It is shown that a superlens with unmatched dielectric medium (termed as “unmatched superlens”, UMSL) can enable super-resolution imaging in a broad frequency range. The broadband UMSL comprises dielectric-metal-dielectric layers with appropriately designed thickness. Numerical simulations demonstrate that the deficiency of imaging due to the mismatched permittivity of the metal and dielectric can be improved with the existence of two waveguide modes of the UMSL structure. The frequency band and quality of super resolution imaging are mainly determined by the two modes, which deliver the amplitude and phase modulation of transmitted evanescent waves in a wide transversal wave-number range.
© 2008 Optical Society of America
In 2000 Pendry  first proposed and deduced mathematically that perfect lens (superlens) can be realized with a thin slab of negative refraction media  (NRM). For TM polarized light, under the electrostatic limit, the metal slab with negative permittivity has the ability to substantially transport a broadband of evanescent waves from the bottom side to the top side of the slab. This is the key point in super lens imaging with a resolution beyond the diffraction limit. Ramakrishna et al. subsequently designed another superlens based on a multilayer metal-dielectric stack with thin metal layers to overcome the losses of metal slab . Cai et al. proposed a tunable superlens based on a metal-dielectric composite film that can practically operate at desired wavelength in the visible and near-infrared ranges . Optical imaging with resolution beyond the diffraction limit was recently experimentally demonstrated using a silver superlens [5–6] and a single-crystalline SiC superlens . However, these superlens are all based on the condition that the superlens and the surrounded dielectric have a matched permittivity (Re(εm)≈-εd) at the working wavelength, where the εm denotes the permittivity of metal and εd is the dielectric. This would confine their applications because the suitable materials in the real world are limited in some cases. Recently some works came down to the permittivity of metal and dielectric unmatched case [8–11]. Wang et al. reported that subwavelength imaging can be obtained even though the permittivity of metal and dielectric are not matched, but this occurs as the effective transversal permittivity tends to zero or the vertical one approaches infinity . Bloemer et al. recently designed a broadband superlens with a metal-dielectric photonic band gap structure, which is based on a series of strongly coupled metal-dielectric Fabry-Pérot cavities . While based on our following work, it seems that there exists another alternative with much simpler structure to obtain a broadband superlens effect. In this paper, we present the demonstration and analysis of the unmatched superlens (UMSL) with dielectric-metal-dielectric layer structure, being capable of imaging subwavelength objects in a broad frequency range.
2. The unmatched superlens
The proposed structure of the UMSL is a dielectric-metal-dielectric (DMD) single layer as shown in Fig. 1. The metal film is surrounded by two films with identical material property and thickness, resembling the SL originally proposed by Pendry . The slit source is positioned closely at the interface of the lens with air and the images are observed at the plane above the end face of the lens. Note that the permittivity of the metal and the dielectric in our work are not matched at the work wavelength λ 0.
At a metal-dielectric interface, surface plasmon polaritons (SPPs) exist due to the coupling of TM wave with k>nk0 and the induced collective excitation of free electrons at the boundary. A metal film can enhance evanescent wave by the SPPs was confirmed in previous experiment work . The field transmission factor of the UMSL has been carried out using the enhanced transmittance matrix approach . In this paper, the metal and dielectric materials are gold and SiC respectively. The Drude model, εr(ω)=ε∞-ω2p[ω (ω+iVc)]-1, was used to describe the permittivity of gold, where the parameters, ε∞=9.0, ωp=1.3673×1016 rad/s, Vc=1.0027×1014 rad/s. The permittivity of SiC refers to Ref. . We firstly discuss thickness influence of the UMSL (SiC/Au/SiC) surrounded by air with a TM polarized light at 633 nm. Figure 2, we calculated the transfer function of the 80nm half-pitch object for various Au film thicknesses with three different SiC thicknesses (10 nm, 15 nm and 20 nm). Clearly, the optimum thickness of the Au is mainly determined by the optical wavelength, period of the object, and the thickness of SiC. With the decrease of the thickness of SiC, the optimum thickness of gold decreases as well while the transmission of the corresponding lens increases simultaneously. This is in good agreement with Ref. . Considering the fabrication limitations and the imaging quality, we choose the thickness of SiC and Au being 15 and 22nm, respectively.
