Self-trapping of polychromatic light in nonlinear photonic lattices

We study dynamical reshaping of polychromatic beams due to collective nonlinear self-action of multiple-frequency components in periodic photonic lattices and predict the formation of polychromatic discrete solitons facilitated by localization of light in spectral gaps. We show that the self-trapping efficiency and structure of emerging polychromatic gap solitons depends on the spectrum of input beams due to the lattice-enhanced dispersion, including the effect of crossover from localization to diffraction in media with defocusing nonlinearity.

We study dynamical reshaping of polychromatic beams due to collective nonlinear self-action of multiple-frequency components in periodic photonic lattices and predict the formation of polychromatic discrete solitons facilitated by localization of light in spectral gaps. We show that the self-trapping efficiency and structure of emerging polychromatic gap solitons depends on the spectrum of input beams due to the lattice-enhanced dispersion, including the effect of crossover from localization to diffraction in media with defocusing nonlinearity. c 2022 Optical Society of America OCIS codes: 190.4420 Nonlinear optics, transverse effects in, 190.5940 Self-action effects The fundamental physics of periodic photonic structures is governed by the wave scattering from periodic modulations of the refractive index and subsequent wave interference. Such a resonant process is sensitive to a variation of the beam frequency and propagation angle 1 . Accordingly, refraction and diffraction of optical beams may depend strongly on the optical wavelength, allowing for construction of superprisms that realize a spatial separation of the frequency components.
In this Letter we address an important question of how the periodicity-enhanced sensitivity of diffraction upon wavelength influences nonlinear self-action of polychromatic light. We show that interaction between multiplefrequency components of an optical beam can lead to a collective self-trapping effect and polychromatic solitons, where spatial diffraction is suppressed simultaneously in a broad spectral region. These solitons can exist in periodic structures with noninstantaneous nonlinear response, such as optically-induced lattices 2,3,4 or waveguide arrays 5,6 in photorefractive materials. We demonstrate that the spectrum of polychromatic solitons possesses a number of distinctive features, related to the structure of the photonic bandgap spectrum. This suggests the possibility to perform nonlinear probing and characterization of the bandgap spectrum in the frequency domain, extending the recently demonstrated approach for nonlinear Bloch-wave spectroscopy with monochromatic light 7 .
We study the dynamics of polychromatic light in planar nonlinear photonic structures with a modulation of the refractive index along the transverse spatial dimension, such as optically-induced lattices 2,3,4 or periodic waveguide arrays 5,6 . Then, the evolution of polychromatic beams in media with slow nonlinearity can be described by a set of normalized nonlinear equations, where A n are the envelopes of the different frequency components of vacuum wavelengths λ n , x and z are the transverse and longitudinal coordinates normalized to x 0 = 10µm and z 0 = 1mm, respectively, I = N n=1 |A n | 2 is the total intensity, N is the number of components, n 0 is the average refractive index, ν(x) is the refractive index modulation in the transverse spatial dimension, and γ is the nonlinear coefficient. We consider the case of a Kerr-type medium response, where the induced change of the refractive index is proportional to the light intensity and neglect higher-order nonlinear effects such as saturation, in order to clearly identify the fundamental phenomena independent on particular nonlinearity. We note that Eq. (1) with λ n = λ describe one-color multigap solitons 8,9,10,11 .
Linear dynamics of optical beams propagating in a periodic photonic lattice is defined through the properties of extended eigenmodes called Bloch waves 1 . We consider an example of lattice with cos 2 refractive index modulation [see Fig. 1(a)] with the period d = 10µm, and calculate dependencies between the longitudinal (β, along z) and transverse (k, along x) wave-numbers for Bloch waves, see Figs. 1(b-d). The top spectral gap is semi-infinite (extends to large β), and it appears due to the effect of the total internal reflection. The effective diffraction of Bloch waves becomes anomalous at the upper edges of Bragg-reflection gaps, where D eff = −∂ 2 β/∂k 2 < 0.
