Abstract

We describe the change of the spatial distribution of the state of polarization occurring during two-dimensional (2D) imaging through a multilayer and in particular through a layered metallic flat lens. Linear or circular polarization of incident light is not preserved due to the difference in the amplitude transfer functions for the TM and TE polarizations. In effect, the transfer function and the point spread function (PSF) that characterize 2D imaging through a multilayer both have a matrix form, and cross-polarization coupling is observed for spatially modulated beams with a linear or circular incident polarization. The PSF in a matrix form is used to characterize the resolution of the superlens for different polarization states. We demonstrate how the 2D PSF may be used to design a simple diffractive nanoelement consisting of two radial slits. The structure 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.

© 2011 Optical Society of America

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 z^ direction. Such an imaging element is an example of an LSI system which realizes linear spatial filtering of the TE (Ez=0)-polarized or TM (Hz=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 Ψ(x,y)z=zinc and an output plane Ψ(x,y)z=zout, with a layered structure in between, are related through a convolution relation with the 2D PSF H(x,y):

Ψ(x,y)z=zout=(2π)1H(x,y)*Ψ(x,y)z=zinc.

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

Ψ^(kx,ky)z=zout=H^(kx,ky)·Ψ^(kx,ky)z=zinc,
where the hat represents the 2D unitary Fourier transform, defined as
Ψ(x,y)z=(2π)1Ψ^(kx,ky)zei(kxx+kyy)dkxdky.
The PSF, related to the TF through the spatial Fourier transform, provides precise information on reshaping of the optical signal due to diffraction combined with multiple reflections and refractions and is direct and convenient to interpret. Resolution of the system depends on the bandwidth of the transmitted spatial spectrum and superresolution in the near-field occurs when the TF of the system does not vanish for the evanescent part of the spatial spectrum (kx2>1).

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 x^z^ plane) of 1D field distributions, when imaging of scalar fields (Hy(x),Ex(x),Ez(x)) and (Ey(x),Hx(x),Hz(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 (Er(r),Hϕ(r),Ez(r)) and angular (Hr(r),Eϕ(r),Hz(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=z0, a monochromatic field is characterized by two polarization components. For instance, for the linear or circular polarization, these are Ex(x,y)z=z0 and Ey(x,y)z=z0, or Eσ+(x,y)z=z0 and Eσ(x,y)z=z0, respectively.

Initially, let us assume knowledge of the 1D TF for the TE and TM polarizations, denoting them as H^TE(kx) and H^TM(kx), 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 xz plane. We introduce denotations for the mean and difference of these two TFs, H^m=(H^TM+H^TE)/2 and H^δ=(H^TMH^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 H^ and PSF H take the form of 2×2 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:

[E^x(kr,kφ)E^y(kr,kφ)]z=zout=H^lin(kr,kφ)·[E^x(kr,kφ)E^y(kr,kφ)]z=zin,
and for circular polarizations,
[E^σ+(kr,kφ)E^σ(kr,kφ)]z=zout=H^circ(kr,kφ)·[E^σ+(kr,kφ)E^σ(kr,kφ)]z=zin,
respectively, where H^lin(kr,kφ) and H^circ(kr,kφ) are TFs in the matrix form which we need to determine. The right sides of Eqs. (4, 5) include matrix-vector multiplications. Notably, using the Jones calculus, it is straightforward to modify Eqs. (4, 5) and obtain expressions for the cross-coupling between linear and circular polarizations as well. Now, we will express these 2D TFs by the TFs for the TE and TM polarizations, by rotating the coordinate system and casting the 3D situation to a planar TE or TM case:
H^lin(kr,kφ)=[H^xx(kr,kφ)H^yx(kr,kφ)H^xy(kr,kφ)H^yy(kr,kφ)]
=R(kφ)·[H^TM(kr)00H^TE(kr)]·R(kφ)
=H^m(kr)·[1001]+H^δ(kr)·[cos(2kφ)sin(2kφ)sin(2kφ)cos(2kφ)],
where R is the rotation matrix and H^xx(kr,kφ) may be understood as the TF linking the x-polarized component of the input signal to the x-polarized component of the output signal, H^xy(kr,kφ) may be understood as the TF linking the x- polarized input signal to the y-polarized component of the output signal, etc. The mean value of the TE and TM TFs H^m(kr) stands for the angularly independent term of the TF H^lin. On the other hand, the difference between the TM and TE TFs, H^δ(kr), appears in the TF H^lin with an angular dependence and is responsible for polarization coupling. This term vanishes when TE and TM polarizations become degenerated, for instance, for diffraction in air. Conversely, plasmon- assisted transmission of evanescent waves is possible for TM polarization only, and this term then results in strong polarization coupling in 2D.

A similar expression for the transfer matrix for circular polarizations is obtained from Eq. (8) using the relations between linear and circular polarizations, Eσ+=(Ex+iEy)/2 and Eσ=(ExiEy)/2:

H^circ(kr,kφ)=[H^σ+σ+(kr,kφ)H^σσ+(kr,kφ)H^σ+σ(kr,kφ)H^σσ(kr,kφ)]
=H^m(kr)·[1001]+H^δ(kr)·[0exp(2ikφ)exp(2ikφ)0].

