Breaking Optical diffraction limitation using Optical ... - OSA Publishing

Breaking Optical diffraction limitation using Optical Hybrid-Super-Hyperlens with Radially Polarized Light Bo Han Cheng,1 Yung-Chiang Lan,2 and Din Ping Tsai1,3,* 1

2

Research Center for Applied Sciences, Academia Sinica, Taipei 115, Taiwan Department of Photonics, National Cheng Kung University, Taiwan 70101, Taiwan 3 Department of Physics, National Taiwan University, Taipei 10617, Taiwan *[email protected]

Abstract: We propose and analyze an innovative device called “HybridSuper-Hyperlens”. This lens is made of two hyperbolic metamaterials with different signs in their dielectric tensor and different isofrequency dispersion curves. The ability of the proposed lens to break the optical diffraction limit is demonstrated using numerical simulations (with the resolution power of about λ/6). Both a pair of nano-slits and a nano-ring can be imaged and resolved by the proposed lens using the radially polarized light source. Such a lens has great potential applications in photolithography and real-time nanoscale imaging. ©2013 Optical Society of America OCIS codes: (110.0180) Microscopy; (350.5730) Resolution; (160.1190) Anisotropic optical materials; (160.3918) Metamaterials.

References and links 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 17.

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Received 4 Feb 2013; revised 1 Mar 2013; accepted 5 Mar 2013; published 17 Jun 2013 17 June 2013 | Vol. 21, No. 12 | DOI:10.1364/OE.21.014898 | OPTICS EXPRESS 14898

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1. Introduction Ernst Abbe first introduced so-called diffraction limit d = λ / 2(n sin θ ) , whereλis the wavelength, d is the distance between two objects, and n sin θ is the numerical aperture (NA) [1]. Generally, features smaller than half wavelength of the light are permanently lost in the image because the waves which carry information of the sub-wavelength details are transverse waves with their wave vectors larger than free space wave vector k 0 = ω / c . Those wave vectors decay exponentially from the surface of the object in free space [2]. This is the key reason for conventional optical microscopy cannot capture the minuscule details of the object in the far field region. In order to improve the optical imaging resolution, and make good use of the optical evanescent wave or near field, photon scanning tunneling microscopy [3–5] and near-field scanning optical microscopy [6–8] are the major paths to achieve nearfield super-resolution image. However, low throughput, poor compatibility with various environment/samples, and inability to obtain the whole image at one scan are the drawbacks need to be overcome. According to Pendry’s conceptual model, using a slab with the refractive index n = −1 , both propagating and evanescent waves excited from the objects can contribute to the resolution image [9]. The perfect lens should be realized by tuning the parameters of constituent elements (thin metal wires and split ring) which provide the potential to fulfill the super-resolution condition as mentioned above [10, 11]. However, it is difficult to materialize at present due to the feasibility of simultaneously reaching negative permittivity and permeability, as well as the impedance mismatch between the perfect lens and surrounding medium [12]. Some other lenses which are similar to perfect lens named superlens have shown the ability of breaking diffraction limit theoretically [13] and experimentally [14]. However, in these so-called superlenses, all the fine features (evanescent waves) cannot be brought to focus by conventional optical devices and instruments.

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Recently, the multi-layered metal-dielectric structures [15–17] and the metal nanorod array structures [18–20] have been proposed to have the ability of delivering evanescent information to far-field space optical microscopy. Due to their unusual optical property, the fine features with evanescent mode can be changed into propagation mode [2]. They can be understood by considering the isofrequency curve (dispersion relation between the frequency and the wave vector) with hyperbolic forms which are known as hyperbolic metamaterials. Especially, the hyperbolic metamaterials with multi-layered metal-dielectric shape used for super-resolution are named hyperlens and have caught the eye and drawn attention to their potential applications since its first demonstration in 2007 [21, 22]. The hyperlens with cylindrically or spherically curved multilayer stacks uses an approach of magnifying the sub-wavelength features [2]. The evanescent waves excited from the objects (placed near or on the curved hyperlens) are magnified and transformed into the propagating waves in such anisotropic medium with a hyperbolic dispersion. However, the cylindrical structure has shortcomings [2, 21, 22], since it is inconvenient to put the objects on the curved platform for practical applications. Further, the semicircle space that is constructed by metal will form a cavity and affect the resolution ability at specific operation wavelength. In this investigation, the proposed hybrid-super-hyperlens [23, 24] with linearly and radially polarized incident light is theoretically investigated. The capability of this lens to break optical diffraction limit is proved by finite element method (FEM) and finite-difference time-domain (FDTD) simulation. The challenge relative to resolve complicated nano pattern is also investigated. Basically, the polarization of light source is an important factor which affects the integrity of the resolution image results from exciting surface plasmon polarions (SPPs) near the patterned region. We demonstrate that the whole magnifying far-field images can be obtained at one scan procedure by using radially polarized light source. That is, superposition of the images under incident light with different polarized directions is unnecessary. The applicability and superiority of hybrid-super-hyperlens for real applications such as photolithography and planar integrated optical devices [25, 26] will also be discussed. 2. Analytical model structure and simulation method Figure 1(a) shows the conceptual objective lens which is composed of two hyperbolic metamaterials i.e., superlens and hyperlens. For TM polarized wave, both of the optical dispersion relations of the superlens and hyperlens can be obtained by the transfer matrix method [27] cos ( k x Λ ) = cos ( k 1x d 1 ) cos ( k 2 x d 2 ) −

