1. Introduction

The interaction of light with matter on a nanometer scale opens up new opportunities which may have far reaching implications in telecommunications, computation, biophotonics, and sensing technologies [1]. Lately, single-mode sub-wavelength waveguides or nanowires have been realized in several material systems [2–11]. Such optical nanowire structures are capable of providing superior light confinement and are thus ideal for nonlinear optics applications [12]. Clearly, of interest will be to fabricate such nanowires using high contrast, highly nonlinear materials, such as AlGaAs-known to exhibit a nonlinearity that is three orders of magnitude higher than that of silica glass [13,14]. Apart from being highly nonlinear, AlGaAs nanowires are also highly promising in terms of applications since they can be integrated with other optoelectronic components on the same wafer and thus can serve as information conduits among miniaturized devices. Quite recently, enhanced spectral broadening or SPM has been observed for the first time in 700μm long AlGaAs nanowires [15]. An important question associated with this particular system is whether optical solitons are possible in AlGaAs nanowaveguides. This is of relevance since solitons can be used to either overcome dispersion effects or to achieve pulse compression in such nanostructures. We note that so far, optical solitons have only been observed in multi-layer AlGaAs structures in which the dispersion can be engineered [16,17]. Yet, in primitive weakly guiding AlGaAs waveguides such solutions are not possible since this material system exhibits appreciable normal dispersion in the spectral region of interest [18], i.e. for 1.1μm < λ < 1.7μm. Here we theoretically demonstrate, that, because of high contrast, the dispersion of an air-clad AlGaAs nanowire can become strongly anomalous (becomes inverted), thus overcoming material dispersion limitations. This in turn may allow optical soliton formation in millimeter long structures. These solitons are possible at very low power levels (at ~5 W) in spite of the fact that the AlGaAs nanowires can exhibit anomalous dispersion that is a thousand times higher than that of silica glass. The intensity and spectral evolution of these solitons is investigated in AlGaAs nanowaveguides in the presence of loss, multi-photon absorption, and higher-order dispersive and nonlinear effects.

2. AlGaAs Nanowire dispersion properties

To analyze the dispersion properties of AlGaAs nanowires, let us consider for example an Al _0.2 Ga _0.8 As nanorod of core radius a as shown in Fig. 1(a). This particular composition (Al _0.2 Ga _0.8 As) is deliberately chosen since in the neighborhood of λ = 1.55μm it is known to exhibit relatively low two-photon absorption [19]. In addition, we assume that the nanowire is air-cladded. We note that even though completely air-clad structures are rather difficult to develop these days, it is yet possible to fabricate waveguides that are mostly surrounded by air, with characteristics very similar to the one analyzed here. At this wavelength the Al _0.2 Ga _0.8 As is approximately 3.27, and the nanorod is operated at or close to the single-mode regime. In general, the nanowire dispersion relation is given by [20]:

In Eq.(1) J_m (u) is a Bessel function of the first kind and of order m and similarly K_m (w) is a modified Bessel function, and primes represent derivatives with respect to the argument. u = ak_o $\sqrt{n_{\mathit{\text{core}}}^2-n_{\mathit{\text{eff}}}^2}$,w = ak_o $\sqrt{n_{\mathit{\text{eff}}}^2-n_{\mathit{\text{clad}}}^2}$ where k ₀ = 2π/λ is the free space wavevector and n_eff is the effective refractive index of the mode [20]. n_core is the core index and n_clad =1 since the exterior medium is air. The wavelength dependence of the refractive index is accounted by using the Sellmeier expansion of Al_xGa_1-x As, i.e [18]:

In general, the constants A, C ₀, C ₁, E ₀ and E ₁ depend on the Al composition (x), the temperature T, and the free space wavelength λ. Here all calculations were carried out at room temperature.

The dispersion properties of the fundamental mode HE ₁₁ of such a nano-structure are obtained by numerically solving Eq. (1) for m = 1, with the core index evaluated at each frequency step from Eq. (2). The dispersion coefficient β″ =d ² β/d ω ² of an Al _0.2 Ga _0.8 As nanowire is shown in Fig.1(b) in units of ps ²/m for different core radii. In this same figure, the bulk Al _0.2 Ga _0.8 As dispersion is also included for comparison.

 

Fig. 1. (a) a nanowire structure; (b) group velocity dispersion β″ of an Al _0.2 Ga _0.8 As nanowire when its radius is (A) 160, (B) 175 and (C) 193 nm. Bulk dispersion of Al _0.2 Ga _0.8 As is also shown.

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It is important to note that because of the strong index contrast of the AlGaAs nanowaveguide and the resulting field confinement, the waveguide dispersion dominates and as a result the dispersion can become inverted and strongly anomalous [4,21]. In fact for_a = 193nm, the dispersion of such a nanorod can reach very high values, as high as -12ps ²/m, which is 10³ times higher than that of a standard silica fiber at 1.55 μm.

Parenthetically, we would like to point out that these same structures can also lead to very high normal dispersion. Figure 2(a) depicts the total dispersion of an Al _0.2 Ga _0.8 As nanorod when its radius is a = 160 nm. As one can see, the anomalous dispersion of this structure (also shown in Fig. 1(b) for λ ≤ 1.4μm) is followed by a region of strong normal dispersion especially at 1.55 μm. In fact around 1.55 μm the dispersion is approximately β″ =80ps ²/m. Thus the dispersion of a 2 cm long AlGaAs nanorod will be sufficient to cancel that arising from a 1.25km long anomalously dispersive fiber with group velocity dispersion D = 1ps/km.nm. In addition Fig. 2(b) shows the group index n = c/vg and the effective index n_eff of an Al _0.2 Ga _0.8 As nanowire of radius a = 175nm as a function of wavelength. Note that the group index in this case can be as high as n_g ≈ 5.52 around 1.55 μm, which indicates “slow” light n_eff =1.56 .

