1. Introduction

With the spreading of nanotechnology, there is an increasing need for ultrasensitive surface imaging techniques. Indeed, “nano” samples are most often supported on solid surfaces. This holds for all domains of application ranging from molecular electronics to diagnostics to biology to electrochemistry. In reflected light microscopy, anti-reflecting surfaces enhance the contrast of tiny objects by turning off the background reflectivity. This serves as a basis for a number of label free optical techniques [1–6]. Among them, the Backside Absorbing Layer Microscopy (BALM), a recent contrast technique in wide field optical microscopy [7,8], is especially promising. This reflected light microscopy lies on the use of special anti-reflecting layers as sample supporting plates. They are named ARA (meaning Anti-Reflecting and made of Absorbing materials). BALM offers the unmatched combination of a 200 nm optical resolution and a SPR-like sensitivity (SPR = Surface Plasmon Resonance [5]). It is also remarkable for its simplicity and for its flexibility, since the top surface of the ARA coating can be of almost any nature and observed in any ambient medium using any reflected light microscope. The only requirement is to image the sample supported by the ARA coating from the opposite side of a thin transparent window on which it is deposited. The performance of the technique was already demonstrated by imaging 2D flakes of various materials [8], by following in situ and in real time their evolution upon small molecules adsorption and by detecting operando the electrochemical reduction of individual sub-10nm nanoparticles [9]. However, up to now the BALM images remained mainly qualitative [10] when characterization and sensing applications demand quantitative information. In the present work, we establish a universal relationship between the BALM reflectivity of a molecular layer sample and the physical thickness of this layer, with the two quantities made adimensional. This relationship can be used without any knowledge of the microscope settings or the ARA coating in order to obtain topographic AFM-like images of the sample, whatever its complex refractive index, in a typical thickness range going from 0 to 30 nm. The thickness unit can be further determined with the help of a single reference measurement.

The most direct approach for a quantitative measurement consists in a comparison of the experimental data with a numerical calculation of the expected optical signal. It requires the adjustment of all the parameters affecting this signal. In the case of BALM, their number exceeds ten. Indeed, BALM is a white light, high aperture technique and there are many benefits to use ARA multi-layers made of two or three superimposed materials [11]. The modeling must take into account the spectral characteristics of the source and of the camera, all incidence angles contributing to the signal on the microscope, and one thickness and one complex refractive index per constitutive layer of the ARA coating. The comparison is heavy and risky. In addition, It must be redone for each BALM surface and each setup of the microscope. At the opposite of this “steam roller” procedure is the empirical approach, based on the comparison of the sample with a ruler made of regular steps imprinted on the same BALM substrate, allowing to map the local thickness. However, the reflectivity also depends on the refractive index of the sample, so it must be the same for the sample and the ruler, and this is a serious limitation. An alternative approach is to calibrate the optical measurement with a few AFM measurements on the same sample [11]. It has other drawbacks such as limitations on the nature of the sample or having to extract the sample from its environment in the optical setup, especially inconvenient when in a liquid. Here we present a new approach, which is equivalent in practice to replace the many above listed parameters by only three consolidated parameters, that we name C-parameters, which are functions of the former ones but independent on the physical thickness of the sample. These C-parameters have a direct physical meaning when considering the BALM reflectivity $R(e )$ as a function of the sample thickness. Our approach is to expand $R(e )$ around its minimum. It takes advantage of two BALM specificities, namely the very low BALM reflectivity and the inverted (“backside”) geometry. The mathematical existence of the minimum ${R_{min}}({{e_{min}}} )$ is granted by the low level, the continuity and the positive sign of $R(e )$. Yet ${e_{min}}$ is not necessary positive. When negative, it is virtual. The inverted geometry allows direct analytical derivation of the sought expansion from a simple Fresnel theory. Limiting ourselves to the lower approximation level (second order), we end up with a universal relationship between an adimensional function of R and a reduced sample thickness, made adimensional as well. The two quantities are ruled case by case by the C-parameters which determine two scale factors acting along, respectively, the e and the R axis. A huge restriction on $R(e )$ is the existence of a fixed point $[{0,{R_0}} ]$, with ${R_0} = R(0 )$. ${R_0}$ is one C-parameter. It is accessible since it corresponds to the bare substrate. Depending on the application, several experimental strategies can be employed in order to determine the two other C-parameters which are ${e_{min}}$ and ${R_{min}}$ . Although their numerical estimate is not forbidden, they are over all also experimentally accessible. Then the thickness of the sample can be mapped without any information about the substrate or the microscope. Even the calibration with a known article becomes optional when the knowledge of the thickness unit is unimportant. As an example of application, we briefly envisage sensor-like kinetic studies.

