Design of an All-Optical Long-Distance Solition-Based Optical Fiber Communication System Using Flouride Fibers

The stability of optical solitons amplified periodically by stimulated Raman scattering process in heavy fluoride fibers, is numerically studied for a range of parameters using computer simulation. An optimum single-mode fluoride fiber was used to design an all-optical soliton based optical fiber communication system. The value of the chromatic dispersion is 0.6 psec/nm/ km with acceptable manufacturer tolerances. The length-bandwidth product is four times that of a soliton-based optical communication system using present day silica fibers. In this design we numerically justify why we neglect the higher order terms in the Taylor series expansion of the propagation constant around the pulse central frequency. The single channel bandwidth-length product for bit error rate (BER) less than 10is nearly 120,000 GHz.km. Typical amplification periods are in the range of 200-300 km; average soliton and pump power are in the range of milliwatts and hence well within the capability of semiconductor lasers.


Introduction
In present-day optical fiber communication systems (OFCS) operating at many Gbits/s, the bit rate between regenerators is limited by the chromatic dispersion characteristics of the fiber material and its design. For a single mode fiber, which has intrinsically higher bit rate, the chromatic dispersion is the deterministic factor in influencing the rate of pulse spread, hence the channel capacity .
A natural choice to minimize the pulse spread is to carefully tune the laser wavelength to the wavelength at which the chromatic dispersion passes through zero. Nevertheless still pulse spreading occurs due to the higher order effects in the dispersion, also thus far it has been proven difficult to control the wavelength match between the laser diode output and the zero chromatic dispersion sufficiently well for this scheme to achieve its full potential.
Also, optical signals are detected and electronically repeated every 20-100 1 km before continuing along the next fiber span, electronic repeaters can limit bit rate of transmission to a few Gbit / s, which means that only a tiny fraction of the tremendous information carrying capacity of the single mode optical fibers is used.
Thus the only sensible way to overcome electronic rise time limitations is by ensuring that the signal remains strictly optical in nature throughout the system.
Hasegawa suggested [1] a high bit rate all-optical communication system could be created through the combined use of loss compensating Raman gain [2] and solitons [3], [4]. Numerical studies [5] showed that solitons can indeed be made to propagate over long fiber spans with little change in shape, when Raman gain is used to compensate for energy loss. Experimental demonstration of soliton propagation over short fiber lengths was reported [6] using Raman gain to exactly compensate fiber intrinsic loss.
Figl shows a schematic of the system configuration. CW Raman pump light is injected periodically into the fiber through wavelength dependent directional couplers spaced L(km) apart. The pump wavelength is determined from the signal wavelength, which is normally chosen to coincide with the wavelength of minimum predicted loss, and the frequency shift that corresponds to the peak of the broad band Raman gain. For fluoride fibers the minimum loss occurs at >. = 2.55 µm and the peak Raman gain shift occurs around 580 cm-1 [7], hence the pump 2 wavelength will be 2.22 µm .

Soliton theory
The single mode optical fiber has small core diameter (about 10 µm). Since light can propagate over long distances, it possible to observe non linear phenomena in fibers at power levels much smaller than that requi,.ed for bulk media. Non-linear effects in single mode fibers have many important applications. Among these are the generation of new frequencies through the Raman process, the compression of optical pulses and the all-optical soliton based optical fiber communication system.
A light propagating through a single mode fiber of length L and refractive index n will acquire a phase shift <f> = 2 { n L . ff the refractive index was. intensity dependent, n(I) = n 0 + n 2 I (r,t), n 2 is the non linear index coefficient, r is the radial distance, t is time and I is the input intensity, And if the input beam has an intensity modulation, the transmitted light will exhibit a temporally varying phase given by [8] (1) Where n2 is in esu units, E is the peak field amplitude in cgs units, p(t) is the transmitted power in watts, 8n is the change in the refractive index n, c is the velocity of light and Aet I is the core effective area of the fiber [see Appendix A] Thus self phase modulation (SPM), can be regarded as a conversion of amplitude modulation to phase modulation through the non-linear refractive index. The instantaneous frequency shift at any point in the pulse is given by the time derivative of the phase modulation w -w 0 = %t (~¢>). According to the rate of change of the input light with respect to time, the instantaneous frequency shift could be either positive or negative, which means that some points in the pulse will be accelerated while others will be retarded with respect to the pulse center. This leads to frequency broadening ( chirp ) and consequently pulse spreading, which limits the bit rate of the system. Group velocity chromatic dispersion causes pulse broadening, in particular, a gaussian pulse of duration 5 psec will double its width after propagating a distance 4 of about 800 m if the value of the dispersion parameter is 16 psec / nm/ km at wavelength of 1.5 µm. The group velocity dispersion (Bv 9 / B>.) may be positive (normal) or negative (anomalous), depending the fiber composition, waveguide geometry and light wavelength.
The presence of group velocity dispersion regardless of its sign, will always lead to broadening of the transform limited pulse. If the pulse was chirped, however, then pulse compression can occur if the dispersion has the correct sign to reverse the chirp. The frequencies in the leading half of the pulse are lowered, while those in the trailing half are raised. When the dispersion is negative, the group velocity increases for increasing frequency, thus the trailing half of the pulse containing the higher frequencies is advanced, while the leading half containing the lowered frequencies is retarded. The pulse then tends to collapse on itself. Thus the combined effects of non linearity and negative group velocity dispersion achieves pulse narrowing. It also becomes possible to have the propagation of the so called "solitons", pulses that either does not change shape or have shapes that change periodically with propagation along the fiber.
Although the above argument can explain simple narrowing, it is not sufficient to account for pulse shaping effects and soliton behavior. To correctly predict the latter, one must solve the non-linear Schrodinger equation, which governs the evo-5 lution of the pulse envelope.

