Abstract

In order to build nanophotonic devices, it is important to understand and ultimately control the optical mode structure within potential components such as nanoscale waveguides. However, experimental characterization of such modes in the optical regime is difficult due to the nanoscale dimensions of such components and the perturbations that would be induced by a near-field probe. Here, we demonstrate a probe-free, far-field method to characterize the optical modes within GaN nanowires (NWs) based on a novel off-axis scanning confocal microscope system. Using this microscope, we observe spectral signatures resulting from lateral leakage of waveguide modes when they exceed their respective cutoff limits. We identify these modes within hyperspectral images using an analytical model coupled with finite element simulations. The model can also be used to predict the spectral signatures for given geometrical and dielectric parameters, which enabled us to deduce the transverse dimension of the NW from hyperspectral images with an accuracy of ${\sim}30\;\text{nm}$.

© 2021 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

1. INTRODUCTION

In recent years, semiconductor nanowires (NWs) have become promising as potential components in nanophotonic devices. Researchers have successfully fabricated numerous NW structures [1,2] and demonstrated various applications including waveguides [3], nano-light sources [47], nano-antennas [8], optical logic gates [9], optoelectronics [7,10], and photodetectors [1113], among others. Development of more complex integrated nanophotonic devices [14] requires detailed knowledge of light generation, propagation, distribution, and the structure and composition of optical modes within component nanostructures over a range of potential wavelengths. In particular, if NWs are to be used to couple different nanophotonic components, the spatial and spectral aspects of guided and lossy modes, including higher-order transverse modes, should be characterized with fidelity. However, since the transverse dimensions of NWs are comparable to or smaller than visible wavelengths, it has been challenging to experimentally characterize the modal structure within a NW waveguide—the precise mode composition across the optical spectrum at each longitudinal position along the NW. To our knowledge, existing methods such as white-light scattering measurements [15,16] and far-field polarization-resolved emission pattern analysis [17] have not been used to spatially resolve the modal structure along the NW length. Furthermore, numerical simulations of semiconductor NWs can produce unreliable results since they often utilize models based on idealized geometries and bulk material properties. Thus, it is important to develop new approaches to measure, and ultimately control, the distribution of light within nanophotonic components and composite devices.

Nanoscale waveguides will play an important role in transporting energy and information between various components within integrated nanophotonic devices. Here, we investigate the waveguide modes inside gallium nitride (GaN) semiconductor NWs with triangular cross sections large enough to support light propagation (Fig. 1). Several optical properties of GaN NWs make them interesting for use as nanophotonic components [4,6] and waveguides in particular. GaN has a relatively high index of refraction (${\sim}2.4$ over most of the visible range [18]) and has a bulk bandgap in the near UV (3.4 eV $\approx 365\;\text{nm} $) [19], which makes it a good dielectric for visible light wavelengths. GaN nanostructures have been shown to exhibit broad fluorescence emission across the entire visible spectrum [19], originating from defects [19] and surface states [2023] within the bandgap. This broad emission is particularly useful here as it provides a mechanism to couple different wavelengths into the waveguide using a single, monochromatic excitation laser.

 

Fig. 1. Schematic of experiment. (a) Scanning electron micrograph of the NW growth substrate. (b) Rendering of an atomic force microscope (AFM) image of a GaN NW lying atop a glass coverslip. The outline of the NW is indicated by red lines. (c) A simplified schematic of the optical setup: a scanning mirror is positioned at a Fourier plane that is conjugate to the back focal plane of the objective lens allowing the confocal collection volume to be displaced from the laser focus. (d) Fluorescence spectrum collected from near the center of the NW.

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In this work, we primarily examine tapered GaN NWs where the cross section decreases from one end of the NW to the other. The NWs vary in length from 1–20 µm and have triangular cross sections with edge lengths of 100–600 nm and small variations in shape. By utilizing a combination of conventional fluorescence microscopy and a novel implementation of scanning confocal hyperspectral imaging, we demonstrate a method to extract modal information from NW waveguides, which is validated with numerical simulations. In particular, we use focused laser light to induce broadband emission from the NW, some of which can propagate along the NW axis in multiple guided modes. We find that when the transverse dimension of the tapered NW becomes too small to support a specific mode at a particular wavelength [2426], some of the fluorescence “leaks” from the waveguide [27], leaving a spectral signature of that mode that can be collected in the far-field. Making use of this information, we are also able to deduce fine geometrical features of the NW structure and its material properties.

2. WORKING PRINCIPLE

To understand how our approach works, we consider total internal reflection (TIR) of light inside the NW [28]. TIR at the interface between a GaN NW and a glass substrate requires that the transverse component of the $k$-vector (in the $x-z$ plane) inside the NW satisfies

$${k_{\text{trans}}} \lt |{\bf k}|\sqrt {1 - \frac{{n_{\text{glass}}^2}}{{n_{\text{GaN}}^2}}} .$$
Due to the uncertainty principle, ${k_{\text{trans}}} \gt \pi /\Delta {r_{\text{mode}}}$, where the effective mode size $\Delta {r_{\text{mode}}}$ is related to the NW edge length, $a$, and mode type/number. Thus, the NW waveguide will support modes with wavelengths shorter than the cutoff wavelength for those modes, which can be approximated by
$${\lambda _c} = 2\sqrt {n_{\text{GaN}}^2 - n_{\text{glass}}^2} \Delta {r_{\text{mode}}}.$$
Emission wavelengths longer than this limit are expected to leak out laterally from the NW rather than being confined as a guided mode [27]. Light propagating in different modes within a tapered NW will encounter the cutoff condition at different locations along the NW length, leading to leakage of the corresponding spectral components into the glass substrate. The spectral pattern of leaked light along the length of the NW thus provides a signature of the waveguide modes within the NW.

As discussed in more detail in Section 3.B, to detect these spectral signatures, we introduce a scanning mirror at a Fourier plane in the collection path, Fig. 1(c), which allows the detection volume to be separated from the laser focus [29]. In this way, emission from different points along the NW can be directed to a fiber-coupled spectrometer for analysis. Scanning the mirror produces an off-axis confocal hyperspectral image where each 2D pixel contains spectral information for the emission collected from a particular position along the NW.

3. METHODS

A. GaN Nanowire Preparation

For this study, GaN NWs were grown using metal–organic chemical vapor deposition leading to a predominantly wurtzite crystalline structure with nearly equilateral triangular cross sections [30,31]. The NWs were grown using a nickel catalyst and various substrates that matched the structure and lattice constants of GaN, and transmission electron microscopy (TEM) analysis showed no crystallographic defects in the bulk of the NW. Additional specifics on the growth and NW structure are discussed elsewhere [30]. Following growth, the NWs were mechanically transferred to clean glass coverslips, allowing isolated, single NWs to settle onto the coverslip [21]. Deposited NWs were characterized using a home-built microscope that combines optical microscopy and atomic force microscopy (AFM) capabilities.

B. Optical Setup

1. Widefield Imaging with a Focused Laser

NWs were excited with a pulsed laser beam ($\lambda = 401\;\text{nm} $, pulse width ${\sim}50\;\text{ps}$) focused to a near diffraction-limited spot using an oil-immersion objective lens ($\text{NA} = 1.4$). The NW position was controlled by an $x{-}y$ piezo-scanning stage, allowing the focal spot to be positioned at different locations along the NW. Both scattered laser light and fluorescence emission were collected via an objective lens, Fig. 1(c). The collected light was separated into “excitation” and “emission” spectral channels using optical filters, and directed to a Hamamatsu C11440 CMOS camera to capture widefield images. Images were obtained using a 50 ms exposure. Images were interpolated to a rectangular window aligned with the NW. The results shown in Fig. 2 were obtained using this widefield imaging setup.

 

Fig. 2. Widefield images of a tapered NW. The NW was measured by AFM to be 7.3 µm long with height 380 nm at the left end and 217 nm at the right end. Images were captured for three positions of the laser focus: the (a)–(c) left end, (d)–(f) center, and (g)–(i) right end of the NW. Panels (a), (d), (g) correspond to backscattered laser light, and (b), (e), (h) to fluorescence emission. Panels (c), (f), (i) show the intensity profiles for the scattered laser light (blue solid lines) and fluorescence (red dashed lines) along the NW, where the signals have been summed perpendicular to the NW axis. The small arrows in panels (c), (f), (i) indicate the estimated position of the focal spot in each case. There was a ${\sim}2\;\text{s}$ delay between the backscatter images and the corresponding fluorescence images for each laser position. No effort was made to individually align the backscatter and fluorescence profiles in (c), (f), (i).

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2. Off-Axis Confocal Hyperspectral Imaging

Using a scanning mirror located at a Fourier plane in the collection path, the optical signal was collected from various locations on the sample plane and directed into a fiber-coupled spectrometer, yielding the off-axis confocal hyperspectral data shown in figures below. The intensity images were created by integrating the spectrum over the wavelength range $\lambda = 405 - 810\;\text{nm} $ for each pixel. The hyperspectral images plot the corresponding spectra as a function of position along the NW axis ($y$), where the pixels spanning the NW width ($x$ direction) were binned together at each position along the NW axis. Each binned spectrum was also self-normalized to reveal additional details that would otherwise be suppressed by large intensity variations. Hyperspectral images (and AFM images) were interpolated to a rectangular window aligned with the NW.

C. Numerical Simulations

This work utilizes both finite difference eigenmode (FDE) [32,33] and finite difference time domain (FDTD) [34] simulations from Lumerical. In the simulations, the glass coverslip is modeled as a semi-infinite dielectric environment with refractive index 1.46 and top surface (glass–GaN interface) at $z = 0$. A NW is modeled as a tapered triangular prism with equilateral bases and overall dimensions extracted from AFM measurements. The refractive index used for wurtzite GaN is obtained from measurements by Lin et al. [18]. In FDTD simulations, the laser is modeled as an unpolarized focused Gaussian beam with $\text{NA} = 1.4$, and broadband fluorescence is modeled as an unpolarized dipole emitter positioned on the surface of the NW with a power spectrum extracted from experiments. Optical fields within a NW are obtained using various $E$-field monitors that intersect different parts of the NW. To compare with experimental hyperspectral measurements, the emission spectrum outside the NW is extracted from a horizontal ($x$-$y$) monitor positioned 300 nm below the GaN–glass interface, where $|{\bf E}{|^2}$ is summed across the transverse dimension ($x$) and normalized.

4. RESULTS AND DISCUSSION

A. Laser Coupling and Delocalized Emission

When using a focused laser for excitation, broadband fluorescence is induced locally within the focal volume and also along the NW from laser light propagating in guided modes. The resulting emission can radiate into the far-field or propagate along the NW in waveguide modes. The guided broadband emission provides a mechanism to study the waveguide modes as a function of wavelength through leakage of various spectral components as described above in Section 2. This leakage signal is, however, convoluted with the emission radiated directly into the far-field by fluorescence induced by laser light propagating within the NW. It is thus important to determine illumination conditions that minimize fluorescence emission along the NW to optimize sensitivity to the small leakage signal we seek.

Figure 2 shows widefield fluorescence and backscattered laser light images of a tapered NW. When the laser is focused near the center of the NW [Figs. 2(d)–2(f)], scattered laser light is observed only near the focal position, while fluorescence is observed both near the focal position and also near the end facets. In contrast, when the laser is focused near the end facets [Figs. 2(a)–2(c) and 2(g)–2(i)], faint laser light is observed at the opposite end of the NW, and fluorescence is observed along its entire length. These observations suggest that laser light does not propagate when the laser is focused near the center of the NW, but the resulting broadband emission is guided from the center to the end facets. When the focus is positioned near an end facet, however, laser light couples into the NW and excites fluorescence while propagating along its length. This delocalized excitation process, which can be inhomogeneous depending on the density of absorbing centers along the NW [Figs. 2(b) and 2(h)], reduces the intensity of the guided laser light and generates broadband emission, some of which is also guided along the NW length.

