Abstract

This work theoretically investigates the frequency noise (FN) characteristics of quantum cascade lasers subject to the optical injection through a set of coupled rate equations with Langevin noise sources. It is shown that the low-frequency FN is completely suppressed by the optical injection, and the suppression bandwidth increases with the increasing injection ratio. The optimal FN peak suppression ratio at an injection ratio of 10 dB reaches 2.9 dB. In addition, it is found that the optical injection at positive frequency detunings close to the locking boundary invokes an additional peak in the FN spectrum, which can be higher than the carrier noise-induced one of free-running lasers. This peak amplitude strongly depends on the value of the linewidth broadening factor. Unlike injection-locked interband lasers, the FN peak does not necessarily exhibit a resonance.

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

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References

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2018 (1)

2017 (1)

2016 (2)

M. F. Pereira, “The linewidth enhancement factor of intersubband lasers: From a two-level limit to gain without inversion conditions,” Appl. Phys. Lett. 109(22), 222102 (2016).
[Crossref]

C. Wang, K. Schires, M. Osiński, P. J. Poole, and F. Grillot, “Thermally insensitive determination of the linewidth broadening factor in nanostructured semiconductor lasers using optical injection locking,” Sci. Rep. 6(1), 27825 (2016).
[Crossref] [PubMed]

2015 (2)

C. Juretzka, H. Simos, A. Bogris, D. Syvridis, W. Elsäßer, and M. Carras, “Intensity noise properties of midinfrared injection locked quantum cascade lasers: II. Experiments,” IEEE J. Quantum Electron. 51(1), 2300208 (2015).
[Crossref]

B. Argence, B. Chanteau, O. Lopez, D. Nicolodi, M. Abgrall, C. Chardonnet, C. Daussy, B. Darquié, Y. Le Coq, and A. Amy-Klein, “Quantum cascade laser frequency stabilization at the sub-Hz level,” Nat. Photonics 9(7), 456–460 (2015).
[Crossref]

2014 (1)

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]

2013 (4)

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]

T. Erneux, V. Kovanis, and A. Gavrielides, “Nonlinear dynamics of an injected quantum cascade laser,” Phys. Rev. E 88(3), 032907 (2013).
[Crossref] [PubMed]

C. Wang, F. Grillot, V. I. Kovanis, J. D. Bodyfelt, and J. Even, “Modulation properties of optically injection-locked quantum cascade lasers,” Opt. Lett. 38(11), 1975–1977 (2013).
[Crossref] [PubMed]

T. Liu, K. E. Lee, and Q. J. Wang, “Importance of the microscopic effects on the linewidth enhancement factor of quantum cascade lasers,” Opt. Express 21(23), 27804–27815 (2013).
[Crossref] [PubMed]

2012 (3)

2011 (2)

S. Borri, S. Bartalini, P. C. Pastor, I. Galli, G. Giusfredi, D. Mazzotti, M. Yamanishi, and P. De Natale, “Natale, “Frequency-noise dynamics of mid-infrared quantum cascade lasers,” IEEE J. Quantum Electron. 47(7), 984–988 (2011).
[Crossref]

Y. Petitjean, F. Destic, J. C. Mollier, and C. Sirtori, “Dynamic modeling of Terahertz quantum cascade lasers,” IEEE J. Sel. Top. Quantum Electron. 17(1), 22–29 (2011).
[Crossref]

2010 (1)

2009 (1)

E. K. Lau, L. J. Wong, and M. C. Wu, “Enhanced Modulation Characteristics of Optical Injection-Locked Lasers: A Tutorial,” IEEE J. Sel. Top. Quantum Electron. 15(3), 618–633 (2009).
[Crossref]

2007 (1)

2006 (2)

T. Aellen, R. Maulini, R. Terazzi, N. Hoyler, M. Giovannini, J. Faist, S. Blaser, and L. Hvozdara, “Direct measurement of the linewidth enhancement factor by optical heterodyning of an amplitude-modulated quantum cascade laser,” Appl. Phys. Lett. 89(9), 091121 (2006).
[Crossref]

J. von Staden, T. Gensty, W. Elsässer, G. Giuliani, and C. Mann, “Measurements of the α factor of a distributed-feedback quantum cascade laser by an optical feedback self-mixing technique,” Opt. Lett. 31(17), 2574–2576 (2006).
[Crossref] [PubMed]

2005 (2)

M. K. Haldar, “A simplified analysis of direct intensity modulation of quantum cascade laser,” IEEE J. Quantum Electron. 41(11), 1349–1355 (2005).
[Crossref]

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

2004 (2)

M. S. Taubman, T. L. Myers, B. D. Cannon, and R. M. Williams, “Stabilization, injection and control of quantum cascade lasers, and their application to chemical sensing in the infrared,” Spectrochim. Acta A Mol. Biomol. Spectrosc. 60(14), 3457–3468 (2004).
[Crossref] [PubMed]

J. Faist, L. Ajili, G. Scalari, M. Giovannini, M. Beck, M. Rochat, H. Beere, A. G. Davies, E. H. Linfield, and D. Ritchie, “Terahertz quantum cascade lasers,” Philos Trans A Math Phys Eng Sci 362(1815), 215–231 (2004).
[Crossref] [PubMed]

2003 (1)

A. Murakami, K. Kawashima, and K. Atsuki, “Cavity resonance shift and bandwidth enhancement in semiconductor lasers with strong light injection,” IEEE J. Quantum Electron. 39(10), 1196–1204 (2003).
[Crossref]

2002 (2)

C. Sirtori, “GaAs Quantum cascade laser: fundamentals and performances,” EDP sciences 7(3), 1– 20 (2002).

F. Rana and R. J. Ram, “Current noise and photon noise in quantum cascade lasers,” Phys. Rev. B 65(12), 125313 (2002).
[Crossref]

2001 (1)

D. J. Higham, “An algorithmic introduction to numerical simulation of stochastic differential equations,” SIAM Rev. 43(3), 525–546 (2001).
[Crossref]

2000 (1)

G. Yabre, H. de Waardt, H. P. A. Van den Boom, and G.-D. Khoe, “Noise characteristics of single-mode semiconductor lasers under external light injection,” IEEE J. Quantum Electron. 36(3), 385–393 (2000).
[Crossref]

