Physics 2C:

Optics

Reflection, Reflection, refraction, Snell’s law,

polarization, images, thin mirrors, thin

lensesJuly 11, 2011 4

Reflection: specular and diffuse• Size of objects a>>λ, treat waves as rays

• Light strikes medium, some transmitted, reflected

• Angle of reflection=angle of incidence

• When surface roughness >> λ, individual light rays

follow law of reflectionfollow law of reflection

• But... when roughness ~ λ ...?

speculardiffuse

θi θr

(specular)

July 11, 2011 5

Example: chapter 35 problem 2• The mirrors in the figure make a 60 degree angle. A light ray enters parallel

to the symmetry axis. (a) How many relfections does it make? (b) Where

and in what direction does it exit the mirror system?

30

60

3060

angle from normal: 60

# reflections: 3# reflections: 3

exits 180degrees from initial direction

July 11, 2011 6

Refraction – Snell’s (Descartes’) Law• Interface b/n 2

(transparent) medium– Reflection

– Refraction• Due to different

speeds

• Frequency must • Frequency must be the same

• sinθi=viT/hypotenuseθ1

θ2

θ2

θ1

λ1=v1T

λ2=v2T

• sinθ2/sinθ1 =v2/v1

• index of refraction: n=c/v

• Snell’s law: n1sinθ1=n2sinθ2

n≥1

July 11, 2011 7

Example Mirages• Ch 35 Problem 14

A meter stick lies on the bottom of the rectangular trough with its zero mark at the left edge of the trough. You look into the long dimension of the trough at a 45 degree angle, with your line of sight just grazing the top edge of the tank. What mark on the meter stick do you see if the trough is (a) empty (b) half full of water (c) full of water

• Index of refraction of air istemperature dependent

n increasing

• Several parallel layers –continuously varying function of frequency

• Lightfollows curved path

Caused by

heated

air due

to sand

a.) empty

tan(45)=L/(40cm)

L=(40cm)tan(45)

L=40 cm

40-cm

b.) ½ full

tan(45)=L/(20cm)

L1=20cm

L2=(20cm)tanθ2

n1sinθ1=n2sinθ2

sin45=1.33 sinθ2

Θ2=32.1, L2=13cm

L=33 cm

c.) full

n1sinθ1=n2sinθ2

sin45=1.33 sinθ2

Θ2=32.1,

L=(40cm)sin32.1

L=26 cm

July 11, 2011 8

Total Internal

Reflection

• n2<n1, rays bent

away from normal

• (n1sinθ1=n2sinθ2)

•Critical angle: sinθ =1, sinθ =n /n•Critical angle: sinθ2=1, sinθc=n2/n1

•Above which 100% reflected•Example: Problem, 35-31: A compound lens is made from crown glass (n=1.52)

bonded to flint glass (n=1.89). What is the critical angle for light incident on the

flint-crown interface?

n2=1.52 n1=1.89

(n2<n1)

sinθc=n2/n1

sinθc=1.52/1.89

sinθc=53.5 degrees

July 11, 2011 9

Dispersion

• Index of refraction – frequency dependent

• Different wavelengths (colors) refracted

through different angles

• (n increases with increasing frequency)

• Basis for spectroscopy• Basis for spectroscopy

July 11, 2011 10

Critical Angle and Rainbows• Observer standing

between sun and rain

sees rainbow

• Max angle ~42degrees

• Index of refraction varies • Index of refraction varies

with λ, so does θmax

• Fainter, larger arc due to

double internal

reflections (color order

reversed)

July 11, 2011 11

Polarizing (Brewster’s) Angle• Reflection depends on interaction of E-

field component of EM wave

with electrons (polariation)

• Recall directional

antennas– No EM radiation in direction of particle

acceleration

90 -

acceleration

• Polarization in plane of reflection, 100% transmission if:– Reflected ray perpendicular to refracted

ray

– Snell’s law: n1sinθB=n2sinθ2

– θB+θ2=90, sinθ2=cosθB�tanθB=n2/n1

• Unpolarized light – reflected polarized

July 11, 2011 12

(Quiz 2)

Quiz 3 Formulas (Ch 37 & 35)(Quiz 1)

