Abstract

We demonstrate a Si photonic crystal waveguide Mach–Zehnder modulator that incorporates meander-line electrodes to compensate for the phase mismatch between slow light and RF signals. We first employed commonized ground electrodes in the modulator to suppress undesired fluctuations in the electro-optic (EO) response due to coupled slot-line modes of RF signals. Then, we theoretically and experimentally investigated the effect of the phase mismatch on the EO response. We confirmed that meander-line electrodes improve the EO response, particularly in the absence of internal reflection of the RF signals. The cut-off frequency of this device can reach 27 GHz, which allows high-speed modulation up to 50 Gbps.

© 2018 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

1. Introduction

Silicon (Si) photonics have become a standard platform for integrating functional photonic devices with optical interconnects owing to the strong optical confinement of Si waveguides, high accuracy, reproducible, and low cost mass production using complementary metal oxide semiconductor (CMOS) processing. For their use as optical interconnects, modulators with high speed, small size, low loss, low power consumption, and large operating wavelength and temperature tolerances are important. Mach–Zehnder modulators (MZMs) exploit carrier depletion in p-n doped Si rib waveguides. These devices allow high speed and large tolerances in wavelength and temperature [1–8] and have been used in commercial systems because of these practical advantages. However, they require a long phase shifter of the order of several millimeters, which increases the device size and power consumption. Our research group has studied photonic crystal waveguide (PCW) MZMs, which enhance the phase shift Δϕ in proportion to the slow-light group index ng. This allows a reduction in the length of the phase shifter, which also reduces power consumption [9–11]. A PCW can produce a wide range of ng from <10 to ~100 by structural tuning [12]. The group index can be set between 20 and 30 to balance the advantages of slow light, a sufficiently wide working spectrum, and moderately low propagation loss. This value is five to seven times larger than ng in Si rib waveguides. Lattice-shifted PCWs (LSPCWs), which feature specific rows of holes that are shifted in the photonic crystal slab along the waveguide, can produce low dispersion slow light over a wide working spectrum of Δλ = 15−20 nm. By employing a wavy p-n junction, a large phase shift Δϕ can be obtained with a fast operating speed. 32 Gbps modulation has been obtained with an extinction ratio ER = 3 dB and an excess modulation loss ML = 1 dB using a phase shifter of length L = 200 μm and drive voltage Vpp = 1.75 V [9].

A large value of the group index potentially reduces the operating voltage, but it simultaneously degrades the frequency response because of the electro-optic (EO) phase mismatch [10]. The modal index for an RF signal, nRF, ranges from 2 to 6 [4,5,7], which is much different from ng ≈20 for slow light. Therefore, the phase of the modulated slow light deviates from the RF signal phase as the waves propagate. The mismatch becomes large as the frequency increases, ultimately degrading the frequency response. Several studies have discussed the compensation of the phase mismatch in MZMs based on Si, LiNbO3, and III-V semiconductors [6–8,13,14]. One may consider to use capacitively-loaded slow-wave electrodes, which has been studied for phase or impedance matching [6–8]. However, nRF in these electrodes is 6 at most, which is not sufficiently high to offset the phase mismatch. An increasingly efficient phase matching scheme is needed to use slow light in MZMs at high operating speeds. In addition, we have previously observed a peculiar local minimum in the frequency response of such devices [10]. This minimum is likely caused by unwanted coupled slot-line (CSL) modes in the RF signal, which are excited owing to the imbalance of the electric potential between the ground (G) electrodes placed on both sides of the central signal (S) electrode [15]. This dip must be suppressed because it obscures the EO phase mismatch, which makes compensating for that mismatch increasingly difficult. Xu et al. suppressed this minimum by commonizing the ground electrodes with wire bonding [3].

The device proposed in this paper employs meander-line electrodes to suppress the phase mismatch and improve the frequency response of a Si photonic crystal slow light MZM. The meander-line electrodes delay the RF signals by simply extending the electrode length. In the device’s preparation, we first discuss an experimental test that confirms that commonized ground electrodes suppress the unwanted spectral local minimum, and then we focus on a design that compensates for the phase mismatch between the RF and slow light signals. We theoretically and experimentally determine the manner in which the frequency response is degraded by the phase mismatch and the manner it is compensated by the meander-line electrodes. We finally optimize the meander-line electrodes considering the reflection of RF signals at the ends of the phase shifters.

2. Device fabrication

We fabricated prototypes using a standard Si photonics CMOS process with a 200-mm-diameter silicon-on-insulator (Si thickness of 210 nm), KrF excimer laser exposure (λ = 248 nm), and phase shift masks with resolution less than 130 nm. Figure 1 shows the p-n doped and SiO2-cladded PCW MZMs that were fabricated. The Al coplanar waveguide (CPW) comprises a signal electrode and a pair of ground electrodes sandwiching the signal electrode. Termination resistors were not included in the design to simplify the device layout. In device (a), the ground electrodes are independent, while in devices (b) and (c), they were commonized. We used PCWs without lattice shifts to take advantage of the gradually increasing ng spectrum, which allows us to test a range of slow light group indices by varying the wavelength. We set the lattice constant a = 400 nm, the hole diameter 2r = 190 nm or 220 nm, and phase shifter length L = 200 μm. The design also employs a simple linear p-n junction. Although the interleaved and wavy junctions we have studied in the past effectively increase Δϕ [9], a linear junction is appropriate for evaluating the phase mismatch because the frequency response is constrained less by the RC time constant. The p- and n-type doping concentrations were NA = 1.05 × 1018 cm−3 and ND = 6.2 × 1017 cm−3, respectively. p+ and n+ doping was performed for the metal contacts at a concentration of NA+ = ND+ = 1.9 × 1020 cm−3. The doping sites were spaced at 4-µm intervals across the p-n junction to avoid strong free carrier absorption. Also, 36.5-μm long TiN heaters were integrated inside the SiO2 cladding above the PCWs so that we could tune the initial phase using the thermo-optic effect.

