## Lossless intensity modulation in integrated photonics |

Optics Express, Vol. 20, Issue 4, pp. 4280-4290 (2012)

http://dx.doi.org/10.1364/OE.20.004280

Acrobat PDF (1564 KB)

### Abstract

We present a dynamical analysis of lossless intensity modulation in two different ring resonator geometries. In both geometries, we demonstrate modulation schemes that result in a symmetrical output with an infinite on/off ratio. The systems behave as lossless intensity modulators where the time-averaged output optical power is equal to the time-averaged input optical power.

© 2012 OSA

## 1. Introduction

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

2. A. Liu, R. Jones, L. Liao, D. Samara-Rubio 1, D. Rubin, O. Cohen, R. Nicolaescu, and M. Paniccia, “A high-speed silicon optical modulator based on a metal-oxide-semiconductor capacitor,” Nature **427**, 615–618 (2004). [CrossRef] [PubMed]

7. H.-W. Chen, Y.-H. Kuo, and J. E. Bowers, “25 Gb/s hybrid silicon switch using a capacitively loaded traveling wave electrode,” Opt. Express **18**, 1070–1075 (2010). [CrossRef] [PubMed]

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

11. S. Manipatruni, K. Preston, L. Chen, and M. Lipson, “Ultra-low voltage, ultra-small mode volume silicon microring modulator,” Opt. Express **18**, 18235–18242 (2010). [CrossRef] [PubMed]

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

12. A. Yariv, “Critical coupling and its control in optical waveguide-ring resonator systems,” IEEE Photon. Technol. Lett. **14**, 483–485 (2002). [CrossRef]

14. T. Ye and X. Cai, “On power consumption of silicon-microring-based optical modulators,” J. Lightwave Technol. **28**, 1615–1623 (2010). [CrossRef]

*critical coupling*state where the ring resonator’s intrinsic loss rate is equal to its waveguide coupling rate. However, operation around such a lossy state can result in a significant loss of optical power in these modulation schemes.

*lossless*intensity modulation. As an illustration, we consider lossless resonant all-pass filters consisting of a waveguide side-coupled to either a single-ring resonator or coupled-ring resonators. For such a system, when we input into the waveguide a continuous-wave (CW) signal, the steady state transmission coefficient is always unity, independent of the resonance frequency or the coupling constants of the system. Nevertheless, we show that significant intensity modulation of the system output can be achieved when the system parameters such as the resonant frequencies are modulated at a rate comparable to the waveguide coupling rate. In fact, the modulation

*on/off ratio*, defined as the ratio of the maximum to minimum output power, can be infinity. This system behaves as a lossless intensity modulator where the time-averaged output optical power is equal to the time-averaged input optical power. Thus, the peak power of the modulated output signal is in fact higher than the input CW signal peak power. We also show that in the case of a coupled-three-ring system, a clear symmetric output pulse shape can be generated by only modulating the ring resonance frequency. Examples of possible applications of our intensity modulation schemes include optical clock signal generation and optical sampling [15

15. Z. Pan, S. Chandel, and C. Yu, “Ultrahigh-speed optical pulse generation using a phase modulator and two stages of delayed Mach-Zehnder interferometers,” Opt. Eng. **46**, 075001 (2007). [CrossRef]

16. C. Schmidt-Langhorst and H.-G. Weber, “Optical sampling techniques,” J. Opt. Fiber Commun. Rep. **2**, 86–114 (2005). [CrossRef]

## 2. Photon dynamics in a modulated system

*T*varies as a function of some parameter

*x*[Fig. 1(a)]. For example, in the simple case of a single-ring modulator [12

12. A. Yariv, “Critical coupling and its control in optical waveguide-ring resonator systems,” IEEE Photon. Technol. Lett. **14**, 483–485 (2002). [CrossRef]

14. T. Ye and X. Cai, “On power consumption of silicon-microring-based optical modulators,” J. Lightwave Technol. **28**, 1615–1623 (2010). [CrossRef]

*x*can either be the ring’s resonance frequency, its radiative loss rate or its waveguide coupling rate. At some operating frequency

*ω*of the system, the steady state transmission spectrum has a value of

*T*

_{max}for some

*x*=

*x*

_{1}and a value of

*T*

_{min}for some

*x*=

*x*

_{2}. Modulating

*x*between

*x*

_{1}and

*x*

_{2}[Fig. 1(b)] at some frequency Ω then results in the intensity modulation of an input optical beam between the

*T*

_{max}state and the

*T*

_{min}state [Fig. 1(c)] at the same frequency Ω. If we return back to our example of the single-ring modulator [Fig. 2(a)], the modulation of

*x*here can be carried out such that the single-ring system is modulated between (i) the critical coupling state where

*T*=

*T*

_{min}= 0, and (ii) away from the critical coupling state where

*T*=

*T*

_{max}≈ 1.

