Critical-Angle Differential Refractometry of Lossy Media: A Theoretical Study and Practical Design Issues

Abstract

At a critical angle of incidence, Fresnel reflectance at an interface between a front transparent and a rear lossy medium exhibits sensitive dependencies on the complex refractive index of the latter. This effect facilitates the design of optical sensors exploiting single (or multiple) reflections inside a prism (or a parallel plate). We determine an empirical framework that captures performance specifications of this sensing scheme, including sensitivity, detection limit, range of linearity and—what we define here as—angular acceptance bandwidth. Subsequently, we develop an optimization protocol that accounts for all relevant optical or geometrical variables and that can be utilized in any application.

1. Introduction

Refractive index sensors are intensely investigated for numerous biomedical [1,2,3], chemical [4,5] and industrial [6,7] applications. An indicative yet far from exhaustive list of sensing mechanisms relies on plasmonic [8,9,10,11], photonic crystal [12,13,14,15], micro-cavity [16,17,18,19], optical fiber [20,21,22,23] and wave-guide [24,25,26,27] configurations. Associated with Fresnel reflectance properties at planar interfaces, differential refractometry offers an alternative path to sensing refractive index changes, by exploitation of interference [28], deflection [29] or (more relevant to the present work) critical-angle [30,31,32,33,34,35] effects. Today, differential refractometry is not only a standard analytical tool that operates routinely in many laboratories, but also infiltrates emerging optofluidic and lab-on-chip technologies [36,37,38].

In critical-angle differential refractometry (CADR), a front transparent medium (commonly, a prism) is interfaced with a sample which, typically, is also assumed to be lossless. The underlying principle of operation is simple: provided that the front medium is optically denser than the sample, there exists a sharp transition from total internal reflection (TIR) to partial internal reflection, taking place at a critical angle which corresponds to the location of an abrupt discontinuity in the derivative of reflectance with respect to incidence angle. Operating the sensing interface at the transition point leads to the generation of an intensity readout signal, as soon as the TIR condition is disturbed by refractive index fluctuations.

With non-transparent samples, CADR interpretation is less straightforward. To begin with, the refractive index of the rear medium becomes a complex quantity, the imaginary part of which incorporates absorption or scattering effects. Furthermore, reflectance never reaches unity (except for the limiting case of incidence at 90° with respect to the surface normal) and the transition from “attenuated” total internal reflection (ATIR) to partial internal reflection is gradual. As a result, the reflectance derivative with respect to incidence angle, now peaking to a finite value, is no longer the proper quantity to conceptualize the sensing principle; this purpose is better served by reflectance derivatives with respect to the real and imaginary index of the sample. These are negative quantities exhibiting local extrema at the vicinity of the transition from ATIR to partial internal reflection, albeit at slightly different “critical” angles. In general, these extrema are stronger for p- than for s-polarization, an observation that indicates the preferential wave orientation for CADR sensing.

In this work, we attempt a theoretical study of CADR with lossy media, accounting for (i) the standard prism configuration, (ii) an alternative geometry that exploits multiple reflections inside a parallel plate. Both schemes are, in principle, compatible with optofluidic technologies and static or real-time monitoring applications. We untangle the perplex dependencies of the sensor’s specifications on various optical and geometrical parameters, revealing among other facts the dominant role of the sample’s loss. In doing so, we introduce the concept of the “angular acceptance bandwidth” which trades-off with sensor’s sensitivity and helps clarify several complexities in terms of light coupling and detection. Our results provide a universal roadmap for rapid performance evaluation and optimization of CADR devices, which might be essential for pushing the technique’s detection limit from the current standard ($\sim 100 \mathsf{\mu }\mathrm{RIU}$) down to the current state-of-the-art ($\sim 1 \mathsf{\mu }\mathrm{RIU}$ for CADR [31], as well as for all noninterferometric methods), or even lower.

