## Lasing threshold control in two-dimensional photonic crystals with gain |

Optics Express, Vol. 22, Issue 16, pp. 19242-19251 (2014)

http://dx.doi.org/10.1364/OE.22.019242

Acrobat PDF (5738 KB)

### Abstract

We demonstrate how the lasing threshold of a two dimensional photonic crystal containing a four-level gain medium is modified, as a result of the interplay between the group velocity and the modal reflectivity at the interface between the cavity and the exterior. Depending on their relative strength and the optical density of states, we show how the lasing threshold may be dramatically altered inside a band or, most importantly, close to the band edge. The idea is realized via self-consistent calculations based on a finite-difference time-domain method. The simulations are in good agreement with theoretical predictions.

© 2014 Optical Society of America

## 1. Introduction

*Q*cavities with small sizes. The Bragg mirror realized the necessary end reflectors in the microscale and the distributed Bragg reflector (DBR) incorporated the feedback mechanism within the gain medium (DFB - Distributed Feedback lasers) [1

1. H. Kogelnik and C. V. Shank, “Stimulated emission in a periodic structure,” Appl. Phys. Lett. **18**(4), 152–154 (1971). [CrossRef]

1. H. Kogelnik and C. V. Shank, “Coupled-wave theory of distributed feedback lasers,” J. Appl. Phys. **43**(5), 2327–2335 (1972). [CrossRef]

2. J. P. Dowling, M. Scalora, M. J. Bloemer, and C. M. Bowden, “The photonic band edge laser: A new approach to gain enhancement,” J. Appl. Phys. **75**(4), 1896–1899 (1994). [CrossRef]

3. M. D. Tocci, M. Scalora, M. J. Bloemer, J. P. Dowling, and C. M. Bowden, “Measurement of spontaneous-emission enhancement near the one-dimensional photonic band edge of semiconductor heterostructures,” Phys. Rev. A **53**(4), 2799–2803 (1996). [CrossRef] [PubMed]

4. M. Meier, A. Mekis, A. Dodabalapur, A. Timko, R. Slusher, J. Joannopoulos, and O. Nalamasu, “Laser action from two-dimensional distributed feedback in photonic crystals,” Appl. Phys. Lett. **74**(1), 7 (1999). [CrossRef]

12. D. Luo, X. W. Sun, H. T. Dai, Y. J. Liu, H. Z. Yang, and W. Ji, “Two-directional lasing from a dye-doped two-dimensional hexagonal photonic crystal made of holographic polymer-dispersed liquid crystals,” Appl. Phys. Lett. **95**(15), 151115 (2009). [CrossRef]

13. W. Cao, A. Muñoz, P. Palffy-Muhoray, and B. Taheri, “Lasing in a three-dimensional photonic crystal of the liquid crystal blue phase II,” Nat. Mater. **1**(2), 111–113 (2002). [CrossRef] [PubMed]

24. C. Fietz and C. M. Soukoulis, “Finite element simulation of microphotonic lasing system,” Opt. Express **20**(10), 11548–11560 (2012). [CrossRef] [PubMed]

*Q*factor [7

7. H.-Y. Ryu, S.-H. Kwon, Y.-J. Lee, Y.-H. Lee, and J.-S. Kim, “Very-low-threshold photonic band-edge lasers from free-standing triangular photonic crystal slabs,” Appl. Phys. Lett. **80**(19), 3476–3478 (2002). [CrossRef]

8. S.-H. Kwon, H.-Y. Ryu, G.-H. Kim, Y.-H. Lee, and S.-B. Kim, “Photonic bandedge lasers in two-dimensional square-lattice photonic crystal slabs,” Appl. Phys. Lett. **83**(19), 3870–3872 (2003). [CrossRef]

17. L. Florescu, K. Busch, and S. John, “Semiclassical theory of lasing in photonic crystals,” J. Opt. Soc. Am. B **19**(9), 2215–2223 (2002). [CrossRef]

