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Problem 15.

Problem 15. Optical Tunnelling. 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. Experiment. Two measurement ranges: Centimeter waves – accurate measuring

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Problem 15.

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  1. Problem 15. Optical Tunnelling

  2. 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.

  3. 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

  4. 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!

  5. Centimeter waves cont. • Apparatus schematic: detector Radiation source Translation system multimeter

  6. Centimeter waves cont.

  7. Centimeter waves cont. Tunelled field:

  8. 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!

  9. 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

  10. Visible light cont. • Apparatus schematic: Slow motor + translation laser prisms Data receiving computer oscilloscope

  11. Visible light cont.

  12. 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

  13. 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 haven’t interfered completely and a decaying field exists

  14. 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

  15. Maxwell equations E – electric field B – magnetic field induction P – polarization c – speed of light in a vacuum ε0 – vacuum permittivity

  16. Plane wave solutions Electrical field: E0 – amplitude ω – frequency t – time k – wave vector r - radiusvector Magnetic field:

  17. Geometry of the problem y Incident wave d E1 k1 Prism 1 Prism 2 φ φ x n1 n0 n0 Er Et' kt' kr Reflected wave Tunnelled wave

  18. 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

  19. 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

  20. Solution • If the incident angle is greater than the reflection angle, Snell’s law gives • x – component of the wave vector is a pure imaginary => the wave propagates along the plane • => the amplitude decays exponentially φ – incident angle

  21. Solution cont. • The field in the second prism: d – prism distance λ – vacuum wavelegth of incident light Θ – decay coefficient

  22. Comparation – decay coefficient • Centimeter waves: • Optical range:

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

  24. 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

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