80 PROPAGATION OF SIGNALS IN OPTICAL FIBER 2.4.3 The intensities of the pump wave Ip and the Stokes wave I~ are related by the coupled-wave equations [Buc95] dis dz gBIpIs + Otis, (2.14) and dip gBIpls otlp. (2.15) dz The intensities are related to the powers as Ps - Aels and Pp - Aelp. For the case where the Stokes power is much smaller than the pump power, we can assume that the pump wave is not depleted. This amounts to neglecting the gBIpls term on the right-hand side of (2.15). With this assumption, (2.14) and (2.15) can be solved (see Problem 5.23) for a link of length L to yield Ps(O) Ps(L)e LegsPP(~ __ Or Ae (2.16) and Pp(L) = Pp(O)e -~ (2.17) Note that the output of the pump wave is at z = L, but the output of the Stokes wave is at z = 0 since the two waves are counterpropagating. Stimulated Raman Scattering If two or more signals at different wavelengths are injected into a fiber, SRS causes power to be transferred from the lower-wavelength channels to the higher-wavelength channels (see Figure 2.16). This coupling of energy from a lower-wavelength signal to a higher-wavelength signal is a fundamental effect that is also the basis of optical amplification and lasers. The energy of a photon at a wavelength k is given by hc/k, where h is Planck's constant (6.63 x 10 -34 J s). Thus, a photon of lower wavelength has a higher energy. The transfer of energy from a signal of lower wavelength to a signal of higher wavelength corresponds to emission of photons of lower energy caused by photons of higher energy. Unlike SBS, SRS is a broadband effect. Figure 2.17 shows its gain coefficient as a function of wavelength spacing. The peak gain coefficient gR is approximately 6 x 10 -14 ~ at 1.55 #m, which is much smaller than the gain coefficient for SBS. However, channels up to 15 THz (125 nm) apart will be coupled with SRS. Also SRS causes coupling in both the direction of propagation and the reverse direction. We will study the system impact of SRS in Section 5.8.3. While SRS between channels in a WDM system is harmful to the system, we can also use SRS to provide 2.4 Nonlinear Effects 81 )21 )22 )23 )24 Fiber )21 )22 )23 )24 Figure 2.16 The effect of SRS. Power from lower-wavelength channels is transferred to the higher-wavelength channels. 7 7~ 5 x ~4 3 "~2 I 0 10 20 30 40 Channel separation (THz) Figure 2.17 SRS gain coefficient as a function of channel separation. (After [Agr97].) amplification in the system, which benefits the overall system performance. We will discuss such amplifiers in Section 3.4.4. 2.4.4 Propagation in a Nonlinear Medium In order to discuss the origin of SPM, CPM, and FWM in the following sections, we need to understand how the propagation of light waves is affected when we relax the linearity assumption we made in Section 2.1.2. This is the subject of this section. We will continue, however, to make the other assumptions of local responsivity, isotropy, homogeneity, and losslessness on the silica medium. The losslessness assumption can be removed by carrying out the remaining discussion using complex variables for the following fields and susceptibilities, as is done, for example, in [Agr95]. However, 82 PROPAGATION OF SIGNALS IN OPTICAL FIBER to keep the discussion simple, we use real variables for all the fields and neglect the effect of fiber loss. For a linear medium, as we saw in Section 2.1.2, we have the relation shown in (2.8)- P(r, co) = ~o,~ (r, co)E(r, co) between the Fourier transforms P and E of the induced dielectric polarization and applied electric field, respectively. Since we are considering nonlinearities in this section, it is no longer as convenient to work in the Fourier transform domain. By taking inverse Fourier transforms, this relation can be written in the time domain as (2.7): Pc(r, t) - eo X(1)(t - t')E(r, t') dt', (2.18) oo where we have dropped the dependence of the susceptibility on r due to the homo- geneity assumption, written PL instead of P to emphasize the linearity assumption used in obtaining this relation, and used x(1)0 instead of X 0 for convenience in what follows. In discussing the effect of nonlinearities, we will assume that the electric field of the fundamental mode is linearly polarized along the x