Abstract

The frequency-noise power spectral density of a room-temperature distributed-feedback quantum cascade laser emitting at λ = 4.36 μm has been measured. An intrinsic linewidth value of 260 Hz is retrieved, in reasonable agreement with theoretical calculations. A noise reduction of about a factor 200 in most of the frequency interval is also found, with respect to a cryogenic laser at the same wavelength. A quantitative treatment shows that it can be explained by a temperature-dependent mechanism governing the transport processes in resonant tunnelling devices. This confirms the predominant effect of the heterostructure in determining shape and magnitude of the frequency noise spectrum in QCLs.

© 2011 OSA

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  1. S. Bartalini, S. Borri, P. Cancio, A. Castrillo, I. Galli, G. Giusfredi, D. Mazzotti, L. Gianfrani, and P. De Natale, “Observing the intrinsic linewidth of a quantum-cascade laser: beyond the Schawlow–Townes limit,” Phys. Rev. Lett. 104, 083904 (2010).
    [CrossRef] [PubMed]
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    [CrossRef]
  10. K. Fujita, S. Furuta, A. Sugiyama, T. Ochiai, T. Edamura, N. Akikusa, M. Yamanishi, and H. Kan, “Room temperature, continuous-wave operation of quantum cascade lasers with single phonon resonance-continuum depopulation structures grown by metal organic vapor-phase epitaxy,” Appl. Phys. Lett. 91, 141121 (2007).
    [CrossRef]
  11. K. Fujita, S. Furuta, A. Sugiyama, T. Ochiai, T. Edamura, N. Akikusa, M. Yamanishi, and H. Kan, “High-performance λ ∼ 8.6 μm quantum cascade lasers with single phonon-continuum depopulation structures,” IEEE J. Quantum Electron. 46, 683–688 (2010).
    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef] [PubMed]
  21. S. Bartalini, S. Borri, and P. De Natale, “Doppler-free polarization spectroscopy with a quantum cascade laser at 4.3 μm,” Opt. Express 17, 7440–7449 (2009).
    [CrossRef] [PubMed]
  22. M. S. Taubman, T. L. Myers, B. D. Cannon, R. M. Williams, F. Capasso, C. Gmachl, D. L. Sivco, and A. Y. Cho , “Frequency stabilization of quantum cascade lasers by use of optical cavities” Opt. Lett. 27, 2164–2166 (2002).
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2011 (1)

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

2010 (4)

G. Giusfredi, S. Bartalini, S. Borri, P. Cancio, I. Galli, D. Mazzotti, and P. De Natale, “Saturated-absorption cavity ring-down spectroscopy,” Phys. Rev. Lett. 104, 110801 (2010).
[CrossRef] [PubMed]

S. Bartalini, S. Borri, P. Cancio, A. Castrillo, I. Galli, G. Giusfredi, D. Mazzotti, L. Gianfrani, and P. De Natale, “Observing the intrinsic linewidth of a quantum-cascade laser: beyond the Schawlow–Townes limit,” Phys. Rev. Lett. 104, 083904 (2010).
[CrossRef] [PubMed]

K. Fujita, S. Furuta, A. Sugiyama, T. Ochiai, T. Edamura, N. Akikusa, M. Yamanishi, and H. Kan, “High-performance λ ∼ 8.6 μm quantum cascade lasers with single phonon-continuum depopulation structures,” IEEE J. Quantum Electron. 46, 683–688 (2010).
[CrossRef]

G. Di Domenico, S. Schilt, and P. Thomann, “Simple approach to the relation between laser frequency noise and laser line shape,” Appl. Opt. 49, 4801–4807 (2010).
[CrossRef] [PubMed]

2009 (2)

S. Bartalini, S. Borri, and P. De Natale, “Doppler-free polarization spectroscopy with a quantum cascade laser at 4.3 μm,” Opt. Express 17, 7440–7449 (2009).
[CrossRef] [PubMed]

K. Fujita, T. Edamura, N. Akikusa, A. Sugiyama, T. Ochiai, S. Furuta, A. Ito, M. Yamanishi, and H. Kan, “Quantum cascade lasers based on single phonon-continuum depopulation structures,” Proc. SPIE 7230, 723016 (2009).
[CrossRef]

2008 (2)

M. S. Vitiello, G. Scamarcio, and V. Spagnolo, “Temperature dependence of thermal conductivity and boundary resistance in THz quantum cascade lasers,” IEEE J. Select Top. Quantum Electron. 14, 431–435 (2008).
[CrossRef]

M. Yamanishi, T. Edamura, K. Fujita, N. Akikusa, and H. Kan, “Theory of the intrinsic linewidth of quantum-cascade lasers: hidden reason for the narrow linewidth and line-broadening by thermal photons,” IEEE J. Quantum Electron. 44, 12–29 (2008).
[CrossRef]

2007 (3)

M. S. Vitiello, V. Spagnolo, G. Scamarcio, A. Lops, Q. Yang, C. Manz, and J. Wagner, “Experimental investigation of the lattice and electronic temperatures in Ga0.47In0.53As/Al0.62Ga0.38As1–xSbx quantum-cascade lasers,” Appl. Phys. Lett. 90, 121109 (2007).
[CrossRef]

K. Fujita, S. Furuta, A. Sugiyama, T. Ochiai, T. Edamura, N. Akikusa, M. Yamanishi, and H. Kan, “Room temperature, continuous-wave operation of quantum cascade lasers with single phonon resonance-continuum depopulation structures grown by metal organic vapor-phase epitaxy,” Appl. Phys. Lett. 91, 141121 (2007).
[CrossRef]

M. S. Vitiello, T. Gresch, A. Lops, V. Spagnolo, G. Scamarcio, N. Hoyler, M. Giovannini, and J. Faist, “Influence of InAs, AlAs δ layers on the optical, electronic, and thermal characteristics of strain-compensated GaInAs/AlInAs quantum-cascade lasers,” Appl. Phys. Lett. 91, 161111 (2007).
[CrossRef]

2006 (1)

