- Email: [email protected]
Pulse Propagation in Optical Fiber
Pulse Propagation in Optical Fiber
N M A T H E M A T I C A L TERMS, chromatic dispersion arises because the propagation constant /3 is not proportional to the angular frequency co, that is, dfl/dco constant (independent of co). dfl/dco is denoted by/31, and/311 is called the group velocity. As we will see, this is the velocity with which a pulse propagates through the fiber (in the absence of chromatic dispersion). Chromatic dispersion is also called
I
group velocity dispersion. If we were to launch a pure monochromatic wave at frequency coo into a length of optical fiber, the magnitude of the (real) electric field vector associated with the wave would be given by IE(r, t ) [ - J(x, y)cos(coot -/3(coo)z).
(E.1)
Here the z coordinate is taken to be along the fiber axis, and J (x, y) is the distribution of the electric field along the fiber cross section and is determined by solving the wave equation. This equation can be derived as follows. For the fundamental mode, the longitudinal component is of the form Ez = 27rJl(x, y)exp(i~Sz). Here Jl(x, y) is a function only of p - v / x 2 - + - y2 due to the cylindrical symmetry of the fiber and is expressible in terms of Bessel functions. The transverse component of the fundamental mode is of the form Ex(Ey) 27rJt(x, y)exp(ijSz), where again Jt(x, y) depends only on v/x2+ y 2 and can be expressed in terms of Bessel functions. Thus, for each of the solutions corresponding to the fundamental mode, we can write E(r, co) - 27r J(x, y)ei~(c~
y),
(E.2) 731
732
PULSE PROPAGATION IN OPTICAL FIBER
where J (x, y) - v/Jl (x, y)2 + ,It ( x , y)2 and the ~ is the unit vector along the direction of l~(r, co). In this equation, we have explicitly written fl as a function of co to emphasize this dependence. In general, J 0 and ~() are also functions of co, but this dependence can be neglected for pulses whose spectral width is much smaller than their center frequency. This condition is satisfied by pulses used in optical communication systems. Equation (E.1) now follows from (E.2) by taking the inverse Fourier transform. This pure monochromatic wave propagates at a velocity o)o/fl(o)o). This is called the phase velocity of the wave. In practice, signals used for optical communication are not monochromatic waves but pulses having a nonzero spectral width. To understand how such pulses propagate, consider a pulse consisting of just two spectral components: one at coo + Ao) and the other at coo - Ao). Further assume that Ao) is small so that we may approximate fl(O)0 -4- AO)) ~ fl0 -+-/~1/NO),
where flo = fl(o)0) and
dt~ 0)=60 0
The magnitude of the electric field vector associated with such a pulse would be given by IE(r, t)l
--
J(x,
y) [cos ((coo + Aco)t -- fl(co0 + Aog)z) + cos ((coo- A~o)t- t~(oJo- A~o)z)]
2J(x, y) cos(Acot -- fil Acoz) cos(coot - floz). This pulse can be viewed in time t and space z as the product of a very rapidly varying sinusoid, namely, cos(co0t - floz), which is also called the phase of the pulse, and a much more slowly varying envelope, namely, cos(Acot - t31Acoz). Note that in this case the phase of the pulse travels at a velocity of co0/ri0, whereas the envelope of the pulse travels at a velocity of 1~ill. The quantity co0/ri0 is called the phase velocity of the pulse, and 1~ill is called the group velocity. In general, pulses used for optical communication can be represented in this manner as the product of a slowly varying envelope function (of z and t), which is usually not a sinusoid, and a sinusoid of the form cos(co0t - floz), where coo is termed the center frequency of the pulse. And just as in the preceding case, the envelope of the pulse propagates at the group velocity, l/ill. This concept can be stated more precisely as follows.
PULSE PROPAGATION IN OPTICAL FIBER
733
Consider a pulse whose shape, or envelope, is described by A(z, t) and whose center frequency is coo. Assume that the pulses have narrow spectral width. By this we mean that most of the energy of the pulse is concentrated in a frequency band whose width is negligible compared to the center frequency coo of the pulse. This assumption is usually satisfied for most pulses used in optical communication systems. With this assumption, it can be shown that the magnitude of the (real) electric field vector associated with such a pulse is IE(r, t)] = J(x, y)gt[A(z, t)e-i(c~176176
(E.3)
where 9t[q] denotes the real part of q (see, for example, [Agr97]). Here/30 is the value of the propagation constant fl at the frequency coo. J(x, y) has the same significance as before. It is mathematically convenient to allow the pulse envelope A(z, t) to be complex valued so that it captures not only the change in the pulse shape during propagation but also any induced phase shifts. Thus if A(z, t) = IA(z, t)lexp(iq)z(z, t)), the phase of the pulse is given by 4)(t) = coot - floz - 4)z (z, t).
