Design and simulation results are presented for an ultralow switching energy, resonator based, silicon-on-insulator (SOI) electro-optical modulator. The nanowire waveguide and Q ~8500 resonator are seamlessly integrated via a high-transmission tapered 1D photonic crystal cavity waveguide structure. A lateral p-n junction of modulation length Lm ~λ is used to alter the index of refraction and, therefore, shift the resonance wavelength via fast carrier depletion. Differential signaling of the device with ΔV ~0.6 Volts allows for a 6dB extinction ratio at telecom wavelengths with an energy cost as low as 14 attojoules/bit.
© 2014 Optical Society of America
Electro-optical modulation in silicon nanobeams produced by the depletion of electrons and holes is investigated in this paper. Silicon nanobeam (NB) components are a silicon-on-insulator (SOI) technology. The NB is a single-mode Si strip channel waveguide or Si “photonic nanowire” that includes a resonant cavity at a telecom wavelength in the present case. The resonance is created by a 1D photonic crystal (PhC) lattice which is a series of cylindrical air holes etched into the Si down to the buried oxide interface. A high transmission, high quality factor waveguide-cavity system is made from a series of quadratically tapered holes with a zero-length cavity region in the middle . The series of tapered holes on each side of the cavity can be thought of as a “modified” Bragg mirror.
Prior examples of resonant nanowire electro-optical (EO) modulators consist mainly of a bus waveguide that is evanescently side-coupled through an air gap to a micro-ring, micro-disk or micro-donut, i.e., the resonator is a separate entity from the nanowire. In contrast, our device has the resonator and optical transmission waveguide seamlessly combined in one wire resulting in a much smaller footprint. Owing to device area considerations, the NB EO modulator offers the possibility of a significantly lower switching energy than the disk/ring approach.
2. Prior-Art NB Modulators
The field of SOI NBs began auspiciously with the pioneering work of Foresi et al . Experiments by Schmidt et al  demonstrated EO modulation in an SOI NB using forward-biased injection of electrons and holes into the intrinsic cavity via an embedded PIN junction. By comparison, the new feature in our device is depletion-layer widening in a pn NB which has the advantage of much higher “inherent” modulation speeds than those attained with carrier injection. A new feature of our active resonator is a “no point-defect cavity” which differs from the more traditional “series-of-defects cavity” used in Schmidt’s Fabry-Perot resonator where the quality factor (Q) was found to be 253. A pathway to higher Q was pointed out clearly in the Si NB works of Kuramochi et al  and Quan et al . For example, extremely high Q’s above 105 were produced in SOI NB devices based upon “ladder cavities” of circular and rectangular holes in which the hole size was uniform. The new feature in our NB is the tapered-diameter circular hole arrays that give a loaded Q of 104 and a cavity that gives improved optical transmission at the fundamental resonance. Other advantages of our device, by comparison with the prior Q ~105 devices, are wider information bandwidth due to an order of magnitude lower Q and enhanced thermal stability from the device being on-substrate. For comparison, a 2D photonic-crystal SOI slab can contain a line-defect waveguide with an integral resonator whose area-footprint is similar when compared to the footprint of our device. A good example of this is the prior work of Tanabe et al  where lateral p- and n-doped regions defined a PIN junction enabling EO injection modulation. Our device, however, features a much stronger confinement of light in the waveguide-width direction, thus our footprint is effectively considerably smaller. Our device also is designed to operate via the faster process of carrier depletion as opposed to injection and the 1D PhC lattice used offers superior waveguide cavity coupling  when compared to conventional strip waveguides coupled to 2D PhCs.
