SpaceâTime Metamaterials - ACS Photonics (ACS Publications)

Article Cite This: ACS Photonics XXXX, XXX, XXX−XXX

Space−Time Metamaterials Andrei Rogov and Evgenii Narimanov*

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School of Electrical and Computer Engineering and Birck Nanotechnology Center, Purdue University, West Lafayette, Indiana 47907, United States ABSTRACT: Despite more than a decade of active research, the fundamental problem of material loss remains a major obstacle in fulfilling the promise of the recently emerged fields of metamaterials and plasmonics to bring in revolutionary practical applications. In the present work, we demonstrate that the problem of strong material absorption that is inherent to plasmonic systems and metamaterials based on plasmonic components, can be addressed by utilizing the time dimension. By matching the pulse profile to the actual response of a lossy metamaterial, this approach allows to offset the effect of the material absorption. The existence of the corresponding solution relies on the fundamental property of causality, that relates the absorption in the medium to the variations in the frequency-dependent time delay introduced by the material via the Kramers−Kronig relations. We demonstrate that the proposed space-time approach can be applied to a broad range of metamaterial-based and plasmonic systems, from hyperbolic media to metal optics and new plasmonic materials. KEYWORDS: metamaterials, plasmonics, super-resolution imaging he fields of metamaterials and plasmonics promise a broad range of exciting applications, from electromagnetic invisibility and cloaking1 to negative refraction and super-resolution imaging2 to subwavelength field localization, confinement, and amplification in plasmonic resonators.3,4 However, the performance of practical plasmonics- and metamaterials-based applications is severely limited by losses,3 for example, it is the material absorption that “confines” the superlens to the near-field operation.5 Despite multiple attempts to remove this stumbling block with new materials6 or incorporating material gain in the composite design,7 the (nearly) “lossless metal”8 that would allow the evanescent field control and amplification promised by metamaterial research for nearly two decades since the seminal work of J. Pendry,2 remains an elusive goal.6 Here we present an alternative approach to address the problem of optical absorption in metamaterials and plasmonics, that builds upon the pioneering work lead by J.-J. Greffet9 that demonstrated a substantial improvement of the superlens resolution when using time-dependent illumination. We show that, by introducing time-variation in the intensity of the incident light with the pattern that is defined by the actual response of the lossy (meta)material, combined with timegating of the transmitted signal, the detrimental effect of the electromagnetic absorption can be dramatically reduced. The general existence of such “matching” solution follows from the fundamental Kramers−Kronig relations,10 and the “residual” absorption is only limited by the signal-to-noise ratio in the system, and thus ultimately set by the quantum fluctuations. Furthermore, our approach is not limited to a narrow frequency band, and can be applied to a wide range of systems,

T

© XXXX American Chemical Society

from wide-bandwidth hyperbolic materials (such as, e.g., tetradymites, where the hyperbolic response covers the entire visible range11) to polaritonic media in the reststrahlen band.12 The fundamental impact of material losses in nanophotonics goes substantially beyond simply loosing a significant fraction of the signal power. In plasmonic media and metamaterials with plasmonic components, optical absorption leads to essentially nonuniform attenuation of different transverse wavenumber components. What makes the matters worse for nanophotonics, is that these high-wavenumber fields that are essential for subwavelength light confinement, experience the stronger attenuation. This results in a deterioration of the performance of nanophotonic devices even in the conditions of the relatively small loss. As an example of this behavior, consider light propagation in a hyperbolic (meta)material, where the real parts of the dielectric permittivity components have opposite signs in two orthogonal directions.13−15 In contrast to light in conventional dielectrics, where the frequency limits the propagation wavelength and the corresponding light focusing and confinement, the wavenumbers of propagating fields in hyperbolic media are unrestricted by the frequency.15 As a result, hyperbolic materials support tightly focused optical beams, see Figure 1a. However, as a function of the propagation distance the relative amplitudes of different wavenumber components in the beam scale (see Methods) as Received: February 19, 2018 Published: June 18, 2018 A

DOI: 10.1021/acsphotonics.8b00233 ACS Photonics XXXX, XXX, XXX−XXX

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Figure 1. Beam broadening in hyperbolic media. (a) Light intensity from a source at the edge of a half-infinite (z > 0) hyperbolic material (sapphire). The corresponding free-space wavelength is λ0 = 20 μm. The subwavelngth source is formed by a metallic mask with a long slit aligned parallel to the y-direction, with the width a = 20 nm. (b) Magnetic field profile at different distances from the surface of the hyperbolic medium, for the same parameters as the panel (a). Note the progressive widening of the beams away from the source at x = 0, z = 0. (c) Schematics illustrating the origin of the degradation of imaging resolution in hyperbolic (meta)materials in the presence of material losses.

