Total Internal Reflection
Thursday, 9/14/2006
Physics 158
Peter Beyersdorf
Document info
7. 1
Class Outline
Conditions for total internal reflection
The evanescent wave
Uses for total internal reflection
Prisms
Beamsplitters
Fiber Optics
Laser slabs
Phase shift on total internal reflection
Reflection from metals
7. 2
Refraction at an interface
Snell’s law tells us light bends towards the normal
when going from low-index to high-index materials
θi
ni
θt
nt
θi
nt
θt
ni
Going from high-index to low-index light must bend
away from the normal
At some critical angle, the transmitted beam in the
low index material will be at 90°
As the incident beam angle increases the transmitted
beam angle cannot increase!
7. 3
Snell’s law
Snell’s law allows us to calculate the angle of the
beam transmitted through an interface. Are there
conditions that prevent there from being a real
mathematical solution?
ni sin θi = nt sin θt
ni
sin θt =
sin θi ≤ 1
nt ! "
nt
−1
θi ≤ sin
ni
What happens when there is no real mathematical
solution?
7. 4
Transmission beyond the critical angle
Consider the Fresnel
reflection coefficients
E0r
ni cos θi − nt cos θt
r⊥ =
=
E0i
ni cos θi + nt cos θt
E0r
nt cos θi − ni cos θt
r! =
=
E0i
ni cos θt + nt cos θi
at the critical angle,
θc=sin-1(nt/ni)
r⊥ = 1
r! = 1
Beyond the critical angle what do we get for
the transmitted angle?
7. 5
Transmission beyond the critical angle
Beyond the critical angle what do we get for
the transmitted field?
eiθ − e−iθ
sin θ =
>1
2i
let θ = π/2 + iα
e−α + eα
eiπ/2 e−α − e−iπ/2 eα
sin θ =
= cosh α
sin θ =
2
2i
eiπ/2 e−α + e−iπ/2 eα
ie−α − ieα
cos θ =
=
= i sinh α
2
2
Et = tE0i eik0 (sin θt x+cos θt y)
Ei
θi θr
Er
ni
Plane of the interface (here the
Interface
yz plane) (perpendicular to page)
y
z
θt
x
Et
nt
Et = tE0i eik0 cosh(αt )x+k0 sinh(αt )y
The transmitted field is a traveling wave
in the direction along the interface
The transmitted field exponentially decays
as it gets further from the interface
7. 6
Complex reflection coefficients
Beyond the critical angle the reflection
coefficients are complex
imaginary part of coefficient implies a phase
shift Er = rE0 ei(ωt+φ)
Magnitude of reflection coefficient is 1,
indicating 100% reflection
Power reflectivity coefficient must be
generalized to allow for complex reflection
coefficients
∗
R = rr
7. 7
Evanescent Wave
Because the transmitted field is an evanescent
wave that decays exponentially to zero, it does
not carry energy away from the interface
The evanescent wave is still necessary to
satisfy the boundary conditions at the interface
100% of the power is contained in the reflected
field, i.e. there is total internal reflection
7. 8
Evanescent Wave
Incident and reflected fields on reflection from
a high-index to low-index material are in-phase
Without a transmitted field the E field would be
discontinuous across the boundary
E
7. 9
Frustrated Total Internal Reflection
By placing a high-index material in the presence of the
evanescent wave power can be coupled through the
low-index gap, frustrating the total internal reflection
total internal reflection
n=1
n
n
frustrated total
n=1 internal reflection
n
n
The prisms must be within a few wavelengths (where
the evanescent field is non-zero) for this to work
This is the principle of operation for cube beamsplitters
7.10
Uses for Total Internal Reflection
zig-zag laser slabs
prisms
fiber optics
fingerprinting
7. 11
Zig-Zag Laser Slabs
The circulating beam in many high-power lasers
is made to zig-zag through the laser crystal to
average over the thermal gradient in the
crystal. Having many reflections requires the
reflectivity at each interface be high
1
0.75
Tef f = RN
0.5
0.25
0
2.5
5
7.5
N
10
12.5
15
17.5
20
7.12
Prisms
Prisms are used for reflecting beams with unit
efficiency via TIR. Various configurations allow
many interesting properties
7.13
Fiber-Optics
Glass fibers are used as
waveguides to transmit light
over great distance
High index “core” guides the light
A low index “cladding” protects the interface
of the core
The acceptance angle of a fiber determines
what light will be guided through the fiber
7.14
Fingerprinting with TIR
fingertip valleys reflect light via TIR, while
finger tip ridges in contact with prism frustrate
the reflection
7.15
Phase Shift on TIR
nt < ni
above the critical angle,
TIR field shows an
interesting phase shift
┴
nt < ni
||
A π phase shift occurs at
Brewster’s angle indicating
a change in the reflection
coefficient sign as it
passes through zero
7.16
Reflection from Ideal Metals
For a perfect conductor, there can be no
internal electric fields, hence the boundary
condition requires E||=0, so for the parallel
component of the field Er=-Ei Et=0
Reflection coefficient is r=1, R=1
Transmission coefficient is t=0, T=0
Does a real metal behave like this?
7.17
Reflection from Real Metals
The free electrons in a metal can be thought of
as a gas or plasma with a plasma frequency
(natural frequency of oscillation) of
ωp =
!
N e2
"0 me
The refractive index of metals is given by
n =1−
2
! ω "2
p
ω
When ω<ωp , the index of refraction is imaginary
and the metal is absorbing - but most of the
incident power is reflected
When ω>ωp, the metal is transparent
typical metals have a value for ωp in the UV
7.18
Reflection from Real Metals
7.19
Summary
When light passes from a dense material to a less dense material it
bends away from the normal
When the incident angle is large enough the transmitted angle if
90° and cannot increase
Beyond the critical angle 100% of the power is reflected
An evanescent wave is present in the transmitted material that
matches the boundary conditions at the interface, but carries no
power away from the interface
A high index material in the presence of the evanescent wave can
couple light through the low index gap causing frustrated total
internal reflection
The reflected field acquires a phase shift upon totally internally
reflecting
Metals reflect light efficiently below their plasma frequency
7.20