A polymer-infiltrated P-S-N diode capacitor configuration is proposed and a high speed electro-optic phase shifter based on a silicon organic hybrid platform is designed and modeled. The structure enables fast carrier depletion in addition to the second order nonlinearity so that a large electro-optic overlapped volume is achievable. Moreover, the device speed can be significantly improved with the introduction of free carriers due to a reduced experienced transient capacitance. The advantages of the diode capacitor structure are highly suitable for application to a class of low aspect ratio slot waveguides where the RC limitation of the radio frequency response is minimized. According to our numerical results, by optimizing both the waveguide geometry and polarization mode, at least 269 GHz 3-dB bandwidth with high efficiency of 5.5 V-cm is achievable. More importantly, the device does not rely on strong optical confinement within the nano-slot, a feature that gives considerable tolerance in the use of nano-fabrication techniques. Finally, the high overlap and energy efficiency of the device can be applied to slow light or optical resonance media for realizing photonic integrated circuits-based green photonics.
© 2011 OSA
Recently, electro-optic (EO) organic materials, especially nonlinear optical (NLO) polymers, have attracted increased research attention for enabling high speed integrated optical modulators due to their strong nonlinear coefficients and instantaneous response to an electrical field (E-field) [1–4]. As a result, the so-called silicon organic hybrid (SOH) is considered a promising platform to provide process and application compatibility with the well developed complementary metal-oxide-semiconductor (CMOS) industry. In particular, the polymer infiltrated or deposited slot waveguide (SWG) system has been studied intensively, where 170 Gb/s all optical signal processing has been demonstrated . However, the devices proposed previously mostly rely on strong optical confinement within a nonlinear slot in such a way that extremely high EO overlap is used for optimizing modulation efficiency [5,6]. This usually involves narrowing down the slot width and reducing the slab thickness to keep a high aspect ratio slot waveguide geometry, which is extremely challenging even for state-of-the-art nano-fabrication techniques. More importantly, the device speeds can therefore be significantly limited by the large slot capacitance and slab resistance due to RC limitations.
In this work, we propose and study a polymer infiltrated P-S-N (“S” refers to the slot) diode capacitor structure for EO phase shifters. In the new configuration, an optical phase shift is realized based on index perturbation both inside the slot via Pockels nonlinearity and within the silicon ridges via the free carrier effect (carrier depletion) [7–9] simultaneously. Therefore, the overall EO overlap can be noticeably improved from that which relies on either of the individual NLO effects alone. Additionally, the device speed can be significantly elevated with the introduction of free carriers as they are found to respond faster to the external driving signal than the E-field due to the smaller experienced transient capacitance. By correctly choosing a suitable polarization type for propagation, the advantage of this improved EO overlapped volume as well as the response speed can be applied, for example, to a class of low aspect ratio slot waveguides, which are also first studied in this work to improve the modulation speed by reducing the time constant of the small signal equivalent circuit. According to our numerical results, the combination of the polymer diode capacitor configuration with the low aspect ratio slot waveguide system leads to a promising method of constructing sub-THz speed optical modulators without sacrificing either modulation efficiency or energy consumption. To be more specific, by optimizing the waveguide geometry in terms of balancing effective index shift and device speed, at least 269 GHz bandwidth can be achieved with a high modulation efficiency of 5.5 V-cm when the diode capacitor is reverse biased by an external radio frequency (RF) voltage signal between the electrodes (optical propagation loss is acceptably low at 4.29 dB). The device figures of merits (FOM) are found to be encouraging enough when compared to either depletion mode or polymer SWG based designs. These are conservative numerical results, and as will be shown, are grounded in well-understood physical phenomena.
Since the device is based on the less tightly confined TM-like mode propagation, its overall EO conversion efficiency may still be considered to be less competitive than that in the slot waveguide only optimized for Pockels nonlinearity. This may lead to a larger device length and/or propagation loss. However, the primary motivation here is to explore and demonstrate a means of constructing a low cost high speed (269 GHz bandwidth) modulation component where, for the first time, the RC limitation is no longer the bottleneck for RF response so that the potential of ultra-high bandwidth polymers can be unleashed. At the same time, by optimizing the waveguide geometry, it is still possible to keep the device length comparable to conventional polymer phase shifters and propagation loss acceptable for optical communication purposes. Additionally, it can be experimentally important that the device does not rely on a high aspect ratio slot, which gives considerable fabrication tolerance in terms of nano-patterning, etching, and cladding (infiltration). Instead, a major portion of the SWG cross section is engaged to explore the ultimate limitations of multiple EO overlaps within the waveguide for highly efficient modulation purposes. Therefore, the usefulness of the electrical configuration shown here in the polymer infiltrated P-S-N profile can also be applied to slow light or optical resonance media, e.g. ring resonators, to realize much more compact silicon modulation/switching  and lasing  components suitable for photonic integrated circuits. Additionally, the device consumes almost zero DC energy and ultra-low AC energy (5.83 picojoule per bit, pJ/bit), which makes it highly appropriate for green photonics-related applications .