(E.4)
To get the description of the actual pulse, we must multiply A(z,t) by exp ( - i (coot - floz)) and take the real part. We will illustrate this in (E.6). Here we have also assumed that the pulse is obtained by modulating a nearly monochromatic source at frequency coo. This means that the frequency spectrum of the optical source has negligible width compared to the frequency spectrum of the pulse. We will consider the effect of relaxing this assumption later in this section. By assuming that the higher derivatives of fi with respect to co are negligible, we can derive the following partial differential equation for the evolution of the pulse shape A(z, t) [Agr97]: 3A
3A
OZ if- fl l
i
32A
--~ -+--~ f12 - - ~
-- O.
(E.5)
Here,
d2fl ] & -
o = oo"
Note that if 13were a linear function of co, that is, f12 --- 0, then A (z, t) = F (t-fllZ), where F is an arbitrary function that satisfies (E.5). Then A(z, t) = A(O, t - f l l z ) for all z and t, and arbitrary pulse shapes propagate without change in shape (and at velocity 1~ill ). In other words,/f the group velocity is independent of co, no broadening of the pulse occurs. Thus f12 is the key parameter governing group velocity or chromatic dispersion. It is termed the group velocity dispersion parameter or, simply, G VD parameter
734
PULSE PROPAGATION IN OPTICAL FIBER
E.1
Propagation of Chirped Gaussian Pulses Mathematically, a chirped Gaussian pulse at z = 0 is described by the equation
G(t)
Aoe
l+ix (TO)2
--
~
=
t( Aoe } (T0)2
cos
e
--icoot ]
o)ot +
K(~O)2)
~
.
(E.6)
The peak amplitude of the pulse is A0. The parameter To determines the width of the pulse. It has the interpretation that it is the half-width of the pulse at the 1/e-intensity point. (The intensity of a pulse is the square of its amplitude.) The chirp factor K determines the degree of chirp of the pulse. From (E.4), the phase of this pulse is
O(t) = coot +
tot2 2#
The instantaneous angular frequency of the pulse is the derivative of the phase and is given by
d,
oo, +
v2
+ Vo2,.
We define the chirp factor of a Gaussian pulse as Tff times the derivative of its instantaneous angular frequency. Thus the chirp factor of the pulse described by (E.6) is K. This pulse is said to be linearly chirped since the instantaneous angular frequency of the pulse increases or decreases linearly with time t, depending on the sign of the chirp factor x. In other words, the chirp factor K is a constant, independent of time t, for linearly chirped pulses. Let A(z, t) denote a chirped Gaussian pulse as a function of time and distance. At z = 0,
A (O, t) - Aoe
(E.7)
If we solve (E.5) for a chirped Gaussian pulse (so the initial condition for this differential equation is that A (0, t) is given by (E.7)), we get
A(z, t) -
AoTo exp ( - (l + iK)(t - fllz)2 ) v/T~ -ifl2z(1 + iK) 2 (T2 -ifi2z(1 + iK)) "
(E.8)
E.2
Nonlinear Effects on Pulse Propagation
735
This can be rewritten in the form A (z, t) --
Aze
k~
(E.9)
] e i4)z
Comparing with (E.6), we see that A(z, t) is also the envelope of a chirped Gaussian pulse for all z > 0, and the chirp factor K remains unchanged. However, the width of this pulse increases as z increases if fl2K > 0. This happens because the parameter governing the pulse width is now -1
T2
-
-
9]
T2
T2_ifl2z(l+ix)
1+
T2
+\~02
,
(E.10)
which monotonically increases with increasing z if fi2K > 0. A measure of the pulse broadening at distance z is the ratio Tz/To. The analytical expression (2.13) for this ratio follows from (E.10).
E.2
Nonlinear Effects on Pulse Propagation So far, we have understood the origins of SPM and CPM and the fact that these effects result in changing the phase of the pulse as a function of its intensity (and the intensity of other pulses at different wavelengths in the case of CPM). To understand the magnitude of this phase change or chirping and how it interacts with chromatic dispersion, we will need to go back and look at the differential equation governing the evolution of the pulse shape as it propagates in the fiber. We will also find that this relationship is important in understanding the fundamentals of solitons in Section 2.5. We will consider pulses for which the magnitude of the associated (real) electric field vector is given by (E.3), which is IE(r, t)l -- J ( x , y ) ~ [ A ( z , t)e-i(~176176 Recall that J (x, y) is the transverse distribution of the electric field of the fundamental mode dictated by the geometry of the fiber, A(z, t) is the complex envelope of the pulse, coo is its center frequency, and ~[.] denotes the real part of its argument. Let A0 denote the peak amplitude of the pulse, and P0 - A2 its peak power.
