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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 ≈