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### Djaafari

1. 1. Chapter 2 Pulse Propagation in Fibers For an understanding of the nonlinear phenomena in optical ﬁbers, it is neces- sary to consider the theory of electromagnetic wave propagation in dispersive nonlinear media. The objective of this chapter is to obtain a basic equation that governs propagation of optical pulses in single-mode ﬁbers. Section 2.1 intro- duces Maxwell’s equations and important concepts such as the linear and non- linear parts of the induced polarization and the frequency-dependent dielectric constant. The concept of ﬁber modes is introduced in Section 2.2 where the single-mode condition is also discussed. Section 2.3 considers the theory of pulse propagation in nonlinear dispersive media in the slowly varying envelope approximation with the assumption that the spectral width of the pulse is much smaller than the frequency of the incident radiation. The numerical methods used to solve the resulting propagation equation are discussed in Section 2.4. 2.1 Maxwell’s Equations Like all electromagnetic phenomena, the propagation of optical ﬁelds in ﬁbers is governed by Maxwell’s equations. In the International System of Units (Syst`eme international d’unit´es or SI), these equations are [1] For Evaluation Only. Copyright (c) by Foxit Software Company, 2004 - 2007 Edited by Foxit PDF Editor
2. 2. 32 Pulse Propagation in Fibers (2.1.4) where E and H are electric and magnetic ﬁeld vectors, respectively, and D and B are corresponding electric and magnetic ﬂux densities. The current density vector J and the charge density ρf represent the sources for the electromagnetic ﬁeld. In the absence of free charges in a medium such as optical ﬁbers, J 0 and ρf 0. The ﬂux densities D and B arise in response to the electric and magnetic ﬁelds E and H propagating inside the medium and are related to them through the constitutive relations given by [1] (2.1.5) (2.1.6) where ε0 is the vacuum permittivity, µ0 is the vacuum permeability, and P and M are the induced electric and magnetic polarizations. For a nonmagnetic medium such as optical ﬁbers, M 0. Maxwell’s equations can be used to obtain the wave equation that describes light propagation in optical ﬁbers. By taking the curl of Eq. (2.1.1) and using Eqs. (2.1.2), (2.1.5), and (2.1.6), one can eliminate B and D in favor of E and P and obtain (2.1.7) where c is the speed of light in vacuum and the relation µ0ε0 1 c2 was used. To complete the description, a relation between the induced polarization P and the electric ﬁeld E is needed. In general, the evaluation of P requires a quantum-mechanical approach. Although such an approach is often necessary when the optical frequency is near a medium resonance, a phenomenological relation of the form (1.3.1) can be used to relate P and E far from medium resonances. This is the case for optical ﬁbers in the wavelength range 0.5– 2 µm that is of interest for the study of nonlinear effects. If we include only the third-order nonlinear effects governed by χ´3µ , the induced polarization consists of two parts such that (2.1.8) where the linear part PL and the nonlinear part PNL are related to the electric ﬁeld by the general relations [2]–[4] (2.1.9) For Evaluation Only. Copyright (c) by Foxit Software Company, 2004 - 2007 Edited by Foxit PDF Editor
3. 3. Maxwell’s Equations 33 These relations are valid in the electric-dipole approximation and assume that the medium response is local. Equations (2.1.7)–(2.1.10) provide a general formalism for studying the third-order nonlinear effects in optical ﬁbers. Because of their complexity, it is necessary to make several simplifying approximations. In a major simpliﬁca- tion, the nonlinear polarization PNL in Eq. (2.1.8) is treated as a small pertur- bation to the total induced polarization. This is justiﬁed because the nonlinear effects are relatively weak in silica ﬁbers. The ﬁrst step therefore consists of solving Eq. (2.1.7) with PNL 0. Because Eq. (2.1.7) is then linear in E, it is useful to write in the frequency domain as where ˜E´r ωµ is the Fourier transform of E´r tµ deﬁned as The frequency-dependent dielectric constant appearing in Eq. (2.1.12) is deﬁned as where ˜χ´1µ ´ωµ is the Fourier transform of χ´1µ ´tµ. As ˜χ´1µ ´ωµ is in general complex, so is ε´ωµ. Its real and imaginary parts can be related to the refractive index n´ωµ and the absorption coefﬁcient α´ωµ by using the deﬁnition From Eqs. (2.1.13) and (2.1.14), n and α are related to χ´1µ by the relations where Re and Im stand for the real and imaginary parts, respectively. The frequency dependence of n and α has been discussed in Section 1.2. For Evaluation Only. Copyright (c) by Foxit Software Company, 2004 - 2007 Edited by Foxit PDF Editor
