## Abstract

When the metallic near-field superlens is to image a planar object, which is itself metallic, such as that in the near-field lithography applications, the object nanometer features will act as the Hertzian dipole sources and launch homogeneous and evanescent waves. The imaging system can be modeled as a dielectric Fabry-Perot cavity with the two surface plasmon resonant mirrors. We show the expressions of the transfer function and optimize the imaging system configuration using the genetic algorithm. The effectiveness of the design is confirmed by the image intensity profile computed with the numerical finite difference in time domain method.

© 2011 OSA

## 1. Introduction

The superlens is a slab of negative index material (NIM) which can restore the phase of propagating waves and the amplitude of the evanescent waves [1] to achieve near-field imaging with sub-wavelength resolution beyond the diffraction limit. The metallic superlens is an alternative to the NIM lens made with noble metals existing in the nature like silver, gold and aluminum, which have a sole negative real part of dielectric function Re{*ε _{m}*}

*<0*at operating optical frequency, but a normal permeability

*μ*1 [2]. The metallic superlens is easier to implement, experimentally demonstrated [3,5] and has potential applications to near-field lithography of nanometre resolution [4–7]. The principle of the metallic near field superlens is based on the amplification of the surface plasmon polaritons (SPPs) by the resonance of the multiply reflected SPP modes between the two interfaces of the superlens. The SP resonant amplification is necessary to compensate for the exponential decay of the evanescent waves away from the object plane [2,8], but enhances disproportionally the evanescent part of spatial spectrum, resulting in narrow peaks in the transfer function and high sidelobes in the images [6]. Research efforts have been made for improving the imaging performance of the near field superlens in terms of its transfer function by optimizing the parameters of the superlens imaging system configuration [9,10]. It has been found that the typical two dominant feature peaks in the transfer function of the metallic near field superlens correspond to the long-range and short-range SPP waveguide modes, respectively, propagating along the interfaces of the superlens [11]. Therefore, the metallic superlens can be designed by approaching the condition of the cut-off of the long-range SPP mode [12] in order to flatten the transfer function and suppress the sidelobes in the image [10].

_{m}~Most papers on the superlens represent the imaging system as a slab of superlens placed in two semi-infinite dielectric media, where the object and image planes are placed. In this case the superlens can be modeled by the modal analysis for the insulator-metal-insulator (IMI) SPP waveguide. For near-field lithography applications, the object mask can be metallic. This fact has been considered recently in modeling the system by the numerical finite-difference in time-domain (FDTD) simulation [6]. Moor et al [7] consider the metal object as a reflection mirror in the transfer-matrix analyses of the metallic superlens. In fact, the nanometric features in the object plane can launch the SPPs along the interfaces of the object layer, and the imaging system can be modeled as the metal-insulator-metal (MIM) SPP waveguide sandwiched between two semi-infinite dielectric media [13]. The transfer function of the metallic near field superlens with a metallic object is significantly different from that with a dielectric object. In this paper we propose a new model of the metallic near field superlens with a planar metal object. Compared with the model proposed in Ref [7], our model is more general by taking into account the SPPs along the interfaces of the object layer and the SP resonance in the object layer. We use the genetic algorithm to optimize the imaging system configuration and show how to re-design a metallic near field superlens imaging system, which was initially designed with the model where the effect of the fact that the object is metallic was not taken into consideration. All the analysis and the design with the analytical expressions will be verified by the numerical simulation with the FDTD method of the near-field imaging.

## 2. Transfer function

Consider a near-field imaging system with a metallic superlens of dielectric function ε_{s} = ε_{4} with Re{ε_{s}}<0 The object is also a metal layer, such as a metal coated lithographic mask, of dielectric function ε_{o} = ε_{2} with Re{ε_{o}}<0, as shown in Fig. 1
. In practice there can be multiple dielectric layers between the object and the superlens, such as index matching layers [6]. We consider a single dielectric layer of dielectric function ε_{d} = ε_{3} and assume that a photosensitive layer is in contact with the superlens, for the sake of simplicity. Although in practice there can be a spacer of thickness d_{5} between the superlens and the image plane, the impact of such a spacer is well known as introducing an exponential decay of the evanescent waves over an additional distance d_{5} away from the superlens interface ε_{m}-ε_{5}. Consider also the dielectric media 1 and 5 as semi-infinite. For the purpose of the image analysis, we consider a sub-wavelength feature slit perforated through the metal object layer, and that the illuminating light is a plane wave of TM polarization incident normally to the object layer from the dielectric medium ε_{1}.