Optimum fluoride fiber design
Because of their ultra low loss in the 2-5 µm region, multicomponent heavy metal fluoride glass fibers appear to be a serious contender as the transmission medium of future long-haul telecommunication networks. It remains to be -seen whether this loss can be achieved in practical fibers. The best loss achieved at 2.55 µmis approximately 1 dB/km whereas the predicted minimum is 0.03 dB / km. As will be indicated, for the sake of pulse stability, it is desirable to have values of the soliton characteristic period (Z 0 ) to be a good fraction of the amplification period length (L). Since the pulse width scales as ( Z 0 D) 1 / 2 , it is necessary to reduce the fiber dispersion parameter D to the smallest possible value so as to increase the bit rate of the system. This represents the first criterion in the fiber design.
As we have explained earlier, soliton propagation in fibers is only possible in the region of negative group velocity dispersion (8vg/8').. ::::; 0). This is the second criterion.
Using the programs developed in [9] to optimize fluoride fiber performance in the 1. 7 -5 µm for a ZBLAN composition, with a simple step refractive index profile and the proper values for the refractive index . difference ( ~) and core radius (a); we were able to tailor the shape of the fiber dispersion parameter (D) as a function of wavelength (>.) and obtained the best possible fiber design for a soliton-based system. In that design : 1-D is minimized to the lowest possible value allowing for tolerances in the manufacturing techniques. µm . This means at >. = 2.55 µm, which corresponds to the signal wavelength, the group velocity dispersion is negative. This is an essential condition to reverse the pulse chirp produced by self phase modulation, hence pulse compression and soliton propagation is possible. On the other hand, at >. = 2.22 µm, which corresponds to the pump wavelength, the group velocity dispersion is positive, and hence solitons cannot be produced at this pump wavelength.
3-The slope of the D vs. >. curve at >. -2.55 µm is small. This means that the third order term in the Taylor series expansion of the propagation constant k around the pulse or soliton frequency , and consequently the other higher order 7 terms, can be neglected. We numerically verified, using t he calculated values of the second and third derivatives of the propagation constant, that the ratio of the third order to the second order term in the Taylor series expansion of the propagation constant around the pulse central frequency is less than 10-3 • It is important to note that : 1-The values of the first, second, and third derivatives of the propagation constants were not obtained by numerical differentiation of the propagation constant , which is calculated by solving the scalar wave equation and matching the boundary conditions of the tangential fields. We obtained these by the method developed by Sharma [10] , which transforms the scaler wave equation into three first order differential equations that are solved using the fourth order Runge-Kutta algorithm [9]. This is more accurate than calculating the derivatives of the propagation constant by numerical differentiation.
2-It is well known that the fractional changes in D resulting from variation m (a) or (~) during fiber manufacture, undoubtly becomes large in the limit of small D, however, sensitivity analysis was done on this design. With a = 3 µm and ~ = 0.95 %, it was found out that variation in (a) ± 1% will change D by ± 0.3 psec/nm/km, while a change in (~) ± 10% can lead to change in ± D = 0.15 psec/nm/ km.

