## Excitation transfer between optically injected microdisk lasers |

Optics Express, Vol. 21, Issue 23, pp. 28922-28932 (2013)

http://dx.doi.org/10.1364/OE.21.028922

Acrobat PDF (1283 KB)

### Abstract

Recently, we have theoretically demonstrated that optically injected microdisk lasers can be tuned in a class I excitable regime, where they are sensitive to both inhibitory and excitatory external input pulses. In this paper, we propose, using simulations, a topology that allows the disks to react on excitations from other disks. Phase tuning of the intermediate connections allows to control the disk response. Additionally, we investigate the sensitivity of the disk circuit to deviations in driving current and locking signal wavelength detuning. Using state-of-the-art fabrication techniques for microdisk laser, the standard deviation of the lasing wavelength is still about one order of magnitude too large. Therefore, compensation techniques, such as wavelength tuning by heating, are necessary.

© 2013 Optical Society of America

## 1. Introduction

1. W. Maass, “Networks of spiking neurons: the third generation of neural network models,” Neural Networks **10**, 1659–1671 (1997). [CrossRef]

2. S. Beri, L. Mashall, L. Gelens, G. Van der Sande, G. Mezosi, M. Sorel, J. Danckaert, and G. Verschaffelt, “Excitability in optical systems close to Z2-symmetry,” Phys. Lett. A **374**, 739–743 (2010). [CrossRef]

7. S. Barbay, R. Kuszelewicz, and A. M. Yacomotti, “Excitability in a semiconductor laser with saturable absorber,” Opt. Lett. **36**, 4476–4478 (2011). [CrossRef] [PubMed]

8. M. Brunstein, A. M. Yacomotti, I. Sagnes, F. Raineri, L. Bigot, and A. Levenson, “Excitability and self-pulsing in a photonic crystal nanocavity,” Phys. Rev. A **85**, 031803 (2012). [CrossRef]

10. T. Van Vaerenbergh, M. Fiers, P. Mechet, T. Spuesens, R. Kumar, G. Morthier, B. Schrauwen, J. Dambre, and P. Bienstman, “Cascadable excitability in microrings,” Opt. Express **20**, 20292–20308 (2012). [CrossRef] [PubMed]

12. J. Van Campenhout, P. Romeo, D. Van Thourhout, C. Seassal, P. Regreny, L. Di Cioccio, J.-M. Fedeli, and R. Baets, “Design and optimization of electrically injected InP-based microdisk lasers integrated on and coupled to a SOI waveguide circuit,” J. Lightwave Technol. **26**, 52–63 (2008). [CrossRef]

_{2}-substrate layer. The evolution of the slowly varying complex field amplitudes of the modes can be described by the rate equations discussed in [13

13. Y. De Koninck, K. Huybrechts, G. Van der Sande, J. Danckaert, R. Baets, and G. Morthier, “Nonlinear dynamics of asymmetrically coupled microdisk lasers,” in *LEOS Annual Meeting Conference Proceedings, 2009. LEOS’09. IEEE* (IEEE, 2009), pp. 503–504. [CrossRef]

14. K. Alexander, T. Van Vaerenbergh, M. Fiers, P. Mechet, J. Dambre, and P. Bienstman, “Excitability in optically injected microdisk lasers with phase controlled excitatory and inhibitory response,” Opt. Express **21**, 20292–20308 (2013). [CrossRef]

13. Y. De Koninck, K. Huybrechts, G. Van der Sande, J. Danckaert, R. Baets, and G. Morthier, “Nonlinear dynamics of asymmetrically coupled microdisk lasers,” in *LEOS Annual Meeting Conference Proceedings, 2009. LEOS’09. IEEE* (IEEE, 2009), pp. 503–504. [CrossRef]

3. W. Coomans, L. Gelens, S. Beri, J. Danckaert, and G. Van der Sande, “Solitary and coupled semiconductor ring lasers as optical spiking neurons,” Phys. Rev. E **84**, 036209 (2011). [CrossRef]

*E*

^{+}) and a suppressed mode (

*E*

^{−}) [14

14. K. Alexander, T. Van Vaerenbergh, M. Fiers, P. Mechet, J. Dambre, and P. Bienstman, “Excitability in optically injected microdisk lasers with phase controlled excitatory and inhibitory response,” Opt. Express **21**, 20292–20308 (2013). [CrossRef]