1995 (1)

T. B. Simpson, J. M. Liu, and A. Gavrielides, “Bandwidth enhancement and broadband noise reduction in injection-locked semiconductor lasers,” IEEE Photonics Technol. Lett. 7(7), 709–711 (1995).
[Crossref]

1985 (4)

P. Gallion, H. Nakajima, G. Debarge, and C. Chabran, “Contribution of spontaneous emission to the linewidth of an injection-locked semiconductor laser,” Electron. Lett. 21(14), 626–628 (1985).
[Crossref]

S. Saito, F. Mogensen, and H. Olesen, “Effective bandwidth for FM noise suppression in an injection-locked semiconductor laser,” Electron. Lett. 21(24), 1173–1175 (1985).
[Crossref]

F. Mogensen, H. Olesen, and G. Jacobsen, “FM noise suppression and linewidth reduction in an injection-locked semiconductor laser,” Electron. Lett. 21(16), 696–697 (1985).
[Crossref]

S. Saito, O. Nilsson, and Y. Yamamoto, “Frequency modulation noise and linewidth reduction in a semiconductor laser by means of a negative frequency feedback technique,” Appl. Phys. Lett. 46(1), 3–5 (1985).
[Crossref]

1982 (3)

C. Harder, J. Katz, S. Margalit, J. Shacham, and A. Yariv, “Noise equivalent circuit of a semiconductor laser diode,” IEEE J. Quantum Electron. 18(3), 333–337 (1982).
[Crossref]

R. Lang, “Injection locking properties of a semiconductor laser,” IEEE J. Quantum Electron. 18(6), 976–983 (1982).
[Crossref]

C. Henry, “Theory of the linewidth of semiconductor lasers,” IEEE J. Quantum Electron. 18(2), 259–264 (1982).
[Crossref]

Abgrall, M.

B. Argence, B. Chanteau, O. Lopez, D. Nicolodi, M. Abgrall, C. Chardonnet, C. Daussy, B. Darquié, Y. Le Coq, and A. Amy-Klein, “Quantum cascade laser frequency stabilization at the sub-Hz level,” Nat. Photonics 9(7), 456–460 (2015).
[Crossref]

Aellen, T.

T. Aellen, R. Maulini, R. Terazzi, N. Hoyler, M. Giovannini, J. Faist, S. Blaser, and L. Hvozdara, “Direct measurement of the linewidth enhancement factor by optical heterodyning of an amplitude-modulated quantum cascade laser,” Appl. Phys. Lett. 89(9), 091121 (2006).
[Crossref]

Ajili, L.

J. Faist, L. Ajili, G. Scalari, M. Giovannini, M. Beck, M. Rochat, H. Beere, A. G. Davies, E. H. Linfield, and D. Ritchie, “Terahertz quantum cascade lasers,” Philos Trans A Math Phys Eng Sci 362(1815), 215–231 (2004).
[Crossref] [PubMed]

Amy-Klein, A.

B. Argence, B. Chanteau, O. Lopez, D. Nicolodi, M. Abgrall, C. Chardonnet, C. Daussy, B. Darquié, Y. Le Coq, and A. Amy-Klein, “Quantum cascade laser frequency stabilization at the sub-Hz level,” Nat. Photonics 9(7), 456–460 (2015).
[Crossref]

Argence, B.

B. Argence, B. Chanteau, O. Lopez, D. Nicolodi, M. Abgrall, C. Chardonnet, C. Daussy, B. Darquié, Y. Le Coq, and A. Amy-Klein, “Quantum cascade laser frequency stabilization at the sub-Hz level,” Nat. Photonics 9(7), 456–460 (2015).
[Crossref]

Atsuki, K.

A. Murakami, K. Kawashima, and K. Atsuki, “Cavity resonance shift and bandwidth enhancement in semiconductor lasers with strong light injection,” IEEE J. Quantum Electron. 39(10), 1196–1204 (2003).
[Crossref]

Bartalini, S.

S. Borri, I. Galli, F. Cappelli, A. Bismuto, S. Bartalini, P. Cancio, G. Giusfredi, D. Mazzotti, J. Faist, and P. De Natale, “Direct link of a mid-infrared QCL to a frequency comb by optical injection,” Opt. Lett. 37(6), 1011–1013 (2012).
[Crossref] [PubMed]

S. Borri, S. Bartalini, P. C. Pastor, I. Galli, G. Giusfredi, D. Mazzotti, M. Yamanishi, and P. De Natale, “Natale, “Frequency-noise dynamics of mid-infrared quantum cascade lasers,” IEEE J. Quantum Electron. 47(7), 984–988 (2011).
[Crossref]

Beck, M.

J. Faist, L. Ajili, G. Scalari, M. Giovannini, M. Beck, M. Rochat, H. Beere, A. G. Davies, E. H. Linfield, and D. Ritchie, “Terahertz quantum cascade lasers,” Philos Trans A Math Phys Eng Sci 362(1815), 215–231 (2004).
[Crossref] [PubMed]

Beere, H.

J. Faist, L. Ajili, G. Scalari, M. Giovannini, M. Beck, M. Rochat, H. Beere, A. G. Davies, E. H. Linfield, and D. Ritchie, “Terahertz quantum cascade lasers,” Philos Trans A Math Phys Eng Sci 362(1815), 215–231 (2004).
[Crossref] [PubMed]

Bielsa, F.

Bismuto, A.

Blaser, S.

T. Aellen, R. Maulini, R. Terazzi, N. Hoyler, M. Giovannini, J. Faist, S. Blaser, and L. Hvozdara, “Direct measurement of the linewidth enhancement factor by optical heterodyning of an amplitude-modulated quantum cascade laser,” Appl. Phys. Lett. 89(9), 091121 (2006).
[Crossref]

Bodyfelt, J. D.

Bogris, A.

C. Juretzka, H. Simos, A. Bogris, D. Syvridis, W. Elsäßer, and M. Carras, “Intensity noise properties of midinfrared injection locked quantum cascade lasers: II. Experiments,” IEEE J. Quantum Electron. 51(1), 2300208 (2015).
[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]

Borri, S.

