Tài liệu Fibre optic communication systems P2 pdf

Maxwell’s equations are used in Section 2.2 to describe wave propagation geometrical-in optical fibers.. Figure 2.1: Cross section and refractive-index profile for step-index and graded-

Chapter 2

Optical Fibers

The phenomenon of total internal reflection, responsible for guiding of light in

opti-cal fibers, has been known since 1854 [1] Although glass fibers were made in the1920s [2]–[4], their use became practical only in the 1950s, when the use of a claddinglayer led to considerable improvement in their guiding characteristics [5]–[7] Before

1970, optical fibers were used mainly for medical imaging over short distances [8].Their use for communication purposes was considered impractical because of highlosses (∼1000 dB/km) However, the situation changed drastically in 1970 when, fol-

lowing an earlier suggestion [9], the loss of optical fibers was reduced to below 20dB/km [10] Further progress resulted by 1979 in a loss of only 0.2 dB/km near the1.55-µm spectral region [11] The availability of low-loss fibers led to a revolution

in the field of lightwave technology and started the era of fiber-optic communications.Several books devoted entirely to optical fibers cover numerous advances made in theirdesign and understanding [12]–[21] This chapter focuses on the role of optical fibers

as a communication channel in lightwave systems In Section 2.1 we use optics description to explain the guiding mechanism and introduce the related basicconcepts Maxwell’s equations are used in Section 2.2 to describe wave propagation

geometrical-in optical fibers The origgeometrical-in of fiber dispersion is discussed geometrical-in Section 2.3, and Section2.4 considers limitations on the bit rate and the transmission distance imposed by fiberdispersion The loss mechanisms in optical fibers are discussed in Section 2.5, andSection 2.6 is devoted to a discussion of the nonlinear effects The last section coversmanufacturing details and includes a discussion of the design of fiber cables

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)

Figure 2.1: Cross section and refractive-index profile for step-index and graded-index fibers.

properties of optical fibers can be gained by using a ray picture based on geometricaloptics [22] The geometrical-optics description, although approximate, is valid when

the core radius a is much larger than the light wavelengthλ When the two becomecomparable, it is necessary to use the wave-propagation theory of Section 2.2

Consider the geometry of Fig 2.2, where a ray making an angleθiwith the fiber axis

is incident at the core center Because of refraction at the fiber–air interface, the raybends toward the normal The angleθrof the refracted ray is given by [22]

n0sinθi = n1sinθr , (2.1.1)

where n1and n0are the refractive indices of the fiber core and air, respectively The fracted ray hits the core–cladding interface and is refracted again However, refraction

re-is possible only for an angle of incidenceφ such that sinφ< n2/n1 For angles larger

than a critical angleφc, defined by [22]

where n2is the cladding index, the ray experiences total internal reflection at the core–cladding interface Since such reflections occur throughout the fiber length, all rayswithφ>φcremain confined to the fiber core This is the basic mechanism behind lightconfinement in optical fibers

2.1 GEOMETRICAL-OPTICS DESCRIPTION 25

Figure 2.2: Light confinement through total internal reflection in step-index fibers Rays for

whichφ < φ care refracted out of the core

One can use Eqs (2.1.1) and (2.1.2) to find the maximum angle that the incidentray should make with the fiber axis to remain confined inside the core Noting that

θr=π/2 −φcfor such a ray and substituting it in Eq (2.1.1), we obtain

n0sinθi = n1cosφc = (n2− n2)1/2 (2.1.3)

In analogy with lenses, n0sinθi is known as the numerical aperture (NA) of the fiber.

It represents the light-gathering capacity of an optical fiber For n1 n2the NA can beapproximated by

NA= n1(2∆)1/2 , ∆ = (n1− n2)/n1, (2.1.4)where∆ is the fractional index change at the core–cladding interface Clearly, ∆ should

be made as large as possible in order to couple maximum light into the fiber ever, such fibers are not useful for the purpose of optical communications because of a

How-phenomenon known as multipath dispersion or modal dispersion (the concept of fiber

modes is introduced in Section 2.2)

Multipath dispersion can be understood by referring to Fig 2.2, where differentrays travel along paths of different lengths As a result, these rays disperse in time atthe output end of the fiber even if they were coincident at the input end and traveled

at the same speed inside the fiber A short pulse (called an impulse) would broaden

considerably as a result of different path lengths One can estimate the extent of pulsebroadening simply by considering the shortest and longest ray paths The shortest pathoccurs forθi = 0 and is just equal to the fiber length L The longest path occurs forθi given by Eq (2.1.3) and has a length L /sinφc By taking the velocity of propagation

v = c/n1, the time delay is given by

The time delay between the two rays taking the shortest and longest paths is a measure

of broadening experienced by an impulse launched at the fiber input

We can relate∆T to the information-carryingcapacity of the fiber measured through the bit rate B Although a precise relation between B and ∆T depends on many details,

such as the pulse shape, it is clear intuitively that∆T should be less than the allocated bit slot (T B = 1/B) Thus, an order-of-magnitude estimate of the bit rate is obtained from the condition B ∆T < 1 By using Eq (2.1.5) we obtain

BL < n2

n21

c

This condition provides a rough estimate of a fundamental limitation of step-index

fibers As an illustration, consider an unclad glass fiber with n1= 1.5 and n2= 1.The bit rate–distance product of such a fiber is limited to quite small values since

BL < 0.4 (Mb/s)-km Considerable improvement occurs for cladded fibers with a small

index step Most fibers for communication applications are designed with∆ < 0.01.

