# Determination of the point spread function of layered metamaterials assisted with the blind deconvolution algorithm

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## Abstract

We propose to use the blind deconvolution and its modification to extract the point spread function (PSF) of layered metamaterials from a SNOM measurement. The measurement results are processed using the blind deconvolution algorithm to reconstruct the real-valued non-coherent PSF, or using the modified blind deconvolution introduced in this paper to reconstruct the complex-valued coherent PSF. The two algorithms are tested in simulations with a layered metamaterial deposited on a thin metallic mask with test apertures. We show that the modified algorithm is capable of recovering the approximate shape of complex PSF with a sub-wavelength full width at half maximum from a measurement in which the apertures are larger than the wavelength.

## Keywords

Point spread function Metamaterial Blind deconvolution## 1 Introduction

Layered metamaterials consisting of metallic and dielectric layers (MDM) sandwiched in a periodic way find applications in imaging with sub-wavelength resolution (Fang et al. 2005), far-field imaging of sub-wavelength objects (Jacob et al. 2006), cloaking (Paul et al. 2012), or design of novel absorbers (Guclu et al. 2012; Pastuszczak et al. 2014). They can be designed to achieve such properties as an extreme anisotropy (Wood et al. 2006), effective-zero permittivity (Castaldi et al. 2012), or hyperbolic dispersion (Ni et al. 2011; Simovski et al. 2013), negative refraction (Scalora et al. 2007), superprism or supercollimation effects (Ceglia et al. 2008; Li et al. 2007), resonant tunneling (Liu and Behdad 2012), sub-wavelength focusing (Zapata-Rodríguez et al. 2012) etc.

MDM can be regarded as linear spatial filters (Schurig and Smith 2003) and the framework of shift invariant systems adapted from Fourier Optics (Goodman 2004) is useful for their characterization. The theory of three-dimensional point spread function (Zapata-Rodríguez et al. 2011) (PSF) as well as a vectorial PSF (Kotyński et al. 2011) have been recently developed for MDM. It is possible to measure the modulation transfer function (MTF) experimentally in a double exposure experiment (Moore and Blaikie 2012). However we are not aware of any attempts to measure the amplitude transfer function, or equivalently the complex PSF for coherent illumination.

In this paper we make an attempt to recover the PSF from the intensity measurement. Our method includes a phase retrieval algorithm, which can be seen as a modification of the blind deconvolution algorithm (Ayers and Dainty 1988; Yu and Paganin 2010). Additionally, we address the difficulty of obtaining a mask with aperture sizes of the same order or smaller than the size of measured PSF. Thanks to the use of the algorithm we propose, the apertures can be significantly larger than the size of PSF. On the other hand, the intensity measurement needs to be done with a high resolution. These assumptions agree well with typical SNOM measurement conditions.

## 2 Image formation models

We will refer to two different image formation models—a non-coherent model involving only real and positive functions, and a coherent model involving a complex-valued point spread function. In practice, the recovery of a complexed-valued PSF is by far more difficult than the recovery of real-valued PSF, and the complex-valued problem is more likely to have a non-unique solution. Only the non-coherent problem can be solved with the classical blind deconvolution algorithm.

### 2.1 Incoherent image formation model

### 2.2 Coherent image formation model

*a priori*knowledge of \(f(x)\). This can be accomplished only with a sufficiently complicated design of apertures \(f(x)\) that results in a measurement \(I(x)\) rich in interference patterns. Moreover, it is not possible to resolve the sign of the phase of \(h(x)\), since both \(h(x)\) and \(h^{*}(x)\) produce the same distribution \(I(x)\).

In the simulations we assume first that \(f(x)\) is estimated from \(I(x)\), so that \(f(x)=\varTheta (\sqrt{I(x)}-0.5\cdot max(\sqrt{I(x)}))\), where \(\varTheta (\cdot )\) is the Heaviside step function. A more sophisticated procedure to determine \(f(x)\) can be used if necessary. In Ref. (Moore and Blaikie 2012), the MDM material was removed in order to make an AFM measurement of the developed photoresist.

### 2.3 Estimation of the PSF broadening due to diffraction on the apertures

## 3 Description of the algorithm

## 4 Numerical results

In this section we demonstrate the recovery of PSF from a measurement which follows non-coherent, and coherent image formation models, given in Eqs. (1), (2), respectively. We also illustrate the problems with the recovery due to non-unique solution of the decompositions (1), (2). The measurement is one-dimensional and corresponds to one (for the non-coherent case) or more (for the coherent case) scans with a SNOM probe in the direction perpendicular to the rectangular apertures in the mask.

Now, let us demonstrate the performance of the novel algorithm proposed in this paper. We apply the coherent image formation model from Eq. (2) and the algorithm presented in Fig. 4. The PSF recovery will be demonstrated for a set of 20 measurements, each of which is obtained using a pair of two apertures. The aperture widths and the distance between them vary randomly in the set of measurements assuring a rich information content of the interference patterns. The apertures and the distance between them are always larger than \(\lambda \). Therefore, the PSF is subwavelength in size, however the apertures are not.

For layered metamaterials, the PSF, i.e. the response to a delta-shaped input signal, may be substantially different than the response to a sub-wavelength, still finite, Gaussian signal (Kotyński and Stefaniuk 2010). In fact, in this paper we address a problem which is often ill-posed i.e. there exist multiple PSF functions that give the same or almost the same interference pattern. Nontheless, the approximate resolution of the metamaterial can be estimated from the recovered PSF.

Altogether, thanks to the proposed algorithm, the resolution required for the fabrication of the mask can be relaxed significantly. In particular the mask may be produced with laser lithographic techniques, and still can be used to measure the PSF with a sub-wavelength resolution.

## 5 Conclusions

We have introduced an algorithm for improved measurement of complex PSF of layered metamaterials. The proposed algorithm is based on the blind deconvolution algorithm, however it operates on complex functions, and it is assumed that an estimate of the mask function is known first. Subsequently, the PSF is iteratively refined, including an apodization and symmetrization operation. The phase distribution of the output wavefront is also being estimated at the same time. Due to the problems with the uniqueness of the decomposition, the algorithm should be used with caution. Its main advantage is that it makes possible to determine the PSF in an experiment with the mask containing apertures significantly broader than the PSF, provided that the wavefront intensity is measured with a high resolution.

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