Ultra-low power modulators using MOS depletion in a high-Q SiO<sub>2</sub>-clad silicon 2-D photonic crystal resonator

doi:10.1364/OE.18.019129

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Ultra-low power modulators using MOS depletion in a high-Q SiO₂-clad silicon 2-D photonic crystal resonator

Sean P. Anderson and Philippe M. Fauchet »View Author Affiliations

Optics Express, Vol. 18, Issue 18, pp. 19129-19140 (2010)
http://dx.doi.org/10.1364/OE.18.019129

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Abstract

In modulators that rely on changing refractive index, switching energy is primarily dependent upon the volume of the active optical mode. Photonic crystal microcavities can exhibit extremely small mode volumes on the order of a single cubic wavelength with Q values above 10⁶. In order to be useful for integration, however, they must be embedded in oxide, which in practice reduces Q well below 10³, significantly increasing switching energy. In this work we show that it is possible to create a fully oxide-clad microcavity with theoretical Q on the order of 10⁵. We further show that by using MOS charge depletion this microcavity can be the basis for a modulator with a switching energy as low as 1 fJ/bit.

1. Introduction

On-chip interconnects are limited in terms of performance by a number of factors including bandwidth density and signal latency. They are also limited in terms of power consumption, which is increasingly becoming the primary limitation on chip performance. All of these limitations stem from the electrical nature of the interconnects, which requires that the entire wire length of a given interconnect segment be charged in order to send a single bit of information. Although a wide variety of architectural, signaling and materials innovations have led to significant improvements in interconnect performance over the past several years [1

T. Gupta, Copper Interconnect Technology (Springer, New York, 2009).

], the fraction of the chip’s total power budget that is devoted to interconnects has continued to increase rapidly [2

D. A. B. Miller, “Device requirements for optical interconnects to silicon chips,” Proc. IEEE 97, 1166–1185 (2009). [CrossRef]

R. G. Beausoleil, P. J. Kuekes, G. S. Snider, S. Y. Wang, and R. S. Williams, “Nanoelectronic and nanophotonic interconnect,” Proc. IEEE 96(2), 230–247 (2008). [CrossRef]

]. Even with continued advances in interconnect technology, it is unlikely that electrical interconnects will be able to keep pace with the performance requirements of future generations of microprocessors. It is therefore necessary to look toward alternative signaling technologies for on-chip use.

Optical interconnects (OIs) have already been proposed [4

J. W. Goodman, F. J. Leonberger, K. Sun-Yuan, and R. A. Athale, “Optical interconnections for VLSI systems,” Proc. IEEE 72(7), 850–866 (1984). [CrossRef]

D. A. B. Miller, “Optics for low-energy communication inside digital processors: quantum detectors, sources, and modulators as efficient impedance converters,” Opt. Lett. 14(2), 146–148 (1989). [CrossRef] [PubMed]

] as a replacement for electrical interconnects, and it has been shown that optics can have performance superior to electrical in terms of both latency and bandwidth density [6

M. Haurylau, G. Chen, H. Chen, J. Zhang, N. A. Nelson, D. H. Albonesi, E. G. Friedman, and P. M. Fauchet, “On-chip optical interconnect roadmap: Challenges and critical directions,” IEEE J. Sel. Top. Quantum Electron. 12(6), 1699–1705 (2006). [CrossRef]

G. Chen, H. Chen, M. Haurylau, N. A. Nelson, D. H. Albonesi, P. M. Fauchet, and E. G. Friedman, “Predictions of CMOS compatible on-chip optical interconnect,” Integr. VLSI J. 40(4), 434–446 (2007). [CrossRef]

]. The issue of power consumption, however, remains an ongoing challenge, and has only relatively recently become a focus of OI research [2

D. A. B. Miller, “Device requirements for optical interconnects to silicon chips,” Proc. IEEE 97, 1166–1185 (2009). [CrossRef]

R. G. Beausoleil, P. J. Kuekes, G. S. Snider, S. Y. Wang, and R. S. Williams, “Nanoelectronic and nanophotonic interconnect,” Proc. IEEE 96(2), 230–247 (2008). [CrossRef]

]. Because the modulators, which convert the electrical signal to the optical domain, are estimated to be the most power-hungry component of OIs, much recent work has focused on reducing their power consumption [6

]. A target switching energy of 10 fJ/bit [2

D. A. B. Miller, “Device requirements for optical interconnects to silicon chips,” Proc. IEEE 97, 1166–1185 (2009). [CrossRef]

] has been identified for practical applications, although the present state of the art is roughly an order of magnitude greater than that [8

M. R. Watts, D. C. Trotter, R. W. Young, A. L. Lentine, and W. A. Zortman, “Limits to silicon modulator bandwidth and power consumption,” Proc. SPIE 7221, 72210M (2009). [CrossRef]

–11

X. Zheng, J. Lexau, Y. Luo, H. Thacker, T. Pinguet, A. Mekis, G. Li, J. Shi, P. Amberg, N. Pinckney, K. Raj, R. Ho, J. E. Cunningham, and A. V. Krishnamoorthy, “Ultra-low-energy all-CMOS modulator integrated with driver,” Opt. Express 18(3), 3059–3070 (2010). [CrossRef] [PubMed]

]. This is due primarily to the volume of the optical mode which, in ring and disk-based resonators, is on the order of several cubic microns. Since the volume of the optical mode determines the volume of material in which an index perturbation must be induced, a large mode volume leads to a high switching energy. Microcavity resonators based on two-dimensional photonic crystals have already been demonstrated that support vastly smaller mode volumes on the order of (λ/n)³ or less [12

B. S. Song, S. Noda, T. Asano, and Y. Akahane, “Ultra-high-Q photonic double-heterostructure nanocavity,” Nat. Mater. 4(3), 207–210 (2005). [CrossRef]

]. Existing microcavity designs, however, rely on the inclusion of air cladding layers above and below the photonic crystal, which severely limits their utility in integrated applications due to the prohibition on void space on-chip. Although one-dimensional photonic crystal microcavities with higher-index (i.e. n ~1.5) cladding have been demonstrated with Q above 10⁶ [13

S. Tomljenovic-Hanic, C. M. de Sterke, and M. J. Steel, “Design of high-Q cavities in photonic crystal slab heterostructures by air-holes infiltration,” Opt. Express 14(25), 12451–12456 (2006). [CrossRef] [PubMed]

,14

Q. Quan, P. B. Deotare, and M. Loncar, “Photonic crystal nanobeam cavity strongly coupled to the feeding waveguide,” Appl. Phys. Lett. 96(20), 203102 (2010). [CrossRef]

], the highest Q demonstrated in SiO₂-clad 2-D microcavities is only 200 [15

I. Märki, M. Salt, H. P. Herzig, R. Stanley, L. El Melhaoui, P. Lyan, and J. M. Fedeli, “Optically tunable microcavity in a planar photonic crystal silicon waveguide buried in oxide,” Opt. Lett. 31(4), 513–515 (2006). [CrossRef] [PubMed]

]. This is due to the greatly reduced index contrast between silicon and SiO₂ as compared to silicon and air, which causes increased coupling of the resonant mode to the cladding.

