## Abstract

Metamaterials and plasmonics potentially offer an ultimate control of light to enable a rich number of non-conventional devices and a testbed for many novel physical phenomena. However, optical loss in metamaterials and plasmonics is a fundamental challenge rendering many conceived applications not viable in practical settings. Many approaches have been proposed so far to mitigate losses, including geometric tailoring, active gain media, nonlinear effects, metasurfaces, dielectrics, and 2D materials. Here, we review recent efforts on the less explored and unique territory of “virtual gain” as an alternative approach to combat optical losses. We define the virtual gain as the result of any extrinsic amplification mechanism in a medium. Our aim is to accentuate virtual gain not only as a promising candidate to address the material challenge, but also as a design concept with broader impacts.

© 2020 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

## 1. Introduction

The problem of overcoming optical losses in metamaterials and plasmonics is perhaps the greatest fundamental challenge, rendering many of the proposed applications not viable in real world scenarios. Progress towards the development of a robust solution has been sluggish even after nearly two decades of efforts. Soukoulis and Wegener noted the gravity of this problem and the desired improvement any potential solution must hope to achieve, especially at optical frequencies [1]. Meanwhile, multiple strategies were proposed including gain medium [2], optical parametric amplification [3], geometric tailoring and optimization [4], and metasurfaces [5]. Although not eliminated completely, with many new devices metasurface approach substantially minimized the attenuation suffered by bulk metamaterials. Alternative low-loss materials such as high-index dielectrics, nitrides [6], oxides [7], and two-dimensional (2D) materials such as graphene, hexagonal boron nitride (h-BN) [8,9], and their 2D analogue transition metal dichalcogenides (TMDCs) [10] have been considered as potential replacements for metals.

The early years of metamaterial research were primarily centered around sub-wavelength metallic resonators. However, the desired strong field enhancement and confinement resulting from the metallic free electron response is accompanied by non-radiative Ohmic losses. Geometric tailoring and optimization [4] of the underlying current distribution within the resonator permits limited reduction of losses.

Replacement of metallic resonators with high-index dielectric resonators exhibiting Mie resonances [11] in silicon [12], germanium, gallium phosphide, and certain perovskites has been considered to be a promising route for overcoming losses [13]. Dielectric resonators with strong displacement currents from bound electron oscillations are free from Ohmic losses. This enables low-loss non-plasmonic metamaterials [14,15] and metasurfaces [16] having a plethora of exotic optical properties and light-matter interactions [17] including linear [18] and nonlinear [19,20] effects. Loss-free negative index dielectric metamaterial was demonstrated at microwaves based on resonant forward scattering [21]. However, this was achieved only for radiative losses using lossless dielectrics. Material losses have been shown to increase at larger operation frequencies, and available maximum range of refractive index is limited [13]. The latter renders dielectric resonators diffraction limited with moderate local field enhancement and makes it unsuitable for applications requiring small-mode volumes and large electric field enhancements, such as surface enhanced Raman spectroscopy [22] and deep sub-wavelength miniaturization [23]. Hydrogenated amorphous silicon (a-Si:H) appears as a relatively good candidate as a low-loss dielectric material for visible spectrum [24]. A difficulty often encountered in the visible spectrum is the fabrication of high aspect ratio dielectric resonators for phase control applications [16,25–27]. This difficulty arises from two competing processes: Minimizing losses in the visible requires the use of larger bandgap materials. However, materials with larger bandgap have smaller refractive index for the same wavelength, significantly limiting the field confinement. In short, the fundamental problem of simultaneous suppression of material losses while preserving field enhancement remains elusive across the electromagnetic spectrum [22].

The use of gain materials involving pumped semiconductor quantum dots and dye molecules in epoxy resins was also originally thought as a promising strategy for active compensation of losses [2]. However, problems with the pump requirement, stability, gain saturation [28], intense noise generation near the field enhancement regions [29–31], causality [30,31], and maintenance [28] have posed stringent constraints and raised fundamental questions about the viability of this approach.

