Backside Absorbing Layer Microscopy: A universal relationship between physical thickness and reflectivity

The Backside Absorbing Layer Microscopy (BALM) is a recently introduced surface imaging technique in reflected light with an unprecedented combination of sensitivity and lateral resolution, hence very promising for the development of imaging sensors. This requires to turn BALM images into quantitative measurements. The usual way to analyze reflectivity measurements is to compare the optical signal and a numerical model with many adjustable parameters. Here we demonstrate a universal relationship between the sample reflectivity and the physical thickness of the sample, ruled by three measurable quantities. Mapping the true sample thickness becomes possible whatever the instrument configuration and the sample refractive index. Application to kinetic measurements is discussed. © 2018 Optical Society of America under the terms of the OSA Open Access Publishing Agreement


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][2][3][4][5][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 antireflecting 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 [3] 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 (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 as a function of the sample thickness. Our approach is to expand 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 is granted by the low level, the continuity and the positive sign of . Yet 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 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 and the axis. A high constraint on is the existence of a fixed point , with . is one C-parameter. It is experimentally 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 and . Although a numerical estimating of these parameters is not forbidden, they are 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.

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.
In the Fresnel theory, the amplitude reflection coefficient of a multilayer [1,2,..., ] sandwiched between two semi-infinite media indexed by and and lighted from material is given by [12,13]: the reflection coefficient of the -1 first layers with the material semi-infinite, the Fresnel coefficient of the interface, the phase factor in layer , the wavelength of light, and the thickness and complex refractive index of layer and the angle of refraction in that layer. In imaging and sensing applications based on reflectivity, the probed layer can be either layer 1, we call it the upright geometry, or layer , we call it the inverted geometry. Eq. (1) gives a special status to the last layer. The BALM measurement envisaged herein consists in extracting the physical thickness of the last layer from the (complex) amplitude coefficient given by Eq. (1). The two terms and are functions of the refractive index of the probed layer, and is a function of both its refractive index and thickness. A complex reflection coefficient cannot be directly probed from a reflectance measurement. What is measured is . In general, is a complex number. On the understanding that we talk about the last layer in the stack, we will most often neglect the index. Setting and expanding to second order in and , we find: , with the bare substrate reflectivity, corresponding to in Eq. (1), and the polynomial coefficients (not made explicit here) independent of . The relative reflectivity change with the presence of the probed layer is: , with and independent of . Eq. (3) shows thickness and index separability, which somehow contrasts with the intuitive belief that the optical thickness obtained from optical interference measurements cannot be split into the two parts. Indeed, Eq. (3) expresses the reflectivity as a parabolic expression of the true thickness , the dependence of with and being confined within the two coefficients and . Even when the probed layer is non absorbing, i.e. , we cannot merge with because depend on via the Fresnel and multilayer reflection coefficients. In a sensing or a sensitive imaging experiment, what is demanded is a maximal optical sensitivity to the presence of the upper layer l. This is obtained when the reflectivity of the substrate is very low. Therefore, the reflectivity given by Eq.
The two parameters and act as two scale factors in a universal relationship. Whatever the nature of the substrate, Eq. (4) will apply. It will only affect and the two unknown scale factors. Once the three C-parameters associated to the BALM surface and the instrument determined, it will deliver thickness measurements everywhere on the same surface, with the thickness unit as the only missing information. When mapping in plane or in time thickness variations of a surface layer, it is a considerable intake when the intensity in the image is proportional to the local thickness. Thus we reformulate Eq. (4) as : (5) , with . When unambiguous, we will refer to this function as . The sign uncertainty in Eq. (5) is resolved with .
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 for a more readable image according to the affine transform , with and arbitrary real numbers. Applied to BALM images, the true reflectivity is thus changed for some displayed reflectivity . The affine transform affects all the terms in the left member of Eq. (4), and we get , 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, may become negative. A very interesting situation is to work with an (initially) negative contrast and to have , hence inversion of the reflectivity variation with , appearing somewhere in the image. Then, suppressing all the levels under in the histogram will set to zero. In that case reduces to and the square root of the image is a true topographic image, with the intensity everywhere proportional to the thickness of the layer, but with a slope inversion at . 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 is positive when the contrast of the burgeoning layer with the bare substrate is negative, and it is negative with a positive contrast, becoming an extrapolated length. With positive, the contrast decreases with up to and increases afterwards. Then, to a given value of correspond two possible values of the thickness, which are symmetric on each part of . The symmetry can be broken by changing the color of the lighting. Indeed, 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 . Assuming incoherent superimposition of all angle and wavelength contributions, we get : The averaged reflectivity is a polynomial of similar to Eq. (3), so we can directly write: (7) with and the effective coordinates of the minimum with the averaged reflectivity. The important point is that Eq. (4) applies again. The measurement of can be performed with any lighting spectrum and with any microscope aperture, provided that the contrast remains sufficient.