Figure 2. (a) Schematics illustrating the origins of the resolution degradation of the superlens in the presence of material losses. The lossy near-field superlens (SL) creates a blurred image of the object (CW point source) due to uneven attenuation of low-kx and high-kx components. (b) kxspectrum at the image plane of a lossless superlens (ϵ = −1, blue dashed line) and a superlens with material loss (ϵ ≈ −1 + 0.5i, orange curve), demonstrating the suppression of high-kx components when illuminated with CW light. The data were obtained for a slab (thickness d = 10 nm) of aluminum-doped zinc oxide (AZO) as a superlens creating an image of the p-polarized point light source with the carrier wavelength of 1617 nm 2π corresponding to Re[ϵ] = −1; kx is the transverse wavenumber component and kd = d .

Hk τ ≫ ω / c

ÅÄÅ ÑÉ | l o ÅÅ ϵτ ÑÑÑÑ o o o Å ∝ expm o −ImÅÅÅÅ − ϵ ÑÑÑÑkτz} o o o o n Ñ ÅÇ Ö o n ~

Light in the superlens shows a similar behavior, while the perfect lossless superlens supports all wavenumbers,2 intro-

(1)

duction of loss effectively suppresses high-kx components,15,16

so that the beam looses high-k components at progressively faster rate, leading to the eventual broadening of the beam in the hyperbolic medium, see Figure 1.

see Figure 2. With a finite noise that is always present in the imaging system, relatively weak high-wavenumber components B


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disproportionally represented in the observed image, with the resulting distortions and loss of resolution. However, in the limit γ → 0, at appropriate frequency an exact cancellation reduces this “transfer function” to exact unity, corresponding to the behavior of a “perfect lens.” A finite amount of the effective loss γ in eq 2 will define the actual resolution. In the super-resolution imaging approach based on the hyperstructured illumination,17 a hyperbolic metamaterial substrate is used to generate subwavelength illumination “spots” that can be used for super-resolution imaging within the structured illumination framework, as described in Methods. In this case, ( in corresponds to the illumination field amplitude at a given wavelength, that is incident on the hyperbolic metamaterial substrate, see Figure 1, while (out represents the field at the other surface of the hyperbolic medium. The spatial variation of (out is defined by the coordinate dependence of Ω, see eq 25 in Methods. In this example, the Lorenzian structure of the “transfer function” of eq 2 with the spatial coordinate-dependent Ω corresponds to the (subwavelength) illumination spot whose central location depends on the illumination frequency. The spatial size of this localized illumination area, set by the effective material loss parameter γ (see eqs 21 and 24), ultimately defines the imaging resolution of this structured illumination imaging approach that is based on hyperbolic metamaterials. As most examples hyperbolic media, both natural and artificial, generally support several frequency bands with hyperbolic dispersion (e.g., tetradymites,11 hexagonal boron nitride,12 and planar metal-dielectric metamaterials18 all support two separate hyperbolic frequency bands, while sapphire19 has four hyperbolic bands), this allows to combine super-resolution imaging based on hyperstructured illumination, with the elements of spectroscopy. In the proposed space-time approach, we consider a metamaterial system that can be described by the general eq 2, whose “input” ( in (such as, e.g., the illumination field in an imaging system) is modulated with a pulse-shaper,

responsible for spatial resolution get lost in the background and the resolution is compromised.5 Note, however, that the essential difference in the relative absorption of different wavenumber components, by virtue of causality and its mathematical representation in terms of Kramers−Kronig relations, implies a difference in the corresponding phase velocity and the resulting time delays. As a result, if instead of coming from a continuous-wave (CW) source the incident field forms a pulse train, with the proper choice of pulse profile, the amplitudes of low- and high-kx components can be matched at t > t2, see Figure 3. While

Figure 3. Key idea behind the effective loss reduction with pulse shaping. If the illuminating field pulse profile is appropriately chosen, with the difference in the time delays accumulated over the propagation in the metamaterial, low-kx (red curve) and high-kx (blue curve) components of the image can be power-matched at some point t2 in time.

detection at an earlier time (e.g., t1) will reveal a signal with mismatched wavenumber components (and resulting realspace broadening), time-gating of the output signal near the time t2 will recover the undistorted high-resolution signal. In order to match the amplitudes of different kx components at the desired time t2, the required pulse profile (and thus its frequency spectrum) depends on both the material parameters and the dimensions of the metamaterial/plasmonic system. When the medium satisfies Kramers−Kronig relations, such solution always exists and, as we show in the next section, in most cases is mathematically straightforward.