µF

v =

vAP 22

2

1 µω=24 r

PI

π=

T

πω 2=λπ2=kλθ md =sin

=λ

θπ sincos4 2

0

dSS

mN=∆λλ ( )λ2

12 += mnda

λθ =min

λθ ma =sin ( )( )

2

0 2

2sin

=

φφ

θ SS θλπφ sin

2 a=

D

λθ 22.1min =

(Quiz 2)

kfv

ωλ ==ργP

v =ρB

v =VV

PB

∆∆−=

smvsound /340=

( )0log10 II=β

2120 /10 mWI −=

( )vs

v

PI 2

02

20

2

1

2ρω

ρ=∆=

2

λnL =

vu

ff

±=′

1( )vuff ±=′ 1

tI E

ntdisplaceme ∂Φ∂= 0ε

2

2

22

2 1

t

E

cx

E

∂∂=

∂∂

vv

00

1

εµ=c

cBE =

smc /100.3 8×=

0µBE

S

vvv ×=

0

00

2µBE

S =

θ20 cosSS =

cUp =

cSPrad =2212

0 /1085.8 mNC ⋅×= −ε

270 /104 AN−×= πµ

∆λ D

ir θθ = vcn = nn λλ =

2211 sinsin θθ nn =

12sin nnc =θ 12tan nnB =θ

July 11, 2011 13

Plane Mirrors• Mirror: image appears to

be behind the mirror, upright, same size

• Consider the reflection of as small object (point)

• Object appears as far behind the mirror as it is

Observer

perceives light

from image

Image is

virtual

object

image

behind the mirror as it is in front

Smallest mirror to show full image?

Draw image reflected, connect (red) rays from

observer’s eye to top of head, bottom of feet,

blue rays show actual path light travels

Image reversed, L to R

Parabolic Mirror• y=ax^2

• Light parallel to the

symmetry axis will all

reflect through a special

point called the FOCUS

or FOCAL POINTor FOCAL POINT

• Light from far away will

converge at the focus

• Point source at focus will

emerge as parallel

Spherical Mirror• Similar to parabola near the

apex

• Spheres are more “regular” so most mirrors are actually part of a spherical curve (spherical aberration)

• Use very small fraction of the • Use very small fraction of the entire sphere

• Focal length >> size of mirror

• Light only strikes the mirror if they are close to and nearly parallel to the mirror axis (called paraxial rays)

Image Formation• Need to find 2 or more rays

from each point on the object that converge to form the image

• Any ray parallel to the mirror axis passes through focus

• Any ray through focus reflects parallel to the axis Any 2 rays is sufficient reflects parallel to the axis

• Any ray through the center of the mirror reflects symmetrically through the mirror axis

• Any ray through the center of curvature strikes the mirror normal to the mirror surface, and is reflected 180 degrees

Any 2 rays is sufficient

to locate an image:

o = distance to object

i = distance to image

f = focal length

o>C, C>o>f, and C<f

Image Formation – (Concave) Spherical

Mirrors

• o>C � image is real,

objectC F

image

f

object

C F

image

f

• C>o>f � image is real, inverted,

i

o

• o>C � image is real,

inverted, and reduced in size

objectC F

image

f

• C>o>f � image is real, inverted,

and magnified

• o<f � image is virtual,

upright, and magnified

Concave vs Convex Mirrors

• Convex – no possibility of real image, f<0

• Reflected rays appear to diverge from a common point behind the mirror

• Image is upright and reduced in size

• (used when image of a broad region needs to be captured in a small space)

Mirror Equation

• Consider the ray reflected

symmetrically about the mirror

axis to derive magnification:

• M=h’/h=-i/o (M<0, inverted)

• Consider the ray reflected

through the focus

h

h'

o

i

o

i

through the focus

• -h’/h=(i-f)/f=i/o

• 1/i+1/o=1/f

• If i<0, image is virtual

• If f<0, mirror is convex

• Using ray thru C, f=C/2 (R/2)

i

Approximations true for C large

compared with mirror size, and

Lenses (Thin, thickness<<radius of

curvature)• Transparent material that uses

refraction to form images

• Can be convex (parallel rays converge to a focal point) or concave (parallel rays appear to diverge from a focal point)

• Thin lenses – light bends just • Thin lenses – light bends just once

• Focal point on each side of lens

• Focal length same on both directions – orientation doesn’t matter

Diverging lenses M<1, can’t use a

diverging lens to start a fire!

