Far-field super-resolution imaging with a planar hyperbolic metamaterial lens beyond the Fabry-Perot resonance condition
aa r X i v : . [ phy s i c s . op ti c s ] J a n Far-field super-resolution imaging with a planar hyperbolic metamaterial lens beyondthe Fabry-Perot resonance condition
Cheng Lv
Key Laboratory of Terahertz Solid-State Technology,Shanghai Institute of Microsystem and Information Technology,Chinese Academy of Sciences, Shanghai 200050, ChinaState Key Laboratory of Functional Materials for Informatics,Shanghai Institute of Microsystem and Information Technology,Chinese Academy of Sciences, Shanghai 200050, China andUniversity of Chinese Academy of Sciences, Beijing 100049, People’s Republic of China
Wei Li ∗ State Key Laboratory of Functional Materials for Informatics,Shanghai Institute of Microsystem and Information Technology,Chinese Academy of Sciences, Shanghai 200050, China
Xunya Jiang
Dept. of Illuminating Engineering and Light Sources,School of Information Science and Engineering, Fudan University, Shanghai 200433, China andEngineering Research Center of Advanced Lighting Technology,Fudan University, Ministry of Education, Shanghai 200433, China
Juncheng Cao † Key Laboratory of Terahertz Solid-State Technology,Shanghai Institute of Microsystem and Information Technology,Chinese Academy of Sciences, Shanghai 200050, China
We demonstrate achieving the far-field super-resolution imaging can be realized by using a planarhyperbolic metamaterial lens (PHML), beyond the Fabry-Perot resonance condition. Although thethickness of the PHML is much larger than wavelength, the PHML not only can transmit radiativewaves and evanescent waves with high transmission, but also can collect all the waves in the imageregion with the amplitudes of them being the same order of magnitude. We present a design for aPHML to realize the far-field super-resolution imaging, with the distance between the sources andthe images 10 times larger than wavelength. We show the superresolution of our PHML is robustagainst losses, and the PHML can be fabricated by periodic stacking of metal and dielectric layers.
I. INTRODUCTION
Remarkable progress has been made over the past decade in achieving optical super-resolution imaging[1] thatbreaks the traditional resolution limit, offering substantial advances for the imaging and lithography systems thatare the corner stones of modern biology and electronics. Metamaterial-based superlenses have shown the super-resolution imaging in the near field[2, 3], but magnification of subwavelength features into the far field has notbeen possible. Recently, super-oscillatory lenses[4] and hyperlenses[5, 6] have demonstrated to the far-field super-resolution imaging. However, both the super-oscillatory lenses and hyperlenses suffer from some limitations: thatsuper-oscillatory lenses come at a price of losing most of the optical energy into diffuse sidebands; and hyperlensesneed complex co-centrically curved-layer structures, which not only bring more challenges in fabrication, but alsofundamentally lack the Fourier transform function due to their inability to focus plane waves[7]. To avoid theselimitations, planar hyperbolic metamaterial lenses (PHMLs) were reported[8, 9] and have demonstrated capable ofproviding not only Fourier transform function by introducing a phase compensation mechanism[7], but also advantagesof super-resolving capabilities. However, so far most of the studies for the super-resolution imaging of PHMLs aredone in the near field [8, 9] (with the distances between source and image are much smaller than unit wavelength). ∗ Electronic address: [email protected] † Electronic address: [email protected] source2hyperbolicmetamaterialairair image1 ( ) b ( ) a source1 image2 xzy d x z FIG. 1: (a)The schematic diagram of our model. (b) The EFCs of hyperbolic metamaterials with | ε z /ε x | = 140 (blue solidline) and | ε z /ε x | = 5 (green dash line), and the EFC of air (red dash-dot line). Recently, P. Belov and co-authors proposed an alternative regime based on the Fabry-Perot resonance effect, andthus it is capable of transmitting images to far-field [10]. Nevertheless, the regime itself assumes that the thicknessof the PHML should be equal to an integer number of half-wavelengths [10]. In this paper, we demonstrate that thefar-field super-resolution imaging