1. INTRODUCTION

In 2000, Pendry [1] showed that a simple silver slab is capable of imaging in the near-field with resolution beyond the diffraction limit for the near-UV wavelength range. The operation of a similar but asymmetric [2] flat lens was later experimentally verified [3, 4]. In these works, the mechanism leading to superresolution is related to the appearance of a surface plasmon polariton (SPP) mode, which enables transfer of the evanescent part of the spatial spectrum through the slab. Another way of achieving superlensing includes the use of photonic crystals showing negative refraction [5, 6, 7, 8, 9]. Both losses and the cut-off wavelength of the SPP mode limit the operation range to subwavelength distances only [10, 11]. This obstacle has been tackled to some degree in several ways. One of them is by breaking the silver slab into a metallic multilayer [2, 12] with effective anisotropic properties and strong coupling of the SPP modes between neighboring layers. Otherwise the loss may be compensated by the use of media with gain [13, 14]. Notably, according to the effective medium model, a layered lens is approximated with a uniaxially anisotropic uniform slab [12]. Uniaxial crystals are useful in the engineering of spatially varying polarization states and optical vortices [15, 16] and are a convenient medium for coupling cylindrical beams [17].

Until the introduction of the concept of a perfect flat lens with either a single layer [1, 3, 4, 18] or with multiple layers [13], it was rather unusual to regard multilayers as spatial imaging systems or linear spatial filters. Instead, the conventional characteristics of a multilayer include the dependence of transmission and reflection coefficients on the angle of incidence and polarization, as well as the photonic band structure in case of periodic stacks. However, in order to describe the resolution of an imaging system consisting of a multilayer in a systematic way, it is convenient to refer to the theory of linear shift-invariant systems (LSI, also termed as linear isoplanatic systems [19, 20]), and such a description has already gained considerable interest [21, 22, 23, 24, 25]. In this paper, metal–dielectric multilayers (MDM) are regarded as LSI systems, and a layered superlens is a one-dimensional (1D) spatial filter characterized by the point spread function (PSF). This approach may facilitate the application of plasmonic elements to optical signal processing, which is currently capturing increasing research interest [26]. In the present paper, we introduce a vectorial description of a linear spatial filter consisting of a multilayer, which enables accounting for polarization coupling that occurs during imaging. Therefore, we examine features specific to two-dimensional (2D) imaging when the description of the system does not trivially decouple into independent TE and TM polarizations.

2. BACKGROUND

We analyze coherent imaging through a multilayer consisting of linear and isotropic materials. Further, we focus on MDM. A schematic of a periodic multilayered superlens is shown in Fig. 1. The slab boundaries are infinite and parallel to each another and perpendicular to the $\widehat{z}$ direction. Such an imaging element is an example of an LSI system which realizes linear spatial filtering of the TE ($E_z=0$)-polarized or TM ($H_z=0$)-polarized incident wave function. Notably, diffraction in air and propagation through a metallic or dielectric slab are simple examples of linear spatial filters of the incident field, while a layered superlens is a more sophisticated one and may be seen as a superposition of such simpler systems; however, it is still a linear spatial filter. From an engineering perspective, multiple degrees of freedom of a multilayer provide some leeway to perform PSF optimization of the response of the spatial filter. The distribution of scalar electric or magnetic field components in the plane of incidence $\mathrm{\Psi }(x,y{)}_{z=z_{\text{inc}}}$ and an output plane $\mathrm{\Psi }(x,y{)}_{z=z_{\text{out}}}$, with a layered structure in between, are related through a convolution relation with the 2D PSF $\mathcal{H}(x,y)$:

Here * denotes a 2D convolution, and we assume that propagation is in the $\widehat{z}$ direction. The same relation transformed to the domain of spatial harmonics provides the definition of the transfer function (TF), which we denote as $\widehat{\mathcal{H}}$:

Until now, we assumed that the system is scalar. This is true only when the field is purely TE-polarized or TM- polarized or when the TF is the same for the TE or TM polarizations, which basically occurs only for simple propagation in air. There exist at least two important situations when the field symmetry in our system decouples Maxwell’s equations into these two polarization states. One takes place for in-plane imaging (e.g., in the $\widehat{x}–\widehat{z}$ plane) of 1D field distributions, when imaging of scalar fields $(H_y(x),E_x(x),E_z(x))$ and $(E_y(x),H_x(x),H_z(x))$ are independent and stand for the TM and TE polarizations, respectively. The other important situation takes place for a cylindrical beam with the angular wavenumber equal to $m=0$, when the radial $(E_r(r),H_{\varphi }(r),E_z(r))$ and angular $(H_r(r),E_{\varphi }(r),H_z(r))$ polarizations are also examples of pure TM and TE polarizations. In both situations, the symmetry of the system requires that the incident field is only 1D with the dependence on either x or r. More complex incident field distributions are neither TE- polarized or TM-polarized and due to the different TF for the TE and TM polarizations, are subject to a change in the polarization state during imaging through a multilayer.

