Coherent perfect absorbers: linear control of light with light

TL;DR

This paper reviews the concept of coherent perfect absorbers, devices that achieve complete electromagnetic energy absorption through wave interference control, highlighting design principles, structures, and potential applications in nanophotonics.

Contribution

It provides a comprehensive overview of the theoretical foundations, various photonic structures, and future prospects of coherent perfect absorbers, integrating recent advances in the field.

Findings

Complete absorption achieved via wave interference control.

Various structures enable coherent perfect absorption, including graphene and 3D systems.

Potential applications in nanophotonics and sensing.

Abstract

Absorption of electromagnetic energy by a material is a phenomenon that underlies many applied problems, including molecular sensing, photocurrent generation and photodetection. Commonly, the incident energy is delivered to the system through a single channel, for example by a plane wave incident on one side of an absorber. However, absorption can be made much more efficient by exploiting wave interference. A coherent perfect absorber is a system in which complete absorption of electromagnetic radiation is achieved by controlling the interference of multiple incident waves. Here, we review recent advances in the design and applications of such devices. We present the theoretical principles underlying the phenomenon of coherent perfect absorption and give an overview of the photonic structures in which it can be realized, including planar and guided-mode structures, graphene-based systems, parity- and time-symmetric structures, 3D structures and quantum-mechanical systems. We then discuss possible applications of coherent perfect absorption in nanophotonics and, finally, we survey the perspectives for the future of this field.

Equations

$$\left(\begin{array}{c}{b}_1\\ b_2\end{array}\right)=\hat{S}\left(\begin{array}{c}{a}_1\\ a_2\end{array}\right),\hat{S}=\left(\begin{array}{cc}{r}_{11}& t_{12}\\ t_{21}& r_{22}\end{array}\right),$$

$$\hat{S}{\mathbf{a}}_{\text{CPA}}=0,$$

$$\mathcal{A}\equiv 1-\frac{{\left|b_1\right|}^2+{\left|b_2\right|}^2}{{\left|a_1\right|}^2+{\left|a_2\right|}^2}.$$

$$r=-{\left(1+\frac{2{\epsilon }_{0}c}{{\sigma }_s}\right)}^{-1},t=r+1,$$

$$|r{|}^2+|r+1{|}^2\le 1.$$

$$\epsilon (\mathbf{r})={\epsilon }^{\ast }(\mathcal{P}\mathbf{r}),$$

$$|\varphi\rangle=\frac{1}{\sqrt{2}}\Big{(}|1_{a},0_{b}\rangle+e^{i\varphi}|0_{a},1_{b}\rangle\Big{)},$$

Full text

Introduction

Interference is a ubiquitous wave phenomenon occurring whenever two or more coherent waves overlap, leading to a spatial redistribution of energy. It underlies the operation of numerous devices, including antennas, interferometers, spectrometers and Bragg reflectors Saleh and Teich (2013); Haus (1984); Ishimaru (1990). One interesting recent discovery in optics and photonics concerns the role that interference can have in regulating light absorption. In particular, it is possible for a device to perfectly absorb an incident electromagnetic wave if all the wave components scattered out by the device destructively interfere Salisbury (1952); Fante and McCormack (1988); Gorodetsky and Ilchenko (1999); Chong et al. (2010). Here, we use the term absorption in a broad sense, meaning any process by which the energy of an electromagnetic wave is transferred to a medium and hence converted, for example, to heat, electrical current or fluorescence. Perfect absorption is of great interest for a range of optical applications Ra’di et al. (2015); Kats and Capasso (2016), including radar cloaking Salisbury (1952); Fante and McCormack (1988); Vinoy and Jha (1996), sensing and molecular detection Liu et al. (2010); Kravets et al. (2013), photovoltaics Luque and Hegedus (2008), and photodetection Konstantatos and Sargent (2010); Knight et al. (2010).

A classic example of interference-assisted absorption is the Salisbury screen Salisbury (1952); Fante and McCormack (1988), an absorber for microwaves consisting of a thin resistive sheet placed above a flat reflector. With a specific sheet resistivity and a spacing of exactly one quarter of the wavelength, the reflectance goes to zero due to destructive interference between the scattering pathways in the device; an incident wave therefore delivers all of its energy into the sheet. Perfect absorption can also be achieved with lossy resonant cavities in the regime in which the cavities act as critically-coupled resonators  Gorodetsky and Ilchenko (1999); Cai et al. (2000); Tischler et al. (2006) at infrared and optical frequencies. Another conceptually distinct route to total absorption is ideal impedance matching between free space and a lossy substrate, for example using metamaterials Nefedov et al. (2013); Baranov et al. (2015).

The Salisbury screen and similar devices are one-port networks: the light enters through a single channel (such as a plane wave incident on one face of a slab with a perfect reflector on the other side) and is perfectly absorbed owing to destructive self-interference. A coherent perfect absorber (CPA) Chong et al. (2010) generalizes this phenomenon to arbitrary incoming waveforms that can consist of two or more waves, such as waves incident on opposite faces of an open slab or film Chong et al. (2010); Wan et al. (2011); Zhang et al. (2012). To achieve perfect absorption, it is essential to appropriately choose the values of the system parameters, the operating wavelength and the input waveform, including the intensities and relative phases of the input beams. The sensitivity of CPAs to the input waveform provides the opportunity to flexibly control light scattering and absorption Zhang et al. (2012); Fang et al. (2015); Papaioannou et al. (2016a). The possibility of achieving perfect absorption at subwavelength lengthscales is extremely interesting for nanophotonic applications.

In this Review, we discuss the theoretical ideas underlying coherent perfect absorption and overview recent advances in realizing various types of CPAs with several platforms, including ultrathin metasurfaces, graphene-based structures, gain/loss structures and waveguides. We also discuss emerging applications for CPAs in sensing and all-optical information processing. Although our emphasis is on classical optics, coherent perfect absorption is a general phenomenon arising from the interplay of interference and dissipation, and can therefore be realized in other interesting contexts, including acoustics Wei et al. (2014); Song et al. (2014); Ma et al. (2014); Duan et al. (2015), polaritonics Zanotto et al. (2014), and quantum optics Roger et al. (2015).

