Semiconductor and Diamond Cavity-QED

Numerical simulations of photonic crystal (PC) defect microcavities have indicated that they can confine light to ultra-small volumes, approaching the theoretical limit of a cubic half-wavelength. Such tight confinement results in extremely large electromagnetic energy densities, with field strengths per photon on the order of ~ 1 kV/cm. If combined with potentially high quality factors (predicted to be ~ 100,000 for certain designs) and hence relatively long photon lifetimes, these cavities are of interest in a number of applications in nonlinear optics, biomolecular sensing, and quantum optics. We are particularly interested in the latter field, where the demonstration of an ultra-small volume, high quality factor (Q) PC microcavity could enable fundamental studies of coherent interactions between single photons and single atoms or quantum dots. Our work on this subject, which has recently led to the experimental demonstration of such a cavity with Q ~ 40,000 and V_eff ~ 0.9(l/n)³ [8], is described briefly below.

Design

The basic geometry we consider is shown in Figure 1, and is termed a two-dimensional (2D) photonic crystal slab waveguide microcavity. The structure consists of a localized defect in a two-dimensional lattice of air holes etched into an optically thin membrane. This configuration provides in-plane modal confinement due to distributed Bragg reflection via the photonic lattice, and vertical optical confinement by total internal reflection at the membrane-air interface. The particular lattice and defect geometry chosen determine many of the salient properties of the cavity modes, such as polarization, emission pattern, and quality factor [1].

Figure 1: Schematics of the 2D PC defect microcavity geometry.

PC microcavities are designed using both analytic and numerical design tools. We have developed a simple group theory based analysis [1,2] which is used to predict the symmetries and dominant Fourier space components of defect modes within 2D PC defect microcavities. When combined with a Wannier equation formalism that describes the localizing effects of the photonic lattice [3], an approximate, non-numerically intensive model for localized resonant states within PC lattices is produced.

A Fourier-space description of the potential sources of loss in PC defect microcavities is used as a starting point in the design of high-Q devices. By understanding that modes comprised of certain in-plane Fourier components will radiate, either due to insufficient in-plane momentum for total internal reflection (leading to vertical radiation loss) or an angular orientation where the photonic lattice does not provide sufficient optical feedback (leading to in-plane radiation loss), the group theory based analysis is used to select specific cavity geometries that support spatially odd modes, where the lack of DC momentum components provides an immediate reduction in vertical radiation losses. We combine this symmetry choice with a tailoring of the photonic lattice through a grade in the hole radius radially outwards from the center of the cavity, used to confine the mode in-plane and reduce in-plane radiative losses. Three-dimensional finite-difference time-domain (FDTD) simulations are used to quantitatively analyze these structures, and cavity designs within both square [4] and hexagonal [5] lattices have predicted Qs of ~ 100,000 with an effective modal volume (V_eff) on the order of a cubic wavelength (~ (l/n)³). The cavity design we choose to employ in experiments is shown in Figure 2; one of the primary advantageous characteristics of this design is its relative insensitivity to variations in the photonic lattice, as fairly significant modifications to the lattice grade and average hole radius do not degrade its Q below 20,000.

Figure 2: Magnetic field amplitude for the graded square lattice cavity mode of interest. Blue lines show the grade in hole radius along the central x and y axes of the cavity, while the red lines show the field component B_z along these directions.

InP-Based 1.3 Micron Lasers

As an initial means to probe the properties of these PC microcavities, we have fabricated the square lattice cavities of Figure 2 in an InP-based 1.3 mm multi-quantum well laser material, grown by our collaborators at Lucent Technologies, Bell Laboratories (please see our Collaborators page). PC device formation is accomplished through electron beam lithography, mask transfer to a silicon dioxide layer through an inductively-coupled plasma reactive ion etch (ICP-RIE), a high-temperature ICP-RIE etch through the waveguide layer into a sacrificial buffer layer, and an undercut wet etch that removes the buffer layer, leaving a free-standing membrane [6]. Scanning electron microscope (SEM) images of a fabricated device are shown in Figure 3(a).

