Silica Nanofibers and Subwavelength-Diameter Fibers
This is a continuation from the previous tutorial - introduction to binary optics.
1. Nanofiber at a glance
Air-clad silica (SiO2) nanofibers, named for their submicrometer diameters, have a large core-cladding index contrast for efficient optical confinement. For single-mode operation, these fibers are usually thinner than the wavelength of the light they carry and are, therefore, also called subwavelength-diameter fibers.
The small diameter of a nanofiber and the large core-cladding index contrast yield a number of interesting optical properties such as tight optical confinement, large evanescent fields, strong field enhancement, and large waveguide dispersions.
The nanofibers are fabricated by taper drawing of standard optical fibers and have extraordinary diameter uniformity and low surface roughness, making them ideal for low-loss optical wave-guiding. They can also be made very long and have high mechanical strength and pliability, facilitating assembly and patterning.
Because of their compactness and their optical and mechanical properties, these nanofibers find applications in a variety of fields, including photonic devices, optical sensors, and nonlinear optics.
2. Introduction
In the past 30 years, optical fibers with diameters larger than the wavelength of the guided light have found broad applications in optical communication, optical sensing, and optical power delivery systems.
Advances in microtechnology and nanotechnology for optoelectronics and photonics and the demand for improved performance, wider applications, and higher integration density, however, have spurred efforts for the miniaturization of photonic devices and waveguides.
A major step toward the miniaturization of devices is reducing the diameter of the optical fiber or waveguide. Therefore, an important motivation for fabricating optical-quality nanowires or nanofibers is their potential usefulness as building blocks in future micrometer- or nanometer-scale photonic devices and as tools for mesoscopic optics research.
There are several methods for fabricating one-dimensional (1D) optical nanostructures, including bottom-up chemical growth and top-down photo or electron beam lithography.
The silica nanowires (referred to as ‘‘nanofibers’’ in the following text) introduced in this tutorial are fabricated from standard optical fibers by a taper drawing method. Taper drawing of glass fiber is a top-down process that permits the fabrication of nanowires with diameters down to 50 nm.
Compared to other techniques, the taper-drawing approach not only provides a simple fabrication method but also yields nanowires with extraordinary diameter uniformity, atomic-level surface smoothness, and ultralow wave-guiding loss that cannot be achieved by subwavelength-width structures obtained by other methods.
Generally, when the diameter of a nanofiber is smaller than the wavelength of the guided light, the fiber can operate as a single-mode subwavelength-diameter waveguide with air cladding.
Because the index difference between the silica core and the surrounding medium (usually air) is large, the small index difference between the high-index center used as core in a standard fiber and the low-index cladding inherited from the starting fiber can be ignored in a nanofiber.
The tight optical confinement, large evanescent fields, strong field enhancement, and large waveguide dispersions of the silica nanofibers have generated broad interest in their potential for applications in a variety of fields such as microscale and nanoscale photonic devices, nanofiber optical sensors, nonlinear interactions and supercontinuum generation, and atom trapping and guidance.
This tutorial begins with a theoretical modeling of the optical wave-guiding properties of nanofibers, followed by a description of the taper-drawing fabrication technique and electron microscopy of the nanofibers, and then an experimental investigation of the nanofibers with an emphasis on micromanipulation and optical losses. Finally, we briefly review current and potential applications of the nanofibers.
3. Modeling of Single-Mode Wave-Guiding Properties of Silica Nanofibers
Although the wave-guiding theory and properties of conventional optical fibers have been investigated extensively, subwavelength-diameter nanofibers have not been modeled until recently.
Theoretically, the optical waveguiding properties of subwavelength-diameter nanofibers can be analyzed using Maxwell’s equation using boundary conditions analogous to those used for standard glass optical fibers.
However, unlike the weakly guiding optical fibers that have a small refractive index difference between the doped core and undoped cladding, the index contrast between the silica core and air cladding of a nanofiber is much higher, so exact analysis (i.e., without approximations) becomes necessary.
This section is devoted to the modeling of single-mode waveguiding properties of subwavelength-diameter nanofibers based on the exact solutions of Maxwell’s equations and numerical calculations.
3.1. Basic Model
The mathematic model of an air-clad nanofiber is shown in Fig. 11.1. The fiber is assumed to have a circular cross-section, a uniform diameter, an infinite air-cladding, and a step-index profile as follows:
\[\tag{11.1}n(r)=\begin{cases}n_1,\quad0\lt{r}\lt{a},\\n_2,\quad{a}\le{r}\lt\infty\end{cases}\]
where a is the radius of the nanofiber, \(n_1\) and \(n_2\) are refractive indices of the fiber material and the air, respectively.
Within their transparent range (250 nm–2.5 μm), silica nanofibers are non-dissipative and source free, so Maxwell’s equations can be reduced to the following Helmholtz equations:
\[\tag{11.2}\begin{align}(\nabla^2+n^2k^2-\beta^2)\vec{e}&=0\\(\nabla^2+n^2k^2-\beta^2)\vec{h}&=0\end{align}\]
where \(k=2\pi/\lambda\) is the wave vector and \(\beta\) is the propagation constant.