4. S. Wieczorek, B. Krauskopf, and D. Lenstra, “Multipulse excitability in a semiconductor laser with optical injection,” Phys. Rev. Lett. **88**, 063901 (2002). [CrossRef] [PubMed]

5. D. Goulding, S. Hegarty, O. Rasskazov, S. Melnik, M. Hartnett, G. Greene, J. McInerney, D. Rachinskii, and G. Huyet, “Excitability in a quantum dot semiconductor laser with optical injection,” Phys. Rev. Lett. **98**, 153903 (2007). [CrossRef] [PubMed]

15. M. Turconi, B. Garbin, M. Feyereisen, M. Giudici, and S. Barland, “Control of excitable pulses in an injection-locked semiconductor laser,” Phys. Rev. E **88**, 022923 (2013). [CrossRef]

*Saddle-Node on an Invariant Circle (SNIC)*bifurcation, to class I excitability, which phenomenologically resembles the well-known leaky integrate-and-fire model of a spiking neuron [16]. Small pulses, preferably out of phase with the locking signal, can be used to perturb the laser state, pushing it across this bifurcation. By controlling the phase, both an excitatory and an inhibitory response can be created. Indeed, the importance of the phase of the trigger pulse was recently demonstrated experimentally in a single-mode semiconductor laser under optical injection [15

15. M. Turconi, B. Garbin, M. Feyereisen, M. Giudici, and S. Barland, “Control of excitable pulses in an injection-locked semiconductor laser,” Phys. Rev. E **88**, 022923 (2013). [CrossRef]

14. K. Alexander, T. Van Vaerenbergh, M. Fiers, P. Mechet, J. Dambre, and P. Bienstman, “Excitability in optically injected microdisk lasers with phase controlled excitatory and inhibitory response,” Opt. Express **21**, 20292–20308 (2013). [CrossRef]

17. M. Fiers, T. Van Vaerenbergh, K. Caluwaerts, D. Vande Ginste, B. Schrauwen, J. Dambre, and P. Bienstman, “Time-domain and frequency-domain modeling of nonlinear optical components at the circuit-level using a node-based approach,” J. Opt. Soc. Am. B **29**, 896–900 (2012). [CrossRef]

## 2. Connection topology

**21**, 20292–20308 (2013). [CrossRef]

*CW*

_{1}-power (

*P*

_{CW1}) and half of the

*CW*

_{2}-power (

*P*

_{CW2}), for disks 1 and 2, respectively. Note as well that the input pulse power is attenuated by a factor of four, before it is felt by disk 1. If the first disk gets excited, its output pulse will travel through the connecting waveguide to the second disk and also this pulse’s power will get attenuated by a factor of four. Despite this power loss, due to the strong coupling between the disks and their bus waveguide, the coupling between the two different disks is still 1.8× stronger than the intermodal coupling. To increase the possibility that the output from the first disk excites the second one, the detuning of the locking signal is slightly different in this paper compared to [14

**21**, 20292–20308 (2013). [CrossRef]

*ω*=

*ω*−

_{in}*ω*

_{disk 1,2}= −20 ns

^{−1}instead of −15 ns

^{−1}). The current is still 2.3 mA. For these parameters, the saddle-node bifurcation lies at

**21**, 20292–20308 (2013). [CrossRef]

*ϕ*, and the relative phase of both locking signals, will thus be decisive in whether or not the first disk will be able to excite the second one. Moreover, the phase of the output pulse of the first disk is not constant as a function of time. To assure that the first pulse can excite the second one, the phase delay Δ

*ϕ*has to be chosen in such a way that the pulse is out of phase with the

*CW*

_{2}input, for a significantly long amount of time. We first assume that the two locking signals,

*CW*

_{1}and

*CW*

_{2}, arrive at the disks having the same phase. Figure 1(b) shows a power and phase trace of the output pulse of first disk (locked with a

*π*-interval, it stays nearly constant during the whole downward stroke of the pulse (around −1.9

*π*). This pulse has to be out of phase with the locking signal at the second disk. This leads to (2

*n*+ 1)

*π*= −1.9

*π*+ Δ

*ϕ*,

*n*∈ , for

*n*= −1: Δ

*ϕ*= 1.9

*π*−

*π*≈ 2.8 rad.

