Abstract

We show, both experimentally and theoretically, that a slave laser injected with an optical frequency comb can undergo two distinct locking mechanisms, both of which decrease the output optical comb’s frequency spacing. We report that, for certain detuning and relative injection strengths, slave laser relaxation oscillations can become undamped and lock to rational frequencies of the optical comb spacing, creating extra comb tones by nonlinear dynamics of the injected laser. We also study the frequency locking of the slave laser at detunings in between the injected comb lines, which add the slave laser’s frequency to the comb. Our results demonstrate the effect of the $\alpha$ parameter and stability of the locked states and indicate how the frequency of the relaxation oscillations affect both of these locking mechanisms. These optical locking mechanisms can be applied to regenerate or multiply optical combs.

© 2019 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

Full Article  |  PDF Article
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References

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    [Crossref]
  5. T. M. Fortier, M. S. Kirchner, F. Quinlan, J. Taylor, J. C. Bergquist, T. Rosenband, N. Lemke, A. Ludlow, Y. Jiang, C. W. Oates, and S. A. Diddams, “Generation of ultrastable microwaves via optical frequency division,” Nat. Photonics 5(7), 425–429 (2011).
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    [Crossref]
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    [Crossref]
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    [Crossref]
  24. F. Mogensen, H. Olesen, and G. Jacobsen, “Locking conditions and stability properties for a semiconductor laser with external light injection,” IEEE J. Quantum Electron. 21(7), 784–793 (1985).
    [Crossref]
  25. I. Petitbon, P. Gallion, G. Debarge, and C. Chabran, “Locking bandwidth and relaxation oscillations of an injection-locked semiconductor laser,” IEEE J. Quantum Electron. 24(2), 148–154 (1988).
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    [Crossref]
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  29. A. S. Tistomo and S. Gee, “Laser frequency fixation by multimode optical injection locking,” Opt. Express 19(2), 1081–1090 (2011).
    [Crossref]
  30. V. I. Arnol’d, “Small denominators. I. Mapping of the circumference onto itself,” AMS Transl., Ser. 2 46, 213–284 (1965).
  31. K. Shortiss, M. Dernaika, L. Caro, M. Seifikar, and F. H. Peters, “Inverse Scattering Method Design of Regrowth-Free Single-Mode Semiconductor Lasers Using Pit Perturbations for Monolithic Integration,” IEEE Photonics J. 10(5), 1–10 (2018).
    [Crossref]
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    [Crossref]
  37. R. E. Ecke, J. D. Farmer, and D. K. Umberger, “Scaling of the Arnold tongues,” Nonlinearity 2(2), 175–196 (1989).
    [Crossref]

2019 (2)

W. Cotter, P. E. Morrissey, H. Yang, J. O’Callaghan, B. Roycroft, B. Corbett, and F. H. Peters, “Integrated demultiplexing and amplification of coherent optical combs,” Opt. Express 27(11), 16012–16023 (2019).
[Crossref]

K. Shortiss, M. Shayesteh, W. Cotter, A. H. Perrott, M. Dernaika, and F. H. Peters, “Mode Suppression in Injection Locked Multi-Mode and Single-Mode Lasers for Optical Demultiplexing,” Photonics 6(1), 27 (2019).
[Crossref]

2018 (4)

P. Trocha, M. Karpov, D. Ganin, M. H. P. Pfeiffer, A. Kordts, S. Wolf, J. Krockenberger, P. Marin-Palomo, C. Weimann, S. Randel, W. Freude, T. J. Kippenberg, and C. Koos, “Ultrafast optical ranging using microresonator soliton frequency combs,” Science 359(6378), 887–891 (2018).
[Crossref]

L. Lundberg, M. Karlsson, A. Lorences-Riesgo, M. Mazur, V. Torres-Company, J. Schröder, and P. Andrekson, “Frequency Comb-Based WDM Transmission Systems Enabling Joint Signal Processing,” Appl. Sci. 8(5), 718 (2018).
[Crossref]

