We report for the first time two typical phase coherence lengths in highly confined silicon waveguides fabricated in a standard CMOS foundry's multi-project-wafer shuttle run in the 220nm silicon-on-insulator wafer with 248nm lithography. By measuring the random phase fluctuations of 800 on-chip silicon Mach-Zehnder interferometers across the wafer, we extracted, with statistical significance, the coherence lengths to be 4.17 ± 0.42 mm and 1.61 ± 0.12 mm for single mode strip waveguide and rib waveguide, respectively. We present a new experimental method to quantify the phase coherence length. The theory model is verified by both our and others' experiments. Coherence length is expected to become one key parameter of the fabrication non-uniformity to guide the design of silicon photonics.
© 2015 Optical Society of America
Silicon photonics makes use of CMOS compatible fabrication technologies to realize large-scale, energy-efficient photonic integrated circuits (PICs) with high yield and low cost [1,2]. Applications have been identified in on-chip optical interconnections [3–6], sensors [7,8] and all optical processing [9–12]. Reliable fabrication processes and process design kits (PDKs) are required to provide the infrastructure for the manufacture of practical, competitive PICs in silicon . One of the significant considerations in system level integration is the fabrication non-uniformity that results in fluctuations in waveguide's effective index. The issue creates large phase uncertainty that makes the peak wavelength distribution in resonators hard to predict [14–18]. Thus, extra power as well as development effort for control systems will be spent on the tuning of these devices' resonance positions. This is undesirable in power budget limited scenarios [3,15,19].
The phase coherence length is a key parameter to characterize these fabrication non-uniformities. Traditionally, coherence length was used to characterize the phase noise of a laser in the interferometric measurement. This concept was then applied to study the phase noise of low confinement non-silicon waveguides. For instance, typical fiber Bragg grating's coherence length is about 10-100 cm  and the silica channel waveguide's coherence length is about 27 m . However, former coherence length reports in integrated optics didn't show strong statistical significance as they were extracted from small numbers of samples, due to fabrication and test limitations [21, 22].
To the best of our knowledge, coherence length has not yet been reported in silicon photonic waveguides. This is because analyzing coherence length requires a large amount of data from special designed interferometer structures across the entire wafer. Unlike many researchers [20–23] in silicon photonics, the authors had access to a complete multi-project-wafer (MPW) Si photonics wafer from a commercial foundry. This gave us a unique opportunity to take the extensive cross-wafer data set needed to accurately measure the coherence length. Developing reliable statistical methods to measure random phase is another hard problem. Researchers have made important progress by stitching a hundred scanning electron microscope (SEM) images of one waveguide to quantify phase noise in integrated silicon Bragg gratings , but the cost would be high if hundreds of devices are characterized using SEM.
In this paper, we report the typical coherence length of silicon waveguides fabricated in a foundry's silicon photonics MPW run. We propose a new analytic method to verify the phase coherence length model. 800 silicon Mach-Zehnder interferometers (MZIs) in clusters were measured across the wafer. MZIs' spectra clearly show the fabrication non-uniformity similar to previous work [14–16]. We cite  as a validation but our work is not the extension of it . is a significant study of the non-uniformity of chip-scale silicon photonics by ring resonators in one die. However , didn't consider the coherence length value because there was only one type of ring. But our work analyzed a large number of MZI structures with different optical path differences across the wafer. The coherence length extracted from MZIs shows the relationship between waveguide's random phase variance as a function of the waveguide length. For rings, this relationship is not clear, because of the small device footprint. We find the coherence lengths are 4.17 ± 0.42 mm and 1.61 ± 0.12 mm for silicon strip and rib waveguides respectively. We hope to use the coherence length as a standard figure of merit to evaluate the fabrication non-uniformity to support system level integration efforts in silicon photonics.