3. Numerical Simulation and Discussion
3.1 Optical transmission of evanescent waves
The optical transmission of the UMSL is presented in Fig. 3(a) as a function of the optical wavelength and the transverse incident wave-number through this UMSL. Two surface p-polarization SPP modes can be distinguished due to the coupling between the surface modes at the two sides of the metal film. The two modes approach closely at a specific optical wavelength (about 560 nm this case with well matched permittivity of Au and SiC). Around the two mode dispersion curves, the Au film enhances the evanescent waves in transmission. It also shows in Fig. 3(a) that within the wavelength band (500nm–650 nm), the amplitude of the propagating waves is not obviously suppressed or enhanced. Generally, the evanescent wave enhancement reduces with the increase of transverse wave-number. But there exists an effective evanescent waves restoration range confined by the wave number position of the two SPP modes. In addition, it is worth mentioning that the broadband of k in which there is an enhancement of evanescent waves can be tuned by changing the thickness of dielectric and metal.
The white-dots lines ((1)–(3)) shown in Fig. 3(a) denote the first order wave-numbers of periodical object with half-pitch of 110 nm, 80 nm, and 60 nm, respectively. The imaging resolution and the corresponding results will be discussed later. Since evanescent waves delivering the object information can be enhanced around the two SPP modes, it seems to be reasonable to believe that resolved images can be obtained in a wide frequency range. In Fig. 3(b) and Fig. 3(c) we gave the distribution of the transmission factor versus of the in-plane wave-number and the phase distribution of the transmission of this UMSL at six different optical wavelengths from 500nm to 700 nm. Obviously, the evanescent wave transmission (corresponding to object pitch of 160nm) with large enhancement factor can be obtained within the frequency range of 500~700 nm. One would naively think that the super-resolution imaging of objects with pitch larger than 160nm can be obtained in this range as well. Our following work shows that this is not true for some reasons of phase modulation and absorption of light.
3.2 Super resolution imaging of single slit
To demonstrate the imaging effect for variant wavelength, numerical simulations are performed by employing the angular spectrum theory as did in Ref . The object is placed at the entrance surface of this UMSL. It is necessary to note that the intensity of light field is 1 at the slit position and 0 elsewhere.
Here we first give the simulation of imaging single slit of width 50nm for variant wavelength ranging from 500nm to 700nm, as shown in Fig. 4. The imaging plane is intuitively set to be 40nm away from the end of the UMSL. It is shown that the full width at half maximum (FWHM) of image is about 150nm for wavelength (500nm ~633nm) with a gauss like profile and small side lobe. In addition, the side lobe for 500nm wavelength is somewhat high due to the excitation of surface plasmon mode close to the wave-number in SiC, shown in Fig. 3(b). For wavelength larger than 650nm, the image profiles widen greatly and the level of side lobe becomes serious. So we can see that broadband super resolution imaging can be effectively obtained through the UMSL.
Then we try to give a discussion of the position of the image plane. As an example, we present the calculated images with UMSL at different planes for a two-line slit sources (50 nm width and center to center separation 160 nm) illuminated by light of wavelength 500 nm and 600 nm, respectively, as shown in Figs. 5(a) and 5(b). The image plane is placed at 0, 10, 30, 50 and 80 nm away from the end face of the lens. Due to the evanescent waves’ contribution of imaging in super lens, it is not justified to give a clear definition of imaging plane, which was demonstrated in Fig. 5. That is to say we can not observe the process of obvious focusing on the imaging plane and defocusing behavior with strongly blurred image as going away from it. The fact is that resolvable nano images can be realized in the region confined by a distance from the end of the lens. The distance is determined by the objects and wavelength. For instance, the slit pair can be resolved in a distance of about 50nm from the end of lens for wavelength of 633nm and slit width of 50nm and separation of 160nm, as shown in Fig. 5. The maximum imaging distance decrease to about 30nm for wavelength 500nm. Usually the plane close to the end face of the lens gives much finer resolution due to the restoration of more evanescent wave information. But it is not desired to have an image closely confined to the lens, from the viewpoint of practical application. Approximately, the imaging distance for the wavelength range from 500nm to 650nm is about 40nm, which is selected as the fixed observation plane in this paper.