Exact solutions for this model have been provided by Snyder and Love, yielding the following eigenvalue equations for the HE\(_{vm}\) and EH\(_{vm}\) modes:
\[\tag{11.3}\left\{\frac{J_v'(U)}{UJ_v(U)}+\frac{K_v'(W)}{WK_v(W)}\right\}\left\{\frac{J_v'(U)}{UJ_v(U)}+\frac{n_2^2K_v'(W)}{n_1^2WK_v(W)}\right\}=\left(\frac{v\beta}{kn_1}\right)^2\left(\frac{V}{UW}\right)^4\]
for the TE\(_{0m}\) modes:
\[\tag{11.4}\frac{J_1(U)}{UJ_0(U)}+\frac{K_1(W)}{WK_0(W)}=0\]
and for the TM\(_{0m}\) modes:
\[\tag{11.5}\frac{n_1^2J_1(U)}{UJ_0(U)}+\frac{n_2^2K_1(W)}{WK_0(W)}=0\]
where \(J_v\) is the Bessel function of the first kind, and \(K_v\) is a modified Bessel function of the second kind, \(U=D(k_0^2n_1^2-\beta^2)^{1/2}/2\), \(W=D(\beta^2-k_0^2n_2^2)^{1/2}/2\), \(V=k_0a(n_1^2-n_2^2)^{1/2}\), and \(D=2a\) is the diameter of the nanofiber.
Numerically solving these eigenvalue equations after substituting the indices of refraction for air (\(n_2=1\)) and silica (\(n_1=1.46\) for \(\lambda=633\) nm), we obtain the propagation constants \(\beta\) for an air-clad nanofiber.
Figure 11.2 shows the diameter-dependent \(\beta\) at a wavelength of 633 nm, where the fiber diameter \(D\) is directly related to the \(V\)-number [\(V=k_0D(n_1^2-n_2^2)^{1/2}/2\)]. The figure clearly shows that at a given wavelength, the number of modes that can be supported by the nanofiber is determined by its diameter. When the fiber diameter is reduced to a certain value (denoted by DSM and corresponding to \(V=2.405\)), we obtain single-mode operation and only the HE\(_{11}\) mode is supported.
The single-mode condition of an air-clad fiber, marked by a dashed line in Fig. 11.2, can be obtained from Eqs. [11.4] and [11.5], yielding
\[\tag{11.6}V=2\pi\frac{a}{\lambda_0}(n_1^2-n_2^2)^{\frac{1}{2}}\approx2.405\]
Figure 11.3 shows the single-mode and multimode regimens of the air-clad silica nanofiber obtained from Eq. [11.6] after substituting \(n_2=1.0\) for the index of refraction of the air and using a Sellmeier-type dispersion formula for the fused silica:
\[\tag{11.7}n^2-1=\frac{0.6961663\lambda^2}{\lambda^2-(0.0684043)^2}+\frac{0.4079426\lambda^2}{\lambda^2-(0.1162414)^2}+\frac{0.8974794\lambda^2}{\lambda^2-(9.896161)^2}\]
with the wavelength \(\lambda\) in units of μm.
The region beneath the solid line in Fig. 11.3 corresponds to single-mode operation. For example, at the He-Ne laser wavelength of 633 nm, a silica nanofiber with a diameter smaller than 457 nm is a single-mode waveguide; in the near infrared at 1.55 μm, the diameter of the silica nanofiber should be less than about 1.1 μm for single-mode operation.
The dashed line in Fig. 11.3, representing the wavelength of the propagating light in silica (that is, \(\lambda=\lambda_0/n_1\)), shows that the nanofiber is always single mode when the fiber diameter is smaller than the wavelength of the light in the silica.
Taking into consideration that the UV absorption edge in silica is around 200 nm, the minimum critical diameter \(D_{SM}\) for silica nanofibers is about 129 nm. Because single-mode operation of silica nanofibers is preferable for most applications, we will concentrate henceforth on the guiding properties of the fundamental modes.
The propagation constant \(\beta\) of the fundamental HE\(_{11}\) mode can be obtained by setting \(v=1\) in Eq. [11.3] and numerically solving the resulting eigenvalue equation
\[\tag{11.8}\left\{\frac{J_1'(U)}{UJ_1(U)}+\frac{K_1'(W)}{WK_1(W)}\right\}\left\{\frac{J_1'(U)}{UJ_1(U)}+\frac{n_2^2K_1'(W)}{n_1^2WK_1(W)}\right\}=\left(\frac{\beta}{kn_1}\right)^2\left(\frac{V}{UW}\right)^4\]
Writing the electromagnetic fields in the form
\[\tag{11.9}\begin{cases}\vec{E}(r,\phi,z)=(e_r\hat{\pmb{r}}+e_\phi\hat{\pmb{\phi}}+e_z\hat{z})e^{i\beta{z}}e^{-i\omega{t}}\\\vec{H}(r,\phi,z)=(h_r\hat{\pmb{r}}+h_\phi\hat{\pmb{\phi}}+h_z\hat{z})e^{i\beta{z}}e^{-i\omega{t}}\end{cases}\]
we obtain for the electric fields of the fundamental modes inside the core (\(0\lt{r}\lt{a}\)):
\[\tag{11.10}e_r=-\frac{a_1J_0(UR)+a_2J_2(UR)}{J_1(U)}\cdot{f_1}(\phi)\]
\[\tag{11.11}e_\phi=-\frac{a_1J_0(UR)-a_2J_2(UR)}{J_1(U)}\cdot{g_1}(\phi)\]
