Broadband Achromatic Anomalous Mirror in Near

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Broadband achromatic anomalous mirror in near-IR and visible frequency range Andrei Nemilentsau, and Tony Low ACS Photonics, Just Accepted Manuscript • DOI: 10.1021/acsphotonics.6b00922 • Publication Date (Web): 12 Jun 2017 Downloaded from http://pubs.acs.org on June 15, 2017

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Broadband achromatic anomalous mirror in near-IR and visible frequency range

Andrei Nemilentsau ∗ and Tony Low ∗

Department of Electrical & Computer Engineering, University of Minnesota, Minneapolis, MN 55455, USA E-mail: [email protected]; [email protected]

Abstract An anomalous achromatic mirror operating in the near-IR and visible frequency ranges was designed using an array of metal-insulator-metal (MIM) resonators. An incident wave interacting with the MIM resonator experiences phase shift that is equal to the optical path travelled by the gap plasmon excited by the wave. A phase gradient along the mirror surface is created through dierence in plasmons optical paths in the resonators of varying lengths. In a frequency region well below the plasma frequency of metal, the phase gradient is a linear function of frequency, and thus the mirror operates in the achromatic regime, i.e. the reection angle does not depend on radiation frequency. Using silver-air-silver resonators, we predicted that the mirror can steer normally incident beam to angles as large as 40◦ with high radiation eciency (exceeding 98 %) and small Joule losses (below 10 %). Our study indicates that it is feasible to have ecient broadband anomalous mirror.

1

ACS Paragon Plus Environment

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Er dielectric

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metal

θr

visible

Einc

near-IR

a

metal

Keywords Anomalous mirror, gradient metasurface, achromatic reection, gap plasmons, metal-insulatormetal resonators Recent progress in the design of gradient metasurfaces allows for unprecedented control of the characteristics of propagating light beams. 1,2 This includes anomalous reection and refraction that dees conventional Snell's law, 311 ecient manipulation of the polarization state of electromagnetic waves, 1216 and polarization dependent steering of light beams. 1720 Electrically tunable graphene 2123 and conducting oxide 24 metasurfaces, as well as thermally tunable dielectric metasurfaces 25 were recently demonstrated. A number of applications utilizing these nontrivial properties of metasurfaces were proposed, such as ultrathin at lenses, 2629 perfect absorbers, 30,31 invisibility cloaks, 32 polarization detectors 33 and waveplates, 34,35 vortex and Bessel beam generators, 3639 and holograms. 4043 Despite successes achieved in the engineering of gradient metasurfaces with non-trivial optical properties, their performance is usually restricted to very narrow frequency range and suers from chromatic aberrations, such as a frequency dependent angle of reection in the case of an anomalous mirror. One can argue that the limitation on the operating frequency range is inherent to the metasurface design. Indeed, basic element of typical metasurface is a sub-wavelength scatterer, dielectric or metallic, which upon interacting

2


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with an electromangetic wave experiences a new scattering phase. Magnitude of the phase change depends on detuning between frequency of the electromagnetic wave and resonance frequency of the scatterer. Hence, spatial gradient of phase change can be created by placing resonators of dierent shapes and sizes along the metasurface. It is this phase gradient that is responsible for the unique properties of gradient metasurfaces. 1,2 However, since the phase gradient originates from resonance coupling between electromagnetic wave and the scatterers, the metasurface performance is therefor limited to only near-resonance frequencies. It was demonstrated that anomalous reection and refraction of light are governed by the generalized Snell's law 3

1 dΦ(ω, x) , k0 dx 1 dΦ(ω, x) nt sin θt − sin θi = k0 dx sin θr − sin θi =

(1) (2)

for the metasurface residing in x − y plane, and with the phase change gradient Φ(ω, x) created in x direction only. The halfspace above the metasurface is vacuum, while that below has a refractive index nt . Here, θi , θr and θt are angles of incidence, reection, and refraction, respectively, k0 = ω/c is free-space wavenumber, and ω is radiation frequency. One can clearly see from Eq. (1) that the necessary condition for achromatic reection and refraction in a broad frequency range is for the phase gradient to be a separable function of the form

