We demonstrate a fully-reconfigurable fourth-order optical lattice filter built by cascading identical unit cells consisting of a Mach-Zehnder interferometer (MZI) and a ring resonator. The filter is fabricated using a commercial silicon complementary metal oxide semiconductor (CMOS) process and reconfigured by current injection into p-i-n diodes with a reconfiguration time of less than 10 ns. The experimental results show full control over the single unit cell pole and zero, switching the unit cell transfer function between a notch filter and a bandpass filter, narrowing the notch width down to 400 MHz, and tuning the center wavelength over the full free spectral range (FSR) of 10 GHz. Theoretical and experimental results show tuning dynamics and associated optical losses in the reconfigurable filters. The full-control of each of the four cascaded single unit cells resulted in demonstrations of a number of fourth-order transfer functions. The multimedia experimental data show live tuning and reconfiguration of optical lattice filters.
© 2011 OSA
RF-photonic processing of microwave signals using passive optical filters can in many cases replace traditional electrical signal processing while providing higher bandwidth and potentially lower power consumption [1, 2]. Moreover, as optical delay lines are characterized by their low-loss that is independent of RF frequency, a much more complex processing (filtering) network can be constructed. Such all-optical signal processing approaches can be beneficial for a broad range of applications [3–5] especially at high bandwidths. All-optical signal processing in optical lattice filters [5‒7] can support the fully reconfigurable optical transfer function with a large number of zeros and poles synthesized by cascading many identical unit cells. Preferably, the optical lattice filter design should accompany recursive algorithms to facilitate synthesis of sophisticated filter functions by reconfiguring many unit cells of known identical transfer functions.
In addition to scalability and reconfigurability, practical implementations of optical lattice filters should be based on low loss and compact integration of many unit cells that can be reconfigured rapidly while consuming low energy. Silicon photonics realization of optical lattice filters exploiting complementary metal oxide semiconductor (CMOS) fabrication process  can offer high levels of yield, uniformity, resolution, and repeatability while keeping the fabrication cost relatively low. Recently, such integrated optical lattice filters have been demonstrated on silicon as well as InP platforms [7–12]. This paper discusses rapid reconfiguration and synthesis of a fourth-order silicon optical lattice filter that is fabricated using a commercial silicon CMOS foundry process. The current injections into p-i-n diodes in silicon photonic unit cells allow lattice filter reconfiguration in less than 10 ns.
2. Device concept and design
Figure 1(a) shows the input coupler (outside the largest rectangle) and the unit cell  (inside the largest rectangle), a basic building block used to construct higher-order filters. The unit cell takes the form of an incomplete Mach-Zehnder interferometer (MZI) where the input coupler is separate from the unit cell. The upper arm of the incomplete MZI is coupled to a ring resonator and the lower arm of the MZI connects directly to the output coupler. All of the couplers are designed as tunable MZI’s with 3-dB directional couplers at the input and the output. Phase shifters (shown as numbered red rectangles in Fig. 1) in each arm control the output splitting ratio irrespective of the device’s initial conditions. The ring resonator’s perimeter is 8.2 mm resulting in a free spectral range (FSR) of 10 GHz (100-ps lattice constant). The output coupler of the first unit cell acts as the input coupler for the next stage. The inset of Fig. 1(a) shows images of the fabricated devices.
The tuning elements used to reconfigure the filter are phase-shifters based on the free-carrier plasma dispersion effect in silicon . Each phase-shifter is fabricated by embedding an optical waveguide between n-type and p-type doped regions which are defined by ion implantation and the required phase shift is induced by injecting current into the resulting p-i-n diode structure. The length of phase shifters 1 and 6 are 700 µm, and the length of phase shifters 2, 3, 4, and 5 are 500 µm. These diodes consistently achieved a uniform threshold voltage of 0.85 V and a DC forward resistance below 12 Ω on measurements on more than hundreds of unit cells.
