1. Introduction

Whispering gallery mode resonators offer in themselves a unique test bed for exploring fundamental physics, as well for designing exotic applications related to the field of photonics and cavity quantum electrodynamics. Total internal reflections resulting from rotational symmetry of whispering gallery mode (WGM) resonators can result in ultrahigh quality factors in the order of $\sim {10}^6-{10}^8$, placing them in equal footing with high-quality photonic crystals as far as optical confinement is concerned [1, 2]. The suitability of WGM resonators have already been studied extensively for sensing [3, 4], ultra-low threshold lasing [2, 5, 6], quantum information processing, probing, imaging and filtering related applications [7–9]. However, unfortunately, the characteristic isotropic emission pattern of WGMs poses a long-standing challenge in designing applications where substantial directional emission is required [2, 5, 6, 10–12].

Previous studies related to the directionality of WGMs have primarily focused on breaking the rotational symmetry of the resonator, employing schemes such as deformation of the cavity shape [10–12], incorporation of scatterer inside the cavity [13], or embedding of scatterers at the periphery of the cavity [14]. These studies have been based on conventional micro-ring, micro-disk, or micro-sphere like WGM resonators. Though there have been a report of directionality enhancement in triangular-shaped deformed cavity comprising of micro-posts [15], the case of circularly symmetric nanowire assembly has remained unexplored in spite of the significant potential of such systems for realizing tunable, high-density, defect-free optoelectronic and photonic systems [16, 17].

Disordered systems designed with non-Hermitian potentials offer the prospect of defining specific behaviors of light, which cannot be otherwise realized in a homogenous environment [18, 19]. Based on this principle, disordered photonic systems have been extensively studied for numerous exotic applications, such as for attaining unusual localization of light [20], and also for enhancing the collimation bandwidth of light [21], to name a few. In the present study, we study the role of disorder on directionality enhancement in a circular nanowire array based on finite-difference time-domain analysis technique. Quality factor in the order of ${10}^5$ is obtained along with quasi-isotropic emission characteristics for the radially symmetric case. However with increasing disorder, in-plane directionality tends to increase and eventually unidirectional emission characteristics are obtained in the Anderson localized regime. By identifying cross-over between the rotational symmetric and multiple scattering mediated regimes, an optimized design of correlated disorder has been proposed such that directionality can be enhanced without substantial degradation of the Q-factor.

 

Fig. 1 (a) Schematic representation of the rotationally symmetric GaN nanowire array (inset shows the hexagonal unit cell); (b) calculated Q-factors of the resonant modes with corresponding mode numbers (inset shows enlarged view of Q-factor vs. resonant wavelength values of the weakly confined modes existing between 400 nm – 700 nm); (c) near-field distribution of E-field of the WGM; (d) far-field emission characteristics exhibiting directionality characteristics of the corresponding WGM.

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2. Radially symmetric nanowire array

The nanowire assembly considered in this work comprises of homogenous GaN nanowires having air medium in between. Experimental growth and fabrication of high quality GaN-nanowire arrays in both patterned and self-organized manner have been reported extensively employing electron-beam lithography, molecular beam epitaxy, focused ion beam milling and chemical vapor deposition techniques [16, 17, 22, 23]. For investigating whispering gallery modes, a uniform distribution of GaN nanowires having experimentally reported diameter of 70 nm is considered [16, 17]. These nanowires are considered to be vertically oriented along z-axis such that they form a radially symmetric pattern along the x-y plane, which is schematically illustrated in Fig. 1(a). Aclose examination of this pattern indicates the presence of a hexagonal unit cell, which is shown as an inset to Fig. 1(a). In this unit cell, the distance between adjacent nanowires along x- and y- directions are defined as h_X and h_Y respectively. For the radial symmetric case, h_X and h_Y are taken to be 85 nm, and the nanowires are oriented such that the overall diameter of the ensemble forming the cavity becomes 3 μm. This overall diameter has been kept constant throughout the study.