To test our interpretation of Fig. 2, we modeled the propagation of laser light ($\lambda = 401\;\text{nm} $) within a NW for different positions of the focus spot using FDTD simulations. As Fig. 3 shows, when focused near the midpoint, very little laser light couples into propagating modes because the GaN–glass interface is parallel to the NW axis (i.e., horizontal), which prohibits laser light from entering the NW at angles that lead to TIR regardless of the angle of incidence. Alternatively, when focused near an end facet, laser light can enter the NW and fill most of its volume, since the high refractive-index contrast means that some laser light incident on the vertical end facet from air will lead to TIR within the NW regardless of the angle of incidence. In addition, the tapered NW has an asymmetric response: when focused at the large end of the NW, the laser propagation pattern is more complicated compared to when it is focused at the small end. This suggests that when focused near the larger end, the laser can couple into more modes dictated by the wider waveguide entrance. These modes can be compressed, scattered, or leaked at different positions as the laser propagates along the NW.

 

Fig. 3. FDTD simulations of a GaN NW atop glass illuminated by a focused laser beam (see Section 3.C). The modeled physical dimensions are based on AFM measurements of the NW shown in Fig. 2. Panels (a)–(d) show $|{\bf E}{|^2}$ on the central $y{-}z$ plane ($x = 0$) that intersects the NW apex for four laser positions: (a) $y = 7.65\;\unicode{x00B5}\text{m}$ (right end facet), (b) $y = 0.35\;\unicode{x00B5}\text{m}$ (left end facet), (c) $y = 0.4\;\unicode{x00B5}\text{m}$, (d) $y = 4\;\unicode{x00B5}\text{m}$. (e) Internal intensity profiles along the NW, where $|{\bf E}{|^2}$ is integrated over cross section planes ($x{-}z$) of the NW (i.e., internal to the NW) at each position, $y$. The images in (a)–(d) are all self-normalized and use the same color scale shown in panel (d), whereas the profiles in (e) are normalized to the maximum internal intensity when the laser is at $y = 0.4\;\unicode{x00B5}\text{m}$.

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The coupling efficiency is extremely sensitive to the position of the focus relative to an end facet, as demonstrated by comparing Figs. 3(b) and 3(c), which show the simulated laser intensity in the NW for two laser focus positions that are only 50 nm apart. In the experiment, the coupling efficiency is also sensitive to the focal position, but somewhat less so compared to simulations (see Supplement 1). This sensitivity to focal position is important for detecting spectral signatures of transverse modes, since it enables us to identify and suppress the signal from emission excited by propagating laser light, thus making us more sensitive to emission leaked from propagating modes.

B. Spectral Signatures of Transverse Modes

When laser light is focused onto the NW, the induced broadband emission can propagate along the NW axis in multiple longitudinal and transverse modes. Information about the longitudinal modes is encoded in spectral fringes in the light emerging from the (uncoated) end facets of the NW [4,5,21], which can be detected with a high-NA objective lens. Here we show that transverse modes imprint a spectral signature in the light that leaks from the basal facet of a triangular waveguide at different points along the NW length. However, these signatures are relatively weak, and are thus only observed in regions of the NW where the competing fluorescence signal is small.

Figure 4 shows off-axis hyperspectral images of a tapered NW that is 13 µm long. For the particular NW shown, two diagonal spectral stripes are observed. As expected, these stripes are observed only in the “dark” region of the NW when each fluorescence spectrum is self-normalized. These spectral stripes do not arise from fluorescence emitted locally at different points along the NW, but rather are signatures of the transverse waveguide modes, as discussed in Section 2. More specifically, each stripe occurs when a specific transverse mode exceeds its cutoff condition imposed by the cross-sectional dimension of the waveguide at a particular wavelength and position along the NW; the slope of the spectral stripes reflects the taper angle along the length of the NW waveguide. As broadband fluorescence light propagates from the larger toward the smaller end of the tapered NW, fewer transverse modes are present. This effect is demonstrated in Fig. 3(b) where the mode structure of the simulated laser light becomes less complex as the light propagates from the larger toward the smaller end of the NW.

 

Fig. 4. AFM and off-axis confocal hyperspectral images of a tapered NW atop a glass coverslip. (a) Height profile along the NW apex extracted from the AFM image in (b). (c) Integrated fluorescence intensity map when the laser is positioned near the left (larger) end of the NW. (d) Hyperspectral data from (c). The color scale shown in panel (c) is also used in panel (d).

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We used FDE to analyze the modal structure in the NW, as summarized in Fig. 5. The inset shows the field intensity patterns for the first eight eigenmodes; the odd and even modes are degenerate and correspond to orthogonal polarization states. For each of these modes, FDE calculates the effective refractive index [35,36] as a function of free-space wavelength, ${n_{\text{eff}}}(\lambda) = c\beta /\omega$, where $c$ is the speed of light in vacuum, $\beta$ is the light propagation constant in the NW [35], and $\omega$ is the optical frequency. The wavelengths where ${n_{\text{eff}}} = {n_{\text{glass}}}$ (${\sim}1.46$) determine the cutoff wavelengths [37], ${\lambda _c}$, and at the GaN–glass interface, we can define a reduced cutoff wavelength, ${\lambda _{\text{r}}}$:

$${\lambda _{r}} = \frac{{{\lambda _c}}}{{\sqrt {n_{\text{GaN}}^2 - n_{\text{glass}}^2}}},$$
which takes into account both indices of refraction. The computed values of ${\lambda _{ r}}$ as a function of triangular edge length are plotted as the solid symbols in Fig. 5 for the first eight eigenmodes. Equation (2) suggests a linear relationship between ${\lambda _{ r}}$ and the triangular edge length, $a$, and indeed the lines in Fig. 5 are linear fits,
$${\lambda _{r}} = 2a/{V_{\text{mode}}},$$
where ${V_{\text{mode}}}$ is a dimensionless number that separates the regions of parameter space where different modes can propagate within the waveguide, in analogy with the well-known $V$-number for optical fibers [2426]. In particular, no propagation of light is allowed for wavelength/edge length combinations that map to points above the solid black line (square symbols in Fig. 5, ${V_{1,2}} = 1.40$), while single-mode propagation (degenerate modes 1,2) occurs between the red line (circular symbols in Fig. 5, ${V_{3,4}} = 2.49$) and black line. The other two lines in Fig. 5 define the parameters that allow for propagation of higher-order modes. In the context of a tapered NW, the lines in Fig. 5 indicate the conditions for which light from different waveguide modes starts leaking into the glass substrate, where it is radiated into the far-field providing a spectral signature of the corresponding transverse modes within the NW.
 

Fig. 5. Reduced cutoff wavelengths and mode profiles for a GaN NW waveguide atop a glass coverslip. Modes were computed using FDE simulations for an equilateral triangular prism geometry with edge length $a$ and refractive index from Lin [18]. The reduced cutoff wavelength ${\lambda _{\text{r}}}$ is defined by Eq. (3), and the lines through the symbols are linear fits to Eq. (4), giving the V-numbers shown for eigenmodes 1–8. The insets of the figure show intensity profiles ($|{\bf E}{|^2}$) for the modes at $\lambda = 400\;\text{nm} $ (${\lambda _{\text{r}}} = 189\;\text{nm} $) for a prism with 400 nm edge length (indicated as violet crosshair).

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To compare directly with experimental hyperspectral measurements, we performed FDTD simulations of the fluorescence emission using geometric models of the NW extracted from AFM measurements, e.g., Fig. 4(b). The lateral pixel size in the AFM image was ${\ge}40\,\text{nm}$, leading to sampling artifacts and an uncertainty in the measured apex height. In addition, the radius-of-curvature of the AFM probe is not known precisely. We account for these measurement uncertainties by modeling two different NW geometries constructed by establishing lower and upper bounds for the size of the triangular cross section at each point along the NW length (see Supplement 1 for more details). The results of simulations using these two geometries can then be compared directly with the measured off-axis hyperspectral images, as shown in Fig. 6.

 

Fig. 6. Off-axis confocal hyperspectral images and FDTD simulations of a NW. This is the same NW from Fig. 4, but the laser spot is ${\sim}500\;\text{nm}$ further away from the end facet. (a) Fluorescence intensity map when the laser is positioned near the left (larger) end of the NW. (b) Hyperspectral data from (a). (c)–(e) Hyperspectral “data” extracted from FDTD simulations where the simulated emitter (dipole; see Section 3.C) is positioned at $y = 0$. In panel (c), the NW cross section from AFM measurements (see inset) is used in the FDTD simulations. In panel (d), an equilateral triangular cross section with overestimated edge length is assumed, while in panel (e), an underestimated triangular cross section is used. The solid/dashed red diagonal lines in (b)–(e) indicate the predicted cutoff wavelengths for modes 3–8 based on the mode analysis in Fig. 5 for the overestimated/underestimated triangular cross section. The spectral stripes correspond to leakage of specific modes above their cutoff wavelengths, e.g.,  the bright stripe between $6 \lt y \lt 8\;\unicode{x00B5}\text{m}$ in panel (b) corresponds to leakage of mode 5,6.

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The image shown in Fig. 6(b) depicts experimental hyperspectral data for the same NW from Fig. 4, but where the laser focus has been shifted ${\sim}500\;\text{nm}$ further away from the end facet. Figures 6(c)–(e) show simulated hyperspectral images for three different model geometries (see insets) that correspond to the AFM data [Fig. 6(c)], an overestimated equilateral triangular cross section [Fig. 6(d)], and an underestimated equilateral triangular cross section [Fig. 6(e)]. The red curves in Fig. 6 correspond to the cutoff wavelength, ${\lambda _c}$, deduced from Eq. (4) for ${V_{3,4}} = 2.49$, ${V_{5,6}} = 2.82$, and ${V_{7,8}} = 3.64$, where the solid curves assume the overestimated triangular geometry and the dashed curves assume the underestimated triangular geometry. Longer wavelengths exceed the cutoff condition so the leaked transverse modes—the observed spectral stripes—are expected to occur above the cutoff lines. Clearly, the cutoff lines extracted from the overestimated triangular cross section agree well with the experimental hyperspectral data [Fig. 6(b)], and with the simulated hyperspectral data using the geometric model deduced from the AFM measurements [Fig. 6(c)]. The extracted cutoff wavelengths also agree with the corresponding over- and underestimated triangular models shown in Figs. 6(d) and 6(e), respectively.

The observed stripes (higher intensity regions) in the experimental and simulated hyperspectral images in Fig. 6 are caused by leakage of propagating modes with corresponding cutoff wavelengths indicated by the red curve immediately below each stripe. This correspondence is particularly clear for Figs. 6(d) and 6(e) where the geometrical model assumes a perfect triangular cross section. The relative intensity of different stripes is associated with the spectral shape of the emission, and also by the modal composition within the NW, which is affected by the polarization direction of the emitter [38] (Supplement 1, Fig. S1). Interestingly, the simulated hyperspectral images in Figs. 6(c) and 6(d) show strong agreement even though they correspond to two different geometric models. This demonstrates that the corners of the triangular cross section do not strongly affect the intensity profile of the waveguide modes, in agreement with the mode profiles shown in Fig. 5 where light hardly penetrates into the corners. It is also important to note that our analysis is based on published refractive indices [18]; conversely, if the detailed geometry of the sample was measured with high accuracy, the refractive index could be deduced.