2.1 Effective index and geometry
In this work, the principal devices were the strip waveguide and the rib waveguide as shown in Fig. 1. The input optical wavelength is 1550 nm. The effective index of the waveguides will change due to variations in waveguide thickness (t), width (w), sidewall angle (α) and slab layer thickness (s). The relationships between the effective index and geometry can be considered to be linear if the standard deviations satisfy that δt<10nm, δw<10nm, δs<10nm and δα<10° as shown in Fig. 2. The proportionality constants that link geometries to the effective index are defined as Cw, Ct, Cα and Cs. Their values are simulated by a finite element mode solver  and summarized in Table 1. From simulations, we find that the effective index is very sensitive to the silicon layer thickness and sidewall angle. The effective index of the rib waveguide is less sensitive to sidewall angle as compared to the strip waveguide.
2.2 Phase noise model
The vertical geometry variations come from the wafer top silicon layer thickness non-uniformity and etching process. The thickness can vary up to ± 20 nm between wafers  even before the wafer processing begins. The lateral geometry variations come from sidewall roughness created during the etching process. Although the standard 248 nm lithography CMOS processes were used in the fabrication of the samples, there were uncertainties in the photoresist thickness/roughness, mask alignment positions, the developing rate, the silicon dry etching rate and the thermal oxidation growth rate. Since a lot of random factors contribute to the change of effective index, based on the central limit theorem , the accumulated output phase of the waveguide is assumed to have a zero-centered component that has a random walk under Gaussian distribution. The randomly introduced phase noises are assumed to be independent. As we use the standard and fixed CMOS-compatible processes , the relationship between the fabrication parameters and the coherence length is not studied in this paper that focuses on the measurements of coherence length in silicon photonics waveguides. But it is an area for future study.
2.3 Coherence length model
The coherence length characterizes the physical phenomena that when the waveguide's length (L) is equal to the coherence length (Lcoh), the random phase (Δϕ(L) is a Gaussian random variable with zero average) inside the waveguide as a result of fabrication non-uniformity will make the signal phase vary in a certain range. The average range (deviation) is expressed as <eiΔϕ∙Lcoh)> = e−1 in order to follow the laser community's tradition. In other words, the random phase noise's standard deviation is (2L/Lcoh)1/2 for the waveguide of length L. If we use the waveguide to build interferometers like rings and MZIs, we can expect the average difference in resonant wavelengths between two as fabricated interferometers will be ± FSR × Δϕ(Lcoh)/2π ≈ ± 0.11 × FSR when the path-difference length equals Lcoh. FSR is the free spectral range (unit: nm). The theory predicts that the random phase noise variances increase linearly with the waveguide length if the coherence length is constant. In order to calculate the coherence length, we propose a new analyzing method to extract optical phase noise from unbalanced MZIs. Although each MZI's waveguide has a different overall effective index after real fabrication processes, we can assume the average effective index (<neff>) to be same for the same type of MZIs. Then for the i-th MZI of the same type, the effective index can be expressed as a function of position (x):
Although ideally Eq. (3-b) should be written:Fig. 3(c), distance (x)< 700 μm), they become statistically correlated in terms of fabrication. (see Appendix). We then calculate the effective index dispersion slope (k) by a mode solver  in order to find the mean of effective index at the wavelength of interest.
We simulated that k = −1.13 × 10−3 nm−1 and −9.81 × 10−4 nm−1 for strip and rib waveguides respectively. From the MZI spectrum, we measured the FSR and the corresponding resonance wavelengths at the spectrum’s null points. We can calculate the average effective index as shown by <neff (λ1)> by the following formula:16]. That is, using measured resonant wavelength (λi), FSR and dL to extract group index () by:
Then we can determine the m from the closely gathered cluster in the scatter plotting of (λi,) as shown in Fig. 3(d). Substituting MZIi's <neff (λ1)> and m into Eq. (3-a), we get the random phase shift Δϕi (dL). Finally, we can calculate the coherence length (Lcoh) from the linear regression of the variance (<Δϕi(dL)2>) and dL. The relationship is shown as:
In our method, we replace L by dL in Eq. (7).
