Abstract

A Stochastic Simulator (SS) is proposed, based on a semiclassical description of the radiation-matter interaction, to obtain an efficient description of the lasing transition for devices ranging from the nanolaser to the traditional “macroscopic” laser. Steady-state predictions obtained with the SS agree both with more traditional laser modeling and with the description of phase transitions in small-sized systems, and provide additional information on fluctuations. Dynamical information can easily be obtained, with good computing time efficiency, which convincingly highlights the role of fluctuations at threshold.

© 2015 Optical Society of America

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References

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  1. H. Soda, K. Iga, C. Kitahara, and Y. Suematsu, “GaInAsP/InP surface emitting injection lasers,” Jpn. J. Appl. Phys. 18 (12), 2329 (1979).
    [Crossref]
  2. G. Björk and Y. Yamamoto, “Analysis of semiconductor microcavity lasers using rate equations,” IEEE J. Quantum Electron. 27(11), 2386 (1991).
    [Crossref]
  3. W.W. Chow, F. Jahnke, and Ch. Gies, “Emission properties of nanolasers during the transition to lasing,” Light Sci. Appl. 3, e201 (2014).
  4. A. Lebreton, I. Abram, N. Takemura, M. Kuwata-Gonokami, I. Robert-Philip, and A. Beveratos, “Stochastically sustained population oscillations in high-β nanolasers,” New J. Phys. 15, 033039 (2013).
    [Crossref]
  5. L. Chusseau, F. Philippe, and F. Disanto, “Monte Carlo modeling of the dual-mode regime in quantum-well and quantum-dot semiconductor lasers,” Opt. Express 22(5), 5312 (2014).
    [Crossref] [PubMed]
  6. H. Risken, The Fokker-Planck Equation, Springer Series in Synergetics, (Springer, 1984, 18).
  7. K. Roy-Choudhury, S. Haas, and A. F. J. Levi, “Quantum fluctuations in small lasers,” Phys. Rev. Lett. 102(5), 053902 (2009).
    [Crossref] [PubMed]
  8. M. Lorke, T. Suhr, N. Gregersen, and J. Mørk, “Theory of nanolaser devices: rate equation analysis versus microscopic theory,” Phys. Rev. B 87(20), 205310 (2013).
    [Crossref]
  9. H. Haken, “Cooperative phenomena in systems far from thermal equilibrium and in nonphysical systems,” Rev. Mod. Phys. 47(1), 67 (1975)
    [Crossref]
  10. A.E. Siegman, Lasers (University Science Books, 1986).
  11. L. M. Narducci and N. B. Abraham, Laser Physics and Laser Instabilities (World Scientific, 1988).
    [Crossref]
  12. L.E. Reichl, A Modern Course in Statistical Physics, (Wiley, 1998).
  13. V. Dohm, “Nonequilibrium phase transition in laser-active media,” Solid State Commun. 11 (9), 1273 (1972).
    [Crossref]
  14. J. R. Tredicce, F. T. Arecchi, G. L. Lippi, and G. P. Puccioni, “Instabilities in lasers with an injected signal,” J. Opt. Soc. Am. B2(1), 173 (1985).
    [Crossref]
  15. M. Galassi, J. Davies, J. Theiler, B. Gough, G. Jungman, P. Alken, M. Booth, and F. Rossi, “GNU scientific library reference manual,” (3rd Ed.), http://www.gnu.org/software/gsl/ .

2014 (2)

W.W. Chow, F. Jahnke, and Ch. Gies, “Emission properties of nanolasers during the transition to lasing,” Light Sci. Appl. 3, e201 (2014).

L. Chusseau, F. Philippe, and F. Disanto, “Monte Carlo modeling of the dual-mode regime in quantum-well and quantum-dot semiconductor lasers,” Opt. Express 22(5), 5312 (2014).
[Crossref] [PubMed]

2013 (2)

M. Lorke, T. Suhr, N. Gregersen, and J. Mørk, “Theory of nanolaser devices: rate equation analysis versus microscopic theory,” Phys. Rev. B 87(20), 205310 (2013).
[Crossref]

A. Lebreton, I. Abram, N. Takemura, M. Kuwata-Gonokami, I. Robert-Philip, and A. Beveratos, “Stochastically sustained population oscillations in high-β nanolasers,” New J. Phys. 15, 033039 (2013).
[Crossref]

2009 (1)

K. Roy-Choudhury, S. Haas, and A. F. J. Levi, “Quantum fluctuations in small lasers,” Phys. Rev. Lett. 102(5), 053902 (2009).
[Crossref] [PubMed]

1991 (1)

G. Björk and Y. Yamamoto, “Analysis of semiconductor microcavity lasers using rate equations,” IEEE J. Quantum Electron. 27(11), 2386 (1991).
[Crossref]

