2. Fundamentals

The guiding mechanism of WGMs in the azimuthal direction of a microresonator (MR) is total internal reflection, just as in the case of axial propagation in a conventional waveguide; see Figure 1a. Resonance occurs when the guided wave travels along the perimeter of the MR, and it drives itself coherently by returning in phase after every revolution. In its way, the wave follows continuously the surface of the MR, and the optical path in a circumnavigation must be equal to an integer multiple of the optical wavelength, λ. When this condition is fulfilled, resonances appear, and a series of discrete modes at specific wavelengths will show up. The resonant condition can be written as [13]

Whispering Gallery Modes for Accurate Characterization of Optical Fibers' Parameters DOI: http://dx.doi.org/10.5772/intechopen.81259

$$
\lambda\_R = \frac{2\pi \mathbf{a} \cdot n\_{\rm eff}}{m} \tag{1}
$$

where λ<sup>R</sup> is the resonant wavelength, a is the radius of the MR, neff is the effective index of the WGM, and m is the azimuthal order of the mode (i.e., the number of wavelengths in the perimeter of the MR). The effective indices of the different modes are calculated, as usual, by solving Maxwell's equations and applying the proper boundary conditions [5]. In our case, we will deal with cylindrical, dielectric MRs with translational symmetry in the axial direction (see Figure 1b). Two zones can be identified, regions I (of radius a) and II (which extends to the infinite), with refractive indices n<sup>1</sup> and n2, respectively, with n1>n2. The magnetic permeability of the material and of the external medium is equal to that of the vacuum, μ0, and both media are homogeneous although, in general, they present an anisotropy in the dielectric permittivity. In the axial direction, we will consider a refractive index of the material n1<sup>z</sup> which is different to the refractive index in the transversal directions, n1<sup>t</sup> (see Eq. (2) for the expression of the tensor of the refractive index):

$$m\_2 = \begin{pmatrix} n\_{1\sharp} & 0 & 0 \\ 0 & n\_{1\sharp} & 0 \\ 0 & 0 & n\_{1\sharp} \end{pmatrix} \tag{2}$$

We do not intend to give a full description of the solution of this problem, which can be found in [14], but we will summarize the main equations and features of WGMs.

If we solve Maxwell's equations with this uniaxial tensor, the modes split in two series of family modes that, analogously to the case of axial waveguides, are denoted as TE-WGMs, which show a transversal electric field (ez ¼ 0), and TM-WGMs, with transversal magnetic field (hz ¼ 0). Each series of modes is ruled by a transcendental equation that must be solved: Eq. (3) for TM modes and Eq. (4) for TE modes. The solutions consist on a series of discrete wavelengths, which correspond to the different radial orders l of each mth value. With these values, it is possible to calculate the effective indices of each WGM resonance using Eq. (1):

$$n\_{1\pm} \frac{J\_{m'}(k\_0 n\_{1\mathbf{z}} \mathbf{a})}{J\_m(k\_0 n\_{1\mathbf{z}} \mathbf{a})} = n\_2 \frac{H\_m^{(2)'}(k\_0 n\_{2\mathbf{z}} \mathbf{a})}{H\_m^{(2)}(k\_0 n\_{2\mathbf{z}} \mathbf{a})} \tag{3}$$

$$\frac{1}{n\_{1\mathfrak{t}}} \frac{J\_{m'}(k\_0 n\_{\mathfrak{t}} \mathfrak{a})}{J\_m(k\_0 n\_{\mathfrak{t}} \mathfrak{a})} = \frac{1}{n\_2} \frac{H\_m^{(2)'}(k\_0 n\_{\mathfrak{t}} \mathfrak{a})}{H\_m^{(2)}(k\_0 n\_{\mathfrak{t}} \mathfrak{a})} \tag{4}$$

Figure 1.

