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Abstract
Light transmission characteristics in a strongly disordered medium of dielectric scatterers, having dimensionalities similar to those of self-organized GaN nanowires, is analyzed employing finite difference time domain analysis technique. While photonic bandgap like transmission gaps have already been reported for several quasi-crystalline and weakly disordered media, the results of this work show that in spite of the lack of any form of quasi-crystallinity, distinct transmission gaps can be attained in a strongly disordered medium of dielectric scatterers. In fact, similar to the case of a two-dimensional photonic crystal, transmission gap of a uniform random medium of GaN nanowires can be tuned from ultra-violet to visible regime of the spectrum by varying diameter and fill-factor of the nanowires. Comparison of transmission characteristics of periodic, weakly disordered, correlated strongly disordered and uniform strongly disordered arrays having nanowires of identical diameters and fill factors suggest that in spite of the dominance of multiple scattering process, the underlying Mie and Bragg processes contribute to the emergence and tunability of transmission gaps in a strongly disordered medium. Without any loss of generality, the findings of this work offer significant design latitude for controlling transmission properties in the strong disorder regime, thereby offering the prospect of designing disorder based novel photonic and optoelectronic devices and systems.
© 2021 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
1. Introduction
A photonic crystal (PC) offers in itself a synthetic medium where propagation of electromagnetic (EM) wave is prohibited over a specific frequency range known as the photonic band gap (PBG) [1]. The unique characteristics of PCs, which arise from periodic variation of dielectric constant of the constituent medium, have inspired researchers to control the propagation and confinement of light in a variety of systems, which include but are not limited to optical switches, microcavities, waveguides and filters [2–5]. Though periodicty is integral to the design of PC based structures, interestingly the requirement of perodicity for the existence of PBG has somewhat been diluted by recent studies, which suggest that PBG-like transmission gaps can exist in quasi-periodic, or even in disordered media. Band gaps for TM- or TE-polarized wave have been theoretically predicted or experimentally demonstrated for photonic quasi-crystals where quasi-periodic variation of dielectric constant results in the scattering required to inhibit the flow of light over a specific range of the spectrum [6–8]. Complete photonic band gaps have also been attained in two-dimensional quasicrystals designed based on the concept of hyperuniformity, which represents an exotic amorphous state of matter that lies between perfect crystals and ordinary fluids [9,10]. A weakly disordered system of dielectric scatterers, derived from their perfectly ordered counterparts, also reportedly exhibit PBG-like transmission gaps for a specific polarization of the EM wave [11,12].
In spite of the referred detailed studies on the formation and characterization of PBG in periodic, quasi-periodic, and hyperuniform structures, there has not been any report on whether similar transmission gaps can be attained in strongly disordered systems, where any form of quasi-crystallinity is non-existent. A two-dimensional (2D) uniform random array of dielectric scatterer deserves particular attention in this regard as such an array is representative of the 2D cross-section of molecular beam epitaxy (MBE) grown self-organized nanowires, which have gained extensive utilization among the photonics and optoelectronics research community owing to their superior electrical properties and light-trapping characteristics [13,14]. From materials perspective, GaN-based nanowires grown on silicon are particularly prospective as lasers and detectors extending from UV to near-infrared regime of the spectrum have been experimentally demonstrated based on these nanostructures [15–19], and more recently a monolithic photonic integrated circuit has been reported based on III-nitride nanowires grown on (001) silicon substrate [20]. Though tunability of Anderson localized resonant modes have been confirmed in such self-organized nanowire arrays [21], it remains to be seen whether PBG-like tunable transmission gaps can also be attained in these systems of uniform random scatterers.