1985 (1)

J. R. Tredicce, F. T. Arecchi, G. L. Lippi, and G. P. Puccioni, “Instabilities in lasers with an injected signal,” J. Opt. Soc. Am. B2(1), 173 (1985).
[Crossref]

1979 (1)

H. Soda, K. Iga, C. Kitahara, and Y. Suematsu, “GaInAsP/InP surface emitting injection lasers,” Jpn. J. Appl. Phys. 18 (12), 2329 (1979).
[Crossref]

1975 (1)

H. Haken, “Cooperative phenomena in systems far from thermal equilibrium and in nonphysical systems,” Rev. Mod. Phys. 47(1), 67 (1975)
[Crossref]

1972 (1)

V. Dohm, “Nonequilibrium phase transition in laser-active media,” Solid State Commun. 11 (9), 1273 (1972).
[Crossref]

Abraham, N. B.

L. M. Narducci and N. B. Abraham, Laser Physics and Laser Instabilities (World Scientific, 1988).
[Crossref]

Abram, I.

A. Lebreton, I. Abram, N. Takemura, M. Kuwata-Gonokami, I. Robert-Philip, and A. Beveratos, “Stochastically sustained population oscillations in high-β nanolasers,” New J. Phys. 15, 033039 (2013).
[Crossref]

Arecchi, F. T.

J. R. Tredicce, F. T. Arecchi, G. L. Lippi, and G. P. Puccioni, “Instabilities in lasers with an injected signal,” J. Opt. Soc. Am. B2(1), 173 (1985).
[Crossref]

Beveratos, A.

A. Lebreton, I. Abram, N. Takemura, M. Kuwata-Gonokami, I. Robert-Philip, and A. Beveratos, “Stochastically sustained population oscillations in high-β nanolasers,” New J. Phys. 15, 033039 (2013).
[Crossref]

Björk, G.

G. Björk and Y. Yamamoto, “Analysis of semiconductor microcavity lasers using rate equations,” IEEE J. Quantum Electron. 27(11), 2386 (1991).
[Crossref]

Chow, W.W.

W.W. Chow, F. Jahnke, and Ch. Gies, “Emission properties of nanolasers during the transition to lasing,” Light Sci. Appl. 3, e201 (2014).

Chusseau, L.

Disanto, F.

Dohm, V.

V. Dohm, “Nonequilibrium phase transition in laser-active media,” Solid State Commun. 11 (9), 1273 (1972).
[Crossref]

Gies, Ch.

W.W. Chow, F. Jahnke, and Ch. Gies, “Emission properties of nanolasers during the transition to lasing,” Light Sci. Appl. 3, e201 (2014).

Gregersen, N.

M. Lorke, T. Suhr, N. Gregersen, and J. Mørk, “Theory of nanolaser devices: rate equation analysis versus microscopic theory,” Phys. Rev. B 87(20), 205310 (2013).
[Crossref]

Haas, S.

K. Roy-Choudhury, S. Haas, and A. F. J. Levi, “Quantum fluctuations in small lasers,” Phys. Rev. Lett. 102(5), 053902 (2009).
[Crossref] [PubMed]

Haken, H.

H. Haken, “Cooperative phenomena in systems far from thermal equilibrium and in nonphysical systems,” Rev. Mod. Phys. 47(1), 67 (1975)
[Crossref]

Iga, K.

H. Soda, K. Iga, C. Kitahara, and Y. Suematsu, “GaInAsP/InP surface emitting injection lasers,” Jpn. J. Appl. Phys. 18 (12), 2329 (1979).
[Crossref]

Jahnke, F.

W.W. Chow, F. Jahnke, and Ch. Gies, “Emission properties of nanolasers during the transition to lasing,” Light Sci. Appl. 3, e201 (2014).

Kitahara, C.

H. Soda, K. Iga, C. Kitahara, and Y. Suematsu, “GaInAsP/InP surface emitting injection lasers,” Jpn. J. Appl. Phys. 18 (12), 2329 (1979).
[Crossref]

Kuwata-Gonokami, M.

A. Lebreton, I. Abram, N. Takemura, M. Kuwata-Gonokami, I. Robert-Philip, and A. Beveratos, “Stochastically sustained population oscillations in high-β nanolasers,” New J. Phys. 15, 033039 (2013).
[Crossref]

Lebreton, A.

A. Lebreton, I. Abram, N. Takemura, M. Kuwata-Gonokami, I. Robert-Philip, and A. Beveratos, “Stochastically sustained population oscillations in high-β nanolasers,” New J. Phys. 15, 033039 (2013).
[Crossref]

Levi, A. F. J.

