Abstract

We investigate the nonlinear propagation of an ultra-short, 150 fs, optical pulse along the waveguide of a quantum dot (QD) laser operating above threshold. We demonstrate that among the various nonlinear processes experienced by the propagating pulse, four-wave mixing (FWM) between the pulse and the two oscillating counter-propagating cw fields of the laser is the dominant one. FWM has two important consequences. One is the creation of a spectral hole located in the vicinity of the cw oscillating frequency. The width of the spectral hole is determined by an effective carrier and gain relaxation time. The second is a modification of the shape of the trailing edge of the pulse. The wave mixing involves first and second order processes which result in a complicated interaction among several fields inside the cavity, some of which are cw while the others are time varying, all propagating in both directions. The nonlinear pulse propagation is analyzed using two complementary theoretical approaches. One is a semi-analytical model which considers only the wave mixing interaction between six field components, three of which propagate in each direction (two cw fields and four time-varying signals). This model predicts the deformation of the tail of the output signal by a secondary idler wave, produced in a cascaded FWM process, which co-propagates with the original injected pulse. The second approach is a finite-difference time-domain simulation, which considers also additional nonlinear effects, such as gain saturation and self–phase modulation. The theoretical results are confirmed by a series of experiments in which the time dependent amplitude and phase of the pulse after propagation are measured using the cross-frequency-resolved optical gating technique.

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  1. K. L. Hall, J. Mark, E. P. Ippen, and G. Eisenstein, “Femtosecond gain dynamics in InGaAsP optical amplifiers,” Appl. Phys. Lett.56(18), 1740–1742 (1990).
    [CrossRef]
  2. P. Borri, W. Langbein, J. M. Hvam, F. Heinrichsdorff, M.-H. Mao, and D. Bimberg, “Ultrafast gain dynamics in InAs-InGaAs quantum-dot amplifiers,” IEEE Photon. Technol. Lett.12(6), 594–596 (2000).
    [CrossRef]
  3. A. Capua, G. Eisenstein, and J. P. Reithmaier, “A nearly instantaneous gain response in quantum dash based optical amplifiers,” Appl. Phys. Lett.97(13), 131108 (2010).
    [CrossRef]
  4. A. Uskov, J. Mork, and J. Mark, “Wave mixing in semiconductor laser amplifiers due to carrier heating and spectral-hole burning,” IEEE J. Quantum Electron.30(8), 1769–1781 (1994).
    [CrossRef]
  5. A. Capua, V. Mikhelashvili, G. Eisenstein, J. P. Reithmaier, A. Somers, A. Forchel, M. Calligaro, O. Parillaud, and M. Krakowski, “Direct observation of the coherent spectral hole in the noise spectrum of a saturated InAs/InP quantum dash amplifier operating near 1550 nm,” Opt. Express16(3), 2141–2146 (2008).
    [CrossRef] [PubMed]
  6. J. Zimmermann, S. T. Cundiff, G. von Plessen, J. Feldmann, M. Arzberger, G. Böhm, M.-C. Amann, and G. Abstreiter, “Dark pulse formation in a quantum-dot laser,” Appl. Phys. Lett.79(1), 18–20 (2001).
    [CrossRef]
  7. C. Sun, B. Golubovic, H. Choi, C. A. Wang, and J. G. Fujimoto, “Femtosecond investigations of spectral hole burning in semiconductor lasers,” Appl. Phys. Lett.66(13), 1650–1652 (1995).
    [CrossRef]
  8. G. P. Agrawal, “Population pulsations and nondegenerate four-wave mixing in semiconductor lasers and amplifiers,” J. Opt. Soc. Am. B5(1), 147–159 (1988).
    [CrossRef]
  9. M. Shtaif and G. Eisenstein, “Analytical solution of wave mixing between short optical pulses in a semiconductor optical amplifier,” Appl. Phys. Lett.66(12), 1458–1460 (1995).
    [CrossRef]
  10. A. Mecozzi and J. Mork, “Saturation effects in nondegenerate four-wave mixing between short optical pulses in semiconductor laser amplifiers,” IEEE J. Sel. Top. Quantum Electron.3(5), 1190–1207 (1997).
    [CrossRef]
  11. A. Bilenca, R. Alizon, V. Mikhelashvili, D. Dahan, G. Eisenstein, R. Schwertberger, D. Gold, J. P. Reithmaier, and A. Forchel, “Broad-band wavelength conversion based on cross-gain modulation and four-wave mixing in InAs-InP quantum-dash semiconductor optical amplifiers operating at 1550 nm,” IEEE Photon. Technol. Lett.15(4), 563–565 (2003).
  12. P. Agrawal and N. A. Olsson, “Self-phase modulation and spectral broadening of optical pulses in semiconductor laser amplifiers,” IEEE J. Quantum Electron.25(11), 2297–2306 (1989).
    [CrossRef]
  13. S. Linden, H. Giessen, and J. Kuhl, “XFROG — A new method for amplitude and phase characterization of weak ultrashort pulses,” Phys. Status Solidi, B Basic Res.206(1), 119–124 (1998).
    [CrossRef]
  14. A. Capua, A. Saal, O. Karni, G. Eisenstein, J. P. Reithmaier, and K. Yvind, “Complex characterization of short-pulse propagation through InAs/InP quantum-dash optical amplifiers: from the quasi-linear to the two-photon-dominated regime,” Opt. Express20(1), 347–353 (2012).
    [CrossRef] [PubMed]
  15. R. Trebino, Frequency-Resolved Optical Gating: The Measurement of Ultrashort Laser Pulses (Kluwer Academic Publishers, Norwell, 2002).
  16. C. H. Henry, “Theory of the linewidth of semiconductor lasers,” IEEE J. Quantum Electron.18(2), 259–264 (1982).
    [CrossRef]
  17. M. Shtaif and G. Eisenstein, “Noise properties of nonlinear semiconductor optical amplifiers,” Opt. Lett.21(22), 1851–1853 (1996).
    [CrossRef] [PubMed]
  18. A. P. Bogatov, P. G. Eliseev, and B. N. Sverdlov, “Anomalous interaction of spectral modes in a semiconductor laser,” IEEE J. Quantum Electron.11(7), 510–515 (1975).
    [CrossRef]
  19. A. Bilenca and G. Eisenstein, “On the noise properties of linear and nonlinear quantum-dot semiconductor optical amplifiers: the impact of inhomogeneously broadened gain and fast carrier dynamics,” IEEE J. Quantum Electron.40(6), 690–702 (2004).
    [CrossRef]
  20. G. M. Slavcheva, J. M. Arnold, and R. W. Ziolkowski, “FDTD simulation of the nonlinear gain dynamics in active optical waveguides and semiconductor microcavities,” IEEE J. Sel. Top. Quantum Electron.10(5), 1052–1062 (2004).
    [CrossRef]
  21. J. E. Kim, E. Malic, M. Richter, A. Wilms, and A. Knorr, “Maxwell–bloch equation approach for describing the microscopic dynamics of quantum-dot surface-emitting structures,” IEEE J. Quantum Electron.46(7), 1115–1126 (2010).
    [CrossRef]
  22. J. Kim, C. Meuer, D. Bimberg, and G. Eisenstein, “Role of carrier reservoirs on the slow phase recovery of quantum dot semiconductor optical amplifiers,” Appl. Phys. Lett.94(4), 041112 (2009).
    [CrossRef]

