## Quantitative verification of ab initio self-consistent laser theory

Optics Express, Vol. 16, Issue 21, pp. 16895-16902 (2008)

http://dx.doi.org/10.1364/OE.16.016895

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### Abstract

We generalize and test the recent “ab initio” self-consistent (AISC) time-independent semiclassical laser theory. This self-consistent formalism generates all the stationary lasing properties in the multimode regime (frequencies, thresholds, internal and external fields, output power and emission pattern) from simple inputs: the dielectric function of the passive cavity, the atomic transition frequency, and the transverse relaxation time of the lasing transition. We find that the theory gives excellent quantitative agreement with full time-dependent simulations of the Maxwell-Bloch equations after it has been generalized to drop the slowly-varying envelope approximation. The theory is infinite order in the non-linear hole-burning interaction; the widely used third order approximation is shown to fail badly.

© 2008 Optical Society of America

*time-independent*methods to find these stationary properties in the multi-mode regime for an open laser cavity did not exist until the recent work of Tureci

*et al*. [2

2. 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]

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

5. 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. 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]

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

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

8. H. Haken and H. Sauermann, “Nonlinear interaction of laser modes,” Z. Phys. **173**, 261–275 (1963). [CrossRef]

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

*γ*

_{‖}≪

*γ*

_{⊥},Δ(where

*γ*

_{⊥}is the transverse (polarization) relaxation rate,

*γ*

_{Δ}is the typical mode spacing, and γ‖ is the longitudinal (inversion) relaxation rate). The only significant approximation in the theory presented below is this approximation of stationary inversion (SIA) (well-satisfied in many lasers of interest). In addition to the excellent agreement we find between the AISC theory and the MB simulations when the ratios

*γ*

_{‖}/

*γ*

_{⊥},

*γ*

_{‖}/Δ are very small, we develop below a perturbative treatment of the beating terms which are neglected in SIA to extend the theory to larger values of these ratios. The key improvements contained in the AISC theory are: 1) Treatment of the openness of the cavity exactly. 2) Inclusion of the space-dependent non-linear modal interactions (spatial hole-burning) to all orders, in contrast to standard third-order treatments [1]. We show below that the third order treatment fails quantitatively and qualitatively even for the simple laser resonator we study here.

2. 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]

3. 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]

*n*

_{0}and length

*L*terminating abruptly on air (see inset, Fig. 1). The MB equations are simulated in time and space using a FDTD approach for the Maxwell equations, while the Bloch equations are discretized using a Crank-Nicholson scheme. To avoid solving a nonlinear system of equations at each spatial location and time step, we adopt the method proposed by Bidégaray [10

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

2. 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]

3. 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]

*after*two standard simplifications, the rotating-wave approximation (RWA) and slowly-varying envelope approximation (SVEA). The SVEA involves factoring out the rapid time-dependence of the electric field and the atomic polarization field at the atomic frequency,

*k*

_{a}, (here and below we set

*c*=1 and use frequency and wavevector interchangeably), and neglecting the second time derivatives in the Maxwell wave equation of the remaining envelope function of the fields. The resulting non-linear system contains only first time-derivatives of the fields and the inversion and is sometimes referred to as the Schrödinger-Bloch (SB) equations [11

11. T. Harayama, P. Davis, and K. S. Ikeda, “Stable oscillations of a spatially chaotic wave function in a microstadium Laser,” Phys. Rev. Lett. **90**, 063901 (2003). [CrossRef] [PubMed]

*n*

_{0}

*k*

_{a}

*L*=60 corresponding to roughly ten wavelengths of radiation inside the cavity, approaching the micro-cavity limit. For the case studied here, we find noticeable discrepancies between MB solutions and the AISC theory of Refs. [2

**74**, 043822(2006). [CrossRef]

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

*E*(

*x,t*)=2Re[

*e*(

*x,t*)]=2Re[∑

_{µ}

*Ψ*

_{µ}(

*x*)exp(-

*ik*

_{µ}

*t*)] and for the polarization fields, which oscillate at unknown lasing frequencies,

*k*

_{µ}. The spatial variation of the field amplitude Ψ

_{µ}(

*x*) is also unknown, and not assumed to be a cavity resonance, but is determined self-consistently. For a high finesse cavity and near the first threshold it was shown [2

**74**, 043822(2006). [CrossRef]

_{µ}(

*x*) is well-approximated by a single cavity resonance, but above threshold and for lower finesse this is not at all the case [3

3. 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]

**320**, 643–646 (2008). [CrossRef] [PubMed]

**74**, 043822(2006). [CrossRef]

_{µ}and the frequencies

*k*

_{µ}are:

*G*(

*x,x*

^{′};

