Subwavelength Imaging Using Phase-Conjugating Nonlinear

LETTER pubs.acs.org/NanoLett

Subwavelength Imaging Using Phase-Conjugating Nonlinear Nanoantenna Arrays Pai-Yen Chen and Andrea Al
u* Deparment of Electrical and Computer Engineering, The University of Texas at Austin, 1 University Station C0803, Austin, Texas 78712, United States ABSTRACT: We investigate the use of nonlinear metasurfaces formed by plasmonic nanoantennas loaded with χ(3) nonlinear elements, in order to realize subwavelength imaging based on phase conjugation and time reversal. The nanoantennas’ plasmonic resonance is used to boost the nonlinear response over an ultrathin surface, meeting the conditions for efficient phase conjugation necessary for imaging applications. Pairing two such surfaces, we put forward a realistic design for a time-reversal ‘perfect lens’, which can overcome the limitations in resolution and sensitivity to losses typical of negative-index lenses. KEYWORDS: Subwavelength imaging, plasmonic nanoantennas, nano-optics, phase conjugation, nonlinear wave mixing

T

he interest in using metamaterials to overcome the diffraction limit in imaging applications has sparked after the seminal discovery that a planar negative-index slab may constitute a lensing device with, in principle, unlimited resolution.1 Negative refraction at the planar boundaries of such a perfect lens can focus propagating waves and support the resonant amplification of the evanescent spectrum of an image, restoring all its unperturbed spectral details at the lens’ focal plane. Subsequent papers have broadened the applicability of these concepts to use perfect lenses for subdiffractive imaging and sensing.2
8 One of the major limitations of this setup, however, consists in its large sensitivity to losses and to the finite granularity of the metamaterial.9,10 This is easily understood by considering that subwavelength resolution is associated with the exponential amplification of evanescent waves inside the metamaterial lens, where losses are not negligible. Maslovski and Tretyakov11 proposed an alternative approach to perfect lensing, based on phase conjugation and time reversal. A pair of ideal phase-conjugating surfaces would realize a system electromagnetically analogous to a negative-index slab, without requiring the presence of lossy metamaterial inclusions in regions where the evanescent spectrum is amplified. Since phase conjugation requires a nonlinear process, these concepts have been experimentally realized at microwaves12
14 employing signal mixers and nonlinearly loaded dipoles. More efficient elementary inclusions based on dipoles loaded with nonlinearities have also been recently proposed in ref 15. In parallel with these findings, Pendry16 has theoretically shown that this imaging setup may be ideally translated to optics using the four-wave mixing (FWM) process.17 In his proposed configuration, an ideal lens may be achieved using a pair of parallel ultrathin layers supporting an infinitely large nonlinear response. Considering that optical thin films have very weak nonlinear properties,18 however, this ideal r 2011 American Chemical Society

Figure 1. Schematic diagram (top view) for: (a) phase conjugation at a single nanoantenna metasurface, through nonlinear four-wave mixing and (b) imaging device formed by two phase-conjugating metasurfaces.

setup may require unrealistic pumping levels to achieve sufficient level of nonlinearity for gaining subwavelength resolution. In this paper, we propose a viable, realistic setup to realize the subwavelength focusing in the near-infrared and visible spectral range, overcoming several of the limitations of previous solutions. We use plasmonic metasurfaces formed by nanoantennas loaded with nonlinear materials, similar to the arrays we employed in ref 21, to realize enhanced optical bistability and alloptical switching. Here, we load the nanoantennas with χ(3) nonlinearities, and we use a degenerate FWM process to obtain large phase conjugation over an ultrathin surface, ideal for Received: September 27, 2011 Revised: November 11, 2011 Published: November 16, 2011 5514

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Nano Letters

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Figure 2. (a) Local field enhancement and electric near-field distribution, (b) amplitude (normalized to 6πε0/k30) and phase of the linear polarizability of an individual nanoantenna element, showing resonant plasmonic features and continuity of the displacement current across the nanoload (Figure 1a). The inset of (b) shows the far-field dipolar radiation pattern.