2012

2010

A. Capua, G. Eisenstein, and J. P. Reithmaier, “A nearly instantaneous gain response in quantum dash based optical amplifiers,” Appl. Phys. Lett.97(13), 131108 (2010).
[CrossRef]

J. E. Kim, E. Malic, M. Richter, A. Wilms, and A. Knorr, “Maxwell–bloch equation approach for describing the microscopic dynamics of quantum-dot surface-emitting structures,” IEEE J. Quantum Electron.46(7), 1115–1126 (2010).
[CrossRef]

2009

J. Kim, C. Meuer, D. Bimberg, and G. Eisenstein, “Role of carrier reservoirs on the slow phase recovery of quantum dot semiconductor optical amplifiers,” Appl. Phys. Lett.94(4), 041112 (2009).
[CrossRef]

2008

2004

A. Bilenca and G. Eisenstein, “On the noise properties of linear and nonlinear quantum-dot semiconductor optical amplifiers: the impact of inhomogeneously broadened gain and fast carrier dynamics,” IEEE J. Quantum Electron.40(6), 690–702 (2004).
[CrossRef]

G. M. Slavcheva, J. M. Arnold, and R. W. Ziolkowski, “FDTD simulation of the nonlinear gain dynamics in active optical waveguides and semiconductor microcavities,” IEEE J. Sel. Top. Quantum Electron.10(5), 1052–1062 (2004).
[CrossRef]

2003

A. Bilenca, R. Alizon, V. Mikhelashvili, D. Dahan, G. Eisenstein, R. Schwertberger, D. Gold, J. P. Reithmaier, and A. Forchel, “Broad-band wavelength conversion based on cross-gain modulation and four-wave mixing in InAs-InP quantum-dash semiconductor optical amplifiers operating at 1550 nm,” IEEE Photon. Technol. Lett.15(4), 563–565 (2003).

2001

J. Zimmermann, S. T. Cundiff, G. von Plessen, J. Feldmann, M. Arzberger, G. Böhm, M.-C. Amann, and G. Abstreiter, “Dark pulse formation in a quantum-dot laser,” Appl. Phys. Lett.79(1), 18–20 (2001).
[CrossRef]

2000

P. Borri, W. Langbein, J. M. Hvam, F. Heinrichsdorff, M.-H. Mao, and D. Bimberg, “Ultrafast gain dynamics in InAs-InGaAs quantum-dot amplifiers,” IEEE Photon. Technol. Lett.12(6), 594–596 (2000).
[CrossRef]

1998

S. Linden, H. Giessen, and J. Kuhl, “XFROG — A new method for amplitude and phase characterization of weak ultrashort pulses,” Phys. Status Solidi, B Basic Res.206(1), 119–124 (1998).
[CrossRef]

1997

A. Mecozzi and J. Mork, “Saturation effects in nondegenerate four-wave mixing between short optical pulses in semiconductor laser amplifiers,” IEEE J. Sel. Top. Quantum Electron.3(5), 1190–1207 (1997).
[CrossRef]

1996

1995

M. Shtaif and G. Eisenstein, “Analytical solution of wave mixing between short optical pulses in a semiconductor optical amplifier,” Appl. Phys. Lett.66(12), 1458–1460 (1995).
[CrossRef]

C. Sun, B. Golubovic, H. Choi, C. A. Wang, and J. G. Fujimoto, “Femtosecond investigations of spectral hole burning in semiconductor lasers,” Appl. Phys. Lett.66(13), 1650–1652 (1995).
[CrossRef]

1994

A. Uskov, J. Mork, and J. Mark, “Wave mixing in semiconductor laser amplifiers due to carrier heating and spectral-hole burning,” IEEE J. Quantum Electron.30(8), 1769–1781 (1994).
[CrossRef]

1990

K. L. Hall, J. Mark, E. P. Ippen, and G. Eisenstein, “Femtosecond gain dynamics in InGaAsP optical amplifiers,” Appl. Phys. Lett.56(18), 1740–1742 (1990).
[CrossRef]

1989

P. Agrawal and N. A. Olsson, “Self-phase modulation and spectral broadening of optical pulses in semiconductor laser amplifiers,” IEEE J. Quantum Electron.25(11), 2297–2306 (1989).
[CrossRef]

1988

1982

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

1975

A. P. Bogatov, P. G. Eliseev, and B. N. Sverdlov, “Anomalous interaction of spectral modes in a semiconductor laser,” IEEE J. Quantum Electron.11(7), 510–515 (1975).
[CrossRef]

Abstreiter, G.

J. Zimmermann, S. T. Cundiff, G. von Plessen, J. Feldmann, M. Arzberger, G. Böhm, M.-C. Amann, and G. Abstreiter, “Dark pulse formation in a quantum-dot laser,” Appl. Phys. Lett.79(1), 18–20 (2001).
[CrossRef]

Agrawal, G. P.

Agrawal, P.

P. Agrawal and N. A. Olsson, “Self-phase modulation and spectral broadening of optical pulses in semiconductor laser amplifiers,” IEEE J. Quantum Electron.25(11), 2297–2306 (1989).
[CrossRef]

Alizon, R.

