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Random Coherent Perfect Absorption with 2D Atomic Materials Mediated by Anderson Localization Judson Ryckman ACS Photonics, Just Accepted Manuscript • DOI: 10.1021/acsphotonics.7b01135 • Publication Date (Web): 30 Nov 2017 Downloaded from http://pubs.acs.org on December 1, 2017
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ACS Photonics
Random Coherent Perfect Absorption with 2D Atomic Materials Mediated by Anderson Localization Judson D. Ryckman* Holcombe Department of Electrical and Computer Engineering, Clemson University, Clemson, S.C. 29634.
KEYWORDS: nanophotonics, disorder, 2D atomic materials, Anderson localization, absorption. ABSTRACT: Tailoring the spatio-temporal confinement of light with two-dimensional (2D) atomic materials is critical to harnessing their unique and attractive optoelectronic properties. Here, we report on the interplay between the localization of light within a disordered medium and the extreme material/loss localization afforded by deeply sub-wavelength (~λ0/5000) single-layer 2D atomic materials. Structures in the regime of Anderson localization are found to support a condition termed ‘random coherent perfect absorption’ (RCPA). This yields an array of optical effects which are unachievable in conventional lossy random media, including: >99.9% absorption with Q-factors ranging from ~102 - 106, Fano-resonance behavior in 1D, co-existence of RCPA and extraordinary transmission, and angle-selective conditions supporting the coalescence of all random modes toward perfect absorption.
Two dimensional (2D) atomic materials present a host of unique properties with significant technological potential in areas spanning optoelectronics, non-linear optics, quantum information processing, and sensing. The atomic scale physical thickness and deeply sub-wavelength optical thickness of these materials presents a unique challenge in the optical domain, requiring careful optimization of both spatial and temporal confinement to achieve the desired degree of light-matter interaction. In this vein, resonant optical devices incorporating 2D atomic materials have been intensely studied within various sub-domains of optics including integrated photonics,1 photonic crystals,2,3 plasmonics,4–6 and metamaterials.7,8 A principle and motivating challenge regarding the use of 2D atomic materials in these platforms is the optical design of passives with tailored spectral properties,2,3 and active components such as lasers and their time reversed counterparts, coherent perfect absorbers.9–11 Existing platforms are generally designed systematically through careful optimization techniques where disorder is considered undesirable. Localization of light in a disordered medium is an alternative and potentially transformative approach for tailoring light-matter interactions.12,13 Progress in this field has enabled extreme manipulation over photon confinement and transport across many spectral regimes and dimensionalities. Remarkably, optoelectronic components derived from random media, such as random lasers14–16 and absorbers,17,18 enable record scale performance metrics to be achieved despite the inherent lack of a well ordered or deterministic design. Furthermore, recent work pertaining to Anderson localization effects with deeply-subwavelength nanometric films continues to test
the boundaries of modern optics,19,20 and promises to open new pathways of physical understanding and technological application. In this article, we introduce random coherent perfect absorption (RCPA) and show how it can be achieved with a single lossy 2D atomic material embedded within, or on-top of, a finite and disordered all-dielectric medium. Although the introduction of a 2D atomic material, with sub-nanometer physical thickness, presents an extreme ultra-subwavelength limit of both material and optical loss localization; we find that Anderson localized resonant modes naturally overcome this challenge and are in fact crucial to achieving the signature of coherent perfect absorption in a finite disordered medium. Through this study, a diverse array of unique resonant phenomena are identified, many of which are not achievable in conventional random systems where loss is spatially distributed.
APPROACH AND METHODS As a prototypical lossy 2D atomic material, we herein consider single-layer graphene and select a near-to-midinfrared (IR) spectral window = 1 − 4 for our analysis. In this region, graphene’s optical response shifts from interband to intraband dominated; and we therefore describe graphene’s sheet conductivity, using a Kubo formula which captures both contributions: = + .8,21 Carrier doping of graphene and modulation of the Fermi level, , provides an effective method for manipulating conductivity and therefore optical losses. This attractively offers the potential to implement high speed active devices, particularly in the mid-IR and into the THz regime, which operate on the imaginary part of a resonator’s complex frequency.
