## Finite element simulation of microphotonic lasing system |

Optics Express, Vol. 20, Issue 10, pp. 11548-11560 (2012)

http://dx.doi.org/10.1364/OE.20.011548

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### Abstract

We present a method for performing time domain simulations of a microphotonic system containing a four level gain medium based on the finite element method. This method includes an approximation that involves expanding the pump and probe electromagnetic fields around their respective carrier frequencies, providing a dramatic speedup of the time evolution. Finally, we present a two dimensional example of this model, simulating a cylindrical spaser array consisting of a four level gain medium inside of a metal shell.

© 2012 OSA

## 1. Introduction

## 2. FEM microphotonic lasing simulation

### 2.1. Field equations of a microphotonic lasing system

**A**is the electromagnetic vector potential,

**P**is a polarization vector describing either a Drude response from a metal inclusion or a Lorentzian response from a four level gain system, and

*ε*is a relative permittivity that is constant with respect to frequency and not included in

_{r}**P**. Here and for the remainder of the paper we have use SI units. Also, we have used the temporal gauge condition

*∂*A

_{0}/

*∂t*= 0 along with the initial condition for the electrostatic potential A

_{0}(

*t*= 0,

**x**) = 0 to ensure that the electrostatic potential is zero for all time, eliminating it from our equations. Given this choice of gauge the electric field and magnetic flux density are defined as Using this definition for

**E**, the Drude response of a metal inclusion is determined by the equation where

*ω*is the plasma frequency and

_{p}*γ*is the damping frequency of the Drude metal.

*ω*. Similarly, the 0 → 3 transition is also an electric dipole transition with frequency

_{a}*ω*. Spontaneous decay between the i-th level to the j-th level occurs at the decay rate of 1/

_{b}*τ*. These decay rates include both radiative (spontaneous photon emission) and non-radiative (spontaneous phonon emission) decay processes. In the case of spontaneous photon emission, our model does not produce a photon. Coupling of the gain medium to the electromagnetic field is only allowed for stimulated photon emission.

_{ij}*and Γ*

_{a}*are the linewidths of these transitions,*

_{b}*σ*and

_{a}*σ*are coupling constants, and N

_{b}_{0}

*, N*

_{i}_{1}

*, N*

_{i}_{2}

*and N*

_{i}_{3}

*are the population number densities for oscillators polarized in the i-th direction for the 0th, 1st, 2nd and 3rd energy levels. Note that Γ*

_{i}*≥ 1/*

_{a}*τ*

_{21}and Γ

*≥ 1/*

_{b}*τ*

_{30}[12].

13. X. Jiang and C. M. Soukoulis, “Time dependent theory for random lasers,” Phys. Rev. Lett. **85**, 70–73 (2000). [CrossRef] [PubMed]

### 2.2. Period averaged approximation

*ω*, and the 0 → 3 transition frequency

_{a}*ω*. Much of the computational effort required in this time domain simulation is spent on these simple, approximately harmonic oscillations. A good approximation is to assume these fields oscillate harmonically, with complex valued amplitudes that are slowly changing in time. We can ignore the fast oscillations and instead simulate the relatively slower time dependence of these amplitudes.

_{b}**A**

_{1}is the complex valued amplitude for an electromagnetic field that oscillates at a frequency close to the 1 → 2 transition (

*ω*

_{1}≈

*ω*), and

_{a}**A**

_{2}is the complex valued amplitude for an electromagnetic field that oscillates close to the 0 → 3 transition (

*ω*

_{2}≈

*ω*). Also,

_{b}*c.c.*indicates the complex conjugate of the preceding terms.

**A**, we derive two new field equations

_{1i}and E

_{2i}are the i-th components of the electric fields associated with the potentials

**A**

_{1}and

**A**

_{2}, and are defined as

**E**

_{1}= −

*∂*

**A**

_{1}/

*∂t*and

**E**

_{2}= −

*∂*

**A**

_{1}/

*∂t*respectively.

