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We perform time-domain measurements of optical transport dynamics in silicon nano-photonic devices. Using pulsed optical excitation the thermal and carrier induced optical nonlinearities of micro-ring resonators are investigated, allowing for identification of their individual contributions. Under pulsed excitation build-up of free carriers and heat in the waveguides leads to a beating oscillation of the cavity resonance frequency. When employing a burst of pulse trains shorter than the carrier life-time, the slower heating effect can be separated from the faster carrier effect. Our scheme provides a convenient way to thermally stabilize optical resonators for high-power time-domain applications and nonlinear optical conversion.
©2010 Optical Society of America
1. Introduction
Integrated optical components based on silicon nano-structures are promising for a multitude of applications, in particular with respect to telecommunication [1,2]. For the most part silicon photonic devices are fabricated from silicon-on-insulator (SOI) substrates, which comprise a thin silicon top layer sitting on top of a buried oxide cladding layer. Due to the large refractive index contrast between silicon and the underlying oxide, light in the telecom spectral window with wavelengths around 1550nm can be tightly confined into sub-micrometer waveguides. The strong confinement cannot only lead to densely packed photonic structures, but also provides strong field enhancement within the waveguide. As a result the energy density within the waveguide may be very large and thus optical nonlinear phenomena can be observed even with moderate optical input power [1–3].
In silicon several nonlinear processes play together to form a complex picture of nonlinear interactions. Light can be absorbed through two-photon absorption (TPA), leading to the generation of free carriers in the waveguide [4–8]. The free carriers lead to the modification of the refractive index of silicon through free-carrier dispersion (FCD) and further light absorption through free-carrier absorption (FCA) [9]. In addition, strong light intensities lead to direct modification of the refractive index through the optical Kerr non-linearity which is two orders of magnitude larger than in silica fibers [10]. Furthermore, the heat generated in the waveguide leads to yet another modification of the refractive index through the thermo-optical effect [11,12]. These features enable efficient nonlinear interaction of optical waves at relatively low input power levels inside waveguide only a few centimeters in length. In consequence, considerable effort has been dedicated towards the investigation of nonlinear phenomena such as self-phase modulation (SPM) [13–15], cross-phase modulation (XPM) [16–18] and four-wave mixing (FWM) [19–22]. These nonlinear effects are being exploited to realize a variety of optical functions in integrated optical devices. The interplay among the various dispersive and nonlinear effects leads to many interesting features that provide new functionalities, but may also induce unwanted side-effects. Therefore it is important to separate these effects with experimental methods that cannot only be used for understanding the underlying physics but also provide guidance for the design of realistic devices [23,24].
Because the nonlinear phenomena are governed by different characteristic time constants, they can be distinguished in a time-domain framework [25,26]. Nonlinearities due to carrier effects occur on a fast time-scale on the order of nano-seconds or even picoseconds. Thermally induced nonlinearities on the other hand occur on a microsecond scale in bulk optical materials. When nano-photonic elements are designed for high thermal speed this may be reduced to a tens of nanoseconds time scale, depending on the material parameters and the waveguide geometry [27,28]. Thus by performing time-domain analysis of nanophotonic waveguides the effect of individual processes can be analyzed and understood [29–32].
In this article we report on the investigation of SOI nanophotonic components under pulsed optical time-domain excitation. Using a wavelength tunable optical pulse source with tailored pulse width and repetition rate we study free-carrier and thermo-optical effects. The free carrier life-time is first extracted in long spiral waveguides. We then apply our measuring scheme to micro-ring resonators. Besides introducing free carrier effects, the short pulses induce photo-thermal shift of the cavity resonance on a nano-second scale. Therefore stroboscopic measurements are designed to sample the thermal shift of the ring optical resonances. By using repetitive pulse trains (“bursts”) we are able to steadily position the cavity resonance at a chosen wavelength. In this stabilized regime free-carrier and photo-thermal effects are separately sampled. The experimental observations are supported by numerical modeling based on coupled mode equations, taking into account the thermal and free-carrier dynamics in a time-domain framework. Close agreement between theory and experiment allow us to extract the time constants governing the different nonlinear processes, which are also confirmed from independent experimental measurements.
