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Coupling of a single dipole with a nanofiber Bragg cavity (NFBC) approximating an actually fabricated structure was numerically analyzed using three dimensional finite-difference time-domain simulations for different dipole positions. For the given model structure, the Purcell factor and coupling efficiency reached to 19.1 and 82%, respectively, when the dipole is placed outside the surface of the fiber. Interestingly, these values are very close to the highest values of 20.2 and 84% obtained for the case when the dipole was located inside the fiber at the center. The analysis performed in this study will be useful in improving the performance of single-photon emitter-related quantum devices using NFBCs.
© 2016 Optical Society of America
1. Introduction
Efficient coupling of photons emitted from light emitters into a single-mode fiber is important for various applications in biosensing [1], microlasers [2], and quantum information devices [3–6]. For this purpose, an optical nanofiber, which is a single-mode fiber with a tapered region, with a diameter as small as approximately half a wavelength is particularly attractive. Due to its small diameter, photons radiated from light emitters close to the surface of the fiber can be efficiently coupled to the nanofiber via a strong evanescent field [7–15]. According to numerical analysis, the coupling efficiency of photon emission to the fundamental mode can reach approximately 30% when the light emitters are placed on the fiber surface [9]. This type of coupling has been experimentally demonstrated using various light emitters, such as quantum dots [10,11], molecules [12], atoms [13], and diamond nanocrystals [14,15].
To further improve the coupling efficiency, a nanofiber with a microcavity embedded inside has been proposed independently by the present authors and Hakuta [16,17]. Hakuta and associates fabricated nanofiber cavity systems by ion beam milling, femtosecond laser ablation, and coupling with an external grating [18–20]. We recently reported the fabrication of nanofiber Bragg cavities (NFBCs) by ion beam milling. The NFBCs have small mode volumes, high quality factors, an ultra-wide tuning range of the resonance, and lossless coupling to a single-mode fiber. Furthermore, we demonstrated the enhancement of photon emission from a quantum dot coupled with an NFBC [21].
A detailed analysis of the dependence of the coupling efficiency on the position of the light emitters is important because the coupling efficiency strongly depends on the position of the emitter in the cavity. Le Kien et al. analytically calculated the dependence of the coupling efficiency on the position of an atom in the vicinity of the nanofiber with two fiber Bragg grating mirrors [17]. In their calculation model, two mirrors are placed inside the nanofiber while maintaining the cavity length L, but the loss induced by the grating due to scattering has been neglected. Hence, their analysis cannot be directly applied to the NFBC. Even though a preliminary analysis of the coupling efficiency has been performed, the analysis considered only four dipole positions at a fixed Bragg cavity length [21]. In this study, the position of the single light emitter that yielded the highest Purcell factor and coupling efficiency in an NFBC model and their dependence on the number of the grating periods were analyzed in detail using the three-dimensional (3D) finite-difference time-domain (FDTD) method.
This paper is organized as follows. Section 2 explains the calculation model used in the 3D FDTD simulations. In Section 3.1, the transmission spectra and effective mode volumes of the NFBC are calculated. Section 3.2 discusses simulations of the spectra of the photoluminescence from a dipole coupled with the NFBC. Section 3.3 demonstrates the dependence of the Purcell factor and the coupling efficiency on the number of grating periods. In Section 3.4, the position of the dipole that optimizes the Purcell factor and the coupling efficiency is analyzed. Finally, Section 4 concludes the paper.
2. Calculation method
To numerically analyze the NFBCs, 3D FDTD simulations (FDTD Solutions, Lumerical) were performed. The geometry of our calculation model is shown in Fig. 1(a). The NFBCs have a grating on the top and transverse sides of the nanofiber and no grating on the bottom side [21]. To approximate this structure, cylindrical grooves were carved on the top and transverse sides of a cylinder (diameter 300 nm). The period of the grooves, the groove depth, and the total number of periods were set to 300 nm, 45 nm, and 160, respectively. To form a cavity, the spacing between the grooves at the center of the structure was made 450 nm, i.e., 1.5 times the length of the grating period. The geometry of the calculation is shown in Fig. 1(b). The time step, simulation time, and calculation region (L × W × Z) were set to 0.039 fs, 5 ps, and 60 µm × 2 µm × 2 μm, respectively. An automatic nonuniform mesh with a high accuracy and the material properties of SiO2 were used in the calculation [22]. To calculate the transmission (Fig. 2), a light source with a pulse duration of approximately 4 fs was placed 24.5 μm from the center of the simulation region in the negative z-direction, and the spectrum was monitored at a position 28 μm from the center of the simulation region in positive z-direction. For the photoluminescence spectra, a single dipole source with a pulse duration of 4 fs was placed in the cavity and monitored at a position 28 μm from the center of the simulation region, as shown in Fig. 3.