2. Universal relationship

In this section, we show that BALM reflectivity can be approximated by a second degree polynomial of the sample thickness. After proper rescaling, it will become a universal function, with its square root a linear function of the thickness, hence a linear sensor. The rescaling involves 3 parameters which can quite easily be determined experimentally. Next we confront this rescaling with common process imaging tools. Finally, we show that the method applies as well with continuous or discrete wavelength and/or incidence angle superposition. Figure 1 recalls the BALM geometry, and gives a visual support of the three parameters introduced.

 

Fig. 1. a) Sketch of the BALM setup and layer numbering; b) Hypothetical reflectivity curves as a function of sample thickness. The minimum reflectivity ${\textrm{R}_{\textrm{min}}}$ may correspond to positive (red curve) or negative (blue curve) values of ${\textrm{e}_{\textrm{min}}}$. They can be simultaneously obtained when probing a sample with two different wavelengths. Examples may be found in further Fig. 3(e); c) When ${\textrm{e}_{\textrm{min}}}$ positive, to a given value of $\textrm{R}$ correspond two values ${e_1}$ and ${e_2}$ of the sample thickness e. They can be separated with the help of a second wavelength.

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In the Fresnel theory, the amplitude reflection coefficient of a multilayer [1,2,…, $l$] made of isotropic materials and sandwiched between two semi-infinite media indexed by $0$ and $l + 1$ and lighted from material $0$ is given by [12,13]:

With optical microscopy, an important issue is image contrast. Optimizing the brightness/contrast of the desired patterns with numerical tools consists in changing the raw intensity distribution $I({x,y} )\; $ for a more readable image $J({x,y} )$ according to the affine transform $(J{x,y} )= {q_1}I({x,y} )+ {q_2}$, with ${q_1}$ and ${q_2}$ arbitrary real numbers. Applied to BALM images, the true reflectivity R is thus changed for some displayed reflectivity $R^{\prime} = {q_1}R + {q_2}$. The affine transform affects all the terms in the left member of Eq. (4), and we get $({R^{\prime} - {R_{min}}^{\prime}} )/({{R_0}^{\prime} - {R_{min}}^{\prime}} )= [{{{({({e - {e_{min}}} )/{e_{min}}} )}^2}} ]$, with the right member unchanged. This remark is of a high practical importance: it says that the measurements can be performed as well on brightness/contrast transformed images, and that display optimization of BALM images or videos and quantitative BALM analysis are fully compatible. Notice that following the affine transform, ${R_{min}}^{\prime}$ may become negative. A very interesting situation is to work with an (initially) negative contrast, and to have the sign inversion of $\partial R/\partial e$, hence ${R_{min}}$, a or in time in the image. Then, suppressing all the levels under ${R_{min}}$ in the histogram will set ${R_{min}}^{\prime}$ to zero. In that case $f(R )$ reduces to $\sqrt {R^{\prime}/{R_0}^{\prime}} $ and the square root of the image is a topographic image, with the intensity everywhere proportional to the thickness of the layer, but with a slope inversion at $[{{e_{min}},\; 0} ]$. Instead of calculating the square root of the image, one can equivalently apply a 0.5 gamma correction.

In order to convert the reflectivity into a thickness, the sign in Eq. (5) must be identified. It relies in general on ordering the local reflectivity values in the image, or simply on the appropriate choice of the BALM substrate. The thickness ${e_{min}}$ is positive when the contrast of the burgeoning layer l with the bare substrate is negative, and it is negative with a positive contrast, becoming an extrapolated length. With ${e_{min}}$ positive, the contrast decreases with e up to $e = {e_{min}}$ and increases afterwards. Then, to a given value of R correspond two possible values of the thickness, which are symmetric on each part of ${e_{min}}$. The symmetry can be broken by changing the color of the lighting. Indeed, ${e_{min}}$ depends on this choice. Here we take advantage of the compatibility of the BALM with white light. It is therefore important to examine the effect of color (wavelength) averaging on the measurements. We will envisage at the same time incidence angle averaging, which conditions lateral resolution. Starting again from Eq. (3), the reflectivity can be expressed for a given wavelength and incidence angle as $R({\theta ,\lambda } )= {R_0}({\theta ,\lambda } )+ [{{R_0}({\theta ,\lambda } )U({\theta ,\lambda } )/\lambda } ]e + [{{R_0}({\theta ,\lambda } )V({\theta ,\lambda } )/{\lambda^2}} ]{e^2}$. Assuming incoherent superimposition of all angle and wavelength contributions, we get:

3. Model cases and discussion

Next we make things more concrete by considering three ARA configurations which are particularly relevant for the design of BALM imaging sensors. The first one is the glass/gold/n2/water stack, or $\textrm{GA}{n_2}\textrm{W}$. It is interesting because the gold sensing surface allows direct comparison with SPR, the reference technique in the study of bio-molecular interactions. The second one is glass/gold/silica/n3/water, or $\textrm{GAS}{n_3}\textrm{W}$ which extends the BALM imaging sensor to the silica surface, a standard for surface chemistry. The third one is glass/gold/graphene oxide/${n_3}$/water, or $\textrm{GAGO}{n_3}\textrm{W}$, one of the most promising combination for BALM ultra-high density multi-array diagnostics.

