Tailoring Infrared Absorption and Thermal Emission with Ultrathin-film Interferences in Epsilon-Near-Zero Media

Engineering nanophotonic mode dispersions in ultrathin, planar structures enables significant control over infrared perfect absorption (PA) and thermal emission characteristics. Here, using simulations, the wavelength and angular ranges over which ultrathin, low loss, epsilon-near-zero (ENZ) films on a reflecting surface most efficiently absorb and re-radiate are identified, and the design parameters that tailor the ENZ mode dispersion within these limits are investigated. While the absorption is spectrally limited to wavelengths where the refractive index ($n$) lies below unity, the angular limits are determined by the ENZ material dispersion in this range. A model of ultrathin-film interference is developed to provide physical insight into the absorption resonances in this regime, occurring well below the conventional quarter-wavelength thickness limit. Driven by non-trivial phase shifts incurred on reflection at the $n<1$ surface, these resonant interferences are shown to be universal wave phenomena in planar structures having appropriate index contrast, extending beyond ENZ materials. Selective choice of material, film thickness and loss allows fine-tailoring the mode dispersions, enabling wide variation in spectral range ($ \sim 0.1 - 1.0 \mu m$) and precise directional control of spectrally and angularly narrow-band PA and thermal radiation, paving the way towards efficient ENZ-based infrared optical and thermal coatings.