Resonant characteristics of the nanowire array have been studied using open-source finite-difference time-domain (FDTD) software package MEEP [24]. A Gaussian pulse-source is placed at the center of the array and the resonant modes are identified by decomposing the electric (E) and magnetic (H) field components of the electromagnetic wave using filter diagonalization method [25]. Because transverse electric (TE) modes are expected to be rather weakly confined in GaN-nanowire based micro-cavities [6], only the cases of transverse magnetic (TM) modes have been considered in this work. Q-factors corresponding to these modes are plotted as a function of wavelength in Fig. 1(b). Near-field distribution of the mode having the highest Q-factor of2.2× 10⁵ is shown in Fig. 1(c), which clearly indicates the presence of whispering gallery like characteristics with a standing wave formed along the cavity boundary and a null of energy at the center. The radial and azimuthal mode numbers obtained for this mode are r $=$1 and m $=$15 respectively. Similar WGM characteristics are observed for the strongly confined modes located at 914.7 nm and 793.2 nm, and also for the relatively weakly confined modes residing within the wavelength range of 400nm – 700 nm. Radial and azimuthal mode numbers obtained from corresponding near-field distributions of the modes are indicated as (r, m) in Fig. 1(b). Directionality characteristics of these resonant modes are obtained from their corresponding far-field distributions. Far-field radiation pattern obtained for the most strongly confined mode of (1, 15) is shown in Fig. 1(d), which clearly indicates the presence of quasi-isotropic emission characteristics. Similar quasi-isotropic directionality profiles are also obtained for the remaining whispering gallery modes of this array, thereby confirming the presence of perfect rotational symmetry in the structure.

 

Fig. 2 (a) Nanowire arrays generated with percentage randomness values of 15%, 50%, and 100% respectively; (b) Q-factors of the most strongly confined modes plotted as a function of disorder strength and percentage randomness; (c) corresponding resonant wavelengths plotted as a function of disorder strength and percentage randomness; inset (i) shows disorder strength plotted as a function of percentage randomness and inset (ii) shows the variation of l/l_c with percentage randomness in regards to the satisfaction of Ioffe-Regel criterion.

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3. Directionality and resonant characteristics in disordered array

In order to disturb rotational symmetry of the array, spatial locations of the nanowires shown in Fig. 1(a) are systematically altered by a percentage randomness parameter, P $\in$ [0, 100]. For a given value of P, arbitrary co-ordinates (x+δx, y+δy) are generated such that δx = (P/100)×(h_X/2)$\times r_x$ and δy = (P/100)×(h_Y/2)$\times r_y$, where $\left\{r_x,r_y\right\}\in$ [-1, 1] are random numbers, and (x, y) represent center positions of the nanowires shown in the symmetric array of Fig. 1(a). Three disordered systems derived from this symmetric structure are schematically shown in Fig. 2(a), where percentage randomness values of 15%, 50% and 100% have been considered. To compare disorder between arrays having different diameters and fill factors of the nanowires, a disorder strength parameter ζ, defined as the correlation between refractive index profiles of the symmetric and disordered arrays, is also utilized. In order to calculate ζ, the two-dimensional refractive index profiles having N elements of the periodic and disordered structures are first transformed into one-dimensional vectors A and B respectively. In this transformation, consecutive rows of a two-dimensional refractive index profile of an array have been cascaded to form the corresponding one-dimensional vector. The mean and standard deviation of the vector A (B) being μ_A (μ_B) and σ_A (σ_B) respectively, the disorder strength is calculated as [26]

Using Eq. (1), ζ values of 0.105, 0.523 and 0.95 are obtained respectively for the patterns shown in Fig. 2(a). Further calculations for patterns obtained for different values of P indicate a positive correlation between P and ζ such that ζ increases from 0 to 0.95 as P is increased from 0 to 100%.

 

Fig. 3 (a) Electric-field distributions of WGM, quasi-WGM and Anderson localized modes obtained for percentage randomness values of 5%, 20% and 50% respectively; (b) corresponding far-field distribution profiles illustrating directionality characteristics of these modes; (c) beam divergence angle plotted as a function of both disorder strength and percentage randomness; the error bars indicate the variation of FWHM obtained for random configurations having identical values of percentage randomness and nanowire diameter.