2. Theoretical Background

The isosceles triangular prism and parallelogram plate configurations under investigation are depicted in Figure 1. The sample is interfaced with the base of the prism, or equivalently, the top side of the plate, the bottom of which is high-reflection-coated. Coming from air, light hits the input facet at an external angle of incidence $\theta$ and reaches the output facet at an angle $\varphi$, after $N=1$ (or $N>1$) reflections at the sensing interface of the prism (plate), respectively; each one of these reflections is at a constant angle ${\theta }_1$. The input and output coupling geometries are determined by the cut angle $\alpha$, which in Figure 1 is taken $\alpha <\pi /2$; however, the upcoming formalism remains valid also for $\alpha \ge \pi /2$. Then, $\theta$, $\varphi$ and ${\theta }_1$ (which are trivially defined with respect to the corresponding normal), as well as plate’s length L and thickness d (which is assumed to be much larger than the light wavelength, so as to avoid interference effects), relate via:

$${\theta }_1=\alpha -\varphi ,\varphi =arcsin\left[\frac{n_{air}}{n_o}·sin\theta \right],L\approx 2d·N·tan\left({\theta }_1\right)$$

(1)

where $n_{air}(\approx 1)$ and $n_o$ are the real indices of air and the transparent prism (or plate), respectively.

Light transfer in the proposed layouts can be accurately simulated via standard Fresnel theory. Let us begin by assuming that a reference medium with a complex refractive index $(n=n_r-i{n}_i)$ is first put to the test. The “reference” prim/plate transmittance $T_p$, defined as the ratio of the output light intensity $I_{ref}$ over incident light intensity $I_{in}$, is:

$$T_p\left(\theta \right)=\frac{I_{ref}}{I_{in}}=\left[1-R_{1/2}\left(\theta \right)\right]·{\left[R_{2/3}\left({\theta }_1\right)\right]}^N·\left[1-R_{2/1}\left(\varphi \right)\right]$$

(2)

where $R_{k/j}$ is the reflectance at the interface between medium k and medium j ($k,j=1,2,3$ with $1=air,2=prism/plate,3=sample$); Reflectance at the metal-coated surface of the plate is taken as unity; Equation (2) is valid for the prism, as well as the plate, by proper substitution of the respective N value. In turn, $R_{k/j}$ is the squared modulus of the corresponding Fresnel amplitude coefficient, which—for the preferred p-polarization—reads:

$$r_{k/j}(\angle )=\frac{cos(\angle )·{(n_j/n_k)}^2-\sqrt[]{{(n_j/n_k)}^2-si{n}^2(\angle )}}{cos(\angle )·{(n_j/n_k)}^2+\sqrt[]{{(n_j/n_k)}^2-si{n}^2(\angle )}}$$

(3)

where $\angle =\theta ,{\theta }_1,\varphi$ for $(k/j=1/2,2/3,2/1)$ and $n_k,n_j$ are the refractive indices of the respective media.

Then, suppose that the reference medium is replaced by a corrupted sample, which exhibits a complex index

$$n^{{}^{\prime }}=(n_r+\delta n_r)-i·(n_i+\delta n_i)=n+(\delta n_r-i·\delta n_i)$$

(4)

that is, equal to the refractive index n of the reference medium, with an added perturbation term that comprises a real ($\delta n_r$) and an imaginary ($\delta n_i$) component. At this point, we may appropriately define a total index perturbation parameter $\delta$ via:

$$\delta =\delta n_r+\delta n_i.$$

(5)

This index perturbation term shifts the plate transmittance (and output intensity) to new values T (and I) that can be straightforwardly calculated by substituting $n^{{}^{\prime }}$ for n in Equation (2). Assuming that the incident light intensity $I_{in}$ remains constant, the relative intensity change $\Delta I/I$ between measurements of the corrupted and reference sample is:

$$\frac{\Delta I}{I}=\frac{I-I_{ref}}{I}=\frac{T-T_p}{T}$$

(6)

Figure 2 depicts the reference transmittance $T_p$, along with the relative intensity change $\Delta I/I$, versus external angle of incidence $\theta$; specifics on parameters used for calculations are given in the figure’s label. (Note that the assumption of unit reflectance at the metal-coated surface may lead to somewhat overestimated $T_p$ values, but does not affect the relative intensity change.) Apparently, $\Delta I/I$ exhibits local extremum at a critical external angle $\theta ={\theta }_c$, which shifts only slightly, when a perturbation in the real index alone ($\delta =\delta n_r$) is replaced by an equal-sized perturbation in the imaginary index ($\delta =\delta n_i$). Interestingly, these two cases are distinguishable by the asymmetric shape of the relative intensity change curve; indeed, when $\delta =\delta n_r$ ($\delta =\delta n_i$), $\Delta I/I$ decays faster towards smaller (larger) incidence angles, respectively. These observations reflect similar properties of the reflectance derivatives with respect to $n_r$, $n_i$ and may be relevant for applications requiring sensor’s selectivity.

