1. INTRODUCTION

Measuring the properties of random systems is ubiquitous. Their characterization can be approached in different ways, but in many situations, the only meaningful depiction is in statistical terms. For instance, in active sensing, information is recovered through a stochastic analysis of the system’s output in response to controlled external stimuli. Examples include the characterization of structural dynamics [1] or individual scattering objects [2,3], to mention just a few. Nevertheless, the external stimuli can be themselves stochastic, like in the case of events of molecular origin [4].

In all these examples, the information about the targeted system is contained in the fluctuations of the measured signals. Therefore, efficient information retrieval requires augmenting such fluctuations, which is reminiscent of stochastic resonance [5]. We note that the traditional stochastic resonance approach relies on (i) some sort of system nonlinearity and (ii) an increase of the interaction energy through the external control of the level of additional noise.

An effective enhancement of the signal fluctuations can be generated not only by selecting a specific type of physical interaction, but also by appropriate manipulations of the probing stimulus. Thus, it can be argued that using stochastic probing, one may be able to enhance the signal variability. This kind of approach may also extend certain practical attributes of the sensing process, such as robustness against noisy and unpredictable environments.

2. FLUCTUATION-BASED SENSING

In this paper, we introduce a stochastic sensing method for characterizing properties of random systems. We will show that the statistical features of a scattering medium can be recovered efficiently by simply controlling the statistical realizations $\xi$ of a stochastic probe $S(\mathit{x};\xi )$ that interrogates the targeted system $V(\mathit{x})$ via a multidimensional interaction in the domain $\{\mathit{x}\}$. In a nutshell, we will show that changes in the statistical parameters of a non-stationary probe $S(\mathit{x};\xi )$ lead to abrupt modifications in the signals’ statistics, which can be related to the target properties. Even though this approach is reminiscent of stochastic resonance, it does not come at the expense of increasing the energy of interaction between the probe and the targeted system and, furthermore, it does not require the presence of any nonlinearity. Moreover, because the non-stationary stochastic probe and the different possible sources of noise are statistically independent, we will demonstrate that this sensing methodology is robust against various external perturbations.

In practice, attaining a reliable signal encoded in $\mathit{x}$ is often problematic because of, among other things, the limited bandwidth of a detection system and the presence of perturbations in both transferring the stimulus (perturbation in) and recovering the response from the target (perturbation out). In these circumstances, a preferable quantity to measure is the average (integrated) signal response:

Without loss of generality, we will consider an optical situation in which the intensity statistics of a two-dimensional illumination field are manipulated to recover information regarding a random scattering target $V(\mathit{\rho };l_V)$ characterized by the correlation length $l_V$. In this case, the interaction is described in the two-dimensional Cartesian domain $\mathit{\rho }$, and $\xi$ represents a simple, one-dimensional realization index. The general setting for this optical situation is illustrated in Fig. 1. We note that the concept is not limited to backscattered light; a similar operation in forward scattering or any other direction can be described by simply folding the setup around the plane $\mathbf{\Omega }$.

 

Fig. 1. Schematic procedure of sensing statistical properties of a random scattering potential $V(\mathit{\rho };l_V)$ interrogated with a controlled stochastic illumination $S(\mathit{\rho },\xi )$, which is affected by the inherent source noise $n(\mathit{\rho };\xi )$, and perturbations $U_i(\mathit{\rho },\xi )$ and $U_o(\mathit{\rho },\xi )$ in the illumination and recording paths, respectively. The measured signal is the integrated intensity $I(\xi )$.

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A generic description of the targeted medium can be advanced using the concept of the scattering potential $V(\mathit{\rho })=-k^2[n^2(\mathit{\rho })-n_0^2(\mathit{\rho })]$, where $k$ is the wave number based on the first Born approximation. This approach is commonly used in reconstructing the distributions of refractive index variations $n^2(\mathit{\rho })$ [6]. In this case, the averaged response $I(\xi )$ in Eq. (1) is, in fact, the average intensity of scattered light from the targeted scattering potential that fluctuates when the configuration $\xi$ of the excitation changes. The proposed stochastic sensing method relies on processing these fluctuations and not the magnitude of the integrated scattered signal itself.

