1. Introduction

Processing and transmission of high-speed data in optical domain is a promising approach to tackle the bandwidth and speed challenges of electronics [1–3]. Realization of complex optical signal processing functionalities seems possible more than any time before, thanks to the recent achievements in silicon photonics [4, 5] such as compact passive components (e.g., filters) [6], low-loss delay lines [7], and high-speed modulators and switches [8–10]. Reconfiguration of optical components is required in many of the signal processing applications and is usually achieved through the modification of the optical path length in different sections of a photonic device or circuit. Among different reconfiguration techniques in Si, reconfiguration through thermo-optic effect [11–21] and electro-optic effect (through free-carrier injection) [8–10, 22] have been the most widely used approaches because of their compatibility with complementary metal-oxide-semiconductor (CMOS) fabrication processes. While reconfiguration through free-carrier injection has a very fast response (less than 1 ns), the excess optical loss and the low tuning range limits its applicability to high-speed switching and modulation [8–10]. On the other hand, thermal reconfiguration has been more widely used for general signal processing applications [11–21] as it provides a larger tuning range with low power consumption without additional optical loss. However, because of the slow heat diffusion process in silicon-on-insulator (SOI) substrates this technique has a slow response time on the order of a few microseconds in the best optimized devices [12, 13]. This slow reconfiguration time is a limiting factor in many RF-photonics applications that require much faster response time [23]. At the same time, if the reconfiguration time can be improved to a few nanoseconds, thermal reconfiguration can be used for switching in network-on-chip (NoC) systems and will become an alternative to electro-optic approach [10].

In this work, we propose and demonstrate that direct pulsed excitation of the Si layer can be used to improve the thermal reconfiguration time to less than 100 nanoseconds. We theoretically show that this approach can also tackle the speed power consumption tradeoff that exists in conventional thermal tuning techniques. In other words, we show that it is possible to retain the reconfiguration speed of pulsed excitation scheme while increasing the thermal resistance of the device for lower power consumption. We experimentally demonstrate 85 ns reconfiguration time in 4 µm diameter microring resonators using the proposed reconfiguration technique. We achieve 2.06 nm/mW resonant wavelength shift in this device, which is on the order of the performance of devices that are optimized for low power operation with much slower reconfiguration time [11]. We also present a system-level model that can accurately describe the response of the demonstrated microheaters. We further use this model to derive an analytic solution for the excitation signal for reconfiguring the optical output in the minimum possible time without the need for a feedback loop. To show the viability of this technique for reconfiguration in opposite directions (e.g., OFF-to-ON and ON-to-OFF in a switch), we also demonstrate a differentially addressed optical switch with sub-100 nanosecond switching time using the proposed pulsed-excitation technique.

2. Theoretical study of heat transport and the modeling results

In this section, we first study heat transport in the more widely used microheaters on cladding architecture (Type I in Fig. 1(a)) in detail and explain the limitations of this device for improving the reconfiguration speed. Then, we theoretically show that by placing the microheater on the Si layer of the device (Type II in Fig. 1(a)), reconfiguration speed can be considerably improved thanks to the high thermal conductivity of Si. To be able to place the microheater directly on the Si layer without introducing optical loss, we will use microdisk resonators, in which the optical mode evanesces towards the center of these devices (see Fig. 1(b)) and the microheater can be directly placed in this area. Microdisk resonators are proven to enable device miniaturization while maintaining high quality factors (Q’s) with single mode operation in a large bandwidth [24]; and are great candidates for the design of low power reconfigurable devices. The small intrinsic bandwidth of these devices can also be overcome by using coupled resonators for wideband applications [25].

 

Fig. 1 (a) Schematic of the microdisk on SOI substrate with Type I and II microheater architectures. (b) H_z field profile of the first radial-order TE-like mode of a 5 µm diameter microdisk on SOI substrate. (c) The structure of the Type I and II device architectures (as shown in (a)) used in the simulation of heat transport. The actual values of the dimensions marked in this figure are tabulated in Table 1.

