4.2.1 Single-Mode Transmission Fibers

A single-mode fiber used for transmission is characterized, from the point of view of electromagnetism, by the condition of being a weakly guiding waveguide, that is, the refraction index variation along the fiber section radius is very small (generally less than 1%).

The fiber section of the simpler transmission fiber, the so-called step index fiber, is reported in Figure 4.1: it is constituted by a central cylindrical core with a refraction index n_c and a coaxial cladding with a refraction index n_e. Examples of indexes values used in telecommunications are n_c = 1.4600 and n_e = 1.4592, so that Δn/n¯=2(nc−ne)/(nc+ne)≈0.55%.

The field propagation in the fiber can be studied by solving the Maxwell equations in the structure [1,2].

Exploiting the weakly guiding approximation [3] it is possible to solve the propagation problem by individuating propagation modes that constitute a complete set for the description of the guided field. In long transmission systems, the optical power that is not guided by the fiber is irradiated in the first fiber segment; thus, a description of the guided field is sufficient.

The generic propagation mode in a perfectly cylindrical step index fiber in the absence of field sources can be described with the following expression of the electrical field:

(4.1)

where

ρ represents the radial coordinate in the fiber section

β(ω) is the dispersion function, that is obviously mode dependent z is the axial coordinate

The individual propagation mode can be identified through one of the three mode classes, represented by the footer ζ in Equation 4.1 and by two progressive numbers: l and m.

The mode classes are generally indicated as follows:

• Transverse Electrical (TE) modes, having the electrical field vector “transverse,” thus orthogonal to the fiber section radius

• Transverse magnetic (TM) modes, having the magnetic field vector “transverse,” thus orthogonal to the fiber section radius

• HE modes, where no field is “transverse”

The propagation description through the modes represented in Equation 4.1 is accurate, but difficult to manage for our purpose.

In order to work with a simpler mode set, the solution of Maxwell equations in an orthogonal coordinate system can be attempted [3].

By exploiting again the weakly guiding property of the fiber, two sets of modes can be derived that due to the adopted coordinate system are linearly polarized (LP).

These modes (called LP) can be divided, looking at the electrical field, into x-polarized and y-polarized modes, where x and y are the orthogonal coordinates in the plane of the fiber section and due to the fiber cylindrical symmetry, every x-polarized mode is degenerate with a y-polarized mode.

Naturally, as far as the adopted approximations are valid, LP modes can be represented as a sum of TE, TM, and HE modes.

An LP mode, having, for example, linear polarization along the x axis, can be represented with the expression

(4.2)

where

x→is the axis unitary vector

l,m are the mode identification numbers

LP modes are much easier to use when the fiber propagation has to be considered for telecommunication purposes.

The description of fiber propagation via LP modes is less accurate than via TE, TM, and HE modes with the effect that LP modes with the same mode index are not exactly orthogonal in real fibers, but a coupling exists.

However, in studying fiber propagation, many other approximations are done (e.g., linear fiber axis, perfect circular section, abrupt index change between the core, and the cladding) that creates much more modes coupling both among LP and among TE, HE, and TM modes, so that the LP approximation can be adopted without any problem.

Different, nondegenerate modes propagate along the fiber with different group velocities due to the dependence of the function β(ω) appearing in Equation 4.2 from the mode numbers.

The dependence of β(ω) on the propagation mode (called mode dispersion [2]) generates signal distortion when transmitting a plurality of modes and greatly limits the possible span of a transmission system.

Even if a single LP mode is excited at the fiber input, the presence of material imperfections excites all the modes that can propagate through the fiber, thus causing multimodal propagation.

In order to solve this problem, a single-mode fiber has to be designed, that is, a fiber having only one mode that propagates at the minimum attenuation wavelength.

As in any kind of waveguide, modes propagating in an optical fiber are characterized by the so-called cutoff frequency, that is, the minimum frequency at which the mode can propagate. At lower frequencies, the mode propagation is impossible since the field rapidly vanishes advancing along the fiber, irradiating the greater part of the optical power out of the waveguide.

The mode cutoff is generally expressed in terms of normalized frequency V, defined as

(4.3)

where

a is the core diameter

c is the light speed in vacuum

λ is the field wavelength

ω is the field angular frequency

It is assumed that the diffraction index n(ρ) is a decreasing function of the radial coordinate ρ so that n(0) is the index at the core center and n(∞) is the index of the cladding far from the core.

At each fiber mode, a cutoff normalized frequency V_c can be associated, but to LP₀₁ modes whose cutoff is zero. That means that these two polarization degenerate modes propagate at every frequency.

The cutoff frequency of the lower cutoff mode depends on the index profile; for a step index fiber with the indexes given in Figure 4.1, the mode with the lower cutoff is the LP₁₁ and V_c = 2.405, corresponding to a wavelength of 1262 nm for n_c = 1.4600, n_e = 1.4592, and the core radius equal to 10 μm. This means that, at 1300 and 1500 nm (the wavelengths at which the fiber loss is minimum), the only modes that can propagate are the LP₀₁.

4.2.2 Fiber Losses

We have seen that the primary reason for the success of optical fiber as communication medium is the extremely low loss. Thus, studying loss mechanisms is very important.

When the light is injected into the fiber from an external light source and propagates along the fiber for a long span, the output light power is reduced with respect to the source emitted power by two kinds of losses: coupling losses and propagation losses.

Coupling losses are due to the fact that the field shape of the incoming light on the input fiber facet does not coincide with the fundamental mode profile, so that both radiation modes and below-threshold propagation modes are excited. After a very short fiber length only the fundamental mode remains, and all the power coupled with other modes is radiated out of the fiber core.

During propagation along the fiber, the fundamental mode power also decreases, due to propagation losses, to produce at the fiber output the attenuated output power.

4.2.3 linear Propagation in an Optical Fiber

In a real fiber, attenuation is not the only phenomenon causing the difference between the field coupled at the fiber input and the field emerging from the output.

In this section, we will assume that the power of the coupled field is sufficiently low that all propagation nonlinear effects are negligible. In this hypothesis, propagation along the fiber is purely linear and the fiber itself can be modeled as a linear distributed system.

In this condition, if the field coupled to the fiber at the input facet is a combination of the two main fiber modes, then

(4.7)

The field at the output will be obtained by the multiplication of the input field in the frequency domain with the frequency transfer function of the fiber.

In real fibers factors breaking fundamental modes degeneracy in terms of the transverse field shape are present, but are not important for telecommunication systems. Thus, we can assume that the transverse shape of the field remains equal to A(ρ) in every fiber section and for every field polarization.

We have also seen that the field attenuation is a slowly varying function of the field frequency. For practical modulation, even at 100 Gbit/s, the attenuation variation in the signal bandwidth is completely negligible; thus, in writing the fiber transfer function we can assume a constant attenuation.

Fiber nonideality influences greatly the field phase evolution. As a matter of fact, in a real fiber it is not possible to assume that the propagation constants of the two LP fundamental modes are equal and that these modes do not couple during propagation.

