Endface output patterns of micro/nanofibers (MNFs) are simulated using a Three-Dimension Finite-Difference Time-Domain (3D-FDTD) method. The intensity distribution and beam widths of near- or far-field output patterns of freestanding silica and tellurite MNFs with flat, angled, spherical and tapered endfaces in air and/or water are obtained. It shows that, for a subwavelength-diameter MNF, highly confined output beam can be obtained in the near field, and the beam width can be tuned by the ratio of fiber diameter and light wavelength with a minimum width smaller than the wavelength. Meanwhile, MNFs with shaped endfaces behave differently from standard fibers in reflection, redirection and focus of light beam at the endfaces. These results may offer valuable references for practical evaluation and application of terminated MNFs with wavelength- or subwavelength-scale endfaces.
©2008 Optical Society of America
Recently, optical micro/nanofibers (MNFs) have attracted much attention owing to their favorable properties such as tight optical confinement, low optical loss, high fraction of evanescent fields, strong field enhancement, large waveguide dispersion and powerful enhancement of optical nonlinearity [1–8]. By utilizing these excellent properties, a variety of micro-/nanophotonic components or devices, such as optical couplers, fiber gratings, Mach-Zehnder interferometers, ring resonators, filters, knot lasers and sensors, have been demonstrated [9–16]. Contrary to the above-mentioned devices that rely on uninterrupted MNFs, it is interesting to explore the possibility of using interrupted MNFs or nanowires for subwavelength-dimension light beams or optical probes [17,18]; also, investigation of light reflection or emission from nanowires at their distal ends are of special interest for subwavelength-dimension point sources [19,20]. Although endface output properties of MNFs or nanowires at their distal ends are critical for these purposes, so far they have not been adequately investigated. In this paper, using a Three-Dimension Finite-Difference Time-Domain (3D-FDTD) method [21,22], we model the endface output patterns (EOPs) and modal fields of freestanding silica and tellurite MNFs with flat, angled, spherical and tapered endfaces in air and water. The dependence of EOPs on fiber parameters (e.g., diameter of the MNF, wavelength of the guided light, indices of the MNF and the surrounding, and shape of the endface) are investigated. Some interesting features of the EOPs of the output beam, including beam divergence, direction, power distribution and end reflection, are revealed. Results presented here may be helpful for seeking appropriate EOPs of MNFs for possible applications in nanoscale lasing, nanoprobe sensing, laser trapping, and laser scalpel for nanosurgery.
2. Numerical Simulation Model
It is noticeable that the weak guidance approximation is not valid for high index-contrast waveguides. The rigorous method to resort is to solve Maxwell’s equations analytically or numerically. For investigating structures like MNFs with relatively small calculation dimensions, FDTD simulation is one of the best numerical methods regarding accuracy and efficiency . In this work, the FDTD simulations are performed by Meep, a freely available software package .
The basic model for numerical analysis is shown in Fig. 1. We assume that the MNF has a circular cross-section with a diameter of D and a length of L, an infinite air or water clad, and a step-index profile. To model the MNF, we use a Cartesian coordinate with its origin located at the center of the output endface of the MNF, as shown in Fig. 1. For simplicity, the source is assumed to be z-polarized, which does not loss generality due to the cylindrical symmetry. The computational domain is discretized into a uniform orthogonal 3-dimensional mesh with cell size of 40 nm for silica MNFs and 20 nm for tellurite MNFs, terminated by perfectly matched layer boundaries . The index of air is assumed to be 1.0, and the indices of water, silica, and tellurite are obtained from their wavelength-dependent dispersion relations (e.g., 1.33, 1.46 and 2.02 for water, silica and tellurite at 633-nm wavelength, respectively) [24–26].
3. Results and analyses
3. 1 MNFs with flat endfaces
At first, we investigate the endface output patterns of single-mode freestanding silica and tellurite MNFs with flat endfaces in air and water with a 633-nm-wavelength light source. Here the diameters of silica and tellurite MNFs are assumed to be 400 nm and 250 nm respectively to ensure the single-mode operation at the wavelength of 633 nm .
Figures 2(a)–2(d) show the calculated output patterns in x-y plane (z=0) of a 4.2-µm-length freestanding silica and tellurite MNFs in air and in water, respectively. Standing wave patterns, similar to those observed in MNFs experimentally , are clearly presented along the length of the MNFs due to the interference between the forward propagating light and backward reflection generated at the output endfaces, which is also confirmed by calculating the period of standing waves theoretically (Period=λ/2neff, where neff=β/k is the effective indices of the air/water-clad MNFs). Compared with MNFs in air, MNFs in water present higher fraction of evanescent fields, smaller divergence in EOPs and weaker standing wave patterns, which can be explained by tighter confinement and stronger endface reflection in air-clad MNFs due to the higher index contrast.
