Spontaneous Optical Fractal Pattern Formation

We report, for the first time, spontaneous nonlinear optical spatial fractals. The proposed generic mechanism employs intrinsic nonlinear dynamics both to generate an initial pattern seed and to fill out structure across decades of spatial scale. We demonstrate this in one of the simplest of nonlinear optical systems, composed of a Kerr slice and a single-feedback mirror. In this case, the smallest pattern scales are limited by either the optical wavelength or the diffusion length of the medium photoexcitation. The dimension characteristics of these particular fractals are also derived.

Figure 1

Schematic diagram of the Kerr slice with single-feedback mirror system. Spatial fluctuations in the carrier density modulate the phase of the field (dashed line) and diffraction changes this into an amplitude modulation (solid line).

Figure 2

(a) Instability threshold and (b) qualitative sketch of power spectrum ($I_0=50$) for the Kerr slice with single-feedback mirror system ($l_D=1$, $d/k_0=1$, $R=0.9$, $\chi L=1$, $K^2\ll k_0^2$). (c) as (a), but $l_D=0$.

Figure 3

Transverse pattern evolution of the system for $l_D=0$, $\tau =0$, $d/k_0=1$, $\chi L=1$, $I_0/I_{\min }=2$, and $R=0.9$. (a) Hexagonal pattern formed by introducing a one-band-pass frequency filter ($t=100T_R$, $k_c=2$). (b), (c), and (d) are patterns after the filter is removed: (b) $t=103T_R$, (c) $t=106T_R$, (d) $t=109T_R$.

Figure 4

Spatial pattern evolution in time (upper row) and the corresponding power spectra (lower row): $l_D=0$, $\tau =0$, $d/k_0=1$, $\chi L=1$, $I_0/I_{\min }=2$, and $R=0.9$. (a) With a one-band-pass frequency filter ($t=100T_R$, $k_c=2$). (b), (c), and (d) are patterns after the filter is removed. (b) $t=102T_R$, (c) $t=113T_R$, (d) $t=150T_R$.

Figure 5

Power spectrum evolution in time: (a) $t=2T_R$, (b) $t=5T_R$, (c) $t=50T_R$, (d) $t=2000T_R$ ($l_D=0.1$, $d/k_0=100$, $\tau =1$, $R=0.9$, $\chi L=1$, $I_0=3.0$).

Figure 6

Variation of equilibrium power spectra with diffusion length $l_D$: (a) $l_D=0.8$, (b) $l_D=0.4$, (c) $l_D=0.2$, (d) $l_D=0.1$ ($d/k_0=100$, $\tau =1$, $R=0.9$, $\chi L=1$, $I_0=3.0$, $t=1500T_R$).

Figure 7

Variation of the slope $b$ of the equilibrium power spectrum vs (a) diffusion length $l_D$ and (b) intensity of the incident plane wave $I_0$. Parameters are $d/k_0=100$, $\tau =1$, $R=0.9$, $\chi L=1$. (a) has $I_0=3.0$; (b) has $l_D=0.1$.

Figure 8

Variation of fractal dimension vs space frequency $K$ for different values of: (a) diffusion length $l_D$ when $I_0=3.0$; (b) intensity of the incident wave $I_0$ when $l_D=0.01$ ($d/k_0=100$, $R=0.9$, $\chi L=1$).

Figure 9

Variogram of the intensity of the backward field vs window interval length $W$ ($I_0=3.0$, $l_D=1$, $d/k_0=100$, $\tau =1$, $R=0.9$, $\chi L=1$). (a) $W=1$, $S=1.996$, $D_v=1.002$; (b) $W=27$, $S=1.002$, $D_v=1.499$; (c) $W=46$, $S=0.49$, $D_v=1.755$; (d) $W=100$, $S=0$, $D_v=2$.