Abstract

We perform a first-principles calculation of the quantum-limited laser linewidth, testing the predictions of recently developed theories of the laser linewidth based on fluctuations about the known steady-state laser solutions against traditional forms of the Schawlow-Townes linewidth. The numerical study is based on finite-difference time-domain simulations of the semiclassical Maxwell-Bloch lasing equations, augmented with Langevin force terms, and includes the effects of dispersion, losses due to the open boundary of the laser cavity, and non-linear coupling between the amplitude and phase fluctuations (α factor). We find quantitative agreement between the numerical results and the predictions of the noisy steady-state ab initio laser theory (N-SALT), both in the variation of the linewidth with output power, as well as the emergence of side-peaks due to relaxation oscillations.

© 2015 Optical Society of America

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References

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  1. A. L. Schawlow and C. H. Townes, “Infrared and optical masers,” Phys. Rev. 112, 1940–1949 (1958).
    [Crossref]
  2. M. Lax, “Quantum noise v: Phase noise in a homogeneously broadened maser,” in “Physics of Quantum Electronics,”, P. L. Kelley, B. Lax, and P. E. Tannenwald, eds. (McGraw-Hill, 1966).
  3. C. Henry, “Theory of the linewidth of semiconductor-lasers,” IEEE J. Quantum Elect. 18, 259–264 (1982).
    [Crossref]
  4. C. Henry, “Theory of spontaneous emission noise in open resonators and its application to lasers and optical amplifiers,” J. Lightwave Technol. 4, 288–297 (1986).
    [Crossref]
  5. K. Petermann, “Calculated spontaneous emission factor for double-heterostructure injection-lasers with gain-induced waveguiding,” IEEE J. Quantum Elect. 15, 566–570 (1979).
    [Crossref]
  6. H. Haus and S. Kawakami, “On the excess spontaneous emission factor in gain-guided laser-amplifiers,” IEEE J. Quantum Elect. 21, 63–69 (1985).
    [Crossref]
  7. A. Siegman, “Excess spontaneous emission in non-hermitian optical-systems .2. laser-oscillators,” Phys. Rev. A 39, 1264–1268 (1989).
    [Crossref] [PubMed]
  8. W. Hamel and J. Woerdman, “Nonorthogonality of the longitudinal eigenmodes of a laser,” Phys. Rev. A 40, 2785–2787 (1989).
    [Crossref] [PubMed]
  9. W. Hamel and J. Woerdman, “Observation of enhanced fundamental linewidth of a laser due to nonorthogonality of its longitudinal eigenmodes,” Phys. Rev. Lett. 64, 1506–1509 (1990).
    [Crossref] [PubMed]
  10. H. Haken, Laser theory (Springer-Verlag, 1984).
  11. M. Kolobov, L. Davidovich, E. Giacobino, and C. Fabre, “Role of pumping statistics and dynamics of atomic polarization in quantum fluctuations of laser sources,” Phys. Rev. A 47, 1431–1446 (1993).
    [Crossref] [PubMed]
  12. S. Kuppens, M. van Exter, and J. Woerdman, “Quantum-limited linewidth of a bad-cavity laser,” Phys. Rev. Lett. 72, 3815–3818 (1994).
    [Crossref] [PubMed]
  13. M. van Exter, S. Kuppens, and J. Woerdman, “Theory for the linewidth of a bad-cavity laser,” Phys. Rev. A 51, 809–816 (1995).
    [Crossref] [PubMed]
  14. S. Kuppens, M. van Exter, M. Vanduin, and J. Woerdman, “Evidence of nonuniform phase-diffusion in a bad-cavity laser,” IEEE J. Quantum Elect. 31, 1237–1241 (1995).
    [Crossref]
  15. D. Meiser, J. Ye, and M. J. Holland, “Spin squeezing in optical lattice clocks via lattice-based QND measurements,” New J. Phys. 10, 073014 (2008).
    [Crossref]
  16. D. Meiser, J. Ye, D. R. Carlson, and M. J. Holland, “Prospects for a Millihertz-Linewidth Laser,” Phys. Rev. Lett. 102, 163601 (2009).
    [Crossref] [PubMed]
  17. D. Meiser and M. J. Holland, “Steady-state superradiance with alkaline-earth-metal atoms,” Phys. Rev. A 81, 033847 (2010).
    [Crossref]
  18. D. Meiser and M. J. Holland, “Intensity fluctuations in steady-state superradiance,” Phys. Rev. A 81, 063827 (2010).
    [Crossref]
  19. J. G. Bohnet, Z. Chen, J. M. Weiner, D. Meiser, M. J. Holland, and J. K. Thompson, “A steady-state superradiant laser with less than one intracavity photon,” Nature 484, 78–81 (2012).
    [Crossref] [PubMed]
  20. H. E. Türeci, A. D. Stone, and B. Collier, “Self-consistent multimode lasing theory for complex or random lasing media,” Phys. Rev. A 74, 043822 (2006).
    [Crossref]
  21. H. E. Türeci, L. Ge, S. Rotter, and A. D. Stone, “Strong interactions in multimode random lasers,” Science 320, 643–646 (2008).
    [Crossref] [PubMed]
  22. L. Ge, Y. D. Chong, and A. D. Stone, “Steady-state ab initio laser theory: generalizations and analytic results,” Phys. Rev. A 82, 063824 (2010).
    [Crossref]
  23. L. Ge, R. J. Tandy, A. D. Stone, and H. E. Türeci, “Quantitative verification of ab initio self-consistent laser theory,” Opt. Express 16, 16895–16902 (2008).
    [Crossref] [PubMed]
  24. S. Esterhazy, D. Liu, M. Liertzer, A. Cerjan, L. Ge, K. G. Makris, A. D. Stone, J. M. Melenk, S. G. Johnson, and S. Rotter, “Scalable numerical approach for the steady-state ab initio laser theory,” Phys. Rev. A 90, 023816 (2014).
    [Crossref]
  25. Y. D. Chong and A. D. Stone, “General linewidth formula for steady-state multimode lasing in arbitrary cavities,” Phys. Rev. Lett. 109, 063902 (2012).
    [Crossref] [PubMed]
  26. J. C. Pillay, Y. Natsume, A. D. Stone, and Y. D. Chong, “Generalized subSchawlow-Townes laser linewidths via material dispersion,” Phys. Rev. A 89, 033840 (2014).
    [Crossref]
  27. A. Pick, A. Cerjan, D. Liu, A. Rodriguez, A. D. Stone, Y. D. Chong, and S. Johnson, “Ab initio multimode linewidth theory for arbitrary inhomogeneous laser cavities,” Phys. Rev. A 91, 063806 (2015).
    [Crossref]
  28. A. Cerjan, Y. D. Chong, L. Ge, and A. D. Stone, “Steady-state ab initio laser theory for n-level lasers,” Opt. Express 20, 474–488 (2012).
    [Crossref] [PubMed]
  29. A. Cerjan, Y. D. Chong, and A. D. Stone, “Steady-state ab initio laser theory for complex gain media,” Opt. Express 23, 6455–6477 (2015).
    [Crossref] [PubMed]
  30. P. Drummond and M. Raymer, “Quantum-theory of propagation of nonclassical radiation in a near-resonant medium,” Phys. Rev. A 44, 2072–2085 (1991).
    [Crossref] [PubMed]
  31. B. Bidégaray, “Time discretizations for maxwell-bloch equations,” Numer. Meth. Partial Differential Equations 19, 284–300 (2003).
    [Crossref]
  32. D. Marcuse, “Computer-simulation of laser photon fluctuations - theory of single-cavity laser,” IEEE J. Quantum Elect. 20, 1139–1148 (1984).
    [Crossref]
  33. D. Marcuse, “Computer-simulation of laser photon fluctuations - single-cavity laser results,” IEEE J. Quantum Elect. 20, 1148–1155 (1984).
    [Crossref]
  34. G. Gray and R. Roy, “Noise in nearly-single-mode semiconductor-lasers,” Phys. Rev. A 40, 2452–2462 (1989).
    [Crossref] [PubMed]
  35. M. Kira, F. Jahnke, W. Hoyer, and S. W. Koch, “Quantum theory of spontaneous emission and coherent effects in semiconductor microstructures,” Prog. Quantum Electron. 23, 189–279 (1999).
    [Crossref]
  36. H. F. Hofmann and O. Hess, “Quantum maxwell-bloch equations for spatially inhomogeneous semiconductor lasers,” Phys. Rev. A 59, 2342–2358 (1999).
    [Crossref]
  37. C. Luo, A. Narayanaswamy, G. Chen, and J. D. Joannopoulos, “Thermal Radiation from Photonic Crystals: A Direct Calculation,” Phys. Rev. Lett. 93, 213905 (2004).
    [Crossref] [PubMed]
  38. J. Andreasen, H. Cao, A. Taflove, P. Kumar, and C.-q. Cao, “Finite-difference time-domain simulation of thermal noise in open cavities,” Phys. Rev. A 77, 023810 (2008).
    [Crossref]
  39. A. W. Rodriguez, A. P. McCauley, J. D. Joannopoulos, and S. G. Johnson, “Casimir forces in the time domain: Theory,” Phys. Rev. A 80, 012115 (2009).
    [Crossref]
  40. J. Andreasen and H. Cao, “Finite-difference time-domain formulation of stochastic noise in macroscopic atomic systems,” J. Lightwave Technol. 27, 4530–4535 (2009).
    [Crossref]
  41. J. Andreasen and H. Cao, “Numerical study of amplified spontaneous emission and lasing in random media,” Phys. Rev. A 82, 063835 (2010).
    [Crossref]
  42. J. Andreasen, “Numerical studies of lasing and electromagnetic fluctuations in open complex systems,” Ph.D. thesis, Northwestern University (2009).
  43. A. Pusch, S. Wuestner, J. M. Hamm, K. L. Tsakmakidis, and O. Hess, “Coherent Amplification and Noise in Gain-Enhanced Nanoplasmonic Metamaterials: A Maxwell-Bloch Langevin Approach,” ACS Nano 6, 2420–2431 (2012).
    [Crossref] [PubMed]
  44. S. Wuestner, J. M. Hamm, A. Pusch, F. Renn, K. L. Tsakmakidis, and O. Hess, “Control and dynamic competition of bright and dark lasing states in active nanoplasmonic metamaterials,” Phys. Rev. B 85, 201406 (2012).
    [Crossref]
  45. T. Pickering, J. M. Hamm, A. F. Page, S. Wuestner, and O. Hess, “Cavity-free plasmonic nanolasing enabled by dispersionless stopped light,” Nat. Commun. 5, 4972 (2014).
    [Crossref] [PubMed]
  46. S. Wuestner, T. Pickering, J. M. Hamm, A. F. Page, A. Pusch, and O. Hess, “Ultrafast dynamics of nanoplasmonic stopped-light lasing,” Faraday Discuss. 178, 307–324 (2015).
    [Crossref] [PubMed]
  47. K. Vahala, C. Harder, and A. Yariv, “Observation of relaxation resonance effects in the field spectrum of semiconductor lasers,” Appl. Phys. Lett. 42, 211–213 (1983).
    [Crossref]
  48. C. H. Henry and R. F. Kazarinov, “Quantum noise in photonics,” Rev. Mod. Phys. 68, 801–853 (1996).
    [Crossref]
  49. H. Haken, Light: Laser Dynamics, vol. 2 (North-Holland Phys. Publishing, 1985).
  50. L. D. Landau and E. M. Lifshitz, Statistical Physics,, 3rd ed. (Butterworth-Heinemann, Oxford, 1980), Part 1, Vol. 5.
  51. K. S. Yee, “Numerical solution of the initial boundary value problems involving maxwell’s equations in isotropic media,” IEEE Trans. Antennas Propag. 14, 302–307 (1966).
    [Crossref]
  52. A. Taflove and S. C. Hagness, Computational Electrodynamics: The Finite-Difference Time-Domain Method (Artech House, Incorporated, 2005).
  53. J. G. Proakis and D. G. Manolakis, Digital Signal Processing (Pearson Prentice Hall, 2007).
  54. M. Frigo and S. G. Johnson, “The design and implementation of FFTW3,” Proc. IEEE 93, 216–231 (2005).
    [Crossref]
  55. J. Ohtsubo, Semiconductor Lasers: Stability, Instability, and Chaos (Springer, 2007).
  56. K. R. Manes and A. E. Siegman, “Observation of Quantum Phase Fluctuations in Infrared Gas Lasers,” Phys. Rev. A 4, 373–386 (1971).
    [Crossref]
  57. J. D. Jackson, Classical Electrodynamics, 3rd ed. (Wiley, 1998).
  58. L. Ge, “Steady-state ab initio laser theory and its applications in random and complex media,” Ph.D. thesis, Yale University (2010).
  59. R. Shankar, Principles of Quantum Mechanics, 2nd ed. (Springer, 2013).
  60. F. T. Arecchi, G. L. Lippi, G. P. Puccioni, and J. R. Tredicce, “Deterministic chaos in laser with injected signal,” Opt. Commun. 51, 308–314 (1984).
    [Crossref]
  61. Y. Huang and S. Ho, “Computational model of solid-state, molecular, or atomic media for fdtd simulation based on a multi-level multi-electron system governed by pauli exclusion and fermi-dirac thermalization with application to semiconductor photonics,” Opt. Express 14, 3569–3587 (2006).
    [Crossref] [PubMed]
  62. K. Böhringer and O. Hess, “A full-time-domain approach to spatio-temporal dynamics of semiconductor lasers. i. theoretical formulation,” Prog. Quantum Electron. 32, 159–246 (2008).
    [Crossref]
  63. K. Böhringer and O. Hess, “A full time-domain approach to spatio-temporal dynamics of semiconductor lasers. II. spatio-temporal dynamics,” Prog. Quantum Electron. 32, 247–307 (2008).
    [Crossref]
  64. K. Ravi, Y. Huang, and S.-T. Ho, “A Highly Efficient Computational Model for FDTD Simulations of Ultrafast Electromagnetic Interactions With Semiconductor Media With Carrier Heating/Cooling Dynamics,” J. Lightwave Technol. 30, 772–804 (2012).
    [Crossref]
  65. R. Buschlinger, M. Lorke, and U. Peschel, “Light-matter interaction and lasing in semiconductor nanowires: A combined finite-difference time-domain and semiconductor Bloch equation approach,” Phys. Rev. B 91, 045203 (2015).
    [Crossref]
  66. M. Osinski and J. Buus, “Linewidth broadening factor in semiconductor lasers–An overview,” IEEE J. Quantum Elect. 23, 9–29 (1987).
    [Crossref]