As an example, BL < 100 (Mb/s)-km for ∆ = 2 × 10 −3 Such fibers can communicate

data at a bit rate of 10 Mb/s over distances up to 10 km and may be suitable for somelocal-area networks

Two remarks are in order concerning the validity of Eq (2.1.6) First, it is obtained

by considering only rays that pass through the fiber axis after each total internal

re-flection Such rays are called meridional rays In general, the fiber also supports skew rays, which travel at angles oblique to the fiber axis Skew rays scatter out of the core at

bends and irregularities and are not expected to contribute significantly to Eq (2.1.6).Second, even the oblique meridional rays suffer higher losses than paraxial meridionalrays because of scattering Equation (2.1.6) provides a conservative estimate since allrays are treated equally The effect of intermodal dispersion can be considerably re-duced by using graded-index fibers, which are discussed in the next subsection It can

be eliminated entirely by using the single-mode fibers discussed in Section 2.2

The refractive index of the core in graded-index fibers is not constant but decreases

gradually from its maximum value n1at the core center to its minimum value n2 atthe core–cladding interface Most graded-index fibers are designed to have a nearlyquadratic decrease and are analyzed by usingα-profile, given by

n(ρ) =



n1[1 − ∆(ρ/a)α]; ρ< a,

n1(1 − ∆) = n2 ; ρ≥ a, (2.1.7)where a is the core radius The parameterαdetermines the index profile A step-indexprofile is approached in the limit of largeα A parabolic-index fiber corresponds to

α= 2

It is easy to understand qualitatively why intermodal or multipath dispersion is duced for graded-index fibers Figure 2.3 shows schematically paths for three differentrays Similar to the case of step-index fibers, the path is longer for more oblique rays.However, the ray velocity changes along the path because of variations in the refractiveindex More specifically, the ray propagating along the fiber axis takes the shortest pathbut travels most slowly as the index is largest along this path Oblique rays have a largepart of their path in a medium of lower refractive index, where they travel faster It istherefore possible for all rays to arrive together at the fiber output by a suitable choice

re-of the refractive-index prre-ofile

2.1 GEOMETRICAL-OPTICS DESCRIPTION 27

Figure 2.3: Ray trajectories in a graded-index fiber.

Geometrical optics can be used to show that a parabolic-index profile leads to

nondispersive pulse propagation within the paraxial approximation The trajectory

of a paraxial ray is obtained by solving [22]

whereρ is the radial distance of the ray from the axis By using Eq (2.1.7) forρ<

a withα = 2, Eq (2.1.8) reduces to an equation of harmonic oscillator and has thegeneral solution

ρ=ρ0cos(pz) + (ρ

0/p)sin(pz), (2.1.9)

where p = (2∆/a2)1/2 andρ0 andρ

0are the position and the direction of the inputray, respectively Equation (2.1.9) shows that all rays recover their initial positions

and directions at distances z = 2mπ/p, where m is an integer (see Fig 2.3) Such a

complete restoration of the input implies that a parabolic-index fiber does not exhibitintermodal dispersion

The conclusion above holds only within the paraxial and the geometrical-optics proximations, both of which must be relaxed for practical fibers Intermodal dispersion

ap-in graded-ap-index fibers has been studied extensively by usap-ing wave-propagation niques [13]–[15] The quantity∆T/L, where ∆T is the maximum multipath delay in

tech-a fiber of length L, is found to vtech-ary considertech-ably withα Figure 2.4 shows this

varia-tion for n1= 1.5 and ∆ = 0.01 The minimum dispersion occurs forα= 2(1 − ∆) and

over distances up to 100 km The BL product of such fibers is improved by nearly

three orders of magnitude over that of step-index fibers Indeed, the first generation

Figure 2.4: Variation of intermodal dispersion∆T/L with the profile parameterα for a index fiber The scale on the right shows the corresponding bit rate–distance product.

graded-of lightwave systems used graded-index fibers Further improvement is possible only

by using single-mode fibers whose core radius is comparable to the light wavelength.Geometrical optics cannot be used for such fibers

Although index fibers are rarely used for long-haul links, the use of

graded-index plastic optical fibers for data-link applications has attracted considerable

atten-tion during the 1990s [24]–[29] Such fibers have a relatively large core, resulting in

a high numerical aperture and high coupling efficiency but they exhibit high losses

(typically exceeding 50 dB/km) The BL product of plastic fibers, however, exceeds

2 (Gb/s)-km because of a graded-index profile [24] As a result, they can be used totransmit data at bit rates>1 Gb/s over short distances of 1 km or less In a 1996

demonstration, a 10-Gb/s signal was transmitted over 0.5 km with a bit-error rate ofless than 10−11[26] Graded-index plastic optical fibers provide an ideal solution fortransferring data among computers and are becoming increasingly important for Eth-ernet applications requiring bit rates in excess of 1 Gb/s

In this section we consider propagation of light in step-index fibers by using Maxwell’sequations for electromagnetic waves These equations are introduced in Section 2.2.1.The concept of fiber modes is discussed in Section 2.2.2, where the fiber is shown tosupport a finite number of guided modes Section 2.2.3 focuses on how a step-indexfiber can be designed to support only a single mode and discusses the properties ofsingle-mode fibers

2.2 WAVE PROPAGATION 29

Like all electromagnetic phenomena, propagation of optical fields in fibers is governed

by Maxwell’s equations For a nonconducting medium without free charges, these

equations take the form [30] (in SI units; see Appendix A)

where E and H are the electric and magnetic field vectors, respectively, and D and B

are the corresponding flux densities The flux densities are related to the field vectors

by the constitutive relations [30]

whereε0is the vacuum permittivity,µ0is the vacuum permeability, and P and M are the induced electric and magnetic polarizations, respectively For optical fibers M= 0because of the nonmagnetic nature of silica glass

