Abstract

The phase-locking, noise, and modulation properties of two face-to-face optically mutual-injected terahertz (THz) quantum cascade lasers (QCLs) are analyzed theoretically. In the phase-locking range, the two THz QCLs are in stationary states working at the same frequency. Outside the phase-locking range, the amplitude and the instantaneous frequency of the optical field oscillate with time, and the power spectrum shows a series of discrete peaks. For strong mutual injection, the optical field of the THz QCL array also exhibits oscillatory behavior. Coherent collapse or chaotic behavior is not observed within the range of the parameters used in this simulation. The spontaneous emission noise of phase-locked THz QCLs is higher than that of THz QCLs at free-running operation, and mutual injection may introduce additional modulation peaks in the noise spectrum. The modulation response of the mutual-injected THz QCLs to an individual modulation is investigated. It is found that the modulation bandwidth and the phase difference are significantly dependent on the modulation parameters. These results are helpful for further understanding the nonlinear dynamic behaviors of THz QCLs under optical injection and provide theoretical support for the development of THz QCL phase-locked arrays.

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

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2017 (1)

X. Qi, G. Agnew, I. Kundu, T. Taimre, Y. Lim, K. Bertling, P. Dean, A. Grier, A. Valavanis, E. Linfield, A. Giles Davies, D. Indjin, and A. Rakić, “Multi-spectral terahertz sensing: proposal for a coupled-cavity quantum cascade laser based optical feedback interferometer,” Opt. Express 20(9), 10153–10165 (2017).
[Crossref]

2016 (8)

J. F. Webb, K. S. C. Yong, and M. K. Haldar, “Harmonic, intermodulation and cross-modulation distortion in directly modulated quantum cascade lasers,” IOP Conf. Ser.: Mater. Sci. Eng. 131(1), 012025 (2016).
[Crossref]

C. Q. Quiroz, J. T. Alsina, J. Roma, M. C. Torrent, and C. Masoller, “Quantitative identification of dynamical transitions in a semiconductor laser with optical feedback,” Sci Rep 6, 37510 (2016).
[Crossref]

X. Wang, C. Shen, T. Jiang, Z. Zhan, Q. Deng, W. Li, W. Wu, N. Yang, W. Chu, and S. Duan, “High-power terahertz quantum cascade lasers with ~ 0.23 W in continuous wave mode,” AIP Adv. 6(7), 075210 (2016).
[Crossref]

H. Zhu, F. Wang, Q. Yan, C. Yu, J. Chen, G. Xu, L. He, L. Li, L. Chen, A. G. Davies, E. H. Linfield, J. Hao, P. B. Vigneron, and R. Colombelli, “Terahertz master-oscillator power-amplifier quantum cascade lasers,” Appl. Phys. Lett. 109(23), 231105 (2016).
[Crossref]

T.-Y. Kao, J. L. Reno, and Q. Hu, “Phase-locked laser arrays through global antenna mutual coupling,” Nat. Photonics 10(8), 541–546 (2016).
[Crossref]

R. Vallon, B. Parvitte, L. Bizet, G. M. D. Naurois, B. Simozrag, G. Maisons, M. Carras, and V. Zeninari, “External cavity coherent quantum cascade laser array,” Infrared Phys. Technol. 76, 415–420(2016).
[Crossref]

L. Junges, A. Gavrielides, and J. A. C. Gallas, “Synchronization properties of two mutually delay-coupled semiconductor lasers,” J. Opt. Soc. Am. B 33(7), C65–C71 (2016).
[Crossref]

M. M. Sheikhey, M. Goudarzi, R. Yadipour, and H. Baghban, “Analytical investigation of relative intensity noise properties of injection-locked mid-IR quantum cascade lasers,” J. Opt. Soc. Am. B 33(11), D57–D64 (2016).
[Crossref]

2015 (3)

2014 (3)

L. Li, L. Chen, J. Zhu, J. Freeman, P. Dean, A. Valavanis, A. G. Davies, and E. H. Linfield, “Terahertz quantum cascade lasers with >1 W output powers,” Electron. Lett. 50(4), 309–311 (2014).
[Crossref]

H. Simos, A. Bogris, D. Syvridis, and W. Elsäßer, “Intensity noise properties of mid-infrared injection locked quantum cascade lasers: I. modeling,” IEEE J. Quantum Electron. 50(2), 98–105 (2014).
[Crossref]

D. Liu, C. Sun, B. Xiong, and Y. Luo, “Nonlinear dynamics in integrated coupled DFB lasers with ultra-short delay,” Opt. Express 22(5), 5614–5622 (2014).
[Crossref] [PubMed]

2013 (7)

Y. F. Li, J. Wang, N. Yang, J. Liu, T. Wang, F. Liu, Z. Wang, W. Chu, and S. Duan, “The output power and beam divergence behaviors of tapered terahertz quantum cascade lasers,” Opt. Express 21(13), 15998–16006 (2013).
[Crossref] [PubMed]

A. D. Rakić, T. Taimre, K. Bertling, Y. L. Lim, P. Dean, D. Indjin, Z. Ikonić, P. Harrison, A. Valavanis, S. P. Khanna, M. Lachab, S. J. Wilson, E. H. Linfield, and A. G. Davies, “Swept-frequency feedback interferometry using terahertz frequency QCLs: a method for imaging and materials analysis,” Opt. Express 21(19), 22194–22205 (2013).
[Crossref]

C. Wang, F. Grillot, V. Kovanis, and J. Even, “Rate equation analysis of injection-locked quantum cascade lasers,” J. Appl. Phys. 113(6), 063104 (2013).
[Crossref]

F. Wang, X. G. Guo, C. Wang, and J. C. Cao, “Ultrafast population dynamics in electrically modulated terahertz quantum cascade lasers,” New J. Phys. 15(15), 075009 (2013).
[Crossref]

M. Ravaro, V. Jagtap, C. Manquest, P. Gellie, G. Santarelli, C. Sirtori, S. P. Khanna, E. H. Linfield, and S. Barbieri, “Spectral properties of THz quantum-cascade lasers: frequency noise, phase-locking and absolute frequency measurement,” J. Infrared Milli. Terahz. Waves 34(5–6), 342–356 (2013).
[Crossref]

M. Brandstetter, C. Deutsch, M. Krall, H. Detz, D. C. MacFarland, T. Zederbauer, A. M. Andrews, W. Schrenk, G. Strasser, and K. Unterrainer, “High power terahertz quantum cascade lasers with symmetric wafer bonded active regions,” Appl. Phys. Lett. 103(17), 171113 (2013).
[Crossref]

O. P. Marshall, S. Chakraborty, M. Khairuzzaman, H. E. Beere, and D. A. Ritchie, “Reversible mode switching in Y-coupled terahertz lasers,” Appl. Phys. Lett. 96(102), 111105 (2013).
[Crossref]

2012 (3)

G. Xu, R. Colombelli, S. P. Khanna, A. Belarouci, X. Letartre, L. Li, E. H. Linfield, A. G. Davies, H. E. Beere, and D. A. Ritchie, “Efficient power extraction in surface-emitting semiconductor lasers using graded photonic heterostructures,” Nat. Commun. 3, 952 (2012).
[Crossref] [PubMed]

