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

Optics Express, Vol. 20, Issue 1, pp. 474-488 (2012)

http://dx.doi.org/10.1364/OE.20.000474

Acrobat PDF (1049 KB)

### 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.

© 2011 OSA

## 1. Introduction

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

*N*atomic levels in which the polarization and level populations obey the equations of motion of the quantum density matrix. The simplest version of the theory, used widely in textbooks, is the two-level Maxwell-Bloch (MB) model [1]; however, most design and characterization simulations of lasers use models with

*N*= 3 or more levels. In addition, most theoretical solutions for the semiclassical laser equations employ a large number of simplifying assumptions in order to make them analytically tractable, most notably neglecting the openness of the cavity and/or treating only simple one-dimensional (1D) or ring cavities, as well as approximating the nonlinear interactions to cubic order. The results are typically not useful for quantitative modeling. Until recently, the only useful way to obtain quantitative results for non-trivial laser structures was to integrate the semiclassical laser equations in space and time. For novel and interesting modern laser structures with non-trivial 2D and 3D cavity geometries, such simulations are at the limits of computational feasibility, making it difficult to study a large parameter space or ensemble of designs.

6. 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]

9. 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]

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

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

11. 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]

12. 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]

13. 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]

6. 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]

7. 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]

7. 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]

15. 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]

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

9. 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]

12. 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]

9. 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]

15. 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]

*N*-level lasers. In the current work, we show analytically that the steady-state equations for an

*N*-level laser can be reduced to those for an effective two-level system, and hence solved using the efficient SALT algorithm with essentially the same degree of computational effort. We also explore how this effective two-level system differs from the ordinary two-level laser. Next, we present a numerical comparison between the results of SALT calculations and exact

*N*-level finite-difference time-domain (FDTD) calculations, for the same simple 1D laser studied in [15

15. 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]

12. 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]

## 2. Effective two-level systems

### 2.1. Four level system analysis

**E**

^{+}and

**P**

^{+}are the positive frequency components of the scalar electric and polarization fields respectively.

*ρ*is the population density of level |

_{ii}*i*〉,

*ω*is the frequency of the gain center,

_{a}*γ*

_{⊥}is the gain width (polarization dephasing rate),

*g*is the dipole matrix element, 𝒫 is the pump rate, and

*γ*is the decay rate from level |

_{ij}*j*〉 to level |

*i*〉. The four levels are labelled from 0 – 3 in order of increasing energy (Fig. 1). The polarization equation, Eq. (2), is obtained from the four-level density matrix equation of motion, assuming that only the level 2 → 1 transition will be inverted and lase. Often in FDTD calculations a real classical oscillating dipole equation is used to describe the polarization [12

**19**, 1539 (2011). [CrossRef] [PubMed]

*D*(

*x,t*) ≡

*ρ*

_{22}(

*x,t*) –

*ρ*

_{11}(

*x,t*), similar to the polarization equation for a two-level laser. By assuming that the non-lasing populations are stationary,

*ρ̇*

_{00}=

*ρ̇*

_{33}= 0, we can show that

*D*(

*x,t*) obeys which is precisely the form of the inversion equation for the two-level medium [1, 18

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

*D*′

_{0}and

*γ*′

_{||}serve as an effective equilibrium inversion and inversion relaxation rate respectively, and are given by [14]: where

*S*= (

*γ*

_{01}–

*γ*

_{12})/

*γ*

_{12}and

*n*= ∑

*is the total density of gain atoms. Equation (9) for the inversion in the absence of laser emission (which here acts as the effective pump parameter) has been discussed by Siegman [3], while Eq. (8) for the effective relaxation rate has been derived for a special case by Khanin [19]. These expressions have not been used previously to solve the four-level lasing equations in terms of the two-level solutions, as we do here. If we use an incoherent pump, the 𝒫(*

_{i}ρ_{ii}*ρ*

_{00}–

*ρ*

_{33}) terms in Eqs. (3) and (6) would be replaced by 𝒫

*ρ*

_{00}; the effect of this would be the removal of the factor of 2 preceding the ratio

*γ*

_{01}/

*γ*

_{23}in the denominators of Eqs. (8) and (9). For typical laser systems, each denominator is dominated by the

*γ*

_{01}/𝒫 term, so the difference between coherently and incoherently pumping the system is negligible [17].

