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We observe, by means of finite element calculations, that some photonic crystals produce negative refraction with almost circular isofrequency lines of their band diagram, so that a slab of this structure presents a large degree of isoplanatism and thus can behave like an imaging system. However, it has aberrations on comparison with a model of ideal lossless left-handed material within an effective medium theory. Further, we see that it does not produce subwavelength focusing. We discuss the limitations and requirements for such photonic crystal slabs to yield superresolved images of extended objects.
©2007 Optical Society of America
1. Introduction
An alternative to left-handed materials (LHM), or negative index media, [1–14] to produce negative refraction has been theoretically proposed with photonic crystals (PC) [15–25] within a certain range of incidence angles. Also, selfwaveguiding has been demonstrated in some arrangements of these crystals as well as localized intensity concentration [16], [17], [18], [19]. In particular, imaging with slabs of PC structures consisting of a triangular 2D arrays of dielectric rods was obtained [26]. A similar process was demonstrated with a triangular array of dielectric bars [27]. However, although in these works imaging was demonstrated on the basis of negative refraction at the slab interfaces, provided by the circular-like isofrequency lines that these structures yield in their band diagram, we have been unable to obtain subwavelength imaging in calculations done in these PC slabs, even though their response to one point source may yield a point focus on the other side. The important aspect in this connection is that in order to be sure that this bright spot indeed corresponds to a true imaging process yielded through negative refraction, subwavelength details should be retrieved when other incident wavefronts corresponding to sources other than just one point, impinge on the slab. This, in turn, should convey a certain degree of isoplanatism of the system. We have obtained a similar conclusion with slabs of other PC arrays, like for instance a square lattice of two rectangular air voids in a lossless dielectric [28]. We notice that in particular the work of Ref. [28] contains an interesting discussion on the role of surface modes, excited at the back surface of the slab, on the spatial details of the image. In this connection it should be mentioned that the role of the cut of the PC slab interfaces, (namely, the edge cut of the first and last row elements of the crystal slab), has been analysed [16], [28], and it has been recently shown [29] that it strongly influences the possibility of acting behavior of the PC slab like a homogeneous medium with an effective negative permittivity and permeability.
Consequently, in this paper we investigate the imaging characteristics of photonic crystals that present a degree of negative refraction. We shall study and illustrate them with a slab of photonic crystal models that have been demonstrated to posses negative refraction [26], [27] and thus acts as a lens. To this end, we shall investigate the important property of imaging that the aforementioned isoplanatism represents [30], [31]. This involves to asses whether one can define a transfer function of the system and, if so, what are its characteristics and influence on the quality of the image of any arbitrary extended wavefront incident on the slab. In this respect, it is illuminating, and hence of interpretative value, to carry out a comparison with the corresponding imaging properties of an ideally low loss effective medium LHM slab possessing a real part of the refractive index equal to -1. In addition, as regards superresolution, we will discuss the conditions under which the surface modes on the exit interface of the PC slab, (playing a role analogous to that of the surface plasmon polaritons (SPP) of a LHM surface), couple with evanescent components of the incident wavefield angular spectrum, and among those, what are requirements that lead to superresolved images.
Fig. 1. (a) Geometry of the numerical simulation for the PC slab of air cylinders in a dielectric matrix with ε = 12.96. The wave source is at distance z 0 from the slab limiting interface. (b) Brillouin zone for the hexagonal PC and symmetry points.
2. Propagation of extended wavefronts in photonic crystal slabs
We first address a 2-D hexagonal lattice of air cylinders in a dielectric matrix with permittivity ε = 12.96, lattice constant a and radius r = 0.4a. The geometry is shown in Figs. 1(a) and 1(b) both for a slab of this crystal and its Brillouin zone. This structure was proposed in [26] and its band diagram is shown in Fig. 2(a). From it, one obtains isofrequency lines that, at the frequency at work, are circle-like centered in the Γ point (Fig. 2(b) (top inset)). In Fig. 2(b) we show the different permitted wavevectors k in this regime of frequencies, which are in the second band, for the directions ΓK and ΓM. We observe that the the magnitude of the permitted wavevectors is different for the ΓK and ΓM directions in this range of frequencies (bottom inset of Fig. 2 (b)), and therefore the effective refractive index is not the same for both directions, which causes the deformation of the isofrequency from a circle one; this deformation being the cause of an aberration in the imaging process by a slab of this crystal, as shown next. We shall denote as the input plane that at which we specify the limiting value of the wavefield illuminating a slab of thickness d of this array. This plane is taken at distance z 0 from the entrance surface of the crystal slab. Let us now consider a point source, in the input plane, at distance z 0 = 3.442a, which emits linearly polarized harmonic waves around a normalized frequency ω = 0.305 × 2πc/a, λ being the wavelength, with the electric vector Ey parallel to the cylinder axis.