3. Signal, Sensitivity and Acceptance Bandwidth

From a practical standpoint, the most important features of the relative intensity change curve are (i) its extreme value at the critical angle ${\left[\Delta I/I\right]}_c$, thereon understood as the sensor’s output signal, (ii) its angular bandwidth $A_{BW}$, defined as the full-width-at-half-maximum and thereon understood as the sensor’s acceptance bandwidth. Due to the entangled dependencies on multiple variables, attempting to evaluate the sensor’s signal and acceptance bandwidth by generating successive plots like those presented in Figure 2, is an elaborate, time-demanding and somewhat confusing task. Fortunately, we found out that it is possible to produce a set of empirical equations that can handily trace both ${\left[\Delta I/I\right]}_c$ and $A_{BW}$ (more details on the heuristic model-building process are given in Appendix A). These equations are:

$${\left[\frac{\Delta I}{I}\right]}_c=S·\delta$$

(7)

$$S\approx -F\left(n_r/n_o\right)·{\left(\frac{n_i}{n_o}\right)}^{-a_o}·\left(\frac{N}{n_o}\right)·{\delta }_1,$$

(8)

$$A_{BW}\approx {10}^5·{\left[\frac{d{\theta }_1}{d\theta }{|}_{\theta ={\theta }_c}\right]}^{-1}·G\left(n_r/n_o\right)·\left(\frac{n_i}{n_o}\right)·{\delta }_2,$$

(9)

where S is sensor’s sensitivity, $A_{BW}$ is given in degrees and:

$${\delta }_1=1-\frac{1}{2}\frac{\delta n_r}{\delta }+\frac{1}{2}{\left(\frac{\delta n_r}{\delta }\right)}^2,{\delta }_2=1-\frac{\delta n_r}{\delta }+{\left(\frac{\delta n_r}{\delta }\right)}^2$$

(10)

are factors acquiring a maximum value of 1 when $\delta =\delta n_r+\delta n_i=\delta n_r$ or $\delta =\delta n_r+\delta n_i=\delta n_i$, and a minimum value when $\delta n_r=\delta n_i=\delta /2$,

$$F\left(n_r/n_o\right)=a_1·exp[a_2{\left(\frac{n_r}{n_o}\right)}^{-1}]+a_3{\left[1-a_4\left(\frac{n_r}{n_o}\right)\right]}^{-1},$$

(11)

$$G\left(n_r/n_o\right)=a_5+a_6·{\left[1-a_7\left(\frac{n_r}{n_o}\right)\right]}^{-1/2},$$

(12)

and

$$\frac{d{\theta }_1}{d\theta }{|}_{\theta ={\theta }_c}=cos{\theta }_c·{\left[{\left(\frac{n_{air}}{n_o}\right)}^{-2}-si{n}^2{\theta }_c\right]}^{-1/2},$$

(13)

with: $a_o=0.49458$, $a_1=1.84$, $a_2=0.695658$, $a_3=0.267344$, $a_4=0.990367$, $a_5=0.000717133$, $a_6=0.00259023$, $a_7=1.00072$.