On the contrary, the propagation constant depends on the field polarization and if a mix of the two fundamental modes is injected into the fiber, they will couple during propagation generating in every fiber section a different elliptical polarization.

Last but not least, microscopic imperfections cause coupling between the fundamental modes so that even if only one fundamental mode is launched into the fiber, that is the input field is perfectly LP, at the output, the field has a generic elliptic polarization revealing the excitation of both the fundamental modes.

Moreover, the output polarization will vary slowly in time, due to time variation of the coupling coefficient driven by phenomena like temperature and material stress variation.

These empirical observations can be summarized in the following expression of the field at a generic fiber section

(4.8)

where the matrix

(4.9)

called Jones matrix, has to be unitary to fulfill energy conservation and represents both birefringence (i.e., the dependence of the propagation constant from the polarization) and mode coupling (through the off diagonal coefficients). It is to be noted that, since α represents the power attenuation, α/2 appears in the field expression.

Equation 4.8 summarizes all the propagation effects characterizing the linear propagation regime in case of limited signal bandwidth.

4.2.4 Fiber Chromatic Dispersion

Chromatic dispersion in single-mode fibers is the phenomenon causing the broadening of a light pulse propagating along the fiber that is independent from the pulse polarization.

In Equation 4.8, chromatic dispersion is caused by the dependence on the angular frequency of the common mode propagation constant β(ω).

In particular, remembering the small signal bandwidth hypothesis and indicating with ω₀ the central angular frequency of the signal spectrum, we can write

(4.10)

The terms of the Taylor series of β(ω) have different physical meanings.

The term β′(ω ₀)z that is obtained substituting Equation 4.10 in Equation 4.8 represents a group delay, as can be easily shown from the property of the Fourier integral. This is the average delay a pulse experiments traversing the fiber span (as a matter of fact it has also the dimension of a time interval). To evidence this meaning, the time delay τ can be written as being v_g = 1/β′(ω₀) the group velocity.

(4.11)

In a first approximation, injecting in the fiber a pulse with bandwidth Δω, the pulse broadening Δt can be evaluated by the group velocity dispersion (GVD) between the frequencies at the spectrum borders. Using Equation 4.10 up to the second order, the following expression is obtained.

(4.12)

Alternatively, considering wavelengths instead that angular frequencies

(4.13)

The parameter D = 2πc/λ²β″(λ₀) is called the fiber dispersion parameter (measured in ps/nm/km).

There are fundamentally two phenomena contributing to the value of the parameter D: material dispersion and waveguide dispersion, so that D can be written as the sum of a material dispersion term D_M, a waveguide dispersion term D_W, and a mixed term D_MW, that in typical fibers is much smaller with respect to the other two.

Accurate expressions of the three terms are obtained as a result of the propagation modal analysis; here we will introduce approximated expressions for the material and guide terms in order to analyze their behavior with respect to the main fiber parameters.

The material term is related to the wavelength dependence of the silica refraction index; thus, it is the same term causing dispersion in bulk silica. It is positive, independently from the structure and design of the fiber.

In order to derive an approximated expression of this term, the propagating wave front can be approximated as a plane. The differential delay between two frequency components of the field can be written as a function of the frequency dependent group velocity as

(4.14)

The expression of the propagation constant in bulk material in the case of a plane wave is simply β(ω) = ω n(ω)/c. Substituting this expression in (4.14) and considering that in the near infrared region that is of interest for telecommunication, the angular frequency is on the order of 10¹⁵, it results as

(4.15)

From (4.15) and the dispersion parameter definition, passing from ω to λ, the following expression is obtained.

(4.16)

The material dispersion, as it results from Equation 4.16 and the measured behavior of silica refraction index, is zero at a wavelength of 1300 nm, positive at longer wavelength, and negative at shorter once. In any case, D_M increases while increasing λ in all the near infrared region.

The waveguide term is due to the fact that the guided mode travels part in the core and part in the cladding. Considering the case of a step index fiber, if we analyze the distribution of optical power at different frequencies we found that it depends critically on the ratio a/λ due to the different effect of the core–cladding interface diffraction on waves with different wavelength.

This effect can also be represented by studying the dependence of the effective refraction index n_e (i.e., the ratio between the group velocity of a monochromatic guided wave and the velocity of light in vacuum) on the wavelength.

In particular, at short wavelengths, the effects of diffraction are smaller and the light is confined well within the core. In this condition, n_e is very close to the refractive index of the core.

As the wavelength increases, the effects of diffraction become more important and the light spreads slightly into the cladding. The effective refractive index decreases toward the refractive index of the cladding.

Finally, at long wavelengths, the effects of diffraction dominate and the light in the fiber spreads well into the cladding, n_e being very close to the refractive index of the cladding.

In order to obtain an approximate expression of the waveguide dispersion parameter, we will assume a zero material dispersion, a condition that can be realized only if the refraction index does not depend on wavelength. In this condition, it is useful to substitute the angular frequency with the normalized frequency V defined in Equation 4.3 that is directly related to the guiding properties of the fiber.

Thus, it is possible to write

(4.17)

Substituting (4.17) in the definition of the dispersion parameter it results, after expressing D_W as a function of λ

(4.18)

The shape of the propagation constant in an ideal dielectric waveguide whose refraction index distribution does not depend on the wavelength can be varied widely by shaping the index profile being in particular either positive or negative. For example, in a step index fiber, there are two wavelength intervals: one in which the two dispersion terms are cumulative, the other in which they have different signs and one tends to compensate the other.

This effect is shown in Figure 4.3 in the case of a step index fiber. In the figure, both the overall dispersion and the individual contributions are shown, evidencing the zero dispersion point (around 1.33 μm) where the two dispersion contributions are opposite.

The zero dispersion wavelength divides the wavelength axis in two intervals, that for historical reasons take the name of normal dispersion wavelengths (β″ > 0) and anomalous dispersion wavelengths (β″ < 0).

The dispersion parameter takes into account the greater contribution to fiber dispersion, but there are several cases in which the other terms of Equation 4.10 cannot be neglected. The most important of these cases is propagation around the zero dispersion frequency.

In this case, dispersion is almost all due to the third-order term, which at high transmission speed cannot be neglected.

As in the case of second-order dispersion, also in the third-order case, the pulse broadening can be approximately expressed as a linear function of the square signal bandwidth, thus defining a third-order dispersion coefficient.

In particular, the following equation is obtained:

(4.19)

The dispersion-related parameters of a standard step index fiber in the second and the third transmission window are reported in Table 4.1.

The fact that waveguide dispersion can be opposite in sign with respect to material dispersion can be used to shape the fiber dispersion curve. To control waveguide dispersion, the refraction index behavior along the fiber diameter is suitably designed.

Depending on the application, different types of dispersion-managed optical fibers can be used. Dispersion-shifted (DS) fibers are designed to move the zero dispersion point from the second to the third transmission window, dispersion flattened fibers are engineered to have a very low (but nonzero) dispersion in a wide wavelength interval, possibly including both second and third transmission window, nonzero dispersion fibers are generally realized to move the zero dispersion wavelength in between the two transmission windows.

