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Optics Express

Optics Express

  • Editor: C. Martijn de Sterke
  • Vol. 20, Iss. 7 — Mar. 26, 2012
  • pp: 7300–7315
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Study on transition from photonic-crystal laser to random laser

Garuda Fujii, Toshiro Matsumoto, Toru Takahashi, and Tsuyoshi Ueta  »View Author Affiliations


Optics Express, Vol. 20, Issue 7, pp. 7300-7315 (2012)
http://dx.doi.org/10.1364/OE.20.007300


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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

Both the lasing phenomena in random systems [1

1. N. M. Lawandy, R. M. Balachandra, A. S. L. Gomez, and E. Sauvain, “Laser action in strongly scattering media,” Nature 368, 436–438 (1994). [CrossRef]

14

14. S. Mujumdar, M. Ricci, R. Torre, and D. S. Wiersma, “Amplified extended modes in random lasers,” Phys. Rev. Lett. 93, 053903 (2004). [CrossRef] [PubMed]

] and those in photonic crystals [15

15. K. Ohtaka, “Energy band of photons and low-energy photon diffraction,” Phys. Rev. B 19, 5057–5067 (1979). [CrossRef]

19

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. 10, 012072 (2010). [CrossRef]

], have hopeful properties that the conventional laser devices cannot give. Photonic crystals are widely used in various optical devices for controlling light waves because of their noticeable abilities caused by their periodic structures. It is known that low-threshold laser action can be realiezed in photonic crystals [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

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]

] due to the extremely low group velocities at band edge frequencies. Random lasers occur from multiple scatterings and interference effects, causing Anderson localizations [23

23. P. W. Anderson, “Absence of diffusion in certain random lattices,” Phys. Rev. 109, 1492–1505 (1958). [CrossRef]

], in disordered structures, thus their laser modes take various and complex forms. Lawandy et al. mentioned that the lasing threshold of random lasers was extremely low [1

1. N. M. Lawandy, R. M. Balachandra, A. S. L. Gomez, and E. Sauvain, “Laser action in strongly scattering media,” Nature 368, 436–438 (1994). [CrossRef]

].

In the present study, we study the transition from photonic-crystal laser to random laser as a new research topic to realize lower threshold laser action. We simulate laser actions numerically to investigate the changes of lasing thresholds under the influence of disorders in two-dimensional dielectric structures. Disorder of dielectric structures is treated by giving positional disorders of dielectric cylinders. The amount of disorder is parameterized with the length between the grid points corresponding to the centers of fictitious cylinders distributed periodically and the centers of randomly distributed cylinders. We reveal the change of lasing threshold against the amount of disorder and find an appropriate one for lower-threshold laser action.

2. Analysis model

We show a model of a dielectric system in Fig. 1(a). Dielectric cylinders are assumed to be infinitely long in vertical direction (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 Cin and Cg, as shown in Fig. 1(b). An oscillating polarizaton is assumed to exist at the center of the circle Cin as a light source. Radii of Cin and Cg are denoted by Rin and Rg, respectively. We compute the fluxes of Poynting vectors of out-flowing light waves on the circle Cout whose radius is Rout. The unit outward normal vectors to Cin and Cout are denoted by nin and nout, respectively. We define three regions: Ωact, Ωcylinder and Ωout, where Ωact is in the interspace among the cylinders inside the circle Cout, Ωcylinder is the union of the regions inside the cylinders, and Ωout is the region outside the circle Cout. The optically active materials are assumed to be filled in the region Ωact.

Fig. 1 Concept of random media. (a) A random structure model. (b) An illustration of whole random system.

2.1. Model parameters

In Table 1 is shown the parameters used to create the analysis models. The radii of the cross sections of the cylinders are assumed to be the same, and the radius, 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 = 1.47735a and 3.47735a, respectively. The coordinate values of the cylinder’s center are specified in single precision numbers.

Table 1. Model parameters.

table-icon
View This Table

2.2. Disordered systems

A parameter used to control disorder of the cylinder arrangement is defined in Fig. 2. The circles drawn by broken lines illustrate fictitious cylinders of a periodic structure. The centers of these circles are used as the datum points to control the cylinders arranged in disorder. Centers of the circles are denoted by xp. A circle drawn by a solid line illustrates a cylinder arranged in disorder, whose center is denoted by xr. The disordered position xr is determined by a sum of xp and a random vector Δxr, as follows:
xr(n,m)=xp(n,m)+Δxr,
(1)
where n and m are latiice-point numbers and (n,m) (0,0). xp are defined as
xp(n,m)=nr1+mr2,
(2)
where r1 and r2 are lattice vectors defined as
r1=[3.47735a×3/2,3.47735a×1/2]T,
(3)
r2=[0.0,3.47735a]T,
(4)
where 3.47735a is the periodic length.

Fig. 2 Parameterization of disorder by Δxr.

