Module 5 – Solitons in Optical Fibers - Login ERC...

1 The authors would like to acknowledge support from the National Science Foundation through CIAN NSF ERC under grant #EEC-0812072

Module 5 Solitons in Optical Fibers

Dr. E. M. Wright Professor, College of Optics, University of Arizona

Dr. E. M. Wright is a Professor of Optical Sciences and Physic sin the University of Arizona. Research activity within Dr. Wright's group is centered on the theory and simulation of propagation in nonlinear optical media including optical fibers, integrated optics, and bulk media. Active areas of current research include nonlinear propagation of light strings in the atmosphere, and the propagation of novel laser fields for applications in optical manipulation and trapping of particles, and quantum nonlinear optics.

Email: [email protected]

Introduction In modules 4 and 5 we considered the effects of linear GVD and nonlinear SPM in isolation and it is now time to look at how these effects combine in optical fibers. In section 4.6 of module 4 an intuitive argument was given that the combination of anomalous GVD and SPM in fibers could give rise to optical solitons which represent nonlinear pulsed solutions that do not spread under propagation. In the current module we shall explore optical solitons in fibers in detail, and in particular we shall use numerical simulations to convey the key ideas. The learning outcomes for this module include

The student will be conversant with the Beam Propagation method that is used to model pulse propagation in the presence of both GVD and SPM.

The student will be able to evaluate the properties of the fundamental bright optical solitons in fibers.

The student will be conversant with the higher-order fiber solitons of the NLSE and to model their propagation numerically.

The student will be conversant with the ideas surrounding the effect of losses on fundamental soliton propagation and approaches to dealing with loss including periodic amplification and dispersion-managed solitons.

The student will be able to asses the use of modulational instability in fibers to create trains of soliton pulses.

mailto:[email protected]


5.1 Slowly varying envelope equation with GVD and SPM Our starting point for discussing optical fiber propagation in the presence of both GVD and SPM is the slowly varying envelope Equation 4.15 from module 4 that we write in the form

222

2 | | ,2 2iA A i A A A

z T = +

(Equation 5.1)

where the first term on the right-hand-side of Equation 5.1 describes linear GVD, the second term SPM, and the third term linear absorption. This is the most general form of the slowly varying envelope equation that we shall consider, and it is often referred to as the nonlinear Schrdinger equation (NLSE) due to its formal similarity to the Schrdinger equation of quantum mechanics. We next give a discussion of how the NLSE in Equation 5.1 is solved numerically using the so-called Beam Propagation Method or Split Step Method. For this purpose we shall neglect linear absorption and set 0 = , in which case we may write Equation 5.1 as

( , ) ( ) ( , ) ,A z T D N A z Tz

= +

(Equation 5.2)

with 2

22

2

iDT

=

and 2 | |N i A= the dispersion and nonlinear operators describing GVD and

SPM introduced in Equations. 3.4 and 4.20. The formal solution of Equation 5.2 can then be written as ( )( , ) ( , ) .D N hA z h T e A z T++ =

(Equation 5.3)

In this formal solution the operator ( )D N he + acts or operates on the initial pulse ( , )A z T to produce the pulse ( , )A z h T+ after a propagation distance dz h= . We have already studied the

actions of the operators Dhe and Nhe separately: Dhe means propagate a distance h under the action of GVD alone, and Nhe means propagate a distance h under the action of SPM alone. It is tempting on the basis of Equation 4.20 to write the formal solution as ( , ) ( , ) ( , ) ,Dh Nh Nh DhA z h T e e A z T e e A z T+ = =

(Equation 5.4) so that propagation under the combined actions of GVD and SPM may be broken into steps involving only GVD and SPM via the operators Dhe and Nhe separately. However, whilst


Equation 5.4 would be correct if D and N were simple numbers it is not correct since they are in fact operators that do not commute, that is, the order in which they appear matters, DN ND . However, it turns out that Equation 5.4 provides an excellent approximation in the limit of small steps h, yielding the propagation algorithm

( , )

( , ) ( , ) .Nh Dh

A z T

A z h T e e A z T

+

(Equation 5.5)

