Elimination of phase singularity to achieve superresolution in lossy metamaterials

The presence of absorption losses softens the singular behavior of transmission resonances and leads to a good image in spite of limited effective spatial frequency range. Nonetheless, we found that the phase singularity does not disappear despite the considerably reduced retardation effects by softening the transmission resonances. Because the phase singularity severely deteriorates the ideal image reconstruction, broad transmission bandwidth in spatial frequency domain is not sufficient enough to achieve superresolution in TiO2 thin film lens. The present work predicts successful elimination of the phase singularity and the achievement of ∼ λ/12.9 superresolution in TiO2 thin film lens through the phase correction method. © 2010 Optical Society of America OCIS codes: (160.3918) Metamaterials; (350.3618) Left-handed materials; (110.4850) Optical transfer functions. References and links 1. J. B. Pendry, “Negative refraction makes a perfect lens,” Phys. Rev. Lett. 85, 3966–3969 (2000). 2. D. R. Smith, W. J. Padilla, D. C. Vier, S. C. Nemat-Nasser, and S. Schultz, “Composite medium with simultaneously negative permeability and permittivity,” Phys. Rev. Lett. 84, 4184–4187 (2000). 3. T. J. Yen, W. J. Padilla, N. Fang, D. C. Vier, D. R. Smith, J. B. Pendry, D. N. Basov, and X. Zhang, “Terahertz magnetic response from artificial materials,” Science 303, 1494–1496 (2004). 4. T. Taubner, D. Korobkin, Y. Urzhumov, G. Shvets, and R. Hillenbrand, “Near-field microscopy through a SiC superlens,” Science 313, 1595 (2006). 5. N. Fang, H. Lee, C. Sun, and X. Zhang, “Sub-Diffraction-limited optical imaging with a silver superlens,” Science 308, 534–537 (2005). 6. X. Yang, Y. Liu, J. Ma, J. Cui, H. Xing, W. Wang, C. Wang, and X. Luo, “Broadband super-resolution imaging by a superlens with unmatched dielectric medium,” Opt. Express 16, 19686–19694 (2008), http://www. opticsinfobase.org/abstract.cfm?uri=oe-16-24-19686. 7. H. Raether, Surface plasmons on smooth and rough surfaces and on gratings (Springer-Verlag, 1988). 8. R. Hillenbrand, T. Taubner, and F. Keilmann, “Phonon-enhanced light-matter interaction at the nanometre scale,” Nature 418, 159–162 (2002). 9. R. J. Blaikie and S. J. McNab, “Simulation study of ‘perfect lenses’ for near-field optical nanolithography,” Microelectron. Eng. 61-62, 97–103 (2002). 10. A. Grbic, L. Jiang, and R. Merlin, “Near-field plates: subdiffraction focusing with patterned surfaces,” Science 320, 511–513 (2008). 11. L. Markley, A. M. H. Wong, Y. Wang, and G. V. Eleftheriades, “Spatially shifted beam approach to subwavelength focusing,” Phys. Rev. Lett. 101, 113901 (2008). #126677 $15.00 USD Received 7 Apr 2010; accepted 11 May 2010; published 25 May 2010 (C) 2010 OSA 07 June 2010 / Vol. 18, No. 12 / OPTICS EXPRESS 12269 12. K. Lee, H. Park, J. Kim, G. Kang, and K. Kim, “Improved image quality of a Ag slab near-field superlens with intrinsic loss of absorption,” Opt. Express 16, 1711–1718 (2008), http://www.opticsinfobase.org/ abstract.cfm?URI=OE-16-3-1711. 13. K. Lee, Y. Jung, G. Kang, H. Park, and K. Kim, “Active phase control of a Ag near-field superlens via the index mismatch approach,” Appl. Phys. Lett. 94, 101113 (2009). 14. K. Lee, Y. Jung, and K. Kim, “Near-field phase correction for superresolution enhancement,” Phys. Rev. B 80, 033109 (2009). 15. D. Korobkin, Y. Urzhumov, and G. Shvets, “Enhanced near-field resolution in midinfrared using metamaterials,” J. Opt. Soc. Am. B 23, 468–478 (2006). 16. S. A. Ramakrishna, J. B. Pendry, D. Schurig, D. R. Smith, and S. Schultz, “The asymmetric lossy near-perfect lens,” J. Mod. Opt. 49, 1747–1762 (2002). 17. D. R. Smith, D. Schurig, M. Rosenbluth, S. Schultz, S. A. Ramakrishna, and J. B. Pendry, “Limitations on subdiffraction imaging with a negative refractive index slab,” Appl. Phys. Lett. 82, 1506–1508 (2003). 18. N. Fang and X. Zhang, “Imaging properties of a metamaterial superlens,” Appl. Phys. Lett. 82, 161–163 (2003). 19. D. O. S. Melville and R. J. Blaikie, “Analysis and optimization of multilayer silver superlenses for near-field optical lithography,” Physica B 394, 197–202 (2007). 20. E. D. Palik, Handbook of optical constants of solids (Academic, New York, 1985) 21. F. Gervais and B. Piriou, “Temperature dependence of transverseand longitudinal-optic modes in TiO2 (rutile),” Phys. Rev. B 10, 1642–1654 (1974). 22. C. P. Moore, M. D. Arnold, P. J. Bones, and R. J. Blaikie, “Image fidelity for single-layer and multi-layer silver superlenses,” J. Opt. Soc. Am. A 25, 911–918 (2008).


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][2][3][4][5][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, TiO 2 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][13][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, TiO 2 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 TiO 2 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.

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 ( , 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 ω ε 1 /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 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 x c 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].
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 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 (ω 2 ε 1 /k 2 x,r c 2 ) 2 = (ε L /ε 1 ) 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 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.

Achievement of superresolution in TiO 2 NFSL imaging system
Based on above analysis, we consider a 400nm-thick TiO 2 thin-film imaging system sandwiched by SiO 2 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 TiO 2 and SiO 2 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, TiO 2 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 TiO 2 lens.Visual inspection of the PTF plot [Fig.3(b)] determines λ =12.9μm for the zero-PTF condition for phase retrieval.Because the TiO 2 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 TiO 2 thin film imaging system [see Fig. 3(e)].This poor resolution of TiO 2 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 TiO 2 thin film.
The superresolution in TiO 2 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  (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.(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 TiO 2 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.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).

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, TiO 2 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 TiO 2 thin film using the phase correction method.

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

Fig. 3 .
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).
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).

Fig. 4 .
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.