(a) Scheme of the WGM propagating azimuthally in the MR. (b) Cylindrical system of coordinates which shows the two regions considered in the problem.

plane electromagnetic waves dispersed by spheres with diameters of the same size as the optical wavelength [2]. Shortly after, Debye stablished the equations for the optical resonances of dielectric and metallic spheres based on Mie's dispersion theory [3]. The detailed study of the mathematical equations of WGMs was

performed by Richtmyer [4] and Stratton [5], who predicted high-quality factors Q for these resonances and led to its implementation in different technologies based on microwave and acoustic waves. In the microscopic world, light can be guided by the same mechanism, when the resonator has dimensions of tens to hundreds of microns, and the wavelength of the light is in the visible-infrared range. In 1989, Braginsky et al. set the beginning of the optical WGMs when reporting the technique to excite optical modes in microresonators with spherical shape [6]. Since then, many researchers have studied the propagation of WGMs in structures with different symmetries [7] and have reported efficient methods based on microtapers

Due to the intrinsic low losses, WGMs show very high Q factors. For example, they can achieve values of 10<sup>10</sup> in spheres [9], 10<sup>8</sup> in silicon microtoroids [10], or

–10<sup>7</sup> in cylindrical microresonators [11]. At the resonance, the light guided by a WGM is recirculated in the microresonator many times, which provides a mechanism for decreasing the detection limit of the sensors based on them. This enhanced detection limit has been demonstrated to be low enough to measure a single mole-

WGM resonances shift in wavelength as the refractive index of the external medium changes. The sensitivity of WGMs as a function of these variations is significant: when considering a silica-cylindrical microresonator of 125 μm in diameter, immersed in water (n ¼ 1:33), the calculated shift in wavelength of the resonance is 77 nm/RIU. For a typical resonance width of 0.5 pm, this leads to a

detection limit of 6 � <sup>10</sup>�<sup>6</sup> RIU. It is worth to note that the light guided by WGMs is mainly confined in the microresonator. Thus, their sensitivity to variations of the material refractive index will be even higher. For example, it can achieve values as high as 1.1 μm/RIU when considering variations of the refractive index of the silica. In this example, the detection limit of the WGM decreases down to 4 � <sup>10</sup>�<sup>7</sup> RIU. In this chapter, we will report the use of WGMs in silica, cylindrical microresonators (an optical fiber) to measure and characterize the properties of the microresonator itself. There are a number of parameters, such as temperature or strain, which modify the refractive index of the material. Thus, this technique allows measuring with accuracy variations of temperature in doped optical fibers, in optical devices as fiber Bragg gratings (FBG), the elasto-optic coefficients of conventional silica fibers, and the absorption coefficient of photosensitive optical fibers, for example. We will report here the fundamentals of the technique, as well as the experimental

The guiding mechanism of WGMs in the azimuthal direction of a

microresonator (MR) is total internal reflection, just as in the case of axial propagation in a conventional waveguide; see Figure 1a. Resonance occurs when the guided wave travels along the perimeter of the MR, and it drives itself coherently by returning in phase after every revolution. In its way, the wave follows continuously the surface of the MR, and the optical path in a circumnavigation must be equal to an integer multiple of the optical wavelength, λ. When this condition is fulfilled, resonances appear, and a series of discrete modes at specific wavelengths will show

to excite these modes in the optical range [8].

Applications of Optical Fibers for Sensing

cule on the surface of a microtoroid [12].

results we obtained for these experiments.

up. The resonant condition can be written as [13]

2. Fundamentals

138

10<sup>6</sup>

In Eqs. (3) and (4), k<sup>0</sup> is the wavenumber in vacuum, k<sup>0</sup> ¼ 2π=λ, Jm is the Bessel function of order m and J 0 <sup>m</sup> is its first derivative, Hð Þ<sup>2</sup> <sup>m</sup> is the second class Hankel function of order m, and Hð Þ<sup>2</sup> <sup>0</sup> <sup>m</sup> is its first derivative. We have considered that the external medium does not present any anisotropy (in our case, it will be air).

Regarding the distribution of the fields, Figure 3a shows the amplitude of the electric field of the first radial order TM-WGM, propagating in a cylindrical, silica MR of 10 μm diameter (the order m of the mode is 40; a low-order mode was considered in order to show the details of the field). As it can be observed, the field is well confined within the MR material (although its evanescent field is high enough to enable the use of these modes for sensing). As the azimuthal order of the WGM gets higher, the field will be more localized near the interface between the MR and the external medium. Also, it should be noted that, as the radial order of the WGM increases, the evanescent tail in the outer medium is larger; thus the quality factor of the correspondent resonance will be poorer. Figure 3b shows the field amplitude along the radial coordinate of the MR. As it can be observed, the optical power is localized in the outer region of the MR, near the interface, and shows a

Whispering Gallery Modes for Accurate Characterization of Optical Fibers' Parameters

low-evanescent field in the outer medium, especially for the l ¼ 1 mode.

ing the WGM resonances in reflection by means of a photodetector (PD).

