1. Introduction

Microsensing systems are essential for portable monitoring and diagnostic applications ranging from healthcare to homeland security. In recent years, a number of integrated sensors and sensing systems have been developed. These include electro-chemical sensors [1], microcantilever sensors [2], carbon nanotube sensors [3], surface plasmon sensors [4], and optical sensors [5–17]. Among these existing sensors, optical sensors offer low noise, immunity to electromagnetic interference, high sensitivity, and compact size. Unlike some sensors, optical sensors can be utilized in broad application environments. There have been developments in different integrated optical sensors that include Mach-Zehnder interferometers [5], surface plasmon sensors [6], Bragg grating sensors [7], and microresonators [9–17]. Among various optical sensors, microresonator sensors are attractive for portable applications due to their ultra-high sensitivity [8] and small footprint. Microresonator sensors have been successfully demonstrated for different applications, such as glucose detection [9–11], biomolecule detection [12–13], strain detection [14], and chemical detection [15].

Microresonator sensors respond to refractive index changes around the resonator cavity and produce the wavelength shifts of resonant modes. Analyte discriminations have been demonstrated by detecting the refractive index changes in bio/chemical transducer layers on the surface of the microresonator sensor [15,17,18]. By monitoring the resonant wavelength shift using spectral interrogation equipment, the induced index change can be accurately estimated. However, two technical issues need to be addressed in the microresonator sensing system for portable applications: 1) bulky and expensive measurement system and 2) reduced measurement accuracy in the conventional spectral interrogation approach, especially in a noisy environment. Recently, the impact of Gaussian noise on the detection limit of optical resonant refractive index sensors has been studied by White et al [19]. Although there have been individual studies on improving the sensitivity of microresonator sensors [11,12] and developing integrated spectral analysis systems [20,21], the overall system performance evaluation, improvement, and implementation have not been investigated.

In order to address existing limitations of the conventional spectral interrogation approach, we propose here a new hybrid approach by incorporating spectral sampling and estimation using digital samples. This hybrid approach significantly reduces the size of the entire optical sensing system and provides methods to estimate any refractive index perturbations. The estimation accuracy of the proposed method is extremely high with only few spectral sampling points, which will reduce the size, cost, and complexity of the sensing system.

The objective of this paper is to develop, for the first time, a novel optical sensing system based on digital signal processing which will significantly reduce the size and cost of the entire sensing system. Due to the nature of high frequencies in optical regime, a Nyquist-rate-sampling in current electronic systems is almost impossible. We provide an innovative optical signal processing technique to overcome this fundamental limitation in conventional sampling approaches. We show in this paper that we can obtain highly accurate results from spectrally undersampled outputs to implement chip-scale optical sensing systems. The proposed hybrid sensing system estimates refractive index from temporal and spectral samples via a maximum likelihood estimation (MLE) algorithm which can be implemented in an integrated digital circuit format. The idea of MLE is to find the most likely value of the refractive index by maximizing a likelihood function for the given spectral and temporal samples. We investigate overall system performance for different noise levels, different quality (Q) factors, and non-ideal microresonator sensors. This hybrid system approach offers cost effective, standalone chip-scale optical sensing systems and can be applicable to miniaturizing other existing sensors.

2. Hybrid optical sensing systems

The hybrid optical sensing system consists of a broadband source such as a light emitting diode, a microresonator sensor, an arrayed-waveguide-grating (AWG), an array of PDs, an analog-to-digital converter (ADC), and a simple digital processor. Figure 1 shows a schematic diagram of the hybrid optical sensing system. The output of the microresonator sensor is split into N-channels by the AWG, forming N spectral sampling points. An array of PDs is integrated in the output channels of the AWG to convert each optical spectral sample into an electrical signal. In this novel approach, we can significantly reduce the number of spectral sampling points, i.e. the number of output channels of the AWG, using MLE algorithm while keeping the estimation accuracy high.

Active optoelectronic devices such as PDs and optical sources integrated with planar lightwave components for chip-scale hybrid microsystem applications have been experimentally demonstrated by the author [22]. This chip-scale photonic integration technique can be directly applied to implement this hybrid optical sensing system.