Φ(ω, x) = k0 X(x),

(3)

as in this case the reection and refraction angles do not depend on frequency, i.e.

dX(x) , dx dX(x) nt sin θt − sin θi = . dx sin θr − sin θi =

(4) (5)

The fact that the gradient, Φ(ω, x), has to be linear function of frequency poses challenge 3


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for creating achromatic metasurface using sub-wavelength near-resonance scatterers, as the phase of the scattered wave varies abruptly around the resonance frequency, and thus the phase gradient, dened by Eq. (3), is not easy to implement. Recently, the problem of broadband achromatic refraction has been addressed in a number of papers. 4446 Particularly, achromatic lens operating at three wavelengths (1300 nm, 1550 nm and 1800 nm) has been designed 44,45 utilizing dielectric resonators supporting dense spectrum of optical modes that allows near-resonant coupling at multiple wavelengths. However, the phase change experienced by the wave still varies abruptly around each of the resonances, and thus the achromatic behavior can be only implemented for discrete number of frequencies. Alternatively, metal-insulator-metal (MIM) resonators were proposed 46 to implement the broadband achromatic refraction in near-IR frequency range. In this case, the incident wave excites gap plasmons in an array of MIM resonators that re-radiate their energy after propagation through the resonator. The phase gradient was implemented by changing the resonators width and thus the gap plasmons phase velocities. The device, proposed in Ref., 46 operates in the transmission mode and covers near-IR frequency range only. In this paper, we address the issue of achromatic anomalous reection, i.e. a mirror that reects incident light beam at an angle that is dened by the mirror geometry only, and does not depend on radiation wavelength. We utilize an aperiodic array of MIM resonators for this purpose and demonstrate that the mirror can operate in near-IR and visible frequency ranges if silver is chosen as a material for the resonators design. Particularly, we demonstrate that the metasurface is capable of steering a normally incident beam to an angle as high as 40 degrees, while having Joule losses not exceeding 10 % and radiation eciency above 98

%.

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Theory In order to design the achromatic anomalous mirror, we use the metal-insulator-metal (MIM) resonators as building blocks (see Fig. 1a,b). The wave impinging on this resonator excites a gap plasmon, e±iβz u(x), where β is the plasmon wavenumber and u(x) is the plasmon electric eld distribution in transversal plane (see Fig. 1b). An additional phase acquired by the plasmon, propagating back and forth inside the resonator before re-radiating its energy back to space, is (see SI and Ref. 46 for details)

s

Φ ≈ 2h(x)Reβ = k0 2h(x) εd (x) 1 +

2c a(x)ωp

(6)

where h and a are the resonator length and width, respectively, εd is a relative permittivity of dielectric inside the resonator gap, ωp is a plasma frequency of the resonator metal walls. p Note that Eq. (6) is only valid if a β 2 − k02 εd 1, and ωp ω, γ , where γ is the electron relaxation frequency in the metal. In what follows we consider silver resonators for which 47

ε∞ = 1, ωp = 1.37 × 1016 s−1 , γ = 2.73 × 1013 s−1 . As the plasma frequency of silver falls into UV frequency range, Eq. (6) is valid in IR and visible frequency ranges (see SI for details). (b) (a) Er

d dielectric

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z eiβzu(x)

θr Einc x

hi

z

x

metal

εm εd εm a

Figure 1: (a, b) Geometry of an aperiodic metasurface for steering a normally incident beam at an angle θr with respect to the surface normal. The metasurface is build from N MIM resonators (panel (b)) of length hi and has width D = N d, where d is a width of the individual resonator ( a is width of a dielectric in each MIM). Relative permittivities of metal and dielectric are εm and εd , respectively. An incident wave, Einc , excites gap plasmons in each of the resonators, with the plasmon wavenumber, β , and electric eld distribution of a form e±iβz u(x). As one can see from Eq. (6), there are several dierent ways to create the phase gradient, 5


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(a)