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To investigate the effect of disorder on resonant conditions of the array, Q-factor of the most strongly confined mode and its corresponding wavelength are next calculated and plotted as a function of both P and ζ in Figs. 2(b) and 2(c). In order to avoid the artifact of a random sample, configurational averaging on several realizations of a disordered array having same parameters has been performed. The relation between disorder strength and percentage randomness for these configurational averaged arrays is shown as an inset to Fig. 2(c), where ζ is plotted as a function of P. The range of variation of ζ for a fixed value of P, nanowire spacing and diameter, is indicated as an error bar in this plot. Higher values of P results in greater variation of ζ, thereby indicating the presence of enhanced spatial disorder. As can be observed in Fig. 2(b), the loss of rotational symmetry with increasing disorder results in degradation of the Q-factor. However, it is noteworthy that for a wide range of the disorder strength, the Q-factor remains above ${10}^4$. Moreover, resonant wavelength of the most tightly confined mode shifts from near-infrared to ultra-violet regime of the EM spectra as disorder is increased. To explain this behavior, near-field pattern of the E-field for P = 5%, 20% and 50% have been examined. As can be seen in Fig. 3(a), though WGM persists for the case of 5% randomness, Q-factor decreases by about one order of magnitude because of the reduced symmetry of the structure. For P = 20%, the peak resonant characteristics changes from being WGM to quasi-WGM, an operation regime also known as the triangular WGM [5, 6]. This regime in fact represents the co-existence of regular and chaotic wave dynamics- an attribute which has been previously described in the framework of Kolmogorov-Arnold-Moser (KAM) theorem [27] and dynamical tunneling [28]. Such regular-to-chaotic wave dynamics have also been exploited to experimentally realize momentum transformation in optical resonators [29], and also to suppress spatiotemporal instabilities in microcavity lasers [30]. The qausi-WGM mode in our case is rather weakly confined compared to the case of WGM and the corresponding Q-factors monotonically decrease to about 7× 10³. As disorder is further increased, wave dynamics inside the cavity shifts from multiple total internal reflection mediated whispering-gallery regime to multiple random scattering mediated strong-localization regime. Such strong-localization of light resulting from interference of multiply scattered wave in a disordered medium is commonly referred to as Anderson localization (AL) effect [17, 31–33]. Transition to Anderson localized regime has been further verified based on the Ioffe-Regel criterion. According to this criterion, Anderson localization of wave in the disordered medium is governed by the condition $l/l_c\le 1$, where l and l_c denote the scattering length and critical scattering length respectively. In our study, the values of $l/l_c$ for each isotropic disordered structure have been calculated using the theoretical formulation reported in [33]. As has been shown as an inset to Fig. 2(c), the value of $l/l_c$ decreases with the increase of percentage randomness, and eventually becomes smaller than 1 when P > 30%. This confirms that Ioffe-Regel criteria is indeed satisfied for such values of percentage randomness and the system consequently operates in the Anderson localized regime. Quality factor of the Anderson localized mode is observed to increase with percentage randomness because of enhanced multiple scattering events in the array. For the considered nanowire diameter of 70 nm, localization wavelength resides within the range of 300nm − 330 nm, which is consistent with our previous findings [33]. Three operation regimes, namely WGM, quasi-WGM and AL regime, have been identified and indicated in Figs. 2(b) and 2(c) based on the observed transitions of wave dynamics as the degree of randomness is gradually changed.

 

Fig. 4 (a) Schematic of the correlated disordered nanowire array having spatially varying density of nanowires; (b) Q-factors of the resonant modes plotted as a function of wavelength; (c) normalized E-field distribution of the highest Q-factor WGM; (d) far-field directionality profiles of the WGM for the nanowire array and also for the equivalent effective medium structure; (e) equivalent effective medium structure for the array shown in Fig. 4(a).

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In order to investigate the emission directionality in these three regimes, far-field distribution of the E-fields are next calculated for arrays of different disorder strengths. Representative far-field patterns for these regimes are shown in Fig. 3(b). As can be observed, in spite of the loss of rotational symmetry, the emission profile is nearly isotropic for 5% randomness, and for the quasi-WGM case, the emission collimates as several narrow beams in different directions. In the AL regime however, complete breakdown of rotational symmetry results in a highly directional emission with a narrow beam angle. To further distinguish directionality characteristics of the three operation regimes, beam divergence angles are calculated as full-width at half maxima (FWHM) of the corresponding far-field distributions. In Fig. 3(c), beam divergence angles plotted as a function of percentage randomness indicate gradual improvement of beam directionality with increasing disorder. A relatively sharp decrease in FWHM, accompanied by a concurrent blue-shift of the resonant wavelength of the highest Q-factor mode, is observed in the quasi-WGM regime. For higher degrees of randomness, in the AL regime, the FWHM value resides within 15${ }^{\circ }-$ 50${ }^{\circ }$ and nearly unidirectional emission characteristics are obtained. The underlying random scattering events in this regime however dictate that the direction of emission is predominantly governed by the local randomness of the specific disordered pattern, and therefore the beam-direction remains random for all values of the disorder strength. The error bars shown in Fig. 3(c) indicate the range of variation of FWHM obtained for random configurations of nanowire arrays having identical values of percentage randomness and nanowire diameter. As can be observed, the FWHM varies over a wider range in the quasi-WGM regime and tends to decrease as the degree of randomness increases. To have an estimate of the fraction of power radiated along a specific direction, output power density at each angle has also been calculated for each resonances and is normalized by incident power density. For the different arrays considered in this study, the maximum fraction of power radiated along the direction of maximum intensity is observed to range from 0.65 − 0.82.