Introduction
Nanophotonics, with its ability to control and manipulate light at the nanoscale, has led to the realization of various novel applications such as sub-diffraction waveguiding, [1] imaging, [2] optical cloaking [3] and single molecule sensing. [4] These unusual optical responses are often derived from the sub-wavelength contrast in optical properties of nanophotonic structures such as metasurfaces [5] and nanoparticle arrays. [6] However, recent advances have shown that optically thin films in planar structures can achieve substantial control in modifying the reflected and transmitted light spectrum exploiting interface phase shifts that differ significantly from 0 or π [7][8][9][10] , which can be tuned to a desired value by engineering parameters like film thickness, doping etc. The absence of nanostructuring, large area scalability, and reduced fabrication and material costs make such ultrathin films (d << , d is the film thickness and  is the wavelength of light) attractive for applications in coloring, filtering, enhanced absorption in photovoltaic applications, infrared absorbers and emitters, and in reconfigurable flat optics. [11][12][13][14][15][16][17][18][19][20] One class of thin film media that has gained much interest recently are epsilon-near-zero (ENZ) materials, [21,22] which have a vanishing real part of permittivity ( ′) at a specific frequency. ENZ materials offer strong light-matter interactions and show exciting effects such as super-coupling, [23] nanoscale field confinement and enhancement, [24,25] radiation control [26,27] and extreme non-linearities. [28] Further, ENZ thin films are known to support radiative modes called Berreman modes, which have a nearly flat dispersion near the frequency where ′  0. [29][30][31] Perfect coupling of p-polarized free space light into these radiative modes has been exploited to design efficient polarization switching, [32] high harmonic generation, [33] and tunable and broadband perfect absorption (PA). [34] This strong absorption in ENZ thin films, originating from the large field enhancements imposed by field continuity and the vanishing permittivity, [25] also results in a high thermal emissivity whose spectral and angular characteristics are determined by the dispersion of the radiative modes. [35] Therefore, by tailoring the modal dispersion, both the spectral and angular response of thermal emission can be controlled, [35,36] varying from spectrally selective, narrow [37] or wide-angle emission, [38,39] to broadband and directional emission. [40] The building block of these structures, the ENZ-metal bilayer, is known to support PA for p-polarized light, which has been explained by various effects such as coherent cancelling, [41] effective impedance matching [42,43] and critical coupling to ENZ modes. [34,44] However, various properties crucial to the engineering of thermal radiation sources remain to be investigated such as the available ranges, both spectral and angular, of thermal emission from an ENZ thin film, which are important in deciding the overall characteristics of a multilayer emitter. [40] Further, the degree to which the dispersion can be engineered by systematically varying the ENZ thickness and controlling the loss has yet to be elucidated, which will determine the fine-tunability of its thermal emission features. [45,46] It is therefore important to obtain a physical picture that not only provides a quantitative insight into the absorption and dispersion in ENZ layers, but also identifies their readily tunable parameters for tailoring the response.
Here, we show that the angular and spectral range of electromagnetic absorption and thermal radiation in ENZ thin films is restricted to a characteristic regime determined by the dielectric function of the ENZ material relative to the ambient superstrate medium. Using doped cadmium oxide (CdO) and SiO2 ENZ layers on a reflecting (metallic) substrate with ambient air superstrate as model systems in the near-IR and mid-IR range respectively, we numerically demonstrate that perfect absorption of light in ultrathin layers is restricted to the spectral range where the ENZ layer is a dielectric ( ′ > 0) with refractive index (n) below unity. Further, the radiative modes giving rise to the absorption features are observed to be restricted to angles above a wavelength-dependent critical angle θc. For practical purposes, the θc obtained is equivalent to that defined for total external reflection (TER) at the air-ENZ interface, for the corresponding lossless ENZ material. Indeed, a critical angle is undefined for lossy media, especially for ENZ media at regimes where ′ ~ 0. Remarkably, the observed trends in ultrathin film absorption are not merely limited to ENZ materials, but are shown to be universal features of structures having an appropriate index contrast between the ambient medium (n1) and thin film (n2 + ik2), such that n1 > n2. This indicates that the driving mechanism of absorption in these systems, generally described in terms of the Berreman mode excitation, can be explained as a more general wave interference phenomenon applicable even in media that do not support polaritonic resonances. To explain these unique features, we develop a model of thin film interference in n < 1 ultrathin films, which predicts that these films can sustain resonant interferences well below the quarter-wave thickness limit. The key to this phenomenon is the external reflection (ER) effect at the air-ENZ interface [47][48][49] arising from the refractive index contrast, due to the simultaneous realization of below unity refractive index and dielectric nature ( ′ > 0) of the ENZ layer. The angle-dependent ER phase shifts provide the required phases to satisfy the conditions for destructive interference of reflected light, making it tolerant to a fixed reflection phase due to the reflecting substrate or the near-zero propagation phase in the ultrathin film. Through straightforward relations connecting the ER phase and magnitude at the interfaces, we explain the spectral and angular confinement of the resonant modes and show how the dispersions and the spectral and angular characteristics of the resulting thermal emission can be finely tailored by choosing appropriate ENZ layer thicknesses and loss. Finally, we show that PA in the low-loss limit in n < 1 media is achieved only in the vicinity of the ENZ regime owing to the strong field enhancements therein, removing the necessity of high material loss in the ultrathin