A. Vasanelli, A. Leuliet, C. Sirtori, A. Wade, G. Fedorov, D. Smirnov, G. Bastard, B. Vinter, M. Giovannini, and J. Faist, “Role of elastic scattering mechanisms in GaInAs/AlInAs quantum cascade lasers,” Appl. Phys. Lett.,  89, 172120 (2006).
[CrossRef]

2003 (1)

J. S. Yu, S. Slivken, A. Evans, L. Doris, and M. Razeghi, “High-power continuous-wave operation of a 6 μm quantum-cascade laser at room temperature,” Appl. Phys. Lett. 83, 2503–2505 (2003).
[CrossRef]

2002 (2)

2001 (1)

D. Hofstetter, M. Beck, T. Aellen, and J. Faist, “High-temperature operation of distributed feedback quantum-cascade lasers at 5.3 μm,” Appl. Phys. Lett. 78, 396–398 (2001).
[CrossRef]

1999 (1)

1993 (1)

K. G. Libbrecht and J. L. Hall, “A low-noise high-speed diode laser current controller,” Rev. Sci. Instrum. 64, 2133–2135 (1993).
[CrossRef]

1982 (2)

D. S. Elliott, R. Roy, and S. J. Smith, “Extracavity laser band-shape and bandwidth modification,” Phys. Rev. A 26, 12–18 (1982).
[CrossRef]

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

Aellen, T.

D. Hofstetter, M. Beck, T. Aellen, and J. Faist, “High-temperature operation of distributed feedback quantum-cascade lasers at 5.3 μm,” Appl. Phys. Lett. 78, 396–398 (2001).
[CrossRef]

Akikusa, N.

K. Fujita, S. Furuta, A. Sugiyama, T. Ochiai, T. Edamura, N. Akikusa, M. Yamanishi, and H. Kan, “High-performance λ ∼ 8.6 μm quantum cascade lasers with single phonon-continuum depopulation structures,” IEEE J. Quantum Electron. 46, 683–688 (2010).
[CrossRef]

K. Fujita, T. Edamura, N. Akikusa, A. Sugiyama, T. Ochiai, S. Furuta, A. Ito, M. Yamanishi, and H. Kan, “Quantum cascade lasers based on single phonon-continuum depopulation structures,” Proc. SPIE 7230, 723016 (2009).
[CrossRef]

M. Yamanishi, T. Edamura, K. Fujita, N. Akikusa, and H. Kan, “Theory of the intrinsic linewidth of quantum-cascade lasers: hidden reason for the narrow linewidth and line-broadening by thermal photons,” IEEE J. Quantum Electron. 44, 12–29 (2008).
[CrossRef]

K. Fujita, S. Furuta, A. Sugiyama, T. Ochiai, T. Edamura, N. Akikusa, M. Yamanishi, and H. Kan, “Room temperature, continuous-wave operation of quantum cascade lasers with single phonon resonance-continuum depopulation structures grown by metal organic vapor-phase epitaxy,” Appl. Phys. Lett. 91, 141121 (2007).
[CrossRef]

Baillargeon, J. N.

Bartalini, S.

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

G. Giusfredi, S. Bartalini, S. Borri, P. Cancio, I. Galli, D. Mazzotti, and P. De Natale, “Saturated-absorption cavity ring-down spectroscopy,” Phys. Rev. Lett. 104, 110801 (2010).
[CrossRef] [PubMed]

S. Bartalini, S. Borri, P. Cancio, A. Castrillo, I. Galli, G. Giusfredi, D. Mazzotti, L. Gianfrani, and P. De Natale, “Observing the intrinsic linewidth of a quantum-cascade laser: beyond the Schawlow–Townes limit,” Phys. Rev. Lett. 104, 083904 (2010).
[CrossRef] [PubMed]

S. Bartalini, S. Borri, and P. De Natale, “Doppler-free polarization spectroscopy with a quantum cascade laser at 4.3 μm,” Opt. Express 17, 7440–7449 (2009).
[CrossRef] [PubMed]

Bastard, G.

A. Vasanelli, A. Leuliet, C. Sirtori, A. Wade, G. Fedorov, D. Smirnov, G. Bastard, B. Vinter, M. Giovannini, and J. Faist, “Role of elastic scattering mechanisms in GaInAs/AlInAs quantum cascade lasers,” Appl. Phys. Lett.,  89, 172120 (2006).
[CrossRef]

Beck, M.

D. Hofstetter, M. Beck, T. Aellen, and J. Faist, “High-temperature operation of distributed feedback quantum-cascade lasers at 5.3 μm,” Appl. Phys. Lett. 78, 396–398 (2001).
[CrossRef]

Borri, S.

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

S. Bartalini, S. Borri, P. Cancio, A. Castrillo, I. Galli, G. Giusfredi, D. Mazzotti, L. Gianfrani, and P. De Natale, “Observing the intrinsic linewidth of a quantum-cascade laser: beyond the Schawlow–Townes limit,” Phys. Rev. Lett. 104, 083904 (2010).
[CrossRef] [PubMed]

G. Giusfredi, S. Bartalini, S. Borri, P. Cancio, I. Galli, D. Mazzotti, and P. De Natale, “Saturated-absorption cavity ring-down spectroscopy,” Phys. Rev. Lett. 104, 110801 (2010).
[CrossRef] [PubMed]

S. Bartalini, S. Borri, and P. De Natale, “Doppler-free polarization spectroscopy with a quantum cascade laser at 4.3 μm,” Opt. Express 17, 7440–7449 (2009).
[CrossRef] [PubMed]

Cancio, P.

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

G. Giusfredi, S. Bartalini, S. Borri, P. Cancio, I. Galli, D. Mazzotti, and P. De Natale, “Saturated-absorption cavity ring-down spectroscopy,” Phys. Rev. Lett. 104, 110801 (2010).
[CrossRef] [PubMed]

S. Bartalini, S. Borri, P. Cancio, A. Castrillo, I. Galli, G. Giusfredi, D. Mazzotti, L. Gianfrani, and P. De Natale, “Observing the intrinsic linewidth of a quantum-cascade laser: beyond the Schawlow–Townes limit,” Phys. Rev. Lett. 104, 083904 (2010).
[CrossRef] [PubMed]

Cannon, B. D.