B. Meng and Q. J. Wang, “Theoretical investigation of injection-locked high modulation bandwidth quantum cascade lasers,” Opt. Express 20(2), 1450–1464 (2012).
[Crossref] [PubMed]

M. Yamanishi, “Theory of intrinsic linewidth based on fluctuation-dissipation balance for thermal photons in THz quantum-cascade lasers,” Opt. Express 20(27), 28465–28478 (2012).
[Crossref] [PubMed]

2011 (2)

G. N. Rao and A. Karpf, “External cavity tunable quantum cascade lasers and their applications to trace gas monitoring,” Appl. Opt. 50(4), A100–A115 (2011).
[Crossref] [PubMed]

T. Liu and Q. J. Wang, “Fundamental frequency noise and linewidth broadening caused by intrinsic temperature fluctuations in quantum cascade lasers,” Phys. Rev. B 84(12), 125322 (2011).
[Crossref]

2010 (5)

2008 (2)

A. G. Davies, A. D. Burnett, W. Fan, E. H. Linfield, and J. E. Cunningham, “Terahertz spectroscopy of explosives and drugs,” Mater. Today 11(3), 18–26 (2008).
[Crossref]

R. P. Green, J. Xu, L. Mahler, A. Tredicucci, F. Beltran, G. Giuliani, H. E. Beere, and D. A. Ritchie, “Linewidth enhancement factor of terahertz quantum cascade lasers,” Appl. Phys. Lett. 92(7), 071106 (2008).
[Crossref]

2006 (1)

A. W. M. Lee, Q. Qin, S. Kumar, B. S. Williams, Q. Hu, and J. L. Reno, “Real-time terahertz imaging over a standoff distance (> 25 meters),” Appl. Phys. Lett. 89(14), 141125 (2006).
[Crossref]

2005 (3)

T. Gensty and W. Elsäßer, “Semiclassical model for the relative intensity noise of intersubband quantum cascade lasers,” Opt. Commun. 256(1), 171–183 (2005).
[Crossref]

J. Xi, Y. Yu, J. F. Chicharo, and T. Bosch, “Estimating the parameters of semiconductor lasers based on weak optical feedback self-mixing interferometry,” IEEE J. Quantum Electron. 41(8), 1058–1064 (2005).
[Crossref]

B. Williams, S. Kumar, Q. Hu, and J. Reno, “Operation of terahertz quantum-cascade lasers at 164 K in pulsed mode and at 117 K in continuous-wave mode,” Opt. Express 13(9), 3331–3339 (2005).
[Crossref] [PubMed]

2004 (3)

S. Kumar, B. S. Williams, S. Kohen, Q. Hu, and J. L. Reno, “Continuous-wave operation of terahertz quantumcascade lasers above liquid-nitrogen temperature,” Appl. Phys. Lett. 84(14), 2494 (2004).
[Crossref]

E. A. Viktorov, A. M. Yacomotti, and P. Mandel, “Semiconductor lasers coupled face-to-face,” J. Opt. B: Quantum Semiclassical Opt. 6(2), L9–L12 (2004).
[Crossref]

E. Wille, M. Peil, I. Fischer, and W. Elsäßer, “Dynamical scenarios of mutually delay-coupled semiconductor lasers in the short coupling regime,” Proc. SPIE 5452, 41–50 (2004).
[Crossref]

2003 (1)

2002 (1)

F. Capasso, R. Paiella, R. Martini, R. Colombelli, C. Gmachl, T. L. Myers, M. S. Taubman, R. M. Williams, C. G. Bethea, K. Unterrainer, H. Y. Hwang, D. L. Sivco, A. Y. Cho, A. M. Sergent, H. C. Liu, and E. A. Whittaker, “Quantum cascade lasers: ultrahigh-speed operation, optical wireless communication, narrow linewidth, and far-infrared emission,” IEEE J. Quantum Electron. 38(6), 511–532 (2002).
[Crossref]

2001 (1)

R. Paiella, R. Martini, F. Capasso, C. Gmachl, H. Y. Hwang, D. L. Sivco, J. N. Baillargeon, A. Y. Cho, E. A. Whittaker, and H. C. Liu, “High-frequency modulation without the relaxation oscillation resonance in quantum cascade lasers,” Appl. Phys. Lett. 79(16), 2526–2528 (2001).
[Crossref]

2000 (1)

M. V. Romalis, “Narrowing of high power diode laser arrays using reflection feedback from an etalon,” Appl. Phys. Lett. 77(8), 1080–1081 (2000).
[Crossref]

1994 (1)

J. Faist, F. Capasso, D. L. Sivco, C. Sirtori, A. L. Hutchinson, and A. Y. Cho, “Quantum cascade laser,” Science 264(5158), 553–556 (1994).
[Crossref] [PubMed]

1991 (1)

1984 (1)

1980 (1)

R. Lang and K. Kobayashi, “External optical feedback effects on semiconductor injection laser properties,” IEEE J. Quantum Electron. 16(3), 347–355 (1980).
[Crossref]

Agnew, G.

X. Qi, G. Agnew, I. Kundu, T. Taimre, Y. Lim, K. Bertling, P. Dean, A. Grier, A. Valavanis, E. Linfield, A. Giles Davies, D. Indjin, and A. Rakić, “Multi-spectral terahertz sensing: proposal for a coupled-cavity quantum cascade laser based optical feedback interferometer,” Opt. Express 20(9), 10153–10165 (2017).
[Crossref]

Akalin, T.

W. Maineult, L. Ding, P. Gellie, P. Filloux, C. Sirtori, S. Barbieri, T. Akalin, J. F. Lampin, I. Sagnes, H. E. Beere, and D. A. Ritchie, “Microwave modulation of THz quantum cascade lasers: a transmission-line approach,” Appl. Phys. Lett. 96(2), 021108 (2010).
[Crossref]

Alsina, J. T.

C. Q. Quiroz, J. T. Alsina, J. Roma, M. C. Torrent, and C. Masoller, “Quantitative identification of dynamical transitions in a semiconductor laser with optical feedback,” Sci Rep 6, 37510 (2016).
[Crossref]

Andrews, A. M.

M. Brandstetter, C. Deutsch, M. Krall, H. Detz, D. C. MacFarland, T. Zederbauer, A. M. Andrews, W. Schrenk, G. Strasser, and K. Unterrainer, “High power terahertz quantum cascade lasers with symmetric wafer bonded active regions,” Appl. Phys. Lett. 103(17), 171113 (2013).
[Crossref]

Baghban, H.

Bai, Y.

Baillargeon, J. N.

R. Paiella, R. Martini, F. Capasso, C. Gmachl, H. Y. Hwang, D. L. Sivco, J. N. Baillargeon, A. Y. Cho, E. A. Whittaker, and H. C. Liu, “High-frequency modulation without the relaxation oscillation resonance in quantum cascade lasers,” Appl. Phys. Lett. 79(16), 2526–2528 (2001).
[Crossref]

Bandyopadhyay, N.

Barbieri, S.