Theoretical background

The simplest yet most illustrative example of a CPA is a two-port linear system. This can represent a variety of photonic structures, such as an absorbing planar slab or film illuminated on both sides by coherent light beams with fixed incidence angles (Fig. 1a). Each input beam is partially reflected and partially transmitted; some of its energy can also be absorbed into the medium. The total outgoing wave on each side, therefore, is a superposition of a reflected wave and of a wave transmitted from the opposite side.

This picture can be described rigorously using the scattering matrix formalism Saleh and Teich (2013); Haus (1984). The relationship between input and output waves is described by

[TABLE]

where the complex scalars $a_{i}$ and $b_{i}$ denote, respectively, input and output wave amplitudes in the $i$-th channel (or port); $r_{ii}$ are the reflection coefficients, and $t_{ij}$ the transmission coefficients. This formalism can be extended to describe more scattering channels and to include polarization degrees of freedom. The scattering matrix $\hat{S}$ is derived from Maxwell’s equations and depends on the geometry of the scatterer, material properties and operating wavelength. It can be calculated in several ways, including the transfer matrix method Saleh and Teich (2013), Green’s function methods Lee and Fisher (1981); Sentenac et al. (2013), finite-element methods with scattering boundary conditions Jin (2014) and coupled-mode theory Haus (1984); Poladian (1996); Suh et al. (2004); Kang et al. (2013, 2014); Zhu et al. (2016).

Coherent perfect absorption occurs when every output wave component vanishes, i.e.,

[TABLE]

where $\mathbf{a}_{\textrm{CPA}}$ is a non-zero vector of input wave amplitudes. This means that at least one eigenvalue of $\hat{S}$ is zero; the eigenvector $\mathbf{a}_{\textrm{CPA}}$, the ’CPA eigenmode’, coincides with a solution to Maxwell’s equations obeying purely incoming boundary conditions. If the structure is supplied with a different set of inputs, not matching a CPA eigenmode, Eq. (2) will not be satisfied and outgoing waves will emerge.

Symmetries of the scatterer result in symmetries of the $\hat{S}$ matrix. If the system is Hermitian — that is, the dielectric function and permeability are real everywhere — it follows from Maxwell’s equations that $\hat{S}$ is unitary, making Eq. (2) impossible to satisfy. In other words, if the material is not lossy or amplifying, the electromagnetic energy entering the system is equal to that leaving it. For non-magnetic lossy media, the dielectric function $\varepsilon(r)$ has a positive imaginary part, which results in eigenvalues of $\hat{S}$ smaller than unity. Note that the scattering matrix formalism distinguishes between intrinsic material losses, which are induced by dissipation of electromagnetic energy, and out-coupling losses, which are caused by the openness of the system. For instance, out-coupling losses are present even in the Hermitian case.

In most cases, $\hat{S}$ is also constrained by optical reciprocity Potton (2004). In the steady-state and linear regime and in the absence of magneto–optical materials, $\hat{S}$ must be symmetric; for the two-port case, this means that $t_{12}=t_{21}=t$. Importantly, adding loss (or gain) does not necessarly violate reciprocity, despite breaking time-reversal symmetry. Unless otherwise specified, in the following we assume that reciprocity holds.

The $\hat{S}$ matrix can also be constrained by geometrical symmetries. For instance, if a two-port structure is symmetric under a mirror reflection that exchanges the ports, $r_{11}=r_{22}=r$. The two eigenvalues of $\hat{S}$ are then $r\pm t$ and the corresponding eigenvectors $\mathbf{a}_{\pm}\propto\left({1,\pm 1}\right)$, with the $+$ and $-$ solution corresponding to a symmetric and antisymmetric input of equal intensity.

To understand how the CPA condition set by Eq. (2) can be satisfied, we analytically continue the frequency $\omega$ (on which $\hat{S}$ depends) to complex values Chong et al. (2010). Zeros of $\hat{S}$ — that is, occurrences of zero eigenvalues — generally occur at discrete points in the complex $\omega$ plane (Fig. 1b). When $\varepsilon(r)$ is real, the zeros lie in the upper half-plane, $\textrm{Im}[\omega]>0$; these have a one-to-one correspondence with the scattering resonances of the system, and the zeros themselves coincide with ’unphysical’ solutions featuring input waves that grow exponentially with time Chong et al. (2010); Weinstein (1969). If loss is increased by adding a positive imaginary part to $\varepsilon(r)$, the zeros move in the plane, typically downwards. For a certain amount of loss, a zero can move onto the real axis, $\rm Im[\omega]=0$, producing a physical CPA eigenmode.

The CPA condition can be interpreted as the time-reverse of the lasing threshold Poladian (1996); Chong et al. (2010). The complex conjugate of a CPA solution, for real $\omega$, is a wave that is purely outgoing rather than incoming. Such a solution describes a laser poised exactly at the lasing threshold Haken (1985); Ge et al. (2010); it satisfies Maxwell’s equations for a dielectric function $\varepsilon^{*}(r)$ that exhibits gain rather than loss (note that above the lasing threshold, the dielectric function becomes intrinsically nonlinear and the CPA-laser correspondence typically breaks down Nireekshan Reddy and Dutta Gupta (2013); Reddy et al. (2013)). In the complex $\omega$ scenario, threshold lasing solutions correspond to poles, rather than zeros, of $\hat{S}$ (that is, points at which an eigenvalue of $\hat{S}$ diverges). For real $\varepsilon(r)$, the poles lie in the lower half-plane, mirroring the CPA zeros; adding gain moves the poles and zeros upward. This relationship between zeros and poles is particularly important for the CPA-laser structures discussed later. Note that this is unrelated to the idea of laser cooling as an ’optically pumped laser running in reverse’ Mungan and Gosnell (1981).