Capasso, F.

Castrillo, A.

S. Bartalini, S. Borri, P. Cancio, A. Castrillo, I. Galli, G. Giusfredi, D. Mazzotti, L. Gianfrani, and P. De Natale, “Observing the intrinsic linewidth of a quantum-cascade laser: beyond the Schawlow–Townes limit,” Phys. Rev. Lett. 104, 083904 (2010).
[CrossRef] [PubMed]

Cho, A. Y.

De Natale, P.

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

G. Giusfredi, S. Bartalini, S. Borri, P. Cancio, I. Galli, D. Mazzotti, and P. De Natale, “Saturated-absorption cavity ring-down spectroscopy,” Phys. Rev. Lett. 104, 110801 (2010).
[CrossRef] [PubMed]

S. Bartalini, S. Borri, P. Cancio, A. Castrillo, I. Galli, G. Giusfredi, D. Mazzotti, L. Gianfrani, and P. De Natale, “Observing the intrinsic linewidth of a quantum-cascade laser: beyond the Schawlow–Townes limit,” Phys. Rev. Lett. 104, 083904 (2010).
[CrossRef] [PubMed]

S. Bartalini, S. Borri, and P. De Natale, “Doppler-free polarization spectroscopy with a quantum cascade laser at 4.3 μm,” Opt. Express 17, 7440–7449 (2009).
[CrossRef] [PubMed]

Di Domenico, G.

Doris, L.

J. S. Yu, S. Slivken, A. Evans, L. Doris, and M. Razeghi, “High-power continuous-wave operation of a 6 μm quantum-cascade laser at room temperature,” Appl. Phys. Lett. 83, 2503–2505 (2003).
[CrossRef]

Edamura, T.

K. Fujita, M. Yamanishi, T. Edamura, A. Sugiyama, and S. Furuta, “Extremely high T0-values (≃450 K) of long-wavelength (≃ 15 μm), low-threshold-current-density quantum-cascade lasers based on the indirect pump scheme,” Appl. Phys. Lett. 97, 201109 (2010).

K. Fujita, S. Furuta, A. Sugiyama, T. Ochiai, T. Edamura, N. Akikusa, M. Yamanishi, and H. Kan, “High-performance λ ∼ 8.6 μm quantum cascade lasers with single phonon-continuum depopulation structures,” IEEE J. Quantum Electron. 46, 683–688 (2010).
[CrossRef]

K. Fujita, T. Edamura, N. Akikusa, A. Sugiyama, T. Ochiai, S. Furuta, A. Ito, M. Yamanishi, and H. Kan, “Quantum cascade lasers based on single phonon-continuum depopulation structures,” Proc. SPIE 7230, 723016 (2009).
[CrossRef]

M. Yamanishi, T. Edamura, K. Fujita, N. Akikusa, and H. Kan, “Theory of the intrinsic linewidth of quantum-cascade lasers: hidden reason for the narrow linewidth and line-broadening by thermal photons,” IEEE J. Quantum Electron. 44, 12–29 (2008).
[CrossRef]

K. Fujita, S. Furuta, A. Sugiyama, T. Ochiai, T. Edamura, N. Akikusa, M. Yamanishi, and H. Kan, “Room temperature, continuous-wave operation of quantum cascade lasers with single phonon resonance-continuum depopulation structures grown by metal organic vapor-phase epitaxy,” Appl. Phys. Lett. 91, 141121 (2007).
[CrossRef]

Elliott, D. S.

D. S. Elliott, R. Roy, and S. J. Smith, “Extracavity laser band-shape and bandwidth modification,” Phys. Rev. A 26, 12–18 (1982).
[CrossRef]

Evans, A.

J. S. Yu, S. Slivken, A. Evans, L. Doris, and M. Razeghi, “High-power continuous-wave operation of a 6 μm quantum-cascade laser at room temperature,” Appl. Phys. Lett. 83, 2503–2505 (2003).
[CrossRef]

Faist, J.

M. S. Vitiello, T. Gresch, A. Lops, V. Spagnolo, G. Scamarcio, N. Hoyler, M. Giovannini, and J. Faist, “Influence of InAs, AlAs δ layers on the optical, electronic, and thermal characteristics of strain-compensated GaInAs/AlInAs quantum-cascade lasers,” Appl. Phys. Lett. 91, 161111 (2007).
[CrossRef]

A. Vasanelli, A. Leuliet, C. Sirtori, A. Wade, G. Fedorov, D. Smirnov, G. Bastard, B. Vinter, M. Giovannini, and J. Faist, “Role of elastic scattering mechanisms in GaInAs/AlInAs quantum cascade lasers,” Appl. Phys. Lett.,  89, 172120 (2006).
[CrossRef]

D. Hofstetter, M. Beck, T. Aellen, and J. Faist, “High-temperature operation of distributed feedback quantum-cascade lasers at 5.3 μm,” Appl. Phys. Lett. 78, 396–398 (2001).
[CrossRef]

J. Faist, F. Capasso, D. L. Sivco, C. Sirtori, A. L. Hutchinson, and A. Y. Cho, “Quantum cascade laser,” Science264, 553–556 (1994).
[CrossRef] [PubMed]

Fedorov, G.

A. Vasanelli, A. Leuliet, C. Sirtori, A. Wade, G. Fedorov, D. Smirnov, G. Bastard, B. Vinter, M. Giovannini, and J. Faist, “Role of elastic scattering mechanisms in GaInAs/AlInAs quantum cascade lasers,” Appl. Phys. Lett.,  89, 172120 (2006).
[CrossRef]

Fujita, K.

K. Fujita, M. Yamanishi, T. Edamura, A. Sugiyama, and S. Furuta, “Extremely high T0-values (≃450 K) of long-wavelength (≃ 15 μm), low-threshold-current-density quantum-cascade lasers based on the indirect pump scheme,” Appl. Phys. Lett. 97, 201109 (2010).

