Steady-state ab initio laser theory for N-level lasers

Alexander Cerjan; Yidong Chong; Li Ge; A. Douglas Stone

doi:10.1364/OE.20.000474

Steady-state ab initio laser theory for N-level lasers

Alexander Cerjan, Yidong Chong, Li Ge, and A. Douglas Stone

Optics Express
Vol. 20,
Issue 1,
pp. 474-488
(2012)
•https://doi.org/10.1364/OE.20.000474

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Alexander Cerjan, Yidong Chong, Li Ge, and A. Douglas Stone, "Steady-state ab initio laser theory for N-level lasers," Opt. Express 20, 474-488 (2012)

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Related Topics
Table of Contents Category
- Lasers and Laser Optics
Optics & Photonics Topics
?

The topics in this list come from the OSA Optics and Photonics Topics applied to this article.
Previously assigned OCIS codes
- Laser theory (140.3430)
- Microcavities (140.3945)

About this Article
History
- Original Manuscript: September 27, 2011
- Revised Manuscript: November 21, 2011
- Manuscript Accepted: November 22, 2011
- Published: December 21, 2011

Accessible

Open Access

Abstract

We show that Steady-state Ab initio Laser Theory (SALT) can be applied to find the stationary multimode lasing properties of an N-level laser. This is achieved by mapping the N-level rate equations to an effective two-level model of the type solved by the SALT algorithm. This mapping yields excellent agreement with more computationally demanding N-level time domain solutions for the steady state.

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References

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|
Year
|
Author
|
Publication

H. Haken, Light: Laser Dynamics (North-Holland Phys. Publishing, 1985), Vol. 2.
W. E. Lamb, “Theory of an optical maser,” Phys. Rev. 134, A1429 (1964).
[Crossref]
A. E. Siegman, Lasers (University Science Books, 1986).
A. S. Nagra and R. A. York, “FDTD analysis of wave propagation in nonlinear absorbing and gain media,” IEEE Trans. Antennas Propag. 46, 334–340 (1998).
[Crossref]
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]
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]
H. E. Türeci, A. D. Stone, and L. Ge, “Theory of the spatial structure of nonlinear lasing modes,” Phys. Rev. A 76, 013813 (2007).
[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, 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. Cao, “Review on the latest developments in random lasers with coherent feedback,” J. Phys. A 38, 10497–10535 (2005).
[Crossref]
O. Painter, R. K. Lee, A. Scherer, A. Yariv, J. D. O’Brien, P. D. Dapkus, and I. Kim, “Two-dimensional photonic band-gap defect mode laser,” Science 284, 1819–1821 (1999).
[Crossref] [PubMed]
S. Chua, Y. D. Chong, A. D. Stone, M Soljac̆ić, and J. Bravo-Abad, “Low-threshold lasing action in photonic crystal slabs enabled by Fano resonances,” Opt. Express 19, 1539 (2011).
[Crossref] [PubMed]
C. Gmachl, F. Capasso, E. E. Narimanov, J. U. Nöckel, A. D. Stone, J. Faist, D. L. Sivco, and A. Y. Cho, “High-power directional emission from microlasers with chaotic resonators,” Science 280, 1556–1564 (1998).
[Crossref] [PubMed]
Ge Li, Yale PhD thesis, 2010.
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 (2008).
[Crossref] [PubMed]
The equations are written for the TM case, the modifications for TE are straightforward.
The observation that coherent and incoherent pumping are nearly equivalent for most systems, is invalid when the coherent pumping is supplied at a similar frequency to the atomic lasing transition and thus interactions between the lasing field and pumping field must be taken into account.
H. Fu and H. Haken, “Multifrequency operations in a short-cavity standing-wave laser,” Phys. Rev. A 43, 2446–2454 (1991).
[Crossref] [PubMed]
Y. I. Khanin, Principles of Laser Dynamics (Elsevier, 1995).
B. Bidégaray, “Time discretizations for Maxwell-Bloch equations,” Numer. Meth. Partial Differential Equations 19, 284–300 (2003).
[Crossref]
For any 1D cavity which is uniformly pumped the TCF states for solving SALT can also be found using a transfer matrix method which does not require discretizing space. We use a more general TCF solver in the calculations presented here which does discretize space.
X. Jiang and C. M. Soukoulis, “Time dependent theory for random lasers,” Phys. Rev. Lett. 85, 70 (2000).
[Crossref] [PubMed]
X. Jiang, S. Feng, C. M. Soukoulis, J. Zi, J. D. Joannopoulos, and H. Cao, “Coupling, competition, and stability of modes in random lasers,” Phys. Rev. B 69, 104202 (2004).
[Crossref]
R. W. Boyd, Nonlinear Optics (Academic Press, 2008).