Fig. 1 (a) Sketch of top and side views of calculation model. The inset is a cross section at the bottom of a groove. The solid and dotted lines indicate the grating structure and original surface of the nanofiber, respectively. (b) Geometry of FDTD simulation. Note that the number of grooves in this sketch is less than that in the actual simulated geometry.
Fig. 2 (a) Calculated resonance spectra for x- and y-polarized modes (black and red lines, respectively). Electric field distributions |E| at the center of the fiber for (b) x- and (c) y-polarized modes. The black lines are sketches of the cavity structure.
Fig. 3 Calculated photoluminescence spectra inside the nanofiber when a single dipole is placed at (a) positions A (black line) and C (red line) and (b) positions B (black, blue, and green lines) and D (red line). The dipole orientation of the black line, the red line and the green line is the radial, tangential, and longitudinal orientation, respectively. Spectra are normalized to the highest value. (c) Dipole positions A and B on the transverse side of the fiber. (d) Dipole positions C and D on the top side of the fiber.
To calculate the Purcell factor Γ and the coupling efficiency to the fundamental mode of the nanofiber η, the center wavelength of the dipole source was matched to the cavity resonance wavelength [9]. Here, Γ is calculated from Γ = Pcav/P0, where Pcav is the power emitted by the dipole in the cavity and P0 is the power radiated in the vacuum. η was calculated from η = Pcouple/Pcav, where Pcouple is the power coupled to the fundamental mode of the nanofiber. The coupling efficiency of one end of the fiber was monitored at the positions of ± 28 μm from the center of the simulation region and then they were summed to give the total coupling efficiency.
The effective cavity mode volume Veff was calculated as [23]
where ε(r) is the dielectric constant and E(r) is the electric field.3. Simulation results
3.1 Calculation of transmission spectra and effective mode volume
First, the transmission spectra and the effective mode volume of the NFBC were calculated [21]. Figure 2(a) shows the calculated transmission spectra for the x- and y-polarized modes. A sharp resonant peak is clearly observable in the middle of the stop band of the Bragg grating for each polarization mode. The resonances of the x- and y-polarized modes are 634.32 nm and 635.82 nm, respectively. The Q factors of the x- and y-polarized modes are 1600 and 1260, respectively. The x-polarized mode has a higher Q factor because it experiences larger index modulation by the Bragg grating.
In a previous experiment, a single resonant peak was observed in the middle of the stop band of a Bragg grating (linewidth of approximately 10 nm) near 640 nm [21]. This may be because these polarization modes overlapped in the observed peak.
Figures 2(b) and 2(c) show cross sections of the normalized electric field distribution |E| at the fiber center for the x- and y-polarized modes, respectively, at the resonance wavelength. The field intensity approaching the center of the cavity increased exponentially with a period of 300 nm. Outside the fiber, strong evanescent fields were observed because the diameter of the fiber is less than the wavelength of the input light [7]. The field distributions of the polarization modes differed slightly. For the x-polarized mode, the field distribution was symmetric about the z-axis, whereas the symmetry was broken for the y-polarized mode because the structure is asymmetric. From Eq. (1), the effective mode volumes for the x- and y-polarized modes were estimated to be 0.72 and 0.77 μm3, respectively.
The next section will discuss photoluminescence spectra when a single light emitter is coupled to the NFBC.
3.2 Spectra of photoluminescence from dipole coupled to NFBC
To study the coupling of a light emitter to the NFBC, the photoluminescence spectra were calculated from a single dipole coupled to the NFBC. In this calculation, the dipole was placed at a center of the cavity (weak electric field) and on an edge of a groove slightly over the surface of the fiber (strong electric field). The positions of the dipole are labeled A, B, C, and D in Figs. 3(c) and 3(d). The orientations of the dipole at the side position (positions A) and the top positions (positions C and D) were set to the radial direction to optimize the coupling of the dipole to each polarization mode. In order to compare different orientations of the dipole, the orientation of the dipole at the side position (position B) was set to the radial (x-axis), tangential (y-axis), and longitudinal (z-axis) orientation. Figures 3(a) and 3(b) show the photoluminescence spectra inside the nanofiber. Each spectrum was normalized with respect to the peak intensity of the spectrum obtained when the dipole with the radial orientation was placed at position B. When the dipole was placed in one of the positions with a weak electric field (positions A and C), the emission was clearly suppressed, as shown in Fig. 3(a). However, when the dipole with the radial orientation was placed in one of the positions with a strong electric field (positions B and D), a sharp emission peak clearly appeared for x- and y-polarization modes, as shown in Fig. 3(b). The peak intensity at the position B was stronger than that at the position D, as expected from its higher Q factor and lower mode volume.