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(b)

λ=400 nm

(c) visible

max

λ=2000 nm min

(d)

near-IR

(e)

(f)

Figure 2: Reection of a normally incident Gaussian beam, dened by Eq. (7), by a metasurface consisting of N = 450 resonators of width d = 80 nm each. The total metasurface width is D = N d = 36µm. The metasurface is designed to steer beam by an angle θr ≈ 17o in the wavelength range from λ = 400 nm till 2000 nm. α = 0.005 µm−2 . (a, b) Spatial distribution of x-component of electric eld intensity in a reected wave for wavelengths λ = 400 nm (a) and λ = 2000 nm (b), assuming that the gap width a = 60 nm. (c) An angle of reection, θr , as a function of wavelength for dierent gap widths a. (d) Angular distribution of intensity of the reected wave for dierent wavelengths. a = 60 nm. (e) Radiation eciency of the metasurface, calculated using Eq. (8) and δθ = 4◦ . (f ) Joules losses in the metasurface (see Eq. (9))

Φ(ω, x). We can use dierent dielectric materials inside the resonators along the mirror surface (i.e. to vary εd (x)). However, this approach is not ecient as the permittivity of the dielectric material should be close to the permittivity of the material above the metasurface in order to suppress specular reection, which imposes considerable restrictions on the variation of εd (x). Alternatively, we can vary gap widths, a(x). The parameter space, we can explore

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in this case, is also quite limited as the gap width should be much smaller than the radiation wavelength. Otherwise, we have plasmons propagating along the resonator walls instead of concentrating their electric eld in the resonator gap, which leads to high losses. Moreover, the resonator width cannot be too small either in order to prevent specular reection from metal. In this paper we pursue the third approach, depicted in Fig. 1a, i.e. we vary the resonators lengths, h.

Results and discussion All the results presented in this section were obtained assuming that the minimum length of the resonators in the mirror is hmin = 200 nm, while the maximum length depends on the number of the resonators N . Simulations were made using COMSOL 5.2. We use a normally incident Gaussian beam, 2

Ei (x, z) = ex E0 e−ikz e−αx ,

(7)

to illuminate the mirror. The electric eld of the beam has to be polarized along the resonator width (i.e. x-axis, see Fig. 1a) in order to excite the gap plasmon, k = k0 n, and the parameter

α denes spatial width of the wave, i.e. the beam spot size. The simulation results are presented in Fig. 2 for the metasurface designed to steer the beam by an angle θr ≈ 17o . We assume the normally incident gaussian beam with α = 0.005

µm−2 (corresponding to a spot size of about 30 µm). The mirror consists of N = 450 silver-air-silver resonators, each d = 80 nm wide, and has total width D = N d = 36 µm. We consider three dierent gap widths, a = 50, 60, and 70 nm. The spatial distribution of x-component of the scattered electric eld is presented in Figs. 2a,b. One can clearly see that wavefront of the reected wave propagates at an angle with respect to the normal to the metasurface 1 and not the scattered electric eld. The spatial distribution of total 1 Note

that scattered electric eld, Esc , in Figs. 2a,b is non-zero inside the metal. This is because it is