D. J. Blumenthal, “Integrated combs drive extreme data rates,” Nat. Photonics 12(8), 447–450 (2018).
[Crossref]

K. Shortiss, M. Dernaika, L. Caro, M. Seifikar, and F. H. Peters, “Inverse Scattering Method Design of Regrowth-Free Single-Mode Semiconductor Lasers Using Pit Perturbations for Monolithic Integration,” IEEE Photonics J. 10(5), 1–10 (2018).
[Crossref]

2017 (1)

S. P. Ó Duill, P. M. Anandarajah, F. Smyth, and L. P. Barry, “Injection-locking criteria for simultaneously locking single-mode lasers to optical frequency combs from gain-switched lasers,” Proc. SPIE 10098, 100980H (2017).
[Crossref]

2015 (2)

A. C. Bordonalli, M. J. Fice, and A. J. Seeds, “Optical injection locking to optical frequency combs for superchannel coherent detection,” Opt. Express 23(2), 1547–1557 (2015).
[Crossref]

R. Zhou, T. Shao, M. D. Gutierrez Pascual, F. Smyth, and L. P. Barry, “Injection Locked Wavelength De-Multiplexer for Optical Comb-Based Nyquist WDM System,” IEEE Photonics Technol. Lett. 27(24), 2595–2598 (2015).
[Crossref]

2014 (1)

A. Gavrielides, “Comb Injection and Sidebands Suppression,” IEEE J. Quantum Electron. 50(5), 364–371 (2014).
[Crossref]

2013 (1)

D. S. Wu, D. J. Richardson, and R. Slavik, “Selective amplification of frequency comb modes via optical injection locking of a semiconductor laser: influence of adjacent unlocked comb modes,” Proc. SPIE 8781, 87810J (2013).
[Crossref]

2011 (2)

T. M. Fortier, M. S. Kirchner, F. Quinlan, J. Taylor, J. C. Bergquist, T. Rosenband, N. Lemke, A. Ludlow, Y. Jiang, C. W. Oates, and S. A. Diddams, “Generation of ultrastable microwaves via optical frequency division,” Nat. Photonics 5(7), 425–429 (2011).
[Crossref]

A. S. Tistomo and S. Gee, “Laser frequency fixation by multimode optical injection locking,” Opt. Express 19(2), 1081–1090 (2011).
[Crossref]

2010 (1)

B. Bernhardt, A. Ozawa, P. Jacquet, M. Jacquey, Y. Kobayashi, T. Udem, R. Holzwarth, G. Guelachvili, T. W. Hánsch, and N. Picqué, “Cavity-enhanced dual-comb spectroscopy,” Nat. Photonics 4(1), 55–57 (2010).
[Crossref]

2008 (1)

2005 (1)

S. Wieczorek, B. Krauskopf, T. Simpson, and D. Lenstra, “The dynamical complexity of optically injected semiconductor lasers,” Phys. Rep. 416(1-2), 1–128 (2005).
[Crossref]

2004 (1)

2002 (1)

T. Udem, R. Holzwarth, and T. W. Hänsch, “Optical frequency metrology,” Nature 416(6877), 233–237 (2002).
[Crossref]

1998 (1)

K. Otsuka, Y. Asakawa, R. Kawai, S.-L. Howng, and J.-L. Chern, “Locking of Relaxation Oscillation Frequencies and Chaos in a Free-Running Two-Mode Nd:YVO 4 Laser,” Jpn. J. Appl. Phys. 37, L1523–L1526 (1998).
[Crossref]

1997 (2)

M. Brøns, P. Gross, and K. Bar-Eli, “Circle Maps and the Devil’s Staircase in a Periodically Perturbed Oregonator,” Int. J. Bifurc. Chaos 07(11), 2621–2628 (1997).
[Crossref]

T. Simpson and J. Liu, “Enhanced modulation bandwidth in injection-locked semiconductor lasers,” IEEE Photonics Technol. Lett. 9(10), 1322–1324 (1997).
[Crossref]

1996 (1)