4. 4. 34 Pulse Propagation in Fibers Two further simpliﬁcations can be made before solving Eq. (2.1.11). First, because of low optical losses in ﬁbers in the wavelength region of interest, the imaginary part of ε´ωµ is small in comparison to the real part. Thus, we can replace ε´ωµ by n2´ωµ in the following discussion of ﬁber modes and include ﬁber loss later in a perturbative manner. Second, as n´ωµ is often independent of the spatial coordinates in both the core and the cladding of step-index ﬁbers, one can use where the relation ∇ ¡D ε∇ ¡E 0 was used from Eq. (2.1.3). With these simpliﬁcations, Eq. (2.1.11) takes the form This equation is solved in the next section on ﬁber modes. 2.2 Fiber Modes At any frequency ω, optical ﬁbers can support a ﬁnite number of guided modes whose spatial distribution ˜E´r ωµ is a solution of the wave equation (2.1.18) and satisﬁes all appropriate boundary conditions. In addition, the ﬁber can support a continuum of unguided radiation modes. Although the inclusion of radiation modes is crucial in problems involving transfer of power between bounded and radiation modes [5], they do not play an important role in the discussion of nonlinear effects. As ﬁber modes are covered in many text- books [5]–[7], they are discussed only brieﬂy in this section. 2.2.1 Eigenvalue Equation Because of the cylindrical symmetry of ﬁbers, it is useful to express the wave equation (2.1.18) in cylindrical coordinates ρ φ and z: where k0 ω c 2π λ and ˜E is the Fourier transform of the electric ﬁeld E, i.e., For Evaluation Only. Copyright (c) by Foxit Software Company, 2004 - 2007 Edited by Foxit PDF Editor
5. 5. Fiber Modes 35 Similar relations exist for the magnetic ﬁeld H´r tµ. As E and H satisfy Maxwell’s equations (2.1.1)–(2.1.4), only two components out of six are inde- pendent. It is customary to choose ˜Ez and ˜Hz as the independent components and express ˜Eρ ˜Eφ ˜Hρ, and ˜Hφ in terms of ˜Ez and ˜Hz. Both ˜Ez and ˜Hz satisfy Eq. (2.2.1). The wave equation for ˜Ez is easily solved by using the method of separation of variables, resulting in the following general form: where A is a normalization constant, β is the propagation constant, m is an integer, and F´ρµ is the solution of where the refractive index n n1 for ρ a for a ﬁber of core radius a but takes the value n2 outside the core (ρ a). Equation (2.2.4) is the well-known differential equation for Bessel func- tions. Its general solution inside the core can be written as where Jm is the Bessel function, Nm is the Neumann function, and The constants C1 and C2 are determined using the boundary conditions. As Nm´κρµ has a singularity at ρ 0, C2 0 for a physically meaningful solution. The constant C1 can be absorbed in A appearing in Eq. (2.2.3). Thus, In the cladding region (ρ a), the solution F´ρµ should be such that it decays exponentially for large ρ. The modiﬁed Bessel function Km represents such a solution. Therefore, where For Evaluation Only. Copyright (c) by Foxit Software Company, 2004 - 2007 Edited by Foxit PDF Editor
