## Study on transition from photonic-crystal laser to random laser |

Optics Express, Vol. 20, Issue 7, pp. 7300-7315 (2012)

http://dx.doi.org/10.1364/OE.20.007300

Acrobat PDF (5383 KB)

### Abstract

The dependence of the lasing threshold on the amount of positional disorder in photonic crystal structures is newly studied by means of the finite element method, not of the finite difference time domain method usually used. A two-dimensional model of a photonic crystal consisting of dielectric cylinders arranged on a triangular lattice within a circular region is considered. The cylinders are assumed to be homogeneous and infinitely long. Positional disorder of the cylinders is introduced to the photonic crystals. Optically active medium is introduced to the interspace among the cylinders. The population inversion density of the optically active medium is modeled by the negative imaginary part of dielectric constant. The ratio between radiative power of electromagnetic field without amplification and that with amplification is computed as a function of the frequency and the imaginary part of the dielectric constant, and the threshold of the imaginary part, namely population inversion density for laser action is obtained. These analyses are carried out for various amounts of disorder. The variation of the lasing threshold from photonic-crystal laser to random laser is revealed by systematic computations with numerical method of reliable accuracy for the first time. Moreover, a novel phenomenon, that the lasing threshold have a minimum against the amount of disorder, is found. In order to investigate the properties of the lasing states within the circular system, the distributions of the electric field amplitudes of the states are also calculated.

© 2012 OSA

## 1. Introduction

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## 2. Analysis model

*z*-direction) and light waves propagate within

*xy*-plane. Figure 1(b) shows the concept of a random system from the top view. Dielectric cylinders are arranged randomly in the region between circles C

_{in}and C

_{g}, as shown in Fig. 1(b). An oscillating polarizaton is assumed to exist at the center of the circle C

_{in}as a light source. Radii of C

_{in}and C

_{g}are denoted by

*R*

_{in}and

*R*

_{g}, respectively. We compute the fluxes of Poynting vectors of out-flowing light waves on the circle C

_{out}whose radius is

*R*

_{out}. The unit outward normal vectors to C

_{in}and C

_{out}are denoted by

**n**

_{in}and

**n**

_{out}, respectively. We define three regions: Ω

_{act}, Ω

_{cylinder}and Ω

_{out}, where Ω

_{act}is in the interspace among the cylinders inside the circle C

_{out}, Ω

_{cylinder}is the union of the regions inside the cylinders, and Ω

_{out}is the region outside the circle C

_{out}. The optically active materials are assumed to be filled in the region Ω

_{act}.

### 2.1. Model parameters

*a*, is treated as the characteristic length. The size of the analysis models is normalized by

*a*. We create the analysis model of the periodic structure of triangular lattice, by giving the transfer mean free path (TMFP), denoted by

*l*, and the periodic length of the periodic structure. The values of the TMFP and the periodic length are given as

*l*= 1.47735

*a*and 3.47735

*a*, respectively. The coordinate values of the cylinder’s center are specified in single precision numbers.

### 2.2. Disordered systems

**x**

_{p}. A circle drawn by a solid line illustrates a cylinder arranged in disorder, whose center is denoted by

**x**

_{r}. The disordered position

**x**

_{r}is determined by a sum of

**x**

_{p}and a random vector Δ

**x**

_{r}, as follows: where

*n*and

*m*are latiice-point numbers and (

*n,m*)

*≠*(0

*,*0).

**x**

_{p}are defined as where

**r**

_{1}and

**r**

_{2}are lattice vectors defined as where 3.47735

*a*is the periodic length.

**x**

_{r}as where

*|*Δ

**x**

_{r}

*|*

_{max}is the maximum length of the random vector. The amounts of disorders in dielectric structures are evaluated with

*|*Δ

**x**

_{r}

*|*

_{max}

*/a*, that is, the maximum length normalized by the radii of cylinders. In the case

*|*Δ

**x**

_{r}

*|*

_{max}

*/a*= 0.00, a dielectric structure becomes periodic. In Fig. 2,

*L*

_{H}, the distance between the edge of the hexagonal lattice and the center of the cylinder periodically distributed and included in the lattice, is equal to 1.73867

*a*, a half of the periodic length. Therefore, when

*L*

_{H}

*< |*Δ

**x**

_{r}

*|*

_{max}, the center of the cylinder distributed randomly is located within the adjacent hexagonal lattice. When

*|*Δ

**x**

_{r}

*|*

_{max}is smaller than

*L*

_{H}−

*a*= 0.73867

*a*, the entire region of the cylinder is included in each lattices.

