Generation, propagation, and amplification of dark solitons

W. Zhao and E. Bourkoff

Department of Electrical and Computer Engineering, The University of South Carolina, Columbia, South Carolina, 29208

Abstract

The technique for generating dark solitons with constant background using guidedwave Mach–Zehnder interferometers is further examined. Under optimal conditions, a reduction of 30% in both the input optical power and the driving voltage can be achieved, as compared with the case of complete modulation. Dark solitons are also found to experience compression through amplification. When the gain coefficient is small, adiabatic amplification is possible. Raman amplification can be used as the gain mechanism for adiabatic amplification, in addition to being used for losscompensation. The frequency and time shifts caused by intrapulse stimulated Raman scattering are both found to be a factor of 2 smaller than those for bright solitons. Finally, the propagation properties of even dark pulses are described quantitatively.

1. INTRODUCTION

Nonlinear optical pulses can propagate in dispersive fibers in the form of bright and dark solitons under certain conditions, as first described by Zakharov and Shabat in 1972^{1} and in 1973,^{2} respectively. They are stationary solutions of the initial boundary value problem of the nonlinear Schrödinger equation (NLSE).^{32} In the anomalous dispersion regime of the fiber, under the boundary condition u(z, t = ±∞) = 0, there exists a class of particle-like, stationary solutions called bright solitons.^{4} In the normal dispersion region, under the

\begin{matrix}c \\ ◯ \end{matrix} Optical Society of America, 1992.

boundary condition |u(z, t = ±∞)| =constant, one can obtain another class of stationary solutions, which are called dark solitons, since a dip occurs at the center of the pulse.^{5} Ever since the pioneering work by Zakharov and Shabat^{1,2} and Hasegawa and Tappert,^{4,5} optical solitons have been an active topic of research. This is particularly true since advances in experimental techniques for generating ultrashort pulses in the picosecond regime have made it possible to observe soliton effects in single-mode optical fibers. The bright soliton was first successfully observed in single-mode optical fibers by Mollenauer et al. in 1980,^{6} and the dark soliton was first observed by Emplit et al. in 1987.^{7}

The characteristics of bright solitons have been studied extensively during the past decade.^{8,9} It was found^{32} that bright solitons are periodic and highly stable against small perturbations, such as fiber loss, background noise, and amplitude variations.^{4,5} Ideally, when fiber loss is neglected, the fundamental bright soliton can propagate inside an anomalously dispersive fiber over an infinitely long distance without changing its pulse shape. This can occur because, for a fundamental soliton, the effect of dispersion on the pulse is exactly balanced by that of the nonlinear refractive index of the fiber, i.e., the self-phase modulation. Solitons can also survive collisions between them. The interaction force between two neighboring solitons is periodic and decreases exponentially with their separation.^{10} Another characteristic of bright solitons is that they can be adiabatically amplified under certain conditions when gain is introduced into the fiber, e.g., through Raman amplification.^{11} The effect of fiber loss on the pulse can thus be compensated for by injecting a cw laser beam at a shorter wavelength into the fiber, whereby stimulated Raman scattering transfers its energy to the soliton.^{12} Therefore solitons are candidates for information carriers for future optical communications. Much research has been done in this area.^{13} The possibility of stable, repeaterless, all-optical soliton transmission at a 10–GHz rate across almost 5000 km has been numerically demonstrated^{14,15} and experimentally realized with a rate-length product of approximately 11,000 GHz km.^{16} More recently, with erbium-doped fiber amplifiers, soliton transmission of 9,000 km at 4 Gbits/s has been realized.^{17}