\[\tag{11.12}e_z=\frac{-iU}{a\beta}\frac{J_1(UR)}{J_1(U)}\cdot{f_1}(\phi)\]
and outside the core (\(a\le{r}\lt\infty\)):
\[\tag{11.13}e_r=-\frac{U}{W}\frac{a_1K_0(WR)-a_2K_2(WR)}{K_1(W)}\cdot{f_1}(\phi)\]
\[\tag{11.14}e_\phi=-\frac{U}{W}\frac{a_1K_0(WR)+a_2K_2(WR)}{K_1(W)}\cdot{g_1}(\phi)\]
\[\tag{11.15}e_z=\frac{-iU}{a\beta}\frac{K_1(WR)}{K_1(W)}\cdot{f_1}(\phi)\]
where \(f_1(\phi)=\sin(\phi)\), \(g_1(\phi)=\cos(\phi)\),
\[\begin{align}a_1&=\frac{F_2-1}{2},\;a_3=\frac{F_1-1}{2},\;a_5=\frac{F_1-1+2\Delta}{2}\\a_2&=\frac{F_2+1}{2},\;a_4=\frac{F_1+1}{2},\;a_6=\frac{F_1+1-2\Delta}{2}\\F_1&=\left(\frac{UW}{V}\right)^2[b_1+(1-2\Delta)b_2],\;F_2=\left(\frac{V}{UW}\right)^2\frac{1}{b_1+b_2}\\b_1&=\frac{1}{2U}\left\{\frac{J_0(U)}{J_1(U)}-\frac{J_2(U)}{J_1(U)}\right\},\;b_2=-\frac{1}{2W}\left\{\frac{K_0(W)}{K_1(W)}+\frac{K_2(W)}{K_1(W)}\right\}\end{align}\]
Because the \(h\)-components can readily be obtained from \(e\)-components with some calculations, they are not presented here.
Figure 11.4 shows the normalized electric components of the fundamental modes in cylindrical coordinates for silica nanofibers at a wavelength of 633 nm. The dashed line in the radial distribution graph shows the Gaussian profile for reference; the dotted lines represent the electric fields in a silica fiber with critical diameter \(D_{SM}\).
As can be seen in the graph, because of the high index contrast between the air and silica, air-clad silica fiber tightly confines the electric field at a diameter of about 400 nm.
When the diameter is reduced further, however, a significant portion of the electric field extends far outside the nanofiber, indicating that the field is no longer tightly confined inside or around the fiber. A similar behavior is obtained at other wavelengths.
3.2. Power Distribution: Fraction of Power Inside the Core and Effective Diameter
For nanofibers with uniform diameters, there is no net flow of energy in the radial (\(r\)) or azimuthal (\(\phi\)) directions, so we need to consider only the energy flow in the \(z\)-direction. The \(z\)-component of the Poynting vector inside the core (\(0\lt{r}\lt{a}\)) is
\[\tag{11.16}\begin{align}S_{z1}&=\frac{1}{2}\left(\frac{\epsilon_0}{\mu_0}\right)^{\frac{1}{2}}\frac{kn_1^2}{\beta{J_1^2}(U)}\left[a_1a_3J_0^2(UR)+a_2a_4J_2^2(UR)\right.\\&\quad+\left.\frac{1-F_1F_2}{2}J_0(UR)J_2(UR)\cos2\phi\right]\end{align}\]
and the one outside the core (\(a\le{r}\lt\infty\)) is
\[\tag{11.17}\begin{align}S_{z2}&=\frac{1}{2}\left(\frac{\epsilon_0}{\mu_0}\right)^{\frac{1}{2}}\frac{kn_1^2}{\beta{K_1^2}(W)}\frac{U^2}{W^2}\left[a_1a_5K_0^2(WR)+a_2a_6K_2^2(WR)\right.\\&\quad-\left.\frac{1-2\Delta-F_1F_2}{2}K_0(WR)K_2(WR)\cos2\phi\right]\end{align}\]
Figure 11.5 shows the Poynting vectors for a 200- and a 400-nm diameter silica nanofiber at a wavelength of 633 nm; the mesh profile represents the fields propagating inside the fiber and the gradient profile stands for the evanescent field.
As one can see, the 400-nm fiber confines most of the light inside the fiber, whereas for the 200-nm fiber, a large amount of light is guided outside the wire in the form of an evanescent wave.
To obtain a more intuitive understanding of the power distribution in the radial direction, we calculate two additional parameters. The first is the fractional power inside the core,
\[\tag{11.18}\eta=\frac{\int_0^aS_{z1}dA}{\int_0^aS_{z1}dA+\int_a^\infty{S}_{z2}dA}\]
where \(dA=a^2RdRd\phi=rdrd\phi\).
The second one is the effective diameter of the light field \(D_\text{eff}\)—the diameter within which \(1-1/e^2\) (86.5%) of the total power is confined—which can be obtained from
\[\tag{11.19}\begin{cases}\frac{\int_0^{D_\text{eff}}S_{z1}dA}{\int_0^aS_{z1}dA+\int_a^{\infty}S_{z2}dA}=86.5\%,\quad(D_\text{eff}\lt{a})\\\frac{\int_0^aS_{z1}dA\int_a^{D_\text{eff}}S_{z1}dA}{\int_0^aS_{z1}dA+\int_a^{\infty}S_{z2}dA}=86.5\%,\quad(D_\text{eff}\gt{a})\end{cases}\]
Figure 11.6 shows the fractional power in the core as a function of the fiber diameter \(D\) for silica nanofibers at wavelengths of 633 nm and 1.5 μm. At the critical diameter \(D_{SM}\) (dashed lines), \(\eta\) is around 80% at both wavelengths.
