1. Introduction

When in 1935, working at Zeiss, Oleksander Smakula invented and patented interference-based optical antireflection coatings [1], quite a few people perceived this as the main invention of the 20th century in the field of optics. In fact, the suggested technique allowed to considerably improve the efficiency of many optical devices, including cameras, binoculars, telescopes, microscopes, etc. The major benefit here is associated with improving the image contrast by elimination of stray light. However, the significance of this invention goes far beyond traditional optical devices. In particular, it is of value when high optical absorption is necessary [2].

Talking about optical absorption in bulk materials, they usually entail the dimensionless absorption index (the imaginary part of the complex refraction index), or the dimensional absorption coefficient of a volume. Here we deal with such a dimensionless quantity as the total absorbance $A$, which for infinite non-scattering species is limited only by the reflectance $R$. Indeed, the radiation is gradually absorbed as it propagates deeper into a semi-infinite lossy substrate, and hence $A=1-R$. In other words, minimizing the reflectance maximizes the absorbance. Thus, "broadband high absorption" can be alternatively considered in terms of "broadband low reflectivity".

Currently, of special interest is the design of optical absorbers with a broadened bandwidth. Although there is a large body of literature on this subject, it is quite challenging to realize broadband absorption based on a simple design. While a detailed review on this subject is beyond the scope of the present paper (see, e.g., [3–6]), to gain insight into the problem, we mention only a few relevant works.

Light absorption above 90% in the visible has been obtained for a thin (160 nm) nanocomposite metal-dielectric film deposited on a glass substrate [7]. A double-layered subwavelength structure comprising a 1d metal grating and a dielectric layer is shown to broaden the absorptance spectrum in the near-IR [8]. Using a self-consistent numerical technique, high optical absorption has been predicted for a composite film with spheroidal metal nanoparticles on a silicon substrate [9]. The bandwidth is shown to be broadened if the spheroids are distributed in shape [10]. Multiple resonances at different wavelengths, which occur in multilayered doped silicon/undoped silicon nanostructures integrated with nanohole silicon gratings, are predicted to provide high broadband absorption in the mid-IR [11]. The average absorption of a nanostructure, comprised of a stack of alternating nanosquares of TiO$_2$ and TiN over a dielectric substrate backed by a metal sheet, is reported to reach 96% within the band of 200-3000 nm [12]. An absorber based on a metallic multisized disk array is shown to provide a bandwidth of 2 $\mu m$ in the mid-IR range [13].

A number of recent papers deal with polarization-dependent broadband absorption, which is of special interest for remote sensing, image reconstruction, biomedical imaging, and broadband polarizers. Examples of design solutions include, in particular, a sawtoothed anisotropic metamaterial slab [14], a metal 1d grid, lying over a planar dielectric-metal stacks and a metal substrate [15], a heavily doped semiconductor grating stacked on a dielectric spacer and an semiconductor ground plane [16], a metal grating, ultrathin metal cubes and a semiconductor layer between them [17], a three-layer structure, which consists of cut-wire resonator array - dielectric layer - subwavelength metal grating [18].

The motivation of this work is to address two issues. First, as we could see, customary techniques to suppress the reflection and to enhance the absorption involve multilayer antireflective coatings or complicated surface nanostructures. This requires expensive and sophisticated equipment. The technique proposed in this paper has the potential to overcome the above difficulty. Second, there are some general constraints on the absorption performance over the desired bandwidth [19]. In particular, the total absorber thickness to bandwidth ratio is known to be limited by the maximal reflection within the actual band [20]. However, a problem remains to choose between different design approaches to optimize absorber performance. Our technique is one of such options.