also arises in well-designed PHMLs when the Fabry-Perot resonance condition doesnot hold.Generally, achieving the far-field super-resolution imaging through a planar lens is very challenging. For thesubwavelength imaging, both evanescent waves and radiative waves are required, and their amplitudes should becomparable with each other in the image. If the amplitude of radiative waves is much larger than that of evanescentwaves, the subwavelength information taken by evanescent waves would be concealed by radiative waves, as thus theimaging system loses the super-resolution capability[11]. Therefore, the amplitude of radiative waves and evanescentwaves should be the same order of magnitude in the image region for superresolution. But satisfying this condition isgenerally very difficult for the far-field imaging, because of the inherent nature of evanescent waves with amplitudesexponential decaying in the air[12, 13]. One of the most striking properties of hyperbolic metamaterials is theirstrong ability to carry evanescent waves to far field [5], which enables PHMLs to be good candidates for far-fieldsuper-resolution imaging. To realize the far-field superresolution, however, a PHML should overcome at least fourdifficulties: First, beyond the Fabry-Perot condition, the PHML should be able to transmit radiative waves andevanescent waves from a source with high transmission and low loss. Second, it should be able to collect the radiativewaves and evanescent waves in the image, with the amplitudes of them being the same order of magnitude. Third,the distance between the source and image should be much larger than unit wavelength. Fourth, the resolution of thePHML must be robust against losses to a certain degree, including radiative losses and material losses.In this work, we present our design for a PHML that can be used to realize the far-field super-resolution imagingbeyond the Fabry-Perot condition, with the resolution less than 0 . II. MODEL
Our model is shown in Fig.1(a), in which a slab-like planar hyperbolic metamaterial with thickness d ( d is muchlarger than the wavelength, but doesn’t hold the Fabry-Perot condition) is positioned at 0 < z < d for imaging,two monochromatic line sources with infinity length along y direction are set very close to the lower interface andtheir images are formed above the upper interface at a distance of ∆ z . The distance between the two sources is ∆ x .The hyperbolic metamaterial has a matrix form permittivity ε = diag ( ε x , ε y , ε z ) with ε x = ε y , and ε x · ε z <
0, andso its dispersion relation k x /ε z + k z /ε x = k is hyperbolic [9, 14], where k = 2 π/λ is the wavevector, with λ thewavelength in air. The equi-frequency contours (EFCs) of hyperbolic metamaterials are hyperbolic. In the case of ε z < ε x >
0, two typical examples are shown in Fig.1(b). In this work, only the transverse electric (TE) mode(i.e., H = H y = 0) is considered. ( ) a ( ) b FIG. 2: (a) The distribution of | ~H | of a single source placed at the interface of PHMLs with | ε z /ε x | = 5 (left panel), and | ε z /ε x | = 140 (right panel). Both lenses’ width are d = 10 λ . (b) The distribution of | ~H | at the interface (along the white dashline in Fig2.(a)) for the PHMLs with | ε z /ε x | = 140 (blue solid line) and | ε z /ε x | = 5 (red dash line), respectively. III. CHOICE OF HYPERBOLIC METAMATERIALS
For a PHML, there are many choices of the transverse axis’ direction of the hyperbolic EFC. In this work we choosethe one that is parallel to the z axis (corresponding to ε z < ε x > z axis (in this case we have ε z > ε x < k x < √ ε z k would exponentially decay in the lens, so this caseis unsuitable for far-field imaging. If the transverse axis is tilted of an angle with z axis[15], this kind of PHML canbe directly used as a Fourier transform lens, but it’s also unsuitable for far-field superresolution because of its largelateral deviation of the image through a long distance.After fixing the transverse axis, i.e., ε z < ε x >
