# Attenuation in Fibers

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This is a continuation from the previous tutorial - graded-index fibers.

Several factors contribute to attenuation of the power of an optical wave propagating in an optical fiber. As discussed in the propagation in an isotropic medium tutorial, when an optical wave propagates in a lossy medium with an attenuation coefficient $$\alpha$$, its intensity decays exponentially with distance according to (103) [refer to the tutorial]. Since the power of an optical wave in a fiber is simply the integration of its intensity over the cross section of the fiber, the attenuation of optical power over a propagation distance l in a fiber having an attenuation coefficient $$\alpha$$ is given by

$\tag{89}P_{out}=P_{in}e^{-\alpha l}$

where Pin and Pout are the input and output power, respectively.

In (89), Pin and Pout are measured in watts or, for example, milliwatts or microwatts in low-power applications or kilowatts or megawatts in high-power applications, while $$\alpha$$ is given per meter. In practical applications, $$\alpha$$ is also measured per centimeter or per kilometer when l is measured in centimeters or kilometers.

In practical engineering applications, it is convenient to use decibels (dB) as a measure of relative changes of quantities. The attenuation coefficient $$\alpha$$ is then measured in decibels per meter. In the case of low-loss fibers, the propagation length in a fiber is usually measured in kilometers, and $$\alpha$$ is conventionally given in decibels per kilometer:

$\tag{90}\alpha(\text{dB km}^{-1})=-\frac{1}{l(km)}10\log\frac{P_{out}}{P_{in}}$

where Pin and Pout are measured in watts, milliwatts, or microwatts. Comparing (90) with (89), we have

$\tag{91}\alpha(\text{dB km}^{-1})=4.34\alpha(\text{km}^{-1})\qquad\text{and}\qquad\alpha(\text{km}^{-1})=0.23\alpha(\text{dB km}^{-1})$

Power can also be measured in decibels and has units of decibel-watts (dBW), decibel-milliwatts (dBm), and decibel-microwatts (dBμ) defined as follows:

$\tag{92}P(\text{dBW})=10\log P(\text{W}),\qquad P(\text{dBm})=10\log P(\text{mW}),\qquad P(\text{dBμ})=10\log P(\text{μW})$

When power is given in decibel-watts or decibel-milliwatts and the attenuation coefficient is in decibels per kilometer, (89) can be expressed as

$\tag{93}P_{out}(\text{dBW})=P_{in}(\text{dBW})-\alpha(\text{dB km}^{-1})l(\text{km})$

or, equivalently,

$\tag{94}P_{out}(\text{dBm})=P_{in}(\text{dBm})-\alpha(\text{dB km}^{-1})l(\text{km})$

A similar formula can be written for power measured in decibel-microwatts. These formulas are very convenient and useful in practical applications as they relate the input power, output power, and attenuation in a simple arithmetic relation.

Example

A fiber of 40 km length has an attenuation coefficient of 0.6 dB km-1 at 1.3 μm and 0.3 dB km-1 at 1.55 μm. If 1 mW of optical power at each wavelength is launched into the fiber, what is the output power at each wavelength?

We can convert the attenuation coefficient given in decibels per kilometer into that measured per kilometer and then use (89) to find the output power. Alternatively, we can convert the input power given in milliwatts into that in decibel-milliwatts or decibel-microwatts and then use (94) to find the output power. The results are the same. Here we use the second approach. Then, Pin = 1 mW is converted to Pin = 0 dBm = 30 dBμ using (92). The output power at 1.3 μm is

Pout = 0 dBm - 0.6 dB km-1 x 40 km = -24 dBm = 6 dBμ,

which is Pout ≈ 4 μW from (92). Similarly, the output power at 1.55 μm is

Pout = 0 dBm - 0.3 dB km-1 x 40 km = -12 dBm = 18 dBμ,

which is Pout ≈ 63 μW. Comparing the results at two wavelengths, we see the importance of reducing the losses in a fiber: a reduction in the attenuation coefficient by a factor of 2 increases the output power by a factor of nearly 16 in this particular example. The effect is even more dramatic at high losses. A doubling of the attenuation coefficient from 0.6 to 1.2 dB km-1 results in an output power of only 15.8 nW, which is a reduction of more than 250 times from the 4 μm output for the 0.6 dB km-1 attenuation.

