**기계연, 음향 메타렌즈 개발로 회절한계 4배 극복**

- 음향 이미징 장치의 성능을 크게 향상시킬 수 있는 원천기술 개발 -

- 초음파 지문인식, 음향 현미경 등 생체인식 시장 활용 기대 -

https://www.kimm.re.kr/pr_kimm_news/articles/view/tableid/news/id/1148

한국기계연구원(원장 박천홍)이 핀테크 기술의 확산 등 생체인식 기술이 활발히 이용되고 있는 가운데 고해상도 이미징이 가능한 음향 메타렌즈 개발에 성공했다. 향후 초음파 지문인식, 근접장 음향현미경, 비파괴검사 등 생활 속 음향 이미징이 필요한 분야에 활용될 전망이다.

기계연 나노자연모사연구실 허신 책임연구원 연구팀은 특정 주파수를 갖는 두 음원을 메타렌즈에 통과시켜 회절한계1)의 4배까지 이미징하는데 성공했다. 물체의 형상정보를 전달하는 소멸파는 거리에 따라 급격히 사라지지만 메타렌즈에 통과시키면 내부에 공진터널링2) 현상이 발생하면서 소멸되지 않고 이미징 할 수 있다.

▲ 한국기계연구원 나노자연모사연구실 허신 책임연구원 연구팀이 3D프린팅으로 제조한 서브파장 이미징 음향 메타렌즈

현재 스마트폰 지문인식에는 지문패턴에 따른 정전용량의 변화를 감지하여 이를 이미징하는 정전용량방식이 주로 쓰이고 있다. 하지만 최근에는 지문복사 등으로 해킹 위험이 제기되고 있어 미국 등 선진국을 중심으로 더욱 보안성이 높은 초음파를 이용한 지문인식 기술 개발이 활발히 이뤄지고 있다.

이번에 개발된 음향 메타렌즈 기술은 초음파 지문인식에 적용할 수 있는 핵심원천기술이다. 연구팀의 독창적인 설계가 적용된 음향 메타렌즈를 스마트폰 지문인식 기술에 적용하면 이미징 해상도를 4배 향상시킬 수 있다.

지문인식 해상도를 높이면 지문의 패턴을 더 자세히 이미징할 수 있어 더 강력한 보안이 가능하고, 음파의 우수한 투과특성 덕분에 지문인식에 필요한 전력소모도 줄일 수 있다.

▲ 서브파장 이미징 음향 메타렌즈의 작동원리

연구팀은 음향 메타렌즈 내부에서 1차 및 2차의 공진터널링 현상이 발생하는 것을 규명하고, 이를 이용해 회절한계를 4배 뛰어넘는 이미지 측정에 성공했다. 음향 메타렌즈의 1차 및 2차 공진터널링 주파수를 이용해 서브파장 이미징3)에 성공한 것은 이번이 처음이다. 또 마치 광학현미경을 사용할 때 렌즈배율을 높여 물체를 확대 관찰하듯이 필요에 따라 1차 및 2차의 다른 공진터널링 현상을 이용해 이미지의 해상도를 높여 관찰할 수 있다.

모든 음파는 임의의 물체에 반사되면 파장이 점점 감쇠하는 소멸파가 생긴다. 이를 음향 메타렌즈에 통과시키면 메타구조에서 발생하는 독특한 공진터널링 현상으로 소멸파의 에너지를 거의 손실 없이 완전히 투과시킨다. 메타렌즈 내부는 다수의 공진터널링 주파수가 발생하지만 지금까지는 모두 1차 공진현상을 이용하는 수준에 머물러왔다.

기존의 음향메타 이미징기술은 필요한 이미징 주파수에 따라 렌즈의 두께를 다르게 만들어야 했지만, 이 기술을 활용하면 개발된 음향 메타렌즈의 두께를 다르게 할 필요가 없다. 또 1차 및 2차 공진터널링 현상을 이용해 현미경의 렌즈배율을 높이듯 물체를 한층 고해상도로 확대 관찰할 수도 있다. 그리고 복잡한 구조물이지만 3D프린팅 기술로 제작하면 제작비용도 크게 절감할 수 있다.

▲ 서브파장 이미징을 위한 음향 메타렌즈 및 실험장치

이번 연구는 과학기술정보통신부 글로벌프론티어사업 파동에너지극한제어연구단 (단장 이학주)의 지원을 받아 수행됐으며 연구결과는 지난 10월 19일 Scientific Reports 온라인에 ‘In-depth study on resonant tunneling for subwavelength imaging(서브파장 이미징을 위한 음향 메타렌즈의 멀티 공진터널링 모드에 관한 연구)’이라는 제목으로 게재됐다.