14. T. Ye and X. Cai, “On power consumption of silicon-microring-based optical modulators,” J. Lightwave Technol. **28**, 1615–1623 (2010). [CrossRef]

17. W. D. Sacher and J. K. S. Poon, “Dynamics of microring resonator modulators,” Opt. Express **16**, 15741–15753 (2008). [CrossRef] [PubMed]

**28**, 1615–1623 (2010). [CrossRef]

17. W. D. Sacher and J. K. S. Poon, “Dynamics of microring resonator modulators,” Opt. Express **16**, 15741–15753 (2008). [CrossRef] [PubMed]

## 3. Single-ring system

*S*

_{in}(

*t*) = exp(

*jωt*) and static coupling rate

*γ*

_{coup}(

*t*) =

*γ*

_{coup}, the transmission spectrum of the single-ring system is: If we further assume the system is lossless (i.e.

*γ*

_{loss}= 0), the power transmission coefficient of the system is |

*T*(

*ω*)| = 1 for all values of the coupling rate

*γ*

_{coup}and ring resonant frequency

*ω*

_{o}. Thus, the conventional description of intensity modulation in Fig. 1, which neglects the system dynamics, predicts that for the lossless ring system in Fig. 2, modulating any parameter at any modulation frequency will not result in the intensity modulation of an input optical beam.

*γ*

_{coup}(

*t*) and CW input

*S*

_{in}(

*t*) = exp(

*jω*) operating at the ring resonance frequency

_{o}t*ω*. From Eq. (1) we can derive the following analytical form of the system output: where the resonator amplitude

_{o}*a*(

*t*) =

*A*(

*t*) exp(

*jω*). The output

_{o}t*S*

_{out}(

*t*) in Eq. (2) can be described as having a carrier frequency

*ω*and an envelope 1 +

_{o}*B*(

*t*). The envelope results from the interference between a direct pathway of unity amplitude and an indirect pathway ring resonance assisted amplitude

*B*(

*t*). The expression for

*B*(

*t*) in Eq. (3) consists of integrals which contains

*memory effects*as discussed in Ref. [17

17. W. D. Sacher and J. K. S. Poon, “Dynamics of microring resonator modulators,” Opt. Express **16**, 15741–15753 (2008). [CrossRef] [PubMed]

*π*(20GHz): where

*γ*

_{o}is the mean coupling rate amplitude and Δ

*γ*is the modulation amplitude. We numerically solve the single-ring system CMT equations [Eq. (1)] for the output

*S*

_{out}(

*t*). Figure 2(b) and 2(c) show the output power solutions at

*γ*

_{o}= 0.069Ω, Δ

*γ*= 0.025Ω) and (

*γ*

_{o}= 6.43Ω, Δ

*γ*= 2.92Ω), respectively. In both of these examples, the output power is modulated between a maximum amplitude state and a zero amplitude state (i.e. infinite on/off ratio) with a modulation frequency equivalent to the coupling rate modulation frequency Ω = 2

*π*(20GHz). Qualitatively, the maximum amplitude state in Fig. 2(b) and (c) occurs when there is constructive interference between the direct pathway amplitude and the resonance assisted indirect pathway amplitude in Eq. (2), while the zero amplitude state occurs when there is destructive interference between the pathways. In general, for any mean coupling rate amplitude

*γ*

_{o}≪

*ω*

_{o}in Eq. (4), an infinite modulation on/off ratio can be achieved by an appropriate choice of the modulation amplitude Δ

*γ*.

*γ*

_{o}, Δ

*γ*≪ Ω and Δ

*γ*<

*γ*

_{o}in Eq. (4). In this weak coupling rate regime, assuming a sinusoidal modulation of the coupling rate [Eq. (4)], the indirect pathway amplitude

*B*(

*t*) in Eq. (3) at

*a*(

*t*) within the original

*B*(

*t*) expression [Eq. (3)] has been approximated by a constant. This constant energy within the resonator results in the modulation of the output envelope [Eq. (2)] being only driven by the

*γ*

_{o}in this weak coupling regime, an infinite on/off ratio can be achieved by using a modulation amplitude Δ

*γ*≈ 0.73

*γ*

_{o}.