It should be noted that, rigorously, the intensity distribution across a detector placed at an axial distance $\zeta$ from the target is $\mathcal{E}(\mathit{\rho },\zeta ){\mathcal{E}}^{*}(\mathit{\rho },\zeta )$, where $\mathcal{E}(\mathit{\rho },\zeta )=\int v({\mathit{\rho }}^{\prime }){\mathcal{E}}_S({\mathit{\rho }}^{\prime },0)G(\mathit{\rho }-{\mathit{\rho }}^{\prime },\zeta )\mathrm{d}{\mathit{\rho }}^{\prime }$ is the field scattered due to the stochastic illumination ${\mathcal{E}}_S(\mathit{\rho },0)$. In this general scattering problem, the scattering potential is characterized by a complex field scattering coefficient $v(\mathit{\rho })$ (${|v(\mathit{\rho })|}^2=V(\mathit{\rho })$), while the field propagation to the detector plane is described by the corresponding Green’s function, $G(\mathit{\rho },\zeta )$. However, one can show that the average scattered intensity can still be modeled with Eq. (1) for any $\zeta$, as long as the propagation of scattered light is lossless, or if the loss is the same for all the realizations $\xi$. The interested reader is referred to Supplement 1 for more details.

In the following proof-of-concept demonstration, we consider only static scattering potentials, but this is not a conceptual limitation. Temporal fluctuations and their evolution can be recovered using dynamic illumination with the appropriate temporal characteristics. We also note that if all the realizations $\xi$ of the stimulus $S(\mathit{\rho };\xi )$ are known, one could, in principle, invert Eq. (1) to recover an “image” of the static target $V(\mathit{\rho };l_V)$ using variants of the so-called “ghost imaging” techniques [7–9]. The problem at hand, however, is different, as it refers to stochastic illumination in noisy ambient conditions. In this situation, the question is: can any information about $V(\mathit{\rho };l_V)$ still be recovered?

In the following, we will first outline how the statistical moments characterizing the scattering potential and the spatial and dynamic properties of the illumination relate to the fluctuations of the recorded intensity. Then, we will exemplify how this stochastic sensing formulation can be used in different practical applications. We will demonstrate the method’s robustness against perturbations in both the structure of illumination and the intensity detection.

3. STOCHASTIC OPTICAL SENSING

The stochastic Fredholm integral equation, Eq. (1), relates the unknown scattering potential $V(\mathit{\rho };l_V)$ to four random processes, $S(\mathit{\rho };\xi )$, $n(\mathit{\rho };\xi )$, $P(\mathit{\rho };\xi )$, and $R(\mathit{\rho };\xi )$. These processes are considered to be statistically stationary spatially and also statistically independent of each other, which is a reasonable practical assumption. For each realization $\xi$, these processes are characterized by probability distributions and corresponding statistical moments with respect to variable $\mathit{\rho }$. Upon averaging over an ensemble of all realizations $\xi$, these functions can be treated as constants in the further analysis.

We will now examine the impact of different types of fluctuating illumination intensity. First, we will discuss the case of a $\delta$-correlated distribution defined by ${〈\tilde{S}(\mathit{\rho };\xi )\tilde{S}(\mathit{\rho }+{\mathit{\rho }}_0;\xi )〉}_{\mathit{\rho }}={\tilde{m}}_S^{(2)}\delta ({\mathit{\rho }}_0)$, with ${\tilde{m}}_S^{(k)}=〈{(\tilde{S}(\mathit{\rho },\xi ))}^k{〉}_{\xi }$ in terms of $\tilde{S}(\mathit{\rho },\xi )=S(\mathit{\rho },\xi )-〈S{(\mathit{\rho },\xi )〉}_{\xi }$. In this case, one can use the statistical properties of the distribution $\sigma (\mathit{\rho };\xi )=(S(\mathit{\rho };\xi )+n(\mathit{\rho };\xi ))P(\mathit{\rho };\xi )$, effectively impinging on the scattering potential to evaluate recursively the moments as follows:

In the practical case where the correlation length $l_S$ of the illumination field is finite, then this length scale can be manipulated to infer information about the spatial properties of the scattering potential. The illumination can be thought of as the convolution $S(\mathit{\rho };\xi ,l_S)=e(\mathit{\rho };l_S)*C(\mathit{\rho };\xi )$ of a positively defined elementary unit $e(\mathit{\rho };l_S)$ with a spatially $\delta$-correlated stochastic process $C(\mathit{\rho };\xi )$. In these conditions, we show in Supplement 1 that

However, information about the two-point correlation of the scattering potential can still be accessed without a priori knowledge by simply examining the fluctuations of the recorded intensity over different realizations $\xi$ of the illumination intensity. In this case, the fluctuations of the integrated intensity become

The procedure described above suggests a direct way to infer $l_V$ from the values of the Fano factor of the recorded intensity as a function of $l_S$. When changing $l_S$, it is evident based on Schwarz’s inequality that this Fano factor attains its maximum when $l_S=l_V$. The interested reader is referred to Supplement 1 for more details.

The same conclusion can be reached by analyzing this sensing process in the spatial frequency domain. It is clear that the dominant frequency components of the spectral distributions (the Fourier transform of the corresponding autocorrelation function) of the illumination intensity and the scattering potential will overlap when $l_V=l_S$. The similar extent of the spectral distributions enhances the magnitude of the intensity fluctuations [6]. Of course, one could have used a deterministic illumination to reconstruct the characteristic length of the targeted potential. However, as follows from Eq. (5), a statistical analysis makes our method more robust to perturbations. Moreover, using random illumination may also lead to faster measurements.

We also note that this procedure is not limited to a single statistical process. A superposition of uncorrelated scattering potentials with different characteristic length scales can be represented, for instance, by a non-monotonic autocorrelation function leading to a Fano spectrum having multiple maxima. The repeatability of this technique is rather high because, as mentioned before, four terms in Eq. (5) are statistically independent. As a result, the variations in the Fano spectrum are independent of the considered perturbations. The repeatability is demonstrated in the Supplement 1 for different targeted potentials in noisy environments.

4 Concept Demonstration

In the following, we present a proof-of-concept demonstration in which we will incorporate the two main requirements for a generic sensing procedure. First, when changing $l_S$, the total incident irradiance may change. Because in some circumstances, the scattering potential could depend on the irradiance level, we impose that $m_S^{(1)}$ remains constant throughout the procedure. Second, sometimes there is simply no a priori information about the targeted potential and, therefore, an effective sensing method must sample it uniformly and isotropically.

A. Random Scattering Potentials

In the first example, we simulate a scattering potential based on a random array of circular disks with $l_V=\mathrm{\Delta }$ and a volume fraction of 15%. Figure 2(a) shows the normalized Fano spectrum of the intensity fluctuations corresponding to illumination patterns with varying $l_S$ and a filing fraction of 10%. The elementary unit used here is spatially isotropic. It is evident that the maximum in the Fano spectrum corresponds to the situation where $l_S=l_V$. This matching condition can be used to describe the targeted potential and the illumination, and is rigorously demonstrated in the Supplement 1.

 

Fig. 2. Numerical calculation of the Fano spectrum corresponding to (a) random targeted potential consisting of circular disks with $l_V=\mathrm{\Delta }$, and (b) the corresponding Fano spectrum. The maximum of the Fano spectrum is normalized to unity.

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B. Multi-Scale Scattering Potentials

In the first example, we considered a random system with only one length scale. The effectiveness of this stochastic sensing technique as outlined in Eq. (5) was also studied in more complex situations where the scattering potential has two characteristic lengths. For example, in Fig. 3, we present the normalized $F_I$ spectrum corresponding to a scattering target consisting of two rectangular areas with width $2\mathrm{\Delta }$, which are separated by a distance of $3\mathrm{\Delta }$. To mimic natural perturbations, the illumination was purposely corrupted by additive noise: $S(\mathit{\rho };\xi )+n(\mathit{\rho };\xi )$. In addition, the scattered light was also distorted by additive and multiplicative white noise to effectively create detection conditions with an SNR of 2.5 dB. As can be seen in Fig. 3(b), even in this highly perturbed condition, the characteristic features of the target can be effectively recovered. This is the benefit of relying on statistical parameters, which makes the information distinguishable from the influence of any stationary perturbations.