As the upper part of Fig. 1(a) indicates, the unit cell design incorporates two types of silicon rib waveguides, a “narrow” waveguide which is 0.5-μm wide and a “wide” waveguide which is 3-μm wide. Figure 1(b) shows the mode field profiles calculated for the two different widths (they have equal rib and slab heights of 250 nm). The “narrow” and “wide” waveguides are connected with a linear taper. The advantages of the narrow waveguide include single-mode confinement, strong lateral evanescent coupling, and relatively large phase shifts with low tuning-current , while the advantages of the wide waveguide include lower propagation loss (~0.3-0.5 dB/cm) and lower optical nonlinearity . The bends (300-μm radius) and the tuning sections use the narrow waveguide and the long runs of waveguide in the ring resonator use the wide waveguide. This design ensures that the filter supports only a single mode at wavelengths near 1.55 μm for the TE-polarization and that the bend and propagation losses are minimized. Figure 1(c) shows a four-unit-cell filter which consists of four cascaded unit cells.
To understand the optical frequency response of the unit cell filter in Fig. 1(a), we analyze the transmission from the input (In 1) to the Out 1 (bar) and Out 2 (cross) outputs (H11 and H21, respectively). The first step is to look at the frequency response of the single-mode waveguide coupled to the ring through the ring coupler [i.e., the transfer function between points ‘A’ and ‘B’ in Fig. 1(a)]. Using the z-transform notation  this can be written as
The two-port transfer function of the unit cell filter is
The poles and zeros of the transfer function are simply the roots of the denominator, Dr, and numerator with respect to z −1. From Eq. (1) and Eq. (2), it is apparent that the unit cell transfer function has a pole equal to which is the same for both the H21 and H11 transmission and two zeros which are different for the H21 and the H11 transmission. In the special case of a lossless device these transfer functions are complementary. The two zeros and the single pole are fully-controllable by four parameters: the ring phase shift (electrode 6), the coupling strength between the MZI upper waveguide and the ring (electrodes 2 and 3), the phase shift in the MZI lower waveguide (electrode 1), and the splitting ratio of the output coupler (electrodes 4 and 5). The magnitude and phase of the unit cell’s pole is fully defined by the ring coupler (including γ) and the ring phase shift, respectively, whereas the two linked zeros are a function of the pole value. They are adjusted by tuning the main MZI’s lower waveguide phase shifter and output coupler. Achieving a near-unity pole requires minimizing the round-trip and excess coupling losses. The general filter tuning procedure is as follows: first, the pole magnitude and phase are set using the ring coupler and ring phase-shifter, and then the zero is adjusted by tuning the main MZI lower phase shifter (ϕ 3) and the output coupler.
The transfer function of a higher-order filter composed of several cascaded unit cells is the multiplication of the two-port transfer function of each stage, which is written as
The denominator of Hfilter is the product of Dr from each stage. Whereas the numerators are a complex function of each stages Nr, Dr, and κ o. Therefore, the poles of the higher-order filter are independently adjustable (i.e., roots of the denominator) since each relies on the ring parameters (ring coupler, and ring phase shifter) of a separate unit cell. However, the zeros depend upon the pole order, all of the output coupler splitting ratios, and all of the lower waveguide phase shifters. This feature is crucial to consider when developing algorithms for filter synthesis and reconfiguration. Typically, a recursion algorithm  can find the tuning values of the lower waveguide phase shifters and output couplers for a desired set of zeros.
Control of the unit cell pole magnitude between 0 and 1 enables a filter transfer function with either a finite impulse response (FIR) or an infinite impulse response (IIR), respectively. A pure FIR filter requires full coupling into and out of the ring resonator corresponding to a pole magnitude of zero. In this case, the ring simply acts as a delay line and the unit cell impulse response contains two impulses spaced by T. When the pole magnitude is greater than zero, IIR functionality occurs and the optical signal coupled to the ring is stored (i.e., circulates the ring many times) resulting in an impulse every T, and hence extending the length of the impulse response to much longer than the delay provided by the physical path length of the device. The duration of the impulse response is proportional to the pole magnitude and the pole magnitude is inversely proportional to the coupling into the ring.
From (2), an IIR filter with a pole magnitude near unity requires a very small coupling to the ring and a very low ring round-trip loss. Then, light inside the ring circulates many times before decaying, thereby producing a very long impulse response. In the frequency or wavelength domain, the pole magnitude controls the finesse of the features in the filter transfer function. If the pole magnitude is held constant, then changes in the filter’s zero engrave filters with very different shapes. The results in Section 5 will demonstrate these concepts.
