A parity-time symmetric coherent plasmonic absorber-amplifier

Modulation of optical signals typically requires wavelength-scale devices, since most materials have relatively weak optical nonlinearities or electro-optic coefficients. Such large dimensions ensure that the optical signal has sufficient interaction volume with the active material. However, recent advances in plasmonics have enabled compact, subwavelength optical modulators based on the strongly enhanced near fields afforded by surface plasmons.¹ These devices offer active control of optical signals based on a variety of elegant schemes, including using non-linear effects² to detune coupled resonators,^3,4 compensating inherent losses with varying gain,^5,6 using destructive interference effects,^7,8 or increasing or decreasing transmission losses via absorption.⁹ While plasmonic losses play an important role in some of these devices, for example, aiding with signal absorption, they also generally limit the maximum signal modulation that can be achieved. Indeed, in many plasmonic modulators, it is challenging to completely extinguish losses, resulting in signal attenuation even when the device should be in a transmissive or amplifying state. Accordingly, ohmic losses are often treated as a detriment—reducing propagation lengths or siphoning energy into wasteful heating.

Loss, however, plays a beneficial role in non-Hermitian parity-time (⁠$PT$⁠)-symmetric optical systems, which are characterized by equal magnitude and spatially symmetric loss and gain profiles. $PT$-symmetry has been a field of exciting growth for over a decade in quantum mechanics,¹⁰ and within the optics community has led to proposals for both active and passive devices based on the isomorphism between the Schrödinger equation and the wave equations for light. In general, $PT$-symmetric potentials allow a non-Hermitian system to retain real eigenvalues below the exceptional point, while the eigenvalues become complex above the exceptional point.

A system may be described as $PT$-symmetric if the system remains unchanged after both a parity (x → −x) and time reversal (t → −t) transformation.^11–13 In the context of optics, this requirement implies that the real component of the refractive index is spatially symmetric across the device (n(x) = n(−x)), while the imaginary component is antisymmetric (κ(x) = −κ(−x)).^14,15 Such $PT$-symmetric optical systems will exhibit real eigenvalues below a critical magnitude of the loss and gain (⁠$|κ|crit$⁠), and complex eigenvalues above. Importantly, below κ_crit, optical modes will neither become amplified nor attenuated as the loss and gain perfectly balance one another. Above κ_crit, however, new optical modes emerge which preferentially interact with either the gain or loss and exhibit either strong amplification or attenuation.¹⁶ This transition plays an important role in enabling unidirectional^17–19 and even non-reciprocal^20,21 propagation in $PT$-symmetric optical devices.

For passive devices, $PT$-symmetry has enabled unidirectional invisibility,²² Bloch power oscillations,²³ and broad spectral responses.^24,25 Active devices based on $PT$-symmetry have been theoretically investigated and are predicted to exhibit substantial modulation depths and strong directionality.^26–28 Lasers have also been proposed and created with $PT$ geometries, and may be characterized by lower thresholds than non-$PT$-symmetric cavities,²⁹ reversed pump dependence,^30,31 and enhanced mode selectivity.^32–34 

For active signal modulation, no greater contrast exists than that between perfect absorption and lasing, which represent zeros and poles in a scattering matrix, respectively. While lasing produces a coherent beam of emitted light, a coherent perfect absorber (CPA) uses the interference of two incoming coherent sources to trap all light in a cavity that contains a threshold amount of loss until it is completely absorbed.^35–39 If the phase between the two sources is offset, the interference effect is modified and total absorption is lost, just as a laser is incapable of lasing incoherent light. The processes therefore represent the time-reverse of one another, (t → −t), as illustrated in Figure 1(a). Furthermore, note that in lasers, gain (−κ_crit) provides amplification and ultimately stimulated emission, while in CPAs, perfect absorption is achieved via equal but opposite loss (+κ_crit). The relationship between time, loss, and gain in both devices motivates the combination of perfect absorption and lasing in one device based on $PT$-symmetry.^40–42 The output of a CPA-laser is determined by its illumination: two coherent illumination sources with the correct phase relation result in CPA, while all other illumination phases produce amplification and lasing.