That is, for a small propagation distance h one takes the initial field ( , )A z T and propagates it a distance h under the action of GVD alone to obtain '( , )A z T as shown in Equation 5.5. The method to evolve the field under the action of GVD alone was discussed in module 3. Next one takes the field '( , )A z T and propagates it a distance h under the action of SPM alone, the method to evolve the field under the action of SPM alone having been discussed in module 4. The net result of these two steps in succession yields an excellent approximation to the propagated field ( , )A z h T+ . This approach to breaking the propagation into small steps of GVD and SPM separately is called the Beam Propagation Method (BPM) or split step method. One can propagate over an extended distance L=Nzdz by breaking the propagation into small steps of length dz=h and repeating the above procedure Nz times. This scheme is illustrated in Figure 5.1. More information on the BPM applied to pulse propagation is given in Sec. 2.4 of Ref. [1].

Figure 5.1 This figure illustrates the scheme used to propagate a field over an extended distance using

the BPM. The propagation is broken up into small steps of length h with only dispersion (GVD) or nonlinearity (SPM).


The code SolitonVisual.m on the next page implements the BPM for propagation in an optical fiber. In many respects it is like the previous code PulseVisual.m for linear propagation and also SPMpulseVisual.m for nonlinear propagation. The key difference is that within the for loop, which runs through the Nz individual small steps of length dz=h, there are steps of both linear propagation involving GVD and nonlinear propagation involving SPM. You should back-engineer this code and make sure that understand the ingredients in it based on the material in modules 3 and 4. For additional reading on the material covered in this section see Sec. 2.4 of Ref. [1].


function [mov] = SolitonVisual; close all % % Input parameters % Tmax = 100.00; % temporal grid size in ps N = 0512; % number of time grid points T0 = 5.0; % pulse duration in ps P0 = 1.0; % peak input power in W gamma = 1.0; % nonlinear parameter in W^{-1}/km beta2 = -25.0; % GVD parameter in W^{-1}/km L = pi/2; % fiber length in km Nz = 400; % number of points along z % % Set up time and frequency grids % v = linspace(0,N-1,N); dT = Tmax/N; T = -Tmax/2 + v*dT; % Time grid dOmega = 2*pi/Tmax; p = find(v > floor(N/2)); % These three lines set up the frequency v(p) = v(p)-N; % grid in the storage form employed by Omega = v*dOmega; % Matlab. % A = sqrt(P0)*2./(exp(T/T0)+exp(-T/T0)); % soliton pulse A0 = A; % copy of the input pulse % zval = linspace(0,L,Nz); zv = linspace(0,L,Nz+1); dz = L/Nz; Prof(:,1) = abs(A).^2; % for iz = 2:Nz+1 A = ifft(exp(1i*beta2*Omega.^2*dz/2).*fft(A)); A = A.*exp(1i*gamma*abs(A).^2*dz); Prof(:,iz) = abs(A).^2; end % % Plot the output pulse % figure imagesc(zv,T,Prof) colorbar set(gca,'FontSize',15); xlabel('z (km)'); ylabel('T (ps)'); title(' '); % figure plot(T,abs(A).^2,T,abs(A0).^2,'--'); set(gca,'FontSize',15); xlabel('T (ps)'); ylabel('|A(z,T)|^2'); xlim([-40,40]); title(' ');


5.2 The fundamental bright soliton in fibers In the study of nonlinear propagation in optical fibers it is advantageous to write the NLSE in Equation 5.1 in scaled or dimensionless form, so that general features can be distinguished from those relying on specific parameter values. In particular, we introduce scaled variables 0 0 ,( , ) ( , ) , , ,D DA z T P U z T T T z L L = = = =

(Equation 5.6)

where 0P is the peak input power, 0T the input pulse width, and 2

0 2/ | |DL T = the corresponding dispersion length. Thus, is the time in units of 0T , is the propagation distance in units of the dispersion length DL , and is the absorption coefficient for a dispersion length. In terms of these dimensionless variables the NLSE in Equation 5.1 becomes

222

22

sg |n( |2

) ,2

UU ii N U U A

=

(Equation 5.7)

where 2( )sgn is +1 for normal dispersion, 0 in the absence of GVD, and -1 for anomalous dispersion, and the coefficient 2N is given by

2

2 0 0

2

.| |

D

NL

PTLNL

= =

(Equation 5.8)

Here we note that the parameter 2N is physically the ratio of the linear dispersion length 2