6. 6. 36 Pulse Propagation in Fibers The same procedure can be followed to obtain the magnetic ﬁeld compo- nent ˜Hz. The boundary condition that the tangential components of ˜E and ˜H be continuous across the core-cladding interface requires that ˜Ez, ˜Hz, ˜Eφ , and ˜Hφ be the same when ρ a is approached from inside or outside the core. The equality of these ﬁeld components at ρ a leads to an eigenvalue equation whose solutions determine the propagation constant β for the ﬁber modes. Since the whole procedure is well known [5]–[7], we write the eigenvalue equation directly: where a prime denotes differentiation with respect to the argument and we used the important relation The eigenvalue equation (2.2.10) in general has several solutions for β for each integer value of m. It is customary to express these solutions by βmn, where both m and n take integer values. Each eigenvalue βmn corresponds to one speciﬁc mode supported by the ﬁber. The corresponding modal ﬁeld distribution is obtained from Eq. (2.2.3). It turns out [5]–[7] that there are two types of ﬁber modes, designated as HEmn and EHmn. For m 0, these modes are analogous to the transverse-electric (TE) and transverse-magnetic (TM) modes of a planar waveguide because the axial component of the electric ﬁeld, or the magnetic ﬁeld, vanishes. However, for m 0, ﬁber modes become hybrid, i.e., all six components of the electromagnetic ﬁeld are nonzero. 2.2.2 Single-Mode Condition The number of modes supported by a speciﬁc ﬁber at a given wavelength de- pends on its design parameters, namely the core radius a and the core-cladding index difference n1  n2. An important parameter for each mode is its cut-off frequency. This frequency is determined by the condition γ 0. The value of κ when γ 0 for a given mode determines the cut-off frequency from Eq. (2.2.11). It is useful to deﬁne a normalized frequency V by the relation where κc is obtained from Eq. (2.2.11) by setting γ 0. For Evaluation Only. Copyright (c) by Foxit Software Company, 2004 - 2007 Edited by Foxit PDF Editor
7. 7. Fiber Modes 37 The eigenvalue equation (2.2.10) can be used to determine the values of V at which different modes reach cut-off. The procedure is complicated, but has been described in many texts [5]–[7]. Since we are interested mainly in single-mode ﬁbers, we limit the discussion to the cut-off condition that allows the ﬁber to support only one mode. A single-mode ﬁber supports only the HE11 mode, also referred to as the fundamental mode. All other modes are beyond cut-off if the parameter V Vc, where Vc is the smallest solution of J0´Vcµ 0 or Vc 2 405. The actual value of V is a critical design parameter. Typically, microbending losses increase as V Vc becomes small. In practice, therefore, ﬁbers are designed such that V is close to Vc. The cut-off wavelength λc for single-mode ﬁbers can be obtained by using k0 2π λc and V 2 405 in Eq. (2.2.12). For a typical value n1  n2 0 005 for the index difference, λc 1 2 µm for a 4 µm, indicating that such a ﬁber supports a single mode only for λ 1 2 µm. In practice, core radius should be below 2 µm for a ﬁber to support a single mode in the visible region. 2.2.3 Characteristics of the Fundamental Mode The ﬁeld distribution E´r tµ corresponding to the HE11 mode has three non- zero components Eρ, Eφ , and Ez, or in Cartesian coordinates Ex, Ey, and Ez. Among these, either Ex or Ey dominates. Thus, to a good degree of approxima- tion, the fundamental ﬁber mode is linearly polarized in either x or y direction depending on whether Ex or Ey dominates. In this respect, even a single-mode ﬁber is not truly single mode because it can support two modes of orthogo- nal polarizations. The notation LPmn is sometimes used to denote the linearly polarized modes, which are approximate solutions of Eq. (2.2.1). The funda- mental mode HE11 corresponds to LP01 in this notation [6]. The two orthogonally polarized modes of a single-mode ﬁber are degen- erate (i.e., they have the same propagation constant) under ideal conditions. In practice, irregularities such as random variations in the core shape and size along the ﬁber length break this degeneracy slightly, mix the two polariza- tion components randomly, and scramble the polarization of the incident light as it propagates down the ﬁber. As discussed in Section 1.2.4, polarization- preserving ﬁbers can maintain the linear polarization if the light is launched with its polarization along one of the principal axes of the ﬁber. Assuming that the incident light is polarized along a principal axis (chosen to coincide with the x axis), the electric ﬁeld for the fundamental ﬁber mode HE11 is approxi- For Evaluation Only. Copyright (c) by Foxit Software Company, 2004 - 2007 Edited by Foxit PDF Editor
