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Optics Express

Optics Express

  • Editor: C. Martijn de Sterke
  • Vol. 19, Iss. 15 — Jul. 18, 2011
  • pp: 14354–14369
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A high speed electro-optic phase shifter based on a polymer-infiltrated P-S-N diode capacitor

Maoqing Xin, Ching Eng Png, Soon Thor Lim, Vivek Dixit, and Aaron J. Danner  »View Author Affiliations


Optics Express, Vol. 19, Issue 15, pp. 14354-14369 (2011)
http://dx.doi.org/10.1364/OE.19.014354


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Abstract

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

1. Introduction

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

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]

4

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]

]. 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 [1

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]

]. 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

5. V. R. Almeida, Q. Xu, C. A. Barrios, and M. Lipson, “Guiding and confining light in void nanostructure,” Opt. Lett. 29(11), 1209–1211 (2004). [CrossRef] [PubMed]

,6

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]

]. 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

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

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]

] 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.

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 (nsi=3.472) 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)
Fig. 1 (a) 2D schematic cross section of the device showing the lateral polymer infiltrated P-S-N diode capacitor configuration. (b) The small signal equivalent circuit corresponding to (a) showing the RC limitation of the phase shifter.
, 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 (npoly=1.54for the AJSP-series EO polymers [13

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]

], Pockels coefficientr33=65pm/V).

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 [14

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]

]. 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 mode
Δneff=ΓSΔnpoly=ngnSSε|E|2dxdyε|E|2dxdyΔnpoly
(1)
whereΓSis the confinement factor of the slot;ngandnSare the group index of the mode and bulk index of the active region (slot) respectively [11

11. K. Preston and M. Lipson, “Slot waveguides with polycrystalline silicon for electrical injection,” Opt. Express 17(3), 1527–1534 (2009). [CrossRef] [PubMed]

,15

15. J. T. Robinson, K. Preston, O. Painter, and M. Lipson, “First-principle derivation of gain in high-index-contrast waveguides,” Opt. Express 16(21), 16659–16669 (2008). [CrossRef] [PubMed]

]. Δnpolyis the refractive index change of the polymer caused by the electrical biasing field
Δnpoly=npoly3r33Ebias/2npoly3r33Vbias/2Wslot
(2)
where Vbiasis the external bias voltage. As one can tell from Eq. (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 to
ΔneffΔneff,poly+Δneff,si=ngnSSε|E|2Δnpolydxdyε|E|2dxdy+ngnRRε|E|2Δnsidxdyε|E|2dxdy
(3)
where nRis the bulk index of the silicon ridges. The total index change consists of two parts, Δneff,polyandΔneff,si, from the Pockels effect inside the nonlinear slot and the free carrier effect within the P/N silicon ridges, respectively. Δnsiis the localized silicon index change from the Drude-Lorenz model and at a wavelength of 1.55 µm [16

16. R. A. Soref and B. R. Bennett, “Electro-optical effects in silicon,” IEEE J. Quantum Electron. 23(1), 123–129 (1987). [CrossRef]

]
Δnsi=8.8×1022Δne8.5×1018(Δnh)0.8
(4)
whereΔneandΔnhare electron and hole contrast, respectively. In Eq. (3), both ΔnpolyandΔnsiare 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 theχ(2)nonlinear polymer is almost instantaneous (~femtoseconds [17

17. H. Cao, T. F. Heinz, and A. Nahata, “Electro-optic detection of femtosecond electromagnetic pulses by use of poled polymers,” Opt. Lett. 27(9), 775–777 (2002). [CrossRef] [PubMed]

], fs), the upper frequency limit of the polymer SWG modulatorfULlies in the transient evolution of the biasing field Ebiasinside the slot, which is partially governed by the RC time constant τRCof the small signal equivalent circuit. According to Fig. 1(b), the speed limitation can be formulated as
fUL=12πτRC=12πi=13(Rip+Rin)C1Wslothslab2πεpoly(ρp+Lp++ρn+Ln+)hWG
(5)
whereρp+,n+and Lp+,n+are the resistivity and length of the P+/N+ region, respectively. Therefore, to unleash speed potential by reducingτRC, the slot capacitance C1 and slab resistance R2p,n need to be further reduced, which can be achieved by increasing the slot width Wslotand/or slab thicknesshslabrelative to the waveguide height hWGas one can tell from Eq. (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

Table 1. List of the major design parameters in the RM device.

table-icon
View This Table
.