3. Experiment and discussion
In this work, the MZIs were fabricated by BAE Systems  through an MPW passive run with 248 nm lithography. The devices were built on a 6” SOI wafer from SOITEC with 220 nm top silicon, 3 μm buried oxide layer with 10 Ω-substrate . The MZIs were made by the waveguides as shown in Fig. 1. In the measurement, a linearly transverse electric (TE) polarized light beam from a tunable laser (Agilent 81980A) centered at a wavelength around of 1550 nm was coupled into the MZI through a fiber array and an on-chip grating coupler (GC). The MZI's output light was coupled out through another GC to the fiber array and measured by a lightwave multimeter (Agilent 8163B) as shown in Fig. 3(a, b). The measured spectra were normalized against the transmission of a reference GC loop connected by the same waveguide. We tracked each resonance wavelength in the spectra of a set of MZIs by sine square function curve fitting . Spectra of strip waveguide based MZIs with dL = 144 μm are shown in Fig. 3(c). The azimuthal mode m was determined by the scatter plot of group index and resonant wavelength as shown in Fig. 3(d).
We studied five groups of MZIs in each die (area size: 2.5 × 3.2 cm2). Each group had a different dL and 20 nominally identical MZIs that were spaced 120 μm apart as shown in Fig. 1(b). The dLs of the MZIs are 50 μm, 144 μm, 444 μm, 744 μm and 1044 μm. The transmission spectra of 800 MZIs from 8 dies across the wafer were measured to guarantee statistical significance. Based on the analyzing method mentioned in the first section, we found that the coherence length of the strip and rib waveguides across the wafer were 4.17 ± 0.42 mm and 1.61 ± 0.12 mm, respectively. The linear regressions of <Δϕ(dL )2> and dL are shown in the Fig. 4. We also did statistical T-test to verify the linear relationship as shown in Figs. 4(a) and 4(c). T-test is used because the degree of freedom is small (4) as we characterized 5 types of dLs. Calculated by the experimental data shown in Figs. 4(a) and 4(c), T value proves that the relationship of Eq. (7) has statistical significance. The type-I error's α level is smaller than 0.1% as shown in the Appendix.
The strip waveguide's coherence length is longer than the rib waveguide's, which indicates that the fabrication process for the strip waveguide has better tolerance to process variations. Both strip and rib waveguides have same waveguide width (500 nm) and the fundamental mode was tightly confined in the center of the waveguide from simulation results. So it can be assumed that the top surface roughness is approximately same. It's clear that the rib waveguide has larger overall non-uniformity due to the additional slab layer. The extra phase error may mainly suffer from the partial etch step that forms the slab layer. Different mechanisms of fabrications uncertainties have been studied by others, including etching, mask aligning, and oxidation step errors. For example, some reports showed the waveguide sidewall roughness as ± 1.8 nm  and ± 2.7 nm  and the waveguide top surfaces roughness as ± 0.45 nm . The thickness and width variations of slab layer of rib waveguide in one die were about ± 0.1 nm and ± 0.4 nm . Compared with the strip waveguide, we think the rib waveguide has larger non-uniformity in the slab layer fabrication. Although these results came from dedicated runs, they could qualitatively show that slab layer variations degrade the phase noise performance indicated by shorter coherence length. We randomly generated 10000 MZIs with slab thicknesses that is a Gaussian random variable centered at 50 nm. The standard deviation was assumed as 0.25 nm. Each waveguide's effective index at desired wavelength was simulated . By Monte-Carlo method, we can find MZIs resonant wavelengths near 1550 nm under the same azimuthal mode m. The simulated phase noise variance as a function of dL is shown in the Appendix. The simulated coherence length was about 1.8 mm that agreed well with the experimental results (1.6 mm). Thus, our coherence length model is proved. Moreover, 0.25 nm is also a reasonable prediction  for the standard deviation of slab thickness. Thus, our method opens a low-cost, high-throughput pathway to characterize fabrication non-uniformity without using SEM test. The measured coherence length was shorter than the simulation results because of other non-uniformities from, for instance, the waveguide thickness, width and sidewall roughness. In this work, we provide a wafer scale variation of coherence length since it is critical to analyze the wafer level fabrication non-uniformities. The small deviation of coherence length across the wafer proves the high uniformity. Depending on the applications, die scale variation may be more important than wafer scale [20, 22]. Therefore, we provide this simple but general analyzing method that can extract the coherence length under different scales.