 

Fig. 1 Fabricated Si PCW MZMs. (a) Normal electrode device without commonizing grounds.In the inset, the p- and n-doped regions are colored. (b) Normal electrode device with commonized grounds. (c) Meander-line electrode device with commonized grounds. Insets show the details of a bend in the meander-line electrode and the separated p-n junction.

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For device (c), we introduced meander-line electrodes of length Ld and RF index nd as the signal electrodes in the CPWs. Similar to the electrodes at the PCW phase shifter, the Si layer was left intact and a metal contact was formed below the meander-line electrode. However, nd is different from nRF because of the no doping in this region; we assumed nd = 2 and nRF = 4, referring to [7]. The phase mismatch at the end of the phase shifter was given by δϕ = 2πf [(ngnRF)LndLd]/c. where f is the frequency of RF signals. We set Ld = 1186 μm to make δϕ = 0 when ng = 15.9 for slow light. Although the device footprint is increased by these electrodes, it is still much smaller than rib waveguide devices, and its width is comparable to the total width of probe pads. In the meander-line electrodes, we placed a corner reflector at each bend, which minimizes the impedance change and suppresses the reflection of RF signals. We separated the p-n junctions before and after the meander-line electrodes to maintain the continuity of the PCW and Si slab so that the waveguide loss is almost the same as it is for devices (a) and (b). The typical on-chip loss of a 200-μm p-n doped PCW is ~5 dB, including the coupling loss from Si-wire waveguides [9,10].

3. Ground commonization

To test the effects of the commonized ground electrodes, we measured the reflection parameter S11 using an RF vector network analyzer (VNA, Anritsu 37269-R), as shown in Fig. 2(a). At low frequencies, S11 is close to 0 dB; this means that the reflection is nearly complete owing to the absence of electrical termination. At high frequencies, S11 decreased owing to the loss of the RF signals. Without ground commonization, a local minimum and maximum were observed at f = 17 GHz and f = 23 GHz, respectively. We previously found that the frequency of this maximum corresponds to a minimum in the EO frequency response [10] and that it is similar to the minimum caused by the CSL mode [3]. In the two ground-commonized devices, the minimum and maximum disappeared and the response was smooth, thus indicating that the CSL mode was suppressed. Comparing the two ground-commonized devices, the S11 of the meander-line electrode device was 0.5–1.5 dB lower than that of the normal electrode device. This difference might be explained by the transmission loss over the round trip of RF signals in the long delay line, including the sum of weak radiation from the 24 bends. We also consider that the leakage current between the signal and ground electrodes in the delay line may contribute to this difference. However, the excess loss of one-way transmission is 1.5 dB/2 = 8% at the most.

 

Fig. 2 Measured S11 of fabricated devices.

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4. Theoretical electro-optical response

Next, we theoretically analyze the EO frequency response of the modulator. Figure 3 shows transmission models of the two types of electrodes that were tested. When we define the voltages of forward and backward propagating waves as Vf and Vb, the voltage V(z,f) at position z and frequency f is expressed as follows:

V(z,f)=Vfeγz+Vbeγz,
γ=α+jβRF,and
βRF=2πfnRFc,
where γ, α and βRF are the complex propagation constant, attenuation constant, and propagation constant of RF signals, and c is the speed of light. Considering the RF reflectivities at the start and end points of the phase shifter, Γg and ΓL, for the model in Fig. 3(a), V(z,f) is represented as follows:
V(z,f)=ZinVg(eγz+ΓLeγz2γL)(Zg+Zin)(1+ΓLe2γL)=Z0Vg(eγz+ΓLeγz2γL)(Z0+Zg)(1ΓgΓLe2γL),
ΓL=ZLZ0ZL+Z0,Γg=ZgZ0Zg+Z0,and
Zin=Z01+ΓLe2γL1ΓLe2γL,
where Zg is the internal impedance of the signal generator, Z0 is the characteristic impedance of the phase shifter, and Zin is the impedance on the right side from z = 0. From the velocity difference between RF signals and slow light pulses, the effective voltage for modulating the slow light, Veff, is represented as follows:
Veff(z,f)=Z0Vg[e(jβoγ)z+ΓLe(jβo+γ)z2γL](Z0+Zg)(1ΓgΓLe2γL)and
βo=2πfngc
To calculate the modulation depth over the entire phase shifter, we derive the average voltage Vave as follows [5,13,14]:
Vave(f)=0LVeffdzL=Z0Vg(ejφ+sincφ++ΓLejφe2γLsincφ)(Z0+Zg)(1ΓgΓLe2γL)and
φ±=(βo±jγ)L2
Furthermore, we consider the p-n doped PCW as a simple RC circuit consisting of the series electrical resistance Rpn and capacitance Cpn at the p-n junction. The appropriate transfer function G(f) is represented as follows:
G(f)=11+j2πf(Zg+Rpn)Cpn
We define the frequency response of the average voltage, η(f), as follows:
η(f)=|Vave(f)G(f)Vave(0)G(0)|
The EO response S21 is represented as follows:
S21(f)[dB]=20log10η(f)
For the meander-line electrode device in Fig. 3(b), we consider the phase changes before (0 ≤ zL/2) and after the middle of the line (L/2zL) separately and assume that in the delay line as ϕd. The effective voltages Veff1 and Veff2 are shown as
Veff1(z,f)=Z0Vg{e(jβoγ)z+ΓLe(jβo+γ)z2γLj2φd}(Z0+Zg)(1ΓgΓLe2γLj2φd)(0zL/2)
Veff2(z,f)=Z0Vg{e(jβoγ)zjφd+ΓLe(jβo+γ)z2γLjφd}(Z0+Zg)(1ΓgΓLe2γLj2φd)(L/2zL)
φd=2πfndLdc
From Eqs. (14) and (15), similar to the normal electrode, we derive the following:

 

Fig. 3 Calculation model of (a) normal electrode device, and (b) meander-line electrode device.