A. Bilenca, R. Alizon, V. Mikhelashvili, D. Dahan, G. Eisenstein, R. Schwertberger, D. Gold, J. P. Reithmaier, and A. Forchel, “Broad-band wavelength conversion based on cross-gain modulation and four-wave mixing in InAs-InP quantum-dash semiconductor optical amplifiers operating at 1550 nm,” IEEE Photon. Technol. Lett.15(4), 563–565 (2003).

Amann, M.-C.

J. Zimmermann, S. T. Cundiff, G. von Plessen, J. Feldmann, M. Arzberger, G. Böhm, M.-C. Amann, and G. Abstreiter, “Dark pulse formation in a quantum-dot laser,” Appl. Phys. Lett.79(1), 18–20 (2001).
[CrossRef]

Arnold, J. M.

G. M. Slavcheva, J. M. Arnold, and R. W. Ziolkowski, “FDTD simulation of the nonlinear gain dynamics in active optical waveguides and semiconductor microcavities,” IEEE J. Sel. Top. Quantum Electron.10(5), 1052–1062 (2004).
[CrossRef]

Arzberger, M.

J. Zimmermann, S. T. Cundiff, G. von Plessen, J. Feldmann, M. Arzberger, G. Böhm, M.-C. Amann, and G. Abstreiter, “Dark pulse formation in a quantum-dot laser,” Appl. Phys. Lett.79(1), 18–20 (2001).
[CrossRef]

Bilenca, A.

A. Bilenca and G. Eisenstein, “On the noise properties of linear and nonlinear quantum-dot semiconductor optical amplifiers: the impact of inhomogeneously broadened gain and fast carrier dynamics,” IEEE J. Quantum Electron.40(6), 690–702 (2004).
[CrossRef]

A. Bilenca, R. Alizon, V. Mikhelashvili, D. Dahan, G. Eisenstein, R. Schwertberger, D. Gold, J. P. Reithmaier, and A. Forchel, “Broad-band wavelength conversion based on cross-gain modulation and four-wave mixing in InAs-InP quantum-dash semiconductor optical amplifiers operating at 1550 nm,” IEEE Photon. Technol. Lett.15(4), 563–565 (2003).

Bimberg, D.

J. Kim, C. Meuer, D. Bimberg, and G. Eisenstein, “Role of carrier reservoirs on the slow phase recovery of quantum dot semiconductor optical amplifiers,” Appl. Phys. Lett.94(4), 041112 (2009).
[CrossRef]

P. Borri, W. Langbein, J. M. Hvam, F. Heinrichsdorff, M.-H. Mao, and D. Bimberg, “Ultrafast gain dynamics in InAs-InGaAs quantum-dot amplifiers,” IEEE Photon. Technol. Lett.12(6), 594–596 (2000).
[CrossRef]

Bogatov, A. P.

A. P. Bogatov, P. G. Eliseev, and B. N. Sverdlov, “Anomalous interaction of spectral modes in a semiconductor laser,” IEEE J. Quantum Electron.11(7), 510–515 (1975).
[CrossRef]

Böhm, G.

J. Zimmermann, S. T. Cundiff, G. von Plessen, J. Feldmann, M. Arzberger, G. Böhm, M.-C. Amann, and G. Abstreiter, “Dark pulse formation in a quantum-dot laser,” Appl. Phys. Lett.79(1), 18–20 (2001).
[CrossRef]

Borri, P.

P. Borri, W. Langbein, J. M. Hvam, F. Heinrichsdorff, M.-H. Mao, and D. Bimberg, “Ultrafast gain dynamics in InAs-InGaAs quantum-dot amplifiers,” IEEE Photon. Technol. Lett.12(6), 594–596 (2000).
[CrossRef]

Calligaro, M.

Capua, A.

Choi, H.

C. Sun, B. Golubovic, H. Choi, C. A. Wang, and J. G. Fujimoto, “Femtosecond investigations of spectral hole burning in semiconductor lasers,” Appl. Phys. Lett.66(13), 1650–1652 (1995).
[CrossRef]

Cundiff, S. T.

J. Zimmermann, S. T. Cundiff, G. von Plessen, J. Feldmann, M. Arzberger, G. Böhm, M.-C. Amann, and G. Abstreiter, “Dark pulse formation in a quantum-dot laser,” Appl. Phys. Lett.79(1), 18–20 (2001).
[CrossRef]

Dahan, D.

A. Bilenca, R. Alizon, V. Mikhelashvili, D. Dahan, G. Eisenstein, R. Schwertberger, D. Gold, J. P. Reithmaier, and A. Forchel, “Broad-band wavelength conversion based on cross-gain modulation and four-wave mixing in InAs-InP quantum-dash semiconductor optical amplifiers operating at 1550 nm,” IEEE Photon. Technol. Lett.15(4), 563–565 (2003).

Eisenstein, G.

A. Capua, A. Saal, O. Karni, G. Eisenstein, J. P. Reithmaier, and K. Yvind, “Complex characterization of short-pulse propagation through InAs/InP quantum-dash optical amplifiers: from the quasi-linear to the two-photon-dominated regime,” Opt. Express20(1), 347–353 (2012).
[CrossRef] [PubMed]

A. Capua, G. Eisenstein, and J. P. Reithmaier, “A nearly instantaneous gain response in quantum dash based optical amplifiers,” Appl. Phys. Lett.97(13), 131108 (2010).
[CrossRef]

J. Kim, C. Meuer, D. Bimberg, and G. Eisenstein, “Role of carrier reservoirs on the slow phase recovery of quantum dot semiconductor optical amplifiers,” Appl. Phys. Lett.94(4), 041112 (2009).
[CrossRef]

A. Capua, V. Mikhelashvili, G. Eisenstein, J. P. Reithmaier, A. Somers, A. Forchel, M. Calligaro, O. Parillaud, and M. Krakowski, “Direct observation of the coherent spectral hole in the noise spectrum of a saturated InAs/InP quantum dash amplifier operating near 1550 nm,” Opt. Express16(3), 2141–2146 (2008).
[CrossRef] [PubMed]

A. Bilenca and G. Eisenstein, “On the noise properties of linear and nonlinear quantum-dot semiconductor optical amplifiers: the impact of inhomogeneously broadened gain and fast carrier dynamics,” IEEE J. Quantum Electron.40(6), 690–702 (2004).
[CrossRef]

A. Bilenca, R. Alizon, V. Mikhelashvili, D. Dahan, G. Eisenstein, R. Schwertberger, D. Gold, J. P. Reithmaier, and A. Forchel, “Broad-band wavelength conversion based on cross-gain modulation and four-wave mixing in InAs-InP quantum-dash semiconductor optical amplifiers operating at 1550 nm,” IEEE Photon. Technol. Lett.15(4), 563–565 (2003).

