1. Introduction

Managing material emissivity, or, equivalently, the spectral absorbance of a given object is a crucial issue that can lead to the camouflage of thermal infrared emission. In the seek of coherent thermal source, narrow antenna-like angular emissivity lobes have been shown from a polar material (SiC) surmounted by a periodic 2D-photonic crystal, supporting surface polaritons [1]. This structure lead to a high degree of spatial coherence for the thermal light, as a result of the surface phonon-polariton diffraction by the grating. However, surface polaritons emission lobe can be observed only when the electric field direction is perpendicular and the magnetic field is parallel to the grooves, respectively, i.e. in TM polarization. As a consequence, the radiation emitted by the source can be confined into a narrow lobe only for TM polarization, while it is isotropically distributed for TE polarization.

Left handed metamaterials composed of periodic metallic structures [2,3] can be exploited to obtain highly directional emission around the surface normal, at those frequencies near plasmon resonance, i.e. when the refractive index of metamaterials approaches zero value however, the restriction to TM waves still holds. The excitation of tunneling modes in a photonic quantum well structure with zero-averaged refractive index (zero-n) coating was shown to enable frequency selective coherent thermal emittance for both TM and TE mode although only theoretically [4].

The use of one-dimensional layered structures to tailor the spectral properties of thermal radiation in both polarizations relies on completely different principles with respect to periodic gratings and includes several works investigating thin films [5,6] and –moreover- several types of multilayers [7,8].

It is well-known that electromagnetic surface waves can be guided by the boundary of a semi-infinite periodic multilayer dielectric medium, thus one-dimensional photonic crystal (PC) can support a surface mode or surface wave for both TM and TE polarizations in the stop band [9]. Being surface waves nonradiative, an attenuated total reflection configuration is usually arranged -using a coupling prism- to excite the surface waves at the interface between a PC and air [10,11]. Alternatively, a metallic layer can be coated onto a PC and surface waves can be directly excited by propagating waves in air, resulting in a sharp reduction in the reflectance that can be detected without prism or grating configuration even at normal incidence [12].

Coherent thermal emission for both TM and TE polarizations has been predicted by exciting surface waves at the interface between a polar material (SiC) and a modified periodic multilayer structure, where the external layers’ thickness are half with respect to inner layers [13]. Furthermore, the latter structure can greatly enhance the emission by the cavity resonance mode and the Brewster mode [14]. The wavelength range corresponding to stop bands of the PC should be scaled by changing the thickness of the unit cell, in order to match the phonon absorption band of the SiC, thus surface waves can be excited at the SiC–PC interface within the SiC phonon absorption band for both TM and TE polarizations.

A somewhat different configuration is exploited in the vertical-cavity enhanced resonant thermal emitter (VERTE) [15], consisting of a lossless cavity (a SiO₂ layer) sandwiched between a partially reflective and lossless PC and a highly reflective metallic mirror. In this configuration the optical cavity resonance, which can be tuned by changing cavity thickness, enhances thermal emission originating from the metallic mirror building up a strong quasimonochromatic field while suppressing nonresonant frequencies.

If the symmetry of the structure is broken by a defect layer, the defect mode associated with the characteristics of the defect layer, i.e. material properties or thickness, can be exploited [16]. If an absorbing defect, i.e. a layer of SiC, is embedded into a photonic crystal, it is possible to couple a surface mode with a defect mode as shown in [17]. The presence of a defect generate two localized modes in the reflectance, and consequently in the absorbance curve. Emissivity close to unity can be reached using five periods per mirror for TE polarization (20 microns thick).

Quasi-periodic fractal periodicity can also give rise to interesting spectral features enabling coherent thermal emission. Thermal radiation from quasi-periodic Cantor multilayers containing negative index metamaterials have been theoretically investigated [18]. Phase compensation, i.e. the partial or full compensation of phase shift of an electromagnetic wave propagating through a stack alternating positive and negative index layers, can be a useful tool to get sharp directional emittance spectra.

In this work we investigate the modulation of thermal radiation emissivity obtainable with 1D multilayer structures encompassing all-dielectric layers, whose thickness display quasi-periodic fractal periodicity [19,20]. We study the angular emissivity of such structures and show that the appropriate choice of layer thickness and fractal periodicity leads to a quasimonochromatic thermal radiation pertaining a high degree of spatial and temporal coherence. Furthermore, the geometrical dispersion introduced by the fractal periodicity allows to find some wavelengths where both the transverse electric wave and the transverse magnetic undergo a narrow angular and spectral emissivity peak.

