All-Dielectric Metasurface for Achieving Perfect Reflection at Visible

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All-Dielectric Metasurface for Achieving Perfect Reflection at Visible Wavelengths Yali Huang, Haixia Xu, Yanxin Lu, and Yihang Chen J. Phys. Chem. C, Just Accepted Manuscript • DOI: 10.1021/acs.jpcc.7b10417 • Publication Date (Web): 17 Jan 2018 Downloaded from http://pubs.acs.org on January 18, 2018

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The Journal of Physical Chemistry

All-Dielectric Metasurface For Achieving Perfect Reflection At Visible Wavelengths Yali Huang1, Haixia Xu2, Yanxin Lu1 and Yihang Chen1,* 1

Guangdong Provincial Key Laboratory of Quantum Engineering and Quantum Materials,

School of Physics and Telecommunication Engineering, South China Normal University, Guangzhou 510006, China. 2

School of Information Science and Technology, Zhongkai University of Agriculture and

Engineering, Guangzhou 510225, China. ABSTRACT: Metamaterials have attracted considerable attention owing to their extraordinary ability in controlling the propagation of electromagnetic waves. These materials can be realized using artificial composites consisting of subwavelength metallic resonators, but losses of the metallic components may significantly degrade the performance of metamaterials, especially in the visible region. Here, we propose low-loss all-dielectric metasurfaces, comprised of a monolayer of titanium dioxide (TiO2) nanoparticles, to achieve perfect reflection band at visible wavelengths. Using the Mie scattering theory, we explore the electromagnetic scattering features of one single TiO2 nanosphere and show that both electric and magnetic dipole resonances can be excited inside the sphere in the visible range. Then, a semi-infinite medium of TiO2 nanospheres is studied using Lewin effective-medium model and we find that the effective permeability or permittivity becomes negative around the magnetic or electric resonance wavelength, leading to the perfect reflection of light. Based on these results, we design a

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monolayer of TiO2 nanocylinder array to achieve a flat-top perfect reflection band by optimizing the wavelength interval between the magnetic and electric resonances. In addition, it is shown that the position of the perfect reflection band can be adjusted across the whole visible spectrum by changing the dimensions and lattice period of the TiO2 nanocylinder array. Our design of alldielectric metamaterial reflectors may find applications in diverse fields such as filter, color printing, spectroscopy, and so on. INTRODUCTION Metamaterials are artificial structures engineered at scales that are smaller than their functional wavelengths. They can be designed to exhibit many unprecedented electromagnetic phenomena such as negative refraction,1,2 near zero refraction,3,4 hyperbolicity,5 and optical chirality.6,7 The most common designed metamaterials are composed of metallic split-ring resonators8 and metallic cut-wire array,9 providing the magnetic and electric dipole resonances, respectively. These metallic constituent elements work well at gigahertz, terahertz, and near-infrared frequencies. However, metals have relatively high absorption loss at optical frequencies, which limits the applications of the metal-based metamaterials in photonics. Recently, it has been demonstrated that dielectric metamaterials composed of low-loss dielectric nanoparticles can replace the metallic counterparts to achieve the electric or magnetic resonances at visible or near-infrared region.10,

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Similar to metallic split-ring resonators, a

dielectric particle with relatively high permittivity can produce a strong magnetic resonance resulted from the excitation of circular displacement currents. The first experimental observation of the strong magnetic dipole resonance at visible frequencies was reported for Si nanospheres with radii in the range of 50-100 nm.12 Manipulation of the optically induced Mie resonances in high-refractive-index dielectric nanoparticles may lead to superior performance which is difficult