Because a dark pulse (with a dip of pulse intensity under constant background),^{7,18,19} especially the so called odd dark pulse (for which the electric field changes sign at the center of the pulse), cannot be easily generated, dark solitons have been studied less than their counterparts, bright solitons. However, as a result of recent developments in techniques for synthesizing short optical pulses with almost arbitrary shapes and phases,^{20} it is possible to observe soliton like propagation of individual dark pulses in single-mode fibers. Because these fibers exhibit normal dispersion over a large spectral region, extending from UV to IR (λ < 1.3μm), many cw and pulsed laser sources can be used to generate dark solitons. As a result, dark solitons have attracted increasing attention. …

In the following discussions, we adopt the normalization convention used in Agrawal’s book.^{26} We normalize the field amplitude A (optical power P_{0} = A^{2}) into u by

u = (\frac{2πn_{2}τ_{0}^{2}}{λA_{eff}|β_{2}|})^{1/2} A,

where A_{eff} is the effective area of the propagating mode, n_{2} = 3.2 × 10^{−16}cm^{2}/W is the nonlinear optical Kerr coefficient of the silica fiber, and β_{2} is a parameter describing the group velocity dispersion of fiber, defined as the second-order derivative of the propagation constant with respect to the radiant frequency evaluated at the signal frequency. The time variable t is normalized by a characteristic time constant τ_{0} (e.g., τ_{FWHM} = 1.76τ_{0} for hyperbolic secant pulses), and the spatial variable z is normalized by the so-called dispersion length,

L_{D} = \frac{τ_{0}^{2}}{β_{2}.}

As an example, at wavelength λ = 1.06 μm with A_{eff} = 40 μm^{2}, for a pulse with τ_{0} = 1 ps, the normalized distance z = 1 corresponds to a real fiber length of L_{D} = 60 m, and u = 1 represents an optical power of P_{0} = 3.5 W. However, when τ_{0} = 0.1 ps, L_{D} = 60 cm, and P_{0} = 350 W.

2. GENERATION OF DARK SOLITONS

In our earlier work^{24,27} we discussed the possibility of using an integrated Mach–Zehnder interferometer (MZI) to generate dark solitons with constant background. The idea is to drive a broad bandwidth MZI with a square-shape electric voltage with picosecond rise time. The applied electric voltage signal introduces a relative phase shift, proportional to the voltage, between the two arms of the interferometer by means of the electro-optic effect of the waveguide material. At the output, the two components of light are recombined, and the resultant optical field is proportional to the cosine of half of the total phase difference, the induced relative phase shift plus any other static (residual) phase differences. Therefore the pulse after the MZI, when properly biased, can have the form

u(0, t) = a sin[δπ/2 tanh(t)],  (1)

where a is the field amplitude of the input cw laser beam and δ the ratio of the applied voltage, approximated by a hyperbolic tangent function of time, to the half-wave voltage of the MZI. …

… The bandwidth requirement of the MZI is determined by the desired pulse duration, which is approximately half of the reciprocal of pulse duration. For a 50 ps dark soliton, 10 GHz is required, and this is achievable by current technology.

3. PROPAGATION AND AMPLIFICATION

As discussed in Section 2, when smaller values of δ are used, pulses of better characteristics are obtained. This can be seen in Fig. 1(d), where a = 1.33 and a pure fundamental dark soliton is generated. . . .. . . . In what follows, we will examine the possibility of amplification and compression of dark solitons with a constant gain and show that the stimulated Raman scattering can be used to amplify dark solitons as well as to compensate for the fiber loss.

We first examine the solution of a modified NLSE with a constant gain:

iu_{z} − 1/2u_{tt} + |u|^{2}u = iΓu,  (2)

where Γ is assumed to be a constant, appropriate for the Raman amplification under strong pumping without depletion. The solution of a similar equation to Eq. (2), but in the anomalous dispersion regime, in which bright solitons are amplified and compressed by the gain, has been analyzed by Blow et al.^{11} To solve the equation, we make the following variable transformation:

t′ = te^{Γz},  (3a)
z′ = \frac{e^{2Γz} − 1}{2Γ},  (3b)
u = ve^{Γz}.  (3c)

