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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