S. Borri, I. Galli, F. Cappelli, A. Bismuto, S. Bartalini, P. Cancio, G. Giusfredi, D. Mazzotti, J. Faist, and P. De Natale, “Direct link of a mid-infrared QCL to a frequency comb by optical injection,” Opt. Lett. 37(6), 1011–1013 (2012).
[Crossref] [PubMed]

S. Borri, S. Bartalini, P. C. Pastor, I. Galli, G. Giusfredi, D. Mazzotti, M. Yamanishi, and P. De Natale, “Natale, “Frequency-noise dynamics of mid-infrared quantum cascade lasers,” IEEE J. Quantum Electron. 47(7), 984–988 (2011).
[Crossref]

Cancio, P.

Cannon, B. D.

M. S. Taubman, T. L. Myers, B. D. Cannon, and R. M. Williams, “Stabilization, injection and control of quantum cascade lasers, and their application to chemical sensing in the infrared,” Spectrochim. Acta A Mol. Biomol. Spectrosc. 60(14), 3457–3468 (2004).
[Crossref] [PubMed]

Cappelli, F.

Carras, M.

C. Juretzka, H. Simos, A. Bogris, D. Syvridis, W. Elsäßer, and M. Carras, “Intensity noise properties of midinfrared injection locked quantum cascade lasers: II. Experiments,” IEEE J. Quantum Electron. 51(1), 2300208 (2015).
[Crossref]

Chabran, C.

P. Gallion, H. Nakajima, G. Debarge, and C. Chabran, “Contribution of spontaneous emission to the linewidth of an injection-locked semiconductor laser,” Electron. Lett. 21(14), 626–628 (1985).
[Crossref]

Chanteau, B.

B. Argence, B. Chanteau, O. Lopez, D. Nicolodi, M. Abgrall, C. Chardonnet, C. Daussy, B. Darquié, Y. Le Coq, and A. Amy-Klein, “Quantum cascade laser frequency stabilization at the sub-Hz level,” Nat. Photonics 9(7), 456–460 (2015).
[Crossref]

Chardonnet, C.

B. Argence, B. Chanteau, O. Lopez, D. Nicolodi, M. Abgrall, C. Chardonnet, C. Daussy, B. Darquié, Y. Le Coq, and A. Amy-Klein, “Quantum cascade laser frequency stabilization at the sub-Hz level,” Nat. Photonics 9(7), 456–460 (2015).
[Crossref]

Darquié, B.

B. Argence, B. Chanteau, O. Lopez, D. Nicolodi, M. Abgrall, C. Chardonnet, C. Daussy, B. Darquié, Y. Le Coq, and A. Amy-Klein, “Quantum cascade laser frequency stabilization at the sub-Hz level,” Nat. Photonics 9(7), 456–460 (2015).
[Crossref]

Daussy, C.

B. Argence, B. Chanteau, O. Lopez, D. Nicolodi, M. Abgrall, C. Chardonnet, C. Daussy, B. Darquié, Y. Le Coq, and A. Amy-Klein, “Quantum cascade laser frequency stabilization at the sub-Hz level,” Nat. Photonics 9(7), 456–460 (2015).
[Crossref]

Davies, A. G.

J. R. Freeman, L. Ponnampalam, H. Shams, R. A. Mohandas, C. C. Renaud, P. Dean, L. Li, A. G. Davies, A. J. Seeds, and E. H. Linfield, “Injection locking of a terahertz quantum cascade laser to a telecommunications wavelength frequency comb,” Optica 4(9), 1059–1064 (2017).
[Crossref]

J. Faist, L. Ajili, G. Scalari, M. Giovannini, M. Beck, M. Rochat, H. Beere, A. G. Davies, E. H. Linfield, and D. Ritchie, “Terahertz quantum cascade lasers,” Philos Trans A Math Phys Eng Sci 362(1815), 215–231 (2004).
[Crossref] [PubMed]

De Natale, P.

S. Borri, I. Galli, F. Cappelli, A. Bismuto, S. Bartalini, P. Cancio, G. Giusfredi, D. Mazzotti, J. Faist, and P. De Natale, “Direct link of a mid-infrared QCL to a frequency comb by optical injection,” Opt. Lett. 37(6), 1011–1013 (2012).
[Crossref] [PubMed]

S. Borri, S. Bartalini, P. C. Pastor, I. Galli, G. Giusfredi, D. Mazzotti, M. Yamanishi, and P. De Natale, “Natale, “Frequency-noise dynamics of mid-infrared quantum cascade lasers,” IEEE J. Quantum Electron. 47(7), 984–988 (2011).
[Crossref]

de Waardt, H.

G. Yabre, H. de Waardt, H. P. A. Van den Boom, and G.-D. Khoe, “Noise characteristics of single-mode semiconductor lasers under external light injection,” IEEE J. Quantum Electron. 36(3), 385–393 (2000).
[Crossref]

Dean, P.

Debarge, G.

P. Gallion, H. Nakajima, G. Debarge, and C. Chabran, “Contribution of spontaneous emission to the linewidth of an injection-locked semiconductor laser,” Electron. Lett. 21(14), 626–628 (1985).
[Crossref]

Destic, F.

Y. Petitjean, F. Destic, J. C. Mollier, and C. Sirtori, “Dynamic modeling of Terahertz quantum cascade lasers,” IEEE J. Sel. Top. Quantum Electron. 17(1), 22–29 (2011).
[Crossref]

Di Domenico, G.

Douillet, A.

Elsäßer, W.

C. Juretzka, H. Simos, A. Bogris, D. Syvridis, W. Elsäßer, and M. Carras, “Intensity noise properties of midinfrared injection locked quantum cascade lasers: II. Experiments,” IEEE J. Quantum Electron. 51(1), 2300208 (2015).
[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]

Elsässer, W.

Erneux, T.

T. Erneux, V. Kovanis, and A. Gavrielides, “Nonlinear dynamics of an injected quantum cascade laser,” Phys. Rev. E 88(3), 032907 (2013).
[Crossref] [PubMed]

Even, J.