Evaluation of the electric polarization P requires a microscopic quantum-mechanical

approach Although such an approach is essential when the optical frequency is near

a medium resonance, a phenomenological relation between P and E can be used far

from medium resonances This is the case for optical fibers in the wavelength region0.5–2µm, a range that covers the low-loss region of optical fibers that is of interest

for fiber-optic communication systems In general, the relation between P and E can

be nonlinear Although the nonlinear effects in optical fibers are of considerable terest [31] and are covered in Section 2.6, they can be ignored in a discussion of fiber

in-modes P is then related to E by the relation

P(r,t) =ε0

 ∞

−∞χ(r,t −t  )E(r,t  )dt  (2.2.7)Linear susceptibilityχis, in general, a second-rank tensor but reduces to a scalar for

an isotropic medium such as silica glass Optical fibers become slightly birefringentbecause of unintentional variations in the core shape or in local strain; such birefrin-gent effects are considered in Section 2.2.3 Equation (2.2.7) assumes a spatially localresponse However, it includes the delayed nature of the temporal response, a featurethat has important implications for optical fiber communications through chromaticdispersion

Equations (2.2.1)–(2.2.7) provide a general formalism for studying wave

propaga-tion in optical fibers In practice, it is convenient to use a single field variable E By

taking the curl of Eq (2.2.1) and using Eqs (2.2.2), (2.2.5), and (2.2.6), we obtain thewave equation

where the speed of light in vacuum is defined as usual by c= (µ0ε0)−1/2 By

introduc-ing the Fourier transform of E(r,t) through the relation

˜

E(r,ω) =

 ∞

−∞E(r,t)exp(iωt )dt, (2.2.9)

as well as a similar relation for P(r,t), and by using Eq (2.2.7), Eq (2.2.8) can be

written in the frequency domain as

∇ × ∇ × ˜E = −ε(r,ω)(ω2/c2) ˜E, (2.2.10)

where the frequency-dependent dielectric constant is defined as

ε(r,ω) = 1 + ˜χ(r,ω), (2.2.11)and ˜χ(r,ω) is the Fourier transform ofχ(r,t) In general,ε(r,ω) is complex Its real

and imaginary parts are related to the refractive index n and the absorption coefficient

where Re and Im stand for the real and imaginary parts, respectively Both n andα

are frequency dependent The frequency dependence of n is referred to as chromatic dispersion or simply as material dispersion In Section 2.3, fiber dispersion is shown

to limit the performance of fiber-optic communication systems in a fundamental way.Two further simplifications can be made before solving Eq (2.2.10) First,εcan

be taken to be real and replaced by n2because of low optical losses in silica fibers

Second, since n(r,ω) is independent of the spatial coordinate r in both the core and the

cladding of a step-index fiber, one can use the identity

∇ × ∇ × ˜E ≡ ∇(∇ · ˜E) − ∇2E˜ = −∇2E˜, (2.2.15)where we used Eq (2.2.3) and the relation ˜D=εE to set˜ ∇ · ˜E = 0 This simplification

is made even for graded-index fibers Equation (2.2.15) then holds approximately aslong as the index changes occur over a length scale much longer than the wavelength

By using Eq (2.2.15) in Eq (2.2.10), we obtain

∇2E˜+ n2(ω)k2

where the free-space wave number k0is defined as

k0=ω/c = 2π/λ, (2.2.17)andλ is the vacuum wavelength of the optical field oscillating at the frequencyω.Equation (2.2.16) is solved next to obtain the optical modes of step-index fibers

2.2 WAVE PROPAGATION 31

The concept of the mode is a general concept in optics occurring also, for example, in

the theory of lasers An optical mode refers to a specific solution of the wave equation

(2.2.16) that satisfies the appropriate boundary conditions and has the property that itsspatial distribution does not change with propagation The fiber modes can be classified

as guided modes, leaky modes, and radiation modes [14] As one might expect, nal transmission in fiber-optic communication systems takes place through the guidedmodes only The following discussion focuses exclusively on the guided modes of astep-index fiber

sig-To take advantage of the cylindrical symmetry, Eq (2.2.16) is written in the drical coordinatesρ,φ, and z as

For simplicity of notation, the tilde over ˜E has been dropped and the frequency

de-pendence of all variables is implicitly understood Equation (2.2.18) is written for the

axial component E zof the electric field vector Similar equations can be written for the

other five components of E and H However, it is not necessary to solve all six

equa-tions since only two components out of six are independent It is customary to choose

E z and H z as the independent components and obtain Eρ, Eφ, Hρ, and Hφ in terms ofthem Equation (2.2.18) is easily solved by using the method of separation of variables

must be periodic inφwith a period of 2π

Equation (2.2.23) is the well-known differential equation satisfied by the Besselfunctions [32] Its general solution in the core and cladding regions can be written as

where A, A  , C, and C  are constants and J m , Y m , K m , and I mare different kinds of Bessel

functions [32] The parameters p and q are defined by

Y m (pρ) has a singularity at ρ= 0, F(0) can remain finite only if A = 0 Similarly

F(ρ) vanishes at infinity only if C = 0 The general solution of Eq (2.2.18) is thus ofthe form

E z=



AJ m (pρ)exp(imφ)exp(iβz) ; ρ≤ a,

CK m (qρ)exp(imφ)exp(iβz); ρ> a. (2.2.27)The same method can be used to obtain H z which also satisfies Eq (2.2.18) Indeed,

the solution is the same but with different constants B and D, that is,

H z=



BJ m (pρ)exp(imφ)exp(iβz) ; ρ≤ a,

DK m (qρ)exp(imφ)exp(iβz); ρ> a. (2.2.28)The other four components Eρ, Eφ, Hρ, and Hφcan be expressed in terms of E z and H z

by using Maxwell’s equations In the core region, we obtain

These equations can be used in the cladding region after replacing p2by−q2

Equations (2.2.27)–(2.2.32) express the electromagnetic field in the core and

clad-ding regions of an optical fiber in terms of four constants A, B, C, and D These

constants are determined by applying the boundary condition that the tangential

com-ponents of E and H be continuous across the core–cladding interface By requiring

the continuity of E z , H z , Eφ, and Hφ atρ = a, we obtain a set of four homogeneous equations satisfied by A, B, C, and D [19] These equations have a nontrivial solution

only if the determinant of the coefficient matrix vanishes After considerable algebraicdetails, this condition leads us to the following eigenvalue equation [19]–[21]:

2.2 WAVE PROPAGATION 33

where a prime indicates differentiation with respect to the argument

For a given set of the parameters k0, a, n1, and n2, the eigenvalue equation (2.2.33)can be solved numerically to determine the propagation constant β In general, it

may have multiple solutions for each integer value of m It is customary to enumerate

these solutions in descending numerical order and denote them byβmn for a given m (n = 1,2, ) Each valueβmncorresponds to one possible mode of propagation of theoptical field whose spatial distribution is obtained from Eqs (2.2.27)–(2.2.32) Sincethe field distribution does not change with propagation except for a phase factor and sat-

isfies all boundary conditions, it is an optical mode of the fiber In general, both E zand

H z are nonzero (except for m= 0), in contrast with the planar waveguides, for which

one of them can be taken to be zero Fiber modes are therefore referred to as hybrid modes and are denoted by HE mnor EHmn , depending on whether H z or E zdominates

In the special case m= 0, HE0nand EH0nare also denoted by TE0nand TM0n,

respec-tively, since they correspond to transverse-electric (E z= 0) and transverse-magnetic

(H z= 0) modes of propagation A different notation LPmn is sometimes used for

weakly guiding fibers [33] for which both E z and H z are nearly zero (LP stands forlinearly polarized modes)

A mode is uniquely determined by its propagation constantβ It is useful to troduce a quantity ¯n=β/k0, called the mode index or effective index and having the

physical significance that each fiber mode propagates with an effective refractive dex ¯n whose value lies in the range n1> ¯n > n2 A mode ceases to be guided when

in-¯

n ≤ n2 This can be understood by noting that the optical field of guided modes decaysexponentially inside the cladding layer since [32]

K m (qρ) = (π/2qρ)1/2exp(−qρ) for qρ (2.2.34)When ¯n ≤ n2, q2≤ 0 from Eq (2.2.26) and the exponential decay does not occur The mode is said to reach cutoff when q becomes zero or when ¯ n = n2 From Eq (2.2.25),

p = k0(n2

1−n2

2)1/2 when q= 0 A parameter that plays an important role in determining

the cutoff condition is defined as

Figure 2.5 shows a plot of b as a function of V for a few low-order fiber modes obtained

by solving the eigenvalue equation (2.2.33) A fiber with a large value of V supports

many modes A rough estimate of the number of modes for such a multimode fiber

is given by V2/2 [23] For example, a typical multimode fiber with a = 25µm and

∆ = 5×10 −3 has V  18 atλ= 1.3µm and would support about 162 modes However,

the number of modes decreases rapidly as V is reduced As seen in Fig 2.5, a fiber with

V = 5 supports seven modes Below a certain value of V all modes except the HE11

mode reach cutoff Such fibers support a single mode and are called single-mode fibers.The properties of single-mode fibers are described next

Figure 2.5: Normalized propagation constant b as a function of normalized frequency V for a

few low-order fiber modes The right scale shows the mode index ¯n (After Ref [34]; c
1981

Academic Press; reprinted with permission.)

Single-mode fibers support only the HE11mode, also known as the fundamental mode

of the fiber The fiber is designed such that all higher-order modes are cut off at the

operating wavelength As seen in Fig 2.5, the V parameter determines the number of

modes supported by a fiber The cutoff condition of various modes is also determined

by V The fundamental mode has no cutoff and is always supported by a fiber.

Single-Mode Condition

The single-mode condition is determined by the value of V at which the TE01and TM01

modes reach cutoff (see Fig 2.5) The eigenvalue equations for these two modes can

be obtained by setting m= 0 in Eq (2.2.33) and are given by

pJ0(pa)K0 (qa) + qJ0 (pa)K0(qa) = 0, (2.2.37)

pn2J0(pa)K 

0(qa) + qn2J0 (pa)K0(qa) = 0. (2.2.38)

A mode reaches cutoff when q = 0 Since pa = V when q = 0, the cutoff condition for both modes is simply given by J0(V) = 0 The smallest value of V for which J0(V) = 0

is 2.405 A fiber designed such that V < 2.405 supports only the fundamental HE11

mode This is the single-mode condition

2.2 WAVE PROPAGATION 35

We can use Eq (2.2.35) to estimate the core radius of single-mode fibers used

in lightwave systems For the operating wavelength range 1.3–1.6µm, the fiber isgenerally designed to become single mode forλ > 1.2µm By takingλ = 1.2µm,

n1= 1.45, and ∆ = 5 × 10 −3 , Eq (2.2.35) shows that V < 2.405 for a core radius

a < 3.2µm The required core radius can be increased to about 4µm by decreasing∆

to 3× 10 −3 Indeed, most telecommunication fibers are designed with a ≈ 4µm.The mode index ¯n at the operating wavelength can be obtained by using Eq (2.2.36),

according to which

¯

n = n2+ b(n1− n2) ≈ n2(1 + b∆) (2.2.39)

and by using Fig 2.5, which provides b as a function of V for the HE11 mode An

analytic approximation for b is [15]

b (V) ≈ (1.1428 − 0.9960/V)2

(2.2.40)

and is accurate to within 0.2% for V in the range 1.5–2.5.

The field distribution of the fundamental mode is obtained by using Eqs (2.2.27)–

(2.2.32) The axial components E z and H zare quite small for 11

mode is approximately linearly polarized for weakly guiding fibers It is also denoted

as LP01, following an alternative terminology in which all fiber modes are assumed to

be linearly polarized [33] One of the transverse components can be taken as zero for

a linearly polarized mode If we set E y = 0, the E xcomponent of the electric field forthe HE11mode is given by [15]

E x = E0



[J0(pρ)/J0(pa)]exp(iβz) ; ρ≤ a, [K0(qρ)/K0(qa)]exp(iβz); ρ> a, (2.2.41)where E0is a constant related to the power carried by the mode The dominant com-

ponent of the corresponding magnetic field is given by H y = n2(ε0/µ0)1/2 E

is broken Degeneracy between the orthogonally polarized fiber modes is removedbecause of these factors, and the fiber acquires birefringence The degree of modalbirefringence is defined by

Figure 2.6: State of polarization in a birefringent fiber over one beat length Input beam is

linearly polarized at 45◦with respect to the slow and fast axes.