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Figure 1(a) shows the schematic view of a microdisk resonator coupled to a waveguide with the two microheater configurations considered in this study. Type I microheater which is more widely used [11, 13–15] is separated from the microdisk by an oxide cladding layer to prevent optical loss. The red curve in Fig. 2(a) shows the modeling result for the temperature rise in the microdisk (ΔT_d) at the location of the maximum of the optical mode (approximately 300 nm inside the outer edge of the microdisk) for 1 mW power dissipation in Type I microheater (Inset shows the steady-state distribution of temperature). Here, heat transport is modeled using finite-element method in a cylindrical coordinate system to account for the symmetry of the device [26]. The details of the numerical modeling technique and the thermal properties of materials are found in [13] and device dimensions (see Fig. 1(c)) are tabulated in Table 1. It is observed that the rise time (t_r) of the microdisk temperature is limited to a few microseconds (~4.1 µs) as a results of the low thermal conductivity of the buried oxide (BOX) layer. In this work, we define t_r as the time it takes the signal to grow from 10% to 90% of the steady-state. The low thermal conductivity of the BOX layer has been the main factor for the limitation of the thermal response time of almost all of the reconfigurable photonic devices on SOI to roughly 0.6 µs to 4 µs [27].

 

Fig. 2 (a) Modeling result of the step response of the microdisk temperature at the location of the maximum of the mode energy for 1 mW power dissipation in the microheater. The red and blue curves show the result for the Type I and Type II microheaters, respectively. The insets show the profile of the temperature at the cross-section of the two devices at stead-state. (b) Modeling result of the impulse response of the microdisk temperature for 1 nJ impulse dissipation. The inset shows the impulse response of the Type II microheater in the first 100 ns.

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Table 1. Dimensions of the structure in Fig. 1(c) used in simulations.

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A known technique to reduce the rise time beyond the step response of the device is through pulsed excitation (or pre-emphasis driving) [13, 19, 20]. In [13], we applied this technique to Type I microheaters to improve the rise time by a factor of 4 down to 1 µs. In [13], we also showed that this “forced” response time is limited to the delay that it takes the heat impulse to reach the optical mode in Si, which we refer to as heat-propagation delay (δ_HPD). The red curve in Fig. 2(b) shows the temperature rise in the Si microdisk for a 1 nJ impulse dissipated in Type I microheater. A δ_HPD = 840 ns is observed in this device. In heat conduction in a semi-infinite plane, it can be shown that δ_HPD is proportional to α⁻¹d², where α is the thermal diffusivity of the material and d is the distance of the heat source to the temperature measurement point (see Chap. 2 of [26] for analytic treatment of this problem). As the distance of the heater from the optical mode (d) cannot be reduced beyond approximately 1 µm because of added optical loss, the only viable solution for reducing δ_HPD is to use a material with higher thermal diffusivity. Si has a thermal diffusivity that is almost two orders of magnitude higher than that of SiO₂ [26]; and therefore, we expect that δ_HPD of Type II microheater to be much smaller than that of Type I device.

The blue curve in Fig. 2(b) shows the simulated impulse response of the microdisk temperature for Type II microheater architecture (impulse energy is 1 nJ). As expected from theoretical predictions, it is observed that Type II device has a much smaller δ_HPD = 19 ns as compared to 840 ns in Type I microheater. The lower δ_HPD is a great advantage of Type II device for achieving faster reconfiguration time compared to Type I device through pulsed excitation, as will be experimentally demonstrated in Section 3. It is also observed from the step response of Type II microheater (blue curve in Fig. 2(a)) that this device has 17% lower power consumption compared to Type I device (temperature rise is roughly 20% higher in Type II compared to Type I device), which is another advantage of this microheater. Direct heating of the Si layer was previously demonstrated and investigated in [12, 16–18]. In Section 5, we will also discuss that how pulse excitation enables to further reduce power consumption without affecting the reconfiguration speed.