In recent years, graphene, h-BN, 2D TMDCs, and layered oxides have fostered significant interest towards low loss optoelectronics and nanophotonics. 2D TMDCs [32] are nanomaterials with a single plane of transition metal atoms (such as Mo, W, Ta etc.) sandwiched (X-M-X) between two planes of chalcogen atoms (such as Se, S, Te, etc.). Individual atomic planes are ambitiously used as building blocks for artificially stacked materials to obtain the desired combined functionality [32]. The sandwich structure provides unique opportunities to directly control material properties by inserting guest species at the van der Waals gap through intercalation [33]. Recently, the possibility of extremely low optical losses with TaS${_2}$ was shown and 2D TDMC-halide structures with engineered band structures and density of states were proposed as alternative plasmonic materials with greatly suppressed intrinsic losses [10]. Similarly, plasmonic and phonon-polaritonic properties exhibiting very low optical losses have been demonstrated [8,9,34] for h-BN. However, problems associated with thermodynamic stability, control of surface construction with low defect densities, charge transfers, and built-in electric fields in heterostructures impose cumbersome fabrication and design challenges for 2D materials [35]. Additionally, continuous or pulsed laser induced thermal or non-thermal damage to the crystal structure is a serious concern [35]. Energetic phonons, hot electrons due to electron-electron and electron-phonon interactions have been shown to damage the crystal structure through ionization and Coulombic explosion process [36,37]. Laser induced damage strongly depends on the underlying defect density [35]. High quality atomic layers fabricated with chemical vapor deposition have higher damage thresholds than other processes such as liquid-phase exfoliation due to their relatively low defect densities [38–40].

It is also worth mentioning the numerous attempts made to find alternative low-loss plasmonic materials especially in the optical regime. Candidates such as alkali-noble intermetallics [41,42], and transition metal nitrides [6], doped semiconductors and transparent conducting oxides [43] were proposed for infrared and optical frequencies. However, the performance of these materials is still lower than that of the noble metal counterparts.

Fortunately, finding the elusive “metal” [44] with losses several orders of magnitude less than naturally occurring noble metals is not the only hope for transitioning plasmonics and metamaterials into a practical field from a purely research enterprise. In this review, we discuss recent efforts on “virtual gain” particularly for the compensation of losses in metamaterials and plasmonics. Virtual gain, here, is fundamentally different than all the above-mentioned approaches and is defined as the result of an amplification in a medium acquired with no intrinsic gain mechanism (i.e., optical gain from parametric, nonlinear, luminescence, fluorescence, or emission processes is not involved). A very recent review on the latter can be found in [22]. The term was coined for the first time in [45] and has been adopted and expanded here with the hope of initiating discussions of alternative explorations in a broader context.

## 2. Plasmon injection scheme

The concept of virtual gain for the full compensation of losses in a plasmonic negative index metamaterial (NIM) was first proposed in [46], where a plasmonic injection process was used to achieve effectively zero imaginary part in the refractive index. It was shown that the concept can be extended to imperfect superlens, in the absence of noise. Figure 1 illustrates a conceptual structure for the implementation of the plasmon injection ($\Pi$) scheme where a central superlattice between input port $P_1$ and output port $P_2$ constitutes a lossy metamaterial, and the auxiliary ports $P_3$ and $P_4$ are used for the injection of externally excited surface plasmons for amplification. The excited surface plasmon polaritons (SPPs) then propagate to central superlattice and amplify the domestic SPP mode of the metamaterial through a coherent constructive interference between fields in the superlattice and the injected ones. Finally, the coherently amplified SPPs are coupled to free-space modes. Figure 2 shows the resultant effective parameters for a single functional layer when the power provided from each auxiliary port is $3$ times the input power provided from port $P_1$. Under this condition, around $552$ and $562THz$, the system operates effectively as an ultralow-loss NIM. The NIM becomes virtually loss-free at $n=-1$ when the auxiliary power reaches about $3.5$ times the input power [46].

The $\Pi$ scheme can be easily generalized to other metamaterials. For example, Fig. 3 describes how it can be used for the loss compensation in a superlens. Here, the object is coherently superimposed with an auxiliary object to compensate the high spatial frequency components, which are attenuated due to losses in the lens. The result is a reconstruction of the original object in the image plane.