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Vave(f)=0L/2Veff1dzL/2+L/2LVeff2dzL/2
=Z0Vg{(ejφ+2+ej3φ+2jφd)sincφ+2+ΓLe2γL(ejφ2j2φd+ej3φ+2jφd)sincφ2}2(Z0+Zg)(1ΓgΓLe2γLj2φd)

The EO frequency responses are calculated from Eqs. (9) and (18) with c = 3.0 × 108 m/s and L = 200 μm. We set Rpn = 60 Ω and Cpn = 50 fF, as obtained from the previously reported simulations [9], nRF = 4 and nd = 2 [4,5,7], considering the doping and un-doping in the phase shifter and delay line, respectively, and neglecting the attenuation, i.e., α = 0, and reflection at z = 0, i.e., Γg = 0. We assume Z0 and Zg to be independent of the frequency so that their effect is eliminated by the normalization in Eq. (11). Figures 4(a) and 4(b) show the results for the normal electrode devices with ΓL = 1 and ΓL = 0, respectively. S21 is degraded as the value of ng increases. A sharp dip appears at f = 38 GHz for ng = 40, and it shifts to a low frequency for a large value of ng. This dip is caused by the inversion of Veff against the modulated light in the phase shifter. No marked differences are observed between ΓL = 1 and ΓL = 0, except the different depths of the dip and the slight improvement of the response with ΓL = 0. Figures 4(c) and 4(d) show the results for the meander-line device with Ld = 1186 μm. The dip appears at f < 40 GHz even if ng = 55. This shows that the delay line effectively compensates for the phase mismatch. The response is particularly improved when ΓL = 0. For example, the response of the meander-line device with ng = 35 appears to be the same as that of the normal electrode devices with ng = 22. Even when ΓL = 1, the response is improved by the meander line at f > 30 GHz, but a slow valley appears at f = 20–30 GHz, which degrades the cut-off frequency f3dB. This slow valley is caused by the cancelation of forward and backward RF signals, which is made increasingly evident owing to the large phase shift in the delay line. Figure 5 summarizes f3dB for various ΓL. When ΓL > 0.25, f3dB decreases as Ld increases; therefore, the meander-line electrodes degrade performance somewhat. When ΓL ≤ 0.25, f3dB shows a maximal value for certain Ld. For ΓL = 0 and Ld = 0.8 mm, f3dB = 27 GHz is possible, which allows for 50 Gbps modulation.

 

Fig. 4 Calculated S21 of the EO frequency response for L = 200 μm, α = 0, Γg = 0, nRF = 4. (a), (b) Normal electrode device. (c), (d) Meander-line electrode device of Ld = 1186 μm, nd = 2. (a), (c) ΓL = 1, (b), (d) ΓL = 0.

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Fig. 5 Calculated f3dB for L = 200 μm, Γg = 0, nd = 2, nRF = 4, ng = 20, and various ΓL.

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5. Measurement of electro-optical response

While measuring the EO frequency responses, we applied small amplitude sinusoidal signals to a single channel of the fabricated ground-commonized devices from the VNA through a bias tee and an RF probe. Modulated light was input to an OE converter (Anritsu MN4765, >70 GHz) after passing through an erbium-doped fiber and a tunable band pass filter (BPF), after which the signal was returned to the VNA. The optical phase difference between the two arms of the device was tuned to 90° by the TiN heaters. Figures 6(a) and 6(b) show the results for the normal electrode device and the corresponding ng spectrum as measured via the modulation phase shift method using a dispersion analyzer (Alnair Labs CDA2100). ng changed from 22 to a value higher than 70 over the measured wavelength range. The spectral dip caused by the phase mismatch did not appear in the range ng < 32, while it appeared for ng > 34, which is consistent with our calculations. Figures 6(c) and 6(d) show the results for the meander-line device. The dip caused by the phase mismatch was suppressed up to ng = 69, while the slow valley appeared at f = 20–30 GHz. These results were consistent with the calculation for ΓL = 1, which is the case in the fabricated device with no electrical terminations.

 

Fig. 6 (a), (c) Measured EO frequency response and (b), (d) corresponding ng spectrum. (a), (b) Normal electrode device, (c), (d) Meander-line electrode device. VDC = −2 V.

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6. Conclusion

In this paper, we proposed a PCW MZM with meander-line electrodes that suppress the phase mismatch between slow light and RF signals. While preparing this device, we first measured the frequency response of devices fabricated with Si photonics CMOS processing and confirmed that the CSL mode of RF signals that deforms the frequency response is suppressed by commonizing the ground electrodes. We also confirmed that the one-way transmission loss of the meander-line electrode is as low as 0.75 dB. Then, we theoretically compared devices with normal and meander-line electrodes and concluded that the meander-line device achieves a high cut-off frequency at 27 GHz, which will allow operation at 50 Gbps. We confirmed a clear correspondence between our theoretical expectation and preliminary experimental results. In the measurements, the emergence of a slow valley caused by the reflection of RF signals slightly degraded the cut-off frequency. We aim to achieve a sufficiently high cut-off frequency by optimizing the electrode terminations, which we leave for future work. Some groups have already demonstrated high speed devices over 40 Gbps in Si photonics, but they used a phase shifter of as long as 1.0‒3.5 mm having a large optical loss of 9‒15 dB [2,3]. We expect our device to achieve 50 Gbps operation with a much shorter phase shifter of 200 μm and a moderate loss of ~6 dB [9,10].

Funding

New Energy and Industrial Technology Development Organization (NEDO) (project #P13004).