Inspired by this concept, we have designed a nanoscale $PT$-symmetric modulator capable of both strong coherent absorption and amplification. The device design is based on a subwavelength plasmonic aperture. As shown in Figure 1(b), we consider two 25 nm-thick dielectric channels separated by a 30 nm metal spacer layer and embedded in a semi-infinite metallic cladding. The length of the aperture is L. The dielectric channels are modeled with Lorentzian material models that have an approximate real refractive index of n ≈ 1.44 corresponding to SiO₂, while the imaginary component is varied for different simulations but always kept equal in magnitude and opposite in sign for the two channels. The metal is a lossy Drude model representative of silver, the details of which are included in the Supporting Information.⁴³ Note that we call the aperture $PT$-symmetric in describing the dielectric channels; the cladding and separating metallic layer do not maintain $PT$-symmetry and are uniformly lossy.

We first explore this $PT$-symmetric system for the case of a large aperture length (L), with L → ∞, to determine the dependence of the modal dispersion on the non-Hermiticity factor (κ). Using both an analytic transfer matrix method⁴⁴ and a finite difference time domain (FDTD) mode solver (Lumerical), we determine the complex wavevectors and the $|E|$ field profiles for different values of κ at a fixed near-infrared wavelength, λ = 1662 nm, for the infinite waveguide. This wavelength corresponds to the peak transmission through a 300 nm long, finite length, absorber-amplifier described in later sections and is near wavelengths of interest to telecom applications. Our analytic and FDTD methods produce highly corroborating results for modes supported by the structure from κ = 0 to κ = 0.2, which encompasses a range of experimentally accessible values.⁴⁵ 

As seen in Figures 2(a) and 2(b), the two lowest order modes in the passive (κ = 0) structure have imaginary wavevectors of a similar magnitude based on their interaction with the lossy metal but distinct real wavevectors, since they represent the even and odd magnetic field (H_y) distributions in a structure. The green point represents the even mode of the five-layer metal-insulator-metal structure,⁴⁶ while the red point represents the odd mode of the κ = 0 structure. For κ > 0, the modes lose their definite parity, and we simply refer to mode 1 (green trace) and mode 2 (red trace). As the non-Hermiticity factor is increased, the real components of the wavevectors approach the same value, while the imaginary components begin to diverge. Above κ = 0.1 and approaching κ = 0.2, mode 2 becomes increasingly lossy (i.e., possesses larger imaginary wavevectors), while mode 1 evolves into a gain mode. This is the transition from unbroken- to broken-symmetry. At κ = 0.11, mode 1 crosses the zero point in Figure 2(b) and becomes lossless, and for greater κ, it experiences amplification.

Figures 2(c) and 2(d) display the time averaged $|E|$-field profiles for the two modes for selected values of κ. At κ = 0, the field intensity is greatest within the dielectric channels, with the odd mode possessing slightly greater field intensity in the metal spacer layer. This profile results in the larger imaginary part of the wavevector. For larger values of κ, the field profiles in the loss and gain channels become unequal for the two modes with greater intensity in the gain channel for mode 1 and greater intensity in the loss channel for mode 2. Note that in our figure, the magnitude of $|E|$ is normalized such that the intensity remains constant in the dominant channel and is diminished in the other channel. As κ increases, the reduction of intensity in the non-dominant channel becomes more pronounced, and at κ = 0.15, it is half the intensity of the dominant channel. The unequal field distribution and clearly defined loss and gain modes indicate a transition to broken-symmetry and a unidirectionality of power flow between the two channels in the structure.

Within ideal $PT$-symmetric optical systems, a singular exceptional point marks the transition between the unbroken- and broken-symmetry regimes of the device. In other words, this exceptional point marks the splitting of the imaginary components of the mode wavevectors and the merging of the real components. Correspondingly, the distribution of the power remains perfectly symmetric up to the exceptional point and becomes asymmetric beyond. Our system does not exhibit such a sharp exceptional point because we include a lossy Drude model for the metal, and thus our system is not perfectly $PT$-symmetric. However, we note that ideal $PT$-symmetric behavior could be restored through “healing” the device⁴⁷ by altering the real part of the index in one or both of the waveguides.