Since the fabrication non-uniformity may bring extra loss to waveguide, we measured waveguide insertion loss to study whether it is related to the phase coherence length. We fitted the measured spectra of GC loops (connected by waveguides of different lengths) with parabola in a 40 nm range centered at their peak wavelengths. The insertion loss obtained from linear regression between waveguide length and maximum power at around 1550nm is −4.8 ± 0.03 dB/cm and −5.2 ± 0.06 dB/cm for the strip waveguide and the rib waveguide, respectively, as shown in Fig. 5. Lower insertion loss could be attributed to the better sidewall roughness in strip waveguide, but the difference is only 0.4 dB, which shows that the low insertion loss and the long coherence length are not strongly related. One reason may be that the insertion loss reflects the averaged random phase-shift variance contributed by all fabrication non-uniformities, but the coherence length stands for the total sum of variance as a result of all non-uniformities.
Compared with the reported coherence lengths of fiber device and silica waveguide [20–22], the coherence length of silicon waveguide is several orders of magnitude shorter. Note that the index contrast of silicon waveguide is much larger than others. Near 1550nm, silicon and silica refractive index are 3.47 and 1.45 respectively. Moreover, the uncertainty of the silicon thickness is high . Therefore, the shorter coherence length is likely due to the high-index-contrast highly confined waveguide. In other words, the effective index is very sensitive to geometry changes as shown in Fig. 2.
To show the fabrication non-uniformity, we also measured destructive resonance wavelengths and FSRs contours of two types of MZIs (dL = 144 μm, strip waveguide and rib waveguide) across the wafer as shown in Fig. 6. Twenty nominally identical samples were measured in each MZI set. FSRs are 3.92 ± 0.09 nm (strip waveguide) and 4.20 ± 0.08 nm (rib waveguide), respectively. Peak resonance wavelengths are 1548.11 ± 0.71 nm (strip waveguide) and 1548.48 ± 0.96 nm (rib waveguide), respectively. We use experimental results of peak resonance wavelengths (λ) and FSRs (nm) to calculate group index (ng) by Eq. (6). We find ng = 4.2 and 4.0 for strip and rib waveguides respectively. The corresponding group indices extracted from effective index simulations using Eq. (4) and Eq. (8) were 4.2 and 3.9 that agree well with experiment results.
For each MZI, the peak resonance wavelength was picked up under the same azimuthal mode index. The rib waveguide's resonance wavelength random shift is larger than the strip waveguide's, which is consistent with our coherence length analysis. Although the uncertainty of resonance wavelength shift is about ten times larger than the FSR's, which is consistent with the reports in [14,15], long coherence length guarantees that the output phase is still under well control.
4. Coherence length validation and application
We use other reported experimental results to verify the coherence length measurements in this study. In , researchers tested 371 racetrack resonators' that have the same designed cross-section of the strip waveguide used in this work. The devices from  were also fabricated in a commercial CMOS foundry similar to us. Thus, we can assume our strip waveguide's coherence length can be applied to . In , when the device separation distance (this is equivalent to dL in our model) is 1mm, the resonant wavelength average shift (<Δλ>) was about 0.75 nm as shown in 's Fig. 3(b). The racetrack resonator's perimeter was 84.36 μm and FSR was about 6.8 nm at around 1550nm. Therefore, the phase noise's standard deviation is 0.69 rad by 2π × <Δλ>/FSR. By our coherence length model, substituting strip waveguide's Lcoh = 4mm and dL = 1mm, we find the phase noise's standard deviation is (2dL/Lcoh)1/2 = 0.71 rad. Therefore 's experiment results agrees quite well with our theory prediction. In general, coherence lengths may vary among different MPW runs.