References and links

1. G. T. Reed, G. Z. Mashanovich, F. Y. Gardes, and D. J. Thomson, “Silicon optical modulators,” Nat. Photonics 4(8), 518–526 (2010). [CrossRef]  

2. D. J. Thomson, F. Y. Gardes, S. Liu, H. Porte, L. Zimmermann, J. M. Fedeli, Y. Hu, M. Nedeljkovic, X. Yang, P. Petropoulos, and G. Z. Mashanovich, “High Performance Mach-Zehnder-Based Silicon Optical Modulators,” IEEE J. Sel. Top. Quantum Electron. 19(6), 85–94 (2013). [CrossRef]  

3. H. Xu, X. Li, X. Xiao, Z. Li, Y. Yu, and J. Yu, “Demonstration and Characterization of High-Speed Silicon Depletion-Mode Mach–Zehnder Modulators,” IEEE J. Sel. Top. Quantum Electron. 20(4), 23–32 (2014). [CrossRef]  

4. X. Tu, K. F. Chang, T. Y. Liow, J. Song, X. Luo, L. Jia, Q. Fang, M. Yu, G. Q. Lo, P. Dong, and Y. K. Chen, “Silicon optical modulator with shield coplanar waveguide electrodes,” Opt. Express 22(19), 23724–23731 (2014). [CrossRef]   [PubMed]  

5. H. Yu and W. Bogaerts, “An Equivalent Circuit Model of the Traveling Wave Electrode for Carrier-Depletion-Based Silicon Optical Modulators,” J. Lightwave Technol. 30(11), 1602–1609 (2012). [CrossRef]  

6. J. Shin, S. R. Sakamoto, and N. Dagli, “Conductor Loss of Capacitively Loaded Slow Wave Electrodes for High-Speed Photonic Devices,” J. Lightwave Technol. 29(1), 48–52 (2011). [CrossRef]  

7. R. Ding, Y. Liu, Y. Ma, Y. Yang, Q. Li, A. E. J. Lim, G. Q. Lo, K. Bergman, T. B. Jones, and M. Hochberg, “High-Speed Silicon Modulator With Slow-Wave Electrodes and Fully Independent Differential Drive,” J. Lightwave Technol. 32(12), 2240–2247 (2014). [CrossRef]  

8. D. Patel, S. Ghosh, M. Chagnon, A. Samani, V. Veerasubramanian, M. Osman, and D. V. Plant, “Design, analysis, and transmission system performance of a 41 GHz silicon photonic modulator,” Opt. Express 23(11), 14263–14287 (2015). [CrossRef]   [PubMed]  

9. Y. Terada, T. Tatebe, Y. Hinakura, and T. Baba, “Si Photonic Crystal Slow-Light Modulators with Periodic p–n Junctions,” J. Lightwave Technol. 35(9), 1684–1692 (2017). [CrossRef]  

10. Y. Hinakura, Y. Terada, T. Tamura, and T. Baba, “Wide spectral characteristics of Si photonic crystal Mach-Zehnder modulator fabricated by complementary metal–oxide–semiconductor process,” Photonics 3(2), 17 (2016).

11. K. Hojo, Y. Terada, N. Yazawa, T. Watanabe, and T. Baba, “Compact QPSK and PAM modulators with Si photonic crystal slow light phase shifters,” IEEE Photonics Technol. Lett. 28(13), 1438–1441 (2016). [CrossRef]  

12. T. Tamura, K. Kondo, Y. Terada, Y. Hinakura, N. Ishikura, and T. Baba, “Silica-clad silicon photonic crystal waveguides for wideband dispersion-free slow light,” J. Lightwave Technol. 33(7), 3034–3040 (2015).

13. K. Kubota, J. Noda, and O. Mikami, “Traveling wave optical modulator using a directional coupler LiNbO3waveguide,” IEEE J. Quantum Electron. 16(7), 754–760 (1980). [CrossRef]  

14. I. Kim, M. R. T. Tan, and S. Y. Wang, “Analysis of a new microwave low-loss and velocity-matched III-V transmission line for traveling-wave electrooptic modulators,” J. Lightwave Technol. 8(5), 728–738 (1990). [CrossRef]  

15. G. E. Ponchak, J. Papapolymerou, and M. M. Tentzeris, “Excitation of coupled slotline mode in finite-ground CPW with unequal ground-plane widths,” IEEE Trans. Microw. Theory Tech. 53(2), 713–717 (2005). [CrossRef]  

References

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  1. G. T. Reed, G. Z. Mashanovich, F. Y. Gardes, and D. J. Thomson, “Silicon optical modulators,” Nat. Photonics 4(8), 518–526 (2010).
    [Crossref]
  2. D. J. Thomson, F. Y. Gardes, S. Liu, H. Porte, L. Zimmermann, J. M. Fedeli, Y. Hu, M. Nedeljkovic, X. Yang, P. Petropoulos, and G. Z. Mashanovich, “High Performance Mach-Zehnder-Based Silicon Optical Modulators,” IEEE J. Sel. Top. Quantum Electron. 19(6), 85–94 (2013).
    [Crossref]
  3. H. Xu, X. Li, X. Xiao, Z. Li, Y. Yu, and J. Yu, “Demonstration and Characterization of High-Speed Silicon Depletion-Mode Mach–Zehnder Modulators,” IEEE J. Sel. Top. Quantum Electron. 20(4), 23–32 (2014).
    [Crossref]
  4. X. Tu, K. F. Chang, T. Y. Liow, J. Song, X. Luo, L. Jia, Q. Fang, M. Yu, G. Q. Lo, P. Dong, and Y. K. Chen, “Silicon optical modulator with shield coplanar waveguide electrodes,” Opt. Express 22(19), 23724–23731 (2014).
    [Crossref] [PubMed]
  5. H. Yu and W. Bogaerts, “An Equivalent Circuit Model of the Traveling Wave Electrode for Carrier-Depletion-Based Silicon Optical Modulators,” J. Lightwave Technol. 30(11), 1602–1609 (2012).
    [Crossref]
  6. J. Shin, S. R. Sakamoto, and N. Dagli, “Conductor Loss of Capacitively Loaded Slow Wave Electrodes for High-Speed Photonic Devices,” J. Lightwave Technol. 29(1), 48–52 (2011).
    [Crossref]
  7. R. Ding, Y. Liu, Y. Ma, Y. Yang, Q. Li, A. E. J. Lim, G. Q. Lo, K. Bergman, T. B. Jones, and M. Hochberg, “High-Speed Silicon Modulator With Slow-Wave Electrodes and Fully Independent Differential Drive,” J. Lightwave Technol. 32(12), 2240–2247 (2014).
    [Crossref]
  8. D. Patel, S. Ghosh, M. Chagnon, A. Samani, V. Veerasubramanian, M. Osman, and D. V. Plant, “Design, analysis, and transmission system performance of a 41 GHz silicon photonic modulator,” Opt. Express 23(11), 14263–14287 (2015).
    [Crossref] [PubMed]
  9. Y. Terada, T. Tatebe, Y. Hinakura, and T. Baba, “Si Photonic Crystal Slow-Light Modulators with Periodic p–n Junctions,” J. Lightwave Technol. 35(9), 1684–1692 (2017).
    [Crossref]
  10. Y. Hinakura, Y. Terada, T. Tamura, and T. Baba, “Wide spectral characteristics of Si photonic crystal Mach-Zehnder modulator fabricated by complementary metal–oxide–semiconductor process,” Photonics 3(2), 17 (2016).
  11. K. Hojo, Y. Terada, N. Yazawa, T. Watanabe, and T. Baba, “Compact QPSK and PAM modulators with Si photonic crystal slow light phase shifters,” IEEE Photonics Technol. Lett. 28(13), 1438–1441 (2016).
    [Crossref]
  12. T. Tamura, K. Kondo, Y. Terada, Y. Hinakura, N. Ishikura, and T. Baba, “Silica-clad silicon photonic crystal waveguides for wideband dispersion-free slow light,” J. Lightwave Technol. 33(7), 3034–3040 (2015).
  13. K. Kubota, J. Noda, and O. Mikami, “Traveling wave optical modulator using a directional coupler LiNbO3waveguide,” IEEE J. Quantum Electron. 16(7), 754–760 (1980).
    [Crossref]
  14. I. Kim, M. R. T. Tan, and S. Y. Wang, “Analysis of a new microwave low-loss and velocity-matched III-V transmission line for traveling-wave electrooptic modulators,” J. Lightwave Technol. 8(5), 728–738 (1990).
    [Crossref]
  15. G. E. Ponchak, J. Papapolymerou, and M. M. Tentzeris, “Excitation of coupled slotline mode in finite-ground CPW with unequal ground-plane widths,” IEEE Trans. Microw. Theory Tech. 53(2), 713–717 (2005).
    [Crossref]

