Fig. 1. Refractive index distribution diagram^[26]. (a) Refractive index distribution diagram of graded-index fiber and step-index fiber; (b) propagation constant distribution diagram of graded-index multimode fiber

Fig. 2. GPI sidebands and corresponding wavelength spot conditions^[6]. (a) Yellow circle represents the frequency position of the analyzed sideband, red curve represents the GPI sideband position of numerical simulation, and blue curve represents the GPI sideband position in the experiment; (b) spatial distribution of the output beam corresponds to the increase of the input power

Fig. 3. Nonlinear dynamics of beam self-cleaning experiment in a GRIN-MMF^[10]. (a)‒(d) Near-field images of the MMF output recorded at 1064 nm show spatial beam self-cleaning, namely the formation of a well-defined bell-shaped transverse beam distribution when the output peak power is increasing; (e)‒(h) corresponding beam profiles versus x (y=0 section), fiber length is 12 m

Fig. 4. Influence of gradient refractive index multimode fiber length on beam self-cleaning experiment in GRIN-MMF nonlinear dynamics^[10]. (a)‒(f) Near-field images recorded at 1064 nm show the development of beam self-cleaning along the propagation distance z, output peak power at z = 12 m is 44 kW

Fig. 5. Influence of pulse energy on the shape of output spot^[11]

Fig. 6. Evolution of energy in different modes withtransmission distance^[11]

Fig. 7. Numerical results for BSC in spatial and spectral domains^[12]. (a) Evolution of energy in indicated modes; (b) spatially integrated power spectrum during propagation

Fig. 8. Experimental supercontinuum spectra obtained in a 28.5 m long graded-index MMF using 185 kW peak pump power at 1064 nm^[34]. (a) Image of the visible component of the dispersed output spectrum; (b) spectrum recorded using two different OSAs covering the spectral range from 350 nm to 2400 nm; (c) near-field beam profiles in the visible and near infrared at the output of multimode optical fiber power

Fig. 9. Experimental result^[23]. (a) Double clad ytterbium doped multimode optical fiber; (b) fiber core refractive index distribution diagram (inset: measured ytterbium ion distribution)

Fig. 10. Experimental result^[23]. (a)‒(f) Near-field distribution at different input peak power levels recorded at the output end of gain free double clad ytterbium doped multimode fiber; (g) measured M² and input peak power of passive ytterbium doped MMF (inset: low power in the upper left and high power output beam in the lower right)

Fig. 11. Amplified pulse output near-field pattern in Yb-MMF (G is the gain, input peak power is 500 W)^[23]

Fig. 12. Acceleration effect of multi-mode collisions in optical fibers^[34]

Fig. 13. Experimental result^[34]. (a) Output supercontinuum spectrum of a 15 m long gradient refractive index multi-mode fiber, when its core radius decreases from 40 μm to 10 μm; (b) photograph of the visible component of the dispersed output spectrum; (c) spot measurements taken at different wavelengths

Fig. 14. Experimental setup for coupling a microchip laser at 1064 nm (signal) and a pump laser diode (LD) into the Yb-doped MMF taper^[35]. (a) Doped tapered multimode optical fiber; (b) (c) refractive index difference distribution between the core and cladding of ytterbium doped multimode fibers with large and narrow ends; (d) relationship between diameter of tapered fiber core and taper length determined by truncation method

Fig. 15. Near-field spatial distribution of conical fiber output end with input power of 19.6 kW^[35]

Fig. 16. Supercontinuum spectrum at peak power of 19.6 kW and maximum pump power (G=1.34) (inset: near-field output beam profiles of different bandpass filters)^[35]

Fig. 17. Characteristics of PBG gradient refractive index multimode fiber^[40]. (a) Attenuation of PBG81 and PBG89 constituent glasses; (b) group refractive index of PBG81 and PBG89

Fig. 18. Experimental setup for SC generation and intensity noise measurement^[40]

Fig. 19. Influence of input beam tilt angle on supercontinuum spectrum and spatial intensity distribution^[40]

Fig. 20. Generation of supercontinuum in graded-index multimode fiber with all-fiber structure^[43]

Fig. 21. 40 W visible supercontinuum generation based on GRIN-MMF^[45-46]. (a) Experimental setup; (b) spectral evolution with pump power using frequency as x-coordinate; (c) final output spectrum (insets: near-field spot profiles of total and filtered supercontinuum at wavelengths of 730, 620, 532, and 470 nm, respectively)