C. Applications of Spectral Signatures

By collecting and analyzing hyperspectral signatures of transverse eigenmodes across the fluorescence bandwidth, we can estimate the modal composition along the NW length and deduce the field pattern in the cross section of the NW. For example, for the NW in Fig. 6, we can expect the presence of mode 1–2 across almost the entire fluorescence spectrum, mode 3–4 for wavelengths below the upper stripe, mode 5–6 below the middle stripe, and mode 7–8 below the lower stripe. Conversely, if the NW cross section is too small to support even the lowest-order waveguide modes for the shortest fluorescence wavelengths, then we would not expect spectral shaping of the broadband emission. Indeed, the NW shown in Fig. 7 (12.6 µm long) is too narrow to guide fluorescence induced at the left (larger) end of the NW toward the smaller end. In this case, the excitation laser has a short enough wavelength ($\lambda = 401\;\text{nm} $) to be guided along most of the NW length, but the emitted fluorescence cannot be guided, and therefore radiates with no obvious spectral shaping.

 

Fig. 7. Off-axis confocal hyperspectral images of a narrow NW atop a glass coverslip. (a) Height profile along the NW apex measured by AFM. (b) Integrated fluorescence intensity map when the laser is positioned near the left (larger) end of the NW. (c) Hyperspectral data from (b). In panel (c), the solid red line is the predicted cutoff line for mode 1–2 assuming an equilateral triangular cross section with overestimated edge length, and the dashed line is for the underestimated edge length (see inset).

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The spectral signatures can, conversely, be used to extract information related to the geometrical properties of the NWs, and more generally to other nanostructures that could be studied using this approach. In particular, inspection of Eqs. (3) and (4) demonstrates that two parameters contribute to the cutoff condition: $a$ (structure) and $n$ (dielectric), so to extract one precisely from the pattern of spectral stripes requires detailed knowledge of the other. In Fig. 6(b), there is a discontinuity in the predominant spectral stripe near $y = 9\;\unicode{x00B5}\text{m}$, indicating a change in the structure and/or dielectric properties of the NW at that position. There is a relatively minor change in the taper angle there, but it does not appear to be more prominent than those occurring at other positions along the NW [cf. Fig. 4(a)], which suggests that this discontinuity might arise more from a change in the intrinsic properties of the NW (e.g., crystal structure, refractive index) rather than its gross structure/topography.

 

Fig. 8. Using spectral signatures to deduce NW height. (a) Integrated fluorescence intensity map when the laser is positioned near the middle of the NW. (b) One-dimensional hyperspectral data from (a), where the red and black lines indicate the estimated cutoff wavelengths of mode 3–4 and 5–6, respectively, as described in the text. (c) Deduced topography of NW sections based on Eq. (4), assuming an equilateral triangular cross section. (d) AFM image of the NW after background flattening. (e) Comparison of apex height deduced from hyperspectral data (orange solid line) and AFM measurements (blue dashed line). The blue dotted line represents the estimated upper bound of the apex height taking into account AFM sampling artifacts and uncertainties in the shape of the AFM probe. Error bars are calculated as described in the text.

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Assuming a NW maintains a triangular cross section along its length, the spectral signatures in the leaked fluorescence can be used to deduce the height of the NW. In particular, Fig. 8 shows a non-tapered NW of height ${\sim}300\;\text{nm}$, which is large enough to support multiple waveguide modes within the broad fluorescence spectrum. To extract the height of the NW, the observed spectral stripes need to be associated with their corresponding mode indices. This is done by assuming particular mode indices and then using Eq. (4) to predict the NW edge length and thus the height; only the correct mode indices will predict the same height. The cutoff wavelengths used in Eq. (4) were obtained by determining the midpoint between the two spectral stripes in Fig. 8, which are separated by ${\sim}60\;\text{nm}$ (see Supplement 1 for more details). We find that the V-numbers corresponding to the cutoff conditions for the neighboring pair of nondegenerate modes 3 (${V_3} = 2.49$) and 5 (${V_5} = 2.82$) predict the same NW height (to within 5 nm—see Supplement 1, Fig. S2), while the V-numbers for the pair, modes 1 and 3, predict radically different heights, as do V-numbers for the pair, modes 5 and 7 (recall that even modes are degenerate with the corresponding odd mode, e.g., 3–4). Note that near the laser focus, radiation modes at longer wavelengths attenuate faster since they are further into the cutoff regime [27]. This causes an upturn in spectral stripes in that region, and a slight overestimation of the deduced height can be expected there.

The orange curve in Fig. 8(e) shows the height extracted from the spectrum in the way described above. We estimate an upper bound on the uncertainty in the cutoff wavelengths to be ${\pm}30\;\text{nm}$, which accounts for the fast attenuation of higher-order modes near the laser spot (see above) and also the case when only the 3–4 or 5–6 mode leakage is visible rather than both. Also, the exact value of the index of refraction for the GaN NW is unknown, but the uncertainty should not exceed the difference between refractive indices of wurtzite and zincblende GaN, i.e., at most ${\sim}4\%$ from 550 nm to 650 nm. Hence, assuming ${\pm}4\%$ refractive index uncertainty and ${\pm}30\;\text{nm}$ cutoff wavelength uncertainty, overall there would be about ${\pm}28\;\text{nm}$ of uncertainty for the deduced height. For comparison, the height was measured independently by AFM, represented by the dashed blue curve in Fig. 8(e). The AFM measurements systematically underestimate the apex height of the NW due to sampling artifacts as the AFM probe scans across NW apex and uncertainties in the sharpness of both the AFM probe and the NW apex, as described in the supplemental information. For this AFM scan, the lateral pixel size is ${\sim}40\;\text{nm}$, and the AFM can underestimate the actual apex height by as much as 30 nm, represented by the dotted blue curve in Fig. 8(e). The agreement between the spectrally deduced height and the height range extracted from AFM measurements demonstrates that spectral signatures can indeed be used to estimate the NW dimensions accurately. Referring again to Fig. 6 above, the difference in height for the overestimated [Fig. 6(d)] and underestimated [Fig. 6(e)] triangular cross sections was only 53 nm on average, but the FDTD simulations and model predictions give very different spectral maps in these two cases. This demonstrates the sensitivity of the spectral maps to the dimensions of the NW. We note that the spectral maps are mainly affected by the gross shape and cross-sectional area of the NW rather than by its detailed geometric features, as demonstrated by comparing Fig. 6(c) with Fig. 6(d).

5. CONCLUSION

In this work, we demonstrated a probe-free, far-field method to visualize the modal structure within GaN NWs. In particular, we used a custom off-axis confocal microscope to collect hyperspectral data along the length of each NW, and identified spectral signatures originating from the leakage of transverse modes from the NW basal facet due to waveguide cutoff. Supported by analytical and numerical studies, we pinpointed the parameter space (NW edge length, wavelength, and refractive indices) for cutoff limits of various waveguide modes. This allows us to identify the exact mode types associated with the spectral signatures shown in the hyperspectral images. Conversely, we can extract structural and dielectric information from hyperspectral data of the NW. In particular, with this optical technique, we successfully deduced topographical characteristics of a GaN NW with ${\sim}30$ nm accuracy. The accuracy can be further improved by developing a more accurate method to find the cutoff wavelengths on a hyperspectral image and utilize a more accurate technique to measure refractive index of GaN NWs. This analysis method based on lateral leakage of transverse waveguide modes can potentially be adapted to provide detailed modal and structural information for various photonic device components, including different geometries (e.g., NWs with hexagonal cross sections) and mode types (e.g., whispering gallery modes). In addition, it could be possible to obtain similar spectral information by aligning isolated NWs with the slit of a spectrometer, thus negating the need for the off-axis confocal system described here and broadening the utility of our approach.

Funding

U.S. Department of Energy (DE-AC02-05CH11231).

Acknowledgment

The authors acknowledge the Molecular Foundry at Lawrence Berkeley National Laboratory for providing GaN NW samples. We would like to especially thank Tevye R. Kuykendall, Nick Borys, P. James Schuck, and Shaul Aloni for the sample growth and the insightful discussions of the sample properties.

Disclosures

The authors declare no conflicts of interest.

Supplemental document

See Supplement 1 for supporting content.

REFERENCES

1. N. P. Dasgupta, J. Sun, C. Liu, S. Brittman, S. C. Andrews, J. Lim, H. Gao, R. Yan, and P. Yang, “25th anniversary article: semiconductor nanowires–synthesis, characterization, and applications,” Adv. Mater. 26, 2137–2184 (2014). [CrossRef]  

2. R. Yan, D. Gargas, and P. Yang, “Nanowire photonics,” Nat. Photonics 3, 569–576 (2009). [CrossRef]  

3. M. Law, D. J. Sirbuly, J. C. Johnson, J. Goldberger, R. J. Saykally, and P. Yang, “Nanoribbon waveguides for subwavelength photonics integration,” Science 305, 1269–1273 (2004). [CrossRef]  

4. S. Gradečak, F. Qian, Y. Li, H.-G. Park, and C. M. Liebera, “GaN nanowire lasers with low lasing thresholds,” Appl. Phys. Lett. 87, 173111 (2005). [CrossRef]  

5. A. M. Schwartzberg, S. Aloni, T. Kuykendall, P. J. Schuck, and J. J. Urban, “Optical cavity characterization in nanowires via self-generated broad-band emission,” Opt. Express 19, 8903–8911 (2011). [CrossRef]  

6. H. Xu, J. B. Wright, T.-S. Luk, J. J. Figiel, K. Cross, L. F. Lester, G. Balakrishnan, G. T. Wang, I. Brener, and Q. Li, “Single-mode lasing of GaN nanowire-pairs,” Appl. Phys. Lett. 101, 113106 (2012). [CrossRef]  

7. X. Duan, Y. Huang, R. Agarwal, and C. M. Lieber, “Single-nanowire electrically driven lasers,” Nature 421, 241–245 (2003). [CrossRef]  

8. G. Grzela, R. Paniagua-Domínguez, T. Barten, Y. Fontana, J. A. Sánchez-Gil, and J. G. Rivas, “Nanowire antenna emission,” Nano Lett. 12, 5481–5486 (2012). [CrossRef]  

9. H. Yang, V. Khayrudinov, V. Dhaka, H. Jiang, A. Autere, H. Lipsanen, Z. Sun, and H. Jussila, “Nanowire network–based multifunctional all-optical logic gates,” Sci. Adv. 4, eaar7954 (2018). [CrossRef]  

10. Z. Wang and B. Nabet, “Nanowire optoelectronics,” Nanophotonics 4, 491–502 (2015). [CrossRef]  

11. S. Han, S.-K. Lee, I. Choi, J. Song, C.-R. Lee, K. Kim, M.-Y. Ryu, K.-U. Jeong, and J. S. Kim, “Highly efficient and flexible photosensors with GaN nanowires horizontally embedded in a graphene sandwich channel,” ACS Appl. Mater. Interfaces 10, 38173–38182 (2018). [CrossRef]  

12. A. A. Abdul-Hameed, M. A. Mahdi, B. Ali, A. M. Selman, H. F. Al-Taay, P. J. Jennings, and W.-J. Lee, “Fabrication of a high sensitivity and fast response self-powered photosensor based on a core-shell silicon nanowire homojunction,” Superlattices Microstruct. 116, 27–35 (2018). [CrossRef]  

13. H.-R. Kim, B.-G. An, Y. W. Chang, M.-J. Kang, J.-G. Park, and J.-C. Pyun, “Highly sensitive in situ-synthesized cadmium sulfide (CdS) nanowire photosensor for chemiluminescent immunoassays,” Enzym. Microb. Technol. 133, 109457 (2019). [CrossRef]  