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to be achieved in conventional metamaterials. For example, metasurfaces with high transmission efficiency were realized using arrays of Si nanodisks in which electric and magnetic dipole resonances are overlapped.13,14 Another example is that low-loss metasurfaces with highreflection were achieved by the periodic arrangement of Si nanospheres or nanocylinders with spectrally separated electric and magnetic dipole resonances at near-infrared communication band.15-17 However, Si has a relatively large extinction coefficient in the visible range, which degrades the performance of the Si-based metamaterial reflectors at visible wavelengths. It is well known that metals, such as gold and silver, can serve as reflectors. But they cannot attain perfect reflection because of the inevitable absorption loss. Bragg reflectors consisting of multilayer stack of high and low-index dielectric films can reflect almost all of the incident light within the photonic band gaps. However, the thicknesses of the Bragg reflectors are required to be larger than the wavelength to achieve functionality.18-20 It is desired to obtain a perfect reflector which not only operates in the visible range but also has a subwavelength scale thickness. Recently, it was demonstrated that TiO2 is almost lossless in the entire visible region and can be used to fabricate metasurfaces in the form of holograms for visible wavelengths with high efficiency.21 The refractive index of TiO2 is shown in Figure 1a.21 It can be seen that TiO2 has a relatively large refractive index n ≈ 2.5 and a near zero extinction coefficient in the visible spectrum, which makes TiO2 suitable for the design of optical metamaterials. Consequently, we propose a dielectric metasurface composed of an array of TiO2 nanoparticles, as shown in Figure 1b, to achieve perfect reflection at visible wavelengths. In this paper, we first theoretically study the scattering properties of single TiO2 spherical particle and demonstrate that both magnetic dipole and electric dipole responses can be excited in the visible region. Next, we use Lewin effective medium theory to calculate the effective


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permeability, effective permittivity and reflectance of a semi-infinite medium of periodic TiO2 nanospheres and we show that perfect reflection can be achieved at the resonance wavelengths. Then, we use a monolayer of TiO2 nanocylinder array to design the metasurface reflector, the thickness of which is much smaller than its operating wavelengths. Finally, we present that the position of the perfect reflection band can be varied within the whole visible spectrum by changing the geometrical parameters of the metasurface.

Figure 1. (a) Measured refractive index of TiO2 from Ref. (21). Blue squares and black circles are, respectively, the refractive index n and the extinction coefficient k at different wavelengths. (b) Schematic of the metasurface perfect reflector composed of TiO2 nanoparticle array.

OPTICAL RESPONSE OF SINGLE TiO2 SPHERICAL PARTICLE Scattering characteristics of single dielectric nanoparticle can be investigated by the analytical solution of Maxwell’s equations (i.e. Mie theory).22 The scattered fields of an isolated dielectric sphere with refractive index n and radius r can be decomposed into electric and magnetic multipolar modes. The 2m-pole mode of the scattered electric field is proportional to22 nψ m (nα )ψ m' (α ) −ψ m (α )ψ m' (nα ) am = nψ m (nα )ξ m' (α ) − ξ m (α )ψ m' (nα )

(1)


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and the 2m-pole mode of the scattered magnetic field is proportional to

ψ m ( nα )ψ m' (α ) − nψ m (α )ψ m' ( nα ) . bm = ψ m ( nα )ξm' (α ) − nξm (α )ψ m' ( nα )

(2)

Here α = 2πr/λ is a size parameter, ψm(α) and ξm(α) are the Riccati-Bessel functions.22 The total scattering efficiency Qscat and backward scattering efficiency QBS can be written as Qscat =

QBS =

2 k r

2 2

∑

∞ m =1

(2m + 1)(|am |2 +|bm |2 ),

∞ 1 |∑ m =1 ( −1)m (2m + 1)( am − bm ) |2 , k r 2 2

(3)

(4)