K. Roy-Choudhury, S. Haas, and A. F. J. Levi, “Quantum fluctuations in small lasers,” Phys. Rev. Lett. 102(5), 053902 (2009).
[Crossref] [PubMed]

Lippi, G. L.

J. R. Tredicce, F. T. Arecchi, G. L. Lippi, and G. P. Puccioni, “Instabilities in lasers with an injected signal,” J. Opt. Soc. Am. B2(1), 173 (1985).
[Crossref]

Lorke, M.

M. Lorke, T. Suhr, N. Gregersen, and J. Mørk, “Theory of nanolaser devices: rate equation analysis versus microscopic theory,” Phys. Rev. B 87(20), 205310 (2013).
[Crossref]

Mørk, J.

M. Lorke, T. Suhr, N. Gregersen, and J. Mørk, “Theory of nanolaser devices: rate equation analysis versus microscopic theory,” Phys. Rev. B 87(20), 205310 (2013).
[Crossref]

Narducci, L. M.

L. M. Narducci and N. B. Abraham, Laser Physics and Laser Instabilities (World Scientific, 1988).
[Crossref]

Philippe, F.

Puccioni, G. P.

J. R. Tredicce, F. T. Arecchi, G. L. Lippi, and G. P. Puccioni, “Instabilities in lasers with an injected signal,” J. Opt. Soc. Am. B2(1), 173 (1985).
[Crossref]

Reichl, L.E.

L.E. Reichl, A Modern Course in Statistical Physics, (Wiley, 1998).

Risken, H.

H. Risken, The Fokker-Planck Equation, Springer Series in Synergetics, (Springer, 1984, 18).

Robert-Philip, I.

A. Lebreton, I. Abram, N. Takemura, M. Kuwata-Gonokami, I. Robert-Philip, and A. Beveratos, “Stochastically sustained population oscillations in high-β nanolasers,” New J. Phys. 15, 033039 (2013).
[Crossref]

Roy-Choudhury, K.

K. Roy-Choudhury, S. Haas, and A. F. J. Levi, “Quantum fluctuations in small lasers,” Phys. Rev. Lett. 102(5), 053902 (2009).
[Crossref] [PubMed]

Siegman, A.E.

A.E. Siegman, Lasers (University Science Books, 1986).

Soda, H.

H. Soda, K. Iga, C. Kitahara, and Y. Suematsu, “GaInAsP/InP surface emitting injection lasers,” Jpn. J. Appl. Phys. 18 (12), 2329 (1979).
[Crossref]

Suematsu, Y.

H. Soda, K. Iga, C. Kitahara, and Y. Suematsu, “GaInAsP/InP surface emitting injection lasers,” Jpn. J. Appl. Phys. 18 (12), 2329 (1979).
[Crossref]

Suhr, T.

M. Lorke, T. Suhr, N. Gregersen, and J. Mørk, “Theory of nanolaser devices: rate equation analysis versus microscopic theory,” Phys. Rev. B 87(20), 205310 (2013).
[Crossref]

Takemura, N.

A. Lebreton, I. Abram, N. Takemura, M. Kuwata-Gonokami, I. Robert-Philip, and A. Beveratos, “Stochastically sustained population oscillations in high-β nanolasers,” New J. Phys. 15, 033039 (2013).
[Crossref]

Tredicce, J. R.

J. R. Tredicce, F. T. Arecchi, G. L. Lippi, and G. P. Puccioni, “Instabilities in lasers with an injected signal,” J. Opt. Soc. Am. B2(1), 173 (1985).
[Crossref]

Yamamoto, Y.

G. Björk and Y. Yamamoto, “Analysis of semiconductor microcavity lasers using rate equations,” IEEE J. Quantum Electron. 27(11), 2386 (1991).
[Crossref]

IEEE J. Quantum Electron. (1)

G. Björk and Y. Yamamoto, “Analysis of semiconductor microcavity lasers using rate equations,” IEEE J. Quantum Electron. 27(11), 2386 (1991).
[Crossref]

J. Opt. Soc. Am. (1)

J. R. Tredicce, F. T. Arecchi, G. L. Lippi, and G. P. Puccioni, “Instabilities in lasers with an injected signal,” J. Opt. Soc. Am. B2(1), 173 (1985).
[Crossref]

Jpn. J. Appl. Phys. (1)

H. Soda, K. Iga, C. Kitahara, and Y. Suematsu, “GaInAsP/InP surface emitting injection lasers,” Jpn. J. Appl. Phys. 18 (12), 2329 (1979).
[Crossref]

Light Sci. Appl. (1)

W.W. Chow, F. Jahnke, and Ch. Gies, “Emission properties of nanolasers during the transition to lasing,” Light Sci. Appl. 3, e201 (2014).

