Nonlinear distortion of optical pulses by self-produced free carriers in short or highly lossy silicon-based waveguides

Hagen Renner

doi:10.1364/OE.20.025718

1. Introduction

Silicon technology provides a promising platform for Integrated Optics since the high index contrast can be used to realize a great variety of useful components and the maturity achieved for the CMOS technology can be drawn on [1–3]. Compact and stable lowloss wavelength-selective linear components acting as filters and demultiplexers have been fabricated, amongst them ring resonators, interferometers, arrayed waveguide gratings and diffraction gratings [4]. However, if the interesting non-linear properties of silicon waveguides [5–8] shall be taken advantage of, for which high intensity levels are required, free carriers generated by two-photon absorption (TPA) often limit the range of useful operation. For instance, free-carrier absorption (FCA) limits the achievable spectral broadening when using the Kerr effect for continuum generation [9,10]. In silicon Raman amplifiers, FCA puts an upper limit on the absolute Stokes power which can be amplified at most [11], and even for low Stokes-power levels FCA restricts the maximum possible gain [12–17]. On the other hand, the attenuation by free carriers has successfully been exploited for compressing and shaping optical pulses in [18, 19].

In a number of papers dealing with non-linear optical pulse propagation in silicon waveguides [9, 10, 18–21] the discussion of the physics is based on the helpful and enlightening concept of an interplay or competition of FCA and TPA, and the individual contributions from TPA, Kerr effect, FCA and free-carrier refraction (FCR) are partially studied in isolation from the others. While the well-known analytical solution [21, 22] for pulse propagation in the presence of TPA and Kerr effect eases this discussion considerably, no such tool for describing explicitly the effects of self-produced free-carriers on the non-linear distortion of short optical pulses in silicon waveguides is available up to now [22–32]. This lack of an analytical tool as the corresponding counterpart in this discussion of FCA versus TPA is addressed in this paper.

In order to derive analytical results for the action of FCA including FCR and the Kerr effect, we intentionally omit the effect of TPA as an attenuation while keeping its function of generating free carriers. We will prove in Section 4 that this is an appropriate approximation for describing the propagation of optical pulses in short or highly lossy silicon-based waveguides such as silicon-metal plasmonic waveguides. In numerical simulations for the latter type of waveguides [30] it was observed that self-induced FCA may have a significantly stronger effect on the pulse propagation than TPA has. However, we also propose this model to serve as a tool in discussing the competition of FCA and TPA in low-loss silicon waveguides such as those based on silicon-on-insulator technology. As a result we will arrive at explicit closed-form formulations for the evolution of the pulse intensity, free-carrier concentration, Kerr-induced phase shift and the phase shift caused by FCR, and we will draw a number of general conclusions on their principal behavior in dependence of the input pulse properties.

We show in Section 5.5 that sech-shaped intensity pulses maintain their original shape and width during propagation all along the waveguide independently of the peak intensity and pulse duration. We find that self-induced FCA is an ideal limiter for sech intensity pulses, as they remain undistorted but cannot exceed a certain peak-intensity level at the waveguide output. The behavior of pulses travelling under self-induced FCA turns out to have some similarities with superluminal pulse propagation.

Further, regarding FCA as a means for pulse compression [18, 19], we discuss in Section 6 the requirements on a pulse’s shape to be compressible at all, show the sech pulse to indicate a compressibility limit and illustrate by means of the Lorentzian pulse that a widespread intensity profile causes the pulse even to be broadened by FCA instead of being compressed.

Finally, we calculate in Section 7 the attenuation a weak probe pulse experiences from the free carriers left by a preceding strong pump pulse and compare it with the CW operation. Interestingly, at high intensities the pulsed pump may exert a higher aggregate FCA attenuation on the probe pulse if the latter follows closely enough.

Since the full problem of optical pulse propagation in silicon waveguides taking both TPA and FCA into account can not be solved analytically, exact results have only been obtained purely numerically [9, 10, 18–32]. Some methods provide a simplification of the mathematical treatment with the benefit of deeper physical insight. For instance, the variational method in [33] gets rid of the time integration, but a set of differential equations with respect to the propagation distance remains to be solved numerically without accounting for the pulse distortion caused by the self-produced free carriers. The explicit analytical results given in [34] take the FCA-induced self-distortion partially into account in the sense of a successive approximation method, however, restricting itself to weak pulse distortions only. In both methods the introduced error has to be interpreted mathematically. In contrast to that, the present model explicitly omits the physical effect of TPA-induced pulse attenuation but fully accounts for the attenuation (FCA) and phase shift (FCR) induced by the TPA-generated free carriers as well as for the Kerr effect, while from that point on all the mathematical formulations carried out are exact. Hence, the error introduced here is unambiguously interpretable physically in terms of the omission of TPA what, as already mentioned, even allows us to identify the propagation of high-energy optical pulses in short or highly lossy silicon-based waveguides to be the physically realistic corresponding situation of useful application.

2. Pulse propagation in silicon waveguides

Pulse propagation along waveguide direction z and time t is assumed to be governed by the system of the nonlinear partial integro-differential equations [21, 22, 24–27]

\frac{\partial E}{\partial z} + \frac{1}{v} \frac{\partial E}{\partial t} + \frac{i β_{2}}{2} \frac{\partial^{2} E}{\partial t^{2}} = - \frac{α}{2} E - \frac{β_{r} + i β_{i}}{2} {| E |}^{2} E - \frac{σ_{r} + i σ_{i}}{2} N E,

N (z, t) = γ \int_{- \infty}^{t} I^{2} (z, t^{'}) G (t - t^{'}) d t^{'},

where the electrical field propagates according to E(z, t)exp[i(β₀z − ωt)] through the silicon waveguide, where ω is the optical angular frequency and β₀ the real propagation constant of the waveguide mode, N = N(z,t) is the free-carrier concentration, v the group velocity of the waveguide mode, α the linear power attenuation coefficient, β_r the two-photon absorption (TPA) coefficient, β_i = −2k₀n₂, where k₀ = 2π/λ and n₂ is the Kerr coefficient, σ_r the coefficient of the free-carrier absorption (FCA), and σ_i is the efficiency of the free-carrier refraction (FCR) or free-carrier plasma dispersion effect [21]. Throughout the paper we consider pulse propagation at the telecom wavelength λ = 1.55μm, where [21] the Kerr coefficient is n₂ ≃ 6 × 10⁻¹⁸m²/W and thus β_i = −5 × 10⁻¹²m/W, β_r ≃ 5 × 10⁻¹²m/W, and further σ_r = 1.45 × 10⁻²¹m² and σ_i = 1.1 × 10⁻²⁰m². For the linear losses α we choose a conservative estimate of 1 dB/cm if not stated otherwise. The group velocity v of the fundamental mode propagating along the silicon waveguide is assumed to be v = 10⁸m/s.

Group-velocity dispersion (GVD) is represented by the third term on the left-hand side of Eq. (1) containing the GVD parameter β₂. For a pulse duration t₀ the latter determines the dispersion length $L_{D} = t_{0}^{2} / | β_{2} |$ after which GVD may have caused a significant distortion of the pulse. In the wavelength range around 1.55 μm, modern silicon nanowires exhibit a GVD parameter |β₂| well below 10 ps²/m while waveguides with a larger cross section come along with even smaller values [26, 29]. For the silicon-based plasmonic waveguides [30] considered here |β₂| is below 5 ps²/m for thicknesses above 50 nm. Thus, for both waveguide types and picosecond pulses the dispersion length is well above 10 cm. On the other hand, silicon waveguide structures have been designed which exhibit chromatic dispersion as low as that in silica optical fibers over a wide wavelength range with a dispersion length for picosecond pulses of tens of meters [35]. For waveguides significantly shorter than L_D as those considered here GVD may be neglected as we do so in the present formulation by setting β₂ = 0. This step is justified even more as the analytical results derived here by neglecting the effect of TPA as a source of pulse attenuation will be shown to apply to short waveguide lengths or to highly lossy waveguides which, in turn, will usually be kept short as well to avoid unnecessarily high linear losses. However, it should be taken into account that pulses significantly shortened by self-induced FCA and distorted in phase by FCR and the Kerr effect during propagation may have a wider spectrum than the input pulse and will possibly be prone to experience a larger GVD.

Further, although we use bulk-silicon values for the waveguide examples studied here these values have to be properly adjusted according to the specific waveguide structure under consideration. While in waveguides with cross sections larger than 1 μm² the parameters may be closer to bulk silicon, in nanowire waveguides the proper effective areas have to be incorporated to modify the nonlinear parameters [22, 25–27] to take the strong mode guidance into account. The intensity has then to be interpreted in terms of the modal power and the corresponding effective mode areas [22, 25–27]. In the metal-silicon-metal plasmonic waveguides [30] considered here the necessary modifications may even get to the point that nonlinearity parameters which would be purely real in dielectric silicon waveguides may become complex-valued in extremely thin plasmonic structures. This latter modification has been taken into account in [30] by introducing a plasmonic attenuation factor. The influence of the latter, however, decreases rapidly for larger waveguide thicknesses.

Stimulated Raman scattering may be neglected for the piscosecond pulses considered here which may have a narrow spectral bandwidth on the order of 1 THz or less, i.e., scarcely spanning the Raman frequency shift of 15.6 THz in silicon [22]. Thus, for all the situations considered here the principal form of the pulse-evolution equation is similar [27] and only the linear and nonlinear parameters have to be properly adjusted [21, 25, 26, 30].

The free-carrier concentration N in Eq. (2) is related to the squared intensity I² via a convolution integral with the decay function G(t) of the concentration of free carriers due to recombination and diffusion out of the propagating-mode area. Defining G(0) = 1 ensures that γI² represents the instantaneous free-carrier generation rate where γ = πβ_r/(hω), and h and ω are Planck’s constant and the optical angular frequency, respectively. We define an effective free-carrier lifetime τ_eff by

τ_{eff} = \int_{0}^{\infty} G (t) d t,

as the definition G(0) = 1 and the assumed monotonous decay makes the integral a measure of the temporal width of G(t). Frequently, a simple exponential decay

G (t) = exp (- t / τ_{exp})

is assumed, e.g. for modelling pure bulk recombination, in which case the free-carrier lifetime τ_eff is equal to τ_exp, while other processes such as free-carrier diffusion may cause a deviation from the simple exponential model [36].