It is known that the presence of Bragg-reflection gaps and associated anomalous diffraction regions allows for the formation of monochromatic spatial gap solitons even in media with self-defocusing nonlinearity 3,6,12 . Results in Figs. 1(b-d) show that the spatial bandgap spectrum depends on the optical wavelength and, in particular, we find that the anomalous diffraction regime is strongly frequency dependent as D eff ∼ λ 3 at large wavelengths, whereas the bulk diffraction coefficient is proportional to λ. Accordingly, the Bragg-reflection gap becomes much narrower at larger wavelengths, limiting the maximum degree of spatial localization that is inversely proportional to the gap width.
The variation of the gap width can have a dramatic effect on self-action of an input Gaussian beam focused at a single site of a defocusing nonlinear lattice 6 , where a sharp crossover from self-trapping to defocusing occurs as the gap becomes narrower. We note that, most remarkably, these distinct phenomena can be observed in the same photonic structure but for different wavelength components. In our numerical simulations, we put γ = −10 −4 and choose the lattice parameters such that the critical wavelength corresponding to the crossover is around 591nm. We confirm that the monochromatic beam with λ = 443nm experiences strong self-trapping, whereas the largest fraction of input beam power becomes delocalized at a shorter wavelength λ = 665nm. We then address a key question of how an interplay between these opposite effects changes the nonlinear propagation of polychromatic beams.
We model the self-action of polychromatic light beams by simulating the propagation of nine components with the wavelengths ranging between 443nm and 665nm. The input corresponds to a narrow Gaussian beam that has the width of one lattice site, i.e. in our case 5µm.  Fig. 2(a). As the input power is increased, we find that the spatial spreading can be compensated in a broad spectral region by self-defocusing nonlinearity. We observe a spatially localized total intensity profile at the output, indicating the formation of a polychromatic gap soliton [ Fig. 2(b)]. We note that the spatial localization of the soliton components strongly depends on the wavelength [Figs. 2(d)], so that the long wavelength component has a much larger spatial extent than the short wavelength component. Hence, the soliton has a blue center and red tails, and this effect is more pronounced than for solitons with the same spectra in bulk media. Additionally, the power spectrum of the soliton becomes blue-shifted at the output. Figure 2(c) shows the so-called 'self-trapping efficiency', which we define here as the percentage of light that remains in the three central waveguides of the optical lattice after the propagation for each wavelength. This value essentially is identical to the trapped fraction of light, as even for longer propagation distances the light would remain localized in these waveguides. We see that more that 40% of the light with the wavelengths between 443nm and 484nm is trapped, whereas for the longer wavelengths that percentage drastically decreases due to the narrowing of the Bragg-reflection gap. However, due to the nonlinear interaction between the different wavelengths, there still is a noticeable amount of light from the red side of the spectrum that is trapped (roughly 8% of the light at 591nm). This is in a sharp contrast to the case of monochromatic red light propagation, where the self-trapping efficiency vanishes. We now study the effect of input frequency spectrum on the nonlinear self-action of polychromatic light. We perform numerical simulations for the same profiles of the input beam as in Fig. 2(b), but considering different power distribution between the frequency components. Figures 3(a-d) show the characteristic propagation results for beams with blue-and red-shifted input spectra.
For the blue-shifted input spectrum [ Figs. 3(a,b)], we observe self-trapping of the polychromatic light beam, and a small percentage of the red light is trapped by the nonlinear index change caused by the blue parts of the spectrum. In fact, the self-trapping efficiency for the red part of the spectrum is almost identical to the case of a white spectrum shown in Fig. 2(c). Fundamentally different behavior is observed for a polychromatic beam with red-shifted spectrum [ Figs. 3(c,d)]. In this case, the beam strongly diffracts and self-trapping does not occur even when the total input intensity is increased several times compared to the case of white spectrum. This happens due to the tendency of red components to experience enhanced diffraction as the effect of defocusing nonlinearity is increased at higher intensities. We note that, according to Fig. 3(c), the blue part of the spectrum is also diffracting.
In conclusion, we have studied the propagation of polychromatic light and the formation of polychromatic solitons in periodic photonic lattices, and demonstrated that light self-action can be used to reshape multiple frequency components of propagating beams in media with noninstantaneous nonlinear response, such as photorefractive materials or liquid crystals. We have demonstrated that self-trapping efficiency and structure of emerging polychromatic gap solitons depends strongly on the spectrum of input beams due to the lattice-enhanced dispersion, and identified the effect of crossover between localization and diffraction in defocusing media.