1. Introduction

In the past years, extensive research and development activities have been devoted to evanescent-field-based optical waveguide sensors, which are now playing important roles in a variety of sensing applications [1–4]. By means of measuring small changes in optical phase or intensity of the guided light, these sensors present excellent properties such as high sensitivity, fast response, immunity to electromagnetic fields, and safety in the detection of combustible and explosive materials. Along with increasing demands and rapid development of nanotechnology in various fields, the combination of nanotechnology, biology, chemistry and photonics opens new opportunities of developing optical sensors with subwavelength or nanometric structures. Recently, subwavelength-diameter silica nanowires have been demonstrated for guiding light within the visible and near infrared spectral ranges [5–8]. Fabricated by taper-drawing of optical fibers, these wires show excellent diameter uniformity and atomic-level sidewall smoothness, making them possible to guide light with low optical losses [6–8]. Light guided along such a nanowire leaves a large fraction of the guided field outside the wire as evanescent waves [5, 9], making it highly sensitive to the index change of the surrounding medium. Here we propose to use silica nanowires as single-mode waveguides in evanescent-wave-based optical sensors. Phase shift of the guided mode caused by index change of aqueous solution is obtained by solving Maxwell’s equation, and is used as a criterion for sensitivity estimation. Nanowire sensor employing a wire-assembled Mach- Zehnder interferometer is proposed. Our simulation shows that optical nanowire waveguides are very promising for developing high-sensitivity optical sensors of significantly reduced sizes.

2. Wave guiding properties of silica nanowires for optical sensing

Optical wave guiding properties of silica nanowires in air have been studied else where [9]. However, in practical applications, e.g. for chemical and biological sensing, the sensitive element is usually immersed in or exposed to a liquid such as aqueous solution [1–4], and single-mode operation is generally required in waveguide-based optical sensing when coherent (rather than intensity) detection is used to achieve high sensitivity. Also, only when the nanowire works under single-mode condition can it leave a large amount of guided field outside the wire. Therefore, here we consider the single-mode wave guiding properties of silica nanowires for optical sensing in aqueous solutions.

Single mode condition for nanowire waveguides discussed here can be obtained as [10]

where n ₁ and n ₂ are the refractive indices of silica and water respectively, D is the diameter of nanowire, and λ₀ the wavelength of the probing light in vacuum. For example, at wavelengths of 325 nm (e.g. of a He-Cd laser) and 650 nm (e.g. of a laser diode), the single-mode cutoff diameters are 415 nm and 847 nm respectively.

Propagation constant (β) of the fundamental mode (HE11 mode) can be obtained by solving the eigenvalue equation of a circular cross-section waveguide in cylindrical coordination [10]

where J ₁ is the Bessel function of the first kind, and K ₁ is the modified Bessel function of the second kind, U=D($k_0^2$ $n_1^2$-β ²)^1/2/2, W=D(β ²-$k_0^2$ $n_2^2$)^1/2/2, and k ₀=2π/λ₀ .

Using propagation constants obtained by numerically solving Eq. (2), we calculate the profile of the evanescent fields and the power distribution around the wire waveguide, which are important properties for understanding and estimating the behavior of a nanowire sensor discussed here. Refractive indices of the silica (1.457 at 650-nm wavelength, and 1.482 at 325-nm wavelength) and water (1.333 at 650-nm wavelength, and 1.355 at 325-nm wavelength) are obtained from their dispersion formulas at room temperature [11,12]. For reference, the z-components (the only non-zero component along the axial direction of the nanowire) of the Poynting vectors (S_z ) of the HE11 mode of a 200- and 400-nm diameter silica wires at 325-nm wavelength are plotted in Fig. 1. The evanescent field (separated from the central peak with a discontinuous gap) outside the guiding core is clearly seen, which is used as an antenna for sensing slight change of the surrounding media in this work. The 200-nm diameter wire provides much higher fraction of evanescent waves than the 400-nm diameter one, indicating that thinner wire is much more sensitive to the surroundings than the thick one. Calculated fractional power of the evanescent wave outside the silica core (η) at 325-and 650-nm wavelengths is shown in Fig. 2, the two dashed lines represent single-mode cutoff diameters at the two wavelengths. It shows that, depends on the diameter, a single-mode silica nanowire can guide light with about 20 to almost 100 percent of energy as evanescent waves, which is difficult to achieve in a conventional waveguide with diameter or width larger than the wavelength.

 

Fig. 1. z-components of the Poynting vectors (S_z ) of the HE₁₁ mode of a (a) 200- and (b) 400-nm diameter silica wires at 325-nm wavelength

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Another concerned property for optical sensing is the guiding loss, which directly determines the intensity and coherence of the guided signal, and sometimes the signal-to-noise ratio due to the interference of the scattered light. Optical loss of a taper-drawn silica nanowire for single-mode wave guiding is now below 0.01dB/mm [7,8], which is low enough and virtually neglectable in the modeling since the wire length required by a nanowire sensor propose here is no longer than millimeters.

 

Fig. 2. Fractional power of the fundamental mode outside the core of silica nanowires at 325-and 650-nm wavelength. Dashed lines: single-mode cutoff diameters.

