Spectral properties of limiting solitons in optical fibers

Sh. Amiranashvili; U. Bandelow; N. Akhmediev

doi:10.1364/OE.22.030251

1. Introduction

Description of short optical pulses is usually based on the generalized nonlinear Schrödinger equation (GNLSE) [1], which yields dynamics of single-cycle [2] and even sub-cycle [3] pulses. The validity of the GNLSE is because of two reasons, (i) the standard slowly varying envelope approximation (SVEA) is relaxed by considering an unidirectional regime of pulse propagation, and (ii) the complex envelope is defined by omitting the negative-frequency components from the real field without any reference to the SVEA. On the other hand, exact analytical solutions of the GNLSE are rare and typically require either a simplified quadratic dispersion law or artificial relations between equation parameters, like in the case of Sasa-Satsuma equation.

An attractive alternative to the GNLSE is to use a special non-envelope propagation model, one of the so-called short pulse equations (SPEs). The latter are derived using a simplified dispersion law. For instance, Schäfer and Wayne [4] noticed that the relative permittivity of bulk fused silica can be precisely approximated by an extremely simple expression

ε (ω) = n_{s}^{2} (1 - ω_{p}^{2} / ω^{2}),

for all frequencies that are relevant for optical solitons. As opposed by the Drude dispersion law, n_s and ω_p are just fit parameters. For instance, selecting them as n_s = 1.4538 and ω_p/(2π) = 21.78THz, one gains the relative accuracy of ≲ 0.15% between 0.9 μm and 3.6 μm (from 83THz to 333THz), i.e., for two octaves in fused silica. From the physical side n_s is just a typical value of the refractive index, whereas ω_p indicates at which frequency the imaginary part of the refractive index cannot be neglected. The rational approximation (1) naturally applies to the transparency region with ω ≫ ω_p.

In contrast to the Taylor expansion of the dispersion relation in the GNLSE, i.e., a local approach that is very accurate but only within the convergence radius at the carrier frequency [5], Eq. (1) pretends to be a global approximation. First, it is bounded for ω → ∞, as opposed by the unphysical behavior of the GNLSE dispersion. Second, one can apply the Kramers Kronig relations to demonstrate that Eq. (1) is a kind of universal representation of the anomalous dispersion domain for any transparent material [6]. Third, simplified propagation equations yield useful analytic solutions for short optical solitons, in some cases even a full solution of the SPE is available [7]. These solutions can be used to test more general models.

In what follows we use the dispersion relation (1), to address the following fundamental question: what happens with an optical soliton when its duration approaches a single oscillation of the carrier field? The known analytical [6, 8, 9] and numerical solutions [10] suggest the following feature: a singular cusp develops at the soliton top, preventing the soliton from being too short. Such peaked solitons have originally been found outside optics, for the shallow water waves [11].

In the frequency domain the limiting shortest soliton is characterized by the following property: an exponential decay of the spectral power is replaced by a rational one. This behavior is of special interest because pulse spectrum is easier to measure than the actual value of the electric field inside the pulse. In the following we first revisit the derivation of the SPE and its limiting soliton [6, 12] and then explicitly calculate corresponding spectra.

2. Complex short pulse equation

In this section we outline the derivation of the real [4] and complex [6, 12] SPEs and demonstrate that the transition to a more simple complex SPE is dictated by the dispersion law (1).

Propagation of a short pulse in a one-dimensional setting, e.g., in a single-mode polarization-preserving optical fiber, is approximated by a nonlinear wave equation for the electric field

\partial_{t}^{2} (\hat{ε} E + χ^{(3)} E^{3}) - c^{2} \partial_{z}^{2} E = 0, where \hat{ε} [\sum_{ω} E_{ω} e^{- i ω t}] = \sum_{ω} ε (ω) E_{ω} e^{- i ω t} .

The nonlinear susceptibility of the third order χ⁽³⁾ is taken constant. Using Eq. (1) one derives the following self-consistent equation

n_{s}^{2} (\partial_{t}^{2} E + ω_{p}^{2} E) - c^{2} \partial_{z}^{2} E + χ^{(3)} \partial_{t}^{2} (E^{3}) = 0,

with the following dispersion law

β (ω) = (n_{s} / c) {(ω^{2} - ω_{p}^{2})}^{1 / 2}

for a linear e^i(βz−ωt) wave.