2011 (1)

S. Chua, Y. D. Chong, A. D. Stone, M Soljac̆ić, and J. Bravo-Abad, “Low-threshold lasing action in photonic crystal slabs enabled by Fano resonances,” Opt. Express 19, 1539 (2011).
[Crossref] [PubMed]

2010 (1)

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]

2008 (2)

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 (2008).
[Crossref] [PubMed]

2007 (1)

H. E. Türeci, A. D. Stone, and L. Ge, “Theory of the spatial structure of nonlinear lasing modes,” Phys. Rev. A 76, 013813 (2007).
[Crossref]

2006 (1)

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]

2005 (1)

H. Cao, “Review on the latest developments in random lasers with coherent feedback,” J. Phys. A 38, 10497–10535 (2005).
[Crossref]

2004 (1)

X. Jiang, S. Feng, C. M. Soukoulis, J. Zi, J. D. Joannopoulos, and H. Cao, “Coupling, competition, and stability of modes in random lasers,” Phys. Rev. B 69, 104202 (2004).
[Crossref]

2003 (1)

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

2000 (1)

X. Jiang and C. M. Soukoulis, “Time dependent theory for random lasers,” Phys. Rev. Lett. 85, 70 (2000).
[Crossref] [PubMed]

1999 (1)

O. Painter, R. K. Lee, A. Scherer, A. Yariv, J. D. O’Brien, P. D. Dapkus, and I. Kim, “Two-dimensional photonic band-gap defect mode laser,” Science 284, 1819–1821 (1999).
[Crossref] [PubMed]

1998 (2)

A. S. Nagra and R. A. York, “FDTD analysis of wave propagation in nonlinear absorbing and gain media,” IEEE Trans. Antennas Propag. 46, 334–340 (1998).
[Crossref]

C. Gmachl, F. Capasso, E. E. Narimanov, J. U. Nöckel, A. D. Stone, J. Faist, D. L. Sivco, and A. Y. Cho, “High-power directional emission from microlasers with chaotic resonators,” Science 280, 1556–1564 (1998).
[Crossref] [PubMed]

1991 (1)

H. Fu and H. Haken, “Multifrequency operations in a short-cavity standing-wave laser,” Phys. Rev. A 43, 2446–2454 (1991).
[Crossref] [PubMed]

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]

1964 (1)

W. E. Lamb, “Theory of an optical maser,” Phys. Rev. 134, A1429 (1964).
[Crossref]

Bidégaray, B.

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

Boyd, R. W.

R. W. Boyd, Nonlinear Optics (Academic Press, 2008).

Bravo-Abad, J.

Cao, H.

H. Cao, “Review on the latest developments in random lasers with coherent feedback,” J. Phys. A 38, 10497–10535 (2005).
[Crossref]

X. Jiang, S. Feng, C. M. Soukoulis, J. Zi, J. D. Joannopoulos, and H. Cao, “Coupling, competition, and stability of modes in random lasers,” Phys. Rev. B 69, 104202 (2004).
[Crossref]

Capasso, F.

Cho, A. Y.

Chong, Y. D.

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]

Chua, S.

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]

Dapkus, P. D.

O. Painter, R. K. Lee, A. Scherer, A. Yariv, J. D. O’Brien, P. D. Dapkus, and I. Kim, “Two-dimensional photonic band-gap defect mode laser,” Science 284, 1819–1821 (1999).
[Crossref] [PubMed]

Faist, J.

Feng, S.

X. Jiang, S. Feng, C. M. Soukoulis, J. Zi, J. D. Joannopoulos, and H. Cao, “Coupling, competition, and stability of modes in random lasers,” Phys. Rev. B 69, 104202 (2004).
[Crossref]

Fu, H.

H. Fu and H. Haken, “Multifrequency operations in a short-cavity standing-wave laser,” Phys. Rev. A 43, 2446–2454 (1991).
[Crossref] [PubMed]

Ge, L.