*|*Δ

**x**

_{r}

*|*

_{max}. The analysis models and radial distribution functions for these values are shown in Figs. 3 and 4, respectively. Radial distribution functions describe the change of the density of dielectric cylinders as the function of the distance from the center of dielectric structures. The functions express how dielectric structures are disordered.

## 3. Formulation

56. J. P. Berenger, “A perfectly matched layer for the absorption of electromagnetic waves,” J. Comput. Phys. **114**, 185–200 (1994). [CrossRef]

57. A. Bermúdez, L. Hervella-Nieto, A. Prieto, and R. Rodríguez, “An exact bounded pml for the helmholtz equation,” C. R. Acad, Sci. Paris, Ser. I **339**, 803–808 (2004). [CrossRef]

59. A. Bermúdez, L. Hervella-Nieto, A. Prieto, and R. Rodríguez, “An optimal perfectly matched layer with unbounded absorbing function for time-harmonic acoustic scattering problems,” J. Comput. Phys. **223**, 469–488 (2007). [CrossRef]

### 3.1. Basic equations of electromagnetic scattering problem

*ωa*t the center of the entire region of the random system,

**x**

_{0}, as a light source (Fig. 1). The electric and magnetic fields,

**E**and

**H**, are assumed to be time-harmonic waves with the same angular frequency as that of the light source. The following non-homogeneos equation is derived from Maxwell’s equations as where

*c*is the speed of light in vacuum,

*ε*

_{0}and

*ε*(

**x**) are the permittivity in vacuum and the position-dependent relative permittivity,

**D**

_{d}is the polarization vector,

*δ*(

**x**) is Dirac’s delta function.

**E**(

**x**) as the sum of the scattering and incident fields as follows: where,

**E**

_{i}(

**x**) is the electric field in the region without scatterers, satisfying the following equation: where

*ε*

_{i}is the constant relative permittivity in Ω

_{out}(Fig. 1). We substitute Eqs. (7) and (8) to Eq. (6), to have When we assume TM mode, the incident field

**E**

_{i}(

**x**) satisfying Eq. (8) can be expressed by 0th-order Hankel function of the first kind,

**E**(

**x**) defined by Eq. (7).

### 3.2. Population inversion density of optically active materials

*γ*(

*γ*

*>*0) [20

20. K. Ohtaka, “Density of states of slab photonic crystals and the laser oscillation in photonic crystals,” J. Lightwave Technol. **17**, 2161–2169 (1999). [CrossRef]

22. K. Sakoda, K. Ohtaka, and T. Ueta, “Low-threshold laser oscillation due to group-velocity anomaly peculiar to two- and three-dimensional photonic crystals,” Opt. Express **4**, 481–489 (1999). [CrossRef] [PubMed]

*γ*is a parameter proportional to the population inversion density of an optically active material. Hence,

*γ*at which a laser action occurs is interpreted as the threshold for the laser action [20

20. K. Ohtaka, “Density of states of slab photonic crystals and the laser oscillation in photonic crystals,” J. Lightwave Technol. **17**, 2161–2169 (1999). [CrossRef]

22. K. Sakoda, K. Ohtaka, and T. Ueta, “Low-threshold laser oscillation due to group-velocity anomaly peculiar to two- and three-dimensional photonic crystals,” Opt. Express **4**, 481–489 (1999). [CrossRef] [PubMed]

*γ*. The relative permittivities in individual regions (Fig. 1) are given as follows:

### 3.3. Definition of amplification factor

**S**〉 is the time-average of Poynting vector

**S**, Re(

**Z**) means the real part of complex vector

**Z**, and

**H**

*denotes the complex conjugate of the magnetic field.*

^{*}*A*by the ratio of the fluxes of the Poynting vectors of light, flowing out from the dielectric system, between the excited state (

*γ*

*>*0) and non-excited state (

*γ*= 0), as follows: The light flux is calculated by a line integral of the Poynting vector along the circle C

_{out}.