736
PULSE PROPAGATION IN OPTICAL FIBER
We have seen that the refractive index becomes intensity dependent in the presence of SPM and is given by (2.23) for a plane monochromatic wave. For nonmonochromatic pulses with envelope A propagating in optical fiber, this relation must be modified so that the frequency and intensity-dependent refractive index is now given by h(co, E) = n(co) + hlAl2/Ae.
(E.11)
Here, n(co) is the linear refractive index, which is frequency dependent because of chromatic dispersion, but also intensity independent, and Ae is the effective cross-sectional area of the fiber, typically 50/zm 2 (see Figure 2.15 and the accompanying explanation). The expression for the propagation constant (2.22) must also be similarly modified, and the frequency and intensity-dependent propagation constant is now given by co hlA] 2 fi(co, E) --/3(o9) + -. c Ae
(E.12)
Note that in (E.11) and (E.12) when we use the Value t7 - 3.2 x 10 -8/zm2/W, the intensity of the pulse IAI2 must be expressed in watts (W). We assume this is the case in what follows and will refer to ]AI2 as the power of the pulse (though, strictly speaking, it is only proportional to the power). For convenience, we denote cob c Ae
27rt7 X Ae
and thus fi = r + yIAI 2. Comparing this with (E.11), we see that y bears the same relationship to the propagation constant/3 as the nonlinear index coefficient h does to the refractive index n. Hence, we call 9/the nonlinear propagation coefficient. At a wavelength )~ - 1.55 #m and taking Ae - 50/zm 2, y - 2.6/W-km. To take into account the intensity dependence of the propagation constant, (E.5) must be modified to read 3A 3A i 32A 0-~ -~- fl l --~- ~t_ -~f12 - ~ " - i y I a l 2 A .
(E.13)
i 2 32A incorporates the effect of chromatic dispersion, as In this equation, the term ~/3 discussed in Section 2.3, and the term i yIAIZA incorporates the intensity-dependent phase shift. Since this equation incorporates the effect of chromatic dispersion also, the combined effects of chromatic dispersion and SPM on pulse propagation can be analyzed using this equation as the starting point. These effects are qualitatively different from that of chromatic dispersion or SPM acting alone.
E.2
Nonlinear Effects on Pulse Propagation
737
In order to understand the relative effects of chromatic dispersion and SPM, it is convenient to introduce the following change of variables: t - fllZ r - ~ ,
~=
Z
=
zlfl2l
and '
U -
A
(E.14)
,/eo
In these new variables, (E.13) can be written as OU sgn(fl2) 02 U N2 2 i~ ~- [U[ U - 0, O~e 2 07,"2
(E.15)
where N 2 -- y P o L D --
yPo 1~21/Zo2"
Equation (E.15) is called the nonlinear Schr6dinger equation (NLSE). The change of variables introduced by (E.14) has the following interpretation. Since the pulse propagates with velocity/31 (in the absence of chromatic dispersion), t -/31z is the time axis in a reference frame moving with the pulse. The variable r is the time in this reference frame but in units of To, which is a measure of the pulse width. The variable ~ measures distance in units of the c h r o m a t i c dispersion length LD -- Tff/lfl21, which we already encountered in Section 2.3. The quantity P0 represents the peak power of the pulse, and thus U is the envelope of the pulse normalized to have unit peak power. Note that the quantity 1/yP0 also has the dimensions of length; we call it the nonlinear length and denote it by L NL. Using ?, = 2.6/W-km and P0 = 1 mW, we get LNL -- 384 kin. If the pulse power P0 is increased to 10 roW, the nonlinear length decreases to 38 km. The nonlinear length serves as a convenient normalizing measure for the distance z in discussing nonlinear effects, just as the chromatic dispersion length does for the effects of chromatic dispersion. Thus the effect of SPM on pulses can be neglected for pulses propagating over distances z << LNL. Then we can write the quantity N introduced in the NLSE as N 2 ---- L D / L N L . Thus it is the ratio of the chromatic dispersion and nonlinear lengths. When N << 1, the nonlinear length is much larger than the chromatic dispersion length so that the nonlinear effects can be neglected compared to those of chromatic dispersion. This amounts to saying that the third term (the one involving N) in the NLSE can be neglected. In this case, the NLSE reduces to (E.5) for the evolution of pulses in the presence of chromatic dispersion alone, with the change of variables given by (E.14). The NLSE serves as the starting point for the discussion of the combined effects of GVD and SPM. For arbitrary values of N, the NLSE has to be solved numerically. These numerical solutions are important tools for the understanding of the combined
738
PULSE PROPAGATION IN OPTICAL FIBER
effects of chromatic dispersion and nonlinearities on pulses and are discussed extensively in [Agr95]. The qualitative description of these solutions in both the normal and anomalous chromatic dispersion regimes is discussed in Section 2.4.5. We can use (E.13) to estimate the SPM-induced chirp for Gaussian pulses. To do this, we neglect the chromatic dispersion term and consider the equation OA OA _ OZ ]- f l l - ~ iyIAI2A. _
(E.16)
By using the variables r and U introduced in (E.14) instead of t and A, and LNL -(y P0) -1, this reduces to OU i = ~IuIzu. Oz LNL
(E.17)
Note that we have not used the change of variable ~ for z since L D is infinite when chromatic dispersion is neglected. This equation has the solution U(Z, rs) - U(O, r)e izlU(O'r)12/LNL .