When there is no auxiliary object, the field distribution at the image plane is [47]

where $d$ is the lens thickness, $O(k_y)$ is the object spectrum, and the fraction corresponds to the transfer function of the lossy system. The 0.04 term in the denominator is particularly important, because it results from the losses in the superlens. The question is, “how can this term be mitigated here to control or even completely eliminate the effect of losses?” One approach is to multiply the expression inside the summation in Eq. (1) with a properly selected filter function $f(k_y)$ such that the loss term in the denominator is effectively reduced or even completely removed. Equation (1) then becomes [46] The loss term in Eq. (2) can be arbitrarily reduced by a factor of $1/p$ (i.e., $0\leq p \leq 1$) as given in provided that the corresponding filter function $f_p(k_y)$ is selected. For example, choosing $p=\frac {1}{2}$ in Eq. (3) gives the filter function Substituting Eq. (4) into Eq. (3) yields where the loss term is clearly reduced by a factor of 2. The multiplication process in Eq. (3), key to the virtual gain approach here, can be implemented either by computational post-processing [48–53] or by using a structured light. The latter can be, for example, achieved with metasurfaces [16,54,55]. The physical implementation is not only more interesting, but also more effective in loss compensation especially in the presence of noise. This is how the physical implementation works: since the product $O(k_y)f(k_y)$ of the object spectrum with the filter function in Eq. (2) can also be interpreted as a new total object field, one can coherently superimpose the original object field with an auxiliary counterpart correlated with the object to construct the total object. Then, the auxiliary counterpart in the total object compensates the losses experienced by the object field during propagation to leave it unscathed in the image plane as described in Fig. 3. In analogy with active gain medium, the original object can be thought of as the “signal,” while the auxiliary field as the “pump.”#### 2.1 Passive implementation of the plasmon injection scheme

### 2.1.1 Coherent light

In [56], the operation of hyperlens was combined with the $\Pi$ scheme to compensate losses and thus improve the resolution. Loss compensation was achieved by deconvolution as post-processing, which was shown to be equivalent to providing an external auxiliary source along with the input object in the absence of noise. This approach was named as “passive” implementation of the $\Pi$ scheme, meaning that no physical external auxiliary was used unlike the original $\Pi$ scheme [46]. The name emphasizes the connection between the computational post-processing and the equivalent physical pre-processing that involves a total input field. Using this passive implementation of the $\Pi$ scheme, an object, not resolvable by the hyperlens alone, was reconstructed with a minimum feature size of one seventh of the free-space wavelength. Figure 4(a) shows the magnetic field magnitude for an original object (blue), truncated object (black), raw image (red), and loss compensated image (green). At high spatial frequencies, the deconvolution also amplifies the noise due to numerical errors, which adversely affect the compensated image. Therefore, to avoid noise amplification, the inverse filter function used in the deconvolution is truncated at the spatial frequency, where the numerical noise floor is reached. The resultant is the truncated object (black), which maintains the major features of the original object beyond diffraction limit. The deconvolution almost perfectly reconstructs this truncated object beyond the diffraction limit (green), while the raw image without deconvolution cannot be resolved. Figure 4(b) compares the images obtained by deconvolution (green) and the $\Pi$ scheme with the physical auxiliary source (black) for the hyperlens. Figure 4(b) also shows the total input (blue), which is a coherent superposition of the auxiliary and the input fields, equivalent to deconvolution.

This “virtual gain” approach was later extended to analytically describe the passive implementation of the $\Pi$ scheme to improve the hyperlens resolution [57], where it was shown that the passive implementation of the $\Pi$ scheme with deconvolution can compensate all major losses, including absorption loss, discretization loss, and impedance mismatch loss, and reconstruct an object with deep subwavelength features of about one tenth of the free-space wavelength.