When the diameter drops below \(0.5D_{SM}\), more than 80% of the energy is guided in the evanescent wave outside the silica core. Because \(\eta\) varies so steeply around the range of diameters of interest, it is easy to tailor the amount of confinement to a particular application.
Tight confinement, obtained at diameters around \(D_{SM}\), is important for reducing the modal width and increasing the integrated density of the optical circuits with less cross-talk, while the weaker confinement obtained at smaller diameters is helpful for exchanging energy between nanofibers within a short interaction length and for improving the sensitivity of evanescent wave–based fiber optic sensors.
Figure 11.7 shows the effective diameter \(D_\text{eff}\) of the fundamental mode in a silica nanofiber at 633-nm wavelength. The dotted line shows the real diameter of the nanofibers for comparison. As expected, \(D_\text{eff}\) is large when the fiber diameter is very small and the two curves intersect near the critical diameter (dashed line).
The intersection point is the minimum diameter for which it is possible to confine 86.5% of the light energy within the wire at the given wavelength. Note that at this point the diameter is smaller than the wavelength of the light (450 vs 633 nm).
For small fiber diameters, \(D_\text{eff}\) becomes very large. For example, for a nanofiber diameter of 200 nm, \(D_\text{eff}\) is about 2.3 μm, which is more than 10 times the fiber diameter.
Maintaining a steady guiding field in such a situation may be difficult; any small deviation (such as surface contamination and/or microbends) from the ideal condition leads to a change in propagating fields and radiation loss. On the other hand, the high sensitivity of such a guiding fiber to small perturbations may be useful in sensing applications that require high sensitivity.
3.3. Group Velocity and Waveguide Dispersion
The diameter-dependent group velocity of the HE\(_{11}\) mode for the air-clad silica nanofiber is given by
\[\tag{11.20}v_g=\frac{c}{n_1^2}\frac{\beta}{k}\frac{1}{1-2\Delta(1-\eta)}\]
and shown in Fig. 11.8 for two wavelengths.
When the fiber diameter \(D\) is very small, \(v_g\) approaches the speed of light in vacuum \(c\) because most of the light energy propagates in air. As \(D\) increases, an increasing fraction of the energy is guided in the silica core and \(v_g\) decreases until it reaches a minimum value that is smaller than \(c/n_1\), the group velocity of a plane wave in silica. As \(D\) continues to increase, \(v_g\) increases again, approaching \(c/n_1\) at large values of \(D\).
Figure 11.8 shows the wavelength dependence of the group velocity for various fiber diameters, also obtained from Eq. [11.20]. For a given fiber diameter \(D\), the group velocity is \(c\) when the wavelength \(\lambda\) is very large and approaches \(c/n_1\) when \(\lambda\) is very small, with a minimum value somewhat smaller than \(c/n_1\). Similarly, the wavelength dependence of the group velocity with fiber diameter can be seen in Fig. 11.9.
From the group velocity in Eq. [11.20], one can obtain the waveguide dispersion
\[\tag{11.21}D_w=\frac{d(v_g^{-1})}{d\lambda}\]
Figures 11.10 and 11.11 illustrate the diameter and wavelength dependence of this waveguide dispersion; the dotted line in Fig. 11.11 also shows the material dispersion of fused silica obtained from Eq. [11.6].
As can be seen, the waveguide dispersion \(D_w\) of the nanofibers can be very large compared with those of weakly guiding fibers and bulk material. For example, for an 800-nm diameter silica fiber at a wavelength of 1.5 μm, \(D_w=-1400\) ps/(nm\(\cdot\)km), which is about 70 times larger than that of the material dispersion.
Note also that the total dispersion (the combined material and waveguide dispersions) of a nanofiber can be made positive, zero, or negative within a given spectral range by choosing the appropriate fiber diameter. Controlling light propagation by tailoring the dispersion is widely used in optical communications and nonlinear optics, so nanofiber waveguides present an opportunity to miniaturize devices in these fields.
4. Fabrication and Microscopic Characterization of Silica Nanofibers
Quite a few techniques can be used to fabricate silica nanowires or nanofibers such as photo or electron beam lithography, chemical growth, and taper drawing of optical fibers.
Among these techniques, the taper-drawing method exhibits not only simplicity in fabrication, but also the ability to fabricate nanofibers with extraordinary diameter uniformities, atomic-level surface smoothness, and long length that are difficult to achieve by any other means.
Diameter uniformity and surface smoothness are particularly critical for low-loss optical wave-guiding in subwavelength–width waveguides. This section focuses on taper-drawing fabrication of silica nanofibers.
4.1. Two-Step Taper Drawing of Silica Nanofibers
The fabrication of thin silica fibers using a high-temperature taper-drawing technique was first reported in the nineteenth century, when the mechanical properties of the fibers were studied, but their optical properties and applications remained uninvestigated.
It was not until a century later, when optical waveguide theory had become well established, that researchers began to investigate the optical applications of very thin silica fibers made by laser- or flame-heated taper drawing of optical fibers.