TABLE 4.1 Typical Values of the Parameters of a Step Index Single-Mode Transmission Fiber

All values are at λ = 1.55 μm.

Different index profiles corresponding to different categories of fibers for long-haul transmission manufactured by the main fiber vendors are represented in Figure 4.4.

In all the cases represented in the figure, the study of the structure-guided modes demonstrate that two degenerate fundamental modes of the LP type can propagate in the fiber and that the spatial profile of this mode is Gaussian with a good approximation, exactly as in a step index fiber. The variance of the field mode shape is different for different fibers causing a difference in the field confinement. As we will see, this will have an impact on the efficiency of nonlinear effects during field propagation.

The dispersion curve of a set of “managed dispersion” fibers is reported in Figure 4.5, and some important parameters of commercial fibers are summarized in Table 4.2.

The various performances are compared with the performances of a standard step index fiber.

The first dispersion-managed fibers to be produced were DS fibers, where the dispersion curve is shifted toward higher wavelengths so that the zero dispersion wavelength is moved around 1.5 μm.

These fibers were designed and deployed in the period in which long-haul single-­ channel systems were designed in order to decrease chromatic dispersion and increase the transmission reach.

With the introduction of dense wavelength division multiplexing (DWDM) systems, this class of fibers was practically abandoned, due to the fact that having the zero dispersion wavelength in the third transmission window causes the maximization of nonlinear channel crosstalk due to the channel phase matching.

In order to allow high channel count DWDM systems to be deployed and contemporary to decrease as much as possible the value of dispersion, nonzero dispersion fibers were designed.

Table 4.2 Comparison between the Properties of a Set of Fibers with Different Index Profiles

All values in the table that depend on wavelength are reported at λ = 1.55 μm.

In this kind of fibers, the dispersion curve is shifted to move the zero dispersion wavelength either in between the second and the third window or beyond the third window around 1570 nm.

In the first case, the fibers are called NZ+ and in the second NZ−.

Using these fibers allows avoiding zero dispersion in the transmission bandwidth while reducing the average dispersion.

Moreover, especially in the case of NZ−, the fiber design can be performed to achieve a greater effective mode ratio to decrease nonlinear effects. This point will be considered in more detail describing nonlinear propagation.

4.2.5 Polarization Mode Dispersion

Polarization mode dispersion (PMD) is the phenomenon causing strong coupling between the fundamental LP modes so that the propagating field polarization evolves during propagation [8,9].

The polarization not only changes from fiber section to fiber section, but also changes in time and wavelength. This is due to the fact that mode coupling is generated by fiber imperfection both at the material level and at a fiber form level. While temperature and mechanical conditions like micro-stresses evolve, mode coupling changes.

The fact that the field polarization changes with wavelength is particularly dangerous for modulated signals when the field polarization has to be recovered at the receiver. As a matter of fact, this becomes impossible if different frequencies of the signal spectrum have different polarizations.

Moreover, since the diagonal terms of the Jones matrix are not equal to one, different polarizations travel through the fiber at slightly different group velocities, thus causing the so-called polarization dispersion. This means that a pulse of finite duration sent into the fiber experiments a widening during propagation even in the absence of chromatic dispersion due to the PMD.

The base for the analysis of PMD impact is the definition of the principal states of polarization (PSPs), which for a fiber with small random birefringence somehow take the place of the birefringence axes in a high birefringence medium.

Given a fiber piece, the PSPs of that fiber in the considered deploying conditions are defined as the input states of polarization that generate at the fiber output a polarization state that does not depend on ω at the first order [8].

The PSPs can be derived directly from their definition and from the general expression Equation 4.8 of the field during linear propagation in the fiber. As a matter of fact, the definition of the PSPs implies that, at first order, their ω derivative has to be equal to zero.

Let us consider the slowly varying envelope of the field in Equation 4.8. To simplify the notation, it is useful to eliminate on both sides of Equation 4.8 the transversal field shape and to write

(4.20)

(4.21)

where p→ is the polarization vector.

If the input field is a PSP, by definition the first-order derivative of the output field has to be zero. Deriving Equation 4.8 and equaling the result to zero, the following equation is obtained:

(4.22)

where

(4.23)

Taking into account that the Jones matrix is unitary, it is easy to recognize that Equation 4.22 always has two solutions, which are the fiber PSPs and whose coordinate expression is easy to evaluate solving Equation 4.22.

This procedure demonstrates that, for a given fiber piece, there exist two orthogonal input polarization states, called PSPs that characterize the polarization behavior of the fiber. In particular, if a field is launched into the fiber along one of the PSPs, the output polarization does not change, at least at first order, changing ω. This means that a finite duration pulse injected in the fiber along a PSPs does not experiment widening due to PMD.

Since a higher-order dependence of the PSP output polarization on the angular frequency exists, if the bandwidth of the input signal rises above a certain value, the PSP output polarization is no more ω independent and the concept of PSPs becomes quite useless.

The PSPs bandwidth is generally on the order of 100 GHz [10]; thus, it is quite larger than the bandwidth of normal modulated signals, but for modulation at 100 Gbit/s. In this case, if pure on–off modulation is used, PSPs probably cannot be used and some other means have to be adopted to describe polarization evolution.

Even if the PSP approximation can be used, the expression of the PSPs depends on the elements of the Jones matrix and thus slowly changes with time due to the change of the random birefringence [11].

Last but not least, macroscopically identical pieces of fiber can have completely different PSPs, due to the different distribution of micro-defects causing coupling between the fundamental modes.

Starting from the PSPs definition, the PMD characteristics can be derived.

Let us call p+→ and p−→ the unit vectors corresponding to the PSPs output polarizations of the considered fiber piece. A generic field injected into the fiber can be decomposed along the input PSPs so that, for the properties of the PSPs, the same decomposition holds for the output field with respect to the output PSPs.

This decomposition can be written in general as follows:

(4.24)

Starting from Equation 4.24, it is possible to evaluate the polarization dispersion contribution with a simple derivation exploiting the PSPs properties [12].

Squaring Equation 4.24 and remembering that the PSPs are orthogonal states, the following equation is obtained for the pulse envelope:

(4.25)

where the envelopes g₊(t) and g₋(t) are both replicas of the input pulse envelope that arrive at the output with different delays due to their travel along the two different PSPs.

Taking into account this consideration, we can imagine the propagation of two identical pulses launched through the fiber along the input linear polarization of the LP states.

The output envelope in the two cases can be written as in Equation 4.25 with the suitable coefficients obtaining

(4.26)

where τ_– and τ₊ are the delays of the PSPs.

The differential delay due to PMD is by definition the time interval separating the instants in which the peaks of two identical pulses launched in the fiber along the LP fundamental states arrive at the fiber output.

Since Equation 4.25 simply gives the expressions of the output pulses envelopes, this can be the starting point for the PMD differential delay calculation.

For a finite duration pulse, it is possible to define the moment generating function and the general theory of the Fourier transform tells us that the abscissa of the pulse maximum (that in our model is the required arrival time) is given by –iG′(0) if G(ω) is the moment generating function of its envelope, while the pulse variance is iG″(0).