The complex field amplitude in Eq. (1) can be written $E (z, t) = \sqrt{I (z, t)} \times exp [i ϕ (z, t)]$ , where I(z,t) is the intensity and ϕ(z,t) the phase of the electrical field. After neglecting the attenuation caused by TPA by omitting the term containing β_r and the GVD term by setting β₂ = 0 as discussed in the Introduction and below Eqs. (1) and (2), respectively, we may separate Eq. (1) into the following differential equations for the intensity I(z,t) and the phase ϕ(z,t) of the propagating field,

\frac{\partial I}{\partial z} + \frac{1}{v} \frac{\partial I}{\partial t} ≃ - α I - σ_{r} N I,

\frac{\partial ϕ}{\partial z} + \frac{1}{v} \frac{\partial ϕ}{\partial t} = - \frac{β_{i}}{2} I - \frac{σ_{i}}{2} N,

where we assume that an optical pulse with intensity I(z = 0, t) = I₀(t) and phase ϕ(z = 0, t) = ϕ₀(t) has been launched at the waveguide input at z = 0. Although we have neglected the attenuation of the wave by the TPA process itself by dropping the β_r term, we keep the β_i term as it describes the strong influence of the Kerr effect on the phase chirp of the pulse which may be highly sensitive to interference experiments and which may cause a non-linear frequency shift in the pulse. The action of the TPA-generated free carriers on the intensity attenuation and on the phase chirp is kept fully included by the σ_r and σ_r terms.

We consider a short propagating pulse essentially confined to a time interval [t_P(z) − Δt, t_P(z) + Δt]. For a pulse launched at the waveguide input z = 0 at time t = t_P(0) and travelling at the group velocity v along the waveguide, t_P(z) = t_P(0) + z/v designates the time at which the center of this time slot passes the waveguide position z. If now this maximal pulse width 2Δt is much smaller than the characteristic decay time τ_eff of the free carriers, the pulse itself only experiences the accumulation of free carriers, and will have travelled on already when the free-carrier concentration starts to perceptibly decrease again. Thus the free-carrier density can be approximated by

N (z, t) ≃ {\begin{array}{l} γ R (z, t), & t \leq t_{P} (z) + Δ t, \\ γ Q (z) G [t - t_{P} (z)], & t > t_{P} (z) + Δ t . \end{array}

where the upper line applies for times before and during the travelling pulse, while the lower line stands for times after the pulse has already passed the position z, and

R (z, t) = \int_{- \infty}^{t} I^{2} (z, t^{'}) d t^{'}, Q (z) = R (z, t \to \infty) = \int_{= \infty}^{+ \infty} I^{2} (z, t^{'}) d t^{'} .

Since in the moment we are interested only in the evolution of an optical pulse with a duration much shorter than the time scale τ_eff of the free-carrier decay function G(t), we focus on a time interval around the pulse only and approximate N(z,t) ≃ γR(z,t) as in Eq. (7a).

In order to arrive at explicit analytical results, we multiply the whole equation (5) by I and express the squared intensity as I²(z,t) = ∂R(z,t)/∂t =: Ṙ(z,t),

\frac{1}{2} \frac{\partial \dot{R}}{\partial z} + \frac{1}{2 v} \frac{\partial \dot{R}}{\partial t} = - α \dot{R} - γ σ_{r} R \dot{R}

We integrate this equation over time t,

\frac{1}{2} \frac{\partial R}{\partial z} + \frac{1}{2 v} \frac{\partial R}{\partial t} = - α R - \frac{γ σ_{r}}{2} R^{2} + K (z)

where the integration constant K(z) is independent of t. Since Eq. (10) must vanish all along the waveguide before any pulse has been launched we have K(z) = 0 for all z. To solve this partial differential equation (10), we introduce two new independent time coordinates

\tilde{x} = t + z / v, \tilde{y} = t - z / v

implying z = v(x̃ − ỹ)/2 and t = (x̃ + ỹ)/2, and obtain

\frac{1}{v} \frac{\partial R}{\partial \tilde{x}} = - α R - \frac{γ σ_{r}}{2} R^{2} .

This ordinary differential equation is easily solved by separation of the variables x̃ and R (or using [37]),

R = \frac{2 α}{C (\tilde{y}) exp (2 α v \tilde{x}) - γ σ_{r}}

where, after re-inserting the original space and time coordinates, the integration constant C(ỹ) = C(t − z/v) is used to fit R(z,t) to its initial value R₀(t) = R(z = 0, t) at the waveguide input z = 0, where

R_{0} (t) = R (z = 0, t) = \int_{- \infty}^{t} I_{0}^{2} (t^{'}) d t^{'}

and I₀(t) = I(z = 0, t) is the input pulse intensity at z = 0. We obtain

R (z, t) = \frac{R_{0} (t - z / v) exp (- 2 α z)}{1 + γ σ_{r} L_{eff}^{(2)} (z) R_{0} (t - z / v)}

where

L_{eff}^{(m)} (z) = [1 - exp (- m α z)] / (m α)

defines two different effective interaction lengths for m = 1 and m = 2. Further, we have

Q (z) = \frac{Q_{0} exp (- 2 α z)}{1 + γ σ_{r} L_{eff}^{(2)} (z) Q_{0}} \leq \frac{exp (- 2 α z)}{γ σ_{r} L_{eff}^{(2)} (z)},

where Q₀ = Q(z = 0) = R₀(t → ∞), and the rightmost term is the limit for Q₀ → ∞ for z > 0. This quantity is of interest since γQ(z) gives us the peak value of the free-carrier density right behind the pulse when it has just passed the position z, and which afterwards decays as in Eq. (7b). Interestingly, Q(z) has the same dependence on z for all pulses with the same Q₀ at the input, no matter what shape they have and how much they are distorted along the waveguide by the self-produced free carriers.

Thus, Eqs. (15) and (17) with Eq. (7) already give us the free-carrier concentration before, during and after the optical pulse at position z. The pulse intensity I(z,t) follows directly from Eqs. (8) and (15) as $I (z, t) = \sqrt{\partial R (z, t) / \partial t}$ ,

I (z, t) = \frac{I_{0} (t - z / v) exp (- α z)}{1 + γ σ_{r} L_{eff}^{(2)} (z) R_{0} (t - z / v)} .

The fact that R₀(t) in Eq. (18) is monotonously increasing with time causes an attenuation increasing towards the rear part of the pulse during propagation as already observed in numerical studies [9, 10, 18–21, 31] and seen experimentally [38].

To summarize, in this Section we have derived explicit analytical expressions for the intensity of short optical pulses propagating in silicon waveguides and for the concentration of TPA-generated free-carriers under omission of TPA as an attenuation effect.

3. Nonlinear phase shifts

In order to calculate how the self-produced free carriers affect the pulse phase, we similarly transform the differential equation (6) for the phase, with N(z,t) = γR(z,t) from (7a),

\frac{1}{v} \frac{\partial ϕ}{\partial \tilde{x}} = - \frac{β_{i}}{4} I - \frac{σ_{i} γ}{4} R .

After inserting the analytical solutions Eqs. (18) and (15) for I(z,t) and R(z,t) using the coordinate transformation z = v(x̃ − ỹ)/2 and t − z/v = ỹ, the integration is easily done by means of the integral relations 2.313.1 and 2.314 given in [39]. Fitting the ỹ-dependent integration constant properly for a given phase function ϕ₀(t) = ϕ(z = 0, t) at the waveguide input z = 0 we can write the total pulse phase in original time-space coordinates in terms of the individual contributions from the Kerr and FCR effects as

ϕ (z, t) = ϕ_{0} (t - z / v) + ϕ_{Kerr} (z, t) + ϕ_{FCR} (z, t) .

Explicit expressions for ϕ_Kerr(z,t) and ϕ_FCR(z,t) will be given in Sections 3.1 and 3.2, respectively. We will also discuss the non-linear frequency shift Δω(z,t) = −∂ϕ/∂t which represents a spectral red shift for ∂ϕ/∂t > 0 and a blue shift for ∂ϕ/∂t < 0 [9, 10, 18–22, 40].

3.1. Kerr-induced phase shift

The Kerr-induced phase shift in Eq. (20) is obtained as

\begin{array}{l} ϕ_{Kerr} (z, t) & = & - \frac{β_{i} I_{0} (t - z / v)}{2 α \sqrt{η R_{0} (t - z / v) [1 + η R_{0} (t - z / v)]}} \\ \times arctanh {\frac{α L_{eff}^{(1)} (z) \sqrt{η R_{0} (t - z / v) [1 + η R_{0} (t - z / v)]}}{1 + α L_{eff}^{(1)} (z) η R_{0} (t - z / v)}} . \end{array}

where η = γσ_r/2α. Like the intensity does, the Kerr phase vanishes for t = ±∞. For negligibly small free-carrier absorption ηR₀ → 0, it reduces to the well-known result for nonlinear pulse propagation in optical fibers [41],

ϕ_{Kerr} (z, t) = - β_{i} I_{0} (t - z / v) L_{eff}^{(1)} (z) / 2,

which is temporally symmetric for symmetric input pulses. Interestingly, this stringent conservation of symmetry between pulse intensity and Kerr phase is destroyed in the presence of free-carriers. This can be shown as follows.

At the leading edge of the propagating pulse (t → −∞), where ηR₀ is still small, the Kerr phase shift is approximately described by Eq. (22), which after sufficiently large propagation distances z ≫ 1/α becomes ϕ_Kerr(z,t) = −β_i I₀(t − z/v)/2α. In the trailing edge, however, ηR₀ approaches its constant asymptotic value ηQ₀ and the Kerr phase becomes proportional to Eq. (22) times a factor different from unity. For example, ϕ_Kerr(z,t → +∞) = −β_i I₀(t − z/v)ln(4ηQ₀)/(4αηQ₀) if ηQ₀ ≫ 1 in addition to z ≫ 1/α. Thus, for symmetric input pulses I₀(t) the Kerr phase will evolve asymmetrically. In particular, even if the asymmetrically attenuated pulse I(z,t) maintains its temporal symmetry during propagation, as it will be shown to be possible for the form-stable sech pulse in Section 5.5, the Kerr phase shift may not keep this symmetry as we discuss now. Using Eq. (18) the Kerr phase at a given position z can also be written

ϕ_{Kerr} (z, t) = - \frac{β_{i}}{2} exp (α z) L_{eff}^{(1)} (z) \cdot I (z, t) \cdot h {u [R_{0} (t - z / v)]} \cdot g [R_{0} (t - z / v)] .

where

h (u) = \frac{arctanh (u)}{u}, u (R_{0}) = \frac{α L_{eff}^{(1)} (z) \sqrt{η R_{0} (1 + η R_{0})}}{1 + α L_{eff}^{(1)} (z) η R_{0}}, g (R_{0}) = \frac{1 + 2 α L_{eff}^{(2)} (z) η R_{0}}{1 + α L_{eff}^{(1)} (z) η R_{0}} .