absorbers and emitters. The general wave interference mechanism of ultrathin film absorption elucidated here will be useful in the design of thermal emitters by providing design principles to engineer the dispersion in sub-wavelength modes and expanding the choice of available materials. Moreover, the identified characteristic spectral and angular ranges of ENZ materials will be useful in controlling thermal radiation in multilayer ENZ films or metasurfaces for spectrally-or angularly-selective infrared emission and absorption. ultrathin resonator requires non-trivial phase shifts at the air-film interface, which is achieved when n2 < 1. (e) Phasor diagrams for the case in (d) with n2 = 0.45 + 0.15i for d = 100 nm (dashed), 10 nm (solid) (see text) (f) Corresponding reflectance goes to zero for the 100 nm film and ≈ 0.1 for the 10 nm film at the respective angles where the phasor sum approaches the origin. Figure 1a shows the schematic of a thin film perfect absorber, with light incident from air (n1 = 1) onto a thin film of refractive index n2 + ik2 of thickness d, on an opaque substrate (n3 + ik3). The reflectance of the system = | | 2 can be written as; [50] = 12 + 23 where is the Fresnel reflection coefficient for incidence from medium i to j, = 2 and 2 = 0 √ 2 − 1 sin 2 is the wave vector component normal to the interface in the thin film, is the complex permittivity, k0 is the free space wave number and θ is the angle of incidence from air. The reflection coefficient r may also be expressed as the coherent sum of infinitely many partial waves rm (Figure 1a). r0 corresponds to the partially reflected wave from the first interface (= r12) and = (1 − 12 2 ) 23 21 for ≥ 1 correspond to those emerging from the cavity after m roundtrips. The partial waves may be conveniently represented as phasors in the complex plane [Re(r), Im(r)]. [8,9] With negligible losses, the interface phase shifts (i.e. on reflection or transmission) are typically either 0 or π, and resonant interference conditions are determined by the phase accumulated during propagation. For example, the simplest dielectric anti-reflection coating has a thickness of d = /4n2 that provides the necessary π propagation phase for destructive interference of reflected waves. [51] Such a quarter-wave thick, lossy film placed on an opaque substrate absorbs incident light completely since both reflection and transmission are simultaneously inhibited. The dashed arrows in Figure 1b show the phasors for such a PEC-backed conventional (n2 < 1) quarter-wave thick absorber. [8] At normal incidence, the first reflection r0 has a phase shift of π (rarer to denser medium), which is cancelled out by the remaining partial waves that acquire π phase from each reflection at the PEC substrate and roundtrip in the film. The corresponding phasor trajectory then ends up at the origin, giving = | | 2 = 0.
To overcome this thickness limit of PA, the primary dependence on film thickness for phase accumulation must be overcome, requiring interface phase shifts that deviate significantly from the lossless case to satisfy the resonance conditions. Such interferences exploiting non-trivial interface phase shifts have been demonstrated in ultrathin films by introducing weakly reflecting metal substrates, [8,9] metamaterial mirrors [52] or perfect magnetic mirrors [18] instead of PECs. If a weak reflector such as Au in visible frequencies is combined with a lossy thin film, the phasors are no longer horizontal (Figure 1c). In this case, the reflection phase at the film-substrate interface differs significantly from π, allowing the partial waves to cancel out r0 even when d << . Solid arrows in Figure 1b shows the phasor trajectory of such a system [9] with parameters d = 10 nm, n2 = 4.3+0.7i and n3 = 0.44+2.24i at  = 532 nm, which returns to the origin tracing out a loop, corresponding to ≈ /12 2 .
However, constraints on the substrate material limits the applicability of these interferences on highly reflecting surfaces, [12,17] e.g. ubiquitous low cost metals such as Al and conventional metals such as Au, Ag at infrared wavelengths that behave increasingly like PECs. In order to achieve PA in ultrathin films backed by PECs (which impart fixed π phase), it is necessary to obtain non-trivial phase shifts at the top interface i.e. the air-film interface. For normal incidence at a lossless top interface with 1 < 2 , e.g. air-glass, r12 is a negative real number determined by the optical constants with a fixed phase of π. Even for highly absorbing dielectrics where r12 becomes complex, the phase deviates by less than ∼ 10 0 from the lossless case. [10,53] However, if 1 > 2 , light can undergo total internal reflection at oblique incidence.
In such cases, r12 becomes complex above the critical angle = sin −1 2 / 1 , signifying the presence of evanescent waves in the rarer medium, and above θc, incurs an angle-dependent reflection phase varying between 0 and π. [50] In most applications where n1 = 1, this cannot be realized with conventional dielectrics. However, lossless ENZ or near-zero-index (NZI) materials [22,54] with refractive indices well below 1 (Figure 1d) would allow TER of light at their surfaces above their critical angles. [47,49] Importantly, non-trivial phase shifts are also available in lossy systems (n2+ik2) where the external reflection is not total, above an equivalent critical angle defined for the corresponding lossless material, i.e. k2 = 0, = sin −1 2 / 1 (Supporting Information Section S1). This is demonstrated by the phasor diagrams in Figure 1e for p-polarized light of  = 532 nm incident on a PEC-backed low-index layer with n2+ik2 = 0.45+0.15i. For d = 100 nm (≈ /12 2 ), the phasors return back to the origin at an incident angle θ = 32 0 (>θc = 27 0 ). If d = 10 nm (≈ /120 2 ), the reflectance reaches a minimum ≈ 0.1 at θ = 68 0 . The corresponding numerically evaluated total reflectance for both thicknesses are plotted against θ in Figure 1f, showing their minima at the angles where the phasor sum reaches closest to the origin. Notably, in contrast to the highly absorbing 10 nm layer in Figure 1b, it takes many more roundtrips for the partial waves to converge to the final reflectance for the low-loss 10 nm layer in Figure 1e, stemming from the low loss nature of absorption that will be discussed later. To illustrate the spectral and angular characteristics of infrared absorption in continuous, planar media, we numerically analyze two low loss, ENZ film on metal (ENZ:M) structures.