18. H.-Y. Ryu, M. Notomi, and Y.-H. Lee, “Finite-difference time-domain investigation of band-edge resonant modes in finite-size two-dimensional photonic crystal slab,” Phys. Rev. B **68**(4), 045209 (2003). [CrossRef]

15. K. Sakoda, “Enhanced light amplification due to group-velocity anomaly peculiar to two- and three-dimensional photonic crystals,” Opt. Express **4**(5), 167–176 (1999). [CrossRef] [PubMed]

18. H.-Y. Ryu, M. Notomi, and Y.-H. Lee, “Finite-difference time-domain investigation of band-edge resonant modes in finite-size two-dimensional photonic crystal slab,” Phys. Rev. B **68**(4), 045209 (2003). [CrossRef]

*Q*factor [14,17

17. L. Florescu, K. Busch, and S. John, “Semiclassical theory of lasing in photonic crystals,” J. Opt. Soc. Am. B **19**(9), 2215–2223 (2002). [CrossRef]

*Q*factor. In order to demonstrate them in realistic structures we choose a 2D PC of a finite amount of layers as testbed. Our simulations show that operation near the band edge can lead to strong reduction of the lasing threshold with respect to a uniform gain slab of the same dimensions, in spite of the lower gain density, even for as few as 10 layers. Moreover, with a slight structural modification, the PC is surprisingly demonstrated to exhibit lower lasing threshold at steeper band edges. Our self-consistent finite-difference time-domain (FDTD) calculations are explained in a straightforward manner by the theoretical model, which may also predict other counter-intuitive behaviors, such as lower lasing threshold inside a band, rather than the band edge.

## 2. The two-dimensional photonic crystal system and the self-consistent model

*w*. The system is assumed to be infinite along the

*y*,

*z*directions and finite along the

*x*direction with a total length of

*n*is the number of layers and

*a*the lattice constant. The permittivity of the dielectric background is chosen to be

16. S. Nojima, “Optical-gain enhancement in two-dimensional active photonic crystals,” J. Appl. Phys. **90**(2), 545–551 (2001). [CrossRef]

*a =*840

*nm*and

*w =*540

*nm*, which give a gain density of

*n*= 10 layers is shown in Fig. 1, where the respective homogeneous gain slab of same length is also depicted. The band structure for the TE polarization of the 2D PC is also shown. The shaded zone denotes the full band gap and the red line marks the edge of the ΓΧ direction, along which the system is studied.

23. A. Fang, T. Koschny, and C. M. Soukoulis, “Lasing in metamaterial nanostructures,” J. Opt. **12**(2), 024013 (2010). [CrossRef]

24. C. Fietz and C. M. Soukoulis, “Finite element simulation of microphotonic lasing system,” Opt. Express **20**(10), 11548–11560 (2012). [CrossRef] [PubMed]

*R*and the system of the Maxwell equations coupled with the atomic rate equations is self-consistently solved. This procedure is repeated for several pump rates and the emitted optical power,

_{p}*P*, is calculated for each input

_{out}*R*. For high pump rates, far above the lasing threshold,

_{p}*P*varies nonlinearly in terms of

_{out}*R*, but as the threshold is approached the variation becomes linear. This validates the linear extrapolation of the calculated data close to the lasing threshold, the determination of which might otherwise demand prohibitively large computational times. In all FDTD calculations the discrete time and space steps are set to

_{p}## 3. Theoretical model for the *Q* factor

*L*, formed by a finite amount of layers of the photonic crystal, in which a pulse with carrier frequency

*iω*

_{0}

*t*) and an envelope function, so that

*k*is the wavenumber of the Bloch mode,

*m*round-trips the pulse is written as

*n*= 0,1,2…). Assuming a high

*Q*cavity, the term

*Q*factor. Last, for pulses beyond the quasi-monochromatic limit and/or highly lossy/dispersive cavities, Eq. (1) will be modified, as higher order terms in the expansion need to be retained [26

26. Y. Pinhasi, A. Yalalom, and G. A. Pinhasi, “Propagation analysis of ultrashort pulses in resonant dielectric media,” J. Opt. Soc. Am. B **26**(12), 2404–2413 (2009). [CrossRef]

*Q*systems we are interested in.