Fig. 2. (a)Band diagram, obtained using a plane wave expansion calculation, for the hexagonal lattice of air cylinders in a dielectric matrix with ε = 12.96. The red line corresponding to the range of frequencies used, cuts the second band in such a way that there is a circle-like isofrequency about the Γ point. The corresponding frequency surface gradient is negative and so is the group velocity. The light line (broken) is also shown. (b) Second band for ΓK (solid line) and ΓM (dashed line) directions, showing the differences, at used frequencies, of wavevector distances from the Γ point to the K and M points, (bottom inset), as well as the corresponding deformation of the corresponding isofrequency from a circle, (top inset) for λ = 3.36a.
This source is realized with a narrow slit whose exit is at z 0 = 3.442a. The thickness of the PC slab is d = 2z 0. The calculation gives the distribution of the field propagated throughout the crystal and out of it. In particular, its response in the image plane at the Veselago distance z 0 from the exit surface is evaluated. This is shown for λ = 3.28a in Fig. 3.
It is well known that any imaging system, and in particular a lens, should be space invariant, at least in a certain field region. This involves the isoplanatic condition, according to which the image field distribution i(x) is given in terms of the object wavefront distribution o(x) by the convolution [30], [31]:
In Eq. (1) P(x) is the point spread function of the system, or response to a point source, (i.e., P(x) constitutes the wavefunction at the focus), and it allows, by Fourier transformation, to define a transfer function of the lens or optical system. This also involves that in the region of isoplanacity the response o(xi)P(x - xi) in the detection plane to each ith sampling point o(xi)δ(x - xi) of the object wavefield distribution in the input plane, only depends on the difference of coordinates x - xi and not on the position xi.
Fig. 3. Map of the modulus of the electric field Ey for a narrow slit which acts like point source (down) with a λ = 3.28a in front of a 2-D hexagonal photonic crystal. We observe a focus both inside and out the crystal. The outer focus is elongated as a consequence of aberrations. (White spots in the color distribution indicate that the value is out of the color scale on the right).
In order to test the degree of isoplanatism of this PC slab, we carry on the following operation: the inverse Fourier transform of this point source image P(x) is performed, then this result which is the transfer function T(u), is multiplied by the Fourier spectrum O(u) of any extended object o(x) placed at the input plane z 0. Subsequently, an inverse Fourier transform of this product should yield a field distribution i(x) in the image plane according to Eq.(1), and this result ought to be very similar to that obtained by direct FE calculation of the propagation of this extended object wavefront through the crystal slab, in the image plane.
Fig. 4 shows the transfer function obtained by this procedure for the wavelengths: λ = 3.24a, 3.28a, 3.32a and 3.36a. On the other hand, Fig. 5 depicts the transfer function at different distances z 0 of the source from the slab. We observe that the widths of these transfer functions do not exceed the Rayleigh limit of resolution 1/λ. The comparison of the images of an extended object that has no subwavelength details, obtained by the above mentioned procedures is displayed in Fig. 6 for λ = 3.28a measured at a distance z 0 from the slab. They indicate that in the range of wavelengths and distances of operation studied here, there is good degree of isoplanatism in the crystal, however due to a diffraction effects and since the isofrequency lines are not perfect circles, the image possesses aberrations.