1. K. Böhringer and O. Hess, “A full time-domain approach to spatio-temporal dynamics of semiconductor lasers. II. spatio-temporal dynamics,” Prog. Quantum Electron. **32**, 247–307 (2008). [CrossRef]

9. J. M. Hamm, S. Wuestner, K. L. Tsakmakidis, and T. Hess, “Theory of light amplification in active fishnet metamaterials,” Phys. Rev. Lett. **107**, 167405 (2011). [CrossRef] [PubMed]

10. J. A. Gordon and R. W. Ziolkowski, “The design and simulated performance of a coated nano-particle laser,” Opt. Express **15**, 2622–2653 (2007). [CrossRef] [PubMed]

11. J. A. Gordon and R. W. Ziolkowski, “CNP optical metamaterials,” Opt. Express **16**, 6692–6716 (2008). [CrossRef] [PubMed]

### 2.3. Finite element formulation

_{A}_{1}, we find that by integrating by parts we obtain a volume integral enforcing the electromagnetic field equation as well as a second boundary integral enforcing a boundary condition on the field,

*∂*Ω is the boundary of that domain,

*dA*is a infinitesimal differential area on that boundary, and

**n**̂ is the direction normal to the boundary. In the absence of any extra boundary terms, the boundary integral in Eq. (14) forces the tangential component of the magnetic field

**H**

_{1}to zero. This

*perfect magnetic conductor*boundary condition is not desirable for our simulation, so we will modify it to allow for a boundary that absorbs and emits plane waves at normal incidence to the boundary.

**a**is the vector potential of a plane wave propagating toward the boundary and

**b**is the vector potential of a plane wave propagating away from the boundary. The boundary condition we desire is one that absorbs

**a**and emits an arbitrarily defined

**b**. If we take the part of the surface integrand from Eq. (14) that is within the brackets and substitute Eq. (15) for

**A**

_{1}we get where

**a**propagating toward the boundary, and

**b**propagating away from the boundary, the sum of which is

**Ã**

_{1}and integrating over the domain boundary gives us a new boundary weak term Adding this additional boundary weak term to specific boundaries enforces a

*matched boundary condition*(referred to as an

*absorbing boundary condition*in Ref. [15]) which allows for plane waves normal to the boundary to be absorbed and for the incident plane wave

**A**

_{2}can be enforced in the same manner.

_{0}

*by taking advantage of the fact that N*

_{i}_{0}

*= N*

_{i}*− N*

_{int}_{1}

*− N*

_{i}_{2}

*− N*

_{i}_{3}

*where N*

_{i}*is the initial value of N*

_{int}_{0}

*when N*

_{i}_{1}

*= N*

_{i}_{2}

*= N*

_{i}_{3}

*= 0.*

_{i}## 3. Cylindrical spaser array

*r*

_{1}= 30nm and an outer shell composed of Ag with an outer radius of

*r*

_{2}= 35nm. A diagram of the simulation domain is provided in Fig. 2. The artificial gain medium is characterized by the lifetimes

*τ*

_{10}= 10

^{−14}s,

*τ*

_{21}= 10

^{−11}s,

*τ*

_{32}= 10

^{−13}s and

*τ*

_{30}= 10

^{−12}s. The coupling constants in Eq. (10) are

*σ*= 10

_{a}^{−4}C

^{2}/kg and

*σ*= 5 · 10

_{b}^{−6}C

^{2}/kg, and the linewidths of their corresponding transitions are Γ

*= 2 · 10*

_{a}^{13}s

^{−1}and Γ

*= 1/*

_{b}*τ*

_{30}= 10

^{12}s

^{−1}. Finally, the initial population density parameter is N

*= 5 · 10*

_{int}^{23}m

^{−3}. The population densities of the four level gain medium obeys the rate equations given in Eq. (11), and the gain medium interacts with the electromagnetic field through the gain polarization which obeys Eq. (10). The Ag layer interacts with the electromagnetic field through the Drude polarization which evolves according to Eq. (9).

