Impulse response of nonlinear Schrödinger equation and its implications for pre-dispersed fiber-optic communication systems

Shiva Kumar; Jing Shao; Xiaojun Liang

doi:10.1364/OE.22.032282

1. Introduction

The propagation dynamics of the pulse in a cubically nonlinear dispersive medium such as an optical fiber is described by the nonlinear Schrödinger equation (NLSE) [1, 2]. Optical soliton is a normal mode of the nonlinear system described by the NLSE, which can be integrated by means of inverse scattering transform (IST) [3, 4]. Zakharov and Shabat [2] solved the NLSE using IST and obtained soliton and breather solutions. The breathers or higher order solitons undergo periodic compression and expansion with a soliton period. Impulse response approach to nonlinear dispersive propagation in fiber has been studied in the past [5, 6]. In Ref. [5], the impulse response approach of linear system is extended to nonlinear system using a self-consistent time-transformation. In Ref. [6], an impulse response approach is used to calculate the multiplicative correction due to the interplay between chromatic dispersion and Kerr nonlinearity. In this paper, we obtain an exact solution of the NLSE for an impulse input. However, we found that there is a singularity in the phase. To remove this singularity, we introduced pre-dispersion which can be added either in electrical domain at the transmitter or in optical domain prior to transmission. The exact solution in this case has a phase factor which is described by the exponential integral. Next, we investigated the nonlinear interaction among pulses in a fiber due to periodically placed impulses at the input and analyzed the conditions under which they propagate over long distances without exchanging energy among them.

When a cluster of CW beams of different frequencies propagate in optical fiber, they exchange energy through the process known as four wave mixing (FWM). Eventually the amplitudes of CW beams reach an equilibrium in which there is no exchange of energy among them and they take secant-hyperbolic shape corresponding to soliton spectrum. There exists an alternate explanation in time domain. The dual of classical FWM is time-domain FWM or intra-channel FWM (IFWM) [7–10] and the dual of CW signal is a Dirac delta function in time domain (CW signal is an impulse function in frequency domain). When a cluster of closely spaced impulses propagate in fiber, they exchange energy through IFWM. However, if the weights of the impulses have secent-hyperbolic shapes, they do not exchange energy and propagate stably as solitons over long distances. In order to have soliton propagation, the impulses have to be infinitesimally closer. In this paper, we have investigated if it is possible to propagate a large number of periodically placed impulses over large distances without exchanging energy among them. We found that if the impulse weights at the input have a secant-hyperbolic shape and a proper chirp factor, they propagate without change in shape over long distances just like the soliton of NLSE. The amplitude of the soliton solution depends on system parameters such as pre-accumulated dispersion, separation between the impulses and the dispersion of the transmission fiber. When the impulses are infinitesimally closer, this solution becomes the classical soliton of the continuous NLSE. We have derived a discrete NLSE which describes the evolution of the discrete Fourier transform of the product of the impulse weights and a chirp factor. We note that the discrete NLSE can be easily obtained by discretizing the continuous NLSE. In such a discrete NLSE, the dispersion term would be directly proportional to fiber dispersion coefficient. However, in the discrete NLSE derived here, the effective dispersion term is inversely proportional to the square of the accumulated dispersion and the effective nonlinear term is inversely proportional to the absolute accumulated dispersion. It is not yet known if the discrete NLSE derived here can be integrated by IST. However, we have numerically found that the discrete NLSE admits higher order soliton solutions which undergo periodic compression and expansion with a certain period, similar to its continuous analogue.

In the context of discrete NLSE, if the effective dispersion length is much longer than the effective nonlinear length, the equation becomes significantly simplified. In this case, intra-channel cross-phase modulation (IXPM) and intra-channel four wave mixing (IFWM) [7–11] vanish in the transformed system. We have obtained nonlinear eigenmodes which form the natural basis for description of signal propagation and signal and noise nonlinear interaction in highly pre-dispersed fiber-optic systems.