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 (2008).
[Crossref] [PubMed]

H. E. Türeci, A. D. Stone, and L. Ge, “Theory of the spatial structure of nonlinear lasing modes,” Phys. Rev. A 76, 013813 (2007).
[Crossref]

Gmachl, C.

Haken, H.

H. Fu and H. Haken, “Multifrequency operations in a short-cavity standing-wave laser,” Phys. Rev. A 43, 2446–2454 (1991).
[Crossref] [PubMed]

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

Jiang, X.

X. Jiang, S. Feng, C. M. Soukoulis, J. Zi, J. D. Joannopoulos, and H. Cao, “Coupling, competition, and stability of modes in random lasers,” Phys. Rev. B 69, 104202 (2004).
[Crossref]

X. Jiang and C. M. Soukoulis, “Time dependent theory for random lasers,” Phys. Rev. Lett. 85, 70 (2000).
[Crossref] [PubMed]

Joannopoulos, J. D.

X. Jiang, S. Feng, C. M. Soukoulis, J. Zi, J. D. Joannopoulos, and H. Cao, “Coupling, competition, and stability of modes in random lasers,” Phys. Rev. B 69, 104202 (2004).
[Crossref]

Khanin, Y. I.

Y. I. Khanin, Principles of Laser Dynamics (Elsevier, 1995).

Kim, I.

O. Painter, R. K. Lee, A. Scherer, A. Yariv, J. D. O’Brien, P. D. Dapkus, and I. Kim, “Two-dimensional photonic band-gap defect mode laser,” Science 284, 1819–1821 (1999).
[Crossref] [PubMed]

Lamb, W. E.

W. E. Lamb, “Theory of an optical maser,” Phys. Rev. 134, A1429 (1964).
[Crossref]

Lee, R. K.

O. Painter, R. K. Lee, A. Scherer, A. Yariv, J. D. O’Brien, P. D. Dapkus, and I. Kim, “Two-dimensional photonic band-gap defect mode laser,” Science 284, 1819–1821 (1999).
[Crossref] [PubMed]

Li, Ge

Ge Li, Yale PhD thesis, 2010.

Nagra, A. S.

A. S. Nagra and R. A. York, “FDTD analysis of wave propagation in nonlinear absorbing and gain media,” IEEE Trans. Antennas Propag. 46, 334–340 (1998).
[Crossref]

Narimanov, E. E.

Nöckel, J. U.

O’Brien, J. D.

O. Painter, R. K. Lee, A. Scherer, A. Yariv, J. D. O’Brien, P. D. Dapkus, and I. Kim, “Two-dimensional photonic band-gap defect mode laser,” Science 284, 1819–1821 (1999).
[Crossref] [PubMed]

Painter, O.

O. Painter, R. K. Lee, A. Scherer, A. Yariv, J. D. O’Brien, P. D. Dapkus, and I. Kim, “Two-dimensional photonic band-gap defect mode laser,” Science 284, 1819–1821 (1999).
[Crossref] [PubMed]

Rotter, S.

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

Scherer, A.

O. Painter, R. K. Lee, A. Scherer, A. Yariv, J. D. O’Brien, P. D. Dapkus, and I. Kim, “Two-dimensional photonic band-gap defect mode laser,” Science 284, 1819–1821 (1999).
[Crossref] [PubMed]

Siegman, A. E.

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

Sivco, D. L.

Soljac?ic, M

Soukoulis, C. M.

X. Jiang, S. Feng, C. M. Soukoulis, J. Zi, J. D. Joannopoulos, and H. Cao, “Coupling, competition, and stability of modes in random lasers,” Phys. Rev. B 69, 104202 (2004).
[Crossref]

X. Jiang and C. M. Soukoulis, “Time dependent theory for random lasers,” Phys. Rev. Lett. 85, 70 (2000).
[Crossref] [PubMed]

Stone, A. D.

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 (2008).
[Crossref] [PubMed]

H. E. Türeci, A. D. Stone, and L. Ge, “Theory of the spatial structure of nonlinear lasing modes,” Phys. Rev. A 76, 013813 (2007).
[Crossref]

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]

Tandy, R. J.

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 (2008).
[Crossref] [PubMed]

Türeci, H. E.

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 (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]

H. E. Türeci, A. D. Stone, and L. Ge, “Theory of the spatial structure of nonlinear lasing modes,” Phys. Rev. A 76, 013813 (2007).
[Crossref]

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]

Yariv, A.