0 2/ | |D TL = to the nonlinear length 1

0( )NLL P= , so that 1N implies that D NLL L>> and linear SPM dominates. Of particular interest here is the parameter region 1N , which means N is of order unity, so that

D NLL L , and GVD and SPM are on similar footing. We are now in a position to introduce the optical soliton solution for propagation in an optical fiber. There is actually a family of soliton solutions and we first concentrate on the fundamental solution and discuss higher-order solitons in the next section. (In common usage in nonlinear fiber optics the term soliton implies the fundamental soliton unless otherwise stated.) The soliton arises as a solution of Equation 5.7 in the limit of zero absorption 0 . In particular, the fundamental soliton occurs for anomalous dispersion 2( ) 0sgn < and 1N = , and is given explicitly by

/2 /2( 2, ) .(

sec ( ))

i i

e eU h e e

= +=

(Equation 5.9)


We first note that the fundamental soliton is a pulsed solution of hyperbolic-secant form, so that it represents a localized pulse with a single peak centered at 0 = , often referred to as a bright soliton. The remarkable feature of the fundamental optical soliton is that if it is launched at the input to the fiber it maintains its shape and peak power for all propagation distances, that is, it is a nonlinear solution that is immune to the usual GVD-induced broadening. The fact that temporal optical solitons do not spread as they propagate explains why they are attractive candidates as information bits in optical communications links. In particular, in conventional linear optical fiber links GVD-induced pulse broadening will set a limit on how closely spaced pulses can be without overlapping for a given propagation distance, and this will limit the bandwidth of the link. (For a discussion of how GVD imposes a limitation on the operation of optical fiber links see undergraduate module 14, Sec. 14.3.2.) In contrast, since solitons are immune to pulse preading the bandwidth of a soliton based optical fiber link will ideally be limited by how short the pulse width of the solitons can be made and not the propagation distance. For purposes of calculating the properties of on optical soliton in a given situation it is necessary to return to the original dimensional quantities A,z,T. To specify when we are considering the fundamental soliton we shall use the subscript s as in 0 sT T= and 0 sP P= . Then the fundamental soliton in Equation 5.9 becomes in dimensional form

0/(4 )( , ) sec ( / ) ,i z Ls sA z T P h T T e=

(Equation 5.10)

where 0 / 2DL L= is called the soliton period. At this point we must recall that for the fundamental soliton 1N = , so that from Equation 5.9 the soliton pulse width 0 sT T= and peak power 0 sP P= must be related by

2

2

1 .| |

s sPT

=

(Equation 5.11)

Thus the soliton width sT may take on a range of values as long as the peak power sP is suitably adjusted according to Equation 5.11. The solid line in Figure 5.2 shows an example of the power profile of the fundamental soliton solution in Equation 5.10 for parameters 2 25 = ps

2/km, 1sP = W, 11 /W km = , 5sT = ps,

so that 1N = and the soliton period is 0 1.57L = km. Here we see that the soliton profile indeed has a single localized peak. You can verify for yourselves that if this soliton profile is used as an initial condition in the Matlab code SolitonVisual.m the pulse profile will not change with propagation distance. In contrast, the dashed line in Figure 5.2 shows the result of running SolitonVisual.m with the soliton initial condition but with the SPM turned of so that the propagation is linear and includes only GVD. The dashed line is computed for a propagation distance corresponding to the soliton period 0 1.57L = km in this example. This example


illustrates two points: First the non-spreading nature of the soliton is truly due to the combination of linear anomalous dispersion and nonlinear SPM, if SPM is removed the pulse will spread. Second, the soliton period is a good characteristic length scale over which to to test if a given pulse is a soliton in an optical fiber with given parameters. More specifically, if I want to test if a pulse is a soliton I should make sure it propagates unchanging over distances greater than the soliton period, since if the pulse was not a soliton it would exhibit pulse-broadening over such distances as illustrated in Figure 5.2. This test is often used in experiments to establish that the pulses involved are indeed solitons. This was the case in the first experiment reporting solitons in optical fibers which is included as Ref. [3] below. There they showed that at low input powers their input pulses suffered pulse broadening, whereas at higher input power the pulse broadening was reduced as expected for a soliton.