14. Y. Huang, X. Duan, and C. M. Lieber, “Nanowires for integrated multicolor nanophotonics,” Small 1, 142–147 (2005). [CrossRef]  

15. P. R. Wiecha, A. Cuche, A. Arbouet, C. Girard, G. Colas des Francs, A. Lecestre, G. Larrieu, F. Fournel, V. Larrey, T. Baron, and V. Paillard, “Strongly directional scattering from dielectric nanowires,” ACS Photon. 4, 2036–2046 (2017). [CrossRef]  

16. J.-P. Kakko, A. Matikainen, N. Anttu, S. Kujala, H. Mäntynen, V. Khayrudinov, A. Autere, Z. Sun, and H. Lipsanen, “Measurement of nanowire optical modes using cross-polarization microscopy,” Sci. Rep. 7, 17790 (2017). [CrossRef]  

17. D. Saxena, F. Wang, Q. Gao, S. Mokkapati, H. H. Tan, and C. Jagadish, “Mode profiling of semiconductor nanowire lasers,” Nano Lett. 15, 5342–5348 (2015). [CrossRef]  

18. M. E. Lin, B. N. Sverdlov, S. Strite, H. Morkoç, and A. E. Drakin, “Refractive indices of wurtzite and zincblende GaN,” Electron. Lett. 29, 1759–1760 (1993). [CrossRef]  

19. M. A. Reshchikov and H. Morkoç, “Luminescence properties of defects in GaN,” J. Appl. Phys. 97, 061301 (2005). [CrossRef]  

20. Q. Li and G. T. Wang, “Spatial distribution of defect luminescence in GaN nanowires,” Nano Lett. 10, 1554–1558 (2010). [CrossRef]  

21. L. R. Richey-Simonsen, N. J. Borys, T. R. Kuykendall, P. J. Schuck, S. Aloni, and J. M. Gerton, “Investigating surface effects of GaN nanowires using confocal microscopy at below-band gap excitation,” J. Mater. Res. 32, 4076–4086 (2017). [CrossRef]  

22. M. Landmann, E. Rauls, W. G. Schmidt, M. D. Neumann, E. Speiser, and N. Esser, “GaN m-plane: atomic structure, surface bands, and optical response,” Phys. Rev. B 91, 035302 (2015). [CrossRef]  

23. M. Himmerlich, L. Lymperakis, R. Gutt, P. Lorenz, J. Neugebauer, and S. Krischok, “GaN(0001) surface states: experimental and theoretical fingerprints to identify surface reconstructions,” Phys. Rev. B 88, 125304 (2013). [CrossRef]  

24. A. Snyder and J. Love, Optical Waveguide Theory, 1st ed. (Chapman and Hall, 1983).

25. R. Paschotta, "V number," in Encyclopedia of Laser Physics and Technology, 1st ed. (Wiley-VCH, 2008), Vol. N-Z, p. 785.

26. G. Agrawal, Nonlinear Fiber Optics, 5th ed. (Elsevier, 2012), Chap. 2.

27. J. Hu and C. R. Menyuk, “Understanding leaky modes: slab waveguide revisited,” Adv. Opt. Photon. 1, 58–106 (2009). [CrossRef]  

28. L. Novotny and B. Hecht, Principles of Nano-Optics, 1st ed. (Cambridge University, 2006).

29. W. Tian, C. Zhao, J. Leng, R. Cui, and S. Jin, “Visualizing carrier diffusion in individual single-crystal organolead halide perovskite nanowires and nanoplates,” J. Am. Chem. Soc. 137, 12458–12461 (2015). [CrossRef]  

30. T. R. Kuykendall, M. V. P. Altoe, D. F. Ogletree, and S. Aloni, “Catalyst-directed crystallographic orientation control of GaN nanowire growth,” Nano Lett. 14, 6767–6773 (2014). [CrossRef]  

31. J. Y. Huang, H. Zheng, S. X. Mao, Q. Li, and G. T. Wang, “In situ nanomechanics of GaN nanowires,” Nano Lett. 11, 1618–1622 (2011). [CrossRef]  

32. Z. Zhu and T. G. Brown, “Full-vectorial finite-difference analysis of microstructured optical fibers,” Opt. Express 10, 853–864 (2002). [CrossRef]  

33. Lumerical, “FDE,” https://support.lumerical.com/hc/en-us/articles/360034917233.

34. D. M. Sullivan, Electromagnetic Simulation Using the FDTD Method, 2nd ed. (Wiley-IEEE, 2013).

35. R. A. Gutierrez-Arenas and D. Mendoza, “Band structure of a two dimensional metallic photonic crystal and the experimental observation of negative refraction in the microwave region,” arXiv:1109.0329 [physical.optics] (2011).

36. J.-M. Jin, The Finite Element Method in Electromagnetics, 3rd ed. (Wiley-IEEE, 2014).

37. Lumerical, “Lossy modes and dB/m to k conversion,” https://support.lumerical.com/hc/en-us/articles/360034917493.

38. Z. Li, K. Bao, Y. Fang, Y. Huang, P. Nordlander, and H. Xu, “Correlation between incident and emission polarization in nanowire surface plasmon waveguides,” Nano Lett. 10, 1831–1835 (2010). [CrossRef]  

References

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  1. N. P. Dasgupta, J. Sun, C. Liu, S. Brittman, S. C. Andrews, J. Lim, H. Gao, R. Yan, and P. Yang, “25th anniversary article: semiconductor nanowires–synthesis, characterization, and applications,” Adv. Mater. 26, 2137–2184 (2014).
    [Crossref]
  2. R. Yan, D. Gargas, and P. Yang, “Nanowire photonics,” Nat. Photonics 3, 569–576 (2009).
    [Crossref]
  3. M. Law, D. J. Sirbuly, J. C. Johnson, J. Goldberger, R. J. Saykally, and P. Yang, “Nanoribbon waveguides for subwavelength photonics integration,” Science 305, 1269–1273 (2004).
    [Crossref]
  4. S. Gradečak, F. Qian, Y. Li, H.-G. Park, and C. M. Liebera, “GaN nanowire lasers with low lasing thresholds,” Appl. Phys. Lett. 87, 173111 (2005).
    [Crossref]
  5. A. M. Schwartzberg, S. Aloni, T. Kuykendall, P. J. Schuck, and J. J. Urban, “Optical cavity characterization in nanowires via self-generated broad-band emission,” Opt. Express 19, 8903–8911 (2011).
    [Crossref]
  6. H. Xu, J. B. Wright, T.-S. Luk, J. J. Figiel, K. Cross, L. F. Lester, G. Balakrishnan, G. T. Wang, I. Brener, and Q. Li, “Single-mode lasing of GaN nanowire-pairs,” Appl. Phys. Lett. 101, 113106 (2012).
    [Crossref]
  7. X. Duan, Y. Huang, R. Agarwal, and C. M. Lieber, “Single-nanowire electrically driven lasers,” Nature 421, 241–245 (2003).
    [Crossref]
  8. G. Grzela, R. Paniagua-Domínguez, T. Barten, Y. Fontana, J. A. Sánchez-Gil, and J. G. Rivas, “Nanowire antenna emission,” Nano Lett. 12, 5481–5486 (2012).
    [Crossref]
  9. H. Yang, V. Khayrudinov, V. Dhaka, H. Jiang, A. Autere, H. Lipsanen, Z. Sun, and H. Jussila, “Nanowire network–based multifunctional all-optical logic gates,” Sci. Adv. 4, eaar7954 (2018).
    [Crossref]
  10. Z. Wang and B. Nabet, “Nanowire optoelectronics,” Nanophotonics 4, 491–502 (2015).
    [Crossref]
  11. S. Han, S.-K. Lee, I. Choi, J. Song, C.-R. Lee, K. Kim, M.-Y. Ryu, K.-U. Jeong, and J. S. Kim, “Highly efficient and flexible photosensors with GaN nanowires horizontally embedded in a graphene sandwich channel,” ACS Appl. Mater. Interfaces 10, 38173–38182 (2018).
    [Crossref]
  12. A. A. Abdul-Hameed, M. A. Mahdi, B. Ali, A. M. Selman, H. F. Al-Taay, P. J. Jennings, and W.-J. Lee, “Fabrication of a high sensitivity and fast response self-powered photosensor based on a core-shell silicon nanowire homojunction,” Superlattices Microstruct. 116, 27–35 (2018).
    [Crossref]
  13. H.-R. Kim, B.-G. An, Y. W. Chang, M.-J. Kang, J.-G. Park, and J.-C. Pyun, “Highly sensitive in situ-synthesized cadmium sulfide (CdS) nanowire photosensor for chemiluminescent immunoassays,” Enzym. Microb. Technol. 133, 109457 (2019).
    [Crossref]
  14. Y. Huang, X. Duan, and C. M. Lieber, “Nanowires for integrated multicolor nanophotonics,” Small 1, 142–147 (2005).
    [Crossref]
  15. P. R. Wiecha, A. Cuche, A. Arbouet, C. Girard, G. Colas des Francs, A. Lecestre, G. Larrieu, F. Fournel, V. Larrey, T. Baron, and V. Paillard, “Strongly directional scattering from dielectric nanowires,” ACS Photon. 4, 2036–2046 (2017).
    [Crossref]
  16. J.-P. Kakko, A. Matikainen, N. Anttu, S. Kujala, H. Mäntynen, V. Khayrudinov, A. Autere, Z. Sun, and H. Lipsanen, “Measurement of nanowire optical modes using cross-polarization microscopy,” Sci. Rep. 7, 17790 (2017).
    [Crossref]
  17. D. Saxena, F. Wang, Q. Gao, S. Mokkapati, H. H. Tan, and C. Jagadish, “Mode profiling of semiconductor nanowire lasers,” Nano Lett. 15, 5342–5348 (2015).
    [Crossref]
  18. M. E. Lin, B. N. Sverdlov, S. Strite, H. Morkoç, and A. E. Drakin, “Refractive indices of wurtzite and zincblende GaN,” Electron. Lett. 29, 1759–1760 (1993).
    [Crossref]
  19. M. A. Reshchikov and H. Morkoç, “Luminescence properties of defects in GaN,” J. Appl. Phys. 97, 061301 (2005).
    [Crossref]
  20. Q. Li and G. T. Wang, “Spatial distribution of defect luminescence in GaN nanowires,” Nano Lett. 10, 1554–1558 (2010).
    [Crossref]
  21. L. R. Richey-Simonsen, N. J. Borys, T. R. Kuykendall, P. J. Schuck, S. Aloni, and J. M. Gerton, “Investigating surface effects of GaN nanowires using confocal microscopy at below-band gap excitation,” J. Mater. Res. 32, 4076–4086 (2017).
    [Crossref]
  22. M. Landmann, E. Rauls, W. G. Schmidt, M. D. Neumann, E. Speiser, and N. Esser, “GaN m-plane: atomic structure, surface bands, and optical response,” Phys. Rev. B 91, 035302 (2015).
    [Crossref]
  23. M. Himmerlich, L. Lymperakis, R. Gutt, P. Lorenz, J. Neugebauer, and S. Krischok, “GaN(0001) surface states: experimental and theoretical fingerprints to identify surface reconstructions,” Phys. Rev. B 88, 125304 (2013).
    [Crossref]
  24. A. Snyder and J. Love, Optical Waveguide Theory, 1st ed. (Chapman and Hall, 1983).
  25. R. Paschotta, "V number," in Encyclopedia of Laser Physics and Technology, 1st ed. (Wiley-VCH, 2008), Vol. N-Z, p. 785.
  26. G. Agrawal, Nonlinear Fiber Optics, 5th ed. (Elsevier, 2012), Chap. 2.
  27. J. Hu and C. R. Menyuk, “Understanding leaky modes: slab waveguide revisited,” Adv. Opt. Photon. 1, 58–106 (2009).
    [Crossref]
  28. L. Novotny and B. Hecht, Principles of Nano-Optics, 1st ed. (Cambridge University, 2006).
  29. W. Tian, C. Zhao, J. Leng, R. Cui, and S. Jin, “Visualizing carrier diffusion in individual single-crystal organolead halide perovskite nanowires and nanoplates,” J. Am. Chem. Soc. 137, 12458–12461 (2015).
    [Crossref]
  30. T. R. Kuykendall, M. V. P. Altoe, D. F. Ogletree, and S. Aloni, “Catalyst-directed crystallographic orientation control of GaN nanowire growth,” Nano Lett. 14, 6767–6773 (2014).
    [Crossref]
  31. J. Y. Huang, H. Zheng, S. X. Mao, Q. Li, and G. T. Wang, “In situ nanomechanics of GaN nanowires,” Nano Lett. 11, 1618–1622 (2011).
    [Crossref]
  32. Z. Zhu and T. G. Brown, “Full-vectorial finite-difference analysis of microstructured optical fibers,” Opt. Express 10, 853–864 (2002).
    [Crossref]
  33. Lumerical, “FDE,” https://support.lumerical.com/hc/en-us/articles/360034917233 .
  34. D. M. Sullivan, Electromagnetic Simulation Using the FDTD Method, 2nd ed. (Wiley-IEEE, 2013).
  35. R. A. Gutierrez-Arenas and D. Mendoza, “Band structure of a two dimensional metallic photonic crystal and the experimental observation of negative refraction in the microwave region,” arXiv:1109.0329 [physical.optics] (2011).
  36. J.-M. Jin, The Finite Element Method in Electromagnetics, 3rd ed. (Wiley-IEEE, 2014).
  37. Lumerical, “Lossy modes and dB/m to k conversion,” https://support.lumerical.com/hc/en-us/articles/360034917493 .
  38. Z. Li, K. Bao, Y. Fang, Y. Huang, P. Nordlander, and H. Xu, “Correlation between incident and emission polarization in nanowire surface plasmon waveguides,” Nano Lett. 10, 1831–1835 (2010).
    [Crossref]