*γ*

_{01},

*γ*

_{12},

*γ*

_{23}, 𝒫} describing the four-level medium of Fig. 2, we can calculate the effective pump and relaxation rates, and feed those (together with the cavity dielectric function, lasing transition frequency

*ω*, and gain linewidth

_{a}*γ*

_{⊥}) into the SALT algorithm. That will yield the steady-state lasing properties of this four-level laser.

### 2.2. Arbitrary number of levels

*N*-level laser can be treated via an analogous procedure. Suppose we have a gain medium with an arbitrary number of levels,

*N*. We assume that there is only a single lasing transition, between two levels which we denote by |

*u*〉 and |

*l*〉. The population of each of the

*N*– 2 non-lasing levels obeys where the sums are taken over all of the non-lasing states and

*γ*is the rate at which atoms transition from state |

_{ij}*j*〉 to |

*i*〉 which is either a decay or pump rate depending on the relative energies of the states. In this way, we can incorporate decay and pump processes between any levels. We again assume that

*ρ̇*= 0 for all non-lasing levels. Then where

_{ii}*s*= ∑

_{i}*and*

_{j}γ_{ji}*δ*is the Kronecker delta. The term in brackets on the left hand side corresponds to an (

_{ij}*N*– 2)×(

*N*– 2) matrix, which we denote as

*R*. Upon inverting Eq. (11) and substituting it into the equations of motion for the lasing levels, we obtain where The details of this calculation are given in Appendix B.

## 3. Physical limits of interest

*γ*

_{23}∼

*γ*

_{01}≫

*γ*

_{12}≫ 𝒫, for which one recovers the expected behavior that the equilibrium inversion increases linearly with the pump and that

*γ*′

_{||}is a constant: In this case, varying the equilibrium inversion and the pump strength are essentially equivalent.

*γ*

_{23}∼

*γ*

_{01}≫

*γ*

_{12}∼ 𝒫, i.e. when the slow decay rate between the lasing levels is on the same order as the pump rate. In this regime,

*γ*′

_{||}increases with increasing pump and

*D*′

_{0}saturates with increasing pump: This regime is also interesting from the viewpoint of SALT. As

*γ*′

_{||}increases with 𝒫, a laser could satisfy the inequality

*γ*′

_{||}≪

*γ*

_{⊥}near threshold, leading to stationary inversion and an accurate solution via SALT, but fail to satisfy the inequality as the pump becomes stronger, leading to a decrease in the accuracy of SALT.

## 4. Brief summary of SALT

*E*

^{+}and

*P*

^{+}fields are assumed to obey a multimode ansatz where the indices

*μ*= 1,2,···,

*M*label the different lasing modes, and the field and polarization are now explicitly scalar quantities. The total number of modes,

*M*, is not given, but increases in unit steps from zero as we increase the pump strength

*D*

_{0}. The values of

*D*

_{0}at which each step occurs are the (interacting) modal thresholds, to be determined self-consistently from the theory. The real numbers

*ω*are the lasing frequencies of the modes (henceforth

_{μ}*c*= 1), which will also be determined self-consistently.

*Ḋ*= 0. The result is a set of coupled nonlinear differential equations, which are the fundamental equations of SALT [9

**82**, 063824 (2010). [CrossRef]

*D*are now dimensionless, measured in their natural units

*D*=

_{c}*h*̄

*γ*

_{⊥}/(4

*πg*

^{2}), and

*ω*. Note that these equations are time-independent; Eq. (21) is a stationary wave equation for the electric field mode Ψ

_{ν}*, with an effective dielectric function consisting of both the “passive” contribution*

_{μ}*ε*(

_{c}*r⃗*) and an “active” contribution from the gain medium. The latter is frequency-dependent, and has both a real part and a negative (amplifying) imaginary part. It also includes infinite-order nonlinear “hole-burning” modal interactions, seen in the |Ψ

*|*

_{ν}^{2}dependence of Eq. (22). In addition, we make the key requirement that Ψ

*must be purely outgoing outside the cavity; it is this condition that makes the problem non-Hermitian. It is worth noting that the SIA is not needed until at least two modes are above threshold, so Eq. (22) is exact for single-mode lasing up to and including the second threshold (aside from the well-obeyed RWA).*

