Electrically controlled modulation in a photonic crystal nanocavity

We describe a compact modulator based on a planar photonic crystal nanocavity whose resonance is electrically controlled. A forward bias applied across a p-i-n diode shifts the cavity into and out of resonance with a continuous-wave laser field in a waveguide. The sub-micron size of the nanocavity promises extremely low capacitance, high bandwidth, and efficient on-chip integration in optical interconnects.


III. EXPERIMENT
The structures are mounted in a confocal microscope setup which allows for independent positioning of two laser beams: a narrow-linewidth continuous-wave (cw) probe beam at a wavelength that is tunable from 1250-1369 nm; and a pump beam for exciting photoluminescence (PL) with a wavelength of 633 nm. A movable pinhole in the image plane of the confocal microscope setup allows us to collect light from different regions of the chip with a size as small as ∼ 3µm.

A. Device Characterization
We first characterize the photonic circuit by QD photoluminescence that is excited using the 633 nm pump laser. Fig.3(a) shows the cavity spectrum observed when the cavity is optically pumped and the observation pinhole is open to collect from the full structure. We observe two cavity modes at wavelengths λ = 1350 nm (Q = 1500) and λ = 1327 (Q = 650), both polarized perpendicular to the long cavity axis. We identify these modes as the fundamental and first-order modes of the L3 cavity [14]. We will from now on concentrate on the fundamental cavity mode at λ ≈ 1350 nm. To characterize the transfer of the cavity emission to the waveguide/grating couplers, we graph in Fig.4 the PL when the cavity is pumped while the pinhole is scanned across the length of the device. The QD PL that is collected directly from the cavity is lower than the PL collected at either end of the waveguides. Assuming no material loss in the waveguides, we can use this measurement to estimate that the coupling efficiency out of the two gratings is about 40%, based on a coupled mode equations model described in Appendix D. Scanning electron micrograph (SEM) of the PPC circuit. The circuit consists of an input grating coupler; input waveguide; cavity; output waveguide; and output grating coupler.
We use the electroluminescence (EL) from the p-i-n diode to characterize the electrical pumping of the structure. Fig.3(e) plots the EL when the pinhole is closed around the outcoupling grating. The voltage is pulsed at 1 kHz with a 1% duty cycle to reduce heating. We observe both the fundamental and higher-order modes at a voltage above 2 V, which corresponds to a current of 18 µA (see Fig.3(d)). From simulations, we estimate that the carriers distribute rather evenly across the whole membrane, as seen in Fig.1(b). The cavity modes are visible above the background EL because the cavity-coupled QDs emit more rapidly through the Purcell effect [15,16]. Above a voltage of 4 V, the EL drops rapidly because of heating of the membrane.
We will now use the EL spectra to estimate the cavity index change due to carriers, ∆n r (n, p), and temperature, ∆n r (T ): ∆n r = ∆n r (n, p) + ∆n r (T ), where n and p are the electron and hole concentrations, respectively, and T is the temperature. For the carrierdependent term, we consider contributions due to bandgap narrowing (∆n r (n, p) BG ), bandgap filling (∆n r (n, p) BF ), and free carrier effects (∆n r (n, p) F C ). As derived in Appendix C, the latter two contributions are dominant, and we approximate their combined effect on n r as ∆n r (n, p) ≈ ∆n r (n, p) BF + ∆n r (n, p) F C (2) = −5.4 · 10 −21 (∆n · cm 3 ) − 2.5 · 10 −21 (∆p · cm 3 ) The temperature-dependent index change is modeled as where ∆T is the temperature change from 300 K [17,18]. Experimentally, we can deduce the dielectric index change ∆n from the frequency shift ∆ω c in the cavity resonance, using second order perturbation theory [19] From FDTD simulations, we note that the cavity field is primarily in the high-index material. We can then approximate, for a small index shift ∆n, that where n 0 = 3.4 is the index of GaAs at a wavelength of 1.3 µm [20]. Eqs. [2,3] show that temperature and carriermediated shifts on the cavity resonance are expected to be competing effects. pulsed diode measurement   Fig. 3(e). As the voltage is increased from 2.2V to above 3V, the cavity resonance separates from λ 0 into two peaks centered at λ 1 , λ 2 , which are split by 2.2 nm (the splitting is not visible for the higher order mode at 1327 nm because of the lower Q). λ 1 appears slightly blue-detuned from λ 0 , which would indicate index modulation by free-carriers and/or band-filling. The carrier shift is expected to occur during the RC-limited response time τ RC ∼ 3 ns, as described later. The tuning is accompanied by heating of the structure which causes the red-shift to λ 2 . We estimate that the thermal effect occurs on the time scale of ∼ 5µs (estimated from varied pulse-length measurements). From the red shift, we calculate a temperature-induced refractive index shift of ∆n r (T ) ≈ 1.3 · 10 −3 and a corresponding ∆T ≈ 1.56 • C. The blue-shift indicates ∆n r (n, p) ≈ −1 · 10 −3 .
We compare these experimentally obtained index variations to numerical simulations. From the carrier simulations in Fig.1(b), we estimate that at the location of the cavity and at V in = 3 V, ∆n r (n, p) ≈ −2 · 10 −3 , averaged over the membrane thickness, which is fairly close to our observation. The temperature simulation predicts ∆n r (T ) ≈ 2.2·10 −3 , which is also reasonably close to our observation. A time-dependent simulation of the carrier and temperature index shifts gives the cavity evolution after the control pulse is turned on (Fig.4(d)). This simulation indicates that the cavity first rapidly shifts to short wavelengths due to the carrier-induced refractive index change, and then shifts to longer wavelength because of heating.
We will now consider how fast the cavity can be shifted. Figs.3(b,c) plot the photoluminescence when the cavity is excited with the 633 nm pump laser and the structure is electrically modulated with a 0-3 V square wave at a modulation rate ν m and duty cycle of 20%. The integration time is 100 ms -much longer than the switching time. At low frequency, we observe a cavity splitting of ∆λ c ≈ 1.30 nm, which indicates a refractive index change of the cavity ∆n/n ≈ 9.6 · 10 −4 . The splitting blurs at a driving frequency of ν m ∼ 150 − 300 kHz; we speculate that this The intensity through the bottom coupler is slightly larger because the second-order grating has four periods, whereas the top grating only has three. (c) When a 10µs square-wave voltage with amplitude 3V is applied across the cavity, the cavity refractive index shifts by both free carrier injection and temperature change, which are estimated in this plot. (d) Expected electroluminescence when the cavity is pumped with the same 10µs long square-wave. The cavity first rapidly blue-shifts away from the cold-cavity resonance λ 0 due to free carrier injection, and then red-shifts over a longer time scale ∼ 5µs due to heating.
occurs because the cavity does not reach steady-state temperature during each pump period. Above 300 kHz, the blurred feature narrows into a single peak that is red-shifted by a constant ∆λ = 0.4 nm from λ 0 . At this range of modulation frequencies, the temperature fluctuations decrease as the modulation is faster than the thermal response time, and the cavity remains at a constant temperature-induced offset. Although we expect there to be a shift in the resonance due to free carrier and/or band-filling effects, this shift is considerably smaller than the cavity linewidth and could not be verified even at the highest modulation amplitude of 4V.

