An imaging mechanism for demagnifing features above wavelength into desired images beyond the diffraction limit is proposed in this letter. The super resolution ability (about two times and even more that of diffraction limit) arises from the surface plasmon wave excitation and amplification associated with metallic grating structure. Two specifically designed masks are projected to the grating surface from both sides, at one of which the superimposed field forms the desired images. Conceptually formalism of the imaging process is presented using spatial Fourier analysis and illustrated with numerical simulations.
© 2008 Optical Society of America
Conventionally, resolution in optical imaging instruments is determined by the wavelength (λ) and numerical aperture (NA) in the form of K*λ/NA, where K is a constant depending on the definite imaging process and resolution criteria. Usually, the minimum resolvable features is about half of a wavelength due to the finite value of NA for technical reasons. This resolution limit can be overcome for optical imaging occurred in the near field region. In 2000, Pendry proposed the concept of super lens with a thin metal film where evanescent waves delivering information of object’s small features can be amplified and account for imaging beyond the diffraction limit . About 1/6 wavelength wide pitches are imaged in experiments [2, 3]. The most unwelcome aspect of super lens may be the fact that objects and images are confined in the near field, usually much smaller than a wavelength. There have been some effects made for extending super resolution imaging process into the far field. Comprising multi layers of metal and dielectric in a cylindrical shape, hyper lens give rise to the magnified imaging between the inner and outer lens’ surface, which can be observed with the help of a microscope [4, 5]. But the aperture of Hyper Lens is limited to wavelength scale for the simple reason of metal absorption and hence decreased resolution. In addition, it’s hard to deal with objects and images localized at two greatly curved surfaces. Stéphane Durant et al. proposed and illustrated both numerically and experimentally a far field subwavlength imaging method with specially designed metallic grating structure [6, 7]. The key point is to acquire the evanescent light which are amplified and converted into propagating state by the metallic grating. Numerical combination of evanescent and propagating waves delivers restoration of subwavelength objects. Up to now, the investigation of super resolution imaging methods are usually characterized with just reproducing the object in the near field or detected in the far field with additional numerical processing. The reversed super resolution process for demagnifying objects, which are preferred in a great deal of applications like lithography, optical storage etc, has not been presented yet.
In this letter, a mechanism of projecting optics for demagnifying large structures into nano scales is proposed by using a subwavelength metallic structure. Conceptual imaging formalism is presented with Fourier analysis method. Numerical simulation of illustrative examples demonstrates the validity for realizing diffraction limited features of electromagnetic field distributions.
2. Formalism and analysis of demagnifing super resolution imaging
2.1 Description of the conceptual imaging setup
The overview of the schematic of demagnifing super resolution imaging optics (Fig. 1) looks like a 4π confocal microscope. Two oil immersion microscope objective lenses are positioned oppositely to each other with a common optical axis. Localized at the object planes of objective lenses are two masks with specifically designed micro-scaled features. They are illuminated with coherent monochromatic light. The light is set to be transverse magnetic polarized with the magnetic field perpendicular to the schematic surface. Transmitted light through the two masks are projected to planes close to a metallic grating with subwavelength-scaled period. The key point for subwavelength resolution is that the metallic structure with specifically designed parameters. Evanescent waves in the form of surface plasmon can be excited by light incident from the left side due to the addition of grating momentum. The superimposing of propagating and evanescent waves close to the metallic grating yields nano images with desired electromagnetic intensity distribution. If photo resist is spun at the imaging plane, it is possible for nano lithography.
2.2 Conceptual formalism of super resolution imaging
We do not intend to present the rigorous full wave analysis of this imaging process, since the light experiences great polarization modifying with a high NA objective lens . The purpose of this paper is just to give a conceptually analytical and numerical illustration of the super resolution imaging for this method. A simplified scheme of far field super resolution imaging process is presented with a one-dimensional metallic grating which extends infinitely in the y direction (Fig. 2(a)). Two TM polarized counter propagating waves, coming from the two masks through objective lens, are incident on the corrugated and smooth grating surfaces, respectively. The magnetic fields of the two incident light at the grating and substrate side are denoted as g(x,z) and f (x,z), represented in Fourier spaces spectrum as g(x,z)=∫G(kx)exp(ikxx+ikzz)dkx and f(x,z)=∫F(kx)exp(ikxx-ikzz)dkx with , and n being the refractive index of surrounding media. For the convenience of analysis, the numerical aperture of objective lens is assumed to be n as well. The total magnetic field at the plane (z=0) just close to the grating’s smooth surface is written as h(x,z=0)=∫H(kx)exp(ikxx)dkx. The Fourier spectrum space is divided into multiple regions (Fig. 2(b)), denoted as Hm with kx ranging from m*nk 0 to (m+1)*nk 0, with m being integer number. So the two incident spatial Fourier spectra G(kx) and F(kx) are confined in the Fourier region from -nk 0 to nk 0. The Fourier spectrum H(kx) of EM field close to the grating, however, can be extended outside this region with evanescent components of surface plasmon polaritons due to the light diffraction of grating .