7


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electric eld is presented in SI. The angle of anomalous reection is presented in Fig. 2c, as determined from the angular distribution of the intensity carried by the reected wave, shown in Fig. 2d. Here we took into account the fact that Poynting vector of a plane wave is proportional to |E|2 . The results presented in Fig. 2 clearly demonstrate that the metasurface is achromatic for the wavelengths between 400 nm and 2000 nm. Indeed, for all three gap widths, dierence in steering angles at 2000 nm and 500 nm is less than 0.5 ◦ (see Fig. 2c). This dierence increases around 400 nm (as the approximation dened by Eq. (6) becomes less reliable (see SI)), but even in this case the dierence does not exceed 1 ◦ for gap widths a = 50 and 60 nm. When the gap width becomes suciently large ( a = 70 nm), however, the dierence in reection angles at 2000 nm and 400 nm becomes as large as 3◦ . The non-zero angular width of the reected beam, that observed in Fig. 2d, is due to the spatial localization of the beam. One can see that the angular spread decreases with the decrease of the wavelength. Indeed, the direction of the energy ow in plane wave is dened by wavevector k. The Fourier decomposition of the Gaussian beam dened by Eq. (7), apart from the main component kez , contains additional components, kez + ∆kex , that carry energy at some angle with respect to the normal to the mirror plane. The angular spread of the incident beam ∆k is proportional to λ/∆x , i.e. the ratio between the radiation wavelength and spot size of the beam, ∆x ∝ 1/α. The reection of these obliquely incident components causes angular spread of the reected beam. In order to characterize performance of the mirror, we calculated absorption losses and radiation eciency. We dene the radiation eciency as the ratio between the intensity of the radiation scattered by the metasurface in a given angle to the total intensity of the scattered radiation, i.e. θrR+δθ

η=

θr −δθ 360 R◦

|Ef (θ)|2 dθ ,

|Ef

(8)

(θ)|2 dθ

0◦

total electric eld, Etot = Ei + Esc , that decays fast when moving away from the metal surface and is almost zero beyond the skin-depth range

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where we took into account that Poynting vector of a plane wave, and thus the intensity of radiation in a given angle, is proportional to |Ef (θ)|2 , where Ef (θ) is the angular distribution of the electric eld intensity in the far-eld. Angle δθ accounts for the angular spread of the reected beam discussed above. Thus, the radiation eciency accounts for the losses due to the specular reection and wave scattering into higher diraction orders. However, we also need to account for the fraction of the incident radiation absorbed by the metal in the mirror, which is converted into heat. We account for this by dening the Joules losses as the ratio between power Q converted into heat and total power delivered to the metasurface by the incident beam, i.e.

J=

Q , Pin

(9)

where Q is calculated numerically using COMSOL, while incident power is estimated as Pin = R (1/2η) S |Ei |2 dS , where η is the wave impedance in the medium above the metasurface, and

S is the surface of the metasurface. Calculated radiation eciency and Joules losses are presented in Fig. 2e,f. One can see (Fig. 2e) that radiation eciency of the metasurface can be as high as 99 % in a broad frequency range, when the gap width, a = 70 nm, is close to the resonator width, d = 80 nm. The radiation eciency, however, gradually decreases with decrease of the gap width (η ≈ 96% for a = 50 nm) due to increase in specular reection from the metal. The absorption losses (see Fig. 2f), on the other hand, increase with the decrease of the gap width. This can be attributed to increase in plasmon losses as, in the case of narrower gaps, electric eld of the plasmon penetrates deeper into the metal. The deeper penetration of the electromagnetic eld into the silver is also responsible for increase of the absorption losses for wavelengths above 800 nm (i.e. for small values of a/λ). The absorption losses increase for shorter wavelengths as silver itself gets more absorptive and less reective in the optical range. Thus, we conclude that in the general case it is benecial to use resonators with large gap width, as they tend to have lower absorption losses and higher radiation eciencies. In particular, for a metasurface operating in the range between 500 nm and 2000 nm, gap width 9


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a = 70 nm is about optimal. However, in order to extend the metasurface functionality up to 400 nm, resonators with the smaller gap width of a = 60 nm provide better trade-o, as they can alleviate high absorption losses.

(a)

(c)

θr = 28o (b)

θr = 39o

(e)

(d)

Figure 3: Reection of a normally incident Gaussian beam, dened by Eq. (7), by a metasurface consisting of N = 450 resonators of width d = 80 nm each. Gap width a = 60 nm. α = 0.005 µm−2 . (a, b) Spatial distribution of x-component of electric eld intensity in a reected wave at the wavelength λ = 2000 nm for angles of reection θr = 28◦ (a) and θr = 39◦ (b). (c,d,e) Wavelength dependence of the reections angle θr (c), the radiation eciency (d), and the absorption losses (e). As a next step we assess the possibility for steering the normally incident beam to angles larger than 17◦ . This requires creating larger phase gradient along the metasurface, which can be implemented by increasing dierence between maximum, hmax , and minimum, hmin , lengths of the MIM resonators, while keeping number of the resonators, N , and their widths,