3. TWO-DIMENSIONAL IMAGING PROPERTIES OF LAYERED SYSTEMS IN 3D

At a plane $z=z_0$, a monochromatic field is characterized by two polarization components. For instance, for the linear or circular polarization, these are $E_x(x,y{)}_{z=z_0}$ and $E_y(x,y{)}_{z=z_0}$, or $E_{{\sigma }_{+}}(x,y{)}_{z=z_0}$ and $E_{{\sigma }_{-}}(x,y{)}_{z=z_0}$, respectively.

Initially, let us assume knowledge of the 1D TF for the TE and TM polarizations, denoting them as ${\widehat{\mathcal{H}}}_{\mathrm{TE}}(k_x)$ and ${\widehat{\mathcal{H}}}_{\mathrm{TM}}(k_x)$, respectively. These TFs may be easily determined using the transfer matrix method from the complex transmission coefficients of propagating and evanescent plane waves with the wave vector components in the $x–z$ plane. We introduce denotations for the mean and difference of these two TFs, ${\widehat{\mathcal{H}}}_m=({\widehat{\mathcal{H}}}_{\mathrm{TM}}+{\widehat{\mathcal{H}}}_{\mathrm{TE}})/2$ and ${\widehat{\mathcal{H}}}_{\delta }=({\widehat{\mathcal{H}}}_{\mathrm{TM}}-{\widehat{\mathcal{H}}}_{\mathrm{TE}})/2$, which are more convenient for the description of 2D imaging.

More generally, the incident and output fields are not scalar but vectorial, and the 3D transmission of 2D field distributions through the multilayer transforms the spatial distribution of the polarization state. Obviously in a planar geometry, the description of the system decouples into TE and TM polarizations in a simple way. In 3D, the polarization-dependent 2D TF $\widehat{\mathcal{H}}$ and PSF $\mathcal{H}$ take the form of $2\times 2$ matrices responsible for the transfer and coupling of both polarizations. We will now derive these functions for linear and circular polarizations. In cylindrical coordinates, the following polarization-dependent relations can be written between the incident and output fields for linear polarizations:

A similar expression for the transfer matrix for circular polarizations is obtained from Eq. (8) using the relations between linear and circular polarizations, $E_{{\sigma }_{+}}=(E_x+i{E}_y)/\sqrt{2}$ and $E_{{\sigma }_{-}}=(E_x-i{E}_y)/\sqrt{2}$:

In order to derive the corresponding 2D PSFs, we will use the following decomposition for the 2D Fourier transform of a function separable in the angular coordinate system. For $g(r,\phi )=g_r(r)·g_{\phi }(\phi )$, the corresponding Fourier transform $\widehat{g}(k_r,k_{\phi })$ is equal to [20]:

Summarizing, the numerical calculation of the PSFs in 2D requires the following steps: 1. finding the 1D TE and TM TFs for a planar geometry using the transfer matrix method, 2. determining the 1D Hankel transforms of the zeroth and second orders of the mean and difference of these two TFs, respectively. 3. Reconstruction of the PSFs (15, 17) including the angular dependence. Therefore, the majority of the numerical calculations is done in 1D. While we evaluate directly the Hankel transforms, it should be noted that there exist fast numerical algorithms which may be applied instead [27]. The zeroth-order Hankel transform links the rotationally invariant part of PSF to the respective rotationally invariant part of TF, which in turn is equal to the average of TE and TM 1D TFs. On the other hand, the difference of the TM and TE 1D TFs is responsible for the angularly dependent polarization coupling and is related to the respective part of the PSF through the second-order Hankel transform.