 

Fig. 1. A schematic diagram of the hybrid optical sensing system (inset: uniformly sampled spectral response of a resonator sensor in the presence of normally distributed random noise).

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The inset in Fig. 1 shows a uniformly sampled spectral response of the microresonator sensor by the AWG in the presence of normally distributed random noise at a signal-to-noise ratio (SNR) of 30dB. The photogenerated current from each PD is converted into a digital format by the ADC. The digital processor estimates the effective refractive index of the resonator sensor from the digitized photocurrent using an MLE algorithm. The MLE presented herein is the optimal estimation method for a given set of PDs.

3. Estimation of effective refractive index

The transfer function (T) of an all-pass type microresonator sensor is given by

where λ is the free space wavelength, R is the ring radius, α is an attenuation factor, r is the transmission coefficient at the evanescent coupling section of the sensor, n is an effective refractive index of a resonant mode, and k ₀ is the propagation constant in vacuum. Thus, the i-th PD corresponding to wavelength λ_i, 1 ≤ i ≤ N, receives an optical intensity proportional to T(n,λ_i), and produces an electrical signal η_iT(n,λ_i), where η_i is a normalization constant that also includes optical-to-electrical conversion efficiency and beam coupling efficiency. We keep η_i=1 for simplicity, although the method can be easily modified for other values of η. Let the output of each PD be sampled M times at a fixed time interval. The l-th received electrical signal in temporal domain for the k-th PD in spectral domain is written as,

where T_k(n) = T(n,λ_k) and w_k[l] is modeled as Gaussian noise. The noise samples are assumed to be uncorrelated for different PDs and over different time samples. The probability density function (PDF) if y_k[l] is given by $1/\sqrt{2\pi {\sigma }_n^2}\exp (-\left(y_k\left[l\right]-T_k{\left[n\right]}^2\right)/\left({2\sigma }_n^2\right))$, where σ ² _n is the noise variance. Since the noise samples are independent, the joint PDF of all signal samples is obtained by multiplying individual PDFs. For a given set of received signal samples, this function is dependent on refractive index, n, and is called the likelihood function. The MLE maximizes the likelihood function [23], which becomes equivalent to estimation of n from samples {y_k[T]} by minimizing the cost function, ∑^N _k=1∑^M _l=1(y_k[l] - T_k(n))², over different values of n. After simplification, we obtain an equivalent cost function, J(n), given by

where x_k = ∑^M _l=1 y_k[l] is the sum of all the M time samples obtained at the k-th PD. We compute Eq. (3) for a limited range of n, starting from a minimum value n_min to a maximum possible value n_max, at a step of δn, and choose the value of n that minimizes J(n) as the estimate n̂.

This approach can also be applied to large-scale distributed sensor networks. In Fig. 1, the digital processor can be just one centralized computer, and each sensor needs to send only N values, x ₁,x ₂,….x_N, to the central station. Besides, since the first term in Eq. (2) does not depend on received samples, it can be pre-computed for various values of n and stored in the memory to reduce on-line numerical computations.

4. Performance results

We perform Monte-Carlo simulation by generating M random samples for each of the N PDs as given by Eq. (2). These MN random samples form one data set, and this data set is used to obtain the estimate n̂. This simulation experiment is repeated K times and the root mean-squared error (RMSE) is computed as

where n is the true effective refractive index and n ^(k̂) is its estimate from the k-th data set. In our study, we use n_min = 2.8, n_max = 3.49, δn = 10^-4, α = 0.95, r = 0.95, R = 5μm, and K = 5000_ unless otherwise stated. We assumed a silicon-on-insulator (SOI) microresonator sensor. The true value of n in a SOI sensor is assumed to be 3.05. The PDs are positioned uniformly in a spectral window from 1.55 μm to 1.6 μm.