 

Fig. 3. (a) Simulated reflective target perturbed by illumination noise and (b) the corresponding normalized Fano spectrum. The measurement conditions are similar to those in Fig. 2. (c) Targeted standard microscopy chart under noisy illumination and (d) the corresponding normalized Fano spectrum.

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In Figs. 3(c) and 3(d), we present the results of an experiment where a reflective target consisting of 2 μm wide strips and a 10 μm center-to-center distance was sequentially illuminated with 200 random realizations for each $l_S$ (in this case, $\xi$ represents time). The illumination was generated using a liquid crystal spatial light modulator with pitch size of 15 μm and $512\mathrm{pixels}\times 512\mathrm{pixels}$. The backscattered light was integrated on a single photodetector with peak quantum efficiency of 0.38. The measurements were conducted in noisy conditions characterized by signal-to-noise ratio (SNR) of 1.8 dB, and more details of the experimental setup are included in Supplement 1. As evident, both scale lengths characterizing this target are clearly identified in the experimental Fano spectrum.

In these examples, the correlation function characterizing the targeted potential is non-monotonic. For observing higher-order peaks in the Fano spectrum, the $l_S$ range of variation must be large enough to cover the spatial domain of interest. As a result, the level of variations in the first term of Eq. (5) is reduced, which determines the decrease in the overall level of fluctuations, as can be seen in the results summarized in Figs. 3(b) and 3(d).

C. Disturbed Illumination

So far, we have shown that perturbations that can be modeled as statistically stationary fluctuations do not degrade the performance of this stochastic sensing procedure. However, perturbations in the illumination path that can disturb the illumination structure may affect the results. In other words, the performance may be influenced by introducing uncertainties in the value of $l_S$, our only tuning parameter. This case is illustrated in Fig. 4, where the size of the elementary function $e(\mathit{\rho };l_S)$ changes randomly according to a Gaussian distribution with a standard deviations of $\mathrm{\Delta }/3$.

 

Fig. 4. (a) Normalized Fano spectrum for an unperturbed condition (-) and for the case of $l_S$ distributed Gaussianly with a variance of $\mathrm{\Delta }/3$ (-.-). (b) Correlation length $l_{\sigma }$ of the effective illumination (red shadow) as a function of the correlation length $l_S$ of the unperturbed illumination (black solid line).

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As suggested in the inset of Fig. 4(b), this is a dramatic variation for the value of $l_S$. The Fano spectra corresponding to a scattering potential consisting of two rectangular objects characterized by length scales of $2\mathrm{\Delta }$ and $5\mathrm{\Delta }$ demonstrate that, even though the maxima broaden because of the uncertainty in $l_S$, the characteristic length scales can still be clearly identified.

D. Non-Sparse Scattering Potentials

The examples illustrated so far prove the efficacy of this stochastic sensing procedure. They were, however, conducted on rather sparse scattering potentials. In the following, we will demonstrate that this is not a conceptual limitation. We will now consider targets consisting of densely packed and randomly shaped objects having slightly different statistical parameters. In general, the low sparsity complicates the detection of such subtle changes, which can be easily masked by experimental noise. To ensure realistic conditions, we considered measurement circumstances where noise was added to both the scattering potential and the illumination pattern to create an overall SNR of 2 dB.

We considered potentials with different spatial symmetries, as shown in Figs. 5(a) and 5(b). Note that, in the asymmetric case of Fig. 5(b), some of the objects are more elongated and, due to the asymmetry in the density-density correlation function, they generate two additional characteristic length scales. As shown in Fig. 5(c), this asymmetry is reflected in the Fano spectrum of the recorded intensity. The inset shows the ratio between the two spectra, $F_{I,\text{Asym.}}/F_{I,\text{Sym}}$ and, as can be seen, for larger values of $l_S$ associated with larger linear dimensions of the asymmetric objects, the difference between them increases because of an increased contribution of larger-sized objects. The autocorrelation functions of these two targets are compared in Figure (S4) of Supplement 1, where an additional discussion is provided.

