A study on refractive index sensors based on optical micro-ring resonators

In this work the behavior of optical micro-ring resonators, especially when functioning as refractive index sensors, is studied in detail. Two configurations are considered, namely a linear waveguide coupled to a circular one and two linear waveguides coupled to each other through a circular one. The optimum coupling conditions are derived and it is shown that in both cases the condition for the resonant wavelength, i.e. the wavelength at which the transmission spectrum exhibits a dip (peak), is the same and depends only on the geometrical characteristics of the circular waveguide and the effective refractive index of the propagating mode. The latter, as well as the corresponding mode profile, can be easily calculated through numerical analysis. The sensitivity of the sensor is defined based on the dependence of the effective refractive index on the refractive index of the environment. Using a result of waveguide perturbation theory, the geometrical characteristics of the core of the circular waveguide that maximize the sensitivity of the system are determined. Both single and dual core configurations are considered. It is found that, when optimally designed, the sensor can detect relative refractive index changes of the order of 10^-4, assuming that the experimental setup can detect relative wavelength shifts of the order of 3x10^-5. Finally, the behavior of the system as bio-sensor is examined by considering that a thin layer of bio-material is attached on the surface of the waveguide core. It is found that, when optimally designed, the system can detect refractive index changes of the order of 10^-3 for a layer thickness of 10 nm, and changes in the layer thickness of the order of 0.24 nm, for a refractive index change of 0.05.


Introduction and theoretical analysis
Optical ring resonators are interesting optical devices with a plethora of applications especially in optical switching [1][2][3][4], routing [4][5][6][7], and sensing [8][9][10]. An optical ring resonator usually consists of a straight waveguide coupled to a circular one, as shown in figure 1(a), or two straight waveguides coupled through a circular one, as shown in figure 1(b). In the first case the transmittance is given by the formula [11] 22 22 2 cos 1 2 cos n a r ar T r a ar      (1) where r is the self-coupling coefficient of light between the straight and circular waveguide, and a is a loss parameter related to the power attenuation coefficient It is obvious that 0 n dT dy  since both , ar are less or equal to unity. Thus, the transmittance is minimized for 1 y  , which implies that the condition for minimum transmittance as far as the phase shift is concerned is where m is an integer. Through equations (2), (5) which implies that for ar  , critical coupling, the transmittance is zero.
As far as the arrangement shown in fig. 1(b) is concerned, the transmittance through the lower and upper waveguides are given by the formulae It is obvious that for 1 a  , 1 d T  . Thus, in the case of a symmetrical optical ring resonator, shown in figure 1(b), the light is perfectly coupled to the upper waveguide, when the wavelength is given by equation (6), the loss parameter is equal to unity   1 a  , and the coupling coefficients are equal to each other   12 rr  . As far as the use of micro-ring resonators as refractive index sensors is concerned, the most important result of the above analysis is that, in both configurations, the resonant wavelength is given by equation (6), which practically implies that it is proportional to the effective refractive index of the propagating mode in the circular waveguide. However, the value of eff n depends on the refractive index of the environment. Therefore, the system can function as a refractive index sensor. In more detail, supposing that the effective refractive index changes by eff n  due to a change of the refractive index of the environment by n  , equation (6) Consequently, the relative change in the resonant wavelength is equal to the relative change in the effective refractive index.
According to the above analysis, the most important parameter regarding the performance of an optical ring resonator when used as a refractive index sensor is the dependence of the effective refractive index of the propagating mode on the refractive index of the environment. Mathematically, this is described by the quantity  (13) where in the second form of equation (13) it is supposed that the refractive index change of the environment is sufficiently small, so that the dependence of eff n  on n  is linear. Using the parameter S , equation (12)  From equation (14) it is obvious that S can be regarded as a measure of the sensitivity of the system when used as a refractive index sensor. In practice, the parameter S can be numerically calculated quite easily as described in the following section.