*a*(

*t*) within the

*B*(

*t*) expression [Eq. (3)] generally oscillates with the same periodicity as the coupling rate. However, in the strong coupling regime, the ratio of the variance to the mean value of |

*a*(

*t*)| is significant. Hence, the modulation of the output envelope is driven by the product of a

12. A. Yariv, “Critical coupling and its control in optical waveguide-ring resonator systems,” IEEE Photon. Technol. Lett. **14**, 483–485 (2002). [CrossRef]

**28**, 1615–1623 (2010). [CrossRef]

**14**, 483–485 (2002). [CrossRef]

**28**, 1615–1623 (2010). [CrossRef]

**14**, 483–485 (2002). [CrossRef]

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

**28**, 1615–1623 (2010). [CrossRef]

## 4. Coupled-three-ring system

*effective waveguide coupling rate*and, hence, output power can be modulated by modulating the resonance frequencies of a pair of resonators. In addition, we show that the resulting modulated output envelope of the system can be symmetrical with an infinite on/off ratio. Our system (Fig. 4) consists of a pair of side ring resonators with modal amplitudes

*p*(

*t*) and

*q*(

*t*), coupled to a central ring resonator with modal amplitude

*a*(

*t*). The coupling rate between each side ring and the central ring is

*κ*, and the side rings are not directly coupled to each other. The central ring has a static resonance fequency

*ω*

_{o}while the two side rings have dynamic resonance frequencies

*ω*

_{o}+ Δ(

*t*) and

*ω*

_{o}– Δ(

*t*), respectively. The central ring is coupled to a waveguide and this central-ring-waveguide part of the system has the same geometry as the single-ring system discussed in Section 3. The coupled-three-ring system can be described by the following CMT equations:

*S*

_{in}(

*t*) = exp(

*jωt*) and a static side ring resonance frequency detuning Δ(

*t*) = Δ, the transmission through the system is: If we further assume the system is lossless (i.e.

*γ*

_{loss}= 0), the absolute transmission of the system is |

*T*(

*ω*)| = 1 for all values of the detuning Δ. On the other hand, the spectra of energy stored in each of the three resonators in Fig. 4 varies with Δ.

*μ*m, such that the waveguide supports only a single mode in the 1.55

*μ*m wavelength range. Each ring resonator waveguide has the same width as the straight waveguide, and a ring radius of 2

*μ*m (measured from the center of the ring to its outer circumference). The center-to-center separation between the central ring and the straight waveguide is 2.417

*μ*m while the center-to-center separation between the central ring and each side ring is 4.653

*μ*m. The center-to-center separation between the side rings is 6.581

*μ*m. The straight waveguide and side rings have a refractive index of 3.5, while the central ring has a refractive index of 3.500491. This results in all three rings having an identical resonance frequency

*ω*

_{o}= 2

*π*(193THz) when the side ring detuning is Δ = 0. The inter-ring coupling rate between the central ring and each side ring is

*κ*= 2

*π*(17.9GHz), the waveguide coupling rate is

*γ*

_{coup}= 2

*π*(18.7GHz), and each ring has a very low amplitude-radiative loss rate of

*γ*

_{loss}= 2

*π*(38.6MHz). The circles in Fig. 5 show the FDTD simulation results for the energy spectra |

*a*(

*ω*)|

^{2}within the central ring at three different side ring detunings. Also shown in Fig. 5 are the spectras (solid lines) from the CMT model of the system [Eq. (6)] with identical values of the system parameters as in the FDTD simulations. Both the analytical CMT plots and FDTD simulation results show excellent agreement.

*ω*, and hence the system at this resonance frequency is at a

_{o}*dark state*that is completely decoupled from the waveguide. When the side ring detuning Δ is non-zero [Fig. 5(b) and 5(c)], the spectrum of the energy in the central ring has a peak centered at its resonance frequency. In addition, the width of this peak increases as Δ is increased. This behavior is similar to varying the waveguide coupling rate in a single-ring system [Section 3] [12

**14**, 483–485 (2002). [CrossRef]

*push-pull*configuration where there is a

*π*phase diferrence between the detunings Δ(

*t*) of the side rings. This push-pull configuration can be shown to result in zero chirp in the output

*S*

_{out}(

*t*) [Eq. (6)] for an input

*S*

_{in}(

*t*) operating at the resonance frequency

*ω*=

*ω*

_{o}. We note that a chirpless output is also a characteristic of a waveguide-coupling modulated single-ring system [Eq. (2)]. We also specialize to a Ω = 2

*π*(20GHz) sinusoidal Δ(

*t*) modulation: where

*δω*is the resonance frequency modulation amplitude.