The rest of this paper is organized as follows: in Section 2, the electrical configuration of the phase shifter, i.e. the polymer infiltrated P-S-N diode capacitor, and the motivation behind it are introduced. Then, the device physics involving optical mode overlap with multiple NLO effects in low aspect ratio slot waveguide systems is explained in detail. In Section 3, a polarization mode study is first performed to select a suitable polarization type for efficient propagation and switching in the low aspect ratio slot waveguide system, based upon which a sophisticated parametric study of the waveguide geometry is carried out to strike a balance between effective index perturbation, modulation speed, and propagation loss. Then in Section 4, the EO modulation behavior of the device is studied both statically and dynamically for the optimized device from Section 3. In particular, a much higher modulation speed is found due to the introduction of free carrier depletion, where ultra-low energy consumption is detected. Also, transmission line design is discussed to address practical concerns regarding phase and impedance matching for microwave propagation throughout the device. Potential fabrication methods are also discussed after that.
2. Device structure and EO overlap
The phase shifter is based on a strip-loaded vertical slot waveguide, where a thin silicon slab () is maintained as the EO active channel at the Si/SiO2 interface. Electrically, a lateral P-S-N diode capacitor is embedded into the vertical slot waveguide by introducing high level anti-symmetric (opposite types of) impurities into the left and right silicon wings. As shown in Fig. 1(a) , the whole P/N silicon wing is divided into three parts with an ascending level of boron/phosphorous impurity from central slot to side electrodes (denoted as P/N, P+/N+, and P++/N++ regions respectively). Finally, the waveguide and the slot are completely covered and infiltrated with the NLO polymer (for the AJSP-series EO polymers , Pockels coefficient).
2.1 Enhanced EO overlap with the P-S-N diode capacitor
Previously, both uniform and symmetric impurity profiles (same type of dopant is used in both left and right silicon wings) have been incorporated into high aspect ratio slot waveguide based polymer modulators to reduce the resistance of the silicon thin slab . This kind of device relies on strong optical confinement for TE-like polarization (E-field mostly in the x direction) in a vertical slot or TM-like polarization (E-field mostly in the y direction) in a horizontal slot. In either case, the effective index change of the fundamental mode is highly sensitive to the overlap integral of the optical intensity within the slot and the group velocity of the guided mode11,15]. is the refractive index change of the polymer caused by the electrical biasing fieldEq. (1), only a minor portion of the total waveguide cross section is EO active and hence responsible for the effective index and phase shift, which results in a limited EO conversion efficiency.
Compared to the symmetric impurity profile, the anti-symmetric dopant scheme, i.e. the P-S-N diode capacitor, not only reduces potential drop within the silicon wings, but more importantly, enables high level fast carrier depletion within the silicon ridge, where the free carrier index shift can be added constructively to the Pockels index shift of the polymer by correctly aligning the direction of the poling field with the diode capacitor (in the negative x direction for the configuration shown in Fig. 1(a)). Hence, the overall effective index change of the guided mode can be increased to16]Eq. (3), both andare inside the integral in the numerator because of the non-uniform biasing field and carrier contrast distribution within the nonlinear slot and silicon ridges, respectively. Consequently, both the P/N regions and the slot are EO active in the P-S-N diode capacitor configuration, where the overwhelming majority of the optical intensity is confined. Therefore, the total EO overlapped volume can be increased significantly due to the presence of multiple EO active media.
2.2 Combination of the P-S-N diode capacitor with the low aspect ratio slot waveguide
Although the EO response of thenonlinear polymer is almost instantaneous (~femtoseconds , fs), the upper frequency limit of the polymer SWG modulatorlies in the transient evolution of the biasing field inside the slot, which is partially governed by the RC time constant of the small signal equivalent circuit. According to Fig. 1(b), the speed limitation can be formulated asEq. (5). This results in a so-called low aspect ratio slot waveguide. To start with, the slot width and slab thickness are kept at 200 and 90 nm respectively in the reference model (RM), and the rest of the geometric and dopant parameters shown in Fig. 1(a) are listed in Table 1 .
