Discrete solitons in optical fiber systems with large pre-dispersion

Shiva Kumar; Masataka Nakazawa

doi:10.1364/OE.25.019923

1. Introduction

Discrete solitons have drawn significant attention in the context of nonlinear waveguide arrays [1–4]. Recently, a discrete nonlinear Schrödinger equation is derived for describing the field evolution in optical fiber system with pre-dispersion [5]. In this paper, we extend the work of [5] and derive conditions for the existence of discrete solitons. When ultrashort pulses co-propagate in a fiber, they nonlinearly interact and exchange energy. However, if the amplitudes of these short pulses are properly chosen, they propagate without exchanging energy. The amplitude distribution to have the undistorted propagation takes the secant-hyperbolic form and may be termed discrete soliton in analogy with its counterpart in spatial domain [1–4]. The discrete soliton refers to the envelope of the peak of periodically placed short pulses of pulse width T₀ separated by T. We have derived an analytical expression for the peak power to form a discrete soliton and our result shows that the peak power is directly proportional to the square of the pulse separation T and it is inversely proportional to the square of the pulse width T₀.The peak power is also inversely proportional to the pre-accumulated dispersion of the pre-dispersive device that precedes the transmission fiber. As the pulse separation T and pre-accumulated dispersion approach zero, discrete soliton proposed here approaches the classical soliton [6,7].

The dynamics of discrete soliton is governed by the discrete nonlinear Schrödinger equation (NLSE) in which the effective dispersion and nonlinear coefficients are varying with distance although the dispersion and nonlinear coefficients of the transmission fiber are constants. The analytically predicted peak power to form the discrete soliton is larger than that found numerically. This is because in our initial analytical calculation, we approximated the effective dispersion and nonlinear coefficients to be constants that are equal to the values at the beginning of the transmission fiber. However, by taking the path-averaged dispersion and nonlinear coefficients in the discrete NLSE, we recalculated the peak power which is found to be in good agreement with the numerical solution. The similar situation arises in the case of classical soliton – effective nonlinear coefficient decreases as a function of distance due to fiber loss. Strictly speaking, soliton solution does not exist for such a system. However, guiding center soliton can exist which is a solution of the NLSE with varying dispersion and nonlinear coefficients replaced by their path-averaged values [8]. In the case of continuous NLSE, the required power to form the guiding center soliton is higher than that of a soliton in a lossless fiber. Hence, they are also called power-enhanced solitons or path-averaged solitons. In contrast, in the case of discrete NLSE, the required power to form the path-averaged discrete soliton is less than that of a soliton solution of the discrete NLSE in which the variations in dispersion and nonlinear coefficients are ignored. This difference is due to the time-frequency interchange caused by the pre-dispersive device. When the peak power is less than the threshold power required to form the discrete soliton, our results show that it undergoes compression. In contrast, in the continuous case, the nonlinear pulse broadens if the launch power is less than the power required to form a soliton since the dispersive effects dominate the nonlinear effects. When the pre-accumulated dispersion is sufficiently large, the effective dispersion and nonlinearity terms appearing in the discrete NLSE are roughly constants and in this case, due to its close resemblance with NLSE, it is likely to be integrable and it may be possible to develop a communication system with inverse nonlinear discrete Fourier transform at the transmitter and nonlinear discrete Fourier transform at the receiver, which would be the discrete analogue (with time-frequency interchange) of the idea proposed in [9,10].

The study of soliton is closely related to modulation instability – the breakup of a continuous wave (CW) into a train of ultrashort pulses. The analogous situation in our case is the stability of an isolated sinc pulse in the anomalous dispersion fiber. Our results show that the sinc pulse is not stable and breaks up into many temporally separated sinc pulses. The pump sinc pulse at t = 0 amplifies the signal sinc pulse centered at nT if the separation nT is less than a certain threshold. The nonlinear interaction of the pump pulse and signal pulse generates an idler pulse at –nT. The interaction of pump, signal and idler pulses leads to pulses at −2nT and 2nT. If the pump is sufficiently strong, this process would continue generating pulses at integral multiples of nT and the end result is likely to be the formation of discrete soliton if this fiber system is enclosed in a cavity. The idler pulse at –nT is the conjugated copy of the signal pulse and hence, this result could be useful for time domain optical amplification and phase conjugation. Typically, four wave mixing in a highly nonlinear fiber is used for optical phase conjugation (OPC). A disadvantage of the conventional OPC scheme employing highly nonlinear fibers is that the phase-conjugated copy is a frequency-shifted copy of the signal which would interfere with the other channels of a wavelength division multiplexed (WDM) system if it is used in midpoint spectral inversion applications. Hence, another OPC to shift the frequency of the phase-conjugated signal is required. In contrast, in the proposed scheme, the phase conjugated copy of the signal is of the same frequency (but separated in time) as the signal.

Before propagating through the nonlinear transmission fiber, we assume that the signal passes through a linear pre-dispersive device which can be either realized in digital domain or optical domain. In this paper, we develop the concept of discrete chirp transform (DChT) and show that the signal samples at the input and output of the pre-dispersive device are related by the DChT. The computational cost of evaluating the output of a linear dispersive fiber using DChT approach is N/2log₂(N) + 2N complex multiplications (N = number of samples), which is nearly half of the conventional scheme based on FFT. The concept of DChT is also helpful to find the amplitudes of short pulses propagating in the nonlinear transmission fiber from the samples of the field envelope at various propagation distances.

2. Discrete channel model

The evolution of optical field envelope in optical fibers is described by the nonlinear Schrödinger equation (NLSE) [8,11],

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

where

α

,

β_{2}

, and

γ

are the loss, dispersion and nonlinear coefficients, respectively. Figure 1 shows the block diagram of a fiber-optic system consisting of a pre-dispersive device, transmission fiber (TF) and dispersion compensating fiber (DCF). We assume that the dispersion and nonlinearity profiles are

\begin{array}{l} β_{2} (z) = β_{2, p r e}, z < 0 \\ = β_{2, t r}, 0 \leq z \leq L_{t r} \\ = β_{2, p o s t}, z > L_{t r}, \end{array}

and

\begin{array}{l} γ (z) = 0, z < 0 \\ = γ_{0}, 0 \leq z \leq L_{t r} \\ = 0, z > L_{t r} . \end{array}

Here,

β_{2, p r e}

is the dispersion coefficient of the pre-dispersive device which can be realized using a real fiber or fiber Bragg grating (FBG) prior to the transmission fiber or a digital filter in the digital signal processing (DSP) unit of the optical transmitter.