Absorption in a CPA is highly sensitive to the illumination conditions, particularly to the relative phase of the input field components. For a two-port structure, we can define the joint absorption Baldacci et al. (2015) as

[TABLE]

This is a convenient figure of merit, because $\mathcal{A}=0$ for a Hermitian structure and $\mathcal{A}=1$ for a CPA. The normalized output intensity $1-\mathcal{A}$ for a uniform dielectric slab near a CPA frequency is plotted in Fig. 1c for symmetric, antisymmetric and incoherent input beams of equal intensity. CPAs may occur both for symmetric or antisymmetric modes, depending on the frequency. Remarkably, at the CPA frequency of the symmetric and antisymmetric mode, the other eigenmode of $\hat{S}$ experiences a lower absorption than incoherent inputs: this is because the outgoing wave components interfere constructively, suppressing absorption. This analysis can be extended to two-port structures lacking mirror symmetry, whose CPA eigenmodes consist of beams with unequal intensities Piper et al. (2014); Piper and Fan (2014); Baldacci et al. (2015); Zanotto et al. (2017).

An important limiting case of a two-port CPA is a flat resistive sheet much thinner than the operating wavelength, such as a metasurface or a graphene layer. For normal incidence, $\hat{S}$ has a simple form. In addition to reciprocity and mirror symmetry, the reflection coefficient $r$ and the transmission coefficient $t$ are expressed through the complex-valued sheet conductivity $\sigma_{s}$ (in SI units), as

[TABLE]

where $\varepsilon_{0}$ is the permittivity of free space and $c$ is the speed of light Ra’di et al. (2015); Luo et al. (2014); Liu et al. (2014). In the absence of gain, energy conservation requires that

[TABLE]

From this, it can be shown that absorption under single-port illumination can only reach a maximum of 50%, which occurs when $\sigma_{s}\,=\,2\varepsilon_{0}c\,\approx\,1/(188.4\,\Omega)$ and $r=-1/2$. It follows from Eq. (4) that the eigenvalues of $\hat{S}$ are $\{-1,1+2r\}$, thus satisfying $r=-1/2$ realizes a CPA: if the sheet is illuminated symmetrically with two coherent beams, it coincides with an anti-node of the electric field and absorption reaches 100% Zhang et al. (2012); Ra’di et al. (2015); Li et al. (2015). Conversely, for antisymmetric beams, the sheet is located at a node and the absorption vanishes. This example demonstrates that perfect absorption can occur even for an absorber of subwavelength thickness; this can be generalized to more complex 2D and 3D geometries, and is one of the most interesting features of CPAs.

Perfect absorption at subwavelength thickness also underlies the operation of the already mentioned Salisbury screen Salisbury (1952); Fante and McCormack (1988). The separation between the resistive sheet and the perfect reflector ensures that the incident wave and the wave returning from the reflector enter the sheet with equal phase; if the sheet conductivity is tuned to $\sigma_{s}\,=\,2\varepsilon_{0}c$, perfect absorption ensues. Interestingly, changing the reflector type (for instance, from a perfect electrical conductor to a perfect magnetic conductor) affects how the incident and reflected waves interfere at the absorbing region, changing the total absorption. Single-port thin-film absorbers and their relationships with CPAs are discussed in detail in Refs. Li et al. (2014, 2015); Ra’di et al. (2015); Kats and Capasso (2016). It is worth emphasizing that one-port devices cannot exhibit coherent control of absorption through beam modulation.

Planar structures

Dielectric slabs

The simplest realization of a CPA is a slab of lossy homogeneous material illuminated by plane waves from both sides Chong et al. (2010); Longhi (2010a); Gmachl (2010), which acts as the time-reverse of a Fabry-Pérot laser Haus (1984). A very similar effect was also discussed in the context of optical fiber gratings Poladian (1996) (the CPA eigenmodes were called ’loss resonances’ and identified as the time-reverse of threshold lasing modes).

A simple experimental demonstration of a slab CPA was realized using a silicon wafer Wan et al. (2011). CPA resonances were observed in the $1000$ nm wavelength range, with measured output intensities showing strong dependence on the relative phase of the two input beams. The intensities approached zero at $\pi$ phase difference (Fig. 2a). Several other planar CPAs have been theoretically studied, including composite metal–dielectric films Dutta-Gupta et al. (2012a), heavily-doped silicon films Pu et al. (2012) and homogeneous lossy slabs bounded by dispersive mirrors Villinger et al. (2015).

It was predicted that dielectric photonic crystal slabs can realize a variant of perfect absorption known as ’degenerate critical coupling’ Piper et al. (2014); Piper and Fan (2014). Typically, the zeros of $\hat{S}$ occur at distinct frequencies; however, it is possible to design a resonant two-port mirror-symmetric cavity supporting degenerate resonances with opposite mirror symmetries, so that both eigenvalues of $\hat{S}$ are zero at a certain frequency. The absorption at that frequency becomes insensitive to the relative phase of the two input beams — the exact opposite of the absorption modulation discussed above. Moreover, perfect absorption occurs regardless of the intensity of the input beams, and even for a single input beam incident on either port. This phenomenon has been demonstrated numerically in a photonic crystal slab incorporating a graphene layer to provide dissipation; the slab contained guided resonances that could be tuned by changing the crystal parameters to achieve the mode degeneracy Piper et al. (2014); Piper and Fan (2014).

Diffraction gratings

Another route to realizing multi-port CPAs is to use a diffraction grating Yoon et al. (2012); Dutta-Gupta et al. (2012b); Jung et al. (2015). A grating of period $D$ couples diffraction orders with lateral wavenumbers differing by $2\pi/D$. The input channels are two beams incident on the same side of the grating, but at different angles (Fig. 2b). The angles are chosen so that the lateral wavenumbers match the grating’s surface plasmon polariton mode at different diffraction orders. This produces a $2\times 2$ scattering problem in which the input and output channels are not reciprocal partners; the effective $\hat{S}$ matrix is asymmetric (the actual reciprocal partners do not satisfy the wavenumber-matching condition, and are thus decoupled). Because the absorption is mediated by surface plasmons, this CPA can be regarded as the time-reverse of a spaser Bergman and Stockman (2003), and is accompanied by a substantial field enhancement around the grating Yoon et al. (2012).