3. W. Coomans, L. Gelens, S. Beri, J. Danckaert, and G. Van der Sande, “Solitary and coupled semiconductor ring lasers as optical spiking neurons,” Phys. Rev. E **84**, 036209 (2011). [CrossRef]

*S*, as defined in [18

_{out}18. W. Coomans, G. Van der Sande, and L. Gelens, “Oscillations and multistability in two semiconductor ring lasers coupled by a single waveguide,” Phys. Rev. A **88**, 033813 (2013). [CrossRef]

## 3. Transfer of the excitation

### 3.1. Symmetrical coupling: oscillations

### 3.2. Asymmetrical coupling: unidirectional excitation transfer

## 4. Sensitivity to parameter variations

### 4.1. Influence of current variations

*I*

_{1}and 2.3 mA + Δ

*I*

_{2}through disk 1 and 2, respectively. Black regions represent the single excitation of the disk, in the white regions, the laser does not get excited, neither does it oscillate. The light grey regions give the parameters for which the laser oscillates permanently. In the dark grey regions, the laser shows more complex multipulse excitability, in which the first disk excites multiple times before it is able to excite the second disk and the system finally decays back to the initial condition. The region for which excitability in both disks exists is rather small, with a cross-section of about 0.1 mA. However, a current stability of 0.1 mA is experimentally achievable, such that the current stability will not be the limiting factor in practice. This sensitivity to current variations can be used to make rough estimates of sensitivity to variations in other parameters. Taking into account that the bifurcation locking amplitude at Δ

*ω*= −20 ns

^{−1}changes roughly

### 4.2. Influence of detuning variations

*ω*approaches zero [14

**21**, 20292–20308 (2013). [CrossRef]

*ω*closer to 0 has an effect similar to decreasing current. One can clearly see that Fig. 5 has a similar structure as Fig. 4, but mirrored around the center. Figure 5 shows that the excitability is very sensitive to deviations of the detuning. Roughly, the excitability can be found in a region of about 1 ns

^{−1}wide, which corresponds to ∼ 1.27 pm in wavelength, whereas variations larger than a picometer already cause completely different behavior. Unfortunately, using current fabrication techniques, the typical variations in microdisk laser wavelength, are on the order of 0.4 nm [19

19. P. Mechet, F. Raineri, A. Bazin, Y. Halioua, T. Spuesens, T. Karle, P. Regreny, P. Monnier, D. Van Thourhout, I. Sagnes, R. Raj, G. Roelkens, and G. Morthier, “Uniformity of the lasing wavelength of heterogeneously integrated inp microdisk lasers on soi,” Opt. Express **21**, 10622–10631 (2013). [CrossRef] [PubMed]

## 5. Compensating for variations in detuning

20. L. Liu, T. Spuesens, G. Roelkens, D. Van Thourhout, P. Regreny, and P. Rojo-Romeo, “A thermally tunable iii–v compound semiconductor microdisk laser integrated on silicon-on-insulator circuits,” IEEE Photonics Technol. Lett., IEEE **22**, 1270–1272 (2010). [CrossRef]

## 6. Conclusion

**21**, 20292–20308 (2013). [CrossRef]

^{−1}(≈ 1.27 pm). One can however compensate for detuning variations by changing other, more controllable, parameters, such as the locking amplitude or the current, making the transfer of excitation robust to variations in lasing wavelength of several tens of pm. However, using the state-of-the-art production techniques for microdisk lasers, the standard deviation of the lasing wavelength is still about one order of magnitude larger. Additional compensation techniques, such as wavelength tuning by heating, will have to be considered.

## A. Rate equation model of two coupled microdisk lasers

**21**, 20292–20308 (2013). [CrossRef]

*E*

^{+}and

*E*

^{−}(|

*E*

_{±}|

^{2}is the number of photons in the mode, while the optical field oscillates with an additional

*e*

^{−jωint}-dependency), and the number of free carriers,

*N*, in the cavity. Caphe, the circuit simulator we use in this paper, converts the equations that describe the coupling of the optical modes to the bus waveguide into the formalism described in [17

17. M. Fiers, T. Van Vaerenbergh, K. Caluwaerts, D. Vande Ginste, B. Schrauwen, J. Dambre, and P. Bienstman, “Time-domain and frequency-domain modeling of nonlinear optical components at the circuit-level using a node-based approach,” J. Opt. Soc. Am. B **29**, 896–900 (2012). [CrossRef]