Considering self-organized nanowire array to be the perfect testbed for exploring the possibility of attaining tunable transmission gaps in the strongly disordered regime, in this work we present a systematic study on the propagation of light in uniform random arrays of dielectric scatterers having dimensonalities and areal densities similar to those of MBE-grown self-organized GaN nanowire arrays on silicon. To obtain a comprehensive understanding of wave propagation in the strong-disorder regime, transmission characteristics of light in perfectly ordered, correlated weakly disordered and correlated strongly disordered arrays have also been assessed and compared. The finite difference time domain (FDTD) based theoretical analysis presented in this work suggests that in spite of the utter lack of periodicity, strongly disordered systems exhibit PBG-like transmission gaps which can be predicted from diameter and areal density of the nanowire arrays. Furthermore, without any loss of generality, we show that such transmission gaps can be tuned by controlling areal density and diameter of the uniform random array of nanowires. Detailed analysis of spectral characteristics suggest that in spite of the high degree of disorder, the underlying Mie and Bragg scattering resonances result in the emergence of band gap and the consequent tunability of transmission properties in the strong-disorder regime.
2. Analyzed structures and methodology
As the present study aims to investigate transmission properties in the strong disorder regime, it is important to define geometrical and structural aspects of the transport medium accordingly. To this end, we resort to the scattering t-matrix based analysis reported by Arya $et~ al.$ [22]. For a random medium of circular scatterers, this theoretical framework defines a characteristic scattering length $l_{c}=\sqrt {3 / \pi }(c / \omega )$, where $\omega$ is angular frequency and $c$ is velocity of EM wave in the background medium. For strong-localization to occur, the condition $l / l_{c} \leq 1$ needs to be satisfied where $l=c /(2 \gamma )$ is known as the diffusion length. Here
Fig. 1. (a) Contour plot of $l/l_c$ for different diameters and fill-factors of the nanowire arrays; (b) cross-sectional view of uniform random and (c) periodic array having identical diameter and fill-factor of 60 nm and 50% respectively; (d) disorder strength ($\zeta$) as a function of allowed variation for different fill-factors of the arrays (inset shows unit cell of periodic array having diameter d and lattice constant a); (e) weakly correlated disordered ($\zeta = 0.1425$), (f) and strongly correlated disordered ($\zeta = 0.9963$) arrays with diameters and fill-factors of 60 nm and 50% respectively.
To compare transmission characteristics of the uniform random, correlated strongly disordered and weakly disordered nanowire arrays, correlated random arrays are systematically generated based on the periodic structure shown in Fig. 1(c). It is to be noted that nanowire diameter and fill-factor of this periodic structure are identical to those of the uniform random array. The unit cell of this periodic structure, which has a lattice constant of 75 nm, is shown as an inset to Fig. 1(d). By defining center co-ordinates of nanowires of the periodic array as (x, y), co-ordinates (x+$\delta$x, y+$\delta$y) of correlated disordered arrays have been generated using the following relation:
Here P is a percentage randomness parameter defined as $\mathrm {P} \in [0, 100]$, $\{r_x,~r_y\}$ are uniform random numbers within the range [-1, 1] and $R$ is the allowed range of variation. To avoid overlapping of nanowires in an array, center-to-center distance between adjacent nanowires is maintained to be larger than their diameters. To have a quantitative estimate of the degree of disorder of an array, the parameter disorder strength ($\zeta$) is defined and calculated as follows [24]:
Here $A$ and $B$ are one dimensional vectors created by converting two-dimensional refractive index profiles of the periodic and correlated-disordered structures respectively, N is the total number of elements in each array, $\mu _A$ ($\mu _B$) and $\sigma _A$ ($\sigma _B$) are mean and standard deviation of vector $A$ ($B$) respectively. In Fig. 1(d), $\zeta$ is plotted as a function of $R$ for different fill factors of the arrays. As can be observed, even for the highest value of the randomness parameter (P = 100), disorder strength can be substantially low if the allowed range of variation is small. It has been reported earlier that an allowed range of variation of $a/5$, which corresponds to $\zeta = 0.20$ according to Fig. 1(d), results in a weakly disordered system. This is in accordance with our previous work where it has been shown that strong-localization of light in GaN nanowires occur in random systems having disorder strengths of about 0.4 or higher [21]. In this work, by considering $R$ = $a/2$, correlated disordered arrays having $\zeta$ values of 0.14 and 0.99 (shown in Figs. 1(e)–1(f)) have been generated as representative weakly- and strongly-disordered systems respectively. It is noteworthy that though both the arrays of Figs. 1(b) and 1(f) represent strongly disordered systems of identical fill factors and nanowire diameters, the former represents a uniform random system whereas the latter is a correlated disordered array generated based on the periodic array of Fig. 1(c).