3. Fabrication and experimental results

The performance of Type II microheater analyzed in the previous section is experimentally tested by implementing the microheater in a reconfigurable add-drop filter with a 5 µm diameter microdisk resonator. Figure 3(a) shows the scanning electron micrograph (SEM) of the device. The add-drop filter is fabricated on an SOI wafer with Si and BOX layer thickness of 230 nm and 1 µm, respectively. After the fabrication of photonic components on the Si layer using electron-beam (e-beam) lithography (EBL) and dry etching using Cl₂ chemistry, 1 µm SiO₂ is deposited using plasma-enhanced chemical vapor deposition (PECVD) as the top cladding layer (gas mixture: 500 sccm SiH₄, 500 sccm N₂O, 560 sccm He). Vias are then etched in the cladding at the center of the microdisk resonator for the placement of the microheaters. ZEP520A e-beam resist (Zeon) that is used for dry etching of the vias is reflowed for 30 minutes at 160°C on a hot plate. This results in a sidewall angle of roughly 60 degrees in the PECVD SiO₂ after dry etching (gas mixture: 15 sccm C₄F₈, 28 sccm CO₂, 5 sccm Ar) and assures that there is good electrical connection between the microheaters on the Si layer and signal connector lines on top of the cladding. Dry etching of vias is stopped at 50 nm above the surface of the Si layer, and the rest of the via is wet-etched using a buffered oxide etchant to avoid damaging of the surface of the Si by the plasma during dry etching. This is critical for having good heat conduction from the metallic heaters to the Si device. Then, microheater patterns are defined using EBL and e-beam evaporation. 100 nm NiCr and 80 nm Au are deposited for microheater and connectors/pads, respectively. In this design, the width of both bus and drop waveguides is 325 nm and the waveguide-resonator gap is 100 nm. This results in a filter bandwidth of approximately 0.5 nm. We used inverse tapers (tip width = 250 nm) at the input and output ports of the device to improve coupling and reduce the Fabry-Perot resonances from the cleaved facets [28].

 

Fig. 3 (a) False color SEM of the add-drop filter using Type II microheater fabricated directly on the Si layer at the center of a microdisk resonator with a diameter of 5 µm. (b) Transmission spectrum at the drop port of the device shown in (a) with (blue curve) and without (red curve) signal applied to the microheater (P_heat). (c) Blue and red curves show the step response of the reconfigurable filter in the rising and falling edges of the applied signal, respectively. (d) Blue curve shows the experimental result of the response of the drop port of the filter to a 25 ns pulse applied to the microheater. Red curve shows the result of the proposed system-level model (shown in 4(a)) fitted to the experimental data.

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Figure 3(b) shows the normalized transmission spectrum of the drop port of this device before and after applying a 0.34 mW signal to the microheater. The details of the characterization technique are found in [13]. The power consumption for tuning the center wavelength of this filter is measured to be 0.68 mW/nm (equivalent to 1.47 nm/mW wavelength shift). We also fabricated a similar add-drop filter with a 4 µm diameter microdisk. The power consumption of this device is measured to be 0.48 mW/nm (equivalent to 2.06 nm/mW wavelength shift). The resistance of the microheater for the 4 µm and 5 µm diameter resonators is 650Ω and 860Ω, respectively. It is observed that the power consumption of this device does not scale with the perimeter of the resonator as in Type I microheaters. This is because in Type II microheater, as a result of direct heating of the Si layer and its high thermal conductivity, the temperature of the whole microdisk is elevated rather than just the area underneath the microheater. We experimentally observed that power consumption is proportional to $r_{disk}^{1.5}$ in our devices ($r_{disk}^{}$ is the radius of the microdisk). The 2.06 nm/mW wavelength shift achieved in the 4 µm diameter mircrodisk is in a par with the best microheaters optimized for low power consumption through increasing device thermal resistance [11]. The low power consumption of our device, however, originates from the miniaturization of the mode volume of the microdisk. Since by miniaturizing of the device the free spectral range (FSR) increases proportionally, the power consumption for tuning of the resonance over one FSR is not improved by miniaturization. For applications where tuning over an FSR is needed the thermal resistance of the device needs to be increased, which results in the increase of the response time of the device [11]. In Section 5, we will show that using pulsed excitation it is possible to reduce the power consumption by increasing device thermal resistance without degrading the response time.