When the dipole has tangential orientation [blue line in Fig. 3(b)], the light is only coupled to the y-polarization mode, and the intensity of the peak became smaller than when the dipole is set to radial direction at the position D (Red line). Also, when the dipole has the longitudinal direction [green line in Fig. 3(b)], the emission from the dipole is slightly coupled to the x-polarization mode.
To evaluate this enhancement and the suppression of the dipole emission, Γ and η were calculated at positions A and B. Γ and η at position A were 1.02 and 1%, respectively, and those at position B were 15.4 and 81%, respectively.
This section discussed the photoluminescence spectra inside the fiber and the Purcell factors and coupling efficiencies at four dipole positions when the number of grating periods was 160. The next sections will discuss the dependence of Γ and η on the number of the grating periods.
3.3 Dependence of Purcell factor and coupling efficiency on number of grating periods
To calculate the dependence of Γ and η on the number of grating periods, the calculation conditions were modified as follows. The simulation region (L × W × Z) was expanded to 120 µm × 2 µm × 2 μm. The simulation time was extended from 5 to 60 ps to ensure the field was completely decayed even for the expected higher quality factors and larger simulation volume (i.e., the field intensity in the simulation region decayed to an order of less than 10−7). The dipole was placed at an edge of a groove, and its orientation was set to the radial orientation, as shown in the inset of Fig. 4(a). η was monitored at a position 56 μm from the center of the simulation region (outside the grating).
Fig. 4 Dependence of (a) Purcell factor, (b) coupling efficiency, and (c) power Pcouple coupled to the fundamental mode on the number of grating periods.
Figure 4(a) shows the grating number dependence of Γ. Γ increased with increasing grating number and reached 153 at 320 grating periods. To calculate η, the dependence of Pcouple on the grating number was calculated [Fig. 4(b)]. η increased exponentially until approximately 160 periods. However, as the number of periods continued to increase, η approached saturation and then decreased at approximately 300 periods. Figure 4(c) shows the grating number dependence of η derived from these calculation results. η increased by approximately 80% with increasing grating number until approximately 160 periods. However, η declined when the grating number increased to more than 160 periods. This result is different from previous analytical results assuming lossless grating mirrors [17].
The reason Pcouple became saturated and then decreased as the number of grating periods increased was then investigated. The scattered power Pscat was then calculated by subtracting Pcouple from the total power coupled to a box (112 µm × 1.8 µm × 1.8 μm) surrounding the fiber. Note that these powers are normalized by the source power. At 160 periods, Pscat and Pcouple were 2.3 and 13.6, respectively. The scattering loss was small, and most of the power from the dipole emission was coupled to the fundamental mode. Moreover, at 320 periods, Pscat and Pcouple became 115.6 and 35.3, respectively. These results clearly suggest that the photons coupled to the NFBC were strongly scattered to outside the fiber with increasing grating number by the grating loss. Hence, the coupling efficiency was maximized at approximately 160 periods.
3.4 Detailed analysis of dependence of Purcell factor and coupling efficiency on dipole position
To obtain more insight into the coupling process, the dependence of Γ and η on the dipole position was analyzed in detail. For this calculation, the number of grating periods was set to 160 again. The polarization of the dipole was set to the x-polarized mode because of its higher Q factor.
Figure 5(b) shows the calculated Γ when the dipole was moved straight along the original surface (solid circles) and the fiber center (empty triangles) starting from zero on the z-axis, as shown in Fig. 5(a). Γ oscillated with a period of 300 nm when the dipole was located at the center and surface of the fiber. This behavior almost agrees with the square of the electric field distribution. When the dipole was placed on the surface 160 nm from zero on the z-axis, Γ increased to 18.5, which is close to Γ for the fiber center (Γ = 19.5). As shown in Fig. 5(a), this position is in a groove, meaning the dipole had no contact with the fiber surface. The peak values of Γ for the surface position decreased slightly with increasing distance from zero on the z-axis, but the enhancement of 17-fold was maintained at z = 760 nm.
Fig. 5 (a) Cross-sectional image of NFBC in the xz plane. The dipoles (solid red circles with arrows) were scanned along the z-axis at intervals of 20 nm. (b) Purcell factor Γ and (c) coupling efficiency η. Solid black dots and empty black triangles indicate when the dipole was located on the surface and at the center of the NFBC with 160 grating periods, respectively. Empty red diamonds indicate when the dipole was located on the surface of the NFBC with 80 grating periods.
Figure 5(c) shows the calculated η, which oscillated with a period of 300 nm but with behavior differing from that of Γ. η exhibited long flat peaks spanning lengths of approximately 100 nm and sharp dips, as shown in a previous study [17]. When z was 160 nm, which corresponds to the maximum value of Γ, the peak values of η when the dipole was located at the center and the surface of the fiber were 84% and 82%, respectively. Furthermore, the values of η at the three peaks (z = 160 nm, 460 nm, and 760 nm) were almost constant. These results suggest that the accurate alignment of a light emitter in the region containing the defect (450 nm length) is not critical for the NFBC. This is a large experimental advantage when coupling emitters to the cavity because it allows coupling experiments to be performed without the use of a high-resolution nanomanipulation system [24,25].
