1. Introduction

Whispering gallery modes are surface modes that propagate azimuthally around resonators with rotational symmetry, generally a dielectric. This phenomenon was first described by Lord Rayleigh in the nineteenth century, when studying the propagation of acoustic waves in interfaces with a curvature [1]. St. Paul's Cathedral (London, UK), the Temple of Heaven (Beijing, China), the Pantheon (Rome, Italy), the Tomb of Agamemnon (Mycenae, Greece), and the Whispering Gallery in the Alhambra (Granada, Spain) are examples of architectonical structures that support acoustic modes which propagate guided by the surface of the walls. It was at the beginning of the twentieth century when the study of this guiding mechanism was extended to the electromagnetic waves, since Mie developed his theory for the

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

<sup>λ</sup><sup>R</sup> <sup>¼</sup> <sup>2</sup>π<sup>a</sup> � <sup>n</sup>eff m

> n1<sup>t</sup> 0 0 0 n1<sup>t</sup> 0 0 0 n1<sup>z</sup>

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

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

Hð Þ<sup>2</sup> <sup>0</sup>

Hð Þ<sup>2</sup>

Hð Þ<sup>2</sup> <sup>0</sup>

Hð Þ<sup>2</sup> <sup>m</sup> ð Þ k0n2a

n2

(a) Scheme of the WGM propagating azimuthally in the MR. (b) Cylindrical system of coordinates which

<sup>m</sup> ð Þ k0n2a

<sup>m</sup> ð Þ k0n2a

<sup>m</sup> ð Þ k0n2a

1

CA (2)

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 refrac-

Whispering Gallery Modes for Accurate Characterization of Optical Fibers' Parameters

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

n<sup>2</sup> ¼

calculate the effective indices of each WGM resonance using Eq. (1):

Jm0ð Þ k0n1<sup>z</sup>a Jmð Þ <sup>k</sup>0n1<sup>z</sup><sup>a</sup> <sup>¼</sup> <sup>n</sup><sup>2</sup>

Jm0ð Þ k0n1<sup>t</sup>a Jmð Þ <sup>k</sup>0n1<sup>t</sup><sup>a</sup> <sup>¼</sup> <sup>1</sup>

n1<sup>z</sup>

1 n1<sup>t</sup> 0

B@

tive index):

WGMs.

Figure 1.

139

shows the two regions considered in the problem.

(1)

(3)

(4)

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>6</sup> –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 molecule on the surface of a microtoroid [12].

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 results we obtained for these experiments.
