Engineering the electromagnetic vacuum for controlling light with light in a photonic-band-gap microchip

We demonstrate a trimodal waveguide architecture in a three-dimensional (3D) photonic-band-gap (PBG) material, in which the local electromagnetic density of states (LDOS) within and adjacent to the waveguide exhibits a forklike wavelength filter characteristic. This facilitates the control and switching of one laser beam with other laser beams ( $\sim 1\mu \mathrm{W}$ steady-state holding power and $\sim 5\mathrm{nW}$ switching power) through mutual coherent resonant interaction with quantum dots. Two waveguide modes provide narrow spectral windows where the electromagnetic LDOS is enhanced by a factor of 100 or more relative to the background LDOS of a third air-waveguide mode with nearly linear dispersion. This “engineered vacuum” can be used for frequency-selective, atomic population inversion and switching (by coherent resonant optical pumping) of an inhomogeneously broadened collection of “atoms” situated adjacent to the waveguide channel. The “inverted” atomic system can then be used to coherently amplify fast optical pulses propagating through the third waveguide mode. This coherent “control of light with light” occurs without recourse to microcavity resonances (involving long cavity buildup and decay times for the optical field). Our architecture facilitates steady-state coherent pumping of the atomic system (on the lower-frequency LDOS peak) to just below the gain threshold. The higher-frequency LDOS peak is chosen to coincide with the upper Mollow sideband of the same atomic resonance fluorescence spectrum. The probing laser is adjusted to the lower Mollow sideband, which couples to the linear dispersion (high group velocity part) of the third waveguide mode. This architecture enables rapid modulation (switching) of light at the lower Mollow sideband frequency through light pulses conveyed by the linear dispersion mode at frequencies corresponding to the central Mollow component (lower LDOS peak). We demonstrate that LDOS jumps of order 100 can occur on frequency scales of $\Delta \omega \approx {10}^{-4}{\omega }_c$ (where ${\omega }_c$ is the frequency of the jump) in a finite-size 3D photonic crystal (PC) consisting of only $10\times 10\times 20$ unit cells. When the semiconductor backbone of the PC has a refractive index of 3.5 and ${\omega }_c$ corresponds to a wavelength of $1.5\mu \mathrm{m}$, this vacuum engineering may be achieved in a sample whose largest dimension is about $12\mu \mathrm{m}$.

Figure 1

Schematic picture of (a) anticrossing of a waveguide mode and a cavity mode, in which the dashed lines are the waveguide (cavity) mode with the presence of the cavity (waveguide); (b) two waveguide modes, in which the dot-dashed line indicates the cutoff frequency of the high-frequency mode. The upper and lower photonic bands are shaded to show the photonic band gaps.

Figure 2

The heterostructure with woodpile structure as the upper and lower 3D PBG cladding sections. The middle part is a W3 waveguide with highlighted rods in which quantum dots (or two-level atoms) are integrated. The ends of the W3 waveguide are connected by W1 waveguides. (a) Bird’s eye view with the upper cladding section removed. (b) Side view with both upper and lower cladding sections.

Figure 3

Band structure of (a) W1 heterostructure when $h=0.4a$, and $r_1=0.12a$; (b) W3 heterostructure when $h=0.3a$, and $r_1=0.12a$. Shaded areas are 2D and 3D bands, respectively. The on-chip PBG is shown by dashed lines. The group velocity of the second mode at the cutoff frequency of the first mode is $0.24c$, where $c$ is the speed of light in the vacuum.

Figure 4

Electric field distributions of the W3 structure with $h=0.3a$, and $r_1=0.12a$, on the $z=0$ plane for (a) the second (group velocity of about $0.25c$) waveguide mode at $\mathbf{k}=(0,0.3,0)(2\pi ∕a)$, where the corresponding frequency is near the cutoff frequency of the second (holding field) waveguide mode; (b) the first (zero group velocity) waveguide mode at $\mathbf{k}=(0,0.5,0)(2\pi ∕a)$. The $y$ axis is shifted to plot the figure. The shaded circles are 2D dielectric rods.

Figure 5

The LDOS of the finite heterostructure with different length in the waveguide direction ( $y$ direction). The dipole is placed at $\mathbf{r}=(-0.7,0.3,0)a$. The origin of the coordinate is the center of the structure.

Figure 6

The emission power when the dipole is placed at different positions inside a $y9$ structure. The line with rectangular symbols is the case in which the dipole is placed at $\mathbf{r}=(-0.7,0.3,0)a$ (same as the dash line with rectangular symbols in Fig. 5); and the line with circular symbols is the case in which the dipole is placed at $\mathbf{r}=(-0.7,-0.7,0)a$.

Figure 7

Analytic functions fitted for (a) peak value and (b) peak width.

Figure 8

Emission power calculated by the FDTD method using two different meshes.

Figure 9

Architecture for an LDOS filter for switching in an inhomogeneously broadened “atomic system.” The two-level atoms or quantum dots are placed at point $P$ and other equivalent points. The 2D photonic crystal has a square lattice with rod radius $r_0$.

Figure 10

The band structure of the 2D photonic crystal with three waveguide modes. Mode 1 (solid line) and mode 3 (dashed line) exhibit cutoffs within the PBG and contribute to sharp LDOS peaks, whereas mode 2 (dotted line) exhibits nearly linear dispersion in the same vicinity. The probe laser frequency (position of lower Mollow sideband) is indicated as ${\omega }_2$, the pump laser frequency is indicated as ${\omega }_1$, and the position of the upper Mollow sideband is indicated as ${\omega }_3$.

Figure 11

Electric-field distributions of the waveguide modes in Fig. 10. (a) Mode 1 at $\mathbf{k}=(0,0)(2\pi ∕a)$; (b) mode 2 at $\mathbf{k}=(0,0.3)(2\pi ∕a)$; and (c) mode 3 at $\mathbf{k}=(0,0)(2\pi ∕a)$.

Figure 12

The dependence of the cutoff frequencies on the rod radius $r_1$.

Figure 13

The local density of states for 2D photonic crystal waveguides with the length $15a$. The radii of rods are $r_0=0.25a$, $r_1=0.166a$, and $r_2=0.145a$, respectively. The dipole is placed at the point $P$ in Fig. 9. To use this waveguide architecture for atomic inversion and switching, pumping laser beams whose frequency coincides with the left LDOS peak are used to drive the upper Mollow sideband of an atom to the peak frequency of the right LDOS peak. Meanwhile, the same pump laser beams drive the lower Mollow sideband of the atom into the low LDOS region to the left of the leftmost LDOS peak. A probe beam whose frequency coincides with the lower Mollow sideband can then be amplified by the inverted atom.

Figure 14

The integration contour $C$ in Eq. (B15).