2. Formalism

We start with the well-known equation for the normal reflectivity of a thin film placed on a substrate. The amplitude reflection coefficient is [21]

Let us now consider what happens if we impose the condition $r=0$ at two closely spaced wavelengths, say, $\lambda _1$ and $\lambda _2$, where $\lambda _1<\lambda _2$, and $n_1$ is frequency dispersive. If so, then $\phi =(2\pi /\lambda _1)n_1(\lambda _1)t=(2\pi /\lambda _2)n_1(\lambda _2)t,$ and herewith $n_1(\lambda _1)>n_1(\lambda _2)$. It means that $dn_1/d\lambda >0$, i.e., the film must have anomalous dispersion. This, in turn, introduces high losses, which cannot be neglected. As a result, in the absence of the dielectric loss, single-layer antireflective coatings are usually narrow-band.

A question arises how to broaden the bandwidth. We show how this can be achieved by using an absorbing composite substrate with a complex-valued refractive index $\tilde n_2=n_s+ik_s$. In this case the reflection coefficient for the boundary between the film and substrate $\tilde r_{12}$ becomes complex, $\tilde r_{12}=\rho _{12}\exp (i\delta ),$ where $\rho _{12}= |\tilde r_{12}|$, and the argument $\delta$ of the complex $\tilde r_{12}$

In the ($n_s,k_s$) coordinates, Eq. (7) yields a semicircle in the upper half plane, centred at $n_s=(n_1^2+1)/2, k_s=0$, with the radius which is equal to $(n_1^2-1)/2$. This implies that the maximum admissible value of $k_s$ is bounded above by $k_s^{max}=(n_1^2-1)/2$ and does not depend on the film thickness $t$.

As we show below, the conditions (7) and (8) can be approximately satisfied within a frequency band, when choosing the substrate with a lattice structure.

3. Narrowband absorption and the mathematical formulation of the problem

It would be useful to consider in more detail the case of the narrowband absorption. With this goal, in Fig. 1 we show the curves of $R=0$ (black solid line), represented at the $(n_s,k_s)$ plane, and the corresponding $t/\lambda$ (black dashed line), obtained from Eqs. (7) and (8), respectively. The film refractive index $n_1$ is taken as $n_1=\sqrt {\epsilon _\infty }=3.42$, that corresponds to undoped silicon in the IR.

 

Fig. 1. The semicircle $R=0$ with the corresponding values of $t/\lambda$ and the circles $R=0.01$ calculated at $n_1=3.42$ for five different values of $t/\lambda$.

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In accordance with [2], by decreasing $t/\lambda$, the point of $R=0$ moves in the ($n_s,k_s$) plane toward $\tilde n_s=1+0i$, and no solutions exist if $n_s<1$. This means that at small $t$ and relatively small values of $n_s$, the condition $R=0$ holds at $k_s<n_s$, i.e. the substrate behaves like a dielectric. In contrast, when $t$ rises and $n_s$ approaches $n_s^{max}=(n_1^2+1)/2$, the same condition holds at $k_s>n_s$, i.e. the substrate behaves like a metal. The transition from the dielectric-like behavior to the metal-like one takes place at $n_s=k_s$, that corresponds to the ENZ (epsilon-near-zero) condition.

Close inspection shows that if $1<n_1<1+\sqrt {2}\simeq 2.414$, the line $n_s=k_s$ does not intersect the semicircle specified by Eq. (7). If so, the condition of $R=0$ occurs when $k_s<n_s$, that corresponds to the dielectric-like behavior of substrate.

If $n_1>1+\sqrt {2}$, the line $n_s=k_s$ intersects the above semicircle at two points, which are the solutions of the quadratic equation $2n_s^2-n_s(1+n_1^2)+n_1^2=0$,

The above analysis has been focused on the condition of $R=0$. It is natural, however, to operate in the vicinity of this point. If so, we could also consider the case of a small but nonzero reflection, $R=R_0>0$. At the ($n_s,k_s$) plane the above condition yields the closed circular curves [23]. At fixed values of $t/\lambda$, the isolines $R=R_0=const$ are the concentric circles with centres lying on the curve of $R=0$. In Fig. 1 we show the circles of $R=R_0=0.01$ for different values of $t/\lambda$. Obviously, the condition $R<R_0$ holds inside the circles. The results evidence: in order to keep $R$ close to zero, generally $n_s$ should decrease with $\lambda$, that corresponds to normal dispersion, but exceptions are possible. At the same time, the absorption coefficient $k_s$ should be high (of the order or larger than $n_s$), that is characteristic of anomalous dispersion. Obviously, if we want to broaden the bandwidth, the region of anomalous dispersion should be broadened, too.