These aberrations have been characterized by means of a ray tracing (see Figs. 7(a)–(d)). In order to understand the different mechanisms that alter the imaging process, we think that it is useful to separate the aberration due to only the deformation of the isofrequencies from circles (fig. 7(a)), (where the exterior medium plays no role), from the actual one that takes place in the real system subjected to the FE simulation, namely, that in which one also takes into account the difference between the refractive index of the surrounding air and the effective refractive index of the PC slab. This additional cause of aberration is produced by the size difference between the isofrequency lines in the exterior and in the crystal. The lateral (∆x) and longitudinal (∆y) aberration distances, defining the vertices of the resulting ray tracing caustic, (cf. Figs. 7(a) and 7(b)), are shown in Fig. 7(d). The distances ∆x and ∆y of the aberration due only to deformations of the isofrequency from a circle, increase with the wavelength, (this is clear by looking at the bottom inset of Fig. 2(b), where details of the separation between the ΓK and ΓM curves are appreciated). On the other hand, the distances ∆x and ∆y of the aberration in the real case, namely that in which we have also had into account that the refractive index of the surrounding air, being different to that of the PC, gives rise to transmission into the PC that varies with the angle of incidence, decrease with the wavelength.
Fig. 4. Modulus of the real part of the transfer function (i.e. Fourier transform of the image of a point source) produced by a hexagonal PC of air holes in a dielectric matrix with permittivity ε = 12.96, lattice constant a and radius r = 0.4a, at different wavelengths. We observe that their widths do not exceed the Rayleigh limit of resolution 1/λ.
It should be noted that there is yet a third cause of aberration provoked by the transmission dependence on the angle of incidence due to the surface impedance Z = √ε/μ, which has been demonstrated to be different from Z = 1 in these PC slabs [29]; this is known to alter the the position of the focus, within an effective medium theory [32].
3. LHM slabs
In order to gain an understanding of the requirements that the PC slab must fulfill to attain imaging with superresolution, it is worth analysing the transfer function of a LHM slab whose constitutive parameters are very close to those of an ideal lens: ε̂ = μ̂ = -1 + i0.001, which involves that the complex refractive index will be n̂ = - 1 + i0.001. A time harmonic dependence exp(iωt) for the wave is assumed. Figs. 8(a)–(c) show the spatial distribution of the electric field modulus when a point source, made by a narrow slit (Fig. 8(d)), is placed at distances z 0 = λ, λ/3 and λ/6 from the entrance surface of the slab.
The corresponding transfer functions, calculated by the procedure described before, are shown in Fig. 9. It is observed how the width of these curves increases as z 0/λ decreases. Also, while at z 0 = λ there is focusing inside the slab (cf. Fig. 5(a)), which manifests the dominant contribution of the propagating components of the wavefield, at subwavelength distances z 0 this focusing is almost imperceptible, showing the predominance of the evanescent components of the wavefield angular spectrum. In addition in Figs. 8(a)–(c) one sees how both the object and the image fields are spatially coupled with intensity enhancements in the entrance and exit surfaces of the slab, respectively. Both enhancements being due to the excitation of surface plasmon polaritons (SPPs) by the evanescent components of the wave, and thus being increasingly prominent as z 0/λ decreases; this is shown in Fig. 10. These SPPs are present for both TE and TM polarization cf.[33], and are extended in their transversal wavevector component at the frequency under study, thus being excited by a large spectrum of evanescent waves. Of course the larger excitation of these surface waves in the back interface as z 0/λ diminishes, is what broadens the bandwidth of the transfer function. Figs. 8(a)–(c) and 9 confirm it.
Fig. 5. Modulus of the real part of the transfer function (Fourier transform of the image of a point source) produced by a hexagonal PC of air holes in a dielectric matrix with permittivity ε = 12.96 for λ = 3.32a, at different distances z 0 and different thickness d = 2z 0 of the source from the slab. Their widths do not exceed those of the Rayleigh limit of resolution 1/λ, although the source is in near field.
Fig. 6. Electric field amplitude distribution of an extended object wavefront for λ = 3.28a measured at a distance z 0 from the slab (red dashed line); which does not demand superresolution details. Its image obtained from the transfer function via by Fourier transform, as explained in the text, is shown in the green dotted line. The image obtained by propagation FE simulation (black solid line) is very similar to the latter.
Figs. 8(a)–(c) and 10 also show that the distance of the focus plane to the exit surface of the slab is not well defined as z 0/λ decreases beyond 1/2. Since the image field is constituted by the distribution expelled out from the standing wave pattern of the SPP at the exit interface. This progressive lack of a focus plane as one approaches the quasi-electrostatic limit is in agreement with a similar observation of lack of intensity maximum in z > 0 in [28] when dealing with the image of a point source through a slab of photonic crystal of small thickness compared to the wavelength λ. The same can be said with respect to the field of the incident wave in front of the entrance surface of the slab.