The realization of a CPA based on the diffraction grating configuration has been reported in REF. Jung et al. (2015). The tuning of the system to the CPA condition can be aided by incorporating a medium with adjustable optical gain Giese et al. (2014); Yoon et al. (2015); Jung et al. (2015) or by varying the temperature (which alters the free-electron collision rate in a metal grating Jung et al. (2015)).

Metasurfaces

There is currently a considerable interest in using planar metamaterials — metasurfaces — to realize CPAs Zhang et al. (2012); Kang et al. (2013); Fang et al. (2014); Kang et al. (2014); Nie et al. (2014); Fang et al. (2015, 2016a); Zhu et al. (2016). The meta-atoms (unit cells of the metasurface) can be designed to tailor the electromagnetic properties at any desired frequency Meinzer et al. (2014). A metasurface-based CPA was first demonstrated using a free-standing plasmonic metasurface (Fig. 2c)  Zhang et al. (2012), which had thickness $\lambda/13$ ($\lambda$ indicates the wavelength) and was tailored to function as a CPA at exactly $\lambda=1550\,\textrm{nm}$ with normally-incident beams on each side. This system acts like the thin-film CPA described in Eq. (5), with the meta-atom parameters determining the effective sheet conductivity. Subsequent experiments demonstrated ultrafast switching between the absorbing and transparent regimes of a metasurface, obtained by controlling the relative input phase Fang et al. (2014).

A polarization-independent metasurface CPA was designed using dipolar meta-atoms Kang et al. (2013). The meta-atoms were optimized with the aid of coupled-mode theory, a flexible technique that can be applied to other CPAs Kang et al. (2014). Other theoretical proposals for metasurface CPAs include a bilayer asymmetric split-ring meta-atom design Nie et al. (2014), ceramic fishnet structures for operation in the microwave regime Zhu et al. (2016) and a self-complementary checkerboard-like metasurface for terahertz waves Urade et al. (2016). Owing to the design flexibility offered by metasurfaces, coherent excitation by multiple beams can be used to achieve a variety of other interesting effects apart from perfect absorption. These include the control of chirality and birefringence Mousavi et al. (2015, 2014), anomalous refraction Shi et al. (2014) and excitation of high-order resonances Fang et al. (2016b); Tseng et al. (2017).

The absorption bandwidth of metasurface CPAs depends on their design. Metasurfaces with strongly resonating meta-atoms maintain their desired properties over a narrow bandwidth, whereas weakly resonating metasurface CPAs can exhibit broad absorption bandwidths, similar to the graphene-based CPAs discussed in the next section.

Graphene-based structures

Another viable method to realize CPAs, particularly at terahertz frequencies, is to use graphene as the absorbing medium Liu et al. (2014); Fan et al. (2014); Rao et al. (2014); Fan et al. (2015); Wang and Hu (2016); Zanotto et al. (2017). Coherent control of absorption has been experimentally demonstrated for a 30-layer graphene film sandwiched between silica substrates, with strong modulation (80%) observed in the absorption Rao et al. (2014). CPAs may also provide a convenient way to access optical nonlinearities in graphene, such as four-wave mixing processes Rao et al. (2014, 2015).

A single sheet of undoped graphene exhibits a frequency-independent single-beam absorption of 2.3%, which is insufficient for a thin-film CPA. There are various methods to reach the required 50% single-beam absorption, such as increasing the carrier concentration through chemical doping Koppens et al. (2011); Fan et al. (2014), applying an electrostatic potential to change the Fermi level Mak et al. (2008); Wang et al. (2008); Koppens et al. (2011); Grigorenko et al. (2012); Liu et al. (2014), patterning the graphene sheet to induce local resonances Fan et al. (2015); Zhang et al. (2014) or using multilayer graphene Rao et al. (2014); Zanotto et al. (2017).

Single-layer graphene doped to around 200 meV (either chemically or electrostatically), exhibits 50% single-beam absorption at terahertz frequencies Liu et al. (2014). Electrostatic doping is appealing because it is easily tunable Liu et al. (2014); Fan et al. (2015), but requires a substrate. For free-standing single-layer graphene CPAs Fan et al. (2014), chemical doping is required. In either case, the sheet conductivity for unpatterned graphene is described by the Drude formula $\sigma(\omega)=\sigma(0)/(1-i\omega\tau)$, where $\tau$ is the DC transport scattering time. Because $\sigma(\omega)$ lacks resonant features, the resulting CPA frequency bandwidth can be very large. Graphene can also be combined with a metasurface in a single CPA device Wang and Hu (2016).

Broadband and narrowband operation

The bandwidth of a CPA is the frequency range over which absorption remains nearly perfect. A broad bandwidth is desirable for many telecommunications-related applications, such as optical modulators and logic elements. As discussed, zeros of $\hat{S}$ correspond to resonances; for a uniform dielectric slab, these are Fabry-Pérot resonances, thus the CPA bandwidth is the Fabry-Pérot free spectral range $\Delta\omega=\pi/nL$, where $n$ is the refractive index and $L$ is the slab thickness Chong et al. (2010, 2013).

In the thin-film CPA limit, $L\rightarrow 0$, the CPA bandwidth can be very high, and is limited principally by the frequency dispersion of the sheet conductivity $\sigma_{s}$. This has been demonstrated experimentally at microwave frequencies (6–8 GHz) using a thin ($L\sim 10^{-4}\lambda$) conductive film Li et al. (2015). High bandwidths are a compelling feature of graphene-based CPAs.