direction. Recall from Section 2.1.2 that the electric field in a single-mode fiber is a linear combination of two modes, linearly polarized along the x and y directions. (Note that the term polarization here refers to the energy distribution of a propagation mode and is different from the dielectric polarization. The linearly polarized modes referred to here have no relation to the linear component of the dielectric polarization.) The following results can be generalized to this case, but the resulting expressions are significantly more complex. Hence we make the assumption of linearly polarized fields. Because of the isotropy assumption, even in the presence of nonlinearities, the dielectric polarization is along the same direction as the electric field, which is the x direction, by assumption. Thus the vector functions E(r, t) and P(r, t) have only one component, which we will denote by the scalar functions E(r, t) and P(r, t), respectively. With this assumption, in the presence of nonlinearities, we show in Appendix F that we can write P (r, t) = T'L (r, t) -t- ~NL (r, t). Here 7)L(r, t) is the linear dielectric polarization given by (2.18) with the vectors PL(, ) and E(, ) replaced by the scalars 79L(, ) and E(, ), respectively, due to the linear 2.4 Nonlinear Effects 83 2.4.5 dielectric polarization assumption. The nonlinear dielectric polarization 7)NL (r, t) is given by 7)NL (r, t) ~oX(3)E3(r, t), (2.19) where X (3) is called the third-order nonlinear susceptibility and is assumed to be a constant (independent of t). (With the assumption of linearly polarized modes, the dielectric polarization can be expanded in a power series in E with coefficients ~0X (i), and the superscript i in X (i) refers to the power of the electric field in each term of such an expansion. Since X (2) - 0 for silica, the dominant term in determining 7)NL(r, t) is not the E 2 term but the E 3 term.) Recall that the refractive index is related to the susceptibility by (2.9). Thus the nonlinear dielectric polarization causes the refractive index to become intensity dependent, which is the root cause of these nonlinear effects. We will use this equation (2.19) as the starting point in understanding three important nonlinear phenomena affecting the propagation of signals in optical fiber" self-phase modulation (SPM), cross-phase modulation (CPM), and four-wave mixing (FWM). For simplicity, we will assume that the signals used are monochromatic plane waves; that is, the electric field is of the form E(r, t) E(z, t) E cos(coot - floz), where E is a constant. The term monochromatic implies the electric field has a single frequency component, namely, coo, and the term plane wave indicates that the electric field is constant in the plane perpendicular to the direction of propagation, z. Hence we have also written E (z, t) for E (r, t). In the case of wavelength division multiplexed (WDM) signals, we assume that the signal in each wavelength channel is a monochromatic plane wave. Thus if there are n wavelength channels at the angular frequencies col con, with the corresponding propagation constants fll fin, the electric field of the composite WDM signal is E(r, t) E(z, t) ~ Ei cos(coit - l~iz). i=1 (Since the signals on each WDM channel are not necessarily in phase, we should add an arbitrary phase 4~i to each of the sinusoids, but we omit this in order to keep the expressions simple.) Self-Phase Modulation SPM arises because the refractive index of the fiber has an intensity-dependent com- ponent. This nonlinear refractive index causes an induced phase shift that is pro- portional to the intensity of the pulse. Thus different parts of the pulse undergo 84 PROPAGATION OF SIGNALS IN OPTICAL FIBER different phase shifts, which gives rise to chirping of the pulses. Pulse chirping in turn enhances the pulse-broadening effects of chromatic dispersion. This chirp- ing effect is proportional to the transmitted signal power so that SPM effects are more pronounced in systems using high transmitted powers. The SPM-induced chirp affects the pulse-broadening effects of chromatic dispersion and thus is im- portant to consider for high-bit-rate systems that already have significant chro- matic dispersion limitations. For