8. 8. 38 Pulse Propagation in Fibers mately given by where A´ωµ is a normalization constant. The transverse distribution inside the core is found to be where ρ ´x2 · y2µ1 2 is the radial distance. Outside the ﬁber core, the ﬁeld decays exponentially as [5] where Km´γρµ in Eq. (2.2.8) was approximated by the leading term in its asymptotic expansion and a constant factor was added to ensure the equality of F´x yµ at ρ a. The propagation constant β´ωµ in Eq. (2.2.13) is ob- tained by solving the eigenvalue equation (2.2.10). Its frequency dependence results not only from the frequency dependence of n1 and n2 but also from the frequency dependence of κ. The former is referred to as material dispersion while the latter is called waveguide dispersion. As discussed in Section 1.3, material dispersion generally dominates unless the light wavelength is close to the zero-dispersion wavelength. The evaluation of β´ωµ generally requires a numerical solution of Eq. (2.2.10) although approximate analytic expressions can be obtained in speciﬁc cases [5]. The effective mode index is related to β by neff β k0. As the use of modal distribution F´x yµ given by Eqs. (2.2.14) and (2.2.15) is cumbersome in practice, the fundamental ﬁber mode is often approximated by a Gaussian distribution of the form where the width parameter w is determined by ﬁtting the exact distribution to a Gaussian form or by following a variational procedure. Figure 2.1 shows the dependence of w a on the ﬁber parameter V deﬁned by Eq. (2.2.12). The com- parison of the actual ﬁeld distribution with the ﬁtted Gaussian is also shown for a speciﬁc value V 2 4. The quality of ﬁt is generally quite good [8], par- ticularly for V values in the neighborhood of 2. Figure 2.1 shows that w a for V 2, indicating that the core radius provides a good estimate of w for telecommunication ﬁbers for which V 2. Note that w can be signiﬁcantly larger than a for V 1 8. The use of Gaussian approximation is of consider- able practical value because of its relative simplicity. For Evaluation Only. Copyright (c) by Foxit Software Company, 2004 - 2007 Edited by Foxit PDF Editor
9. 9. Pulse-Propagation Equation 39 Figure 2.1 Variation of mode-width parameter w with V obtained by ﬁtting the fun- damental ﬁber mode to a Gaussian distribution. Traces on the right show the quality of ﬁt for V 2 4. (After Ref. [8].) 2.3 Pulse-Propagation Equation The study of most nonlinear effects in optical ﬁbers involves the use of short pulses with widths ranging from 10 ns to 10 fs. When such optical pulses propagate inside a ﬁber, both dispersive and nonlinear effects inﬂuence their shape and spectrum. In this section we derive a basic equation that governs propagation of optical pulses in nonlinear dispersive ﬁbers. The starting point is the wave equation (2.1.7). By using Eqs. (2.1.8) and (2.1.17), it can be written in the form where the linear and nonlinear parts of the induced polarization are related to the electric ﬁeld E´r tµ through Eqs. (2.1.9) and (2.1.10), respectively. 2.3.1 Nonlinear Pulse Propagation It is necessary to make several simplifying assumptions before solving Eq. (2.3.1). First, PNL is treated as a small perturbation to PL. This is justiﬁed because nonlinear changes in the refractive index are 10 6 in practice. Sec- ond, the optical ﬁeld is assumed to maintain its polarization along the ﬁber length so that a scalar approach is valid. This is not really the case, unless For Evaluation Only. Copyright (c) by Foxit Software Company, 2004 - 2007 Edited by Foxit PDF Editor
10. 10. 40 Pulse Propagation in Fibers polarization-maintaining ﬁbers are used, but the approximation works quite well in practice; it will be relaxed later in Chapter 6. Third, the optical ﬁeld is assumed to be quasi-monochromatic, i.e., the pulse spectrum, centered at ω 0, is assumed to have a spectral width ∆ω such that ∆ω ω0 1. Since ω 0 1015 s 1, the last assumption is valid for pulses as short as 0.1 ps. In the slowly varying envelope approximation adopted here, it is useful to separate the rapidly varying part of the electric ﬁeld by writing it in the form where ˆx is the polarization unit vector, and E´r tµ is a slowly varying