M. Ravaro, V. Jagtap, C. Manquest, P. Gellie, G. Santarelli, C. Sirtori, S. P. Khanna, E. H. Linfield, and S. Barbieri, “Spectral properties of THz quantum-cascade lasers: frequency noise, phase-locking and absolute frequency measurement,” J. Infrared Milli. Terahz. Waves 34(5–6), 342–356 (2013).
[Crossref]

W. Maineult, L. Ding, P. Gellie, P. Filloux, C. Sirtori, S. Barbieri, T. Akalin, J. F. Lampin, I. Sagnes, H. E. Beere, and D. A. Ritchie, “Microwave modulation of THz quantum cascade lasers: a transmission-line approach,” Appl. Phys. Lett. 96(2), 021108 (2010).
[Crossref]

Beere, H. E.

Y. Halioua, G. Xu, S. Moumdji, L. Li, J. Zhu, E. H. Linfield, A. G. Davies, H. E. Beere, D. A. Ritchie, and R. Colombelli, “Phase-locked arrays of surface-emitting graded-photonic-heterostructure terahertz semiconductor lasers,” Opt. Express 23(5), 6915–6923 (2015).
[Crossref] [PubMed]

O. P. Marshall, S. Chakraborty, M. Khairuzzaman, H. E. Beere, and D. A. Ritchie, “Reversible mode switching in Y-coupled terahertz lasers,” Appl. Phys. Lett. 96(102), 111105 (2013).
[Crossref]

G. Xu, R. Colombelli, S. P. Khanna, A. Belarouci, X. Letartre, L. Li, E. H. Linfield, A. G. Davies, H. E. Beere, and D. A. Ritchie, “Efficient power extraction in surface-emitting semiconductor lasers using graded photonic heterostructures,” Nat. Commun. 3, 952 (2012).
[Crossref] [PubMed]

W. Maineult, L. Ding, P. Gellie, P. Filloux, C. Sirtori, S. Barbieri, T. Akalin, J. F. Lampin, I. Sagnes, H. E. Beere, and D. A. Ritchie, “Microwave modulation of THz quantum cascade lasers: a transmission-line approach,” Appl. Phys. Lett. 96(2), 021108 (2010).
[Crossref]

R. P. Green, J. Xu, L. Mahler, A. Tredicucci, F. Beltran, G. Giuliani, H. E. Beere, and D. A. Ritchie, “Linewidth enhancement factor of terahertz quantum cascade lasers,” Appl. Phys. Lett. 92(7), 071106 (2008).
[Crossref]

Belarouci, A.

G. Xu, R. Colombelli, S. P. Khanna, A. Belarouci, X. Letartre, L. Li, E. H. Linfield, A. G. Davies, H. E. Beere, and D. A. Ritchie, “Efficient power extraction in surface-emitting semiconductor lasers using graded photonic heterostructures,” Nat. Commun. 3, 952 (2012).
[Crossref] [PubMed]

Beltran, F.

R. P. Green, J. Xu, L. Mahler, A. Tredicucci, F. Beltran, G. Giuliani, H. E. Beere, and D. A. Ritchie, “Linewidth enhancement factor of terahertz quantum cascade lasers,” Appl. Phys. Lett. 92(7), 071106 (2008).
[Crossref]

Bertling, K.

X. Qi, G. Agnew, I. Kundu, T. Taimre, Y. Lim, K. Bertling, P. Dean, A. Grier, A. Valavanis, E. Linfield, A. Giles Davies, D. Indjin, and A. Rakić, “Multi-spectral terahertz sensing: proposal for a coupled-cavity quantum cascade laser based optical feedback interferometer,” Opt. Express 20(9), 10153–10165 (2017).
[Crossref]

A. D. Rakić, T. Taimre, K. Bertling, Y. L. Lim, P. Dean, D. Indjin, Z. Ikonić, P. Harrison, A. Valavanis, S. P. Khanna, M. Lachab, S. J. Wilson, E. H. Linfield, and A. G. Davies, “Swept-frequency feedback interferometry using terahertz frequency QCLs: a method for imaging and materials analysis,” Opt. Express 21(19), 22194–22205 (2013).
[Crossref]

Bethea, C. G.

F. Capasso, R. Paiella, R. Martini, R. Colombelli, C. Gmachl, T. L. Myers, M. S. Taubman, R. M. Williams, C. G. Bethea, K. Unterrainer, H. Y. Hwang, D. L. Sivco, A. Y. Cho, A. M. Sergent, H. C. Liu, and E. A. Whittaker, “Quantum cascade lasers: ultrahigh-speed operation, optical wireless communication, narrow linewidth, and far-infrared emission,” IEEE J. Quantum Electron. 38(6), 511–532 (2002).
[Crossref]

Bizet, L.

R. Vallon, B. Parvitte, L. Bizet, G. M. D. Naurois, B. Simozrag, G. Maisons, M. Carras, and V. Zeninari, “External cavity coherent quantum cascade laser array,” Infrared Phys. Technol. 76, 415–420(2016).
[Crossref]

Bogris, A.

H. Simos, A. Bogris, D. Syvridis, and W. Elsäßer, “Intensity noise properties of mid-infrared injection locked quantum cascade lasers: I. modeling,” IEEE J. Quantum Electron. 50(2), 98–105 (2014).
[Crossref]

Bosch, T.

J. Xi, Y. Yu, J. F. Chicharo, and T. Bosch, “Estimating the parameters of semiconductor lasers based on weak optical feedback self-mixing interferometry,” IEEE J. Quantum Electron. 41(8), 1058–1064 (2005).
[Crossref]

Brandstetter, M.

M. Brandstetter, C. Deutsch, M. Krall, H. Detz, D. C. MacFarland, T. Zederbauer, A. M. Andrews, W. Schrenk, G. Strasser, and K. Unterrainer, “High power terahertz quantum cascade lasers with symmetric wafer bonded active regions,” Appl. Phys. Lett. 103(17), 171113 (2013).
[Crossref]

Burnett, A. D.

A. G. Davies, A. D. Burnett, W. Fan, E. H. Linfield, and J. E. Cunningham, “Terahertz spectroscopy of explosives and drugs,” Mater. Today 11(3), 18–26 (2008).
[Crossref]

Cao, J.

C. Wang, J. Cao, L. Gu, Q. Wu, and Z. Tan, “20 Mbps wireless communication demonstration using terahertz quantum devices,” Chin. Opt. Lett. 13(8), 81402 (2015).
[Crossref]

Cao, J. C.

F. Wang, X. G. Guo, C. Wang, and J. C. Cao, “Ultrafast population dynamics in electrically modulated terahertz quantum cascade lasers,” New J. Phys. 15(15), 075009 (2013).
[Crossref]

Capasso, F.

F. Capasso, R. Paiella, R. Martini, R. Colombelli, C. Gmachl, T. L. Myers, M. S. Taubman, R. M. Williams, C. G. Bethea, K. Unterrainer, H. Y. Hwang, D. L. Sivco, A. Y. Cho, A. M. Sergent, H. C. Liu, and E. A. Whittaker, “Quantum cascade lasers: ultrahigh-speed operation, optical wireless communication, narrow linewidth, and far-infrared emission,” IEEE J. Quantum Electron. 38(6), 511–532 (2002).
[Crossref]

R. Paiella, R. Martini, F. Capasso, C. Gmachl, H. Y. Hwang, D. L. Sivco, J. N. Baillargeon, A. Y. Cho, E. A. Whittaker, and H. C. Liu, “High-frequency modulation without the relaxation oscillation resonance in quantum cascade lasers,” Appl. Phys. Lett. 79(16), 2526–2528 (2001).
[Crossref]

J. Faist, F. Capasso, D. L. Sivco, C. Sirtori, A. L. Hutchinson, and A. Y. Cho, “Quantum cascade laser,” Science 264(5158), 553–556 (1994).
[Crossref] [PubMed]

Carras, M.