1 ε1k 2 x ε 2 k 1x ( ) sin ( k 1x d 1 ) sin ( k 2 x d 2 ) (1) + 2 ε 2 k 1x ε1k 2 x

where Λ = d 1 + d 2 is the period of one pair of alternately stacked metal and dielectric ( d 1 and d 2 are the thickness of metal and dielectric region with relative permittivity ε1 and

ε 2 respectively); k 1x = (ε1k 0 2 − k z 2 ) and k 2 x = (ε 2 k 0 2 − k z 2 ) denote the x-directional wave vector in the metal and dielectric regions, respectively. The relations between the wavevectors (k x , k z ) and the effectively parallel and perpendicular relative permittivities ( ε xeff , ε zeff ) can be further established.

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Fig. 1. Schematic of the hybrid-super-hyperlens which is composed of two hyperbolic metamaterials (i.e., superlens and hyperlens) with different dielectric tensor under TM wave illumination at 405 nm. The upper part is superlens which consists of 6 paris of alternately stacked Ag (30 nm) and Al2O3 (30 nm) layers, the lower part is hyperlens that is composed of 8 pairs of alternately stacked Ag (30 nm) and HfO2 (30 nm) layers. The orange parts are observed sample with thickness t = 60 nm. The width of grooves carved on the Chromium (Cr) is 50 nm.

the

Considering the long-wavelength approximation (λ>>Λ) and using Taylor expansion to first-order term, the terms cos ( k x Λ ) , cos ( k 1x d 1 ) cos ( k 2 x d 2 ) , and in

sin ( k 1x d 1 ) sin ( k 2 x d 2 )

Eq.

(1)

will

be

replaced

by

1 − (k x Λ) 2 / 2

,

1 − (k 1x d 1 ) / 2 − (k 2 x d 2 ) / 2 , and (k 1x d 1 )(k 2 x d 2 ) ,respectively. Then Eq. (1) is transformed to the following form: 2

2

kx2

ε zeff ε1d 1 + ε 2d 2

+

kz2

ε xeff

= k 02

(2) −1

 d1 1 d2 1  + and ε zeff =   . Equation (2) denotes an d d d ε + d1 + d 2 2 1 1 + d2 ε2   1 anisotropic metamaterial with extraordinary relative permittivity in different propagating directions.

where ε xeff =

Fig. 2. Isofrequency dispersion curves for light propagating in superlens structure with λ = 405 nm. The solid red and dashed blue lines denote the isofrequency curves obtained from Eq. (1) and Eq. (2), respectively. The constructed parameters of superlens are listed in caption of Fig. 1.

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Based on the superresolution conditions of the hybrid-super-hyperlens that is shown in Ref. 23 and 24, the calculated isofrequency dispersion curves of the uper planar-superlens for the incident light of 405 nm are shown in Fig. 2. Figure 2 exhibits that the isofrequency dispersion curves obtained from Eq. (1) and Eq. (2) are essentially the same in their characters. Hence, under the long-wavelength approximation, it is plausible to apply Eq. (2) to design the multi-layered metal-dielectric shape superlens and hyperlens. The numerical results shown in this paper are obtained by computational electromagnetism program Lumerical and COMSOL MultiphysicsTM 3.5a which are based on the FDTD and FEM numerical methods respectively. The silver in the visible region is described by the Lorentz -Drude model [28]