A non-ideal Pendry’s negative index flat lens (NIFL) with an appreciable loss was studied in [58]. Similar to hyperlens work in [56], deconvolution emulating the original $\Pi$ scheme [46] was applied to the images. A perfect reconstruction with a sub-diffraction-limited resolution was demonstrated for an image previously unresolved with the NIFL. As evidenced in Fig. 5(a), for noise-free or low-noise systems, it is not necessary to “pre-process” the imaging system physically, since excitation with the coherent superposition of the auxiliary input and the original object (i.e., total input field) is equivalent to the deconvolution of the raw image in post-processing. It is clearly seen that the images from deconvolution and the total input (green) are equivalent. The small discrepancy likely results from accumulated numerical error related to the total input. Figure 5(b) shows the auxiliary input electric field source corresponding to the deconvolution.

### 2.1.2 Incoherent light

Conceding the use of incoherent light allows for a compact illumination source such as a light-emitting diode and enables deconvolution of the resulting image based on only intensity data, which remains robust in the presence of additive noise. Incoherent imaging with a silver superlens was analyzed in [59]. The result was a reconstruction of a super-resolved image of double-slit metallic mask objects, with increased contrast and reduced fullwidth half-maximum. A previously unresolved double-slit was recovered with a resolution better than one eighteenth of the free-space wavelength [59]. Figure 6 shows results for noisy data from double-slit objects using nonlinear Richardson-Lucy deconvolution [52,53]. Structured light illumination is unified with the linear deconvolution procedure (see Fig. 7). This is analogous to computational and physical loss compensation schemes, based on coherent light, reported in the previous works [56–58]. The results obtained using incoherent light in the $\Pi$ scheme shows the possibility of building an ultracompact superresolution imaging systems requiring only intensity information for image reconstruction [59].

#### 2.2 Active implementation of the plasmon injection scheme for coherent light

The active physical implementation of the $\Pi$ loss compensation scheme was first described, as an example, on Pendry’s non-ideal NIFL with realistic material losses and signal-dependent noise [60]. “Active” here is not related to active gain medium or nonlinear effects, rather refers to adding energy to the system through an external physical auxiliary source as conceived originally in [46]. Moreover, the term “active” is used to distinguish the physical implementations of the $\Pi$ scheme from passive approaches based on post-processing [56–59]. In the active implementation in [60], an external auxiliary source via a physical convolution of the object was proposed, leading to the construction of the total input field described by the product $O(k_y)f(k_y)$ in Eq. (2), for simultaneous signal amplification and noise suppression in the imaging system. In contrast with the passive implementations of the $\Pi$ scheme, the active implementation allowed for enhanced sub-diffraction imaging far beyond the passive deconvolutional scheme by pushing the loss compensation to much higher spatial frequencies. Figure 8 shows that the passive compensation significantly amplifies the noise whereas the active one does not (compare light blue and red lines). This was an important achievement for metamaterial-based imaging systems, which was later shown to be applicable to conventional imaging systems too [61]. It was discovered in [60] that noise amplification can be significantly suppressed through a mechanism of “selective amplification” of the object spectrum using a convolved auxiliary source.

In the active implementation, the noise-free image spectrum corresponding to the total input field can be written in a simplified form as [60,62]

where $O(k_y)$ and $T(k_y)$ are the object spectrum and passive transfer function, respectively. The selective amplification is performed over the pass-band function $G(k_y)$ with the field amplification factor $A_0$. The second term inside the square bracket describes the auxiliary source, which is correlated with the object. Equation (6) can be rearranged as where the expression inside the curly brackets is defined as the active transfer function, which is similar to the fraction inside the summation in Eq. (2). Thus, in the presence of noise [62], the reconstructed image spectrum using inverse filter becomes where $I_{n,A}(k_y)$ is the noisy version of the image spectrum in Eq. (6).Figure 9(a) illustrates the contribution of the signal-dependent (yellow) and signal-independent (blue) noise to the total distortion of an image in a reference imaging system based on a Pendry’s non-ideal NIFL with a relatively weak coherent illumination. As shown in Fig. 9(b), under a strong illumination case, where the illumination intensity is uniformly increased over the spectrum, the signal-dependent noise is amplified proportionally with the illumination amplitude throughout the spectrum, while the contribution of the signal-independent noise is very small. In the strong illumination case, the amount of accessible information about the object can be only barely increased. However, in Fig. 9(c), where a type of structured illumination is employed to implement the $\Pi$ scheme, the signal-dependent noise is approximately at the same level as the reference illumination case except that the noise is redistributed. Therefore, with the $\Pi$ scheme, significant amount of additional information can be learned about the object [compare green lines in Figs. 9(a)-(c)]. Since the auxiliary source was constructed via a physical convolution process with the object (see Eq. (6)), it was also discovered in [60] that correlations [63] play an essential role in the $\Pi$ scheme. Figure 10 shows that a superposition with several auxiliaries uncorrelated with the object does not provide desired amplification and hence the feature at, for example, $k_y=3k_0$, where $k_0$ is the free-space wavenumber, is not recovered (see blue line).