### 3.4. Finite element analysis

60. A. C. Cangellaris and D. B. Wright, “Analysis of the numerical error caused by the stair-stepped approximation of a conducting boundary in FDTD simulations of electromagnetic phenomena,” IEEE Trans. Antennas Propag. **39**, 1518–1525 (1991). [CrossRef]

62. K. H. Dridi, J. S. Hesthaven, and A. Ditkowski, “Staircase-free finite-difference time-domain formulation for general materials in complex geometries,” IEEE Trans. Antennas Propag. **49**, 749–756 (2001). [CrossRef]

*a*. The computational accuracy is reduced due to the rotation of the electric field, ∇ ×

**E**, which is needed to compute the complex conjugate of magnetic field,

**H**

*in Eq. (12). Hence, we discretize the regions in the neighborhood of the circle C*

^{*}_{out}into more smaller elements 0.01

*a*. Since the absorbing function of PML is nonlinear, it is difficult to integrate the components of the stiffness matrix for this region analytically. Therefore, numerical integration scheme, Gauss-Legendre quadrature formula, is employed to evaluate numerically the integrals including the absorbing function for the area discretized with the quadrilateral elements shown in Fig. 5(b). The numbers of nodes and elements are approximately one million and two million, respectively.

## 4. Results

### 4.1. Lasing frequency

*|*Δ

**x**

_{r}

*|*

_{max}

*/a*= 0.00. We compute the amplification factor for the frequency range 0.1

*≤*

*ω*

*a/*2

*π*

*c ≤*0.3 with fixed population inversion density

*γ*= 0.002 to investigate lasing frequencies of the periodic structure corresponding to

*|*Δ

**x**

_{r}

*|*

_{max}

*/a*= 0.00. We also show the band structure of the periodic structure, computed by the plane wave expansion method, in Fig. 7. In the result shown in Fig. 6, we cannot achieve sufficient computational accuracy in red-colored frequency ranges corresponding to band gaps in Fig. 7.

*≤*

*ω*

*a/*2

*π*

*c ≤*0.240. We observe this frequency range, in which laser action occurs, corresponds to that of fourth-lowest band in Fig. 7. We show in Fig. 8 the electric field intensity distributions corresponding to the wave vector,

**k**= (3

*π*

*/*4

*a,*

### 4.2. Laser action

*A*defined by Eq. (13) for the ranges of 0.222

*≤*

*ω*

*a/*2

*π*

*c ≤*0.240 and 0.000

*≤*

*γ*

*≤*0.021, corresponding to the fourth lowest band. Figure 9 shows the laser action in the dielectric structures for each value of the disorder index

*|*Δ

**x**

_{r}

*|*

_{max}

*/a*normalized by

*a*. We calculate the amplification factors for 127,041 grid points for

*ω*

*a/*2

*π*c and

*γ*, by dividing

*ω*

*a/*2

*π*c and

*γ*directions uniformly into 900 an 140 intervals, respectively, and seek the steep peaks of the surface of amplification factor

*A*.

*ωa*

*/*2

*πc*and

*γ*. From this result, we find specific combinations of the values of

*ωa*

*/*2

*πc*and

*γ*at which lasing phenomena occur in the periodic structure. Excitation within the fourth lowest band in the dispersion relation occurs as the surface of the amplification factor

*A*and the lasing phenomena occur at their peaks. Figure 9(i) shows a laser action occurring at band-edge frequency. This laser action occurs at the smallest value of

*γ*among the results of the periodic structure. Therefore, this laser action is considered to be the lowest threshold one. The value of the amplification factor of this laser action is greater than 10

^{4}.

