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 θi nt

nt ni θt θt

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 ≤ 1 nt θ sin− i ≤ n ! i " What happens when there is no real mathematical solution?

7. 4 Transmission beyond the critical angle

Consider the Fresnel reflection coefficients E n cos θ n cos θ r = 0r = i i − t t ⊥ E0i ni cos θi + nt cos θt E n cos θ n cos θ r = 0r = t i − i t ! E0i ni cos θt + nt cos θi at the critical angle, -1 θc=sin (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?

iθ iθ e e− sin θ = − > 1 let θ = π/2 + iα 2i n E i iπ/2 α iπ/2 α α α i Er e e− e− e e− + e θi θr sin θ = − sin θ = = cosh α InPlaneterf aofc ethe interface (here the 2i 2 yz plane) (perpendicular to page) iπ/2 α iπ/2 α α α y e e− + e− e ie− ie cos θ = = − = i sinh α θt 2 2 nt z x Et ik0(sin θtx+cos θty) Et = tE0ie

ik0 cosh(αt)x+k0 sinh(αt)y Et = tE0ie

The transmitted field is a traveling wave The transmitted field exponentially decays in the direction along the interface 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 i(ωt+φ) shift Er = rE0e 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

frustrated total total internal reflection n=1 n=1 internal reflection n n 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 N Teff = R 0.5

0.25

0 2.5 5 7.5 10 12.5 15 17.5 20 N 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

n < n t i above the critical angle, ┴ TIR field shows an interesting phase shift

n < n A π phase shift occurs at t i 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

Ne2 ωp = !"0me The refractive index of metals is given by ω 2 n2 = 1 p − ω When ω<ωp , the inde!x 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

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