*α*is the line broadening factor,

*τ*the photon lifetime in the cavity,

_{p}*τ*is the roundtrip time of the cavity, Δ

*ω*=

*ω*−

_{in}*ω*the detuning between the input light

_{i}*ω*and the free-running cavity frequency

_{in}*ω*of disk

_{i}*i*(

*i*∈ {1, 2}, unless otherwise mentioned

*ω*

_{1}=

*ω*

_{2}=

*ω*

_{0}),

*C*is the complex intermodal coupling coefficient.

*κ*is the coupling between the disks and the waveguide.

*E*are the complex amplitudes of the optical inputs used for the locking of both disks, while

_{CW,i}*E*is the complex amplitude of the input pulse (in both cases |

_{tr}*E*|

_{α}^{2}is the power in the waveguide). Δ

*ϕ*is the phase difference due to the interconnecting waveguide. The factors

*N*.

_{i}*I*is the injected current to each disk,

_{i}*q*the elementary charge,

*η*a current efficiency factor, and

*τ*the carrier lifetime.

_{c}*g*is the differential gain,

_{N}*N*

_{0}the transparency threshold of free carriers and Γ the confinement factor. The denominator in Eq. (5) includes cross- and self-gain modulation,

*ε*is called the nonlinear gain suppression coefficient. The complex amplitude of the output of disk

_{NL}*i*towards the connecting waveguide can be calculated using: Similar to the intermodal coupling

*C*, this results in an intercavity coupling term

3. W. Coomans, L. Gelens, S. Beri, J. Danckaert, and G. Van der Sande, “Solitary and coupled semiconductor ring lasers as optical spiking neurons,” Phys. Rev. E **84**, 036209 (2011). [CrossRef]

## B. Numerical details of the simulations

## Acknowledgments

## References and links

1. | W. Maass, “Networks of spiking neurons: the third generation of neural network models,” Neural Networks |

2. | S. Beri, L. Mashall, L. Gelens, G. Van der Sande, G. Mezosi, M. Sorel, J. Danckaert, and G. Verschaffelt, “Excitability in optical systems close to Z2-symmetry,” Phys. Lett. A |

3. | W. Coomans, L. Gelens, S. Beri, J. Danckaert, and G. Van der Sande, “Solitary and coupled semiconductor ring lasers as optical spiking neurons,” Phys. Rev. E |

4. | S. Wieczorek, B. Krauskopf, and D. Lenstra, “Multipulse excitability in a semiconductor laser with optical injection,” Phys. Rev. Lett. |

5. | D. Goulding, S. Hegarty, O. Rasskazov, S. Melnik, M. Hartnett, G. Greene, J. McInerney, D. Rachinskii, and G. Huyet, “Excitability in a quantum dot semiconductor laser with optical injection,” Phys. Rev. Lett. |

6. | A. Hurtado, K. Schires, I. D. Henning, and M. J. Adams, “Investigation of vertical cavity surface emitting laser dynamics for neuromorphic photonic systems,” Appl. Phys. Lett. |

7. | S. Barbay, R. Kuszelewicz, and A. M. Yacomotti, “Excitability in a semiconductor laser with saturable absorber,” Opt. Lett. |

8. | M. Brunstein, A. M. Yacomotti, I. Sagnes, F. Raineri, L. Bigot, and A. Levenson, “Excitability and self-pulsing in a photonic crystal nanocavity,” Phys. Rev. A |

9. | A. M. Yacomotti, P. Monnier, F. Raineri, B. Ben Bakir, C. Seassal, R. Raj, and J. A. Levenson, “Fast thermo-optical excitability in a two-dimensional photonic crystal,” Phys. Rev. Lett. |

10. | T. Van Vaerenbergh, M. Fiers, P. Mechet, T. Spuesens, R. Kumar, G. Morthier, B. Schrauwen, J. Dambre, and P. Bienstman, “Cascadable excitability in microrings,” Opt. Express |

11. | M. A. Nahmias, B. J. Shastri, A. N. Tait, S. Member, and P. R. Prucnal, “A leaky integrate-and-fire laser neuron for ultrafast cognitive computing,” IEEE J. Sel. Top. Quantum Electron. |