C. Wang, F. Grillot, V. I. Kovanis, J. D. Bodyfelt, and J. Even, “Modulation properties of optically injection-locked quantum cascade lasers,” Opt. Lett. 38(11), 1975–1977 (2013).
[Crossref] [PubMed]

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.

S. Borri, I. Galli, F. Cappelli, A. Bismuto, S. Bartalini, P. Cancio, G. Giusfredi, D. Mazzotti, J. Faist, and P. De Natale, “Direct link of a mid-infrared QCL to a frequency comb by optical injection,” Opt. Lett. 37(6), 1011–1013 (2012).
[Crossref] [PubMed]

T. Aellen, R. Maulini, R. Terazzi, N. Hoyler, M. Giovannini, J. Faist, S. Blaser, and L. Hvozdara, “Direct measurement of the linewidth enhancement factor by optical heterodyning of an amplitude-modulated quantum cascade laser,” Appl. Phys. Lett. 89(9), 091121 (2006).
[Crossref]

J. Faist, L. Ajili, G. Scalari, M. Giovannini, M. Beck, M. Rochat, H. Beere, A. G. Davies, E. H. Linfield, and D. Ritchie, “Terahertz quantum cascade lasers,” Philos Trans A Math Phys Eng Sci 362(1815), 215–231 (2004).
[Crossref] [PubMed]

Freeman, J. R.

Galli, I.

S. Borri, I. Galli, F. Cappelli, A. Bismuto, S. Bartalini, P. Cancio, G. Giusfredi, D. Mazzotti, J. Faist, and P. De Natale, “Direct link of a mid-infrared QCL to a frequency comb by optical injection,” Opt. Lett. 37(6), 1011–1013 (2012).
[Crossref] [PubMed]

S. Borri, S. Bartalini, P. C. Pastor, I. Galli, G. Giusfredi, D. Mazzotti, M. Yamanishi, and P. De Natale, “Natale, “Frequency-noise dynamics of mid-infrared quantum cascade lasers,” IEEE J. Quantum Electron. 47(7), 984–988 (2011).
[Crossref]

Gallion, P.

P. Gallion, H. Nakajima, G. Debarge, and C. Chabran, “Contribution of spontaneous emission to the linewidth of an injection-locked semiconductor laser,” Electron. Lett. 21(14), 626–628 (1985).
[Crossref]

Gavrielides, A.

T. Erneux, V. Kovanis, and A. Gavrielides, “Nonlinear dynamics of an injected quantum cascade laser,” Phys. Rev. E 88(3), 032907 (2013).
[Crossref] [PubMed]

T. B. Simpson, J. M. Liu, and A. Gavrielides, “Bandwidth enhancement and broadband noise reduction in injection-locked semiconductor lasers,” IEEE Photonics Technol. Lett. 7(7), 709–711 (1995).
[Crossref]

Gensty, T.

Giovannini, M.

T. Aellen, R. Maulini, R. Terazzi, N. Hoyler, M. Giovannini, J. Faist, S. Blaser, and L. Hvozdara, “Direct measurement of the linewidth enhancement factor by optical heterodyning of an amplitude-modulated quantum cascade laser,” Appl. Phys. Lett. 89(9), 091121 (2006).
[Crossref]

J. Faist, L. Ajili, G. Scalari, M. Giovannini, M. Beck, M. Rochat, H. Beere, A. G. Davies, E. H. Linfield, and D. Ritchie, “Terahertz quantum cascade lasers,” Philos Trans A Math Phys Eng Sci 362(1815), 215–231 (2004).
[Crossref] [PubMed]

Giuliani, G.

Giusfredi, G.

S. Borri, I. Galli, F. Cappelli, A. Bismuto, S. Bartalini, P. Cancio, G. Giusfredi, D. Mazzotti, J. Faist, and P. De Natale, “Direct link of a mid-infrared QCL to a frequency comb by optical injection,” Opt. Lett. 37(6), 1011–1013 (2012).
[Crossref] [PubMed]

S. Borri, S. Bartalini, P. C. Pastor, I. Galli, G. Giusfredi, D. Mazzotti, M. Yamanishi, and P. De Natale, “Natale, “Frequency-noise dynamics of mid-infrared quantum cascade lasers,” IEEE J. Quantum Electron. 47(7), 984–988 (2011).
[Crossref]

Grillot, F.

X. G. Wang, F. Grillot, and C. Wang, “Rate equation modeling of the frequency noise and the intrinsic spectral linewidth in quantum cascade lasers,” Opt. Express 26(3), 2325–2334 (2018).
[Crossref] [PubMed]

C. Wang, K. Schires, M. Osiński, P. J. Poole, and F. Grillot, “Thermally insensitive determination of the linewidth broadening factor in nanostructured semiconductor lasers using optical injection locking,” Sci. Rep. 6(1), 27825 (2016).
[Crossref] [PubMed]

C. Wang, F. Grillot, V. I. Kovanis, J. D. Bodyfelt, and J. Even, “Modulation properties of optically injection-locked quantum cascade lasers,” Opt. Lett. 38(11), 1975–1977 (2013).
[Crossref] [PubMed]

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]

Haldar, M. K.

M. K. Haldar, “A simplified analysis of direct intensity modulation of quantum cascade laser,” IEEE J. Quantum Electron. 41(11), 1349–1355 (2005).
[Crossref]

Harder, C.

C. Harder, J. Katz, S. Margalit, J. Shacham, and A. Yariv, “Noise equivalent circuit of a semiconductor laser diode,” IEEE J. Quantum Electron. 18(3), 333–337 (1982).
[Crossref]

Henry, C.

C. Henry, “Theory of the linewidth of semiconductor lasers,” IEEE J. Quantum Electron. 18(2), 259–264 (1982).
[Crossref]

Higham, D. J.

D. J. Higham, “An algorithmic introduction to numerical simulation of stochastic differential equations,” SIAM Rev. 43(3), 525–546 (2001).
[Crossref]

Hilico, L.

Hoyler, N.