The foregoing implies that to provide a broad antireflection band, the complex refractive index of substrate, $\tilde n_2$, should exhibit a nontrivial frequency dispersion admitting a few close resonances. As an example, the region of admissible values of $\tilde n_2$ in the $(n_s,k_s)$ plane to provide $R<0.01$, when $t/\lambda$ ranges from 0.04 to 0.0291, is shown in Fig. 1 by the blue-gray color.

Mathematically, the problem of the bandwidth broadening reduces to minimizing the objective function of the form $\int _{\omega _1}^{\omega _2}R(\omega )d\omega$ with the imposed condition $R(\omega )\leq R_{max}$. As it easy to see, such a formulation imposes constraints on the reflection coefficient at the band edges $R(\omega _1)=R(\omega _2)=R_{max}$.

Thus, here we are dealing with the problem of the design of metamaterials with on-demand complex refractive index [24]. Its geometrical interpretation could be formulated as follows. The initial point of the $(n_s,k_s)$ trajectory should lie on the circle of $R=R_{max}$ with its centre, specified by Eq. (7), where $\tilde n_2(\omega )=n_s(\omega )+ik_s(\omega )$ is the complex refractive index of the substrate at $\omega =\omega _1=2\pi \lambda _1/c$. The final point of the $(n_s,k_s)$ trajectory should lie on the circle of $R=R_{max}$ with its centre, specified by Eq. (7), where $\tilde n_2(\omega )=n_s(\omega )+ik_s(\omega )$ is the complex refractive index of the substrate at $\omega =\omega _2=2\pi \lambda _2/c$. The Argand diagram of the needed refractive index (the curve of $k_s\ vs\ n_s$) should lie within a cigar-like region, similar to that shown in Fig. 1 by the blue-gray color.

4. Geometry and materials

Figure 2 shows a typical example of our design. The unit cell includes a thin (subwavelength) film of the thickness $t$, made of an undoped semiconductor, and a semi-infinite substrate, which consists of several variably doped subwavelength semiconductor layers oriented normally to the film. The permittivities and filling factors of the layers are $\epsilon _i$ and $f_i$, respectively. Such a substrate can be considered as effectively an uniaxial hyperbolic metamaterial. The plane electromagnetic wave is normally incident on the film with the electric film, oriented normally to the layers (TM polarization) or along them (TE polarization).

 

Fig. 2. An example of the unit cell of a subwavelength structure which consists of an undoped semiconductore film of the thickness $t$ (blue) and five variably doped semi-infinite semiconductor nanolayers with the corresponding permittivities $\epsilon _i$ and filling factors $f_i$ (yellow). The direction of the electric field is shown as for TM polarization.

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In the long-wavelength limit, neglecting nonlocal effects, the dielectric response of the substrate is described by a diagonal tensor of the form $\epsilon _s=diag(\epsilon _s^{TE},\epsilon _s^{TM})$ with $\epsilon _s^{TM}=[\sum _i (f_i/\epsilon _i)]^{-1}$ and $\epsilon _s^{TE}=\sum _if_i\epsilon _i$. In the following, only the TM-component of the tensor will be involved to get needed broadband absorption. Because of this we take that $\tilde n_2=\sqrt {\epsilon _s}=\sqrt {\epsilon _s^{TM}}$. It has been earlier noticed that this component of the tensor $\epsilon _s$ for such microgeometry can be tuned in wide limits, that holds promise for various applications [24].