We restrict the length of the random vector Δxr as
0|Δxr||Δxr|max
(5)
where |Δxr|max is the maximum length of the random vector. The amounts of disorders in dielectric structures are evaluated with |Δxr|max/a, that is, the maximum length normalized by the radii of cylinders. In the case |Δxr|max/a = 0.00, a dielectric structure becomes periodic. In Fig. 2, LH, 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 1.73867a, a half of the periodic length. Therefore, when LH < |Δxr|max, the center of the cylinder distributed randomly is located within the adjacent hexagonal lattice. When |Δxr|max is smaller than LHa = 0.73867a, the entire region of the cylinder is included in each lattices.

We analyze lasing phenomena in dielectric structures for various |Δxr|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.

Fig. 3 Analysis models. (a) |Δxr|max/a = 0.00. (b) |Δxr|max/a = 0.250 (c) |Δxr|max/a = 0.500 (d) |Δxr|max/a = 1.00 (e) |Δxr|max/a = 2.00. (f) |Δxr|max/a = 4.00.
Fig. 4 Radial distribution functions. (a) |Δxr|max/a = 0.00. (b) |Δxr|max/a = 0.250. (c) |Δxr|max/a = 0.500. (d) |Δxr|max/a = 1.00. (e) |Δxr|max/a = 2.00. (f) |Δxr|max/a = 4.00.

3. Formulation

In this study, we simulate lasing phenomena in dielectric structures consisting of homogeneous cylinders in the case of TM mode by using the node-base FEM. We use perfectly matched layers (PMLs) [56

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

] to simulate scatterings in the open region, and employ an optimized absorbing function [57

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

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]

] that minimizes numerical reflections.

3.1. Basic equations of electromagnetic scattering problem

We assume an electric polarization oscillating with angular frequency ωat the center of the entire region of the random system, x0, 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
×[×E(x)]ω2c2ε(x)E(x)=ω2ε0c2Ddδ(xx0),
(6)
where c is the speed of light in vacuum, ε0 and ε(x) are the permittivity in vacuum and the position-dependent relative permittivity, Dd is the polarization vector, δ(x) is Dirac’s delta function.

We define total electric field E(x) as the sum of the scattering and incident fields as follows:
E(x)=Es(x)+Ei(x),
(7)
where, Ei(x) is the electric field in the region without scatterers, satisfying the following equation:
×[×Ei(x)]ω2c2εiEi(x)=ω2ε0c2Ddδ(xx0),
(8)
where εi is the constant relative permittivity in Ωout (Fig. 1). We substitute Eqs. (7) and (8) to Eq. (6), to have
×[×Es(x)]ω2c2ε(x)Es(x)=ω2c2[ε(x)εi]Ei(x).
(9)
When we assume TM mode, the incident field Ei(x) satisfying Eq. (8) can be expressed by 0th-order Hankel function of the first kind, H0(1) as follows:
Ei(x)=ω2ε0c2Ddi4H0(1)(ωcεi|xx0|),
(10)
where i is the imaginary unit. We solve Eq. (9) by using FEM formulated based on Galerkin’s method and obtain the total electric field E(x) defined by Eq. (7).

3.2. Population inversion density of optically active materials

Population inversion density of optically active material can be modeled by a negative imaginary part of relative permittivity, −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

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

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]

]. We assume a system whose interspace among dielectric cylinders are filled with an optically active material, and set the imaginary part of relative permittivity in interspace among cylinders to −γ. The relative permittivities in individual regions (Fig. 1) are given as follows:
ε(x)={1.0+i(γ)xΩact4.0xΩcylinder1.0xΩout.
(11)

In the above modeling of active material, uniform gain media are assumed because the excitation by an external source is considered. We consider the population inversion as an independent parameter of electric field. The above parameterization of the population inversion may become well-approximated the model when the parameter corresponds to the population inversion in localization space in the case of spatially localized modes. In the case of spatially extended modes, the approximation becomes more appropriate because electric field is homogeneously extended. We can consider such a simple modeling of population inversion as the zeroth-order approximation.

3.3. Definition of amplification factor

Because of the assumption that electric and magnetic waves are time-harmonic, we need to compute Poynting vectors in the following time-averaged form:
S=Re(E×H*2)
(12)
where 〈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.

We define the amplification factor 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:
A=CoutSnoutdl|γ0CoutSnoutdl|γ=0.
(13)
The light flux is calculated by a line integral of the Poynting vector along the circle Cout.

3.4. Finite element analysis

In most numerical analyses of random lasers, lasing phenomena are simulated using finite-difference time-domain (FDTD) method. FDTD is effective for numerical analyses of electromagnetic (EM) waves in homogeneous optical media. In the analyses of EM waves in heterogeneous ones, FDTD simulations suffer from staircasing errors [60

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

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]

] due to spatial discretizations with square grids which are unfitted to curved interfaces between different optical materials. On the other hand, FEM, one of the most effective numerical methods for the analyses of EM waves, can model arbitrary shaped fields and create their mesh very easily. Therefore, staircasing errors may not arise if FEM is used in the analyses of random lasers because the FEM meshes are well-fitted to the interfaces.