The step response of the fabricated Type II microheater is measured by applying a low-voltage (0.2V) square signal to the microheater and by observing the optical output of the drop port on an oscilloscope. The blue and red curves in Fig. 3(c) show the response of the device in the heating (rising) and cooling (falling) cycles, respectively. A rise/fall time of approximately 2.9 µs is measured for this device, which is in agreement with the simulation results presented in Section 2. We also characterized the response of the drop port of this device to a 0.5V 25 ns pulse applied to the microheater. The blue curve in Fig. 3(d) shows the normalized output of the resonator. We did not observe any change in the response of the drop port for excitation pulses shorter than 25 ns. Therefore, the measured response is a good approximation of the impulse response of the device. It is observed that the fabricated device has a δ_HPD of approximately 85 ns. This is almost an order of magnitude smaller compared to the Type I microheaters that were previously demonstrated [13]. The deviation of the experimental result (85 ns) and modeling result (19 ns, shown in Fig. 2(b)) of δ_HPD is attributed to the thermal contact resistance between the metallic heater and the Si layer [29].

The short δ_HPD of Type II microheater can be exploited for fast reconfiguration of the device. This requires an appropriate excitation signal starting with a high-energy pulse for over-driving the microheater. In order to have a fast transition of temperature for reconfiguration without an over-shoot, we have developed a system-level model that facilitates evaluating the optimal excitation without the need for a feedback loop. Figure 4(a) shows this model that is composed of a delay-like response (to account for heat propagation delay) followed by a double-exponential response (to account for heat diffusion). Appendix A describes the physics behind this model. This model is fitted to the experimental impulse response and the result is shown by the red curve in Fig. 3(d). It is observed that the model agrees well with the characterization results. The actual values of the parameters in the model are tabulated in Table 3 in Appendix A.

 

Fig. 4 (a) The system-level model for the thermal response of the device shown in Fig. 3(a). (b) Excitation signal for fast reconfiguration of the add-drop filter derived using Eq. (7). (c) The experimental response of the device to the pulsed excitation signal (shown in (b)). Inset shows a close up of the response in the first 200 ns.

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Table 3. Parameters of the system-level model fitted to the experimental data in Fig. 4(d).

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The optimal excitation signal for reconfiguration is derived by compensating for slow heat diffusion processes in the device using a pre-emphasis filter. In Appendix A, we explain the details of this pre-emphasis filter and we derive an analytic expression for the output signal of this filter (i.e., excitation signal) as a function of the parameters in the model (Eq. (7)). In the experiment, we applied this excitation signal (shown in Fig. 4(b)) to the microheater using an arbitrary waveform generator (Tektronix AFG3252). The transmission response of the through port of the filter is measured using an oscilloscope and the result is shown in Fig. 4(c) (inset shows the response in the first 200 ns). It is observed that the state of the device is reconfigured within 85 ns with very small overshoot, which is equal to the δ_HPD of this device. The demonstrated sub-100-ns reconfiguration time is one order of magnitude faster than the previously reported results in low loss Si photonic devices [12, 13, 20]. Table 2 compares the performance of the microheater in this work with some of the best devices reports in the literature.

Table 2. Comparison of the demonstrated microheater with the literature.

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4. Differential architecture for reconfiguration in opposite directions

One of the limitations of the pulse-excitation scheme for thermal reconfiguration is its applicability to only the heating cycle. Therefore, the device should still be cooled down with the natural time constant of the device, which is on the order of a few microseconds. One of the approached to resolve this limitation is to use differential (or balanced) device architectures, in which the fast reconfiguration of the device in opposite directions can be achieved using the pulsed excitation of two differential arms. Here, we demonstrate the feasibility of this approach by implementing a switching device using a Mach-Zehnder interferometer (MZI) with two 5 µm diameter microdisk phase shifters in each arm. The optical micrograph in Fig. 5(a) shows the structure of this device. The bottom plot in Fig. 5(b) shows the optical output of this switch for consecutive switching operations (top plot shows the power dissipation in the phase shifters). At the beginning (t < 0) no signal is applied to either of the microheaters and the state of the switch is at OFF. Then, at t = 0 a 0.2 mW pulsed (pre-emphasis) signal is applied to the top microheater (H₁), and the state of the output is changed in less than 100 ns to ON. At t = 5 µs another 0.2 mW signal is applied to the bottom microheater (H₂), and the state of the output is reconfigured in the opposite direction back to OFF within 100 ns. The slight variation in the initial (t < 0) and the final (5 µs < t < 10 µs) states of the device is due to a slight difference in the phase shift per applied power in the two phase shifters, which can be compensated by the calibration of microheaters. This shows that a differential architecture can be effectively used to achieve sub-100-ns thermal reconfiguration in opposite directions.