The way in which the Kerr phase ϕ_Kerr(z,t) in Eq. (23) is distorted in comparison to the pulse intensity I(z,t) can be seen by investigating the monotonicity of the function g · h with respect to time t. Since arctanh(u) ≃ u for u ≃ 0 and d²arctanh(u)/du² ≥ 0 the function h(u) is monotonously increasing with u starting from h(0) = 1. Obviously, u(R₀) is monotonously increasing with R₀ from u(0) = 0 since

α L_{eff}^{(1)} (z) \leq 1

, while g(R₀) is monotonously increasing with R₀ from g(0) = 1 because

2 L_{eff}^{(2)} (z) \geq L_{eff}^{(1)} (z)

. Finally, R₀(t − z/v) is monotonously increasing in t because of its definition (14). Therefore we know that the function g · h is monotonously increasing with respect to t. This property of g · h causes ϕ_Kerr(z,t) to be elevated towards later times as compared to the propagating pulse intensity I(z,t). This behavior is caused by the fact that the Kerr phase already started to accumulate at the waveguide input where the intensity in the rear part of the pulse was even stronger than the current peak of the propagated pulse. Consequently the Kerr phase can only peak after or at the occurrence of an intensity peak. Thus, the latter will always be given a Kerr-induced red-shift

Δ ω_{Kerr} (z, t) = (β_{i} / 2) L_{eff}^{(1)} (z) exp (α z) I (z, t) \partial (g \cdot h) / \partial t \leq 0

which, however, is too lengthy to display here explicitly. The trailing pulse edge will get a blue shift after the Kerr phase has peaked for the last time.

3.2. Free-carrier induced phase shift

The additional phase shift induced by FCR in Eq. (20), i.e. by the refractive-index change in the presence of self-produced free carriers, is obtained as

ϕ_{FCR} (z, t) = - \frac{σ_{i}}{2 σ_{r}} \times ln {1 + γ σ_{r} L_{eff}^{(2)} (z) R_{0} (t - z / v)}

Since γσ_r ≥ 0, the logarithmic function is always non-negative and ϕ_FCR(z,t) ≤ 0 as σ_i/σ_r > 0. The FCR-induced phase shift is zero for t → −∞, decreases monotonously with time and reaches its maximum (negative) value at t → +∞, which is ϕ_FCR = −σ_i ln(1 + ηQ₀)/2σ_r for z ≫ 1/2α. Thus, FCR always causes a blue shift, i.e. its corresponding frequency shift Δω_FCR(z,t) = −∂ϕ_FCR(z,t)/∂t is non-negative. The latter can be written

Δ ω_{FCR} (z, t) = \frac{σ_{i}}{2} \times \frac{γ L_{eff}^{(2)} (z) I_{0}^{2} (t - z / v)}{1 + γ σ_{r} L_{eff}^{(2)} (z) R_{0} (t - z / v)}

= \frac{σ_{i} γ}{2} \times exp (+ 2 α z) L_{eff}^{(2)} (z) I^{2} (z, t) [1 + γ σ_{r} L_{eff}^{(2)} (z) R_{0} (t - z / v)],

and its time dependence at a position z is proportional to the squared intensity I²(z,t) of the travelling pulse times a monotonously increasing function of t. This means that Δω_FCR(z,t) can only peak after or at the occurrence of a peak in I(z,t). For smoothly single-peaked pulses this lag may make the net blue shift less efficient in the spectrum of the complete pulse. The logarithmic function in Eq. (25) results from the fact that the accumulation of FCR phase at the end of the pulse is mitigated by the simultaneous FCA-induced attenuation of the rear part of the pulse as already observed in numerical simulations [9]. Consequently, the FCR phase shift is smaller than its upper limit

ϕ_{FCR} = - σ_{i} γ L_{eff}^{(2)} (z) R_{0} (t - z / v) / 2

obtained for σ_r → 0 in Eq. (25), i.e., when free carriers would fictitiously cause FCR only and no FCA. The corresponding upper limit for the frequency blue shift would the be

Δ ω_{FCR} = σ_{i} γ L_{eff}^{(2)} (z) I_{0}^{2} (t - z / v) / 2

.

Summarizing, the Kerr- and FCR-induced phase shifts can be expressed explicitly as well. For symmetric input pulses any asymmetry in the total phase arises from the asymmetric history of FCA attenuation across the pulse when contributed by the Kerr effect, and from the accumulation of free carriers towards the end of the pulse when contributed by FCR.

4. Interpretation of omitting TPA as a cause of attenuation

We now illustrate how omitting TPA as an attenuation as done in this paper may be interpreted to model FCA distortion of short optical pulses in short or highly lossy silicon-based waveguides with particular use for high input pulse energies. This is in agreement with the experimental observation of a predominance of FCA over TPA in pulses propagating large energies [18]. A direct comparison of these two attenuation mechanisms is hampered by their different temporal behavior. While TPA is an instantaneous effect, FCA only accumulates during the passage of the pulse. Therefore we define a shape function

f_{0} (ξ) = f_{0} (t / t_{0}) = I_{0} (t) / {\hat{I}}_{0}

of the input pulse with a peak value of unity, where ξ = t/t₀, and t₀ determines the temporal width of the pulse. Thus, any input pulse I₀(t) = Î₀f₀(t/t₀) is solely described by its peak intensity Î₀, its duration t₀, and a dimensionless shape function f₀(ξ). Accordingly, we define

F_{0} (ξ) = \int_{- \infty}^{ξ} f_{0}^{2} (ξ^{'}) d ξ^{'}, W_{0} = t_{0} {\hat{I}}_{0} J, J = \int_{- \infty}^{+ \infty} f_{0} (ξ^{'}) d ξ^{'},

such that

R_{0} (t) = t_{0} {\hat{I}}_{0}^{2} F_{0} (t / t_{0})

,

Q_{0} = t_{0} {\hat{I}}_{0}^{2} F_{0} (\infty)

, and W₀ is the input pulse energy per unit cross-sectional area. To compare first the local relevance of TPA versus FCA we consider the ratio α_TPA/α_FCA of the non-linear attenuation due to TPA and FCA, respectively, for general input pulses,

\frac{α_{TPA}}{α_{FCA}} = \frac{β_{r} {\hat{I}}_{0} f_{0} (t / t_{0})}{σ_{r} γ t_{0} {\hat{I}}_{0}^{2} F_{0} (t / t_{0})} = \frac{1}{W_{0}} \times \frac{β_{r} J f_{0} (t / t_{0})}{σ_{r} γ F_{0} (t / t_{0})} .

Since by definition f₀(ξ), F₀(ξ) and J depend on neither the peak intensity Î₀ nor the pulse width t₀, Eq. (30) shows that the ratio of TPA to FCA attenuation at any time t of the pulse duration is inversely proportional to the pulse energy W₀, and the relevance of TPA against FCA locally reduces more and more for increasing W₀.

However, in order to see the more important aggregate effect of TPA and FCA after propagation through the waveguide, we re-write the pulse intensity evolution Eq. (18) in a normalized way,

f (z, t) = \frac{I (z, t)}{{\hat{I}}_{0} exp (- α z)} = \frac{f_{0} (ξ - z / v t_{0})}{1 + Θ_{FCA} F_{0} (ξ - z / v t_{0})},

where

Θ_{FCA} = σ_{r} γ t_{0} {\hat{I}}_{0}^{2} L_{eff}^{(2)} (z)

is a dimensionless distortion parameter due to FCA. On the other hand, in a model accounting for TPA only [21, 22], i.e., when σ_r = 0 or γ = 0 in Eq. (67) in Appendix A, pulse propagation can be described by

f (z, t) = \frac{I (z, t)}{{\hat{I}}_{0} exp (- α z)} = \frac{f_{0} (ξ - z / v t_{0})}{1 + Θ_{TPA} f_{0} (ξ - z / v t_{0})},

where the dimensionless distortion parameter due to TPA is

Θ_{TPA} = β_{r} {\hat{I}}_{0} L_{eff}^{(1)} (z) .

An important result of the present paper is derived in Appendix A: If TPA as a source of attenuation is found to be too small to affect pulse propagation perceptibly in a TPA-only model according to Eq. (33) then it can be neglected all the more when solving the full problem actually admitting of both TPA and FCA, and the FCA-only formulation presented in this paper applies. Therefore, such a consideration of FCA and TPA in separate models allows us to formulate two criteria for identifying physically reasonable situations in which the omission of TPA is justified while nevertheless the pulses are significantly distorted by self-produced free carriers, i.e., when Θ_TPA be small while Θ_FCA remain finite.

The first condition requires

Θ_{TPA} = β_{r} {\hat{I}}_{0} L_{eff}^{(1)} ≪ 1,

which ensures that the denominator in Eq. (33) is close to constant and unity since f₀(ξ) ≤ 1, and the pulse remains unaffected by TPA. Trivially, Eq. (35) is fulfilled for small input intensities representing the quasi-linear case in which TPA can be neglected and our formulation applies, but no FCA-induced distortion arises at the same time either. In the more interesting case of non-vanishing input peak intensities Î₀, Eq. (35) is equivalent with requiring a small TPA-effective propagation length

L_{eff}^{(1)}

, i.e., short or highly lossy waveguides for which

L_{eff}^{(1)} = \frac{1 - exp (- α z)}{α} = \frac{Θ_{TPA}}{β_{r} {\hat{I}}_{0}} ≪ \frac{1}{β_{r} {\hat{I}}_{0}} .

In accordance with Appendix A the conditions displayed in Eq. (35) and hence in Eq. (36) which have been derived from a simple TPA-only model are already sufficient to justify omitting TPA and applying our FCA-only model, no matter whether FCA has a significant effect or not.

