Incoherent white-light solitons in nonlinear periodic lattices

We predict the existence of lattice solitons made of incoherent white light: lattice solitons made of light originating from an ordinary incandescent light bulb. We find that the intensity structure and spatial power spectra associated with different temporal frequency constituents of incoherent white-light lattice solitons (IWLLSs) arrange themselves in a characteristic fashion, with the intensity structure more localized at higher frequencies, and the spatial power spectrum more localized at lower frequencies; the spatial correlation distance is larger at lower frequency constituents of IWLLSs. This characteristic shape of incoherent white-light lattice solitons reflects the fact that diffraction is stronger for lower temporal frequency constituents, while higher frequencies experience stronger effective nonlinearity and deeper lattice structure.

Figure 1

(Color online) The effective potentials and band gap structure at different temporal frequencies. (a) The effective potential $V_{\mathrm{eff}}(\omega ;x)$ for $\lambda =400\mathrm{nm}$ (violet line) and $\lambda =700\mathrm{nm}$ (red line) of the self-consistently calculated IWLLS (the potential value for the violet light is shifted upwards by 1 for better comparison). (b) The band gap structure of the lattice (shaded regions denote bands), and the effective propagation constants ${\kappa }_m^{\mathrm{eff}}\left(\omega \right)$ of the populated soliton modes (red lines).

Figure 2

(Color online) Intensity profile and spatial power spectra of the IWLLS example in a nonlinear waveguide array displaying a self-focusing nonlinearity. Left column: soliton quantities integrated over all frequency constituents. (a) Intensity profile of the IWLLS (dot-dashed line), and broadened intensity profile of the IWLLS input beam after linear diffraction for $42\mathrm{mm}$ (solid line) in a linear lattice. (b) Intensity structure of the self-consistently calculated IWLLS (solid line), the experimentally accessible (simple) input beam (dot-dashed line), and the structure of the simple input beam after evolution for $42\mathrm{mm}$ (dotted line) under the nonlinear conditions that support the self-consistently calculated IWLLS. (c) Same as (b) for the Fourier power spectrum, and (d) same as (b) for the Floquet-Bloch power spectrum (the spectra are in arbitrary units). Right column: soliton quantities of different wavelength constituents, $\lambda =400\mathrm{nm}$ ($4.71\bullet {10}^{15}\mathrm{Hz}$, violet dotted line), $\lambda =509\mathrm{nm}$ ($3.70\bullet {10}^{15}\mathrm{Hz}$, green dashed line), $\lambda =700\mathrm{nm}$ ($2.69\bullet {10}^{15}\mathrm{Hz}$, red solid line): (e) Intensity profiles after linear diffraction. (f) Intensity profiles of the IWLLS. (g) Fourier, and (h) Floquet-Bloch power spectra of the IWLLS.

Figure 3

(Color online) Contour plots of ${\mu }_{\omega }(x_1,x_2)$ expressing the coherence properties of the IWLLS at different frequency constituents. (a) ${\mu }_{\omega }(x_1,x_2)$ for the red constituent $\lambda =700\mathrm{nm}$, and (b) for the violet component $\lambda =400\mathrm{nm}$. Cross sections of the coherence factor, ${\mu }_{\omega }(x_1,0)$ (solid line), and ${\mu }_{\omega }(x_1,2D)$ (dotted line) for the red (c), and the violet (d) frequency constituent.

Figure 4

(Color online) The intensity profiles, Fourier, and Floquet-Bloch spatial power spectra of a simple bell-shaped incoherent beam shown at different frequencies: $\lambda =400\mathrm{nm}$ ($4.71\bullet {10}^{15}\mathrm{Hz}$, violet dotted line), $\lambda =509\mathrm{nm}$ ($3.70\bullet {10}^{15}\mathrm{Hz}$, green dashed line), $\lambda =700\mathrm{nm}$ ($2.69\bullet {10}^{15}\mathrm{Hz}$, red solid line). (a) The above quantities at the input $z=0$, and (b) after propagation for $42\mathrm{mm}$. These properties at different frequency components differ from one another after propagation through the system.

Figure 5

The evolution of the total intensity profile (a), total Fourier power spectrum (b), and total Floquet-Bloch power spectrum (c) of an incoherent, bell-shaped, incoherent white-light beam launched into the nonlinear waveguide array (see text for details).