18. E. F. Kuester, M. A. Mohamed, M. Piket-May, and C. L. Holloway, “Averaged transition conditions for electromagnetic fields at a metafilm,” IEEE Trans. Antennas Propag. **51**, 2641–2651 (2003). [CrossRef]

19. C. Fietz and G. Shvets, “Homogenization theory for simple metamaterials modeled as one-dimensional arrays of thin polarizable sheets,” Phys. Rev. B **82**, 205128 (2010). [CrossRef]

*α*̂ is defined from the scattering matrix

*S*. The S matrix is defined from the amplitude of the electric field of the scattered waves and is adjusted so that the effective thickness of the characterized array is zero [19

19. C. Fietz and G. Shvets, “Homogenization theory for simple metamaterials modeled as one-dimensional arrays of thin polarizable sheets,” Phys. Rev. B **82**, 205128 (2010). [CrossRef]

*S*

_{11}=

*S*

_{22}are the reflection amplitude of a scattered wave and

*S*

_{12}=

*S*

_{21}are the transmitted amplitude of the scattered wave.

*ε*= 9. We see from the surface polarizabilities that there is an electric resonance near

_{G}*λ*

_{0}= 1220nm and a magnetic resonance near

*λ*

_{0}= 830nm. Figure 2 also show fields profiles for each of these resonances calculated using a FEM eigenfrequency simulation. Also shown are the wavelengths of the corresponding resonances

*λ*= 2

_{r}*πc*/Re(

*ω*), and a resonance quality factor Q = 2

_{r}*π*Re(

*ω*)/Im(

_{r}*ω*), where

_{r}*ω*is a complex eigenfrequency returned by the same FEM eigenfrequency simulation.

_{r}*ω*= 2

_{b}*πc*/830nm, and the 1 → 2 transition approximately matches the lower frequency electric resonance

*ω*= 2

_{a}*πc*/1221nm. The presence of a electronic transition will modify the spectrum of the cylindrical array for frequencies near that transition. Figure 3 plots the surface polarizability near the magnetic resonance at

*λ*

_{0}= 831nm for the cylindrical array where the gain medium now has the relative permittivity

*λ*

_{0}= 826nm. For our lasing simulations this will be the pump frequency. There is no way to know exactly what the lasing frequency will be without first running the time domain lasing simulation, except to say that it will be approximately equal to the frequency of the 1 → 2 transition

*ω*. A good initial guess is to set

_{a}*ω*

_{1}=

*ω*, but after running the lasing simulation this can be adjusted to better match true lasing frequency. In what follows, we have used

_{a}*ω*

_{1}= 2

*πc*/1219.3nm.

**A**

_{2}, with an intensity of 8W/mm

^{2}, while the incident probe field is set to

**A**

_{1}= 0. The pump beam is turned on slowly with

**A**

_{2}having the profile where A

*is the amplitude of the pump beam,*

_{pu}*τ*= 1.0 · 10

_{pu}^{−12}

*s*is the pump rise time. The pump beam excites oscillators in the gain medium to the third energy level, which decays to the second energy level at the rate of 1/

*τ*

_{32}. After

*t*≫

*τ*

_{21}, the system is in steady state population inversion, but cannot lase since our model does not allow for generation of light due to spontaneous emission. The time is then reset, and the simulation shown in Fig. 4 begins in this steady state population inversion. Shortly after t=0, a short probe pulse is emitted into the simulation domain. This excites the polarization field

_{2}and N

_{1}for the system beginning in population inversion.

*τ*. Once the pump is at a maximum the time step can be increased while the gain system approaches steady state. When the time is reset and a probe pulse is introduced the time step must be made smaller than the width of the probe pulse, and must remain small to resolve the resulting oscillations of the interaction between the probe pulse and the resonators as well as the initial exponential growth of the lasing beam. As the laser approaches steady state the time step can again be increased. At all times the time step must be smaller than the inverse rate of change of any transient beams (pump, probe or lasing beams). If

_{pu}*ω*

_{1}is not close to the resulting lasing frequency the phase of

**A**

_{1}will rapidly change and will require a correspondingly small time step. Once the system begins lasing, the actual lasing frequency can be inferred from this oscillation in the phase of

**A**

_{1}, and

*ω*

_{1}can be changed in the middle of the simulation. This causes the phase of

**A**

_{1}to slowly change allowing for a larger time step.