2019 (1)

H.-R. Kim, B.-G. An, Y. W. Chang, M.-J. Kang, J.-G. Park, and J.-C. Pyun, “Highly sensitive in situ-synthesized cadmium sulfide (CdS) nanowire photosensor for chemiluminescent immunoassays,” Enzym. Microb. Technol. 133, 109457 (2019).
[Crossref]

2018 (3)

S. Han, S.-K. Lee, I. Choi, J. Song, C.-R. Lee, K. Kim, M.-Y. Ryu, K.-U. Jeong, and J. S. Kim, “Highly efficient and flexible photosensors with GaN nanowires horizontally embedded in a graphene sandwich channel,” ACS Appl. Mater. Interfaces 10, 38173–38182 (2018).
[Crossref]

A. A. Abdul-Hameed, M. A. Mahdi, B. Ali, A. M. Selman, H. F. Al-Taay, P. J. Jennings, and W.-J. Lee, “Fabrication of a high sensitivity and fast response self-powered photosensor based on a core-shell silicon nanowire homojunction,” Superlattices Microstruct. 116, 27–35 (2018).
[Crossref]

H. Yang, V. Khayrudinov, V. Dhaka, H. Jiang, A. Autere, H. Lipsanen, Z. Sun, and H. Jussila, “Nanowire network–based multifunctional all-optical logic gates,” Sci. Adv. 4, eaar7954 (2018).
[Crossref]

2017 (3)

P. R. Wiecha, A. Cuche, A. Arbouet, C. Girard, G. Colas des Francs, A. Lecestre, G. Larrieu, F. Fournel, V. Larrey, T. Baron, and V. Paillard, “Strongly directional scattering from dielectric nanowires,” ACS Photon. 4, 2036–2046 (2017).
[Crossref]

J.-P. Kakko, A. Matikainen, N. Anttu, S. Kujala, H. Mäntynen, V. Khayrudinov, A. Autere, Z. Sun, and H. Lipsanen, “Measurement of nanowire optical modes using cross-polarization microscopy,” Sci. Rep. 7, 17790 (2017).
[Crossref]

L. R. Richey-Simonsen, N. J. Borys, T. R. Kuykendall, P. J. Schuck, S. Aloni, and J. M. Gerton, “Investigating surface effects of GaN nanowires using confocal microscopy at below-band gap excitation,” J. Mater. Res. 32, 4076–4086 (2017).
[Crossref]

2015 (4)

M. Landmann, E. Rauls, W. G. Schmidt, M. D. Neumann, E. Speiser, and N. Esser, “GaN m-plane: atomic structure, surface bands, and optical response,” Phys. Rev. B 91, 035302 (2015).
[Crossref]

W. Tian, C. Zhao, J. Leng, R. Cui, and S. Jin, “Visualizing carrier diffusion in individual single-crystal organolead halide perovskite nanowires and nanoplates,” J. Am. Chem. Soc. 137, 12458–12461 (2015).
[Crossref]

D. Saxena, F. Wang, Q. Gao, S. Mokkapati, H. H. Tan, and C. Jagadish, “Mode profiling of semiconductor nanowire lasers,” Nano Lett. 15, 5342–5348 (2015).
[Crossref]

Z. Wang and B. Nabet, “Nanowire optoelectronics,” Nanophotonics 4, 491–502 (2015).
[Crossref]

2014 (2)

N. P. Dasgupta, J. Sun, C. Liu, S. Brittman, S. C. Andrews, J. Lim, H. Gao, R. Yan, and P. Yang, “25th anniversary article: semiconductor nanowires–synthesis, characterization, and applications,” Adv. Mater. 26, 2137–2184 (2014).
[Crossref]

T. R. Kuykendall, M. V. P. Altoe, D. F. Ogletree, and S. Aloni, “Catalyst-directed crystallographic orientation control of GaN nanowire growth,” Nano Lett. 14, 6767–6773 (2014).
[Crossref]

2013 (1)

M. Himmerlich, L. Lymperakis, R. Gutt, P. Lorenz, J. Neugebauer, and S. Krischok, “GaN(0001) surface states: experimental and theoretical fingerprints to identify surface reconstructions,” Phys. Rev. B 88, 125304 (2013).
[Crossref]

2012 (2)

G. Grzela, R. Paniagua-Domínguez, T. Barten, Y. Fontana, J. A. Sánchez-Gil, and J. G. Rivas, “Nanowire antenna emission,” Nano Lett. 12, 5481–5486 (2012).
[Crossref]

H. Xu, J. B. Wright, T.-S. Luk, J. J. Figiel, K. Cross, L. F. Lester, G. Balakrishnan, G. T. Wang, I. Brener, and Q. Li, “Single-mode lasing of GaN nanowire-pairs,” Appl. Phys. Lett. 101, 113106 (2012).
[Crossref]

2011 (2)

2010 (2)

Q. Li and G. T. Wang, “Spatial distribution of defect luminescence in GaN nanowires,” Nano Lett. 10, 1554–1558 (2010).
[Crossref]

Z. Li, K. Bao, Y. Fang, Y. Huang, P. Nordlander, and H. Xu, “Correlation between incident and emission polarization in nanowire surface plasmon waveguides,” Nano Lett. 10, 1831–1835 (2010).
[Crossref]

2009 (2)

J. Hu and C. R. Menyuk, “Understanding leaky modes: slab waveguide revisited,” Adv. Opt. Photon. 1, 58–106 (2009).
[Crossref]

R. Yan, D. Gargas, and P. Yang, “Nanowire photonics,” Nat. Photonics 3, 569–576 (2009).
[Crossref]

2005 (3)

S. Gradečak, F. Qian, Y. Li, H.-G. Park, and C. M. Liebera, “GaN nanowire lasers with low lasing thresholds,” Appl. Phys. Lett. 87, 173111 (2005).
[Crossref]

Y. Huang, X. Duan, and C. M. Lieber, “Nanowires for integrated multicolor nanophotonics,” Small 1, 142–147 (2005).
[Crossref]

M. A. Reshchikov and H. Morkoç, “Luminescence properties of defects in GaN,” J. Appl. Phys. 97, 061301 (2005).
[Crossref]

2004 (1)

M. Law, D. J. Sirbuly, J. C. Johnson, J. Goldberger, R. J. Saykally, and P. Yang, “Nanoribbon waveguides for subwavelength photonics integration,” Science 305, 1269–1273 (2004).
[Crossref]

2003 (1)

X. Duan, Y. Huang, R. Agarwal, and C. M. Lieber, “Single-nanowire electrically driven lasers,” Nature 421, 241–245 (2003).
[Crossref]

2002 (1)

1993 (1)

M. E. Lin, B. N. Sverdlov, S. Strite, H. Morkoç, and A. E. Drakin, “Refractive indices of wurtzite and zincblende GaN,” Electron. Lett. 29, 1759–1760 (1993).
[Crossref]

Abdul-Hameed, A. A.

A. A. Abdul-Hameed, M. A. Mahdi, B. Ali, A. M. Selman, H. F. Al-Taay, P. J. Jennings, and W.-J. Lee, “Fabrication of a high sensitivity and fast response self-powered photosensor based on a core-shell silicon nanowire homojunction,” Superlattices Microstruct. 116, 27–35 (2018).
[Crossref]

Agarwal, R.

X. Duan, Y. Huang, R. Agarwal, and C. M. Lieber, “Single-nanowire electrically driven lasers,” Nature 421, 241–245 (2003).
[Crossref]

Agrawal, G.

G. Agrawal, Nonlinear Fiber Optics, 5th ed. (Elsevier, 2012), Chap. 2.

Ali, B.

A. A. Abdul-Hameed, M. A. Mahdi, B. Ali, A. M. Selman, H. F. Al-Taay, P. J. Jennings, and W.-J. Lee, “Fabrication of a high sensitivity and fast response self-powered photosensor based on a core-shell silicon nanowire homojunction,” Superlattices Microstruct. 116, 27–35 (2018).
[Crossref]

Aloni, S.

L. R. Richey-Simonsen, N. J. Borys, T. R. Kuykendall, P. J. Schuck, S. Aloni, and J. M. Gerton, “Investigating surface effects of GaN nanowires using confocal microscopy at below-band gap excitation,” J. Mater. Res. 32, 4076–4086 (2017).
[Crossref]

T. R. Kuykendall, M. V. P. Altoe, D. F. Ogletree, and S. Aloni, “Catalyst-directed crystallographic orientation control of GaN nanowire growth,” Nano Lett. 14, 6767–6773 (2014).
[Crossref]

A. M. Schwartzberg, S. Aloni, T. Kuykendall, P. J. Schuck, and J. J. Urban, “Optical cavity characterization in nanowires via self-generated broad-band emission,” Opt. Express 19, 8903–8911 (2011).
[Crossref]

Al-Taay, H. F.

A. A. Abdul-Hameed, M. A. Mahdi, B. Ali, A. M. Selman, H. F. Al-Taay, P. J. Jennings, and W.-J. Lee, “Fabrication of a high sensitivity and fast response self-powered photosensor based on a core-shell silicon nanowire homojunction,” Superlattices Microstruct. 116, 27–35 (2018).
[Crossref]

Altoe, M. V. P.

T. R. Kuykendall, M. V. P. Altoe, D. F. Ogletree, and S. Aloni, “Catalyst-directed crystallographic orientation control of GaN nanowire growth,” Nano Lett. 14, 6767–6773 (2014).
[Crossref]

An, B.-G.