3.1 Gold substrate

Figure 2(a) displays the reflectivity curves $R({{e_2}} )$ expected with a $\textrm{GA}{n_2}\textrm{W}$ BALM substrate with fixed ${n_0} = 1.52$ (glass) and ${n_2} = 1.5$ (sample). They were numerically computed for three different values of the gold layer thickness, for three different wavelengths and for the two lighting configurations of a normal incidence and of a uniform illumination cone with a 30 degrees apex semi-angle (angle averaging). We also included wavelength average, referred to as “white”, by adding with equal weight the contributions of three wavelengths. 450 nm, 480 nm and 550 nm. Superimposing all cases, we obtain in Fig. 2(b) a forest of 24 different graphs without apparent correlation between them. In each case we extract the value of the three C-parameters $[{{R_0},\; {e_{min}},\; {R_{min}}} ]$ and we plot the graph $[{e/{e_{min}},\; f(R )} ]$. The result is shown in Fig. 2(c). All previous graphs merge in a single curve made of two orthogonal line segments with slope -1 and + 1, crossing at $[{1,0} ]$. This figure illustrates how the entire reflectivity analysis resumes in the determination of the three C-parameters. Figure 2(d) clarifies with the help of color correspondences the partial covering of the universal curve specific to each configuration. We previously mentioned that ${e_{min}}$ can be positive as it the case in Fig. 2 or negative, see Fig. 1. It may also be zero, making Eq. (4) and subsequent Eqs. ill-defined. It corresponds to $U = 0$ in Eq. (3), ${R_0} = {R_{min}}$, and $({R - {R_{min}}} )/{R_{min}} = V{e^2}$. In this special case, $f(R )= \; \sqrt {R/{R_{min}} - 1} $ and the scaling factor for e is $\sqrt V $ in place of ${e_{min}}$. In the left part of Fig. 2(c), small deviations with respect to Eq. (4) (or equivalent Eq. (7)) are observed. They correspond to the case ${e_1} = 5\ \textrm{nm}$ where $\; {e_{min}}$ is large and where the second order approximation reaches its limits. When needed, the expansion of $f(R )$ can be extended to next orders in $({e - {e_{min}}_{eff}} )/{e_{min}}_{eff}$ while keeping all variables adimensional, allowing to reproduce curve asymmetry and inflexion points for larger sample thickness. These developments are beyond our present scope.

 

Fig. 2. a): Reflectivity as a function of the sample thickness ${e_2}$ for 24 configurations : [${e_1}$ = 3, 4 and 5 nm (gold)] x [λ=450, 480, 550 nm and addition of the three wavelengths (“white”)] x [$\theta = 0$ or 30 deg. aperture angle], with one graph per thickness; b): same with the 24 superimposed results; c): transformed reflectivity $f({R/{R_{min}},{R_0}/{R_{min}}} )$, noted $f(R )$, as a function of the reduced sample thickness $e/{e_{min}}$, with different ${e_{min}},\; {R_{min}}$ and ${R_0}$ for each parameter configuration; d): same curve as c) with each thickness isolated and the corresponding $e/{e_{min}}$ range compared to each other using dotted lines; Figure insert : color code for each (λ, aperture) setting. The refractive indices ${n_0}$, ${n_2},$ ${n_3}$, weakly dependent of λ, are respectively fixed to 1.52, 1.5 and 1.34, and ${n_1}\; $ (λ) was taken from [14].