(a) Scheme of the experimental setup. (b) Typical reflection spectrum of a WGM.

The MR will consist on a section of the bare optical fiber under test (FUT). Depending on the experiment, it will be a conventional telecom fiber, a rare-earth doped fiber, a photosensitive fiber, or a fiber where a grating has been previously inscribed. It is carefully cleaned and mounted on a three-axis flexure stage. WGMs are excited around the FUT by using the evanescent optical field of an auxiliary microtaper with a waist of 1–2 μm in diameter and a few millimeters in length. This is not the only method that allows exciting WGMs in MRs: for example, one of the first techniques consisted on using a prism to excite the resonances in a spherical MR [15], but the efficiency was very poor. More recently, a fused-tapered fiber tip fabricated using a conventional fiber splicer was demonstrated to be capable of exciting WGMs in a cylindrical MR [16]. However, the highest efficiencies are achieved by using microtapers, with coupling efficiencies higher than 99% [8]. These microtapers are fabricated by the fuse-and-pull technique from conventional

The general setup used in the experiments is shown in Figure 4a. The light source is a tunable diode, linearly polarized laser (TDL) with a narrow linewidth (<300 kHz). The tuning range covers from 1515 to 1545 nm. The laser integrated a piezoelectric-based fine frequency tuning facility that allows continuous scanning of the emitted signal around a given wavelength, with subpicometer resolution. A polarization controller (PC) after the laser allows rotating the polarization of the light, and, as a consequence, it allows exciting TE- and TM-WGMs separately. The optical signal is then launched through an optical circulator, which enables measur-

3. Experimental setup

DOI: http://dx.doi.org/10.5772/intechopen.81259

Figure 4.

141

By following this procedure, it is possible to calculate the dispersion curves of several WGMs propagating in a cylindrical, silica MR of 125 μm diameter (the parameters of conventional optical fibers). Sellmeier dispersion of the silica was taken into account for the refractive index of the material. It is worth to note that the dispersion curves are not truly a curve, but a series of discrete solutions that have a particular radial order l and azimuthal order m. For a standard optical fiber and 1550 nm optical wavelength, the azimuthal orders will be relatively high (m � 300). Figure 2 shows the calculations of the resonant wavelengths for the first radial orders, as a function of the azimuthal order m for the TM polarization. The curves for the TE polarization follow the same trend, but the values of the resonant wavelengths are slightly different. By using Eq. (1), it is possible to relate the resonant wavelength with the effective index of the WGM resonance. For the azimuthal order m ¼ 360 and the first radial order, l ¼ 1, the resonant wavelengths and the effective indices of both polarization families are λTM <sup>R</sup> ¼ 1508:25 nm, nTM eff <sup>¼</sup> <sup>1</sup>:3826 and <sup>λ</sup>TE <sup>R</sup> <sup>¼</sup> <sup>1505</sup>:39 nm, <sup>n</sup>TE eff ¼ 1:3800. Thus, the resonances for each polarization are not overlapped in wavelength.

### Figure 2.

Resonant wavelength of WGMs with azimuthal orders from 250 to 370. Only a selection of the solutions to highlight their discrete nature is shown.

Figure 3.

(a) Optical field of a m ¼ 40 and l ¼ 1 WGM in a silica, cylindrical MR. (b) Field amplitude of the WGM as a function of the radial coordinate, for m ¼ 40 and l ¼ 1; 2; 3.

Whispering Gallery Modes for Accurate Characterization of Optical Fibers' Parameters DOI: http://dx.doi.org/10.5772/intechopen.81259

Regarding the distribution of the fields, Figure 3a shows the amplitude of the electric field of the first radial order TM-WGM, propagating in a cylindrical, silica MR of 10 μm diameter (the order m of the mode is 40; a low-order mode was considered in order to show the details of the field). As it can be observed, the field is well confined within the MR material (although its evanescent field is high enough to enable the use of these modes for sensing). As the azimuthal order of the WGM gets higher, the field will be more localized near the interface between the MR and the external medium. Also, it should be noted that, as the radial order of the WGM increases, the evanescent tail in the outer medium is larger; thus the quality factor of the correspondent resonance will be poorer. Figure 3b shows the field amplitude along the radial coordinate of the MR. As it can be observed, the optical power is localized in the outer region of the MR, near the interface, and shows a low-evanescent field in the outer medium, especially for the l ¼ 1 mode.