K. Fujita, S. Furuta, A. Sugiyama, T. Ochiai, T. Edamura, N. Akikusa, M. Yamanishi, and H. Kan, “High-performance λ ∼ 8.6 μm quantum cascade lasers with single phonon-continuum depopulation structures,” IEEE J. Quantum Electron. 46, 683–688 (2010).
[CrossRef]

K. Fujita, T. Edamura, N. Akikusa, A. Sugiyama, T. Ochiai, S. Furuta, A. Ito, M. Yamanishi, and H. Kan, “Quantum cascade lasers based on single phonon-continuum depopulation structures,” Proc. SPIE 7230, 723016 (2009).
[CrossRef]

M. Yamanishi, T. Edamura, K. Fujita, N. Akikusa, and H. Kan, “Theory of the intrinsic linewidth of quantum-cascade lasers: hidden reason for the narrow linewidth and line-broadening by thermal photons,” IEEE J. Quantum Electron. 44, 12–29 (2008).
[CrossRef]

K. Fujita, S. Furuta, A. Sugiyama, T. Ochiai, T. Edamura, N. Akikusa, M. Yamanishi, and H. Kan, “Room temperature, continuous-wave operation of quantum cascade lasers with single phonon resonance-continuum depopulation structures grown by metal organic vapor-phase epitaxy,” Appl. Phys. Lett. 91, 141121 (2007).
[CrossRef]

Furuta, S.

K. Fujita, M. Yamanishi, T. Edamura, A. Sugiyama, and S. Furuta, “Extremely high T0-values (≃450 K) of long-wavelength (≃ 15 μm), low-threshold-current-density quantum-cascade lasers based on the indirect pump scheme,” Appl. Phys. Lett. 97, 201109 (2010).

K. Fujita, S. Furuta, A. Sugiyama, T. Ochiai, T. Edamura, N. Akikusa, M. Yamanishi, and H. Kan, “High-performance λ ∼ 8.6 μm quantum cascade lasers with single phonon-continuum depopulation structures,” IEEE J. Quantum Electron. 46, 683–688 (2010).
[CrossRef]

K. Fujita, T. Edamura, N. Akikusa, A. Sugiyama, T. Ochiai, S. Furuta, A. Ito, M. Yamanishi, and H. Kan, “Quantum cascade lasers based on single phonon-continuum depopulation structures,” Proc. SPIE 7230, 723016 (2009).
[CrossRef]

K. Fujita, S. Furuta, A. Sugiyama, T. Ochiai, T. Edamura, N. Akikusa, M. Yamanishi, and H. Kan, “Room temperature, continuous-wave operation of quantum cascade lasers with single phonon resonance-continuum depopulation structures grown by metal organic vapor-phase epitaxy,” Appl. Phys. Lett. 91, 141121 (2007).
[CrossRef]

Galli, I.

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

G. Giusfredi, S. Bartalini, S. Borri, P. Cancio, I. Galli, D. Mazzotti, and P. De Natale, “Saturated-absorption cavity ring-down spectroscopy,” Phys. Rev. Lett. 104, 110801 (2010).
[CrossRef] [PubMed]

S. Bartalini, S. Borri, P. Cancio, A. Castrillo, I. Galli, G. Giusfredi, D. Mazzotti, L. Gianfrani, and P. De Natale, “Observing the intrinsic linewidth of a quantum-cascade laser: beyond the Schawlow–Townes limit,” Phys. Rev. Lett. 104, 083904 (2010).
[CrossRef] [PubMed]

Gianfrani, L.

S. Bartalini, S. Borri, P. Cancio, A. Castrillo, I. Galli, G. Giusfredi, D. Mazzotti, L. Gianfrani, and P. De Natale, “Observing the intrinsic linewidth of a quantum-cascade laser: beyond the Schawlow–Townes limit,” Phys. Rev. Lett. 104, 083904 (2010).
[CrossRef] [PubMed]

Giovannini, M.

M. S. Vitiello, T. Gresch, A. Lops, V. Spagnolo, G. Scamarcio, N. Hoyler, M. Giovannini, and J. Faist, “Influence of InAs, AlAs δ layers on the optical, electronic, and thermal characteristics of strain-compensated GaInAs/AlInAs quantum-cascade lasers,” Appl. Phys. Lett. 91, 161111 (2007).
[CrossRef]

A. Vasanelli, A. Leuliet, C. Sirtori, A. Wade, G. Fedorov, D. Smirnov, G. Bastard, B. Vinter, M. Giovannini, and J. Faist, “Role of elastic scattering mechanisms in GaInAs/AlInAs quantum cascade lasers,” Appl. Phys. Lett.,  89, 172120 (2006).
[CrossRef]

Giusfredi, G.

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

S. Bartalini, S. Borri, P. Cancio, A. Castrillo, I. Galli, G. Giusfredi, D. Mazzotti, L. Gianfrani, and P. De Natale, “Observing the intrinsic linewidth of a quantum-cascade laser: beyond the Schawlow–Townes limit,” Phys. Rev. Lett. 104, 083904 (2010).
[CrossRef] [PubMed]

G. Giusfredi, S. Bartalini, S. Borri, P. Cancio, I. Galli, D. Mazzotti, and P. De Natale, “Saturated-absorption cavity ring-down spectroscopy,” Phys. Rev. Lett. 104, 110801 (2010).
[CrossRef] [PubMed]

Gmachl, C.

Gresch, T.

M. S. Vitiello, T. Gresch, A. Lops, V. Spagnolo, G. Scamarcio, N. Hoyler, M. Giovannini, and J. Faist, “Influence of InAs, AlAs δ layers on the optical, electronic, and thermal characteristics of strain-compensated GaInAs/AlInAs quantum-cascade lasers,” Appl. Phys. Lett. 91, 161111 (2007).
[CrossRef]

Hall, J. L.

Hartman, J. S.

Henry, C.

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

Hofstetter, D.