(E.18)
Thus the SPM causes a phase change but no change in the envelope of the pulse. Note that the initial pulse envelope U(0, r) is arbitrary; so this is true for all pulse shapes. Thus SPM by itself leads only to chirping, regardless of the pulse shape; it is chromatic dispersion that is responsible for pulse broadening. The SPM-induced chirp, however, modifies the pulse-broadening effects of chromatic dispersion.
E.3
Soliton Pulse Propagation In the anomalous chromatic dispersion regime (1.55 #m band for standard single-mode fiber and most dispersion-shifted fibers), the GVD parameter f12 is negative. Thus sgn(fl2) - - 1 , and the NLSE of (E.15) can be written as OU 1 O2U N2 i O--~-+ ~ - ~ 2 + IUI2U - 0 .
(E.19)
An interesting phenomenon occurs in this anomalous chromatic dispersion regime when N is an integer. In this case, the modified NLSE (E.19) can be solved analytically, and the resulting pulse envelope has an amplitude that is independent of ~e (for N - 1) or periodic in ~e (for N _> 2). This implies that these pulses propagate with no change in their widths or with a periodic change in their widths. The solutions of this equation are termed solitons, and N is called the order of the soliton.
References
/39
It can be verified that the solution of (E.19) corresponding to N = 1 is U(~, r) = ei~/2sechr.
(E.20)
The pulse corresponding to this envelope is called the fundamental sol#on. The fundamental soliton pulse and its envelope are sketched in Figure 2.25(a) and (b), respectively. (As in the case of chirped Gaussian pulses in Section 2.3, the frequency of the pulse is shown vastly diminished for the purposes of illustration.) Note that (in a reference frame moving with the pulse) the magnitude of the fundamental soliton pulse envelope, or the pulse shape, does not change with the distance coordinate z. However, the pulse acquires a phase shift that is linear in z as it propagates. Recall that the order of the soliton, N, is defined by N 2 = yPoLD =
zP0 1,821/ T~ "
Since Z and/32 are fixed for a given fiber and operating wavelength, for a fixed soliton order, the peak power P0 of the pulse increases as the pulse width To decreases. Since operation at very high bit rates requires narrow pulses, this also implies that large peak powers are necessary in soliton communication systems. It can also be verified that the solution of (E.19) corresponding to N = 2 is U(~, r) - 4e i~/2 cosh3r + 3 c~ . cosh 4r + 4 cosh 2r + 3 cos 4~
(E.21)
The magnitude of this normalized pulse envelope is sketched in Figure E. 1 as a function of ~ and r. The periodicity of the pulse envelope with respect to ~ can be clearly seen from this plot. In each period, the pulse envelope first undergoes compression due to the positive chirping induced by SPM and then undergoes broadening, finally regaining its original shape.
Further Reading Pulse propagation is covered in detail in [Agr95]. The classic papers by Marcuse [Mar80, Mar81] are a must-read for anyone wishing to dig deeper into the mathematics of Gaussian and chirped Gaussian pulse propagation.
References [Agr95] G.P. Agrawal. Nonlinear Fiber Optics, 2nd edition. Academic Press, San Diego, CA, 1995.
740
PULSE PROPAGATION IN OPTICAL FIBER
~
/2
~/8 ~/ 0
Figure E.1 The magnitude of the pulse envelope of the second-order soliton. [Agr97] G.P. Agrawal. Fiber-Optic Communication Systems. John Wiley, New York, 1997. [Mar80] D. Marcuse. Pulse distortion in single-mode fibers. Applied Optics, 19:1653-1660, 1980. [Mar81] D. Marcuse. Pulse distortion in single-mode fibers. 3: Chirped pulses. Applied Optics, 20:3573-3579, 1981.