_{μ}*k*, equal to the lasing frequencies. We refer to these states as the

_{μ}*threshold constant flux*(TCF) states, because one member of the basis set is always equal to the (non-interacting) threshold lasing mode, leading to very rapid convergence of the basis expansion above threshold [9

**82**, 063824 (2010). [CrossRef]

## 5. Numerical comparison

*N*-level generalizations, we studied 1D microcavity lasers for which the FDTD calculations are tractable and fast enough to generate extensive steady-state data. We first consider the same simple edge emitting uniform-index laser treated in Refs. [6

6. 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]

7. 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]

**16**, 16895 (2008). [CrossRef] [PubMed]

*L*terminating abruptly in air (see schematic, Fig. 2). The simulations were carried out using standard FDTD for the electromagnetic field, and Crank-Nicholson discretization for the polarization and rate equations based on the method of Bidégaray [20

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

*ω*is chosen so that

_{a}*n*

_{0}

*k*= 60, corresponding to roughly ten wavelengths of radiation within the cavity. Physical quantities are reported in terms of their natural scales,

_{a}L*D*=

_{c}*h̄γ*

_{⊥}/(4

*πg*

^{2}) and

*c*=

*h̄*= 1 and measure rates in dimensionless units, i.e.

*γ*

_{meas}=

*γ*

_{real}

*L/c*. We note that the parameters chosen accurately reflect those of real microcavities at optical frequencies [12

**19**, 1539 (2011). [CrossRef] [PubMed]

*D*′

_{0}= 0.488

*D*, the mode intensities produced by SALT differ from those of the four- and six-level FDTD simulations by ∼ 1%, while the frequencies differ by < 0.1%. The difference in mode frequencies between SALT and FDTD also exists at the first lasing threshold, for which an analytical value can be calculated. There, we find that the FDTD simulation has a 0.2% error in the first mode frequency, while SALT has a 0.08% error; this error arises from the spatial discretization of the cavity employed in both approaches [21]. It is worth emphasizing that SALT treats the non-linearity to infinite order; in the earlier work on the Maxwell-Bloch model [15

_{c}**16**, 16895 (2008). [CrossRef] [PubMed]

*D*′

_{0}and

*γ*′

_{||}, and otherwise have the same polarization relaxation rate and atomic transition frequency, the cavities are equivalent from the electromagnetic point of view, and will have identical lasing properties.

*γ*′

_{||}is a linear function and

*D*′

_{0}a non-linear function of 𝒫. However, the unscaled modal intensity leaving the cavity is still, to leading order, linear in 𝒫. This can be seen by rearranging Eq. (22), inserting the expressions for

*γ*′

_{||}and

*D*′

_{0}, and noting that at the end of the cavity the inversion is roughly independent of the pump strength. This result is discussed further in Appendix C.

*N*-level laser and two-level SALT breaks down at large pump strengths, when the condition

*γ*′

_{||}≪

*γ*

_{⊥}is violated due to the increase of

*γ*′

_{||}with 𝒫. In Ref. [15

**16**, 16895 (2008). [CrossRef] [PubMed]

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

*γ*′

_{||}= 0.1,

*γ*

_{⊥}= 4.0 and accuracy is already lost for the third lasing mode. For the six-level data of Fig. 3, which is in the non-linear parameter regime, SIA is satisfied and SALT agrees with the FDTD simulations for small values of the normalized equilibrium inversion; for larger values of

*D*′

_{0}, the SALT and FDTD results begin to diverge.

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

23. 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]

*n*

_{1}= 1.25 and

*n*

_{2}= 1. Each random layer was generated according to the formula

*d*

_{1,2}= 〈

*d*

_{1,2}〉(1+

*ηζ*) where 〈

*d*

_{1}〉 = (1/3)(

*L*/30) and 〈

*d*

_{2}〉 = (2/3)(

*L*/30) are the average thicknesses of the layers,

*η*= 0.9 represents the degree of randomness of the cavity, and

*ζ*∈ [−1,1] is a randomly generated number. The gain medium was added uniformly to the entire cavity, and the coherent pump was likewise uniform. The transition frequency was chosen such that

*n*

_{1}

*k*= 120, corresponding to roughly 20 wavelengths inside of the cavity. We find only small discrepancies between the SALT and FDTD results, with ∼ 1.1% difference in the modal intensities. These differences did not vary significantly between different realizations of the random laser.