4. POLARIZATION COUPLING FOR LINEARLY OR CIRCULARLY POLARIZED INCIDENT BEAMS

In this section, we calculate the TF matrix and PSF matrix for 2D vectorial spatial filtering realized with a layered lens. We assume that the lens consists of silver and strontium titanate with a filling fraction of silver equal to 0.37 and operates at a wavelength of $\lambda =430\mathrm{nm}$. The structure is presented in Fig. 1. It is a low-loss self-guiding superlens with a subwavelength FWHM of the PSF for the TM polarization of the order of $\lambda /10$, and it has been already thoroughly investigated in terms of transmission efficiency as well as of resolution for in-plane imaging [23, 25]. The elementary cell consists of an Ag layer symmetrically coated with ${\mathrm{SrTiO}}_3$. Strontium titanate is an isotropic material with a high refractive index $n=2.674+0.027i$ at $\lambda =430\mathrm{nm}$ [28]. The refractive index of silver at the same wavelength is equal to $n_{\mathrm{Ag}}=0.04+2.46i$ [29]. Further, we assume that the multilayer consists of $N=10$ periods with a thickness of $\mathrm{\Lambda }=57.5\mathrm{nm}$ each.

In Fig. 2 (top), we show the phase (dashed curves) and amplitude (solid curves) of the TFs, ${\widehat{\mathcal{H}}}_{\mathrm{TM}}(x)$, ${\widehat{\mathcal{H}}}_{\mathrm{TE}}(x)$, ${\widehat{\mathcal{H}}}_m(k_r)$, ${\widehat{\mathcal{H}}}_{\delta }(r)$, and the TF for propagation in air at the same distance. In Fig. 2 (bottom), the corresponding 1D PSF and the radial part of the 2D PSF are also presented. Our example illustrates some of the general properties of the TF—for normal incidence ($k_x=0$), we have ${\widehat{\mathcal{H}}}_{\mathrm{TM}}(0)={\widehat{\mathcal{H}}}_{\mathrm{TE}}(0)={\widehat{\mathcal{H}}}_m(0)$, and ${\widehat{\mathcal{H}}}_{\delta }(r)=0$. On the other hand, for evanescent harmonics, SPPs are only supported for the TM polarization, and therefore, for a sufficiently large $k_x$, we have $|{\widehat{\mathcal{H}}}_{\mathrm{TM}}(k_x)|\gg |{\widehat{\mathcal{H}}}_{\mathrm{TE}}(k_x)|$, and in effect $2{\widehat{\mathcal{H}}}_{\mathrm{TM}}\approx {\widehat{\mathcal{H}}}_m\approx {\widehat{\mathcal{H}}}_{\delta }$. Therefore, the bandwidth and asymptotic behavior of ${\widehat{\mathcal{H}}}_m$ and ${\widehat{\mathcal{H}}}_{\delta }$ are the same as ${\widehat{\mathcal{H}}}_{\mathrm{TM}}$ [see the solid curve in Fig. 2b (top) for $k_r/k_0>1.8$]. If the bandwidth of ${\widehat{\mathcal{H}}}_{\mathrm{TM}}$ extends to $k_x\gg 1$ and its phase is approximately constant, which are the necessary conditions for superresolution, the same properties are also satisfied by ${\widehat{\mathcal{H}}}_m$ and ${\widehat{\mathcal{H}}}_{\delta }$. Therefore, we expect to see superresolution in 2D, even though 2D imaging depends on both TE and TM polarizations and the TE polarization itself does not allow for superresolution. This observation is confirmed by the shape of the 2D PSF presented in Fig. 2b (bottom). We further see [the violet line in Fig. 2b, bottom] that the polarization coupling determined by ${\mathcal{H}}_{\delta }$ is zero at $r=0$ and smoothly increases with $k_r$, reaching a maximum at $r/\lambda \sim 0.18$, i.e., at (approximately) the first minimum of ${\mathcal{H}}_m$.

In Figs. 3, 4 we show the same 2D TF and PSF as in Fig. 2b, plotted, however, in the 2D space, including their angular dependence according to Eqs. (15, 17). The polarization-preserving and polarization-coupled elements of the TF and PSF matrix are drawn for both linear and circular polarizations. Notably, the size of the PSF is subwavelength, and the polarization state is preserved in the central part of the PSF because the second-order Hankel transform vanishes at the origin and the strongest coupling to the orthogonal polarization is observed at some distance from the center. For circular polarization, the term ${\mathcal{H}}_{{\sigma }_{+}{\sigma }_{+}}(x,y)$ depends on ${\mathcal{H}}_m(r)$ and is rotationally symmetric, while the term ${\mathcal{H}}_{{\sigma }_{+}{\sigma }_{-}}(x,y)$ depends on ${\mathcal{H}}_{\delta }(r)$ and has a simple angular phase dependence [see Eq. (17) and Fig. 4b]. For linear polarization, the term ${\mathcal{H}}_{xx}(x,y)$ depends on both ${\mathcal{H}}_m(r)$ and ${\mathcal{H}}_{\delta }(r)$ [see Eq. (15) and Fig. 4a] and is not rotationally symmetric.