Spectral and angular characteristics of infrared absorption
The permittivities of the ENZ medium ( 2 ( )) are described by those of CdO in the near-IR [55] and SiO2 in the mid-IR [56] , and that of the substrate ( 3 ( )) by Au, as outlined in SI Section S2. For the model parameters of CdO, the 'zero-epsilon wavelength' (ZE) where 2 ′ = 0, lies at ZE = 1900 nm (Figure 2a). Figure 2b shows that n2 ≈ 0.3 at ZE and increases towards shorter , reaching 1 at n=1 ≈ 1720 nm. In the wavelength window Δ between n=1 and ZE, the thin film is a dielectric with n2 < 1, [57] where ER with the non-trivial phase shifts are realized. Figure   2c plots the dispersion relations for transverse magnetic modes in the CdO:Au system for d varying from 20 to 400 nm (see SI Section S3 for details of numerical calculations).
Interestingly, the dispersions are limited on the short wavelength side by n=1, and on the long wavelength side by ZE, above which the layer becomes metallic in nature i.e. 2 ′ < 0. [43,46,58] Figure S2 in SI plots the reflectance spectra for a CdO:PEC system demonstrating a close match with the data above, indicating that Au is well approximated by a PEC in the investigated wavelength range. [59] The flat dispersion of the film with the lowest thickness (20 nm) is the well-known Berreman ENZ mode seen in very thin films nearly independent of its optical environment. [46,60] In thicker films, the ENZ mode dispersion moves away from ZE and the optical environment (symmetric or asymmetric, metallic or dielectric) may be expected to play a more prominent role in deciding the nature of supported modes. Since the highest thickness is 400 nm (< /4), the modes at higher thicknesses evidenced above are not conventional Fabry-Perot modes, which only start appearing for shorter wavelengths as shown in SI Figure S3. Furthermore, the wavelength-dependent critical angle, sin ( ) = 2 ( ) (grey solid line in Figure 2c), overlaid on the dispersion relations shows that the dispersions lie to its right irrespective of the film thickness, i.e. the modes are supported only in the regime θ > θc, where light undergoes external reflection at the ENZ surface. The dispersion profiles gradually shift towards θc() with increasing thickness, with the thickest layer dispersion nearly lying on θc() (see Figure S4 for plots showing this for d > 400 nm and for a detailed discussion on this limiting behavior). The critical angle approaches 90 o as n2 approaches 1 (at n=1) and the radiative modes vanish at larger values of refractive index for  < n=1.
The corresponding plots for the SiO2:Au system in the mid-IR are shown in Figure 3. Here, n2 goes below 1 when  > 7.25 m and the presence of the reststrahlen band above 8 m induces the ENZ condition. [58] Consequently, in the range Δ = 7.25 -8 m, SiO2 possesses a low-loss,  It is worth noting that the gradual shift in dispersion with thickness and the limiting behavior at higher thicknesses where the dispersions approach θc() are present in both ENZ systems, within their n < 1 spectral regimes. For the CdO ENZ system, this spectral range spans ≈ 200 nm in the vicinity of ZE whereas for SiO2, it is ≈ 1m. In both systems, the angular range θ > θc is determined by the ENZ material dispersion, sin ( ) = 2 ( ). To verify that these features arise from a general wave interference phenomenon not necessarily restricted to ENZ or polaritonic media, the reflectance of a system where the ENZ layer is replaced by a lossy dielectric with n2 = 1 and the ambient medium has a refractive index greater than the dielectric layer is presented in SI Section S5. Strong absorption above the critical angle similar to those shown in Figure 2   To understand these universal features in the absorption of thin films in their n < 1 regime, we write the three medium reflection coefficient in Equation 1 in the form = 12 − 2 1− 12 2 by setting 23 1 r  , corresponding to a PEC substrate. [50] Here, e 2 gives the phase advance (and attenuation) for a round trip in the film. For strong absorption, we require that the terms in the numerator of r cancel each other, giving two conditions to be satisfied simultaneously viz. | 12 | ≈ | 2 | and a phase difference ≈ 0. [12] Physically, this signifies complete destructive interference between the reflected wave at the first interface and the resultant of the partial waves emerging from the thin film. Figure 4a plots the phases of r12 (12, reflection phase shift at the air-ENZ interface) and e 2 (d, propagation phase in the film) for CdO:Au with d = 400 nm, at a representative  = 1800 nm lying in the Δ range (solid lines). A comparison of these plots at three different wavelengths is shown in Figure S9. For the equivalent lossless ENZ (dashed lines), 12 varies continuously in the range 0 to π above θc. On the other hand, = 2 Re( 2 ) is non-zero at low angles (< π as d < /4n2), and falls to zero above θc. Introducing loss in the film perturbs the phases as shown by solid lines in Figure 4a. The plot indicates that the phase shifts determining resonant interferences originate dominantly from the reflection phase at the top interface, in contrast to those in Fabry-Perot resonances (determined by propagation phase in the film) and the previously investigated ultrathin film interferences [11][12][13][14][15][16][17][18][19][20] (reflection phase at the bottom interface). Figure 4b plots the variation in magnitudes of r12 and e 2 for real and lossless ENZ. In the absence of loss (dashed lines), | r12| rises up to unity at θc while |e 2 | = −2 Im( 2 ) has unit magnitude below θc and drops off sharply above it as Im( 2 ) becomes non-zero. Both magnitudes are limited by loss as shown by solid lines when kz2 becomes complex valued. Evidently, the crossover between the two magnitudes occurs near θc for both lossless and lossy media. Note that these features are generally valid at wavelengths where external reflection occurs i.e. in the Δ window. Together, Figure 4a,b show that r will be minimum near θc corresponding to an optimal match between the magnitudes and phases for d = 400 nm, explaining why its dispersion in Figure 2c lies close to θc() throughout the Δ range.
In order to explain the thickness and polarization dependence of the dispersions, 12 for both s and p polarizations at  = 1800 nm are plotted along with d for d in the range 20 -400 nm in Figure 4c. Evidently, 12 for s-polarized light cannot match d for any thickness in the investigated range, explaining why interferences effects are observed only for p-polarization. It is worth noting that using the appropriate Fresnel reflection coefficients for magnetic media, similar effects may be expected in the case of materials having a near zero magnetic susceptibility. In contrast to its phase, |e 2 | in Figure 4d has a pronounced thickness dependence originating from its exponential dependence on d. In general, the phases match in two angle ranges: one at low angles and one at larger angles. Importantly, |e 2 | and |r12| match only at angles above θc, due to the steep drop off in |e 2 | above the critical angle as the wave vector in the medium becomes dominantly imaginary and the increase in |r12| due to the effect of external reflection. Further, the two magnitudes match at larger angles for lower thicknesses.
Together, these observations qualitatively explain the confinement to angles above θc and the shift in reflectance minima with thickness in Figure 2  can act as a useful design parameter for tailoring dispersion in ENZ media, in addition to the well explored carrier density tunability of the ENZ spectral range. [57]  To illustrate the potential application of these results in tailoring thermal emission, we analyze the emissivity of the CdO:Au structure in its Δ range, calculated as 1-R, where R is the reflectance of the structure. [35]  In order to explore the angular dependence of emissivity of the ENZ structure, polar plots in Figure 5(d-f) compare emissivity as a function of angle of incidence at three different wavelengths in the Δ range of CdO: 1750, 1800 and 1850 nm. A notable feature at each of these wavelengths is the thickness dependent angular selectivity of the emissivity, which can be tailored from grazing angles for the lowest thicknesses up to the respective θc for larger thicknesses. For example, the emissivity is restricted to lie above θc = 64 0 at 1750 nm and θc = 33 0 at 1850 nm for all the ENZ thicknesses, calculated from the refractive index of the ENZ medium. Such control over angular emission can be useful in thermal camouflaging and directional beaming applications. [35] The angle-dependent features of thermal emissivity can also be straightforwardly understood from the dispersion relations in Figure 2c, where at any given wavelength in the Δ range, the modes are constrained to lie towards the right of θc(),