N M A T H E M A T I C A L TERMS, chromatic dispersion arises because the propagation constant /3 is not proportional to the angular frequency co, that is, dfl/dco constant (independent of co). dfl/dco is denoted by/31, and/311 is called the group velocity. As we will see, this is the velocity with which a pulse propagates through the fiber (in the absence of chromatic dispersion). Chromatic dispersion is also called
I
group velocity dispersion. If we were to launch a pure monochromatic wave at frequency coo into a length of optical fiber, the magnitude of the (real) electric field vector associated with the wave would be given by IE(r, t ) [ - J(x, y)cos(coot -/3(coo)z).
(E.1)
Here the z coordinate is taken to be along the fiber axis, and J (x, y) is the distribution of the electric field along the fiber cross section and is determined by solving the wave equation. This equation can be derived as follows. For the fundamental mode, the longitudinal component is of the form Ez = 27rJl(x, y)exp(i~Sz). Here Jl(x, y) is a function only of p - v / x 2 - + - y2 due to the cylindrical symmetry of the fiber and is expressible in terms of Bessel functions. The transverse component of the fundamental mode is of the form Ex(Ey) 27rJt(x, y)exp(ijSz), where again Jt(x, y) depends only on v/x2+ y 2 and can be expressed in terms of Bessel functions. Thus, for each of the solutions corresponding to the fundamental mode, we can write E(r, co) - 27r J(x, y)ei~(c~
y),
(E.2) 731
732
PULSE PROPAGATION IN OPTICAL FIBER
where J (x, y) - v/Jl (x, y)2 + ,It ( x , y)2 and the ~ is the unit vector along the direction of l~(r, co). In this equation, we have explicitly written fl as a function of co to emphasize this dependence. In general, J 0 and ~() are also functions of co, but this dependence can be neglected for pulses whose spectral width is much smaller than their center frequency. This condition is satisfied by pulses used in optical communication systems. Equation (E.1) now follows from (E.2) by taking the inverse Fourier transform. This pure monochromatic wave propagates at a velocity o)o/fl(o)o). This is called the phase velocity of the wave. In practice, signals used for optical communication are not monochromatic waves but pulses having a nonzero spectral width. To understand how such pulses propagate, consider a pulse consisting of just two spectral components: one at coo + Ao) and the other at coo - Ao). Further assume that Ao) is small so that we may approximate fl(O)0 -4- AO)) ~ fl0 -+-/~1/NO),
where flo = fl(o)0) and
dt~ 0)=60 0
The magnitude of the electric field vector associated with such a pulse would be given by IE(r, t)l
--
J(x,
y) [cos ((coo + Aco)t -- fl(co0 + Aog)z) + cos ((coo- A~o)t- t~(oJo- A~o)z)]
2J(x, y) cos(Acot -- fil Acoz) cos(coot - floz). This pulse can be viewed in time t and space z as the product of a very rapidly varying sinusoid, namely, cos(co0t - floz), which is also called the phase of the pulse, and a much more slowly varying envelope, namely, cos(Acot - t31Acoz). Note that in this case the phase of the pulse travels at a velocity of co0/ri0, whereas the envelope of the pulse travels at a velocity of 1~ill. The quantity co0/ri0 is called the phase velocity of the pulse, and 1~ill is called the group velocity. In general, pulses used for optical communication can be represented in this manner as the product of a slowly varying envelope function (of z and t), which is usually not a sinusoid, and a sinusoid of the form cos(co0t - floz), where coo is termed the center frequency of the pulse. And just as in the preceding case, the envelope of the pulse propagates at the group velocity, l/ill. This concept can be stated more precisely as follows.
PULSE PROPAGATION IN OPTICAL FIBER
733
Consider a pulse whose shape, or envelope, is described by A(z, t) and whose center frequency is coo. Assume that the pulses have narrow spectral width. By this we mean that most of the energy of the pulse is concentrated in a frequency band whose width is negligible compared to the center frequency coo of the pulse. This assumption is usually satisfied for most pulses used in optical communication systems. With this assumption, it can be shown that the magnitude of the (real) electric field vector associated with such a pulse is IE(r, t)] = J(x, y)gt[A(z, t)e-i(c~176176
(E.3)
where 9t[q] denotes the real part of q (see, for example, [Agr97]). Here/30 is the value of the propagation constant fl at the frequency coo. J(x, y) has the same significance as before. It is mathematically convenient to allow the pulse envelope A(z, t) to be complex valued so that it captures not only the change in the pulse shape during propagation but also any induced phase shifts. Thus if A(z, t) = IA(z, t)lexp(iq)z(z, t)), the phase of the pulse is given by 4)(t) = coot - floz - 4)z (z, t).