The second condition requires that FCA shall still perceptibly affect the pulse and thus Θ_FCA shall remain finite even when Θ_TPA becomes very small. This latter condition requires

\frac{Θ_{FCA}}{Θ_{TPA}} = \frac{σ_{r} γ t_{0} {\hat{I}}_{0}^{2} L_{eff}^{(2)}}{β_{r} {\hat{I}}_{0} L_{eff}^{(1)}} ≫ 1

In order to translate these two conditions Eqs. (35) and (37) into requirements on the waveguide and on the launching conditions, we combine Θ_TPA and Θ_FCA of Eqs. (32) and (34) to eliminate

{\hat{I}}_{0}^{2}

and use Eq. (29) for W₀ to write

L_{eff}^{(1)} (z) = \frac{σ_{r} γ t_{0}}{β_{r}^{2} Ξ (α z)} \frac{Θ_{TPA}^{2}}{Θ_{FCA}}, W_{0} = \frac{J β_{r} Ξ (α z)}{σ_{r} γ} \frac{Θ_{FCA}}{Θ_{TPA}},

where

Ξ (α z) = L_{eff}^{(1)} (z) / L_{eff}^{(2)} (z) = 2 / [1 + exp (- α z)] = 2 / [2 - α L_{eff}^{(1)} (z)]

can range only between unity and two, 1 = Ξ(αz ≪ 1) ≤ Ξ(αz) ≤ Ξ(αz ≫ 1) = 2. Thus, a vanishing small TPA-distortion parameter Θ_TPA and a finite and hence significantly larger FCA-distortion parameter Θ_FCA require short TPA-effective propagation lengths

L_{eff}^{(1)}

and large input pulse energies W₀, and the two conditions in Eqs. (35) and (37) read

L_{eff}^{(1)} (z) ≪ \frac{σ_{r} γ t_{0}}{β_{r}^{2} Ξ (α z) Θ_{FCA}}, W_{0} ≫ \frac{J β_{r} Ξ (α z) Θ_{FCA}}{σ_{r} γ} .

Each of these two conditions is equivalent with requiring Θ_TPA ≪ 1 according to Eq. (35) as long as the desired finite Θ_FCA is realized via Eq. (32). Now aiming at decreasing Θ_TPA while keeping Θ_FCA fixed and finite, we have to take into account that the non-linear parameters of silicon are not available to being varied significantly. Although the pulse duration t₀ may vary in a wide range it can not be arbitrarily increased as we assume t₀ ≪ τ_eff. Finally, the pulse shape determining J is reserved to the specific application. Therefore we re-arrange Eqs. (38) as a function of the linear attenuation α,

z = \frac{2}{α} arctanh (\frac{α σ_{r} γ t_{0}}{2 β_{r}^{2}} \cdot \frac{Θ_{TPA}^{2}}{Θ_{FCA}}), W_{0} = J \cdot \frac{2 β_{r}^{2} Θ_{FCA} + α σ_{r} γ t_{0} Θ_{TPA}^{2}}{2 β_{r} σ_{r} γ Θ_{TPA}}

where, depending on the choice of z, the ratio

Θ_{TPA}^{2} / Θ_{FCA} = 2 β_{r}^{2} tanh (α z / 2) / (α σ_{r} γ t_{0})

can range only within

0 \leq Θ_{TPA}^{2} / Θ_{FCA} \leq 2 β_{r}^{2} / (α σ_{r} γ t_{0})

independently of the input peak intensity Î₀. Hence, two extreme situations to realize the conditions in Eq. (39) can be distinguished.

First, if the attenuation α cannot be changed (e.g. when using the same waveguide technology), then the waveguide length would have to be small and the input pulse energy to be large according to

z ≃ \frac{σ_{r} γ t_{0} Θ_{TPA}^{2}}{β_{r}^{2} Θ_{FCA}} ≪ \frac{σ_{r} γ t_{0}}{β_{r}^{2} Θ_{FCA}}, W_{0} ≃ \frac{J β_{r} Θ_{FCA}}{σ_{r} γ Θ_{TPA}} ≫ \frac{J β_{r} Θ_{FCA}}{σ_{r} γ}, for α fixed, z ≪ 1 / α .

The parameter values given in Section 2 yield

σ_{r} γ / β_{r}^{2} = 1.13

m/ns and β_r/σ_rγ = 0.177 Wns/μm². For Θ_FCA = 1, Θ_TPA = 0.01, t₀ = 0.1ns and

J = \sqrt{π}

for a Gaussian input pulse (see Section 5.1) we obtain z ≃ 11.3μm and W₀ = 31.4 Wns/μm² corresponding to a peak intensity of Î₀ = 177W/μ²=17.7GW/cm². This means, for Gaussian 0.1ns input pulses, Θ_TPA never exceeds 0.01 up to an input intensity of Î₀ = 177W/μ², at which Θ_FCA already reaches unity. For a more relaxed TPA suppression of Θ_TPA = 0.05 the results are z ≃ 283μm and W₀ = 6.27 Wns/μm² corresponding to Î₀ = 35.4W/μ²=3.54GW/cm².

Second, for a fixed geometric length z, on the other hand, the conditions in Eq. (39) require an increased attenuation α ≫ 2/z compelling the arctanh function in Eq. (40) to approach unity [as tanh(αz/2) → 1], and both the linear losses and the input pulse energy are required to be large enough according to

α ≃ \frac{2 β_{r}^{2} Θ_{FCA}}{σ_{r} γ t_{0} Θ_{TPA}^{2}} ≫ \frac{2 β_{r}^{2} Θ_{FCA}}{σ_{r} γ t_{0}}, W_{0} ≃ \frac{2 J β_{r} Θ_{FCA}}{σ_{r} γ Θ_{TPA}} ≫ \frac{2 J β_{r} Θ_{FCA}}{σ_{r} γ}, for z fixed, α ≫ 2 / z,

ensuring that even the maximum possible effective propagation lengths

L_{eff}^{(1)} (z ≫ 1 / α) = 2 L_{eff}^{(2)} (z ≫ 1 / α) = 1 / α

are small. With the parameters of Section 2 and Θ_FCA = 1, Θ_TPA = 0.01, t₀ = 0.1ns and

J = \sqrt{π}

for the Gaussian pulse again, we find α ≃ 0.177/μm according to 768dB/mm and W₀ = 62.7 Wns/μm² corresponding to Î₀ = 354W/μ²=35.4GW/cm². For Θ_TPA = 0.05 the results are α ≃ 7.08/mm according to 30.7dB/mm and W₀ = 12.5 Wns/μm² corresponding to Î₀ = 70.8W/μ²=7.08GW/cm².

The requirement of large linear attenuation is frequently met in silicon-based plasmonic waveguides with their typically high linear propagation losses as the mode field is partially guided in metal layers [1]. Pulse propagation in metal-silicon-metal waveguides has been proposed in [30] to be modelled by a set of integro-differential equations analogous to Eqs. (1) and (2) with the non-linear parameters modified by a plasmonic attenuation factor. This factor becomes negligibly small in the silver-silicon-silver waveguide analyzed there for silicon layers thicker than 150nm, and the non-linear parameters approach the values valid for silicon waveguides. The propagation losses, however, are still high and only start to decrease below 1dB/μm, restricting the effective propagation lengths to a few micrometers. For example, $L_{eff}^{(1)} (z) \leq 2 L_{eff}^{(2)} (z) \leq 1 / α ≃ 4 μ m$ at 150nm silicon thickness. In this latter case Θ_TPA ≤ 0.01 for input peak intensities up to 500W/μm² at which intensity level a significant FCA-distortion with Θ_FCA = 2.82 is reached when t₀ = 0.1ns. Thus, a practical approach could be as follows. If the error caused by a TPA-distortion parameter of Θ_TPA ≤ 0.01 has been found to be tolerable for the desired application and the silicon-based waveguide has such a high attenuation of α = 0.25/μm= 1.086dB/μm, then all calculations using input peak intensities up to 500W/μm² can be carried out by the present FCA-only formulation without checking the accuracy again. It should be noted that a smaller FCA-distortion parameter Θ_FCA then merely relaxes the requirements on W₀ and z or α in Eqs. (41) or (42) while the validity of the FCA-only formulation is maintained. We will illustrate the range of applicability of our FCA-only formulation in Section 5.1.

Summarizing we have shown that our FCA-only model for pulse propagation in silicon-based waveguides applies as soon as the condition in Eq. (35) or (36) is met, i.e. for short or highly lossy waveguides. A significant distortion by FCA is then possible for high-energy input pulses while TPA remains negligible.

5. Propagation of pulses with various input shapes

We now consider input pulses I₀(t) = Î₀f(t/t₀) of different shapes f(t/t₀), which may experience different distortions of intensity and phase.

5.1. Gaussian input pulse

Assuming a Gaussian input pulse of duration 2t₀, we write

I_{0} (t) = {\hat{I}}_{0} exp (- t^{2} / t_{0}^{2}), R_{0} (t) = \sqrt{π / 8} t_{0} {\hat{I}}_{0}^{2} [1 + erf (\sqrt{2} t / t_{0})]

and obtain

J = \sqrt{π}

and

Q_{0} = \sqrt{π / 2} t_{0} {\hat{I}}_{0}^{2}

. The evolution for Gaussian input pulses along the silicon waveguide according to Eqs. (18) is shown in Fig. 1. It can clearly be seen that the rear part of the pulse is attenuated more and more causing a temporally asymmetric pulse shape as already observed experimentally in Ref. [38] and calculated numerically by a full model including both FCA and TPA, e.g., in [14, 20]. The error function from Eq. (43) in the pulse evolution Eq. (18) makes the trailing edge of the Gaussian input pulse flatter and flatter while the remaining front edge becomes steeper and steeper. The plots in Figs. 1(b), 1(d), 1(f) and 1(h) are versus the reference time t − z/v for a better visualization of the distortion with respect to the input pulse.

Fig. 1 (a) Pulse intensity I(z,t) at waveguide input z = 0 (dotted red curve) and after z = 1, 10, 20 and 50mm (black solid curves) propagation distance in a silicon waveguide with α = 1dB/cm for a Gaussian input pulse [2t₀ = 0.2ns in Eq. (43)] with input peak intensity Î₀ = 3W/μm². The green dash-dotted curve indicates the theoretical temporal peak intensity of an undistorted pulse at z = vt in the presence of linear attenuation exp(−αz) only. The center times of the pulses at positions z = 1, 10, 20 and 50 mm are 0.01, 0.1, 0.2 and 0.5 ns, respectively. (b) Same as (a) in a reference time frame t − z/v for better visualization of asymmetric pulse attenuation. (c) Non-linear phase shifts ϕ_Kerr (dashed red curve), ϕ_FCR (dotted green curve), ϕ = ϕ_Kerr + ϕ_FCR (solid black curve), and normalized free-carrier density N′ = 10⁻¹⁵cm³ × N with N = γR (blue dash-dotted curve) for the pulse after z = 50mm propagation distance. (d) Total non-linear phase shifts ϕ = ϕ_Kerr + ϕ_FCR for the pulses shown in (b). (e)–(h) Same as (a)–(d), respectively, for Î₀ = 30W/μm².

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Interestingly, in Fig. 1(e), the FCA-induced attenuation of the rear part of the pulse is so fast that the attenuated pulse seems to move backwards in time and to peak earlier at the waveguide end than at the input. This seeming superluminal propagation behavior [42] will be discussed in more detail using the analytical expressions which can be derived for the sech pulse in Section 5.5.