Laser heating provides highly stable and repeatable conditions for fiber drawing, but the laser power required for drawing silica fibers with uniform diameters smaller than 1 μm is impractically large.
As discussed in Section 2, for single-mode operation in the optically transparent range of silica (250–2000 nm), the fiber diameter has to be smaller than 1 μm. Therefore, flame heating is the only practical technique for the taper drawing of single-mode silica nanofibers.
Silica nanofibers have been obtained by one-step and two-step taper-drawing methods. The onestep approach is simple and convenient. However, when drawing fibers directly from a flame-heated melt, turbulence and convection usually make it difficult to control the temperature gradient in the drawing region and to maintain stable drawing conditions.
Consequently silica nanofibers with diameters of less than 200 nm are difficult to obtain with a one-step draw. A two-step technique circumvents these difficulties, making it possible to draw silica nanofibers with diameters as small as 20 nm.
A schematic view of the two-step taper-drawing method is shown in Fig. 11.12a.
As in the one-step method, a bare silica fiber is flame-heated and first drawn down to a micrometer-sized diameter taper. A low-carbon fuel such as CH3OH or hydrogen is recommended to avoid contamination of the fiber with incompletely burned carbon particles.
To obtain sufficiently steady to reduce the fiber diameter below 1 μm, we use a tapered sapphire fiber with a tip diameter around 100 μm to absorb the thermal energy from the flame. The sapphire fiber taper (fabricated using a laser-heating growth method) confines the heating to a small volume and helps maintain a steady temperature distribution during the drawing.
As long as the working temperature is kept below the melting temperature of sapphire (~2320 K), the sapphire tip can be used repeatedly. One end of the previously drawn fiber with a micrometer-sized diameter is placed horizontally on the sapphire tip, and the flame is adjusted until the temperature of the sapphire tip is just above the drawing temperature (~2000 K).
The sapphire tip then is rotated around its axis to wind the silica fiber around it and the resulting fiber coil is moved about 0.5 mm out of the flame to prevent melting, as shown in Fig. 11.12b.
Finally, a nanofiber is drawn from the coil at a speed of 1–10 mm/sec in the horizontal plane in a direction perpendicular to the axis of the sapphire tip. With this two-step technique, the diameter of a silica fiber can be reduced to about 50 nm, thinner than required for most optical applications.
To obtain even thinner nanofibers for the investigation of the structural, dynamic, and catalytic properties of silica nanowires, we used a self-modulated drawing force instead of the constant drawing force in the two-step taper-drawing process described earlier. The self-modulated force is obtained by the technique illustrated in Fig. 11.13.
The silica fiber is held parallel to the sapphire taper and the elastic bend in the taper area of the fiber generates the tensile force in the microfiber between the silica and sapphire tapers. During the initial stage of the drawing, when the fiber is still thick, the bending center occurs in the thicker part of the taper, causing a relatively large force; as the fiber is elongated and its diameter is reduced, the bending center moves towards the thin end of the taper, reducing the tensile force that causes the drawing.
This self-modulation not only permits the drawing of fibers with diameters as small as 20 nm, but also counteracts the effects of temperature fluctuations by buffering the drawing force and avoids any sudden changes in fiber diameter. To monitor the drawing process, we launch a continuous-wave He-Ne laser (633-nm wavelength) along the silica fiber to illuminate the taper and nanofiber, as illustrated in Fig. 11.13b. When the drawing is completed, the nanofiber is connected to the starting fiber at one end and freestanding on the other end.
4.2. Electron Microscope Study of Silica Nanofibers
The nanofibers obtained with the two-step taper-drawing techniques described in the previous section consist of three parts: a millimeters-long taper that is connected to the starting microfiber, a uniform nanofiber with a length up to tens of millimeters, and an abruptly tapered end that is usually several to tens of micrometers in length.
Figure 11.14a shows an SEM image of the uniform parts of a nanofiber with a uniform diameter of about 200 nm; Fig. 11.14b shows the abruptly tapered end.
In the remainder of this tutorial, we focus on the uniform part of the nanofiber, which can be used as a subwavelength-diameter waveguide for low-loss optical wave-guiding.
Depending on the experimental conditions such as drawing temperature, force, and speed, the diameter of the taper-drawn silica nanofiber ranges from tens of nanometers to one micrometer.
Figure 11.15 shows SEM images of silica nanofibers with diameters ranging from 50 to 400 nm, illustrating the range of dimensions and uniformity of the silica nanofibers.
The surface tension of the molten silica during the drawing process ensures that the cross-sections of the taper-drawn nanofibers are perfectly circular. Figure 11.16 shows an SEM image of the cross-section of a 480-nm diameter nanofiber.
The resulting cylindrical geometry of the nanofibers makes it possible to obtain exact expressions of the guided modes by solving Maxwell’s equations analytically (see Section 2).
The length of the nanofibers depends on their diameter. Typically, nanofibers with diameters smaller than 200 nm can have length up to 1 mm; nanofibers with larger diameters can be as long as several hundreds of millimeters. For example, Fig. 11.17 shows an SEM image of a 4-mm long nanofiber with a diameter of 260 nm; the nanofiber is coiled up on the surface of a silicon wafer to show its length.