*k*

_{µ}) is the Green function of the open cavity, Γ

_{ν}=Γ(

*k*

_{ν}) is the gain profile evaluated at

*k*

_{ν},

*D*

_{0}(

*x*)=

*D*

_{0}(1+

*d*

_{0}(

*x*)) is the spatial pump, and

*ε*(

*x*)=

*n*

^{2}(

*x*) is the dielectric function of the cavity (for the micro-cavity edge emitting laser

*n*(

*x*)=

*n*

_{0}and the pump is assumed uniform (

*d*

_{0}(

*x*)=0)). Electric field and pump strength are measured in units

*g*is the dipole matrix element of the gain medium. With a slight change in notation this equation differs from that derived in Ref. [2

**74**, 043822(2006). [CrossRef]

*k*

^{2}

_{µ}/

*k*

^{2}

_{a}multiplying the integral. This is consistent with the expectation that the SVEA is good when the lasing frequency is very close to the atomic frequency; incorporating this change into the iterative algorithm for solving the system is trivial, but crucial for quantitative agreement with the current MB simulations. However we do not find qualitative changes from dropping the SVEA, either for this simple laser or for the more complicated two-dimensional random laser studied elsewhere [4

**320**, 643–646 (2008). [CrossRef] [PubMed]

*n*

_{0}=1.5, 3, finding remarkable agreement between MB and AISC approaches

*with*no fitting parameters for the case of three mode and four mode lasing respectively. Both thresholds, modal intensities and frequencies are found correctly by the AISC approach. Note that all of these quantities are direct outputs of the AISC theory, whereas they must be found by numerical Fourier analysis of the MB outputs, which can introduce some numerical error. If the earlier AISC approach is used with the SVEA then significant discrepancies are found, for example for

*n*

_{0}=3 the threshold of the third mode is found to be higher than from MB while the fourth threshold is too low. Note however that the theory with the SVEA does get the right number of modes and the correct linear behavior for large pumps. We believe that this original AISC approach

*does*solve very accurately the Schrödinger-Bloch equations and that the same discrepancies would arise between MB and SB simulations, although we haven’t confirmed this.

_{ν}|

^{2}. This third-order version of the AISC theory then becomes similar to standard treatments of Haken [8

8. H. Haken and H. Sauermann, “Nonlinear interaction of laser modes,” Z. Phys. **173**, 261–275 (1963). [CrossRef]

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

**74**, 043822(2006). [CrossRef]

**76**, 013813 (2007). [CrossRef]

**74**, 043822(2006). [CrossRef]

12. H. E. Türeci and A. D. Stone, “Mode competition and output power in regular and chaotic dielectric cavity lasers,” Proc. SPIE **5708**, 255–270 (2005). [CrossRef]

**74**, 043822(2006). [CrossRef]

*D*

_{0}=10 in Fig. 2), and shows the same spurious saturation as found earlier [2

**74**, 043822(2006). [CrossRef]

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

*γ*

_{‖}≪

*γ*

_{⊥},Δ, where

*γ*

_{‖}is the relaxation rate of the inversion and Δ is the frequency difference of lasing modes. For a typical semiconductor laser

*γ*

_{⊥}≃10

^{12}-10

^{13}

*s*

^{-1}and

*γ*

_{‖}≃10

^{8}-10

^{9}

*s*

^{-1}, or equivalently

*γ*

_{‖}/

*γ*

_{⊥}≃10

^{-3}-10

^{-5}. For the micro-cavity edge-emitting laser we are modeling we took

*γ*

_{⊥},Δ~1 and

*γ*

_{‖}=10

^{-3}, and we found the excellent agreement shown in Figs. 1a, 1b. In addition, direct analysis of the inversion vs. time obtained from the MB simulations confirm very weak time-dependence in the steady state, justifying the use of SIA. The previous work did not develop a systematic theory in which

*γ*

_{‖}/

*γ*

_{⊥},

*γ*

_{‖}/Δ appear as small parameters in the lasing equations, allowing perturbative treatment of corrections to the SIA; we are now able to do this within the AISC formalism.

*γ*

_{⊥},

*γ*

_{‖}does not appear explicitly. Hence the solutions of Eq. (1) depend on

*γ*

_{‖}only through this scale factor, and Eq. (1) makes the strong prediction of a universal overall scaling of the field intensities: |

*E*(

*x*)|

^{2}~

*γ*

_{‖}when dimensions are restored. The perturbative corrections to Eq. (1) are obtained by including the leading effects of the beating terms between the different lasing modes which lead to time-dependence of the inversion at multiples of the beat frequencies. These population oscillations non-linearly mix with the electric field and polarization to generate all harmonics of the beat frequencies in principle, but the multimode approximation assumes all the newly generated Fourier components of the fields are negligible. The leading correction to this approximation is to evaluate the effect of the lowest sidebands of population oscillation on the polarization at the lasing frequencies and on the static part of the inversion, both of which will enter Eq. (1). For simplicity we present a sketch of the correction to Eq. (1) in the two-mode regime; details and the straightforward generalization to more modes will be given elsewhere.