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3.2 Silica substrate

In Fig. 3, the gold layer thickness was fixed to 3 nm. In Figs. 3(a) and 3(c), we envisage the second case, $\textrm{GAS}{n_3}\textrm{W}$ and we chose normal incidence. According to what precedes, the detailed reflectivity curves are unnecessary. Only matters the value of the three C-parameters. With a given substrate, $\textrm{GAS}$ and W are fixed in $\textrm{GAS}{n_3}\textrm{W}$, thus ${R_0}$ is fixed. It is obtained for ${e_3} = 0\; $ and does not depend on ${n_3}$, the zero thickness meaning no layer. Figure 3(a) shows the variation of ${e_{3min}}$ with the thickness ${e_2}$ of the silica layer for a number of refractive indices ${n_3}$. It is monotonic. A straight line with a slope -1 is obtained when ${n_3} = \; {n_2}$, meaning that a variation of ${e_3}$ and the opposite variation of ${e_2}$ are equivalent. With this stack, ${e_{3min}}$ goes from positive to negative, depending on the ${e_2}$ value. It covers the case ${e_{3min}} = 0$, marked by a fixed point in the batch of ${n_3}$-curves in Fig. 3(a) because the absence of layer is insensitive to its refractive index. The corresponding value of ${e_2}$ is $24\ \textrm{nm}$. Figure 3(c) shows the variation with ${e_2}\; $ of ${R_{3min}}$ (minimum of R vs ${e_3}$). It also exhibits a fixed point with a zero slope in correspondence with ${e_{3min}}({{e_2}} )= 0$, i.-e. when ${e_2} = 24\ \textrm{nm}$. As for ${e_{3min}}$, the existence of the fixed point corresponds to the absence of the ${n_3}$ layer. A first order transition on the sign of the curvature around this extremum is observed for ${n_3} = 1.46$. It corresponds to the change of the phase sign at the ${n_2}:{n_3}$ interface. Figures 3(b) and 3(d) show equivalent variations ${e_{3min}} = f({{e_2}} )$ and ${R_{min}} = f({{e_2}} )$ in the case of the $\textrm{GAGO}{\textrm{n}_3}\textrm{W}$ stack. In Figs. 3(b) and 3(d) we chose a 30 deg. microscope aperture, aiming to illustrate the flexibility of our approach. They present a number of common features with Figs. 3(a) and 3(c), in particular the existence of the fixed points, here for ${\textrm{e}_2} = 5,4 \ \textrm{nm}$ . The deviation with the straight line on ${e_{3min}} = f({{e_2}} )$ and the deviation with the zero slope on ${R_{min}} = f({{e_2}} )$ are due the 0.3 absorption coefficient of the GO layer, making ${n_3} = {n_2}$ impossible. Due to the same, the value of ${R_{min}}$ becomes too high (> 10⁻²) when the GO thickness reaches 10 or 15 nm, while it is of order 10⁻³ when ${e_2}$ is under 8 nm. Figures 3(a)–3(d) underline with indexed color curves the important variations of ${e_{min}}$ and ${R_{min}}$ with the sample refractive index away from the fixed point, bringing interesting possibilities for estimating ${n_3}$. Figures 3(e) and 3(f) at last illustrate the strong dependence of ${e_{min}}({{e_2}} )$ and ${R_{min}}({{e_2}} )$ with wavelength expected with $\textrm{GAGO}{\textrm{n}_3}\textrm{W}$ for ${e_1} = 3\ \textrm{nm}$, which opens promising possibilities for multispectral measurements.

 

Fig. 3. a),c) : GASn3W, ${\theta} = 0$ ; b),d): GAGOn3W and illumination cone $\theta = 0 - 30 \deg.$ a),b): ${e_{3min}}$ (probed layer) as a function of ${e_2}$ for a number of ${n_3}$ refractive indices ranging from 1.4 to 1.7; c),d): ${R_{min}}$ as a function of ${e_2}$ for the same ${n_3}$ values; e),f) GAGOn3W with $\theta = 0$ and ${e_1}$ = 3nm (gold). ${e_{3min}}$ and ${R_{min}}$ as a function of ${e_2}$ (GO) for four different wavelengths; inserts: color codes for the sample refractive index in 2a-d and for the wavelength in 2e-f. The GO refractive index as a function of λ was taken from [15].