In Fig. 2(a) and 2(b), the fundamental mode of the RM waveguide is calculated for both TE-like and TM-like polarizations using a 3D semi-vectorial beam propagation method (BPM) [18,19], which is accurate enough for calculating each polarization separately instead of the coupling between them . One can tell that in the case of either polarization, the E-field intensity is weakly confined in the central slot and penetrates extensively into the silicon ridges and slabs. In Fig. 2(c), an intensity cutline at the waveguide center (y=0.14 µm)indicates that the nonlinear slot only confines 16.1% of the total intensity for the TE mode and 14.9% for the TM mode, while 28.6% and 55% of the total intensity are distributed in the left and right silicon ridges for TE- and TM-like modes, respectively. Therefore, it is clear that by incorporating a P-S-N diode capacitor configuration, a larger portion of the optical intensity within the silicon ridges can be engaged in the free carrier based EO medium to compensate for the degraded slot confinement factor associated with the low aspect ratio slot waveguide design.
To illustrate this point, the device is reverse biased by an external DC voltage signal (−10 V is used as a benchmark voltage in the following study unless stated otherwise) between the electrodes, and the E-field and free carrier distributions are then predicted by physically-based device modeling from ATLAS , which includes concentration dependent mobility, field dependent mobility, Shockley-Read-Hall (SRH) and Auger recombination, concentration dependent lifetime, and bandgap narrowing models . In Fig. 3(a) , a highly confineddistribution is found within the nonlinear nanogap (only the x component is plotted and is responsible for the EO effect) although a higher degree of non-uniformity is detected due to the widened slot. On the other hand, a sizable depletion region is found in the carrier profile in Fig. 3(b) (only the hole profile is plotted due to the dopant anti-symmetry). The width of the carrier depletion layer in Fig. 3(b) is ~100 nm and is larger than that of the accumulation layer reported previously in forward biased operation , which is limited to.
The corresponding EO index shift profiles due to the Pockels and free carrier effects are calculated and are shown in Fig. 3(c) and 3(d), respectively. In Fig. 3(d) in particular, the profile shows a combined index shift from both hole and electron depletion (electron depletion is not plotted here since it is the mirror image of hole depletion), where an asymmetricprofile is found in the left and right silicon ridges due to the larger free carrier coefficient for holes, shown in Eq. (4). The profile is calculated from the carrier profile under bias since the depletion width at zero bias is negligibly small (discussed in detail in Section 4.1). However, unlike in rib waveguide based EO active components [7,9], it is not viable to take advantage of the higher index perturbation efficiency of the holes by extending the P region into the N region because of the presence of the polymer nanogap, which acts as an optical barrier for mode intensity to change continuously in the x direction. As a result, the confinement factor would decrease quickly while propagation loss would increase for an unbalanced P/N region. Finally, although the localized index change from free carrier effect is much smaller than that from the Pockels effect (comparing Fig. 3(c) and 3(d)), the relative contribution of these two effects to the overall effective index change also depends on the mode distribution according to Eq. (3) and therefore can be very different from the respective localized index change within each governing domain (explained in detail in Section 3).
3. Parameter study and optimization
The two most sensitive geometrical parameters of the low aspect ratio slot waveguide, i.e. slot width and slab thickness, are next studied for optimized tradeoff between individual FOM of the device while other parameters are fixed according to Table 1. Although TE-like polarization is usually preferred for strong optical confinement in vertical slot waveguide systems, the low aspect ratio design proposed in this work may cause significant deformation in its mode shape such that its suitability for propagation and switching can be highly dependent on the respective geometrical dimensions. Because of this, we first consider both TE-like and TM-like polarizations when studying the effect of on the effective index change, device speed, and propagation losses. The same physically based modeling in Section 2 is used here to predict both static carrier distribution and dynamic carrier movement within the silicon waveguide as well as the biasing field profile within the nonlinear slot. The 2D E-field and free carrier profiles are then interpreted intoandprofiles using Eq. (2) and (4), which are inserted into the 3D BPM simulator for the calculation of and losses. In this section, the dynamic carrier movement is used to represent device speed since similar trends are found in the E-field evolution when geometric parameters are varied.