Recall, that typical carrier frequencies of optical solitons are considerably larger than the fit parameter ω_p from Eq. (1), as discussed in the Introduction. Therefore $β (ω) \approx (n_{s} / c) [ω - ω_{p}^{2} / (2 ω)]$ and one concludes that the third harmonics generation (THG) process is a non-resonant one for our system. Indeed, the corresponding resonance conditions ω₁ + ω₂ + ω₃ = ω and $ω_{1}^{- 1} + ω_{2}^{- 1} + ω_{3}^{- 1} = ω^{- 1}$ are incompatible for any four positive frequencies.

The SPE results from the asymptotic expansion of Eq. (3) with respect to μ = ω_p/ω₀ ≪ 1, e.g., μ = 0.1 for a standard carrier wave at 1.4 μm. Actually, we will keep O(μ²) terms and neglect O(μ⁴) ones. To derive the SPE we first consider a reference carrier wave (the SVEA is not assumed) e^{i(β₀z−ω₀t)} and calculate the phase

ω_{0} t - β_{0} z = ω_{0} [t - \frac{n_{s}}{c} z {(1 - μ^{2})}^{1 / 2}] = ω_{0} (t - \frac{n_{s}}{c} z) + μ^{2} \frac{n_{s} ω_{0}}{2 c} z + O (μ^{4}) .

Motivated by this expression we introduce the normalized field F(ζ, τ) and assume the following scaling of the new variables

τ = ω_{0} (t - \frac{n_{s}}{c} z), ζ = μ^{2} \frac{n_{s} ω_{0}}{c} z, {(\frac{3 χ^{(3)}}{8})}^{1 / 2} E (z, t) = μ n_{s} F (ζ, τ) .

A straightforward calculation yields that Eq. (3) takes the following form

2 \partial_{ζ} \partial_{τ} F + F + \frac{8}{3} \partial_{τ}^{2} (F^{3}) = μ^{2} \partial_{ζ}^{2} F .

After neglecting the last term one obtains the real SPE as derived in [4]. One should mention that a more general SPE with an additional term accounting for transition to the normal dispersion domain first appeared in [13]. To derive the complex SPE we introduce a complex-valued electric field ℱ (ζ, τ) such that by construction

\partial_{ζ} \partial_{τ} ℱ + \frac{1}{2} ℱ + \frac{1}{3} \partial_{τ}^{2} (3 {| ℱ |}^{2} ℱ + ℱ^{3}) = 0 .

It is easy to see that F = (ℱ + ℱ^*)/2 is a valid solution of the real SPE. The clear advantage of the complex representation is that the self-phase modulation (SPM) term and the THG term are now separated. Neglecting the non-resonant THG term we obtain the complex SPE

\partial_{ζ} \partial_{τ} ℱ + \frac{1}{2} ℱ + \partial_{τ}^{2} ({| ℱ |}^{2} ℱ) = 0 .

Another physical situation that directly leads to the complex SPE is for a wave with circular polarization. Two components of the real electric field are just combined into a singe complex field and one obtains the complex SPE (8) without referring to the real equation [6, 12].

To conclude this section we note that strictly speaking Eq. (8) presupposes that ℱ contains mainly positive (and ℱ^* mainly negative) frequencies. That is, the initial condition ℱ (0, τ) is chosen to be exactly the positive frequency part of F(0, τ) in accord with the most general definition of the complex envelope [1,14]. In the course of pulse propagation, the complex field ℱ (z, τ) accrues some small negative-frequency part generated by the SPM term, however, this process is non-resonant such that the negative frequencies are not further amplified.

3. Solitary solutions of the complex SPE

For a linear ℱ ∼ exp[i(ℵζ − ντ)] wave, in which the dimensionless frequency ν corresponds to the physical frequency νω₀ and the reference wave corresponds to ν = 1, Eq. (8) yields ℵ = −1/(2ν). Therefore the phase and group velocities read V_ph = −2ν² and V_gr = 2ν², in particular, V_ph + V_gr = 0. Although both velocities slightly change for a solitary wave, they still have equal magnitudes. Therefore we introduce a positive parameter γ such that

V_{phase nonlinear} = γ V_{ph} = - 2 γ ν^{2} and V_{group nonlinear} = γ V_{gr} = 2 γ ν^{2},

and assume that soliton’s phase and amplitude depend on τ − ζ/(γV_ph) and τ − ζ/(γV_gr) respectively. In accord with Eq. (9) we then try the following substitution

ℱ (ζ, τ) = f (ν τ - \frac{ζ}{2 γ ν}) exp [- i (ν τ + \frac{ζ}{2 γ ν})],

where f (ξ) is the complex shape function that depends also on γ and ν. Substituting ℱ (ζ, τ) into Eq. (8) we obtain the following ordinary differential equation for f (ξ)

{(f - 2 γ ν^{2} {| f |}^{2} f)}^{″} - (γ - 1) f + 4 i γ ν^{2} {({| f |}^{2} f)}^{'} + 2 γ ν^{2} {| f |}^{2} f = 0,

where prime denotes derivative with respect to ξ. At the soliton tails, where f (ξ) → 0, we have f″ = (γ − 1) f, such that localized solitary solutions require γ > 1 and the soliton duration is proportional to (γ − 1)^−1/2.