Under this transformation, the NLSE has the new form

iv_{z′} − ^{1}/_{2}v_{t′t′} − |v|^{2}v = −\frac{Γt′}{2Γz′ + 1}v_{t′}.  (4)

The solution of Eq. (2) when Γ = 0 is well known and has the form exp[iσ(z, t)]κ tanh κt, where κ is the form factor and the phase variable satisfies ∂σ/∂z = κ^{2}.^{1} Therefore, when the right-hand-side of Eq.(4) is zero, an exact solution for v(z′, t) can be obtained from Eq. (4). On the other hand, when z → ∞ and hence z′ → ∞ or Γ → 0, the right-hand side of Eq. (4) becomes infinitely small. Under these conditions, we can treat the right-hand side of Eq. (4) as a perturbation to the NLSE. …

u(z, t) = exp (i\frac{e^{2Γz} − 1}{2Γ}) e^{Γz} tanh(te^{Γz}),  (5)
Γ = g(e^{−2Γ_{p}z} + e^{−2Γ_{p}(L−z)}) − Γ_{s},  (6)
g = \frac{Γ_{p}(Γ_{s} + β)L}{sinh(Γ_{p}L)}e^{Γ_{p}L},  (7)
κ(z) = κ_{0} exp(βz).  (8)

…

In summary, we have studied the propagation properties of dark solitons under the influence of gain. The dark soliton can be amplified and compressed adiabatically when the gain coefficient remains small, e.g., Γ < 0.1. As the gain increases above this value, secondary gray solitons will be generated. Stimulated Raman scattering can be utilized to provide the gain. When the product Γ_{p}L is kept small, dark solitons can be amplified adiabatically with high quality. Such a property of dark solitons enable us to obtain dark solitons with short durations for the ease of observation and transmission.

4. EFFECTS OF INTRAPULSE STIMULATED RAMAN SCATTERING

The properties of dark solitons considered thus far are based on the simplified propagation equation (2). When the pulse duration reaches the subpicosecond regime, it becomes necessary to include higher-order nonlinear and dispersive effects.^{28} These effects represent higher-order terms in the derivation of wave equation from the Maxwell’s equations. Intrapulse stimulated Raman scattering (ISRS) is one of the dominating effects. It causes soliton self-frequency shift for both bright solitons^{29,30} as well as for dark solitons.^{31} Since its discovery for bright solitons,^{28} considerable attention has been paid to such effects. …

The effect of ISRS on bright solitons is to shift in both the temporal and spectral domains. It has been demonstrated that the frequency red shift of bright solitons is linear with propagation length, at a rate of −8t_{d}/15τ_{0}^{4} per unit propagation distance, where t_{d} is the delay time the nonlinear response of the medium (typically 6 fs) and τ_{0} is the normalized soliton duration. The temporal shift is a direct consequence of the group velocity dispersion of the fiber. The temporal shift was found to be 4t_{d}z^{2}/15τ_{0}^{4}.^{33} Note that the shifting rate is proportional to |u|^{4} because τ_{0} is the inverse of the normalized amplitude. ISRS is especially pronounced for high peak power pulses. Therefore, when a higher-order soliton is launched, the ISRS will cause soliton fission.^{34} Because of such effects, the initially bound state ceases to exist and solitons of different amplitudes are separated from one another. The energies of these separating solitons are distributed in such way to ensure conservation of momentum. …

iu_{z} − 1/2u_{tt} + |u|^{2}u = τ_{d} \frac{∂|u|^{2}}{∂t} u,  (9)

… We introduce a simple model of the shift for fundamental dark solitons:

\frac{dω}{dz} = \frac{4τ_{d}}{15} κ^{4},  (10a)
\frac{dθ}{dz} = ω.  (10b)