기계연 허신 책임연구원은 “음향 메타렌즈는 근접한 거리에서 이미징이 필요한 **초음파 지문인식, ****근접장 음향현미경**, **생체 이미징, 비파괴검사 기술 등에 활용이 가능**하다”며 “우리 생활에 많이 쓰이고 있는 음향 이미징 장치의 성능을 크게 향상시켜 신뢰성을 높일 수 있는 기술”이라고 말했다.

▲ 음향 메타렌즈 실험장치를 살펴보는 연구진

***용어설명**

**1. 회절한계 (Diffraction Limit)**

음파의 회절한계는 이미징 장치의 공간 분해능 한계를 결정한다. 어떤 물체를 이미징하기 위해서는 물체의 크기가 이미징 주파수의 절반(λ/2) 이상이 되어야 한다. 그 이하가 되면 물체를 구분할 수 없으며 이때 회절한계를 넘어섰다고 한다. 예를 들어 이미징 주파수 1 ㎑의 파장(λ·남다)이 약 30 ㎝라면, 15 ㎝(λ/2)보다 작은 물체는 회절한계 이하이기 때문에 이미징이 불가능하다.

**2. 공진터널링 (Resonant Tunneling)**

공진은 외부의 에너지 없이도 고유의 주파수와 특정 주파수가 만나 파장이 증폭되는 현상이다. 공진터널링은 음향 메타 구조체의 터널구조 안으로 특정 음파가 들어올 때 그 음파 고유의 공진 주파수와 터널 내부의 공진 주파수가 일치할 경우 음파가 에너지의 손실없이 음향 메타 구조체의 터널을 통과하는 현상을 의미한다.

**3. 서브파장 이미징 (Subwavelength Imaging)**

일반적으로 회절한계 이하의 크기를 갖는 물체는 이미징할 수 없다. 예를 들어 특정한 이미징 주파수를 갖는 초음파로 우리 몸 속의 종양을 찾을 때 이미징 주파수 파장의 절반 이하 크기를 갖는다면 이미징할 수 없다. 때문에 초음파 화면으로는 어떤 이미지도 구별할 수 없다. 하지만 특정한 이미징 주파수를 갖는 초음파가 음향메타 구조체를 통과하여 종양을 이미징할 경우에는 회절한계를 극복하고 이미징이 가능하다. 이를 서브파장 이미징이라고 한다.

https://www.nature.com/articles/s41598-018-33653-y

*Scientific Reports*volume 8, Article number: 15535 (2018)

We report new frequency bands for subwavelength imaging by using the resonant tunneling method which have not been explored previously. As per the existing theory of resonant tunneling, imaging frequency is limited for a certain number of crystals. However, after conducting an analytical analysis over a wide range of frequencies, we observed that higher frequencies do exist for subwavelength imaging. We verified this observation both numerically and experimentally. We extended our study to observe the effect of lattice periodicity on image resolution. By reducing periodicity during experiment, we achieved a resolution of λ/9.5 at the conventional region and λ/2.45 at the higher band region.

Abbe limit1, which is commonly known as the diffraction limit, is a natural constraint on image resolution that roughly implies that the maximum achievable resolution corresponds to half of the wavelength. To obtain a good-quality image, it is necessary to overcome this limit. Since Pendry’s successful work on perfect lens2, several methods have been proposed to overcome the diffraction limit in both the electromagnetic and acoustic wave regimes, such as the time reversal technique3,4,5,6, Brag Scattering7,8,9, superlens10,11,12,13,14,15,16,17,18,19,20,21,22,23,24, and hyperlens25,26,27,28,29,30. According to our ken, the best resolution reported for acoustic waves is λ/5020. The technique behind the lens was strong coupling of evanescent wave through Fabry‒Pérot (FP) resonance. As evanescent waves contain finer feature information of an object, by recovering them at the image plane by the lens, it was possible to obtain subwavelength imaging. However, in this method, the imaging frequency is dependent on lens thickness as the FP resonance is dependent on it. To reduce the dependency on lens thickness, zero-mass19,31,32 and Bloch-wave-based anisotropic resonant tunneling33,34metamaterials have been anticipated. For the resonant tunneling case, both evanescent and propagating waves will be tunneled if the lens thickness is equal to the integer multiple of half of the Bloch wavelength. This implies that the imaging frequency can be altered without changing the lens thickness by modifying the micro-structure of the lens. However, it has been observed that the imaging frequency tends to decrease on increasing the diameter modulation factor, and the highest imaging frequency is obtained33 when the modulation factor is unity, which is in fact the FP resonance condition. As subwavelength imaging has potential application at higher frequency regime, an FP-based lens seems more suitable than resonant tunneling. However, the frequency of FP-based lens is also limited owing to the requirement that the wavelength should be much higher than the lattice periodicity and hole diameter20. Besides this, except refs14,21, most previous studies in the acoustic regime were based on low frequency. Thus, we studied resonant tunneling and discovered that higher bands of frequency exist for it, which was not possible according to previous works33,34. Moreover, unlike an FP-based lens, the wavelength of a resonant tunneling lens need not be very higher than the lattice periodicity and hole diameter. Therefore, we hope that our work will open new avenues for practical application of subwavelength imaging.