L B Figure 2.6 shows schematically such a periodic change in the state of polarization

for a fiber of constant birefringence B The fast axis in this figure corresponds to the axis along which the mode index is smaller The other axis is called the slow axis.

In conventional single-mode fibers, birefringence is not constant along the fiber butchanges randomly, both in magnitude and direction, because of variations in the coreshape (elliptical rather than circular) and the anisotropic stress acting on the core As

a result, light launched into the fiber with linear polarization quickly reaches a state

of arbitrary polarization Moreover, different frequency components of a pulse acquiredifferent polarization states, resulting in pulse broadening This phenomenon is called

polarization-mode dispersion (PMD) and becomes a limiting factor for optical

com-munication systems operating at high bit rates It is possible to make fibers for whichrandom fluctuations in the core shape and size are not the governing factor in determin-

ing the state of polarization Such fibers are called polarization-maintaining fibers A

large amount of birefringence is introduced intentionally in these fibers through designmodifications so that small random birefringence fluctuations do not affect the light

polarization significantly Typically, B m ∼ 10 −4for such fibers.

Spot Size

Since the field distribution given by Eq (2.2.41) is cumbersome to use in practice, it is

often approximated by a Gaussian distribution of the form

E x = Aexp(−ρ2/w2)exp(iβz ), (2.2.44)

where w is the field radius and is referred to as the spot size It is determined by fitting

the exact distribution to the Gaussian function or by following a variational

proce-dure [35] Figure 2.7 shows the dependence of w /a on the V parameter A comparison

2.3 DISPERSION IN SINGLE-MODE FIBERS 37

Figure 2.7: (a) Normalized spot size w /a as a function of the V parameter obtained by fitting the

fundamental fiber mode to a Gaussian distribution; (b) quality of fit for V = 2.4 (After Ref [35];

c

1978 OSA; reprinted with permission.)

of the actual field distribution with the fitted Gaussian is also shown for V = 2.4 The quality of fit is generally quite good for values of V in the neighborhood of 2 The spot size w can be determined from Fig 2.7 It can also be determined from an analytic

approximation accurate to within 1% for 1.2 < V < 2.4 and given by [35]

w/a ≈ 0.65 + 1.619V −3/2 + 2.879V −6 (2.2.45)

The effective core area, defined as Aeff=πw2, is an important parameter for opticalfibers as it determines how tightly light is confined to the core It will be seen later that

the nonlinear effects are stronger in fibers with smaller values of Aeff

The fraction of the power contained in the core can be obtained by using Eq

(2.2.44) and is given by the confinement factor

Equations (2.2.45) and (2.2.46) determine the fraction of the mode power contained

inside the core for a given value of V Although nearly 75% of the mode power resides

in the core for V = 2, this percentage drops down to 20% for V = 1 For this reason most

telecommunication single-mode fibers are designed to operate in the range 2<V < 2.4.

It was seen in Section 2.1 that intermodal dispersion in multimode fibers leads to siderable broadening of short optical pulses (∼ 10 ns/km) In the geometrical-optics

con-description, such broadening was attributed to different paths followed by differentrays In the modal description it is related to the different mode indices (or group ve-locities) associated with different modes The main advantage of single-mode fibers

is that intermodal dispersion is absent simply because the energy of the injected pulse

is transported by a single mode However, pulse broadening does not disappear together The group velocity associated with the fundamental mode is frequency de-pendent because of chromatic dispersion As a result, different spectral components

al-of the pulse travel at slightly different group velocities, a phenomenon referred to as

group-velocity dispersion (GVD), intramodal dispersion, or simply fiber dispersion.

Intramodal dispersion has two contributions, material dispersion and waveguide persion We consider both of them and discuss how GVD limits the performance oflightwave systems employing single-mode fibers

Consider a single-mode fiber of length L A specific spectral component at the

fre-quencyωwould arrive at the output end of the fiber after a time delay T = L/v g, where

v g is the group velocity, defined as [22]

By usingβ = ¯nk0= ¯nω/c in Eq (2.3.1), one can show that v g = c/ ¯n g, where ¯n gis the

group index given by

¯

The frequency dependence of the group velocity leads to pulse broadening simply cause different spectral components of the pulse disperse during propagation and donot arrive simultaneously at the fiber output If∆ω is the spectral width of the pulse,

be-the extent of pulse broadening for a fiber of length L is governed by

In some optical communication systems, the frequency spread∆ω is determined

by the range of wavelengths∆λ emitted by the optical source It is customary to use

∆λ in place of∆ω By usingω= 2πc /λand∆ω= (−2πc /λ2)∆λ, Eq (2.3.3) can bewritten as

2.3 DISPERSION IN SINGLE-MODE FIBERS 39

The effect of dispersion on the bit rate B can be estimated by using the criterion

B ∆T < 1 in a manner similar to that used in Section 2.1 By using ∆T from Eq (2.3.4)

this condition becomes

BL |D|∆λ< 1. (2.3.6)

Equation (2.3.6) provides an order-of-magnitude estimate of the BL product offered

by single-mode fibers The wavelength dependence of D is studied in the next two subsections For standard silica fibers, D is relatively small in the wavelength region

near 1.3µm [D ∼ 1 ps/(km-nm)] For a semiconductor laser, the spectral width ∆λ is

2–4 nm even when the laser operates in several longitudinal modes The BL product

of such lightwave systems can exceed 100 (Gb/s)-km Indeed, 1.3-µm nication systems typically operate at a bit rate of 2 Gb/s with a repeater spacing of

telecommu-40–50 km The BL product of mode fibers can exceed 1 (Tb/s)-km when

single-mode semiconductor lasers (see Section 3.3) are used to reduce∆λ below 1 nm

