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.
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Both the lasing phenomena in random systems [1–14] and those in photonic crystals [15–19], 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 [20–22] due to the extremely low group velocities at band edge frequencies. Random lasers occur from multiple scatterings and interference effects, causing Anderson localizations , 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 .
In our previous study [24, 25], we discussed lasing threshold of random lasers. The effects of positions of active medium  and filling factor of dielectric cylinders  on the lasing threshold were investigated and appropriate dielectric structures for lower threshold laser action were proposed.
Photonic crystals including small amounts of disorders have been investigated actively. There are many numerical and experimental studies on the effects of disorders on the optical properties of photonic crystals, such as band structures [26–28], light localizations [26, 29, 30], transmissions [28, 31–40], and reflectance . These previous studies treated various types of disorders, for example, in size [32, 38, 40, 42–44], positions [5, 7–9, 33, 36, 39, 42, 45–47], shapes of dielectric material [26, 48, 49], refractive indices [34, 38, 43], surface roughness [33, 36, 46, 47, 50–52], and fabrication errors [53, 54]. Additionally, we can also find a paper showing the effects of disorder on lasing phenomena in dielectric structures. Kwan, et al.  studied the effect of position and size-disorder on the emission spectra, and the lasing frequency at the highest intensity emission with a fixed pumping rate. However, no studies are found on the influence of disorder on lasing threshold.
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.
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.
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:
We restrict the length of the random vector Δxr asFig. 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 LH − a = 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.
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)  to simulate scatterings in the open region, and employ an optimized absorbing function [57–59] 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
We define total electric field E(x) as the sum of the scattering and incident fields as follows:Fig. 1). We substitute Eqs. (7) and (8) to Eq. (6), to have Eq. (8) can be expressed by 0th-order Hankel function of the first kind, as follows: 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) [20–22]. 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 [20–22]. 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:
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:
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:
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 [60–62] 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.
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.
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, ), 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.
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.
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.
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.
We show next the electric amplitude distributions (EADs) of tightly confined laser mode in Fig. 11. The angular distributions of group velocities of light waves in a periodic array are computed with tight-binding approximation and are shown in Fig. 12. From the comparison between the results shown in Figs. 11 and 12, we confirm that such tightly confined modes occur from the periodic array structure of dielectric cylinders. We observe that light wave confinement becomes weak as the disorder index increases, particularly in Fig. 11(d). Such leakage of light is considered to lead to the rise in the lasing threshold of the laser action with the tight confinement of light wave.
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.
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: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.
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 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.
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 , we find that the influence of discarding some of the random numbers occurs forFig. 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.
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.
G. Fujii acknowledges the funding and support by Global COE Program of Nagoya University.
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