R. Vallon, B. Parvitte, L. Bizet, G. M. D. Naurois, B. Simozrag, G. Maisons, M. Carras, and V. Zeninari, “External cavity coherent quantum cascade laser array,” Infrared Phys. Technol. 76, 415–420(2016).
[Crossref]

Chakraborty, S.

O. P. Marshall, S. Chakraborty, M. Khairuzzaman, H. E. Beere, and D. A. Ritchie, “Reversible mode switching in Y-coupled terahertz lasers,” Appl. Phys. Lett. 96(102), 111105 (2013).
[Crossref]

Chen, J.

H. Zhu, F. Wang, Q. Yan, C. Yu, J. Chen, G. Xu, L. He, L. Li, L. Chen, A. G. Davies, E. H. Linfield, J. Hao, P. B. Vigneron, and R. Colombelli, “Terahertz master-oscillator power-amplifier quantum cascade lasers,” Appl. Phys. Lett. 109(23), 231105 (2016).
[Crossref]

Chen, L.

H. Zhu, F. Wang, Q. Yan, C. Yu, J. Chen, G. Xu, L. He, L. Li, L. Chen, A. G. Davies, E. H. Linfield, J. Hao, P. B. Vigneron, and R. Colombelli, “Terahertz master-oscillator power-amplifier quantum cascade lasers,” Appl. Phys. Lett. 109(23), 231105 (2016).
[Crossref]

L. Li, L. Chen, J. Zhu, J. Freeman, P. Dean, A. Valavanis, A. G. Davies, and E. H. Linfield, “Terahertz quantum cascade lasers with >1 W output powers,” Electron. Lett. 50(4), 309–311 (2014).
[Crossref]

Chicharo, J. F.

J. Xi, Y. Yu, J. F. Chicharo, and T. Bosch, “Estimating the parameters of semiconductor lasers based on weak optical feedback self-mixing interferometry,” IEEE J. Quantum Electron. 41(8), 1058–1064 (2005).
[Crossref]

Cho, A. Y.

F. Capasso, R. Paiella, R. Martini, R. Colombelli, C. Gmachl, T. L. Myers, M. S. Taubman, R. M. Williams, C. G. Bethea, K. Unterrainer, H. Y. Hwang, D. L. Sivco, A. Y. Cho, A. M. Sergent, H. C. Liu, and E. A. Whittaker, “Quantum cascade lasers: ultrahigh-speed operation, optical wireless communication, narrow linewidth, and far-infrared emission,” IEEE J. Quantum Electron. 38(6), 511–532 (2002).
[Crossref]

R. Paiella, R. Martini, F. Capasso, C. Gmachl, H. Y. Hwang, D. L. Sivco, J. N. Baillargeon, A. Y. Cho, E. A. Whittaker, and H. C. Liu, “High-frequency modulation without the relaxation oscillation resonance in quantum cascade lasers,” Appl. Phys. Lett. 79(16), 2526–2528 (2001).
[Crossref]

J. Faist, F. Capasso, D. L. Sivco, C. Sirtori, A. L. Hutchinson, and A. Y. Cho, “Quantum cascade laser,” Science 264(5158), 553–556 (1994).
[Crossref] [PubMed]

Chu, W.

X. Wang, C. Shen, T. Jiang, Z. Zhan, Q. Deng, W. Li, W. Wu, N. Yang, W. Chu, and S. Duan, “High-power terahertz quantum cascade lasers with ~ 0.23 W in continuous wave mode,” AIP Adv. 6(7), 075210 (2016).
[Crossref]

Y. F. Li, J. Wang, N. Yang, J. Liu, T. Wang, F. Liu, Z. Wang, W. Chu, and S. Duan, “The output power and beam divergence behaviors of tapered terahertz quantum cascade lasers,” Opt. Express 21(13), 15998–16006 (2013).
[Crossref] [PubMed]

Y. Xie, Y. Li, J. Wang, N. Yang, W. Chu, and S. Duan, Chapter 5 in Quantum Cascade Lasers (IntechOpen, 2017).

Colombelli, R.

H. Zhu, F. Wang, Q. Yan, C. Yu, J. Chen, G. Xu, L. He, L. Li, L. Chen, A. G. Davies, E. H. Linfield, J. Hao, P. B. Vigneron, and R. Colombelli, “Terahertz master-oscillator power-amplifier quantum cascade lasers,” Appl. Phys. Lett. 109(23), 231105 (2016).
[Crossref]

Y. Halioua, G. Xu, S. Moumdji, L. Li, J. Zhu, E. H. Linfield, A. G. Davies, H. E. Beere, D. A. Ritchie, and R. Colombelli, “Phase-locked arrays of surface-emitting graded-photonic-heterostructure terahertz semiconductor lasers,” Opt. Express 23(5), 6915–6923 (2015).
[Crossref] [PubMed]

G. Xu, R. Colombelli, S. P. Khanna, A. Belarouci, X. Letartre, L. Li, E. H. Linfield, A. G. Davies, H. E. Beere, and D. A. Ritchie, “Efficient power extraction in surface-emitting semiconductor lasers using graded photonic heterostructures,” Nat. Commun. 3, 952 (2012).
[Crossref] [PubMed]

F. Capasso, R. Paiella, R. Martini, R. Colombelli, C. Gmachl, T. L. Myers, M. S. Taubman, R. M. Williams, C. G. Bethea, K. Unterrainer, H. Y. Hwang, D. L. Sivco, A. Y. Cho, A. M. Sergent, H. C. Liu, and E. A. Whittaker, “Quantum cascade lasers: ultrahigh-speed operation, optical wireless communication, narrow linewidth, and far-infrared emission,” IEEE J. Quantum Electron. 38(6), 511–532 (2002).
[Crossref]

Cunningham, J. E.

A. G. Davies, A. D. Burnett, W. Fan, E. H. Linfield, and J. E. Cunningham, “Terahertz spectroscopy of explosives and drugs,” Mater. Today 11(3), 18–26 (2008).
[Crossref]

Davies, A. G.

