1. Introduction

Along with the experimental proof of existence of bright solitons in semiconductor resonators [1] which were predicted earlier [2], dark solitons were also observed [3]. Their existence was subsequently also shown numerically [4]. Whereas bright solitons appear as precursors of the full plane wave switching (Fig. 1), the dark solitons reside on a switched background and exist at much higher illumination intensity than bright solitons (Fig. 1).

 

Fig. 1. The domains of soliton existence in the predominantly absorptive regime (wavelength is approximately 870 nm). MI: Modulation Instability domain. E_h is the illuminating field amplitude inside cavity, θ is is the cavity detuning parameter. Parameters D_N = 2.32∙10^-3, ξ′= 7.96, ξ″= 20.58, N_tr = 1.694 (see model description). Area limited by the dashed line is the plane wave bistability domain. The insets show the calculated intracavity amplitude profile of dark and bright solitons.

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Figure 2 shows experimentally observed dark solitons. For the observations we used the same set-up as used before [3]. Observations are done in reflection, as explained below. The resonator is illuminated by a laser beam of Gaussian intensity distribution. In the central part of the illumination the intensity is high enough to switch the bistable resonator. Thus the central part is switched. The reflectivity here is consequently smaller than that of the unswitched area of the resonator. Dark solitons can exist in the switched area. Figure 2(a) shows a single soliton and Fig. 2(b) shows four solitons within the area switched by a laser beam of elongated cross sections.

 

Fig. 2. Dark solitons in switched areas observed in reflection (see text for details).

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Due to the relatively high intensities necessary to make a semiconductor substantially nonlinear (and a semiconductor resonator bistable), which are in the range of ~kW/cm₂, heating effects are not negligible in the experiments with semiconductor resonator solitons. Heating effects lead among other to the spontaneous switch-on of solitons (experimentally found in [5] and numerically reproduced in [6]). A further interesting effect of heating was pointed out in [4,7]: dark solitons move spontaneously under temperature influence, since they “cool” at their position, but heated material is more favorable than cool material for the soliton’s existence. We have called this peculiar kind of instability of the location of a localized structure “restless” motion [8]. In [7] it was called “fugitive” and in [4] the term “self-propelled” was used.

Thus, there are substantial differences in the properties of bright and dark solitons and in this paper we discuss further characteristics of dark solitons which have not received as much attention as bright solitons, so far.

2. Experiment

The semiconductor resonator used for the experiments consists of flat Bragg mirrors of about 99.8% reflectivity, with 18 GaAs/Ga_0.5Al_0.5As quantum wells between them. The optical resonator length is about 3μm so that with a diameter of the illuminating beam of ~45 μm, as used experimentally, a Fresnel number of ~200 is excited, sufficient for complex structure to form. Such resonators show optical bistability when driven by a coherent field [9] so that they can support spatial solitons, which coexist with a uniform plane wave structure and are thus possibly useful as information carriers. This system has been shown experimentally and theoretically to support hexagonal patterns and bright and dark solitons [10–13]. Light of wavelengths near the semiconductor band edge (860 nm), generated by a continuous TiAl₂O₃-laser, illuminates the semiconductor sample (Fig. 3) through a mechanical chopper for durations of a few μs to limit thermal phenomena. The illumination is repeated at a rate of 1 kHz. Observations are made in reflection because the sample substrate is absorbing at the wavelengths used. The light reflected from the sample is imaged onto a CCD camera for recording two-dimensional images. A fast intensity modulator, placed in front of the camera and triggered with a variable delay with respect to the start of the illumination, allows to take snapshots with exposure times down to a few ns, within the period of the illumination, as required by the fast dynamics of the semiconductor resonator.

 

Fig. 3. Experimental set-up. PBS: polarizer beam splitter, λ/2: halfwave plate, λ/4: quarter-wave plate, L: lens, BS: beam splitter, EOM: electro-optical modulator, PD: photodiode to follow intensity in time in certain points of the patterns. Beam waist placed on the sample surface is 45 μm FWHM in intensity.

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In some cases, an additional semiconductor laser (with wavelength of 808 nm, near one of the minima of the resonator mirror’s stop band) was used for optical pumping the semiconductor resonator. The pump generates carriers and allows conversion from absorption to gain. When pumping below the transparency point of the material and with the illuminating laser wavelength near the semiconductor quantum well band edge, bright and dark solitons form similarly to the unpumped case. If pumping is strong enough the semiconductor microresonator emits light as a laser.