New J. Phys. (1)

A. Lebreton, I. Abram, N. Takemura, M. Kuwata-Gonokami, I. Robert-Philip, and A. Beveratos, “Stochastically sustained population oscillations in high-β nanolasers,” New J. Phys. 15, 033039 (2013).
[Crossref]

Opt. Express (1)

Phys. Rev. B (1)

M. Lorke, T. Suhr, N. Gregersen, and J. Mørk, “Theory of nanolaser devices: rate equation analysis versus microscopic theory,” Phys. Rev. B 87(20), 205310 (2013).
[Crossref]

Phys. Rev. Lett. (1)

K. Roy-Choudhury, S. Haas, and A. F. J. Levi, “Quantum fluctuations in small lasers,” Phys. Rev. Lett. 102(5), 053902 (2009).
[Crossref] [PubMed]

Rev. Mod. Phys. (1)

H. Haken, “Cooperative phenomena in systems far from thermal equilibrium and in nonphysical systems,” Rev. Mod. Phys. 47(1), 67 (1975)
[Crossref]

Solid State Commun. (1)

V. Dohm, “Nonequilibrium phase transition in laser-active media,” Solid State Commun. 11 (9), 1273 (1972).
[Crossref]

Other (5)

A.E. Siegman, Lasers (University Science Books, 1986).

L. M. Narducci and N. B. Abraham, Laser Physics and Laser Instabilities (World Scientific, 1988).
[Crossref]

L.E. Reichl, A Modern Course in Statistical Physics, (Wiley, 1998).

H. Risken, The Fokker-Planck Equation, Springer Series in Synergetics, (Springer, 1984, 18).

M. Galassi, J. Davies, J. Theiler, B. Gough, G. Jungman, P. Alken, M. Booth, and F. Rossi, “GNU scientific library reference manual,” (3rd Ed.), http://www.gnu.org/software/gsl/ .

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Figures (3)

Fig. 1
Fig. 1 Scheme of principle of the intracavity processes included in the SS: each dipole is pumped by an external source and may decay by emitting a photon in one of the three following channels – a. the off-axis spontaneous “mode”, grouping all modes other than the lasing one (green photons); b. the on-axis spontaneous mode (blue photons); c. the (on-axis) stimulated mode (red photons). All on-axis photons are recycled by the cavity and are transmitted by the coupling mirror; the off-axis (green) ones exit the interaction volume laterally. All processes (including the mirror transmission) are described stochastically. In the schematic representation, we summarize the ensemble of processes leading to the population of the upper state – with possible population inversion – as a single upwards magenta arrow. The large bracket denotes the interaction processes occurring between each individual dipole and the radiation. Absorption is neglected as we consider an ideal system (e.g., perfect four-level laser). Its inclusion is expected to mainly raise the laser threshold.
Fig. 2
Fig. 2 Left panel: Photon number vs. injection current i computed from the steady-states of the rate equations (eq. (25) in [2]) with the following parameter values: γ = 1 × 10−10s−1, ξ = 0.1, τsp = 1×10−9s, τnr = 1×10−10s. Right panel: Average photon number (<S>) as a function of pump (P) obtained from the SS. <S> is computed as the average of the temporal average photon number S ¯ = 1 N = 1 N S over ten series of events (issued from different values input into the random number generator). The corresponding error bars represent the standard deviation of the individual averages and give a measure of the variability of the photon number, due to the stochastic nature of the conversion process. Parameter values: γ = 2.5 × 109s−1, Γc = 1 × 1011s−1, Γo = 5 × 1013s−1. The time step used for evaluating the probabilities (last column of Table 1) is Δ τ = 1 10 × Γ o = 2 fs, thus τr = r Δτ (r already defined). Notice that the curve with β = 10−7 is rescaled by a factor 10−2 both in the horizontal and vertical axes for graphical purposes.
Fig. 3
Fig. 3 Static and dynamical predictions for β = 0.01. Left panel: Average photon number for stimulated photons (blue curve), on-axis (brown) and off-axis (magenta) spontaneous photons as a function of pump. Center panel: dynamical evolution of <S> for three different values of pump, marked by the corresponding colors on the steady-state response (right panel).

Tables (1)

Tables Icon

Table 1 Synoptic of the processes handled by the SS. The off-axis spontaneous emission results from the difference between the population relaxation events (Nd) and the on-axis spontaneous emissions (DL); thus no explicit process is associated to this channel (we assume, for simplicity, all relaxations to be radiative). Each event (column 3) is computed from the probability distribution (column 4) whose first argument is defined in column 5.

Equations (4)

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N q + 1 = N q + N P N d E S ,
S q + 1 = S q + E S L S + S s p ,
R L , q + 1 = R L , q + D L L L S s p ,
R o , q + 1 = R o , q + ( N d D L ) L o ,

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