We have also calculated the same elements of the PSF matrix using the body-of-revolution finite-difference time-domain (FDTD) method and obtained good agreement with the previous results calculated using the transfer matrix method and Hankel transforms. For comparison, the FDTD simulations are shown in Fig. 5, where we plot the axial energy flux resulting from a linearly x-polarized incident point source. Figure 5a illustrates the contributions from the two linear polarizations to the Poynting vector $P_z$ at the output plane. These contributions may be directly matched to the 2D PSF presented in Fig. 4a. Figure 5b shows the Poynting vector $P_z$ in the $x–z$ and $y–z$ cross-sections of the structure and shows that the direction of polarization has a major influence on the diffraction rate in orthogonal directions inside the structure. The spatial distribution of $P_z$ shown in Fig. 5a, with contributions from both polarizations, is in good agreement with Fig. 4a.

Further, we demonstrate how the 2D PSF may be used to design a simple diffractive nanoelement. Using predicted properties of the 2D PSF, we design a pattern that could be imprinted into a metallic mask attached to the layered superlens. The pattern consists of two narrow radial slits, and we use the PSF to determine the distance between the slits so that the interference between the polarization-coupled elements of two images of slits is enhanced. From the 2D PSF shown in Fig. 2b, we find that, for the radial separation of $r=0.18\lambda$, the maximum of the polarization cross-coupling term ${\mathcal{H}}_m(r)$ coincides with a minimum of the direct coupling term ${\mathcal{H}}_{\delta }(r)$. The FDTD simulation showing the operation of the proposed element is presented in Figs. 6, 7. A linearly polarized Gaussian beam is diffracted on the two circular slits in the mask. For the purpose of simplicity, the mask is made of a perfect conductor (PEC) and the width of slits is equal to $4\mathrm{nm}$, which corresponds to eight grid points of the FDTD 2D mesh. Inside the slits, the wave becomes polarized perpendicularly to the slits. This is shown in Fig. 6a. It should be noted that inside the slits the field is neither TE-polarized or TM-polarized, and $E_z\ne 0$, $H_z\ne 0$. The polarization at the output plane is shown in Fig. 6b. In between the images of two slits, there is a local maximum of intensity resulting from the interference of the 2D PSFs with the cross-polarized term eliminated. At the external side of the images of the slits, without interference, the cross-polarized terms of PSF are still present and the polarization state varies considerably with respect to that within the slits. At the same time, in between the slits, the phase shift of ${\mathcal{H}}_m(r)$ reaches approximately π with respect to its central value, resulting in the reversal of the direction of power flow $P_z$. This reversal is demonstrated in Fig. 7, which shows the cross-section of the multilayer along the $x–z$ axes. An inverted direction of $P_z$ may be seen along most of the propagation distance within the multilayer, although within the mask itself the direction of the power is clearly defined in the forward direction only.

5. CONCLUSION

Two-dimensional beams generally do not preserve their initial spatial distribution of the state of polarization after passing through a layered flat lens. This is the direct result of different TFs corresponding to the TM and TE components of the spatial spectrum and the presence of both TM and TE components in the spatial spectrum of a 2D beam. We use the framework of the theory of LSI systems to describe the imaging system and express the PSF $2\times 2$ matrix for imaging of vectorial 2D wavefronts using the zeroth- and second-order Hankel transforms of planar TFs for the TE and TM polarizations.

We calculate the polarization-preserving and cross- polarization 2D PSF matrix elements of a layered superlens showing that the resolution in 2D is of the same order as the resolution for in-plane imaging with the TM polarization. This comes in spite of the fact that both the TE-polarized and TM-polarized spatial harmonics take part in the 2D imaging. Moreover, cross-polarization coupling is observed for spatially modulated beams with a linear or circular incident polarization.