2. Fabrication and device design
We fabricate our nano-photonic circuits from standard silicon-on-insulator (SOI) wafers (smart-cut by Soitec). The thickness of the top silicon layer is thinned down to 110nm. The waveguiding layer sits on top of a buried oxide layer of 3μm thickness.Photonic circuitry as shown in Fig. 1 is defined by electron-beam lithography and subsequent reactive ion etching using inductively coupled chlorine plasma.
Fig. 1 Device layout: (a) A long silicon nanowire is arranged in a spiral pattern to preserve chip area. The waveguide has total length of 1cm using and bend radius of 30μm. (b) An optical micrograph of a fabricated photonic circuit including optical grating couplers, waveguides and a ring resonator.
For the measurement of the free carrier dynamics we pattern long nano-photonic waveguides. In order to reduce chip area the waveguides are fabricated in a spiral pattern as shown in Fig. 1(a). The total length of the device is designed to be roughly 1cm. The radius of the spiral is chosen to be 30μm in order to reduce bending loss at each turn of the spiral. Each fabricated device then covers less than 300μm2 of chip area. For convenient optical input and output on-chip grating couplers are used with a typical coupling loss of 7dB. The couplers feature a usable optical bandwidth of 30nm in the L- or C telecom bands.
For the investigation of the competition between thermal and free-carrier effects we design optical ring resonators with a radius of 132μm. Given a simulated and measured group index of 3.6, the resonators feature a free-spectral range of 0.8nm, which is closely aligned to the channel spacing of standard DWDM systems in the telecom C-band. The waveguides have an optimized waveguide width of 600nm for low propagation loss. A typical fabricated device is presented in Fig. 1(b), showing the on-chip grating couplers, waveguides and the ring resonator. The resonator has a loaded optical quality factor of 26,500 under critical coupling conditions. In this case the extinction ratio is almost 20dB. The resonator can be readout from four grating coupler terminals, where the central two ports allow for transmission measurement in the through port and the outer two ports provide measurement capability in the drop port. A four-port design is chosen in order to provide calibration capability for the grating couplers and ring resonator performance.
3. Time-domain measurement implementation
In order to characterize the fabricated sample described in the previous section, we employ the measurement setup shown schematically in Fig. 2 . The setup includes a flexible pulsed optical source, in which the optical wavelength, the pulse repetition rate and the pulse width can be chosen arbitrarily.
Fig. 2 The measurement setup used for the time-domain analysis. Wavelength tunable optical pulse of variable width and repetition rate are generated using a tunable laser source combined with an electro-optical modulator and an electrical pulse generator. The sample under test is mounted in a vacuum chamber on a temperature stabilized stage. The optical signal is readout using an optical oscilloscope.
As shown in Fig. 2, a wavelength tunable laser diode provides the input optical signal and is launched into an optical fiber. The optical input is modulated with a high-speed electro-optical modulator, driven by a 3.3GHz pulse generator, through which the pulse width and repetition rate are programmed.
The modulated signal is amplified with an EDFA pre-amplifier providing 28dB of amplification. The signal is the further amplified by another 33dB using an optical power amplifier. In order to suppress the ASE background from the amplifier stages the modulated signal is cleaned spectrally with two cascaded DWDM filters, which provide better than 60dB filter extinction. The modulated signal is launched into the nanophotonic device through grating couplers. The sample is mounted on a temperature stabilized positioning stage in a vacuum chamber. Under vacuum conditions with 10−4mTorr the temperature stability is better than 1mK. After passing through the device, the transmitted optical signal is detected by a fast optical oscilloscope (HP83340).
The 100GHz DWDM filter has a typical 3dB bandwidth of 0.5nm. Within the filter window the pump wavelength can be fine tuned by adjusting the wavelength of the pump diode laser. This fine-tuning capability is particularly important when aligning the pulsed source to high-Q optical resonators.
