Optical Networks: A Practical Perspective - Part 14 pps

100 PROPAGATION OF SIGNALS IN OPTICAL FIBER called higher-order solitons.. The significance of solitons for optical communication is that they overcome the detrimental effects of chromat

100 PROPAGATION OF SIGNALS IN OPTICAL FIBER

called higher-order solitons A brief quantitative discussion of soliton propagation

in optical fiber appears in Section E.3

The significance of solitons for optical communication is that they overcome the detrimental effects of chromatic dispersion completely Optical amplifiers can be used

at periodic intervals along the fiber so that the attenuation undergone by the pulses

is not significant, and the higher powers and the consequent soliton properties of the pulses are maintained Solitons and optical amplifiers, when used together, offer the promise of very high-bit-rate, repeaterless data transmission over very large distances

By the combined use of solitons and erbium-doped fiber amplifiers (Section 3.4.3), repeaterless data transmission at a bit rate of 80 Gb/s over a distance of 10,000 km has been demonstrated in the laboratory [NSK99]

The use of soliton pulses is key to the realization of the very high bit rates required

in OTDM systems These aspects of solitons will be explored in Chapter 12 The main advantage of soliton systems is their relative immunity to fiber disper- sion, which in turn allows transmission at high speeds of a few tens of gigabits per second On the other hand, in conventional on-off-keyed systems, dispersion can be managed in a much simpler manner by alternating fibers with positive and negative dispersion We encountered this in Section 2.4.9 and we will study this further in Chapter 5 Such systems, when using special pulses called chirped RZ pulses, can also be viewed as soliton systems, albeit of a different kind, and we discuss this in the next section

2.5.1 Dispersion-Managed Solitons

Solitons can also be used in conjunction with WDM, but significant impairments arise when two pulses at different wavelengths overlap in time and position in the fiber Such collisions, which occur frequently in the fiber, add timing jitter to the pulses Although methods to overcome this timing jitter have been devised, commercial deployment of soliton-based systems has not been widespread for two main reasons First, solitons require new disperson-shifted fiber with a small value

of anomalous dispersion (0 < D < 1 ps/nm-km) Thus soliton-based systems cannot

be used on existing fiber plants, whether based on SMF or on the popular NZ-DSF fibers Second, solitons require amplification about every 20 km or so, which is an impracticably small spacing compared to today's WDM systems, which work with amplifier hut spacings of the order of 60-80 kin Larger values of dispersion lead

to higher levels of timing jitter, higher peak pulse powers, and even closer amplifier spacings

High-bit-rate transmission on widely deployed fiber plants, with reasonable am- plifier spacings, has been achieved through a combination of (1) using pulses nar- rower than a bit period but much wider than solitons, and (2) dispersion compensa- tion of the fiber plant at periodic intervals to keep the average dispersion low The pulses used in such systems are called chirped return-to-zero (RZ) pulses and will

be discussed in Section 4.1 When the characteristics of such a dispersion-managed

system are mathematically analyzed, it can be shown that such a system is indeed

"soliton-like" in the sense that a specific chirped Gaussian pulse shape will be trans- mitted through such a system with only periodic changes in shape, that is, with no net broadening due to dispersion, in the absence of loss Such pulses are also called

dispersion-managed (DM) solitons We will discuss the performance of systems em-

ploying such pulses in Chapter 5 By the use of chirped RZ pulses, repeaterless data transmission in a 25-channel WDM system at a bit rate of 40 Gb/s per channel, over

a distance of 1500 kin, has been demonstrated in the laboratory [SKN01]

Summary

The understanding of light propagation in optical fiber is key to the appreciation of not only the significant advantages of using optical fiber as a propagation medium but also of the problems that we must tackle in designing high-bit-rate WDM systems We started by understanding how light propagates in multimode fibers using a simple ray theory approach This introduced the concept of pulse broadening due to multimode dispersion and motivated the use of single-mode fibers After describing the elements

of light propagation in single-mode fibers, we studied the limitations imposed on optical communication systems due to the pulse-broadening effects of chromatic dispersion

Although dispersion is the most important phenomenon limiting the performance

of systems at bit rates of 2.5 Gb/s and below, nonlinear effects become important at higher bit rates The main nonlinear effects that impair high-speed WDM transmis- sion are self-phase modulation and four-wave mixing We studied the origin of these,

as well as other nonlinear effects, and briefly outlined the constraints on optical com- munication systems imposed by them We will return to the system limitations of both dispersion and nonlinearities when we discuss the design of optical transmission systems in Chapter 5