β_{2, t r}

is the dispersion coefficient of the transmission fiber. We have chosen the origin of z- coordinate to be the input end of the transmission fiber. We assume that the signs of dispersions of the pre-dispersive device and that of the transmission fiber are the same and hence, it is different from the dispersion managed system widely studied in the past [12,13]. The dispersions of pre-dispersion device and the transmission fiber are fully compensated using the dispersion compensating fiber (DCF) in optical domain or using the dispersion compensating filter in digital domain.

Fig. 1 Block diagram of a fiber-optic system. Tx = transmitter, TF = transmission fiber, DCF = dispersion compensating fiber and Rx = Receiver.

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Let us first consider the signal propagation in the pre-dispersive device which is linear. Suppose the input of the pre-dispersive device is a train of impulses,

u (t, - L_{p r e}) \equiv u_{i n} (t) = \sum_{n = - N / 2}^{N / 2 - 1_{}} A_{n} δ (t - n T),

where L_pre is the length of the pre-dispersive device, N is the number of impulses, T is the separation between impulses and

A_{n}

is the amplitude of the impulse located at nT. We assume that the sequence

{A_{n}}

is periodic, i.e.

A_{n + N} = A_{n} .

Let

s_{0} = \int_{- L_{p r e}}^{0} β_{2, p r e} (z) d z,

be the pre-accumulated dispersion. The output of the pre-dispersive device is given by [5],

u (t, 0) \equiv u_{o u t}^{p r e} (t) = \frac{1}{\sqrt{- i 2 π s_{0}}} \sum_{n = - N / 2}^{N / 2 - 1_{}} A_{n} \exp [- i {(t - n T)}^{2} / (2 s_{0})] .

For a practical implementation, the impulse function in Eq. (4) could be replaced by a suitably chosen sampling function,

u_{i n} (t) = \sum_{n = - N / 2}^{N / 2 - 1_{}} A_{n} δ (t - n T) \to \sum_{n = - N / 2}^{N / 2 - 1_{}} \frac{A_{n}}{T_{0}} sinc [(t - n T) / T_{0}],

where T₀ is the pulse width and

sinc (t) = \frac{\sin (π t)}{π t} .

If the signal at the output of the pre-dispersion fiber is known,

A_{n}

can be extracted by the inverse transformation (see Appendix A)

A_{n} = \frac{T}{\sqrt{i 2 π s_{0}}} \int_{- N T / 2}^{N T / 2} u_{o u t}^{p r e} (t) \exp [i {(t - n T)}^{2} / (2 s_{0}) d t,

if

s_{0} = \frac{- N T^{2}}{2 π | k |},

where k is a non-zero integer. In this paper, we choose k to be unity. The inverse transformation is possible only if the pre-accumulated dispersion is chosen such that it satisfies Eq. (9). In the rest of the paper, we assume that the pre-accumulated dispersion,

s_{0}

satisfies Eq. (9).

2.1 Discrete chirp transform (DChT)

Let us discretize the time t as mT, where m is an integer and the output of pre-dispersive device, $u_{o u t}^{p r e} (m T) = u_{m} .$ Replacing the integral in Eq. (8) by the rectangular rule, we define the discrete chirp transform as (DChT) as

\begin{array}{l} A_{n} = DChT{u_{m}; m \to n}, \\ = \frac{T^{2}}{\sqrt{i 2 π s_{0}}} \sum_{m = - N / 2}^{N / 2 - 1_{}} u_{m} \exp {- i π {(m - n)}^{2} / N} . \end{array}

Equation (10) is obtained from Eq. (8) with the assumption that

s_{0}

satisfies Eq. (9). By discretizing the time in Eq. (6), inverse discrete chirp transform (IDChT) can be obtained as

\begin{array}{l} u_{m} = IDChT{A_{n}; n \to m}, \\ = \frac{1}{\sqrt{- i 2 π s_{0}}} \sum_{m = - N / 2}^{N / 2 - 1_{}} A_{n} \exp {i π {(m - n)}^{2} / N} . \end{array}

Here,

u_{m}

is the signal sample at the output of the pre-dispersive device at

t = m T

and

A_{n}

is the signal sample at the input of pre-dispersive device at

t = n T

; The discrete chirp transform pairs (Eqs. (10) and (11)) provide the relation between the two. The computational cost of Eq. (11) (or Eq. (10)) is N/2log₂(N) + 2N complex multiplications (See Appendix B). In contrast, the conventional approach of taking the Fourier transform, multiplying the spectrum by the fiber transfer function and inverse Fourier transforming requires N log₂(N) + N complex multiplications. The proposed approach may also be used for dispersion compensation in the receiver or to implement the dispersion step in digital back propagation (DBP) leading to about 50% savings in computational cost.

To verify Eq. (10), we simulated the response of the pre-dispersive device using Eq. (1) for $- L_{p r e} < z < 0.$ The input of the pre-dispersive device is

u_{i n} (t) = \sum_{n = - N / 2}^{N / 2 - 1_{}} \frac{A_{n}}{T} \sin c [(t - n T) / T],

where N = 256 and T = 10 ps. The accumulated dispersion of pre-dispersive device is s₀ = −4000 ps². The value of

A_{n}

is randomly chosen between

- \sqrt{P_{0}} T

and

\sqrt{P_{0}} T

with equal probability, i.e.,

A_{n} = \pm \sqrt{P_{0}} T .

The peak power P₀ is 15 mW. Figures 2(a) and 2(b) show the normalized field envelopes at the input and output of the pre-dispersive device, respectively, obtained by solving Eq. (1) using the fast Fourier transform (FFT) approach. Taking the signal samples of the output of the pre-dispersive device at t = mT as u_m (i.e. discretizing the field envelope shown in Fig. 2(b), DChT of u_m is performed using Eq. (10). $A_{n}$ calculated in this way is shown as ‘ + ’ in Fig. 3. From Eq. (12), we see that $A_{n}$ is simply u_in(nT)T and this $A_{n}$ is shown as circles in Fig. 3. As can be seen from Fig. 3, $A_{n}$ extracted from the output of the pre-dispersive device using DChT matches quite well with that calculated from the input.