We used a holey structured metamaterial introduced by H. Su *et al*.33, in a periodic array, which is shown in Fig. 1. Here, the unit cell diameter of a crystal has been modulated along the X direction and its length, s = 2s1 + s2, is shown in Fig. 1a. It should be noted that we have used s2 = 2s1. The diameter of the first and third layers of the single hole are d1 and that of the second layer is d2. The diameter modulation factor is δ = d1/d2 = 4. The lattice periodicity of the crystal is denoted by Λ. The holes are filled with air, and the boundary of the holes are made of hard material for acoustic waves, except at the opening and end. The three layer (i = 1, 2, 3) effective mass density of a unit cell can be expressed in tensor format as35

(1)

Here, and di = diameter of the respective layer.

If the number of crystals be N, the transmission equation can be given as33,34

(2)

where,

(3)

(4)

(5)

q = wavenumber of the Bloch Wave along the X-direction, *c*0 = sound velocity in air = 343 m/s,

incident wave number,

The dispersion relation for the propagating Bloch wave along the X-direction can be determined as33

(6)

We know at the resonant tunneling condition, 100% transmission will occur for both propagating and evanescent waves34, i.e. .

From equation (2), we can see that tunneling can be achieved by satisfying the following condition34

(7)

From equations (2) and (7), we realize that for tunneling, the minimum number of crystals (N) required are two. The diameters are taken as d1 = 4d2 = 8 mm. By putting the value of *δ* and replacing s1 and s2 in terms of s, we can simplify the dispersion equation (6) for N = 2 as follows:

(8)

For s = 40 mm33, we can deduce the tunneling frequency from equation (8) as f = 926 Hz. It should be noted here that this is the highest and only frequency for the given number of crystals, N = 2, as per equation (7). However, we have analytically examined all the values of qs from 100 Hz‒18490 Hz according to equation (6) and found that for certain bands of frequency, its value becomes indefinite as −1 > cos qs > 1 and is repetitive in nature (see Fig. 2a). For example, for our current parameters, until 1335 Hz, the value of qs is definite as

. Then, for a bandwidth (BW) of 1340‒7235 Hz (=5895 Hz), it becomes indefinite. Again, the value of qs becomes definite for a BW of 7240‒9910 Hz (=2670 Hz), and then, it becomes indefinite again for a BW of 9915‒15810 Hz (=5895 Hz), and this process continues. We have determined the transmission co-efficient for a periodicity of Λ = 10 mm33 and found some interesting results at higher definite band regions which indicated that bands of frequencies exist for tunneling, as shown in Fig. 2.

(a) Contour plot of transmission co-efficent amplitudes |T| for a wide band frequency ranging from 100 Hz‒18490 Hz with respect to different parallel wavenumbers. (b) Amplified view of the contour plot for the first definite region of the frequency band up to 1335 Hz. (c)Amplified view of the second definite region for a bandwidth (BW) of 7240‒9910 Hz (=2670 Hz). (d) Amplified view of the third definite region for a BW of 15815‒18485 Hz (=2670 Hz).

Full size image

From Fig. 2b, we see the first tunneling frequency (926 Hz) in the first definite region which was determined by equation (8). From Fig. 2c,d, we can see that at the second and third definite regions, bands of frequencies exist for imaging. Therefore, analytically, we can realize frequencies higher than the first tunneling frequency of 926 Hz. These findings indicate that we can extend the subwavelength imaging to higher frequency regimes to detect tiny objects with a smaller feature size. It should also be noted that at the third definite region, the bands of tunneling frequencies are narrower than those at the second definite region.