We also studied the new types of fibers that have been introduced to mitigate the effects of dispersion and nonlinearities Finally, we discussed solitons, which are special pulses designed to play off dispersion and nonlinearities against each other

to achieve high-bit-rate, ultra-long-haul transmission

102 PROPAGATION OF SIGNALS IN OPTICAL FIBER

Further Reading

The propagation of light in optical fiber is treated in several books at varying levels

of detail One of the earliest books on this subject is by Marcuse [Mar74] The book by Green [Gre93] starts with the fundamentals of both geometrical optics and electromagnetics and describes the propagation of light using both the ray and wave theory approaches The concepts of polarization and birefringence are also treated in some detail However, the effects of dispersion and nonlinearities are described only qualitatively The book on fiber optic communication by Agrawal [Agr97] focuses

on the wave theory approach and treats the evolution of chirped Gaussian pulses

in optical fiber and the pulse-broadening effects of chromatic dispersion in detail Chromatic dispersion and intermodal dispersion are also treated at length in the books edited by Miller and Kaminow [MK88] and Lin [Lin89] We recommend the book by Ramo, Whinnery, and van Duzer [RWv93] for an in-depth study of electromagnetic theory leading up to the description of light propagation in fiber The books by Jeunhomme Ueu90] and Neumann [Neu88] are devoted to the propagation

of light in single-mode fibers Jeunhomme treats fiber modes in detail and has a more mathematical treatment We recommend Neumann's book for its physical explanations of the phenomena involved The paper by Gloge [Glo71] on fiber modes is a classic

In all these books, nonlinear effects are only briefly mentioned The book by Agrawal [Agr95] is devoted to nonlinear fiber optics and contains a very detailed description of light propagation in optical fiber, including all the nonlinear effects

we have discussed Soliton propagation is also discussed One of the earliest papers

on four-wave mixing is [HJKM78] Note that cgs units are used in this paper The units used in the description of nonlinear effects are a source of confusion The relationships between the various units and terminologies used in the description of nonlinear effects are described in the book by Butcher and Cotter [BC90] This book also contains a particularly clear exposition of the fundamentals of nonlinear effects The system impact of dispersion and nonlinearities and their interplay is discussed

in detail in [KK97, Chapter 8]

Information on the new types of fibers that have been introduced to combat dispersion and nonlinearities can be found on the Web pages of the manufacturers: Corning and Lucent Much of the data on the new fiber types for this chapter was gathered from these Web pages The ITU has standardized three fiber types ITU-T recommendation (standard) G.652 specifies the characteristics of standard single-mode fiber, G.653 that of DSF, and G.655 that of NZ-DSE

A nice treatment of the basics of solitons appears in [KBW96] Issues in the design

of WDM soliton communication systems are discussed at length in [KK97, Chapter

12] A summary of soliton field trials appears in [And00] DM solitons are discussed

in [Nak00]

2.1

2.2

2.3

2.4

2.5

2.6

2.7

Problems

Note that some of these problems require an understanding of the material in the appendices referred to in this chapter

Derive (2.2)

A step-index multimode glass fiber has a core diameter of 50 /,m and cladding refractive index of 1.45 If it is to have a limiting intermodal dispersion aT of 10 ns/km, find its acceptance angle Also calculate the maximum bit rate for transmission over a distance of 20 km

Derive equation (2.11) for the evolution of the magnetic field vector I2I

Derive an expression for the cutoff wavelength ~-cutoff of a step-index fiber with core radius a, core refractive index n l, and cladding refractive index n2 Calculate the cutoff wavelength of a fiber with core radius a = 4 # m and A = 0.003

Consider a step-index fiber with a core radius of 4 / , m and a cladding refractive index of 1.45

(a) For what range of values of the core refractive index will the fiber be single moded for all wavelengths in the 1.2-1.6 # m range?

(b) What is the value of the core refractive index for which the V parameter is 2.0 at k = 1.55/,m? What is the propagation constant of the single mode supported by the fiber for this value of the core refractive index?

Assume that, in the manufacture of single-mode fiber, the tolerance in the core radius

a is +5% and the tolerance in the normalized refractive index difference A is +10%, from their respective nominal values If the nominal value of A is specified to be 0.005, what is the largest nominal value that you can specify for a while ensuring that the resulting fiber will be single moded for )~ > 1.2/,m even in the presence of the worst-case (but within the specified tolerances) deviations of a and A from their nominal values? Assume that the refractive index of the core is 1.5

In a reference frame moving with the pulse, the basic propagation equation that governs pulse evolution inside a dispersive fiber is

8A i 82A

104 PROPAGATION OF SIGNALS IN OPTICAL FIBER

2.8

2.9

2.10

2.11

2.12

2.13

where A(z, t) is the pulse envelope If A(0, t) = A o e x p ( - t 2 / 2 T ~ ) for some constants