P. C. D. Jagher, W. A. van der Graaf, and D. Lenstra, “Relaxation-oscillation phenomena in an injection-locked semiconductor laser,” Quantum Semiclassical Opt. J. Eur. Opt. Soc. Part B 8(4), 805–822 (1996).
[Crossref]

1989 (1)

R. E. Ecke, J. D. Farmer, and D. K. Umberger, “Scaling of the Arnold tongues,” Nonlinearity 2(2), 175–196 (1989).
[Crossref]

1988 (1)

I. Petitbon, P. Gallion, G. Debarge, and C. Chabran, “Locking bandwidth and relaxation oscillations of an injection-locked semiconductor laser,” IEEE J. Quantum Electron. 24(2), 148–154 (1988).
[Crossref]

1985 (1)

F. Mogensen, H. Olesen, and G. Jacobsen, “Locking conditions and stability properties for a semiconductor laser with external light injection,” IEEE J. Quantum Electron. 21(7), 784–793 (1985).
[Crossref]

1982 (2)

R. Wyatt, D. Smith, and K. Cameron, “Megahertz linewidth from a 1.5 $\mu$μm semiconductor laser with HeNe laser injection,” Electron. Lett. 18(7), 292–293 (1982).
[Crossref]

W. W. Chow, “Phase locking of lasers by an injected signal,” Opt. Lett. 7(9), 417–419 (1982).
[Crossref]

1973 (1)

K. Kurokawa, “Injection locking of microwave solid-state oscillators,” Proc. IEEE 61(10), 1386–1410 (1973).
[Crossref]

1965 (1)

V. I. Arnol’d, “Small denominators. I. Mapping of the circumference onto itself,” AMS Transl., Ser. 2 46, 213–284 (1965).

1946 (1)

R. Adler, “A Study of Locking Phenomena in Oscillators,” Proc. IRE 34(6), 351–357 (1946).

Adler, R.

R. Adler, “A Study of Locking Phenomena in Oscillators,” Proc. IRE 34(6), 351–357 (1946).

Anandarajah, P. M.

S. P. Ó Duill, P. M. Anandarajah, F. Smyth, and L. P. Barry, “Injection-locking criteria for simultaneously locking single-mode lasers to optical frequency combs from gain-switched lasers,” Proc. SPIE 10098, 100980H (2017).
[Crossref]

Andrekson, P.

L. Lundberg, M. Karlsson, A. Lorences-Riesgo, M. Mazur, V. Torres-Company, J. Schröder, and P. Andrekson, “Frequency Comb-Based WDM Transmission Systems Enabling Joint Signal Processing,” Appl. Sci. 8(5), 718 (2018).
[Crossref]

Arnol’d, V. I.

V. I. Arnol’d, “Small denominators. I. Mapping of the circumference onto itself,” AMS Transl., Ser. 2 46, 213–284 (1965).

Asakawa, Y.

K. Otsuka, Y. Asakawa, R. Kawai, S.-L. Howng, and J.-L. Chern, “Locking of Relaxation Oscillation Frequencies and Chaos in a Free-Running Two-Mode Nd:YVO 4 Laser,” Jpn. J. Appl. Phys. 37, L1523–L1526 (1998).
[Crossref]

Bar-Eli, K.

M. Brøns, P. Gross, and K. Bar-Eli, “Circle Maps and the Devil’s Staircase in a Periodically Perturbed Oregonator,” Int. J. Bifurc. Chaos 07(11), 2621–2628 (1997).
[Crossref]

Barry, L. P.

S. P. Ó Duill, P. M. Anandarajah, F. Smyth, and L. P. Barry, “Injection-locking criteria for simultaneously locking single-mode lasers to optical frequency combs from gain-switched lasers,” Proc. SPIE 10098, 100980H (2017).
[Crossref]

R. Zhou, T. Shao, M. D. Gutierrez Pascual, F. Smyth, and L. P. Barry, “Injection Locked Wavelength De-Multiplexer for Optical Comb-Based Nyquist WDM System,” IEEE Photonics Technol. Lett. 27(24), 2595–2598 (2015).
[Crossref]

M. D. Gutierrez, J. Braddell, F. Smyth, and L. P. Barry, “Monolithically integrated 1x4 comb de-multiplexer based on injection locking,” in Proceedings of European Conference of Integrated Optics, pp. 1–2, (2016).