Tailoring thermal emission
i.e. at angles above the critical angle.
Finally, we analyze the locations of zero reflectance, i.e. PA, along the dispersions and its dependence on material loss. (see SI Section S7 for numerical methods). Figure 5g shows the PA locations for varying d (20 -400 nm) and for two values of loss (scattering rate): low, typical value for doped CdO (squares) and high, typical value for indium tin oxide (circles) (see SI Section S2 for material parameters). In general, there are two PA points for a given value of loss and thickness, one at lower angles and another at higher angles. [43] The low angle PA points lie closer to ZE for lower losses. Figure 5h shows the corresponding emissivities calculated for the low system for the various thicknesses at their respective PA wavelengths that vary between 1850 -1900 nm (low angle PA). Unit emissivities, corresponding to perfect absorption of light, are observed with high angular and spectral selectivity depending on wavelength and ENZ thickness, illustrating highly efficient tailored thermal emission in these sub-wavelength ENZ layers. Although the tunability and limiting ranges of ENZ thermal emissivity are demonstrated here at near-IR wavelengths, the results presented earlier for the case of SiO2, a mid-IR ENZ material, shows that these results are equally applicable in the infrared regions of interest for thermal emission such as the atmospheric transparency window (8-13 m).
A combination of low loss and low thickness in a perfect absorber may be counter-intuitive considering that a reduced absorber volume is expected to be compensated by sufficiently high loss. However, as remarked earlier, low loss PA in metal-backed ENZ layers has been predicted, which requires lower loss for thinner films. [42,61] To explain this, we note that the power dissipated per unit volume in an absorbing medium is given by = 0 ′′ | | 2 /2, where 0 is the free space permittivity, ′′ is the imaginary part of permittivity and E is the electric field.
Clearly, the effect of low loss ( ′′ ) may be compensated by increased electric field strengths.
Considering a PEC-backed low loss ENZ layer (Figure 1d) that satisfies the PA condition for p-polarized light, the absence of a reflected field implies that the incident and transmitted fields are related by the continuity relation 1 = 2 2 and the field enhancement in the ENZ medium is | 2 |/| 1 | ≈ 1/ 2 ′′ . Thus, the field in the ENZ layer scales as | 2 | ∝ 1/ 2 ′′ . This gives ∝ 1/ 2 ′′ , leading to stronger absorption for lower loss in ENZ systems. Additionally, this explains why the PA points for the low loss case lie closer to ZE than the high loss case.
The material loss plays an important role in determining infrared absorption and emission linewidths as well as the achievable field enhancements in the structures. The electric field profiles and enhancement factors are compared for the two losses and varying thicknesses at and away from PA conditions in SI Section S8. The high field enhancement at PA may be effectively exploited for a variety of applications e.g. designing high efficiency photodetectors. [25,62]