Thus, the differential delay can be written as

(4.27)

From Equation 4.27, it is evident that the delay is maximum for θ = 0, that is, if the input pulses are launched along the PSPs, while it is minimum if the states along which the two input pulses are launched are inclined of π/4 with respect to the PSPs. In this case, both the input pulses are equally distributed among the PSPs and no differential delay exists.

The pulse spreading due to PMD can be evaluated starting from the definition

(4.28)

Exploiting the properties of the moment generating function and starting from (4.26), the following expression is found for the pulse spreading:

(4.29)

Equation 4.29 confirms the fact that, if θ = 0, that is, the input pulse is launched along a PSP, the pulse spreading is zero, while it is maximum if the pulse energy is equally divided between the PSPs at the fiber input.

Equation 4.29, even if it gives a very insightful expression of the pulse spreading, does not permit immediately to evaluate it starting from fiber parameters.

However, the evaluation of the average value of the pulse spread during propagation along a PMD-affected fiber can be carried out starting from Equation 4.29 with a further step.

We can imagine that the fiber link can be decomposed in several fiber pieces, each of which is completely independent from the others from a PSP point of view.

Since the output PSPs of the jth fiber piece are completely independent from the input PSPs of the (j + 1)th, the final pulse broadening is the sum of a great number of independent contributions. We can consider the average pulse spreading given by Equation 4.29 also as the time delay standard deviation, due to the fact that in the presence of PMD the time delay experienced by a pulse during the propagation along a fiber piece is a random variable.

Thus, since the variance of a sum of independent random variables is the sum of the variances, the square of the pulse broadenings experienced by the pulse while traveling along each fiber piece can be added to derive the square of the final pulse spreading, obtaining

(4.30)

where j is the index running over the different cascaded pieces of fiber. Since the injection angle is completely independent from the delay difference, the average of the product can be calculated as the product of the average.

Assuming the injection angle is uniformly distributed, in the limit of infinite pieces of negligible length, the variance of the delay difference between the PSPs is proportional to N, where N is the number of fiber pieces.

Thus, if we have to evaluate the pulse broadening for a certain fiber length L, we have to multiply and divide for the square of the length of the single piece l_j of fiber and make the limit N → ∞ and l_j → 0 in such a way that N l_j → L. By calculating the limit the following relation is obtained

(4.31)

where the D_PMD is the PMD parameter and it is evident that, due to its statistical nature the pulse broadening due to PMD increases with the square root of the fiber length.

The same result can be derived from a statistical approach on the Jones matrix (or on the equivalent representation in the Stokes space).

A statistical polarization dispersion theory has been carried out both in the Jones and in the Stokes space [13,14] and a very elegant relationship has been developed between the two approaches based on the use of the σ spin matrixes [15].

The rigorous statistical theory arrives at the conclusion that, considering a very large ensemble of macroscopically identical fibers, the PSPs of every fiber piece are different. A statistical average of the delay can be done considering the link as a cascade of a large number of identical pieces and studying polarization evolution in this case.

The propagation along one piece of fiber generates a dispersion given by Equation 4.29, while the overall propagation, can be viewed as a random walk [14,15].

Due to the properties of the random walk, the standard deviation of the propagation time depends on the square root of the number of jumps. Since the jumps are equally probable in every fiber piece, depending on the probability on the fiber length L, the average pulse broadening σ_Δτ, evaluated as time delay standard deviation, is given exactly by Equation 4.31.

4.2.6 Nonlinear Propagation in Optical Fibers

Even if the optical power launched in a fiber transmission system is not huge, the core radius is so small that it is not difficult to generate so high fields that they change the equilibrium of the dielectric composing the fiber [6,7].

In this case, the constants describing the material characteristics in the wave equation depend on the field, and nonlinear effects appear.

Nonlinear propagation effects can be divided in two categories: scattering effect and the Kerr effect.

Scattering effect includes the phenomena in which a propagating field photon is scattered by some material alteration caused by the field itself.

Kerr effect includes all the phenomena due to the dependence of the real part of dielectric susceptibility on the field.

Both the classes of effects are important in telecommunication systems, because they are either exploited to improve the system performance or avoided to avoid a too high system penalty.

The starting point for the propagation problem is always the wave equation with the boundary conditions corresponding to the fiber structure.

In particular, indicating with P→(E→) the polarization vector of the glass composing the fiber, the wave equation writes [16]

(4.32)

where P→(E→) is the polarization vector.

In a nonlinear propagation regime, the polarization is not proportional to the field, but it depends on the field in a more complex way. The consequence of this more complex dependence is the presence of wavelengths in the output signal that were not present in the input signal.

If the more powerful of these nonlinear products is much smaller than the linear component (i.e., the signal power remaining along propagation in the input bandwidth) the propagation regime is told to be weakly nonlinear. In the weak nonlinear propagation regime a few approximations can be used that simplify the problem.

In normal propagation conditions, the impact of nonlinear propagation can be classified into three main effects:

1. Brillouin effect: It is caused by photons scattering from coherent phonons. This is a backscattering effect.

2. Raman effect: It is caused by photon scattering from glass molecules in an excited state.

3. Kerr effect: It is caused by incoherent interaction between molecules’ external orbitals and the traveling field.

In order to set up a mathematical model of nonlinear propagation, an expression has to be adopted for the nonlinear polarization.

Assuming to be in the weak nonlinear propagation regime, series approximation of the nonlinear susceptibility can be adopted.

In general, the relationship between the susceptibility and the field is not a local and instantaneous relationship, due to the presence of distributed interactions as in the case of the Brillouin scattering. In similar cases, the susceptibility depends on the field not only in a single point and in a single instant, but incorporates a dependence extended to a finite area and to a finite time interval.

Formally, this property can be coded in mathematical terms considering the polarization vector as a convolution between the nonlinear susceptibility and the field [17], but the resulting integral–differential wave equation is really difficult to manage.

For this reason, we will assume that all fiber nonlinearities are characterized by a scalar nonlinear susceptibility. We know that in this way we will have a unified theory of Kerr and Raman effect, but we will be unable to describe Brillouin effect. For this last case, an ad hoc model will be developed.

In the local and instantaneous approximation, the Taylor series of the polarization reads

(4.33)

but for χ⁽¹⁾ that represents the linear susceptibility, the generic term [χ^(k)] of the power expansion is an Euclidean tensor with rank k + 1 so that the generic term results to be a vector, whose components are forms of degree k in the electric field components.

In particular, since the glass composing the fiber is an isotropic medium at molecular level, it can be demonstrated that [χ⁽²⁾] = 0; thus, the nonlinear propagation is mainly due to the third term of the expansion (4.33) [7].

The tensor nature of χ⁽³⁾ is responsible for nonlinear polarization evolution, that is, a quite weak phenomenon, but important in some kinds of long-haul systems [17] (compare Chapter 9, Ultra Long Haul 100 Gbit/s systems).

If nonlinear polarization evolution can be neglected, the third term can be simplified introducing a scalar third-order susceptibility, so that (4.33) rewrites

(4.34)

χ⁽³⁾ is a complex term, whose real part χR(3) causes the Kerr effect and whose imaginary part χI(3) causes the Raman effect. A typical value of the real part of χ⁽³⁾ is reported in Table 4.1.