4. Experimental results
The above setup allows us to obtain the dynamic response of nanophotonic devices in the time-domain. Here we are interested in the interplay between free-carrier induced and temperature-induced optical non-linearity. By measuring the dynamic response of both long waveguides and optical ring resonators, we are able to distinguish the effects of carrier-induced and temperature-induced effects in silicon waveguide structures.
4.1 Free-carrier dynamics in long waveguides
We first investigate the spiral waveguide device shown in Fig. 1(a). Because the waveguide has a total length of 1cm, the cumulative nonlinear effects are sufficiently strong to be detectable in the transmittal signal. We employ a strong pulse of 10ns length with peak power of up to 650μW. The wavelength of the pulse is centered at 1554.134nm at one of the DWDM channels in the 100GHz grid.
At low optical input powers the transmitted optical signal is not significantly affected by the free-carrier effects. When the optical power within the pulse is sufficiently high, carriers generated from two-photon absorption lead to extra loss in addition to the propagation loss in the waveguide as shown in Fig. 3(a) .
Fig. 3 (a) The time-domain response of the spiral waveguide shown in Fig. 1(a). The total length of the waveguide is 1cm in order to allow optical nonlinear effects to manifest. The waveguide is excited with pulses of 10ns length with a pulse power of up to 650μW. The repetition rate is set to 100kHz. For high pulse powers free-carrier absorption during the initial period of the pulse limits the peak pulse power in the stabilized regime. (b) Zoom-in from (a) in the high-power regime. From the fit the free-carrier life-time in the waveguide is estimated to be ~1.9ns.
The pulse front is not exposed to the absorption effect, only the pulse body sees the carrier absorption. The pulse profile changes from a rectangular shape to a nonlinear response due to free-carrier effects, when the pulse power is increased. Therefore the overall profile of the pulse provides a time-domain representation of the carrier profile along the waveguide. Upon further increase of pulse power, the free-carrier induced loss becomes significant and the transmitted pulse power saturates below the initial peak power. The time-domain response allows us to extract the free-carrier life-time. From the zoom-in of the higher power regime in Fig. 3(b) it is obvious that the carriers build up with in the initial period of the pulse. From the fit of the exponential power decay we can extract the free-carrier life-time as ~1.9ns. The extracted free-carrier life-time provides the time-constant for the subsequent measurements of optical resonators and is also used in the time-domain modeling described in the next section.
4.2 Measurement of ring resonators with short pulses
Having obtained the free-carrier life-time of the typical waveguides, we investigate the optical transport dynamics in microring resonators, using devices as shown in Fig. 1(b). The measured spectral response, in both through and drop ports of the ring resonator is shown in Fig. 4(a) and 4(b). In the DWDM window of interest we obtain the designed resonances with a FSR of 0.8nm. In order to overlap the resonances with the optical filters we slightly heat the sample stage to 304K. From the measurements in the through and drop port we find good extinction of roughly 20dB, illustrating that the ring resonator is indeed critically coupled to the bus waveguide. The quality factor of the ring is obtained by fitting the resonance to a Lorentzian as shown in Fig. 4(b), revealing an optical Q of 26,500. From Fig. 4(b) we also find that the insertion loss into the drop port is small, confirming the critical coupling condition.
Fig. 4 (a) The spectral response of the ring resonator from Fig. 1(b) in the DWDM regime. The radius of the resonator is chosen such that the free-spectral range coincides with the DWDM channels. The sample is heated to 304K in order to position the resonator in close vicinity of the filter channels. Shown are the measured transmission spectra in the through port (red) and the drop port (black). The ring resonator is close to critically coupled with an extinction ratio of almost 20dB. (b) Zoom into one resonance close to DWDM channel 30 at 1553.329nm (linear scale). The fitted optical Q is 26,500. Low insertion loss into the drop port is achieved, illustrating that the ring is almost critically coupled to the bus waveguide.