The Kerr phase ϕ_Kerr plotted by red dashed lines in Figs. 1(c) and 1(g) has a temporal asymmetry with the slope steeper in the leading edge than in the trailing edge. Thus the Kerr effect results in a spectral extension Δω_Kerr deeper into the ”red” than into the ”blue”. The FCR phase shift ϕ_FCR (green dotted lines) decreases monotonously with time. It can be seen comparing Figs. 1(c) and 1(g) for the propagation distance z =50mm, that the FCR-induced phase shift ϕ_FCR grows much stronger than N does when increasing Î₀ from 3 to 30W/μm², because ϕ_FCR is accumulating all along the waveguide [see Figs. 1(d) and 1(h)] and has already seen the larger carrier densities N near the input of the waveguide. The sum of ϕ_Kerr and ϕ_FCR shown by black solid lines in Figs. 1(c–d) and 1(g–f) first forms a positive peak, decreases afterwards and becomes negative, thus causing a redshift in the front part of the pulse and a blueshift in the rear part.

5.2. Effect of omitting TPA illustrated with the Gaussian input pulse

We use the Gaussian input pulse to discuss the meaning and the range of applicability of our FCA-only formulation of pulse propagation in silicon-based waveguides according to Section 4. In Fig. 2, red dashed lines show the normalized intensity of the Gaussian input pulse with t₀ = 0.1ns which is launched with different peak intensities Î₀ at the waveguide input z = 0. Green dotted lines represent the pulse intensities after a propagation distance z calculated by solving numerically the full problem Eq. (67) taking both FCA and TPA into account. The black solid lines stand for the pulse intensities obtained from our analytical FCA-only formulation Eq. (18) omitting TPA. Note that all intensities are normalized by dividing them by the product Î₀ exp(−αz) to better visualize the distortion and the introduced error, i.e., in the absence of any non-linear effects all curves would coincide with that for the input intensity.

Fig. 2 Interpretation of omitting TPA. See explanation in the main text of Section 5.2.

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Figure 2(a) shows the results for a typical silicon-on-insulator waveguide with a conservative loss estimate of α = 1dB/cm and a length of 10cm, i.e. for a long low-loss waveguide. Obviously omitting TPA causes an error of more than 80 percent for 30W/μm² for which case a TPA-distortion parameter Θ_TPA as large as 5.87 obtained from Eq. (34) prohibits the omission of TPA from the outset. Even for an input peak intensity as small as 0.3W/μm² TPA still depresses the pulse intensity by about 6 percent while FCA has almost no effect. In this situation the FCA-only formulation presented in this paper as well as the TPA-only model of Refs. [21, 22], are actually restricted to serve as a mere tool in estimating the individual contributions of FCA and TPA, e.g., in studying their competition in pulse compression [18, 19] or continuum generation [9, 10]. Thus, the results shown in Fig. 1 illustrate the contribution from FCA only while TPA actually would add another concurrent non-negligible contribution to the pulse distortion which would have to be calculated numerically by the full model of Eq. (67) as done in Fig. 2(a).

On the other hand, Fig. 2(b) illustrates the situation for the highly lossy silicon-based plasmonic waveguide from [30] already considered at the end of Section 4 with a 150nm thick silicon layer sandwiched between two silver layers. The high losses of α = 0.25/μm=1.086dB/μm limit the TPA-effective and the FCA-effective propagation length to no more than 4μm and 2μm, respectively. In this regard, the geometrical waveguide length of 20μm may be considered to be relatively large and not to terminate any non-linear propagation effects, and the linear loss accumulates up to almost 22 dB at the waveguide end. Thus, the results in Fig. 2(b) illustrate the situation required in Eqs. (42). Up to 500W/μm² the TPA-distortion parameter stays below one percent, and the intensity curves obtained by the exact numerical method and by the present FCA-only model almost coincide even up to Î₀ = 3kW/μm² or 300GW/cm² while FCA significantly distorts the pulse.

Figure 2(c) shows the results for the low-loss silicon-on-insulator waveguide with α = 1dB/cm again [as already used in Figs. 1 and 2(a)] when, however, its length of only 50μm is relatively small as compared to the maximum possible TPA- and FCA-effective propagation length of 43.5mm and 21.7mm, respectively, corresponding to the situation required by Eqs. (41). For instance, at Î₀ = 100W/μm² the pulse is significantly distorted by FCA, while Θ_TPA is no larger than 2.5 percent. Again, the results obtained by the present FCA-only formulation are very close to those from the numerical integration of the exact equations (67) plotted by green dotted lines. Even up to Î₀ = 1kW/μm² or 100GW/cm² the curves almost coincide although the TPA-distortion parameter Θ_TPA already reaches 0.25 at this input intensity. This proves even more that requiring Θ_TPA to be small is a sufficient criterion for the applicability of the FCA-only model developed in this paper.

5.3. Rectangular input pulse

In case of a rectangular input pulse of duration 2t₀, we have

\begin{array}{l} t < - t_{0} : I_{0} (t) = 0, R_{0} (t) = 0, \\ | t | < t_{0} : I_{0} (t) = {\hat{I}}_{0}, R_{0} (t) = (t + t_{0}) {\hat{I}}_{0}^{2}, \\ t > t_{0} : I_{0} (t) = 0, R_{0} (t) = 2 t_{0} {\hat{I}}_{0}^{2}, \end{array}

while J = 2 and

Q_{0} = 2 t_{0} {\hat{I}}_{0}^{2}

. The evolution of this pulse is shown in Fig. 3 where again a clear attenuation of the rear part of the pulse can be seen. Due to the absence of free carriers in front of the leading edge of the rectangular pulse all along the propagation, the intensity step there is only attenuated by the linear losses α and can be arbitrarily large by properly choosing the input intensity. This is in contrast to smoothly rising pulses such as the Gaussian or the shape-maintaining sech input pulse which we will discuss in more detail in Section 5.5. Since R₀(t − z/v) in the denominator of Eq. (18) is a continuous function of time t − z/v, any intensity step in general pulses will maintain its relative height during propagation as compared to the intensity at neighboring times and will propagate at the group velocity v. We will return to this observation when discussing the resemblance to superluminal pulse propagation in Section 5.5.

Fig. 3 Same as Fig. 1 but for a rectangular input pulse [2t₀ = 0.2ns in Eq. (44)].

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The Kerr phase has a maximum just coinciding with the intensity maximum because the latter is infinitely peaked and Eq. (23) cannot act as a delay, while it decreases towards the end of the pulse. Thus, besides forming a frequency spike at either end of the pulse due to the phase steps there, the Kerr phase causes a pure blue shift all along the pulse even enhancing the blue shift already induced by FCR.

5.4. Lorentzian input pulse

For a Lorentzian input pulse of duration 2t₀, we write

I_{0} (t) = \frac{{\hat{I}}_{0}}{1 + {(t / t_{0})}^{2}}, R_{0} (t) = \frac{t_{0} {\hat{I}}_{0}^{2}}{2} [\frac{t_{0} t}{t_{0}^{2} + t^{2}} + arctan (\frac{t}{t_{0}}) + \frac{π}{2}]

and obtain J = π and

Q_{0} = π t_{0} {\hat{I}}_{0}^{2} / 2

. The pulse evolution during propagation is illustrated in Fig. 4. For the propagating pulse intensity we observe a flipped temporal asymmetry as compared to the Gaussian pulse, see Figs. 4(f) and 1(f), although both pulses are exactly symmetric at the waveguide input. The trailing edge is now steeper than the leading edge. The asymmetry of the Kerr-induced phase shift is also inverted as compared to the Gaussian pulse as it is shown by the red dashed lines in Figs. 4(g) and 1(g). Here, in contrast to the Gaussian pulse, the Kerr-induced blue shift in the rear part is larger than the red shift of the leading edge including the peak.

Fig. 4 Same as Fig. 1 but for a Lorentzian input pulse [2t₀ = 0.2ns in Eq. (45)].

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5.5. Sech input pulse

A particularly interesting input shape is the sech pulse. For an assumed input pulse of duration 2t₀, we have

I_{0} (t) = \frac{{\hat{I}}_{0}}{cosh (t / t_{0})}, R_{0} (t) = t_{0} {\hat{I}}_{0}^{2} [1 + tanh (t / t_{0})]

and find J = π and

Q_{0} = 2 t_{0} {\hat{I}}_{0}^{2}

. The corresponding results for the propagating pulse are shown in Fig. 5, while the free-carrier density after the pulse can be calculated from Q(z) in Fig. 6 according to Eq. (7b). For this sech pulse, Eqs. (18) and (15) can be re-written

I (z, t) = \frac{{\hat{I}}_{0} exp [- M (z) - α z]}{cosh [M (z) + (t - z / v) / t_{0}]} = exp [- M (z) - α z] \cdot I_{0} [t_{0} M (z) + t - z / v]

and

\begin{array}{l} R (z, t) & = & t_{0} {\hat{I}}_{0}^{2} exp [- 2 M (z) - 2 α z] \cdot {1 + tanh [M (z) + (t - z / v) / t_{0}]} \\ = & exp [- 2 M (z) - 2 α z] \cdot R_{0} [t_{0} M (z) + t - z / v] \end{array}

where

M (z) = \frac{1}{2} ln [1 + 2 γ σ_{r} t_{0} {\hat{I}}_{0}^{2} L_{eff}^{(2)} (z)] .

Obviously, the shape and the duration of the sech pulse is maintained all along the propagation down the waveguide, while it is attenuated by a modified damping function

\frac{I [z, t_{peak} (z)]}{{\hat{I}}_{0}} = exp [- M (z) - α z] = \frac{exp (- α z)}{\sqrt{1 + 2 γ σ_{r} t_{0} {\hat{I}}_{0}^{2} L_{eff}^{(2)} (z)}} .

The attenuation profile Eq. (50) for the sech pulse of Eq. (47) along the propagation distance z is the same as that for the time-invariant intensity of the CW case given by Eq. (71) in Appendix B, when

τ_{eff} {\bar{I}}_{0}^{2}

is replaced by

t_{0} {\hat{I}}_{0}^{2}

. This means, that the pulse peak intensity may be by a factor Î₀/Ī₀ = (τ_eff/t₀)^1/2 larger than the CW intensity to induce the same FCA self-attenuation at any position z. For an effective free-carrier lifetime of 10ns and a pulse duration of 100ps this factor amounts to 10.

Fig. 5 Same as Fig. 1 but for a sech-shaped input pulse [2t₀ = 0.2ns in Eq. (46)].

