Figure 1

(Color online) (a) Unit cell of the square lattice of silicon (Si, ${ϵ}_a=12.1$) cylindrical rods (radius $r=0.3a$) in glass matrix $(\mathrm{Si}{\mathrm{O}}_2,{ϵ}_b=2.1)$. Dots (red online) represent the uniform distribution (random) of the dopant atoms within the glass component. (b) Unit cell of the square lattice of air cylindrical pores (radius $r=0.45a$, ${ϵ}_a=1.0$) etched in silicon matrix (Si, ${ϵ}_b=12.1$). Light (red online) regions represent the coating of the silicon matrix with close packed colloidal quantum dots. The thickness of the coated shell is $0.04a$. The average, frequency-independent dielectric constant of the coating is ${ϵ}_c=6.0$. (c) The same as (b) but now the coating has a thickness $0.12a$.Reuse & Permissions

Figure 2

(Color online) (a) Photonic band structure of the square lattice of silicon cylindrical rods $(r=0.3a,{ϵ}_a=12.1)$ in a glass matrix $({ϵ}_b=2.1)$. (b) The structure of (a) but now the glass matrix is modified by the addition of a susceptibility $4\pi \overline{\chi }\left(\vec{r}\right)={\widetilde{ϵ}}_b-{ϵ}_b$, with ${\widetilde{ϵ}}_b=3.3$. Solid lines represent an exact calculation of the new structure using the PWE method with 3000 plane waves. The dots represent the calculation using our method. 156 first Bloch modes of the BPC are considered coupled from the perturbation.Reuse & Permissions

Figure 3

(Color online) 156 first Bloch modes of the BPC [see Fig. 2a] are coupled from the perturbation [read Fig. 2b caption]. The coupling coefficients are $h_{\vec{\mathrm{q}},l}\equiv {\omega }_{\vec{\mathrm{q}},l}∕c{f}_{\vec{\mathrm{q}},l}$. Only the first 20 of the coupling coefficients $\left\{h_{\vec{\mathrm{q}},l}\right\}$ for $\vec{\mathrm{q}}=X$ point and the second photonic band of the doped BPC are shown.Reuse & Permissions

Figure 4

(Color online) (a) Glass matrix with dielectric constant ${ϵ}_b=2.1$ is doped with atoms contributing a real, frequency-dependent susceptibility $\overline{\chi }\left({\omega }_s\right)$ given by Eq. (17a) with $g_0=0.17,{\omega }_{0s}=0.36,\tau =15$ (solid blue line). The dielectric constant of the doped regions is positive over all frequencies and its maximum value is within the range where a number of 156 coupled Bloch modes of the BPC gave an excellent result as demonstrated in Sec. III A. Two additional lines represent the frequencies of the first band for the $M$ point (dash-dotted black line) and the second band for the $X$ point (dashed red line), respectively, as functions of the dielectric constant of the glass regions. The intersection of the frequency-dependent dielectric function curve with the bands (pointed to by arrows) determines the frequencies of the new bands at the $M$ point and the $X$ point. (b) and (c) show the convergence of an iterative method for two particular cases of the positioning of the center frequency, ${\omega }_{0s}$, and different linewidths $\tau$ of the frequency-dependent susceptibility of the dopant atoms with regard to the band structure of the BPC. Clearly, the convergence trajectory depends sensitively on the slope of the susceptibility (solid blue line) relative to the slope of the band (dashed red line).Reuse & Permissions

Figure 5

(Color online) Comparison of the results obtained with our self-consistent, iterative method and these obtained by the CSM are shown on the graph (b). Graph (a) represents the photonic band structure of the BPC. The enhancement of the fundamental band gap is apparent.Reuse & Permissions

Figure 6

(Color online) (a) Schematic graph of the imaginary part of the susceptibility of the pumped erbium ions. The frequency of the band edge [drawn in (b)] is detuned by $\Delta =\delta {\omega }_{0s}$, where $\delta {\omega }_{0s}$ is the FWHM of the resonance curve, from the resonance frequency ${\omega }_{0s}=0.266323$ $({\lambda }_0=1535\mathrm{nm})$ of the ${\mathrm{Er}}^{3+}$ ions. (b) The response of the pumped erbium ions is drawn on top of the photonic band structure of the square lattice of silicon cylindrical rods $(r=0.3a,{ϵ}_a=12.1)$ in glass matrix $({ϵ}_b=2.1)$. The ions absorb the pump power resonantly at ${\omega }_{ps}=0.276364$ $({\lambda }_p=1480\mathrm{nm})$. The frequency of ${}^{4}I_{13∕2}\to {}^{4}I_{15∕2}$ transition is situated slightly $(\Delta =\delta {\omega }_{0s})$ within the first band gap by choosing $a=409\mathrm{nm}$. The frequency of ${}^{4}I_{11∕2}\to {}^{4}I_{15∕2}$ transition is well within the second band gap, and may be an effective factor in suppressing the excitation lost by upconversion for the excited dopant atoms at high field amplitudes.Reuse & Permissions