Fig. 7. Aberrations analyzed by means of a ray tracing for: (a) A PC illuminated by a point source with λ = 3.32a, where we have considered only the aberration due to the deformation of the isofrequency from a circle one, (namely, the surrounding medium has a refractive index that equals the averaged effective one of the crystal slab). (b) The same illuminated PC where we have also taken into account that the surrounding medium is air. This is the real situation of the FE simulation. We consider both the aberration caused by deformation in the isofrequency from a circle and that caused by difference between effective refractive index of the PC and the surrounding medium (air) (λ = 3.36a). (c) Same as (b) with the numeric simulation and λ = 3.32a; we observe agreement between ray tracing and FE simulation. This agreement is clear both in the position and depth of the focus obtained by both methods. (d) Distances ∆x (white symbols) and ∆y (black symbols) corresponding to the lateral and longitudinal aberration, respectively, for the case in which the configuration is that of Fig. (a)(circles and red line) and for the case in which is that of Fig. (b) (squares and blue line). These distances are referred to the caustic vertex in the paraxial focus.
Fig. 8. (a)-(c) Map of the modulus |Ey| of the electric field for a slit emitting in front of left handed slab with n̂ = -1 + i0.001 for different distances z 0 in terms of the wavelength λ. (d) Modulus of the electric field measured on the exit plane of the slit, namely, at distance z 0 from the slab entrance interface. This magnitude is evaluated both with and without the presence of the LHM slab, both profiles are indistinguishable from each other.
One may even wonder how the effect of the standing wave in the entrance surface actually distorts the field distribution of the field in the input plane when the stationary field state is reached. However, a cross section of the field distribution below the entrance surface, as shown for the wavefield modulus in Fig. 8(d) for a slit source, and in Fig. 11(d) for an extended object, shows that for these parameters used, the resulting field pattern in this region accommodates so that the incident wavefront distribution in the input plane is not appreciably distorted by the presence of the slab interface.
On the other hand, Figs. 11(a)–(c) show the image distribution at distance z 0 from the exit surface of the slab for an incident extended object wavefront, on the input plane at distance z 0 from the entrance surface. The superresolution attained as z 0 decreases, in association with that of the corresponding transfer functions (cf Fig. 9) is evident. Also, in these Figs. 11(a)–(c) we have performed the same operation as in the PC slab, namely, a comparison between the image obtained by FE simulation of the propagation process throughout the slab and that obtained by Fourier transformation via the transfer function. The agreement shown, even in those cases demanding superresolution, demonstrates the isoplanatism of this system in the range of distances and wavelengths employed.
Fig. 9. Modulus of the real part of the transfer function (Fourier transform of the image of a point source) produced by a LHM with real part of the refractive index n 1 = -1 and imaginary part n 2 = 0.05 at different distances z 0 of the source from the slab. We observe that their widths exceed those limited by the Rayleigh limit of resolution 1/λ.
Fig. 10. Electric field modulus |E| (normalized to the value in the exit of the slit) versus z coordinate for different distances z 0 from the slit to the slab. We observe an increasing of the modulus |E| on the interfaces and a suppression of the resolution in the focus along oz as z 0 decreases.
4. Discussion on the conditions for superresolution
Based on the previous discussion for LHMs, in this section we analyze the conditions for superresolution in a photonic crystal. First, it is convenient to begin with an easy model, that is frequently on use, illustrating the amplification of evanescent waves and superlensing, namely, both in a LHM and a silver plate (SL). This will provide an example and guidance for the behavior of the PC slab.
Fig. 11. (a)-(c) Electric field amplitude Ey of an extended object containing subwavelength details (red solid line) and its image by a LHM slab for different distances z 0 and different thickness of the slab d = 2z 0. The image obtained via the transfer function is shown by the blue dashed line. The image obtained from the propagation simulation by the FE calculation is displayed by the black dotted line. (d) Electric field amplitude measured at distance z 0 from the slab entrance interface. This magnitude is evaluated both with and without the presence of the LHM slab (black solid line and red dashed line, respectively), both profiles are indistinguishable from each other except the slight differences in the tails.
It is illustrative to consider first the constitutive parameters of the LHM slab to be ε′ = μ′ = n′ = - 1. These values are not strictly physical, since some absorption should be present [9], however they help to interpret the behavior of the evanescent waves which grow with the slab depth.