T. Aellen, R. Maulini, R. Terazzi, N. Hoyler, M. Giovannini, J. Faist, S. Blaser, and L. Hvozdara, “Direct measurement of the linewidth enhancement factor by optical heterodyning of an amplitude-modulated quantum cascade laser,” Appl. Phys. Lett. 89(9), 091121 (2006).
[Crossref]

Hvozdara, L.

T. Aellen, R. Maulini, R. Terazzi, N. Hoyler, M. Giovannini, J. Faist, S. Blaser, and L. Hvozdara, “Direct measurement of the linewidth enhancement factor by optical heterodyning of an amplitude-modulated quantum cascade laser,” Appl. Phys. Lett. 89(9), 091121 (2006).
[Crossref]

Jacobsen, G.

F. Mogensen, H. Olesen, and G. Jacobsen, “FM noise suppression and linewidth reduction in an injection-locked semiconductor laser,” Electron. Lett. 21(16), 696–697 (1985).
[Crossref]

Juretzka, C.

C. Juretzka, H. Simos, A. Bogris, D. Syvridis, W. Elsäßer, and M. Carras, “Intensity noise properties of midinfrared injection locked quantum cascade lasers: II. Experiments,” IEEE J. Quantum Electron. 51(1), 2300208 (2015).
[Crossref]

Karr, J. P.

Katz, J.

C. Harder, J. Katz, S. Margalit, J. Shacham, and A. Yariv, “Noise equivalent circuit of a semiconductor laser diode,” IEEE J. Quantum Electron. 18(3), 333–337 (1982).
[Crossref]

Kawashima, K.

A. Murakami, K. Kawashima, and K. Atsuki, “Cavity resonance shift and bandwidth enhancement in semiconductor lasers with strong light injection,” IEEE J. Quantum Electron. 39(10), 1196–1204 (2003).
[Crossref]

Khoe, G.-D.

G. Yabre, H. de Waardt, H. P. A. Van den Boom, and G.-D. Khoe, “Noise characteristics of single-mode semiconductor lasers under external light injection,” IEEE J. Quantum Electron. 36(3), 385–393 (2000).
[Crossref]

Kovanis, V.

T. Erneux, V. Kovanis, and A. Gavrielides, “Nonlinear dynamics of an injected quantum cascade laser,” Phys. Rev. E 88(3), 032907 (2013).
[Crossref] [PubMed]

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]

Kovanis, V. I.

Krauskopf, B.

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

Lang, R.

R. Lang, “Injection locking properties of a semiconductor laser,” IEEE J. Quantum Electron. 18(6), 976–983 (1982).
[Crossref]

Lau, E. K.

E. K. Lau, L. J. Wong, and M. C. Wu, “Enhanced Modulation Characteristics of Optical Injection-Locked Lasers: A Tutorial,” IEEE J. Sel. Top. Quantum Electron. 15(3), 618–633 (2009).
[Crossref]

Le Coq, Y.

B. Argence, B. Chanteau, O. Lopez, D. Nicolodi, M. Abgrall, C. Chardonnet, C. Daussy, B. Darquié, Y. Le Coq, and A. Amy-Klein, “Quantum cascade laser frequency stabilization at the sub-Hz level,” Nat. Photonics 9(7), 456–460 (2015).
[Crossref]

Lee, K. E.

Lenstra, D.

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

Li, L.

Linfield, E. H.

J. R. Freeman, L. Ponnampalam, H. Shams, R. A. Mohandas, C. C. Renaud, P. Dean, L. Li, A. G. Davies, A. J. Seeds, and E. H. Linfield, “Injection locking of a terahertz quantum cascade laser to a telecommunications wavelength frequency comb,” Optica 4(9), 1059–1064 (2017).
[Crossref]

J. Faist, L. Ajili, G. Scalari, M. Giovannini, M. Beck, M. Rochat, H. Beere, A. G. Davies, E. H. Linfield, and D. Ritchie, “Terahertz quantum cascade lasers,” Philos Trans A Math Phys Eng Sci 362(1815), 215–231 (2004).
[Crossref] [PubMed]

Liu, J. M.

T. B. Simpson, J. M. Liu, and A. Gavrielides, “Bandwidth enhancement and broadband noise reduction in injection-locked semiconductor lasers,” IEEE Photonics Technol. Lett. 7(7), 709–711 (1995).
[Crossref]

Liu, T.

Lopez, O.

B. Argence, B. Chanteau, O. Lopez, D. Nicolodi, M. Abgrall, C. Chardonnet, C. Daussy, B. Darquié, Y. Le Coq, and A. Amy-Klein, “Quantum cascade laser frequency stabilization at the sub-Hz level,” Nat. Photonics 9(7), 456–460 (2015).
[Crossref]

Mann, C.

Margalit, S.

C. Harder, J. Katz, S. Margalit, J. Shacham, and A. Yariv, “Noise equivalent circuit of a semiconductor laser diode,” IEEE J. Quantum Electron. 18(3), 333–337 (1982).
[Crossref]

Maulini, R.

T. Aellen, R. Maulini, R. Terazzi, N. Hoyler, M. Giovannini, J. Faist, S. Blaser, and L. Hvozdara, “Direct measurement of the linewidth enhancement factor by optical heterodyning of an amplitude-modulated quantum cascade laser,” Appl. Phys. Lett. 89(9), 091121 (2006).
[Crossref]

Mazzotti, D.

S. Borri, I. Galli, F. Cappelli, A. Bismuto, S. Bartalini, P. Cancio, G. Giusfredi, D. Mazzotti, J. Faist, and P. De Natale, “Direct link of a mid-infrared QCL to a frequency comb by optical injection,” Opt. Lett. 37(6), 1011–1013 (2012).
[Crossref] [PubMed]

S. Borri, S. Bartalini, P. C. Pastor, I. Galli, G. Giusfredi, D. Mazzotti, M. Yamanishi, and P. De Natale, “Natale, “Frequency-noise dynamics of mid-infrared quantum cascade lasers,” IEEE J. Quantum Electron. 47(7), 984–988 (2011).
[Crossref]

Meng, B.

Mogensen, F.