( in(t ) = ((t ) ·exp( −iω0t )

(3)

in such a wave that the resulting “input−output” relation in the time domain (out(t ) =

ceff ( in(t ) ω0 − Ω + iγeff

(4)

has the same mathematical form as eq 2 but with a new (possibly time-dependent) effective value γeff. A substantial reduction of γeff as compared to its original value γ, together with appropriate time-gating of the “output” (out is then an equivalent to using a new class of metamaterials with reduced loss. While somewhat counterintuitive, this objective can be achieved with a relatively simple pulse shape of the form

■

RESULTS In many applications of metamaterials, the observable (out , in the frequency domain can be related to the original “input” ( in via the linear relation c0 A in (ω) Aout (ω) = ω − Ω + iγ (2)

(N (t ) = a0θ(t )t N exp( −γt )

For example, in the case of subwavelength imaging with a metamaterial superlens,2 ( in represents the original object pattern, while (out corresponds to the electromagnetic field amplitude measured in the image plane, see Methods. In this case, while γ relates to the loss in the metamaterial, the corresponding parameters c0 and Ω explicitly depend on the Fourier wavenumber kτ (see eqs 58−63 in Methods) so that the distinct Fourier components of the original pattern are

(5)

where γ is the material loss parameter as defined in eq 2, N ≥ 0 does not need to be an integer, and θ(t) is the Heaviside function (zero for a negative argument and unity otherwise). Note that the frequency spectrum of eq 5 is a simple power law, so that its actual implementation with a practical pulseshaper should be straightforward. With eq 5, for (out in the time domain we find C


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Figure 4. Pulse shaping for structured illumination imaging with hyperbolic media (HM). (a) Schematics of the imaging system and magnetic field (Hy) intensity profile of the shaped light source. (b) Magnetic field intensity profiles of the light at the right side surface of the slab obtained by illumination with CW light (blue dashed curve) and shaped light (green curve corresponding to the detection time t* = 65 ps), normalized to the unity at their respective maxima. The data were obtained for a 3 μm slab of sapphire, one side of which is covered with an opaque mask with a small aperture in it that effectively creates a point light source on the surface of the slab. The mask surface is illuminated with p-polarized light at the carrier frequency f 0 = 15 THz (λ0 = 20 μm), corresponding to the permittivities ϵe = −2.2 + 0.17i and ϵo = 7.1 + 0.29i along the extraordinary and ordinary axes of the sapphire crystal, respectively. The pulse profile is shaped according to eq 5, with the modulation parameter N = 3, but the signal bandwidth strictly limited to the full spectral width f w = 100 GHz, and the free spectral range FSR = 7.8 GHz. The extraordinary axis of the sapphire crystal is normal to the slab surfaces.

Figure 5. Pulse shaping for imaging for a superlens (SL). (a) Schematics of the imaging system and magnetic field (Hy) intensity profile of the shaped light source. (b) Magnetic field intensity of the images obtained by illuminating the target with CW light (blue dashed curve) and with the pulse train “tuned” to the material response (orange and green curves corresponding to the detection times t1 = 0.09 ps and t2 = 0.14 ps, respectively), normalized to the unity at their respective maxima. The imaging data were calculated for a 10 nm slab of aluminum-doped zinc oxide (AZO) as a near-field superlens creating an image of two point objects located 20 nm from each other. The SL is illuminated with p-polarized light source at the carrier frequency f 0 = 185 THz (λ0 = 1617 nm), corresponding to Re[ϵ] = −1. The pulse modulation parameter N = 3. In the simulation, the superlens is illuminated with a period pulse train, with free spectral range FSR = 3.5 THz. The pulse spectrum is limited to full bandwidth f w = 44 THz.