Using the above device we perform the single-pulse time-domain measurements to investigate the carrier and thermal dynamics within the ring resonator. The measured results are presented in Fig. 5(a) –5(c).
Fig. 5 (a) The time-domain response of the ring resonator shown in Fig. 1(b). The waveguide is excited with pulses of 40ns length with a pulse power of 300μW. Shown is the response in dependence of wavelength and time. The blue regions correspond to low optical transmission when the pulse wavelength overlaps with the ring resonance. (b) A cut through the measured profile of for a time delay of 27ns showing the optical resonance dip of the ring resonator. (c) The time-wavelength response of the ring resonator at an optical input power of 650μW, close to the ring resonance. The measured response is shown in dependence of wavelength, showing the thermal resonance tuning of the ring resonator. The generation of free-carriers leads to a rapid change of the resonance frequency which is then compensated by thermal heating.
In the first experiment we determine the pulse response of the resonator in dependence of the input wavelength around 1560.3nm. The time-domain dynamic response of the ring is then investigated within the passband of the DWDM filter (roughly 0.5nm). In addition to the pulse wavelength the pulse power is adjusted by increasing the driving current of the EDFA. In Fig. 5(a) we show the result for the through port with low input pulse power of 300μW. The pulse width is set to 40ns and the pulse repetition rate to 100kHz, corresponding to a pulse period of 10μs. Therefore the pulse duty cycle is very small (0.4%), which implies that the pulses arrive with sufficient time interval for the device to reset in both carrier-density and temperature.
At low pulse power, we find that the resonance wavelength is not affected significantly when the pulse wavelength is scanned through the resonance. When the resonant wavelength is reached all optical light is filtered by the ring and the transmission of the device is low. This corresponds to the dark blue valley in Fig. 5(a), signifying minimum optical transmission. When the pulse wavelength is tuned far out of the resonance, the transmission is high which is shown as the red areas (high transmission regime) in Fig. 5(a). The time-domain snapshot thus reproduces the optical response of the resonator as shown in Fig. 5(b). Here we plot the cross-section at a fixed delay of 27ns in dependence of wavelength, as indicated by the dashed green line in Fig. 5(a).The situation changes when the pulse power is increased. Depending on the initial wavelength detuning condition of the pulse with respect to the ring resonance, the free carrier and thermal effects compete and dynamically change the resonance wavelength during the duration of the pulse. In Fig. 5(c), we show the measurement results with input power of 650 µW and varying wavelengths.
When the pulse wavelength is initially blue-detuned, as shown in the upper traces, a smooth dip is observed in the transmitted pulse. The first drop in power can be attributed to free carrier dispersion (FCD) which blue-shifts the resonance of the ring and thus reduces the transmission of the device. With reduced detuning, the circulating power inside the ring increases and the temperature of the device rises. Since silicon has a positive thermo-optical coefficient of 1.86 × 10−4/K, the resonance of the ring is red-shifted and the transmission recovers to an even higher level. As the blue-detuning of pulse wavelength is further reduced, the free carrier effect induced drop in pulse transmission becomes increasingly more abrupt and appears as a spike. Because the optical power inside the ring is high at the beginning of the pulse, a large amount of free carriers is excited and free carrier absorption (FCA) also plays a significant role.
When the pulse wavelength is tuned to the ring resonance, shown in the middle traces, the power inside the ring is maximal and the competing FCD and FCA are both strong, resulting in a slower blue-shift of the resonance and rise in transmitted pulse power at the beginning. Later as the free carriers decays, the photo-thermal effect dominates and red-shifts the resonance. Since the resonance shifts past the pulse wavelength, the transmitted pulse power first drops and then rises again as the temperature of the ring increases.
When the pulse wavelength is far red-detuned from the ring resonance, shown in the lower traces, only the thermal effect is dominant. From above results it is apparent that the thermal shift of the resonance occurs on a 10s of nanosecond time-scale. When the pulse wavelength is moved further beyond the ring resonance the transmission through the ring is unperturbed as in the situation with lower pulse power excitation.
