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• Abstract

SECTION 1

## Introduction

Tapering of optical fibers is a very important technique to fabricate some crucial optical fiber devices such as directional couplers. The tapering technique was intensively studied in the late 1980s and the early 1990s [1], [2], [3], [4], [5]. In 2003, Tong et al. employed a novel tapering technique to fabricate nanowires from standard single-mode optical fibers tapered down to a few hundred nanometers [6], [7]. The core diameter of the fiber is so small that the optical propagation field spreads out to the cladding and is thus guided by the silica and air. The evolution of modes in microfibers in the diameters ranging from 8 to 35 $\mu\hbox{m}$ was reported by Fielding et al. [8] to study the mode patterns for different microfiber diameters and demonstrated experimentally that the optical guiding becomes multimode.

Recently, fiber Bragg gratings (FBGs) inscribed in microfibers using different lasers have been reported [9], [10], [11], [12]. Microfiber Bragg gratings (MFBGs) fabricated by femtosecond laser pulses [9], KrF excimer laser [10] and 193-nm laser [11] were investigated for refractive index sensing and liquid level variation sensing [12]. It is important to note that MFBGs exhibit multiple peaks [9], [10], [11] as the diameters of the microfiber were too large to ensure single-mode propagation. The multimode characteristic is not desirable for demodulation and demultiplexing and, thus, limits its application.

In this paper, we report a technique to recouple the fundamental mode reflected from an MFBG and couple the higher order modes to the cladding via the fiber taper with an appropriate length. To the best of our knowledge, this is the first demonstrated technique to obtain effectively single reflection peak from an FBG written in a multimode microfiber. The main advantage of using such MFBGs is that the fiber diameter (30- $\mu\hbox{m}$ in our work) is much larger than the single-mode microfiber (typically $\sim\!\! 1\hbox{-}\mu\hbox{m}$) and is thus easier to handle and fabricate.

SECTION 2

## Experiment

### 2.1. Taper Fabrication

The microfiber was fabricated by a glass-processing machine made by Vytran (model GPX-3400). An illustration of the taper method of pulling conventional optical fiber into microfiber is shown in Fig. 1. The optical fiber with diameter of 125 $\mu\hbox{m}$ is fed into the filament at the velocity of $v_{f}$ and pulled from the opposite side of the filament at the velocity of $v_{p}$. Since the volume of the fiber entering and existing from the heating element remains constant, the diameter of the pulled fiber can be determined by the following relationship: TeX Source $$v_{f} = {d_{p}^{2}\over d_{f}^{2}}\times v_{p}\eqno{\hbox{(1)}}$$ where $d_{f}$ and $d_{p}$ are the diameters of the feeding and pulled fiber. In our experiment, $v_{p}$ was set to a constant speed of 1-mm/s with initial delay of 0.1 s to allow the filament to heat up. By varying the feeding speed, tapers with the desired diameters can be made. The heating filament is made of tungsten and its operation temperature is greater than 1900 °C at an input electrical power of 29-W. The upside down $\Omega$ shape of the heating filament facilitates a uniform heating zone around the fiber being pulled. The interior loop diameter and width of the heating element are 0.035 and 0.025 in, respectively. The fiber was placed at the center of the loop. During the pulling process, zero tension was applied and argon was supplied to the heating filament region at the flow rate of 0.65 L/min to prevent oxidation of the heating filament. Tungsten has a melting point of 3422 °C but oxidizes at 400 to 500 °C. The taper region was designed to be linear. Using this method, we could make microfibers with different taper and waist lengths, and diameter smaller than 1 $\mu\hbox{m}$.

Fig. 1. Tapering method for making microfiber.

### 2.2. FBG Inscription

A microfiber with a waist diameter of 30 $\mu\hbox{m}$ and taper and waist lengths of 25 mm and 20 mm, respectively, was fabricated for our experiment. In order to enhance the taper's photosensitivity for FBG inscription, the optical fiber was hydrogen loaded at 25 °C for over 60 h. A bend-insensitive G.657B optical fiber from Silitec which has high germanium concentration than SMF28 fiber was used. Since fiber diameter is small, hydrogen diffuses out so fast that the FBG inscription has to be made within 4-h after the optical fiber was withdrawn from the hydrogen chamber.

The FBG was fabricated using standard phase mask technique with a 1061.5-nm pitch phase mask. A KrF laser with a wavelength of 248 nm and a 2-mm beam width was used as the writing beam. The pulse energy and pulse rate of the laser were 13 mJ and 200 Hz, respectively. The reflection spectrum of an FBG written in the taper is shown in Fig. 2. The dimensions of the taper are also shown in the inset of Fig. 2. The length of the apodized FBG is 10 mm long. The apodization profile was achieved by scanning the writing beam along 8 mm at the taper waist using a Hamming profile with maximum velocity of 5 mm/s. The grating spectrum was measured by an optical fiber sensor interrogator from Micron Optic, Inc. (model SM-125) which has a 5-pm resolution. The peak wavelength and 3-dB bandwidth of the FBG are 1534.895 nm and 0.135 nm, respectively. The reflection of the FBG was slightly less than 10%. The reflections of other modes are significantly weaker than the main peak and have only 0.1% or −30-dB reflection. Reflections of the higher order modes are at least 20-dB smaller than the fundamental mode, making it an effectively single-mode FBG.