systems operating at 10 Gb/s and above, or for lower-bit-rate systems that use high transmitted powers, SPM can significantly in- crease the pulse-broadening effects of chromatic dispersion. In order to understand the effects of SPM, consider a single-channel system where the electric field is of the form E(z, t) = E cos(coot - floz). In the presence of fiber nonlinearities, we want to find how this field evolves along the fiber. For the monochromatic plane wave we have assumed, this means finding the propagation constant/30. Using (2.19), the nonlinear dielectric polarization is given by J)NL (r, t) = eoX(3)E 3 cos3(coot - floz) = ~oX(3)E3(~cos(ooot-floz) 1 ) + ~ cos(3coot - 3floz) 9 (2.20) Thus the nonlinear dielectric polarization has a new frequency component at 3oo0. The wave equation for the electric field (2.10) is derived assuming only the linear component of the dielectric polarization is present. In the presence of a nonlinear dielectric polarization component, it must be modified. We omit the details of how it should be modified but just remark that the solution of the modified equation will have, in general, electric fields at the new frequencies generated as a result of nonlinear dielectric polarization. Thus, in this case, the electric field will have a component at 3o)0. The fiber has a propagation constant at the angular frequency 30)0 of the gener- ated field, which we will denote by fl(3~o0). From (2.20), the electric field generated as a result of nonlinear dielectric polarization at 3co0 has a propagation constant 3fi0, where/30 = fl(oo0) is the propagation constant at the angular frequency coo. In an ideal, dispersionless fiber, I3 = tonic, where the refractive index n is a constant independent of co so that fl(3co0) = 3fl(co0). But in real fibers that have dispersion, n is not a constant, and fl(3oo0) will be very different from 3fl(co0). Because of this mismatch between the two propagation constants~which is usually described as a lack of phase match~ the electric field component at 3o)0 becomes negligible. This 2.4 Nonlinear Effects 85 phase-matching condition will be important in our discussion of four-wave mixing in Section 2.4.8. Neglecting the component at 30)0, the nonlinear dielectric polarization can be written as 3 (3)E2) J)NL(r, t) ~60X E cos(co0t -/30z). (2.21) When the wave equation (2.10) is modified to include the effect of nonlinear dielec- tric polarization and solved for/30 with this expression for the nonlinear dielectric polarization, we get /30- coo / 3 1 + + x(3)E2. From (2.9), n 2 1 + )~(1). Hence ~/ 3 /30 coon 1 + X (3) E 2. c Since X (3) is very small for silica fibers (as we will see), we can approximate this by /30 co0 n+8n3 (3))X E2 9 (2.22) Thus the electric field E (z, t) = E cos(co0t-/30z) is a sinusoid whose phase changes as E2z. This phenomenon is referred to as self-phase modulation. The intensity of the electric field corresponding to a plane wave with amplitude E is I - 89 2. Thus the phase change due to SPM is proportional to the intensity of the electric field. Note that this phase change increases as the propagation distance z increases. Since the relation between/3 and the refractive index n in the linear regime is/3 = con~c, we can also interpret (2.22) as specifying an intensity-dependent refractive index h(E) = n + hi (2.23) for the fiber, in the presence of nonlinearities. Here, I = 89 2 is the intensity of the field, and is measured in units of W/#m 2. The quantity ~ - E0c~2 8n3 X (3) is called the nonlinear index coefficient and varies in the range 2.2-3.4 x 10 -8 #m2/W in silica fiber. We will assume the value 3.2 x 10 -8 #m2/W in the numerical examples we compute. Pulses used in optical communication systems have finite temporal widths, and hence are not monochromatic. They are also not plane waves~that is, they have a transverse ((x, y)-plane) distribution of the electric field that is not constant but 86 PROPAGATION OF SIGNALS IN OPTICAL FIBER dictated by the geometry of the fiber. Nevertheless, the same qualitative effect of self-phase modulation holds for these pulses. In this section, we will give an intuitive explanation of the effect of SPM on pulses. A more quantitative explanation can be found in