A0 and To, solve this propagation equation to find an expression for A(z, t)

Note: You may use the following result without proof:

_ ~ e x p ( - ( x m)2/2~) dx

for all complex m and ~ provided ~t(~) > 0

Hint: Consider the Fourier transform A (z, co) of A (z, t)

Starting from (E.8), derive the expression (2.13) for the width Tz of a chirped Gaus-

sian pulse with initial width To after it has propagated a distance z

Show that an unchirped Gaussian pulse launched at z - 0 remains Gaussian for all

z but acquires a distance-dependent chirp factor

sgn (fl2 ) z / L o K(z) =

1 + (Z/LD) e"

Show that the rms width of a Gaussian pulse whose half-width at the 1/e-intensity point is To is given by To/~/~

Consider a chirped Gaussian pulse for which the product xfi2 is negative that is launched at z = 0 Let x = 5

(a) For what value of z (as a multiple of LD) does the launched pulse attain its minimum width?

(b) For what value of z is the width of the pulse equal to that of an unchirped pulse, for the same value of z? (Assume the chirped and unchirped pulses have the same initial pulse width.)

Show that in the case of four-wave mixing, the nonlinear polarization is given by terms (2.28)through (2.32)

You want to design a soliton communication system at 1.55 ~m, at which wavelength the fiber has f12 = - 2 psZ/km and y = 1/W-km The peak power of the pulses you can generate is limited to 50 mW If you must use fundamental solitons and the bit period must be at least 10 times the full width at half-maximum (T~VHM) of the soliton pulses, what is the largest bit rate you can use? (This problem requires familiarity with the material in Appendix E.)

References

[Agr95] G.P Agrawal Nonlinear Fiber Optics, 2nd edition Academic Press, San Diego,

CA, 1995

[Agr97] G.P Agrawal Fiber-Optic Communication Systems John Wiley, New York, 1997

[And00] P.A Andrekson High speed soliton transmission on installed fibers In OFC 2000

Technical Digest, pages TuP2-1/229-231, 2000

[BC90] P.N Butcher and D Cotter The Elements of Nonlinear Optics, volume 9 of

Cambridge Studies in Modern Optics Cambridge University Press, Cambridge,

1990

[Buc95] J.A Buck Fundamentals of Optical Fibers John Wiley, New York, 1995

[BW99] M Born and E Wolf Principles of Optics: Electromagnetic Theory of Propagation,

Diffraction and Interference of Light Cambridge University Press, 1999

[Glo71] D Gloge Weakly guiding fibers Applied Optics, 10:2252-2258, 1971

[Gre93] P.E Green Fiber-Optic Networks Prentice Hall, Englewood Cliffs, NJ, 1993 [HJKM78] K.O Hill, D C Johnson, B S Kawasaki, and R I MacDonald CW three-wave

mixing in single-mode optical fibers Journal of Applied Physics,

49(10):5098-5106, Oct 1978

[Jeu90] L.B Jeunhomme Single-Mode Fiber Optics Marcel Dekker, New York, 1990

[Kan99] J Kani et al Interwavelength-band nonlinear interactions and their suppression in

multiwavelength-band WDM transmission systems IEEE/OSA Journal on

Lightwave Technology, 17:2249-2260, 1999

[KBW96] L.G Kazovsky, S Benedetto, and A E Willner Optical Fiber Communication

Systems Artech House, Boston, 1996

[KK97] I.P Kaminow and T L Koch, editors Optical Fiber Telecommunications IIIA

Academic Press, San Diego, CA, 1997

[Lin89] C Lin, editor Optoelectronic Technology and Lightwave Communications

Systems Van Nostrand Reinhold, New York, 1989

[Liu98] Y Liu et al Advanced fiber designs for high capacity DWDM systems In

Proceedings of National Fiber Optic Engineers Conference, 1998

[Mar74] D Marcuse Theory of Dielectric Optical Waveguides Academic Press, New York,