Spatial, temporal and spectral aspects of light-transmission characteristics in the nanowire arrays have been analyzed based on finite difference time domain (FDTD) analysis technique. Open-source software package MEEP has been utilized for FDTD simulation of both random and periodic arrays [25]. Bandstructures of the periodic arrays have been estimated using open-source eigen-frequency solver MPB [26]. Throughout this study a Gaussian pulse source having spectral extent of 200 nm to 600 nm is end-fired from one end of the array, and transmittance is measured from the opposite end. Phase matched layer (PML) having thickness greater than the highest wavelength of the source is utilized around the computational region to model an open-system. Transmitted and reflected flux have been spectrally resolved and normalized to have an estimate of transmission properties of the arrays.
3. Results and discussions
Prior to investigating transmission properties of strongly-disordered systems, wave propagation characteristics of periodic nanowire arrays are analyzed. To have an estimate of the transmission gap of the periodic system of Fig. 1(c), dispersion relations corresponding to both transverse electric (TE) and transverse magnetic (TM) modes have been computed based on iterative eigensolving techniques [26]. As can be observed from Fig. 2(a), though a complete PBG is non-existent in the periodic system, distinct stop-band is observed for TM mode over the wavelength range of interest (250 nm 600 nm) of the present study. Signature of photonic bandgap becomes evident in the periodic system’s transmission and reflection spectra as well (Fig. 2(b)). As can be observed, clear transmission (reflection) gap (band) exist in the transmission (reflection) spectrum for TM modes, whereas TE mode exhibits near-unity transmission over the considered wavelength range. Nearly equally spaced spikes are observed in the flux spectra of Fig. 2(b) for wavelengths larger than 400 nm. These spikes are in fact related to Fabry-Perot (FP) resonance between boundaries of the nanowire array. By varying length (L) of the array, we notice that the spacing ($\Delta \lambda _{m}$) between adjacent spikes vary in accordance with the relation $\Delta \lambda _{m}=\frac {2 L}{m}-\frac {2 L}{m+1}$, where $m$ is the mode number of FP resonant mode. It is also noteworthy that the PBG calculated in Fig. 2(a) is narrower than the transmission gap shown in Fig. 2(b). Such difference arises from finite dimensionality of the nanowire array. The bandstructure shown in Fig. 2(a) is for a photonic crystal of infinite extent, whereas the transmittance spectra of Fig. 2(b) are computed for a $3 \times 3$ $\mu$m$^2$ periodic array. In fact, transmittance of this finite array is not exactly zero over the transmission gap, though the value remains negligibly small (from $10^{-6}$ to $10^{-3}$) for all practical purposes. Nevertheless, bandstructure calculations and FDTD simulation results suggest that transmission gap of the finite dimensional periodic system is directly correlated to PBG of the photonic crystal, and for the considered dimensions of GaN-nanowire arrays, the transmission gap varies from UV to visible regime of the spectra for TM-polarized waves. Considering controllability of transmission characteristics over this wavelength range in GaN-based systems, this study henceforth investigates transmission characteristics for TM-polarized incident waves.
Fig. 2. (a) Photonic bandstructure of both TE and TM modes of periodic array having 70 nm diameter and 50 % fill-factor (the PBG under consideration is highlighted in green), and corresponding (b) transmittance (T) and reflectance (R) spectra; (c) contour plot of transmittance for different diameters with a constant fill-factor of 50 %, and (d) transmittance spectra for varying fill-factors of periodic arrays having nanowire diameters of 70 nm (solid lines) and 60 nm (dotted lines).