2. Impulse response

The evolution of optical field envelope is described by NLSE [12]

i \frac{\partial u}{\partial z} - \frac{β_{2}}{2} \frac{\partial^{2} u}{\partial t^{2}} + γ_{0} e^{- α z} {| u |}^{2} u = 0,

where α, β₂, and γ₀ are the loss, dispersion and nonlinear coefficients, respectively. In a linear fiber (γ₀ = 0), when an impluse is launched,

u (t, 0) = A δ (t),

the optical field in the fiber is

u (t, z) = \frac{A}{\sqrt{- i 2 π β_{2} z}} e^{- i \frac{t^{2}}{2 β_{2} z}} .

In the presence of nonlinearity, we look for a solution of Eq. (1) in the form,

u (t, z) = \frac{A}{\sqrt{- i 2 π β_{2} z}} e^{- i \frac{t^{2}}{2 β_{2} z} + i v (z)},

Substituting Eq. (4) in Eq. (1), we obtain

\frac{A}{\sqrt{- i 2 π β_{2} z}} {i [(- \frac{1}{2}) z^{- 1} + \frac{i t^{2}}{2 β_{2} z^{2}} + i \frac{d v (z)}{d z}] - \frac{β_{2}}{2} (- \frac{i}{β_{2} z} - \frac{t^{2}}{β_{2}^{2} z^{2}}) + γ_{0} e^{- α z} \frac{{| A |}^{2}}{2 π | β_{2} | z}} = 0 .

Simplifying Eq. (5), we obtain

v (z) = \frac{γ_{0} {| A |}^{2}}{2 π | β_{2} |} \int_{0}^{z} \frac{e^{- α x}}{x} d x .

The integrand of Eq. (6) has a singularity, which should be expected due to the impulse input. The singularity can be avoided by using pre-dispersion. Suppose

β_{2} (z) = {\begin{matrix} β_{2 -}, & for z < 0 \\ β_{2 +}, & for z < 0, \end{matrix}

γ = {\begin{matrix} 0, & for z < 0 \\ γ_{0}, & for z < 0 . \end{matrix}

Let

s_{0} = \int_{- L}^{0} β_{2 -} (z) d z

be the pre-accumulated dispersion. The pre-dispersion can be realized using a high dispersion fiber prior to transmission fiber or a digital dispersion filter in the digital signal processing (DSP) unit of the optical transmitter [11, 13]. Now for z > 0, Eqs. (4) and (6) are modified as

u (t, z) = \frac{A}{\sqrt{- i 2 π s (z)}} e^{- i \frac{t^{2}}{2 s (z)} + i γ_{0} \frac{{| A |}^{2}}{2 π} θ (z)},

θ (z) = \int_{0}^{z} \frac{e^{- α x}}{s (x)} d x .

s (z) = s_{0} + β_{2 +} z .

θ(z) in Eq. (9) does not diverge only if s₀ + β₂₊z does not cross 0 for any z. In this paper, we assume that pre-accumulated dispersion s₀ has the same sign as β₂₊ so that s(z) does not cross 0. Under this condition, Eq. (10) can be written in a closed form as [14]

θ (z) = e^{α s_{0} / β_{2 +}} [Ei (\frac{- α s (z)}{β_{2 +}}) - Ei (\frac{α s_{0}}{β_{2 +}})],

where Ei(−x) is the exponential integral.

Ei (- x) = - \int_{x}^{\infty} \frac{e^{- t}}{t} d t .

Equation (9) is an exact solution of the NLSE when the input (at z = −L) is a single impulse. To our knowledge, this solution is not known in literature. Suppose the input consists of a train of impluses,

u_{in} (t) = \sum_{n = - N / 2}^{N / 2 - 1} A_{n} δ (t - n T),

where N is the number of impulses, which is assumed to be large. The optical field in the transmission fiber for this input may be written as

u (t, z) = \sum_{n = - N / 2}^{N / 2 - 1} \frac{A_{n} (z) e^{- i {(t - n T)}^{2} / 2 s (z)}}{\sqrt{- i 2 π s (z)}}, for z \geq 0 .

In the absence of nonlinear interaction with the neighboring pulses, we have

A_{n} (z) = A_{n} (0) e^{i γ_{0} {| A_{n} (0) |}^{2} θ (z) / 2 π} .