First, is scanned from a lower fabrication limit of 120 nm up to 300 nm whileis kept at 90 nm. It is shown in Fig. 4(a) that a higher(ON state is defined by −10 V) can be achieved in TE-like modes when the slot width is pushing the lower limit, where high EO overlap is available through strong optical confinement within the slot as suggested in previous studies . However, asincreases, its index change reduces quickly and becomes smaller than that of the TM-like modes whenexceeds 160 nm. This reduction in the effective index of the TE-like mode can be partially explained by the reduced overlap between mode intensity and the nonlinear slot. In Fig. 4(b), it is shown that the intensity fraction within the slot decreases from 20% to only 8% for the TE-like polarization while the same fraction increases from 10% to ~20% for the other polarization. Since the nonlinear coefficient is higher for the polymer than the free carriers at the same bias voltage, the overall effect is a degraded. As a result, it is clear that the TM-like polarization is more suitable for high efficiency EO modulation in low aspect ratio slot waveguide systems, which is somewhat counter-intuitive for vertical slot waveguide systems. Still in Fig. 4(a), the polymer component of the effective index change is also plotted which is derived from a symmetric dopant profile waveguide (same geometrical dimensions) with the absence of carrier movement. One can tell, for either polarization, that the total index is improved noticeably with introduction of carrier depletion by the P-S-N diode capacitor profile. In particular, for the TM-like polarization, is optimized at 1.78 × 10−4 whenis kept at 240 nm, where the free carrier effect contributes 45.2% of the total index change due to the large portion (49.3%) of the optical intensity confined within the left/right silicon ridges. For above 240 nm, the optical mode becomes more and more weakly confined within the silicon ridges as an increasing portion of the evanescent field penetrates and interacts with the cladding polymer, which causes the overall decrease inalthough the intensity fraction in slot keeps increasing gradually, as shown in Fig. 4(b). It is also worth noting thatdoes not follow exactly the same trend as intensity fraction within the slot during the slot width scan. This is because effective index shift from the polymer is also proportional to the group index of the guided mode, Eq. (3). As a result, it is shown in Fig. 4(a) that thedecreases whengoes beyond 270 nm (TM-like) due to the faster decrease of the group index (widened slot with lower index) although the intensity fraction continues to increase.
On the other hand, the device speed is predicted by one carrier movement cycle (depletion and recovery) within the silicon waveguide. The carrier concentration (only hole concentration is plotted as the electron concentration is its mirror image) is probed inside the depletion region (x=, y=0.14, in µm) to make sure the dynamic behavior of free carriers is monitored at the same location relative to the edge of the slot, where complete carrier depletion can be achieved and therefore a fair speed comparison can be drawn at the same ON/OFF concentration contrast level. According to Fig. 5 , the 10% to 90% carrier rise time (during which carriers deplete from 90% to 10% of their unbiased level) is almost fixed at 1.1 ps while fall time (during which carriers recover from 10% to 90% of their unbiased level) reduces significantly from 4.3 to 2 ps whenincreases from 120 to 300 nm. The improvement in the carrier response time agrees well with Eq. (5) and therefore confirms that the RC speed limitation can be relieved by reducing the slot capacitance in the low aspect ratio design. The fact that the fall time improvement is much larger than the rise time through the geometric modification reflects the different carrier mechanisms involved within one operation cycle. When the reverse voltage is released at the falling edge, the device undergoes a process akin to the accumulation phase . This will inevitably cause a slower response compared to the rise time, during which the device experiences the depletion phase. The unbalanced effect is more pronounced when the geometry approaches the high aspect ratio end where the RC limitation is more significant.
Considering both optical and electrical performance, theis kept at 240 nm as a tradeoff between speed and effective index change. Then the slab thicknessis varied from 50 to 170 nm for TM-like mode propagation. Unlike the relationship found in the, it is shown in Fig. 6(a) thatdecreases monotonically as the slab thickness increases and more quickly whenexceeds 130 nm, which is around half the waveguide height. It can be told from the mode profile that an increasing portion of the total intensity is found in the silicon slab as it becomes more and more prominent, where the localized index changeis at minimum according to Fig. 3(d). Hence, the EO overlap between optical mode and free carrier perturbation is reduced. On the other hand, the total propagation loss consists of two parts: free carrier absorption and radiation loss . In our BPM calculations, the absorption effect due to the high dopant levels is included in the material property through the imaginary part of the refractive index, where the absorption coefficient is derived from the Drude-Lorenz relationship
The radiation loss mainly results from the coupling of the evanescent field of the guided mode into the polymer cladding and/or the buried oxide layer, which is take into consideration by specifying fully transparent boundary conditions to effectively let the radiated intensity pass the boundary and leave the computational domain . As a result, it is shown in Fig. 6(a) that the total propagation loss increases rapidly withpartially due to the fact that a larger portion of the optical intensity is distributed within the silicon slab (P+/N+ region), where the impurity absorption is much higher according to Eq. (6). The larger propagation loss also comes from the higher radiated intensity into the polymer cladding due to weaker lateral confinement inherent in the lower aspect ratio geometry. On the dynamic side, the rise time changes only slightly with an increase of while the fall time continues to decrease from 2.2 to 1.3 ps, which comes from the reduced slab resistance R2p,n formulated in Eq. (5). As a tradeoff betweenand speed, is kept at 130 nm where the propagation loss is acceptably low at 0.78 dB/mm.
