Generating a sub-nanometer-confined optical field in a nanoslit waveguiding mode

Abstract. We propose to generate a sub-nanometer-confined optical field in a nanoslit waveguiding mode in a coupled nanowire pair (CNP). We show that, when a conventional waveguide mode with a proper polarization is evanescently coupled into a properly designed CNP with a central nanoslit, it can be efficiently channeled into a high-purity nanoslit mode within a waveguiding length <10  μm. The CNP can be either freestanding or on-chip by using a tapered fiber or planar waveguide for input-coupling, with a coupling efficiency up to 95%. Within the slit region, the output diffraction-limited nanoslit mode offers an extremely confined optical field (∼0.3  nm  ×  3.3  nm) with a peak-to-background ratio higher than 25 dB and can be operated within a 200-nm bandwidth. The group velocity dispersion of the nanoslit mode for ultrafast pulsed operation is also briefly investigated. Compared with the previous lasing configuration, the waveguiding scheme demonstrated here is not only simple and straightforward in structural design but is also much flexible and versatile in operation. Therefore, the waveguiding scheme we show here may offer an efficient and flexible platform for exploring light–matter interactions beyond the nanometer scale, and developing optical technologies ranging from superresolution nanoscopy and atom/molecule manipulation to ultra-sensitivity detection.


Introduction
Generating a light beam or an optical field with tighter spatial confinement is always of great interest in both fundamental research and technological applications ranging from ultratight field confinement, 1-12 nanoscale light-matter interaction, [13][14][15][16][17] to superresolution optical imaging. [18][19][20][21] Due to the optical diffraction limit, 22 optical confinement with a conventional dielectric component (e.g., a lens) cannot be better than λ∕2n, where λ is the vacuum wavelength of light and n is the refractive index of the medium. Recently, relying on optical resonance or interference, field confinement down to deep-subwavelength level has been realized in dielectric nanostructures. [1][2][3][4][5][6][7][8][10][11][12][13] However, limited by the feature size and the fabrication precision of the functional structure, confining optical field to sub-10 nm (i.e., λ∕100 level for visible or near-infrared light) remains challenging. On the other hand, optical-frequency nanoplasmonics, relying on the collective oscillation of free electrons in nanoscale metallic structures, can now confine an optical field down to 10-nm (i.e., λ∕100 level 23 ) or even 1-nm level (i.e., λ∕1000 level 24,25 ), and has been explored for various applications, including nanowaveguides, nanolasers, nonlinear optical effects enhancement, and optical sensing. [26][27][28][29] However, at optical frequency, when the plasmon mode is confined down to a certain degree (e.g., λ∕100), the optical loss of oscillating free electrons will increase drastically with the increasing confinement, 30 reducing the coherence of the confined field, regardless of the geometries of the structure. 31 Also, the optical-loss-generated heat in the material will bring thermal noise and damage and limit the available optical intensity either. Moreover, from the uncertainty relation (Δx · Δp ≥ αℏ; where ℏ is the reduced Planck constant, α is a constant determined by the field characteristics 32,33 ) between the photon position (x) and its momentum (p), to confine a light field to an extremely small spatial size (e.g., reduce Δx to 1-nm level), a huge momentum uncertainty (Δp) is required for compensation, which will significantly increase the momentum mismatch between the freespace propagating wave and the confined optical field.
To circumvent the above-mentioned issues, recently, relying on a nanoslit assembled with bottom-up grown single-crystal nanowires with atomic surface roughness, we realized a subnanometer-confined (sub-nm-confined) field in coupled nanowire pair (CNP) lasers. 34 In such a scheme, the probability of photon occurrence (i.e., the field intensity) within the 1-nm-slit was significantly increased by the oscillation of bound electrons in the CdSe nanowire interfaces around the slit, resulting in a sub-nm-confined optical field with high peak-to-background intensity ratio. Meanwhile, since the whole mode area remains large, the momentum of the confined photon, averaged by its occurrence in the background field, is not necessarily to be much larger than the photon momentum in free space. Such a confinement scheme can generate a coherent field with unprecedented sub-nm optical confinement and avoid a large photon momentum mismatch between the confined field and the free-space light. However, as it uses a lasing configuration, the selectivity and tunability of the confined field in time and frequency domains, which are highly desired for practical applications, are limited due to the critical and complicated lasing and pumping conditions in a nanolaser.