2. Thermal radiation from pre-fractal multilayers

Fractal structures differ by the algorithm used for the stack construction. In order to generate a self-similar fractal, a recursive mathematical operation is defined on an object, the so-called initiator [19]. Among them, Cantor fractals are largely diffused due to their simplicity and realization easiness. In the case of Cantor fractals 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 L/3² 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. Theoretically, for a full fractal set the division process should be repeated infinitely long, but for practical reasons a truncation occurs giving a so-called pre-fractal set. If layers exhibit a non-zero imaginary part of the refractive index, a Cantor structure with a very large number of layers would, in fact, lead to intense attenuation and thus it would be useless for practical applications [21].

A fractal layered structure can be practically realized by filling the empty spaces with a second material, resulting in the alternation of two dielectric layers of different refractive indices, as depicted in the inset of Fig. 1. The result is an alternation of two dielectric layers of refractive index n₁ and n₂ (n₁>n₂), of such thickness that the optical path of the smallest segments is the same for both materials.

 

Fig. 1 (a) Calculated spectral emittance for the third generation of Cantor multilayer structures composed by TiO₂ and SiO₂ alternating layers. SiO₂ was chosen as initiator layer (gray layer in the inset). Dashed line represent the blackbody radiation (calculated at 580K) normalized by its value at Wien’s frequency. Inset: Schematic of proposed fractal layered structure (arranged as triadic Cantor set) for thermal emission control.

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Due to their structural self-similarity, the consecutive generations of fractal multilayers show some peculiarities in their transmission/reflection spectra, as well as in their absorbance curve. First evident property is spectral scalability, i.e. the whole spectrum of a given generation repeats scaled as a part of the next generation spectrum. Another marked feature is given by sequential splitting, i.e. spectral peaks from one generation split into doublets in the next [22,23]. Most importantly, the sharp resonances in spectral dependences of fractal multilayers enable to localize the electromagnetic field in a very efficient way. We exploited this strong localization for tailoring the infrared, thermal, radiation in both on-axis and off-axis direction.

The spectral and directional emission properties of pre-fractal multilayer structures were calculated using the well established numerical technique of the transfer matrix formalism [24], assuming the incident light to be a plane wave whose propagation direction forms an angle, θ, with respect to the surface normal. In order to understand the conditions causing large emission in a narrow spectral and angular range, a wavelength/angle map is developed. The effects of geometrical parameters as well as the distribution of the light inside the high- and low-index dielectrics in the pre-fractal structure are investigated.

A layer of a low refractive index material was chosen as the initiator, SiO₂ (n₁) while TiO₂ was chose to fill the empty spaces. For both materials a frequency-dependent complex refractive index was employed [25]. We studied a third generation Cantor set where the smallest constitutive layers have an equal optical thickness, that of a quarter-wavelength slabs n₁d₁ = n₂d₂ = λ₀/4, while thicker layers scale consequently, i.e. three and nine quarter- wavelength. The overall structure is less than 18 microns thick or equivalently 27 quarter-wavelength.

The reference wavelength to scale layer thickness was chosen to be λ₀ = 5 μm, corresponding to the emission peak of a black body at ~580K. The multilayer is assumed to be deposited on a thick substrate (d >> λ) with an index n_S, and surrounded by a semi-infinite medium n₀, which in our case for the sake of simplicity is air or vacuum. As a starting point, we chose a SiO₂ substrate, while later on we discuss the effect of using different substrates.

In Fig. 1 we show the structure absorbance as a function of wavelength, calculated in the wavelength range 2-14 μm, for the third generation of a Cantor multilayer which is schematically represented in the inset. This particular fractal structure, i.e. the third generation of a Cantor set, is very close to a resonant cavity with embedded defect having a restricted layers’ number and showing some spectral advantages that will be described in the following.

We now focus onto the angular dependence of the sharpest peak of spectral absorbance and plot, in Fig. 2(a), the angular emittance calculated at λ = 4.5 μm. The three different colours correspond to TM, TE and average polarization, respectively (see Fig. 2 caption). The emissivity lobes are confined in a very narrow angular region, for both TM and TE polarizations. It’s worth to note that emissivity is here deliberately plotted as a polar plot in order to highlight the narrow angular lobe in a well definite emission angle, occurring for both polarization directions. However, considering the actual geometry of the pre-fractal Cantor multilayer, due to its intrinsic azimuthal symmetry, coherent emission radiate in circular patterns, instead of the antenna shape obtainable with grating surfaces [1].