Tunability of transmission characteristics for TM mode, obtained by adjusting diameters and fill-factors of periodic nanowire arrays, are shown in Figs. 2(c)–2(d). As nanowire-diameter increases from 60 nm to 90 nm at a constant fill-factor of 50 %, the transmission-gap red-shifts from $\sim$300 nm to 450 nm. The bandwidth of the transmission-gap also increases from about 52 nm to 82 nm for such variation of diameter. Opposite trends are observed for fill-factor variation while the diameter is kept constant (Fig. 2(d)). As the fill-factor increases from 30% to 50%, the transmission-gap shifts towards smaller wavelength, and also the bandwidth decreases gradually from about 100 nm to 62 nm. Such dependence of transmission-gap on diameter and fill-factor can be explained in terms of lattice constant ($a$), which is related to diameter and fill-factor of the array according to the relation $a=\sqrt {0.25 \pi d^{2} / f}$. This relation suggests that the increase (decrease) of diameter (fill-factor) while keeping fill-factor (diameter) constant effectively increases lattice constant of the array. Therefore according to the dispersion relation, a red-shift of the PBG is expected for the increase (decrease) of diameter (fill-factor) of the periodic structure. It is also noteworthy from Fig. 2(d) that upper-edge of the transmission-gap is governed by fill-factor of the array, whereas nanowire dimension defines lower-edge of the gap. Therefore significant tunability of transmission characteristics can be attained by varying fill-factor and diameter of nanowires of the periodic array.
Having analyzed transmission properties of periodic nanowire arrays, wave propagation characteristics of correlated- and uniform-disordered structures are analyzed and compared in what follows. In Fig. 3(a), transmission and reflection spectra of correlated and uniform disordered nanowire arrays are compared with those of the equivalent periodic array. As can be observed, in spite of the lack of periodicity, transmission gaps similar to the photonic gap of a periodic system are obtained for the disordered arrays. Though such characteristics have earlier been reported for weakly disordered systems operating in the microwave regime [12], here it is noteworthy that stop-bands of near-zero transmission are obtained in the strong-disordered regime as well. In addition to the transmission gap, correlated disordered systems having a low degree of disorder (e.g. with $\zeta$=0.14) tends to retain some of the FP resonant peaks, which have been observed earlier in the transmission spectra of the periodic structure. However in both uniform random and strongly-disordered systems, such resonant peaks are non-existent because of the dominance of random scattering events. The log-scale plot of transmission spectra (shown as an inset of Fig. 3(a)) of the correlated strongly disordered system (with $\zeta$=0.99) and uniform random system suggests that transmittance is about an order of magnitude smaller in the uniform random one. It may be noted that uniform random systems are spatially uncorrelated to the periodic ones and are generated based on uniform random distributions. Consequently, a disorder strength based on Eq. (3) cannot be evaluated for a uniform random array. Nonetheless, it is quite obvious that the emergence of transmission gap is expected to be more pronounced in uniform random systems than in strongly disordered arrays having near-unity $\zeta$ values. Also transmission-gap shrinkage appears to be more pronounced in the uniform random system. As will be discussed later, this is possibly related to the higher fraction of Bragg process in such media. It may also be noted that though transmission characteristics of periodic and disordered systems over the gap region appear to be similar in terms of absolute value of transmittance, transmission coefficients and therefore transmission phases are expected to be different. A theoretical study on the comparison between transmission coefficients of photonic crystals and disordered arrays was beyond the scope of the present work.
Fig. 3. (a) Transmittance spectra of periodic, weakly correlated ($\zeta = 0.1425$) disordered, strongly correlated disordered ($\zeta = 0.9963$), and uniform random systems having identical diameter and fill-factor of 60 nm and 50 %, respectively (FP resonant modes appearing at same wavelengths for periodic and weakly correlated arrays have been marked with circles); inset shows log-scale plot of transmittance of random and strongly disordered arrays; (b) configurational average of transmittance and reflectance spectra for four different uniform random arrays having 70 nm diameter and 50 % fill-factor; (c) contour plot of transmittance for different diameters with a constant fill-factor of 50 %, and (d) transmittance spectra for different fill-factors of uniform random arrays having nanowire diameter of 70 nm (inset shows the relative bandgap ($\chi$) as a function of disorder strength ($\zeta$)).