12. | J. Van Campenhout, P. Romeo, D. Van Thourhout, C. Seassal, P. Regreny, L. Di Cioccio, J.-M. Fedeli, and R. Baets, “Design and optimization of electrically injected InP-based microdisk lasers integrated on and coupled to a SOI waveguide circuit,” J. Lightwave Technol. |

13. | Y. De Koninck, K. Huybrechts, G. Van der Sande, J. Danckaert, R. Baets, and G. Morthier, “Nonlinear dynamics of asymmetrically coupled microdisk lasers,” in |

14. | K. Alexander, T. Van Vaerenbergh, M. Fiers, P. Mechet, J. Dambre, and P. Bienstman, “Excitability in optically injected microdisk lasers with phase controlled excitatory and inhibitory response,” Opt. Express |

15. | M. Turconi, B. Garbin, M. Feyereisen, M. Giudici, and S. Barland, “Control of excitable pulses in an injection-locked semiconductor laser,” Phys. Rev. E |

16. | E. M. Izhikevich, |

17. | M. Fiers, T. Van Vaerenbergh, K. Caluwaerts, D. Vande Ginste, B. Schrauwen, J. Dambre, and P. Bienstman, “Time-domain and frequency-domain modeling of nonlinear optical components at the circuit-level using a node-based approach,” J. Opt. Soc. Am. B |

18. | W. Coomans, G. Van der Sande, and L. Gelens, “Oscillations and multistability in two semiconductor ring lasers coupled by a single waveguide,” Phys. Rev. A |

19. | P. Mechet, F. Raineri, A. Bazin, Y. Halioua, T. Spuesens, T. Karle, P. Regreny, P. Monnier, D. Van Thourhout, I. Sagnes, R. Raj, G. Roelkens, and G. Morthier, “Uniformity of the lasing wavelength of heterogeneously integrated inp microdisk lasers on soi,” Opt. Express |

20. | L. Liu, T. Spuesens, G. Roelkens, D. Van Thourhout, P. Regreny, and P. Rojo-Romeo, “A thermally tunable iii–v compound semiconductor microdisk laser integrated on silicon-on-insulator circuits,” IEEE Photonics Technol. Lett., IEEE |

**OCIS Codes**

(130.4310) Integrated optics : Nonlinear

(140.5960) Lasers and laser optics : Semiconductor lasers

(200.4700) Optics in computing : Optical neural systems

(230.1150) Optical devices : All-optical devices

**ToC Category:**

Integrated Optics

**History**

Original Manuscript: September 23, 2013

Revised Manuscript: November 4, 2013

Manuscript Accepted: November 4, 2013

Published: November 15, 2013

**Citation**

Thomas Van Vaerenbergh, Koen Alexander, Joni Dambre, and Peter Bienstman, "Excitation transfer between optically injected microdisk lasers," Opt. Express **21**, 28922-28932 (2013)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-21-23-28922