Figure 3: (a) SEM micrograph of fabricated laser cavity. (b) Emitted power vs. pump power (L-L) curve for device pumped with a broadly focused pump beam. (inset) Emission spectrum near estimated material transparency for this device. (c) L-L curve for a device pumped with a tightly focused pump beam.

Devices are optically pumped (room temperature) at 830 nm (pulsed, typically 10 ns pulse width, 300 ns period) and the emitted photoluminescence (PL) normal to the sample surface is collected and resolved using an optical spectrum analyzer [7]. Devices pumped with a relatively broad pump beam (Figure 3(b)) show a characteristic laser threshold at ~ 360 mW, where the power measured is the peak external pump power incident on the sample surface, without any correction factor included for actual absorption by the quantum wells. The measured sub-threshold (pump level close to our estimate for material transparency) emission linewidth of 0.10 nm is near the resolution limit of our experiment, and gives us an estimate for the cold-cavity Q of ~ 13,000. Using a more focused pump beam (Figure 3(c)), laser thresholds as low as ~ 100 mW are measured.

Passive Measurements of Q and V_eff In Si-Based PC Microcavities

The graded square lattice cavities of Figure 2 have also been fabricated in Si membranes, where the devices are passively probed with an external waveguide that can map both the spectral and spatial properties of the devices [8]. Thus, in addition to being able to measure modal linewidths and hence Q factors (in a configuration with significantly better resolution than the PL measurements), the in-plane localization of the cavity modes can be quantitatively investigated. Devices are fabricated in a silicon-on-insulator wafer where the insulating (oxide) layer serves as a sacrificial buffer layer which is removed to form a freestanding membrane. SEM micrographs of a fabricated device are shown in Figure 4(a).

Figure 4: (a) SEM micrograph of fabricated Si PC microcavity. (b) Schematic illustrating the fiber taper probe measurement setup.

Devices are probed with an optical fiber taper formed by heating and stretching a standard optical fiber to a diameter of ~ 1-2 mm. The taper is mounted (Figure 4(b)) above and parallel to an array of PC cavities, in a configuration so that its lateral and vertical positions can be easily adjusted. The wavelength-dependent transmission through the taper when it is positioned ~ 500 nm over one of the PC cavities is shown in Figure 5(a), and shows a number of resonance features. By studying the spectral shifts in the positions of these resonances between devices in which the average hole radius is varied (for a given lattice constant), we can identify our mode of interest, which is the fundamental (lowest frequency) resonance within the in-plane bandgap under consideration.

Figure 5: (a) Normalized taper transmission for a cavity with lattice spacing a=409 nm, highlighting the mode of interest. (b) Measured linewidth (blue dots) vs. taper-PC gap for the mode of interest in a sample with a=425 nm The red curve is a fit to the experimental data. (Inset) Normalized taper transmission for this device when the taper-PC gap is 650 nm (c) Measured normalized taper transmission (dots) as a function of taper displacement along the (c) x and (d) y axes of the cavity. The dashed lines are Gaussian fits to the data and the solid lines are numerically calculated coupling curves based on the FDTD-generated cavity field and analytically determined fiber taper field. The insets in (c)-(d) are optical micrographs of the taper aligned along the y and x axes of the cavity, respectively.

A wavelength scan through the taper for the mode of interest in one of our devices is shown in Figure 5(b), for the case when the taper is positioned 650 nm above the surface of the cavity. The linewidth of the resonance (estimated by a Lorentzian fit) is g ~ 0.047 nm in this case. The taper, however, can have a loading effect on the cavity so that g is larger than the cold-cavity linewidth g₀. By examining g as a function of the taper-PC gap, we can estimate the taper loading effects. This data, given in Figure 5(b), shows that the taper does indeed load the cavity, though the effects are minimized when the taper is moved sufficiently far away from the cavity. In this regime, g asymptotically approaches g₀. A fit to this data gives an estimate g₀ ~ 0.041 nm, close to the measured linewidth when the taper is ~ 800 nm above the cavity, and corresponding to a cold cavity Q ~ 40,000. Note that the measured device in Figure 5(b) has an average hole radius ~ 20 % larger (as measured by the SEM) than the design simulated in Figure 2. Repeating those FDTD simulations to account for this yields a mode with predicted Q ~ 56,000 and V_eff ~ 0.88(l/n)³.