function of time (relative to the optical period). The polarization components PL and PNL can also be expressed in a similar way by writing The linear component PL can be obtained by substituting Eq. (2.3.3) in Eq. (2.1.9) and is given by where ˜E´r ωµ is the Fourier transform of E´r tµ and is deﬁned similarly to Eq. (2.1.12). The nonlinear component PNL´r tµ is obtained by substituting Eq. (2.3.4) in Eq. (2.1.10). Considerable simpliﬁcation occurs if the nonlinear response is assumed to be instantaneous so that the time dependence of χ´3µ in Eq. (2.1.10) is given by the product of three delta functions of the form δ´t  t1µ. Equation (2.1.10) then reduces to The assumption of instantaneous nonlinear response amounts to neglecting the contribution of molecular vibrations to χ´3µ (the Raman effect). In gen- eral, both electrons and nuclei respond to the optical ﬁeld in a nonlinear man- ner. The nuclei response is inherently slower compared with the electronic For Evaluation Only. Copyright (c) by Foxit Software Company, 2004 - 2007 Edited by Foxit PDF Editor
11. 11. Pulse-Propagation Equation 41 response. For silica ﬁbers the vibrational or Raman response occurs over a time scale 60–70 fs. Thus, Eq. (2.3.6) is approximately valid for pulse widths 1 ps. The Raman contribution is included later in this section. When Eq. (2.3.2) is substituted in Eq. (2.3.6), PNL´r tµ is found to have a term oscillating at ω0 and another term oscillating at the third-harmonic frequency 3ω 0. The latter term requires phase matching and is generally neg- ligible in optical ﬁbers. By making use of Eq. (2.3.4), PNL´r tµ is given by where the nonlinear contribution to the dielectric constant is deﬁned as To obtain the wave equation for the slowly varying amplitude E´r tµ, it is more convenient to work in the Fourier domain. This is generally not possible as Eq. (2.3.1) is nonlinear because of the intensity dependence of εNL. In one approach, ε NL is treated as a constant during the derivation of the propagation equation [9], [10]. The approach is justiﬁed in view of the slowly varying envelope approximation and the perturbative nature of PNL. Substituting Eqs. (2.3.2)–(2.3.4) in Eq. (2.3.1), the Fourier transform ˜E´r ω  ω0µ, deﬁned as is found to satisfy the Helmholtz equation where k0 ω c and is the dielectric constant whose nonlinear part εNL is given by Eq. (2.3.8). Similar to Eq. (2.1.14), the dielectric constant can be used to deﬁne the refrac- tive index ˜n and the absorption coefﬁcient ˜α. However, both ˜n and ˜α become intensity dependent because of εNL. It is customary to introduce Using ε ´ ˜n · i ˜α 2k0µ2 and Eqs. (2.3.8) and (2.3.11), the nonlinear-index coefﬁcient n2 and the two-photon absorption coefﬁcient α2 are given by For Evaluation Only. Copyright (c) by Foxit Software Company, 2004 - 2007 Edited by Foxit PDF Editor
12. 12. 42 Pulse Propagation in Fibers The linear index n and the absorption coefﬁcient α are related to the real and imaginary parts of ˜χ´1µ xx as in Eqs. (2.1.15) and (2.1.16). As α2 is relatively small for silica ﬁbers, it is often ignored. The parameter n2 should not be confused with the cladding index of Section 2.2 even though the same notation has been used. From here onward, n2 is a measure of the ﬁber nonlinearity. Equation (2.3.10) can be solved by using the method of separation of vari- ables. If we assume a solution of the form where ˜A´z ωµ is a slowly varying function of z and β0 is the wave number to be determined later, Eq. (2.3.10) leads to the following two equations for F´x yµ and ˜A´z ωµ: In obtaining Eq. (2.3.16), the second derivative ∂2 ˜A ∂z2 was neglected since ˜A´z ωµ is assumed to be a slowly varying function of z. The wave number ˜β is determined by solving the eigenvalue equation (2.3.15) for the ﬁber modes using a procedure similar to that used in Section 2.2. The dielectric constant ε´ωµ in Eq. (2.3.15) can be approximated by where ∆n is a small perturbation given by Equation (2.3.15) can be solved using ﬁrst-order perturbation theory [11]. We ﬁrst replace ε with n2 and obtain the modal distribution F´x yµ, and the corresponding wave number β´ωµ. For a single-mode ﬁber, F´x yµ corre- sponds to the modal distribution of the fundamental ﬁber mode HE11 given by Eqs. (2.2.14) and (2.2.15), or by the Gaussian approximation (2.2.16). We then include the effect of ∆n in Eq. (2.3.15). In the ﬁrst-order perturbation theory, ∆n does not affect the modal distribution F´x yµ. However, the eigenvalue ˜β becomes For Evaluation Only. Copyright (c) by Foxit Software Company, 2004 - 2007 Edited by Foxit PDF Editor