The dispersion parameter D can vary considerably when the operating wavelength

is shifted from 1.3µm The wavelength dependence of D is governed by the frequency

dependence of the mode index ¯n From Eq (2.3.5), D can be written as

where Eq (2.3.2) was used If we substitute ¯n from Eq (2.2.39) and use Eq (2.2.35),

D can be written as the sum of two terms,

Here n 2g is the group index of the cladding material and the parameters V and b are

given by Eqs (2.2.35) and (2.2.36), respectively In obtaining Eqs (2.3.8)–(2.3.10)the parameter∆ was assumed to be frequency independent A third term known as

differential material dispersion should be added to Eq (2.3.8) when d∆/dω = 0 Its

contribution is, however, negligible in practice

Material dispersion occurs because the refractive index of silica, the material used forfiber fabrication, changes with the optical frequencyω On a fundamental level, theorigin of material dispersion is related to the characteristic resonance frequencies atwhich the material absorbs the electromagnetic radiation Far from the medium reso-

nances, the refractive index n(ω) is well approximated by the Sellmeier equation [36]

Figure 2.8: Variation of refractive index n and group index n gwith wavelength for fused silica.

whereωj is the resonance frequency and B j is the oscillator strength Here n stands for

n1or n2, depending on whether the dispersive properties of the core or the cladding areconsidered The sum in Eq (2.3.11) extends over all material resonances that contribute

in the frequency range of interest In the case of optical fibers, the parameters B j and

ωj are obtained empirically by fitting the measured dispersion curves to Eq (2.3.11)

with M= 3 They depend on the amount of dopants and have been tabulated for several

kinds of fibers [12] For pure silica these parameters are found to be B1= 0.6961663,

B2= 0.4079426, B3= 0.8974794,λ1= 0.0684043µm,λ2= 0.1162414µm, andλ3=

9.896161µm, whereλj = 2πc/ωj with j = 1–3 [36] The group index n g = n +

ω(dn/dω) can be obtained by using these parameter values

Figure 2.8 shows the wavelength dependence of n and n gin the range 0.5–1.6µm

for fused silica Material dispersion D M is related to the slope of n gby the relation

D M = c −1 (dn g /dλ) [Eq (2.3.9)] It turns out that dn g /dλ = 0 atλ= 1.276µm This

wavelength is referred to as the zero-dispersion wavelengthλZD, since D M= 0 atλ =

λZD The dispersion parameter D Mis negative belowλZDand becomes positive abovethat In the wavelength range 1.25–1.66µm it can be approximated by an empiricalrelation

D M ≈ 122(1 −λZD/λ). (2.3.12)

It should be stressed that λZD= 1.276µm only for pure silica It can vary in therange 1.27–1.29µm for optical fibers whose core and cladding are doped to vary therefractive index The zero-dispersion wavelength of optical fibers also depends on the

core radius a and the index step∆ through the waveguide contribution to the totaldispersion

2.3 DISPERSION IN SINGLE-MODE FIBERS 41

Figure 2.9: Variation of b and its derivatives d (Vb)/dV and V [d2(Vb)/dV2] with the V

param-eter (After Ref [33]; c
1971 OSA; reprinted with permission.)

The contribution of waveguide dispersion D W to the dispersion parameter D is given

by Eq (2.3.10) and depends on the V parameter of the fiber Figure 2.9 shows how

d (Vb)/dV and Vd2(Vb)/dV2change with V Since both derivatives are positive, D W

is negative in the entire wavelength range 0–1.6µm On the other hand, D Mis negativefor wavelengths belowλZDand becomes positive above that Figure 2.10 shows D M,

D W , and their sum D = D M + D W, for a typical single-mode fiber The main effect ofwaveguide dispersion is to shiftλZDby an amount 30–40 nm so that the total dispersion

is zero near 1.31µm It also reduces D from its material value D M in the wavelengthrange 1.3–1.6µm that is of interest for optical communication systems Typical values

of D are in the range 15–18 ps/(km-nm) near 1.55µm This wavelength region is ofconsiderable interest for lightwave systems, since, as discussed in Section 2.5, the fiberloss is minimum near 1.55µm High values of D limit the performance of 1.55-µmlightwave systems

Since the waveguide contribution D W depends on fiber parameters such as the core

radius a and the index difference∆, it is possible to design the fiber such that λZD

is shifted into the vicinity of 1.55µm [37], [38] Such fibers are called shifted fibers It is also possible to tailor the waveguide contribution such that the total dispersion D is relatively small over a wide wavelength range extending from

dispersion-1.3 to 1.6 µm [39]–[41] Such fibers are called dispersion-flattened fibers Figure 2.11 shows typical examples of the wavelength dependence of D for standard (conven-

tional), shifted, and flattened fibers The design of

dispersion-Figure 2.10: Total dispersion D and relative contributions of material dispersion D Mand

wave-guide dispersion D Wfor a conventional single-mode fiber The zero-dispersion wavelength shifts

to a higher value because of the waveguide contribution

modified fibers involves the use of multiple cladding layers and a tailoring of the

refractive-index profile [37]–[43] Waveguide dispersion can be used to produce sion-decreasing fibers in which GVD decreases along the fiber length because of ax- ial variations in the core radius In another kind of fibers, known as the dispersion- compensating fibers, GVD is made normal and has a relatively large magnitude Ta-

disper-ble 2.1 lists the dispersion characteristics of several commercially availadisper-ble fibers