For illustrative purposes we consider an undoped silicon film and a substrate made of four variably doped silicon layers. Our choice is motivated by the fact that silicon is well studied and widely used for various applications. Besides, it can be heavily doped, above $10^{20}$ cm$^{-3}$. According to Drude model, its refractive index in the IR is

In what follows we consider the free carrier concentrations in the layers $N_i$, filling factors $f_i$, and film thickness $t$ as fitting parameters. To determine them we solve the above optimization problem. For further convenience, we introduce the dimensionless relative bandwidth $\Delta$ defined as

5. Results and discussion

To illustrate the solution to the optimization problem, in Fig. 3 we show the reflection coefficient obtained for three different values of $R_{max}$, where the substrate unit cell has been taken as composed of four variably doped silicon layers.

 

Fig. 3. The optimized reflection coefficient for different values of $R_{max}$. The dashed curves show the corresponding reflection coefficient for the TE-polarization.

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In all the above cases the broadband absorption is achieved at a subwavelength film thickness ($t/\lambda <0.04$), whose optimal values only weakly depend on the bandwidth and are $t=0.135 \ \mu m$ at $R_{max}=0.0014$, $t=0.128\ \mu m$ at $R_{max}=0.0095$, and $t=0.123\ \mu m$ at $R_{max}=0.0177$. The other values of the fitted parameters of the substrate are listed in Table 1.

Table 1. Fitted geometrical and material parameters of the substrate.

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As can be seen, broadening bandwidth correlates with an increase in the filling factor of the lowest doped layers and a decrease in the filling factor of the highest doped ones. The refractive index $n_s$ and the absorption coefficients $k_s$ for all three sets of fitted parameters are represented in Fig. 4 as wavelength dependencies (left panel) and in the form of the Argand diagrams (right panel). In contrast to the case of fixed $t/\lambda$, the Argand plots $k_s(n_s)$ are not now closed curves. Typically, $n_s$ rises with $\lambda$ first, then it oscillates and drops. In contrast, $k_s$ tends to drop first, then to oscillate and finally to grow.

 

Fig. 4. Left panel: the dependencies of $n_s(\lambda )$ (solid curves) and $k_s(\lambda )$ (dashed curves) for different values of $R_{max}$ and $\Delta$. Right panel: the corresponding Argand diagrams.

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It was recently noted that "$\ldots$completely suppressed reflection, and thus near-perfect absorption, for near-normal incidence using optically this coatings almost exclusively occurs on substrates with $n_s\le 1$" [2]. This is characteristic of ENZ metamaterials. However, it is not our case. The matter is that the above paper dealt mainly with absorbing coatings, that sufficiently shifts the region of $R \approx 0$ toward smaller values of $n_s$ as compared to transparent ones. Indeed, looking at Fig. 1 we can see that the condition $R=0$ at $n_s<1$ is fulfilled only when $k_s<0$, i.e., for materials with gain.

As has been noted above for the narrowband absorption, the closer to unity is $n_s$, the smaller $t$ is necessary to get $R=0$. In practice, of course, our options are limited by our choice of constituent materials. Figure 4 shows that the refractive index of designed substrate exhibits metal-like behavior within the bandwidth, which occurs when $n_s<k_s$. Indeed, in our example $n_1=3.42>1+\sqrt {2}$. According to Eq. (9), $n_s^-\simeq 1.12$ and $n_s^+\simeq 5.23$, and the real part of the refractive index, $n_s$, lies between these points within the actual frequency bands.

