Influence of the retarded response on an ultrafast nonlinear optofluidic fiber coupler

Sebastian Pricking; Marius Vieweg; Harald Giessen

doi:10.1364/OE.19.021673

With the recent advances in preparing selectively liquid-filled photonic crystal fibers (PCFs) [1,2] it has become possible to fabricate new coupling devices [3] based on two or more nearby waveguides filled with highly nonlinear liquids embedded in the photonic structure of a PCF [4] (see Fig. 1a). These devices are of high interest for the studies of the nonlinear coupling behavior since they offer the opportunity to tune the coupling strength, the dispersion, as well as the nonlinear parameter by adjusting the fiber geometry and the used materials accordingly. In addition to these advantages the PCF forms a convenient scaffold which ensures a constant core-to-core distance over several centimeters of propagation length, hardly achievable by usual liquid-filled capillaries put in close vicinity of each other.

Fig. 1 a) Photonic crystal fiber with the two waveguides filled with the hypothetic medium. In our simulations the hole diameter is d = 2.7 μm, the hole-to-hole distance Λ = 5.6 μm, and the length L = 36 mm. b) Measured retarded responses of commonly used media, normalized to their maximum value (CS₂ [11], toluene [12], CCl₄ [13], chloroform [14], and fused silica [15]). The gray lines show the retarded responses of our hypothetic medium for different values of t_R, rising quadratically from 2 fs (narrowest) to 200 fs (broadest). The inset is a zoom for t < 0.2 ps.

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The Kerr index of the highly nonlinear liquids, among them CS₂, CCl₄, nitrobenzene, or toluene, is up to 100 times larger than in the case of fused silica, which reduces the required nonlinear switching power so that low-power nonlinear coupling becomes possible.

Earlier experimental or theoretical investigations of nonlinear fiber couplers have mainly focused on a continuous-wave (cw) input [3, 5] or on nonlinear soliton switching between two waveguides made of fused silica [6, 7]. In the work presented in [8] and [9], the retarded response of fused silica due to the Raman effect and its influence on the coupling was investigated. However, when using highly nonlinear liquids as the carrier medium, their retarded responses differ considerably from each other and that of fused silica in strength as well as in their characteristic time scales [10]. Due to this reason a general analysis of the effects induced by the retarded response without focussing on a specific medium is desirable for the design of an optofluidic coupler.

In this paper we investigate the influence of the input pulse duration and the retarded material response on the properties of an ultrafast nonlinear optofluidic fiber coupler. We find that in particular the critical input peak power P_C, above which the coupling is suppressed, may increase dramatically, depending strongly on both parameters. We present a simple model capable to explain the findings.

By introducing a hypothetic medium with fixed dispersion and nonlinearity but with a variable retarded response we are able to separate the effects induced by the retarded response from other influences [10]. The nonlinear propagation in two identical nearby waveguides filled with this hypothetic medium is known to be well described by two coupled generalized nonlinear Schrödinger equations (NLSEs) [15]:

\begin{array}{l} \frac{\partial A_{j}}{\partial z} & = & \sum_{k = 2}^{7} \frac{i^{k + 1}}{k!} β_{k} \frac{\partial^{k}}{\partial T^{k}} A_{j} \\ + & i κ A_{3 - j} \\ + & i γ (1 + \frac{i}{ω_{0}} \frac{\partial}{\partial T}) A_{j} \int_{- \infty}^{+ \infty} R (t^{'}) \\ \times & ({| A_{j} (T - t^{'}) |}^{2} + f_{j (3 - j)} {| A_{3 - j} (T - t^{'}) |}^{2}) d t^{'} \end{array}

with A_j (j = 1,2) being the electric field envelope in the jth waveguide, z the propagation distance, and T the time in the co-moving reference frame.

