1. Introduction

The interest in WGMs can be traced back to a century ago when Lord Rayleigh studied the sound propagation in a gallery with a curved surface [1]. In 1989, Braginsky and coworkers demonstrated that silica microspheres can support optical resonant modes with high quality (Q) factor [2]. Since then, silica microspheres have been considered as the key element for making modulators and sensors in fiber communication. For spherical resonators with size much larger than the wavelength of light, the WGM resonant frequencies can be predicted by using geometric optics [3]. The Q-factor of a WGM mode is defined as the ratio of the resonant frequency to the peak width, which is proportional to the energy dissipation rate of the mode [4]. To determine the Q-factor of individual WGM mode, it is better to use wave mechanics to find the solution to Maxwell equations for the wave function that satisfies the boundary condition of a dielectric sphere and find the complex root for the resonance frequency as illustrated by Little [5]. The imaginary of the complex root then describes the width of the resonance peak, which can be used to evaluate the Q factor of the mode.

The Q factor so determined is for an ideal resonator and its value can be very high, particularly for high frequency modes. For realistic resonators, there are extrinsic factors that limit the value of the net Q factor of the resonance mode. The net Q factor ($Q$) is related to the intrinsic Q factor ($Q_0 $) and extrinsic Q factor ($Q_{ex} $) by $Q^{ - 1} = Q_0^{ - 1} + Q_{ex}^{ - 1}$. As pointed out in [2,6–7], $Q_{\textrm{ex}}$ consists of the following contributions from absorption loss of the microsphere material, the defects inside the microsphere, and the surface roughness. For a dielectric medium of silica or ZnO, the absorption loss is very weak for photon energy far below the band gap. Thus, $Q_{ex} $ is dominated by defects and the surface roughness.

In a previous work [8], we applied the Green’s function method to calculate the optical emission spectra of ZnO microspheres. By writing the full Green’s function of the microsphere in terms of a sum over resonant poles in the complex ω plane weighted by the Purcell factor [9], the optical emission spectra (including both spontaneous emission and stimulated emission) can be evaluated quantitatively. The lineshapes of emission spectra (including principal and secondary modes) agree well with the experimental data obtained by photoluminescence. We found that the relative strengths of principal and secondary modes with TE or TM polarization are quite sensitive to the net Q factor of each resonance mode. Thus, by studying the relative strengths of WGM peaks in the emission spectrum of a microsphere, we can characterize the surface roughness of the microsphere, since it is related to $Q_{ex} $ of the microsphere. In this paper, we carry out systematic experimental and theoretical studies of ZnO microspheres of various degrees of defect density and surface roughness to gain better understanding of the relation between $Q_{ex} $ and the spectral features of optical emission spectra.

In our theoretical analyses, we use Mie basis functions to describe the full Green’s function of the microsphere, which is obtained by solving the Dyson equation. We calculate the Purcell factor of each resonance mode, and use the pole expansion method to evaluate the optical emission spectra as described in [8]. Our calculated emission spectra contains three main empirical parameters, the porosity (F), extrinsic Q factor ($Q_{\textrm{ex}}$), and the ratio of stimulated emission to spontaneous emission (${f_r}$), which are adjusted to fit the experimental spectra obtained by photoluminescence (PL).

2. Theoretical model

The current model follows a previous work [8] with some modifications. The key ingredients of the theory are summarized below. To describe the electromagnetic waves in a microsphere (MS), we express the full Green’s function (for $r < R < r^{\prime}$) within the Mie basis functions as $\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\leftrightarrow$}} \over G} ({{k_0},\textbf{r},\textbf{r}^{\prime}} )= {\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\leftrightarrow$}} \over G} ^{TE}}({{k_0},\textbf{r},\textbf{r}^{\prime}} )+ {\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\leftrightarrow$}} \over G} ^{TM}}({{k_0},\textbf{r},\textbf{r}^{\prime}} )$ with

The zeros of $D_l^{TE}({{k_0}} )$ and $D_l^{TM}({{k_0}} )$ determine the TE and TM resonance frequencies.