Figure 5.2 The solid line shows the soliton power profile for parameters 2 25 = ps2/km, 1sP = W,

11 /W km = , 5sT = ps. The dashed line shows the result of a calculation using the soliton initial condition but including only linear GVD for 0 1.57L L= = km.

For additional reading on the material covered in this section see Sec. 5.2 of Ref. [1], and Sec. 12.7 of and Appendix B of Ref. [2], and Ref. [3].


5.3 Soliton stability and higher-order solitons One of the amazing properties of optical solitons is that they are stable solutions of the nonlinear Schrdinger equation in Equation 5.1. What this means practically speaking is that even if the condition 1N = is not precisely satisfied for an initial hyperbolic-secant pulse of width 0T and peak power 0P the pulse will propagate stably with some oscillation around the ideal soliton. This is reassuring given the imperfections and absorption that are always present in real fibers. The animation in Figure 5.2 shows the evolution of the pulse power profile with propagation distance over the range z=0-10 km for parameters 2 25 = ps

2/km, 0 1.2P = W, 11 /W km = ,

0 5T = ps, so that according to Equation 5.8 1.1N = , and the initial pulse has a power 1.2 times that of the soliton in Figure 5.2. What you will see when you run the animation is that the pulse breaths as it propagates, in that it goes through cycles of narrowing while increasing its peak power and broadening while decreasing its peak power. The pulse may then be seen as oscillating around the exact and stable fundamental soliton for 1N = .

Figure 5.3 This animation shows the evolution of the pulse power profile with propagation distance over the range z=0-10 km for parameters 2 25 = ps

2/km, 0 1.2P = W, 11 /W km = , 0 5T = ps, and

1.1N = , and illustrates soliton stability.

The previous discussion is intended to make the point that for initial pulses with 1N the input pulses will evolve close to ideal solitons. This raises the question of what happens to the pulse propagation if N in Equation 5.8 deviates significantly from unity. For example, for N=0, which corresponds to linear propagation with only GVD, the initial pulse will undergo pulse spreading. Indeed for N


(Equation 5.12) It turns out that higher-order solitons occur for integer values of N. To illustrate this the animation in Figure 5.4 shows the evolution of the pulse power profile 2| ( , ) |A z T with propagation distance over a distance corresponding to the soliton period 0 / 2 1.57L = = km for parameters 2 25 = ps

2/km, 0 4 4sP P= = W, 11 /W km = , 0 5T = ps, and N=2. This

animation shows that for the N=2 soliton the pulse power profile distorts with propagation distance but returns to its initial hyperbolic-secant profile in Equation 5.12 after the soliton period.

Figure 5.4 Animation of the N=2 higher-order soliton over a soliton period.

As a second example Figure 5.5 shows the animation for the N=3 higher-order soliton with the same parameters as Figure 5.4 but peak input power 0 9P = W, and again the pulse initially distorts but returns to its initial hyperbolic-secant form after a soliton period. The periodicity of the pulse dynamics applies for integer N: For non-integer values of N the dynamics is no longer periodic and can be quite complex depending on how large N>1 is.

Figure 5.5 Animation for the N=3 higher-order soliton over a soliton period.


To further reinforce the idea of the periodic dynamics of the higher-order solitons Figure 5.6 shows a color coded plot of the pulse power 2| ( , ) |A z T with z along the horizontal axis and T along the vertical axis, the color code being shown on the right. This figure corresponds to the N=3 higher-order soliton in Figure 5.5 but for a propagation distance of two soliton periods, and the fact that the wave is periodic is evident. From these examples we see that the higher-order solitons with integer N are indeed periodic with propagation distance with period equal to the soliton period, and that the oscillations in the pulse profile become more complex as N increases, and this a general property of the higher-order solitons.