Another important approach to EO modulation outlined by Qi et al , is the investigation of a nanoscale slot etched into the midline of a high Q nanohole array, allowing for the introduction of an EO polymer. With doped silicon contact regions straddling the slot, it is proposed that an applied voltage perturbing the polymer will shift the nanobeam resonance. Although the slot NB resonator offers the smallest possible cavity volume and switching energy this work estimates what can be done without a slot using free-carrier dispersion effects. Further, these effects are exploited in their “fastest embodiment”, namely through carrier depletion as opposed to carrier injection. In many ways, this specific depletion architecture emulates the aforementioned slot design because the vertical depletion layer of this work closely resembles a slot in both cross-sectional dimensions and active length. Therefore, looking at the slot and our slot free, depletion mode NB collectively, we believe that these Si EO NB structures offer the fastest and lowest-energy modulation currently available in silicon photonics (not considering here the sub-wavelength class of plasmon based photonic modulators ). Comparing the EO NB modulator of this work directly to that of the slotted EO modulator of Qi, consider the disadvantage of the extrinsic nature of the EO polymer, as being a “guest” inserted inside of the silicon host. The pn depletion mode device of this work, on the other hand, is completely intrinsic, meaning that the active material is seamlessly and monolithically integrated directly into the NB. This results in a device that is easier to manufacture and offers improved CMOS compatibility.
3. Device design
Our design consists of a 1D silicon-on-insulator (SOI) photonic crystal nanobeam cavity strongly coupled to a feeding waveguide. Such devices offer high optical transmission as well as a high quality factor and a small mode volume. The fundamental mode optical transmission consists of a Lorentzian peak situated midway in the PhC’s low background forbidden gap transmission spectrum.
Several design algorithms have recently been proposed; in our example we follow that outlined by the Loncar group  where the photonic crystal hole size (or equivalently the fill factor) is quadratically tapered down from the center of a zero length cavity region to either edge of the photonic crystal. This creates a linear change in the photonic crystal mirror strength, allowing for efficient coupling between the waveguide structure and the photonic crystal structure.
The device consists of a 500nm wide by 220nm high silicon strip waveguide on top of a silicon dioxide layer. While performance of 1D photonic crystal nanobeam structures can be increased by creating suspended structures through undercutting of the buried oxide layer beneath the device, we have chosen to leave our device on the substrate for maximum robustness. The oxide is assumed to be optically thick. The number of mirror pairs in the TE0-mode photonic crystal was chosen to be 12 (a total array of 24 holes) which results in a quality factor slightly below 104. We found this to be a fair tradeoff between the resonance linewidth and the cavity lifetime which are inversely proportional quantities. If the quality factor is too high, the speed of the device is compromised since higher Q results in longer photon storage time in the photonic crystal cavity. However, if the quality factor is too low then very large modulation shifts of the wavelength are required due to the large resonant linewidth of low Q resonators. A hole-array periodicity of 350nm was selected in order to get device operation in the telecom-friendly 1500nm range. We note that one could easily adjust this parameter in order to tweak the actual wavelength of operation to any other desired value, such as 1550nm. The fill factor of the center two holes was chosen to be 0.2 and was quadratically tapered down to 0.1 at the outer two holes. This results in a quadratic tapering of the hole radii from roughly 125nm to 88nm. For electrical contact and for defining the p-n junction, initially undoped 50nm-thick silicon “wings” extending the entire length of the photonic crystal device were placed on either side of the nanobeam. The Si wings-on-oxide integrate seamlessly with the nanowire and a cross-section view in the wing region shows the geometry of a rib waveguide. Voltage contacts for electro-optic modulation are made in doped wing areas.