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,φ)=gr(r)·gφ(φ), the corresponding Fourier transform g^(kr,kφ) is equal to [20]:

g^(kr,kφ)=n=+(i)ncn·exp(inkφ)·Hn{gr(r)},
where the expansion coefficients cn are equal to
cn=02πgφ(φ)·exp(inφ)dφ,
and Hn denotes the nth-order Hankel transform:
Hn{gr(r)}=0gr(r)·Jn(krr)·r·dr.
This decomposition leads to the following expressions for the polarization-dependent 2D PSF of the layered system:
Hlin(r,φ)=[Hxx(r,φ)Hyx(r,φ)Hxy(r,φ)Hyy(r,φ)]
=Hm(r)·[1001]Hδ(r)·[cos(2φ)sin(2φ)sin(2φ)cos(2φ)],
Hcirc(r,φ)=[Hσ+σ+(r,φ)Hσσ+(r,φ)Hσ+σ(r,φ)Hσσ(r,φ)]
=Hm(r)·[1001]Hδ(r)·[0exp(2iφ)exp(2iφ)0],
where Hm(r)=H0{H^m(kr)} and Hδ(r)=H2{H^δ(kr)}.

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 λ=430nm. 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 λ/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 SrTiO3. Strontium titanate is an isotropic material with a high refractive index n=2.674+0.027i at λ=430nm [28]. The refractive index of silver at the same wavelength is equal to nAg=0.04+2.46i [29]. Further, we assume that the multilayer consists of N=10 periods with a thickness of Λ=57.5nm each.

In Fig. 2 (top), we show the phase (dashed curves) and amplitude (solid curves) of the TFs, H^TM(x), H^TE(x), H^m(kr), H^δ(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 (kx=0), we have H^TM(0)=H^TE(0)=H^m(0), and H^δ(r)=0. On the other hand, for evanescent harmonics, SPPs are only supported for the TM polarization, and therefore, for a sufficiently large kx, we have |H^TM(kx)||H^TE(kx)|, and in effect 2H^TMH^mH^δ. Therefore, the bandwidth and asymptotic behavior of H^m and H^δ are the same as H^TM [see the solid curve in Fig. 2b (top) for kr/k0>1.8]. If the bandwidth of H^TM extends to kx1 and its phase is approximately constant, which are the necessary conditions for superresolution, the same properties are also satisfied by H^m and H^δ. 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 Hδ is zero at r=0 and smoothly increases with kr, reaching a maximum at r/λ0.18, i.e., at (approximately) the first minimum of Hm.

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 Hσ+σ+(x,y) depends on Hm(r) and is rotationally symmetric, while the term Hσ+σ(x,y) depends on Hδ(r) and has a simple angular phase dependence [see Eq. (17) and Fig. 4b]. For linear polarization, the term Hxx(x,y) depends on both Hm(r) and Hδ(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 Pz at the output plane. These contributions may be directly matched to the 2D PSF presented in Fig. 4a. Figure 5b shows the Poynting vector Pz in the xz and yz 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 Pz 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=0.18λ, the maximum of the polarization cross-coupling term Hm(r) coincides with a minimum of the direct coupling term Hδ(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 4nm, 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 Ez0, Hz0. 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 Hm(r) reaches approximately π with respect to its central value, resulting in the reversal of the direction of power flow Pz. This reversal is demonstrated in Fig. 7, which shows the cross-section of the multilayer along the xz axes. An inverted direction of Pz 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 2×2 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 λ=430nm, Λ=57.5nm, N=10, dAg=0.37Λ, dSrTiO3=0.63Λ, nAg=0.04+2.46i, nSrTiO3=2.674+0.027i.

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Fig. 2 (a) One-dimensional TF and PSF of the multilayer for in-plane propagation: H^TM, H^TE, TF for the TM and TE polarizations; (b) radial cross-section of 2D TF and PSFs of the multilayer for 3D propagation H^m=(H^TM+H^TE)/2, H^δ=(H^TMH^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 H^xx(kx,ky) and H^xy(kx,ky) for linearly polarized light; (b) matrix elements H^σ+σ+(kx,ky) and H^σ+σ(kx,ky) for circularly polarized light. Phase isolines are drawn at the distances of π/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 Hxx(x,y) and Hxy(x,y) for linearly polarized light; (b) matrix elements Hσ+σ+(x,y) and H^σ+σ(x,y) for circularly polarized light; phase isolines are drawn at the distances of π/2 and are changed between solid and dashed lines every π.

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Fig. 5 (a) Contributions to the axial Poynting vector Pz from both polarizations at the output plane, resulting from a linearly x polarized incident point source; (b) axial component of the Poynting vector Pz in the xz and yz cross-sections inside the structure normalized with respect to Pz(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 Pz in the xz cross-section of the structure (where y=0). The red lines separate areas with the positive and negative values of Pz and the white arrows show the orientation of the energy flow in the vertical direction.