Bergquist, J. C.

T. M. Fortier, M. S. Kirchner, F. Quinlan, J. Taylor, J. C. Bergquist, T. Rosenband, N. Lemke, A. Ludlow, Y. Jiang, C. W. Oates, and S. A. Diddams, “Generation of ultrastable microwaves via optical frequency division,” Nat. Photonics 5(7), 425–429 (2011).
[Crossref]

Bernhardt, B.

B. Bernhardt, A. Ozawa, P. Jacquet, M. Jacquey, Y. Kobayashi, T. Udem, R. Holzwarth, G. Guelachvili, T. W. Hánsch, and N. Picqué, “Cavity-enhanced dual-comb spectroscopy,” Nat. Photonics 4(1), 55–57 (2010).
[Crossref]

Blumenthal, D. J.

D. J. Blumenthal, “Integrated combs drive extreme data rates,” Nat. Photonics 12(8), 447–450 (2018).
[Crossref]

Bordonalli, A. C.

Braddell, J.

M. D. Gutierrez, J. Braddell, F. Smyth, and L. P. Barry, “Monolithically integrated 1x4 comb de-multiplexer based on injection locking,” in Proceedings of European Conference of Integrated Optics, pp. 1–2, (2016).

Brøns, M.

M. Brøns, P. Gross, and K. Bar-Eli, “Circle Maps and the Devil’s Staircase in a Periodically Perturbed Oregonator,” Int. J. Bifurc. Chaos 07(11), 2621–2628 (1997).
[Crossref]

Cameron, K.

R. Wyatt, D. Smith, and K. Cameron, “Megahertz linewidth from a 1.5 $\mu$μm semiconductor laser with HeNe laser injection,” Electron. Lett. 18(7), 292–293 (1982).
[Crossref]

Caro, L.

K. Shortiss, M. Dernaika, L. Caro, M. Seifikar, and F. H. Peters, “Inverse Scattering Method Design of Regrowth-Free Single-Mode Semiconductor Lasers Using Pit Perturbations for Monolithic Integration,” IEEE Photonics J. 10(5), 1–10 (2018).
[Crossref]

Chabran, C.

I. Petitbon, P. Gallion, G. Debarge, and C. Chabran, “Locking bandwidth and relaxation oscillations of an injection-locked semiconductor laser,” IEEE J. Quantum Electron. 24(2), 148–154 (1988).
[Crossref]

Chang-Hasnain, C. J.

Chen, Y.-K.

Chern, J.-L.

K. Otsuka, Y. Asakawa, R. Kawai, S.-L. Howng, and J.-L. Chern, “Locking of Relaxation Oscillation Frequencies and Chaos in a Free-Running Two-Mode Nd:YVO 4 Laser,” Jpn. J. Appl. Phys. 37, L1523–L1526 (1998).
[Crossref]

Chow, W. W.

Corbett, B.

Cotter, W.

W. Cotter, P. E. Morrissey, H. Yang, J. O’Callaghan, B. Roycroft, B. Corbett, and F. H. Peters, “Integrated demultiplexing and amplification of coherent optical combs,” Opt. Express 27(11), 16012–16023 (2019).
[Crossref]

K. Shortiss, M. Shayesteh, W. Cotter, A. H. Perrott, M. Dernaika, and F. H. Peters, “Mode Suppression in Injection Locked Multi-Mode and Single-Mode Lasers for Optical Demultiplexing,” Photonics 6(1), 27 (2019).
[Crossref]

Debarge, G.

I. Petitbon, P. Gallion, G. Debarge, and C. Chabran, “Locking bandwidth and relaxation oscillations of an injection-locked semiconductor laser,” IEEE J. Quantum Electron. 24(2), 148–154 (1988).
[Crossref]

Dernaika, M.