Then, we study the scattering properties of a single TiO2 sphere using the Mie theory, as shown in Figure 2. Here, the refractive index of TiO2 is assumed as n = 2.5. It can be seen from the Figure 2a that the total scattering efficiency Qscat approximately equals to the sum of the contributions from electric dipole (Qed), magnetic dipole (Qmd), electric quadrupole (Qeq), and magnetic quadrupole (Qmq). Two dominant peaks of Qscat are observed and identified as the resonant excitation of the magnetic dipolar and magnetic quadrupolar modes. The dependence of the backward scattering efficiency QBS on the size parameter α is also calculated, as shown in Figure 2b. It is seen that QBS reaches maximum when α = 2πr/λ = 1.33. From this result, it is obtained that large backward scattering occurs at about 500 nm when the radius of the TiO2 sphere is set to be around 100 nm. Figures 2c and 2d show the scattering properties of the TiO2 sphere with r = 100 nm as a function of wavelength. It can be noted from Figure 2c that an electric dipole resonance arises at 441 nm and a magnetic dipole resonance exists at 525 nm. Moreover, there are two intersections between the two curves of the electric dipole coefficient a1 and the magnetic dipole coefficient b1, as depicted in Figure 2d. The red intersection point corresponds to the first Kerker condition, meaning that the electric and magnetic dipoles have the


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same amplitude and they oscillate in-phase. On the other hand, the black intersection point corresponds to the second Kerker condition, for which the electric and magnetic dipoles have the same amplitude but they oscillate out-of-phase.23,24 In fact, at the second Kerker point at λ = 477.9 nm, the size parameter is α = 1.31, which means that the backscattering cross section of the TiO2 sphere reaches nearly maximum according to the result in Figure 2b. To realize perfect reflection at visible spectrum, we then focus on the array of TiO2 spheres with the radius round 100 nm.

Figure 2. (a) Total scattering efficiency Qscat and partial scattering efficiency (contributed from electric dipole Qed, magnetic dipole Qmd, electric quadrupole Qeq, and magnetic quadrupole Qmq) as a function of size parameter α =2πr/λ for a TiO2 sphere. (b) Backscattering efficiency QBS vs size parameter α, exhibiting a maximum at α = 1.33. (c) The scattering efficiencies and (d) the dipole amplitude coefficient a1 and the magnetic dipole amplitude coefficient b1 vs wavelengths for a TiO2 sphere with r = 100 nm. The red circle at λ


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= 572.5 nm corresponds to Re(a1) = Re(b1) and Im(a1) = Im(b1), meaning that the first Kerker condition is satisfied. The black circle at λ = 477.9 nm where Re(a1) = Re(b1) and Im(a1) = -Im(b1), corresponds to the second Kerker condition.

SEMI-INFINITE MEDIUM COMPOSED OF TiO2 NANOSPHERES We then consider a semi-infinite dilute medium composed of a cubic lattice of TiO2 spheres in air background. The effective permittivity εeff and permeability µeff of such a composite can be calculated by Lewin’s effective medium theory. It should be noted that the Lewin effectivemedium model, which considers the optical response of a periodic array of lossless dielectric sphere (with permittivity ε2 and permeability µ2) embedded inside a background media (with permittivity ε1 and permeability µ1), is valid for small dielectric particles and dilute loading.25 Using the Lewin effective-medium model, the effective parameters of the considered composite can be obtained by25

ε eff = ε（ 1

µeff = µ（ 1

3v f ）, F (θ ) + 2be − vf F (θ ) − be

3v f ）, F (θ ) + 2bm − vf F (θ ) − bm

(5)

(6)

where F (θ ) =

2( sinθ − θ cosθ ) , (θ − 1)sin θ + θ cosθ 2

be = ε1 / ε 2 , bm = µ1 / µ2 .

(7) (8)

4 r The ratio of volume is v f = π ( )3 , θ = k0 r ε 2 µ 2 , a is the lattice constant. In case of normal 3 a incidence, the reflectance for the semi-infinite medium can be written as


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(Z '− 1)2 + Z "2 R= , (Z '+ 1)2 + Z "2

where the complex impedance Z = Z '+ iZ " =

(9)

