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Appendix A

System of Units

The international system of units (known as the SI, short forSysteme International) is

used in this book. The three fundamental units in the SI are meter (m), second (s), and

kilogram (kg). A prefix can be added to each of them to change its magnitude by a

multiple of 10. Mass units are rarely required in this book. Most common measures

of distance used are km (10 3 m) and Mm (106 m). On the other hand, common time

measures are ns (109 s), ps (1012 s), and fs (1015 s). Other common units in this

book are Watt (W) for optical power and W/m2 for optical intensity. They can be related

to the fundamental units through energy because optical power represents the rate of

energy flow (1 W = 1 J/s). The energy can be expressed in several other ways using

E= h= kBT= mc2, where h is the Planck constant, kB is the Boltzmann constant, andcis the speed of light. The frequency is expressed in hertz (1 Hz = 1 s 1). Of course,

because of the large frequencies associated with the optical waves, most frequencies in

this book are expressed in GHz or THz.

In the design of optical communication systems the optical power can vary over

several orders of magnitude as the signal travels from the transmitter to the receiver.

Such large variations are handled most conveniently using decibel units, abbreviated

dB, commonly used by engineers in many different fields. Any ratioRcan be converted

into decibels by using the general definition

R(in dB) =10 log10R. (A.1)

The logarithmic nature of the decibel allows a large ratio to be expressed as a much

smaller number. For example, 10 9 and 109 correspond to 90 dB and 90 dB, respec-

tively. SinceR = 1 corresponds to 0 dB, ratios smaller than 1 are negative in the decibel

system. Furthermore, negative ratios cannot be written using decibel units.The most common use of the decibel scale occurs for power ratios. For instance,

the signal-to-noise ratio (SNR) of an optical or electrical signal is given by

SNR=10 log10(PS/PN), (A.2)

wherePSand PNare the signal and noise powers, respectively. The fiber loss can also

be expressed in decibel units by noting that the loss corresponds to a decrease in the

518

Fiber-Optic Communications Systems, Third Edition. Govind P. Agrawal

Copyright 2002 John Wiley & Sons, Inc.

ISBNs: 0-471-21571-6 (Hardback); 0-471-22114-7 (Electronic)

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APPENDIX A. SYSTEM OF UNITS 519

optical power during transmission and thus can be expressed as a power ratio. For

example, if a 1-mW signal reduces to 1 W after transmission over 100 km of fiber,

the 30-dB loss over the entire fiber span translates into 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

(by about 20%) when the signal passes through the connector. The bandwidth of an

optical filter is defined at the 3-dB point, corresponding to 50% reduction in the signal

power. The modulation bandwidth of ight-emitting diodes (LEDs) in Section 3.2 and

of semiconductor lasers in Section 3.5 is also defined at the 3-dB point, at which the

modulated powers drops by 50%.

Since the losses of all components in a fiber-optic communication systems are ex-

pressed in dB, 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 (in dBm) =10 log10power

1 mW, (A.3)

where the reference level of 1 mW is chosen simply because typical values of the

transmitted power are in that range (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 below 1 mW are expressed as negative numbers. For example,

a 10-W power corresponds to 20 dBm. The advantage of decibel units becomes

clear when the power budget of lightwave systems is considered in Chapter 5. Because

of the logarithmic nature of the decibel scale, the power budget can be made simply by

subtracting various losses from the transmitter power expressed in dBm units.