Broadband reflected wavefronts manipulation using structured phase gradient metasurfaces

Broadband reflected wavefronts manipulation using structured phase gradient metasurfaces Broadband reflected wavefronts manipulation using structured phase gradient metasurfaces Xiao Peng Wang, Le Le[.]

Xiao-Peng Wang, Le-Le Wan, Tian-Ning Chen, Ai-Ling Song, and Xiao-Wen Du

Citation: AIP Advances 6, 065320 (2016); doi: 10.1063/1.4954750

View online: http://dx.doi.org/10.1063/1.4954750

View Table of Contents: http://aip.scitation.org/toc/adv/6/6

Published by the American Institute of Physics

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AIP ADVANCES 6, 065320 (2016)

Broadband reflected wavefronts manipulation using

structured phase gradient metasurfaces

Xiao-Peng Wang,aLe-Le Wan, Tian-Ning Chen, Ai-Ling Song,

and Xiao-Wen Du

School of Mechanical Engineering and State Key Laboratory for Strength

and Vibration of Mechanical Structures, Xi’ an Jiaotong University, Xi’ an 710049,

People’s Republic of China

(Received 26 March 2016; accepted 13 June 2016; published online 20 June 2016)

Acoustic metasurface (AMS) is a good candidate to manipulate acoustic waves due

to special acoustic performs that cannot be realized by traditional materials In this paper, we design the AMS by using circular-holed cubic arrays The advantages

of our AMS are easy assemble, subwavelength thickness, and low energy loss for manipulating acoustic waves According to the generalized Snell’s law, acoustic waves can be manipulated arbitrarily by using AMS with different phase gradients

By selecting suitable hole diameter of circular-holed cube (CHC), some interesting phenomena are demonstrated by our simulations based on finite element method, such as the conversion of incoming waves into surface waves, anomalous reflections (including negative reflection), acoustic focusing lens, and acoustic carpet cloak Our results can provide a simple approach to design AMSes and use them in wave-front manipulation and manufacturing of acoustic devices C 2016 Author(s) All article content, except where otherwise noted, is licensed under a Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/) [http://dx.doi.org/10.1063/1.4954750]

I INTRODUCTION

The wavefronts manipulation of electromagnetic and acoustic waves has attracted more and more researchers’ attention in recent years Some previous studies have demonstrated that the bulk metamaterial can be used to manipulate electromagnetic and acoustic waves However, there are some disadvantages by using the bulk metamaterial, such as large size, high energy loss, and diffi-cult assembly Recently, researchers have focused on designing metasurfaces1 5to manipulate the electromagnetic waves at will By designing the structure of metasurface, we can obtain some novel phenomena, such as extraordinary refraction and reflection,2 , 4 , 5the conversion of incoming waves into surface waves,3and fabricate a flat lens and axicons.6

The acoustic waves are also investigated all the time Fang et al.7experimentally demonstrated that 1D acoustic waveguide composed of Helmholtz resonators can realize an effective negative modulus In order to further research this similar system, Wang et al.8numerically investigated the defect states in different parameters and demonstrated that point defect and line defect can play an important role in Bragg gap Lemoult et al.9 , 10experimentally demonstrated that acoustic waves can

be controlled and focused arbitrarily by using an array of soda cans as Helmholtz resonators, and found that both electromagnetic and acoustic waves can be manipulated by tailoring resonant unit cells The manipulation of acoustic waves has attracted increasing attentions with the emergence of AMS According to the generalized Snell’s law, the metasurfaces can easily manipulate the wave-front of electromagnetic waves Therefore, if the generalized Snell’s law and the theory of metasur-faces can be successfully applied in acoustics, it will improve the performance of existing acoustic devices We can use metallic antennas, e.g., ‘V’ antenna and ‘H’ antenna, as artificial resonant

a The e-mail address of the corresponding author: xpwang@mail.xjtu.edu.cn

structure to design electromagnetic metasurface.1 5But similar counterpart has not yet appeared

to design AMS So it is very difficult to design AMS Ma et al.11 proposed a method to realize AMS to absorb acoustic waves totally based on the impedance-matching concept by using acoustic metamaterial to design membrane structure which can result in hybrid resonance in a certain fre-quency range Besides, some researchers have demonstrated that the structure of coiling-up space or subwavelength corrugated surface can be used to design AMS with the phase change covering 0∼2π range.12–22 And then some novel phenomena are demonstrated, such as the anomalous reflection and refraction,12–18and the planar acoustic axicon and lens.19–22Ding et al.23,24have proposed that the split hollow sphere and double-split hollow sphere, as acoustic resonator, can be also used

to construct AMS However, they may not be effective in a broad frequency range or at a given frequency In practice, they are also difficult to fabricate and assemble