Attenuation of light in a fiber is primarily caused by absorption and scattering. In addition, there are mechanical losses and losses due to nonlinear optical effects. The effects of these loss mechanisms vary, but they all add up to the total loss in a fiber. Since the majority of optical fibers are silica fibers, we discuss the loss mechanisms and their effects in silica fibers below.

1. Electronic absorption.

The bandgap of fused silica is about 8.9 eV, which corresponds to the photon energy of light at the ultraviolet wavelength of approximately 140 nm. This causes strong absorption of light in the ultraviolet spectral region due to electronic transitions across the bandgap.

Light in the visible and infrared regions has photon energies less than the bandgap energy and is not expected to be absorbed through direct electronic transitions across the bandgap. However, in practice, the bandgap of a material is not sharply defined but usually has bandtails extending from the conduction and valence bands into the bandgap due to a variety of reasons, such as thermal vibrations of the lattice ions and microscopic imperfections of the material structure. In particular, an amorphous material like fused silica generally has very long bandtails. These bandtails lead to an absorption tail extending into the visible and infrared regions. Empirically, it is found that the absorption tail at photon energies below the bandgap falls off exponentially with photon energy.

2. Molecular absorption.

In the infrared region, the absorption of photons is accompanied by transitions between different vibrational modes of silica molecules. The fundamental vibrational transition of fused silica causes a very strong absorption peak at about 9 μm wavelength. Nonlinear effects contribute to important harmonics and combination frequencies corresponding to minor absorption peaks at 4.4, 3.8, and 3.2 μm wavelengths. The result is a long absorption tail extending into the near infrared, causing a sharp rise in absorption at optical wavelengths longer than 1.6 μm. Molecular absorption is the major cause of attenuation in the infrared spectral region for a silica fiber.

3. Impurity absorption.

Impurity absorption could be very important in the near infrared region because most impurity ions such as OH-, Fe2+, and Cu2+ form absorption bands in this region where both electronic and molecular absorption losses of the host silica glass are very low.

Near the peaks of the impurity absorption bands, an impurity concentration as low as one part per billion can contribute to an absorption loss as high as 1 dB km-1. In fact, fiber-optic communications were not considered possible until it was realized in 1966 that most losses in earlier fibers were caused by impurity absorption and then ultra-pure fibers were produced in the early 1970s.

Today, impurities in fibers have been reduced to levels where losses associated with their absorption are negligible, with the exception of the OH- radical. The OH- radical results from the presence of water, which can enter a fiber through the manufacturing process or as humidity in the environment. Therefore, fibers are manufactured in ultra-dry conditions and are protected by plastic coating from water in the environment to reduce the loss caused by OH- absorption.

The absorption peak due to the fundamental vibration of the OH- ions appear at 2.73 μm wavelength where intrinsic molecular absorption of silica is strong. The most important absorption peaks are those at the harmonics and combination frequencies of 1.39, 1.25 and 0.95 μm wavelengths.

4. Rayleigh scattering.

The intrinsic Rayleigh scattering in a fiber is caused by variations in density and composition that are built into the fiber during the manufacturing process. They are primarily a result of thermal fluctuations in the density of silica glass and variations in the concentration of dopants before silica passes its glass transition point to become a solid.

These variations are a fundamental thermo-dynamic phenomenon and cannot be completely removed. They create microscopic fluctuations in the index of refraction, which scatter light in the same manner as microscopic fluctuations of the density of air scatter sunlight.

This elastic Rayleigh scattering process creates a loss given by

$\tag{95}\alpha_R=\frac{8\pi^2}{3\lambda^4}(n^2-1)\beta k_BT$

where n is the index of refraction, kB is the Boltzmann constant, T is the glass transition temperature, and β is isothermal compressibility.

Note that $$\alpha_R\propto\lambda^{-4}$$. The loss due to Rayleigh scattering is very important in the short-wavelength region but falls off rapidly as the wavelength increases.