Figure 5.6 This color coded plot shows the evolution of the pulse power profile for an N=3 higher-order

soliton over a distance of two soliton periods. We finish this section with a discussion of the first experiment by Mollenauer et al [3] to report optical solitons in fibers. For that experiment the parameters were 2 20 = ps

2/km, 11.3 /W km = , 0 4T = ps, which yields a soliton power of 1sP W. The fiber length in the

experiment was 700 m which is close to half a soliton period. The results from the experiment are shown in Figure 5.7, and the inset on the top left shows the input pulse profile (right) and spectrum (left). The bottom line shows the output pulse profile with input peak power

0 0.3,1.2,5.0,11.4,22.5P = W increasing from left to right. For the input power 0 0.3P = W the pulse broadens upon propagation as the power is below the soliton power, whereas for 0 1.2P =


W the pulse maintains its width in a manner consistent with a fundamental soliton. For an input power of 0 5P = W which is close to the N=2 higher-order soliton the output pulse profile is seen to have developed modulations that are suggestive of those seen in the animation in Figure 5.4. Simarly the results for peak input powers 011.4,22.5P W are suggestive of higher-order soliton propagation with N>2. The top line in Figure 5.7 shows the spectra corresponding to the power profiles shown. This seminal experiment therefore provides evidence of both fundamental and higher-order soliton propagation.

Figure 5.7 Experimental results from the first experiment on solitons in fibers in Ref. [3]. For additional reading on the material covered in this section see Sec. 5.2.3 of Ref. [1].


5.4 Fiber losses and soliton amplification We now turn to the effects of fiber losses on the N=1 fundamental soliton as described by Equation 5.1. It turns out that for weak absorption as occurs in fibers, if a soliton is launced into the fiber at z=0 then the pulse changes its peak power P(z) and width 1( )T z as it propagates so that it dominantly remains an N=1 soliton. In this case Equation 5.8 is replaced by

22 1

2

( ) ( )( ) 1 ,| |

P z T zN z

= =

(Equation 5.13)

and in order for N to remain equal to unity we require 2 21( ) ( ) s sP z T z PT= for all z. It can be shown that the peak pulse power evolves according to the Beers law ( ) exp( 2 )sP z P z= , in which case the pulse width must vary as 21 .( ) , ( )

z zs sT z T e P z P e

= =

(Equation 5.14)

According to these equations, as the soliton peak power reduces the soliton width increases in such a way that the pulse remains an N=1 soliton. An approximate form for the optical soliton in the presence of loss is therefore

( , )( , ) sec ( / ) ,z z i z Ts sA z T P e h Te T e =

(Equation 5.15)

where ( , )z T is a phase factor. We remark that this approximate solution for the optical soliton in the presence of absorption is only valid for propagation distances such that 1z


so there is little absorption between amplifiers. To reduce the effects of pulse spreading between amplifiers the amplifier spacing is typically chosen such that

2

0

2

.| |a DTL L


5.5 Dispersion-managed solitons The broadening of the soliton width due to fiber absorption discussed in the last section is clearly a limitation to soliton based communications systems, and it would be nice to have a way to counteract the spreading. One solution that has been proposed is to change the fiber dispersion along the fiber length 2 ( )z , so that Equation 5.13 becomes

22 1

2

( ) ( )( ) 1 .| ( ) |

P z T zN zz

= =

(Equation 5.18)

The idea is that if the fiber can be manufactured such that 2 2( ) exp( 2 )z z = , then the variation in peak soliton power ( ) exp( 2 )sP z P z= due to absorption can be compensated using the variation in the dispersion parameter, and the soliton width can remain constant

1( ) sT z T= . Such fibers are termed dispersion-decreasing fibers (DDFs) and they have been fabricated in the lab. Although DDFs are not commercially avalaible they are an nice illustration of dispersion-management in which the dispersion of the fiber is varied along its length to improve performance. The key to dispersion-management is the choice of dispersion variation 2 ( )z along the z-axis. To highlight an example of this we return to the periodic amplification of solitons in Sec. 5.4. For this purpose we require 20 2/ | |a DL L T


5.6 Cross-phase modulation So far the discussion has assumed that the field propagating in the fiber has been linearly polarized and described by the field envelope ( , ) ( , )xA z t A z t= for an x-polarized field. We next briefly discuss the extension where we include the field envelope ( , )yA z T corresponding to the y-polarized state, so that now we can have a general state of polarization. Without proof we state that the field envelope equations including GVD and SPM are ( 0 = )

22

22

21 ( ) (| | ,2

| | )x x x xy x xx

yA A i Aib z A i A Az v t

Bt

A + += +

(Equation 5.19) 2

2 2 22

1 ( ) (| | | | .2

)y y yx y x yy

yA A iib z A i A B A Az v t t

A

+ = + +

(Equation 5.20)