Equations (7)–(9) reproduce the sensor’s signal, sensitivity and acceptance bandwidth with a typical accuracy of second significant digit, which is sufficient for evaluation and optimization purposes, as long as the respective variables remain within the following ranges:

$$\delta <n_i,N<50,0.5<\frac{n_r}{n_o}<1,{10}^{-2}<\frac{n_i}{n_o}<{10}^{-8},$$

(14)

The information contained in these empirical equations is graphically exposed in Figure 3. Therein, it is clear that ${\left[\Delta I/I\right]}_c$ depends explicitly on $\delta$, $\delta n_r/\delta$, $N/n_o$, $n_r/n_o$ and $n_i/n_o$. These dependencies are linear with respect to $\delta$ and $N/n_o$, but nonlinear with respect to the remaining three variables, such that the signal maximizes when: $\delta n_r/\delta =0or1$, $n_r/n_o\to 1$ and, more rapidly, as $n_i/n_o$ reduces. It is noted that sensitivity S is by definition a direct measure of the output signal—after normalization by the input stimuli $\delta$—and thus, it behaves likewise.

Comparatively, $A_{BW}$ exhibits nonlinear dependencies on $\delta n_r/\delta$, $n_r/n_o$ and scales linearly with $n_i/n_o$; however, it is not affected by $\delta$ or N. Contrary to ${\left[\Delta I/I\right]}_c$, which is geometry-independent, $A_{BW}$ varies also with the critical angle ${\theta }_c$. This nonlinear dependence emerges from the derivative $d{\theta }_1/d\theta {|}_{\theta ={\theta }_c}$, which translates the internal angular bandwidth (in terms of ${\theta }_1$) into an external one (in terms of $\theta$). Note that the critical angle ${\theta }_c$ is tunable by means of the plate’s angle $\alpha$, as is indicated by Equation (1). Derivative $d{\theta }_1/d\theta {|}_{\theta ={\theta }_c}$ introduces also a relatively weak dependence on $n_{air}/n_o$, which can be neglected at first instance and, thus, is not included in Figure 3. To sum up, angular acceptance bandwidth maximizes when: $\delta n_r/\delta =0or1$, $n_r/n_o\to 1$, ${\theta }_c\to {90}^{°}$ and, more rapidly, as $n_i/n_o$ increases.

Remarkably, the main regulator of the sensor’s sensitivity (or equivalently, signal) and acceptance bandwidth is the sample’s imaginary index, which introduces a trade-off between these quantities. Indeed, at the limit of transparency $(n_i=0)$ Equations (8) and (9) return an infinite S and a null $A_{BW}$; this result conforms with the standard CARD interpretation, conveying features of the reflectance derivative with respect to incidence angle, at the point of discontinuity. Inversely, as loss increases, sensitivity tends to zero and acceptance bandwidth grows larger.

The only means available for simultaneous increase in S and $A_{BW}$, which is an obvious-seeming operational advantage, is to stretch parameters $\{n_r/n_o,N,{\theta }_c\}$ close to their limiting values $\{1,\gg 1,{90}^{°}\}$. Although appealing in principle, these limits should be avoided in practise, because they diminish the overall prism/plate transmittance, an effect that intensifies with increasing loss and complicates output light detection. There exists two more reasons to avoid $N\gg 1$. First, large N values deteriorate sensor’s performance if the plate is not perfectly parallel, a repercussion that is otherwise minor, considering the sub-$\mu rad$ tolerances that are nowadays available with state-of-the-art plane-parallel optics. Second, $N\gg 1$ corresponds to large plate lengths, compromising the device’s compactness and increasing sample volume demands.

4. Detection Limit Considerations

The trade-off between sensitivity and angular acceptance bandwidth raises some interesting questions; does an optimal value for $n_i$ and by extension, a preferred wavelength of operation, exist? If so, how can it be determined? The compromise solution to these problems emerges via detection limit ($DL$) considerations.

The detection limit, i.e., the minimum index perturbation that can be reliably sensed, is an instrument, and application-specific quantity that can be estimated, provided that significant sources of noise are distinguishable. For example, let us assume that non-negligible noise contributions originate from (i) the detection stage $\left({\sigma }_{det}\right)$, (ii) power fluctuations of the laser source $({\sigma }_{laser}$), (iii) random fluctuations in the real index of the sample, caused by temperature instabilities $\Delta T$ and the nonzero spectral bandwidth $\Delta \lambda$ of the laser source, which introduce noise variance components ${\sigma }_T=S·[d{n}_r/dT]·\Delta T$ and ${\sigma }_{\lambda }=S·[d{n}_r/d\lambda ]·\Delta \lambda$, respectively. These assumptions are reasonable, as long as the thermooptic and dispersion coefficients for the real index of the sample (i.e., $d{n}_r/dT$ and $d{n}_r/d\lambda$) are much larger than the respective coefficients for $n_i$, $n_o$, and $n_{air}$.