0, the value | ε z /ε x | determining the eccentricity of thehyperbola is the core parameter of our system. We choose large | ε z /ε x | for our PHML. There are two reasons forour choice. First, if | ε z /ε x | is not large enough, the image will be formed at a long distance ∆ z from the interfaceof the lens[19]. In this case the evanescent waves would be too small to carry subwavelength information for thelong-distance imaging. Second, for a PHML, larger | ε z /ε x | corresponds to better self-collimation of light, which isvery significant for the super-resolution imaging. Hyperbolic metamaterials with larger | ε z /ε x | have flatter EFCs, andso the light diffraction can be lower when lights propagate in the hyperbolic metamaterials with larger | ε z /ε x | . Toshow this, we present two typical examples, i.e., one case is | ε z /ε x | = 5 (with ε x = 1 . , ε z = − . | ε z /ε x | = 140 (with ε x = 1 . , ε z = − x = 0, d = 10 λ . Their magnetic field intensity distributionscalculated by Green’s function method[12, 16, 17] are shown in Fig.2, in which Fig.2(a) shows the distributions inthe x - z plane, and Fig.2(b) shows the distributions in an imaging region along the upper interface of the hyperbolicmetamaterials. Comparing the left panel with the right panel in Fig.2(a), we can see the right one (corresponding tothe latter one with | ε z /ε x | = 140) has much better capable of self-collimation. Also, from Fig.2(b), we can see theintensity of the latter one’s image is more than 35 times larger than that of the former one, which indicates the latterone can focus more waves in the image. Therefore, hyperbolic metamaterials with larger | ε z /ε x | are more suitable forsuper-resolution imaging. IV. FAR-FIELD SUPERRESOLUTION
In Fig.2(b), the full width half maximum (FWHM) of the intensity distribution for the case of | ε z /ε x | = 140 (bluesolid line) is about 0 . λ that is much smaller than half of wavelength. In addition, the distance between the sourceand the image are 10 λ that is much larger than unit wavelength, suggesting our system has the ability for the far-fieldsubwavelength imaging.To show it more clearly, we use such a PHML for imaging. As shown in Fig.1(a), the two sources are spaced at adistance of ∆ x = 0 . λ , the thickness of the PHML is d = 10 λ . For the hyperbolic metamaterial, the real part of the ( ) d ( ) a ( ) b ( ) c ( ) e z ( ) f FIG. 3: (a)-(c) The field distributions along the upper interface of our PHML. (a) The real part of ~H tot (blue solid line), ~H rad (red dash line) and ~H eva (green dash-dot line). (b) The imaginary part of ~H tot ] (blue solid line), ~H rad (red dash line) and ~H eva (green dash-dot line). (c) The distribution of | ~H tot | (blue solid line), | ~H rad | (red dash line) and | ~H eva | (green dash-dot line).(d)-(f) are the distribution of | ~H rad | , | ~H eva | and | ~H tot | in the image region, respectively. In (a)-(f), γ = 0 . relative permittivity is Re [ ε x ] = 1 . Re [ ε z ] = −
210 (i.e., Re [ ε z ] / Re [ ε x ] = 140), and the imaginary part is muchsmaller than the real part, which is phenomenologically introduced by Im [ ε x ( z ) ] = γ | Re [ ε x ( z ) ] | , (1)where γ is the loss coefficient. Based on the Green’s function method, we calculated the field ( H rad , H eva and H tot ) distributions and the intensity distributions in the image region, as shown in Fig.3(a)-(e). Here H rad , H eva and H tot = H rad + H eva represent the radiative waves, the evanescent waves, and the total field, respectively. Notethat H rad , H eva and H tot are all complex. Figure 3(c) shows our PHML’s ability for the far-field super-resolutionimaging. In this figure, we show the intensity distributions of | H rad | , | H eva | , and | H tot | in the image region thatis very close to the upper interface of the PHML (∆ z ∼ x = 0 . λ is smaller than 0 . λ . Furthermore, we stress that thedistance between the sources and the images is d + ∆ z ∼ λ , which is much larger than λ . So our PHML does havethe ability for the far-field super-resolution imaging.Physically, the reason that our PHML exhibits the far-field super-resolution capability can be explained as follows.First, as shown in Fig.3(a) and Fig.3(b), both radiative waves and evanescent waves from sources can be transmittedthrough our PHML with high transmission (although the thickness of the PHML is very large), and they can becollected at the image region. Second, also from Fig.3(a)-(c), we can see the amplitudes of the radiative waves and theevanescent waves are comparable with each other, and they are constructive at the image peaks but simultaneouslydestructive at the side peaks. m i n x ( ) d ( ) c ( ) a ( ) b FIG. 4: (a) The transmission rate of the PHML with different thicknesses. (b) The | H tot | distribution at the interface of thePHMLs with different thicknesses. Other parameters in (a) and (b) are the same as Fig.3. (c) The transmission of the PHMLwith ε x = 1 . ε x = 1 . ε x = 2 . x min vs ∆ z with ε x = 1 . ε x = 1 . ε x = 2 . d = 10 λ ,and eccentricity | ε z /ε x | = 140 are fixed. Evanescent waves play a very important role on the super-resolution. Without evanescent waves, our PHML wouldlose its super-resolution capability. In Fig.3(c), the images formed only by the radiative waves become a large spotwithout any subwavelength information that totally cannot be resolved. This fact can also be found in Fig.3(d)-(f). InFig.3(d), the images formed only by the radiative waves cannot be resolved at all in the image region; while in Fig.3(f),the two images can be resolved well in the image region within a certain range of ∆ z with the help of the evanescentwaves. As ∆ z increasing, the amplitudes of evanescent waves would exponentially decay that lessens the resolution ofthe PHML. As shown in Fig.3(f), when ∆ z > . λ , the two images cannot be resolved at all. To show the increaseof ∆ z can impair the resolution of the PHML (i.e., the minimal distinguishable distance ∆ x min would become largeras ∆ z increases), ∆ x min versus ∆ z are plotted in Fig.4(d) (blue line). When ∆ z = 0, our PHML can achieve itshighest resolution, with the minimal distinguishable distance ∆ x min = 0 . λ . In Fig.4(d), whether the images canbe resolved or not is following the Rayleigh criterion[18]. In this work, we choose the criterion of distinguishable twoimages should be smaller than the critical contrast I sad /I max = 75%, where I sad and I max represent the saddle pointintensity and maximum intensity, respectively. Our criterion is a little bit stricter than that used in ref[4]. V. FAR-FIELD SUPERRESOLUTION BEYOND THE FABRY-PEROT RESONANCE CONDITION
Our PHML can be capable of far-field superresolution beyond the Fabry-Perot resonance condition. To see it, herewe would like to show the far-field superresolution ability of several our PHMLs with different thicknesses (i.e., d = 9 . . ... , 10 . λ ) as shown in Fig.4, in which the Fabry-Perot resonance condition is satisfied only when d = 9 . λ , whileother thickness values correspond to the nonresonant case. From Fig.4(b), by comparing with the | H tot | distributionsin the resonant case and those in the nonresonant case, we can see their | H tot | distributions are very similar and allthe PHMLs are able to superresolution. Also, from Fig.4(a), we can see that the transmissions (here we only plot the o ( ) b ( ) a m i n x FIG. 5: (a) The normalized intensity distribution at z = 10 λ with γ = 0 .
01 (blue solid line), γ = 0 .
05 (red dash line) and γ = 0 . x min vs | ε z /ε x | with γ = 0 .
01 (blue solid line), γ = 0 .
05 (red dash line) and γ = 0 . radiative waves) of the PHMLs with different thicknesses are similar, too. These similarities of superresolution andtransmission between the PHMLs with different thicknesses indicate that the far-field super-resolution ability of ourPHMLs is not linked to the Fabry-Perot resonance effect, but related to the nonresonant impedance matching.We find in our case, better (nonresonant) impedance matching corresponds to better superresolution, as shown inFig.4(c) and (d). For the PHMLs in Fig.4(c) and (d), the eccentricity | ε z /ε x | = 140 is fixed, but the values of ε x aredifferent, i.e., with ε x = 1 . ε x = 1 .
5, and ε x = 2 .
0, respectively. Other parameters are the same as Fig.3. Clearly,when ε x becomes closer to unit, the nonresonant impedance matching will become better. From Fig.4(d) we canfind when the impedance is matching better, the minimal distinguishable distance ∆ x min becomes smaller, and thelargest ∆ z becomes larger, which corresponds to better superresolution. In the following, we will show that to realizea PHML with quasi-perfect nonresonant impedance matching is not very difficult. VI. ROBUST AGAINST LOSSES
The superresolution of our PHML has robustness against losses, which is also important for imaging system. Tosee it, three PHMLs with three different losses are respectively used for imaging. The losses are phenomenologicallyintroduced by Eq.(1), with the loss coefficient γ = 0 .
01, 0 .
05 and 0 .