*≤*

*ωa*

*/*2

*πc*

*≤*0.24, in the results for small amounts of disorder,

*|*Δ

**x**

_{r}

*|*

_{max}

*/a*= 0.0625, 0.125, and 0.250, shown in Figs. 9(b), 9(c), and 9(d), respectively. Laser actions occurring in the periodic structure are also found even in the results of small amounts of disorder. However, in the case of higher disorder, i.e. 0.500

*≤ |*Δ

**x**

_{r}

*|*

_{max}

*/a*, lasing phenomena caused by random light scatterings become noticeable also in the lower frequency range, and lasing phenomena found in the periodic structure disappear.

### 4.3. Lasing threshold

*γ*, interpreted as the imaginary part of relative permittivity, is proportional to population inversion density of optically active material. Hence, steep peaks of the amplification factors in the region with small

*γ*are interpreted as occurrences of low-threshold laser generation. To find how lasing threshold changes in accordance with the increase of the disorder index

*|*Δ

**x**

_{r}

*|*

_{max}, we investigate the smallest value of

*γ*at which laser action occurs. To seek the more precise values of

*ωa*

*/*2

*πc*and

*γ*at which the lowest-threshold laser action occurs, we scan the neighborhood of the amplification peak using finer numerical steps.

*γ*. We analyze 10 different types of cylinder arrangement for each disorder index other than

*|*Δ

**x**

_{r}

*|*

_{max}

*/a*= 0.00 to check the lasing threshold tendency. We find two types of lowest-threshold laser modes: those with tight confinement and spacial extension of light wave.

*≤ |*Δ

**x**

_{r}

*|*

_{max}

*/a*.

*|*Δ

**x**

_{r}

*|*

_{max}

*/a <*0.250, then decreases in 0.250

*< |*Δ

**x**

_{r}

*|*

_{max}

*/a <*1.00. In Fig. 10, the average threshold of 10 different cylinders arrangements rises again in 1.00

*< |*Δ

**x**

_{r}

*|*

_{max}

*/a*. The average threshold becomes minimum at

*|*Δ

**x**

_{r}

*|*

_{max}

*/a*= 1.00.

*< |*Δ

**x**

_{r}

*|*

_{max}

*/a <*1.00. In view of the EADs shown in Figs. 13(d) and 13(f), such decrease of lasing threshold is caused by increase in multiple scatterings.

*|*Δ

**x**

_{r}

*|*

_{max}

*/a*= 1.00, and another increase of the lasing threshold for 1.00

*< |*Δ

**x**

_{r}

*|*

_{max}

*/a*in Fig. 10. We compute mode volumes of the lasing peaks oscillating at the smallest

*γ*in each sample to investigate how the optical properties of the laser mode affect on lasing threshold. The mode volume is defined as follows: where

*λ*= 2

*πc*

*/*

*ω*is the wavelength in vacuum. Figure 14 shows the relation between the mode volume and the amount of positional disorder. The mode volume decreases as the amount of disorder increases in the state of spatially extended modes. In the case of tightly confined modes, the mode volume increases because the periodic structure is disordered and the increase indicates leaks of the confined light caused by disorder. The average values of the mode volumes of 10 samples in 0.75

*≤ |*Δ

**x**

_{r}

*|*

_{max}

*/a ≤*1.5 become smaller than those in 2

*≤ |*Δ

**x**

_{r}

*|*

_{max}

*/a ≤*4. Based on the comparison between Figs. 10 and 14, we find that lasing threshold tends to become lower as the mode volume becomes smaller. The above comparison indicates that strong light confinements are needed in order that low-threshold random lasings are realized.

*≤ |*Δ

**x**

_{r}

*|*

_{max}

*/a*. In such a case, the random number value is discarded and a regenerated random number is used to determine the shifted position of the cylinder. Therefore, the average of the random numbers employed to generate the random arrangement of the cylinders may differ from 0.5 of the admissible shift range.

*|*Δ

**x**

_{r}

*|*are given by random numbers of a uniform distribution between 0 and 1. In the range 0.73867

*≤ |*Δ

**x**

_{r}

*|*

_{max}

*/a ≤*2.00, the effect of discard of some random numbers are observed clearly. The average values tend to decrease as

*|*Δ

**x**

_{r}

*|*

_{max}increases.