Inspired from the results mentioned above, we have designed a new lens with a crystal thickness of s = 2s1 + s2 = 20 mm with s2 = 2s1 = 10 mm, d1 = 4d2 = 4 mm, and Λ = 6 mm. We keep the total number of crystals as N = 2 which implies a lens thickness of 2 s = 40 mm. Equation (8) can still be applied to the current parameters of the lens for deducing the first tunneling frequency, and it should be 1852 Hz. During analytical computations, we have observed that decreasing the crystal thickness (s) by half, the definite and indefinite region of frequency has been doubled. By using a step size of 5 Hz in our calculation, we obtain a BW of 5340 Hz (except for the first definite region, which is 2670 Hz) and 11800 Hz for the definite and indefinite regions, respectively. The first definite region exists until 2670 Hz (Fig. 3a), which is half of the next definite region, 14480‒19820 Hz (=5340 Hz) (Fig. 3b). We have performed analytical calculations up to 21000 Hz which comprises only the first two definite regions of frequency. As tunneling is possible only in the definite regions, we have focused on these regions. The transmission characteristics are shown in Fig. 3.

From Fig. 3a, the first tunneling frequency can be observed at 1852 Hz. on the other hand, as shown in Fig. 3b, three bands of frequencies are observed for tunneling. At second band in Fig. 3b, we see a straight line is appeared slightly above 17000 Hz. This line corresponds to a special frequency which does not pertain for subwavelength imaging as no tunneling is possible for this frequency. Analytically this phenomenon can be understood. If we rewrite equation (6) for our current parameters, may be written as

(9)

or,

(10)

If cos *qs* = 1, equation (2) becomes indefinite because of the “sin(Nqs)/sin(qs)” term at the denominator. Therefore, for any λ = s/n. For our case, if n = 1, tunneling will not be possible at λ = 20 mm, which corresponds to 17150 Hz. The fact can hardly be distinguished from Fig. 3(b) as it is a single frequency inside the tunneling band which has such non- tunneling characteristic. Therefore, it is necessary to mention them separately. We term these frequencies as prohibited frequencies.

On a different note but not unrelated to this topic, we were also interested to investigate the fact that what happens if we interchange the diameters of the layers such as 4d1 = d2 = 4 mm which means the narrow tunnel will be at the both end and wide tunnel at the middle of the unit cell. Although the modulation factor, δ (=d1/d2 = 1/4) has been reduced to 1/4, according to equation (6) the definite band and indefinite band size will be the same as factor remains unchanged. From equations (2), (7) and (8), we understand the tunneling frequency at the first definite band will be the same as previous which is 1852 Hz. But, from equation (2), it is not possible to predict the case about the second definite band, as T1 and T2 has been changed due to interchange of diameters. However, we obtained the transmission behavior for both of the definite bands analytically (see Supplementary Information, Fig. S7) and found for second definite regions, there are no band for tunneling. But, interestingly there was one more tunneling band at the end of the first definite region. At our next work, we will investigate about it rigorously. The facts of tunneling at the first definite region and not tunneling at the second definite region has been demonstrated numerically and represented at the Supplementary Information (see Fig. S8).

To corroborate our analytical findings, we have used a commercial multi-physics software COMSOL for numerical analysis. We have made the lens with an array of hole 30 × 5 along the (Y, Z) directions with a periodicity of Λ = 6 mm. All other parameters were identical to those of the previously developed lens: s = 2s1 + s2 = 20 mm, s2 = 2s1 = 10 mm, d1 = 4d2 = 4 mm. A perfectly matched layer (PML) having a thickness of 20 mm was used around the model. only the air part was meshed to determine the transmission behavior. Acoustic Pressure, Frequency Domain (acpr) Physics was used in our simulation model. The model can be seen in Fig. 4.

As shown in Fig. 4, the monopole point sources were taken as speakers and placed in front of the lens at a distance of 1 mm from it. The flow rate of the point source was 10 m3/s, and distance between them was 24 mm. The position of the scanning line was at the same height as the sources along the Z-direction. Scanning was performed 1 mm behind the lens along the Y direction. Subwavelength imaging can be realized at the frequencies shown in Fig. 5.

From Fig. 5a, it is evident that our crystal can detect point sources at a subwavelength distance (24 mm ≈ λ/8), whereas without lens, no sources can be detected. A small discrepancy was found between the theoretical imaging frequency (1852 Hz) and the best simulation frequency (1780 Hz). We will discuss about it at next at the chapter “Effect of periodicity on subwavelength imaging”.

