 0 oh f

eh f 0m

o em f f