3.3 Imaging slit pairs
Here we discuss the imaging property with slit pairs. First we present a group of simulation results for slit pairs with fixed slit width and variant center to center distance, as shown in Fig. 6. The objects are all 50 nm wide, the center to center slit separation is 220 nm (a), 160 nm (b) and 120 nm (c). As a comparison, the calculated intensity profile at the same plane without UMSL is plotted as well.
As it can be noted in Fig. 5, the position of maximum intensity depends on the wavelength, and it is not always matched with the position of the slit source center. This shift aberration would deliver some wrong information as we observe the images, but the definite information of the number and approximate position of resolvable slits (the shift value is usually smaller than 1/5 that of slit separation) can be concluded. So the criterion of resolving two points here can refer to the rule of Rayleigh method. That is to say, the two adjacent slits can be resolved as the level of the dip is less than 0.81 times that of the maximum intensity of the two slit image profile.
First, slit pairs with large separation (220nm) can be well resolved in a wide working wavelength range from 480nm to 650nm. Due to the failure of finding the permittivity of SiC at wavelength below 480nm, the correspondent image profiles are not given. But it is believed that small wavelength down to 450nm can distinguish the two slits as well. As the two slits separation get to 160nm, the imaging property decreases with lower contrast. The approximate imaging wavelength is about lying in a narrower band from 500nm to 633 nm. From Figs. 3(a) and 3(b), we can see the reasons. Evanescent waves corresponding to features of 120nm can not be restored effectively with very low intensity. This delivers the decrease of contrast of the images. For much closer line source with 120nm separation, the two slits become nearly irresolvable simply because that required evanescent waves are hardly restored.
In fact, a convenient way for approximately evaluate the frequency band of imaging is to see the cross positions of SPP dispersion curves and the transverse wave-number line of the required evanescent waves, as denoted in Fig. 3(a). For instance, the frequency band for imaging two-line source with 220nm separation is about from 450nm to 650nm. This agrees well with the white-dot line (1) between the two modes presented in Fig. 3(a). We can also conclude from Fig. 3(a) that narrower wavelength band occurs for slits pairs with smaller separation.
One important point worth to note is that only restoration of evanescent waves does not promise resolvable images. For instance, the imaging performance of wavelength slightly larger than 633nm gives bad results for 160nm separation case, contradicting with the transmission behavior in Fig. 3(b). This occurs because the phase information of the optical transmitted function of the UMSL also plays an important role for super-resolution imaging. Usually, high quality imaging requests the phase aberration of the optical system’s wave front less than π/2. The phase modulation in Fig. 3(c) indicates that great phase aberration occur around 4k0 for light at 650nm. This great phase modulation occurs as the transverse wave-number just passes the SPP mode position. This can be clearly seen from the analysis in the appendix.
One obvious phenomenon of the image shown in Fig. 6 is that position of maximum intensity usually differs from the correspondent slit’s central position. This occurs simply because the influence of adjacent objects. Usually, the maximum intensity position of the image profile does not match exactly with the central position of correspondent object. But this position shift effect seems to be obvious in the near field imaging proposed here, with great position shift value up to 20nm for slit width. We believe the great shift effect arises from the imperfect property of optical transmission function. As shown in Fig. 3(b), the UMSL does not give uniform transmission for object spatial information. A transmission peak appears for spatial frequency approaching the first SPP mode. Evanescent waves are effectively restored but often in a manner of over amplification. All these features would result in the deleterious effects for the image profile of a single slit object, such as the great side lobe and the fat profiles of main lobe which delivers the great shift of intensity peak’ position.