D. Hofstetter, M. Beck, T. Aellen, and J. Faist, “High-temperature operation of distributed feedback quantum-cascade lasers at 5.3 μm,” Appl. Phys. Lett. 78, 396–398 (2001).
[CrossRef]

Hoyler, N.

M. S. Vitiello, T. Gresch, A. Lops, V. Spagnolo, G. Scamarcio, N. Hoyler, M. Giovannini, and J. Faist, “Influence of InAs, AlAs δ layers on the optical, electronic, and thermal characteristics of strain-compensated GaInAs/AlInAs quantum-cascade lasers,” Appl. Phys. Lett. 91, 161111 (2007).
[CrossRef]

Hutchinson, A. L.

J. Faist, F. Capasso, D. L. Sivco, C. Sirtori, A. L. Hutchinson, and A. Y. Cho, “Quantum cascade laser,” Science264, 553–556 (1994).
[CrossRef] [PubMed]

Ito, A.

K. Fujita, T. Edamura, N. Akikusa, A. Sugiyama, T. Ochiai, S. Furuta, A. Ito, M. Yamanishi, and H. Kan, “Quantum cascade lasers based on single phonon-continuum depopulation structures,” Proc. SPIE 7230, 723016 (2009).
[CrossRef]

Kan, H.

K. Fujita, S. Furuta, A. Sugiyama, T. Ochiai, T. Edamura, N. Akikusa, M. Yamanishi, and H. Kan, “High-performance λ ∼ 8.6 μm quantum cascade lasers with single phonon-continuum depopulation structures,” IEEE J. Quantum Electron. 46, 683–688 (2010).
[CrossRef]

K. Fujita, T. Edamura, N. Akikusa, A. Sugiyama, T. Ochiai, S. Furuta, A. Ito, M. Yamanishi, and H. Kan, “Quantum cascade lasers based on single phonon-continuum depopulation structures,” Proc. SPIE 7230, 723016 (2009).
[CrossRef]

M. Yamanishi, T. Edamura, K. Fujita, N. Akikusa, and H. Kan, “Theory of the intrinsic linewidth of quantum-cascade lasers: hidden reason for the narrow linewidth and line-broadening by thermal photons,” IEEE J. Quantum Electron. 44, 12–29 (2008).
[CrossRef]

K. Fujita, S. Furuta, A. Sugiyama, T. Ochiai, T. Edamura, N. Akikusa, M. Yamanishi, and H. Kan, “Room temperature, continuous-wave operation of quantum cascade lasers with single phonon resonance-continuum depopulation structures grown by metal organic vapor-phase epitaxy,” Appl. Phys. Lett. 91, 141121 (2007).
[CrossRef]

Kelly, J. F.

Leuliet, A.

A. Vasanelli, A. Leuliet, C. Sirtori, A. Wade, G. Fedorov, D. Smirnov, G. Bastard, B. Vinter, M. Giovannini, and J. Faist, “Role of elastic scattering mechanisms in GaInAs/AlInAs quantum cascade lasers,” Appl. Phys. Lett.,  89, 172120 (2006).
[CrossRef]

Libbrecht, K. G.

K. G. Libbrecht and J. L. Hall, “A low-noise high-speed diode laser current controller,” Rev. Sci. Instrum. 64, 2133–2135 (1993).
[CrossRef]

Lops, A.

M. S. Vitiello, V. Spagnolo, G. Scamarcio, A. Lops, Q. Yang, C. Manz, and J. Wagner, “Experimental investigation of the lattice and electronic temperatures in Ga0.47In0.53As/Al0.62Ga0.38As1–xSbx quantum-cascade lasers,” Appl. Phys. Lett. 90, 121109 (2007).
[CrossRef]

M. S. Vitiello, T. Gresch, A. Lops, V. Spagnolo, G. Scamarcio, N. Hoyler, M. Giovannini, and J. Faist, “Influence of InAs, AlAs δ layers on the optical, electronic, and thermal characteristics of strain-compensated GaInAs/AlInAs quantum-cascade lasers,” Appl. Phys. Lett. 91, 161111 (2007).
[CrossRef]

Manz, C.

M. S. Vitiello, V. Spagnolo, G. Scamarcio, A. Lops, Q. Yang, C. Manz, and J. Wagner, “Experimental investigation of the lattice and electronic temperatures in Ga0.47In0.53As/Al0.62Ga0.38As1–xSbx quantum-cascade lasers,” Appl. Phys. Lett. 90, 121109 (2007).
[CrossRef]

Mazzotti, D.

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

S. Bartalini, S. Borri, P. Cancio, A. Castrillo, I. Galli, G. Giusfredi, D. Mazzotti, L. Gianfrani, and P. De Natale, “Observing the intrinsic linewidth of a quantum-cascade laser: beyond the Schawlow–Townes limit,” Phys. Rev. Lett. 104, 083904 (2010).
[CrossRef] [PubMed]

G. Giusfredi, S. Bartalini, S. Borri, P. Cancio, I. Galli, D. Mazzotti, and P. De Natale, “Saturated-absorption cavity ring-down spectroscopy,” Phys. Rev. Lett. 104, 110801 (2010).
[CrossRef] [PubMed]

Myers, T. L.

Ochiai, T.

K. Fujita, S. Furuta, A. Sugiyama, T. Ochiai, T. Edamura, N. Akikusa, M. Yamanishi, and H. Kan, “High-performance λ ∼ 8.6 μm quantum cascade lasers with single phonon-continuum depopulation structures,” IEEE J. Quantum Electron. 46, 683–688 (2010).
[CrossRef]

K. Fujita, T. Edamura, N. Akikusa, A. Sugiyama, T. Ochiai, S. Furuta, A. Ito, M. Yamanishi, and H. Kan, “Quantum cascade lasers based on single phonon-continuum depopulation structures,” Proc. SPIE 7230, 723016 (2009).
[CrossRef]

K. Fujita, S. Furuta, A. Sugiyama, T. Ochiai, T. Edamura, N. Akikusa, M. Yamanishi, and H. Kan, “Room temperature, continuous-wave operation of quantum cascade lasers with single phonon resonance-continuum depopulation structures grown by metal organic vapor-phase epitaxy,” Appl. Phys. Lett. 91, 141121 (2007).
[CrossRef]

Razeghi, M.