S. Saito, F. Mogensen, and H. Olesen, “Effective bandwidth for FM noise suppression in an injection-locked semiconductor laser,” Electron. Lett. 21(24), 1173–1175 (1985).
[Crossref]

F. Mogensen, H. Olesen, and G. Jacobsen, “FM noise suppression and linewidth reduction in an injection-locked semiconductor laser,” Electron. Lett. 21(16), 696–697 (1985).
[Crossref]

Mohandas, R. A.

Mollier, J. C.

Y. Petitjean, F. Destic, J. C. Mollier, and C. Sirtori, “Dynamic modeling of Terahertz quantum cascade lasers,” IEEE J. Sel. Top. Quantum Electron. 17(1), 22–29 (2011).
[Crossref]

Murakami, A.

A. Murakami, K. Kawashima, and K. Atsuki, “Cavity resonance shift and bandwidth enhancement in semiconductor lasers with strong light injection,” IEEE J. Quantum Electron. 39(10), 1196–1204 (2003).
[Crossref]

Myers, T. L.

M. S. Taubman, T. L. Myers, B. D. Cannon, and R. M. Williams, “Stabilization, injection and control of quantum cascade lasers, and their application to chemical sensing in the infrared,” Spectrochim. Acta A Mol. Biomol. Spectrosc. 60(14), 3457–3468 (2004).
[Crossref] [PubMed]

Nakajima, H.

P. Gallion, H. Nakajima, G. Debarge, and C. Chabran, “Contribution of spontaneous emission to the linewidth of an injection-locked semiconductor laser,” Electron. Lett. 21(14), 626–628 (1985).
[Crossref]

Nicolodi, D.

B. Argence, B. Chanteau, O. Lopez, D. Nicolodi, M. Abgrall, C. Chardonnet, C. Daussy, B. Darquié, Y. Le Coq, and A. Amy-Klein, “Quantum cascade laser frequency stabilization at the sub-Hz level,” Nat. Photonics 9(7), 456–460 (2015).
[Crossref]

Nilsson, O.

S. Saito, O. Nilsson, and Y. Yamamoto, “Frequency modulation noise and linewidth reduction in a semiconductor laser by means of a negative frequency feedback technique,” Appl. Phys. Lett. 46(1), 3–5 (1985).
[Crossref]

Olesen, H.

F. Mogensen, H. Olesen, and G. Jacobsen, “FM noise suppression and linewidth reduction in an injection-locked semiconductor laser,” Electron. Lett. 21(16), 696–697 (1985).
[Crossref]

S. Saito, F. Mogensen, and H. Olesen, “Effective bandwidth for FM noise suppression in an injection-locked semiconductor laser,” Electron. Lett. 21(24), 1173–1175 (1985).
[Crossref]

Osinski, M.

C. Wang, K. Schires, M. Osiński, P. J. Poole, and F. Grillot, “Thermally insensitive determination of the linewidth broadening factor in nanostructured semiconductor lasers using optical injection locking,” Sci. Rep. 6(1), 27825 (2016).
[Crossref] [PubMed]

Pastor, P. C.

S. Borri, S. Bartalini, P. C. Pastor, I. Galli, G. Giusfredi, D. Mazzotti, M. Yamanishi, and P. De Natale, “Natale, “Frequency-noise dynamics of mid-infrared quantum cascade lasers,” IEEE J. Quantum Electron. 47(7), 984–988 (2011).
[Crossref]

Pereira, M. F.

M. F. Pereira, “The linewidth enhancement factor of intersubband lasers: From a two-level limit to gain without inversion conditions,” Appl. Phys. Lett. 109(22), 222102 (2016).
[Crossref]

Petitjean, Y.

Y. Petitjean, F. Destic, J. C. Mollier, and C. Sirtori, “Dynamic modeling of Terahertz quantum cascade lasers,” IEEE J. Sel. Top. Quantum Electron. 17(1), 22–29 (2011).
[Crossref]

Ponnampalam, L.

Poole, P. J.

C. Wang, K. Schires, M. Osiński, P. J. Poole, and F. Grillot, “Thermally insensitive determination of the linewidth broadening factor in nanostructured semiconductor lasers using optical injection locking,” Sci. Rep. 6(1), 27825 (2016).
[Crossref] [PubMed]

Ram, R. J.

F. Rana and R. J. Ram, “Current noise and photon noise in quantum cascade lasers,” Phys. Rev. B 65(12), 125313 (2002).
[Crossref]

Rana, F.

F. Rana and R. J. Ram, “Current noise and photon noise in quantum cascade lasers,” Phys. Rev. B 65(12), 125313 (2002).
[Crossref]

Renaud, C. C.

Ritchie, D.

J. Faist, L. Ajili, G. Scalari, M. Giovannini, M. Beck, M. Rochat, H. Beere, A. G. Davies, E. H. Linfield, and D. Ritchie, “Terahertz quantum cascade lasers,” Philos Trans A Math Phys Eng Sci 362(1815), 215–231 (2004).
[Crossref] [PubMed]

Rochat, M.

J. Faist, L. Ajili, G. Scalari, M. Giovannini, M. Beck, M. Rochat, H. Beere, A. G. Davies, E. H. Linfield, and D. Ritchie, “Terahertz quantum cascade lasers,” Philos Trans A Math Phys Eng Sci 362(1815), 215–231 (2004).
[Crossref] [PubMed]

Saito, S.

S. Saito, O. Nilsson, and Y. Yamamoto, “Frequency modulation noise and linewidth reduction in a semiconductor laser by means of a negative frequency feedback technique,” Appl. Phys. Lett. 46(1), 3–5 (1985).
[Crossref]

S. Saito, F. Mogensen, and H. Olesen, “Effective bandwidth for FM noise suppression in an injection-locked semiconductor laser,” Electron. Lett. 21(24), 1173–1175 (1985).
[Crossref]

Scalari, G.

J. Faist, L. Ajili, G. Scalari, M. Giovannini, M. Beck, M. Rochat, H. Beere, A. G. Davies, E. H. Linfield, and D. Ritchie, “Terahertz quantum cascade lasers,” Philos Trans A Math Phys Eng Sci 362(1815), 215–231 (2004).
[Crossref] [PubMed]

Schilt, S.

Schires, K.