2015 (4)

A. Pick, A. Cerjan, D. Liu, A. Rodriguez, A. D. Stone, Y. D. Chong, and S. Johnson, “Ab initio multimode linewidth theory for arbitrary inhomogeneous laser cavities,” Phys. Rev. A 91, 063806 (2015).
[Crossref]

A. Cerjan, Y. D. Chong, and A. D. Stone, “Steady-state ab initio laser theory for complex gain media,” Opt. Express 23, 6455–6477 (2015).
[Crossref] [PubMed]

S. Wuestner, T. Pickering, J. M. Hamm, A. F. Page, A. Pusch, and O. Hess, “Ultrafast dynamics of nanoplasmonic stopped-light lasing,” Faraday Discuss. 178, 307–324 (2015).
[Crossref] [PubMed]

R. Buschlinger, M. Lorke, and U. Peschel, “Light-matter interaction and lasing in semiconductor nanowires: A combined finite-difference time-domain and semiconductor Bloch equation approach,” Phys. Rev. B 91, 045203 (2015).
[Crossref]

2014 (3)

T. Pickering, J. M. Hamm, A. F. Page, S. Wuestner, and O. Hess, “Cavity-free plasmonic nanolasing enabled by dispersionless stopped light,” Nat. Commun. 5, 4972 (2014).
[Crossref] [PubMed]

S. Esterhazy, D. Liu, M. Liertzer, A. Cerjan, L. Ge, K. G. Makris, A. D. Stone, J. M. Melenk, S. G. Johnson, and S. Rotter, “Scalable numerical approach for the steady-state ab initio laser theory,” Phys. Rev. A 90, 023816 (2014).
[Crossref]

J. C. Pillay, Y. Natsume, A. D. Stone, and Y. D. Chong, “Generalized subSchawlow-Townes laser linewidths via material dispersion,” Phys. Rev. A 89, 033840 (2014).
[Crossref]

2012 (6)

Y. D. Chong and A. D. Stone, “General linewidth formula for steady-state multimode lasing in arbitrary cavities,” Phys. Rev. Lett. 109, 063902 (2012).
[Crossref] [PubMed]

J. G. Bohnet, Z. Chen, J. M. Weiner, D. Meiser, M. J. Holland, and J. K. Thompson, “A steady-state superradiant laser with less than one intracavity photon,” Nature 484, 78–81 (2012).
[Crossref] [PubMed]

A. Cerjan, Y. D. Chong, L. Ge, and A. D. Stone, “Steady-state ab initio laser theory for n-level lasers,” Opt. Express 20, 474–488 (2012).
[Crossref] [PubMed]

A. Pusch, S. Wuestner, J. M. Hamm, K. L. Tsakmakidis, and O. Hess, “Coherent Amplification and Noise in Gain-Enhanced Nanoplasmonic Metamaterials: A Maxwell-Bloch Langevin Approach,” ACS Nano 6, 2420–2431 (2012).
[Crossref] [PubMed]

S. Wuestner, J. M. Hamm, A. Pusch, F. Renn, K. L. Tsakmakidis, and O. Hess, “Control and dynamic competition of bright and dark lasing states in active nanoplasmonic metamaterials,” Phys. Rev. B 85, 201406 (2012).
[Crossref]

K. Ravi, Y. Huang, and S.-T. Ho, “A Highly Efficient Computational Model for FDTD Simulations of Ultrafast Electromagnetic Interactions With Semiconductor Media With Carrier Heating/Cooling Dynamics,” J. Lightwave Technol. 30, 772–804 (2012).
[Crossref]

2010 (4)

J. Andreasen and H. Cao, “Numerical study of amplified spontaneous emission and lasing in random media,” Phys. Rev. A 82, 063835 (2010).
[Crossref]

L. Ge, Y. D. Chong, and A. D. Stone, “Steady-state ab initio laser theory: generalizations and analytic results,” Phys. Rev. A 82, 063824 (2010).
[Crossref]

D. Meiser and M. J. Holland, “Steady-state superradiance with alkaline-earth-metal atoms,” Phys. Rev. A 81, 033847 (2010).
[Crossref]

D. Meiser and M. J. Holland, “Intensity fluctuations in steady-state superradiance,” Phys. Rev. A 81, 063827 (2010).
[Crossref]

2009 (3)

D. Meiser, J. Ye, D. R. Carlson, and M. J. Holland, “Prospects for a Millihertz-Linewidth Laser,” Phys. Rev. Lett. 102, 163601 (2009).
[Crossref] [PubMed]

A. W. Rodriguez, A. P. McCauley, J. D. Joannopoulos, and S. G. Johnson, “Casimir forces in the time domain: Theory,” Phys. Rev. A 80, 012115 (2009).
[Crossref]

J. Andreasen and H. Cao, “Finite-difference time-domain formulation of stochastic noise in macroscopic atomic systems,” J. Lightwave Technol. 27, 4530–4535 (2009).
[Crossref]

2008 (6)

J. Andreasen, H. Cao, A. Taflove, P. Kumar, and C.-q. Cao, “Finite-difference time-domain simulation of thermal noise in open cavities,” Phys. Rev. A 77, 023810 (2008).
[Crossref]

K. Böhringer and O. Hess, “A full-time-domain approach to spatio-temporal dynamics of semiconductor lasers. i. theoretical formulation,” Prog. Quantum Electron. 32, 159–246 (2008).
[Crossref]

K. Böhringer and O. Hess, “A full time-domain approach to spatio-temporal dynamics of semiconductor lasers. II. spatio-temporal dynamics,” Prog. Quantum Electron. 32, 247–307 (2008).
[Crossref]

D. Meiser, J. Ye, and M. J. Holland, “Spin squeezing in optical lattice clocks via lattice-based QND measurements,” New J. Phys. 10, 073014 (2008).
[Crossref]

L. Ge, R. J. Tandy, A. D. Stone, and H. E. Türeci, “Quantitative verification of ab initio self-consistent laser theory,” Opt. Express 16, 16895–16902 (2008).
[Crossref] [PubMed]

H. E. Türeci, L. Ge, S. Rotter, and A. D. Stone, “Strong interactions in multimode random lasers,” Science 320, 643–646 (2008).
[Crossref] [PubMed]

2006 (2)

2005 (1)

M. Frigo and S. G. Johnson, “The design and implementation of FFTW3,” Proc. IEEE 93, 216–231 (2005).
[Crossref]

2004 (1)

C. Luo, A. Narayanaswamy, G. Chen, and J. D. Joannopoulos, “Thermal Radiation from Photonic Crystals: A Direct Calculation,” Phys. Rev. Lett. 93, 213905 (2004).
[Crossref] [PubMed]

2003 (1)

B. Bidégaray, “Time discretizations for maxwell-bloch equations,” Numer. Meth. Partial Differential Equations 19, 284–300 (2003).
[Crossref]

1999 (2)

M. Kira, F. Jahnke, W. Hoyer, and S. W. Koch, “Quantum theory of spontaneous emission and coherent effects in semiconductor microstructures,” Prog. Quantum Electron. 23, 189–279 (1999).
[Crossref]

H. F. Hofmann and O. Hess, “Quantum maxwell-bloch equations for spatially inhomogeneous semiconductor lasers,” Phys. Rev. A 59, 2342–2358 (1999).
[Crossref]

1996 (1)

C. H. Henry and R. F. Kazarinov, “Quantum noise in photonics,” Rev. Mod. Phys. 68, 801–853 (1996).
[Crossref]

1995 (2)

M. van Exter, S. Kuppens, and J. Woerdman, “Theory for the linewidth of a bad-cavity laser,” Phys. Rev. A 51, 809–816 (1995).
[Crossref] [PubMed]

S. Kuppens, M. van Exter, M. Vanduin, and J. Woerdman, “Evidence of nonuniform phase-diffusion in a bad-cavity laser,” IEEE J. Quantum Elect. 31, 1237–1241 (1995).
[Crossref]

1994 (1)

S. Kuppens, M. van Exter, and J. Woerdman, “Quantum-limited linewidth of a bad-cavity laser,” Phys. Rev. Lett. 72, 3815–3818 (1994).
[Crossref] [PubMed]

1993 (1)

M. Kolobov, L. Davidovich, E. Giacobino, and C. Fabre, “Role of pumping statistics and dynamics of atomic polarization in quantum fluctuations of laser sources,” Phys. Rev. A 47, 1431–1446 (1993).
[Crossref] [PubMed]

1991 (1)

P. Drummond and M. Raymer, “Quantum-theory of propagation of nonclassical radiation in a near-resonant medium,” Phys. Rev. A 44, 2072–2085 (1991).
[Crossref] [PubMed]

1990 (1)

W. Hamel and J. Woerdman, “Observation of enhanced fundamental linewidth of a laser due to nonorthogonality of its longitudinal eigenmodes,” Phys. Rev. Lett. 64, 1506–1509 (1990).
[Crossref] [PubMed]

1989 (3)

A. Siegman, “Excess spontaneous emission in non-hermitian optical-systems .2. laser-oscillators,” Phys. Rev. A 39, 1264–1268 (1989).
[Crossref] [PubMed]

W. Hamel and J. Woerdman, “Nonorthogonality of the longitudinal eigenmodes of a laser,” Phys. Rev. A 40, 2785–2787 (1989).
[Crossref] [PubMed]

G. Gray and R. Roy, “Noise in nearly-single-mode semiconductor-lasers,” Phys. Rev. A 40, 2452–2462 (1989).
[Crossref] [PubMed]

1987 (1)

M. Osinski and J. Buus, “Linewidth broadening factor in semiconductor lasers–An overview,” IEEE J. Quantum Elect. 23, 9–29 (1987).
[Crossref]

1986 (1)

C. Henry, “Theory of spontaneous emission noise in open resonators and its application to lasers and optical amplifiers,” J. Lightwave Technol. 4, 288–297 (1986).
[Crossref]

1985 (1)

H. Haus and S. Kawakami, “On the excess spontaneous emission factor in gain-guided laser-amplifiers,” IEEE J. Quantum Elect. 21, 63–69 (1985).
[Crossref]

1984 (3)

D. Marcuse, “Computer-simulation of laser photon fluctuations - theory of single-cavity laser,” IEEE J. Quantum Elect. 20, 1139–1148 (1984).
[Crossref]

D. Marcuse, “Computer-simulation of laser photon fluctuations - single-cavity laser results,” IEEE J. Quantum Elect. 20, 1148–1155 (1984).
[Crossref]

F. T. Arecchi, G. L. Lippi, G. P. Puccioni, and J. R. Tredicce, “Deterministic chaos in laser with injected signal,” Opt. Commun. 51, 308–314 (1984).
[Crossref]

1983 (1)

K. Vahala, C. Harder, and A. Yariv, “Observation of relaxation resonance effects in the field spectrum of semiconductor lasers,” Appl. Phys. Lett. 42, 211–213 (1983).
[Crossref]

1982 (1)

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

1979 (1)

K. Petermann, “Calculated spontaneous emission factor for double-heterostructure injection-lasers with gain-induced waveguiding,” IEEE J. Quantum Elect. 15, 566–570 (1979).
[Crossref]

1971 (1)

K. R. Manes and A. E. Siegman, “Observation of Quantum Phase Fluctuations in Infrared Gas Lasers,” Phys. Rev. A 4, 373–386 (1971).
[Crossref]

1966 (1)

K. S. Yee, “Numerical solution of the initial boundary value problems involving maxwell’s equations in isotropic media,” IEEE Trans. Antennas Propag. 14, 302–307 (1966).
[Crossref]

1958 (1)

A. L. Schawlow and C. H. Townes, “Infrared and optical masers,” Phys. Rev. 112, 1940–1949 (1958).
[Crossref]

Andreasen, J.