d, the same. The simulation results are presented in Figure 3 for three dierent steering angles, θr = 17◦ (hmax = 4.6µm), θr = 28◦ (hmax = 7.6µm), and θr = 39◦ (hmax = 10µm). The minimum length of the resonators was the same in all three cases, i.e. hmin = 0.2µm. One can see from Fig. 3 that for wavelengths between 600 nm and 2000 nm the mirror is achromatic, has the radiation eciency as high as 95 % and the Joules losses smaller than 10


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10% even if the reection angle is as high as 40◦ . At the shorter wavelengths the reection angle deviates signicantly from that at 2000 nm (around 5 ◦ for θr = 39◦ ), moreover, the absorption losses increase drastically (more than 20 % around 400-500 nm for θr = 39◦ ). In general, the absorption losses tend to increase with the increase of the reection angle as the achromatic mirror for large angles is built from longer resonators. This leads to longer travel distances for the gap plasmons, and correspondingly to higher attenuation. We can, however, conclude that the proposed design for the mirror can be used for ecient achromatic light steering at angles as large as 40 ◦ in the near-IR frequency ranges. Finally, let us dene total conversion eciency from incident beam to anomalously steered beam as η(1 − J). Here we consider radiation eciency as dened with respect to the total scattered intensity, and not to intensity of incident beam. Thus the term 1 − J accounts for amount of incident beam power that was not lost to Joule heating. In the frequency range between 500 and 2000 nm, and for steering angles up to 40 degrees, the conversion eciency of the mirror is above 90%. It is instructive to compare this eciency with the results available in the literature. For example, the conversion eciency of about 80 % was reported 6 in the frequency range between 750 and 900 nm for steering angles around 40 degrees. The device reported in Ref. 6 suers from chromatic aberrations, as the steering angle depends on the radiation frequency. Comparable eciency (50-80 %, frequency range 750-900 nm) was also reported 7 for plasmonic mirror that focuses linear polarized light with focal length between 11 and 15 um. Alternatively, reectarrays based holograms operating at wavelength 825 nm with diraction eciencies as high as 80 % were reported in Ref. 43 High Joule losses, especially in the visible frequency range, is signicant drawback of plasmonic metasurfaces. As a viable alternative, all-dielectric metasurfaces were proposed 8 with steering eciency in the transmission regime as high as 75 % at a wavelength 500 nm. Linear polarization conversion with the eciency as high as 98 % in the frequency range between 1500 and 1600 nm was also implemented 14 using array of Si resonators on silver plane. Thus conversion eciency of the anomalous mirror proposed in the paper is at least as high as that of those

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demonstrated in the literature, while exhibiting much broader operating frequency range. The design of achromatic mirror proposed so far relied on utilizing aperiodic metasurface for achromatic beam steering. The natural question, however, is whether it is possible to create periodic achromatic mirror utilizing aperiodic chunks as building blocks. Indeed, in the case of mirror that steers beam at 17◦ angle (see Fig. 2), the smallest resonator had length 0.2µm and a wave impinging on this resonator experiences phase change of about

2Reβhmin ≈ 0.53π (at the wavelength 2000 nm). On the other hand, the wave impinging on the longest resonator of length 4.6µm experiences phase change of about 12.14π . Thus the total phase change along the mirror signicantly exceeds 2π and it seems that periodic mirror can be created from smaller aperiodic chunks providing 2π phase dierence. The problem, however, arises from the fact that the phase change experienced by the wave interacting with a single resonator is wavelength dependent (see Eq. (6)). In order to clarify this point, let us assume that we have an aperiodic mirror (consisting of N resonators of width d) that provides phase change equals to 2π at a wavelength λ1 . This condition obviously imposes restriction on the maximum and minimum lengths of the resonators in the metasurface such p εd (1 + 2c/aωp ). It is obvious, that for a given as k1 2(hmax − hmin )l = 2π , where l = geometry the above condition can be satised only at a single wavelength. However, we can reformulate this condition in more general form kn 2(hmax − hmin )l = 2πn, or

λn = 2(hmax − hmin )l/n.