Numerical analysis
The profile and effective refractive indexes of the propagating modes in the circular waveguide are calculated using finite element analysis [12,13]. The simulations were assisted by the Comsol Multiphysics software package [14]. The analysis is made on a two dimensional cross-section of the waveguide, assuming that its profile and properties remain constant throughout its length. First, the electromagnetic field is written in the form Here are matrix differential operators with respect to   xy  for the electric and magnetic field respectively, and , , , , , Thus, solving equation (17)  It should be noted that in most applications of practical interest, single mode operation is preferable, because the existence of more than one propagating modes would complicate the behavior of the system. For example, in the case of multimode operation, equation (6) implies that more than one resonant wavelengths exist for each configuration, which would result to multiple dips (or peaks, if measured through the upper waveguide) in the transmittance spectrum, complicating both the theoretical and experimental analysis of the system. Therefore, in the simulations, special care is taken in order to ensure single mode operation.

Simulations, results and discussion regarding design optimization
According to a result of waveguide perturbation theory [15], the change in the effective refractive index of the propagating mode due to a refractive index change in some part of the waveguide is proportional to the fraction of the mode power in this specific part of the waveguide, defined as  (18) where p S is the area where the refractive index has changed and tot S is the total area of the waveguide. Further, according to equation (13) the sensitivity S is proportional to eff n  and consequently it will also be proportional to p  . This practically means that the sensitivity increases as the overlapping of the propagating mode with its environment becomes more extensive.
Therefore in the simulations the core of the waveguide is considered to be in direct contact with the environment. Further the core dimensions are chosen to be small, close to the limit of waveguiding, in order to increase the mode spreading and maximize the parameter p  . The cross section of the core of the waveguide was considered to be rectangular, as shown in figure 2, where the mode profile is also depicted, for two different heights of the waveguide core. The material of the core was chosen to be silicon nitride (SiN), which has the advantage that its refractive index is relatively insensitive to temperature fluctuations [16,17]. It is supposed that the core is developed on a layer of SiO 2 , as usually happens in practice. The environment was chosen to be water, which is the most common solvent, and it is also suitable for biological applications. The free space wavelength was supposed to be 0 1550 nm   , which is a common wavelength emitted by fiber lasers, and further corresponds to the minimum of the fiber losses [18].    . It should be noted that although these optimal values refer to an incident wavelength of1550 nm , they can be used as guidelines for other wavelengths. This is due to the fact that the Maxwell equations are invariant under scale transformations and consequently the behavior of the system does not change if its dimensions are scaled according to the wavelength ratio. For example, if the wavelength is multiplied by a factor x and all the dimensions of the waveguide are multiplied by the same factor, then its behavior will remain unchanged. Of course, in practice this cannot be achieved because the optical properties of the materials change due to dispersion. However, in any case, these values can be used as good starting points for design optimization. Next, the sensitivity is calculated for the three optimal configurations, specifically  Table 1. It should be noted that the gain in the sensitivity for a dual core design is not that significant in order to justify the complexity of the design and the manufacturing challenges that entails. Therefore, I guess that in most practical situations, the design of choice would be a single core waveguide with dimensions 0.175 for the core dimensions provides better sensitivity than the dual core design as far as the behavior of the system as biosensor is considered.

Study of the bio-sensing properties of the system
In the following, the behavior of the system as biosensor is considered. For this purpose, a thin layer of biomaterial is supposed to be attached on the surface of the waveguide core. The dependence of the parameter p  , namely the fraction of the mode power propagating within the layer, as function of the layer thickness is shown in figure 7. It is clear that p  , and consequently the sensitivity, increases linearly for small values of the layer thickness, but exhibits some saturation as t increases beyond 40 nm . It should also be noted that in this case the sensitivity of the single core waveguide with height 32 env h   is greater than that of the dual core one.
In order to study the sensing ability of the system as the refractive index of the biolayer changes, the relative change in the effective refractive index has been calculated as function of the effective index difference of the layer material, for a layer thickness of for a layer thickness of 10 nm , which is reduced by a factor of almost two when the layer thickness is doubled. Consequently, the system is able to detect very subtle changes in the composition or in the environment of the bio-layer.  . This impressive result leads to the conclusion that the system can detect even the slightest changes in the configuration of the biomolecules.

Conclusions
In this work the behavior of optical micro-ring resonators when functioning as refractive index sensors has been studied in detail. The sensitivity of the sensors has been defined and design guidelines in order to maximize its value are provided. It is found that, when optimally designed, the system can detect relative refractive index changes of the order of 4 10 nn   , assuming that the experimental setup is able to resolve relative wavelength shifts of the order of of the systems as bio-sensor has also been examined. It is found that, when optimally designed, the system can detect refractive index changes of the order of