*e*(

*x,t*)= 2Re[∑

^{2}

_{µ=1}Ψ

_{µ}(

*x*)exp(-

*ik*

_{µ}

*t*)] and

*p*(

*x,t*)=2Re[∑

^{2}

_{µ=1pµ}(

*x*)exp(-

*ik*

_{µ}

*t*)], and allow for the first two side-bands at the beat frequency Δ=

*k*

_{1}-

*k*

_{2}in the inversion, so that the total inversion is

*D*(

*x,t*)=

*D*

_{e}(

*x*)+

*d*+(

*x*)exp(-

*i*Δ

*t*)+

*d*-(

*x*)exp(+

*i*Δ

*t*) where the real quantity

*D*

_{s}(

*x*) is the time-independent part of the inversion and

*d*+(

*x*)=

*d*-(

*x*)*. This ansatz is inserted into the Bloch equation for the inversion [2

**74**, 043822(2006). [CrossRef]

*i*Δ

*t*) relates

*d*+(

*x*) to the product of the field and polarization and then substitution of the zeroth order result for the polarization in terms of the zeroth order static inversion

*D*

^{(0)}

_{s}gives

*f*(

*k*

_{1},

*k*

_{2})=-(

*i*+Δ/(2

*γ*

_{⊥}))/(1+

*k*̃

_{1}

*k*̃

_{2}-

*i*Δ/

*γ*

_{⊥}),

*k*̃

_{ν}=(

*k*

_{ν}-

*k*

_{a})/

*γ*⊥ and the fields are not yet measured in units of

*e*

_{c}. The component

*d*

_{+}will mix with the field Ψ

_{2}to yield a contribution to the polarization

*p*

_{1}at frequency

*k*

_{1}, (and similarly

*d*- and Ψ

_{1}mix to contribute to

*p*

_{2}),

*p*

^{(1)}

_{2}(

*x*) is obtained by interchanging subscripts 1, 2. The correction to the AISC formalism is the term in the numerator explicitly proportional to the small parameter

*γ*

_{‖}/Δ. However, having found a correction to the polarizations

*p*

_{1},

*p*

_{2}we must then evaluate their contribution to the static inversion. We find

*g*(

*k*

_{1},

*k*

_{2})=(2+

*k*̃

^{2}

_{1}+

*k*̃

^{2}

_{2})(1-

*k*̃

_{1}

*k*̃

_{2})/[(1+

*k*̃

^{2}

_{1})(1+

*k*̃

^{2}

_{2})]

^{2}and now and below electric fields have been scaled by

*e*. Note that the correction term in Eq. (4) is explicitly proportional to the second small parameter,

_{c}*γ*

_{‖}/

*γ*

_{⊥}. For our simulations

*γ*

_{⊥}≈Δ and the functions

*f*(

*k*

_{1},

*k*

_{2}),

*g*(

*k*

_{1},

*k*

_{2}) are order unity. The full correction to the non-linear polarization in Eq. (3) is obtained by replacing

*D*

^{(0)}

_{s}with

*D*

_{s}of Eq. (4). The corrected polarization leads to corrected version of Eq. (1) of the AISC theory:

_{1}(

*x*) satisfies the same equation with the subscripts 1 and 2 interchanged.

*E*(

*x*)|

^{2}~

*γ*

_{‖}, found in Fig. 1. There is no correction to the first mode below the second threshold as the correction terms all vanish (there needs to be two modes to have beats). However, the theory predicts a non-trivial correction to the threshold of the second mode. Note that the correction to the numerator in Eq. (5) does not vanish below the second threshold but contributes self-consistently to that threshold. This correction can be regarded as modifying the dielectric function of the micro-cavity to take the form

*k*

_{2}<

*k*

_{1}, the imaginary part of the effective index is always amplifying and tends to decrease the second threshold; we find this effect dominates over the change in the real part and increasing

*γ*

_{‖}uniformly decreases the thresholds. The opposite effect is possible and observed in other cases we have studied (not shown). In Fig. 3 we show the results of MB simulations as

*γ*

_{‖}is varied from 0.001 to 0.1 (with

*γ*

_{⊥}=4). Note that the universal behavior (in units scaled by

*γ*

_{‖}) is well obeyed until

*γ*

_{‖}=0.1, encompassing most lasers of interest. The qualitative effect predicted by the perturbation theory is clearly seen, the higher thresholds are reduced as

*γ*

_{‖}is increased. The effect is small for the 2nd threshold but large for the third as we expect as the corrections scale with the product of the intensities of lower modes. The inset to Fig. 3 shows that the perturbation theory for the third threshold (a suitable generalization of the two-mode Eq. (5)) yields semiquantitative agreement with the threshold shifts found from the simulations. Detailed comparisons between multimode perturbation theory and simulations above threshold will be given elsewhere. Note that the simulations also find additional modes turning on for

*γ*

_{‖}=0.1 but their intensities are very small and they are not shown in Fig. 3.