J. S. Yu, S. Slivken, A. Evans, L. Doris, and M. Razeghi, “High-power continuous-wave operation of a 6 μm quantum-cascade laser at room temperature,” Appl. Phys. Lett. 83, 2503–2505 (2003).
[CrossRef]

Roy, R.

D. S. Elliott, R. Roy, and S. J. Smith, “Extracavity laser band-shape and bandwidth modification,” Phys. Rev. A 26, 12–18 (1982).
[CrossRef]

Scamarcio, G.

M. S. Vitiello, G. Scamarcio, and V. Spagnolo, “Temperature dependence of thermal conductivity and boundary resistance in THz quantum cascade lasers,” IEEE J. Select Top. Quantum Electron. 14, 431–435 (2008).
[CrossRef]

M. S. Vitiello, T. Gresch, A. Lops, V. Spagnolo, G. Scamarcio, N. Hoyler, M. Giovannini, and J. Faist, “Influence of InAs, AlAs δ layers on the optical, electronic, and thermal characteristics of strain-compensated GaInAs/AlInAs quantum-cascade lasers,” Appl. Phys. Lett. 91, 161111 (2007).
[CrossRef]

M. S. Vitiello, V. Spagnolo, G. Scamarcio, A. Lops, Q. Yang, C. Manz, and J. Wagner, “Experimental investigation of the lattice and electronic temperatures in Ga0.47In0.53As/Al0.62Ga0.38As1–xSbx quantum-cascade lasers,” Appl. Phys. Lett. 90, 121109 (2007).
[CrossRef]

Schilt, S.

Sharpe, S. W.

Sirtori, C.

A. Vasanelli, A. Leuliet, C. Sirtori, A. Wade, G. Fedorov, D. Smirnov, G. Bastard, B. Vinter, M. Giovannini, and J. Faist, “Role of elastic scattering mechanisms in GaInAs/AlInAs quantum cascade lasers,” Appl. Phys. Lett.,  89, 172120 (2006).
[CrossRef]

J. Faist, F. Capasso, D. L. Sivco, C. Sirtori, A. L. Hutchinson, and A. Y. Cho, “Quantum cascade laser,” Science264, 553–556 (1994).
[CrossRef] [PubMed]

Sivco, D. L.

Slivken, S.

J. S. Yu, S. Slivken, A. Evans, L. Doris, and M. Razeghi, “High-power continuous-wave operation of a 6 μm quantum-cascade laser at room temperature,” Appl. Phys. Lett. 83, 2503–2505 (2003).
[CrossRef]

Smirnov, D.

A. Vasanelli, A. Leuliet, C. Sirtori, A. Wade, G. Fedorov, D. Smirnov, G. Bastard, B. Vinter, M. Giovannini, and J. Faist, “Role of elastic scattering mechanisms in GaInAs/AlInAs quantum cascade lasers,” Appl. Phys. Lett.,  89, 172120 (2006).
[CrossRef]

Smith, S. J.

D. S. Elliott, R. Roy, and S. J. Smith, “Extracavity laser band-shape and bandwidth modification,” Phys. Rev. A 26, 12–18 (1982).
[CrossRef]

Spagnolo, V.

M. S. Vitiello, G. Scamarcio, and V. Spagnolo, “Temperature dependence of thermal conductivity and boundary resistance in THz quantum cascade lasers,” IEEE J. Select Top. Quantum Electron. 14, 431–435 (2008).
[CrossRef]

M. S. Vitiello, T. Gresch, A. Lops, V. Spagnolo, G. Scamarcio, N. Hoyler, M. Giovannini, and J. Faist, “Influence of InAs, AlAs δ layers on the optical, electronic, and thermal characteristics of strain-compensated GaInAs/AlInAs quantum-cascade lasers,” Appl. Phys. Lett. 91, 161111 (2007).
[CrossRef]

M. S. Vitiello, V. Spagnolo, G. Scamarcio, A. Lops, Q. Yang, C. Manz, and J. Wagner, “Experimental investigation of the lattice and electronic temperatures in Ga0.47In0.53As/Al0.62Ga0.38As1–xSbx quantum-cascade lasers,” Appl. Phys. Lett. 90, 121109 (2007).
[CrossRef]

Sugiyama, A.

K. Fujita, M. Yamanishi, T. Edamura, A. Sugiyama, and S. Furuta, “Extremely high T0-values (≃450 K) of long-wavelength (≃ 15 μm), low-threshold-current-density quantum-cascade lasers based on the indirect pump scheme,” Appl. Phys. Lett. 97, 201109 (2010).

K. Fujita, S. Furuta, A. Sugiyama, T. Ochiai, T. Edamura, N. Akikusa, M. Yamanishi, and H. Kan, “High-performance λ ∼ 8.6 μm quantum cascade lasers with single phonon-continuum depopulation structures,” IEEE J. Quantum Electron. 46, 683–688 (2010).
[CrossRef]

K. Fujita, T. Edamura, N. Akikusa, A. Sugiyama, T. Ochiai, S. Furuta, A. Ito, M. Yamanishi, and H. Kan, “Quantum cascade lasers based on single phonon-continuum depopulation structures,” Proc. SPIE 7230, 723016 (2009).
[CrossRef]

K. Fujita, S. Furuta, A. Sugiyama, T. Ochiai, T. Edamura, N. Akikusa, M. Yamanishi, and H. Kan, “Room temperature, continuous-wave operation of quantum cascade lasers with single phonon resonance-continuum depopulation structures grown by metal organic vapor-phase epitaxy,” Appl. Phys. Lett. 91, 141121 (2007).
[CrossRef]

Taubman, M. S.

Thomann, P.

Vasanelli, A.