2017 (1)

2016 (2)

Y. Hinakura, Y. Terada, T. Tamura, and T. Baba, “Wide spectral characteristics of Si photonic crystal Mach-Zehnder modulator fabricated by complementary metal–oxide–semiconductor process,” Photonics 3(2), 17 (2016).

K. Hojo, Y. Terada, N. Yazawa, T. Watanabe, and T. Baba, “Compact QPSK and PAM modulators with Si photonic crystal slow light phase shifters,” IEEE Photonics Technol. Lett. 28(13), 1438–1441 (2016).
[Crossref]

2015 (2)

2014 (3)

2013 (1)

D. J. Thomson, F. Y. Gardes, S. Liu, H. Porte, L. Zimmermann, J. M. Fedeli, Y. Hu, M. Nedeljkovic, X. Yang, P. Petropoulos, and G. Z. Mashanovich, “High Performance Mach-Zehnder-Based Silicon Optical Modulators,” IEEE J. Sel. Top. Quantum Electron. 19(6), 85–94 (2013).
[Crossref]

2012 (1)

2011 (1)

2010 (1)

G. T. Reed, G. Z. Mashanovich, F. Y. Gardes, and D. J. Thomson, “Silicon optical modulators,” Nat. Photonics 4(8), 518–526 (2010).
[Crossref]

2005 (1)

G. E. Ponchak, J. Papapolymerou, and M. M. Tentzeris, “Excitation of coupled slotline mode in finite-ground CPW with unequal ground-plane widths,” IEEE Trans. Microw. Theory Tech. 53(2), 713–717 (2005).
[Crossref]

1990 (1)

I. Kim, M. R. T. Tan, and S. Y. Wang, “Analysis of a new microwave low-loss and velocity-matched III-V transmission line for traveling-wave electrooptic modulators,” J. Lightwave Technol. 8(5), 728–738 (1990).
[Crossref]

1980 (1)

K. Kubota, J. Noda, and O. Mikami, “Traveling wave optical modulator using a directional coupler LiNbO3waveguide,” IEEE J. Quantum Electron. 16(7), 754–760 (1980).
[Crossref]

Baba, T.

Y. Terada, T. Tatebe, Y. Hinakura, and T. Baba, “Si Photonic Crystal Slow-Light Modulators with Periodic p–n Junctions,” J. Lightwave Technol. 35(9), 1684–1692 (2017).
[Crossref]

K. Hojo, Y. Terada, N. Yazawa, T. Watanabe, and T. Baba, “Compact QPSK and PAM modulators with Si photonic crystal slow light phase shifters,” IEEE Photonics Technol. Lett. 28(13), 1438–1441 (2016).
[Crossref]

Y. Hinakura, Y. Terada, T. Tamura, and T. Baba, “Wide spectral characteristics of Si photonic crystal Mach-Zehnder modulator fabricated by complementary metal–oxide–semiconductor process,” Photonics 3(2), 17 (2016).

T. Tamura, K. Kondo, Y. Terada, Y. Hinakura, N. Ishikura, and T. Baba, “Silica-clad silicon photonic crystal waveguides for wideband dispersion-free slow light,” J. Lightwave Technol. 33(7), 3034–3040 (2015).

Bergman, K.

Bogaerts, W.

Chagnon, M.

Chang, K. F.

Chen, Y. K.

Dagli, N.

Ding, R.

Dong, P.

Fang, Q.

Fedeli, J. M.

D. J. Thomson, F. Y. Gardes, S. Liu, H. Porte, L. Zimmermann, J. M. Fedeli, Y. Hu, M. Nedeljkovic, X. Yang, P. Petropoulos, and G. Z. Mashanovich, “High Performance Mach-Zehnder-Based Silicon Optical Modulators,” IEEE J. Sel. Top. Quantum Electron. 19(6), 85–94 (2013).
[Crossref]

Gardes, F. Y.

D. J. Thomson, F. Y. Gardes, S. Liu, H. Porte, L. Zimmermann, J. M. Fedeli, Y. Hu, M. Nedeljkovic, X. Yang, P. Petropoulos, and G. Z. Mashanovich, “High Performance Mach-Zehnder-Based Silicon Optical Modulators,” IEEE J. Sel. Top. Quantum Electron. 19(6), 85–94 (2013).
[Crossref]

G. T. Reed, G. Z. Mashanovich, F. Y. Gardes, and D. J. Thomson, “Silicon optical modulators,” Nat. Photonics 4(8), 518–526 (2010).
[Crossref]

Ghosh, S.