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Fig. 6 (a) Peaking time t_peak(z) of sech pulses along the propagation distance z for input peak intensities 0, 1, and 2 times the threshold intensity I_inv of Eq. (55) and a pulse duration of 2t₀ = 0.2ns. (b) Q(z)/Q₀ for sech pulses as a function of z for different input peak intensities and 2t₀ = 0.2ns. (c) Upper limiting peak intensity I_lim(z) (black solid curves) for sech pulses of different durations 2t₀, and z-dependent peak intensity Î₀ exp [−M(z) −αz] (dashed red curves) of the travelling sech pulses for an input peak intensity Î₀ = 10 W/μm² and the given pulse lengths.

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We may conclude from Eq. (47), that attenuation by self-induced FCA is an ideal limiter for sech pulses since for large intensities or large effective interaction lengths, i.e. for ${\hat{I}}_{0}^{2} L_{eff}^{(2)} (z) ≫ 1 / (2 γ σ_{r} t_{0})$ , the output pulse will be purely sech but simultaneously be limited in strength to the asymptotic form

I (z, t) \leq \frac{I_{lim} (z)}{cosh [M (z) + (t - z / v) / t_{0}]}, I_{lim} (z) = \frac{exp (- α z)}{\sqrt{2 γ σ_{r} t_{0} L_{eff}^{(2)} (z)}},

where I_lim(z) is the maximum possible peak intensity of a sech pulse of width 2t₀ at a position z in the waveguide. Except of the waveguide input z = 0 where I(0, t) = I₀(t) may assume any launched initial condition and thus I_lim(z = 0) → ∞, the sech-pulse intensity inside the waveguide is restricted by the upper limit I_lim(z) as illustrated in Fig. 6(c). This is in clear contrast to the rectangular input pulse of Section 5.3, where the front peak of the propagating pulse does not see any self-induced FCA and can thus be arbitrarily increased in proportion to the input intensity.

Note that unlike solitons [41, 43], the sech input pulse here does not need to fulfil any fixed relation between pulse duration and peak intensity to propagate without changing its shape. Any sech pulse propagating according to the present model described by Eqs. (5), (7a) and (8) will keep its original shape. In addition, any input pulse having an exponentially rising leading edge will at least asymptotically turn into a sech pulse after enough self-induced FCA has shaped it.

In Figs. 1, 4 and 5, i.e. for smooth pulses, it can be observed that the pulses despite having already propagated into the waveguide may peak earlier than the input pulse at z = 0 itself. For the sech pulse of Eq. (47) we can quantify this peaking time at an arbitrary position z explicitly as

t_{peak} (z) = \frac{z}{v} - t_{0} M (z) .

Its dependence on the propagation distance z is illustrated in Fig. 6(a). Its slope along z,

\frac{d t_{peak} (z)}{d z} = \frac{1}{v_{peak} (z)} = \frac{1}{v} - \frac{γ σ_{r} t_{0}^{2} {\hat{I}}_{0}^{2} exp (- 2 α z)}{1 + 2 γ σ_{r} t_{0} {\hat{I}}_{0}^{2} L_{eff}^{(2)} (z)},

becomes zero at the position z = z_inv where

z_{inv} = \frac{1}{2 α} ln [\frac{γ σ_{r} t_{0} {\hat{I}}_{0}^{2} (1 + α v t_{0})}{α + γ σ_{r} t_{0} {\hat{I}}_{0}^{2}}] .

Thus, if z_inv > 0, then the apparent group velocity v_peak(z) = [dt_peak(z)/dz]⁻¹ of the FCA-attenuated sech pulse seems to be ±∞ at z = z_inv ± 0, negative for z < z_inv and positive only for z > z_inv (while, nevertheless, all energy or photons coming through unabsorbed up to this point have quite regularly been travelling at the group velocity v). Thus, z_inv is the position in the waveguide where the pulse peaks first and from which the latter seems to travel into both directions afterwards, seemingly amplified towards the input and obviously attenuated towards the end of the waveguide. This point z_inv shifts deeper into the waveguide with increasing peak intensity Î₀ and pulse width t₀, but no further than z_inv ≤ ln(1 + αvt₀)/2α. The seeming behavior of a group-velocity inversion requires a large enough input energy W₀ (or peak intensity Î₀ for a given t₀) for which z_inv ≥ 0,

W_{0} = π t_{0} {\hat{I}}_{0} \geq W_{inv} = π t_{0} I_{inv} = π / \sqrt{γ σ_{r} v} .

For the waveguide parameters given in Section 2, these conditions correspond to W₀ ≥ 1.87 × 10⁻⁹Ws/μm² or, e.g., to Î₀ ≥ 5.95W/μm² for t₀ = 0.1ns. The peaking time of the propagating sech pulse as a function of the position z is plotted in Fig. 4(a) for different input intensities Î₀ = 0 × I_inv, 1 × I_inv, and 2 × I_inv. It is easy to see that the pulse may peak inside the waveguide (z > 0) before the input pulse does when Eq. (55) is fulfilled.

At the waveguide input the velocity of the peak is $v_{peak} (z = 0) = v / (1 - v γ σ_{r} t_{0}^{2} {\hat{I}}_{0}^{2})$ , where it is larger than v for input energies W₀ up to W_inv, while it becomes infinite and changes sign when W₀ = W_inv, and finally reduces to zero for further increasing input energies. Together with the observation made in Section 5.3 that any intensity step always propagates at the group velocity v, the occurrence of the peak of smooth pulses inside the waveguide before the peaking at the input and the seeming inversion of the peak velocity v_peak have some similarity with the superluminal pulse propagation in media with extreme group-velocity modifications [42]. However, it should be noted that in the present paper no special requirements have been put on the group velocity v and the seeming superluminal propagation behavior only results from the attenuation of the rear part of the pulse by self-induced free carriers.

An explicit expression for Kerr phase shift of the sech pulse can be given by inserting Eqs. (46) into (21) but it is too lengthy to display here. A temporal delay of the Kerr phase against the propagating pulse intensity profile I(z,t) was predicted from Eq. (23). Indeed, the peak of the intensity of the propagating sech pulse in Fig. 4(e) after 50mm propagation distance occurs 47ps earlier than the peak of the corresponding Kerr phase shift in Fig. 4(g).

Recalling the observation that a temporally asymmetric Kerr phase should originate from the FCA-induced asymmetry of the propagating pulse intensity [21] it may seem paradoxical that the Kerr phase of the form-stable and hence always symmetric sech pulse also happens to become asymmetric. However, applying the general discussion of the destroyed symmetry relation between the propagating intensity profile and the Kerr phase from Section 4.1, it is clear that despite its maintained shape even the always symmetric sech pulse may develop an asymmetric Kerr phase as it is the history of the propagating pulse which determines the accumulated Kerr phase. While the trailing edge of the pulse contains the remainder of the peak of the input pulse, the Kerr phase could accumulate a significant amount there and thus the Kerr phase lags the intensity profile during propagation.

Finally, the FCR-induced phase and frequency shifts of the sech pulse follow directly from Eqs. (25) to (27) by inserting the corresponding expression for I₀(t − z/v), I(z,t) and R₀(t − z/v) from Eqs. (46) to (47). The FCR frequency shift may be written

Δ ω_{FCR} (z, t) = \frac{σ_{i} γ L_{eff}^{(2)} (z) {\hat{I}}_{0}^{2} exp [- M (z)]}{cosh [M (z)] + cosh [M (z) + 2 (t - z / v) / t_{0}]} .

Thus, the FCR-induced phase ϕ_FCR of the sech pulse, which is the dominant non-linear phase shift in Figs. 4(g) and 4(h), causes a maximum spectral blueshift

Δ ω_{FCR}^{(max)} (z) = \frac{σ_{i}}{σ_{r} t_{0}} \times \frac{\sqrt{1 + 2 γ σ_{r} L_{eff}^{(2)} (z) t_{0} {\hat{I}}_{0}^{2}} - 1}{\sqrt{1 + 2 γ σ_{r} L_{eff}^{(2)} (z) t_{0} {\hat{I}}_{0}^{2}} + 1},

which increases with input peak intensity and effective propagation length but never exceeds Δω_FCR ≤ σ_i/σ_rt₀. This maximum blueshift occurs at time

t_{Δ ω_{FCR}}^{(max)} (z) = \frac{z}{v} - \frac{t_{0}}{4} ln [1 + 2 γ σ_{r} L_{eff}^{(2)} (z) t_{0} {\hat{I}}_{0}^{2}]

= \frac{z}{v} - \frac{t_{0} M (z)}{2} = t_{peak} (z) + \frac{t_{0} M (z)}{2},

i.e., it lags the intensity peak at time t_peak(z) given by Eq. (52) by a delay t₀M(z)/2 steadily growing up to

t_{0} ln (1 + 2 η t_{0} {\hat{I}}_{0}^{2}) / 4

in agreement with the discussion below Eq. (27). Thus, the maximum FCR-induced blue shift occurs later and later with respect to the pulse peak. Interestingly, even the relative spread of the peaking times of the intensity and of the frequency shift grows with intensity and pulse width itself as

[t_{Δ ω_{FCR}}^{(max)} (z) - t_{peak} (z)] / t_{0} = ln [1 + 2 γ σ_{r} L_{eff}^{(2)} (z) t_{0} {\hat{I}}_{0}^{2}] / 4

. The FCR blueshift at the intensity peak t = t_peak(z) becomes

Δ ω_{FCR} [z, t_{peak} (z)] = \frac{σ_{i} γ L_{eff}^{(2)} (z) {\hat{I}}_{0}^{2}}{2 [1 + σ_{r} γ t_{0} {\hat{I}}_{0}^{2} L_{eff}^{(2)} (z)]},

and stabilizes at its maximum value σ_i/(2σ_rt₀) for large effective propagation distances and input intensities, i.e. at half of the limit of Eq. (57).

6. Asymptotic compressibility of pulses by self-induced free-carrier absorption

The self-attenuation of the pulse’s tail by the accumulating free carriers has been proposed as a means for compressing optical pulses in silicon waveguides, i.e., for reducing the pulse duration [18, 19]. Naturally, the discussion of pulse compressibility depends on the detailed definition of the pulse width, especially when the compression goes along with a change of the pulse shape. Here we compare pulses by normalizing and shifting them to make their peaks coincide. The spreading of the edges will then be seen to be a sufficiently unique criterion for the simple pulses considered here.

Some estimates about the necessary predominance of FCA over TPA for the compression of pulses have already been made in [19]. Here, however, we show that even in the absence of the attenuation by TPA only pulses with certain features of the initial pulse shape can be compressed.

