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Abstract
Optical nanofiber-based single-photon source has attracted considerable interest due to its property of seamless integration with a single-mode fiber. With nanostructure engraved in the nanofiber, the single-photon collection efficiency can be greatly boosted with enhanced interaction between the single quantum emitter and the guided light. However, the prerequisite nanofabrication processes introduce complexities and extra loss. Here, we demonstrate that by simply placing a quantum emitter in the gap of two parallel nanofibers, single-photon coupling efficiency may reach 54.2%. Our numerical simulation results indicate that photon coupling efficiency of such simple structure is insensitive to the discrepancy in nanofiber radii, which further reduces the difficulties in device fabrication.
© 2022 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement
1. Introduction
Solid-state quantum emitters provide an ideal way to generate single photons for quantum information technology [1–4]. However, high-efficient coupling of created photons into a single-mode fiber remains a challenging work after decades of research. Among various cavity-based [5–10] and waveguide-based [11–16] photon collection schemes, nanofiber that interacts with the quantum emitter attached to the surface via strong evanescent field [17–25] features several advantages. First of all, nanofiber can be directly pulled from a single-mode fiber. Adiabatic tapering minimizes transmission loss between the nanofiber and the single-mode fiber [26]. Secondly, nanofiber is much easier to fabricate comparing to those e-beam lithography processed on-chip devices. Thirdly, the cylindrical waveguide nature of nanofiber makes it insensitive to the emission wavelength and location of the quantum emitter. However, the optimal single-photon coupling efficiency close to 30% [17,27] is far lower than the minimum efficiency required for photonic quantum computation as 50% [28,29].
To enhance the photon collection efficiency, nanostructures have been introduced into the nanofibers. Grating-based microcavities or grooves have been applied to enhance the interaction between the quantum emitter and the light field [30–36]. Complex structure such as combination of periodic air-nanohole arrays and air-filled groove predicts a coupling efficiency as high as 80% [34], while a single hole in the nanofiber demonstrates an efficiency of 62.8% numerically [36]. Nevertheless, engraving nanostructures in nanofiber usually requires complicate lithography processes with expensive facilities. Besides, the fabrication processes will inevitably cause damage to the nanofiber with additional loss.
Here we propose a simple coupling structure based on nanofibers that doesn’t require any complicate nanofabrication process yet having optimal coupling efficiency > 50%. Basically, two parallelly placed nanofibers naturally form a slot-waveguide structure with drastically enhance field in the gap region. The optical properties of such twin-nanofiber structure have been previously studied, however mainly concentrated on the optical potential for atom trapping [37–39]. Here we place the quantum emitter in the middle of the gap to utilize the strongly localized slot-waveguide mode to enhance the coupling efficiency. Our numerical simulation result reveals that when two nanofibers with identical diameters are attached with each other, which has been realized experimentally [40,41], the coupling efficiency reaches a maximum of 54.2%.
2. Numerical simulation results
The coupling structure consists of two parallelly placed identical nanofibers with radii of r and a gap width of d, as shown in Fig. 1. A quantum emitter is place in the center of the gap as the origin of the coordinates. The twin-nanofiber forms a slot-waveguide structure that comprises two strips with high refractive index separated by a slot region with low refractive index. Strongly confined optical mode can be guided in the slot region, drastically enhancing the coupling between the light field and the quantum emitter. Consequently, the single-photon collection efficiency is boosted comparing to single nanofiber structure [18–22,25,27].
To investigate the properties of guided modes, numerical simulations based on finite-difference time-domain (FDTD) method (Lumerical, Ansys Canada Ltd.) are carried out. Figure 2(a)-(d) plot the intensity distributions of different modes when the propagation light is at 630 nm, the radii of the nanofibers are 165 nm and the gap width is 10 nm. The wavelength is based on the emission spectrum of typical ZnSe/ZnS core/shell colloidal quantum dots. Mode I shown in Fig. 2(a) is a typical slot-waveguide mode that has strongly confined field localized in the nanometer-scale low-index gap region. The intensity distributions along x- and y-axes plotted in Fig. 2(e) and (f) clearly illustrate that the field is drastically enhanced in the gap region with the maximum localized at the origin. Besides this mode, there exists three other guided modes that have the maximum E-field away from the gap.