It would be certainly of interest to explore how the relative bandwidth $\Delta$ is related to $R_{max}$. The dependence of $\Delta (R_{max})$ has been calculated numerically by solving the above optimization problem at a few values of $R_{max}$, keeping $t$ constant. First the relative bandwidth $\Delta$ rapidly grows at small $R_{max}$ and then its growth slows down gradually. As we have made sure, with a reasonable accuracy the function $\Delta (R_{max})$ demonstrates a power law dependence of the form $\Delta =CR_{max}^\alpha$ with parameters of $C\approx 1$ and $\alpha \approx 0.285$. This is in contrast to the results by Rozanov, who has derived that the maximum bandwidth for a multilayer metal-backed slab exhibits an inverse logarithmic dependence on $R_{max}$ [20]. In the general case, specific bandwidth limits are known to depend on substrate parameters [26]. The plot of $\Delta$ vs $R_{max}$ and the corresponding fitting are shown in Fig. 5.

 

Fig. 5. The dependence $\Delta (R_{max})$ obtained from simulations (blue circles) at $t=0.125\ \mu m$ and the corresponding fitting (red curve).

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Above we have considered the substrate as semi-infinite. How critical is this assumption? To answer, let us make a simple estimation. A slab of the thickness $2\ \mu m$, made of material with $k_s=2.5$ (this is typical for our design), transmits only about $0.67\%$ of radiation at $\lambda =3.8\ \mu m$. Thus, if substrate is one wavelength thick, it provides almost complete light absorption.

One more assumption we have made is a uniform doping profile. As a rule, single ion implantation results in a Gaussian-like doping profile. However, modern implantation techniques are known to generate exactly defined profiles, including very uniform ones [27].

A question arises when and why the proposed design might provide better outcomes. To address this issue, the following points deserve to be noticed.

Unlike most previous approaches to the broadband absorption, our set-up is relatively easy to fabricate using ion implantation with a mask. One more option involves photogenerating a grating where the carrier density variations are caused by two incident beams [28].

Although the optimization problem can involve a lot of fitting parameters and in general admits multiple solutions, in essence, it is simple, because the initial values of these parameters can be preliminary estimated. A lot of degrees of freedom is thus rather an advantage than otherwise, because it allows for unprecedented flexibility of our design.

Varying geometrical and material parameters of the substrate, the bandwidth can be tuned within wide limits, providing broadband and narrowband absorption as well. Moreover, although we have not considered such an option in this paper, enhanced absorption can be achieved within two or more bands by minimizing a properly chosen objective function. This could be of interest, in particular, for obtaining spectrally-selective, strong thermal emission [16] or for sensor applications [29].

When dealing with heavily-doped silicon, the bandwidth can range from the mid-IR to the microwave region. Dealing with heavily doped transparent conducting oxides, such as indium tin oxide, Al-doped and Ga-doped zinc oxide could potentially shift the bandwidth to the near-IR, including the telecommunication region.

Finally, although we have dealt with 1d lattice-based design which imposes the polarization-dependent dielectric response, in fact, such kind of design is not a necessary condition for obtaining broadband absorption with the use of direct fitting. An example of macroscopically homogeneous isotropic substrate to realize the above broadened bandwidth using a similar approach will be presented elsewhere.

6. Conclusion

In the presented paper we have concerned ourselves with the design of a broadband polarization-sensitive light absorber for the mid-IR, based on a transparent film and vertically stacked and variably doped semiconductor multilayers, which form a semi-infinite superlattice. The complex refractive index of the multilayers has been tailored by the dopant concentration. Setting a frequency band and varying multilayer parameters, we have shown that a low polarization-dependent reflection and correspondingly high absorption within this band can be obtained. Specifically, we have dealt with an inverse problem which may be recast as the determination of unknown geometrical and material parameters of the unit cell using the brutal-force method.

For the illustrative purposes, we have chosen undoped silicon and heavily doped silicon as the constituent materials for the film and substrate, respectively. Near-perfect absorption at near-normal incidence has been demonstrated for the subwavelength thickness of the film and even moderately thick (wavelength-sized) substrate, making the proposed design experimentally feasible. The substrate refractive index trajectories $n_s(k_s)$, needed to realize this effect, have been presented. Finally, as we have found, the relative bandwidth $\Delta$ approximately follows the power-law dependence on the maximum value of the reflection coefficient $R_{max}$ within the actual band.