Figure 5 illustrates the finite elements of an analysis model. A blue circular area in Fig. 5(a) shows a cross section of a dielectric cylinder. The edge size of the elements is approximately given as 0.08a. 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 Cout into more smaller elements 0.01a. 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.

Fig. 5 Finite elements. (a) Mesh in cylinder. (b) Mesh in PML.

4. Results

4.1. Lasing frequency

Figure 6 shows the result of laser action in the periodic structure corresponding to |Δxr|max/a = 0.00. We compute the amplification factor for the frequency range 0.1 ωa/2πc ≤ 0.3 with fixed population inversion density γ = 0.002 to investigate lasing frequencies of the periodic structure corresponding to |Δxr|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.

Fig. 6 Amplification factor versus normalized frequency ωa/2πc for γ = 0.002.
Fig. 7 Dispersion relation of light waves in the periodic structure corresponding to |Δxr|max/a = 0.00.

We observe laser action in the frequency range colored blue in Fig. 6, 0.222 ωa/2πc ≤ 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 = (3π/4a, 3π/4a), of each band in Fig. 7. Figure 8(a) shows the unit cell of the periodic structure. We find in the distribution of the fourth-lowest band shown in Fig. 8(e), the electric field intensity in interspace among dielectric cylinders becomes high, i.e., the electric field becomes intensive within an optically active material. The laser action in Fig. 6 is thought to be caused by such light localization in an optically active material.

Fig. 8 Electric intensity distributions corresponding to the wave vector, k = (3π/4a, 3π/4a) in the unit cell of the periodic structure. (a) The unit cell of the periodic structure. (b) Lowest frequency band. (c) 2nd lowest frequency band. (d) 3rd lowest frequency band. (e) 4th lowest frequency band. (f) 5th lowest frequency band.

4.2. Laser action

We compute the amplification factors A defined by Eq. (13) for the ranges of 0.222 ωa/2πc ≤ 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 |Δxr|max/a normalized by a. We calculate the amplification factors for 127,041 grid points for ωa/2πc and γ, by dividing ωa/2πc and γ directions uniformly into 900 an 140 intervals, respectively, and seek the steep peaks of the surface of amplification factor A.

Fig. 9 Amplification factor A versus normalized frequency ωa/2πc and γ for each |Δxr|max/a. (a) |Δxr|max/a = 0.00 (periodic structure). (b) |Δxr|max/a = 0.0625. (c) |Δxr|max/a = 0.125. (d) |Δxr|max/a = 0.250. (e) |Δxr|max/a = 0.500. (f) |Δxr|max/a = 1.00. (g) |Δxr|max/a = 2.00. (h) |Δxr|max/a = 4.00. (i) A peak at the bottom edge frequency ωa/2πc = 0.22196531 of the 4th band for |Δxr|max/a = 0.00 (periodic structure).

Figure 9(a) shows the distribution of the amplification factor corresponding to the combinations of ωa/2πc and γ. From this result, we find specific combinations of the values of ωa/2πc 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 104.

In Figs. 9(b) to 9(h), we observe that surface of amplification factor reflects the effect of disorder on laser action. The lasing phenomena caused by random light scatterings noticeably emerge as the disorder of the structure increases. Such lasing phenomena are found more in higher frequency range 0.23 ωa/2πc 0.24, in the results for small amounts of disorder, |Δxr|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 ≤ |Δxr|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

Parameter 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 |Δxr|max, we investigate the smallest value of γ at which laser action occurs. To seek the more precise values of ωa/2πc and γ at which the lowest-threshold laser action occurs, we scan the neighborhood of the amplification peak using finer numerical steps.

Average values of the lowest thresholds for different disorder index values are plotted in Fig. 10 with error bars showing the ranges of γ. We analyze 10 different types of cylinder arrangement for each disorder index other than |Δxr|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.

Fig. 10 Relation between normalized disorder index |Δxr|max/a and the average value of lasing thresholds, namely, lowest γ with error bar.

The threshold of the tightly confined mode, plotted by red squares and line in Fig. 10, rises as the disorder index increases. Such laser modes finally disappear when 0.500 ≤ |Δxr|max/a.

Fig. 11 Electric amplitude distributions of lasing states with tight confinement (sample 1) (a) |Δxr|max/a = 0.00, ωa/2πc = 0.22196531, γ = 0.0004965. (b) |Δxr|max/a = 0.0625, ωa/2πc = 0.2221812, γ = 0.0007350. (c) |Δxr|max/a = 0.125, ωa/2πc = 0.2223220, γ = 0.001140. (d) |Δxr|max/a = 0.250, ωa/2πc = 0.2222716, γ = 0.002490.
Fig. 12 Polar plots of the group velocity of tight-binding model. (a) ωa/2πc = 0.3650. (b) ωa/2πc = 0.3675.