K. Shortiss, M. Shayesteh, W. Cotter, A. H. Perrott, M. Dernaika, and F. H. Peters, “Mode Suppression in Injection Locked Multi-Mode and Single-Mode Lasers for Optical Demultiplexing,” Photonics 6(1), 27 (2019).
[Crossref]

K. Shortiss, M. Dernaika, L. Caro, M. Seifikar, and F. H. Peters, “Inverse Scattering Method Design of Regrowth-Free Single-Mode Semiconductor Lasers Using Pit Perturbations for Monolithic Integration,” IEEE Photonics J. 10(5), 1–10 (2018).
[Crossref]

Diddams, S. A.

T. M. Fortier, M. S. Kirchner, F. Quinlan, J. Taylor, J. C. Bergquist, T. Rosenband, N. Lemke, A. Ludlow, Y. Jiang, C. W. Oates, and S. A. Diddams, “Generation of ultrastable microwaves via optical frequency division,” Nat. Photonics 5(7), 425–429 (2011).
[Crossref]

Dubois, F.

B. Lingnau, K. Shortiss, F. Dubois, F. H. Peters, and B. Kelleher, “The Devil’s Staircase in the Frequency and Amplitude Locking of Nonlinear Oscillators with Continuous Modulated Forcing,” arXiv:1905.01122 (2019).

Ecke, R. E.

R. E. Ecke, J. D. Farmer, and D. K. Umberger, “Scaling of the Arnold tongues,” Nonlinearity 2(2), 175–196 (1989).
[Crossref]

Farmer, J. D.

R. E. Ecke, J. D. Farmer, and D. K. Umberger, “Scaling of the Arnold tongues,” Nonlinearity 2(2), 175–196 (1989).
[Crossref]

Fice, M. J.

Fortier, T. M.

T. M. Fortier, M. S. Kirchner, F. Quinlan, J. Taylor, J. C. Bergquist, T. Rosenband, N. Lemke, A. Ludlow, Y. Jiang, C. W. Oates, and S. A. Diddams, “Generation of ultrastable microwaves via optical frequency division,” Nat. Photonics 5(7), 425–429 (2011).
[Crossref]

Freude, W.

P. Trocha, M. Karpov, D. Ganin, M. H. P. Pfeiffer, A. Kordts, S. Wolf, J. Krockenberger, P. Marin-Palomo, C. Weimann, S. Randel, W. Freude, T. J. Kippenberg, and C. Koos, “Ultrafast optical ranging using microresonator soliton frequency combs,” Science 359(6378), 887–891 (2018).
[Crossref]

Gallion, P.

I. Petitbon, P. Gallion, G. Debarge, and C. Chabran, “Locking bandwidth and relaxation oscillations of an injection-locked semiconductor laser,” IEEE J. Quantum Electron. 24(2), 148–154 (1988).
[Crossref]

Ganin, D.

P. Trocha, M. Karpov, D. Ganin, M. H. P. Pfeiffer, A. Kordts, S. Wolf, J. Krockenberger, P. Marin-Palomo, C. Weimann, S. Randel, W. Freude, T. J. Kippenberg, and C. Koos, “Ultrafast optical ranging using microresonator soliton frequency combs,” Science 359(6378), 887–891 (2018).
[Crossref]

Gardiner, C. W.

C. W. Gardiner, Stochastic methods : a handbook for the natural and social sciences (Springer, 2009).

Gavrielides, A.

A. Gavrielides, “Comb Injection and Sidebands Suppression,” IEEE J. Quantum Electron. 50(5), 364–371 (2014).
[Crossref]

Gee, S.

Gohle, C.

Gross, P.

M. Brøns, P. Gross, and K. Bar-Eli, “Circle Maps and the Devil’s Staircase in a Periodically Perturbed Oregonator,” Int. J. Bifurc. Chaos 07(11), 2621–2628 (1997).
[Crossref]

Guelachvili, G.