## 4. Lasing in dielectric slab with gain

*Q*factor may cast a purely analytical form. The reflectivity at the interface between two semi-infinite homogeneous media labeled

*i*,

*j*is given by

*c*is the light velocity in vacuum. Ignoring internal losses, Eq. (1) now yields

*Q*and

*ω*and the Fabry-Perot resonances

_{a}*ω*referring to the unpumped system. When lasing starts, the overall permittivity, which is nonlinearly dependent on the pump, is slightly changed within a frequency range of Γ

_{FP}*, around the frequency*

_{a}*ω*. This change alone, when countable, is sufficient to invalidate the conclusions drawn by the simple form of Eq. (2). Moreover, if

_{a}*ω*does not coincide with a certain

_{a}*ω*, then the cavity resonance will shift, rendering lasing more power demanding. Fortunately, as the lasing threshold is approached, the overall permittivity approaches the host material permittivity and the resonances converge to those of the unpumped system: the lasing system lases approximately at

_{FP}*ω*. Given the experimental feasibility, tuning

_{a}*ω*=

_{a}*ω*should thus be a good starting point.

_{FP}*L*

_{1}= 10

*a*,

*L*

_{2}= 20

*a*and

*L*

_{3}= 30

*a*. In the simulations we pump the system, let it radiate and, after the lasing amplitude is stabilized, we measure the output EM field at a certain distance from the cavity and extract the output lasing power for this specific pump rate. The output lasing power versus the input pump rate for the examined cases is presented in Fig. 3.

*L*

_{1}= 10

*a*is examined at successive Fabry-Perot resonances, as shown in Fig. 3(a), where, for each case, the emission peak of the gain material is set to coincide with the respective

*Q*factor among these operating frequencies, the lasing threshold changes only slightly. On the other hand it is clear that as the operating frequency increases less pumping is needed in order to achieve the same output power. In Fig. 3(b) the three systems are compared at a common

*THz*. It is evident that the lasing threshold is reduced with increasing slab length, as expected. The

*Q*factor in these configurations is rather low due to the poor reflectivity (

*ε*=

_{r}*ε*(

_{r}*ω*), and their contributions to

*Q*do not exhibit anything dramatic as the operating frequency changes.

## 5. Lasing in dielectric 2D photonic crystal with gain

25. C. Fietz, “Absorbing boundary condition for Bloch-Floquet eigenmodes,” J. Opt. Soc. Am. B **30**(10), 2615–2620 (2013). [CrossRef]

*Q*factor envelope, as depicted with a white line in Fig. 4(c). This envelope serves as the locus of the

*Q*points for any system of arbitrary length, regardless of the exact frequency position of the resonances. This fact is verified with another set of FEM eigenfrequency simulations, where the complex frequency

*ω*of the field (simulation eigenvalue) expresses the radiative losses. The results of the simulations are interpreted as

*Q*= Re(

*ω*)/2Im(

*ω*), consistently with the definition in section 3, i.e. Im(

*ω*) = 1/2

*τ*. They are depicted in Figs. 4(d)–4(f) for three (unpumped) systems of length

_{c}*L*

_{1}= 10

*a*,

*L*

_{2}= 20

*a*and

*L*

_{3}= 30

*a*, respectively. These results, normalized to each system’s number of layers

*n = L/a*and mapped on the same scale, coincide with the semi-analytical

*Q/n*envelope (Fig. 4(c)).

*Q*is improved in longer systems, simply because the optical power traveling at

*Q*can grow dramatically. In essence, as a band edge is approached,

*Q*, although not arbitrarily high, since in a finite system the band edge cannot be approached arbitrarily close. The contribution from both

*Q*is higher at those with the higher

*Q*. Inversely, for points of the same

*Q*is indeed higher at those with lower

*L*

_{1}= 10

*a*,

*L*

_{2}= 20

*a*and

*L*

_{3}= 30

*a*. The results are separated into two sets and summarized in Fig. 5. One set of simulations regards the examination of the lasing threshold in a certain system (we chose

*L*

_{1}= 10

*a*) as the frequency is swept throughout the resonances (Figs. 5(a) and 5(b)). A second set regards comparison among different systems, at frequencies close to the band edges of the 1st band gap (Figs. 5(c) and 5(d)), namely at ~60THz (below the gap) and ~100THz (above). As mentioned, for the finite systems these frequencies are better approached with increasing cavity length. As with the slab, the gain material emission frequency

*ω*was tuned for each case exactly at the examined cavity resonance.

