1. Introduction

There have been extensive research efforts into negative index materials to realize imaging above the diffraction limit of light. In 1968, Veselago suggested double negative materials (DNG) with negative values of both permittivity and permeability (ε<0,µ<0) [1]. DNG are experimentally demonstrated in microwaves using split ring resonators with wire medium [2] and optical metamaterials are realized using a periodic structure [3]. Single negative materials (SNG) for permeability based on artificial magnetism are reported in the THz [4] and RF [5] regime. Negative refraction is also possible in a photonic crystal with the presence of a negative index [6]. In 2000, Pendry suggested the possibility of a near-field superlens (NFSL) made of nano-structured Ag slab, which has both high resolution beyond the diffraction limit and the amplification of evanescent waves in the UV region [7]. This is SNG with negative permittivity and positive permeability (ε<0,µ>0) only at the near field zone based on the electrostatic limit of surface plasmon resonance in a metal slab. Many theoretical and experimental studies concerning NFSL have been conducted for their potential application to optical nanolithography and optical nano-imaging [8, 9, 10].

Silver slab NFSL is simply feasible with conventional fabrication processes compared to metamaterials with complicated periodic structures. However, NFSL are usually made of metals which have significant intrinsic loss of absorption (ε″>0). This absorption loss of metal slab blurs the NFSL image, which is an ultimate limitation to demonstrate a near-field perfect lens [8]. Typically, the lens condition of NFSL has been given as the impedance match, ε′=-ε_h, where ε′ is the real part of the permittivity of the metal and ε_h is the permittivity of the host medium interfacing the lens [11]. In this report, we optimized a variable parameter (ε′) for the best quality of near-field image using a Ag slab lens to compensate for blurring due to the intrinsic absorption loss (ε″>0). Through detailed numerical FDTD studies, the improved depth of field and image quality of NFSL are achieved in the impedance mismatch case (ε′=-0.8).

2. Calculation technique

In the UV region, the permittivity of Ag can be approximated by the Drude model as follows

$\epsilon =\epsilon \prime +i·\epsilon ″={\epsilon }_{\infty }-\frac{{\omega }_p^2}{\omega \left(\omega +i\Gamma \right)}$ ,

where ω_p is the plasma frequency (9.01 eV for Ag), the contribution of bounded electrons ε _∞, the relaxation constant Γ, and the angular frequency ω of the UV source. Appropriate values of ε _∞ and Γ from the experimental measurement of Ag thin film are used to calculate the optical properties [12]. We used the conventional geometry of Ag NFSL suggested by Pendry with a grating mask, which is 40 nm thick and located 20 nm behind the 40 nm-thick tungsten mask (see Fig. 1(a))[7]. The lateral period of the tungsten mask is 140 nm and the width of the aperture is 70 nm. A p-polarized sinusoidal plane wave (TM mode) is incident through the mask with an electric field parallel to the plane of incidence between 330 nm to 340 nm. Our FDTD calculation is based on the numerical analysis of K.S. Kunz and R.J. Luebbers [13].

 

Fig. 1. (a) The typical geometry of Ag NFSL, which is suggested by Pendry. t ₁=40 nm, t ₂=40 nm, and f ₁=20 nm. A tungsten mask has the grating period of 140 nm (w ₂) and the aperture width of 70 nm(w ₁). (b) Light intensity distribution calculated from the FDTD method in the impedance match case at 341 nm.

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We take over 25 periods of source to avoid observing any transient behavior at early times: a periodic boundary condition (PBC) is used to reduce the computing memory. The cell size of the mesh grid is 2 nm and the time step is less than 5 attoseconds. Figure 1(b) shows the result of our FDTD calculation when the impedance is matched at 341 nm.