Modeling the metallic near field superlens with a metallic object is significantly different from modeling the superlens with a dielectric object. When the object features are represented as sub-wavelength slits or grooves in a metal layer, the slit or groove will launch the SPPs along the object layer and along the superlens, with the coupled SPP modes. The illuminating plane wave of TM polarization is the physical source of the light, which passes through the slit, and is also transmitted through the metal part of the object layer, the consecutive dielectric layer and the superlens to reach the image plane, where the slit is imprinted with a DC background, which is the contribution of the plane wave transmitted through the structure. More importantly, the incident light will introduce two dipole sources *B* and *A*, as shown in Fig. 1. The illuminating light is first scattered by the slit inducing an electrical dipole at the two corners of the slit, which is the source *B* located on the slit entrance side at the first interface *ε _{1}-ε_{o}*. The oscillating dipole

*B*launches the SPPs propagating along the interfaces of the metal object layer in the

*x*-direction transversal to the imaging direction, and radiates homogeneous waves in all directions. The light radiated by source

*B*and the transmitted incident plane wave passing through the slit and the object layer will be scattered again by the slit at the exit side of the slit, inducing an electrical dipole at the two corners of the slit on the slit exit side in the second interface

*ε*. This is the source

_{o}-ε_{d}*A,*which launches SPPs along the interface of the object layer and radiates homogeneous waves towards and away from the superlens. The SPPs at the two interfaces of the object layer are in general mutually coupled, and furthermore coupled with the SPPs at the two interfaces of the superlens, giving rise to the SPP waveguide modes. As the metallic near field superlens with a metallic object depicted in Fig. 1 has a similar structure as the MIM SPP waveguide with the 3-asymmetrical MIM layers of finite thickness sandwiched between two semi-infinite dielectric media, the SPP modal analysis based on the dispersion relation [13] may be performed in order to identify and reduce the respective SPP resonance peaks in the superlens transfer function [11].

The transmission coefficient of the imaging system shown in Fig. 1 can be written by considering the multiple reflections at the interfaces of each of the 3 layers of the media of *ε _{o}, ε_{d}* and

*ε*, respectively, as

_{s}This is the transmission coefficient of a Fabry-Perot resonator, which consists of the dielectric medium *ε _{d}* sandwiched by two SP resonant mirrors: the metal object layer of ε

_{o}and the superlens of

*ε*The propagation factor

_{s}.*e*(

_{i}= exp*ik*) with the sub-index

_{zi}d_{i}*i*=

*o, d, s*describing the phase shift of the homogeneous waves propagating over the distance

*d*and the exponent decay in amplitude of the evanescent waves over

_{i}*d*, respectively. Equation (1) can describe the impact of both homogeneous and inhomogeneous waves on the transmission coefficient, where the Fresnel reflection and transmission coefficients from medium

_{i}*i*to medium

*j*are

*z*, and ${r}_{ij}=-{r}_{ji}$, ${t}_{ij}={t}_{ji}{\epsilon}_{j}{k}_{zi}/{\epsilon}_{i}{k}_{zj}$. The reflection coefficient of the superlens${\rho}_{s}^{+}$and that of the metal object ${\rho}_{o}^{-}$ expressed respectively as:

In fact, ${\rho}_{s}^{+}$ is the reflected wave from the superlens to the layer *ε _{d}* with the Fresnel reflection coefficient

*r*, summed with the waves, which are multiply reflected between two interfaces of the superlens and transmitted through the interface

_{ds}*ε*-

_{s}*ε*to the layer

_{d}*ε*, and ${\rho}_{o}^{-}$ is the reflected wave from the object to the layer

_{d}*ε*with the Fresnel reflection coefficient

_{d}*r*summed with the waves, which are multiply reflected between two interfaces of the object layer and transmitted through the interface

_{do}*ε*-

_{o}*ε*to the layer

_{d}*ε*. The superscripts + and – of the reflection coefficients in Eqs. (3) indicate the propagation direction of the incident wave relative to the

_{d}*z*axis. Moreover, application of the Maxwell’s equations and the boundary conditions at the interfaces to the electromagnetic fields in each layer in the system would result in the same transfer function as expressed in Eq. (1) [11]. Equation (1) can also be obtained as a transfer function using the transfer matrix approach [14].