Figures 2(a) and 2(b) show RMSE values for different SNRs with N and M respectively, as the parameters. Both figures clearly demonstrate that using sufficiently large N and M values, highly accurate estimate of n can be obtained. Figure 2(a) shows that RMSE for N = 15 and N = 20 are similar at a given SNR. Also, since we are calculating Eq. (3) for various values of n at a step of δn, the RMSE improves dramatically after a certain SNR as the noise is not strong enough to produce false estimation. However, the RMSE at a given SNR is significantly reduced with 25 PDs. For example, the calculated RMSE is in the order of 1×10^-6 for more than 20 dB SNR with 25 PDs. In a statistical average sense, the estimated index is thus within ±10^-6 of the true value. The estimated refractive index is obtained for a given output spectrum. In the context of reduced sensitivity of microresonator sensors due to reduced field overlap [24], the proposed approach can provide highly accurate estimation to detect spectral shifts for sensing applications. Figure 2(b) shows improved performance with an increase in the number of time samples. As we increase M from 10 to 50, RMSE becomes significantly reduced at a given SNR. The number of time samples increases the cost of the system (e.g. speed, memory). The number of sampling points in temporal domain (M) and spectral domain (N) can be optimally determined by the system requirements for a certain application.

 

Fig. 2. (a). RMSE performance for different numbers of PDs with M=100, (b). RMSE performance for different numbers of time samples per PD with 25 PDs.

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Effective refractive index of an optical sensing system with a non-ideal AWG is also estimated to demonstrate the feasibility of the proposed approach. The passband spectral response of a uniform AWG is a sharp Gaussian profile [25,26]. However, a number of approaches have been developed to achieve wideband flat-top AWGs [26–28]. Therefore, in this study, we also assume a flat-top AWG with 1 dB loss uniformity and 75 GHz passband. The calculated RMSE at 27 dB SNR is 1.05×10^-5 with 15 PDs and 100 time samples.

5. Effects of different spectral characteristics of microresonator sensors

The spectral response of a microresonator sensor can be altered by a number of factors, such as Fabry-Pérot resonance from sensor facets [29] and coupling conditions [30]. To investigate the impact of different spectral characteristics of resonator sensors on estimation performance, two cases are considered: (1) asymmetric spectral responses caused by undesired Fabry-Pérot resonance and (2) different quality factors due to different coupling conditions. We choose a reference curve (N=15, M=100) to compare with each case.

Fabry-Pérot resonance is caused by the undesired reflection from sensor facets, introducing asymmetric responses with ripples in output spectrum. To study the effect of Fabry-Pérot resonance on estimation performance, we derive the overall transfer function, TF(n,λ), using the optical transfer matrix approach [31, 32].

where ρ is the amplitude reflection coefficient at the sensor facet, δ is the phase shift due to propagation between the sensor and one of the facets, and t is the transmission coefficient of a microresonator sensor where the transfer function, T, in Eq. (1) is defined by T=∣t∣².

In Fig. 3(a), we compare the RMSE performance for an ideal microresonator sensor and a non-ideal sensor. The non-ideal sensor has an asymmetric response with ripples in the output spectrum due to unwanted Fabry-Pérot resonance. For simulation, ρ is assumed to be 0.1. It is noticed that as the SNR increases, RMSE for asymmetric case also reduces significantly as in the symmetric case. As noticed in Section 4, RMSE is expected to improve with increased sampling points and number of PDs. To investigate estimation performance for different Q factors, two resonator sensors with different coupling conditions are considered. Figure 3(b) shows estimation performance for different coupling conditions. The calculated Q factors for the first coupling condition, r=α=0.95, and the second coupling condition, r=α=0.7, are 1871 and 291, respectively. This result shows that the proposed method provides highly accurate refractive index estimation for different Q resonator sensors.

 

Fig. 3. RMSE performance for different spectral characteristics of microresonator sensors: (a) asymmetric spectral responses caused by undesired Fabry-Pérot resonance and (b) different quality factors due to different coupling conditions.

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6. Conclusion

We propose a novel standalone chip-scale hybrid optical sensing system based on highly sensitive optical microresonator sensors. We investigate overall system performance of the hybrid sensing system in a noisy environment with reasonable number of spectral and temporal sampling points for different noise levels. For more than 20 dB SNR, the estimated effective refractive index is within ± 10^-6 of the true value or better with 25 spectral sampling points in the hybrid sensing system. The proposed method provides highly accurate refractive index estimation even for non-ideal resonator sensors. This approach allows fully functional, chip-scale hybrid sensing systems offering great potential in a number of applications including distributed sensors and sensor on a chip.