H.-R. Kim, B.-G. An, Y. W. Chang, M.-J. Kang, J.-G. Park, and J.-C. Pyun, “Highly sensitive in situ-synthesized cadmium sulfide (CdS) nanowire photosensor for chemiluminescent immunoassays,” Enzym. Microb. Technol. 133, 109457 (2019).
[Crossref]

Andrews, S. C.

N. P. Dasgupta, J. Sun, C. Liu, S. Brittman, S. C. Andrews, J. Lim, H. Gao, R. Yan, and P. Yang, “25th anniversary article: semiconductor nanowires–synthesis, characterization, and applications,” Adv. Mater. 26, 2137–2184 (2014).
[Crossref]

Anttu, N.

J.-P. Kakko, A. Matikainen, N. Anttu, S. Kujala, H. Mäntynen, V. Khayrudinov, A. Autere, Z. Sun, and H. Lipsanen, “Measurement of nanowire optical modes using cross-polarization microscopy,” Sci. Rep. 7, 17790 (2017).
[Crossref]

Arbouet, A.

P. R. Wiecha, A. Cuche, A. Arbouet, C. Girard, G. Colas des Francs, A. Lecestre, G. Larrieu, F. Fournel, V. Larrey, T. Baron, and V. Paillard, “Strongly directional scattering from dielectric nanowires,” ACS Photon. 4, 2036–2046 (2017).
[Crossref]

Autere, A.

H. Yang, V. Khayrudinov, V. Dhaka, H. Jiang, A. Autere, H. Lipsanen, Z. Sun, and H. Jussila, “Nanowire network–based multifunctional all-optical logic gates,” Sci. Adv. 4, eaar7954 (2018).
[Crossref]

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M. E. Lin, B. N. Sverdlov, S. Strite, H. Morkoç, and A. E. Drakin, “Refractive indices of wurtzite and zincblende GaN,” Electron. Lett. 29, 1759–1760 (1993).
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H. Yang, V. Khayrudinov, V. Dhaka, H. Jiang, A. Autere, H. Lipsanen, Z. Sun, and H. Jussila, “Nanowire network–based multifunctional all-optical logic gates,” Sci. Adv. 4, eaar7954 (2018).
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H. Yang, V. Khayrudinov, V. Dhaka, H. Jiang, A. Autere, H. Lipsanen, Z. Sun, and H. Jussila, “Nanowire network–based multifunctional all-optical logic gates,” Sci. Adv. 4, eaar7954 (2018).
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S. Han, S.-K. Lee, I. Choi, J. Song, C.-R. Lee, K. Kim, M.-Y. Ryu, K.-U. Jeong, and J. S. Kim, “Highly efficient and flexible photosensors with GaN nanowires horizontally embedded in a graphene sandwich channel,” ACS Appl. Mater. Interfaces 10, 38173–38182 (2018).
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S. Han, S.-K. Lee, I. Choi, J. Song, C.-R. Lee, K. Kim, M.-Y. Ryu, K.-U. Jeong, and J. S. Kim, “Highly efficient and flexible photosensors with GaN nanowires horizontally embedded in a graphene sandwich channel,” ACS Appl. Mater. Interfaces 10, 38173–38182 (2018).
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S. Han, S.-K. Lee, I. Choi, J. Song, C.-R. Lee, K. Kim, M.-Y. Ryu, K.-U. Jeong, and J. S. Kim, “Highly efficient and flexible photosensors with GaN nanowires horizontally embedded in a graphene sandwich channel,” ACS Appl. Mater. Interfaces 10, 38173–38182 (2018).
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Y. Huang, X. Duan, and C. M. Lieber, “Nanowires for integrated multicolor nanophotonics,” Small 1, 142–147 (2005).
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S. Gradečak, F. Qian, Y. Li, H.-G. Park, and C. M. Liebera, “GaN nanowire lasers with low lasing thresholds,” Appl. Phys. Lett. 87, 173111 (2005).
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H. Yang, V. Khayrudinov, V. Dhaka, H. Jiang, A. Autere, H. Lipsanen, Z. Sun, and H. Jussila, “Nanowire network–based multifunctional all-optical logic gates,” Sci. Adv. 4, eaar7954 (2018).
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N. P. Dasgupta, J. Sun, C. Liu, S. Brittman, S. C. Andrews, J. Lim, H. Gao, R. Yan, and P. Yang, “25th anniversary article: semiconductor nanowires–synthesis, characterization, and applications,” Adv. Mater. 26, 2137–2184 (2014).
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J.-P. Kakko, A. Matikainen, N. Anttu, S. Kujala, H. Mäntynen, V. Khayrudinov, A. Autere, Z. Sun, and H. Lipsanen, “Measurement of nanowire optical modes using cross-polarization microscopy,” Sci. Rep. 7, 17790 (2017).
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J.-P. Kakko, A. Matikainen, N. Anttu, S. Kujala, H. Mäntynen, V. Khayrudinov, A. Autere, Z. Sun, and H. Lipsanen, “Measurement of nanowire optical modes using cross-polarization microscopy,” Sci. Rep. 7, 17790 (2017).
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Mokkapati, S.

D. Saxena, F. Wang, Q. Gao, S. Mokkapati, H. H. Tan, and C. Jagadish, “Mode profiling of semiconductor nanowire lasers,” Nano Lett. 15, 5342–5348 (2015).
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M. E. Lin, B. N. Sverdlov, S. Strite, H. Morkoç, and A. E. Drakin, “Refractive indices of wurtzite and zincblende GaN,” Electron. Lett. 29, 1759–1760 (1993).
[Crossref]

Nabet, B.

Z. Wang and B. Nabet, “Nanowire optoelectronics,” Nanophotonics 4, 491–502 (2015).
[Crossref]

Neugebauer, J.

M. Himmerlich, L. Lymperakis, R. Gutt, P. Lorenz, J. Neugebauer, and S. Krischok, “GaN(0001) surface states: experimental and theoretical fingerprints to identify surface reconstructions,” Phys. Rev. B 88, 125304 (2013).
[Crossref]

Neumann, M. D.

M. Landmann, E. Rauls, W. G. Schmidt, M. D. Neumann, E. Speiser, and N. Esser, “GaN m-plane: atomic structure, surface bands, and optical response,” Phys. Rev. B 91, 035302 (2015).
[Crossref]

Nordlander, P.

Z. Li, K. Bao, Y. Fang, Y. Huang, P. Nordlander, and H. Xu, “Correlation between incident and emission polarization in nanowire surface plasmon waveguides,” Nano Lett. 10, 1831–1835 (2010).
[Crossref]

Novotny, L.

L. Novotny and B. Hecht, Principles of Nano-Optics, 1st ed. (Cambridge University, 2006).

Ogletree, D. F.

T. R. Kuykendall, M. V. P. Altoe, D. F. Ogletree, and S. Aloni, “Catalyst-directed crystallographic orientation control of GaN nanowire growth,” Nano Lett. 14, 6767–6773 (2014).
[Crossref]

Paillard, V.

P. R. Wiecha, A. Cuche, A. Arbouet, C. Girard, G. Colas des Francs, A. Lecestre, G. Larrieu, F. Fournel, V. Larrey, T. Baron, and V. Paillard, “Strongly directional scattering from dielectric nanowires,” ACS Photon. 4, 2036–2046 (2017).
[Crossref]

Paniagua-Domínguez, R.

G. Grzela, R. Paniagua-Domínguez, T. Barten, Y. Fontana, J. A. Sánchez-Gil, and J. G. Rivas, “Nanowire antenna emission,” Nano Lett. 12, 5481–5486 (2012).
[Crossref]

Park, H.-G.

S. Gradečak, F. Qian, Y. Li, H.-G. Park, and C. M. Liebera, “GaN nanowire lasers with low lasing thresholds,” Appl. Phys. Lett. 87, 173111 (2005).
[Crossref]

Park, J.-G.

H.-R. Kim, B.-G. An, Y. W. Chang, M.-J. Kang, J.-G. Park, and J.-C. Pyun, “Highly sensitive in situ-synthesized cadmium sulfide (CdS) nanowire photosensor for chemiluminescent immunoassays,” Enzym. Microb. Technol. 133, 109457 (2019).
[Crossref]

Paschotta, R.

R. Paschotta, "V number," in Encyclopedia of Laser Physics and Technology, 1st ed. (Wiley-VCH, 2008), Vol. N-Z, p. 785.

Pyun, J.-C.

H.-R. Kim, B.-G. An, Y. W. Chang, M.-J. Kang, J.-G. Park, and J.-C. Pyun, “Highly sensitive in situ-synthesized cadmium sulfide (CdS) nanowire photosensor for chemiluminescent immunoassays,” Enzym. Microb. Technol. 133, 109457 (2019).
[Crossref]

Qian, F.

S. Gradečak, F. Qian, Y. Li, H.-G. Park, and C. M. Liebera, “GaN nanowire lasers with low lasing thresholds,” Appl. Phys. Lett. 87, 173111 (2005).
[Crossref]

Rauls, E.

M. Landmann, E. Rauls, W. G. Schmidt, M. D. Neumann, E. Speiser, and N. Esser, “GaN m-plane: atomic structure, surface bands, and optical response,” Phys. Rev. B 91, 035302 (2015).
[Crossref]

Reshchikov, M. A.

M. A. Reshchikov and H. Morkoç, “Luminescence properties of defects in GaN,” J. Appl. Phys. 97, 061301 (2005).
[Crossref]

Richey-Simonsen, L. R.

L. R. Richey-Simonsen, N. J. Borys, T. R. Kuykendall, P. J. Schuck, S. Aloni, and J. M. Gerton, “Investigating surface effects of GaN nanowires using confocal microscopy at below-band gap excitation,” J. Mater. Res. 32, 4076–4086 (2017).
[Crossref]

Rivas, J. G.

G. Grzela, R. Paniagua-Domínguez, T. Barten, Y. Fontana, J. A. Sánchez-Gil, and J. G. Rivas, “Nanowire antenna emission,” Nano Lett. 12, 5481–5486 (2012).
[Crossref]

Ryu, M.-Y.

S. Han, S.-K. Lee, I. Choi, J. Song, C.-R. Lee, K. Kim, M.-Y. Ryu, K.-U. Jeong, and J. S. Kim, “Highly efficient and flexible photosensors with GaN nanowires horizontally embedded in a graphene sandwich channel,” ACS Appl. Mater. Interfaces 10, 38173–38182 (2018).
[Crossref]

Sánchez-Gil, J. A.

G. Grzela, R. Paniagua-Domínguez, T. Barten, Y. Fontana, J. A. Sánchez-Gil, and J. G. Rivas, “Nanowire antenna emission,” Nano Lett. 12, 5481–5486 (2012).
[Crossref]

Saxena, D.

D. Saxena, F. Wang, Q. Gao, S. Mokkapati, H. H. Tan, and C. Jagadish, “Mode profiling of semiconductor nanowire lasers,” Nano Lett. 15, 5342–5348 (2015).
[Crossref]

Saykally, R. J.

M. Law, D. J. Sirbuly, J. C. Johnson, J. Goldberger, R. J. Saykally, and P. Yang, “Nanoribbon waveguides for subwavelength photonics integration,” Science 305, 1269–1273 (2004).
[Crossref]

Schmidt, W. G.

M. Landmann, E. Rauls, W. G. Schmidt, M. D. Neumann, E. Speiser, and N. Esser, “GaN m-plane: atomic structure, surface bands, and optical response,” Phys. Rev. B 91, 035302 (2015).
[Crossref]

Schuck, P. J.