Sections 2.4.6 and E.2. Because of SPM, the phase of the electric field contains a term that is proportional to the intensity of the electric field. However, because of their finite temporal extent, such pulses do not have a constant intensity for the electric field. Thus the phase shift undergone by different parts of the pulse is different. Note that the sign of the phase shift due to SPM is negative because of the minus sign in the expression for the phase, namely, coot - fl0z. The peak of the pulse undergoes the maximum phase shift in absolute value, and its leading and trailing edges undergo progressively smaller phase shifts. Since the frequency is the derivative of the phase, the trailing edges of the pulse undergo a negative frequency shift, and the leading edges a positive frequency shift. Since the chirp is proportional to the derivative of the frequency, this implies that the chirp factor K is positive. Thus SPM causes positive chirping of pulses. Because of the relatively small value of the nonlinear susceptibility X (3) in optical fiber, the effects of SPM become important only when high powers are used (since E 2 then becomes large). Since the SPM-induced chirp changes the chromatic dispersion effects, at the same power levels, it becomes important to consider SPM effects for shorter pulses (higher bit rates) that are already severely affected by chromatic dispersion. These two points must be kept in mind during the following discussion. We quantify the required powers and pulse durations in Section E.2. The effect of this positive chirping depends on the sign of the GVD parameter f12. Recall that when f12 > 0, the chromatic dispersion is said to be normal, and when f12 < 0, the chromatic dispersion is said to be anomalous (see Figure 2.12). We have seen in Section 2.3 that if the product Kfl2 > 0, the chirp significantly en- hances the pulse-broadening effects of chromatic dispersion. Since the SPM-induced chirp is positive, SPM causes enhanced, monotone, pulse broadening in the normal chromatic dispersion regime. In the anomalous chromatic dispersion regime even the qualitative effect of SPM depends critically on the amount of chromatic dispersion present. When the effects of SPM and chromatic dispersion are nearly equal, but chromatic dispersion dominates, SPM can actually reduce the pulse-broadening ef- fect of chromatic dispersion. This phenomenon can be understood from Figure 2.10, where we saw that a positively chirped pulse undergoes initial compression in the anomalous chromatic dispersion regime. The reason the pulse doesn't broaden con- siderably after this initial compression as described in Problem 2.11 is that the chirp factor is not constant for the entire pulse but dependent on the pulse amplitude (or intensity). This intensity dependence of the chirp factor is what leads to qualitatively different behaviors in the anomalous chromatic dispersion regime, depending on the amount of chromatic dispersion present. When the effects of chromatic dispersion 2.4 Nonlinear Effects 87 and SPM are equal (we make this notion precise in Section E.2), the pulse remains sta- ble, that is, doesn't broaden further, after undergoing some initial broadening. When the amount of chromatic dispersion is negligible, say, around the zero-dispersion wavelength, SPM leads to amplitude modulation of the pulse. 2.4.6 SPM-Induced Chirp for Gaussian Pulses Consider an initially unchirped Gaussian pulse with envelope U(0, r) - e -r2/2. We have assumed a normalized envelope so that the pulse has unit peak amplitude and 1/e-width To = 1. For such a pulse, the parameter ~,Ae LNL 2zchPo is called the nonlinear length. Here P0 is the peak power of the pulse, assumed to be unity in this case. If the link length is comparable to, or greater than, the nonlinear length, the effect of the nonlinearity can be quite severe. In the presence of SPM alone (neglecting chromatic dispersion), this pulse ac- quires a distance-dependent chirp. The initially unchirped pulse and the same pulse with an SPM-induced chirp after the pulse has propagated a distance L = 5LNL are shown in Figure 2.18. In this figure, the center frequency of the pulse is greatly diminished for the purposes of illustration. Using (E.18), the SPM-induced phase change can be calculated to be (L/LNL)e -~2. Using the definition of the instantaneous frequency and chirp factor from Section 2.3, we can calculate the instantaneous frequency of this pulse to be (a) (b) Figure 2.18 Illustration of the SPM-induced chirp. (a) An unchirped Gaussian pulse. (b) The pulse in (a) after it has propagated a distance L 5LNL under the effect of SPM. (Dispersion has been neglected.) 