Coherent absorption of light by graphene and other optically conducting surfaces in realistic on-substrate configurations

Optics of conducting surfaces (i.e., surfaces displaying mainly real ac conductivity at the frequencies considered) has experienced in the past decade major breakthroughs. Initially, researchers established the concept of metasurface, where subwavelength conducting elements (traditionally metals) enable to shape the incident beam in the spatial and temporal domains.^1–3 Afterwards, thanks to graphene and other two dimensional (2D) materials, which exhibit large values of the optical conductivity on a broad frequency band, remarkable non-linear optical response and tunability properties were demonstrated.^4–8 In addition, the conjunction of the two concepts above is expected to further boost the research field, thanks to the huge potential offered by 2D materials patterned as a metasurface^9–11 and by hybrid devices where a metallic metasurface is coupled to 2D conductors.^12–14 

Whether based on 2D materials, traditional metals, or innovative alloys,¹⁵ a (structured) optically conducting surface (OCS) shows a certain degree of energy loss, due to the very nature of the conductors which constitute the device. Such losses may be either detrimental or functional, depending on the intended application. For instance, wavefront-shaping devices and nonlinear optical components would perform better in the absence of losses, while wave filters, sensors, thermal emitters, and memory surfaces have opposite requirements. Hence, in general, it is of paramount importance to have a precise control over the absorption of an OCS.

Due to the extremely thin nature of two-dimensional materials and/or the implementation of patterns that are geometrically disconnected, in most cases OCSs are placed on top of a substrate. If the substrate is transparent and the back surface is optically flat, the absorption of the surface can be tailored interferometrically via a second beam incident on the substrate back surface. By properly calibrating its phase and amplitude, absorption regimes otherwise not reachable in a single-beam arrangement can be achieved, such as coherent perfect absorption (CPA) and coherent perfect transparency (CPT).¹⁶ In the present article, we will derive analytical formulas describing coherent absorption of light in a realistic sample arrangement consisting of an OCS placed on top of a transparent, optically flat substrate. From the theory, two key points stand out: first, CPT can theoretically always be achieved. Second, the measurement of the ellipse minimum in the imbalanced two-beam coherent absorption setup is able to reveal the conductance of an OCS also when the thickness of the supporting substrate is unknown. A technique for nondestructive wafer scale diagnosis of optically conducting surfaces is hence enabled. This technique has a remarkable potential for instance when considering multilayer graphene sheets, whose thickness determination is necessary for applications in optics^17–20 and electronics.^21–23 Currently, several methods are used for estimating the multilayer graphene thickness. However, some limitations and/or drawbacks characterize these methods. For example, low-energy electron microscopy (LEEM) determines the graphene number of layers from the quantized oscillations in the electron reflectivity, but its application is generally restricted to conductive substrates and its reliability is limited up to 10 layers.²⁴ Thickness evaluation by ordinary micro-Raman spectroscopy can be based on the determination of the changes occurring to the width, shape, position, and intensity of the 2D band for samples containing up to 10 layers,²⁵ on monitoring the power of the laser beam reflected from the sample in a slightly modified micro-Raman setup,²⁶ or on quantifying the attenuation factor of the substrate Raman signal in the presence of the graphene layers.²⁷ Nevertheless, the latter two micro-Raman methods cannot be applied on graphene sheets thicker than 40 monolayers. Alternatively, atomic force microscopy (AFM) is a method that can be applied in case of graphene on smooth substrates.²⁶ Although this technique would allow the investigation of very thick graphene, it is mainly appropriate to the case of flakes, whereas it fails for large area graphene, as the multiple layers cover the entire substrate surface. This imposes a destructive patterning of the graphene sheets to create the necessary graphene/substrate steps for the graphene thickness determination. Moreover, the surface morphology is another fundamental aspect that has to be considered when using AFM. High roughness of substrates like silicon carbide, where terraces and steps are created during the graphitization process, may prevent to distinguish graphene areas with different number of layers. Hence, although the present technique seems at par with the aforementioned traditional methods when considering only few-layer graphene characterization, it would constitute a simple tool for assessing in a non-destructive way the properties of films composed by a high number of graphene layers, furthermore overcoming substrate morphology limitations. In addition, the concept is generally applicable to all those OCSs having mostly real ac conductivity.

