Functional Tapered Fiber Devices Using Polymeric Coatings

A wide variety of fiber devices can be created by adding special coatings on tapered sections of optical fibers. In this work we present the fundamentals for the fabrication of tapered optical fibers coated with functional polymers. The required aspects of light propagation in tapered sections of optical fibers are introduced and the relevant parameters enabling light interaction with external media are discussed. A special case of interest is the addition of polymeric coatings with prescribed thicknesses in the tapered sections allowing for adjusting the light propagation features. We assess the use of liquid polymer coatings with varying thicknesses along the taper profile that can be tailored for tuning the transmission features of the devices. Hence, we introduce a methodology for obtaining coatings with predefined geometries whose optical properties will depend on the polymer functionality. As demonstrated with numerical simulations, the use of functional polymer coatings in tapered optical fibers allows for obtaining a wide variety of functionalities. Thus, controlled polymer coating deposition may provide a simple means to fabricate fiber devices with adjustable transmission characteristics.


Introduction
Optical fiber-based devices and sensors have been widely used in many fields of science and technology for different applications. In most applications, the transmission of light through in-line fiber devices is modified by an external perturbation and this can be quantified through variations in one or more characteristic features of the guided optical wave. Although extrinsic interaction of the light with the surrounding media is possible, the use of the evanescent portion of the guided wave offers some advantages. Exposure of the optical wave to the surrounding media in specific sections of the optical fibers is usually done by two alternatives [1][2][3][4][5]: removing the cladding material, or upon tapering a section of the optical fiber. The latter is the preferred approach to expose the evanescent wave since it involves a simple and reproducible process yielding low-loss devices. Although the description and basics of tapering optical fibers have been described since early 1990's by Birks et al. [1], new applications for these devices have been a subject of research due to their potential use in many fields of science and technology. Some of these applications include physical and biomedical sensing [6][7][8][9][10], interferometry [11], excitation of surface plasmons [12,13], atom detection [14,15], and light coupling to micro-resonators [8,16], just to name a few.
Tapering techniques and systems have evolved over the years resulting in optimized devices and standardized processes. As a result, tapering of optical fibers has been employed for modifying the light propagation conditions in fibers, and further allowing for the guided light to interact with other structures or materials. The latter capabilities allow for creating fiber-based devices incorporating coatings with controlled thicknesses and different optical properties. In this chapter we present the main aspects and basic guidelines for achieving a proper guidance of light through tapered devices providing also adequate interaction with the surrounding medium.

Basics of light propagation in optical fibers
Light propagation in optical fibers occurs within the fiber core (radius a) following the total internal reflection condition from Snell's law. Propagation conditions are defined by the physical characteristics of the core and cladding sections of the fiber: refractive indices and diameters. Due to the cylindrical geometry of the optical fibers and the small refractive index difference between the core and cladding materials, the propagating modes are obtained in a cylindrical coordinate system (r, φ, ϕ) and in terms of Bessel equations [17,18]. Rigorous analysis and proper boundary conditions are used to obtain mathematical expressions describing the components of the electric and magnetic fields for the core (r ≤ a) and the cladding (r ≥ a) regions. The combination of these propagating fields allows for an alternative description in terms of linearly polarized modes, denoted as LP lm ,in which l and m are respectively the axial and radial indices. As an example, the electric field for the set of LP 0m modes can be shown to be given by: where E 0 is the field amplitude, J 0 is the zeroth-order Bessel function of the first kind, and K 0 is the zeroth-order modified Bessel function of the second kind. The parameters u and w are respectively the normalized propagation and attenuation constants, defined as: involving the propagation constant of the guided wave (β), the wave number (k) and the refractive indices of the core (n core ) and the cladding (n cladd ). The description in terms of the LP lm modes is very useful as these can be experimentally observed as intensity patterns (I lm ) whose mathematical representations are [17]: : The normalized frequency is typically used as an indicator of how many modes are supported by the fiber. It can be shown that standard single mode fibers (SMFs) supporting only the fundamental (LP 01 ) mode have a V < 2:405. Hence, this condition is typically used as an indicator of single-mode propagation in an optical fiber.