In this work we describe the design of a silicon-based two-dimensional photonic crystal microcavity that is clad in SiO₂ and exhibits a record Q above 10⁵. The design is based on the rationale proposed by Noda et al for use in air-clad cavities [12

B. S. Song, S. Noda, T. Asano, and Y. Akahane, “Ultra-high-Q photonic double-heterostructure nanocavity,” Nat. Mater. 4(3), 207–210 (2005). [CrossRef]

], but instead makes use of a ‘graded’ cavity that successfully reduces vertical light leakage for an oxide-clad cavity. The reduced vertical leakage allows efficient coupling to a waveguide, with loaded Q estimated to be on the order of 10⁴, allowing high speed operation (>10 Gbps) while maintaining an ultrasmall mode volume on the order of (λ/n)³. Finally, we describe how a MOS charge depletion geometry that tailors the gate dimensions to match the size of the optical mode can be used to modulate the microcavity, achieving a theoretical switching energy as low as 1 fJ/bit, well below the 10 fJ/bit target.

2. Design considerations

The primary limitation on the switching energy of electro-optic modulators is the volume of the optical mode being perturbed. By minimizing the mode volume, the volume of active material in which an index perturbation must be induced is also minimized. Among the variety of modulation approaches already demonstrated, there is a clear empirical trend relating the optical mode volume in each device to its switching energy. Figure 1 illustrates this trend for several representative devices. Interferometric approaches, for example, distribute the optical mode over the entire length of both Mach-Zehnder arms, resulting in estimated switching energies of 5–10 pJ/bit [16

A. Liu, L. Liao, D. Rubin, H. Nguyen, B. Ciftcioglu, Y. Chetrit, N. Izhaky, and M. Paniccia, “High-speed optical modulation based on carrier depletion in a silicon waveguide,” Opt. Express 15(2), 660–668 (2007). [CrossRef] [PubMed]

,17

W. M. Green, M. J. Rooks, L. Sekaric, and Y. A. Vlasov, “Ultra-compact, low RF power, 10 Gb/s silicon Mach-Zehnder modulator,” Opt. Express 15(25), 17106–17113 (2007). [CrossRef] [PubMed]

]. More recently, approaches based on ring [9

K. Preston, S. Manipatruni, A. Gondarenko, C. B. Poitras, and M. Lipson, “Deposited silicon high-speed integrated electro-optic modulator,” Opt. Express 17(7), 5118–5124 (2009). [CrossRef] [PubMed]

,10

L. Chen, K. Preston, S. Manipatruni, and M. Lipson, “Integrated GHz silicon photonic interconnect with micrometer-scale modulators and detectors,” Opt. Express 17(17), 15248–15256 (2009). [CrossRef] [PubMed]

] or disk resonators [8

M. R. Watts, D. C. Trotter, R. W. Young, A. L. Lentine, and W. A. Zortman, “Limits to silicon modulator bandwidth and power consumption,” Proc. SPIE 7221, 72210M (2009). [CrossRef]

] have been demonstrated with significantly smaller mode volumes on the order of several cubic microns, and having commensurately smaller switching energies as low as 120 fJ/bit [10

], though still far above the 10 fJ/bit target.

Fig. 1 Empirical relationship between the optical mode volumes and resulting switching energies of several modulation techniques. MZI-based [16

,17

W. M. Green, M. J. Rooks, L. Sekaric, and Y. A. Vlasov, “Ultra-compact, low RF power, 10 Gb/s silicon Mach-Zehnder modulator,” Opt. Express 15(25), 17106–17113 (2007). [CrossRef] [PubMed]

]; ring and disk [8

M. R. Watts, D. C. Trotter, R. W. Young, A. L. Lentine, and W. A. Zortman, “Limits to silicon modulator bandwidth and power consumption,” Proc. SPIE 7221, 72210M (2009). [CrossRef]

–10

]; and the present work.

Using existing two-dimensional photonic crystal (PhC) microcavity designs it is possible to confine photons to volumes on the order of a cubic wavelength, equivalent to several tenths of a cubic micron [12

B. S. Song, S. Noda, T. Asano, and Y. Akahane, “Ultra-high-Q photonic double-heterostructure nanocavity,” Nat. Mater. 4(3), 207–210 (2005). [CrossRef]

,18

Y. Takahashi, Y. Tanaka, H. Hagino, T. Sugiya, Y. Sato, T. Asano, and S. Noda, “Design and demonstration of high-Q photonic heterostructure nanocavities suitable for integration,” Opt. Express 17(20), 18093–18102 (2009). [CrossRef] [PubMed]

–20

T. Yamamoto, M. Notomi, H. Taniyama, E. Kuramochi, Y. Yoshikawa, Y. Torii, and T. Kuga, “Design of a high-Q air-slot cavity based on a width-modulated line-defect in a photonic crystal slab,” Opt. Express 16(18), 13809–13817 (2008). [CrossRef] [PubMed]

]. These microcavities are normally designed as thin, symmetric air-clad silicon membranes in order to maximize the size of the photonic bandgap and also to minimize coupling to ‘leaky’ or cladding modes. This reliance upon an air cladding, however, makes such microcavities incompatible with on-chip applications where void space is prohibited.

It is possible to replace the air cladding with SiO₂, although this has the effect of shrinking the bandgap (see Fig. 2 ) and increasing leakage into lossy cladding modes due to the lower light line (see Fig. 3 ). The requirement of keeping the resonant mode below the SiO₂ light line creates a major constraint on the design of the microcavity by shrinking the range of wavevectors available for a tightly confined (low-leakage) mode. To date, the highest theorized Q in a SiO₂-clad 2-D PhC microcavity is only 200 [15

]. To obtain a high Q resonance, it is necessary to engineer the mode to be below the SiO₂ light line (Fig. 3).

Fig. 2 Available bandgaps TE-like modes for air- and SiO₂-clad silicon photonic crystals. Calculated for triangular lattice of low-index holes in a silicon slab of thickness h = 0.5a.

Fig. 3 Dispersion diagram showing upper and lower bandedges and the relevant light lines for air- and SiO₂-clad silicon photonic crystals. Shading corresponds to regions with low vertical leakage and thus high potential Q.