3. Optical lattice filter die fabrication and preparation for testing
All device fabrication took place at the BAE Systems CMOS foundry following the various process steps illustrated in Fig. 2 . The fabrication process starts with a commercial 6-inch silicon-on-insulator (SOI) wafer that has a 3-μm-thick buried oxide (BOX) layer and a 0.5-μm-thick upper silicon layer. The lithography uses a deep ultraviolet (DUV) scanner, development, and photoresist reflow  to smooth the resist profile. The pattern is transferred to the dielectric hard mask, and the waveguides are formed by reactive ion etching (RIE) using the hard mask. A second lithography step defines the waveguide trenches in which silicon is etched down to the BOX layer for thermally isolating individual p-i-n diodes designed for current tuning. The trenching regions are filled by silicon dioxide (SiO2) whose thermal conductivity is two orders of magnitude lower than that of silicon .
After the trenches are etched using a RIE process, a thermal oxidation step reduces the waveguide corrugations and then an oxide deposition step deposits over a micron of oxide. The wafer surface is then planarized by chemical mechanical polishing (CMP) to facilitate the subsequent metallization steps. To obtain an accurate 50/50 splitting ratio in the directional couplers, the fabrication process was optimized through several iterations to find the best compromise between the counteracting effects of the photoresist reflow, the waveguide oxidation and the waveguide rib etching depth parameters. These optimizations and adjustments were necessary to produce directional couplers with splitting errors below 3%, which helped realize a full range tuning of the pole magnitude.
To form the p-i-n diodes required for tuning, via openings are made in the cladding SiO2 by a RIE process followed by boron and phosphorus ion implantations to form the p + and n + doped regions, respectively. The electrodes comprise aluminum on top of a thin layer of Ti/TiN (Titanium/Titanium Nitride). The wafer is prepared for testing by diamond saw dicing followed by facet polishing. The waveguides meet the facets at a 7° tilt angle with respect to the facet surface normal to suppress the formation of a Fabry-Pérot cavity that would disturb the filter response. Also, an anti-reflection (AR) coating reduces optical coupling losses and further suppresses any residual Fabry-Pérot effects. The optical lattice filter die is then mounted on a chip-carrier and their electrodes were wire-bonded to facilitate simultaneous tuning of many electrodes on the die. However, this basic wiring technique limits high-frequency modulations beyond several hundred megahertz.
4. Single unit cell characterization and parameter extraction
This section describes the amplitude and phase measurement technique, pole-zero curve fitting, estimation of internal circuit parameters (e.g., coupling ratio, phase shifter values) from the pole-zero fit, and characterization of the phase shifters.
Measurement technique and pole-zero fitting
An accurate and fast transmission measurement technique offers both visual feedback and a means to quickly determine the pole and zero information from the phase and amplitude transmission. A frequency-domain swept coherent interferometer [20, 21] enables simultaneous complex spectral transmission measurements (i.e., intensity and phase) across 10 nm with a 100 dB dynamic range and an update rate of 10 Hz. Lensed fibers couple light into, and out of, the device while preserving only the TE-polarization. The experimental arrangement provides simultaneous measurement of both outputs (i.e., Out 1 and Out 2 in Fig. 1(a)).
Figure 3(a) shows a phase (red) and amplitude (blue) measurement (grey lines) of a bandpass filter shape with a 400-MHz, 3-dB bandwidth displayed across four FSRs. We use the MATLAB System Identification Toolbox to find the best-fit pole and zero to the measured data . The pole and zero fits [Fig. 3(b)] are overlaid on the measured data in Fig. 3(a) (red and blue curves). The excellent match between the measurement and fit indicates that the unit cell provides a pure single pole and single zero without any undesired features (e.g., Fabry-Pérot fringes). Using the equations that describe the transfer function that were discussed earlier, we can estimate many of the internal circuit parameters of the filter (i.e., those that cannot be directly measured) including the ring loss, the directional coupler coupling ratio, phase shift and attenuation versus injection current.
Figure 3(c) shows the extracted unit cell pole magnitude versus separate sweeps of the drive current to each phase-shifter in Fig. 1(a). As expected, the data show that the phase shifters inside the ring coupler [electrodes 2 and 3 from Fig. 1(a)] primarily control the pole magnitude over a range of 0.08–0.93. This corresponds to 99.4% and 8% power coupled into the ring, respectively and demonstrates the filter changing from FIR to IIR. It is apparent from Fig. 3(c), that tuning the zero (electrodes 1, 4) does not affect the pole magnitude and this indicates low crosstalk between the electrodes. However, due to loss associated with phase changes from free-carrier absorption, operating the ring phase shifter (electrode 6) and the ring coupler electrodes simultaneously (electrodes 2 and 3) decreases the pole magnitude. Although not shown here, similar plots are available for the pole’s angle, the zero’s magnitude, and the zero’s phase. The detailed analyses of the relationship between these and the injection current levels will be the subject of a future publication.