 

Fig. 5 (a) Optical micrograph of the switch with thermally tunable phase shifters in the two arms of a MZI. In this architecture, the input and output couplers are 3dB; the diameters of both microdisk resonators are 5 µm; and the rest of the parameters are the same as those in the device shown in Fig. 3(a). (b) Top plot: The power dissipated in each of the microheaters shown in (a). The initial pulses are not shown and only the steady-state value is plotted. Bottom plot: The output of the switch in (a) as power (shown in the top plot) is dissipated in the two microheaters.

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In the demonstrated switch the two arms of the differential architecture should be cooled simultaneously before the next switching operation in order to avoid continuous heating of the device. This can be implemented using a feedback loop to assure the optical output remains unchanged while cooling the two arms of the device. Here, we have simply let the two heaters cool down without any feedback loop and the results are observed in Fig. 5(b) for t > 10 µs. Again, because of the slight difference in the thermal response of the two phase shifters there is a slight distortion in the response of the device at t = 10 µs, which can be avoided by using a feedback loop. After waiting for the heaters to cool down for 10 µs, the same switching operation as that explained above (OFF-ON-OFF) is repeated starting at t = 20 µs. The limitation of this technique is in the wait time for cooling of the heaters (~3 µs).

5. Discussion: Breaking speed power consumption tradeoff in microheaters

In this section, we theoretically demonstrate that it is possible to take advantage of the small heat-propagation delay of Type II microheaters to tackle the speed power consumption tradeoff that exists in conventional thermal reconfiguration approaches. This tradeoff originates from the fact that the step response time is proportional to R_TC_T and the steady-state power consumption is proportional to 1/R_T, where R_T and C_T are the thermal resistance and capacitance of the device, respectively. Figure 6(a) shows the trend of rise/fall time (t_r) and temperature rise in Type II device as the BOX thickness (t_BOX) is increased (for 1 mW power dissipation). Heat transport modeling technique and all of the device parameters are the same as those in Section 2. Because of the increase in the thermal resistance and capacitance by increasing t_BOX, response time is slowed down while temperature is increased (i.e., power consumption is reduced). The important observation is that for the whole range of t_BOX studied here (i.e., 1-6 µm), we did not observe any change in δ_HPD in the simulation results. This is illustrated in Fig. 6(b) which shows the impulse response of Type II reconfigurable device for the two extreme values of t_BOX considered in this study (i.e., 1 µm and 6 µm). It is observed that the impulse response of the device is unchanged in the first few hundreds of nanoseconds when the BOX thickness is increased. This is because the thermal response of the device in the short time-scales is dictated by the fast heat diffusion through Si because of its high thermal conductivity. This is a unique advantage of Type II microheaters as it enables to lower the power consumption (e.g., through increasing BOX thickness) while reconfiguring the device in less than 100 ns as experimentally demonstrated in this work. To the best of our knowledge, this is the first method proposed to this date that enables simultaneous enhancement of the speed and power consumption of thermally reconfigured devices in Si. Here, we only considered the effect of the BOX thickness in the performance of the device. However, there are numerous other techniques such as partial removal of the Si substrate [11] that can be used for improving the power consumption of thermally tuned devices. The placement of microheater on the Si layer with pulsed-excitation scheme can potentially be applied to these architectures for improving the reconfiguration speed and lifting the speed power consumption tradeoff.

 

Fig. 6 (a) The rise/fall time (blue curve) and the steady-state temperature rise (red curve) of a 5 µm diameter microdisk for 1 mW power dissipation in Type II microheater vs. t_BOX. (b) The impulse response of the microdisk temperature to a 1 nJ impulse dissipated in the microheater for t_BOX = 1 µm (blue curve) and t_BOX = 6 µm (red curve).