Raman gain and pulse energy
Optical solitons can propagate free of distortion in a fiber with group dispersion, however the pulse width increases in the course of propagation because of loss [3] in the soliton power due to fiber attenuation.
If the Raman amplification is distributed such that it exactly compensates for the fiber loss, the soliton can propagate free of distortion practically for an unlimited distance. Although the Raman gain cannot be kept constant because of the fiber loss and pump depletion, the amplification process is still adiabatic, hence allowing the soliton to keep its characteristic property as a soliton (width x amplitude) = constant. 9 In contrast, if the amplification is made locally, only the amplitude is increased (without reducing the width) and the soliton radiates away excess energy in the form of dispersive waves [ 11]. In this case it has been found necessary to amplify the soliton again before the dispersive waves escapes from the soliton. This means that the amplifier spacing is determineded by the distance controlled by the group velocity dispersion.
Since the Raman gain is non uniform with each period L, the corresponding effective gain(loss) coefficient and pulse energy variation must be established first before any further calculations of the soliton properties.
Consider soliton pulses at wavelength A 8 propagating down a fiber span of length L, while subject to the simultaneous effects of fiber loss and bidirectional CW Raman pumping at wavelength Ap . The loss(gain) is given by (2) aa is the fiber loss coefficient (km-1 ) at the soliton wavelength,ae/f is the effective fiber loss coefficient and ag is given by : 10

R =g Pp
Aetf ap is the fiber loss coefficient (km-1 ) at the pump wavelength , R is the Raman gain factor (km-1 ) , Pp is the Raman pump power (mW), and Aeff is the effective area [see Appendix A] of the fiber core (µm 2 ).
If we choose ag such that the pulse energy, at the end of one amplification period L, is equal to the pulse input energy E 0 , then ag becomes Using (2) and (5)

Soliton propagation in a lossless medium
The envelope A(z,t) of the electric field E(z,t) = A(z,t)exp(i(kz -wt)) in an optical fiber satisfies the following dimensionless non linear Schrodinger (NLS)

(7)
The first term on the right hand side describes the effect of dispersion while the second term describes the effect of non linearity. In the absence of these two terms the left hand side alone would describe the distortionless propagation of the pulse envelope, thus one can imagine that when non linearity balances dispersion similar distortionless propagation might occur.
If a pulse shape of the form : is inserted in eqn. (7), then it will propagate as a pure soliton for integer N (which we refer to as the order of the soliton). Explicit expressions for A(c,s) have been obtained for N = 1,2 [13], [14] and these are 13 We see that IA11 2 = sech 2 s is independent of e ' however' this is not true for IA2 1 2 · which develops periodical structure in e with period 71" /2. This, in fact, is the same for all higher order solitons (N : ; : : : : : 2) .
We can also generate a continuous set of solutions from these by using the invariance of the NLS equation under the following transformations : where r is the amplitude gain(loss) coefficient, € and s are the dimensionless versions of real world quantities Z and t respectively.
To make r consistent with the definition of cr.,ff, as in eqn. (2), the sign of the first and last term in (12) should be the same, such that positive r applies t o gain and negative r applies to loss; the quantities € and r are related to their real world counter parts [see Appendix A] by the following 15 (14) The factor 2 in eqn. (14) converts r into an energy coefficient. Unit propagation of dimensionless length(€= 1) corresponds to the real world quantity Zc, "the soliton characteristic length". For the fundamental soliton it is given [see Appendix A] by the following where r is the full width at half intensity maximum (FWHM) of the pulse and D (psec /nm/ km) is the fiber dispersion parameter.
To avoid the appearance of of the factor 7r in some formulae we shall more often use the closely related quantity Z 0 , known as the soliton period, and given by To obtain pulse behavior in the fiber with gain (loss), we have solved eqn. (12) numerically on a DG/Unix computer. Each solution corresponds to a set of values for the parameters L, Z 0 (orZc), aa and ap .Once these quantities have been chosen, f(€) was computed using eqns. (14), (2) and (5). With the R (Raman gain factor) adjusted such that the pulse energy is conserved at the end of each amplification period, f(€) was then inserted in the the NLS equation (12) which is then ready to be solved numerically.
The NLS equation is a parabolic, initial value, boundary value problem in one space variable. It is a well posed problem if-the initial condition, A(€ = 0, s) and the two boundary conditions, are known. Note that the initial condition is related to the space variable and the boundary conditions are in time variables.
The initial condition is A(€ = 0, s) = sech( s), which is the fundamental soliton, and the boundary conditions are periodic, i.e. the envelope function and its first derivative are continuous at the boundaries.
The pulse envelope amplitude wings are truncated at isJ2:10 where it reaches a negligibly small value of approximately 9 x 10-5 , i.e., the boundaries are at s = ±10. With the boundary defined, initial and boundary conditions determined, the solution proceeded over a span of € corresponding to one or more amplification periods A .
The NLS equation was solved numerically, in the complex plane using the FRANKEL-DUFORT 3-level finite difference scheme [15]. The first level only was obtained using a 2-level explicit scheme, the second level was obtained using the first level and the initial condition, the third level was obtained using the second Reference [ 15] shows that this scheme is stable for any value of r = k / h 2 since the absolute value of the eigenvalues of the associated block tridiagonal matrix 18 are less than or equal to one. In our case r is taken to be one, since for large values of r the solution becomes less accurate.
The scheme stability and convergence were tested numerically by changing Before proceeding with the calculations, the following checks on self consistency and accuracy were done : 1-With f(€) temporarily set to zero and for A= j (L = Zo), the program was run for an input function of 2 sech (s) which is the second order soliton. Since the 19 pulse width changes by more than a factor of 4 at midpoint ( ~ = 7r / 4 ) , the N = In real world quantities, the fundamental soliton peak power P 1 is given by [see