Extraction of circuit parameters from the pole-zero fit
From the pole magnitude versus current injected into electrodes 2 and 3 [see Fig. 3(c)], we can estimate two device circuit parameters: the splitting ratios of the directional couplers which make up the tunable MZI and the ring round-trip loss. The pole magnitude is the product of the ring-round trip loss, γ, and where κr is the ring coupler coupling ratio [see denominator of Eq. (2)]. We will show the upper limit of the pole magnitude is proportional to the ring round-trip loss and the lower limit is proportional to imperfect 50/50 splitting ratios of the direction couplers that comprise the tunable MZI.
The tuning range of the coupling coefficient of the tunable MZI, κr, and thus the pole magnitude is a function of the splitting ratios of its two directional couplers. For example, 100% power coupling into the ring occurs only when the two directional couplers in the MZI have either a 50/50 splitting ratio or opposite splitting ratios (e.g., 60/40 for the first coupler and 40/60 for the second). When the two couplers in the MZI have equal but imperfect 50/50 splitting ratios, the coupling into the ring can always tune to 0% (i.e., a pole magnitude of 1.0) but can never tune to 100%. Since the two directional couplers are fabricated identically they will have near equal coupling ratios and the MZI can always weakly couple to ring (i.e., achieve pole magnitude near unity). Therefore, the lower limit of the pole magnitude (i.e., less than 100% coupling into the ring) is primarily due to an imperfect 50/50 splitting ratio of the direction couplers in the MZI. In our case, 99.4% max power coupling into the ring corresponds to a 3% directional coupler splitting ratio error (i.e, 47/53 or 53/47 splitting ratio).
Likewise, since κr can be tuned to 0, the upper limit of the pole magnitude is limited by losses (i.e., γ < 1). In the measured device, the total ring round trip loss is estimated at 0.6 dB which includes excess coupler, bending, and waveguide losses. This is consistent with our measured waveguide loss of 0.3-0.5 dB/cm.
Characterization of the phase shifters phase and amplitude response
The total phase shift resulting from current injection into each tuning element (phase-shifter) in the device is the result of two counteracting physical effects; the free-carrier plasma dispersion  and the thermo-optic effect . The plasma dispersion effect decreases the refractive index with increasing carrier density and it is accompanied by free-carrier absorption (undesired), whereas the thermo-optic effect increases the refractive index and depends on the power absorbed in the diode structure.
The phase shift and amplitude response of the phase shifter are extracted from measurement of the pole magnitude and pole phase versus sweep of the ring phase shifter (electrode 6) [see Fig. 3(c)] and the mathematical form of the pole, . The phase shifter phase response and amplitude response versus injection current are simply the angle of the pole and magnitude of the pole, respectively, normalized to the pole value without injection current. Figure 4 shows the phase induced in a typical phase-shifter and the associated optical loss versus the drive current. At low current densities, the plasma dispersion effect is dominant and at high currents the effect is weakened—most likely due to thermal power dissipation. The loss versus current curve shows an almost exponential behavior since it depends only on free-carrier absorption and not heating. Despite the increase of the loss versus phase shifter current, it is evident from Fig. 3(c) that the unit cell provides a pole with a large tuning range due to the low-loss waveguides and the large range κr.
Characterization of the phase-shifters speed
Figure 5(a) shows the modulator test structure used to examine the device tuning speed. The modulator is fabricated on the same chip with the filter and its input and output waveguides are extended for direct access at the device facets. For the measurement, a single-frequency laser is coupled into the structure using a lensed fiber. The phase shifter is driven by a 0–5 mA 15-MHz square wave and the output light is measured with a 20-GHz sampling oscilloscope. Figure 5(b) shows the optical response of the modulator where the measured optical output rise and fall times are less than 10 ns (10%–90%). The limitation in speed is partly due to the electrical connection to the chip and the carrier lifetime inside the p-i-n structure. Figure 5(c) shows the measured optical modulator response when it is driven by a 1-kHz square wave current signal. This shows the device’s slower heating and cooling effects which have a time constant on the order of 20 μs. For fast reconfiguration of the filter shape, the drive signals must include pre-emphasis that account for the time dependent response of the phase shifter.
