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The
law
of reflection
P
P
mirror
or
dielectric
interface
MIT
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2 6 8wkl
b 29
A
ray
departing
from
P
in the direction
,withrmpBecto the mirror
normal
isre cted symmefr i c e at t
same
a g k
8.
This
is
because
the
symmetric path POP'
has
minimum length.
Campxe,
for example
the
altanative POP .
Clearly,
IPOI
JUP I
<
I P ~ I +~ P I .
Comider tbe
c~di32wtion
af OP backwards
through
the
mirror.
To
o server
in the
direction
f
the ray
w ll
appew
to b v e
originated
t
P .
=
NUS
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MIT 2.71/2.71002/06/08 wk1-b-
Reflection from mirrors
30
In:right-handed
triad
Out:left -handed
triad
mirror
Projected:left -handed
triad
(front view)
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The law of refraction (Snell's Law)
Taking
derivatives wit
respe t
to
x
a(cP ) s b - z
=~&4-22 '&4 =Q
las;
=+-
n a
a's @'
bis re
is know s s
ew,
or e
MIT
2 7112 710
NUS
21 61 8
wkl-b-31
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Snell's Law as minimum time principle
Beach,
runnin speed
v
Which path should the lifeguard follow
to reach the drowning person in minimum time
MIT 2 7112 710
21 61 8 wkl -b-32
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Snell's Law as momentum conservation principle
M T 2.7112.710 NUS
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w
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Two types of refraction
air glass glass air
from lower to higher index(towards optically denser material)
angle wrt normal
from higher to lower index(towards optically less dense material)
angle wrt normal
34
if the combination of andis such that
then Snell’s law in the less dense mediumcannot be satisfied. This situation is
known as Total Internal Reflection (TIR)
the maximum anglethat can enter the optically dense medium
is such that
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glass
air
MIT 2.71/2.71002/06/08 wk1-b-
Total Internal Reflection (TIR)
35
glass
air
at critical angle, the refracted light propagatesparallel to the interface
the result is called a “surface wave”
at angles of incidence higher than critical,the light is totally internally reflected
almost as ifthe glass-air interface were a mirror
in both cases of surface wave and TIR,there is an exponential tail of electric field
leaking into the medium of lower optical density;this is called the evanescent wave.
We will learn more about evanescent waves after we coverthe electromagnetic nature of light
Typically, the evanescent wave is a few wavelengths longbut it can be much longer near the critical angle
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Frustrated Total Internal Reflection (FTIR)
6 glass
n c
If another dielectric ppso cbes within ew distant canatants
from the TLR interface, the tail of the exponenti lly evmescent wave
becomes propagating;
i e the
light couples
out
of
the
medium.
This
situat i ~
s known
as
h s t r a t e d Total injernol
&e j b t i o n ( F T B ) .
n
qu ntum mechanics, there is
n
analogous
ect
b o r n as tunneling.
We will compute the amount
of
energy coupled out of the medium
of
incidence
later, after we study the ekctramgnetic nature of light.
lllii
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Prisms
glassair air
TIR TIR
glass
air air
air air
45° 45°
air
glass
TIR
TIRretro-reflector
pentaprism
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Fingerprint sensors
laser
illumination
finger
right-angleprism
digitalcamera
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Optical waveguides
“planar” waveguide: high-index dielectric materialsandwiched between lower-index dielectrics
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Numerical Aperture (NA) of a waveguide
high index contrast (n1/n2) " high NA
NA is the sine of the largest angle that is waveguided
air(n=1)
i.e., NA is the incident angle of acceptance of the waveguide
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GRadient INdex (GRIN) waveguide
41
TIRlowest index
highest index
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Optical fibers: step cladding
cladding(lower index)
core(higher index)
Core diameter = 8-10µm (commercial grade)
Cladding diameter = 250µm (commercial grade)Index contrast $n = 0.007 (very low NA)attenuation = 0.25dB/km
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Optical fibers: GRIN
cladding(lower index)
core ofgradually
decreasingindex
optical rays “swirl” in a helical trajectory
around the axis of the core
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GRIN waveguides in nature: insect eyes
Image by Thomas Bresson at Wikimedia Commons.
Images removed due to copyright restrictions.
Please see
http://commons.wikimedia.org/wiki/File:Superposkils.gif
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Dispersion
3.5
3.0
2.5
2.0
1.5
1.0
0.5
010
-110
010
1
A b s o r
p t i o n c o n s t a n t k ( λ )
Wavelength λ / µm
Dispersion = the dependence of refractive index n on wavelength λ .
R e f r
a c t i v e i n d e x n ( λ )
10-4
10-3
10
-2
10-1
100
101
Fused silica (SiO2)
n
k
Figure by MIT OpenCourseWare. Adapted from Fig. 2.4 in Bach, Hans, and Norbert Neuroth. The Properties of Optical Glass. New York, NY: Springer, 2004.
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Dispersion from a prism
incidentwhite light(all visible
wavelengths)
Newton’s prism
red
green
blue
glassair
rainbowexit
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Dispersion measures
Reference color lines
C (H- #
=656.3nm, red),D (Na- #=589.2nm, yellow),
F (H- #=486.1nm, blue)
e.g., crown glass has
Dispersive power
Dispersive index aka Abbe’s number
e.g. for crown glass: V =0.01704 or v =58.698
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Paraboloidal reflector: perfect focusing
What should the shape function s( x) bein order for the incoming parallel ray bundle
to come to perfect focus?
The easiest way to find the answer is
to invoke Fermat’s principle: since the rays from infinity follow
the minimum path before they meet at P ,
it follows that they must follow the same path.
focus at F
A paraboloidal reflector focuses
a normally incident plane wave
to a point
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2.71 / 2.710 OpticsSpring 2009
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