Apart from using thin films, there are several other methods to boost the CPA bandwidth. One is based on white-light cavities Wu and Xiao (2008), which use superluminal phase elements to compensate the phase accumulated by light propagating through the cavity; it has been predicted that an absorbing layer surrounded by two white-light cavities can act as a CPA for an operating wavelength of 1.5 $\mu$m, with a bandwitdth of $\approx 40\,\textrm{nm}$  Kotlicki and Scheuer (2014). Another design for a broad-band CPA uses epsilon-near-zero (ENZ) materials, which are materials with permittivity close to zero Kim et al. (2016). At the ENZ material interface, the normal component of the electric field obeys the boundary condition ${\varepsilon_{1}}E_{1}^{\bot}={\varepsilon_{{\rm{ENZ}}}}E_{{\rm{ENZ}}}^{\bot}$. Therefore, the electric field can become very large in the ENZ material, enabling strong light absorption. Finally, broadband CPA in a full octave (800–1600 nm) has been experimentally realized by tailoring aperiodic dielectric mirrors on both sides of a silicon cavity Pye et al. (2016).

Very narrow-band CPAs might also be useful for certain applications, such as sensors and narrow-band optical switches Vasic and Gajic (2014). One strategy for achieving a narrow-band CPA exploits the ability of a Fano-resonant system Miroshnichenko et al. (2010); Luk’yanchuk et al. (2010) to host narrow asymmetric resonances Yu et al. (2015). This idea has been explored through simulations of a photonic crystal cavity waveguide system; the bandwidth of the resulting CPA was found to be on the order of one hundredth of the operating frequency Yu et al. (2015).

Another important property is the range of incident angles over which a system exhibits near-perfect absorption. This is particularly important when the incident illumination is non-planar, as for light emitted by nearby point sources. It was shown that light from two coherent point sources can be completely absorbed by locating a small sink in a slab of material with negative refractive index Klimov et al. (2012). A more advanced approach is to design a nonlocal transversely homogeneous multilayer planar slab Monticone et al. (2016); such a structure operates as an omnidirectional CPA, able to perfectly absorb light emitted by coherent point sources located symmetrically on both sides of the slab.

Polarization effects

Although many CPA designs operate at a fixed linear polarization or respond equally to any incident polarization Kang et al. (2013), the concept can be extended to structures with nontrivial polarization characteristics. For instance, a chiral slab can have separate $\hat{S}$ matrices for clockwise and counterclockwise polarization states, with the CPA conditions for clockwise and counterclockwise modes satisfied at different frequencies Ye et al. (2016). The CPA eigenmodes then consist of input waves of the same circular polarization but different intensities, in contrast to the non-chiral case, in which CPA eigenmodes consist of two beams of equal intensities. Chiral CPA structures may have applications in coherent polarization control Kang and Chong (2015); Ye et al. (2016).

An unusual phenomenon known as ’polarization standing waves’ arises in CPA-like systems with incident circularly-polarized waves Fang et al. (2016a). Unlike conventional energy standing waves, polarization standing waves have constant electromagnetic energy densities, but a periodically modulated polarization state along the axial direction. Light absorption in an anisotropic sample becomes dependent on the sample orientation relative to the polarization. On the other hand, the absorption is insensitive to the sample’s axial position, in sharp contrast to thin-film absorption in a conventional energy standing wave, which depends on the axial position of the film relative to the nodes and antinodes of the wave.

$\mathcal{PT}$-symmetric structures

Parity- and time-reversal ($\mathcal{PT}$) symmetric optics has attracted considerable interest Bender and Boettcher (1998); Bender et al. (2002); Makris et al. (2008); Guo et al. (2009); Rüter et al. (2010). Among other intriguing properties, $\mathcal{PT}$-symmetric structures can be used to realize CPAs with unique features. The concept of $\mathcal{PT}$ symmetry originated in the theoretical quantum physics literature Bender and Boettcher (1998); Bender et al. (2002), in which a Hamiltonian is said to be $\mathcal{PT}$-symmetric if it is invariant under consecutive time ($\mathcal{T}$) and parity ($\mathcal{P}$) reversal operations Bender and Boettcher (1998); Bender et al. (2002); Zyablovsky et al. (2014). The concept can be extended to electromagnetics Makris et al. (2008); Guo et al. (2009); Rüter et al. (2010), where time reversal interchanges gain and loss. An optical structure is $\mathcal{PT}$-symmetric if the dielectric function satisfies the condition

[TABLE]

where $\mathcal{P}$ is a parity operation (for example a mirror reflection). An example of such a system are adjacent gain/loss layers (Fig. 3a). Despite being non-Hermitian, $\mathcal{PT}$-symmetric systems exhibit features similar those of Hermitian systems; for instance, they can have real eigenfrequencies within a parameter domain called the $\mathcal{PT}$-unbroken phase. They can also undergo symmetry-breaking transitions, causing the eigenvalues to coalesce and become complex Bender and Boettcher (1998); Bender et al. (2002); Makris et al. (2008); Guo et al. (2009); Rüter et al. (2010).

A one-dimensional $\mathcal{PT}$-symmetric structure can act as a CPA and threshold laser simultaneously Longhi (2010b); Chong et al. (2011); Longhi and Feng (2014). If the system is not separately $\mathcal{P}$- and $\mathcal{T}$- symmetric, $\hat{S}$ is not unitary. However, $\hat{S}$ can exhibit a $\mathcal{PT}$-unbroken phase in which its eigenvalues $\{s_{i}\}$ are unimodular, $|s_{i}|=1$, implying a net conservation of energy. Alternatively, the eigenvalues can pair up and undergo $\mathcal{PT}$-symmetry breaking, such that the eigenvalues $s_{a}$ and $s_{b}$ are not unimodular, but satisfy $s_{a}s_{b}^{*}=1$ (Fig. 3a). Effectively, one eigenmode is amplified and its partner correspondingly damped by the system. If an eigenvalue goes to zero ($s_{a}=0$), indicating a CPA, the other eigenvalue simultaneously becomes infinite ($s_{b}=1/s_{a}=\infty$), indicating threshold lasing. This is illustrated for a two-port gain/loss bilayer in Fig. 3b. Notably, CPA and lasing are allowed even though the underlying structure has zero net gain or loss.