O. Painter, R. K. Lee, A. Scherer, A. Yariv, J. D. O’Brien, P. D. Dapkus, and I. Kim, “Two-dimensional photonic band-gap defect mode laser,” Science 284, 1819–1821 (1999).
[Crossref] [PubMed]

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]

York, R. A.

A. S. Nagra and R. A. York, “FDTD analysis of wave propagation in nonlinear absorbing and gain media,” IEEE Trans. Antennas Propag. 46, 334–340 (1998).
[Crossref]

Zi, J.

X. Jiang, S. Feng, C. M. Soukoulis, J. Zi, J. D. Joannopoulos, and H. Cao, “Coupling, competition, and stability of modes in random lasers,” Phys. Rev. B 69, 104202 (2004).
[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]

IEEE Trans. Antennas Propag. (1)

A. S. Nagra and R. A. York, “FDTD analysis of wave propagation in nonlinear absorbing and gain media,” IEEE Trans. Antennas Propag. 46, 334–340 (1998).
[Crossref]

J. Phys. A (1)

H. Cao, “Review on the latest developments in random lasers with coherent feedback,” J. Phys. A 38, 10497–10535 (2005).
[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. Express (2)

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 (2008).
[Crossref] [PubMed]

Phys. Rev. (1)

W. E. Lamb, “Theory of an optical maser,” Phys. Rev. 134, A1429 (1964).
[Crossref]

Phys. Rev. A (4)

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]

H. E. Türeci, A. D. Stone, and L. Ge, “Theory of the spatial structure of nonlinear lasing modes,” Phys. Rev. A 76, 013813 (2007).
[Crossref]

H. Fu and H. Haken, “Multifrequency operations in a short-cavity standing-wave laser,” Phys. Rev. A 43, 2446–2454 (1991).
[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]

Phys. Rev. B (1)

X. Jiang, S. Feng, C. M. Soukoulis, J. Zi, J. D. Joannopoulos, and H. Cao, “Coupling, competition, and stability of modes in random lasers,” Phys. Rev. B 69, 104202 (2004).
[Crossref]

Phys. Rev. Lett. (1)

X. Jiang and C. M. Soukoulis, “Time dependent theory for random lasers,” Phys. Rev. Lett. 85, 70 (2000).
[Crossref] [PubMed]

Science (3)

O. Painter, R. K. Lee, A. Scherer, A. Yariv, J. D. O’Brien, P. D. Dapkus, and I. Kim, “Two-dimensional photonic band-gap defect mode laser,” Science 284, 1819–1821 (1999).
[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]

Other (8)

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

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

Ge Li, Yale PhD thesis, 2010.

Y. I. Khanin, Principles of Laser Dynamics (Elsevier, 1995).

The equations are written for the TM case, the modifications for TE are straightforward.

The observation that coherent and incoherent pumping are nearly equivalent for most systems, is invalid when the coherent pumping is supplied at a similar frequency to the atomic lasing transition and thus interactions between the lasing field and pumping field must be taken into account.

For any 1D cavity which is uniformly pumped the TCF states for solving SALT can also be found using a transfer matrix method which does not require discretizing space. We use a more general TCF solver in the calculations presented here which does discretize space.

R. W. Boyd, Nonlinear Optics (Academic Press, 2008).

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Fig. 1

Schematic of a four-level gain medium.

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Fig. 2

Modal intensities as functions of the normalized equilibrium inversion D′₀/D_c (effective pump) in a 1D microcavity edge emitting laser (schematic inset). The cavity is bounded on one side by a perfect mirror and on the other side by air, and has uniform refractive index n = 1.5. Solid lines are results obtained by the time-independent SALT method; open circles are results of FDTD simulations with a coherently pumped four-level medium (Fig. 1); solid triangles are results of FDTD simulations with a coherently pumped six-level medium with a lasing transition between |3〉 and |1〉. Simulation parameters are given in Appendix D. Both the four-level and six-level media are chosen to satisfy SIA. The dephas-ing rate is γ_⊥ = 4.0. The four-level system is in the linear regime described by Eqs. (16) and (17). The six-level system is calculated using the formula in the appendix B, but is in the non-linear regime described by Eqs. (18) and (19). The spectra at D₀/D_c = 0.488, and the gain curve, are shown in the upper left inset.