J. Andreasen and H. Cao, “Numerical study of amplified spontaneous emission and lasing in random media,” Phys. Rev. A 82, 063835 (2010).
[Crossref]

J. Andreasen and H. Cao, “Finite-difference time-domain formulation of stochastic noise in macroscopic atomic systems,” J. Lightwave Technol. 27, 4530–4535 (2009).
[Crossref]

J. Andreasen, H. Cao, A. Taflove, P. Kumar, and C.-q. Cao, “Finite-difference time-domain simulation of thermal noise in open cavities,” Phys. Rev. A 77, 023810 (2008).
[Crossref]

J. Andreasen, “Numerical studies of lasing and electromagnetic fluctuations in open complex systems,” Ph.D. thesis, Northwestern University (2009).

Arecchi, F. T.

F. T. Arecchi, G. L. Lippi, G. P. Puccioni, and J. R. Tredicce, “Deterministic chaos in laser with injected signal,” Opt. Commun. 51, 308–314 (1984).
[Crossref]

Bidégaray, B.

B. Bidégaray, “Time discretizations for maxwell-bloch equations,” Numer. Meth. Partial Differential Equations 19, 284–300 (2003).
[Crossref]

Bohnet, J. G.

J. G. Bohnet, Z. Chen, J. M. Weiner, D. Meiser, M. J. Holland, and J. K. Thompson, “A steady-state superradiant laser with less than one intracavity photon,” Nature 484, 78–81 (2012).
[Crossref] [PubMed]

Böhringer, K.

K. Böhringer and O. Hess, “A full-time-domain approach to spatio-temporal dynamics of semiconductor lasers. i. theoretical formulation,” Prog. Quantum Electron. 32, 159–246 (2008).
[Crossref]

K. Böhringer and O. Hess, “A full time-domain approach to spatio-temporal dynamics of semiconductor lasers. II. spatio-temporal dynamics,” Prog. Quantum Electron. 32, 247–307 (2008).
[Crossref]

Buschlinger, R.

R. Buschlinger, M. Lorke, and U. Peschel, “Light-matter interaction and lasing in semiconductor nanowires: A combined finite-difference time-domain and semiconductor Bloch equation approach,” Phys. Rev. B 91, 045203 (2015).
[Crossref]

Buus, J.

M. Osinski and J. Buus, “Linewidth broadening factor in semiconductor lasers–An overview,” IEEE J. Quantum Elect. 23, 9–29 (1987).
[Crossref]

Cao, C.-q.

J. Andreasen, H. Cao, A. Taflove, P. Kumar, and C.-q. Cao, “Finite-difference time-domain simulation of thermal noise in open cavities,” Phys. Rev. A 77, 023810 (2008).
[Crossref]

Cao, H.

J. Andreasen and H. Cao, “Numerical study of amplified spontaneous emission and lasing in random media,” Phys. Rev. A 82, 063835 (2010).
[Crossref]

J. Andreasen and H. Cao, “Finite-difference time-domain formulation of stochastic noise in macroscopic atomic systems,” J. Lightwave Technol. 27, 4530–4535 (2009).
[Crossref]

J. Andreasen, H. Cao, A. Taflove, P. Kumar, and C.-q. Cao, “Finite-difference time-domain simulation of thermal noise in open cavities,” Phys. Rev. A 77, 023810 (2008).
[Crossref]

Carlson, D. R.

D. Meiser, J. Ye, D. R. Carlson, and M. J. Holland, “Prospects for a Millihertz-Linewidth Laser,” Phys. Rev. Lett. 102, 163601 (2009).
[Crossref] [PubMed]

Cerjan, A.

A. Cerjan, Y. D. Chong, and A. D. Stone, “Steady-state ab initio laser theory for complex gain media,” Opt. Express 23, 6455–6477 (2015).
[Crossref] [PubMed]

A. Pick, A. Cerjan, D. Liu, A. Rodriguez, A. D. Stone, Y. D. Chong, and S. Johnson, “Ab initio multimode linewidth theory for arbitrary inhomogeneous laser cavities,” Phys. Rev. A 91, 063806 (2015).
[Crossref]

S. Esterhazy, D. Liu, M. Liertzer, A. Cerjan, L. Ge, K. G. Makris, A. D. Stone, J. M. Melenk, S. G. Johnson, and S. Rotter, “Scalable numerical approach for the steady-state ab initio laser theory,” Phys. Rev. A 90, 023816 (2014).
[Crossref]

A. Cerjan, Y. D. Chong, L. Ge, and A. D. Stone, “Steady-state ab initio laser theory for n-level lasers,” Opt. Express 20, 474–488 (2012).
[Crossref] [PubMed]

Chen, G.

C. Luo, A. Narayanaswamy, G. Chen, and J. D. Joannopoulos, “Thermal Radiation from Photonic Crystals: A Direct Calculation,” Phys. Rev. Lett. 93, 213905 (2004).
[Crossref] [PubMed]

Chen, Z.

J. G. Bohnet, Z. Chen, J. M. Weiner, D. Meiser, M. J. Holland, and J. K. Thompson, “A steady-state superradiant laser with less than one intracavity photon,” Nature 484, 78–81 (2012).
[Crossref] [PubMed]

Chong, Y. D.

A. Cerjan, Y. D. Chong, and A. D. Stone, “Steady-state ab initio laser theory for complex gain media,” Opt. Express 23, 6455–6477 (2015).
[Crossref] [PubMed]

A. Pick, A. Cerjan, D. Liu, A. Rodriguez, A. D. Stone, Y. D. Chong, and S. Johnson, “Ab initio multimode linewidth theory for arbitrary inhomogeneous laser cavities,” Phys. Rev. A 91, 063806 (2015).
[Crossref]

J. C. Pillay, Y. Natsume, A. D. Stone, and Y. D. Chong, “Generalized subSchawlow-Townes laser linewidths via material dispersion,” Phys. Rev. A 89, 033840 (2014).
[Crossref]

A. Cerjan, Y. D. Chong, L. Ge, and A. D. Stone, “Steady-state ab initio laser theory for n-level lasers,” Opt. Express 20, 474–488 (2012).
[Crossref] [PubMed]

Y. D. Chong and A. D. Stone, “General linewidth formula for steady-state multimode lasing in arbitrary cavities,” Phys. Rev. Lett. 109, 063902 (2012).
[Crossref] [PubMed]

L. Ge, Y. D. Chong, and A. D. Stone, “Steady-state ab initio laser theory: generalizations and analytic results,” Phys. Rev. A 82, 063824 (2010).
[Crossref]

Collier, B.

H. E. Türeci, A. D. Stone, and B. Collier, “Self-consistent multimode lasing theory for complex or random lasing media,” Phys. Rev. A 74, 043822 (2006).
[Crossref]

Davidovich, L.

M. Kolobov, L. Davidovich, E. Giacobino, and C. Fabre, “Role of pumping statistics and dynamics of atomic polarization in quantum fluctuations of laser sources,” Phys. Rev. A 47, 1431–1446 (1993).
[Crossref] [PubMed]

Drummond, P.

P. Drummond and M. Raymer, “Quantum-theory of propagation of nonclassical radiation in a near-resonant medium,” Phys. Rev. A 44, 2072–2085 (1991).
[Crossref] [PubMed]

Esterhazy, S.

S. Esterhazy, D. Liu, M. Liertzer, A. Cerjan, L. Ge, K. G. Makris, A. D. Stone, J. M. Melenk, S. G. Johnson, and S. Rotter, “Scalable numerical approach for the steady-state ab initio laser theory,” Phys. Rev. A 90, 023816 (2014).
[Crossref]

Fabre, C.

M. Kolobov, L. Davidovich, E. Giacobino, and C. Fabre, “Role of pumping statistics and dynamics of atomic polarization in quantum fluctuations of laser sources,” Phys. Rev. A 47, 1431–1446 (1993).
[Crossref] [PubMed]

Frigo, M.

M. Frigo and S. G. Johnson, “The design and implementation of FFTW3,” Proc. IEEE 93, 216–231 (2005).
[Crossref]

Ge, L.

S. Esterhazy, D. Liu, M. Liertzer, A. Cerjan, L. Ge, K. G. Makris, A. D. Stone, J. M. Melenk, S. G. Johnson, and S. Rotter, “Scalable numerical approach for the steady-state ab initio laser theory,” Phys. Rev. A 90, 023816 (2014).
[Crossref]

A. Cerjan, Y. D. Chong, L. Ge, and A. D. Stone, “Steady-state ab initio laser theory for n-level lasers,” Opt. Express 20, 474–488 (2012).
[Crossref] [PubMed]

L. Ge, Y. D. Chong, and A. D. Stone, “Steady-state ab initio laser theory: generalizations and analytic results,” Phys. Rev. A 82, 063824 (2010).
[Crossref]

L. Ge, R. J. Tandy, A. D. Stone, and H. E. Türeci, “Quantitative verification of ab initio self-consistent laser theory,” Opt. Express 16, 16895–16902 (2008).
[Crossref] [PubMed]

H. E. Türeci, L. Ge, S. Rotter, and A. D. Stone, “Strong interactions in multimode random lasers,” Science 320, 643–646 (2008).
[Crossref] [PubMed]

L. Ge, “Steady-state ab initio laser theory and its applications in random and complex media,” Ph.D. thesis, Yale University (2010).

Giacobino, E.

M. Kolobov, L. Davidovich, E. Giacobino, and C. Fabre, “Role of pumping statistics and dynamics of atomic polarization in quantum fluctuations of laser sources,” Phys. Rev. A 47, 1431–1446 (1993).
[Crossref] [PubMed]

Gray, G.

G. Gray and R. Roy, “Noise in nearly-single-mode semiconductor-lasers,” Phys. Rev. A 40, 2452–2462 (1989).
[Crossref] [PubMed]

Hagness, S. C.

A. Taflove and S. C. Hagness, Computational Electrodynamics: The Finite-Difference Time-Domain Method (Artech House, Incorporated, 2005).

Haken, H.

H. Haken, Light: Laser Dynamics, vol. 2 (North-Holland Phys. Publishing, 1985).

H. Haken, Laser theory (Springer-Verlag, 1984).

Hamel, W.

W. Hamel and J. Woerdman, “Observation of enhanced fundamental linewidth of a laser due to nonorthogonality of its longitudinal eigenmodes,” Phys. Rev. Lett. 64, 1506–1509 (1990).
[Crossref] [PubMed]

W. Hamel and J. Woerdman, “Nonorthogonality of the longitudinal eigenmodes of a laser,” Phys. Rev. A 40, 2785–2787 (1989).
[Crossref] [PubMed]

Hamm, J. M.