Fig. 2. (a)-(d) Normalized intensity distributions of four guided modes in twin-nanofiber structure. (e), (f) The normalized field intensity distributions along x- and y-axis of mode I, respectively.
Figure 3(a)-(l) plot electric field components of these four modes that are consistent with the results reported in Ref. 38. One can easily find that the Ex component of mode I, the Ey component of mode II and the Ez component of mode III have symmetric distributions, while other components are all anti-symmetric. A distinct difference between these two types lies in that the intensity at the origin is non-zero for symmetric distribution yet zero for anti-symmetric one. Thus a quantum emitter located at the origin can only interact with the optical modes via symmetric components if it has proper dipole orientation. It's clear that mode I has the strongest light-matter interaction for an emitter with x-oriented dipole moment, as Ex is the dominant component of mode I which is strongly localized in the gap region with field maximum at the origin. A quantum emitter with y-oriented dipole moment have sufficient interaction with the Ey component of mode II, as well as z-oriented quantum emitter with Ez component of mode III. In the two latter cases, the interaction strength is not as strong as that of the first one, as the effective mode areas are much larger with expanded field distributions. On the other hand, mode IV has no interaction with an emitter no matter what the dipole orientation is, as the fields of all three components are zero at the origin.
Fig. 3. (a)-(l) Electric field components of guided modes I, II, III and IV, respectively. The color bars indicate the relative magnitude of the components in one mode.
The collection efficiency of photons from the emitter to the twin-nanofiber structure can be estimate via FDTD simulation by $\beta = \frac{{{P_{guided}}}}{{{P_{emitter}}}}$ , where ${P_{guided}}$ is the overall light power coupled into guided modes, ${P_{emitter}}$ is the total emission power from the quantum emitter [42]. Here, the Purcell factor that enhances spontaneous emission rate via the twin-nanofiber structure has been taken into account. To validate this method, we perform simulations on a single nanofiber structure with a quantum emitter attached to the surface. Taking the wavelength as 785 nm, the collection efficiency maximizes around 32% when nanofiber radius is 180 nm, which agrees with the results reported in Ref. [27] and Ref. [36].
To figure out the optimal configuration for photon collection, we calculate the coupling efficiency with various nanofiber radii and dipole orientations of the quantum emitter. The wavelength of photons radiated from the emitter is set as 630 nm. To simplify the calculation, the emitter is assumed to have negligible size here. The gap width is set as 0 to mimic the natural state that two nanofibers are attached to each other via the van de Waals force. Figure 4(a) shows the curves of coupling efficiency of mode I as functions of nanofiber radius for three orthogonally-oriented dipoles. Not surprisingly, the emission from x-oriented dipole exhibits high coupling efficiency, as the dipole orientation coincides with the polarization of the slot-waveguide mode. When the radius of the nanofiber is 135 nm, the coupling efficiency reaches a maximum value as 54.2%. Though the coupling efficiency is not as large as twice of the optimal coupling efficiency for a single nanofiber, the collective effect of double nanofiber is still constructive in the sense that the magnitude of the field at the origin is boosted more than two times. On the other hand, the y- and z-oriented dipoles can’t excite the slot-waveguide mode due to the vanishing Ey and Ez components at origin, as indicated in Fig. 3(b) and (c). Similarly, only the emission from y-oriented dipole may be coupled to guided mode II, as shown in Fig. 4(b). When the radius of the nanofiber is 120 nm, the coupling efficiency reaches a maximum of 34.9%, which is close to the maximum value in single nanofiber structure. The mode III only exists when the radius of the nanofiber is larger than 140 nm. The maximum coupling efficiency between a z-oriented dipole and mode III approaches 15.4% when the radius of the nanofiber is 175 nm. The coupling efficiency of mode IV is not plotted here because the efficiency is always zero with vanishing field at the origin. The discussion above mainly focuses on dipoles with fixed orientations. In certain circumstances, the quantum emitters orient at random. The coupling efficiency in such case is calculated and plotted in Fig. 4(d), with a maximum around 32.4%.
Fig. 4. (a)-(c) Coupling efficiency of quantum emitter with particular orientations vs. nanofiber radius for mode I, II and III, respectively. (d) Coupling efficiency of a quantum emitter with random orientation vs. nanofiber radius.