In propagating wave devices, confining light below the light line can be trivially accomplished by tuning the frequency of the light to correspond to a desired point in k-space. In PhC microcavities, however, the defect band is highly localized in frequency space and thus delocalized in k-space, so tuning is ineffective. It has been demonstrated, however, that it is possible to modify the location of the resonance in k-space by engineering the geometry of the microcavity [21

O. Painter, K. Srinivasan, J. D. O'Brien, A. Scherer, and P. D. Dapkus, “Tailoring of the resonant mode properties of optical nanocavities in two-dimensional photonic crystal slab waveguides,” J. Opt. A, Pure Appl. Opt. 3(6), 161–170 (2001). [CrossRef]

–25

Y. Akahane, T. Asano, B. S. Song, and S. Noda, “High-Q photonic nanocavity in a two-dimensional photonic crystal,” Nature 425(6961), 944–947 (2003). [CrossRef] [PubMed]

]. This can be understood by decomposing the resonant mode into its constituent plane waves, each having perpendicular (out-of-plane) and parallel (in-plane) components, k_⊥ and k_||, respectively. Plane waves for which k_|| < 2πn/λ, where n is the index of the cladding material, are able to escape from the cavity and into the cladding. Equivalently, in the ray optics picture, rays in the cavity propagating with k_|| > 2πn/λ have a large enough angle relative to the normal that they are confined within the cavity by total internal reflection. Since the k-vector components of the resonant mode are determined by the spatial profile of the electric field inside the cavity, it is necessary to engineer the resonant mode in order to optimize vertical confinement.

As explained by Akahane et al [25

Y. Akahane, T. Asano, B. S. Song, and S. Noda, “High-Q photonic nanocavity in a two-dimensional photonic crystal,” Nature 425(6961), 944–947 (2003). [CrossRef] [PubMed]

], the spatial distribution of the electric field inside the cavity can be thought of as a sinusoid modulated by an envelope. The sinusoid corresponds to delta functions in k-space with k_|| > 2πn/λ when the frequency of the light is below (to the right of) the light line. The envelope function introduces additional wavevector components, however, due to that fact that its spatial frequency spectrum is convolved with that of the sinusoid. If the envelope function has high spatial frequency components, then the convolution introduces new components that are above (to the left of) the light line, increasing leakage. This leads to the somewhat counterintuitive conclusion that strong, abrupt confinement in the x- and y-directions leads to increased vertical leakage and thus decreased Q.

3. Oxide-clad microcavities

In order to improve vertical confinement, the confinement in the x- and y-directions must be made more gradual so that the profile of the resonant mode more closely approaches a gaussian shape, which minimizes the extent of the spatial frequency spectrum of the mode. A number of PhC microcavity designs have already been proposed or demonstrated that utilize this concept. The concept has also been extended to one-dimensional microcavities with higher-index cladding (n ~1.5), where Q values on the order of 10⁵ have been demonstrated [14

Q. Quan, P. B. Deotare, and M. Loncar, “Photonic crystal nanobeam cavity strongly coupled to the feeding waveguide,” Appl. Phys. Lett. 96(20), 203102 (2010). [CrossRef]

], although no such high Q values have been demonstrated in two-dimensional oxide-clad microcavities. We have previously shown, however, that it is possible to use the concept of momentum space engineering to modestly increase the Q of an oxide-clad single-defect two-dimensional PhC microcavity [26

S. P. Anderson and P. M. Fauchet, “Ultra-low energy switches based on silicon photonic crystals for on-chip optical interconnects,” Proc. SPIE 7606, 76060R (2010). [CrossRef]

]. This was accomplished by leveraging a design from Vuckovic et al [24

J. Vuckovic, M. Loncar, H. Mabuchi, and A. Scherer, “Optimization of the Q factor in photonic crystal microcavities,” IEEE J. Quantum Electron. 38(7), 850–856 (2002). [CrossRef]

] in which the resonant cavity is elongated in one direction in order to allow the resonant mode to occupy a slightly larger volume. In contrast to the air-clad version of this cavity, which achieved a theoretical Q over 10⁴, our oxide-clad version exhibited a Q value of only 550. While this is still a relatively low Q value from the point of view of switching applications, it represents a significant improvement over the Q of 200 theorized in previous work [15

]. Further, it shows that the momentum space engineering approach has merit for 2-D oxide-clad photonic crystals.

Because this microcavity is still limited by vertical leakage, in order to further improve Q it is necessary to relax the x- and y-direction confinement even further. More recent microcavity designs have relied on inducing small lattice constant variations (normally less than 3%) to produce a resonant defect in the photonic crystal [12

B. S. Song, S. Noda, T. Asano, and Y. Akahane, “Ultra-high-Q photonic double-heterostructure nanocavity,” Nat. Mater. 4(3), 207–210 (2005). [CrossRef]

,19

E. Kuramochi, M. Notomi, S. Mitsugi, A. Shinya, T. Tanabe, and T. Watanabe, “Ultrahigh-Q photonic crystal nanocavities realized by the local width modulation of a line defect,” Appl. Phys. Lett. 88(4), 041112 (2006). [CrossRef]

]. The resulting modes are much less abruptly confined in x and y and therefore contain fewer leaky components. This has been successfully exploited to produce 2-D microcavities with theoretical Q values above 10⁷ [18

We have found that the double heterostructure design proposed by Takahashi et al [18

] is highly amenable to embedding in SiO₂. The microcavity is based on a double heterostructure design that concentrates the optical field within a central cavity region that is on the order of several lattice constants in length by one lattice constant in width. The cavity is created by abruptly varying the lattice constant of the photonic crystal in the x-direction. Although this design increases the volume of the resonant mode relative to that of a point-defect microcavity (1.3(λ/n)³ compared to < 0.5(λ/n)³), the large increase in Q and subsequent reduction in the magnitude of the index perturbation required for switching more than offsets the increase in mode volume. Three-dimensional finite difference time domain (FDTD) simulations reveal resonances with Q values as high as 6,000 for a SiO₂-clad version of the “NC1” microcavity of ref [18

]. This design has not been optimized for use in an oxide cladding, however, and further improvements to the design in that setting are possible.

4. Improved microcavity design

Here we propose the use of a graded photonic crystal microcavity in which the lattice constant varies smoothly in the region of the cavity. In such a cavity the confinement is less abrupt and thus exhibits reduced vertical leakage. Figure 4 shows a diagram of the microcavity. A single row of 11 holes is removed from the center to form the defect region. The lattice constant in the x-direction is varied in the shaded region such that it reaches a maximum of 1.025a at the center, and tapers linearly down to the bulk lattice constant a at the edge (where a = 410 nm for operation at λ = 1.55 µm). The lattice constant in the y-direction remains unchanged throughout the structure in order to maintain row alignment and also to restrict the discontinuity to one dimension. In this way, the lattice constant of the photonic crystal is a linear function of position in x. In contrast to the heterostructure approach, which has abrupt lattice constant changes of 1.2 or 2.5%, the linear grade relies on a series of <0.5% changes in lattice constant. This leads to a much smoother mode profile having fewer high-spatial-frequency components in the resonant mode and thus lower vertical leakage.

Fig. 4 Microcavity geometry used in this work. The microcavity is symmetric in x and y, with lateral spacing between columns of holes greatest at the center, decreasing linearly from 1.025a down to a at the edge of the graded region. The linear grade is actually made up of a series of finite steps due to the discrete nature of photonic crystals.