 

Fig. 2 (a) Directional spectral emittance calculated at 4.5 μm for the third generation of a pre-fractal multilayer composed by SiO₂ and TiO₂ in TE polarization (blue curve) TM polarization (red curve) and average polarization (black curve). (b) The corresponding field profile inside the structure for both TE (blue curve) and TM polarization (red curve) calculated for an angle of 34.4°, i.e. where the emissivity maximum occurs.

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Furthermore, in Fig. 2.(b) we plot the corresponding field profile inside the structure calculated at an angle of 34.4°, i.e. where the emissivity maximum overlap between TE and TM polarizations occurs, showing that the electric field distribution is localized into the internal layer.

The angular plots show that the Cantor pre-fractal multilayer structure behaves as a polarized quasimonochromatic thermal source with a spatial coherence length of several wavelengths, resulting from the calculated lobe width at half maximum, Δθ = 43 mrad, thus the spatial coherence length l_s of the thermal radiation can be easily retrieved being l_s = λ/Δθ ~114 μm. At the same wavelength the corresponding coherence time τ_c calculated for an incidence angle of θ = 33° is τ_c = λ²(cΔλ)⁻¹ ~1.5 ps.

We then consider a set of wavelength and incidence angle conditions and show in Fig. 3 the calculated spectral and directional emissivity in the plane (λ, θ) for both TE, Fig. 3(a), and TM, Fig. 3(b), polarization respectively.

 

Fig. 3 Calculated emissivity as a function of wavelength and incidence angle, for the third generation of a pre-fractal multilayer composed by SiO₂ (initiator) and TiO₂. (a) TE polarization and (b) TM polarization.

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Well defined forbidden band appears in spectral absorbance/emittance of periodic structures, while pre-fractal Cantor multilayers exhibit sharp resonances inside the band even for low generation numbers. As a result, high emissivity peaks due to cavity resonance modes can be found in the wavelength region between 4.2 and 6 μm for both polarizations (see Fig. 1).

Interesting conditions develop when materials and geometrical dispersion allow to find out a wavelength where both the transverse electric and the transverse magnetic waves undergo an emissivity angular peak, as already shown in Fig. 1 for λ = 4.5 μm. By looking at the spectral and directional emissivity curves, one can select other wavelengths where the polar plot of angular emissivity display rather similar TE and TM curves. So far, any time these conditions are fulfilled, the investigated pre-fractal structure may support quasi-coherent thermal emission in a well-defined direction, thus the corresponding operational wavelength, is not only one but span within a certain range, given by the cavity resonance band. In Fig. 4 the geometrical dispersion of the pre-fractal multilayer structure is shown in the ω-k_x domain, for both TE and TM polarization, being k_x the parallel component of the wave vector, i.e. $k_x=k\sin \left(\theta \right)$. Choosing a different λ₀ for multilayer design has the effect to shift the cavity resonance modes, and thus the emissivity peak wavelengths consequently.

 

Fig. 4 Geometrical dispersion of the pre-fractal multilayer structure shown in Fig. 1, in the ω-k_x domain: (a) TE polarization and (b) TM polarization.

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Owing to the low generation number of the pre-fractal multilayer, these cavity resonance modes are similar to those appearing in photonic crystals having a defect layer inside, i.e. so called defect-modes. In other words, the investigated pre-fractal multilayer is similar to a defect layer sandwiched between two mirrors-like stacks. For example, a Brag stack that has the internal layer replaced by a different material can support a defect-mode. Consequently, the resonant electromagnetic wave is confined inside the internal layer at the stop band due to high reflection from boundaries, resulting in a large absorption peak at the resonance condition if the constitutive layers, as well as the substrate, are absorbing media.

For comparison, in Fig. 5 we plot the corresponding geometrical dispersion for a cavity with defect (single-defect Bragg stack) composed of SiO₂ and TiO₂ mirrors and a TiO₂ defect layer, having the same total optical thickness of the pre-fractal multilayer, i.e. 27 quarter-wavelength. The same angular and wavelength step was employed as well as the same substrate.