The first physical system for an implementation of the active $\Pi$ scheme was described in detail in [64] for the enhancement of superlens imaging in the presence of noise and absorption losses. The auxiliary source was constructed automatically during a high-intensity illumination of the object, in the object plane of a near-field spatial filter integrated superlens [see Fig. 11(a)]. The resultant field in the image plane is deconvolved using the active transfer function (see Eqs. (7) and 8) to reconstruct the high-resolution image. This process amounts to coherently superimposing a loss compensating auxiliary field with the object field in the object plane in 11(b), as in Fig. 3. The integrated system enabled reconstruction of an object previously not resolvable with the superlens alone. Figure 12 illustrates the superiority of the active $\Pi$ scheme over passive post-processing. The work in [64] elevated the viability of the active $\Pi$ scheme for loss compensation in near-field imaging systems without nonlinear amplification or gain media.

More details on how to physically realize the coherent active $\Pi$ loss compensation scheme was provided in [65]. It was described in detail that the characteristics of the auxiliary source used in the active $\Pi$ scheme, such as selective amplification via convolution over a tunable narrow spectral band, could be realized by using an HMM functioning as a near-field spatial filter. Figure 13 plots the passive transfer function of the lossy metamaterial and the active transfer function of the HMM integrated with the lossy metamaterial [see Fig. 11(a)]. Losses progressively degrade the transmission for greater than $k_0$ in the passive transfer function. However, the transmission spectrum for the active transfer function of the integrated system displays an improvement within the pass band of the HMM.

The implementation of the coherent active $\Pi$ scheme in [64,65] is summarized schematically in Fig. 14. In general, a desired input [see upper left in Fig. 14(a)] cannot be efficiently transferred through a passive metamaterial due to noise and optical losses. However, in the active $\Pi$ scheme, the input is superimposed with a correlated auxiliary source [see lower left in Fig. 14(a)] to compensate the losses and noise. Thus, the output can be produced with a high fidelity. As displayed in Fig. 14(b), this superposition process amounts to integrating the lossy metamaterial with the HMM while simply increasing the illumination intensity. Clearly, in all the cases in Fig. 14, transmission through passive metamaterials is linear.

#### 2.3 Active convolved illumination for coherent and incoherent light

More recently, the first comprehensive theory of the active $\Pi$ scheme for coherent imaging has been developed in [62], where the name has been changed to more encompassing “active convolved illumination (ACI),” as a step toward a generalized linear systems theory. Recognizing the key role of physical deconvolution, the ACI term was first introduced in [66]. Even though significant improvement in the performance of imaging systems was predicted prior to [62], the origin of the enhancement was not fully understood. In [62], the noise variance in the image spectrum was analytically derived for coherent light, leading to a surprisingly simple and yet remarkable mathematical principle, underlying the enhancement, that noise variance in the image spectrum is constant and proportional to the power contained in the noise-free image. It was shown in [62] that ACI utilizes this principle through selective spectral amplification with convolution and correlations. Since the amplification occurs within a narrow spectral region (see Fig. 13), the total power contained in the signal does not increase much, hence the noise. The same principle for incoherent light was used in [67] to beat the SNR limit based on a split-pupil optimization technique, and in [61] to demonstrate improved spectral SNR and higher tolerance to pixel saturation.