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As illustrated in Figure 1b, our disordered medium consists of a 1D all-dielectric and lossless multilayer stack comprised of / total layers with a single 2D atomic layer introduced above the 01 dielectric layer. Disorder is introduced by assigning each layer an index 2, from a random choice in a binary refractive index system 5 2 , 2 . Each layer’s thickness is chosen be 4 = 6 , 7
Figure 1. Configuration. (a) Temporal coupled mode picture of our generalized cavity and available decay channels when excited from the superstrate. (b) 1D disordered medium comprised of N layers, with the 2D atomic material located above the Mth layer.
Although a detailed investigation of active modulation is considered outside the scope of this particular work, we point the reader to related demonstrations in this area.1,22– 25 Unless otherwise noted, single layer graphene is modelled under a constant Fermi level, = 0.3 .
Here we pause to briefly summarize the requisite conditions for achieving coherent perfect absorption,10,21 and by extension RCPA arising in a disordered medium. An optical resonator oscillating at frequency , dissipates energy at a total rate = , which is the sum of damping rates associated with radiation coupling and internal losses respectively. For the one-dimensional (1D) scenario under consideration, describes the net radiation coupling rate into port 1 (i.e. reflection) and port 2 (i.e. transmission), each occurring at partial radiation coupling rates and respectively. The internal loss rate can be expressed as a sum of rates associated with all scattering and absorptive loss channels existing within the structure. All sources of loss other than the loss attributed to the 2D atomic material are considered parasitic, and we therefore express the internal damping rate as = ! . The partial absorptance associated with the 2D atomic material can then be expressed as21 " =
4 − # $ ! %
.
1
From this expression, it is clear that perfect absorption arising strictly from the 2D atomic material may occur when a critical coupling condition is achieved on resonance such that = and ! = 0 are both satisfied. For single-port illumination from port 1, this further necessitates = and = 0, such that the resonant mode is decoupled from the output and radiative coupling is entirely associated with the single-port of illumination. Elimination of parasitic losses is readily achievable when constructing the multilayer from all-dielectric layers and by assuming optically smooth interfaces with negligible scattering losses. Decoupling resonant modes from the transmission channel and driving → 0, is also achievable in disordered and highly scattering media which commonly feature broadband regions of high reflectivity,26,27 and an average transmission which decays exponentially according to 〈ln *〉 = −,/., where , is the sample thickness and . is the localization length.28,29
where # = 2.5 . The choice of eighth-wave thickness layers provides the ability to reduce the localization length in the spectral window of interest, and will enable RCPA to be achieved with an efficient number of layers. We have also examined alternative approaches for introducing disorder, such as an array of nanometric scale binary index layers, and nanometric scale arbitrary index layers where 2 is uniformly distributed over a set index interval. Regardless of the type of disorder, RCPA is achievable when the localization length is smaller than the sample length and structures are operating in the Anderson localized regime. Here we will consider three scenarios of binary refractive index contrast, :2 = |2 − 2 | = (0.2, 0.4, 0.8), which all maintain the same average index 299.9% maximum absorption.
Figure 5. Stochastic behavior inside and outside the Anderson regime. Maximum recorded absorption value from each spectra plotted as a function of 2D layer location, M, for varying refractive index contrasts: (a) :2 = 0.8, (b) :2 = 0.4, and (c) :2 = 0.2. Cumulative distribution function (cdf) for selected values of M under :2 = 0.8. Colored bars illustrate scale of localization length compared to sample.
