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Document info 7.
Total Internal ReflectionThursday, 9/14/2006
Physics 158Peter Beyersdorf
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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
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Refraction at an interface
Snell’s law tells us light bends towards the normal when going from low-index to high-index materials
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!
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θi
θtnt
ni θi
θtni
nt
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?
What happens when there is no real mathematical solution?
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Snell’s law
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ni sin !i = nt sin !t
sin !t =ni
ntsin !i ! 1
!i ! sin!1
!nt
ni
"
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Transmission beyond the critical angle
Consider the Fresnel reflection coefficients
at the critical angle, θc=sin-1(nt/ni)
Beyond the critical angle what do we get for the transmitted angle?
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r! =E0r
E0i=
ni cos !i ! nt cos !t
ni cos !i + nt cos !t
r! =E0r
E0i=
nt cos !i ! ni cos !t
ni cos !t + nt cos !i
r! = 1 r! = 1
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Transmission beyond the critical angle
Beyond the critical angle what do we get for the transmitted field?
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sin ! =ei!/2e!" ! e!i!/2e"
2isin ! =
e!! + e!
2= cosh "
Et = tE0ieik0(sin !tx+cos !ty)
cos ! =ei!/2e!" + e!i!/2e"
2=
ie!" ! ie"
2= i sinh"
Et = tE0ieik0 cosh(!t)x+k0 sinh(!t)y
sin ! =ei! ! e!i!
2i> 1 let ! = "/2 + i#
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
Plane of the interface (here the yz plane) (perpendicular to page)
ni
nt
θi θr
θt
Ei Er
Et
Interface
x
y
z
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Complex reflection coefficients
Beyond the critical angle the reflection coefficients are complex
imaginary part of coefficient implies a phase shift
Magnitude of reflection coefficient is 1, indicating 100% reflection
Power reflectivity coefficient must be generalized to allow for complex reflection coefficients
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R = rr!
Er = rE0ei(!t+")
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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
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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
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E
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Frustrated Total Internal Reflection
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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
nn
nn
n=1 n=1total internal reflection frustrated total
internal reflection
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
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Uses for Total Internal Reflection
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zig-zag laser slabs
fiber optics
prisms
fingerprinting
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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
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0 2.5 5 7.5 10 12.5 15 17.5 20
0.25
0.5
0.75
1
Teff = RN
N
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Prisms
Prisms are used for reflecting beams with unit efficiency via TIR. Various configurations allow many interesting properties
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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
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Fingerprinting with TIR
fingertip valleys reflect light via TIR, while finger tip ridges in contact with prism frustrate the reflection
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Phase Shift on TIR
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nt < ni
┴
||
above the critical angle, TIR field shows an
interesting phase shift
A π phase shift occurs at Brewster’s angle indicating a change in the reflection
coefficient sign as it passes through zero
nt < ni
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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?
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7.
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
The refractive index of metals is given by
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 18
!p =
!Ne2
"0me
n2 = 1!!!p
!
"2
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Reflection from Real Metals
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Summary
When light passes from a dense material to a less dense material it bends away from the normalWhen the incident angle is large enough the transmitted angle if 90° and cannot increaseBeyond the critical angle 100% of the power is reflectedAn evanescent wave is present in the transmitted material that matches the boundary conditions at the interface, but carries no power away from the interfaceA high index material in the presence of the evanescent wave can couple light through the low index gap causing frustrated total internal reflectionThe reflected field acquires a phase shift upon totally internally reflectingMetals reflect light efficiently below their plasma frequency
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