In addition to being long, taper-drawn nanofibers also provide excellent diameter uniformity and surface roughness (see, e.g., Figs. 11.14b and 11.15). We determined the diameter uniformity of the nanofibers by measuring the diameter variation \(\Delta{D}\) along the entire length \(L\) with a scanning or transmitting electron microscope. Figure 11.18 shows the measured diameter \(D\) and diameter uniformity \(U_D=\Delta{D}/L\) of a thin nanofiber along its length (starting from the thin end).
Although the nanofiber exhibits an overall monotonic tapering, the central part of the nanofiber shows a very high uniformity. For example, in the region where \(D=30\) nm, \(U_D=1.2\times10^{-5}\), which means that in more than an 80-μm length of nanofiber the maximum diameter difference between the two ends is less than 1 nm.
Thicker nanofibers show even better uniformities. For example, the 260-nm diameter nanofiber shown in Fig. 11.17 has a maximum diameter variation \(\Delta{D}\) of about 8 nm over its 4-mm length, giving \(U_D=2\times10^{-6}\).
The small diameter of the nanofiber makes it possible to investigate the surface roughness with a TEM. The TEM image in Fig. 11.19 shows that the edge of a 330-nm diameter nanofiber has no irregularities or defects.
The typical sidewall root-mean-square roughness of these fibers can be as small as 0.2 nm, approaching the intrinsic roughness of melt-formed glass surfaces. Considering that the length of Si-O bond is about 0.16 nm, such a roughness represents an atomic-level smoothness of the nanofiber surface and is much lower than those of silica nanowires, tubes, or strips obtained using other fabrication methods.
5. Properties of Silica Nanofibers
For optical applications, the most important properties of silica nanofibers are optical loss, mechanical strength, and pliability. In this section, we review the mechanical properties and optical losses of silica nanofibers and discuss techniques for micromanipulation and assembly.
5.1. Micromanipulation and Mechanical Properties
The ability to manipulate nanofibers individually is critical to their characterization and application. Because of their long length, silica nanofibers obtained by the taper-drawing method can be seen under an optical microscope, even when the fiber diameter is less than 100 nm.
For example, Fig. 11.20 shows a photograph of a 60-nm diameter silica nanofiber taken under an optical microscope in dark-field reflection mode. The nanofibers can also clearly be seen when they are supported on a silicon wafer. This optical visibility makes it possible to manipulate single nanofibers, greatly facilitating the handling, tailoring, and assembly of these nanofibers.
A typical experimental setup for the micromanipulation of silica nanofibers is shown in Fig. 11.21. An optical microscope objective is used to image the nanofiber onto a CCD camera for real-time monitoring. To hold and manipulate the nanofibers, probes from a scanning tunneling microscope (STM) are mounted on micromanipulators and placed as shown in Fig. 11.21a; Fig. 21b shows the bending of a nanofiber using the tips of two probes.
The nanofibers can be either freestanding in air or supported by high-index substrates for better visibility (e.g., silicon or sapphire wafers). Using micromanipulation under an optical microscope, the nanofibers can be cut, positioned, bent, and twisted with high precision. To cut a nanofiber to a desired length, a bend-to-fracture method can be applied by holding the fiber with two STM probes on silicon or sapphire substrate and using a third probe to bend the nanofiber to fracture at the desired point. This process leaves flat end faces at the fracture point, as shown in Fig. 11.22.
Because of their excellent uniformity, taper-drawn silica nanofibers show high mechanical strength and pliability. By rubbing the two ends of nanofibers on a finely polished substrate using a tilted probe, they can be twisted together without breaking.
Figure 11.23 shows the ropelike twist obtained with a 480 nm diameter silica nanofiber on a silicon wafer. The ‘‘nanorope’’ retains its shape when it is lifted up from the substrate, indicating that the nanofiber can withstand shear deformation.
To position and bend a nanofiber, it is first placed on a finely polished substrate such as a silicon wafer, where it is tightly held in place by the van der Waals or electrostatic attraction between the nanofiber and the substrate, and then pushed by STM probes to a desired bending radius or position. The shape of elastic bends is retained after removing the STM probes because of the attraction between the nanofiber and the substrate.
Figure 11.24 shows a 280 nm diameter nanofiber bent to a radius of 2.7 μm. The sharp bend in Fig. 11.24 indicates that the nanofiber has excellent flexibility and mechanical properties. Using the Young modulus of ordinary silica fibers (73.1 GPa), we find that the tensile strength of the bent nanofiber in Fig. 11.24 is at least 4.5 GPa. Silica nanofibers can also be shaped into more complex forms and tied in knots.
Figure 11.25 shows a 15-μm diameter knot assembled with a 520-nm diameter silica nanofiber. Such a ringlike structure can be used as a ring resonator in micro-optical components (see Section 5).
To avoid long-term fatigue and fracture due to bending stress, the elastically bent nanofibers can be annealed around 1400 K to form permanent plastic deformation, without change in surface smoothness or diameter uniformity.
Because the elastically bent fiber is held tightly on the substrate, the geometry of the assembly is not affected by the annealing, making it possible to lay out a final design before annealing. The annealing-after-bending process can also be performed repeatedly to obtain very tight bends or multiple bends as shown (see Figs. 11.26 and 11.27).
5.2. Wave-Guiding and Optical Loss
To investigate the optical and wave-guiding properties of silica nanofibers, it is necessary to couple light into and out of them. If a nanofiber still is connected to the starting fiber, one can couple light into the starting fiber in the standard manner to launch a guided wave into the nanofiber through the tapered region.