5. Single unit cell measurement and tuning
This section shows amplitude and phase measurement examples of a single-unit cell’s complex transfer function which demonstrate large pole tuning range, and control over the zero. Attached movies show how the filters change from one shape to another.
Bandpass filter bandwidth tuning
Figure 6 shows simultaneous measurements of the transfer function from the ‘In 1’ port of Fig. 1(a) to both outputs (Out 1 and Out 2 or H 11(f) and H 21(f), respectively) of a single-unit cell when it is tuned to act as a bandpass filter optimized for Out 2 with pole magnitudes of 0.11, 0.53, and 0.88, respectively. The filter’s impulse response’s (i.e., h 11(t) and h 21(t)) are displayed along the top of Fig. 6 and they are equal to the inverse Fourier transform of the corresponding measured complex spectral transmission [i.e., H 11(f) and H 21(f)]. The increase in duration of the filter’s impulse response, and the corresponding narrowing of the filter shape in the frequency domain as the pole’s magnitude increases is evident in Fig. 6.
The change in the filter’s bandwidth is inversely proportional to the filter’s pole magnitude and the zero helps adjust the filter shape (e.g., bandpass vs. notch). To create a symmetric bandpass filter shape there must be a relative π rad phase difference between the unit cell’s pole and zero and the filter rejection is maximized (approaching infinite) when the filter’s zero is located on the unit circle. The complimentary output transmission, H 11(f), shows a notch filter shape and has the same pole as H 21(f). The shape difference between H 21(f) and H 11(f) is from the position of its zero which shares the same angle as its pole.
Figure 6(a) illustrates the impulse response and spectral transmission of the filter when it is configured for a near ideal FIR response. Such a filter is equivalent to a delay interferometer and has sinusoidal transmission versus frequency. The impulse response contains two peaks of almost the same magnitude; the first peak comes from the lower arm of the main MZI and second peak is delayed by T and comes from the ring. The remaining unwanted peaks in the FIR filter’s response constitute less than 1% of the energy passed by the filter’s transfer function.
Figure 6(b,c) show the impulse response and transmission functions of the filter when it is configured as IIR bandpass filters with −3 dB bandwidths of 2 GHz and 400 MHz, respectively. The transfer functions for the cross and bar states (H 21 and H 11) are bandpass and notch, and the impulse responses have the same duration for both outputs. The difference in the impulse response between the two outputs is that the energy of the first peak is smaller for the bandpass filter and higher for the notch filter. In the spectral domain, the energy of the first peak corresponds to the background level of the filter which is large for a notch filter and small for the bandpass filter.
The narrow bandpass filter impulse response in Fig. 6(c) rolls off at about 1 dB per peak and extends beyond 5 ns before reaching the measurement noise floor. The 14-dB decrease in peak transmission between the 400-MHz filter and 2-GHz filter occurs because of the resonant enhancement of the ring loss. However, this filter still achieves a clean bandpass shape with a pole and zero near unity. Media 1 associated with Fig. 6 shows the single-unit cell filter switching between FIR [see Fig. 6(a)] and IIR [see Fig. 6(b,c)] including intermediate stages when the zero is not positioned optimally. In addition to the filter shapes shown in Fig. 6, Media 1 includes bandpass filters with pole magnitudes of 0.11, 0.28, 0.53, 0.62, 0.71, 0.8, 0.85, and 0.89.
Notch to bandpass filter tuning
Figure 7 shows a tuning example optimized for the H 21 transmission where the filter shape is changed from notch to bandpass without changing the filter’s pole. The difference between the two shapes is the location of the zero at either −1 or +1. Changing the phase of the main MZI phase shifter (electrode 1) by π rad changes the filter from notch to bandpass. Adjusting the output coupler (electrode 4) helps to position the zero on the unit circle in presence of additional losses. Fine adjustments to the zero in Fig. 7(a) (Media 2) shows the shape of the filter in intermediate states between bandpass and notch. Note that only the magnitude of the first impulse changes.