H. Zhu, F. Wang, Q. Yan, C. Yu, J. Chen, G. Xu, L. He, L. Li, L. Chen, A. G. Davies, E. H. Linfield, J. Hao, P. B. Vigneron, and R. Colombelli, “Terahertz master-oscillator power-amplifier quantum cascade lasers,” Appl. Phys. Lett. 109(23), 231105 (2016).
[Crossref]

Y. Halioua, G. Xu, S. Moumdji, L. Li, J. Zhu, E. H. Linfield, A. G. Davies, H. E. Beere, D. A. Ritchie, and R. Colombelli, “Phase-locked arrays of surface-emitting graded-photonic-heterostructure terahertz semiconductor lasers,” Opt. Express 23(5), 6915–6923 (2015).
[Crossref] [PubMed]

L. Li, L. Chen, J. Zhu, J. Freeman, P. Dean, A. Valavanis, A. G. Davies, and E. H. Linfield, “Terahertz quantum cascade lasers with >1 W output powers,” Electron. Lett. 50(4), 309–311 (2014).
[Crossref]

A. D. Rakić, T. Taimre, K. Bertling, Y. L. Lim, P. Dean, D. Indjin, Z. Ikonić, P. Harrison, A. Valavanis, S. P. Khanna, M. Lachab, S. J. Wilson, E. H. Linfield, and A. G. Davies, “Swept-frequency feedback interferometry using terahertz frequency QCLs: a method for imaging and materials analysis,” Opt. Express 21(19), 22194–22205 (2013).
[Crossref]

G. Xu, R. Colombelli, S. P. Khanna, A. Belarouci, X. Letartre, L. Li, E. H. Linfield, A. G. Davies, H. E. Beere, and D. A. Ritchie, “Efficient power extraction in surface-emitting semiconductor lasers using graded photonic heterostructures,” Nat. Commun. 3, 952 (2012).
[Crossref] [PubMed]

A. G. Davies, A. D. Burnett, W. Fan, E. H. Linfield, and J. E. Cunningham, “Terahertz spectroscopy of explosives and drugs,” Mater. Today 11(3), 18–26 (2008).
[Crossref]

Dean, P.

X. Qi, G. Agnew, I. Kundu, T. Taimre, Y. Lim, K. Bertling, P. Dean, A. Grier, A. Valavanis, E. Linfield, A. Giles Davies, D. Indjin, and A. Rakić, “Multi-spectral terahertz sensing: proposal for a coupled-cavity quantum cascade laser based optical feedback interferometer,” Opt. Express 20(9), 10153–10165 (2017).
[Crossref]

L. Li, L. Chen, J. Zhu, J. Freeman, P. Dean, A. Valavanis, A. G. Davies, and E. H. Linfield, “Terahertz quantum cascade lasers with >1 W output powers,” Electron. Lett. 50(4), 309–311 (2014).
[Crossref]

A. D. Rakić, T. Taimre, K. Bertling, Y. L. Lim, P. Dean, D. Indjin, Z. Ikonić, P. Harrison, A. Valavanis, S. P. Khanna, M. Lachab, S. J. Wilson, E. H. Linfield, and A. G. Davies, “Swept-frequency feedback interferometry using terahertz frequency QCLs: a method for imaging and materials analysis,” Opt. Express 21(19), 22194–22205 (2013).
[Crossref]

Deng, Q.

X. Wang, C. Shen, T. Jiang, Z. Zhan, Q. Deng, W. Li, W. Wu, N. Yang, W. Chu, and S. Duan, “High-power terahertz quantum cascade lasers with ~ 0.23 W in continuous wave mode,” AIP Adv. 6(7), 075210 (2016).
[Crossref]

Detz, H.

M. Brandstetter, C. Deutsch, M. Krall, H. Detz, D. C. MacFarland, T. Zederbauer, A. M. Andrews, W. Schrenk, G. Strasser, and K. Unterrainer, “High power terahertz quantum cascade lasers with symmetric wafer bonded active regions,” Appl. Phys. Lett. 103(17), 171113 (2013).
[Crossref]

Deutsch, C.

M. Brandstetter, C. Deutsch, M. Krall, H. Detz, D. C. MacFarland, T. Zederbauer, A. M. Andrews, W. Schrenk, G. Strasser, and K. Unterrainer, “High power terahertz quantum cascade lasers with symmetric wafer bonded active regions,” Appl. Phys. Lett. 103(17), 171113 (2013).
[Crossref]

Ding, L.

W. Maineult, L. Ding, P. Gellie, P. Filloux, C. Sirtori, S. Barbieri, T. Akalin, J. F. Lampin, I. Sagnes, H. E. Beere, and D. A. Ritchie, “Microwave modulation of THz quantum cascade lasers: a transmission-line approach,” Appl. Phys. Lett. 96(2), 021108 (2010).
[Crossref]

Duan, S.

X. Wang, C. Shen, T. Jiang, Z. Zhan, Q. Deng, W. Li, W. Wu, N. Yang, W. Chu, and S. Duan, “High-power terahertz quantum cascade lasers with ~ 0.23 W in continuous wave mode,” AIP Adv. 6(7), 075210 (2016).
[Crossref]

Y. F. Li, J. Wang, N. Yang, J. Liu, T. Wang, F. Liu, Z. Wang, W. Chu, and S. Duan, “The output power and beam divergence behaviors of tapered terahertz quantum cascade lasers,” Opt. Express 21(13), 15998–16006 (2013).
[Crossref] [PubMed]

Y. Xie, Y. Li, J. Wang, N. Yang, W. Chu, and S. Duan, Chapter 5 in Quantum Cascade Lasers (IntechOpen, 2017).

Elsäßer, W.

H. Simos, A. Bogris, D. Syvridis, and W. Elsäßer, “Intensity noise properties of mid-infrared injection locked quantum cascade lasers: I. modeling,” IEEE J. Quantum Electron. 50(2), 98–105 (2014).
[Crossref]

T. Gensty and W. Elsäßer, “Semiclassical model for the relative intensity noise of intersubband quantum cascade lasers,” Opt. Commun. 256(1), 171–183 (2005).
[Crossref]

E. Wille, M. Peil, I. Fischer, and W. Elsäßer, “Dynamical scenarios of mutually delay-coupled semiconductor lasers in the short coupling regime,” Proc. SPIE 5452, 41–50 (2004).
[Crossref]

Even, J.

C. Wang, F. Grillot, V. Kovanis, and J. Even, “Rate equation analysis of injection-locked quantum cascade lasers,” J. Appl. Phys. 113(6), 063104 (2013).
[Crossref]

Faist, J.

J. Faist, F. Capasso, D. L. Sivco, C. Sirtori, A. L. Hutchinson, and A. Y. Cho, “Quantum cascade laser,” Science 264(5158), 553–556 (1994).
[Crossref] [PubMed]

Fan, W.

A. G. Davies, A. D. Burnett, W. Fan, E. H. Linfield, and J. E. Cunningham, “Terahertz spectroscopy of explosives and drugs,” Mater. Today 11(3), 18–26 (2008).
[Crossref]

Filloux, P.

W. Maineult, L. Ding, P. Gellie, P. Filloux, C. Sirtori, S. Barbieri, T. Akalin, J. F. Lampin, I. Sagnes, H. E. Beere, and D. A. Ritchie, “Microwave modulation of THz quantum cascade lasers: a transmission-line approach,” Appl. Phys. Lett. 96(2), 021108 (2010).
[Crossref]

Fischer, I.

E. Wille, M. Peil, I. Fischer, and W. Elsäßer, “Dynamical scenarios of mutually delay-coupled semiconductor lasers in the short coupling regime,” Proc. SPIE 5452, 41–50 (2004).
[Crossref]

Freeman, J.

L. Li, L. Chen, J. Zhu, J. Freeman, P. Dean, A. Valavanis, A. G. Davies, and E. H. Linfield, “Terahertz quantum cascade lasers with >1 W output powers,” Electron. Lett. 50(4), 309–311 (2014).
[Crossref]

Gallas, J. A. C.

