1. Introduction

Recently, optical nanowires and nanofibers have been widely used in assembling microphotonic devices such as couplers, filters, interferometers, resonators, and lasers, with indices spanning over a wide range [1–14]. In particular, nanoscale light emitting devices relying on high-index semiconductor nanowires have been demonstrated [1,2]; interrupted nanowires with flat endfaces have shown possible applications in subwavelength-dimension illuminations, photon-momentum-induced effects, all-optical switching, nanoprobe sensing, laser trapping, and laser scalpel for nanosurgery [15–21]. In all these applications, the endface effects are critical to the behaviour of the nanowires. In nanowire lasers, for example, the endfaces are believed to function as the two mirrors of Fabry–Perot cavity. The reflectivities at the endfaces of nanowires play important roles in determining the quality factor Q, and thus the lasing threshold. However, due to the difficulty of handling nanowires, experimental measurement of reflectivities of a single nanowire has not yet been reported. Recently, numerical and analytical studies of reflections of the lowest-order guided modes (HE₁₁, TE₀₁, and TM₀₁) at the endface of a semiconductor nanowire (ε = 6) have been carried out [22–24]. These results offer helpful references for understanding lasing mechanism and improving the design of nanolasers. But the dependence of endface reflectivity on nanowire parameters (e.g., diameter and index) and wavelength of the guided light has not been adequately investigated.

In this paper, we studied the endface reflectivities (ERs) of optical nanowires using the three-dimensional (3D) finite-difference time-domain (FDTD) method [25,26]. Typical ERs of silica, tellurite, PMMA and semiconductor nanowires are calculated, and the dependences of ERs on the diameter and index of the nanowire, and wavelength of the light are also computed. Results presented here may offer valuable references for estimating ERs of optical nanowires for practical applications, including nanowire or nanofiber-based resonators and lasers, or other endface-reflection-based devices.

2. Numerical simulation model

FDTD simulation is considered as one of the best numerical methods to study light propagation in small dimensions, regarding the accuracy and efficiency [26]. In this work, the FDTD simulations are performed by Meep [27]. We assume that the nanowire has a circular cross-section, and a step-index profile which is general for many types of nanowires. The diameter and the length of the nanowire are assumed to be D and L, respectively. We have considered two cases, free-standing nanowire, and nanowire supported by a low index substrate (Fig. 1 ). Cartesian coordinate is used with its origin located at the center of the left endface of the nanowire. The nanowires we investigated are operated in single mode. Eigenmodes obtained by the mode analysis are launched along the nanowires. The source is assumed to be x-polarized for free-standing nanowire. Instead, sources of two polarization states (x- and y-polarization) are used separately for nanowire with a substrate. The computational domain is divided into a uniform orthogonal 3D mesh with the length of cell size less than one twentieth of the light wavelength, and terminated by perfectly matched layer (PML) boundaries [25].

 

Fig. 1 Schematic of a nanowire with substrate.

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To obtain the ERs of nanowires, we use two steps to run the simulation. In the first step, we set the left and right PML boundaries next to the left and right endfaces of nanowire, respectively. In this way, only incident wave is propagating along the fiber with no reflections at both endfaces. A vertical plane one wavelength away from the right side of the source plane is chosen to calculate the total incident energy flux. For the second step, the right side PML is set ten wavelengths away from the right endface of nanowire so that ER exists at the right endface. Energy flux is calculated again at the same plane as the first step. Finally the ER is calculated based on the two energy fluxes from these two steps.

3. Results and analysis

To verify our codes, we repeated the Ning’s simulation [22]. A free-standing semiconductor nanowire is used, with diameter of 120 nm and index of refraction as 2.45, the ERs of which are calculated for the wavelengths from 340 nm to 500 nm. As is shown in Fig. 2(a) , the ER decreases with the increasing of wavelength, which is due to the higher fractional diffraction at the endface. It is noticed that the ERs of nanowires are much different from those of conventional fibers, which are usually estimated by the ERs at normal incidence (n-1)²/(n + 1)². Interestingly, for single-mode nanowire, the ERs of fundamental mode (HE₁₁ mode) are less than the one of normal incidence on an infinitely large dielectric/air interface. Our results show good agreement with the Ning’s simulations (red dots in Fig. 2(a)).

 

Fig. 2 (a) ERs of 120-nm-diameter and 2.45-index semiconductor nanowire. (b) ERs of PMMA, tellurite, and silica nanowires with various diameters.

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Using the same method, we investigate the ERs of free-standing silica, PMMA and tellurite nanowires. The indices of air, PMMA and tellurite are assumed to be 1.0, 1.59 and 2.05, and the index of silica is obtained from the chromatic dispersion relations [28]. Figure 2(b) shows the calculated ERs of free-standing 250-nm-diameter tellurite, 250- and 400-nm-diameter PMMA and 400-nm-diameter silica nanowires with the wavelengths scanning from 500 nm to 850 nm. The ER dependence on wavelength shows similar trend as the semiconductor nanowire’s shown in Fig. 2(a). Comparing between the nanowires with the same diameter (e.g. 250-nm-diameter PMMA and tellurite nanowires), we find that higher index results in larger ER, as discussed below.