The physical situation under analysis is represented in Fig. 1(a). An OCS, here schematized as a graphene layer, is placed on top of a substrate having thickness L and refractive index n. The structure is excited by two coherent parallel-propagating plane wave beams of amplitudes $s 1 +$ and $s 2 +$⁠, while $s 1 −$ and $s 2 −$ represent the amplitude of the output beams. The scheme in Fig. 1(b) represents the experimental setup employed to study a sample consisting of multilayer graphene grown on a silicon carbide substrate. In the following, we will provide analytical expressions concerning the CPA/CPT related quantities for the general model structure of Fig. 1(a), while the experimental results obtained through the setup schematized in Fig. 1(b) will be detailed in the second part of the article.

This analytical expression allows for some insightful observations. First, we focus the attention on the single-beam absorbance (A) by a free-standing OCS. Following from the above equation, it straightforwardly reads $A = 4 Re ( G ) η 0 / | 2$ + $G η 0 | 2$⁠. In the case here considered of an optically conducting surface, we can assume G to be approximately real (see the supplementary material for the slight modifications introduced when G becomes complex). The function A(G) is monotonic in the two intervals $G η 0$ ∈ (0,2) and $G η 0$ ∈ (2,∞), hence the conductance determines univocally the absorption. On one side, absorption can be tuned by a proper manipulation of the OCS conductance; on the other, the conductance can be measured relying upon an absorption measurement (provided of course one knows at least in which of the two above intervals G is expected). However, this result relies on a quite abstract arrangement, namely, that of a suspended OCS. Instead, most the practical situations involve an OCS placed on top of a transparent substrate: in this case, if the back surface of the substrate is polished, multiple reflections may occur, and the observations given so far are no more valid. The presence of multiple reflections ends up in Fabry–Pérot resonances, hence the single-beam absorption A will depend on the substrate round trip phase $2 φ$ and it will no more be possible to extrapolate the conductance from an ordinary absorption measurement.

According to Equation (2), the joint absorbance reaches extreme values when sin (⁠$ψ + δ$⁠) = ±1; those extremes are elliptically dependent on the imbalance factor x, as generically depicted in Fig. 2(a). The joint absorbance can be thus described in terms of the amplitude (i.e., the ellipticity) and of the inclination angle of the ellipse. In fact, the ellipse amplitude is quantified by means of the joint absorbance modulation A_mod, whereas the inclination angle of the ellipse is determined by the single-beam absorbances A₁ and A₂. Other important parameters are the minimum and maximum joint absorbance, A_min and A_max. These quantify the amount of transparency (A_min) and of absorption enhancement (A_max) that can be achieved by properly modulating the amplitudes and the phases of the input beams. To connect with the notation given above, we recall that A_max = 1 means CPA, where complete absorption of the incident light occurs. Contrarily, when A_min = 0, the system is in CPT, which means that it behaves as a perfectly transparent element.

In the present model of an OCS supported by a transparent layer, all the joint absorption parameters entering Equation (2) depend on three quantities: G, n, and φ. We have performed a systematic analysis of how the joint absorption parameters depend upon G and φ, while keeping fixed n = 2.6. The latter choice is motivated by the experimental situation that will be analyzed later on, i.e., that of graphene supported by a silicon carbide slab. Again, being interested in studying the coherent absorption properties of 2D systems in which the optical response is mainly dominated by the ac conductivity, we consider first the case of a real conductance G, i.e., when the imaginary part of G assumes sufficiently small values compared to Re(G) (typically Im(G)/Re(G) <≈ 0.1). Notice that, at optical frequencies, graphene lies safely within that bound.^31,32 This is also true at room temperature and for quite large doping levels (E_F ≈ 300 meV), as well as for decoupled multilayers, where the conductance scales linearly with the number of layers and the ratio Im(G)/Re(G) remains constant.^32,33

Panels ((b)–(d)) in Fig. 2 represent the “primitive” parameters entering Equation (2), and reveal that their dependence on G and φ is entangled. The vertical axes report both the conductance in $Ω − 1$ for a generic conducting surface and the corresponding number N_G of graphene monolayers, calculated assuming that the conductance of such a graphene multilayer³³ is G = N_GG0 = N_G × e²/4ℏ ≈ N_G × 6.08 × 10⁻⁵ $Ω − 1$⁠.

