Single quantum-dot Purcell factor and $\beta$ factor in a photonic crystal waveguide

A theoretical formalism to calculate the spontaneous emission rate enhancement (Purcell factor) and propagation mode $\beta$ factor from single quantum dots in a planar-photonic-crystal waveguide is presented. Large Purcell factors for slow light modes, and enormous $\beta$ factors ($>0.85$) over a broadband (10 THz) spectral range are subsequently predicted. The local density of photon states is found to diverge at the photonic band edge, but we discuss why this divergence will always be broadened in real samples, most notably due to structural disorder. Applications towards “on-chip” single photon sources are highlighted.

Figure 1

Schematic showing a portion of the planar photonic crystal waveguide with a missing row of air holes creating a W1 waveguide along the $x$ axis and an embedded single quantum dot (QD). The structure parameters to be used for calculations later are as follows: hole radius $r=0.275a$, slab thickness $0.5a$, refractive index $n=\sqrt{12}$, and lattice constant (pitch) $a=420\mathrm{nm}$.

Figure 2

Photonic crystal waveguide band structure within the TE-like band gap, showing the bound modes below the light line (solid curves) and the continuum of radiation modes (dark shaded regions above the light line).

Figure 3

(Color online) (a) Example of electric-field profile of $y$ component $\left({\mid {\mathbf{e}}^y\left(\mathbf{r}\right)\mid }^2\right)$ of a slow-light mode on the plane parallel to the slab and located at midplane of the slab $(z=0)$ shown for one unit cell; the wave vector is $k=0.47$$(2\pi ∕a)$ and the corresponding group index $n_g=154$. The superimposed white circles show boundaries of air holes in the slab. (b) Purcell factor (PF) as a function of QD position on the same plane as in (a) for a slow-light mode with dipole orientation along the $y$ axis. (c) $\beta$ factor as a function of fundamental mode frequency for the case of $y$-oriented dipole positioned at the field antinode. (d) The corresponding Purcell factor.

Figure 4

(Color online) (a) Example of electric-field profile of the $x$ component $\left({\mid {\mathbf{e}}^x\left(\mathbf{r}\right)\mid }^2\right)$ using identical parameters as in Fig. 3. (b) Purcell factor (PF) as a function of QD position on the same plane as in (a) for a slow-light mode with the dipole orientation along the $x$ axis. (c) $\beta$ factor as a function of fundamental mode frequency for the case of $x$-oriented dipole positioned at one of the field antinodes. (d) The corresponding Purcell factor.

Figure 5

(Color online) (a) Spatial map of the Purcell factor as function of QD position on the same plane as in Fig. 3b (center of the slab) for the case when the dipole orientation is 45° with respect to the $x$ or $y$ axes. The other parameters remain the same as those used in Fig. 3b or 4b. (b) $\beta$ factors as a function of fundamental mode frequency for two different dot positions (center antinode and corner antinode shown by the crosses and circles, respectively). (c) The corresponding Purcell factors.