Optical Fiber Properties

Orientational Polarizability • No restoring force: analogous to conductivity O - p H + θ O O p=0 Analogous to conductivity, the molecules collide after a certain time t, giving: CDα = oα τωi−1 +q C H -q For a group of many molecules at some temperature: f =e −U k bT =e pE cosθ k bT After averaging over the polarization of the ensemble molecules (valid for low E-fields): α DC p2 ~ 3kbT 1 Dielectric Loss • • • For convenience, imagine a low density of molecules in the gas phase C-M can be ignored for simplicity There will be only electronic and orientational polarizability ε r = 1+ χ e + χ o = n 2 + Nα DC 3ε o (1− iωτ ) ωτ << 1, , ε r = ε so = n 2 + Nα DC 3ε o ε so − n 2 ∴ εr = n + 1− iωτ 2 ε’,ε’’ αe+αi εso We can write this in terms of a real and imaginary dielectric constant if we choose: ε r = ε '+iε '' ε so − n 2 ε so − n 2 ; ε '' = ε'= n + ωτ 2 2 2 2 1+ ω τ 1+ ω τ 2 αe n2 -2 0 logωτ Water molecule: τ=9.5x10-11 sec, ω~1010 microwave oven, transmission of E-M waves +2 2 Dielectric Constant vs. Frequency • Completely general ε due to the localized charge in materials ε molecules αo ions αi n2 1 electrons αe 1/τ ωT ωoe ω Dispersion-free regions, vg=vp 3 Dispersion • Dispersion can be defined a couple of ways (same, just different way) – when the group velocity ceases to be equal to the phase velocity – when the dielectric constant has a frequency dependence (i.e. when dε/dω not 0) ω Dispersion-free Dispersion ω= vp = vp = ω k ω k = = c εr c εr k = ∂ω = vg ∂k ∂ω c ≠ = vg ε r (ω ) ∂k k 4 Characteristics of Optical Fiber • Snell’s Law θ1 n1 Boundary conditions for E-M wave gives Snell’s Law: n2 Refraction n1 sin θ1 = n2 sin θ 2 θ2 θ1 θ2 n1 n2 Internal Reflection: θ1=90° θ 2 = θ c = sin −1 n1 n2 Glass/air, θc=42° 5 Characteristics of Optical Fiber • • Attenuation – Absorption • OH- dominant, SiO2 tetrahedral mode – Scattering • Raleigh scattering (density fluctuations) αR~const./λ4 (<0.8 μm not very useful!) Dispersion – material dispersion (see slide i13) x •Light source always has Δλ •parts of pulse with different l propagate at – modal dispersion different speeds Black wave arrives later than red wave y n1 Solution: grade index n2 n 6 Characteristics of Optical Fiber Egyptian 106 Venetian Optical loss dB/km 105 Bohemian 104 Window Glass 103 100 10 1 0.1 3000 BC 1000 AD 1964 1974 1988 Figure by MIT OpenCourseWare. 7 Characteristics of Optical Fiber Image removed due to copyright restrictions. Please see any diagram of the typical optical attenuation curve, such as http://www.nikhef.nl/~nooren/dispersie3_files/image010.jpg 8 Characteristics of Optical Fiber Pluse spreading (ns/km) -8 -6 LED ∆λ = 30nm -4 -2 -0 2 Laser ∆λ = 3nm Narrow Width Laser ∆λ = 0.1nm 0.7 0.8 0.9 1.0 1.1 Wavelength (µm) 1.2 1.3 1.4 Figure by MIT OpenCourseWare. 9 Ferroelectrics • ‘Confused’ atom structure creates metastable relative positions of positive and negative ions Barium titanate (BaTiO3) Titanium (Ti4+) Barium (Ba2+) Oxygen (O2-) Figure by MIT OpenCourseWare. 10 Ferroelectrics • • • Each unit cell a dipole! Very large Ps (saturated polarization, P(E=0) No iron involved; ‘Ferro’ since hysterisis loop analogous to magnetic materials E Two equal-energy atom positions Can flip cell polarization by applying large enough reverse E-field to get over barrier P Ps ΔE ‘normal’ dielectric Ro Ec E R 11