_{a}L## 6. Computational efficiency of SALT for N-level systems

*below*threshold via a modified threshold matrix [8

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

*N*

^{2}where

*N*is the number of lasing modes. As SALT was implemented in this and previous works, it automatically calculates the entire lasing fields and spectrum up to a specific pump level, using the results from the lower pump values iteratively to expedite convergence. Thus it is not optimized to produce numerical data at a single given pump level, well above threshold, as one can do easily with FDTD. However studies of the convergence of SALT with an initial guess far away from the final solution have shown that SALT is rather robust and flows to the correct stable solution [14], and codes optimized for this different type of calculation should be possible As we see in Fig. 5, even a non-optimal implementation of SALT is substantially more efficient than FDTD even when calculating the steady state of a single pump value.

*k*values, which does increase the full computation compared to a single resonance calculation. Once one has a TCF basis library, using the SALT algorithm to iterate above threshold does not directly scale with the dimensionality of the system.

## 7. Conclusion

*N*-level laser can be mapped to an effective two-level model, for which the steady-state multimode lasing properties are efficiently solvable using Steady-state Ab initio Laser Theory (SALT). Using this mapping, we found excellent agreement between SALT and FDTD simulations for the modal frequencies, thresholds and above-threshold intensities, in the expected domain of validity. SALT is typically several orders of magnitude more efficient computationally than time domain solution of the laser-rate equations, assuming only steady-state properties are needed.

## Appendix A: Classical polarization in the laser-rate equations

*u*〉 and |

*l*〉 are the upper and lower lasing states,

*ρ*is the density matrix element

_{ij}*ij*and

*g*=

_{ul}*g*=

_{lu}*g*is the coupling constant of the lasing states to the electric field, which allows for the definition

*P*

^{+}=

*gρ*. Alternatively, following the notation of Boyd [24], we could consider the equation of motion for

_{ul}*M*=

*gρ*+ c.c. =

_{ul}*P*

^{+}+

*P*

^{−}, which is the expectation value of the dipole moment induced by the applied field, i.e. the classical oscillating dipole field. Thus using Eq. (A2), and This can be rewritten as where

*D*=

*ρ*–

_{uu}*ρ*is the inversion density of the lasing states. For

_{ll}*σ*= 2

*ω*

_{a}g^{2}/

*h̄*[3].

## Appendix B: Effective two-level parameters from N-level rate equations

*u*〉 and |

*l*〉, which need not be adjacent. The rate equation for an arbitrary

*non-lasing*level in the system is where the summations are taken over all non-lasing levels. Here we do not distinguish between decay rates and pumping rates;

*γ*is simply interpreted as the rate at which level |

_{ij}*j*〉 transitions into level |

*i*〉, regardless of the energies of those states. If the populations of all the non-lasing transitions are stationary, i.e.

*ρ̇*= 0, then we can rewrite Eq. (B1) as Here,

_{ii}*s*≡ ∑

_{i}*and*

_{j}γ_{ji}*δ*is the Kronecker delta. Inverting Eq. (B2) gives Hence, we can express the total number density of gain atoms as where Noting that

_{ij}*D*=

*ρ*–

_{uu}*ρ*, we can write the populations of the lasing states as

_{ll}## Appendix C: Inversion as a function of the pump

*r⃗*=

*L*, all of the lasing modes have a maximum in their fields, and thus the inversion is effectively clamped at this point beyond the first lasing threshold, which can be seen in Fig. 6(b).

*E*and

_{c}*D*gives Substituting in Eqs. (18) and (19), which are valid for this simulation, gives The inversion

_{c}*D*is a function of both position and the pump. However, for

*r⃗*=

*L*corresponding to the cavity edge,

*D*should be mostly independent of the pump, as at this location every mode is at its maximum intensity and the effect of spatial hole-burning is most pronounced. The FDTD simulation results, shown in Fig. 6, demonstrate that

*D*indeed varies very weakly with 𝒫 at the cavity edge. Since the left hand side of Eq. (C2) evaluated at

*r*=

*L*is proportional to the total output power, and this must hold for any number,

*N*, of lasing modes, pump-independence of

*D*(

*r*) implies a linear behavior of the unscaled output power of each mode with pump, as observed in Fig. 6(a).

## Appendix D: Simulation constants

*γ*denoting the decay rate from |

_{ij}*j*〉 to |

*i*〉. These values are given in their dimensionless form, i.e.