The coherence length theory can be applied in PIC system design. For example, considering an on-chip system with 4cm long routing strip waveguide using the waveguide geometry shown in this work, we can find the standard deviation of random phase shift to be 4.47 radians by Eq. (7). The equivalent waveguide length uncertainty's standard deviation is:
Then we can calculate the time uncertainty by
Therefore, the uncertainty in time domain due to phase coherence length (assume neff = 2.5) is about 3.7 fs. Another example is to apply the coherence length to analyze MZ modulator wavelength tuning range. If the modulator is made by the rib waveguide as shown in Fig. 1(b), the waveguide's coherence length (Lcoh) is 1.6mm. Take experimental results from . The Vπ is 9 V for a 3 mm (L) length phase shifter. The MZI's dL is 100μm. From a conservative viewpoint, the random phase variance is 3.75 rad2 by Eq. (7). Actually, the L in the Eq. (7) could be less than 3 mm due to phase correlation when two arms of the MZI are closely placed (e.g. separation<1mm). The average resonance wavelength difference between two MZ modulators working near 1550nm is about 0.3FSR by:
In order to make one MZ modulator work on the same wavelength of the other, the maximum tuning bias (Vtune) can be expected as 5.4 V by:
Phase coherence lengths are extracted by a new experimental method and reported in silicon photonics platform for the first time to the best of our knowledge. The measured coherence lengths are 4.17 ± 0.42 mm and 1.61 ± 0.12 mm for strip and rib waveguides respectively. These results show statistical significance and high consistence based on 800 samples of MZIs across a 6” SOI wafer. The phase variation of the strip waveguide has better fabrication tolerance than rib waveguide, which is not strongly correlated to the waveguide insertion loss. Moreover, the coherence length method can be applied to other researchers' study of the non-uniformity on chip-scale silicon photonic integrated circuits. Our prediction of phase noise agrees well with the experimental results of . This work provides theoretical and experimental support of using coherence length as a guideline to evaluate the fabrication non-uniformity of silicon photonics platform.
In our experiment, the total arm lengths (L) of MZIs are: 68.8 μm, 306.2 μm, 606.2 μm, 906.2 μm and 1206.2 μm corresponding to arm length differences (dL) of 50 μm, 144 μm, 444 μm, 744 μm and 1044 μm. Although the condition dL >>L1 is not always satisfied as shown in Table 2. But the condition Lcoh>>L1 always stands for both waveguides. We think that when Lcoh>>L1, the dL can replace the total length's influence to the Lcoh. Therefore, replacing Eq. (3-b') by of Eq. (3-b) is a good approximation.
We also studied the relationship between the random phase shift's variance and the MZI's total arm length (L = L1 + L2) as shown in Fig. 7. The extracted coherence lengths for strip and strip loaded strip waveguides are 4.63 ± 0.35 mm and 1.72 ± 0.11 mm, respectively. The coherence lengths are almost the same as the results extracted from the dL because Lcoh>> L1. However, there is one problem: the random phase shift's variance (<(Δϕ)2>) is not zero when L = 0 by the linear regression as shown in Fig. 7. In contrast, <(Δϕ)2> extract from dL is almost 0 as shown in Fig. 4. So using dL is suitable for our experiment condition.
To prove our method's feasibility, we also designed and measured about 800 MZIs which have the same dL (110 µm) but different total length (L = 1400 µm and 290 µm, respectively). By the same method, the <Δϕ(L)2> are 0.035 rad2 when L = 1400 µm and 0.033 rad2 when L = 290 µm, which is very close. The small difference in the results proves our analyzing that using dL instead of L to extract coherence length is reasonable. In this experiment, the extra L was added in the x dimension of the layout plane not in the orthogonal dimension as shown in the Fig. 3(c) inset. So we infer that there are correlations in the non-uniformities in the x dimension due to this foundry's fabrication conditions. We think it is possible that other errors in thickness or mask alignment will be correlated to reduce L's influence on the phase errors. The simulation of the random phase noise variance as a function of the rib MZI's optical path difference is shown in Fig. 8 by Monte-Carlo method. We assume that the rib MZI's non-uniformity is only in the slab thickness with a standard deviation of 0.25 nm. We also did t-test to prove the coherence length model's strong statistical significance as shown in Fig. 9.