Light propagation in tapered optical fibers
Tapered optical fibers are fabricated such that the physical dimensions of the core and cladding are reduced, thereby modifying the light propagation conditions [8,19,20]. The implication of this reduction in the physical dimensions of the fiber is a decrease in the effective refractive index of the core and thus a change in the light confinement. An effect caused by the core reduction is the compression of the light inside the core, although the evanescent wave increases in magnitude. However, there is a physical limit for the core reduction since the light cannot be effectively guided inside this region under certain conditions. After reaching this "guiding threshold" in the dimensions of the core, light escapes to the cladding material, which acts as the new core of the optical fiber while the surrounding media becomes the new cladding. As a consequence, the evanescent wave is effectively exposed to the external medium, granting the possibility of light interaction with different materials or structures.
Typically, tapered optical fibers are segmented in three sections for their analysis [1]: the non-tapered segment of the optical fiber, the transition and the waist sections, as depicted in Figure 1a. The waist section of the fiber is where the fiber is tapered down to its final diameter after which it remains constant across all its length. Usually, the interaction or coupling to other structures or materials takes place at this section. The transition sections of the fiber are those at which the diameter goes from the original diameter to the tapered section, and vice versa. Although the transition regions are commonly not used for sensing or coupling, they are significantly important as disturbances or non-desired alterations to light can occur in these sections. Such effects appear when the transition between diameters is not smooth or non-adiabatic.
Adiabaticity criteria comes from the tapering angle, which must be small enough for the fundamental mode to smoothly propagate across all the sections with minimal power loss [19]. Any sudden change in the dimensions or geometry of the fiber will provoke imminent light leaking from the core. This is avoided by performing the tapering process at a very slow rate. Considering the case of tapering a SMF, an adequate transition should maintain the fundamental mode through all the sections of the tapered fiber with negligible losses. Conversely, a non-adiabatic tapering process will produce a "leakage" of light into the cladding exciting higher order modes.
The parameter used to define the adiabaticity of the tapered fiber is the local length-scale of the taper (z t ), resulting from the ratio of the core radius (rÞ over the tapering angle Ω ðÞ , expressed as z t ¼ r Ω . Notice that both variables (r and ΩÞ vary with length, i.e., r ¼ rz ðÞand Ω ¼ Ω z ðÞ [19]. To achieve adiabaticity, the length of the taper must be larger than the length required to couple the fundamental mode to the predominant cladding mode. This coupling length is given by the beat length (z b ), which in turn depends on the propagation constant, β ¼ kn eff , with k ¼ 2π λ . Thus, this variable is given by the expression [19]: where β 1 is the propagation constant of the fundamental LP 01 mode, and β CM corresponds to the propagation constant of the excited cladding mode. In an axially symmetric tapered fiber, the fundamental mode will only couple to azimuthally symmetric higher order modes (i.e., the LP 0m modes) supported by the cladding structure (see Figure 1b).
To illustrate the phenomena related to tapering optical fibers, we will use as an example a step-index standard single-mode silica fiber with core radius of 4.1 μm, cladding radius of 62.5 μm and refractive index difference Δn ¼ 0:061. These are typical dimensions for the standard telecommunication fibers that are commonly used for the fabrication of tapered fibers for evanescent coupling of light to other photonic devices. Upon considering an adiabatic tapering process, firstly can be noticed that by reducing the core radius the V number is also modified. This implies that the effective refractive index (n eff ) of the waveguide will be modified as well and thus the propagation constant. The solutions in terms of the Bessel approximations yield the values of the propagation constants (β) resulting from these changes in geometry [17,18]. Figure 2 shows results from numerical calculations illustrating that a reduction in the propagation constant is obtained as the core radius decreases. Notice also that for V < 1, a sudden decrease in β occurs indicating that light cannot be longer confined in the core and hence it couples to the cladding material. Such phenomenon implies that the whole material of the optical fiber, core and cladding, acts together as the new core of the waveguide, while the surrounding medium serves as the new cladding. The first reports on this phenomenon mention values for this guiding threshold V TH near 0.8 [8,19] to achieve this condition. Our numerical results indicate that light confinement in the fiber core is sustained until V TH ≈0:77, which corresponds to an approximate core radius of 1.55 μm, as illustrated in Figure 2.
The intensity distribution of the fundamental mode is modified as the fiber is tapered. For core diameters above the guiding threshold, a tighter confinement of light within the core is achieved. However, the amount of light in the evanescent portion of the wave increases and so its extension inside the cladding. Once the guiding threshold is surpassed, light confinement shifts from the core to the cladding, distributing across the entire waist material. This effect is illustrated in Figure 3, showing light confinement at different tapering ratios. Instantly, as the guided light is exposed to the external media, light confinement and propagation are determined by the surrounding materials. At this stage, interaction with the surrounding media occurs via the evanescent wave. Nonetheless, the amount of light exposed to the external media can be tailored by defining the final waist diameter. For a fiber surrounded by air, due to the large refractive index difference with respect to the silica, the V number substantially increases allowing to reduce the fiber core down to very small diameters. Subwavelength waist diameters have been reported for tapered fiber devices in sensing and other applications [8,15,21].
The wavelength of light is also relevant in this phenomenon as in every waveguiding structure. Light confinement in the fundamental mode will change depending on the wavelength, showing a larger amount of evanescent wave for longer wavelengths. Assuming an adiabatically tapered SMF with a 5 μm waist radius with air as the surrounding medium, the extension of the evanescent wave will present a significant increase as illustrated in Figure 4. Hence, for a given tapered device with specific dimensions, wavelength selection is essential as this will change the exposure and hence the interaction of the evanescent wave with the external medium.
Light exposure through evanescent wave is useful for the creation of functional devices by means of adding different materials as coatings. The amount of evanescent wave exposed to the surrounding media determines the range of coating thicknesses that can be effectively used for efficient interaction depending on the materials as well. The fundamentals of fiber coating and main aspects to consider for the creation of devices using diverse materials as coatings are discussed in the following sections.