The first active implementation of the $\Pi$ loss compensation scheme for incoherent light was introduced in [66], along with the name change to ACI, relying on the convolution of the object with a high spatial frequency passband function for the construction of the auxiliary source to increase the resolution of a plasmonic superlens. Implementation with incoherent light eliminated the phase retrieval challenge. Numerical results with artificial noise indicated enhanced resolution of a few tens of nanometers. Figure 15 considers a superlens incoherent imaging example of four arbitrarily positioned point sources. The two sources near the origin are separated by $25nm$ and cannot be resolved with this reference superlens system even after deconvolution. In contrast, as shown in Fig. 16, plasmonic superlens imaging enhanced with incoherent ACI method fully resolves the objects, under about $60dB$ signal-to-noise ratio (SNR), with resolution better than one fifteenth of the free-space wavelength after the deconvolution of the active image. A physical implementation of the method was designed to provide a proof-of-principle toward an experimental realization.

The incoherent counterpart of the ACI theory was presented in [61], showing how to improve the spectral SNR for the high spatial frequencies. Experimental images were collected using a low numerical aperture imaging system to confirm the predictions. The end result was an image with higher resolution and improved contrast compared to the control image. It has also been shown that the ACI can prevent pixel saturation for longer exposures. This work in [61] presented the first proof-of-concept experimental verification of the ACI as well as the first far-field extension. Even though the experiment was performed with a conventional 4f imaging system instead of metamaterials, it is clear from the theory of ACI, developed in [61,62], that from the perspective of losses and noise mechanisms these two systems are not fundamentally different.

## 3. Other virtual gain techniques

There are two other virtual gain techniques that have been introduced so far by other groups. The first one is the coherent pulse amplification that was proposed theoretically [68] in 2002 and demonstrated experimentally [69] in the year after. This technique is illustrated in Fig. 17. In the time domain picture, the repetition period of a pulse train is matched to cavity round-trip rate to achieve coherent addition of the pulses inside the cavity. Once the sufficient amount of energy is built up coherently inside the cavity, the intracavity pulse is periodically switched out, leading to coherently amplified pulses with lower repetition rate. The pulse amplification factor is dependent on the cavity decay time. Using this technique in an experiment, energy of the picosecond pulses were amplified by a factor of $30$ from about $5nJ$ to $150nJ$ at a repetition rate of $253kHz$ [69]. Intracavity resonant enhancement in this virtual amplification technique is somewhat akin to mode-locked lasers [68]. Also, it is interesting to note that this technique resembles the $\Pi$ scheme at nanoscale (see Fig. 1).

The second technique has been very recently introduced in [45] and it is where the term “virtual gain” comes from. In this technique, virtual gain is achieved by exciting the passive system with a decaying signal. This is dual of the virtual absorption [70–72], which was also introduced by the same group. Virtual gain can be easily described in, for example, a transmission line [22] with uniform loss coefficient $\alpha >0$. The transmission line is excited by a decaying signal $V(z,t)=V_0 e^{j(\Omega t-\beta z)}$, where $\Omega =\omega ^\prime +j\omega ^{\prime \prime }$ is a complex frequency such that $0<\omega ^{\prime \prime }\ll \omega ^\prime .$ Then, the power flow in the transmission line becomes $S(z,t)=\frac {|V_0|^2}{2Z_0} e^{-2\omega ^{\prime \prime }t+2(\omega ^{\prime \prime }-\alpha )z/v}$, where $Z_0$ and $v$ are the characteristic impedance and phase velocity, respectively. When the decay rate $\omega ^{\prime \prime }>\alpha$, the power flow grows along the direction of propagation for all times as if the signal is amplified. This amplification is acquired via reactive energy stored at earlier times due to the decaying signal. Note that gain mechanism here is extrinsic to the medium, because the same effect can also be observed even in active gain medium. In such a case, $\omega ^{\prime \prime }>0$ and $\alpha <0$ independently introduce gain. Therefore, the part of the gain will be still virtual. Similar arguments can be made to classify other gain mechanisms. This virtual gain is illustrated in Fig. 18 and compared with other cases, where $\omega ^{\prime \prime }=0$ and $\omega ^{\prime \prime }=\alpha$. The latter case may be called virtual transparency. Some of the landmark phenomena in ${\cal PT}$-symmetric systems, such as broken phase transitions, anisotropic transmission resonances, and laser-absorber pairs have been proposed based on this virtual gain and virtual absorption [45].