µeff µ 'eff + µ "eff = . To achieve perfect ε eff ε 'eff + ε "eff

reflection, i.e. R=1, two conditions should be satisfied,16 they are

ε 'eff < 0 and µ 'eff

ε "eff µ 'eff − µ "eff ε 'eff = 0 . The first condition can be met by single-negative (negative permittivity or negative permeability) metamaterial which is complete impedance mismatch with air. For dielectric metamaterials, negative effective permeability can be achieved near the magnetic dipole resonance frequency while negative effective permittivity can be observed near the electric dipole resonance frequency.26, 27 The second condition is conveniently satisfied by lossless material. As discussed above, TiO2 nanoparticles are suitable for constructing metamaterial with perfect reflection because they are nearly lossless and may possess electric or magnetic responses at visible region. Here, we assume that TiO2 spheres with radius of 100 nm are arranged in a cubic lattice with a period a = 360 nm in air background. Using Equations (5) – (9), the effective parameters and the reflectance of this composite are calculated and shown in Figure 3. It is seen that resonances of εeff and µeff occur at 420 nm and 470 nm, respectively. The reflectance can reach almost 1 near the resonance frequencies, which means that TiO2-based metamaterial has the capability to achieve perfect reflection at visible frequency range.


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Figure 3. (a) Calculated effective permittivity, effective permeability, and (b) reflectance as a function of wavelength for a semi-infinite medium composed of TiO2 sphere with r = 100 nm, arranged in cubic lattice with periodicity a = 360 nm.

MONOLAYER OF TiO2 NANOPARTICLE ARRAY For the potential applications in compact optical devices, we next investigate the reflection properties of a monolayer of TiO2 nanoparticles. We first perform Finite-Difference TimeDomain (FDTD) simulations on a monolayer of TiO2 sphere array embedded in air. Here, the TiO2 spheres with r = 100 nm are arranged in a square array with lattice constant a = 360 nm. FDTD SOLUTIONS, a commercial 3D FDTD-method Maxwell solver, was used to conduct the simulations. Here, the incident light propagates along the z direction and the polarization is along x direction. The simulated reflection spectrum of the TiO2 sphere array is shown in Figure 4a. It is seen that two perfect reflection peaks appear at 430 nm and 490 nm, respectively. Recent studies have showed that Si nanospheres can exhibit strong magnetic and electric dipolar resonances at near-infrared and visible wavelengths.12,

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Large-scale Si-based metamaterial

perfect reflector has been fabricated and it works well in the telecommunication band where the


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absorption of Si is negligible.15 However, the absorption loss of the Si-based nanostructures increases significantly at visible wavelengths. For comparison, we calculated the reflection spectrum for an infinite 2D tetragonal lattice array of Si spheres using simplified coupled-dipole model29-31 and FDTD method, respectively, as shown in Figure 4b. Here, the radius of the Si spheres is 55 nm and the array lattice constant is 200 nm. Although there is some difference between the theoretical and the FDTD simulated results, both of them have similar trends. As shown in Figure 4b, electric and magnetic resonances emerge at 420 nm and 490 nm, respectively. However, the Si sphere array cannot achieve perfect reflection in the visible range because it absorbs a part of the optical fields. It is seen from Figure 4b that the maximum reflectance of the Si sphere array can only reach about 0.55. In comparison, the reflectance of the TiO2 nanospheres is significantly higher than that of the Si nanospheres within the range from 430 to 490 nm. It should be noted that the lattice constant of the TiO2 sphere array is close to the resonance wavelengths, which makes the simplified coupled-dipole model invalid. Thus, the theoretical result for the reflectance of the monolayer of TiO2 spheres is not given.


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Firure 4. (a) Simulated reflection spectra of a monolayer of TiO2 sphere array with r = 100 nm and a = 360 nm. (b) Reflection spectra of Si sphere array with r = 55 nm and a = 200 nm obtained from couple-dipole approach and FDTD simulation, respectively.