To quantify the effectiveness of the near-field imaging, the depth of field (DOF) in optical lithography can be a decent evaluation method for the quality of an image [14]. DOF is defined as the length of the region where the intensity visibility (contrast) in the image is larger than a specified value (k) at the distance (z) behind the mask exit as follows

$k\left(z\right)\equiv \frac{I_{max}\left(z\right)-I_{min}\left(z\right)}{I_{max}\left(z\right)+I_{min}\left(z\right)}.$

Because DOF_k is the depth at which sufficient contrast (k) is available for an optical exposure in optical lithography, DOF is an appropriate criterion for the possible application of NFSL in lithography.

3. Regime of parameters to optimize image quality

For DNG without absorption loss, the impedance match case with the host material generates the best-quality image because there is no reflection loss on the interfaces and no aberration depending on the incident angle of light [15]. The presence of inherent material loss in DNG causes an ultimate limitation for the subwavelength imaging performance [16, 17]. Since Pendry suggested a NFSL based on surface plasmon resonance (SPR), the real part (ε′) of permittivity of Ag slab has been preferably impedance-matched to the host material (air in this case, ε_h=ε′_air=1) because only ε″ prevents ideal reconstruction of image [7]. Recently, Blaikie et al. reported that the near-field image quality and focal position of NFSL are strongly dependent on the absorption loss in materials, which is characterized by parameter ε″(>0) [8]. They calculated the focal length of Ag slab while ‘virtually’ changing ε″ with a fixed ε′ for the impedance match case. They recognized that higher ε″ gives shorter focal length (see Fig. 5 of Ref. [8]). In the UV region (330 nm to 341 nm) for SPR regime of Ag slab, however, ε″(=0.3) is intrinsically constant while ε′ is dominantly changing. As the existence of constant loss from non-zero ε″ has been considered as an unavoidable blurring source of a NFSL image, we propose to optimize the variable parameter (ε′) for the improvement of the near-field image by extending to the impedance mismatch cases. The change of ε′ can be realized by tuning the wavelength of the incident light for imaging as shown in Table 1. A tunable light source from 330 nm to 341 nm is feasible by using the second harmonic generation of commercial tunable diode lasers (660 nm to 700 nm) popularly used for optical data storage [18]. To estimate the performance of a NFSL, various parameters can be used such as focal spot size, intensity visibility (contrast), confocal parameter, and transmission. DOF is a useful criterion to evaluate the contrast which is important for optical lithography.

 

Fig. 2. In the absorptive material of ε″=0.3, we calculated (a) the visibilities and (b) the mean intensities of near-field imaging, depending on the distance from the mask exit in the case of various values of ε′. These results indicate that the impedance mismatch cases (ε′=-0.8) give higher visibility while the intensity is slightly lower.

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Table 1. The wavelength, focal spot sizes, and confocal parameters versus various ε′ of Ag NFSL including impedance mismatch cases.

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4. Results and discussion

Using FDTD calculation, we obtained an 84-nm focal spot size in the impedance match case, which is smaller than the diffraction limit. Visibility is a critical parameter in optical lithography and we used DOF to optimize the quality of the image. Figure 2(a) is the calculated intensity visibility of the NFSL image versus the distance behind the mask exit depending on differing ε′ of -1.0,-0.8,-0.7, and -0.5 of Ag. The focal positions are decided by the

Table 2. Visibility and relative intensity at focal positions versus various ε′.

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Fig. 3. Near field light intensity profiles calculated from FDTD method in the impedance mismatch cases of (a) ε′=-0.8 and (b) ε′=-0.5.

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maximum visibility and the focal spot sizes are calculated from the FWHM for different values of ε′ as shown in Table 1 [8]. This indicates that the focal spot sizes and confocal parameters of impedance mismatch cases are as good as those of the impedance match case. The integrated mean intensity of the NFSL image versus the distance is also shown in Fig. 2(b) for various ε′. Table 2 is the number of visibilities and relative intensities at focal positions for various ε′. The visibility (0.91) of ε′=-0.8 is the largest and ~69% higher than that (0.54) of ε′=-1.0. In contrast, visibility without NFSL degrades monotonically along with the distance from the mask. The relative intensity at the focal position of ε′=-0.8 is ~9% lower than that of ε′=-1.0. These results signify that we can get ~69% higher visibility at the cost of ~9% intensity loss if we use the impedance mismatch case instead of the impedance match case. Figure 3(a) and 3(b) are the light intensity distributions with NFSL for the impedance mismatch cases of ε′=-0.8 and ε′=-0.5.