ε (ω ) = 1 −

k ωp 2 f jωj 2 +  ω 2 + i γ p ω j =1 ω j 2 − ω2 2 − i γ j ω

(3)

with ω p the plasma frequency and γ p the damping constant. ω j and γ j are the resonant frequency and the damping constants of the j-th Lorentz oscillator, respectively. The perfectly matched layers are applied outside the hybrid-super-hyperlens device. The incident wave with linearly/radially polarization is launched from the top of the simulation region. For the linearly polarized incident wave, the electric field perpendicular to the groove-shaped pattern (i.e. y-direction in Fig. 1) is adopted. Since the SPPs only exist for TM polarization, only the grooves that aligned to x axis can be resolved (the corresponding simulation results are shown in Section 3). In contrary, for the beams with radially polarized mode, every position in the beam has the polarization vector (electric field) pointing towards the center of the beam. In cylindrical coordinates system, the electric field distributions (on xy-plane) of the radially beam can be expressed by [29] E x (r , θ ) = e − r

2

/ 4σ 2

cos θ

(4)

2

/ 4σ 2

sin θ

(5)

E y (r ,θ ) = e − r

where θ = arc tan( y / x ) , cos θ and sin θ are used to create radially polarized beam; the 2

2

exponential term e − r / 4σ represents the Gaussian envelope of the radial profile of the beam, σ = FW HM / 2 2 ln 2 with the chosen full width at half maximum FW HM = 1.95 μ m . The imported radially-polarized source is depicted in Fig. 3, in which the arrows indicate the polarization direction of the electric fields on xy-plane.

Fig. 3. Plots of polarization directions of imported radially-polarized beam used in simulation.

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In general, the radially-polarized source has been used to produce a smaller focused spot and in other applications such as optical trapping [30]. Here, we employ the characteristics of the radially polarized beam that possesses electric components with arbitrary polarization directions. Using the radially-polarized incident wave, the SPPs along all directions of the pattern carved on the sample plate can be excited. Therefore, the whole pattern is expected to be resolved on the image plane. (The corresponding simulation results are shown below in Section 3). 3. Numerical simulation results

The abilities of the hybrid-super-hyperlens to break optical diffraction limit are demonstrated first. Here the simulated object is a pair of nano-slits and the two-dimensional FEM (COMSOL) is utilized in the simulation. The metal in the hybrid-super-hyperlens is silver. The other simulation parameters are listed in Table 1. The relative permittivity values are designed based on Eq. (2), and the thickness ratio ( d 1 / d 2 ) is 1 in the simulation. The corresponding material can be prepared by using available nano-fabrication techniques. (Note that the constituent parameters of the planar superlens and cylindrical (sphere) hyperlens fulfilled the requirement given in [23]).

Fig. 4. Simulated time-averaged power flow contours (left) and normalized power intensity versus x position measured at the cross section dashed line (right) for incident wavelengths of (a) 532 nm (b) 632.8 nm and (c) 650 nm. Green dotted lines in right figures indicate positions of slits with 50 nm width.

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Figures 4(a)-4(c) plot the simulated time-averaged power flows (left) and the normalized power intensities as a function of x position measured along the dotted line (right) with the incident wavelengths of 532 nm, 632.8 nm and 650 nm, respectively. The figures show that the pair of slit pair whose center-to-center distance smaller than λ / 4 can be resolved by the hybrid-super-hyperlens. The resolution powers for various incident wavelengths are also listed in Table 1. The figures also display that the signals extracted from the slits can propagate in the planar superlens and in the cylindrical hyperlens, and finally transfer into the far field. This is the reason why the hybrid-super-hyperlens can break the optical diffraction limit. Table 1. Parameters of the hybrid-super-hyperlens at variant incident wavelength Wave length 405 nm 532 nm 632.8 nm

AlGaInP ( ε

650 nm

AlSb ( ε

Upper dielectric Al2O3 ( ε

= 3.217)

TiO2 ( ε

= 8.892) = 11.71)

= 14.59)

Lower dielectric HfO2 ( ε Cu2O ( ε

= 3.9)

Center to center 100 nm

= 9.625)

100 nm

~λ

/ 5.3

110 nm

~λ

/ 5.7

120 nm

~λ

/ 5.4

SiGe ( ε = 15.1) (SOPRA 2% Ge) SiGe( ε = 16.85) (SOPRA 35% Ge)

Resolution power ~λ

/4

Fig. 5. Simulated Power flow images of (a) a pair of nano-slits and (b) a nano-ring that covered on the Chromium with linearly polarized incident light. The center to center distance of the nano-slits pattern is 140 nm, and the width of the slit is 50 nm. The inner and outer radius of the nano-ring is 70 and 120 nm, respectively. Images are recorded at 20 nm below the outer surface of sphere-shape hyperlens.