)

discussion, this large field enhancement is supported over an ultrathin profile, almost a mathematical surface compared to the incident wavelength, which ensures the equivalence between reflected and transmitted conjugate signals, necessary for the subwavelength imaging. Figure 2b shows the calculated amplitude and the phase of the nanoantenna linear polarizability α(ω), with a distinct dipolar far-field radiation pattern and sharp resonant features. We now consider a planar array of such nanoantennas in the y
z plane with transverse periods dy = 100 and dz = 240 nm. Due to the inherent anisotropy of these inclusions, we assume transverse-electric (TE) impinging waves, linearly polarized with an electric field along the nanoantenna axis ^z (Figure 1a). The signal field is Es = ^zE0 exp(ik 3 r + kxx) exp(
iωt), with r = y^y + z^z, k is the projection of the wave vector ks on the array, and E0 is arbitrary amplitude coefficients. After the degenerate FWM, each nanoantenna gap supports, in addition to their linear dipole moment, a phase-conjugated dipole moment, stemming from the mixing frequency ω = ω + ω
ω. The total dipole moment at frequency ω induced on each nanoantenna is then given by )

subwavelength imaging.16 In our setup, each metasurface is composed of an array of optical nanodipoles, with square lattice and periods dy, dz, consisting of two silver nanorods of length l separated by a small gap with height h, loaded with nonlinear optical materials (NOM) (Figure 1a). The plasmonic features of each nanoantenna ensure that, at resonance, we achieve drastic enhancement of the third-order nonlinear response at the nanoload, which may efficiently generate phase-conjugated signals through the degenerate FWM process, as experimentally verified for gold nanoparticles in ref 22. Alternative nonlinear imaging techniques have been suggested in recent times.23 One of the key advantages of the proposed setup consists in the large field enhancement associated with the plasmonic features of the periodic array, which is directly translated into giant amplification of its collective nonlinear response. This advantage appears to be unique to optical nanoantennas, and it cannot be directly translated to lower frequencies, as the nonlinear circuit elements employed in refs 12
15 typically respond to the applied voltage across the gap of RF antennas, rather than to the local electric field. Figure 1a illustrates the principle of operation of one of such proposed phase-conjugating metasurfaces: The FWM process is operated by exciting the metasurface with a monochromatic signal wave at frequency ω and two strong counterpropagating pump waves at the same frequency (so that kp1 + kp2 = 0). Due to the phase matching,17 a nonlinear dipole moment with conjugate phase (kpc r =
ks) is then generated at the nonlinear load embedded in the nanogap. We tune the metasurface to collectively resonate at the signal frequency ω, causing large reflections and very weak transmission of the impinging signal in its linear operation. For large and almost undepleted pump levels, the drastically enhanced fields at the nanogap produce a conjugate signal with amplitude significantly larger than the signal wave. The subwavelength periodicity of the metasurface ensures that the scattered fields are combined into equal amplitude reflected and transmitted phase-conjugated plane waves, propagating at negative angles (see Figure 1a). We have designed each individual nanodipole element to support its linear resonance around the design frequency ω0/2π = 330 THz, with l = 100 nm, h = 10 nm, and a = 5 nm (Figure 2a). We assume a linear permittivity of the loading material εL = 12, ensuring a moderately large χ(3) nonlinear response, following the Miller rule.18 Silver is modeled with a Drude-type permittivity as εAg = ε∞
ω2p/[ω(ω + iγ)], with ωp/2π = 2175 THz, γ/2π = 4.35 THz, and ε∞ = 5.24 Figure 2a shows the magnitude of local field enhancement at the gap and the electric near-field distribution at the designed nanoantenna resonance (ω0/2π = 330 THz), using full-wave simulations.25 Very relevant to the following

ppc ðωÞ ¼ αðωÞEloc ðωÞ þ γðω : ω, ω,
ωÞEloc ðωÞEloc ðωÞEloc ð
ωÞ

ð1Þ where Eloc is the local electric field at the nanoantenna location and γ(ω:ω,ω,
ω) is the second hyperpolarizability, accounting for hyper-Rayleigh scattering γ(ω:ω,ω,
ω) = χ(3)ε0πa2hf4g (ω),18,26,27 fg(ω) = |Egap/E0| represents the field enhancement factor compared to the local field (Figure 2a). The local fields are given by the superposition of the impinging wave and the array coupling, whose interaction has been analytically calculated within the dipolar approximation, extending the formulation in ref 28 to nonlinear operation. After some algebraic manipulation, we may write the phase-conjugate component in ref 28 as " # Ep1 ðωÞ γðω : ω, ω,
ωÞ pc p ðωÞ ¼
1 1
ε
1 0 αðωÞCðωÞ 1
ε0 αðωÞCðωÞ " # Ep2 ðωÞ  1
ε
1 0 αðωÞCðωÞ   Es ð
ωÞ  1
ε
1 0 αð
ωÞCð
ωÞ ¼ γðω : ω, ω,
ωÞLðωÞLðωÞLðωÞLð
ωÞ  Ep1 ðωÞEp2 ðωÞEs ð
ωÞ 5515

ð2Þ

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Figure 4. Field distribution at the source (black solid) and image plane for the case of no metasurfaces (gray dash) and double phase-conjugating metasurfaces (colored solid).