From Fig. 3b, it can be seen that, theoretically, the first band of frequency at the second definite region for which imaging is possible ranges from 15000‒16000 Hz. To obtain subwavelength imaging in this region, we need to reduce the distance between the sources to approximately 10 mm. However, we noticed that imaging can be obtained until a distance of 2Λ, which is related to periodicity and will be discussed at next (see section “effect of periodicity on subwavelength imaging”). From Fig. 3b, a small band of frequency (approximately 300 Hz) is also observed at which no imaging occurs. However, we found some discrepancy between the theoretical first band of the second definite region (15000‒16000 Hz), as the simulation frequency for imaging has been found for a BW of 13400‒15300 Hz. Anyway, a portion of the imaging BW of 14000‒14550 Hz has been represented in Fig. 5b. The remaining frequencies in this BW can be found in Supplementary Information (Figs S1 and S2). During the simulation, we also observed that no imaging region exists between 15350 and 15750 Hz (400 Hz) which is similar to the theoretical prediction.

From Fig. 3b, we can see that the second band is the widest band among all the three bands and ranges from 16300‒18000 Hz. It should be noted that the prohibited frequency (17150 Hz) lies inside this band. During simulation, we also found that a wide BW of 15800‒18700 Hz for imaging, in addition to a narrow band (16950‒17350 Hz) of non-imaging frequencies around the prohibited frequency of 17150 Hz (See Figs S3and S4). The non-imaging nature of the frequency 17150 Hz can be seen in Fig. 5b.

According to Fig. 3b, after a narrow band of no-imaging region, we should observe the third band which approximately ranges from 18300 to 19800 Hz. The gap between the second and third bands is almost 300 Hz. According to our simulation result, the gap was found to be 150 Hz, ranging from 18750 to 18900 Hz. In the simulation, imaging was observed in the third band from 18950 to 20000 Hz (See Fig. S5). Thus, from the above discussion, it can be said that with some inconsistencies, the simulation result coincide well with the theoretical model. The next section will describe the experiment conducted by the designed lens.

The metamaterial was prepared by mechanical machining of the Polymethyl methacrylate (PMMA), bars, which can be regarded as hard material for sound propagation with respect to air. As shown in Fig. 6a, the lens was prepared by placing two crystals together tightly. The speaker diameter was 22 mm and placed at a distance of 1 mm in front of the lens. The centre to centre distance between the two speakers was 66 mm. The microphone was placed at the same height as the centre of the speakers. The microphone was moved along the Y direction at a step size of 6 mm. The two sources separated by 66 mm (≈λ/3) can be easily discernible from Fig. 6c; however, without the meta-lens, it cannot be identified. Although we get the resolution beyond the diffraction limit, but still it is far away from the numerical result. This might stem from the air leakage due to direct coupling without any sealing process between one crystal and another. However, we overcome the problem using adhesive material to couple the crystals and 1.5 mm diameter MEMS speakers. The distance between the speakers was 24 mm. Now, as shown in Fig. 6d, we get higher resolution (≈λ/8) which is similar to the simulation result (Fig. 5a).

Here, d denotes the center to center distance between the speakers. (a) Experimental set-up for verifying subwavelength imaging. (b) Conceptual 2D set-up of the experiment. (c)Normalized pressure amplitudes at a distance 1 mm behind the lens where black line and red line represent the data with lens and without lens respectively. (d) Normalized pressure amplitudes at a distance 1 mm behind the lens with a step size 3 mm. (e) Normalized pressure amplitudes for the second definite region. Solid lines and dotted lines represent data with lens and without lens, respectively.

Full size image

Next, for obtaining the image at the 2nd definite region, we used speakers with 22 mm diameter. We maintained a 2 mm distance between the two speakers which yielded a 24 mm center to center distance. As shown in Fig. 6e, the two speakers which were separated by only 2 mm can be easily resolved. This result provides us the experimental evidence that much higher frequencies exist for tunnelling, which is not governed by equation (7).

From Fig. 6e, it can be observed that all the three bands are covered in the 2nd definite region with a step size 500 Hz starting from 14000 to 18500 Hz. Because the no-imaging band is very narrow compared to our step size, no such frequency appeared in our graph. Although we have shown only the better imaging frequency region, in the experiment, we found imaging for a BW 13500–19000 Hz, which is approximately close to both theoretical and simulation results. Furthermore, Fig. S6differentiates the imaging region from the non-imaging region. However, for without lens case at Fig. 6e, two small peaks were appeared around ± 40 mm distance. We assume this might be inherited from interference.

