WGM frequencies in an optical fibre, or indeed any cylindrical resonator, are reasonably well approximated by matching an integral number of wavelengths to the resonator circumference, i.e.
is an integer (the azimuthal mode number, discussed further below), λ
is the free space wavelength and ν
the frequency of the light used to excite the resonator, n
is the refractive index of the resonator, and d
is the resonator diameter. This treatment becomes more accurate as m
increases and the path taken by the light within the resonator becomes more closely approximated by a circle. A key result is that both the WGM frequencies and the frequency spacing between adjacent modes are inversely proportional to the resonator diameter. However, in reality the true path is ‘polygonal’ in nature, such that the path length is shorter, and therefore the resonant frequency higher, than predicted by this simple model.
The true mode frequencies within a cylindrical resonator are eigenvalues of the appropriate wave equation. The solutions to such an equation take the form of Bessel functions (also known as cylindrical harmonics) and the whispering gallery modes are defined by three mode numbers: the radial mode number, l
; the azimuthal mode number, m
; and the slab mode number, p
]. A WGM described by mode numbers m
field maxima around the cylinder azimuth (i.e.
, around the circumference of the cylinder), l
− 1 nodes along the radial coordinate of the cylinder, and p
nodes along the cylinder axis. Outside the resonator, the field decays exponentially with a decay constant d
of approximately [26
is the wavelength, na
is the refractive index of the resonator, nb
is the refractive index of the surroundings, and θ
is the angle at which total internal reflection takes place at the internal surface of the resonator. We see that the evanescent field from the resonator decays over a distance scale on the order of a wavelength. This has the important consequence that WGM sensing techniques are sensitive only to chemical species in close proximity to the sensor surface.
The measure of an optical resonator or cavity’s ability to confine light of a given frequency ν0
is usually expressed in terms of a ‘quality factor’ or ‘Q-factor’, Q
are the power stored within the cavity and lost from the cavity, respectively, and δν
is the spectral linewidth of the cavity mode at frequency ν0
. Gorodetsky et al.
] have calculated that the ultimate Q
of a perfect microsphere resonator is around 1010
. Experimentally measured Q
factors are generally much lower than this due to scattering from surface roughness and impurities, absorption by the material, and simultaneous excitation of multiple closely spaced modes. Cylindrical resonators have been characterised with Q
factors of the order of 105
or higher [28
The response of the whispering gallery mode spectrum to molecular binding at the resonator surface has been explained in terms of a change in optical path length caused by direct interactions between the resonant photons and the bound molecules. When a molecule binds to the resonator surface, it displaces water and creates an excess dipole moment at the binding point. Arnold et al
] considered the interaction energy between this excess dipole and the evanescent field of the WGM resonances, and converted the resulting energy loss from the WGM field into a fractional shift in the corresponding resonance frequency. According to this model, the shift δν
associated with binding of N
randomly located molecules to the resonator surface is given by
is the excess polarisability, σp
is the surface density of adsorbed molecules, E0
) is the electric field at position r
is the permittivity of free space, and rs
is the relative permittivity of the resonator. The integrals are over the surface area and volume of the whispering gallery mode, with the volume integral yielding the total energy of the mode within the resonator. Of key importance is that the fractional shift on binding depends on both the fraction of the mode energy found at the positions of the bound molecules, and on the excess polarisability of the molecules. For proteins, αex
is approximately proportional to the molecular mass, such that larger shifts are predicted for heavier molecules.
White and Fan [29
] have generalised this result. For a sensor with a sensitivity S
to changes in the refractive index of the bulk, where δλ
is the observed wavelength shift accompanying a change δn
in refractive index, the fractional wavelength shift on molecular binding to the sensor surface is
is the refractive index of the sensor material and ns
is the refractive index of the solvent. This expression allows the surface density of detected molecules to be determined from the measured wavelength shift.
The detection limit is highly sensitive to the Q
factor of the cavity, since a higher Q
leads to sharper cavity mode spectra and detection of smaller shifts in resonance frequency. A consequence of this is that the detection limit of a fibre-based whispering gallery mode sensor is likely to be somewhat lower than for microsphere-based sensors, due to the lower Q factor of cylindrical relative to spherical cavities. However, there are a number of potential advantages that make fibre-based sensors worth exploring. Some of these have been pointed out previously by Farca et al.
]. Optical fibres are highly uniform in diameter, allowing large numbers of identical resonators to be fabricated and providing a high degree of repeatability. An additional advantage is the straightforward optical setup for fibre-based experiments. Fibres are easily mounted, and alignment for optimal coupling of the excitation light into the fibre WGMs depends on only one angular degree of freedom, as opposed to two for experiments involving microspheres.