*γ*

_{‖}/Δ,

*γ*

_{‖}/

*γ*

_{⊥}and leading corrections in these small parameters can be evaluated and understood qualitatively.

_{µ}(

*x*), construction of threshold matrices to find the modal thresholds from a linear analysis, and solution of the non-linear eigenvalue problem posed by Eq. (1) itself. Some details are given in ref. [4

**320**, 643–646 (2008). [CrossRef] [PubMed]

*D*

_{0}as in Fig. 1a takes roughly 60 times longer with the MB code as compared to the AISC code, and the AISC code works roughly 130 times more efficiently to capture the entire range of

*D*

_{0}.

*γ*

_{‖}

^{-1}, whereas the field oscillates at the rapid rate, ~

*k*

_{a}. Full wave MB simulations must bridge these two time scales and hence are quite time-consuming. That is why the SVEA is employed, to transform to the Schrodinger-Bloch equations [11

11. T. Harayama, P. Davis, and K. S. Ikeda, “Stable oscillations of a spatially chaotic wave function in a microstadium Laser,” Phys. Rev. Lett. **90**, 063901 (2003). [CrossRef] [PubMed]

**320**, 643–646 (2008). [CrossRef] [PubMed]

## Acknowledgments

## References and links

1. | H. Haken, |

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

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

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

5. | 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 |

6. | 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 |

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

8. | H. Haken and H. Sauermann, “Nonlinear interaction of laser modes,” Z. Phys. |

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

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

11. | T. Harayama, P. Davis, and K. S. Ikeda, “Stable oscillations of a spatially chaotic wave function in a microstadium Laser,” Phys. Rev. Lett. |

12. | H. E. Türeci and A. D. Stone, “Mode competition and output power in regular and chaotic dielectric cavity lasers,” Proc. SPIE |

13. | L. Ge, H. E. Türeci, and A. D. Stone (unpublished). |

**OCIS Codes**

(140.3430) Lasers and laser optics : Laser theory

**ToC Category:**

Lasers and Laser Optics

**History**

Original Manuscript: August 11, 2008

Revised Manuscript: September 23, 2008

Manuscript Accepted: September 28, 2008

Published: October 8, 2008

**Citation**

Li Ge, Robert J. Tandy, A. D. Stone, and Hakan E. Türeci, "Quantitative verification of ab initio self-consistent laser theory," Opt. Express **16**, 16895-16902 (2008)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-16-21-16895

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### References

- H. Haken, Light: Laser Dynamics Vol. 2 (North-Holland Phys. Publishing, 1985).
- 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]
- C. Gmachl, F. Capasso, E. E. Narimanov, J. U. Nöckel, A. D. Stone, J. Faist, D. L. Sivco, and A. Y. Cho, "Highpower directional emission from microlasers with chaotic resonators," Science 280, 1556-1564 (1998). [CrossRef] [PubMed]
- 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. Cao, "Review on latest developments in random lasers with coherent feedback," J. Phys. A 38, 10497-10535 (2005). [CrossRef]
- H. Haken and H. Sauermann, "Nonlinear interaction of laser modes," Z. Phys. 173, 261-275 (1963). [CrossRef]
- H. Fu and H. Haken, "Multifrequency operations in a short-cavity standing-wave laser," Phys. Rev. A 43, 2446-2454 (1991). [CrossRef] [PubMed]
- B. Bidégaray, "Time discretizations for Maxwell-Bloch equations," Numer. Meth. Partial Differential Equations 19, 284-300 (2003). [CrossRef]
- T. Harayama, P. Davis, and K. S. Ikeda, "Stable oscillations of a spatially chaotic wave function in a microstadium Laser," Phys. Rev. Lett. 90, 063901 (2003). [CrossRef] [PubMed]
- H. E. Türeci and A. D. Stone, "Mode competition and output power in regular and chaotic dielectric cavity lasers," Proc. SPIE 5708, 255-270 (2005). [CrossRef]
- L. Ge, H. E. Türeci, and A. D. Stone (unpublished).

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