We propose an innovative active imaging device named gain-assisted hybrid-superlens hyperlens and examine its resolving power theoretically. This semi-cylindrical device consists of a core of semi-cylindrical super-lens and a half cylindrical outer shell of hyperlens. Both the superlens and hyperlens parts of the device are appropriately designed multi-layered metal-dielectric structures having indefinite eigenvalues of dielectric tensors. The dielectric layers of the hyperlens are doped with Coumarin, which play the role of gain medium. The gain medium is analyzed thoroughly using a generic four-level system model, and the permittivity of the gain medium is extracted from this analysis for simulating the imaging characteristics of the device. According to our simulation at wavelength of 365 nm, an excellent resolution power much better than the diffraction limit value can be achieved.
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In 2000, Pendry realized that the diffraction limit in conventional optics is not an unbreakable restriction; instead, this limit is mainly caused by the fact that the sub-wavelength information of the light source encoded in the evanescent waves is lost once they leave the source. He found that a slab made of a metamaterial having both negative-permittivity and negative-permeability would also have negative refractive index, and these properties help to enhance the evanescent waves, making the sub-wavelength imaging possible . When only the permittivity is negative, sub-wavelength imaging is still possible if quasi-static conditions are satisfied, thus a thin silver slab can make a superlens. Thereafter, studies related to superlens, sub-wavelength imaging, and negative refractive index materials grew rapidly, and corresponding publications increased exponentially [2–7]. Superlens, however, is a ‘near-sighted device’ and works only in the near-field region, which constrains its practical application . For this reason, a modified structure named hyperlens which can break diffraction limit and make images of sub-wavelength sized objects in the far-field zone was proposed and received great attention [9–12]. A hyperlens is a cylindrical or spherical structure consisting of cylindrically or spherically arranged metal-dielectric layers [13–17]. Effectively, this kind of structure is an indefinite medium, which means the eigenvalues of the dielectric tensor of the device along the radial and non-radial directions have different signs, and the dispersion curve relating the different components of the wave vector of the propagating mode at a specific frequency is of hyperbolic form. This property implies that the evanescent waves which contain the information of the sub-wavelength features of the objects can be transferred into propagating waves inside this device. A thorough analysis indicates that for a properly designed hyperlens the propagating waves inside the device can be transferred back to the propagating waves in the surrounding medium when they leave the outer surface of the hyperlens. Magnified images of the objects in the far-field zone can thus be reconstructed.
On the other hand, usually a superlens or hyperlens cannot be fabricated without using metallic materials. It always accompanies with significant energy loss when operating at terahertz frequency, even at optical frequency. This disadvantage results in that metamaterials are hardly utilized in practical manner since the throughput of the incident waves is extremely low. Consequently, in order to overcome the dissipation problem, some different materials, such as graphenes and superconductors are used [18,19]. It has, however, other limitations such as the extremely low operating temperature and fragility; all of these restrict their utilities. In 2009, two different types of metamaterials with gain were demonstrated numerically [20,21]. It had been proved that gain material is capable of reducing or compensating the loss of the wave medium.
In this Letter, we propose an active device named gain-assisted hybrid-superlens hyperlens and demonstrate its super-resolution ability. The structure and material parameters of this device are based on our previous research of hybrid-superlens hyperlens . By doping a dye molecule Coumarin 2 into PMMA, which is a gain medium with absorption wavelength at 365 nm, it can be made as a thin gain layer. Base on this concept, active hybrid-superlens hyperlens with gain can be designed. We model the gain medium as a four-level system, and derive an analytic formula for calculating the effective permittivity of the medium under the condition of weak field intensity (|E|<<104). It is not necessary using numerical calculations, such as retrieval method, to define the permittivity of the gain medium when this condition is satisfied. With our equation, the permittivity of the four-level active medium can be calculated directly, and it is used for simulating the imaging characteristics of this active device. Our simulation at wavelength 365 nm reveals that an excellent resolution power much better than the diffraction limit value can indeed be achieved when the gain layers are used.