Fiber coating
There are multiple reports using diverse techniques for coating optical fibers with tapered sections. We will focus on a method based on the deposition of liquid materials over specific sections of optical fibers since these are convenient for postprocessing fibers. We consider in particular the use of liquid polymers as coating materials, since they offer a wide variety of functionalities. Polymers have been used for recoating fiber devices such as Bragg gratings, amplifiers or splices, just to mention the most important. Various works have demonstrated the effectiveness of using this technique for diverse purposes [22][23][24][25], and recoating systems to perform such task have been readily reported [26].
The process for deposition of controlled layer coatings on optical fibers using liquid materials, such as polymers, are suitably described by the wire coating technique [27][28][29]. This involves the immersion of the fiber into the liquid and retrieve it at a prescribed speed to obtain a desired thickness, as illustrated in Figure 5. The main factors involved in this process include the characteristics of the liquid such as viscosity η ðÞand surface tension γ ðÞ , and the velocity (v) at which the process takes place. The ratio of the viscosity to the surface tension, multiplied by the velocity of the coating process, is defined as the capillary number, which allows to compare the behavior of different liquids. This ratio is a dimensionless number defined explicitly as Ca ¼ ηv = γ . For liquids with Ca <<1, the resulting coating thickness (h) deposited on a fiber of radius (a) can be obtained as [29]: This expression holds only for Ca < 1 since the fluid dynamics are different for larger values of the capillary number. Nonetheless, proper adjustments to this theory can be made yielding a correction factor for the thickness [30]. This wire coating theory has been largely investigated and described in the field of fluidics describing in detail multiple scenarios and materials.
Given the nature of the liquids, an additional factor to consider for their use as fiber coatings is the Rayleigh-Plateau instability [31]. This effect leads to the breakdown of the uniform layer into a periodic array of droplets. Such behavior will always occur at characteristic time (t 0 ) given by: This is an important effect that must be considered when applying liquid coatings because it indicates the maximum time in which the liquid layer will remain with a uniform thickness. Therefore, the coating must be solidified before this characteristic time in order to preserve its shape. Hence, different polymeric materials cured by means of chemical, thermal or photo-active processes must account for this breakdown time.