(out(t ) = −ic0t ·- N [(ω0 − Ω)t ]·( in(t )

(6)

γeff =

where

N (ix)k zy N! jijj zz - N (x ) = · − exp( ix ) ∑ j zz z ! k (ix)N + 1 jjk k=0 {

=

exp(ix) (ix)N + 1

(N! − Γ(N + 1, ix))

1 N + 1 − ix

(7)

(8)

(9)

which together with eq 6 yields the desired eq 4, with the effective parameters ceff = c0

(11)

Time-gating the output (out(t ) at t > 1/(N + 1)γ will then reduce the effective width γeff beyond the original value γ that was defined by the actual material parameters. Furthermore, as long as the photonic system is adequately described by the classical Maxwell equations (so that the quantum noise can be neglected), there is no limit on the degree of the reduction of the “effective” loss. At the fundamental level, the proposed space-time approach therefore solves the loss problem that plagued the field of metamaterials for the past decade. However, with the pulse profile (eq 5), time-gating the output signal at t ≫ 1/γ implies operating at progressively lower powers. As a result, there is a practical limit to the “loss mitigation” in the proposed space-time approach that is defined by the actual signal-to-noise power ratio SNR in the system. From eqs 5 and 11, we obtain

and Γ(n,x) is the incomplete Gamma-function. Using the Padé approximation21 for the function -(x) (see Methods), we obtain - N (x ) ≃

N+1 t

(10) D


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0.29 to Im[ϵeff] ≈ 0.13 in the ordinary direction, and from Im[ϵ] ≈ 0.17 to Im[ϵeff] ≈ 0.075 in the extraordinary one. While the optimal pulse shape is defined by the electromagnetic properties of the hyperbolic medium, no a priori knowledge is required on the frequency response of the object and its consituents, as it is the light propagation in the hyperbolic material that defines the spatial localization of the illumination field, see eqs 46 and 47 in Methods. This behavior is illustrated in Figure 6, where we consider the effect of a thin

N+1 N+

1 2

log

Ppeak

(12)

P(t )

where Ppeak is the peak power of the pulse, which defines the limit to the effective loss reduction in the proposed space-time approach γmin =

2γ log SNR

(13)

For super-resolution imaging, this result can also be interpreted in terms of the effective loss, for example, the imaginary part of the dielectric permittivity of the (meta)material that would allow the same resolution power with CW illumination as our approach: ϵeff ″ ≃ ϵ″

N+1 N+

1 2

log

Ppeak P

(14)

where ϵ″ corresponds to the actual value of the permittivity. In the case of a large signal-to-noise ratio, our approach therefore offers an intriguing alternative to the search of new and better metamaterials and plasmonic media, instead of trying to reduce the actual material absorption, the same result can be achieved with the temporal modulation of the incident light that is “tuned” to the response of the existing media (note that the pulse parameter γ in eq 5 is defined by the material response in eq 2).

■

DISCUSSION Based on the general linear response formulation of eq 2, our proposed space-time approach can be applied to a broad range of nanophonic systems that involve metamaterial or plasmonic elements. In the first example, we apply it to super-resolution imaging with the near-field superlens based on aluminumdoped zinc oxide (AZO), operating at the telecommunication wavelength of 1617 nm, see Figure 5. While this superlens is originally unable to resolve two point objects at 20 nm spacing due to the material loss of the AZO, using the pulse-shaped illumination with full spectral width f w = 44 THz and free spectral range FSR = 3.5 THz immediately solves the problem, see Figure 5b. These results are consistent with the original work of ref 9, which demonstrated an improvement of the superlens focusing properties for time-dependent illumination. As our main example of the application of the proposed spate-time approach to super-resolution imaging, we consider the hyperstructured illumination imaging,17 using sapphire as the hyperbolic medium. Note that sapphire has four distinct hyperbolic bands19 (11.9−13.1 THz, 14.4−15.3 THz, 17.1− 17.5 THz, and 26.6−27.2 THz), which allows to use this material platform for super-resolution imaging combined with the elements of spectroscopy. In our simulation of Figure 4, we consider the 14.4−15.3 THz hyperbolic band (corresponding to the free-space wavelenght near 20 μm). To adequately represent a practical experimental setup, we furthermore assume that the pulseshaper is only operating in a finite bandwidth window, with all the frequency components of the band-limited signal within its range. With the pulse-shaping of the incident field and the time-gating of the transmitted light at t*=65 ps, we achieve the same field localization as if the imaginary parts of the dielectric permittivity at both ordinary and extraordinary axes were effectively reduced by approximately 50% − from Im[ϵ] ≈

Figure 6. Hyperstructured illumination for materials with resonant absorption. (a) Dielectric permittivity of PTFE polymer (Teflon), with a strong resonance at 15.5 THz.20 (b, c) Dielectric polarization induced in a thin film of the polymer (red lines) deposited on the “object” interface of the hyperbolic medium (see the schematics in Figure 4a) and a material with frequency-independent refractive index (blue curve). Solid and dashed curves correspondingly show the magnitude and the phase of the (complex) polarization, (b) in the normal (Pn) and (c) the tangential (Pτ) directions to the hyperbolic medium interface. As in Figure 4, the spectrum of the illumination pulse train is centered at the frequency of 15 THz, well within the absorption resonance in the Teflon target (see (a)). Note that while the phase of the induced polarization is indeed modified by the presence of the absorption resonance in the object, the spatial localization of the induced polarization is essentially unaffected.