4. Modulation performance and transmission line design
In this section, we demonstrate the EO modulation and microwave (driving signal) propagation behavior of the device based on the optimized phase shifter from Section 3. First in Section 4.1, we find that free carriers respond faster to the external RF voltage signal than the E-field due to the smaller depletion layer capacitance for the former, which reemphasizes the importance of the introduction of free carriers to further improve device speed as well as modulation efficiency. Then in Section 4.2, both static (DC) and dynamic (AC) performance of the device are investigated, where the tradeoff relationship between phase shift and bias voltage as well as the dynamic energy consumption of the device are analyzed in detail. Finally in Section 4.3, practical concerns regarding microwave propagation (impedance and phase matching conditions) are addressed by proper design of the transmission line (TL). Fabrication related issues are also discussed at the end of the section.
4.1 Speed improvement via free carrier effects
The modulation mechanism of the device proposed in this work involves the combination of two different kinds of NLO effects: the E-field based Pockels effect and the free carrier effect. Although the response time of the two respective effects are quite uniform within each domain (polymer slot for Pockels effect and silicon ridge for free carrier effect), the response time of the two effects is not found to be equal. In Fig. 7(a) , a non-return-to-zero (NRZ) voltage pulse (peak-to-peak voltage Vpp=10 V) is applied between the electrodes. The 10% to 90% rise and fall time of (at the center of the slot, i.e. x=0, y=0.14, in µm) are measured to be 2.5 and 2.8 ps respectively, which are slightly more than twice the rise/fall time for free carriers (at the same location as in Section 3). The reason behind this is that the E-field and free carrier experience different transient capacitance and resistance when externally driven by the same voltage pulse. The slot and depletion layer capacitance per unit length can be formulated as follows for E-field and free carrier respectively22]Equation (8) is modified from the depletion region modeling of a biased metal-insulator-semiconductor (MIS) capacitor, where it can be calculated that the depletion width under zero bias is negligibly small due to the abrupt discontinuity of the work function between doped silicon and polymer insulator (therefore, built-in potential is ~0). In Fig. 7(b), a carrier level cutline is made across the center of the waveguide (−0.365<x<0.365, y=0.14, in µm) where optical intensity is at maximum to demonstrate the depletion width shift of the diode capacitor under bias. It is confirmed that the depletion width at zero bias is only negligible and therefore the total carrier contrast between ON and OFF states can be approximated by the depletion width under reverse bias. Due to the same dopant level being used in both P and N regions, a symmetric depletion region is found where the depletion width at the P/N side (at −10 V) is around 120 nm and the total depletion width is comparable to the slot width. Therefore, for free carriers, the depletion layer capacitanceis noticeably reduced compared tofor the E-field. Similarly, for free carriers, the silicon ridge resistance is also reduced to
The overall effect is a reduced time constant and hence improved device speed associated with the free carriers, which was previously limited by the E-field. According to the rise/fall time of the free carriers, the device 3-dB bandwidth is estimated to be at least 269 GHz using [25,26], where the E-field ON/OFF contrast suffers by 34% from its maximum value according to Fig. 7(a) and the resultantdecreases from 1.65×10−4 to 1.4×10−4 by BPM calculation. Therefore, the device length associated with a π-phase shift () increases from 4.7 to 5.5 mm, which results in a phase modulation efficiency. The modulation efficiency shown here is comparable to state-of-the art polymer phase shifters while the speed potential of our device is almost 2 orders of magnitude higher . It is important to point out that the abovementioned device figures of merit are highly conservative to illustrate the idea of a phase shifter working with multiple NLO effects. Therefore, only a modest Pockels coefficient is used here while recent progress in polymer science has demonstrated much higher nonlinear coefficients with high thermal stability: in hybrid polymer/sol-gel  and in self-assembled molecular glasses . In either case, the modulation efficiency can be significantly improved without sacrificing the high speed demonstrated in the low aspect ratio slot waveguide design. Total propagation loss is calculated as 4.29 dB from Section 3.2.
4.2 Modulation performance investigation
The performance of the phase shifter is investigated both statically and dynamically when its active length is kept at 5.5 mm. First, a DC voltage scan is applied from 0 to −12 V to study the tradeoff relationship between phase shift and bias voltage. As shown in Fig. 8(a) , an almost linear dependence of the phase shift on the bias voltage is found at the two ends of the voltage scan where carrier depletion is either well below its turn-on threshold or reaches full capacity and the phase shift increases only with an increase of within the nonlinear slot according to Eq. (3). At the center part of the scan (2<<10), the phase shift increases nonlinearly due to the combined effects of E-field and free carrier contrast improvement. The S-shaped curve indicates the bias voltage can be further reduced at the expense of a longer device length to practically reduce the difficulty in designing the driver circuitry for sub-THz modulation.