Here, we propose a waveguiding scheme to generate a sub-nm-confined optical field in a nanoslit mode. Unlike the lasing approach that actively generates coherent fields via lasing oscillation inside the CNP, here the coherent field comes from light sources outside the CNP and is evanescently coupled into the CNP with high efficiency, making it possible to generate extremely confined optical fields with great flexibilities, such as narrow linewidth, broadband tunability, and ultrafast pulsed operation. We show that, with matched effective refractive indices (n eff ), a waveguiding mode from a fiber taper can be evanescently coupled into a CNP with an efficiency of up to 95%, and channeled into the desired nanoslit mode with a purity up to 99%. The confined field also has a sub-nm confinement and a high peak-to-background ratio within a spectral range over 100 nm. In addition, we show that, such a waveguiding scheme also works in the mid-infrared (MIR) spectral range, while maintaining an optical confinement of ∼0.2 nm, corresponding to ∼1∕20,000 vacuum wavelength.

Configuration of the CNP Waveguiding Scheme
The CNP waveguiding scheme we proposed here is schematically illustrated in Fig. 1. Light from an outer light source is first coupled into a standard glass fiber and adiabatically guided through a fiber taper fabricated at the other end of the fiber. Then, relying on an evanescent coupling process, it is coupled into a freestanding CNP waveguide from one end, propagating along the properly designed CNP waveguide, in which the waveguided field is evoluted into the nanoslit mode, and finally outputs at the opposite end of the CNP [ Fig. 1(a)]. Benefiting from the tapering profile of the fiber taper, the matching of n eff between the fiber taper and the CNP can be achieved within a broadband spectral range, 35,36 ensuring a high efficiency when coupling light from the fiber taper to the CNP. Similarly, in an on-chip configuration [ Fig. 1(b)], a tapered planar waveguide can be used for coupling light into one end of a silicon-oninsulator-supported CNP with a high efficiency and high-purity nanoslit mode with extreme optical confinement at its center can be obtained at the other end. Practically, the CNP waveguide can be assembled by two identical single-crystal nanowires, which are cut from one nanowire with a hexagonal cross section and atomic-level smoothness (see Sec. S1 in the Supplemental Material). Based on the sub-nm sidewall roughness, an ultrafine central slit (∼1 nm) can be naturally formed along the central axis of the CNP. 34 Besides, material-rich semiconductor nanowires, such as ZnO, CdS, and CdTe, exhibiting high refractive index (e.g., n ∼ 2.37 @650 nm for CdS 37 ) and negligible absorption loss (e.g., absorption coefficient ∼1.1 × 10 −12 cm −1 @650 nm for CdS 37 ) below the absorption bandgap, can be used for generating sub-nm-confined field from ultraviolet, visible to MIR or even far-infrared bands. It is worth mentioning that in recent years, individual high-quality semiconductor nanowire has been widely used for low-loss optical waveguiding (typical waveguiding loss ∼1 dB mm −1 ). [38][39][40] 3 Mode Evolution in the CNP For a nanowire with a noncircular symmetrical cross section, the lowest-order waveguiding mode contains two nondegenerate modes with different polarizations: a horizontal-polarizationdominated TE 0 -like mode and a vertical-polarization-dominated TM 0 -like mode (see Sec. S3 in the Supplemental Material). 41 When two identical nanowires with a proper wavelength-todiameter ratio are assembled into a CNP, strong mode coupling occurs, resulting in a new set of strong coupling-induced waveguiding modes of the CNP. In general, the lowest four modes can be considered as the symmetric/antisymmetric combinations of the two lowest modes. Since the diameter of nanowires (d, defined as the diagonal of the hexagonal cross section of the nanowire) discussed here is relatively small, we plot only the four lowest eigenmodes with corresponding spatial distributions of polarized bound electrons supported in the CNP [Figs. 2(a)-2(d)]. From the spatial distribution of the charge density [right column in Figs. 2(a)-2(d)], the field confinement and enhancement can be clearly seen. For example, in the TE 0 -like mode, around the two central opposite vertices, the polarized charge density (with opposite signs) reaches the maximum, leading to the extreme confinement of optical field within the gap. For comparison, the charge density is not concentrated around the central gap in other modes. Therefore, the existence of modes other than TE 0 -like mode may not only split a fraction of total mode power but may also serve as an undesired background field. However, n eff of TE 0 -and TM 0 -like modes are relatively close [Figs. 2(e)-2(h)], especially in the on-chip case [Figs. 2(g) and 2(h)], which is experimentally difficult to separate by selecting the nanowire diameter, due to the error in diameter measurement 42 and the varying refractive index of the nanowire material under different waveguiding powers and temperatures. 43,44 Fortunately, the polarizations of the TE 0and TM 0 -like modes are almost orthogonal, making it possible to selectively launch the TE 0 -like mode by controlling the polarization of the input field. For the higher-order modes (here the TE 1 -and TM 1 -like modes), the much larger cut-off diameters make it possible to separate from the TE 0 -like mode, and thus can be easily eliminated by selecting a proper nanowire diameter. Thus, in the following text, we consider CNP that only supports the lowest two modes (i.e., TE 0 -and TM 0 -like modes), and selectively launch the TE 0 -like mode by selecting the polarization of the input field.