Equation (16) includes the effect of self-phase modulation (SPM) only. However, due to IXPM and IFWM [7–10], the pulses undergo amplitude/phase shifts. Substituting Eq. (15) in Eq. (1), we find

i \sum_{n} \frac{d A_{n}}{d z} e^{- i \frac{{(t - n T)}^{2}}{2 s (z)}} + \frac{γ_{0} e^{- α z}}{2 π | s (z) |} \sum_{k} \sum_{l} \sum_{m} A_{k} A_{l} A_{m}^{*} F_{k l m} = 0,

where F_klm = e^{−i[(t−kT)²+(t−lT)²−(t−mT)²]/2s(z)}. Multiplying Eq. (17) by e^{i(t−jT)²/2s(z)} and integrating from −t to t with t → ∞, we find

i \sum_{n} \frac{d A_{n}}{d z} δ_{j n} + \frac{γ_{0} e^{- α z}}{2 π | s (z) |} \sum_{k} \sum_{l} \sum_{m} A_{k} A_{l} A_{m}^{*} Y_{k l m, j} = 0,

where δ_jn is a Kronecker delta function and

\begin{array}{l} Y_{k l m, j} & = lim_{t \to \infty} \frac{1}{2 t} \int_{- t}^{t} F_{k l m} e^{i {(τ - j T)}^{2} / 2 s (z)} d τ, \\ = lim_{t \to \infty} \frac{1}{2 t} e^{- i (k^{2} + l^{2} - m^{2} - j^{2}) T^{2} / 2 s (z)} \int_{- t}^{t} e^{i (k + l - m - j) τ T / s (z)} d τ . \end{array}

Y_klm,j will be non-zero only if m = k + l − j. In this case,

Y_{k l j} \equiv Y_{k l m, j} = e^{- i [k^{2} + l^{2} - {(k + l - j)}^{2} - j^{2}] T^{2} / 2 s (z)} .

So, now Eq. (18) becomes

i \frac{d A_{j}}{d z} + \frac{γ_{0} e^{- α z}}{2 π | s (z) |} \sum_{k} \sum_{l} A_{k} (z) A_{l} (z) A_{k + l - j}^{*} Y_{k l j} = 0 .

In the absence of nonlinear effects (γ₀ = 0), from Eq. (21) we find

\frac{d A_{j}}{d z} = 0,

which indicates that there is no interaction among pulses in a linear medium. Let

U_{k} (z) = e^{- i k^{2} T^{2} / 2 s (z)},

where k is an integer. Equation (20) may be written as

Y_{k l j} = U_{k} U_{l} U_{k + l - j}^{*} e^{i j^{2} T^{2} / 2 s (z)} .

Let

B_{k} (z) = A_{k} (z) U_{k} (z) .

Using Eqs. (23)–(25) in Eq. (21), we find

i \frac{d B_{j}}{d z} + \frac{j^{2} T^{2} β_{2 +}}{2 s^{2} (z)} B_{j} + \frac{γ_{0} e^{- α z}}{2 π | s (z) |} \sum_{k} \sum_{l} B_{k} B_{l} B_{k + l - j}^{*} = 0 .

The second term is similar to dispersion in NLSE. If we take the Fourier transform of Eq. (1), the second term would be β₂ω²ũ(ω, z)/2, where ũ(ω, z) = ℱ{u(t, z)}, ℱ denotes the Fourier transformation. Therefore, in Eq. (26), β₂₊/s²(z) may be interpreted as the effective dispersion. However, unlike u(t, z), B_j(z) is a discrete variable and hence, we consider the discrete Fourier transform,

DFT {B_{j}; j \to m} = {\tilde{B}}_{m} = \sum_{j = - N / 2}^{N / 2 - 1} B_{j} e^{- i 2 π j m / N} .