B. Bernhardt, A. Ozawa, P. Jacquet, M. Jacquey, Y. Kobayashi, T. Udem, R. Holzwarth, G. Guelachvili, T. W. Hánsch, and N. Picqué, “Cavity-enhanced dual-comb spectroscopy,” Nat. Photonics 4(1), 55–57 (2010).
[Crossref]

Gutierrez, M. D.

M. D. Gutierrez, J. Braddell, F. Smyth, and L. P. Barry, “Monolithically integrated 1x4 comb de-multiplexer based on injection locking,” in Proceedings of European Conference of Integrated Optics, pp. 1–2, (2016).

Gutierrez Pascual, M. D.

R. Zhou, T. Shao, M. D. Gutierrez Pascual, F. Smyth, and L. P. Barry, “Injection Locked Wavelength De-Multiplexer for Optical Comb-Based Nyquist WDM System,” IEEE Photonics Technol. Lett. 27(24), 2595–2598 (2015).
[Crossref]

Hánsch, T. W.

B. Bernhardt, A. Ozawa, P. Jacquet, M. Jacquey, Y. Kobayashi, T. Udem, R. Holzwarth, G. Guelachvili, T. W. Hánsch, and N. Picqué, “Cavity-enhanced dual-comb spectroscopy,” Nat. Photonics 4(1), 55–57 (2010).
[Crossref]

Hänsch, T. W.

T. Udem, R. Holzwarth, and T. W. Hänsch, “Optical frequency metrology,” Nature 416(6877), 233–237 (2002).
[Crossref]

Holzwarth, R.

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IEEE Photonics J. (1)

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IEEE Photonics Technol. Lett. (2)

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Supplementary Material (2)

NameDescription
» Visualization 1       Animation displaying how the output comb spacing varies in the two dimensional space spanned by optical injection power and detuning. The animation shows the Arnol’d tongues varying in size as the pump current in the slave laser is increased, for the
» Visualization 2       Animation displaying how the output comb spacing varies in the two dimensional space spanned by optical injection power and detuning. The animation shows the Arnol’d tongues varying in size as the pump current in the slave laser is increased, for the

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Figures (8)