We next consider a finite-length resonator with L = 300 nm for both κ = 0 and κ = 0.1. Using FDTD simulations,⁴³ the resonator is illuminated with a single plane wave at normal incidence and the transmitted and reflected intensities are collected in a two-port configuration (see supplemental material). Figure 3 shows the transmission and reflection of both the κ = 0 and κ = 0.1 resonators. As seen, the maximum transmission of the passive resonator peaks at a wavelength of 1542 nm and is approximately 6% of the incident intensity. A minimum in reflection occurs at slightly blue-shifted wavelengths (1470 nm) and dips below 3%. Except at this minimum, the passive structure reflects substantially more than it transmits, reaching a maximum of 50% of the incident intensity at a wavelength of 1770 nm.

In contrast to the κ = 0 configuration, peak transmission through the cavity with κ = 0.1 is 40 times greater than the peak with κ = 0, exhibits substantial line-width narrowing, and reaches a maximum at 1662 nm. The reflection and transmission peaks also occur at 1662 nm, and for both reflection and transmission, the signal is amplified by more than two times. In other words, the cavity with κ = 0.1 has greater-than-unity reflection and transmission coefficients on resonance. For wavelengths longer than the peak transmission, much of the incident intensity is reflected. However, for wavelengths shorter than the peak transmission, there is little reflection or transmission (< 30%) with a minimum in reflection of less than 4% at a wavelength of 1645 nm. From these simulations, we see that the κ = 0.1 configuration shows unique transmission and reflection properties from κ = 0, and has promise as a device with signal amplification.

To realize phase-modulated absorption and amplification, we illuminate the resonator with two plane wave sources of equal intensity.⁴⁸ As illustrated in Figure 4(a), two sources of intensity I_in impinge on the aperture from opposite sides. The right side source possesses a phase lead, $ϕ$⁠, between 0° and 360°. The scattering from the aperture is collected as I_out,l and I_out,r, with I_out,r corresponding to the side with the variable phase lead. As shown in Figure 3, the κ = 0 and κ = 0.1 resonators have different spectral profiles, and therefore exhibit different phase-dependent behavior at different wavelengths. For κ = 0, we consider a wavelength of λ = 1767 nm, which is the reflection peak. We leave out the transmission peak wavelength for κ = 0 from this discussion as the transmission intensity is far less than the reflection intensity and the phase dependent behavior is accordingly similar but less remarkable. For κ = 0.1, we look at the peak transmission and reflection wavelengths, λ = 1662 nm, as well as the minimum reflection wavelength, λ = 1645 nm. The peak transmission wavelength exhibits large modulation depths, while the minimum in reflection wavelength shows strong directionality. The total intensity scattered from the structures, as well as I_out,l and I_out,r, is plotted as a function of phase offset for these wavelengths in Figures 4(b)–4(d).

The phase dependent scattering for κ = 0 at λ = 1767 nm is shown in Figure 4(b). The total scattering and directional scattering are normalized to the combined amplitudes of the two plane waves constructively interfering in free space. The total scattering (I_out,l + I_out,r) reaches a maximum of 63% when the two sources are perfectly out of phase and a minimum of 47% when both sources are in-phase. Between these two extremes, the aperture exhibits minimal directionality, with the difference in scattering between both sides differing by no more than 10% of the incident intensity.

The device's scattering when κ = 0.1 at a wavelength of λ = 1662 nm is shown in Figure 4(c). Sweeping the phase offset reveals far greater modulation in the total scattered intensity compared to the κ = 0 configuration. Notably, a phase offset near 0° generates a total output that is less than half of the input intensity, while a phase offset of 180° generates a total output that amplifies the input intensity by 5.7 times. Such modulation corresponds to over 10 dB in the total output of the aperture, compared to 3 dB achieved in a similar planar plasmonic aperture.⁴ Alternatively, the output from each side (I_out,l or I_out,r) can also vary between roughly three times to less than 15% of the input intensity. Furthermore, the devices with κ = 0.1 also exhibit more pronounced directionality than an aperture with κ = 0. Almost complete directional absorption of the incident field is observed at a wavelength of λ = 1645 nm, shown in Figure 4(d). With an incident phase offset of 80°, the scattering from I_out,r can be reduced to a minimum of 0.46% of the incident intensity. Maximum scattering occurs at a phase offset of 260° with I_out,r reaching 55% of the input intensity. The modulation depth at this wavelength is therefore greater than 99%.