L. R. Richey-Simonsen, N. J. Borys, T. R. Kuykendall, P. J. Schuck, S. Aloni, and J. M. Gerton, “Investigating surface effects of GaN nanowires using confocal microscopy at below-band gap excitation,” J. Mater. Res. 32, 4076–4086 (2017).
[Crossref]

A. M. Schwartzberg, S. Aloni, T. Kuykendall, P. J. Schuck, and J. J. Urban, “Optical cavity characterization in nanowires via self-generated broad-band emission,” Opt. Express 19, 8903–8911 (2011).
[Crossref]

Schwartzberg, A. M.

Selman, A. M.

A. A. Abdul-Hameed, M. A. Mahdi, B. Ali, A. M. Selman, H. F. Al-Taay, P. J. Jennings, and W.-J. Lee, “Fabrication of a high sensitivity and fast response self-powered photosensor based on a core-shell silicon nanowire homojunction,” Superlattices Microstruct. 116, 27–35 (2018).
[Crossref]

Sirbuly, D. J.

M. Law, D. J. Sirbuly, J. C. Johnson, J. Goldberger, R. J. Saykally, and P. Yang, “Nanoribbon waveguides for subwavelength photonics integration,” Science 305, 1269–1273 (2004).
[Crossref]

Snyder, A.

A. Snyder and J. Love, Optical Waveguide Theory, 1st ed. (Chapman and Hall, 1983).

Song, J.

S. Han, S.-K. Lee, I. Choi, J. Song, C.-R. Lee, K. Kim, M.-Y. Ryu, K.-U. Jeong, and J. S. Kim, “Highly efficient and flexible photosensors with GaN nanowires horizontally embedded in a graphene sandwich channel,” ACS Appl. Mater. Interfaces 10, 38173–38182 (2018).
[Crossref]

Speiser, E.

M. Landmann, E. Rauls, W. G. Schmidt, M. D. Neumann, E. Speiser, and N. Esser, “GaN m-plane: atomic structure, surface bands, and optical response,” Phys. Rev. B 91, 035302 (2015).
[Crossref]

Strite, S.

M. E. Lin, B. N. Sverdlov, S. Strite, H. Morkoç, and A. E. Drakin, “Refractive indices of wurtzite and zincblende GaN,” Electron. Lett. 29, 1759–1760 (1993).
[Crossref]

Sullivan, D. M.

D. M. Sullivan, Electromagnetic Simulation Using the FDTD Method, 2nd ed. (Wiley-IEEE, 2013).

Sun, J.

N. P. Dasgupta, J. Sun, C. Liu, S. Brittman, S. C. Andrews, J. Lim, H. Gao, R. Yan, and P. Yang, “25th anniversary article: semiconductor nanowires–synthesis, characterization, and applications,” Adv. Mater. 26, 2137–2184 (2014).
[Crossref]

Sun, Z.

H. Yang, V. Khayrudinov, V. Dhaka, H. Jiang, A. Autere, H. Lipsanen, Z. Sun, and H. Jussila, “Nanowire network–based multifunctional all-optical logic gates,” Sci. Adv. 4, eaar7954 (2018).
[Crossref]

J.-P. Kakko, A. Matikainen, N. Anttu, S. Kujala, H. Mäntynen, V. Khayrudinov, A. Autere, Z. Sun, and H. Lipsanen, “Measurement of nanowire optical modes using cross-polarization microscopy,” Sci. Rep. 7, 17790 (2017).
[Crossref]

Sverdlov, B. N.

M. E. Lin, B. N. Sverdlov, S. Strite, H. Morkoç, and A. E. Drakin, “Refractive indices of wurtzite and zincblende GaN,” Electron. Lett. 29, 1759–1760 (1993).
[Crossref]

Tan, H. H.

D. Saxena, F. Wang, Q. Gao, S. Mokkapati, H. H. Tan, and C. Jagadish, “Mode profiling of semiconductor nanowire lasers,” Nano Lett. 15, 5342–5348 (2015).
[Crossref]

Tian, W.

W. Tian, C. Zhao, J. Leng, R. Cui, and S. Jin, “Visualizing carrier diffusion in individual single-crystal organolead halide perovskite nanowires and nanoplates,” J. Am. Chem. Soc. 137, 12458–12461 (2015).
[Crossref]

Urban, J. J.

Wang, F.

D. Saxena, F. Wang, Q. Gao, S. Mokkapati, H. H. Tan, and C. Jagadish, “Mode profiling of semiconductor nanowire lasers,” Nano Lett. 15, 5342–5348 (2015).
[Crossref]

Wang, G. T.

H. Xu, J. B. Wright, T.-S. Luk, J. J. Figiel, K. Cross, L. F. Lester, G. Balakrishnan, G. T. Wang, I. Brener, and Q. Li, “Single-mode lasing of GaN nanowire-pairs,” Appl. Phys. Lett. 101, 113106 (2012).
[Crossref]

J. Y. Huang, H. Zheng, S. X. Mao, Q. Li, and G. T. Wang, “In situ nanomechanics of GaN nanowires,” Nano Lett. 11, 1618–1622 (2011).
[Crossref]

Q. Li and G. T. Wang, “Spatial distribution of defect luminescence in GaN nanowires,” Nano Lett. 10, 1554–1558 (2010).
[Crossref]

Wang, Z.

Z. Wang and B. Nabet, “Nanowire optoelectronics,” Nanophotonics 4, 491–502 (2015).
[Crossref]

Wiecha, P. R.

P. R. Wiecha, A. Cuche, A. Arbouet, C. Girard, G. Colas des Francs, A. Lecestre, G. Larrieu, F. Fournel, V. Larrey, T. Baron, and V. Paillard, “Strongly directional scattering from dielectric nanowires,” ACS Photon. 4, 2036–2046 (2017).
[Crossref]

Wright, J. B.

H. Xu, J. B. Wright, T.-S. Luk, J. J. Figiel, K. Cross, L. F. Lester, G. Balakrishnan, G. T. Wang, I. Brener, and Q. Li, “Single-mode lasing of GaN nanowire-pairs,” Appl. Phys. Lett. 101, 113106 (2012).
[Crossref]

Xu, H.

H. Xu, J. B. Wright, T.-S. Luk, J. J. Figiel, K. Cross, L. F. Lester, G. Balakrishnan, G. T. Wang, I. Brener, and Q. Li, “Single-mode lasing of GaN nanowire-pairs,” Appl. Phys. Lett. 101, 113106 (2012).
[Crossref]

Z. Li, K. Bao, Y. Fang, Y. Huang, P. Nordlander, and H. Xu, “Correlation between incident and emission polarization in nanowire surface plasmon waveguides,” Nano Lett. 10, 1831–1835 (2010).
[Crossref]

Yan, R.

N. P. Dasgupta, J. Sun, C. Liu, S. Brittman, S. C. Andrews, J. Lim, H. Gao, R. Yan, and P. Yang, “25th anniversary article: semiconductor nanowires–synthesis, characterization, and applications,” Adv. Mater. 26, 2137–2184 (2014).
[Crossref]

R. Yan, D. Gargas, and P. Yang, “Nanowire photonics,” Nat. Photonics 3, 569–576 (2009).
[Crossref]

Yang, H.

H. Yang, V. Khayrudinov, V. Dhaka, H. Jiang, A. Autere, H. Lipsanen, Z. Sun, and H. Jussila, “Nanowire network–based multifunctional all-optical logic gates,” Sci. Adv. 4, eaar7954 (2018).
[Crossref]

Yang, P.

N. P. Dasgupta, J. Sun, C. Liu, S. Brittman, S. C. Andrews, J. Lim, H. Gao, R. Yan, and P. Yang, “25th anniversary article: semiconductor nanowires–synthesis, characterization, and applications,” Adv. Mater. 26, 2137–2184 (2014).
[Crossref]

R. Yan, D. Gargas, and P. Yang, “Nanowire photonics,” Nat. Photonics 3, 569–576 (2009).
[Crossref]

M. Law, D. J. Sirbuly, J. C. Johnson, J. Goldberger, R. J. Saykally, and P. Yang, “Nanoribbon waveguides for subwavelength photonics integration,” Science 305, 1269–1273 (2004).
[Crossref]

Zhao, C.

W. Tian, C. Zhao, J. Leng, R. Cui, and S. Jin, “Visualizing carrier diffusion in individual single-crystal organolead halide perovskite nanowires and nanoplates,” J. Am. Chem. Soc. 137, 12458–12461 (2015).
[Crossref]

Zheng, H.

J. Y. Huang, H. Zheng, S. X. Mao, Q. Li, and G. T. Wang, “In situ nanomechanics of GaN nanowires,” Nano Lett. 11, 1618–1622 (2011).
[Crossref]

Zhu, Z.

ACS Appl. Mater. Interfaces (1)

S. Han, S.-K. Lee, I. Choi, J. Song, C.-R. Lee, K. Kim, M.-Y. Ryu, K.-U. Jeong, and J. S. Kim, “Highly efficient and flexible photosensors with GaN nanowires horizontally embedded in a graphene sandwich channel,” ACS Appl. Mater. Interfaces 10, 38173–38182 (2018).
[Crossref]

ACS Photon. (1)

P. R. Wiecha, A. Cuche, A. Arbouet, C. Girard, G. Colas des Francs, A. Lecestre, G. Larrieu, F. Fournel, V. Larrey, T. Baron, and V. Paillard, “Strongly directional scattering from dielectric nanowires,” ACS Photon. 4, 2036–2046 (2017).
[Crossref]

Adv. Mater. (1)

N. P. Dasgupta, J. Sun, C. Liu, S. Brittman, S. C. Andrews, J. Lim, H. Gao, R. Yan, and P. Yang, “25th anniversary article: semiconductor nanowires–synthesis, characterization, and applications,” Adv. Mater. 26, 2137–2184 (2014).
[Crossref]

Adv. Opt. Photon. (1)

Appl. Phys. Lett. (2)

S. Gradečak, F. Qian, Y. Li, H.-G. Park, and C. M. Liebera, “GaN nanowire lasers with low lasing thresholds,” Appl. Phys. Lett. 87, 173111 (2005).
[Crossref]

H. Xu, J. B. Wright, T.-S. Luk, J. J. Figiel, K. Cross, L. F. Lester, G. Balakrishnan, G. T. Wang, I. Brener, and Q. Li, “Single-mode lasing of GaN nanowire-pairs,” Appl. Phys. Lett. 101, 113106 (2012).
[Crossref]

Electron. Lett. (1)

M. E. Lin, B. N. Sverdlov, S. Strite, H. Morkoç, and A. E. Drakin, “Refractive indices of wurtzite and zincblende GaN,” Electron. Lett. 29, 1759–1760 (1993).
[Crossref]

Enzym. Microb. Technol. (1)

H.-R. Kim, B.-G. An, Y. W. Chang, M.-J. Kang, J.-G. Park, and J.-C. Pyun, “Highly sensitive in situ-synthesized cadmium sulfide (CdS) nanowire photosensor for chemiluminescent immunoassays,” Enzym. Microb. Technol. 133, 109457 (2019).
[Crossref]

J. Am. Chem. Soc. (1)

W. Tian, C. Zhao, J. Leng, R. Cui, and S. Jin, “Visualizing carrier diffusion in individual single-crystal organolead halide perovskite nanowires and nanoplates,” J. Am. Chem. Soc. 137, 12458–12461 (2015).
[Crossref]

J. Appl. Phys. (1)

M. A. Reshchikov and H. Morkoç, “Luminescence properties of defects in GaN,” J. Appl. Phys. 97, 061301 (2005).
[Crossref]

