## Abstract

The control and tailoring of infrared absorbance/emittance is a crucial task for all those applications involving thermal radiation management and detection. We theoretically investigated the peculiar absorbing/emitting behaviour of pre-fractal Cantor multilayers, in order to design a polarization-insensitive multilayer stack absorbing over a wide angular lobe in the mid wavelength infrared range (8-10 μm). Using transfer matrix method, we explored the spectral properties arising from both the material and the geometrical dispersion. We considered several combinations of the constituent materials: SiO_{2} was combined with TiO_{2} and Si, respectively.

© 2014 Optical Society of America

## 1. Introduction

The ideal thermal absorbers/emitters display high absorbance/emittance values at wide angles for polarised as well as unpolarised radiation [1] and properly fit all those applications regarding energy conversion and infrared (IR) radiation detection. For example, thermophotovoltaic (TPV) cells may gain from the design and realization of a so-called perfect, polarization insensitive, absorber that may have a great impact on the overall device efficiency, increasing the absorbed power for energy harvesting as well as the reprocessing of wasted heat in industrial processes [2]. Furthermore, in the far IR wavelength range (8-12 μm), the tuning and managing of IR absorbance/emittance of a given object is a useful tool for all those applications involving thermal radiation detection. In particular, the realization of a wide angle, broad band, polarization insensitive absorbing device is highly sought [3] for the development of efficient microbolometers, optical attenuators, sensors [4] as well as IR cameras and thermal sources [5].

In order to design a polarization insensitive device (emittance/absorbance angular curves displaying similar behaviour for both TE and TM polarizations), several different approaches were developed, including one-dimensional (1D) [6] and two-dimensional (2D) periodical structures [7], metamaterials [8] and nanostructured materials [9]. In this work we exploited one-dimensional quasi-periodical structures, obtained by recursively applying a fractal algorithm. We show that the geometrical dispersion introduced by the quasi-periodicity, allows to get the spatial modulation of spectral and angular absorbance, by only exploiting interference effects among the consecutive layers. We also considered the effect of combinations of different materials in the design of the structure. In particular, TiO_{2} and Si have been considered because of good transparency in the IR range and appropriateness of the refractive index behaviour with respect to the SiO_{2} index.

## 2. Pre-fractal multilayers

In order to generate a self-similar fractal, a recursive mathematical operation is defined on a given object, the so-called initiator [10]. Among all possibilities, Cantor fractals are largely diffused due to their simplicity and easiness of realization: the initiator is a straight line of a given initial length, L. A segment of length L/3 is then erased in the middle of the initiator. This operation can be further performed at a reduced scale: a line of length L/3^{2} is erased again in the middle of each of the two (remaining) segments. The Cantor fractal is obtained by further iterating this operation, and stopping the iterations at the N^{th} order, corresponding to the N^{th} generation, with a scale factor of 3. Fractal-like periodicity represents an efficient strategy for thermal radiation tuning and management, allowing to determine different interesting spectral features, such as broadband absorption [11] or thermal emission with a high coherence degree [12].

We here investigate the modulation of thermal radiation absorbance obtainable using a 1D
multilayer structure encompassing all-dielectric layers, whose thicknesses display
quasi-periodic fractal periodicity [10]. We study the
angular absorbance of such structures and show that the appropriate choice of layer thicknesses
and fractal periodicity leads to polarization-insensitive, wide angle absorbance curves, i.e. to
super-absorbing behavior at some wavelengths in the long wavelength IR range. For all
calculations, substrate was chosen to be silicon, due to its transparency in the IR range. A
structure following a fractal sequence can be designed starting from a thick layer of a given
material (the initiator) and by filling the empty spaces with a second material. In practical
situations, the structure is realized by stacking layers, whose optical thickness is
proportional to the length of each segment, according to Cantor’s sequence. The outcome
is an alternation of two dielectric layers of refractive index n_{A} and
n_{B}≠n_{A}, such that the optical path of the layers corresponding to
the smallest segment is the same for both materials, i.e. a quarter-wavelength with respect to a
reference wavelength λ_{0}. Thicker layers scale consequently, i.e. three and
nine quarter-wavelengths (QW). Repeating the Cantor algorithm for three times, will generate an
overall structure composed by 15 consecutive layers of total optical thickness equal to 27 QW.
In particular, a thick (9QW) layer (B) is sandwiched between two quasi periodic stacks, each
composed according to the following sequence (see Fig.
1(a)):