The presence of a nonlinear part of the susceptibility implies the dependence of the refraction index on the propagating field intensity ℐ. In particular, from the definition of the diffraction index, the following equation is derived:

(4.35)

From what we have seen up to now, it is evident that nonlinear effect is a distributed cause of modification of the input pulses and it is more and more effective increasing the transmission fiber length.

However, by increasing the transmission length, attenuation tends to decrease the nonlinear contribution to the index, which is proportional to the local field intensity. On the other hand, increasing the dimension of the mode, the peak intensity decreases and the nonlinear effect should be less effective.

It is thus intuitive that some characteristic length has to exist, depending on fiber attenuation, mode area, and launched power, such that when the transmitted signal goes beyond this characteristic length, it is so attenuated that the nonlinear effect is no more effective.

Let us define an effective length L_e such that the overall optical power present along the link is equal to the injected power multiplied by the characteristic length. The definition can be converted in a simple equation for L_e that reads P₀ Le=∫0LP0e−αz and whose solution is

(4.36)

where L is the fiber link length. If the fiber length is much greater than 1/α, L_e results to be of the order of 1/α, that is about 20 km in standard transmission fibers.

Equation 4.36 is a pure empirical definition, but provides a useful idea of what is the fiber length causing relevant nonlinear effects.

The definition does not include the mode area. In order to also include this element, It can be suitably modified starting from the definition of the mode effective area that, in the hypothesis of perfectly symmetric mode, reads:

(4.37)

At this point, Equation 4.36 could be modified as follows:

(4.38)

where A_c is the geometrical core area. In the case of a step index fiber with a core radius of 5.2 μm, A_c/A_e = 0.95, so that the results of Equations 4.36 and 4.38 practically do not differ. In other cases however, there is quite a difference. For example, in the case of a DS fiber with the same geometrical core radius, A_c/A_e = 0.6 clearly indicating the greater efficiency of nonlinear effects in this kind of fiber due to the greater field confinement.

However, it is the length defined by Equation 4.36 that appears in several equations regarding nonlinear effects and generally it is the quantity called characteristic length of fiber nonlinear effects.

4.2.7 Kerr Effect

Under the hypotheses set in the previous section, to analyze the impact of Kerr effect, we can neglect in a first approximation the polarization evolution and imagine dealing with ideal LP modes.

Several effects induced by the Kerr nonlinearity are much more efficient if the interacting signals polarizations are aligned so that a scalar model results to enhance the nonlinear effect with respect to the real vector case. A discussion on the applicability limits of the scalar model for the analysis of transmission systems is reported in [18].

If transmission penalties have to be evaluated, this is a reasonable approach, providing a worst case system performance, if the nonlinear effect is exploited to design some fiber-based device, the correctness of the scalar assumption has to be explicitly verified due to the risk of overestimating the device performances.

In these conditions, it is possible to derive a propagation equation in the presence of Kerr effect by applying the slowly varying approximation and considering the Kerr effect response time about zero so to deal with an instantaneous effect (being the Kerr characteristic time of the order of 100 fs this, approximation is always valid in telecommunications). This equation, called nonlinear Schrödinger equation, writes [6,19]

(4.39)

where τ is the time in the pulse reference frame, thus τ = t − z/v_g = t − β′z, and the nonlinear coefficient given by γ = n₂ω_k/(Cπr_m²) with r_m representing the modal radius.

Also within this approximation, the evolution in the presence of a wideband signal is complex and, in order to understand from a qualitative point of view the signal distortions introduced by Kerr effect, signal propagation is generally studied assuming a few simple expressions for the fiber input signal.

These particular signals are conceived to put in evidence particular aspects of the Kerr effect, aspects that are called, for historical reasons “effects.” Thus, we speak about self phase modulation (SPM) effect, cross phase modulation (XPM) effect, four wave mixing (FWM) effect, and so on. However, all these phenomena are Kerr effect–induced evolutions of particular input signals.

If the input signal is complex, the evolution can be qualitatively understood by considering the input as a superposition of a certain number of simple signals and using the fact that the nonlinear effect is small to try to model the final result as the superimposition of the effects on each simple signal (compare in Chapter 6 the rule of penalties addition and Appendix C).

This approach is quite useful to understand complex phenomena also, but has clear limits either when the power is very high and the superposition of the individual effects cannot be assumed or when the bandwidth is very large, so that parameters dependence on wavelength cannot be neglected. These conditions are not so far from the operational conditions of long-haul high-capacity WDM systems; thus, the validity of all the approximations has always to be verified when in front of a real system.

To access more quantitative results, several methods have been devised to numerically solve Equation 4.39 and their study can start from bibliographies [18,20].

4.2.8 Raman Scattering

Raman scattering is the photon scattering caused by local optical phonons of the glass and in Equation 4.39 it is taken into account via the imaginary part of the third-order nonlinear susceptibility [6,7].

When a monochromatic light beam propagates in an optical fiber, spontaneous Raman scattering occurs. It transfers some of the photons to new frequencies. The scattered photons may lose energy (Stokes shift) or gain energy (anti-Stokes shift). If photons at other frequencies are already present then the probability of scattering to those frequencies is enhanced. This process is known as stimulated Raman scattering and rely on stimulated emission due to the excited optical phonons.

In the Stokes scattering process, the only process happening in guided propagation due to the overall momentum conservation, the energy of incident photon is reduced to lower level and energy is transferred to the molecules of silica in the form of kinetic energy, inducing stretching, bending or rocking of the molecular bonds. The Raman shift ω_R = ω_P − ω_S (when P indicates the injected wavelength and S the Stokes wavelength produced by the nonlinearity) is dictated by the vibrational energy levels of silica.

The Stokes Raman process is also known as the forward Raman process since the Stokes wave propagates in the same direction of the input wave and the energy conservation for the process is

(4.47)

where ε_g and ε_f are ground state and final state energies, respectively, and in silica fiber ω_P − ω _S ≈ 12 THz (that in the third windows are about 100 nm).

Since Raman scattering is not only a potential cause of channel crosstalk in WDM systems, but also the principle of operation of Raman amplifiers that are used in several transmission system architectures, we will delay the discussion of a detailed physical model of the Raman amplification to the next section.

Here it is interesting to note that the gain bandwidth of the Raman effect, whose experimental measure is reported in Figure 4.8 [24], is extremely wide and that, as all nonlinear effects generate new waves, the Stokes wave has an exponential growth in the limit of nondepleted pump, when the effect is thus very low.

In correspondence to the slope change of the exponential, a Raman threshold can be defined whose value is about 600 mW in conventional standard single-mode fibers (SSMFs).

4.2.9 Brillouin Scattering

Brillouin scattering is a nonlinear process that can occur in optical fibers at large field intensity. The large intensity produces compression in the material of the core of the fiber through the process known as electrostriction. This phenomenon produces density fluctuations in the fiber medium, which in turn modulates the refractive index of the medium and results in an electrostrictive nonlinearity [25].

