Refractometry is one of the classical workhorses among a variety of optical techniques in analytical chemistry. In its basic principle, it allows the quantification of concentration changes by means of an associated refractive-index change leading to frequency shifts of optical resonances. Optofluidics integration [1–3] holds promises for refractometric analysis of minute sample volumes by the aid of optical resonators integrated in microfluidic architectures. Integrated nanophotonic resonators are beginning to show promising potential for high sensitivity and detection of minute concentrations [4–8] and the sensing performance is under active consideration in the research community [9–11]. More recent photonic crystal resonator designs emphasize the optimization of the light-matter overlap [12], which serves to increase the sensitivity [11]. However, the detection limit is of equal importance in many applications [9]. How small a refractive-index change can a resonator-based sensor setup quantify reliably? In this paper we explore the material absorption limitations. This allows us to estimate the ultimate detection limit, provided that resonators with sufficiently high intrinsic Q₀ value are available. For silicon based sensors operating in an aqueous environment, we find that for λ ≳ 1100 nm, the ultimate detection limit is given by min {Δn} ∼ η with η being the strongly wavelength-dependent imaginary index of water, i.e., the extinction coefficient. As a consequence, it should in principle be possible to detect refractive-index changes down to 10⁻⁵ at λ ∼ 1200 nm, while at wavelength of λ ∼ 1500 nm the ultimate detection limit is increased by an order of magnitude to 10⁻⁴. On the other hand, in the visible it is difficult to go below 10⁻². According to our knowledge, the limitation of material absorption is a central, but overlooked issue that was only pointed out very recently in independent work (on photonic crystal resonators) by Tomljenovic-Hanic et al. [13].

The remaining part of the manuscript is organized as follows. In Section 2. we use electromagnetic perturbation theory to calculate the sensitivity and the associated detection limit. In Section 3. we discuss our results in the context of various resonator examples and as a particular example we consider the ultimate detection limit for silicon-based sensors in an aqueous environment. Finally, in Section 4. discussions and conclusions are given.

Consider an electromagnetic resonance with a density of states (or power spectrum) which for simplicity could be given by a Lorentzian line shape: (1) ρ ( ω ) = 1 π δ ω / 2 ( ω − Ω ) 2 + ( δ ω / 2 ) 2where Ω is the resonance frequency and δω is the line width corresponding to a quality factor Q = Ω/δω. The sensitivity of a resonator is a measure of the resonance wavelength shift as function of the refractive-index change. For applications in refractometry, first order perturbation theory is adequate and gives (e.g., see [11]): (2) Δ Ω = − Ω 2 〈 E | Δ ε | E 〉 2 〈 E | ε | E 〉

This expression can be used to calculate the resonance frequency shift caused by a small change in the real part of the complex refractive index for materials in proximity with the cavity mode. We label the different material constituents by the index j so that [11] (3) Δ Ω = − Ω ∑ j f j Δ n j n jwhere n_j is the real part of the complex refractive index n_j + iη_j and the filling fraction is given by (4) f j = 〈 E | ε | E 〉 j 〈 E | ε | E 〉with Σ_j f_j = 1. The subscript in the numerator indicates that the integral is restricted to the volume fraction where the perturbation is present, while the integral in the denominator is unrestricted.

Next, consider refractometry where a small change in the real part of the refractive index in, say, material j = 1 causes a shift in the resonance frequency One first important question is of course what is the sensitivity (or the responsivity) of the system. The answer is given by Equation (3) and basically the higher is the f₁ value, the higher is the sensitivity. However, in many applications, the detection limit is of equal concern. How small changes may one quantify? As discussed in [13, 14], the resonance line-width δω = Ω/Q represents an ultimate measure of the smallest frequency shift that can be quantified accurately. Equation (3) consequently leads to a bound on the smallest refractive-index change that can be quantified accurately. In this way we arrive at (5) min { Δ n j } ≳ n j 2 f j Q

Obviously, the higher a quality factor the lower a detection limit. In the following we consider the general situation with (6) Q − 1 = Q 0 − 1 + Q abs − 1where the first term corresponds to the intrinsic quality factor in the absence of absorption, and the second term accounts for material absorption. For weak absorption (η ≪ n) we apply Equation (2), so that η gives a small imaginary frequency shift. In the framework of Equation (1) this causes an additional broadening corresponding to [11, 15] (7) Q abs − 1 = ∑ j 2 f j η j n j

The detection limit, Equation (5), now becomes (8) min { Δ n j } ≳ n j 2 f j Q 0 + ∑ i f i f j n j n i η i .

This is the main result of this section. For a two-component structure with f₁ + f₂ = 1, the result further simplifies to (9) min { Δ n 1 } ≳ n 1 2 f Q 0 + η 1 + 1 − f f n 2 n 1 η 2where we have introduced f ≡ f₁ and where the perturbation is assumed to be applied to medium j = 1. We emphasize that a much similar result was reported recently by Tomljenovic-Hanic et al. [13] in a study of silicon photonic crystal resonators, though min {Δn_j} was emphasized less explicitly. In this work we arrive at the simple result for min {Δn_j} by using the conventional limit, that the smallest detectable frequency shift is limited by the resonance linewidth, as also discussed in [13]. If the signal-to-noise ratio is adequate, detection of sub-linewidth shifts is in principle possible, but this would call for more advanced data analysis, e.g., locking onto the resonant frequency in order to compensate for noise and fluctuations in the spectrum. As pointed out in [14], the above classical limit is the most desirable one to consider when balancing detection limits and experimental complexity. We note that our assumption of a homogeneous sample greatly simplifies the calculation while a general expression can not be derived for a heterogeneous sample. In that case, one would need detailed information on the spatial variations of both the sample concentration as well as the field profiles. In the following we analyze the consequences of Equation (9) for a number of optical sensing architectures as well as in the context of material parameters.

In conclusion, with the aid of perturbation theory, we have explored the fundamental limitations on the detection limit due to material absorption. As a general result, it is difficult to detect refractive-index changes smaller than the associated imaginary part of the refractive index. The main assumption behind this quite intuitive result is that the smallest detectable frequency shift is limited by the resonance linewidth. For liquid-infiltrated silicon-based optical resonators in photonic crystal architectures, we find an optimal wavelength range resulting from the strong wavelength dependence of η for both water and silicon. Our results may be extended to also other material platforms. As an example, for visible applications one may take advantage of transparent glasses, such as SiON or SiO₂ where η can be of the order of 10⁻⁴, though with a real part of the refractive index much reduced compared with semiconductor materials like silicon. In the quest for refractometric sensors with yet better detection limits, our results are central to the general design of liquid-infiltrated dielectric resonators. In addition to the efforts put into the cavity structural design (e.g., the optimization of Q₀), the present analysis illustrates the importance of choosing the appropriate combination of light-sources (λ), resonator designs (f), materials (η₂), and bio-chemical buffer media (η₁).