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Fig. 3

Breakdown of the equivalence between SALT and FDTD when SIA is not valid is shown here in two different ways. Here, modal intensities as a function of the normalized equilibrium inversion D′₀/D_c (effective pump) are shown for a 1D microcavity edge emitting laser with γ_⊥ = 4.0 and n = 1.5. Solid lines again represent results obtained from SALT, while open circles represent FDTD simulations of a simple four-level system with γ′_|| = 0.1. Triangles represent FDTD simulations of a six-level system in the non-linear parameter regime in which γ′_|| ∼ 0.001 for D′₀ ≤ 0.1, and thus satisfying SIA, but γ′_|| ∼ 0.01 for D′₀ ≥ 0.45, and consequently no longer satisfying SIA.

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Fig. 4

SALT and FDTD results for a 1D random laser. Modal intensities are plotted against the normalized equilibrium inversion D′₀/D_c (effective pump). Solid lines represent SALT results, and circles represent FDTD simulations for a four-level system with γ′_|| = 0.001. The refractive index distribution of the edge emitting random laser is described in the text. The gain medium has γ_⊥ = 4.0 and is in the regime described by Eqs. (16) and (17). Left inset: log-log plot of the indicated region where three modes turn on in close proximity. Right inset: schematic of the cavity structure.

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Fig. 5

Comparison of SALT and FDTD run-times. Modal intensities are shown as a function of the run-time for SALT (squares) and four-level FDTD simulations (circles), using the parameters of Fig. 2. FDTD simulations that have not begun to lase are marked as crosses. Plot (a) shows data for D₀/D_c = 0.071, just above the first lasing threshold. SALT determined the steady-state single modal intensity in under three minutes, while the FDTD required ∼ 5000 minutes to reach steady state. Plot (b) shows data for D₀/D_c = 0.486, well above the third lasing threshold. SALT calculated all data up to and including this pump value in under 90 minutes, whereas FDTD required > 500 minutes for the first two modes to reach steady-state, with the third mode intensity (green circles) still fluctuating after 5000 minutes (not shown).

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Fig. 6

(a) Unscaled modal intensity of the six-level simulations from Fig. 2 as a function of the pump. A cross sectional area of 1m² is assumed to calculate the power. (b) Inversion as a function of the pump at the cavity boundary. Dashed lines in plots a and b correspond to the pump values shown in plot c. (c) Inversion as a function of position within the cavity for three different pump values, cyan corresponds with �� = 3.75 × 10⁸s⁻¹, magenta with �� = 1.65 × 10⁹s⁻¹, and orange with �� = 4.85 × 10⁹s⁻¹ to show the evolution of the inversion within the cavity as a function of the pump strength.

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Equations (50)

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

4 π {\ddot{P}}^{+} = c^{2} \nabla^{2} E^{+} - ε_{c} (r) {\ddot{E}}^{+}

{\dot{P}}^{+} = - (i ω_{a} + γ_{⊥}) P^{+} + \frac{g^{2}}{i \bar{h}} E^{+} (ρ_{22} - ρ_{11})

{\dot{ρ}}_{33} = 𝒫 (ρ_{00} - ρ_{33}) - γ_{23} ρ_{33}

{\dot{ρ}}_{22} = γ_{23} ρ_{33} - γ_{12} ρ_{22} - \frac{1}{i \bar{h}} E^{+} \cdot ({(P^{+})}^{*} - P^{+})

{\dot{ρ}}_{11} = γ_{12} ρ_{22} - γ_{01} ρ_{11} + \frac{1}{i \bar{h}} E^{+} \cdot ({(P^{+})}^{*} - P^{+})

{\dot{ρ}}_{00} = γ_{01} ρ_{11} - 𝒫 (ρ_{00} - ρ_{33}) .