In the numerical analysis chapter, it was discussed that periodicity is the prevailing factor for obtaining subwavelength imaging, as the wavelength becomes comparable to it at a higher frequency region. Therefore, we decided to reduce the periodicity by keeping all the other parameters same. We reduced periodicity from 6 mm to 5 mm and conducted the contour plot of transmission amplitudes, which is depicted in Fig. 7. Though we have reduced the periodicity (Λ), according to equation (7), for s = 20 mm, our first tunnelling frequency remains the same 1852 Hz as with the periodicity 6 mm. However, the tunnelling frequency according to the simulation is 1810 Hz (see Fig. 7c). From Fig. 7b, we can observe the band regions have been changed slightly than from Fig. 3b as effective masses (see equation (1)) of the respective layers have been changed, which have effect on T1 and T2,affected the transmission behaviour. From Figs 5b and 7d, we observe that the image can be generated until the 2Λ distance. However, as Λ has been decreased to 5 mm, the object can be detected for a 10-mm distance, which is less than the previous lens with 6 mm periodicity. Figure 7e represents the data for 1Λ distance between the speakers; it is realised that it is difficult to produce a reliable image with this distance.

Here, d = distance between the sources. Contour plot of transmission co-efficient amplitudes |T| for a lens with 5 mm periodicity (a) for the first definite region; (b) for the second definite region; Numerical study: Normalized pressure amplitudes at a distance 1 mm behind the lens (c) for a lens with Λ = 5 mm and distance between the sources 20 mm; (d) for distance between the sources 10 mm (=2Λ) at 2nd definite region; (e) for distance between the sources 5 mm (=1Λ); (f) for a lens with Λ = 2.5 mm and distance between the sources 10 mm; (g) for distance between the sources 5 mm (=2Λ) at 2nd definite region; Experimental result with Λ = 2.5 mm: (h) at the first definite region (i) for distance between sources 10 mm (=4Λ) at 2nd definite region.

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Next, we decreased the periodicity (Λ) further to 2.5 mm with d1 = 2 mm and d2 = 0.5 mm. From equation (1), it can be realized that the effective mass of the respective layer would remain same with the crystal with periodicity 5 mm as Λ/di remains the same. Thus, analytically, there should not be any difference in the transmission behaviour between the two crystals. However, from Fig. 7f,h, the first tunnelling frequencies obtained numerically and experimentally were 1840 Hz and 1800 Hz, respectively. This is closer to the theoretical frequency 1852(Hz). From this point of view, by observing Figs 5a, and 7(c,f,h), we may conclude that by reducing the periodicity, the tunnelling frequency tends to approach towards the theoretical frequency. By reducing the periodicity and hole diameters, we can image at the first definite region down to 20 mm or λ/9.5 (see Fig. 7h), which is much better than earlier (Fig. 6c).

The numerical result at Fig. 7g shows that for a 2Λ(=5 mm) separation between the speaker, the new lens can produce the image similar to the previous lenses at the second definite region. However, as the periodicity reduced, for a frequency 14500 Hz, the resolution reduces to approximately λ/4.7. From Fig. 7i, up to 4Λ(=10 mm) resolution was obtained experimentally by the lens which is also below the diffraction limit and equal to λ/2.45. Therefore, we could prove both numerically and experimentally that by reducing the periodicity and hole diameters, we can increase the resolution of same lens thickness.

In this work, we explored the higher bands of frequency for subwavelength imaging using resonant tunnelling. Primarily, we presented the layout of analytical evidence and then bolstered it with numerical and experimental data. Some inconsistencies were found among analytical, numerical, and experimental data. However, it was observed that by increasing the periodicity, the difference between analytical and numerical data tends to decrease. Furthermore, the difference in experimental data probably arose from the imperfection in coupling between two crystals which was demonstrated once. However, the data obtained were close enough to point out the fact that for a certain number of crystals, much higher band of frequencies exist, which was not found earlier. Moreover, we showed the effect of periodicity that with decreasing periodicity and consequently the hole diameters, the resolution can be increased. Furthermore, it was also found that the resonant tunnelling method has the advantage over FP resonance-based lens that the periodicity and hole diameter need not be much smaller than the wavelength of imaging frequency. Therefore, we expect our finding will draw more attention to resonant tunnelling method for its working capability in higher frequency regions and intrigue the research with it on ultrasonic subwavelength imaging for medical and non-destructive testing area.