One remarkable feature of the present system is that CPT can theoretically always be obtained (see Fig. 2(a)). In CPT condition, the imbalance factor x assumes the value x_min; it is interesting to note that x_min is strictly independent of G (Fig. 2(e)), and its knowledge provides then straightforward independent access to the value of φ. It is then easy to determine G directly from any of the absorption parameters. If G were not exactly real, the maximum relative error on its determined value, and hence in the numbers of layers, is of the same order of the ratio Im(G)/Re(G) (see the supplementary material for more details and for a calculation of such error as a function of Re(G) and φ for the exemplary case when Im(G)/Re(G) = 0.1 is assumed.). Note, however, that CPT and the x_min value are strictly independent of G, even when G is complex (see the supplementary material).

Finally, the model reveals that CPA only occurs for certain combinations of the conductance (G ≈ 6 × 10⁻³ $Ω − 1$⁠, corresponding to a graphene thickness N_G ≈ 100) and optical phase $( φ = n π$⁠, with n = 0, 1, 2 …), as pointed out in Fig. 2(f). In the more general picture of a complex-valued G, the behavior of CPA is less obvious. The numeric analysis shows that CPA points may still exist, albeit for different values of Re(G) and φ. Intriguingly, CPA points eventually disappear as the ratio Im(G)/Re(G) is increased (Figs. 3(a)–3(c)). To better quantify this phenomenon we reported in Fig. 3(d) the maximum value of A_max which can be achieved as Re(G) and φ are varied, for a given ratio Im(G)/Re(G) and a given substrate refractive index. Notably, there exists a clear threshold of the ratio Im(G)/Re(G) above which, for a given n, CPA cannot be achieved for any combination of Re(G), Im(G), and φ. This phenomenon can be referred to as inductively induced CPA suppression, since it originates from the strength of the inductive response of the optically conducting surface. This result contradicts the common perception that CPA can take place independently of how small Re(G), and hence the absorption coefficient, is (a concept applicable instead to 3D systems whose thickness can be arbitrarily increased).

Coherent absorption as a probe for the properties of multilayer graphene. Inspired by the present formalism, the coherent absorption of a real two-dimensional material on substrate was experimentally studied. The system consisted in multilayer graphene grown via chemical vapor deposition (CVD) on the carbon-terminated surface of an insulating silicon carbide substrate (4H-SiC (000-1)); more details on the growth parameters are reported elsewhere.¹⁸ The graphene is composed of 30 ± 5 electronically decoupled layers, as confirmed by the single-Lorentzian shape of the 2D peak in the Raman spectrum (Fig. 4(a)). The decoupled nature ensures that the optical properties of the sample arise from the superposition of the monolayer-like optical response of each layer. The number of layers was estimated by considering the flat graphene sheet transmittance in the mid-infrared measured by Fourier transform infrared spectrometer (not shown). The same number of layers was also evaluated by quantifying the attenuation factor of the substrate Raman signal in the presence of graphene in the visible region.²⁷ In both cases, the 97.7% transmittance (2.3% absorption) was assumed per single layer.

In order to measure the coherent absorption of the system, a 633 nm He-Ne laser was split into two beams. These were then focused with a beam waist of about $30 μ m$ onto the graphene (beam 1) and SiC (beam 2) surface, as shown in Fig. 1(b). With this setup, the spatial resolution can be tailored by selecting the proper focusing lenses. The phase delay $ψ$ between the two beams was realized by inserting a mechanical delay stage in the optical path of beam 2. The intensity imbalance of the two beams was instead controlled by neutral filter wheels. A silicon-based detector coupled to an oscilloscope was employed to measure the intensity of the output signals when illuminating the sample with a single or double beam.

By dephasing the input beams, the absorption properties of the graphene-based system were coherently modulated. First, balanced input beams were used. As shown in Fig. 4(b), the result was a sinusoidal output intensity, whose mean value and amplitude are related to the joint absorbance and joint absorbance modulation. Remarkably, no similar behaviour was observed when investigating the bare SiC substrate (magenta line in Figure 4(b)).

