1. Introduction

For the applications in photolithography and optical imaging, there have been extensive efforts to overcome the diffraction limit by developing negative index materials in various wavelength region from microwave to visible light [1–6]. To acquire the superresolution with the near-field superlens (NFSL), the real part (ε′) of permittivity is required to be negative for single negative index materials. The negative ε′ for superlensing effect is obtained by plasmonic oscillations of metals in the visible wavelength [7] and by lattice vibrations (i.e. phonon resonance) of dielectric polar crystals such as ZnSe, SiC, TiO₂ and InP [8] in the mid infrared wavelength. In the wavelength region around the plasmon or phonon resonance, the absorption loss is unavoidable because the negative ε′ leads to nonzero imaginary part (ε″) of permittivity as derived by Kramers-Kronig relation. Unfortunately, this inevitable loss of NFSL ultimately limits ideal image reconstruction in the image plane [1,9]. Therefore, superlensing effects have been observed only in the narrow wavelength region for the limited few metamaterials when both smaller loss and negative ε′ are properly satisfied at the same time.

For these purposes, two noticeable methods have been proposed. The field synthesis method combined the spatially shifted beams after the phase changes, as well as the transmission, using a near-field plate or transmission screen [10, 11]. Recently, the active phase correction method was introduced to accomplish phase retrieval for the improved near-field image quality of lossy NFSL system by tuning the wavelength of incident light, based on the fact that the phase change due to absorption loss is essential for the image resolution [12–14].

In addition to resistive losses, another important limiting factor for superresolution is the retardation effects due to a finite frequency of incident light [15, 16]. In the electrostatic (magnetostatic) limit of large transverse wave number, the change of µ(ε) is independent of the behavior of p(s)-polarized incident light. In realistic metamaterials, however, this decoupling is hindered by the deviation from the electrostatic (magnetostatic) limit due to the non-zero frequency of the incident light. A delicate change in µ(ε) excites the coupled surface modes and leads to large transmission resonances in the spatial frequency domain, which cause Fourier components to be a broad background ‘noise’ in the image [16, 17]. These retardation effects severely limit the image resolution and also stimulate the excitation of slab resonances that degrade the performance of the lens.

It has been noted that the presence of absorption losses softens all the singular behavior at the transmission resonances and produces a good image although the range of spatial frequencies with effective amplification is limited. Thus, absorption loss has contradicting double role of limiting the resolution and yet vitalizing the image formation. These bottlenecks restrict our choices of appropriate metamaterials at desired operating wavelength. In this paper, we recognize that the phase singularity persists despite the reduced retardation effects by softening the singular behavior of transmission resonances. Because this phase singularity significantly prevents the ideal image reconstruction, TiO₂ thin-film lens, as an example, has not been considered as a NFSL candidate even though the transmission has enough broad bandwidth for superresolution in the spatial frequency domain. Using the phase correction method [13, 14], we successfully eliminated the phase singularity and achieved ~ λ/12.9 superresolution. Aforementioned TiO₂ thin film phase correction illustrates that lossy metamaterials with phase singularity can accomplish superresolution if the phase singularity is eliminated by the phase correction method.

 

Fig. 1. The schematic geometry of the imaging system used in this work.

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2. The limited superresolution by a phase singularity in lossy metamaterial system

Figure 1 shows the schematic geometry of the near-field imaging system investigated in this work. p-polarized light of specified wavelength is incident into z-direction on the thin-film lens sandwiched between two layers of symmetrical host materials through a double slit with a given geometry of peak-to-peak (p-p) separation and slit width in x-direction. The thicknesses of thin film lens (d_L) and symmetrical two host materials (d ₁ = d ₂) are d_L = d, d ₁ = d ₂ = d/2, and the permittivities(permeabilities) are, in general, complex numbers of ε_L, ε ₁ = ε ₂(µ_L, µ ₁ = µ ₂), respectively. The conventional optical transfer function (OTF) from an object to an image is used to quantify the transmission and phase change of the imaging system. The OTF is the multiplication of the transmission coefficient of the slab thin film and the wave propagation factors through the host materials.