The compression of optical pulses by self-induced FCA [18, 19] relies on the attenuation of the rear part of the pulse by the accumulated free carriers. However, the form stability of the sech pulses discussed in Section 5.5 raises the question what shape features are required to make a pulse compressible.

Obviously, after the original trailing edge and the pulse peak have been attenuated by FCA, the remainder of the propagating pulse will be comprised of the former leading edge and the shape of the latter becomes most important for the evolving pulse shape. To discuss this in more detail we refer to the normalized presentation of the pulse evolution in Eq. (31) where f₀(ξ) and F₀(ξ) depend only on the shape but neither on the peak intensity Î₀ nor the duration t₀ of the pulse. If the relative slope (∂f₀/∂t)/f₀ of the leading edge increases and hence the latter becomes steeper and steeper towards negative times t → −∞, or the intensity is even zero before a certain time t < 0, the pulse may be compressible at least asymptotically for large FCA-distortion parameters Θ_FCA. This may be visualized as follows. After the rear part of a pulse has been taken away by self-induced FCA the pulse width may be judged by re-normalizing the attenuated pulse, e.g., to its now-reduced power or peak intensity. If then the remaining leading edge is more flat because it has a smaller relative slope it will be found to stretch further towards negative times and the re-normalized pulse appears to be broadened. Vice versa, if the relative slope of the remaining leading edge is found to be larger and the latter has become steeper then the pulse may have been compressed.

This is obvious for the Gaussian pulse in Eq. (43) which is compressible since $(\partial f_{0} / \partial t) / f_{0} = - 2 t / t_{0}^{2}$ and the leading edge becomes steeper and steeper for t → −∞, where the pulse energy is not widely enough distributed to prevent compression [see Fig. 7(a)]. In other words, the mechanism which cuts away the rear part of pulse faces a steeper and steeper relative slope of the remaining front part of the pulse as it progresses towards the leading edge. The rectangular pulse of Eq. (44) is even easier to compress since it has an infinitely steep front edge and is zero for t < −t₀, compare Fig. 7(b). Thus, for pulses having an intensity step in the leading edge down to zero such as the rectangular pulse in Section 5.3, pulse compression by self-produced free carriers is most efficient.

Fig. 7 Pulse compressibility: Normalized pulse intensity after 10 mm propagation distance (solid black curves), where arrows indicate the increase of the input peak intensities Î₀ from 5 to 50 to 500 W/μm². Dashed red curves indicate the input intensity for (a) Gaussian, (b) rectangular, (c) sech, and (d) Lorentzian input pulse.

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On the other hand, the relative front slope of the Lorentzian pulse (∂f₀)/∂t/f₀ ≃ −2/t decreases monotonously towards t → −∞. The more energy is cut away from the rear end the more important becomes the remaining energy spreading far into the front edge, and self-induced FCA even broadens the Lorentzian pulse as shown in Fig. 7(d).

The limiting shape between pulse compression and broadening is the exponentially rising front edge since its relative slope is the same everywhere and cutting away some rear part of the pulse by self-produced free carriers asymptotically shapes the pulse into the sech form. Thus any pulse having an exponentially rising leading edge can at least asymptotically be compressed or broadened until it has assumed the sech shape. The resulting incompressible sech pulse, then, remains stable and does not change its shape as shown in Section 5.5, see Fig. 7(c). Moreover, the sech pulse can be compressed even less if TPA is present. TPA tends to broaden the pulse [18,21] while FCA would tend to attenuate its rear part and shape it back into sech as the leading edge remains exponential.

The FCA-distortion parameter $Θ_{FCA} (z) = γ σ_{r} t_{0} {\hat{I}}_{0}^{2} L_{eff}^{(2)} (z)$ in Eq. (31) approaches $η t_{0} {\hat{I}}_{0}^{2}$ for large propagation distances z ≫ 1/(2α), after which the pulse distortion comes to an end and the pulse shape stabilizes. Thus, pulse distortion is eased when η is large, i.e., when α is small or $t_{0} {\hat{I}}_{0}^{2}$ is large. For zero attenuation α → 0, the pulse distortion parameter $Θ_{FCA} (z) = γ σ_{r} t_{0} {\hat{I}}_{0}^{2} z$ may increase arbitrarily large along z. Thus if a pulse has a compressible shape such as the rectangular or the Gaussian one, small linear losses α help achieving highly compressed pulses. Equation (31) also implies that a pulse with a given input shape at z = 0 will go through the same distortion no matter what pulse duration t₀ or peak intensity Î₀ it has, as long as Θ_FCA(z) reaches the same final value at the waveguide output. For Θ_FCA(z) = 0 the pulse shape remains unchanged.

Thus, one of the particular advantages of the present FCA-only formulation is that it allows us to check easily whether a pulse is compressible by self-induced FCA at all, even when TPA actually has such a significant effect on the pulse distortion that it may not be neglected. In contrast to this, with the alternative and more accurate full model the lack of analytical solutions would frustrate the attempt for such an explicit judgement, and the numerical simulations as the only left alternative would end up with less clear insight.

The principal tendency of TPA to broaden a pulse even in the presence of FCA can clearly be seen in Fig. 2(a). As compared to the FCA-only results from Eq. (18) shown by solid black lines, the full inclusion of TPA according to Eq. (67) flattens the propagating pulse down to the green dotted curvesd which essentially have lost the pronounced peak and are therefore significantly broader. Thus, if a pulse is found to be not compressible in a FCA-only model then it is even less so after inclusion of TPA.

7. Attenuation and phase shift of a weak probe pulse following an intense pump pulse

We now consider a weak second pulse, which we call the ”probe pulse”, following an intense first pulse, which we call the ”pump pulse”, in a temporal distance T at nearly the same wavelength and group velocity. Its pulse duration shall again be much shorter than the free-carrier lifetime τ_eff. The knowledge of this attenuation may be useful in experiments where pulsed instead of CW operation is chosen to mitigate FCA such as pulsed Raman experiments [13, 14, 44, 45], pulse compression [18, 19] or continuum generation [9, 10]. Because of its small intensity assumed here the probe pulse does not suffer any non-linear self-distortion. However, it experiences an excess attenuation and phase shift caused by the free carriers generated by the strong pump pulse ahead, which has been launched a time T ≫ Δt earlier. Both the free-carrier-induced attenuation and the phase shift vary only slightly across the short probe pulse as its duration is assumed much shorter than the free-carrier lifetime. We may write the field amplitude of the probe pulse e(z,t) = e₀(t − z/v)exp{iψ_FCR(z) − [αz + a_FCA(z)]/2}, where e(z = 0, t) = e₀(t) is the input field amplitude of the probe pulse at z = 0 and

a_{FCA} (z) = \int_{0}^{z} α_{FCA} (z^{'}) d z^{'}, ψ_{FCR} (z) = - \frac{σ_{i}}{2 σ_{r}} a_{FCA} (z)

are the accumulated attenuation and phase, respectively, induced via FCA and FCR by the free-carriers generated by the pump pulse ahead. With Eqs. (7b) and (17) the local FCA-induced attenuation coefficient is

α_{FCA} (z) = σ_{r} N [z, t_{P} (z) + T] = σ_{r} γ Q (z) G (T) = \frac{G (T) σ_{r} γ Q_{0} exp (- 2 α z)}{1 + γ σ_{r} Q_{0} L_{eff}^{(2)} (z)} .

Comparison with the analogous attenuation for the probe pulse caused by the free carriers generated by a CW pump as derived in Eq. (71) in Appendix B reveals that pulsed and CW operation result in the same probe-pulse attenuation α_FCA(z) = ᾱ_FCA(z) at all positions z only if G(T) = 1/2 and

Q_{0} = t_{0} {\hat{I}}_{0}^{2} F_{0} (\infty) = 2 τ_{eff} {\bar{I}}_{0}^{2}

, where Ī₀ is the input intensity in case of CW operation at z = 0 as defined in Appendix B. For a Gaussian input pump pulse and an exponential decay according to Eq. (4), this happens for Î₀ = (2/π)^1/4(2τ_exp/t₀)^1/2Ī₀ ≃ 1.26 (τ_exp/t₀)^1/2Ī₀ and T = ln(2) × τ_exp = 0.69 τ_exp. For a discussion of sech-shaped pump pulses see Section 5.5.

The accumulated attenuation the probe pulse will have experienced after following the pump pulse all along the waveguide length L follows from Eqs. (61) and (62),

a_{FCA} (L) = G (T) ln [1 + σ_{r} γ Q_{0} L_{eff}^{(2)} (L)] .

For a sufficiently long waveguide L ≫ 1/2α, it approaches the constant asymptotic value a_FCA ≃ G(T)ln(1 + ηQ₀). In even longer waveguides the probe pulse finally travels in a region where the preceding pump pulse had no longer been strong enough to leave a perceptible amount of free carriers. The attenuation can be arbitrarily reduced by increasing the time T between the two pulses according to the detailed shape of the decay function G(t).

In order to see what is gained by using pulse operation instead of CW, we now seek to find out up to which input peak intensities Î₀ of the pump pulse the aggregate attenuation for the probe pulse at the waveguide output z = L is smaller than in the CW case, i.e.,

a_{FCA} (L) < {\bar{a}}_{FCA} (L) .

The aggregate FCA attenuation ā_FCA(L) for the CW operation is given by Eq. (72) in Appendix B. Thus, the relation in Eq. (64) is fulfilled when

Q_{0} = t_{0} F_{0} (\infty) {\hat{I}}_{0}^{2} < \frac{{[1 + 2 σ_{r} γ L_{eff}^{(2)} (L) τ_{eff} {\bar{I}}_{0}^{2}]}^{1 / [2 G (T)]} - 1}{σ_{r} γ L_{eff}^{(2)} (L)}

It is easy to see that for G(T) = 1/2 this relation is true for

Q_{0} < 2 τ_{eff} {\bar{I}}_{0}^{2}

independently of the waveguide length. Thus, the input peak intensity of the pump pulse can be up to a factor

{\hat{I}}_{0} / {\bar{I}}_{0} \leq \sqrt{2 τ_{eff} / t_{0} F_{0} (\infty)}

larger than the CW pump intensity before the subsequent probe pulse suffers the same FCA-induced attenuation. For G(T) = 1/4, this relation becomes length dependent,

Q_{0} < 4 τ_{eff} {\bar{I}}_{0}^{2} [1 + σ_{r} γ τ_{eff} {\bar{I}}_{0}^{2} L_{eff}^{(2)} (L)]

. In waveguides of short length L pulse-induced FCA for the probe pulse is weaker than CW-induced FCA as long as

{\hat{I}}_{0} / {\bar{I}}_{0} < 2 \sqrt{τ_{eff} / t_{0} F_{0} (\infty)}

while long waveguides leave even more room, namely

{\hat{I}}_{0} / {\bar{I}}_{0} < 2 {[τ_{eff} (1 + η τ_{eff} {\bar{I}}_{0}^{2}) / t_{0} F_{0} (\infty)]}^{1 / 2}

, as the latter limit increases with

{\bar{I}}_{0}^{2}

. In case of an exponential decay function according to Eq. (4), G(T) = 1/2 and G(T) = 1/4 corresponds to T = ln(2) × τ_exp and T = ln(4) × τ_exp, respectively. In both cases the pump pulse can have a peak intensity larger than the CW pump intensity in proportion to

\sqrt{τ_{eff} / t_{0}}

before causing more FCA for the probe pulse.