Teflon film that supports a sharp (Q-factor ∼ 50) resonance close to the carrier frequency of the incident pulse train, deposited on the “object” interface of the hyperbolic medium. While the resonance in the polymer layer does affect the phase of the electromagnetic filed in the “object plane” (dashed lines in Figure 6b,c), the intensity distribution and especially its spatial localization length that defines the imaging resolution, remain essentially unaffected by the Teflon resonance. E


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Figure 7. Full-width at half-maximum (fwhm) of the high-intensity peaks in the spatial distribution of the illuminating field I(x) (see Figure 4a), for hyperstructured illumination imaging with sapphire, at different values of the pulse shape parameter γ. (a) fwhm of the peaks when illuminated with pulse (green curve) and CW light (blue dashed line). The minimum in fwhm at ω0/γopt = 576 corresponds to the optimal value of γ given by the electromagnetic response of the hyperbolic material, via eqs 21 and 24. (b) The fwhm for the pulse as a function of γ in the vicinity of γopt shows that even estimates of γopt based on the individual permittivities ϵn and ϵτ (γn and γτ, respectively) provide near-optimal fwhm. fwhm of the peaks in the magnetic field (Hy) intensity profiles along the x-axis is measured at the detection time corresponding to the input pulse power level −30 dB and given in the units of λ0 = 20 μm. The data were obtained for a 3 μm slab of sapphire with the symmetry axis normal to its side surfaces, one of which is covered with an opaque mask with a small aperture in it that effectively creates a narrow light source on the surface of the slab, see Figure 4a. The mask surface is illuminated with p-polarized light at the carrier frequency f 0 = 15 THz, and the pulse modulation parameter N = 3, similar to the calculation in Figure 4.

■

METHODS In this section, we provide the theoretical background for eq 2 that describes electromagnetic field in a range of nanophotonic

The proposed approach is also robust with respect to the variations in the temporal profile of the illuminating pulse, as illustrated in Figure 7. There we consider the effect of the variations in the pulse duration 1/γ on the resulting spatial intensity pattern. While the optimal value γopt is defined by the hyperbolic material response, via eqs 21 and 24, a substantial variation in γ still yields an improved imaging resolution, see Figure 7. This high degree of tolerance to variations in the temporal pulse profile and to the spectral composition of the object, should therefore substantially simplify the implementation of the proposed space-time metamaterial imaging approach to actual biomedical imaging. In the practical realization of the proposed imaging system, the key challenges lie in the actual realization of ultrafast pulse shaping and ultrafast temporal measurement and characterization. With active research in these areas over the last two decades, the necessary tools for ultrafast pulse shaping and arbitrary waveform generation on picosecond scale and beyond are already available.22,23 However, the time-gating at the relatively low signal powers that is essential for our approach, remains a challenging task. For example, in the case of the hyper-structured illumination with sapphire, one needs ultrafast measurement of optical pulses in the far-infrared region. While the standard GaAs and InGaAs detectors are primarily limited to the near-infrared spectral range, there have been proposed several mechanisms to achieve ultrafast detection in the far-infrared, including rectification in field-effect transistors,24 Schottky diodes,25 or superlattice detectors.26 For picosecond and subpicosecond optical pulse measurement that would be required for implementation of the suggested approach for the superlens imaging with AZO (λ0 = 1617 nm) there are also a number of techniques that are already available, such as temporal magnification (or “time-lens”)27−30 and time-to-frequency conversion.31,32 We therefore conclude that our space−time metamaterial approach can be implemented entirely within the constraints of the currently available technology.