…

We next study the behavior of dark solitons when both adiabatic amplification and ISRS are present. Figure 6(a) shows the pulse shape of a fundamental dark soliton in such a case. In this case the fundamental dark soliton loses its amplitude contrast, as it does in Fig. 4(a), and the ISRS temporal shift is enhanced by the effect of adiabatic gain. In the simple model described by Eqs. (5) and (10a), the temporal shift by ISRS has the functional form

θ = \frac{τ_{d}}{60Γ^{2}} (e^{4Γz} − 1 − 4Γz).  (11)

…

In summary, the ISRS causes a shift of dark solitons. A salient feature of dark solitons is that the rate of such shift is half the value for bright solitons, when the slow loss of contrast is neglected. This leads to a better stability of a fundamental dark soliton against such perturbations. However, the situation for higher-order dark solitons is more complicated because there are amplitude changes associated with each soliton. The symmetry of higherorder solitons is broken. Red secondary solitons gain energy at the expense of blue ones. The primary soliton ceases to be a fundamental dark soliton and suffers energy losses and frequency blue shift.

5. EVEN DARK PULSES

Even dark pulses,^{18,19} which are symmetric functions of time centered around the pulse, can be simply generated by driving the MZI with a short electric pulse. In this case, only an intensity modulation that gives a dip of optical power under the constant background is required.

The propagation characteristics of even dark pulses are different from those of odd dark solitons. Generally, even dark pulses split into pairs of secondary dark solitons without the formation of a primary dark soliton. The energy of the input dark pulse is then redistributed into a certain number of paired secondary dark solitons. Secondary dark solitons, which are called gray solitons^{35} have nonzero intensity of pulse centers. …

If we define the amplitudes of the secondary soliton pairs as

κ_{n} = κ_{0} − Δ_{n},  (12)

then the nth order secondary pulse shape (n = 1, 2, 3, …) has the form

u_{n}(z, t) = κ_{0} \frac{(λ_{n} − iν_{n})^{2} − ν_{n} exp[2ν_{n}(t − t_{n0} − λ_{n}z)]}{1 + ν_{n} exp[2ν_{n}(t − t_{n0} − λ_{n}z)]}e^{iz},  (13)

…

6. CONCLUSIONS

We have discussed the possibility of using the waveguide Mach–Zehnder interferometer to generate a variety of dark solitons under constant background. Under optimal operation, 30% less input power and driving voltage are required than for complete modulation. The generated solitons can have good pulse quality and stimulated Raman scattering process can be utilized to compensate for fiber loss and even to amplify and compress the dark solitons. Generally speaking, when a constant gain coefficient is included in the NLSE, adiabatic amplification of the dark soliton is possible, as long as the gain Γ is kept small …

When a fundamental dark soliton is adiabatically amplified in the presence of ISRS, the spectral shift and thus the temporal shift follow a simple rule, Eq. (11), which takes into consideration the exponentially increasing amplitude and linear dependence of the shift on the propagation distance. We find that such a simple model can accurately describe the behavior of fundamental dark solitons subject to adiabatic amplification and ISRS. The propagation properties of even dark pulses are also studied, with special attention to the distribution of energies among secondary gray solitons. Despite their more complicated nature, our results demonstrate that the partition of the energy is similar for quite different input pulse shapes, as long as they have the same background intensity and total energy for the input pulse. One can use the partition rule obtained here to predict the amplitude of secondary solitons produced from an input even dark pulse.

ACKNOWLEDGMENTS

The authors thank the reviewers for their constructive comments. This research was supported by National Science Foundation grant ECS-91960-64.

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FIGURES

Fig. 1. The dark solitons generated by the waveguide Mach-Zehnder interferometer. The amplitude of the input cw light is chosen to be a = π/2 for (a)-(c). The parameter δ is (a) 0.8, (b) 0.5, and (c) 0.2. Part (d) is the case of optimal operation when a = 1.33, and δ = 0.7. In all cases, the output pulse shapes are plotted as solid curves while the dashed curves are input pulse shapes. The pulses shown here are at a propagation distance of z = 4.