Figure 5 shows the electric field distribution (E_y ) of the resonant mode, calculated using the 3D FDTD method [27

A. Farjadpour, D. Roundy, A. Rodriguez, M. Ibanescu, P. Bermel, J. D. Joannopoulos, S. G. Johnson, and G. W. Burr, “Improving accuracy by subpixel smoothing in the finite-difference time domain,” Opt. Lett. 31(20), 2972–2974 (2006). [CrossRef] [PubMed]

,28

Meep, http://ab-initio.mit.edu/meep

]. This mode has a Q value of approximately 180,000, computed using a harmonic inversion technique [29

V. A. Mandelshtam and H. S. Taylor, “Harmonic inversion of time signals and its applications,” J. Chem. Phys. 107(17), 6756–6769 (1997). [CrossRef]

,30

Harminv, http://ab-initio.mit.edu/harminv

]. Here, the hole radius is set to 0.285a, the slab thickness is set to 0.60a and the size of the computation area is 30 lattice periods in x and 20 in y.

Fig. 5 Resonant mode of the graded microcavity, showing side-coupling to bus and drop waveguides, plotting E_y .

The large increase in Q over the double heterostructure geometry is due to the decrease in vertical radiation and not to any increase in in-plane confinement, which is substantially the same in both structures. This is a desirable property for integrated applications because it reduces vertical emission without sacrificing in-plane coupling efficiency, which is important when coupling to waveguides. Placing the graded microcavity in proximity to a rib waveguide effectively loads the cavity resonance and reduces Q due to the increased in-plane coupling to the waveguide. The extent of reduction in Q is dependent upon the separation between the cavity and waveguide as well as upon their relative orientations. For the configuration shown in Fig. 5, the loaded Q is approximately 20,000. This reduction in Q is critical to allow the bandwidth of the resonance to be large enough to support high data rates. Whereas a Q of 180,000 has a FWHM of only 1.1 GHz, the loaded Q of 20,000 has a FWHM bandwidth of 10 GHz, which is sufficient for a 10 Gbps data rate when NRZ signaling is used. It should be noted that the reduction in Q is due only to the waveguide coupling and not to any increase in vertical leakage. In fact, vertical leakage is negligibly affected by waveguide coupling and so remains small. Furthermore, because a large fraction (>90%) of the energy of the mode is inside the silicon slab, the cavity is well-suited to modulation involving the free carrier plasma dispersion effect in silicon [31

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

5. Switching using charge injection

A number of modulators have been demonstrated that rely on either a pn or p-i-n junction [32

J. P. Lorenzo and R. A. Soref, “1.3 µm electro-optic silicon switch,” Appl. Phys. Lett. 51(1), 6–8 (1987). [CrossRef]

–35

Y. Jiang, W. Jiang, L. Gu, X. Chen, and R. T. Chen, “80-micron interaction length silicon photonic crystal waveguide modulator,” Appl. Phys. Lett. 87(22), 221105 (2005). [CrossRef]

], where switching time is limited by carrier diffusion. In contrast, charge injection using a MOS capacitor approach is governed by carrier drift due to electric field which allows higher speed operation [36

S. M. Sze, Semiconductor Devices: Physics and Technology (Wiley, New York, 1985).

,37

A. Liu, R. Jones, L. Liao, D. Samara-Rubio, D. Rubin, O. Cohen, R. Nicolaescu, and M. Paniccia, “A high-speed silicon optical modulator based on a metal-oxide-semiconductor capacitor,” Nature 427(6975), 615–618 (2004). [CrossRef] [PubMed]

]. Additionally, leakage current in MOS capacitors is negligible due to the insulating gate oxide layer. Finally, the gate electrode can be easily tailored to suit the size and shape of the optical mode, allowing maximum overlap between free carriers and the active region of the device without the need for complex carrier confinement structures. Previous attempts at using a MOS capacitor for switching applications have focused on varying the carrier concentration in silicon strip waveguides [37

–39

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]

] for the purpose of interferometric modulation. By applying a MOS capacitor to the graded photonic crystal microcavity described above, however, gate capacitance can be minimized, enabling both higher switching speed and lower switching energy due to the compact size of the active region.

In general, the operation of a MOS capacitor falls into three modes: accumulation, depletion and inversion. In this work we rely upon the depletion mode of operation because it allows a larger change in free carrier concentration and also lower capacitance than the accumulation mode. Further, the depletion mode has more favorable optical loss characteristics due to its removal of free carriers from the optically active region. We eliminate the inversion mode from consideration because it relies upon diffusion of carriers through the depletion region, which is inherently a slower process. Additionally, it requires operation at higher gate voltages (above the depletion-inversion threshold), making it less suitable for low-energy operation.

Figure 6 shows a cross-sectional view of the device geometry used is this work. A highly conducting layer of n⁺ polysilicon is used as the gate electrode, and is separated from the p-doped resonant cavity by a thin gate dielectric of SiO₂ on the order of 100 nm in thickness.

Fig. 6 Cross-sectional schematic of device configuration on SOI. The size of the n⁺ polysilicon gate electrode is tailored to overlap only the area of the optical mode in the microcavity.

Due to the small size of the optical mode, the gate occupies an area of only 0.5 µm by 5 µm, or 2.5 µm². The gate capacitance is thus on the order of 1 fF, with the total device capacitance being slightly less due to the series capacitance of the depletion layer. We choose p-type doping for the microcavity because of the simultaneous higher available index shift and lower optical absorption of free holes as compared to free electrons [31

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

6. Device operation

In order to model the electrical and optical performance of the device, we first assume operation in full depletion mode. Depletion depth, modal index shift, extinction ratio and switching energy were computed for a range of substrate doping levels. Figure 7 shows the modal index change ∆n_mode available for varying levels of substrate doping. The kink at N_A ~10¹⁶ cm⁻³ corresponds to the point at which the depletion depth is equal to the thickness of the photonic crystal slab. Thus, at low doping levels, the electro-optic effect is felt in the entire 250 nm thickness of the cavity, while at higher doping levels the increasing change in material index competes with the decreasing depth of the depletion region.

Fig. 7 Change in the modal index as a function of substrate doping, assuming full depletion. The kink near N_A = 10¹⁶ cm⁻³ occurs when the depletion depth is equal to the 250 nm thickness of the silicon slab. Depletion depth decreases with doping.

In practice, the close proximity of the gate electrode to the microcavity results in some perturbation of the optical mode. Specifically, as the gate oxide is thinned, the resonance is redshifted and the mode is increasingly pulled up toward the gate electrode, resulting in lower mode confinement within the cavity. Thus, thinning the gate oxide has the same effect as increasing the index of the cladding material. For this reason, thinner gate oxides cause increased vertical leakage and thus degrade cavity Q. These trends are displayed in Fig. 8 , which shows that Q and confinement are substantially decreased below t_ox ~200 nm. It may be possible to mitigate the reduction in Q by further modifying the design of the photonic crystal, however that analysis was outside the scope of this work.