Hinakura, Y.

Hochberg, M.

Hojo, K.

K. Hojo, Y. Terada, N. Yazawa, T. Watanabe, and T. Baba, “Compact QPSK and PAM modulators with Si photonic crystal slow light phase shifters,” IEEE Photonics Technol. Lett. 28(13), 1438–1441 (2016).
[Crossref]

Hu, Y.

D. J. Thomson, F. Y. Gardes, S. Liu, H. Porte, L. Zimmermann, J. M. Fedeli, Y. Hu, M. Nedeljkovic, X. Yang, P. Petropoulos, and G. Z. Mashanovich, “High Performance Mach-Zehnder-Based Silicon Optical Modulators,” IEEE J. Sel. Top. Quantum Electron. 19(6), 85–94 (2013).
[Crossref]

Ishikura, N.

Jia, L.

Jones, T. B.

Kim, I.

I. Kim, M. R. T. Tan, and S. Y. Wang, “Analysis of a new microwave low-loss and velocity-matched III-V transmission line for traveling-wave electrooptic modulators,” J. Lightwave Technol. 8(5), 728–738 (1990).
[Crossref]

Kondo, K.

Kubota, K.

K. Kubota, J. Noda, and O. Mikami, “Traveling wave optical modulator using a directional coupler LiNbO3waveguide,” IEEE J. Quantum Electron. 16(7), 754–760 (1980).
[Crossref]

Li, Q.

Li, X.

H. Xu, X. Li, X. Xiao, Z. Li, Y. Yu, and J. Yu, “Demonstration and Characterization of High-Speed Silicon Depletion-Mode Mach–Zehnder Modulators,” IEEE J. Sel. Top. Quantum Electron. 20(4), 23–32 (2014).
[Crossref]

Li, Z.

H. Xu, X. Li, X. Xiao, Z. Li, Y. Yu, and J. Yu, “Demonstration and Characterization of High-Speed Silicon Depletion-Mode Mach–Zehnder Modulators,” IEEE J. Sel. Top. Quantum Electron. 20(4), 23–32 (2014).
[Crossref]

Lim, A. E. J.

Liow, T. Y.

Liu, S.

D. J. Thomson, F. Y. Gardes, S. Liu, H. Porte, L. Zimmermann, J. M. Fedeli, Y. Hu, M. Nedeljkovic, X. Yang, P. Petropoulos, and G. Z. Mashanovich, “High Performance Mach-Zehnder-Based Silicon Optical Modulators,” IEEE J. Sel. Top. Quantum Electron. 19(6), 85–94 (2013).
[Crossref]

Liu, Y.

Lo, G. Q.

Luo, X.

Ma, Y.

Mashanovich, G. Z.

D. J. Thomson, F. Y. Gardes, S. Liu, H. Porte, L. Zimmermann, J. M. Fedeli, Y. Hu, M. Nedeljkovic, X. Yang, P. Petropoulos, and G. Z. Mashanovich, “High Performance Mach-Zehnder-Based Silicon Optical Modulators,” IEEE J. Sel. Top. Quantum Electron. 19(6), 85–94 (2013).
[Crossref]

G. T. Reed, G. Z. Mashanovich, F. Y. Gardes, and D. J. Thomson, “Silicon optical modulators,” Nat. Photonics 4(8), 518–526 (2010).
[Crossref]

Mikami, O.

K. Kubota, J. Noda, and O. Mikami, “Traveling wave optical modulator using a directional coupler LiNbO3waveguide,” IEEE J. Quantum Electron. 16(7), 754–760 (1980).
[Crossref]

Nedeljkovic, M.

D. J. Thomson, F. Y. Gardes, S. Liu, H. Porte, L. Zimmermann, J. M. Fedeli, Y. Hu, M. Nedeljkovic, X. Yang, P. Petropoulos, and G. Z. Mashanovich, “High Performance Mach-Zehnder-Based Silicon Optical Modulators,” IEEE J. Sel. Top. Quantum Electron. 19(6), 85–94 (2013).
[Crossref]

Noda, J.

K. Kubota, J. Noda, and O. Mikami, “Traveling wave optical modulator using a directional coupler LiNbO3waveguide,” IEEE J. Quantum Electron. 16(7), 754–760 (1980).
[Crossref]

Osman, M.

Papapolymerou, J.

G. E. Ponchak, J. Papapolymerou, and M. M. Tentzeris, “Excitation of coupled slotline mode in finite-ground CPW with unequal ground-plane widths,” IEEE Trans. Microw. Theory Tech. 53(2), 713–717 (2005).
[Crossref]

Patel, D.

Petropoulos, P.

D. J. Thomson, F. Y. Gardes, S. Liu, H. Porte, L. Zimmermann, J. M. Fedeli, Y. Hu, M. Nedeljkovic, X. Yang, P. Petropoulos, and G. Z. Mashanovich, “High Performance Mach-Zehnder-Based Silicon Optical Modulators,” IEEE J. Sel. Top. Quantum Electron. 19(6), 85–94 (2013).
[Crossref]

Plant, D. V.

Ponchak, G. E.

G. E. Ponchak, J. Papapolymerou, and M. M. Tentzeris, “Excitation of coupled slotline mode in finite-ground CPW with unequal ground-plane widths,” IEEE Trans. Microw. Theory Tech. 53(2), 713–717 (2005).
[Crossref]

Porte, H.

D. J. Thomson, F. Y. Gardes, S. Liu, H. Porte, L. Zimmermann, J. M. Fedeli, Y. Hu, M. Nedeljkovic, X. Yang, P. Petropoulos, and G. Z. Mashanovich, “High Performance Mach-Zehnder-Based Silicon Optical Modulators,” IEEE J. Sel. Top. Quantum Electron. 19(6), 85–94 (2013).
[Crossref]

Reed, G. T.