Then, we may adopt standard convention $DL=3\sigma /S$, where $\sigma \approx \sqrt{{\sigma }_{laser}^2+{\sigma }_T^2+{\sigma }_{\lambda }^2+{\sigma }_{det}^2}$ is the total noise variance, so as to deduce the following useful approximation:

$$DL\approx 3\sqrt{{\sigma }_{laser}^2+{\left[S\frac{d{n}_r}{dT}\Delta T\right]}^2+{\left[S\frac{d{n}_r}{d\lambda }\Delta \lambda \right]}^2+{\sigma }_{det}^2}/S.$$

(15)

It is appropriate at this point to parenthetically note that Equation (15) combined with the “rule of the thumb” articulated in Equation (14), specify the range of sensor’s linear response, which is the last remaining major specification that depends also on sample’s loss:

$$Linearrange:DL<\delta <n_i.$$

(16)

Under the assumptions validating Equation (15), the detection limit decreases monotonically with increasing sensitivity, reaching its minimum theoretical value $D{L}_{min}$ at the transparency limit (that is, where $S\to \infty$):

$$D{L}_{min}=\underset{S\to \infty }{lim}DL=3\sqrt{{\left[\frac{d{n}_r}{dT}\Delta T\right]}^2+{\left[\frac{d{n}_r}{d\lambda }\Delta \lambda \right]}^2}$$

(17)

However, reduction in the detection limit by means of sensitivity increase is meaningful only up to the point that $S=S_{opt}$, where

$$S_{opt}=\sqrt{\frac{{\sigma }_{laser}^2+{\sigma }_{det}^2}{{\left[\left(d{n}_r/dT\right)\Delta T\right]}^2+{\left[\left(d{n}_r/d\lambda \right)\Delta \lambda \right]}^2}}.$$

(18)

Indeed, when $S=S_{opt}$, the detection limit reaches its “optimum” value $D{L}_{opt}$ that is equal to:

$$D{L}_{opt}=\sqrt{2}D{L}_{min}=3\sqrt{2}\sqrt{{\left[\frac{d{n}_r}{dT}\Delta T\right]}^2+{\left[\frac{d{n}_r}{d\lambda }\Delta \lambda \right]}^2}.$$

(19)

Further increase in sensitivity towards infinity (by decreasing $n_i$ towards the transparency limit) has a negligible effect on $DL$, while drastically reducing the acceptance bandwidth. Besides, higher sensitivities also compromise sensor’s range of linearity, as may be seen by inspection of Equation (15). Therefore, operating the sensing interface with $S>S_{opt}$ is not only unnecessary but also detrimental.

5. Spatially-Unresolved vs. Spatially-Resolved Detection

Setting $S=S_{opt}$ by choosing the proper value for $n_i$ automatically locks angular acceptance bandwidth to its own respective value, which is easily determinable, as well as decisive for the design of the detection stage. The latter can be based either on a spatially-unresolved, or on a spatially-resolved, scheme.

Accurate spatially-unresolved detection (SUD) by use of a standard photodiode is the most straightforward option; it requires that the input light is stably coupled into the prism/plate at the critical angle ${\theta }_c$, with its entire optical power contained within an angular spread that does not exceed $A_{BW}$. Assuming an input beam with high pointing-stability, waist radius $w_o$ and quality factor $M^2$, the full-angle divergence ${\theta }_d$ at the diffraction-limit is:

$${\theta }_d=\frac{2M^2\lambda }{\pi w_o}$$

(20)

and the implied condition for SUD detection reads:

$$SUDcondition:\frac{A_{BW}}{{\theta }_d}\ge 1.$$

(21)

As $A_{BW}$ becomes narrower (i.e., $S_{opt}$ increases), the SUD condition necessitates beams with large waists. When Equation (21) is not fulfilled, the signal hides inside a fraction of the beam profile and, hence, spatially-resolved detection (SRD) by use of a diode array should be opted. Accurate SRD requires that the “active area” of the output beam profile (i.e., the fraction containing the sensing information) should be larger than the surface of at least one pixel of the diode array, that is:

$$SRDcondition:\frac{d_{pixel}}{2w}<\frac{A_{BW}}{{\theta }_d},$$

(22)

where $d_{pixel}$ is the pixel’s diameter and w is the beam radius at the detector.