Fig. 2. Reflection spectrum of an effectively single reflective mode FBG written in the taper waist of a 30-$\mu\hbox{m}$ microfiber. The inset shows the dimensions of the microfiber with an FBG inscribed at the waist center.
SECTION 3

## Discussion

### 3.1. Multimode Characteristic of Microfiber

To analyze the FBG in the microfiber, we consider the $V$-number of the microfiber. For conventional optical fiber, the $V$-number is obtained by the following equation: TeX Source $$V = {2\pi a\over \lambda}\sqrt{n_{1}^{2} - n_{2}^{2}}\eqno{\hbox{(2)}}$$ where $a$ is the core radius, and $n_{1}$ and $n_{2}$ are the refractive indices of the core and cladding, respectively. The waist of the taper has a diameter of 30 $\mu\hbox{m}$, and the core diameter is reduced to about 2 $\mu\hbox{m}$. Since the core diameter is significantly smaller than the mode field diameter, the optical field is no longer bounded by the core but rather by the 30-$\mu\hbox{m}$ cladding [7]. Using $a = 15 \ \mu\hbox{m}$ and $n_{1} = 1.444$ and $n_{2} = 1$, the $V$-number was calculated to be $\sim$64. Therefore, the FBG written in the microfiber supports many modes. It should be noted that the LP mode approximation cannot be applied here because the microfiber is no longer “weakly guiding” as $n_{1} - n_{2}$ is 0.44 and, thus, not significantly smaller than 1.

The coupling between different modes of an FBG written on a multimode optical fiber was reported in [14]. There are two conditions that have to be satisfied for the mode coupling in an FBG. First, the overlap of coupling coefficient has to be large enough. Since the diameter of the FBG is very small, small amount of power in the modes $\hbox{HE}_{1{\rm M}}$ and $\hbox{EH}_{1{\rm M}}$ are guided within the core while TE and TM modes almost have zero power in the core. As a result, MFBG cannot couple those modes. The other higher order HE modes, e.g., $\hbox{HE}_{2{\rm M}}$ and $\hbox{HE}_{3{\rm M}}$, also have zero coupling coefficients between forward $\hbox{HE}_{1{\rm M}}$ modes. Therefore, they cannot be coupled to $\hbox{HE}_{1{\rm M}}$. The second condition is that the mode coupling has to satisfy the phase matching condition. Since different modes have different propagation constants, the phase matching wavelength is different. In the general case, the Bragg wavelength of the mode coupling from mode A to mode B is given by the equation TeX Source $$\lambda_{{A} \rightarrow {B}} = (n_{A} + n_{B}) \Lambda = 2n_{eff}\Lambda\eqno{\hbox{(3)}}$$ where $\underline{n}_{\underline{A}}$ and $\underline{n}_{\underline{B}}$ are the effective refractive indices of mode A and B, respectively, and $\Lambda$ is the pitch of the grating.

### 3.2. Short Taper Length MFBG

Fig. 3(a) illustrates the mechanism of the mode coupling in the taper from an FBG written in the microfiber. The taper length between the 125-$\mu \hbox{m}$-diameter standard fiber and the 30-$\mu\hbox{m}$-diameter microfiber is 5 mm.

Fig. 3. (a) Mode coupling of an FBG written in the waist of a taper. The dimensions of the fiber taper are also indicated. (b) Measured reflection spectrum of the grating. (c) A table listing the simulated $n_{eff}$ of modes having power guided within the core and the corresponding labels for the mode. (d) Simulated and experimental results of $n_{eff}$ of modes coupling in the MFBG.

Fig. 3(c) shows a table listing the labels, the corresponding modes which have power guided in the core and the simulated $n_{eff}$ using COMSOL. It should be noted that EH11 mode (with $n = 1.44161$) has zero power guided within the core and thus it is not shown in the table. In addition, the $n_{eff}$ of EH12 and HE13 are very close to each other, and hence, they are grouped as #3. The label of the modes in the table will be used throughout the paper for clarity. Fig. 3(a) illustrates the mode coupling mechanism of the grating device. The coupling of the modes can happen as follows. At first, fundamental mode is launched to the taper device (indicated by the big black arrow in Fig. 3(a), and then, higher order modes are excited in the taper region (illustrated by two small arrows). The modes are then coupled backwards by the MFBG. The coupling is not only limited by the self-coupling, e.g., $\#1 \rightarrow \#1$, but occurs mutually as well, e.g., $\#1\rightarrow \#2$ and $\#1 \rightarrow \#3$. The resulted $n_{eff}$ can be obtained by (3). Finally, the reflected modes from the MFBG are recoupled by the taper into the fundamental mode. Even though the modes are eventually recoupled to the fundamental mode, they appear as several peaks because the $n_{eff}$ of the modes are different as the result of the coupling in the MFBG.