Subsequently, the sample was illuminated with asymmetric beam intensities (−1 < x < 1 and x ≠ 0) and the elliptical dependence on the imbalance factor x predicted by Eq. (2) was found. Figs. 4(c) and 4(d) show the experimental results for two probed areas of the sample. Notably, the two ellipses differ in amplitude and inclination angles. The difference can be understood if the theoretical model is considered. As already described, the amplitude is parametrically dependent on both graphene thickness and optical phase. Instead, the joint absorbance minimum (x_min) is univocally determined only by φ. Based on this consideration, we were able to estimate the φ values by extracting x_min from the two measured ellipses (see Table I). A φ variation of about 50% was quantified for the two areas. It is worth noting that for the sake of simplicity, the model assumed a generic φ ranging between 0 and $2 π$ (without loss of generality). Recomputing all theoretical results by assuming the actual substrate thickness (L = 500 $μ m )$⁠, the optical phase variation of the two areas is about 5.5%. This value is consistent with the 5% uncertainty of SiC thickness reported by the producer. When the optical phase is known, the model allows the graphene thickness to be quantified per each probed area. In fact, the number of layers N_G is identified by a unique combination of A_mod and φ values (see Fig. 2(d)). By taking into account the noise level of our present experimental setup that establishes the detectable lower bound of A_mod, the minimum estimable number of layers can be quantified to be as low as N ≈ 3. From the measured joint absorbance modulations, 28 and 30 layers were estimated in area 1 and 2, respectively. These values are consistent with those obtained by FTIR and micro-Raman spectroscopy. It is worth mentioning that although different areas were investigated with the three techniques, here the graphene thickness is homogeneously distributed over the whole sample, as revealed by micro-Raman spectroscopy within various $30 × 30 μ m 2$ scanned areas (lateral resolution ≈ $1 μ m )$⁠.

A final remark concerns coherent perfect transparency. According to the model, independently on the graphene thickness and optical phase, the minimum of the joint absorbance A_min is expected to be always equal to zero. However, this was not experimentally observed. Both measured ellipses exhibited A_min ≈ 0.2. Taking into account the single-beam optical properties of the system and considering reliable the values of N_G and φ previously discussed, experimental reflectance and transmittance each was found to be lower than the theoretical ones by about 0.1. This discrepancy might be ascribed to two causes: first, an imperfect spatial overlap of the two incident beams could have occurred. Additionally, the incident waves might have not had a perfectly plane wavefront, contrarily to the model assumptions. Moreover, an imperfect polishing of the SiC surface might also have been responsible of further loss of detected signals. Nevertheless, the number of graphene layers estimated by means of coherent absorption agrees with that assessed by the other two spectroscopic techniques and is not influenced by these issues. The coherent absorption technique can hence be considered as a confident method for non-invasive assessment of the conductance of a sheet placed on top of a substrate, finally connected, in the case of multilayer graphene, to the number of its layers.

In conclusion, we derived analytical formulas describing the coherent absorptive properties of a realistic multilayer structure composed by a generic optical conducting surface (OCS) lying on a supporting substrate. The model pointed out that the absorption regime known as coherent perfect transparency can theoretically always be reached, independently of the sample parameters, and that the optical conductance of the OCS can be derived even when the substrate thickness is unknown. The particular case consisting of multilayered graphene sheets on a silicon carbide substrate was also analyzed. The single- and double-beam absorbances were numerically evaluated as a function of the graphene number of layers and of the substrate thickness, indicating the requisites to reach the CPA regime. To support the model, experiments of coherent absorption were performed on multilayer graphene grown on silicon carbide substrate. From the analysis of the measured ellipses, we were able to quantify the number of layers of the graphene sheet. The layer number resulted to be in good agreement with the value obtained by means of two other different techniques. The present results pave the way for a new method of characterizing the optical conductance of graphene and, more in general, of nearly 2D systems with mostly imaginary ac dielectric constant. Thanks to the potentially high lateral resolution, coherent-absorption spectroscopy represents a non-destructive diagnostic tool for the surface mapping of two-dimensional materials and of metasurfaces on a wafer scale.