Fig. 2 Normalized field envelope at the input (a) and output (b) of the pre-dispersive device.

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Fig. 3 Comparison of A_n obtained by DChT and that at the input ( = u_in(nT)T).

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2.2 Discrete solitons

So far we considered the linear propagation in the pre-dispersive device. Next, we investigate the nonlinear propagation in the transmission fiber. Suppose the input to the transmission fiber is given by Eq. (6). The optical field envelope in the transmission fiber may be written as [5]

u (t, z) = \frac{1}{\sqrt{- i 2 π s (z)}} \sum_{n = - N / 2}^{N / 2 - 1_{}} A_{n} (z) \exp {- i {(t - n T)}^{2} / [2 s (z)]},

where

s (z) = s_{0} + β_{2, t r} z,

is the accumulated dispersion.

A_{n} (0) = A_{n} (- L_{p r e}),

since the amplitudes of pulses do not change in the linear pre-dispersive device. Let

U_{n} (z) = \exp {- i n^{2} T^{2} / [2 s (z)]},

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

Substituting Eq. (14) in Eq. (1) and using Eq. (17), we find [5]

i \frac{d B_{n}}{d z} + \frac{β_{2, t r} {(2 π n T)}^{2}}{8 π^{2} s^{2} (z)} B_{n} + \frac{γ_{0} e^{- α z}}{2 π | s (z) |} \sum_{l} \sum_{m} B_{l} B_{m} B_{l + m - n}^{*} = 0, 0 \leq z \leq L_{t r}

The last term in Eq. (18) is a double convolution which corresponds to two products in a Fourier domain. Besides, if we interchange time and frequency and identify

2 π n T

as the angular frequency

ω = 2 π f,

we see that the second term is analogous to the dispersion term in NLSE in the Fourier domain. However, unlike u(t,z), B_n(z) is a discrete variable and hence, we take the discrete Fourier transform (DFT) of Eq. (18) to find

i \frac{d {\tilde{B}}_{m}}{d z} - \frac{β_{2, t r}}{8 π^{2} s^{2} (z)} \sum_{k = - N / 2}^{N / 2 - 1} {\tilde{x}}_{k} {\tilde{B}}_{m - k} + \frac{γ_{0} e^{- α z}}{2 π | s (z) |} | {\tilde{B}}_{m} |^{2} {\tilde{B}}_{m} = 0, 0 \leq z \leq L_{t r}

where

\begin{array}{l} {\tilde{B}}_{m} = DFT {B_{n}; n \to m}, \\ = \sum_{n = - N / 2}^{N / 2 - 1} B_{n} \exp (i 2 π n m / N), \end{array}

and

{\tilde{x}}_{k} = - DFT {{(2 π n T)}^{2}; n \to k} / N .

Equation (19) provides the discrete channel model for the fiber optic system shown in Fig. 1. Equation (19) does not contain any new physics that is not already present in Eq. (1). However, it gives a different perspective, as will be illustrated with examples below. In addition, in order for Eq. (19) to be valid, a non-zero pre-accumulated dispersion s₀ is required. If s₀ is zero, the second and third terms would become singular at z = 0. Also, the accumulated dispersion s(z) should not be zero in the transmission fiber as it causes singularity in Eq. (19). Since we have assumed that the sign of dispersion of pre-dispersive device is the same as that of transmission fiber, s(z) is not zero in the transmission fiber. It is possible to have the signs of s₀ and

β_{2, t r}

to be opposite as long as

| β_{2, t r} | L_{t r} < | s_{0} | .

The effective dispersion and nonlinear coefficients in this discrete channel model are given by

β_{2, e f f} = \frac{β_{2, t r}}{4 π^{2} s^{2} (z)}, γ_{e f f} = \frac{γ_{0}}{2 π | s (z) |} .

The effective dispersion and nonlinear coefficients are varying as a function of distance and the integrability of Eq. (19) is not yet known. However, due to its close resemblance with NLSE, one would expect Eq. (19) to be integrable when

α = 0

and the pre-accumulated dispersion is large, i.e.

| s_{0} | > > | β_{2, t r} | L_{t r}

so that

s (z) \approx s_{0} .

If $\partial^{2} u / \partial t^{2}$ in the NLSE (Eq. (1)) is approximated by the central difference, $(u_{m + 1} - 2 u_{m} + u_{m - 1}) / T^{2}$ , where $u_{m} = u (m T),$ one obtains a discrete nonlinear Schrödinger equation (DNLS) [14,15] which is quite different from Eq. (19) since the dispersion and nonlinear terms in the DNLS do not depend on the accumulated dispersion. The DNLS exhibits chaotic behavior and is likely to be not integrable [15].

The discrete channel model (Eq. (19)) has been derived under the assumption that signal pulses are impulses, which require infinite bandwidth. However, for a practical realization (numerical simulation or experiment), the available bandwidth is limited. For example, if the simulation bandwidth is B, it amounts to approximating the impulse by a sinc pulse of width T₀ = 1/B. We first show the equivalence of the discrete NLSE (Eq. (19)) and continuous NLSE (Eq. (1)) under this condition through numerical examples. Let the input of pre-dispersive device be

u_{i n} (t) = \sum_{n = - N / 2}^{N / 2 - 1_{}} \frac{A_{n, i n}}{T} \sin c [(t - n T) / T] .

Due to our choice of sampling function, the signal sample at nT,

A_{n, i n}

is simply

u_{i n} (n T) T .

The energy of the sinc pulse centered at nT is proportional to

| A_{n, i n} |^{2} .

Similarly, the output of the fiber optic link (i.e. the output of DCF) is

u_{o u t} (t) = \sum_{n = - N / 2}^{N / 2 - 1_{}} \frac{A_{n, o u t}}{T} \sin c [(t - n T) / T],

and

A_{n, o u t}

can be calculated as

A_{n, o u t} = u_{o u t} (n T) T .

However, this simple relation is valid only when the dispersion is fully compensated. At any point z in the transmission fiber,

A_{n} (z)

may be extracted from the signal sample at z using Eq. (8) with s₀ replaced by s(z). In order to validate the discrete NLSE, we first calculate

B_{n} (0)

as (See Eq. (17))

B_{n} (0) = A_{n, i n} U_{n} (0) .