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References

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  1. J. B. Pendry, “Negative refraction makes a perfect lens,” Phys. Rev. Lett. 85, 3966–3969 (2000).
    [CrossRef] [PubMed]
  2. S. A. Ramakrishna, J. B. Pendry, D. Schurig, and D. R. Smith, “The asymmetric lossy near-perfect lens,” J. Mod. Opt. 49, 1747–1762 (2002).
    [CrossRef]
  3. D. O. Melville and R. J. Blaikie, “Super-resolution imaging through a planar silver layer,” Opt. Express 13, 2127–2134(2005).
    [CrossRef] [PubMed]
  4. N. Fang, H. Lee, C. Sun, and X. Zhang, “Sub-diffraction-limited optical imaging with a silver superlens,” Science 308, 534–537(2005).
    [CrossRef] [PubMed]
  5. M. Notomi, “Theory of light propagation in strongly modulated photonic crystals: refractionlike behavior in the vicinity of the photonic band gap,” Phys. Rev. B 62, 10696–10705(2000).
    [CrossRef]
  6. H. Zhang, L. Shen, L. Ran, Y. Yuan, and J. Kong, “Layered superlensing in two-dimensional photonic crystals,” Opt. Express 14, 11178–11183 (2006).
    [CrossRef] [PubMed]
  7. C. Luo, S. G. Johnson, J. D. Joannopoulos, and J. B. Pendry, “Subwavelength imaging in photonic crystals,” Phys. Rev. B 68, 045115 (2003).
    [CrossRef]
  8. A. Grbic and G. V. Eleftheriades, “Overcoming the diffraction limit with a planar left-handed transmission-line lens,” Phys. Rev. Lett. 92, 117403 (2004).
    [CrossRef] [PubMed]
  9. H. Zhang, H. Zhu, L. Qian, and D. Fan, “Collimations and negative refractions by slabs of 2D photonic crystals with periodically-aligned tube-type air holes,” Opt. Express 15, 3519–3530 (2007).
    [CrossRef] [PubMed]
  10. K. J. Webb, M. Yang, D. W. Ward, and K. A. Nelson, “Metrics for negative-refractive-index materials,” Phys. Rev. E 70, 035602(2004).
    [CrossRef]
  11. D. R. Smith, D. Schurig, M. Rosenbluth, S. Schultz, S. A. Ramakrishna, and J. B. Pendry, “Limitations on subdiffraction imaging with a negative refractive index slab,” Appl. Phys. Lett. 82, 1506–1508 (2003).
    [CrossRef]
  12. B. Wood, J. B. Pendry, and D. P. Tsai, “Directed subwavelength imaging using a layered metal–dielectric system,” Phys. Rev. B 74, 115116 (2006).
    [CrossRef]
  13. S. A. Ramakrishna and J. B. Pendry, “Removal of absorption and increase in resolution in a near-field lens via optical gain,” Phys. Rev. B 67, 201101 (2003).
    [CrossRef]
  14. E. V. Ponizovskaya and A. M. Bratkovsky, “Metallic negative index nanostructures at optical frequencies: losses and effect of gain medium,” Appl. Phys. A 87, 161–165 (2007).
    [CrossRef]
  15. T. A. Fadeyeva, V. G. Shvedov, Y. V. Izdebskaya, A. V. Volyar, E. Brasselet, D. N. Neshev, A. S. Desyatnikov, W. Krolikowski, and Y. S. Kivshar, “Spatially engineered polarization states and optical vortices in uniaxial crystals,” Opt. Express 18, 10848–10863 (2010).
    [CrossRef] [PubMed]
  16. X. L. Wang, Y. Li, J. Chen, C. S. Guo, J. Ding, and H. T. Wang, “A new type of vector fields with hybrid states of polarization,” Opt. Express 18, 10786–10795 (2010).
    [CrossRef] [PubMed]
  17. Q. Zhan, “Cylindrical vector beams: from mathematical concepts to applications,” Adv. Opt. Photon. 1, 1–57(2009).
    [CrossRef]
  18. S. A. Ramakrishna, J. B. Pendry, D. Schurig, D. R. Smith, and S. Schultz, “The asymmetric lossy near-perfect lens,” J. Mod. Opt. 49, 1747–1762 (2002).
    [CrossRef]
  19. B. Saleh and M. Teich, Fundamentals of Photonics, 2nd ed.(Wiley, 2007).
  20. J. W. Goodman, Introduction to Fourier Optics, 3rd ed.(Roberts, 2005).
  21. C. P. Moore, M. D. Arnold, P. J. Bones, and R. J. Blaikie, “Image fidelity for single-layer and multi-layer silver superlenses,” J. Opt. Soc. Am. A 25, 911–918 (2008).
    [CrossRef]
  22. N. Mattiucci, D. Aguanno, M. Scalora, M. J. Bloemer, and C. Sibilia, “Transmission function properties for multi-layered structures: application to super-resolution,” Opt. Express 17, 17517–17529 (2009).
    [CrossRef] [PubMed]
  23. R. Kotynski and T. Stefaniuk, “Multiscale analysis of subwavelength imaging with metal–dielectric multilayers,” Opt. Lett. 35, 1133–1135 (2010).
    [CrossRef] [PubMed]
  24. R. Kotynski and T. Stefaniuk, “Comparison of imaging with sub-wavelength resolution in the canalization and resonant tunnelling regimes,” J. Opt. A Pure Appl. Opt. 11, 015001(2009).
    [CrossRef]
  25. R. Kotynski, “Fourier optics approach to imaging with sub-wavelength resolution through metal–dielectric multilayers,” Opto-Electron. Rev. 18, 366–375 (2010).
    [CrossRef]
  26. B. Lee, Ph. Lalanne, and Y. Fainman, eds., “Feature Issue on Plasmonic Diffractive Optics and Imaging,” Appl. Opt. 49(7), PD01, A01–A41 (2010).
    [CrossRef]
  27. A. W. Norfolk and E. J. Grace, “Reconstruction of optical fields with the quasi-discrete Hankel transform,” Opt. Express 18, 10551–10556 (2010).
    [CrossRef] [PubMed]
  28. E.Palik, ed., Handbook of Optical Constants of Solids(Academic, 1998).
  29. P. Johnson and R. Christy, “Optical constants of the noble metals,” Phys. Rev. B 6, 4370–4379 (1972).
    [CrossRef]