*γ*

_{coup}and

*κ*. Figure 6 shows the FDTD result (circles) of the system output power at

*t*≫ 1/

*γ*

_{coup}for the case of a CW input

*S*

_{in}(

*t*) = exp(

*jω*

_{o}

*t*) operating at the resonance frequency

*ω*

_{o}= 2

*π*(193THz) of the central ring, and a side ring modulation amplitude

*δω*= 2

*π*(27.65GHz) in Eq. (7). We emphasize that the time-averaged output optical power in Fig. 6 is equal to the time-averaged input optical power. In addition, the modulated output waveform is symmetrical with an infinite on/off ratio, and an output modulation frequency of 40GHz that is twice the side ring modulation frequency. Also shown in Fig. 6 is the result (solid line) from numerically solving the system’s CMT equations [Eq. (6)]. The CMT simulation has identical values of the system paramaters as in the FDTD simulation, except for a slight adjustment of the side ring detuning modulation amplitude to

*δω*= 2

*π*(26.3GHz) in order to fit the FDTD results. We also note that the phase of

*S*

_{out}(

*t*) in Eq. (6) is zero at all times during the modulation.

*S*

_{out}(

*t*) of the coupled-three-ring system (Fig. 4) is the interference between a direct-path amplitude and an indirect-path amplitude, where the latter amplitude is now a coupled-three-ring resonance assisted indirect-path amplitude. Starting from any maximum output point in Fig. 6, the modulated output power trajectory in half a modulation period consists of the following three characteristic states whose electric field plots are shown in Fig. 7: (a) a maximum output power state, (b) a dark state and (c) a zero output power state. The maximum output power state [Fig. 7(a)] occurs when the direct-path amplitude interferes constructively with the indirect-path amplitude, while the zero output power state [Fig. 7(c)] occurs when there is destructive interference between the two pathways. In between this maximum and zero output power states is the dark state shown in Fig. 7(b) where the central ring amplitude

*a*(

*t*) is zero. At this dark state, the central ring is completely decoupled from the waveguide, and the system ouput consists of only the direct-path amplitude [

*S*

_{out}(

*t*) =

*S*

_{in}(

*t*)].

*π*/Ω of the side ring detuning Δ(

*t*), the coupled-three-ring system states at times

*t*=

*t*

_{1}and

*t*=

*t*

_{2}=

*t*

_{1}+

*π*/Ω are identical up to a flip in the sign of the detunings in both side resonators. Consequently, the system output has identical values at both times

*t*

_{1}and

*t*

_{2}within a modulation period, resulting in two identical output pulses for every one modulation cycle of the side ring detuning. This frequency doubling can be avoided by using a modulation Δ(

*t*) in Eq. (6) that is always positive, for example.

## 5. Conclusion

*γ*

_{loss}= 0.65GHz [20

20. Q. Reed, P. Dong, and M. Lipson, “Breaking the delay-bandwidth limit in a photonic structure,” Nat. Phys. **3**, 406–410 (2007). [CrossRef]

*t*) around the dark state of the system. In our numerical simulation example (Fig. 6), modulation of Δ(

*t*) requires a fractional refractive index tuning of ∼ 10

^{−4}which can be implemented using free carrier injection/depletion in silicon [20

20. Q. Reed, P. Dong, and M. Lipson, “Breaking the delay-bandwidth limit in a photonic structure,” Nat. Phys. **3**, 406–410 (2007). [CrossRef]

22. Q. Xu, “Silicon dual-ring modulator,” Opt. Express **17**, 20783–20793 (2009). [CrossRef] [PubMed]

22. Q. Xu, “Silicon dual-ring modulator,” Opt. Express **17**, 20783–20793 (2009). [CrossRef] [PubMed]

## Acknowledgments

## References and links

1. | G. T. Reed, G. Mashanovich, F. Y. Gardes, and D. J. Thomson, “Silicon optical modulators,” Nat. Photonics |

2. | A. Liu, R. Jones, L. Liao, D. Samara-Rubio 1, D. Rubin, O. Cohen, R. Nicolaescu, and M. Paniccia, “A high-speed silicon optical modulator based on a metal-oxide-semiconductor capacitor,” Nature |

3. | L. Liao, D. Samara-Rubio, M. Morse, A. Liu, D. Hodge, D. Rubin, U. Keil, and T. Franck, “High speed silicon Mach-Zehnder modulator,” Opt. Express |