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### References

- W. Maass, “Networks of spiking neurons: the third generation of neural network models,” Neural Networks10, 1659–1671 (1997). [CrossRef]
- S. Beri, L. Mashall, L. Gelens, G. Van der Sande, G. Mezosi, M. Sorel, J. Danckaert, and G. Verschaffelt, “Excitability in optical systems close to Z2-symmetry,” Phys. Lett. A374, 739–743 (2010). [CrossRef]
- W. Coomans, L. Gelens, S. Beri, J. Danckaert, and G. Van der Sande, “Solitary and coupled semiconductor ring lasers as optical spiking neurons,” Phys. Rev. E84, 036209 (2011). [CrossRef]
- S. Wieczorek, B. Krauskopf, and D. Lenstra, “Multipulse excitability in a semiconductor laser with optical injection,” Phys. Rev. Lett.88, 063901 (2002). [CrossRef] [PubMed]
- D. Goulding, S. Hegarty, O. Rasskazov, S. Melnik, M. Hartnett, G. Greene, J. McInerney, D. Rachinskii, and G. Huyet, “Excitability in a quantum dot semiconductor laser with optical injection,” Phys. Rev. Lett.98, 153903 (2007). [CrossRef] [PubMed]
- A. Hurtado, K. Schires, I. D. Henning, and M. J. Adams, “Investigation of vertical cavity surface emitting laser dynamics for neuromorphic photonic systems,” Appl. Phys. Lett.100, 103703 (2012). [CrossRef]
- S. Barbay, R. Kuszelewicz, and A. M. Yacomotti, “Excitability in a semiconductor laser with saturable absorber,” Opt. Lett.36, 4476–4478 (2011). [CrossRef] [PubMed]
- M. Brunstein, A. M. Yacomotti, I. Sagnes, F. Raineri, L. Bigot, and A. Levenson, “Excitability and self-pulsing in a photonic crystal nanocavity,” Phys. Rev. A85, 031803 (2012). [CrossRef]
- A. M. Yacomotti, P. Monnier, F. Raineri, B. Ben Bakir, C. Seassal, R. Raj, and J. A. Levenson, “Fast thermo-optical excitability in a two-dimensional photonic crystal,” Phys. Rev. Lett.97, 143904 (2006). [CrossRef] [PubMed]
- T. Van Vaerenbergh, M. Fiers, P. Mechet, T. Spuesens, R. Kumar, G. Morthier, B. Schrauwen, J. Dambre, and P. Bienstman, “Cascadable excitability in microrings,” Opt. Express20, 20292–20308 (2012). [CrossRef] [PubMed]
- M. A. Nahmias, B. J. Shastri, A. N. Tait, S. Member, and P. R. Prucnal, “A leaky integrate-and-fire laser neuron for ultrafast cognitive computing,” IEEE J. Sel. Top. Quantum Electron.16, 1–12 (2013).
- J. Van Campenhout, P. Romeo, D. Van Thourhout, C. Seassal, P. Regreny, L. Di Cioccio, J.-M. Fedeli, and R. Baets, “Design and optimization of electrically injected InP-based microdisk lasers integrated on and coupled to a SOI waveguide circuit,” J. Lightwave Technol.26, 52–63 (2008). [CrossRef]
- Y. De Koninck, K. Huybrechts, G. Van der Sande, J. Danckaert, R. Baets, and G. Morthier, “Nonlinear dynamics of asymmetrically coupled microdisk lasers,” in LEOS Annual Meeting Conference Proceedings, 2009. LEOS’09. IEEE (IEEE, 2009), pp. 503–504. [CrossRef]
- K. Alexander, T. Van Vaerenbergh, M. Fiers, P. Mechet, J. Dambre, and P. Bienstman, “Excitability in optically injected microdisk lasers with phase controlled excitatory and inhibitory response,” Opt. Express21, 20292–20308 (2013). [CrossRef]
- M. Turconi, B. Garbin, M. Feyereisen, M. Giudici, and S. Barland, “Control of excitable pulses in an injection-locked semiconductor laser,” Phys. Rev. E88, 022923 (2013). [CrossRef]
- E. M. Izhikevich, Dynamical Systems in Neuroscience: The Geometry of Excitability and Bursting (Computational Neuroscience), 1 (MIT, 2006).
- M. Fiers, T. Van Vaerenbergh, K. Caluwaerts, D. Vande Ginste, B. Schrauwen, J. Dambre, and P. Bienstman, “Time-domain and frequency-domain modeling of nonlinear optical components at the circuit-level using a node-based approach,” J. Opt. Soc. Am. B29, 896–900 (2012). [CrossRef]
- W. Coomans, G. Van der Sande, and L. Gelens, “Oscillations and multistability in two semiconductor ring lasers coupled by a single waveguide,” Phys. Rev. A88, 033813 (2013). [CrossRef]
- P. Mechet, F. Raineri, A. Bazin, Y. Halioua, T. Spuesens, T. Karle, P. Regreny, P. Monnier, D. Van Thourhout, I. Sagnes, R. Raj, G. Roelkens, and G. Morthier, “Uniformity of the lasing wavelength of heterogeneously integrated inp microdisk lasers on soi,” Opt. Express21, 10622–10631 (2013). [CrossRef] [PubMed]
- L. Liu, T. Spuesens, G. Roelkens, D. Van Thourhout, P. Regreny, and P. Rojo-Romeo, “A thermally tunable iii–v compound semiconductor microdisk laser integrated on silicon-on-insulator circuits,” IEEE Photonics Technol. Lett., IEEE22, 1270–1272 (2010). [CrossRef]

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