M. Shtaif and G. Eisenstein, “Noise properties of nonlinear semiconductor optical amplifiers,” Opt. Lett.21(22), 1851–1853 (1996).
[CrossRef] [PubMed]

M. Shtaif and G. Eisenstein, “Analytical solution of wave mixing between short optical pulses in a semiconductor optical amplifier,” Appl. Phys. Lett.66(12), 1458–1460 (1995).
[CrossRef]

K. L. Hall, J. Mark, E. P. Ippen, and G. Eisenstein, “Femtosecond gain dynamics in InGaAsP optical amplifiers,” Appl. Phys. Lett.56(18), 1740–1742 (1990).
[CrossRef]

Eliseev, P. G.

A. P. Bogatov, P. G. Eliseev, and B. N. Sverdlov, “Anomalous interaction of spectral modes in a semiconductor laser,” IEEE J. Quantum Electron.11(7), 510–515 (1975).
[CrossRef]

Feldmann, J.

J. Zimmermann, S. T. Cundiff, G. von Plessen, J. Feldmann, M. Arzberger, G. Böhm, M.-C. Amann, and G. Abstreiter, “Dark pulse formation in a quantum-dot laser,” Appl. Phys. Lett.79(1), 18–20 (2001).
[CrossRef]

Forchel, A.

A. Capua, V. Mikhelashvili, G. Eisenstein, J. P. Reithmaier, A. Somers, A. Forchel, M. Calligaro, O. Parillaud, and M. Krakowski, “Direct observation of the coherent spectral hole in the noise spectrum of a saturated InAs/InP quantum dash amplifier operating near 1550 nm,” Opt. Express16(3), 2141–2146 (2008).
[CrossRef] [PubMed]

A. Bilenca, R. Alizon, V. Mikhelashvili, D. Dahan, G. Eisenstein, R. Schwertberger, D. Gold, J. P. Reithmaier, and A. Forchel, “Broad-band wavelength conversion based on cross-gain modulation and four-wave mixing in InAs-InP quantum-dash semiconductor optical amplifiers operating at 1550 nm,” IEEE Photon. Technol. Lett.15(4), 563–565 (2003).

Fujimoto, J. G.

C. Sun, B. Golubovic, H. Choi, C. A. Wang, and J. G. Fujimoto, “Femtosecond investigations of spectral hole burning in semiconductor lasers,” Appl. Phys. Lett.66(13), 1650–1652 (1995).
[CrossRef]

Giessen, H.

S. Linden, H. Giessen, and J. Kuhl, “XFROG — A new method for amplitude and phase characterization of weak ultrashort pulses,” Phys. Status Solidi, B Basic Res.206(1), 119–124 (1998).
[CrossRef]

Gold, D.

A. Bilenca, R. Alizon, V. Mikhelashvili, D. Dahan, G. Eisenstein, R. Schwertberger, D. Gold, J. P. Reithmaier, and A. Forchel, “Broad-band wavelength conversion based on cross-gain modulation and four-wave mixing in InAs-InP quantum-dash semiconductor optical amplifiers operating at 1550 nm,” IEEE Photon. Technol. Lett.15(4), 563–565 (2003).

Golubovic, B.

C. Sun, B. Golubovic, H. Choi, C. A. Wang, and J. G. Fujimoto, “Femtosecond investigations of spectral hole burning in semiconductor lasers,” Appl. Phys. Lett.66(13), 1650–1652 (1995).
[CrossRef]

Hall, K. L.

K. L. Hall, J. Mark, E. P. Ippen, and G. Eisenstein, “Femtosecond gain dynamics in InGaAsP optical amplifiers,” Appl. Phys. Lett.56(18), 1740–1742 (1990).
[CrossRef]

Heinrichsdorff, F.

P. Borri, W. Langbein, J. M. Hvam, F. Heinrichsdorff, M.-H. Mao, and D. Bimberg, “Ultrafast gain dynamics in InAs-InGaAs quantum-dot amplifiers,” IEEE Photon. Technol. Lett.12(6), 594–596 (2000).
[CrossRef]

Henry, C. H.

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

Hvam, J. M.

P. Borri, W. Langbein, J. M. Hvam, F. Heinrichsdorff, M.-H. Mao, and D. Bimberg, “Ultrafast gain dynamics in InAs-InGaAs quantum-dot amplifiers,” IEEE Photon. Technol. Lett.12(6), 594–596 (2000).
[CrossRef]

Ippen, E. P.

K. L. Hall, J. Mark, E. P. Ippen, and G. Eisenstein, “Femtosecond gain dynamics in InGaAsP optical amplifiers,” Appl. Phys. Lett.56(18), 1740–1742 (1990).
[CrossRef]

Karni, O.

Kim, J.

J. Kim, C. Meuer, D. Bimberg, and G. Eisenstein, “Role of carrier reservoirs on the slow phase recovery of quantum dot semiconductor optical amplifiers,” Appl. Phys. Lett.94(4), 041112 (2009).
[CrossRef]

Kim, J. E.

J. E. Kim, E. Malic, M. Richter, A. Wilms, and A. Knorr, “Maxwell–bloch equation approach for describing the microscopic dynamics of quantum-dot surface-emitting structures,” IEEE J. Quantum Electron.46(7), 1115–1126 (2010).
[CrossRef]

Knorr, A.

J. E. Kim, E. Malic, M. Richter, A. Wilms, and A. Knorr, “Maxwell–bloch equation approach for describing the microscopic dynamics of quantum-dot surface-emitting structures,” IEEE J. Quantum Electron.46(7), 1115–1126 (2010).
[CrossRef]

Krakowski, M.

Kuhl, J.

S. Linden, H. Giessen, and J. Kuhl, “XFROG — A new method for amplitude and phase characterization of weak ultrashort pulses,” Phys. Status Solidi, B Basic Res.206(1), 119–124 (1998).
[CrossRef]

Langbein, W.