G. T. Reed, G. Z. Mashanovich, F. Y. Gardes, and D. J. Thomson, “Silicon optical modulators,” Nat. Photonics 4(8), 518–526 (2010).
[Crossref]

Sakamoto, S. R.

Samani, A.

Shin, J.

Song, J.

Tamura, T.

Y. Hinakura, Y. Terada, T. Tamura, and T. Baba, “Wide spectral characteristics of Si photonic crystal Mach-Zehnder modulator fabricated by complementary metal–oxide–semiconductor process,” Photonics 3(2), 17 (2016).

T. Tamura, K. Kondo, Y. Terada, Y. Hinakura, N. Ishikura, and T. Baba, “Silica-clad silicon photonic crystal waveguides for wideband dispersion-free slow light,” J. Lightwave Technol. 33(7), 3034–3040 (2015).

Tan, M. R. T.

I. Kim, M. R. T. Tan, and S. Y. Wang, “Analysis of a new microwave low-loss and velocity-matched III-V transmission line for traveling-wave electrooptic modulators,” J. Lightwave Technol. 8(5), 728–738 (1990).
[Crossref]

Tatebe, T.

Tentzeris, M. M.

G. E. Ponchak, J. Papapolymerou, and M. M. Tentzeris, “Excitation of coupled slotline mode in finite-ground CPW with unequal ground-plane widths,” IEEE Trans. Microw. Theory Tech. 53(2), 713–717 (2005).
[Crossref]

Terada, Y.

Y. Terada, T. Tatebe, Y. Hinakura, and T. Baba, “Si Photonic Crystal Slow-Light Modulators with Periodic p–n Junctions,” J. Lightwave Technol. 35(9), 1684–1692 (2017).
[Crossref]

K. Hojo, Y. Terada, N. Yazawa, T. Watanabe, and T. Baba, “Compact QPSK and PAM modulators with Si photonic crystal slow light phase shifters,” IEEE Photonics Technol. Lett. 28(13), 1438–1441 (2016).
[Crossref]

Y. Hinakura, Y. Terada, T. Tamura, and T. Baba, “Wide spectral characteristics of Si photonic crystal Mach-Zehnder modulator fabricated by complementary metal–oxide–semiconductor process,” Photonics 3(2), 17 (2016).

T. Tamura, K. Kondo, Y. Terada, Y. Hinakura, N. Ishikura, and T. Baba, “Silica-clad silicon photonic crystal waveguides for wideband dispersion-free slow light,” J. Lightwave Technol. 33(7), 3034–3040 (2015).

Thomson, D. J.

D. J. Thomson, F. Y. Gardes, S. Liu, H. Porte, L. Zimmermann, J. M. Fedeli, Y. Hu, M. Nedeljkovic, X. Yang, P. Petropoulos, and G. Z. Mashanovich, “High Performance Mach-Zehnder-Based Silicon Optical Modulators,” IEEE J. Sel. Top. Quantum Electron. 19(6), 85–94 (2013).
[Crossref]

G. T. Reed, G. Z. Mashanovich, F. Y. Gardes, and D. J. Thomson, “Silicon optical modulators,” Nat. Photonics 4(8), 518–526 (2010).
[Crossref]

Tu, X.

Veerasubramanian, V.

Wang, S. Y.

I. Kim, M. R. T. Tan, and S. Y. Wang, “Analysis of a new microwave low-loss and velocity-matched III-V transmission line for traveling-wave electrooptic modulators,” J. Lightwave Technol. 8(5), 728–738 (1990).
[Crossref]

Watanabe, T.

K. Hojo, Y. Terada, N. Yazawa, T. Watanabe, and T. Baba, “Compact QPSK and PAM modulators with Si photonic crystal slow light phase shifters,” IEEE Photonics Technol. Lett. 28(13), 1438–1441 (2016).
[Crossref]

Xiao, X.

H. Xu, X. Li, X. Xiao, Z. Li, Y. Yu, and J. Yu, “Demonstration and Characterization of High-Speed Silicon Depletion-Mode Mach–Zehnder Modulators,” IEEE J. Sel. Top. Quantum Electron. 20(4), 23–32 (2014).
[Crossref]

Xu, H.

H. Xu, X. Li, X. Xiao, Z. Li, Y. Yu, and J. Yu, “Demonstration and Characterization of High-Speed Silicon Depletion-Mode Mach–Zehnder Modulators,” IEEE J. Sel. Top. Quantum Electron. 20(4), 23–32 (2014).
[Crossref]

Yang, X.

D. J. Thomson, F. Y. Gardes, S. Liu, H. Porte, L. Zimmermann, J. M. Fedeli, Y. Hu, M. Nedeljkovic, X. Yang, P. Petropoulos, and G. Z. Mashanovich, “High Performance Mach-Zehnder-Based Silicon Optical Modulators,” IEEE J. Sel. Top. Quantum Electron. 19(6), 85–94 (2013).
[Crossref]

Yang, Y.

Yazawa, N.

K. Hojo, Y. Terada, N. Yazawa, T. Watanabe, and T. Baba, “Compact QPSK and PAM modulators with Si photonic crystal slow light phase shifters,” IEEE Photonics Technol. Lett. 28(13), 1438–1441 (2016).
[Crossref]

Yu, H.

Yu, J.

H. Xu, X. Li, X. Xiao, Z. Li, Y. Yu, and J. Yu, “Demonstration and Characterization of High-Speed Silicon Depletion-Mode Mach–Zehnder Modulators,” IEEE J. Sel. Top. Quantum Electron. 20(4), 23–32 (2014).
[Crossref]

Yu, M.

Yu, Y.

H. Xu, X. Li, X. Xiao, Z. Li, Y. Yu, and J. Yu, “Demonstration and Characterization of High-Speed Silicon Depletion-Mode Mach–Zehnder Modulators,” IEEE J. Sel. Top. Quantum Electron. 20(4), 23–32 (2014).
[Crossref]

Zimmermann, L.

