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

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References

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  1. H. Haken, Light: Laser Dynamics Vol. 2 (North-Holland Phys. Publishing, 1985).
  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]
  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, "Highpower 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]
  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]
  10. B. Bidégaray, "Time discretizations for Maxwell-Bloch equations," Numer. Meth. Partial Differential Equations 19, 284-300 (2003).
    [CrossRef]
  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]
  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]
  13. L. Ge, H. E. Türeci, and A. D. Stone (unpublished).

2008 (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]

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 (2)

H. Cao, "Review on latest developments in random lasers with coherent feedback," J. Phys. A 38, 10497-10535 (2005).
[CrossRef]

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]

2003 (2)

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]

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 (1)

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]

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]

1963 (1)

H. Haken and H. Sauermann, "Nonlinear interaction of laser modes," Z. Phys. 173, 261-275 (1963).
[CrossRef]

Bidégaray, B.

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

Cao, H.

H. Cao, "Review on latest developments in random lasers with coherent feedback," J. Phys. A 38, 10497-10535 (2005).
[CrossRef]

Capasso, F.

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]

Cho, A. Y.

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]

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]

Davis, P.

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]

Faist, J.

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]

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.

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]

Gmachl, C.

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]

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 and H. Sauermann, "Nonlinear interaction of laser modes," Z. Phys. 173, 261-275 (1963).
[CrossRef]

Harayama, T.

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]

Ikeda, K. S.

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]

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]

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]

Narimanov, E. E.

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]

Nöckel, J. U.

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???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]

Sauermann, H.

H. Haken and H. Sauermann, "Nonlinear interaction of laser modes," Z. Phys. 173, 261-275 (1963).
[CrossRef]

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]

Sivco, D. L.

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]

Stone, A. D.

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]

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]

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]

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]

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]

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]

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]

J. Phys. A (1)

H. Cao, "Review on 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]

Phys. Rev. A (3)

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

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]

Phys. Rev. Lett. (1)

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]

Proc. SPIE (1)

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]

Science (3)

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]

Z. Phys. (1)

H. Haken and H. Sauermann, "Nonlinear interaction of laser modes," Z. Phys. 173, 261-275 (1963).
[CrossRef]

Other (2)

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

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

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Figures (3)

Fig. 1.
Fig. 1.

Modal intensities as functions of the pump strength D 0 in a one-dimensional micro-cavity edge emitting laser of γ =4.0 and γ =0.001. (a) n=1.5, k a L=40. (b) n=3, k a L=20. Square data points are the result of MB time-dependent simulations; solid lines are the result of time-independent ab initio calculations (AISC) of Eq. (1). Excellent agreement is found with no fitting parameter. Colored lines represent individual modal output intensities; the black lines the total output intensity. Dashed lines are results of AISC calculations when the slowly-varying envelope approximation is made as in Ref. [3] showing significant quantitative discrepancies. For example, in the n=3 case the differences of the third/fourth thresholds between the MB and AISC approaches are 46%/63%, respectively, but are reduced to 3% and 15% once the SVEA is removed. The spectra at D 0=10 and the gain curve are shown as insets in (a) and (b) with the solid lines representing the predictions of the AISC approach (Eq. (1)) and with the diamonds illustrating the height and frequency of each lasing peak. The schematic in (a) shows a uniform dielectric cavity with a perfect mirror on the left and a dielectric-air interface on the right.

Fig. 2.
Fig. 2.

Modal intensities as functions of the pump strength D 0 in a one-dimensional micro-cavity edge emitting laser of n=3, k a L=20, γ =4.0 and γ‖=0.001; the solid lines and data points are the same as in Fig. 1b. The dashed lines are the results of the third order approximation to Eq. (1). The frequently used third order approximation is seen to fail badly at a pump level roughly twice the first threshold value, exhibiting a spurious saturation not present in the actual MB solutions or the AISC theory. In addition, the third order approximation predicts too many lasing modes at larger pump strength. For example, it predicts seven lasing modes at D 0=10, while both the MB and AISC show only four. Right inset just shows the same data on a larger vertical scale.

Fig. 3.
Fig. 3.

Modal intensities for the micro-cavity edge emitting laser of Fig. 1a as γ is varied (n=1.5, k a L=20, γ =4.0 and mode-spacing Δ≈1.8. Solid lines are AISC results from Fig. 1a (with γ =0.001); dashed lines are γ =0.01 and dot-dashed are γ =0.1. The color scheme is the same as in Fig. 1a. The inset shows the shifts of the third threshold as a function of γ . The perturbation theory (squares, with the line to guide the eyes) predicts semiquantitatively the decrease of the threshold as γ increases found in the MB simulations. The MB threshold is not sharp and we add an error bar to denote the size of the transition region.

Equations (5)

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Ψ μ ( x ) = i γ γ i ( k μ k a ) k μ 2 k a 2 d x D 0 ( x ) G ( x , x ; k μ ) Ψ μ ( x ) ε ( x ' ) ( 1 + ν Γ ν Ψ ν ( x ' ) 2 ) .
d + ( x ) = 2 i h ̅ [ Ψ 1 p 2 * Ψ 2 * p 1 ] ( i Δ γ ) = γ Δ f ( k 1 , k 2 ) D s ( 0 ) ( x ) Ψ 1 ( x ) Ψ 2 * ( x ) / e c 2
p 1 ( 1 ) ( x ) = g 2 i h ̅ [ 1 + ( γ / Δ ) f ( k 1 , k 2 ) Ψ 2 ( x ) 2 / e c 2 ] γ i ( k 1 k a ) D s ( 0 ) ( x ) Ψ 1 ( x ) ,
D s ( x ) = D 0 1 + ν Γ ν Ψ ν ( x ) 2 + ( γ / γ ) g ( k 1 , k 2 ) Ψ 1 ( x ) Ψ 2 ( x ) 2 ,
Ψ 2 ( x ) = i γ D 0 γ i ( k 2 k a ) k 2 2 k a 2 d x ε ( x ) ( 1 + γ Δ f ( k 1 , k 2 ) Ψ 1 ( x ) 2 ) G ( x , x ; k 2 ) Ψ 2 ( x ) ( 1 + ν Γ ν Ψ ν ( x ) 2 + γ γ g ( k 1 , k 2 ) Ψ 1 ( x ) Ψ 2 ( x ) 2 ) .

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