We now split the amplitude and the phase by setting γ^1/2νf (ξ) = a(ξ)e^iϕ(ξ) and obtain two real differential equations for a(ξ) and ϕ(ξ). The phase equation can be integrated and yields

ϕ^{'} = - \frac{(3 - 4 a^{2}) a^{2}}{{(1 - 2 a^{2})}^{2}} .

The amplitude equation A″ − (γ − 1)a − ϕ′²A − 4ϕ′a³ + 2a³ = 0 with A ≡ a − 2a³ can be multiplied by A′ and then integrated as well. Restricting ourselves to the localized solutions we obtain a “mechanical” equation for some effective potential U(a)

{a^{'}}^{2} + U (a) = 0, U (a) = - (γ - 1) \frac{(1 - 3 a^{2}) a^{2}}{{(1 - 6 a^{2})}^{2}} + \frac{(1 - 7 a^{2} + 12 a^{4}) a^{4}}{{(1 - 2 a^{2})}^{2} {(1 - 6 a^{2})}^{2}} .

An exemplary shape of U(a) for γ slightly smaller than 9/8 is shown in Fig. 1(a). The target soliton starts from the equilibrium position a = 0 for ξ → −∞ and returns to it for ξ → +∞. Physically acceptable solutions require U(a) ≤ 0. The solution of the latter inequality that is “connected” to the a = 0 state is given by

0 \leq a \leq {[\frac{4 γ - 3 - {(9 - 8 γ)}^{1 / 2}}{8 γ}]}^{1 / 2},

the right limit is labelled by the red point in Fig. 1(a). The target solitary solution exists for 1 ≤ γ ≤ 9/8. Classical solitons appear for γ = 1 + ε² with ε ≪ 1, then a = O(ε) and in the leading term Eq. (13) yields that a(ξ) = ε/cosh(εξ). As γ increases the soliton becomes shorter, the final limiting soliton is obtained for γ → 9/8 and is yielded by the equations

{a^{'}}^{2} = \frac{a^{2} (1 - 3 a^{2})}{8 {(1 - 2 a^{2})}^{2}} with a (0) = \frac{1}{\sqrt{6}} \Rightarrow a e^{B (a)} = Λ e^{- \frac{| ξ |}{2 \sqrt{2}}},

B (a) \equiv {[\frac{2}{3} {(1 - 3 a^{2})}^{1 / 2} - ln [1 + {(1 - 3 a^{2})}^{1 / 2}]]}_{0}^{a}, Λ \equiv \frac{e^{B (\frac{1}{\sqrt{6}})}}{\sqrt{6}} \approx 0.3935 .

The phase dependence of the limiting soliton is calculated from Eq. (12). Without loss of generality we take ϕ(0) = 0, then ϕ(−ξ) = −ϕ(ξ). The shape of ϕ(ξ) is shown in Fig. 2(a). It is of interest that the limiting values ϕ(±∞) = ±ϕ_∞ can be calculated analytically from (12) and (15)

ϕ_{\infty} = \int_{0}^{\infty} ϕ^{'} (ξ) d ξ = - \int_{0}^{\infty} \frac{(3 - 4 a^{2}) a^{2}}{{(1 - 2 a^{2})}^{2}} (\frac{d ξ}{d a} d a) = - \frac{4 (\sqrt{2} - 1)}{3} - arcsin \frac{1}{3},

the quantity ϕ_∞ ≈ −0.8921 appears later in the equation for the soliton spectrum.

Fig. 1 (a) An exemplary potential U(a) from Eq. (13) for γ = 9/8 − δ with δ = 10⁻³. The red point labels the upper value of a(ξ) from the inequality (14). As δ → 0, an infinite wall is formed at $a = 1 / \sqrt{6}$ , resulting in cusp formation at the top of the soliton. (b) Shape of the shortest soliton calculated from Eq. (15).

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Fig. 2 (a) Phase of the limiting soliton and (b) its power spectrum. Blue line: numerical solution, red line: expression (22). The two auxiliary thin lines show the spectrum of the fundamental soliton (dashed), and the Ω⁻⁴ power law (solid).