2010 (6)

2009 (3)

2008 (1)

2007 (2)

H. Zhang, H. Zhu, L. Qian, and D. Fan, “Collimations and negative refractions by slabs of 2D photonic crystals with periodically-aligned tube-type air holes,” Opt. Express 15, 3519–3530 (2007).
[CrossRef] [PubMed]

E. V. Ponizovskaya and A. M. Bratkovsky, “Metallic negative index nanostructures at optical frequencies: losses and effect of gain medium,” Appl. Phys. A 87, 161–165 (2007).
[CrossRef]

2006 (2)

B. Wood, J. B. Pendry, and D. P. Tsai, “Directed subwavelength imaging using a layered metal–dielectric system,” Phys. Rev. B 74, 115116 (2006).
[CrossRef]

H. Zhang, L. Shen, L. Ran, Y. Yuan, and J. Kong, “Layered superlensing in two-dimensional photonic crystals,” Opt. Express 14, 11178–11183 (2006).
[CrossRef] [PubMed]

2005 (2)

N. Fang, H. Lee, C. Sun, and X. Zhang, “Sub-diffraction-limited optical imaging with a silver superlens,” Science 308, 534–537(2005).
[CrossRef] [PubMed]

D. O. Melville and R. J. Blaikie, “Super-resolution imaging through a planar silver layer,” Opt. Express 13, 2127–2134(2005).
[CrossRef] [PubMed]

2004 (2)

A. Grbic and G. V. Eleftheriades, “Overcoming the diffraction limit with a planar left-handed transmission-line lens,” Phys. Rev. Lett. 92, 117403 (2004).
[CrossRef] [PubMed]

K. J. Webb, M. Yang, D. W. Ward, and K. A. Nelson, “Metrics for negative-refractive-index materials,” Phys. Rev. E 70, 035602(2004).
[CrossRef]

2003 (3)

D. R. Smith, D. Schurig, M. Rosenbluth, S. Schultz, S. A. Ramakrishna, and J. B. Pendry, “Limitations on subdiffraction imaging with a negative refractive index slab,” Appl. Phys. Lett. 82, 1506–1508 (2003).
[CrossRef]

S. A. Ramakrishna and J. B. Pendry, “Removal of absorption and increase in resolution in a near-field lens via optical gain,” Phys. Rev. B 67, 201101 (2003).
[CrossRef]

C. Luo, S. G. Johnson, J. D. Joannopoulos, and J. B. Pendry, “Subwavelength imaging in photonic crystals,” Phys. Rev. B 68, 045115 (2003).
[CrossRef]

2002 (2)

S. A. Ramakrishna, J. B. Pendry, D. Schurig, D. R. Smith, and S. Schultz, “The asymmetric lossy near-perfect lens,” J. Mod. Opt. 49, 1747–1762 (2002).
[CrossRef]

S. A. Ramakrishna, J. B. Pendry, D. Schurig, and D. R. Smith, “The asymmetric lossy near-perfect lens,” J. Mod. Opt. 49, 1747–1762 (2002).
[CrossRef]

2000 (2)

J. B. Pendry, “Negative refraction makes a perfect lens,” Phys. Rev. Lett. 85, 3966–3969 (2000).
[CrossRef] [PubMed]

M. Notomi, “Theory of light propagation in strongly modulated photonic crystals: refractionlike behavior in the vicinity of the photonic band gap,” Phys. Rev. B 62, 10696–10705(2000).
[CrossRef]

1972 (1)

P. Johnson and R. Christy, “Optical constants of the noble metals,” Phys. Rev. B 6, 4370–4379 (1972).
[CrossRef]

Aguanno, D.

Arnold, M. D.

Blaikie, R. J.

Bloemer, M. J.

Bones, P. J.

Brasselet, E.

Bratkovsky, A. M.

E. V. Ponizovskaya and A. M. Bratkovsky, “Metallic negative index nanostructures at optical frequencies: losses and effect of gain medium,” Appl. Phys. A 87, 161–165 (2007).
[CrossRef]

Chen, J.

Christy, R.

P. Johnson and R. Christy, “Optical constants of the noble metals,” Phys. Rev. B 6, 4370–4379 (1972).
[CrossRef]

Desyatnikov, A. S.

Ding, J.

Eleftheriades, G. V.

