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white. n. t. t = 400 nm, n = 1.515, q = 0. R. Wavelength (nm). Q t. Multiple Beam Interference. for internal:. Reflection coefficient:. for internal:. Transmission coefficient:. Amplitude of each reflection:. rE o. tt’r’E o. tt’r’ 7 E o. tt’r’ 5 E o. tt’r’ 3 E o. E o. n.

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  1. white n t t = 400 nm, n = 1.515, q = 0 R Wavelength (nm)

  2. Qt Multiple Beam Interference for internal: Reflection coefficient: for internal: Transmission coefficient: Amplitude of each reflection: rEo tt’r’Eo tt’r’7Eo tt’r’5Eo tt’r’3Eo Eo n tr’Eo d tEo

  3. Phase lag between reflections: (difference between the first and the rest will be covered by reflection amplitudes r and r’) Modify amplitude and phase for each reflected ray: For N >= 2

  4. Total reflected field: factor combine

  5. This makes a geometric series: Which converges to: for |x|<1 By the way: Stoke’s Relations

  6. …but when using complex notation for fields: …which means… …where * means the complex conjugate ( j -> -j )

  7. Use:

  8. t = 500 nm, n = 1.515, q = 0 R Wavelength (nm) t = 500 nm, n = 1.515, q = 0 R Double Beam Multiple Beam

  9. t = 500 nm, n = 1.515, q = 0 1.2 1 0.8 R 0.6 0.4 0.2 0 400 450 500 550 600 650 700 Wavelength (nm) t = 500 nm, n = 1.515, q = 0 R Add 50% reflective surfaces! Double Beam Multiple Beam

  10. Phasors: Describe amplitude and phase with vector length and direction. destructive constructive A A B B A+B A+B

  11. A B A+B Phasors: Describe amplitude and phase with vector length and direction. in between

  12. l/20 off constructive: constructive: destructive:

  13. R destructive l/20 off destructive

  14. Double Beam Changing phase, not time Multiple Beam

  15. Multiple beam interference also causes sharp peaks with angle of incidence: t = 500 nm, n = 1.515, l = 600 nm R 0 45 90 qi

  16. t = 500 nm, n = 1.515, q = 0 T

  17. Multiple beams interfere constructively and destructively with a much sharper phase dependence than double beam interference since the multiple interfering components dephase at different rates.

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