The metallic slab superlens does not focus the propagating waves, so that the image is formed by the projection of the plane wave passing through the slit. If we remove the constant Fresnel transmission coefficient of the interface ε_{1}-ε_{o} for the incident plane wave, t_{1o}, in Eq. (1), then we obtain the transfer function of the system due to the source B located on the first interface of metal object layer:

In the same way, the transfer function of the system due to the source *A* is written as

Equation (5) alone describes a particular case, where the object layer is very thick with *d _{o}* much larger than the SPP penetration depth in the metal, so that the SPP at the first interface

*ε*

_{1}

*-ε*is decoupled from the SPP at the second interface

_{o}*ε*. Furthermore, as the homogeneous waves radiated from the source

_{o}-ε_{d}*B*are not transmitted through the thick object layer, but only through the slit, the system is considered as illuminated by a single source

*A*, and as the multi-reflections within the thick metal object layer cannot be maintained for both homogeneous and evanescent waves,${\rho}_{o}^{-}$ becomes a simple Fresnel reflection coefficient of the metal object. In this case, our model is reduced to the model proposed by Moore [7]. In the case, where the object is dielectric, the reflection coefficient ${\rho}_{o}^{-}$ is small, so that there is no resonance related to the multi-reflections between the two interfaces of the dielectric layer,

*ε*and

_{o}-ε_{d}*ε*, so that the dielectric layer is reduced to a simple spacer for the propagation over a distance

_{d}-ε_{s}*d*

_{3}, and

*τ*~

_{d}*e*in Eq. (1). Our model is general, including the case of moderate thickness

_{d}*d*

_{o}of the object layer. The dipole sources

*B*and

*A*emit both homogeneous and evanescent waves. In Eqs. (4) and (5) the propagation factors

*e*describe both propagating and evanescent waves. The losses by absorption in the media also are taken into account by the complex-valued

_{i}*ε*and

_{o}*ε*in Eqs. (2) and (3). The imaging property of the superlens is quite different, depending on the presence of both point sources

_{s}*B + A*or only

*A*described by Eq. (4) or (5), respectively.

The transfer function of the near field superlens is defined as the transmission coefficients of the system described in Eqs. (4) and (5) as a function of the spatial spectral frequency. According to the classical Fourier optics theory, in a linear and space invariant imaging system the point spread function is the Fourier transform of the transfer function. However, in the near field imaging one models an object point as a feature slit. Thus the space invariance condition is compromised, if the finite width of the slit is taken into account. In this case the image of an isolated two-nanoslit pattern would be more appropriate than the point spread function to describe the imaging property of the near field imaging system. In the next section we design the near field imaging system by optimizing the transfer function computed with Eqs. (4) and (5), and we check the results of the optimization by analyzing the image of a two-nanoslit pattern, which is computed with the FDTD method.

## 3. Optimal design with genetic algorithm

When the object is metallic, the transfer function of the metallic near field superlens, as expressed in Eqs. (4) and (5), is significantly different from that with a dielectric object, so that the superlens needs to be redesigned. In this paper we propose an approach for designing the metallic near field superlens with metallic object by optimizing the metal-superlens/metal-object configuration using the genetic algorithm (GA). The GA is a powerful optimization tool based on the evolution principles found in nature. In the GA a population of randomly generated potential solutions, *i.e.* chromosomes, evolves through recursive processes of selection, reproduction and mutation until a globally optimal solution is found. In each iteration in the GA the quality of every chromosome in the population is first evaluated by computing the cost function. Then, the chromosomes are ranked according to their quality and randomly selected for reproduction with a probability of selection following a cumulative Gaussian normal probability distribution, which favors high-ranked chromosomes [15]. To ensure the monotonic progression of the search, the best chromosomes, *i.e.* elites, are kept intact. They are cloned from one generation to the next generation. To introduce more genetic diversity, mutation of the chromosomes is also performed in each iteration. The search is stopped after a given number of iterations or when the evolution of the best chromosome stagnates.