In this paper, we present a new method by using some CHCs to design AMS, which can manipulate the acoustic waves at a broadband frequency range near the designed center frequency Compared with Refs.7and8, our AMS is comprised of a two-dimensional array of subwavelength Helmholtz resonators and our study is focused on the phase gradient What is more, compared with Refs 9and10, our broadband reflected wavefronts manipulation is performed by controlling the phase change according to the generalized Snell’ law Besides, compared with Refs 23 and24, our CHC is easy to fabricate and assemble due to its simple geometric profile (the surface is a plane and the inside is a cylinder) In the following sections, we first introduce one CHC unit cell and use ten cells to realize discrete phase shifts from 0 to 2π with a step of π/5 in sectionII In section III, some interesting phenomena, such as the conversion of incoming waves into surface waves, anomalous reflections (including negative reflection), acoustic focusing lens, and acoustic carpet cloak, will be demonstrated by selecting suitable phase gradients Finally, a brief conclusion

is given in sectionIV

II MODELING AND SIMULATIONS

In this section, we investigate one unit cell whose surface is a cube and inside is a cylinder, so

we call the unit cell circular-holed cube A CHC is designed as shown in Fig.1, which can be used

to construct AMS The hole diameter of CHC should be selected in terms of designed frequrency

1500 Hz If the designed frequency changes, these hole diameter will be calculated again to match designed frequrency If the frequency of incident waves is closed to the natural frequency of CHC, a

FIG 1 Three-dimensional structure of CHC The length of the cube is w = 20 mm It has a cylindrical cavity with diameter

D = w-2t = 19 mm and height h = w-2t = 19 mm The diameter of the hole is d (d = 7 mm in this example) The minimum thickness of the cube shell is t = 0.5 mm.

065320-3 Wang et al. AIP Advances 6, 065320 (2016)

FIG 2 (a) Cross-sectional diagram of one CHC (b) The reflected phase with di fferent hole diameters of CHC at 1500 Hz The blue dots refer to selected d for ten cells with the phase step of π/5 (c) The designed AMS made of periodic reflected arrays with ten kinds of CHCs (d) The pressure field distributions of the reflected waves by the ten cells.

strong resonance will appear, which is similar with a single degree of freedom vibration system by mechanical equivalent principle as shown in Fig.2(a) The air in the cavity is equivalent to a spring, which has an acoustic capacitor with the capacitance of Ca= V/(ρ0c02) and the air in the hole acts as

an acoustic mass Ma= ρ0l s When the acoustic waves are incident onto the hole, frictional damping will appear So the oscillatory system has a certain acoustic resistance Ra Where V is the volume of the cavity; t is the thickness of the cube shell; ρ0is the density of air; c0is the sound speed in air;

s= πd2/4 is the cross-sectional area of the opening hole; and l=t+0.85d is the effective length of the hole Therefore, we can get Helmholtz equation

and the resonant frequency

where P is acoustic pressure in air and λ0is the wavelength of incident waves in air

It is necessary to point out that our simulations are done by commercial software COMSOL Multiphysics based on Finite Element Method The simulations were calculated by the module Pressure Acoustics, Frequency Domain When the strong resonance happen, the phase of the re-flected waves will change 2π continuously It is clear that the natural frequency of CHC is related

to the hole diameter d So we can adjust d to obtain the desired natural frequency at will, and then control the phase of the reflected waves In order to get the relationship between the phase of the reflected waves and d, we designed a simulation as shown in Fig.2(a) The background materials designed in the simulation are air and the CHC was fabricated by steel The left face is taken as plane wave radiation boundaries, the right face is set as sound hard boundaries, and the other four faces are set as periodic boundaries as shown in Fig 2(a) The blue arrow indicates the incident waves and the red indicates the reflected waves, respectively The values of w, h and D are fixed

in the simulation When only changing the values of d from 0.01 to 8 mm at a given frequency

1500 Hz, we will find that the reflected phases cover a range of 2π as shown in Fig.2(b) So it is easy to get ten cells that could realize discrete phase shifts covering the full 2π span with a step of

π/5 The exact values of d for achieving these discrete phase shifts are also illustrated with blue dots

in Fig.2(b) In the simulation, a reflected array composed of ten cells is regarded as one super unit cell, and then the reflected array is periodically arranged to realize the AMS as shown in Fig.2(c)

In order to further explain the phase gradient, we show the reflected waves by ten cells in Fig.2(d) When the incident waves are cast onto AMS, the high maps of pressure field are applied to show the different phase shifts of each unit cells It is therefore reasonable to believe that the AMS can

be fabricated by these CHCs The AMS has a thickness equal to λ/11.3, meaning that the AMS is

a good candidate to design acoustic devices, such as small footprint sonar and medical ultrasonic scanners It is no doubt that the anomalous reflection will obey the generalized Snell’ law

sin(θr) − sin (θi)= (λ/2πni)(dφ/dy), (3) where θi is the incident angle; θr is the reflected angle; λ is the wavelength; ni is the refractive index; and dφ/d y is the phase gradient in the y-axis