Note that

A_{n, i n}

does not change in the pre-dispersive device as a function of distance since it is assumed to be linear. Equation (19) is solved using the symmetric split-step Fourier scheme [16]. In the dispersion step,

γ_{0}

is set to zero and inverse discrete Fourier transform (IDFT) of Eq. (19) yields

\begin{array}{l} \frac{d B_{n}^{}}{d z} = i β_{2, e f f} (z) Ω_{n}^{2} B_{n} / 2, \\ Ω_{n} = 2 π n T . \end{array}

The solution of Eq. (27) is

\begin{array}{l} B_{n} (Δ z / 2) = B_{n} (0) \exp [i Ω_{n}^{2} \int_{0}^{Δ z / 2} β_{2, e f f} (z) d z / 2], \\ = B_{n} (0) \exp [\frac{i Ω_{n}^{2}}{8 π^{2}} (\frac{1}{s (0)} - \frac{1}{s (Δ z / 2)})] . \end{array}

The nonlinear step is done by setting

β_{2, e f f} (z)

= 0 in Eq. (19).

{\tilde{B}}_{m} (Δ z) = {\tilde{B}}_{m} (0) \exp [i θ_{N L} (Δ z)],

where

θ_{N L} (Δ z) = \frac{γ_{0} | {\tilde{B}}_{m} (0) |^{2}}{2 π} \int_{0}^{Δ z} \frac{\exp (- α z) d z}{| s_{0} + β_{2, t r} z |} .

When

α

= 0, Eq. (30) is simplified as

θ_{N L} (Δ z) = \frac{γ_{0} | {\tilde{B}}_{m} (0) |^{2}}{2 π | β_{2, t r} |} ln (1 + \frac{β_{2, t r} Δ z}{s_{0}}) .

At the end of the transmission fiber,

A_{n} (L_{t r}) = A_{n, o u t}

is calculated as

A_{n, o u t} = B_{n} (L_{t r}) U_{n}^{- 1} (L_{t r}) .

For the simulation, the following parameters are assumed throughout unless otherwise specified. $β_{2, t r} = - 20 {ps}^{2} /km, γ_{0} = 1 .1 W^{- 1} /km, L_{t r} = 80 km, s_{0} = - 12787 {ps}^{2}, T = T_{0} = 10 ps,$ and the fiber loss is ignored. The number of samples is 7232 and N = 904. This means that there are 904 symbols (‘-1’,’1’ or ‘0’) and the number of samples per symbol is 8. The peak power, P₀ is 15 mW. The value of $A_{n}$ is randomly chosen between $- \sqrt{P_{0}} T$ (for symbol ‘-1’) and $\sqrt{P_{0}} T$ (for symbol ‘1’) with equal probability for $n \in [- 32, 32]$ and the rest of $A_{n}$ are chosen to be zero.

Figure 4 shows the real part of the field envelope at the input and output of the transmission fiber obtained by solving Eq. (1) using the symmetric split-step Fourier scheme. $A_{n, o u t}$ is calculated by solving the discrete NLSE (Eq. (19)) using Eqs. (26)-(32) and is shown as ‘o’ in Fig. 5. The output of DCF $u_{o u t} (t)$ is calculated numerically solving the NLSE (Eq. (1)) and $A_{n, o u t}$ is extracted from $u_{o u t} (t)$ using Eq. (25) and is shown as ‘ + ’ in Fig. 5. For comparison, $A_{n, i n}$ ( = $u_{i n} (n T) T)$ is shown as dotted line in Fig. 5. From Fig. 4, we see that the field envelopes at the input and output of the transmission fiber look like noise due to the interference of sinc pulses broadened significantly by the pre-dispersive device. However, $A_{n, i n}$ at the input of the transmission fiber (shown in Fig. 5) has a well-defined structure. In the transmission fiber, the sinc pulses transfer energy among themselves leading to fluctuations in A_n shown in Fig. 5. It would be of interest to see if there exists a distribution of amplitudes A_n such that there would be no exchange of energy among the pulses and $| A_{n} |^{2}$ would remain invariant as a function of distance, even if the field envelope may look noisy. The discrete NLSE (Eq. (19)) facilitates to find such a distribution as explained below.

Fig. 4 Real part of the optical field envelope at the input and output of the transmission fiber.

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Fig. 5 Comparison of the real part of A_n obtained by solving the NLSE and discrete NLSE. The dotted line, ‘ + ’ and ‘o’ show $A_{n, i n}$ , $A_{n, o u t}$ extracted from NLSE (Eq. (1)) and $A_{n, o u t}$ obtained using the discrete NLSE (Eq. (19)), respectively.

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When the absolute of pre-accumulated dispersion |s₀| is much larger than $| β_{2, t r} | L_{t r},$ s(z) in Eq. (19) may be approximated by s₀. In this case, effective dispersion $β_{2, e f f}$ and nonlinear coefficient $γ_{e f f}$ become constants and Eq. (19) admits a solution if N >> M and $α = 0$ [5]

{\tilde{B}}_{m} (z) = {\tilde{B}}_{0} sech (m / M) \exp (i μ z),

{\tilde{B}}_{0}^{2} = \frac{| β_{2, e f f} | N^{2} T^{2}}{γ_{e f f} M^{2}} .

Taking the IDFT of Eq. (33), we find

B_{k} (z) = B_{0} sech (k / K) \exp (i μ z),

\begin{array}{l} B_{0} = \frac{{\tilde{B}}_{0} M π}{N}, \\ K = \frac{N}{M π^{2}} . \end{array}

Using Eqs. (17) and (35), we find

\begin{array}{l} A_{k} (z) = B_{0} sech (k / K) \exp [i μ z + i k^{2} T^{2} / (2 s_{0})], 0 \leq z \leq L_{t r} \\ = A_{k} (0), z < 0. \end{array}

Equation (37) represents the discrete soliton. If the peaks of the pulses are prescribed in accordance with Eq. (37) at the input of the pre-dispersive device, they would propagate undistorted in the transmission fiber. Although the optical field envelope may be distorted due to the dispersion of pre-dispersion device and that of the transmission fiber,

| A_{k} |^{2}

would be invariant as a function of z. Using Eq. (37), we find that the optical field envelope at the input of pre-dispersion device to excite the discrete soliton should be of the form:

u_{i n} (t) = \frac{B_{0}}{T_{0}} \sum_{k = - N / 2}^{N / 2_{}} sech (k / K) \sin c [(t - n T) / T_{0}] \exp [i k^{2} T^{2} / (2 s_{0})] .