D. J. Thomson, F. Y. Gardes, S. Liu, H. Porte, L. Zimmermann, J. M. Fedeli, Y. Hu, M. Nedeljkovic, X. Yang, P. Petropoulos, and G. Z. Mashanovich, “High Performance Mach-Zehnder-Based Silicon Optical Modulators,” IEEE J. Sel. Top. Quantum Electron. 19(6), 85–94 (2013).
[Crossref]

IEEE J. Quantum Electron. (1)

K. Kubota, J. Noda, and O. Mikami, “Traveling wave optical modulator using a directional coupler LiNbO3waveguide,” IEEE J. Quantum Electron. 16(7), 754–760 (1980).
[Crossref]

IEEE J. Sel. Top. Quantum Electron. (2)

D. J. Thomson, F. Y. Gardes, S. Liu, H. Porte, L. Zimmermann, J. M. Fedeli, Y. Hu, M. Nedeljkovic, X. Yang, P. Petropoulos, and G. Z. Mashanovich, “High Performance Mach-Zehnder-Based Silicon Optical Modulators,” IEEE J. Sel. Top. Quantum Electron. 19(6), 85–94 (2013).
[Crossref]

H. Xu, X. Li, X. Xiao, Z. Li, Y. Yu, and J. Yu, “Demonstration and Characterization of High-Speed Silicon Depletion-Mode Mach–Zehnder Modulators,” IEEE J. Sel. Top. Quantum Electron. 20(4), 23–32 (2014).
[Crossref]

IEEE Photonics Technol. Lett. (1)

K. Hojo, Y. Terada, N. Yazawa, T. Watanabe, and T. Baba, “Compact QPSK and PAM modulators with Si photonic crystal slow light phase shifters,” IEEE Photonics Technol. Lett. 28(13), 1438–1441 (2016).
[Crossref]

IEEE Trans. Microw. Theory Tech. (1)

G. E. Ponchak, J. Papapolymerou, and M. M. Tentzeris, “Excitation of coupled slotline mode in finite-ground CPW with unequal ground-plane widths,” IEEE Trans. Microw. Theory Tech. 53(2), 713–717 (2005).
[Crossref]

J. Lightwave Technol. (6)

Nat. Photonics (1)

G. T. Reed, G. Z. Mashanovich, F. Y. Gardes, and D. J. Thomson, “Silicon optical modulators,” Nat. Photonics 4(8), 518–526 (2010).
[Crossref]

Opt. Express (2)

Photonics (1)

Y. Hinakura, Y. Terada, T. Tamura, and T. Baba, “Wide spectral characteristics of Si photonic crystal Mach-Zehnder modulator fabricated by complementary metal–oxide–semiconductor process,” Photonics 3(2), 17 (2016).

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Figures (6)

Fig. 1
Fig. 1 Fabricated Si PCW MZMs. (a) Normal electrode device without commonizing grounds.In the inset, the p- and n-doped regions are colored. (b) Normal electrode device with commonized grounds. (c) Meander-line electrode device with commonized grounds. Insets show the details of a bend in the meander-line electrode and the separated p-n junction.
Fig. 2
Fig. 2 Measured S11 of fabricated devices.
Fig. 3
Fig. 3 Calculation model of (a) normal electrode device, and (b) meander-line electrode device.
Fig. 4
Fig. 4 Calculated S21 of the EO frequency response for L = 200 μm, α = 0, Γg = 0, nRF = 4. (a), (b) Normal electrode device. (c), (d) Meander-line electrode device of Ld = 1186 μm, nd = 2. (a), (c) ΓL = 1, (b), (d) ΓL = 0.
Fig. 5
Fig. 5 Calculated f3dB for L = 200 μm, Γg = 0, nd = 2, nRF = 4, ng = 20, and various ΓL.
Fig. 6
Fig. 6 (a), (c) Measured EO frequency response and (b), (d) corresponding ng spectrum. (a), (b) Normal electrode device, (c), (d) Meander-line electrode device. VDC = −2 V.

Equations (18)

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V(z,f)= V f e γz + V b e γz ,
γ=α+j β RF , and
β RF = 2πf n RF c ,
V(z,f)= Z in V g ( e γz + Γ L e γz2γL ) ( Z g + Z in )(1+ Γ L e 2γL ) = Z 0 V g ( e γz + Γ L e γz2γL ) ( Z 0 + Z g )(1 Γ g Γ L e 2γL ) ,
Γ L = Z L Z 0 Z L + Z 0 , Γ g = Z g Z 0 Z g + Z 0 ,and
Z in = Z 0 1+ Γ L e 2γL 1 Γ L e 2γL ,
V eff (z,f)= Z 0 V g [ e ( j β o γ )z + Γ L e ( j β o +γ )z2γL ] ( Z 0 + Z g )(1 Γ g Γ L e 2γL ) and
β o = 2πf n g c
V ave (f)= 0 L V eff dz L = Z 0 V g ( e j φ + sinc φ + + Γ L e j φ e 2γL sinc φ ) ( Z 0 + Z g )(1 Γ g Γ L e 2γL ) and
φ ± = ( β o ±jγ )L 2
G(f)= 1 1+j2πf( Z g + R pn ) C pn
η(f)=| V ave (f)G(f) V ave (0)G(0) |
S 21 (f)[dB]=20 log 10 η(f)
V eff1 ( z,f )= Z 0 V g { e ( j β o γ )z + Γ L e ( j β o +γ )z2γLj2 φ d } ( Z 0 + Z g )(1 Γ g Γ L e 2γLj2 φ d ) (0zL/2)
V eff2 ( z,f )= Z 0 V g { e ( j β o γ )zj φ d + Γ L e ( j β o +γ )z2γLj φ d } ( Z 0 + Z g )(1 Γ g Γ L e 2γLj2 φ d ) (L/2zL)
φ d = 2πf n d L d c
V ave (f)= 0 L/2 V eff1 dz L/2 + L/2 L V eff2 dz L/2
= Z 0 V g { ( e j φ + 2 + e j3 φ + 2 j φ d )sinc φ + 2 + Γ L e 2γL ( e j φ 2 j2 φ d + e j3 φ + 2 j φ d )sinc φ 2 } 2( Z 0 + Z g )(1 Γ g Γ L e 2γLj2 φ d )

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