4.3 Time-domain measurement of ring resonators with long pulses
During the excitation with short pulses the ring resonator cannot reach thermal equilibrium because the thermal time-constant is larger than the pulse width. Therefore in a further experiment we investigate the temporal behavior when the ring is excited with long pulses in order to allow the ring resonator to reach thermo-optical equilibrium. Here we use pulses of 900ns length at a fixed repetition period of 10μs. We exploit a resonance which is tuned thermally to a DWDM wavelength of 1553.9nm. The pulse wavelength is fixed at a red detuned wavelength of 1554.08nm while the pulse power is gradually increased. The results are presented in Fig. 6 for amplifier currents varying from 900mA to 1400mA, corresponding to an optical input power from 280μW to 560μW. When the pulse power is relatively low, we observe the complete thermal resonance shift. Due to the thermal shift the resonance is moved beyond the pulse wavelength during the pulse. Additional free-carrier absorption and two-photon absorption causes the final pulse amplitude to be lower than the initial pulse amplitude.
Fig. 6 Shown is the time-domain response of the ring resonator under excitation with long single pulses. The repetition period is kept at 10μs while the amplifier current is varied from 900mA to 1400mA. At lowest pulse power the thermal shift of the resonance during the pulse is apparent. When the pulse power is increased, competition between free-carrier absorption and thermal heating leads to oscillations during the pulse train. The oscillation amplitude increases with increasing pulse power. Simultaneously the onset of the oscillations shifts towards the beginning of the pulse, which is a signature of increased free-carrier effects. Inset: The origin of the oscillations is shown schematically. Thermal drift shifts the resonance wavelength to the right to a blue detuned wavelength. FCD leads to a backward shift and resulting oscillating thermal drift due to power modulation when the resonance passes through the pulse wavelength.
When the pulse power is increased we observe oscillatory behavior after the thermal shift moved the ring resonance beyond the pulse wavelength. The oscillations occur on the blue side of the resonance. This behavior is schematically illustrated in the inset in Fig. 6, showing the thermal shift by the dashed red line and the oscillation cycle by the green dashed oval. The oscillations are due to the competing refractive index changes of the free-carrier effects and the photo-thermal effect. When the pulse power is further increased the amplitude of the oscillations increases as well. Furthermore it is notable that the free-carrier effects are enhanced, which leads to an earlier onset of the oscillations after the resonance has passed the pulse wavelength. It is also notable that the total duration of the oscillations reduces with increasing pulse power, since at higher power the competition between free-carrier effects and the photo-thermal effects takes less time to settle to a steady state.
At higher pulse powers two time-constants can be identified during the oscillatory behavior: a fast scale, which leads to the sharp dips in the transmission profile and a slower thermal relaxation. Thus in this time domain measurement with high pulse power, the interplay between free-carrier dispersion and the thermo-optical refractive index change can be directly observed.
4.4 Burst excitation
In addition to single pulse excitation we consider the resonator response under burst excitation. In this configuration, trains of pulses are sent into the device. The burst repetition rate is fixed at 10kHz (100μs train period). In burst mode, the delay between the individual pulses, the pulse width and the number of burst pulses are programmed. After the resonator device is excited with repeated pulses, the photonic device can be stabilized at a given temperature and carrier density. Because the thermal response of the resonator is governed by a longer time-constant the burst trains need to be sufficiently long to allow the photonic device to reach thermal equilibrium. A particular attractive feature of the burst excitation is the avoidance of accumulated free-carrier effects. As mentioned in the previous section the free-carrier dynamics occur on a nanosecond scale, whereas the thermal response happens on a longer time-scale. By using short burst pulses it is thus possible to avoid accumulating free carriers while only sampling the overall thermal drift profile of the photonic device.