S. Wuestner, T. Pickering, J. M. Hamm, A. F. Page, A. Pusch, and O. Hess, “Ultrafast dynamics of nanoplasmonic stopped-light lasing,” Faraday Discuss. 178, 307–324 (2015).
[Crossref] [PubMed]

T. Pickering, J. M. Hamm, A. F. Page, S. Wuestner, and O. Hess, “Cavity-free plasmonic nanolasing enabled by dispersionless stopped light,” Nat. Commun. 5, 4972 (2014).
[Crossref] [PubMed]

S. Wuestner, J. M. Hamm, A. Pusch, F. Renn, K. L. Tsakmakidis, and O. Hess, “Control and dynamic competition of bright and dark lasing states in active nanoplasmonic metamaterials,” Phys. Rev. B 85, 201406 (2012).
[Crossref]

A. Pusch, S. Wuestner, J. M. Hamm, K. L. Tsakmakidis, and O. Hess, “Coherent Amplification and Noise in Gain-Enhanced Nanoplasmonic Metamaterials: A Maxwell-Bloch Langevin Approach,” ACS Nano 6, 2420–2431 (2012).
[Crossref] [PubMed]

Harder, C.

K. Vahala, C. Harder, and A. Yariv, “Observation of relaxation resonance effects in the field spectrum of semiconductor lasers,” Appl. Phys. Lett. 42, 211–213 (1983).
[Crossref]

Haus, H.

H. Haus and S. Kawakami, “On the excess spontaneous emission factor in gain-guided laser-amplifiers,” IEEE J. Quantum Elect. 21, 63–69 (1985).
[Crossref]

Henry, C.

C. Henry, “Theory of spontaneous emission noise in open resonators and its application to lasers and optical amplifiers,” J. Lightwave Technol. 4, 288–297 (1986).
[Crossref]

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

Henry, C. H.

C. H. Henry and R. F. Kazarinov, “Quantum noise in photonics,” Rev. Mod. Phys. 68, 801–853 (1996).
[Crossref]

Hess, O.

S. Wuestner, T. Pickering, J. M. Hamm, A. F. Page, A. Pusch, and O. Hess, “Ultrafast dynamics of nanoplasmonic stopped-light lasing,” Faraday Discuss. 178, 307–324 (2015).
[Crossref] [PubMed]

T. Pickering, J. M. Hamm, A. F. Page, S. Wuestner, and O. Hess, “Cavity-free plasmonic nanolasing enabled by dispersionless stopped light,” Nat. Commun. 5, 4972 (2014).
[Crossref] [PubMed]

A. Pusch, S. Wuestner, J. M. Hamm, K. L. Tsakmakidis, and O. Hess, “Coherent Amplification and Noise in Gain-Enhanced Nanoplasmonic Metamaterials: A Maxwell-Bloch Langevin Approach,” ACS Nano 6, 2420–2431 (2012).
[Crossref] [PubMed]

S. Wuestner, J. M. Hamm, A. Pusch, F. Renn, K. L. Tsakmakidis, and O. Hess, “Control and dynamic competition of bright and dark lasing states in active nanoplasmonic metamaterials,” Phys. Rev. B 85, 201406 (2012).
[Crossref]

K. Böhringer and O. Hess, “A full time-domain approach to spatio-temporal dynamics of semiconductor lasers. II. spatio-temporal dynamics,” Prog. Quantum Electron. 32, 247–307 (2008).
[Crossref]

K. Böhringer and O. Hess, “A full-time-domain approach to spatio-temporal dynamics of semiconductor lasers. i. theoretical formulation,” Prog. Quantum Electron. 32, 159–246 (2008).
[Crossref]

H. F. Hofmann and O. Hess, “Quantum maxwell-bloch equations for spatially inhomogeneous semiconductor lasers,” Phys. Rev. A 59, 2342–2358 (1999).
[Crossref]

Ho, S.

Ho, S.-T.

Hofmann, H. F.

H. F. Hofmann and O. Hess, “Quantum maxwell-bloch equations for spatially inhomogeneous semiconductor lasers,” Phys. Rev. A 59, 2342–2358 (1999).
[Crossref]

Holland, M. J.

J. G. Bohnet, Z. Chen, J. M. Weiner, D. Meiser, M. J. Holland, and J. K. Thompson, “A steady-state superradiant laser with less than one intracavity photon,” Nature 484, 78–81 (2012).
[Crossref] [PubMed]

D. Meiser and M. J. Holland, “Steady-state superradiance with alkaline-earth-metal atoms,” Phys. Rev. A 81, 033847 (2010).
[Crossref]

D. Meiser and M. J. Holland, “Intensity fluctuations in steady-state superradiance,” Phys. Rev. A 81, 063827 (2010).
[Crossref]

D. Meiser, J. Ye, D. R. Carlson, and M. J. Holland, “Prospects for a Millihertz-Linewidth Laser,” Phys. Rev. Lett. 102, 163601 (2009).
[Crossref] [PubMed]

D. Meiser, J. Ye, and M. J. Holland, “Spin squeezing in optical lattice clocks via lattice-based QND measurements,” New J. Phys. 10, 073014 (2008).
[Crossref]

Hoyer, W.

M. Kira, F. Jahnke, W. Hoyer, and S. W. Koch, “Quantum theory of spontaneous emission and coherent effects in semiconductor microstructures,” Prog. Quantum Electron. 23, 189–279 (1999).
[Crossref]

Huang, Y.

Jackson, J. D.

J. D. Jackson, Classical Electrodynamics, 3rd ed. (Wiley, 1998).

Jahnke, F.

M. Kira, F. Jahnke, W. Hoyer, and S. W. Koch, “Quantum theory of spontaneous emission and coherent effects in semiconductor microstructures,” Prog. Quantum Electron. 23, 189–279 (1999).
[Crossref]

Joannopoulos, J. D.

A. W. Rodriguez, A. P. McCauley, J. D. Joannopoulos, and S. G. Johnson, “Casimir forces in the time domain: Theory,” Phys. Rev. A 80, 012115 (2009).
[Crossref]

C. Luo, A. Narayanaswamy, G. Chen, and J. D. Joannopoulos, “Thermal Radiation from Photonic Crystals: A Direct Calculation,” Phys. Rev. Lett. 93, 213905 (2004).
[Crossref] [PubMed]

Johnson, S.

A. Pick, A. Cerjan, D. Liu, A. Rodriguez, A. D. Stone, Y. D. Chong, and S. Johnson, “Ab initio multimode linewidth theory for arbitrary inhomogeneous laser cavities,” Phys. Rev. A 91, 063806 (2015).
[Crossref]

Johnson, S. G.

S. Esterhazy, D. Liu, M. Liertzer, A. Cerjan, L. Ge, K. G. Makris, A. D. Stone, J. M. Melenk, S. G. Johnson, and S. Rotter, “Scalable numerical approach for the steady-state ab initio laser theory,” Phys. Rev. A 90, 023816 (2014).
[Crossref]

A. W. Rodriguez, A. P. McCauley, J. D. Joannopoulos, and S. G. Johnson, “Casimir forces in the time domain: Theory,” Phys. Rev. A 80, 012115 (2009).
[Crossref]

M. Frigo and S. G. Johnson, “The design and implementation of FFTW3,” Proc. IEEE 93, 216–231 (2005).
[Crossref]

Kawakami, S.

H. Haus and S. Kawakami, “On the excess spontaneous emission factor in gain-guided laser-amplifiers,” IEEE J. Quantum Elect. 21, 63–69 (1985).
[Crossref]

Kazarinov, R. F.

C. H. Henry and R. F. Kazarinov, “Quantum noise in photonics,” Rev. Mod. Phys. 68, 801–853 (1996).
[Crossref]

Kira, M.

M. Kira, F. Jahnke, W. Hoyer, and S. W. Koch, “Quantum theory of spontaneous emission and coherent effects in semiconductor microstructures,” Prog. Quantum Electron. 23, 189–279 (1999).
[Crossref]

Koch, S. W.

M. Kira, F. Jahnke, W. Hoyer, and S. W. Koch, “Quantum theory of spontaneous emission and coherent effects in semiconductor microstructures,” Prog. Quantum Electron. 23, 189–279 (1999).
[Crossref]

Kolobov, M.

M. Kolobov, L. Davidovich, E. Giacobino, and C. Fabre, “Role of pumping statistics and dynamics of atomic polarization in quantum fluctuations of laser sources,” Phys. Rev. A 47, 1431–1446 (1993).
[Crossref] [PubMed]

Kumar, P.

J. Andreasen, H. Cao, A. Taflove, P. Kumar, and C.-q. Cao, “Finite-difference time-domain simulation of thermal noise in open cavities,” Phys. Rev. A 77, 023810 (2008).
[Crossref]

Kuppens, S.

M. van Exter, S. Kuppens, and J. Woerdman, “Theory for the linewidth of a bad-cavity laser,” Phys. Rev. A 51, 809–816 (1995).
[Crossref] [PubMed]

S. Kuppens, M. van Exter, M. Vanduin, and J. Woerdman, “Evidence of nonuniform phase-diffusion in a bad-cavity laser,” IEEE J. Quantum Elect. 31, 1237–1241 (1995).
[Crossref]

S. Kuppens, M. van Exter, and J. Woerdman, “Quantum-limited linewidth of a bad-cavity laser,” Phys. Rev. Lett. 72, 3815–3818 (1994).
[Crossref] [PubMed]

Landau, L. D.

L. D. Landau and E. M. Lifshitz, Statistical Physics,, 3rd ed. (Butterworth-Heinemann, Oxford, 1980), Part 1, Vol. 5.

Lax, M.

M. Lax, “Quantum noise v: Phase noise in a homogeneously broadened maser,” in “Physics of Quantum Electronics,”, P. L. Kelley, B. Lax, and P. E. Tannenwald, eds. (McGraw-Hill, 1966).

Liertzer, M.

S. Esterhazy, D. Liu, M. Liertzer, A. Cerjan, L. Ge, K. G. Makris, A. D. Stone, J. M. Melenk, S. G. Johnson, and S. Rotter, “Scalable numerical approach for the steady-state ab initio laser theory,” Phys. Rev. A 90, 023816 (2014).
[Crossref]

Lifshitz, E. M.

L. D. Landau and E. M. Lifshitz, Statistical Physics,, 3rd ed. (Butterworth-Heinemann, Oxford, 1980), Part 1, Vol. 5.

Lippi, G. L.

F. T. Arecchi, G. L. Lippi, G. P. Puccioni, and J. R. Tredicce, “Deterministic chaos in laser with injected signal,” Opt. Commun. 51, 308–314 (1984).
[Crossref]

Liu, D.

A. Pick, A. Cerjan, D. Liu, A. Rodriguez, A. D. Stone, Y. D. Chong, and S. Johnson, “Ab initio multimode linewidth theory for arbitrary inhomogeneous laser cavities,” Phys. Rev. A 91, 063806 (2015).
[Crossref]

S. Esterhazy, D. Liu, M. Liertzer, A. Cerjan, L. Ge, K. G. Makris, A. D. Stone, J. M. Melenk, S. G. Johnson, and S. Rotter, “Scalable numerical approach for the steady-state ab initio laser theory,” Phys. Rev. A 90, 023816 (2014).
[Crossref]

Lorke, M.

R. Buschlinger, M. Lorke, and U. Peschel, “Light-matter interaction and lasing in semiconductor nanowires: A combined finite-difference time-domain and semiconductor Bloch equation approach,” Phys. Rev. B 91, 045203 (2015).
[Crossref]

Luo, C.

C. Luo, A. Narayanaswamy, G. Chen, and J. D. Joannopoulos, “Thermal Radiation from Photonic Crystals: A Direct Calculation,” Phys. Rev. Lett. 93, 213905 (2004).
[Crossref] [PubMed]

Makris, K. G.

S. Esterhazy, D. Liu, M. Liertzer, A. Cerjan, L. Ge, K. G. Makris, A. D. Stone, J. M. Melenk, S. G. Johnson, and S. Rotter, “Scalable numerical approach for the steady-state ab initio laser theory,” Phys. Rev. A 90, 023816 (2014).
[Crossref]

Manes, K. R.

K. R. Manes and A. E. Siegman, “Observation of Quantum Phase Fluctuations in Infrared Gas Lasers,” Phys. Rev. A 4, 373–386 (1971).
[Crossref]

Manolakis, D. G.

J. G. Proakis and D. G. Manolakis, Digital Signal Processing (Pearson Prentice Hall, 2007).

Marcuse, D.

D. Marcuse, “Computer-simulation of laser photon fluctuations - theory of single-cavity laser,” IEEE J. Quantum Elect. 20, 1139–1148 (1984).
[Crossref]

D. Marcuse, “Computer-simulation of laser photon fluctuations - single-cavity laser results,” IEEE J. Quantum Elect. 20, 1148–1155 (1984).
[Crossref]

McCauley, A. P.