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3.3 Graphene Oxide spot

Finally, Fig. 4 focuses on a major BALM application which is investigation of (bio)molecular interactions through capture, dissociation or exchange kinetic studies. For illustration, we consider the $GAGO{n_3}W$ stack, where GO flakes are considered the probe spots. For the sake of simplicity, we envisage only one layer fixing on the GO surface, with refractive index 1.5. We chose 3 nm and 2 nm for respectively the Au and the GO thickness because it makes a good compromise between negative contrast (${e_{3min}} > 0)$, high sensitivity and ease of manufacture. We also fix $\lambda = 480\ \textrm{nm}$, the most sensitive wavelength with the gold layer [7] and we consider both a normal incidence and a 30 degrees microscope aperture. In presence of the target solution, the growth of the target layer $({{e_3},{n_3}} )$ on GO is revealed by the time dependent reflectivity $R[{{e_3}(t )} ]$. It is linked to the layer thickness by Eq. (4). ${e_{min}}$ positive is an essential asset in the kinetic studies because, in addition to bare substrate reflectivity ${R_0}$, the parameter ${R_{min}}$ is experimentally determined, the curve $R[{{e_3}(t )} ]$ covered during the kinetics passing through the minimum $[{t({{e_{min}}} ),{R_{min}}} ]$. Figure 4(c) shows a canonical kinetics ${e_3}(t )= {e_3}(\infty )({1 - {e^{ - t/\tau }}} )$, with ${e_3}(\infty )$ the saturation thickness fixed to 25 nm and ${\tau}$ the undefined characteristic time of the capture process. Figures 4(a) and 4(e) show the corresponding evolution of the BALM reflectivity, respectively calculated for a normal incidence and for a 30 deg. aperture. In each case, ${R_{min}}$ and $t({{e_{min}}} )$ are extracted. Figures 4(b) and 4(f) are obtained from Figs. 4(a) and 4(e) by applying the transform $R \to f({R/{R_{min}},{R_0}/{R_{min}}} )$, with a careful attention paid to the sign. Although R is a function of t, the transform automatically yields the two straight lines displayed in Figs. 4(b) and 4(f), a universal function of $e(t )/{e_{min}}(t )$ independent of t. In other words, the curve in Figs. 4(b) or 4(f) is always the same, parameterized by t. t rules the speed at which the curve is explored but does not impact its shape. From Figs. 4(b) or 4(f), we get the result $t({e/{e_{min}}} )$ shown in Fig. 4(d) when replacing $f[{R(t )} ]$ by the corresponding value of t.

 

Fig. 4. Schematic sequence of a kinetic experiment. The green arrows show two measurement and analysis sequences with two different illuminations. c) postulated kinetics ${e_3}({t/\tau } )$; a) Simulated experimental reflectivity $R({t/\tau } )$ for a normal incidence; e) Simulated reflectivity $R({t/\tau } )$ for a 30 deg. aperture; b) $f(R )$ as a function of $\textrm{e}/{\textrm{e}_{\textrm{min}}}$ resulting from the transform of a); e) same with the transform of e); d) measured kinetics $({t/\tau } )({{e_3}} )$ obtained from a) and b) or from e) and f); fixed values $e1$=3 nm (gold) and $e2$ = 2 nm (GO).

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In Fig. 4, the two sequences corresponding to the two aperture settings are highlighted by the green arrows. The C-parameters ${R_0}$, ${e_{min}}$ and ${R_{min}}$ are different in each case but the process and the result are the same, apart for the unknown thickness unit, which case by case determination requires a reference step or a single external measurement. The comparison of the two sequences illustrates with the aperture example how we save the adjustment of experimental parameters. Incidentally, it also illustrates the weak sensitivity of BALM to numerical aperture, which is the cornerstone of the BALM potential in imaging sensor applications. To be practical, nothing but the presence of noise would be different with a real kinetic in Fig. 4(c). The BALM experiment would deliver Figs. 4(a) or 4(e), and Fig. 4(d) would be directly obtained from the transform ${({f^\circ R} )^{ - 1}}(t )$. For a 90° rotation, it reproduces the real kinetics of Fig. 4(b). Thus, Figs. 4(b) or 4(f) have mainly a pedagogical virtue. Whatever the experimental conditions, the characteristic time ${\tau}$ would be obtained from the time dependent reflectivity without any calibration. Measurement of the saturation level ${e_3}(\infty )$ would require the help of a single thickness caliber.

4. Conclusion

To step back from the above examples, the aim of our approach is to introduce three consolidated “C-parameters” in replacement of the multiple parameters to consider when fitting the experimental curve with a model. The former ones are functions of the latter, and constitute an intermediate stage between influencing parameters and analysis. Studying the dependence of the C-parameters with some others is a useful route for many purposes such as instrument optimization, substrate design and characterization or refractive index measurement. On the other hand, using the C-parameters in order to obtain the “AFM-like” topography of a sub-nanometer thick sample film is immediate and effective with very little effort. In the numerical examples presented above, we have only envisaged purely transparent, i.e. non absorbing sample layers. The case of absorbing layers was addressed in [11]. The high sensitivity of ${e_{min}}$ on ${k_3}$ was demonstrated, which confirms possibilities for molecular layer characterization. We do not insist here on the refractive index analysis, since our present focus is to introduce the universal relationship between reflectivity and physical thickness as a basis for quantitative BALM measurements. Application to real experiments will follow with SPRi-like kinetic studies and monolayer counting in 2D material flakes.