Note that, in addition to secant-hyperbolic envelope, a quadratic chirp factor is required at the input to excite a discrete soliton. Using Eqs. (34) and (36), the peak power is

P_{t h} \equiv {(\frac{B_{0}}{T_{0}})}^{2} = \frac{π | β_{2, t r} | T^{2}}{2 γ_{0} | s_{0} | T_{0}^{2}} .

The peak power is directly proportional to the square of the separation between sinc pulses and inversely proportional to the square of the pulse width T₀. If T = T₀, the peak power is independent of the separation between the pulses. The peak power is inversely proportional to the absolute of pre-accumulated dispersion s₀ and it would become infinite if s₀ = 0. So, it is essential to have

s_{0} \neq 0

for the existence of this type of soliton. In the case of classical soliton, the peak power is inversely proportional to the square of the pulse width. However, in the case of discrete soliton, the peak power is independent of the pulse width parameter K.

Figure 6 shows the evolution of $| A_{n} |^{2}$ as a function of distance in the transmission fiber, obtained by solving Eq. (1) with Eq. (38) as input. A_n is obtained from the field envelope using Eq. (10) with u_m representing discretized field envelope at z. In this simulation, we assumed that T = T₀ and K = 5. For this case, according to the analytical estimate given by Eq. (39), the peak power required to form the discrete soliton is 2.23 mW. In the simulation, we optimized the peak power to have nearly undistorted propagation (for A_n) and found that the optimum peak power is 2.12 mW which is less than the analytical prediction. The reason for the discrepancy is due to the fact that in Eq. (19), the effective dispersion and nonlinear coefficients are functions of distance and under the assumption that the pre-accumulated dispersion is large, we approximated s(z) by s₀ so that we could have a soliton solution of the form given by Eq. (33), and Eq. (38) is obtained based on this approximation. When the dispersion and nonlinear coefficients change as a function of propagation distance, guiding center soliton or path-averaged soliton can exist which is a solution of the NLSE (Eq. (1)) with dispersion and nonlinearity coefficients are replaced by their respective path-averaged values [8]. Using this concept, we find that the path-averaged discrete soliton can exist if we replace the varying dispersion and nonlinearity profiles in Eq. (19) by their path-averaged values, i.e.

i \frac{d {\tilde{B}}_{m}}{d z} - \frac{< β_{2} >}{2} \sum_{k = - N / 2}^{N / 2} {\tilde{x}}_{k} {\tilde{B}}_{m - k} + < γ > | {\tilde{B}}_{m} |^{2} {\tilde{B}}_{m} = 0,

where

\begin{array}{l} < β_{2} > = \frac{1}{L_{t r}} \int_{0}^{L_{t r}} β_{2, e f f} (z) d z, \\ = \frac{β_{2, t r}}{4 π^{2} s_{0} s (L_{t r})}, \end{array}

and

\begin{array}{l} < γ > = \frac{1}{L_{t r}} \int_{0}^{L_{t r}} γ_{e f f} (z) d z, \\ = \frac{γ_{0}}{2 π | β_{2, t r} | L_{t r}} \ln [s (L_{t r}) / s_{0}] . \end{array}

Replacing

β_{2, e f f}

and

γ_{e f f}

in Eq. (34) by and

< β_{2} >

and

< γ >

, respectively, the peak power required to form the path-averaged discrete soliton is

Fig. 6 Propagation of discrete soliton. T = T₀. K = 5.

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P_{t h} = \frac{π β_{_{2, t r}}^{2} L_{t r} T^{2}}{2 γ_{0} | s_{0} | | s (L_{t r}) | T_{0}^{2} \ln [s (L_{t r}) / s_{0}]} .

In our parameter space, we find $P_{t h}$ = 2.11 mW using Eq. (43) which is close to the value of 2.12 mW obtained by numerically optimizing the launch power to have undistorted propagation. In the case of classical soliton, the power required to form the guiding center soliton is higher than that in the lossless case [8] and hence they are also called power enhanced solitons. In contrast, in the case of discrete soliton, we find that the power required to form the path-averaged discrete soliton is less than that in the case in which the effective dispersion and nonlinearity (see Eq. (22)) are approximated by their values at z = 0. The reason is due to the time and frequency interchange as illustrated in the examples corresponding to Figs. 11 to 14.

The red lines in Fig. 7 show the distribution of $| A_{n} |^{2}$ at the output of transmission fiber and the blue line shows the envelope of $| A_{n} |^{2}$ at the transmission fiber output which is obtained by using the curve fitting function ${sech}^{2} [t / (K T)],$ which matches quite well with the distribution of $| A_{n} |^{2} .$

Fig. 7 Distribution of $| A_{n} |^{2}$ at the output of transmission fiber (red line). The envelope (blue line) is obtained by using a curve fitting function ${sech}^{2} [t / (K T)] .$ The parameters are the same as in Fig. 6.

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The peak power required to form the discrete soliton is independent of the pulse width parameter K (See Eq. (39) or Eq. (43)). So far we assumed that K = 5. Now we choose K = 50 and T₀ = T, and numerically solve Eq. (1) with the launch power of 2.12 mW which is the same as that in Fig. 7. Figure 8 shows $| A_{n} |^{2}$ at the input and output of the transmission fiber for this case and it can be seen that the peak of envelope of $| A_{n} |^{2}$ is the same as that in Fig. 7, but the width is larger.

Fig. 8 $| A_{n} |^{2}$ at the input and output of the transmission fiber when K = 50. The rest of the parameters are the same as that of Fig. 6.