Let k = (kx,kz) and k′ = (k′x,k′z) be the wavevectors of one evanescent plane wave component, incident and transmitted into the slab, respectively. Namely, ae ikxx-kzz is refracted in the first interface of the LHM slab into ae ik′x-k′zZ, where
Let s and s′ be the directional sines of k and k′ for kx and k′x, respectively, namely, kx = k 0 s, k′x = k 0 n′s, then the Snell law derived from the continuity equation: k′x = kx implies that s′ = -s and thus k′z = -kz which is usually interpreted as the amplification of the incident evanescent wave component by the LHM slab.
On the other hand, in a silver slab we assume again approximately that : ε′ = -1, μ = 1 and thus n′ = i, then the Snell law yields is′ = s, i.e.: s′2 = -s 2; thus the z-component of the wavevector transmitted into the slab is in this case
which is an amplified wave, even though with the sign difference: + instead of: - in the square root, namely: instead of . In the electrostatic limit, however, s >> 1 which yields exactly k′z = -kz. Thus the silver slab acts in a similar way as the LHM slab in this limit which involves scales of the slab thickness, and distances outside it, much smaller than λ.
On the other hand, in a photonic crystal the restrictions to amplify incident evanescent waves are stringent [34]. For example, in Fig. 12 we show the evolution of an incident evanescent wave on a slab of the photonic crystal addressed in section II. This evanescent wave has been created by total internal reflection of a plane wave incident on a dielectric prism, placed below the slab. No amplification is observed. The question however is: under what circumstances can some evanescent components couple to surface waves of the exit interface of the PC slab?. In other words, can we design a photonic crystal with a band diagram in the evanescent region, that contains evanescent modes that are flat enough and involve a large enough number of wavevectors so that there is coupling with a significant number of evanescent components of the incident wavefield, and thus amplify them at a given working frequency?
To search an answer, let us first look at this amplification in either a LHM and a silver slab. To this end, one has to search for the excitation of surface waves, plasmon-polariton-like ones, on the exit surface of the slab, and these waves must couple to as many as possible evanescent components of the wavefront propagated along the slab. This was seen in Fig. 8 for a LHM and was further illustrated in Fig. 10 where an intensity distribution along the z-direction showed the corresponding increase of the intensity at the slab interfaces as the the source becomes closer to the slab and thus more evanescent components of the incident wave are retrieved. SPP in LHM exist for both TE and TM waves [33]. On the other hand, in a silver slab the Drude model gives the permittivity:
ωP being the plasma frequency in the bulk. If we look into the dispersion relation:
Fig. 12. (a) Total internal reflection at 45 degrees in a dielectric prism with n=3.5 below the photonic crystal slab. An evanescent wave is transmitted and incides on the crystal (b) Variation of the evanescent wave intensity along the depth (z-coordinate) of the crystal.
When the real part of the permittivity is ε ≈ - 1 then from Eq. (6) one sees that the surface plasmon satisfies that kx(ω) → ∞, this is the essence of the electrostatic limit: kx(ω) >> k 0 at subwavelength scales, and thus from Eq. (5) we have that its excitation frequency ωsp is:
Fig. 8 shows the importance of the evanescent components in the electrostatic limit, namely when the source is at a distance from the slab much smaller than the wavelength, then the contribution of the propagating components is not so important and, as a result, no focus inside the slab, as dictated by Veselago’s geometry, is formed.
In order that each evanescent component of the incident wavefront couples with a surface wave of the exit interface of the slab, namely, it should follow the hyperbolic isofrequency line of the (kx,kz)-complex plane in the evanescent zone, given by:
If we refer to Fig. 13, one sees that Eq. (8) would impose that each evanescent component of the slab follows that isofrequency line at the same plasmon frequency ωsp. However, The SPP plasmon dispersion relation shown in Fig. 10 shows that this is only possible in the flat region of this curve which corresponds to large values of kx at which Eq. (8) takes on the asymptotic form given by the straight line:
This is the reason why the silver slab works yielding superfocusing at such small distances in the electrostatic limit: all evanescent components at the same frequency excite the corresponding value kx in the flat zone of the dispersion relation, all of them satisfying it at the plasmon frequency ωsp.This could not happen, however, in the curved portion of the dispersion relation, since there only one kx corresponds to each ωsp, and thus only one evanescent component of the incident wavefront would excite it.