 

Fig. 2. (a) group velocity dispersion β″ when a = 160nm , (b) group and effective refractive indices of an Al _0.2 Ga _0.8 As nanowire when a = 175nm .

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For the subsequent discussion, we chose two operating points based on the designs (C) and (B) shown in Fig. 1(b). In particular the design (C) is used at 1.55 μm where the third-order dispersion is very small whereas design (B) is used at 1.5 μm where cubic dispersive effects become appreciable. The nanowire of case (C) is single-moded for λ ≥ 1.57μm whereas that of (B) is monomode for λ ≥ 1.42μm.

3. Soliton effects in AlGaAs nanowires

Nonlinear pulse propagation in such nanowire structures is modeled using the evolution equation [22]:

In Eq.(3), ϕ represents the pulse envelope, T = t - z/V_g ,α is the linear loss coefficient, β ^(m) = ∂^m/∂ω^m is the m^th order dispersion coefficient evaluated at the carrier angular frequency ω ₀ = 2πc/λ, n ₂ = (n̑ ₂ n)/(2η_o ) where n̑ ₂ = 1.3×10^-13 cm ²/W is the nonlinear Kerr coefficient and n is the linear refractive index of AlGaAs (η_o is the free space wave impedance). α ₂ = 6.5×10^-15 m/V ² and α ₃ = 5.63×10^-31 m ³/V ⁴ are the two-photon and three-photon absorption coefficients respectively at λ ≈ 1.55μm [19]. In all cases, the dispersion curve is incorporated in Eq.(3) using a dispersion Taylor series over a broad spectral range (greatly exceeding the pulse spectrum). Here the linear loss for the field is taken α = 25cm ^-1.

Figure 3(a) shows the intensity evolution of a 200fs (FWHM) hyperbolic secant optical pulse when is launched into a 5mm long Al _0.2 Ga _0.8 As nanowire of core radius a = 193nm (corresponding to curve (C) of Fig. 1(b)) at 1.55μm. For this design, the quadratic dispersion (β ^″ =-12ps ²/m) dominates the propagation process (the higher-order dispersion terms are negligible) and the dispersion length is ≈ 1.5mm, i.e., is very small. The pulse peak power in this case is approximately 5.5 W (with a 75% confinement factor), corresponding to the fundamental soliton in this nanowire structure. As Fig. 3(a) clearly indicates, this soliton can be sustained over approximately 3-4 dispersion lengths, i.e. up to a distance that is ultimately determined by the linear loss of the system. Fig. 3(b) on the other hand depicts soliton compression under similar conditions when the input power of the pulse is 8 W. For this power level, at 2 dispersion lengths, the FWHM pulsewidth becomes 175fs. Figure 3(c) and (d) demonstrate similar results when an AlGaAs nanowire corresponding to curve (B) of Fig. 1(b), is used at 1.5 μm. Even in this case, in spite of appreciable third-order dispersion effects and multi-photon absorption, a 200 fs soliton forms at 2W, as shown in Fig. 3(c). Soliton compression is also shown for this case in Fig. 3(d) when the input power is 2.8 W. It is important to note that in all cases ultrashort solitons in these nanowires can form within millimeters and at very low power levels.

 

Fig. 3 Intensity evolution of a 200 fs soliton propagating in a 193 nm (radius) AlGaAs nanowire when the peak power is (a) 5.5 and (b) 8 W. Similarly, (c) and (d) depict soliton propagation and compression in a nanowire of radius 175 nm when the peak power is 2 and 2.8 W respectively.

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Figure 4 depicts the power spectra that may result in an Al _0.2 Ga _0.8 As nanowaveguide of radius a = 193 nm when is excited at 1.55μm with a 200fs FWHM optical pulse. When the input peak power is enough to establish an optical soliton (5.5 W) the power spectrum remains essentially invariant during propagation as shown in Fig. 4(a). At a higher power (8W) the pulse undergoes compressions and thus its spectrum broadens (Fig. 4(b)). Figures 4(c) and (d) on the other hand depict the expected spectra and intensity profile of a higher order soliton at L ≈ 5 mm. In this latter case, the peak power is 50 W and thus corresponds to an N = 3 higher order soliton solution. The splitting behavior observed in Fig. 4(d) is attributed to multi-photon absorption.

 

Fig. 4. (a) Input and output soliton power spectra corresponding to the case shown in (a) Fig.3 (a) ; (b) Fig.3.(b) ; (c) and (d) spectral generation and intensity profile for an N =3 soliton (50 W peak power) at L ≈ 5mm

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In conclusion we have shown that optical solitons can exist in dispersion-inverted highly-nonlinear AlGaAs nanowires. These soliton waves are possible at very low power levels in spite of strong dispersion and can form in millimeter long nanowire structures. The intensity and spectral evolution of solitons propagating in such AlGaAs nanowaveguides was investigated in the presence of loss, multiphoton absorption and higher-order dispersion.