Fig. 8 Reduction in both Q factor (left) and mode confinement (right) in the waveguide-coupled microcavity as the gate oxide thickness is reduced. Q is limited to approximately 20,000 due to coupling to the bus and drop waveguides.

The reduction in mode confinement, and thus the increase in the fraction of the mode exposed to the gate electrode, can lead to excess losses due to optical absorption if the gate oxide is too thin [39

]. In our simulations we calculated the effect of changing the gate oxide thickness on the vertical confinement of the resonant mode, and found that a gate oxide thickness of just 100 nm is sufficient to reduce optical absorption in the gate electrode to a minimal level. For a gate oxide thickness of 100 nm, the portion of the optical mode exposed to the gate electrode is less than 1%. Assuming a gate electrode doping level of 1 × 10¹⁹ cm⁻³, the net optical loss is below 1 dB and the reduction in Q is negligible.

It is also important to include the effect of optical absorption due to substrate doping, which in principle also reduces the Q of the cavity. This results in a broadened transmission spectrum whenever the cavity is not depleted. The broadened Q can be calculated as 1/Q_net = 1/Q_optical + 1/Q_loss, where Q_optical is the Q value calculated from 3D FDTD, and Q_loss = 4πn/λα, where n is the index of silicon and α is the material absorption [40

D. Englund and J. Vucković, “A direct analysis of photonic nanostructures,” Opt. Express 14(8), 3472–3483 (2006). [CrossRef] [PubMed]

]. For a doping level of N_A = 6 × 10¹⁶ cm⁻³, Q_loss = 7 × 10⁵ and broadening is thus negligible. For higher levels of substrate doping, broadening can become important. However, because applying a bias voltage depletes the cavity of free carriers, a logical ‘1’ (full transmission) always corresponds to the narrowest and lowest-loss resonance (Fig. 9 ). This is in contrast to p-i-n-based modulators, which often rely on free carrier injection such that an applied bias broadens the resonance.

Fig. 9 Normalized transmission spectra present on drop waveguide for fully depleted (biased) and undepleted (unbiased) cases, for the device displayed in Fig. 5.

For a doping level of N_A = 3.6 × 10¹⁶ cm⁻³, and assuming a gate oxide layer 250 nm in thickness, the full depletion depth is approximately d = 158 nm, which occurs at a gate bias of 6.6 V, and the index shift in this region is ∆n_Si = 2.2 × 10⁻⁴ (Fig. 7). Figure 10 shows a cross-sectional view of the optical mode as a function of depth, overlaid with the region in which ∆n_Si is present. The modal index shift ∆n_mode can then be computed by evaluating the overlap of ∆n_Si with the optical mode. For these conditions, the mode confinement, calculated as the fraction of optical energy ε| E | ² inside the silicon region of the microcavity [41

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]

], is approximately 95%. Therefore, ∆n_mode = 1.0 × 10⁻⁴, which is sufficient to realize a 6 dB extinction ratio for a Q factor of 18,000 (Fig. 8). The switching energy required in this case can be estimated as U = CV_g ²/2, where C is the series capacitance of the gate dielectric and the depletion region. Here, C = 0.285 fF and V_g = 7.6 V, thus the switching energy for 6 dB extinction is 8.3 fJ. The extinction ratio is determined by assuming an input signal centered at the logical ‘1’ state with bandwidth equal to the FWHM of the resonator, and calculating the reduction in optical energy in the drop waveguide when the device is switched from the logical ‘1’ state to the ‘0’ state. It should be noted that a higher extinction ratio can be achieved if a narrower bandwidth input signal is used.

Fig. 10 Cross-sectional view of the simulated optical mode (red) present in the cavity when the gate oxide thickness is t = 250 nm. The blue regions indicate the p-type silicon slab and the n⁺-type gate electrode, and white indicates SiO₂. Brown indicates the depletion region, in which the index shift is induced.

Because the energy required to charge the MOS capacitor scales as the square of gate voltage, it is possible to further reduce switching energy by allowing operation at voltages below that required for full depletion. Additionally, since the level of depletion is negatively related to the gate oxide thickness, a thinner gate oxide results in a greater depletion depth for the same gate voltage. Intuitively, then, a thinner gate oxide should always result in a lower switching energy. However, because both Q and the fraction of the optical mode intensity inside the silicon slab fall off rapidly as the gate oxide is thinned below approximately 250 nm (Fig. 8), switching energy is minimized at some finite value of t_ox . Thus, below t_ox ~250 nm, Q and confinement are degraded as t_ox is reduced, thus requiring a larger ∆n_mode, and thus higher gate voltage, to achieve a 6 dB extinction ratio. Above t_ox ~500 nm, Q and confinement have plateaued, and as t_ox is increased, a higher gate voltage is required to achieve the same extinction ratio. Thus, the slope of the curve below t_ox ~250 nm is determined mainly by the falloff in Q with decreasing t_ox . The slope above t_ox ~500 nm is given by the linear relationship between V and t_ox and the inverse relationship between C and t_ox , resulting in a linear increase in switching energy as the gate oxide grows in thickness.

Figure 11 shows the switching energy that yields an extinction ratio of 6 dB as a function of gate oxide thickness t_ox . This figure was generated by computing the device characteristics for a wide range of values of both gate voltage V_g and substrate doping level N_A and selecting those that resulted in the lowest switching energy for each value of t_ox . We allowed N_A to vary between 10¹⁶ and 10¹⁹ cm⁻³ and V_g to vary from 0 to 20 V. We find that the overall minimum switching energy to achieve 6 dB extinction occurs at t_ox = 400 nm (96% confinement, Q = 20,000) when N_A = 8x10¹⁶ cm⁻³. This corresponds to a depletion depth of 109 nm, occurring at a gate voltage of V_g = 3 V, and leads to a switching energy of 0.9 fJ/bit.

Fig. 11 Minimum switching energy required to achieve 6 dB extinction for a range of values of gate oxide thickness, determined by optimizing over gate voltage and substrate doping. For thin gate oxide, switching energy increases due to the degraded Q and lower confinement, while thicker gate oxides require a higher gate voltage.

The capacitance in this case is just 0.195 fF, and therefore the electrical time constant of the device is extremely small. Assuming a 50 Ω driver impedance, the RC time constant τ_RC = RC = 8 fs, corresponding to a frequency on the order of 10 THz. Practically, the RC time constant of a modulator based on the present device would be limited by the driver electronics, with typical values estimated below 10 ps in on-chip environments [7

]. Device performance is thus limited by the photon lifetime τ_ph = λ²/(2πcΔλ_FWHM) = 16 ps, which corresponds to a frequency of 10 GHz. Because the electrical time constant is much shorter than the photon lifetime, this device may have potential as an electrically-driven adiabatic wavelength converter [42

S. F. Preble, Q. Xu, and M. Lipson, “Changing the colour of light in a silicon resonator,” Nat. Photonics 1(5), 293–296 (2007). [CrossRef]

A number of tradeoffs can be made in order to further reduce the switching energy of the device. For example, switching energy can be lowered even further below those values shown in Fig. 11 if we are willing to reduce the extinction ratio below 6 dB. Alternately, increasing the loaded cavity Q, which can be accomplished by increasing its separation from the waveguides, will result in greater sensitivity to shifts in index. The device can then be operated at lower drive voltages, at the price of decreased operational bandwidth.