Next, the images of three-dimensional nano-patterns formed by the proposed hybridsuper-hyperlens are examined by FDTD (Lumerical) simulations. Figures 5(a) and 5(b) plot the simulated time-averaged power flows images below the bottom facet of the spherical hyperlens for a pair of nano-slits and a nano-ring, respectively, carved on the chromium sample stage and illuminated by linearly polarized light with the incident wavelength of 405 nm. (Note that, the sample stage design such as the thickness (width) of slits and type of material can be found in Ref [31, 32].) The materials and geometry parameters of the proposed lens are also listed in Table 1. Both the nano-slits and nano-ring have the widths of 50 nm and are filled with a dielectric of refractive index n = 1. (The structures are depicted in the insets of Figs. 5). Figure 5(a) displays that the far-field magnified image of the nano-slits pair can be obtained via the hybrid-super-hyperlens. Since the z component of electric field of the incident linearly light is perpendicular to the metal (chromium) surface, SPPs in the vicinity of the pair (nano-slits) are excited [33]. The Fabry-Perot-like resonance inside the slits would induce the SPPs coupling between the upper and lower surface of chromium stage. When the multi-layered superlens is close to the chromium, the excited higher order

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signals (fine features) with evanescent form can be transferred to propagation mode [16]. After the signals pass through the superlens and arrive at the interface between superlens and hyperlens, they can be received and delivered by spherical hyperlens from the interface to the outer surface. Because they become a propagating mode, the energy can be detected in the far field through a conventional optical microscope with a magnification factor ( rout / rin ) [17]. Conversely, Fig. 5(b) shows that the resolvable image of the nano-ring cannot be obtained by the lens with a linearly polarized incident light. This result is attributed to the fact that SPPs are excited at parts of the nano-ring only. That is, the polarization direction of the incident light plays an important role in the ability of the lens to resolve a complicated geometrical pattern. To resolve the nano-ring pattern, all the radial directions of this structure should be “seen”. SPPs along the groove of the nano-ring should be excited and all the corresponding fine feature signals should be delivered to the superlens at the same time. Figures 6(a) and 6(b) depict the time-averaged power flow images of a pair of nano-slits and a nano-ring that are irradiated with radially polarization source (using Lumerical). The images are also extracted at 20 nm below the hybrid-super-hyperlen outer facet. Figures 6(a) and 6(b) exhibit that both of the nano-slits and the nano-ring are resolvable by the radiallypolarized incident light, which are significantly different from those in Figs. 5. As we have mentioned, the formation of the images are caused by excitation of SPPs along the groove and transfer of fine-structure signals into the far-field. Notably, an unbroken and completed image can be obtained at one scan procedure by using the proposed hybrib-super-hyperlens. To image more complex nano patterns, the unpolarized light source should be required. Furthermore, all parameters of material presented here are currently existent and available in industry, so that they can be applied to design real and operable devices. The choice of parameters of the material in the hybrid-super-hyperlens is flexible; basically, any values that suit the requirement shown in section 2 will also work well.

Fig. 6. Simulated power flow images of (a) a pair of nano-slits and (b) a nano-ring with radially polarized incident light. Simulation parameters and observation plane are the same as in Fig. 5.

4. Conclusion

The hybrid-super-hyperlens is proposed and numerically investigated. In this proposed lens, by using the anisotropic feature of superlens and hyperlens, the high order signals that reveal the fine feature of the resolved object can be transferred and delivered into far field region. The proposed lens has demonstrated the resolution aboutλ/6 that breaks the diffraction limit. Both the nano-slit pair and the nano-ring are successfully resolved using radially incident source. The feasibility of the proposed lens is assured since all the materials are available. Acknowledgment

This work is supported by the National Science Council of Taiwan under grants NSC1002923-M-002-007-MY3, 101-2112-M-006-002-MY3, 101-3113-P-002-021, 101-2911-I-002107 and 101-2112-M-002-023. We also thank National Center for High-Performance #184807 - $15.00 USD (C) 2013 OSA


Computing, Research Center for Applied Sciences, Academia Sinica, Taiwan, National Center for Theoretical Sciences, Taipei Office, and Molecular Imaging Center of National Taiwan University for their kind support.

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