The generalized relation between H(kx) in the mth Fourier spectrum region and G(kx) and F(kx) can be written in matrix formalism as
|X〉 (X denotes G, F and Hm) means vertical vector of Fourier spectrum series [X(k x1),X(k x2), …, X(kxu), …, X(kxN)] with kxu=unk 0/N for |G〉 and |F〉, and kxu=(m+u/N)nk 0 for |Hm〉. I is the unit matrix and δ 0m is Kronecker function. Rm and Tm are matrix operators for transmission and reflection imposed on |G〉 and |F〉 with the conversion from the 0th region to the mth region. Usually, Rm and Tm are sparse matrix and the nonzero elements ruv and tuv are diffracted amplitude of transmission and reflection happened around the grating who meets the relation Kxu+qkg=kxv+mnk 0 with q being the diffraction order, kg=2π/d and d the grating period. They just mean the transmitted and reflected complex amplitude for incident light with transversal wave vector kxu diffracted to kxv+mnk 0.
Assume that the desired electromagnetic field distribution at the image plane is confined in the Fourier space ranging from -2nk0 to 2nk0. So, we can precisely determine the incident Fourier spectrum vector |G〉 and |F〉 for the desired image spectrum vector |H 0〉 and |H 1〉 by the 2N×2N matrix equation
The generation of the two incident lights defined by |G〉 and |F〉 be realized by designing appropriate binary masks with the computer generated hologram (CGH) method. The feature size of masks is in wavelength scale and filtered for removing noise accompanied in CGH . Therefore, images with subwavelength features can be obtained with the help of two immersion objective lens and metallic grating structure. In fact, the process of projecting the masks to the metallic grating structure maybe requires full wave analysis due to the polarization changing effect in a high NA objective lens . The detailed analysis would be published elsewhere.
2.3 Imaging aberration analysis
But Eq. (2) does not promise evanescent waves with kx outside the region [-2nk0, 2nk0] can be controlled and contribute positively to the imaging process. The best and direct solution seems to be suppressing them as small as possible by designing appropriate grating parameters. Generally, this implies high conversion to useful Fourier regions and slight excitation of waves outside this region. It’s reasonable only considering the nearest two Fourier spectrum regions with m=2 and 3. For desired imaging Fourier spectrum vector |H 0〉 and |H 1〉, the Fourier amplitude in the two regions can be calculated by
To give a general evaluation of the aberration from larger kx, the Fourier spectrum error function is defined as the maximum amplitude occurred in the 2th and 3th Fourier resgion for imaging EM field defined by single wave vector kx. This is just the maximum element in each column of the multiplied matrix in Eq. (3).
Usually, the grating vector kg should be smaller than 2nk 0, promising that all transversal wave vector components in 1th region can be generated. Thus the power of imaging two closely positioned lines can be doubled, compared with the greatest resolution available in conventional imaging optics. Imaging with kg>2nk 0 is also possible with the evanescent imaging waves being shifted to region from kg-nk 0 to kg. This would result in greater resolution due to the larger maximum transversal wave vector. But the existence of blind spectrum region from nk 0 to kg is deleterious for imaging some structures.
Now, we summarize the process of designing masks for super resolution imaging here. First, metallic grating with appropriate geometric parameters is designed and the plane wave conversion matrix in Eq. (2) and Eq. (3) can be calculated with rigorous coupled wave analysis. The aberration value can be obtained with Eq. (3) and checked to promise the imaging property. Then, the spatial Fourier spectrum of the desired field distribution function at the image plane is divided into two regions and sampled in representation of vector |H 0〉 and |H 1〉. Using Eq. (2), the two incident light |G〉 and |F〉 can be readily obtained. Finally, two masks can be designed with CGH technique. It is particularly worth to note that the patterns on the mask are not like the desired pattern projected to the image plane. The improved resolution beyond the diffraction limit comes from two objective lens and metallic grating structure rendering conversion from propagating waves to evanescent ones. It seems much more like the optical version of the synthetic aperture technology employed in the radar field.
3. Numerical illustrative examples
Here is a designed example for numerical illustration of super resolution imaging. Figure 3 plots the transmitted and reflected magnetic amplitudes of light converted to transversal wave vector kx for light incident from corrugated and smooth side of grating, calculated by rigorous coupled wave analysis (RCWA). The inset in Fig. 3(b) is the metallic structure with period 247nm, grating depth 30nm, metallic ridge’s width 100nm and substrate depth 55nm. The illuminating light wavelength is 376nm and the permittivity of the metal and dielectric are -3.16+0.8i and 2.31, respectively. For the convenience of analysis, we assume the grating structure and incident light field are both symmetric with respect to the x=0 plane. The cases of imaging arbitrary objects can be expressed as the summation of symmetrical and asymmetrical functions. Only the Fourier component with positive kx is considered here. The grating vector is set to be nk0. So the element Tm in Eq.(1–3) is crossed diagonal matrix with diagonal element vectors [tm(k x1),tm(k x2),…,tm(kxN)] and [t -m-1(kxN),…,t -m-1(k x2),t -m-1(k x1)]. Similar form is exhibited for Rm. Element tm(kx) is the transmitted complex amplitude for incident plane wave from the grating side and rm(kx) is the reflected complex amplitude for plane wave incident from the substrate side. Figure 3(a) and Fig. 3(b) show that the transmitted and reflected light is mainly localized in the Fourier region from 0 to 2nk0, which can be manipulated for the generation of desired spatial electromagnetic field distribution. We can also clearly see the role of surface plasmon behavior in metallic film. Plotted in the top inset of Fig. 3(a) are transmitted amplitudes diffracted into variant kx by the free standing metallic grating. Evanescent waves with comparable amplitude are excited in the 2th and 3th Fourier region, bringing the unwelcome source of large imaging aberration. This problem can be greatly relieved using a metallic thin film with proper thickness, through which evanescent waves can be amplified for those in the imaging Fourier region and damped in the aberration region, as shown in the bottom inset.