For larger separation in Fig. 6(a), this effect is nearly unnoticeable. As the image property gets worse with low contrast for some wavelength and slit pairs, the shift becomes obvious. Generally, the shift direction is not constant for variant objects, for instance, the images of 633nm in Figs. 6(a) and 6(b). But the shift changes slightly as the imaging plane moves (i.e. Fig. 5). It is believed that the influence of changing the image plane distance in the near field mainly delivers the broadening of slits image and decrease of visibility, but with little changes to the characteristic features of image profiles.
Another phenomenon in the imaging is that extra maxima, like a bump, maybe appear between the two maxima of intensity profile, which correspond to the two slits. Just like the position shift effect of the maximum image profile intensity, the bump also arises from the imperfect transmission of evanescent waves with over amplification and phase modulation. Fortunately, as an artifact of imaging, the level of bump usually is not high enough to give considerable influence for resolving the two close silts.
Now we analyze the line width of slit pairs’ influence to the super-resolution imaging. Figure 7 presented the image constructed by this UMSL for three different objects at four different wavelengths of 500, 560, 633, and 650 nm, respectively. The objects have the same slit separation (160 nm) while with different slit widths (a) 30nm, (b) 50nm and (c) 70nm and (d) 110nm.
The apparent imaging behavior is that the image profile keeps almost unchanged as we reduce the slit width. So the two slits can be still resolved with nearly the same contrast and similar imaging wavelength band. This can be well understood that evanescent waves with larger transversal wave-number would not contribute to the imaging process due to the finite transmission power of UMSL. But as the two slits is so wide that the images of each slit overlap greatly, the two slits become almost undistinguishable for some wavelength (Fig. 7(d)). Due to the same reason, the relative shift of the position of the intensity maximum gets larger. This implies that the details of object information far beyond the minimum resolvable features become increasingly blurred.
As a final note, the imaging frequency band is mainly determined with the two SPP dispersion curves. So thinner thickness of the dielectric and the metal yields much broader imaging frequency band. However, this would bring challenges for fabrication techniques and practical application as well.
In summary, we analyzed the UMSL with suitable thicknesses capable of super-resolution imaging in a broadband optical wavelength. The optimum thicknesses of the metal (Au) and its surrounded dielectric (SiC) can compensate for the deficiency to imaging caused by the mismatch of the permittivity of the metal and dielectric. The frequency band and quality of super resolution imaging are mainly determined by the two modes and the accompanied effect of amplitude and phase modulation of transmitted evanescent waves in a wide transversal wave-number range. The influences arising by the slit width and it separation for super-resolution are also discussed. The UMSL may find applications in subwavelength imaging, nanolithography, bimolecular sensing etc, where light with broad spectral line width is employed.
Here we present the demonstration of phase and transmission factor influence for super-resolution imaging with the same object presented in Fig. 6(b), as plotted in Figs. 8(a)–8(d) at the 670 and 700 nm wavelengths. Figures 8(a) and 8(c) are the original and suppressed phase curve of transmitted light through the super lens at 670 and 700 nm. The corresponding images in free space and by the lens with and without the phase correction are presented in Figs. 8(b) and 8(d). Figures 8(a) and 8(b) show that the transmitted evanescent waves with slight phase aberrations can deliver well resolved images in free space. Figures 8(c) and 8(d) demonstrate that the less range of enhanced evanescent waves limits the super-resolution image even phase was suitably modulated. These unambiguously demonstrate the phase and the transmission factor roles in the super-resolution imaging system mentioned above, which is not fully emphasized in many investigations.
The authors would be grateful to the reviewer’s advice for this work. This work was supported by 973 Program of China (No.2006-CB302900) and the Chinese Nature Science Grant (No.60778018 and No. 60736037).
References and links
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