A. W. Rodriguez, A. P. McCauley, J. D. Joannopoulos, and S. G. Johnson, “Casimir forces in the time domain: Theory,” Phys. Rev. A 80, 012115 (2009).
[Crossref]

Meiser, D.

J. G. Bohnet, Z. Chen, J. M. Weiner, D. Meiser, M. J. Holland, and J. K. Thompson, “A steady-state superradiant laser with less than one intracavity photon,” Nature 484, 78–81 (2012).
[Crossref] [PubMed]

D. Meiser and M. J. Holland, “Intensity fluctuations in steady-state superradiance,” Phys. Rev. A 81, 063827 (2010).
[Crossref]

D. Meiser and M. J. Holland, “Steady-state superradiance with alkaline-earth-metal atoms,” Phys. Rev. A 81, 033847 (2010).
[Crossref]

D. Meiser, J. Ye, D. R. Carlson, and M. J. Holland, “Prospects for a Millihertz-Linewidth Laser,” Phys. Rev. Lett. 102, 163601 (2009).
[Crossref] [PubMed]

D. Meiser, J. Ye, and M. J. Holland, “Spin squeezing in optical lattice clocks via lattice-based QND measurements,” New J. Phys. 10, 073014 (2008).
[Crossref]

Melenk, J. M.

S. Esterhazy, D. Liu, M. Liertzer, A. Cerjan, L. Ge, K. G. Makris, A. D. Stone, J. M. Melenk, S. G. Johnson, and S. Rotter, “Scalable numerical approach for the steady-state ab initio laser theory,” Phys. Rev. A 90, 023816 (2014).
[Crossref]

Narayanaswamy, A.

C. Luo, A. Narayanaswamy, G. Chen, and J. D. Joannopoulos, “Thermal Radiation from Photonic Crystals: A Direct Calculation,” Phys. Rev. Lett. 93, 213905 (2004).
[Crossref] [PubMed]

Natsume, Y.

J. C. Pillay, Y. Natsume, A. D. Stone, and Y. D. Chong, “Generalized subSchawlow-Townes laser linewidths via material dispersion,” Phys. Rev. A 89, 033840 (2014).
[Crossref]

Ohtsubo, J.

J. Ohtsubo, Semiconductor Lasers: Stability, Instability, and Chaos (Springer, 2007).

Osinski, M.

M. Osinski and J. Buus, “Linewidth broadening factor in semiconductor lasers–An overview,” IEEE J. Quantum Elect. 23, 9–29 (1987).
[Crossref]

Page, A. F.

S. Wuestner, T. Pickering, J. M. Hamm, A. F. Page, A. Pusch, and O. Hess, “Ultrafast dynamics of nanoplasmonic stopped-light lasing,” Faraday Discuss. 178, 307–324 (2015).
[Crossref] [PubMed]

T. Pickering, J. M. Hamm, A. F. Page, S. Wuestner, and O. Hess, “Cavity-free plasmonic nanolasing enabled by dispersionless stopped light,” Nat. Commun. 5, 4972 (2014).
[Crossref] [PubMed]

Peschel, U.

R. Buschlinger, M. Lorke, and U. Peschel, “Light-matter interaction and lasing in semiconductor nanowires: A combined finite-difference time-domain and semiconductor Bloch equation approach,” Phys. Rev. B 91, 045203 (2015).
[Crossref]

Petermann, K.

K. Petermann, “Calculated spontaneous emission factor for double-heterostructure injection-lasers with gain-induced waveguiding,” IEEE J. Quantum Elect. 15, 566–570 (1979).
[Crossref]

Pick, A.

A. Pick, A. Cerjan, D. Liu, A. Rodriguez, A. D. Stone, Y. D. Chong, and S. Johnson, “Ab initio multimode linewidth theory for arbitrary inhomogeneous laser cavities,” Phys. Rev. A 91, 063806 (2015).
[Crossref]

Pickering, T.

S. Wuestner, T. Pickering, J. M. Hamm, A. F. Page, A. Pusch, and O. Hess, “Ultrafast dynamics of nanoplasmonic stopped-light lasing,” Faraday Discuss. 178, 307–324 (2015).
[Crossref] [PubMed]

T. Pickering, J. M. Hamm, A. F. Page, S. Wuestner, and O. Hess, “Cavity-free plasmonic nanolasing enabled by dispersionless stopped light,” Nat. Commun. 5, 4972 (2014).
[Crossref] [PubMed]

Pillay, J. C.

J. C. Pillay, Y. Natsume, A. D. Stone, and Y. D. Chong, “Generalized subSchawlow-Townes laser linewidths via material dispersion,” Phys. Rev. A 89, 033840 (2014).
[Crossref]

Proakis, J. G.

J. G. Proakis and D. G. Manolakis, Digital Signal Processing (Pearson Prentice Hall, 2007).

Puccioni, G. P.

F. T. Arecchi, G. L. Lippi, G. P. Puccioni, and J. R. Tredicce, “Deterministic chaos in laser with injected signal,” Opt. Commun. 51, 308–314 (1984).
[Crossref]

Pusch, A.

S. Wuestner, T. Pickering, J. M. Hamm, A. F. Page, A. Pusch, and O. Hess, “Ultrafast dynamics of nanoplasmonic stopped-light lasing,” Faraday Discuss. 178, 307–324 (2015).
[Crossref] [PubMed]

S. Wuestner, J. M. Hamm, A. Pusch, F. Renn, K. L. Tsakmakidis, and O. Hess, “Control and dynamic competition of bright and dark lasing states in active nanoplasmonic metamaterials,” Phys. Rev. B 85, 201406 (2012).
[Crossref]

A. Pusch, S. Wuestner, J. M. Hamm, K. L. Tsakmakidis, and O. Hess, “Coherent Amplification and Noise in Gain-Enhanced Nanoplasmonic Metamaterials: A Maxwell-Bloch Langevin Approach,” ACS Nano 6, 2420–2431 (2012).
[Crossref] [PubMed]

Ravi, K.

Raymer, M.

P. Drummond and M. Raymer, “Quantum-theory of propagation of nonclassical radiation in a near-resonant medium,” Phys. Rev. A 44, 2072–2085 (1991).
[Crossref] [PubMed]

Renn, F.

S. Wuestner, J. M. Hamm, A. Pusch, F. Renn, K. L. Tsakmakidis, and O. Hess, “Control and dynamic competition of bright and dark lasing states in active nanoplasmonic metamaterials,” Phys. Rev. B 85, 201406 (2012).
[Crossref]

Rodriguez, A.

A. Pick, A. Cerjan, D. Liu, A. Rodriguez, A. D. Stone, Y. D. Chong, and S. Johnson, “Ab initio multimode linewidth theory for arbitrary inhomogeneous laser cavities,” Phys. Rev. A 91, 063806 (2015).
[Crossref]

Rodriguez, A. W.

A. W. Rodriguez, A. P. McCauley, J. D. Joannopoulos, and S. G. Johnson, “Casimir forces in the time domain: Theory,” Phys. Rev. A 80, 012115 (2009).
[Crossref]

Rotter, S.

S. Esterhazy, D. Liu, M. Liertzer, A. Cerjan, L. Ge, K. G. Makris, A. D. Stone, J. M. Melenk, S. G. Johnson, and S. Rotter, “Scalable numerical approach for the steady-state ab initio laser theory,” Phys. Rev. A 90, 023816 (2014).
[Crossref]

H. E. Türeci, L. Ge, S. Rotter, and A. D. Stone, “Strong interactions in multimode random lasers,” Science 320, 643–646 (2008).
[Crossref] [PubMed]

Roy, R.

G. Gray and R. Roy, “Noise in nearly-single-mode semiconductor-lasers,” Phys. Rev. A 40, 2452–2462 (1989).
[Crossref] [PubMed]

Schawlow, A. L.

A. L. Schawlow and C. H. Townes, “Infrared and optical masers,” Phys. Rev. 112, 1940–1949 (1958).
[Crossref]

Shankar, R.

R. Shankar, Principles of Quantum Mechanics, 2nd ed. (Springer, 2013).

Siegman, A.

A. Siegman, “Excess spontaneous emission in non-hermitian optical-systems .2. laser-oscillators,” Phys. Rev. A 39, 1264–1268 (1989).
[Crossref] [PubMed]

Siegman, A. E.

K. R. Manes and A. E. Siegman, “Observation of Quantum Phase Fluctuations in Infrared Gas Lasers,” Phys. Rev. A 4, 373–386 (1971).
[Crossref]

Stone, A. D.

A. Cerjan, Y. D. Chong, and A. D. Stone, “Steady-state ab initio laser theory for complex gain media,” Opt. Express 23, 6455–6477 (2015).
[Crossref] [PubMed]

A. Pick, A. Cerjan, D. Liu, A. Rodriguez, A. D. Stone, Y. D. Chong, and S. Johnson, “Ab initio multimode linewidth theory for arbitrary inhomogeneous laser cavities,” Phys. Rev. A 91, 063806 (2015).
[Crossref]

J. C. Pillay, Y. Natsume, A. D. Stone, and Y. D. Chong, “Generalized subSchawlow-Townes laser linewidths via material dispersion,” Phys. Rev. A 89, 033840 (2014).
[Crossref]

S. Esterhazy, D. Liu, M. Liertzer, A. Cerjan, L. Ge, K. G. Makris, A. D. Stone, J. M. Melenk, S. G. Johnson, and S. Rotter, “Scalable numerical approach for the steady-state ab initio laser theory,” Phys. Rev. A 90, 023816 (2014).
[Crossref]

Y. D. Chong and A. D. Stone, “General linewidth formula for steady-state multimode lasing in arbitrary cavities,” Phys. Rev. Lett. 109, 063902 (2012).
[Crossref] [PubMed]

A. Cerjan, Y. D. Chong, L. Ge, and A. D. Stone, “Steady-state ab initio laser theory for n-level lasers,” Opt. Express 20, 474–488 (2012).
[Crossref] [PubMed]

L. Ge, Y. D. Chong, and A. D. Stone, “Steady-state ab initio laser theory: generalizations and analytic results,” Phys. Rev. A 82, 063824 (2010).
[Crossref]

H. E. Türeci, L. Ge, S. Rotter, and A. D. Stone, “Strong interactions in multimode random lasers,” Science 320, 643–646 (2008).
[Crossref] [PubMed]

L. Ge, R. J. Tandy, A. D. Stone, and H. E. Türeci, “Quantitative verification of ab initio self-consistent laser theory,” Opt. Express 16, 16895–16902 (2008).
[Crossref] [PubMed]

H. E. Türeci, A. D. Stone, and B. Collier, “Self-consistent multimode lasing theory for complex or random lasing media,” Phys. Rev. A 74, 043822 (2006).
[Crossref]

Taflove, A.

J. Andreasen, H. Cao, A. Taflove, P. Kumar, and C.-q. Cao, “Finite-difference time-domain simulation of thermal noise in open cavities,” Phys. Rev. A 77, 023810 (2008).
[Crossref]

A. Taflove and S. C. Hagness, Computational Electrodynamics: The Finite-Difference Time-Domain Method (Artech House, Incorporated, 2005).

Tandy, R. J.

Thompson, J. K.

J. G. Bohnet, Z. Chen, J. M. Weiner, D. Meiser, M. J. Holland, and J. K. Thompson, “A steady-state superradiant laser with less than one intracavity photon,” Nature 484, 78–81 (2012).
[Crossref] [PubMed]

Townes, C. H.

A. L. Schawlow and C. H. Townes, “Infrared and optical masers,” Phys. Rev. 112, 1940–1949 (1958).
[Crossref]

Tredicce, J. R.

F. T. Arecchi, G. L. Lippi, G. P. Puccioni, and J. R. Tredicce, “Deterministic chaos in laser with injected signal,” Opt. Commun. 51, 308–314 (1984).
[Crossref]

Tsakmakidis, K. L.

A. Pusch, S. Wuestner, J. M. Hamm, K. L. Tsakmakidis, and O. Hess, “Coherent Amplification and Noise in Gain-Enhanced Nanoplasmonic Metamaterials: A Maxwell-Bloch Langevin Approach,” ACS Nano 6, 2420–2431 (2012).
[Crossref] [PubMed]

S. Wuestner, J. M. Hamm, A. Pusch, F. Renn, K. L. Tsakmakidis, and O. Hess, “Control and dynamic competition of bright and dark lasing states in active nanoplasmonic metamaterials,” Phys. Rev. B 85, 201406 (2012).
[Crossref]

Türeci, H. E.