J. Mater. Res. (1)

L. R. Richey-Simonsen, N. J. Borys, T. R. Kuykendall, P. J. Schuck, S. Aloni, and J. M. Gerton, “Investigating surface effects of GaN nanowires using confocal microscopy at below-band gap excitation,” J. Mater. Res. 32, 4076–4086 (2017).
[Crossref]

Nano Lett. (6)

Q. Li and G. T. Wang, “Spatial distribution of defect luminescence in GaN nanowires,” Nano Lett. 10, 1554–1558 (2010).
[Crossref]

T. R. Kuykendall, M. V. P. Altoe, D. F. Ogletree, and S. Aloni, “Catalyst-directed crystallographic orientation control of GaN nanowire growth,” Nano Lett. 14, 6767–6773 (2014).
[Crossref]

J. Y. Huang, H. Zheng, S. X. Mao, Q. Li, and G. T. Wang, “In situ nanomechanics of GaN nanowires,” Nano Lett. 11, 1618–1622 (2011).
[Crossref]

D. Saxena, F. Wang, Q. Gao, S. Mokkapati, H. H. Tan, and C. Jagadish, “Mode profiling of semiconductor nanowire lasers,” Nano Lett. 15, 5342–5348 (2015).
[Crossref]

G. Grzela, R. Paniagua-Domínguez, T. Barten, Y. Fontana, J. A. Sánchez-Gil, and J. G. Rivas, “Nanowire antenna emission,” Nano Lett. 12, 5481–5486 (2012).
[Crossref]

Z. Li, K. Bao, Y. Fang, Y. Huang, P. Nordlander, and H. Xu, “Correlation between incident and emission polarization in nanowire surface plasmon waveguides,” Nano Lett. 10, 1831–1835 (2010).
[Crossref]

Nanophotonics (1)

Z. Wang and B. Nabet, “Nanowire optoelectronics,” Nanophotonics 4, 491–502 (2015).
[Crossref]

Nat. Photonics (1)

R. Yan, D. Gargas, and P. Yang, “Nanowire photonics,” Nat. Photonics 3, 569–576 (2009).
[Crossref]

Nature (1)

X. Duan, Y. Huang, R. Agarwal, and C. M. Lieber, “Single-nanowire electrically driven lasers,” Nature 421, 241–245 (2003).
[Crossref]

Opt. Express (2)

Phys. Rev. B (2)

M. Landmann, E. Rauls, W. G. Schmidt, M. D. Neumann, E. Speiser, and N. Esser, “GaN m-plane: atomic structure, surface bands, and optical response,” Phys. Rev. B 91, 035302 (2015).
[Crossref]

M. Himmerlich, L. Lymperakis, R. Gutt, P. Lorenz, J. Neugebauer, and S. Krischok, “GaN(0001) surface states: experimental and theoretical fingerprints to identify surface reconstructions,” Phys. Rev. B 88, 125304 (2013).
[Crossref]

Sci. Adv. (1)

H. Yang, V. Khayrudinov, V. Dhaka, H. Jiang, A. Autere, H. Lipsanen, Z. Sun, and H. Jussila, “Nanowire network–based multifunctional all-optical logic gates,” Sci. Adv. 4, eaar7954 (2018).
[Crossref]

Sci. Rep. (1)

J.-P. Kakko, A. Matikainen, N. Anttu, S. Kujala, H. Mäntynen, V. Khayrudinov, A. Autere, Z. Sun, and H. Lipsanen, “Measurement of nanowire optical modes using cross-polarization microscopy,” Sci. Rep. 7, 17790 (2017).
[Crossref]

Science (1)

M. Law, D. J. Sirbuly, J. C. Johnson, J. Goldberger, R. J. Saykally, and P. Yang, “Nanoribbon waveguides for subwavelength photonics integration,” Science 305, 1269–1273 (2004).
[Crossref]

Small (1)

Y. Huang, X. Duan, and C. M. Lieber, “Nanowires for integrated multicolor nanophotonics,” Small 1, 142–147 (2005).
[Crossref]

Superlattices Microstruct. (1)

A. A. Abdul-Hameed, M. A. Mahdi, B. Ali, A. M. Selman, H. F. Al-Taay, P. J. Jennings, and W.-J. Lee, “Fabrication of a high sensitivity and fast response self-powered photosensor based on a core-shell silicon nanowire homojunction,” Superlattices Microstruct. 116, 27–35 (2018).
[Crossref]

Other (9)

A. Snyder and J. Love, Optical Waveguide Theory, 1st ed. (Chapman and Hall, 1983).

R. Paschotta, "V number," in Encyclopedia of Laser Physics and Technology, 1st ed. (Wiley-VCH, 2008), Vol. N-Z, p. 785.

G. Agrawal, Nonlinear Fiber Optics, 5th ed. (Elsevier, 2012), Chap. 2.

L. Novotny and B. Hecht, Principles of Nano-Optics, 1st ed. (Cambridge University, 2006).

Lumerical, “FDE,” https://support.lumerical.com/hc/en-us/articles/360034917233 .

D. M. Sullivan, Electromagnetic Simulation Using the FDTD Method, 2nd ed. (Wiley-IEEE, 2013).

R. A. Gutierrez-Arenas and D. Mendoza, “Band structure of a two dimensional metallic photonic crystal and the experimental observation of negative refraction in the microwave region,” arXiv:1109.0329 [physical.optics] (2011).

J.-M. Jin, The Finite Element Method in Electromagnetics, 3rd ed. (Wiley-IEEE, 2014).

Lumerical, “Lossy modes and dB/m to k conversion,” https://support.lumerical.com/hc/en-us/articles/360034917493 .

Supplementary Material (1)

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Figures (8)

Fig. 1.
Fig. 1. Schematic of experiment. (a) Scanning electron micrograph of the NW growth substrate. (b) Rendering of an atomic force microscope (AFM) image of a GaN NW lying atop a glass coverslip. The outline of the NW is indicated by red lines. (c) A simplified schematic of the optical setup: a scanning mirror is positioned at a Fourier plane that is conjugate to the back focal plane of the objective lens allowing the confocal collection volume to be displaced from the laser focus. (d) Fluorescence spectrum collected from near the center of the NW.
Fig. 2.
Fig. 2. Widefield images of a tapered NW. The NW was measured by AFM to be 7.3 µm long with height 380 nm at the left end and 217 nm at the right end. Images were captured for three positions of the laser focus: the (a)–(c) left end, (d)–(f) center, and (g)–(i) right end of the NW. Panels (a), (d), (g) correspond to backscattered laser light, and (b), (e), (h) to fluorescence emission. Panels (c), (f), (i) show the intensity profiles for the scattered laser light (blue solid lines) and fluorescence (red dashed lines) along the NW, where the signals have been summed perpendicular to the NW axis. The small arrows in panels (c), (f), (i) indicate the estimated position of the focal spot in each case. There was a ${\sim}2\;\text{s}$ delay between the backscatter images and the corresponding fluorescence images for each laser position. No effort was made to individually align the backscatter and fluorescence profiles in (c), (f), (i).
Fig. 3.
Fig. 3. FDTD simulations of a GaN NW atop glass illuminated by a focused laser beam (see Section 3.C). The modeled physical dimensions are based on AFM measurements of the NW shown in Fig. 2. Panels (a)–(d) show $|{\bf E}{|^2}$ on the central $y{-}z$ plane ($x = 0$) that intersects the NW apex for four laser positions: (a) $y = 7.65\;\unicode{x00B5}\text{m}$ (right end facet), (b) $y = 0.35\;\unicode{x00B5}\text{m}$ (left end facet), (c) $y = 0.4\;\unicode{x00B5}\text{m}$, (d) $y = 4\;\unicode{x00B5}\text{m}$. (e) Internal intensity profiles along the NW, where $|{\bf E}{|^2}$ is integrated over cross section planes ($x{-}z$) of the NW (i.e., internal to the NW) at each position, $y$. The images in (a)–(d) are all self-normalized and use the same color scale shown in panel (d), whereas the profiles in (e) are normalized to the maximum internal intensity when the laser is at $y = 0.4\;\unicode{x00B5}\text{m}$.
Fig. 4.
Fig. 4. AFM and off-axis confocal hyperspectral images of a tapered NW atop a glass coverslip. (a) Height profile along the NW apex extracted from the AFM image in (b). (c) Integrated fluorescence intensity map when the laser is positioned near the left (larger) end of the NW. (d) Hyperspectral data from (c). The color scale shown in panel (c) is also used in panel (d).
Fig. 5.
Fig. 5. Reduced cutoff wavelengths and mode profiles for a GaN NW waveguide atop a glass coverslip. Modes were computed using FDE simulations for an equilateral triangular prism geometry with edge length $a$ and refractive index from Lin [18]. The reduced cutoff wavelength ${\lambda _{\text{r}}}$ is defined by Eq. (3), and the lines through the symbols are linear fits to Eq. (4), giving the V-numbers shown for eigenmodes 1–8. The insets of the figure show intensity profiles ($|{\bf E}{|^2}$) for the modes at $\lambda = 400\;\text{nm} $ (${\lambda _{\text{r}}} = 189\;\text{nm} $) for a prism with 400 nm edge length (indicated as violet crosshair).
Fig. 6.
Fig. 6. Off-axis confocal hyperspectral images and FDTD simulations of a NW. This is the same NW from Fig. 4, but the laser spot is ${\sim}500\;\text{nm}$ further away from the end facet. (a) Fluorescence intensity map when the laser is positioned near the left (larger) end of the NW. (b) Hyperspectral data from (a). (c)–(e) Hyperspectral “data” extracted from FDTD simulations where the simulated emitter (dipole; see Section 3.C) is positioned at $y = 0$. In panel (c), the NW cross section from AFM measurements (see inset) is used in the FDTD simulations. In panel (d), an equilateral triangular cross section with overestimated edge length is assumed, while in panel (e), an underestimated triangular cross section is used. The solid/dashed red diagonal lines in (b)–(e) indicate the predicted cutoff wavelengths for modes 3–8 based on the mode analysis in Fig. 5 for the overestimated/underestimated triangular cross section. The spectral stripes correspond to leakage of specific modes above their cutoff wavelengths, e.g.,  the bright stripe between $6 \lt y \lt 8\;\unicode{x00B5}\text{m}$ in panel (b) corresponds to leakage of mode 5,6.
Fig. 7.
Fig. 7. Off-axis confocal hyperspectral images of a narrow NW atop a glass coverslip. (a) Height profile along the NW apex measured by AFM. (b) Integrated fluorescence intensity map when the laser is positioned near the left (larger) end of the NW. (c) Hyperspectral data from (b). In panel (c), the solid red line is the predicted cutoff line for mode 1–2 assuming an equilateral triangular cross section with overestimated edge length, and the dashed line is for the underestimated edge length (see inset).
Fig. 8.
Fig. 8. Using spectral signatures to deduce NW height. (a) Integrated fluorescence intensity map when the laser is positioned near the middle of the NW. (b) One-dimensional hyperspectral data from (a), where the red and black lines indicate the estimated cutoff wavelengths of mode 3–4 and 5–6, respectively, as described in the text. (c) Deduced topography of NW sections based on Eq. (4), assuming an equilateral triangular cross section. (d) AFM image of the NW after background flattening. (e) Comparison of apex height deduced from hyperspectral data (orange solid line) and AFM measurements (blue dashed line). The blue dotted line represents the estimated upper bound of the apex height taking into account AFM sampling artifacts and uncertainties in the shape of the AFM probe. Error bars are calculated as described in the text.

Equations (4)

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k trans < | k | 1 n glass 2 n GaN 2 .
λ c = 2 n GaN 2 n glass 2 Δ r mode .
λ r = λ c n GaN 2 n glass 2 ,
λ r = 2 a / V mode ,