A. Grbic and G. V. Eleftheriades, “Overcoming the diffraction limit with a planar left-handed transmission-line lens,” Phys. Rev. Lett. 92, 117403 (2004).
[CrossRef] [PubMed]

Fadeyeva, T. A.

Fainman, Y.

B. Lee, Ph. Lalanne, and Y. Fainman, eds., “Feature Issue on Plasmonic Diffractive Optics and Imaging,” Appl. Opt. 49(7), PD01, A01–A41 (2010).
[CrossRef]

Fan, D.

Fang, N.

N. Fang, H. Lee, C. Sun, and X. Zhang, “Sub-diffraction-limited optical imaging with a silver superlens,” Science 308, 534–537(2005).
[CrossRef] [PubMed]

Goodman, J. W.

J. W. Goodman, Introduction to Fourier Optics, 3rd ed.(Roberts, 2005).

Grace, E. J.

Grbic, A.

A. Grbic and G. V. Eleftheriades, “Overcoming the diffraction limit with a planar left-handed transmission-line lens,” Phys. Rev. Lett. 92, 117403 (2004).
[CrossRef] [PubMed]

Guo, C. S.

Izdebskaya, Y. V.

Joannopoulos, J. D.

C. Luo, S. G. Johnson, J. D. Joannopoulos, and J. B. Pendry, “Subwavelength imaging in photonic crystals,” Phys. Rev. B 68, 045115 (2003).
[CrossRef]

Johnson, P.

P. Johnson and R. Christy, “Optical constants of the noble metals,” Phys. Rev. B 6, 4370–4379 (1972).
[CrossRef]

Johnson, S. G.

C. Luo, S. G. Johnson, J. D. Joannopoulos, and J. B. Pendry, “Subwavelength imaging in photonic crystals,” Phys. Rev. B 68, 045115 (2003).
[CrossRef]

Kivshar, Y. S.

Kong, J.

Kotynski, R.

R. Kotynski and T. Stefaniuk, “Multiscale analysis of subwavelength imaging with metal–dielectric multilayers,” Opt. Lett. 35, 1133–1135 (2010).
[CrossRef] [PubMed]

R. Kotynski, “Fourier optics approach to imaging with sub-wavelength resolution through metal–dielectric multilayers,” Opto-Electron. Rev. 18, 366–375 (2010).
[CrossRef]

R. Kotynski and T. Stefaniuk, “Comparison of imaging with sub-wavelength resolution in the canalization and resonant tunnelling regimes,” J. Opt. A Pure Appl. Opt. 11, 015001(2009).
[CrossRef]

Krolikowski, W.

Lalanne, Ph.

B. Lee, Ph. Lalanne, and Y. Fainman, eds., “Feature Issue on Plasmonic Diffractive Optics and Imaging,” Appl. Opt. 49(7), PD01, A01–A41 (2010).
[CrossRef]

Lee, B.

B. Lee, Ph. Lalanne, and Y. Fainman, eds., “Feature Issue on Plasmonic Diffractive Optics and Imaging,” Appl. Opt. 49(7), PD01, A01–A41 (2010).
[CrossRef]

Lee, H.

N. Fang, H. Lee, C. Sun, and X. Zhang, “Sub-diffraction-limited optical imaging with a silver superlens,” Science 308, 534–537(2005).
[CrossRef] [PubMed]

Li, Y.

Luo, C.

C. Luo, S. G. Johnson, J. D. Joannopoulos, and J. B. Pendry, “Subwavelength imaging in photonic crystals,” Phys. Rev. B 68, 045115 (2003).
[CrossRef]

Mattiucci, N.

Melville, D. O.

Moore, C. P.

Nelson, K. A.

K. J. Webb, M. Yang, D. W. Ward, and K. A. Nelson, “Metrics for negative-refractive-index materials,” Phys. Rev. E 70, 035602(2004).
[CrossRef]

Neshev, D. N.

Norfolk, A. W.

Notomi, M.

M. Notomi, “Theory of light propagation in strongly modulated photonic crystals: refractionlike behavior in the vicinity of the photonic band gap,” Phys. Rev. B 62, 10696–10705(2000).
[CrossRef]

Pendry, J. B.

B. Wood, J. B. Pendry, and D. P. Tsai, “Directed subwavelength imaging using a layered metal–dielectric system,” Phys. Rev. B 74, 115116 (2006).
[CrossRef]

C. Luo, S. G. Johnson, J. D. Joannopoulos, and J. B. Pendry, “Subwavelength imaging in photonic crystals,” Phys. Rev. B 68, 045115 (2003).
[CrossRef]

D. R. Smith, D. Schurig, M. Rosenbluth, S. Schultz, S. A. Ramakrishna, and J. B. Pendry, “Limitations on subdiffraction imaging with a negative refractive index slab,” Appl. Phys. Lett. 82, 1506–1508 (2003).
[CrossRef]

S. A. Ramakrishna and J. B. Pendry, “Removal of absorption and increase in resolution in a near-field lens via optical gain,” Phys. Rev. B 67, 201101 (2003).
[CrossRef]

S. A. Ramakrishna, J. B. Pendry, D. Schurig, and D. R. Smith, “The asymmetric lossy near-perfect lens,” J. Mod. Opt. 49, 1747–1762 (2002).
[CrossRef]

S. A. Ramakrishna, J. B. Pendry, D. Schurig, D. R. Smith, and S. Schultz, “The asymmetric lossy near-perfect lens,” J. Mod. Opt. 49, 1747–1762 (2002).
[CrossRef]

J. B. Pendry, “Negative refraction makes a perfect lens,” Phys. Rev. Lett. 85, 3966–3969 (2000).
[CrossRef] [PubMed]

Ponizovskaya, E. V.