For further illustrating the optical confinement of the near- and far-field output patterns shown in Fig. 2(a)–2(d), the normalized intensity distributions along the y-axis in x-y plane (z=0) with distances of 100 nm (x=100 nm, near-field) and 3000 nm (x=3000 nm, far-field) departed from the end facets of MNFs are shown in Fig. 3. As is shown, the intensity peaks at the central axis of the MNF (y=0), and decreases monotonously when departing from the central axis. The spatial concentration of the endface output from a MNF can be estimated by defining a “beam width” as the full widths at half maximum (FWHMs) of the intensity distributions, as shown in Fig. 3. For example, the near-field beam width of a 633-nm light output from a 250-nm-diameter tellurite MNF in water is about 290 nm, indicating that in the near-field region (i.e., 100-nm departure from the output endface) the majority of the light energy is confined within a wavelength-scale span. In a relatively far field, e.g., 3000-nm departure from the endface, the beam width spread to 1.6 µm that is still promising for microscale photophysical or photochemical applications.
To search the minimum beam width of the near-field output in MNFs, we calculate the beam width with respect to the normalized fiber diameter (D/λ), in which the wavelength of the light is assumed to be 633 nm and the beam width is measured 100-nm departure from the output endface, as shown in Fig. 4. It shows that, for a MNF with given index and clad, there always exists a certain value of (D/λ) with which the MNF emits light with the lowest divergence in near field. For example, for a silica MNF, either in air or water, when D/λ=0.822 (i.e., D=520 nm with 633-nm-wavelength light), the beam width goes to a minimum of 418 nm in air or 480 nm in water. For a high-index tellurite MNF, much smaller beam width (e.g., 320 nm in water) is obtained with a smaller fiber diameter (367 nm in water).
Similar to the diameter of the MNF, wavelength of the light is another dimension that directly determines the features of the EOP of a MNF. To illustrate the evolvement of the EOP with varying wavelength in the MNF, we simulate the output intensity distribution of a 2.8-µm-length 400-nm-diameter silica MNF in air, with the wavelength scanning from 250–800 nm, as shown in Fig. 5. At short wavelength, the MNF operates in mutli-mode, resulting in multiple lobes in the EOP that varies dramatically with the wavelength; when the wavelength increases to about 556 nm, the MNF becomes a single-mode waveguide with a single maximum in EOP that varies rather slowly with the wavelength.
3.2 MNFs with angled endfaces
Propagation behavior of a light beam incident on an angled endface in a conventional optical fiber can be well defined by ray optics, and has been widely used for eliminating back-reflection or redirecting output beams in applications including optical communication, sensing, imaging and laser surgery [28,29]. However, in an MNF with diameter below the wavelength of the light, light behaves differently at the endface due to the large fraction of diffraction and evanescent fields. Here we investigate the output patterns of freestanding silica MNFs with angled endfaces. We assume that the wavelength of the probing light is 633 nm, and the fiber diameter is 400 nm.
Figure 6(a)–6(f) show the calculated output patterns in x-y plane (z=0) of the 400-nm-diameters silica MNF in air with 15°, 30°, 45°, 60°, 75°, and 90° (flat endface) -angled endfaces, respectively. It is not surprise to see that the standing wave patterns are more evident in the MNFs with large endface angles (e.g., 75° and 90°), while become dim with decreased angles (e.g., 45° and 60°), and finally disappear with small angles (e.g., 15°). Calculated backward reflectances are 1.6×10-4, 2.9×10-4, 0.0012, 0.006, 0.017, and 0.022 for 15°, 30°, 45°, 60°, 75°, and 90°-angled endfaces respectively, indicating the feasibility for reducing or eliminating back reflection in MNFs by shaping the fiber with angled endfaces, a similar technique used in conventional fibers. However, it is noticed that the reflectance in a MNF is considerably lower than in a conventional fiber (e.g., with a 90°-angled endface, the reflectance is about 2% in a MNF and 4% in a conventional fiber), which can be attributed to the large fraction of diffraction and evanescent fields in a MNF. For reference, calculated percentages of launched energy escaping in 15°-, 30°-, 45°-, 60°-, 75°-, and 90°-angled endface MNFs are 0.9998, 0.9997, 0.9988, 0.994, 0.983, and 0.978, respectively. Meanwhile, the output patterns in Fig. 6 show that, the angled endfaces in MNFs can no longer efficiently redirect the light propagation, but slightly shift the output pattern in x-y plane. To quantify this effect, we calculated the intensity distribution along the y-axis in x-y plane (z=0) with a distance of 100 nm (x=100 nm, near-field) and 3000 nm (x=3000 nm, far-field) from the apex of the angled endface, as shown in Fig. 7(b). It shows that, in both near- and far-field patterns, the angled endface shift the profile of the output patterns towards the opposite y-direction (that is upwards in Fig. 6). The magnitude of the shift is inverse-proportional to the angled degree of the endface. For example, in a 75°-angled MNF, the shift is about 14 nm (near-field) and 23 nm (far-field); in a 60°-angled MNF, the shift is about 28 nm (near-field) and 51 nm (far-field), while in a 15 °-angled MNF, the shift increases to about 138 nm (near-field) and 288 nm (far-field).