When either of these conditions is fulfilled, detection noise may become negligible and thus, it is eliminated from Equation (15) and (18). Indeed, removing measurement complications associated with the finite acceptance bandwidth can reduce ${\sigma }_{det}$ down to levels that relate only to fundamental noise mechanism, such as thermal and quantum noise. Then, we may reasonably assume that (i) the noise-equivalent-power at the photodiode will be in the sub-$nW$ range, (ii) output light power reaching the detector will far exceed $\sim 1\mathsf{\mu }W$. These assumptions confirm the estimate that ${\sigma }_{det}\ll {10}^{-3}$, which is indeed well-below the power noise ${\sigma }_{laser}$ of common laser sources.

6. Optimization Protocol

Following the preceding analysis, it is now possible to formulate a simple optimization process that can be always adopted, as long as the optical constants of the reference sample (i.e., $n_r,n_i,d{n}_r/dT,d{n}_r/d\lambda$) and their wavelength dependence are known. For the sake of simplicity, we may select ${\delta }_1={\delta }_2=1$, since in most practical cases an extraneous stimulus will affect primarily either the real or the imaginary index of the sample. The proposed protocol comprises the following steps:

Step 1: Impose strict temperature regulation and narrow-band laser emission (or spectral filtering), so as to reduce $\Delta T$ and $\Delta \lambda$; for any given sample, these are the only free variables affecting the detection limit, which can be calculated by use of Equation (19).

Step 2: Compute the optimum sensitivity $S_{opt}$ via Equation (18); it is at this stage that laser power noise ${\sigma }_{laser}$ becomes relevant, while as previously explained, detection noise ${\sigma }_{det}$ can be considered practically negligible.

Step 3: Select a transparent solid with known index $n_o>n_r$ that can be shaped into a prism. Initially, avoid close-matching between the refractive indices of the front medium and the sample, so as to ensure substantial total transmittance. For the same reason, the number of reflections is initially set to $N=1$.

Step 4: Given the light extinction spectrum of the sample, identify the laser wavelength that corresponds to the value of $n_i$ which, when substituted to Equation (8), allows sensitivity to reach its optimal value $S_{opt}$.

Step 5: With all involved variables being determined, calculate the respective angular acceptance bandwidth $A_{BW}$ using Equation (9); to sustain high total transmittance, select initially the external critical angle ${\theta }_c$ for grazing incidence.

Step 6: By use of the criteria shown in Equations (21) and (22), evaluate the feasibility of spatially-unresolved and spatially-resolved detection geometries. If found practical, the optimization process is complete and the setup fully drawn accordingly. Otherwise, go on to the next step:

Step 7: Repeat the process so as to increase $A_{BW}$, by selecting an optically denser transparent medium, a parallel plate with multiple reflections instead of a prism, and oblique incidence, in order to approach the operational limits $\{n_r/n_o,N,{\theta }_c\}\to \{1,\gg 1,{90}^{°}\}$. Ensure that the total transmittance is high enough to empower output light detection and the resulting plate length is small enough to comply with application’s needs.

7. Further Discussion and Concluding Remarks

To demonstrate the susceptibility of the sensor’s specifications to the various parameters involved,

Table 1. Results from the application of the optimization protocol to three hypothetical scenarios. Calculations assume ${\delta }_1={\delta }_2=1$, $M^2=1.2$ and ${\sigma }_{det}=0$. Units: T in °C; $\lambda$ in $\mathsf{\mu }$ m; $A_{BW}$ in degrees; $w_o$ in mm; $DL$ in $\mathsf{\mu }$RIU. Other quantities are dimensionless. Sensor’s linear range can be deduced from available data by use of Equation (16). Minimum useful signal (i.e., relative intensity change) at the detector amounts to $S·DL\approx 4\%,2\%$ and $0.4\%$, for Cases 1, 2 and 3, respectively.