_{a}*Q*factor for the system of length

*L*

_{1}= 10

*a*already presented in Fig. 4 is shown again for easier comparison with the lasing threshold depicted in Fig. 5(a). It is evident that the lasing threshold varies consistently with the

*Q*factor. The same observation holds for the 2nd set of simulations among the three systems. The lasing threshold reduces as the cavity is elongated, both above the band gap (Fig. 5(c)) and below (Fig. 5(d)). Moreover, it drops one order of magnitude when going from lower to upper band edge. The PC performs better than the slab, despite the lower gain density, except at some frequencies deep in the 1st band. This is not a surprise, because the PC reflectivity in the 1st band is lower than that of the slab (except close to the band edge) and, also, below ~50THz the group velocity in the PC becomes greater than in the slab. On the other hand a PC is more likely to operate at the band edges, where the dramatic increase of the DOS offers a unique design versatility. Since the exact emission frequency

*ω*of the gain material may have a countable frequency tolerance due to experimental limitations during manufacturing, a system with dense resonances

_{a}*ω*increases the probability of

_{R}*ω*,

_{a}*ω*matching and consequently of lower lasing threshold.

_{R}## 6. Discussion

*L*= 10

*a*long and their schematics together with the simulations are shown in Fig. 6.

*Q*improves in the 1st band and drops significantly in the 2nd band (compare with Fig. 6(a) which corresponds to the original system). This has the expected impact on the

*Q*envelope and is reflected accordingly in the lasing threshold, which now drops below the band gap (~60THz) but becomes higher at the bottom and top of the 2nd band (~100THz and ~136THz respectively), as shown in Fig. 6(d). The great increase of the threshold in band #2 is a result of the great change in

*Q*(by a factor of 5~12), as compared to the change in the 1st band, where

*Q*is enhanced by a factor of 3~4. The 1st system, in particular, manifests a huge dip towards the bottom of the 2nd band. As a result, the lasing threshold at those frequencies increases to levels even higher than those of the 1st band. On the other hand, for the 2nd system the dip is situated towards the top of the band, where the lasing threshold is pushed to a level higher than that of the bottom of the same band. These results express a more general remark: a dip in the reflectivity may affect the overall performance of a system severely, even if the group velocity is very low, as for example close to a band edge. Hence, lower lasing thresholds at sharper band edges or even inside some band should not be a surprise, especially in narrow bands with sparse resonances. Of course, longer systems with enhanced DOS will gradually eliminate this paradox, as more resonances will shift closer to the band edges. The results from Figs. 5(a) and 5(b) regarding the lasing threshold and

*Q*factor at the edges of the 1st and 2nd band are reproduced in Figs. 6(d) and 6(e) respectively, for easier comparison.

## 7. Conclusion

## Acknowledgments

## References and links

1. | H. Kogelnik and C. V. Shank, “Stimulated emission in a periodic structure,” Appl. Phys. Lett. H. Kogelnik and C. V. Shank, “Coupled-wave theory of distributed feedback lasers,” J. Appl. Phys. |

2. | J. P. Dowling, M. Scalora, M. J. Bloemer, and C. M. Bowden, “The photonic band edge laser: A new approach to gain enhancement,” J. Appl. Phys. |

3. | M. D. Tocci, M. Scalora, M. J. Bloemer, J. P. Dowling, and C. M. Bowden, “Measurement of spontaneous-emission enhancement near the one-dimensional photonic band edge of semiconductor heterostructures,” Phys. Rev. A |