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Figures 9(a) and 9(b) show the optical power distribution at the input and output of the transmission fiber when the pulse width T₀ = T and T₀ = T /2, respectively. When T₀ = T /2, the required launch power to form the discrete soliton is four times that of the case with T = T₀ (see Eq. (39) or Eq. (43)). When T₀ = T (T₀ = T/2), the launch power to the pre-dispersive device is 2.12 mW (8.48 mW), but the peak power at the input of the transmission fiber is 65 mW (102 mW), which is due to the compression caused by chirping and propagation in a dispersive medium. For both cases, the power distribution at the output of the transmission fiber differs from that at its input. Comparing Figs. 8 and 9(a), we see that $| A_{n} |^{2}$ remains invariant as a function of transmission distance whereas the power distribution is not. The circles and ‘ + ’ in Fig. 10 shows $| A_{n} |^{2}$ extracted from the field envelope at the output of the transmission fiber for the case of T₀ = T/2 and T₀ = T. Although the optical power distribution at the transmission fiber output does not have the secant hyperbolic form when T₀ = T/2 (see Fig. 9(b)), $| A_{n} |^{2}$ extracted from the field envelope has the secant hyperbolic form and it matches with that obtained for the case of T₀ = T. Since an impulse can be approximated by different kinds of pulses, the discrete NLSE (Eq. (19)) should be valid for any type of short pulses. Our simulation with short Gaussian pulses have provided qualitatively similar results as with sinc pulses. However, we found that simulations with sinc pulses matched with the theory better than that with short Gaussian pulses. Also, any sinc pulse with T₀ = T/l, where l is a positive integer provides results that matches well with the theory.

Fig. 9 The power distribution at the input and output of the transmission fiber. (a) T₀ = T (b) T₀ = T/2. The other parameters are the same as that of Fig. 8.

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Fig. 10 Comparison of $| A_{n} |^{2}$ for the case of T₀ = T/2 and T₀ = T. The other parameters are the same as that of Fig. 8.

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The physical interpretation of the discrete soliton is as follows. First consider the nonlinear interaction between short signal pulses (sinc or any other type) located at –T, 0 and T leading to first order ghost pulses centered at −2T and 2T due to intra-channel FWM [17–19]. The nonlinear interaction of signal pulses and first order ghost pulses leads to second order ghost pulses at −3T and 3T, and so on. The ghost pulses pick energy from the signal pulses and could transfer energy back to signal pulses. After sufficiently longer distances, equilibrium will be established such that there would be no transfer of energy among signal and ghost pulses. The energy distribution of pulses centered at nT take the secant hyperbolic form under such equilibrium condition. If there are a large number of signal pulses centered at nT at the fiber input and if their energy distribution is properly chosen, there would be no energy transfer among them right from the beginning and hence $| A_{n} |^{2}$ would be invariant as a function of distance, corresponding to discrete soliton shown in Figs. 6 and 7 or Figs. 8 and 10. When the signal and ghost pulses interact, the phase matching condition dictates that the ghost pulses to be centered at nT, where n is an integer and if the pulses are very short, the temporal spacing between the pulses is empty. Hence, discrete soliton corresponds to the envelope of the peak of short pulses. As the pulse separation $T \to 0,$ and $s_{0} \to 0,$ the discrete soliton approaches the classical soliton.

Figure 11 shows $| A_{n} |^{2}$ at the input and output of the transmission fiber when the peak power P = 1.5 mW < P_th and T = T₀. Figure 12 shows the corresponding 3-D plot. The input excitation is the same as that in Fig. 8 except that the peak power is now 1.5 mW. From Figs. 11 and 12, we see that the envelope of $| A_{n} |^{2}$ is compressed at the output. In contrast, in the case of classical soliton, when the signal power of a secant hyperbolic pulse is less than the soliton power, the pulse broadens. The reason for this difference is as follows. When P < P_th, the nonlinear mechanism is such that the pulses at the edges transfer energy to those near the center. Another explanation is that the pre-dispersive device acts as an optical Fourier transformer [20,21] leading to interchange of time and frequency, which can be seen in the second term of Eq. (18). The broadening in time domain leads to compression in frequency domain if we interpret nT as frequency. In other words, in the continuous limit, $A_{n}$ corresponds to classical soliton in frequency domain rather than in time domain.

Fig. 11 Compression of $| A_{n} |^{2}$ when the peak power P = 1.5 mW < P_th. The other parameters are the same as that of Fig. 8.

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Fig. 12 Evolution of $| A_{n} |^{2}$ in the transmission fiber when the peak power P < P_th. The other parameters are the same as that of Fig. 8.

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Figure 13 shows $| A_{n} |^{2}$ at the input and output of the transmission fiber when the peak power P = 2.5 mW > P_th and T = T₀. Figure 14 shows the corresponding 3-D plot. As can be seen, the envelope of $| A_{n} |^{2}$ is broadened.

Fig. 13 Broadening of $| A_{n} |^{2}$ when P = 2.5 mW > P_th. The other parameters are the same as that of Fig. 8.

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Fig. 14 Evolution of $| A_{n} |^{2}$ when the peak power P > P_th. The other parameters are the same as that of Fig. 8.

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Just like classical solitons, discrete solitons may find applications in nonlinear signal processing, logic gates and communications. The additional degree of freedoms provided by the pulse separation T, pulse width T₀ and the pre-accumulated dispersion s₀ may alleviate the design constraints in some systems. If the discrete NLSE (Eq. (40)) is integrable, it may be possible to develop a communication system with inverse discrete nonlinear Fourier transform at the transmitter and discrete nonlinear Fourier transform at the receiver similar to the proposals in [9,10].

The periodically placed sinc pulses form a temporal lattice and it would be interesting to see the impact of defect in such a lattice and the possibility of trapping a weak light pulse at the defect site, which would be the subject of future research.

3. Modulation instability

In the context of continuous NLSE (Eq. (1)), modulation instability refers to the breakup of cw beam into ultrashort pulses and it has been studied widely [22–26]. Due to the time-frequency interchange caused by the pre-dispersive device, in our case, the analogous scenario is the instability of a sinc pulse. In this section, we investigate the stability of a single sinc pulse launched to the fiber system shown in Fig. 1. The input to the pre-dispersive device is

u_{i n} (t) = \sqrt{P_{0}} \sin c(t / T_{0}),

where

P_{0} = A_{0}^{2} / T_{0}^{2} .

The stability of this pulse can be conveniently analyzed using the discrete NLSE (Eq. (40)). First we look for the solution of Eq. (40) in the following form

{\tilde{B}}_{m} = {\tilde{B}}_{0} \exp (i μ z), m \in [- N / 2 N / 2 - 1],

where

{\tilde{B}}_{0}

is assumed to be real. Equation (45) is the discrete analogue of the CW solution of the NLSE [16]. Substituting Eq. (45) in Eq. (40), we find

μ = < γ > {\tilde{B}}_{0}^{2} .