The modulated refractive index behaves as an index grating, which is pump-induced and the scattered light is frequency shifted (Brillouin shift) by the frequency of the sound wave.

Quantum mechanically, the Brillouin shift originates from the coherent scattering between photon and local acoustic phonon, and the Brillouin shift is due to the Doppler displacement that is a consequence of the phonon momentum [26].

Brillouin scattering may be spontaneous or stimulated. In spontaneous Brillouin scattering, there is annihilation of a pump photon, which results in creation of Stokes photon and an acoustic phonon simultaneously. The conservation laws for energy and momentum must be followed in such scattering processes. For energy conservation, the Stokes shift ω_B must be equal to (ω_P − ω_S), where ω_P and ω_S are frequencies of pump and Stokes waves.

The momentum conservation requires

(4.48)

where

k→p and k→S are the pump photon and the phonon wave vectors j is an integer

k→lattice takes into account the pseudo-particle nature of the phonon and accounts for other Stokes orders after the fundamental

In transmission systems, Brillouin effect has to be as small as possible; thus, only the first-order Stokes wave is to be taken into account, so that Equation 4.48 can be simplified as k→A=(k→p−k→S).

If v_A is the acoustic velocity, then the dispersion relation Equation 4.48 can be written as

(4.49)

where ϑ is the angle between the pump and Stokes momentum vectors and the modules of the pump and the Stokes wave vectors are assumed almost equal due to Equation 4.48 and to the fact that the light speed is much greater than the sound speed. From the aforementioned expression, it is clear that the frequency shift depends on angle ϑ. For ϑ = 0, the shift is zero, that is, there is no frequency shift in forward direction that means no Brillouin scattering. The only other possible direction in guided propagation is ϑ = π that represents backward direction and gives the maximum shift. The backward Stokes shift can be evaluated from Equation 4.48 obtaining

(4.50)

where n is the effective mode index, that is, the ratio between the light speed in vacuum and the mode group velocity.

The dependence of gain on frequency can be described evaluating the Brillouin gain spectrum. The finite life time T_B of acoustic phonons is the cause of the frequency dependence of the gain and of the small spectral width of the gain spectrum.

Solving the coupled wave equations of the input optical wave, the Stokes wave, and the acoustic wave, the following expression can be obtained for the Brillouin gain g_B

(4.51)

The peak value of the Brillouin gain occurs at ω = ω_B. The gain g_B0 depends on many parameters like concentration of dopants in the fiber, inhomogeneous distribution of dopants, and the electrostrictive coefficient. Figure 4.9 describes the Brillouin gain spectra at pump wavelength 1.525 mm for a silica-core fiber, a depressed-cladding fiber, and a DS fiber.

The inhomogeneous distribution of germania within the core of the depressed-cladding fiber used for the experiment is responsible for the double peak in the Brillouin gain spectrum of the figure, while difference in Brillouin shift among various fibers is due to the different percentage of germania in the core.

Exploiting the expression of the Brillouin gain, it is possible to develop a simple model of the evolution of the pump and Stokes wave intensities during propagation.

As a matter of fact, separating the equation for the pump and the equation for the Stokes from the general wave equation and applying the rotating wave and the slowly varying envelope approximations, the following coupled equations are obtained for the two optical waves intensities:

(4.52)

In the condition of very small nonlinear effect, the Stokes intensity can be neglected in the equation for the pump and the system (4.52) can be solved exactly.

Since the optical power is proportional to the optical intensity via the simple relationship ℐA_e = P, where the proportionality coefficient is the effective area defined in (4.37), and since the Stokes is counter-propagating with respect to the pump, the solution for the Stokes power exiting from the fiber input facet is

(4.53)

From Equation 4.53, it is possible to evaluate the Brillouin threshold as the pump power at which the Stokes power at the input facet is equal to the pump power at the output facet.

Assuming P_S(L) equal to the average spontaneous emission noise at the wavelength of 1.55 μm, thus representing the case of the absence of a coherent Stokes signal at the far end fiber facet and describing the pump evolution in the nondepleted pump approximation, the Brillouin threshold can be expressed as [27]

(4.54)

The value of polarization factor δ lies between 1 and 2 depending on relative polarization of pump and Stokes waves. Typically, A_e ≈ 50 μm², L _e ≈ 20 km, and g_B = 4 × 10⁻¹¹ m/W for a fiber system at 1550 nm.

With these values and taking δ = 1, P_th ≈ 1.3 mW. The threshold power becomes just double if polarization factor is taken equal to 2. When threshold is reached, the effect of stimulated Brillouin scattering (SBS) on the signal power is described by the Figure 4.10, where the power exiting from the far and the near end of 13 km of DS fiber is plotted versus the input power at the near end of the fiber. Up to the threshold power, the transmitted power increases linearly. When scattered power attains the value equal to threshold power, the transmitted power becomes almost constant and independent of input signal power.

4.2.10 ITU-T Fiber Standards

The International Standardization Organization (the ITU-T) has standardized several types of fibers to allow system designers to rely on well-known set of fiber parameters.

Table 4.3 Summary of the Main ITU-T and IEC Standards Regarding Transmission Fibers

Besides ITU-T, the most important organization that has done an extensive optical fiber standardization is International Electrotechnical Commission (IEC), whose influence is mainly present in North America.

A summary of the main recommendations of ITU-T and IEC for standardization is reported in Table 4.3.

The most common fiber in the world, deployed almost by all the carriers is the nondispersion-shifted fiber (NDSF) or G.652 fiber, also called SSMF.

This is a step index fiber optimized for low dispersion in the third windows. Even if SSMFs are the oldest among all the installed fibers, their characteristics are evolving on the push of the increasing speed of transmitted channels.

In particular, manufacturing processes are continuously improved to decrease PMD and to control the attenuation in the extreme parts of the spectrum.

Due to the relatively high value of the dispersion parameter, SSMFs allow a very good control of FWM and XPM interference in WDM systems.

Moreover, a particular class of G.652 fiber exists where the HO content in the fiber glass is so reduced that the HO attenuation peak almost disappears widening the available spectrum. This property is useful especially in coarse wavelength division multiplexing (CWDM) systems, where two of the standard channels are superimposed to the HO attenuation peak and cannot be used with SSMFs.

A few key specifications from ITU-T G.652 are reported in Table 4.4.

Second from the point of view of deployed length are the G.655 fibers. They are NZ+ and NZ− fibers, which trade a bit of attenuation with a substantially low dispersion in all the third window.

In reality, G.655 fibers have never demonstrated their capability to support 10 Gbit/s DWDM transmission systems with performance superior to that of G.652 fibers.

The continuous increase of the value of D at 1550 nm of the most recent G.655 fibers has demonstrated that the optimal trade-off between D, A_e, and α has not been found yet.

DS fibers, standardized in ITU-T recommendation G.653, have on the contrary almost disappeared from the market today.

Avoiding dispersion in single-channel transmission at 1550 nm resulted in exacerbation of both XPM and FWM. Moreover, DS fibers have a higher PMD with respect to SSMF because of the core–cladding index difference two times higher than that of G.652 fibers.