Then dynamically, the device is biased with a NRZ signal with Vpp=10 V. The period and duty cycle of the signal are 2.4 ps and 46% respectively to match the rise/fall time of free carriers to maximize EO conversion efficiency, as shown in Fig. 8(b). Although the NRZ signal with frequency up to ~400 GHz is still quite a challenge for state-of-the-art RF signal generators, the rectangular pulse has been a well-acknowledged way of numerically predicting dynamic device performance with speeds up to 1 THz , and therefore serves as a reliable guide to related experiments that will become achievable in the near future. The resultant transient evolution of the phase shift is shown in Fig. 8(c) where smooth transitions are found at the voltage transit points as carrier depletion approaches its saturated level with a gradually reduced depletion rate, which agrees with previous study of a depletion mode phase shifter . Although the diode capacitor experiences negligibly small DC leakage current and therefore consumes almost zero DC energy due to the insulating slot, noticeable AC current flow is found in capacitive characteristics as the RF energy flows back and forth between the electrodes. In Fig. 8(d), sharp turning points of the AC current are found at each 0-1 voltage transit point as the free carriers are accelerated with soaring momentum upon the sudden potential drop between the electrodes. Since energy consumption only happens at 0-1 transitions where both current and voltage are non-zero, the averaged energy consumption per unit length is calculated as low as 1.06 fJ/µm·bit through Eq. (10) , while the total dynamic (AC) energy consumption of the device is 5.83 pJ/bit by multiplying by the device active length.
4.3 Microwave propagation analysis
Generally, for EO active devices such as the phase shifter studied in this work, there are both electrical and optical waves propagating co-directionally through the device. Therefore, the transmission line needs to be properly designed to avoid undesired interference between them. The interference may come from three aspects. First, the aluminum electrodes can be a significant source of optical absorption for the guided mode. Here in this work, this potential metallic loss is avoided by keeping a large horizontal clearance between electrodes and waveguide (1 µm as shown in Tab. 1), where according to Fig. 2(c), the optical intensity drops by more than 25 dB for TM-like polarization and 14 dB for TE-like polarization (similar results are found for the optimized geometry). Vertically, the electrodes are placed higher than the waveguide by an amount that equals waveguide height, where the total intensity drop can be even larger. Therefore, the spacing (horizontal and vertical) is safe enough to keep metallic loss negligibly small as also suggested in previous studies of EO modulators based on a similar structure [14,31].
Second, since the travelling-wave electrodes extend much longer in the propagation direction than in the transverse direction, the device speed can be further limited by the walk-off bandwidth that is determined by the group velocity difference between electrical and optical waves in the following relationship  (assuming a π phase difference in the electrical and optical signal envelops)
An equivalent circuit for infinitesimal TL segments is shown in Fig. 9 , where RTL, CTL, and LTL are the resistance, capacitance, and inductance of the unloaded TL, respectively . From Fig. 9, the effective index of the TL can therefore be calculated as14].
Experimentally, refractive index difference has been reported for nonlinear polymers before and after poling, and several methods have been proposed to measure the exact index of poled polymers [32,33]. Therefore, even for the optimized TL design (), the walk-off bandwidth may be reduced due to the change from the polymer index shift from the poling effect, denoted. Based on previous studies, here we studied a probable index variation range from −0.1 to 0.1 while the exact amount of index change is highly dependent on the combination of the poling field, time, and temperature, and therefore yet to be measured in a real experiment. The effective index change of the guided mode is calculated accordingly by the BPM simulator to further calculate the walk-off bandwidth using Eq. (11) andin a low dispersion environment. As shown in Fig. 10 , the walk-off bandwidth for both ON and OFF states overlaps, and is reduced to 20 and 13 THz when goes up to −0.1 and 0.1 respectively, which is at least 48 times larger than the operating speed of the device. Therefore, the device speed is still primarily governed by the RC effect of the RF circuitry and highly tolerant to polymer index shift due to the poling effect.
For the slot waveguide with optimized TL design (phase matching condition satisfied), the TL impedance increases almost linearly with the slot width . As a result, for the optimized waveguide geometry (), the is estimated to be around 42 Ω (extrapolated from the finite element solver for Eq. (13)). Therefore, the optimized phase shifter can be connected to a 42 Ω RF terminator (or a more commonly available 50 Ω one) in order to minimize the RF rejection (reflection) resulting from possible impedance mismatch.