The situation is different if the probe pulse follows the intense pump pulse very closely and Eq. (65) has to be fulfilled for G(T ≪ τ_eff) ≃ 1. For short enough waveguides or low intensities, this condition reqiures ${\hat{I}}_{0} / {\bar{I}}_{0} < \sqrt{τ_{eff} / t_{0} F_{0} (\infty)}$ . However, for sufficiently long waveguides and large intensities, the admissible ratio ${\hat{I}}_{0} / {\bar{I}}_{0} < {[\sqrt{2 τ_{eff}} / η t_{0} F_{0} (\infty) {\bar{I}}_{0}]}^{1 / 2}$ for guaranteeing the condition in Eq. (64) decreases for an increasing CW input intensity and asymptotically becomes zero for Ī₀ → ∞. In particular, if

{\bar{I}}_{0} > \frac{1}{t_{0} F_{0} (\infty)} \times \sqrt{\frac{2 [τ_{eff} - t_{0} F_{0} (\infty)]}{σ_{r} γ L_{eff}^{(2)} (L)}},

the input peak intensity Î₀ of the pump pulse must be smaller than the input CW intensity Ī₀ to exert a lower accumulated FCA on the closely following probe pulse.

This inverted behavior can be understood as follows. For the same input intensity the CW pump is attenuated by self-induced FCA more rapidly than the pulsed pump, see for example the discussion below Eq. (50) for the sech pulse. When the CW pump is exhausted after a certain propagation distance it can no longer generate enough free carriers to attenuate the probe pulse. The pulsed pump with the same input peak intensity, however, has a much longer reach to produce free carriers and the probe pulse following closely experiences a larger aggregate FCA when propagating to the end of the waveguide.

The upper limit on G(T) above which pulse operation is no longer prior to CW operation for all input intensities is G(T) = 1/2. Since for large input intensities $exp [a_{FCA} (L)] \propto {\hat{I}}_{0}^{G (T)}$ and $exp [{\bar{a}}_{FCA} (L)] \propto {\bar{I}}_{0}^{1 / 2}$ , a_FCA(L) will always be able to exceed ā_FCA(L) for G(T) > 1/2 even for Î₀ = Ī₀, if only these two input intensities are large enough. Vice versa, for G(T) < 1/2 pulsed operation may reduce the aggregate FCA-induced attenuation on a following probe pulse. For an exponential free-carrier decay according to Eq. (4) this is true when the probe pulse follows in a temporal distance larger than T > 0.693τ_exp.

For the phase ψ_FCR(z) of the probe pulse analogous conclusions can be drawn using Eq. (61).

8. Conclusions

In conclusion we have derived explicit analytical solutions for the propagation of optical pulses and their attenuation by self-produced free carriers in silicon waveguides allowing us to quantify the pulse distortion and to calculate explicitly the free-carrier density and the nonlinear phase shifts caused by the Kerr effect and by free-carrier refraction. We have shown that the omission of two-photon absorption as a cause of attenuation as done in the derivation and accounting only for free-carrier absorption appropriately models the situation in short or highly lossy silicon-based waveguides especially for high-energy input pulses. On the other hand this formulation may also serve as a tool in understanding the role of TPA and FCA in the interplay during propagation when aiming at continuum generation or pulse compression in low-loss waveguides. Sech-shaped intensity pulses have been shown to maintain their shape independently of the intensity or pulse width with a resemblance to superluminal light propagation.

9. Appendix A

In this Appendix we show that TPA as a source of attenuation can be neglected in a full model actually admitting of both TPA and FCA, as soon as it has been found to be negligible already in a simpler TPA-only model. The well-known analytical solution of the latter allows us to make this decision of omitting TPA explicitly. The propagation equation analogous to Eqs. (5), (7a) and (8) resulting from the full problem described by Eq. (1) with TPA included but GVD neglected may be written

\frac{\partial I}{\partial z} + \frac{1}{v} \frac{\partial I}{\partial t} = \frac{2}{v} \frac{\partial I}{\partial \tilde{x}} = - (α + σ_{r} N) I - β_{r} I^{2}, N (z, t) = γ \int_{- \infty}^{t} I^{2} (z, t^{'}) d t^{'},

where N = N(z,t) = N [v(x̃ − ỹ)/2, (x̃ + ỹ)/2] is understood according to Eq. (11). Using [37] and assuming I(0, t) = Î₀f₀(t/t₀) at the waveguide input with f₀(t/t₀) ≤ 1 as in Eq. (28), we derive the solution accounting for the accumulated non-linear attenuation by TPA and FCA after propagation distance z,

I (z, t) = \frac{{\hat{I}}_{0} f_{0} [(t - z / v) / t_{0}] exp (- α z) U (z, t)}{1 + β_{r} {\hat{I}}_{0} {\tilde{L}}_{eff}^{(1)} (z, t) f_{0} [(t - z / v) / t_{0}]},

where

U (z, t) = exp {- σ_{r} \int_{0}^{z} N [ζ, t + (ζ - z) / v] d ζ}, N (z, t) = γ \int_{- \infty}^{t} I^{2} (z, t^{'}) d t^{'} .

Importantly,

{\tilde{L}}_{eff}^{(1)} (z, t) = \int_{0}^{z} exp (- α χ) U (χ, t) d χ

is a TPA-effective propagation length modified by the presence of U(χ, t) in the integral as compared to the original definition (16). The system of Eqs. (68)–(70) is complete to describe pulse propagation in the presence of both TPA and FCA. Although N can not be given explicitly, we will later use its property of always being non-negative, N ≥ 0.

For the moment, we only focus on the denominator in Eq. (68). If FCA is mathematically switched off by setting γ = 0, then N ≡ 0, U(z,t) ≡ 1, and ${\tilde{L}}_{eff}^{(1)} (z, t) \equiv L_{eff}^{(1)} (z) = [1 - exp (- α z)] / α$ becomes the TPA-effective propagation length according to (16), and hence the TPA-only model of Eqs. (33) and (34) is obtained. We assume further that, within this TPA-only model, $L_{eff}^{(1)} (z)$ has already been found to be small enough that the denominator can be approximated by unity because the condition in Eq. (35) be valid. If now FCA would be taken into account by setting γ > 0 again, then N > 0, and consequently U(z,t) < 1 would ensure that ${\tilde{L}}_{eff}^{(1)} (z, t)$ becomes even smaller than $L_{eff}^{(1)} (z)$ , and approximating the denominator in Eq. (68) by unity would be justified even more. Finally, since this denominator is the only place where β_r enters explicitly the complete system of the modelling equations (68)–(70), approximating it by unity is equivalent with setting β_r = 0 in the whole model and hence with fully ignoring TPA as a source of attenuation. This becomes obvious when checking that Eqs. (68) through (70) with the denominator set to unity exactly solve the FCA-only problem Eq. (5). The range of applicability of our FCA-only formulation is illustrated in Section 5.1.

Thus, we have shown that it is sufficient to check in a simple TPA-only model as described by Eqs. (33) and (34) whether the TPA-distortion parameter Θ_TPA is small enough that TPA can be neglected in the full problem of Eq. (67) actually admitting of both TPA and FCA, and hence whether the FCA-only model presented here applies.

10. Appendix B

In this Appendix we derive the solution of the problem described by Eqs. (5) and (2) under CW operation. In this case we have ∂/∂t = 0, and I(z,t) = Ī(z), N(z,t) = N̄(z) = γτ_effĪ²(z) with the initial conditions Ī₀ = Ī(z = 0) and ${\bar{N}}_{0} = \bar{N} (z = 0) = γ τ_{eff} {\bar{I}}_{0}^{2}$ , where τ_eff is defined by Eq. (3). For example, we have τ_eff = τ_exp for the exponential decay function in Eq. (4). We derive the CW intensity Ī(z) and the local FCA coefficient ᾱ_FCA(z) = σ_rN̄(z) from Eq. (5) by separation of variables,

\bar{I} (z) = \frac{{\bar{I}}_{0} exp (- α z)}{\sqrt{1 + 2 σ_{r} γ τ_{eff} {\bar{I}}_{0}^{2} L_{eff}^{(2)} (z)}}, {\bar{α}}_{FCA} (z) = \frac{σ_{r} γ τ_{eff} {\bar{I}}_{0}^{2} exp (- 2 α z)}{1 + 2 σ_{r} γ τ_{eff} {\bar{I}}_{0}^{2} L_{eff}^{(2)} (z)} .

The latter results in an accumulated attenuation for a weak probe pulse after propagation to the waveguide output z = L,

{\bar{a}}_{FCA} (L) = \int_{0}^{L} {\bar{α}}_{FCA} (z^{'}) d z^{'} = \frac{1}{2} ln [1 + 2 σ_{r} γ τ_{eff} {\bar{I}}_{0}^{2} L_{eff}^{(2)} (L)] .

These results allow us to compare the pulse propagation with the CW situation in Sections 5.5 and 7.

Acknowledgments

This work was funded by the Deutsche Forschungsgemeinschaft (DFG) under Grant FOR 653.

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Nonlinear distortion of optical pulses by self-produced free carriers in short or highly lossy silicon-based waveguides

Abstract

1. Introduction

2. Pulse propagation in silicon waveguides

3. Nonlinear phase shifts

3.1. Kerr-induced phase shift

3.2. Free-carrier induced phase shift

4. Interpretation of omitting TPA as a cause of attenuation

5. Propagation of pulses with various input shapes

5.1. Gaussian input pulse

5.2. Effect of omitting TPA illustrated with the Gaussian input pulse

5.3. Rectangular input pulse

5.4. Lorentzian input pulse

5.5. Sech input pulse

6. Asymptotic compressibility of pulses by self-induced free-carrier absorption

7. Attenuation and phase shift of a weak probe pulse following an intense pump pulse

8. Conclusions

9. Appendix A

10. Appendix B

Acknowledgments

References and links

Cited By

Figures (7)

Equations (72)

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