Taking the discrete Fourier transform of Eq. (26) and noting that a convolution becomes product in spectral domain (and vice versa), we find

i \frac{d {\tilde{B}}_{m}}{d z} - \frac{β_{2 +} T^{2}}{2 s^{2} (z)} \sum_{k = - N / 2}^{N / 2 - 1} {\tilde{B}}_{m - k} {\tilde{x}}_{k} + \frac{γ e^{- α z}}{2 π | s (z) |} {| {\tilde{B}}_{m} |}^{2} {\tilde{B}}_{m} = 0,

where

{\tilde{x}}_{k} = DFT {j^{2}; j \to k} .

Equation (28) may be interpreted as a discrete analogue of the NLSE. Since A_n may be interpreted as signal sample at nT, a discrete NLSE can be easily obtained for A_n [15, 16]. In such a discrete NLSE, the dispersion term would be directly proportional to fiber dispersion coefficient. However, in Eq. (28), the effective dispersion term is inversely proportional to the square of accumulated dispersion and the effective nonlinear term in inversely proportional to the absolute accumulated dispersion. The discrete NLSE in Eq. (28) does not describe A_n, instead it discribes the evolution of the DFT of B_n which is the product of A_n and U_n. In the absence of pre-dispersion (s₀ = 0), the effective dispersion term and the effective nonlinear term of Eq. (28) diverge at z = 0 and hence, pre-dispersion is essential for the solution of Eq. (28). In the terminology of Ref. [16], Eq. (28) is a discrete self-trapping (DST) equation of the form [17],

i \frac{d {\tilde{B}}_{m}}{d z} + ε \sum_{k} m_{j k} {\tilde{B}}_{k} + γ {| {\tilde{B}}_{m} |}^{2} {\tilde{B}}_{m} = 0,

where [m_jk] is a f × f coupling matrix. In Eq. (1), when α = 0, dispersion and nonlinear coefficients are constants for z > 0 and hence, it admits soliton solutions. However, in Eq. (28), the effective dispersion and nonlinear coefficients are varying with distance due to s(z). If we choose the pre-dispersion such that s₀ >> β₂₊L_tr where L_tr is the length of the transmission fiber, we can approximate s(z) as s₀. In this case with α = 0 km⁻¹, we look for a soliton solution of Eq. (28) in the form

{\tilde{B}}_{m} (z) = {\tilde{B}}_{0} sech (\frac{m}{M}) e^{i μ (z)} .

Equation (28) is numerically solved using the split-step Fourier method with the initial condition,

{\tilde{B}}_{m} (0) = {\tilde{B}}_{0} sech (\frac{m}{M}) .

Figure 1 shows the evolution of |B̃_m|² in the transmission fiber. As can be seen, when B̃₀ is less than a threshold B̃_th, we see the broadening effect and when B̃₀ = B̃_th, the pulse shape is retained throughout. Figure 2 shows the evolution of |B_n|² obtained by taking the inverse discrete Fourier transform (IDFT) of B̃_m. As can be seen, when B̃₀ < B̃_th (Fig. 2a), the envelope of |B_n|²(= |A_n|²) becomes narrower which indicates that the pulses exchange energy among them resulting in the pulse at the center (n = 0) becoming stronger. When B̃₀ = B̃_th, pulses propagate long distances without exchanging energy among them.

Fig. 1 Evolution of B̃_m in the transmission fiber, (a) B̃₀ < B̃_th, ${\tilde{B}}_{0} = 10 \sqrt{m W} ps$ , (b) B̃₀ = B̃_th. ${\tilde{B}}_{th} = 14.9 \sqrt{m W} ps$ , M = 28, α = 0 km⁻¹, s₀ = −1.28 × 10⁴ ps², γ₀ = 1.1 W⁻¹km⁻¹.

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Fig. 2 Evolution of B_m in the transmission fiber, (a) B̃₀ < B̃_th, (b) B̃₀ = B̃_th. The parameters are the same as in Fig. 1.