If a slab of metamaterial consists of lossless host materials with real valued dielectric constant ε ₁ = ε′₁, the thin film lens is supposed to have nonzero imaginary number such as ε_L = −ε′₁+iε_L″ in order to qualify a typical NFSL. For the near field contribution ( $k_x\gg \frac{\omega \sqrt{{\epsilon }_1^{\prime }}}{c}$ ) which is responsible for high resolution imaging, the OTF is approximated as follows [15, 18]:

where k_x is the transverse wave vector. The mismatch term (ε″_L/ε′₁−iω ² ε′₁/k ² _xc ²) of the denominator in Eq. (1) indicates that two contributions from the normalized losses (ε″_L/ε′₁) and the retardation effect(ω ² ε′₁/k ² _xc ²) cause OTF(k_x) to cumulatively deviate from unity [15]. The magnitude and phase of a complex OTF function are defined as the modulation transfer function (MTF) and the phase transfer function (PTF) and derived from Eq. (1) as follows, respectively [19].

 

Fig. 2. The MTF and PTF of the Ag slab NFSL imaging system are plotted versus k_x/k ₀ when (a) virtually ε″_L = 0.001 and (b) realistically ε″_L = 0.4. These data are cited from Fig. 2 of Ref. [13]

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If the thin film lens(ε″_L = 0) has no absorption loss, the MTF in Eq. (3) has diverging resonances at the specific spatial frequency, k_x, which satisfies the relation of ${\left(\frac{{\omega }^2{\epsilon }_1^{\prime }}{k_x^2{c}^2}\right)}^2=4e^{-2k_{x}d}$ . The transmission, as well as the MTF, has diverging resonances, because the transmission is the square of MTF. The sharp transmission resonances, originating from the retardation effects, are undesirable for imaging because the Fourier components of corresponding spatial frequency will be introduced as background ‘noise’ in the image. To reduce these effects of retardation, a finite absorption loss in the lens is essentially useful because it prevents transmission from diverging and alleviates the effects of resonances in the image [16]. If the thin film lens(ε″_L ≠ 0) has finite absorption loss, the resonance peaks become finite values at the spatial frequency, k_x,r, which satisfies a relation of ${\left(\frac{{\omega }^2{\epsilon }_1^{\prime }}{k_{x,r}^2{c}^2}\right)}^2={\left(\frac{{\epsilon }_L^{″}}{{\epsilon }_1^{\prime }}\right)}^2+4e^{-2k_{x,r}d}$ , as follows:

By reducing the peak values of MTF resonances with absorption loss, the transmission resonances can be softened.

It is noted that the phase, as well as transmission, is also critical to the resolution and the phase retrieval (PTF=0) is the best condition in the superlens imaging systems [14]. The phase of OTF can be corrected if ε′_L is changed by the index mismatch approach through tuning the wavelength of incident light for imaging [13, 14]. At the spatial frequencies of transmission resonance, k_x,r, the denominator of the argument for tan⁻¹ in Eq. (4) crosses zero to change the argument value of tan⁻¹ from +∞ to −∞. Hence, the PTF becomes singular and changes from 90° to −90° discontinuously. The rapid phase change at the singularity disproportionately contributes to Fourier components of corresponding spatial frequencies and severely limits the image resolution.

In Fig. 2, for incident light of λ=341nm and d=40nm, the MTF and PTF of a Ag slab NFSL imaging system with vacuum host material are depicted when the loss of Ag is unrealistically small [Fig. 2(a); ε″_L = 0.001] and when the realistic absorption [Fig. 2(b); ε″_L = 0.4] is considered. At k_x/k ₀ = 5.4 where k ₀(=ω/c) is the wave number of incident light [see Fig. 2(a)], the transmission resonance appears as a finite peak and the PTF is singular. In Fig. 2(b), the singular behavior of the transmission (MTF) resonances at k_x/k ₀ = 1.2 is well-softened because of higher loss, but the phase singularity of PTF persists. Fortuitously, this PTF singularity in realistic Ag appears at the spatial frequency of k_x/k ₀ = 1.2 which does not make significant effect on superresolution.