Figure 4 shows structures observed in the resonator as they form spontaneously at different irradiation intensities (without pump). At small intensity (Fig. 4(a)) a dark structure (bright soliton) forms on the unswitched background. Figure 4(b) is a domain switched to high transmission in the central part of the Gaussian illumination beam at moderate irradiation intensity. At large intensity (Fig. 4(c)) a sharp bright (in reflection) structure (dark soliton) forms, in contrast to bright solitons on the switched background.

Dark solitons at photon energy below band gap can form hexagonal patterns (Fig. 4(d)). These range from coherent patterns at small nonlinearity to arrangements of loosely bound spatial solitons at large nonlinearity, which can be individually switched [3]. Such loosely bound solitons show thus multistability of large numbers of spatial arrangements and can therefore carry substantial amounts of information, possibly useful in optical parallel information processing.

 

Fig. 4. Switched structures observable in experiment: intensity reflectivity (reflected intensity/incident intensity) of the sample: (a) bright soliton on the unswitched background (dark spot in reflection); (b) switched area; (c) dark soliton in the switched area (bright spot in reflection); (d) hexagonal pattern of loosely bound individual dark solitons.

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A striking feature of the dark solitons is immediately evident from Fig. 5. It shows the measured reflectivity (reflected intensity/incident intensity) as a function of illuminating light intensity (without additional pumping). As one can see, the reflectivity of a soliton can be larger than 1. In other words, the dark soliton can amplify incident radiation. It does so in a way not too different from a laser or other amplifiers; it collects energy from its surroundings: the surrounding acts as an energy reservoir (just like the inverted energy states in a laser amplifier or a power supply in an electronic amplifier). The energy is then reemitted in the soliton field.

 

Fig. 5. Measured reflectivity in the center of dark soliton depending on intensity of the illuminating beam for dispersive regime (882 nm curve) and mixed absorptive-dispersive regime (curve for 871 nm).

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Figure 5 shows that in addition to the differential gain predicted for bright solitons in [14] and experimentally observed in [15], the dark solitons experimentally observed here permit “DC” or “absolute” gain, the reflected intensity being higher than the incident intensity. Whereas differential gain requires only a characteristic with a slope larger than 1 (while it can be purely absorptive and thus attenuating light), DC- or absolute gain means real amplification of light. We have observed gain up to 200% for isolated dark solitons (Fig. 5). It appears that in the absorptive-dispersive regime a dark soliton located at the switching front (the latter connecting the switched area with the unswitched area, see Fig. 5, upper curve), has a substantially higher light amplification than a dark soliton, in the dispersive regime, located in the middle of the switched area, well separated from the switching front (Fig. 5, lower curve).

Pumping with the additional semiconductor laser, a strong influence of pump on the soliton reflectivity is observable. Figure 6 shows the measured reflectivity at the peak of a soliton with increasing illumination intensity for different pump powers. For pump power below lasing threshold (curves for 0, 0.425, 1.2 W) increasing the pump leads to increasing the reflectivity and the dependence of reflectivity on incident light intensity is linear. For large pump powers (curve for 1.82 W) the dependence of soliton reflectivity on illumination intensity becomes nonlinear which may be related to the fact that the laser threshold is exceeded at these pump powers.

 

Fig. 6. Measured reflectivity at the center of dark soliton in dependence on illuminating light intensity at different pump powers. Measurements at 860 nm wavelength of illuminating light. The solid lines serve to guide the eye.

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The observed light amplification can have at its root the dissipative part or the reactive part of the nonlinearity. A clue to the dominant mechanism could be the wavelength dependence of the soliton peak reflectivity. In general the soliton peak reflectivity comes out higher in the calculations for the longer wavelengths. However, there are conditions where the shorter wavelength yields the higher reflectivity (see Fig. 8b) theoretically, as it is also found experimentally. Thus at present a clear interpretation of the light amplification in terms of either the reactive or the dissipative part of the nonlinearity is not possible.

3. Model

For the theoretical calculations the standard equations of [16,17] are used:

here E is the intracavity field, N is the carrier density, θ = τ _ph (ω _res - ω₀) is the cavity detuning parameter, ω _res is the resonance frequency, ω₀ is the illuminating field frequency, τ _ph = 2l/v(1-ρ _u ρ _l ) is the photon lifetime (l is the effective length between mirrors, v is the group velocity, ρ _u,l are the reflectivity coefficients of upper and low mirrors, respectively). D_E , D_N are the diffraction and carrier diffusion coefficients, $E_h=E_{\mathit{inp}}\frac{\sqrt{1-{\rho }_u^2}}{\left(1-{\rho }_u{\rho }_l\right)}$ is the illuminating amplitude of the intracavity field, E_inp is the field amplitude on the input mirror. δ _N is the ratio of the photon lifetime in the resonator to the carrier recombination time, β is the radiative carrier recombination coefficient.