Fig. 13. Condition for superresolution of a superlens with real part of the permittivity ε ≈ - 1 at a working frequency ωp/√2, ωp being the plasma frequency. Due to the fact that excited surface plasmons (green dashed line) have high value of the wavevector kx, the hyperbolic condition Eq. (8) in the evanescent zone (blue solid line) is satisfied for the wavevector kx of all evanescent components of the incident wavefront, since for high kx this condition becomes the asymptotic straight line kz = ikx (red dotted line) which corresponds to the flat zone of the dispersion relation (green dashed line)
Fig. 14. Surface mode dispersion curve (red line) in the second band for TE polarization (Electric field parallel to the prisms). The surface mode is ”flat”, which implies that all those kx wavevectors at the the right of the light line included in the definition domain of this red segment are permitted in the surface at practically the same frequency of excitation.
Now, turning back to a PC slab, we can create surfaces modes by introducing a cut in the last row elements, namely those forming the exit surface of the slab. In this work we present a hexagonal array of rectangular prisms 0.4a × 0.8a with permittivity ε = 9.61 in air, like those used in reference [27]. We have excited a surface mode by cutting 0.1a the first and last rows of prisms, and we have calculated it by the supercell method [35–37] in Fig. 14. However, this mode does not conveys superresolution because the wavevectors k x are very close to the light cone ω/c and thus they are not sufficiently high to reach the hyperbolic isofrequency curve posed in our Eq. (8) in the evanescent zone, or conversely the corresponding dispersion relation line, (green curve in Fig. 13). Furthermore we have a limited spectrum of permitted k x, with a maximum kM that does not allow this subwavelength detail.
Fig. 15. (a) Map of the modulus of an extended object wavefront in the bottom inciding on a photonic crystal (b) Corresponding intensity of extended object wavefront (blue solid line) and its image by a crystal whose exit interface is cut such that a surface mode is either not excited (red dotted line) or excited (black dashed line) at the frequency ω = 0.345 × 2πc/a (i.e.λ = 2.898a)
Finally, in order to show and illustrate that the excitation of surface modes itself, is not enough to imaging with superresolution, unless they fulfill the requirements discussed in the above analysis for left handed materials and silver slabs, we show in Fig. 15 the image obtained from an extended object in a slab of this crystal, either with or without introducing the aforementioned cut in the first and last rows. In other words, either with or without the existence of surface modes. One observes that the creation of this surface mode is not enough, and hence it does not yield any improvement in the quality of the subwavelength detail of such an image of this extended object that demands superresolution. This lack of superresolution is due to the fact that the length of the range of values of the transversal wavevector of the surface waves is only , (cf. Fig. 14); and hence, if one looks to the dispersion curve ω(kx) of Fig. 13, one sees that this range lies inside the zone in which the hyperbola of Eq. (8) does not take the asymptotic form Eq. (9); namely, that range lies in the curved region of the plasmon dispersion relation which is between the light line (value ω/c) and the flat zone region ω = constant of the dispersion curve. Therefore, the excited surface waves do not follow the required isofrequency line in the evanescent zone.
5. Conclusions
We have studied the propagation of extended object wavefronts in some photonic crystal slabs capable of yielding negative refraction. We have found by means of an analysis that introduces the transfer function, that this system is isoplanatic, at least in a wide range of distances of the source and frequencies as those addressed here. However, due to imperfections in the isofrequencies of their band diagram, and to angle dependent transmissions at their interfaces, they present aberrations that we have characterized. We have compared this photonic crystal behavior with that of a left handed material and we have shown the differences in the transfer functions of both as regards the resolution of images. Finally, we have discussed the necessary conditions to achieving superresolved images by PC slabs by taking guidance of the equivalent problem for both left handed materials and silver slabs . We have concluded that the created surface waves in the exit interface of a PC should have a wavevector that either follows the correct hyperbolic isofrequency line in the complex plane of its Cartesian components, or the experimental conditions should be such that the electrostatic limit, where a flat dispersion relation is held for enough values of the transverse wavevector, applies. We are currently aiming to design a photonic crystal that yields such surface waves.
Acknowledgments
Work supported by the Spanish DGICYT and the European Union. J.L. G.-P. acknowledges I3P grant program.
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