B. Optical modulation through the photonic crystal cavity
We will now operate the cavity as a switchable drop filter. As illustrated in Fig.5(a), an external laser is coupled through the input grating into the waveguide. If it is on resonance with the cavity frequency, it is transmitted to the output waveguide and scattered by the grating towards the objective lens. To reduce stray light, we observe the transmission in the crossed polarization: the input beam is polarized at 45 • to the waveguide, while the output is observed at −45 • . The output is also spatially filtered using the pinhole to select the output grating. Fig.5(c) shows the transmission observed on the spectrometer when the signal beam's wavelength λ s = 1351 nm, which is red-detuned 1 nm from the cavity frequency at zero bias field. We also measured the transmission when the laser was blue-detuned by ∼ 0.5 nm from the zero-bias cavity wavelength. Because the transmitted signal intensity was small due to low coupling through the input grating, it was necessary to measure the transmitted amplitude using a lock-in amplifier. This limited the measurement to the lock-in amplifier's cut-off frequency of 100 kHz. Fig.5(b) shows the modulation amplitude observed on the lock-in amplifier. In addition to the lock-in amplifier restriction, the modulation rate is limited by the frequency-dependent thermal effects of the device, as discussed above. The thermal stability could be greatly improved in future experiments by placing the cavities on top of a low-index substrate such as sapphire or silicon dioxide for improved thermal conductivity [21,22,23,24], or by using line coding schemes [25]. The thermal stabilization in the PL under modulation exceeding 200 kHz suggests operating beyond this frequency with a faster detection technique. To estimate the ultimate modulation speed, we measured the RC time constant. The capacitance was directly measured to be only 3 pF, while the resistance was estimated at 1.1 kΩ from the forward bias part of the I-V curve. The RC time constant of ∼ 3ns suggests that a fast modulation speed is possible.