To ascertain that such observation of PBG-like near-zero transmission is independent of spatial distribution of the array, configurational averaged transmitted and reflected flux for four different random arrays having identical diameters and fill-factors are calculated and plotted in Fig. 3(b). The flux spectra confirm that distinct and repeatable transmission gaps are obtained for the strongly-disordered random arrays irrespective of spatial configuration of the scatterers. Besides exhibiting transmission-gaps, strongly disordered systems demonstrate tunability of transmission characteristics similar to the case of a periodic system. As shown in Fig. 3(c), the center of the TM transmission-gap of a uniform random array having $f$ = 50 % can be conveniently tuned from about 370 nm to 410 nm by varying nanowire diameter from 60 nm to 90 nm. Also similar to the case of a periodic system, transmission-gap of the uniform random system can be tuned towards smaller wavelength by increasing the fill-factor, while maintaining a constant diameter of the nanowires. Such characteristics are shown in Fig. 3(d), where fill factor of uniform random arrays having $d = 70$ nm is varied from 30 % to 50 %.
In the context of perturbation theory, it is understandable that transmission properties of a correlated weakly-disordered array can be predicted from transmission characteristics of the corresponding periodic system. However it is quite remarkable that in spite of the lack of spatial-correlation with the periodic system, a uniform random array exhibits tunable transmission characteristics over the same wavelength regime as that of the periodic one. From these observations it can be inferred that transmission characteristics of a periodic system are retained even in an uncorrelated strongly-disordered one, if identical dimensions (represented here by diameter) and areal-density (represented here by fill-factor) of scatterers are maintained within the two media. Such similarity in transmission properties possibly emanate from the underlying Bragg process in these structures. To gain further insight into this phenomenon, transmission gaps of the nanowire arrays are carefully examined. A closer observation of the transmittance spectra of Fig. 3(a) suggest that transmission-gap of the arrays gradually shrink from 52 nm to 35 nm with increasing disorder in the arrays. This is further illustrated in the inset of Fig. 3(d) where relative transmission-gap, $\chi =\Delta \lambda (\zeta )/\Delta \lambda (0)$ is plotted as a function of disorder strength. Here $\Delta \lambda (\zeta )$ is transmission-gap of an array having disorder strength of $\zeta$. As can be observed, transmission-gap of the array decreases by about 50 % as the system transits from a perfectly ordered to a disordered one.
Whereas sustenance of band-gap in a disordered array is attributed to the Mie process, band-gap shrinkage is considered to be a signature of Bragg process in the medium [12,27]. As the Mie process is governed by coupling of modes between adjacent scatteres, it can contribute to band-gap formation even in the lack of periodicity in the medium. The randomness however debilitates the Bragg process, which is dependent on long-range periodicity of the array, thereby resulting in shrinkage of the transmission-gap. It has been shown in [27] that the Mie and Bragg processes co-exist in weakly disordered dielectric media having $\zeta$ values of 0.15 or smaller. Here we observe that such co-existence extends into correlated, as well as uncorrelated strongly disordered regime. It is therefore likely that the Bragg process, inherent in both the periodic and random systems, is giving rise to the tunable transmission characteristics. To have a quantitative estimate of the contributions of Bragg and Mie scattering towards the emergence of transmission gap, we have adopted the formalism described by Nojima $et~ al.$ [27]. In this approach, the contribution of Bragg scattering towards bandgap formation in a disordered medium of dielectric scatterers is estimated using the following relation:
Here $\Gamma _{Bragg}$ is the relative contribution of the Bragg process, $\chi _{dielectric}$ is relative transmission gap of a disordered system of dielectric scatterers, and $\chi _{metal}$ is relative transmission gap of the disordered system of metallic scatterers having identical spatial co-ordinate, diameter and fill factor as of the dielectric system. Based on Eq. (4), Bragg contribution towards the formation of transmission gap is plotted as a function of disorder strength in Fig. 4. The error bars shown in this plot indicate the range of $\chi$ and $\Gamma _{Bragg}$ obtained from statistical variation of spatial co-ordinates of the arrays. As can be observed, the Bragg contribution tends to increase with disorder strength. For weakly disordered systems, the average Bragg contribution tends to be around 30%, which is consistent with the results reported in [27]. On the other hand, for strongly disordered systems having $\zeta$ > 0.4, the average Bragg contribution is obtained to be 57%. Such contributions of Bragg process in GaN based disordered medium is consistent with the results reported in [27,28], which show that for dielectric disordered medium having refractive index of 3.5 or smaller, Bragg process is expected to be dominant. It is also noteworthy that upon statistical simulation, Bragg contribution in uniform random arrays is obtained to be higher (around 80%) compared to the value of $\Gamma _{Bragg}$ obtained for correlated strongly disordered systems. This explains our previous observation in regards to Fig. 3(a) that transmission-gap shrinks more in uniform random systems, compared to the case of a correlated disordered system having near-unity disorder strength.
Fig. 4. Relative bandgap and Bragg contribution plotted as a function of disorder strength for correlated disordered arrays of different disorder strengths
To have a comparative estimate of tunability in the strongly disordered and perfectly ordered regimes, center wavelength ($\lambda _c$) of transmission-gap and gap-midgap ratio ($\Delta \lambda /\lambda _c$) are plotted in Figs. 5(a)–5(b) as a function of diameter and fill-factor of the periodic and uniform random arrays. It is quite obvious that both systems exhibit similar characteristic trends, though some fluctuations are observed for the random arrays because of configurational dependence of the flux spectrum. It is also noteworthy that $\lambda _c$ values obtained for the random arrays are smaller than those of the periodic ones because of the corresponding bandgap shrinkage. Bandgap shrinkage also results in a reduced value of $\Delta \lambda /\lambda _c$ for the random systems. Notwithstanding this, similar tunability of transmission characteristics for the random and periodic systems become evident from these results and analysis. It is quite interesting to note that tunability of Anderson localized resonant modes has been reported earlier in [21] for GaN-based self-organized nanowire arrays having dimensionalities similar to those reported herein. Upon careful examination of the results reported in [21], we observe that for identical diameter and fill-factor of GaN nanowire arrays, the Anderson localized resonant mode invariably lies within the transmission gaps reported in this work. This is quite similar to the concept of designing high quality factor resonant modes by employing point or line defects in photonic crystals. Just as the resonant mode of a PC-based resonator can be tuned by controlling PBG of the underlying structure, we notice that the Anderson localized resonant modes can be tuned by controlling transmission gap of the corresponding strongly-disordered scattering medium.
Fig. 5. (a) Dependence of center wavelength and gap-midgap ratio on nanowire diameter (for a constant fill-factor of 50 %) and (b) fill-factor (for a constant diameter of 70 nm), for both periodic and uniform random systems.