P-type and n-type doping of the respective left and right wings is then performed as shown in the NB perspective view of Fig. 1(a), orange for p and dark green for n—with doping length centered on the cavity. The next design consideration consisted of choosing the appropriate hole and carrier doping concentrations and the exact lateral position of the abrupt p-n junction . While higher doping concentrations can lead to larger changes in the refractive index they also result in higher levels of loss due to free carrier absorption. A compromise of these two parameters led us to choose a p-type doping concentration of 5x1017 cm−3 and an n-type doping concentration of 1x1018 cm−3, p-type being slightly less since holes have a greater effect on the optical index of refraction. Using different concentrations, the index of refraction can, therefore, be made nearly equivalent in both regions, and in our case is roughly 3.474. Both the nanobeam and the “wings” are doped by the same amount and to the same length, Lm, which is the modulator’s interaction length. Note that the 24-hole on-axis length of the lateral wings Lw overlapping the array is always held constant and that only a fraction Lm/Lw of the wing is doped. The reason for this is that wings with a length shorter than that of the photonic crystal structure resulted in significant transmission loss of the device. As illustrated in the cross-section cavity view of Fig. 1(b), the width of the p-type doped region in the nanobeam was set to 285nm and the width of the n-type doped region was set to 215nm. The position of the p-n junction is, therefore, not actually centered in the nanobeam but is offset by 35nm. Our simulation results, which follow, show that this is an optimized junction location for our device. A top view of the EO NB is shown in Fig. 2.
Optical simulations were carried out using a commercially available finite difference time domain software package (Lumerical FDTD). A transverse electric waveguide mode was input into one end of the structure and the intensity transmission through the device was calculated at the other end. Within the photonic crystal structure area, a mesh size of 10nm along the length and height and 5nm across the width was used. The simulation time was set to 80ps. For the doped regions a complex refractive index of n = 3.474148 and k = 0.000112 was used for the n-type regions and n = 3.473960 and k = 0.000031 for the p-type regions [10,11]. In the depleted regions a value of n = 3.475 and k = 0 was used.
For an abrupt silicon pn junction the built in potential is given by:
The total width of the depletion region can then be calculated using the following equation:
For this work, Eq. (2), when valid, was used to obtain depletion widths at all voltages up to the built in voltage of 0.935V. At an applied forward bias equal to the built in voltage of 0.935V, where increased charge injection makes Eq. (2) invalid, band diagram calculations (not shown) established a minimum depletion width of 10nm. Table 1 lists the depletion widths along with their corresponding bias voltages. Specific depletion widths were chosen such that they would be located exactly on lines of the simulation mesh grid.
For a given modulation length, the strength of the wavelength shift for each applied voltage (the initial voltage and the final voltage) is dependent upon the effective index of refraction that the resonant mode sees at each voltage; the larger the change in the effective indices of refraction, the larger the wavelength shift. For strong forward bias the depletion region width is minimized and the effective index is predominately an admixture of the n-type and p-type indices of refraction which, for our device, are not equal (nn = 3.474148, np = 3.473960). When reverse bias is applied the depletion width widens and the index of refraction of the depletion layer (nd = 3.475) must now be taken into account. If the mode predominately sees np when forward biased and predominately sees nd when reverse biased, the largest change in the index of refraction is achieved and the wavelength shift will be maximized. This condition will not necessarily be met when the p-n junction is located in the center of the device and, therefore, in order to determine the optimal p-n junction location, a series of simulations was performed.
Using an arbitrary modulation length of 2 microns, the resonance shift between + 0.935V and −1.14V was determined for six different p-n junction locations. Figure 3 shows a plot of wavelength shift as a function of junction location, given as a ratio of the p-type doped region width to the n-type doped region width. One can easily see that the greatest shift occurs when the p-type region is 285nm and the n-type region is 215nm, which is a 35nm p-n junction location offset from the center of the 500nm device. The optimal junction location gives a fundamental resonance peak wavelength of 1517.4nm and provides a 0.77nm blueshift when reversed biases at −1.14V. Note that the 260/240 p-type width to n-type width ratio, which results in a junction location that is only 10nm off center, is actually the least efficient.