The goal of our GA optimization is to obtain the transfer function of the superlens near-field imaging system as smooth and as flat as possible over the broadest spectral range. We focus the search over a spatial bandwidth in which the impact of the structure parameter variations is critical. Interesting results have been obtained with a search over the bandwidth |*k _{x}*

_{0}| ≤ 5

*n*

_{3}

*k*

_{0}, where ${n}_{3}=\sqrt{{\epsilon}_{3}}$. When the metallic object layer with a perforated feature slit is illuminated by a TM polarization plane wave the electrical charges accumulated at the corners of the slit form two electrical dipoles in the entrance and exit sides of the slit, which are the sources,

*B*and

*A,*of evanescent and homogeneous waves in the structure. The dipole source

*B*is in general much stronger than source

*A*as it is illuminated by the direct incident wave. However, the quantitative relation between the dipoles

*B*and

*A*is in general unknown, so that we define a cost function which includes both transfer functions for

*B*and

*A*expressed in Eqs. (4) and (5), respectively. For source

*A*, we seek a transfer function of unitary amplitude, whereas for source

*B*we only seek a transfer function with constant amplitude. In this way, we limit ourselves at avoiding excessive excitation of SPPs for the signal emitted from source

*B*. Thus the cost function is defined as

*A*and

*B*, expressed in Eqs. (5) and (4) respectively, calculated over a frequency comb of

*N*=

*M*= 501 points evenly distributed within the bandwidth |

*k*

_{x}_{0}| ≤ 5

*n*

_{3}

*k*

_{0}and $\left|\overline{TFB}\right|$ is the average amplitude of ∣

*TFB*∣ over the bandwidth |

*k*

_{x}_{0}| ≤ 5

*n*

_{3}

*k*

_{0}. Note that Eq. (6) gives equal weight to the transmission of the two sources, even though the contribution from source

*B*is more important than that from source

*A*.

In most GA implementations, the chromosomes are encoded with strings of binary numbers. However, the parameters to optimize in the structure of the metallic near field superlens with metallic object are the thickness and dielectric functions of the layers, in which *d _{o}*,

*d*,

_{d}*d*and

_{s}*ε*are real valued, while the real part of the dielectric function,

_{d}*ε*and

_{o},*ε*, depends on the wavelength. The real and imaginary parts of the complex-valued

_{s}*ε*and

_{o},*ε*may not be optimized separately. Thus, we use the GA, in which the chromosomes are encoded in vectors with the real valued parameters to be optimized as the vector elements (genes).

_{s}Once the parent chromosomes were selected, our GA with real parameter genes performed the reproduction with two distinct crossover mechanisms with equal probability [15]. The first mechanism generated an offspring by randomly selecting genes from its two parents. The offspring $c$ is reproduced by randomly selecting genes from the parents $a$ and $b$ as:

*x*” with

_{i}*1≤ i ≤ n*in $c$ has an equal probability of being an “

*a*” or a “

_{i}*b*”. The second mechanism generating an offspring

_{i}**c**is a linear crossover of the real-valued chromosomes by performing a linear sum of the two parents,

**a**and

**b**, usingwhere g is a random parameter within the interval 0 ≤ g ≤ 2. The probability of the reproduction was set to 0.9. In the case where reproduction does not occur, the offspring is cloned from its second parent.

Once the reproduction process was done, new genes were introduced into the population. The mutations were performed on the newly generated chromosomes with a probability of 0.5. This value is higher than typical values of 0.15-0.3 for real-valued GA [15] in order to increase genetic diversity. The mutations consist in randomly choosing a single gene inside a chromosome and changing the gene randomly. A real valued chromosome element $m$ is mutated to generate ${m}_{m}$ by

where*h’*is a random value, 0<

*h’*<1, such that the mutated gene is kept close to the original value of the gene within an interval controlled by

*h’*. Note that two elites were cloned for each new generation and were never mutated to avoid losing precious genetic material.

For design example we have chosen deliberately the superlens studied in Ref [6], which is a five-layer structure with an Al superlens of thickness *d*_{s} = 13 *nm*. The object mask is also in Al and of thickness *d*_{o} = 20 *nm*. Thus, *ε*_{o} = *ε*_{s} = −4.42 + 0.43*i* at *λ =* 193 *nm*. In between the object mask and superlens, there are two layers of SiO_{2} with *ε _{d}*

_{1}= 2.4 and MgO with Re(

*ε*

_{d}_{2}) = 4.08, which serves as an index matching layer. Both layers are of thickness

*d*

_{d}_{1}=

*d*

_{d}_{2}= 10

*nm*. There is a spacer layer of thickness

*d*

_{5}= 8

*nm*and

*ε*

_{5}= 2.89 between the superlens and photoresist layer of

*ε*

_{6}= 2.89. The system showed a nice transfer function, as shown in Fig. (1) in Ref [6], with a high SPP resonance peak in the region of frequency 1.5 <

*k/k*< 2.0. The superlens showed a resolution of 20

_{0}*nm*for a periodic object. However, for an object of two-slit structure, which had a larger spatial spectrum bandwidth, there were two high side lobes lying outside of the image, which can be suppressed only by placing assistant features on both sides of the slits in the object plane, as shown in Fig. (5) in Ref [6]. The results in Ref [6]. have been obtained by numerical FDTD simulation.