Parameters	Case 1	Case 2	Case 3
$\Delta T$	1	$0.1$	$0.01$
$\Delta \lambda$	${10}^{-3}$	${10}^{-4}$	${10}^{-5}$
${\sigma }_{laser}$	$1\%$	$0.5\%$	$0.1\%$
$n_r$	$1.5$	$1.5$	$1.5$
$n_i$	$3\times {10}^{-3}$	$1.3\times {10}^{-4}$	$1.0\times {10}^{-5}$
$d{n}_r/dT$	${10}^{-4}$	$5\times {10}^{-5}$	${10}^{-5}$
$d{n}_r/d\lambda$	$0.1$	$0.05$	$0.01$
$n_o$	2	$1.8$	$1.6$
N	1	2	4
${\theta }_c$	15	45	75
$A_{BW}$	$1.82$	$0.12$	$0.03$
$A_{BW}/{\theta }_d$	$41.5·w_o/\lambda$	$2.7·w_o/\lambda$	$0.8·w_o/\lambda$
$S=S_{opt}$	71	710	7100
$DL$	600	30	$0.6$

presents results from the application of the proposed protocol to three hypothetical scenarios. Chosen values of variables remain within realistic ranges, being very tolerant in Case 1 and becoming more and more rigid as we move to Cases 2 and 3. Top rows in Table 1 group together noise-related parameters $\Delta T,\Delta \lambda$ and ${\sigma }_{laser}$ (${\sigma }_{det}$ is considered negligible). The second set accounts for sample’s optical properties, namely $n_r,n_i,d{n}_r/dT,d{n}_r/d\lambda$, while the third set incorporates information regarding the transparent medium and the prism/plate geometry, that is $n_o,N$ and ${\theta }_c$. Based on these values, acceptance bandwidth $A_{BW}$ and ratios $A_{BW}/{\theta }_d$ are then evaluated. At the bottom of Table 1, corresponding values for the sensitivity S, optimal sensitivity $S_{opt}$ and detection limit $DL$ are given. In accordance with the protocol’s guidelines, the sample’s imaginary index $n_i$ is always chosen such that sensitivity S, quantified via Equation (8), equals its optimal value $S_{opt}$ which, in its own turn, is determined via Equation (18).

Several points of interests emerge from data in Table 1. Depending on the magnitude of $\Delta T$ and $\Delta \lambda$, the detection limit varies from $600\mathsf{\mu }$RIU down $30\mathsf{\mu }$RIU and sub-$\mathsf{\mu }$RIU levels, as we move on from Case 1 via Case 2 to Case 3; $DL$ reduction is accompanied by (i) increase in $S_{opt}$ from 71 to 710 and 7100, (ii) decrease in $n_i$ from $3\times {10}^{-3}$ to $1.3\times {10}^{-4}$ and $1\times {10}^{-5}$, (iii) narrowing of $A_{BW}$ from ${1.82}^{°}$ to ${0.12}^{°}$ and ${0.03}^{°}$, respectively. Corresponding $A_{BW}/{\theta }_d$ values indicate that spatially-unresolved detection is possible with beam waists $w_o\ge$ 0.02 mm, 0.37 mm and 1.25 mm, respectively, assuming for the sake of argument an operational wavelength in the $1\mathsf{\mu }$m range. To meet these specifications, parameters $\{n_r/n_o,N,{\theta }_c\}$ changed from initial values $\{0.75,1,{15}^{°}\}$ in Case 1, to $\{0.83,2,{45}^{°}\}$ in Case 2 and $\{0.89,4,{75}^{°}\}$ in Case 3; corresponding values for the prism/plate cut angle are $\alpha ={55.95}^{°},{79.57}^{°}$ and ${106.77}^{°}$. As a final check, these changes cause no severe cutback to the overall plate transmittance, which always exceeds $50\%$, requiring, however, plate lengths as large as $L\approx$ 3 cm in Case 2 and $L\approx$ 11 cm in Case 3, for an assumed plate thickness $d=0.5$ cm.