The behavior of the wavelength shift as a function of junction location can be understood as an interdependence between the overlap of the depletion region with both the photonic mode and the dielectric material in the photonic crystal waveguide. When the pn junction is located far from the center of the device (very large and very small Lp/Ln values beyond those shown in Fig. 3) one can expect the wavelength shift to approach zero since there is no overlap between the depletion region and the photonic mode. On the other hand, when the depletion region is centered in the device, as is the case for Lp/Ln = 260/240 at −1.14V bias voltage, the overlap between the photonic mode and the depletion region is maximized. However, in this case, the overlap between the depletion region and the photonic crystal air holes is now also maximized which limits the change in the effective index induced by depletion. By slightly offsetting the junction location one can increase the change in effective index while still achieving good overlap with the photonic mode. Since the p-type material depletes faster and has a lower index of refraction than the n-type material, a junction location offset to larger Lp/Ln ratios provides the greatest wavelength shift.
After determining the optimal p-n junction offset, a number of simulations were carried out in order to characterize the functionality of our device. Transmission spectra were obtained for three different modulation lengths: 0.5μm, 1μm, and 2μm. Each of these modulation lengths was simulated for four different bias voltages: + 0.3V, −0.315V, −1.14V, and −2.16V. These voltage values were chosen to avoid charge injection and maintain the depletion mode characteristics of the device for improved switching speed. All twelve transmission spectra are shown in Fig. 4. For completeness, transmission spectra for + 0.935V, where the depletion width is minimized, are also presented in Fig. 4. As expected, the minimal depletion width resonance shifts to shorter wavelengths as the modulation length is increased, due to the lower refractive index of the doped regions as compared to the intrinsic/depleted areas. On the other hand, larger depletion width resonances are redshifted; a non-intuitive behavior that we speculate is due to the complex cavity effects introduced in a zero cavity length photonic crystal waveguide.
The most important aspects of our device, namely the extinction E, the modulation shift Δλ of the resonance wavelength λo, the quality factor Q, and the peak optical power transmission value T are all shown in Fig. 5 as a function of modulation length Lm. For short modulation lengths the transmission, Fig. 5(a), hovers up around 80%, decreasing down to ~65% for the longest modulation length of 2 μm. This is to be expected as longer modulation lengths result in increased free carrier absorption. Also, as expected, narrower depletion width results in a slightly lower transmission value when compared to wider depletion widths which is simply a result of the decreased number of free carriers in the depletion region. This effect is also mirrored in the quality factor values shown in Fig. 5(b), with Q’s in the 10k range for a 0.5 μm modulation length, decreasing down to the 8.5k range for a 2μm modulation length. Figure 5(c) shows the wavelength shift and Fig. 5(d) the extinction E as referenced from the + 0.3 forward bias resonance position; E = 10* log (T(λo,Vo) / T(λo,VR)). The wavelength shift ranges from a minimum of 0.04nm for a + 0.3V to −0.315V symmetric differential voltage swing at a 0.5 μm modulation length with 1.13dB of extinction to a maximum of 0.33nm for a + 0.3V to −2.16V asymmetric differential voltage swing at a 2 μm modulation length and 11.7dB of extinction.
Field profiles for a 2μm modulation length, −2.16V reverse biased device are shown in Fig. 6. A cross sectional view of the fundamental TE mode in the bare waveguide portion of the device is shown in Fig. 6(a). Within the photonic crystal cavity portion of the device, this waveguide mode is converted into a Bloch mode whose mode profile can be seen in Fig. 6(b). Figure 6(c) shows a view of the electric field magnitude in the mid-plane of the photonic crystal nanobeam, above which is shown the device layout. Here one can see that the strongest part of the field is located in the center, cavity region of the photonic crystal, which is also the center location of the depletion region. Note that the mode concentrates in the silicon material areas rather than in the air hole regions. These field profiles are essentially consistent upon change of applied bias and modulation length; e.g. there is no discernible difference in the field profiles for + 0.3V versus −2.16V.