The gain medium in this paper is analyzed by using the same model of a four-level system presented in . The occupation numbers per unit volume vary according to the following equations:
To solve these coupled equations, a proper approximation will be applied. Note that in the four-level system, in the interest time scale, the occupation numbers at energy level 2 and 4 tend to zero (i.e.,) because both of them have extremely short relaxation time: τ21~τ43~10−14 sec. Conversely, the relaxation time τ32 (~10−12 sec.) for the transition from energy level 3 to 2 is much longer than τ21 and τ43, making the population inversion possible. Subtracting Eq. (3) from Eq. (2) and Eq. (1) from Eq. (4), defining and then using the conditions N1 >>N4, N3>> N2, N ~N1 + N3 ~N1 + ΔΝ, we getEq. (5), we getEq. (6) stands for the expectation value of, as can be verified by doing time-average of Eq. (5). The time-varying (AC) term gives the first-order perturbation correction to. It is obvious to observe that the magnitude of the non-linear term (the integral part) increases with the applied electric field intensity.
Referring to , the dynamical equation for the induced electric polarization is given byEq. (7) can be approximated asEq. (8), the non-linear term gives the first-order perturbation correction to the population numbers. Nevertheless, no well-developed method can be readily implemented to solve this differential equation for finding its exact solution. Though we are not able to solve it exactly, approximate solution is still possible to obtain if the electric field is weak enough. Hereafter we assume that the weak field condition is fulfilled, thus the nonlinear term on the right hand side of Eq. (8) can be dropped. The extracted effective permittivity under this assumption is still of Lorentz form, written as
Here is a characteristic frequency, and the unusual plus sign appearing in the right hand side of the formula between 1 and the remaining term indicates that the gain material indeed provides energy compensation for the absorption of the medium. The validity of the above approximations can be verified numerically using FDTD method . However, if the electric field intensity increases to 104 or higher orders, the non-linear effect cannot be neglected. This argument had already been mentioned in  and we get a good match for its conclusion.
3. Simulation result
In this section we introduce the structure of the gain-assisted hybrid-superlens hyperlens and implement numerical simulations to examine its performance. As shown in Fig. 1 , this device consists of two kinds of multi-layered metal-dielectric anisotropic metamaterials: the semi-cylindrical superlens core as the upper part, surrounded by the cylindrical-hyperlens shell as the lower part. Both lens structures can be treated as effective indefinite media, which means their dielectric tensors have eigenvalues of different signs along the two principal axes (horizontal and vertical for the superlens; radial and azimuzal for the hyperlens). This property also implies that their isofrequency dispersion curves are of hyperbolic form. With dispersion curves of this type, the evanescent waves containing the high spatial frequency information of sub-wavelength sized objects can excite the propagating modes inside the lens media. As mentioned before, for a properly designed hyperlens the propagating waves inside the structure can be transferred back into the propagating waves in the surrounding medium when passing through the outer surface, and these transmitted waves reconstruct the magnified images of the objects in the far-field zone. However, the original design ofhyperlens is not very convenient to use because its inner surface is curved. We thus combine it with a semi-cylindrical shaped superlens with one flat top surface to form this hybrid structure. Sub-wavelength imaging ability of this hybrid-superlens hyperlens has been demonstrated in our previous work . High performance for resolution ability of this hybrid device can be achieved by using different materials [8,9]. Numerical simulation provides further solid evidences to our designed structure. The chosen parameters for the geometry and materials are realizable using today’s nano-fabrication techniques. Furthermore, this device can be combined with conventional lens system to form a novel imaging system for making magnified real-time images of sub-wavelength sized objects.