4. | L. Liao, A. Liu, J. Basak, H. Nguyen, M. Paniccia, D. Rubin, Y. Chetrit, R. Cohen, and N. Izhaky, “40 Gbit/s silicon optical modulator for highspeed applications,” Electron. Lett. |

5. | A. Liu, L. Liao, D. Rubin, H. Nguyen, B. Ciftcioglu, Y. Chetrit, N. Izhaky, and M. Paniccia, “High-speed optical modulation based on carrier depletion in a silicon waveguide,” Opt. Express |

6. | X. Chen, Y.-S. Chen, Y. Zhao, W. Jiang, and R. T. Chen, “Capacitor-embedded 0.54 pj/bit silicon-slot photonic crystal waveguide modulator,” Opt. Lett. |

7. | H.-W. Chen, Y.-H. Kuo, and J. E. Bowers, “25 Gb/s hybrid silicon switch using a capacitively loaded traveling wave electrode,” Opt. Express |

8. | Q. Xu, B. Schmidt, S. Pradhan, and M. Lipson, “Micrometre-scale silicon electro-optic modulator,” Nature |

9. | D. M. Gill, M. Rasras, K.-Y. Tu, Y.-K. Chen, A. E. White, S. S. Patel, D. Carothers, A. Pomerene, R. Kamocsai, C. Hill, and J. Beattie, “Internal bandwidth equalization in a CMOS-compatible Si-ring modulator,” IEEE Photon. Technol. Lett. |

10. | T. Tanabe, K. Nishiguchi, E. Kuramochi, and M. Notomi, “Low power and fast electro-optic silicon modulator with lateral p-i-n embedded photonic crystal nanocavity,” Opt. Express |

11. | S. Manipatruni, K. Preston, L. Chen, and M. Lipson, “Ultra-low voltage, ultra-small mode volume silicon microring modulator,” Opt. Express |

12. | A. Yariv, “Critical coupling and its control in optical waveguide-ring resonator systems,” IEEE Photon. Technol. Lett. |

13. | W. D. Sacher and J. K. S. Poon, “Characteristics of microring resonators with waveguide-resonator coupling modulation,” J. Lightwave Technol. |

14. | T. Ye and X. Cai, “On power consumption of silicon-microring-based optical modulators,” J. Lightwave Technol. |

15. | Z. Pan, S. Chandel, and C. Yu, “Ultrahigh-speed optical pulse generation using a phase modulator and two stages of delayed Mach-Zehnder interferometers,” Opt. Eng. |

16. | C. Schmidt-Langhorst and H.-G. Weber, “Optical sampling techniques,” J. Opt. Fiber Commun. Rep. |

17. | W. D. Sacher and J. K. S. Poon, “Dynamics of microring resonator modulators,” Opt. Express |

18. | M. F. Yanik and S. Fan, “Stopping light all optically,” Phys. Rev. Lett. |

19. | A. Taflove and S. C. Hagness, |

20. | Q. Reed, P. Dong, and M. Lipson, “Breaking the delay-bandwidth limit in a photonic structure,” Nat. Phys. |

21. | M. Lipson, “Guiding, modulating, and emitting light on silicon-challenges and opportunities,” J. Lightwave Technol. |

22. | Q. Xu, “Silicon dual-ring modulator,” Opt. Express |

**OCIS Codes**

(140.4780) Lasers and laser optics : Optical resonators

(230.0230) Optical devices : Optical devices

(350.4238) Other areas of optics : Nanophotonics and photonic crystals

**ToC Category:**

Integrated Optics

**History**

Original Manuscript: November 4, 2011

Revised Manuscript: January 20, 2012

Manuscript Accepted: January 30, 2012

Published: February 7, 2012

**Citation**

Sunil Sandhu and Shanhui Fan, "Lossless intensity modulation in integrated photonics," Opt. Express **20**, 4280-4290 (2012)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-20-4-4280

