Quantitative analysis of directional spontaneous emission spectra from light sources in photonic crystals

We have performed angle-resolved measurements of spontaneous-emission spectra from laser dyes and quantum dots in opal and inverse opal photonic crystals. Pronounced directional dependencies of the emission spectra are observed: angular ranges of strongly reduced emission adjoin with angular ranges of enhanced emission. It appears that emission from embedded light sources is affected both by the periodicity and by the structural imperfections of the crystals: the photons are Bragg diffracted by lattice planes and scattered by unavoidable structural disorder. Using a model comprising diffuse light transport and photonic band structure, we quantitatively explain the directional emission spectra. This work provides detailed understanding of the transport of spontaneously emitted light in real photonic crystals, which is essential in the interpretation of quantum optics in photonic-band-gap crystals and for applications wherein directional emission and total emission power are controlled.

Figure 1

(Color online) Scheme of the experimental setup. The pump beam is focused onto the sample at incident angle ${\theta }_p$ by the objective $O_1$ ($f=7.3\mathrm{cm}$, numerical aperture 0.05). Luminescence within a cone centered at detection angle ${\theta }_e$ relative to the surface normal is collected by the lens $L_1$ $(f=12\mathrm{cm})$ and imaged on the spectrometer slit by the lens $L_2$ $(f=12\mathrm{cm})$. A color filter $F$ prevents scattered pump light from entering the prism spectrometer. The angle ${\theta }_e$ is varied by rotating the rotation stage, which carries the sample holder, the fiber, and the objective $O_1$, whereas the incident angle ${\theta }_p$ is kept fixed.

Figure 2

(Color online) (a) Photonic band structure for polystyrene opals [(blue) dashed curves], center frequency of the stop band vs detection angle ${\theta }_e$ according to Bragg’s law [(red) solid curve], and the frequency range of R6G emission for opals with lattice parameters $a=178$ and $365\mathrm{nm}$ [hatched regions between the (orange) dash-dot-dotted lines and the (green) dash-dotted lines, respectively]. (b) Normal-incidence reflectivity as a function of frequency for polystyrene opals with a lattice parameter $a=365\mathrm{nm}$ and refractive-index contrast $m=1.59$. (Blue) circles are measured values, the (black) solid curve is a fit with the modified Gaussian reflectivity peak [Eq. (5)], the (red) dashed curve is a fit with a Gaussian reflectivity peak.

Figure 3

(Color online) Emission spectra of R6G in polystyrene opals with lattice parameters $a=365$ (a) and $178\mathrm{nm}$ (b). The (black) solid curves are obtained at ${\theta }_e=0°$, the (red) dashed curves at ${\theta }_e=30°$, the (green) dotted curves at ${\theta }_e=37.5°$, and the (blue) dash-dotted curves at ${\theta }_e=60°$. The (magenta) short-dashed curve in (a) indicates the total emission spectrum $I_{\mathrm{tot}}\left(\omega \right)$.

Figure 4

(Color online) Angular distribution of the R6G emission from a polystyrene opal with a lattice parameter $a=178\mathrm{nm}$. The (black) pentagons indicate the intensity measured at the spectral maximum ($\omega =17860{\mathrm{cm}}^{-1}$ or $\lambda =560\mathrm{nm}$); the (blue) solid curve is the calculated escape function with Fresnel internal-reflection coefficient. Measured intensities corrected for the detection efficiency of the setup are displayed as (red) stars. All data are normalized at ${\theta }_e=0°$.

Figure 5

(Color online) R6G emission from polystyrene opal with lattice parameter $a=365\mathrm{nm}$. The scatter plots represent the measured spectra corrected for the angular aperture of the collecting lens ${\Omega }_L$ and divided by the total emission spectrum $I_{\mathrm{tot}}\left(\omega \right)$. The calculated escape functions are plotted with solid curves. The (black) solid curve and squares are for ${\theta }_e=0°$, (red) dashed curve and triangles are for ${\theta }_e=30°$, (green) inverted triangles and solid curve are for ${\theta }_e=45°$, and (blue) circles with dash-dotted curve are for ${\theta }_e=60°$.

Figure 6

(Color online) Intensity ratios $I_c\left({\theta }_e\right)∕I_{\mathrm{tot}}$ corrected for the lens aperture ${\Omega }_L$ as a function of the exit angle ${\theta }_e$ for a polystyrene opal with a lattice parameter $a=365\mathrm{nm}$ for frequencies $\omega =15000$, 16 500, 17 400, and $18350{\mathrm{cm}}^{-1}$ [(blue) inverted triangles, (black) squares, (red) circles, and (green) triangles]. The corresponding curves represent the calculated escape distributions (for reduced frequencies $\omega a∕2\pi c=0.55$, 0.6, 0.64, and 0.67, respectively).

Figure 7

(Color online) Emission spectra of CdSe quantum dots in a titania inverse opal with a lattice parameter $a=500\mathrm{nm}$. The (black) solid curve is obtained at ${\theta }_e=0°$, the (red) dashed curve at ${\theta }_e=15°$, the (green) dash-dotted curve at ${\theta }_e=25°$, and the (blue) short-dashed curve at ${\theta }_e=50°$.

Figure 8

(Color) (a) Escape distribution as a function of the exit angle ${\theta }_e$ for titania inverse opals with lattice parameters $a=370$ and $420\mathrm{nm}$ at $\omega =16390{\mathrm{cm}}^{-1}$ ($\lambda =610\mathrm{nm}$, black squares and red circles, respectively), and with a lattice parameter $a=500\mathrm{nm}$ at $\omega =15870{\mathrm{cm}}^{-1}$ ($\lambda =630\mathrm{nm}$, green triangles). The corresponding curves represent the calculated distributions. (b) Photonic band structure for the inverse opals (black dotted curves). The hatched regions indicate the stop band caused by Bragg diffraction by (111) lattice planes. The horizontal bars represent the reduced center frequencies of the quantum-dot emission from the crystals with lattice parameters (bottom to top) $a=370$, 420, 500, 580, and $650\mathrm{nm}$. The colors of the bars indicate the values of the escape function $P(\omega ,{\theta }_e)$ obtained from the measurements at exit angles shown by the black stars.

Figure 9

(Color online) The angle-averaged internal reflectivity $\overline{R}\left(\omega \right)$ for the polystyrene opals [(black) solid curve] and for the titania inverse opals [(red) dashed curve] according to the diffusion model, in which only the first-order Bragg diffraction is taken into account. $\overline{R}\left(\omega \right)$ determines the enhancement of the escape probability outside a stop-band direction.

Figure 10

(Color online) Real-space cartoon of the projection of the spectrometer slit (striped rectangle) on the sample surface overlapped with the emission spot (blue). The quarter of the slit projection [(red) dashed rectangle] has one of the corners in the emission-spot center. The width of the projection (along $x$) increases at larger detection angles ${\theta }_e$.