From the discussions so far it is quite obvious that scattering is expected to play a dominant role to reduce output transmission in disordered systems comprising of dielectric scatterers. To have an estimate of the degree of scattering in these systems, transmittance of perfectly ordered nanowire arrays and uniform random nanowire arrays of identical diameters ($d$ = 60 nm) and fill factor ($f$ = 50 %) are compared for different lengths ($L$) of the waveguides. For all these waveguides, the width is kept fixed at 3 $\mu$m. The transmittance plot of Fig. 6(a) suggests that transmittance of the periodic system remains invariant with the system length. However in the uniform random system, the transmittance tends to decreases with increasing propagation length. This is further illustrated in Fig. 6(b), which plots the light output at a transmission wavelength of 550 nm. As can be observed, the output of the random system decreases linearly with system length, whereas in the periodic system it almost remains constant. Similar characteristic trends have been experimentally reported for a self-organized nanowire array based green laser [29]. To have a quantitative estimate of the degree of scattering in periodic and random arrays, light scattering in both these systems have been analyzed by computing the in-plane scattered flux, perpendicular to the direction of wave transmission. The flux planes along which scattering is calculated are denoted as ‘Scattering Monitors’ (Fig. 6(c)). As can be observed, at wavelengths above the transmission gap, scattering is minimal in the periodic systems for all lengths of the waveguide. However in the random system, the fraction of scattered light tends to decrease with increasing length of the waveguide. As shown in Fig. 6(d), the fraction of in plane scattered light can be as high as 40 % in a random system having $L=7\,\mu$m. This explains the considerably low output transmission of the random system. For both periodic and random systems, scattering appears to be small within 260 nm -290 nm as this wavelength range falls within the transmission gap of the array and consequently a very small fraction of the incident light is collected by the scattering monitors. To further clarify this point, we have included as insets electric field distributions in these arrays for an incident Gaussian flux having center wavelength of $\lambda = 277$ nm. As can be observed, almost all the light is reflected back to the source for this incident flux, and consequently the light collected by the scattering monitor comes out to be significantly small. Similar characteristics are obtained for any wavelength of the input flux residing within spectral range of the transmission gap. As expected, beyond the transmission gap, in-plane scattering appears to be significantly higher for the random system. In spite of such high degree of scattering, the random system offers the tunability of transmission gap in accordance with characteristics of periodic system having identical diameter and fill-factor of the nanowires, thereby offering the prospect of designing disorder based photonic systems, including photonic integrated circuits, where disorder can be exploited to obtain predictable output characteristics within a degree of uncertainty. The findings presented here can also be extended to modulate the behavior of light in a variety of nano-engineered disordered media, which are becoming increasingly important for applications related to biological imaging, biosensing, non-invasive medical diagnosis, nanophotonics and quantum information processing.
Fig. 6. (a) Transmittance spectra of periodic and uniform random systems having nanowire diameter, array width and fill-factor values of 60 nm, 3 $\mu$m and 50 %, respectively, while the system lengths (L) is varied from 3 $\mu$m to 7 $\mu$m, and (b) the corresponding output (in dB) at 550 nm wavelength; (c) the resultant scattering spectra (normalized by input flux) of the periodic, and (d) uniform random arrays (detection plane of scattered flux is shown as an inset of (c)); electric field distributions in periodic and random arrays for an incident wavelength of $\lambda$=277 nm have been shown as insets of (c) and (d) respectively.
4. Conclusion
To summarize, two-dimensional uniform random scatterers having dielectric constant and dimensionalities similar to those of self-organized GaN nanowire arrays have been considered to investigate wave propagation in the strong-disorder regime. The finite-difference time domain based numerical analysis of this study suggests that similar to the case of perfectly ordered and weakly disordered systems, strongly-disordered arrays exhibit transmission gaps which can be controlled by varying diameter and areal density of the scatterers. Configurational averaging based light transmission characteristics of the random arrays suggest that band gaps in such systems emanate from the complex interplay between Bragg scattering and Mie resonance in the media. Though in-plane multiple scattering significantly diminishes light output at the receiving end of these arrays, it is quite remarkable that the underlying Bragg scattering remains strong enough for the transmission gap to be tuned over a pre-defined range of the EM spectrum. Such observance of transmission gap and its subsequent tunability offers the prospect of designing self-assemebled, disordered array based transmission medium, where disorder can be tailored to meet specific requirements of the photonic systems.
Acknowledgments
The authors acknowledge the support and facilities obtained from the Department of Electrical and Electronic Engineering (EEE), Bangladesh University of Engineering and Technology (BUET). M. I. Tahmid also acknowledges the facilities received from the Department of EEE, Shahjalal University of Science and Technology (SUST) during the course of this work.
Disclosures
The authors declare that there are no conflicts of interest related to this article.
Data availability
Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.
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