The above result is obtained by noting that the second term of Eq. (40) with

{\tilde{B}}_{m}

given by Eq. (45) is zero since

B_{n} = IDFT{{\tilde{B}}_{m}, m \to n} = {\tilde{B}}_{0} \exp (i μ z) δ_{n 0,}

IDFT {\sum_{k}^{} {\tilde{B}}_{m - k} {\tilde{x}}_{k}, m \to n}} = - B_{n} {(2 π n T)}^{2} δ_{n 0} = 0.

In order to investigate the stability of the solution given by Eq. (45), we perturb it slightly, i.e.

{\tilde{B}}_{m} = ({\tilde{B}}_{0} + {\tilde{ε}}_{m}) \exp (i < γ > B_{0}^{2} z), m \in [- N / 2 N / 2 - 1] .

Here,

{\tilde{ε}}_{m}

is a small perturbation with

| {\tilde{ε}}_{m} | < < | {\tilde{B}}_{m} | .

Eq. (49) is equivalent to studying the stability of the sinc pulse, i..e.,

u (t, z) = \sqrt{P_{0}} \sin c(t / T_{0})exp[i θ (z)] + ε (t, z) .

However, it is hard to investigate the stability of the sinc pulse analytically using the NLSE Eq. (1). Instead, the modulation instability of the discrete NLSE is analytically tractable. We follow the approach similar to that used for the stability analysis of CW beam [16]. Substituting Eq. (49) in Eq. (40) and retaining only the terms that are of first order in

{\tilde{ε}}_{m}

, we find

i \frac{d {\tilde{ε}}_{m}}{d z} - \frac{< β_{2} >}{2} \sum_{k = - N / 2}^{N / 2} {\tilde{x}}_{k} {\tilde{ε}}_{m - k} + < γ > {\tilde{B}}_{0}^{2} ({\tilde{ε}}_{m} + {\tilde{ε}}_{m}^{*}) = 0.

Let

{\tilde{ε}}_{m} = a \exp {i [2 π m n / N - K_{n} z]} + b \exp {- i [2 π m n / N - K_{n} z]} .

Without loss of generality, we assume that a and b are real. Substituting Eq. (52) in Eq. (51), we find

[\begin{matrix} δ + K_{n} & < γ > {\tilde{B}}_{0}^{2} \\ < γ > {\tilde{B}}_{0}^{2} & δ - K_{n} \end{matrix}] [\begin{matrix} a \\ b \end{matrix}] = 0,

where

δ = < γ > {\tilde{B}}_{0}^{2} + < β_{2} > 2 (π n T) {}^{2}.

To obtain Eq. (52), we have made use of Eq. (21) and the following convolution relation

\sum_{k = - N / 2}^{N / 2 - 1} \exp [i 2 π (m - k) n / N] {\tilde{x}}_{k} = - {(2 π n T)}^{2} \exp (i 2 π m n / N) .

Equation (53) has a nontrivial solution only if the determinant is zero which leads to

K_{n} = \pm | < β_{2} > n | π T {[{(2 π n T)}^{2} + sgn (< β_{2} >) Ω_{c}^{2}]}^{1 / 2},

where

sgn ()

denotes the Signum function and

Ω_{c}^{2} = 4 < γ > {\tilde{B}}_{0}^{2} / | < β_{2} > | .

If

2 π n T < Ω_{c} and < β_{2} > < 0,

K_n becomes imaginary and hence, these components grow exponentially with distance. The gain coefficient is

g_{n} = 2 Im (K_{n}) = | < β_{2} > n | π T {[Ω_{c}^{2} - {(2 π n T)}^{2}]}^{1 / 2} .

This means that the sinc pulse centered at t = 0 becomes unstable and generates pulses centered at t = nT if

2 π n T < Ω_{c} .

The maximum gain occurs at

n_{\max} = \pm round [\frac{1}{\sqrt{2} π T} {(\frac{< γ > {\tilde{B}}_{0}^{2}}{| < β_{2} > |})}^{1 / 2}],

where round() gives the nearest integer. Using Eqs. (41) and (42), Eq. (59) may be rewritten as

n_{\max} = \pm round [\frac{T_{0}}{\sqrt{π} T} {(\frac{P_{0} γ_{0} s_{0} s (L) \ln [s (L) / s_{0}]}{| β_{2}_{, t r} |^{2} L_{t r}})}^{1 / 2}] .

In this section, we assume the following parameters:

β_{2, t r} = - 20 {ps}^{2} /km, γ_{0} = 11 W^{- 1} /km,

L_{t r} = 10 km, s_{0} = - 12787 {ps}^{2}, and T = T_{0} = 10 ps .

When P₀ = 18 W, from Eq. (60), we find

n_{\max}

= 202. Equation (1) is numerically solved with the following initial condition:

\begin{array}{l} u_{i n} (t) = \sqrt{P_{0}} \sin c(t / T) + ε_{n} \sin c[(t - n T) / T], \\ ε_{n} = 0.05 \sqrt{P_{0}} \exp [i (^{n T) 2} / (2 s_{0})], \end{array}

and n = 202.The red and blue lines in Fig. 15 show the input power (before the pre-dispersive device) and the output power (i.e. after the DCF) distributions, respectively. As can be seen, the signal sinc pulse centered at nT is amplified. The nonlinear interaction of the pump pulse centered at t = 0 and signal pulse centered at t = nT generates an idler pulse centered at t = -nT. In addition, time domain FWM of pulses centered at –nT, 0 and nT leads to pulses at −2nT and 2nT. If the pump power is sufficiently large, this process would continue generating pulses at integral multiples of nT and the end result is likely to be the formation of discrete soliton if this fiber system is enclosed in a cavity. In the continuous case, the end result of modulation instability is the formation of periodic train of classical solitons [24,26]. The idler pulse shown in Fig. 15 at t = −202T (see the blue curve) is the phase conjugated copy of the signal pulse. Hence, this result could have potential applications for time domain optical amplification and phase conjugation. Typically, optical phase conjugation (OPC) is achieved by making use of four wave mixing in a highly nonlinear fiber. A disadvantage of the conventional OPC scheme is that the phase-conjugated copy is a frequency-shifted copy of the signal which would interfere with the other channels of a WDM system if it is used in midpoint spectral inversion applications. In the proposed scheme, the phase conjugated copy of the signal is of the same frequency (but separated in time) as the signal.