(E.4)
To get the description of the actual pulse, we must multiply A(z,t) by exp ( - i (coot - floz)) and take the real part. We will illustrate this in (E.6). Here we have also assumed that the pulse is obtained by modulating a nearly monochromatic source at frequency coo. This means that the frequency spectrum of the optical source has negligible width compared to the frequency spectrum of the pulse. We will consider the effect of relaxing this assumption later in this section. By assuming that the higher derivatives of fi with respect to co are negligible, we can derive the following partial differential equation for the evolution of the pulse shape A(z, t) [Agr97]: 3A
3A
OZ if- fl l
i
32A
--~ -+--~ f12 - - ~
-- O.
(E.5)
Here,
d2fl ] & -
o = oo"
Note that if 13were a linear function of co, that is, f12 --- 0, then A (z, t) = F (t-fllZ), where F is an arbitrary function that satisfies (E.5). Then A(z, t) = A(O, t - f l l z ) for all z and t, and arbitrary pulse shapes propagate without change in shape (and at velocity 1~ill ). In other words,/f the group velocity is independent of co, no broadening of the pulse occurs. Thus f12 is the key parameter governing group velocity or chromatic dispersion. It is termed the group velocity dispersion parameter or, simply, G VD parameter
734
PULSE PROPAGATION IN OPTICAL FIBER
E.1
Propagation of Chirped Gaussian Pulses Mathematically, a chirped Gaussian pulse at z = 0 is described by the equation
G(t)
Aoe
l+ix (TO)2
--
~
=
t( Aoe } (T0)2
cos
e
--icoot ]
o)ot +
K(~O)2)
~
.
(E.6)
The peak amplitude of the pulse is A0. The parameter To determines the width of the pulse. It has the interpretation that it is the half-width of the pulse at the 1/e-intensity point. (The intensity of a pulse is the square of its amplitude.) The chirp factor K determines the degree of chirp of the pulse. From (E.4), the phase of this pulse is
O(t) = coot +
tot2 2#
The instantaneous angular frequency of the pulse is the derivative of the phase and is given by
d,
oo, +
v2
+ Vo2,.
We define the chirp factor of a Gaussian pulse as Tff times the derivative of its instantaneous angular frequency. Thus the chirp factor of the pulse described by (E.6) is K. This pulse is said to be linearly chirped since the instantaneous angular frequency of the pulse increases or decreases linearly with time t, depending on the sign of the chirp factor x. In other words, the chirp factor K is a constant, independent of time t, for linearly chirped pulses. Let A(z, t) denote a chirped Gaussian pulse as a function of time and distance. At z = 0,
A (O, t) - Aoe
(E.7)
If we solve (E.5) for a chirped Gaussian pulse (so the initial condition for this differential equation is that A (0, t) is given by (E.7)), we get
A(z, t) -
AoTo exp ( - (l + iK)(t - fllz)2 ) v/T~ -ifl2z(1 + iK) 2 (T2 -ifi2z(1 + iK)) "
(E.8)
E.2
Nonlinear Effects on Pulse Propagation
735
This can be rewritten in the form A (z, t) --
Aze
k~
(E.9)
] e i4)z
Comparing with (E.6), we see that A(z, t) is also the envelope of a chirped Gaussian pulse for all z > 0, and the chirp factor K remains unchanged. However, the width of this pulse increases as z increases if fl2K > 0. This happens because the parameter governing the pulse width is now -1
T2
-
-
9]
T2
T2_ifl2z(l+ix)
1+
T2
+\~02
,
(E.10)
which monotonically increases with increasing z if fi2K > 0. A measure of the pulse broadening at distance z is the ratio Tz/To. The analytical expression (2.13) for this ratio follows from (E.10).
E.2
Nonlinear Effects on Pulse Propagation So far, we have understood the origins of SPM and CPM and the fact that these effects result in changing the phase of the pulse as a function of its intensity (and the intensity of other pulses at different wavelengths in the case of CPM). To understand the magnitude of this phase change or chirping and how it interacts with chromatic dispersion, we will need to go back and look at the differential equation governing the evolution of the pulse shape as it propagates in the fiber. We will also find that this relationship is important in understanding the fundamentals of solitons in Section 2.5. We will consider pulses for which the magnitude of the associated (real) electric field vector is given by (E.3), which is IE(r, t)l -- J ( x , y ) ~ [ A ( z , t)e-i(~176176 Recall that J (x, y) is the transverse distribution of the electric field of the fundamental mode dictated by the geometry of the fiber, A(z, t) is the complex envelope of the pulse, coo is its center frequency, and ~[.] denotes the real part of its argument. Let A0 denote the peak amplitude of the pulse, and P0 - A2 its peak power.