III RESULTS AND DISCUSSIONS

A Reflected waves modulation with different phase gradient

In this section, we demonstrate that the AMS can turn the incident waves into surface waves by selecting suitable phase gradients If each CHC is arrayed with the spacing distance of 25 mm, we will obtain a phase gradient of π/125 rad/mm as shown in Fig.3(a) When ten cells are sufficiently close together with the spacing distance of 25 mm, we calculate the phases of ten cells, which agree well with those of individual cells as shown in Fig.3(a) Therefore, we can simply regard the abnormal reflected wave as a new acoustic radiation by a line of secondary sources with different phases modulated by the CHC with varying hole diameters According to the generalized Snell’s law, the reflected angle at 1360 Hz will be 90◦ The simulation result is shown in Fig.3(b), which is

in a good agreement with theoretical result The white and black arrows represent the incident and reflected waves, respectively

In order to investigate the influence of different phase gradients of the reflected waves, we simulate a normal material with the phase gradient of ξ= 0 rad/mm as shown in Fig.4(a) From Fig.4(a), it can be seen that the acoustic waves are normally reflected, which obeys the traditional Snell’ law As a comparison, we calculate another three kinds of AMSes (labeled as sample A,

B and C) with the phase gradients of π/125 rad/mm, π/250 rad/mm, and π/375 rad/mm, respec-tively The anomalous reflection phenomenon not only appears at a designed frequency, but at other frequencies as well When the acoustic waves are vertically incident onto three kinds of AMSes

at 1600 Hz, the pressure field distributions of the reflected waves are shown in Fig 4(b)–4(d) From the simulation results, we can get that the reflected angles are 60◦, 27◦and 17◦, which agree

FIG 3 (a) The AMS designed with the phase gradient of ξ = π/125 rad/mm in the y-axis (b) The pressure field distribution

at 1360 Hz with the reflected angle of 90◦ The white and black arrows represent incident and reflected waves, respectively.

065320-5 Wang et al. AIP Advances 6, 065320 (2016)

FIG 4 (a) The reflected acoustic field distribution of normal material with the phase gradient of ξ = 0 rad/mm (b)-(d) The reflected acoustic field distributions of AMS with the phase gradient of ξ = π/125 rad/mm, ξ = π/250 rad/mm, and

ξ = π/375 rad/mm, respectively.

well with the theoretical results 58.2◦, 25.2◦ and 16.5◦ according to the generalized Snell’s law, respectively

We have discussed that the incident angle is 0◦and the phase gradient of AMS is a positive value In this part, we will discuss the situation that the acoustic waves are obliquely incident onto the AMS and the phase gradident is a negative value When the phase gradient dφ/dy is a negative value, different incident angles will lead to different reflected angles as shown in Fig.5(a)–5(c)

In Ref.24, Ding et al demonstrated that negative reflection appears in the left side of the normal

We disagree with this view, because negative reflection only appears in the right side of the normal

as shown in Fig 5(b) According to traditional Snell’ law, the incident and reflected waves are

on both sides of the normal and the reflected angle is equal to the incident angle However, our designed AMS will break this law The AMS was fabricated by using ten kinds of CHCs as shown

in Fig.2(b), which can realize the phase change covering 0∼2π with a step of −π/5 The distance

of adjacent CHCs is 22 mm, which means the phase gradient of AMS is ξ= −π/110 rad/mm

In order to realize negative reflection, the incident angle should be less than 52◦when we select incident frequency from 1450 Hz to 1950 Hz The bandwidth will be illustrated later Figure 6(a)

shows that the theoretical reflected angle changes with the frequency of the incident waves from

1450 Hz to 1950 Hz and with the incident angle from 5◦to 52◦ The simulation results are shown

in Fig.6(b), which agrees well with theoretical analysis When the incident angle is 15◦at 1500 Hz, the distribution of the reflected waves can be seen in Fig 6(c) with the reflected angle of −51◦ According to the generalized Snell’s law, the reflected angle

θr= arcsin (sin (θi)+ (λ/2πni) (dφ/dy))= arcsin(sin(15◦

) − 34/33) = −50.5◦ The simulation result is in a good agreement with the theoretical result The white and black arrows represent the incident and reflected waves, respectively

FIG 5 (a)-(c) The distributions of the reflected angles versus different incident angles when the phase gradient is a negative value along the y-axis.