We also show the lowest threshold lasing phenomena with spacial extension of light wave in Fig. 10. Blue triangles and line in the figures show the dependence of the threshold of the lasing phenomena with spatial extension on the disorder index. As this index increases, the threshold rises in |Δxr|max/a < 0.250, then decreases in 0.250 < |Δxr|max/a < 1.00. In Fig. 10, the average threshold of 10 different cylinders arrangements rises again in 1.00 < |Δxr|max/a. The average threshold becomes minimum at |Δxr|max/a = 1.00.

We show in Fig. 13 the EADs of the spatially extended modes. We observe in Figs. 13(a), 13(b), and 13(c) the collapse of the localization forms from that of the periodic structure as the disorder index increases. Lasing threshold of the laser action with spatial extension is found to decrease in 0.250 < |Δxr|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.

Fig. 13 Electric amplitude distributions of lasing states with spatial extension (sample 1). (a) |Δxr|max/a = 0.00, ωa/2πc = 0.2310212, γ = 0.001380. (b) |Δxr|max/a = 0.125, ωa/2πc = 0.2308372, γ = 0.001470. (c) |Δxr|max/a = 0.250, ωa/2πc = 0.2300848, γ = 0.001995. (d) |Δxr|max/a = 0.500, ωa/2πc = 0.2297800, γ = 0.001785. (e) |Δxr|max/a = 0.750, ωa/2πc = 0.2280060, γ = 0.001515. (f) |Δxr|max/a = 1.00, ωa/2πc = 0.2238840, γ = 0.001080. (g) |Δxr|max/a = 1.25, ωa/2πc = 0.2276240, γ = 0.001575. (h) |Δxr|max/a = 4.00, ωa/2πc = 0.2321304, γ = 0.002430.

We observe the average of lasing threshold becomes locally minimum at |Δxr|max/a = 1.00, and another increase of the lasing threshold for 1.00 < |Δxr|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:
Ωact+Ωcylinderε(x)|E(x)|2dΩmax[ε(x)|E(x)|2]λ2,
(14)
where λ = 2πc/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 ≤ |Δxr|max/a ≤ 1.5 become smaller than those in 2 ≤ |Δxr|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.

Fig. 14 Mode volume versus normalized disorder index |Δxr|max/a.

When shifting the positions of the cylinders using the random numbers, there is a possibility that the shifted cylinder may overlaps for 0.73867 ≤ |Δxr|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.

Figure 15 shows the averages of |Δxr||Δxr|max of all samples analyzed in the present study. The average becomes 0.5 theoretically if all the generated random numbers are used to rearrange the cylinders because the distributions of |Δxr| are given by random numbers of a uniform distribution between 0 and 1. In the range 0.73867 ≤ |Δxr|max/a ≤ 2.00, the effect of discard of some random numbers are observed clearly. The average values tend to decrease as |Δxr|max increases.

Fig. 15 Average values of |Δxr||Δxr|max for each |Δxr|max.

The serious influence occurs when the average of |Δxr| corresponds to 0.73867a. Considering the average of the random number m = 0.5, its dispersion ρ = 1/12, and the standard deviation σ=1/12, we find that the influence of discarding some of the random numbers occurs for
(mσ)|Δxr|max<0.73867a<(m+σ)|Δxr|max,
(15)
resulting in
0.93660<|Δxr|max/a<3.49544.
(16)
This range agrees well with the results shown in Fig. 15. Hence, the local minimum of the lasing threshold observed in Fig. 10 may be the result of the employment of non-uniform random numbers to avoid the overlapping to the cylinders when shifting them from the hexagonal grids.

We investigate lasing threshold in random samples created by another positioning algorithm. In order to determine the center of the cylinder, we used the coordinate of the center (xr, yr) directly instead of using (|Δxr|,θ). Again, random numbers of uniform distribution are used to determine the values of (xr, yr) so that the generated cylinder is located with the circular are bounded by the certain radius |Δxr|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.

Fig. 16 Lasing thresholds versus normalized disorder index |Δxr|max/a (5 samples with another positioning algorithm).

5. Conclusion

Investigations of lasing phenomena and lasing thresholds of disordered structures are presented based on FE analyses by changing the disorder index. The threshold of the laser action in the periodic structure is extremely low because of the zero-group velocity at the band edge frequency. However, the lasing phenomena are sensitive to the degrees of disorder, and the threshold of the photonic crystal laser rises by the effect of a small amount of disorder. As the disorder increases, lasing phenomena shift from photonic crystal lasers to random ones, and the threshold of laser action once rises, then it decreases. However, a further increase in the disorder causes the rise of the lasing threshold again. Lasing threshold of random lasing tends to become locally minimum.