\dot{D} = - γ_{| |}^{'} (D - D_{0}^{'}) - \frac{2}{i \bar{h}} E^{+} \cdot ({(P^{+})}^{*} - P^{+}),

γ_{| |}^{'} = 2 γ_{12} (1 + \frac{S}{2 + \frac{γ_{01}}{𝒫} + 2 \frac{γ_{01}}{γ_{23}}})

D_{0}^{'} = \frac{S 𝒫 n}{γ_{01} + (S + 2 + 2 \frac{γ_{01}}{γ_{23}}) 𝒫},

{\dot{ρ}}_{i i} = \sum_{j} γ_{i j} ρ_{j j} - \sum_{j} γ_{j i} ρ_{i i} + γ_{i u} ρ_{u u} + γ_{i l} ρ_{l l} - (γ_{u i} + γ_{l i}) ρ_{i i},

\sum_{j} [(s_{i} + γ_{u i} + γ_{l i}) δ_{i j} - γ_{i j}] ρ_{i j} = γ_{i u} ρ_{u u} + γ_{i l} ρ_{l l},

γ_{| |}^{'} = \frac{B_{l} T_{u} - B_{u} T_{l}}{T_{u} + T_{l}}, D_{0}^{'} = \frac{B_{u} + B_{l}}{T_{u} + T_{l}} \frac{n}{γ_{| |}^{'}},

T_{u / l} = 1 + \sum_{i j} {[R^{- 1}]}_{i j} γ_{j, u / l}

B_{u} = - s_{u} + \sum_{i j} (γ_{u i} - γ_{l i}) {[R^{- 1}]}_{i j} γ_{j u},

B_{l} = s_{l} + \sum_{i j} (γ_{u i} - γ_{l i}) {[R^{- 1}]}_{i j} γ_{j l} .

γ_{| |}^{'} \approx 2 γ_{12},

D_{0}^{'} \approx \frac{𝒫}{γ_{12}} n .

γ_{| |}^{'} \approx 2 (γ_{12} + 𝒫),

D_{0}^{'} \approx \frac{1}{1 + \frac{𝒫}{γ_{12}}} (\frac{𝒫}{γ_{12}}) n .

\begin{array}{l} E^{+} (\vec{r}, t) = \sum_{μ = 1}^{M} Ψ_{μ} (\vec{r}) e^{- i ω_{μ} t}, \\ P^{+} (\vec{r}, t) = \sum_{μ = 1}^{M} p_{μ} (\vec{r}) e^{- i ω_{μ} t}, \end{array}

[\nabla^{2} + (ε_{c} (\vec{r}) + \frac{γ_{⊥} D (\vec{r})}{k_{μ} - k_{a} + i γ_{⊥}}) k_{μ}^{2}] Ψ_{μ} (\vec{r}) = 0,

D (\vec{r}) = D_{0} (\vec{r}) {[1 + \sum_{ν = 1}^{M} Γ_{ν} {| Ψ_{ν} (\vec{r}) |}^{2}]}^{- 1} .

{\dot{P}}^{+} = - (i ω_{a} + γ_{⊥}) P^{+} + \frac{g^{2}}{i \bar{h}} E^{+} (ρ_{u u} - ρ_{l l})

{\dot{ρ}}_{u l} = (i ω_{a} + γ_{⊥}) ρ_{u l} - \frac{i}{\bar{h}} g E (ρ_{u u} - ρ_{l l})

\dot{M} = g {\dot{ρ}}_{u l} + c.c. = (i ω_{a} + γ_{⊥}) ρ_{u l} - \frac{i}{\bar{h}} g E D + c.c.,

\ddot{M} = (- ω_{a}^{2} + 2 i ω_{a} γ_{⊥} + γ_{⊥}^{2}) g ρ_{u l} - \frac{ω_{a}}{\bar{h}} g^{2} E D + c.c. .

\ddot{M} + 2 γ_{⊥} \dot{M} + ω_{a}^{2} M = - γ_{⊥}^{2} M - \frac{2 ω_{a} g^{2}}{\bar{h}} E D,

\dot{D} = - γ_{| |} (D - D_{0}) + \frac{2 E}{\bar{h} ω_{a}} \dot{M},

{\dot{ρ}}_{i i} = \sum_{j} γ_{i j} ρ_{j j} - \sum_{j} γ_{j i} ρ_{i i} + γ_{i u} ρ_{u u} + γ_{i l} ρ_{l l} - (γ_{u i} + γ_{l i}) ρ_{i i},

\sum_{j} R_{i j} ρ_{j j} = γ_{i u} ρ_{u u} + γ_{i l} ρ_{l l},

R_{i j} \equiv (s_{i} + γ_{u i} + γ_{l i}) δ_{i j} - γ_{i j} .