B The broadband of anomalous reflection

At first, we consider that the anomalous reflection only happens at a designed frequency

1500 Hz as shown in Fig.7(c) But this phenomenon also appears near 1500 Hz In order to study the AMS at a broad frequency range, the sample A is selected as a research object The distributions

of the reflected waves are calculated at a broad frequency range from 1400 Hz to 2000 Hz as shown

in Fig 7(a)–7(f) It is clear that the anomalous reflection exists at the frequencies of 1450 Hz,

1500 Hz, 1600 Hz, and 1950 Hz with different reflected angles of 70◦, 66◦, 60◦ and 45◦, which agrees well with the theoretical values 69.7◦, 65.0◦, 58.2◦and 44.2◦according to the generalized

FIG 6 (a) Theoretical reflection angles versus the frequency of the incident waves and incident angle (b) The reflected angle from the simulation versus the frequency of incident waves and incident angle (c) The left picture indicates the incident acoustic pressure field distribution with the incident angle of 15◦and the right picture indicates the reflected acoustic pressure field distribution of AMS with the phase gradient of ξ = −π/110 rad/mm.

065320-7 Wang et al. AIP Advances 6, 065320 (2016)

FIG 7 Reflected pressure field distribution of AMS with the phase gradient of ξ = π/125 rad/mm (a) Acoustic pressure field distribution of normal reflection at 1400 Hz (b) Reflected acoustic pressure field distribution at 1450 Hz with 70◦reflection, (c) 1500 Hz with 66◦reflection, (d) 1600 Hz with 60◦reflection and (e) 1950 Hz with 45◦reflection (f) The field distribution

of the acoustic scattering at 2000 Hz.

Snell’ law, respectively The acoustic waves will be vertically reflected at 1400 Hz as shown in Fig.7(a), which obeys the traditional Snell’ law In Fig 7(f), the reflected waves are irregular at

2000 Hz We can obtain that the AMS with the phase gradient of ξ= π/125 rad/mm can realize anomalous reflection at a broad frequency range from 1450 Hz to 1950 Hz with a bandwidth of

500 Hz, which will open a new door to for our AMS to acoustic applications When the frequency

is near the designed frequency 1500 Hz, a strong resonance of CHCs appears, which means that the reflected amplitude of CHCs is equal to 1 and the reflected phases of CHCs are the designed values by the generalized Snell’ law Therefore, the anomalous manipulation of the reflected waves can be realized near 1500 Hz The similation results indicate that our designed AMS is effective in a broadband frequency range with the bandwidth of 500 Hz

C Acoustic focusing lens constructed by AMS

We have proved that the CHC can be used to design an AMS with an ultrathin structure which can manipulate the reflected waves at will In this section, we will show that an acoustic focusing lens can be designed successfully by our AMS In order to achieve acoustic focusing at a given position (0, f ), the hyperboloidal phase gradient is applied along the y-axis, as shown in Fig.8(a)

FIG 8 (a) Schematic diagram of acoustic focusing lens (b) The theoretical continuous phase shifts (red line) and the discrete phase gradient provided by AMS (blue dots) along the y-axis (c) The distribution of the acoustic pressure field for the designed lens (d) Transverse cross-section of intensity distribution at y = 0 mm along the z-axis (e) Spatial intensity distribution |p| 2 (f) Transverse cross-section of intensity distribution at z = 634 mm along the y-axis The blue dash line refers to the intensity of the incident waves.

065320-9 Wang et al. AIP Advances 6, 065320 (2016)

The phase shift φ(y) is required as follows:

φ (y) = k ·(y2+ f2− f

)

where k is the wave vector and f is the focal length According to the generalized Snell’ law, the desirable continuous phase gradient (red line) and the discrete phase gradient (blue dots) provided

by the AMS are plotted in Fig.8(b) The acoustic focusing lens can be designed by 29 CHCs ac-cording to the phase gradient The acoustic pressure field distribution with f = 3λ (λ = 21.25 cm)

is shown in Fig 8(c) The transverse cross-section intensity distribution at y = 0 mm along the z-axis is shown in Fig 8(d) Spatial intensity distribution |p|2 of acoustic focusing is shown in Fig.8(e) In order to further quantify the focusing effect of the acoustic focusing lens, the transverse cross-section intensity distribution at z = 634 mm along the y-axis is shown in Fig.8(f) The inten-sity of pressure at the focal spot is nearly 15.1 times larger than that of the incident waves as shown

in Fig.8(f)with blue dash line, which shows a excellent focusing effect

D Acoustic carpet cloak designed by AMS

Yang et al.25 have demonstrated that a metasurface carpet cloak constructed by AMS can be used to hide large objects In this section, we demonstrate that acoustic carpet cloak can also be

FIG 9 (a)-(c) Reflected pressure field distribution when the acoustic waves are incident onto a flat wall, onto the object without cloak, and onto the object with cloak vertically, respectively (d)-(f) Total pressure field distribution when the acoustic waves are incident onto a flat wall, onto the object without cloak, and onto the object with cloak vertically, respectively (g) Local reflected phase of each unit cell.