For MNFs in water, similar results are obtained, as provided in the appendix.
3.3 MNFs with spherical and tapered endfaces
Spherical or tapered fiber tips are usually used for focusing or dispersing light in conventional fibers [30,31], similarly, when the tip size goes down to wavelength or subwavelength level, they funtion differently. Here we investigate the output pattern of a MNF with spherical or tapered tip, with the fiber diameter and the light wavelength assumed to be 400 nm and 633 nm, respectively.
Figure 8(a) and 8(b) show the output patterns in x-y plane (z=0) of MNFs with a 400-nm and 800-nm-diameter spherical tips, respectively. The light output from the tip spreads out as it propagates along the x-direction, indicating that the spherical tip has no evident focusing effect. In the MNF shown in Fig. 8(a), where the diameter of the sphere equals to that of the MNF, obvious standing wave pattern inside the MNF is observed. Figure 8(c)–8(f) show the output patterns in x-y plane (z=0) of MNFs with tapered tips; for simplicity, the shape of the taper is assumed to be an isoceles triangle with vertex angle of 15°, 30°, 60°, and 120° respectively. It shows that, light output from the tapered tip spreads out symmetrically along the x-axis, similar to those presented in MNFs with flat or spherical endfaces. For tips with large vertex angle (e.g., 120°), evident standing wave pattern is formed in the MNF. For reference, Fig. 9(b) provides intensity distribution of the MNF output along the y-axis in x-y plane (z=0) with a distance of 100 nm (x=100 nm, near-field) and 3000 nm (x=3000 nm, farfield) from the apex of the tip, showing that the tapered tips produce slightly larger divergence than the spherical tips do, and the smaller vertex angle yields larger divergence in MNFs with tapered ends.
In conclusion, we have numerically calculated the output patterns of MNFs using a 3D-FDTD method. We show that in a subwavelength-diameter MNF, highly concentrated output can be obtained in the near field, and the beam width of near-field output can be tuned by the ratio of fiber diameter and light wavelength with a minimum width smaller than the wavelength. We also show that, because of the large fraction of diffraction and evanescent fields, light output from a MNF with wavelength or subwavelength diameter behaves differently with standard optical fibers. Since terminated MNFs, or more generally terminated optical-quality nanowires, are promising structures for manipulating light with micro-/nanometer scale endfaces, quantitative investigation of output patterns of MNFs presented in this work may offer valuable references for possible applications of MNFs in a variety of areas such as nanofiber or nanowire lasing, sensing, optical trapping, and laser surgery.
We investigate the output patterns of freestanding silica MNFs in water with angled endfaces. The wavelength of the probing light is assumed to be 633 nm.
Figure 10(a)–10(f) show the calculated output patterns in x-y plane (z=0) of the 400-nm-diameter silica MNF in air with 15°, 30°, 75°, and 90° (flat endface) angled endfaces, respectively. Similar to the MNFs in air, the standing wave patterns exist in MNFs with large endface angles. However, because of the lower index contrast between silica and water, the standing wave patterns are much less evident. For reference, calculated backward reflectances are 8.1×10-7, 1.5×10-6, 1.3×10-4, and 2.2×10-4 for 15°, 30°, 75° and 90°-angled endfaces respectively, which are over two orders of magnitude lower than those in air-clad MNFs with the same endface angles. Fig. 11 provides the intensity distributions along the y-axis in x-y plane (z=0) with distances of 100 nm (x=100 nm, near-field) and 3000 nm (x=3000 nm, far-field) from the apex of the angled endfaces. It shows that, the lateral shift of the EOP due to the angled endface is smaller in water than that in air. For example, in the 30°-angled MNF, the shift is about 38 nm (near-field) in water and 87 nm (near-field) in air.
This work is supported by the the National Basic Research Program 973 of China (2007CB307003), National Natural Science Foundation of China (No.60728309), China Postdoctoral Science Foundation (No.20060401033), and the 111 Project (No.B07031). M. Qiu also thanks the supports from the Swedish Foundation for Strategic Research (SSF) through the future research leader program, and from the Swedish Research Council (VR).
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