The near-field superlens could be designed with the numerical simulation, but by scanning the configuration parameters or by trials with randomly chosen parameters. The close-to-cutoff of the long-range SPP waveguide mode technique has been proposed as a rule of thumb for the design [10,11]. The technique was applied to the superlens of Shi [6] with the modification in the configuration that first put *ε _{d}*

_{1}=

*ε*

_{d}_{2}= 4.08, for simplicity, and then decreased the permittivity of the spacer layer to

*ε*

_{5}= 1.93 in order to approach the cutoff condition [10]. Figure 2 shows the transfer function of this superlens structure computed with Eqs. (4) and (5) where the presence of the metal object mask was considered, and the thickness of the metal object layer was set to

*d*

_{o}= 20

*nm*, and the image intensity profile computed with the FDTD simulation. We see that the high SPP resonance peak shown in Ref [6] was effectively suppressed, as shown in Fig. 2a, that led to a significant reduction of the high side lobes lying outside the image of the two-slit, as shown in Fig. 2b, although the DC component and the central lobe in the image plane were still high. Note that the close-to-cutoff technique is based on the three-layer IMI structure without considering the presence of metal object mask, so that several trials were still needed to make the design useful for the real structure with the presence of metal object.

In the design using the GA the parameters to be considered were the permittivities of the dielectric media *ε*_{1}, *ε*_{d}, *ε*_{5} and the thickness of the metal object layer *d*_{o}. Other parameters remained the same as that of the superlens of Shi [6], which were Al superlens and object with *ε _{o}* =

*ε*= −4.42 + 0.43

_{s}*i*at

*λ*= 193

*nm*. The superlens thickness is

*d*

_{s}= 13

*nm*. There are one single dielectric layer of thickness

*d*= 20

_{d}*nm*between the object and the superlens and one dielectric spacer of thickness

*d*

_{5}= 8

*nm*and

*ε*

_{5}= 2.89 between the superlens and the photoresist PMMA layer of

*ε*

_{6}= 2.89, into which the image is inscribed. These choices limiting the number of degrees of freedom in the optimization process were deliberately made to facilitate the search and to show that the structure previously optimized by Shi [6] and by the close-to-the-cutoff of the long-range SPP waveguide mode technique [10,11] can be significantly improved without excessive modifications of the structure dimensions and material parameters. Our GA was performed with a population of 200 chromosomes randomly generated with the constraints for the dielectric permittivities 2 ≤

*ε*

_{1},

*ε*

_{d},

*ε*

_{5}≤ 5 and the thickness of object layer 13

*nm*≤

*d*

_{o}≤ 33

*nm*. The search was performed over 50 generations in the GA. The optimal parameters obtained were as

*ε*

_{1}= 2.5215,

*ε*= 3.1832,

_{d}*ε*

_{5}= 3.3191 and

*d*

_{o}= 31.5

*nm*. The performances of the optimized structure in imaging a pattern of 40

*nm*-spaced two 20

*nm*slits are illustrated in Fig. 3 .

The intensity profile of the image in Fig. 3b shows a ratio of the maximal image intensity of the two slits over the background “DC” noise of about 10, which is a great improvement compared to the results shown in Fig. 3b, for which this ratio never exceeds 2. Also, the sidelobes are much weaker. The reduction of the background noise is due to the 31.5 nm thickness of the Al object layer that attenuates most of the incident signal. The improvement of the image quality is due to the optimization of the transfer functions, shown in Fig. 3a, where, for source *A* the blue line shows better field amplification for the frequency range *k _{x}*

_{0}> 2 and a significant reduction of the peak at the frequency of the peak

*k*

_{x}_{0}~1 as compared with Fig. 2a. For source

*B*, the transfer function is free of any feature peaks with a profile broader than that illustrated in Fig. 2a.

## 4. Conclusion

We have proposed a general theoretical model of the metallic near-field superlens with a metallic object layer as a Fabry-Perot dielectric cavity with two SPP resonant mirrors, and introduced expressions of the transfer function, which are valid for both homogeneous and evanescent waves. We have proposed to design the superlens system configuration by optimizing the transfer function of the system using the GA, which is advantageous over the numerical simulations based on random trials and the close-to-cutoff of long-range SPP mode technique

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