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6. Conclusion

We experimentally demonstrated that by exploiting the high thermal conductivity of Si and by pulsed excitation of metallic microheaters on ultra-compact Si microdisk resonators it is possible to achieve sub-100-ns thermal reconfiguration time. We also achieved 0.48 mW/nm power consumption for center wavelength tuning of 4 µm diameter microdisks, which is comparable to the performance of the best microheaters optimized for low-power operation that considerably sacrifice reconfiguration time. An optical switch in a differential architecture is also demonstrated as a proof-of-concept for the capability of the proposed techniques for reconfiguration in opposite directions. We also proposed a system-level model that describes the temporal response of this type of microheaters very well. This model enables us to derive the optimal signal for fast reconfiguration of the device without the need for a feedback loop. We also discussed that direct pulsed heating of the Si device lifts the speed power consumption tradeoff in thermally reconfigured devices.

Appendix A: System-level model for Type II microheater response

Here, we elaborate on the system-level model (Fig. 4(a)) proposed for the response of the Type II microheater. As explained in Section 2, there is a delay associated with the propagation of heat in our device at very short time-scales (approximately 20 ns). We heuristically model this through a simple delay-like response, h_d(t), as shown in Fig. 4(a) and given by

(1)$$h_d\left(t\right)=\{\begin{array}{l}\frac{t}{t_d} 0<t<t_d\\ 1-\frac{t}{t_d} t_d<t<2t_d\\ \\ 0 otherwise\end{array},$$

where t_d is a delay parameter that models heat propagation delay. After this short time scale, heat flux diffuses in the Si layer and reaches the optical mode. At the same time, the distributed heat in the Si layer diffuses into the top cladding and BOX SiO₂ layers. Since the thermal conductivity of Si is two orders of magnitude larger than that of SiO₂, heat diffuses through the Si layer much faster than into the surrounding SiO₂. This is illustrated in Figs. 7(a) and 7(b) which depict the temperature distribution in the microdisk at t = 0.4 µs and t = 4 µs, respectively. In these simulations a 1 mW step signal is applied to the microheater at t = 0 and the rest of the simulation parameters are the same as in Section 2. It is observed in Fig. 7(a) that at the first few hundreds of nanoseconds heat is mostly diffused in the Si disk. This heat is later diffused in the surrounding cladding and BOX layers as observed in Fig. 7(b). The presence of two different heat diffusion mechanisms with different time-scales in this device requires a second-order system (or a double-exponential) to accurately model its temporal response. This double-exponential is given by

(2)$$h_c\left(t\right)=\exp \left(-\frac{t}{{\tau }_f}\right)+\alpha \exp (-\frac{t}{{\tau }_s}),$$

where τ_f and τ_s are the time constants for the fast and slow heat diffusion in the device, and α is the ratio of the slow to fast sources of heat diffusion. This model is fitted to the experimental data and the result is shown in Fig. 4(d). It is observed that very good agreement between the model and experimental results is achieved. The fitted parameters in the model are tabulated in Table 3.

 

Fig. 7 (a) and (b) Distribution of temperature in Type II microheater at t = 0.4 µs and t = 4 µs for a 1 mW step signal applied to the heater at t = 0, respectively. The black arrows show the heat flux in the device. Heat flux is scales by a factor of 5 in the BOX and cladding layers for better visualization. Here, we have also considered a sloped via to exactly model the actual devices experimentally demonstrated in this work.

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Using the proposed model, we derive the excitation signal that can reconfigure the device without any overshoot in the minimum possible time. This is achieved by compensating for the sources of heat diffusion in our device (represented by the double-exponential in the device model) through a pre-emphasis filter before the excitation of the microheater. This is depicted in the system block diagram shown in Fig. 8(a). Here, $H_d(j\omega )$ and $H_c(j\omega )$ are the transfer functions of the delay and double-exponential blocks in the device model, and $H_{pre}(j\omega )$is the transfer function of the pre-emphasis filter. Since the pre-emphasis filter is designed to compensate for the slow response of the double-exponential, we have

 

Fig. 8 (a) The block diagram of the system-level representation of the reconfigurable microdisk device including the pre-emphasis filter that is used to compensate for the slow response of heat diffusion. (b) The block diagram of the system-level representation of the excitation signal including the seed input pulse, p(t), and the pre-emphasis filter. The three components of the pre-emphasis filter (i.e., differentiator, proportional term, and first-order system) are shown in the dashed box.