IV. CONCLUSIONS
In conclusion, we have demonstrated a modulator that relies on a photonic crystal nanocavity as the active component. The small size allows for a low capacitance and promises operation at high bandwidth. While we measured and out-coupled through through the bottom grating, which is selected with a pinhole. (b) The signal is modulated at vm and measured using lock-in detection, giving voltage V LI . We measured V LI for λs red-detuned and blue-detuned from the zero-voltage cavity frequency.
(c) Cavity transmission measured in time on a spectrometer.
1/RC ∼ 300MHz, the modulation speed in future designs could be increased significantly by changing the refractive index only in the photonic crystal cavity whose area is less than 100 times smaller than the full membrane in this study. Lateral dopant implantation could allow a small junction with sub-fF capacitance and a time constant of RC < 10 ps [26]. The frequency-selective modulation of the cavity is suited for wavelength division multiplexing, which greatly increases the total interconnect bandwidth and may become essential in off-chip optical interconnects [1]. Electrically controlled photonic crystal networks furthermore have promise in applications including biochemical sensing [27,28] and quantum information processing in on-chip photonic networks [13,29].

A. Acknowledgements
This work was supported by the MARCO Interconnect Focus Center and the and DARPA Young Faculty Award.

APPENDIX A: SAMPLE GROWTH
The sample is grown by molecular beam epitaxy on an n-type GaAs substrate and consists of a 1 µm n-doped Al 0.8 Ga 0.2 As sacrificial layer, a 40 nm n-doped GaAs layer, a 160 nm intrinsic GaAs membrane that contains three layers of InAs quantum dots (QDs) separated by 50 nm GaAs spacers, a 25 nm p-doped GaAs layer and a 15 nm highly p-doped GaAs layer to ensure low-resistance contacts. The QD layers were grown by depositing 2.8 monolayers (ML) of InAs at 510 • C at a growth rate of 0.05 ML/s. To achieve emission at ∼ 1.3 µm, the dots were capped with a 6 nm In 0.15 Ga 0.85 As strain-relaxing layer.

APPENDIX B: FABRICATION
First, metal contacts are deposited on the n-type substrate and annealed to form the bottom contacts. These contacts consist of Au/Ge/Ni/Au layers. Then the photonic crystal structure, isolating layer, trenches, and the topcontact PMMA mask are fabricated by a combination of electron beam lithography and dry/wet etching steps, which are outlined in Ref. [15]. The p-contact consists of Pt/Ti/Pt/Au layers. The free-standing membrane is created by removing the sacrificial AlGaAs layer using a hydrofluoric acid-based selective wet etch.

APPENDIX C: INDEX DEPENDENCE ON CARRIERS AND TEMPERATURE
The primary sources of refractive index change in the GaAs membrane are bandgap narrowing, bandgap filling, free carrier effects, and temperature effects. We neglect band-gap narrowing as it only changes slowly with carrier density at high carrier densities and is far weaker than band-filling effects at 1350 nm [30]. Because the carrier effects are independent, we can express ∆n r as the sum due to band filling, free carrier effects, and thermal changes, respectively: ∆n(n, p, T ) = ∆n(n, p) BF + ∆n(n, p) F C + ∆n(T ) (C1) We follow the discussion in Ref. [30] to estimate the carrier contributions ∆n(n, p) BF +∆n(n, p) F C . Using fundamental constants [30], a DC refractive index n g = 3.6, normalized electron and hole masses m e = 0.066, m h = 0.45, and assuming equal ∆n and ∆p, ∆n r (n, p) BF ≈ −2 × 10 −21 (∆n + ∆p) (C2) We note also that larger refractive index changes are possible if working with photon energies nearer to the band edge [31] or by exploiting the index change in quantum confined structures [32]. The temperature dependence of the refractive index follows where T 0 =300K is the initial temperature and T is the operating temperature. We chose n r =3.410 for GaAs at λ = 1.3µm T=300 • C, after [20]. α is the thermo-optic constant, which we take to be 2.5 × 10 −4 / • C as an interpolation between Refs. [17,18] for λ = 1.35µm. Therefore, the total expected change in refractive index is Let a, b, c, d, e represent, respectively, the input grating; first waveguide; cavity; second waveguide; and output grating. With κ g , κ w,g , κ w,c , κ representing, respectively, the vertical coupling rate through the grating; the cavitygrating coupling rate; the cavity-waveguide coupling rate, and the cavity loss rate, we have where p(t) is the pump rate. We solve this system of equations in the steady state, since the coupling rates are much faster than changes in p(t). Then we can solve for the transfer efficiency to the two grating couplers as η c = 2(κ g a) 2 /(2(κ g a) 2 + (κc) 2 ) = (1 + (κκ w,g / √ 2κ g κ w,c ) 2 ) −1 . Assuming negligible loss the waveguide, we can estimate from the measured intensities in Fig.4(b) that η c ≈ 0.4 and κκ w,g /κ g κ w,c = 1.8. Assuming κ/κ w,c ∼ 0.85 from simulation, we estimate κ g /κ w,g ∼ 0.5. An increase in κ w,g , which could be achieved by fabricating larger grating couplers, is therefore expected to improve the coupling efficiency to the input/output ports.