# FiberOptics_02_FiberOpticsExplained

Theory

The entire Electro-magnetic spectrum spans from a wavelength of 100 km at 3 kHz frequency (audio waves) to a wavelength of 10-4 nm at 1010 Hz frequency (gamma rays). The wavelength, λ, is defined as speed of light in a vacuum/ frequency, or

\begin{align} \lambda = \frac{c}{f} where\ c = 3.0 \times 10^8 m/s\\ \end{align}

The actual visible range of the Electro-magnetic spectrum is between the color violet, at a wavelength of 455 nm and the color red, at a wavelength of 750 nm. Light may be faintly seen up to wavelengths of 820 nm. Just because light may not be visible does not mean that it is safe to stare into the source.

Communications are conducted at wavelengths of 850 nm, 1300 nm, and 1550 nm, none of which are visible. These wavelengths in which most communication applications occur in industry are known as “windows”. Windows are the places in which light is transmitted in a fiber at low loss (attenuation) or without serious compromise of the bandwidth (dispersion). The bandwidth of the system defines how quickly information can be transmitted.

The index of refraction is the optical density of a material. It is the ratio of the velocity of light in free space (i.e. a vacuum) to the velocity of light through the transparent material. It is the quantity that measures how much light will slow down when travelling through an optically transparent material when compared to air which has an index of one, n = 1. The larger the index of refraction is, the slower light will travel through the material. The index of refraction depends on the wavelength of light. Propagating light of one wavelength may slow down more or less than light at another wavelength in the same material. Glass has an index of refraction of about 1.5.

\begin{align} n &= \frac{speed\ of\ light\ in\ a\ vacuum}{speed\ of\ light\ in\ the\ material} = \frac{c}{v} \\ \end{align}

The following is a graphical representation of ray refraction through two materials:

Bandwidth

Fiber dispersion, sources, and detectors effect the rate of transmission.

Graded index fiber has a larger bandwidth due to the fact that it limits dispersion and therefore preserves the integrity of a pulse over a longer distance than step index fiber does.

The bandwidth-length product is a number that tells what the bandwidth of the fiber is for the given length. For example, if a fiber is rated at 30 MHz-km, it mean that communications at a bandwidth of 30 MHz can occur without decay of the signal from dispersion causing signal errors over a distance of 1 km. It also means that if the transmission requires 120 MHz that it can be successfully achieved for distances of 250 m.

Dispersion

Dispersion is a broadening of a pulse as it propagates down the fiber and is due to several properties. The causes of dispersion depend on the type of application: single mode or multi mode.

## Single-mode fiber:

The main source of dispersion in single-mode fiber is material dispersion, which is in relation to the spectral width of the source. Material dispersion comes from a frequency-dependent response of a material to waves3, so different wavelengths of light will cause the material to respond differently. This is due to the relationship between the wavelength of the transmitted light, and as a function of that light, the index of refraction.

Waveguide dispersion effects single mode applications as well. Waveguide dispersion occurs when the speed of a wave in a waveguide depends on its frequency for geometric reasons, independent of any frequency dependence of the materials from which it is constructed. More generally, "waveguide" dispersion can occur for waves propagating through any inhomogeneous structure, whether or not the waves are confined to some region.3 Any source will have a line width associated with it or a range of light frequencies it produces. The pulse transmitted will spread as a result of the pulse being created by several frequencies of light. Therefore, even if the index of refraction were completely the same for all frequencies, they would still travel at a slightly different velocity.

## Multi-mode fiber:

Multi-mode fiber is limited by modal dispersion due to its large core-size. Modal dispersion is a distortion mechanism in which the signal is spread in time because the propagation velocity of the optical signal is not the same for all modes.4 The signal is spread in time because different modes or paths of light travel at different speeds as they propagate through the waveguide.

Having a graded index in a fiber greatly reduces this dispersion effect by causing modes that take a longer path to travel faster along the edges than they would if they had traveled up the center or by the shortest path possible. In a step index multi-mode fiber, modal dispersion can result in a delay time between modes of 50 ns/km while the delay in a graded index fiber would be .5 ns/km.

Attenuation

The amount of loss incurred as light propagates through a fiber is known as fiber attenuation. The attenuation depends on materials used as dopants and wavelength of operation. A typical value of attenuation in step index multimode fiber is 4 dB/km at 850 nm wavelength operation with a bandwidth of 100 Hz-km, and .15 dB/km for single mode at 1500 nm wavelength operation with a bandwidth of 88 GHz-km.  The large-core plastic fibers (approximate core sizes are around 200 microns) have 1000 dB/km attenuation.

An excessive bend radius will increase attenuation.

\begin{align} Attenuation &= -10 \times \log_{10} \frac{I_{out}}{I_{in}} & (dB/km) \\ \end{align}

Numeric Aperture

To maintain the “weakly guiding” waveguide condition, the difference between the index in the core and the cladding must be much less than one or

\begin{align} \Delta \ll 1\ & where\ \Delta = \frac{(n_{core} - n_{clad})}{n_{core}} \\ \end{align}

The weakly guiding condition is necessary for quantifying the light that propagates down the fiber.

The numeric aperture, NA of a fiber is basically the “cone of acceptance” of the fiber in that all light entering the waveguide within the cone area defined by the NA will be guided down the fiber or will successfully propagate.

\begin{align} NA\ = \sqrt{(n_{core}^2 - n_{clad}^2)} & which\ is\ approximated\ by\ NA\ = n_{core} \sqrt{(2\Delta)} \\ \end{align}

Modes and V Number

A mode is a possible path that light can take when propagating down a fiber optic. The geometry of the fiber and the wavelength of light propagating determine how many modes will propagate. Fiber that allows light to follow one single propagation path is called “single-mode” fiber.