Jpc av ¼


iωppc ðωÞ dy dz

¼ ^z


iω γðω : ω, ω,
ωÞLðωÞLðωÞLðωÞLð
ωÞ dy dz

Ep1, 0 Ep2, 0 E0 e
ik| r e
iωt

ð3Þ

EðxÞ ¼ ^zE0

This averaged current distribution emits phase-conjugate plane waves on either side of the array, as illustrated in Figure 1a: qffiffiffiffiffiffiffiffiffiffiffiffiffiffi pc
ωμ0 Jav expð ( i k20
k2|| xÞ E(, pc ¼ ^z qffiffiffiffiffiffiffiffiffiffiffiffiffiffi ð4Þ k20
k2|| 2

)

  4Ip1 Ip2 1=2  Lð
ωÞ c2 ε20

EðxÞ ¼ ^zE0

pffiffiffiffiffiffiffiffiffiffi ffi 2 0 2 qffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðR pc Þ2 eik| 3 r| þ i k0
k| jx
x j
iωt pffiffiffiffiffiffiffiffiffiffi ffi , for imaginary k20
k 2|| 2 2 2 1
ðR pc Þ ei2 k0
k| d

ð7Þ which, in the limit R f ∞ "k , corresponds to an ideal imaging system in the region d < x < 2d, for an object placed at x0 (
d < x0 < 0). The plasmonic metasurface design proposed here is an ideal setup to achieve these strict requirements with very good approximation. Figure 3 shows the transmission coefficient Timg on the image plane for a single and a pair of phase-conjugating metasurfaces, which are calculated using full-wave numerical simulations to determine the coefficients in (eq 5). It is seen that the phase-conjugated waves radiated by a single metasurface are largely amplified by the pump waves for a wide range of transverse wave vectors k , both in the propagation (|k |/k0 < 1) and evanescent (|k |/k0 > 1) portion of the spatial spectrum. In the evanescent region, a pole is obtained at the location of a surface-wave supported by the array, in a regime for which the periodicity becomes comparable to the transverse variation of the evanescent spectrum, analogous to a Wood’s anomaly.28 This pole effectively represents the cutoff of the phase-conjugation pc

)

iω2 μ0 Rpc ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffi γðω : ω, ω,
ωÞLðωÞLðωÞLðωÞ 2 k20
k 2|| dy dz

ð6Þ

ð5Þ

where Ip1 and Ip2 are the impinging flux intensity of the pumping waves. The transmitted wave has equal amplitude Tpc = Rpc, due to the low array profile. For large pump levels, and considering

)

)

Phase conjugation is achieved (eqs 3 and 4) for two conterpropagating pump waves, which corresponds to radiated plane pc waves with wave vectors kpc r =
k
kx and kt =
k + kx on the two sides of the array; here the subscripts r and t indicate reflected and transmitted fields, respectively. We may define the reflection coefficient for phase-conjugated waves Rpc as the ratio between the amplitude of the backward propagating phaseconjugate wave to that of the incident signal wave:

pffiffiffiffiffiffiffiffiffiffi ffi 2 0 2 qffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðR pc Þ2 eik| 3 r|
i k0
k| ðjx
x j
2dÞ
iωt , for real k20
k2|| 2 pc 1
ðR Þ

)

where C is the array interaction constant, which relates the induced dipoles to the local field in each unit cell, and it can be calculated in form of rapidly converging series,29 and L = [1

1 is the local field correction factor.30,31 Since the array ε
1 0 α 3 C] periods are subwavelength, the polarization fields collectively radiate the fundamental Floquet mode, effectively sustained by the averaged current density

the large enhancement factor due to the interaction coefficients, Rpc in eq 5 may be larger than unity and dominate the metasurface response at frequency ω (compared with the linearly reflected signal associated with α(ω)Eloc(ω) in eq 1). We now consider a practical design scenario with two equalintensity pump waves with power density Ip1 = Ip2 = 100 MW/ cm2 and a nanoload with εL = 12 and χ(3) = 10
16m2/V2 (consistent with realistic values of χ(3) for some III
V compounds (i.e., AlInAs/GaInAs) or polydiacetylenes).18
20 Although a single phase-conjugating metasurface supports a large Rpc = Tpc, its value is nonuniform for different spatial frequencies, causing image distortions and nonoptimal resolution. Therefore, following the theoretical proposal in ref 16, we combine two of such highly reflective metasurfaces separated by a distance d (see sketch in Figure 1b). This combination can effectively produce the same effect as a negative-refraction planar slab, in the limit of large Rpc = Tpc. The total transmitted field of phase-conjugate waves behind the second metasurface (Figure 3) may be written as16

)

Figure 3. Transmission of the phase-conjugated signal for a single (red solid line) and a pair (blue dashed line) of phase-conjugating metasurfaces.