It appears from Eq (2.3.6) that the BL product of a single-mode fiber can be increased

indefinitely by operating at the zero-dispersion wavelengthλZD where D= 0 Thedispersive effects, however, do not disappear completely atλ =λZD Optical pulsesstill experience broadening because of higher-order dispersive effects This feature

can be understood by noting that D cannot be made zero at all wavelengths contained

within the pulse spectrum centered atλZD Clearly, the wavelength dependence of D

will play a role in pulse broadening Higher-order dispersive effects are governed by the

dispersion slope S = dD/dλ The parameter S is also called a differential-dispersion

parameter By using Eq (2.3.5) it can be written as

S= (2πc/λ2)2β3+ (4πc/λ3)β2, (2.3.13)whereβ3= dβ2/dω≡ d3β/dω3is the third-order dispersion parameter Atλ =λZD,

β2= 0, and S is proportional toβ3

The numerical value of the dispersion slope S plays an important role in the design

of modern WDM systems Since S > 0 for most fibers, different channels have slightly

2.3 DISPERSION IN SINGLE-MODE FIBERS 43

Figure 2.11: Typical wavelength dependence of the dispersion parameter D for standard,

dispersion-shifted, and dispersion-flattened fibers

different GVD values This feature makes it difficult to compensate dispersion for allchannels simultaneously To solve this problem, new kind of fibers have been devel-

oped for which S is either small (reduced-slope fibers) or negative (reverse-dispersion

fibers) Table 2.1 lists the values of dispersion slopes for several commercially able fibers

avail-It may appear from Eq (2.3.6) that the limiting bit rate of a channel operating at

λ =λZDwill be infinitely large However, this is not the case since S orβ3becomesthe limiting factor in that case We can estimate the limiting bit rate by noting thatfor a source of spectral width∆λ, the effective value of dispersion parameter becomes

D = S∆λ The limiting bit rate–distance product can now be obtained by using Eq

(2.3.6) with this value of D The resulting condition becomes

BL |S|(∆λ)2< 1. (2.3.14)For a multimode semiconductor laser with∆λ = 2 nm and a dispersion-shifted fiber

with S = 0.05 ps/(km-nm2) atλ = 1.55µm, the BL product approaches 5 (Tb/s)-km.

Further improvement is possible by using single-mode semiconductor lasers

A potential source of pulse broadening is related to fiber birefringence As discussed

in Section 2.2.3, small departures from perfect cylindrical symmetry lead to gence because of different mode indices associated with the orthogonally polarizedcomponents of the fundamental fiber mode If the input pulse excites both polariza-tion components, it becomes broader as the two components disperse along the fiber

birefrin-Table 2.1 Characteristics of several commercial fibers

In fibers with constant birefringence (e.g., polarization-maintaining fibers), pulsebroadening can be estimated from the time delay ∆T between the two polarization components during propagation of the pulse For a fiber of length L, ∆T is given by

quite large (∼ 1 ns/km) when the two components are equally excited at the fiber input

but can be reduced to zero by launching light along one of the principal axes

The situation is somewhat different for conventional fibers in which birefringencevaries along the fiber in a random fashion It is intuitively clear that the polarizationstate of light propagating in fibers with randomly varying birefringence will generally

be elliptical and would change randomly along the fiber during propagation In thecase of optical pulses, the polarization state will also be different for different spectralcomponents of the pulse The final polarization state is not of concern for most light-wave systems as photodetectors used inside optical receivers are insensitive to the state

of polarization unless a coherent detection scheme is employed What affects suchsystems is not the random polarization state but pulse broadening induced by random

changes in the birefringence This is referred to as PMD-induced pulse broadening.

The analytical treatment of PMD is quite complex in general because of its tical nature A simple model divides the fiber into a large number of segments Boththe degree of birefringence and the orientation of the principal axes remain constant

statis-in each section but change randomly from section to section In effect, each fiber tion can be treated as a phase plate using a Jones matrix [44] Propagation of each

sec-2.4 DISPERSION-INDUCED LIMITATIONS 45

frequency component associated with an optical pulse through the entire fiber length isthen governed by a composite Jones matrix obtained by multiplying individual Jonesmatrices for each fiber section The composite Jones matrix shows that two principalstates of polarization exist for any fiber such that, when a pulse is polarized along them,the polarization state at fiber output is frequency independent to first order, in spite ofrandom changes in fiber birefringence These states are analogous to the slow and fastaxes associated with polarization-maintaining fibers An optical pulse not polarizedalong these two principal states splits into two parts which travel at different speeds.The differential group delay∆T is largest for the two principal states of polarization.

The principal states of polarization provide a convenient basis for calculating themoments of∆T The PMD-induced pulse broadening is characterized by the root-

mean-square (RMS) value of∆T, obtained after averaging over random birefringence

changes Several approaches have been used to calculate this average The variance

σ2≡ (∆T )2 turns out to be the same in all cases and is given by [46]

For short distances such that z c,σT = (∆β1)z from Eq (2.3.16), as expected for a polarization-maintaining fiber For distances z > 1 km, a good estimate of pulse broadening is obtained using z c For a fiber of length L,σT in this approximationbecomes

σT ≈ (∆β1)2l c L ≡ D p

√

where D p is the PMD parameter Measured values of D pvary from fiber to fiber in the

range D p = 0.01–10 ps/ √km Fibers installed during the 1980s have relatively large

PMD such that D p > 0.1 ps/ √km In contrast, modern fibers are designed to have low

PMD, and typically D p < 0.1 ps/ √km for them Because of the√

L dependence,

PMD-induced pulse broadening is relatively small compared with the GVD effects Indeed,

σT ∼ 1 ps for fiber lengths ∼100 km and can be ignored for pulse widths >10 ps.

However, PMD becomes a limiting factor for lightwave systems designed to operateover long distances at high bit rates [48]–[55] Several schemes have been developedfor compensating the PMD effects (see Section 7.9)

Several other factors need to be considered in practice The derivation of Eq.(2.3.16) assumes that the fiber link has no elements exhibiting polarization-dependentloss or gain The presence of polarization-dependent losses can induce additionalbroadening [50] Also, the effects of second and higher-order PMD become impor-tant at high bit rates (40 Gb/s or more) or for systems in which the first-order effectsare eliminated using a PMD compensator [54]

The discussion of pulse broadening in Section 2.3.1 is based on an intuitive nomenological approach It provides a first-order estimate for pulses whose spectral

phe-width is dominated by the spectrum of the optical source In general, the extent ofpulse broadening depends on the width and the shape of input pulses [56] In thissection we discuss pulse broadening by using the wave equation (2.2.16).