To launch light into nanofibers with freestanding ends, one can use evanescent coupling between a pair of fibers, as shown in Fig. 11.28a. Light is first sent into the core of a single-mode fiber that is tapered down to a nanofiber and the nanotaper is then used to evanescently couple the light into another nanofiber by overlapping the two in parallel. Because of electrostatic and van der Waals forces, nanofibers attract one another, making a parallel contact connection.
Fig. 11.28b shows an optical micrograph of the coupling of light between a 390-nm diameter launching taper and a 450-nm diameter nanofiber. The coupling efficiency of this evanescent coupling can be as high as 90% when the fiber diameter and overlap length are properly selected. This method can also be used to couple light out of a nanofiber.
Because of the nanofibers’ extraordinary uniformity, their optical losses are low. Figure 11.29 shows a 360-nm diameter nanofiber guiding light of 633-nm wavelength from the left. The scattering of light along its length is due to nanoparticles that are stuck to the wire and that scatter the evanescent wave. The light guided by the nanowire is intercepted at the right by a supporting 3-μm diameter taper to show qualitatively that the amount of light scattered by the fiber is small compared to that guided by it.
To quantify the optical losses of silica nanofibers, we measure the nanofibers’ transmission as a function of the length. The optical loss for freestanding nanofibers in air is shown in Fig. 11.30.
In single-mode operation, the optical loss can be as low as 0.0014 dB/mm, which is much lower than the optical loss of other subwavelength structures such as metallic plasmon waveguides, nanowires, or nanoribbons. The increasing loss with decreasing fiber diameter can be attributed to surface contamination; as the fiber diameter is reduced below the wavelength, more light is guided outside the fiber as an evanescent wave and becomes susceptible to scattering by surface contamination and/or microbends.
Because of their low-loss optical wave-guiding properties, these nanofibers are ideal building blocks for microphotonic applications. For example, low optical loss is essential to obtain a high Q-factor in an optical microcavity resonator, to maintain the coherence of the guided light in optical waveguide/ fiber sensors using coherent detection, to reduce the noise or crosstalk in high-density optical integration, and to reduce energy consumption when many devices are connected in series.
6. Applications and Potential Uses of Silica Nanofibers
As shown in the previous section, taper-drawn silica nanofibers can serve as low-loss subwavelength-diameter single-mode optical waveguides. Their high mechanical strength and pliability allows assembly into complex structures.
Because silica is one of the fundamental materials for photonics, taper-drawn silica nanofibers hold great promise for nanoscale optical sensors, for low-energy nonlinear interactions and supercontinuum generation, and for atom trapping and guiding. In this section, we discuss these applications in further detail.
6.1. Microscale and Nanoscale Photonic Components
Silica nanofibers have been used as building blocks in the assembly of a variety of microscale or nanoscale photonic components or devices such as linear waveguides, waveguide bends, optical couplers, and ring resonators. Because of their small dimensions, low optical losses, evanescent wave-guiding, and mechanical flexibility, nanofiber-assembled photonic devices have a number of advantages over conventional photonic devices.
To assemble microphotonic devices from silica nanofibers, the fibers must be supported by a substrate. The relatively low index of silica (~1.45) necessitates a substrate with an index much lower than 1.45. A microphotonic device consisting of an assembly of silica nanofibers on a silica aerogel substrate has been reported.
Silica aerogel is a tenuous porous network of silica nanoparticles with a diameter of about 5 nm, much smaller than the wavelength of the guided light, and has a transparent optical spectral range similar to that of silica. Because the aerogel is mostly composed of air, its refractive index is similar to that of air (1.03–1.08). Figure 11.31 shows a close-up view of a 450-nm diameter silica fiber supported on a substrate of silica aerogel.
Because the index difference between the silica aerogel and air (0.03–0.08) is much lower than the index difference between the silica nanofiber and air (~0.45), the optical guiding properties of aerogel-supported nanofibers are virtually identical to those of air-clad ones.
Figure 11.32 shows a 380-nm diameter silica nanofiber guiding 633-nm wavelength light on the surface of a silica aerogel substrate. The uniform and virtually unattenuated scattering along the 0.5-mm length of the fiber and the strong output at the end face show that the scattering is small relative to the guided intensity.
Figure 11.33 shows the measured optical loss of silica nanofibers supported by an aerogel substrate. The low loss provides further evidence that the aerogel substrate does not degrade the guiding of light through the nanofibers.
For fibers with a diameter near the single-mode cutoff diameter, the loss is less than 0.06 dB/mm, much lower than the optical loss in other subwavelength structures and acceptable for most photonic applications. These data show that silica aerogel supported silica nanofibers can be used as low-loss single-mode linear waveguides, as well as building blocks for assembling microphotonic devices.
By transferring plastic nanofibers bends (i.e., nanofibers that have been bent and annealed) onto a silica aerogel substrate, one can fabricate microscale waveguide bends with subwavelength diameters. Figure 11.34 shows a plastically bent 530-nm diameter silica nanofiber supported by a silica aerogel substrate.
The nanofiber was first bent to a radius of about 8 μm on a sapphire wafer, annealed, and then transferred to silica aerogel. The aerogel-supported plastic bends show excellent optical wave-guiding with good confinement of the light.