However, hybrid-superlens hyperlens still suffers from the drawback of extremely high energy loss. In fact, when passing through this device, only less than 1% of the incident wave energy remains. In order to resolve this disadvantage, a combined structure consisting of a hybrid-superlens hyperlens and a laser dye thin film is proposed, as shown in Fig. 1. In this letter, the laser dye we choose is Coumarin, which is an excellent candidate of gain material because: 1its absorption wavelength is in the violet-blue region and there is indeed laser operating at the same wavelength , 2it is almost non-toxin to the human body since Coumarin and its derivatives are commonly used as food additive and ingredient in perfume, and 3Coumarin can be easily doped into PMMA to a high concentration and its knowledge for being gain medium is well-established . To fit our designing parameters, Coumarin 2 (7-(Ethylamino)-4,6-dimethylcoumarin) is chosen to be the gain material because its absorption wavelength (λa = 365 nm) is readily acquirable using commercial laser products. We also assume that the PMMA has been doped with Coumarin 2 to the concentration 1.0 × 10−2 M . Other experimental parameters of Coumarin 2 are as follows: the total electron density N is 6.0 × 1026 per cubic meter, the lifetime τ32 is 4.2 × 10−9 second , the coupling strength σa is 10−8 C2/kg , the bandwidth of the absorption wavelength Δλ is 50 nm, and the pumping rate r is 1.0 × 109. According to these chosen parameters, the real and imaginary parts of the effective permittivity of pure Coumarin 2 are shown in Fig. 2(a) . Based on the results shown in Fig. 2, the permittivity of Coumarin 2 at λ = 365 nm is 0.9057 + 94.5550i. As has been mentioned before, the concentration of Coumarin 2 in PMMA is 1.0 × 10−2 M, corresponding to a ratio of weight (denoted as f) about 0.002. This value is derived from the data that the density of PMMA is 1.35 g/cm3 and the molecular weight of Coumarin 2 is 217.26 g/mol. The dielectric constant of the gain medium layer thus becomes 2.222 + 0.1887i. The commercial solver COMSOL MultiphysicsTM 3.5a based on finite element method (FEM) is utilized insimulating the two-dimensional structure, and perfectly matched layers surrounding the simulation region are used. Transverse magnetic (TM) polarized incident light is considered with the incident electric field (x-direction) being perpendicular to the axis of the cylindrical structure (along the y-direction), as shown in Fig. 1. Figure 2(b) plots the isofrequency dispersion relations for the superlens and hyperlens structures of the hybrid-superlens hyperlens at the incident wavelength of 365 nm.
In order to satisfy the requirement in , the selection of isofrequency curves should take a specific set of dispersive relations. For the ability of resolution and transferred the signal extracted from the slits, this device should use two different anisotropic media. There are opposite signs for dielectric tensor between the upper planar-superlens and the lower cylindrical-hyperlens resulting from different dielectric formations. This fact would result in different hyperbolic dispersion between the upper and lower anisotropic media. In our work, according to ref , the upper planar-superlens with East–west opening hyperbolic form can be regarded as the resolution component of hybrid-superlens hyperlens excited by TM wave. In addition, the lower anisotropic medium needs to be in the form of North–south opening hyperbolic form in order to receive and spread subwavelength signals from the upper planar-superlens .
The magnetic field amplitudes and the images of the two slits evaluated at the cross-section lines using the hybrid-superlens hyperlens with and without gain are shown in Fig. 3 . In Fig. 3(a), magnetic field amplitudes with/without gain layers are demonstrated. It is obvious that this device is capable of resolving two silts having a center-to-center distance smaller than one-third of the incoming wavelength. The position of the cross-section line is located at the tangential plane of the outer hyperlens. As is shown in Fig. 3(b), before adding gain layers into the device, poor resolution is observed. With gain layers in our device, the field intensity is magnified approximately three times as we set the structure parameters properly. Moreover, without obvious distortion, the magnified images of the two slits are clearly observed in the far-field region outside the gain-assisted hybrid-superlens hyperlens. This exciting result shows that the combination of gain and device can definitely compensate energy loss. With regards to our achievement, it is simple to improve the imaging quality by doping dye molecule since laser dye is very convenient to obtain.
We have derived an analytic formula for evaluating the permittivity of four-level-system gain medium theoretically. Similar analytical formulas for other kinds of gain medium under weak field condition can also be derived using similar approximation approaches. In addition, we have demonstrated the imaging ability of the active device named gain-assisted hybrid-superlens hyperlens, which consists of metal-dielectric multilayer structures of superlens core and a hyperlens shell, and is assisted with gain. This active device not only has the ability of resolving the sub-wavelength fine features of objects, forming magnified images in the far field, but also enhances the field intensity of the images without distortion. Our numerical simulations further verified these statements. Finally, since the parameters of geometry and the materials we chose are practically realizable by using today’s nano-fabrication technology, we expect this innovative device can be realized in the near future.
The authors acknowledge financial support from National Science Council, Taiwan, under grant numbers 100-2923-M-002-007-MY3, NSC 101-3113-P-002-021- and NSC101-2112-M-002-023-. We are also grateful to the National Center for Theoretical Sciences, Taipei Office, Molecular Imaging Center of National Taiwan University and National Center for High-Performance Computing, Taiwan and Research Center for Applied Sciences, Academia Sinica for their kind support.
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