Fig. 1.
Fig. 1. Experimental setup for performing optical comb injection. TLS: Tunable laser source, MZM: Mach zehnder modulator, PC: Polarisation controller, EDFA: Erbium doped fibre amplifier, VOA: Variable optical attenuator, OSA: Optical spectrum analyser, PD: Photodiode, ESA: Electrical spectrum analyser.
Fig. 2.
Fig. 2. Optical spectra from a slave laser with $-4.2~{\,\textrm {dBm}}$ free-running power, under injection from a $14~{\textrm {GHz}}$ optical comb with optical power $+4.1~{\,\textrm {dBm}}$. In each case, vertical dashed lines indicate the free-running frequency of the slave laser. (a) Stable locking to the centre comb line, at $-7.5~{\textrm {GHz}}$ detuning. (b) At a detuning of $-6.1~{\textrm {GHz}}$, relaxation oscillations become undamped. (c) The undamped relaxation oscillations lock to $\tfrac 13$ the optical comb spacing, creating a new optical comb with spacing $4.66~{\textrm {GHz}}$.
Fig. 3.
Fig. 3. (a) Calculated RF spectrum of a slave laser under optical injection from a $10~{\textrm {GHz}}$ comb, as the pump current of the slave is swept. The dashed white line indicates the frequency of the free-running ROs. The injection strength of $K = 0.06$ was fixed. (b) Measured free-running ROs of the slave laser from $55~{\textrm {mA}}$ to $120~{\textrm {mA}}$, in steps of $5~{\textrm {mA}}$ up until $90~{\textrm {mA}}$ and steps of $10~{\textrm {mA}}$ thereafter, offset vertically from top to bottom by $-10~{\,\textrm {dB}}$ for clarity. Solid black lines indicate fits from which we determined parameter values for our model.
Fig. 4.
Fig. 4. Experimental and theoretical results from a frequency detuning sweep. In the experimental results, the slave laser was initially biased at $1.75$ times threshold current with free-running power -4.5 dBm, and was injected by 3 line comb of total power -7 dBm, with spacing $\Delta _{{\textrm {comb}}} =10~{\textrm {GHz}}$. The slave laser’s temperature was varied to perform the frequency detuning sweep, and the vertical dashed white lines show where the free-running laser frequency is resonant to one of the optical comb lines. (a) Intensity plot of measured optical spectra as the slave laser was tuned across the injected optical comb. The frequency axis has been scaled such that $0~{\textrm {GHz}}$ coincides with the centre comb line. (b) Electrical power spectra for the corresponding frequency sweep in (a). (c) Intensity plot of simulated optical spectra with $K=0.06$, for the same parameters as in (a). (d) Comparison between the optical spectra of the injected comb, and the output comb as the slave is frequency locked between the optical comb lines (at a detuning of $-5.6~{\textrm {GHz}}$).
Fig. 5.
Fig. 5. Two dimensional maps of the parameter space spanned by detuning and injected power, calculated with $J = 1.82J_{\textrm {th}}$ and $\Delta _{{\textrm {comb}}} = 10~{\textrm {GHz}}$. (a) Map of the output comb frequency spacing. White regions indicate unlocked states. The triangular region enclosed within the dashed lines shows the parameter region used to calculate the relative size of the harmonic locking areas in Fig. 6. (b) RF linewidth of the dominant harmonic’s beat note, over the same parameter space as in (a).
Fig. 6.
Fig. 6. Plots showing the relative size of the Arnol’d resonances as a function of slave laser pump current, for (a) $\alpha =3$, and (b) $\alpha =0$. For each current, a two dimensional map over injection strength and detuning was calculated as in Fig. 5 with $\Delta _{{\textrm {comb}}} =$ 10 GHz, and only the area contained in each resonant Arnol’d tongue was counted. The vertical dotted lines indicate the pump currents at which the free-running RO frequency matches that of a resonance. Animations showing the two dimensional maps used to calculate the above results are presented in the supplementary material submitted with this manuscript (see Visualization 1 for the $\alpha = 3$ case, and Visualization 2 for the $\alpha = 0$ case).
Fig. 7.
Fig. 7. (a) Intensity plot showing the electrical spectra measured as the frequency of a single mode slave laser was swept across the injected $6.5~{\textrm {GHz}}$ optical comb. The optical comb had a power of $-5.0~$dBm, and the slave laser was initially biased at $1.75$ times threshold current, with $-4.4~$dBm coupled to the lensed fibre. The vertical dashed white lines show the parameter values where the free-running laser frequency is resonant to one of the optical comb lines. (b) Corresponding simulated experiment for the parameters in (a). (c) Two dimensional map of the parameter space spanned by detuning and injected power, showing the output comb frequency spacing, calculated with $J = 1.75J_{\textrm {th}}$ and $\Delta _{{\textrm {comb}}} = 6.5~{\textrm {GHz}}$. The vertical dotted line indicates the injection strength used in (b). White regions indicate unlocked states.
Fig. 8.
Fig. 8. Two dimensional maps of the parameter space spanned by detuning and injected power, which show the output frequency spacing of the slave laser, calculated with $J = 1.82J_{th}$ (corresponding to an RO frequency of 4.375 GHz). The comb spacings used were (a) 10 GHz, (b) 15 GHz, (c) 20 GHz, (d) 25 GHz, (e) 30 GHz, and (f) 35 GHz. White regions indicate unlocked states.

Tables (1)

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Table 1. Simulation parameters unless noted otherwise.

Equations (5)

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d d t E = ( 1 + i α ) ( N N 0 2 κ ) E + E t | inj + β   ( ξ s p ( t ) + i ξ s p ( t ) ) ,
d d t N = J N T ( N N 0 ) | E | 2 .
E t | inj = K E 0 ( 1 + 2 m cos ( ϕ comb ) ) e i ϕ inj ( t ) i 2 π ν inj E ,
ϕ comb t = 2 π Δ comb + 2 π Δ ν comb   ξ comb ( t ) ,
ϕ inj t = 2 π Δ ν inj   ξ inj ( t ) .

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