The effective index of refraction of the fundamental mode in the waveguide region, , is 2.40. Introduction of the silicon “wings” causes the effective index to increase to 2.46. Within the photonic crystal cavity portion of the device, the Bloch mode effective index can be calculated as where is the resonant wavelength and is the photonic crystal lattice spacing, which we have set to 350nm in all cases. For the 2μm modulation length, −2.16V reverse biased device, the resonant wavelength is 1518.5nm, resulting in a Bloch mode effective index of .
According to the Fig. 6(b) mode profile in the photonic crystal cavity region, the mode intensity profile is strongly bunched at the vertical midline of the NB cross section, near the vertical line location of the abrupt PN junction. This bunching at the cavity region is inherent in the nanobeam PhC design and, therefore, also exists for undoped, all intrinsic devices. By contrast, in the region of the Si strip waveguide where the vertical air holes are absent, we find that this mode profile is more uniform across the strip cross section. The lateral mode “bunching” that exists within the 1D PhC lattice region means generally that there is a strong overlap between the mode and the laterally concentrated depletion-layers that widen during reverse bias. The increased overlap found here accounts for the ultra-sensitivity of this modulator. For example, our modulator is more sensitive than a state of the art silicon photonic resonant microdisk pn junction modulator ; for Lm = 2 μm, our frequency shift in Fig. 5(c) is 17 GHz per volt.
In order to calculate an estimate for the required switching energies for our different device parameters, a commercially available software package (Lumerical DEVICE) was employed to solve drift-diffusion equations and determine the charge carriers within the device for various bias voltages. For each bias point an additional total charge calculation offset by 25mV is made. The junction capacitance, C, can then be calculated from:
As Fig. 7 indicates, and as expected, the junction capacitance of the device decreases for increasing reverse bias. Larger device lengths predictably lead to correspondingly larger values of junction capacitance.
Table 2 lists the energy per bit required for various modulation lengths when switching from forward bias of 0.3V to a reverse bias of −0.315V, −1.14V, and −2.16V. For a 2μm modulation length and a 0.3V to −0.315V modulation-voltage swing, where C0.3 = 359 aF and C-0.315 = 245 aF, the Eb required for 6dB of extinction is only 14 attojoules per bit. We would like to note that our simulations for capacitance and, therefore, calculations for energy, do take into account the air holes in the photonic crystal structure.
The small footprint of this device, along with its extremely small cross sectional junction area, leads to extremely small device capacitance. By maintaining depletion mode characteristics, improved modulator performance can be realized through the use of differential signaling in order to minimize diffusion capacitance . This device drive architecture, along with the designed Q of less than 104 and the hundreds of atto-farad device capacitance suggests both short carrier lifetime and a reasonable photon lifetime when compared to other resonant cavity based modulators exhibiting 25Gb/s modulation speeds . Although direct comparison with other resonant cavity modulators is difficult, 50Gb/s modulation has been achieved with larger ring resonator based modulators suggesting the device of this work to be able to achieve similar performance in the range of 25-50 Gb/s with properly designed high frequency contact pads . As is typical with most of the prior art referenced, parasitic capacitance has not been addressed although it is expected to be similar to those devices based on the similarity of device size, geometry and mode of operation. Note that parasitic capacitance could increase energy consumption values. An appropriate RF voltage feed for this device has also not been addressed as being beyond the scope of this work. Table 3 provides a summary of important device parameters for a number of state of the art silicon based electro-optic modulators as compared to our device.
Most of the published implementations of 1550 nm SOI electro-optical modulators consist of a Si strip waveguide side coupled to a Si micro-ring or micro-disk resonator containing a circumferential p-n junction for carrier depletion in the cavity. By comparison, our Si NB EO modulator has a smaller footprint because it foregoes the micro-ring or disk. In addition, the NB modulator has a smaller mode volume within the active region than the ring modulator, allowing lower switching energy in the NB. The resonance characteristics are also different because the NB, with its 1D PhC cavity, possesses one or two high transmission peaks within the low transmission gap of the photonic crystal, while the micro-disk offers a long periodic series of resonances. In addition to coupling several NBs cavities serially along one wire , it is feasible to arrange an evanescent wave side coupling of a NB to one or more parallel NBs or to one or more non-resonant SOI channel waveguides. We expect to study EO versions of those NB directional couplers in the future.