The radiation-matter interaction is described in Eq. (1) by the linearized complex susceptibility ${\chi }_{\mathit{re}}\left(\tilde{N}\right)=\frac{\xi \prime }{{\mathit{CN}}_0}\left(\tilde{N}-{\tilde{N}}_{\mathit{tr}}\right),{\chi }_{\mathit{im}}\left(\tilde{N}\right)=\frac{\xi ″}{{\mathit{CN}}_0}\left(\tilde{N}-{\tilde{N}}_{\mathit{tr}}\right),$ where the so-called bistability constant C = L_A ω₀/v(1-ρ _u ρ _l ) (L_A is the length of multiple quantum wells) [16]. N͂_tr is the transparency value of carrier density. We use the transparency value for wavelength 880 nm as the normalization constant for the carrier density (N ₀ ≈0.8×10¹⁸ cm ^-3). The complex susceptibility (ξ′, ξ″) is taken from absorption and refractive index spectra measured in similar multiple quantum well structures in [18]. ξ′>ξ″ for the dispersive regime (for long wavelengths far from the band edge) and ξ′<ξ″ for the absorptive regime (near the band edge). The optical fields are normalised to $E_0=\sqrt{\frac{\mathit{ħ}{\mathit{CN}}_0}{{\epsilon }_0}.}$ Typical constants can be found, for instance, in [16]. According with our normalization D_E = 6.53, δ _N = 5.8∙10^-4, β = 3.8∙10^-4, and time was normalized to photon life time in the resonator (without absorption) which is approximately τ _ph ≈ 4ps . Transverse coordinates (x, y) are expressed in μm.

Figures 1 and 7 show the domains of soliton existence together with the bistability domain for plane uniform waves for different wavelengths of incident light. For the short wavelengths (870 nm) close to the semiconductor band edge (Fig. 1) the absorptive nonlinearity is dominant. This means that the losses in the quantum wells decrease nonlinearly with increasing optical field intensity. The dark soliton domain is very narrow in one direction and connects with the modulationally unstable plane wave background. As a consequence the dark solitons are surrounded by distinctive intensity rings.

 

Fig. 7. Domains of soliton existence in the predominantly dispersive regime (wavelength is approximately 890 nm). MI: Modulation Instability domain. E_h is the illuminating field amplitude inside the cavity, θ is the cavity detuning parameter. Area limited by the thick dashed line is the plane wave bistability domain. Inset shows the calculated range of unstable ring dark solitons. Parameters: D_N = 2.32∙10^-3, ξ∙ = 9.44, ξ″ = 0.81 , N_tr =7.76 .

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The preferable nonlinearity for dark solitons is defocusing. When increasing the wavelength of the illuminating light (890 nm) the dispersive nonlinearity becomes defocusing. The domain of dark soliton existence extends then to large values of cavity detuning (Fig. 7). The soliton brunch on the left side of this domain (negative detuning) bifurcates subcritically from the modulationally unstable background, resulting in the above mentioned intensity rings surrounding the central peak of the soliton (the top amplitude profile of dark soliton in Fig. 7). Unlike, for positive detunings, the solitons formed are quite different. In this case, the localized structure (bottom amplitude soliton profile in Fig. 7) does not bifurcate from the modulation unstable background. It shows a dark ring around a bright center (in the intracavity field). Numerical calculations show that this type of solitons is generally stable. However, in some range of parameters (see the inset of Fig. 7), these “ring solitons” destabilise due to a Hopf instability, with temporal oscillations of the optical field.

Simulations with the phenomenological model (1) for a semiconductor microresonator, accounting for the interference between intracavity field E and field reflected from the input mirror E_inp , ρ _u , allow calculating the reflected fields for comparison with the experimental observations. The sum of reflected and transmitted field is not 1 since the resonator is driven by strong external field and behaves in a strongly nonlinear way. Therefore, transmitted (intracavity) and reflected fields have to be calculated separately. Figure 8a shows the calculated reflectivity $\left(R=\frac{{\mid {\mathit{E}}_{\mathit{inp}}{\rho }_u+E\sqrt{1-{\rho }_u^2}\mid }^2}{{\mid E_{\mathit{inp}}\mid }^2}\right)$ corresponding to the dark ring solitons. It is interesting to note that in reflection the shape of the dark ring soliton is not substantially different from the fundamental dark solitons. Thus, experimentally in reflection the two soliton types are not distinguishable.