This excitation scheme is investigated in the following. We employ a burst train of 40μs total length, in which the pulse period is varied. The width of the individual pulses is set to 1ns, shorter than the free-carrier life time. The temporal response of the resonator to the burst is shown in Fig. 7 , for both the through port and the drop port. The pulse wavelength is tuned close to the resonator wavelength. Since the pulse width is set to be shorter than the free-carrier life-time, the free-carrier density can be approximated to be constant within each pulse. During the pulse train the ring resonator is heated and the resonance is red shifted away from the pulse wavelength. As a result the transmitted burst amplitude profile is modulated by the thermal drift. This leads to an amplitude increase in the through port [Fig. 7(a)] and an amplitude decrease in the drop port [Fig. 7(b)]. The shape of the individual pulses is shown in Fig. 7(c) and 7(d) for both ports.
Fig. 7 (a) Shown is the ring resonator response under burst excitation in the through port. The pulse wavelength is tuned to the optical resonance, therefore the pulse amplitude increases as the resonator heats up during the pulse train and the resonance wavelength is shifted. (b) The complementary response of the ring in the drop port. Here the amplitude decreases as the resonance wavelength is shifted away from the pulse wavelength. (c) A zoom into the pulse train showing the pulse profile of the individual pulses in the through port. (d) The equivalent measurement of the pulse profile in the drop port.
The thermal drift of the resonance is usually not a desirable property, because it prevents stable operation of the ring resonator under pulsed excitation. Fortunately, the free carrier dispersion and thermal optical dispersion have opposite sign. Therefore it is possible to null the thermal dispersion by modifying the relative weight of the free-carrier dispersion through the adjustment of the burst train duty cycle, either by reducing the pulse width or increasing the pulse delay within each pulse train. In our proof of principle demonstration, we adjust the thermal heating by varying the pulse period (Fig. 8 ) while the pulse width is set by the free-carrier life-time.
Fig. 8 The response of the drop port of the ring resonator under burst excitation with trains of 40μs length. The width of the individual pulses is set to 1ns. The wavelength of the pulse is tuned into the resonance. The pulse period is tuned from 5ns to 200ns from top to bottom. When the duty cycle is high, thermal drift of the resonance leads to reduced pulse amplitudes towards the end of the pulse train. When the duty cycle is reduced, the temperature of the ring resonator is stabilized and the pulse amplitude remains constant.
We choose a fixed pulse width of 1ns at constant input power while the pulse duty cycle is varied. The heating effect causes an increase in the refractive index, which corresponds to a red-shift of the resonance curve. In contrast, the free-carrier dispersion produces a blue-shift of the resonance curve. Because the pulse width is constant the same amount of free carriers is generated during each pulse cycle and as a result the FCD is kept fixed. This implies that the resonance blue shift due to FCD provides a fixed resonance offset during each pulse cycle. The competing effect of photo-thermal resonance red shift is used to counteract the blue shift. When the duty cycle is high and therefore high thermal heating occurs, the photo-thermal red shift over-compensates the FCD blue shift. When the duty cycle is reduced, the thermal power introduced into the ring per period is also reduced, thus the overall thermal heating is reduced. It is thus possible to obtain an equilibrium position, in which the ring resonance is fixed with the whole burst duration. This situation is investigated in Fig. 8. Due to the high number of data points, only the envelope of the burst train can be shown. Therefore we add the profile of the individual pulses in the inset, for three cases at 5ns, 10n and 50ns pulse period. The pulse wavelength is tuned into the ring resonance so the transmission in the through port is minimal. For clarity, we show the response of the resonator in the drop port. When the pulse density is high (high duty cycle), the ring resonator experiences thermal heating and therefore the resonator drifts out of the pulse wavelength. As shown in the top scans in Fig. 8, the transmitted pulse amplitude decays with time at a rate related to the effective thermal time constant of the ring. When the duty cycle is reduced the thermal drift is increasingly reduced and therefore the resonance wavelength as well as the overall transmitted pulse amplitude remains stable, as shown in the lower curves in Fig. 8. In this way, we demonstrate stabilization of the device performance through time-domain programmed burst patterns to balance two dispersion effects.