E. V. Ponizovskaya and A. M. Bratkovsky, “Metallic negative index nanostructures at optical frequencies: losses and effect of gain medium,” Appl. Phys. A 87, 161–165 (2007).
[CrossRef]

Qian, L.

Ramakrishna, S. A.

D. R. Smith, D. Schurig, M. Rosenbluth, S. Schultz, S. A. Ramakrishna, and J. B. Pendry, “Limitations on subdiffraction imaging with a negative refractive index slab,” Appl. Phys. Lett. 82, 1506–1508 (2003).
[CrossRef]

S. A. Ramakrishna and J. B. Pendry, “Removal of absorption and increase in resolution in a near-field lens via optical gain,” Phys. Rev. B 67, 201101 (2003).
[CrossRef]

S. A. Ramakrishna, J. B. Pendry, D. Schurig, and D. R. Smith, “The asymmetric lossy near-perfect lens,” J. Mod. Opt. 49, 1747–1762 (2002).
[CrossRef]

S. A. Ramakrishna, J. B. Pendry, D. Schurig, D. R. Smith, and S. Schultz, “The asymmetric lossy near-perfect lens,” J. Mod. Opt. 49, 1747–1762 (2002).
[CrossRef]

Ran, L.

Rosenbluth, M.

D. R. Smith, D. Schurig, M. Rosenbluth, S. Schultz, S. A. Ramakrishna, and J. B. Pendry, “Limitations on subdiffraction imaging with a negative refractive index slab,” Appl. Phys. Lett. 82, 1506–1508 (2003).
[CrossRef]

Saleh, B.

B. Saleh and M. Teich, Fundamentals of Photonics, 2nd ed.(Wiley, 2007).

Scalora, M.

Schultz, S.

D. R. Smith, D. Schurig, M. Rosenbluth, S. Schultz, S. A. Ramakrishna, and J. B. Pendry, “Limitations on subdiffraction imaging with a negative refractive index slab,” Appl. Phys. Lett. 82, 1506–1508 (2003).
[CrossRef]

S. A. Ramakrishna, J. B. Pendry, D. Schurig, D. R. Smith, and S. Schultz, “The asymmetric lossy near-perfect lens,” J. Mod. Opt. 49, 1747–1762 (2002).
[CrossRef]

Schurig, D.

D. R. Smith, D. Schurig, M. Rosenbluth, S. Schultz, S. A. Ramakrishna, and J. B. Pendry, “Limitations on subdiffraction imaging with a negative refractive index slab,” Appl. Phys. Lett. 82, 1506–1508 (2003).
[CrossRef]

S. A. Ramakrishna, J. B. Pendry, D. Schurig, and D. R. Smith, “The asymmetric lossy near-perfect lens,” J. Mod. Opt. 49, 1747–1762 (2002).
[CrossRef]

S. A. Ramakrishna, J. B. Pendry, D. Schurig, D. R. Smith, and S. Schultz, “The asymmetric lossy near-perfect lens,” J. Mod. Opt. 49, 1747–1762 (2002).
[CrossRef]

Shen, L.

Shvedov, V. G.

Sibilia, C.

Smith, D. R.

D. R. Smith, D. Schurig, M. Rosenbluth, S. Schultz, S. A. Ramakrishna, and J. B. Pendry, “Limitations on subdiffraction imaging with a negative refractive index slab,” Appl. Phys. Lett. 82, 1506–1508 (2003).
[CrossRef]

S. A. Ramakrishna, J. B. Pendry, D. Schurig, and D. R. Smith, “The asymmetric lossy near-perfect lens,” J. Mod. Opt. 49, 1747–1762 (2002).
[CrossRef]

S. A. Ramakrishna, J. B. Pendry, D. Schurig, D. R. Smith, and S. Schultz, “The asymmetric lossy near-perfect lens,” J. Mod. Opt. 49, 1747–1762 (2002).
[CrossRef]

Stefaniuk, T.

R. Kotynski and T. Stefaniuk, “Multiscale analysis of subwavelength imaging with metal–dielectric multilayers,” Opt. Lett. 35, 1133–1135 (2010).
[CrossRef] [PubMed]

R. Kotynski and T. Stefaniuk, “Comparison of imaging with sub-wavelength resolution in the canalization and resonant tunnelling regimes,” J. Opt. A Pure Appl. Opt. 11, 015001(2009).
[CrossRef]

Sun, C.

N. Fang, H. Lee, C. Sun, and X. Zhang, “Sub-diffraction-limited optical imaging with a silver superlens,” Science 308, 534–537(2005).
[CrossRef] [PubMed]

Teich, M.

B. Saleh and M. Teich, Fundamentals of Photonics, 2nd ed.(Wiley, 2007).

Tsai, D. P.