Acknowledgments

G. Fujii acknowledges the funding and support by Global COE Program of Nagoya University.

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P. Sebbah and C. Vanneste, “Random laser in the localized regime,” Phys. Rev. B 66, 144202 (2002). [CrossRef]

6.

P. Sebbah, B. Hu, J. K. Klosner, and A. Z. Genack, “Extended quasimodes within nominally localized random waveguides,” Phys. Rev. Lett. 96, 183902 (2006). [CrossRef] [PubMed]

7.

C. Vanneste and P. Sebbah, “Selective excitation of localized modes in active random media,” Phys. Rev. Lett. 87, 183903 (2001). [CrossRef]

8.

C. Vanneste and P. Sebbah, “Localized modes in random arrays of cylinders,” Phys. Rev. E 71, 026612 (2005). [CrossRef]

9.

C. Vanneste, P. Sebbah, and H. Cao, “Lasing with resonant feedback in weakly scattering random systems,” Phys. Rev. Lett. 98, 143902 (2007). [CrossRef] [PubMed]

10.

C. Vanneste and P. Sebbah, “Complexity of two-dimensional quasimodes at the transition from weak scattering to anderson localization,” Phys. Rev. A 79, 041802 (2009). [CrossRef]

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. 82, 2278–2281 (1999). [CrossRef]

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. 84, 5584–5587 (2000). [CrossRef] [PubMed]

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 61, 1985–1989 (2000). [CrossRef]

14.

S. Mujumdar, M. Ricci, R. Torre, and D. S. Wiersma, “Amplified extended modes in random lasers,” Phys. Rev. Lett. 93, 053903 (2004). [CrossRef] [PubMed]

15.

K. Ohtaka, “Energy band of photons and low-energy photon diffraction,” Phys. Rev. B 19, 5057–5067 (1979). [CrossRef]

16.

E. Yablonovitch, “Inhibited spontaneous emission in solid-state physics and electronics,” Phys. Rev. Lett. 58, 2059–2062 (1987). [CrossRef] [PubMed]

17.

E. Yablonovitch and T. J. Gmitter, “Photonic band structure: the face-centered-cubic case,” Phys. Rev. Lett. 63, 1950–1953 (1989). [CrossRef] [PubMed]

18.

E. Yablonovitch, “Photonic band-gap structures,” J. Opt. Soc. Am. B 10, 283–295 (1993). [CrossRef]

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. 10, 012072 (2010). [CrossRef]

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]

21.

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

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]

23.

P. W. Anderson, “Absence of diffusion in certain random lattices,” Phys. Rev. 109, 1492–1505 (1958). [CrossRef]

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. 10, 117–122 (2010).

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. 30, 3192–3194 (2005). [CrossRef] [PubMed]

27.

Z.-Y. Li and Z.-Q. Zhang, “Fragility of photonic band gaps in inverse-opal photonic crystals,” Phys. Rev. B 62, 1516–1519 (2000). [CrossRef]

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 61, 13458–13464 (2000). [CrossRef]

29.

T. Schwartz, G. Bartal, S. Fishman, and M. Segev, “Transport and Anderson localization in disordered two-dimensional photonic lattices,” Nature 446, 52–55 (2007). [CrossRef] [PubMed]

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. 100, 013906 (2008). [CrossRef] [PubMed]

31.

H. Li, B. Cheng, and D. Zhang, “Two-dimensional disordered photonic crystals with an average periodic lattice,” Phys. Rev. B 56, 10734–10736 (1997). [CrossRef]

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 66, 113101 (2002). [CrossRef]

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 15, 16954–16965 (2007). [CrossRef] [PubMed]

34.

Y. A. Vlasov, M. A. Kaliteevski, and V. V. Nikolaev, “Different regimes of light localization in disordered photonic crystal,” Phys. Rev. B 60, 1555–1562 (1999). [CrossRef]

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. 102, 253903 (2009). [CrossRef] [PubMed]

36.

R. Ferrini, D. Leuenberger, R. Houdré, H. Benisty, M. Kamp, and A. Forchel, “Disorder-induced losses in planar photonic crystals,” Opt. Lett. 31, 1426–1428 (2006). [CrossRef] [PubMed]

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 59, 12767–12770 (1999). [CrossRef]

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 62, 5711–5720 (2000). [CrossRef]

39.

W. R. Frei and H. T. Johnson, “Finite-element analysis of disorder effects in photonic crystals,” Phys. Rev. B 70, 165116 (2004). [CrossRef]

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 60, 6118–6127 (1999). [CrossRef]

41.

H. Li, H. Chen, and X. Qiu, “Band-gap extension of disordered 1d binary photonic crystals,” Physica B 279, 164–167 (2000). [CrossRef]

42.