Table 4.4 A Few Key Specifications from ITU-T G.652

A few key requirements from G.653 and from G.655 are reported in Tables 4.5 and 4.6, respectively.

4.2.11 Polarization Maintaining and Other Special Telecom Fibers

A polarization maintaining (PM) fiber is an optical fiber in which if the input field is injected along a predefined linear polarization (the polarization of the only LP fundamental mode of the fiber), this polarization is maintained during propagation [28].

Several different designs of PM fiber are used. Most work by inducing stress in the core via a noncircular cladding cross section, or via rods of another material included within the cladding. Several different shapes of rods are used, and the resulting fiber design has the standardized name such as “PANDA” and “Bow-tie,” as it is shown in Figure 4.11. There are small differences between different types of PM fiber design, more in the manufacturing process than in the final product performance.

Table 4.5 A Few Key Requirements from G.653

Table 4.6 A Few Key Requirements from G.655

It is to underline that a PM fiber does not polarize light as a polarizer does. Rather, PM fiber maintains the input linear polarization if it is launched into the fiber along the correct direction. If the polarization of the input light is not aligned with the stress direction in the fiber, the output will vary between linear and circular polarization (and generally will be elliptically polarized).

Polarization maintaining fibers are generally characterized by a minimum loss greater than 1 dB/km (typical for a PANDA fiber is 1.5 dB/km) that, besides the high cost, makes them unsuitable for transmission. However, there are several applications requiring small pieces of a PM fiber, such as optical components pigtailing, fiber optic sensing, interferometry, and quantum key distribution. They are also commonly used in telecommunications for the connection between a source laser and a modulator, since the modulator requires polarized light at the input.

Single-polarization fibers [28,29] are special optical fibers that can transmit light with a certain linear polarization direction, whereas light with the other polarization directions is either not guided or experiences strong optical losses. Such fibers should not be confused with PM fibers, which guide light with any polarization state, but can preserve a linear polarization state when the polarization direction is properly aligned with the birefringence axis.

In many cases, single-polarization guidance occurs in only a limited wavelength range. Outside that range, both polarization directions or no light at all may be guided. Also, some fibers exhibit a limited extinction ratio.

Different principles of operation can be utilized for single-polarization fibers. A common approach is the use an elliptical core, which introduces strong birefringence and also a polarization dependence of a cutoff wavelength, so that only light with one polarization direction is guided, whereas the fiber is a leaky waveguide for the other polarization.

After the fibers having particular polarization properties, a great importance in the design of DWDM systems is the dispersion compensating fibers (DCFs) [30,31].

This kind of fiber is designed to have a strong dispersion in the third transmission window so that a relatively short fiber piece can compensate an entire SSMF span from 70 to 100 km.

This result is achieved by suitably shaping the refraction index in the core and in the cladding of the fiber.

A typical index shape of a DCF is the segmented core profile represented in Figure 4.4. Two figures of merit can be given for DCFs:

1. Efficiency (ɛ) that is dispersion divided by the dispersion slope for a standard SSMF divided by the same ratio but for the DCF.

2. Figure of merit (FM) that is minimum dispersion in the C band divided by maximum attenuation in C band.

Efficiencies greater than 60% and FM as high as 150 can be obtained from commercial products that are completely sufficient for the field applications in DWDM systems. For example, a DCF length having 11.5 dB of loss can compensate 1640 ns dispersion in the C band (1545–1606 nm).

Photonic crystal fibers (PCFs) [32] are another type of special fibers that could find wide application in future telecommunication system mainly in active and passive fiber devices. They are an attempt to incorporate the bandgap ideas of photonic crystals into the fiber structure by stacking periodically a regular array of channels and drawing into fiber form (see Figure 4.11).

Two classes of PCFs exist [33,34], according to their mechanism for field confinement. Those with a solid core, or a core with a higher average index than the microstructured cladding, operates on the same indexguiding principle as conventional optical fiber.

However, they can have a much higher effective refractive index contrast between core and cladding, and therefore, they can have much stronger confinement for applications in nonlinear optical devices or, polarization-maintaining fibers. Alternatively, a “photonic bandgap” fiber can be created, in which the light is confined by a photonic bandgap created by the microstructured cladding—such a bandgap, properly designed, can confine light in a lower-index core and even a hollow (air) core [35].

In general, regular structured fibers such as PCFs, have a cross section (normally uniform along the fiber length) microstructured from one, two or more materials. Most commonly, the microstructure is periodically arranged over much of the cross section, usually as a “cladding” surrounding a core where light is confined. For example, the fibers demonstrated in [36] consisted of a hexagonal lattice of air holes in a silica fiber, with a solid or hollow core at the center where light is guided. Other arrangements include concentric rings of two or more materials.

Recently, it has been recognized that the periodic structure is actually not the best solution for many applications. This idea has been applied to reduce bend losses, and for achieving structured optical fibers with propagation losses below that of the SSMF.

PCFs with different designs of the hole pattern can have very remarkable properties, strongly depending on the design details:

• Single-mode guidance over very wide wavelength regions (endlessly single-mode fiber) is obtained for small ratios of hole size and hole spacing [37].

• Extremely small or extremely large mode areas are possible. These lead to very strong or very weak optical nonlinearities. PCFs can be made with a low sensitivity to bend losses even for large mode areas [38].

• Certain hole arrangements result in a photonic bandgap, where guidance is possible even in a hollow core, as a higher refractive index in the inner part is no longer required.

• Asymmetric hole patterns can lead to extremely strong birefringence for polarization-maintaining fibers [39]. This property can also be combined with large mode areas.

• Strongly polarization-dependent attenuation (polarizing fibers) [40] can be obtained in different ways. For example, there can be a polarization-dependent fundamental mode cutoff, so that the fiber guides only light with one polarization in a certain wavelength range.

• Similarly, it is possible to suppress Raman scattering [41] by strongly attenuating longer-wavelength light.

• Multicore designs are possible, for example, with a regular pattern of core structures in a single fiber, where there may or may not be some coupling between the cores.

• In the field of telecommunications however, probably the most important application of the holey fibers is in optical amplifiers.

Laser-active PCFs for fiber lasers and amplifiers can be fabricated by using, for example, a rare-earth-doped rod as the central element of the preform assembly. Rare earth dopants (e.g., ytterbium or erbium) tend to increase the refractive index, but this can be precisely compensated, for example, with additional fluorine doping, so that the guiding ­properties are determined by the photonic microstructure only and not by a conventional-type refractive index difference.

Generally, PCFs are constructed by the same methods as other optical fibers: first, one constructs a “preform” on the scale of centimeters in size, and then heats the preform and draws it down to the fiber diameter, shrinking the preform cross section but maintaining the same features. In this way, kilometers of fiber can be produced from a single preform.

Most PCFs have been fabricated in silica glass, but other glasses have also been used to obtain particular optical properties (such has high optical nonlinearity). Using this principle, for example, fibers optimized to have a very high Raman efficiency have been fabricated, which are conceived for the design of high-gain low-noise Raman amplifiers.

