Propagation in Optical Fibers

There are two primary system parameters which determine the characteristics of optical communication systems. Specifically, data is transmitted by a sequence of pulses, and the system must ensure these pulses are received with a sufficiently low probability of error, also called the bit-error rate (BER). Given a particular receiver, achieving a specified BER requires a minimum received power and a maximum data rate or signal bandwidth. An optical fiber introduces attenuation and dispersion in the system. Whereas attenuation tends to increase the power requirements of the transmitter needed to meet the power requirements at the receiver, dispersion limits the bandwidth of the data which may be transmitted over the fiber. We first examine dispersion.

We recall the definition of index of refraction n as:

 

and hence n=1 for free-space. For silica glass , . By placing impuriti es (dopants) in the material we can modify n. Optical fibers have a small core surrounded by a (relatively) thick cladding whose index of refraction is slightly less than that in the core. Let denote the (nominal) value of n in the core, and in the cladding. Let us denote the radius of the core as a.

If n is constant in the core, it is a step-index fiber, and is otherwise a graded-index fiber. The value of n as a function of r for a step-index fiber is:

 

and is graphed in Figure , while the value of n as a function of r for a graded-index fiber is:

 

where and is graphed in Figure .

 

 

If we define , typically ; it is usually no more than a few percent. The numerical aperture can be approximated as:

 

A typical value is NA=0.2, which corresponds to a maximum entry angle .

Dispersion refers to the distortion of a propagating wave. Dispersionless transmission in general requires a constant group velocity. There are several types of dispersion.

Modal dispersion is caused when a propagating wave is comprised of several different modes with different propagation characteristics. Modal dispersion can be eliminated (in principle) using single-mode fibers have cross-section close to , and are common at 1300nm and 850nm. The core typically has a diameter of and the cladding a diameter of . Multimode fibers have core diameter and a cladding diameter of . The purpose of graded index fibers is to reduce modal dispersion in multimode fibers. If we use the ray model for propagating waves, different modes correspond to different angles of incidence at the core-cladding boundary; those with steeper angles have lower group velocity. However, in a graded-index fibers, the index of refraction decreases away from the center, hence the speed of light increases as the cladding is approached, and this tends to compensate for the different paths taken by different modes.

Single mode fibers have and multimode fibers have .

Material dispersion is caused by imperfect materials whose n depends on . This effect is usually an order of magnitude smaller than modal dispersion, and is typically quantified in terms of:

 

A third type of dispersion is waveguide dispersion, caused by non-constant group velocity as a function of for a fixed mode.

One of the main consequences of dispersion is that a propagating pulse will broaden. We can quantify the pulse broadening in terms of the variance of the waveform. That is, consider a pulse of light whose intensity as a function of time is ; normalize the intensity so that:

 

Define the mean time as:

 

and the variance as:

 

If we input a pulse with variance to a dispersive system, then the output has variance given by:

 

In quantifying dispersion, an important consideration is the spectral width of the light signal. The spectral width is a measure of the purity of the light as a function of wavelength. If we consider the intensity as a function of wavelength, , then the spectral width is defined as:

 

where:

 

and the intensity is normalized as:

 

Since dispersion causes pulse broadening, if we attempt to place too many pulses per second, they will spread and interfere with each other. Thus, a practical limitation is the available bandwidth, measured in MHz or Mbps. As we shall soon see, the bandwidth is inversely proportional to distance, so the bandwidth-distance product (in units of ) is approximately constant. For a multimode step index fiber, this is , for a step index multimode fiber it is , and it can be larger ( or more) for single mode fibers.

To understand the effect of dispersion, consider the group delay. This is the time delay per unit length of energy propagating through a transmission system. We can assume each spectral component travels independently and undergoes its own time delay, . Let L be the transmission distance and the group velocity. Then:

 

where:

 

If the spectral width is not too big, the delay difference over the range of wavelengths comprising the light energy can be approximated by at wavelengths . Thus, the delay difference between two such spectral components is:

 

We define the dispersion constant D as:

 

Its units are typically given as picoseconds per kilometer per nanometer.

To quantify material dispersion, we use the ray model approximation, which in particular yields where is a function of wavelength. This yields:

 

and hence:

 

If we define the material dispersion constant as:

 

Then:

 

We now quantify waveguide dispersion. We first write:

 

For small , we get:

 

If is not a function of , then:

 

Now, and hence:

 

A detailed computation yields:

 

Then:

 

For the type of material used in fibers, at lower wavelengths:

 

and:

 

and hence material dispersion dominates. At higher wavelengths (about ), waveguide dispersion dominates.

Modal dispersion in multimode fibers can be approximated as:

 

Now we consider attenuation. Any optical fiber will attenuate a propagating signal. Given input power over a fiber of length L and output power , the mean attenuation constant of the fiber, in units of dB/km, is defined as:

 

The decibel unit (dB) is used to represent power ratios. However, it is sometimes convenient to represent absolute power levels on a logarithmic scale. The most commonly used unit is dBm, which corresponds to power referenced to 10mW:

 

One of the main causes of attenuation is absorption of energy (or photons). Absorption is caused by atomic defects which result when the fiber is exposed to radiation, extrinsic absorption by impurity atoms, and intrinsic absorption by constituent atoms of the material. The dominant mechanism is extrinsic absorption, primarily by metallic ions (iron, cobalt, etc.) and ions.

In early optical fibers, the transmission distance was primarily limited by absorption by ions. These ions were introduced in the material from the presence of water or water vapor during the manufacture process. Attenuation caused by this ion is greatest at 1400, 950 and 725nm, leaving ``windows'' for transmission between these wavelengths. The advent of the vapor phase axial deposition (VAD) manufacture method led to tremendous reduction in the concentration in fibers.

Losses in modern fibers are caused by ultraviolet absorption, infrared loss and scattering losses. The scattering losses, modeled by Rayleigh scattering, are caused by the interaction of the light wave with the constituent molecules which are on the order of the light wavelength. Rayleigh scattering loss is , so it can be reduced by increasing the wavelength. On the other hand, infrared absorption loss tends to increase with , and is usually worst above . The point where this loss starts to increase to unacceptably large levels can be pushed out by doping the with halides. In general, the combined effect of such losses is minimum at about .

There are also losses caused by bends and microbends; a microbend is a tiny ``crinkle'' or imperfection in the surface of the fiber, on the dimensions of several wavelengths, and causes a perturbation in the field. Thus, microbends lead to coupling to higher order modes, which do not have the desired transmission characteristics, and also causes power loss. Bending and microbending can occur while the fiber is being manufactured, specifically during the spooling process. Spooling a fiber to minimize bends and microbends is not trivial when we consider that very long continuous fibers, of lengths 1km or more, are manufactured. The reason for manufacturing such long fibers is that splicing or coupling fiber segments together can introduce significant losses. The basic reason for loss when splicing fibers is the faces of the two segments are not properly aligned, so not all the output power of one segment is inserted to the other. Losses in modern fibers can be kept down to as little as 0.01dB/km.

If BW denotes the signal bandwidth and L the length of the fiber, then the bandwidth-length product is approximately constant. This constant depends on overall system parameters, such as total power loss, BER, etc. More precisely, an empirical result is:

 

is approximately constant, and the parameter is some value between 0.5 to 0.9. Usually, for L<1km, whereas for L>1km, . Actually, this is not an entirely empirical result. What happens is that, as light travels over a longer and longer distance, energy at one modes tends to couple or induce energy at other modes, so that over long distances the modes are strongly coupled to each other, and do not propagate independently. In any case, the length 1km usually separates the applications of optical fiber transmission into short-haul and long-haul links.