Figure 7

(Color online) Photonic crystal of Fig. 1a, with lossy silicon rods $(4\pi {\overline{\chi }}_{\mathit{\text{loss}}}=i{10}^{-6})$ and dopant erbium ions under pumping $\rho$. Shown are (a) the number of photons per unit cell, $n_{\mathit{ph}}$, corresponding to the self-consistent field of the second band at the $X$ point, (b) average inversion $\overline{w}$ of the dopant atoms, (c) the real part of the frequency of the second band at the $X$ point, and (d) the imaginary part of the frequency of the second band at the $X$ point—as functions of the pumping parameter $\rho$.Reuse & Permissions

Figure 8

Threshold pump ${\rho }_{\mathit{th}}$ increases as the resonance frequency of the erbium ions is moved deeper into the band gap. The detuning $\Delta$ of the resonance frequency of the erbium ions from the band edge is measured in terms of the FWHM, $\delta {\omega }_{0s}$, of that resonance.Reuse & Permissions

Figure 9

(Color online) Coupling coefficients $\left\{h_{\vec{\mathrm{q}},l}\right\}$ for $\vec{\mathrm{q}}$ at the $X$ point and the second photonic band of the doped BPC as the pump parameter is increased and the convergence is reached.Reuse & Permissions

Figure 10

(Color online) The absolute value of the electric field of Bloch modes corresponding to bands 2, 6, 8, 12, and with the Bloch vector ending at the $X$ point of the Brillouin zone are drawn. The white dashed lines show the dielectric boundaries.Reuse & Permissions

Figure 11

(Color online) If the normalized nonlinear Bloch field obtained at the end of the convergence procedure, step 3 (as described in Sec. 5) is drawn for different values of the pump parameter ($\rho =1.053$—at the threshold, 1.103—above threshold, 2.000—well above threshold), a very minor change from the eigenmode of the backbone photonic crystal can be observed. Nevertheless the absolute value of the difference of the normalized fields with the BPC eigenmode, for different pumps, reveals the differences. The absolute value of the difference of the normalized nonlinear Bloch field with the BPC eigenmode (upper left picture) for $\rho =1.053$ (upper right picture), $\rho =1.103$ (lower left picture), and $\rho =2.000$ (lower right picture) has a small magnitude of the order of $2\times {10}^{-5}$. The white dashed lines show the dielectric boundaries.Reuse & Permissions

Figure 12

(Color online) Dependence of the threshold for the pump parameter and the slope of the number of photons per unit cell versus the pump parameter, when the losses are varied.Reuse & Permissions

Figure 13

(Color online) (a) The resonance frequency of the homogeneously broadened transition of two level atoms is positioned at different frequencies of the second band of the backbone photonic crystal. (b) Transition threshold for the pump parameter and the slope of the dependency of the number of photons per unit cell versus the pump parameter differs for different positioning of the resonance frequency with regard to the second band. Average slopes are calculated at 45 points (15 Bloch vectors per each direction) around the boundaries of the first Brillouin zone. For clarity, only four slopes, corresponding to the resonance frequency positioned close to $\Gamma$ points $(\vec{\mathrm{q}}=2,44)$ and exactly at the $X$ and $M$ band edges $(\vec{\mathrm{q}}=15,30)$, are drawn. The inset shows how the slopes of the photon number per unit cell versus pumping vary as the resonance frequency is positioned on the boundary of the first Brillouin zone. The singularity and therefore the numerical accuracy of the calculations at the $\Gamma$ point may explain why the slopes there show discontinuities.Reuse & Permissions

Figure 14

(Color online) This picture complements the inset of Fig. 13b. Backbone photonic crystal fields for the second photonic band and Bloch vectors 2, 15, and 44 are drawn in the first column. The respective absolute values of the difference of the electric vector of the converged fields under pump $\rho =12.1$ with the electric vector of the BPC is drawn in the second column.Reuse & Permissions