Garanovich, I. L.

García-Ojalvo, J.

Gavrielides, A.

Gellie, P.

M. Ravaro, V. Jagtap, C. Manquest, P. Gellie, G. Santarelli, C. Sirtori, S. P. Khanna, E. H. Linfield, and S. Barbieri, “Spectral properties of THz quantum-cascade lasers: frequency noise, phase-locking and absolute frequency measurement,” J. Infrared Milli. Terahz. Waves 34(5–6), 342–356 (2013).
[Crossref]

W. Maineult, L. Ding, P. Gellie, P. Filloux, C. Sirtori, S. Barbieri, T. Akalin, J. F. Lampin, I. Sagnes, H. E. Beere, and D. A. Ritchie, “Microwave modulation of THz quantum cascade lasers: a transmission-line approach,” Appl. Phys. Lett. 96(2), 021108 (2010).
[Crossref]

Gensty, T.

T. Gensty and W. Elsäßer, “Semiclassical model for the relative intensity noise of intersubband quantum cascade lasers,” Opt. Commun. 256(1), 171–183 (2005).
[Crossref]

Giles Davies, A.

X. Qi, G. Agnew, I. Kundu, T. Taimre, Y. Lim, K. Bertling, P. Dean, A. Grier, A. Valavanis, E. Linfield, A. Giles Davies, D. Indjin, and A. Rakić, “Multi-spectral terahertz sensing: proposal for a coupled-cavity quantum cascade laser based optical feedback interferometer,” Opt. Express 20(9), 10153–10165 (2017).
[Crossref]

Giuliani, G.

R. P. Green, J. Xu, L. Mahler, A. Tredicucci, F. Beltran, G. Giuliani, H. E. Beere, and D. A. Ritchie, “Linewidth enhancement factor of terahertz quantum cascade lasers,” Appl. Phys. Lett. 92(7), 071106 (2008).
[Crossref]

Gmachl, C.

F. Capasso, R. Paiella, R. Martini, R. Colombelli, C. Gmachl, T. L. Myers, M. S. Taubman, R. M. Williams, C. G. Bethea, K. Unterrainer, H. Y. Hwang, D. L. Sivco, A. Y. Cho, A. M. Sergent, H. C. Liu, and E. A. Whittaker, “Quantum cascade lasers: ultrahigh-speed operation, optical wireless communication, narrow linewidth, and far-infrared emission,” IEEE J. Quantum Electron. 38(6), 511–532 (2002).
[Crossref]

R. Paiella, R. Martini, F. Capasso, C. Gmachl, H. Y. Hwang, D. L. Sivco, J. N. Baillargeon, A. Y. Cho, E. A. Whittaker, and H. C. Liu, “High-frequency modulation without the relaxation oscillation resonance in quantum cascade lasers,” Appl. Phys. Lett. 79(16), 2526–2528 (2001).
[Crossref]

Goudarzi, M.

Green, R. P.

R. P. Green, J. Xu, L. Mahler, A. Tredicucci, F. Beltran, G. Giuliani, H. E. Beere, and D. A. Ritchie, “Linewidth enhancement factor of terahertz quantum cascade lasers,” Appl. Phys. Lett. 92(7), 071106 (2008).
[Crossref]

Grier, A.

X. Qi, G. Agnew, I. Kundu, T. Taimre, Y. Lim, K. Bertling, P. Dean, A. Grier, A. Valavanis, E. Linfield, A. Giles Davies, D. Indjin, and A. Rakić, “Multi-spectral terahertz sensing: proposal for a coupled-cavity quantum cascade laser based optical feedback interferometer,” Opt. Express 20(9), 10153–10165 (2017).
[Crossref]

Grillot, F.

C. Wang, F. Grillot, V. Kovanis, and J. Even, “Rate equation analysis of injection-locked quantum cascade lasers,” J. Appl. Phys. 113(6), 063104 (2013).
[Crossref]

Gu, L.

C. Wang, J. Cao, L. Gu, Q. Wu, and Z. Tan, “20 Mbps wireless communication demonstration using terahertz quantum devices,” Chin. Opt. Lett. 13(8), 81402 (2015).
[Crossref]

Guo, X. G.

F. Wang, X. G. Guo, C. Wang, and J. C. Cao, “Ultrafast population dynamics in electrically modulated terahertz quantum cascade lasers,” New J. Phys. 15(15), 075009 (2013).
[Crossref]

Haldar, M. K.

J. F. Webb, K. S. C. Yong, and M. K. Haldar, “Harmonic, intermodulation and cross-modulation distortion in directly modulated quantum cascade lasers,” IOP Conf. Ser.: Mater. Sci. Eng. 131(1), 012025 (2016).
[Crossref]

Halioua, Y.

Hao, J.

H. Zhu, F. Wang, Q. Yan, C. Yu, J. Chen, G. Xu, L. He, L. Li, L. Chen, A. G. Davies, E. H. Linfield, J. Hao, P. B. Vigneron, and R. Colombelli, “Terahertz master-oscillator power-amplifier quantum cascade lasers,” Appl. Phys. Lett. 109(23), 231105 (2016).
[Crossref]

Harrison, P.

He, L.

H. Zhu, F. Wang, Q. Yan, C. Yu, J. Chen, G. Xu, L. He, L. Li, L. Chen, A. G. Davies, E. H. Linfield, J. Hao, P. B. Vigneron, and R. Colombelli, “Terahertz master-oscillator power-amplifier quantum cascade lasers,” Appl. Phys. Lett. 109(23), 231105 (2016).
[Crossref]

Heydari, D.

Hu, Q.

T.-Y. Kao, J. L. Reno, and Q. Hu, “Phase-locked laser arrays through global antenna mutual coupling,” Nat. Photonics 10(8), 541–546 (2016).
[Crossref]

T. Y. Kao, Q. Hu, and J. L. Reno, “Phase-locked arrays of surface-emitting terahertz quantum-cascade lasers,” Appl. Phys. Lett. 96(10), 101106 (2010).
[Crossref]

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H. Zhu, F. Wang, Q. Yan, C. Yu, J. Chen, G. Xu, L. He, L. Li, L. Chen, A. G. Davies, E. H. Linfield, J. Hao, P. B. Vigneron, and R. Colombelli, “Terahertz master-oscillator power-amplifier quantum cascade lasers,” Appl. Phys. Lett. 109(23), 231105 (2016).
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J. Xi, Y. Yu, J. F. Chicharo, and T. Bosch, “Estimating the parameters of semiconductor lasers based on weak optical feedback self-mixing interferometry,” IEEE J. Quantum Electron. 41(8), 1058–1064 (2005).
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M. Brandstetter, C. Deutsch, M. Krall, H. Detz, D. C. MacFarland, T. Zederbauer, A. M. Andrews, W. Schrenk, G. Strasser, and K. Unterrainer, “High power terahertz quantum cascade lasers with symmetric wafer bonded active regions,” Appl. Phys. Lett. 103(17), 171113 (2013).
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R. Vallon, B. Parvitte, L. Bizet, G. M. D. Naurois, B. Simozrag, G. Maisons, M. Carras, and V. Zeninari, “External cavity coherent quantum cascade laser array,” Infrared Phys. Technol. 76, 415–420(2016).
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X. Wang, C. Shen, T. Jiang, Z. Zhan, Q. Deng, W. Li, W. Wu, N. Yang, W. Chu, and S. Duan, “High-power terahertz quantum cascade lasers with ~ 0.23 W in continuous wave mode,” AIP Adv. 6(7), 075210 (2016).
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AIP Adv. (1)