We studied the dependence of ER on material index of nanowire with certain diameter and working wavelength. The indices of the nanowire span over a range of 1.38-2.65. As shown in Fig. 3 , the ERs demonstrate monotonous increasing with the indices, due to the better optical confinement when the index is higher. For example, with index as 1.45, the ER of 600-nm-diameter nanowire at 1550-nm wavelength is as low as 0.002, which indicates ER from silica nanowire of the same parameters is negligible small. While the index increases to 2.05 as the index of tellurite, the ER increases to 0.068, and cannot be ignored. Additionally, comparing the nanowires with the same diameter of 600 nm, the nanowire at 1000 nm wavelength has relatively higher ER than the one at 1550 nm wavelength for the same refractive index, mainly due to its relatively higher effective index [29].

 

Fig. 3 ERs vs. indices of nanowires with some typical parameters.

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Similarly, diameter of the nanowire is another factor that directly affects the ER. Figure 4 shows the calculated ERs of free-standing silica nanowire at wavelengths of 633 nm and 1000 nm, tellurite nanowire at wavelengths of 1000 nm and 1550 nm, and PMMA nanowire at wavelengths of 532 nm and 980 nm, respectively. It is evident that the ERs increase with the diameters due to the stronger optical confinement and weaker endface diffraction.

 

Fig. 4 ERs vs. diameters of nanowires at various wavelengths.

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Experimentally, nanofibers and nanowires are usually placed on low index substrates (e. g. MgF₂ substrate with refractive index of 1.38) for supporting. Figure 5 shows the calculated ERs of MgF₂-supported silica and tellurite nanowires at wavelengths of 633 nm and 1550 nm, respectively. The interaction length between the nanowire and the substrate are assumed to be 3.5 μm for silica and 9.5 μm for tellurite. Two polarization states have been taken into account due to the breaking of cylindrical symmetry. It is noticeable that the ERs are considerably lower than those of free-standing nanowires, which is due to that the presence of the substrate shifts the mode profile further towards the substrate and a fraction of light guided by nanowire will leak into the substrate.

 

Fig. 5 ERs of MgF₂-supported 1.46-index silica nanowire at 633 nm wavelength and 2.05-index tellurite nanowire at 1550 nm with two polarizations. The interaction lengths are assumed to be 3.5 μm for silica nanowire, and 9.5 μm for tellurite nanowire.

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Figure 6(a) and 6(b) gives better illustration. We plotted the intensity distributions of 450-nm-diameter silica and 650-nm-diameter tellurite nanowires on y = 0 plane, as well as the output patterns on z = L plane. Due to the small index-contrast between silica nanowire and substrate, no guided mode exists in the nanowire shown in Fig. 6(a), as also demonstrated experimentally [30]. Thus a large fraction of the energy is leaked to the substrate over a short interaction length of 3.5 μm, resulting in a negligible ER of 0.002 (blue open and half-open circles in Fig. 5). Tellurite, on the contrary, has higher index. As shown in Fig. 6(b), little fraction of light guided by 650-nm-diameter tellurite nanowire is attracted into the substrate and no obvious mode profile shift are observed on z = L plane. As results, the ERs of tellurite nanowires are considerably large (red open and half-open squares in Fig. 5). Besides, ERs of x-polarization are smaller than those of the y-polarization, due to better coupling of the x-polarized mode to the substrate.

 

Fig. 6 The intensity distributions in y = 0 plane and output patterns in z = L plane of (a) 450-nm-diameter silica nanowire (point a in Fig. 5), and (b) 650-nm-diameter tellurite nanowire (point b in Fig. 5) on y = 0 and z = L planes. The incident waves are y-polarized.

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It’s also interesting to study the ER dependence on the interaction length. Taking the similar silica nanowire structure as shown in Fig. 6(a), we tried to vary the interaction length from zero to 7 μm. The ER shows a monotonous decreasing till reaching a negligible value. Similar simulations were carried out for tellurite nanowire as shown in Fig. 6(b). The ER first shows a decrease when the interaction length between tellurite nanowire and substrate is short. While the interaction length exceeds 1.9 μm, ER eventually stabilizes around 0.06. The change of ER with interaction length mainly comes from the effect of energy leakage induced by the substrate. For the tellurite nanowire with 9.5μm interaction length, we placed the energy flux calculation plane one wavelength away on the left side of the right endface to calculate the modified ERs. The optical mode is stable close to the flux plan, and thus the energy leakage effect is eliminated. Take x-polarization for instance, we plot the dependence of the modified ER as a function of wire diameter (black curve in Fig. 5). The modified ERs deviate from the ERs with substrate to the ERs of free-standing tellurite nanowire when the diameters increase.

4. Conclusions

In conclusion, we have numerically studied the ERs of optical nanowires via 3D-FDTD method. Typical ERs of both free-standing and substrate-supported silica, tellurite, PMMA and semiconductor nanowires are calculated. The nanowire parameters effects on ERs are investigated. It shows that, ERs of nanowires strongly depend on the wire diameter and the wavelength of the propagation light. This is significantly different from conventional fibers. Although the cross sections of a number of nanowires are non-circular (e.g., hexagon), considering the diameter used here is close to or below the wavelength of the guided light, the behaviour of ERs should be similar as the circular cases discussed above. Our results may offer valuable references for estimating ERs of optical nanowires for practical applications, including nanowire-based resonators and lasers, or other endface-reflection-based devices.