7. Conclusion

We have studied an electro-optic modulator based on charge depletion in a high-Q two-dimensional photonic crystal microcavity. The microcavity is fully embedded in SiO₂ and is thus suitable for integration into on-chip applications. It is based on a graded microcavity design that allows smooth in-plane confinement of the optical mode in order to minimize vertical leakage, and exhibits a theoretical Q on the order of 10⁵, the highest yet reported for a fully oxide-clad silicon-based 2-D photonic crystal microcavity. When coupled to bus and drop waveguides the loaded Q is on the order of 10⁴, appropriate for high-bandwidth switching. By relying on a MOS capacitor for charge depletion, the switching energy can be reduced to as little as 1 fJ/bit by tailoring the gate dimensions to match those of the optical mode. Such a low switching energy makes the oxide-embedded microcavity a viable candidate for integration into optical interconnects and other on-chip systems.

Acknowledgements

This work was supported by the Air Force Office of Scientific Research (G. Pomrenke). Numerical simulations were performed at the Center for Research Computing at the University of Rochester.

References and links

1.	T. Gupta, Copper Interconnect Technology (Springer, New York, 2009).
2.	D. A. B. Miller, “Device requirements for optical interconnects to silicon chips,” Proc. IEEE 97, 1166–1185 (2009). [CrossRef]
3.	R. G. Beausoleil, P. J. Kuekes, G. S. Snider, S. Y. Wang, and R. S. Williams, “Nanoelectronic and nanophotonic interconnect,” Proc. IEEE 96(2), 230–247 (2008). [CrossRef]
4.	J. W. Goodman, F. J. Leonberger, K. Sun-Yuan, and R. A. Athale, “Optical interconnections for VLSI systems,” Proc. IEEE 72(7), 850–866 (1984). [CrossRef]
5.	D. A. B. Miller, “Optics for low-energy communication inside digital processors: quantum detectors, sources, and modulators as efficient impedance converters,” Opt. Lett. 14(2), 146–148 (1989). [CrossRef] [PubMed]
6.	M. Haurylau, G. Chen, H. Chen, J. Zhang, N. A. Nelson, D. H. Albonesi, E. G. Friedman, and P. M. Fauchet, “On-chip optical interconnect roadmap: Challenges and critical directions,” IEEE J. Sel. Top. Quantum Electron. 12(6), 1699–1705 (2006). [CrossRef]
7.	G. Chen, H. Chen, M. Haurylau, N. A. Nelson, D. H. Albonesi, P. M. Fauchet, and E. G. Friedman, “Predictions of CMOS compatible on-chip optical interconnect,” Integr. VLSI J. 40(4), 434–446 (2007). [CrossRef]
8.	M. R. Watts, D. C. Trotter, R. W. Young, A. L. Lentine, and W. A. Zortman, “Limits to silicon modulator bandwidth and power consumption,” Proc. SPIE 7221, 72210M (2009). [CrossRef]
9.	K. Preston, S. Manipatruni, A. Gondarenko, C. B. Poitras, and M. Lipson, “Deposited silicon high-speed integrated electro-optic modulator,” Opt. Express 17(7), 5118–5124 (2009). [CrossRef] [PubMed]
10.	L. Chen, K. Preston, S. Manipatruni, and M. Lipson, “Integrated GHz silicon photonic interconnect with micrometer-scale modulators and detectors,” Opt. Express 17(17), 15248–15256 (2009). [CrossRef] [PubMed]
11.	X. Zheng, J. Lexau, Y. Luo, H. Thacker, T. Pinguet, A. Mekis, G. Li, J. Shi, P. Amberg, N. Pinckney, K. Raj, R. Ho, J. E. Cunningham, and A. V. Krishnamoorthy, “Ultra-low-energy all-CMOS modulator integrated with driver,” Opt. Express 18(3), 3059–3070 (2010). [CrossRef] [PubMed]
12.	B. S. Song, S. Noda, T. Asano, and Y. Akahane, “Ultra-high-Q photonic double-heterostructure nanocavity,” Nat. Mater. 4(3), 207–210 (2005). [CrossRef]
13.	S. Tomljenovic-Hanic, C. M. de Sterke, and M. J. Steel, “Design of high-Q cavities in photonic crystal slab heterostructures by air-holes infiltration,” Opt. Express 14(25), 12451–12456 (2006). [CrossRef] [PubMed]
14.	Q. Quan, P. B. Deotare, and M. Loncar, “Photonic crystal nanobeam cavity strongly coupled to the feeding waveguide,” Appl. Phys. Lett. 96(20), 203102 (2010). [CrossRef]
15.	I. Märki, M. Salt, H. P. Herzig, R. Stanley, L. El Melhaoui, P. Lyan, and J. M. Fedeli, “Optically tunable microcavity in a planar photonic crystal silicon waveguide buried in oxide,” Opt. Lett. 31(4), 513–515 (2006). [CrossRef] [PubMed]
16.	A. Liu, L. Liao, D. Rubin, H. Nguyen, B. Ciftcioglu, Y. Chetrit, N. Izhaky, and M. Paniccia, “High-speed optical modulation based on carrier depletion in a silicon waveguide,” Opt. Express 15(2), 660–668 (2007). [CrossRef] [PubMed]