1, respectively. The intensity distributions of thethree PHMLs’ image region at ∆ z = 0 are shown in Fig.5(a). In Fig.5(a), the parameters are the same as Fig.3, exceptthe loss coefficient γ . In this figure, the maximum intensity value of each intensity distribution has been normalizedto unit. From Fig.5(a), we can see even for a big loss ( γ = 0 . | ε z /ε x | , the robustness against losses are also valid, as shown in Fig.5(b).From Fig.5(b), we can also see that the larger | ε z /ε x | , the higher resolution of our PHML, which agrees with ourprevious discussion.The fact that the superresolution ability of the PHML is robust against losses doesn’t mean the losses has nothingto do with the transmitted light waves. Actually, the losses play a very important role in the amplitude of thetransmitted light waves. For such a thick PHML (e.g. d ∼ λ ), the intensities of the transmitted light waves withdifferent losses can differ by orders of magnitude. For instance, in Fig.5(a), the light’s intensity magnitude in the casewith γ = 0 .
01 can be ∼ times larger than that with γ = 0 .
1. Here we would like to emphasize that, althoughthe losses can lead to large attenuation of signal strength, the losses don’t impair the superresolution. This result ismeaningful. According to it, when using some techniques such as loss-compensation to overcome the losses, we canpredict that the PHML will retain its superresolution ability.
VII. ANISOTROPIC LAYERED STRUCTURE FOR PHMLS
Our PHMLs can be realized by anisotropic structure comprising alternately layered metal and dielectric films, asshows in Fig.6(a). The thickness of the cell (a cell means a combination of one metal layer and one dielectric layer)is much smaller than the wavelength. Using the transfer-matrix method and imposing the Bloch theorem[14], the zxy
MetalDielectric o ( ) b ( ) a FIG. 6: (a) The schematic diagram of the effective hyperbolic lens consisting of alternately layered metal and dielectric layerswith total thickness d . (b) The normalized distribution of | ~H tot | at z = 10 λ with N = 200 , , , , effective permittivity of such a layered structure can be obtained as[8] ε x = ε m f m + ε d f d and ε − z = f m /ε m + f d /ε d .Here ε m and ε d are the permittivities of the metal and dielectric, respectively. f m and f d = 1 − f m are the fillingfactors for metal and dielectric layer, respectively. In realization, the number of cells of the layered structure is finite.For a practical PHML, the number of cells has much influence on the super-resolution imaging. To see it, here wepresent a comparison of the | ~H tot | distributions at the upper interface of an ideal PHML with ε x = 1 . ε x = − N in Fig.6(b). All the PHMLs in Fig.6(b) have a fixed thicknesses d = 10 λ . For the practical PHMLs with different cell number N , the thicknesses of each cell d/N are different, butin each cell the filling factors f d = 0 . f m = 0 .
48 and the permittivities ε d = 12 . ε m = − .
16 + 0 . i are fixed.In order to show the resolution ability of each PHML, the maximum intensity value of each intensity distribution inFig.6(b) is also normalized to unit. From Fig.6(b), we can see that the superresolution can be reached with the cellnumber N = 300, and when N increases, the distribution of | ~H tot | of the layered structure becomes more similar tothat of the ideal one, which is expected.Next, we will discuss the feasibility of the practical PHML achieves higher resolution. According to Fig.5(b), wecan see that the resolution can be 0 . λ or even smaller when | ε z /ε x | > f m , f d , ε m , ε d ) for the multilayered structure as shown in Fig.6(a). For example, for the 1000-cellmultilayered structure shown in Fig.6(b), with the parameters remaining unchanged except the filling factors, i.e., theybecome f m = 0 .
545 and f d = 1 − f m = 0 . ε x and ε z of the 1000-cell multilayeredstructure become about 0.037 and -60.31, respectively, and the eccentricity | ε z /ε x | is about 1638. Using such 1000-cellmultilayered structure as the PHML, we find the resolution is about 0 . λ . VIII. CONCLUSION
In conclusion, we present an approach to achieve superresolution at a long-distance (10 λ ) through a PHML beyondthe Fabry-Perot resonance condition, which can distinguish two sources with distance 0 . λ . The resolution of ourimaging system is robust against losses. The PHMLs can be structured by periodic stacking of metal and dielectriclayers. Acknowledgments
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