Figure 8. Exact values of -N (symbols) and the corresponding Padé approximation (solid lines), for N = 2 (a) and N = 5 (b). Red color shows the values of the real parts of -N and - NW , while blue represents the corresponding imaginary components.

systems, from hyperbolic media to plasmonic systems and negative index metamaterials. Light in Hyperbolic Media. With the opposite signs of the dielectric permittivity components in two orthogonal directions, hyperbolic media can support TM-polarized electromagnetic fields with the wavenumbers that are only limited by the size of the unit cell of the material. In a uniaxial hyperbolic medium with the permittivities ϵx = ϵy and ϵz, for the relation between the wavevector k and the frequency ω we find13 kx2 + k y2 ϵz

+

kz2 ω2 = 2 ϵx c

(15)

For a thin (subwavelength) illumination slit at the surface of the hyperbolic medium, corresponding to the configuration in Figure 1, the TM-polarized magnetic field in the hyperbolic material, F


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∫ dωHω(x , z)exp(−iωt )ŷ

where rs and r0 are the reflection coefficients at the z = 0 and z = d interfaces. For |kx| ≫ ω/c

(16)

where its time-harmonic amplitudes Hω(x,z) can be expressed as Hω(x , z) =

Hω 2π

ÄÅ ÅÅ Å s = signÅÅÅIm ÅÅ ÅÇ

∞

∫−∞ dkx exp(ikxx)exp(i

with

ϵ − x ϵz

rs =

ϵx(ω /c)2 − (ϵx /ϵz)kx2 sz)

r0 = (18)

kz ≡

ϵx(ω/c) −

(ϵx /ϵz)kx2

≃

−ϵx /ϵz |kx|

Hω 2πi

∑ ±

ϵx ≃ η′0 + η′1(ω − ωc) + iη″ ϵz

kx2 − ϵs(ω/c)2 ϵx

ikz ϵ0 +


≃

i ϵ0 / −ϵxϵz − 1 i ϵ0 / −ϵxϵz + 1

+ r0 exp[ikz(2d(S + 1) − z)]}

(29)

For |kx| ≫ ω/c, eq 29 reduces to

l o o o Hω ∞ 1 So Hω(x , z) = ∑ (r0rs) moo ϵx 2πi S = 0, ± o −s − ϵ (z + 2d S) ± x o o z n | o o o r0 o + } o ϵx o −s − ϵ (2d(S + 1) − z) ± x o o o z ~

(19)

(20)

If the frequency bandwidth of the “signal” H(t), centered around the frequency ωc, is much smaller than the characteristic scale corresponding to a substantial variation of any of the components of the permittivity tensor of the hyperbolic medium, we can approximate s −

ikz ϵ0 −

S=0

ϵ

z

i ϵs / −ϵxϵz + 1

where ϵs and ϵ0 are the permittivities at z < 0 and z > d, respectively. Equivalently, ∞ Hω(x , z) = Hω ∑ (r0rs)S {exp[ikz(z + 2d S)]

1 −s − ϵx z ± x


(28)

which corresponds to the asymptotic behavior of the dispersion relation eq 15. The integral eq 17 can then be calculated analytically, which yields Hω(x , z) =

ikz ϵs +

i ϵs / −ϵxϵz − 1

≃

and

The expression eq 17 is exact for a natural hyperbolic medium, and is an accurate approximation for a hyperbolic metamaterial when its unit cell size is on the same order or smaller than the width of the illumination slit. Within the absorption distance from a narrow (compared to the free-space wavelength) illumination slit, the integral eq 17 is dominated by the wavenumbers kx ≫ ω/c. The zcomponent of the wavevector can then be approximated by 2


(27) (17)

ÉÑ ÑÑ ÑÑ ÑÑ ÑÑ ÑÖ

ikz ϵs −

(30)

which again corresponds to a special case of our general expression (eq 2). In particular, at the “object plane” z = d, Hω(x , d) =

(21)

∞ (r0rs)S Hω (1 + r0) ∑ ϵx 2πi S = 0, ± − s − ϵ d(2S + 1) ± x z

which reduces eq 20 to a special case of our general expression eq 2, ch Hω(x , z) ≃ Hω ∑ ω − Ω±(x , z) + iγh (22) ±

(31)

∞

=Hω

S = 0, ±

γh =

cS ω − Ω±(x , d(2S + 1)) + iγh

(32)

where

where ch =

∑

i 2πzη′1

cS =

(23)

(24)

and η′ x Ω±(x , z) = ωc − 0 ± η′1 η′1z

Px =

(25)

exp(2ikzz) + r0 exp(ikz(2d − z)) 1 − r0rs exp(2ikzd)

ÄÅ ÉÑ ÅÅ ÑÑ Å ÑÑ ∞ ÑÑ ± (r0rs)S χ ∂ ÅÅÅÅ Hω ÑÑ (1 + r0) ∑ ÅÅ ÑÑ ϵ Å x −ϵxϵz ∂x ÅÅ 2πi ÑÑ − − + ± S s d x (2 1) S = 0, ± ÅÅÅ ÑÑÑ ϵz Ç Ö