X. Wang and K. Kempa, “Effects of disorder on subwavelength lensing in two-dimensional photonic crystal slabs,” Phys. Rev. B 71, 085101 (2005). [CrossRef]

43.

T. N. Langtry, A. A. Asatryan, and L. C. Botten, “Effects of disorder in two-dimensional photonic crystal waveguides,” Phys. Rev. E 68, 026611 (2003). [CrossRef]

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. 85, 5508–5510 (2004). [CrossRef]

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. 82, 4414–4416 (2003). [CrossRef]

46.

D. P. Fuell, S. Hughes, and M. M. Dignam, “Effect of disorder strength on the fracture pattern in heterogeneous networks,” Phys. Rev. B 76, 144201 (2008).

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. 3, 120–128 (2005). [CrossRef]

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. 85, 2454–2456 (1999).

49.

J. Topolancik and F. Vollmer, “Random high-q cavities in disordered photonic crystal waveguides,” Appl. Phys. Lett. 91, 201102 (2007). [CrossRef]

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. 73, 1943–1945 (1998). [CrossRef]

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. 81, 798–800 (2002). [CrossRef]

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. 29, 2917–2919 (2004). [CrossRef]

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. 78, 1415–1418 (1995). [CrossRef]

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 15, 13129–13138 (2007). [CrossRef]

55.

K. C. Kwan, X. M. Tao, and G. D. Peng, “Transition of lasing modes in disordered active photonic crystals,” Opt. Lett. 32, 2720–2722 (2007). [CrossRef] [PubMed]

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]

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. 13, 58–71 (2006).

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]

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]

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. 9, 444–446 (1999). [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]

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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References