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 . It is interesting because the gold sensing surface allows direct comparison with SPR, the gold technique in the study of bio-molecular interactions. The second one is glass/gold/silica/n3/water, or which extends the BALM imaging sensor to the silica surface, the gold standard in surface chemistry. The third one is glass/gold/graphene oxide/ /water, or , one of the most promising combination for BALM ultra-high density multi-array diagnostics.  Figure 1a displays the reflectivity curves expected with a BALM substrate with fixed (glass) and (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 Figure 1b a forest of 24 different graphs without apparent correlation between them. In each case we extract the value of the three C-parameters and we pl . The result is shown in Figure 1c. All previous graphs merge in a single curve made of two orthogonal line segments with slope -1 and +1, crossing at . This figure illustrates how the entire reflectivity analysis resumes in the determination of the three C-parameters. Figure  1d clarifies with the help of color correspondences the partial covering of the universal curve specific to each configuration. We previously mentioned that can be positive as it the case in Figure 1 or negative. It may also be zero, making Eq. (4) and subsequent equations illdefined. It corresponds to in Eq. (3), , and . In this special case, and the scaling factor for is in place of . In the left part of Figure 1c, small deviations with respect to Eq. (4) (or equivalent Eq. (7)) are observed. They correspond to the case where is large and where the second order approximation reaches its limits. When needed, the expansion of can be extended to next orders in while keeping all variables adimensional, allowing to reproduce curve asymmetry and inflexion points for larger sample thickness. These developments are beyond our present scope. In Figure 2, the gold layer thickness was fixed to 3 nm. In Figures 2a and 2c, we envisage the second case, and we chose normal incidence. According to what precedes, the detailed reflectivity curves are useless. Only matters the value of the three C-parameters. With a given substrate, and W are fixed in , thus is fixed. It is obtained for and does not depend on , the zero thickness meaning no layer. Figure 2a shows the variation of with the thickness of the silica layer for a number of refractive indices . It is monotonic. A straight line with a slope -1 is obtained when , meaning that a variation of and the opposite variation of are equivalent. With this stack, goes from positive to negative, depending on the value. It covers the case , marked by a fixed point in the batch of -curves in Figure 2a because the absence of layer is insensitive to its refractive index. The corresponding value of is . Figure 2c shows the variation with of (minimum of vs ). It also exhibits a fixed point with a zero slope in correspondence with , i.-e. when . As for , the existence of the fixed point corresponds to the absence of the layer. The zero slope is explained by similar yet less direct arguments as for the existence of the fixed point. We do not dwell on it.  Figure 3 at least 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 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 ( , high sensitivity and ease of manufacture. We also fix , 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 on GO is revealed by the time dependent reflectivity . It is linked to the true layer thickness by Eq. (4). positive is an essential asset in the kinetic studies because, in addition to bare substrate reflectivity , the parameter is experimentally determined, the curve covered during the kinetics passing through the minimum . Figure 3c shows a canonical kinetics , with the saturation thickness fixed to 25 nm and the undefined characteristic time of the capture process. Figures 3a and 3e show the corresponding evolution of the BALM reflectivity, respectively calculated for a normal incidence and for a 30 deg. aperture. In each case, and are extracted . Figures 3b  and 3f are obtained from Figures 3a and 3e by applying the transform , with a careful attention paid to the sign. Although is a function of , the transform automatically yields the two straight lines displayed in Figures 3b and 3f, a universal function of independent of . In other words, the curve in Figure 3b or 3f is always the same, parameterized by . rules the speed at which the curve is explored but does not impact its shape. From Figures 3b or 3f, we get the result shown in Figure 3d when replacing by the corresponding value of . Figure 3 Schematic sequence of a kinetic experiment. The green arrows show two measurement and analysis sequences with two different illuminations. c) postulated kinetics ; a) Simulated experimental reflectivity for a normal incidence; e) Simulated reflectivity for a 30 deg. aperture; b) as a function of resulting from the transform of a); e) same with the transform of e); d) measured kinetics obtained from a) and b) or from e) and f); fixed values =3 nm (gold) and = 2 nm (GO).

Graphene Oxide spot
In Figure 3, the two sequences corresponding to the two aperture settings are highlighted by the green arrows. The C-parameters , and 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 caliber 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 Figure  3c

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 true "AFMlike" 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 on 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.