C. Wang, K. Schires, M. Osiński, P. J. Poole, and F. Grillot, “Thermally insensitive determination of the linewidth broadening factor in nanostructured semiconductor lasers using optical injection locking,” Sci. Rep. 6(1), 27825 (2016).
[Crossref] [PubMed]

Seeds, A. J.

Shacham, J.

C. Harder, J. Katz, S. Margalit, J. Shacham, and A. Yariv, “Noise equivalent circuit of a semiconductor laser diode,” IEEE J. Quantum Electron. 18(3), 333–337 (1982).
[Crossref]

Shams, H.

Simos, H.

C. Juretzka, H. Simos, A. Bogris, D. Syvridis, W. Elsäßer, and M. Carras, “Intensity noise properties of midinfrared injection locked quantum cascade lasers: II. Experiments,” IEEE J. Quantum Electron. 51(1), 2300208 (2015).
[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]

Simpson, T. B.

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

T. B. Simpson, J. M. Liu, and A. Gavrielides, “Bandwidth enhancement and broadband noise reduction in injection-locked semiconductor lasers,” IEEE Photonics Technol. Lett. 7(7), 709–711 (1995).
[Crossref]

Sirtori, C.

Y. Petitjean, F. Destic, J. C. Mollier, and C. Sirtori, “Dynamic modeling of Terahertz quantum cascade lasers,” IEEE J. Sel. Top. Quantum Electron. 17(1), 22–29 (2011).
[Crossref]

C. Sirtori, “GaAs Quantum cascade laser: fundamentals and performances,” EDP sciences 7(3), 1– 20 (2002).

Syvridis, D.

C. Juretzka, H. Simos, A. Bogris, D. Syvridis, W. Elsäßer, and M. Carras, “Intensity noise properties of midinfrared injection locked quantum cascade lasers: II. Experiments,” IEEE J. Quantum Electron. 51(1), 2300208 (2015).
[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]

Taubman, M. S.

M. S. Taubman, T. L. Myers, B. D. Cannon, and R. M. Williams, “Stabilization, injection and control of quantum cascade lasers, and their application to chemical sensing in the infrared,” Spectrochim. Acta A Mol. Biomol. Spectrosc. 60(14), 3457–3468 (2004).
[Crossref] [PubMed]

Terazzi, R.

T. Aellen, R. Maulini, R. Terazzi, N. Hoyler, M. Giovannini, J. Faist, S. Blaser, and L. Hvozdara, “Direct measurement of the linewidth enhancement factor by optical heterodyning of an amplitude-modulated quantum cascade laser,” Appl. Phys. Lett. 89(9), 091121 (2006).
[Crossref]

Thomann, P.

Valenzuela, T.

Van den Boom, H. P. A.

G. Yabre, H. de Waardt, H. P. A. Van den Boom, and G.-D. Khoe, “Noise characteristics of single-mode semiconductor lasers under external light injection,” IEEE J. Quantum Electron. 36(3), 385–393 (2000).
[Crossref]

von Staden, J.

Wang, C.

X. G. Wang, F. Grillot, and C. Wang, “Rate equation modeling of the frequency noise and the intrinsic spectral linewidth in quantum cascade lasers,” Opt. Express 26(3), 2325–2334 (2018).
[Crossref] [PubMed]

C. Wang, K. Schires, M. Osiński, P. J. Poole, and F. Grillot, “Thermally insensitive determination of the linewidth broadening factor in nanostructured semiconductor lasers using optical injection locking,” Sci. Rep. 6(1), 27825 (2016).
[Crossref] [PubMed]

C. Wang, F. Grillot, V. I. Kovanis, J. D. Bodyfelt, and J. Even, “Modulation properties of optically injection-locked quantum cascade lasers,” Opt. Lett. 38(11), 1975–1977 (2013).
[Crossref] [PubMed]

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]

Wang, Q. J.

Wang, X. G.

Wieczorek, S.

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

Williams, R. M.

M. S. Taubman, T. L. Myers, B. D. Cannon, and R. M. Williams, “Stabilization, injection and control of quantum cascade lasers, and their application to chemical sensing in the infrared,” Spectrochim. Acta A Mol. Biomol. Spectrosc. 60(14), 3457–3468 (2004).
[Crossref] [PubMed]

Wong, L. J.

E. K. Lau, L. J. Wong, and M. C. Wu, “Enhanced Modulation Characteristics of Optical Injection-Locked Lasers: A Tutorial,” IEEE J. Sel. Top. Quantum Electron. 15(3), 618–633 (2009).
[Crossref]

Wu, M. C.

E. K. Lau, L. J. Wong, and M. C. Wu, “Enhanced Modulation Characteristics of Optical Injection-Locked Lasers: A Tutorial,” IEEE J. Sel. Top. Quantum Electron. 15(3), 618–633 (2009).
[Crossref]

Yabre, G.

G. Yabre, H. de Waardt, H. P. A. Van den Boom, and G.-D. Khoe, “Noise characteristics of single-mode semiconductor lasers under external light injection,” IEEE J. Quantum Electron. 36(3), 385–393 (2000).
[Crossref]

Yamamoto, Y.

S. Saito, O. Nilsson, and Y. Yamamoto, “Frequency modulation noise and linewidth reduction in a semiconductor laser by means of a negative frequency feedback technique,” Appl. Phys. Lett. 46(1), 3–5 (1985).
[Crossref]

Yamanishi, M.

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]

S. Borri, S. Bartalini, P. C. Pastor, I. Galli, G. Giusfredi, D. Mazzotti, M. Yamanishi, and P. De Natale, “Natale, “Frequency-noise dynamics of mid-infrared quantum cascade lasers,” IEEE J. Quantum Electron. 47(7), 984–988 (2011).
[Crossref]

Yariv, A.