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Figure 3 shows the similar result by solving Eq. (1). The impulses of Eq. (14) are approximated by Gaussian pulses of short pulse widths,

A_{n} δ (t - n T) \to \frac{A_{n}}{\sqrt{2 π} T_{0}} e^{- \frac{{(t - n T)}^{2}}{2 T_{0}^{2}}},

and Eq. (1) is solved with the following initial condition

u_{in} (t) = \sum_{n = - N / 2}^{N / 2 - 1} \frac{A_{n} (0) e^{- \frac{{(t - n T)}^{2}}{2 T_{0}^{2}}}}{\sqrt{2 π} T_{0}},

where

A_{n} (0) = B_{n} (0) e^{i \frac{n^{2} T^{2}}{2 s_{0}}},

B_{n} (0) = IDFT {{\tilde{B}}_{m} (0); m \to n},

and B̃_m(0) is given by Eq. (32). To obtain Fig. 3, the pre-accumulated dispersion is fully compensated at the receiver so that the pulse width of the Gaussian pulses at the output is the same as that at the input. As can be seen from Fig. 3, the envelope of Gaussian pulses propagate undistorted over the transmission fiber. If we had not properly chosen the input power, the nonlinear interaction among Gaussian pulses would broaden/compress the shape of the envelope. “×” in Fig. 3 show the power obtained by numerically solving Eq. (28) after converting B̃_m to A_n using Eqs. (23) and (25). The power required to form fundamental soliton is found to be

P_{s} = \frac{β_{2 +} T^{2}}{4 s_{0} γ_{0} T_{0}^{2}},

where T is the pulse separation and T₀ is the pulse width of the Gaussian pulses. Strictly speaking, the approximation of impulses by ultra-short Gaussian pulses is not really necessary. To test the validity of Eq. (28), in principle, Eq. (1) can be solved with the initial condition u(t, 0) given by Eq. (15). However, the extraction of A_n from the transmission fiber output becomes hard.

Fig. 3 Comparison of discrete NLSE (Eq. (28)) and continuous NLSE (Eq. (1)). Peak power = 35.5 mw, T = 10 ps, T₀ = 1 ps, s₀ = −1.28 × 10⁴ ps², β₂₊ = −20 ps²/km, γ₀ = 1.1 W⁻¹km⁻¹, transmission distance = 240 km.

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As pointed in Ref. [16], DST is not typically integrable when f > 2. The integrability of Eq. (28) is not known yet, and to test if it admits high order soliton solutions, we solved Eq. (28) with the initial condition

{\tilde{B}}_{m} (0) = 2 {\tilde{B}}_{th} sech (\frac{m}{M}) .

Figure 4a shows the evolution of the second order soliton. As can be seen, it undergoes periodic compression just like its continuous analogue. The soliton period is found to be

z_{0} = \frac{2 s_{0}^{2}}{π M^{2} T^{2} | β_{2 +} |} .

Figure 4b shows the evolution of |B_n|². When the envelope of B̃_m is compressed, the corresponding envelope of B_n is broadened and vice versa.

Fig. 4 Evolution of second order soliton. ${\tilde{B}}_{0} = 29.8 \sqrt{m W} ps$ . The rest of the parameters are the same as in Fig. 1.

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3. Nonlinear eigenmodes

When the effective dispersive effects are much weaker than the effective nonlinear effects in Eq. (28), i.e.,

γ P T_{0}^{2} / π > > β_{2 +} T^{2} / | s (z) |,

where P is the peak power and T₀ is the half-width at 1/e-intensity point of the Gaussian pulse that approximates the impulse, the second term in Eq. (28) can be ignored and we obtain

\frac{\partial {\tilde{B}}_{m}}{\partial z} = i \frac{γ e^{- α z}}{2 π | s (z) |} {| {\tilde{B}}_{m} |}^{2} {\tilde{B}}_{m} .

In Eq. (26), the last term is responsible for nonlinear interactions such as IXPM and IFWM among pulses. However, in Eq. (41), in the transformed system, these terms are absent and hence, the description of the nonlinear interactions becomes significantly simplified. Let

{\tilde{B}}_{m} = Y_{m} e^{i θ_{m}} .

Substituting Eq. (42) in Eq. (41), we find

Y_{m} = const,

θ_{m} (z) = θ_{m} (0) + γ {| Y_{m} |}^{2} \int_{0}^{z} \frac{e^{- α x}}{2 π | s (x) |} d x .