All the numerical simulations were conducted by using COMSOL multiphysics software, which is based on the Finite Element Method (FEM). Unit cells of 30 × 5 size were used to make a single crystal for the lens with periodicity (Λ) 5 and 6 mm; the results are shown in Figs 5, 7(c,d,e). For lens with periodicity Λ = 2.5 mm, 30 × 10 unit cells were used; the results can be seen in Fig. 7(f,g). Each lens contains two single crystal i.e. N = 2 and for every simulation, the sources were positioned 1 mm in front of the lens; the data were acquired 1 mm behind the lens.

The data of Figs 6c,e and S6 were obtained with the speakers with diameter of 22 mm and Figs 6d and 7h,i with MEMS speakers with 1.5 mm diameter of its opening. The sound wave was generated by exciting the speakers with an Agilent 33522 A function generator. 10Vp-p and 2Vp-p was supplied via the function generator to the 22-mm speaker and MEMS speaker, respectively for this purpose. An (1/4″) G.R.A.S Type 26 CB microphone was placed behind the lens at a distance of 1 mm. A National Instruments DAQ (M/N:c DAQ-9171) card was used to convert the analog signal of the microphone to digital signal. Finally, a LabVIEW based software ‘NI Signal Express 2013’ was used for data processing. The output signal was acquired in DB for which the reference pressure was 20 µPa. Therefore, we converted the gain output into µPa using the formula, DB = 20 log(P/Pref). Then, all data were normalized.

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- 1.
E., A. VII. on the Estimation of Aperture in the Microscope.

*J. R. Microsc. Soc.*1, 388–423 (1881). - 2.
Pendry, J. B. Negative Refraction Makes a Perfect Lens.

*Phys. Rev. Lett.*85, 3966–3969 (2000). - 3.
Lemoult, F., Lerosey, G., de Rosny, J. & Fink, M. Resonant Metalenses for Breaking the Diffraction Barrier.

*Phys. Rev. Lett.*104, 203901 (2010). - 4.
de Rosny, J. & Fink, M. Overcoming the Diffraction Limit in Wave Physics Using a Time-Reversal Mirror and a Novel Acoustic Sink.

*Phys. Rev. Lett.*89, 124301 (2002). - 5.
Lerosey, G., de Rosny, J., Tourin, A. & Fink, M. Focusing Beyond the Diffraction Limit with Far-Field Time Reversal.

*Science (80-).*315, 1120 LP–1122 (2007). - 6.
Lemoult, F., Fink, M. & Lerosey, G. Acoustic Resonators for Far-Field Control of Sound on a Subwavelength Scale.

*Phys. Rev. Lett.*107, 64301 (2011). - 7.
Cervera, F.

*et al*. Refractive Acoustic Devices for Airborne Sound.*Phys. Rev. Lett.*88, 23902 (2001). - 8.
Sukhovich, A., Jing, L. & Page, J. H. Negative refraction and focusing of ultrasound in two-dimensional phononic crystals.

*Phys. Rev. B*77, 14301 (2008). - 9.
Yang, S.

*et al*. Focusing of Sound in a 3D Phononic Crystal.*Phys. Rev. Lett.*93, 24301 (2004). - 10.
Belov, P. A. & Hao, Y. Subwavelength imaging at optical frequencies using a transmission device formed by a periodic layered metal-dielectric structure operating in the canalization regime.

*Phys. Rev. B*73, 113110 (2006). - 11.
Fang, N., Lee, H., Sun, C. & Zhang, X. Sub–Diffraction-Limited Optical Imaging with a Silver Superlens.

*Science (80-.).*308, 534 LP–537 (2005). - 12.
Leonhardt, U. Optical Conformal Mapping.

*Science (80-.).*312, 1777 LP–1780 (2006). - 13.
Jia, H.

*et al*. Subwavelength imaging by a simple planar acoustic superlens.*Appl. Phys. Lett.*97, 173507 (2010). - 14.
Amireddy, K. K., Balasubramaniam, K. & Rajagopal, P. Holey-structured metamaterial lens for subwavelength resolution in ultrasonic characterization of metallic components.

*Appl. Phys. Lett.*108, 224101 (2016). - 15.
Pendry, J. B., Schurig, D. & Smith, D. R. Controlling Electromagnetic Fields.

*Science (80-.).*312, 1780 LP–1782 (2006). - 16.
Yang, X., Yin, J., Yu, G., Peng, L. & Wang, N. Acoustic superlens using Helmholtz-resonator-based metamaterials.

*Appl. Phys. Lett.*107, 193505 (2015). - 17.
Ambati, M., Fang, N., Sun, C. & Zhang, X. Surface resonant states and superlensing in acoustic metamaterials.