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(3)$$H_{pre}\left(j\omega \right)= {\left[H_d\left(j\omega \right)\right]}^{-1}={\left[ \frac{1}{j\omega +{\tau }_f^{-1}}+ \frac{\alpha }{j\omega +{\tau }_s^{-1}}\right]}^{-1}.$$

With some algebraic manipulations Eq. (3) can be rewritten as

(4)$$H_{pre}\left(j\omega \right)= A\left[ j\omega +B+ \frac{C}{j\omega {\tau }_{eff}+1}\right],$$

where

(5)$$\{\begin{array}{l}\frac{1}{{\tau }_{eff}}=\frac{1}{1+\alpha }(\frac{1}{{\tau }_s}+\frac{\alpha }{{\tau }_f})\\ A=\frac{1}{1+\alpha }\\ B=\frac{1}{{\tau }_s}+\frac{1}{{\tau }_f}-\frac{1}{{\tau }_{eff}}\\ C=\left(\frac{1}{{\tau }_{eff}}-\frac{1}{{\tau }_s}\right)\left(\frac{1}{{\tau }_{eff}}-\frac{1}{{\tau }_f}\right)\end{array}.$$

From Eq. (4) it is observed that the pre-emphasis is composed of a differentiator, a proportional term, and a first-order filter. All of the calculations in the above are based on the assumption that$H_d(j\omega )H_c(j\omega )$ is the transfer function of the system. However, in practice, we measure the impulse response of the device by applying a finite-width pulse p(t). Therefore, the actual transfer function of the device is given by $H_d(j\omega )H_c(j\omega )/P(j\omega )$ where $P(j\omega )$ is the Fourier transform of the input pulse. The effect of the finite width of p(t) can be compensated by using the exact same pulse for the reconfiguration of the device. In this approach, instead of applying a step signal to the pre-emphasis filter as considered in the simplified picture shown in Fig. 8(a), p(t) is passed through an integrator and then applied to the pre-emphasis (Fig. 8(b)). Therefore, the response of the system $T(j\omega )$ (temperature of the microdisk) is given by

(6)$$\begin{array}{l}T\left(j\omega \right)= \stackrel{\text{Output}\text{of}\text{the}\text{pre}-\text{emphasis}\text{filter}}{\overbrace{\left[P\left(j\omega \right).\frac{1}{j\omega }. H_{pre}\left(j\omega \right)\right]}}{\stackrel{\text{System}\text{transfer}\text{function}}{\overbrace{\left[\frac{H_d\left(j\omega \right)H_c\left(j\omega \right)}{P\left(j\omega \right)}\right]}}}^{},\\ =\frac{H_d\left(j\omega \right)}{j\omega }\end{array}$$

which is the same as the output of the system in the ideal case shown in Fig. 8(a) (notice that the Fourier transform of a step signal is 1/jω). By taking inverse Fourier transform from the output of the pre-emphasis filter (Eq. (6)), we arrive at the following analytic expression for the excitation signal:

(7)$$s\left(t\right)=A [p\left(t\right) + \frac{{\tau }_{eff}}{{\tau }_s{\tau }_f} p\left(t\right)*u\left(t\right) - {\tau }_{eff}^{2}C\left(e^{\frac{t_p}{{\tau }_{eff}}}-1\right)e^{- \frac{t_{}}{{\tau }_{eff}}} u\left(t\right)],$$

where t_p is the pulse-width of p(t), $\ast$ denotes the convolution operation and u(t) represents a step signal. It is observed that the excitation signal is composed of three components: 1) a short pulse, 2) a steady-state component, and 3) a decaying exponential. The combination of all of these components is observed in the excitation signal shown in Fig. 5(a). It is also observed in Fig. 5(b) that this excitation pulse reconfigures the output of the device with a very small over-shoot in approximately 85 ns. The significance of knowing the exact time dependence of the excitation signal is that it eliminates the need for a feedback loop.

Acknowledgment

This work was supported by the Air Force Office of Scientific Research (AFOSR), contract No. FA9550-09-1-0572 (G. Pomrenke) and by Defense Advanced Research Project Agency (DARPA) Microsystems Technology Office (MTO) under Si-PhASER program, contract No. HR0011-09-1-0014 (S. Rodgers).

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