*|*Δ

**x**

_{r}

*|*corresponds to 0.73867

*a*. Considering the average of the random number

*m*= 0.5, its dispersion

*ρ*= 1

*/*12, and the standard deviation

*x*

_{r}*, y*

*) directly instead of using (*

_{r}*|*Δ

**x**

_{r}

*|,*

*θ*). Again, random numbers of uniform distribution are used to determine the values of (

*x*

_{r}*, y*

*) so that the generated cylinder is located with the circular are bounded by the certain radius*

_{r}*|*Δ

**x**

_{r}

*|*

_{max}. The number of samples computed is five. We show the results for lasing threshold in Fig. 16, in which a similar behavior of lasing threshold is observed. We find the tendency in the relation between the lasing threshold and the amount of disorder is independent of positioning algorithms.

## 5. Conclusion

## Acknowledgments

## References and links

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2. | D. S. Wiersma, M. P. van Albada, and A. Lagendijk, “Random laser ?” Nature |

3. | D. S. Wiersma, “The physics and applications of random lasers,” Nat. Phys. |

4. | P. Sebbah, R. Pnini, and A. Z. Genack, “Field and intensity correlation in random media,” Phys. Rev. E |

5. | P. Sebbah and C. Vanneste, “Random laser in the localized regime,” Phys. Rev. B |

6. | P. Sebbah, B. Hu, J. K. Klosner, and A. Z. Genack, “Extended quasimodes within nominally localized random waveguides,” Phys. Rev. Lett. |

7. | C. Vanneste and P. Sebbah, “Selective excitation of localized modes in active random media,” Phys. Rev. Lett. |

8. | C. Vanneste and P. Sebbah, “Localized modes in random arrays of cylinders,” Phys. Rev. E |

9. | C. Vanneste, P. Sebbah, and H. Cao, “Lasing with resonant feedback in weakly scattering random systems,” Phys. Rev. Lett. |

10. | C. Vanneste and P. Sebbah, “Complexity of two-dimensional quasimodes at the transition from weak scattering to anderson localization,” Phys. Rev. A |

11. | H. Cao, Y. G. Zhao, S. T. Ho, E. W. Seelig, Q. H. Wang, and R. P. H. Chang, “Random laser action in semiconductor powder,” Phys. Rev. Lett. |

12. | H. Cao, J. Y. Xu, D. Z. Zhang, S. H. Chang, S. T. Ho, E. W. Seeling, X. Liu, and R. P. H. Chang, “Spatial confinement of laser light in active random media,” Phys. Rev. Lett. |

13. | H. Cao, J. Y. Xu, S. H. Chang, and S. T. Ho, “Transition from amplified spontaneous emission to laser action in strongly scattering media,” Phys. Rev. E |

14. | S. Mujumdar, M. Ricci, R. Torre, and D. S. Wiersma, “Amplified extended modes in random lasers,” Phys. Rev. Lett. |

15. | K. Ohtaka, “Energy band of photons and low-energy photon diffraction,” Phys. Rev. B |

16. | E. Yablonovitch, “Inhibited spontaneous emission in solid-state physics and electronics,” Phys. Rev. Lett. |

17. | E. Yablonovitch and T. J. Gmitter, “Photonic band structure: the face-centered-cubic case,” Phys. Rev. Lett. |

18. | E. Yablonovitch, “Photonic band-gap structures,” J. Opt. Soc. Am. B |

19. | G. Fujii, T. Matsumoto, T. Takahashi, and T. Ueta, “A study on optical properties of photonic crystals consisting of hollow rods,” IOP Conf. Ser.: Mater. Sci. Eng. |

20. | K. Ohtaka, “Density of states of slab photonic crystals and the laser oscillation in photonic crystals,” J. Lightwave Technol. |

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

22. | K. Sakoda, K. Ohtaka, and T. Ueta, “Low-threshold laser oscillation due to group-velocity anomaly peculiar to two- and three-dimensional photonic crystals,” Opt. Express |

23. | P. W. Anderson, “Absence of diffusion in certain random lattices,” Phys. Rev. |

24. | G. Fujii, T. Matsumoto, T. Takahashi, and T. Ueta, “Finite element analysis for laser oscillation in random system consisting of heterogeneous dielectric materials,” Trans. Jpn. Soc. Comput. Methods Eng. |