13. 13. Pulse-Propagation Equation 43 where This step completes the formal solution of Eq. (2.3.1) to the ﬁrst order in perturbation PNL. Using Eqs. (2.3.2) and (2.3.12), the electric ﬁeld E´r tµ can be written as where A´z tµ is the slowly varying pulse envelope. The Fourier transform ˜A´z ω  ω0µ of A´z tµ satisﬁes Eq. (2.3.16), which can be written as where we used Eq. (2.3.19) and approximated ˜β2  β2 0 by 2β0´ ˜β  β0µ. The physical meaning of this equation is clear. Each spectral component within the pulse envelope acquires, as it propagates down the ﬁber, a phase shift whose magnitude is both frequency and intensity dependent. At this point, one can go back to the time domain by taking the inverse Fourier transform of Eq. (2.3.22), and obtain the propagation equation for A´z tµ. However, as an exact functional form of β´ωµ is rarely known, it is useful to expand β´ωµ in a Taylor series about the carrier frequency ω0 as The cubic and higher-order terms in this expansion are generally negligible if the spectral width ∆ω ω0. Their neglect is consistent with the quasi- monochromatic assumption used in the derivation of Eq. (2.3.22). If β2 0 for some speciﬁc values of ω0 (in the vicinity of the zero-dispersion wavelength of the ﬁber, as discussed in Section 1.3.3), it may be necessary to include the cubic term. We substitute Eq. (2.3.23) in Eq. (2.3.22) and take the inverse Fourier transform by using For Evaluation Only. Copyright (c) by Foxit Software Company, 2004 - 2007 Edited by Foxit PDF Editor
14. 14. 44 Pulse Propagation in Fibers During the Fourier-transform operation, ω  ω0 is replaced by the differential operator i´∂ ∂tµ. The resulting equation for A´z tµ becomes The term with ∆β includes the effect of ﬁber loss and nonlinearity. By using Eqs. (2.3.18) and (2.3.20), ∆β can be evaluated and substituted in (2.3.26). The result is where the nonlinear parameter γ is deﬁned as In obtaining Eq. (2.3.27) the pulse amplitude A is assumed to be normal- ized such that A 2 represents the optical power. The quantity γ A 2 is then measured in units of m 1 if n2 is expressed in units of m2/W (see Appendix B). The parameter Aeff is known as the effective core area and is deﬁned as Its evaluation requires the use of modal distribution F´x yµ for the fundamen- tal ﬁber mode. Clearly Aeff depends on ﬁber parameters such as the core ra- dius and the core-cladding index difference. If F´x yµ is approximated by a Gaussian distribution as in Eq. (2.2.16), Aeff πw2. The width parameter w depends on the ﬁber parameters and can be obtained using Fig. 2.1 and Eq. (2.2.12). Typically, Aeff can vary in the range 20–100 µm2 in the 1.5-µm re- gion depending on the ﬁber design. As a result, γ takes values in the range 1–10 W 1/km if n2 2 6¢10 20 m2/W is used (see Appendix B). In a large- effective-area ﬁber (LEAF), Aeff is increased intentionally to reduce the impact of ﬁber nonlinearity. Equation (2.3.27) describes propagation of picosecond optical pulse in single-mode ﬁbers. It is often referred to as the nonlinear Schr¨odinger (NLS) equation because it can be reduced to that form under certain conditions. It includes the effects of ﬁber losses through α, of chromatic dispersion through β1 and β2, and of ﬁber nonlinearity through γ. The physical signiﬁcance of For Evaluation Only. Copyright (c) by Foxit Software Company, 2004 - 2007 Edited by Foxit PDF Editor