Conclusion
In conclusion, we have shown that resonant interferences on a highly reflecting surface can be sustained employing an ultrathin film of n < 1, surpassing the quarter-wave thickness limit for perfect absorption on a PEC surface. The continuous range of phase shifts available on external reflection at the air-film interface provides the necessary phases to satisfy the destructive interference conditions. This is in contrast with conventional interferences that depend on propagation phase or previously investigated ultrathin film interferences exploiting substrate reflection phases. The established conditions for ultrathin film interferences such as weakly reflecting substrates and/or large intrinsic material loss are overcome here, allowing absorption resonances even on PEC substrates, albeit at oblique incidence. The low loss realization, a unique feature distinct from previous demonstrations that used highly lossy films or substrate materials, is achieved only in the ENZ limit due to the enhanced ENZ field-assisted

Supporting Information
Supporting Information is attached at the end of this document.

Conflict of interest
The authors declare no conflict of interest.

Tailoring Infrared Absorption and Thermal Emission with Ultrathin-film Interferences in Epsilon-Near-Zero Media
Ben Johns*, Shashwata Chattopadhyay and Joy Mitra* School of Physics, Indian Institute of Science Education and Research, Thiruvananthapuram, India 695551 E-mail: ben16@iisertvm.ac.in, j.mitra@iisertvm.ac.in Section S1. Analyzing reflection coefficient in the case of external reflection The polarization-dependent Fresnel reflection coefficients at an interface of media i and j are given by [1]  For a PEC substrate, 23 = −1. This simplifies the above two equations to: For the case of 2 < 1 such as an ENZ or NZI medium, in the lossless case the reflection phase shifts are angle dependent above the critical angle [ Fig. S1(a), dashed curves], given by: In the presence of loss ( 2 + 2 ), analogous phase shifts are still obtained as seen from solid curves in Fig. S1(a). Fig. S1(b) plots the magnitude of 12 showing the effect of loss on total external reflection. In the lossy case, this is referred to as external reflection due to the fact that reflection is not total anymore. The validity of TER in the limit of low losses can be inferred by plotting the magnitude of the reflection coefficient for decreasing values of loss, as shown in the figure S1(c), plotted for varying from 0.018 eV to 0.0 eV. In the low loss limit, the external reflection tends to become total. The ENZ material here is doped CdO.