As practical fabrication of two identical nanofibers is a challenge, we investigate the effect of discrepancy in nanofiber radii on coupling efficiency when these two nanofibers are attached to each other. Figure 5(a) plots the relationship between the overall coupling efficiency and the radius (r) of one nanofiber while keeping the other one fixed to 135 nm. The dipole orientation of the quantum emitter is along x-axis. Surprisingly, the coupling efficiency retains a value around 50% even when the discrepancy is larger than 100 nm. When r reaches 260 nm, there exists a secondary maximum of coupling efficiency as 51.3%. A detailed analysis reveals that the coupling efficiency of mode I monotonically decreases as r increases, as shown in Fig. 5(b). This is because that the field is no longer strongly localized in the gap region, but with expanded mode area, as illustrated in Fig. 5(c). However, due to the breaking of symmetry, the Ex component of mode III in the gap region gradually increases, as shown in Fig. 5(d). Eventually the coupling efficiency of mode III excesses that of mode I.
Fig. 5. (a) The overall coupling efficiency of x-oriented emitter vs the radius of one nanofiber. The other nanofiber has a fixed radius of 135 nm. The dashed line indicates the position of a secondary maximum where r = 260 nm. (b) Coupling efficiency of mode I and III vs the radius of nanofiber. (c), (d) The Ex components of mode I and III when r = 260 nm, respectively.
In reality, the size of a solid state quantum emitter can’t be ignore. For example, the diameter of a typical colloidal quantum dot is around 10 nm, and the size of nano-diamond with nitrogen vacancy may extend to 100 nm. Thus the gap width can’t go down to zero with a physical quantum emitter in between. Figure 6 plots the coupling efficiency as a function of gap width when the radii of both nanofibers are 135 nm. Here the dipole is x-oriented so that only mode I is excited. The monotonic decreasing curve indicates that the smaller the gap, the higher the coupling efficiency could be. When the gap width is smaller than 16 nm that can host a typical colloidal quantum dot, the coupling efficiency remains above 50%.
Fig. 6. Coupling efficiency as a function of gap width when the radii of both nanofibers are 135 nm.
3. Discussions
The physical realization of the coupling structure discussed above is feasible. We have reported the fabrication of two parallelly placed nanofibers with identical diameters and the application of such structure as a tunable single-mode fiber coupler [41]. Based on such attached-nanofiber structure, one can apply a tiny drop of colloidal quantum dot solution on one end of the structure. Then the solution will fill the gap region automatically via capillary action. Once the solution is dried, there is a certain chance to find quantum dot with proper location and dipole orientation. Figure 7 plots the coupling efficiency of an x-oriented quantum emitter located at different positions within the gap region. It can be clearly seen that efficiency is decent and higher than that in single nanofiber case, even when the emitter is quite far away from the contact point.
Fig. 7. Color coded coupling efficiency of an x-oriented quantum emitter located at different position within the gap region. The dashed curves indicate the boundaries of two attached nanofibers with radii of 135 nm.
Another deterministic way to achieve high coupling efficiency could be processed as follows. Take colloidal quantum dots for example, one may apply diluted solution on one of the nanofibers via dip-coating in the first step so that quantum dots are well isolated from each other along the nanofiber [19,20]. Then the three-dimensional dipole orientation of individual quantum dot can be determined by methods such as polarization analysis [43] or using polarization microscopy [44]. By scanning the nanofiber, one may find a quantum dot that has the right orientation and azimuthal angle respect to nanofiber. This can also be realized by nano-manipulation technique with the help of atomic force microscope [45]. After that, one can carefully attach the other nanofiber from the right direction. Such procedure can produce relatively reliable results, however auxiliary setups are needed.
4. Conclusion
To summarize, we have demonstrated a photon coupling efficiency of 54.2% based on twin-nanofiber structure. The effects of discrepancy in nanofiber radii and gap width on coupling efficiency have been studied, and the feasibility of the coupling structure is discussed. Our simple structure paves a way for practical fiber-integrated single-photon sources with high-brightness.
Funding
National Key Research and Development Program of China (2018YFB2200400); National Natural Science Foundation of China (62035013, 62075192); Major Scientific Research Project of Zhejiang Laboratory (2019MC0AD01); Quantum Joint Funds of the Natural Foundation of Shandong Province (ZR2020LLZ007); Fundamental Research Funds for the Central Universities.
Acknowledgments
The authors would like to thank Mr. Jue Gong for valuable discussions.
Disclosures
The authors declare no conflicts of interest.
Data availability
Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.
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