For spherical particles, the wavelengths of the dipole responses can only be varied by changing the sphere radius. In comparison, cylindrical particles can provide an additional freedom to adjust the wavelength interval between the electric and magnetic resonances by altering their aspect ratio. So next we investigate the reflection properties of a monolayer of TiO2 cylinder array, as shown in Figure 5a. The simulated reflection spectrum for a square array of TiO2 cylinders with radius rc = 100 nm, height H = 150 nm, lattice constant a = 360 nm under normal incidence is shown in Figure 5b. It is seen that perfect reflection peaks appear at 448 nm and 486 nm, respectively. According to our simulated results, electric response occurs at 448 nm while magnetic response exists at 510 nm. Figures 5c and 5d, respectively, show the electric field

E and magnetic field H distributions at λ = 510 nm inside one periodic unit cell of the proposed metasurface. We can see that both the electric and magnetic fields are mainly localized inside the nanocylinders. The electric field loop forms in xz-plane, resulting in a strong magnetic field along y axis. Figure 5e and 5f show the field distributions at 448 nm. It can be seen that a large electric field along x axis and a corresponding ring-shaped magnetic field vector on yz-plane exist inside the nanocylinders. These are clearly equivalent to an electric dipole mode. These results verify that the perfect reflection peaks are mainly due to the electric dipole and magnetic dipole resonances of the TiO2 nanocylinders. The difference between the frequencies of the reflection peaks and those of the dipole resonances are attributed to the influence of the multipole resonances. These results verify that we can realize nearly perfect reflection using TiO2 nanocylinder array.


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Figure 5. (a) Schematic of an (infinite) array of TiO2 nanocylinders arranged in square lattice embedded in air.(b) Simulated reflection spectrum of the TiO2 nanocylinder array with radius rc = 100 nm, hight H = 150


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nm, and lattice constant a = 360 nm under normal incidence. (c) The electric field distribution in xz-plane and (d) the magnetic field distribution in yz-plane at λ = 510 nm for the TiO2 nanocylinders. (e) The electric field distribution in xz-plane and (f) the magnetic field distribution in yz-plane at λ = 448 nm for the TiO2 nanocylinders,

Finally, we investigate the reflection properties of the TiO2 nanocylinder array with different structural parameters. Here, we fix the aspect ratio ϒ= H / (2rc) = 0.59 and the lattice period a = 2rc + 140 nm so that the electric and magnetic dipole resonances are separated with a suitable wavelength interval, leading to a spectrally flat reflection band, as shown in Figure 6. It is seen from Figure 6a that a perfect reflection band exists from 460 nm to 476 nm when the diameter of the TiO2 nanocylinder is rc = 110 nm. As rc increases, the reflection band shifts to longer wavelengths. As shown in Figures 6b, 6c, and 6d, the perfect reflection band appears at around 545 nm, 610 nm, and 675 nm when rc is chosen is 130 nm, 150 nm, and 170 nm. It means that one can design the TiO2 cylinder array to realize perfect reflection band at any position within the visible region.


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Figure 6. Numerical simulated reflection spectra for a monolayer of TiO2 nanocylinder array with an aspect ratio ϒ = H / (2rc) = 0.59 and a lattice period a = (2rc) + 140 nm. The structural parameters are (a) rc = 110 nm, H = 130 nm, a = 360 nm; (b) rc = 130 nm, H = 153 nm, a = 400 nm; (c) rc = 150 nm, H = 177 nm, a = 440 nm: (d) rc = 170 nm, H =201 nm, a = 480 nm.

CONCOUSIONS In summary, we reported a novel design of all-dielectric perfect reflector composed of TiO2 nanoparticles. The optical scattering characteristics of the TiO2 nanoparticles were investigated using the Mie scattering theory and Lewin effective-medium model, respectively. We showed that electric dipole and magnetic dipole resonances can both e excited in the visible region and they can significantly enhance the backscattering efficiency of the TiO2 nanoparticles, leading to the perfect reflection of the incident light. By adjusting the dimensions and lattice constant of a monolayer of TiO2 nanocylinders, flat-top reflection band could form and its frequency position


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can be tuned across the visible spectrum. Our results offer a new way for the design of ultrathin perfect reflectors which can be used for filter, spectroscopy, subwavelength color printing, and so on. METHODS