Figure 11.35 shows an optical microscope image of 633-nm wavelength light guided through such an aerogel-supported plastic bend. The measured bending losses through a 90-degree bend in a 530-nm diameter nanofiber are shown in Fig. 11.36 as a function of bending radius.
For example, the optical loss around a 5-μm radius bend in a 530-nm diameter fiber is less than 1 dB—acceptable for use in photonics devices. In contrast, the bending of light by planar photonic crystal structures not only requires much more complex fabrication techniques, but also suffers from inevitable out-of-plane losses.
Aerogel-supported nanofiber bends thus offer the advantage of compact overall size, low coupling loss, simplicity, and easy fabrication. Furthermore, contrary to wavelength-specific photonic crystal structures, nanofiber bends can be used over a broad range of wavelengths, from the near-infrared to ultraviolet wavelengths.
Using these waveguide bends as building blocks, one can readily assemble an optical coupler. Figure 11.37 shows an X-coupler assembled from two 420-nm diameter silica fiber bends.
When 633-nm wavelength light is launched into the bottom left arm, the coupler splits the flow of light in two. By changing the overlap between the two bends, it is possible to tune the splitting ratio of the coupler. With an overlap of less than 5 μm, the device works as a 3-dB splitter with an excess loss of less than 0.5 dB.
In contrast, microscopic couplers such as fused couplers made from fiber tapers using conventional methods require an interaction length on the order of 100 μm. Couplers assembled with silica nanofibers, thus, reduce the device size by more than an order of magnitude.
Another microphotonic device that can readily be fabricated from taperdrawn silica nanofibers is a microring resonator. Figure 11.38 shows a 150-μm diameter microring made by tying a knot in an 880-nm diameter nanofiber. The measured transmittance of this microring for wavelengths near 1.55 μm is shown in Fig. 11.39.
The transmittance clearly shows optical resonances with an extinction ratio that corresponds to a Q-factor of more than 1000. Microcoil/loop resonators with Q-factors as high as 95,000 have been realized, and a proposed microcoil resonator with self-coupling turns is expected to display a Q-factor as high as \(10^{10}\).
6.2. Nanofiber Optical Sensors
As discussed in Section 2, one of the prominent optical properties of a subwavelength-diameter silica nanofiber is its ability to guide light with a large fraction of power propagating outside the solid core. This evanescent wave is highly sensitive to index changes in the environment and to microscale bending of the nanofiber. At the same time, the coherence of the guided light can be maintained over a considerable length because of the low wave-guiding loss of these nanofibers. These properties make silica nanofibers ideal for high-sensitivity nanoscale optical sensing.
A schematic diagram of a silica nanofiber sensing element is illustrated in Fig. 11.40. A single-mode silica nanofiber is exposed to or immersed in a gaseous or liquid environment containing the molecules to be detected. The nanowire can readily be functionalized with the appropriate receptors for the molecules to be detected. If the nanowire is guiding light, any index change around the fiber due to the binding of molecules to the receptors or a temperature change affects the guided light’s optical phase and intensity.
By detecting the signal at the output, one can thus obtain information about the environment of the nanofiber. Numerical simulations show that if a Mach-Zehnder interferometer is used to detect phase shifts in the guided light, the sensitivity of a nanofiber sensor can be more than one magnitude higher than that of a conventional fiber/waveguide optical sensor. Most importantly, the size of the sensing element is greatly reduced using nanofibers.
Several optical sensors based on subwavelength- or nanometer-diameter silica fibers have been experimentally realized. A nanofiber optical sensor for measuring the refractive index of liquids propagating in microfluidic channels was made from a 700-nm diameter fiber taper that was tapered from a standard single-mode fiber and immersed in a transparent curable soft polymer.
A channel for the liquid analyte was created in the immediate vicinity of the taper waist. Light propagating through the nanotaper extends into the channel, making the optical loss in the system sensitive to the refractive-index difference between the polymer and the liquid. The estimated sensitivity of this refractive-index sensor is about \(5\times10^{-4}\).
A miniature hydrogen sensor consisting of a subwavelength diameter tapered optical fiber coated with an ultrathin palladium film has also been reported. The hydrogen changes the optical properties of the palladium layer and consequently the absorption of the evanescent waves.
Measurements at a wavelength of 1550 nm show that the sensor’s response time (~10 seconds) is several times faster than that of several optical and electrical hydrogen sensors reported so far. Moreover, the sensor is small, reversible, and suitable for detection of hydrogen in the lower explosive limit.
6.3. Additional Applications
Besides integration into microphotonic devices and optical sensing, silica nanofibers have been applied in nonlinear optics and supercontinuum generation and in atom trapping and guidance. Nonlinear optical interactions in nanofibers have been extensively investigated.
Supercontinuum generation was reported in submicrometer fibers and in microstructured optical fibers with subwavelength core diameter. Because of the tight mode confinement and strong waveguide dispersion, subwavelength-diameter nanotapers or fibers exhibit nonlinear optical properties at relatively low-power and short interaction length.
In another promising application, silica nanofibers were used to trap and guide atoms by the optical force of the evanescent field around the fibers. It was shown that the gradient force of a red-detuned evanescent-wave field in the fundamental mode of a silica nanofiber can balance the centrifugal force of the atoms. Likewise, using a two-color evanescent light field around a nanofiber a net potential with large depth, coherence time, and trap lifetime can be produced.
The next tutorial discusses in detail about nonlinear optical pulse propagation.