H. E. Türeci, L. Ge, S. Rotter, and A. D. Stone, “Strong interactions in multimode random lasers,” Science 320, 643–646 (2008).
[Crossref] [PubMed]

L. Ge, R. J. Tandy, A. D. Stone, and H. E. Türeci, “Quantitative verification of ab initio self-consistent laser theory,” Opt. Express 16, 16895–16902 (2008).
[Crossref] [PubMed]

H. E. Türeci, A. D. Stone, and B. Collier, “Self-consistent multimode lasing theory for complex or random lasing media,” Phys. Rev. A 74, 043822 (2006).
[Crossref]

Vahala, K.

K. Vahala, C. Harder, and A. Yariv, “Observation of relaxation resonance effects in the field spectrum of semiconductor lasers,” Appl. Phys. Lett. 42, 211–213 (1983).
[Crossref]

van Exter, M.

M. van Exter, S. Kuppens, and J. Woerdman, “Theory for the linewidth of a bad-cavity laser,” Phys. Rev. A 51, 809–816 (1995).
[Crossref] [PubMed]

S. Kuppens, M. van Exter, M. Vanduin, and J. Woerdman, “Evidence of nonuniform phase-diffusion in a bad-cavity laser,” IEEE J. Quantum Elect. 31, 1237–1241 (1995).
[Crossref]

S. Kuppens, M. van Exter, and J. Woerdman, “Quantum-limited linewidth of a bad-cavity laser,” Phys. Rev. Lett. 72, 3815–3818 (1994).
[Crossref] [PubMed]

Vanduin, M.

S. Kuppens, M. van Exter, M. Vanduin, and J. Woerdman, “Evidence of nonuniform phase-diffusion in a bad-cavity laser,” IEEE J. Quantum Elect. 31, 1237–1241 (1995).
[Crossref]

Weiner, J. M.

J. G. Bohnet, Z. Chen, J. M. Weiner, D. Meiser, M. J. Holland, and J. K. Thompson, “A steady-state superradiant laser with less than one intracavity photon,” Nature 484, 78–81 (2012).
[Crossref] [PubMed]

Woerdman, J.

S. Kuppens, M. van Exter, M. Vanduin, and J. Woerdman, “Evidence of nonuniform phase-diffusion in a bad-cavity laser,” IEEE J. Quantum Elect. 31, 1237–1241 (1995).
[Crossref]

M. van Exter, S. Kuppens, and J. Woerdman, “Theory for the linewidth of a bad-cavity laser,” Phys. Rev. A 51, 809–816 (1995).
[Crossref] [PubMed]

S. Kuppens, M. van Exter, and J. Woerdman, “Quantum-limited linewidth of a bad-cavity laser,” Phys. Rev. Lett. 72, 3815–3818 (1994).
[Crossref] [PubMed]

W. Hamel and J. Woerdman, “Observation of enhanced fundamental linewidth of a laser due to nonorthogonality of its longitudinal eigenmodes,” Phys. Rev. Lett. 64, 1506–1509 (1990).
[Crossref] [PubMed]

W. Hamel and J. Woerdman, “Nonorthogonality of the longitudinal eigenmodes of a laser,” Phys. Rev. A 40, 2785–2787 (1989).
[Crossref] [PubMed]

Wuestner, S.

S. Wuestner, T. Pickering, J. M. Hamm, A. F. Page, A. Pusch, and O. Hess, “Ultrafast dynamics of nanoplasmonic stopped-light lasing,” Faraday Discuss. 178, 307–324 (2015).
[Crossref] [PubMed]

T. Pickering, J. M. Hamm, A. F. Page, S. Wuestner, and O. Hess, “Cavity-free plasmonic nanolasing enabled by dispersionless stopped light,” Nat. Commun. 5, 4972 (2014).
[Crossref] [PubMed]

S. Wuestner, J. M. Hamm, A. Pusch, F. Renn, K. L. Tsakmakidis, and O. Hess, “Control and dynamic competition of bright and dark lasing states in active nanoplasmonic metamaterials,” Phys. Rev. B 85, 201406 (2012).
[Crossref]

A. Pusch, S. Wuestner, J. M. Hamm, K. L. Tsakmakidis, and O. Hess, “Coherent Amplification and Noise in Gain-Enhanced Nanoplasmonic Metamaterials: A Maxwell-Bloch Langevin Approach,” ACS Nano 6, 2420–2431 (2012).
[Crossref] [PubMed]

Yariv, A.

K. Vahala, C. Harder, and A. Yariv, “Observation of relaxation resonance effects in the field spectrum of semiconductor lasers,” Appl. Phys. Lett. 42, 211–213 (1983).
[Crossref]

Ye, J.

D. Meiser, J. Ye, D. R. Carlson, and M. J. Holland, “Prospects for a Millihertz-Linewidth Laser,” Phys. Rev. Lett. 102, 163601 (2009).
[Crossref] [PubMed]

D. Meiser, J. Ye, and M. J. Holland, “Spin squeezing in optical lattice clocks via lattice-based QND measurements,” New J. Phys. 10, 073014 (2008).
[Crossref]

Yee, K. S.

K. S. Yee, “Numerical solution of the initial boundary value problems involving maxwell’s equations in isotropic media,” IEEE Trans. Antennas Propag. 14, 302–307 (1966).
[Crossref]

ACS Nano (1)

A. Pusch, S. Wuestner, J. M. Hamm, K. L. Tsakmakidis, and O. Hess, “Coherent Amplification and Noise in Gain-Enhanced Nanoplasmonic Metamaterials: A Maxwell-Bloch Langevin Approach,” ACS Nano 6, 2420–2431 (2012).
[Crossref] [PubMed]

Appl. Phys. Lett. (1)

K. Vahala, C. Harder, and A. Yariv, “Observation of relaxation resonance effects in the field spectrum of semiconductor lasers,” Appl. Phys. Lett. 42, 211–213 (1983).
[Crossref]

Faraday Discuss. (1)

S. Wuestner, T. Pickering, J. M. Hamm, A. F. Page, A. Pusch, and O. Hess, “Ultrafast dynamics of nanoplasmonic stopped-light lasing,” Faraday Discuss. 178, 307–324 (2015).
[Crossref] [PubMed]

IEEE J. Quantum Elect. (7)

M. Osinski and J. Buus, “Linewidth broadening factor in semiconductor lasers–An overview,” IEEE J. Quantum Elect. 23, 9–29 (1987).
[Crossref]

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

K. Petermann, “Calculated spontaneous emission factor for double-heterostructure injection-lasers with gain-induced waveguiding,” IEEE J. Quantum Elect. 15, 566–570 (1979).
[Crossref]

H. Haus and S. Kawakami, “On the excess spontaneous emission factor in gain-guided laser-amplifiers,” IEEE J. Quantum Elect. 21, 63–69 (1985).
[Crossref]

S. Kuppens, M. van Exter, M. Vanduin, and J. Woerdman, “Evidence of nonuniform phase-diffusion in a bad-cavity laser,” IEEE J. Quantum Elect. 31, 1237–1241 (1995).
[Crossref]

D. Marcuse, “Computer-simulation of laser photon fluctuations - theory of single-cavity laser,” IEEE J. Quantum Elect. 20, 1139–1148 (1984).
[Crossref]

D. Marcuse, “Computer-simulation of laser photon fluctuations - single-cavity laser results,” IEEE J. Quantum Elect. 20, 1148–1155 (1984).
[Crossref]

IEEE Trans. Antennas Propag. (1)

K. S. Yee, “Numerical solution of the initial boundary value problems involving maxwell’s equations in isotropic media,” IEEE Trans. Antennas Propag. 14, 302–307 (1966).
[Crossref]

J. Lightwave Technol. (3)

Nat. Commun. (1)

T. Pickering, J. M. Hamm, A. F. Page, S. Wuestner, and O. Hess, “Cavity-free plasmonic nanolasing enabled by dispersionless stopped light,” Nat. Commun. 5, 4972 (2014).
[Crossref] [PubMed]

Nature (1)

J. G. Bohnet, Z. Chen, J. M. Weiner, D. Meiser, M. J. Holland, and J. K. Thompson, “A steady-state superradiant laser with less than one intracavity photon,” Nature 484, 78–81 (2012).
[Crossref] [PubMed]

New J. Phys. (1)

D. Meiser, J. Ye, and M. J. Holland, “Spin squeezing in optical lattice clocks via lattice-based QND measurements,” New J. Phys. 10, 073014 (2008).
[Crossref]

Numer. Meth. Partial Differential Equations (1)

B. Bidégaray, “Time discretizations for maxwell-bloch equations,” Numer. Meth. Partial Differential Equations 19, 284–300 (2003).
[Crossref]

Opt. Commun. (1)

F. T. Arecchi, G. L. Lippi, G. P. Puccioni, and J. R. Tredicce, “Deterministic chaos in laser with injected signal,” Opt. Commun. 51, 308–314 (1984).
[Crossref]

Opt. Express (4)

Phys. Rev. (1)

A. L. Schawlow and C. H. Townes, “Infrared and optical masers,” Phys. Rev. 112, 1940–1949 (1958).
[Crossref]

Phys. Rev. A (18)

A. Siegman, “Excess spontaneous emission in non-hermitian optical-systems .2. laser-oscillators,” Phys. Rev. A 39, 1264–1268 (1989).
[Crossref] [PubMed]

W. Hamel and J. Woerdman, “Nonorthogonality of the longitudinal eigenmodes of a laser,” Phys. Rev. A 40, 2785–2787 (1989).
[Crossref] [PubMed]

M. Kolobov, L. Davidovich, E. Giacobino, and C. Fabre, “Role of pumping statistics and dynamics of atomic polarization in quantum fluctuations of laser sources,” Phys. Rev. A 47, 1431–1446 (1993).
[Crossref] [PubMed]

H. E. Türeci, A. D. Stone, and B. Collier, “Self-consistent multimode lasing theory for complex or random lasing media,” Phys. Rev. A 74, 043822 (2006).
[Crossref]

D. Meiser and M. J. Holland, “Steady-state superradiance with alkaline-earth-metal atoms,” Phys. Rev. A 81, 033847 (2010).
[Crossref]

D. Meiser and M. J. Holland, “Intensity fluctuations in steady-state superradiance,” Phys. Rev. A 81, 063827 (2010).
[Crossref]

P. Drummond and M. Raymer, “Quantum-theory of propagation of nonclassical radiation in a near-resonant medium,” Phys. Rev. A 44, 2072–2085 (1991).
[Crossref] [PubMed]

J. C. Pillay, Y. Natsume, A. D. Stone, and Y. D. Chong, “Generalized subSchawlow-Townes laser linewidths via material dispersion,” Phys. Rev. A 89, 033840 (2014).
[Crossref]

A. Pick, A. Cerjan, D. Liu, A. Rodriguez, A. D. Stone, Y. D. Chong, and S. Johnson, “Ab initio multimode linewidth theory for arbitrary inhomogeneous laser cavities,” Phys. Rev. A 91, 063806 (2015).
[Crossref]

S. Esterhazy, D. Liu, M. Liertzer, A. Cerjan, L. Ge, K. G. Makris, A. D. Stone, J. M. Melenk, S. G. Johnson, and S. Rotter, “Scalable numerical approach for the steady-state ab initio laser theory,” Phys. Rev. A 90, 023816 (2014).
[Crossref]

M. van Exter, S. Kuppens, and J. Woerdman, “Theory for the linewidth of a bad-cavity laser,” Phys. Rev. A 51, 809–816 (1995).
[Crossref] [PubMed]

L. Ge, Y. D. Chong, and A. D. Stone, “Steady-state ab initio laser theory: generalizations and analytic results,” Phys. Rev. A 82, 063824 (2010).
[Crossref]

G. Gray and R. Roy, “Noise in nearly-single-mode semiconductor-lasers,” Phys. Rev. A 40, 2452–2462 (1989).
[Crossref] [PubMed]

H. F. Hofmann and O. Hess, “Quantum maxwell-bloch equations for spatially inhomogeneous semiconductor lasers,” Phys. Rev. A 59, 2342–2358 (1999).
[Crossref]

J. Andreasen, H. Cao, A. Taflove, P. Kumar, and C.-q. Cao, “Finite-difference time-domain simulation of thermal noise in open cavities,” Phys. Rev. A 77, 023810 (2008).
[Crossref]