4. | M. Meier, A. Mekis, A. Dodabalapur, A. Timko, R. Slusher, J. Joannopoulos, and O. Nalamasu, “Laser action from two-dimensional distributed feedback in photonic crystals,” Appl. Phys. Lett. |

5. | M. Imada, S. Noda, A. Chutinan, T. Tokuda, M. Murata, and G. Sasaki, “Coherent two-dimensional lasing action in surface-emitting laser with triangular-lattice photonic crystal structure,” Appl. Phys. Lett. |

6. | S. Riechel, C. Kallinger, U. Lemmer, J. Feldmann, A. Gombert, V. Wittwer, and U. Scherf, “A nearly diffraction limited surface emitting conjugated polymer laser utilizing a two-dimensional photonic band structure,” Appl. Phys. Lett. |

7. | H.-Y. Ryu, S.-H. Kwon, Y.-J. Lee, Y.-H. Lee, and J.-S. Kim, “Very-low-threshold photonic band-edge lasers from free-standing triangular photonic crystal slabs,” Appl. Phys. Lett. |

8. | S.-H. Kwon, H.-Y. Ryu, G.-H. Kim, Y.-H. Lee, and S.-B. Kim, “Photonic bandedge lasers in two-dimensional square-lattice photonic crystal slabs,” Appl. Phys. Lett. |

9. | X. Wu, A. Yamilov, X. Liu, S. Li, V. P. Dravid, R. P. H. Chang, and H. Cao, “Ultraviolet photonic crystal laser,” Appl. Phys. Lett. |

10. | F. Raineri, G. Vecchi, A. M. Yacomotti, C. Seassal, P. Viktorovitch, R. Raj, and A. Levenson, “Doubly resonant photonic crystal for efficient laser operation: Pumping and lasing at low group velocity photonic modes,” Appl. Phys. Lett. |

11. | C. Karnutsch, M. Stroisch, M. Punke, U. Lemmer, J. Wang, and T. Weimann, “Laser diode-pumped organic semiconductor lasers utilizing two-dimensional photonic crystal resonators,” Phot. Tech. Lett. |

12. | D. Luo, X. W. Sun, H. T. Dai, Y. J. Liu, H. Z. Yang, and W. Ji, “Two-directional lasing from a dye-doped two-dimensional hexagonal photonic crystal made of holographic polymer-dispersed liquid crystals,” Appl. Phys. Lett. |

13. | W. Cao, A. Muñoz, P. Palffy-Muhoray, and B. Taheri, “Lasing in a three-dimensional photonic crystal of the liquid crystal blue phase II,” Nat. Mater. |

14. | B. D’Urso, O. Painter, J. Brien, T. Tombrello, A. Yariv, and A. Scherer, “Modal refletivity in finite-depth two-dimensional photonic crystal microcavities,” J. Opt. Soc. Am. B |

15. | K. Sakoda, “Enhanced light amplification due to group-velocity anomaly peculiar to two- and three-dimensional photonic crystals,” Opt. Express |

16. | S. Nojima, “Optical-gain enhancement in two-dimensional active photonic crystals,” J. Appl. Phys. |

17. | L. Florescu, K. Busch, and S. John, “Semiclassical theory of lasing in photonic crystals,” J. Opt. Soc. Am. B |

18. | H.-Y. Ryu, M. Notomi, and Y.-H. Lee, “Finite-difference time-domain investigation of band-edge resonant modes in finite-size two-dimensional photonic crystal slab,” Phys. Rev. B |

19. | N. Susa, “Threshold gain and gain-enhancement due to distributed-feedback in two-dimensional photonic-crystal lasers,” J. Appl. Phys. |

20. | P. Bermel, E. Lidorikis, Y. Fink, and J. D. Joannopoulos, “Active materials embedded in photonic crystals and coupled to electromagnetic radiation,” Phys. Rev. B |

21. | S. V. Zhukovsky and D. N. Chigrin, “Numerical modelling of lasing in microstructures,” Phys. Status Solidi |

22. | X. Hu, J. Cao, M. Li, Z. Ye, M. Miyawaki, and K.-M. Ho, “Modeling of three-dimensional photonic crystal lasers in a frequency domain: A scattering matrix solution,” Phys. Rev. B |