We consider the case of a freestanding CNP that is excited by a waveguiding mode from a silica fiber taper. The CdS CNP is assumed to sit symmetrically on the sidewall of the fiber taper, largely overlapped with the fiber taper in length, with a certain length protruding outward [ Fig. 3(a)]. To ensure an adiabatic mode transition in the fiber taper, the tapering angle is assumed to be 3 deg. 36 As shown in Fig. 3(b), the input field is assumed to be a TE-polarized HE 11 mode at 650-nm wavelength, which is efficient in exciting TE-like modes in the CNP, while almost isolated from the TM-like mode due to the orthogonal polarization. To avoid coupling into higher-order modes [Figs. 2(e) and 2(f)], we select a nanowire diameter of 140 nm, which is below the cut-off diameter of the TE 1 ∕TM 1 -like modes, while sufficiently larger than the cut-off diameter of the TE 0 -like mode.
When the input mode from a standard fiber (i.e., a TE-polarized HE 11 mode) enters the tapering area, its n eff is initially larger than that of the TE 0 -like mode of the nanoslit [the leftmost region in Fig. 3(f)]. As it propagates towards the thin end of the fiber taper, the n eff of the input mode decreases with decreasing taper diameter [ Fig. 3(f)]. When the n eff of the input mode approaches that of the TE 0 -like mode, evident light coupling occurs [ Fig. 3(c)]. Benefiting from the properly designed tapering profile of the fiber taper, the coupling length with approximately matched n eff between the two waveguides is optimized for high-efficiency transferring of light from the fiber taper mode into the nanoslit mode [red-colored area in Fig. 3(f)], while almost no light will couple from the nanoslit mode back into the fiber taper mode beyond the n eff -matching area. Eventually, after overlapping area (∼9 μm in length), the high-purity nano-slit mode will output from the right-most end of the CNP [ Fig. 3(e)].
Owing to a tapering profile and thus a wide-range n eff of the input fiber taper, the waveguiding scheme can be operated within a broad spectral range. For example, when the fiber taper diameter decreases from 1.1 to 0.2 μm, the n eff of the input fiber taper changes from 1.41 to ∼1.0 [ Fig. 3(f)], which is broad enough for covering the whole range of n eff of the nanoslit mode [Figs. 2(e) and 3(f)], leading to the possibility of efficient excitation of the nanoslit mode within a broad spectral range. Figure 3(g) gives calculated broadband coupling efficiency and mode purity of the excited nanoslit mode output at the right-most end face, showing a broadband high coupling efficiency (>80%) from 550-to 750-nm wavelength, with a maximum efficiency approaching 95% from 569 to 694 nm wavelength.
Within the 200-nm bandwidth (from 550-to 750-nm wavelength), the mode purity of the nanoslit mode (see Sec. S2 in the Supplemental Material) is higher than 93%, with a maximum of 99.8%. The slight impurity may come from the forwardscattering fields (due to the breakage of the symmetry of the waveguiding structure) and/or a very weak TM 0 -like mode (due to the non-strictly orthogonal polarization with TE-polarized mode) that may be excited during the mode coupling and evolution processes.
Also, we have carried out similar calculations for the case of an on-chip CNP, with similar results given in Sec. S5 in the Supplemental Material.