Coherent perfect absorption in a system with an effective $\mathcal{PT}$ symmetry was first demonstrated experimentally using a pair of coupled microwave resonators Sun et al. (2014). Recently, two independent papers reported the realization of $\mathcal{PT}$-symmetric laser-absorbers based on optical-frequency waveguides Gu et al. (2016); Wong et al. (2016). In the experiments described in Ref. 108, a halving of the emission peaks was observed, indicating the occurrence of the $\mathcal{PT}$-broken phase, although there was no direct evidence for CPA-laser behaviour. For the system discussed in Ref. 98, the input and output intensities were directly measured, and for different phase offsets of the inputs both a maximum and a minimum of the relative output intensity were observed, with a contrast of around 30 dB (Fig. 3c). The pump intensity was maintained near the lasing threshold to prevent above-threshold gain nonlinearity from distorting the $\mathcal{PT}$ symmetry.

Other $\mathcal{PT}$-symmetric CPA-laser designs have been theoretically studied, including adjacent atomic cells filled with $\Lambda$-type three-level atoms Hang et al. (2016), plasmonic phase-controlled absorber-amplifier systems Baum et al. (2015) and non-$\mathcal{PT}$-symmetric systems with purely imaginary effective permittivity and permeability Bai et al. (2016a).

Guided mode structures

For many applications, it is desirable to realize CPAs in guided mode configurations, such as optical fibers or plasmonic waveguides. These CPAs could serve as slow-light waveguide couplers Gutman et al. (2013) or as components in integrated optical circuits for signal processing and computing. In one possible implementation, resonant absorbing nanoantennas are placed within or near the waveguide (Fig. 4a). When two counter-propagating modes excite the nanoantennas, the transmitted waves destructively interfere, increasing the total absorption. If the nanoantennas are designed so that the single-beam absorption is 50%, the structure can serve as a CPA. This approach has been numerically demonstrated for silicon-on-insulator waveguides at the telecom wavelength Bruck and Muskens (2013) and for metal-insulator-metal plasmonic waveguides in the near-infrared regime Park et al. (2015).

An alternative technique to coherently control absorption in guided mode systems is to exploit the near-field interaction between a waveguide and a dissipative coupler (Fig. 4b). The coupler can be a ring resonator, a 1D photonic crystal cavity Grote et al. (2013) or simply a plasmonic element coupled to the waveguide Zanotto and Melloni (2016). Also in this case, the light interference within the lossy component can increase the absorption, producing a CPA if the parameters are appropriately tuned.

In reality, whenever a waveguide is coupled to a nanoantenna (or any other element that breaks the waveguide’s translational symmetry), part of the loss can consist of off-axis scattering into free space modes Stegeman et al. (1981). This is undesirable for applications in which absorption into the medium serves a purpose, such as photocurrent generation. One solution was theoretically proposed for a metal-insulator plasmonic waveguide (Fig. 4c): it was shown that a scattering-free coherent surface plasmon absorber may be realized at the interface between two uniaxial media with specific dielectric tensors Ignatov et al. (2016). Parasitic scattering of surface plasmons into free-space modes is eliminated by the extreme anisotropy of the absorbing layer, leading to ideal matching of plasmon fields in the two regions Elser and Podolskiy (2008). The incident plasmons are reflected and refracted at the boundary, similarly to plane waves, without out-of-plane scattering.

Advances towards the use of CPAs in integrated nanophotonic structures have recently been reported Rothenberg et al. (2016); Malara et al. (2016). A CPA was realized using a photonic silicon racetrack resonator coupled to four input/output ports Rothenberg et al. (2016) (Fig. 4d): absorption was achieved by Si-ion implantation, and an extinction of almost 25 dB was measured at the 1.54 $\mu$m operating wavelength. Similar results were obtained for a ring cavity with an integrated variable coupler Malara et al. (2016).

2D, 3D and more complex structures

It should also be possible to achieve coherent perfect absorption in 2D and 3D geometries, such as spheres or rods suspended in free space. The main difficulty is that the CPA eigenmodes are not, in general, planar; for instance, the CPA eigenmodes of a sphere are spherical waves. The theoretical conditions for realizing plasmonic nanoparticle and nanorod CPAs were investigated, Noh et al. (2012, 2013) and it was found that for a metallic nanosphere with radius far below the free-space wavelength, CPA eigenmodes exist only for incident transverse magnetic spherical waves, and coincide with plasmonic resonances. Under irradiation with a CPA eigenmode, perfect absorption is accompanied by strong field enhancement near the nanoparticle surface, resulting from perfect coupling of the incident light to a lossy surface plasmon.

Coherent perfect absorption of a partial spherical harmonic by a core-shell nanoparticle has also been studied Mostafazadeh and Sarisaman (2012). Interestingly, the requirement for a highly coherent spherical wave input may be loosened with the use of multilayer nanoparticles, allowing for efficient absorption of random wavefronts Bai et al. (2016b). For non-spherical nanoparticles in a complex environment, the CPA eigenmode can be derived using Green’s tensor techniques Sentenac et al. (2013).

In complex structures with a large number of coupled scattering channels, very strong coherent absorption enhancement may be possible, even if perfect absorption is not achieved. An example is a random structure containing a strongly scattering (optically diffusive) medium with weak material losses, or a network of coupled optical cavities Li et al. (2017). The system may appear white under ordinary illumination, but nearly-perfect absorption can occur if the multi-channel inputs are optimized so that they form an appropriate complex wavefront Chong and Stone (2011). However, the achievable total absorption strongly degrades if not all available scattering channels can be precisely controlled Goetschy and Stone (2013).

Quantum CPAs

With the rapid advancement of quantum optics technology and its promising applications in telecommunications and quantum computing Gisin et al. (2002); Ladd et al. (2010), it is important to investigate how CPAs function in the quantum regime, such as few-photon or entangled-photon states.