C. Harder, J. Katz, S. Margalit, J. Shacham, and A. Yariv, “Noise equivalent circuit of a semiconductor laser diode,” IEEE J. Quantum Electron. 18(3), 333–337 (1982).
[Crossref]

Appl. Opt. (1)

Appl. Phys. Lett. (3)

T. Aellen, R. Maulini, R. Terazzi, N. Hoyler, M. Giovannini, J. Faist, S. Blaser, and L. Hvozdara, “Direct measurement of the linewidth enhancement factor by optical heterodyning of an amplitude-modulated quantum cascade laser,” Appl. Phys. Lett. 89(9), 091121 (2006).
[Crossref]

M. F. Pereira, “The linewidth enhancement factor of intersubband lasers: From a two-level limit to gain without inversion conditions,” Appl. Phys. Lett. 109(22), 222102 (2016).
[Crossref]

S. Saito, O. Nilsson, and Y. Yamamoto, “Frequency modulation noise and linewidth reduction in a semiconductor laser by means of a negative frequency feedback technique,” Appl. Phys. Lett. 46(1), 3–5 (1985).
[Crossref]

EDP sciences (1)

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Figures (7)

Fig. 1
Fig. 1 Stable locking boundaries of the QCL for different LBFs.
Fig. 2
Fig. 2 Injection strength effect on the photon number at zero detuning. The dashed line indicates that of the free-running laser.
Fig. 3
Fig. 3 (a) Injection strength effect on the FN at zero detuning for αH = 0.5. The dashed curve is –3 dB of the free-running laser’s FN, and the arrow indicates the suppression bandwidth. (b) The suppression ratio of the FN peak (squares) and the suppression bandwidth (circles) as a function of the injection ratio.
Fig. 4
Fig. 4 Frequency detuning effect on the photon number at Rinj = 0 dB. The dashed line indicates that of the free-running laser.
Fig. 5
Fig. 5 Frequency detuning effect on the FN for (a) αH = 0, (b) αH = 0.5, and for (c) αH = 2.0 at Rinj = 0 dB. The dashed curves indicate that of the free-running laser.
Fig. 6
Fig. 6 (a) FN peak suppression ratio (in dB), the dashed line indicates the maximum suppression ratio, and the thick-white curve highlights the 0-dB suppression ratio. (b) FN suppression bandwidth (in GHz), the dashed line indicates the maximum suppression bandwidth. The LBF is set at αH = 0.5.
Fig. 7
Fig. 7 Simulated optical spectra of the injection-locked QCL at Rinj = 0 dB for (a) αΗ = 0, (b) αΗ = 0.5, and (b) αΗ = 2.0. The optical power is normalized to that of the spectral peak. The arrows indicate the side peaks.

Tables (1)

Tables Icon

Table 1 QCL material and optical parameters [23–27]

Equations (16)

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d N 3 dt =η I q N 3 τ 32 N 3 τ 31 G 0 ΔNS+ F 3 (t)
d N 2 dt = N 3 τ 32 N 2 τ 21 + G 0 ΔNS+ F 2 (t)
d N 1 dt = N 3 τ 31 + N 2 τ 21 N 1 τ out + F 1 (t)
dS dt =( m G 0 ΔN 1 τ p )S+mβ N 3 τ sp +2 k c S S inj cosϕ+ F S (t)
dϕ dt = α H 2 ( m G 0 ΔN 1 τ p )2πΔ F inj k c S inj S sinϕ+ F ϕ (t)
F i (t) F j (t') = U i j δ(tt')
U 33 =2( G 0 N 3 S+ N 3 / τ 32 + N 3 / τ 31 ); U 22 =2( G 0 N 3 S+ N 3 / τ 32 ) U SS =2m( G 0 N 3 S+β N 3 / τ sp ); U ϕϕ =2m( G 0 N 3 S+β N 3 / τ sp )/(4 S 2 ) U 32 =( G 0 N 3 S+ G 0 N 2 S+ N 3 / τ 32 ); U 3S =( G 0 N 3 S+ G 0 N 2 S+β N 3 / τ sp ) U 2S =( G 0 N 3 S+ G 0 N 2 S+β N 3 / τ sp ); U 3ϕ = U 2ϕ = U Sϕ =0
S= k c 2 R inj S 0 1 4 ( 1 τ p m G 0 ΔN ) 2 + [ α H 2 ( m G 0 ΔN 1 τ p )2πΔ F inj ] 2
ΔN= 1 m G 0 ( 1 τ p 2 k c R inj S 0 S cosϕ )
ϕ= sin 1 ( 2πΔ F inj k c 1+ α H 2 S R inj S 0 ) tan 1 α H
FN(f)= f 2 | δφ(f) | 2
FN(f0)=0
FN(f)= m( G 0 N 3 S+ β N 3 / τ sp ) 8 π 2 S 2 m G 0 N 3 8 π 2 S
N 3 (t)= N 3 0 +δ n 3 (ω) e jωt ; N 2 (t)= N 2 0 +δ n 2 (ω) e jωt , N 1 (t)= N 1 0 +δ n 1 (ω) e jωt , S(t)= S 0 +δs(ω) e jωt , ϕ(t)= ϕ 0 +δφ(ω) e jωt .
[ jω+ γ 11 γ 12 0 γ 14 0 γ 21 jω+ γ 22 0 γ 24 0 γ 31 γ 32 jω 0 0 γ 41 γ 42 0 jω+ γ 44 γ 45 γ 51 γ 52 0 γ 54 jω+ γ 55 ][ δ n 3 (ω) δ n 2 (ω) δ n 1 (ω) δs(ω) δϕ(ω) ]=[ F 3 F 2 F 1 F s F ϕ ]
γ 11 = τ 32 1 + τ 31 1 + G 0 S 0 , γ 12 = G 0 S 0 , γ 14 = G 0 Δ N 0 , γ 21 = τ 32 1 + G 0 S 0 , γ 22 = τ 21 1 + G 0 S 0 , γ 24 = G 0 Δ N 0 , γ 31 = τ 31 1 , γ 32 = τ 21 1 , γ 41 =m( β τ sp 1 + G 0 S 0 ), γ 42 =m G 0 S 0 , γ 44 = τ p 1 m G 0 Δ N 0 k c S inj / S 0 cos ϕ 0 , γ 45 =2 k c S inj S 0 sin ϕ 0 , γ 51 = α H m G 0 /2 , γ 52 = α H m G 0 /2 , γ 54 = k c S inj / S 0 sinϕ/ 2 S 0 , γ 55 = k c S inj / S 0 cos ϕ 0 .

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