*γ*

_{meas}=

*γ*

_{real}

*L/c*. Unlisted entries are zero. We also note that throughout this paper |0〉 denotes the ground state, so these matrices are 0 indexed.

*g*= 2.3 · 10

^{−12}m

^{3/2}, and the number of gain atoms is

*n*= 5 · 10

^{23}m

^{−3}. The pump 𝒫 was varied between 3 · 10

^{−6}and 3 · 10

^{−5}.

*λ*= 628nm, the requirement in Fig. 2 that

*n*

_{0}

*kL*= 60 means that

*L*= 4

*μ*m. Using this length, the decay rates can be converted to their unit-full values as

*γ*

_{⊥}= 3 · 10

^{14}s

^{−1},

*γ*

_{23}=

*γ*

_{01}= 6 · 10

^{13}s

^{−1},

*γ*

_{12}= 3.75 · 10

^{10}s

^{−1}, and the pump at threshold is 𝒫 = 3 · 10

^{8}s

^{−1}. Similarly, the dipole matrix element also acquires units of inverse time, and can be expressed as

*g*

^{2}/

*h̄*= 3.98 · 10

^{−9}m

^{3}/s, which corresponds to a coupling constant in the classical oscillating dipole picture of

*σ*= 10

^{−4}C

^{2}/kg. These constants can be seen to be similar to those used in other studies of optical microcavities [4

4. 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]

**19**, 1539 (2011). [CrossRef] [PubMed]

*γ*

_{15}= 10

^{−4}, and the lasing transition is between levels |3〉 and |1〉 (where the ground state is again |0〉 and the states are numbered in order of increasing energy).

## Acknowledgments

## References and links

1. | H. Haken, |

2. | W. E. Lamb, “Theory of an optical maser,” Phys. Rev. |

3. | A. E. Siegman, |

4. | A. S. Nagra and R. A. York, “FDTD analysis of wave propagation in nonlinear absorbing and gain media,” IEEE Trans. Antennas Propag. |

5. | K. S. Yee, “Numerical solution of the initial boundary value problems involving Maxwell’s equations in isotropic media,” IEEE Trans Antennas Propag . |

6. | H. E. Türeci, A. D. Stone, and B. Collier, “Self-consistent multimode lasing theory for complex or random lasing media,” Phys. Rev. A |

7. | H. E. Türeci, A. D. Stone, and L. Ge, “Theory of the spatial structure of nonlinear lasing modes,” Phys. Rev. A |

8. | H. E. Türeci, L. Ge, S. Rotter, and A. D. Stone, “Strong interactions in multimode random lasers,” Science |

9. | L. Ge, Y. D. Chong, and A. D. Stone, “Steady-state ab initio laser theory: generalizations and analytic results,” Phys. Rev. A |

10. | H. Cao, “Review on the latest developments in random lasers with coherent feedback,” J. Phys. A |

11. | 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 |

12. | 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 |

13. | 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 |

14. | Ge Li, Yale PhD thesis, 2010. |

15. | L. Ge, R. J. Tandy, A. D. Stone, and H. E. Türeci, “Quantitative verification of ab initio self-consistent laser theory,” Opt. Express |

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

17. | 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. |

18. | H. Fu and H. Haken, “Multifrequency operations in a short-cavity standing-wave laser,” Phys. Rev. A |

19. | Y. I. Khanin, |

20. | B. Bidégaray, “Time discretizations for Maxwell-Bloch equations,” Numer. Meth. Partial Differential Equations |

21. | 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. |

22. | X. Jiang and C. M. Soukoulis, “Time dependent theory for random lasers,” Phys. Rev. Lett. |

23. | 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 |

24. | R. W. Boyd, |

**OCIS Codes**

(140.3430) Lasers and laser optics : Laser theory

(140.3945) Lasers and laser optics : Microcavities

**ToC Category:**

Lasers and Laser Optics

**History**

Original Manuscript: September 27, 2011

Revised Manuscript: November 21, 2011

Manuscript Accepted: November 22, 2011

Published: December 21, 2011

**Citation**

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)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-20-1-474

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### References

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- 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.
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- B. Bidégaray, “Time discretizations for Maxwell-Bloch equations,” Numer. Meth. Partial Differential Equations19, 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. B69, 104202 (2004). [CrossRef]
- R. W. Boyd, Nonlinear Optics (Academic Press, 2008).

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