The solution of Eq. (41) may be written as

{\tilde{B}}_{m} (z) = {\tilde{B}}_{m} (0) e^{λ_{m} z^{'}},

where

λ_{m} = i γ {| {\tilde{B}}_{m} |}^{2}, z^{'} = \frac{1}{2 π} \int_{0}^{z} \frac{e^{- α x}}{| s (x) |} d x .

When s₀ and β₂₊ have the same sign, z′ can be written as

z^{'} = \frac{e^{α | s_{0} / β_{2 +} |}}{2 π} [Ei (- α | \frac{s (z)}{β_{2 +}} |) - Ei (- α | \frac{s_{0}}{β_{2 +}} |)] .

B̃_m may be interpreted as the nonlinear eigenmode of the fiber-optic system in the presence of pre-dispersion with the eigenvalue λ_m. These eigenmodes form a natural basis for the description of signal propagation, and signal and noise nonlinear interaction in highly pre-dispersed fiber-optic transmission systems. We note that using a different approach with stationary phase approximation, it has been shown that propagation equations can be considerably simplified in the presence of high pre-dispersion [18]. We found a few similarities and differences between Ref. [19] and our work. In this paper, we introduce a transformation B_k(z) = A_k(z)exp (−ik²T²/2s(z)) in time domain, whereas in Ref. [19], the transformation û(z, ω) ∼ Û(z, ω)exp(−iCω²/2) is used in frequency domain. In Ref [19], when the system has a small value of path-average dispersion, the average dynamic of the pulse transmission is characterized only by the nonlinear phase shift. In contrast, from Eq. (28), it follows that when the system has a very large pre-accumulated dispersion, the pulse transmission is characterized only by the nonlinear phase shift given by Eq. (45). Even when the condition given by Eq. (40) is not met, i.e. when the pre-accumulated dispersion is moderate, the nonlinear eigenmode could serve as the unperturbed solution and a first order perturbation theory could be developed for the discrete NLSE of Eq. (28). An interesting fact is that the square of the accumulated dispersion appears in the denominator of the second term in Eq. (28). This means that the effect of the second term becomes smaller for the fiber spans closer to receiver in a long haul system. Typically, in quasilinear fiber optic systems, dispersion length is much shorter than the nonlinear length. Hence linear solution (including dispersive effects) is treated as the unperturbed solution and first order correction due to nonlinear effects are calculated [8]. However, the computational complexity of the first order calculations scales as M² per sample where M is the number of neighbors with which the nonlinear interaction is significant and as a result, the digital compensation of fiber nonlinearities using first order perturbation theory is time-consuming [20, 21]. In contrast, if the nonlinear eigenmodes are treated as the unperturbed solution with the second term of Eq. (28) being treated as perturbation, the computational complexity is expected to be much smaller.

4. Conclusions

In conclusion, we have derived an exact solution of NLSE for an impulse input in the presence of pre-dispersion. The exact solution has a phase factor that is described by the exponential integral. Next, we considered the nonlinear interaction among pulses in a fiber due to periodically placed impulses at the input. We found that these pulses will propagate stably over long distances if the complex weights of impulses at the input has a secant-hyperbolic envelope and a proper chirp factor. We have derived the discrete version of the NLSE under the condition that the input of an optical fiber is a periodic train of impluses. When the accumulated pre-dispersion is large, the discrete NLSE admits soliton and breather solutions similar to its continuous analogue. In the discrete NLSE derived here, the effective dispersion term is inversely proportional to the square of the accumulated dispersion and the effect nonlinear term is inversely proportional to absolute of accumulated dispersion. The derived discrete NLSE has a solution only if the pre-accumulated dispersion is non-zero. In the context of discrete NLSE, if the effective dispersion length is much longer than the effective nonlinear length, we have obtained the nonlinear eigenmodes of the highly pre-dispersed fiber-optic system which may be useful for the description of signal propagation, and signal and noise interaction.

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Impulse response of nonlinear Schrödinger equation and its implications for pre-dispersed fiber-optic communication systems

Abstract

1. Introduction

2. Impulse response

3. Nonlinear eigenmodes

4. Conclusions

References and links

Cited By

Figures (4)

Equations (47)

Optics Express