Figure 15

(Color online) (a) The BPC photonic band structure for the case of Fig. 1b. A band gap for the $E$-polarized fields extends from ${\omega }_s=0.2232$ (first photonic band, $M$ point) to ${\omega }_s=0.2432$ (second photonic band, $X$ point). (b) The coating shell of Fig. 1b has a susceptibility ${\overline{\chi }}_{\mathit{\text{coat}}}\left({\omega }_s\right)$ defined by Eq. (33a) with coefficients ${\omega }_{0s}=0.2328$, ${\tau }_2=5100$, $g_0=0.44$ (infinite pumping). The real (solid blue line) and the imaginary (dotted green line, plotted close to the real part for clarity) part of the susceptibility of the ensemble of quantum dots are drawn for the very weak field. Two additional lines represent the frequencies of the first band for the $M$ point (dash-dotted black line) and the second band for the $X$ point (dashed red line), respectively, as the dielectric constant of the coating shell is varied.Reuse & Permissions

Figure 16

(Color online) (a) The number of photons per unit cell $n_{\mathit{ph}}$, (b) average inversion $\overline{w}$ of the dopant atoms, (c) the real part of the frequency of the second band at the $X$ point, and (d) the imaginary part of the frequency of the second band at the $X$ point—as functions of the pumping parameter $\rho$.Reuse & Permissions

Figure 17

(Color online) Threshold and photon density behavior as the background loss is varied. 156 lowest (in frequency) plane waves of the BPC are considered coupled from the spatial distribution of the resonant colloidal quantum dots and the background loss.Reuse & Permissions

Figure 18

(Color online) Photonic band structure of the $E$-polarized field in the BPC with unit cell as shown in the gray scale inset. A tiny band gap extends from ${\omega }_s=0.2186$ (first photonic band, $M$ point) to ${\omega }_s=0.2209$ (second photonic band, $X$ point). The absolute value of the electric field for the normalized Bloch eigenmodes is drawn for three first bands at points $X$ and $M$ of the first Brillouin zone.Reuse & Permissions

Figure 19

(Color online) The shell formed from quantum dots coating the inner surfaces of the air pores, shown in Fig. 1c, has a susceptibility ${\overline{\chi }}_{\mathit{\text{coat}}}\left({\omega }_s\right)$ defined by Eq. (33) with coefficients ${\omega }_{0s}=0.2197$, ${\tau }_2=5100$, $g_0=0.44(\rho -1)∕(\rho +1)$. The real part (solid blue line) and the imaginary part (dotted green line, shifted close to the real part for clarity) of the susceptibility of the shell of quantum dots are drawn for pump $\rho =1.2$ and $n_{\mathit{ph}}=0$. The real part of susceptibility is drawn (solid gray line) for another pump value, $\rho =2.0$ and $n_{\mathit{ph}}=0$. Two additional lines represent the frequencies of the first band for the $M$ point (dash-dotted black line) and the second band for the $X$ point (dashed red line), respectively, for a frequency independent BPC, as the dielectric constant of the coating shell is varied.Reuse & Permissions

Figure 20

(Color online) The spatial normalization of the quasimonochromatic electromagnetic field, as defined in Eq. (28), requires a positive value for the function $\partial \left[\omega {ϵ}_{\mathrm{R}}\left(\omega \right)\right]∕\partial \omega$ [see Eq. (C5)]. This function is plotted above for the case of very weak fields $(n_{\mathit{ph}}\simeq 0)$ and pump values $\rho =0$, 1.2, and 2. The red vertical line represents the frequency position (roughly) of the second band at the $X$ point. As shown in this graph, the normalization of the band edge field may not be performed for very weak fields and pumps above a certain value, due to $\partial \left[\omega {ϵ}_{\mathrm{R}}\left(\omega \right)\right]∕\partial \omega$ being negative. This problem is overcome using the modified iteration procedure.Reuse & Permissions

Figure 21

(Color online) (a) The number of photons per unit cell $n_{\mathit{ph}}$, (b) average inversion $\overline{w}$ of the dopant atoms, (c) the real part of the frequency of the second band at the $X$ point, and (d) the imaginary part of the frequency of the second band at the $X$ point—as functions of the pumping parameter $\rho$.Reuse & Permissions

Figure 22

(Color online) The absolute value of the difference of the normalized nonlinear Bloch field for $\rho =1.068$ (upper right picture), $\rho =1.1$ (lower left picture), and $\rho =1.2$ (lower right picture) with the BPC eigenmode (upper left picture) has a small magnitude of the order of ${10}^{-3}$. The white dashed lines show the dielectric boundaries.Reuse & Permissions

Figure 23

(Color online) Transition threshold for the number of photons per unit cell, $n_{\mathit{ph}}$, occurs for a lower pump power at the upper band edge ($X$ point).Reuse & Permissions

Figure 24

(Color online) Dependency of the threshold for the pump parameter and the slope of the number of photons per unit cell versus the pump parameter, when the losses are varied.Reuse & Permissions