ÅÄÅ ÑÉÑ ÅÅ ÑÑ S ∞ Å ÑÑ ( ) r r H χ ∂ ÅÅ ω 0s ÑÑ ÅÅ (1 ) + r Pz = ∑ ÑÑ 0 ϵ Å x Ñ ϵ ∂x ÅÅ 2πi S = 0, ± − s − ϵ d(2S + 1) ± x Ñ ÅÅ ÑÑ z ÅÇ ÑÖ

For a hyperbolic (meta)material slab of thickness d, with a (metallic) mask at z = 0 that has a thin (subwavelength) slit at x = 0, we find Hω(x , z) = Hω

(33)

Hyperstructured Illumination. When a thin film with the dielectric permittivity ϵ ≡ 1 + 4πχ is deposited on the hyperbolic medium surface at z = d, the illumination field (eq 32) will induce the polarization P such that

η ′′h η′1

1 + r0 2πiη′1d(2S + 1)

(26)

(34)

(35) G


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li yz ωp;2 x , z o ∂ o ojjj (0) zz m z jjαx , z + o 2ωr; x , z(Ω − ωr; x , z + i(γ − γh)) zz ∂x o ojk { n 1

If the film supports a sharp resonance in proximity to the illumination frequency,

Px , z(t ) = μx , z ((t )

ωp2

ϵ(ω) = ϵ∞ +

ω0 − Ω + i

ωr2 − ω 2 − 2iγω

While the presence of a sharp resonance in the target material does affect the induced polarization (note the last term in eqs 46 and 47), it is still strongly localized close to the point where Ω(x) = ω0, and super-resolution imaging based on structured illumination with such field, is still possible. Electromagnetic Field in the Superlens. In its simple realization,2 the superlens is essentially a parallel slab of a metamaterial with simultaneously negative values of the (isotropic) dielectric permittivity ϵ and magnetic permeability μ, that “match” the corresponding parameters of the surrounding medium ϵ0, μ0: ϵ = −ϵ0 (48)

Hω (1 + r0)(r0rs)S H (1 + r0)(r0rs)S , μz = ω 2πi dη′1(2S + 1) 2πi dη′1(2S + 1) (38)

and ωp;2 x , z (39)

μ = −μ0

Here 1 ϵ∞ − 1 αx(0) = 4π −ϵxϵz

(40)

ωr; x = ωr

(41)

ωp, x =

and αz(0)

ωr; z = ωp, z =

TωTM(kτ ) =

(42)

1 ijj 1 yzz = jjj1 − z ϵ∞ zz{ 4π k

= (43)

ωr2 + ωp2 /ϵ∞

4 exp[(κ − κ0)d ]

(2 +

κ ϵκ0

+

)exp(2κd) + (2 −

ϵκ0 κ

κ ϵκ0

−

ϵκ0 κ

)

2ϵκ0κ exp[(κ − κ0)d ] ϵκ0 − κ (50)

×∑

(44)

±

1 (κ − ϵκ0) ± (κ + ϵκ0)exp(κd)

(51)

for TM-polarized fields and

ωp 4π ϵ∞

(45)

TωTE(kτ ) =

For illumination with the pulse defined by eq 5, at the distance x in proximity to the illumination maximum of the given order S , we obtain Px , z(t ) = μx , z ((t )

(49)

A perfect resonance of this kind is however unattainable, due to inevitable finite amount of loss in the metamaterial, leading to nonzero imaginary parts of ϵ and μ. The planar superlens produces a (nearly) perfect image when the separation between the “object”’ and “image” plane is equal to twice the thickness of the superlens d, with the corresponding transmission coefficient as a function of the inplane wavenumber kτ, equal to

ωp −4π ϵxϵz

( N +t 1 + γ − γh)) ωr;x ,z

| o o o o 1 o } o − Ω + i(γ − γh) o o o o ~

(47)

where

ωr;2x , z − ω 2 − 2iγω

(

2ωr; x , z ω0 − ωr; x , z + i

μx , z ∂ ∞ ∑ ∂x S = 0, ± ω − Ω±(x , d(2S + 1)) + iγh

αxeff, z = αx(0) ,z +

ωp;2 x , z

+

(37)

μx = ±

N+1 t

(36)

the induced polarization can be expressed as Px , z = αxeff, z(ω)