X. Wang, C. Shen, T. Jiang, Z. Zhan, Q. Deng, W. Li, W. Wu, N. Yang, W. Chu, and S. Duan, “High-power terahertz quantum cascade lasers with ~ 0.23 W in continuous wave mode,” AIP Adv. 6(7), 075210 (2016).
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Appl. Opt. (1)

Appl. Phys. Lett. (10)

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M. Brandstetter, C. Deutsch, M. Krall, H. Detz, D. C. MacFarland, T. Zederbauer, A. M. Andrews, W. Schrenk, G. Strasser, and K. Unterrainer, “High power terahertz quantum cascade lasers with symmetric wafer bonded active regions,” Appl. Phys. Lett. 103(17), 171113 (2013).
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T. Y. Kao, Q. Hu, and J. L. Reno, “Phase-locked arrays of surface-emitting terahertz quantum-cascade lasers,” Appl. Phys. Lett. 96(10), 101106 (2010).
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R. P. Green, J. Xu, L. Mahler, A. Tredicucci, F. Beltran, G. Giuliani, H. E. Beere, and D. A. Ritchie, “Linewidth enhancement factor of terahertz quantum cascade lasers,” Appl. Phys. Lett. 92(7), 071106 (2008).
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S. Kumar, B. S. Williams, S. Kohen, Q. Hu, and J. L. Reno, “Continuous-wave operation of terahertz quantumcascade lasers above liquid-nitrogen temperature,” Appl. Phys. Lett. 84(14), 2494 (2004).
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R. Paiella, R. Martini, F. Capasso, C. Gmachl, H. Y. Hwang, D. L. Sivco, J. N. Baillargeon, A. Y. Cho, E. A. Whittaker, and H. C. Liu, “High-frequency modulation without the relaxation oscillation resonance in quantum cascade lasers,” Appl. Phys. Lett. 79(16), 2526–2528 (2001).
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Chin. Opt. Lett. (1)

C. Wang, J. Cao, L. Gu, Q. Wu, and Z. Tan, “20 Mbps wireless communication demonstration using terahertz quantum devices,” Chin. Opt. Lett. 13(8), 81402 (2015).
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Electron. Lett. (1)

L. Li, L. Chen, J. Zhu, J. Freeman, P. Dean, A. Valavanis, A. G. Davies, and E. H. Linfield, “Terahertz quantum cascade lasers with >1 W output powers,” Electron. Lett. 50(4), 309–311 (2014).
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IEEE J. Quantum Electron. (4)

H. Simos, A. Bogris, D. Syvridis, and W. Elsäßer, “Intensity noise properties of mid-infrared injection locked quantum cascade lasers: I. modeling,” IEEE J. Quantum Electron. 50(2), 98–105 (2014).
[Crossref]

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[Crossref]

Infrared Phys. Technol. (1)

R. Vallon, B. Parvitte, L. Bizet, G. M. D. Naurois, B. Simozrag, G. Maisons, M. Carras, and V. Zeninari, “External cavity coherent quantum cascade laser array,” Infrared Phys. Technol. 76, 415–420(2016).
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Figures (6)

Fig. 1
Fig. 1 The phase-locking range of face-to-face optically mutual-injected THz QCLs. The shadow region is the phase-locked range of the mutual-injected THz QCLs. Inset: the schematic of two THz QCLs with mutual injection. The red dashed line indicates the limitation of κ due to stability requirements.
Fig. 2
Fig. 2 Time evolution of |EA|, |EB | (first column), the corresponding power spectral density (second column), and instantaneous frequency (third column) with different coupling strength κ and detuning frequency ΔΩ/2π. The coupling strength in the first three rows are set as κ = 9.87 × 10−3, which is the case of moderate coupling. (a)–(c) within the phase-locking regime, ΔΩ/2π = 0.5GHz, (d)–(f) out of the pahse-locking regime, ΔΩ/2π = 5GHz, (g)–(i) out of the pahse-locking regime, ΔΩ/2π = 0.55GHz. The fourth row is the case of strong coupling with κ = 0.247. (j)–(l) out of the pahse-locking regime, ΔΩ/2π = 0.2GHz. The effective injection current is 1.5Ith in all simulations.
Fig. 3
Fig. 3 The comparison of RINs for mutual-injected THz QCLs and the free-running one. Black line: moderate coupling within the pahse-locking regime with κ = 9.87 × 10−3 and ΔΩ/2π = 0.2GHz; Red line: moderate coupling out of the phase-locking regime with κ = 9.87 × 10−3 and ΔΩ/2π = 20GHz; Blue line: the RIN of a free-running THz QCL. And in all cases, Iin = 1.5Ith.
Fig. 4
Fig. 4 RINs of two mutual-injected THz QCLs with different coupling strengths and injection currents. (a) κ = 9.87 × 10−4 , 9.87 × 10−2, with the effective injection current and detuning frequency set as Iin = 1.5Ith and ΔΩ/2π = 0.01GHz, respectively. (b) Iin = 1.5Ith, 2Ith, with the coupling strength and detuning frequency set as κ = 9.87 × 10−3, ΔΩ/2π = 0.2GHz, respectively.
Fig. 5
Fig. 5 Within the phase-locking regime, the normalized frequency response |HAA| and |HAB | with different coupling strengths and injection currents. The frequency detunning ΔΩ/2π is set as 0.01GHz. (a) Iin = 1.5Ith, different coupling strength with κ = 9.87×10−4, 9.87 × 10−2. (b) κ = 9.87 × 10−3, different injection current Iin = 1.5Ith,2.0Ith . The −3 dB modulation renponse is indicated by the blue dashed line.
Fig. 6
Fig. 6 The phase difference between input and output signals of the mutual-injected THz-QCLs under frequency modulation with d = 0.05m, 0.1m, 0.15m. (a) self-modulation ∠HAA(), (b) mutual-modulation ∠HAB(). The coupling strength, detuning frequency, and effective injection current are set as κ = 9.87 × 10−3, ΔΩ/2π = 0.01GHz, and 1.5Ith , respectively.

Tables (1)

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Table 1 Parameters values used in the simulations.