Fig. 15 Input and output power distributions. $β_{2, t r} = - 20 {ps}^{2} /km, γ_{0} = 11 W^{- 1} /km,$ $L_{t r} = 10 km, s_{0} = - 12787 {ps}^{2}, T = T_{0} = 10 ps, and P_{0} = 18 W .$

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To show that Eq. (60) predicts the location of the maximum gain, we perform the following numerical experiment. First, Eq. (1) is solved with the initial condition given by Eq. (61) for the fixed n. The gain is calculated by dividing the signal power of the sinc pulse (located at nT) at the output by that at the input. This process is repeated by changing n from 5 to 300. The pump sinc pulse at t = 0s is left unchanged as the location of signal sinc pulse is changed. Figure 16 shows the gain as a function of n for various pump powers. As the pump power increases, the peak of the gain increases and the gain occurs over a wide time region. From Fig. 16, we see that the gain maximum occurs at n = 170, 200 and 230 when the pump power P₀ is 12W, 18W and 24W, respectively. Equation (61) predicts that the location of the maximum gain is 170, 202 and 233 when P₀ is 12W, 18W and 24W, respectively.

Fig. 16 Gain vs location of the signal pulse. The parameters are the same as that of Fig. 15.

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4. Conclusions

The periodically placed short pulses exchange energy as they propagate in an optical fiber. However, if the amplitudes of these pulses are properly chosen, they would propagate undistorted and the discrete soliton refers to the envelope of the peak of these pulses. An analytical expression for the peak power of a discrete soliton is derived. The analytical result predicts that the peak power is directly proportional to the square of the pulse separation and inversely proportional to the square of the pulse width, which is verified by numerical simulation. The modulation instability in discrete NLSE revealed that an isolated single pulse is unstable in the anomalous dispersion fiber and it generates several temporally separated sinc pulses. The pump pulse amplifies the signal pulse if the temporal separation between the pump and signal is less than a certain threshold. This result could have a potential application for time domain optical amplification. The nonlinear interaction of the pump pulse and signal pulse generates an idler pulse which is a phase-conjugated copy of the signal, which could be used for time domain optical phase conjugation. An advantage of this scheme is that the phase-conjugated signal is of the same frequency as that of the signal, but separated in time.

In linear dispersive device, we have shown that the input and output are related by the discrete chirp transform (DChT). The computational cost of evaluating the output of a linear dispersive fiber using DChT approach is N/2log₂(N) + 2N complex multiplications, which is nearly half of the conventional scheme based on FFT. The DChT is also used to extract the amplitudes of the short pulses from the optical field envelope in the case of nonlinear transmission.

Appendix A

Let

X_{m} = \int_{- N T / 2}^{N T / 2} u_{o u t}^{p r e} (t) \exp [i {(t - m T)}^{2} / (2 s_{0})] d t .

Using Eq. (6) in Eq. (62), we find

\begin{array}{l} X_{m} = \frac{1}{\sqrt{- i 2 π s_{0}}} \sum_{n = - N / 2}^{N / 2 - 1_{}} A_{n} e^{- i (n^{2} - m^{2}) T^{2} / (2 s_{0})} \int_{- N T / 2}^{N T / 2} \exp [- i 2 π t (m - n) T / (2 π s_{0})] d t, \\ = \frac{1}{\sqrt{- i 2 π s_{0}}} \sum_{n = - N / 2}^{N / 2 - 1_{}} A_{n} e^{- i (n^{2} - m^{2}) T^{2} / (2 s_{0})} Y_{m n}, \end{array}

where

Y_{m n} = \frac{\exp [- i π N T (m - n) f_{0}] - \exp [i π N T (m - n) f_{0}]}{i 2 π f_{0} (n - m)},

f_{0} = \frac{T}{2 π s_{0}} .

When m = n,

Y_{m n} = N T .

When

m \neq n, Y_{m n}

could be zero if

N T f_{0} = k,

where k is a non-zero integer. We assume that

s_{0} < 0.

Substituting Eq. (66) in Eq. (65), we find

s_{0} = \frac{- N T^{2}}{2 π | k |} .

Thus, we have

Y_{m n} = N T δ {}_{m n}.

Using Eq. (68) in Eq. (63) and rearranging, we obtain

\begin{array}{l} A_{m} = \frac{X_{m} T}{\sqrt{i 2 π s_{0}}}, \\ = \frac{T}{\sqrt{i 2 π s_{0}}} \int_{- N T / 2}^{N T / 2} u_{o u t} (t) \exp [i {(t - m T)}^{2} / (2 s_{0})] d t . \end{array}

Appendix B

Let

U_{n} = \exp (i π n^{2} / N) .

From Eq. (11), we have

\begin{array}{l} u_{m} = \frac{1}{\sqrt{- i 2 π s_{0}}} \sum_{n = - N / 2}^{N / 2 - 1_{}} A_{n} \exp {i π (m^{2} + n^{2} - 2 m n) / N}, \\ = \frac{1}{\sqrt{- i 2 π s_{0}}} U_{m} \sum_{n = - N / 2}^{N / 2 - 1_{}} A_{n}^{'} \exp {- i 2 π m n / N}, \end{array}

where

A_{n}^{'} = A_{n} U_{n} .

The summation in Eq. (71) is the DFT of the sequence

{A_{n}^{'}},

and hence it can be computed using the fast Fourier transform (FFT) with N/2log₂N complex multiplications. The computational cost of multiplying with

U_{n}

or

U_{m}

is N complex multiplications. So, the total cost of calculating u_m is N/2log₂N + 2N complex multiplications.

Acknowledgment

Shiva Kumar acknowledges the support of Japan Society for the Promotion of Science (JSPS), Japan for this research work.

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Discrete solitons in optical fiber systems with large pre-dispersion

Abstract

1. Introduction

2. Discrete channel model

2.1 Discrete chirp transform (DChT)

2.2 Discrete solitons

3. Modulation instability

4. Conclusions

Appendix A

Appendix B

Acknowledgment

References and links

Cited By

Figures (16)

Equations (74)

Optics Express