We demonstrate how the 2D PSF may be used to design a simple diffractive nanoelement. The spatial separation of the PSF matrix elements for polarization-preserving and polarization-coupled imaging is used to design a mask that forms a wavefront which, during propagation, exhibits destructive interference in the cross-polarization term, assures the separation of nondiffracting radial beams originating from two slits in the mask, and exhibits an interesting property of a backward power flow in between the two rings. This opens a possibility for further PSF engineering where the polarization effects are accounted for.

ACKNOWLEDGMENTS

We acknowledge support from the Polish Ministry of Science and Higher Education research project N N202 033237, the National Centre for Research and Development research project N R15 0018 06, and the framework of European Cooperation in Science and Technology–COST action MP0702.

 

Fig. 1 Schematic of a layered metal–dielectric periodic lens with $N=4$ periods. It is further assumed that $\lambda =430\mathrm{nm}$, $\mathrm{\Lambda }=57.5\mathrm{nm}$, $N=10$, $d_{\mathrm{Ag}}=0.37\mathrm{\Lambda }$, $d_{{\mathrm{SrTiO}}_3}=0.63\mathrm{\Lambda }$, $n_{\mathrm{Ag}}=0.04+2.46i$, $n_{{\mathrm{SrTiO}}_3}=2.674+0.027i$.

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Fig. 2 (a) One-dimensional TF and PSF of the multilayer for in-plane propagation: ${\widehat{\mathcal{H}}}_{\mathrm{TM}}$, ${\widehat{\mathcal{H}}}_{\mathrm{TE}}$, TF for the TM and TE polarizations; (b) radial cross-section of 2D TF and PSFs of the multilayer for 3D propagation ${\widehat{\mathcal{H}}}_m=({\widehat{\mathcal{H}}}_{\mathrm{TM}}+{\widehat{\mathcal{H}}}_{\mathrm{TE}})/2$, ${\widehat{\mathcal{H}}}_{\delta }=({\widehat{\mathcal{H}}}_{\mathrm{TM}}-{\widehat{\mathcal{H}}}_{\mathrm{TE}})/2$. One-dimensional and 2D PSFs and TF corresponding to propagation in air for the same distance are also shown for comparison. Solid curves and dashed curves represent the amplitudes and phases of the complex functions, respectively.

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Fig. 3 Two-dimensional TF matrix of the multilayer. (a) Matrix elements ${\widehat{\mathcal{H}}}_{xx}(k_x,k_y)$ and ${\widehat{\mathcal{H}}}_{xy}(k_x,k_y)$ for linearly polarized light; (b) matrix elements ${\widehat{\mathcal{H}}}_{{\sigma }_{+}{\sigma }_{+}}(k_x,k_y)$ and ${\widehat{\mathcal{H}}}_{{\sigma }_{+}{\sigma }_{-}}(k_x,k_y)$ for circularly polarized light. Phase isolines are drawn at the distances of $\pi /2$ and are changed between solid and dashed lines every π.

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Fig. 4 Two-dimensional PSF matrix of the multilayer. (a) Matrix elements ${\mathcal{H}}_{xx}(x,y)$ and ${\mathcal{H}}_{xy}(x,y)$ for linearly polarized light; (b) matrix elements ${\mathcal{H}}_{{\sigma }_{+}{\sigma }_{+}}(x,y)$ and ${\widehat{\mathcal{H}}}_{{\sigma }_{+}{\sigma }_{-}}(x,y)$ for circularly polarized light; phase isolines are drawn at the distances of $\pi /2$ and are changed between solid and dashed lines every π.

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Fig. 5 (a) Contributions to the axial Poynting vector $P_z$ from both polarizations at the output plane, resulting from a linearly x polarized incident point source; (b) axial component of the Poynting vector $P_z$ in the $x–z$ and $y–z$ cross-sections inside the structure normalized with respect to $P_z(x=0,y=0,z=L)$. The simulations were performed using FDTD. The red lines separate areas with positive and negative direction of the power flow.

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Fig. 6 Intensity of the electric field and the state of polarization (a) inside the mask and (b) at the output plane of the multilayer. The mask consists of a PEC with two circular slits and is illuminated with a linearly x polarized Gaussian beam. FDTD simulation, linear intensity mapping.

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Fig. 7 Time-averaged Poynting vector component $〈P_z〉$ in the $x–z$ cross-section of the structure (where $y=0$). The red lines separate areas with the positive and negative values of $P_z$ and the white arrows show the orientation of the energy flow in the vertical direction.

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