15. 15. Pulse-Propagation Equation 45 the parameters β1 and β2 has been discussed in Section 1.2.3. Brieﬂy, the pulse envelope moves at the group velocity vg 1 β1 while the effects of group-velocity dispersion (GVD) are governed by β2. The GVD parameter β2 can be positive or negative depending on whether the wavelength λ is be- low or above the zero-dispersion wavelength λD of the ﬁber (see Fig. 1.5). In the anomalous-dispersion regime (λ λD), β2 is negative, and the ﬁber can support optical solitons. In standard silica ﬁbers, β2 50 ps2/km in the visi- ble region but becomes close to  20 ps2/km near wavelengths 1 5 µm, the change in sign occurring in the vicinity of 1.3 µm. 2.3.2 Higher-Order Nonlinear Effects Although the propagation equation (2.3.27) has been successful in explaining a large number of nonlinear effects, it may need modiﬁcation depending on the experimental conditions. For example, Eq. (2.3.27) does not include the effects of stimulated inelastic scattering such as SRS and SBS (see Section 1.3.2). If peak power of the incident pulse is above a threshold level, both SRS and SBS can transfer energy from the pulse to a new pulse, which may propagate in the same or the opposite direction. The two pulses interact with each other through the Raman or Brillouin gain and XPM. A similar situation occurs when two or more pulses at different wavelengths (separated by more than individual spectral widths) are incident on the ﬁber. Simultaneous propagation of multi- ple pulses is governed by a set of equations similar to Eq. (2.3.27), modiﬁed suitably to include the contributions of XPM and the Raman or Brillouin gain. Equation (2.3.27) should also be modiﬁed for ultrashort optical pulses whose width is close to or 1 ps [12]–[22]. The spectral width of such pulses becomes large enough that several approximations made in the deriva- tion of Eq. (2.3.27) become questionable. The most important limitation turns out to be the neglect of the Raman effect. For pulses with a wide spectrum ( 0 1 THz), the Raman gain can amplify the low-frequency components of a pulse by transferring energy from the high-frequency components of the same pulse. This phenomenon is called intrapulse Raman scattering. As a result of it, the pulse spectrum shifts toward the low-frequency (red) side as the pulse propagates inside the ﬁber, a phenomenon referred to as the self-frequency shift [12]. The physical origin of this effect is related to the delayed nature of the Raman (vibrational) response [13]. Mathematically, Eq. (2.3.6) cannot be used in the derivation of Eq. (2.3.27); one must use the general form of the nonlinear polarization given in Eq. (2.1.10). For Evaluation Only. Copyright (c) by Foxit Software Company, 2004 - 2007 Edited by Foxit PDF Editor
16. 16. 46 Pulse Propagation in Fibers The starting point is again the wave equation (2.3.1). Equation (2.1.10) de- scribes a wide variety of third-order nonlinear effects, and not all of them are relevant to our discussion. For example, nonlinear phenomena such as third- harmonic generation and four-wave mixing are unlikely to occur unless an appropriate phase-matching condition is satisﬁed (see Chapter 10). Nonres- onant, incoherent (intensity-dependent) nonlinear effects can be included by assuming the following functional form for the third-order susceptibility [17]: where R´tµ is the nonlinear response function normalized in a manner similar to the delta function, i.e., Ê ∞  ∞ R´tµdt 1. By substituting Eq. (2.3.30) in Eq. (2.1.10) the nonlinear polarization is given by where it is assumed that the electric ﬁeld and the induced polarization vectors point along the same direction. The upper limit of integration in Eq. (2.3.31) extends only up to t because the response function R´t  t1µ must be zero for t1 t to ensure causality. The analysis of Section 2.3.1 can still be used by working in the frequency domain. Using Eqs. (2.3.2)–(2.3.4), ˜E is found to satisfy [18] where ˜R´ωµ is the Fourier transform of R´tµ. As before, one can treat the terms on the right-hand side as a small perturbation and ﬁrst obtain the modal distri- bution by neglecting them. The effect of perturbation terms is to change the propagation constant for the fundamental mode by ∆β as in Eq. (2.3.19) but with a different expression for ∆β. One can then deﬁne the slowly varying am- plitude A´z tµ as in Eq. (2.3.21) and obtain, after some algebra, the following equation for pulse evolution inside a single-mode ﬁber [18]: For Evaluation Only. Copyright (c) by Foxit Software Company, 2004 - 2007 Edited by Foxit PDF Editor