P. Borri, W. Langbein, J. M. Hvam, F. Heinrichsdorff, M.-H. Mao, and D. Bimberg, “Ultrafast gain dynamics in InAs-InGaAs quantum-dot amplifiers,” IEEE Photon. Technol. Lett.12(6), 594–596 (2000).
[CrossRef]

Linden, S.

S. Linden, H. Giessen, and J. Kuhl, “XFROG — A new method for amplitude and phase characterization of weak ultrashort pulses,” Phys. Status Solidi, B Basic Res.206(1), 119–124 (1998).
[CrossRef]

Malic, E.

J. E. Kim, E. Malic, M. Richter, A. Wilms, and A. Knorr, “Maxwell–bloch equation approach for describing the microscopic dynamics of quantum-dot surface-emitting structures,” IEEE J. Quantum Electron.46(7), 1115–1126 (2010).
[CrossRef]

Mao, M.-H.

P. Borri, W. Langbein, J. M. Hvam, F. Heinrichsdorff, M.-H. Mao, and D. Bimberg, “Ultrafast gain dynamics in InAs-InGaAs quantum-dot amplifiers,” IEEE Photon. Technol. Lett.12(6), 594–596 (2000).
[CrossRef]

Mark, J.

A. Uskov, J. Mork, and J. Mark, “Wave mixing in semiconductor laser amplifiers due to carrier heating and spectral-hole burning,” IEEE J. Quantum Electron.30(8), 1769–1781 (1994).
[CrossRef]

K. L. Hall, J. Mark, E. P. Ippen, and G. Eisenstein, “Femtosecond gain dynamics in InGaAsP optical amplifiers,” Appl. Phys. Lett.56(18), 1740–1742 (1990).
[CrossRef]

Mecozzi, A.

A. Mecozzi and J. Mork, “Saturation effects in nondegenerate four-wave mixing between short optical pulses in semiconductor laser amplifiers,” IEEE J. Sel. Top. Quantum Electron.3(5), 1190–1207 (1997).
[CrossRef]

Meuer, C.

J. Kim, C. Meuer, D. Bimberg, and G. Eisenstein, “Role of carrier reservoirs on the slow phase recovery of quantum dot semiconductor optical amplifiers,” Appl. Phys. Lett.94(4), 041112 (2009).
[CrossRef]

Mikhelashvili, V.

A. Capua, V. Mikhelashvili, G. Eisenstein, J. P. Reithmaier, A. Somers, A. Forchel, M. Calligaro, O. Parillaud, and M. Krakowski, “Direct observation of the coherent spectral hole in the noise spectrum of a saturated InAs/InP quantum dash amplifier operating near 1550 nm,” Opt. Express16(3), 2141–2146 (2008).
[CrossRef] [PubMed]

A. Bilenca, R. Alizon, V. Mikhelashvili, D. Dahan, G. Eisenstein, R. Schwertberger, D. Gold, J. P. Reithmaier, and A. Forchel, “Broad-band wavelength conversion based on cross-gain modulation and four-wave mixing in InAs-InP quantum-dash semiconductor optical amplifiers operating at 1550 nm,” IEEE Photon. Technol. Lett.15(4), 563–565 (2003).

Mork, J.

A. Mecozzi and J. Mork, “Saturation effects in nondegenerate four-wave mixing between short optical pulses in semiconductor laser amplifiers,” IEEE J. Sel. Top. Quantum Electron.3(5), 1190–1207 (1997).
[CrossRef]

A. Uskov, J. Mork, and J. Mark, “Wave mixing in semiconductor laser amplifiers due to carrier heating and spectral-hole burning,” IEEE J. Quantum Electron.30(8), 1769–1781 (1994).
[CrossRef]

Olsson, N. A.

P. Agrawal and N. A. Olsson, “Self-phase modulation and spectral broadening of optical pulses in semiconductor laser amplifiers,” IEEE J. Quantum Electron.25(11), 2297–2306 (1989).
[CrossRef]

Parillaud, O.

Reithmaier, J. P.

A. Capua, A. Saal, O. Karni, G. Eisenstein, J. P. Reithmaier, and K. Yvind, “Complex characterization of short-pulse propagation through InAs/InP quantum-dash optical amplifiers: from the quasi-linear to the two-photon-dominated regime,” Opt. Express20(1), 347–353 (2012).
[CrossRef] [PubMed]

A. Capua, G. Eisenstein, and J. P. Reithmaier, “A nearly instantaneous gain response in quantum dash based optical amplifiers,” Appl. Phys. Lett.97(13), 131108 (2010).
[CrossRef]

A. Capua, V. Mikhelashvili, G. Eisenstein, J. P. Reithmaier, A. Somers, A. Forchel, M. Calligaro, O. Parillaud, and M. Krakowski, “Direct observation of the coherent spectral hole in the noise spectrum of a saturated InAs/InP quantum dash amplifier operating near 1550 nm,” Opt. Express16(3), 2141–2146 (2008).
[CrossRef] [PubMed]

A. Bilenca, R. Alizon, V. Mikhelashvili, D. Dahan, G. Eisenstein, R. Schwertberger, D. Gold, J. P. Reithmaier, and A. Forchel, “Broad-band wavelength conversion based on cross-gain modulation and four-wave mixing in InAs-InP quantum-dash semiconductor optical amplifiers operating at 1550 nm,” IEEE Photon. Technol. Lett.15(4), 563–565 (2003).

Richter, M.

J. E. Kim, E. Malic, M. Richter, A. Wilms, and A. Knorr, “Maxwell–bloch equation approach for describing the microscopic dynamics of quantum-dot surface-emitting structures,” IEEE J. Quantum Electron.46(7), 1115–1126 (2010).
[CrossRef]

Saal, A.

Schwertberger, R.

A. Bilenca, R. Alizon, V. Mikhelashvili, D. Dahan, G. Eisenstein, R. Schwertberger, D. Gold, J. P. Reithmaier, and A. Forchel, “Broad-band wavelength conversion based on cross-gain modulation and four-wave mixing in InAs-InP quantum-dash semiconductor optical amplifiers operating at 1550 nm,” IEEE Photon. Technol. Lett.15(4), 563–565 (2003).

Shtaif, M.