Although the absorption properties can be optimized and tailored by taking advantage of the geometrical dispersion, the starting point for the design of an efficient polarization insensitive absorber is the proper choice of the materials composing the stack. Indeed, if we are looking for a device operating at a given wavelength range with a certain bandwidth, the first rule of thumb is that at least one of the materials that we are using for the stack must display absorption in the desired wavelength range.

For this purpose we considered SiO_{2} [13]
as absorbing material, because it exhibits a strong absorption band located at around 9
μm (k>2.5 at 9 μm), which is attributed to the phonon vibration mode [14]. In Fig. 1(b) we
show the average polarization absorbance as a function of wavelength and angle of incidence for
a single SiO_{2} layer of total optical thickness of 27 QW with respect to a reference
wavelength λ_{0} = 2 μm, on Si substrate. Although the imaginary part of
the refractive index of the SiO_{2} exhibits a broad peak in the range 8-10 μm
(see inset of Figs. 2(b)), the absorption is limited by
the sharp increase of the real part of the refractive index leading to a significant reflection
when the incident field is impinging from air, for example. A multilayer stack is commonly used
as antireflection coating, to minimize reflection, increasing the absorption of the effective
layered medium [15].

Here we propose to use an optimized fractal layered structure performing a comparison with the most commonly used periodic stacks [16]. As a first step we investigated the spectral and angular absorption features arising from a third generation Cantor set, where TiO_{2} (n_{A}) represents the initiator, and the second material (n_{B}), is SiO_{2}, as previously mentioned. Using the transfer matrix method we performed a set of numerical computations of transmittance and reflectance values of the quasi-periodical stack vs wavelength and incidence angle. According to Kirchoff’s law, we then retrieved the absorbance value (*A = 1-T-R*) and analyzed the obtained patterns. As an example we calculated the absorbance patterns setting the reference wavelength λ_{0} to 2 μm. The resulting QW thicknesses, calculated at this wavelength are QW_{A} = 206 nm (TiO_{2}) and QW_{B} = 347 nm (SiO_{2}). The choice of λ_{0} = 2 μm assures a good angular behavior in the 8-12 μm IR range, in terms of high absorbance values over a wide angle range. Indeed for longer wavelengths compared to the reference wavelength the Cantor structure typically exhibits broad bands of near zero reflection [17].

In Fig. 2 we show the absorbance pattern calculated for TE (Fig. 2(a)) and TM (Fig. 2(b)) polarization. The investigated structure would allow to get three high absorption bands over a wide angle range, centered at λ = 8.5 μm, λ = 10 μm, and λ = 12 μm respectively. Unfortunately there are two wavelength regions showing low absorbance around λ = 9 μm and λ = 11 μm. The first low absorption region is due to the fact that the real part of the refractive indices of SiO_{2} and TiO_{2} are very similar (see inset of Fig. 2(a)) leading to weak interference effects inside the structure and weak impedance matching with the external medium. On the other hand, the low absorption band around 11 μm is due to the small value of the imaginary part of the refractive index of SiO_{2} (see inset of Fig. 2(b)).

Since we are interested in the design of a polarization insensitive multilayer stack working in the far IR wavelength range, i.e. 8-10 μm, we introduced a merit function that takes into account the simultaneous maximization of the average absorbance value for both polarization states, $\u3008{A}_{TE}\u3009$ and $\u3008{A}_{TM}\u3009$in a given wavelength and angular range:

being:

The merit function (2) ranges between 0 and 4 and it allows an easier and more accurate evaluation of the performances of the device with respect to the bare sum of the TE and TM absorptions (which in turn would range between 0 and 2, being one the maximum value for the absorption coefficients). Indeed the introduced FOM allows to better visualize if one absorption coefficient is low with respect to the other, thanks to the double of the product of the two terms in the square of a binomial expression. The merit function (2) was calculated by varying the reference wavelength, λ_{0}, which in turns scales the layer thicknesses. The angular range was set to *θ*_{MIN} = 0°and *θ*_{MAX} = 70°, while for wavelength range we considered *λ*_{MIN} = 8 μm and *λ*_{MAX} = 10 μm, because of the interest in the enhancement of performances of microbolometers and IR sensors.