Pump power and pump depletion
In ordinary single mode fibers, the relative pump and signal polarizations change rapidly with propagation. This makes the average gain ag just half of its value given by eqn. (3). Also in a bidirectional pumped system, the quantity where Pa is the average signal power and is given by where T is the time separation between soliton pulses; the factor 1/2 accounts for the average occupation of pulse slots in a typical data stream. It is required that this pump depletion be small, such that the Raman pump power will not be signif-22 icantly dependent on the signal power. As an illustrative example, for r = 16 psec, which corresponds to Zo = 62.5 km, T /r = 10, Aeff = 75 µm 2 , P1 = 30 mw, the additional attenuation at the pump wavelength is 2.4 x 10-3 km-1 , which is only 17 % of the fiber intrinsic attenuation at Ap.

Optimum system design
For the sake of soliton stability, it is desirable to use large values of Z 0 such that ; 0 « 8 , [16]. This fact will be proven numerically by showing that the change (6S) in the pulse area S, for different values of Z 0 , is small for large Z 0 and peaks when L = 8Z 0 • In this domain ( of large Z 0 ) the soliton itself is not preserved everywhere, but nevertheless recovered along with the pulse energy at the end of the amplification period. As we will also show, the soliton peak power is reduced significantly over that required for small Z 0 • Note that increasing Z 0 will increase r and hence decreasing the bit rate for certain pulse spacing (T). The steps of the system design are in the next section. , this point represents a peak for 8S and corresponds to a resonance between the perturbation period L and the soliton phase.
As discuss. ed earlier, for the sake of stability, one would then pick the largest possible value of L/ Z 0 (smallest possible Z 0 ) on the left side of the resonance peak 24 and consistent with a certain maximum allowable value for 68 . Once Z 0 is determined, r can be calculated. As an example, for 1=250 km and L/ Z 0 = 4 , Z 0 = 62.5 km gives r approximately equal to 16 psec for .Xs = 2.55 µm and fiber dispersion parameter D= 0.6 psec/nm/km. Figures 21 to 25 show , for this ·specific example, the soliton amplitude envelope, energy envelope, the soliton area S, intensity peak and the soliton energy E as a function of propagation distance along one amplification period span produced by numerical solution of the NLS equation.

Determination of the period (T) between adjacent pulse slots
There are two major limiting factors that impose an upper bound on the bandwidth of a soliton-based optical communication system. These are soliton interaction and the random walk of the coherently amplified solitons caused by spontaneous Raman emission.

Soliton interaction
In the NLS equation, the term IAl 2 corresponds to the potential function in the classical quantum mechanical form of the Schrodinger equation. This potential 25 causes both attractive and repulsive forces among soliton pairs. Starting from the general 2-soliton solution, it has been shown [17] that solitons in fibers exert forces on their neighbors, which decrease exponentially with increasing the spacing between them, and depend sinusoidally on their relative phases. These forces account for the displacement suffered by solitons during collisions, and their effects must be taken into account in system design. Equation (18) in [17], if transformed into real world quantities [18] gives where CTin and C1 0 u are the initial and final pulse separation respectively, f is the cosine function for the attractive case and hyperbolic cosine function for the repulsive case. Fig 26 shows CTin vs. C1 0 u for the specific example we gave earlier when CTin changes from 0 to 200 psec.
The interaction between solitons can lead to a significant reduction in the bandwidth. However, it .can be neglected when the separation (T)between pulses is ~ 10 r. For large values of Z 0 (which is our case), T can be as small as 5r. We shall soon see that it is desirable to have a bit of extra spacing between adjacent pulses to allow for other effects which we will discuss in the next section; therefore let us conservatively set T / r = 10 .

