Very Large Spontaneous-Emission $\beta$ Factors in Photonic-Crystal Waveguides

Through a new rigorous Bloch-mode formalism, we theoretically study the generation of photons in single-row-defect photonic-crystal waveguides. In contrast with previous related works relying on a reinforcement of the spontaneous emission (SE) through microcavity effects, we explore situations for which the SE into radiation modes is reduced to a very low level while the SE into the guided mode is maintained at a level comparable to that in the bulk material. Remarkably large SE $\beta$ factors in excess of 95% are obtained, and since no resonance effect is involved, this efficiency is achieved over a 40-nm-large spectral interval at $\lambda \approx 950\mathrm{nm}$.

Figure 1

(a) PC waveguide considered in this work and formed by removing a single row in a triangular lattice of air holes in a semiconductor membrane (thickness $h$) with air claddings. (b) Associated computational geometry obtained by surrounding the actual waveguide with PMLs (in blue) in the transverse translation-invariant directions. The dark purple part helps visualizing the surfaces $S_1$ and $S_2$. The PML thickness is not shown for the sake of clarity.

Figure 2

Dipole radiation into a PC waveguide with $a=240\mathrm{nm}$, $h=0.75a$, and a hole radius of $0.29a$. (a) Dispersion diagram of the guided Bloch modes for $s=0\mathrm{nm}$ (solid blue curves) and for $s=40\mathrm{nm}$ (dashed black curves). (b) Normalized SE decay rate $P_M$ into the fundamental Bloch modes. (c) Normalized SE decay rates into the radiation modes for the two in-plane dipole moment orientations. (d) Associated $\beta$ factors. The thin solid blue curve is obtained for an off-axis dipole location ($x_0=100\mathrm{nm}$ and $z_0=a/2$). In all plots, circles, squares, and stars are reference marks locating the fundamental Bloch-mode group velocities equal to $c/20$, $c/10$, and $c/5$, respectively. From (b) to (d), three on-axis dipole locations, $z_0=a/2$ (solid blue line), $a/4$ (dash-dotted red line), and 0 (dashed black line), are considered.

Figure 3

Engineered PC waveguides for minimizing $\gamma$. (a) Definition of parameters $s$ and $L$. (b) Effect of transversal shifts $s$ on ${\gamma }_x$ and ${\gamma }_z$ for $L=\sqrt{3}a$ and $z_0=a/2$. ${\gamma }_x$ and ${\gamma }_z$ are shown on the spectral domains of monomode operation for the PC waveguides, between the air-light line cutoff and the band-edge cutoff ${\lambda }_0$. Circles, squares, and stars as in Fig. 2.

Figure 4

Broadband photon emission into a single guided mode ${\mathbf{\Phi }}^{(1)}$ for the semiclosed geometry (inset). (a) SE decay rate $P_c$ into ${\mathbf{\Phi }}^{(1)}$ for $a/\lambda =0.255$, $s=40\mathrm{nm}$, and $L=\sqrt{3}a$. Solid red curve: predictions of Eq. (4). Blue circles: data calculated with the fully vectorial formalism. (b) $\beta$ factor of the open (solid blue curve) and semiclosed (dashed red curve) geometries for the engineered waveguide ($s=40\mathrm{nm}$). The blue curve is obtained for $z_0=a/2$, and the red curve for $z_0=0.65a$. Circles, squares, and stars as in Fig. 2.