To calculate the emission spectra, we need to consider the steady state population of photo-excited carriers in the PL process. In ZnO microspheres, there exist many light-emitting centers related to oxygen vacancies or other defect complexes. These emitting centers have distributed energies that lead to a broad-band emission covering the entire visible range [10]. As illustrated in Fig. 1, when the incident photon gets absorbed, it excites an electron from the valance band to the conduction band. This electron emits a phonon and relaxes to the upper defect energy level $|\Psi _i^U$. Next, the electron emits a photon and falls into the lower level $|\Psi _i^L$. The steady-state carrier distribution in these defect levels can be obtained by the rate equations as depicted in Fig. 1 [11].

 

Fig. 1. Schematic diagram for the photoluminescence process in the microsphere. The horizontal bars denote emitting centers (defects).

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In Fig. 1, ${f_i}^\textrm{U}({f_i}^\textrm{L})$ is the carrier population in the upper (lower) level. $F_i^U(F_i^L)$ and $\Gamma _i^U(\Gamma _i^L)$ represent the feeding and relaxation rates, ${\tau _i}$ is the photon lifetime of the cavity mode with the resonant frequency ${\omega _i}$, and ${R_{sp}}({{\omega_i}} )$ is the spontaneous rate for mode i which is proportional to the Purcell factor (${P_F})$. ${P_F}$ is given by the ratio of photon density of states (DOS) of the cavity to its corresponding value in the uniform medium [8,9]. We define ${g_i} \propto {\gamma _i}({f_i^U - f_i^L} )$ as the gain coefficient, where ${\mathrm{\gamma }_\textrm{i}}$ denotes the corresponding radiative recombination rate. The photon number in mode i is given by ${n_i} = {\tau _i}{R_{sp}}({{\omega_i}} )/({1 - {\tau_i}{g_i}} )$ in steady state. The photon lifetime ${\tau _i} \equiv {\tau _0}{Q_i}$ is proportional to the quality factor ${Q_\textrm{i}}$, as can be understood from the fact that the higher the quality factor, the longer the photon lives. Here ${\tau _0}$ is a characteristic time scale. In our current study, the emission spectrum is due to recombination at defect sites, thus the gain coefficient is small (i.e. ${\tau _i}{g_i} \ll 1$), and we have ${n_i} \approx {\tau _i}{R_{sp}}({{\omega_i}} )$. Given a spontaneous emission rate ${R_{sp}}({{\omega_i}} )\propto {P_F}{\gamma _i}f_i^U({1 - f_i^L} )$ for each emitter, the corresponding stimulated emission rate becomes ${R_{st}}({{\omega_i}} )\propto {n_i}{g_i} \propto {Q_i}{R_{sp}}({{\omega_i}} )$. The spontaneous ($I_{sp}^\mu )$ and stimulated ($I_{st}^\mu )$ emission spectra (for $\mu $ =TE, TM) can then be related by $I_{st}^\mu = {f_r}{Q_i}I_{sp}^\mu $, where ${f_r}$ is treated as an empirical parameter.

The photon number with mode frequency ${\omega _i}$ increases with ${Q_i}$. However, the effective ${Q_i}$ may be reduced due to the unavoidable surface roughness and other external scattering mechanisms (such as defects and contact with the substrate). We write $Q_i^{ - 1} = Q_{0i}^{ - 1} + Q_{ex}^{ - 1}$, where $Q_{0i}$ denotes the Q factor of in an ideal resonator, and $Q_{ex}$ denotes the limiting Q factor due to external scattering mechanisms.