B. Wood, J. B. Pendry, and D. P. Tsai, “Directed subwavelength imaging using a layered metal–dielectric system,” Phys. Rev. B 74, 115116 (2006).
[CrossRef]

Volyar, A. V.

Wang, H. T.

Wang, X. L.

Ward, D. W.

K. J. Webb, M. Yang, D. W. Ward, and K. A. Nelson, “Metrics for negative-refractive-index materials,” Phys. Rev. E 70, 035602(2004).
[CrossRef]

Webb, K. J.

K. J. Webb, M. Yang, D. W. Ward, and K. A. Nelson, “Metrics for negative-refractive-index materials,” Phys. Rev. E 70, 035602(2004).
[CrossRef]

Wood, B.

B. Wood, J. B. Pendry, and D. P. Tsai, “Directed subwavelength imaging using a layered metal–dielectric system,” Phys. Rev. B 74, 115116 (2006).
[CrossRef]

Yang, M.

K. J. Webb, M. Yang, D. W. Ward, and K. A. Nelson, “Metrics for negative-refractive-index materials,” Phys. Rev. E 70, 035602(2004).
[CrossRef]

Yuan, Y.

Zhan, Q.

Zhang, H.

Zhang, X.

N. Fang, H. Lee, C. Sun, and X. Zhang, “Sub-diffraction-limited optical imaging with a silver superlens,” Science 308, 534–537(2005).
[CrossRef] [PubMed]

Zhu, H.

Adv. Opt. Photon. (1)

Appl. Opt. (1)

B. Lee, Ph. Lalanne, and Y. Fainman, eds., “Feature Issue on Plasmonic Diffractive Optics and Imaging,” Appl. Opt. 49(7), PD01, A01–A41 (2010).
[CrossRef]

Appl. Phys. A (1)

E. V. Ponizovskaya and A. M. Bratkovsky, “Metallic negative index nanostructures at optical frequencies: losses and effect of gain medium,” Appl. Phys. A 87, 161–165 (2007).
[CrossRef]

Appl. Phys. Lett. (1)

D. R. Smith, D. Schurig, M. Rosenbluth, S. Schultz, S. A. Ramakrishna, and J. B. Pendry, “Limitations on subdiffraction imaging with a negative refractive index slab,” Appl. Phys. Lett. 82, 1506–1508 (2003).
[CrossRef]

J. Mod. Opt. (2)

S. A. Ramakrishna, J. B. Pendry, D. Schurig, and D. R. Smith, “The asymmetric lossy near-perfect lens,” J. Mod. Opt. 49, 1747–1762 (2002).
[CrossRef]

S. A. Ramakrishna, J. B. Pendry, D. Schurig, D. R. Smith, and S. Schultz, “The asymmetric lossy near-perfect lens,” J. Mod. Opt. 49, 1747–1762 (2002).
[CrossRef]

J. Opt. A Pure Appl. Opt. (1)

R. Kotynski and T. Stefaniuk, “Comparison of imaging with sub-wavelength resolution in the canalization and resonant tunnelling regimes,” J. Opt. A Pure Appl. Opt. 11, 015001(2009).
[CrossRef]

J. Opt. Soc. Am. A (1)

Opt. Express (7)

Opt. Lett. (1)

Opto-Electron. Rev. (1)

R. Kotynski, “Fourier optics approach to imaging with sub-wavelength resolution through metal–dielectric multilayers,” Opto-Electron. Rev. 18, 366–375 (2010).
[CrossRef]

Phys. Rev. B (5)

P. Johnson and R. Christy, “Optical constants of the noble metals,” Phys. Rev. B 6, 4370–4379 (1972).
[CrossRef]

M. Notomi, “Theory of light propagation in strongly modulated photonic crystals: refractionlike behavior in the vicinity of the photonic band gap,” Phys. Rev. B 62, 10696–10705(2000).
[CrossRef]

B. Wood, J. B. Pendry, and D. P. Tsai, “Directed subwavelength imaging using a layered metal–dielectric system,” Phys. Rev. B 74, 115116 (2006).
[CrossRef]

S. A. Ramakrishna and J. B. Pendry, “Removal of absorption and increase in resolution in a near-field lens via optical gain,” Phys. Rev. B 67, 201101 (2003).
[CrossRef]

C. Luo, S. G. Johnson, J. D. Joannopoulos, and J. B. Pendry, “Subwavelength imaging in photonic crystals,” Phys. Rev. B 68, 045115 (2003).
[CrossRef]

Phys. Rev. E (1)

K. J. Webb, M. Yang, D. W. Ward, and K. A. Nelson, “Metrics for negative-refractive-index materials,” Phys. Rev. E 70, 035602(2004).
[CrossRef]

Phys. Rev. Lett. (2)

A. Grbic and G. V. Eleftheriades, “Overcoming the diffraction limit with a planar left-handed transmission-line lens,” Phys. Rev. Lett. 92, 117403 (2004).
[CrossRef] [PubMed]

J. B. Pendry, “Negative refraction makes a perfect lens,” Phys. Rev. Lett. 85, 3966–3969 (2000).
[CrossRef] [PubMed]

Science (1)

N. Fang, H. Lee, C. Sun, and X. Zhang, “Sub-diffraction-limited optical imaging with a silver superlens,” Science 308, 534–537(2005).
[CrossRef] [PubMed]

Other (3)

E.Palik, ed., Handbook of Optical Constants of Solids(Academic, 1998).