In Fig. 2(a)
Fig. 2 The fundamental mode profile cross section of the RM SWG: (a) TE-like polarization and (b) TM-like polarization. (c) E-field intensity cutlines of the TE-like and TM-like mode profiles in the x direction across the waveguide center, which show a large portion of the total intensity located within the silicon ridges.
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

18. F. Y. Gardes, G. T. Reed, N. G. Emerson, and C. E. Png, “A sub-micron depletion-type photonic modulator in Silicon On Insulator,” Opt. Express 13(22), 8845–8854 (2005). [CrossRef] [PubMed]

,19

19. P. Muellner, M. Wellenzohn, and R. Hainberger, “Nonlinearity of optimized silicon photonic slot waveguides,” Opt. Express 17(11), 9282–9287 (2009). [CrossRef] [PubMed]

], which is accurate enough for calculating each polarization separately instead of the coupling between them [20

20. BeamProp 7.0 user guide, RSoft Design Group, Inc., Ossining, NY.

]. 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 [21

21. ATLAS user’s manual, SILVACO International, Santa Clara, CA.

], which includes concentration dependent mobility, field dependent mobility, Shockley-Read-Hall (SRH) and Auger recombination, concentration dependent lifetime, and bandgap narrowing models [22

22. S. M. Sze and K. K. Ng, Physics of Semiconductor Devices, 3rd ed. (John Wiley & Sons, Inc., New Jersey, 2007).

]. In Fig. 3(a)
Fig. 3 (a) E-field (x component) profile within the nonlinear slot when the diode capacitor is reverse biased. (b) Free carrier (hole) concentration profile within the silicon ridges under the same bias. (c) Localized index perturbation corresponding to (a) via Pockels effect. (d) Localized index perturbation corresponding to a combined effort of hole and electron depletion via free carrier effect (values below the lower end of the color scale are shown in black).
, a highly confinedEbiasdistribution 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 [23

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]

], which is limited toC1Vbias/Δne,h.

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 asymmetricΔnsiprofile is found in the left and right silicon ridges due to the larger free carrier coefficient for holes, shown in Eq. (4). The Δnsiprofile 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

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

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]

], 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

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=0.05Wslot/2, 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
Fig. 5 The carrier concentration evolution with time indicating a reduced device response time with an increase in Wslot.
, 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 whenWslotincreases 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 [23

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]

]. 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, theWslotis kept at 240 nm as a tradeoff between speed and effective index change. Then the slab thicknesshslabis varied from 50 to 170 nm for TM-like mode propagation. Unlike the relationship found in theWslot, it is shown in Fig. 6(a)
Fig. 6 (a) ∆neff decreases as propagation loss increases with an increase of hslab. (b) The device response time is further reduced with an increase of hslab.
thatΔneffdecreases monotonically as the slab thickness increases and more quickly whenhslabexceeds 130 nm, which is around half the waveguide heighthWG. 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 changeΔnsiis 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 [24

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]

]. 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

Δαsi=8.5×1018Δne+6.0×1018Δnh.
(6)

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 [20

20. BeamProp 7.0 user guide, RSoft Design Group, Inc., Ossining, NY.

]. As a result, it is shown in Fig. 6(a) that the total propagation loss increases rapidly withhslabpartially 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 hslabwhile 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 betweenΔneffand speed, hslabis kept at 130 nm where the propagation loss is acceptably low at 0.78 dB/mm.

4. Modulation performance and transmission line design

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)
Fig. 7 (a) Faster response is found for free carriers than E-field due to the reduced transient capacitance. (b) A carrier cutline along the x direction indicating the depletion width shift under reverse bias.
, 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 Ebias (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 respectively
Cslot=εpolyhWGWslot1/CD=1/Cslot+1/CDp+1/CDn=WslotεpolyhWG+WDεsihWG
(7)
whereWDis the biased depletion width following the relationship [22

