Tailoring Correlations of the Local Density of States in Disordered Photonic Materials

We present experimental evidence for the different mechanisms driving the fluctuations of the local density of states (LDOS) in disordered photonic systems. We establish a clear link between the microscopic structure of the material and the frequency correlation function of LDOS accessed by a near-field hyperspectral imaging technique. We show, in particular, that short- and long-range frequency correlations of LDOS are controlled by different physical processes (multiple or single scattering processes, respectively) that can be—to some extent—manipulated independently. We also demonstrate that the single scattering contribution to LDOS fluctuations is sensitive to subwavelength features of the material and, in particular, to the correlation length of its dielectric function. Our work paves a way towards complete control of statistical properties of disordered photonic systems, allowing for designing materials with predefined correlations of LDOS.

Figure 1

Near-field hyperspectral imaging of QD photoluminescence. (a) Three-dimensional equi-intensity surface plot of PL signal $I_{\mathrm{PL}}(\mathbf{r},\nu )$ in a typical experimental scan. The total number of voxels of the three-dimensional image is of the order of $4\times 10^6$. (b) Sketch of the experiment. (c) and (d) show PL intensity as a function of position for a given, randomly chosen frequency ${\nu }_i$, and as a function of frequency for a given, randomly chosen position $(x_i,y_i)$, respectively.

Figure 2

Frequency-resolved correlation analysis. (a) Frequency correlation matrix of QD photoluminescence for a system with $k{\ell }^{*}=4$, which in terms of structural parameters corresponds to an average hole diameter $⟨d_h⟩=210\mathrm{nm}$ and a hole surface filling fraction $f=0.35$. (b) The diagonal elements of $C_{\mathrm{PL}}(\nu ,{\nu }^{\prime })$ equal to the normalized variance of PL intensity fluctuations (upper curve). The gray lower curve shows the fluctuations of off-diagonal terms ($\nu \ne {\nu }^{\prime }$) at large detuning ${\nu }^{\prime }\gg \nu$. It was evaluated along the gray line in the panel (a).

Figure 3

Frequency correlation function of PL (black solid line) and the corresponding theoretical fit with $⟨C_{\rho }(\mathrm{\Delta }\nu )⟩$ (red dashed line) for samples with different scattering strengths $k{\ell }^{*}$. The inset shows scattering diagrams yielding different contributions to $⟨C_{\rho }(\mathrm{\Delta }\nu )⟩$. The classical (diffuson) and coherent (cooperon) diagrams are multiple scattering contributions that occur on large length and time scales and determine the behavior of $⟨C_{\rho }(\mathrm{\Delta }\nu )⟩$ at small detuning $\mathrm{\Delta }\nu$ [18]. The single scattering is the fastest process that determines the asymptotic tail of $⟨C_{\rho }(\mathrm{\Delta }\nu )⟩$ at large $\mathrm{\Delta }\nu$. The top of the image shows the SEM images of samples with different scattering strengths $k{\ell }^{*}$.

Figure 4

Values of $F_1$ (black points) obtained from the fits to the measured $C_{\mathrm{PL}}(\mathrm{\Delta }\nu )$ by Eq. (2). The two continuous lines and the dashed line show the behavior of $F_1$ expected from the theory for $ka=0.9$, 1.1, and 1, respectively, and for $k\ell =10$ that is typical for the whole set of samples (the dependence on $k\ell$ is very weak). The two insets show the sensitivity of $F_1$ to the average minimum distance between adjacent scatterers $2{\ell }_{\epsilon }$ [18].

Figure 5

(a) Renormalization of the diffusion constant $D(0)$ as a function of $k{\ell }^{*}$. Full black circles are the values obtained from the theoretical fits to the measured autocorrelation function of PL. The two solid lines show the marginal values of $D(0)/D_B$ for our set of samples. The inset shows the values of the dimensionless parameter $3D_B\tau /{\ell }^{*2}$ that roughly corresponds to the number of scattering events experienced by a photon before leaving the sample. (b) Characteristic length scale $L$ as a function of $k{\ell }^{*}$. The four empty circles are the direct measurements of the decay length of photonic modes taken from Ref. [17]. The two insets show the schematic representation of $L$ for two values of $k{\ell }^{*}$. $L$ shrinks with decreasing $k{\ell }^{*}$.