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

In this work, the behavior of refractive index sensors based on optical micro-ring resonators is studied in detail. Using a result of waveguide perturbation theory in combination with numerical simulations, the optimum design parameters of the system, maximizing the sensitivity of the sensor, are determined. It is found that, when optimally designed, the sensor can detect relative refractive index changes of the order of Δn/n≈3×10−4, assuming that the experimental setup can detect relative wavelength shifts of the order of Δλ/λ≈3×10−5. The behavior of the system 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 Δn≈10−3 for a layer thickness of t=10 nm, and changes in the layer thickness of the order of λt≈0.24 nm, for a refractive index change of Δn=0.05.

In the first case the transmittance is given by the formula [ (1) where r is the self-coupling coefficient of light between the straight and circular waveguides, and a is a loss parameter related to the power attenuation coefficient  through the equation a 2 =exp(L), with L being the length of the circular waveguide.
Here,  is the single-pass phase shift, defined as eff 0 2 n L L       (2) where  is the propagation constan, ff e n the effective refractive index of the propagating mode in the circular waveguide, and 0  the free space wavelength. Setting cos  y , (1) takes the form: The derivative of n T with respect to y is as follows: It is obvious that dT n / dy0 since both a and r are less than or equal to unity. Thus, the transmittance is minimized for y=1, which implies that the condition for minimum transmittance, as far as the phase shift is concerned, is cos 1 where m is an integer. From (2) and (5) (6) where R is the radius of the circular waveguide. In this case the transmittance takes the form which implies that for  a r , 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 respectively. Here 1 r and 2 r are the coupling coefficients between the lower-circular and uppercircular waveguides respectively and cos  y . Following a similar analysis it can be shown that the condition for the minimum transmittance through the lower waveguide, and simultaneously the maximum transmittance through the upper waveguide, is described again by (6). In this case (8a) and (8b)  . Thus, in the case of a symmetrical optical ring resonator, shown in Fig. 1(b), the light is perfectly coupled to the upper waveguide, when the wavelength is given by (6), the loss parameter is equal to unity   1  a , and the coupling coefficients are equal to each other 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 (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 in the refractive index of the environment by n , (6) takes the form (11) where 0,res   is the corresponding change in the resonant wavelength. Dividing (11) and (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 as where in the second form of (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 , (12) takes the form: From (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 the finite element analysis [12]. The simulations were assisted by the Comsol Multiphysics software package. The analysis is made on a two dimensional cross-section of the waveguide, assuming that its profile and properties remain constant throughout its length. Firstly, the electromagnetic field is written in the form where ( , , ) E x y z and ( , , ) H x y z are the electric and magnetic field components of the propagating wave, respectively. The two-dimensional functions ( , ) A x y and ( , ) B x y describe the mode profile, assuming that the electromagnetic wave is propagating in the z direction. Here 0  is the linear loss coefficient and  is the propagation constant, related to the effective refractive index through the formula as 0 eff The parameter  corresponds to the generalized propagation constant, defined as Writing the Maxwell equations in matrix form and eliminating the longitudinal field components, an eigenvalue problem of the form  It should be noted that in most applications of practical interest, the 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 multi-mode operation, (6) implies that more than one resonant wavelengths exist for each configuration, which would result in 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 [13], 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 (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 is considered to be rectangular, as shown in Fig. 2, where the mode profile is also depicted, for two different heights of the waveguide core. The material of the core is chosen to be silicon nitride (SiN), which has the advantage that its refractive index is relatively insensitive to temperature fluctuations [14,15]. It is supposed that the core is developed on a layer of SiO 2 , as usually happens in practice. The environment is chosen to be water, which is the most common solvent, and it is also suitable for biological applications. Thus, the simulation model consists of a vertical column of SiN with small dimensions (the core of the waveguide) developed on a horizontal layer of SiO 2 , while all the system is submersed in water. The free space wavelength is 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 [16].
The  . The results are shown in Fig. 4 A double core configuration has also been studied, as shown in Fig. 5, where the mode profile is depicted. In this case, the dimensions of each core are chosen to be . Higher values of h have not been considered because the system exhibits multi-mode operation. In order to find the optimum distance between the fiber cores, the parameter  p has been calculated as function of the core separation d . The results are shown in Fig. 6.  . It should be noted that although these optimal values refer to an incident wavelength of 1 550 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 the design optimization. Next, the sensitivity is calculated for the three optimal configurations, specifically   3 2   h and 1.120 for a dual core configuration. This practically means that if the experimental setup is able to resolve a wavelength shift of the order of 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, it is suggested that in most practical situations, the design of choice would be a single core waveguide with dimensions 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 recent years, the use of photonic nano-sensors as bio-sensors is a hot topic of research, with a plethora of designs and applications [18][19][20][21][22][23]. Therefore, in the following, the behavior of the system described in the previous section 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 Fig. 7. It is assumed that the refractive index within the bio-layer increases by n=0.01 compared to the environment, although the parameter  p is not very sensitive to the refractrive index of the biolayer. It is clear that  p and consequently the sensitivity increase linearly for small values of the layer thickness but exhibit a trend for 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 h=3 env /2 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 bio-layer varies, the relative change in the effective refractive index has been calculated as function of the refractive index difference of the layer material. Two values of the layer thickness are considered, namely t=10 nm and t=20 nm. The results are shown in Fig. 8.
It is clear that the dependence is linear, and further the single core waveguide with height env 3 2   h exhibits the highest sensitivity, with a slope of 0.026 in the case of t=10 nm and 0.050 when the layer thickness doubles. This practically means that if the experimental setup can resolve a relative change in the resonant wavelength of the order of 5 3 10   , then the system can sense refractive index changes within the bio-layer of the order of 3 min 1.15 10 n     for a layer thickness of 10 nm, which is reduced by a factor of two when the layer thickness doubles. Consequently, the system is able to detect very subtle changes in the composition or in the environment of the bio-layer. Finally, the sensing ability of the system regarding the thickness of the bio-layer has been examined. For this purpose, the values of the relative change in the effective refractive index have been calculated as function of the layer thickness, supposing that the refractive index change within the layer is 0.01   n and 0.05   n . The results are shown in Fig. 9.
(a) (b) Fig. 8 Relative changes in the effective refractive index, and consequently in the resonant wavelength, as function of the refractive index change n in the bio-layer, when its thickness is (a) t=10 nm and (b) t=20 nm. It is clear that the dependence is again linear, and the highest sensitivity is exhibited by the env 3 2   h single core waveguide. Indeed, the slope in this case is 5  n . 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. Detailed guidelines for maximizing the sensitivity of the sensors are provided. It is found that, when optimally designed, the system can detect relative refractive index changes of the order of 4 10    n n , assuming that the experimental setup is able to resolve relative wavelength shifts of the order of 5 3 10       . The performance 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 n10 3 for a layer thickness of t=10 nm, and changes in the layer thickness of the order of t0.24 nm, for a refractive index change of n0.05.
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