M. Shtaif and G. Eisenstein, “Noise properties of nonlinear semiconductor optical amplifiers,” Opt. Lett.21(22), 1851–1853 (1996).
[CrossRef] [PubMed]

M. Shtaif and G. Eisenstein, “Analytical solution of wave mixing between short optical pulses in a semiconductor optical amplifier,” Appl. Phys. Lett.66(12), 1458–1460 (1995).
[CrossRef]

Slavcheva, G. M.

G. M. Slavcheva, J. M. Arnold, and R. W. Ziolkowski, “FDTD simulation of the nonlinear gain dynamics in active optical waveguides and semiconductor microcavities,” IEEE J. Sel. Top. Quantum Electron.10(5), 1052–1062 (2004).
[CrossRef]

Somers, A.

Sun, C.

C. Sun, B. Golubovic, H. Choi, C. A. Wang, and J. G. Fujimoto, “Femtosecond investigations of spectral hole burning in semiconductor lasers,” Appl. Phys. Lett.66(13), 1650–1652 (1995).
[CrossRef]

Sverdlov, B. N.

A. P. Bogatov, P. G. Eliseev, and B. N. Sverdlov, “Anomalous interaction of spectral modes in a semiconductor laser,” IEEE J. Quantum Electron.11(7), 510–515 (1975).
[CrossRef]

Uskov, A.

A. Uskov, J. Mork, and J. Mark, “Wave mixing in semiconductor laser amplifiers due to carrier heating and spectral-hole burning,” IEEE J. Quantum Electron.30(8), 1769–1781 (1994).
[CrossRef]

von Plessen, G.

J. Zimmermann, S. T. Cundiff, G. von Plessen, J. Feldmann, M. Arzberger, G. Böhm, M.-C. Amann, and G. Abstreiter, “Dark pulse formation in a quantum-dot laser,” Appl. Phys. Lett.79(1), 18–20 (2001).
[CrossRef]

Wang, C. A.

C. Sun, B. Golubovic, H. Choi, C. A. Wang, and J. G. Fujimoto, “Femtosecond investigations of spectral hole burning in semiconductor lasers,” Appl. Phys. Lett.66(13), 1650–1652 (1995).
[CrossRef]

Wilms, A.

J. E. Kim, E. Malic, M. Richter, A. Wilms, and A. Knorr, “Maxwell–bloch equation approach for describing the microscopic dynamics of quantum-dot surface-emitting structures,” IEEE J. Quantum Electron.46(7), 1115–1126 (2010).
[CrossRef]

Yvind, K.

Zimmermann, J.

J. Zimmermann, S. T. Cundiff, G. von Plessen, J. Feldmann, M. Arzberger, G. Böhm, M.-C. Amann, and G. Abstreiter, “Dark pulse formation in a quantum-dot laser,” Appl. Phys. Lett.79(1), 18–20 (2001).
[CrossRef]

Ziolkowski, R. W.

G. M. Slavcheva, J. M. Arnold, and R. W. Ziolkowski, “FDTD simulation of the nonlinear gain dynamics in active optical waveguides and semiconductor microcavities,” IEEE J. Sel. Top. Quantum Electron.10(5), 1052–1062 (2004).
[CrossRef]

Appl. Phys. Lett.

K. L. Hall, J. Mark, E. P. Ippen, and G. Eisenstein, “Femtosecond gain dynamics in InGaAsP optical amplifiers,” Appl. Phys. Lett.56(18), 1740–1742 (1990).
[CrossRef]

A. Capua, G. Eisenstein, and J. P. Reithmaier, “A nearly instantaneous gain response in quantum dash based optical amplifiers,” Appl. Phys. Lett.97(13), 131108 (2010).
[CrossRef]

J. Zimmermann, S. T. Cundiff, G. von Plessen, J. Feldmann, M. Arzberger, G. Böhm, M.-C. Amann, and G. Abstreiter, “Dark pulse formation in a quantum-dot laser,” Appl. Phys. Lett.79(1), 18–20 (2001).
[CrossRef]

C. Sun, B. Golubovic, H. Choi, C. A. Wang, and J. G. Fujimoto, “Femtosecond investigations of spectral hole burning in semiconductor lasers,” Appl. Phys. Lett.66(13), 1650–1652 (1995).
[CrossRef]

M. Shtaif and G. Eisenstein, “Analytical solution of wave mixing between short optical pulses in a semiconductor optical amplifier,” Appl. Phys. Lett.66(12), 1458–1460 (1995).
[CrossRef]

J. Kim, C. Meuer, D. Bimberg, and G. Eisenstein, “Role of carrier reservoirs on the slow phase recovery of quantum dot semiconductor optical amplifiers,” Appl. Phys. Lett.94(4), 041112 (2009).
[CrossRef]

IEEE J. Quantum Electron.

J. E. Kim, E. Malic, M. Richter, A. Wilms, and A. Knorr, “Maxwell–bloch equation approach for describing the microscopic dynamics of quantum-dot surface-emitting structures,” IEEE J. Quantum Electron.46(7), 1115–1126 (2010).
[CrossRef]

A. Uskov, J. Mork, and J. Mark, “Wave mixing in semiconductor laser amplifiers due to carrier heating and spectral-hole burning,” IEEE J. Quantum Electron.30(8), 1769–1781 (1994).
[CrossRef]

P. Agrawal and N. A. Olsson, “Self-phase modulation and spectral broadening of optical pulses in semiconductor laser amplifiers,” IEEE J. Quantum Electron.25(11), 2297–2306 (1989).
[CrossRef]

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

A. P. Bogatov, P. G. Eliseev, and B. N. Sverdlov, “Anomalous interaction of spectral modes in a semiconductor laser,” IEEE J. Quantum Electron.11(7), 510–515 (1975).
[CrossRef]

A. Bilenca and G. Eisenstein, “On the noise properties of linear and nonlinear quantum-dot semiconductor optical amplifiers: the impact of inhomogeneously broadened gain and fast carrier dynamics,” IEEE J. Quantum Electron.40(6), 690–702 (2004).
[CrossRef]

IEEE J. Sel. Top. Quantum Electron.