4. Theory and modeling
In order to model the measurement results we employ a coupled-mode theory framework, including both thermal and free-carrier time-domain effects. Thermal heating increases the refractive index of silicon; free-carriers generated by two-photon absorption (TPA) introduce FCD and FCA, affecting both the refractive index of the waveguide and the circulating optical power in the ring resonator.
Starting with the coupled-mode equations for the ring-resonator, we use the following equation to model the circulating fields in the ring
where a is the mode amplitude, γ is the loss rate for the mode in the resonator, κ is the coupling rate into the ring and s is the amplitude of the driving optical field. Δω is the detuning from the resonance frequency of the ring. The energy U stored in the ring is then given by . The optical loss inside the ring resonator is composed of linear propagation loss, the loss due to two-photon absorption and the loss due to free-carrier absorption. The total loss is given as . According to reference [31], the loss due to TPA is given as , where is the TPA absorption coefficient. Similarly, the loss due to FCA is given as, where is the FCA absorption coefficient and N(t) is the time-dependent carrier density. Absorbing both effects into one expression, the total power absorbed in the ring resonator is given as .Accompanying the equation for the mode amplitude are time-dependent differential equations for the temperature rise in the waveguide T(t), with respect to the equilibrium temperature when the optical power is turned off. The governing differential equation is then given as
where is the temperature decay time and is the thermal absorption rate. In addition, we have to consider the temporal behavior of the free-carriers governed by the differential equationwhere is the FCA decay rate and is the FCA absorption rate. Combining the above Eqs. (1)–(3), the detuning of the ring resonator from the cavity resonance frequency is then given asIn Eq. (4) is the thermo-optical coefficient of silicon and is the coefficient describing the free carrier-induced change of the refractive index of silicon. is the initial detuning of the driving laser wavelength, with respect to the ring resonance.
Using the framework outlined above, the measured time-domain response of the ring resonator can be analyzed. The simulation is driven by a time-dependent input field s(t), which is set to comply with the pulse shape chosen in the actual measurements. We employ a predictor-corrector time integration scheme to solve the above system of coupled equations. The thermal time constants of the free-carriers and the temperature decay are used as fitting parameters and are compared to the measured results from the previous section. The material parameters passed to the time-domain solver are taken from reference [31].
The simulated results for amplitude, temperature and free-carrier density are shown in Fig. 9 . We show the simulated response of the ring resonator to pulsed excitation with a pulse period of 7ns and a pulse width of 1ns. The response is obtained for a burst of 350ns length, for comparison with the measured results shown in Fig. 7. In Fig. 9(a) the simulated amplitude response to the driving pulses is shown. The response is measured in the through port of the ring resonator when the pulse wavelength is tuned into the ring resonance. Clearly visible is the exponential increase of the pulse amplitude, which corresponds to the behavior observed experimentally in Fig. 7. Because the ring resonator is heated up due to TPA and FCA, the resonance wavelength shifts away from the driving pulse wavelength and therefore the transmitted amplitude increases. When looking at the ring temperature in Fig. 9(b) this behavior is confirmed by the identical time-constant. During each pulse the ring temperature fluctuates only slightly. Because the pulse width is on the order of the free-carrier life-time and the dead time is larger than the free-carrier life-time, the same amount of carriers is accumulated in each pulse. The carriers recombine almost completely during the dead time between pulses as shown in Fig. 9(c). The carriers take only a few cycles to settle into a steady-state pulse pattern. This is illustrated in the inset in Fig. 9(c), showing the initial four periods of the burst. As a result we confirm the experimental observation that the long term fluctuation of the resonance wavelength can be prevented by stabilizing the thermal heating rate.
Fig. 9 (a) The simulated amplitude response of a ring resonator over 350ns in the through port. The pulsed driving input field is shown by the light blue profile. The input wavelength is tuned into the cavity resonance. Due to thermal heating the resonance shifts away from the pulse wavelength and thus the pulse amplitude increases. (b) The simulated temperature response of the device. The overall temperature increases during the pulse duration and drops only slightly during the dead time. (c) The simulated free-carrier density inside the ring resonator, showing free-carrier stabilization after 3 cycles. Inset: the first through cycles of the burst showing the free-carriers settling into steady-state amplitude. After three cycles the carriers have achieved equilibrium dynamics.