The main modifications to the previous theory are twofold. (1) We incorporate the wavelength dependence in $Q_{\textrm{ex}}$ according to the Rayleigh scattering theory. (2) Instead of replacing $Q_0$ by the effective Q for the Q factor appearing in ${P_F}$ and the stimulated emission intensity ($I_{st}^\mu )$, here we include the effect of diffusive scattering by surface roughness and impurities as a “self-energy” correction to the full Green’s function. Namely, the energy loss due to diffusive scattering can be incorporated into the imaginary part of the frequency via the relation $\omega \to ({n + i{\kappa_{ex}}} ){k_0}$ with ${\kappa _{ex}} = n/Q_{ex}$. In this way, the effect of external loss is automatically included in the Q factor as well as in the full Green’s function, which influences not only the resonant part in the cavity but also the leak-out part outside the cavity. The defects due to doping and porosity of the MS can also lead to a change of the real part of refractive index n, which can be included via the effective-medium theory by using a porosity factor (F) in our simulation. In the Maxwell Garnett approximation [12], F is related to the dielectric constants of the host (${\varepsilon _{ZnO}})$ and inclusions (${\varepsilon _{inc}})$ by

Here the inclusions are air holes, so we use ${\varepsilon _{inc}} = 1$. The introduction of corrections to both the imaginary part (through ${\kappa _{ex}})$ and real part of ${n_{\textrm{eff}}}$ may violate the Kramers-Kronig relation. However, the imaginary part ${\kappa _{ex}}$ needed to fit the emission spectra is quite small [${\kappa _{ex}}$ < ${10^{ - 5}} \times {({\lambda /\mu m} )^3}$], and using the Kramers-Kronig relation to calculate the change of the real part of refractive index leads to a negligible correction ($\Delta {n_{\textrm{eff}}} < {10^{ - 5}})$.

The $Q_{ex}$ of spherical cavity has been investigated in previous studies [13–16]. Generally speaking, we can divide the total ${Q^{ - 1}}$ into four parts: (1) Intrinsic radiative loss (2) Surface roughness (3) Contaminants and (4) Material loss, that is,

The dependence on the wavelength of the quality factor induced by the surface roughness ad defects can be obtained as follows. First, the attenuation coefficient ${\mathrm{\alpha }_{\textrm{ss}}}$ derived based on the concept of Born approximation is proportional to ${\mathrm{\lambda }^{ - 4}}$ [6,7]. On the other hand, the quality factor is equal to ${Q_{ss}} = 2\pi n/{\alpha _{ss}}\lambda $. So, the quality factor induced by the surface roughness and defects takes the form $Q_{ex} = {f_{ss}}{\lambda ^3}R$ [6,7], where the prefactor ${f_{ss}}$ is determined by fitting the experimental emission spectrum.

3. Experimental method

Here, we adopted the chemical route to synthesis 2 mol.% of Li doped ZnO microsphere as reported in Ref. [16]. A sample of the distinct individual doped ZnO microspheres was prepared on Si substrate for detail characterization and measurements. The size and surface morphology of the chosen individual microsphere was recorded using field emission scanning electron microscope (FESEM) (FEI, Nano-Nova), and its position was located via an optical microscope by referring the FESEM image, which is shown in the inset of the Fig. 2(b). The crystalline structure, chemical compositions along with the valence state of the dopant of the microsphere were studied through XRD, µ-Raman and XPS analysis as discussed in ref. [17]. Furthermore, WGM spectra of the doped ZnO microsphere was collected using a commercial µ-Raman setup (Horiba Jobin Yvon HR-800) attached with He-Cd laser (325 nm) as excitation source. The pumping power of the laser light used for excitation of the microsphere was about 9.13mW/cm², which can be varied by using ND filters. The desired single Li-doped ZnO microspheres (size ∼ 3.8µm and 3.9µm) were identified by using 10X and 40X objective lenses, and finally µ-PL spectra were taken using 40X objective lens. The WGM coupled µ-PL spectra were recorded in the spectral range from 500 nm to 700 nm. The defects such as Zn and O vacancies assisted luminescence in doped ZnO microsphere resulted multiple sharp resonance peaks of WGM in the µ-PL spectra similar to previous studies [18–20].

 

Fig. 2. Quality factors of the principal and secondary WGMs for the four ZnO microspheres considered.

