Optical Fibers Feb. 8, 2018

• Effect of microcracks, microbending and temperature: Microcracks and microbending cause abrupted change in refractive index at certain locations$\text{ }\to$ leakage of signal, i.e. attenuation.
Temperature can cause gradual changes in refractive index along a fiber -- introduce phase distortion, dispersion, i.e. spreading of pulse and limiting speed.
Fiber structure: Buffer coating > cladding > core
index of cladding is$\text{ }{n}_{2}$ , index of core is$\text{ }{n}_{1}$ and core radius is$\text{ }a$
• Index profiles: Common ones -- step and grade index profiles with percentage change in refractive index,$\text{ }\Delta$ :
General expression - n= n_1 [1 - 2 Delta ( r / a )^ alpha ] ^(1/2) for r < a
and n = n_2 = n_1 (1 - 2 Delta ) ^ (1/2) for r > a
Note:$\text{ }{n}_{1}\text{\hspace{0.17em}}>\text{\hspace{0.17em}}{n}_{2}$

• Type of fibers:
Materials -- glass fiber, lambda = 800nm, 1300nm and 1500nm, low losses (i.e. windows of transmission), used in long distance; plastic fiber, lambda = 580nm, high losses, short distance (computer applications).
Core size -- single mode, a = 5 mu m , only 1 mode, reduced dispersion; multimode, a = 50 - 85 mu m , many modes, large dispersion

Mode description: can be thought as harmonics in space for a light pipe.
- Each mode corresponds to a ray along a certain path.
- only a finite number of incidence angles results in constructive interference and forms mode pattern.
- Number of mode M = "Int" ( V^2 / 2) where normalized freq. parameter V = (2 pi) / lambda a sqrt ( n_1^2 - n_2^2 ) where sqrt( n_1^2 - n_2^2 ) is the numerical aperture (NA) and Int(.) means integer part.
This formula is good when M is large.
- This formula explains that a fiber with very small a as single mode fiber.
- Mode designation, e.g. TE_(mn) or TM_(mn) where m and n are the mode indices (harmonic number), TE means transverse electric (i.e. s-polarized), TM transverse magnetic (i.e. p-polarized).
Assuming wave propagating along z, TE -> E_z =0 and TM -> H_z =0
Each mode is a eigenfunction of the wave equation and its eigenvalue is the transverse wave number.
- Fundamental mode that cannot be cut-off is called HE_{11} or LP_(01) Its distribution matches Gaussian beam. Next mode appear when V > 2.405

• Common fibers: Grade index fiber - dispersion is medium => data rate times distance = RL is medium; difficulty in coupling of light is medium.
Step index multimode - dispersion is large, RL small, difficulty in coupling low.
Step index single mode - dispersion is low, RL large, difficulty in coupling high => require laser as source for single mode.