736
PULSE PROPAGATION IN OPTICAL FIBER
We have seen that the refractive index becomes intensity dependent in the presence of SPM and is given by (2.23) for a plane monochromatic wave. For nonmonochromatic pulses with envelope A propagating in optical fiber, this relation must be modified so that the frequency and intensity-dependent refractive index is now given by h(co, E) = n(co) + hlAl2/Ae.
(E.11)
Here, n(co) is the linear refractive index, which is frequency dependent because of chromatic dispersion, but also intensity independent, and Ae is the effective cross-sectional area of the fiber, typically 50/zm 2 (see Figure 2.15 and the accompanying explanation). The expression for the propagation constant (2.22) must also be similarly modified, and the frequency and intensity-dependent propagation constant is now given by co hlA] 2 fi(co, E) --/3(o9) + -. c Ae
(E.12)
Note that in (E.11) and (E.12) when we use the Value t7 - 3.2 x 10 -8/zm2/W, the intensity of the pulse IAI2 must be expressed in watts (W). We assume this is the case in what follows and will refer to ]AI2 as the power of the pulse (though, strictly speaking, it is only proportional to the power). For convenience, we denote cob c Ae
27rt7 X Ae
and thus fi = r + yIAI 2. Comparing this with (E.11), we see that y bears the same relationship to the propagation constant/3 as the nonlinear index coefficient h does to the refractive index n. Hence, we call 9/the nonlinear propagation coefficient. At a wavelength )~ - 1.55 #m and taking Ae - 50/zm 2, y - 2.6/W-km. To take into account the intensity dependence of the propagation constant, (E.5) must be modified to read 3A 3A i 32A 0-~ -~- fl l --~- ~t_ -~f12 - ~ " - i y I a l 2 A .
(E.13)
i 2 32A incorporates the effect of chromatic dispersion, as In this equation, the term ~/3 discussed in Section 2.3, and the term i yIAIZA incorporates the intensity-dependent phase shift. Since this equation incorporates the effect of chromatic dispersion also, the combined effects of chromatic dispersion and SPM on pulse propagation can be analyzed using this equation as the starting point. These effects are qualitatively different from that of chromatic dispersion or SPM acting alone.
E.2
Nonlinear Effects on Pulse Propagation
737
In order to understand the relative effects of chromatic dispersion and SPM, it is convenient to introduce the following change of variables: t - fllZ r - ~ ,
~=
Z
=
zlfl2l
and '
U -
A
(E.14)
,/eo
In these new variables, (E.13) can be written as OU sgn(fl2) 02 U N2 2 i~ ~- [U[ U - 0, O~e 2 07,"2
(E.15)
where N 2 -- y P o L D --
yPo 1~21/Zo2"
Equation (E.15) is called the nonlinear Schr6dinger equation (NLSE). The change of variables introduced by (E.14) has the following interpretation. Since the pulse propagates with velocity/31 (in the absence of chromatic dispersion), t -/31z is the time axis in a reference frame moving with the pulse. The variable r is the time in this reference frame but in units of To, which is a measure of the pulse width. The variable ~ measures distance in units of the c h r o m a t i c dispersion length LD -- Tff/lfl21, which we already encountered in Section 2.3. The quantity P0 represents the peak power of the pulse, and thus U is the envelope of the pulse normalized to have unit peak power. Note that the quantity 1/yP0 also has the dimensions of length; we call it the nonlinear length and denote it by L NL. Using ?, = 2.6/W-km and P0 = 1 mW, we get LNL -- 384 kin. If the pulse power P0 is increased to 10 roW, the nonlinear length decreases to 38 km. The nonlinear length serves as a convenient normalizing measure for the distance z in discussing nonlinear effects, just as the chromatic dispersion length does for the effects of chromatic dispersion. Thus the effect of SPM on pulses can be neglected for pulses propagating over distances z << LNL. Then we can write the quantity N introduced in the NLSE as N 2 ---- L D / L N L . Thus it is the ratio of the chromatic dispersion and nonlinear lengths. When N << 1, the nonlinear length is much larger than the chromatic dispersion length so that the nonlinear effects can be neglected compared to those of chromatic dispersion. This amounts to saying that the third term (the one involving N) in the NLSE can be neglected. In this case, the NLSE reduces to (E.5) for the evolution of pulses in the presence of chromatic dispersion alone, with the change of variables given by (E.14). The NLSE serves as the starting point for the discussion of the combined effects of GVD and SPM. For arbitrary values of N, the NLSE has to be solved numerically. These numerical solutions are important tools for the understanding of the combined
738
PULSE PROPAGATION IN OPTICAL FIBER
effects of chromatic dispersion and nonlinearities on pulses and are discussed extensively in [Agr95]. The qualitative description of these solutions in both the normal and anomalous chromatic dispersion regimes is discussed in Section 2.4.5. We can use (E.13) to estimate the SPM-induced chirp for Gaussian pulses. To do this, we neglect the chromatic dispersion term and consider the equation OA OA _ OZ ]- f l l - ~ iyIAI2A. _
(E.16)
By using the variables r and U introduced in (E.14) instead of t and A, and LNL -(y P0) -1, this reduces to OU i = ~IuIzu. Oz LNL
(E.17)
Note that we have not used the change of variable ~ for z since L D is infinite when chromatic dispersion is neglected. This equation has the solution U(Z, rs) - U(O, r)e izlU(O'r)12/LNL .
