Scaling analysis of transverse Anderson localization in a disordered optical waveguide

The intention of this paper is twofold. First, the mode-width probability density function (PDF) is introduced as a powerful statistical tool to study and compare the transverse Anderson localization properties of a disordered quasi-one-dimensional optical waveguide. Second, by analyzing the scaling properties of the mode-width PDF with the transverse size of the waveguide, it is shown that the mode-width PDF gradually converges to a terminal configuration. Therefore, it may not be necessary to study a real-sized disordered structure in order to obtain its statistical localization properties and the same PDF can be obtained for a substantially smaller structure. This observation is important because it can reduce the often demanding computational effort that is required to study the statistical properties of Anderson localization in disordered waveguides. Using the mode-width PDF, substantial information about the impact of the waveguide parameters on its localization properties is extracted. This information is generally obscured when disordered waveguides are analyzed using other techniques such as the beam propagation method. As an example of the utility of the mode-width PDF, it is shown that the cladding refractive index can be used to quench the number of extended modes, hence improving the contrast in image transport properties of disordered waveguides.

Figure 1

Sample refractive index profiles of (a) ordered and (b) disordered slab waveguides are shown.

Figure 2

Typical mode profiles for (a) an ordered slab waveguide where each mode extends over the entire waveguide, and (b) a disordered slab waveguide, where the modes are localized.

Figure 3

Mode-width PDF of an ordered waveguide defined as $\mathrm{\Delta }{n}_{\mathrm{core}}=0.01$ and $\mathrm{\Delta }{n}_{\mathrm{clad}}=0$, for $N=20$, 40, 60, 80, and 100 slabs. Mode width increases as the number of slabs increases, so the average mode width scales proportional to the size of the structure.

Figure 4

Mode-width PDF of a disordered waveguide defined by $\mathrm{\Delta }{n}_{\mathrm{core}}=0.01,\mathrm{\Delta }{n}_{\mathrm{clad}}=0$, for $N=20$, 40, 60, 80, and 100 slabs. The inset is the magnified version of the localized peaks of the PDFs. Saturation of the PDF beyond $N_{\mathrm{sat}}\approx 60$ is clear in this figure.

Figure 5

$\mathrm{\Delta }{n}_{\mathrm{core}}=0.02$ is increased in comparison to Fig. 4. The small-mode-width peak of the PDF shifts towards smaller mode-width values indicating a stronger localization for a larger index contrast in the disordered waveguide. A magnified version is shown as an inset. $N_{\mathrm{sat}}\approx 40$ is smaller for a stronger transverse scattering.

Figure 6

Mode-width PDF of a disordered waveguide defined as $\mathrm{\Delta }{n}_{\mathrm{core}}=0.01,\mathrm{\Delta }{n}_{\mathrm{clad}}=0.01$, and $N=40$, 80, 120, 160, 200 slabs. A lower cladding index significantly changes the distribution of the extended modes and PDF saturates at a much larger value of N $\left(N_{\mathrm{sat}}\approx 160\right)$. The inset represents a magnified version. A smaller cladding index only affects the extended mode-width distribution, while the localized modes remain nearly unaffected.

Figure 7

The same as Fig. 6 except $\mathrm{\Delta }{n}_{\mathrm{clad}}=0.02$. The PDF saturates to its terminal shape at a larger number of slabs $\left(N_{\mathrm{sat}}\approx 200\right)$. The inset is a magnified version.

Figure 8

The same as Fig. 6 except $\mathrm{\Delta }{n}_{\mathrm{clad}}=0.04$. Further decreasing the cladding index delays the saturation of the PDF to larger values of N $\left(N_{\mathrm{sat}}\gg 200\right)$. The inset is a magnified version.

Figure 9

The same as Fig. 4 except $\mathrm{\Delta }{n}_{\mathrm{clad}}=-0.005$. A cladding index larger than $n_0$ not only reduces the probability density of the extended modes, but also diminishes the second peak of the localized regime. The inset represents a magnified version.

Figure 10

Normalized PDF for disordered waveguides defined by $\mathrm{\Delta }{n}_{\mathrm{core}}=0.01,N=200$, and $\mathrm{\Delta }{n}_{\mathrm{clad}}=-0.005$, 0.01, 0.02, 0.03, and 0.04. The magnified inset clearly shows that the statistics of the localized modes is independent of the cladding index unless for a negative $\mathrm{\Delta }{n}_{\mathrm{clad}}$.

Figure 11

Normalized mode-width PDF of disordered waveguides defined by $\mathrm{\Delta }{n}_{\mathrm{core}}=0.01,\mathrm{\Delta }{n}_{\mathrm{clad}}=0,N=200$, and unit slab thickness of $d=0.5,1.0,$ and $1.5\lambda$. Overall, the waveguides show stronger localization for a thicker unit slab.

Figure 12

The same as Fig. 11, except that the thickness of the unit slabs in the structure of the disordered waveguides is larger. The formation of local waveguides leads to sharp discrete peaks in the normalized PDF.

Figure 13

The unit slab thickness is fixed at $d=\lambda$ but $\mathrm{\Delta }{n}_{\mathrm{core}}=0.01,0.03,$ and 0.05. As $\mathrm{\Delta }{n}_{\mathrm{core}}$ increases, more local waveguides form in the structure and the PDF becomes more discrete.

Figure 14

Propagation of a Gaussian beam $\left(\omega =3\lambda \right)$ along a disordered waveguide defined by $\mathrm{\Delta }{n}_{\mathrm{core}}=0.01,\mathrm{\Delta }{n}_{\mathrm{clad}}=0,$ and $N=200$. The beam eventually localizes to a relatively stable width after an initial expansion.

Figure 15

The refractive index profile of the disordered waveguide is exactly the same as Fig. 14, except $\mathrm{\Delta }{n}_{\mathrm{clad}}=0.04$. Formation of the extended modes generates a background noise.