The evaluations of spectrum error function are presented in Fig. 3(c). We are glad to see that the spectrum error amplitude is usually smaller than 0.2. There is an obvious fact that image in the propagating wave region usually displays much better performance. This is because reflected light dominates the imaging process in this case and the excitation of waves in higher Fourier regions is very small, as can be seen from Fig. 3(b). The difference of imaging aberration value occurred in the 0th and 1th Fourier region can be further understood for the sinusoidal fringes imaging. Plotted in Fig. 3(c) inset are two fringes with period of 154nm and 77nm (kx=0.8nk 0 and kx=1.6nk 0), corresponding to features above and beyond the diffraction limit respectively. The imaging of 154nm period fringe is very successful, reproducing the sinusoidal profile with high fidelity. For the fringe period far beyond the wavelength, it looks not so good for its greater imaging error. But the fringes are clearly resolved with high contrast and accurate positions. This is very important to image arbitrary objects with high fidelity. In addition, there is a large error near nk0. The intrinsic noise for the huge spike error comes from the decreased transfer amplitude for kx near to nk0, nearly to zero for -2th order as can be seen from Fig. 3(a). The magnitude of excited light for this range of kx turn almost nearly down to the level of that in the aberration range with kx larger than 2nk0. So the error goes up rapidly. This may deliver some regular lattice patterns in the background of the image, which can be deleted by eliminate the input light around nk0.
Now we see the imaging of two spikes extending infinity in the y direction and spaced with distance much smaller than wavelength. As shown in Fig. 4(a), the Fourier spectrum of the two nano spikes with full width at half maximum (FWHM) of 30nm and distance of about 140nm (plotted in Fig. 4(b)) goes far beyond the propagating waves region and can not be imaged with diffraction limited optics. The calculated spatial spectrum of illuminating light G and F are inherently confined in the propagating region. At the imaging plane close to the metallic grating, the spatial Fourier spectrum agrees with that of object for wave vector smaller than 2nk 0 (Fig. 4(a)). Outside the imaging Fourier region is aberration spectrum with much smaller amplitude. Two clearly resolved spikes can be realized with increased FWHM width of about 54nm and slightly shifted positions, due to the incomplete transfer of all the Fourier components. Assuming diffraction limited imaging optics with numerical aperture being the refractive index n, the calculated images of the two spikes are fully overlapped (control image in Fig. 4(b)). In Fig. 4(c) and Fig. 4(b), we can also see the image evolution of the two spikes as going away from the grating substrate surface. The effective image depth is about 30nm. This occurs because of the strong decaying property of evanescent waves contributing to the super resolution imaging. Considering the good visibility of the simulated image (Fig. 4(c)), closer spikes can be resolved as well if practical criterion of resolution is employed. According to our simulation, imaging two spikes with distance down to 90nm is available with Rayleigh criterion. And higher resolution power is possible by taking grating with smaller period. In addition, the distance between two spike images is a bit larger than that of the object (the ideal image). It is believed the finite length of the spatial spectrum of the ideal image accounts for this aberration (see Fig.4(a)). It can be expected that the images would appear at the right positions if we can reproduce much more Fourier components or the features of the objects are not so far beyond the wavelength. You can also pre-shift the designed object to compensate this effect.
In summary, a mechanism for super resolution imaging in the far field is proposed, which renders demagnification of larger mask features in wavelength scale to subwavelength images with advantages of large imaging area, projecting operation and the most important, greatly improved resolution of two times and even larger that of diffraction limit. Conceptual analytical formalism of imaging process is presented and illustrated with numerical simulations for clearly imaging sinusoidal fringes (period 77nm) and two subwavelength spikes (30nm wide and 140nm separation) with illuminating light at 376nm. This method is believed to have many potential and exciting applications including nano lithography, optical storage, biomedical sensing etc. In addition, the super resolution optics presented here is presented and illustrated for imaging one-dimensional objects. But the extension to two-dimensional cases is possible, but seems much more challenging due to the polarization issues and two-dimensional diffraction analysis.
This work was supported by 973 Program of China (No.2006CB302900) and 863 Program of China (2006AA04Z310).
References and links
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