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4. Comparison of theoretical results with experiment

We compare our theoretical results with experimental PL data on four ZnO MSs with different degrees of inhomogeneity. Figure 2 shows the calculated quality factors $Q_{0i}$ (intrinsic) and $Q_i$ (net) of the TE principal and secondary WGMs for the four MSs as functions of wavelength, $\lambda $. The behaviors of TM principal modes (not shown) are similar to the TE principal modes. The principal modes have a higher quality factor compared to secondary modes. For a given radial quantum number, WGMs with higher angular quantum number (l) have shorter resonance wavelength and higher quality factor. As seen in Fig. 2, the effect of $Q_{ex}$ causes the net $Q_i$ to saturate once the WGM wavelength is shorter than a critical value (${\lambda _c})$, since $Q_i$ is limited by $Q_{ex}$. The value of ${\lambda _c}$ is shorter for secondary modes (TE2) than for the principal modes (TE1). When $Q_{0i}$ of a secondary mode exceeds $Q_{ex}$, its net $Q_i$ becomes comparable to that of the principal mode at similar wavelength, and its peak intensity in the PL spectrum can become close to that of the principal mode. The value of ${\lambda _c}$ is shorter for secondary modes (TE2) than for the principal modes (TE1). For the 1.8 µm pristine MS, the saturation effect is insignificant, since ${\lambda _c}$ for TE2 occurs outside the wavelength range considered. For the other three chemically synthesized MSs, ${\lambda _c}$ for TE1 (TE2) occurs near 640 (420) nm, 660 (530) nm, and 700 (560) nm for the 1.84 µm, 3.9 µm, and 3.8 µm MSs, respectively.

In Fig. 3, we compare our simulation results with the experimental data for the pristine 1.8µm ZnO MS taken from ref. [21] and an undoped MS of similar size (1.84µm) grown by chemical synthesis. The mode numbers (N,l) are labeled in blue (red) for TE (TM) modes. N=1,2 for principal and secondary modes. For the 1.8µm MS the effect from the inhomogeneity on this MS is very small and the porosity found is tiny (F ∼ 0.018) as it is made by laser ablation. From our fitting, its prefactor of the external quality factor ($Q_{ex}$) is ${f_{ss}} = 5.56 \times {10^7}\mathrm{\mu }{\textrm{m}^{ - 4}}$ and stimulated emission ratio is ${f_r} = 3 \times {10^{ - 6}}$. ${f_{ss}}$ is determined by fitting the average ratio of strengths of secondary modes to those of principal modes. Since the secondary TM modes are quite weak in the samples considered, we only adopted a single value of ${f_{ss}}$ to do the fitting of the whole spectrum. In principle, if prominent secondary TM modes are observed, it would be justifiable to use a different value of ${f_{ss}}$ for TM secondary modes to improve the fitting. ${f_r}$ is determined by the ratio of principal TE and TM modes in the short-wavelength regime. Without ${f_r}$, the calculated TM peaks can be stronger than the TE peaks for short wavelengths.

 

Fig. 3. Experimental data of PL spectrum (orange) and simulation (green) for (a) a pristine ZnO MS with a diameter of 1.8µm (b) a chemically-synthesized ZnO MS with a diameter of 1.84µm. The theoretically predicted emission spectrum (green) for (a) is obtained with a porosity F_1e = F_1m = 0.018, for (b) with F_1e = 0.12 (TE modes) ${F_{1\textrm{m}}}$=0.145 (TM modes) and ${F_{2\textrm{e}}} = {F_{2\textrm{m}}} = $0.13 for secondary modes.

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For the 1.84µm ZnO MS grown by chemical synthesis, we see quite different emission spectrum (Fig. 3(b)) in comparison to the pristine MS (Fig. 3(a)). The main difference here is the much larger porosity (F ∼ 0.12) introduced in the chemical synthesis and rougher surface. The best fit to the spectra gives ${f_{\textrm{ss}}} = 5.43 \times {10^4}\mathrm{\mu }{\textrm{m}^{ - 4}}$ and ${f_r} = 0$, which indicates negligible contribution from the stimulated emission in the 1.84µm MS, suggesting a lossy resonator. The noticeable change in spectral feature is the emergence of stronger secondary peaks at ${\lambda}$ between ${\lambda _c}({TE2} )$ and ${\lambda _c}({TE1} )$ [see peaks labeled by (2,12) (2,13), and (2,14) in Fig. 3(b)] due to the reduced $Q_{ex}$ for the 1.84µm MS. Because of the rougher surface of this MS, the spectral features are broader and the principal TE and TM modes cannot be resolved.