Thin lens

•

objectf

h (o)

+ object + focus

radius

image

h’ (i)< 0

• Any ray through the

center of the lens �

undeflected

• Rays parallel to lens

axis � refracted

through focus

Equation: again, similar triangles

Essentially, the same triangles as for mirror

object

f

h

image

h’ < 0

' 'l f h

f h

− = −

l’

objectf

h

h’ < 0

h’ < 0

' '

1 1 1

'

l h

l h

l l f

= −

⇒ + =fio

111 =+

Lens Image Formation

objectf

h (o)

h’ (i)< 0

• o>f, image is • o>f, image is

inverted and real

• o<f, image is

upright and

virtual object

f

h (o)

h’ (i)> 0

f

Lens Equation

• Same as that for mirrors!

• M=-i/o (real images are

inverted)

• 1/i+1/0=1/f

• o>0 always

Focal

length

Object

distance

Image

distance

Image

+ +, o>f + Real,

inverted

+ +, o<f - Virtual,

uprighto>0 always

• i<0 = image virtual (image

on same side as lens), i>0

image real (not on same

side as object)

• Diverging lenses: f<0

- + - Virtual,

upright

fio

111 =+Find focal

length by

changing i until

the image

upright

Refraction at a curved surface• Light rays diverging from point source are

refracted to a common image point

• Restricted to small angles, Snell’s n1θ1=n2θ2

• n1(α+β)=n2(β -γ), treat AB as vertical

• α~BA/o, β~BA/Rθ1 A

• α~BA/o, β~BA/R

γ~BA/in1 n2

o radius R> 0object

image

this will become the

second surface of thin

lens

i

θ1

θ2

β γα

A

B

R

nn

i

n

o

n 1221 −=+

True for any α, as long as

small angle approximation is

valid

Refraction at a curved surface• True for real, virtual images (i<0)

• If light incident from opposite side, R<0

• Works for flat surfaces, R�infinity

• Used for understanding thick lenses

An1 n2

oobject

image

i

γα

A

B

R

nn

i

n

o

n 1221 −=+Left hand formula

Thick lenses• 2 interfaces: image from first serves as the

object of the other

• First: Second:

• Express in terms of thickness

• Lensmaker’s formula

1

2

1

2

1

11

R

n

i

n

o

−=+2

2

22

2 11

R

n

io

n −=+

12 oto −=

Left hand formula

( )

−−==+ 111

111n• Lensmaker’s formula

R1 R2

o1

n1=1

n2i1, 02,

i2t

( )

−−==+21

1RR

nfoiRi<0 for diverging lenses

Ri=infinity for flat surface

Conversion/diversion is reversed

if surround medium has a higher

index of refraction

Lens Aberrations• Spherical (multiple

foci) – can be reduced by making incident rays parallel– For close up objects,

include only the central portion of the lens (camera)(camera)

• Chromatic – due to index of refraction varying with wavelength (color)– Reduced using

composite lenses

– Unique to lenses

Multiple Mirrors

• Single mirror

• 2 mirrors @ θ? (infinite)

• 2 mirrors @ 60 degrees

• 2 mirrors @ 90 degrees

• Single mirror

6060

60

606060

Curved Mirrors• 1/i+1/o=1/f

• Magnification M=-i/o

• i>0 – real, inverted

• i<0 – virtual, upright

• Concave (f>0)• Concave (f>0)

– i>0 if o>f (|M|<1 for o>R=2f)

– i<0 if o<f (|M|>1)

• Convex (f<0)

– i<0 ONLY

– |M|<1 only

Virtual images appear to be

behind the mirror, while light

from a real image cannot be

distinguished from the object

Thin Lenses• 1/o+1/i=1/f

• M=-i/o

• Convex (converging)

– f>0

– o>f, i>0 (real, inverted)

• o>2f |M|<1

– o<f, i<0 (virtual, upright) |M|>1

• Concave (diverging)