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

24. | C. Fietz and C. M. Soukoulis, “Finite element simulation of microphotonic lasing system,” Opt. Express |

25. | C. Fietz, “Absorbing boundary condition for Bloch-Floquet eigenmodes,” J. Opt. Soc. Am. B |

26. | Y. Pinhasi, A. Yalalom, and G. A. Pinhasi, “Propagation analysis of ultrashort pulses in resonant dielectric media,” J. Opt. Soc. Am. B |

**OCIS Codes**

(140.3410) Lasers and laser optics : Laser resonators

(140.3490) Lasers and laser optics : Lasers, distributed-feedback

(050.1755) Diffraction and gratings : Computational electromagnetic methods

(230.5298) Optical devices : Photonic crystals

**ToC Category:**

Photonic Crystals

**History**

Original Manuscript: April 15, 2014

Revised Manuscript: June 29, 2014

Manuscript Accepted: July 2, 2014

Published: August 1, 2014

**Citation**

Sotiris Droulias, Chris Fietz, Peng Zhang, Thomas Koschny, and Costas M. Soukoulis, "Lasing threshold control in two-dimensional photonic crystals with gain," Opt. Express **22**, 19242-19251 (2014)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-22-16-19242

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### References

- H. Kogelnik and C. V. Shank, “Stimulated emission in a periodic structure,” Appl. Phys. Lett.18(4), 152–154 (1971).H. Kogelnik and C. V. Shank, “Coupled-wave theory of distributed feedback lasers,” J. Appl. Phys.43(5), 2327–2335 (1972). [CrossRef]
- J. P. Dowling, M. Scalora, M. J. Bloemer, and C. M. Bowden, “The photonic band edge laser: A new approach to gain enhancement,” J. Appl. Phys.75(4), 1896–1899 (1994). [CrossRef]
- M. D. Tocci, M. Scalora, M. J. Bloemer, J. P. Dowling, and C. M. Bowden, “Measurement of spontaneous-emission enhancement near the one-dimensional photonic band edge of semiconductor heterostructures,” Phys. Rev. A53(4), 2799–2803 (1996). [CrossRef] [PubMed]
- M. Meier, A. Mekis, A. Dodabalapur, A. Timko, R. Slusher, J. Joannopoulos, and O. Nalamasu, “Laser action from two-dimensional distributed feedback in photonic crystals,” Appl. Phys. Lett.74(1), 7 (1999). [CrossRef]
- M. Imada, S. Noda, A. Chutinan, T. Tokuda, M. Murata, and G. Sasaki, “Coherent two-dimensional lasing action in surface-emitting laser with triangular-lattice photonic crystal structure,” Appl. Phys. Lett.75(3), 316 (1999). [CrossRef]
- S. Riechel, C. Kallinger, U. Lemmer, J. Feldmann, A. Gombert, V. Wittwer, and U. Scherf, “A nearly diffraction limited surface emitting conjugated polymer laser utilizing a two-dimensional photonic band structure,” Appl. Phys. Lett.77(15), 2310–2312 (2000). [CrossRef]
- H.-Y. Ryu, S.-H. Kwon, Y.-J. Lee, Y.-H. Lee, and J.-S. Kim, “Very-low-threshold photonic band-edge lasers from free-standing triangular photonic crystal slabs,” Appl. Phys. Lett.80(19), 3476–3478 (2002). [CrossRef]
- S.-H. Kwon, H.-Y. Ryu, G.-H. Kim, Y.-H. Lee, and S.-B. Kim, “Photonic bandedge lasers in two-dimensional square-lattice photonic crystal slabs,” Appl. Phys. Lett.83(19), 3870–3872 (2003). [CrossRef]
- X. Wu, A. Yamilov, X. Liu, S. Li, V. P. Dravid, R. P. H. Chang, and H. Cao, “Ultraviolet photonic crystal laser,” Appl. Phys. Lett.85(17), 3657–3659 (2004). [CrossRef]