Article

=

l o ∂ o o − it - [(ω − Ω)t ] m N 0 o ∂x o o n ωp;2 x , z

(2 +

+

μκ0 κ

)exp(2κd) + (2 −

κ μκ0

−

μκ0 κ

)

2μκ0κ exp[(κ − κ0)d ] μκ0 − κ (52)

yz jij (0) zz jjαx , z + z jj 2ωr; x , z(Ω − ωr; x , z + i(γ − γh)) zz { k

+ it -N [(ω0 − ωr; x , z + i(γ − γh))t ]

4 exp[(κ − κ0)d ] κ μκ0

ωp;2 x , z 2ωr; x , z(Ω − ωr; x , z

×∑ ±

| o o o } o + i(γ − γh)) o o ~

1 (κ − μκ0) ± (κ + μκ0)exp(κd)

(53)

for the TE-polarization, where κ0 ≡

(46)

kτ2 − ω 2 /c 2

(54)

kτ2 − ϵμω 2 /c 2

(55)

and

where the function - N is defined in eqs7 and 8. Using the Padé approximation (eq 9) for γ > γh, we finally obtain

κ≡ H


Article

ACS Photonics In the limit ϵ/ϵ0, μ/μ0 → −1, we find Tω(kτ) → 1 for any kτ, which implies the formation of a perfect image. Note that the concept of super-resolution relies on the accurate representation of high-k Fourier components of the object pattern, kτ ≫ ω/c. In this limit, for a small loss ϵ″ ≪ |ϵ|, μ′′ ≪ |μ| and the signal bandwidth smaller than the frequency scale corresponding to a substantial variation of ϵ and μ, we find TωTM(kτ ) ≃

∑ ±

ω−

c±TM Ω±TM(kτ )

+ iγ

TM

AN 1 1 = · lim [x - N (x)] CN x x x →∞

which yields AN 1 A = , N =i BN N CN

∑ ±

- NW (x) = (56)

(57)

where c±TM = ±

exp( −|kτ|d) |ω = ωc d ϵ′/dω

Ω±TM(kτ ) = ωc ∓

γ

TM

2 exp( −|kτ|d) |ω = ωc d ϵ′/dω

ϵ″ = |ω = ωc d ϵ′/dω

■

(58)

(68)

AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected]. ORCID

(59)

Evgenii Narimanov: 0000-0003-2448-6482 Author Contributions

Both authors equally contributed to all aspects of the work.

(60)

Notes

and c±TE

1 N + 1 − ix

which immediately leads to eq 9. In Figure 8 we compare the exact values of - N for N = 2 (a) and N = 5 (b) with the corresponding Padé approximation of eq 68. Note that our first order Padé approximant (eq 68) offers an accurate representation of the exact function -(x) in the entire range 0 ≤ x < ∞.

c±TE ω − Ω±TE(kτ ) + iγ TE

(67)

and therefore

and TωTE(kτ ) ≃

(66)

The authors declare no competing financial interest. exp( −|kτ|d) =± |ω = ωc dμ′/dω

Ω±TE(kτ ) = ωc ∓ γ TE =

2 exp( −|kτ|d) |ω = ωc dμ′/dω

μ″ |ω = ωc dμ′/dω

■

ACKNOWLEDGMENTS This work was partially supported by Gordon and Betty Moore Foundation and NSF Center for Photonic and Multiscale Nanomaterials (C-PHOM). The authors thank Prof. A. M. Weiner for helpful discussions and comments on the article.

(61)

(62)

■

(63)

As a result, the transmission function of a superlens can also be represented by our general expression eq 2. Note that, for super-resolution imagining in the near field, the desired functionality can be also achieved by the so-called “poor man’s superlens”, that is essentially a metallic layer used for TM-polarized light near the plasmon resonance frequency that corresponds to eq 48. The corresponding transmission coefficient, also given by eq 51, but with the value of κ taken at μ = 1, in the subwavelength resolution resolution limit kτ ≫ ω/c then also reduces to eq 56 that can be treated as a special case of eq 2. Padé Approximation. For a given function f(x) in a specified interval (a,b), the Padé approximant is the rational function whose power series expansions near x = a and x = b agree with the corresponding power series expansions of f(x) to the highest possible order.21 In our case, we seek to approximate - N (x) in the entire range 0 ≤ x < ∞, so that with Padé approximant of the first order, - NW (x) =

AN BN + CN x

(64)

we impose the conditions

AN = - N (0) BN

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