## Abstract

The sensitivity of the resonant optical sensors, which are based on measurement of the transmission and reflection spectra of optical resonators, is investigated. The following problem is addressed: When the losses of the resonator are known, what is the sharpest possible and the steepest possible shape of the resonant peaks that can be achieved experimentally? This optimization problem is solved for the case of a separated peak, which corresponds to a nondegenerated eigenvalue of the resonator. It is shown that the reflection spectrum possesses better sensitivity than the transmission spectrum. The model of the resonant sensor consisting of two coupled resonators is also considered. This model demonstrates that the sensitivity of transmission spectrum can be significantly increased by modification of the resonator structure. However, for the reflection spectrum, the best sensitivity is still given by a separated resonant peak.

©2007 Optical Society of America

## 1. Introduction

A resonant optical sensor (ROS) is a sensor based on optical resonators such as Fabry-Perot [1,2], sphere [3,4], ring [5–9], disk [10,11], and others. Usually, the ROS spectrum consists of positive and/or negative resonant peaks. The ROS monitors changes in the tested object by measuring and processing variations in the ROS spectrum. It is often necessary to detect very small variations of the spectrum, comparable with or much smaller than the dimensions of a resonant peak. In order to magnify the ROS response and to increase the accuracy, it is desirable to have the resonant peak as sharp and/or as steep as possible. However, the FWHM of a resonant peak is restricted by the losses of the resonator. In fact, at wavelength, λ, the FWHM of a resonant peak, Δ*λ*
_{FWHM}, cannot be smaller than λ/*Q*
* _{int}*, where the intrinsic Q-factor,

*Q*

*, is determined by the internal losses of the ROS. This paper addresses the following question:*

_{int}*When the losses of the resonator are known, what is the sharpest possible and the steepest possible shape of the resonant peaks that can be achieved experimentally?*To the best of the author’s knowledge, this problem has not been addressed until recently [12]. The optimization problem is formulated in Section 2. The approach for optimization of ROSs is based on the model of coupled resonators. Within this model, the transmission and reflection spectrum can be calculated by the generalized Breit-Wigner formula [13–16], which is described in Section 3. Section 4 solves the optimization problem for the case of separated peaks in the ROS spectrum, which correspond to a nondegenerated eigenvalue of an ROS. It is shown that the reflection spectrum provides a better sensitivity than the transmission spectrum. In some instances, the distance between resonant peaks in the ROS spectrum can be comparable or smaller than the width of an individual peak. It is interesting to know if the sharpness and steepness of the ROS spectrum could be increased by increasing the number of coupled resonators. Section 5 addresses this problem for an ROS consisting of two coupled resonators. Optimization of such ROS is done numerically. It is shown that the sensitivity of transmission spectrum can be significantly increased by modification of an ROS. However, for the reflection spectrum, the best sensitivity is still given by a separated resonant peak.

## 2. Principles of ROS optimization

Optimization of the ROS parameters is performed in order to determine the maximum possible slope and/or sharpness of resonant peaks. The shape of resonances in the ROS spectrum depends on the wavelength of light, *λ*, as well as on the other parameters of the ROS, *γ*=(*γ*
_{1},*γ*
_{2},..., *γ*
* _{L}*). In practice, not all of these parameters can vary. Some of the parameters are predetermined by the material properties. Other parameters can vary within certain limits determined by geometrical conditions, fabrication quality, etc. Thus, the monitored profile of the resonant spectrum should be optimized by varying the parameters that can be changed in the actual design and fabrication of an ROS. Therefore, the set of ROS parameters,

*γ*, should be divided into the subset of variable parameters,

*γ*

*, and the subset of parameters, which should be considered as constants,*

_{var}*γ*

*:*

_{const}*γ*=(

*γ*

*,*

_{var}*γ*

*). In the case, when one is looking for the maximum possible slope,*

_{const}*S*

*, of an ROS spectrum,*

_{max}*P*(

*λ*,

*γ*), the optimization problem is written in the form:

Similarly, the sharpest peak in the ROS spectrum, Θ* _{max}*, is found from the solution of the optimization problem:

This paper considers a model of an ROS, which is described by parameters *γ*
* _{var}* that can vary with geometric configuration of the ROS and also by parameters