## 4. Conclusions

In this manuscript, we have reviewed virtual amplification process and discussed its performance in various settings. We have broadly defined the term “virtual gain” from its original use such that it includes any extrinsic amplification mechanism in a medium. This way, we hope to draw attention to “virtual gain” not only as a promising approach to overcome the material challenge but also as an enabler with broad implications.

After its introduction in 2015 as an effective solution for controlling and compensating losses in metamaterials and plasmonics, virtual optical amplification (initially called as the $\Pi$ scheme [46] and later as the ACI [66]) has emerged as a versatile technique that provides substantial performance improvement in near-field imaging settings even in the presence of noise. It offers new possibilities not only for many previously conceived plasmonic and meta-devices, but also for conventional imaging systems [61]. With the theoretical framework of ACI developed in [61,62], one can envision applications to numerous scenarios beyond loss compensation in metamaterials and plasmonics.

The theory of ACI can be potentially generalized to the compensation of photon or information loss in various linear systems, such as those in atmospheric imaging in the presence of turbulence and scattering, time-domain spectroscopy, ${\cal PT}$-symmetric systems, photolithography, and quantum information processing. In addition to ACl, coherent pulse amplification with a passive cavity and decaying signal excitation of a passive medium have also been proposed as virtual gain methods. It was shown recently that virtual gain can provide more accurate balance of gain and loss In ${\cal PT}$-symmetric systems [45].

We think that an extension of the concepts of “virtual gain” to quantum optics and quantum information science [73,74] and to nonlinear systems will open a very exciting and fruitful direction. We envision that the extension of the “virtual gain” approach to quantum information sciences can benefit the field by providing a path to mitigating losses (photon or information). Such an extension may build on recent studies that fundamental notions such as interference, coherence and entanglement are present both in quantum and classical optics, and the study of these in one of the regimes will benefit the other. For example, studies in the quantum domain gave us a better picture of classical light and its interaction with the matter, and the techniques inspired from quantum concepts and technologies have shown to improve the performance of metrology, communication, imaging and computing. In a similar way, the techniques developed in the classical domain could potentially be useful for quantum technologies. The question is whether the “virtual gain” concepts developed for classical systems can be used to improve the performance of quantum tasks that are hindered by losses and noise. How one can translate virtual gain concept to quantum domain is not clear yet. An immediate application of virtual gain may be its use in hybrid classical-quantum processors [75–83] where the performance of the classical part is improved to optimally harvest classical and quantum correlations for the best practical performance [84]. It is known that gain and amplification in quantum systems always come with added quantum noise, it is thus crucial to investigate how quantum states and correlations will be affected by the inclusion of “virtual gain.” A comparison of the noise performances of quantum and virtual amplification schemes is thus necessary. Another question that may be of interest to investigate is whether “virtual gain” can be used to improve the fidelity or the success probability of quantum processes where these are limited by the losses and noises. With the same spirit as its application in classical systems, can we use virtual gain, for example, to improve the performance of quantum imaging systems in the presence of loss or noise? Many questions as those listed above are waiting to be answered.

Further work on the theory of ACI and its potential connections with other imaging techniques, such as edge detection [61,85], dark-field imaging [86], structured light microscopy [87], superoscillatory imaging [88–91], and super-gain [92,93] can foster further understanding of imaging beyond fundamental boundaries. When integrated with the ACI, the resolution limit of computational superresolution techniques [94–99] can be extended. Coupled with data-driven approaches [94] it may be possible to enable dynamic optical visualization at nanoscale. Since ACI operates in the physical layer, all of the above scenarios may benefit for improved performance. Finally, we believe that introduction of virtual gain and virtual amplification schemes in a variety of classical and quantum optical systems will open an exciting research direction where many classical and quantum optical phenomena can be studied without the stringent requirements and conditions imposed by gain, amplification and noise.

## Funding

National Science Foundation (ECCS-1202443); Office of Naval Research (N00014-15-1-2684).

## Disclosures

The authors declare no conflicts of interest.

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