^{2}. This threshold intensity depends on a number of variables, including all of the parameters of the gain medium, as well as the cylinder plasmon resonances used to enhance both the pump and lasing transition (Figs. 2 and 3). These resonances in turn depend on the geometry and material parameters of the cylindrical array.

**A**

_{2}for a long period of time (

*t*≫

*τ*

_{21}), and then injecting a Gaussian probe field

**A**

_{1}with a much weaker intensity. Applying a Fourier transform to the resulting time domain reflected and transmitted probe fields gives us the reflection and transmission amplitudes in the frequency domain, allowing us to calculate the surface polarizability according to Eq. (21).

*γ*is made smaller, narrowing the lineshape. We see in Fig. 2(b) that at even higher pump intensities,

_{α}*γ*continues to shrink, passing through zero, and the imaginary part of

_{α}*γ*continues to grow more negative and the lineshape begins to broaden.

_{α}## 4. Conclusion

*ω*

_{1}or

*ω*

_{2}, with slowly changing complex valued amplitudes. A demonstration of this simulation was provided with a two dimensional model of a one dimensional cylindrical spaser array as an example. The threshold pump intensity for this array was determined. Finally, we have shown how the linewidth of the lasing transition changes for various pump intensities.

## Acknowledgments

## References and links

1. | K. Böhringer and O. Hess, “A full time-domain approach to spatio-temporal dynamics of semiconductor lasers. II. spatio-temporal dynamics,” Prog. Quantum Electron. |

2. | A. Fang, T. Koschny, M. Wegener, and C. M. Soukoulis, “Self-consistent calculation of metamaterials with gain,” Phys. Rev. B |

3. | A. Fang, “Reducing the losses of optical metamaterials,” Ph.D. thesis, Iowa State University (2010). |

4. | A. Fang, T. Koschny, and C. M. Soukoulis, “Lasing in metamaterial nanostructures,” J. Opt. |

5. | A. Fang, T. Koschny, and C. M. Soukoulis, “Self-consistent calculations of loss-compensated fishnet metamaterials,” Phys. Rev. B |

6. | S. Wuestner, A. Pusch, K. L. Tsakmakidis, J. M. Hamm, and O. Hess, “Overcoming losses with gain in a negative refractive index metamaterial,” Phys. Rev. Lett. |

7. | A. Fang, Z. Huang, T. Koschny, and C. M. Soukoulis, “Overcoming the losses of a split ring resonator array with gain,” Opt. Express |

8. | S. Wuestner, A. Pusch, K. L. Tsakmakidis, J. M. Hamm, and O. Hess, “Gain and plasmon dynamics in active negative-index metamaterials,” Phil. Trans. R. Soc. London Ser. A |

9. | J. M. Hamm, S. Wuestner, K. L. Tsakmakidis, and T. Hess, “Theory of light amplification in active fishnet metamaterials,” Phys. Rev. Lett. |

10. | J. A. Gordon and R. W. Ziolkowski, “The design and simulated performance of a coated nano-particle laser,” Opt. Express |

11. | J. A. Gordon and R. W. Ziolkowski, “CNP optical metamaterials,” Opt. Express |

12. | A. E. Siegman, |

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

14. | W. B. J. Zimmerman, |

15. | J. Jin, |

16. | D. J. Bergman and M. I. Stockman, “Surface plasmon amplification by stimulated emission of radiation: quantum generation of coherent surface plasmons in nanosystems,” Phys. Rev. Lett. |

17. | M. I. Stockman, “Spasers explained,” Nat. Photon. |

18. | E. F. Kuester, M. A. Mohamed, M. Piket-May, and C. L. Holloway, “Averaged transition conditions for electromagnetic fields at a metafilm,” IEEE Trans. Antennas Propag. |

19. | C. Fietz and G. Shvets, “Homogenization theory for simple metamaterials modeled as one-dimensional arrays of thin polarizable sheets,” Phys. Rev. B |