Also, consider that germanium becomes transparent for λ > 1.8 μm and possesses a free-carrier depletion effect much stronger than that of silicon. Therefore, a Ge nanobeam EO modulator corresponding to the present device design is anticipated to be even more sensitive than a Si device.
The Si NB can also serve as a refractometer if the Si NB resonance wavelength λo is chosen to correspond to the fundamental absorption line of a target molecular species and if this analyte is infused into the pores of the Si modulator, serving as a new upper cladding material instead of air. Then the complex cladding index of the NB will change substantially in the presence of the analyte, thereby shifting the NB resonance. Since the NB is EO, it is possible to “dither” the cavity resonance at audio frequencies, allowing for an extremely precise measurement of λo and its shift. This measurement method has been described by Qiu et al  who show that the resonance of the refractometer can be measured to within 0.16 pm, thus enhancing the refractometer sensitivity.
As a final comment, we note that this NB modulator is highly versatile because it can be actuated by several available physical mechanisms for actively shifting λo of the Si NB. These are: a p-i-n or p-n junction for electron and hole depletion or injection, an MOS high-K gate on the NB for carrier depletion or accumulation , a micro-heater-induced thermo-optic effect , the Franz-Keldysh field effect operative around 1100 nm, and two nonlinear optical effects; (1) generation of e-h pairs in the resonator due to an above gap light beam incident on the Si and (2) an intensity dependent refractive index n2 of the cavity induced by strong optical pumping of the third-order nonlinear Si. We have selected p-n depletion as the most practical technique. Furthermore, we had a choice of depleting either a lateral p-n junction or a vertical p-n junction. The vertical junction is thought to be almost twice as efficient as the horizontal junction , although an air bridge contact may be required in the vertical case. We chose to use the lateral junction which has been applied quite successfully in the Si modulator literature . Fairly abrupt lateral junctions, with control of junction location, are feasible in practice . The vertical junction NB awaits future exploration.
We have simulated an SOI thick-oxide lateral p-n junction modulator compatible with the low-voltage symmetric and asymmetric differential signaling described by Zortman et al . By examining the case of 300mV forward bias combined with 315mV reverse bias (asymmetric signaling), we achieved a 6dB extinction ratio (Lm = 2 μm) at the 1D PhC cavity resonance λo ~1518 nm. In the 500nm x 220nm TE0-mode Si nanobeam, a point-defect resonator was not used, and instead the cavity consisted of a zero-length defect midway in an array of 24 cylindrical air holes whose diameter was quadratically tapered in and out from 250 to 176 nm giving an Lm-dependent cavity Q in the 8000 to 10,000 range. The Si strip waveguide became a fixed-length rib waveguide over the Lw = 8.2 μm PhC array length where 50nm high Si lateral wings, initially undoped, extended out from both sides of the NB. To create a horizontal abrupt p-n junction in the NB, and to provide external electrical contacts for bias voltage, the left and right portions of the NB were locally doped over the central region (length Lm) of the hole array, as were the left and right wings. The p type doping was 5x1017 cm−3 and the n type doping was 1x1018 cm−3 . Simulation tests showed that a 35 nm offset of the junction from the NB center (p width = 285nm, n width = 215nm) produced the best modulation.