Fig. 2. Dark solitons under constant gain. Pulse shapes (solid) when Γ=0.05 (a) and 1(b), after certain propagation distance, Γz=1.6, as compared to input pulse shapes (dashed). (c): The pulse duration, relative to its input, as a function of Γz at various Γ. The solid curve is the adiabatic approximation obtained by perturbation method. Three values of Γ are used: Γ = 0.05 (dotted); 0.2 (dash-dotted); and 1 (dashed). Negative Γz depicts the case of loss.

Fig. 3. The pulse shapes of amplified dark solitons. (a) δ = 0.5, β = 2ln1.05, Γ_{p}L = 2, after 8 amplifying cycles (solid); (b) δ = 0.5, β = 2ln1.02, Γ_{p}L = 2, after 16 amplifying cycles (solid);

(c) δ = 0.5, β = 2ln1.02, Γ_{p}L = 0.5, after 16 amplifying cycles (solid); (d) The input pulse is the same as in Fig. 1(c), β = 2ln1.05, after 8 amplification periods (solid). The input pulse shapes are plotted as dashed curves.

Fig. 4. (a) The shape of a fundamental dark soliton after a propagation distance of 40 (solid). The normalized time delay τ_{d} = 0.01. The dashed curve is the input pulse shape. (b) The trace of the soliton (solid) as a function of propagation distance for the situation described by (a). The dotted curve represents the case for a fundamental bright soliton under similar conditions.

Fig. 5. The shape of a higher-order dark soliton [2 tanh(t)] after a propagation distance of 12 for τ_{d} = 0.01 (solid). The dotted curve is the pulse if τ_{d} = 0, i.e., without ISRS.

Fig. 6. (a) The shape of an adiabatically amplified fundamental dark soliton (solid). Γ = 0.05, z = 16, and τ_{d} = 0.01. The dotted curve corresponds to the pulse shape without ISRS; (b) The trace of the soliton (solid) for the case of (a). The dotted curve is a fit as described by Eq. (11) in the text.

Fig. 7. Even dark pulses when the input pulse (dashed curve) is κ_{0}| tanh(t)|. (a) κ_{0} = 1.56, and z = 8 (solid curve), (b) κ_{0} = 4 and z = 3.75 (solid curve). In (c), three different input pulses are assumed: 8| tanh(t)| (solid curve), 8[1 − exp(−t^{2}/τ_{g}^{2})]^{1/2} (dotted curve), and 8[1 − sech(t/τ_{s})] (dashed curve). The propagation distance is z = 8.

Fig. 8. Even dark pulses generated from MZI. The pulse after MZI is 2 cos(π/2sech^{2}t) (dashed curve) and the shape of secondary dark solitons is shown by the solid curve for z = 4.

Fig. 9. The loss compensated even dark pulses. The input pulse is 2 cos(π/2sech^{2}t) (dotted curve), the secondary solitons with fiber losses compensated by stimulated Raman scattering is shown by the solid curve. For comparison, the pulse shape without fiber losses is shown by the dashed curve (same as Fig. 8). The propagation distance is 4.

TABLES

Table 1. Amplitudes of Secondary Even Dark Pulses

Input Pulse Shape

Δ_{n}Values  κ_{0}|tanht|  κ_{0}[1 − exp(−t^{2}/τ_{g}^{2})]^{1/2}  κ_{0}[1 − sech(t/τ_{s})]  Avg.  Range
Δ_{1}  0.34  0.30  0.21  0.28  ±25%
Δ_{2}  1.56  1.41  1.26  1.41  ±11%
Δ_{3}  2.47  2.26  2.28  2.34  ±6%
Δ_{4}  3.52  3.25  3.31  3.36  ±6%
Δ_{5}  4.45  4.26  4.42  4.38  ±6%
Δ_{6}  5.52  5.35  5.50  5.50  ±5%