5. Waveguide scattering.

Imperfections in the waveguide structure of a fiber, such as nonuniformity in the size and shape of the core, perturbations in the core-cladding boundary, and defects in the core or cladding, can be generated in the manufacturing process.

In addition, environmentally induced effects, such as stress and temperature variations, also cause imperfections. The imperfections in a fiber waveguide result in additional scattering losses. They sometimes also induce coupling between different guided modes. Losses caused by waveguide scattering due to imperfections can be measured experimentally.

6. Nonlinear losses.

In an optical fiber, because light is confined over long distances, nonlinear optical effects can become important even at a relatively moderate optical power.

Nonlinear optical processes such as stimulated Brillouin scattering and stimulated Raman scattering can cause significant attenuation in the power of an optical signal.

Other nonlinear processes can induce mode mixing or frequency shift, all contributing to the loss of a particular guide mode at a particular frequency.

Because nonlinear effects are intensity dependent, they can become very important at high optical powers.

Figure 8 below summarizes the contributions of various loss mechanisms, except those of waveguide scattering and nonlinear losses, to the total attenuation in a fiber as a function of wavelength.

The limiting effect at short wavelengths is the Rayleigh scattering, which dominates the electronic absorption of fused silica in this spectral region.

In the infrared region beyond 1.6 μm, attenuation is completely dominated by intrinsic absorption due to molecular vibrations of silica.

In the near-infrared region, attenuation strongly depends on the concentration of the OH- impurity. In addition, in this low-loss region, any amount of loss caused by waveguide scattering would be relatively significant. Therefore, attenuation in this spectral region varies with the quality of the fiber.

The attenuation coefficient is also mode dependent. The fundamental mode generally has lower attenuation than high-order modes because its power is more confined to the core. Therefore, single-mode fibers usually have lower attenuation than multimode fibers.

Among multimode fibers of a fixed outer diameter, such as the standard 125-μm size, the ones with larger cores, and simultaneously thinner claddings, typically have higher attenuation because the intensity distribution spreads farther out. A graded-index multimode fiber usually has lower attenuation than a comparable step-index multimode fiber because the intensity in a graded-index fiber is more concentrated at the center of the fiber.

There are three wavelength windows for applications in the transmission of light with fibers. They are the 850-nm window, corresponding to the wavelengths of GaAs/AlGaAs lasers, and the 1.3- and 1.55-μm windows, corresponding to the wavelengths of InGaAsP/InP lasers.

It can be seen from figure 8 that the lowest attenuation in the entire spectral range occurs at 1.55 μm while the attenuation at 1.3 μm is slightly higher. At present, the best fibers have attenuation as low as 0.15 dB/km at 1.55 μm and 0.3 dB/km at 1.3 μm, while attenuation at 850 nm is typically 2 dB/km.

This is the reason why the wavelength of 1.55 μm is chosen for long-distance optical communication systems and the wavelength of 1.3 μm is suitable for metropolitan and wide-area networks, while wavelengths in the 850-nm window are only useful for local optical links.

In addition to the losses discussed above, there are also bending losses caused by macrobends and microbends in a fiber and connection losses incurred at the junctions of fibers. Macrobends are bends visible from outside and are encountered in the looping or routing of fibers. Microbends are not visible from outside and are typically created by mechanical stresses associated with bundling, packaging, and handling of the fiber.

Bending loss can be understood from the viewpoint of ray optics or that of wave optics. For simplicity, consider the fact that the evanescent field of a guide mode actually extends to infinity in all radial directions. When a fiber is bent, the evanescent field on the outside of the bent curve has to travel along a path that has a larger radius of curvature than that travelled by the field in the core of the fiber. Because different parts of a mode field have to stay in phase as a single entity, this evanescent field has to travel faster in order to keep up with the field in the core. The farther outside the field is, the faster it has to travel. At a critical radius, the required speed would exceed the speed of light. At this point, the field cannot keep up and radiates away, resulting in bending loss.

Loss caused by controlled bending can be quantified. Fiber sensors based on bending-induced losses can be constructed for many useful applications.

The next part continues with the Dispersion in Fibers tutorial.