G. M. Slavcheva, J. M. Arnold, and R. W. Ziolkowski, “FDTD simulation of the nonlinear gain dynamics in active optical waveguides and semiconductor microcavities,” IEEE J. Sel. Top. Quantum Electron.10(5), 1052–1062 (2004).
[CrossRef]

A. Mecozzi and J. Mork, “Saturation effects in nondegenerate four-wave mixing between short optical pulses in semiconductor laser amplifiers,” IEEE J. Sel. Top. Quantum Electron.3(5), 1190–1207 (1997).
[CrossRef]

IEEE Photon. Technol. Lett.

A. Bilenca, R. Alizon, V. Mikhelashvili, D. Dahan, G. Eisenstein, R. Schwertberger, D. Gold, J. P. Reithmaier, and A. Forchel, “Broad-band wavelength conversion based on cross-gain modulation and four-wave mixing in InAs-InP quantum-dash semiconductor optical amplifiers operating at 1550 nm,” IEEE Photon. Technol. Lett.15(4), 563–565 (2003).

P. Borri, W. Langbein, J. M. Hvam, F. Heinrichsdorff, M.-H. Mao, and D. Bimberg, “Ultrafast gain dynamics in InAs-InGaAs quantum-dot amplifiers,” IEEE Photon. Technol. Lett.12(6), 594–596 (2000).
[CrossRef]

J. Opt. Soc. Am. B

Opt. Express

Opt. Lett.

Phys. Status Solidi, B Basic Res.

S. Linden, H. Giessen, and J. Kuhl, “XFROG — A new method for amplitude and phase characterization of weak ultrashort pulses,” Phys. Status Solidi, B Basic Res.206(1), 119–124 (1998).
[CrossRef]

Other

R. Trebino, Frequency-Resolved Optical Gating: The Measurement of Ultrashort Laser Pulses (Kluwer Academic Publishers, Norwell, 2002).

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Figures (15)

Fig. 1
Fig. 1

Illustration of the various signals accounted for in the analytical model. Their directions of propagation are marked by the colored arrows, where their central angular frequencies are noted as well.

Fig. 2
Fig. 2

Calculated propagation of the pulse and the generation of the FWM products along the waveguide. (a) near the input facet, (b) at the center and (c) near the output facet. The traces show in logarithmic scale the features at low intensity relative to the pulse peak. The color code of the plot matches the one used in Fig. 1.

Fig. 3
Fig. 3

Calculated time-dependent amplitudes of (a) the output signal at the right output facet, (b) the signals exiting the left input facet. The inset shows a magnification of the trailing edge of the output signal, as composed of the right propagating pulse and idler.

Fig. 4
Fig. 4

Calculated time-dependent amplitude and inverted phase of the output signal.

Fig. 5
Fig. 5

Calculated FWM signatures in the spectra of the output signal for gain recovery times of 1 ps, 2 ps and 5 ps. The inset presents an overview of the spectrum for the case of 5 ps.

Fig. 6
Fig. 6

Calculated sum-frequency spectrogram of the output pulse gated by the input pulse. The trace is magnified by power of 1/5 for visualization of details. Clear dips are visible near the trailing edge, followed by the spectrally-narrow wave mixing tail (circled in red).

Fig. 7
Fig. 7

A schematic description of the energy levels and interactions considered in the numerical model. The electron and hole reservoir is fed by the electrical bias.

Fig. 8
Fig. 8

Calculated spectrum of the output pulse (blue), with respect to the spectrum of the initial pulse (green). The spectrum of the cw multi-mode oscillations is plotted in red as a frequency reference. The FWM induced spectral hole is visible on the left of the cw trace. The inset zooms into the cw spectrum, noting its multi-mode nature. The traces are shifted in intensity for clarity.

Fig. 9
Fig. 9

Calculated time-dependent amplitude and phase of the output pulse. The phase on the tail of the pulse decreases linearly due to the spectral shift of the FWM generated tail from the pulse carrier frequency.

Fig. 10
Fig. 10

Calculated spectra of the output pulse for different input pulse energies. The spectral hole is less distinct as the input energy increases.

Fig. 11
Fig. 11

Calculated sum-frequency spectrogram of the FDTD simulation output pulse gated by the input pulse. The trace is enhanced by a power of 1/5 for clarity. A clear dip is visible near the trailing edge, followed by the spectrally-narrow wave mixing tail (circled in red). The other holes are SPM footprints.

Fig. 12
Fig. 12

Measured bias dependent ASE spectra. The gain peak experiences a blue shift with increasing bias. The oscillation threshold is around 75 mA.

Fig. 13
Fig. 13

(a) Measured X-FROG trace (spectrogram). (b) Retrieved X-FROG trace. The images are enhanced by a power of 1/3 for clarity. The wave mixing-generated tail and dip are circled in red.

Fig. 14
Fig. 14

Retrieved time-dependent amplitudes and spectra of the amplified pulse at different bias levels. (a) and (b) show time-dependent pulses initially at 1470 nm and 1510 nm, respectively. The inset in (a) and the left inset in (b) show the instantaneous phase of the amplified pulse at 100 mA bias. The right inset in (b) is a magnification of the trailing edge of the amplified pulse at 1510 nm. (c) and (d) are the spectra of the 1470 nm and 1510 nm pulses, respectively. The wave mixing hole is visible in the 1485 nm region in (c), and near 1495 nm in (d).

Fig. 15
Fig. 15

Retrieved spectra of 1510 nm, 50 pJ pulses for different bias levels. The wave mixing signature is masked by other effects.

Tables (2)

Tables Icon

Table 1 List of parameters used for the semi-analytical FWM evaluation.

Tables Icon

Table 2 List of parameters used for the FDTD simulation.