Finally, device fabrication can be completely based on well-developed CMOS compatible processing techniques. The fabrication starts with a silicon-on-insulator (SOI) wafer where the dopant regions can first be patterned and defined by deep ultraviolet (DUV) lithography and ion implantation (followed by rapid thermal annealing, RTA). Then, the SWG is defined by reactive ion etching (RIE) where the thin silicon slab can be maintained by controlling the etching rate and time of the process. Another RIE process is needed after that to open a complete slot to the silicon/insulator interface. Then, the aluminum contacts can be deposited by e-beam evaporation followed by annealing to enhance conductivity. After that, the device is spin-coated with EO polymer and the polymer needs to be cleared from the contact pad by laser ablation . As a final step, the polymer is poled by applying a certain combination of poling field, time, and temperature.
In conclusion, we study a polymer-infiltrated P-S-N diode capacitor structure for high speed electro-optic phase shifters. The structure explores a way of constructing a low cost high speed modulation component where the RC limitation is effectively alleviated in a group of low aspect ratio slot waveguides. By incorporating multiple nonlinear effects within the waveguide cross section, our results show that a much higher 3-dB bandwidth of 269 GHz modulation is achievable at a high efficiency of 5.5 V-cm for the optimized waveguide geometry than can be accomplished with any single nonlinear effect alone. The encouraging device performance relies on the high EO overlap within a major portion of the waveguide cross section rather than strong optical confinement of the nano-slot which formed the basis of previous studies. Therefore, fabrication tolerance for nano-patterning, etching, and infiltration has been significantly relieved. Additionally, the device consumes almost zero DC energy and ultra-low AC energy and hence is suitable for green photonics related applications.
This work was supported in part by Singapore’s A*Star Science and Engineering Research Council (SERC) grant 0921010049.
References and links
1. C. Koos, P. Vorreau, T. Vallaitis, P. Dumon, W. Bogaerts, R. Baets, B. Esembeson, I. Biaggio, T. Michinobu, F. Diederich, W. Freude, and J. Leuthold, “All-optical high-speed signal processing with silicon-organic hybrid slot waveguides,” Nat. Photonics 3(4), 216–219 (2009). [CrossRef]
2. J.-M. Brosi, C. Koos, L. C. Andreani, M. Waldow, J. Leuthold, and W. Freude, “High-speed low-voltage electro-optic modulator with a polymer-infiltrated silicon photonic crystal waveguide,” Opt. Express 16(6), 4177–4191 (2008). [CrossRef] [PubMed]
3. T. Gorman, S. Haxha, H. Ademgil, and J. J. Ju, “Ultra-high-speed deeply etched electrooptic polymer modulator,” IEEE J. Quantum Electron. 44(12), 1180–1187 (2008). [CrossRef]
4. T. B. Jones, B. Penkov, J. Huang, P. Sullivan, J. Davis, J. Takayesu, J. Luo, T.-D. Kim, L. Dalton, A. Jen, M. Hochberg, and A. Scherer, “Nonlinear polymer-clad silicon slot waveguide modulator with a half wave voltage of 0.25 V,” Appl. Phys. Lett. 92(16), 163303 (2008). [CrossRef]
6. C. Koos, L. Jacome, C. Poulton, J. Leuthold, and W. Freude, “Nonlinear silicon-on-insulator waveguides for all-optical signal processing,” Opt. Express 15(10), 5976–5990 (2007). [CrossRef] [PubMed]
7. F. Y. Gardes, A. Brimont, P. Sanchis, G. Rasigade, D. Marris-Morini, L. O’Faolain, F. Dong, J. M. Fedeli, P. Dumon, L. Vivien, T. F. Krauss, G. T. Reed, and J. Martí, “High-speed modulation of a compact silicon ring resonator based on a reverse-biased pn diode,” Opt. Express 17(24), 21986–21991 (2009). [CrossRef] [PubMed]
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 0.25µm silicon-on-insulator waveguides,” Opt. Express 18(8), 7994–7999 (2010). [CrossRef] [PubMed]
12. B. Jalali, S. Fathpour, and K. Tsia, “Green silicon photonics,” Opt. Photon. News 20(6), 18–23 (2009). [CrossRef]
13. R. Ding, T. Baehr-Jones, Y. Liu, R. Bojko, J. Witzens, S. Huang, J. Luo, S. Benight, P. Sullivan, J. M. Fedeli, M. Fournier, L. Dalton, A. Jen, and M. Hochberg, “Demonstration of a low V π L modulator with GHz bandwidth based on electro-optic polymer-clad silicon slot waveguides,” Opt. Express 18(15), 15618–15623 (2010). [CrossRef] [PubMed]
14. J. Witzens, T. Baehr-Jones, and M. Hochberg, “Design of transmission line driven slot waveguide Mach-Zehnder interferometers and application to analog optical links,” Opt. Express 18(16), 16902–16928 (2010). [CrossRef] [PubMed]
16. R. A. Soref and B. R. Bennett, “Electro-optical effects in silicon,” IEEE J. Quantum Electron. 23(1), 123–129 (1987). [CrossRef]
