Nonlinear Fiber Optics - 4 ed. Agrawal

Appendix A 515
converted into decibels using the general definition
R (in dB)=10 log 10 R.
(A.1)
The logarithmic nature of the decibel scale allows a large ratio to be expressed as a
much smaller number. For example, 10 9 and 10 −9 correspond to 90 dB and −90 dB,
respectively. As R = 1 corresponds to 0 dB, ratios smaller than 1 are negative on the
decibel scale. Furthermore, negative ratios cannot be expressed using decibel units.
The most common use of the decibel scale occurs for power ratios. For instance,
the fiber-loss parameter α appearing in Eq. (1.2.3) can be expressed in decibel units by
noting that fiber losses decrease the optical power launched into an optical fiber from
its value at the input end, and thus can be written as a power ratio. Equation (1.2.4)
shows how fiber losses can be expressed in units of dB/km. If a 1-mW signal reduces to
1 μW after transmission over 100 km of fiber, the power reduction by a factor of 1000
translates into a 30-dB loss from Eq. (A.1). Spreading this loss over the 100-km fiber
length produces a loss of 0.3 dB/km. The same technique can be used to define the
insertion loss of any component. For instance, a 1-dB loss of a fiber connector implies
that the optical power is reduced by 1 dB (≈20%) when the signal passes through the
connector. Examples of other quantities that are often quoted using the decibel scale
include the signal-to-noise ratio and the amplification factor of an optical amplifier.
If optical losses of all components in a fiber-optic communication system are expressed
in decibel units, it is useful to express the transmitted and received powers also
by using a decibel scale. This is achieved by using a derived unit, denoted as dBm and
defined as
( power
)
power (in dBm)=10 log 10 , (A.2)
1mW
where the reference level of 1 mW is chosen for convenience; the letter m in dBm is
a reminder of the 1-mW reference level. In this decibel scale for the absolute power,
1 mW corresponds to 0 dBm, whereas powers