Equations (34)

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d ε A ( t ) d t = i Ω A 0 ε A ( t ) + 1 2 ( 1 + i α ) { Z g [ N A ( t ) N T ] 1 τ p } ε A ( t ) + κ c τ l e i C p ε B ( t τ e x t ) ,
d ε B ( t ) d t = i Ω B 0 ε B ( t ) + 1 2 ( 1 + i α ) { Z g [ N B ( t ) N T ] 1 τ p } ε B ( t ) + κ c τ l e i C p ε A ( t τ e x t ) ,
d N A ( t ) d t = η I i n , A q γ e N A ( t ) g [ N A ( t ) N T ] | ε A ( t ) | 2 ,
d N B ( t ) d t = η I i n , B q γ e N B ( t ) g [ N B ( t ) N T ] | ε B ( t ) | 2 ,
E A ( s ) = ε A exp i Ω A 0 t E c , n A ( s ) = N A N c 0 N c ,
E B ( s ) = ε B exp i Ω B 0 t E c , n B ( s ) = N B N c 0 N c ,
d E A ( s ) d s = 1 2 ( 1 + i α ) n A ( s ) E A ( s ) + κ e i C p e i ω B 0 θ E B ( s θ ) e i Δ ω s ,
d E B ( s ) d s = 1 2 ( 1 + i α ) n B ( s ) E B ( s ) + κ e i C p e i ω A 0 θ E A ( s θ ) e i Δ ω s ,
T d n A ( s ) d s = p A n A ( s ) [ 1 + n A ( s ) ] | E A ( s ) | 2 ,
T d n B ( s ) d s = p B n B ( s ) [ 1 + n B ( s ) ] | E B ( s ) | 2 .
E A L = F A L e i ( ω L ω A 0 ) s + i ϕ L , n A L = p F A L 2 F A L 2 + 1 ,
E B L = F B L e i ( ω L ω B 0 ) s , n B L = p F B L 2 F B L 2 + 1 ,
F A L = p n A L 1 + n A L , n A L = 2 κ F B L F A L cos ( ω L θ + C p + ϕ L ) ,
F B L = p n B L 1 + n B L , n B L = 2 κ F A L F B L cos ( ω L θ + C p ϕ L ) ,
ω L Ω A 0 = κ F B L F A L 1 + α 2 sin ( ω L θ + C p + arctan α + ϕ L ) ,
ω L Ω B 0 = κ F A L F B L 1 + α 2 sin ( ω L θ + C p + arctan α ϕ L ) .
Δ ω = 2 κ 1 + α 2 cos ( ω L θ + C p + arctan α ) sin ( ϕ L ) .
| Δ ω | max = 2 κ 1 + α 2 .
d E A d s = 1 2 ( 1 + i α ) n A E A + κ e i C p e i ω B 0 θ E B ( s θ ) e i Δ ω s + X A ( s ) ,
d E B d s = 1 2 ( 1 + i α ) n B E B + κ e i C p e i ω A 0 θ E A ( s θ ) e i Δ ω s + X B ( s ) .
< X A ( s ) X A ( s ) ¯ > = β ( n A + n A 0 ) δ ( s s ) ,
< X B ( s ) X B ( s ) ¯ > = β ( n B + n B 0 ) δ ( s s ) ,
R I N A ( ω ) = lim T 1 T | 0 T [ P A ( s ) < P A ( s ) > ] e i ω s d s | 2 < P A ( s ) > 2 .
{ F A = Δ F A + F A L , F B = Δ F B + F B L ϕ A = Δ ϕ A + ϕ A L , ϕ B = Δ ϕ B + ϕ B L n A = Δ n A + n A L , n B = Δ n B + n B L ,
d Δ F A ( s ) d s = 1 2 n A L Δ F A ( s ) + 1 2 Δ n A ( s ) F A L κ F B L [ Δ ϕ A ( s ) Δ ϕ B ( s θ ) ] sin ( ω L θ + C p + ϕ L ) + κ Δ F B ( s θ ) cos ( ω L θ + C p + ϕ L ) ,
d Δ F B ( s ) d s = 1 2 n B L Δ F B ( s ) + 1 2 Δ n B ( s ) F B L κ F A L [ Δ ϕ B ( s ) Δ ϕ A ( s θ ) ] sin ( ω L θ + C p + ϕ L ) + κ Δ F A ( s θ ) cos ( ω L θ + C p + ϕ L ) ,
d Δ ϕ A ( s ) d s F A L = ( ω L ω A 0 ) Δ F A ( s ) + 1 2 α n A L Δ F A ( s ) κ F B L [ Δ ϕ A ( s ) Δ ϕ B ( s θ ) ] cos ( ω L θ + C p + ϕ L ) κ Δ F B ( s θ ) sin ( ω L θ + C p + ϕ L ) ,
d Δ ϕ B ( s ) d s F B L = ( ω L ω B 0 ) Δ F B ( s ) + 1 2 α n B L Δ F B ( s ) κ F A L [ Δ ϕ B ( s ) Δ ϕ A ( s θ ) ] cos ( ω L θ + C p ϕ L ) κ Δ F A ( s θ ) sin ( ω L θ + C p ϕ L ) ,
d Δ n A ( s ) d s T A = p A 2 F A L Δ F A ( s ) ( n A L + 1 ) Δ n A ( s ) ( F A L 2 + 1 ) ,
d Δ n B ( s ) d s T B = p B 2 F B L Δ F B ( s ) ( n B L + 1 ) Δ n B ( s ) ( F B L 2 + 1 ) .
( Q 11 Q 12 Q 13 Q 14 Q 15 Q 16 Q 21 Q 22 Q 23 Q 24 Q 25 Q 26 Q 31 Q 32 Q 33 Q 34 Q 35 Q 36 Q 41 Q 42 Q 43 Q 44 Q 45 Q 46 Q 51 Q 52 Q 53 Q 54 Q 55 Q 56 Q 61 Q 62 Q 63 Q 64 Q 65 Q 66 ) ( Δ F A Δ F B Δ ϕ A Δ ϕ B Δ n A Δ n B ) = Z g q γ γ e ( 0 0 0 0 Δ J A Δ J B ) .
{ Q 11 = 1 2 n A L λ Q 12 = κ e λ θ cos ψ A L Q 13 = κ F B L sin ( ψ A L ) Q 14 = e λ θ κ F B L sin ( ψ A L ) Q 15 = 1 2 F A L Q 16 = 0 Q 21 = κ e λ θ cos ψ B L Q 22 = 1 2 n B L λ Q 23 = e λ θ κ F A L sin ( ψ B L ) Q 24 = κ F A L sin ( ψ B L ) Q 25 = 0 Q 26 = 1 2 F B L Q 31 = 1 2 α n A L ( ω L ω A 0 ) Q 32 = κ e λ θ sin ( ψ A L ) Q 33 = F B L λ F A L Q 34 = e λ θ κ F B L Q 35 = 1 2 α F A L Q 36 = 0 Q 41 = κ e λ θ sin ( ψ B L ) Q 42 = 1 2 α n B L ( ω L ω B 0 ) Q 43 = e λ θ κ F A L Q 44 = κ F A L λ F B L Q 45 = 0 Q 46 = 1 2 α F B L Q 51 = 2 F A L ( n A L + 1 ) Q 52 = 0 Q 53 = 0 Q 54 = 0 Q 55 = ( F A L 2 + 1 ) T λ Q 56 = 0 Q 61 = 0 Q 62 = 2 F B L ( n B L + 1 ) Q 63 = 0 Q 64 = 0 Q 65 = 0 Q 66 = ( F B L 2 + 1 ) T λ
H A A ( λ ) = q γ γ e Z g Δ F A Δ J A = Q 1 ( 000010 ) .
H A B ( λ ) = q γ γ e Z g Δ F A Δ J B = Q 1 ( 000001 ) .

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