1974

[MK88] S.D Miller and I P Kaminow, editors Optical Fiber Telecommunications II

Academic Press, San Diego, CA, 1988

[Nak00] M Nakazawa et al Ultrahigh-speed long-distance TDM and WDM soliton

transmission technologies IEEE Journal of Selected Topics in Quantum

Electronics, 6:363-396, 2000

[Neu88] E.-G Neumann Single-Mode Fibers Springer-Verlag, Berlin, 1988

106 PROPAGATION OF SIGNALS IN OPTICAL FIBER

[NSK99] M Nakazawa, K Suzuki, and H Kubota Single-channel 80 Gbit/s soliton

transmission over 10000 km using in-line synchronous modulation Electronics

[RN76] H.-D Rudolph and E.-G Neumann Approximations for the eigenvalues of the

fundamental mode of a step-index glass fiber waveguide Nachrichtentechnische

[RWv93] S Ramo, J R Whinnery, and T van Duzer Fields and Waves in Communication

[SKN01] K Suzuki, H Kubota, and M Nakazawa 1 Tb/s (40 Gb/s x 25 channel) DWDM

quasi-DM soliton transmission over 1,500 km using dispersion-managed

single-mode fiber and conventional C-band EDFAs In OFC 2001 Technical Digest,

pages TUN7/1-3, 2001

N T H I S CHAPTER, we will discuss the physical principles behind the operation

I of the most important components of optical communication systems For each component, we will give a simple descriptive treatment followed by a more detailed mathematical treatment

The components used in modern optical networks include couplers, lasers, pho- todetectors, optical amplifiers, optical switches, and filters and multiplexers Cou- plers are simple components used to combine or split optical signals After describing couplers, we will cover filters and multiplexers, which are used to multiplex and de- multiplex signals at different wavelengths in WDM systems We then describe various types of optical amplifiers, which are key elements used to overcome fiber and other component losses and, in many cases, can be used to amplify signals at multiple wavelengths Understanding filters and optical amplifiers is essential to understand- ing the operation of lasers, which comes next Semiconductor lasers are the main transmitters used in optical communication systems Then we discuss photodetec- tors, which convert the optical signal back into the electrical domain This is followed

by optical switches, which play an important role as optical networks become more agile Finally, we cover wavelength converters, which are used to convert signals from one wavelength to another, at the edges of the optical network, as well as inside the network

107

108 COMPONENTS

Figure 3.1 A directional coupler The coupler is typically built by fusing two fibers together It can also be built using waveguides in integrated optics

A directional coupler is used to combine and split signals in an optical network

A 2 x 2 coupler consists of two input ports and two output ports, as is shown in Figure 3.1 The most commonly used couplers are made by fusing two fibers together

in the middle these are called fused fiber couplers Couplers can also be fabricated using waveguides in integrated optics A 2 x 2 coupler, shown in Figure 3.1, takes a fraction o~ of the power from input 1 and places it on output 1 and the remaining fraction 1 - ot on output 2 Likewise, a fraction 1 - ot of the power from input 2 is distributed to output 1 and the remaining power to output 2 We call ot the coupling ratio

The coupler can be designed to be either wavelength selective or wave- length independent (sometimes called wavelength flat) over a usefully wide range

In a wavelength-independent device, ot is independent of the wavelength; in a wavelength-selective device, ot depends on the wavelength

A coupler is a versatile device and has many applications in an optical network The simplest application is to combine or split signals in the network For example,

a coupler can be used to distribute an input signal equally among two output ports

if the coupling length, l in Figure 3.1, is adjusted such that half the power from each input appears at each output Such a coupler is called a 3 dB coupler An n x n star coupler is a natural generalization of the 3 dB 2 x 2 coupler It is an n-input, n-output device with the property that the power from each input is divided equally among all the outputs An n x n star coupler can be constructed by suitably interconnecting

a number of 3 dB couplers, as shown in Figure 3.2 A star coupler is useful when multiple signals need to be combined and broadcast to many outputs However, other constructions of an n x n coupler in integrated optics are also possible (see, for example, [Dra89])

Figure 3.2 A star coupler with eight inputs and eight outputs made by combining 3 dB couplers The power from each input is split equally among all the outputs

Couplers are also used to tap off a small portion of the power from a light stream for monitoring purposes or other reasons Such couplers are also called taps and are designed with values of ~ close to 1, typically 0.90-0.95

Couplers are the building blocks for several other optical devices We will explore the use of directional couplers in modulators and switches in Sections 3.5.4 and 3.7 Couplers are also the principal components used to construct Mach-Zehnder interferometers, which can be used as optical filters, multiplexers/demultiplexers, or

as building blocks for optical modulators, switches, and wavelength converters We will study these devices in Section 3.3.7

So far, we have looked at wavelength-independent couplers A coupler can be made wavelength selective, meaning that its coupling coefficient will then depend

on the wavelength of the signal Such couplers are widely used to combine signals

at 1310 nm and 1550 nm into a single fiber without loss In this case, the 1310 nm signal on input 1 is passed through to output 1, whereas the 1550 nm signal on input

2 is passed through also to output 1 The same coupler can also be used to separate the two signals coming in on a common fiber Wavelength-dependent couplers are