  1. N. M. Lawandy, R. M. Balachandra, A. S. L. Gomez, E. Sauvain, “Laser action in strongly scattering media,” Nature 368, 436–438 (1994). [CrossRef]
  2. D. S. Wiersma, M. P. van Albada, A. Lagendijk, “Random laser ?” Nature 373, 203–204 (1995). [CrossRef]
  3. D. S. Wiersma, “The physics and applications of random lasers,” Nat. Phys. 4, 359–367 (2008). [CrossRef]
  4. P. Sebbah, R. Pnini, A. Z. Genack, “Field and intensity correlation in random media,” Phys. Rev. E 62, 7348–7352 (2000). [CrossRef]
  5. P. Sebbah, C. Vanneste, “Random laser in the localized regime,” Phys. Rev. B 66, 144202 (2002). [CrossRef]
  6. P. Sebbah, B. Hu, J. K. Klosner, A. Z. Genack, “Extended quasimodes within nominally localized random waveguides,” Phys. Rev. Lett. 96, 183902 (2006). [CrossRef] [PubMed]
  7. C. Vanneste, P. Sebbah, “Selective excitation of localized modes in active random media,” Phys. Rev. Lett. 87, 183903 (2001). [CrossRef]
  8. C. Vanneste, P. Sebbah, “Localized modes in random arrays of cylinders,” Phys. Rev. E 71, 026612 (2005). [CrossRef]
  9. C. Vanneste, P. Sebbah, H. Cao, “Lasing with resonant feedback in weakly scattering random systems,” Phys. Rev. Lett. 98, 143902 (2007). [CrossRef] [PubMed]
  10. C. Vanneste, P. Sebbah, “Complexity of two-dimensional quasimodes at the transition from weak scattering to anderson localization,” Phys. Rev. A 79, 041802 (2009). [CrossRef]
  11. H. Cao, Y. G. Zhao, S. T. Ho, E. W. Seelig, Q. H. Wang, R. P. H. Chang, “Random laser action in semiconductor powder,” Phys. Rev. Lett. 82, 2278–2281 (1999). [CrossRef]
  12. H. Cao, J. Y. Xu, D. Z. Zhang, S. H. Chang, S. T. Ho, E. W. Seeling, X. Liu, R. P. H. Chang, “Spatial confinement of laser light in active random media,” Phys. Rev. Lett. 84, 5584–5587 (2000). [CrossRef] [PubMed]
  13. H. Cao, J. Y. Xu, S. H. Chang, S. T. Ho, “Transition from amplified spontaneous emission to laser action in strongly scattering media,” Phys. Rev. E 61, 1985–1989 (2000). [CrossRef]
  14. S. Mujumdar, M. Ricci, R. Torre, D. S. Wiersma, “Amplified extended modes in random lasers,” Phys. Rev. Lett. 93, 053903 (2004). [CrossRef] [PubMed]
  15. K. Ohtaka, “Energy band of photons and low-energy photon diffraction,” Phys. Rev. B 19, 5057–5067 (1979). [CrossRef]
  16. E. Yablonovitch, “Inhibited spontaneous emission in solid-state physics and electronics,” Phys. Rev. Lett. 58, 2059–2062 (1987). [CrossRef] [PubMed]
  17. E. Yablonovitch, T. J. Gmitter, “Photonic band structure: the face-centered-cubic case,” Phys. Rev. Lett. 63, 1950–1953 (1989). [CrossRef] [PubMed]
  18. E. Yablonovitch, “Photonic band-gap structures,” J. Opt. Soc. Am. B 10, 283–295 (1993). [CrossRef]
  19. G. Fujii, T. Matsumoto, T. Takahashi, T. Ueta, “A study on optical properties of photonic crystals consisting of hollow rods,” IOP Conf. Ser.: Mater. Sci. Eng. 10, 012072 (2010). [CrossRef]
  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]
  21. K. Sakoda, “Enhanced light amplification due to group-velocity anomaly peculiar to two- and three-dimensional photonic crystals,” Opt. Express 4, 167–176 (1999). [CrossRef] [PubMed]
  22. K. Sakoda, K. Ohtaka, 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]
  23. P. W. Anderson, “Absence of diffusion in certain random lattices,” Phys. Rev. 109, 1492–1505 (1958). [CrossRef]
  24. G. Fujii, T. Matsumoto, T. Takahashi, T. Ueta, “Finite element analysis for laser oscillation in random system consisting of heterogeneous dielectric materials,” Trans. Jpn. Soc. Comput. Methods Eng. 10, 117–122 (2010).
  25. G. Fujii, T. Matsumoto, T. Takahashi, 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, S. G. Johnson, “Disorder-immune confinement of light in photonic-crystal cavities,” Opt. Lett. 30, 3192–3194 (2005). [CrossRef] [PubMed]
  27. Z.-Y. Li, Z.-Q. Zhang, “Fragility of photonic band gaps in inverse-opal photonic crystals,” Phys. Rev. B 62, 1516–1519 (2000). [CrossRef]
  28. E. Lidorikis, M. M. Sigalas, E. N. Economou, C. M. Soukoulis, “Gap deformation and classical wave localization in disordered two-dimensional photonic-band-gap materials,” Phys. Rev. B 61, 13458–13464 (2000). [CrossRef]
  29. T. Schwartz, G. Bartal, S. Fishman, M. Segev, “Transport and Anderson localization in disordered two-dimensional photonic lattices,” Nature 446, 52–55 (2007). [CrossRef] [PubMed]
  30. Y. Lahini, A. Avidan, F. Pozzi, M. Sorel, R. Morandotti, D. N. Christodoulides, Y. Silberberg, “Anderson localization and nonlinearity in one-dimensional disordered photonic lattices,” Phys. Rev. Lett. 100, 013906 (2008). [CrossRef] [PubMed]
  31. H. Li, B. Cheng, D. Zhang, “Two-dimensional disordered photonic crystals with an average periodic lattice,” Phys. Rev. B 56, 10734–10736 (1997). [CrossRef]
  32. M. A. Kaliteevski, J. M. Martinez, D. Cassagne, J. P. Albert, “Disorder-induced modification of the transmission of light in a two-dimensional photonic crystal,” Phys. Rev. B 66, 113101 (2002). [CrossRef]
  33. T. Prasad, V. L. Colvin, D. M. Mittleman, “The effect of structural disorder on guided resonances in photonic crystal slabs studied with terahertz time-domain spectroscopy,” Opt. Express 15, 16954–16965 (2007). [CrossRef] [PubMed]
