Problem 15'

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Problem 15.
Optical Tunnelling
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Problem
Take two glass prisms separated by a small gap.
Investigate under what conditions light incident
at angles greater than the critical angle is not
totally internally reflected.
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Experiment

• Two measurement ranges
• Centimeter waves accurate measuring
• Visile light obtaining the effect

• Parameters
• Waveelngth of the light used
• Refraction index of prisms and medium in the gap
• Polarization
• Distance between prisms

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1. Centimeter waves

• Wavelength 3 cm
• Polarization linear, electrical field
  perpendicular to plane of propagation
• Prism refraction index 1.5 (paraffin)
• Measurements
• Intensity of tunneled waves
• Intenzity of reflected waves

in dependence on prism distance

• Measured voltage in the detector
• Voltage proportional to field!

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Centimeter waves cont.

• Apparatus schematic

detector
Radiation source
Translation system
multimeter
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Centimeter waves cont.
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Centimeter waves cont.
Tunelled field
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2. Visible light

• Wavelenght 780 nm
• Polarization linear
• electric field perpendicular to plane of
  propagation
• Prism index of refraction 1.48 (measured)
• Measurements
• Intensity of tunneled waves
• Intensity of reflected waves

• Intensity measurement photodiode
• Voltage proportional to square of field!

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2. Visible light

• Measurement in time
• A slow motor (0.5 r/min) moves the translator
• Voltage sampling at the diode every 1/50 of a
  second

• The signal grows in time
• Change of prism distance

v translator speed t elapsed time
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Visible light cont.

• Apparatus schematic

Slow motor translation
laser
prisms
Data receiving computer
oscilloscope
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(No Transcript)
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Visible light cont.
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Explanation

• Huygens principle

Every atom ?through? which light passes is a
source of light identical to the incident light
? Electromagnetic waves in dielectrics the
resultant of interference of the initial wave and
all scattered waves
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Explanation cont.

• At total reflection the reflected ray is the
  only interference maximum
• Behind the reflection plane destructive
  interference, but only far away from the plane
• Close to the plane (distances of the order of the
  wavelength) the waves havent interfered
  completely and a decaying field exists

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Explanation cont.

• That field decays fast due to interference
• If a prism is put into the field a new
  interference maximum can be formed in the prism
• A new, tunnelled wave is formed in the prism
• The energy of the reflected wave becomes smaller

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Maxwell equations
E electric field B magnetic field induction P
polarization c speed of light in a vacuum e0
vacuum permittivity
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Plane wave solutions
Electrical field
E0 amplitude ? frequency t time k wave
vector r - radiusvector
Magnetic field
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Geometry of the problem
y
Incident wave
d
E1
k1
Prism 1
Prism 2
f
f
x
n1
n0
n0
Er
Et'
kt'
kr
Reflected wave
Tunnelled wave
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Boundary conditions

• If the electric field is perpendicular to the
  wave vector plane

E10, Er0, Et0 amplitudes of the incident,
reflected and transmitted waves
k1x, krx, ktx x components of the wave vectors
of the incident, reflected and transmitted waves
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Boundary conditions cont.

• For the wave vectors

k1y, kty y components of the incident and
transmitted wave vectors n0 prism index of
refraction n1 medium between prisms index of
refraction
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Solution

• If the incident angle is greater than the
  reflection angle, Snells law gives
• x component of the wave vector is a pure
  imaginary gt the wave propagates along the plane
• gt the amplitude decays exponentially

f incident angle
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Solution cont.

• The field in the second prism

d prism distance ? vacuum wavelegth of
incident light T decay coefficient
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Comparation decay coefficient

• Centimeter waves
• Optical range

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Comparation cont.

• Agreement is relatively good
• Error causes
• Imprecise prism refraction index values
• In optical range
• Prism surface defects and dust
• Motor precision ...

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Conclusion

• We have obtained, measured and modelled optical
  tunnelling
• It may be said

The only condition for light incident on a prism
plane with an angle greater than the critical
angle not reflecting completely is to put another
prism plane next to the original plane to a
distance of the order of the wavelength used