To better illustrate how far-field modulation arises from near-field properties, Figures 4(e)–4(g) plot 2D near-field maps of the Poynting vector for the κ = 0 and κ = 0.1 resonators at the same wavelengths in Figures 4(b)–4(d). The overlaid arrows indicate the direction of the Poynting vector and the underlying color maps show the magnitude of the Poynting vector and are normalized such that the maximum magnitude in the κ = 0 plot is unity. As seen in Figure 4(e), when κ = 0 and the structure is illuminated by a phase offset of 180°, the Poynting vector from each aperture facet is symmetric. The Poynting vector points towards the aperture and into the metal with generally low intensities as would be expected for a structure without gain. When κ = 0.1 and the structure is illuminated with a 180° phase offset, the Poynting vector behavior is again symmetric along the y-axis (Figure 4(f)). However, the maximum intensity is more than two orders of magnitude greater than the κ = 0 aperture and the distribution is no longer symmetric along the x-axis. The Poynting vector in the gain (top) and loss (bottom) channels now differs by roughly a factor of 1.5, and point from the gain channel to the loss channel. Since the aperture is near but not beyond the exceptional point at κ = 0.1 and the length of the aperture is small in comparison to the wavelength, we expect this directionality between the two channels. The arrows in Figure 4(f) also show that the scattering from the aperture is dominated by the loss channel, indicating that the destructive interference at the ends of the channel necessary for complete CPA is not present although nascent intensity nodes do appear at the ends of the gain channel.

A near-field Poynting vector map is also plotted at a phase offset of 80° (Figure 4(g)) for λ = 1645 nm. As with the 180° phase offset at λ = 1662 nm, the Poynting vector points from the gain channel to the loss channel inside of the aperture, while on the aperture's left, the Poynting vector mainly emerges from the loss channel. Unlike panels (e) and (f), however, the color map indicates an unequal distribution of the magnitude of the Poynting vector at the two ends of the aperture. Intensity is greater on the left side and the scattering from the right side is hindered by interference effects visible as a more pronounced node in intensity in the gain channel. The right side of the loss channel does not possess an equally apparent node, but intensity is still substantially reduced in comparison to the intensity from the left side matching the asymmetry in Figure 4(d).

In summary, we have designed a nanoscale plasmonic aperture capable of phase-controlled coherent absorption and amplification. While the structure does not possess ideal $PT$-symmetry, as evidenced by the lack of a singular exceptional point, clear emergence of gain and loss modes are still observed as κ is increased. When illuminated by a single plane wave, the finite-length $PT$-symmetric plasmonic resonator exhibits simultaneously greater-than-unity reflection and transmission coefficients as well as near-zero reflection coefficients. When illuminated by two counter-propagating plane waves, the device exhibits strong modulation as the illumination phase is varied. On resonance (λ = 1662 nm), modulation depths of over 10 dB are achieved from coherent to out-of-phase illumination with output power varying from less than half of the input to nearly six times amplification. Off resonance (λ = 1645 nm) and at the minimum in reflection, the device exhibits coherent directional absorption with phase-controlled modulation exceeding 99%. An integrated nanoscale all-optical modulator with gain could be realized by coupling the aperture to an external plasmonic phase modulator⁴⁹ and could make use of the strong signal modulation and directionality. Although we have not performed switching time simulations, the small aperture volume and phase-controlled interference could offer femtosecond-scale modulation times. Ultimately, the device may also provide a promising platform for future compact coherent perfect absorber-lasers and nanophotonic logic.

The authors thank Amr Saleh and Aitzol Garcia-Etxarri for many fruitful discussions. Funding from a Presidential Early Career Award administered through the Air Force Office of Scientific Research is gratefully acknowledged, as are funds from Northrop Grumman. This material is based upon work supported by the National Science Foundation Graduate Research Fellowship under Grant No. DGE-114747. Any opinion, findings, and conclusions or recommendations expressed in this material are those of the authors(s) and do not necessarily reflect the views of the National Science Foundation.