25. | G. Fujii, T. Matsumoto, T. Takahashi, and T. Ueta, “A study on the effect of filling factor for laser action in dielectric random media,” (2012). Appl. Phys. A DOI: [CrossRef] . |

26. | A. Rodriguez, M. Ibanescu, J. D. Joannopoulos, and S. G. Johnson, “Disorder-immune confinement of light in photonic-crystal cavities,” Opt. Lett. |

27. | Z.-Y. Li and Z.-Q. Zhang, “Fragility of photonic band gaps in inverse-opal photonic crystals,” Phys. Rev. B |

28. | E. Lidorikis, M. M. Sigalas, E. N. Economou, and C. M. Soukoulis, “Gap deformation and classical wave localization in disordered two-dimensional photonic-band-gap materials,” Phys. Rev. B |

29. | T. Schwartz, G. Bartal, S. Fishman, and M. Segev, “Transport and Anderson localization in disordered two-dimensional photonic lattices,” Nature |

30. | Y. Lahini, A. Avidan, F. Pozzi, M. Sorel, R. Morandotti, D. N. Christodoulides, and Y. Silberberg, “Anderson localization and nonlinearity in one-dimensional disordered photonic lattices,” Phys. Rev. Lett. |

31. | H. Li, B. Cheng, and D. Zhang, “Two-dimensional disordered photonic crystals with an average periodic lattice,” Phys. Rev. B |

32. | M. A. Kaliteevski, J. M. Martinez, D. Cassagne, and J. P. Albert, “Disorder-induced modification of the transmission of light in a two-dimensional photonic crystal,” Phys. Rev. B |

33. | T. Prasad, V. L. Colvin, and D. M. Mittleman, “The effect of structural disorder on guided resonances in photonic crystal slabs studied with terahertz time-domain spectroscopy,” Opt. Express |

34. | Y. A. Vlasov, M. A. Kaliteevski, and V. V. Nikolaev, “Different regimes of light localization in disordered photonic crystal,” Phys. Rev. B |

35. | M. Patterson, S. Hughes, S. Combrié, N.-V.-Q. Tran, A. D. Rossi, R. Gabet, and Y. Jaouën, “Disorder-induced coherent scattering in slow-light photonic crystal waveguides,” Phys. Rev. Lett. |

36. | R. Ferrini, D. Leuenberger, R. Houdré, H. Benisty, M. Kamp, and A. Forchel, “Disorder-induced losses in planar photonic crystals,” Opt. Lett. |

37. | M. M. Sigalas, C. M. Soukoulis, C. T. Chan, R. Biswas, and K. M. Ho, “Effect of disorder on photonic band gaps,” Phys. Rev. B |

38. | A. A. Asatryan, P. A. Robinson, L. C. Botten, R. C. McPhedran, N. A. Nicorovici, and C. M. de Sterke, “Effects of geometric and refractive index disorder on wave propagation in two-dimensional photonic crystals,” Phys. Rev. E |

39. | W. R. Frei and H. T. Johnson, “Finite-element analysis of disorder effects in photonic crystals,” Phys. Rev. B |

40. | A. A. Asatryan, P. A. Robinson, L. C. Botten, R. C. McPhedran, N. A. Nicorovici, and C. M. de Sterke, “Effects of disorder on wave propagation in two-dimensional photonic crystals,” Phys. Rev. E |

41. | H. Li, H. Chen, and X. Qiu, “Band-gap extension of disordered 1d binary photonic crystals,” Physica B |

42. | X. Wang and K. Kempa, “Effects of disorder on subwavelength lensing in two-dimensional photonic crystal slabs,” Phys. Rev. B |

43. | T. N. Langtry, A. A. Asatryan, and L. C. Botten, “Effects of disorder in two-dimensional photonic crystal waveguides,” Phys. Rev. E |

44. | V. N. Astratov, J. P. Franchak, and S. P. Ashili, “Optical coupling and transport phenomena in chains of spherical dielectric microresonators with size disorder,” Appl. Phys. Lett. |