6. Four-unit-cell filters
Figure 9 displays the transmission of the four-unit-cell filter optimized for the H 21 (cross) transmission [(a)] and the H 11 (bar) transmission [(b)]. The four-unit-cell structure has 26 phase shifters to control 18 independent filter parameters: the amplitude and phase of the four poles, the four zeros, and the complex gain of the filter. (a) shows the filter manually tuned for a flat-top bandpass shape with a sharp roll-off for the H 21 transmission. The filter has up to 30-dB of rejection and its flat-top −1-dB bandwidth is 600 MHz. Figure 9 shows this filter shaped measured across 95 FSRs. Across this range the filter has uniform transmission and shape and near 30-dB extinction. (b) shows a filter optimized for the H 11 transmission. As expected, the four unit-cell devices achieved more complex and versatile filter shapes when compared to the single unit cell filter.
Currently, these filter shapes are obtained through manual adjustment of the electrodes. The tuning procedure follows the theory presented in Section 2. First, since each unit cell uniquely defines a pole, the poles were positioned in their proper locations using the pole tuning electrodes. The zero values are a function of the pole values, the pole order (i.e., which unit cell provides the pole value) and all of the zero tuning electrodes. The adjustment of the zeros tuning electrodes finalizes the filter shape. This procedure is challenging because adjusting a single zero tuning phase shifter changes every zero’s value (14 zero phase shifters total). Currently on-going studies include an automated procedure to switch between filter shapes including a detailed calibration of the four-unit cell filters.
In this work we demonstrate a fully-reconfigurable fourth-order silicon optical lattice filter built by cascading identical unit cells consisting of a Mach-Zehnder interferometer (MZI) and a ring resonator. The filter is fabricated using a silicon (CMOS) foundry process and its reconfiguration is achieved by current injection into p-i-n diodes. This demonstration of higher-order optical lattice filters with a reconfiguration speed of under 10 ns will enable many new applications. Employing the recent achievements which provide optical gain in silicon photonic devices would allow the implementation of similar filters with much higher orders; a key step in realizing all-optical processing system for a broad range of RF and microwave applications.
This work was supported in part by DARPA MTO Si-PhASER project Grant No. HR0011-09-1-0013.
References and links
1. E. M. Dowling and D. L. MacFarlane, “Lightwave lattice filters for optically multiplexed communication systems,” J. Lightwave Technol. 12(3), 471–486 (1994). [CrossRef]
2. J. Capmany, B. Ortega, and D. Pastor, “A tutorial on microwave photonic filters,” J. Lightwave Technol. 24(1), 201–229 (2006). [CrossRef]
3. R. A. Minasian, “Photonic signal processing of microwave signals,” IEEE Trans. Microw. Theory Tech. 54(2), 832–846 (2006). [CrossRef]
4. K. W. Ang, T. Y. Liow, Q. Fang, M. B. Yu, F. F. Ren, S. Y. Zhu, J. Zhang, J. W. Ng, J. F. Song, Y. Z. Xiong, G. Q. Lo, and D. L. Kwong, “Silicon photonics technologies for monolithic electronic-photonic integrated circuit (EPIC) applications: Current progress and future outlook,” in IEEE International Electron Devices Meeting (IEDM), (2009), 1–4.