Consider for the moment the linear case 0 = : Here we have generally included different group velocities ,x yv and dispersion parameters 2x and 2 y for the two orthogonal polarizations states, and also a coupling term proportional to ( )b z between the x- and y-polarized fields. The coupling between the two polarizations may come from a systematic source, for example periodic twisting of an elliptical core fiber, or from random fluctuations in the fiber profile along its length. We are interested in the coupling due to random fluctuations, and even although they are weak they can cause polarization coupling that accumulates over large fiber lengths and can scramble the polarization state of the field propagating in the fiber. This effect is called polarization mode dispersion (PMD) and was discussed in detail in undergraduate module 5, Sec. 5.1.4. PMD can cause the polarization sate of a well defined input polarization state to be randomized or scrambled, and this can be a limitation in optical fiber links. Returning to Eqs. 5.19 and 5.20 we note that if we set 0yA = and b=0 we recover our previous single polarization slowly varying envelope Equation 5.1 for xA A= including just linear GVD and nonlinear SPM, the nonlinear term being described by the term 2| |x xi A A on the right-hand-side of Equation 5.19. When both polarization states are included the nonlinear term becomes 22 |(| | )|x y xB Ai A A + , this describing both SPM and cross-phase modulation (XPM) between the polarization states. The parameter B characterizes the relative strengths of the SPM and XPM in the fiber. Equations 5.19 and 5.20 incorporate the fact that in the presence of both orthogonal polarization states the nonlinear changes in refractive-index 2NLn n I = for x- and y-polarized fields are of the form

2 22 (| | | | ) ,

xNL x yn n E B E = +

(Equation 5.21)


2 2

2 (| | | | ) ,yyNL xn n E B E = +

(Equation 5.22) which generalizes the single polarization case given in module 4. There are many important linear and nonlinear polarization effects in fibers and these are discussed in Chap. 6 of Ref. [1]. Here we wish to point out one important result, namely, that using soliton input pulses of well defined polarization state can be used to greatly reduce the effects of PMD. It is well known that PMD leads to considerable pulse broadening for non-soliton pulses in fibers. However, studies have shown that solitons are relatively immune to the effects of PMD, and one way to intuit this is to remember that solitons are stable solutions of the NLSE. Thus we can think that the nonlinearity enables soliton pulses to hold together in the presence of the perturbing effects of PMD, and solitons are observed to maintain uniform their polarization state over the whole pulse. This is a very important aspect of soliton propagation if one wishes to perform polarization-division multiplexing. For additional reading on the material covered in this section see Secs. 1.2.4 and 6.2.1 of Ref. [1], and Sec. 3.2.6 of Ref. [2]. 5.7 Modulational instability and the generation of soliton trains In this final section we describe the phenomenon of modulation instability (MI) in optical fibers and its use for generating trains of solitons. So far we have considered the soliton solutions of the NLSE in Equation 5.1 which represent temporally localized solutions with the prescribed relation between the soliton peak power and width in Equation 5.11. The next question is what happens if we input a pulse of a given peak power 0P into the fiber whose width is many times the corresponding soliton width? To address this question we consider the limit of a continuous wave (CW) field with constant power 0P . In the CW limit with zero absorption ( 0 = ) the NLSE 5.1 has the solution ( )0( ,) NL

i zA z P e =

(Equation 5.23)

where 0( )NL z P z = is the nonlinear phase-shift due to self-phase modulation. Modulational instability analysis addresses the issue of whether weak side-bands representing a modulation added to the strong CW solution ( )A z will exhibit spatial growth or gain under propagation along the z-axis. More specifically, one considers a perturbed solution of the form ( , ) ( )[1 ( ) ( ) ] ,i T i TA z T A z a z e b z e = + +

(Equation 5.24) with |a|,|b|


probe beams at 0( )p = . The details of the MI analysis are given in Secs. 5.1.1 and 5.1.2 of Ref. [1], and here we summarize the key results. In particular, one finds that the weak side-bands of frequency experience gain given by 2 22( ) | | .cg =

(Equation 5.25)

where

02

4 .| |c

P

=

(Equation 5.26)

The gain only exists for | | c < , the maximum gain occurring for / 2max c = . An example of the gain due to MI is shown in Figure 5.8 for parameters 1.57 = W-1/km, 0 7.1P = W, and

2 3 = ps2/km, yielding 2.7max = rad/s.