The analysis of fiber modes in Section 2.2.2 showed that each frequency component ofthe optical field propagates in a single-mode fiber as

˜

E(r,ω) = ˆxF(x,y) ˜B(0,ω)exp(iβz ), (2.4.1)

where ˆx is the polarization unit vector, ˜B (0,ω) is the initial amplitude, andβ is the

propagation constant The field distribution F (x,y) of the fundamental fiber mode can

be approximated by the Gaussian distribution given in Eq (2.2.44) In general, F (x,y)

also depends onω, but this dependence can be ignored for pulses whose spectral width

∆ωis much smaller thanω0—a condition satisfied by pulses used in lightwave systems.Hereω0is the frequency at which the pulse spectrum is centered; it is referred to as thecarrier frequency

Different spectral components of an optical pulse propagate inside the fiber ing to the simple relation

accord-˜

B (z,ω) = ˜B(0,ω)exp(iβz ). (2.4.2)The amplitude in the time domain is obtained by taking the inverse Fourier transformand is given by

tude B(0,t).

Pulse broadening results from the frequency dependence ofβ For pulses for which

∆ω ω0, it is useful to expandβ(ω) in a Taylor series around the carrier frequency

ω0and retain terms up to third order In this quasi-monochromatic approximation,

β(ω) = ¯n(ω)ω

c ≈β0+β1(∆ω) +β2

2(∆ω)2+β3

6 (∆ω)3, (2.4.4)where∆ω=ω−ω0andβm = (d mβ/dωm)ω=ω0 From Eq (2.3.1)β1= 1/v g, where

v gis the group velocity The GVD coefficientβ2is related to the dispersion parameter

D by Eq (2.3.5), whereasβ3is related to the dispersion slope S through Eq (2.3.13).

We substitute Eqs (2.4.2) and (2.4.4) in Eq (2.4.3) and introduce a slowly varying amplitude A (z,t) of the pulse envelope as

2.4 DISPERSION-INDUCED LIMITATIONS 47

where ˜A (0,∆ω) ≡ ˜B(0,ω) is the Fourier transform of A(0,t).

By calculating∂A /∂z and noting that∆ω is replaced by i(∂A /∂t) in the time main, Eq (2.4.6) can be written as [31]

As a simple application of Eq (2.4.9), let us consider the propagation of chirped sian pulses inside optical fibers by choosing the initial field as

where A0 is the peak amplitude The parameter T0 represents the half-width at 1/e

intensity point It is related to the full-width at half-maximum (FWHM) of the pulse

by the relation

TFWHM= 2(ln2)1/2 T

0≈ 1.665T0. (2.4.11)

The parameter C governs the frequency chirp imposed on the pulse A pulse is said to

be chirped if its carrier frequency changes with time The frequency change is related

to the phase derivative and is given by

δω(t) = −∂φ

∂t = C

whereφ is the phase of A(0,t) The time-dependent frequency shiftδω is called the

chirp The spectrum of a chirped pulse is broader than that of the unchirped pulse This

can be seen by taking the Fourier transform of Eq (2.4.10) so that

The spectral half-width (at 1/e intensity point) is given by

∆ω0= (1 +C2)1/2 T −1

In the absence of frequency chirp (C= 0), the spectral width satisfies the relation

∆ω0T0= 1 Such a pulse has the narrowest spectrum and is called transform-limited.

The spectral width is enhanced by a factor of(1+C2)1/2in the presence of linear chirp,

con-A (z,t) = A0

Q (z)exp



− (1 + iC)t22T02Q (z)

where T1 is the half-width defined similar to T0 Figure 2.12 shows the broadening

factor T1/T0as a function of the propagation distance z /L D , where L D = T2

0/|β2| is the dispersion length An unchirped pulse (C = 0) broadens as [1 + (z/L D)2]1/2 and

its width increases by a factor of√

2 at z = L D The chirped pulse, on the other hand,may broaden or compress depending on whetherβ2and C have the same or opposite

signs Forβ2C > 0 the chirped Gaussian pulse broadens monotonically at a rate faster

than the unchirped pulse Forβ2C < 0, the pulse width initially decreases and becomes

minimum at a distance

zmin= |C|/(1 +C2) L D (2.4.18)The minimum value depends on the chirp parameter as

T1min= T0/(1 +C2)1/2 (2.4.19)Physically, whenβ2C < 0, the GVD-induced chirp counteracts the initial chirp, and the effective chirp decreases until it vanishes at z = zmin

2.4 DISPERSION-INDUCED LIMITATIONS 49

Figure 2.12: Variation of broadening factor with propagated distance for a chirped Gaussian

input pulse Dashed curve corresponds to the case of an unchirped Gaussian pulse Forβ2< 0

the same curves are obtained if the sign of the chirp parameter C is reversed.

Equation (2.4.17) can be generalized to include higher-order dispersion governed

byβ3in Eq (2.4.15) The integral can still be performed in closed form in terms of

an Airy function [57] However, the pulse no longer remains Gaussian on propagationand develops a tail with an oscillatory structure Such pulses cannot be properly char-acterized by their FWHM A proper measure of the pulse width is the RMS width ofthe pulse defined as

2

where L is the fiber length.

The foregoing discussion assumes that the optical source used to produce the put pulses is nearly monochromatic such that its spectral width satisfies∆ωL ω0

in-(under continuous-wave, or CW, operation), where∆ω0is given by Eq (2.4.14) This

2

where L is the fiber length.

The foregoing discussion assumes that the optical source used to produce the put pulses is nearly monochromatic such that its spectral width