To explain the higher visibility in the impedance mismatch, we calculated the relative phases of the electric field in x-direction (E_x) as a function of the distance behind the center of the mask aperture (see Fig. 4(a)) and the center of the mask line (see Fig. 4(b)). The distances behind the centers of the mask aperture and mask line are indicated as the lines of AB and CD in the schematic of Fig. 1(a), respectively. Behind the center of the mask aperture, the relative phases of E_x show similar behavior for the various values of ε′ from -0.6 to -1.0. Behind the center of the mask line, the relative phase change (~180°) over the distance is dramatic around the focal position (~80 nm) when ε′=-0.8. On the other hand, the phase changes at the focal position are slower for the cases of ε′=-1 and -0.6. Because this significant phase change over the distance generates a more efficient shadow on the image plane behind the mask line, visibility becomes highest in the impedance mismatch case of ε′=-0.8.

 

Fig. 4. Relative phase changes of the electric field in x-direction (E_x) are calculated as a function of the distance behind (a) the center of the mask aperture and (b) the center of the mask line.

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Table 3. The ε′ for optimized DOF and corresponding wavelength versus various ε″. For real silver, ε″=0.3

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We also calculated DOF depending on ε′ because the improvement of DOF is an important issue in optical lithography. DOF for real silver (ε″=0.3) is shown in Fig. 5(a) and the highest DOF is achieved when ε′=-0.8, which is consistent with Fig. 2(a). For example, DOF_k=0.5 for the intensity contrast k=0.5 is ~138% enhanced by our optimization technique from 13 nm for ε′=-1.0 to 31 nm for ε′=-0.8. This artificial impedance mismatch for our optimization is possible if we tune the wavelength of the incident light from 341 nm to 335 nm. To get a clear picture of this behavior, we ‘virtually’ changed ε″ of silver (ε″=0.02, 0.6) and calculated DOF depending on ε′. Figure 5(b) shows that the optimized DOF is achieved when ε′=-0.85 if the virtual ε″ is 0.02. If ε″=0.6, the DOF is highest when ε′=-0.75 as shown in Fig. 5(c). Table 3 is ε″ vs. ε′ values for the optimized DOF and corresponding wavelength of the incident light. Table 3 indicates that for a NFSL with the higher absorption loss, the larger impedance mismatch (the smaller absolute value of negative ε′) is needed to obtain optimized image quality, which can be realized by changing the wavelength of the incident light for imaging. For example, if we use 335 nm instead of 341 nm, we can get >69% higher visibility and >138% increased DOF_k for k=0.5 at the cost of 9% intensity loss.

5. Conclusion

We optimized a variable parameter (ε′) for the best quality of image using a Ag slab NFSL to compensate for the blurring due to intrinsic absorption loss (ε″>0) using FDTD calculation. A ~138% increase in depth of field and ~69% higher visibility are achieved in the impedance mismatch case (ε′=-0.8) compared to the impedance match case. To optimize the image quality of a NFSL with higher absorption loss, larger impedance mismatch is needed. This optimization technique of the near-field image is useful for the design of NFSL in the infrared and Terahertz region with significant loss of absorption.

 

Fig. 5. In the absorptive materials (ε″≠=0), the best visibilities of a NFSL image are achieved when ε′ is not -1 but in the range of -0.7~-0.9 depending on the value of ε″. DOF for a different value of the intensity contrast (k) are calculated when (a) ε″=0.3, (b) ε″=0.02, and (c) ε″=0.6. This artificial impedance mismatch for our optimization is possible if we change the incident light from 341 nm to 335 nm for real silver as shown in (a).

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