*γ*

*that are predetermined by the internal losses of the ROS. The input power and accuracy of measurement of the transmission and reflection powers are assumed to be constant. The transmission and reflection power spectra of an ROS are expressed as a function of these parameters using the generalized Breit-Wigner formula considered in Section 3.*

_{const}As opposed to formulation of the optimization problem with Eq. (1) or (2), it is commonly accepted that the most appropriate resonances for sensing applications are those that have a maximum possible Q-factor. The latter is not entirely correct for two reasons. First, as it is noted in section 4, the maximum Q-factor of a separated resonance corresponds to a resonance peak with vanishing height, useless for sensing. Secondly, a resonant peak composed of several adjacent peaks may have a complex shape, for which the Q-factor cannot be introduced at all.

## 3. Transmission and reflection of an ROS

Consider a case of a well pronounced resonance behavior of transmission or reflection spectrum of the ROS. The spectral resonances are formed by scattering of light at a single or a few adjacent eigenmodes of this resonator. The resonator is modeled as a set of single-mode resonators with wavelength eigenvalues *λ*
* _{n}*. These resonators are coupled to each other, to the input and output waveguide, and to the environment as illustrated in Fig. 1. The coupling coefficient between resonators

*n*and

*m*is

*δ*

*and the coupling of a resonator*

_{mn}*n*to an input/output waveguide,

*k*, is defined by the transmission coefficient

*γ*

^{(k)}

*. The internal losses are modeled using the virtual output waveguides, such as the vertically-directed waveguides shown in Fig. 1(a). The resonant transmission and reflection spectrum of such optical resonator can be calculated using the generalized Breit-Wigner formula [13–16]. In particular, the resonant transmission power from the input waveguide p to the output waveguide*

_{n}*q*is:

Here *P*
^{(0)}
* _{p}* is the power input into the waveguide

*q*and the sum is taken over all resonators, which couple to the waveguides

*p*and

*q*, and Parameters

*γ*

*in Eq. (5) determine the widths of the uncoupled eigenvalues*

_{m}*λ*

*:*

_{m}The reflection power into waveguide *p*, *P*
* _{pp}*, can be found from the power conservation law:

## 4. Optimization of the transmission and reflection spectrum for a separated resonance

Figure 1(b) illustrates the model of transmission through a resonator eigenmode. The inset in this Fig. shows a realistic example that can be investigated using this model: a photonic crystal waveguide with a built-in microresonator (see e.g. [17,18]). From Eq. (3)–(6), the transmission pick is described by the Breit-Wigner formula [19,13]:

where the internal loss of the resonator, *γ*=*γ*
^{(3)}
_{1} is introduced. With Eq. (8) and (7), the reflected power is determined in the form:

Note that Eq. (9) is known as the expression for a negative transmission peak in the theory of a ring resonator (see e.g. [12]). The analogy between the reflection specrum defined by Eq.(9) and transmission spectrum of a ring resonator becomes more obvious if the output waveguide is absent, *γ*
^{(2)}
_{1}=0, and the transmitted light in a ring resonator is conceived as the unfolded reflected light in our resonator.

The Q-factor of the transmission and reflection resonances is

and it cannot exceed the intrinsic Q-factor:

The smallest FWHM of the resonances, Δλ* _{FWHM}*, is achieved for the largest possible Q-factor,

*Q*=

*Q*

*, which corresponds to*

_{int}*γ*

^{(1)}

_{1}=

*γ*

^{(2)}

_{1}=0. In this case,

*P*

_{12}=0 and

*R*

_{11}=1, i.e. the peak is absent and no sensing is possible (column 1 in Table 1 (a) and (b)). Alternatively, the relative height of the resonance achieves its maximum equal to 1 when max(

*γ*

^{(1)}

_{1},

*γ*

^{(2)}

_{1})=∞ and

*Q*=0. Then

*P*

_{12}=1 and

*P*

_{11}=0, i.e. the peak becomes infinitely broad and, again, no sensing is possible (see column 2 in Table 1 (a) and (b)). Thus, the maximum slope and sharpness of the resonance is achieved at an intermediate value of Q-factor, 0<