Defining the critical angle
The critical angle is defined as sin = 2 / 1 , where 2 + 2 = √ 2 ′ + 2 ′′ . Although total external reflection is not well defined for a lossy medium, the is calculated for a lossless material equivalent with the same real refractive index to demarcate the external reflection region of interest that contribute to the observed absorption features. The use of this nominal critical angle in the case of lossy media is motivated by the following reasons: 1. The reflection minima in Fig 2 and 3 clearly reach a limiting case with increasing ENZ layer thickness (ref ( ) curve in Fig 2c and 3d). The discussion on Fig 4 and Fig S4-S6 provide further evidence that the limiting case is the defined for the lossless limit.
2. The significance of however is more general, which is demonstrated in Fig S8, where resonant interferences are demonstrated in dielectric structures without ENZ or polaritonic resonances. The numerical results show that absorption features restricted to lie above the nominal critical angle at the top interface are found in these structures as well.
These observations guide us to the conclusion that the absorption in the lossy system is confined to angles above , defined for the lossless system.

Section S2. Thin film and substrate permittivities
Dielectric function of the ENZ layer is given by a modified Drude permittivity model, where is the angular frequency, ∞ = 5.3 is the background (high frequency) permittivity, = 2.28 × 10 15 rad/s (1.5 eV) is the plasma frequency, = 1900 nm (zero-epsilon wavelength), and = 2.8 × 10 13 rad/s (0.018 eV) is the scattering rate, corresponding to typical values for doped CdO in this wavelength range [2] .
corresponds to typical value for CdO in this wavelength range [2] .
ℎ ℎ corresponds to typical value for indium tin oxide (ITO) in this wavelength range [4] .

Section S3. Eigenmode analysis
The dispersion relation for TM modes in a three medium structure is obtained as a solution ( , ) of the equation [5] 1 + 1 3 3 1 = tan( 2 ) ( 2 3 3 2 where is the in-plane wave number and the transverse wave number in medium i = 1,2,3, which are related by 2 = 2 / 2 − 2 . A real and complex representation is chosen here, in accordance with previous numerical studies on such systems [6,7] .   In ENZ media, the Brewster's angle is approximately equal to the critical angle [8] . To verify that the limiting behavior is not a Brewster angle absorption, Fig S6 compares S6a) and 5000 nm SiO2 (Fig S6b) on Au. Evidently, the absorption approaches the critical angle, and not the Brewster's angle, in the limiting case.
The Brewster's angle is calculated numerically as the angle of minimum reflectance at the air-ENZ interface to take into account the ENZ loss.

Section S5. Interference in the absence of ENZ or polaritonic media
As evident from the reflectance maps in Fig. 2

Section S7. Analysis of perfect absorption
While various complementary descriptions may be used to analyze perfect absorption in the system, we use the formalism of Luk et al [6] to identify PA locations from the eigenmode analysis described in SI section 3. From the real wave number-complex frequency solutions of the dispersion relation in eqn. S10, the condition for PA is a zero imaginary part of , i.e.
( ) = 0. A positive (negative) imaginary part of the mode frequency corresponds to growing (decaying) fields in the structure. The sign of 1 is chosen such that it always corresponds to an incoming wave in medium 1 (air). A zero imaginary part thus corresponds to an incoming plane wave in the first medium that is completely absorbed in the structure without any reflection. Figure S11a,b shows the trajectory of modes in the complex frequency plane i.e.
Im ( ) vs Re ( ) for thicknesses from 20 nm to 400 nm for = 2.8 × 10 13 and ℎ ℎ = 6.0 × 10 13 rad/s respectively. The PA locations are identified where the trajectories cross the x-axis, which agree well with the reflectance minima extracted numerically from eqn. 1 in the main text. Section S8. Field profiles at and away from the PA condition Figure S12 (a-e) shows color maps of the local field enhancement for the z-component of the field ( ) with respect to the incident (background) field, at the respective PA wavelengths for three CdO film thicknesses and two values of . Realization of PA is characterized by the absence of any reflected wave in air. The plots further show the variation of field enhancement across the layers (solid blue/cyan lines), quantifying the field enhancement within the ENZ layer. It is worth noting that for equal thickness, the film with higher exhibits weaker field enhancement compared to that with lower . Fig S12(f) plots the percentage of light absorbed in the Au substrate as a function of ENZ film thickness. Interestingly, even for the low-loss thinnest film (20 nm), only 4% of the incident light is dissipated in the substrate with the remaining 96% absorbed by the thin film, which rises to 99% in the thicker films. Fig. S13 similarly plots the field enhancement for samples at = 1750 nm, away from their PA conditions. Here, the non-zero reflectance manifests itself in the field pattern in air and the subdued field enhancement in the ENZ medium.