The possibility of achieving complete absorption of single photons has long been understood in the context of lossy beam splitters Caves (1982); Barnett et al. (1998); Jeffers (2009). Consider a beam splitter with two input ports and two output ports, denoted $a$ and $b$. For a single-photon state, such as

[TABLE]

incident on the beam splitter, the state leaving the beam splitter is $\hat{S}|\varphi\rangle$, where $\hat{S}$ is unitary if the beam splitter is lossless. If the beam splitter is lossy, $\hat{S}$ can have a zero eigenvalue; the corresponding single-photon input state is completely absorbed into the beam splitter itself, yielding no output state Chong et al. (2013); Huang and Agarwal (2014); Ra’di et al. (2015). The process is non-unitary because it describes the transfer of quantum state amplitudes into parts of the Hilbert space not included in the model, that is, the ’unmonitored’ degrees of freedom responsible for lossiness Caves (1982).

An experimental demonstration of coherent perfect absorption for entangled single-photon states was realized Roger et al. (2015), starting with the preparation of the single-photon state $|\varphi\rangle$ using spontaneous parametric down-conversion of a laser source together with a lossless 50/50 beam splitter (Fig. 5a). The state $|\varphi\rangle$ was then directed onto a lossy metasurface with 50% single-sided absorption (equivalent to a classical thin-film CPA). The sample position was modulated to vary the phase $\varphi$; the resulting nearly-perfect absorption was similar to that of a classical CPA (Fig. 5a). In another recent experiment, coherent absorption was demonstrated for surface plasmon-polaritons in the quantum limit Vest et al. (2017), although non-entangled input states $\left|{{1_{a}},{1_{b}}}\right\rangle$ were used and perfect absorption was not achieved.

The situation is more complicated for multiple-photon states, such as a symmetric ’N00N state’ of the form $2^{-1/2}\left(\left|N_{a},0_{b}\right\rangle+\left|0_{a},N_{b}\right\rangle\right)$, which is a superposition of $N$ photons in one interferometer arm and $N$ photons in the other. For a two-photon N00N state, either both photons are completely absorbed by the beam splitter, or both photons pass through Jeffers (2009). This phenomenon has been observed using a multilayer graphene film as a lossy beam splitter Roger et al. (2016).

Another extension of the CPA concept into the quantum regime involves quantum nonlocality. In a recent experiment Altuzarra et al. (2017), a polarization-entangled photon pair was generated, and one photon directed into a two-arm interferometer containing a thin-film polarization-sensitive CPA. Performing a polarization measurement on the external ’idle’ photon collapsed the polarization state of the ’signal’ photon inside the interferometer, allowing for nonlocal control over the photon absorption.

Several researchers have employed quantum models to describe physical limitations to coherent perfect absorption. First, the performance of a CPA is limited by noise Clerk et al. (2010), which may be of thermal or quantum nature; even at zero temperature, there can be substantial quantum noise within the CPA’s absorption bandwidth, originating from spontaneous emission induced by the incident coherent radiation Chong et al. (2013). Second, the simple picture of a CPA as a linear absorber is accurate only if the absorbing medium is not saturable. If the absorbing medium is implemented as an ensemble of two-level systems, then absorption saturation may play a role. Owing to the dispersive nature of a two-level medium, when gain is replaced by absorption, not only does the imaginary part of $\varepsilon(r)$ change sign, but so does the real part, whereas time-reversal symmetry requires the transformation $\varepsilon\to{\varepsilon^{*}}$. As a result, the CPA frequency is generally different from the lasing frequency in the equivalent system Longhi (2011a). The inherent nonlinearity of a two-level medium also leads to bistability and hysteresis in the input-output characteristic of the system. Qualitatively similar behaviour was observed for a cavity quantum electrodynamic treatment of the problem Agarwal et al. (2016).

Although the absorption processes giving rise to CPAs typically lie within the regime of weak light-matter interaction, CPAs can also be realized in strongly coupled systems, such as polaritonic cavities (Fig. 5b). Using temporal coupled mode theory, it can be shown that coherent perfect absorption can be achieved by satisfying the strong critical coupling condition $\gamma_{r}=\gamma_{nr}+\gamma_{m}$, where $\gamma_{r}$ and $\gamma_{nr}$ are the radiative and non-radiative decay rates of the cavity mode, and $\gamma_{m}$ is the decay rate for the matter subsystem. A strong-coupling CPA was experimentally demonstrated Zanotto et al. (2014) using a metallo-dielectric photonic crystal (patterned with gold stripes) coupled to a multi-quantum-well heterostructure with a resonant intersubband transition (Fig. 5b).

Applications

The phenomenon of coherent perfect absorption may be useful in a variety of applications, ranging from all-optical data processing to the exploitation of enhanced photovoltaic effects. In the context of data processing, a CPA allows an optical signal to be strongly modulated by another coherent optical signal, without the need of a material nonlinearity. Consider a two-port system satisfying the scattering relations in Eq. (1). If both input amplitudes are scaled by a factor $\alpha$, the output fields are also scaled by $\alpha$. However, if one input amplitude (a control beam) is fixed while the other (a probe beam) is varied, the response becomes essentially nonlinear. This suggests a route towards the realization of linear optical switches, modulators and logical gates Fang et al. (2014, 2015); Papaioannou et al. (2016a, b). CPAs have also been proposed for pulse restoration and coherence filtering of optical signals Zhang et al. (2012).

A prototype CPA-based device performing different boolean operations (AND, OR, XOR, and NOT) has been developed using a CPA setup to construct a desired truth table exploiting various phase shifts between the two incident beams Fang et al. (2015); Papaioannou et al. (2016a). The realization of the AND and OR operations is illustrated in Fig. 6a.

By exploiting the fact that the output from the probe port varies nonlinearly with the control beam, it is possible to design a small-signal amplifier without incorporating nonlinear elements Fang et al. (2015) (Fig. 6b). The effective differential gain, $G=dI_{\rm out}/dI_{\rm in}$, varies as a function of the relative phase between the two input beams and can reach large values as the signal beam intensity approaches zero.

The idea of controlling light with light was further developed, and processing of binary images and pattern recognition were demonstrated Papaioannou et al. (2017). Similarities and differences between binary images were detected by exploiting the coherent interaction of input test and target images, resulting in either transmisison or absorption of specific image elements. CPAs may also be employed for novel imaging schemes, such as volumetric imaging with reduced aberrations Monticone et al. (2016).