- F. Raineri, G. Vecchi, A. M. Yacomotti, C. Seassal, P. Viktorovitch, R. Raj, and A. Levenson, “Doubly resonant photonic crystal for efficient laser operation: Pumping and lasing at low group velocity photonic modes,” Appl. Phys. Lett.86(1), 011116 (2005). [CrossRef]
- C. Karnutsch, M. Stroisch, M. Punke, U. Lemmer, J. Wang, and T. Weimann, “Laser diode-pumped organic semiconductor lasers utilizing two-dimensional photonic crystal resonators,” Phot. Tech. Lett.19(10), 741–743 (2007). [CrossRef]
- D. Luo, X. W. Sun, H. T. Dai, Y. J. Liu, H. Z. Yang, and W. Ji, “Two-directional lasing from a dye-doped two-dimensional hexagonal photonic crystal made of holographic polymer-dispersed liquid crystals,” Appl. Phys. Lett.95(15), 151115 (2009). [CrossRef]
- W. Cao, A. Muñoz, P. Palffy-Muhoray, and B. Taheri, “Lasing in a three-dimensional photonic crystal of the liquid crystal blue phase II,” Nat. Mater.1(2), 111–113 (2002). [CrossRef] [PubMed]
- B. D’Urso, O. Painter, J. Brien, T. Tombrello, A. Yariv, and A. Scherer, “Modal refletivity in finite-depth two-dimensional photonic crystal microcavities,” J. Opt. Soc. Am. B15, 1155–1159 (1998).
- K. Sakoda, “Enhanced light amplification due to group-velocity anomaly peculiar to two- and three-dimensional photonic crystals,” Opt. Express4(5), 167–176 (1999). [CrossRef] [PubMed]
- S. Nojima, “Optical-gain enhancement in two-dimensional active photonic crystals,” J. Appl. Phys.90(2), 545–551 (2001). [CrossRef]
- L. Florescu, K. Busch, and S. John, “Semiclassical theory of lasing in photonic crystals,” J. Opt. Soc. Am. B19(9), 2215–2223 (2002). [CrossRef]
- H.-Y. Ryu, M. Notomi, and Y.-H. Lee, “Finite-difference time-domain investigation of band-edge resonant modes in finite-size two-dimensional photonic crystal slab,” Phys. Rev. B68(4), 045209 (2003). [CrossRef]
- N. Susa, “Threshold gain and gain-enhancement due to distributed-feedback in two-dimensional photonic-crystal lasers,” J. Appl. Phys.89(2), 815–823 (2001). [CrossRef]
- P. Bermel, E. Lidorikis, Y. Fink, and J. D. Joannopoulos, “Active materials embedded in photonic crystals and coupled to electromagnetic radiation,” Phys. Rev. B73(16), 165125 (2006). [CrossRef]
- S. V. Zhukovsky and D. N. Chigrin, “Numerical modelling of lasing in microstructures,” Phys. Status Solidi244(10), 3515–3527 (2007). [CrossRef]
- X. Hu, J. Cao, M. Li, Z. Ye, M. Miyawaki, and K.-M. Ho, “Modeling of three-dimensional photonic crystal lasers in a frequency domain: A scattering matrix solution,” Phys. Rev. B77(20), 205104 (2008). [CrossRef]
- A. Fang, T. Koschny, and C. M. Soukoulis, “Lasing in metamaterial nanostructures,” J. Opt.12(2), 024013 (2010). [CrossRef]
- C. Fietz and C. M. Soukoulis, “Finite element simulation of microphotonic lasing system,” Opt. Express20(10), 11548–11560 (2012). [CrossRef] [PubMed]
- C. Fietz, “Absorbing boundary condition for Bloch-Floquet eigenmodes,” J. Opt. Soc. Am. B30(10), 2615–2620 (2013). [CrossRef]
- Y. Pinhasi, A. Yalalom, and G. A. Pinhasi, “Propagation analysis of ultrashort pulses in resonant dielectric media,” J. Opt. Soc. Am. B26(12), 2404–2413 (2009). [CrossRef]

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