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Optic Fiber

The optic fiber in the modern era of fiber-optic communication has its origin as wave guides in early 20th century for radio wave transmission. Application was expended to microwave during WWII. Expansion into higher frequencies in the optical range became feasible with the dielectric wave guide in the 1970s when fiber with very small diameter (the size of human hair ~ 0.005 cm thickness) can be manufactured. The reason will become obvious at the end of the following mathematical derivation.

The solution of the wave equation for optical fiber (see Figure 01) is obtained under several assumptions : The wave equation in cylindrical coordinates is :

BTW, the Maxwell's equations in medium after the separation of the variable t by e-it become : xB = (-i/v)E, xE = (i/v)B, B = 0, and E = 0, giving, for example, B = (v/)2 xxB. Also see Bessel Function in its various forms.

Im Bessel Function

Figure 03 Im Bessel Function
[view large image]

Optic Fiber Upgrade
    As the use of fiber optics for data transmission over the internet has been skyrocketed especially for video viewing (Figure 06), the capacity of conventional optic fiber would reach its limit by 2020. There are upgrades to pump more data through existing fibers as summarized below. Eventually, it would have to be replaced by an entirely new design.

  • At present, one fiber can carry up to 160 laser beams with different frequency. The beams start to interfere with each other as the number increases. Software can be programmed to correct the distortion until the lasers become powerful enough to melt down the fiber.

Figure 06 Optic Fiber Upgrade [view large image]

  • By planting an optical conjugator halfway along the fiber, the distortion can be corrected by itself at the end. The problem is with variable length which requires a large amount of implementations.

  • Twisted light beams provide more ways to encode data depending on its twisted appearance. The number of patterns is potentially infinite, but it is still in experimental stage.

  • Reflection and Transmission
  • Ultimately, the unlimited demand could be radically satisfied by re-designing the fiber with a hollow core. There will not be total reflection (Figure 07), the glass cladding instead has to be micro-engineered - a difficult process that involves etching intricate marks onto the inner surface. Paradoxically, the cost would be reduced because the quality of the glass could be lowered to those among the everyday variety.
  • Figure 07 Reflection and Transmission [view large image]

    Since there would be no total reflection in this case, the fiber has to be constructed such that the incident angle is close to grazing (~ 90o) to maximize reflection (Figure 07).
    As shown in Figure 07, the reflection coefficients of both the and polarized beams attain the maximum value of 1.0 at 1 = 90o. The formulas for the reflection and transmission are also shown in the same picture. These are just the amplitude, the total intensity is the squared sum (Figure 07). The formulas can be expressed as a function of 1 by using the Snell's Law, also taking into account the narrower flux size for the transmitted beam in calculating the intensity. Theoretically, the value should be 1 (relative to the incident intensity) in case of no absorption and internal reflection. The Brewster's angle is unique for transmission, which becomes total at that angle making the reflected beam polarized from un-polarized light.