ρ_{i i} = \sum_{j} {[R^{- 1}]}_{i j} (γ_{j u} ρ_{u u} + γ_{j l} ρ_{l l}) .

n = \sum_{i} ρ_{i i} + ρ_{u u} + ρ_{l l}

= T_{u} ρ_{u u} + T_{l} ρ_{l l}

T_{u} = 1 + \sum_{i j} {[R^{- 1}]}_{i j} γ_{j u},

T_{l} = 1 + \sum_{i j} {[R^{- 1}]}_{i j} γ_{j l} .

ρ_{u u} = \frac{n + T_{l} D}{T_{l} + T_{u}},

ρ_{l l} = \frac{n - T_{u} D}{T_{l} + T_{u}} .

\dot{D} = {\dot{ρ}}_{u u} - {\dot{ρ}}_{l l} = \sum_{i} (γ_{u i} - γ_{l i}) ρ_{i i} - s_{u} ρ_{u u} + s_{l} ρ_{l l} - \frac{2}{i \bar{h}} E^{+} \cdot ({(P^{+})}^{*} - P^{+}),

\dot{D} = B_{u} ρ_{u, u} + B_{l} ρ_{l, l} - \frac{2}{i \bar{h}} E^{+} \cdot ({(P^{+})}^{*} - P^{+}),

B_{u} \equiv - s_{u} + \sum_{i j} (γ_{u i} - γ_{l i}) {[R^{- 1}]}_{i j} γ_{j u},

B_{l} \equiv s_{l} + \sum_{i j} (γ_{u i} - γ_{l i}) {[R^{- 1}]}_{i j} γ_{j l} .

γ_{| |}^{'} = \frac{B_{l} T_{u} - B_{u} T_{l}}{T_{u} + T_{l}},

D_{0}^{'} = \frac{B_{u} + B_{l}}{B_{l} T_{u} - B_{u} T_{l}} n .

\frac{2 g^{2}}{{\bar{h}}^{2}} \sum_{ν = 1}^{N} Γ_{ν} {| Ψ_{ν} (\vec{r}) |}^{2} = γ_{| |}^{'} (\frac{D_{0}^{'}}{D (\vec{r})} - 1) .

\frac{g^{2}}{{\bar{h}}^{2}} \sum_{ν = 1}^{N} Γ_{ν} {| Ψ_{ν} (\vec{r}) |}^{2} = 𝒫 (\frac{N}{D (\vec{r})} - 1) - γ_{12} .

γ_{4 lv, fig 2} = (\begin{matrix} \cdot & 0.8 & \cdot & \cdot \\ \cdot & \cdot & 5 \times 10^{- 4} & \cdot \\ \cdot & \cdot & \cdot & 0.8 \\ \cdot & \cdot & \cdot & \cdot \end{matrix}) .

γ_{6 lv, fig 2} = (\begin{matrix} \cdot & 0.8 & 10^{- 5} & 10^{- 5} & 10^{- 5} & 10^{- 5} \\ \cdot & \cdot & 0.8 & 10^{- 5} & 10^{- 5} & 10^{- 4} \\ \cdot & \cdot & \cdot & 5 \times 10^{- 5} & 10^{- 5} & 10^{- 5} \\ \cdot & \cdot & \cdot & \cdot & 0.8 & 10^{- 5} \\ \cdot & \cdot & \cdot & \cdot & \cdot & 0.8 \\ \cdot & \cdot & \cdot & \cdot & \cdot & \cdot \end{matrix}) .

γ_{4 lv, fig 3} = (\begin{matrix} \cdot & 0.8 & \cdot & \cdot \\ \cdot & \cdot & 5 \times 10^{- 2} & \cdot \\ \cdot & \cdot & \cdot & 0.8 \\ \cdot & \cdot & \cdot & \cdot \end{matrix}) .

γ_{6 lv, fig 3} = (\begin{matrix} \cdot & 0.8 & 10^{- 5} & 10^{- 5} & 10^{- 5} & 10^{- 5} \\ \cdot & \cdot & 0.8 & 10^{- 5} & 10^{- 5} & 10^{- 4} \\ \cdot & \cdot & \cdot & 5 \times 10^{- 5} & 10^{- 5} & 10^{- 5} \\ \cdot & \cdot & \cdot & \cdot & 0.8 & 10^{- 5} \\ \cdot & \cdot & \cdot & \cdot & \cdot & 0.8 \\ \cdot & \cdot & \cdot & \cdot & \cdot & \cdot \end{matrix}) .

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