A. Vasanelli, A. Leuliet, C. Sirtori, A. Wade, G. Fedorov, D. Smirnov, G. Bastard, B. Vinter, M. Giovannini, and J. Faist, “Role of elastic scattering mechanisms in GaInAs/AlInAs quantum cascade lasers,” Appl. Phys. Lett.,  89, 172120 (2006).
[CrossRef]

Vinter, B.

A. Vasanelli, A. Leuliet, C. Sirtori, A. Wade, G. Fedorov, D. Smirnov, G. Bastard, B. Vinter, M. Giovannini, and J. Faist, “Role of elastic scattering mechanisms in GaInAs/AlInAs quantum cascade lasers,” Appl. Phys. Lett.,  89, 172120 (2006).
[CrossRef]

Vitiello, M. S.

M. S. Vitiello, G. Scamarcio, and V. Spagnolo, “Temperature dependence of thermal conductivity and boundary resistance in THz quantum cascade lasers,” IEEE J. Select Top. Quantum Electron. 14, 431–435 (2008).
[CrossRef]

M. S. Vitiello, T. Gresch, A. Lops, V. Spagnolo, G. Scamarcio, N. Hoyler, M. Giovannini, and J. Faist, “Influence of InAs, AlAs δ layers on the optical, electronic, and thermal characteristics of strain-compensated GaInAs/AlInAs quantum-cascade lasers,” Appl. Phys. Lett. 91, 161111 (2007).
[CrossRef]

M. S. Vitiello, V. Spagnolo, G. Scamarcio, A. Lops, Q. Yang, C. Manz, and J. Wagner, “Experimental investigation of the lattice and electronic temperatures in Ga0.47In0.53As/Al0.62Ga0.38As1–xSbx quantum-cascade lasers,” Appl. Phys. Lett. 90, 121109 (2007).
[CrossRef]

Wade, A.

A. Vasanelli, A. Leuliet, C. Sirtori, A. Wade, G. Fedorov, D. Smirnov, G. Bastard, B. Vinter, M. Giovannini, and J. Faist, “Role of elastic scattering mechanisms in GaInAs/AlInAs quantum cascade lasers,” Appl. Phys. Lett.,  89, 172120 (2006).
[CrossRef]

Wagner, J.

M. S. Vitiello, V. Spagnolo, G. Scamarcio, A. Lops, Q. Yang, C. Manz, and J. Wagner, “Experimental investigation of the lattice and electronic temperatures in Ga0.47In0.53As/Al0.62Ga0.38As1–xSbx quantum-cascade lasers,” Appl. Phys. Lett. 90, 121109 (2007).
[CrossRef]

Williams, R. M.

Yamanishi, M.

K. Fujita, M. Yamanishi, T. Edamura, A. Sugiyama, and S. Furuta, “Extremely high T0-values (≃450 K) of long-wavelength (≃ 15 μm), low-threshold-current-density quantum-cascade lasers based on the indirect pump scheme,” Appl. Phys. Lett. 97, 201109 (2010).

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

K. Fujita, S. Furuta, A. Sugiyama, T. Ochiai, T. Edamura, N. Akikusa, M. Yamanishi, and H. Kan, “High-performance λ ∼ 8.6 μm quantum cascade lasers with single phonon-continuum depopulation structures,” IEEE J. Quantum Electron. 46, 683–688 (2010).
[CrossRef]

K. Fujita, T. Edamura, N. Akikusa, A. Sugiyama, T. Ochiai, S. Furuta, A. Ito, M. Yamanishi, and H. Kan, “Quantum cascade lasers based on single phonon-continuum depopulation structures,” Proc. SPIE 7230, 723016 (2009).
[CrossRef]

M. Yamanishi, T. Edamura, K. Fujita, N. Akikusa, and H. Kan, “Theory of the intrinsic linewidth of quantum-cascade lasers: hidden reason for the narrow linewidth and line-broadening by thermal photons,” IEEE J. Quantum Electron. 44, 12–29 (2008).
[CrossRef]

K. Fujita, S. Furuta, A. Sugiyama, T. Ochiai, T. Edamura, N. Akikusa, M. Yamanishi, and H. Kan, “Room temperature, continuous-wave operation of quantum cascade lasers with single phonon resonance-continuum depopulation structures grown by metal organic vapor-phase epitaxy,” Appl. Phys. Lett. 91, 141121 (2007).
[CrossRef]

Yang, Q.

M. S. Vitiello, V. Spagnolo, G. Scamarcio, A. Lops, Q. Yang, C. Manz, and J. Wagner, “Experimental investigation of the lattice and electronic temperatures in Ga0.47In0.53As/Al0.62Ga0.38As1–xSbx quantum-cascade lasers,” Appl. Phys. Lett. 90, 121109 (2007).
[CrossRef]

Yu, J. S.

J. S. Yu, S. Slivken, A. Evans, L. Doris, and M. Razeghi, “High-power continuous-wave operation of a 6 μm quantum-cascade laser at room temperature,” Appl. Phys. Lett. 83, 2503–2505 (2003).
[CrossRef]

Appl. Opt. (1)

Appl. Phys. Lett. (7)

K. Fujita, S. Furuta, A. Sugiyama, T. Ochiai, T. Edamura, N. Akikusa, M. Yamanishi, and H. Kan, “Room temperature, continuous-wave operation of quantum cascade lasers with single phonon resonance-continuum depopulation structures grown by metal organic vapor-phase epitaxy,” Appl. Phys. Lett. 91, 141121 (2007).
[CrossRef]

J. S. Yu, S. Slivken, A. Evans, L. Doris, and M. Razeghi, “High-power continuous-wave operation of a 6 μm quantum-cascade laser at room temperature,” Appl. Phys. Lett. 83, 2503–2505 (2003).
[CrossRef]

A. Vasanelli, A. Leuliet, C. Sirtori, A. Wade, G. Fedorov, D. Smirnov, G. Bastard, B. Vinter, M. Giovannini, and J. Faist, “Role of elastic scattering mechanisms in GaInAs/AlInAs quantum cascade lasers,” Appl. Phys. Lett.,  89, 172120 (2006).
[CrossRef]

K. Fujita, M. Yamanishi, T. Edamura, A. Sugiyama, and S. Furuta, “Extremely high T0-values (≃450 K) of long-wavelength (≃ 15 μm), low-threshold-current-density quantum-cascade lasers based on the indirect pump scheme,” Appl. Phys. Lett. 97, 201109 (2010).

