Broadband coherent backscattering spectroscopy of the interplay between order and disorder in three-dimensional opal photonic crystals

We present an investigation of coherent backscattering of light that is multiple scattered by a photonic crystal by using a broadband technique. The results significantly extend on previous backscattering measurements on photonic crystals by simultaneously accessing a large frequency and angular range. Backscatter cones around the stop gap are successfully modeled with diffusion theory for a random medium. Strong variations of the apparent mean free path and the cone enhancement are observed around the stop band. The variations of the mean free path are described by a semiempirical three-gap model including band structure effects on the internal reflection and penetration depth. A good match between theory and experiment is obtained without the need of additional contributions of group velocity or density of states. It is found that the internal reflection contribution does not improve the semiempirical fits. We argue that the cone enhancement reveals additional information on directional transport properties that are otherwise averaged out in diffuse multiple scattering.

Figure 1

(a) Experimental setup for coherent backscattering spectroscopy. (b) Photonic crystal effects influencing the apparent mean free path: (1) modification of source of diffuse light by Bragg diffraction and (2) modification of angle-dependent escape probability due to internal reflection.

Figure 2

(a) Wigner-Seitz cell of an fcc lattice with its irreducible part, showing symmetry points $\Gamma$, $L$, $X$, $U$, $K$, and $W$. (b) Calculated band structure for a polystyrene opal, with total density of states (DOS) of photonic crystal (line) compared to effective medium (dashed line). Shaded area indicates the reduced frequency range accessed in our experiments, with horizontal lines indicating individual ranges of the 130-nm (dashed line) and 180-nm (dash-dotted line) opals

Figure 3

(a) Experimental reflectivity spectra of a $r=130$ nm opal for angles of incidence from $\varphi =4^{°}$ to ${50}^{°}$. Subsequent curves scaled by a decade for visibility. (b) Dispersion relations for $r=130$ nm opal, (circles) half-maximum values of the reflectivity peaks of (a), (lines) calculations for fcc crystal along $\mathit{LU}$ (dashed lines) and $\mathit{LW}$ (full lines) ($a=362$ nm). [(c) and (d)] Same for $r=180$ nm opal (calculation, $a=510$ nm).

Figure 4

(a) Experimental CBS spectrum of a $r=130$ nm diameter opal at $\varphi =8^{°}$ and (b) cross sections at frequencies of 15 500 cm${}^{-1}$ (red circles), 17 000 cm${}^{-1}$ (black dots), and 18 000 cm${}^{-1}$ (blue diamonds), with (lines) fits using a diffuse scattering model. [(c) and (d)] Values of apparent mean free path ${\ell }_W$ (dots) (c) and CBS enhancement $E$ (d) obtained from fits to (a). Lines denote $\omega {}^{-0.96}$ dependence (green dash-dotted line), obtained from experimental data at $\varphi ={50}^{°}$ (open triangles), and calculations using Eq. (1) with (red line) and without (blue dashed line) internal reflection correction.

Figure 5

[(a)–(f)] Values of ${\ell }_W$ [(a)–(c)] and $E$ [(d)–(f)] obtained from fits of CBS spectra for the 180-nm opal. Error bars denote 95$\%$ confidence interval of the fits. (Lines) calculations using Eq. (1) with (red line) and without (blue dashed line) internal reflection correction.

Figure 6

(a) (Left) Color density maps of ${\ell }_W$ obtained from fits to CBS cones for the $r=130$ nm opal. (Right) Calculated ${\ell }_W$ using Eq. (1). (b) Same for the $r=180$ nm opal. Lines indicate band structure from Fig. 2. [(c) and (d)] Calculations made using Eq. (1). Numbers indicate contributions from three bands of Eq. (3).

Figure 7

Fitted CBS enhancement factor for the $r=130$ nm opal (a) and the $r=180$ nm opal (b). Lines indicate band structure from Fig. 2.

Figure 8

Internal reflection coefficient calculated using the three-gap model of Eq. (3).

Figure 9

(a) CBS cone-narrowing factor ${\ell }_W/\ell$ calculated using Eq. (1) including effect of internal reflection $\stackrel{̲}{R}$ from Fig. 8. (b) Calculated penetration depth parameter $\xi$ and (c) CBS cone narrowing resulting from combined effects of $\stackrel{̲}{R}$ and $\xi$.

Figure 10

Scattering intensity from the 130-nm (a) and 180-nm (b) opals for angles of incidence from 8${}^{°}$ to 50${}^{°}$ measured at a detection angle of 5${}^{°}$ using a helicity-conserving channel. Subsequent spectra are offset by 0.5 for visibility, starting from $\varphi ={50}^{°}$ (no offset).