The authors thank Mark S. Mirotznik, Lukas Chrostowski, Christophe Galland, Ari Novack, and Nicholas C. Harris for helpful discussions. The authors would like to thank Steven Danziger and Stewart Ocheltree of BAE systems, for their support of device fabrication. All authors gratefully acknowledge support from AFOSR STTR grants, numbers FA9550-12-C-0079 and FA9550-12-C-0038. The authors would like to thank Gernot Pomrenke, of AFOSR, for his support of the OpSIS effort, though both a PECASE award (FA9550-13-1-0027) and ongoing funding for OpSIS (FA9550-10-1-0439), Brett Pokines and AFOSR SOARD office, for their support under grant FA9550-13-1-0176.
References and links
1. G. T. Reed, Silicon Photonics: The State of the Art (WILEY, 2008).
2. M. Hochberg and T. Baehr-Jones, “Towards fabless silicon photonics,” Nat. Photonics 4(8), 492–494 (2010). [CrossRef]
3. C. Gunn, “CMOS photonics for high-speed interconnectors,” IEEE Micro 26(2), 58–66 (2006). [CrossRef]
4. T. Seok, N. Quack, S. Han, and M. Wu, “50x50 digital silicon photonic switches with MEMS-actuated adiabatic couplers,” in Optical Fiber Communication Conference, OSA Technical Digest (online) (2015), paper M2B.4. [CrossRef]
5. P. Dong, Y. Chen, and L. Buhl, “Reconfigurable four-channel polarization diversity silicon photonic WDM receiver,” in Optical Fiber Communication Conference, OSA Technical Digest (online) (2015), paper W3A.2. [CrossRef]
6. X. Xiao, H. Xu, X. Li, Z. Li, Y. Yu, and J. Yu, “High-speed on-chip photonic link based on ultralow-power microring modulator,” in Optical Fiber Communication Conference, OSA Technical Digest (online) (2014), paper Tu2E.6. [CrossRef]
8. M. Soltani, J. Lin, R. A. Forties, J. T. Inman, S. N. Saraf, R. M. Fulbright, M. Lipson, and M. D. Wang, “Nanophotonic trapping for precise manipulation of biomolecular arrays,” Nat. Nanotechnol. 9(6), 448–452 (2014). [CrossRef] [PubMed]
9. B. Jalali, “Silicon photonics: nonlinear optics in the mid-infrared,” Nat. Photonics 4(8), 506–508 (2010). [CrossRef]
10. N. C. Harris, D. Grassani, A. Simbula, M. Pant, M. Galli, T. Baehr-Jones, M. Hochberg, D. Englund, D. Bajoni, and C. Galland, “Integrated source of spectrally filtered correlated photons for large-scale quantum photonic systems,” Phys. Rev. X 4, 041047(1)-041047(10) (2014). [CrossRef]
11. K. Petermann, A. Gajda, G. Dziallas, M. Jazayerifar, L. Zimmermann, B. Tillack, F. Da Ros, D. Vukovic, K. Dalgaard, M. Galili, and C. Peucheret, “Phase-sensitive optical processing in silicon waveguides,” in Optical Fiber Communication Conference, OSA Technical Digest (online) (OSA, 2015), paper Tu2F.4. [CrossRef]
12. Q. Chen, F. Zhang, L. Zhang, Y. Tian, P. Zhou, J. Ding, and L. Yang, “1 Gbps directed optical decoder based on two cascaded microring resonators,” Opt. Lett. 39(14), 4255–4258 (2014). [CrossRef] [PubMed]