– f<0

– i<0 (virtual, upright)

– |M|<1

Virtual images can only be

seen through the lens, while

light actually emanates from

the real image location

(Thick Lenses)

• Contact lenses

• (R<0 if concave to object)

• Thicker on the edge than in

( )

−−==+

21

111

111

RRn

foi

n>1

n>1

21 RR >

21 RR <

0<f

<

21

11

RR01 >R

0>f∞~2R

∞~1R

0>f02 <R

• Thicker on the edge than in

the center – concave

(diverging)

• Thicker in the center than on

the edges - convex

(converging)

0>f

Simple Optical Instruments• Film projector

– f>0, i>0, M>1 (f<o<2f)

• Magnifying glass

– f>0, if i<0, |M|>1

– If f<0, |M|<1– If f<0, |M|<1

• Side-view mirrors

– f<0, |M|<1

– (objects may appear closer…)

The Eye

• Light passes through lens

(adjustable focal length)

• Focused, real image formed on retina

• Myopic (near-sighted): image forms in front of retina (concave lenses)retina (concave lenses)

• Hyperopic (far-sighted): image forms behind the retina (convex lenses)

• Near point: min distance below which the eye cannot focus sharply (~25cm)

• corrective power (diopters)=1/f

Eyeglasses• Reading glasses: can’t focus closer than x cm?

– Want lenses with focal length to produce virtual

image @ x cm for an object located at the

standard near point, 25 cm

– Since x>25cm, f>0 (converging lenses)fio

111 =+fx

11

25

1 =−

+

Make sure that

lens power is

1/f, where f is

in meters

– Since x>25cm, f>0 (converging lenses)

• Nearsighted person cannot see clearly beyond y cm?

• Prescribe lens power produce virtual image @ y cm for objects very far away

fy

111 =−

+∞ yf −

= 11

Camera

• Similar to eye – image focused on film

• (adjustable lens)

• Adjustable diaphragm (reduces spherical aberrations)– Broader range of distances in focus

• Zoom lenses: adjust focal length (magnification of the same subject)

Ch 36 Problem 51

Camera’s zoom lens covers focal length range from 38 mm to 110 mm. You photograph

an object first with 38mm focal length then with 110mm focal length. Compare the

size of the object.

111

111

fio=+

222

111

fio=+

21 oo =Same object position

2211

11111

ififo−=−=

111 oiM =

222 oiM =

Simple magnifier (converging lens)• You can get the object closer, and still focus on it

• Angular magnification (m) must be defined relative to some observing distance

25 cm

angular size, αobject

naked eye

Angle subtended by

object in lensAngle subtended by

Maximum m

w/image at

near point

objecto ~ fe

far

eyepiece

lens

25( )

e

cmangular m

f

βα

≡ =

β

virtual image

(absolute magnification

is irrelevant)

object in lensAngle subtended by

object in lens

fofo

~111 →=

∞+

Image formed

“far”

(close)

object

h

Compound Microscope• Single lens: spherical aberrations m>4

intermediate

imagel ≈ fo

slightly

< fe

L

'

o

h LM

h f≡ ≈ −

objective

lens

eyepiece

lens

(Small fo)

(Magnified,

real image)

(Simple

magnifier)

Object placed just beyond

focal length of objective Image just inside focal

(fe<<L)

Lens spacing

final virtual image

(absolute magnification

is irrelevant)

25Total angular: total

o e

L cmm Mm

f f= = −

25( )

e

cmangular m

f=

focal length of objective Image just inside focal

length of eyepiece

As long as object size >> λ (geometric requirement)

Telescope• Only angular magnification, m, is meaningful

• Overall magnification= ratio of m’s

object

h ?

intermediatl ≈ 1,000,000 slightly

< fe

fo

α β

Images distant

objects at focal lengthRefracting or reflecting

intermediat

e image

l ≈ 1,000,000

light years

< fe

'

o

h

fα ≈ −

final virtual image

(absolute magnification

is irrelevant)

( ) o

e

fangular m

f

βα

≡ = −

objectiv

e lenseyepiece

lens

α β

'

e

h

fβ ≈