 

Fig. 5. Simulated reflective targets consisting of randomly shaped objects having, on average, symmetric (a) and asymmetric (b) correlation functions. (c) Normalized Fano spectra corresponding to the symmetric (continuous line) and asymmetric (dashed line) targets, respectively. The inset shows the ratio of the two spectra as a function of the correlation length $l_S$ of the illumination pattern.

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Clearly, only an averaged characterization is meaningful for the asymmetry of the targets illustrated in Fig. 5. When the overall asymmetry increases, the corresponding Fano spectrum tends to develop maxima corresponding to the additional scales defined by the asymmetry and, as a result, the spectrum displays a longer tail. As clearly seen in the inset, this departure from the symmetric signature evolves nonlinearly because of the way the variation of the first term in Eq. (5) reduces when the characteristic scale $l_S$ increases under conditions of constant total reflectivity.

E. Weak Scattering Targets

Up to this point, we have shown that the stochastic optical sensing operates efficiently in harshly perturbed conditions and for different levels of target sparsity. Also important for a sensing method is its ability to recover information from targets with different scattering potential strengths. To examine this property experimentally, we examined two different targets with very different backscattering reflectivities. First, we examined the high-contrast element five of the group 5 in the positive standard USAF 1951 chart with $l_V=9.8\mathrm{\mu m}$. This is close to the case already shown in Fig. 3(c). As a second example, we used a significantly less reflective target, an H2c9 cell placed in an aqueous medium ($\mathrm{DME}\mathrm{M}+10\%\mathrm{FPS}+1\times \mathrm{p}/\mathrm{s}$). The cell has an almost symmetrical shape with a characteristic diameter of $\mathrm{\Delta }\approx 10\mathrm{\mu m}$, similar to the length scale characterizing the USAF chart. Similar illumination conditions were used, and the typical results of these two experiments are illustrated Fig. 6. We note that during the 0.25 second illumination, the cumulative intensity of light used for the cells’ characterization was eight orders of magnitudes smaller than the sunlight intensity. As clearly seen, the stochastic sensing technique works quite well even in the case of such very low-contrast targets.

 

Fig. 6. Average over 100 Fano spectra corresponding to (a) element five of group 5 of USAF 1951 positive resolution chart, and (b) a live H2c9 cell placed in an aqueous medium.

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Moreover, the repeatability of the procedure is rather high, and the interested reader can find an ample discussion in Supplement 1.

5. CONCLUSION

Fluctuations are not always artifacts. In this paper, we proposed and demonstrated a stochastic sensing technique that relies on enhancing the fluctuations measured in a scattering experiment under structured illumination. Using the dispersion of the integrated scattered intensities’ fluctuations, we showed that the characteristic lengths of the interaction potential can be consistently recovered. We have also demonstrated the efficiency of this method in extremely perturbed conditions, either in illumination or detection. In addition, we proved that this method is efficient over a large range of practical conditions. We successfully characterized targets with both very low and very high scattering coefficients.

Our stochastic sensing procedure can be regarded as a type of “noise spectroscopy,” where abrupt changes in the non-stationary properties of a signal are controlled by adjusting the stochastic properties of the excitation field. Varying the correlation length of the illumination can be exploited to recover structural information about the interaction potentials.

We also present a model that essentially relies on the fact that, even for a Gaussian distribution of a random scattering potential, the scattered intensity is not necessarily Gaussian. By tailoring the illumination, one can therefore enforce non-Gaussian scattering processes, leading to measurable departures from the central limit theorem [17–20]. Being non-universal, these deviations are measurable signatures of the specific properties of the scattering potential.

Finally, our experiments demonstrate that, in addition to being robust, this stochastic sensing approach is capable of providing spatial measurements with low errors in highly noisy conditions (SNR below 2 dB). This sensing method is not limited to the optical situations illustrated here and it should be appealing for a range of biomedical applications [21,22], material sciences [23,24], and atmospheric characterization [25,26]. Finally, although the examples presented here addressed static scattering potentials, a similar stochastic sensing approach can be extended to characterizing dynamic phenomena [24,27].