**OCIS Codes**

(140.3460) Lasers and laser optics : Lasers

(050.1755) Diffraction and gratings : Computational electromagnetic methods

(160.3918) Materials : Metamaterials

**ToC Category:**

Lasers and Laser Optics

**History**

Original Manuscript: April 9, 2012

Revised Manuscript: April 27, 2012

Manuscript Accepted: April 30, 2012

Published: May 4, 2012

**Citation**

Chris Fietz and Costas M. Soukoulis, "Finite element simulation of microphotonic lasing system," Opt. Express **20**, 11548-11560 (2012)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-20-10-11548

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### References

- K. Böhringer and O. Hess, “A full time-domain approach to spatio-temporal dynamics of semiconductor lasers. II. spatio-temporal dynamics,” Prog. Quantum Electron.32, 247–307 (2008). [CrossRef]
- A. Fang, T. Koschny, M. Wegener, and C. M. Soukoulis, “Self-consistent calculation of metamaterials with gain,” Phys. Rev. B79, 241104 (2009). [CrossRef]
- A. Fang, “Reducing the losses of optical metamaterials,” Ph.D. thesis, Iowa State University (2010).
- A. Fang, T. Koschny, and C. M. Soukoulis, “Lasing in metamaterial nanostructures,” J. Opt.12, 024013 (2010). [CrossRef]
- A. Fang, T. Koschny, and C. M. Soukoulis, “Self-consistent calculations of loss-compensated fishnet metamaterials,” Phys. Rev. B82, 121102 (2010). [CrossRef]
- S. Wuestner, A. Pusch, K. L. Tsakmakidis, J. M. Hamm, and O. Hess, “Overcoming losses with gain in a negative refractive index metamaterial,” Phys. Rev. Lett.105, 127401 (2010). [CrossRef] [PubMed]
- A. Fang, Z. Huang, T. Koschny, and C. M. Soukoulis, “Overcoming the losses of a split ring resonator array with gain,” Opt. Express19, 12688–12699 (2011). [CrossRef] [PubMed]
- S. Wuestner, A. Pusch, K. L. Tsakmakidis, J. M. Hamm, and O. Hess, “Gain and plasmon dynamics in active negative-index metamaterials,” Phil. Trans. R. Soc. London Ser. A369, 3525–3550 (2011). [CrossRef]
- J. M. Hamm, S. Wuestner, K. L. Tsakmakidis, and T. Hess, “Theory of light amplification in active fishnet metamaterials,” Phys. Rev. Lett.107, 167405 (2011). [CrossRef] [PubMed]
- J. A. Gordon and R. W. Ziolkowski, “The design and simulated performance of a coated nano-particle laser,” Opt. Express15, 2622–2653 (2007). [CrossRef] [PubMed]
- J. A. Gordon and R. W. Ziolkowski, “CNP optical metamaterials,” Opt. Express16, 6692–6716 (2008). [CrossRef] [PubMed]
- A. E. Siegman, Lasers (University Science Books, 1986). See chapters 2, 3, and 6.
- X. Jiang and C. M. Soukoulis, “Time dependent theory for random lasers,” Phys. Rev. Lett.85, 70–73 (2000). [CrossRef] [PubMed]
- W. B. J. Zimmerman, Process Modelling and Simulation with Finite Element Methods (World Scientific, 2004).
- J. Jin, The Finite Element Method in Electromagnetics, 2nd ed. (John Wiley & Sons, Inc., 2002).
- D. J. Bergman and M. I. Stockman, “Surface plasmon amplification by stimulated emission of radiation: quantum generation of coherent surface plasmons in nanosystems,” Phys. Rev. Lett.90, 027402 (2003). [CrossRef] [PubMed]
- M. I. Stockman, “Spasers explained,” Nat. Photon.2, 327–329 (2008). [CrossRef]
- E. F. Kuester, M. A. Mohamed, M. Piket-May, and C. L. Holloway, “Averaged transition conditions for electromagnetic fields at a metafilm,” IEEE Trans. Antennas Propag.51, 2641–2651 (2003). [CrossRef]
- C. Fietz and G. Shvets, “Homogenization theory for simple metamaterials modeled as one-dimensional arrays of thin polarizable sheets,” Phys. Rev. B82, 205128 (2010). [CrossRef]

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