Figure 5.8 Gain due to modulational instability versus frequency detuning for parameters 1.57 = W-1/km, 0 7.1P = W, and 2 3 = ps

2/km, yielding 2.7max = rad/s.


The animation in Figure 5.9 illustrates the basic MI for an initial condition

0 max(0, ) [1 0.01 cos( )] ,A T P T= +

(Equation 5.27)

which represents the CW solution of power 0P plus a one percent modulation at the frequency

max of maximum gain. Parameters employed were 1.57 = W-1/km, 0 7.1P = W, and 2 3 =

ps2/km, yielding 2.7max = rad/s. The animation shows the evolution of the field power profile 2| ( , ) |A z T over a time window of 11 ps as a function of the propagation distance z=0-0.5 km. At

z=0 km the power profile is near flat but weakly modulated. However, as the propagation distance increases the initial modulation grows noticeably until at z=0.5 km the propagating power profile resembles a train of pulses.

Figure 5.9 Animation of a train of soliton pulses evolving from a periodically perturbed CW input due to an induced modulational instability.

This is the basic phenomenon of MI in which a CW solution is rendered unstable against the development of temporal modulation, and in the example of Figure 5.9 we refer to an induced modulational instability since the initial field in Equation 5.27 includes an explicit initial small modulation. What is very interesting about the pulse train that appears at 0.5z = km in Figure 5.9 is that is corresponds very closely to a train of fundamental soliton pulses! In particular, for the parameters above and a peak soliton power 40sP W (estimated from the peak power at z=0.5 km in the animation in Figure 5.9) we obtain 0.22sT = ps from Equation 5.11, and this is close to the width of the pulses in the train that appear for z=0.5 km in the animation in Figure 5.9. Furthermore, the spacing between solitons is around 3 ps, which translates to a repetition rate of around 0.3 THz. The above example is intended to demonstrate that induced MI can be used to convert an input CW signal into an output soliton train with repetition rates in the THz range by injecting a weak coherent probe field of frequency detuning along with the CW signal. The properties and


quality of the soliton train can be optimized by choice of pulse and fiber properties and choice of the fiber length. Figure 5.10 shows experimental results for the soliton trains produced by induced MI in a fiber, the two traces corresponding to the different pulse trains obtained for different choices of the modulation frequency . These results were reported in Ref. [4] the parameters for which correspond closely to the example in Figure 5.9, and this experiment reported repetition rates on the order of 0.3 THz, in agreement with the simulations.

Figure 5.10 Experimental demonstration of induced modulational instability in a fiber showing generation of solitonlike pulses at 0.3 THz repetition rate. The two different traces are for different

perturbation frequencies

For additional reading on the material covered in this section see Sec. 5.1 of Ref. [1], and Ref. [4]. Some contemporary engineering problems that require a knowledge of the material taught in this module are

Modeling and design of soliton based optical fiber links. Modeling and design of soliton based ultrashort pulsed laser sources. The design of supercontinuum light sources using fiber solitons.


References 1. G. P. Agrawal, Nonlinear Fiber Optics, 3rd Ed. (Academic Press, San Diego 2001). 2. G. Keiser, Optical Fiber Communications, 3rd Ed. (McGraw Hill, Boston, 2000). 3. L. F. Mollenauer, R. H. Stolen, and J. P. Gordon, Experimental observation of picosecond pulse narrowing and solitons in opticsl fibers, Phys. Rev. Letts. 45, 1095-1098 (1980). 4. K. Tai, A. Tomita, J. L. Jewell, and A. Hasegawa, Generation of subpicosecond solitonlike optical pulses at 0.3 THz repetition rate by induced modulational instability, Appl. Phys. Letts. 49, 236-238 (1986).

Introduction5.1 Slowly varying envelope equation with GVD and SPM5.2 The fundamental bright soliton in fibers5.3 Soliton stability and higher-order solitons5.4 Fiber losses and soliton amplification5.5 Dispersion-managed solitons5.6 Cross-phase modulation5.7 Modulational instability and the generation of soliton trains

Module 5 – Solitons in Optical Fibers - Login ERC...

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