  34. Y. A. Vlasov, M. A. Kaliteevski, V. V. Nikolaev, “Different regimes of light localization in disordered photonic crystal,” Phys. Rev. B 60, 1555–1562 (1999). [CrossRef]
  35. M. Patterson, S. Hughes, S. Combrié, N.-V.-Q. Tran, A. D. Rossi, R. Gabet, Y. Jaouën, “Disorder-induced coherent scattering in slow-light photonic crystal waveguides,” Phys. Rev. Lett. 102, 253903 (2009). [CrossRef] [PubMed]
  36. R. Ferrini, D. Leuenberger, R. Houdré, H. Benisty, M. Kamp, A. Forchel, “Disorder-induced losses in planar photonic crystals,” Opt. Lett. 31, 1426–1428 (2006). [CrossRef] [PubMed]
  37. M. M. Sigalas, C. M. Soukoulis, C. T. Chan, R. Biswas, K. M. Ho, “Effect of disorder on photonic band gaps,” Phys. Rev. B 59, 12767–12770 (1999). [CrossRef]
  38. A. A. Asatryan, P. A. Robinson, L. C. Botten, R. C. McPhedran, N. A. Nicorovici, C. M. de Sterke, “Effects of geometric and refractive index disorder on wave propagation in two-dimensional photonic crystals,” Phys. Rev. E 62, 5711–5720 (2000). [CrossRef]
  39. W. R. Frei, H. T. Johnson, “Finite-element analysis of disorder effects in photonic crystals,” Phys. Rev. B 70, 165116 (2004). [CrossRef]
  40. A. A. Asatryan, P. A. Robinson, L. C. Botten, R. C. McPhedran, N. A. Nicorovici, C. M. de Sterke, “Effects of disorder on wave propagation in two-dimensional photonic crystals,” Phys. Rev. E 60, 6118–6127 (1999). [CrossRef]
  41. H. Li, H. Chen, X. Qiu, “Band-gap extension of disordered 1d binary photonic crystals,” Physica B 279, 164–167 (2000). [CrossRef]
  42. X. Wang, K. Kempa, “Effects of disorder on subwavelength lensing in two-dimensional photonic crystal slabs,” Phys. Rev. B 71, 085101 (2005). [CrossRef]
  43. T. N. Langtry, A. A. Asatryan, L. C. Botten, “Effects of disorder in two-dimensional photonic crystal waveguides,” Phys. Rev. E 68, 026611 (2003). [CrossRef]
  44. V. N. Astratov, J. P. Franchak, S. P. Ashili, “Optical coupling and transport phenomena in chains of spherical dielectric microresonators with size disorder,” Appl. Phys. Lett. 85, 5508–5510 (2004). [CrossRef]
  45. K. C. Kwan, X. Zhang, Z. Q. Zhang, C. T. Chan, “Effects due to disorder on photonic crystal-based waveguides,” Appl. Phys. Lett. 82, 4414–4416 (2003). [CrossRef]
  46. D. P. Fuell, S. Hughes, M. M. Dignam, “Effect of disorder strength on the fracture pattern in heterogeneous networks,” Phys. Rev. B 76, 144201 (2008).
  47. D. Gerace, L. C. Andreani, “Effects of disorder on propagation losses and cavity q-factors in photonic crystal slabs,” Photon. Nanostruct. Fundam. Appl. 3, 120–128 (2005). [CrossRef]
  48. A. Golshani, H. Pier, E. Kapon, M. Moser, “Photon mode localization in disordered arrays of vertical cavity surface emitting lasers,” Appl. Phys. Lett. 85, 2454–2456 (1999).
  49. J. Topolancik, F. Vollmer, “Random high-q cavities in disordered photonic crystal waveguides,” Appl. Phys. Lett. 91, 201102 (2007). [CrossRef]
  50. T. A. Leskova, A. A. Maradudin, I. V. Novikov, A. V. Schegrov, E. R. Méndez, “Design of one-dimensional band-limited uniform diffusers of light,” Appl. Phys. Lett. 73, 1943–1945 (1998). [CrossRef]
  51. E. R. Méndez, E. E. García, T. A. Leskova, A. A. Maradudin, J. Muñoz-Lopez, I. Simonsen, “Design of one-dimensional random surfaces with specified scattering properties,” Appl. Phys. Lett. 81, 798–800 (2002). [CrossRef]
  52. E. R. Méndez, T. A. Leskova, A. A. Maradudin, J. Muñoz-Lopez, “Design of two-dimensional random surfaces with specified scattering properties,” Opt. Lett. 29, 2917–2919 (2004). [CrossRef]
  53. S. Fan, P. R. Villeneuve, J. D. Joannopoulos, “Theoretical investigation of fabrication-related disorder on the properties of photonic crystals,” J. Appl. Phys. 78, 1415–1418 (1995). [CrossRef]
  54. L. O. Faolain, T. P. White, D. O. Brien, X. Yuan, M. D. Settle, T. F. Krauss, “Dependence of extrinsic loss on group velocity in photonic crystal waveguides,” Opt. Express 15, 13129–13138 (2007). [CrossRef]
  55. K. C. Kwan, X. M. Tao, G. D. Peng, “Transition of lasing modes in disordered active photonic crystals,” Opt. Lett. 32, 2720–2722 (2007). [CrossRef] [PubMed]
  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, R. Rodríguez, “An exact bounded pml for the helmholtz equation,” C. R. Acad, Sci. Paris, Ser. I 339, 803–808 (2004). [CrossRef]
  58. A. Bermúdez, L. Hervella-Nieto, A. Prieto, R. Rodríguez, “Numerical simulation of time-harmonic scattering problems with an optimal PML,” Var. Formul. Mech.:Theory Appl. 13, 58–71 (2006).
  59. A. Bermúdez, L. Hervella-Nieto, A. Prieto, 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]
  60. A. C. Cangellaris, 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]
  61. A. Akyurtlu, D. H. Werner, V. Veremey, D. J. Steich, K. Aydin, “Staircasing errors in FDTD at an air-dielectric interface,” IEEE Microwave Guided Wave Lett. 9, 444–446 (1999). [CrossRef]
  62. K. H. Dridi, J. S. Hesthaven, A. Ditkowski, “Staircase-free finite-difference time-domain formulation for general materials in complex geometries,” IEEE Trans. Antennas Propag. 49, 749–756 (2001). [CrossRef]

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