$\mathcal{PT}$-symmetric structures operating near the $\mathcal{PT}$-breaking point exhibit various interesting behaviours, such as strongly asymmetric reflection Lin et al. (2011); Feng et al. (2012); Wu et al. (2016). This regime is associated with strongly asymmetric CPA eigenmodes, which allow the manipulation of a strong signal beam by a much weaker control beam (Fig. 6c). Such asymmetric switching has been experimentally demonstrated in a periodically modulated Si waveguide, with observed extinction ratios of up to 60 dB for a 3:1 intensity ratio between signal and control beam Zhao et al. (2016).

Coherent absorption effects may prove very useful for improving photocurrent generation in photoelectrochemical systems Liew et al. (2016). In such systems, the collection efficiency depends on the position at which charges are created. Using the principle of coherent enhancement of absorption, incident waveforms can be tailored to enhance or suppress absorption. This effect was demonstrated in a dye-sensitized solar cell, in which a six-fold modulation of the photocurrent was observed Liew et al. (2016) (Fig. 6d).

Similarly, metasurface-based CPAs can be used to to control photoexcitation in photonic nanostructures. An optical-frequency CPA was realized with a 2D periodic array of metallic nanopyramids on a silica substrate, and used to demonstrate coherent control over the photoluminescence intensity of a dye layer covering the nanoantennas  Pirruccio et al. (2016). Also, because coherent absorption experiments involve a delicate interplay between geometry, material absorption and incident waveform, they may provide a useful technique for non-destructive measurements of material absorptivity and other sample characteristics Zanotto et al. (2017); Zhao et al. (2016).

The field of acoustic metamaterials, which enable flexible control of sound waves, has been rapidly developing in recent years Cummer et al. (2016). Following a few theoretical suggestions Wei et al. (2014); Song et al. (2014), an acoustic CPA was recently experimentally demonstrated using decorated membrane resonators Meng et al. (2017). Acoustic CPAs have a host of intriguing potential applications, such as noise control.

Discussion and Outlook

We have discussed the fundamental principles of coherent perfect absorption, how CPAs can be implemented in various types of photonic structures and some examples of applications. CPAs have been realized in a wide variety of geometries, including slabs, gratings, metasurfaces, thin films, complex multi-port cavities and guided-mode structures compatible with integrated nanophotonics applications. Different techniques have been devised for increasing or decreasing the absorption bandwidth and for achieving perfect absorption under strongly asymmetric inputs or non-planar incident waves. CPAs have also been extended to the quantum optics regime.

Several of these advances were inspired by the concept of CPA as the time-reverse of a laser at threshold. As a corollary to the principle of laser physics that “anything will lase if you hit it hard enough” Chu and Townes (2003), practically any photonic structure can act as a CPA, once an appropriate amount of loss is introduced. It may be interesting to look for CPA counterparts of various complex laser structures, such as photonic crystal slab lasers and random lasers. However, unlike a laser, where the lasing mode emerges through self-organization Haken (1985), perfect absorption requires the supply of exactly the right incident waveform. It is unclear whether there can be any useful CPA analog of laser self-organization — it could be perhaps found in the presence of nonlinear absorptive media, as suggested by several theoretical studies Longhi (2011a); Agarwal et al. (2016); Nireekshan Reddy and Dutta Gupta (2013); Reddy et al. (2013); Reddy and Gupta (2014). Furthermore, interesting variants of the time-reversal concept have been formulated for nonlinear processes such as wave mixing and parametric oscillations Longhi (2011b); Schackert et al. (2013); Zheng et al. (2013). For instance, the idea of a coherent perfect absorber for waves of different frequencies, acting as the time-reverse of an optical parametric oscillator, was put forward Longhi (2011b).

As we have seen, promising applications for coherent perfect absorption are beginning to be explored. Looking ahead, we expect CPAs or CPA-like devices to be useful is situations where coherent optical signals need to be compactly and efficiently processed. Consider, for instance, the prospect of a complicated photonic circuit that employs coherent light for linear low-power optical signal processing or computing. If only non-absorptive interferometric effects are used, whenever an unwanted signal product has to be removed from the system — for example an optical bit that plays no further role in a computation — an unmonitored output port must be inserted. This may complicate the circuit design if the unwanted signal occurs deep within the circuit. Integrated CPA devices could resolve this difficulty by serving as efficient ’sinks’ for optical signals Bruck and Muskens (2013); Park et al. (2015); Rothenberg et al. (2016); Malara et al. (2016). In the quantum optics limit, the ability to control and remotely trigger perfect absorption of single- or few-photon states could likewise be useful for quantum information processing Altuzarra et al. (2017).

Another broad class of applications for CPAs involves the exploitation of the sharp intensity contrast between CPA and non-CPA eigenmodes. The most extreme examples are CPA-laser devices Longhi (2010b); Chong et al. (2011); Gu et al. (2016); Wong et al. (2016), in which the output intensity is zero for the CPA eigenmode and very high for other inputs (limited only by gain saturation). CPA-lasers have been demonstrated in simple two-port configurations; in the future, such devices could allow for sensitive interferometric measurements, particularly of low-intensity optical signals.

Acknowledgements.

D.G.B. and T.S. acknowledge support from the Knut and Alice Wallenberg Foundation. D.G.B. acknowledges support from the Russian Foundation for Basic Research (project no. 16-32-00444). T.S. acknowledges financial support from Swedish Research Council (Vetenskapsområdet, grant no. 2012-0414). A.A. and A.K. acknowledge support from the Air Force Office of Scientific Research with grant no. FA9550-17-1-0002 and the Welch Foundation with grant no. F-1802. C.Y.D. is grateful to A. D. Stone, L. Ge, and A. Cerjan for numerous stimulating and deep discussions, and acknowledges support from the Singapore MOE Academic Research Fund Tier 2 grant no. MOE2015-T2-2-008, and the Singapore MOE Academic Research Fund Tier 3 grant no. MOE2011-T3-1-005.