In order to further validate the numerical model we consider the response of the ring to longer pulses of 40ns length (pulse period: 80ns). The simulated results in the through port are compared with measured results as shown in Fig. 10(a) and 10(b). We consider 4 pulses in the stabilized regime of a burst, when the temperature of the ring has reached equilibrium. The pulse wavelength is slightly blue detuned from the ring resonance.
Fig. 10 (a) Shown is the simulated ring response in the through port to excitation with pulses of 40ns length and a period of 80ns. The resonator is simulated in a regime, where the temperature drift has stabilized. The driving wavelength is slightly red detuned from the cavity resonance. (b) The measured response of the ring resonator, agreeing well with the modeled behavior.
Rapid generation of free-carriers in the initial stage of the pulse leads to strong FCD and FCA. Because the pulse wavelength is initially blue-detuned, FCD and FCA combined to make a sharp drop of the transmitted pulse power. Later, the ring temperature rises and the photo-thermal effect dominates, causing a red-shift of the resonance and restoring the transmitted pulse power. As shown in Fig. 10(b) the measured response agrees well with the modeled prediction. From the response which best corresponds to the measured date we extract a free-carrier life-time of 2.2ns and a thermal time-constant of 50ns, which is in good agreement with the numbers extracted in the previous sections.
In a final simulation we compare the response of the ring resonator to excitation with a long pulse of 450ns length as shown in Fig. 11 . From the measured result shown in Fig. 11(a) we find oscillatory behavior as described in the experimental section due to the competing resonance shifts caused by thermal heating and FCD. In the experiment, the driving wavelength is again red-detuned from the cavity resonance. Initial thermal drift shifts the resonance shifts to the driving wavelength, so the power inside the ring increases rapidly, generating free-carriers which blue-shift the resonance. This process repeats in an oscillatory way due to the competition between the two effects. This oscillatory phenomenon is qualitatively well reproduced by the simulation as shown in Fig. 11(b).
Fig. 11 (a) The measured profile for the excitation of a ring resonator with a 450ns long pulse. The oscillations are measured in the through port. (b) The corresponding simulated result. Good qualitative agreement is observed between the measurement and the simulation.
5. Conclusion
In conclusion we have presented time-domain measurements of the thermal and free-carrier dynamics in silicon ring resonators and long silicon waveguides. A wavelength tunable pulsed source allows us to investigate the interplay between waveguide heating and free-carrier generation in single pulse or burst mode.
We find that during the pulse duration free-carriers are generated with a life-time of 1.9ns at high pulse power. From simulation results we conclude that the build-up of carriers occurs in the initial stage of the pulse, accompanied by arapid rise in temperature due to free-carrier absorption. Subsequently thermal heating leads to a refractive index change which moves the resonance frequency of ring resonators through the pulse wavelength within 10s of nanoseconds. Thermal equilibrium can be established by driving the photonic circuit in burst mode with trains of single pulses. This allows for temperature stabilization and therefore precise control of the position of the ring resonance.
Acknowledgements
This work was supported by a seedling program from the Defense Advanced Research Projects Agency DARPA/MTO and the DARPA/MTO ORCHID program through a grant from the Air Force Office of Scientific Research (AFOSR). W.H.P. Pernice would like to thank the Alexander-von-Humboldt foundation for providing a postdoctoral fellowship. H.X.T acknowledges support from a Packard Fellowship in Science and Engineering and a CAREER award from the National Science Foundation. The authors wish to thank Dr. Mike Rooks and Michael Power for assistance in device fabrication. The project makes use of Yale cleanroom and electron beam lithography facilities at Brookhaven national laboratory, which is supported by the U.S. Department of Energy, Office of Basic Energy Sciences, under Contract No. DE-AC02-98CH10886.
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