3. Achievement of superresolution in TiO NFSL imaging system

Based on above analysis, we consider a 400nm-thick TiO₂ thin-film imaging system sandwiched by SiO₂ host materials(d=400nm), as illustrated in Fig. 1, with incident light of wavelength(λ) from 11.5µm to 15.5µm. The complex dielectric permittivities of TiO₂ and SiO₂ are determined by the experimental measurement [20, 21]. To evaluate the image quality, we calculated the transmission field passing through double slits aligned with x-axis. The field distribution at the image plane is obtained by the inverse Fourier transform of the object field [13, 14].

Without the phase correction, TiO₂ thin film lens does not provide superlensing effect in the typical index match case (ε′_L = −ε′₁) at λ=14.3µm (this case will be denoted as ‘no phase correction’). Figures 3(a) and 3(b) depict the MTF and PTF versus k_x/k ₀ depending on the wavelength of incident light. The k_x/k ₀ bandwidth of MTF is broader in 12.6µm< λ <14.3µm wavelength, while it is relatively narrower in 12.1µm< λ <12.6µm wavelength [see Fig. 3(a)]. It should be noted that phase behavior, as well as transmission behavior, is also essential for the image resolution. In order to perform the phase correction by tuning the wavelength of the incident light, we chose the range between 12.1µm (which will be denoted as lower limit or ‘blue limit’) and 13.6µm (which will be denoted as upper limit or ‘red limit’), so that superresolution better than < λ/6 could be obtained from TiO₂ lens. Visual inspection of the PTF plot [Fig. 3(b)] determines λ=12.9µm for the zero-PTF condition for phase retrieval. Because the TiO₂ imaging system is lossy, the phase retrieved case at λ=12.9µm is expected to offer the best image resolution due to the retardation reduction. Figures 3(c) and 3(d) show the MTF and PTF as a function of k_x/k ₀ at the index-matched case(λ=14.3µm), the phase retrieved case(λ=12.9 µm), the blue limit(λ=12.1µm), and the red limit(λ=13.6µm).

Figures 3(e), 3(f), 3(g), and 3(h) depict the visibilities (V ≡ (I_max−I_min)/(I_max+I_min)) [22] of lateral intensity distribution through a double slit in the image plane versus slit width and p-p separation (e) for no phase correction(λ = 14.3µm), (f) the phase-retrieved case (λ = 12.9µm), (g) blue limit (λ = 12.1µm), and (h) red limit (λ = 13.6µm), respectively. Without phase correction at λ=14.3µm, the absorption loss effectively reduced the effects of retardation and the MTF resonance peak was well-softened at k_x/k ₀ = 5.0 [see Fig. 3(c)]. Interestingly, the MTF has enough broad bandwidth for superresolution, but the superlensing effect is not clearly observed with TiO₂ thin film imaging system [see Fig. 3(e)]. This poor resolution of TiO₂ lens, in spite of broad MTF bandwidth, can be explained by the effect of PTF. Even after the retardation effects are considerably reduced by softening the MTF resonances, the phase singularity still persist in PTF at k_x/k ₀=5.0 [the black solid line in Fig. 3(d)] [18]. Because this drastic phase change in k_x/k ₀ domain severely distorts image reformation, the superresolution has not been observed in TiO₂ thin film.