The ideal polarization insensitive perfect absorber is characterized by a FOM = 4 meaning that
both the averaged absorption coefficients are equal to 1. In the optimization procedure we find
the best parameters to maximize the FOM. As a comparison we performed the same optimization
procedure on a symmetric periodic stack obtained by alternating the same QW layers of the Cantor
structure for 13 periods plus one layer in order to match the total optical thickness of 27 QW
of the Cantor scheme (see Fig. 3(a)). In order to
evaluate the improvement with respect to the single SiO_{2} layer we also calculated the
FOM for a 27 QW single SiO_{2} layer. We note that the optimization procedure allows us
to find a value for λ_{0} (2.55 μm) corresponding to the best performances
(see Fig. 3(a)). The fractal scheme allows to reach
higher values of the FOM (2.55) with respect to both the periodic stack and the SiO_{2}
single layer.

In order to design a more efficient device operating in the same wavelength range we replaced the
TiO_{2} layers of the Cantor stack with Si layers. The results of the optimization
procedure are reported in Fig. 3(b). Because of the high
refractive index value of the Si, an adequate index contrast between the two materials in all
the considered wavelength range is assured (see inset of Fig.
4(a)).The Cantor structure exhibits higher performances with respect to the periodic one. Also,
the maximum FOM achieved is higher with respect to the one obtained with
SiO_{2}/TiO_{2} stack, reaching a maximum value of 2.9, i.e. more than twice
the value achievable by the single SiO_{2} layer. The optimum reference wavelength that
maximizes the merit function, FOM, is λ_{0} = 1.75 μm. Finally we
calculated the averaged absorbance for the optimized structure (Fig. 4(a)) showing that a broad absorption band over a wide angular range is obtained
in the range 8-10 μm. For this structure, the QW thicknesses, calculated at
λ_{0} = 1.75 μm are QW_{A} = 127 nm (Si) and QW_{B} = 303
nm (SiO_{2}).

The polarization insensitive behavior of the optimized multilayer structure with respect to the bulk SiO_{2} and to the periodic stack, calculated using the same reference wavelength, is evidenced in Fig. 4(b), where the polar plot of the angular absorbance curves for a λ = 9.05 μm (located in the middle of the absorption band) are reported. Remarkably, the absorbance/emittance lobes, for TM and TE polarizations, nearly overlap, showing that this structure is rather insensitive to the angle of incidence, despite the usual intrinsic anisotropy introduced by geometrical dispersion in layered structures. Furthermore, it allows to achieve high absorption values over a wide angular range, remaining higher than 80% for angles up to 60°.

## 3. Conclusions

We studied the polarization insensitive absorbing behaviour of pre-fractal Cantor multilayers, in order to maximize (using a defined optimization procedure) both absorbance value and angle, so to design a multilayer stack absorbing over a wide angular lobe in the long wavelength IR range (8-10 μm). The same optimization procedure has been performed on a periodic stack of the same optical thickness of the Cantor structure showing lower performances. Thus, pre fractal Cantor multilayer stacks offer a valid and competitive alternative to periodic multilayer structures for the control and tailoring of IR absorbance and thermal radiation management applications. Finally, other possible improvements, such as an extended absorption band up to 12-14 μm, could be achieved adding impurities in the Si layers or properly designing quantum dots absorbing in the far IR range. Optical constants of doped silicon, in fact, change dramatically with the dopant type and concentration [18]. Concerning quantum dots, it is possible to work at 7-14 μm although low operating temperatures are required [19]. However these options deserve further investigation and will be subject of future works.

## Acknowledgments

This work has been performed in the framework of the project “FISEDA” granted by Italian Ministry of Defence.

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