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Optics Express

  • Editor: C. Martijn de Sterke
  • Vol. 16, Iss. 21 — Oct. 13, 2008
  • pp: 16895–16902
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Quantitative verification of ab initio self-consistent laser theory

Li Ge, Robert J. Tandy, A. Douglas Stone, and Hakan E. Türeci  »View Author Affiliations


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

The Maxwell-Bloch (MB) equations provide the foundation of semiclassical laser theory [1

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

] and are the simplest description which captures the full space-dependent non-linear behavior of a laser. These time-dependent equations can be simulated to determine the stationary lasing state. However 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

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]

] presented an “ab initio” self-consistent (AISC) formalism which generates all of the lasing properties including the output power and emission pattern from a few simple inputs. The laser cavity can be of arbitrary complexity and openness, including, e.g., chaotic dielectric disk lasers [5

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]

], photonic lattice defect mode lasers [6

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]

] and random lasers [4

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

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

]. Here we generalize this infinite order non-linear theory by extending it beyond the slowly-varying envelope approximation. With this improvement it gives remarkably good agreement with time-dependent simulations of the Maxwell-Bloch (MB) equations, while the standard third order approximation to the non-linear hole-burning interaction fails badly.

The AISC theory consists of a set of coupled non-linear time-independent integral equations of size equal to the number of lasing modes at a given pump. The AISC theory presented in Refs. [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]

, 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]

] is a solution to the MB equations [1

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

] 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]

]. The SVEA works well when the cavity frequencies are negligibly shifted from the atomic frequency. For microcavities these shifts need not be negligible; our simulated cavities have 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

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

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]

] which incorporates the SVEA (see Fig. 1). This motivated us to generalize the AISC theory to drop the SVEA (the RWA is found to be well-satisfied in all cases).

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.

The generalization was as follows. Again stationary periodic solutions are assumed for the electric field 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 Ψµ(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

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]

] that Ψµ(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]

, 4

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]

]. The treatment of the matter equations does not involve the SVEA and is exactly the same as in Ref. [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]

], where the key assumption is stationary inversion, which allows the non-linear polarization in the Maxwell equation to be replaced by a non-linear function of the electric field itself. The new element is that we keep the second time-derivative of the polarization in the Maxwell equation and evaluate it by differentiating the polarization equation. The resulting improved AISC/MB equations for the mode functions Ψµ and the frequencies k µ are:

Ψμ(x)=iγγi(kμka)kμ2ka2dxD0(x)G(x,x;kμ)Ψμ(x)ε(x')(1+νΓνΨν(x')2).
(1)

Here G(x,x ;k µ) is the Green function of the open cavity, Γν=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 ec=h¯γγ2g,D0c=4πka2g2h¯γ , where 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

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]

] only by the additional factor 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

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]

].

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.

Note first that in Eq. (1) the electric field is measured in units ec~r|| but, unlike 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.

We start by writing the electric field and the polarization in the multiperiodic forms e(x,t)= 2Re[∑2 µ=1Ψµ (x)exp(-ik µ t)] and p(x,t)=2Re[∑2 µ=1(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(-iΔt)+d-(x)exp(+iΔt) 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

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]

]; solving for the component of the inversion equation which oscillates at exp(-iΔt) 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

d+(x)=2ih̅[Ψ1p2*Ψ2*p1](iΔγ)=γΔf(k1,k2)Ds(0)(x)Ψ1(x)Ψ2*(x)/ec2
(2)

where the dimensionless function f (k 1,k 2)=-(i+Δ/(2γ ))/(1+k̃1 k̃2 -iΔ/γ ), k̃ν=(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 Ψ2 to yield a contribution to the polarization p 1 at frequency k 1, (and similarly d- and Ψ1 mix to contribute to p 2),

p1(1)(x)=g2ih̅[1+(γ/Δ)f(k1,k2)Ψ2(x)2/ec2]γi(k1ka)Ds(0)(x)Ψ1(x),
(3)

where 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

Ds(x)=D01+νΓνΨν(x)2+(γ/γ)g(k1,k2)Ψ1(x)Ψ2(x)2,
(4)

where the dimensionless function g(k 1,k 2)=(2+k̃2 1+k̃2 2)(1-k̃1 k̃2)/[(1+k̃2 1)(1+k̃2 2)]2 and now and below electric fields have been scaled by ec. Note that the correction term in Eq. (4) is explicitly proportional to the second small parameter, 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:

Ψ2(x)=iγD0γi(k2ka)k22ka2dxε(x)(1+γΔf(k1,k2)Ψ1(x)2)G(x,x;k2)Ψ2(x)(1+νΓνΨν(x)2+γγg(k1,k2)Ψ1(x)Ψ2(x)2).
(5)

Ψ1(x) satisfies the same equation with the subscripts 1 and 2 interchanged.

Equation (5) predicts corrections to the universal behavior, |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 ε'(x)=ε(x)[1+γ||Δf(k2,k1)|ψ1(x)|2] ; the effective dielectric function then becomes complex and varying in space according to the intensity of the first mode. This in turn changes the threshold for the second mode. If the modes are on opposite sides of the atomic frequency and 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.

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.

In conclusion, for the case studied, the recently developed ab initio self-consistent laser theory without the slowly varying envelope approximation presents a very accurate solution of the steady-state Maxwell-Bloch semiclassical lasing equations without solving the time-dependent problem. Third order treatments fail badly and our infinite order treatment is essential. The theory is a well-controlled expansion in the small parameters γ /Δ,γ /γ and leading corrections in these small parameters can be evaluated and understood qualitatively.

However the qualitative advantage of the AISC approach from the numerical (as opposed to the conceptual) point of view arises for more realistic two-dimensional and three-dimensional laser structures. Note that the reason we can employ the stationary inversion approximation in deriving Eq. (1) is that there is a separation of time scales in typical lasers. The approach to steady-state is controlled by the long time scale, 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]

]. However our results demonstrate that the SVEA is not very accurate. Full wave MB simulations in two and three dimensions over the full range of time scales necessary to describe multi-mode lasing are probably not yet feasible, and have not yet been done. In contrast, two and three dimensional AISC calculations are quite feasible; two-dimensional random laser simulations have already been performed [4

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]

], and two-dimensional micro-cavity laser simulations are in progress [13

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

]. Therefore the AISC method can give access to accurate laser simulations which are currently not possible by any other method. This method also provides more physical insight into lasing properties. The accuracy of the method suggests it can be useful in the analysis and design of novel laser systems.

Acknowledgments

We would like to thank Stefan Rotter, Hui Cao and Jonathan Andreasen for helpful discussions. This work was supported in part by NSF grant no. DMR-0408636.

References and links

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, “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]

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).

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

  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).

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