*Q*<

*Q*

*. Eq. (8) and (9) allow simple solution of the optimization problem. The parameters λ*

_{int}_{1},

*λ*

^{(1)}

_{1},

*γ*

^{(2)}

_{1}, and are variable because they can be modified by changing the geometry of the resonator and, in particular, by changing the distance between the input/output waveguides and the cavity. The parameter

*γ*is a constant because it determines the internal loss of the ROS. The following calculations allow to find the optimized transmission and reflection resonant peaks depicted in Fig. 2(a) and (b), respectively. The sharpest peak in transmission is achieved at ${\gamma}_{1}^{\left(1\right)}={\gamma}_{1}^{\left(2\right)}=\frac{1}{2}\gamma $, which correspond to $Q=\frac{2}{3}{Q}_{\mathrm{int}}$ and the relative peak height equal to 1/4 (column 3 in Table 1 (a)). The steepest slope in transmission is achieved at $\lambda ={\lambda}_{1}\pm \frac{{3}^{\frac{1}{2}}}{2}\gamma $ and γ(1)1=γ(2)1=γ, which corresponds to $Q=\frac{1}{2}{Q}_{\mathrm{int}}$ and the relative peak height equal to 4/9 (column 4 in Table 1 (a)). The sharpest peak in reflection is achieved at γ(2)1=0, ${\gamma}_{1}^{\left(1\right)}=\frac{1}{3}\gamma $, which correspond to $Q=\frac{3}{4}{Q}_{\mathrm{int}}$ and the relative peak height equal to 3/4 (column 3 in Table 1 (b)). The steepest slope in reflection is achieved at $\lambda ={\lambda}_{1}\pm \frac{{3}^{\frac{1}{2}}}{4}\gamma $, γ(2)1=0, and ${\gamma}_{1}^{\left(1\right)}=\frac{1}{2}\gamma $, which corresponds to $Q=\frac{2}{3}{Q}_{\mathrm{int}}$ and the relative peak height equal to 8/9 (column 4 in Table 1 (b)).

It is also interesting to compare the values of maximum possible slope and sharpness of resonance that can be achieved in transmission and in reflection. For transmission:

For reflection:

Comparison of Eqs. (12), (13) and Eqs. (14), (15) shows that sensing in reflection is more favorable than sensing in transmission. The maximum possible steepness and sharpness of the reflection peak is, respectively, 4 and 6.75 times greater than those of the transmission peak.

The obtained results are particularly useful for a simple visual analysis of experimentally observed spectra of ROSs. The relative heights of the optimized peak shapes have the universal values that are independent of the system parameters. For example, for transmission spectrum, the peak with surprisingly small relative height ¼ is the sharpest possible peak that can be produced.

## 5. Optimization of transmission spectrum for two coupled resonators of equal loss

The generalized Breit-Wigner formulae, Eqs.(3)–(7) allows to optimize more complex ROSs composed of more than one elementary resonator. This Section considers an ROS shown in Fig. 1(c), which consists of two elementary resonators. Resonator 1 is coupling to the input and output waveguides, 1 and 2. Resonator 2 is coupling to resonator 1 and is not coupling to the waveguides. The output waveguides 3 and 4 model the internal losses of resonators 1 and 2, respectively. Resonators 1 and 2 have the same loss *γ*. An example of this type of resonator, which can be created in photonic crystals, is illustrated in the inset of Fig. 1(c). There are five variable parameters of this ROS: the wavelength eigenvalues *λ*
_{1} and *λ*
_{2}, coupling between resonators, *δ*
_{12}, and also transmission coefficients between resonators 1 and waveguides 1and 2: *γ*
^{(1)}
_{1} and *γ*
^{(2)}
_{1}. From Eq.(3)–(6), the transmission spectrum of the ROS shown in Fig. 1(c) is found in the form:

${\gamma}_{1}={\gamma}_{1}^{(1)}+{\gamma}_{1}^{(2)}+\gamma $

Optimization of the transmission spectrum defined by Eq. (16) was performed numerically by variation of parameters *λ*
_{1}, *λ*
_{2}, *δ*
_{12}, *γ*
^{(1)}
_{1}, and *γ*
^{(2)}
_{1}. It was found that the transmission slope reaches the maximum for very large *γ*
^{(1)}
_{1}=*γ*
^{(2)}
_{1}>>*γ*. The latter condition means that the resonance 1 should be very broad. In other words, the waveguides 1 and 2 should be very strongly coupled to each other and should practically compose a single waveguide coupled to Resonator 2. The whole device becomes a single resonance ROS consisting of Resonator 2 side-coupled to the waveguide. In contrast to this device, the ROS in Fig. 1(b) consists of the resonator positioned inline with the waveguides. Numerical simulation showed that the slope can achieve maximum only if *δ*
^{2}
_{12}=*γγ*
^{(1)}
_{1}/2 and *λ*
_{2}-*λ*
_{1}=*γ*
^{(1)}
_{1}/2. With these relations, Eq. (16) yields the optimized transmission spectrum:

which is independent of the parameters of Resonator 1. Fig. 3 compares the plot of this spectrum (curves 2 in Fig. 3(a) and (b)) with the plot of optimized transmission spectrum of the inline resonator (curves 1), which are also shown in Fig. 2(a). From Eq. (17), the maximum possible slope of the transmission spectrum of the side-coupled resonator is

This equation yields a slope that is three times greater than does Eq. (12) for the inline ROS structure. However, it is still smaller than the slope of the reflection spectrum of the inline ROS defined by Eq. (14).

Similar numerical simulation shows that the resonance of the ROS shown in Fig. 1(c) is the sharpest for very large *γ*
^{(1)}
_{1}=*γ*
^{(2)}
_{1}>>*γ*.and *γ*
^{(1)}
_{1}=*γ*
^{(2)}
_{1}>>|*λ*-*λ*
_{1}|. Thus, again, the optimized ROS is a side-coupled single resonance device. In contrast to Eq. (17), in this case *γ*
^{(1)}
_{1}=*γ*
^{(2)}
_{1}>>|*λ*-*λ*
_{1}| and the transmission spectrum is symmetric with respect to the position λ_{2} of the second resonance:

This expression resembles the reflection spectrum of the inline resonator considered in Section 4. The maximum sharpness of this spectrum

is achieved for ${\delta}_{12}^{2}\u2044{\gamma}_{0}^{\left(1\right)}=\frac{1}{2}\left({2}^{1\u20442}-1\right)\gamma =0.207\gamma $. The corresponding plot of the transmission spectrum with the maximum possible sharpness is shown in Fig. 3(b) (curve 2) where it is compared with the similar plot for the inline ROS (curve 1) from Fig. 2(a). The sharpness of the peak for the side-coupled ROS is four times greater than for the inline ROS.

Numerical simulation shows that the maximum steepness and sharpness of the reflection spectrum of the double-resonance ROS cannot exceed those of the inline ROS shown in Fig. 2(b). In other words, the numerically solved optimization problem yields *δ*
_{12}=0. Then the performance of the double resonance structure is equivalent to that of the inline single resonance structure and the optimum parameters of the resonator 1 should be equal to the corresponding parameters determined in Section 4.

Thus, addition of the second resonance allows to significantly increase the optimum steepness and sharpness of the transmission spectrum. However, the maximum values of these parameters in the reflection spectrum are still achieved for the simplest inline ROS of Section 4.

## 6. Summary

The parameters of optical resonators used for sensing applications can be divided into those that can be varied in practice (e.g. resonator dimensions) and those that are constant (e.g. material loss). The optimization of the ROS is performed by variation of the variable parameters. For the case of a separated resonance it is possible to determine the resonator/waveguide coupling parameters, which correspond to the maximum steepness and sharpness of transmission and reflection resonant peaks. For more complex ROS, the optimization can be performed numerically based on the generalized Breit-Wigner formula, as illustrated on the example of a double-resonance ROS. It is shown that modification of ROS allows to increase the steepness and sharpness of the transmission spectrum. However, it is found that the optimized reflection spectrum of the inline ROS with a separated resonance possesses the largest steepness and sharpness among the considered examples. The results presented in this paper can be useful for optimizing the design of the resonant optical sensors.

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