 

Fig. 5 Geometrical dispersion of a cavity with a defect composed by TiO₂ defect layer sandwiched between two SiO₂ and TiO₂ mirrors, resulting in a total optical thickness of 27 quarter-wavelength, in the ω-k_x domain: (a) TE polarization and (b) TM polarization.

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Some similarities are immediately clear, since the defect layer determines two narrow cavity modes in band structure, thus this optimized cavity can support a similar behaviour as observed in pre-fractal multilayer. By plotting the directional spectral emittance, in Fig. 6(a), it is possible to observe that the maximum emittance value is lower in the defect cavity, with respect to the pre-fractal structure, and the undesired emittance at high angles, i.e. between 80°and 90°, is rather significant. On the other hand, the plot of the field profile inside the different layers of the structure, Fig. 6(b), show that the electric filed distribution is similar to the investigated low generation pre-fractal structure, reaching a somewhat higher value due to decreased resonance width.

 

Fig. 6 (a) Directional spectral emittance calculated at 4.5 μm for the defect cavity of Fig. 5 in TE polarization (blue curve), TM polarization (red curve) and average polarization (black curve). (b) The corresponding field profile inside the structure for both TE (blue curve) and TM polarization (red curve) calculated for an incidence angle of 32.4°.

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An advantage of the pre-fractal multilayer over the defect cavity is that the high frequency resonance mode, for TE and TM display a larger capability for spectral and angular overlap, resulting in a wider wavelength range of polarization insensitive quasi-coherent thermal emission in a well-defined direction.

At the same time, we also investigated the complementary pre-fractal Cantor multilayer, where the high refractive index TiO₂ plays the initiator role and the low refractive index material is SiO₂. In this configuration we find that, beside a slight increase of the total physical thickness to 22 μm, the cavity resonance modes for both TE and TM polarization partially overlap thus it is not possible to find a wavelength with a single and well defined angular peak.

It is also interesting to investigate how the spectral emissivity develops if pre-fractal is further developed. In Fig. 7 we show the spectral emissivity calculated for the fourth and the fifth pre-fractal generation, respectively. The corresponding curve for the third pre-fractal generation is also included for comparison. The spectral resonances survive, although significantly broadened. As a consequence of this broadening, the directionality of angular emissivity is reduced as well as the electric field profile calculated at both wavelength and angular peak.

 

Fig. 7 Calculated spectral emittance for the three consecutive generations of Cantor multilayer structures composed by TiO₂ and SiO₂ (initiator) alternating layers.

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As a final remark, we wish to address the effect of the substrate on the absolute emissivity peak value. As a matter of fact, the highest is substrate absorbance at a certain wavelength, the highest is the corresponding emissivity peak. We therefore investigated different types of substrate going from IR-absorbing (SiO₂, SiC and Ag) to IR-transparent (CaF₂ or air) ones. In Fig. 8 we show that while the emissivity peak at λ = 5μm is close to unity for SiO₂ substrate (see Fig. 1), its value decreases as the substrate reflectivity (for SiC and Ag) or transparency (for CaF₂ and air) is increased. Given the substrate dispersion law, other regime can be exploited, as for instance the excitation of surface waves [14]. Depending on the chose operational wavelength is thus possible to choose the most appropriate substrate.

 

Fig. 8 Angular emissivity for the pre-fractal multilayer of Fig. 1, calculated at 5 μm for TE polarization and for several different substrates (see arrows).

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3. Conclusions

We investigated one-dimensional pre-fractal Cantor multilayer structures supporting quasi-coherent thermal emission at selected infrared wavelengths and in a well-defined direction, due to the excitation of cavity resonance mode. Using two ordinary optical materials as SiO₂ and TiO₂ and for a total physical thickness as small as 18 μm, we find that it is possible to obtain a polarization-insensitive quasimonochromatic infrared source having both spectral and angular narrow divergence. Choosing a reference wavelength λ₀ = 5μm for scaling multilayer design, a spatial coherence length as high as l_s~25·λ is predicted at λ = 4.5 μm, together with a coherence time of τ_c ~1.5 ps. A change in λ₀ would in turn shift the cavity resonance modes, and thus the emissivity peak wavelengths would scale consequently. Tailoring and handling thermal radiation is a compulsory feature for a wide variety of applications ranging from infrared camouflage to selective and coherent thermal emission.