It is worth noting that, when the gap size reduces to 1-nm level, in a metal structure, the nonlocal effect of free electrons (e.g., quantum tunneling, 45 spillover of electrons, 46 and Landau damping 47 ) may seriously degrade the field confinement; while in an all-dielectric structure used here, owing to the much stronger localization of the bound electrons, the above-mentioned effects are absent, and the classical electromagnetic field theory is still valid. 48

Sub-nm-Confined Optical Fields in the Waveguiding Nanoslit Mode
To investigate the field distribution of the nanoslit mode in a freestanding CdS CNP, we use a 3-D FDTD simulation for the overall structure and a 2-D COMSOL simulation around the slit area for ultrafine calculation (see Sec. S1 in the Supplemental Material), with results given in Fig. 4. When a 550-nm wavelength light is coupled into and waveguided through a CdS CNP with a nanowire diameter of 140 nm, a slit width (w) of 1 nm, and a corner radius (r) of 5 nm, the generated TE 0 -like slit mode offers a maximum field intensity around the central slit [ Fig. 4(a)], with ultratight field confinements of 0.26-nm (x axis) and 3.05-nm (y axis) in full width at halfmaximum (FWHM) of the field intensity [ Fig. 4(b)]. With increasing wavelength, the FWHM of intensity distribution increases [ Fig. 4(c)], mainly due to the decreasing refractive index of the CdS material. Within the broadband from 550 to 750 nm, the field confinement remains a sub-0.3-nm scale (x axis) and a sub-3.3-nm scale (y axis). The calculated spot size is about 1.26 × 10 −5 λ 2 (∼4 nm 2 ) at 550-nm wavelength [ Fig. 4(d)], a similar size of a small molecule (e.g., glucose molecule). 49 Excepting the ultraconfined central field, the overall nanoslit mode is a diffraction-limited optical field [Figs. 4(e) and 4(f)] with an effective mode area 34 of ∼0.95ðλ∕nÞ 2 , which can be considered as a low-intensity large-area background field for averaging the photon momentum of the ultraconfined central field [ Fig. 4(f)].
Due to the dielectric noncontinuity of the CNP at the CdS-air interface, besides the central field maximum, there are a series of field maxima in the background field. Figure 4(g) shows the calculated two second-highest peaks (dashed line), which are 23 dB lower in intensity compared with the central peak. By defining a peak-to-background intensity ratio R P1∕B between the central peak and the averaged background, and R P2∕B between the second-highest peak and the averaged background, 34 we calculate the broadband peak-to-background ratio of the nanoslit mode, with results given in Fig. 4(h). It shows that both ratios increase with increasing wavelength, and R P1∕B keeps 27 dB higher than the background and ∼20 dB higher than the second-highest peak, which is favorable for using the central peak in applications such as ultraresolution optical nanoscopy and manipulations. In addition, the dependence of the field confinement on the slit width and the corner radius of a CdS CNP waveguide has also been investigated, as shown in Sec. S6 in the Supplemental Material.
It is worth mentioning that, due to its ultralow fraction (e.g., ∼0.008% at 550-nm wavelength) of the total mode area, despite its ultrastrong field intensity, the central peak concentrates only a very small fraction (e.g., ∼0.19% at 550-nm wavelength) of the total mode power (see Sec. S7 in the Supplemental Material), agreeing well with the low momentum mismatch between the confined field and the free space.

Sub-nm-Confined Optical Fields at MIR
Benefiting from the ultrabroadband polarization characteristics (e.g., refractive index, transparency window) of the semiconductor materials such as CdS and CdTe, the operation wavelength of the nanoslit mode in a CNP can be extended to a much broader spectral range. For example, here we investigate field confinement in a CNP operating at the MIR spectrum. We keep the slit width of 1 nm but increase the nanowire diameter to 1 μm according to the much larger wavelength. As shown in Fig. 5, the TE 0 -like nanoslit mode can offer an ultratight field confinement in the central slit [ Fig. 5(a)], with ultratight field confinements of 0.3-nm (x axis) and 3.2-nm (y axis) in FWHM of the field intensity [ Fig. 5(b)], and a spot size of about 3.78 × 10 −7 λ 2 at 3.7-μm wavelength [ Fig. 5(c)]. The calculated diffraction-limited mode area is about 1.14ðλ∕nÞ 2 [ Fig. 5(d)]. Interestingly, compared to the visible-spectrum results in Fig. 4(h), here R P1∕B is much higher [Fig. 5(e)], owing to the much smaller ratio of the slit size to the wavelength. In addition, CNPs assembled with CdTe nanowires (with material index n CdTe of ∼2.68 versus n CdS ∼ 2.27 at 3.7-μm wavelength) 50 have also been investigated [Figs. 5(b) and 5(e)], showing the possibility of obtaining a tighter field confinement and a higher R P1∕B by using higher-index materials. For reference, a 0.23-nm (x axis) FWHM of the field intensity [ Fig. 5(b), blue solid line] corresponds to ∼λ∕20,000 confinement.