If such plate lengths are inappropriate, parameters $\{n_r/n_o,N,{\theta }_c\}$ can be kept constant to their initial values (i.e., $\{0.75,1,{15}^{°}\}$). The detection limit would then be unaffected, but acceptance bandwidth would be reduced in Cases 2 and 3, such that spatially-unresolved detection would require beam waists $w_o\ge$ 2.5 mm and 250 mm, respectively. Therefore, spatially-resolved detection would become a reasonable option in Case 2 and an inevitable choice in Case 3 for which, a maximum pixel diameter of $16\mathsf{\mu }$m is specified, when plugging into the SRD criterion indicative values $w_o\approx$ 1 mm and $w\approx$ 2 mm.

As a further practical demonstration on the relevance of our analysis to real-life applications, Figure 4 depicts the wavelength scaling of S and $A_{BW}$ for two interfaces comprising water as rare medium and (i) calcium fluoride, (ii) standard $SF10$ glass, as front media. Water is chosen as an exemplary test medium since it is the main constituent of biofluids/tissues and a highly relevant medium for environmental sensing and quality control purposes. Calculations adopt values for the required optical constants (that is, the wavelength-dependent complex index of water and the wavelength-dependent real indices of $Ca{F}_2$ and $SF10$, which can be considered transparent throughout the spectral range of interest) from references [39,40,41]. Within the near-infrared band $0.9\mathsf{\mu }$m–$1.6\mathsf{\mu }$m, S and $A_{BW}$ can be tuned by approximately two and three orders of magnitude, respectively, for the SF10/water interface and indicative values of $N=1$, ${\theta }_c={15}^{°}$. This behavior reflects primarily the absorption profile of water, which exhibits a minimum at $0.9\mathsf{\mu }$m ($n_i\approx 4.5\times {10}^{-7}$) and a maximum at $1.46\mathsf{\mu }$m ($n_i\approx 3.2\times {10}^{-4}$). Further tuning is possible by increasing the values of N, ${\theta }_c$ and simultaneously using $Ca{F}_2$ as front medium, whose index matches better the real index of water, thus corresponding to larger values of $n_r/n_o$. Figure 4 reveals the heavy impact of laser wavelength on the sensor’s specifications, an effect that does not apply only to water, but practically extends to all non-black, liquid or solid media that exhibit absorption profiles containing nearly-transparent regions, along with absorption peaks. Therefore, multipurpose critical angle refractometers, which are nowadays commercially available and typically equipped with just one or few single-wavelength lasers, would benefit significantly from exploiting tunable sources of coherent radiation, such as continuous-wave optical parametric oscillators [42].

To conclude, in this technical note we thoroughly examined the sensing properties of transparent/lossy interfaces. An empirical framework was developed to describe performance specifications of corresponding sensing geometries that exploit either a single reflection inside a prism, or multiple reflections inside a parallel plate. The dominant role of sample’s loss was revealed and quantified through a trade-off between sensitivity on one hand and angular acceptance bandwidth on the other. These observations enabled the establishment of a generically applicable optimization’s protocol that is based on two main realizations. First, a noise-dependent optimal value $S_{opt}$ does exist, so that the quest for ever-increasing sensitivities is not just pointless, but also undesirable. Second, operating the sensing interface with $S=S_{opt}$ directly locks acceptance bandwidth $A_{BW}$ to a calculable value, which then imposes definable conditions for the spatially-unresolved—or spatially-resolved—detection of the signal. The exemplary application of the proposed optimization protocol to three hypothetical scenarios revealed the critical dependence of sensor’s specifications on the relevant optical and geometrical variables; this fact was further supported by calculations of the—widely varying—sensitivity S and acceptance bandwidth $A_{BW}$, as a function of wavelength, in the representative cases of the $Ca{F}_2$/water and $SF10$/water interfaces. In this sense, the present note clarifies several design issues in the area of optical refractometry, relating to the pump source (e.g., choice of wavelength), the mounting medium (e.g., choice of prism material and geometry), as well as the detection stage (that is, spatially unresolved versus spatially resolved approach). We thus anticipate that our study will serve as a valuable tool for the realization of optimized CADR setups, which will push the limit of detection towards the sub-$\mathsf{\mu }$RIU regime.