The superresolution in TiO₂ thin film can be successfully accomplished by the phase correction method. Compared with the result without phase correction, the visibility obtained by the phase correction is remarkably enhanced in the overall region [see Fig. 3(f)]. The resolvable p-p separation is significantly improved to 1.0µm (~ λ/12.9) for a given slit width of 0.3µm(~λ/43.0). This highest resolution is achieved by virtue of both uniform zero-PTF and the broadest bandwidth MTF. In the region between λ=12.1µm (blue limit) and λ=13.6µm (red limit), we are able to eliminate the phase singularity over the spatial frequency range (1 < k_x/k ₀ < 7) which is the most crucial region to achieve the superresolution of < λ/6. In the red limit, although the MTF has a sufficient bandwidth for ~ λ/10 [see Fig. 3(c)], the existence of a PTF singularity at k_x/k ₀=7.8 makes the phase to be partially distorted over the spatial frequency which is critical for superresolution [see Fig. 3(d)]. The image blurring due to the phase distortion decreases the visibility at higher resolution and the resolvable separation is limited to ~ λ/6 in spite of broader bandwidth of MTF [see Fig. 3(h)]. In the blue limit, although the MTF bandwidth is much narrower than the index-match case shown in Fig. 3(c), the resolvable separation is as good as ~ λ/6 [see Fig. 3(g)]. This superresolution originates from nearly constant phase over broad spatial frequency region because the phase singularity is eliminated [Fig. 3(d)]. Although TiO₂ thin film imaging system is not a NFSL with conventional index matching condition, the elimination of phase singularity provides the imaging system with superlensing effect to qualify a NFSL in the range from 12.1µm to 13.6µm.

 

Fig. 3. In the plane of wavelength (λ) and k_x/k ₀, (a) the MTF and (b) PTF are plotted and the white dashed lines represent the index-matched wavelength. As a function of k_x/k ₀, (c) the MTF and (d) PTF are depicted for the index matched case (14.3µm), the phase-retrieved case (12.9µm), blue limit (λ=12.1µm), and red limit (λ=13.6µm). The visibilities (V) of lateral intensity distribution in the image plane through a double slit with given slit width and p-p separation (e) for the index matched case (14.3µm), (f) the phase-retrieved case (12.9µm), (g) blue limit (λ=12.1µm), and (h) red limit (λ=13.6µm).

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Fig. 4. (a) The visibility versus p-p separation when slit width is 0.3µm, (b) the resolvable separation (or p-p separation) between two slits with a 0.3µm slit width, and the lateral intensity distributions through a double slit with p-p separation of (c) 1.7µm and (d) 3.8µm, as well as 0.3µm slit width.

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The visibility versus p-p separation is shown in Fig. 4(a) when slit width is 0.3µm. We obtained superresolution with resolvable separation of 1.0 µm (λ/12.9 at λ = 12.9µm) for V = 0.4 after the phase correction, compared to 5.8µm (λ/2.5 at λ = 14.3µm) without phase correction. The resolvable separation (or p-p separation) between two slits of 0.3µm slit width is predicted to obtain images of various quality (visibility) in Fig. 4(b). The phase-singularity elimination enables us to achieve a higher quality image of V>0.5 with resolvable separation of ~λ/10.8, which is not possible even for ~λ/2.0 resolution without it. Figures 4(c) and 4(d) plots intensity of the transmitted light on the image plane after the double slit of 0.3µm slit width and p-p separation of 1.7µm and 3.8µm, respectively. The intensity peaks obtained for two slits are not resolved (the blue line: λ = 14.3µm) without the phase correction because they are beyond the diffraction limit [Fig. 4(a) ~λ/8.4; 4(b) ~λ/3.8]. On the other hand, they are well separated and resolved with higher intensity in the phase-retrieved case (the red line: λ = 12.9µm).

4. Conclusion

In conclusion, the phase singularity persists although the retardation effects are reduced by softening the transmission resonances. Because this phase singularity degrades the image resolution severely, TiO₂ thin film lens can not achieve superresolution even though it has broad transmission bandwidth in spatial frequency domain. We eliminated this phase singularity and accomplished ~ λ/12.9 superresolution from TiO₂ thin film using the phase correction method.