Dispersion of the Nanoslit Mode
Benefiting from the waveguiding scheme, it is also possible to generate broadband or ultrafast-pulsed sub-nm-confined optical fields in the nanoslit by selecting appropriate input light sources. In such a case, group velocity (V g ) and dispersion (D) should be considered. 51 Figure 6 gives the calculated V g and D of the TE 0 -like nanoslit mode in visible and MIR spectral ranges, respectively. Due to the relatively small nanowire diameter (i.e., d · n∕λ < 1), the increasing λ leads to the increasing fractional power in the air, thus increasing V g with negative dispersion in both spectral ranges. The larger D in the visible spectrum is due to the larger material dispersion near the bandgap of CdS (∼520 nm).
Although the dispersion of the nanoslit mode is orders of magnitude larger than those of conventional waveguides (e.g., ∼0.01 fs · nm −1 · mm −1 at 1550 nm for single-mode fiber 51,52 ), the short length of the CNP used for in-coupling and mode evolution (e.g., <10 μm) has little influence on the pulse width. For example, after waveguiding through a 10-μm-length CdS CNP, a 100-fs pulse with a central wavelength of 650 nm and a bandwidth of 30 nm will be broadened ∼6 fs in pulse width.

Conclusion and Discussion
Based on the numerical calculation, we have demonstrated a waveguiding scheme to generate a sub-nm-confined optical field in a TE 0 -like nanoslit mode. We show that waveguiding mode from a standard optical fiber (or on-chip waveguide) can be efficiently coupled into a CNP waveguide and evoluted into a high-purity TE 0 -like nanoslit mode within a propagation length less than 10 μm. Using a tapered fiber (or on-chip waveguide) for matching the n eff of the TE 0 -like nanoslit mode within a broad spectral range, the waveguiding scheme can work in broadband with a coupling efficiency up to 95% and a bandwidth up to 200 nm. At both visible and MIR bands, within the slit region, the diffraction-limited mode can offer an optical confinement down to ∼0.3 nm × 3.3 nm (i.e., ∼λ∕2000 in visible and ∼λ∕20,000 in MIR in the tightest-confinement dimension) and a peak-to-background ratio ∼30 dB. Benefiting from the short length required for mode coupling and evolution, ultrafast pulses from outer sources can also be converted into a TE 0 -like nanoslit mode to offer a sub-nm-confined ultrafast pulsed field with mild dispersion.
Compared with the previous lasing configuration, the waveguiding scheme proposed here is basically a linear optical system, which not only simplifies the complicated lasing system into a linear waveguiding system but also offers great flexibilities for broadband and/or ultrafast pulsed operation, as well as linear combination of multiple sub-nm optical fields with engineerable spatial, spectral, and time sequence in a single output. As the field confinement demonstrated here reaches the same scale of a single small molecule, such a field can offer a spatially inhomogeneous and asymmetric optical field with a large field gradient at the scale of a chemical bond or even an atom, and may thus offer an efficient and flexible platform for exploring light-matter interactions on single molecule or atom level, and developing optical technologies ranging from superresolution nanoscopy, 21 atom/molecule manipulation, 53 to ultrasensitivity detection. 54 Fig. 6 λ-dependent V g and D of the nanoslit mode in a CdS CNP at (a) visible spectrum with d ¼ 140 nm and (b) MIR spectrum with d ¼ 1 μm. In all calculations, w is assumed to be 1 nm and r is assumed to be 5 nm. Jiaxin Gao is currently an undergraduate student in the College of Optical Science and Engineering at Zhejiang University, China. Her current research interest is nanophotonics.
Xin Guo is currently an associate professor in the College of Optical Science and Engineering at Zhejiang University, China. Her main research interests include nanofiber, nanoplasmonics, and nanophotonic devices.
Pan Wang is currently a professor in the College of Optical Science and Engineering at Zhejiang University, China. His main research interests include nanoplasmonics and quantum plasmonics.
Limin Tong is currently a professor in the College of Optical Science and Engineering at Zhejiang University, China. His main research interests include nanophotonics, nanoplasmonics, and fiber optics, with emphases on nanowaveguides and nanophotonic devices. He is an OPTICA fellow.