Of equal importance is the ability of the taper to map the in-plane localization of the cavity modes by examining the depth of the resonance transmission as the taper is offset along the central x and y axes of the cavity (Figure 5(c)-(d)). These measurements show the mode to be localized to a micron-scale central region of the cavity. By considering a simple picture of taper-PC coupling where the coupling coefficient is calculated from the FDTD-generated cavity fields and the analytically determined taper fields, the solid curves in Figure 5(c)-(d) are produced, and correspond closely with the measured results, indicating that V_eff ~ 0.9(l/n)³ for this high-Q cavity (assuming, as seen experimentally, modal localization in the out-of-plane dimension).

Future Interests

As described in more detail in Ref. [8], cavities with these measured Q and V_eff values are particularly exciting due to their potential applications in quantum optics. In particular, simple calculations of the coherent coupling rate of the cavity field to a cesium atom or an InAs quantum dot indicate that it is larger than the decoherence rates of the constituent elements of the system (cavity and atomic or quantum dot decay rates), so that the condition of strong coupling could be achieved in both of these configurations. Such a demonstration could be of potential importance for applications in quantum computing, where strongly coupled atom-photon systems have been proposed as candidates to produce the required quantum bit (qubit) states. Alternately, in the regime of weak coupling, a microcavity-enhanced spontaneous emission rate for an embedded emitter species (Purcell factor) of ~ 3,500 is calculated using these experimentally demonstrated Q and V_eff values, and is a very high value for a semiconductor microcavity.

Future (long-term) research will focus on working towards experimental observation of some of these exciting phenomena in quantum optics. Other interests include integration with fiber-coupled photonic crystal waveguides, studies in nonlinear optics, sensing, and novel lasers, and further use of the optical fiber taper probe to study the properties of other wavelength-scale resonators.

References

[1] Painter O, Srinivasan K, O'Brien JD, Scherer A, Dapkus PD, "Tailoring of the resonant mode properties of optical nanocavities in two-dimensional photonic crystal slab waveguides," JOURNAL OF OPTICS A-PURE AND APPLIED OPTICS, v3(6): pp. S161-S170 NOV 2001 (pdf).

[2] Painter O, Srinivasan K, "Localized defect states in two-dimensional photonic crystal slab waveguides: a simple model based upon symmetry analysis," PRB v68, 035110: July 2003 (pdf).

[3] Painter O, Srinivasan K, Barclay PE, "A Wannier-like Equation for Localized Resonant Cavity Modes of Locally Perturbed Photonic Crystals", PRB v68, 035214: July. 2003 (pdf).

[4] Srinivasan K, Painter O, "Momentum space design of high-Q photonic crystal optical cavities," OPTICS EXPRESS, v10(15): pp. 670-684 JUL 29 2002 (pdf).

[5] Srinivasan K, Painter O, "Fourier space design of high-Q cavities in standard and compressed hexagonal lattice photonic crystals," OPTICS EXPRESS, v11(6): pp. 579-593 MAR 24 2003 (pdf).

[6] Srinivasan K, Barclay PE, Painter O, Chen J, Cho AY, "Fabrication of high quality factor photonic crystal microcavities in InAsP/InGaAsP membranes," submitted to JVST-B, Jun. 2003.

[7] Srinivasan K, Barclay PE, Painter O, Chen J, Cho AY, Gmachl C, "Experimental demonstration of a high-Q photonic crystal microcavity," APL, v83(10): pp. 1915-1917 Sept. 2003 (pdf).

[8] Srinivasan K, Barclay PE, Borselli M, Painter O, "Optical-fiber based measurement of an ultra-small volume high-Q photonic crystal microcavity," submitted to PRL Sept. 25 2003, (http://arxiv.org/abs/quant-ph/0309190).

Acknoweldgements

Kartik Srinivasan thanks the Hertz Foundation for its graduate fellowship support.

Questions?

Please contact Oskar Painter if there are any questions.