Fig. 3(b) shows the reflection spectrum of an FBG written in the 30- $\mu\hbox{m}$ microfiber. When the fundamental mode passes though the taper region, higher orders modes are also excited. The MFBG couples the forward #1 and #2 mode to backward #1, #2, and #3 modes, and then these three modes are recoupled back to backward #1 via the linear shaped taper region. In addition to these three modes, other higher order modes (higher than #3), which are also coupled from forward #1 or #2 mode and annotated by blue arrow, are reflected and coupled to the cladding and eventually lost. The length of the taper determines the number of modes recoupled into the single-mode fiber via the taper. From taper end, only #1 couples backwards to the single-mode fiber.

There are five peaks in the reflection spectrum shown in Fig. 3(b). The peaks are highlighted in different colors. The corresponding modes with same color are indicated on the left side of the spectrum. The first peak, located at the longest wavelength, was a result of forward #1 to backward #1. The second peak has the highest reflection power and it was a consequence of #1 and #2 mutual coupling, i.e., $\#1 \rightarrow \#2$ and $\#2 \rightarrow \#1$. The third peak was the result of self-coupling of #2. The fourth and last peaks were the coupling result of $\#1\rightarrow \#3$ and $\#2\rightarrow \#3$, respectively. In the microfiber, only the modes from #1 and #2 were recoupled to the core via the taper and therefore no reflection aroused from coupling of the other higher order modes. That is, $\#3\rightarrow \#1$, $\#3\rightarrow \#2$, $\#3 \rightarrow \#3$, etc.

Fig. 3(d) shows the comparison between the simulated mode refractive index and the $n_{eff}$ obtained from experiment. A three-layer model of the step index profile was used. The refractive indices of the core, cladding and air were set to 1.44998, 1.44402, and 1, respectively. The diameters of the core, cladding and air were 1.992 (converted from 8.3 $\mu\hbox{m}$ core diameter as in original fiber specification: $8.3\ast 30/125 = 1.992\hbox{-}\mu\hbox{m}$), 30 and 1000 $\mu\hbox{m}$ (large enough to treat as infinity), respectively. The effective refractive indices resulted from the corresponding coupling modes in the MFBG were then calculated using (3). The results are in excellent agreement with each other. It should be noted that other higher order modes ($\hbox{EH}_{1{\rm M}}$ and $\hbox{HE}_{1{\rm M}}$ where ${\rm M} \geq 3$) theoretically can also be coupled as they also have some power guided in the core; however, their mode refractive indices are far from 1.44 and, therefore, cannot be coupled to the taper.

Fig. 4 shows the transmission and reflection spectra of an effectively single reflective mode MFBG. In the transmission spectrum, there are several notches that are the combined results of the both modes that are recoupled to the fundamental mode via the taper region and coupled to cladding. Those cladding modes are not present in the reflection spectrum.

Fig. 4. Transmission and reflection spectra of an effectively single reflective mode taper FBG. The insert shows the dimensions of the taper and FBG.

Fig. 5 shows the measured reflection spectra of 20-mm-long microfiber with taper lengths of 5, 10, 15, 20, and 25 mm. From the reflection spectra, it is clear that the taper length determines the numbers of peaks that can be recoupled and propagate in the core of the standard 125- $\mu\hbox{m}$ diameter fiber. For those with taper lengths of 5, 10, and 15 mm, the number of peaks that can be detected were 5, 4, and 2, respectively. For taper lengths longer than 15 mm, the result indicated that effectively one mode was obtained.

Fig. 5. Reflection spectra of MFBGs with different taper lengths. The inset shows the dimensions of the MFBGs.

The experimental results demonstrated that fiber taper with sufficiently long taper length can effectively recouple the fundamental mode reflected back to the core and filter out the undesirable higher order modes excited in the MFBG.

SECTION 4

## Conclusion

A novel technique to attain effective single reflective mode in MFBGs and their experimental results are reported. The study of reflection modes coupling via fiber tapers with different taper lengths was also conducted. The results demonstrated experimentally that the length of a linear taper can be used to achieve an effectively single reflective mode from the FBG written in the multimode microfiber. Single reflective mode MFBGs permit ease of multiplexing and demodulation and, thus, are highly desirable for sensing applications in particular.

## Footnotes

This work was supported by the University Grants Council's Matching Grant of the Hong Kong Special Administrative Region Government under the Niche Areas project J-BB9J. Corresponding author: K. M. Chung (e-mail: chungkitman@gmail.com).

K. M. Chung, Z. Liu, and H.-Y. Tam are with the Photonics Research Centre, Department of Electrical Engineering, The Hong Kong Polytechnic University, Kowloon, Hong Kong.

C. Lu is with the Photonics Research Centre, Department of Electronics and Information Engineering, The Hong Kong Polytechnic University, Kowloon, Hong Kong.

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