Coatings on tapered optical fibers
In most cases, the coatings on the tapered sections of the fiber are sought to be with uniform thickness in order to obtain interaction with the evanescent wave. Ideally, only the waist section of the taper should be coated, as indicated in the green region in Figure 6a. However, there are two aspects that must be also considered when designing the coating: the actual exposition of the evanescent wave and the coating process itself. In practice, coating only the waist section of the tapered fiber might represent a challenging task since the liquid to be deposited is usually contained within a reservoir with prescribed dimensions that do not match the length of the waist. Extraction of the fiber from the reservoir can also be a potential factor to generate a non-uniform coating; this task is usually performed upon pulling the fiber from the non-tapered section and hence liquid is also deposited in other regions of the tapered fiber. The transition zones of the tapered fiber are therefore coated as well, and this minimizes any perturbations generated by the reservoir.
Non-uniformities in the coating thickness are a potential source of perturbation that may result in light leaking in the optical fiber. Given the variable radius profile of the tapered fiber and the radius dependency for the resultant thickness given by (6), the profile of the coating can be modified depending on the velocity of the process. Then, two scenarios can result from the coating process: a uniform coating along the entire fiber obtained upon adjusting the speed (Figure 6b), or a variable coating layer using a constant speed during the process (Figure 6c). In all cases, it is also important to consider the cross-linking mechanism of the coating polymer. Once the coating is deposited, it might require a longer period of time for achieving solidification.
To exemplify these possibilities, we consider a tapered optical fiber coated with polydimethylsiloxane elastomer (PDMS, Sylgard 184 from Dow), which has a η ¼ 3:5 Pa • s ½ and γ ¼ 0:0198 N • m À1 ½ [26,32]. For the SMF tapered down to a waist radius of 5 μm, the lengths of the waist and transitions are 10 and 25 mm, respectively; the transitions are assumed to have an exponential profile as described in [1]. For this specific fiber light couples into the cladding, and thus evanescent wave exposition into the surrounding media occurs along the transition section (red region in Figure 6). The radii of the fiber core and the cladding are compared with the modification of the normalized frequency in Figure 6d. The nature of the curing or solidifying process of the polymer is crucial and should be considered due to the possible appearance of instabilities. To avoid this, both the coating and curing processes needs to be shorter than the characteristic time (t 0 ) in the waist section.
PDMS is a polymer cured by heating, requiring from several minutes to hours for complete solidification, depending on the temperature of the curing process. For achieving a 200 nm thick uniform PDMS coating at the waist section, a speed coating process of around ν ¼ 0:62 mm • min À1 ÂÃ is required, and the critical time is t 0 ¼ 1:4 hr ½ . The conditions required for generating a coating with constant or variable thickness layers are analyzed in Figure 6e. For obtaining a coating with constant thickness (solid line), the required coating speed increases inversely with the fiber diameter. Although the instabilities mostly appear at larger times along the tapered fiber, the total duration of the process will be in general longer. In contrast, the variable coating (dashed line) implicates overall shorter times for instabilities to appear, and the process must be performed much faster.
Depending on the application, fiber coatings can be designed to optimize the interaction of the evanescent wave with the surrounding media. This means that the thickness of the coating can be extended to completely enclose the evanescent wave, or simply to favor some interaction and leave a remaining portion of light still interacting with the external media. Having this in mind, one must also consider the refractive index of the material: while low refractive index materials will require thinner coatings to completely isolate the evanescent wave, high refractive index Effective refractive index at the waist section of the fiber as a function of the coating thickness for materials commonly used for taper coating and for sensing applications: PTFE [33], PBS [34], water [34], 1-propanol [35], isobutanol [35] and PDMS [26,32]. materials will allow to use thicker coatings before isolating the evanescent wave, as illustrated in Figure 7. Of course, to have efficient and low loss light propagation in the tapered fiber, the refractive index of the coating material must be lower than the refractive index of the optical fiber. Otherwise, this will induce losses since the light propagated in the cladding will never been coupled again the core.

Applications of coatings for devices
Different effects can be generated in the guided wave depending on the optical properties of the coating material. For instance, the sensitivity of the tapered device to physical parameters can be tailored up to some extent. Using again as an example PDMS coatings, the effective refractive index of the fundamental mode can be adjusted with the coating thickness. It is well known that PDMS experiences changes with temperature due to its thermal expansion coefficient and to its thermo-optic coefficient ( ∆n = ∆T K À1 ÂÃ ¼À1:8 Â 10 À4 ) [32]. As the thickness of the PDMS coating is increased, the effective refractive index will increase its variation with temperature, as shown in Figure 8. Evidently, this "tunability" in sensitivity is limited, and it further depends on the thermo-optical properties of the polymer. To illustrate this, we have also included in Figure 8 an example using 1-Butanol as the coating material [35]. Notice that in this case, lower effective indices are obtained and their variations with coating thickness are not as pronounced as obtained for PDMS. However, the sensitivity of the effective index to temperature changes for this coating is larger than that of PDMS.

Conclusions
We have introduced guidelines for coating tapered sections of optical fibers with liquid materials. Specifically, we undertake the subject of polymeric thin layers deposited by the wire coating technique and the interaction of the coating with the evanescent wave. Firstly, the propagation considerations to achieve light interaction with the surrounding media via evanescent wave were described. This aspect depends on the physical dimensions, refractive indices of the materials and propagating wavelength. As shown upon analyzing the wire coating technique, it is possible to define parameter for the coating process that will allow for controlling the coating thickness and avoid coating instabilities. Essentially, once the properties of the liquid polymer and the dimensions of the taper are known, the coating velocity may be calculated in order to obtain a prescribed coating thickness. Finally, light propagation for different coating thickness with different materials was discussed in terms of the influence on the effective refractive index in the waist section of the fiber. These aspects should be of interest for obtaining photonic devices with functional polymers coatings that may be useful for sensing applications.