45. | K. C. Kwan, X. Zhang, Z. Q. Zhang, and C. T. Chan, “Effects due to disorder on photonic crystal-based waveguides,” Appl. Phys. Lett. |

46. | D. P. Fuell, S. Hughes, and M. M. Dignam, “Effect of disorder strength on the fracture pattern in heterogeneous networks,” Phys. Rev. B |

47. | D. Gerace and L. C. Andreani, “Effects of disorder on propagation losses and cavity q-factors in photonic crystal slabs,” Photon. Nanostruct. Fundam. Appl. |

48. | A. Golshani, H. Pier, E. Kapon, and M. Moser, “Photon mode localization in disordered arrays of vertical cavity surface emitting lasers,” Appl. Phys. Lett. |

49. | J. Topolancik and F. Vollmer, “Random high-q cavities in disordered photonic crystal waveguides,” Appl. Phys. Lett. |

50. | T. A. Leskova, A. A. Maradudin, I. V. Novikov, A. V. Schegrov, and E. R. Méndez, “Design of one-dimensional band-limited uniform diffusers of light,” Appl. Phys. Lett. |

51. | E. R. Méndez, E. E. García, T. A. Leskova, A. A. Maradudin, J. Muñoz-Lopez, and I. Simonsen, “Design of one-dimensional random surfaces with specified scattering properties,” Appl. Phys. Lett. |

52. | E. R. Méndez, T. A. Leskova, A. A. Maradudin, and J. Muñoz-Lopez, “Design of two-dimensional random surfaces with specified scattering properties,” Opt. Lett. |

53. | S. Fan, P. R. Villeneuve, and J. D. Joannopoulos, “Theoretical investigation of fabrication-related disorder on the properties of photonic crystals,” J. Appl. Phys. |

54. | L. O. Faolain, T. P. White, D. O. Brien, X. Yuan, M. D. Settle, and T. F. Krauss, “Dependence of extrinsic loss on group velocity in photonic crystal waveguides,” Opt. Express |

55. | K. C. Kwan, X. M. Tao, and G. D. Peng, “Transition of lasing modes in disordered active photonic crystals,” Opt. Lett. |

56. | J. P. Berenger, “A perfectly matched layer for the absorption of electromagnetic waves,” J. Comput. Phys. |

57. | A. Bermúdez, L. Hervella-Nieto, A. Prieto, and R. Rodríguez, “An exact bounded pml for the helmholtz equation,” C. R. Acad, Sci. Paris, Ser. I |

58. | A. Bermúdez, L. Hervella-Nieto, A. Prieto, and R. Rodríguez, “Numerical simulation of time-harmonic scattering problems with an optimal PML,” Var. Formul. Mech.:Theory Appl. |

59. | A. Bermúdez, L. Hervella-Nieto, A. Prieto, and R. Rodríguez, “An optimal perfectly matched layer with unbounded absorbing function for time-harmonic acoustic scattering problems,” J. Comput. Phys. |

60. | A. C. Cangellaris and D. B. Wright, “Analysis of the numerical error caused by the stair-stepped approximation of a conducting boundary in FDTD simulations of electromagnetic phenomena,” IEEE Trans. Antennas Propag. |

61. | A. Akyurtlu, D. H. Werner, V. Veremey, D. J. Steich, and K. Aydin, “Staircasing errors in FDTD at an air-dielectric interface,” IEEE Microwave Guided Wave Lett. |

62. | K. H. Dridi, J. S. Hesthaven, and A. Ditkowski, “Staircase-free finite-difference time-domain formulation for general materials in complex geometries,” IEEE Trans. Antennas Propag. |

**OCIS Codes**

(140.3460) Lasers and laser optics : Lasers

(160.3380) Materials : Laser materials

**ToC Category:**

Lasers and Laser Optics

**History**

Original Manuscript: December 5, 2011

Revised Manuscript: February 5, 2012

Manuscript Accepted: February 15, 2012

Published: March 15, 2012

**Citation**

Garuda Fujii, Toshiro Matsumoto, Toru Takahashi, and Tsuyoshi Ueta, "Study on transition from photonic-crystal laser to random laser," Opt. Express **20**, 7300-7315 (2012)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-20-7-7300

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