However, special care is required in various respects:

• Ends of PCFs may not be cleaned with liquid solvents, such as ethanol, as capillary forces may pull them into the holes.

• Pulling very long length of microstructured fibers is not easy due to the need of maintaining the cross section constant and in general there is a limit to the length of a piece of such fibers.

• Cleaving and fusion splicing PCFs is in principle possible, but can be more difficult, particularly for fibers with large air content. During fusion splicing, the air may expand and distort the fiber structure.

• Connections between fibers are also possible with a variety of mechanical splices, fiber connectors, protected patch cables, beam expansion units, etc.

• Even when the splicing process works well, there may be a substantial coupling loss due to a mismatch of mode areas, for example, when a small-core PCF is copled to a standard single-mode fiber. There are special tapered single-mode fibers and tapered PCFs for enhancing the coupling efficiency, but these may not be easily available.

4.2.12 Fiber Cables

Simply covered by the polymer jacket, an optical fiber is impossible to deploy in the field. In order to be used, an optical fiber has to be packaged in a fiber cable [42].

The major benefits of fiber optic cabling are as follows:

• Easy handling: Some communication systems require tens or even hundreds of fibers (such as a metro backbone system). However, fibers in a cable make it very easy to install and maintain.

• Protection from damaging forces: Fiber optic cables have to be pulled into place through ducts (outdoor) or conduits (indoor). Pulling eyes are attached to the strength members or cable outer jackets. This is critical for isolating the fibers from the applied pulling forces. Glass fibers cannot endure more than 0.1%–0.2% elongation during installation.

• Protection from harsh environment factors: Cable structures protect fibers from moisture (outdoor cables), extreme temperature (aerial cables) and influx of hydrogen into the fiber (which causes light absorption peak at 1380 nm, which in turn impair fibers’ transmission properties).

For current telecommunication applications, practically only single-mode fibers are used, also in the access area. Thus, all the telecommunication cables include single-mode fibers, even if multimode fiber cables and even mixed fiber cables are still used for other applications or for compatibility with legacy systems generally in the access area.

The first element in determining the characteristics of a fiber cable is the choice of materials used to package the cable.

Depending on the environment in which the cable has to be deployed, different requirements are imposed to the design and different materials have to be chosen.

For indoor cables, used for LANs or internal cabling of carriers central offices, fire safety is the number one factor. Moreover, especially in the LAN application, protection of the inner fibers against excessive bending and easy installation in the typical LAN conduit is another key requirement.

In the case of outdoor cables, moisture resistance and temperature tolerance are the major factors when choosing materials. They also need to be ultraviolet resistant and capable of sustaining the stresses related to typical installation procedures without damage.

Finally, aerial self-supporting cables must endure extreme temperature ranges from sunlight heat to freezing snow. They also must survive high wind loading.

The structure of a fiber cable, schematically reported in Figure 4.12, is designed to allow these requirements to be fulfilled.

The fibers are sustained by an internal strength member made from steel, Aramid yarn [43], or some other material with high resistance to traction and high elasticity. The role of the strength member is to give mechanical resistance to the cable and to allow cable traction during installation. When a cable is pulled into a duct, the tension is applied to the strong member instead of the fibers.

Aramid yarn (Kevlar is its commercial name that is a product trademark) is a yellow color, fiber-looking material. It is strong and is used to bundle and protect the loose tubes or fibers in the cable.

Every fiber is individually protected with a polymer tube. These polymer tubes are also colored so that the fibers can be recognized at the two extremes of the cable. This is an important cable characteristic, since the circular symmetry make it impossible to recognize the single fiber from its position.

The set of fibers is protected as a whole by different concentric tubes, each of which has a different role in assuring either cable resistance to mechanical, chemical or thermal aggressions or other cable characteristics, for example, resistance to burning or nonpoisonous smoke if flaming.

A gel compound fills buffer tubes and cable interiors, making the cable impervious to water. It needs to be completely cleaned off when the cable end is stripped for termination.

The outer tube is called jacket and has a particular importance since it is the part of the cable that is exposed to the outside environment.

For this reason, the jacket material is quite important; in a typical fiber cable the jacket is composed of the following materials:

• Polyethylene (PE) is the standard jacket material for outdoor fiber optic cables. PE has excellent moisture and weather-resistance properties. It has very stable dielectric properties over a wide temperature range. It is also abrasion resistant.

• Polyvinyl chloride (PVC) is the most common material for indoor cables; however, it can also be used for outdoor cables. It is flexible and fire-retardant. PVC is more expensive than PE and that has an impact for very long cables.

• Polyvinyl difluoride (PVDF) is used for plenum cables because it has better fire-retardant properties than PE and produces little smoke.

• Low smoke zero halogen (LSZH) plastics are used for a special kind of cable called LSZH cables. They produce little smoke and no toxic halogen compounds. But they are the most expensive jacket material.

Embedded just below the cable jacket there is the ripcord: a thin but very strong thread. Its role is to split the cable easily without harming cable interiors.

Fiber optic cables are available in a wide variety of physical constructions. They can be anything from simple simplex or duplex (zipcord) cables used for jumpers to 144-fiber cable for intercity transmission.

Most of the fibers used in these cables come down to two basic configurations—900 μm tight-buffered fibers or 250 μ m coated fibers (also called bare fibers). Actually, tight-­ buffered fibers cover a coated fiber (the coating is soft plastic) with a thick layer of harder plastic, making it easier to handle and providing physical protection.

There are two basic types of fiber optic cable constructions: Tight-buffered cable and loose tube cable. Their structure is shown in Figure 4.12a and b.

Multiple color-coded 900 μm tight-buffered fibers can be packed tightly together in a compact cable structure, an approach widely used indoors; these cables are called tight-buffered cables. Tight-buffered cables are used to connect outside plant cables to TE, and also for linking various devices in a premises network.

Multifiber, tight-buffered cables often are used for intrabuilding, risers, general building, and plenum applications.

On the other hand, multiple (up to 12) 250 μ m coated fibers can be put inside a color coded, flexible plastic tube, which usually is filled with a gel compound that prevents moisture from seeping through the hollow tube. Buffer tubes are stranded around a dielectric or steel central member. Aramid yarn is used as primary strength member. Then an outer PE jacket is extruded over the core. These cables are called loose tube cables.

Loose tube structure isolates the fibers from the cable structure. This is a big advantage in handling thermal and other stresses encountered outdoors, which is why most loose tube fiber optic cables are built for outdoor applications.

Loose tube cables typically are used for outside-plant installation in aerial, duct, and direct-buried applications.

4.3.1 Basic Theory of Optical Amplifiers

Due to the specific properties of optical amplifiers, before dealing with particular devices and technologies it is important to fix a few characteristics that are common to all amplifiers working in a regime where thermal noise is negligible and the amplifier signal to noise ratio is determined by quantum noise.

These characteristics can be derived directly from the base principles of quantum mechanics applied to a system that has the characteristic to multiply the input signal by a constant greater than one, called gain.

Thus, in this section we will review the basic theory of quantum amplifiers [44], thus placing the bases for the analysis of specific amplifiers design.