For a 24-hole device simulated using Lumerical FDTD, we estimate the following performance features: (1) the information bandwidth is approximately 20 GHz as calculated from the “worst case” cavity linewidth of 152 pm, (2) the active modulator length of 0.5 to 2 μm of this “all-dielectric Si wire” is truly wavelength-scale and ranges from 0.33λ to 1.32λ, (3) the λo shift per unit applied-voltage swing per unit length Δλ/ΔV Lm is a figure of merit (FOM) and this FOM is maximum at ~0.13nm/Vμm when ΔV = Vo-VR ~0.6V and Lm ~2 μm, (4) the infrared transmission at λo is ~80% for optimum cases, meaning an insertion loss of ~1dB, and (5) the minimum forward/reverse switching energy for 6dB of extinction is extremely low; for example Eb = 14 attojoules per bit at VR = 315mV and Lm = 2 μm. These specifications, taken as a whole, indicate very high modulation sensitivity, a performance that exceeds the performance reported for Si micro ring and disk modulators.
JH and JS would like to acknowledge support from the Air Force Office of Scientific Research (Program Manager Dr. Gernot Pomrenke) under contract number 12RY05COR. RS appreciates AFOSR support under grant FA9550-10-1-0417. All authors thank Matthew Grupen for helpful discussions.
References and links
2. J. S. Foresi, P. R. Villeneuve, J. Ferrera, E. R. Thoen, G. Steinmeyer, S. Fan, J. D. Joannopoulos, L. C. Kimerling, H. I. Smith, and E. P. Ippen, “Photonic-bandgap microcavities in optical waveguides,” Nature 390(6656), 143–145 (1997). [CrossRef]
3. B. Schmidt, Q. Xu, J. Shakya, S. Manipatruni, and M. Lipson, “Compact electro-optic modulator on silicon-on-insulator substrates using cavities with ultra-small modal volumes,” Opt. Express 15(6), 3140–3148 (2007). [CrossRef] [PubMed]
4. E. Kuramochi, H. Taniyama, T. Tanabe, K. Kawasaki, Y.-G. Roh, and M. Notomi, “Ultrahigh-Q one-dimensional photonic crystal nanocavities with modulated mode-gap barriers on SiO2 claddings and on air claddings,” Opt. Express 18(15), 15859–15869 (2010). [CrossRef] [PubMed]
5. T. Tanabe, E. Kuramochi, H. Taniyama, and M. Notomi, “Electro-optic adiabatic wavelength shifting and Q switching demonstrated using a p-i-n integrated photonic crystal nanocavity,” Opt. Lett. 35(23), 3895–3897 (2010). [CrossRef] [PubMed]
6. B. Qi, P. Yu, Y. Li, X. Jiang, M. Yang, and J. Yang, “Analysis of electrooptic modulator with 1-D slotted photonic crystal nanobeam cavity,” IEEE Photon. Technol. Lett. 23(14), 992–994 (2011). [CrossRef]
7. V. J. Sorger, “λ-size silicon-based modulator,” invited paper 8629–23, SPIE Proceedings vol. 8629, Silicon Photonics VIII, SPIE Photonics West, San Francisco, 5 Feb 2013.
8. Q. Quan, P. B. Deotare, and M. Loncar, “Photonic crystal nanobeam cavity strongly coupled to the feeding waveguide,” Appl. Phys. Lett. 96(20), 203102 (2010). [CrossRef]
9. N.-N. Feng, S. Liao, D. Feng, P. Dong, D. Zheng, H. Liang, R. Shafiiha, G. Li, J. E. Cunningham, A. V. Krishnamoorthy, and M. Asghari, “High speed carrier-depletion modulators with 1.4V-cm VπL integrated on 0.25microm silicon-on-insulator waveguides,” Opt. Express 18(8), 7994–7999 (2010). [CrossRef] [PubMed]
10. M. Nedeljkovic, R. A. Soref, and G. Z. Mashanovich, “Free-carrier electrorefraction and electroabsorption modulation predictions for silicon over the 1-14 μm infrared wavelength range,” IEEE Photonics Journal 3(6), 1171–1180 (2011). [CrossRef]
11. M. Nedeljkovic, R. A. Soref, and G. Z. Mashanovich, “Free-carrier electro-absorption and electro-refraction modulation in group IV materials at mid-infrared wavelengths,” SPIE Photonics West, paper 8266–31, San Jose, CA (25 Jan 2012).