Equations (34)

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2 E n b 2 c 2 2 E t 2 = 1 ε 0 c 2 2 P t 2
2 E n b 2 c 2 2 E t 2 1 c 2 2 ( χ E ) t 2 =0
χ t = χ ss χ τ χ τ | E | 2 I s
E=F( x,y ) 1 2 { E f ( z ) e i( ω 0 t k 0 z ) + E b ( z ) e i( ω 0 t+ k 0 z ) +c.c. + A pf ( z,t ) e i( ω p t k p z ) + A pb ( z,t ) e i( ω p t+ k p z ) +c.c. + A if ( z,t ) e i( ω i t k i z ) + A ib ( z,t ) e i( ω i t+ k i z ) +c.c.}
χ= χ 0 +( χ long_f ( z,t ) e i( Ωt( k 0 k p )z ) + χ ˜ long_f ( z,t ) e i( Ωt( k 0 k p )z ) ) +( χ long_b ( z,t ) e i( Ωt+( k 0 k p )z ) + χ ˜ long_b ( z,t ) e i( Ωt+( k 0 k p )z ) ) +( χ short_fp ( z,t ) e i( Ωt( k 0 + k p )z ) + χ ˜ short_fp ( z,t ) e i( Ωt( k 0 + k p )z ) ) +( χ short_bp ( z,t ) e i( Ωt+( k 0 + k p )z ) + χ ˜ short_bp ( z,t ) e i( Ωt+( k 0 + k p )z ) ) +( χ short_fi ( z,t ) e i( Ωt( k 0 + k i )z ) + χ ˜ short_fi ( z,t ) e i( Ωt( k 0 + k i )z ) ) +( χ short_bi ( z,t ) e i( Ωt+( k 0 + k i )z ) + χ ˜ short_bi ( z,t ) e i( Ωt+( k 0 + k i )z ) )
χ 0 = χ ss 1+ I 0
χ long_f t = χ long_f 1+ I 0 +iΩτ τ χ 0 4τ ( E b A ib * I s + A pb E b * I s ) 1 4τ ( E f E b * I s χ short_bp + E f * E b I s χ short_fi * )
χ long_b t = χ long_b 1+ I 0 +iΩτ τ χ 0 4τ ( E f A if * I s + A pf E f * I s ) 1 4τ ( E f E b * I s χ short_bi + E f * E b I s χ short_fp * )
χ short_fp t = χ short_fp 1+ I 0 +iΩτ τ χ 0 4τ A pf E b * I s χ long_b 4τ E f E b * I s
χ short_bp t = χ short_bp 1+ I 0 +iΩτ τ χ 0 4τ A pb E f * I s χ long_f 4τ E b E f * I s
χ short_fi t = χ short_fi 1+ I 0 +iΩτ τ χ 0 4τ A ib * E f I s χ long_f 4τ E f E b * I s
χ short_bi t = χ short_bi 1+ I 0 +iΩτ τ χ 0 4τ A if * E b I s χ long_b 4τ E b E f * I s
2 F ω 0 2 c 2 ( n ¯ 2 n b 2 )F=0
E f z = i 2 Γ ω 0 n ¯ c χ 0 E f
E b z = i 2 Γ ω 0 n ¯ c χ 0 E b
A pf z + 1 v g A pf t =i ω 0 Ω ω p ( n ¯ c 1 v g ) A pf i 2 Γ ω p n ¯ c χ 0 A pf i 2 Γ ω p n ¯ c [ ( χ long_b + χ long_f e iΔkz ) E f +( χ short_fp + χ short_fi e iΔkz ) E b ] Γ n ¯ c [ ( χ long_b t + χ long_f t e iΔkz ) E f +( χ short_fp t + χ short_fi t e iΔkz ) E b ]
A pb z 1 v g A pb t =i ω 0 Ω ω p ( n ¯ c 1 v g ) A pb + i 2 Γ ω p n ¯ c χ 0 A pb + i 2 Γ ω p n ¯ c [ ( χ short_bp + χ short_bi e iΔkz ) E f +( χ long_f + χ long_b e iΔkz ) E b ] + i 2 Γ n ¯ c [ ( χ short_bp t + χ short_bi t e iΔkz ) E f +( χ long_f t + χ long_b t e iΔkz ) E b ]
A if z + 1 v g A if t =i ω 0 Ω ω i ( n ¯ c 1 v g ) A if i 2 Γ ω i n ¯ c χ 0 A if i 2 Γ ω i n ¯ c [ ( χ ˜ long_b + χ ˜ long_f e iΔkz ) E f +( χ ˜ short_bi + χ ˜ short_bp e iΔkz ) E b ] Γ n ¯ c [ ( χ ˜ long_b t + χ ˜ long_f t e iΔkz ) E f +( χ ˜ short_bi t + χ ˜ short_bp t e iΔkz ) E b ]
A ib z 1 v g A ib t =i ω 0 Ω ω i ( n ¯ c 1 v g ) A ib + i 2 Γ ω i n ¯ c χ 0 A ib + i 2 Γ ω i n ¯ c [ ( χ ˜ short_fi + χ ˜ short_fp e iΔkz ) E f +( χ ˜ long_f + χ ˜ long_b e iΔkz ) E b ] Γ n ¯ c [ ( χ ˜ short_fi t + χ ˜ short_fp t e iΔkz ) E f +( χ ˜ long_f t + χ ˜ long_b t e iΔkz ) E b ]
χ( z,t )= c n ¯ ω 0 ( αi )g( z,t )
ρ 11 t = N es 2 N D τ es11 ( 1 ρ 11 ) ρ 11 τ 11es ( 1 N es 2 N D ) γ c ρ 11 i μ x ( ρ 12 ρ 21 ) E x
ρ 22 t = 1 τ esc h ( 1 ρ 22 )( 1 P res D res ) 1 τ cap h P res 2 N D ρ 22 γ v ρ 22 +i μ x ( ρ 12 ρ 21 ) E x
ρ 12 t =( iω+ γ in ) ρ 12 i μ x ( ρ 11 ρ 22 ) E x
N res t = η i I qV N res τ res N res τ cap ( 1 N es 2 N D )+ N es τ esc ( 1 N res D res )
N es t = N res τ cap ( 1 N es 2 N D ) N es τ esc ( 1 N res D res ) N es τ es11 ( 1 ρ 11 ) + 2 N D ρ 11 τ 11es ( 1 N es 2 N D ) N es τ es
P res t = η i I qV N res τ res N es τ es γ c 2 N D ρ 11 P res τ cap h ρ 22 + 1 τ esc h 2 N D ( 1 ρ 22 )( 1 P res D res ) γ v 2 N D ρ 22
E x z = μ 0 H y t
H y z =σ E x + D x t
D x = ε 0 E x + P x
P x = P mat + P dot
P mat = ε 0 χ e E x
P dot =2 N D μ x ( ρ 12 + ρ 12 * )Γ
n 2 = ε r = ε r 0 ε( N res ,2 N D ρ 11 )= ε r 0 C res N res C QD 2 N D ρ 11
E x ( t )= 1 2 A( t )( e i ω 0 tiϕ( t ) + e i ω 0 t+iϕ( t ) )

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