(background) superimposed with a higher Q-factor Fano type peak, both of which achieve >99.9% peak absorption. The formation of this Fano-resonance is enabled by the interplay between two competing loss channels (i.e. reflection and absorption). On the long wavelength side of the Fano-resonance the high Q-factor mode field constructively interferes with low Q-factor mode resulting in a small but finite confinement factor W and a non-zero . On the short wavelength side of the Fano resonance, the high Q-factor mode field destructively interferes with the low Q-factor mode. This coincides with a null in the electric field located at the 2D atomic layer, driving both W and to zero and returning the reflectivity to unity. Notably, this Fano behavior cannot occur in a conventional lossy disordered medium where the loss is distributed throughout the sample. In such samples, the absorption loss rate is strictly non-zero for all mode profiles and therefore it cannot flip between zero and a finite value on either side of a resonance. The distributed loss scenario would instead produce Lorentzian peaks for both resonances. Here, the extreme physical localization of loss, achieved through the integration of a 2D atomic material, breaks this limitation and allows Fano-resonant behavior to be observed in disordered media. Stochastic Behavior and Effect of Localization. Next, we examine the stochastic behavior of peak absorption analyzed over many iterations (i = 45) of random dielectric stacks with a single lossy 2D atomic layer embedded within. We perform this analysis while sweeping the parameter 0 which determines the location of the 2D atomic layer within the / = 500 layered random medium, and we also examine the effect of the refractive index contrast :2. Figure 5a reveals a pattern of maximum absorption which indicates that for :2 = 0.8 the onset of RCPA
As the refractive index contrast, :2, decreases, the localization length . grows exponentially. For :2 = 0.4, the localization length increases to . E 10.4 # though it remains several times smaller than the sample thickness ,. As shown in Fig. 5b RCPA is still observable in this regime but only for a reduced range of 0 and at a lower probability. For :2 = 0.2 however (Fig. 5c), the localization length increases to . E 50# which is comparable to, or slightly larger than, , and the resonant modes are fully out of the Anderson regime. Due to the large ., these modes effectively over-couple to transmission pathways which destroys the possibility for observing RCPA. In principle it is possible to achieve RCPA with lower :2 and a lesser degree of disorder; however, it would require the sample , to be correspondingly increased. In effect, Anderson localization plays a key role in enabling perfect absorption to be achieved in disordered media, as this localization is critical to tailoring the coupling rates into both radiation pathways. Specifically, it is necessary to entirely decouple resonant modes from the transmission channel ( = 0, and to establish a localized resonance which achieves a low reflection port coupling rate which can be matched to the absorption rate to establish critical coupling ( = . RCPA with 2D Atomic Layers at the Surface (M=1). To this point, we have studied RCPA absorption arising from a lossy 2D atomic layer embedded within an open one-dimensional disordered medium. In this analysis, we have considered single-layer doped graphene ( = 0.3 ) as our prototypical lossy 2D atomic layer, a spectral range of = 1 − 4 , and illumination from normal incidence. Under these conditions RCPA is achievable when operating in the Anderson localized regime and when the 2D atomic material is embedded appropriately within the medium. In many cases however, it may be desirable to realize a platform capable of supporting perfect absorption when the 2D atomic layer is placed on top of, and not embedded within, the physical structure. This naturally leads us to consider if RCPA is possible in such a configuration. Although many studies have demonstrated perfect absorption phenomena from lossy ultra-thin films (~10 nm) on planar substrates,35,36 achieving a similar response from lossy 2D atomic materials is significantly more difficult owing to their sub-nanometer physical thicknesses. At
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normal incidence, the radiation coupling rate into reflection is significantly higher than the absorption loss rate of even highly doped lossy 2D atomic materials ( ≫ ). However, a recent study by Zhu et al. has shown that perfect absorption may be achieved by breaking the constraint of normal incidence and instead moving toward a grazing incidence angle.37 By moving toward grazing incidence the radiation coupling rate can be arbitrarily reduced, approaching zero as ? approaches 90°, while the internal loss rate is not significantly altered. In Figure 6a-c we compare the spectral response of a random multilayer (/ = 500, :2 = 0.8) with a single graphene layer located on the top surface (0 = 1) for three separate angles of incidence ? = 0°, 75°, 89° under TE polarization. As before, each absorption spectrum contains many peaks associated with the many resonant modes supported by the disorder. However, unlike prior characteristics, where the loss channel could be over, under, and/or critically coupled in the same sample (i.e. Fig. 3), the responses shown in Fig. 6 a & b, away from grazing incidence, are indicative of strictly under coupled behavior. As the angle increases, the maximum absorption value rises dramatically and reaches 100% at the RCPA threshold, which is found near ? ~ 88.5°. Figure 6d shows the maximum absorption as a function of incidence angle for samples of this type evaluated over ten random substrate iterations. The data is found to fit extremely well with a resonant cavity model accounting for the angle dependence of the radiation decay rate, =
@
[\\
]
^ @ A_`
a^bcdeI _`
f ;37 where 4hh is a constant represent-
ing the effective cavity depth. From the observed absorptance " ? , we apply Eq. 1 to extract the ratio of coupling For this rates / as shown in Fig. 6e. ture / = 500, 0 = 1), near normal incidence the radiation coupling rate is initially stronger than the 2D atomic material’s absorption loss rate by more than two orders of magnitude, yet this large disparity vanishes entirely at the RCPA threshold. For any angle above this threshold, the resonant energy decay into absorption may once again be over, under, and/or critically coupled in the same sample. Remarkably, for incidence near ? ~ 88.5° corresponding to the RCPA threshold, all modes in our spectral window (where the effect of material dispersion is small) are found to exhibit critical or very near-critical coupling.