5. D. J. Richardson, “Silicon photonics: Beating the electronics bottleneck,” Nat. Photonics 3(10), 562–564 (2009). [CrossRef]
6. L. C. Kimerling, D. Ahn, A. B. Apsel, M. Beals, D. Carothers, Y.-K. Chen, T. Conway, D. M. Gill, M. Grove, C.-Y. Hong, M. Lipson, J. Liu, J. Michel, D. Pan, S. S. Patel, A. T. Pomerene, M. Rasras, D. K. Sparacin, K.-Y. Tu, A. E. White, and C. W. Wong, ““Electronic-photonic integrated circuits on the CMOS platform,” in Silicon Photonics, (,” Proc. SPIE 6125, 612502, 612502-10 (2006), http://dx.doi.org/10.1117/12.654455. [CrossRef]
7. S. S. Djordjevic, L. W. Luo, S. Ibrahim, N. K. Fontaine, C. B. Poitras, B. Guan, L. Zhou, K. Okamoto, Z. Ding, M. Lipson, and S. J. B. Yoo, “Fully reconfigurable silicon photonic lattice filters with four cascaded unit cells,” IEEE Photon. Technol. Lett. 23(1), 42–44 (2011). [CrossRef]
8. P. Dong, N.-N. Feng, D. Feng, W. Qian, H. Liang, D. C. Lee, B. J. Luff, T. Banwell, A. Agarwal, P. Toliver, R. Menendez, T. K. Woodward, and M. Asghari, “GHz-bandwidth optical filters based on high-order silicon ring resonators,” Opt. Express 18(23), 23784–23789 (2010). [CrossRef] [PubMed]
9. N. K. Fontaine, S. Ibrahim, S. Djordjevic, B. Guan, T. Su, S. Cheung, R. Yu, R. P. Scott, A. T. Pomerene, L. Gunter, S. Danziger, Z. Ding, K. Okamoto, and S. J. B. Yoo, “Fully reconfigurable silicon CMOS photonic lattice filters,” in European Conference and Exhibition on Optical Communication (ECOC), (2010), 1–3. http://dx.doi.org/10.1109/ECOC.2010.5621287
10. L.-W. Luo, S. Ibrahim, A. Nitkowski, Z. Ding, C. B. Poitras, S. J. Ben Yoo, and M. Lipson, “High bandwidth on-chip silicon photonic interleaver,” Opt. Express 18(22), 23079–23087 (2010). [CrossRef] [PubMed]
11. E. J. Norberg, R. S. Guzzon, S. C. Nicholes, J. S. Parker, and L. A. Coldren, “Programmable photonic lattice filters in InGaAsP–InP,” IEEE Photon. Technol. Lett. 22(2), 109–111 (2010). [CrossRef]
12. M. S. Rasras, T. Kun-Yii, D. M. Gill, C. Young-Kai, A. E. White, S. S. Patel, A. Pomerene, D. Carothers, J. Beattie, M. Beals, J. Michel, and L. C. Kimerling, “Demonstration of a tunable microwave-photonic notch filter using low-loss silicon ring resonators,” J. Lightwave Technol. 27(12), 2105–2110 (2009). [CrossRef]
13. K. Jinguji, “Synthesis of coherent two-port optical delay-line circuit with ring waveguides,” J. Lightwave Technol. 14(8), 1882–1898 (1996). [CrossRef]
15. L. Zhou, S. S. Djordjevic, N. K. Fontaine, Z. Ding, K. Okamoto, and S. J. B. Yoo, “Silicon microring resonator-based reconfigurable optical lattice filter for on-chip optical signal processing,” in IEEE LEOS Annual Meeting Conference Proceedings, (2009), 501–502. http://dx.doi.org/10.1109/LEOS.2009.5343162
16. H. Fukuda, K. Yamada, T. Shoji, M. Takahashi, T. Tsuchizawa, T. Watanabe, J.-i. Takahashi, and S.-i. Itabashi, “Four-wave mixing in silicon wire waveguides,” Opt. Express 13(12), 4629–4637 (2005). [CrossRef] [PubMed]
17. A. V. Oppenheim, R. W. Schafer, and J. R. Buck, Discrete-time signal processing (Prentice Hall, 1999).
18. M. Romagnoli and S. Ghidini, “Pirelli’s roadmap on silicon photonics,” in European Conference on Optical Communication (ECOC), Workshop 6: Silicon Photonics in Telecom/Datacom: from Basic Research to Industrial Deployment, (EPIC (CD), 2007).
19. L. T. Su, J. E. Chung, D. A. Antoniadis, K. E. Goodson, and M. I. Flik, “Measurement and modeling of self-heating in SOI nMOSFET's,” IEEE Trans. Electron. Dev. 41(1), 69–75 (1994). [CrossRef]
20. D. K. Gifford, B. J. Soller, M. S. Wolfe, and M. E. Froggatt, “Optical vector network analyzer for single-scan measurements of loss, group delay, and polarization mode dispersion,” Appl. Opt. 44(34), 7282–7286 (2005). [CrossRef] [PubMed]
21. K. Takada and S.-i. Satoh, “Method for measuring the phase error distribution of a wideband arrayed waveguide grating in the frequency domain,” Opt. Lett. 31(3), 323–325 (2006). [CrossRef] [PubMed]
22. L. Ljung, System identification: theory for the user (Prentice Hall PTR, 1999).
23. G. T. Reed, Silicon photonics: the state of the art (Wiley, 2008).