(10)

Thus periodic metasurface can operate at multiple discrete wavelengths. In order to illustrate this point in more details we did simulations for the metasurface built from two aperiodic identical arrays of resonators placed side by side (see Fig. 4). Each of the arrays contained N = 300 resonators and was designed to steer beam at an angle

θr = 17◦ . Simulations results are presented in Fig. 4. One can clearly see distortion in the reected wavefront at some wavelengths ( λ = 1250 nm, Fig. 4a). This distortion is due to

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(a)

(b)

λ=1250nm

(c)

(d)

(e)

(f)

λ=1350nm

Figure 4: Reection of a normally incident Gaussian beam, dened by Eq. (7), by a metasurface consisting of two aperiodic arrays of the resonators containing N = 300 resonators each. The resonators width d = 80 nm each. Gap width a = 60 nm. α = 0.005 µm−2 . Each of two arrays is designed to steer beam by angle θr ≈ 17o . (a, b) Spatial distribution of x-component of electric eld intensity of reected wave at wavelengths λ = 1250 nm (a) and λ = 1350 nm (b). (c) Angle of reection, θr , as a function of wavelength. (d) Angular distribution of intensity of reected wave for dierent wavelengths. (e) Radiation eciency of the metasurface, calculated using Eq. (8) and δθ = 4◦ . (f ) Joules losses in the metasurface (see Eq. (9)). the fact that at these wavelength total phase dierence across the aperiodic array of 300 resonators is not multiple of 2π . On the other hand there is no distortion of the wavefront at

λ = 1350 nm, where the phase dierence is multiple of 2π . The distortion of the wavefront leads to the splitting of the reected beam (see Fig. 4d, λ = 1250 and 1500 nm) into two slightly detuned angular directions (Fig. 4c). This in turn leads to strong oscillations of the radiation eciency of the metasurface. For example, at λ = 1500 nm, the eciency drops to 13


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60% as the 40% of the radiation is carried by the second beam (see Fig. 4b). Nevertheless, as was predicted by Eq. (10), there is a discrete set of frequencies for which the metasurface produces the single beam ( λ = 1700, 1350, 1150 and 1000 nm) and the radiation eciency of the metasurface exceeds 95% at these frequencies.

Conclusions Concluding, we presented the design of an achromatic anomalous mirror operating in the near-IR and visible frequency ranges. We used an array of metal-insulator-metal (MIM) resonators as building blocks for the anomalous mirror. An electromagnetic wave impinging on the MIM resonator launches plasmons propagating inside the resonator gap and phase change experienced by the wave is proportional to the plasmon optical path. A phase gradient can then be created along the mirror by using array of resonators of dierent lengths. When the frequency of the electromagnetic wave is much smaller than the plasmon frequency in the metal, the plasmon propagation constant is a linear function of frequency, which is a sucient condition for the achromatic anomalous reection. We demonstrated that the mirror comprised from silver-air-silver resonators can sustain achromatic operation in the visible and near-IR frequency ranges. In the case of a normally incident Gaussian beam, Joule losses in such a mirror do not exceed 10 %, while radiation eciency exceeds 98 % for steering angles as high as 40 ◦ .

Acknowledgement This work is supported by a DARPA grant award FA8650-16-2-7640. We acknowledge useful discussions with Predrag Milojkovic, Jay Lewis and Ramzi Zahreddine.

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Supporting Information Available Derivation of Eq. (6) for the phase accumulated by gap plasmon while propagating in metaldielectric-metal resonators; distribution of the total electric eld for the cases presented in Fig. 2; performance characteristics of the metasurface accounting for the fabrication induced uctuations in the resonators lengths; performance characteristics of the metasurface for steering angles exceeding 40 ◦ .

This material is available free of charge via the Internet at

http://pubs.acs.org/ .

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