Theory and simulation methods. The scattering efficiency Qscat and the dipole coefficient of a single TiO2 nanosphere in Figure 2 were calculated using the Mie scattering theory. The effective permittivity, effective permeability, and reflectance of a semi-infinite dilute medium of TiO2 spheres in Figure 3 were calculated using Lewin effective-medium model, following the method detailed in Ref. (25). The results in Figure 4 were obtained from couple-dipole approach in Ref. (29) and FDTD simulation, respectively. The results in Figure 4, Figure 5 and Figure 6 were obtained by FDTD simulations. AUTHOR INFORMATION

Corrseponding Author * E-mail : [email protected]

Notes The authors declare no competing financial interest ACKNOWLEDGMENTS This work is supported by National Natural Science Foundation of China (Grant No. 11274126) and Natural Science Foundation of Guangdong Province (Grant Nos. 2015A030311018 and 2017A030313035). Y.-H.C. acknowledges financial support from Program for Guangdong Excellent Young Teacher.


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Figure 1. (a) Measured refractive index of TiO2 from Ref [21]. Blue squares and black circles are, respectively, the refractive index n and extinction coefficient k at different wavelengths. (b) Schematic of the metasurface perfect reflector composed of TiO2 nanoparticle array. 93x39mm (300 x 300 DPI)



Figure 2. (a) Total scattering efficiency Qscat and partial scattering efficiency (contributed from electric dipole Qed, magnetic dipole Qmd, electric quadrupole Qeq, and magnetic quadrupole Qmq) as a function of size parameter α =2πr/λ for a TiO2 sphere. (b) Backscattering efficiency QBS vs size parameter α, exhibiting a maximum at α = 1.33. (c) The scattering efficiencies and (d) the dipole amplitude coefficient a1 and the magnetic dipole amplitude coefficient b1 vs wavelengths for a TiO2 sphere with r = 100 nm. The red circle at λ = 572.5 nm corresponds to Re(a1) = Re(b1) and Im(a1) = Im(b1), meaning that the first Kerker condition is satisfied. The black circle at λ = 477.9 nm where Re(a1) = Re(b1) and Im(a1) = -Im(b1), corresponds to the second Kerker condition. 169x143mm (300 x 300 DPI)


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Figure 3. (a) Calculated effective permittivity, effective permeability, and (b) reflectance as a function of wavelength for a semi-infinite medium composed of TiO2 sphere with r = 100 nm, arranged in cubic lattice with periodicity a = 360 nm. 120x50mm (300 x 300 DPI)



Firure 4. (a) Simulated reflection spectra of a monolayer of TiO2 sphere array with r = 100 nm and a = 360 nm. (b) Reflection spectra of Si sphere array with r = 55 nm and a = 200 nm obtained from couple-dipole approach and FDTD simulation, respectively. 120x50mm (300 x 300 DPI)


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Figure 5. (a) Schematic of an (infinite) array of TiO2 nanocylinders arranged in square lattice embedded in air.(b) Simulated reflection spectrum of the TiO2 nanocylinder array with radius rc = 100 nm, hight H = 150 nm, and lattice constant a = 360 nm under normal incidence. (c) The electric field distribution in xz-plane and (d) the magnetic field distribution in yz-plane at λ = 510 nm for the TiO2 nanocylinders. (e) The electric field distribution in xz-plane and (f) the magnetic field distribution in yz-plane at λ = 448 nm for the TiO2 nanocylinders, 457x656mm (96 x 96 DPI)



Figure 6. Numerical simulated reflection spectra for a monolayer of TiO2 nanocylinder array with an aspect ratio ϒ = H / (2rc) = 0.59 and a lattice period a = (2rc) + 140 nm. The structural parameters are (a) rc = 110 nm, H = 130 nm, a = 360 nm; (b) rc = 130 nm, H = 153 nm, a = 400 nm; (c) rc = 150 nm, H = 177 nm, a = 440 nm: (d) rc = 170 nm, H =201 nm, a = 480 nm. 239x207mm (300 x 300 DPI)


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TOC graphic 668x307mm (72 x 72 DPI)


All-Dielectric Metasurface for Achieving Perfect Reflection at Visible

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