A. W. Rodriguez, A. P. McCauley, J. D. Joannopoulos, and S. G. Johnson, “Casimir forces in the time domain: Theory,” Phys. Rev. A 80, 012115 (2009).
[Crossref]

K. R. Manes and A. E. Siegman, “Observation of Quantum Phase Fluctuations in Infrared Gas Lasers,” Phys. Rev. A 4, 373–386 (1971).
[Crossref]

J. Andreasen and H. Cao, “Numerical study of amplified spontaneous emission and lasing in random media,” Phys. Rev. A 82, 063835 (2010).
[Crossref]

Phys. Rev. B (2)

S. Wuestner, J. M. Hamm, A. Pusch, F. Renn, K. L. Tsakmakidis, and O. Hess, “Control and dynamic competition of bright and dark lasing states in active nanoplasmonic metamaterials,” Phys. Rev. B 85, 201406 (2012).
[Crossref]

R. Buschlinger, M. Lorke, and U. Peschel, “Light-matter interaction and lasing in semiconductor nanowires: A combined finite-difference time-domain and semiconductor Bloch equation approach,” Phys. Rev. B 91, 045203 (2015).
[Crossref]

Phys. Rev. Lett. (5)

C. Luo, A. Narayanaswamy, G. Chen, and J. D. Joannopoulos, “Thermal Radiation from Photonic Crystals: A Direct Calculation,” Phys. Rev. Lett. 93, 213905 (2004).
[Crossref] [PubMed]

Y. D. Chong and A. D. Stone, “General linewidth formula for steady-state multimode lasing in arbitrary cavities,” Phys. Rev. Lett. 109, 063902 (2012).
[Crossref] [PubMed]

S. Kuppens, M. van Exter, and J. Woerdman, “Quantum-limited linewidth of a bad-cavity laser,” Phys. Rev. Lett. 72, 3815–3818 (1994).
[Crossref] [PubMed]

D. Meiser, J. Ye, D. R. Carlson, and M. J. Holland, “Prospects for a Millihertz-Linewidth Laser,” Phys. Rev. Lett. 102, 163601 (2009).
[Crossref] [PubMed]

W. Hamel and J. Woerdman, “Observation of enhanced fundamental linewidth of a laser due to nonorthogonality of its longitudinal eigenmodes,” Phys. Rev. Lett. 64, 1506–1509 (1990).
[Crossref] [PubMed]

Proc. IEEE (1)

M. Frigo and S. G. Johnson, “The design and implementation of FFTW3,” Proc. IEEE 93, 216–231 (2005).
[Crossref]

Prog. Quantum Electron. (3)

K. Böhringer and O. Hess, “A full-time-domain approach to spatio-temporal dynamics of semiconductor lasers. i. theoretical formulation,” Prog. Quantum Electron. 32, 159–246 (2008).
[Crossref]

K. Böhringer and O. Hess, “A full time-domain approach to spatio-temporal dynamics of semiconductor lasers. II. spatio-temporal dynamics,” Prog. Quantum Electron. 32, 247–307 (2008).
[Crossref]

M. Kira, F. Jahnke, W. Hoyer, and S. W. Koch, “Quantum theory of spontaneous emission and coherent effects in semiconductor microstructures,” Prog. Quantum Electron. 23, 189–279 (1999).
[Crossref]

Rev. Mod. Phys. (1)

C. H. Henry and R. F. Kazarinov, “Quantum noise in photonics,” Rev. Mod. Phys. 68, 801–853 (1996).
[Crossref]

Science (1)

H. E. Türeci, L. Ge, S. Rotter, and A. D. Stone, “Strong interactions in multimode random lasers,” Science 320, 643–646 (2008).
[Crossref] [PubMed]

Other (11)

H. Haken, Laser theory (Springer-Verlag, 1984).

M. Lax, “Quantum noise v: Phase noise in a homogeneously broadened maser,” in “Physics of Quantum Electronics,”, P. L. Kelley, B. Lax, and P. E. Tannenwald, eds. (McGraw-Hill, 1966).

H. Haken, Light: Laser Dynamics, vol. 2 (North-Holland Phys. Publishing, 1985).

L. D. Landau and E. M. Lifshitz, Statistical Physics,, 3rd ed. (Butterworth-Heinemann, Oxford, 1980), Part 1, Vol. 5.

J. Andreasen, “Numerical studies of lasing and electromagnetic fluctuations in open complex systems,” Ph.D. thesis, Northwestern University (2009).

J. D. Jackson, Classical Electrodynamics, 3rd ed. (Wiley, 1998).

L. Ge, “Steady-state ab initio laser theory and its applications in random and complex media,” Ph.D. thesis, Yale University (2010).

R. Shankar, Principles of Quantum Mechanics, 2nd ed. (Springer, 2013).

A. Taflove and S. C. Hagness, Computational Electrodynamics: The Finite-Difference Time-Domain Method (Artech House, Incorporated, 2005).

J. G. Proakis and D. G. Manolakis, Digital Signal Processing (Pearson Prentice Hall, 2007).

J. Ohtsubo, Semiconductor Lasers: Stability, Instability, and Chaos (Springer, 2007).

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Figures (10)

Fig. 1
Fig. 1

(a) Optical intensity spectrum of the output electric field of an n = 3 dielectric slab cavity, shown in the schematic. The simulation parameters for the cavity are γ = .5, ωa = 42.4, γ = .01, θ = 2 × 10−9, NA = 1010, and the cavity is uniformly pumped at D0 = 0.275 which is close to 5 times the threshold lasing pump of D0,thr = 0.0488. The rates quoted here are given in units of c/L, while the intensity is given in SALT units of 4θ2/(2γγ), and the number and inversion of gain atoms are given in the SALT units of 4πθ2/(h̄γ). (b) Plot of the fitted Lorentz error function (red line) and numerically integrated FDTD data (blue dots) of the simulation shown in (a). The spectral resolution for the simulated data in (a) and (b) is = 1.96 × 10−5. The analytic curve fit parameters are found using MATLAB’s curve fitting algorithms.

Fig. 2
Fig. 2

(Left panel) Plot showing the linewidth predictions given by the N-SALT given in Eq. (2) (green), corrected Schawlow-Townes theory given in Eq. (42) (blue), integral form of the Chong-Stone linewidth formula given in Eq. (43) (orange), and FDTD simulations (magenta) for a uniformly pumped, dielectric slab cavity with n = 3, ωa = 42.4, γ = .5, γ = .01, θ = 2 × 10−9, and NA = 1010. All of the linewidth formulas are evaluated using the spatially dependent integral definition of the power given by Eq. (46). (Right panel) Plot of the same data shown on a log-log scale, with reference lines for strict inverse power dependence, P−1, provided for comparison (black dashed). Schematic inset shows the cavity geometry. The rates and frequency are given in units of c/L, the number of atoms in the cavity is given in terms of the SALT units of 4πθ2/(h̄γ), and the output power is given in the SALT units of 4θ2/(2γγ).

Fig. 3
Fig. 3

(Left panel) Plot of the steady-state inversion, D(x), as a function of the location in the cavity for three different values of the output power, P = 0.524 (blue), P = 1.252 (green), and P = 3.116 (red). These values correspond to the first, sixth, and eighteenth data points shown in Fig. 2. Strong spatial hole-burning is seen in the inversion due to the lasing mode. Schematic depicts the cavity from Fig. 2. (Right panel) Plot of the normalized spatial profile of the lasing mode, |ψ0(x)|, as a function of position in the cavity for the same three values of the output power shown in the left panel. The output power is given in dimensionless SALT units of 4θ2/(2γγ).

Fig. 4
Fig. 4

(Left panel) Plot showing the linewidth predictions given by the N-SALT (green line), corrected Schawlow-Townes theory (blue line), and FDTD simulations (red diamonds and magenta triangles) for a uniformly pumped, dielectric slab cavity with n = 3, ωa = 42.4, γ = .5, γ = .04, θ = 4 × 10−9, and NA = 1010, as shown in the schematic. The results of the new FDTD simulations are shown as red diamonds, and are plotted alongside the FDTD results from Fig. 2, shown as magenta triangles. (Right panel) Plot showing the linewidth predictions given by the N-SALT (green line), rescaled N-SALT prediction from Fig. 2 (magenta dashed line), corrected Schawlow-Townes theory (blue line), and FDTD simulations (cyan squares) for a uniformly pumped, dielectric slab cavity with n = 3, ωa = 42.4, γ = .25, γ = .02, θ = 2 × 10−9, and NA = 1010. The rates and frequency are given in units of c/L, the number of atoms in the cavity is given in terms of the SALT units of 4πθ2/(h̄γ), and the output power is given in the SALT units of 4θ2/(2γγ).

Fig. 5
Fig. 5

Plots showing a comparison between the N-SALT prediction (red) and FDTD simulations (blue) of the optical intensity spectrum for increasing values of the pump, D0, for a single-sided, dielectric slab cavity with n = 1.5, ωa = 40.7, γ = 1, γ = 0.0025, θ = 6 × 10−10, and NA = 1010. (a) D0 = 0.18, (b) D0 = 0.28, (c) D0 = 0.38. As can be seen, increasing the pump value increases the rate of stimulated emission, increasing γ(x), Eq. 53, resulting in increasing separation between the relaxation oscillation side peaks and the central lasing frequency. In all three panels of Fig. 5, the central frequency, ω0, chosen to evaluate Eq. (55) is the central frequency found by the FDTD simulations. Intensity is plotted on a log scale in arbitrary units, rates are given in units of c/L, and the inversion and total number of atoms are given in SALT units of 4πθ2/h̄γ.

Fig. 6
Fig. 6

Plot of the linewidth versus the output power for a two-sided dielectric slab cavity, n = 3.5, showing the comparison between the N-SALT linewidth prediction (green line), the N-SALT linewidth without an α factor (cyan line), the N-SALT linewidth using Lax’s α factor (blue line), and the FDTD simulation results (magenta triangles). Excellent quantitative agreement is seen between the FDTD simulations and the correct N-SALT linewidth prediction, confirming the form of the α factor derived by Pick et al. [27]. For the two-level gain medium used here, ωa = 18.3, γ = 0.05, γ = 0.01, θ = 4 × 10−9, and NA = 1010, and results in the total system having α 0 2 = 2.56, while α̃2 ≈ 0.66. Frequencies and rates are given in units of c/L, while the atomic values are given in SALT units of 4πθ2/h̄γ.

Fig. 7
Fig. 7

Plot showing the modal output intensity as a function of the gain medium pump strength D0, for a two-sided system consisting of two coupled dielectric cavities, n = 3, with different lengths, L1 = .42L0, and L2 = .5L0, joined together by a region of air, n = 1, with length Lair = .08L0, where L0 is the total size of the system, and as shown in the schematic. This cavity has up to two active lasing modes (red and orange) for the pump values simulated here, and quantitative agreement is seen between the SALT simulations (solid lines) and noisy FDTD simulations (squares). A slight offset in the interacting threshold for the second lasing mode is seen between the two simulations, with D SALT ( 2 ) = 0.5077, while D FDTD ( 2 ) = 0.5282. The inset plot shows the FDTD simulated intensity of the second lasing mode through its lasing threshold, first showing amplified spontaneous emission, then super-linear behavior at threshold, and finally linear behavior above threshold, as expected. The gain medium was chosen to have ωa = 15, γ = 0.4, γ = 0.01, θ = 10−9, and NA = 1010. Frequencies and rates are given in units of c/L0, while the field quantities and inversion values are given in SALT units of 4θ2/2γγ and 4πθ2/h̄γ, respectively.

Fig. 8
Fig. 8

Comparison of the single mode N-SALT linewidth prediction (green line), two-mode N-SALT linewidth prediction (red line), and FDTD simulations (magenta triangles) for the first lasing mode in coupled cavity system from Fig. 7. Inset shows a zoom in of the same quantities close to the interacting threshold of the second lasing mode. The two slightly different second mode thresholds are marked in the inset, D SALT ( 2 ) (dashed blue line), and D FDTD ( 2 ) (dashed cyan line). While the data is too noisy, and the difference between the single mode and two-mode predictions too small, for the resolution of their differences, we do observe increased linewidth and variance in our simulations close to the threshold of the second lasing mode, as expected.