14. S. P. Anderson and M. Philippe, Fauchet, “Conformal P-N Junctions for Low Energy Electro-optic Switching,” OSA/CLEO/IQEC (2009).
15. S. Meister, H. Rhee, A. Al-Saadi, B. A. Franke, S. Kupijai, C. Theiss, L. Zimmermann, B. Tillack, H. H. Richter, H. Tian, D. Stolarek, T. Schneider, U. Woggon, and H. J. Eichler, “Matching p-i-n-junctions and optical modes enables fast and ultra-small silicon modulators,” Opt. Express 21(13), 16210–16221 (2013). [CrossRef] [PubMed]
16. T. Baba, S. Akiyama, M. Imai, N. Hirayama, H. Takahashi, Y. Noguchi, T. Horikawa, and T. Usuki, “50-Gb/s ring-resonator-based silicon modulator,” Opt. Express 21(10), 11869–11876 (2013). [CrossRef] [PubMed]
17. D. Marris-Morini, C. Baudot, J.-M. Fédéli, G. Rasigade, N. Vulliet, A. Souhaité, M. Ziebell, P. Rivallin, S. Olivier, P. Crozat, X. Le Roux, D. Bouville, S. Menezo, F. Bœuf, and L. Vivien, “Low loss 40 Gbit/s silicon modulator based on interleaved junctions and fabricated on 300 mm SOI wafers,” Opt. Express 21(19), 22471–22475 (2013). [CrossRef] [PubMed]
18. J. Ding, H. Chen, L. Yang, L. Zhang, R. Ji, Y. Tian, W. Zhu, Y. Lu, P. Zhou, R. Min, and M. Yu, “Ultra-low-power carrier-depletion Mach-Zehnder silicon optical modulator,” Opt. Express 20(7), 7081–7087 (2012). [CrossRef] [PubMed]
19. H. Yu, M. Pantouvaki, J. Van Campenhout, D. Korn, K. Komorowska, P. Dumon, Y. Li, P. Verheyen, P. Absil, L. Alloatti, D. Hillerkuss, J. Leuthold, R. Baets, and W. Bogaerts, “Performance tradeoff between lateral and interdigitated doping patterns for high speed carrier-depletion based silicon modulators,” Opt. Express 20(12), 12926–12938 (2012). [CrossRef] [PubMed]
20. T. Baehr-Jones, R. Ding, Y. Liu, A. Ayazi, T. Pinguet, N. C. Harris, M. Streshinsky, P. Lee, Y. Zhang, A. E.-J. Lim, T.-Y. Liow, S. H.-G. Teo, G.-Q. Lo, and M. Hochberg, “Ultralow drive voltage silicon traveling-wave modulator,” Opt. Express 20(11), 12014–12020 (2012). [CrossRef] [PubMed]
21. K. Debnath, L. O’Faolain, F. Y. Gardes, A. G. Steffan, G. T. Reed, and T. F. Krauss, “Cascaded modulator architecture for WDM applications,” Opt. Express 20(25), 27420–27428 (2012). [CrossRef] [PubMed]
24. R. A. Soref, J. Guo, and G. Sun, “Low-energy MOS depletion modulators in silicon-on-insulator micro-donut resonators coupled to bus waveguides,” Opt. Express 19(19), 18122–18134 (2011). [CrossRef] [PubMed]
25. W. S. Fegadolli, J. E. B. Oliveira, V. R. Almeida, and A. Scherer, “Compact and low power consumption tunable photonic crystal nanobeam cavity,” Opt. Express 21(3), 3861–3871 (2013). [CrossRef] [PubMed]
26. M. R. Watts, W. A. Zortman, D. C. Trotter, R. W. Young, and A. L. Lentine, “Vertical junction silicon microdisk modulators and switches,” Opt. Express 19(22), 21989–22003 (2011). [CrossRef] [PubMed]