β_{k} = \frac{\partial^{k} β}{\partial ω^{k}} |_{ω_{0}}

denote the dispersion coefficients with β being the propagation constant. γ = n₂ω₀/(cA_eff) represents the nonlinear parameter with n₂ as the nonlinear index of refraction, ω₀ the central frequency of the pulse, c the vacuum speed of light and A_eff the effective mode area. κ is the linear coupling strength and f_j_(3–_j₎ the cross-phase modulation coefficient. For simplicity we neglect the influence of noise. The NLSEs in this form are well known to accurately simulate the propagation of ultrafast pulses with durations as short as 10 fs, if sufficiently high orders for the dispersion coefficients β_k are taken into account [15].

The response of the medium is governed via R(t), defined by

R (t) = (1 - f_{R}) δ (t) + f_{R} \frac{h_{R} (t / t_{R})}{t_{R}} .

The first term describes the instantaneous response, and the second covers the retarded response via h_R, which we define with a unitless argument as

h_{R} (x) = \frac{1}{N_{0}} (exp (- x) + \frac{1}{x + 1}) (1 - exp (- x)) exp (- \frac{x}{100}) Θ (x)

with N₀ being a normalization constant and Θ(x) the Heaviside function that ensures causality. The fraction of the retarded response is given by f_R.

This phenomenological design of R(t) has been shown to represent a good approximation [10] to measured retarded responses of commonly used media (Fig. 1b), allowing us to model all regarded liquids with only one free parameter apart from the fraction f_R to keep the parameter space small. This free parameter we call t_R, namely the characteristic time, and use it to scale the time axis.

The PCF in which we embed the hypothetic medium has a hole diameter of 2.7 μm and a hole-to-hole distance of Λ = 5.6 μm. This leads to a waveguide separation of $\sqrt{3} Λ = 9.7 μ m$ . For the calculation of the dispersion coefficients [15] we use the Kerr-corrected refractive indices of CCl₄ [16], a central wavelength of λ₀ = 1040 nm and include the fiber geometry. This results in β₂ = 5.5787 × 10⁻²⁷ s²/m, β₃ = 9.6736 × 10⁻⁴² s³/m, β₄ = −3.8758 ×10⁻⁵⁷ s⁴/m, and β₅ = 1.6628 × 10⁻⁷² s⁵/m for the linear coefficients. Since β₂ is positive, we operate in the normal dispersion regime, and soliton formation is not expected. With the Kerr index of CCl₄ being n₂ = 15 × 10⁻²⁰ m²/W [17], we obtain for the nonlinear parameter γ = 0.0265 (Wm)⁻¹. The coupling strength κ is set to 174.53 m⁻¹, leading to a linear coupling length L_C = π/2κ = 9 mm [3]. The propagation length z₀ is set to 4L_C = 36 mm. The cross-phase modulation coefficient becomes f_j_(3–_j₎ = 8 ×10⁻⁴ [18].

In principle, the waveguides can also be operated in the anomalous dispersion regime. However, in that case due to the high nonlinearity of the regarded liquids, soliton fission and supercontinuum generation is likely. A full analysis of the propagation and switching behavior in the anomalous dispersion regime including the supercontinuum generation requires a more detailed discussion which we will present in a future publication.

To solve Eq. 1 we implement the symmetrized split-step Fourier method [15], which is well suited for the pulse durations and spectral widths we expect for the results. As the initial pulse we choose a sech-pulse given by

A_{1} (0, T) = \sqrt{P_{0}} sech (\frac{T}{T_{0}}), A_{2} (0, T) = 0,

where P₀ denotes the peak power and T₀ the pulse width. As central wavelength we choose λ₀ = 1040 nm for the following simulations.