22. S. M. Sze and K. K. Ng, Physics of Semiconductor Devices, 3rd ed. (John Wiley & Sons, Inc., New Jersey, 2007).

]
WD=WDp+WDnWDp,nεsi2Cslot2+2εsi|Vbias|qNP,NεsiCslot
(8)
whereNP,Nis the dopant concentration of the P/N region. 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 capacitanceCDis noticeably reduced compared toCslotfor the E-field. Similarly, for free carriers, the silicon ridge resistance is also reduced to

R1p,n=ρp,nWWGWDp,nhWG.
(9)

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 usingf3dB=0.35/max(tr,tf) [25

25. G. P. Agrawal, Fiber-Optic Communication Systems, 2nd ed. (John Wiley & Sons, Inc., New Jersey, 1997).

,26

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]

], where the E-field ON/OFF contrast suffers by 34% from its maximum value according to Fig. 7(a) and the resultantΔneffdecreases from 1.65×10−4 to 1.4×10−4 by BPM calculation. Therefore, the device length associated with a π-phase shift (Lπ=λ/2Δneff) increases from 4.7 to 5.5 mm, which results in a phase modulation efficiencyVπLπ=5.5V-cm. 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 [13

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]

]. 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: r33=170pm/Vin hybrid polymer/sol-gel [27

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]

] and r33=300pm/Vin self-assembled molecular glasses [28

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]

]. 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)
Fig. 8 (a) The tradeoff relationship between bias voltage and phase shift. (b) The RF driving signal with Vpp=10 V and period of 2.4 ps. (c) The modulated phase shift in response to (b). (d) The transient current flow within the P-S-N diode capacitor in response to (b).
, 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 Ebiaswithin the nonlinear slot according to Eq. (3). At the center part of the scan (2<|Vbias|<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.

Ebit=1/401VbiasIdt
(10)

4.3 Microwave propagation analysis

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 [2

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]

] (assuming a π phase difference in the electrical and optical signal envelops)
fwalkoff,3dB=0.5vg,optLπ|1vg,opt/vg,elec|
(11)
wherevg,optandvg,elecare the group velocity of optical and electrical waves respectively. Therefore, to maximize walk-off bandwidth, the group index of the microwave TL needs to be designed to equal that of the SWG to satisfy the so-called phase matching condition, where the device bandwidth is mostly limited by the intrinsic RC time constant of the SWG, CDi=13(Rip+Rin).

An equivalent circuit for infinitesimal TL segments is shown in Fig. 9
Fig. 9 Equivalent circuit of the TL loaded with the P-S-N diode capacitor.
, where RTL, CTL, and LTL are the resistance, capacitance, and inductance of the unloaded TL, respectively [14

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]

]. From Fig. 9, the effective index of the TL can therefore be calculated as
nTL,eff=c0CTLLTL+CDLTL11+ifBW
(12)
wherec0is the speed of light in vacuum; f and BW are the operating frequency and intrinsic bandwidth of the SWG (1/2πCDi=13(Rip+Rin)), respectively. For the TL loaded with the P-S-N diode capacitor, the RF signal can be significantly slowed down due to the large depletion capacitanceCDand TL self-inductanceLTL so thatnTL,effcan be much larger than the effective index of the waveguide (neff =2.02 for the optimized waveguide geometry). In this work,nTL,effis lowered by first employing a low aspect ratio slot waveguide design to reduce the depletion layer capacitanceCD and operating the device at a much higher speed (higher RF frequency). Then, the width and height of the electrodes are increased to 50 and 10 µm respectively to reduce theLTLso that the totalnTL,effcan be reduced to matchneffaccording to the extrapolated finite element calculation performed in ref [14

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]

].

Third, the characteristic impedance for the loaded TL can be formulated in Eq. (13) using the equivalent circuit given in Fig. 9.

Zc=121CTLLTL+CDLTL(1+ifBW)
(13)

For the slot waveguide with optimized TL design (phase matching condition satisfied), the TL impedance Zcincreases almost linearly with the slot width [14

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]

]. As a result, for the optimized waveguide geometry (Wslot=240nm), theZc 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 [34

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]

]. As a final step, the polymer is poled by applying a certain combination of poling field, time, and temperature.

5. Conclusion

Acknowledgments

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]

5.

V. R. Almeida, Q. Xu, C. A. Barrios, and M. Lipson, “Guiding and confining light in void nanostructure,” Opt. Lett. 29(11), 1209–1211 (2004). [CrossRef] [PubMed]

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]

8.