13. L. Chrostowski and M. Hochberg, Silicon Photonics Design: From Devices to Systems (Cambridge University, 2015).
15. A. V. Krishnamoorthy, X. Zheng, G. Li, J. Yao, T. Pinguet, A. Mekis, H. Thacker, I. Shubin, Y. Luo, K. Raj, and J. Cunningham, “Exploiting CMOS manufacturing to reduce tuning requirements for resonant optical devices,” IEEE Photonics J. 3(3), 567–579 (2011). [CrossRef]
16. L. Chrostowski, X. Wang, J. Flueckiger, Y. Wu, Y. Wang, and S. Talebi Fard, “Impact of fabrication non-uniformity on chip-scale silicon photonic integrated circuits,” in Optical Fiber Communication Conference, OSA Technical Digest (online) (2014), paper Th2A.37. [CrossRef]
17. S. Nakamura, S. Yanagimachi, H. Takeshita, A. Tajima, T. Katoh, T. Hino, and K. Fukuchi, “Compact and low-loss 8x8 silicon photonic switch module for transponder aggregators in CDC-ROADM application,” in Optical Fiber Communication Conference, OSA Technical Digest (online) (2015), paper M2B.6. [CrossRef]
18. Y. Liu, R. Ding, Q. Li, X. Zhe, Y. Li, Y. Yang, A. Lim, P. Lo, K. Bergman, T. Baehr-Jones, and M. Hochberg, “Ultra-compact 320 Gb/s and 160 Gb/s WDM transmitters based on silicon microrings,” in Optical Fiber Communication Conference, OSA Technical Digest (online) (2014), paper Th4G.6. [CrossRef]
19. A. V. Krishnamoorthy, R. Ho, X. Zheng, H. Schwetman, J. Lexau, P. Koka, G. Li, I. Shubin, and J. Cunningham, “Computer systems based on silicon photonic interconnects,” Proc. IEEE 97(7), 1337–1361 (2009). [CrossRef]
20. R. Feced and M. Zervas, “Effects of random phase and amplitude errors in optical fiber Bragg gratings,” IEEE/OSA J. Lightwave Technol. 18(1), 90–101 (2000). [CrossRef]
21. R. Adar, C. Henry, M. Milbrodt, and R. Kistler, “Phase coherence of optical waveguides,” IEEE/OSA J. Lightwave Technol. 12(4), 603–606 (1994). [CrossRef]
22. T. Goh, S. Suzuki, and A. Sugita, “Estimation of waveguide phase error in silica-based waveguides,” IEEE/OSA J. Lightwave Technol. 15(11), 2107–2113 (1997). [CrossRef]
23. A. D. Simard, G. Beaudin, V. Aimez, Y. Painchaud, and S. Larochelle, “Characterization and reduction of spectral distortions in Silicon-on-Insulator integrated Bragg gratings,” Opt. Express 21(20), 23145–23159 (2013). [CrossRef] [PubMed]
24. Multi-Physics Software Comsol Corporation, “RF module - software for microwave and RF design,” http://www.comsol.com/rf-module (3.1.2015).
25. SOITEC Corporation, “Soitec silicon-on-isolator products,” http://www.soitec.com/en/products-and-services/microelectronics/wave-soi(3.1.2015).
26. M. Beals, J. Michel, J. Liu, D. Ahn, D. Sparacin, R. Sun, C. Hong, L. Kimerling, A. Pomerene, D. Carothers, J. Beattie, A. Kopa, A. Apsel, M. Rasras, D. Gill, S. Patel, K. Tu, Y. Chen, and A. White, “Process flow innovations for photonic device integration in CMOS,” Proc. SPIE 6898, 1–14 (2008).
27. K. Okamoto, Fundamentals of Optical Waveguides (Academic, 2000).
28. C. Qiu, Z. Sheng, H. Li, W. Liu, L. Li, A. Pang, A. Wu, X. Wang, S. Zou, and F. Gan, “Fabrication, characterization and loss analysis of silicon nanowaveguides,” IEEE/OSA J. Lightwave Technol. 32(13), 2303–2307 (2014). [CrossRef]
29. R. Ding, Y. Liu, Q. Li, Y. Yang, Y. Ma, K. Padmaraju, A. E.-J. Lim, G.-Q. Lo, K. Bergman, T. Baehr-Jones, and M. Hochberg, “Design and characterization of a 30-GHz bandwidth low-power silicon traveling-wave modulator,” Opt. Commun. 321, 124–133 (2014). [CrossRef]