17.	W. M. Green, M. J. Rooks, L. Sekaric, and Y. A. Vlasov, “Ultra-compact, low RF power, 10 Gb/s silicon Mach-Zehnder modulator,” Opt. Express 15(25), 17106–17113 (2007). [CrossRef] [PubMed]
18.	Y. Takahashi, Y. Tanaka, H. Hagino, T. Sugiya, Y. Sato, T. Asano, and S. Noda, “Design and demonstration of high-Q photonic heterostructure nanocavities suitable for integration,” Opt. Express 17(20), 18093–18102 (2009). [CrossRef] [PubMed]
19.	E. Kuramochi, M. Notomi, S. Mitsugi, A. Shinya, T. Tanabe, and T. Watanabe, “Ultrahigh-Q photonic crystal nanocavities realized by the local width modulation of a line defect,” Appl. Phys. Lett. 88(4), 041112 (2006). [CrossRef]
20.	T. Yamamoto, M. Notomi, H. Taniyama, E. Kuramochi, Y. Yoshikawa, Y. Torii, and T. Kuga, “Design of a high-Q air-slot cavity based on a width-modulated line-defect in a photonic crystal slab,” Opt. Express 16(18), 13809–13817 (2008). [CrossRef] [PubMed]
21.	O. Painter, K. Srinivasan, J. D. O'Brien, A. Scherer, and P. D. Dapkus, “Tailoring of the resonant mode properties of optical nanocavities in two-dimensional photonic crystal slab waveguides,” J. Opt. A, Pure Appl. Opt. 3(6), 161–170 (2001). [CrossRef]
22.	K. Srinivasan and O. Painter, “Momentum space design of high-Q photonic crystal optical cavities,” Opt. Express 10(15), 670–684 (2002). [PubMed]
23.	K. Srinivasan and O. Painter, “Fourier space design of high-Q cavities in standard and compressed hexagonal lattice photonic crystals,” Opt. Express 11(6), 579–593 (2003). [CrossRef] [PubMed]
24.	J. Vuckovic, M. Loncar, H. Mabuchi, and A. Scherer, “Optimization of the Q factor in photonic crystal microcavities,” IEEE J. Quantum Electron. 38(7), 850–856 (2002). [CrossRef]
25.	Y. Akahane, T. Asano, B. S. Song, and S. Noda, “High-Q photonic nanocavity in a two-dimensional photonic crystal,” Nature 425(6961), 944–947 (2003). [CrossRef] [PubMed]
26.	S. P. Anderson and P. M. Fauchet, “Ultra-low energy switches based on silicon photonic crystals for on-chip optical interconnects,” Proc. SPIE 7606, 76060R (2010). [CrossRef]
27.	A. Farjadpour, D. Roundy, A. Rodriguez, M. Ibanescu, P. Bermel, J. D. Joannopoulos, S. G. Johnson, and G. W. Burr, “Improving accuracy by subpixel smoothing in the finite-difference time domain,” Opt. Lett. 31(20), 2972–2974 (2006). [CrossRef] [PubMed]
28.	Meep, http://ab-initio.mit.edu/meep
29.	V. A. Mandelshtam and H. S. Taylor, “Harmonic inversion of time signals and its applications,” J. Chem. Phys. 107(17), 6756–6769 (1997). [CrossRef]
30.	Harminv, http://ab-initio.mit.edu/harminv
31.	R. Soref and B. Bennett, “Electrooptical effects in silicon,” IEEE J. Quantum Electron. 23(1), 123–129 (1987). [CrossRef]
32.	J. P. Lorenzo and R. A. Soref, “1.3 µm electro-optic silicon switch,” Appl. Phys. Lett. 51(1), 6–8 (1987). [CrossRef]
33.	C. A. Barrios, V. Rosa de Almeida, and M. Lipson, “Low-power-consumption short-length and high-modulation-depth silicon electrooptic modulator,” J. Lightwave Technol. 21(4), 1089–1098 (2003). [CrossRef]
34.	Q. Xu, B. Schmidt, S. Pradhan, and M. Lipson, “Micrometre-scale silicon electro-optic modulator,” Nature 435(7040), 325–327 (2005). [CrossRef] [PubMed]
35.	Y. Jiang, W. Jiang, L. Gu, X. Chen, and R. T. Chen, “80-micron interaction length silicon photonic crystal waveguide modulator,” Appl. Phys. Lett. 87(22), 221105 (2005). [CrossRef]
36.	S. M. Sze, Semiconductor Devices: Physics and Technology (Wiley, New York, 1985).
37.	A. Liu, R. Jones, L. Liao, D. Samara-Rubio, D. Rubin, O. Cohen, R. Nicolaescu, and M. Paniccia, “A high-speed silicon optical modulator based on a metal-oxide-semiconductor capacitor,” Nature 427(6975), 615–618 (2004). [CrossRef] [PubMed]
38.	S. R. Giguere, L. Friedman, R. A. Soref, and J. P. Lorenzo, “Simulation studies of silicon electro-optic waveguide devices,” J. Appl. Phys. 68(10), 4964–4970 (1990). [CrossRef]
39.	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]
40.	D. Englund and J. Vucković, “A direct analysis of photonic nanostructures,” Opt. Express 14(8), 3472–3483 (2006). [CrossRef] [PubMed]
41.	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]
42.	S. F. Preble, Q. Xu, and M. Lipson, “Changing the colour of light in a silicon resonator,” Nat. Photonics 1(5), 293–296 (2007). [CrossRef]