Table 1. Comparison of FDTD Analysis Results with the Effective Medium Theory Based TIR Relation

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In order to achieve improved directionality while retaining control over the beam-direction, a correlated disordered structure is next proposed and analyzed such that it can be conveniently fabricated employing experimental techniques such as focused ion-beam milling, electron beam lithography, or patterned epitaxy. This structure, which is illustrated in Fig. 4(a), is designed by gradually reducing h_X from 135 nm to 85 nm, while keeping h_Y constant. The geometry, albeit rotationally asymmetric, is symmetric with respect to x-axis. Resonant characteristics obtained for this array is shown in Fig. 4(b) along with corresponding radial and azimuthal mode numbers. A maximum Q-factor of 3853 is obtained at 719.5 nm for the (1, 20) WGM, the near-field distribution of which is shown in Fig. 4(c). Unidirectional emission characteristics of this mode is confirmed by the normalized far-field pattern (Fig. 4(d)), which exhibits a directional emission along the axis of symmetry (x-axis). A beam divergence angle of 140${ }^{\circ }$ is obtained in this case. It may be noted that in order to have an estimate of the minimum FWHM attainable with the graded-index-like structure, directionality characteristics of several such structures were also explored by gradually varying the value of h_X. However the lower limit of FWHM for such structures appeared to be 140${ }^{\circ }$, which is the case already illustrated in Fig. 4(d). It is envisaged that there is scope of further tuning the value of FWHM for graded-index-like structures by adopting other means, such as by non-linearly reducing the value of h_X. However such a detailed study was beyond the scope of the present work. An important aspect of the graded-index-like structure is that the emission direction for this structure remains controllable by all means as it is governed by the axis along which disorder is introduced by gradually changing the nanowire spacing. The maximum fraction of radiated power along this controlled direction is calculated to be 0.75. This value is well within the range of previously obtained fraction of power radiated along maximum directionalities for arrays having different disorder strengths.

Furthermore, the correlated disordered array offers in itself a system for which resonant wavelengths can be predicted in spite of the inherent disorder of the structure. To justify this claim, ray-optics based multiple total internal reflection (TIR) relation, κL = 2πm is considered, where κ is the wavenumber (κ = ${\eta }_{EM}\times \left(2\pi /\lambda \right)$), L is the optical path length, m is the azimuthal mode number, ${\eta }_{EM}$ is the effective refractive index, and λ is the resonant wavelength corresponding to the WGM. Here effective refractive index of the composite medium is calculated employing the following Maxwell-Garnett model [34]

To compare directionality profile of the effective medium with that of the actual structure shown in Fig. 4(a), its equivalent composite medium is also estimated based on Eq. (2). As indicated by the grayscale bar in Fig. 4(e), refractive index of the equivalent medium gradually varies from 1.623 to 1.839 along the x-axis. The far-field distribution of this medium is shown in Fig. 4(d), which is in good agreement with the directionality profile obtained for the actual nanowire array. These results indicate that in spite of rotational symmetry breaking, resonant modes, as well as directionality characteristics of a disordered structure having gradually varying refractive index can be estimated from its underlying whispering gallery mode-like wave dynamics. Such a design holds promise for realizing nanowire-based photonic and optoelectronic systems where spatial orientation of the nanowires can be tailored to attain specific near- and far-field characteristics of the whispering gallery modes, therefore making them suitable for exotic applications related to lasing, sensing, probing and quantum information processing.

4. Conclusion

In summary, whispering gallery modes in GaN nanowire based circular cavity have been investigated with particular focus on the role of disorder on its directionality properties. FDTD based analysis in the near- and far-field regime shows that notwithstanding the presence of disorder, a circular array of nanowires retains its WGM-like isotropic emission behavior up to a certain degree of spatial randomness. As the disorder is increased, high-Q Anderson localized resonant modes are obtained with unidirectional light output, though the beam direction remains random because of the underlying random scattering process. This limitation can be overcome by designing a system of correlated disorder, where the gradual change of refractive index predefines the beam direction, as well as manifests in itself predictable WGM-like characteristics.