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### References

- G. T. Reed, G. Mashanovich, F. Y. Gardes, and D. J. Thomson, “Silicon optical modulators,” Nat. Photonics4, 518–526 (2010). [CrossRef]
- A. Liu, R. Jones, L. Liao, D. Samara-Rubio, D. Rubin, O. Cohen, R. Nicolaescu, and M. Paniccia, “A high-speed silicon optical modulator based on a metal-oxide-semiconductor capacitor,” Nature427, 615–618 (2004). [CrossRef] [PubMed]
- L. Liao, D. Samara-Rubio, M. Morse, A. Liu, D. Hodge, D. Rubin, U. Keil, and T. Franck, “High speed silicon Mach-Zehnder modulator,” Opt. Express13, 3129–3135 (2005). [CrossRef] [PubMed]
- L. Liao, A. Liu, J. Basak, H. Nguyen, M. Paniccia, D. Rubin, Y. Chetrit, R. Cohen, and N. Izhaky, “40 Gbit/s silicon optical modulator for highspeed applications,” Electron. Lett.43, 1196–1197 (2007). [CrossRef]
- A. Liu, L. Liao, D. Rubin, H. Nguyen, B. Ciftcioglu, Y. Chetrit, N. Izhaky, and M. Paniccia, “High-speed optical modulation based on carrier depletion in a silicon waveguide,” Opt. Express15, 660–668 (2007). [CrossRef] [PubMed]
- X. Chen, Y.-S. Chen, Y. Zhao, W. Jiang, and R. T. Chen, “Capacitor-embedded 0.54 pj/bit silicon-slot photonic crystal waveguide modulator,” Opt. Lett.34, 602–604 (2009). [CrossRef] [PubMed]
- H.-W. Chen, Y.-H. Kuo, and J. E. Bowers, “25 Gb/s hybrid silicon switch using a capacitively loaded traveling wave electrode,” Opt. Express18, 1070–1075 (2010). [CrossRef] [PubMed]
- Q. Xu, B. Schmidt, S. Pradhan, and M. Lipson, “Micrometre-scale silicon electro-optic modulator,” Nature435, 325–327 (2005). [CrossRef] [PubMed]
- D. M. Gill, M. Rasras, K.-Y. Tu, Y.-K. Chen, A. E. White, S. S. Patel, D. Carothers, A. Pomerene, R. Kamocsai, C. Hill, and J. Beattie, “Internal bandwidth equalization in a CMOS-compatible Si-ring modulator,” IEEE Photon. Technol. Lett.21, 200–202 (2009). [CrossRef]
- T. Tanabe, K. Nishiguchi, E. Kuramochi, and M. Notomi, “Low power and fast electro-optic silicon modulator with lateral p-i-n embedded photonic crystal nanocavity,” Opt. Express17, 22505–22513 (2009). [CrossRef]
- S. Manipatruni, K. Preston, L. Chen, and M. Lipson, “Ultra-low voltage, ultra-small mode volume silicon microring modulator,” Opt. Express18, 18235–18242 (2010). [CrossRef] [PubMed]
- A. Yariv, “Critical coupling and its control in optical waveguide-ring resonator systems,” IEEE Photon. Technol. Lett.14, 483–485 (2002). [CrossRef]
- W. D. Sacher and J. K. S. Poon, “Characteristics of microring resonators with waveguide-resonator coupling modulation,” J. Lightwave Technol.27, 3800–3811 (2009). [CrossRef]
- T. Ye and X. Cai, “On power consumption of silicon-microring-based optical modulators,” J. Lightwave Technol.28, 1615–1623 (2010). [CrossRef]
- Z. Pan, S. Chandel, and C. Yu, “Ultrahigh-speed optical pulse generation using a phase modulator and two stages of delayed Mach-Zehnder interferometers,” Opt. Eng.46, 075001 (2007). [CrossRef]
- C. Schmidt-Langhorst and H.-G. Weber, “Optical sampling techniques,” J. Opt. Fiber Commun. Rep.2, 86–114 (2005). [CrossRef]
- W. D. Sacher and J. K. S. Poon, “Dynamics of microring resonator modulators,” Opt. Express16, 15741–15753 (2008). [CrossRef] [PubMed]
- M. F. Yanik and S. Fan, “Stopping light all optically,” Phys. Rev. Lett.92, 083901 (2004). [CrossRef] [PubMed]
- A. Taflove and S. C. Hagness, Computational Electrodynamics: The Finite-Difference Time-Domain Method, 3rd ed. (Artech House, 2005).
- Q. Reed, P. Dong, and M. Lipson, “Breaking the delay-bandwidth limit in a photonic structure,” Nat. Phys.3, 406–410 (2007). [CrossRef]
- M. Lipson, “Guiding, modulating, and emitting light on silicon-challenges and opportunities,” J. Lightwave Technol.23, 4222 – 4238 (2005). [CrossRef]
- Q. Xu, “Silicon dual-ring modulator,” Opt. Express17, 20783–20793 (2009). [CrossRef] [PubMed]

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