20. BeamProp 7.0 user guide, RSoft Design Group, Inc., Ossining, NY.
21. ATLAS user’s manual, SILVACO International, Santa Clara, CA.
22. S. M. Sze and K. K. Ng, Physics of Semiconductor Devices, 3rd ed. (John Wiley & Sons, Inc., New Jersey, 2007).
23. C. A. Barrios and M. Lipson, “Modeling and analysis of high-speed electro-optic modulation in high confinement silicon waveguides using metal-oxide-semiconductor configuration,” J. Appl. Phys. 96(11), 6008–6015 (2004). [CrossRef]
24. A. G. Rickman, G. T. Reed, and F. Namavar, “Silicon-on-insulator optical rib waveguide loss and mode characteristics,” J. Lightwave Technol. 12(10), 1771–1776 (1994). [CrossRef]
25. G. P. Agrawal, Fiber-Optic Communication Systems, 2nd ed. (John Wiley & Sons, Inc., New Jersey, 1997).
26. C. E. Png, G. H. Park, S. T. Lim, E. P. Li, A. J. Danner, K. Ogawa, and Y. T. Tan, “Electrically controlled silicon-based photonic crystal chromatic dispersion compensator with ultralow power consumption,” Appl. Phys. Lett. 93(6), 061111 (2008). [CrossRef]
27. Y. Enami, C. T. Derose, D. Mathine, C. Loychik, C. Greenlee, R. A. Norwood, T. D. Kim, H. Luo, Y. Tian, A. K.-Y. Jen, and N. Peyghambarian, “Hybird polymer/sol-gel waveguide modulators with exceptionally large electro-optic coefficients,” Nat. Photonics 1(3), 180–185 (2007). [CrossRef]
28. T. D. Kim, J. W. Kang, J. Luo, S. H. Jang, J. W. Ka, N. Tucker, J. B. Benedict, L. R. Dalton, T. Gray, R. M. Overney, D. H. Park, W. N. Herman, and A. K. Jen, “Ultralarge and thermally stable electro-optic activities from supramolecular self-assembled molecular glasses,” J. Am. Chem. Soc. 129(3), 488–489 (2007). [CrossRef] [PubMed]
29. M. Hochberg, T. Baehr-Jones, G. X. Wang, M. Shearn, K. Harvard, J. Luo, B. Chen, Z. Shi, R. Lawson, P. Sullivan, A. K. Y. Jen, L. Dalton, and A. Scherer, “Terahertz all-optical modulation in a silicon-polymer hybrid system,” Nat. Mater. 5(9), 703–709 (2006). [CrossRef] [PubMed]
30. M. R. Watts, D. C. Trotter, R. W. Young, and A. L. Lentine, “Ultralow power silicon microdisk modulators and switches,” in Proceedings of 5th IEEE International Conference on Group IV Photonics (IEEE 2008), pp. 4 - 6.
31. J. Leuthold, W. Freude, J.-M. Brosi, R. Baets, P. Dumon, I. Biaggio, M. L. Scimeca, F. Diederich, B. Frank, and C. Koos, “Silicon organic hybrid technology-a platform for practical nonlinear optics,” Proc. IEEE 97(7), 1304–1316 (2009). [CrossRef]
32. H. S. Lee, T. D. Kang, H. Lee, S. K. Lee, J. H. Kim, and D. H. Choi, “Ellipsometric study of the poling effect on nonlinear-optical side-chain polymers containing disperse red 1,” J. Appl. Phys. 102(1), 013514 (2007). [CrossRef]
33. J. T. Gallo, T. Kimura, S. Ura, T. Suhara, and H. Nishihara, “Method for characterizing poled-polymer waveguides for electro-optic integrated-optical-circuit applications,” Opt. Lett. 18(5), 349–351 (1993). [CrossRef] [PubMed]
34. M. Gould, T. Baehr-Jones, R. Ding, S. Huang, J. Luo, A. K. Jen, J. M. Fedeli, M. Fournier, and M. Hochberg, “Silicon-polymer hybrid slot waveguide ring-resonator modulator,” Opt. Express 19(5), 3952–3961 (2011). [CrossRef] [PubMed]