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3. Optical sensing with silica nanowire waveguides

3.1 Sensor system

Based on wave guiding properties of optical nanowires discussed above, we propose to functionalize the nanowire waveguides as sensing elements for detecting specimens in aqueous solutions. As shown in Fig. 3(a), a certain length of a silica nanowire is exposed to or immersed in the solution to be detected (highlighted in light green), and the wire guides light as a single-mode waveguide. When there is an index change (for example, caused by the addition of specimens, or just a temperature rise) around the wire, the guided light is changed in optical phase and intensity. To detect slight changes with high sensitivity, we propose to use a Mach-Zehnder interferometer to coherently measuring the phase shift of the probing light [13]. As shown in Fig. 3(b), a Mach-Zehnder interferometer can be assembled with two uniform silica nanowires. One nanowire (nanowire 1 in the figure) is used as sensing arm with a certain length of sensitive area exposed to the measurand, and the other (nanowire 2) is used as reference arm that is kept in constant condition and is isolated from the measurand. Two 3-dB couplers are formed by evanescent coupling of the two closely placed nanowires, which has been shown achievable with sub-5-µm size [14,15]. When the probing light is launched from the left end of nanowire 1, it propagates along the wire and is bifurcated by the first 3-dB coupler (working as an optical splitter). After traveling through the sensing arm where measurand has access to the evanescent field at the sensitive area, the signal meets with the reference (traveling along the reference arm) by the second 3-dB coupler (working as an optical combiner), where the phase shift of the probing light is measured by the interferometer, and the information of the specimens can thus be retrieved.

 

Fig. 3. Schematic diagram of (a) a silica nanowire sensing element and (b) a proposed sensor with a Mach-Zehnder interferometer.

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3.2 Sensitivity

The phase shift (ΔΦ) of the sensing arm can be obtained as

where L is the effective length of the sensitive area, β ₀ and β are the initial and instant propagate constants of the light in the sensitive area, respectively. For given indices of silica core (n ₁) and cladding (n ₂), β ₀ and β can be obtained by numerically solving Eq. (2), and ΔΦ can then be obtained with Eq. (3).

At a given temperature and pressure, when a certain amount of the specimen is added into a solution with an initial refractive index of n_C0 , the overall index (n_C ) afterwards can be approximately obtained as

where n_s is the index of the specimen, and C the molar concentration of the specimen in the solution.

For quantitative estimation, as a typical case, we use a diode laser (650-nm wavelength) as probing light, and assume n_C0 =1.333 (e.g. pure water [12]), n_s =1.501 (e.g. benzene [16]). We obtain a C-dependent Δβ shown in Fig. 4, in which three curves corresponding to wire diameters of 400, 600 and 800 nm respectively. It shows that Δβ increases with the decreasing of the wire diameter, indicating that thinner wires provide higher sensitivity. Also, C-Δβ curves show good linearity within the concentration we concerned (from 0.5 ppm to 10%), which may be favorable for signal processing and calibration in practical applications.

 

Fig. 4. Changes in propagation constant (Δβ) as a function of molar concentration (C) of the specimen.

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Generally, the sensitivity of sensors investigated here can be normalized as

Calculated S_N of the nanowire sensor we proposed is about 7.5/µm. For comparison, S_N of conventional Mach-Zehnder sensors based on integrated planar waveguides is much lower, e.g., below 0.7/µm [17,18], showing that much higher sensitivity, or equivalently much smaller size can be achieved when sensing with optical nanowires.

By silanizing and bio-modifying the surface of the wire with specific receptors, nanowire waveguides can also be functionalized for highly selective detection. In such cases, the specimen collected by the receptor usually forms a thin layer wrapping around the wire as shown in Fig. 5, and propagation constants of the composite waveguide can be obtained by solving Maxwell’s equations with boundary conditions of a 3-layer cylindrical structure [19,20]. Assuming the specimen to be nanoparticles with size of 12 nm and index of 1.40 (e.g. polystreptavidin [21]), we calculate Δβ induced by coating the nanowire with a monolayer (12-nm thickness) of the nanoparticles; for a 400-nm diameter silica nanowire, calculated Δβ is about 0.015/µm at the wavelength of 650 nm. If we assume the detection limit of ΔΦ to be 2×10^-3π with a Mach-Zehnder inteferometer [21], we find that the sensor is sensitive enough to feel a monolayer coating on a 400-nm-diameter wire with sensitive length of 420 nm; or a 1-mm-length sensitive nanowire is capable of detecting 4.2×10^-4 of a monolayer. In comparison, biosensors built with conventional integrated waveguides require a sensitive length of 15 mm to achieve a detection limit of 6×10^-4 of a monolayer [21]. In addition, our simulation shows that, with the same fractional power of the evanescent field, e.g. scale down the wire diameter and light wavelength simultaneously, the sensitivity of the nanowire sensor increases rapidly with the decreasing of the wavelength of the probing light. For reference, when we use a 200-nm-diameter nanowire at 325-nm-wavelength, Δβ goes up to 0.043/µm, providing an ability to feel a monolayer with sensitive length of only 147 nm; or a detection limit of 6×10^-4 of a monolayer can be obtained with a sensitive length of 245 µm, showing the potential for developing nanowire sensors with significantly reduced sizes.

 

Fig. 5. Schematic diagram of (a) a silica nanowire for selective sensing and (b) a cross section view of the composite waveguide.

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4. Conclusion

In conclusion, our simulation shows that, by using nanowires as single-mode subwavelength-diameter optical waveguides, it is possible to use a high fraction of the guided energy as evanescent wave for optical sensing. Comparing to conventional sensors relying on micrometer-scale waveguides, optical sensing with nanowires may suggest a new approach to miniaturized optical sensors with high sensitivity, and optical-quality nanowires of many other materials may be used for optical sensing [22,23]. Also, the reduced footprint of the nanowire sensor may allow sensing in environment of smaller scale, support integration of sensor array with higher density, and require fewer samples.

Finally, to practically realize a nanowire sensor modeled in this work, some important challenges such as instability of the nanowire-assembled couplers and oversensitivity of the whole system must be overcome, which require further experimental investigation in the future.