The V number is used to roughly estimate how many modes will propagate given fiber geometry and the wavelength of light. To determine how many modes will propagate in a fiber, the V number is calculated by:

\begin{align} V\ = \frac{\pi \text{ D (NA)}}{\lambda}\ \text{ where } \lambda \text{ is wavelength, D is diameter, and NA is the numeric aperture.} \\ \end{align}

To achieve single mode operation or a single path of propagating light, the fiber system must have a V number less than 2.405. Therefore, the diameter will have the following constraints:

\begin{align} D\ \lt \frac{(2.405 \lambda)}{( \pi \text{NA})} \\ \end{align}

Wavelength vs. Core Size: Typical NA can be in the range .1 to .15 for a steeped index single mode fiber.

To maintain single mode:

• NA = 0.2, Diameter must be no more than 3.8 times the wavelength
• NA = 0.15, Diameter must be no more than 5 times the wavelength
• NA = 0.1, Diameter must be no more than 7.6 times the wavelength

Therefore at 8 or 9-micron diameters, the fiber can operate at 1300 nm for a NA of .15.

### Core Size, NA and D:

• Since NA is related to how easy it is to couple light in to a fiber, it makes sense that large a NA fiber is sought. However, the core size must be of a manageable enough size. By adjusting the index of refraction in the core and in the cladding the NA is altered, but if you are going to use wavelengths around 1300 or 1550 nm this will adjust the core size as well.
• The above is not true for multimode fibers. In multimode, there is no limit on the core size since the amount of modes traveling does not matter. However, the larger the D, the larger the modal dispersion will be due the time delay caused between the highest and lowest order modes. For single mode and multimode, D << 1.

Types of Losses

### Alignment Losses

A loss due to lateral displacement results when the center axes of two fiber cores are not aligned. If a 10% displacement results in .5 dB of loss, this would require each fiber end to only contribute to .5 microns of misalignment for a fiber of 10 microns. In a fiber of 100-micron core diameter, this 10% displacement would translate to a tolerance of +/- 5 microns per fiber end. For smaller diameter fiber or single mode fiber in harsh environments the tolerances are not easily maintained, as seen from the example above.

The tolerances of connectors should be checked such that these misalignments are minimized. When the end faces of two fiber surfaces are not completely parallel to one another upon mating, they are said to be angularly misaligned. Proper fiber end polishing should ensure that the fiber end face is indeed perpendicular to the fiber center axis and therefore, parallel to the mated fiber endface.

### Microbend Losses:

Microbend losses are caused by either internal imperfections in the fiber surfaces or external forces outside of the fiber. Whichever the cause, light deviates from the normal propagation path by escaping through the cladding. A loss in power output results from the light not staying confined to the core.
When choosing a fiber optic connector, it is important to minimize the amount of microbend losses incurred as a result of the design. For example, the crimped portion of the connector should not inhibit the transmission signal.

### Macrobend Attenuation:

Macrobend attenuation is caused by large bends in the fiber that allow light to escape through the cladding.

Bend Radius: The bend radius should be no less than five times the cable diameter for unstressed conditions and ten times the cable diameter for stressed conditions. Besides increasing attenuation, sharp bends in the cable will also decrease the fiber tensile strength. Under normal conditions, the tensile strength of fiber is greater than that of a steel fiber. A copper wire would have to be twice the diameter of a fiber to match it in tensile strength.

### Cracks in the Fiber:

The output power will be well below acceptable levels if there is a crack along a transmission fiber.

If a system is functioning adequately, it is rare that a crack is present. Fiber cracks are not usually visible by magnified termination end inspections unless they are close to the terminated surface end.

### Air Gap:

Air gaps (or end separations) are a source of loss in connector systems.  These gaps are present in flat polish systems where the light leaving the transmitter fiber experiences a change in index of refraction upon entering the gap and then again upon entering the receiver fiber. The result is back reflections (or Fresnel’s reflections) from the end face of the transmitter fiber and the end face entrance of the receiver fiber.

One way to avoid the air gap losses of some connector designs is to use a physical contact polish in which the end of the fiber is terminated in a curved fashion as opposed to a flat.

### Fiber Mismatch:

When two fibers are joined such that light from one core can couple into the core of the other, properties such as fiber geometry or numeric aperture (NA) must be taken into account.

If the numerical aperture of the second fiber is smaller than that of the first a NA-mismatch loss is incurred. The calculation for NA-mismatch loss is:

\begin{align} Loss_{\text{NA Mismatch}}\ = 10 \times \log_{10} \frac{{NA}_{rec}}{{NA}_{trans}}^2 \\ \end{align}

“rec” stands for receiving fiber and “trans” stands for transmitting fiber.

Losses that occur as a result of a mismatch in fiber core diameter can be calculated by:

\begin{align} Loss_{\text{Core Diameter Mismatch}}\ = 10 \times \log_{10} \frac{{Diam}_{rec}}{{Diam}_{trans}}^2 \\ \end{align}

Other types of fiber geometry mismatch losses are not as easily quantified without insertion loss testing; cladding diameter mismatch, elliptically or ovality losses (in which both cores have elliptical geometry and are misaligned), and concentricity losses in which the core is not placed directly in the center of the cladding.

### End face obstructions:

Any dirt or damage on the end face of a fiber will inhibit the coupled signal. Magnified inspection of the fiber endface should ensure that the end face is free from imperfections and debris.