In Fig. 4, we compare our simulation with experiment for two Li-doped MSs of similar size (with diameters 3.9µm and 3.8µm) made by chemical synthesis, but having different porosities (F ∼ 0.19 and 0.13, respectively). Because of their larger size, the intrinsic quality factor $Q_0$ can exceed $Q_{ex}$ at some critical wavelength (${\lambda _c})$. Thus, the peak strength of secondary modes becomes comparable with that of principal modes. As a consequence, the emission spectrum exhibits closely-spaced multiple-peak structures due to the emergence of secondary peaks with enhanced relative strength, for $\lambda$ < 550nm [e.g. modes (2,31) and (2,32) and secondary modes with angular quantum number larger than 32 ] in Fig. 4(a) and for $\lambda$ < 650nm [e.g. mode (2,25)] in Fig. 4(b). We note that in Fig. 4(b) the secondary modes with quantum number larger than 25 are merged with the TM principal modes, so they are not resolved. These transition wavelengths (marked by vertical dashed lines in Fig. 4(a) and 4(b)) are close to the mid-point between ${\lambda _c}({TE2} )$ and ${\lambda _c}({TE1} )$. For wavelengths below this transition point, the TM secondary modes also pop out [e.g. TM modes (2,27) and (2,28) in Fig. 4(b)]. Such features allow us to give a good estimate of $Q_{ex}$, which can facilitate the simulation. The best fit to the spectra gives ${f_{ss}} = 2.56 \times {10^7}\; ({7.89 \times {{10}^6}} ){\; }\mathrm{\mu }{\textrm{m}^{ - 4}}$ and ${f_r} = 1 \times {10^{ - 7}}$ ($2 \times {10^{ - 6}})$ for the 3.9 (3.8) µm MS. We note that peak positions of TM modes and secondary modes can shift due to their different radial dependence and we need to adjust their corresponding porosity, ${F_{1\textrm{m}}}$ (for TM) and $F_2$ (for secondary modes) slightly to fit the peak positions. We also found that MSs with higher porosity lead to lower ${f_r}$, meaning less contribution from the stimulated emission.

 

Fig. 4. Experimental data of PL spectrum (orange) and simulation (green) for two Li doped ZnO MSs with diameter of (a) 3.9µm and (b) 3.8µm. The theoretically predicted emission spectrum (green) for (a) is obtained with fitting parameters for (a) ${F_{1\textrm{e}}}$=${F_{1\textrm{m}}} = {F_{2\textrm{m}}}$=0.19, ${F_{2\textrm{e}}} = $0.18, and (b) ${F_{1\textrm{e}}}$=0.127, ${F_{1\textrm{m}}}$=0.14, ${F_{2\textrm{e}}} = $0.126 ${F_{2\textrm{m}}} = $0.128.

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5. Conclusion

In this work, we introduced a wavelength-dependent $Q_{\textrm{ex}}$ to mimic effects of Rayleigh scattering from defects and surface roughness. We then incorporate this effect in the imaginary part of the refractive index used to calculate the full Green’s function of ZnO microspheres with inhomogeneity and the resultant WGM emission spectra. By using only a few fitting parameters, we can simulate the relative strengths of many WGM peaks in various ZnO microspheres of different inhomogeneity. Some discrepancies remain between the experimental spectra and the theoretical simulations. They are likely caused by some features not fully taken into account in the present model, including the substrate effect and inhomogeneity in distribution of air holes and defects that may alter the relative strengths between WGMs of different modal profiles. Nonetheless, our current studies provide a better understanding of the relation between lineshapes of WGM peaks and $Q_{\textrm{ex}}$ for microspheres. The analyses could help the future design of microcavity-based optical emission devices.