88 PROPAGATION OF SIGNALS IN OPTICAL FIBER 1; , , , | , , , , , , , if 3 o.7 o. ~-Irac~ ~ 0"251002 I O. 4 \o.6 f -3 - (a) f K 3 , | | . . , , , rad/s -3~ -0"5 rad/~ 3 (b) (c) Figure 2.19 The phase (a), instantaneous frequency (b), and chirp (c) of an initially unchirped Gaussian pulse after it has propagated a distance L = L NL. 2L _r2 o)(r) coo q- ~re LNL and the chirp factor of this pulse to be 2L _r2 KSPM(Z') = ~e (1 - 2r2). (2.24) LNL Here coo is the center frequency of the pulse. The SPM-induced phase change, the change, co- coo, in the instantaneous frequency from the center frequency, and the chirp factor are plotted in Figure 2.19, for L LNL. Note that the SPM-induced chirp depends on r. Near the center of the pulse when r ~ 0, KSp M ,~ 2L/LNL. The SPM-induced chirp is thus positive around the center of the pulse and is significant if L is comparable to LNL. For example, if L LNL, the chirp factor at the pulse center is equal to 2. The SPM-induced chirp appears to increase linearly with distance from (2.24). However, this is true only when losses are neglected. To take into account the effect of fiber loss, the expression (2.24) for the SPM-induced chirp should be modified by replacing L by the effective length Le, given by def 1 e-~ Le (2.25) O/ and discussed in Section 2.4.1. Here a is the fiber loss discussed in Section 2.2. Note that Le < 1/or and Le > 1/oe for large L. Thus the SPM-induced chirp at the pulse center is bounded above by 2/LNLO~. At 1.55 #m, a ~ 0.22 dB/km and 1/a ~ 20 km. Thus, regardless of the propagated distance L, the SPM-induced chirp is significant only if LNL is comparable to 20 km. Since we calculated that the nonlinear length 2.4 Nonlinear Effects 89 LNL 384 km for a transmitted power of 1 mW, the SPM-induced effects can be neglected at these power levels. At a transmitted power level of 10 mW, LNL = 38 km so that SPM effects cannot be neglected. 2.4.7 Cross-Phase Modulation In WDM systems, the intensity-dependent nonlinear effects are enhanced since the combined signal from all the channels can be quite intense, even when individual channels are operated at moderate powers. Thus the intensity-dependent phase shift, and consequent chirping, induced by SPM alone is enhanced because of the inten- sities of the signals in the other channels. This effect is referred to as cross-phase modulation (CPM). To understand the effects of CPM, it is sufficient to consider a WDM system with two channels. For such a system, E(r, t) - E1 c0s(091 t -/~lZ) + E2 COS(092t fl2Z). Using (2.19), the nonlinear dielectric polarization is given by 79NL (r, t) ~0X (3) (El cos(091t -/31z) 4- E2 cos(092t - fl2z)) 3 = ~0X (3) [(3e -T+~ (3E2 3E2E12 cos(091t -/31z) 4- ~ -+- + 3E2E 4 cos((2091 -092)t - (2ill fl2)Z) + 4 COS((2092 091)t (2fl2 ill)Z) + 3E E2 4 cos((2091 + 092)t (2/31 + fl2)Z) + 3E E1 4 cos((2092 + 091)t - (2fl2 + ill)Z) +TE~ cos(3~Olt- 3t31z)+ q- E23 cos(3co2t - 3/32z)]. 3E2E21 \ cos(092t -/32z) 2 l (2.26) The terms at 2o91 + 092, 20)2 + 091, 3o91, and 3o92 can be neglected since the phase-matching condition will not be satisfied for these terms owing to the presence of fiber chromatic dispersion. We will discuss the terms at 2o91 - 092 and 2o92 - o91 in . energy from a lower-wavelength signal to a higher-wavelength signal is a fundamental effect that is also the basis of optical amplification and lasers. The energy of a photon at a wavelength. generated as a result of nonlinear dielectric polarization at 3co0 has a propagation constant 3fi0, where/30 = fl(oo0) is the propagation constant at the angular frequency coo. In an ideal,. chromatic dispersion and thus is im- portant to consider for high-bit-rate systems that already have significant chro- matic dispersion limitations. For systems operating at 10 Gb/s and above,