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Figure 5. Field distribution at the source (black solid) and image plane for the case of no metasurfaces (gray dash) and phase-conjugating metasurfaces with different distances (colored solid lines). The dash-dot light blue line is for a larger pump intensity of 1 GW/cm2.

imaging effect, as for larger spatial frequencies Timg rapidly tends to zero. It is visible that for a pair of phase-conjugating metasurfaces, the transmission on the image plane Timg is quite uniform over a wide range of transverse wave numbers. In contrast, for a single metasurface Timg is nonuniform and amplified for larger wave numbers, which may cause image distortions. Next, we study the focusing effect of such pair of metasurfaces at ω0/2π = 330 THz, considering the realistic presence of losses and array granularity. We show the transmission at the image plane as a function of transverse wavenumber in Figure 3, which indeed supports the expected subwavelength imaging and focusing properties. Figure 4 shows the corresponding image plane field distribution at x = 3d/2, for a subwavelength source placed at x =
d/2. We show the different cases of: an object plane (solid black line); no device between object and image plane (gray dashed line), for which diffraction dominates the image; and a pair of metasurfaces (red solid line), which are able to restore to a large degree the subwavelength resolution of the source in the image plane. A single metasurface (not shown) would also support some partial focusing but combined with serious distortion of the original image due to the nonuniform amplification of finer image details. In Figure 4 we also show the frequency response for different frequencies around ω0, showing a moderate bandwidth of subwavelength focusing, despite some expected deterioration due to lower levels of Rpc off-resonance. We stress that we have considered here realistic levels of absorption in silver, and yet we predict subwavelength resolution over a moderately broad bandwidth of operation, which is tunable to a large degree as a function of the nanoantenna geometry. It may be relevant to compare this performance to the one of a silver superlens, as the one realized in ref 4. First of all, a uniform silver lens may operate only at one wavelength, dictated by the ultraviolet (UV) frequency for which Re[εAg] =
ε0, with no room for tunability, consistent with ref 4. In addition, due to its much larger sensitivity to absorption, reasonable subwavelength imaging may be achieved only over small distances, d = λ0/10 in ref 4. Here, however, it is possible to increase the distance among metasurfaces and obtain similar subwavelength imaging properties also for significantly longer distances. Figure 5 shows the imaging properties of a pair of metasurfaces with distance varied between d = 0.5λ0 and d = 3λ0 at the frequency of operation. Although losses play a role at the second metasurface, a robust subwavelength imaging performance is obtained, even for larger distances. We have also considered a larger, but still

LETTER

realistic, pump intensity Ip1 = Ip2 = 1 GW/cm2 for the case d = 2λ0 (dash-dot light-blue line), verifying that resolution may be enhanced over larger distances by increasing the pump levels and the corresponding value of Rpc. To conclude, we have proposed here a realistic implementation of a super-resolving imaging system based on ultrathin nonlinear metasurfaces. The plasmonic metasurface design may significantly enhance third-order nonlinear effects at the nanoload, enabling an efficient degenerate FWM process for the generation of phase-conjugated waves. Compared to metamaterial slabs and conventional perfect lenses restricted to the nearfield due to sensitivity to losses, subwavelength imaging is proven to be more robust, thus making relatively long-distance and even far-field focusing realistic. We believe that these concepts may be realized within current e-beam or nanoimprint lithography processes and may pave the way to novel imaging devices. The proposed conjugating metasurface design, with its unique properties of large nonlinear response over an ultrathin profile, may have direct application in high-density data writing and storage, novel microscopy devices and direct observation of nanometersized structures and organisms in vivo. This giant nonlinearity may also be applied to the realization of nanoscale optical multipliers and mixers.

’ AUTHOR INFORMATION Corresponding Author

*E-mail: [email protected].

’ ACKNOWLEDGMENT This work was supported by the AFOSR YIP award no. FA9550
11
1
0009 and the ONR MURI grant no. N00014
10
1
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