M. Xin, C. E. Png, and A. J. Danner, “Breakdown delay-based depletion mode silicon modulator with photonic hybrid-lattice resonator,” Opt. Express 19(6), 5063–5076 (2011). [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]

10.

Q. Xu, B. Schmidt, S. Pradhan, and M. Lipson, “Micrometre-scale silicon electro-optic modulator,” Nature 435(7040), 325–327 (2005). [CrossRef] [PubMed]

11.

K. Preston and M. Lipson, “Slot waveguides with polycrystalline silicon for electrical injection,” Opt. Express 17(3), 1527–1534 (2009). [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]

15.

J. T. Robinson, K. Preston, O. Painter, and M. Lipson, “First-principle derivation of gain in high-index-contrast waveguides,” Opt. Express 16(21), 16659–16669 (2008). [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]

17.

H. Cao, T. F. Heinz, and A. Nahata, “Electro-optic detection of femtosecond electromagnetic pulses by use of poled polymers,” Opt. Lett. 27(9), 775–777 (2002). [CrossRef] [PubMed]

18.

F. Y. Gardes, G. T. Reed, N. G. Emerson, and C. E. Png, “A sub-micron depletion-type photonic modulator in Silicon On Insulator,” Opt. Express 13(22), 8845–8854 (2005). [CrossRef] [PubMed]

19.

P. Muellner, M. Wellenzohn, and R. Hainberger, “Nonlinearity of optimized silicon photonic slot waveguides,” Opt. Express 17(11), 9282–9287 (2009). [CrossRef] [PubMed]

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]

OCIS Codes
(160.4330) Materials : Nonlinear optical materials
(230.2090) Optical devices : Electro-optical devices
(250.5300) Optoelectronics : Photonic integrated circuits
(250.7360) Optoelectronics : Waveguide modulators

ToC Category:
Optoelectronics

History
Original Manuscript: May 9, 2011
Revised Manuscript: June 29, 2011
Manuscript Accepted: July 1, 2011
Published: July 12, 2011

Citation
Maoqing Xin, Ching Eng Png, Soon Thor Lim, Vivek Dixit, and Aaron J. Danner, "A high speed electro-optic phase shifter based on a polymer-infiltrated P-S-N diode capacitor," Opt. Express 19, 14354-14369 (2011)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-19-15-14354


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References

  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]
  5. V. R. Almeida, Q. Xu, C. A. Barrios, and M. Lipson, “Guiding and confining light in void nanostructure,” Opt. Lett. 29(11), 1209–1211 (2004). [CrossRef] [PubMed]
  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]
  8. M. Xin, C. E. Png, and A. J. Danner, “Breakdown delay-based depletion mode silicon modulator with photonic hybrid-lattice resonator,” Opt. Express 19(6), 5063–5076 (2011). [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]
  10. Q. Xu, B. Schmidt, S. Pradhan, and M. Lipson, “Micrometre-scale silicon electro-optic modulator,” Nature 435(7040), 325–327 (2005). [CrossRef] [PubMed]
  11. K. Preston and M. Lipson, “Slot waveguides with polycrystalline silicon for electrical injection,” Opt. Express 17(3), 1527–1534 (2009). [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]
  15. J. T. Robinson, K. Preston, O. Painter, and M. Lipson, “First-principle derivation of gain in high-index-contrast waveguides,” Opt. Express 16(21), 16659–16669 (2008). [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]
  17. H. Cao, T. F. Heinz, and A. Nahata, “Electro-optic detection of femtosecond electromagnetic pulses by use of poled polymers,” Opt. Lett. 27(9), 775–777 (2002). [CrossRef] [PubMed]
  18. F. Y. Gardes, G. T. Reed, N. G. Emerson, and C. E. Png, “A sub-micron depletion-type photonic modulator in Silicon On Insulator,” Opt. Express 13(22), 8845–8854 (2005). [CrossRef] [PubMed]
  19. P. Muellner, M. Wellenzohn, and R. Hainberger, “Nonlinearity of optimized silicon photonic slot waveguides,” Opt. Express 17(11), 9282–9287 (2009). [CrossRef] [PubMed]
  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]

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