Fig. 9
Fig. 9

Comparison of the single mode N-SALT linewidth prediction (green line), two-mode N-SALT linewidth prediction (red line), and FDTD simulations (magenta triangles) for the second lasing mode in coupled cavity system from Fig. 7. Inset shows the same data except with the FDTD simulations plotted at shifted pump values (cyan triangles) to account for the slightly different second lasing mode thresholds seen in Fig. 7. Quantitative agreement between the FDTD and N-SALT linewidth predictions is seen in both versions of the plot, but the inset demonstrates that most of the discrepancy seen in the outer plot is due to differences in the output power of the cavity due to the SALT simulations being further above threshold than the FDTD simulations for the same value of the pump D0.

Fig. 10
Fig. 10

Plot of the autocorrelation of the electric field simulated numerically for the same parameter used in Fig. 1 (blue line) and the analytic prediction for the envelope of the autocorrelation given in the second factor in Eq. (66) (green line). The fast oscillations in the numerically simulated electric field are at the lasing frequency ω0, which is much faster than the other time scales in the problem and leads to the densely packed curve shown in blue. Quantities are normalized, and plotted in units of δωδt.

Equations (66)

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δ ω ST = h ¯ ω 0 γ c 2 2 P
δ ω N-SALT = h ¯ ω 0 2 P ω 0 2 Im [ ε ( x , ω 0 ) ] | ψ 0 ( x ) | 2 d x Im [ ε ( x , ω 0 ) ] N 2 ( x ) D ( x ) | ψ 0 ( x ) | 2 d x | ψ 0 2 ( x ) ( ε ( x , ω 0 ) + ω 0 2 d ε d ω | ω 0 ) d x | 2 ( 1 + α ˜ 2 ) ,
[ × × ω 2 ε ( ω , E 0 ) ] E = ω 2 ( ε ( ω , E ) ε ( ω , E 0 ) ) E + F S ,
F S ( x , ω ) F S ( x , ω ) = 2 h ¯ ω 4 Im [ ε ( ω , E 0 ) ] coth ( h ¯ ω β ( x ) 2 ) δ ( x x ) δ ( ω ω ) ,
1 2 [ coth ( h ¯ ω 0 β ( x ) 2 1 ) ] = N 2 ( x ) D ( x ) ,
F S ( x , ω ) F S ( x , ω ) = 4 h ¯ ω 4 Im [ ε ( x , ω ) ] [ 1 2 coth ( h ¯ ω 0 β ( x ) 2 1 2 ) ] δ ( x x ) δ ( ω ω ) .
t ρ 21 ( α ) ( t ) = ( γ + i ω a ) ρ 21 ( α ) ( t ) + i d ( α ) h ¯ θ E ( x ( α ) , t ) + Γ ( ρ ) ( α ) ( t ) ,
t d ( α ) = γ ( d 0 ( α ) d ( α ) ) + 2 i h ¯ θ E ¯ ( x ( α ) , t ) ( ρ 21 ( α ) * ρ 21 ( α ) ) + Γ ( d ) ( α ) ( t )
[ × × ω 0 2 ε c ] E ( x , ω ) = 4 π ω 0 2 θ α δ ( x x ( α ) ) ρ 21 ( α ) ,
ρ 21 ( α ) = d ( α ) h ¯ ( ω 0 ω a + i γ ) θ E ˜ ( x ( α ) , ω ) + i e i ω 0 t ω 0 ω a + i γ Γ ( ρ ) ( α ) ,
[ × × ω 0 2 ε c ] E ( x , ω ) = 4 π ω 0 2 θ α δ ( x x ( α ) ) × [ d ( α ) ( θ E ( x ( α ) , ω ) h ¯ ( ω 0 ω a + i γ ) + i e i ω 0 t ω 0 ω a + i γ Γ ( ρ ) ( α ) ] .
P N ( x , ω ) = α δ ( x x ( α ) ) i θ e i ω 0 t ω 0 ω a + i γ Γ ( ρ ) ( α ) ( ω ) .
Γ ( ρ ) ( α ) ( t ) Γ ( ρ ) ( β ) ( t ) = [ γ ( 1 + d ( α ) ) + γ 2 ( d 0 α d ( α ) ) ] δ α β δ ( t t ) ,
Γ ( ρ ) ( α ) ( ω ) Γ ( ρ ) ( β ) ( ω ) = γ ( 1 + d ( α ) ) δ α β ,
P N ( x , ω ) P N ( x , ω ) = 2 θ 2 γ ( ω 0 ω a ) 2 + γ 2 N 2 ( x ) δ ( x x ) ,
N 2 ( x ) = 1 2 α δ ( x x ( α ) ) ( 1 + d ( α ) ) .
Im [ ε ] = 4 π θ 2 h ¯ γ D ( x ) ( ω ω a ) 2 + γ 2 ,
F S ( x , ω ) F S ( x , ω ) = 8 π ω 0 4 h ¯ Im [ ε ] N 2 ( x ) D ( x ) δ ( x x ) .
F S ( x , ω ) F S ( x , ω ) = 1 2 F S ( x , ω ) F S ( x , ω ) δ ( ω ω )
d d t E n = c 2 ε c [ d d x B n + 4 π ( θ V 0 ) d d t ( J n + ( J n ) * ) ] ,
d d t B n = d d x E n ,
d d t J n = ( γ + i ω a ) J n θ i h ¯ E n D n + F n ( J ) ,
d d t D n = γ ( D n D 0 , n ) + 2 θ i h ¯ E n ( ( J n ) * J n ) + F n ( D ) ,
J n ( x ) = α ρ 21 ( α ) δ ( x x ( α ) ) = N n ρ 21 ( x ) .
F n ( J ) = ξ n ( J ) 2 i θ E n J n + ξ n ( P ) γ P ( D n + N n ) + ξ n ( N ) γ 21 , n N n ,
F n ( D ) = 2 ξ n ( D ) [ γ 2 ( N n D 0 , n N n D n ) + i θ ( J n E n J n + E n ) 2 γ 21 , n J n + J n N n ] ( 1 / 2 ) 2 [ ξ n ( N ) J n + + ξ n ( N ) * J n ] γ 21 , n N n ,
γ 21 , n = γ 2 ( 1 + D 0 , n N n ) ,
ξ n ( i ) ( t ) ξ m ( j ) ( t ) = δ ( t t ) δ n m δ i j .
F n ( J ) = ξ n ( P ) γ P ( D n + N n ) + ξ n ( N ) γ 21 , n N n ,
F n ( D ) = 2 ξ n ( D ) γ 2 ( N n D 0 , n N n D n ) .
E n ( t i + 1 ) = E n ( t i ) + c 2 Δ t ε c [ 8 π ( θ V 0 ) ( ω a j n ( 2 ) ( t i + 1 2 ) γ j n ( 1 ) ( t i + 1 2 ) ) + B n + 1 2 ( t i + 1 2 ) B n 1 2 ( t i + 1 2 ) Δ x ] ,
B n + 1 2 ( t i + 1 2 ) = B n + 1 2 ( t i 1 2 ) + Δ t Δ x ( E n + 1 ( t i ) E n ( t i ) ) ,
u n ( t i + 1 2 ) = ( 1 Δ t I 1 2 M ) 1 [ d n + f n + ( 1 Δ t I + 1 2 M ) u n ( t i 1 2 ) ] ,
M = ( γ 0 4 θ h ¯ E n ( t i ) 0 γ ω a θ h ¯ E n ( t i ) ω a γ ) ,
f n , 1 = 2 ξ n ( 1 ) γ 2 ( N n D 0 , n N n D n ( t i 1 2 ) ) ,
f n , 2 = ξ n ( 2 ) γ P ( D n ( t i 1 2 ) + N n ) + ξ n ( 3 ) γ 21 , n N n ,
f n , 3 = ξ n ( 4 ) γ P ( D n ( t i 1 2 ) + N n ) + ξ n ( 5 ) γ 21 , n N n .
ξ n ( k ) ( t i ) ξ m ( l ) ( t j ) = 1 Δ t δ i j δ n m δ k l .
L ( ω ) = ( 2 A π ) s 2 ( ω ω 0 ) 2 + s 2
L EF ( ω ) = ω 0 ω L ( ω ) d ω = ( 2 A s π ) arctan ( ω ω 0 s ) .
L EF ( ω ) = ( 2 A s π ) arctan ( ω ω 0 + d s ) + c ,
δ ω ST ( corr ) = h ¯ ω 0 γ c 2 2 P ( N ¯ 2 D ) | | ϕ 0 ( x ) | 2 d x ϕ 0 2 ( x ) d x | 2 | 1 1 + ω 0 2 ε ε ω | ω 0 | 2 ( 1 + α 2 ) ,
δ ω CS = h ¯ ω 0 2 P ( N ¯ 2 D ) ( ω 0 Im [ ε ( x , ω 0 ) ] | ψ 0 ( x ) | 2 d x ) 2 | ψ 0 2 ( x ) ( ε + ω 0 2 d ε d ω | ω 0 ) d x | 2 ( 1 + α 2 ) ,
δ ω CS δ ω N-SALT = N ¯ 2 D ¯ D ( x ) | ψ 0 ( x ) | 2 d x N 2 ( x ) | ψ 0 ( x ) | 2 d x .
δ ω CS δ ω N-SALT = D ( x ) | ψ 0 ( x ) | 2 d x | ψ 0 ( x ) | 2 d x D ( x ) d x .
P = ω 0 2 π Im [ ε ( x ) ] | E 0 ( x ) | 2 d x ,
P ST = γ c n ¯ h ¯ ω 0 ,
P = ( h ¯ 2 γ γ 4 θ 2 ) ω 0 2 π Im [ ε ( x ) ] | E SALT ( x ) | 2 d x .
δ ω N-SALT = ( 4 θ 2 h ¯ 2 γ γ ) h ¯ ω 0 2 P SALT ω 0 2 Im [ ε ] | ψ 0 | 2 d x Im [ ε ] N 2 D | ψ 0 | 2 d x | ψ 0 2 ( ε + ω 0 2 d ε d ω | ω 0 ) d x | 2 ( 1 + α ˜ 2 ) ,
γ = γ spon + γ nr ,
γ spon = 4 α fs ω a 3 n θ 2 3 c 2 ,
B = 1 | ψ 0 2 ( x ) ( ε + ω 0 2 d ε d ω | ω 0 ) d x | | 1 1 + γ c 2 γ | .
γ ( x ) = γ ( 1 + γ 2 ( ω 0 ω a ) 2 + γ 2 | E SALT ( x ) | 2 ) ,
A ( x ) = 2 I Re [ i ω 0 ψ 0 2 ( x ) ε ( ω 0 ) I 2 ψ 0 2 ( x ) ( ε + ω 0 2 d ε d ω | ω 0 ) d x ] ,
S N-SALT ( ω ) = δ ω N-SALT ω 2 + ( δ ω N-SALT 2 ) 2 + δ ω N-SALT ω 2 ( 1 R ( ω ) ) 2 + R ˜ ( ω ) 2 ,
R ( ω ) = A ( x ) γ ( x ) ω 2 + ( δ ω N-SALT 2 + γ ( x ) ) 2 d x ,
R ˜ ( ω ) = A ( x ) γ ( x ) ( δ ω N-SALT 2 + γ ( x ) ) ω 2 + ( δ ω N-SALT 2 + γ ( x ) ) 2 d x .
α 0 = ω 0 ω a γ .
α ˜ = Im [ C 11 ] Re [ C 11 ] ,
C μ ν = [ i ω μ ψ μ 2 ( x ) ε ( ω μ ) I ν d x 2 ψ μ 2 ( x ) ( ε + ω μ 2 d ε d ω | ω μ ) d x ] ,
δ ω N-SALT ( two mode ) = δ ω N-SALT ( 1 ) [ 1 + C 11 I C 22 R C 21 I C 21 R C 11 R C 22 R C 12 R C 21 R ] + δ ω N-SALT ( 2 ) [ C 11 R C 12 I C 11 I C 12 R C 11 R C 22 R C 12 R C 21 R ] ,
E ( t ) = C cos ( ω t + ϕ ( t ) ) ,
R EE ( δ t ) = E 2 ( t ) cos ( ω δ t + δ ϕ ( δ t ) ) C 2 2 sin ( 2 ω t + 2 ϕ ( t ) ) sin ( ω δ t + δ ϕ ( δ t ) ) ,
R EE ( δ t ) = C 2 2 [ cos ( ω δ t ) cos ( δ ϕ ( δ t ) ) sin ( ω δ t ) sin ( δ ϕ ( δ t ) ) ] .
δ ϕ 2 ( δ t ) = δ ω δ t ,
R EE ( δ t ) = C 2 2 cos ( ω δ t ) [ 1 δ ω δ t 2 + O ( δ t 2 ) ] ,

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