Fig. 2 shows a selection of evolutions of the pulse energy contained in waveguide 1, normalized to the input pulse energy, for different input pulse durations and retarded response parameters. Common for all plots is the following behavior: Starting from low input powers, the coupling length increases for increasing input peak power. This corresponds to a decreasing coupling strength, which can be explained by a higher confinement and hence lower overlap of the fields. This higher confinement is caused by the increasing index contrast between the fiber core and cladding due to the Kerr effect. Once a certain input peak power is reached, namely the critical power P_C, the coupling is suppressed and the pulse energy remains in waveguide 1. In case of cw input, the critical power can be calculated by P̃_C = 4κ/γ ≈ 26.4 kW [3], which in contrast to the results of Fig. 2 is independent of T₀ and the retarded response. We observe a shift to higher values of P_C if the pulse duration T₀ is decreased. Furthermore, the shift of the critical power to higher values becomes stronger for a more distinctive retarded response, i.e., for increasing f_R and t_R.

Fig. 2 Pulse energy contained in waveguide 1, normalized to the input pulse energy, in dependence on input peak power P₀ and propagation distance z: a,b,c) T₀ = 50 fs, d,e,f) T₀ = 5 ps; a,d) f_R = 0, b,c,d,e) f_R = 0.8; b,e) t_R = 2 fs, c,f) t_R = 200 fs.

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Performing more simulations with varying input pulse durations and retarded response parameters, we show in Fig. 3a the critical powers P_C extracted manually from the simulation results and the corresponding evolution plots. Clearly a strong deviation from the cw-case is observed. P_C may exceed the cw-case P̃_C by a factor of more than 3 for short pulses and high retardation.

Fig. 3 The critical power P_C in dependence of the input pulse duration T₀, the characteristic response time t_R, and the fraction f_R: a) P_C extracted from the simulations, b) P_C calculated with our model.

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To quantify our findings we expand the cw-case critical power by taking the retarded response into account:

P_{C} (T) = \frac{4 κ}{γ} {(\int_{- \infty}^{\infty} R (t^{'}) \frac{{| A_{1} (0, T - t^{'}) |}^{2}}{P_{0}} d t^{'})}^{- 1} = \frac{{\tilde{P}}_{C}}{D_{C} (T)}

Inserting Eq. 2, we obtain for the correction divisor

D_{C} (τ) = (1 - f_{R}) {sech}^{2} (τ) + \frac{f_{R}}{τ_{R}} \int_{0}^{\infty} h_{R} (\frac{τ^{'}}{τ_{R}}) {sech}^{2} (τ - τ^{'}) d τ^{'}

with τ′ = t′/T₀, τ = T/T₀, and τ_R = t_R/T₀. We define the maximum of D_C(τ) as the appropriate correction divisor so that P_C = P̃_C/D_C. D_C can be derived easily by numerical integration. We note that t_R and T₀ enter D_C only as ratio τ_R. For f_R = 0, t_R = 0, or T₀ → ∞, the correction divisor becomes D_C = 1, and hence the cw-case is reproduced. Fig. 3b shows P_C obtained by this simple model, which exhibits an excellent agreement with the critical powers extracted by the simulations for a small retarded contribution, and gives still a good qualitative agreement for larger retarded contributions. We attribute the occurring deviations to the fact that our model does not include dispersion terms and self-steepening, and that it takes into account only the input field envelope at z = 0.The correction divisor D_C is plotted in dependence of t_R/T₀ and f_R in Fig. 4.

Fig. 4 The correction divisor D_C in dependence of the input pulse duration T₀, the characteristic response time t_R, and the fraction f_R.

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To summarize, we have shown by numerical simulations that the input pulse duration as well as the retarded material response have a large influence on the properties of an ultrafast nonlinear optofluidic fiber coupler. In particular, the critical power may have to be corrected by a factor larger than 3 for commonly used laser sources and liquids. These findings are supported by a simple model and might aid further investigations on design and experimental realization of an ultrafast nonlinear optofluidic fiber coupler.

Acknowledgments

The authors would like to thank the Landesgraduiertenförderung of Baden-Württemberg and DFG (Open Access Publishing) for support of this work.

References and links

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Influence of the retarded response on an ultrafast nonlinear optofluidic fiber coupler

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