Discretized Conical Waves in Multimode Optical Fibers

Multimode optical fibers are essential in bridging the gap between nonlinear optics in bulk media and single-mode fibers. The understanding of the transition between the two fields remains complex due to intermodal nonlinear processes and spatiotemporal couplings, e.g., some striking phenomena observed in bulk media with ultrashort pulses have not yet been unveiled in such waveguides. Here we generalize the concept of conical waves described in bulk media towards structured media, such as multimode optical fibers, in which only a discrete and finite number of modes can propagate. Such propagation-invariant optical wave packets can be linearly generated, in the limit of superposed monochromatic fields, by shaping their spatiotemporal spectrum, whatever the dispersion regime and waveguide geometry. Moreover, they can also spontaneously emerge when a rather intense short pulse propagates nonlinearly in a multimode waveguide, their finite energy is also associated with temporal dispersion. The modal distribution of optical fibers then provides a discretization of conical emission (e.g., discretized $X$ waves). Future experiments in multimode fibers could reveal different forms of dispersion-engineered conical emission and supercontinuum light bullets.

Figure 1

Example of conical waves in a step-index MMF. (a) Full spatiotemporal power (over 3 decades). (b) Modal distribution of phase-matched frequencies (red squares). Black curve: number of guided modes. (c)–(e) Evolution of a realistic (i.e., dispersive) ultrashort (80-fs duration components) conical wave with propagation distance (for $z=0$, 1, and 3 mm, respectively).

Figure 2

Nonlinear propagation of 100-fs pulses (300-nJ energy at 800 nm) in a 4-cm-long segment of step-index MMF. (a) Output modal distribution of the power spectrum. White squares: theoretical phase-matching region of the discretized conical wave. Black curve: number of guided modes. Magenta circles: zero-dispersion frequency for each guided mode. (b)–(d) Evolution of the normalized full power spectrum, instantaneous power, and fluence with propagation distance, respectively. The white dashed line in panel (c) shows the group-velocity of the shock front described in the main text. (e)(f) Snapshots of temporal and spectral power profiles showing the optical shock formation and strong spectral broadening. (g1)–(i1) Evolution of the full spatiotemporal power distribution (over 6 decades) for distinct propagation distances. (g2)–(i2) Corresponding evolution for the filtered tails of the x pattern from the ($m$, $\omega$) plane.

Figure 3

(a) Same step-index MMF as in Fig. 1 with 450-nJ 100-fs pumping at 1400 nm. Output modal distribution of the power spectrum after 4 cm of propagation. (b) Output modal distribution of the power spectrum obtained from nonlinear propagation of 100-fs pulses (210-nJ energy at 800 nm) coupled into the ${\mathrm{LP}}_{0,1}$ mode of a 2-cm-long GRIN fiber.