OCIS Codes
(230.5298) Optical devices : Photonic crystals
(130.4110) Integrated optics : Modulators

ToC Category:
Photonic Crystals

History
Original Manuscript: May 28, 2010
Revised Manuscript: August 8, 2010
Manuscript Accepted: August 18, 2010
Published: August 25, 2010

Citation
Sean P. Anderson and Philippe M. Fauchet, "Ultra-low power modulators using MOS depletion in a high-Q SiO₂-clad silicon 2-D photonic crystal resonator," Opt. Express 18, 19129-19140 (2010)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-18-18-19129

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References

T. Gupta, Copper Interconnect Technology (Springer, New York, 2009).
D. A. B. Miller, “Device requirements for optical interconnects to silicon chips,” Proc. IEEE 97, 1166–1185 (2009). [CrossRef]
R. G. Beausoleil, P. J. Kuekes, G. S. Snider, S. Y. Wang, and R. S. Williams, “Nanoelectronic and nanophotonic interconnect,” Proc. IEEE 96(2), 230–247 (2008). [CrossRef]
J. W. Goodman, F. J. Leonberger, K. Sun-Yuan, and R. A. Athale, “Optical interconnections for VLSI systems,” Proc. IEEE 72(7), 850–866 (1984). [CrossRef]
D. A. B. Miller, “Optics for low-energy communication inside digital processors: quantum detectors, sources, and modulators as efficient impedance converters,” Opt. Lett. 14(2), 146–148 (1989). [CrossRef] [PubMed]
M. Haurylau, G. Chen, H. Chen, J. Zhang, N. A. Nelson, D. H. Albonesi, E. G. Friedman, and P. M. Fauchet, “On-chip optical interconnect roadmap: Challenges and critical directions,” IEEE J. Sel. Top. Quantum Electron. 12(6), 1699–1705 (2006). [CrossRef]
G. Chen, H. Chen, M. Haurylau, N. A. Nelson, D. H. Albonesi, P. M. Fauchet, and E. G. Friedman, “Predictions of CMOS compatible on-chip optical interconnect,” Integr. VLSI J. 40(4), 434–446 (2007). [CrossRef]
M. R. Watts, D. C. Trotter, R. W. Young, A. L. Lentine, and W. A. Zortman, “Limits to silicon modulator bandwidth and power consumption,” Proc. SPIE 7221, 72210M (2009). [CrossRef]
K. Preston, S. Manipatruni, A. Gondarenko, C. B. Poitras, and M. Lipson, “Deposited silicon high-speed integrated electro-optic modulator,” Opt. Express 17(7), 5118–5124 (2009). [CrossRef] [PubMed]
L. Chen, K. Preston, S. Manipatruni, and M. Lipson, “Integrated GHz silicon photonic interconnect with micrometer-scale modulators and detectors,” Opt. Express 17(17), 15248–15256 (2009). [CrossRef] [PubMed]
X. Zheng, J. Lexau, Y. Luo, H. Thacker, T. Pinguet, A. Mekis, G. Li, J. Shi, P. Amberg, N. Pinckney, K. Raj, R. Ho, J. E. Cunningham, and A. V. Krishnamoorthy, “Ultra-low-energy all-CMOS modulator integrated with driver,” Opt. Express 18(3), 3059–3070 (2010). [CrossRef] [PubMed]
B. S. Song, S. Noda, T. Asano, and Y. Akahane, “Ultra-high-Q photonic double-heterostructure nanocavity,” Nat. Mater. 4(3), 207–210 (2005). [CrossRef]
S. Tomljenovic-Hanic, C. M. de Sterke, and M. J. Steel, “Design of high-Q cavities in photonic crystal slab heterostructures by air-holes infiltration,” Opt. Express 14(25), 12451–12456 (2006). [CrossRef] [PubMed]
Q. Quan, P. B. Deotare, and M. Loncar, “Photonic crystal nanobeam cavity strongly coupled to the feeding waveguide,” Appl. Phys. Lett. 96(20), 203102 (2010). [CrossRef]
I. Märki, M. Salt, H. P. Herzig, R. Stanley, L. El Melhaoui, P. Lyan, and J. M. Fedeli, “Optically tunable microcavity in a planar photonic crystal silicon waveguide buried in oxide,” Opt. Lett. 31(4), 513–515 (2006). [CrossRef] [PubMed]
A. Liu, L. Liao, D. Rubin, H. Nguyen, B. Ciftcioglu, Y. Chetrit, N. Izhaky, and M. Paniccia, “High-speed optical modulation based on carrier depletion in a silicon waveguide,” Opt. Express 15(2), 660–668 (2007). [CrossRef] [PubMed]
W. M. Green, M. J. Rooks, L. Sekaric, and Y. A. Vlasov, “Ultra-compact, low RF power, 10 Gb/s silicon Mach-Zehnder modulator,” Opt. Express 15(25), 17106–17113 (2007). [CrossRef] [PubMed]
Y. Takahashi, Y. Tanaka, H. Hagino, T. Sugiya, Y. Sato, T. Asano, and S. Noda, “Design and demonstration of high-Q photonic heterostructure nanocavities suitable for integration,” Opt. Express 17(20), 18093–18102 (2009). [CrossRef] [PubMed]
E. Kuramochi, M. Notomi, S. Mitsugi, A. Shinya, T. Tanabe, and T. Watanabe, “Ultrahigh-Q photonic crystal nanocavities realized by the local width modulation of a line defect,” Appl. Phys. Lett. 88(4), 041112 (2006). [CrossRef]
T. Yamamoto, M. Notomi, H. Taniyama, E. Kuramochi, Y. Yoshikawa, Y. Torii, and T. Kuga, “Design of a high-Q air-slot cavity based on a width-modulated line-defect in a photonic crystal slab,” Opt. Express 16(18), 13809–13817 (2008). [CrossRef] [PubMed]
O. Painter, K. Srinivasan, J. D. O'Brien, A. Scherer, and P. D. Dapkus, “Tailoring of the resonant mode properties of optical nanocavities in two-dimensional photonic crystal slab waveguides,” J. Opt. A, Pure Appl. Opt. 3(6), 161–170 (2001). [CrossRef]
K. Srinivasan and O. Painter, “Momentum space design of high-Q photonic crystal optical cavities,” Opt. Express 10(15), 670–684 (2002). [PubMed]
K. Srinivasan and O. Painter, “Fourier space design of high-Q cavities in standard and compressed hexagonal lattice photonic crystals,” Opt. Express 11(6), 579–593 (2003). [CrossRef] [PubMed]
J. Vuckovic, M. Loncar, H. Mabuchi, and A. Scherer, “Optimization of the Q factor in photonic crystal microcavities,” IEEE J. Quantum Electron. 38(7), 850–856 (2002). [CrossRef]
Y. Akahane, T. Asano, B. S. Song, and S. Noda, “High-Q photonic nanocavity in a two-dimensional photonic crystal,” Nature 425(6961), 944–947 (2003). [CrossRef] [PubMed]
S. P. Anderson and P. M. Fauchet, “Ultra-low energy switches based on silicon photonic crystals for on-chip optical interconnects,” Proc. SPIE 7606, 76060R (2010). [CrossRef]
A. Farjadpour, D. Roundy, A. Rodriguez, M. Ibanescu, P. Bermel, J. D. Joannopoulos, S. G. Johnson, and G. W. Burr, “Improving accuracy by subpixel smoothing in the finite-difference time domain,” Opt. Lett. 31(20), 2972–2974 (2006). [CrossRef] [PubMed]
Meep, http://ab-initio.mit.edu/meep
V. A. Mandelshtam and H. S. Taylor, “Harmonic inversion of time signals and its applications,” J. Chem. Phys. 107(17), 6756–6769 (1997). [CrossRef]
Harminv, http://ab-initio.mit.edu/harminv
R. Soref and B. Bennett, “Electrooptical effects in silicon,” IEEE J. Quantum Electron. 23(1), 123–129 (1987). [CrossRef]
J. P. Lorenzo and R. A. Soref, “1.3 µm electro-optic silicon switch,” Appl. Phys. Lett. 51(1), 6–8 (1987). [CrossRef]
C. A. Barrios, V. Rosa de Almeida, and M. Lipson, “Low-power-consumption short-length and high-modulation-depth silicon electrooptic modulator,” J. Lightwave Technol. 21(4), 1089–1098 (2003). [CrossRef]
Q. Xu, B. Schmidt, S. Pradhan, and M. Lipson, “Micrometre-scale silicon electro-optic modulator,” Nature 435(7040), 325–327 (2005). [CrossRef] [PubMed]
Y. Jiang, W. Jiang, L. Gu, X. Chen, and R. T. Chen, “80-micron interaction length silicon photonic crystal waveguide modulator,” Appl. Phys. Lett. 87(22), 221105 (2005). [CrossRef]
S. M. Sze, Semiconductor Devices: Physics and Technology (Wiley, New York, 1985).
A. Liu, R. Jones, L. Liao, D. Samara-Rubio, D. Rubin, O. Cohen, R. Nicolaescu, and M. Paniccia, “A high-speed silicon optical modulator based on a metal-oxide-semiconductor capacitor,” Nature 427(6975), 615–618 (2004). [CrossRef] [PubMed]
S. R. Giguere, L. Friedman, R. A. Soref, and J. P. Lorenzo, “Simulation studies of silicon electro-optic waveguide devices,” J. Appl. Phys. 68(10), 4964–4970 (1990). [CrossRef]
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]
D. Englund and J. Vucković, “A direct analysis of photonic nanostructures,” Opt. Express 14(8), 3472–3483 (2006). [CrossRef] [PubMed]
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]
S. F. Preble, Q. Xu, and M. Lipson, “Changing the colour of light in a silicon resonator,” Nat. Photonics 1(5), 293–296 (2007). [CrossRef]

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