B. Saleh and M. Teich, Fundamentals of Photonics, 2nd ed.(Wiley, 2007).

J. W. Goodman, Introduction to Fourier Optics, 3rd ed.(Roberts, 2005).

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Figures (7)

Fig. 1
Fig. 1

Schematic of a layered metal–dielectric periodic lens with N = 4 periods. It is further assumed that λ = 430 nm , Λ = 57.5 nm , N = 10 , d Ag = 0.37 Λ , d SrTiO 3 = 0.63 Λ , n Ag = 0.04 + 2.46 i , n SrTiO 3 = 2.674 + 0.027 i .

Fig. 2
Fig. 2

(a) One-dimensional TF and PSF of the multilayer for in-plane propagation: H ^ TM , H ^ TE , TF for the TM and TE polarizations; (b) radial cross-section of 2D TF and PSFs of the multilayer for 3D propagation H ^ m = ( H ^ TM + H ^ TE ) / 2 , H ^ δ = ( H ^ TM H ^ 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.

Fig. 3
Fig. 3

Two-dimensional TF matrix of the multilayer. (a) Matrix elements H ^ x x ( k x , k y ) and H ^ x y ( k x , k y ) for linearly polarized light; (b) matrix elements H ^ σ + σ + ( k x , k y ) and H ^ σ + σ ( k x , k y ) for circularly polarized light. Phase isolines are drawn at the distances of π / 2 and are changed between solid and dashed lines every π.

Fig. 4
Fig. 4

Two-dimensional PSF matrix of the multilayer. (a) Matrix elements H x x ( x , y ) and H x y ( x , y ) for linearly polarized light; (b) matrix elements H σ + σ + ( x , y ) and H ^ σ + σ ( x , y ) for circularly polarized light; phase isolines are drawn at the distances of π / 2 and are changed between solid and dashed lines every π.

Fig. 5
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.

Fig. 6
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.

Fig. 7
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.

Equations (17)

Equations on this page are rendered with MathJax. Learn more.

Ψ ( x , y ) z = z out = ( 2 π ) 1 H ( x , y ) * Ψ ( x , y ) z = z inc .
Ψ ^ ( k x , k y ) z = z out = H ^ ( k x , k y ) · Ψ ^ ( k x , k y ) z = z inc ,
Ψ ( x , y ) z = ( 2 π ) 1 Ψ ^ ( k x , k y ) z e i ( k x x + k y y ) d k x d k y .
[ E ^ x ( k r , k φ ) E ^ y ( k r , k φ ) ] z = z out = H ^ lin ( k r , k φ ) · [ E ^ x ( k r , k φ ) E ^ y ( k r , k φ ) ] z = z in ,
[ E ^ σ + ( k r , k φ ) E ^ σ ( k r , k φ ) ] z = z out = H ^ circ ( k r , k φ ) · [ E ^ σ + ( k r , k φ ) E ^ σ ( k r , k φ ) ] z = z in ,
H ^ lin ( k r , k φ ) = [ H ^ x x ( k r , k φ ) H ^ y x ( k r , k φ ) H ^ x y ( k r , k φ ) H ^ y y ( k r , k φ ) ]
= R ( k φ ) · [ H ^ TM ( k r ) 0 0 H ^ TE ( k r ) ] · R ( k φ )
= H ^ m ( k r ) · [ 1 0 0 1 ] + H ^ δ ( k r ) · [ cos ( 2 k φ ) sin ( 2 k φ ) sin ( 2 k φ ) cos ( 2 k φ ) ] ,
H ^ circ ( k r , k φ ) = [ H ^ σ + σ + ( k r , k φ ) H ^ σ σ + ( k r , k φ ) H ^ σ + σ ( k r , k φ ) H ^ σ σ ( k r , k φ ) ]
= H ^ m ( k r ) · [ 1 0 0 1 ] + H ^ δ ( k r ) · [ 0 exp ( 2 i k φ ) exp ( 2 i k φ ) 0 ] .
g ^ ( k r , k φ ) = n = + ( i ) n c n · exp ( i n k φ ) · H n { g r ( r ) } ,
c n = 0 2 π g φ ( φ ) · exp ( i n φ ) d φ ,
H n { g r ( r ) } = 0 g r ( r ) · J n ( k r r ) · r · d r .
H lin ( r , φ ) = [ H x x ( r , φ ) H y x ( r , φ ) H x y ( r , φ ) H y y ( r , φ ) ]
= H m ( r ) · [ 1 0 0 1 ] H δ ( r ) · [ cos ( 2 φ ) sin ( 2 φ ) sin ( 2 φ ) cos ( 2 φ ) ] ,
H circ ( r , φ ) = [ H σ + σ + ( r , φ ) H σ σ + ( r , φ ) H σ + σ ( r , φ ) H σ σ ( r , φ ) ]
= H m ( r ) · [ 1 0 0 1 ] H δ ( r ) · [ 0 exp ( 2 i φ ) exp ( 2 i φ ) 0 ] ,

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