*Phys. Rev. B*75, 195447 (2007). - 18.
Zhou, X., Badreddine Assouar, M. & Oudich, M. Subwavelength acoustic focusing by surface-wave-resonance enhanced transmission in doubly negative acoustic metamaterials.

*J. Appl. Phys.*116, 194501 (2014). - 19.
Park, J. J., Park, C. M., Lee, K. J. B. & Lee, S. H. Acoustic superlens using membrane-based metamaterials.

*Appl. Phys. Lett.*106, 51901 (2015). - 20.
Zhu, J.

*et al*. A holey-structured metamaterial for acoustic deep-subwavelength imaging.*Nat. Phys.*7, 52 (2010). - 21.
Amireddy, K. K., Balasubramaniam, K. & Rajagopal, P. Deep subwavelength ultrasonic imaging using optimized holey structured metamaterials.

*Sci. Rep.*7, 7777 (2017). - 22.
Cheng, Y., Zhou, C., Wei, Q., Wu, D. & Liu, X. Acoustic subwavelength imaging of subsurface objects with acoustic resonant metalens.

*Appl. Phys. Lett.*103, 224104 (2013). - 23.
Taubner, T., Korobkin, D., Urzhumov, Y., Shvets, G. & Hillenbrand, R. Near-Field Microscopy Through a SiC Superlens.

*Science (80-.).*313, 1595 LP–1595 (2006). - 24.
Melville, D. O. S. & Blaikie, R. J. Super-resolution imaging through a planar silver layer.

*Opt. Express*13, 2127–2134 (2005). - 25.
Jacob, Z., Alekseyev, L. V. & Narimanov, E. Optical Hyperlens: Far-field imaging beyond the diffraction limit.

*Opt. Express*14, 8247–8256 (2006). - 26.
Zhang, H., Zhou, X. & Hu, G. Shape-adaptable hyperlens for acoustic magnifying imaging.

*Appl. Phys. Lett.*109, 224103 (2016). - 27.
Liu, Z., Lee, H., Xiong, Y., Sun, C. & Zhang, X. Far-Field Optical Hyperlens Magnifying Sub-Diffraction-Limited Objects.

*Science (80-.).*315, 1686 LP–1686 (2007). - 28.
Lee, H., Liu, Z., Xiong, Y., Sun, C. & Zhang, X. Development of optical hyperlens for imaging below the diffraction limit.

*Opt. Express*15, 15886–15891 (2007). - 29.
Li, J., Fok, L., Yin, X., Bartal, G. & Zhang, X. Experimental demonstration of an acoustic magnifying hyperlens.

*Nat. Mater.*8, 931 (2009). - 30.
Gu, Y., Cheng, Y. & Liu, X. Acoustic planar hyperlens based on anisotropic density-near-zero metamaterials.

*Appl. Phys. Lett.*107, 133503 (2015). - 31.
Zhou, X. & Hu, G. Superlensing effect of an anisotropic metamaterial slab with near-zero dynamic mass.

*Appl. Phys. Lett.*98, 263510 (2011). - 32.
Hu, X. X., Pei, L., Zhou, X. & Hu, G. Experimental study on acoustic subwavelength imaging based on zero-mass metamaterials.

*EPL Europhysics Lett.*109, 28001 (2015). - 33.
Su, H., Zhou, X., Xu, X. & Hu, G. Experimental study on acoustic subwavelength imaging of holey-structured metamaterials by resonant tunneling.

*J. Acoust. Soc. Am.*135, 1686–1691 (2014). - 34.
Liu, A., Zhou, X., Huang, G. & Hu, G. Super-resolution imaging by resonant tunneling in anisotropic acoustic metamaterials.

*J. Acoust. Soc. Am.*132, 2800–2806 (2012). - 35.
Torrent, D. & Sánchez-Dehesa, J. Anisotropic Mass Density by Radially Periodic Fluid Structures.

*Phys. Rev. Lett.*105, 174301 (2010).

p.s

내가 보고 싶었던 기술이다.

기존의 Piezo소재가 아니고 비싼 MEMS공정도 필요도 없으며 스마트폰의 LCD 백라이트등의 빛의 간섭영향도 크게 받지않는 기술

Flexible, Strechable 디스플레이 지원은 아직 모르겠지만....

소리, 음향, 진동을 이용한 기술이 접근방법이 대안일 수 있었는데 상용화 가능여부는 전문가가 아니어서 아직 판단이 서질 않는다.