(E.18)
Thus the SPM causes a phase change but no change in the envelope of the pulse. Note that the initial pulse envelope U(0, r) is arbitrary; so this is true for all pulse shapes. Thus SPM by itself leads only to chirping, regardless of the pulse shape; it is chromatic dispersion that is responsible for pulse broadening. The SPM-induced chirp, however, modifies the pulse-broadening effects of chromatic dispersion.
E.3
Soliton Pulse Propagation In the anomalous chromatic dispersion regime (1.55 #m band for standard single-mode fiber and most dispersion-shifted fibers), the GVD parameter f12 is negative. Thus sgn(fl2) - - 1 , and the NLSE of (E.15) can be written as OU 1 O2U N2 i O--~-+ ~ - ~ 2 + IUI2U - 0 .
(E.19)
An interesting phenomenon occurs in this anomalous chromatic dispersion regime when N is an integer. In this case, the modified NLSE (E.19) can be solved analytically, and the resulting pulse envelope has an amplitude that is independent of ~e (for N - 1) or periodic in ~e (for N _> 2). This implies that these pulses propagate with no change in their widths or with a periodic change in their widths. The solutions of this equation are termed solitons, and N is called the order of the soliton.
References
/39
It can be verified that the solution of (E.19) corresponding to N = 1 is U(~, r) = ei~/2sechr.
(E.20)
The pulse corresponding to this envelope is called the fundamental sol#on. The fundamental soliton pulse and its envelope are sketched in Figure 2.25(a) and (b), respectively. (As in the case of chirped Gaussian pulses in Section 2.3, the frequency of the pulse is shown vastly diminished for the purposes of illustration.) Note that (in a reference frame moving with the pulse) the magnitude of the fundamental soliton pulse envelope, or the pulse shape, does not change with the distance coordinate z. However, the pulse acquires a phase shift that is linear in z as it propagates. Recall that the order of the soliton, N, is defined by N 2 = yPoLD =
zP0 1,821/ T~ "
Since Z and/32 are fixed for a given fiber and operating wavelength, for a fixed soliton order, the peak power P0 of the pulse increases as the pulse width To decreases. Since operation at very high bit rates requires narrow pulses, this also implies that large peak powers are necessary in soliton communication systems. It can also be verified that the solution of (E.19) corresponding to N = 2 is U(~, r) - 4e i~/2 cosh3r + 3 c~ . cosh 4r + 4 cosh 2r + 3 cos 4~
(E.21)
The magnitude of this normalized pulse envelope is sketched in Figure E. 1 as a function of ~ and r. The periodicity of the pulse envelope with respect to ~ can be clearly seen from this plot. In each period, the pulse envelope first undergoes compression due to the positive chirping induced by SPM and then undergoes broadening, finally regaining its original shape.
Further Reading Pulse propagation is covered in detail in [Agr95]. The classic papers by Marcuse [Mar80, Mar81] are a must-read for anyone wishing to dig deeper into the mathematics of Gaussian and chirped Gaussian pulse propagation.
References [Agr95] G.P. Agrawal. Nonlinear Fiber Optics, 2nd edition. Academic Press, San Diego, CA, 1995.
740
PULSE PROPAGATION IN OPTICAL FIBER
~
/2
~/8 ~/ 0
Figure E.1 The magnitude of the pulse envelope of the second-order soliton. [Agr97] G.P. Agrawal. Fiber-Optic Communication Systems. John Wiley, New York, 1997. [Mar80] D. Marcuse. Pulse distortion in single-mode fibers. Applied Optics, 19:1653-1660, 1980. [Mar81] D. Marcuse. Pulse distortion in single-mode fibers. 3: Chirped pulses. Applied Optics, 20:3573-3579, 1981.