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4. The limiting spectrum

In this section we explicitly calculate the spectrum of the shortest soliton. At first glance, the key Eq. (15) looks too complicated and numerical treatment of the soliton spectrum seems to be the only choice. Actually, this is not the case. Our analytic approach is based on the observation that for all relevant amplitudes, $0 \leq a \leq 1 / \sqrt{6}$ , the factor $e^{B (a)}$ from Eq. (15) is a slow function that is close to 1. This happens just because $B$ (0) = 0 and $B (1 / \sqrt{6}) \approx - 0.037$ . Actually $a (ξ) \approx Λ e^{- | ξ | / (2 \sqrt{2})}$ is already a very good approximation to the full solution of Eq. (15) that is shown in Fig. 1(b). As a natural generalization we consider ξ > 0 and expand the amplitude a(ξ) and phase ϕ(ξ) in a power series with respect to $X = Λ e^{- ξ / (2 \sqrt{2})}$ .

With the help of any computer algebra system it is easy to obtain that

X = a e^{B (a)} = a - \frac{a^{3}}{4} + \frac{a^{5}}{8} + \frac{49 a^{7}}{192} + \dots \Leftrightarrow a = X + \frac{X^{3}}{4} + \frac{X^{5}}{16} - \frac{61 X^{7}}{192} + \dots

Inserting a(X) in Eq. (12) and performing direct integration we get the full solution for ξ > 0

ϕ = ϕ_{\infty} + 3 \sqrt{2} X^{2} + \frac{19 X^{4}}{2 \sqrt{2}} + \frac{457 X^{6}}{24 \sqrt{2}} + \dots,

a e^{i ϕ} = e^{i ϕ_{\infty}} (X + \frac{1 + 12 i \sqrt{2}}{4} X^{3} + \frac{- 143 + 88 i \sqrt{2}}{16} X^{5} + \dots),

where

X = Λ e^{- ξ / (2 \sqrt{2})}

, Λ and ϕ_∞ are yielded by Eq. (16) and Eq. (17). Recall that the solution for ξ < 0 follows from the fact that a(ξ) and ϕ(ξ) are even and odd functions respectively.

Finally Eq. (20), which can be extended to any power, is used to obtain the spectrum

{(a e^{i ϕ})}_{Ω} = \int_{- \infty}^{\infty} a (ξ) e^{i ϕ (ξ)} e^{i Ω ξ} d ξ = \int_{0}^{\infty} a (ξ) e^{i ϕ (ξ)} e^{i Ω ξ} d ξ + c . c .,

where Ω refers to the normalized frequency. The result reads

{(a e^{i ϕ})}_{Ω} = Λ \frac{R_{1} + S_{1} Ω}{1 + 8 Ω^{2}} + Λ^{3} \frac{R_{3} + S_{3} Ω}{9 + 8 Ω^{2}} + Λ^{5} \frac{R_{5} + S_{5} Ω}{25 + 8 Ω^{2}} + \dots,

where

R_{1} = 4 \sqrt{2} C

, S₁ = −16S,

R_{3} = 3 \sqrt{2} C - 72 S

,

S_{3} = - 48 \sqrt{2} C - 4 S

,

R_{5} = - (715 / 4) \sqrt{2} C - 220 S

,

S_{5} = - 88 \sqrt{2} C + 143 S

, and we set C = cosϕ_∞ and S = sinϕ_∞ for the sake of brevity. For Ω ≫ 1 Eq. (22) should be replaced by the power law, (ae^iϕ)_Ω ∼ Ω⁻², which should be observed for any generic cusp solution [Fig. 2(b)]. Actually, such frequencies are beyond validity of Eq. (1) and SPE, such that Eq. (22) is the only reasonable result for the soliton spectrum.

5. Conclusions

Our main result is Eq. (22) for the spectrum of the shortest soliton. As expected, the spectral power has an algebraic decay, as opposed by the exponential one for ordinary solitons. The spectral shape is non-trivial, such a function is difficult to reconstruct from a purely numerical solution. Figure 2(b) shows predictions of Eq. (22) and compares them with the numerical results for the soliton spectrum. The agreement is reasonably good for all frequencies of interest and can be easily improved by calculating further terms in Eq. (22).

Acknowledgments

Sh.A. gratefully acknowledges support by The Einstein Center for Mathematics Berlin.

References and links

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Spectral properties of limiting solitons in optical fibers

Abstract

1. Introduction

2. Complex short pulse equation

3. Solitary solutions of the complex SPE

4. The limiting spectrum

5. Conclusions

Acknowledgments

References and links

Cited By

Figures (2)

Equations (22)

Optics Express