D. Hofstetter, M. Beck, T. Aellen, and J. Faist, “High-temperature operation of distributed feedback quantum-cascade lasers at 5.3 μm,” Appl. Phys. Lett. 78, 396–398 (2001).
[CrossRef]

M. S. Vitiello, T. Gresch, A. Lops, V. Spagnolo, G. Scamarcio, N. Hoyler, M. Giovannini, and J. Faist, “Influence of InAs, AlAs δ layers on the optical, electronic, and thermal characteristics of strain-compensated GaInAs/AlInAs quantum-cascade lasers,” Appl. Phys. Lett. 91, 161111 (2007).
[CrossRef]

M. S. Vitiello, V. Spagnolo, G. Scamarcio, A. Lops, Q. Yang, C. Manz, and J. Wagner, “Experimental investigation of the lattice and electronic temperatures in Ga0.47In0.53As/Al0.62Ga0.38As1–xSbx quantum-cascade lasers,” Appl. Phys. Lett. 90, 121109 (2007).
[CrossRef]

IEEE J. Quantum Electron. (4)

K. Fujita, S. Furuta, A. Sugiyama, T. Ochiai, T. Edamura, N. Akikusa, M. Yamanishi, and H. Kan, “High-performance λ ∼ 8.6 μm quantum cascade lasers with single phonon-continuum depopulation structures,” IEEE J. Quantum Electron. 46, 683–688 (2010).
[CrossRef]

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

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

M. Yamanishi, T. Edamura, K. Fujita, N. Akikusa, and H. Kan, “Theory of the intrinsic linewidth of quantum-cascade lasers: hidden reason for the narrow linewidth and line-broadening by thermal photons,” IEEE J. Quantum Electron. 44, 12–29 (2008).
[CrossRef]

IEEE J. Select Top. Quantum Electron. (1)

M. S. Vitiello, G. Scamarcio, and V. Spagnolo, “Temperature dependence of thermal conductivity and boundary resistance in THz quantum cascade lasers,” IEEE J. Select Top. Quantum Electron. 14, 431–435 (2008).
[CrossRef]

Opt. Express (1)

Opt. Lett. (3)

Phys. Rev. A (1)

D. S. Elliott, R. Roy, and S. J. Smith, “Extracavity laser band-shape and bandwidth modification,” Phys. Rev. A 26, 12–18 (1982).
[CrossRef]

Phys. Rev. Lett. (2)

S. Bartalini, S. Borri, P. Cancio, A. Castrillo, I. Galli, G. Giusfredi, D. Mazzotti, L. Gianfrani, and P. De Natale, “Observing the intrinsic linewidth of a quantum-cascade laser: beyond the Schawlow–Townes limit,” Phys. Rev. Lett. 104, 083904 (2010).
[CrossRef] [PubMed]

G. Giusfredi, S. Bartalini, S. Borri, P. Cancio, I. Galli, D. Mazzotti, and P. De Natale, “Saturated-absorption cavity ring-down spectroscopy,” Phys. Rev. Lett. 104, 110801 (2010).
[CrossRef] [PubMed]

Proc. SPIE (1)

K. Fujita, T. Edamura, N. Akikusa, A. Sugiyama, T. Ochiai, S. Furuta, A. Ito, M. Yamanishi, and H. Kan, “Quantum cascade lasers based on single phonon-continuum depopulation structures,” Proc. SPIE 7230, 723016 (2009).
[CrossRef]

Rev. Sci. Instrum. (1)

K. G. Libbrecht and J. L. Hall, “A low-noise high-speed diode laser current controller,” Rev. Sci. Instrum. 64, 2133–2135 (1993).
[CrossRef]

Other (1)

J. Faist, F. Capasso, D. L. Sivco, C. Sirtori, A. L. Hutchinson, and A. Y. Cho, “Quantum cascade laser,” Science264, 553–556 (1994).
[CrossRef] [PubMed]

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Figures (3)

Fig. 1
Fig. 1

The schematic shows the simple experimental setup used for the measurement of the QCL frequency-noise PSD. In the inset, an acquisition of the molecular absorption profile is also shown, and the working principle of the conversion of frequency fluctuations into amplitude fluctuations is depicted.

Fig. 2
Fig. 2

a) Comparison between the FNPSDs of a cryogenic QCL (light gray, QCL2, operating current Io 2 = 219 mA) and the RT QCL studied in this work (dark gray, QCL1, operating current Io 1 = 776 mA). The level of the flicker noise is about 200 times lower for the RT device. The plot also shows how the contribution from the current noise of our home-made driver, that was negligible for the cryogenic QCL, now becomes relevant. b) The same measurement is repeated with the commercial current driver. Despite the larger noise contribution below 10 MHz, the strong filtering at higher frequencies allows a clear observation of the white intrinsic noise. The dashed blue line, showing the 1/f trend, is the same for both graphs.

Fig. 3
Fig. 3

The plot shows in detail the high-frequency portion of the measured FNPSD, where a flattening down to a white-noise level of 84 Hz2/Hz occurs. The sensitivity of our system is also represented by the detector noise level.

Equations (3)

Equations on this page are rendered with MathJax. Learn more.

δ ν = 1 4 π γ β eff ( 1 ɛ ) [ 1 ( I o / I th 1 ) + ɛ ] ( 1 + α e 2 ) ,
β eff = β τ t τ r , ɛ = τ 21 τ 31 η τ t ( τ 21 + τ 31 ) τ 21 η τ t , α e 0 .
( δ f 2 δ f 1 ) 2 ( R therm 2 R therm 1 ) 2 × ( δ I 2 δ I 1 ) 2

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