The calculation assumes a plane wave uniform intensity illumination. Therefore, in Fig. 8(a) the entire background of the dark ring soliton is switched (high intracavity field, reflectivity 0.21). One finds the reflectivity at the center of the dark soliton larger than 1, roughly as observed experimentally.

 

Fig. 8. Calculated reflectivity of dark solitons for plane wave illumination. (a) Reflectivity profile of dark ring soliton (wavelength is ~890 nm), E_h = 0.33 , θ = 10 . (b) Maximal reflectivity of dark soliton vs. illumination intensity for different wavelengths λ.

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Figure 8(b) shows the maximal reflectivity values for dark solitons as calculated as a function of illumination intensity for different wavelengths. As expected the maximal reflectivity is smaller for short wavelengths (curve for 870 nm in Fig. 8(b)) due to the large light absorption within the semiconductor band. Contrary to the absorptive case, in the dispersive regime (890 nm) the reflectivity of dark solitons reaches a value of almost 3. In the mixed absorptive-dispersive regime (curve for 885 nm, D_N =2.32∙-10^-3, ξ′13.5, ξ″=1.7, N_tr = 2.7) the soliton reflectivity exhibits a maximum. The decrease of reflectivity with illuminating field intensity corresponds roughly to the experimentally measured dark soliton reflectivity in Fig. 5(curve for 871 nm). When the illumination intensity is small ($E_h^2$ <0.027) the reflectivity of dark solitons, at this wavelength, increases rapidly with increasing illumination intensity. Thus, the calculations yield both decrease and increase of dark soliton reflectivity with illumination intensity for different wavelengths. For $E_h^2$ = 0.05 - 0.1 the reflectivity of solitons, for the dispersive and absorptive-dispersive regimes, is in qualitative agreement with the respective dark soliton reflectivities observed experimentally (Fig. 5). For the quantitative agreement between numerical simulation and experiment the thermal effects should be taken into account. Many parameters (such as carrier recombination time, diffusion coefficient and susceptibility) are also not well known for the sample material, so that we can only expect qualitative equivalences.

Characteristic features of the experiment such as switching fronts and finite switched area are not reproduced in a homogeneous plane wave model description. Therefore, calculations with Gaussian beams with finite diameter (and plane wave fronts) were carried out. Figure 9 shows the reflectivity in the center of a dark soliton calculated both for plane wave uniform illumination and for illumination of Gaussian form with different amplitude beam radii.

 

Fig. 9. Calculated reflectivity at the center of a dark soliton for different amplitudes of the Gaussian illumination at a fixed detuning (λ=885 nm). Radii are in μm units. Dashed line is peak reflectivity of the soliton which is unstable at the center of illuminating beam. Parameters: D_N = 0.5∙10^-3, ξ′ = 13.5, ξ″=1.7, N_tr =2.7.

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It is seen from Fig. 9 that the illumination intensity range for soliton existence depends on the amplitude radius of the illuminating beam. This intensity range increases several times from small beam radius (80 μm) to a wide beam (110 μm of beam radius). This happens because at large holding beam radii the dark soliton has the freedom to move from the center of the illuminating beam to a position with beam amplitude appropriate for its existence, if the amplitude in the center of the beam is too large for soliton existence. Then the soliton looses its stability and moves towards the periphery of the switched area. Figure 10 shows typical pictures of solitons in reflection for Gaussian distribution of illumination intensity.

 

Fig. 10. Calculated reflectivity of dark solitons placed at the center (a) and near the boundary (b) of a switched area illuminated by a beam of Gaussian form with amplitude radius r = 110 μm. (a) E_h =0.357, (b) E_h =0.3642. Parameters: D_N = 0.5∙10^-3, ξ′ = 13.5, ξ″ = 1.7, N_tr = 2.7.

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From Fig. 9 one can also see that the maximum of reflectivity is smaller for smaller radii of the illuminating beam at a fixed detuning. Changing the detuning leads to changing the soliton radius and the holding beam intensity supporting solitons. The ratio between soliton radius and illuminating beam radius influences the reflectivity in the center of soliton.

As the intensity distribution of the stable ring solitons is not qualitatively different in reflection from the intensity distribution of the fundamental dark soliton, and being limited to reflection measurements, we have not attempted to search for the ring soliton experimentally.

4. Summary

We have shown that dark solitons can amplify light. The dependence of the amplification on the illumination intensity, wavelength and on (optical) pumping of the semiconductor material were investigated experimentally.

Model calculations confirm the experimental results at least qualitatively.

Along with the dark solitons, as observed, a new type of soliton (dark-ring soliton) was found by the calculations.

Due to limited knowledge and control of experimental parameters, precise quantitative comparisons of experimental observations with model calculations are not possible.