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Since each resonance, R, within the sample features its own unique Q-factor and therefore unique absorption loss rate j2i , the nearly universal observation of critical coupling suggests that the external radiation coupling rate for each respective resonance, j1 , has become parametrically tied to the internal absorption rate of the same resonance. The source of this phenomenon appears to originate from a common inverse dependence on the effective cavity depth, j1 = j2i ∝ 1/4ll , and is similarly predicted to occur for multimode Fabry-Pérot type cavities. In other words, deeper penetrating modes have both a higher passive Q-factor and a smaller confinement factor, W, which forces internal and external decay rates to coalesce.
CONCLUSION In conclusion, we have studied the absorption phenomena exhibited when a single lossy 2D atomic material is integrated with a finite all-dielectric 1D random medium. Optical resonances, only in the regime of Anderson localization, are found to support the opportunity for random coherent perfect absorption (RCPA). Owing to the wide span of spatio-temporal confinement exhibited by the random resonators, RCPA modes with >99.9% absorption are identified over a wide range of Q-factors from ~102 – 106. Additionally, unique phenomena which are unachievable in conventional random systems, with distributed loss, are unlocked by the ultra-subwavelength material and loss localization provided by the subnanometric 2D atomic layer, including: (1) co-existence of extraordinarily transmissive modes and necklace states with perfectly reflecting and absorbing modes in the same medium, and (2) Fano-resonance in a 1D medium. Lastly, we demonstrated that RCPA may be guaranteed for a 2D atomic layer placed on top of a disordered all-dielectric substrate by exploiting angle-selectivity and approaching grazing incidence. At a particular angle termed the ‘RCPA threshold’, every resonant mode supported by the structure coalesces toward perfect absorption. Finally, the reader should note that RCPA is phenomenologically the time-reverse equivalent to a random laser operating at threshold,10 and the concepts emphasized here for lossy 2D materials can be adapted through time-reversal symmetry to 2D materials exhibiting gain.
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Figure 6. Angle-selectivity for a 2D atomic surface layer (M=1). Reflection, transmission, and absorption spectra for the same random structure from Fig. 2 & 3 (:2 = 0.8, / = 500), with doped graphene placed at the top surface (M = 1), under TE polarization for various incident angles ? : (a) 0°, (b) 75° , and (c) 89°. (d) Stochastic behavior of maximum absorption recorded over 10 iterations plotted vs. incident angle, and comparison to fitted model. (e) Extracted trend of decay rate ratio / vs. incident angle.
This study highlights the unique optical characteristics which can be achieved from the interplay between optical localization from disordered media and extreme material/loss localization from 2D atomic materials. Opportunities for follow up study and potential application of this platform are wide ranging and include new regimes of active/passive spectral filters, random absorbers/lasers, and sensors.
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AUTHOR INFORMATION Corresponding Author
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[email protected] ACKNOWLEDGMENT Clemson University is acknowledged for both financial support and generous allotment of compute time on the Palmetto cluster.
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Title: Random Coherent Perfect Absorption with 2D Atomic Materials Mediated by Anderson Localization Author: Judson D. Ryckman Synopsis: The left image depicts the optical platform under consideration: namely a 2D atomic layer embedded in disordered optical substrate. Right images depict example absorption spectra including the first demonstration of Fano resonance in disordered 1D media and the coalescence of all resonant modes toward perfect absorption.
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