Random walk of coherently amplified solitons
Periodic Raman amplification is needed to maintain the energy of the solitons.
Coherent amplification is always accompanied by the generation of spontaneous emission noise. Because the system is non-linear, its behavior is not simply additive; some of the noise field will be incorporated into the soliton. The most troublesome resulting effect is a random shift in the soliton carrier frequency with a corresponding change in its velocity. This in turn, through dispersion, leads to random pulse arrival time. This random walk effect causes timing errors and puts an upper limit on the length-bandwidth product of the system.
In reference [19] Gordon and Haus have calculated the mean square inverse velocity shift (60) due to a coherent amplifier of gain G (similar results have been obtained using the inverse scattering theory [13] and [14]).
For a system of overall length L, consisting of n amplifiers, the overall shift in arrival time is calculated by summing up the contribution of each individual amplifier.
Assuming gaussian distribution of the change in pulse arrival time, probability of error less than 10-9 , and with a detection window W at the receiver to allow for the finite pulse width, and the detector response time, such that W /T = 1/3 . Then the product of the length of the system L and the bit rate R ( R is taken as the 27 inverse of T ) is given by : To complete the picture, we note that this system would use solitons of peak power P, = 30 mW and therefore containing 3. 7 x 10 6 photons. Because the system has an overall gain of unity, the mean number of noise photons per mode at the receiver will be [19] a,L = 134 . This is small compared to the number in a soliton and hence should cause no problem for the detection process.

g Stability with repeated amplification
The system is said to be stable if the pulse distortion 88 does not grow monotonically with repeated amplification, but instead is always contained within certain limits, no matter how many amplification periods are involved.
The program was run for 20 consecutive amplification periods, which corresponds to 5000 km, for different values of Z 0 • The pulse shape obtained at the end of each amplification period become the input to the next and so on ( as a closed loop ) . The pulse envelope and area were examined at the end of each fifth period.
The change in the pulse area was never more than 4 times the change in the initial pulse area at the end of the first amplification period span.   Comparison between the normalized soliton energy obtained by solving the NLSE and the theoritical values obtained using (6) in the text at different values of propagation along one amplification period L of length 250 km when L/Zo is 6 , and using the same fiber loss coefficients mentioned earlier      such that the power P in the optical fiber is given by (2) where Pc is the proportionality constant in watts

Design examples of a soliton-based OFCS
The above equation is the Taylor series expansion of k(w,P) in the neighborhood of (w 0 ,0). The higher order terms (e.g., k"'0 3 , k~OP, etc.) can be neglected or sufficiently treated as perturbation, the term k 2 P represents the non linear effect (self phase modulation), resulting from the intensity dependent refractive index of the fiber. Zc can be expressed in terms of the fiber dispersion parameter D rather than k", the expression for the reciprocal group velocity is Bk v-1 = -= k' + k"(w -Wo) g aw (9) The dispersion parameter D used widely to describe fibers, it is the wavelength derivative of v;1, hence is related to k" by D = -21rc k" ).2 To calculate the value of the non linear refractive index coefficient n 2 , The refractive index of fibers , can be expressed as (12) For ZBLAN fluoride fibers, n 0 = 1.4837 at >. = 2.55 µm, and the intensity dependent part ri 2 = 0.85 x 10-13 cm 2 /statvolt 2 in cgs units [21 ]. If we translate this to the form (13) where n 2 is the nonlinear refractive index coefficient, and I is the intensity in W / cm 2 • In cgs units, I = (87r)-1 cn1 El 2 , so that (14) To derive an expression for the effective area Aef f and the intensity dependent propagation constant k 2 : The wave vector of in a single mode fiber can be expressed in the form k = wne!f/c, where neff is the effective index of the fi ber. The linear part of neff varies depending on frequency, in a range between the indexes of the cladding and the core. The nonlinear part is independent on frequency. Its perturbing effect on the effective index of the fiber, can be sufficiently evaluated, as an average of n 2 1 over the fiber cross section weighted by the intensity distribution I. If we thus write nef 1 = neffo + nzP / Aeff, where P = J ldA is the power, we find the effective area given by p2 Aeff = J J2dA (15) For the fiber design presented, the core radius a = 3 µm and the effective area Finally, the fundamental soliton peak power P 1 , can be calculated using eqn.