To understand the cause of the flat peaks in η for the NFBC with 160 grating periods, Γ and η are calculated for an NFBC with 80 grating periods. The peak values of Γ and η at 80 grating periods were smaller than those at 160 grating periods. Furthermore, the shape of the peaks in η was sharper at 80 grating periods [empty red diamonds in Fig. 5(c)]. These differences clearly indicate why η exhibited flat peaks at 160 grating periods. At 80 grating periods, because the enhancement of the photon emission from the dipole was relatively small, η was more sensitive to the position of the dipole than at 160 grating periods and showed clear peaks. However, at 160 grating periods, Γ was much greater than 1 for the majority of each peak, and the photons emitted from the dipole in this range could be selectively coupled to the fundamental mode of the fiber. As a result, η reached saturation, and the peaks became flat.
Next, the radial dependence of Γ and η were investigated at the position along the z-axis where Γ for the surface dipole was maximized (160 nm away from zero on the z-axis). This position is in a groove where the diameter of the fiber was reduced to 240 nm, as shown in Fig. 6(a). The dipole was oriented parallel to the x-axis and moved from the center of the fiber to outside the fiber.
Fig. 6 (a) Cross-sectional image of NFBC in xz plane. The dipole (solid red circle with arrows) was moved along the x-axis at intervals of 20 nm. (b) Purcell factor Γ and (c) coupling efficiency η.
Figure 6(b) shows the radial dependence of Γ. When the dipole was located in the center of the fiber, Γ was 20.2. After decreasing with increasing distance from the center, Γ then increased to 19.1 at a position about 20 nm from the surface of the fiber (x = 140 nm). After that, it decreased again. The shape of this radial dependence is similar to the distribution of the electric field of a normal nanofiber [7,26]. However, the large enhancement is unique to the NFBC.
Figure 6(c) shows the radial dependence of η. In contrast to Γ, η reduced monotonically with increasing distance from the center of the fiber. However, it remained as high as 82% at the location where Γ when the dipole is outside the fiber was maximized (x = 140 nm) and exceeded 30% even 500 nm from the fiber center. It is important to note that the nearly maximal Γ and η was realized even when the dipole was placed outside the fiber. This feature makes NFBC especially suited for coupling to quantum emitters using hybrid integration techniques [27,28]. In addition to the large regions with great enhancement, this is another experimental advantage for the coupling of light emitters to NFBCs.
4. Conclusion
The Purcell factor and coupling efficiency were analyzed in detail for different positions of a single dipole using the 3D FDTD method. For the model structure approximating the actual NFBC, the dipole position with the highest Purcell factor and the coupling efficiency was inside the fiber. However, a Purcell factor of 19.1 and a coupling efficiency of 82%, which were the nearly highest values, were achieved when the dipole is outside the fiber 20 nm from the surface and 160 nm from the center of the cavity region along the length of the fiber. It was also revealed that the scattering induced by the structure of the grating caused the coupling efficiency to decrease when the number of gratings was increased.
As we described in the main text, the orientation of the dipole is important factor for the coupling of the emitter to the NFBC. The orientation of molecules on the substrate has been controlled using an atomic force microscope (AFM) [29]. Then, the manipulation of the light emitter using the AFM would be useful for the control of the orientation of the dipole as well as the position of the dipole.
There remains room to improve the Q factor of this device. The current design of the NFBC was determined for some technical issues of our milling system. However, by using an advanced milling system with higher performance, it would be possible to fabricate a low-loss cavity structure, such as sinusoidal or tapering grooves [30–32], and recent nanobeam photonic crystal cavities [33,34]. The development of an NFBC with such a structure would be beneficial in the realization of quantum information devices, such as fiber-integrated single photon sources [4], quantum phase gates [5], and quantum memories [6].
Acknowledgments
We gratefully acknowledge financial support from JSPS KAKENHI Grants (Nos. 21101007, 26220712, 23244079, 25620001, 23740228, 26706007, and 26610077), JST CREST, JSPS FIRST, the Project for Developing Innovation Systems of MEXT, the G-COE Program, and the Research Foundation for Opto-Science and Technology. This project was supported by JSPS and DFG under the Japan–Germany Research Cooperative Program. Bilateral Joint Research Project by JSPS/DAAD. A.W.S. acknowledges support as an International Research Fellow of the JSPS. A portion of this work was supported by “Nanotechnology Platform Project (Nanotechnology Open Facilities in Osaka University)” of MEXT, Japan [F-15-OS-0026, F-15-OS-0044].
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